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International Journal of Modern Physics A
c© World Scientific Publishing Company
CP VIOLATION IN BEAUTY DECAYS∗
MICHAEL GRONAU
Physics Department, Technion – Israel Institute of Technology
32000 Haifa, Israel
gronau@physics.technion.ac.il
Precision tests of the Kobayashi-Maskawa model of CP violation are discussed, pointing
out possible signatures for other sources of CP violation and for new flavor-changing
operators. The current status of the most accurate tests is summarized.
Keywords: CP violation; CKM; New Physics.
PACS Nos.: 13.25.Hw, 11.30.Er, 12.15.Ji, 14.40.Nd
1. Introduction
It took thirty-seven years from the discovery of a tiny CP violating effect of order
10−3 inKL → π+π− 1 to a first observation of a breakdown of CP symmetry outside
the strange meson system. A large CP asymmetry of order one between rates of
initial B0 and B̄0 decays to J/ψKS was measured in summer 2001 by the Babar and
Belle Collaborations.2 A sizable however smaller asymmetry had been anticipated
twenty years earlier 3 in the framework of the Kobayashi-Maskawa (KM) model
of CP violation,4 in the absence of crucial information on b quark couplings. The
asymmetry was observed in a time-dependent measurement as suggested,5 thanks
to the long B0 lifetime and the large B0-B̄0 mixing.6 The measured asymmetry,
fixing (in the standard phase convention7) the sine of the phase 2β (≡ 2φ1) ≡
2arg(VtbV
td) of the top-quark dominated B
0-B̄0 mixing amplitude, was found to
be in good agreement with other determinations of Cabibbo-Kobayasi-Maskawa
(CKM) parameters,8,9 including a recent precise measurement of Bs-B̄s mixing.
This showed that the CKM phase γ (≡ φ3) ≡ arg(V ∗ub), which seems to be unable
to account for the observed cosmological baryon asymmetry,11 is the dominant
source of CP violation in flavor-changing processes. With this confirmation the next
pressing question became whether small contributions beyond the CKM framework
occur in CP violating flavor-changing processes, and whether such effects can be
observed in beauty decays.
One way of answering this question is by over-constraining the CKM unitarity
triangle through precise CP conserving measurements related to the lengths of the
∗Based partially on review talks given at recent conferences.
http://arxiv.org/abs/0704.0076v2
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2 M. Gronau
sides of the triangle. An alternative and more direct way, focusing on the origin
of CP violation in the CKM framework, is to measure β and γ in a variety of B
decay modes. Different values obtained from asymmetries in several processes, or
values different from those imposed by other constraints, could provide clues for
new sources of CP violation and for new flavor-changing interactions. Such phases
and interactions occur in the low energy effective Hamiltonian of extensions of the
Standard Model (SM) including models based on supersymmetry.12
In this presentation we will focus on the latter approach based primarily on
CP asymmetries, using also complementary information on hadronic B decay rates
which are expected to be related to each other in the CKM framework. In the
next section we outline several of the most relevant processes and the theoretical
tools applied for their studies, quoting numerous papers where these ideas have
been originally proposed and where more details can be found.13 Sections 3, 4 and
5 describe a number of methods in some detail, summarizing at the end of each
section the current experimental situation. Section 6 discusses several tests for NP
effects, while Section 7 concludes.
2. Processes, methods and New Physics effects
Whereas testing the KM origin of CP violation in most hadronic B decays requires
separating strong and weak interaction effects, in a few “golden modes” CP asym-
metries are unaffected by strong interactions. For instance, the decay B0 → J/ψKS
is dominated by a single tree-level quark transition b̄ → c̄cs̄, up to a correction
smaller than a fraction of a percent.14,15,16,17 Thus, the asymmetries measured
in this process and in other decays dominated by b̄ → c̄cs̄ have already provided a
rather precise measurement of sin 2β,18,19,20
sin 2β = 0.678± 0.025 . (1)
This value permits two solutions for β at 21.3◦ and at 68.7◦. Time-dependent an-
gular studies of B0 → J/ψK∗0,21 and time-dependent Dalitz analyses of B0 →
Dh0 (D → KSπ+π−, h0 = π0, η, ω)22 measuring cos 2β > 0 have excluded the
second solution at a high confidence level, implying
β = (21.3± 1.0)◦ . (2)
Since B0 → J/ψKS proceeds through a CKM-favored quark transition, contribu-
tions to the decay amplitude from physics at a higher scale are expected to be
very small, potentially identifiable by a tiny direct asymmetry in this process or in
B+ → J/ψK+.23
Another process where the determination of a weak phase is not affected
by strong interactions is B+ → DK+, proceeding through tree-level amplitudes
b̄ → c̄us̄ and b̄ → ūcs̄. The interference of these two amplitudes, from D̄0 and D0
which can always decay to a common hadronic final state, leads to decay rates and
a CP asymmetry which measure very cleanly the relative phase γ between these
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CP violation in beauty decays 3
amplitudes.24,25 The trick here lies in recognizing the measurements which yield
this fundamental CP-violating quantity. Physics beyond the SM is expected to have
a negligible effect on this determination of γ which relies on the interference of two
tree amplitudes.
B decays into pairs of charmless mesons, such as B → ππ (or B → ρρ) and
B → Kπ (or B → K∗ρ), involve contributions of both tree and penguin ampli-
tudes which carry different weak and strong phases.14,26,27 Contrary to the case
of B → DK, the determination of β and γ using CP asymmetries in charmless B
decays involves two correlated aspects which must be considered: its dependence
on strong interaction dynamics and its sensitivity to potential New Physics (NP)
effects. This sensitivity follows from the CKM and loop suppression of penguin am-
plitudes, implying that new heavy particles at the TeV mass range, replacing the
W boson and the top-quark in the penguin loop, may have sizable effects.28. In
order to claim evidence for physics beyond the SM from a determination of β and
γ in these processes one must handle first the question of dynamics. There are two
approaches for treating the dynamics of charmless hadronic B decays:
(1) Study systematically strong interaction effects in the framework of QCD.
(2) Identify by symmetry observables which do not depend on QCD dynamics.
The first approach faces the difficulty of having to treat precisely long distance
effects of QCD including final state interactions. Remarkable theoretical progress
has been made recently in proving a leading-order (in 1/mb) factorization formula
for these amplitudes in a heavy quark effective theory approach to perturbative
QCD.29,30,31 However, there remain differences between ways of treating in differ-
ent approaches power counting, the scale of Wilson coefficients, end-point quark dis-
tribution functions of light mesons, and nonperturbative contributions from charm
loops.32 Also, the nonperturbative input parameters in these calculations involve
non-negligible uncertainties. These parameters include heavy-to-light form factors
at small momentum transfer, light-cone distribution amplitudes, and the average
inverse momentum fraction of the spectator quark in the B meson. The resulting
inaccuracies in calculating magnitudes and strong phases of amplitudes prohibit a
precise determination of γ from measured decay rates and CP asymmetries. Also,
the calculated rates and asymmetries cannot provide a clear case for physics be-
yond the SM in cases where the results of a calculation deviate slightly from the
measurements.
In the second approach one applies isospin symmetry to obtain relations among
several decay amplitudes. For instance, using the distinct behavior under isospin of
tree and penguin operators contributing in B → ππ, a judicious choice of observ-
ables permits a determination of γ or α (≡ φ2) = π − β − γ. 33 The same analysis
applies in B decays to pairs of longitudinally polarized ρ mesons. In case that an
observable related to the subdominant penguin amplitude is not measured with
sufficient precision, it may be replaced in the analysis by a CKM-enhanced SU(3)-
related observable, in which a large theoretical uncertainty is translated to a small
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4 M. Gronau
error in γ. The precision of this method is increased by including contributions of
higher order electroweak penguin amplitudes, which are related by isospin to tree
amplitudes.34,35 With sufficient statistics one should also take into account isospin-
breaking corrections of order (md−mu)/ΛQCD ∼ 0.02,36,37 and an effect caused by
the ρ meson width.38 A similar analysis proposed for extracting γ in B → Kπ 39,40
requires using flavor SU(3) instead of isospin for relating electroweak penguin con-
tributions and tree amplitudes.35,41 While flavor SU(3) is usually assumed to be
broken by corrections of order (ms − md)/ΛQCD ∼ 0.3, in this particular case a
rather precise recipe for SU(3) breaking is provided by QCD factorization, reducing
the theoretical uncertainty in γ to only a few degrees.42
Charmless B decays, which are sensitive to physics beyond the SM 28, provide a
rich laboratory for studying various signatures of NP. A large variety of theories have
been studied in this context, including supersymmetric models, models involving
tree-level flavor-changing Z or Z ′ couplings, models with anomalous three-gauge-
boson couplings and other models involving an enhanced chromomagnetic dipole
operator.43,44 The following effects have been studied and will be discussed in
Section 6 in a model-independent manner:
(1) Within the SM, the three values of γ extracted from B → ππ, B → Kπ and
B+ → DK+ are equal. As we will explain, these three values are expected to be
different in extensions of the SM involving new low energy four-fermion operators
behaving as ∆I = 3/2 in B → ππ and as ∆I = 1 in B → Kπ.
(2) Other signatures of anomalously large ∆I = 1 operators contributing to
B → Kπ are violations of isospin sum rules, holding in the SM for both decay rates
and CP asymmetries in these decays.45,46,47
(3) Time-dependent asymmetries in B0 → π0KS , B0 → φKS and B0 → η′KS
and in other b → s penguin-dominated decays may differ substantially from the
asymmetry sin 2β sin∆mt, predicted approximately in the SM.26,43,48 Significant
deviations are expected in models involving anomalous |∆S| = 1 operators behaving
as ∆I = 0 or ∆I = 1.
(4) An interesting question, which may provide a clue to the underlying New
Physics once deviations from SM predictions are observed, is how to diagnose the
value of ∆I in NP operators contributing to |∆S| = 1 charmless B decays. We will
discuss an answer to this question which has been proposed recently.49
3. Determining γ in B → DK
In this section we will discuss in some length a rather rich and very precise method
for determining γ in processes of the form B → D(∗)K(∗), which uses both charged
and neutral B mesons and a large variety of final states. It is based on a broad idea
that any coherent admixture of a state involving a D̄0 from b̄ → c̄us̄ and a state
with D0 from b̄ → ūcs̄ can decay to a common final state.24,25 The interference
between the two channels, B → D(∗)0K(∗), D0 → fD and B → D̄(∗)0K(∗), D̄0 →
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CP violation in beauty decays 5
fD, involves the weak phase difference γ, which may be determined with a high
theoretical precision using a suitable choice of measurements. Effects of D0-D̄0
mixing are negligible.50 While some of these processes are statistically limited,
combining them together is expected to reduce the experimental error in γ. In
addition to (quasi) two-body B decays, the D or D∗ in the final state may be
accompanied by any multi-body final state with quantum numbers of a kaon.25
Each process in this large class of neutral and charged B decays is characterized
by two pairs of parameters, describing complex ratios of amplitudes for D0 and D̄0
for the two steps of the decay chain (we use a convention rB , rf ≥ 0, 0 ≤ δB, δf <
A(B → D(∗)0K(∗))
A(B → D̄(∗)0K(∗))
= rBe
i(δB+γ) ,
A(D0 → fD)
A(D̄0 → fD)
= rfe
iδf . (3)
In three-body decays ofB andD mesons, such asB → DKπ andD → Kππ, the two
pairs of parameters (rB , δB) and (rf , δf ) are actually functions of two corresponding
Dalitz variables describing the kinematics of the above three-body decays. The
sensitivity of determining γ depends on rB and rf because this determination relies
on an interference of D0 and D̄0 amplitudes. For D decay modes with rf ∼ 1 (see
discussion below) the sensitivity increases with the magnitude of rB.
For each of the eight sub-classes of processes, B+,0 → D(∗)K(∗)+,0, one may
study a variety of final states in neutral D decays. The states fD may be divided
into four families, distinguished qualitatively by their parameters (rf , δf ) defined in
Eq. (3):
(1) fD = CP-eigenstate
24,25,51 (K+K−,KSπ
0, etc.); rf = 1, δf = 0, π.
(2) fD = flavorless but non-CP state
52 (K+K∗−,K∗+K−, etc.); rf = O(1).
(3) fD = flavor state
53 (K+π−,K+π−π0, etc.); rf ∼ tan2 θc.
(4) fD = 3-body self-conjugate state
54 (KSπ
+π−); rf , δf vary across the Dalitz
plane.
In the first family, CP-odd states occur in Cabibbo-favored D0 and D̄0 decays,
while CP-even states occur in singly Cabibbo-suppressed decays. The second family
of states occurs in singly Cabibbo-suppressed decays, the third family occurs in
Cabibbo-favored D̄0 decays and in doubly Cabibbo-suppressed D0 decays, while
the last state is formally a Cabibbo-favored mode for both D0 and D̄0.
The parameters rB and δB in B → D(∗)K(∗) depend on whether the B meson
is charged or neutral, and may differ for K vs K∗,55 and for D vs D∗, where a
neutral D∗ can be observed in D∗ → Dπ0 or D∗ → Dγ.56 The ratio rB involves
a CKM factor |VubVcs/VcbVus| ≃ 0.4 in both B+ and B0 decays, and a color-
suppression factor in B+ decays, while in B0 decays both b̄ → c̄us̄ and b̄ → ūcs̄
amplitudes are color-suppressed. A rough estimate of the color-suppression factor in
these decays may be obtained from the color-suppression measured in corresponding
CKM-favored decays, B → Dπ,D∗π,Dρ,D∗ρ, where the suppression is found to
be in the range 0.3 − 0.5.57 Thus, one expects rB(B0) ∼ 0.4, rB(B+) = (0.3 −
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6 M. Gronau
0.5)rB(B
0) in all the processes B+,0 → D(∗)K(∗)+,0. We note that three-body B+
decays, such as B+ → D0K+π0, are not color-suppressed, making these processes
advantageous by their potentially large value of rB which varies in phase space.
58,59
The above comparison of rB(B
+) and rB(B
0) may be quantified more precisely
by expressing the four ratios rB(B
0)/rB(B
+) in B → D(∗)K(∗) in terms of recip-
rocal ratios of known magnitudes of amplitudes:60
0 → D(∗)K(∗)0)
rB(B+ → D(∗)K(∗)+)
B+ → D̄(∗)0K(∗)+)
B0 → D̄(∗)0K(∗)0)
. (4)
This follows from an approximation,
A(B0 → D(∗)0K(∗)0) ≃ A(B+ → D(∗)0K(∗)+) , (5)
where the B0 and B+ processes are related to each other by replacing a spectator
d quark by a u quark. While formally Eq. (5) is not an isospin prediction, it may
be obtained using an isospin triangle relation,61
A(B0 → D(∗)0K(∗)0) = A(B+ → D(∗)0K(∗)+) +A(B+ → D(∗)+K(∗)0), (6)
and neglecting the second amplitude on the right-hand-side which is “pure
annihilation”.62 This amplitude is expected to be suppressed by a factor of four or
five relative to the other two amplitudes appearing in (6) which are color-suppressed.
Evidence for this kind of suppression is provided by corresponding ratios of CKM-
favored amplitudes,57 |A(B0 → D−s K+)/
2A(D̄0π0)| = 0.23 ± 0.03, |A(B0 →
D∗−s K
2A(D̄∗0π0)| < 0.24.
