/home/www/ftp/data/hep-ph/dir_0302063/0302063.dvi


NEW PHYSICS IN CP-VIOLATING
OBSERVABLES FOR BEAUTY

J.Bernabéu

Department of Theoretical Physics and IFIC,
University of Valencia and CSIC

Abstract

After the present establishment of CP-Violation in Bd-physics,
consistency tests of unitarity in the Standard Model and the search
of new phenomena are compulsory. I illustrate the way to look for
T-violation, without contamination of absorptive parts, in correlated
decays in B-factories. Bs-mixing and penguin-mediated Bs-decays are
of prime importance in hadronic machines to look for new physics.



1 Introduction

As exemplified by the presentations of this 2002 edition of the ”Beauty”
Conference, CP violation is currently the focus of a great deal of attention.
The results of the B-factories [1] measure sin(2β), where β is the CP-phase
between the top and charm sides of the (bd) unitarity triangle [2]. To within
the experimental sensitivity, the CKM mechanism [3] of the standard theory
is verified. In this description, all the CP-violating observables depend on a
unique phase in the quark mixing matrix. Any inconsistency between two
independent determinations of the CP-phase is an indication for new physics.
Two main reasons for the search of new physics in CP-violating observables
are:

i) The dynamic generation of the baryonic asymmetry in the Universe
requires CP-violation, and its magnitude in the standard model looks insuf-
ficient.

ii) Essentially all extensions of the standard model introduce new sectors
with additional sources of CP-phases.

The examples that I will consider here to search for new physics are based
on the following attitude: the decays dominated by standard model diagrams
at tree level with W-exchange allow the extraction of |Vub|, |Vus| and |Vcb|.
On the contrary, new physics will be apparent in processes which, for the
standard model, are described by loop diagrams, like Bd −Bd and Bs −Bs
mixing or penguin amplitudes. In so doing, I will cover cases which are
appropriate for B-factories, as well as others which need hadronic machines.

In Section 2 I discuss coherent correlated decays of Bd’s in order to build
(besides CP-odd asymmetries) T-odd and CPT-odd observables. Section 3
is devoted to temporal asymmetries which need absorptive parts and are a
signal of a non-vanishing ∆Γ in Bd decays. In Section 4 I discuss Bs-mixing.
The decays Bs → J/ψφ and B → φKS are presented in Section 5 as ways to
search for new physics. Some conclusions are drawn in Section 6.

2 Genuine Asymmetries from Entangled States

In a B-factory operating at the Υ(4S) peak, correlated pairs of neutral B-
mesons are produced. This permits the performance of either a flavour tag
or a CP tag. To O(λ3), where λ is the Wolfenstein parameter [4] in the
quark mixing matrix, the determination of the single B-state is possible
and unambiguous [5]. Any final configuration (X,Y), where X,Y are decay
channels which are either flavour or CP conserving, corresponds to a single

1



particle mesonic transition. The intensity for the final configuration

I(X,Y, ∆t) ≡ 1
2

∫ ∞
∆t

dt′ |(X,Y )|2 (1)

is thus proportional to the time dependent probability for the meson transi-
tion.

The case (l+, l+) associated with the B
o → Bo transition is shown in

Figure 1.
 

   �
+
 

    

            �
o
   �   

Υ ����           �
+
 

   �                   

                        �
o
                                                      � 

                                 �
o
 

       ∆	 

Figure 1: The flavour tag of B
o

and its decay as Bo after a time ∆t.

We consider genuine asymmetries for CP, T and CPT operations, in the
sense that a non-vanishing value is a proof of the violation of the symmetry.
Two cases are of particular interest:

i) Flavour-to-flavour transitions
The final configuration denoted by (l, l), with flavour definite (for exam-

ple, semileptonic) decays detected on both sides of the detector, corresponds
to flavour-to-flavour transitions at the meson level. The equivalence is shown
in Table 1. The first two processes in the Table are conjugated under CP
and also under T.

