CP Violation: The CKM Matrix and New Physics

16 Pages·0.59 MB·English

by  Yosef Nir

1
CP Violation: The CKM Matrix and New Physics
3
0 Yosef Nira∗†
0
2 aDepartment of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel

n
a Recent measurements of CP violating asymmetries have led to a significant progress in our understanding of
J CP violation. The implications of the experimental results for the Kobayashi-Maskawa mechanism and for new

2 physicsare explained.
1

3
v
1. Introduction 2. Standard Model Lessons
0
8
The study of CP violation is, at last, experi- Within the Standard Model, the only source
0
8 mentdriven. Experimentshavemeasuredtodate of CP violation is the Kobayashi-Maskawa(KM)
0 three independent CP violating parameters: phase[5]. ThisphaseappearsintheCKMmatrix
2 • Indirect CP violation in K → ππ [1] and in which describes the charged current interactions
0
K →πℓν decays is given by of quarks.
/
h
p |ε|=(2.28±0.02)×10−3. (1) 2.1. Unitarity Triangles
- The CKM matrix gives the couplings of the
p • Direct CP violation in K → ππ decays is
W+-boson to u¯ d quark pairs:
e given by i j
h
: Re(ε′/ε)=(1.66±0.16)×10−3. (2) Vud Vus Vub
v
V = V V V . (4)
i (The world average given in eq. (2) includes the  cd cs cb
X V V V
new result from NA48 [2], Re(ε′/ε) = (1.47 ± td ts tb
ar 0.22)× 10−3, and previous results from NA31, The unitarity of the matrix leads to various rela-
E731 and KTeV.) tions among its elements, e.g.,
•The CP asymmetryinB →ψK decay(and
S
other, related, modes) has been measured: VudVu∗b+VcdVc∗b+VtdVt∗b =0. (5)
ImλψK =0.734±0.054. (3) The unitarity triangle is a geometrical presenta-
tion in the complex plane of the relation (5). It
(The world average given in eq. (3) includes the
provides a convenient tool in the study of flavor
new results from Belle [3], Imλ = 0.719 ±
ψK
physics and CP violation:
0.074±0.035, and Babar [4], Imλ = 0.741±
ψK
1. Flavorchangingprocessesandunitaritygive
0.067 ± 0.033, and previous results from Opal,
rather precise information on the magnitudes of
Aleph and CDF.)
allelementsexceptfor|V |and|V |,whichhave
In addition, CP asymmetries in many other ub td
large uncertainties. The unitarity triangle is a
modes(neutralBdecaysintofinalCPeigenstates
pictorialwayofcombiningthevariousconstraints
and non-CP eigenstates and charged B decays)
ontheseelementsandoftestingwhetherthecon-
have been searchedfor. We describe the implica-
straints can be explained consistently within the
tions of the new data for our theoretical under-
CKM framework.
standing of CP violation.
2. The angles of the triangle are related to CP
∗supported by the Israel Science Foundation founded by
violation. In particular, measurements of vari-
theIsraelAcademyofSciencesandHumanities.
†plenarytalkgivenatthe31stinternationalconferenceon ous CP asymmetries in B decays can be used to
highenergyphysics(Amsterdam,24−31July,2002). constrain the values of these angles. Conversely,
2
1 ∆m 1
d
∆m & ∆m
s d
ε
K sin 2β
WA
η 0 η 0
|V /V | |V /V |
ub cb ub cb
ε
K
-1 -1
CKM CKM
f i t t e r f i t t e r
p a c k a g e p a c k a g e
-1 0 1 2 -1 0 1 2
ρ ρ
Figure 1. Constraints on the (ρ¯,η¯) parameters from tree processes and from (left) CP conserving loop
processes (∆m , ∆m ) and (right) CP violating processes (ε, Imλ ).
B Bs ψK
the consistency of the various constraints tests
whether CP violation can be accounted for by ((cid:26)(cid:22);(cid:17)(cid:22))
the Kobayashi-Maskawamechanism.