Applying Eq. (4) to measured branching ratios,57,63 one finds
rB(B+)
B → DK B → DK∗ B → D∗K B → D∗K∗
2.9± 0.4 3.7± 0.3 > 2.2 > 3.0 (7)
This agrees with values of rB(B
0) near 0.4 and rB(B
+) between 0.1 and 0.2. Note
that in spite of the expected larger values of rB in B
0 decays, from the point
of view of statistics alone (without considering the question of flavor tagging and
the efficiency of detecting a KS in B
0 → D(∗)K0), B+ and B0 decays may fare
comparably when studying γ. This follows from (5) because the statistical error on
γ scales roughly as the inverse of the smallest of the two interfering amplitudes.
We will now discuss the actual manner by which γ can be determined using
separately three of the above-mentioned families of final states fD. We will men-
tion advantages and disadvantages in each case. For illustration of the method we
will consider B+ → fDK+. We will also summarize the current status of these
measurements in all eight decay modes B+,0 → D(∗)K(∗)+,0.
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CP violation in beauty decays 7
3.1. fD = CP-eigenstates
One considers four observables consisting of two charge-averaged decay rates for
even and odd CP states, normalized by the decay rate into a D0 flavor state,
RCP± ≡
Γ(DCP±K
−) + Γ(DCP±K
Γ(D0K−)
, (8)
and two CP asymmetries for even and odd CP states,
ACP± ≡
Γ(DCP±K
−)− Γ(DCP±K+)
Γ(DCP±K−) + Γ(DCP±K+)
. (9)
In order to avoid dependence of RCP± on errors in D
0 and DCP branching ratio
measurements one uses a definition of RCP± in terms of ratios of B decay branching
ratios intoDK andDπ final states.59 The four observablesRCP± and ACP± provide
three independent equations for rB, δB and γ,
RCP± = 1 + r
B ± 2rB cos δB cos γ , (10)
ACP± = ±2rB sin δB sin γ/RCP± . (11)
While in principle this is the simplest and most precise method for extracting
γ, up to a discrete ambiguity, in practice this method is sensitive to r2B , because
(RCP+ + RCP−)/2 = 1 + r
B . This becomes very difficult for charged B decays
where one expects rB ∼ 0.1− 0.2, but may be feasible for neutral B decays where
rB ∼ 0.4. An obvious signature for a non-zero value of rB would be observing a
difference between RCP+ and RCP− which is linear in this quantity.
Studies of B+ → DCPK+, B+ → DCPK∗+ and B+ → D∗CPK+ have been car-
ried out recently,64,65,66 each consisting of a few tens of events. A nonzero difference
RCP+ −RCP− at 2.6 standard deviations, measured in B+ → DCPK∗+,64 is prob-
ably a statistical fluctuation. A larger difference is anticipated in B0 → DCPK∗0,
as the value of rB in this process is expected to be three or four times larger than
in B+ → DK∗+. [See Eq. (7).] Higher statistics is required for a measurement of γ
using this method.
3.2. fD = flavor state
Consider a flavor state fD in Cabibbo-favored D̄
0 decays, accessible also to doubly
Cabibbo-suppressed D0 decays, such that one has rf ∼ tan2 θc in Eq. (3). One
studies the ratio of two charge-averaged decay rates, for decays into f̄DK and fDK,
Γ(fDK
−) + Γ(f̄DK
Γ(f̄DK−) + Γ(fDK+)
, (12)
and the CP asymmetry,
Γ(fDK
−)− Γ(f̄DK+)
Γ(fDK−) + Γ(f̄DK+)
. (13)
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8 M. Gronau
These observables are given by
Rf = r
B + r
f + 2rB rf cos(δB − δf ) cos γ , (14)
Af = 2r rf sin(δB − δf ) sin γ/Rf , (15)
where a multiplicative correction 1 +O(rBrf ) ∼ 1.01 has been neglected in (14).
These two observables involve three unknowns, rB , δB − δf and γ. One assumes
rf to be given by the measured ratio of doubly Cabibbo-suppressed and Cabibbo-
favored branching ratios. Thus, one needs at least two flavor states, fD and f
for which two pairs of observables (Rf , Af ) and (Rf ′ , Af ′) provide four equations
for the four unknowns, rB, δB − δf , δB − δf ′ , γ. The strong phase differences δf , δf ′
can actually be measured at a ψ′′ charm factory,67 thereby reducing the number of
unknowns to three.
While the decay rate in the numerator of Rf is rather low, the asymmetry Af
may be large for small values of rB around 0.1, as it involves two amplitudes with
a relative magnitude rf/rB.
So far, only upper bounds have been measured for Rf implying upper limits
on rB in several processes, rB(B
+ → DK+) < 0.2,68,69,70 rB(B+ → D∗K+) <
0.2,68 r(B+ → DK∗+) < 0.4,71 and rB(B0 → DK∗0) < 0.4.63,72 Further con-
straints on rB in the first three processes have been obtained by studying D decays
into CP-eigenstates and into the state KSπ
+π−. Using rB(B
0 → DK∗0)/rB(B+ →
DK∗+) = 3.7 ± 0.3 in (7) and assuming that rB(B+ → DK∗+) is not smaller
than about 0.1, one may conclude that a nonzero measurement of rB(B
0 → DK∗0)
should be measured soon. The signature for B0 → D0K∗0 events would be two
kaons with opposite charges.
3.3. fD = KSπ
The amplitude for B+ → (KSπ+π−)DK+ is a function of the two invariant-mass
variables, m2
≡ (pKS + pπ±)2, and may be written as
A(B+ → (KSπ+π−)DK+) = f(m2+,m2−) + rBei(δB+γ)f(m2−,m2+) . (16)
In B− decay one replaces m+ ↔ m−, γ → −γ. The function f may be written as a
sum of about twenty resonant and nonresonant contributions modeled to describe
the amplitude for flavor-tagged D̄0 → KSπ+π− which is measured separately.73,74
This introduces a model-dependent uncertainty in the analysis. Using the measured
function f as an input and fitting the rates for B± → (KSπ+π−)DK± to the
parameters, rB , δB and γ, one then determines these three parameters.
The advantage of using D → KSπ+π− decays over CP and flavor states is
being Cabibbo-favored and involving regions in phase space with a potentially large
interference between D0 and D̄0 decay amplitudes. The main disadvantage is the
uncertainty introduced by modeling the function f .
Two recent analyses of comparable statistics by Belle and Babar, combining
B± → DK±, B± → D∗K± and B± → DK∗±, obtained values 73 γ = [53+15
−18 ± 3±
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CP violation in beauty decays 9
9(model)]◦ and γ = [92±41±11±12(model)]◦.74 [This second value does not use the
process B+ → D(KSπ+)K∗ , also studied by the same group,75.] The larger errors in
the second analysis are correlated with smaller values of the extracted parameters rB
in comparison with those extracted in the first study. The model-dependent errors
may be reduced by studying at CLEO-c the decays DCP± → KSπ+π−, providing
further information on strong phases in D decays.67
Conclusion: The currently most precise value of γ is γ = [53+15
−18 ± 3± 9(model)]◦,
obtained from B± → D(∗)K(∗)± using D → KSπ+π−. These errors may be reduced
in the future by combining the study of all D decay modes in B+,0 → D(∗)K(∗)+,0.
The decay B0 → DK∗0 seems to carry a high potential because of its expected
large value of rB. Decays B
0 → D(∗)K0 may also turn useful, as they have been
shown to provide information on γ without the need for flavor tagging of the initial
B0.60,76
4. The currently most precise determination of γ: B → ππ, ρρ, ρπ
4.1. B → ππ
The amplitude for B0 → π+π− contains two terms, conventionally denoted “tree”
(T ) and “penguin” (P ) amplitudes, 14,26 involving a weak CP-violating phase γ
and a strong CP-conserving phase δ, respectively:
A(B0 → π+π−) = |T |eiγ + |P |eiδ . (17)
Time-dependent decay rates, for an initial B0 or a B
, are given by
Γ(B0(t)/B
(t) → π+π−) = e−ΓtΓπ+π− [1± C+− cos∆mt∓ S+− sin∆mt] , (18)
where
S+− =
2Im(λππ)
1 + |λππ|2
, C+− =
1− |λππ |2
1 + |λππ |2
, λππ ≡ e−2iβ
0 → π+π−)
A(B0 → π+π−)
. (19)
One has14
S+− = sin 2α+ 2|P/T | cos2α sin(β + α) cos δ +O(|P/T |2) ,
C+− = 2|P/T | sin(β + α) sin δ +O(|P/T |2) . (20)
This tells us two things:
(1) The deviation of S+− from sin 2α and the magnitude of C+− increase with
|P/T |, which can be estimated to be |P/T | ∼ 0.5 by comparing B → ππ rates with
penguin-dominated B → Kπ rates.77
(2) Γπ+π− , S+− and C+− are insufficient for determining |T |, |P |, δ and γ (or α).
Further information on these quantities may be obtained by applying isospin sym-
metry to all B → ππ decays.
In order to carry out an isospin analysis,33 one uses the fact that the three
physical B → ππ decay amplitudes and the three B̄ → ππ decay amplitudes,
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10 M. Gronau
depending each on two isospin amplitudes, obey triangle relations of the form,
A(B0 → π+π−)/
2 +A(B0 → π0π0)−A(B+ → π+π0) = 0 . (21)
Furthermore, the penguin amplitude is pure ∆I = 1/2; hence the ∆I = 3/2 am-
plitude carries a week phase γ, A(B+ → π+π0) = e2iγA(B− → π−π0). Defin-
ing sin 2αeff ≡ S+−/(1 − C2+−)1/2, the difference αeff − α is then determined by
an angle between corresponding sides of the two isospin triangles sharing a com-
mon base, |A(B+ → π+π0)| = |A(B− → π−π0)|. A sign ambiguity in αeff − α is
resolved by two model-independent features which are confirmed experimentally,
|P |/|T | ≤ 1, |δ| ≤ π/2. This implies α < αeff .78
Table I. Branching ratios and CP asymmetries in B → ππ, B → ρρ.
Decay mode Branching ratio (10−6) ACP = −C S
B0 → π+π− 5.16± 0.22 0.38 ± 0.07 −0.61± 0.08
B+ → π+π0 5.7± 0.4 0.04 ± 0.05
B0 → π0π0 1.31± 0.21 0.36
+0.33
−0.31
B0 → ρ+ρ− 23.1
0.11 ± 0.13 −0.06± 0.18
B+ → ρ+ρ0 18.2± 3.0 −0.08± 0.13
B0 → ρ0ρ0 1.07± 0.38
Current CP-averaged branching ratios and CP asymmetries for B → ππ and
B → ρρ decays are given in Table I,20 where ACP ≡ −C for decays to CP eigen-
states. An impressive experimentally progress has been achieved in the past two
years in extracting a precise value for αeff , αeff = (110.6
−3.2)
◦. However, the er-
ror on αeff − α using the isospin triangles is still large. An upper bound, given by
CP-averaged rates and a direct CP asymmetry in B0 → π+π−,79,80
cos 2(αeff − α) ≥
Γ+− + Γ+0 − Γ00
)2 − Γ+−Γ+0
Γ+−Γ+0
1− C2+−
, (22)
leads to 0 < αeff − α < 31◦ at 1σ. Adding in quadrature the error in αeff and the
uncertainty in α−αeff , this implies α = (95± 16)◦ or γ = (64± 16)◦ by . A similar
central value but a smaller error, α = (97± 11)◦, has been reported recently by the
Belle Collaboration.81 The possibility that a penguin amplitude in B0 → π+π−
may lead to a large CP asymmetry S for values of α near 90◦ where sin 2α = 0 was
anticipated fifteen years ago.82
The bound on αeff − α may be improved considerably by measuring a nonzero
direct CP asymmetry in B0 → π0π0. This asymmetry can be shown to be large and
positive (see Eq. (46) in Sec. 5.2), implying a large rate for B̄0 but a small rate for
B0. Namely, the triangle (21) is expected to be squashed, while the B̄ triangle is
roughly equal sided.
An alternative way of treating the penguin amplitude in B0 → π+π− is by
combining within flavor SU(3) the decay rate and asymmetries in this process with
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CP violation in beauty decays 11
rates and asymmetries in B0 → K0π+ or B0 → K+π−.77 The ratio of ∆S = 1 and
∆S = 0 tree amplitudes in these processes, excluding CKM factors, is taken to be
given by fK/fπ assuming factorization, while the ratio of corresponding penguin
amplitudes is allowed to vary by ±0.22 around one. A current update of this rather
conservative analysis obtains 83
γ = (73± 4+10
◦ , (23)
where the first error is experimental, while the second one is due to an uncertainty
in SU(3) breaking. A discussion of SU(3) breaking factors relating B0 → π+π− and
B0 → K+π− is included in Section 5.2.
4.2. B → ρρ
Angular analyses of the pions in ρ decays have shown that B0 → ρ+ρ− is dominated
almost 100% by longitudinal polarization 20. This simplifies the isospin analysis of
CP asymmetries in these decays to becoming similar to B0 → π+π−. The advantage
of B → ρρ over B → ππ is the relative small value of (
ρ0ρ0) in comparison with
ρ+ρ−) and (
ρ+ρ0) (see Table I), indicating a smaller |P/T | in B → ρ+ρ− (|P/T | <
0.3 8) than in B0 → π+π− (|P/T | ∼ 0.5 77). Eq. (22) leads to an upper bound on
αeff − α in B → ρρ, 0 < αeff − α < 17◦ (at 1σ). The asymmetries for longitudinal
ρ’s given in Table I imply αeff = (91.7
−5.2)
◦. Thus, one finds α = (83 ± 10)◦ or
γ = (76± 10)◦ by adding errors in quadrature.
A stronger bound on |P/T | in B0 → ρ+ρ−, leading to a more precise value of γ,
may be obtained by relating this process to B+ → K∗0ρ+ within flavor SU(3). 84
One uses the branching ratio and fraction of longitudinal rate measured for this
process 20, (
K∗0ρ+) = (9.2 ± 1.5) × 10−6, fL(K∗0ρ+) = 0.48 ± 0.08, to normalize
the penguin amplitude in B0 → ρ+ρ−. Including a conservative uncertainty from
SU(3) breaking and smaller amplitudes, one finds a value
γ = (71.4+5.8
−1.7)
◦ , (24)
where the first error is experimental and the second one theoretical.
The current small theoretical error in γ requires including isospin breaking effects
in studies based on isospin symmetry. The effect of electroweak penguin amplitudes
on the isospin analyses of B → ππ and B → ρρ has been calculated and was found
to move γ slightly higher by an amount ∆γEWP = 1.5
◦.34,35 Other corrections,
relevant to methods using π0 and ρ0, includng π0-η-η′ mixing, ρ-ω mixing, and a
small I = 1 ρρ contribution allowed by the ρ-width, are each smaller than one
degree.36,37,38
Conclusion: Taking an average of the two values of γ in (23) and (24) obtained
from B0 → π+π− and B0 → ρ+ρ−, and including the above-mentioned EWP
correction, one finds
γ = (73.5± 5.7)◦ . (25)
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12 M. Gronau
A third method of measuring γ (or α) in time-dependent Dalitz analyses of B0 →
(ρπ)0 involves a much larger error,85 and has a small effect on the overall averaged
value of the weak phase. We note that sin γ is close to one and its relative error
is only 3%, the same as the relative error in sin 2β and slightly smaller than the
relative error in sinβ.
5. Rates, asymmetries, and γ in B → Kπ
5.1. Extracting γ in B → Kπ
The four decays B0 → K+π−, B0 → K0π0, B+ → K0π+, B+ → K+π0 involve
a potential for extracting γ, provided that one is sensitive to interference between
a dominant isoscalar penguin amplitude and a small tree amplitude contributing
to these processes. This idea has led to numerous suggestions for determining γ in
these decays starting with a proposal made in 1994.86,87 An interference between
penguin and tree amplitudes may be identified in two ways:
(1) Two different properly normalized B → Kπ rates.
