1

13. CP Violation in the Quark Sector

13. CP Violation in the Quark Sector
Revised August 2023 by T. Gershon (Warwick U.) and Y. Nir (Weizmann Inst.).
The CP transformation combines charge conjugation C with parity P . Under C, particles
and antiparticles are interchanged, by conjugating all internal quantum numbers, e.g., Q → −Q forelectromagnetic charge. Under P , the handedness of space is reversed, ~x → −~x. Thus, for example,+
a left-handed electron e−
L is transformed under CP into a right-handed positron, eR .
If CP were an exact symmetry, the laws of nature would be the same for matter and for antimatter. We observe that most phenomena are C- and P -symmetric, and therefore, also CP -symmetric.
In particular, these symmetries are respected by the electromagnetic and strong interactions. The
weak interactions, on the other hand, violate C and P in the strongest possible way. For example,
the W bosons couple to left-handed electrons, e−
L , and to their CP -conjugate right-handed positrons,+eR , but to neither their C-conjugate left-handed positrons, e+
L , nor their P -conjugate right-handed
electrons, e−
.
While
weak
interactions
violateC
andP
separately,
CP is still preserved in mostRweak interaction processes. The CP symmetry is, however, violated in certain processes involving
interference effects, as discovered in neutral K decays in 1964 [1], and established later in B (2001)
and D (2019) decays. For example, as discovered in 1967, a KL meson decays more often to π − e+ νethan to π + e− ν e , thus allowing electrons and positrons to be unambiguously distinguished, but the
decay-rate asymmetry is only at the 0.003 level. The CP -violating effects observed in the B system
are larger: the parameter describing the CP asymmetry in the decay time distribution of B 0 /B 0
meson transitions to CP eigenstates like J/ψKS is about 0.7 [2, 3]. These effects are related toK 0 –K 0 and B 0 –B 0 mixing, but CP violation arising solely from decay amplitudes has also been
observed, first in K → ππ decays [4–6], subsequently in B 0 [7, 8], B + [9–11], and Bs0 [12] decays,
and most recently in charm decays [13]. All of these observed CP asymmetries are consistent with
the Standard Model predictions. Similar effects could also occur in decays of baryons, but have
not yet been observed. Given that neutrino masses and lepton mixing have been established, it isexpected that CP is violated also in the lepton sector [14]. Discovering CP violation in the lepton
sector is one of the main goals of current and near-future experiments. CP violation has not yet
been observed in processes involving the top quark, nor in flavor-conserving processes such as electric dipole moments; for these, any significant observation would be a clear indication of physics
beyond the Standard Model.
In addition to parity and to continuous Lorentz transformations, there is one other spacetime
operation that could be a symmetry of the interactions: time reversal T , t → −t. Violations ofT symmetry have been observed in neutral K decays [15]. More recently, T violation has been
observed between states that are not CP -conjugate [16], exploiting the fact that for neutral B
mesons both flavor tagging and CP tagging can be used [17]. Moreover, T violation is expected
as a corollary of CP violation if the combined CP T transformation is a fundamental symmetry ofnature [18]. All observations indicate that CP T is indeed a symmetry of nature [15]. Furthermore,
one cannot build a locally Lorentz-invariant quantum field theory with a Hermitian Hamiltonianthat violates CP T . (At several points in our discussion, we avoid assumptions about CP T , in order
to identify cases where evidence for CP violation relies on assumptions about CP T .)Within the Standard Model, CP symmetry is broken by complex phases in the Yukawa couplings(that is, the couplings of the Higgs scalar to quarks). When all transformations to remove unphysicalphases in this model are exhausted, a single CP -violating parameter remains [19]. In the basis ofmass eigenstates, this single phase appears in the 3 × 3 unitary matrix that gives the W -boson
couplings to an up-type antiquark and a down-type quark. (If the Standard Model is supplemented
with Majorana mass terms for the neutrinos, the analogous mixing matrix for leptons has three
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:13am

2

13. CP Violation in the Quark Sector

CP -violating phases.) The beautifully consistent and economical Standard-Model description of CP
violation in terms of Yukawa couplings, known as the Kobayashi-Maskawa (KM) mechanism [19],agrees with all measurements to date. Furthermore, one can fit the data allowing contributions
from beyond the Standard Model (referred to subsequently as new physics) to loop processes tocompete with, or even dominate over, the Standard Model amplitudes [20, 21]. Such an analysis
provides model-independent proof that the KM phase is different from zero, and that the matrixof three-generation quark mixing is the dominant source of CP violation in meson decays.
The current level of experimental accuracy and the theoretical uncertainties involved in theinterpretation of the various observations leave room, however, for additional subdominant sourcesof CP violation from new physics. Indeed, almost all extensions of the Standard Model imply that
there are such additional sources. Moreover, CP violation is a necessary condition for baryogenesis, the process of dynamically generating the matter-antimatter asymmetry of the Universe [22].
Despite the phenomenological success of the KM mechanism, it fails (by several orders of magnitude) to accommodate the observed asymmetry [23]. This discrepancy strongly suggests thatnature provides additional sources of CP violation beyond the KM mechanism. The evidence forneutrino masses implies that CP can be violated also in the lepton sector. This situation makes
leptogenesis [24,25], a scenario where CP -violating phases in the Yukawa couplings of the neutrinos
play a crucial role in the generation of the baryon asymmetry, a very attractive possibility. The
expectation of new sources motivates the large ongoing experimental effort to find deviations fromthe predictions of the KM mechanism.
CP violation can be experimentally searched for in a variety of processes, such as hadron decays,
electric dipole moments of neutrons, electrons and nuclei, and neutrino oscillations. Hadron decays
via the weak interaction probe flavor-changing CP violation. The search for electric dipole moments
may find (or constrain) sources of CP violation that, unlike the KM phase, are not related to flavorchanging couplings. Following the discovery of the Higgs boson [26, 27], searches for CP violationin the Higgs sector are becoming feasible. Future searches for CP violation in neutrino oscillations
might provide further input on leptogenesis.
The present measurements of CP asymmetries provide some of the strongest constraints on the
weak couplings of quarks. Future measurements of CP violation in K, D, B, and Bs0 meson decays
will provide additional constraints on the flavor parameters of the Standard Model, and can probe
new physics. In this review, we give the formalism and basic physics motivations that are relevant
to present and near future measurements of CP violation in the quark sector.