The corresponding Kabir asymmetry [6], to linear order in the CPT vio-
lating parameter δ of the meson mixing, is given by

A(l+, l+) '
4

Re(ε)

1+|ε|2

1 + 4
Re(ε)

1+|ε|2
(2)

where ε is the rephasing invariant [7] CP-odd, T-odd parameter in the neutral

meson mixing. The asymmetry (2) does not depend on time. However, in
the exact limit ∆Γ = 0, Re(ε) also vanishes and A will be zero. For the Bd

2



Table 1: Flavour-to-flavour transitions

(X,Y ) Meson Transition

(l+, l+) B
o → Bo

CP,T

(l−, l−) Bo → Bo ←↩
(l+, l−) B

o → Bo
CP,CPT

(l−, l+) Bo → Bo ←↩

system, experimental limits on Re(ε) are of few parts in a thousand [8, 9]. A
second asymmetry arises from the last two processes in Table 1, related by
a CP or a CPT operation.

A(l+, l−) '−2
Re

(
δ

1−ε2
)
sinh ∆Γ∆t

2
− Im

(
δ

1−ε2
)
sin (∆m∆t)

cosh ∆Γ∆t
2

+ cos (∆m∆t)
(3)

which is an odd function of ∆t. This asymmetry also vanishes with ∆Γ = 0,

because then Im(δ) = 0 as well. Present limits on Im(δ) are at the level of
few percent [8].

One discovers the weakness of these asymmetries to look for T-, and CPT-
violation in the Bd-system. The reason is that one needs both the violation
of the symmetry and ∆Γ 6= 0.

ii) CP-to-flavour transitions
Alternative asymmetries can be constructed making use of the CP eigen-

states, which can be identified in this system by means of a CP tag. If the
first decay product, X, is a CP eigenstate produced along the CP-conserving
direction [5], the decay is free of CP violation. If Y is a flavour definite chan-
nel, then the mesonic transition corresponding to the configuration (X,Y) is
of the type CP-to-flavour.

In Table 2 we show the mesonic transitions, with their related final con-
figurations, connected by genuine symmetry transformations to B+ → Bo.

Comparing the intensities of the four processes, we may construct three
genuine asymmetries, namely A(CP), A(T) and A(CPT) [10]:

A(CP) = −2 Im(ε)
1 + |ε|2

sin(∆m∆t) +
1 −|ε|2
1 + |ε|2

2Re(δ)

1 + |ε|2
sin2

(
∆m∆t

2

)
(4)

3



Table 2: Transitions connected to (J/ψKS, l
+)

(X,Y) Transition Transformation

(J/ψKS, l
−) B+ → Bo CP

(l−,J/ψKL) Bo → B+ T
(l+,J/ψKL) B

o → B+ CPT

The CP-odd asymmetry, contains both T-violating and CPT-violating
contributions, which are, respectively, odd and even functions of ∆t. This
asymmetry corresponds to the ”gold plate” decay [11] and has been measured
recently [1]. The result is interpreted in terms of the standard model sin(2β)
with neither CPT-violation nor ∆Γ [12]. One finds [7]

− 2Im(ε)
1 + |ε|2

= sin(2β) (5)

The two T- and CPT-violating terms in Eq. (4) can be separated out by
constructing other asymmetries

A(T) = −2 Im(ε)
1 + |ε|2

sin(∆m∆t)

[
1 − 1 −|ε|

2

1 + |ε|2
2Re(δ)

1 + |ε|2
sin2

(
∆m∆t

2

)]
(6)

the T-asymmetry needs ε 6= 0 and turns out to be purely odd in ∆t in
the limit we are considering.

A(CPT) =
1 −|ε|2
1 + |ε|2

2Re(δ)

1 + |ε|2
sin2

(
∆m∆t

2

)
1 − 2 Im(ε)

1+|ε|2 sin(∆m∆t)
(7)

is the CPT asymmetry. It needs δ 6= 0 and includes both even and odd
time dependences.

The expressions (4), (6) and (7) correspond to the limit ∆Γ = 0, but,
being genuine observables, a possible absorptive part could not induce by
itself a non-vanishing asymmetry.