(2) Nonzero direct CP asymmetries.
Table II. Branching ratios and asymmetries in B → Kπ.
Decay mode Branching ratio (10−6) ACP
B0 → K+π− 19.4± 0.6 −0.097± 0.012
B+ → K+π0 12.8± 0.6 0.047± 0.026
B+ → K0π+ 23.1± 1.0 0.009± 0.025
B0 → K0π0 10.0± 0.6 −0.12± 0.11
Current branching ratios and CP asymmetries are summarized in Table II.20 Three
ratios of rates, calculated using the ratio of B+ and B0 lifetimes, τ+/τ0 = 1.076±
0.008,20 are:
R ≡ Γ(B
0 → K+π−)
Γ(B+ → K0π+)
= 0.90± 0.05 ,
2Γ(B+ → K+π0)
Γ(B+ → K0π+)
= 1.11± 0.07 ,
Γ(B0 → K+π−)
2Γ(B0 → K0π0) = 0.97± 0.07 . (26)
The largest deviation from one, observed in the ratio R at 2σ, is insufficient for
claiming unambiguous evidence for a non-penguin contribution. An upper limit,
R < 0.965 at 90% confidence level, would imply γ ≤ 79◦ using sin2 γ ≤ R,88
which neglects however “color-suppressed” EWP contributions.89 As we will argue
now, these contributions and “color-suppressed” tree amplitudes are actually not
suppressed as naively expected.
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CP violation in beauty decays 13
The nonzero asymmetry measured in B0 → K+π− provides first evidence for
an interference between penguin (P ′) and tree (T ′) amplitudes with a nonzero rel-
ative strong phase. Such an interference occurs also in B+ → K+π0 where no
asymmetry has been observed. An assumption that other contributions to the lat-
ter asymmetry are negligible has raised some questions about the validity of the
CKM framework. In fact, a color-suppressed tree amplitude (C′), also occurring in
B+ → K+π0,86 resolves this “puzzle” if this amplitude is comparable in magnitude
to T ′. Indeed, several studies have shown that this is the case,90,91,92,93,94 also im-
plying that color-suppressed and color-favored EWP amplitudes are of comparable
magnitudes.35 For consistency between the two CP asymmetries in B0 → K+π−
and B+ → K+π0, the strong phase difference between C′ and T ′ must be negative
and cannot be very small.95 This seems to stand in contrast to QCD calculations
using a factorization theorem.29,31,94
The small asymmetry ACP (B
+ → K+π0) implies bounds on the sine of the
strong phase difference δc between T
′ +C′ and P ′. The cosine of this phase affects
Rc − 1 involving the decay rates for B+ → K0π+ and B+ → K0π+. A question
studied recently is whether the two upper bounds on | sin δc| and | cos δc| are con-
sistent with each other or, perhaps, indicate effects of NP. Consistency was shown
by proving a sum rule involving ACP (B
+ → K+π0) and Rc − 1, in which an elec-
troweak penguin (EWP) amplitude plays an important role. We will now present a
proof of the sum rule, which may provide important information on γ.95
The two amplitudes for B+ → K0π+,K+π0 are given in terms of topological
contributions including P ′, T ′ and C′,
A(B+ → K0π+) = (P ′ − 1
P ′cEW ) +A
A(B+ → K+π0) = (P ′ − 1
P ′cEW ) + (T
′ + P ′cEW ) + (C
′ + P ′EW ) +A
′ , (27)
where P ′EW and P
EW are color-favored and color-suppressed EWP contributions.
The small annihilation amplitude A′ and a small u quark contribution to P ′ involv-
ing a CKM factor V ∗ubVus will be neglected (|V ∗ubVus|/|V ∗cbVcs| = 0.02). Evidence for
the smallness of these terms can be found in the small CP asymmetry measured for
B+ → K0π+. Large terms would require rescattering and a sizable strong phase
difference between these terms and P ′.
Flavor SU(3) symmetry relates ∆I = 1, I(Kπ) = 3/2 electroweak penguin and
tree amplitudes through a calculable ratio δEW
35,41,
T ′ + C′ + P ′EW + P
EW = (T
′ + C′)(1 − δEW e−iγ) ,
δEW = −
c9 + c10
c1 + c2
|V ∗tbVts|
|V ∗ubVus|
= 0.60± 0.05 . (28)
The error in δEW is dominated by the current uncertainty in |Vub|/|Vcb| = 0.104±
0.007 57, including also a smaller error from SU(3) breaking estimated using QCD
factorization. Eqs. (27) and (28) imply 96
Rc = 1− 2rc cos δc(cos γ − δEW) + r2c (1− 2δEW cos γ + δ2EW) , (29)
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14 M. Gronau
ACP (B
+ → K+π0) = −2rc sin δc sin γ/Rc , (30)
where rc ≡ |T ′ + C′|/|P ′ − 13P
EW | and δc is the strong phase difference between
T ′ + C′ and P ′ − 1
P ′cEW .
The parameter rc is calculable in terms of measured decay rates, using bro-
ken flavor SU(3) which relates T ′ + C′ and T + C dominating B+ → π+π0 by a
factorization factor fK/fπ (neglecting a tiny EWP term in B
+ → π+π0),87
|T ′ + C′| =
|A(B+ → π+π0)| . (31)
Using branching ratios from Tables I and II, one finds
B+ → π+π0)
B+ → K0π+)
= 0.198± 0.008 . (32)
The error in rc does not include an uncertainty from assuming factorization for
SU(3) breaking in T ′ + C′. While this assumption should hold well for T ′, it may
not be a good approximation for C′ which as we have mentioned is comparable in
magnitude to T ′ and carries a strong phase relative to it. Thus one should allow a
10% theoretical error when using factorization for relating B → Kπ and B → ππ
T + C amplitudes, so that
rc = 0.20± 0.01 (exp)± 0.02 (th) . (33)
Eliminating δc in Eqs. (29) and (30) by retaining terms which are linear in rc,
one finds
Rc − 1
cos γ − δEW
ACP (B
+ → K+π0)
sin γ
= (2rc)
2 +O(r3c ) . (34)
This sum rule implies that at least one of the two terms whose squares occur on
the left-hand-side must be sizable, of the order of 2rc = 0.4. The second term,
|ACP (B+ → K+π0)|/ sin γ, is already smaller than ≃ 0.1, using the current 2σ
bounds on γ and |ACP (B+ → K+π0)|. Thus, the first term must provide a dominant
contribution. For Rc ≃ 1, this implies γ ≃ arccos δEW ≃ (53.1± 3.5)◦. This range is
expanded by including errors in Rc and ACP (B
+ → K+π0). For instance, an upper
bound Rc < 1.1 would imply an inportant upper limit, γ < 70
◦. Currently one only
obtains an upper limit γ ≤ 88◦ at 90% confidence level.95 This bound is consistent
with the value obtained in (25) from B → ππ and B → ρρ, but is not competitive
with the latter precision.
Conclusion: The current constraint obtained from Rc and ACP (B
+ → K+π0) is
γ ≤ 88◦ at 90% confidence level. Further improvement in the measurement of Rc
(which may, in fact, be very close to one) is required in order to achieve a precision
in γ comparable to that obtained in B → ππ, ρρ. (A conclusion concerning the
different CP asymmetries measured in B0 → K+π− and B+ → K+π0 will be given
at the end of the next subsection.)
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CP violation in beauty decays 15
5.2. Symmetry relations for B → Kπ rates and asymmetries
The following two features imply rather precise sum rules in the CKM framework,
both for B → Kπ decay rates and CP asymmetries:
(1) The dominant penguin amplitude is ∆I = 0.
(2) The four decay amplitudes obey a linear isospin relation,39
A(K+π−)−A(K0π+)−
2A(K+π0) +
2A(K0π0) . (35)
An immediate consequence of these features are two isospin sum rules, which
hold up to terms which are quadratic in small ratios of non-penguin to penguin
amplitudes,45,46,47
Γ(K+π−) + Γ(K0π+) = 2Γ(K+π0) + 2Γ(K0π0) , (36)
∆(K+π−) + ∆(K0π+) = 2∆(K+π0) + 2∆(K0π0) , (37)
where
∆(Kπ) ≡ Γ(B̄ → K̄π̄)− Γ(B → Kπ) . (38)
Quadratic corrections to (36) have been calculated in the SM and were found to
be a few percent.97,98,99 This is the level expected in general for isospin-breaking
corrections which must therefore also be considered. The above two features imply
that these ∆I = 1 corrections are suppressed by a small ratio of non-penguin to
penguin amplitudes and are therefore negligible.100 Indeed, this sum rule holds
experimentally within a 5% error.101 One expects the other sum rule (37) to hold
at a similar precision.
The CP rate asymmetry sum rule (37), relating the four CP asymmetries, leads
to a prediction for the asymmetry in B0 → K0π0 in terms of the other three
asymmetries which have been measured with higher precision,
ACP (B
0 → K0π0) = −0.140± 0.043 . (39)
While this value is consistent with experiment (see Table II), higher accuracy in
this asymmetry measurement is required for testing this straightforward prediction.
Relations between CP asymmetries in B → Kπ and B → ππ following from
approximate flavor SU(3) symmetry of QCD 102 are not expected to hold as pre-
cisely as isospin relations, but may still be interesting and useful. An important
question relevant to such relations is how to include SU(3)-breaking effects, which
are expected to be at a level of 20-30%. Here we wish to discuss two SU(3) rela-
tions proposed twelve years ago,103,104 one of which holds experimentally within
expectation, providing some lesson about SU(3) breaking, while the other has a an
interesting implication for future applications of the isospin analysis in B → ππ.
A most convenient proof of SU(3) relations is based on using a diagramatic
approach, in which diagrams with given flavor topologies replace reduced SU(3)
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16 M. Gronau
matrix elements.86 In this language, the amplitudes for B0 decays into pairs of
charged or neutral pions, and pairs of charged or neutral π and K, are given by:
−A(B0 → π+π−) = T + (P + 2P cEW /3) + E + PA ,
2A(B0 → π0π0) = C − (P − PEW − P cEW /3)− E − PA ,
−A(B0 → K+π−) = T ′ + (P ′ + 2P ′cEW /3) ,
2A(B0 → K0π0) = C′ − (P ′ − P ′EW − P ′cEW /3) . (40)
The combination E + PA, representing exchange and penguin annihilation topolo-
gies, is expected to be 1/mb-suppressed relative to T and C,
31,62 as demonstrated
by the small branching ratio measured for B0 → K+K−.20 This term will be
neglected.
Expressing topological amplitudes in terms of CKM factors, SU(3)-invariant
amplitudes and SU(3) invariant strong phases, one may write
T ≡ V ∗ubVud|T + Puc| , P + 2P cEW /3 ≡ V ∗tbVtd|Ptc|eiδ ,
T ′ ≡ V ∗ubVus|T + Puc| , P ′ + 2P ′cEW /3 ≡ V ∗tbVts|Ptc|eiδ , (41)
C ≡ V ∗ubVud|C − Puc| , P − PEW − P cEW /3 ≡ V ∗tbVtd|P̃tc|eiδ̃ ,
C′ ≡ V ∗ubVus|C − Puc| , P ′ − P ′EW − P ′cEW /3 ≡ V ∗tbVts|P̃tc|eiδ̃ .
Unitarity of the CKM matrix, V ∗cbVcd(s) = −V ∗tbVtd(s) − V ∗ubVud(s), has been used
to absorb in T (
′) and C(
′) a penguin term Puc ≡ Pu − Pc multiplying V ∗ubVud(s),
while Ptc ≡ Pt − Pc and P̃tc ≡ P̃t − P̃c contain two distinct combinations of EWP
contributions. Using the identity
Im (V ∗ubVudVtbV
td) = −Im (V ∗ubVusVtbV ∗ts) , (42)
one finds103,104
∆(B0 → K+π−) = −∆(B0 → π+π−) (43)
∆(B0 → K0π0) = −∆(B0 → π0π0) , (44)
where ∆ is the CP rate difference defined in (38).
Quoting products of branching ratios and asymmetries from Tables I and II,
Eq. (43) reads
− 1.88± 0.24 = −1.96± 0.37 . (45)
This SU(3) relation works well and requires no SU(3)-breaking. An SU(3) breaking
factor fK/fπ in T but not in P , or in both T and P , are currently excluded at a
level of 1.0σ, or 1.75σ. More precise CP asymmetry measurements in B0 → K+π−
and B0 → π+π− are required for determining the pattern of SU(3) breaking in tree
and penguin amplitudes.
Using the prediction (39) of the B → Kπ asymmetry sum rule, Eq. (44) predicts
ACP (B
0 → π0π0) = 1.07± 0.38 . (46)
The error is dominated by current errors in CP asymmetries for B+ → K0π+
and B+ → K+π0, and to a less extent by the error in (
π0π0). SU(3) breaking in
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CP violation in beauty decays 17
amplitudes could modify this prediction by a factor fπ/fK if this factor applies to
C, and less likely by (fπ/fK)2. A large positive CP asymmetry, favored in all three
cases, will affect future applications of the isospin analysis in B → ππ. It implies
that while the B̄ isospin triangle is roughly equal-sided, the B triangle is squashed.
A twofold ambiguity in the value of γ disappears in the limit of a flat B triangle.24
Conclusion: The isospin sum rule for B → Kπ decay rates holds well, while the
CP asymmetry sum rule predicts ACP (B
0 → K0π0) = −0.140±0.043. The different
asymmetries in B0 → K+π− and B+ → K+π0 can be explained by an amplitude
C′ comparable to T ′ and involving a relative negative strong phase, and should
not be considered a “puzzle”. An SU(3) relation for B0 → ππ and B0 → Kπ CP
asymmetries works well for charged modes. The corresponding relation for neutral
modes predicts a large positive asymmetry in B0 → π0π0. Improving asymmetry
measurements can provide tests for SU(3) breaking factors.
6. Tests for small New Physics effects
6.1. Values of γ
We have described three ways for extracting a value for γ relying on interference
of distinct pairs of quark amplitudes, (b → cūs, b → uc̄s), (b → cc̄s, b → uūs) and
(b → cc̄d, b → uūd). The three pairs provide a specific pattern for CP violation in
the CKM framework, which is expected to be violated in many extensions of the
SM. The rather precise value of γ (25) extracted from B → ππ, ρρ, ρπ is consistent
with constraints on γ from CP conserving measurements related to the sides of the
unitarity triangle.8,9 The values of γ obtained in B → D(∗)K(∗) and B → Kπ
are consistent with those extracted in B → ππ, ρρ, ρπ, but are not yet sufficiently
precise for testing small NP effects in charmless B decays. Further experimental
improvements are required, in particular in the former two types of processes.
While the value of γ in B → D(∗)K∗) is not expected to be affected by NP,
the other two classes of processes involving penguin loops are susceptible to such
effects. The extraction of γ in B → ππ ρρ assumes that γ is the phase of a ∆I =
3/2 tree amplitude, while an additional ∆I = 3/2 EWP contribution is included
using isospin. The extracted value could be modified by a new ∆I = 3/2 effective
operator originating in physics beyond the SM, but not by a new ∆I = 1/2 operator.
Similarly, the value of γ extracted in B → Kπ is affected by a potential new ∆I = 1
operator, but not by a new ∆I = 0 operator, because the amplitude (28), playing
an essential role in this method, is pure ∆I = 1.