Since the length of one side, |VcdVcb|, is well Ru (cid:11) Rt
known, it is convenient to re-scale the unitarity
triangle by the length of this side and put it on (cid:13) (cid:12)
the real axis. When doing so, the coordinates of
the remaining vertex correspond to the ρ and η (0;0) (1;0)
parametersintheWolfensteinparametrization[6]
of the CKM matrix (or, in an improved version
[7], to ρ¯=(1− λ22)ρ and η¯=(1− λ22)η): The lengths Rt and Ru are defined as follows:
1− λ2 λ Aλ3(ρ−iη) V V∗ V V∗
2 R ≡ td tb , R ≡ ud ub . (8)
V = −λ 1− λ22 Aλ2 .(6) t (cid:12)VudVu∗b(cid:12) u (cid:12)VcdVc∗b(cid:12)
Aλ3(1−ρ−iη) −Aλ2 1 (cid:12) (cid:12) (cid:12) (cid:12)
Inwh(cid:12)atfollow(cid:12)s,wepresen(cid:12)tthecon(cid:12)straintsonthe
  (cid:12) (cid:12) (cid:12) (cid:12)
The angles α,β and γ (also known as, respec- (ρ¯,η¯) parameters coming from various classes of
tively, φ ,φ and φ ) are defined as follows: processes. (The plots have been produced using
2 1 3
the CKMFitter package [8].)
V V∗ V V∗
α ≡ arg − td tb , β ≡arg − cd cb , In Figure 1 we compare the constraints from
(cid:18) VudVu∗b(cid:19) (cid:18) VtdVt∗b(cid:19) CPconservingprocessestothosefromCPviolat-
V V∗ ing ones. The CP conserving observables are the
γ ≡ arg − ud ub . (7)
(cid:18) VcdVc∗b(cid:19) mass difference in the neutral B system, ∆mB,
3
1 ∆m 1 ∆m
d d
∆m & ∆m ∆m & ∆m
s d s d
ε ε
K sin 2β K sin 2β
WA WA
η 0 η 0
|V /V | |V /V |
ub cb ub cb
ε ε
K K
-1 -1
CKM CKM
f i t t e r f i t t e r
p a c k a g e p a c k a g e
-1 0 1 2 -1 0 1 2
ρ ρ
Figure2. Constraintsonthe (ρ¯,η¯)parametersfrom(left)CPconservingandtheεobservablescompared
to the Imλ constraint, and (right) from all observables.
ψK
and the lower bound on the mass difference in oretically interpreted with practically zero
the B system, ∆m . The CP violating observ- hadronic uncertainties.
s Bs
ables are the indirect CP violation in K → ππ
decays, ε, and the CP asymmetry in B → ψK Another way to see the consistency of the KM
S
decays,Imλ . Therearetwoimportantlessons picture of CP violation is the following. Within
ψK
to be drawn from this comparison: the Standard Model, there is a single CP violat-
ing parameter. Therefore, roughly speaking, a
• Since there is a significant overlap between measurement of a single CP violating observable
the allowed regions in the two panels, we simply determines the value of this parameter.
learn that the two sets of constraints are This situation is demonstrated in the left panel
consistent with each other. Thus it is very of Figure 2, where the constraints from all but
likelythattheKMmechanismisindeedthe the Imλ -measurementareusedtoproducean
ψK
source of the observed CP violation. allowed range in the (ρ¯,η¯) plane. A second mea-
surement of a CP violating observable tests this
• The constraints from the CP violating ob- mechanism, as demonstrated in the same Figure
servables are stronger than those from the by overlaying the constraint from the measure-
CP conserving ones. While the allowed ment of Imλ . (It is amusing to note that
ψK
rangesarerelatedtotheexperimentalaccu- in ref. [9], the allowed range for Imλ from
ψK
racy,animportantfactorinthissituationis the fit to all other observables is quoted to be
the factthat CP is a goodsymmetry ofthe 0.734+0.055, to be compared with the range from
−0.045
strong interactions. (The effects of θ the directmeasurementsineq. (3).) The allowed
QCD
are irrelevant to meson decays.) Conse- region in the (ρ¯,η¯) plane from the combination
quently, some CP asymmetries can be the- of all observables is shown in the right panel of
4
1 ∆m 1
d K+ → π+νν
∆m & ∆m
s d
ε
K
sin 2β
WA
η 0 η 0
|V /V | |V /V |
ub cb ub cb
ε
K
-1 -1
CKM CKM
f i t t e r f i t t e r
p a c k a g e p a c k a g e
-1 0 1 2 -1 0 1 2
ρ ρ
Figure3. Constraintsonthe(ρ¯,η¯)parametersfrom(left)B physics(∆m , ∆m , Imλ ),and(right)
B Bs ψK
K physics (ε, B(K+ →π+νν¯)).
