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DESY-15-085

Charm, Beauty and Top at HERA

O. Behnke, A. Geiser, M. Lisovyi∗,
DESY, Hamburg, Germany

∗ now at Physikalisches Institut, Universität Heidelberg, Heidelberg, Germany

June 26, 2015

Abstract

Results on open charm and beauty production and on the search for top production in high-
energy electron-proton collisions at HERA are reviewed. This includes a discussion of relevant
theoretical aspects, a summary of the available measurements and measurement techniques, and
their impact on improved understanding of QCD and its parameters, such as parton density func-
tions and charm- and beauty-quark masses. The impact of these results on measurements at the
LHC and elsewhere is also addressed.

Contents

1 Introduction 3

2 Theory of heavy-flavour production at HERA 7
2.1 HERA kinematic variables and phase space . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Perturbative QCD calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Heavy-quark production at HERA in “leading order” . . . . . . . . . . . . . . . . . . . 10
2.4 Quark-mass definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 The zero-mass variable-flavour-number scheme . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 The massive fixed-flavour-number scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.7 The general-mass variable-flavour-number scheme . . . . . . . . . . . . . . . . . . . . . 14
2.8 Proton structure functions in DIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.9 QED corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.10 Fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.11 Choice of renormalisation scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 The HERA collider and experiments 23
3.1 HERA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 H1 and ZEUS Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Event reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1

http://arxiv.org/abs/1506.07519v1


4 Charm and Beauty detection at HERA 30
4.1 Charm tagging using full reconstruction of charm hadrons. . . . . . . . . . . . . . . . . 31
4.2 Heavy-flavour tagging with lepton + prel

T
. . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Charm and Beauty with inclusive lifetime tagging . . . . . . . . . . . . . . . . . . . . . 34
4.4 Charm and Beauty with double tagging . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Search for single top-quark production 40

6 Charm photoproduction 42
6.1 D∗ inclusive measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.1.1 Charm total cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.1.2 D∗ single-differential cross sections . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.1.3 D∗ double-differential cross sections . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.2 Inclusive measurements using other tagging methods . . . . . . . . . . . . . . . . . . . 47
6.3 Studies with a D∗ and one other hard parton . . . . . . . . . . . . . . . . . . . . . . . . 49
6.4 Parton-parton-correlation studies in charm-tagged events . . . . . . . . . . . . . . . . . 50

6.4.1 xobsγ studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.4.2 Azimuthal correlations ∆φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.4.3 Study of hard-scattering angle cos θ∗ . . . . . . . . . . . . . . . . . . . . . . . . 54

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7 Beauty photoproduction 58
7.1 Total cross section for beauty production . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.2 Single-differential cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.3 Measurements of bb̄ and jet-jet correlations . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

8 Charm and beauty production in DIS 63
8.1 Production mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8.2 Single-differential cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8.3 Double-differential cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8.4 Proton structure functions and reduced cross sections . . . . . . . . . . . . . . . . . . . 71
8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

9 Measurement of QCD parameters, proton structure, and impact on LHC and other
experiments 76
9.1 Measurement of charm fragmentation functions and fragmentation fractions . . . . . . . 76

9.1.1 Charm fragmentation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.1.2 Charm fragmentation fractions and ratios . . . . . . . . . . . . . . . . . . . . . 79

9.2 Measurement of parton density functions . . . . . . . . . . . . . . . . . . . . . . . . . . 82
9.3 Proton flavour composition and W/Z/H production at LHC . . . . . . . . . . . . . . . 84
9.4 Measurements of the charm-quark mass and its running . . . . . . . . . . . . . . . . . . 86
9.5 Measurement of the beauty-quark mass and its running . . . . . . . . . . . . . . . . . . 88
9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

10 Summary and outlook 91

2



1 Introduction

HERA was the first and so far only high energy electron1-proton collider. The production of heavy-quark
final states in deeply inelastic scattering (DIS) and photoproduction (γp) from ep interactions at HERA
(Fig. 1) originally was [1] and still is (this review) one of the main topics of interest of HERA-related
physics, and of Quantum Chromodynamics (QCD) in general.

(a) (b)

Figure 1: (a) The dominant production process for charm and beauty quarks in ep collisions at HERA,
the boson-gluon fusion (BGF) reaction. (b) The simplest Quark-Parton-Model diagram for deeply
inelastic scattering on a light quark.

A quark is defined to be “heavy” if its mass is significantly larger than the QCD scale parameter
ΛQCD ∼ 250 MeV. The heavy quarks kinematically accessible at HERA are the charm and beauty
quarks, which are the main topic of this review. At the time of the proposal of the HERA collider and
experiments in the 1980’s [2], a search for the top quark was one of the major goals [3]. This influenced
parts of the detector design: if at all, top quarks would be produced boosted into the proton direction,
and top-quark mass reconstruction from hadronic final states would profit from an excellent hadronic
energy resolution. As we know today, top-quark pair production was out of the kinematic reach of
the HERA collider. Single top-quark production is kinematically possible, but strongly suppressed by
Standard Model couplings. This allows the search for non-Standard Model top-production processes
which will be covered in Section 5.

Charm production at HERA, in particular in deeply inelastic scattering, was realised from very
early on to be of particular interest for the understanding of QCD [1, 4]. Up to one third of the HERA
cross section is expected to originate from processes with charm quarks in the final state: assuming
“democratic” contributions from all quark flavours, which is a reasonable assumption at very high
momentum transfers, this fraction f(c) can be approximated by the ratio of photon couplings in Fig.
1, which are proportional to the square of the charges Qq, q = u, d, s, c, b of the kinematically accessible
quark flavours:

f(c) ∼ Q
2
c

Q2d + Q
2
u + Q

2
s + Q

2
c + Q

2
b

=
4

11
≃ 0.36, (1)

while a similar approximation for beauty yields f(b) ∼ 1
11

≃ 0.09. In general, the impact of beauty on
inclusive cross sections at HERA is thus smaller than the impact of charm.

At momentum transfers large enough for these approximations to be meaningful, charm and beauty
can be treated as an integral part of the “quark-antiquark sea” inside the proton (Fig. 2), similar to the

1Throughout this document, the term “electron” includes positrons, unless explicitly stated otherwise.

3



light quarks in Fig. 1b, originating from the initial state splitting of virtual gluons. Since the proton has
no net charm and beauty flavour number, charm and beauty quarks in the proton can only arise in pairs
of quarks and anti-quarks (Fig. 1(a) and shaded part of Fig. 2). However, due to the large charm- and
beauty-quark masses of about 1.5 GeV and 5 GeV, respectively, such a pair is considerably heavier than
the mass of the proton. From purely kinematic considerations, it can thus not exist as a “permanent”
contribution to the proton in the low-energy limit. Considerations of so-called “intrinsic charm” [5]
have been challenging this simple point of view. Since there is no evidence for such a contribution
from HERA data [6], this will not be pursued further in this review. Thus, charm and beauty “in the
proton”, as depicted in Fig. 2, are always considered to be virtual, and to arise as fluctuations from
the perturbative splitting of gluons inside the proton. This establishes heavy quark production as a
primary probe of the gluon content of the proton.

e+

p

γ c, b

Figure 2: Quark-parton-model view of heavy flavour production in ep collisions at HERA.

Different approaches to the theoretical treatment of charm- and beauty-quark production at HERA
are discussed in Section 2. All these treatments crucially make use of the fact that the heavy-quark
mass acts as a kinematic cut-off parameter in most of the QCD processes in which heavy quarks occur.
Furthermore, the fact that the heavy-quark mass is “large” compared to the QCD scale ΛQCD ∼
0.25 GeV allows the usage of this mass as a “hard scale” in QCD perturbation theory, appropriately
taking into account quark mass effects in perturbative calculations (Fig. 3). On the other hand,
the “smallness” in particular of the charm-quark mass with respect to other scales appearing in the
perturbative expansion, such as the virtuality of the photon, Q2, or the transverse momentum of a jet
or a quark, pT , can give rise to potentially large logarithmic corrections, e.g. of the form

∼ [αs ln(p2T /m2Q)]n or ∼ [αs ln(Q2/m2Q)]n (2)

where n is the order of the logarithmic expansion, αs is the strong coupling constant, and mQ, Q=c,b is
the heavy-quark mass. The size and treatment of these corrections is one of the issues to be investigated.

During the lifetime of the HERA collider, of order 109 charm and 107 beauty events should have
been produced in the H1 and ZEUS detectors, of which O(10%) have been recorded to tape via both
inclusive and dedicated triggers. HERA can thus truly be considered to be a charm factory. Further-
more, the fact that charm- and beauty-quark production at HERA can be studied essentially over its
complete kinematic range, from the cc̄- or bb̄-mass threshold up to squared momentum transfers of
order 1000 GeV2, offers the opportunity to treat HERA as a “QCD laboratory” to test the different
possible theoretical approaches to heavy-quark production against experimental data. Many such tests
are presented in Sections 6 and 8.

In particular, the charm-production measurements can be used to constrain important QCD pa-
rameters, such as the charm-quark mass and its running, and has important consequences for the

4



Figure 3: Possible hard scales in the boson-gluon-fusion process. For the explanation of the symbols
see text.

determination of other parameters like the QCD strong coupling constant, αs. The measurements can
also be used to constrain the charm fragmentation parameters, to constrain the flavour composition
of virtual quarks in the proton, and to determine or cross-check the gluon distribution inside the pro-
ton. Such measurements and results are discussed in Section 9. Finally, the HERA charm results have
a significant impact on measurements and theoretical predictions for many QCD-related processes at
hadron colliders, such as the LHC. For instance, the resulting constraints on the flavour composition of
quarks in the proton reduce the uncertainties of the LHC W- and Z-production cross sections, and the
constraints on the gluon content of the proton are an important ingredient for the determination of the
Higgs Yukawa coupling to top quarks from the dominant gluon-gluon-fusion Higgs-production process
via an intermediate top quark loop. Such cross-correlations are also discussed in detail in Section 9.

Beauty production at HERA (Sections 7 and 8) offers further complementary insight into the the-
oretical intricacies of heavy-flavour production in QCD. Due to its higher mass (mb ∼ 5 GeV) and a
correspondingly smaller value of the strong coupling constant, its perturbative QCD behaviour is some-
what better than the one of charm. However, the beauty mass remains non-negligible over essentially
the full accessible phase space of HERA, and a large fraction of the cross section is close to the kine-
matic bb̄-mass threshold. This offers a particularly sensitive handle on the treatment of mass effects in
QCD but also requires a particularly careful treatment of these mass effects in order to obtain reliable
predictions. Finally, the coupling of the photon to b quarks is four times smaller than the coupling to
charm quarks (Eq. (1)), and the higher b-quark mass yields a strong kinematic suppression. Therefore
in practice, depending on the region of phase space probed, the b-production cross section at HERA
is about 1 – 2 orders of magnitude smaller than the cross section for charm production. This makes
separation from the background and accumulation of a significant amount of statistics experimentally
much more challenging. Also, the experimental analyses of beauty are often not fully separable from
those of charm production. One of the highlights is the measurement of the beauty-quark mass (Section
9). Others are the measurement of the total beauty-production cross section at HERA (Section 7), and
the potential impact of HERA measurements on b-quark-initiated production processes at the LHC
(Section 9).

Last but not least, most of the results presented depend on a good understanding of the performance
of the HERA machine and the HERA detectors, as well as on mastering the heavy-flavour detection
techniques. Unfortunately, only a small fraction of the original data can make it through the various
event filtering and reconstruction stages. These aspects will be addressed in Sections 3 and 4.

Except for the shortest ones, each section will start with a brief introduction and close with a
summary, such that a reader less interested in the details may decide to skip the reading of the more

5



detailed parts of the section.
Some of the material in this review has been adapted from an earlier unpublished review [7] of one

of the authors. Further complementary information, in particular on charmonium and bottomonium
production or diffractive charm production, which are not covered by this review, is available elsewhere
[8, 9, 10, 11, 12]. The broader context of other physics topics can be explored in a more general review
on collider physics at HERA [13].

6



2 Theory of heavy-flavour production at HERA

This section describes the different theoretical approaches to charm and beauty cross-section predictions,
which will be needed later in the discussion of

• the Monte-Carlo (MC) based acceptance corrections for the data sets used to obtain cross sections;

• the extrapolation of different measurements to a common phase space, such that they can be
compared or combined;

• the comparison of different theory predictions to the measured cross sections;

• the parton-density fits including the heavy-flavour data;

• the fits of the charm and beauty masses and their running.

Since there is a large overlap between the theoretical approaches for these different purposes they will
be discusssed in a common framework in the following.

2.1 HERA kinematic variables and phase space

The measurements of heavy-quark production at HERA have been restricted, for statistical reasons,
to neutral current events (exchange of a neutral boson) and to the kinematic region of the negative
four-momentum transfer squared Q2 <∼ 2000 GeV

2, where photon exchange dominates and Z0 exchange
can be neglected. Figure 4 illustrates the event kinematic variables for ep scattering with heavy-quark
production via the boson (i.e. photon) gluon fusion process (see also Fig. 1).

e(k)

e(k’)

γ(q)

c, b
xP

c, b

xgP
p(P)

X

s

Q
2

W γ p

Figure 4: Illustration of event kinematic variables for ep scattering at HERA with heavy-quark pro-
duction via the boson-gluon-fusion process.

The four-momenta of the incoming electron k, the outgoing electron k′ and the proton P can be
used to define the following Lorentz-invariant variables:

s = (k + P)2 (3)

Q2 = −q2 = −(k − k′)2 (4)

7



x =
Q2

2P · q
(5)

y =
P · q
P · k

(6)

W 2γp = (P + q)
2 (7)

Here
√
s is the centre-of-mass energy of the ep system and Q2 is the photon virtuality. Wγp is the

centre-of-mass energy of the γ(∗)p system. In the simple Quark Parton Model [14] (QPM) the Bjorken
scaling variable x describes the proton momentum fraction carried by the scattered parton (Figs. 1(b)
and 2). The inelasticity, y, gives the fraction of the electron energy taken by the photon in the proton
rest frame. Only three of these five kinematic variables are independent. Neglecting the masses of the
electron and the proton the following relations between these quantities hold:

Q2 = s · x · y (8)
W 2γp = y · s − Q2 (9)

In the full QCD case this picture becomes more complicated, as illustrated in Fig. 4, where the proton
momentum fraction xg carried by the gluon does not coincide any longer with Bjorken x. However,
Eqs. (3) – (9) remain mathematically valid.

The ep scattering events are classified by the photon virtuality Q2. The regime of small Q2 ≈ 0 GeV2
is called photoproduction (γp) and the regime Q2 ∼> 1 GeV

2 is called Deeply Inelastic Scattering (DIS).
More details on inclusive DIS results and proton structure can be found elsewhere [15, 16, 17, 18, 19].

2.2 Perturbative QCD calculations

In fixed-order perturbative QCD (pQCD) the calculation of any parton-level cross section in ep, γp, p̄p
or pp collisions can be expressed as

σ(ab) =
∫

dxadxbf
a
pa
(xa, µa)f

b
pb
(xb, µF )σ̂papb(xaPa, xbPb, µa, µF , αs(µR)) (10)

where a = e, γ, p̄ or p is one incoming beam particle, and the other, b, is a proton. pa is a “parton”
taken from a, e.g. the electron or photon itself (DIS and real photons), a slightly virtual photon
radiated from the electron (photoproduction), or a gluon or quark from the structure of a real photon
or (anti)proton. pb is a parton taken from the proton, i.e. a gluon or quark. xa and xb represent the
respective momentum fractions of these partons with respect to their “parent” momenta Pa and Pb.
Note that these correspond to Bjorken x and y as defined in the previous section2 only in the case
of the quark-parton-model approximation to deeply inelastic ep scattering, while they have a different
meaning in other cases. For example, in Fig. 4, the quantity xg (rather than x) corresponds to xb as
defined in Eq. (10), while xa = y in the photoproduction interpretation, and xa = 1 in the hard DIS
interpretation. fapa and f

b
pb

are the probability density functions, or parton density functions (PDFs),
which give the probability, e.g. in the fbpb case, to find a parton of type pb with momentum fraction xb in
a proton. σ̂papb represents the cross section for the partonic hard scattering reaction. This is sometimes
split into the so-called hard process, i.e. the part of the reaction with the highest momentum transfer,
and so-called initial state (i.e. occurring before the hard process) or final state (i.e. occurring after the
hard process) radiation (Fig. 5). Part of the initial state radiation can also be absorbed into the parton
density definition. Three energy scales µa, µF and µR appear in the expansion given by Eq. (10), which
is also called factorisation, because the cross section is separated into semi-independent factors.

The renormalisation scale µR determines the scale at which the value of the strong coupling constant
αs is evaluated, i.e. it is the reference point around which the perturbative Taylor expansion of QCD

2and using the improved Weizsäcker-Williams approximation [20] in the case of y.

8



Figure 5: Example for the factorisation of heavy-flavour production in QCD into proton structure,
photon structure, hard matrix element and fragmentation.

matrix element (ME) calculations is performed. If the expansion is done to all orders in αs, the result
does not depend on the choice of this scale. After truncation of the series to finite order, the neglected
higher-order corrections arising from the contribution of a particular subprocess are minimised if this
scale is chosen to be close to the physical scale of the momentum transfer in this subprocess. Since at
high enough perturbative order there are always different subprocesses with differing physical scales (see
e.g. Figs. 3, 5), no single scale choice can universally cover all such scales. The variation of the cross
section with respect to a variation of the renormalisation scale is used to estimate the uncertainty due
to the finite-order truncation of this perturbative series. Some further aspects concerning the choice of
this scale are discussed in Section 2.11.

The factorisation scale µF = µb determines at which scale the proton PDFs are evaluated. By
default, any initial state radiation (lower blob in Fig. 5) with a momentum transfer smaller than the
factorisation scale will be absorbed into the (usually collinear) PDF definition. In contrast, any initial
state radiation with a momentum transfer larger than this scale, and all final state radiation down
to the fragmentation scale (see Section 2.10), will be considered as part of the matrix element, with
correct (noncollinear) kinematics. On one hand, the choice of a lower factorisation scale therefore gives
a more detailed description of the initial state radiation kinematics at a given order. On the other
hand, the explicit treatment of initial state QCD radiation in the matrix element “uses up” a power
of αs that would otherwise have been available for a real radiation elsewhere in the process, or for a
virtual correction. This effectively reduces the order of the calculation with respect to the case where
the same radiation is absorbed into the PDF definition, and therefore reduces the overall accuracy of the
calculation. Empirically, choosing a factorisation scale equal to or at least similar to the renormalisation
scale has been found to be a good compromise.

The third scale, µa, is conceptually the same as µF in the p̄, p and resolved γ cases, and therefore

9



taken to be equal to it, while in the electron and direct photon case it is the scale at which the electro-
magnetic coupling α is evaluated for the electromagnetic part of the matrix element (see Section 2.9).

2.3 Heavy-quark production at HERA in “leading order”

In general, the terminology “leading order” (LO), next-to-leading order (NLO), etc. for a perturbative
QCD expansion is not unique. It can either refer to a specific power of the strong coupling constant
αs or to a specific number of loops in the perturbative expansion of the matrix elements and/or parton
splitting functions contributing to a given process. In order to be precise, this additional information
thus needs to be quoted explicitly.

At leading (0-loop) order in QCD, as implemented in the form of tree-level 2 → 2 hard matrix
elements in most Monte Carlo generators, charm and beauty production in ep collisions is dominated
by boson-gluon fusion (Fig. 6(a)), complemented by other diagrams (Figs. 6(b-d)). Since a cc̄ or bb̄
pair is being produced (collectively referred to as QQ̄), there is a natural lower cut-off 2mQ for the mass
of the hadronic final state.

(a)

p

γ
b

b

(b)

(
) (d)

(a) (b)

p

γ
g

b

(
) (d)

(a) (b)

(
)

p

γ
b

g

(d)

γ

p

e

b

b

(a) (b)

(
) (d)

direct-γ resolved-γ

(a) γg-fusion (b) hadron-like (c) excitation (d) excitation

Figure 6: Beauty production processes in leading order (0 loop) QCD as implemented e.g. in PYTHIA
[21]. Note that in (c) and (d) the “photon remnant” (arrow arising from the photon) contains a b̄ quark.

In the so-called massless approach (Zero-Mass Variable-Flavour-Number Scheme, ZMVFNS), in
which the heavy quark mass is set to 0 for the computation of the matrix elements and kinematics, this
natural cut-off is replaced by an artificial cut-off (“flavour threshold”) at Q2 ∼ m2Q for deeply inelastic
scattering, or pT ∼ mQ for photoproduction. Below this threshold, which is often applied at the level of
the factorisation scale, the heavy-flavour production cross section is (unphysically) assumed to vanish.
Above this threshold, heavy quarks are assumed to occur as massless partons in the proton, like the
u, d, and s quarks (Figs. 1(b) and 2). Except for cases in which both final state charm quarks have
large transverse momenta p2T > µ

2
F , the gluon splitting to QQ̄ in Fig. 6(a) is thus assumed to happen

inside the proton, and to be part of the evolution of the parton density functions. The running of αs
is calculated using 3 flavours (u, d, s) below the renormalisation scale mc, using 4 flavours (including
charm) between mc and mb, and using 5 flavours above the scale mb. This results in a quark-parton-
model-like scattering of the electron off a heavy quark “in the proton” (Fig. 2), defining the concept
of the heavy-quark PDF. In this picture the leading-order process is now an O(α0s) process, while the
boson-gluon-fusion graph (Fig. 1(a), with both heavy quarks at high pT ) is treated as part of the O(αs)
next-to-leading order corrections. This illustrates the partially ambigous meaning of terms like LO,
NLO, etc., discussed at the start of this subsection.

Higher order corrections can be applied either by explicitly including them into the calculation of the
matrix elements and/or splitting functions, or, if they are to be applied at tree level only, by adding an
additional so-called parton shower step. In the latter case, also referred to as leading order plus leading

10



log parton shower (LO+PS), the outgoing and incoming partons of the core “hard” matrix elements are
evolved forward or backwards using splitting functions as they are applied during the PDF evolution.
Most MCs (e.g. PYTHIA [21], HERWIG [22] and RAPGAP [23]) use the standard DGLAP evolution,
as implemented e.g. in JETSET [24] for this purpose. However, in contrast to the PDF evolution, finite
transverse momenta are assigned to the partons. Some MCs (e.g. ARIADNE [25]) use a colour dipole
model for this evolution, while others use BFKL [26] or CCFM [27] inspired so-called kt factorisation
(e.g. CASCADE [28]). In the context of such MCs, the first diagram in Fig. 6 is referred to as direct
production or flavour creation, and the third and fourth are referred to as flavour excitation (in the
photon). Either the second only (e.g. PYTHIA) or collectively the last three (e.g. HERWIG) are
being referred to as resolved photon processes. The second can uniquely be referred to as a hadron-like
resolved-photon process.

For the explicit generation of heavy-flavour final states in such LO+PS MCs, the boson-gluon-fusion
diagram (Fig. 1(a)) is optionally treated using massive matrix elements, while for all other diagrams
the massless treatment remains the only available option.

2.4 Quark-mass definition

The heavy-quark masses appear in theoretical QCD calculations in several ways. Their physical defini-
tion arises from their appearance as parameters in the QCD Lagrangian. The exact value of the masses
depends on the renormalisation scheme applied. In the MS scheme, the masses are defined as pertur-
bative scale-dependent running parameters (MS running mass), similar to the running strong coupling
constant. In the on-shell mass renormalisation scheme, the masses are defined as the poles of the quark
propagator (pole mass), similar to the usual definition of the lepton masses. This is also the definition
which one would naively expect to enter phase space calculations. However, since quarks do not exist
as free particles, and since the definition of the propagator pole inevitably involves contributions from
the nonperturbative region, the pole mass definition has an intrinsic uncertainty of order ΛQCD [29].
At next-to-leading (one loop) order in perturbation theory, the relation between the pole and running
mass definitions can be expressed as [30]

mQ(mQ) = m
pole
Q (1 −

4αs(mQ)

3π
), (11)

where the running mass has been expressed in terms of its value at “its own scale”. Its scale dependence
can be expressed as [30]

mQ(µ) = mQ(mQ)(1 −
αs(µ)

π
ln

µ2

m2Q
), (12)

or alternatively as [31]

mQ(µ) = mQ(mQ)
(
αs(µ)
π

)
1
β0

(
αs(mQ)

π
)

1
β0

, (13)

with β0 =
9
4
. Higher order expressions can also be found in the quoted references.

At leading (0 loop) order, the difference between the two definitions vanishes. Finally, in the context
of so-called massless schemes, the “mass” is defined as a kinematic cutoff parameter in certain parts of
the theory calculations.

The pole-mass definition has been used in most QCD calculations relevant for this review. In recent
variants of the ABKM [32] and ACOT [33] schemes, the MS-running-mass definition is used instead.
The latter has the advantage of reducing the sensitivity of the cross sections to higher order corrections,
and improving the theoretical precision of the mass definition [32].

11



2.5 The zero-mass variable-flavour-number scheme

In its “NLO” variant, including one-loop virtual corrections (Fig. 7)(b)), the ZMVFNS has been used
for most3 NLO variable-flavour parton-density fits up to a few years ago, such as CTEQ6M [35], ZEUS-S
[36], H1 [37], NNPDF2.0 [38].

(a) (b) (c) (d)

Figure 7: Leading order (O(α0s)) (a) and selection of next to leading order (O(α1s)) (b)–(d) processes
for heavy flavour production in DIS in the massless scheme. For (b), only the interference term with
(a) contributes at this order.

One of its advantages is that e.g. next-to-leading-log (NLL) resummation of terms proportional to
log(Q2/m2Q) can be applied to all orders, avoiding the problem that such logs could spoil the convergence
of the perturbation series at high momentum transfers. However, it is clear that this simplified approach
can not give the correct answer for processes near the “flavour threshold”. This has been verified
experimentally e.g. for the DIS case [39] (Section 8.2). Also, it was found that neglecting the charm
mass in the cross section calculations used for the PDF extraction can result in untolerably large effects
on theoretical predictions even at high scales, such as W and Z production at the LHC [40]. All
more recent PDF approaches [41, 42, 43, 44, 45, 46, 47] therefore include at least a partial explicit
consideration of the charm mass in the matrix elements (Sections 2.6 and 2.7). Nevertheless, since
higher orders are more easily calculable in this scheme, the massless approach can offer advantages e.g.
in high-energy charm-photoproduction processes [48, 49] in which the consideration of an extra order
of αs in the final state allows a reduction of the theoretical uncertainty (Section 6).

2.6 The massive fixed-flavour-number scheme

The fixed-flavour-number scheme (FFNS) treats the heavy-quark masses explicitly and follows a rigourous
quantum field theory ansatz. Full NLO (one loop) calculations of heavy-flavour production in this
scheme exist for DIS [50, 51, 52, 53, 54, 55, 46, 47], for photoproduction [56, 57, 58, 59] and for
hadroproduction [60]. Some partial NNLO (two-loop) calculations are also available [61, 62]. In this
scheme, heavy flavours are treated as massive at all scales, and never appear as an active flavour in the
proton. In the case in which all heavy flavours are treated as massive, the number of light flavours in
the PDFs is thus fixed to 3, and charm as well as beauty are always produced in the matrix element
(Fig. 8). So-called flavour excitation processes (Fig. 6(c,d)), which are often classified as leading order
(O(αs)) QCD in partially massless MC approaches [21, 22, 23] of charm or beauty production, appear
as O(α2s) NLO corrections in the fully massive approach (Fig. 8).

There are several variants of the FFNS for heavy-flavour production in DIS (see e.g. remarks in
appendix of [63]). In one approach, here called FFNS A, the αs evolution used together with the 3 flavour
PDFs is also restricted to 3 flavours. Thus, the small contribution from heavy flavour loops (Fig. 9(a))

3 MRST98 [34] is a notable early exception.

12



(a) (b) (c) (d) (e)

Figure 8: Leading order (a) and selection of next to leading order (b)–(e) processes for heavy flavour
production at HERA in the massive scheme.

is either treated explicitly in the matrix elements (FFNS A) [46], or, somewhat incorrectly, neglected
completely (FFNS A′) [52, 54]. In either variant, this leads to a lower effective value of αs than in the
massless scheme when evolved to high reference scales, e.g. αs(MZ). This is one of the consequences
of the non-resummation of log(Q2/m2Q) terms, and is partially compensated e.g. by a conceptually
larger gluon PDF. Despite the conceptual disadvantage of not allowing all order resummation of mass
logarithms, this scheme yields very reasonable agreement with charm and beauty data at HERA up to
the highest Q2 and p2T (Sections 6–8). At HERA energies, the numerical differences between schemes
A and A′ are of order 1%, and therefore almost negligible compared to the current data precision.

(a) (b) (c)

Figure 9: Heavy Flavour loop correction (a) and gluon splitting (b,c) processes in the massive scheme.
The thick (thin) lines indicate heavy (light) flavours.

In the FFNS B approach4, which was widely used in the early nineties [50, 51, 56, 60, 68], the running
of αs is calculated by incrementing the number of flavours when crossing a flavour threshold, like in
the variable-flavour-number scheme. The class of logs corresponding to this running is thus resummed
both in the αs and in the PDF evolution, and the “missing” heavy flavour log resummation is restricted
to other cases like gluon splitting and vertex corrections in the matrix elements. This approach is
possible since most of the loop and leg corrections which diverge in the massless case, but compensate
each other to yield finite contributions, remain separately finite in the massive case. They can thus
be separated. At one-loop order, the A and B approaches differ by the way a virtual heavy flavour
correction in the BGF matrix element (Fig. 9(a)) (which is missing in the A′ approach) is treated. The
FFNS B scheme conceptually yields a value of αs at high scales which is the same as the one from the
variable-flavour approach, and is probably less sensitive to “missing logs” at very high scales than the
FFNS A approach. Most NLO photoproduction [56] and hadroproduction [60] calculations, as well as

4Elsewhere [64] this is sometimes called the mixed flavour number scheme. For a discussion see [65][66]. A recent new
variant of it [67] is referred to as the “doped” scheme.

13



the electroproduction code HVQDIS [51] have been originally designed to be used with the FFNS B
scheme.

The A and B approaches both converge to the exact QCD result at infinite order if implemented
consistently. The A′ scheme can not converge to the exact result since heavy flavour loop corrections
are completely missing, but, as stated earlier, the practical consequences at HERA energies are small.
It can however serve as a useful ingredient to variable flavour number scheme calculations (Section 2.7).

There are also other differences. In the ABKM approach, final state gluon splitting (Fig. 9(b,c)) is
conceptually treated as part of the light-flavour contribution, while it is treated as part of heavy flavour
production in many others [50, 51, 56, 60] (see Section 2.8), as well as in the measured cross sections,
since it can hardly be distinguished experimentally. This difference is small [69] in most regions of phase
space, but might need to be accounted for when comparing data and theory.

