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FZ-IKP(TH)-2000-30

Diffractive beauty photoproduction as a short distance
probe of QCD pomeron

V. R. Zoller

Institute for Theoretical and Experimental Physics, Moscow 117218, Russia
E-mail: zoller@heron.itep.ru

Abstract

High-energy open beauty photoproduction probes the vacuum exchange at distances ∼ 1/mb
and detects significant corrections to the BFKL asymptotics coming from the subleading
vacuum poles. We show that the interplay of leading and subleading vacuum exchanges gives
rise to the cross section σbb̄(W ) growing much faster than it is prescribed by the exchange
of the leading pomeron trajectory with intercept αIP(0) − 1 = ∆IP = 0.4. Our calculations
within the color dipole BFKL model are in agreement with the recent determination of
σbb̄(W ) by the H1 collaboration. The comparative analysis of diffractive photoproduction
of beauty, charm and light quarks exhibits the hierarchy of pre-asymptotic pomeron inter-
cepts which follows the hierarchy of corresponding hardness scales. We comment on the
phenomenon of decoupling of soft and subleading BFKL singularities at the scale of elastic
Υ(1S) -photoproduction which results in precocious color dipole BFKL asymptotics of the
process γp → Υp.

In this communication we address the issue of open beauty photo- and electroproduction

γ∗p → bb̄X (1)

at large values of Regge parameter

x−1 =
W 2 + Q2

4m2b + Q
2
≫ 1 , (2)

where W is the c.m.s. collision energy, Q2 is the photon virtuality and the mass of beauty
quark mb sets the natural scale of the process (1).

In color dipole (CD) approach the excitation of heavy flavors at small-x is described in
terms of interaction of small size quark-antiquark bb̄ color dipoles in the photon. This makes
the reaction (1) a sensitive probe of short distance properties of vacuum exchange in QCD.
The interaction of color dipole r with the target proton is described by the beam, target and
flavor independent color dipole-proton cross section σ(x, r). The contribution of excitation of
open beauty to photo-absorption cross section is given by color dipole factorization formula
[1, 2, 3]

σbb̄(x, Q2) =
∫

1

0

dz
∫

d2r
[

|Ψbb̄L (z, r)|
2 + |Ψbb̄T (z, r)|

2
]

σ(x, r) , (3)

1

http://arXiv.org/abs/hep-ph/0104093v1


where |Ψ bb̄L,T (z, r)|
2 is a probability to find in the photon the bb̄ color dipole with the beauty

quark carrying fraction z of the photon’s light-cone momentum. The well known result of
[2] for the transverse (T) and longitudinal (L) photons is

|Ψbb̄T (z, r)|
2 =

αem
6π2

{[

z2 + (1 − z)2
]

ε2K1(εr)
2 + m2b K0(εr)

2
}

, (4)

|Ψbb̄L (z, r)|
2 =

2αem
3π2

Q2z2(1 − z)2K2
0
(εr) , (5)

where K0,1(y) are the modified Bessel functions, ε
2 = z(1 − z)Q2 + m2b and mb = 4.7 GeV

is the b-quark mass. Here we focus on the open beauty photoproduction cross section. At
Q2 ≪ m2b the eq.(3) takes the form

σbb̄(x) ≃
αem
6π

∫

dr2m2b

[

2

3
K1(mbr)

2 + K0(mbr)
2

]

σ(x, r) (6)

Because for small dipoles σ(x, r) ∝ r2, the dipole size integration in (6) is well convergent
at small r and the integrand (6) has a peak at r ≃ 1/mb. At very high Q

2

∼> 4m
2
b the peak

develops a plateau for dipole sizes in the interval (m2b + Q
2/4)−1 ∼< r

2

∼< m
−2
b . So, for

moderate photon virtualities Q2 ≪ 4m2b excitation of open beauty probes (scans) the dipole
cross section at a special dipole size rS, rS ∼ 1/mb.

The CD cross section σ(x, r) at x ≪ 1 satisfies the CD BFKL equation

∂σ(x, r)

∂ log(1/x)
= K ⊗ σ(x, r) , (7)

with the kernel K [4]. The basis of the CD BFKL-Regge expansion for σ(x, r) [5, 6]

σ(x, r) =
∑

n

σn(r)
(

x0
x

)∆n

. (8)

form the solutions of the eigen-value problem

K ⊗ σn = ∆nσn(r) (9)

with Regge behavior, σn(x, r) = σn(r) (x0/x)
∆n .

The properties of the CD BFKL equation and the choice of physics motivated boundary
condition at x0 = 0.03 were discussed in detail elsewhere [7, 8, 9, 10, 11], here we only
recapitulate features relevant to the considered problem.