Figure 2. We canagaindrawseveralconclusions: 2.2. Testing the KM mechanism
Since,bytheconsistencybetweenthepredicted
• TheCKMmatrixprovidesaconsistentpic- range and the measured value of the CP asym-
ture of all the measured flavor and CP vio- metry in B → ψK , the KM mechanism of CP
S
lating processes. violationhassuccessfullypasseditsfirstprecision
test,weareabletomakethefollowingstatement:
• The recent measurement of Imλ adds a
ψK Very likely, the KM mechanism is the domi-
significant constraint.
nant source of CP violation in flavor changing
processes.
In Figure 3 we make one final compari-
Thirty eight years have passed since the dis-
son, between observables related to B physics
covery of CP violation [1] and twenty nine years
(∆m , ∆m , Imλ , left panel) and to K
physiBcs (ε, BB(sK+ →πψ+Kνν¯) [10], right panel). have passed since the KM mechanism has been
proposed[5]. Butonlynow,followingtheimpres-
The conclusions that we draw from this com-
sively precise measurements by Belle and Babar
parison are the following:
that yield (3), we can make the above statement
• There is no signal of new flavor physics. based on experimental evidence. This is a very
important step forward in our theoretical under-
• At present, the constraints from B physics
standing of CP violation.
are much stronger. Future measurements
Wewouldliketoemphasize,however,threeim-
of B(K → πνν¯) (for both the charged and
portant points in the above statement:
theneutralmodes)willbeessentialtomake
this comparison into a useful probe of new 1. ‘Very likely:’ since we are using only two
physics. CPviolatingobservables,theconsistencyof
5
the KM picture could be accidental. Addi- 1. In decay:
tional measurements are crucial to make a
A¯ A¯ A¯ +A¯
more convincing case for the validity of the 6=1 = 1 2 . (9)
KM mechanism. (cid:12)A(cid:12) (cid:20)A A1+A2(cid:21)
(cid:12) (cid:12)
2. ‘Dominant:’ the accuracy of the Standard 2. I(cid:12)(cid:12)n m(cid:12)(cid:12)ixing:
Model prediction for Imλ is of O(20%).
ψK q q 2 2M∗ −iΓ∗
Therefore, it is quite possible that new 6=1 = 12 12 . (10)
physics contributes to mesondecays at this (cid:12)p(cid:12) "(cid:18)p(cid:19) 2M12−iΓ12#
level. (cid:12) (cid:12)
(cid:12) (cid:12)
3. I(cid:12)n(cid:12)interference (of decays with and without
3. ‘Flavor changing:’ while we have good rea- mixing):
sonstothinkthatflavorchangingCPviola-
tion is dominated by Standard Model pro- qA¯
Imλ6=0 λ= . (11)
cesses, the situation could be very differ- pA
(cid:20) (cid:21)
ent for flavor diagonal CP violation. Here,
One ofthe beautifulfeaturesofCPviolationis
the KM mechanism predicts unobservably
thatexperimentscanmeasureeachofthesethree
small CP violation, while new physics can
types separately. Take for example B meson de-
dominatesuchobservablesbyseveralorders
cays:
ofmagnitude. FuturesearchesofEDMsare
1. The CP asymmetry in charged B decays is
crucial to clarify this point.