Finally, FFNS calculations in DIS are currently available in leading order (O(αs)), or NLO (O(α2s))
[46, 50, 51]. Partial NNLO (two-loop, O(α3s)) calculations also exist, based on a full calculation of
the O(α3s)log and the O(α3s)log2 terms, and the leading term from threshold resummation for the
O(α3s) constant term [61]. Further NNLO corrections for the high scale limit [16] have not yet been
implemented in practice. Actually, both the NLO and partial NNLO ABKM heavy flavour calculations
use the PDFs from their NNLO (two loop, O(α2s) in the matrix elements) fit to the inclusive data [46].

Some of the differences between the calculations discussed in this section and in the next two sections
are also summarised in Table 1, using the example of reduced charm cross sections in DIS.

2.7 The general-mass variable-flavour-number scheme

An alternative to the fixed-flavour-number approach is given by the so-called general-mass variable-
flavour-number schemes (GMVFNS) [70, 71, 72, 73]. In these schemes, charm production is treated
in the FFNS approach in the low-Q2 region, where the mass effects are largest, and in the massless
approach at very high scales, where the effect of resummation is most noticeable. At intermediate scales
(in practice often at all scales above the “flavour threshold”), an interpolation is made between the two
schemes, avoiding double-counting of common terms, while making a continous interpolation between
differing terms. This scheme combines the advantages of the two previous schemes, while introducing
some level of arbitrariness in the treatment of the interpolation.

γ∗
c

c
c -cX(  )

γ∗
c

c

X

-c

G

c

-c

X

γ∗

G

- +

Figure 10: Leading order (0-loop) diagrams for charm production in DIS in the variable flavour number
scheme: On the left the QPM diagram is shown, on the right the BGF diagram and in the middle the
“subtraction diagram” [74]. The vertex correction loop diagram (Fig. 7(b)), which also contributes to
this order in αs, is not shown.

One of the most constrained schemes is the BMSN scheme [73] used by the VFNS approach of
ABKM [62]. At NLO, it interpolates between O(α2s) charm matrix elements in the FFNS part using the
ABKM FFNS scheme, and O(αs) matrix elements in the massless part. It has no tuneable parameters,
and (currently) uses the pole-mass definition for the FFNS part. In contrast to most other GMVFNS
schemes, the switch to a larger number of flavours should not be made at the “flavour threshold”,
but at a scale which is high enough that additional semi-arbitrary kinematic correction terms are not
required. In practice, the 3-flavour scheme is used for processes at HERA energies, while the 4- or
5-flavour schemes are recommended for applications at the LHC.

14



Theory Scheme Ref. F2(L) mc PDF Massive / FL Massless F2 αs(mZ) Scale

def. [GeV] (Q2 ∼< m
2
c) (Q

2 ≫ m2c) (nf = 5)
MSTW08 NLO RT standard [75] F c

2(L)
1.4 (pole) O(α2s) O(α2s) O(αs) 0.12108 Q

MSTW08 NNLO O(α3s) approx.-O(α3s) O(α2s) 0.11707
MSTW08 NLO (opt.) RT optimised [76] O(α2s) O(α2s) O(αs) 0.12108
MSTW08 NNLO (opt.) O(α3s) approx.-O(α3s) O(α2s) 0.11707
HERAPDF1.5 NLO RT standard [42] F c

2(L)
1.4 (pole) O(α2s) O(α2s) O(αs) 0.1176 Q

NNPDF2.1 FONLL A FONLL A [77] n.a.
√
2 O(α2s) O(αs) O(αs) 0.119 Q

NNPDF2.1 FONLL B FONLL B F c
2(L)

√
2 (pole) O(α2s) O(α2s) / O(αs) O(αs)

NNPDF2.1 FONLL C FONLL C F c
2(L)

√
2 (pole) O(α3s) O(α2s) O(α2s)

CT10 NLO S-ACOT-χ [55] n.a. 1.3 O(α2s) O(αs) O(αs) 0.118
√

Q2 + m2c

CT10 NNLO [78] F cc̄
2(L)

1.3 (pole) O(α2s) O(α2s) O(α2s)
ABKM09 NLO FFNS A [46] F cc̄

2(L)
1.18 (MS) O(α2s) O(α2s) - 0.1135

√

Q2 + 4m2c

ABKM09 NNLO O(α3s) approx.-O(α3s) -
HVQDIS+ZEUS S FFNS B [51] F c

2(L)
1.5 (pole) O(α2s) O(α2s) - 0.118

√

Q2 + 4m2c

Table 1: Selected calculations for reduced charm cross sections in DIS from different theory groups as used in this review. The table shows
the heavy flavour scheme used and the corresponding reference, the respective F2(L) definition (Section 2.8), the value and type of charm
mass used (Section 2.4), the order in αS of the PDF part and the massive and massless parts of the calculation (and of the massless part
of FL, which, except for FONLL B, is usually taken to be the same as for the massive part), the value of αs, the renormalisation and
factorisation scale. The distinction between the two possible F2(L) definitions is not applicable (n.a.) for O(αs) calculations, or in photo-
or hadroproduction. Usually, the order of the PDF part is used to define the label LO, NLO, or NNLO.

1
5



The NLO version of the TR5 scheme [71] combines the O(α2s) charm matrix elements in the FFNS
A′ scheme with the O(αs) matrix elements of the massless scheme, requiring continuity of the physical
observables in the threshold region. In this case the usage of the A′ scheme is fully appropriate, since the
missing terms will be taken care of by the massless and interpolation terms. Several variants exist for the
interpolation, including the so-called standard scheme used e.g. in MSTW08 [43] and HERAPDF1.0
[41], and the optimised scheme preferred for more recent versions, since it avoids a kink in the Q2

dependence of the cross section [76]. Both of these variants also exist in a partial NNLO approach [75],
including approximate O(α3s) threshold resummation terms for the FFNS part, and a full O(α2s) NNLO
calculation for the massless part. They all use the pole mass definition for the FFNS part.

The ACOT [70] scheme, used by CTEQ [44], also exists in several variants. At NLO, O(αs) (i.e.
leading order) FFNS matrix elements are interpolated to O(αs) (now NLO) massless matrix elements6.
Due to the LO FFNS treatment, there is no difference between the pole-mass and running-mass schemes.
The interpolation is made in two variants: the S-ACOT approach [79], and the ACOT-χ approach [80].
The NNLO variant of CT10 [81] uses both FFNS and massless matrix elements at O(α2s), in the S-ACOT
scheme.

The FONLL scheme [69] has 3 variants. The FONLL A approach, used by NNPDF2.1 [45] is
equivalent [69] to the CTEQ S-ACOT approach, and uses O(αs) FFNS heavy-quark matrix elements
at NLO. FONLL B and C both use O(α2s) FFNS heavy-quark matrix elements. FONLL B uses O(αs)
matrix elements for light quarks, like MSTW, while FONLL C uses O(α2s) matrix elements for light
quarks like ABKM. However, they differ from the latter in the way they treat the interpolation terms.
The FONLL C scheme is also similar [69] to the CTEQ S-ACOT NNLO scheme. A full NNLO version
of the FONLL A approach also exists [77, 82]. Final state gluon splitting is not included in the charm
cross-section predictions for any of these schemes.

The ABM group uses the BSMN approach [73] to generate a GMVFNS scheme out of their FFNS
3-, 4- and 5-flavour PDFs [62].

For photoproduction, a GMVFNS calculation [83] exists for single inclusive cross sections.
All GMVFNS variants use the variable-flavour approach for the running of αs. Although the mass

is unambigously defined in the massive part of the calculation (usually the pole mass), the partial
arbitrariness in the treatment of the interpolation terms (Fig. 10) prevents a clean interpretation
of the charm and beauty quark masses in terms of a single renormalisation scheme. Therefore, in
contrast to the pure FFNS treatment, the charm mass appearing in VFNS schemes can be treated
as an effective mass parameter [84]. We will use the symbols Mc and Mb for these effective mass
parameters. Alternatively, the presence of the interpolation terms can be included as an additional
uncertainty on the respective mass definition [33].

2.8 Proton structure functions in DIS

In analogy to the inclusive neutral current DIS cross section, the cross sections for heavy-quark pro-
duction in DIS can be expressed in terms of the heavy-quark contributions to the inclusive structure

functions [17] F2, FL and F3, denoted by F
QQ̄
2 , F

QQ̄
L and F

QQ̄
3 (Q = c, b):

dσQQ̄(e±p)

dx dQ2
=

2πα2

x Q4

(

(1 + (1 − y)2) F QQ̄2 − y2 F
QQ̄
L ∓ x (1 − (1 − y)2) F

QQ̄
3

)

, (14)

where α is the electromagnetic coupling constant. The structure function F
QQ̄
2 makes the dominant

contribution to the neutral current scattering in the kinematic regime accessible at HERA. F
QQ̄
3 contains

5also referred to as RT
6The fact that similar matrix elements are denoted by different labels concerning their effective order in different

context is very confusing, but unavoidable due to different definitions of the truncation of the perturbative QCD series.

16



contributions only from γZ0 interference and Z0 exchange, therefore for the region Q2 ≪ M2Z, which was
studied at HERA, this contribution is suppressed and can be neglected. The longitudinal heavy-quark

structure function F
QQ̄
L parametrises the contribution from coupling to the longitudinally polarised

photons. The contribution of F
QQ̄
L to the ep cross section is suppressed for y

2 ≪ 1, but can be up to a
few percent in the kinematic region of the heavy-quark measurements at HERA and thus can not be
neglected.

For both electron and positron beams, neglecting the F
QQ̄
3 contribution, the reduced heavy-quark

cross section, σ
QQ̄
red , is defined as

σ
QQ̄
red (x, Q

2) =
dσQQ̄(e±p)

dx dQ2
· x Q

4

2πα2Y+
= F

QQ̄
2 −

y2

Y+
F

QQ̄
L , (15)

where Y± = (1 ± (1 − y)2). Thus, σQQ̄red and F
QQ̄
2 only differ by a small F

QQ̄
L correction at high y [85].

In the Quark-Parton Model, the structure functions depend on Q2 only and can be directly related
to the parton density functions. In the QCD case, and in particular for heavy flavour production, this
correlation is strongly diluted, and the structure functions depend on both x and Q2. More information
on the general case can be found e.g. in [17, 41].

Using the example of the charm case [84], the above definition of F cc̄2(L)(x, Q
2) (also denoted as F̃c [69]

or Fc,SI [82]) is suited for measurements in which charm is explicitly detected. It differs from what is
sometimes used in theoretical calculations in which F c2(L)(x, Q

2) [69, 75, 86] is defined as the contribution

to the inclusive F2(L)(x, Q
2) in which the virtual photon couples directly to a c or c̄ quark. The latter

excludes contributions from final state gluon splitting to a cc̄ pair in events where the photon couples
directly to a light quark, and contributions from events in which the photon is replaced by a gluon
from a hadron-like resolved photon. As shown in table 1 of [69], the gluon splitting contribution is
expected to be small enough to allow a reasonable comparison of the experimental results to theoretical
predictions using this definition. The hadron-like resolved photon contribution is expected to be heavily
suppressed at high Q2, but might not be completely negligible in the low Q2 region. From the point of
view of pQCD it appears at O(α3s) and it is neglected in all theoretical DIS calculations used in this
review.

2.9 QED corrections

In addition to the different QCD schemes discussed above, predictions of charm production can also
differ through their treatment of QED corrections. Some of these corrections, e.g. collinear photon
radiation from the initial state electron before the hard interaction (Fig. 11) can actually be large (of

order α ln
Q2max
m2e

), and can influence the definition of the Q2, x and y variables [17]. For photoproduction

calculations, the improved Weizsäcker-Williams approximation [20] can be used to parametrise the
photon spectrum arising from the incoming electron.

For acceptance corrections (and partially for visible cross sections), predictions including full LO
QED radiative corrections, as implemented e.g. in HERACLES [87], are used. For more sophisticated
purposes, an NLO version of these corrections is available in the HECTOR package [88]. At the level of
the DIS structure functions or σr it is customary to translate the measured cross sections to so-called
Born-level cross sections, i.e. cross sections in which all QED corrections have been removed, to ease
comparison of the data with pure QCD predictions. There is one potential exception: the fine structure
“constant” α can be used in two different ways.

• as a genuine atomic scale constant α = 1
137.036

, i.e. all virtual QED corrections are removed, too;

• as running α in the MS scheme, i.e. the respective relevant virtual corrections are kept, and a
typical value for HERA kinematics is then α ≃ 1

134
[89].

17



e+

p

γ

γ c, b

c, b√
αs

g

Figure 11: BGF diagram with initial state photon radiation.

The difference between these two approaches in QED has some remote similarity to the difference
between the FFNS A and B schemes in perturbative QCD (Section 2.6), treating all quarks and leptons
as “heavy” with respect to the atomic scale.

2.10 Fragmentation

Equation (10) allows one to make predictions of heavy-quark production with partons in the final state.
However, cross sections are measured and reported mostly in terms of heavy-flavour hadrons, leptons
from their decay, or collimated jets of hadrons. Therefore such predictions have to be supplemented
with a fragmentation or hadronisation model.

In analogy to e+e− collisions [90], the factorised cross section for the production of a heavy-quark
hadron H as a function of transverse momentum pHT can be written as:

dσH

dpHT
(pHT ) =

∫

dp
Q
T

p
Q
T

dσQ

dp
Q
T

(p
Q
T , µf) D

H
Q (

pHT
p
Q
T

, µf) · f(Q → H), (16)

where σQ(x, µf) is the production cross section for heavy quarks (Eq. (10)), D
H
Q is the fragmentation

function, µf is the fragmentation scale and f(Q → H) in the fragmentation fraction. The latter is
defined as the probability of the given hadron H to originate from the heavy quark Q. The fragmentation
function defines the probability for the final-state hadron to carry the fraction z = pHT /p

Q
T of the heavy-

quark momentum. The fragmentation function is defined similarly to the PDFs. In the “massless”
approximation, it is defined at a starting scale and has to be evolved to a characteristic scale µf
of the process using perturbative QCD. In the massive fixed flavour approach, this evolution can be
conceptually absorbed into the pole mass definition.

Fragmentation fractions as well as the starting parametrisation of the fragmentation function can
not be calculated perturbatively. Thus they have to be extracted from data. Comprehensive phe-
nomenological analyses of the charm and beauty fragmentation functions in e+e− collisions have been
performed [91, 92, 93]. While the QCD evolution is process-dependent, the non-perturbative ingredients
of the fragmentation model are assumed to be universal7. Comparing measurements from HERA and
results from e+e− colliders one can test this universality.

However, the tools that are available for O(α2s) fixed-order calculations of the heavy-quark produc-
tion cross sections in ep collisions, which are mostly used for exclusive final states, do not comprise a

7The non-perturbative fragmentation function is universal only if it is accompanied by appropriate evolution.

18



perturbative component of the fragmentation function. Therefore, an “independent” non-perturbative

fragmentation function DNP(z) is used in conjunction with the parton-level cross sections. The para-
metric forms of the independent fragmentation functions most commonly used at HERA are due to
Peterson [94]:

DNP(z) ∝
1

z(1 − 1/z − ε/(1 − z))2
,

Kartvelishvili [95]:

DNP(z) ∝ zα(1 − z)
and to the Bowler modification of the symmetric Lund [96] parametrisation:

DNP(z) ∝ 1/z1+rQbm
2
Q (1 − z)a exp(−b(m2H + p2T )/z),

where ǫ, α, a, b and rQ are free parameters that depend on the heavy-flavour hadron species and have
to be extracted from data. Since no QCD evolution is applied, the corresponding parameters might be
scale- and process-dependent.

The recent GMVFNS NLO predictions for charm photoproduction [83] incorporate a perturbative
fragmentation function and have been tested against data (Section 6).

2.11 Choice of renormalisation scale

For many cross-section predictions the dominant contribution to the theoretical uncertainty arises from
the variation of the renormalisation and factorisation scales by a factor 2 around some suitably chosen
default scale. Such a variation is intended to reflect the uncertainty due to uncalculated higher orders. It
might therefore be useful to consider some phenomenological aspects of these scale choices as considered
in a mini-review on beauty production at HERA and elsewhere [97], focusing in particular on the choice
of the renormalisation scale.

Ideally, in a QCD calculation to all orders, the result of the perturbative expansion does not depend
on the choice of this scale. In practice, a dependence arises from the truncation of the perturbative
series. Since this is an artefact of the truncation, rather than a physical effect, the optimal scale can
not be “measured” from the data. Thus, it must be obtained phenomenologically.

Traditionally, there have been several options to choose the “optimal” scale, e.g.

• The “natural” scale of the process. This is usually taken to be the transverse energy, ET , of the
jet for jet measurements, the mass, m, of a heavy particle for the total production cross section of

this particle, or the combination
√

m2 + p2T for differential cross sections of such a particle. Often,
this is the only option considered. The choice of this natural scale is based on common sense, and
on the hope that this will minimise the occurrance of large logs of the kind described above, for
the central hard process. However, higher order subprocesses such as additional gluon radiation
often occur at significantly smaller scales, such that this choice might not always be optimal.

• The principle of fastest apparent convergence (FAC) [98]. The only way to reliably evaluate
uncalculated higher orders is to actually do the higher-order calculation. Unfortunately, this is
often not possible. Instead, one could hope that a scale choice which makes the leading-order
prediction identical to the next-to-leading-order one would also minimise the NNLO corrections.
This principle, which can be found in many QCD textbooks, can not be proven. However, recent
actual NNLO calculations might indicate that it works phenomenologically after all (see below).

• The principle of minimal sensitivity (PMS) [99]. The idea is that when the derivative of the cross
section with respect to the NLO scale variation vanishes, the NNLO corrections will presumably

19



0

20

40

60

80

1
µr /

 
MH

0.2 0.5 2 3

σ(pp → H+X) [pb]
MH = 120 GeV

LO

NLO

N2LO

N3LOapprox

Figure 12: Scale dependence of the total cross section for beauty production at HERA-B [100] (left) and for
Higgs production at the LHC [101] (right).

also be small. Again, there is no proof that this textbook principle should work, but actual NNLO
calculations might indicate that it does (see below).

To illustrate these principles, consider two examples. First, the prediction for the total cross section
for beauty production at HERA-B [100] (Fig. 12). The natural scale for this case is the b-quark mass,
µ0 = mb, and all scales are expressed as a fraction of this reference scale. Inspecting Fig. 12, one finds
that both the PMS and FAC principles, applied to the NLO prediction and to the comparison with
LO (NLO stability), would yield an optimal scale of about half the natural scale. The same conclusion
would be obtained by using the NLO+NLL prediction, including resummation, and comparing it to
either the LO or the NLO prediction (NLO+NLL stability).

Second, the prediction for Higgs production at the LHC [101] (Fig. 12). The reference scale is now
the Higgs mass (µ0 = mH). However, inspecting the behaviour of the LO and NLO predictions, neither
the FAC nor the PMS principle would yield a useful result in this case, since the two predictions do not
cross, and the NLO prediction does not have a maximaum or minimum. This situation occurs rather
frequently, and is also true for b production at HERA. Fortunately, in the case of Higgs production, the
NNLO and even approximate NNNLO predictions have actually been calculated (Fig 12). Applying the
FAC and PMS prescriptions to these instead (NNLO stability), again a scale significantly lower than
the default scale would be favoured. This might indicate that choosing a scale which is smaller than
the default one makes sense even if the FAC and PMS principles do not yield useful values at NLO.

Beyond these examples, a more general study is needed to phenomenologically validate this approach.
To avoid additional complications arising from a multiple-scale problem caused by e.g. the scale Q2 at
HERA or the scale MZ at LEP, the study was limited to cross sections for photoproduction at HERA, or
hadroproduction at fixed-target energies, the Tevatron, and LHC. The somewhat arbitrary selection of
processes includes beauty production at the Spp̄S [102, 103], the Tevatron [103], and HERA-B [100], top
production at the Tevatron [100, 103], direct photon production at fixed target [104], Z [105] and Higgs
[101] production at the LHC, jets at HERA [106] and at the Tevatron [107]. This selection is obviously
not complete, and many further calculations, in particular NNLO calculations, have been achieved since
this study [97] was originally made. However, it is not biased in the sense that all processes that were
originally considered were included, and none were discarded. Clearly, a quantitative update of this
study would be useful, but was not yet done. Qualitatively, all newer predictions which the authors
have been made aware of either confirm this conclusion, or at least do not significantly contradict it.

In each case the natural scale as defined above was used as a reference. In addition, wherever

20



Figure 13: Summary of optimised scales derived as described in the text.

possible, the optimal scales from both the FAC and PMS principles, evaluated at NLO (NLO stability),
NLO+NLL (NLO+NLL stability), and/or NNLO/NNNLO (NNLO stability) were evaluated separately.
Figure 13 shows the result of this evaluation. Each crossing point, maximum, or minimum in Fig. 12
yields one entry into this figure, and similarly for all the other processes. The conclusion is that the FAC
and PMS principles tend to favour scales which are around 25-60% of the natural scale. Amazingly, this
seems to be independent of whether these principles are applied at NLO, NLO+NLL, or NNLO level.
For the jet [107] or b-jet [108] cross sections at the Tevatron, it has in part already become customary
to use half the natural scale as the central scale.

Using the natural scale as the default and varying it by a factor two, which is the choice adopted
for most data/theory comparisons, covers only about half the entries, while the other half lies entirely
below this range. Instead, using half the natural scale as the default and varying it by a factor two,
thus still including the natural scale in the variation, covers about 95% of all the entries.

This yields the following conclusions.

• Obviously, whenever an NNLO calculation is available, it should be used.

• Whenever possible, a dedicated scale study should be made for each process for the kinematic
range in question. Although there is no proof that the FAC and PMS principles should work,
in practice they seem to give self-consistent and almost universal answers for processes at fixed
target energies, HERA, the Tevatron, and the LHC.

• In the absence of either of the above, the default scale should be chosen to be half the natural
scale, rather than the natural scale, in particular before claiming a discrepancy between data and
theory. Empirically, this should enhance the chance that the NNLO calculation, when it becomes
available, will actually lie within the quoted error band. To evaluate uncertainties, the customary
variation of the central scale by a factor 2 up and down remains unaffected by this choice.

The latter principle has already been applied to a few of the results covered in this review. Of
course, choosing the natural scale as the central value, which is still the default for most calculations
(or making any other reasonable scale choice), is perfectly legitimate and should also describe the data
within the theoretical uncertainties. However, if it does not, it might be useful to consider alternative
choices as discussed above before claiming evidence for the failure of QCD, and hence for new physics.

21



Further complementary information, in particular on the related theory aspects, is available else-
where [109].

2.12 Summary

The theory of heavy flavour production in the framework of perturbative QCD, and in particular the
occurrence of different possibilities to treat the heavy quark masses in the PDF, matrix element and
fragmentation parts of the calculation, introduces a significant level of complexity into the corresponding
QCD calculations, in addition to the usual scheme and scale choices. Confronting different choices with
data can be helpful to understand the effects of different ways to truncate the perturbative series.

The majority of the available MC calculations for the analysis of HERA data is based on leading
order (plus parton shower) approaches, combining a massive approach for the core boson-gluon fusion
process, and the massless approach for tree level higher order corrections. It will be demonstrated in the
later chapters that this is fully adequate for acceptance corrections. For comparisons of the measured
differential cross sections with QCD predictions, a next-to-leading order massive approach (fixed flavour
scheme) is the state of the art.

In some cases massless calculations are still in use, e.g. to facilitate the perturbative treatment of
fragmentation, or to implement resummation of some of the logarithms arising when the mass competes
with other hard scales occurring in a process. In particular for the prediction of the inclusive heavy
flavour structure functions in DIS, a variety of so-called general-mass variable-flavour schemes are
available, merging massive calculations at low scales with massless calculations at high scales. These
are particularly useful for the extraction of PDFs over very large ranges in energy scale. Partial NNLO
calculations are also available for such inclusive quantities, both in the fixed and variable flavour number
schemes. Due to the absence of extra semi-arbitrary parameters, the fixed flavour number scheme is
particularly well suited for the extraction of QCD parameters like the heavy quark masses.

In general, QED corrections are nonnegligible, and available both at leading and next-to-leading
order. Since α is much smaller than αs, the leading order precision is often sufficient. Several compet-
ing fragmentation models are in use, and the perturbative treatment of fragmentation in the massive
approach is still in its infancy.

Since higher order corrections are large, the uncertainties reflected by the QCD scale variations are
often dominant. Until full NNLO calculations become available, a careful consideration of the choice of
these scales can be helpful to avoid premature conclusions concerning potential discrepancies between
the theory predictions and the data.

22



3 The HERA collider and experiments

In this section, the HERA collider, the H1 and ZEUS experiments, as well as the reconstruction of the
data from these experiments will be briefly described, with focus on aspects relevant for heavy flavour
production.

3.1 HERA

HERA (German: Hadron-Elektron-Ring-Anlage) was the first and so far the only electron–proton
collider. It was located at DESY in Hamburg, Germany. The circumference of the HERA ring (see
Fig. 14) was 6.4 km. The accelerator was in operation from 1992 to 2007. Electrons or positrons and

Figure 14: A schematic view of the HERA ep storage rings with the H1 and ZEUS experiments

protons were accelerated in two separate rings to final energies of 27.5−27.7 GeV and 920 GeV (820 GeV
before 1998), respectively, leading to a centre-of-mass energy of

√
s = 318 − 319 GeV (300 GeV before

1998). Both beams were stored in 180 bunches. The bunch-crossing rate was 10 MHz. Electrons and
protons collided in two interaction regions, where the H1 and ZEUS detectors were located.

In the years 2001 – 2002 the HERA collider was upgraded to increase the instantaneous luminosity.
At the same time a number of upgrades of the H1 and ZEUS detectors were put in place, as described
below. Therefore, the data taking was subdivided into two phases: “HERA I” and “HERA II” corre-
sponding to the data taking periods 1992 – 2000 and 2003 – 2007, respectively. In 2007, a few months
were dedicated to data taking at lower centre-of-mass energies.

3.2 H1 and ZEUS Detectors

The H1 and ZEUS detectors were typical modern multi-purpose collider experiments and are described
in detail in [110, 111, 112, 113]. Figure 15 visualises the layout of the H1 and ZEUS detectors8. Due to
significantly higher energy of the protons there was more detector hardware installed in the direction

8The right-handed Cartesian coordinate system used at H1 and ZEUS has the Z axis pointing in the nominal proton
beam direction, referred to as the “forward direction”, and the X axis pointing towards the centre of HERA. Its coordinate

23



e± p

❄

❈
❈
❈
❈
❈❲

LAR calor.

❄

❄

Central Jet Ch.

✁
✁
✁
✁
✁
✁
✁✕

Forward tracker

✄
✄
✄
✄
✄
✄✎ ❄

SpaCal

ZR View

e± p

✻

MVD

�
�

�
�

�
�

��✒

CTD

❄

✑
✑

✑
✑

✑
✑

✑✑✰

❍
❍
❍

❍
❍

❍
❍
❍

❍
❍❥

CAL

✘✘
✘✘

✘✘
✘✘✿

STT

Figure 15: The rz-view of the tracking system and calorimeters of the H1 (top) and ZEUS (bottom)
detectors. The layout of the central silicon tracker (CST) and the microvertex detector (MVD) are
shown separately below. The electron beam enters from the left, while the proton beam enters from
the right.

24



H1 ZEUS

Silicon vertex detector: CST MVD

# layers 2 3

θ-coverage [30◦, 150◦] [7◦, 150◦]

η-coverage [−1.3, 1.3] [−1.3, 2.8]
σ(δ) 43 ⊕ 51/pT µm 46 ⊕ 122/pT µm

Drift chambers: CTD CTD

θ-coverage [20◦, 160◦] [15◦, 164◦]

η-coverage [−1.74, 1.74] [−2.0, 2.0]
σ(pT )/pT 0.002pT ⊕ 0.015 0.0029pT ⊕ 0.0081 ⊕ 0.0012/pT

Calorimeters: LAr CAL (HERA I)

θ-coverage [4◦, 154◦] [2.6◦, 176.2◦]

η-coverage [−1.46, 3.35] [−3.4, 3.8]
El.-magn. σ(E)/E 0.12/

√
E ⊕ 0.01 0.18/

√
E

Hadronic σ(E)/E 0.50/
√
E ⊕ 0.02 0.35/

√
E

SpaCal

θ-coverage [153◦, 178◦]

η-coverage [−3.95, −1.43]
El.-magn. σ(E)/E 0.07/

√
E ⊕ 0.01

Muon systems: CMD R/B/FMUON+BAC

θ-coverage [4◦, 171◦] [5◦, 171◦]

η-coverage [−2.5, 3.4] [−2.5, 3.1]

Table 2: Parameters and performance of the H1 and ZEUS subdetectors which are relevant for the
heavy-flavour physics analyses presented in this review. The benchmarking is shown for the HERA II
run conditions, unless stated otherwise. Transverse momenta pT and energies E are in units of GeV. δ
is the transverse distance of closest approach of tracks to the nominal vertex. σ(δ) is the resolution of
δ, averaged over the azimuthal distribution of tracks.

of the outgoing proton beam. The figures show the key parts of the main detectors that were used for
tagging and reconstruction of heavy-flavour events. Some of the most important benchmarks of the
H1 and ZEUS detectors, such as polar angle coverage and momentum, the energy resolution and the
resolution of the impact parameter δ9, are listed in Table 2. In the following the main components of
the H1 and ZEUS detectors are discussed with emphasis on the advantages of the respective designs:

Tracking Chambers: Tracks from charged particles were reconstructed based on the position mea-
surements in the large Central Drift Chambers. The pulse height on the sense wires was used to
measure the energy loss in the detector medium, dE/dx. The dE/dx measurements were used for par-
ticle identification, distinguishing between electrons, pions, kaons and protons in a limited momentum

origin is at the nominal center of the respective detector, which coincided with the nominal interaction point in the HERA I
period. The pseudorapidity is defined as η = − ln (tan(θ/2)), where the polar angle, θ, is measured with respect to the
proton beam direction. The x-y- or r-φ-plane is also denoted as the transverse plane.

9Also referred to as the transverse distance of closest approach to the nominal vertex

25



range. The important differences between H1 and ZEUS are:

• The H1 tracking detector comprised two chambers CJC1 and CJC2 [111] while the ZEUS CTD [114]
was a single chamber that was divided into nine superlayers.

• For ZEUS a superconducting coil surrounded the tracking detectors and provides a magnetic field
of 1.43 Tesla. This is considerably higher than the 1.15 Tesla delivered by the H1 superconducting
coil, situated outside the calorimeter. In both experiments the magnetic field within the tracking
system was parallel to the Z axis.