The leading eigen-function σ0(r) for ground state i.e., for the rightmost hard BFKL pole,
is node free. The subleading eigen-function for excited state σn(r) has n nodes. We find
σn(r) numerically [9, 10], for the semi-classical analysis see Lipatov [12]. The intercepts
follow to a good approximation the law ∆n = ∆0/(n + 1). Within our specific infrared
regularization we find ∆0 = 0.402, ∆1 = 0.220 and ∆2 = 0.148 [9]. The node of σ1(r)
is located at r = r1 ≃ 0.056 fm (see Fig.1), for larger n the rightmost node moves to a
somewhat larger r = r1 ∼ 0.1 fm. As we discussed elsewhere [9], for still higher solutions,
n ≥ 3, all intercepts are very small anyway, ∆n ≪ ∆0, For this reason, for the purposes
of practical phenomenology we can truncate expansion (11) at n = 3 lumping in the term
n = 3 contributions of still higher singularities with n ≥ 3 (see Fig.1). The term n = 3 is
endowed with the effective intercept ∆3 = 0.06 [13]. The truncated expansion reproduces

2



the numerical solution σ(x, r) of CD BFKL equation (7) in the wide range of dipole sizes
10−3 ∼< r ∼< 10 fm with accuracy ≃ 10% even at moderately small x.

Each CD BFKL eigen-cross section σn via equation

f bn(Q
2) =

Q2

4π2αem

∫

1

0

dz
∫

d2r
[

|Ψbb̄L (z, r)|
2 + |Ψbb̄T (z, r)|

2
]

σn(r) . (10)

defines the corresponding eigen-structure function (SF) f bn(Q
2) and we arrive at the CD

BFKL-Regge expansion for the beauty SF of the proton (n = 0, 1, 2, 3, soft)

F b2 (x, Q
2) =

Q2

4π2αem
σbb̄(x, Q2) =

∑

n

f bn(Q
2)

(

x0
x

)∆n

. (11)

Analytical parameterization for eigen-SF f bn(Q
2) is presented in Appendix. Notice that

numerically r1 > rS and the rightmost nodes of subleading eigen-cross sections σn(r) are
located to the right of the peak of the integrand (6) (see Fig.1). Consequently, in the
calculation of open beauty eigen-SFs f bn(Q

2) one scans the eigen-cross section in between of
the first and the second node. Hence, subleading f bn(Q

2) which are negative valued in a wide
range of Q2. This point is illustrated in Fig. 2 in which the subleading BFKL-to-rightmost
BFKL and soft-pomeron-to-rightmost BFKL ratio of eigen-SFs rn(Q

2) = f bn(Q
2)/f b

0
(Q2) is

shown. Because a probability to find large color dipoles in the photon decreases rapidly
with the quark mass, the contribution from energy-independent soft-pomeron exchange to
open beauty excitation is very small down to Q2 = 0.

We comment first on the results on σbb̄(W ). Because the CD BFKL-Regge expansion for
color dipole-dipole cross section has already been fixed from the related and highly successful
phenomenology of light flavor and charm contributions to the proton SF [6, 9, 10, 11] the
CD BFKL predictions for the beauty SF of the proton are parameter free.

In Fig.3 our evaluation of the W -dependence of open charm (upper solid curve) and
open beauty photoproduction cross sections (lower solid curve) is confronted to the HERA
data [14, 15]. The H1 analysis of the data on σ(γp → bb̄X) ≡ σbb̄ for the collision energy in
the range 94 < W < 266 GeV with 〈W 〉 ≃ 180 GeV gives [15]

σbb̄ = 111 ± 10+16
−11

± 17 nb.

The CD BFKL-Regge approach results in (Fig.3)

σbb̄ ≃ 81 nb

at W ≃ 180 GeV. For an alternative interpretation of beauty photoproduction see [16].
We differ from [16] in estimate of the effect of log(1/x)-evolution. The latter seems to be
strongly underestimated in [16].

As we have emphasized above, the characteristic feature of the QCD pomeron dynamics
at distances ∼ m−1b is large negative valued contribution to σ

bb̄ coming from subleading
BFKL singularities. Consequences of this observation for the exponent of the energy de-
pendence of the cross section

σbb̄(W ) ∝ W 2∆eff (12)

are quite interesting. In terms of the ratio rn(Q
2) (Fig.2) taken at Q2 ≪ 4m2b and denoted

by rn(0) the exponent ∆eff reads (n=1,2,3,soft)

∆eff = ∆0
1 + ∆−10

∑

n ∆nrn(0)(x0/x)
∆n−∆0

1 +
∑

n rn(0)(x0/x)
∆n−∆0

(13)

3



Because all coefficients rn(0) in eq.(13) are negative, except rsoft(0) > 0, at HERA energies
the effective intercept overshoots the asymptotic value ∆IP ≡ ∆0 = 0.402 (see Fig.4). At
still higher collision energies both the soft and subleading hard BFKL exchanges become
rapidly Regge suppressed. This results in decreasing ∆eff down to ∆IP.