sensitive to only CP violation in decay:
3. General Lessons Γ(B− →f−)−Γ(B+ →f+)
A ≡
f∓ Γ(B− →f−)+Γ(B+ →f+)
CPviolationinmesondecaysisacomplexphe-
nomenon. This is best demonstrated by examin- |A¯f/Af|2−1
= . (12)
ing CP violation in neutral meson decay into a |A¯f/Af|2+1
finalCPeigenstate. (P standshereforanyofthe
2. The CP asymmetry in semileptonic neutral
K, D, B and B mesons.)
s B decaysissensitivetoonlyCPviolationinmix-
ing:
Γ(B¯0 →ℓ+X)−Γ(B0 →ℓ−X)
phys phys
A ≡
SL Γ(B¯0 →ℓ+X)+Γ(B0 →ℓ−X)
A1 phys phys
P0 1 fCP = 1−|q/p|4. (13)
A2 1+|q/p|4
M12 3 A(cid:22)1 3. The CP asymmetry in neutral B decays
2 1
probes separately |λ| (a combination of CP vi-
(cid:0)12 A(cid:22)2
olationinmixing andindecay)andImλ (purely
CP violation in the interference of decays with
0
P
and without mixing) [11,12,13]:
Γ(B¯0 →f )−Γ(B0 →f )
phys CP phys CP
A (t) ≡
fCP Γ(B¯0 →f )+Γ(B0 →f )
phys CP phys CP
= −C cos(∆m t)+S sin(∆m t)
Each arrow in this figure stands for an am- fCP B fCP B
plitude that carries an independent CP violating C = −A = 1−|λfCP|2,
phase. The variousinterferencesbetweenthe dif- fCP fCP 1+|λfCP|2
ferent paths from P0 to f yield three distinct 2Imλ
CP S = fCP . (14)
manifestations of CP violation: fCP 1+|λ |2
fCP
6
Note that Babar’s C corresponds to Belle’s ent language has been used to parametrize the
fCP
−A . Further note that the latter notation various CP violating observables. The transla-
fCP
suggests that this CP asymmetry is analogous tion between that language and our language is
to A measured in charged B decays. For- straightforward:
f∓
mally,A measuresdeviationsof|λ|fromunity
while AffC∓P measures deviations of |A¯/A| from ε = 1−λ0,
1+λ
one. However, the deviation of |q/p| from unity 0
is knownto be <∼O(10−2) and, giventhe present ε′ = 1(λ00−λ+−), (16)
sensitivityofsearchesforA 6=0,canbe safely 6
fCP
neglected. In practice, therefore, the analogy is where the subindex 0 refers to final two pions in
justified. an isospin-zero state, while the +− and 00 sub-
There is a class of CP asymmetries in neutral indicesreferto,respectively,π+π− andπ0π0final
meson decays into final CP eigenstates that the- states.
orists love most. It involves decays where the
direct decay amplitude is dominated by a single
weakphase,sothatCP violationindecaycanbe
neglected(|A¯/A|=1)andwherethe effectofCP
0 0
violation in mixing can be neglected (|q/p| = 1). K Re" (cid:25)(cid:25)
In this case |λ| = 1, and the only remaining CP
violating effect is SfCP 6=0. Re Im"
"
0 A 0
B KS;(cid:30)KS K
M12 SfCP
A(cid:22) The experimental values of the ε and ε′ pa-
0 rameters are given in eqs. (1) and (2). The mea-
B
surement of ε in 1964 constituted the discovery
of CP violation and drove, through the work of
KobayashiandMaskawa,tothepredictionthata
The reason that this case is the theorists’
thirdgenerationexists. Theprecisemeasurement
favorite is that the theoretical interpretation, of Re(ε′/ε) has important implications:
in terms of Lagrangian parameters, is uniquely
clean. Explicitly, the asymmetry can be ex- • Direct CP violation has been observed.
pressed purely in terms of the (CP violating)
phase difference between the mixing amplitude • The superweak scenario [17] is excluded.
and twice the decay amplitude:
• TheresultisconsistentwiththeSMpredic-
S =Imλ =±sin[arg(M∗ )−2arg(A )],(15)
fCP fCP 12 fCP tions.
where the sign depends on the CP eigenvalue of
• Largehadronicuncertaintiesmakeitimpos-
thefinalstate. Amongthefewmodesthatbelong
sible to extract a useful CKM constraint.
to this classarethe B →ψK , B →φK [14,15]
S S
This is the reason that no ε′ constraint ap-
and K →πνν¯ [16] decays.
pears in our unitarity triangles.