3 ∆Γ and Non-genuine Asymmetry

The construction of the quantities described above requires to tag both B+
and B− states, and thus the reconstruction of both B → J/ψKS and

4



Table 3: Final configurations with only J/ψKS

(X,Y) Transition Transformation
(J/ψKS, l

+) B+ → Bo CP
(l+,J/ψKS) B

o → B− ∆t
(l−,J/ψKS) Bo → B− CP∆t

B → J/ψKL decays. One can consider non-genuine asymmetries from
B → J/ψKS only: they involve [10] the discrete transformation, denoted
by ∆t, consisting in the exchange in the order of appearance of decay prod-
ucts X and Y, which cannot be associated with any fundamental symmetry.

Table 3 shows the different transitions we may study from such final
states. Besides the genuine CP asymmetry, there are two new quantities
that can be constructed from the comparison between (J/ψKS, l

+) and the
processes in the Table 3.

In the exact limit ∆Γ = 0, ∆t and T operations, although different (com-
pare the second lines of Tables 2 and 3), are found to become equivalent,
so that the temporal asymmetries satisfy A(∆t) = A(T) and A(CP∆t) =
A(CPT).

The asymmetries A(∆t) and A(CP∆t) are non-genuine, so that the pres-
ence of ∆Γ 6= 0 may induce non-vanishing values for them, even in the ab-
sence of true T or CPT violation. These effects can be calculated and are thus
controllable. This reasoning leads to an interesting suggestion: there are
linear terms in ∆Γ inducing a non-vanishing asymmetry A(CP∆t).
This last asymmetry is particularly clean, under the reasonable assumption
that A(CPT) = 0. Explicit calculations [10] show that, even in the limit
of perfect symmetry, i.e., ε = 0 besides δ = 0, one finds a non-vanishing
(CP∆t)-asymmetry, given by

A(l−,J/ψKS) =
∆Γ∆t

2
; ε = δ = 0 (8)

The simple result (8) is modified under the realistic ε 6= 0 situation. One
has, if only δ = 0, the result

A(l−,J/ψKS) =
1

1 − 2Im(ε)
1+|ε|2 sin(∆m∆t)

(9)

5



{
∆Γ∆t

2

1 −|ε|2
1 + |ε|2

+
4Re(ε)

1 + |ε|2
sin2

(
∆m∆t

2

)
− 2Im(ε)

1 + |ε|2
2Re(ε)

1 + |ε|2
sin(∆m∆t)

}

The three terms of Eq. (9) contain different ∆t-dependences, so that a
good time resolution will allow the determination of the parameters. Taking
into account that Re(ε) = x∆Γ, all of the three terms are linear in ∆Γ.
We conclude that the comparison between the channels (l−,J/ψKS) and
(J/ψKS, l

+) is a good method to obtain information on ∆Γ, due to the
absence of any non-vanishing difference when ∆Γ = 0.

4 Bs Mixing

Bo and B̄o are not mass eigenstates, so that their oscillation frequency is
governed by their mass-difference. The measurement by the UA1 collabora-
tion [13] of a large value of ∆Md was historically the first indication of the
heavy top quark mass. This is so because of non-decoupling effects of the
heavy-mass exchange in the Box Diagram. For Bs −mixing , it is shown in
Figure 2.

                 � 

�         �   

 

                   �       � 

 

�          � 

       � 

Figure 2: Box Diagram responsible of the neutral meson mixing

To avoid many hadronic uncertainties, it is interesting to consider the
ratio between ∆Ms and ∆Md, given by [14]

∆Ms
∆Md

=

∣∣∣∣VtsVtd
∣∣∣∣
2
MBd
MBs

ξ2 (10)

6



where ξ2 ≡ f
2
Bs

BBs
f 2

Bd
BBd

is the flavour-SU(3) breaking parameter in terms

of the meson decay constants and the bag factors. BBq = 1 if one uses a
vacuum saturation of the hadronic matrix element. The great advantage of
Eq. (10) is that, in the ratio, different systematics in the evaluation of the
matrix element tends to cancel out. However, unlike ∆Md = 0.479(12) ps

−1,
which is measured with a good precision [15], the determination of ∆Ms is
an experimental challenge due to the rapid oscillation of the Bs − system.
At present [15], ∆Ms > 13.1ps

−1, with 95% C.L., but this bound already
provides a strong constraint on |Vtd|. The use of QCD spectral sum rules
leads to [14].