13.1 Formalism
The phenomenology of CP violation for neutral flavored mesons is particularly interesting, since
many of the observables can be cleanly interpreted. Although the phenomenology is superficiallydifferent for K 0 , D0 , B 0 , and Bs0 decays, this is primarily because each of these systems is governedby a different balance between decay rates, oscillations, and lifetime splitting. However, the generalconsiderations presented in this section are identical for all flavored neutral pseudoscalar mesons.
The phenomenology of CP violation for neutral mesons that do not carry flavor quantum numbers(such as the η (0) state) is quite different: such states are their own antiparticles and have definite
CP eigenvalues, so the signature of CP violation is simply the decay to a final state with the
opposite CP . Such decays are mediated by the electromagnetic or (OZI-suppressed [28–30]) strong
interaction, where CP violation is not expected and has not yet been observed. In the remainderof this review, we restrict ourselves to considerations of weakly decaying hadrons.
In this section, we present a general formalism for, and classification of, CP violation in the
decay of a weakly decaying hadron, denoted M . We pay particular attention to the case that M
is a K 0 , D0 , B 0 , or Bs0 meson. Subsequent sections describe the CP -violating phenomenology,
31st May, 2024

3

13. CP Violation in the Quark Sector

approximations, and alternative formalisms that are specific to each system.
13.1.1 Charged- and neutral-hadron decays
We define decay amplitudes of M (which could be charged or neutral) and its CP conjugate M
to a multi-particle final state f and its CP conjugate f as
Af = hf |H| M i ,Af = hf |H| M i ,(13.1a)Af = hf |H| M i ,Af = hf |H| M i ,(13.1b)where H is the Hamiltonian governing weak interactions. The action of CP on these states introduces phases ξM and ξf that depend on their flavor content, according to
CP |M i = e+iξM |M i ,CP |M i = e−iξM|M i ,

CP |f i = e+iξf |f i ,CP |f i = e−iξf|f i ,(13.2a)(13.2b)so that (CP )2 = 1. The phases ξM and ξf are arbitrary and unobservable because of the flavor
symmetry of the strong interaction. If CP is conserved by the dynamics, [CP, H] = 0, then Af and
Af have the same magnitude and an arbitrary unphysical relative phase
Af = ei(ξf −ξM ) Af .

(13.3)

13.1.2 Neutral-meson mixing
A state that is initially a superposition of M 0 and M 0 , say|ψ(0)i = a(0)|M 0 i + b(0)|M 0 i ,(13.4)

will evolve in time acquiring components that describe all possible decay final states {f1 , f2 , . . .},that is,|ψ(t)i = a(t)|M 0 i + b(t)|M 0 i + c1 (t)|f1 i + c2 (t)|f2 i + · · · .
(13.5)
If we are interested in computing only the values of a(t) and b(t) (and not the values of all ci (t)),
and if the times t under study are much larger than the typical strong interaction scale, then we
can use a much simplified formalism [31]. The simplified time evolution is determined by a 2 × 2
effective Hamiltonian H that is not Hermitian, since otherwise the mesons would only oscillate andnot decay. Any complex matrix, such as H, can be written in terms of Hermitian matrices M andΓ asi
H=M− Γ.
(13.6)
2
M and Γ are associated with (M 0 , M 0 ) ↔ (M 0 , M 0 ) transitions via off-shell (dispersive), and onshell (absorptive) intermediate states, respectively. Diagonal elements of M and Γ are associatedwith the flavor-conserving transitions M 0 → M 0 and M 0 → M 0 , while off-diagonal elements areassociated with flavor-changing transitions M 0 ↔ M 0 .
The eigenvectors of H have well-defined masses and decay widths. To specify the components ofthe strong interaction eigenstates, M 0 and M 0 , in the light (ML ) and heavy (MH ) mass eigenstates,
we introduce three complex parameters: p, q, and, for the case that both CP and CP T are violated
in mixing, z. Then√√
|ML i ∝ p 1 − z |M 0 i + q 1 + z |M 0 i ,(13.7a)√√00|MH i ∝ p 1 + z |M i − q 1 − z |M i ,(13.7b)
31st May, 2024