3.1. K →ππ
AllthreetypesofCPviolationhavebeenmea- • New physics (e.g. Supersymmetry [18])
sured in K → ππ decays. Historically, a differ- may contribute significantly.
7
3.2. B →ψK Together, eqs. (19) and (21) give
S
The B → ψK decay [14,15], related to the
S
¯b → c¯cs¯quark transition, is dominated by a sin- |λψK|=1.007±0.026. (22)
gleweak phase. An amplitude carryinga second, This is a much strongerconstraint than (17) and
differentphase,issuppressedbybothaCKMfac- allows us to confidently use |λ | = 1 from here
ψK
tor of O(λ2) and a loop factor. Consequently, on. Thisexercisecarriesanimportantlesson: the
the deviation of |A¯ψK/AψK| from unity is pre- two measurements employed here provide only
dicted to be below the percent level. It is then upper bound on asymmetries that are not sub-
expected that |λψK| = 1 to an excellent approx- ject to a clean theoretical interpretation. Yet,
imation. This golden mode belongs then to the they take us a step forwardin our understanding
‘clean’classdescribedabove: SψK =ImλψK and ofCPviolation. Indeed,eachandeverymeasure-
can be cleanly interpreted in terms of the differ- mentcontributesitsparttosolvingthebigpuzzle
ence between the phase of the B0 −B0 mixing of CP violation.
amplitude and twice the phase of the b → cc¯s What do we learn from the measurement of
decay amplitude. Imλ =0.734±0.054?
ψK
It would be nice to confirm this expectation in • CP violation has been observed in B decays.
a model independent way. The present average • The Kobayashi-Maskawa mechanism of CP
over the Babar [4] and Belle [3] measurements is violationhassuccessfullypasseditsfirstprecision
qA¯ test. Figure 2 makes a convincing case for this
ψK
|λψK|= =0.949±0.039. (17) statement.
pA
(cid:12) ψK(cid:12) • A significant constraint on the CKM param-
(cid:12) (cid:12)
The cent(cid:12)ralvalue(cid:12) is five percent awayfrom unity eters(ρ¯,η¯)hasbeenadded. Withinthe Standard
(cid:12) (cid:12)
with an error of order four percent. While this
Model,
result is certainly consistent with |λ | = 1, it
ψK
does not yet confirm it. We would like to ar- λ =∓ Vt∗bVtd VcbVc∗d =∓e−2iβ. (23)
guethatthe informationfromtwoother,entirely ψKS,L (cid:18)VtbVt∗d(cid:19)(cid:18)Vc∗bVcd(cid:19)
independent measurements constrains the devia-
The first CKM factor in (23) comes from (q/p)
tionof|λ |fromonetoamuchbetteraccuracy.
ψK and the secondfromA¯ /A . The ∓ signs are
The first measurement is that of the CP asym- ψK ψK
related to the CP eigenvalue of the final state.
metry in semileptonic B decays. The worldaver-
One obtains [11,13,23]
age over the results from Opal, Cleo, Aleph and
Babar(see[19]andreferencestherein)isgivenby 2η¯(1−ρ¯)
S =Imλ =sin2β = . (24)
ψKS ψKS η¯2+(1−ρ¯)2
A =0.002±0.014. (18)
SL
Througheq. (13)wecanconstrainthe sizeofCP This is the constraint that has been used in
violation in mixing: our unitarity triangles. Note that there are no
hadronic parameters involved in the translation
|q/p|=0.999±0.007. (19)
of the experimental value of S to an allowed
ψKS
ThesecondmeasurementisthatoftheCPasym- regionin the (ρ¯,η¯) plane. Hadronic uncertainties
metry in the charged mode, B∓ →ψK∓: arise only below the percent level.
