Calculation of Direct CP Violation in B Decays

9 Pages·0.14 MB·

by  Cai-Dian Lu

HUPD-9923
hep-ph/0001321
0
0
0
2 CALCULATION OF DIRECT CP

n VIOLATION IN B DECAYS
a
J

1
3

Cai-Dian Lu¨a
1
v Department of Physics, Hiroshima University, Higashi-Hiroshima 739-8526,
1 Japan
2 E-mail: [email protected]
3
1
0
0
0
Abstract
/
h
Usingthegeneralizedfactorizationapproach,wecalculatetheCPasym-
p
- metries of charmless B decays. A number of decay channels has large
p
CP asymmetries, which can be measured in the B factories.
e
h
:
v
i
X
r
a
Invited talk given at the
3rd International Conference on B Physics and CP Violation (BCONF99)
Taipei, Taiwan, 3–7 December 1999
To appear in the Proceedings
aJSPSresearchfellow.
CALCULATION OF DIRECT CP VIOLATION IN B DECAYS
Cai-Dian Lu¨
Department of Physics, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
E-mail: [email protected]
Usingthegeneralizedfactorizationapproach,wecalculatetheCPasymmetriesof
charmlessBdecays. AnumberofdecaychannelshaslargeCPasymmetries,which
canbemeasuredintheBfactories.
1 Introduction
The sourse of CP violation is one of the unsolved problem in the standard
model(SM). The richnessofcharmlesshadronicdecaysofB mesonprovidesa
good place for study of CP violation1. When B0 and B¯0 decay to a common
finalstatef,B0-B¯0 mixingplaysacrucialrole. Itinterferenceswiththedirect
CP asymmetries. For other decays, B0 and B¯0 decay to different final states,
forexampleB0 →K+π−,B¯0 →K−π+. Nomixingisinvolvedhere. Theyare
similar to charged B± decays. CP asymmetry has no time dependence. The
direct CP asymmetry is important even if for neutral B meson decays.
If there is only one amplitude contributing to the decay, both the strong
phase and weak phase can be factored out. we have Γ = Γ. So there is no
directCPasymmetry. ThatisthecaseforDmesondecaysandBmesongoing
to heavy meson decays, like B → J/ψK . If there are two amplitudes, the
S
decay rate ofΓ and Γ may be different. If the strongphase difference between
the two amplitudes M and M is not zero (δ 6= 0) and the weak phase
1 2 12
difference of the two amplitudes is also non zero (φ 6= 0), we have Γ 6= Γ.
12
The direct CP asymmetry is
2rsinδ sinφ
12 12
A = , (1)
CP 1+r2+2rcosδ cosφ
12 12
where r=|M |/|M |. A depends on 2r/(1+r2), sinδ and sinφ . If one
2 1 CP 12 12
ofthethreeparametersissmall,thenA issmall. Inmanydecays,wedonot
CP
havealltheseconditions,thenthereisnosizabledirectCPviolation. However,
most charmless decays have large values for 2r/(1 + r2), where M is tree
1
amplitude and M is penguin amplitude. Furthermore, the CKM parameters
2
for the tree diagramand penguin diagramsare different providing weak phase
differences.
Direct CP asymmetries require an interference between two amplitudes
involving both a CKM phase and a final state strong interaction phase dif-
1
ference. The weak phase difference arises from the superposition of pen-
guin contributions and the tree diagrams. The strong-phase difference arises
through the perturbative penguin diagrams (hard final state interaction), or
non-perturbatively(softfinalstateinteraction). Thesoftpartisnotimportant
whichisshowninsomemodelcalculations2. Therearealsosomeothercontri-
butions, such as annihilation diagramand Soft final state interaction. Mostly,
theircontributionstobranchingratiosaresmall3. Probablytheircontribution
to A is also small. This is also shown in some model calculations2.
CP
The method of Isospin or SU(3) symmetry4 which requires a set of mea-
surements to solve the uncertainties is sometimes difficult for experiments.
We estimate these strong phases in specific models, such like the generalized
factorization approach, which can be tested by experiments.