413. CP Violation in the Quark Sector

with the normalization |q|2 + |p|2 = 1 when z = 0. (Another possible choice of labeling, which isin standard usage for K mesons, defines the mass eigenstates according to their lifetimes: KS forthe short-lived and KL for the long-lived state. The KL is experimentally found to be the heavier
state. Yet another choice is often used for the D mesons [32]: the eigenstates are labeled according
to their dominant CP content.)
The real and imaginary parts of the eigenvalues ωL,H corresponding to |ML,H i represent theirmasses and decay widths, respectively. The mass and width splittings are∆m ≡ mH − mL = Re(ωH − ωL ) ,(13.8a)∆Γ ≡ ΓH − ΓL = −2 Im(ωH − ωL ) .

(13.8b)Note that here ∆m is positive by definition, while the sign of ∆Γ must be experimentally determined. The sign of ∆Γ has not yet been established for B 0 mesons, while ∆Γ < 0 is established
0 mesons; for this reason,
for K and Bs0 mesons. The Standard Model predicts ΓL > ΓH for B(s)0 literature and is the∆Γ = ΓL − ΓH , which is still a signed quantity, is often used in the B(s)convention used in the PDG experimental summaries.
Solving the eigenvalue problem for H yields
 2M∗12 − (i/2)Γ∗12
M12 − (i/2)Γ12

q
p

=

z≡

δm − (i/2)δΓ,∆m − (i/2)∆Γ(13.9)

and(13.10)whereδm ≡ M11 − M22 ,δΓ ≡ Γ11 − Γ22(13.11)

are the differences in effective mass and decay-rate expectation values for the strong interaction
states M 0 and M 0 .
If either CP or CP T is a symmetry of H (independently of whether T is conserved or violated),then the values of δm and δΓ are both zero, and hence z = 0. We also find thatsωH − ωL = 2iM12 − Γ122iM∗12 − Γ∗12



2



.

(13.12)If either CP or T is a symmetry of H (independently of whether CP T is conserved or violated),then Γ12 /M12 is real, leading to
 2
q

p

= e2iξM

⇒

q
=1,
p

(13.13)where ξM is the arbitrary unphysical phase introduced in Eq. (13.2). If, and only if, CP is asymmetry of H (independently of CP T and T ), then both of the above conditions hold, with theresult that the mass eigenstates are orthogonal
hMH |ML i = |p|2 − |q|2 = 0 .
31st May, 2024

(13.14)

13. CP Violation in the Quark Sector

513.1.3 CP -violating observables
All CP -violating observables in M and M decays to final states f and f can be expressed interms of phase-convention-independent combinations of Af , Af , Af , and Af , together with, forneutral meson decays only, q/p. CP violation in charged meson and all baryon decays depends only
on the combination |Af /Af |, while CP violation in flavored neutral meson decays is enriched by
M 0 ↔ M 0 oscillations, and depends, additionally, on |q/p| and on λf ≡ (q/p)(Af /Af ).
The decay rates of the two neutral kaon mass eigenstates, KS and KL , are different enough(ΓS /ΓL ∼ 500) that one can, in most cases, actually study their decays independently. For D0 ,B 0 , and Bs0 mesons, however, values of ∆Γ/Γ (where Γ ≡ (ΓH + ΓL )/2) are relatively small, andso both mass eigenstates must be considered in their evolution. We denote the state of an initially0pure |M 0 i or |M 0 i after an elapsed proper time t as |Mphys(t)i or |M 0phys (t)i, respectively. Using
the effective Hamiltonian approximation, but not assuming CP T to be a good symmetry, we obtainqg− (t)|M 0 i ,ppp|M 0phys (t)i = (g+ (t) − z g− (t)) |M 0 i − 1 − z 2 g− (t)|M 0 i ,q
0|Mphys(t)i = (g+ (t) + z g− (t)) |M 0 i −

where

p

1 − z2(13.15a)(13.15b)1
1
1
g± (t) ≡exp −imH t − ΓH t ± exp −imL t − ΓL t
(13.16)2
22
and z = 0 if either CP T or CP is conserved.
Defining x ≡ ∆m/Γ and y ≡ ∆Γ/(2Γ ), and assuming z = 0, one obtains the following timedependent decay rates:










0



dΓ Mphys(t) → f /dt 
2
2
2
2
=
|A
|
+
|(q/p)A|cosh(yΓt)+|A|−|(q/p)A|cos(xΓ t)ffffe−Γ t Nf

+ 2 Re((q/p)A∗f Af ) sinh(yΓ t) − 2 Im((q/p)A∗f Af ) sin(xΓ t) ,(13.17a)dΓ M 0phys (t) → f /dt 
2
2
2
2
=
|(p/q)A|+|A|
cosh(yΓt)−|(p/q)A|−|A|
cos(xΓ t)ffffe−Γ t Nf