Our CKMfit yields, for example,the following
A =0.008±0.025. (20)
ψK∓ allowed ranges (at CL>32%):
(ThisistheaverageoverCleo[20]andBabar[21]
0.12≤ρ¯≤0.35, 0.28≤η¯≤0.41,
measurements.) Through eq. (12), and using
isospin symmetry to relate this measurement to −0.82≤sin2α≤0.24, 40o ≤γ ≤73o. (25)
the neutral mode [22], we can constrain the size
• Approximate CP is excluded. Approximate
of CP violation in decay:
CP has been one of the more intriguing alterna-
|A¯ /A |=1.008±0.025. (21) tives to the KM mechanism. It assumes that ε
ψK ψK
8
and ε′ are small not because of flavor suppres- In spite of being related to a different quark
sion, as is the case with the KM mechanism, but transition, the CKM dependence of these two
because all CP violating phases are small. This modesis thesame[upto O(λ2)effects]asthatof
ideacanberealized(andiswellmotivated)within λ of eq. (23). Consequently, within the Stan-
ψK
the supersymmetric framework [24,25,26,27,28]. dard Model,
However, the observation of a CP asymmetry of
order one excludes this idea. (One can still write S ≃S ≃S . (28)
φKS η′KS ψKS
viable models of approximate CP, but these in-
volve fine-tuning.) Similarly, minimal left-right-
A difference (larger than a few percent) between
symmetricmodelswithspontaneousCPviolation the CP asymmetries in the φK or η′K modes
S S
[29,30,31] are excluded.
and in the ψK mode is a clear signal of new
S
• New, CP violating physics that contributes
physics [34]. More specifically, such a differ-
>20%to B0−B0 mixing is disfavored. As men-
encerequiresthatnew,CPviolatingphysicscon-
tioned above, it is still possible that there is a
tributes significantly to b→s transitions.
significant new physics in B0 −B0 mixing, but
Themeasurementsofthetwomodessufferfrom
the new phase and the Standard Model β (dif-
large statistical errors. At present, there is no
ferent from the Standard Model fit) conspire to
evidence (i.e., an effect ≥ 3σ) for either S 6=
give an asymmetry that is the same as predicted φKS
S or S 6= S . We conclude that, at
by the Standard Model. This situation is rather ψKS η′KS ψKS
present, there is no evidence for new physics in
unlikely, but is not rigorously excluded.
these measurements.
3.3. B →φK and B →η′K One might be tempted to interpret the 2.7σ
S S
The B →φKS and B → η′KS decays, related difference between SφKS and SψKS as a hint for
to the ¯b → s¯ss¯ quark transition, are dominated new physics. It would be rather puzzling, how-
ever, (though, perhaps, not impossible) if new,
by a single weak phase. An amplitude carrying a
CP violating physics affects B → φK in a dra-
second, different phase, is suppressed by a CKM S
factor of O(λ2). Consequently, the deviation of matic way but gives only a very small effect in
|A¯/A| from unity is predicted to be at the few B →η′KS. Furthermore, Belle has also searched
for CP violation in B →K+K−K decays (with
(< 4) percent level [32,33]. It is then expected S
∼
the φ-resonance contributions removed) [3]:
that |λ | = |λ | = 1 to a good approxima-
φK η′K
tion. Thesemodesbelongthentothe‘clean’class
describedabove: S ≃Imλandcanbecleanlyin- CK+K−KS = +0.42±0.37+−00..2023,
terpreted in terms of the difference between the S = −0.52±0.47+0.03. (29)
phaseoftheB0−B0 mixingamplitudeandtwice K+K−KS −0.27
the phase of the b→ss¯s decay amplitude.