• Four of the nine superlayers of the ZEUS chambers were equipped with stereo wires, which were
tilted ∼ 5◦ with respect to the beam axis. This provided z-measurement points for tracks with
a resolution of ∼ 1.5 mm. At H1 the sense wires were strung parallel to the beam axis and the
track Z-position measurement was obtained by the division of the charges recorded at both wire
ends, yielding a moderate resolution of a few centimetres. Two additional Z-drift-chambers were
installed to provide for each track a few Z-measurement points with typically 300 µm resolution.

In the forward region H1 and ZEUS have installed a Forward Tracking Detector (FTD) [115]
and a Straw Tube Tracker (STT) [116], respectively, that are based on drift-chambers. Their main
purpose is to extend the polar angular coverage to angles smaller than 20◦, outside the acceptance
of the central drift chambers. However, for both experiments these detectors have not been used for
momentum reconstruction due to a large amount of dead material in front of them. Nevertheless, the
forward detectors were partially used in the pattern recognition.

Vertex-detector: The Central Silicon Tracker (CST) [117] and the Micro Vertex Detector
(MVD) [118] were located in the heart of the H1 and ZEUS experiments. The MVD was installed only
for the HERA II data taking. The vertex detectors allowed the determination of trajectories of charged
particles in the vicinity of the primary vertex. The achieved precision was sufficient to resolve vertices
from secondary decays. This is essential for the tagging of weakly-decaying heavy-flavour hadrons with
a typical lifetime cτ ≃ 100 – 300 µm.

The CST (MVD) consisted of two (three) 36 cm (63 cm) long concentric cylindrical layers of double-
sided silicon-strip detectors The innermost layer of the CST and MVD was located at 57.6 mm and
∼ 45 mm, respectively. The most important benchmarking parameters are given in Table 2. The
following intrinsic hit resolutions were achieved: ∼ 11 µm for the CST and ∼ 24 µm for the MVD.
The CST had a somewhat better average transverse impact parameter resolution mainly due to less
material, hence less multiple scattering, but was essentially restricted to track reconstruction in the
transverse plane. The MVD contained four wheels of double-sided silicon-strip detectors in the forward
region that extended the polar-angle coverage from 20◦ to 7◦. Furthermore, it allowed 3D standalone
pattern recognition. At H1 the CST was supplemented with the forward and backward strip detectors,
FST and BST, that extended the polar-angle coverage of CST to [7◦, 173◦]. However, these were not
used in H1 heavy-flavour analyses.

Calorimeters: The tracking detectors were surrounded by calorimeter systems, which covered almost
the full solid angle. Their main tasks were to identify and measure the scattered electron, to reconstruct
the hadronic final state (e.g. jets) and photons and to separate electrons from hadrons. At H1 a fine-
grain liquid-argon (LAr) sandwich calorimeter [111, 119] was installed in the central and forward region.
It was supplemented in the backward region with the lead-scintillating fibre calorimeter SpaCal [112,
120]. In the ZEUS detector the solenoid was surrounded by a high-resolution compensating uranium–
scintillator calorimeter (CAL) [121]. The calorimeters had inner electromagnetic and outer hadronic
sections. The electron and hadron energy scales of the calorimeters were known at the level of 1% and

26



2%, respectively. The calorimeters were calibrated from the data using kinematic constraints. Overall,
both calorimeter systems performed very well.

Additionally, for detection of very-low-Q2 events, where the electron is scattered at a small angle,
ZEUS installed the beampipe calorimeter (BPC) [122]. This calorimeter was in operation in HERA I
and was located just 4.4 cm from the beam line. It allowed the extension of the phase-space coverage
to 0.05 < Q2 < 0.7 GeV2

Electron taggers: Both H1 and ZEUS were equipped with special detectors, called electron taggers,
which were able to detect electrons scattered at very small angles. Especially during the HERA I period,
these could be used to explicitly identify photoproduction events in specific ranges of W .

Muon systems: To identify muons both experiments installed large arrays of limited-streamer tubes [111,
123] inside and outside the magnetic return yoke (not shown), which covered a wide range in polar an-
gle and measured muons efficiently for transverse momenta above ∼ 2 GeV, with significant partial
acceptance also at lower pT . The return yoke of the ZEUS detector was also equipped with drift tubes
providing complementary muon identification and serving as a backing calorimeter (BAC). A forward
muon system completed the coverage of the tracking detectors. The H1 detector had a similar muon
coverage, including the usage of the liquid argon calorimeter as a tracking calorimeter.

Luminosity measurement: In both experiments the luminosity was measured using the photon
bremsstrahlung process ep → eγp. The photons were detected by dedicated detectors [124, 125] about
100 m away from the interaction points in the e-beam direction. In addition, H1 also used the SpaCal
to measure the large-angle QED compton scattering [126]. The ultimate precision of the luminosity
measurement by H1 (ZEUS) is 2.3% (1.8%) for the HERA II period and 1.5% (2.2%) for the HERA I
period.

Trigger and readout system: Both H1 [127] and ZEUS [113, 128] have used a multi-level trigger
system to select interesting ep events online and to suppress background from beam–gas interactions.
The H1 trigger system consisted of two hardware layers and one software filter. It was supplemented for
the HERA II period by an additional track trigger. The ZEUS trigger was based on one hardware and
two software levels. The first two levels mostly operated with the energy sums in the calorimeter, timing
and limited tracking information. On the third level a complete reconstruction of the event is performed,
using a simplified version of the offline reconstruction software, to select more sophisticated objects like
jets, tracks and even D-meson candidates. The triggers reduced the rate from the nominal HERA
bunch-crossing rate 10 MHz to the storage rate ∼ 10 Hz. While the topology of DIS events allowed
triggering on the scattered electron inclusively already at the first level of the trigger chain, triggering
on heavy-flavour photoproduction was more challenging and required reconstruction of leptons, tracks,
hadronic activity in the calorimeter or even explicitly charm hadrons. Both experiments had capabilities
to include limited tracking information already at the first trigger level, for instance on the number of
tracks and the fraction which originates from the ep interaction vertex.

3.3 Event reconstruction

Various heavy-flavour tagging techniques (see Section 4) exploit different measured quantities, like
tracks, vertices, energy-flow objects, jets and muons. The reconstruction of these quantities is described
in the following:

27



Figure 16: Helix parameterisation in the track-fit procedure. The initial parametrisation at the point
of closest approach as well as effects of multiple scattering are shown.

Tracking: Tracks were reconstructed combining hits from the central tracking chambers, silicon-strip
detectors and forward/backward trackers for high |η|. In both experiments tracks were parametrised
with a helix defined by 5 parameters (Fig. 16 shows the r-φ-view): the curvature κ = Q/R, which is
the signed inverse radius, D0, the dca distance of closest approach in the XY plane, φ0, the azimuth
angle, z0, the distance of closest approach along the Z axis, and the polar angle θ0. To account for
multiple scattering and energy loss in the material along the track trajectory and for inhomogeneities of
the magnetic field, the track parametrisation was refined in a track-refit process based on a broken-lines
algorithm [129] in H1 and a Kalman filter [130] in ZEUS. Additionally, the parameterisation of all tracks
that were fitted to the primary or to a secondary vertex (see below) was further improved by a track
refit using the vertex position as a constraint.

Analyses that aimed at the best tracking precision used tracks with typically |η| < 1.7. This coverage
corresponds to the region where efficiencies and resolutions are high and well known. Performance
benchmarking for the two tracking systems is shown in Table 2.

At low momenta, the dE/dx measurement for each track allowed the separation of pions, kaons and
protons, while at high momenta electron/hadron separation was possible to some degree.

Vertexing: Reconstructed tracks were used as an input for the primary vertex in each event. In
addition, if silicon-strip information was available, the time-averaged mean XY position of the ep
interaction region, the beam spot, was used to further confine the position of the primary vertex in the
event. The beam spot was measured by the experiments as a function of time for each ∼ 1000 events.
The beam spot size in the transverse plane was measured to be σx = 145 µm and σy = 25 µm in H1
HERA I data, σx = 110 µm and σy = 30 µm in H1 HERA II data, and σx = 85 µm and σy = 23 µm in
ZEUS HERA II data. The beam-spot size along the Z axis was much larger, σZ ∼ 10 cm, and therefore
was not used as a constraint. In H1 the vertex fitting was performed in the XY plane10, while ZEUS
did a full vertex fit in XY Z. Nevertheless, both experiments used only the XY projections of decays
in heavy-flavour analyses due to superior resolution.

In addition, in the context of some ZEUS heavy-flavour analyses, selected tracks were removed from
the primary vertex fit and the fit was re-done. Combinations of such tracks were fitted to a displaced
secondary vertex that was associated with a decay of a heavy-flavour hadron. Procedures similar to
those used in the primary-vertex fit11 are used to fit the secondary vertices as well. Alternatively,
combinations of impact parameters of several tracks were used in H1. Secondary vertices give an
important handle:

• to test the hypothesis that selected tracks originate from a decay of the same particle by evaluation
of the χ2 of the secondary vertex;

10An iterative procedure to determine the Z position of the primary vertex was used.
11For obvious reasons, no beamspot constraint is used.

28



• to evaluate the flight distance of that particle, which is related to the particle cτ.

This will play an essential role (together with the track impact parameter) in the heavy-flavour lifetime
tagging (cf. Section 4).

Electron reconstruction: Electron identification was needed to reconstruct DIS events as well as
to measure semi-leptonic decays of beauty and charm quarks. Electrons were separated from hadrons
using the shape of clusters in the calorimeter and dE/dx information from the central drift chambers.
A typical phase-space coverage for electrons in a beauty-production measurement in the semi-leptonic
electron decay channel was pT > 1 GeV (pT > 0.9 GeV) and −1 < η < 1.7 (|η| < 1.5) for the H1 (ZEUS)
measurements.

Muon reconstruction: Muons were identified by combining information from the tracking systems,
calorimeters and muon chambers. For pT > 1.5 − 2 GeV (p > 2 GeV) the information from the muon
chambers was exploited and the fraction of hadrons which were misidentified as muons was typically less
than 1%. For momenta 1 – 2 GeV isolated muons can be efficiently identified in the calorimeters, using
“minimum ionising particle” (mip) signatures, which however leads to reduced purity. The efficiency to
identify high momentum isolated muons in the H1 and ZEUS muon systems was 90% and 55% [131],
respectively.

The wide coverage of the muon chambers allowed the extension of the phase space of the measure-
ments up to −1.6 < η < 2.3.

Hadronic system: Energy flow objects (EFOs) were used in both experiments to reconstruct the
hadronic final state [132, 133]. These objects were based on a combination of information from the
calorimeter and the tracking system optimising energy resolution. Track information is superior for
low-energy EFOs, while the calorimeter measurement is preferred at high energy as well as to measure
neutral particles.

Jets at HERA have been reconstructed with the inclusive kT clustering algorithm [134, 135]
12. The

R parameter was set to R = 1, which is larger than the values used typically at pp and pp̄ experiments
(0.4 – 0.7), since at HERA jets with relatively low transverse momenta were analysed. The chosen jet
algorithm is infrared and collinear safe to all orders in perturbation theory. It was checked [137] that
at HERA the kT , anti-kT [138] and SIScone [139] algorithms produce very similar measurement results
and that the precision of NLO QCD calculations for the anti-kT algorithm is very similar to that of the
kT algorithm.

The final precision of the jet energy scale uncertainty of the H1 and ZEUS calorimeters was 1 – 2%
(see [140] for a recent review of jet results from HERA).

3.4 Summary

HERA was the first and so far only high energy ep collider. The results discussed in this review were
obtained with the H1 and ZEUS detector in two different running periods, denoted “HERA I” and
“HERA II”. The main detector parts relevant for the detection and reconstruction of heavy flavour
events were the electromagnetic part of the calorimeters for the reconstruction of the scattered electron
(if detectable), the calorimeters and tracking systems for the reconstruction of the decay products of
heavy flavoured particles, and the muon systems for the detection of semileptonic decay final states.

12Mostly in the longitudinally-invariant mode with the massless PT and massive ET recombination schemes [136] in
the H1 and ZEUS experiments, respectively

29



4 Charm and Beauty detection at HERA

The large charm- and beauty-quark masses result in kinematical suppression of their production com-
pared to the light-flavour cross sections. Therefore, special techniques have to be employed to separate
charm and beauty “signal” from the dominating light-flavour “background”. These techniques utilise
distinct properties of the charm and beauty hadrons (see Fig. 17 for illustration):

)  (MeV)π)-M(KsππM(K
140 150 160 170

C
o

m
b

in
a

tio
n

s 
 /
  
0

.6
 M

e
V

0

500

1000

 -1ZEUS  372 pb
2 < 1 GeV2130 < W < 300 GeV, Q

)| < 1.6    and+(D*η) > 3.8 GeV, | +(D*
T

p

)| > 1.60(Dη) < 3.8 GeV or | 0(D
T

p

 + BackgroundmodGauss
Wrong charge Background
Background

 59±) = 2139 +(D*addN

ZEUS

 / cm δ
-0.1 0 0.1

E
n

tr
ie

s

10

210

310

410

510
  
H1

 / cm δ
-0.1 0 0.1

E
n

tr
ie

s

10

210

310

410

510
  
H1

 Data
 b
 c
 uds
 sum

 / cm δ
-0.1 0 0.1

E
n

tr
ie

s

10

210

310

410

510

 / cm δ
-0.1 0 0.1

E
n

tr
ie

s

10

210

310

410

510
  
H1

 Data
 b
 c
 uds
 sum

 / cm δ
-0.1 0 0.1

E
n

tr
ie

s

10

210

310

410

510

  
(a)

50

100

150

200

250

0 1 2 3
p

rel
t   [ GeV ]

E
n

tr
ie

s

   Data

bb
−

cc
−

uds
Sum

H1

Q2 < 1 GeV2

Figure 17: Overview of tagging methods for heavy-flavour events. Each method is accompanied by an
illustrative distribution [141, 142, 143].

1. Flavour tagging. The tagging of the quark flavour is done either by full reconstruction of
decays of heavy-flavoured hadrons or by lepton tagging from the semi-leptonic decays of those
hadrons. The former was used at HERA only for charm tagging, since low production rates and
small branching ratios for useful decays led to insufficient statistics for fully-reconstructed beauty
hadrons. The latter was mostly used for beauty tagging, since b-hadron decays produce leptons
with sufficiently high momenta for efficient identification in the detectors.

2. Lifetime tagging. This method exploits the relatively long lifetimes of weakly-decaying heavy-
flavour hadrons through the reconstruction of tracks with large impact parameter δ or displaced
secondary vertices. In addition, the information about the flight direction, extracted from either
the track, hadron or jet momentum, can be used to construct a signed impact parameter or a signed
decay length (see later). This is a powerful tool to separate charm and beauty from light-flavour
events, in which tracks mostly originate from the primary vertex.

30



3. Mass tagging. The tagging using the mass of the heavy quark or meson is performed either
explicitly by a full reconstruction of the mass from all decay products (also see flavour tagging), by
a partial reconstruction via the mass of a jet or of all tracks at a secondary vertex, or indirectly by
measuring the relative transverse momentum of a particle with respect to the axis of the associated
jet, prelT . The latter was mostly used to separate beauty events from production of other flavours,
since the large quark mass produces large prelT values.

The above methods are all based on measuring the decay particles of one heavy quark (single tag).
Several methods can be combined to increase the purity at the cost of statistics. Both heavy quarks
(double tag) in an event can be tagged, by applying one method to tag one heavy quark and another (or
the same) for the other heavy quark. This allows a more detailed study of the heavy-quark production
mechanisms, but the double tagging efficiencies are low.

In the following the different tagging methods are discussed in more detail with emphasis on the
advantages and disadvantages of each method.

4.1 Charm tagging using full reconstruction of charm hadrons.

Most of the HERA charm results have been made using the golden decay channel D∗+ → D0π+s →
(K−π+)π+s (see Fig. 18)

13. Occasionally, also the D0 → K−π+π−π+ decay channel was used to increase
statistics. Due to the small energy release in the decay D∗+ → D0π+s (M(D∗+) − M(D0) − M(π+) ≈
6 MeV) the phase space for combinatorial background is suppressed, and the resolution for the D∗ − D
mass difference is strongly enhanced, providing an excellent signal to background ratio. This also leads
to a small momentum of the produced pion, which is often called the “slow” pion, πs. The capability of
a detector to measure very-low-momentum tracks defines the accessible region of pT (πs) and therefore
the phase space of the D∗ : a typical pT (πs) > 0.1 GeV restriction leads to pT (D

∗) > 1.5 GeV. For some
analyses the pT (D

∗) cut was raised to suppress the combinatorial background, which rises at low pT (cf.
Fig. 18 (a) and (b)). For the signal extraction, the observable ∆M = M(K−π+π+s ) − M(K−π+) was
chosen. The number of signal events is determined either by counting the number of events in the peak
region after subtracting the combinatorial background, which is estimated from the ∆M distribution
of “wrong charge” K+π+π−s combinations, by fitting the spectrum with a Gaussian-like shape for the
signal and a phenomenological function for the background, or by a combination of these two methods.

Besides the D∗ golden decay channel, the D0 → K−π+, D+ → K−π+π+, D+s → K+K−π+ and
Λ+c → pK−π+ decays of charm hadrons were used to tag charm in the events. These charm hadrons
feature much larger background (see Fig. 18(c) for an example of a D+ measurement). However, a
lifetime tag can be added, exploiting the relatively large values cτ ∼ 100 – 300 µm for the weakly-
decaying charm hadrons. The decay length is reconstructed by fitting a displaced secondary vertex to
selected tracks of decay products (see Section 4.3 for more details). This combined approach allowed to
significantly reduce combinatorial background, most noticeably for the D+ that has the largest lifetime
(cf. Fig. 18(c) and (d)), to a level that is still somewhat worse than but comparable to the D∗ .
Nevertheless, due to the low boost, lifetime tagging is inefficient in the phase space pT (D) < m(D) that
is also not accessible with D∗ ’s. In this region it is beneficial to study particular decay channels with
neutral strange hadrons K0S or Λ, for example D

+ → K0Sπ+.
The H1 and ZEUS experiments achieved a similarly good signal mass-peak resolution for charm

hadrons. Note that the distributions in Fig. 18 can not be compared directly due to different kinematic
regions.

In summary, the advantages (+) and disadvantages (−) of full hadron reconstruction are:
13Throughout the paper the D∗+ is mostly referred to as D∗ . The full reconstruction of D∗0’s at HERA was impossible

since the resulting decay photon or π0 could not be reliably measured.

31



) [GeV]π) - m(Kππm(K
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 + BackgroundsumGauss

Background

 324±) = 18917 +N(D (d)

Figure 18: Full reconstruction of mass spectra for charm mesons in the data. The reconstructed mass
difference ∆M for D∗ (a) [144] and (b) [143] and the D+ mass (c) [145] and (d) [143] are shown.

+ The full reconstruction allows an accurate determination of the momentum of the charm hadron,
which is correlated with the kinematics of the charm quark and can be used to study the frag-
mentation process.

+ An excellent signal to background ratio of ∼ 1 : 1 can be achieved using the D∗ golden decay
channel or the reconstruction of other weakly-decaying mesons including a lifetime tag (Fig. 18).

+ The combinatorial background can be parametrised with an empirical function and does not
depend on Monte Carlo simulations of the light-flavour background.

+ The signal mass peak is a clear signature which can be used in the online filtering of events to
reduce the trigger rates. This requires the usage of advanced tracking information in the trigger
logic.

− The typical probabilities for a charm quark to hadronise into a specific D meson are ≈ 0.15 – 0.25
and the branching ratios, BR, for the commonly used decay channels are ≈ 0.05 – 0.10; thus, only
∼ 1 – 2% of all c quarks can be tagged. The kinematic and geometric acceptances reduce the
visible fraction even further, in particular in the case of the additional use of lifetime tagging,
which reduces the detection efficiency by a factor ∼ 2 – 5. Therefore, only a very small fraction of
charm quarks can be tagged with the full reconstruction method.

32



1

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Figure 19: Distributions of the signed impact parameter δ of the muon track (left) and the transverse
muon momentum prelT relative to the axis of the associated jet (right) for the photoproduction event
sample of the H1 beauty analysis [142]. The data (dots) are compared to Monte Carlo predictions
(solid line). Separate contributions of events arising from b-quark (dashed line), c-quarks (dotted line)
and the light-quark (dash-dotted line) production are shown.

4.2 Heavy-flavour tagging with lepton + prelT

A well established method to identify beauty14 quarks is to select a muon with high transverse mo-
mentum of typically above 1.5 − 2 GeV from semileptonic b-quark decay, which is associated to a jet
that represents the beauty quark and consists of the muon and further final-state particles. The back-
ground to this signature is composed of charm production with a genuine muon from semi-leptonic
decays and of light-flavour events with a hadron misidentified as a muon (mainly due to in-flight π+

and K+ decays and hadronic energy leakage). To separate beauty from the background contributions
for a single tag, the prelT observable is used which, due to the large beauty-quark mass, extends to much
larger values than for the other sources. Additionally, a lifetime tag can be added by reconstructing the
signed impact parameter, δ, of the muon track (see Section 4.3 for more details). This further improves
beauty separation and also allows charm-event tagging, since long-lived heavy-flavour hadrons lead to
larger δ values than in light-flavour events. Figure 19 shows the distribution of both variables. The
beauty component has a distinct shape in both that allows one to disentangle it from the others. The
fractions of light-flavour, charm and beauty events in the data are determined from template fits to
the discriminating variables extracting template shapes from Monte Carlo simulations with data-driven
corrections. In some analyses also the missing energy in the detector, which is associated with the
undetected neutrino, was considered to improve charm/light-flavour separation.

A similar technique can be followed using electrons. This allows going down to pT (e) > 1 GeV,
however the available η region is narrower than for muons (see Section 3) and the lepton signature is
more complex.

In summary, the advantages (+) and disadvantages (−) of the use of semileptonic hadron decays
are:

14Also charm, but less efficiently.

33



+ The relatively large branching ratio BR(b → ℓX) ∼ 21% [146], which includes b → cX → ℓX and
other cascade decays, provides a reasonable tagging efficiency for b quarks.

+ Muon tagging extends the phase space of heavy-quark measurements, due to additional coverage
outside of the polar acceptance of the central tracking systems. Thus measurements of beauty
production in the forward and backward regions are possible with muons.

+ Semi-isolated leptons provide a clean experimental signature for the trigger system. They allow
one to efficiently select beauty events suppressing charm and light-flavour production by pT (ℓ)
cuts.

− The requirement of a jet associated with the lepton in order to use prelT or lifetime tagging cuts
into the low-pT phase space of the b quarks.

− The usage of semi-leptonic tagging for charm studies is very complicated due to weak separation
power.

4.3 Charm and Beauty with inclusive lifetime tagging

The aforementioned tagging methods suffer from the fact that only a fraction of the charm or beauty
quark decays ends up in the selected final state. This can be avoided by using an inclusive tagging
method, based on the long lifetime of charm and beauty quarks: cτc ≃ 100 – 300 µm and cτb ≃ 500 µm.
This approach relies on silicon-strip detectors to accurately measure the track parametrisation in the
vicinity of the interaction vertex (see Section 3). Therefore, it was not yet available in ZEUS for the
HERA I data set and was pioneered at HERA by the H1 collaboration. Lifetime tagging can be based
either on impact parameters of individual tracks or on decay lengths of displaced secondary vertices
fitted to selected tracks. Both techniques typically use tracks with transverse momenta pT > 0.5 GeV
to limit multiple scattering and hits in at least two layers of the vertex detectors. The lifetime tagging
is often applied based on information in the transverse XY plane, since the profile of the interaction
region and also the detector layouts do not allow for sufficiently high resolution of the tracks in the
coordinate along the beam line.

The impact parameter distribution allows the separation of long-lived heavy-flavour hadrons from
short-lived light-flavour hadrons. Figure 20 illustrates how the signed impact parameter is defined with
respect to the jet to which the track is associated. A positive sign is assigned to δ if the angle α between
the jet axis and the line joining the primary vertex and the point of closest approach is less than 90◦,
and a negative sign otherwise. Figure 19(left) shows the distribution of the signed impact parameter of
muon tracks in the data and the flavour decomposition in the Monte Carlo simulations. The light-flavour
component is characterised by a very small lifetime and the observed symmetric distribution is caused
by the detector resolution. In contrast, the charm and beauty contributions exhibit a pronounced tail
for large positive δ values. To further improve the separation power, the impact parameter significance
S = δ/σ(δ) can be used. This allows the rejection of candidates with an insignificant measurement of
the impact parameter.

In the vertexing approach, a displaced secondary vertex is fitted to all tracks that are associated
to a selected jet15. The distance between the primary and the secondary vertex is sensitive to the
lifetime of the hadron that initiated the jet. Similar to the impact parameter, the flight direction can
be introduced to form the signed decay length as used by H1 or the projected decay length in ZEUS.
The former is defined similarly to the signed impact parameter, whereas the latter is defined as

l =
(~rSV − ~rPV) · ~p

|~p |
15Here, a jet can be either a real jet in the detector or a set of tracks corresponding to a D-meson candidate.

34



jet axis

track

jet axis

track

Figure 20: Illustration of the positive and negative signed impact parameter (left). A reconstructed
vertex for a heavy-flavour decay shown in the XY plane [147] (right). The errors of the primary
and secondary vertex positions (shaded ellipses) have been blown up by a factor of 10 for illustrative
purposes.

(see Fig. 20), i.e. the projection of the vector from the primary to the secondary vertex on the jet
momentum. Finally, the ratio of such a quantity over its uncertainty, S, provides the optimal sepa-
ration power. A kinematic reconstruction of the mass of the vertex that corresponds to the jet, mvtx,
provides an additional handle on flavour separation, since contributions of light-flavours, charm and
beauty are expected to populate predominantly the small (mvtx ≪ mD), medium (mvtx <∼ mD) and
large (mD < mvtx <∼ mB) mass domains, respectively. Figure 21 illustrates the lifetime tagging with
secondary vetrices. The light-flavour contribution is symmetric around zero, while the charm and
beauty components exhibit a pronounced asymmetry in the region of large decay-length significance.
The beauty contribution dominates at large vertex mass and large significance values.

Lifetime tagging can be used either as an add-on to other tagging techniques (as described before) or
as a separate tagging tool. The dominant background is light-flavour production, which is symmetric in
the signed impact parameter significance or the projected decay length significance. The contents of the
negative bins of the significance distribution can be subtracted from the contents of the corresponding
positive bins, yielding a subtracted significance distribution. This way, the contribution from light-
flavour quarks is minimised.

The H1 collaboration has chosen to use a combination of the signed impact parameter significance
of individual tracks and the signed decay-length significance to tag heavy-flavour production. Events
are exclusively categorised according to the number of tracks in the event. The significances S1, S2
and S3 are defined as the significance of the track with the highest, second highest and the third
highest absolute significance, respectively. The S1 and S2 significance distributions (Fig. 22) are used
for events with one and two selected tracks, respectively. For events with three or a higher number
of tracks various sensitive variables including S1, S2, S3 and the signed decay-length significance of
the reconstructed secondary vertex are combined using an artificial neural network. In general, S2
has a better discrimination between light- and heavy-flavour contributions than S1, since the chance of
reconstructing two high significance tracks is further reduced for light-flavour. The neural network (Fig.
22(c)) provides separation between c and b events. For all distributions the negative part was subtracted
from the positive one to minimise the light-flavour component and a least-squares fit was performed
simultaneously to all three distribution. The charm and residual light-flavour components were found to
be very strongly anti-correlated in such fits (typical correlation coefficients are Clc < −0.95), while the

35



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Beauty

Figure 21: Distributions of the decay-length significance for different ranges of the vertex mass mvtx.
[148]. The data (points) are compared to Monte Carlo simulations (filled area). The individual contri-
butions of beauty (dashed line), charm (dotted line) and light flavours (solid line) are shown.

correlation with beauty is weaker due to the more distinct shape of the beauty distributions (Ccb ≈ −0.65
and Clb ≈ 0.55). Also alternative approaches to lifetime tagging have been studied, but were found to
be more sensitive to systematics from track resolution and efficiency.

The ZEUS collaboration, on the other hand, made inclusive charm and beauty measurements ex-
ploiting vertexing for tagging. The projected decay-length significance and the reconstructed mass of
the fitted secondary vertices (Fig. 21) were used as discriminating variables. A χ2 fit of the subtracted
significance distribution was performed in the three vertex-mass bins simultaneously. The correlation
pattern between components was found to be very similar to the one in the H1 analyses. Figure 23
shows the subtracted significance distributions. With optimised cuts after subtraction one can get sam-
ples with very high charm and beauty enrichment of roughly 80% and 90%, respectively. Such selection
resulted in ∼ 26000 charm and ∼ 1500 beauty events after negative subtraction in the recent ZEUS
measurement [148].

In summary, the advantages (+) and disadvantages (−) of this inclusive lifetime tagging method
are:

+ This tagging method gives access to the largest statistics due to the inclusive selection of the final
state.

+ The technique provides strong discrimination power and is often combined with other tagging
methods.

+ With the applied track minimal transverse momentum cut of 0.5 GeV one obtains a good accep-
tance for low heavy-quark momenta, which is of high importance for measuring the charm and
beauty contributions to inclusive ep scattering. The additional typical jet cut E

jet
T > 5 GeV ∼ mb

retains a high acceptance for beauty production near threshold.

− The total achieved effective signal to background ratio is typically not better than 1:10 for both
charm and beauty. This can be estimated from the numbers of charm and beauty events in the
positive subtracted significance spectra, which effectively represent the numbers of tagged events,

36



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The beauty- (dashed line), charm- (dotted line) and light-flavour (dashed-dotted line) contributions are
shown.

and from the errors achieved for the charm and beauty components in the fit.

− The method requires the track resolutions and efficiencies to be thoroughly scrutinised.
− With the typical cuts on the jet transverse momentum one actually cuts strongly into the kinematic

phase space for charm. A requirement of a jet with E
jet
T > 4 GeV corresponds to a cut p

D
T > 2.4 GeV

assuming that ∼ 60% of the quark transverse momentum is transfered to the D meson.

4.4 Charm and Beauty with double tagging

The various heavy-flavour tagging methods outlined above can be combined, aiming towards tagging
both heavy quarks in the event. At HERA, D∗ µ combinations were used to tag both charm and beauty
events, while a di-lepton tag was used for beauty only. The usage of two flavour tags significantly reduces
the light-flavour background, which allows omitting any additional mass or lifetime tags. Furthermore,
it gives access to correlations between the quarks of heavy flavour pair.

In the example of the photon-gluon fusion process, γg → cc̄ or bb̄, the two heavy quarks are produced
back-to-back in the γg frame as illustrated in Fig. 24 for the beauty case. Unlike-sign combinations such
as µ+µ− or D∗+µ− can be produced from either the same or different b quarks, while like-sign combina-
tions originate always from different b quarks (combination of b and c̄ decays + charged conjugate, or
B0−B̄0 mixing). In charm events only oppositely-charged combinations are produced. In analyses using
D∗ µ tags the charm and beauty components were separated16 based on the sign combination and the
angular distance in azimuth. In charm events mostly back-to-back unlike-sign D∗ and µ are produced,
while for beauty both like- and unlike-sign combinations are possible and more complicated angular
distributions arise. The di-lepton analyses have also used the information about charge combination
and angular separation between leptons, and additionally the mass of the lepton pair.