∗

∗

+ 2 Re((p/q)Af Af ) sinh(yΓ t) − 2 Im((p/q)Af Af ) sin(xΓ t) ,(13.17b)where Nf is a common, time-independent, normalization factor that can be determined bearing
in mind that the range of t is 0 < t < ∞. Decay rates to the CP -conjugate final state f are obtained analogously, with Nf = Nf and the substitutions Af → Af and Af → Af in Eqs. (13.17a)and (13.17b). Terms proportional to |Af |2 or |Af |2 are associated with decays that occur without
any net M 0 ↔ M 0 oscillation, while terms proportional to |(q/p)Af |2 or |(p/q)Af |2 are associated with decays following a net oscillation. The sinh(yΓ t) and sin(xΓ t) terms of Eqs. (13.17a)and (13.17b) are associated with the interference between these two cases. Note that, in multi-bodydecays such as D0 → KS π + π − or B 0 → π + π − π + π − , amplitudes are functions of variables that
describe the phase-space of the final state. Interference may be present in some regions but notothers, and is strongly influenced by resonant substructure.
When neutral pseudoscalar mesons are produced coherently in pairs from the decay of a vector
resonance, V → M 0 M 0 (for example, Υ (4S) → B 0 B 0 , ψ(3770) → D0 D0 or φ → K 0 K 0 ), the timedependence of their subsequent decays to final states f1 and f2 has a similar form to Eqs. (13.17a)
31st May, 2024

13. CP Violation in the Quark Sector6and (13.17b):







dΓ Vphys (t1 , t2 ) → f1 f2 /d(∆t) 
2
2
2
2
=
|a
|
+
|a
|
cosh(yΓ
∆t)
+
|a
|
−
|a
|
cos(xΓ ∆t)+−+−e−Γ |∆t| Nf1 f2− 2 Re(a∗+ a− ) sinh(yΓ ∆t) + 2 Im(a∗+ a− ) sin(xΓ ∆t) ,(13.18)where ∆t ≡ t2 − t1 is the difference in the production times, t1 and t2 , of f1 and f2 , respectively,
and the dependence on the average decay time and on decay angles has been integrated out. The
normalization factor Nf1 f2 can be evaluated, noting that the range of ∆t is −∞ < ∆t < ∞. The
coefficients in Eq. (13.18) are determined by the amplitudes for no net oscillation from t1 → t2 ,Af1 Af2 , and Af1 Af2 , and for a net oscillation, (q/p)Af1 Af2 and (p/q)Af1 Af2 , via
a+ ≡ Af1 Af2 − Af1 Af2 ,




p
q
p
2
a− ≡ − 1 − z
Af1 Af2 − Af1 Af2 + z Af1 Af2 + Af1 Af2 .
pq(13.19a)(13.19b)

Assuming CP T conservation, z = 0, and identifying ∆t → t and f2 → f , we find that
Eqs. (13.18) and (13.19) reduce to Eq. (13.17a) with Af1 = 0, Af1 = 1, or to Eq. (13.17b) with
Af1 = 0, Af1 = 1. Indeed, this plays an important role in experiments that exploit the coherence of
V → M 0 M 0 production. Final states f1 with Af1 = 0 or Af1 = 0 are called tagging states, because
they identify the decaying pseudoscalar meson as, respectively, M 0 or M 0 . Before one of M 0 orM 0 decays, they evolve in phase, so that there is always one M 0 and one M 0 present. A tagging
decay of one meson sets the clock for the time evolution of the other: it starts at t1 as purely M 0or M 0 , with time evolution that depends only on t2 − t1 .
When f1 is a state that both M 0 and M 0 can decay into, then Eq. (13.18) contains interferenceterms proportional to Af1 Af1 6= 0 that are not present in Eqs. (13.17a) and (13.17b). Even when
f1 is dominantly produced by M 0 decays rather than M 0 decays, or vice versa, Af1 Af1 can be
non-zero owing to doubly-CKM-suppressed decays (with amplitudes suppressed by at least two
powers of λ relative to the dominant amplitude, in the language of Section 13.3), and these terms
should be considered for precision studies of CP violation in coherent V → M 0 M 0 decays [33]. The
correlations in V → M 0 M 0 decays can also be exploited to determine phase differences between
favored and suppressed decay amplitudes [34, 35].
13.1.4 Classification of CP -violating effects
We distinguish three types of CP -violating effects that can occur in the quark sector:
I. CP violation in decay is defined by|Af /Af | =6 1.

(13.20)In charged meson (and all baryon) decays, where mixing effects are absent, this is the only
possible source of CP asymmetries:
Af ± ≡|Af − /Af + |2 − 1
Γ (M − → f − ) − Γ (M + → f + )=.
Γ (M − → f − ) + Γ (M + → f + )|Af − /Af + |2 + 1(13.21)

Note that the usual sign convention for CP asymmetries of hadrons is for the difference
between the rate involving the particle that contains a heavy quark and that which contains
an antiquark. Hence, Eq. (13.21) corresponds to the definition for B ± mesons, but the±
opposite sign is used for D(s)decays.
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13. CP Violation in the Quark Sector

7

II. CP (and T ) violation in mixing is defined by|q/p| =
6 1.

(13.22)In charged-current semileptonic neutral meson decays M, M → `± X ∓ (taking |A`+ X − | =|A`− X + | and A`− X + = A`+ X − = 0, as is the case in the Standard Model, to lowest orderin GF , and in most of its extensions), this is the only source of CP violation, and can bemeasured via the asymmetry of “wrong-sign” decays induced by oscillations:0dΓ/dt M 0phys (t) → `+ X − − dΓ/dt Mphys(t) → `− X +