6.2. B → Kπ sum rule
Charmless |∆S| = 1 B and Bs decays are particularly sensitive to NP effects, as
new heavy particles at the TeV mass range may replace the the W boson and top-
quark in the penguin loop dominating these amplitudes.28 The sum rule (36) for
B → Kπ decay rates provides a test for such effects. However, as we have argued
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
18 M. Gronau
from isospin considerations, it is only affected by quadratic ∆I = 1 amplitudes
including NP contributions. Small NP amplitudes, contributing quadratically to
the sum rule, cannot be separated from SM corrections, which are by themselves at
a level of a few percent. This is the level to which the sum rule has already been
tested. We will argue below for evidence showing that potential NP contributions
to |∆S| = 1 charmless decays must be suppressed by roughly an order of magnitude
relative to the dominant b→ s penguin amplitudes.
6.3. Values of S,C in |∆S| = 1 B0 → fCP decays
A class of b → s penguin-dominated B0 decays to CP-eigenstates has recently at-
tracted considerable attention. This includes final statesXKS andXKL, whereX =
φ, π0, η′, ω, f0, ρ
0,K+K−,KSKS , π
0π0, for which measured asymmetries −ηCPS
and C are quoted in Table III. [The asymmetries S and C = −ACP were de-
fined in (18) for B0 → π+π−. Observed modes with KL in the final states obey
ηCP (XKL) = −ηCP (XKS).] In these processes, a value S = −ηCP sin 2β (for states
Table III. Asymmetries S and C in B0 → XKS .
X φ π0 η′ ω f0(980)
−ηCP S 0.39± 0.18 0.33± 0.21 0.61± 0.07 0.48± 0.24 0.42 ± 0.17
C 0.01± 0.13 0.12± 0.11 −0.09± 0.06 −0.21± 0.19 −0.02± 0.13
X ρ0 K+K− KSKS π
−ηCP S 0.20± 0.57 0.58
+0.18
−0.13
0.58± 0.20 −0.72± 0.71
C 0.64± 0.46 0.15± 0.09 −0.14± 0.15 0.23± 0.54
with CP-eigenvalue ηCP ) is expected approximately.
26,43 These predictions involve
hadronic uncertainties at a level of several percent, of order λ2, λ ∼ 0.2. It has been
pointed out some time ago105 that it is difficult to separate these hadronic uncer-
tainties within the SM from NP contributions to decay amplitudes if the latter are
small. In the next subsection we will discuss indirect experimental evidence showing
that NP contributions to S and C must be small. Corrections to S = −ηCP sin 2β
and values for the asymmetries C have been calculated in the SM using methods
based on QCD factorization106,107 and flavor SU(3),90,108,109 and were found to
be between a few percent up to above ten percent within hadronic uncertainties.
Whereas the deviation of S from −ηCP sin 2β is process-dependent, a generic
result has been proven a long time ago for both S and C, to first order in |c/p|,14
∆S ≡ −ηCPS − sin 2β = 2
cos 2β sin γ cos∆ ,
C = 2
sin γ sin∆ . (47)
Here p and c are penguin and color-suppressed tree amplitudes involving a small ra-
tio and relative weak and strong phases γ and ∆, respectively. This implies ∆S > 0
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CP violation in beauty decays 19
for |∆| < π/2, which can be argued for several of the above decays using QCD
arguments106,107 or SU(3) fits.109 (Note that while |p| is measurable in certain
decay rates up to first order corrections, |c| and ∆ involve sizable hadronic uncertain-
ties in QCD calculations.) In contrast to this expectation, the central values mea-
sured for ∆S are negative for all decays. (See Table III.) Consequently, one finds an
averaged value sin 2βeff = 0.53±0.05,20 to be compared with sin 2β = 0.678±0.025.
Two measurements which seem particularly interesting are−ηCPSφKS = 0.39±0.18,
where a positive correction of a few percent to sin 2β is expected in the SM,106,107
and −ηCPSπ0KS = 0.33± 0.21, where a rather large positive correction to sin 2β is
expected shifting this asymmetry to a value just above 0.8.90
While the current averaged value of sin 2βeff is tantalizing, experimental errors
in S and C must be reduced further to make a clear case for physics beyond the
SM. Assuming that the discrepancy between improved measurements and calcu-
lated values of S and C persists beyond theoretical uncertainties, can this pro-
vide a clue to the underlying New Physics? Since many models could give rise to
a discrepancy,28,43,44 one would seek signatures characterizing classes of models
rather than studying the effects in specific models. One way of classifying extensions
of the SM is by the isospin behavior of the new effective operators contributing to
b→ sqq̄ transitions.
6.4. Diagnosis of ∆I for New Physics operators
Four-quark operators in the effective Hamiltonian associated with NP in b → sqq̄
transitions can be either isoscalar or isovector operators. We will now discuss a study
proposed recently in order to isolate ∆I = 0 or ∆I = 1 operators, thus determining
corresponding NP amplitudes and CP violating phases.49 We will show that since S
and C in the above processes combine ∆I = 0 or ∆I = 1 contributions, separating
these contributions requires using also information from other two asymmetries,
which are provided by isospin-reflected decay processes.
Two |∆S| = 1 charmless B (or Bs) decay processes, related by isospin reflection,
RI : u ↔ d, ū ↔ −d̄, can always be expressed in term of common ∆I = 0 and
∆I = 1 amplitudes B and A in the form:
A(B+ → f) = B +A , A(B0 → RIf) = ±(B −A) . (48)
A proof of this relation uses a sign change of Clebsch-Gordan coefficients underm↔
−m.49 The description (48) applies, in particular, to pairs of processes involving
all the B0 decay modes listed in Table III, and B+ decay modes where final states
are obtained by isospin reflection from corresponding B0 decay modes. Decay rates
for pairs of isospin-reflected B decay processes, and for B̄ decays to corresponding
charge conjugate final states are therefore given by (we omit inessential common
kinematic factors),
Γ+ = |B +A|2 , Γ0 = |B − A|2 ,
Γ− = |B̄ + Ā|2 , Γ0̄ = |B̄ − Ā|2 . (49)
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20 M. Gronau
The amplitudes B̄ and Ā are related to B and A by a change in sign of all weak
phases, whereas strong phases are left unchanged.
For each pair of processes one defines four asymmetries: an isospin-dependent
CP-conserving asymmetry,
Γ+ + Γ− − Γ0 − Γ0̄
Γ+ + Γ− + Γ0 + Γ0̄
, (50)
two CP-violating asymmetries for B+ and B0,
A+CP ≡
Γ− − Γ+
Γ− + Γ+
, − C ≡ A0CP ≡
Γ0̄ − Γ0
Γ0̄ + Γ0
, (51)
and the time-dependent asymmetry S in B0 decays,
1 + |λ|2
, λ ≡ ηCP
B̄ − Ā
e−2iβ , (52)
In the Standard Model, the isoscalar amplitude B contains a dominant penguin
contribution, BP , with a CKM factor V
cbVcs. The residual isoscalar amplitude,
∆B ≡ B −BP , (53)
and the amplitude A, consist each of contributions smaller than BP by about an
order of magnitude.29,30,31,32,86 These contributions include terms with a much
smaller CKM factor V ∗ubVus, and a higher order electroweak penguin amplitude with
CKM factor V ∗tbVts. Thus, one expects
|∆B| ≪ |BP | , |A| ≪ |BP | . (54)
Consequently, the asymmetries AI , A
CP and ∆S are expected to be small, of or-
der 2|A|/|B| and 2|∆B|/|BP |. In contrast, potentially large contributions to ∆B
and A from NP, comparable to BP , would most likely lead to large asymmetries
of order one. An unlikely exception is the case when both ∆B/BP and A/BP are
purely imaginary, or almost purely imaginary. This would require very special cir-
cumstances such as fine-tuning in specific models. Excluding cancellations between
NP and SM contributions in both CP-conserving and CP violating asymmetries,
tests for the hierarchy (54) become tests for the smallness of corresponding potential
NP contributions to B and A.
There exists ample experimental information showing that asymmetries A+CP
are small in processes related by isospin reflection to the decay modes in Table III.
Upper limits on the magnitudes of most asymmetries are at a level of ten or fifteen
percent [e.g., A+CP (K
+φ) = 0.034±0.044,A+CP (K+η′) = 0.031±0.026], while others
may be as large as twenty or thirty percent [A+CP (K
+ρ0) = 0.31+0.11
−0.10]. Similar values
have been measured for isospin asymmetries AI [e.g., AI(K
+φ) = −0.037± 0.077,
+η′) = −0.001± 0.033, AI(K+ρ0) = −0.16± 0.10].49 Since these two types
of asymmetries are of order 2|∆B|/|BP | and 2|A|/|BP |, this confirms the hierarchy
(54), which can be assumed to hold also in the presence of NP.
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
CP violation in beauty decays 21
We will take by convention the dominant penguin amplitude BP to have a zero
weak phase and a zero strong phase, referring all other strong phases to it. Writing
B = BP +∆B , B̄ = BP +∆B̄ , (55)
and expanding the four asymmetries to leading order in ∆B/BP or A/BP , one has
∆S = cos 2β
Im(Ā−A)
− Im(∆B̄ −∆B)
, (56)
Re(Ā+A)
, (57)
A+CP =
Re(Ā−A)
Re(∆B̄ −∆B)
, (58)
A0CP = −
Re(Ā−A)
Re(∆B̄ −∆B)
. (59)
The four asymmetries provide the following information:
• The ∆I = 0 and ∆I = 1 contributions in CP asymmetries are separated
by taking sums and differences,
A∆I=0CP ≡
(A+CP +A
CP ) =
Re(∆B̄ −∆B)
, (60)
A∆I=1CP ≡
(A+CP −A
CP ) =
Re(Ā−A)
. (61)
• ReA/BP and ReĀ/BP may be separated by using information from A∆I=1CP
and AI .
• ∆S is governed by an imaginary part of a combination of ∆I = 0 and ∆I =
1 terms which cannot be separated in B decays. Such a separation is possible
in Bs decays to pairs of isospin-reflected decays, e.g. Bs → K+K−,KSKS
or Bs → K∗+K∗−,K∗0K̄∗0, where 2β in the definition of ∆S (47) is now
replaced by the small phase of Bs-B̄s mixing.
One may take one step further under the assumption that strong phases as-
sociated with NP amplitudes are small relative to those of the SM and can be
neglected.110 This assumption, which must be confronted by data, is reasonable
because rescattering from a leading b → scc̄ amplitude is likely the main source
of strong phases, while rescattering from a smaller b → sqq̄ NP amplitude is then
a second-order effect. In the convention (55), where the strong phase of BP is set
equal to zero, ∆B and A have the same CP-conserving strong phase δ, and involve
CP-violating phases φB and φA, respectively,
∆B = |∆B|eiδeiφB , A = |A|eiδeiφA . (62)
Since the four asymmetries (56)-(59) are first order in small ratios of amplitudes,
one may take BP in their expression to be given by the square root of Γ+ or Γ0,
thereby neglecting second order terms. These four observables can then be shown to
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
22 M. Gronau
determine |A|, φA and |∆B| sinφB .49 The combination |∆B| cosφB adds coherently
to BP and cannot be fixed independently.
The amplitudes ∆B and A consist of process-dependent SM and potential NP
contributions. Assuming that the former are calculable, either using methods based
on QCD-factorization or by fitting within flavor SU(3) these and other B decay
rates and asymmetries, the four asymmetries determine the magnitude and CP
violating phase of a ∆I = 1 NP amplitude and the imaginary part of a ∆I = 0 NP
amplitude. In certain cases, e.g., B → φK or B → η′KS , stringent upper bounds on
SM contributions to ∆B and Amay suffice if some of the four measured asymmetries
are larger than permitted by these bounds. In the pair B+ → K+π0, B0 → K0π0,
the four measured asymmetries [using the predicted value (39)] are AI = 0.087 ±
0.038, A∆I=0CP = −0.047± 0.025, A∆I=1CP = 0.094± 0.025,∆S = −0.35± 0.21. Some
reduction of errors is required for a useful implementation of this method.
Conclusion: There exists ample experimental evidence in pairs of isospin-reflected
b → s penguin-dominated decays that potential NP amplitudes must be small.
Assuming that these amplitudes involve negligible strong phases, and assuming that
small SM non-penguin contributions are calculable or can be strictly bounded, one
may determine the magnitude and CP violating phase of a NP ∆I = 1 amplitude,
and the imaginary part of a NP ∆I = 0 amplitude in each pair of isospin-reflected
decays.
6.5. Null or nearly-null tests
We have not discussed null tests of the CKM framework.111 Evidence for physics
beyond the Standard Model may show-up as (small) nonzero asymmetries in pro-
cesses where they are predicted to be extremely small in the CKM framework. A
well-known example is B+ → π+π0, where the CP asymmetry is expected to be a
small fraction of a percent including EWP amplitudes.34,35 We have only discussed
exclusive hadronic B decays, where QCD calculations involve hadronic uncertain-
ties. A more robust calculation exists for the direct CP asymmetry in inclusive
radiative decays B → Xsγ, found to be smaller than one percent.112 The current
upper limit on this asymmetry is at least an order of magnitude larger.113
Time-dependent asymmetries in radiative decays B0 → KSπ0γ, for a KSπ0
invariant-mass in the K∗ region and for a larger invariant-mass range including
this region, are interesting because they test the photon helicity, predicted to be
dominantly right-handed in B0 decays and left-handed in B̄0 decays.105,114 The
asymmetry, suppressed by ms/mb, is expected to be several percent in the SM,
and can be very large in extensions where spin-flip is allowed in b → sγ. While
dimensional arguments seem to indicate a possible larger asymmetry in the SM,
of order ΛQCD/mb ∼ 10%,115 calculations using perturbative QCD116 and QCD
factorization117 find asymmetries of a few percent. The current averaged values,
for the K∗ region and for a larger invariant-mass range including this region, are
S((KSπ
0)K∗γ) = −0.28 ± 0.26 and S(KSπ0γ) = −0.09 ± 0.24.20,118 These mea-
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
CP violation in beauty decays 23
surements must be improved in order to become sensitive to the level predicted in
the SM, or to provide evidence for physics beyond the SM.
7. Summary
The Standard Model passed with great success numerous tests in the flavor sector,
including a variety of measurements of CP asymmetries related to the CKM phases
β and γ. Small potential New Physics corrections may occur in ∆S = 0 and |∆S| =
1 penguin amplitudes, affecting the extraction of γ and modifying CP-violating
and isospin-dependent asymmetries in |∆S| = 1 B0 decays and isospin-related B+
decays. Higher precision than achieved so far is required for claiming evidence for
such effects and for sorting out their isospin structure.
Similar studies can be performed with Bs mesons produced at hadron colliders
and at e+e− colliders running at the Υ(5S) resonance. Time-dependence in Bs →
D−s K
+ and Bs → J/ψφ or Bs → J/ψη measures γ and the small phase of the
Bs-B̄s mixing amplitude.
119 Comparing time-dependence and angular analysis in
Bs → J/ψφ with b → s penguin-dominated processes including Bs → φφ,Bs →
K∗+K∗−, Bs → K∗0K̄∗0 provides a methodic search for potential NP effects. Work
on Bs decays has just begun at the Tevatron.
120 One is looking forward to first
results from the LHC.
Acknowledgments
I am grateful to numerous collaborators, in particular to Jonathan Rosner whose
collaboration continued without interruption for many years. This work was sup-
ported in part by the Israel Science Foundation under Grant No. 1052/04 and by
the German-Israeli Foundation under Grant No. I-781-55.14/2003.