For comparison, in photoproduction of open charm which scans the color dipole cross
section at distances ∼ 1/mc, in the vicinity of the rightmost node (see Fig.1), there is a strong
cancellation between soft and subleading contributions to σcc̄(W ) [11, 13]. Consequently, for
this dynamical reason in open charm photoproduction we have an effective one-pole picture
and the effective pomeron intercept ∆eff ≃ ∆IP.

In photoproduction of light flavors the CD cross section σn(r) is close to the saturation
regime σn(r) ∝ const and all subleading and soft terms of the CD BFKL-Regge expansion
are positive valued and numerically important (see [6] for more details). This is the dynam-
ical reason for smallness of a pre-asymptotic pomeron intercept in photoproduction of light
flavors (see Fig.4). Notice that it must not be taken at face value for W ∼ 1 TeV because
of likely strong absorption corrections. However, the hierarchy of pre-asymptotic intercepts
which brings to light the internal dynamics of leading-subleading cancellations at different
hardness scales should withstand unitarity effects.

In Fig.5 we presented our predictions for the beauty structure functions in DIS on protons
as a function of the Bjorken variable xBj = Q

2/(W 2+Q2). The solid curve corresponds to the
complete expansion (11) while the long-dashed curve is the leading hard pole approximation,
F b

2
(x, Q2) ≃ f b

0
(Q2) (x0/x)

∆0. In agreement with the nodal structure of subleading eigen-
SFs the latter over-predicts F b2 significantly because the negative valued contribution from
subleading hard BFKL exchanges overtakes the soft-pomeron exchange, see Fig. 2, and
the background from subleading hard BFKL exchanges is substantial for all Q2. We do
not stretch the theoretical curves to x > x0 = 0.03 beyond the applicability region of CD
BFKL-Regge expansion (11).

Recently the cross section of elastic Υ(1S) meson photoproduction has been measured at
HERA [17]. Quarks in Υ meson are nonrelativistic and |γ〉 ∝ mbK0(mbr). The forward γ →
Υ transition matrix element 〈Υ|σn(r)|γ〉 is controlled by the product σ0(r)K0(mbr) [18] and
the amplitude of elastic of Υ(1S) photoproduction is dominated by the contribution from the
dipole sizes r ∼ rΥ = A/mΥ with A = 5. The crucial observation is that at distances r ∼ rΥ
cancellation between soft and subleading contributions to the elastic photoproduction cross
section results in the exponent ∆ in

dσ(γp → Υp)

dt
|t=0 ∝ W

4∆ (14)

which is very close to ∆IP, ∆ = 0.38 [19]. This observation appears to be in agreement with
the cross section rise observed by ZEUS&H1 [17].

Appendix. For the practical applications it is convenient to have analytical parame-
terization for eigen-SFs f bn(Q

2), which for the rightmost hard BFKL pole n = 0 is of the
form

f b0 (Q
2) =

aR2Q2

1 + R2Q2

[

1 + c log(1 + r2Q2)
]γ

, γ =
4

3∆0
(15)

where a = 3.877·10−2, R2 = 1.036·10−2 GeV−2, c = 0.1482, r2 = 1.004·10−2 GeV−2. For the
subleading pole with n = 1 the eigen-SF f b1 (Q

2) is of the form (15) with a = −4.684 · 10−3,
R2 = 1.352 · 10−2 GeV−2, c = 0.2002, r2 = 1.400 · 10−2 GeV−2, γ = 4δ/3∆0 and δ = 1.621.

4



For still higher n we have

f b2 (Q
2) = af0(Q

2) (1 − z/z1) (1 + z/z2) / (1 + z/z3) , (16)

where z = log(1 + r2Q2)γ, γ = 1.935, r2 = 5.727 · 10−2 GeV−2, a = −0.153, z1 = 47.46,
z2 = 11.56, z3 = 4944.7 and

f b
3
(Q2) = af0(Q

2) (1 − z/z1) (1 − z/z2) , (17)

where z = [1 + c log(1 + r2Q2)]
γ
−1, γ = 4δ/3∆0, δ = 1.362, c = 4.92·10

−2, r2 = 5.002·10−3

GeV−2, a = −0.3386, z1 = 0.6844 and z2 = 3.03. The soft component of the beauty SF
with ∆soft = 0 was derived from eq.(3) with σsoft(r) taken from [19].

Acknowledgments: The author is grateful to J.Speth for hospitality at IKP(Theory)
FZ-Juelich. This work was supported partly by the grant INTAS-97-30494.