ASL (t) ≡
=





0
dΓ/dt M 0phys (t) → `+ X − + dΓ/dt Mphys(t) → `− X +








,

1 − |q/p|4.
1 + |q/p|4(13.23a)(13.23b)Note that this asymmetry of time-dependent decay rates is actually time-independent.
III. CP violation in interference between a decay without mixing, M 0 → f , and a decay withmixing, M 0 → M 0 → f (such an effect occurs only in decays to final states that are common
to M 0 and M 0 , including all CP eigenstates), is defined byarg(λf ) + arg(λf¯) 6= 0 ,

with

λf ≡

q Af
.
p Af

(13.24)For final CP eigenstates, fCP , the condition Eq. (13.24) simplifies toIm(λfCP ) 6= 0 ,(13.25)This form of CP violation can be observed, for example, using the asymmetry of neutralmeson decay rates into CP eigenstates0dΓ/dt M 0phys (t) → fCP − dΓ/dt Mphys(t) → fCP



AfCP (t) ≡0dΓ/dt M 0phys (t) → fCP + dΓ/dt Mphys(t) → fCP








 .

(13.26)

If ∆Γ = 0, as expected to a good approximation for B 0 mesons but not for K 0 and Bs0
mesons, and |q/p| = 1, then AfCP has a particularly simple form (see Eq. (13.75), below).
If, in addition, the decay amplitudes fulfill |AfCP | = |AfCP |, the interference between decays
with and without mixing is the only source of asymmetry and AfCP (t) = Im(λfCP ) sin(xΓ t).
Examples of these three types of CP violation will be given in Sections 13.4, 13.5, and 13.6.

13.2 Theoretical Interpretation: General Considerations
Consider the M → f decay amplitude Af , and the CP conjugate process, M → f , with decay
amplitude Af . There are two types of phases that may appear in these decay amplitudes. Complex
parameters in any Lagrangian term that contributes to the amplitude will appear in complex
conjugate form in the CP -conjugate amplitude. Thus, their phases appear in Af and Af with
opposite signs. In the Standard Model, these phases occur only in the couplings of the W ± bosons,and hence, are often called “weak phases.” The weak phase of any single term is conventiondependent. However, the difference between the weak phases in two different terms in Af isconvention-independent. A second type of phase can appear in scattering or decay amplitudes, evenwhen the Lagrangian is real. This phase originates from the possible contribution from intermediateon-shell states in the decay process. Since such phases are generated by CP -invariant interactions,
31st May, 2024

813. CP Violation in the Quark Sector

they are the same in Af and Af . Usually the dominant rescattering is due to strong interactions;hence the designation “strong phases” for the phase shifts so induced. Again, only the relativestrong phases between different terms in the amplitude are physically meaningful.
The “weak” and “strong” phases discussed here appear in addition to the spurious CP -transformation
phases of Eq. (13.3). Those spurious phases are due to an arbitrary choice of phase convention,and do not originate from any dynamics or induce any CP violation. For simplicity, we set themto zero from here on.
It is useful to write each contribution ai to Af in three parts: its magnitude |ai |, its weak phase
φi , and its strong phase δi . If, for example, there are two such contributions, Af = a1 + a2 , we have
Af = |a1 |ei(δ1 +φ1 ) + |a2 |ei(δ2 +φ2 ) ,(13.27a)

Af = |a1 |ei(δ1 −φ1 ) + |a2 |ei(δ2 −φ2 ) .

(13.27b)

Similarly, for neutral mesons, it is useful to write
M12 = |M12 |eiφM ,Γ12 = |Γ12 |eiφΓ .

(13.28)Each of the phases appearing in Eqs. (13.27) and (13.28) is convention-dependent, but combinations
such as δ1 − δ2 , φ1 − φ2 , φM − φΓ , and φM + φ1 − φ1 (where φ1 is a weak phase contributing toAf ) are physical.
It is now straightforward to evaluate the various asymmetries in terms of the theoretical parameters introduced here. We will do so with approximations that are often relevant to the mostinteresting measured asymmetries.
1. The CP asymmetry in charged meson and all baryon decays [Eq. (13.21)] is given by
Af = −

2|a1 a2 | sin(δ2 − δ1 ) sin(φ2 − φ1 ).
2
|a1 | + |a2 |2 + 2|a1 a2 | cos(δ2 − δ1 ) cos(φ2 − φ1 )(13.29)The quantity of most interest to theory is the weak phase difference φ2 − φ1 . Its extraction fromthe asymmetry requires, however, that the amplitude ratio |a2 /a1 | and the strong phase differenceδ2 − δ1 are known. Both quantities depend on non-perturbative hadronic parameters that are
difficult to calculate, but in some cases can be obtained from experiment.
2. In the approximation that |Γ12 /M12 |  1 (valid for B 0 and Bs0 mesons), the CP asymmetry
in semileptonic neutral-meson decays [Eq. (13.23)] is given by
ASL = −

Γ12
sin(φM − φΓ ) .
M12(13.30)The quantity of most interest to theory is the weak phase φM − φΓ . Its extraction from theasymmetry requires, however, that |Γ12 /M12 | is known. State of the art calculations of this quantity
for the B 0 and Bs0 mesons have uncertainties of around 10% [36].
3. In the approximations that only a single weak phase contributes to decay, Af = |af |ei(δf +φf ) ,and that |Γ12 /M12 | = 0, we obtain |λf | = 1, and the CP asymmetries in decays to a final CP
eigenstate f [Eq. (13.26)] with eigenvalue ηf = ±1 are given by
AfCP (t) = Im(λf ) sin(∆mt) with Im(λf ) = ηf sin(φM + 2φf ) .