In summary, the advantages (+) and disadvantages (−) of double tagging are:
+ For a large fraction of the events double tagging gives access to the kinematics of both heavy

quarks. This information can be used to investigate the cc̄ and bb̄ production processes in detail.

+ Since light flavours are efficiently suppressed by the requirement of two flavour tags, the leptons
can be selected in transverse momentum down to ∼ 1 GeV with a reasonable purity. For beauty
this gives access to much lower quark momenta than the lepton + prelT tag that was discussed in
Section 4.2.

16 Light-flavour production is suppressed by D∗ reconstruction

37



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Figure 23: The subtracted distributions of the decay-length significance for different ranges of the
vertex mass mvtx [148]. The data (points) are compared to Monte Carlo simulations (filled area).
Individual contributions of beauty (dashed line), charm (dotted line) and light flavours (solid line) are
shown.

D* +

−

+

_−

_

Figure 24: Various possible ways to produce D∗ µ or a µµ pair from the decays of the b- and the b̄-quark
produced in the photon gluon fusion process at HERA.

− The total tagging efficiency is very low.
− The lepton tagging is well suited for the measurement of beauty production but has relatively

small acceptance for charm, where, due to softer fragmentation, the leptons take a smaller fraction
of the quark transverse momentum than in the beauty case.

− Due to their low pT , the correlation of the D∗ and the µ momenta with those of the parent quarks
is not as good as for jets.

4.5 Summary

Various heavy-flavour tagging methods have been used at HERA. Each of them has advantages and
disadvantages, which results in different tags being optimal for different purposes. The most commonly
used tags have been D∗ reconstruction and inclusive lifetime tags for charm and lepton + prelT and inclu-
sive lifetime tags for beauty. Whenever possible, a comparison between (and potentially a combination

38



of) measurements performed with different techniques allows improved constraints on the measurements,
due to cross-calibration of systematics of different nature for independent tags. Often a combination
of tags yields an increased purity of the heavy-flavour sample, at the cost of reduced efficiency and
additional systematics. In general, the choice of the tagging method(s) is a trade-off between statistical
and systematic uncertainties.

39



5 Search for single top-quark production

Already before the start of HERA data taking, it became clear from the lower limits of order 70 – 80 GeV
on the top-quark mass obtained from p̄p collisions by the UA1/UA2 [150] and CDF [151] collaborations,
that top-quark pair production would probably be outside the kinematic reach of HERA. This was
confirmed by indirect constraints from LEP [152] and by the direct observation of the top quark at the
Tevatron at a mass of 174 GeV [153]. Single top quark production in the charged current reaction [154]

e+ + b → ν̄e + t

(and its charged conjugate) remained kinematically possible, but the expected Standard Model cross
section of less than 1 fb [155] is too small to be experimentally accessible. This is due to the fact that
the occurrence of initial state b quarks is strongly suppressed at high x, since it would need to originate
from the splitting of ultra-high-x gluons in the proton (in analogy to Fig. 2), which are known to be
very rare [15]. Charged current reactions on light initial state quarks are strongly suppressed by the
very small corresponding CKM matrix elements [156].

If at all, single top quarks could thus be produced at HERA only via a process beyond the Standard
Model [157]. One such process is the transition of a u quark into a t quark via a flavour-changing
neutral current [158] (Fig. 25) caused by non-Standard Model couplings of the photon or Z boson.

Figure 25: Feynman graph for anomalous single top production [159].

This possibility was investigated in particular due to an excess observed by the H1 collaboration in
the single isolated lepton + jets final state [160], which was however not confirmed by a corresponding
ZEUS analysis [161], and greatly reduced in significance by a common analysis of the ZEUS and H1
data [162].

γuBr
-310 -210 -110

u
Z

B
r

-210

-110

1

excluded by ZEUS

ZEUS
-1ZEUS 0.5 fb

,u(c)Zγ u(c)→ALEPH t 

,u(c)Zγ u(c)→CDF t 
γ u→H1 t 

 u(c)Z→D0 t 

γuBr
-310 -210 -110

u
Z

B
r

-210

-110

1

Figure 26: Limits on anomalous couplings for single top production, translated to branching fractions
(Br) for top decay into uZ or uγ [159].

40



No significant signal was observed. Fig. 26 shows the exclusion contours obtained by the H1 [163]
and ZEUS [159] collaborations for the anomalous couplings to photons or Z bosons for this process,
translated to branching ratios for anomalous top quark decays. The H1 limit is a bit looser than the
the one from ZEUS due to the small excess mentioned above. These limits are competitive with limits
obtained from other colliders, also shown in Fig. 26, and currently represent the best limit for the
anomalous photon coupling. They can be improved further in future analyses using data from the LHC
[164]. Single top quarks were also searched for in the full-hadronic top decay channel, with a similar
sensitivity as for the lepton channel [165]. However, the corresponding analysis was performed on the
HERA I data set only, so the resulting limit is no longer competititive.

41



6 Charm photoproduction

The study of open charm- and beauty-production cross sections provides stringent tests of perturbative
QCD from several perspectives. On one hand, the size of the charm- and beauty-quark masses ensures
that for all such final states the production cross sections are in the perturbatively calculable regime,
since mc, mb ≫ ΛQCD. On the other hand, the QCD scales obtained from these masses compete with
other potential scales like the quark tranverse momentum or, in the DIS case, the virtuality of the
exchanged photon (Fig. 3). The treatment of such a multi-scale problem is theoretically challenging
and a comparison of different theoretical schemes to data can shed light on the strengths and weak-
nesses of the respective perturbative approximations. Furthermore, from the theoretical point of view,
the presence of final-state heavy-flavour hadrons ensures that these processes will not interfere with
corresponding final states involving only gluons and light quarks, and that they can thus be treated
independently for each flavour. Thus, in the following, charm and beauty production will be treated
separately.

In general, photoproduction processes have many similarities to corresponding processes in hadropro-
duction, “simply” interchanging the incoming quasi-real photon with a quasi-real gluon. Since the Spp̄S
and Tevatron pp̄ colliders went into operation almost a decade before the HERA collider, perturbative
NLO QCD calculations have often first been obtained for hadroproduction, although the incoming pho-
ton diagrams are somewhat easier to calculate. Furthermore, since the virtuality of the incoming photon
(Q2 < 1 GeV2) is of the order of typical hadron masses or lower, the photon can have a hadron-like
structure (“hadron-like resolved photon”, Fig. 6b). Thus, the cross sections get a contribution from the
convolution of perturbative hadroproduction diagrams with this photon structure. This in turn com-
plicates the photoproduction cross-section calculations. However, this hadron-like photon contribution
is small (of order 10% or less) in the case of the treatment of heavy-flavour production in the massive
scheme.

An important variable for photoproduction analyses at HERA is the event kinematic observable y,
which represents the fraction of the electron energy entering the hard interaction (Section 2.1). This
variable can be reconstructed from the hadronic final state in the main detector (Section 3). An overview
of all charm photoproduction measurements in H1 and ZEUS is given in Table 3.

6.1 D∗ inclusive measurements

Despite the large loss in statistics through fragmentation fractions and branching ratios, it is clear from
Table 3 that the very clean explicit reconstruction of D∗-meson final states offers the best effective
signal sensitivity for charm photoproduction.

6.1.1 Charm total cross sections

In the very first ZEUS [166] and H1 [167] measurements on open-charm production (entries 1 and 2 in
Table 3) the inclusive D∗ results were extrapolated to obtain total charm-photoproduction cross sections.
The results are shown in Fig. 27(left) as a function of the photon-proton centre-of-mass energy Wγp.
Measurements from fixed-target experiments from the pre-HERA era are also shown. At HERA, both
Wγp values and the observed cross sections are roughly one order of magnitude larger. The steep cross-
section rise reflects the fact that with increasing Wγp gluons with smaller and smaller proton momentum
fractions are accessible for charm production via the photon-gluon-fusion process (Fig. 4). The data
in Fig. 27 are compared to a massive scheme NLO prediction [58], which is able to describe both the
fixed-target data at lower Wγp and the HERA data at higher Wγp. Despite the large uncertainties,
this demonstrated early on that the basic charm-production mechanism in photoproduction is at least
reasonably well understood.

42



No. Analysis c-Tag Ref. Exp. Data L [pb−1] Q2 [GeV2] y Particle pT [GeV] η Events
effect.

s:b

bgfree

events

1 D∗ incl. Kππs [166] ZEUS 93 0.5 < 4 [0.15, 0.84] D
∗ > 1.7 [−1.5, 1.5] 48 ± 11 1 : 1.5 19

2 D∗ tagged Kππs [167] H1 94 2.8 < 0.01 [0.28, 0.65] D
∗ > 2.5 [−1.5, 1.0] 119 ± 16 1 : 1.2 55

incl. 1.3 < 4 [0.10, 0.80] 97 ± 15 1 : 1.3 42
3 D∗ incl. Kππs [168] ZEUS 94 3.0 < 4 [0.15, 0.87] D

∗ > 3 [−1.5, 1.0] 152 ± 16 1 : 0.7 90
K3ππs 199 ± 29 1 : 3.2 17

4 D∗ tagged Kππs [169] H1 95-96 10.2 < 0.009 [0.02, 0.32] D
∗ > 2 [−1.5, 1.5] 299 ± 75 n.a. 16

94-96 10.7 < 0.01 [0.29, 0.62] > 2.5 (ŷ(D∗)) 489 ± 92 n.a. 28
5 D∗ incl. Kππs [170] ZEUS 96-97 37 < 1 [0.19, 0.87] D

∗ > 2 [−1.5, 1.5] 3702 ± 136 1 : 4.0 741
K3ππs > 4 1397 ± 108 1 : 7.3 167

D∗ + dijet Kππs
D∗

Jet1(2)

> 3

> 7(6)

[−1.5, 1.5]
[−2.4, 2.4] 587 ± 41 1 : 1.9 205

6 D* incl. Kππs [171] ZEUS 98-00 79 < 1 [0.17, 0.77] D
∗ [1.9, 20] [−1.6, 1.6] 10350 ± 190 1 : 2.5 2970

7 D∗ tagged Kππs [172] H1 99-00 51 < 0.01 [0.29, 0.65] D
∗ > 2 [−1.5, 1.5] 1166 ± 82 1 : 4.8 202

+jet Jet > 3 [−1.5, 1.5] 592 ± 57 1 : 4.5 108
+dijet Jet 1(2) > 4(3) [−1.5, 1.5] 496 ± 53 1 : 4.7 88

8 D∗ + dijet Kππs [173] ZEUS 96-00 120 < 1 [0.17, 0.77]
D∗

Jet1(2)

> 3

> 7(6)

[−1.5, 1.5]
[−1.9, 1.9] 1092 ± 43 1 : 0.7 650

9 D∗ + jet Kππs [174] ZEUS 98-00 79 < 1 [0.17, 0.77] D
∗ > 3 [−1.5, 1.5] 4891 ± 113 1 : 1.6 1870

+ dijet Jet1(2) > 6(7) [−1.5, 2.4] 1692 ± 70 1 : 1.6 584

10 lifet.+dijet imp.par. [141] H1 99-00 57 < 1 [0.15, 0.80]
Track

Jet1(2)

> 0.5

> 11(8)

[−1.3, 1.3]
[−0.9, 1.3] 4600 ± 460 1 : 45 100

11 D∗ + µ
Kππs
+ µ

[175] H1 98-00 89 < 1 [0.05, 0.75]
D∗

µ

> 1.5

p > 2

[−1.5, 1.5]
[−1.74, 1.74] 53 ± 13 1 : 2.2 17

12 e + dijet e + 6 E
T

[176] ZEUS 96-00 120 < 1 [0.2, 0.8]
e

Jet1(2)

> 0.9

> 7(6)

[−1.5, 1.5]
[−2.5, 2.5] ∼ 8000 n.a. 70

13 lifet.+dijet sec. vtx. [177] ZEUS 05 133 < 1 [0.2, 0.8]
tracks

Jet1(2)

> 0.5

> 7(6)

[−1.6, 1.4]
[−2.5, 2.5] ∼ 20000 n.a. 2320

14 µ + dijet
µ +

imp.par.
[178] H1 06-07 179 < 2.5 [0.2, 0.8]

µ

Jet1(2)

> 2.5

> 7(6)

[−1.3, 1.5]
[−1.5, 2.5] 3315 ± 170 1 : 7.7 380

15 D∗ incl Kππs [179] H1 06-07 31 – 93 < 2 [0.1, 0.8] D
∗ > 1.8 [−1.5, 1.5] 8232 ± 164 1 : 2.3 2520

+dijet Jet 1(2) > 3.5 [−1.5, 2.9] 3937 ± 114 1 : 2.3 1200
16 D∗ incl Kππs [180] ZEUS 06-07 144 < 1 [0.167, 0.802] D

∗ [1.9, 20] [−1.6, 1.6] 12256 ± 191 1 : 2.0 4120
MER 07 6.3 417 ± 37 1 : 2.3 127
LER 07 13.4 859 ± 49 1 : 1.8 307

Table 3: Charm photoproduction cross-section measurements at HERA. Information is given for each analysis on the charm tagging method, the
experiment, the data taking period, integrated luminosity, Q2 and y ranges and the cuts on transverse momenta and pseudorapidities of selected final state
particles. The last three columns provide information on the number of tagged charm events, the effective signal-to-background ratio and the equivalent number of
background-free events. The centre-of-mass energy of all data taken up to 1997 (6th column) was 300 GeV, while it was 318 – 319 GeV for all subsequent runs, with
the exception of the analyses marked “MER” and “LER” (entry 16), for which the data were taken at 251 and 225 GeV.

4
3



10
-1

1

10

10 10
2

Wγp  (GeV)

σ
(γ

p 
→

 c
c−  

X
) 

 (
µb

)

E691

PEC
SLAC

WA4

EMC

BFP

CIF

H1 tagged data

H1 untagged data

ZEUS

 QCD (MRSG + GRV)

 upper µ/mc = 0.5
 lower  µ/mc = 2

 (GeV)s
240 260 280 300 320

σ
R

0

0.2

0.4

0.6

0.8

1

1.2
ZEUS

0.167 < y < 0.802
 < 20 GeV  D*

  T
1.9 < p

| < 1.6D*η|

  2 < 1 GeV2Q

 =〉W〈
136 GeV 152 GeV

192 GeV

 D* X→ZEUS ep

NLO QCD

Figure 27: Left: Total charm-photoproduction cross section as a function of centre-of-mass energy Wγp
[167]. The data shown are from the first H1 and ZEUS publications on open charm production and from
previous fixed-target experiments. Right: Inclusive charm-photoproduction cross section as a function
of ep centre-of-mass energy [180], normalised to the cross section at 318 GeV.

Figure 27(right) shows the latest HERA measurement in photoproduction [180] (entry 16 in Table
3), focusing on the dependence of the inclusive visible cross section on the centre-of-mass energy. This
makes use of the very last HERA running period, in which the proton beam energy was lowered. This
result was obtained and published based on a ZEUS master thesis [182], which was made possible by the
strong simplification of the data format and calibration procedure implemented as part of a long-term
high energy physics data preservation project [183]. The result is presented as a ratio to the highest
centre-of-mass energy cross section, such that both experimental and theoretical correlated uncertainties
cancel. While the data uncertainties remain dominated by statistical uncertainties (inner error bars),
the theoretical uncertainties are dramatically reduced with respect to the absolute predictions in Fig.
27. The massive NLO prediction [59] agrees well with the data, indicating that the extrapolation of the
energy dependence to even higher centre-of-mass energies such as those at a future LHeC collider [184]
can be reliably predicted. In addition, since different centre-of-mass energies correspond to different x
ranges, such a ratio potentially provides constraints on the gluon PDF in the proton.

6.1.2 D∗ single-differential cross sections

Figure 28 shows the results for the ZEUS HERA I [171] and H1 HERA II [179] D∗ analyses (entries 6
and 15 of Table 3) as a function of the D∗ transverse momentum and pseudorapidity. These results have
been selected since the data samples used in these analyses are among those with the highest statistical
significance of all heavy flavour measurements at HERA, as can be seen from the last column of Table 3
(entry 6). The data span a large kinematic range from pT (D

∗) = 1.8 GeV ∼ mc to pt = 20 GeV ≫ mc.
Over this range the cross section falls off by about four orders of magnitude. The measurements are com-
pared to five NLO predictions: massive fixed-flavour scheme (NLO,FMNR) calculations from Frixione
et al. [58], a variant of these calculations matched to parton showers (MC@NLO) [185], massless scheme
(NLL) predictions from Kniehl et al. [48], general mass variable flavour scheme (FONLL) calculations
from Cacciari et al. [181], and a different GMVFNS variant (GMVFNS) from Kniehl et al. [83]. Both
calculations from Kniehl et al. include a perturbative treatment of the charm fragmentation function.
At first glance, all five predictions are able to describe the spectrum over the complete pT (D

∗) range
within a factor of two. However, looking more in detail, one observes:

44



2 4 6 8 10 12

 [
n

b
/G

e
V

]
T

/d
p

σ
d

-110

1

10

210

H1
Data
FMNR
GMVFNS
MC@NLO

(D*) [GeV]
T

p
2 4 6 8 10 12

n
o

rm
R

0.5

1

1.5

2

 [
n

b
]

η
/d

σ
d

0

10

20

30

40

H1
Data
FMNR
GMVFNS
MC@NLO

(D*)η
-1.5 -1 -0.5 0 0.5 1 1.5

n
o

rm
R

0.5

1

1.5

Figure 28: D∗ single differential cross sections in photoproduction as function of the D∗ transverse
momentum (left) and pseudorapidity (right), from ZEUS [171] (top and center) and H1 [179] (bottom).
The measurements are compared to five NLO predictions: the massive scheme calculations from Frixione
et al. [58] without (NLO, FMNR) and with (MC@NLO) [185] interface to LL parton showering, the
massless scheme predictions from Kniehl et al. [48] (NLL) and the general mass variable flavour scheme
calculations from Cacciari et al. [181] (FONLL) and Kniehl et al. [83] (GMVFNS). The NLL and
GMVFNS predictions include a perturbative treatment of D∗ fragmentation.

45



1. The uncertainty of the measurements is generally much smaller than those of the theory, domi-
nated by QCD scale variations, the variation of the charm mass, and the variation of the charm
fragmentation parameters. Especially for low transverse momenta pT (D

∗) < 3 GeV the scale
uncertainties reach a factor of two. This indicates that in this kinematic region the hard scales
provided by the charm mass and the transverse momentum of the charm quarks are not yet large
enough to ensure a fast convergence of the QCD perturbation series at next-to-leading order.

2. As to be expected, the FONLL prediction is very close to the massive NLO/FMNR prediction
at low pT , and for the η distribution, which is dominated by the low pT D

∗ contribution. The
measured cross sections are higher than the central prediction, but the predictions are consistent
with the data within the large uncertainties. At high transverse momenta, contrary to many
people’s expectations originally based on leading-order studies [186], the FONLL prediction is
actually lower than the NLO prediction. Thus the final state resummation corrections originating
from higher-order log terms in the massless part of the calculation reduce the prediction, rather
than enhancing it. The data are closer to the pure NLO prediction. At least within the HERA
kinematic regime there is thus no evidence for the claim [186] that the massive fixed-order calcu-
lation should fail at large values of charm transverse momentum unless final state resummation
corrections are applied. Both predictions give a reasonable but not perfect description of the
shape of the η distribution.

3. The massless NLL prediction, which, in contrast to the massive predictions discussed in the
previous item, incorporates a proper perturbative treatment of charm fragmentation [49], fits the
data well at low pT , while it is a bit too low for high pT , where it is expected to work best.
As expected, it is similar to the FONLL prediction in this region. The theoretically superior
treatment of fragmentation does not lead to a smaller uncertainty, as can be seen from the η
distribution. Also, the shape of the η distribution is a bit less well described than with the
massive prediction. In this approximation, a large fraction of the cross section arises from the
(massless) charm contribution to the photon parton density function (using the AFG [187] or GRV
[188] parametrisations), in contrast to the “direct” contribution, which is also shown separately.

4. The partially massive GMVFNS prediction, which incorporates a perturbative treatment of the
charm fragmentation function, has a larger uncertainty than the traditional massive predictions,
similar to the NLL prediction. The shape of this prediction describes the data better than the
NLL prediction.

5. The MC@NLO prediction has the same core parton-level cross section as the NLO/FMNR pre-
dictions by definition. The differences seen w.r.t. the latter must thus arise from the addition of
the HERWIG-type parton showers and the different fragmentation treatment. It exhibits slighty
smaller uncertainties, but, surprisingly, fits the data less well than the original NLO/FMNR pre-
dictions. This offers room for potential retuning of some of the MC parameters entering this
calculation.

A similar inclusive D∗ photoproduction measurement as the above is available from H1 [172], per-
formed in a more restricted Wγp region (entry 7 in table 3) with a roughly ten times smaller data
sample. The narrower kinemetic range and smaller statistics are due to explicit detection of the elec-
tron scattered at very low angles in dedicated forward electron taggers (section 3.2), which was part of
the trigger requirement. The conclusions are very similar.

ZEUS has also recorded such tagged photoproduction samples, but they were found not to be
statistically competitive with results from data sets triggered on inclusive D∗ production.

46



6.1.3 D∗ double-differential cross sections

Double-differential cross-section measurements as a function of the D∗ transverse momentum and pseu-
dorapidity have been performed by ZEUS in [171] and also in a previous charm milestone paper [168]
(entries 6 and 5 in Table 3), and by H1 [179] (entry 15 in Table 3). The results of the latter are
shown in Fig. 29, together with some of the predictions already discussed for the single-differential
case. In general, the conclusions are similar to those from the single-differential cross sections. At high

(D*)η
-1 0 1

 (
D

*)
[n

b
/G

e
V

]
η

d
T

/d
p

σ
2

d

0

5

10

15

20

25

30
(D*) < 2.5 GeV

T
 p≤1.8 

(D*)η
-1 0 1

 (
D

*)
[n

b
/G

e
V

]
η

d
T

/d
p

σ
2

d

0

2

4

6
H1

(D*) < 4.5 GeV
T

 p≤2.5 

(D*)η
-1 0 1

 (
D

*)
[n

b
/G

e
V

]
η

d
T

/d
p

σ
2

d

0

0.1

0.2

0.3

0.4
(D*) < 12.5 GeV

T
 p≤4.5 Data

FMNR
GMVFNS
MC@NLO

Figure 29: D∗ double differential cross sections in photoproduction as function of the D∗ transverse
momentum and pseudorapidity from H1 [179]. The measurements are compared to three out of the five
NLO predictions also shown in Fig. 28.

pT the uncertainty of the theory predictions reduces as expected, such that the comparisons become
more meaningful. Reasonable agreement with the data is observed for all predictions in this high-pT
region, while MC@NLO undershoots the data at low pT and η, similar to what was observed in the
single-differential case (Fig. 28).

6.2 Inclusive measurements using other tagging methods

Although the D∗ channel generally yields the best signal-to-background ratio and therefore the best
effective overall statistics (last column of Table 3), the small branching ratio limits the statistics in
regions in which the cross section is small. In such regions, more inclusive tagging techniques can be an
advantage. Furthermore, the consistency of results obtained with different tagging methods enhances
confidence in the results.

H1 has performed a measurement based on inclusive lifetime tagging [141] (entry 10 in table 3),
which extends to the highest charm transverse momenta pcT = 35 GeV reached so far. Here events

47



with two jets in the central rapidity region are used (cuts are listed in table 3). Due to the high
jet transverse momenta the events are efficiently triggered using the deposits of the jet particles in
the calorimeter. An inclusive lifetime tagging is applied, based on the displaced impact parameters
of jet-associated charged tracks from charm and beauty decays. Details of the tagging method are
discussed in Section 4. Figure 30(left) shows the measured charm-production cross sections as function
of the transverse momentum of the leading jet. The data are compared to a massive scheme NLO

 / GeV1
jet

t
p

15 20 25 30 35

 [
p

b
/G

eV
]

1
je

t

t
/d

p
σ

d

1

10

210

310 Data
Pythia

Cascade
 had⊗NLO QCD 

  
 ejjX→X c ec→ep 

 / GeV1
jet

t
p

15 20 25 30 35

 [
p

b
/G

eV
]

1
je

t

t
/d

p
σ

d

1

10

210

310 Data
Pythia

Cascade
 had⊗NLO QCD 

  
 ejjX→X c ec→ep 

CHARM

 / GeV1
jet

t
p

15 20 25 30 35

 [
p

b
/G

eV
]

1
je

t

t
/d

p
σ

d

1

10

210

310 Data
Pythia

Cascade
 had⊗NLO QCD 

  
 ejjX→X c ec→ep 

CHARM(a)

 / GeV1
jet

t
p

15 20 25 30 35

 [
p

b
/G

eV
]

1
je

t

t
/d

p
σ

d

1

10

210

310 Data
Pythia

Cascade
 had⊗NLO QCD 

  
 ejjX→X c ec→ep 

CHARM(a)

  
H1

 / GeV1
jet

t
p

15 20 25 30 35

 [
p

b
/G

eV
]

1
je

t

t
/d

p
σ

d

1

10

210

310 Data
Pythia

Cascade
 had⊗NLO QCD 

  
 ejjX→X c ec→ep 

CHARM(a)

  
H1

 / GeV1
jet

t
p

15 20 25 30 35

 [
p

b
/G

eV
]

1
je

t

t
/d

p
σ

d

1

10

210

310

 (
p

b
/G

e
V

)
c

-j
e

t

T
/d

p
σ

d
1

10

210

3
10

410

 (
p

b
/G

e
V

)
c

-j
e

t

T
/d

p
σ

d
1

10

210

3
10

410

 e→ZEUS 96-00 c 

ZEUS 98-00 D* jets

 c jet-1ZEUS 133 pb

NLO QCD
)/42

T
+p2

c
=(m2µ     

2
T

+p2c=m
2µ     

ZEUS

 ecX)→ (ep 
c-jet

T
/dpσd

| < 1.5c-jetη, 0.2 < y < 0.8, |
2 < 1 GeV2Q

 (GeV)c-jet
T

p

5 10 15 20 25

th
 

σ/
m

e
a

s
σ

0

1

2

3

 (GeV)c-jet
T

p

5 10 15 20 25

th
 

σ/
m

e
a

s
σ

0

1

2

3

Figure 30: Left: Differential cross sections for the process ep → ecc̄X → ejjX as function of the
transverse momentum p

jet1
t of the leading jet, from the H1 analysis [141]. The data are compared to

an NLO calculation [58] in the massive scheme, and to LO+PS MC predictions from PYTHIA [21]
and CASCADE [28]. Right: Summary of differential c-quark jet cross sections as a function of the jet
transverse momentum, as measured by the ZEUS collaboration [177]. The data are compared to an
NLO calculation [58] in the massive scheme, for two different QCD scale choices.

prediction [58], which describes the data reasonably and equally well up to the highest jet transverse
momenta. To compare this result with the above D∗ measurement (Fig. 28) one has to take into
account that the jet gives a direct approximation of the charm quark kinematics, while on average
the D∗ takes only about 70% of the charm quark momentum in the fragmentation (after cuts). Thus,
the kinematic range tested with the leading jet pT from 11 to 35 GeV roughly corresponds to a D

∗

transverse-momentum region from 8 to 25 GeV. For D∗ transverse momenta from 8 GeV up to the
highest covered value of 20 GeV the D∗ data are similarly well described by the NLO calculation as
the dijet data at their correspondingly higher momenta. So the two independent measurements using
different tagging techniques give consistent results.

A similar and more direct comparison is shown in Fig. 30(right) for several measurements from ZEUS
(entries 8,12,13 in table 3). Here, the measurements have already been translated to cross sections for
inclusive c-jet production. The results obtained from D∗ and inclusive-vertex tagging agree well with
each other and with theory. Since in the core of a jet electrons are not easily separated from π/π0

overlaps, charm tagging using semileptonic decays into electrons is experimentally difficult17 and the
corresponding c → e result, which was a byproduct of an analysis focusing on beauty production, might

17Several other such charm analyses were eventually not published due to insufficient control of systematics.

48



not include all relevant systematic uncertainties. To compare with Fig. 30(left), the prediction with
scale choice m2 + p2T (dashed line in Fig. 30(right)) should be considered. Good agreement is observed
between the results of the two experiments.

6.3 Studies with a D∗ and one other hard parton

To obtain more information on the charm-photoproduction process, one possibility is to require the
presence of a jet in the final state in addition to the D∗, which is not associated to the D∗. This means
that the jet and the D∗ are well separated in their directions and that the jet tags another hard parton
in the process. This parton can be the other charm quark or a gluon or light quark. In one analysis
[172] a very soft jet momentum cut pT > 3 GeV was applied. The jets were restricted to the central
pseudorapidity region |η| < 1.5, thus covering the same kinematic range as the D∗s. In Fig. 31 the
differential cross sections are shown as function of the pseudorapidities of the D∗ and the jet. For the

(D
*)

 [
n

b
]

η
/dσ

d

0

0.5

1

1.5

2

2.5

p:  D* + other jetγ

H1
Data

 Had⊗FMNR 

FMNR

Had⊗ZMVFNS

p:  D* + other jetγ

H1

(D*)η
-1.5 -1 -0.5 0 0.5 1 1.5

R

0.5

1

1.5

Data

 Had⊗FMNR 

FMNR

Had⊗ZMVFNS (j
e

t)
 [

n
b

]
η

/d
σ

d

0

0.5

1

1.5

2

2.5

p:  D* + other jetγ

H1
Data

 Had⊗FMNR 

FMNR

Had⊗ZMVFNS

p:  D* + other jetγ

H1

(jet)η
-1.5 -1 -0.5 0 0.5 1 1.5

R

0.5

1

1.5

Data

 Had⊗FMNR 

FMNR

Had⊗ZMVFNS

Figure 31: D∗ + jet cross sections as function of the pseudorapidities of the D∗ (left) and the jet
(right), from the H1 analysis [172]. The measurements are compared to two NLO predictions, the
massive scheme (FMNR) calculations from Frixione et al. [58] and the massless scheme (ZMVFNS)
predictions from Heinrich and Kniehl [48].

leading-order boson-gluon-fusion process it is expected that the D∗ tags one charm quark and the jet the
other. Since similar momentum cuts are applied for the D∗ and the jet, one would expect very similar
pseudorapidity distributions for the D∗ and the jet. However, the observed pseudorapidity spectrum for
the jet (Fig. 31) is significantly shifted towards the more forward direction compared to that of the D∗.
This indicates that, as expected from higher-order contributions, the jet often tags another parton, i.e.
a gluon or a light quark. This effect is predicted by the massive and massless scheme NLO calculations
to which the data are compared in Fig. 31, and these calculations describe the data reasonably well.
Also, the additional jet requirement significantly reduces the theoretical uncertainties w.r.t. Fig. 28.