2 CP Violation Classification and Formulae
For charged B± decays the CP-violating asymmetries are defined as1
Γ(B+ →f+)−Γ(B− →f−)
A = . (2)
CP Γ(B+ →f+)+Γ(B− →f−)
The charged modes are self-tagged decay channels for experiments. They are
easy to be measured. For B0 decays, more complication is from the B0−B0
mixing. The CP-asymmetries may be time-dependent, if the final states are
the same for B0 and B0
Γ(B0(t)→f)−Γ(B0(t)→f)
A (t) = (3)
CP 0
Γ(B0(t)→f)+Γ(B (t)→f)
(4)
≃aǫ′cos(∆mt)+aǫ+ǫ′sin(∆mt). (5)
Here the direct CP violation parameter aǫ′ is defined as
Γ(B0 →f)−Γ(B¯0 →f)
aǫ′ =AdCiPr = Γ(B0 →f)+Γ(B¯0 →f), (6)
which is the same defination as the charged B decays. And aǫ+ǫ′ is mixing-
induced CP violation1. In this note we will concentrate on the direct CP
asymmetries.
2
2.1 b→s (¯b→s¯), transitions
First we parametrize the decay amplitude like this way1
M =Tξ −P ξ eiδt −P ξ eiδc −P ξ eiδu,
u t t c c u u
M =Tξ∗−P ξ∗eiδt −P ξ∗eiδc −P ξ∗eiδu, (7)
u t t c c u u
where ξ = V V∗. T and P are the tree and i (i = u,c,t) quark penguin
i ib is i
contributions, respectively. Working in SM, we can use the unitarity relation
ξ =−ξ −ξ to simplify the above equation,
c u t
M =Tξ −P ξ eiδtc −P ξ eiδuc,
u tc t uc u
M =Tξ∗−P ξ∗eiδtc −P ξ∗eiδuc, (8)
u tc t uc u
where we define
P eiδtc = P eiδt −P eiδc,
tc t c
P eiδuc = P eiδu −P eiδc. (9)
uc u c
Thus, the direct CP-violating asymmetry is
ACP ≡aǫ′ =(cid:0)|M|2−|M|2(cid:1)/(cid:0)|M|2+|M|2(cid:1)=A−/A+ , (10)
where
A− =2TP |ξ∗ξ |sinφ sinδ +2P P |ξ∗ξ |sinφ sin(δ −δ ), (11)
tc u t 3 tc tc uc u t 3 uc tc
A+ = (T2+P2 )|ξ |2+P2|ξ |2−2P P |ξ∗ξ |cosφ cos(δ −δ )
uc u tc t tc uc u t 3 uc tc
−2TP |ξ |2cosδ +2TP |ξ∗ξ |cosφ cosδ . (12)
uc u uc tc u t 3 tc
First, we note that |ξ | ≪ |ξ | ≃ |ξ |, with an upper bound |ξ |/|ξ | ≤
u t c u t
0.025. In some channels, such as B+ →K+π0, K∗+π0, K∗+ρ0, B0 →K+π−,
K∗+π−, K∗+ρ−, |P /T| is of O(0.1), |P /P | = O(0.3). The CP-violating
tc uc tc
asymmetry in this case is
2z sinδ sinφ
12 tc 3
A ≃ , (13)
CP 1+2z cosδ cosφ +z2
12 tc 3 12
where z = |ξ /ξ |×T/P . We show the CP asymmetry of B → K∗+π−
12 u t tc
as an example in Figure 1. It is easy to see that, there may be large CP
asymmetries in this decay channel. Besides the CKM parameter ρ and η, the
CP asymmetry is also sensitive to the gluon momentum k2, which is related
3
Figure1: CP-violatingAsymmetryACP inB0→K∗+π− decayasafunctionoftheCKM
parameter ρ. (a) k2 =m2/2. The dotted, dashed-dotted and dashed curves correspond to
b
theCKMparametervaluesη=0.42,η=0.34andη=0.26,respectively. (b)η=0.34. The
dotted, dashed-dotted and dashed curves correspond to k2 =m2/2+2 GeV2, k2 =m2/2
b b
andk2=m2/2−2GeV2,respectively.
b
to the size of strong phase. If k2 is known, the strong phase is predictable, we
mayuseA todeterminesinφ . The first6channelsofTable1arethis kind
CP 3
of decays. Two of them are reported from CLEO Collaboration with large
error-bars5. The central values are far away from the theoretical predictions.
If more data suport this, we may expect new physics signals here.