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http://arxiv.org/abs/hep-ex/0607104
http://arxiv.org/abs/hep-ex/0507101
http://arxiv.org/abs/hep-ph/0702011
http://arxiv.org/abs/hep-ex/0701015
http://arxiv.org/abs/hep-ex/0703008
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
CP violation in beauty decays 27
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http://arxiv.org/abs/hep-ex/0702047
http://arxiv.org/abs/hep-ex/0703029

  Precision tests of the Kobayashi-Maskawa model of CP violation are discussed,
pointing out possible signatures for other sources of CP violation and for new
flavor-changing operators. The current status of the most accurate tests is
summarized.

Introduction
It took thirty-seven years from the discovery of a tiny CP violating effect of order
10−3 inKL → π+π− 1 to a first observation of a breakdown of CP symmetry outside
the strange meson system. A large CP asymmetry of order one between rates of
initial B0 and B̄0 decays to J/ψKS was measured in summer 2001 by the Babar and
Belle Collaborations.2 A sizable however smaller asymmetry had been anticipated
twenty years earlier 3 in the framework of the Kobayashi-Maskawa (KM) model
of CP violation,4 in the absence of crucial information on b quark couplings. The
asymmetry was observed in a time-dependent measurement as suggested,5 thanks
to the long B0 lifetime and the large B0-B̄0 mixing.6 The measured asymmetry,
fixing (in the standard phase convention7) the sine of the phase 2β (≡ 2φ1) ≡
2arg(VtbV
td) of the top-quark dominated B
0-B̄0 mixing amplitude, was found to
be in good agreement with other determinations of Cabibbo-Kobayasi-Maskawa
(CKM) parameters,8,9 including a recent precise measurement of Bs-B̄s mixing.
This showed that the CKM phase γ (≡ φ3) ≡ arg(V ∗ub), which seems to be unable
to account for the observed cosmological baryon asymmetry,11 is the dominant
source of CP violation in flavor-changing processes. With this confirmation the next
pressing question became whether small contributions beyond the CKM framework
occur in CP violating flavor-changing processes, and whether such effects can be
observed in beauty decays.
One way of answering this question is by over-constraining the CKM unitarity
triangle through precise CP conserving measurements related to the lengths of the
∗Based partially on review talks given at recent conferences.
http://arxiv.org/abs/0704.0076v2
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
2 M. Gronau
sides of the triangle. An alternative and more direct way, focusing on the origin
of CP violation in the CKM framework, is to measure β and γ in a variety of B
decay modes. Different values obtained from asymmetries in several processes, or
values different from those imposed by other constraints, could provide clues for
new sources of CP violation and for new flavor-changing interactions. Such phases
and interactions occur in the low energy effective Hamiltonian of extensions of the
Standard Model (SM) including models based on supersymmetry.12
In this presentation we will focus on the latter approach based primarily on
CP asymmetries, using also complementary information on hadronic B decay rates
which are expected to be related to each other in the CKM framework. In the
next section we outline several of the most relevant processes and the theoretical
tools applied for their studies, quoting numerous papers where these ideas have
been originally proposed and where more details can be found.13 Sections 3, 4 and
5 describe a number of methods in some detail, summarizing at the end of each
section the current experimental situation. Section 6 discusses several tests for NP
effects, while Section 7 concludes.
2. Processes, methods and New Physics effects
Whereas testing the KM origin of CP violation in most hadronic B decays requires
separating strong and weak interaction effects, in a few “golden modes” CP asym-
metries are unaffected by strong interactions. For instance, the decay B0 → J/ψKS
is dominated by a single tree-level quark transition b̄ → c̄cs̄, up to a correction
smaller than a fraction of a percent.14,15,16,17 Thus, the asymmetries measured
in this process and in other decays dominated by b̄ → c̄cs̄ have already provided a
rather precise measurement of sin 2β,18,19,20
sin 2β = 0.678± 0.025 . (1)
This value permits two solutions for β at 21.3◦ and at 68.7◦. Time-dependent an-
gular studies of B0 → J/ψK∗0,21 and time-dependent Dalitz analyses of B0 →
Dh0 (D → KSπ+π−, h0 = π0, η, ω)22 measuring cos 2β > 0 have excluded the
second solution at a high confidence level, implying
β = (21.3± 1.0)◦ . (2)
Since B0 → J/ψKS proceeds through a CKM-favored quark transition, contribu-
tions to the decay amplitude from physics at a higher scale are expected to be
very small, potentially identifiable by a tiny direct asymmetry in this process or in
B+ → J/ψK+.23
Another process where the determination of a weak phase is not affected
by strong interactions is B+ → DK+, proceeding through tree-level amplitudes
b̄ → c̄us̄ and b̄ → ūcs̄. The interference of these two amplitudes, from D̄0 and D0
which can always decay to a common hadronic final state, leads to decay rates and
a CP asymmetry which measure very cleanly the relative phase γ between these
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
CP violation in beauty decays 3
amplitudes.24,25 The trick here lies in recognizing the measurements which yield
this fundamental CP-violating quantity. Physics beyond the SM is expected to have
a negligible effect on this determination of γ which relies on the interference of two
tree amplitudes.
B decays into pairs of charmless mesons, such as B → ππ (or B → ρρ) and
B → Kπ (or B → K∗ρ), involve contributions of both tree and penguin ampli-
tudes which carry different weak and strong phases.14,26,27 Contrary to the case
of B → DK, the determination of β and γ using CP asymmetries in charmless B
decays involves two correlated aspects which must be considered: its dependence
on strong interaction dynamics and its sensitivity to potential New Physics (NP)
effects. This sensitivity follows from the CKM and loop suppression of penguin am-
plitudes, implying that new heavy particles at the TeV mass range, replacing the
W boson and the top-quark in the penguin loop, may have sizable effects.28. In
order to claim evidence for physics beyond the SM from a determination of β and
γ in these processes one must handle first the question of dynamics. There are two
approaches for treating the dynamics of charmless hadronic B decays:
(1) Study systematically strong interaction effects in the framework of QCD.
(2) Identify by symmetry observables which do not depend on QCD dynamics.
The first approach faces the difficulty of having to treat precisely long distance
effects of QCD including final state interactions. Remarkable theoretical progress
has been made recently in proving a leading-order (in 1/mb) factorization formula
for these amplitudes in a heavy quark effective theory approach to perturbative
QCD.29,30,31 However, there remain differences between ways of treating in differ-
ent approaches power counting, the scale of Wilson coefficients, end-point quark dis-
tribution functions of light mesons, and nonperturbative contributions from charm
loops.32 Also, the nonperturbative input parameters in these calculations involve
non-negligible uncertainties. These parameters include heavy-to-light form factors
at small momentum transfer, light-cone distribution amplitudes, and the average
inverse momentum fraction of the spectator quark in the B meson. The resulting
inaccuracies in calculating magnitudes and strong phases of amplitudes prohibit a
precise determination of γ from measured decay rates and CP asymmetries. Also,
the calculated rates and asymmetries cannot provide a clear case for physics be-
yond the SM in cases where the results of a calculation deviate slightly from the
measurements.
In the second approach one applies isospin symmetry to obtain relations among
several decay amplitudes. For instance, using the distinct behavior under isospin of
tree and penguin operators contributing in B → ππ, a judicious choice of observ-
ables permits a determination of γ or α (≡ φ2) = π − β − γ. 33 The same analysis
applies in B decays to pairs of longitudinally polarized ρ mesons. In case that an
observable related to the subdominant penguin amplitude is not measured with
sufficient precision, it may be replaced in the analysis by a CKM-enhanced SU(3)-
related observable, in which a large theoretical uncertainty is translated to a small
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
4 M. Gronau
error in γ. The precision of this method is increased by including contributions of
higher order electroweak penguin amplitudes, which are related by isospin to tree
amplitudes.34,35 With sufficient statistics one should also take into account isospin-
breaking corrections of order (md−mu)/ΛQCD ∼ 0.02,36,37 and an effect caused by
the ρ meson width.38 A similar analysis proposed for extracting γ in B → Kπ 39,40
requires using flavor SU(3) instead of isospin for relating electroweak penguin con-
tributions and tree amplitudes.35,41 While flavor SU(3) is usually assumed to be
broken by corrections of order (ms − md)/ΛQCD ∼ 0.3, in this particular case a
rather precise recipe for SU(3) breaking is provided by QCD factorization, reducing
the theoretical uncertainty in γ to only a few degrees.42
Charmless B decays, which are sensitive to physics beyond the SM 28, provide a
rich laboratory for studying various signatures of NP. A large variety of theories have
been studied in this context, including supersymmetric models, models involving
tree-level flavor-changing Z or Z ′ couplings, models with anomalous three-gauge-
boson couplings and other models involving an enhanced chromomagnetic dipole
operator.43,44 The following effects have been studied and will be discussed in
Section 6 in a model-independent manner:
(1) Within the SM, the three values of γ extracted from B → ππ, B → Kπ and
B+ → DK+ are equal. As we will explain, these three values are expected to be
different in extensions of the SM involving new low energy four-fermion operators
behaving as ∆I = 3/2 in B → ππ and as ∆I = 1 in B → Kπ.
(2) Other signatures of anomalously large ∆I = 1 operators contributing to
B → Kπ are violations of isospin sum rules, holding in the SM for both decay rates
and CP asymmetries in these decays.45,46,47
(3) Time-dependent asymmetries in B0 → π0KS , B0 → φKS and B0 → η′KS
and in other b → s penguin-dominated decays may differ substantially from the
asymmetry sin 2β sin∆mt, predicted approximately in the SM.26,43,48 Significant
deviations are expected in models involving anomalous |∆S| = 1 operators behaving
as ∆I = 0 or ∆I = 1.
(4) An interesting question, which may provide a clue to the underlying New
Physics once deviations from SM predictions are observed, is how to diagnose the
value of ∆I in NP operators contributing to |∆S| = 1 charmless B decays. We will
discuss an answer to this question which has been proposed recently.49
3. Determining γ in B → DK
In this section we will discuss in some length a rather rich and very precise method
for determining γ in processes of the form B → D(∗)K(∗), which uses both charged
and neutral B mesons and a large variety of final states. It is based on a broad idea
that any coherent admixture of a state involving a D̄0 from b̄ → c̄us̄ and a state
with D0 from b̄ → ūcs̄ can decay to a common final state.24,25 The interference
between the two channels, B → D(∗)0K(∗), D0 → fD and B → D̄(∗)0K(∗), D̄0 →
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
CP violation in beauty decays 5
fD, involves the weak phase difference γ, which may be determined with a high
theoretical precision using a suitable choice of measurements. Effects of D0-D̄0
mixing are negligible.50 While some of these processes are statistically limited,
combining them together is expected to reduce the experimental error in γ. In
addition to (quasi) two-body B decays, the D or D∗ in the final state may be
accompanied by any multi-body final state with quantum numbers of a kaon.25
Each process in this large class of neutral and charged B decays is characterized
by two pairs of parameters, describing complex ratios of amplitudes for D0 and D̄0
for the two steps of the decay chain (we use a convention rB , rf ≥ 0, 0 ≤ δB, δf <
A(B → D(∗)0K(∗))
A(B → D̄(∗)0K(∗))
= rBe
i(δB+γ) ,
A(D0 → fD)
A(D̄0 → fD)
= rfe
iδf . (3)
In three-body decays ofB andD mesons, such asB → DKπ andD → Kππ, the two
pairs of parameters (rB , δB) and (rf , δf ) are actually functions of two corresponding
Dalitz variables describing the kinematics of the above three-body decays. The
sensitivity of determining γ depends on rB and rf because this determination relies
on an interference of D0 and D̄0 amplitudes. For D decay modes with rf ∼ 1 (see
discussion below) the sensitivity increases with the magnitude of rB.
For each of the eight sub-classes of processes, B+,0 → D(∗)K(∗)+,0, one may
study a variety of final states in neutral D decays. The states fD may be divided
into four families, distinguished qualitatively by their parameters (rf , δf ) defined in
Eq. (3):
(1) fD = CP-eigenstate
24,25,51 (K+K−,KSπ
0, etc.); rf = 1, δf = 0, π.
(2) fD = flavorless but non-CP state
52 (K+K∗−,K∗+K−, etc.); rf = O(1).
(3) fD = flavor state
53 (K+π−,K+π−π0, etc.); rf ∼ tan2 θc.
(4) fD = 3-body self-conjugate state
54 (KSπ
+π−); rf , δf vary across the Dalitz
plane.
In the first family, CP-odd states occur in Cabibbo-favored D0 and D̄0 decays,
while CP-even states occur in singly Cabibbo-suppressed decays. The second family
of states occurs in singly Cabibbo-suppressed decays, the third family occurs in
Cabibbo-favored D̄0 decays and in doubly Cabibbo-suppressed D0 decays, while
the last state is formally a Cabibbo-favored mode for both D0 and D̄0.
The parameters rB and δB in B → D(∗)K(∗) depend on whether the B meson
is charged or neutral, and may differ for K vs K∗,55 and for D vs D∗, where a
neutral D∗ can be observed in D∗ → Dπ0 or D∗ → Dγ.56 The ratio rB involves
a CKM factor |VubVcs/VcbVus| ≃ 0.4 in both B+ and B0 decays, and a color-
suppression factor in B+ decays, while in B0 decays both b̄ → c̄us̄ and b̄ → ūcs̄
amplitudes are color-suppressed. A rough estimate of the color-suppression factor in
these decays may be obtained from the color-suppression measured in corresponding
CKM-favored decays, B → Dπ,D∗π,Dρ,D∗ρ, where the suppression is found to
be in the range 0.3 − 0.5.57 Thus, one expects rB(B0) ∼ 0.4, rB(B+) = (0.3 −
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
6 M. Gronau
0.5)rB(B
0) in all the processes B+,0 → D(∗)K(∗)+,0. We note that three-body B+
decays, such as B+ → D0K+π0, are not color-suppressed, making these processes
advantageous by their potentially large value of rB which varies in phase space.
58,59
The above comparison of rB(B
+) and rB(B
0) may be quantified more precisely
by expressing the four ratios rB(B
0)/rB(B
+) in B → D(∗)K(∗) in terms of recip-
rocal ratios of known magnitudes of amplitudes:60
0 → D(∗)K(∗)0)
rB(B+ → D(∗)K(∗)+)
B+ → D̄(∗)0K(∗)+)
B0 → D̄(∗)0K(∗)0)
. (4)
This follows from an approximation,
A(B0 → D(∗)0K(∗)0) ≃ A(B+ → D(∗)0K(∗)+) , (5)
where the B0 and B+ processes are related to each other by replacing a spectator
d quark by a u quark. While formally Eq. (5) is not an isospin prediction, it may
be obtained using an isospin triangle relation,61
A(B0 → D(∗)0K(∗)0) = A(B+ → D(∗)0K(∗)+) +A(B+ → D(∗)+K(∗)0), (6)
and neglecting the second amplitude on the right-hand-side which is “pure
annihilation”.62 This amplitude is expected to be suppressed by a factor of four or
five relative to the other two amplitudes appearing in (6) which are color-suppressed.
Evidence for this kind of suppression is provided by corresponding ratios of CKM-
favored amplitudes,57 |A(B0 → D−s K+)/
2A(D̄0π0)| = 0.23 ± 0.03, |A(B0 →
D∗−s K
2A(D̄∗0π0)| < 0.24.