(This is a-priori not a CP eigenstate, but a com-
Averagingoverthe new Belle [3] andBabar [4]
binationofexperimentaldataandisospinconsid-
results for the φK mode, we obtain the present
S erations allows Belle to conclude that the final
world average,
stateisdominantlyCP-even. Thelast,asymmet-
ric, error is related to the fractions of CP-even
C = +0.56±0.43,
φKS andCP-oddcomponents.) We learnthatthereis
S = −0.39±0.41. (26)
φKS no observeddeviationfrom−SK+K−KS =SψKS.
The new Belle resultsfor the η′K mode [3]read We would like to suggest then that, when spec-
S
ulating on the source of the difference between
C = −0.26±0.22, S and S , one should also explain the dif-
η′KS φKS ψKS
ference between the S and the other b→ss¯s
Sη′KS = +0.76±0.36. (27) processes, S andφ−KSS . (Of course,
η′KS K+K−KS
Thus,CPviolationinb→ss¯stransitionshasnot the large statistical errors make any conclusion
yet been observed. premature).
9
3.4. B →ψπ0 and B →D∗+D∗− we obtain
The B → ψπ0 and B → D∗+D∗− decays, re- 1+(P R /T )e+iβ
lated to the ¯b → c¯cd¯quark transition, get con- λf±(c¯cd¯d) =±e−2iβ 1+(PfRt/Tf)e−iβ . (33)
tributions from both tree and penguin diagrams, (cid:18) f t f (cid:19)
wheretheCKMcombinationsareofsimilarmag- Consequently,aviolationofeither−S =S
ψπ ψKS
nitude, O(λ3), but carry different phases. Con- or −S = S will signal direct CP viola-
(DD)+ ψKS
sequently, the deviation of |A¯/A| from unity can tionand,inparticular,a significantpenguincon-
be large. Since both C and, in particular, S can tribution to the decay. (If the violation is very
be large, CP violation in mixing can be safely strong, it might signal new physics. This state-
neglected. mentisparticularlyvalidfortheDDmode,where
thepenguincontributionisexpectedtobesmall.)
Themeasurementsofthetwomodessufferfrom
largestatisticalerrors. Atpresent,thereisnoev-
B0 C (cid:25);DD;(cid:25)(cid:25) idence for either −Sψπ 6= SψKS or −SD∗D∗ 6=
S . We conclude that, at present, there is no
ψKS
evidencefor‘penguinpollution’inthesemeasure-
S ments.
3.5. B →ππ
The B → ππ decay, related to the ¯b → u¯ud¯
B0 quark transition, gets contributions from both
treeandpenguindiagrams,wheretheCKMcom-
binations are of similar magnitude, O(λ3), but
carry different phases. Consequently, the devia-
Averagingoverthe new Belle [3] andBabar [4] tionof|A¯/A|fromunitycanbe large. Since both
results for the ψπ0 mode, we obtain the present C and,inparticular,S canbelarge,CPviolation
world average, in mixing can be safely neglected.
The results of Belle (based on 41.8 fb−1 [35]
Cψπ = +0.31±0.29, and not updated in this conference) and Babar
S = −0.46±0.49. (30) [4] for CP violation in B → ππ suffer from large
ψπ
statistical errors and are inconsistent with each
The D∗+D∗− state is not a CP eigenstate. The other. It is, therefore, more prudent at present
Babar collaboration has, however, performed an to quote the separate results rather than average
angularanalysiswhichseparatestheCP-evenand over them:
CP-oddcomponents,withthefollowingresultfor
−0.94+0.31±0.09 Belle,
the CP-even final state [4]: C = −0.25
ππ
−0.30±0.25±0.04 Babar,
(cid:26)
|λ(D∗+D∗−)+| = 0.98±0.27, Sππ = −1.21−+00..3287+−00..1163 Belle, (34)
Imλ = 0.31±0.46. (31) +0.02±0.34±0.05 Babar.