In addition to jets not associated to the D∗, the corresponding ZEUS measurement [174] also selected
events in which the D∗ is associated to the jet. In the latter case, one does obtain information only about
one hard parton in the event, which is a charm quark. Furthermore, the jet tranverse momentum cut
p
jet
T > 6 GeV is much harder and a much wider pseudorapidity range −1.5 < ηjet < 2.4 is covered. Good
agreement with NLO predictions is observed for all single-jet distributions (not shown). In particular,
the ET spectra for D

∗ -tagged jets (from charm quarks) and untagged jets (from charm, gluons, or light
quarks) are similar. The pseudorapity distributions for D∗ -tagged and untagged jets (Fig. 32) show

49



differences consistent with those of the H1 analysis. As expected, the average jet pseudorapity increases
with increasing jet ET . Again, the theoretical uncertainties are reduced with respect to those of Fig.
29. At high jet ET , the shape of the massive calculation describes the data somewhat better than the
massless one.

ZEUS
-tagged jet*D >6 GeVjetTE

ZEUS 98-00
Jet energy scale uncertainty

<9 GeV
jet
T6<E

>9 GeVjetTE

Untagged jets >6 GeVjetTE

NLO QCD (massive)
 had.⊗NLO QCD (massive) 

Beauty

Untagged jets >6 GeVjetTE

NLO QCD (massless)

 had.⊗NLO QCD (massless) 
* D→NLO QCD (massless) no b

<9 GeV
jet
T6<E <9 GeV

jet
T6<E

>9 GeVjetTE >9 GeV
jet
TE

 e
’+

D
*+

je
t+

X
) 

(n
b)

→
(e

p
je

t
η

/d
σ

d

0

1

2

3

0

0.5

1

1.5

0

0.2

0.4

0.6

0

0.5

1

1.5

2

0

0.5

1

0

0.2

0.4

0.6

jetη
-1 0 1 2

jetη
-1 0 1 2

jetη
-1 0 1 2

Figure 32: D∗ + jet cross sections as a function of the pseudorapidities of D∗-tagged (left) and untagged
(center and right) jets, from the ZEUS analysis [174]. The measurements are compared to two NLO
predictions, the massive scheme calculations from Frixione et al. [58] and the massless scheme predictions
from Heinrich and Kniehl [48].

The selection of events with a D∗ and a muon from a semileptonic charm decay, not associated with
the D∗ [175], allows explicit tagging of both charm quarks. The small statistics (entry 11 in Table 3,
where also the visible phase space cuts are given) do not allow differential distributions, but the observed
total visible cross section for D∗µ production from double-tagged cc̄ final states of 250 ± 57 ± 40 pb is
consistent with the prediction from the massive NLO calculation [56] of 256+159−59 pb.

6.4 Parton-parton-correlation studies in charm-tagged events

Analyses using tagged charm events with two identified hard partons in the final state and studying the
correlations of the two partons [170, 172, 173, 174, 141, 178, 179] provide the most detailed informa-
tion on the charm-production mechanism. Similar to the previous subsections, there are two different
experimental approaches:

• The D∗ tag is used for charm tagging. For the two hard partons either the reconstructed D∗ plus
an additional non-associated jet are used [172, 174], or alternatively two jets are identified, one
of which is tagged by the D∗ [170, 173, 174, 179].

50



• Alternatively, dijet events are selected and one jet is tagged as a charm jet using the displaced
impact parameters of jet-associated charged tracks [141] or by a muon from a charm semileptonic
decay [178].

With the two identified partons three correlation observables are constructed, which will be discussed
in the following:

1. The observable xobsγ , which allows the separation, in the leading order picture, of direct- and
resolved-photon interactions. In the NLO picture, it separates 3-parton from 2-parton final states.

2. The azimuthal correlation ∆φ of the two partons, which is sensitive to higher-order effects. Com-
bined with xobsγ , it can distinguish between 2-parton, 3-parton and 4-parton final states.

3. The hard-scattering angle cosθ∗ of the two partons, which allows the distinction of contributions
with quark or gluon propagators in the hard scattering.

6.4.1 xobsγ studies

The case of two jets is assumed in the following for the two hardest partons. The observable xobsγ is
defined as

xobsγ =

∑

Jet1
(E − pZ) +

∑

Jet2
(E − pZ)

∑

h(E − pZ)
. (17)

The sums in the numerator run over the particles associated with the two jets and those in the denom-
inator over all detected hadronic final state particles. E and pZ denote the particle energy, and the
momentum parallel to the proton beam, respectively.

In the leading-order pQCD picture (Section 2.3, 2 partons + potential photon remnant + proton
remnant) this variable is an estimator of the fraction of the photon energy entering the hard interaction.
For the direct boson-gluon-fusion process (Fig. 6(a)) xobsγ approaches unity, as the hadronic final state
consists of only the two hard jets and the proton remnant in the forward region, which contributes little
to

∑

h(E − pZ). In resolved processes (Figs. 6(b-d)) the photon remnant significantly contributes to
the denominator but not to the numerator, so xobsγ can be small. The addition of parton showering can
somewhat dilute this simple picture.

xobsγ is also smaller than unity for next-to-leading-order processes with a third hard outgoing parton
(Fig. 8). In the massive NLO case for charm production this often coincides with the other quark
originating from initial-state photon splitting into a cc̄ pair, which would be classified as a photon
remnant in the leading-order picture. Since in the fixed-flavour NLO case there are at most three
partons, xobsγ separates 2-parton from 3-parton final states. In the variable-flavour NLO case the two
pictures described above get mixed, since in the case of an initial-state c quark from the photon the
other c quark can be a fourth hard parton. Thus, in general, the observable xobsγ is sensitive to the
resolved-photon structure (if any) and to tree-level higher-order processes (if any).

One of the milestone papers on charm photoproduction at HERA was the ZEUS analysis [170],
where xobsγ studies are performed using events with a D

∗ and two jets. The jets are required to have

transverse momenta p
jet1(2)
T > 7(6) GeV and are selected in a wide rapidity range |ηjet| < 2.4. In most

events the D∗ is associated to one of the two jets. Figure 33 shows the measured single-differential
cross section as a function of xobsγ . A peak at large x

obs
γ > 0.75 is observed, which reflects the direct-

photon/2-hard-parton component. Roughly 50% of the data are observed at xobsγ < 0.75, indicating
large contributions from resolved-photon/3-hard-parton or other higher-order contributions.

In the lower plot in Fig. 33 the data are compared to predictions from a massive scheme NLO
calculation [58]. Not all theoretical uncertainties are shown here. The calculation has a tendency to
underestimate the data cross sections at xobsγ < 0.75, where it is effectively a leading order calculation.
This might indicate the need for even higher-order corrections.

51



0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1

ZEUS 1996+97

xγOBS
d

σ
/d

x γ
O

B
S
  

(n
b

)

(a)

Data
Herwig: direct + resolved
Herwig: direct
Herwig: resolved
Herwig: resolved without

charm excitation

xγOBS

d
σ

/d
x γ

O
B

S
  

(n
b

)

(b)(b)

Data

Massive NLO, parton level, ε=0.02

µR = 1.0 m⊥ , mc = 1.5 GeV

µR = 0.5 m⊥ , mc = 1.2 GeV

Figure 33: Differential cross section as a function of xobsγ for dijet events with an associated D
∗ meson,

from a ZEUS analysis [170]. The shaded band indicates the energy scale uncertainty. The same data
are compared in the lower plot to an NLO calculation [58], and in the upper plot to Monte Carlo
predictions from HERWIG [22] with direct and resolved photon contributions shown separately. The
latter is dominated by the charm excitation component.

A much better shape description is obtained with the LO+PS HERWIG [22] Monte Carlo program
as shown in the upper plot of Fig. 33. In this calculation a large part of the NLO photon splitting
diagram in Fig. 7(c) is included in the form of a charm excitation component, where the charm quark is
treated as a massless constituent of the resolved photon, as shown in Figs. 6(c) and 6(d). This gives the
dominant contributions for xobsγ < 0.75. Combined with parton showering, which also pulls the “direct”
contribution towards lower xobsγ values, this provides a reasonable data description. This LO+PS MC
approach thus provides an effective way to describe the small xobsγ region, although the charm quark is
treated as massless in a kinematic region where this is probably not a good approximation. Note that
the total cross section with D∗ + dijets is only about 18% of the D∗ cross section without the dijets,
also measured in [170], for the same D∗ cuts applied (pT (D

∗) > 3 GeV and |η(D∗)| < 1.5). Thus the
problematic (for NLO) xobsγ < 0.75 region in the D

∗ + dijet sample contributes only a relatively small
part to the inclusive-D∗ cross section.

Another ZEUS analysis [174] using events with a D∗ and at least one jet compares the measured
xobsγ cross sections to both massive and massless scheme NLO calculations. Here the D

∗ and a jet, to
which the D∗ is not associated, are taken as estimators for the two leading partons and used for the
xobsγ reconstruction

18 in Eq. (17). The jet is required to have transverse momentum pT > 6 GeV in a
pseudorapidity range −1.5 < η < 2.4. Figure 34 shows the differential cross sections as a function of

18 Note that the available massless scheme calculations [48] provide only cross sections for a D∗ + jet final state but
not for two jets.

52



xobsγ . In the left (right) plot the data are compared to the massive (massless) scheme NLO predictions.
Both predictions are a bit too low for the 3-or-more-parton final state region xobsγ < 0.75, but are still
compatible with the data within their uncertainties. Note that in the massless calculation, which absorbs
the initial state photon splitting to cc̄ into the photon PDF, this contribution is effectively calculated to
NLO (one-loop virtual corrections), while it is only calculated to LO (0 loop) in the massive case. This
partially explains why the uncertainty of the massless calculation is much smaller in this region. At
high xobsγ both calculations are effectively NLO (1-loop) calculations, and the uncertainties are similar.

Figure 34: Differential cross sections as a function of xobsγ for events with a jet and a D
∗ meson,

which is not associated to the jet, from the ZEUS analysis [174]. In the left (right) upper plots the
data are compared to a NLO calculation in the massive [58] (massless [48]) scheme. The bottom plot
shows a comparison to the PYTHIA and HERWIG MC, which were used to calculate the hadronisation
corrections in the upper left plot.

6.4.2 Azimuthal correlations ∆φ

In a ZEUS analysis [174] using events with a D∗ and two jets (entry 9 in Table 3), and in the H1
measurement [172] with a D∗ and a non-associated jet (entry 7), the azimuthal correlation of the
two hard partons is investigated. In the leading-order picture of direct-photon interactions (Fig. 6a),
the two charm quarks are produced back-to-back in the azimuthal plane of the lab frame, i.e. with
∆φ = 180◦. Smaller ∆φ can be due to higher-order processes, such as gluon radiation, or due to a non-
zero transverse momentum of the partons that enter the hard interaction, e.g. from a flavour-excitation
process in which the c quark gets a finite transverse momentum in the backwards parton showering step.

53



In the ZEUS analysis jets were selected with harder transverse-momentum requirements but in a wider
η range than in the H1 analysis (cf. entries 9 and 7 in Table 3). Figure 35 shows the differential cross
sections as a function of the azimuthal difference ∆φ between the D∗ and the jet for the H1 analysis
and between the two jets for the ZEUS measurement. The H1 result is shown in the two rightmost
plots. The cross sections are highest for ∆φ = 180◦, i.e. for the back-to-back configuration, and drop off
towards smaller angles. NLO calculations in the massive scheme [58] and in the massless scheme [48] are
compared to the data. Both calculations drop off more steeply than the data towards smaller opening
angles. Below ∆φ ≈ 100◦ the two calculations predict very small contributions, while there are still
sizeable ones in the data. A better description is obtained with the LO+PS programs PYTHIA [21]
and CASCADE [28]. PYTHIA includes charm excitation processes in resolved-photon events, which
give the dominant contribution for ∆φ < 140◦ and provide a reasonable data description in this region.

The results of the ZEUS analysis [174] are shown in the left and central plots of Fig. 35. Here
the azimuthal correlation is measured separately in the 2-parton region xobsγ > 0.75 and in the 3-or-
more-parton region xobsγ < 0.75. The data are compared to an NLO calculation [58] in the massive
scheme. For the high-xobsγ region the description is satisfactory. However, in the low-x

obs
γ region the

NLO calculation is clearly falling below the data for ∆φ < 120◦. This is straightforward to understand
since a 3-parton final state can not produce an angle between the two leading pT partons of less than
120◦ 19. Correspondingly, at least four partons are needed to populate this region. A massive NLO
calculation produces at most three, so an NNLO calculation is needed to fill the gap. Again, a better
shape description is obtained by PYTHIA and HERWIG, which can provide several extra partons
through parton showering (of which flavour excitation is a part). Thus, the conclusion is again very
similar to the above studies with the xobsγ observable: the NLO calculation is missing a component in
the data, which can be effectively described by a LO+PS calculation. As to be expected from this
explanation, in the two lower plots the MC@NLO calculation, which complements the 2- and 3-parton
NLO matrix elements by parton showering, is able to describe these data well.

An H1 analysis using µ+dijet final states [178] (entry 14 in table 3) further supports these conclu-
sions.

6.4.3 Study of hard-scattering angle cos θ∗

In a dedicated analysis [173], using events with a D∗ and two jets (entry 8 in Table 3), ZEUS has
investigated the scattering angle θ∗ of the charm quark with respect to the proton direction in the dijet
rest frame. The charm quark is identified by the jet to which the reconstructed D∗ is associated. The
cos θ∗ distribution strongly reflects the type of the propagator particle exchanged in the 2 → 2 hard
interaction:

• For a charm quark propagator cos θ∗ should follow a (1 − |cos(θ∗)|)−1 distribution. The direct
photon (Fig. 6(a)) and the resolved process with a gluon from the photon structure (Fig. 6(b))
belong to this class of processes and also one of the charm excitation diagrams (Fig. 6(c)).

• For a gluon propagator cos θ∗ should follow a (1 − |cos(θ∗)|)−2 distribution, i.e. a much steeper
rise for |cos θ∗| → 1. For leading-order processes only the charm-excitation mechanism provides
such a contribution (Fig. 6(d)).

The main idea of the analysis is to look for such effects directly in the data. Special cuts are applied
in order to ensure a flat acceptance for the cos θ∗ distribution over a wide range, extending to as large
values of |cos θ∗| as possible. The invariant mass of the two jets is required to be above 18 GeV. The

19However, in the 3-parton topology one of the leading jets can escape outside of the kinematic region of the mea-
surement. Thus, the softest jet is used instead, which leads to strongly suppressed but non-zero charm contribution for
∆φ < 120◦

54



ZEUS

0 1 2 3
-410

-3
10

-210

-110

1
>0.75obsγx

ZEUS 98-00

Jet energy scale uncertainty

0 1 2 3
-410

-3
10

-210

-110

1
<0.75obsγx

NLO QCD (massive)

 had.⊗NLO QCD (massive) 
Beauty

0 1 2 3
-410

-3
10

-210

-110

1
>0.75obsγx

 2.5×HERWIG
 1.5×PYTHIA

0 1 2 3
-410

-3
10

-210

-110

1
<0.75obsγx

 e
’+

D
*+

jj
+

X
) 

(n
b/

ra
d.

)
→

(e
p

jj φ∆
/d

σ
d

 (rad.)
jj

φ∆  (rad.)
jj

φ∆
]°

(D
*,

 j
e

t)
  
[n

b
/

φ 
∆

/d
σ

d

-310

-210

-110

p:  D* + other jetγ

H1
Data

Cascade 1.2

Pythia 6.2

Pythia 6.2 (dir.)

p:  D* + other jetγ

H1

]°(D*, jet) [φ ∆
0 30 60 90 120 150 180

R
0.5

1

1.5

Data

Cascade 1.2

Pythia 6.2

Pythia 6.2 (dir.)

 [deg]
jj

φ∆
0 50 100 150

 [
p

b
/d

e
g

]
jj φ∆

/d
σ

d

-110

1

Xµ ejj→Xc ec→H1 Cross section: ep

>0.75obsγx

 [deg]
jj

φ∆
0 50 100 150

 [
p

b
/d

e
g

]
jj φ∆

/d
σ

d

-110

1

H1 Data 06/07

PYTHIA
CASCADE

HERWIG

MC@NLO

 [deg]
jj

φ∆
0 50 100 150

 [
p

b
/d

e
g

]
jj φ∆

/d
σ

d

-210

-110

1

Xµ ejj→Xc ec→H1 Cross section: ep

0.75≤obsγx

 [deg]
jj

φ∆
0 50 100 150

 [
p

b
/d

e
g

]
jj φ∆

/d
σ

d

-210

-110

1

H1 Data 06/07

PYTHIA
CASCADE

HERWIG

MC@NLO

]°
(D

*,
je

t)
 [

n
b

/
φ 

∆
/d

σ
d

-310

-210

-110

p:  D* + other jetγ

H1
Data

 Had⊗FMNR 

FMNR

Had⊗ZMVFNS

p:  D* + other jetγ

H1
Data

 Had⊗FMNR 

FMNR

Had⊗ZMVFNS

p:  D* + other jetγ

H1

]°(D*, jet) [φ ∆
0 30 60 90 120 150 180

R

0.5
1

1.5

Data

 Had⊗FMNR 

FMNR

Had⊗ZMVFNS

Figure 35: Azimuthal differences of two outgoing hard partons in charm events. The upper row
shows the D∗-tagged jet-jet azimuthal difference from ZEUS [174] separately for the high (left) and
low (middle) xobsγ region and the D

∗-other jet azimuthal difference from H1 [172] without cut on xobsγ
(right). Also shown are massive (NLO/FMNR)[56] and massless (ZMVFNS)[48] NLO predictions. The
middle row shows the same data compared to HERWIG[22], PYTHIA[21] and CASCADE[28] LO+PS
MC predictions. The bottom row shows the muon-tagged jet-jet azimuthal difference from H1 [178]
compared to these same MCs and to the NLO+PS MC@NLO [185] prediction.

55



average pseudorapidity of the two jets, defined as η
jet1+ηjet2

2
is required to be smaller than 0.7. Note

that these cuts yield a much smaller (but still sizeable) contribution at xobsγ < 0.75 than the one shown
in Fig. 33, mainly because they implicitly restrict the two jets to the pseudorapidity region ηjet < 1.9.
As intended, this analysis mainly probes 2-parton final states, plus a potential 3rd parton from the
photon remnant. In Fig. 36 the differential cross sections are shown as a function of cosθ∗, separately
for xobsγ < 0.75 and x

obs
γ > 0.75. In the lower half of Fig. 36 the data are compared to massive scheme

Figure 36: Differential cross sections as function of cos θ∗ for dijet events with an associated D∗

meson [173]. Results are given separately for samples enriched in direct (xobsγ > 0.75) and resolved
photon events (xobsγ < 0.75). The data are compared in the lower two plots to NLO predictions [58]
in the massive scheme using the parameter settings and variations listed in table 3. The predictions of
the CASCADE [28] MC are also shown. In the upper two plots the data are compared to predictions
using the HERWIG [22] and PYTHIA [21] MCs.

NLO predictions [58]. For the xobsγ > 0.75 region the NLO calculation provides a good description of
the data over the whole range of cos θ∗, with reasonably small uncertainty. The relatively shallow cos θ∗

dependence is consistent with the expectation that this region is dominated by the boson-gluon-fusion
graph, where the propagator particle is a charm quark, and for which the prediction is stabilised at
NLO by 1-loop virtual corrections. In the xobsγ < 0.75 region a much stronger rise is visible towards
more negative cos θ∗ values, and the central region is more strongly depleted. This can be interpreted as
a direct proof for sizeable contributions from gluon propagator exchanges such as the charm excitation
process (Fig. 6(d)), which at NLO is a tree-level process (Fig. 8(d,e) ). Correspondingly, the NLO

56



uncertainty is much larger. The upper edge of the uncertainty band describes the data reasonably. The
strong asymmetry can be attributed to the fact that the charm jet will preferentially be correlated with
the incoming photon direction.

The plots in the upper half of Fig. 36 show that the PYTHIA and HERWIG LO+PS MCs with
their large excitation contributions are able to describe the data well everywhere. This is particularly
true for PYTHIA. For the NLO calculation this means that contributions beyond NLO would probably
further improve the description. On the other hand, the CASCADE MC, which attempts to describe
hard higher-order topologies by allowing initial state partons to have sizeable transverse momentum,
reasonably describes the shapes, but fails to describe the relative normalisation of the low- and high-xobsγ
regions.

6.5 Summary

The charm mass provides a semi-hard QCD scale which already allows the application of perturbative
calculations to all of phase space, but which also competes with other, often even harder perturbative
scales. Total cross sections for charm photoproduction are reasonably described by such calculations.
Single-differential cross sections already provide a good handle to test the applicability of different QCD
approximations, although the theoretical uncertainties are mostly much larger than the experimental
ones. The theory predictions agree with the data up to the highest accessible transverse momenta,
showing no indications that final state resummation corrections are needed for massive calculations.

Double-differential cross sections, in particular those including jets, reveal a partial failure of the
massive scheme NLO predictions for the three independent parton-parton kinematic observables xobsγ ,
cos θ∗ and ∆φ, which were studied in charm events with a D∗ and one or two jets. For certain kinematic
regions this can be traced back to the absence of 4-or-more-parton final states in the calculation. The
partially large theoretical uncertainties can be explained by the absence of stabilizing virtual corrections
for 3-parton final states at this order. The NLO calculations in the massless scheme, where available, do
mostly not provide a better description for the observables. The LO+PS MCs PYTHIA and HERWIG,
which are often used for acceptance corrections, are able to describe all topologies reasonably, often
even very well. The CASCADE MC performs somewhat less well on average.

57



7 Beauty photoproduction

From the theoretical point of view, the only differences between charm photoproduction as discussed in
the previous section and beauty photoproduction are the beauty-quark mass and electric charge. The
first suppresses the cross section w.r.t. charm at low values of transverse momentum, while the second
suppresses it by about a factor 4 everywhere (Eq. 1). Experimentally, the signal-to-background ratio is
thus more challenging, and the available statistics is smaller. Together with the small branching ratio
to specific final states, this precludes any attempt to use fully-reconstructed beauty-hadron final states
at HERA. On the other hand, the higher mass and longer lifetime compared to charm hadrons increases
the tagging efficiency for inclusive tagging methods.

Table 4 summarises all H1 and ZEUS beauty photoproduction measurements. For the reasons
explained above, the first such measurements (entries 1 and 2) came several years after the first mea-
surements of charm, focused on beauty jet production, and were severely limited by statistics.

7.1 Total cross section for beauty production

Due to their high mass, even beauty quarks at rest in the centre-of-mass system of the partonic in-
teraction still produce reasonably high-momentum muons or electrons, which can be detected, when
decaying semileptonically. The forward and backward muon systems allow the detection of such beauty
quarks even when they are strongly boosted along the beam direction. Furthermore, the requirement
of two such muons, i.e. a double tag, strongly reduces both the light flavour and charm backgrounds.

In a ZEUS analysis [194] these properties were used to measure the total cross section for beauty
production in ep collisions without any cuts, i.e. including both photoproduction and DIS, by pushing
the measureable muon phase space to the limit (entry 12 of Table 4). After correcting for muon
acceptance and semileptonic branching ratios the resulting total cross section for bb̄ pair production in
ep collisions at HERA for

√
s = 318 GeV was determined to be

σtot(ep → bb̄X) = 13.9 ± 1.5(stat.)+4.0−4.3(syst.) nb, (18)

where the first uncertainty is statistical and the second systematic. The total cross section predicted by
next-to-leading-order QCD calculations was obtained in the massive approach by adding the predictions
from FMNR [56] and HVQDIS [51] for Q2 less than or larger than 1 GeV2, respectively. The resulting

cross section for
√
s = 318 GeV, using the scale choice µ =

√

m2b + p
2
Tb

σNLOtot (ep → bb̄X) = 7.5+4.5−2.1 nb (19)

is a factor 1.8 lower than the measured value, although compatible within the large uncertainties.
Compareable measurements were obtained in reduced regions of phase space from D∗ + muon and

dielectron final states (entries 10, 11 and 13 in table 4), and similar results were obtained for the ratio
of measured to predicted cross sections (see also corresponding entries in Fig. 37). Since D∗ mesons
and semi-isolated electrons could only be measured in the more central rapidity range, total bb̄ cross
sections were not extracted.

7.2 Single-differential cross sections

In order to make them compareable with each other, in Fig. 37 almost all available beauty-photopro-
duction cross sections20 have been translated, using NLO massive QCD calculations, into cross sections
for inclusive b-quark production as a function of pTb in the kinematic range Q

2 < 1 GeV2, 0.2 < y < 0.8

20entries 10,5,4,13,2,11,3,6,7,12,8 in table 4, following the order in the figure legend

58



No. Analysis b Tag Ref. Exp. Data L [pb−1] Q2 [GeV2] y Particle pT [GeV] η Events
bgfree

events

1 µ + dijets µ + prel
T

[189] H1 96 6.6 < 1 [0.1, 0.8]
µ

jet1(2)

> 2

> 6(6)

[−0.9, 1.1]
470 ± 43 120

2 e + dijets e + prel
T

[190] ZEUS 96-97 38.5 < 1 [0.2, 0.8]
e

jet1(2)

> 1.6

> 7(6)

[−1.1, 1.1]
[−2.4, 2.4] 140 ± 35 16

3 µ + dijets µ + prel
T

[191] ZEUS 96-00 110 < 1 [0.2, 0.8]
µ

jet1(2)

> 2.5

> 7(6)

[−1.6, 2.3]
[−2.5, 2.5] 834 ± 65 165

4 µ + dijets µ + prel
T

+ δ [142] H1 99-00 50 < 1 [0.2, 0.8]
µ

jet1(2)

> 2.5

> 7(6)

[−0.55, 1.1]
[−2.5, 2.5] 1745 128

5 lifet.+dijets imp. par. [141] H1 99-00 57 < 1 [0.15, 0.8]
Track

Jet1(2)

> 0.5

> 11(8)

[−1.3, 1.3]
[−0.9, 1.3] ∼ 80000 78

6 e + dijets e + prel
T

+ 6 E
T

[176] ZEUS 96-00 120 < 1 [0.2, 0.8]
e

Jet1(2)

> 0.9

> 7(6)

[−1.5, 1.5]
[−2.5, 2.5] ∼ 6000 129

7 µ+dijets µ + prel
T

+ δ [192] ZEUS 05 126 < 1 [0.2, 0.8]
µ

Jet1(2)

> 2.5

> 7(6)

[−1.6, 1.3]
[−2.5, 2.5] 7351 122

8 lifet.+dijets sec. vtx. [177] ZEUS 05 133 < 1 [0.2, 0.8]
tracks

Jet1(2)

> 0.5

> 7(6)

[−1.6, 1.4]
[−2.5, 2.5] ∼ 70000 1050

9 µ + dijets µ+imp.par. [178] H1 06-07 179 < 2.5 [0.2, 0.8]
µ

Jet1(2)

> 2.5

> 7(6)

[−1.3, 1.5]
[−1.5, 2.5] 6807 425

10 D∗ + µ Kππs + µ [175] H1 98-00 89 < 1 [0.05, 0.75]
D∗

µ

> 1.5

p > 2

[−1.5, 1.5]
[−1.74, 1.74] 56 ± 17 15

11 D∗ + µ Kππs + µ [193] ZEUS 96-00 114 < 1 [0.05, 0.85]
D∗

µ

> 1.9

> 1.4

[−1.5, 1.5]
[−1.8, 1.3] 232 16

12 dimuon µ + µ [194] ZEUS 96-00 114 all all µ1(2) > 1.5(0.75) [−2.2, 2.5] 4146 86
13 dielectron e + e [195] H1 07 48 < 1 [0.05, 0.65] e > 1 [−1.0, 1.74] ∼ 1500 51

Table 4: Beauty photoproduction cross-section measurements at HERA. Information is given for each analysis on the beauty tagging method, the
experiment, the data taking period, integrated luminosity, Q2 and y ranges and the cuts on transverse momenta and pseudorapidities of selected final state
particles. The last two columns provide information on the number of events in the analysis (number of signal events if an uncertainty is given) and the equivalent
number of background-free events. The centre-of-mass energy of all data taken up to 1997 (6th column) was 300 GeV, while it was 318 – 319 GeV for all subsequent
runs.

5
9



 (
p

b
/G

e
V

)
b T

/d
p

σ
d

-110

1

10

210

3
10

 (
p

b
/G

e
V

)
b T

/d
p

σ
d

-110

1

10

210

3
10

µH1 97-00 D*
H1 99-00 b jet

 jetµ →H1 99-00 b 
 ee→H1  b 

 e→ZEUS 96-97 b 
µ D* →ZEUS 96-00 b 

µ →ZEUS 96-00 b 
 e→ZEUS 96-00 b 

 jetµ →ZEUS 05 b 
µµ → bb -1ZEUS 114 pb

 b jet-1ZEUS 133 pb

NLO QCD
)/42

T
+p2

b
=(m2µ     

2
T

+p2b=m
2µ     

HERA

 ebX)→ (ep b
T

/dpσd

| < 2
b

η, 0.2 < y < 0.8, |2 < 1 GeV2Q

 (GeV)b
T

p

0 5 10 15 20 25 30

th
 

σ/
m

e
a

s
σ

1

2

3

 (GeV)b
T

p

0 5 10 15 20 25 30

th
 

σ/
m

e
a

s
σ

1

2

3

Figure 37: Differential cross section as a function of the transverse momentum of b quarks for the
kinematic range indicated in the figure. The bottom inset shows the ratio of the measured cross
sections to the central NLO QCD prediction. For more details see the text.

and beauty pseudorapidity |ηb| < 2. Each data point is displayed at the centroid of the pT distribution
of the b quarks entering the measurement bin of the respective analysis, which is mostly a bin in b-jet
ET , or, where not available, a bin in muon or electron pT . The b-quark pT rather than the b-jet pT
has been chosen here because the measurements extend down to very low pT at which jets can not be
usefully defined any more. Two massive NLO [56] theory predictions are given: one with scale choice

µ =
√

m2b + p
2
Tb (dashed), and one with scale choice µ =

√

m2b + p
2
Tb/2. The full theory uncertainty band

is shown for the latter (for the scale choice see also Section 2.11). It is dominated by the scale variations
(independent variation of renormalisation and factorisation scales by factor 2) and by the variation of
the pole mass (mb = 4.75±0.25 GeV). Where not provided directly in the original publications, the data
points were obtained using the data/theory ratio of the respective original measurements and rescaling
them to the theory prediction in Fig. 37, properly accounting for differences in the respective theory
calculation settings.