Applying Eq. (4) to measured branching ratios,57,63 one finds
rB(B+)
B → DK B → DK∗ B → D∗K B → D∗K∗
2.9± 0.4 3.7± 0.3 > 2.2 > 3.0 (7)
This agrees with values of rB(B
0) near 0.4 and rB(B
+) between 0.1 and 0.2. Note
that in spite of the expected larger values of rB in B
0 decays, from the point
of view of statistics alone (without considering the question of flavor tagging and
the efficiency of detecting a KS in B
0 → D(∗)K0), B+ and B0 decays may fare
comparably when studying γ. This follows from (5) because the statistical error on
γ scales roughly as the inverse of the smallest of the two interfering amplitudes.
We will now discuss the actual manner by which γ can be determined using
separately three of the above-mentioned families of final states fD. We will men-
tion advantages and disadvantages in each case. For illustration of the method we
will consider B+ → fDK+. We will also summarize the current status of these
measurements in all eight decay modes B+,0 → D(∗)K(∗)+,0.
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
CP violation in beauty decays 7
3.1. fD = CP-eigenstates
One considers four observables consisting of two charge-averaged decay rates for
even and odd CP states, normalized by the decay rate into a D0 flavor state,
RCP± ≡
Γ(DCP±K
−) + Γ(DCP±K
Γ(D0K−)
, (8)
and two CP asymmetries for even and odd CP states,
ACP± ≡
Γ(DCP±K
−)− Γ(DCP±K+)
Γ(DCP±K−) + Γ(DCP±K+)
. (9)
In order to avoid dependence of RCP± on errors in D
0 and DCP branching ratio
measurements one uses a definition of RCP± in terms of ratios of B decay branching
ratios intoDK andDπ final states.59 The four observablesRCP± and ACP± provide
three independent equations for rB, δB and γ,
RCP± = 1 + r
B ± 2rB cos δB cos γ , (10)
ACP± = ±2rB sin δB sin γ/RCP± . (11)
While in principle this is the simplest and most precise method for extracting
γ, up to a discrete ambiguity, in practice this method is sensitive to r2B , because
(RCP+ + RCP−)/2 = 1 + r
B . This becomes very difficult for charged B decays
where one expects rB ∼ 0.1− 0.2, but may be feasible for neutral B decays where
rB ∼ 0.4. An obvious signature for a non-zero value of rB would be observing a
difference between RCP+ and RCP− which is linear in this quantity.
Studies of B+ → DCPK+, B+ → DCPK∗+ and B+ → D∗CPK+ have been car-
ried out recently,64,65,66 each consisting of a few tens of events. A nonzero difference
RCP+ −RCP− at 2.6 standard deviations, measured in B+ → DCPK∗+,64 is prob-
ably a statistical fluctuation. A larger difference is anticipated in B0 → DCPK∗0,
as the value of rB in this process is expected to be three or four times larger than
in B+ → DK∗+. [See Eq. (7).] Higher statistics is required for a measurement of γ
using this method.
3.2. fD = flavor state
Consider a flavor state fD in Cabibbo-favored D̄
0 decays, accessible also to doubly
Cabibbo-suppressed D0 decays, such that one has rf ∼ tan2 θc in Eq. (3). One
studies the ratio of two charge-averaged decay rates, for decays into f̄DK and fDK,
Γ(fDK
−) + Γ(f̄DK
Γ(f̄DK−) + Γ(fDK+)
, (12)
and the CP asymmetry,
Γ(fDK
−)− Γ(f̄DK+)
Γ(fDK−) + Γ(f̄DK+)
. (13)
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8 M. Gronau
These observables are given by
Rf = r
B + r
f + 2rB rf cos(δB − δf ) cos γ , (14)
Af = 2r rf sin(δB − δf ) sin γ/Rf , (15)
where a multiplicative correction 1 +O(rBrf ) ∼ 1.01 has been neglected in (14).
These two observables involve three unknowns, rB , δB − δf and γ. One assumes
rf to be given by the measured ratio of doubly Cabibbo-suppressed and Cabibbo-
favored branching ratios. Thus, one needs at least two flavor states, fD and f
for which two pairs of observables (Rf , Af ) and (Rf ′ , Af ′) provide four equations
for the four unknowns, rB, δB − δf , δB − δf ′ , γ. The strong phase differences δf , δf ′
can actually be measured at a ψ′′ charm factory,67 thereby reducing the number of
unknowns to three.
While the decay rate in the numerator of Rf is rather low, the asymmetry Af
may be large for small values of rB around 0.1, as it involves two amplitudes with
a relative magnitude rf/rB.
So far, only upper bounds have been measured for Rf implying upper limits
on rB in several processes, rB(B
+ → DK+) < 0.2,68,69,70 rB(B+ → D∗K+) <
0.2,68 r(B+ → DK∗+) < 0.4,71 and rB(B0 → DK∗0) < 0.4.63,72 Further con-
straints on rB in the first three processes have been obtained by studying D decays
into CP-eigenstates and into the state KSπ
+π−. Using rB(B
0 → DK∗0)/rB(B+ →
DK∗+) = 3.7 ± 0.3 in (7) and assuming that rB(B+ → DK∗+) is not smaller
than about 0.1, one may conclude that a nonzero measurement of rB(B
0 → DK∗0)
should be measured soon. The signature for B0 → D0K∗0 events would be two
kaons with opposite charges.
3.3. fD = KSπ
The amplitude for B+ → (KSπ+π−)DK+ is a function of the two invariant-mass
variables, m2
≡ (pKS + pπ±)2, and may be written as
A(B+ → (KSπ+π−)DK+) = f(m2+,m2−) + rBei(δB+γ)f(m2−,m2+) . (16)
In B− decay one replaces m+ ↔ m−, γ → −γ. The function f may be written as a
sum of about twenty resonant and nonresonant contributions modeled to describe
the amplitude for flavor-tagged D̄0 → KSπ+π− which is measured separately.73,74
This introduces a model-dependent uncertainty in the analysis. Using the measured
function f as an input and fitting the rates for B± → (KSπ+π−)DK± to the
parameters, rB , δB and γ, one then determines these three parameters.
The advantage of using D → KSπ+π− decays over CP and flavor states is
being Cabibbo-favored and involving regions in phase space with a potentially large
interference between D0 and D̄0 decay amplitudes. The main disadvantage is the
uncertainty introduced by modeling the function f .
Two recent analyses of comparable statistics by Belle and Babar, combining
B± → DK±, B± → D∗K± and B± → DK∗±, obtained values 73 γ = [53+15
−18 ± 3±
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
CP violation in beauty decays 9
9(model)]◦ and γ = [92±41±11±12(model)]◦.74 [This second value does not use the
process B+ → D(KSπ+)K∗ , also studied by the same group,75.] The larger errors in
the second analysis are correlated with smaller values of the extracted parameters rB
in comparison with those extracted in the first study. The model-dependent errors
may be reduced by studying at CLEO-c the decays DCP± → KSπ+π−, providing
further information on strong phases in D decays.67
Conclusion: The currently most precise value of γ is γ = [53+15
−18 ± 3± 9(model)]◦,
obtained from B± → D(∗)K(∗)± using D → KSπ+π−. These errors may be reduced
in the future by combining the study of all D decay modes in B+,0 → D(∗)K(∗)+,0.
The decay B0 → DK∗0 seems to carry a high potential because of its expected
large value of rB. Decays B
0 → D(∗)K0 may also turn useful, as they have been
shown to provide information on γ without the need for flavor tagging of the initial
B0.60,76
4. The currently most precise determination of γ: B → ππ, ρρ, ρπ
4.1. B → ππ
The amplitude for B0 → π+π− contains two terms, conventionally denoted “tree”
(T ) and “penguin” (P ) amplitudes, 14,26 involving a weak CP-violating phase γ
and a strong CP-conserving phase δ, respectively:
A(B0 → π+π−) = |T |eiγ + |P |eiδ . (17)
Time-dependent decay rates, for an initial B0 or a B
, are given by
Γ(B0(t)/B
(t) → π+π−) = e−ΓtΓπ+π− [1± C+− cos∆mt∓ S+− sin∆mt] , (18)
where
S+− =
2Im(λππ)
1 + |λππ|2
, C+− =
1− |λππ |2
1 + |λππ |2
, λππ ≡ e−2iβ
0 → π+π−)
A(B0 → π+π−)
. (19)
One has14
S+− = sin 2α+ 2|P/T | cos2α sin(β + α) cos δ +O(|P/T |2) ,
C+− = 2|P/T | sin(β + α) sin δ +O(|P/T |2) . (20)
This tells us two things:
(1) The deviation of S+− from sin 2α and the magnitude of C+− increase with
|P/T |, which can be estimated to be |P/T | ∼ 0.5 by comparing B → ππ rates with
penguin-dominated B → Kπ rates.77
(2) Γπ+π− , S+− and C+− are insufficient for determining |T |, |P |, δ and γ (or α).
Further information on these quantities may be obtained by applying isospin sym-
metry to all B → ππ decays.
In order to carry out an isospin analysis,33 one uses the fact that the three
physical B → ππ decay amplitudes and the three B̄ → ππ decay amplitudes,
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10 M. Gronau
depending each on two isospin amplitudes, obey triangle relations of the form,
A(B0 → π+π−)/
2 +A(B0 → π0π0)−A(B+ → π+π0) = 0 . (21)
Furthermore, the penguin amplitude is pure ∆I = 1/2; hence the ∆I = 3/2 am-
plitude carries a week phase γ, A(B+ → π+π0) = e2iγA(B− → π−π0). Defin-
ing sin 2αeff ≡ S+−/(1 − C2+−)1/2, the difference αeff − α is then determined by
an angle between corresponding sides of the two isospin triangles sharing a com-
mon base, |A(B+ → π+π0)| = |A(B− → π−π0)|. A sign ambiguity in αeff − α is
resolved by two model-independent features which are confirmed experimentally,
|P |/|T | ≤ 1, |δ| ≤ π/2. This implies α < αeff .78
Table I. Branching ratios and CP asymmetries in B → ππ, B → ρρ.
Decay mode Branching ratio (10−6) ACP = −C S
B0 → π+π− 5.16± 0.22 0.38 ± 0.07 −0.61± 0.08
B+ → π+π0 5.7± 0.4 0.04 ± 0.05
B0 → π0π0 1.31± 0.21 0.36
+0.33
−0.31
B0 → ρ+ρ− 23.1
0.11 ± 0.13 −0.06± 0.18
B+ → ρ+ρ0 18.2± 3.0 −0.08± 0.13
B0 → ρ0ρ0 1.07± 0.38
Current CP-averaged branching ratios and CP asymmetries for B → ππ and
B → ρρ decays are given in Table I,20 where ACP ≡ −C for decays to CP eigen-
states. An impressive experimentally progress has been achieved in the past two
years in extracting a precise value for αeff , αeff = (110.6
−3.2)
◦. However, the er-
ror on αeff − α using the isospin triangles is still large. An upper bound, given by
CP-averaged rates and a direct CP asymmetry in B0 → π+π−,79,80
cos 2(αeff − α) ≥
Γ+− + Γ+0 − Γ00
)2 − Γ+−Γ+0
Γ+−Γ+0
1− C2+−
, (22)
leads to 0 < αeff − α < 31◦ at 1σ. Adding in quadrature the error in αeff and the
uncertainty in α−αeff , this implies α = (95± 16)◦ or γ = (64± 16)◦ by . A similar
central value but a smaller error, α = (97± 11)◦, has been reported recently by the
Belle Collaboration.81 The possibility that a penguin amplitude in B0 → π+π−
may lead to a large CP asymmetry S for values of α near 90◦ where sin 2α = 0 was
anticipated fifteen years ago.82
The bound on αeff − α may be improved considerably by measuring a nonzero
direct CP asymmetry in B0 → π0π0. This asymmetry can be shown to be large and
positive (see Eq. (46) in Sec. 5.2), implying a large rate for B̄0 but a small rate for
B0. Namely, the triangle (21) is expected to be squashed, while the B̄ triangle is
roughly equal sided.
An alternative way of treating the penguin amplitude in B0 → π+π− is by
combining within flavor SU(3) the decay rate and asymmetries in this process with
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CP violation in beauty decays 11
rates and asymmetries in B0 → K0π+ or B0 → K+π−.77 The ratio of ∆S = 1 and
∆S = 0 tree amplitudes in these processes, excluding CKM factors, is taken to be
given by fK/fπ assuming factorization, while the ratio of corresponding penguin
amplitudes is allowed to vary by ±0.22 around one. A current update of this rather
conservative analysis obtains 83
γ = (73± 4+10
◦ , (23)
where the first error is experimental, while the second one is due to an uncertainty
in SU(3) breaking. A discussion of SU(3) breaking factors relating B0 → π+π− and
B0 → K+π− is included in Section 5.2.
4.2. B → ρρ
Angular analyses of the pions in ρ decays have shown that B0 → ρ+ρ− is dominated
almost 100% by longitudinal polarization 20. This simplifies the isospin analysis of
CP asymmetries in these decays to becoming similar to B0 → π+π−. The advantage
of B → ρρ over B → ππ is the relative small value of (
ρ0ρ0) in comparison with
ρ+ρ−) and (
ρ+ρ0) (see Table I), indicating a smaller |P/T | in B → ρ+ρ− (|P/T | <
0.3 8) than in B0 → π+π− (|P/T | ∼ 0.5 77). Eq. (22) leads to an upper bound on
αeff − α in B → ρρ, 0 < αeff − α < 17◦ (at 1σ). The asymmetries for longitudinal
ρ’s given in Table I imply αeff = (91.7
−5.2)
◦. Thus, one finds α = (83 ± 10)◦ or
γ = (76± 10)◦ by adding errors in quadrature.
A stronger bound on |P/T | in B0 → ρ+ρ−, leading to a more precise value of γ,
may be obtained by relating this process to B+ → K∗0ρ+ within flavor SU(3). 84
One uses the branching ratio and fraction of longitudinal rate measured for this
process 20, (
K∗0ρ+) = (9.2 ± 1.5) × 10−6, fL(K∗0ρ+) = 0.48 ± 0.08, to normalize
the penguin amplitude in B0 → ρ+ρ−. Including a conservative uncertainty from
SU(3) breaking and smaller amplitudes, one finds a value
γ = (71.4+5.8
−1.7)
◦ , (24)
where the first error is experimental and the second one theoretical.
The current small theoretical error in γ requires including isospin breaking effects
in studies based on isospin symmetry. The effect of electroweak penguin amplitudes
on the isospin analyses of B → ππ and B → ρρ has been calculated and was found
to move γ slightly higher by an amount ∆γEWP = 1.5
◦.34,35 Other corrections,
relevant to methods using π0 and ρ0, includng π0-η-η′ mixing, ρ-ω mixing, and a
small I = 1 ρρ contribution allowed by the ρ-width, are each smaller than one
degree.36,37,38
Conclusion: Taking an average of the two values of γ in (23) and (24) obtained
from B0 → π+π− and B0 → ρ+ρ−, and including the above-mentioned EWP
correction, one finds
γ = (73.5± 5.7)◦ . (25)
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12 M. Gronau
A third method of measuring γ (or α) in time-dependent Dalitz analyses of B0 →
(ρπ)0 involves a much larger error,85 and has a small effect on the overall averaged
value of the weak phase. We note that sin γ is close to one and its relative error
is only 3%, the same as the relative error in sin 2β and slightly smaller than the
relative error in sinβ.
5. Rates, asymmetries, and γ in B → Kπ
5.1. Extracting γ in B → Kπ
The four decays B0 → K+π−, B0 → K0π0, B+ → K0π+, B+ → K+π0 involve
a potential for extracting γ, provided that one is sensitive to interference between
a dominant isoscalar penguin amplitude and a small tree amplitude contributing
to these processes. This idea has led to numerous suggestions for determining γ in
these decays starting with a proposal made in 1994.86,87 An interference between
penguin and tree amplitudes may be identified in two ways:
(1) Two different properly normalized B → Kπ rates.