(13.31)

Note that the phase measured is purely a weak phase, and no hadronic parameters are involved in
the extraction of its value from Im(λf ) .
The discussion above allows us to introduce another classification of CP -violating effects:
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13. CP Violation in the Quark Sector

9

1. Indirect CP violation is consistent with taking φM 6= 0 and setting all other CP violatingphases to zero. CP violation in mixing (type II) belongs to this class.
2. Direct CP violation cannot be accounted for by just φM 6= 0. CP violation in decay (type I)
belongs to this class.
The historical significance of this classification is related to theory. In superweak models [37], CP
violation appears only in diagrams that contribute to M12 , hence predicting no direct CP violation.
In most models and, in particular, in the Standard Model, CP violation is both direct and indirect.
As concerns type III CP violation, a single observation of such an effect would be consistent withindirect CP violation, but observing ηf1 Im(λf1 ) 6= ηf2 Im(λf2 ) (for the same decaying meson andtwo different final CP eigenstates f1 and f2 ) would establish direct CP violation. The experimental
observation of 0 6= 0, which was achieved by establishing that Im(λπ+ π− ) 6= Im(λπ0 π0 ) (seeSection 13.4), excluded the superweak scenario.

13.3 Theoretical Interpretation: The KM Mechanism
Of all the Standard Model quark parameters, only the Kobayashi-Maskawa (KM) phase is CP violating. Having a single source of CP violation, the Standard Model is very predictive for CP
asymmetries: some vanish, and those that do not are correlated.
To be precise, CP could be violated also by strong interactions. The experimental upper boundon the electric-dipole moment of the neutron [38] implies, however, that θQCD , the non-perturbative
parameter that determines the strength of this type of CP violation, is tiny, if not zero [39]. The
smallness of θQCD constitutes a theoretical puzzle, known as “the strong CP problem.” This,
however, is irrelevant to our discussion of hadron decays.
The charged current interactions (that is, the W ± interactions) for quarks are given byg− LW ± = √ uLi γ µ (VCKM )ij dLj Wµ+ + h.c.
2

(13.32)

Here i, j = 1, 2, 3 are generation numbers. The Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix
for quarks is a 3 × 3 unitary matrix [40]. Ordering the quarks by their masses, i.e., (u1 , u2 , u3 ) →(u, c, t) and (d1 , d2 , d3 ) → (d, s, b), the elements of VCKM are written as follows:

Vud Vus Vub

VCKM =  Vcd Vcs Vcb  .
Vtd Vts Vtb(13.33)

While a general 3 × 3 unitary matrix depends on three real angles and six phases, the freedom toredefine the phases of the quark mass eigenstates can be used to remove five of the phases, leaving
a single physical phase, the Kobayashi-Maskawa phase, that is responsible for all CP violation in
the Standard Model.
The fact that one can parameterize VCKM by three real and only one imaginary physical parameters can be made manifest by choosing an explicit parametrization. The Wolfenstein parametrization [41, 42] is particularly useful:

1 − 12 λ2 − 18 λ4λAλ3 (ρ − iη)
1 − 12 λ2 − 81 λ4 (1 + 4A2 )Aλ2
VCKM =  −λ + 12 A2 λ5 [1 − 2(ρ + iη)]
 . (13.34)
Aλ3 [1 − (1 − 12 λ2 )(ρ + iη)] −Aλ2 + 12 Aλ4 [1 − 2(ρ + iη)] 1 − 12 A2 λ4
Here λ ≈ 0.23 (not to be confused with λf ), the sine of the Cabibbo angle, plays the role of an
expansion parameter, and η represents the CP -violating phase. Terms of O(λ6 ) have been neglected.
31st May, 2024

13. CP Violation in the Quark Sector

10

The unitarity of the CKM matrix, (V V † )ij = (V † V )ij = δij , leads to twelve distinct complex
relations among the matrix elements. The six relations with i 6= j can be represented geometricallyas triangles in the complex plane. Two of these,∗
Vud Vub+ Vcd Vcb∗ + Vtd Vtb∗ = 0 ,(13.35a)∗∗∗Vtd Vud+ Vts Vus
+ Vtb Vub=0,

(13.35b)have terms of equal order, O(Aλ3 ), and so have corresponding triangles whose interior angles are
all O(1) physical quantities that can be independently measured. The angles of the first triangle(see Fig. 13.1) are given by
V V∗
α ≡ ϕ2 ≡ arg − td tb∗
Vud Vub!

V V∗
β ≡ ϕ1 ≡ arg − cd cb∗
Vtd Vtb!