Within the large uncertainties, reasonable agreement between theory and data is observed over the
complete pT range covering 3 orders of magnitude in the cross section. In particular, as in the charm
case, there is no indication for a failure of the predictions at large pT . There might be a trend that
on average, the measurements of the double tagging analyses (D∗µ, ee and µµ), which were already
briefly discussed in the total cross-section subscetion, tend to lie a bit above the other measurements
which typically require dijet final states. The effect is not very significant, but if taken serious, might
indicate that the contribution of b quarks not associated to jets might be underestimated by the theory.
Unfortunately, currently no measurement is available which directly tests this hypothesis by considering
both topologies in a single analysis framework.

60



All available beauty photoproduction results are represented in this plot, except the results of the
very first H1 analysis [189] (entry 1 in Table 4), which has been declared superseded by a more recent
analysis [142], and the results of one of the latest H1 analyses [178] (entry 9), for which no comparison
to pure NLO predictions was provided.

Double-differential cross sections have not been measured so far. In the future they could best be
extracted using the inclusive vertexing approach [177], which offers the best effective statistics (entry 8
in Table 4).

Figure 38: Single-differential cross sections for muons from bb̄ decays to dimuons, as a function of
the muon transverse momentum (left), pseudorapidity (center), and dimuon azimuthal angle difference
(right), from ZEUS [194]. The measurements are compared to massive NLO predictions with the same
settings as the band in Fig. 37, and to the PYTHIA [21] prediction scaled to the data.

7.3 Measurements of bb̄ and jet-jet correlations

Several results give insight into correlations between two final state partons in bb̄ events. The ZEUS
analysis of dimuon final states (Fig. 38, entry 11 in Table 4) studies the azimuthal angle difference
between muons originating from different b quarks, in addition to single-differential distributions. Both
the massive NLO predictions and the PYTHIA MC predictions used for acceptance correction show
reasonable agreement with the data, in particular in shape.

An H1 analysis of dijet final states in which one of the jets is tagged by a muon from a semilep-
tonic b decay (Fig. 39, entry 9 in Table 4) studies the xobsγ and ∆φ variables described earlier in the
charm section. The MC@NLO prediction describes the data, except in the high xobsγ , high ∆φ bin.

obs
γx

0 0.2 0.4 0.6 0.8 1

 [
p

b
]

γ
/d

x
σ

d

50

100

150

obs
γx

0 0.2 0.4 0.6 0.8 1

 [
p

b
]

γ
/d

x
σ

d

50

100

150
Xµ ejj→Xb eb→H1 Cross section: ep

H1 Data 06/07

PYTHIA
CASCADE

HERWIG

MC@NLO

obs
γx

0 0.2 0.4 0.6 0.8 1

 [
p

b
]

γ
/d

x
σ

d

50

100

150

 [deg]
jj

φ∆
0 50 100 150

 [
p

b
/d

e
g

]
jj φ∆

/d
σ

d

-210

-110

1

 [deg]
jj

φ∆
0 50 100 150

 [
p

b
/d

e
g

]
jj φ∆

/d
σ

d

-210

-110

1

Xµ ejj→Xb eb→H1 Cross section: ep

>0.75obsγx
H1 Data 06/07

PYTHIA
CASCADE

HERWIG

MC@NLO

 [deg]
jj

φ∆
0 50 100 150

 [
p

b
/d

e
g

]
jj φ∆

/d
σ

d

-210

-110

1

 [deg]
jj

φ∆
0 50 100 150

 [
p

b
/d

e
g

]
jj φ∆

/d
σ

d

-210

-110

1

 [deg]
jj

φ∆
0 50 100 150

 [
p

b
/d

e
g

]
jj φ∆

/d
σ

d

-210

-110

1

Xµ ejj→Xb eb→H1 Cross section: ep

0.75≤obsγx
H1 Data 06/07

PYTHIA
CASCADE

HERWIG

MC@NLO

 [deg]
jj

φ∆
0 50 100 150

 [
p

b
/d

e
g

]
jj φ∆

/d
σ

d

-210

-110

1

Figure 39: Beauty cross sections as a function of xobsγ (left) and as a function of the jet-jet azimuthal
angle difference ∆φ for high (center) and low (right) values of xobsγ , from a recent H1 analysis [178].
The measurements are compared to MC@NLO [185] predictions, as well as to predictions from the
PYTHIA [21], HERWIG [22] and CASCADE [28] LO+PS Monte Carlos.

61



The agreement is thus slightly worse than in the charm case (Fig. 35). PYTHIA agrees everywhere,
HERWIG describes the shape but not the normalisation, and CASCADE fails for both. The latter
finding is again in qualitative agreement with the charm result. Thus the PYTHIA or HERWIG MCs
should preferentially be used for acceptance corrections.

A similar analysis by ZEUS [192] (not shown) compares the data directly to the massive NLO
predictions. Not surprisingly, these predictions fail in the same kinematic regions as for charm (Fig.
35), for the same reasons as discussed there.

Several other analyses [190, 191, 142] also studied xobsγ , with less statistics than but similar conclu-
sions as for charm.

7.4 Summary

Due do the suppression of the cross section by mass and charge, and small branching ratios to exclusive
final states, only inclusive or semi-inclusive tagging methods can be used at HERA to measure beauty
production. The reasonable acceptance for the detection of b hadron decays down to 0 transverse mo-
mentum and the coverage of almost the full physically relevant rapidity range allowed the measurement
of the total beauty production cross section at HERA. This cross section is higher than, but still com-
patible with, NLO QCD predictions. Several single differential beauty photoproduction cross sections
have also been measured. The measurements from H1 and ZEUS and from different final states agree
reasonably well with each other and with QCD predictions from threshold up to the highest accessible
transverse momenta. Double differential cross sections have not yet been measured.

62



8 Charm and beauty production in DIS

In the previous Sections 6 and 7 heavy-flavour production in ep collisions with the exchange of quasi-
real photons was discussed. The production of charm and beauty quarks was also studied in the
deeply inelastic scattering regime, which corresponds to photon virtualities Q2 ∼> 1 GeV

2. Large photon
virtuality provides an additional hard scale in the calculations and allows probing the parton dynamics
inside the proton with high resolution. An overview of all measurements is given in Table 5 and Table 6
for charm and beauty production, respectively.

8.1 Production mechanism

Already in the first H1 [196] and ZEUS [197] measurements in DIS (entries 1 and 2 in Table 5) boson-
gluon fusion was clearly identified to be the dominant production mechanism for charm quarks. This
was investigated using the distribution of the fractional momentum of D∗ mesons in the γp system,
xD =

2p∗
D

Wγp
, where p∗D denotes the D

∗ momentum measured in the γ∗p frame. The data were used to

disentangle between BGF and QPM-like models (cf. Fig. 1(a) and Fig. 2). The BGF process produces
a cc̄ pair that recoils against the proton remnant in the γ∗p frame, while the (massless) QPM produces
a single charm quark recoiling against the proton remnant (which contains the other charm quark).
Since the D meson carries a large fraction xD of the charm quark momentum, the former model should
lead to significantly softer distribution in xD. Figure 40 shows a comparison of the two models against
the ZEUS data. The observed shape of the cross section in the data proves that BGF is the dominant

Figure 40: Normalised differential D∗ -production cross section as a function of x∗D [197]. The mea-
surement was performed for 5 < Q2 < 100 GeV2. The points show the data, while solid and dashed
lines show the BGF (PGF) and QPM predictions.

charm-production process in DIS at HERA. This was quantified in [196] in the leading order QCD
picture by setting an upper limit for the fraction of the QPM-like contribution f(QPM) to charm DIS
production to be below 0.05 at 95% C.L.

8.2 Single-differential cross sections

Transition from photoproduction to DIS: The ZEUS collaboration has studied [202] (entry 9 in
Table 5) charm production in the intermediate Q2 region between photoproduction and DIS: 0.05 <
Q2 < 0.7 GeV2. The scattered electron was detected with the beampipe calorimeter (BPC) at very small
scattering angles. Figure 41(a) shows a comparison of the massive-scheme NLO QCD predictions [51]

63



No. Analysis c-Tag Ref. Exp. Data L [ pb−1] Q2 [GeV2] y Particle pT [GeV] η Events
effect.

s:b

bgfree

events

1 D∗ incl. Kππs [196] H1 94 3 [10, 100] < 0.53 D
∗ > 1.5 [−1.5, 1.5] 103 ± 13 1 : 0.7 64

D0 incl. Kπ D0 > 2.0 144 ± 19 1 : 1.5 57
2 D∗ incl. Kππs [197] ZEUS 94 3 [5, 100] < 0.7 D

∗ [1.3, 9.0] [−1.5, 1.5] 122 ± 17 1 : 1.4 52
3 D∗ incl. Kππs [169] H1 95-96 10 [2, 100] [0.05, 0.7] D

∗ [1.5, 15] [−1.5, 1.5] 583 ± 35 1 : 1.1 278
4 D∗ incl. Kππs [198] ZEUS 96-97 37 [1, 600] [0.02, 0.7] D

∗ [1.5, 15] [−1.5, 1.5] 2064 ± 72 1 : 1.5 822
Kππππs [2.5, 15] 1277 ± 124 1 : 11 106

5 D∗ incl. Kππs [199] H1 97 18 [1, 100] [0.05, 0.7] D
∗ > 1.5 [−1.5, 1.5] 973 ± 40 1 : 0.6 590

6 D∗ incl. Kππs [200] ZEUS 98-00 82 [1.5, 1000] [0.02, 0.7] D
∗ [1.5, 15] [−1.5, 1.5] 5545 ± 129 1 : 2 1850

7 D incl. D mes. + S [147] H1 99-00 48 [2, 100] [0.05, 0.7] D mesons > 2.5 [−1.5, 1.5] n.a. n.a. 263
8 D∗ incl. Kππs [201] H1 99-00 47 [2, 100] [0.04, 0.7] D

∗ [1.5, 15] [−1.5, 1.5] 2604 ± 77 1 : 1.3 1140
+ dijet Jet1(2) > 4(3) [−1, 2.5] 668 ± 49 1 : 2.5 186

9 D∗ incl. Kππs [202] ZEUS 98-00 82 [0.05, 0.7] [0.02, 0.85] D
∗ [1.5, 9] [−1.5, 1.5] 253 ± 25 1 : 1.5 100

10 D incl. D mes. [203] ZEUS 98-00 82 [1.5, 1000] [0.02, 0.7] D mesons > 3 [−1.6, 1.6] n.a. n.a. 1100
11 D+ incl. Kππ [204] ZEUS 96-00 120 [1.5, 1000] [0.02, 0.7] D+ [0, 10] [−1.6, 1.6] 691 ± 107 1 : 16 42

Λc incl. pK
0
S

Λc 79 ± 25 1 : 7 10
Λπ+ 84 ± 34 1 : 13 6

12 incl. lifet. imp. par. [205] H1 99-00 57 > 150 [0.1, 0.7] Track > 0.5 [−1.3, 1.3] ∼ 2300 1 : 22 100
13 incl. lifet. imp. par. [206] H1 99-00 57 [6, 120] [0.07, 0.7] Track > 0.5 [−1.3, 1.3] ∼ 50000 1 : 48 1024
14 D0 incl. Kπ + S [207] ZEUS 05 134 [5, 1000] [0.02, 0.7] D0 [1.5, 15] [−1.6, 1.6] 8274 ± 352 1 : 14 550

15 µ + jet
µ + prel

T
+

δ + 6 E
T

[208] ZEUS 05 126 > 20 [0.01, 0.7] µ > 1.5 [−1.6, 2.3] ∼ 5100 1 : 20 250

16 D∗ incl. Kππs [209] H1 04-07 351 [100, 1000] [0.02, 0.7] D
∗ [1.5, 15] [−1.5, 1.5] ∼ 600 1 : 7 260

17 D∗ incl. Kππs [144] H1 04-07 348 [5, 100] [0.02, 0.7] D
∗ > 1.25 [−1.8, 1.8] 24705 ± 343 1 : 3.8 5200

18 D∗ incl. Kππs [210] ZEUS 04-07 363 [5, 1000] [0.02, 0.7] D
∗ [1.5, 20] [−1.5, 1.5] 12893 ± 185 1 : 2.7 4860

19 D+ incl. Kππ + S [211] ZEUS 04-07 354 [5, 1000] [0.02, 0.7] D+ [1.5, 15] [−1.6, 1.6] 8356 ± 198 1 : 3.7 1800
20 incl. lifet. δ + S [149] H1 06-07 189 [5, 2000] n.a. Track > 0.3 [−1.3, 1.3] ∼ 210000 n.a. n.a.
21 incl. lifet. jet + δ + S [212] H1 06-07 189 > 6 [0.07, 0.625] Jet > 6 [−1.0, 1.5] ∼ 85000 1 : 17 4800
22 incl. lifet. jet + S [148] ZEUS 04-07 354 [5, 1000] [0.02, 0.7] Jet > 4.2 [−1.6, 2.2] ∼ 55000 1 : 11 4400

Table 5: Charm DIS measurements at HERA. Information is given for each analysis on the charm tagging method, the experiment,
the data taking period, integrated luminosity, Q2 and y ranges and the cuts on transverse momenta and pseudorapidities of selected final
state particles. The last three columns provide information on the number of tagged charm events, the effective signal-to-background ratio
and the equivalent number of background-free events. The centre-of-mass energy of all data taken up to 1997 (6th column) was 300 GeV,
while it was 318 – 319 GeV for all subsequent runs.

6
4



No. Analysis c-Tag Ref. Exp. Data L [ pb−1] Q2 [GeV2] y Particle pT [GeV] η Events
effect.

s:b

bgfree

events

1 µ + jet µ + prel
T

[213] ZEUS 99-00 72 > 2 [0.05, 0.7]
µ

Jet

> 2

EBr
T

> 6

[−1.6, 1.3]
[−2, 2.5] ∼ 290 1 : 4.5 70

2 µ + jet µ + prel
T

[142] H1 99-00 50 [2, 100] [0.1, 0.7]
µ

Jet

> 2.5

pBr
T

> 6

[−0.75, 1.15]
[−2.5, 2.5] ∼ 230 1 : 2.5 64

3 µ + jet µ + prel
T

[214] ZEUS 96-00 114 > 2 [0.05, 0.7]
µ

Jet

> 1.5

> 5

> −1.6
[−2, 2.5] ∼ 3000 1 : 18 160

4 µ + jet
µ + prel

T
+

δ + 6 E
T

[208] ZEUS 05 126 > 20 [0.01, 0.7] µ > 1.5 [−1.6, 2.3] ∼ 1300 1 : 14 90

5 e + jet
e + prel

T
+

δ + 6 E
T

[215] ZEUS 04-07 363 > 10 [0.05, 0.7] e [0.9, 8] [−1.5, 1.5] ∼ 2700 1 : 15 170

6 D∗ + µ Kππs + µ [193] ZEUS 96-00 114 > 2 [0.05, 0.7]
D∗

µ

> 1.5

> 1.4

[−1.5, 1.5]
[−1.75, 1.3] ∼ 11 1 : 1 4

7 incl. lifet. imp. par. [205] H1 99-00 57 > 150 [0.1, 0.7] Track > 0.5 [−1.3, 1.3] ∼ 760 1 : 16 45
8 incl. lifet. imp. par. [206] H1 99-00 57 [6, 120] [0.07, 0.7] Track > 0.5 [−1.3, 1.3] ∼ 5800 1 : 60 100
9 incl. lifet. δ + S [149] H1 06-07 189 [5, 2000] n.a. Track > 0.3 [−1.3, 1.3] ∼ 12000 n.a. n.a.
10 incl. lifet. jet + δ + S [212] H1 06-07 189 > 6 [0.07, 0.625] Jet > 6 [−1.0, 1.5] ∼ 10000 1 : 12 400
11 incl. lifet. jet + S [148] ZEUS 04-07 354 [5, 1000] [0.02, 0.7] Jet > 5 [−1.6, 2.2] ∼ 13000 1 : 14 800

Table 6: Beauty DIS measurements at HERA. Information is given for each analysis on the beauty tagging method, the experiment, the
data taking period, integrated luminosity, Q2 and y ranges and the cuts on transverse momenta and pseudorapidities of selected final state
particles. The last three columns provide information on the estimated number of tagged beauty events, the effective signal-to-background
ratio and the equivalent number of background-free events. The centre-of-mass energy of all data taken up to 1997 (6th column) was
300 GeV, while it was 318 – 319 GeV for all subsequent runs. The ”Br” label in the superscript refers to measurements in the Breit frame
(see text).

6
5



ZEUS

10
-4

10
-3

10
-2

10
-1

1

10

10
-1

1 10 10
2

10
3

Q2   (GeV2)

d
σ

/d
Q

2
  
 (

n
b

/G
e

V
2
)

ZEUS  BPC

ZEUS  DIS  98-00

HVQDIS

 e p → D* X

(a)

ZEUS

10 2

10 3

10
-3

10
-2

10
-1

1 10 10
2

10
3

Q2  (GeV 2)

σ
γp

(W
=

1
6

0
 G

e
V

) 
 (

n
b

)

ZEUS BPC
ZEUS DIS  98-00
ZEUS Photoproduction 96-97

 γ p → D* X

(b)

Figure 41: (a) Differential ep cross section for D∗ production as a function of Q2 [202] in the kinematic
region indicated in Table 5. The measurements [202, 200] are compared to massive-scheme NLO QCD
predictions (HVQDIS) [51]. (b) Differential γp cross section for D∗ production as a function of Q2 [202].
The D∗ data are shown in the photoproduction [170], transition [202] and DIS [200] regions. The curve
shows a fit to the data (see text).

to these very-low-Q2 as well as Q2 > 1.5 GeV2 data [200] (entry 6 in Table 5). The calculations provide
a remarkable description of the drop of the measured cross sections over 5 orders of magnitude from
Q2 = 0.05 GeV2 ≪ 4m2c to Q2 = 100 GeV2 ≫ 4m2c. The slope of dσ/dQ2 changes with Q2: it is
steeper at high Q2 > 4m2c, where it is mainly dictated by the photon-propagator dependence 1/Q

4,
than at low Q2 < 4m2c, where an asymptotic 1/Q

2 dependence is expected. To study this further, the
measured D∗ electroproduction cross sections were converted into γ∗p cross sections using the photon
flux in the improved Weizsäcker-Williams approximation (see Section 2.9). Figure 41(b) shows the
converted DIS as well as the photoproduction cross sections [170] (entry 5 in Table 3). The very-low-
Q2 measurements are consistent with the photoproduction cross section. The data were fitted with a
function σγp(Q

2) ∝ M2/(Q2 +M2). The extracted value was M2 = 13±2 GeV2, which is close21 to 4m2c
and is significantly larger than the value obtained from inclusive data, M20 = 0.52±0.04 GeV2 ≃ m2ρ [122].

Performance of the ZMVFNS: D∗ -production single-differential cross sections in DIS have also
been used to test available calculations in the massive and massless schemes. Figure 42 shows a com-
parison of the most precise measurements from H1 [209, 144] (entries 16 and 17 in Table 5) to NLO
QCD calculations. Since the ZMVFNS calculation is only valid in the regime where the charm-quark
mass can be neglected, an additional restriction is needed on the D∗ transverse momentum in the γ∗p
frame, p∗T (D

∗) > 2 GeV, on top of the selection outlined in Table 5. This is compareable to the cuts
used in the photoproduction analysis, for which the laboratory frame approximately coincides with the
γp system (see Fig. 28). The inelasticity is correlated with the centre-of-mass energy in the γ∗p frame,
Wγp (see Eq. (9)), thus the low-y region corresponds to the low-Wγp region. Therefore, as expected,
the ZMVFNS predictions deviate significantly from the massive-scheme calculations at low y, where
Wγp is not ≫ 4m2c, and come close to the FFNS calculations at high y. At low y the massless-scheme
calculations clearly fail to describe the data, while massive predictions are in agreement with the mea-
surement in the whole y range. Also for Q2 > 100 GeV2 the massive predictions describe the data well
within uncertainties, whereas the massless approach significantly overestimates the charm cross section.

21The actual kinematic threshold for a D∗ −D meson pair with only the D∗ detected is (
√

m(D∗)2 + pT (D∗)2 +mD)
2 −

pT (D
∗)2 = 17 GeV2.

66



0.1 0.2 0.3 0.4 0.5 0.6 0.7

 /
 d

y
 [

n
b

]
σ

d

0

5

10

15

20

0.1 0.2 0.3 0.4 0.5 0.6 0.7

 /
 d

y
 [

n
b

]
σ

d

0

5

10

15

20
H1 data
ZM-VFNS (CTEQ6.6M)
HVQDIS (CT10f3)

0.1 0.2 0.3 0.4 0.5 0.6 0.7

 /
 d

y
 [

n
b

]
σ

d

0

5

10

15

20
D* in DIS

H1

y
0.1 0.2 0.3 0.4 0.5 0.6 0.7

n
o

rm
R

0

1

2

3

y
0.1 0.2 0.3 0.4 0.5 0.6 0.7

n
o

rm
R

0

1

2

3

(a)

Q2 [GeV2]

d
σ

v
is

/d
Q

2
 [

n
b

 G
e

V
-2

]

H1
10

-5

10
-4

10
-3

10
2

10
3

H1 data

HVQDIS

ZMVFNS

p*T(D*) >2 GeV

(b)

Figure 42: Differential D∗-production cross section as a function of y for Q2 < 100 GeV2 [144] (a) and as
a function of Q2 for Q2 > 100 GeV2 [209] (b). Also shown are the massive NLO prediction (HVQDIS) [51]
and the massless NLO prediction (ZMVFNS) [48, 216]. The ratio shown on the left, Rnorm, represents
the ratio of individually-normalised distributions to the data, thus allowing a comparison of shapes
only.

The data also clearly establish that the ZMVFNS fails to describe heavy-flavour production in DIS at
HERA. Similar conclusions were drawn in [201] (entry 8 in Table 5), but with a lower precision of the
data.

Event and heavy-flavour kinematics: Most of the analyses summarised in Tables 5 and 6 studied
event, charm and beauty kinematics differentially in the respective fiducial phase spaces. The most
precise D∗ measurements [209, 144, 210] (entries 16–18 in Table 5) were combined [217] to obtain the
most precise charm differential cross sections with essentially no theory uncertainty due to extrapolation
to a common phase space. The combination was done with a careful treatment of correlations. As
expected, the individual measurements were found to be consistent. The uncorrelated uncertainties were
reduced due to effective doubling of statistics, while the correlated systematic uncertainties were reduced
through cross-calibration effects between the two experiments. Figure 43(a)–(c) shows a comparison of
massive-scheme NLO QCD predictions [51] to the D∗ combined single-differential cross sections. The
predictions describe the data very well within uncertainties. However, the data reach 5% precision over
a large fraction of the measured phase-space, whereas the typical theory uncertainty ranges from 30%
at low Q2 to 10% at high Q2. The theory uncertainty is dominated by the independent variation of the
µR and µF scales, the uncertainty on the charm-quark pole mass and variations of the fragmentation
model. Therefore, higher-order massive-scheme NNLO calculations and an improved fragmentation
model for these predictions are needed to fully exploit the potential of these data. In addition, theory
uncertainties were studied in detail and a “customised” prediction was obtained by a variation of the
theory parameters within their uncertainties, to show that the calculations can simultanously provide
a good description of the shape and normalisation of all measured distributions with a single set of
parameters. This led to a renormalisation scale reduced by a factor 2 (see also Section 2.11), the
charm-quark pole mass reduced to mc = 1.4 GeV and to a change of fragmentation parameters, all
within the nominal uncertainties.

Moreover, differential cross sections of other D mesons as well as of leptons from heavy-flavour
decays and of heavy flavour jets were measured. In particular, Fig. 43(d) shows the D+ differential

67



 (D*)η
-1.5 -1 -0.5 0 0.5 1 1.5

 (
D

*)
 (

n
b

)
η

/d
σ

d

0

1

2

HERA-II
NLO QCD
NLO QCD customised

± D*→NLO QCD b 

2 < 1000 GeV25 < Q
0.02 < y < 0.7

(D*) > 1.5 GeV
T

p
(D*)| < 1.5η|

X           H1 and ZEUS± eD*→    ep 

(a)

 (
D

*)
 (

n
b

/G
e

V
)

T
/d

p
σ

d

-410

-310

-210

-110

1

 (D*) (GeV)
T

p
2 3 4 5 6 7 8 9 10 20

ra
ti

o
 t

o
 H

E
R

A

0.6

0.8

1

1.2

HERA-II
NLO QCD
NLO QCD customised

± D*→NLO QCD b 

2 < 1000 GeV25 < Q
0.02 < y < 0.7

(D*) > 1.5 GeV
T

p
(D*)| < 1.5η|

X           H1 and ZEUS± eD*→    ep 

(b)

)
2

 (
n

b
/G

e
V

2
/d

Q
σ

d

-510

-410

-310

-210

-110

1

)2 (GeV2Q
10 210 310

ra
ti

o
 t

o
 H

E
R

A

0.6

0.8

1

1.2

1.4

HERA-II
NLO QCD
NLO QCD customised

± D*→NLO QCD b 

2 < 1000 GeV25 < Q
0.02 < y < 0.7

(D*) > 1.5 GeV
T

p
(D*)| < 1.5η|

X           H1 and ZEUS± eD*→    ep 

(c)

(d)

Figure 43: Differential D∗ -production cross section [217] as a function of (a) η(D∗), (b) pT (D
∗) and

(c) Q2. The data points are shown with uncorrelated (inner error bars) and total (outer error bars)
uncertainties. Also shown are the NLO QCD predictions (HVQDIS) [51] with theory uncertainties
indicated by the band. The beauty-production contribution is included in the cross section definition
and is plotted separately. A customised NLO calculation (see text) is also shown. (d) Differential
D+ cross section as a function of p2T (D

+) down to pT (D
+) = 0 GeV [204].

cross section [204] (entry 11 in Table 5) measured down to pT (D
+) = 0 GeV. The measurement was

done in the D+ → K0Sπ+ decay channel. The presence of a neutral strange hadron in the decay resulted
in a reasonable signal-to-background ratio even at very low transverse momentum of the D+ . The
data were found to be described by the massive NLO QCD calculations within about two standard
deviations.

Furthermore, parton-parton correlations have been studied in D∗ -tagged events [201] (entry 8 in
Table 5). The conclusions are similar to those obtained from the respective photoproduction measure-
ments. In general the massive QCD calculations provide a good description apart from the region of
small ∆φ and very large |∆η| between the two leading jets in the event (not shown).

68



 (GeV)
jet
TE

5 10 15 20 25 30 35

 (
p

b
 /

 G
e

V
)

je
t

T
 /

 d
E

σ
d

-110

1

10

210

-1ZEUS 354 pb
had

 C×HVQDIS+ZEUS-S 
had

 C×HVQDIS+ABKM 
Rapgap x 1.49

ZEUS

 e jet X→X b e b→ep 

 (GeV)
jet
TE

5 10 15 20 25 30 35

D
a

ta
 /

 H
V

Q
D

IS

0.5

1

1.5

2

′ ′ ′

(a)

jetη

-1.5 -1 -0.5 0 0.5 1 1.5 2

 (
p

b
)

je
t

η
 /

 d
σ

d

100

200

300

400

500

-1ZEUS 354 pb
had

 C×HVQDIS+ZEUS-S 
had

 C×HVQDIS+ABKM 
Rapgap x 1.49

ZEUS

 e jet X→X b e b→ep 

jetη
-1.5 -1 -0.5 0 0.5 1 1.5 2

D
a

ta
 /
 H

V
Q

D
IS

0.5

1

1.5

2

′ ′ ′

(b)

Figure 44: Differential cross section for inclusive-jet production in beauty DIS events as a function
of E

jet
T (a) and η

jet (b) [148]. The data points are shown with statistical (inner error bars) and total
(outer error bars) uncertainties. Also shown are the NLO QCD predictions (HVQDIS) [51], corrected
for hadronisation effects, with theory uncertainties indicated by the band. The dashed line shows the
prediction from the RAPGAP MC generator [23] scaled to the measured integrated cross section.

Inclusive lifetime tagging (entries 21, 22 in Table 5) allowed the extension of the kinematic range of
charm measurements up to E

jet
T = 35 GeV (not shown), which roughly corresponds to pT (D) ≈ 20 GeV,

where the statistics of fully reconstructed charm mesons becomes poor. Good agreement is again
observed.

Figure 44 shows the corresponding single-differential jet cross sections for beauty production in
DIS [148] (entries 11 and 22 in Tables 6 and 5, respectively). The lifetime-tagging technique together
with the reconstruction of the vertex mass were used to extract charm- and beauty-jet cross sections
simultaneously. This measurement was selected since it has the highest statistical significance for
beauty-quark production, as can be seen from the last column of Table 6. The typical precision reached
in the data is 10 – 20% and is comparable to the theory uncertainties. The massive-scheme NLO QCD
calculations provide a good description of the shape and normalisation of the measured cross sections.

8.3 Double-differential cross sections

The large collected data samples allowed measurements of double-differential heavy-flavour cross sec-
tions, to study the correlations between various kinematic variables.

The H1 collaboration has studied [201] (entry 8 in Table 5) the cross section as a function of xobsγ
in different Q2 ranges, complementing the measurements in the photoproduction regime discussed in
Section 6.4.1. It was shown that the amount of higher order contributions included in the massive
NLO calculations, including topologies which would be called “flavour excitation” in the leading order
picture, is enough to describe the data for different Q2, while the BGF-only component of the RAPGAP

69



(D*)η
-1.5 -1 -0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

(D*)η
-1.5 -1 -0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1 *(D*) < 1.25 GeV
T

0.30 < p

* 
[n

b
/G

e
V

]
T

 d
p

η
 /

 d
σ

2
d

H1

(D*)η
-1.5 -1 -0.5 0 0.5 1 1.50

0.5

1

1.5

(D*)η
-1.5 -1 -0.5 0 0.5 1 1.50

0.5

1

1.5 *(D*) < 2.00 GeV
T

1.25 < p

(D*)η
-1.5 -1 -0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

(D*)η
-1.5 -1 -0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1 *(D*) < 3.00 GeV
T

2.00 < p

(D*)η
-1.5 -1 -0.5 0 0.5 1 1.50

0.05

0.1

0.15

0.2

(D*)η
-1.5 -1 -0.5 0 0.5 1 1.50

0.05

0.1

0.15

0.2 *(D*) < 6.00 GeV
T

3.00 < p

(D*)η
-1.5 -1 -0.5 0 0.5 1 1.5

0

0.002

0.004

0.006

0.008

0.01

(D*)η
-1.5 -1 -0.5 0 0.5 1 1.5

0

0.002

0.004

0.006

0.008

0.01 *(D*) < 20.00 GeV
T

6.00 < p
H1 data

HVQDIS (MSTW2008f3)

HVQDIS (CT10f3)

D* in DIS

Figure 45: Double-differential D∗ cross section as a function of p∗T (D
∗) and η(D∗) [144]. The data

points are shown with statistical (inner error bars) and total (outer error bars) uncertainties. NLO
QCD calculation (HVQDIS) [51] with two different proton parton densities are compared to the data.
Theoretical uncertainties are indicated by the bands.