ξ ' 1.18 ± 0.03 → ∆Ms ' 18.6(2.1)ps−1 (11)

in agreement with the present experimental lower bound and within the
reach of the proposed experiments.

Ali and London [16] have examined the situation for SUSY theories with
minimal flavour violation. In this class of models, the SUSY contributions
to ∆Md and ∆Ms can both be described by a single common parameter f.

∆Md = ∆Md(SM)[1 + f] (12)

∆Ms = ∆Ms(SM)[1 + f] (13)

The parameter f is positive definite, so that the SUSY contributions add
constructively to the SM contributions in the entire allowed supersymmetric
parameter space. The size of f depends, in general, on the parameters of
the SUSY model. They conclude that, if Ms is measured to be near its lower
limit, SUSY with large f is disfavoured.

With respect to the values of the CP phases α, β and γ of the unitarity
triangle, the key observation is that a measurement of β will not distinguish
among the various values of f, i.e., β is rather independent of f. If one
wants to distinguish among the various SUSY models, it will be necessary to
measure γ and/or α independently.

Contrary to these SUSY models, in which the ratio ∆Ms/∆Md remains
that of the Standard Model, Left-Right-Symmetric Models with Spontaneous
CP Violation modify ∆Ms and ∆Md with different phases relative to the SM
contribution. One has [17]

∆Ms
∆Md

=
∆Ms(SM)

∆Md(SM)

∣∣∣∣ 1 + κe
iσs

1 + κeiσd

∣∣∣∣ (14)
7



As a consequence, the ratio is modified with respect to the SM. In Ref.
[18], an analysis of the joint constraints imposed by ∆MK and ∆MB is per-
formed, with the conclusion that the Left-Right Model favours opposite signs
of �K and sin(2β) and it would be disfavoured for sin(2β) > 0.1. A test of
the Bs −mixing would be crucial in this context.

5 Two Decays: Bs → J/ψφ, Bd → φKS
The general argument of considering CP-Violation in Bs−mixing as a prime
candidate for New Physics is well defined. In Bd → J/ψKS, the SM ampli-
tudes of mixing (dominated by the virtual top quark) VbtV

∗
td ∼ λ3 and decay

VbcV
∗
cd ∼ λ3 define a relative phase β of the order of one, because the corre-

sponding unitarity triangle satisfies the scales of figure 3.

λ3

λ3

λ3

o
β

Figure 3: The (bd) unitarity triangle.

Contrary to this standard β, the SM amplitudes for Bs → J/ψφ satisfy
that VbtV

∗
ts ∼ λ2, VbcV ∗cs ∼ λ2, so that the corresponding unitarity triangle is

shown in figure 4 and the relative phase χ is tiny, of the order of λ2. It does
not take much New Physics to change the tiny standard χ!

k
χ

λ2

λ2
λ4

Figure 4: The (bs) unitarity triangle.

In hadronic machines, substantial number of Bs → J/ψφ events are ex-
pected. The decay process is described by three Lorentz invariant terms, two

8



CP-even terms and one CP-odd term. The joint angular distibution with
J/ψ → l+l− and φ → K+K− has been described in Ref. [19], with the
aim of separating out the definite CP eigenstates and thus recovering a CP
asymmetry free of cancellations.

Following the argument of the Introduction, candidates for New Physics
are processes which, in the SM, are described by either mixing or penguin
amplitudes in the decay. In Left-Right models, the gluonic penguin con-
tribution to b → sss transition is enhanced by mt/mb due to the presence
of right-handed currents. This may overcome the suppression due to small
left-right mixing angle. Two new phases [20] in the B → φKS decay ampli-
tude may therefore modify the time dependent CP asymmetry in this decay
mode by O(1). This scenario implies also large CP asymmetry in the decay
Bs → φφ which can be tested in hadronic machines.