V V∗
γ ≡ ϕ3 ≡ arg − ud ub
Vcd Vcb∗!

1 − ρ − iη
' arg −
ρ + iη


1' arg
1 − ρ − iη,

(13.36a)' arg (ρ + iη) .

,(13.36b)(13.36c)

The angles of the second triangle are equal to (α, β, γ) up to corrections of O(λ2 ). The notations(α, β, γ) and (ϕ1 , ϕ2 , ϕ3 ) are both in common usage but, for convenience, we only use the first
convention in the following.

VtdVtb*
VudVub*

α=ϕ2

β=ϕ1

γ = ϕ3

VcdVcb*∗ + V V ∗ + V V ∗ = 0 as
Figure 13.1: Graphical representation of the unitarity constraint Vud Vub
cd cb
td tb
a triangle in the complex plane.

Another relation that can be represented as a triangle,∗
Vus Vub+ Vcs Vcb∗ + Vts Vtb∗ = 0 ,(13.37)

and, in particular, its small angle, of O(λ2 ),V V∗
βs ≡ arg − ts tb∗
Vcs Vcb!

is convenient for analyzing CP violation in the Bs0 sector.
31st May, 2024,

(13.38)

11

13. CP Violation in the Quark Sector

All unitarity triangles have the same area, commonly denoted by J/2 [43]. If CP is violated, J
is different from zero and can be taken as the single CP -violating parameter. In the Wolfensteinparametrization of Eq. (13.34), J ' λ6 A2 η.

13.4 Kaons
CP violation was discovered in K → ππ decays in 1964 [1]. The same mode provided the firstobservation of direct CP violation [4–6].
The decay amplitudes actually measured in neutral K decays refer to the mass eigenstates KL
and KS , rather than to the K and K states referred to in Eq. (13.1). The final π + π − and π 0 π 0states are CP -even. In the CP conservation limit, KS (KL ) would be CP -even (odd), and therefore
would (would not) decay to two pions. We define CP -violating amplitude ratios for two-pion final
states,hπ + π − |H|KL ihπ 0 π 0 |H|KL i, η+− ≡ + −.
(13.39)η00 ≡ 0 0hπ π |H|KS ihπ π |H|KS i
Another important observable is the asymmetry of time-integrated semileptonic decay rates:δL ≡Γ (KL → `+ ν` π − ) − Γ (KL → `− ν ` π + ).
Γ (KL → `+ ν` π − ) + Γ (KL → `− ν ` π + )(13.40)CP violation has been observed as an appearance of KL decays to two-pion final states [44],|η00 | = (2.220 ± 0.011) × 10−3 ,(13.41a)|η+− | = (2.232 ± 0.011) × 10−3 ,(13.41b)|η00 /η+− | = 0.9950 ± 0.0007 ,(13.41c)where the CP -conserving phases φij of the amplitude ratios ηij have been determined both assuming
CP T invariance:
φ00 = (43.52 ± 0.05)◦ ,φ+− = (43.51 ± 0.05)◦ ,(13.42)and without assuming CP T invariance:
φ00 = (43.7 ± 0.6)◦ ,φ+− = (43.4 ± 0.5)◦ .

(13.43)CP violation has also been observed in semileptonic KL decays [44]
δL = (3.32 ± 0.06) × 10−3 ,(13.44)

where δL is a weighted average of muon and electron measurements, as well as in KL decays toπ + π − γ and π + π − e+ e− [44]. CP violation in K → 3π decays has not yet been observed [44, 45].
Historically, CP violation in neutral K decays has been described in terms of the complex
parameters  and 0 . The observables η00 , η+− , and δL are related to these parameters, and to
those of Section 13.1, by
1 − λπ0 π0=1 + λπ 0 π 0
1 − λπ+ π−
η+− ==1 + λπ+ π− − 20 ,(13.45a) + 0 ,(13.45b)

1 − |q/p|22Re(),
2 =1 + |q/p|
1 + ||2(13.45c)

η00 =

δL =

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1213. CP Violation in the Quark Sector

where, in the last line, we have assumed that A`+ ν` π− = A`− ν ` π+ and A`− ν ` π+ = A`+ ν` π− = 0.
(The convention-dependent parameter ˜ ≡ (1 − q/p)/(1 + q/p), sometimes used in the literature,is, in general, different from  but yields a similar expression, δL = 2Re(˜)/(1 + |˜
|2 ).) A fit to the
K → ππ data yields [44]|| = (2.228 ± 0.011) × 10−3 ,0−3Re( /) = (1.66 ± 0.23) × 10

(13.46a).

(13.46b)

In discussing two-pion final states, it is useful to express the amplitudes Aπ0 π0 and Aπ+ π− interms of their isospin components via
r

r

1
2
Aπ 0 π 0 =|A0 | ei(δ0 +φ0 ) −|A2 | ei(δ2 +φ2 ) ,
3
3
r
r
2
1
i(δ0 +φ0 )Aπ + π − =|A0 | e+|A2 | ei(δ2 +φ2 ) ,3
3(13.47a)(13.47b)where we parameterize the amplitude AI (AI ) for K 0 (K 0 ) decay into two pions with total isospin
I = 0 or 2 asAI ≡ h(ππ)I |H| K 0 i = |AI | ei(δI +φI ) ,i(δI −φI )0AI ≡ h(ππ)I |H| K i = |AI | e.