Monte Carlo can describe the measurement for Q2 > 5 GeV2 after rescaling, but fails to describe
the shape observed in the data at lower Q2 (not shown here). This is to be expected, since in this
“photoproduction-like” region (see Fig. 41) the “flavour excitation” component will then be missing.

Using the full HERA II data sample, cross sections have been measured by H1 [144] (entry 17
in Table 5) as a function of the D∗ pseudorapidity in the laboratory frame, η(D∗), in bins of the
D∗ transverse momentum in the γp centre-of-mass frame, p∗T (D

∗). Figure 45 shows a comparison of the
massive NLO QCD predictions to the data. At large p∗T (D

∗), D∗ production in the backward region
is very suppressed, while at low p∗T (D

∗) the η(D∗) distribution is rather flat in the phase space of
the measurement. The massive-scheme NLO predictions provide a good description of the data. The
predictions depend only very little on the proton PDFs used for the calculation.

In most of the analyses summarised in Tables 5 and 6 the double-differential cross sections in Q2 and
y or Q2 and x were also measured. These measurements allowed dedicated studies of the inclusive heavy-
flavour event kinematics which can be expressed in terms of the charm reduced cross sections, or of the
charm contribution to the structure function F2 (see Section 8.4). Figure 46 shows the combined double-
differential D∗ cross sections as a function of Q2 and y [217]. Massive-scheme NLO QCD predictions
provide a good description of these cross sections in the full range in Q2 between 1.5 GeV2 and 1000 GeV2.

70



)2
d

y
 (

n
b

/G
e

V
2

/d
Q

σ2
d

y

                                             H1 and ZEUS

-2-1 -1
0

2

4

6

-2-1 -1
0

2

4

6
2<3.5 GeV21.5<Q

0.03 0.10 0.30

-2-1 -1
0

1

2

3

-2-1 -1
0

1

2

3
2<5.5 GeV23.5<Q

0.03 0.10 0.30

-2-1 -1
0

1

2

-2-1 -1
0

1

2 2<9 GeV25.5<Q

0.03 0.10 0.30

-2-1 -1
0

0.5

1

-2-1 -1
0

0.5

1
2<14 GeV29<Q

0.03 0.10 0.30

-2-1 -1
0

0.2

0.4

0.6

-2-1 -1
0

0.2

0.4

0.6
2<23 GeV214<Q

0.03 0.10 0.30

-2-1 -1
0

0.1

0.2

-2-1 -1
0

0.1

0.2
2<45 GeV223<Q

0.03 0.10 0.30

-2-1 -1
0

0.02

0.04

-2-1 -1
0

0.02

0.04 2<100 GeV245<Q

0.03 0.10 0.30

-2-1 -1
0

0.005

-2-1 -1
0

0.005

2<158 GeV2100<Q

0.03 0.10 0.30

-2-1 -1
0

0.001

0.002

0.003

-2-1 -1
0

0.001

0.002

0.003
2<251 GeV2158<Q

0.03 0.10 0.30

-2-1 -1
0

0.1

0.2

0.3

-3
10×

-2-1 -1
0

0.1

0.2

0.3

-3
10×

2<1000 GeV2251<Q

0.03 0.10 0.30

 D*X→HERA ep 

NLO QCD

NLO QCD customised

 D*→NLO QCD b 

y

2 < 1000 GeV21.5 < Q
0.02 < y < 0.7

(D*) > 1.5 GeV
T

p
(D*)| < 1.5η|

y

Figure 46: Double-differential D∗ cross section as a function of Q2 and y [217]. The data points
are shown with uncorrelated (inner error bars) and total (outer error bars) uncertainties. Also shown
are the NLO QCD predictions (HVQDIS) [51] with theory uncertainties indicated by the band. The
beauty-production contribution is included in the cross-section definition and is plotted separately.

The theoretical uncertainties decrease with increasing Q2. For Q2 ∼< 50 GeV
2 the theoretical uncertainties

are larger than those of the measured cross sections. Similar to the single-differential distributions shown
in Fig. 43, the theoretical uncertainties are dominated by the scale variations, the uncertainty on the
charm-quark pole mass and the variation of the fragmentation model. A higher-order calculation with
improved fragmentation model is needed to achieve a theoretical precision similar to the data.

8.4 Proton structure functions and reduced cross sections

The measured double-differential DIS cross sections of heavy-flavour production as a function of Q2

and y or Q2 and x were used to extract the heavy-flavour reduced cross sections, σ
QQ̄
red , or the heavy-

flavour contribution to the proton structure function F2, F
QQ̄
2 , where Q is either c or b. As discussed in

Section 2.8, the inclusive double-differential cross sections of heavy-flavour production can be expressed

in terms of σ
QQ̄
red or F

QQ̄
2 and F

QQ̄
L . In measurements of F

QQ̄
2 the small contribution arising from F

QQ̄
L was

subtracted relying on theory, corresponding to a correction of up to 4%. The extraction from the
measured cross sections requires an extrapolation from the experimentally accessible kinematic region

71



in pT and η and a particular final state to the full phase space of heavy quarks. The extrapolation
was done either using the massive-scheme NLO QCD calculations or LO+PS Monte Carlo simulations.
Since this procedure relies on the description of kinematic distributions by predictions, a non-negligible
theoretical uncertainty was introduced. This additional uncertainty was estimated by varying the
parameters in the calculations which affect the shapes of the kinematic distributions.

The σ
QQ̄
red and F

QQ̄
2 values extracted from measurements performed with different experimental tech-

niques and different detectors can be directly compared. Such measurements are complementary to
each other due to different dominant sources of systematics, mostly independent statistics and different
kinematic coverage, resulting in somewhat different theoretical uncertainties due to extrapolation. For
instance, for the σcc̄red measurements the dominant systematics in the H1 inclusive vertexing analysis (en-
try 20 in Table 5) is due to the treatment of the light-flavour component, while in the H1 D∗ HERA II
measurement (entry 17 in Table 5) the dominant systematics is due to the modelling of the tracking
efficiency. The ZEUS analysis of charm semileptonic decays (entry 15 in Table 5) has yet completely
different systematics. Therefore, a combination of measurements with such different techniques allows
a significant reduction not only of statistical and uncorrelated but also of correlated systematic and
extrapolation uncertainties.

0

0.2

0

0.5

0

0.5

0

0.5

σ
re

d

 c
c_

H1 VTX

H1 D* HERA-II

H1 D* HERA-I

ZEUS c → µ X

ZEUS D* 98-00

ZEUS D* 96-97

ZEUS D0

ZEUS D+ H1 and ZEUS
Q2=2.5 GeV2 Q2=5 GeV2 Q2=7 GeV2

Q2=12 GeV2 Q2=18 GeV2 Q2=32 GeV2

Q2=60 GeV2 Q2=120 GeV2 Q2=200 GeV2

Q2=350 GeV2

10-4 10-3 10-2

Q2=650 GeV2

10-4 10-3 10-2

Q2=2000 GeV2

HERA

10-4 10-3 10-2

x

Figure 47: Combined reduced cross sections [84] σcc̄red (filled circles) as a function of x for fixed values of
Q2. The input data are shown with various other symbols as explained in the legend. The error bars
represent the total uncertainty including uncorrelated, correlated and procedural uncertainties added
in quadrature. For presentation purposes each individual measurement was shifted in x.

72



Figure 47 shows a comparison of H1 and ZEUS measurements of the charm reduced cross sections22

as well as the milestone result of their combination [84]. The combination accounts for correlations of the
systematic uncertainties among the different input data sets. The individual σcc̄red measurements show
good consistency, with a χ2 value of 62 for 103 degrees of freedom. The combined data are significantly
more precise than any of the input data sets. Figure 47 also highlights the advantages of different
tagging techniques: while D∗ has superior precision at low Q2 due to better signal-to-background ratio,
the inclusive vertexing analysis with lifetime tagging dominates at high Q2 due to the larger accessible
statistics. The final total precision of the combined charm reduced cross sections is 10% on average and
reaches 6% at low x and medium Q2. This corresponds to a factor 2 improvement over the most precise
data set in the combination.

0

0.2

2=2.5 GeV2Q

0

0.2

2= 5 GeV2Q

0

0.2

2= 7 GeV2Q

0

0.5
2=12 GeV2Q

0

0.5
2=18 GeV2Q

0

0.5
2=32 GeV2Q

0

0.5
2=60 GeV2Q

0

0.5
2=120 GeV2Q

0

0.5
2=200 GeV2Q

0

0.5
2=350 GeV2Q

0

0.5
2=650 GeV2Q

0

0.5
2=2000 GeV2Q

re
dc

 c
σ

0

0.2

0

0.5

0

0.5

0

0.5

-410 -310 -210 -410 -310 -210
x

-410 -310 -210

HERA  
HERAPDF1.5

H1 and ZEUS

)2 (GeV2Q
10 210

310

 +
 0

.0
3

 i
b

b 2
F

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22 -1ZEUS vtx 354 pb
-1ZEUS e 363 pb
-1 114 pbµZEUS 

-1+vtx 126 pbµZEUS 
-1H1 vtx 246 pb

HERAPDF 1.5

ABKM NNLO

MSTW08 NLO

MSTW08 NNLO

CTEQ6.6 NLO

JR09

x=0.032 i=0

x=0.013 i=1

x=0.005 i=2

x=0.002 i=3

x=0.0013 i=4

x=0.0005 i=5

x=0.0002 i=6

x=0.00013 i=7

Figure 48: Left: Combined σcc̄red [84] (filled circles) as a function of x for fixed values of Q
2. The error

bars represent the total uncertainty including uncorrelated, correlated and procedural uncertainties
added in quadrature. The data are compared to the NLO predictions based on HERAPDF1.5 [42]
in the TR standard GMVFNS [71]. The line represents the prediction using Mc = 1.4 GeV. The
uncertainty band shows the full PDF uncertainty which is dominated by the variation of Mc in the
range 1.35 < Mc < 1.65 GeV. Right: Measurements of F

bb̄
2 [148, 149, 208, 214, 215] (various symbols)

as a function of Q2 at fixed values of x. The inner error bars are the statistical uncertainties, while
the outer error bars are the statistical, systematic and extrapolation uncertainties added in quadrature.
The data are compared to several NLO and NNLO predictions, including HERAPDF1.5 [42] in the TR
standard GMVFNS [71]. The uncertainty band shows the full PDF uncertainty which is dominated by
the variation of Mb.

22entries 4,6,8,12,13,14,15,16,17,20 in Table 5

73



      H1 CHARM AND BEAUTY CROSS SECTION FRACTION

10
-3

10
-2

10
-1f 

q
q_

x = 0.0002

H1 Data
f cc

_

f bb
_

x = 0.00032

MSTW08 NNLO

f cc
_

f bb
_

x = 0.0005

10
-3

10
-2

10
-1

x = 0.0008 x = 0.0013 x = 0.002

10
-3

10
-2

10
-1

x = 0.0032 x = 0.005 x = 0.013

10
-3

10
-2

10
-1

10 10
2

10
3

Q2 / GeV2

x = 0.02

10 10
2

10
3

Q2 / GeV2

x = 0.032

10 10
2

10
3

Q2 / GeV2

x = 0.05

- - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - – - - - - - - - - - - - - - - - - -

- - - - - - - - - – - - - - - - - - - - - - - - - - -

- - - - - - - - - – - - - - - - - - - - - - - - - - -

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

0

0.1

0.2

0.3

0

0.1

0.2

0.3

0

0.1

0.2

0.3

10
-4

10
-3

10
-2

0

0.1

0.2

0.3

10
-4

10
-3

10
-2

10
-4

10
-3

10
-2

Q2 = 2.5 GeV2

σ
c

c re
d  

/ 
σ

  
  re
d

Q2 = 5 GeV2 Q2 = 7 GeV2

Q2 = 12 GeV2 Q2 = 18 GeV2 Q2 = 32 GeV2

Q2 = 60 GeV2 Q2 = 120 GeV2 Q2 = 200 GeV2

x
Q2 = 350 GeV2 Q2 = 650 GeV2

x

HERA

HERAPDF1.5

- - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Figure 49: Fraction of charm and beauty contributions to the inclusive DIS cross section as a func-
tion of x and Q2 [206, 84, 218]. Also shown (curves) is a GMVFNS prediction by MRST [219] and
HERAPDF1.5. The dashed and dash-dotted lines are the asymptotic limits for charm and beauty from
Eq. (1).

Additionally, new statistically independent measurements of charm production have been published
(entries 18,19 and 22 in Table 5), in particular the ZEUS D∗ measurement with HERA II data. In
Fig. 48 the combined σcc̄red [84] and individual measurements of F

bb̄
2 (entries 3–5, 7–9, 11 in Table 6) are

compared to NLO and NNLO QCD predictions.

The beauty measurements are all in good agreement with each other and the most precise data were
obtained with inclusive lifetime tagging. The NLO QCD prediction in the GMVFNS approach based on
HERAPDF1.5 [42] is common for the two comparisons. The good agreement between these predictions
and the heavy-flavour data shows that the gluon density, which in HERAPDF1.5 is extracted from the
scaling violations of F2 , is adequate for the description of these gluon-induced production processes.
Other GMVFNS predictions were also compared to the combined charm reduced cross sections (not
shown). The best description of the data was provided by predictions including partial O(α3s) cor-
rections, while predictions including O(α2s) terms agreed well with the data and predictions including
O(αs) have shown the largest deviations [84]. The theoretical uncertainty for σcc̄red and F bb̄2 increases at
low Q2 and is dominated by the Mc variation. This indicates that the low-Q

2 data are sensitive to
the value of the heavy-quark mass used in the calculation, which was exploited to extract the optimal
Mc values for different GMVFNS schemes as well as to measure the running heavy-quark masses (see
Section 9.2).

Fig. 49 shows the fraction of the heavy-flavour component in the total inclusive DIS cross section:
fqq̄ = F

qq̄
2 /F2 and σ

cc̄
red /σred . As expected, the heavy-flavour fractions increase with increasing Q

2.
For x ∼< 0.01, the asymptotic limit is approached towards Q

2 ∼ 50 m2c ∼ 100 GeV2 for charm and
Q2 ∼ 50 m2b ∼ 1000 GeV2 for beauty. The charm and beauty fractions in the high-Q2 data come close to
4/11 and 1/11, respectively, stressing the importance of the heavy-flavour component for the description
of inclusive DIS. The observed suppression of the heavy-flavour fractions for x ∼> 0.01 originates from the

74



rising importance of the valence-quark contribution to the inclusive DIS cross section in this kinematic
domain.

8.5 Summary

Large photon virtuality Q2 provides an additional hard scale in the QCD calculations of heavy flavour
production and allows probing the parton dynamics inside the proton more directly than photoproduc-
tion. The dominant contribution to the charm and beauty cross sections arises from photon-gluon fusion.
For Q2 ≫ 4m2Q, where the photon virtuality is the dominant hard scale, the cross-section behaviour is
similar to the one of the inclusive cross section for deeply inelastic scattering. At high Q2 and low x,
the naively expected charm and beauty contributions of 4/11 and 1/11 are asymptotically approached.
For Q2 ≪ 4m2Q, where the quark mass is the dominant hard scale, the cross section behaves essentially
like photoproduction, i.e. the photon can be approximated to be quasi-real. NLO QCD predictions
using zero-mass schemes (ZMVFNS) fail to describe the data in the vicinity of or below the so-called
“flavour threshold” at Q2 ∼ m2Q. NLO QCD predictions in the massive scheme (FFNS) give a good
description of heavy flavour production at HERA over the complete accessible kinematic range. NLO
predictions in variable flavour number schemes (GMVFNS) are only available for inclusive quantitities,
and perform about equally well. There is no indication for the need of resummation of ln Q2/m2Q terms
at HERA energies. In particular for charm, the uncertainties from QCD corrections beyond NLO and
from the modelling of fragmentation are considerably larger than the experimental uncertainties of the
measured cross sections. Improved QCD calculations would therefore be highly welcome.

75



9 Measurement of QCD parameters, proton structure, and

impact on LHC and other experiments

So far the emphasis was on direct cross-section measurements from the HERA data and on the compar-
ison to and performance of different theoretical approximations for the perturbative QCD expansion.
In this section the extraction of more fundamental QCD parameters and parametrisations will be dis-
cussed, which are of direct relevance to all high energy physics processes and to the Standard Model of
particle physics in general.

9.1 Measurement of charm fragmentation functions and fragmentation
fractions

As outlined in Section 2.10, fragmentation fractions, i.e. the probability of a quark of a given flavour to
form a specific final state hadron, and fragmentation functions, parametrising the fraction of the energy
or momentum of the final state quark which will be taken by the final state hadron, are essential to
relate theoretical QCD calculations at parton level to measurable hadronic final states.

Studies of the fragmentation process are based on a complete reconstruction of the final-state hadron.
The statistics accessible at HERA for fully-reconstructed beauty hadrons is extremely low due to the
moderate beauty-production cross section and small branching ratios. On the other hand, HERA is effec-
tively a charm factory, with about 108 charm events recorded to tape. Therefore, only the fragmentation
of charm quarks has been studied by H1 and ZEUS. Charm fragmentation has been studied in both the
DIS and photoproduction regimes. A comparison between these results and e+e− measurements pro-
vides a so far unique test of the fragmentation universality in the heavy-flavour sector for colour-neutral
(electromagnetic) vs. coloured (strongly interacting) initial states.

9.1.1 Charm fragmentation function

The explicit reconstruction of a D∗ meson in the final state has the optimal signal sensitivity of all
fully reconstructed charm final states (cf. Table 3 and Table 5). Thus, it has been used for studies of
the non-perturbative charm fragmentation function (Section 2.10). Since the momentum of the charm
quark is not measured in the detector, the fragmentation function is not a directly accessible quantity.
It can be approximated either by jets to which a reconstructed D∗ meson is associated (for high-pT
events) or by the overall energy flow in the event hemisphere around the D∗ (for production close to
the kinematic threshold). Parameters of the fragmentation function were extracted from the data by
fitting corresponding predictions to the measured normalised differential cross sections as a function of

zjet =
(E + P)D
(E + P)jet

and

zhem =
(E + P)D

∑

hem(E + P)

for the jet and hemisphere methods, respectively. The tuning of the fragmentation parameters was
done based on Monte Carlo simulations [21, 23] with similar JETSET [24] settings or on NLO QCD
calculations using the same “heavy quark” definition and similar schemes for the cancellation of collinear
and infrared divergences23 in photoproduction and DIS [56, 51].

23Due to the heavy quark masses most of these terms are not really divergent. Nevertheless, events with “similar”
topologies and large but almost cancelling weights are produced in correlated groups.

76



ZEUS

0

1

2

3

0.2 0.4 0.6 0.8 1
z

1
/σ

d
σ

/d
z

ZEUS (120 pb-1)

FMNR×C
PYT
had(Kartvelishvili α=2.67

+0.25
- 0.31)

FMNR×C
PYT
had(Kartvelishvili α=1.2)

FMNR×C
PYT
had(Kartvelishvili α=4.0)

(a)

je
t

/d
z

σ
 d

σ
1

/

0.5

1

1.5

2

2.5

3

3.5

4

je
t

/d
z

σ
 d

σ
1

/

0.5

1

1.5

2

2.5

3

3.5

4
 jet sample

±
D*

H1 Data (parton level) 

 = 3.8 + 0.3      αHVQDIS  
 = 3.8 - 0.3 αHVQDIS  

jetz
0.2 0.4 0.6 0.8 1

R
  

0.5

1

1.5

(b)

h
e

m
/d

z
σ

 d
σ

1
/

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

h
e

m
/d

z
σ

 d
σ

1
/

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5
 jet sample

±
No D*

H1 Data (parton level) 

 = 6.1 + 0.9      αHVQDIS  
 = 6.1 - 0.8 αHVQDIS  

hem
z

0.2 0.4 0.6 0.8 1

R
  

0.5

1

1.5

(c)
h

e
m

/d
z

σ
 d

σ
1

/

0.5

1

1.5

2

2.5

3

3.5

4

h
e

m
/d

z
σ

 d
σ

1
/

0.5

1

1.5

2

2.5

3

3.5

4
 jet sample

±
No D*

H1 Data 
 = 10.3 + 1.9      αRAPGAP  
 = 10.3 - 1.6 αRAPGAP  
 = 4.4αRAPGAP  

hem
z

0.2 0.4 0.6 0.8 1

R
  

0.5

1

1.5

(d)

Figure 50: The normalised D∗ cross sections as a function of z from the (a) ZEUS [221] and (b), (c), (d)
H1 [220] analyses. The hard (b) and the threshold (c-d) regions are shown from the H1 measurement.
The statistical (inner error bars) and the statistical and systematic uncertainties added in quadrature
(outer error bars) are shown separately. The data are compared to the NLO QCD predictions (a-c) from
HVQDIS [51] and FMNR [56] as well as to the RAPGAP MC [23] (d) based on the string fragmentation
model. The parameter of the Kartvelishvili fragmentation function was tuned to the data in each case.

The H1 [220] and ZEUS [221] measurements were done in the DIS and photoproduction regimes,
respectively, utilising HERA I data sets. The H1 experiment investigated both the high pT and the
threshold regions, in order to cover the largest possible phase space, while ZEUS restricted the measure-
ment to the high-pT regime, in order to reach small z values without strongly biasing the distributions,
and in order to minimise perturbative fragmentation factorisation effects within the data set. This led
to H1 selecting jets with E∗T > 3 GeV in the γ

∗p rest-frame and ZEUS cutting on ET > 9 GeV for jets
in the laboratory frame. Figure 50 shows the measured normalised differential cross sections as well as
predictions after tuning of the fragmentation model. The extracted parameters of the fragmentation

77



H1 [220] ZEUS [221]

kinematics threshold high pT high pT
ŝ, GeV2 ∼ 36 ∼ 100 ∼ 350
z method hem hem jet jet

Kartvelishvili [95]

α 6.1+0.9−0.8 3.3
+0.4
−0.4 3.8

+0.3
−0.3 2.7

+0.2
−0.3

(χ2/ndof) (37.6/4) (4.4/4) (4.9/3) n.a.

Peterson [94]

ε 0.007+0.001−0.001 0.068
+0.015
−0.013 0.034

+0.004
−0.004 0.079

+0.013
−0.009

(χ2/ndof) (38.6/4) (18.3/4) (23.3/3) n.a.

Table 7: Parameters of fragmentation function extracted for the NLO QCD predictions by H1 and
ZEUS.

functions for the HVQDIS and FMNR NLO calculations, which should also be applicable to the concep-
tually similar MNR [60] calculations in hadroproduction, are presented in Table 7. A few observations
can be made from the distributions:

• As expected, due to the high quark mass, a charm meson retains a large fraction of the momentum
of a charm quark. Therefore, the charm fragmentation is much harder than that of light hadrons.

• The fragmentation of charm quarks to D∗ mesons near to the kinematic threshold is harder than in
the region away from the threshold (cf. Fig. 50 (b) and (c)). This can be qualitatively understood
as a consequence of the fact that the phase space available for the production of additional
particles is smaller near threshold. As a result, the fragmentation parameters extracted in the
two kinematic regions are significantly different. This leads to the conclusion that the different
kinematic regimes can not be described simultaneously within the framework of the independent
fragmentation function. This is to be expected, since the nonperturbative phase space suppression
is incompletely modeled in this approach.

• NLO QCD calculations in conjunction with an independent fragmentation fail to describe the
data close to the kinematic threshold: χ2/ndof ≈ 38/4 (see Fig. 50 (c) and Table 7). However,
in the same phase space MC simulations can be tuned to provide a reasonable description of the
z distribution in the data: χ2/ndof ≈ 3/4 (see Fig. 50 (d)). This might be due to the proper
treatment of phase space effects in the MC, which are missing in the independent fragmentation
approach used for the NLO predictions.

• The Peterson fragmentation provides a much worse description of the data than the Kartvelishvili
function. This has also been observed elsewhere [222].

• The jet and hemisphere methods in the region where both are applicable, i.e. away from the
threshold, yield similar results for the Kartvelishvili parametrisation, while they remain some-
what different in the Peterson case. Thus again, the Kartvelishvili parameterisation seems to be
preferred.

For some recent measurements [84, 144, 204, 209, 210, 211, 217], these results have been used
explicitly to model the fragmentation for the comparison of theoretical predictions to the charm HERA
data (see Section 8) and for the extrapolation to the full phase space in the context of the extraction
of the charm reduced cross sections (see Section 8 and Section 9.2). This has shown that there are

78



H1 [220] ZEUS [221]

kinematics threshold high pT high pT
ŝ, GeV2 ∼ 36 ∼ 100 ∼ 350
z method hem hem jet jet

Kartvelishvili

α 7.5+1.3−1.2 3.3
+0.4
−0.4 3.1

+0.3
−0.3 n.a.

(χ2/ndof) (37.6/4) (4.4/4) (4.9/3) n.a.

Peterson

ε 0.010+0.003−0.003 0.049
+0.012
−0.010 0.061

+0.011
−0.009 0.062

+0.011
−0.008

(χ2/ndof) (38.6/4) (18.3/4) (23.3/3) n.a.

Table 8: Parameters of fragmentation function extracted by H1 and ZEUS for the fragmentation model
in PYTHIA with other parameter settings set to the default values. Note, that also a set of fragmen-
tation parameters was extracted by H1 [220] for the ALEPH PYTHIA tune [223].

significant theory uncertainties due to fragmentation. A consistent phenomenological reanalysis of the
H1 and ZEUS data is needed in order to resolve the differences observed in different kinematic domains,
which originate from neglecting perturbative evolution and phase space effects, hopefully resulting in
an important reduction of the related theory uncertainties. It is worth mentioning that the complete
HERA II dataset, which has not yet been analysed in this context, is available in principle for this
purpose.

Table 8 shows the equivalent results extracted from LO+PS MCs using the “default” JETSET
settings as used e.g. by the PYTHIA and RAPGAP MCs. In this case a perturbative evolution of the
fragmentation function is partially included through the parton showering, and phase space corrections
are applied. Despite the poor χ2, the Peterson parameters extracted from the intermediate and high
pT jet samples now agree with each other, as well as with the corresponding default parameter 0.05
extracted from e+e− collisions [221]. This confirms the universality of the nonperturbative part of
fragmentation. Kartvelishvili parameters are unfortunately not available for all data sets and can
hence not be compared. Near threshold, even the MC model does not yield the same fragmentation
parameters, and the χ2 is generally bad. This indicates that still not all threshold effects might have
been fully accounted for.

9.1.2 Charm fragmentation fractions and ratios

The fractions of c quarks hadronising into a particular charm hadron, f(c → D, Λc), have been measured
by H1 and ZEUS in the DIS [147, 203, 204] and photoproduction [145, 143] regimes. The measure-
ments were done for D+, D0, D∗+, D+s and Λc based on full reconstruction of the charm-hadron decays.
The fragmentation fractions were extracted from integrated visible cross sections. The typical fidu-
cial phase space of the charm hadrons was defined by pT (D, Λc) > 3 GeV and η(D, Λc) < 1.6. The
fragmentation fractions were extracted with the additional constraint that the sum of the fractions for
all weakly-decaying open-charm hadrons (i.e. the ground states from the point of view of strong and
electromagentic interactions) has to be equal to unity. This was done by a constrained fit in H1 [147]
and by an advanced procedure called equivalent phase space treatment in ZEUS [203]. In addition to
direct production, such experimentally measured fragmentation fractions include also all possible decay
chains of excited charm hadrons.

Figure 51 shows a compilation of all available charm fragmentation fraction measurements. The
HERA data are compared to an average of e+e− measurements [224, 225]. To allow a direct comparison,

79



 

C
h

a
rm

 f
ra

g
m

e
n

ta
tio

n
 f
ra

ct
io

n
s

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

pγ

ZEUS
HERA II

pγ

ZEUS
HERA I

ep DIS

ZEUS
HERA I

ep DIS

H1

-e+e

 )0 D→f (c 
 )+ D→f (c 
)+ D*→f (c 

 )s D→f (c 
 )cΛ →f (c 

Figure 51: Fractions of charm quarks hadronising into a particular charm hadron [143]. Measurements
from HERA are compared to the combined e+e− data. Different hadron species are shown with different
marker types.

all measurements have been corrected [226] to the decay branching fractions from PDG 2010 [227].
The HERA data reach very high precision, benefiting from a partial cancellation of some systematic
uncertainties in the ratio. In particular, the recent ZEUS measurement [143] is based on the full
HERA II data sample and made use of the finalised tracking with lifetime tagging for D0, D+ and D+s .
This allowed to reduce both statistical and systematic uncertainties. The ultimate precision achieved
with ZEUS HERA II data alone is fully competitive with the precision of the e+e− average from several
experiments. All data from DIS, photoproduction and e+e− collisions are in agreement within the
high accuracy of the data. This demonstrates that the charm fragmentation fractions are independent
of the production mechanism, and therefore supports the hypothesis of universality of heavy-quark
fragmentation. The agreement between the fragmentation fractions has been checked quantitatively in
the context of a combination [226].

In addition to the fragmentation fractions, various charm fragmentation ratios were extracted: the
ratio of the neutral to charged D-meson production rates, Ru/d , the fraction of the charged D mesons
produced in a vector state, P dV , and the strangeness-suppression factor, γs . Figure 52 shows a compar-
ison of HERA measurements [143, 145, 147, 203, 204] with results obtained in e+e− collisions (numbers
quoted in [203] using average from [224]) and hadroproduction by CDF [228], ALICE [229, 230] and AT-
LAS [225]. Also shown is a global average of P dV results from e

+e− , photo- and hadroproduction [231]24,
which also includes some e+e− , ep and CDF data shown separately. Note, that the uncertainty of the
average is driven by the π− – A result from the WA92 experiment [232], which is quoted with statistical
uncertainty only, i.e. treating all systematic uncertainties, including the branching-ratio uncertainty, as
correlated between D+ and D∗+. The fragmentation ratios extracted from HERA, e+e− and hadropro-

24The paper as well as other recent measurements [228, 229, 225] report PV values, which correspond to the fraction
of charged and neutral D mesons produced in the vector state. However the measurements rely on isospin symmetry
assumption, which makes PV identical to P

d

V
.