6 Conclusions

The prospects for an experimental study of the Flavour Problem in the
next coming years are much interesting, from CP-violating observables in B-
factories and hadronic machines. The (bd) Unitarity Triangle will be tested,
with separate determinations of the CP-phases (α, β, γ), after the present
establishment of CP-Violation in Bd -physics.

New phenomena are probably around the corner. Discoveries like T-
violation in B-physics, without any contamination of absorptive parts, and
sensitive limits (or spectacular surprises) for CPT-violation are expected.
Temporal asymmetries in B-decays are a good method to search for linear
terms in ∆Γ/Γ. The intensities I(l−, J/ψKS) and I(J/ψKS, l+) are pre-
dicted to be equal under CPT invariance and ∆Γ = 0. Linear terms in ∆Γ
induce a non-vanishing asymmetry for this CP ∆t transformation.

Bs − mixing is considered to be of prime importance for the search of
new physics, particularly in its CP-violating component. Extended models
modify the tiny phase between the top and charm sides of the standard (bs)
unitarity triangle. The non-decoupling effects of new physics can be put
under control by the cancellation of hadronic matrix element uncertainties
in the ratio ∆Ms/∆Md. The interest in a detailed analysis of Bs → J/ψφ is
apparent in this context.

New physics in penguin-mediated decays, like Bd → φKS and Bs → φφ,
is also expected, with information complementary to that of mixing.

All in all, we can expect a beautiful future in front of us!

9



Acknowledgements

I would like to thak E. Alvarez, P. Ball, D. Binosi, F.Botella, J. Matias,
M.Nebot and J. Papavassiliou for interesting discussions and to the Orga-
nizers of ”Beauty 2002” for their invitation to such a beautiful event in
Santiago. This work has been supported by the Grant AEN-99/0692 of the
Spanish Ministry of Science and Technology.

References

[1] B.Aubert et al., Phys.Rev.Lett. 86(2001) 2515;
B.Abe et al., Phys.Rev.Lett. 87(2001) 091802.

[2] L.-L.Chau, W.-Y.Keung, Phys.Rev.Lett. 53(1984) 1802.

[3] N.Cabibbo, Phys.Rev.Lett. 10(1963) 531;
M.Kobayashi, K.Maskawa, Prog.Theor.Phys. 49(1973) 652.

[4] L.Wolfenstein, Phys.Rev.Lett. 51(1983) 1945.

[5] M.C.Bañuls and J.Bernabéu, JHEP9906 (1999) 032.

[6] P.K.Kabir, The CP Puzzle, Academic Press (1968) 99.

[7] M.C.Bañuls and J.Bernabéu, Phys.Lett. B423 (1998) 151.

[8] K.Ackerstaff et al., Z.Phys. C76 (1997) 401;
F.Abe et al., Phys.Rev. D55 (1997) 2546;
J.Bartelt et al., Phys.Rev.Lett. 71(1993) 1680.

[9] B.Aubert et al., SLAC-PUB-927 (2001), hep-ex/0107059.

[10] M.C.Bañuls and J.Bernabéu, Phys.Lett. B464 (1999) 117;
M.C.Bañuls and J.Bernabéu, Nucl.Phys. B590 (2000) 19.

[11] I.I.Bigi and A.I.Sanda, Nucl.Phys. B193 (1981) 85.

[12] V.Khoze et al., Yad.Fiz. 46 (1987) 181;
A.Acuto and D.Cocolicchio, Phys.Rev. D47 (1993) 3945.

[13] C.Albajar et al., Phys.Lett. B186 (1987) 237, 247.

[14] K.Hagiwara et al., hep-ph/0205092

10



[15] D.E.Groom et al. (PDG), Euro.Phys. J. C15 (2000) 1

[16] A.Ali and D.London, Eur.Phys. J. C18 (2001) 665.

[17] G.Barenboim, J.Bernabéu and M.Raidal, Nucl.Phys. B511 (1998)577.

[18] P.Ball, J.-M.Frère and J.Matias, Nucl.Phys. B572 (2000) 3.

[19] I.Dunietz et al., Phys.Rev. D43 (1991) 2193.

[20] M.Raidal, hep-ph/0208091.

11