There are some decays with vanishing tree contributions (T = 0), such
as B+ → π+K0, π+K∗0, ρ+K∗0. Then for these decays, the CP-violating
S
asymmetry is
Puc (cid:12)ξu(cid:12)
ACP ≃2P (cid:12)(cid:12)ξ (cid:12)(cid:12)sin(δuc−δtc)sinφ3. (14)
tc (cid:12) t(cid:12)
Withoutthetreecontribution,thesuppressionduetobothP /P and|ξ /ξ |
uc tc u t
is much stronger. The CP-violating asymmetries are only around −(1-2)%.
Some estimates of the channel B+ → πK0 show that even including the
S
annihilation and soft final state interaction, the CP asymmetry of this decay
is still small2. This means that this channel is clean for new physics to show
up. In table 1, we can see that CLEO’s central value of this decay indicates a
large CP asymmetry maybe possible.
2.2 b→d (¯b→d¯) transitions
Similarly to the b→s transition, we can define the CP asymmetry as
A =A−/A+, (15)
CP
4
Table 1: CP-rate asymmetries ACP and branching ratios for some B → h1h2 decays, up-
datedforthecentralvalues oftheCKMfitsρ=0.20,η=0.37andthefactorizationmodel
parametersξ=0.5andk2=m2/2±2GeV2.
b
Decay Modes A -Exp.(%) A (%) BR(×10−6)
CP CP
B± →K±π0 29±23 −7.7−2.2 10.0
+4.0
B± →K∗±π0 − −14.4−4.4 4.3
+8.2
B± →K∗±ρ0 − −13.5−4.0 4.8
+7.5
B0 →K+π− 4±16 −8.2−2.3 14.0
+4.3
B0 →K∗+π− − −17.2−5.5 6.0
+9.8
B0 →K∗+ρ− − −17.2−5.5 5.4
+9.8
B± →K0π± −18±24 −1.4−0.1 14.0
S +0.1
B± →ηπ± − 9.3+1.9 5.5
−4.1
B± →η′π± − 9.4+2.1 3.7
−4.5
B± →ηρ± − 3.1+0.7 8.6
−1.7
B± →η′ρ± − 3.1+0.7 6.2
−1.8
B± →ρ±ω − 7.0+1.5 21.0
−3.4
B± →η′K± −3±12 −4.9−1.2 23.0
+2.1
B± →π±ω 34±25 7.7+1.7 9.5
−3.7
where
A− = −2TP |ζ∗ζ |sinφ sinδ −2P P |ζ∗ζ |sinφ sin(δ −δ ), (16)
tc u t 2 tc tc uc u t 2 uc tc
A+ =(T2+P2 )|ζ |2+P2|ζ |2−2P P |ζ∗ζ |cosφ cos(δ −δ )
uc u tc t tc uc u t 2 uc tc
−2TP |ζ |2cosδ +2TP |ζ∗ζ |cosφ cosδ , (17)
uc u uc tc u t 2 tc
withζ =V V∗,andagainwehaveusedCKMunitarityrelationζ =−ζ −ζ .
i ib id c t u
For the tree-dominated decays, such as B+ → π+η(′), ρ+η(′), ρ+ω, the
relation P <P ≪T holds. The CP asymmetry is
uc tc
−2z sinδ sinφ
1 tc 2
A ≃ , (18)
CP
1+2z cosδ cosφ
1 tc 2
with z =|ζ /ζ |×TP /T′2,andT′2 ≡T2−2TP cosδ . The CP asymme-
1 t u tc uc uc
tries are proportional to sinφ . They are large enough for the experiments to
2
detect them. The theoretical predictions of these decays are shown in table 1.
For the decays with a vanishing tree contribution (T =0), such as B+ →
K+K0, K+K¯∗0, K∗+K¯∗0, the CP-violatingasymmetry is approximatelypro-
S
5
portional to sinφ again,
2
−2z sin(δ −δ )sinφ
3 uc tc 2
A = , (19)
CP 1−2z cos(δ −δ )cosφ +z2
3 uc tc 2 3
with z = |ζ /ζ |×P /P . As the suppressions from |ζ /ζ | and |P /P |
3 u t uc tc u t uc tc
are not very big, the CP-violatingasymmetry can againbe of order(10-20)%.
Unfortunately, these channels have smaller branching ratios1,3.
More charmless decay channels are discussed in ref.1. Some of them are
morecomplicatedthantheoneswediscussedabove. Therearealsosomeother
interesting decays like B → K∗γ, B → Dπℓν, B → ππℓν6, etc. They have
small CP asymmetries in SM. They are sensitive to new physics.