(D∗+D∗−)+ (cid:26)
Defining T and P through
Thus,CPviolationinb→cc¯dtransitionshasnot
yet been observed. A ≡T V∗V +P V∗V , (35)
ππ ππ ub ud ππ cb cd
The CKM dependence of the tree contribution
to λ in b → cc¯d transitions is the same as that we obtain
of λψK of eq. (23). Loop contributions, however, λ =e2iα 1+(Pππ/TππRu)e+iγ . (36)
modify the CKM dependence. Defining T and P ππ 1+(P /T R )e−iγ
through (cid:18) ππ ππ u (cid:19)
The (expected) violation of −S = S or
ππ ψKS
A ≡T V∗V +P V∗V , (32) of C = 0 will signal direct CP violation. At
f f cb cd f tb td ππ
10
Table 1
CP asymmetries in B →f .
CP
f b→qq¯q′ SM S C −η S 6=S ?(1)
CP CP ψK
ψK b→cc¯s sin2β +0.734±0.054 |λ|=0.95±0.04
S
φK b→ss¯s sin2β −0.39±0.41 +0.56±0.43 2.7σ
S
η′K b→ss¯s sin2β +0.76±0.36 −0.26±0.22 −
S
K+K−K (2) b→ss¯s sin2β −0.52±0.47 +0.42±0.37 −
S
ψπ0 b→cc¯d sin2β −0.46±0.49 +0.31±0.29 −
eff
D∗+D∗−(3) b→cc¯d sin2β Imλ=+0.31±0.46 |λ|=0.98±0.27 2.7σ
eff
π+π− b→uu¯d sin2α −0.48±0.60 −0.54±0.31 −
eff
(1)η =+(−)1 for CP even (odd) states.
CP
(2)Isospin analysis was used to argue that K+K−K is dominated by CP-even states.
S
(3)Angular analysis was used to separate CP-even and CP-odd D∗+D∗− states.
present,however,thereisnoevidenceforeitherof prediction of [39] ([41]).
these options. (The average of the two measure- Inthe future, measurementsofCP violationin
ments, with errors reflecting the inconsistency, is B → ρπ decays will contribute to a model inde-
C =−0.54±0.31 and S =−0.48±0.60.) pendent determination of α [42,43,44]. The first
ππ ππ
TheCPasymmetriesinB →ππdecaysareone experimental steps towards this goal have been
ofthemostinterestingmeasurementsanticipated reported in this conference [4].
in the B-factories because it can potentially de-
3.6. Summary
termine the angle α, thus providing yet another
TheresultsofthesearchesforCPasymmetries
independent constraint on the unitarity triangle.
inB decaysintofinalCPeigenstatesaresumma-
As canbe seenfrom eq. (36), if the penguin con-
rized in Table 1. We can describe the emerging
tributions werenegligible,onewouldsimplyhave
picture as follows:
Imλ =sin2α. The ratio P /T is, however,
ππ ππ ππ
non-negligible. Toproceed,onecanchooseoneof
• CP violation has not yet been observed in
the following options:
BdecaysotherthanB →ψK. (Thelargest
1. A model independent determination of α
effect is at the 2.1σ level in S .)
from isospin analysis [36]. This method requires η′KS
measurements of various branching fractions and • Direct CP violation has not yet been ob-
CP asymmetries and does not yet yield useful served in B decays. (The largest effects
constraints. are at the 2.7σ level in S −S and in
φK ψK
2. A model independent upper bound on the S +S .)
DD ψK
deviationofsin2α ≡ Imλππ fromsin2α[37,38]:
eff |λππ| • There is no evidence of new physics. (The
B(B →π0π0) largest effect that is inconsistent with the
sin2(α −α)≤ ≤0.6. (37)
eff B(B+ →π+π0) Standard Model prediction in a 2.7σ viola-
tion of S =S .)
φK ψK
Thus(α −α)intherange50o−130oisexcluded,
eff
a useful though not very strong bound. • The measurements of branching ratios and
3. A model dependent determination of α with CP asymmetries in B → ππ decays are at
a theoretical value for P /T [39,40]. In this a stage where restrictions on the CKM pa-
ππ ππ
context we would like to mention that a large rameters and on hadronic parameters be-
(small)valueof|C |would(might)giveevidence gintoemerge. Themodelindependentcon-
ππ
foralarge(small)strongphase,incontrasttothe straints are still mild.