References

[1] J.F.Gunion and D.E.Soper, Phys.Rev. D15 (1977) 2617

[2] N.N. Nikolaev and B.G. Zakharov, Z. Phys. C49 (1991) 607; J.Exp.Theor.Phys. 78
(1994) 598

[3] A.H.Mueller, Nucl.Phys. B415 (1994) 373

[4] N.N. Nikolaev, B.G. Zakharov, V.R. Zoller, JETP Letters 59 (1994) 8

[5] N.N. Nikolaev, B.G. Zakharov, V.R. Zoller, JETP 105 (1994) 1498

[6] N.N. Nikolaev, J. Speth and V.R. Zoller, Phys.Lett. B473 (2000) 157

[7] N.N.Nikolaev and B.G.Zakharov, Phys. Lett. B333 (1994) 250

[8] N.N. Nikolaev, B.G. Zakharov, Phys. Lett. B327 (1994) 157.

[9] N.N. Nikolaev, B.G. Zakharov and V.R. Zoller,JETP Lett. 66 (1997) 138

[10] N.N. Nikolaev and V.R. Zoller, JETP Lett. 69 (1999) 103

[11] N.N. Nikolaev and V.R. Zoller,JETP Lett. 69 (1999) 187

[12] L.N.Lipatov, Sov. Phys. JETP 63 (1986) 904.

[13] N.N. Nikolaev and V.R. Zoller, ITEP-PH-6-99, FZ-IKP-TH-1999-34, hep-ph/0001084

[14] ZEUS Coll., M.Derrick et al. Phys.Lett. B349 (1995) 225; H1 Coll., S.Aid, Nucl.Phys.
B472 (1996) 32;

[15] Th. Kuhr, Open Beauty Cross Section at HERA, Talk given at DIS2000, April 2000

[16] S.Frixione, Nucl.Phys.Proc.Suppl. 79 (1999) 399; S.Frixione, M.L.Mangano, P.Nason
and G.Ridolfi, Nucl.Phys. B431 (1994) 453

5

http://arXiv.org/abs/hep-ph/0001084


[17] ZEUS Collab., J.Breitweg et al., Phys.Lett. B437 (1998) 432; H1 Collab., C.Adloff et
al., Phys.Lett. B483 (2000) 23

[18] N.N.Nikolaev, Comments on Nucl. Part. Phys 21 (1992) 41; B.Z.Kopeliovich,
J.Nemchik, N.N.Nikolaev and B.G.Zakharov, Phys.Lett. 309 (1993) 179; ibid. 324
(1994) 469; J.Nemchik, N.N.Nikolaev and B.G.Zakharov, Phys.Lett. 341 (1994) 228

[19] J. Nemchik, N.N. Nikolaev, E. Predazzi, B.G. Zakharov and V.R. Zoller, JETP 86
(1998) 1054

6



Figure captions

Fig.1 The CD BFKL eigen functions plotted as σn(r)/r for n=0,1,2. The background term,
n = 3, which is a combination of higher CD BFKL solutions with n ≥ 3 is also shown.

Fig.2 The subleading hard-to-rightmost hard and soft-pomeron-to-rightmost hard ratio of
eigen-structure functions rn = f

b
n(Q

2)/f b
0
(Q2) as a function Q2 .

Fig.3 Predictions from CD BFKL-Regge factorization for the heavy flavor QQ̄ photoproduc-
tion cross section. The lower curve is a result of the complete CD BFKL-Regge ex-
pansion for σbb̄(W ). The upper curve corresponds to σcc̄(W ) calculated with mc = 1.3
GeV. The data points are from H1&ZEUS HERA experiments [14, 15].

Fig.4 The W-dependence of the exponent ∆eff in parameterization of the photoproduction
cross section σqq̄(W ) ∝ W 2∆eff for different quark flavors.

Fig.5 Prediction from CD BFKL-Regge factorization for the beauty structure function of
the proton F b

2
(x, Q2) as a function of the Bjorken variable xBj .

7



10
−3

10
−2

10
−1

10
0

10
1

r, fm

−0.2

−0.1

0.0

0.1

0.2

0.3

σ
n
(r

)/
r,

 f
m

n=0
n=1
n=2
n=3



10
0

10
1

10
2

10
3

10
4

Q
2
, GeV

2

−1.0

−0.5

0.0

r n
(Q

2
)

n=1
n=2
n=3
soft



10
2

W,  GeV

0.1

1

10

100
σ

(γ
p

−
>

Q
Q

X
) 

, 
µb

H1 charm
ZEUS charm
H1 beauty



10
2

10
3

W, GeV

0.0

0.2

0.4

0.6

0.8
∆

e
ff

beauty

charm

light flavors



10
−2

10
−1

F
2

b
(x

,Q
2
)

10
−5

10
−4

10
−3

10
−2

xBj

10
−2

10
−1

10
0

10
−5

10
−4

10
−3

10
−2

xBj

Q
2=10 GeV

2

Q
2
=40 GeV

2

Q
2
=100 GeV

2

Q
2
=250 GeV

2