80



0

0.2

0.4

0.6

0.8

1

1.2

1.4

u/dR V
dP sγ

ZEUS HERA I DIS ZEUS HERA I PHP -e+e

H1 HERA I DIS ZEUS HERA II PHP  combinationdVP

ALICE 7 TeV ATLAS 7 TeV CDF

Figure 52: Fragmentation ratios Ru/d , P
d
V , γs measured at HERA and elsewhere [147, 203, 204, 145,

143, 224, 228, 229, 230, 225]. Different measurements are shown with different marker types. The error
bars indicate the statistical and systematic uncertainties added in quadrature. The branching-ratio
uncertainties are not shown due to the high degree of correlation between experiments and can be
found in the original papers. The filled band shows the result of the P dV combination [231].

duction data agree within experimental uncertainties. The ultimate precision achieved with the full
HERA II data set is competitive with the most precise measurements in other experiments. Vari-
ous simple theory expectations can be tested against the data. The Ru/d measurements are slightly
above, but still in agreement within uncertainties, with the isospin invariance expectation of unity. The
P dV measurements are smaller than the naive spin-counting expectation 0.75 and the string fragmentation
prediction 0.66 [233, 234].

Excited charm mesons have also been studied with the ZEUS detector using the HERA I [235]
and HERA II [236] datasets. Some parameters of the orbitally-excited charm states D1(2420)

0,+ with
JP = 1+ and D∗2(2460)

0,+ with JP = 2+ as well as charm-strange state Ds1(2536)
+ were measured. The

masses and widths were found to be in good agreement between the two measurements and with the
PDG average. The helicity parameters h for D1(2420)

0 and Ds1(2536)
+ were found to be in agreement

with e+e− measurements. The measured D1(2420)
0 parameter was found to prefer a mixture of S and

D waves in the decay to D∗+π−, although it is also consistent with a pure D wave. In addition,
fragmentation fractions and ratios of branching ratios were extracted. For some parameters HERA
can provide important or even so far unique information. For example, the fragmentation fractions
for the studied excited mesons are so far very poorly experimentally determined. The D1(2420)

+ and
D1(2420)

+ fragmentation fractions were measured for the first time [236]:

f(c → D+1 ) = 4.6 ± 1.8(stat.)+2.0−0.3(syst.)%,

f(c → D∗+2 ) = 3.2 ± 0.8(stat.)+0.5−0.2(syst.)%.

81



9.2 Measurement of parton density functions

The gluon PDF at low- and medium-x values is mostly constrained by the scaling violations of the inclu-
sive structure function F2. In contrast, heavy-quark production at HERA provides a direct probe of the
gluon momentum distribution in the proton through the γ∗g → cc̄ process. Such direct measurements
are complementary to the indirect approach.

Already the very early charm measurements were used to directly extract the gluon PDF, as was
done by the H1 collaboration in [169]. The gluon densities extracted from the charm data were found
to be in agreement with the result of a QCD analysis of inclusive F2 measurements, although the charm
measurement was limited by statistics.

The recent combined charm DIS data [84] were also used in a QCD analysis [84] together with the
combined inclusive HERA I DIS cross sections [218]. Only the data with Q2 > 3.5 GeV2 were used in
the analysis to assure applicability of pQCD calculations. The analysis was performed at NLO using the
HERAFitter package [237, 218, 238] and closely followed the HERAPDF1.0 prescription [218]. Various
implementations of the NLO GMVFNS approach were used and the role of the value of the charm quark
mass parameter (see Section 2.7), Mc, was studied. For each heavy-flavour scheme a number of PDF
fits was performed to scan χ2 of the PDF fit as a function of Mc. From the scan the optimal value,
Moptc , of the charm-quark mass parameter in a given scheme was determined by the minimum of the
χ2 and the corresponding fit uncertainty25 was evaluated from the ∆χ2 = 1 variation. The procedure
is illustrated in Fig. 53(a) which shows the fit to the inclusive DIS data alone and together with σcc̄red .
The inclusive DIS cross sections alone only weakly constrain Moptc , indicated by the shallowness of the
χ2(Mc) distribution. The charm DIS cross sections provide the required constraint to extract M

opt
c .

Additionally, for each GMVFNS approach the model and parametrisation assumptions in the fits were
varied one-by-one and the corresponding χ2 scan as a function of Mc was repeated. The difference
between Moptc obtained with the default assumptions and the result of each variation was taken as the
corresponding source of uncertainty. The dominant contribution to the uncertainty was found to come
from the variation of the minimum Q2 value for inclusive DIS data used in the fit.

Figure 53(b) shows the χ2 distributions as a function of Moptc obtained from fits to the inclusive
HERA I data and the combined σcc̄red for all variable-flavour-number schemes considered. All schemes
yield similar minimal χ2 values, however at quite different values of Moptc . The resulting values of M

opt
c

are given in Table 9 together with the evaluated uncertainties, the minimal total χ2 values and the
χ2 contribution from the charm data. The ACOT-full scheme provides the best global description of
the inclusive and charm data together, while the RT optimised scheme yields the best description of
the charm data alone. The fits in the S-ACOT-χ scheme result in a very low value of Moptc compared
to other approaches. Since this scheme only includes a leading-order approximation of heavy-flavour
production at the order considered here (see Table 1), effectively no distinction is made between pole
or running mass. All NLO VFNS predictions using corresponding Moptc values for each scheme provide
a similarly good description of the σcc̄red data [84].

Figure 54 shows the PDFs extracted from the fit to the inclusive DIS data alone and together with
the σcc̄red data in the RT optimised VFNS

26. A comparison of the extracted PDF uncertainties yields the
following conclusions about the impact of the σcc̄red data [84]:

• the uncertainty on the gluon PDF was reduced, mostly due to a reduction of the parametrisation
uncertainty due to the additional constraints that the charm data introduce due to the BGF
process;

25This minimisation uncertainty is usually referred to as the “experimental” uncertainty in the HERAPDF context [218,
84]. However, it can absorb some other sources of uncertainties, e.g. variations of PDF parametrisation. Therefore, the
more general term “fit uncertainty” is used here.

26 Similar observations were made with other schemes.

82



 [GeV]cM
1.2 1.4 1.6 1.8

)
c

 (
M

2 χ

550

600

650

700

750

800
HERA-I inclusive

Charm + HERA-I inclusive

H1 and ZEUS

RT standard

 0.06 GeV±=1.50 
opt
cM

(a)

 [GeV] cM
1.2 1.4 1.6 1.8

)
c

 (
M

2 χ

600

650

700

750

RT standard
RT optimised
ACOT-full

χS-ACOT-
ZM-VFNS

 
opt
C  M

H1 and ZEUS

Charm + HERA-I inclusive

(b)

Figure 53: (a) The values of χ2(Mc) for the PDF fit to the combined HERA DIS data [84] in the RT
standard scheme [71]. The open symbols indicate the results of the fit to inclusive DIS data only. The
results of the fit including the combined charm data are shown by filled symbols. (b) The values of
χ2(Mc) for the PDF fit to the combined HERA inclusive DIS and charm measurements [84]. Different
heavy flavour schemes are used in the fit and presented by lines with different styles. The values of
Moptc for each scheme are indicated by the stars.

0

0.2

0.4

0.6

0.8

1

-410 -310 -210 -110 1

 HERAPDF1.0 13p

 HERAPDF1.0 + charm 

 

x

xf

2 = 10 GeV2Q

vxu

vxd

 0.05)×xg (

H1 and ZEUS

(a)

0

0.2

0.4

0.6

0.8

1

-410 -310 -210 -110 1

 HERAPDF1.0 13p

 HERAPDF1.0 + charm 

 

x

xf

2 = 10 GeV2Q

cx

ux

sx

 1.1)× (dx

H1 and ZEUS

(b)

Figure 54: Parton density functions [84] x · f(x, Q2) with f = g, uv, dv, u, d, s, c for (a) valence quarks
and gluon and for (b) sea anti-quarks obtained from the combined QCD analysis of the inclusive DIS
data and σcc̄red (dark shaded bands) in the RT optimised scheme as a function of x at Q

2 = 10 GeV2. Note
that, somewhat confusingly but following common practice, here the variable x refers to xb in Eq. 10,
rather than to Bjorken x. For comparison the results of the QCD analysis of the inclusive DIS data
only are also shown (light shaded bands). The gluon distribution function is scaled by a factor 0.05 and
the xd distribution function is scaled by a factor 1.1 for better visibility. The total PDF uncertainties
include fit, model and parametrisation uncertainties.

83



Scheme Moptc χ
2/ndof χ

2/ndp
[GeV] σ

NC,CC
red +σ

cc̄
red σ

cc̄
red

RT standard 1.50 ± 0.06fit ± 0.06mod⊕param⊕αs 630.7/626 49.0/47
RT optimised 1.38 ± 0.05fit ± 0.03mod⊕param⊕αs 623.8/626 45.8/47
ACOT-full 1.52 ± 0.05fit ± 0.12mod⊕param⊕αs 607.3/626 53.3/47
S-ACOT-χ 1.15 ± 0.04fit ± 0.01mod⊕param⊕αs 613.3/626 50.3/47
ZMVFNS 1.60 ± 0.05fit ± 0.03mod⊕param⊕αs 631.7/626 55.3/47

Table 9: The values of the charm mass parameter Moptc as determined from the Mc scans in different
heavy flavour schemes [84]. The uncertainties of the minimisation procedure are denoted as “fit”,
the model, parametrisation and αs uncertainties were added in quadrature and are represented by
“mod ⊕ param ⊕ αs”. The corresponding global and partial χ2 are presented per degrees of freedom,
ndof, and per number of data points, ndp, respectively.

• the uncertainty on the charm-quark PDF is considerably reduced due to the constrained range
of Mc. The Mc variation was set to 1.35 < Mc < 1.65 GeV for the fit to the inclusive data only
and was defined by the evaluated total uncertainties as given in Table 9 for the fit including the
charm data;

• the uncertainty on the up-quark sea PDF was correspondingly reduced, because the inclusive data
constrain the sum of up- and charm-quark sea;

• the uncertainty on the down-quark sea was also reduced because it was constrained to be equal
to the up-quark sea at low x;

• the uncertainties on the valence-quark and strange-quark sea PDFs were almost unaffected;

• the central PDFs were not altered significantly and were found to be within the uncertainties of
the PDFs based on inclusive data only. This reflects the good description of the charm data by
the default PDFs (section 8.4).

By now, the combined charm reduced cross sections [84] have been used in QCD analyses by various
PDF-fitting groups [239, 240, 33, 241, 242, 243, 244]. They are an important ingredient to constrain
the proton flavour composition (see also next section) and to stabilise its gluon content. The latter is
especially important for Higgs production at the LHC, for which the dominant process is gluon-gluon
fusion via an intermediate top-quark loop. Measuring this process precisely, in combination with a
precise knowledge of the gluon content of the proton, allows the extraction of a precise measurement of
the Higgs-top Yukawa coupling.

Instead focusing on the low-x range, the HERA charm and beauty data have recently been used
in conjunction with charm and beauty data from LHCb to constrain the gluon distribution down to
x ∼ 5 × 10−6 [245]. This region is particularly relevant for the prediction of cross sections for processes
occurring in cosmic ray interactions.

9.3 Proton flavour composition and W/Z/H production at LHC

In the previous section it was outlined how the inclusion of charm data into GMVNFS PDF fits, and
in particular the constraint on the charm quark mass parameter derived from these data, imposes
constraints on the gluon content (relevant e.g. for Higgs production) and on the flavour composition of

84



the quarks in the proton. This, in turn, affects theoretical predictions for processes which are sensitive
to this flavour composition, such as the production for W and Z bosons at LHC.

Figure 55 shows NLO predictions for W and Z production at LHC for PDFs extracted in different
heavy flavour schemes as discussed in Section 9.2, as a function of the charm quark mass parameter
Mc used in the PDF fit. For fixed Mc, these predictions differ by about 7%. The dependence on Mc is
opposite to what one would naively expect (see also Fig. 54). A higher charm mass leads to less charm
in the proton (fewer gluons split) but to a higher gluon density. This in turn increases the amount of u
and d sea quarks in the proton, even more so since the total sea is constrained by the inclusive proton
structure functions. The larger number of u and d quarks overcompensates the smaller number of c
quarks and leads to an increase of the W and Z cross sections as shown in Fig. 55. The fit in Fig. 54
actually led to a smaller charm mass than the default, therefore the effect on the PDF was opposite.

 [GeV]cM
1.2 1.4 1.6 1.8

 [
n

b
]

+
W

σ

54

56

58

60

62

64

opt
C  M

 RT standard
 RT optimised

 ACOT-full
χ S-ACOT-

 ZM-VFNS 

 = 7 TeVs

Charm + HERA-I inclusive

H1 and ZEUS

(a)
 [GeV]cM

1.2 1.4 1.6 1.8

 [
n

b
]

-
W

σ

38

40

42

44

opt
C  M

 RT standard
 RT optimised

 ACOT-full
χ S-ACOT-

 ZM-VFNS 

 = 7 TeVs

Charm + HERA-I inclusive

H1 and ZEUS

(b)

 [GeV]cM
1.2 1.4 1.6 1.8

 [
n

b
]

Z
σ

27

28

29

30

31

32
opt
C  M

 RT standard
 RT optimised

 ACOT-full
χ S-ACOT-

 ZM-VFNS 

 = 7 TeVs

Charm + HERA-I inclusive

H1 and ZEUS

(c)

Figure 55: NLO predictions for (a) W +, (b) W − and (c) Z production cross sections at the
LHC for

√
s = 7 TeV as a function of Mc used in the corresponding PDF fit [84]. The different

lines represent predictions for different implementations of the VFNS. The predictions obtained
with PDFs evaluated with the Moptc values for each scheme are indicated by the stars. The
horizontal dashed lines show the resulting spread of the predictions when choosing Mc = M

opt
c .

The stars in Fig. 55 indicate the cross section predictions for the optimal mass for each heavy flavour
scheme, as extracted from the charm data in Fig. 53(b). All predictions then coincide to within 2%,

85



independent of the heavy flavour scheme used. This demonstrates that using the optimal mass for each
scheme which best fits the HERA charm data stabilises the flavour composition in the proton, and leads
to a reduction of this contribution to the cross section uncertainty by about a factor 3. To minimise
the uncertaities arising from the charm and beauty masses, for GMVFNS schemes it is thus strongly
recommended to use the optimal mass parameters as derived from the heavy-flavour structure-function
data rather than a mass obtained from external considerations.

A similar analysis for beauty, which remains to be done, will in addition yield experimental con-
straints on the b PDF in the framework of 5-flavour PDFs for LHC, which are so far constrained by
theory only. This in turn will be relevant e.g. for a future measurement of the Higgs-b Yukawa coupling
from associated Higgs-bb̄ production.

9.4 Measurements of the charm-quark mass and its running

The sensitivity of the HERA reduced charm cross sections to the charm-quark mass, already partially
studied in Section 9.2, can be used to measure the charm quark mass appearing in perturbative QCD,
whose value depends on the renormalisation scheme within which it is being evaluated. The two mass
definitions which are most commonly used are the pole mass and the MS running mass (Section 2.4).
Since the MS mass is perturbatively better defined, recent charm mass measurements concentrate on
this renormalization scheme. The FFNS scheme (Section 2.6) is most suited for this evaluation, since
it fully accounts for mass effects without any additional free parameters. FFNS calculations of the
reduced cross section in this scheme exist at NLO and partial NNLO [62, 32]. All results quoted in the
following are obtained from these calculations unless otherwise quoted.

The first determination of the MS charm-quark mass [246], from a subset of D∗ charm data from
the H1 collaboration, in which also the details of the theoretical framework are given, obtained

mc(mc) = 1.27 ± 0.04(fit) +0.06−0.01(scale) GeV (20)

at NLO, and
mc(mc) = 1.36 ± 0.04(fit) +0.04−0.00(scale) ± 0.1(theory) GeV (21)

at partial NNLO, where the last term reflects a very conservative estimate of the evaluation of the
uncertainties of the NNLO approximation.

Figure 56 shows the comparison of predictions of the ABKM group [46], using the above mass values
as central values, to the combined HERA charm data [84] discussed in Section 8.4, which have smaller
uncertainties than the data used for the initial measurement. Very good agreement is observed for both
NLO and partial NNLO. The H1 and ZEUS collaborations have used a fit to these data (Fig. 57(left)),
using the kinematic region Q2 > 3.5 GeV2, to obtain the NLO measurement [84]

mc(mc) = 1.26 ± 0.05(exp) ± 0.03(mod) ± 0.02(param) ± 0.02(αs) GeV. (22)

This result has a slightly more elaborate evaluation of the uncertainties related to the data extrapolation
as well as other model and parametrisation uncertainties, while it does not include uncertainties on the
normalisation of the cross section predictions due to QCD scale variations.

The same data where then used by the ABM group and collaborators [239] to reobtain similar
evaluations,

mc(mc) = 1.15 ± 0.04(exp) +0.04−0.00(scale) GeV (23)
at NLO, and

mc(mc) = 1.24 ± 0.03(exp) +0.03−0.02(scale) +0.00−0.07(theory) GeV (24)
at partial NNLO. The smaller central NLO value and the smaller uncertainty are mainly due to the
fact that the charm data from the lowest Q2 bin were included. A correlated measurement of mc(mc)

86



0

0.2

0

0.5

0

0.5

0

0.5

σ
re

d

 c
c_

H1 and ZEUS
Q2=2.5 GeV2 Q2=5 GeV2 Q2=7 GeV2

Q2=12 GeV2 Q2=18 GeV2 Q2=32 GeV2

Q2=60 GeV2 Q2=120 GeV2 Q2=200 GeV2

Q2=350 GeV2

10-4 10-3 10-2

Q2=650 GeV2

10-4 10-3 10-2

Q2=2000 GeV2

HERA

ABM09NNLO MS


ABM09NLO MS


10-4 10-3 10-2

x

Figure 56: Combined reduced cross sections [84] filled circles as a function of x for fixed values of
Q2. The error bars represent the total uncertainty including uncorrelated, correlated and procedural
uncertainties added in quadrature. The data are compared to predictions of the ABM group at NLO
(hashed band) and NNLO (shaded band) in FFNS using the MS definition for the charm quark mass.

) [GeV]
c

(mcm
1 1.2 1.4 1.6

2 χ

620

640

660

680

700

FF (ABM)

H1 and ZEUS

Charm + HERA-I inclusive

 0.05 GeV±)=1.26 
c

(mcm

Figure 57: (left) The values of χ2 for the PDF fit [84] to the combined HERA DIS data including charm
measurements as a function of the running charm quark mass mc(mc). The FFNS ABM scheme is used,
where the charm quark mass is defined in the MS scheme. (right) The values of mc(mc) obtained in
the NLO and NNLO variants of the ABM analysis [247] with the value of αs(MZ) fixed. The position
of the star displays the result with the value of αs(MZ) fitted.

87



and the strong coupling constant [247] was also obtained (Fig. 57(right)). In particular for the NLO
case, the correlation between mc(mc) and αs is non-negligible.

They were also used by the CTEQ group [33] to derive the MS mass in the context of the S-ACOT-χ
VFNS, using charm matrix elements to one-loop order in the massive part of the calculation. The result

mc(mc) = 1.19
+0.08
−0.15 GeV (25)

exhibits a larger uncertainty than the previous extractions due to the additional uncertainty from the
variation of the (single) free parameter of this VFNS scheme, and due to conversions between the pole
and running masses in the extraction process.

All these results from a predominantly space-like perturbative process are consistent with each other
and with the world average [146]

mc(mc) = 1.275 ± 0.025 GeV (26)

obtained from lattice QCD and time-like processes. This is a highly nontrivial triumph of QCD. Some
of the above measurements are now included in the latest world average [30], and further improvements
on both the experimental and theoretical sides have the potential to further improve the corresponding
precision.

In a recent preliminary result [248] the same data have again been used to determine the actual
scale dependence (‘running’) of the charm-quark mass in the MS scheme, according to Eq. (13). For
this purpose, the charm data were subdivided into 6 different Q2 ranges, for which the charm mass was
extracted separately at the scale < Q2 > +4m2c, where < Q

2 > is the average of each range. The result
is shown in Fig. 58. This is the first explicit measurement of the scale dependence of the charm quark
mass.

 [GeV]µ
1 10

) 
[G

e
V

]
µ(

c
m

0.4

0.6

0.8

1

1.2

1.4

1.6

H1 and ZEUS preliminary

HERA (prel.)

PDG with evolved uncertainty

Figure 58: Measured charm mass mc(µ) in the MS running mass scheme as a function of the scale µ
as defined in the text (black points). The red point at scale mc is the PDG world average [146] and the
band is its expected running [31].

9.5 Measurement of the beauty-quark mass and its running

Using the same approach as outlined above for charm, the ZEUS collaboration has used a fit (Fig. 59)
to the beauty reduced-cross-section data [148] to extract the value of the beauty-quark MS running
mass at NLO,

mb(mb) = 4.07 ± 0.14(fit) +0.01−0.07(mod) +0.02−0.00(param) +0.08−0.05(theo) GeV, (27)

88



0

0.005

0.01

0.015

0.02

0

0.005

0.01

0.015

0.02

0

0.005

0.01

0.015

0.02

0

0.02

0.04

0

0.02

0.04

0

0.02

0.04

0

0.01

0.02

0.03

x 
-410

-3
10 -210

x 
-410 -310 -210

x 
-410 -310 -210

b
b r

σ

0

0.005

0.01

0.015

0.02

0

0.02

0.04

0

0.01

0.02

0.03
-1ZEUS 354 pb

=4.07 GeV (best fit)
b

QCD fit, m

=3.93 GeV
b

QCD fit, m

=4.21 GeV
b

QCD fit, m

ZEUS
2 = 6.5 GeV2Q 2 = 12 GeV2Q 2 = 25 GeV2Q

2 = 30 GeV2Q 2 = 80 GeV2Q 2 = 160 GeV2Q

2 = 600 GeV2Q

Figure 59: Reduced beauty cross section (filled symbols) as a function of x for seven different values of
Q2 [148]. Also shown are the results of a QCD fit for different values of the MS running mass mb(mb).

where the theoretical uncertainty is dominated by the scale variation uncertainty. This is the first such
extraction from HERA data, and agrees well with the world average [146]

mb(mb) = 4.18 ± 0.03 GeV. (28)

Figure 60 shows this result, translated to the scale 4m2b, compared to the PDG value and its expected
running and to values extracted from LEP data at the scale MZ. The expected running of the MS
beauty-quark mass is confirmed. This is a nontrivial test of the basics of QCD.

 [GeV]µ
10 210

) 
[G

e
V

]
µ(

b
m

2

2.5

3

3.5

4

4.5

5
ZEUS

PDG with evolved uncertainty

ZEUS
DELPHI 3-jets
DELPHI 4-jets NLO

ALEPH
OPAL
SLD

Figure 60: Measured beauty mass mb(µ) in the MS running mass scheme as a function of the scale µ
from HERA [148] and LEP [249] data. The red point at scale mb is the PDG world average [146] and
the band is its expected running [31].

89



9.6 Summary

Heavy flavour physics at HERA yields many results which are of interest for particle physics in general.
The usage of HERA as a “charm factory” generates world-class information on charm fragmentation
functions and fragmentation fractions and allows tests of the fragmentation universality. The constraints
on PDFs from charm data, and to a lesser extent also from beauty data, help to reduce uncertainties
for important cross sections at LHC, such as heavy flavour, W/Z and Higgs production. Constraints
on the latter are important for the measurement of the Higgs Yukawa couplings. More directly, the
charm and beauty DIS data have been used to extract well defined measurements of the charm- and
beauty-quark masses, which enter the world average. The running of the charm-quark mass has been
measured for the first time ever. By comparing with LEP data, the running of the beauty quark mass
has also been confirmed. In general, the good agreement of QCD predictions with the data support the
applicability of the HERA results to all particle physics applications for which they might be directly
or indirectly relevant.

90



10 Summary and outlook

Charm and beauty production at HERA are a great laboratory to test the theory of heavy flavour pro-
duction in the framework of perturbative QCD and to measure some of its parameters. The occurrence
of different possibilities to treat the heavy quark masses in the PDF, matrix element and fragmenta-
tion parts of the calculation introduces a significant level of complexity into the corresponding QCD
calculations, in addition to the usual scheme and scale choices. Confronting such different choices with
data can be helpful to understand the effects of different ways to truncate the perturbative series and
to evaluate their impact on the measurement of fundamental parameters, both at HERA and at other
colliders.

HERA was the first and so far only high energy ep collider. The heavy flavour results discussed in
this review were obtained with the H1 and ZEUS detectors which were well suited for the detection of
heavy flavoured particles. Adding the luminosities from the two collider experiments, a total luminosity
of about 1 fb−1 was collected.

The availability of many different charm and beauty tagging methods allows results to be obtained
through several different final states with different systematics. In addition to the statistical benefit from
combining different samples, such combinations also profit from cross calibrations of the systematics
from different methods and experiments.

Due to the high top mass, the only top final state which might have been detectable at HERA is
single top production with non-Standard-Model couplings. No signal is seen, and the coupling limits
derived are competitive.

The charm (beauty) quark masses provide semi-hard (hard) QCD scales which allow the succesful
application of perturbative calculations over the complete phase space. However, these masses also
compete with other, often even harder perturbative scales. Total cross sections for charm photoproduc-
tion and the total cross section for beauty production (including photoproduction and deeply inelas-
tic scattering) are reasonably described by perturbative calculations at next-to-leading order (NLO).
Single-differential cross sections already provide a good handle to test the applicability of different QCD
approximations, although the theoretical uncertainties are mostly much larger than the experimental
ones. The theory predictions agree with the data up to the highest accessible transverse momenta
or photon vitualities, showing no indications that final state resummation corrections are needed for
massive calculations in the HERA kinematical domain. Double-differential cross sections, in particu-
lar those including jets, reveal a partial failure of the massive scheme NLO predictions for kinematic
observables which would need final states with four or more partons in the calculation. Although
statistics and therefore precision is higher for photoproduction, qualitatively very similar conclusions
are obtained for photoproduction and deeply inelastic scattering (DIS). The NLO calculations in the
massless scheme, where available, do mostly not provide a better description for the observables, and
clearly fail for some DIS observables. The LO+PS MCs PYTHIA and HERWIG, which are often used
for acceptance corrections, are able to describe all topologies reasonably, often even very well. The
CASCADE kt-factorisation MC performs somewhat less well on average.

In DIS, the large photon virtuality Q2 provides an additional hard scale in the QCD calculations of
heavy flavour production and allows probing the parton dynamics inside the proton more directly than in
photoproduction. The dominant contribution to the charm and beauty cross sections arises from photon-
gluon fusion. For Q2 ≫ 4m2Q, where the photon virtuality is the dominant hard scale, the cross-section
behaviour is similar to the one of the inclusive cross section for deeply inelastic scattering. At high Q2

and low x, the naively expected charm and beauty contributions of 4/11 and 1/11 are asymptotically
approached. NLO QCD predictions in the massive scheme (FFNS) give a good description of heavy
flavour production at HERA in DIS over the complete accessible kinematic range. NLO predictions
in variable-flavour-number schemes (GMVFNS) are only available for inclusive quantities, and perform
about equally well.

91



In particular for charm, the uncertainties from QCD corrections beyond NLO and from the modelling
of fragmentation are considerably larger than the experimental uncertainties of the measured cross
sections. Improved QCD calculations would therefore be highly welcome.

Heavy flavour physics at HERA yields many results which are of interest for particle physics in
general. The usage of HERA as a “charm factory” generates world-class information on charm frag-
mentation functions and fragmentation fractions and allows tests of the fragmentation universality. The
constraints on proton parton distribution functions (PDFs) from charm and beauty data help to reduce
uncertainties for important cross sections at the LHC, such as heavy flavour, W/Z and Higgs produc-
tion. Constraints on the latter are important for the measurement of the Higgs Yukawa couplings.
More directly, the charm and beauty DIS data have been used to extract well defined measurements of
the charm- and beauty-quark masses, which enter the world average. The running of the charm-quark
mass has been measured for the first time ever, and the running of the beauty quark mass has been
confirmed.

In general, the good agreement of QCD predictions with the HERA data support the applicability
of the QCD results derived from these data to all particle physics applications for which they might
be directly or indirectly relevant. Some of the most important HERA heavy flavour results have been
obtained during the last 2-3 years. Even 8 years after the end of data taking the potential of the HERA
heavy flavour data has still not been fully used in all cases, so there is room for significant further
improvements, in particular also on the theory side, hoping e.g. for differential NNLO calculations in
ep collisions, similar to those which have recently started to appear for the pp case.

Acknowledgements

This review is a partial summary of the work of perhaps a thousand technicians, engineers and physicists
for more than two decades. It is our great pleasure to thank the many collegues who have contributed
to the results presented here. We thank O. Kuprash and L. Schalow for technical contributions to this
review.

92



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103


	1 Introduction
	2 Theory of heavy-flavour production at HERA
	2.1 HERA kinematic variables and phase space
	2.2 Perturbative QCD calculations
	2.3 Heavy-quark production at HERA in ``leading order''
	2.4 Quark-mass definition
	2.5 The zero-mass variable-flavour-number scheme
	2.6 The massive fixed-flavour-number scheme
	2.7 The general-mass variable-flavour-number scheme
	2.8 Proton structure functions in DIS
	2.9 QED corrections
	2.10 Fragmentation
	2.11 Choice of renormalisation scale
	2.12 Summary

	3 The HERA collider and experiments
	3.1 HERA
	3.2 H1 and ZEUS Detectors
	3.3 Event reconstruction
	3.4 Summary

	4 Charm and Beauty detection at HERA
	4.1 Charm tagging using full reconstruction of charm hadrons.
	4.2 Heavy-flavour tagging with lepton + pTrel
	4.3 Charm and Beauty with inclusive lifetime tagging
	4.4 Charm and Beauty with double tagging
	4.5 Summary

	5 Search for single top-quark production
	6 Charm photoproduction
	6.1 D* inclusive measurements
	6.1.1 Charm total cross sections
	6.1.2 D* single-differential cross sections
	6.1.3 D* double-differential cross sections

	6.2 Inclusive measurements using other tagging methods
	6.3 Studies with a D* and one other hard parton
	6.4 Parton-parton-correlation studies in charm-tagged events
	6.4.1 xobsstudies
	6.4.2 Azimuthal correlations 
	6.4.3 Study of hard-scattering angle cos*

	6.5 Summary

	7 Beauty photoproduction
	7.1 Total cross section for beauty production
	7.2 Single-differential cross sections
	7.3 Measurements of b"7016b and jet-jet correlations
	7.4 Summary

	8 Charm and beauty production in DIS
	8.1 Production mechanism
	8.2 Single-differential cross sections
	8.3 Double-differential cross sections
	8.4 Proton structure functions and reduced cross sections
	8.5 Summary

	9 Measurement of QCD parameters, proton structure, and impact on LHC and other experiments
	9.1 Measurement of charm fragmentation functions and fragmentation fractions
	9.1.1 Charm fragmentation function
	9.1.2 Charm fragmentation fractions and ratios

	9.2 Measurement of parton density functions
	9.3 Proton flavour composition and W/Z/H production at LHC
	9.4 Measurements of the charm-quark mass and its running
	9.5 Measurement of the beauty-quark mass and its running
	9.6 Summary

	10 Summary and outlook