3 Models of Calculation
Inthe FactorizationApproach3,7, the twobody B mesondecayscanbe factor-
ized as two products:
C hP P |O |Bi=C hP |J |0i hP |Jµ|Bi,
i 1 2 i i 1 µ 2
where C is the corresponding Wilson coefficients. The second factor on the
i
right side of the equation is proportional to the meson decay constant. The
last term is the corresponding form factors.
The strong-phase differences arise through Bander-Silverman-Soni Mech-
anism (BSS)8. In this picture, the perturbative penguin diagrams involving
charm and up quark loops, where the light quarks can be on mass shell, pro-
viding the strong phases. They are mostly sensitive to the gluon momentum
k2. For numerical calculations, we use k2 =m2/2±2GeV2.
b
In the perturbative QCD approach (pQCD)9, we need one hard gluon
connecting the spectator quark. Strong phases are from the non-factorizable
diagramandannihilationdiagram,wheretheinnnerquarkorgluonpropagator
canbe onmassshell. The pQCDapproachis basedonfactorization,andgoes
one step further. In pQCD, we can calculate annihilation diagrams and also
the non-factorizable contributions. The k2 of gluon is well defined in this
approach. We have calculated the B → ππ decays in this approach, and the
results compared with the factorization approachin Table 2.
4 Summary
The recently measureddirect CP asymmetries for B0 →K+π−, B+ →K+η′,
B0 → K+π−, B+ → π+K0, and B+ → ωπ+ are encouraging news for di-
rect CP violation in B decays, although the signal is not significantly excess
background.
6
Table2: DirectCPasymmetriesofB→ππ decaysinfactorizationapproachandperturba-
tiveQCDapproach(Preliminary)10.
Channel Adirect (Factorization) Adirect (pQCD)
CP CP
B →π+π− 6.9%+2 -1.4%
−4
B+ →π+π0 0.1%+0.1 0.02%
−0.1
B0 →π0π0 −15%−7 -60%
+14
( )
CP-asymmetriesofA (K±η′),A (π±K0)andA (ρ±K∗0)aresmall,
CP CP S CP
but stable against variation in N , k2 and µ. CP-asymmetries well over 10%
c
in these decay modes will be a sign of new physics. The decay channels of
B → K∗±π∓, K∗±π0, K∗±η, K∗±η′, K∗±ρ∓ and K∗±ρ0, have measurably
large CP-violating asymmetries. A good measurement of the CP-asymmetry
in any one of these decays could be used to determine k2. Such that the
theoretical predictions of all other channels make sense. We also hope that
the perturbativeQCDapproachcouldsolvetheremaininguncertaintiesinthe
factorizationapproach. WiththetwoBfactoriesandotherhadronicmachines,
a number of decays is going to be measured soon.
Acknowledgments
The author thanks A. Ali and G. Kramer for collaboration on the main topic
discussed in this report. We thank the organizer H.Y. Cheng and W.S. Hou
forafruitfulBCP3conference. WeacknowledgetheGrant-in-AidforScientific
Research on Priority Areas (Physics of CP violation) and JSPS for support.
References
1. A. Ali, G. Kramer, C.D. Lu¨, Phys. Rev. D59, 014005 (1999).
2. H.Y. Jin, hep-ph/9805235; D.S. Du , X.Q. Li, Z.T. Wei, B.S. Zou, Eur.
Phys. J.A4,91(1999);M.Neubert, Phys. Lett. B424,152(1998);A,N.
Kamal, Phys. Rev. D60, 094018 (1999).
3. A. Ali, G. Kramer, C.D. Lu¨, Phys. Rev. D58, 094009 (1998); C.D. Lu¨,
Nucl. Phys. Proc. Suppl. 74, 227 (1999).
4. M. Gronau, in these proceedings.
5. S. Chen et al., CLEO Collaboration, hep-ex/0001009.
6. C.S. Kim, J. Lee, W. Nagung, Phys. Rev. D60, 094019 (1999); Phys.
Rev. D59, 114006(1999); Phys. Rev. D59, 114005 (1999).
7. H.Y. Cheng, in these proceedings, hep-ph/9912372.
8. M. Bander, D. Silverman, A. Soni, Phys. Rev. Lett. 43, 242 (1979).
7
9. H.N. Li, in these proceedings.
10. C.D. Lu¨, K. Ukai, M.Z. Yang, in progress.
8