(2) Nonzero direct CP asymmetries.
Table II. Branching ratios and asymmetries in B → Kπ.
Decay mode Branching ratio (10−6) ACP
B0 → K+π− 19.4± 0.6 −0.097± 0.012
B+ → K+π0 12.8± 0.6 0.047± 0.026
B+ → K0π+ 23.1± 1.0 0.009± 0.025
B0 → K0π0 10.0± 0.6 −0.12± 0.11
Current branching ratios and CP asymmetries are summarized in Table II.20 Three
ratios of rates, calculated using the ratio of B+ and B0 lifetimes, τ+/τ0 = 1.076±
0.008,20 are:
R ≡ Γ(B
0 → K+π−)
Γ(B+ → K0π+)
= 0.90± 0.05 ,
2Γ(B+ → K+π0)
Γ(B+ → K0π+)
= 1.11± 0.07 ,
Γ(B0 → K+π−)
2Γ(B0 → K0π0) = 0.97± 0.07 . (26)
The largest deviation from one, observed in the ratio R at 2σ, is insufficient for
claiming unambiguous evidence for a non-penguin contribution. An upper limit,
R < 0.965 at 90% confidence level, would imply γ ≤ 79◦ using sin2 γ ≤ R,88
which neglects however “color-suppressed” EWP contributions.89 As we will argue
now, these contributions and “color-suppressed” tree amplitudes are actually not
suppressed as naively expected.
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CP violation in beauty decays 13
The nonzero asymmetry measured in B0 → K+π− provides first evidence for
an interference between penguin (P ′) and tree (T ′) amplitudes with a nonzero rel-
ative strong phase. Such an interference occurs also in B+ → K+π0 where no
asymmetry has been observed. An assumption that other contributions to the lat-
ter asymmetry are negligible has raised some questions about the validity of the
CKM framework. In fact, a color-suppressed tree amplitude (C′), also occurring in
B+ → K+π0,86 resolves this “puzzle” if this amplitude is comparable in magnitude
to T ′. Indeed, several studies have shown that this is the case,90,91,92,93,94 also im-
plying that color-suppressed and color-favored EWP amplitudes are of comparable
magnitudes.35 For consistency between the two CP asymmetries in B0 → K+π−
and B+ → K+π0, the strong phase difference between C′ and T ′ must be negative
and cannot be very small.95 This seems to stand in contrast to QCD calculations
using a factorization theorem.29,31,94
The small asymmetry ACP (B
+ → K+π0) implies bounds on the sine of the
strong phase difference δc between T
′ +C′ and P ′. The cosine of this phase affects
Rc − 1 involving the decay rates for B+ → K0π+ and B+ → K0π+. A question
studied recently is whether the two upper bounds on | sin δc| and | cos δc| are con-
sistent with each other or, perhaps, indicate effects of NP. Consistency was shown
by proving a sum rule involving ACP (B
+ → K+π0) and Rc − 1, in which an elec-
troweak penguin (EWP) amplitude plays an important role. We will now present a
proof of the sum rule, which may provide important information on γ.95
The two amplitudes for B+ → K0π+,K+π0 are given in terms of topological
contributions including P ′, T ′ and C′,
A(B+ → K0π+) = (P ′ − 1
P ′cEW ) +A
A(B+ → K+π0) = (P ′ − 1
P ′cEW ) + (T
′ + P ′cEW ) + (C
′ + P ′EW ) +A
′ , (27)
where P ′EW and P
EW are color-favored and color-suppressed EWP contributions.
The small annihilation amplitude A′ and a small u quark contribution to P ′ involv-
ing a CKM factor V ∗ubVus will be neglected (|V ∗ubVus|/|V ∗cbVcs| = 0.02). Evidence for
the smallness of these terms can be found in the small CP asymmetry measured for
B+ → K0π+. Large terms would require rescattering and a sizable strong phase
difference between these terms and P ′.
Flavor SU(3) symmetry relates ∆I = 1, I(Kπ) = 3/2 electroweak penguin and
tree amplitudes through a calculable ratio δEW
35,41,
T ′ + C′ + P ′EW + P
EW = (T
′ + C′)(1 − δEW e−iγ) ,
δEW = −
c9 + c10
c1 + c2
|V ∗tbVts|
|V ∗ubVus|
= 0.60± 0.05 . (28)
The error in δEW is dominated by the current uncertainty in |Vub|/|Vcb| = 0.104±
0.007 57, including also a smaller error from SU(3) breaking estimated using QCD
factorization. Eqs. (27) and (28) imply 96
Rc = 1− 2rc cos δc(cos γ − δEW) + r2c (1− 2δEW cos γ + δ2EW) , (29)
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14 M. Gronau
ACP (B
+ → K+π0) = −2rc sin δc sin γ/Rc , (30)
where rc ≡ |T ′ + C′|/|P ′ − 13P
EW | and δc is the strong phase difference between
T ′ + C′ and P ′ − 1
P ′cEW .
The parameter rc is calculable in terms of measured decay rates, using bro-
ken flavor SU(3) which relates T ′ + C′ and T + C dominating B+ → π+π0 by a
factorization factor fK/fπ (neglecting a tiny EWP term in B
+ → π+π0),87
|T ′ + C′| =
|A(B+ → π+π0)| . (31)
Using branching ratios from Tables I and II, one finds
B+ → π+π0)
B+ → K0π+)
= 0.198± 0.008 . (32)
The error in rc does not include an uncertainty from assuming factorization for
SU(3) breaking in T ′ + C′. While this assumption should hold well for T ′, it may
not be a good approximation for C′ which as we have mentioned is comparable in
magnitude to T ′ and carries a strong phase relative to it. Thus one should allow a
10% theoretical error when using factorization for relating B → Kπ and B → ππ
T + C amplitudes, so that
rc = 0.20± 0.01 (exp)± 0.02 (th) . (33)
Eliminating δc in Eqs. (29) and (30) by retaining terms which are linear in rc,
one finds
Rc − 1
cos γ − δEW
ACP (B
+ → K+π0)
sin γ
= (2rc)
2 +O(r3c ) . (34)
This sum rule implies that at least one of the two terms whose squares occur on
the left-hand-side must be sizable, of the order of 2rc = 0.4. The second term,
|ACP (B+ → K+π0)|/ sin γ, is already smaller than ≃ 0.1, using the current 2σ
bounds on γ and |ACP (B+ → K+π0)|. Thus, the first term must provide a dominant
contribution. For Rc ≃ 1, this implies γ ≃ arccos δEW ≃ (53.1± 3.5)◦. This range is
expanded by including errors in Rc and ACP (B
+ → K+π0). For instance, an upper
bound Rc < 1.1 would imply an inportant upper limit, γ < 70
◦. Currently one only
obtains an upper limit γ ≤ 88◦ at 90% confidence level.95 This bound is consistent
with the value obtained in (25) from B → ππ and B → ρρ, but is not competitive
with the latter precision.
Conclusion: The current constraint obtained from Rc and ACP (B
+ → K+π0) is
γ ≤ 88◦ at 90% confidence level. Further improvement in the measurement of Rc
(which may, in fact, be very close to one) is required in order to achieve a precision
in γ comparable to that obtained in B → ππ, ρρ. (A conclusion concerning the
different CP asymmetries measured in B0 → K+π− and B+ → K+π0 will be given
at the end of the next subsection.)
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CP violation in beauty decays 15
5.2. Symmetry relations for B → Kπ rates and asymmetries
The following two features imply rather precise sum rules in the CKM framework,
both for B → Kπ decay rates and CP asymmetries:
(1) The dominant penguin amplitude is ∆I = 0.
(2) The four decay amplitudes obey a linear isospin relation,39
A(K+π−)−A(K0π+)−
2A(K+π0) +
2A(K0π0) . (35)
An immediate consequence of these features are two isospin sum rules, which
hold up to terms which are quadratic in small ratios of non-penguin to penguin
amplitudes,45,46,47
Γ(K+π−) + Γ(K0π+) = 2Γ(K+π0) + 2Γ(K0π0) , (36)
∆(K+π−) + ∆(K0π+) = 2∆(K+π0) + 2∆(K0π0) , (37)
where
∆(Kπ) ≡ Γ(B̄ → K̄π̄)− Γ(B → Kπ) . (38)
Quadratic corrections to (36) have been calculated in the SM and were found to
be a few percent.97,98,99 This is the level expected in general for isospin-breaking
corrections which must therefore also be considered. The above two features imply
that these ∆I = 1 corrections are suppressed by a small ratio of non-penguin to
penguin amplitudes and are therefore negligible.100 Indeed, this sum rule holds
experimentally within a 5% error.101 One expects the other sum rule (37) to hold
at a similar precision.
The CP rate asymmetry sum rule (37), relating the four CP asymmetries, leads
to a prediction for the asymmetry in B0 → K0π0 in terms of the other three
asymmetries which have been measured with higher precision,
ACP (B
0 → K0π0) = −0.140± 0.043 . (39)
While this value is consistent with experiment (see Table II), higher accuracy in
this asymmetry measurement is required for testing this straightforward prediction.
Relations between CP asymmetries in B → Kπ and B → ππ following from
approximate flavor SU(3) symmetry of QCD 102 are not expected to hold as pre-
cisely as isospin relations, but may still be interesting and useful. An important
question relevant to such relations is how to include SU(3)-breaking effects, which
are expected to be at a level of 20-30%. Here we wish to discuss two SU(3) rela-
tions proposed twelve years ago,103,104 one of which holds experimentally within
expectation, providing some lesson about SU(3) breaking, while the other has a an
interesting implication for future applications of the isospin analysis in B → ππ.
A most convenient proof of SU(3) relations is based on using a diagramatic
approach, in which diagrams with given flavor topologies replace reduced SU(3)
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16 M. Gronau
matrix elements.86 In this language, the amplitudes for B0 decays into pairs of
charged or neutral pions, and pairs of charged or neutral π and K, are given by:
−A(B0 → π+π−) = T + (P + 2P cEW /3) + E + PA ,
2A(B0 → π0π0) = C − (P − PEW − P cEW /3)− E − PA ,
−A(B0 → K+π−) = T ′ + (P ′ + 2P ′cEW /3) ,
2A(B0 → K0π0) = C′ − (P ′ − P ′EW − P ′cEW /3) . (40)
The combination E + PA, representing exchange and penguin annihilation topolo-
gies, is expected to be 1/mb-suppressed relative to T and C,
31,62 as demonstrated
by the small branching ratio measured for B0 → K+K−.20 This term will be
neglected.
Expressing topological amplitudes in terms of CKM factors, SU(3)-invariant
amplitudes and SU(3) invariant strong phases, one may write
T ≡ V ∗ubVud|T + Puc| , P + 2P cEW /3 ≡ V ∗tbVtd|Ptc|eiδ ,
T ′ ≡ V ∗ubVus|T + Puc| , P ′ + 2P ′cEW /3 ≡ V ∗tbVts|Ptc|eiδ , (41)
C ≡ V ∗ubVud|C − Puc| , P − PEW − P cEW /3 ≡ V ∗tbVtd|P̃tc|eiδ̃ ,
C′ ≡ V ∗ubVus|C − Puc| , P ′ − P ′EW − P ′cEW /3 ≡ V ∗tbVts|P̃tc|eiδ̃ .
Unitarity of the CKM matrix, V ∗cbVcd(s) = −V ∗tbVtd(s) − V ∗ubVud(s), has been used
to absorb in T (
′) and C(
′) a penguin term Puc ≡ Pu − Pc multiplying V ∗ubVud(s),
while Ptc ≡ Pt − Pc and P̃tc ≡ P̃t − P̃c contain two distinct combinations of EWP
contributions. Using the identity
Im (V ∗ubVudVtbV
td) = −Im (V ∗ubVusVtbV ∗ts) , (42)
one finds103,104
∆(B0 → K+π−) = −∆(B0 → π+π−) (43)
∆(B0 → K0π0) = −∆(B0 → π0π0) , (44)
where ∆ is the CP rate difference defined in (38).
Quoting products of branching ratios and asymmetries from Tables I and II,
Eq. (43) reads
− 1.88± 0.24 = −1.96± 0.37 . (45)
This SU(3) relation works well and requires no SU(3)-breaking. An SU(3) breaking
factor fK/fπ in T but not in P , or in both T and P , are currently excluded at a
level of 1.0σ, or 1.75σ. More precise CP asymmetry measurements in B0 → K+π−
and B0 → π+π− are required for determining the pattern of SU(3) breaking in tree
and penguin amplitudes.
Using the prediction (39) of the B → Kπ asymmetry sum rule, Eq. (44) predicts
ACP (B
0 → π0π0) = 1.07± 0.38 . (46)
The error is dominated by current errors in CP asymmetries for B+ → K0π+
and B+ → K+π0, and to a less extent by the error in (
π0π0). SU(3) breaking in
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CP violation in beauty decays 17
amplitudes could modify this prediction by a factor fπ/fK if this factor applies to
C, and less likely by (fπ/fK)2. A large positive CP asymmetry, favored in all three
cases, will affect future applications of the isospin analysis in B → ππ. It implies
that while the B̄ isospin triangle is roughly equal-sided, the B triangle is squashed.
A twofold ambiguity in the value of γ disappears in the limit of a flat B triangle.24
Conclusion: The isospin sum rule for B → Kπ decay rates holds well, while the
CP asymmetry sum rule predicts ACP (B
0 → K0π0) = −0.140±0.043. The different
asymmetries in B0 → K+π− and B+ → K+π0 can be explained by an amplitude
C′ comparable to T ′ and involving a relative negative strong phase, and should
not be considered a “puzzle”. An SU(3) relation for B0 → ππ and B0 → Kπ CP
asymmetries works well for charged modes. The corresponding relation for neutral
modes predicts a large positive asymmetry in B0 → π0π0. Improving asymmetry
measurements can provide tests for SU(3) breaking factors.
6. Tests for small New Physics effects
6.1. Values of γ
We have described three ways for extracting a value for γ relying on interference
of distinct pairs of quark amplitudes, (b → cūs, b → uc̄s), (b → cc̄s, b → uūs) and
(b → cc̄d, b → uūd). The three pairs provide a specific pattern for CP violation in
the CKM framework, which is expected to be violated in many extensions of the
SM. The rather precise value of γ (25) extracted from B → ππ, ρρ, ρπ is consistent
with constraints on γ from CP conserving measurements related to the sides of the
unitarity triangle.8,9 The values of γ obtained in B → D(∗)K(∗) and B → Kπ
are consistent with those extracted in B → ππ, ρρ, ρπ, but are not yet sufficiently
precise for testing small NP effects in charmless B decays. Further experimental
improvements are required, in particular in the former two types of processes.
While the value of γ in B → D(∗)K∗) is not expected to be affected by NP,
the other two classes of processes involving penguin loops are susceptible to such
effects. The extraction of γ in B → ππ ρρ assumes that γ is the phase of a ∆I =
3/2 tree amplitude, while an additional ∆I = 3/2 EWP contribution is included
using isospin. The extracted value could be modified by a new ∆I = 3/2 effective
operator originating in physics beyond the SM, but not by a new ∆I = 1/2 operator.
Similarly, the value of γ extracted in B → Kπ is affected by a potential new ∆I = 1
operator, but not by a new ∆I = 0 operator, because the amplitude (28), playing
an essential role in this method, is pure ∆I = 1.
6.2. B → Kπ sum rule
Charmless |∆S| = 1 B and Bs decays are particularly sensitive to NP effects, as
new heavy particles at the TeV mass range may replace the the W boson and top-
quark in the penguin loop dominating these amplitudes.28 The sum rule (36) for
B → Kπ decay rates provides a test for such effects. However, as we have argued
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
18 M. Gronau
from isospin considerations, it is only affected by quadratic ∆I = 1 amplitudes
including NP contributions. Small NP amplitudes, contributing quadratically to
the sum rule, cannot be separated from SM corrections, which are by themselves at
a level of a few percent. This is the level to which the sum rule has already been
tested. We will argue below for evidence showing that potential NP contributions
to |∆S| = 1 charmless decays must be suppressed by roughly an order of magnitude
relative to the dominant b→ s penguin amplitudes.
6.3. Values of S,C in |∆S| = 1 B0 → fCP decays
A class of b → s penguin-dominated B0 decays to CP-eigenstates has recently at-
tracted considerable attention. This includes final statesXKS andXKL, whereX =
φ, π0, η′, ω, f0, ρ
0,K+K−,KSKS , π
0π0, for which measured asymmetries −ηCPS
and C are quoted in Table III. [The asymmetries S and C = −ACP were de-
fined in (18) for B0 → π+π−. Observed modes with KL in the final states obey
ηCP (XKL) = −ηCP (XKS).] In these processes, a value S = −ηCP sin 2β (for states
Table III. Asymmetries S and C in B0 → XKS .
X φ π0 η′ ω f0(980)
−ηCP S 0.39± 0.18 0.33± 0.21 0.61± 0.07 0.48± 0.24 0.42 ± 0.17
C 0.01± 0.13 0.12± 0.11 −0.09± 0.06 −0.21± 0.19 −0.02± 0.13
X ρ0 K+K− KSKS π
−ηCP S 0.20± 0.57 0.58
+0.18
−0.13
0.58± 0.20 −0.72± 0.71
C 0.64± 0.46 0.15± 0.09 −0.14± 0.15 0.23± 0.54
with CP-eigenvalue ηCP ) is expected approximately.
26,43 These predictions involve
hadronic uncertainties at a level of several percent, of order λ2, λ ∼ 0.2. It has been
pointed out some time ago105 that it is difficult to separate these hadronic uncer-
tainties within the SM from NP contributions to decay amplitudes if the latter are
small. In the next subsection we will discuss indirect experimental evidence showing
that NP contributions to S and C must be small. Corrections to S = −ηCP sin 2β
and values for the asymmetries C have been calculated in the SM using methods
based on QCD factorization106,107 and flavor SU(3),90,108,109 and were found to
be between a few percent up to above ten percent within hadronic uncertainties.
Whereas the deviation of S from −ηCP sin 2β is process-dependent, a generic
result has been proven a long time ago for both S and C, to first order in |c/p|,14
∆S ≡ −ηCPS − sin 2β = 2
cos 2β sin γ cos∆ ,
C = 2
sin γ sin∆ . (47)
Here p and c are penguin and color-suppressed tree amplitudes involving a small ra-
tio and relative weak and strong phases γ and ∆, respectively. This implies ∆S > 0
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
CP violation in beauty decays 19
for |∆| < π/2, which can be argued for several of the above decays using QCD
arguments106,107 or SU(3) fits.109 (Note that while |p| is measurable in certain
decay rates up to first order corrections, |c| and ∆ involve sizable hadronic uncertain-
ties in QCD calculations.) In contrast to this expectation, the central values mea-
sured for ∆S are negative for all decays. (See Table III.) Consequently, one finds an
averaged value sin 2βeff = 0.53±0.05,20 to be compared with sin 2β = 0.678±0.025.
Two measurements which seem particularly interesting are−ηCPSφKS = 0.39±0.18,
where a positive correction of a few percent to sin 2β is expected in the SM,106,107
and −ηCPSπ0KS = 0.33± 0.21, where a rather large positive correction to sin 2β is
expected shifting this asymmetry to a value just above 0.8.90
While the current averaged value of sin 2βeff is tantalizing, experimental errors
in S and C must be reduced further to make a clear case for physics beyond the
SM. Assuming that the discrepancy between improved measurements and calcu-
lated values of S and C persists beyond theoretical uncertainties, can this pro-
vide a clue to the underlying New Physics? Since many models could give rise to
a discrepancy,28,43,44 one would seek signatures characterizing classes of models
rather than studying the effects in specific models. One way of classifying extensions
of the SM is by the isospin behavior of the new effective operators contributing to
b→ sqq̄ transitions.
6.4. Diagnosis of ∆I for New Physics operators
Four-quark operators in the effective Hamiltonian associated with NP in b → sqq̄
transitions can be either isoscalar or isovector operators. We will now discuss a study
proposed recently in order to isolate ∆I = 0 or ∆I = 1 operators, thus determining
corresponding NP amplitudes and CP violating phases.49 We will show that since S
and C in the above processes combine ∆I = 0 or ∆I = 1 contributions, separating
these contributions requires using also information from other two asymmetries,
which are provided by isospin-reflected decay processes.
Two |∆S| = 1 charmless B (or Bs) decay processes, related by isospin reflection,
RI : u ↔ d, ū ↔ −d̄, can always be expressed in term of common ∆I = 0 and
∆I = 1 amplitudes B and A in the form:
A(B+ → f) = B +A , A(B0 → RIf) = ±(B −A) . (48)
A proof of this relation uses a sign change of Clebsch-Gordan coefficients underm↔
−m.49 The description (48) applies, in particular, to pairs of processes involving
all the B0 decay modes listed in Table III, and B+ decay modes where final states
are obtained by isospin reflection from corresponding B0 decay modes. Decay rates
for pairs of isospin-reflected B decay processes, and for B̄ decays to corresponding
charge conjugate final states are therefore given by (we omit inessential common
kinematic factors),
Γ+ = |B +A|2 , Γ0 = |B − A|2 ,
Γ− = |B̄ + Ā|2 , Γ0̄ = |B̄ − Ā|2 . (49)
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
20 M. Gronau
The amplitudes B̄ and Ā are related to B and A by a change in sign of all weak
phases, whereas strong phases are left unchanged.
For each pair of processes one defines four asymmetries: an isospin-dependent
CP-conserving asymmetry,
Γ+ + Γ− − Γ0 − Γ0̄
Γ+ + Γ− + Γ0 + Γ0̄
, (50)
two CP-violating asymmetries for B+ and B0,
A+CP ≡
Γ− − Γ+
Γ− + Γ+
, − C ≡ A0CP ≡
Γ0̄ − Γ0
Γ0̄ + Γ0
, (51)
and the time-dependent asymmetry S in B0 decays,
1 + |λ|2
, λ ≡ ηCP
B̄ − Ā
e−2iβ , (52)
In the Standard Model, the isoscalar amplitude B contains a dominant penguin
contribution, BP , with a CKM factor V
cbVcs. The residual isoscalar amplitude,
∆B ≡ B −BP , (53)
and the amplitude A, consist each of contributions smaller than BP by about an
order of magnitude.29,30,31,32,86 These contributions include terms with a much
smaller CKM factor V ∗ubVus, and a higher order electroweak penguin amplitude with
CKM factor V ∗tbVts. Thus, one expects
|∆B| ≪ |BP | , |A| ≪ |BP | . (54)
Consequently, the asymmetries AI , A
CP and ∆S are expected to be small, of or-
der 2|A|/|B| and 2|∆B|/|BP |. In contrast, potentially large contributions to ∆B
and A from NP, comparable to BP , would most likely lead to large asymmetries
of order one. An unlikely exception is the case when both ∆B/BP and A/BP are
purely imaginary, or almost purely imaginary. This would require very special cir-
cumstances such as fine-tuning in specific models. Excluding cancellations between
NP and SM contributions in both CP-conserving and CP violating asymmetries,
tests for the hierarchy (54) become tests for the smallness of corresponding potential
NP contributions to B and A.
There exists ample experimental information showing that asymmetries A+CP
are small in processes related by isospin reflection to the decay modes in Table III.
Upper limits on the magnitudes of most asymmetries are at a level of ten or fifteen
percent [e.g., A+CP (K
+φ) = 0.034±0.044,A+CP (K+η′) = 0.031±0.026], while others
may be as large as twenty or thirty percent [A+CP (K
+ρ0) = 0.31+0.11
−0.10]. Similar values
have been measured for isospin asymmetries AI [e.g., AI(K
+φ) = −0.037± 0.077,
+η′) = −0.001± 0.033, AI(K+ρ0) = −0.16± 0.10].49 Since these two types
of asymmetries are of order 2|∆B|/|BP | and 2|A|/|BP |, this confirms the hierarchy
(54), which can be assumed to hold also in the presence of NP.
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
CP violation in beauty decays 21
We will take by convention the dominant penguin amplitude BP to have a zero
weak phase and a zero strong phase, referring all other strong phases to it. Writing
B = BP +∆B , B̄ = BP +∆B̄ , (55)
and expanding the four asymmetries to leading order in ∆B/BP or A/BP , one has
∆S = cos 2β
Im(Ā−A)
− Im(∆B̄ −∆B)
, (56)
Re(Ā+A)
, (57)
A+CP =
Re(Ā−A)
Re(∆B̄ −∆B)
, (58)
A0CP = −
Re(Ā−A)
Re(∆B̄ −∆B)
. (59)
The four asymmetries provide the following information:
• The ∆I = 0 and ∆I = 1 contributions in CP asymmetries are separated
by taking sums and differences,
A∆I=0CP ≡
(A+CP +A
CP ) =
Re(∆B̄ −∆B)
, (60)
A∆I=1CP ≡
(A+CP −A
CP ) =
Re(Ā−A)
. (61)
• ReA/BP and ReĀ/BP may be separated by using information from A∆I=1CP
and AI .
• ∆S is governed by an imaginary part of a combination of ∆I = 0 and ∆I =
1 terms which cannot be separated in B decays. Such a separation is possible
in Bs decays to pairs of isospin-reflected decays, e.g. Bs → K+K−,KSKS
or Bs → K∗+K∗−,K∗0K̄∗0, where 2β in the definition of ∆S (47) is now
replaced by the small phase of Bs-B̄s mixing.
One may take one step further under the assumption that strong phases as-
sociated with NP amplitudes are small relative to those of the SM and can be
neglected.110 This assumption, which must be confronted by data, is reasonable
because rescattering from a leading b → scc̄ amplitude is likely the main source
of strong phases, while rescattering from a smaller b → sqq̄ NP amplitude is then
a second-order effect. In the convention (55), where the strong phase of BP is set
equal to zero, ∆B and A have the same CP-conserving strong phase δ, and involve
CP-violating phases φB and φA, respectively,
∆B = |∆B|eiδeiφB , A = |A|eiδeiφA . (62)
Since the four asymmetries (56)-(59) are first order in small ratios of amplitudes,
one may take BP in their expression to be given by the square root of Γ+ or Γ0,
thereby neglecting second order terms. These four observables can then be shown to
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
22 M. Gronau
determine |A|, φA and |∆B| sinφB .49 The combination |∆B| cosφB adds coherently
to BP and cannot be fixed independently.
The amplitudes ∆B and A consist of process-dependent SM and potential NP
contributions. Assuming that the former are calculable, either using methods based
on QCD-factorization or by fitting within flavor SU(3) these and other B decay
rates and asymmetries, the four asymmetries determine the magnitude and CP
violating phase of a ∆I = 1 NP amplitude and the imaginary part of a ∆I = 0 NP
amplitude. In certain cases, e.g., B → φK or B → η′KS , stringent upper bounds on
SM contributions to ∆B and Amay suffice if some of the four measured asymmetries
are larger than permitted by these bounds. In the pair B+ → K+π0, B0 → K0π0,
the four measured asymmetries [using the predicted value (39)] are AI = 0.087 ±
0.038, A∆I=0CP = −0.047± 0.025, A∆I=1CP = 0.094± 0.025,∆S = −0.35± 0.21. Some
reduction of errors is required for a useful implementation of this method.
Conclusion: There exists ample experimental evidence in pairs of isospin-reflected
b → s penguin-dominated decays that potential NP amplitudes must be small.
Assuming that these amplitudes involve negligible strong phases, and assuming that
small SM non-penguin contributions are calculable or can be strictly bounded, one
may determine the magnitude and CP violating phase of a NP ∆I = 1 amplitude,
and the imaginary part of a NP ∆I = 0 amplitude in each pair of isospin-reflected
decays.
6.5. Null or nearly-null tests
We have not discussed null tests of the CKM framework.111 Evidence for physics
beyond the Standard Model may show-up as (small) nonzero asymmetries in pro-
cesses where they are predicted to be extremely small in the CKM framework. A
well-known example is B+ → π+π0, where the CP asymmetry is expected to be a
small fraction of a percent including EWP amplitudes.34,35 We have only discussed
exclusive hadronic B decays, where QCD calculations involve hadronic uncertain-
ties. A more robust calculation exists for the direct CP asymmetry in inclusive
radiative decays B → Xsγ, found to be smaller than one percent.112 The current
upper limit on this asymmetry is at least an order of magnitude larger.113
Time-dependent asymmetries in radiative decays B0 → KSπ0γ, for a KSπ0
invariant-mass in the K∗ region and for a larger invariant-mass range including
this region, are interesting because they test the photon helicity, predicted to be
dominantly right-handed in B0 decays and left-handed in B̄0 decays.105,114 The
asymmetry, suppressed by ms/mb, is expected to be several percent in the SM,
and can be very large in extensions where spin-flip is allowed in b → sγ. While
dimensional arguments seem to indicate a possible larger asymmetry in the SM,
of order ΛQCD/mb ∼ 10%,115 calculations using perturbative QCD116 and QCD
factorization117 find asymmetries of a few percent. The current averaged values,
for the K∗ region and for a larger invariant-mass range including this region, are
S((KSπ
0)K∗γ) = −0.28 ± 0.26 and S(KSπ0γ) = −0.09 ± 0.24.20,118 These mea-
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
CP violation in beauty decays 23
surements must be improved in order to become sensitive to the level predicted in
the SM, or to provide evidence for physics beyond the SM.
7. Summary
The Standard Model passed with great success numerous tests in the flavor sector,
including a variety of measurements of CP asymmetries related to the CKM phases
β and γ. Small potential New Physics corrections may occur in ∆S = 0 and |∆S| =
1 penguin amplitudes, affecting the extraction of γ and modifying CP-violating
and isospin-dependent asymmetries in |∆S| = 1 B0 decays and isospin-related B+
decays. Higher precision than achieved so far is required for claiming evidence for
such effects and for sorting out their isospin structure.
Similar studies can be performed with Bs mesons produced at hadron colliders
and at e+e− colliders running at the Υ(5S) resonance. Time-dependence in Bs →
D−s K
+ and Bs → J/ψφ or Bs → J/ψη measures γ and the small phase of the
Bs-B̄s mixing amplitude.
119 Comparing time-dependence and angular analysis in
Bs → J/ψφ with b → s penguin-dominated processes including Bs → φφ,Bs →
K∗+K∗−, Bs → K∗0K̄∗0 provides a methodic search for potential NP effects. Work
on Bs decays has just begun at the Tevatron.
120 One is looking forward to first
results from the LHC.
Acknowledgments
I am grateful to numerous collaborators, in particular to Jonathan Rosner whose
collaboration continued without interruption for many years. This work was sup-
ported in part by the Israel Science Foundation under Grant No. 1052/04 and by
the German-Israeli Foundation under Grant No. I-781-55.14/2003.
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