(13.48a)(13.48b)

The smallness of |η00 | and |η+− | allows us to approximate1 ' (1 − λ(ππ)I=0 ) ,
2

0 '

1
(λ 0 0 − λπ+ π− ) .
6 π π(13.49)The parameter  represents indirect CP violation, while 0 parameterizes direct CP violation: Re(0 )measures CP violation in decay (type I), Re() measures CP violation in mixing (type II), and Im()and Im(0 ) measure the interference between decays with and without mixing (type III).
The following expressions for  and 0 are useful for theoretical evaluations:
eiπ/4 Im(M12 )
' √
,
∆m
2

i A2 i(δ2 −δ0 )0 = √esin(φ2 − φ0 ) .
2 A0(13.50)The expression for  is only valid in a phase convention where φ2 = 0, corresponding to a real
∗ , and in the approximation that also φ = 0. The phase of , arg() ≈ arctan(−2∆m/∆Γ ),Vud Vus0
is determined by non-perturbative QCD dynamics and is experimentally determined to be about
π/4. The calculation of  benefits from the fact that Im(M12 ) is dominated by short distancephysics. Consequently, the main sources of uncertainty in theoretical interpretations of  are thevalues of matrix elements, such as hK 0 |(sd)V −A (sd)V −A | K 0 i. The expression for 0 is valid to firstorder in |A2 /A0 | ∼ 1/20. The phase of 0 is experimentally determined, π/2 + δ2 − δ0 ≈ π/4, andis independent of the model of electroweak interactions. Note that, accidentally, 0 / is real to agood approximation. Determination of weak phase information from the measurement of Re(0 /)given in Eq. (13.46) has until now been precluded by uncertainties in the hadronic parameters, but
recent advances in lattice QCD calculations and other theoretical approaches [46–48] suggest thatit may become possible.
A future measurement of much interest is that of CP violation in the rare K → πνν decays.
The signal for CP violation is simply observing the KL → π 0 νν decay. The effect here is that ofinterference between decays with and without mixing (type III) [49]:iΓ (KL → π 0 νν)
1h
2
=
1
+
|λ
' 1 − Re(λπνν ) ,
|
−
2
Re(λ
)
πνν
πνν
Γ (K + → π + νν)2
31st May, 2024

(13.51)

13. CP Violation in the Quark Sector

13

where in the last equation we neglect CP violation in decay and in mixing (expected, modelindependently, to be of order 10−5 and 10−3 , respectively). Such a measurement is experimentallyvery challenging but would be theoretically very rewarding [50]. Similar to the CP asymmetry inB 0 → J/ψKS , the CP violation in K → πνν decay is predicted to be large (that is, the ratio in
Eq. (13.51) is neither CKM- nor loop-suppressed) and can be very cleanly interpreted. In particular,
the independent determinations of the CKM parameters via B-meson and K-meson decays andmixing will over-constrain the unitarity triangle and provide a stringent text of the KM mechanism.
Within the Standard Model, the KL → π 0 νν decay is dominated by an intermediate top quark
contribution and, consequently, can be interpreted in terms of CKM parameters [51]. (For the
charged mode, K + → π + νν, the contribution from an intermediate charm quark is not negligible, and constitutes a source of hadronic uncertainty.) In particular, B(KL → π 0 νν) provides atheoretically clean way to determine the Wolfenstein parameter η [52]:
B(KL → π 0 νν) = κL [X(m2t /m2W )]2 A4 η 2 ,(13.52)where the hadronic parameter κL ∼ 2 × 10−10 incorporates the value of the four-fermion matrix
element which is deduced, using isospin relations, from B(K + → π 0 e+ νe ), and X(m2t /m2W ) is aknown function of the top mass. An explicit calculation gives B(KL → π 0 νν) = (3.00 ± 0.30) ×
10−11 [53].
Currently the most stringent experimental limit is B(KL → π 0 νν) < 3.0 × 10−9 [54, 55], which
does not yet reach the bound that can be derived from Eq. (13.51), B(KL → π 0 νν) < 4.4×B(K + →+++π νν) [49], with the most precise result for the charged kaon decay being B(K → π νν) =+4.0
10.6 −3.4± 0.9 ×10−11 [56]. Significant further progress is anticipated from experiments searchingfor K → πνν decays in the next few years.

13.5 Charm
The existence of D0 –D0 mixing is well established [57–61], with the latest experimental constraints giving [62, 63] x ≡ ∆m/Γ = (0.407 ± 0.044) × 10−2 and y ≡ ∆Γ/(2Γ ) = (0.647 ± 0.024) ×
10−2 . Long-distance contributions make it difficult to calculate Standard Model predictions for the
D0 –D0 mixing parameters. Therefore, the goal of the search for D0 –D0 mixing is not to constrain
the CKM parameters, but rather to probe new physics. Here CP violation plays an important role.
Within the Standard Model, the CP -violating effects are predicted to be small, since the mixing
and the relevant decays are described, to an excellent approximation, by the physics of the firsttwo generations only. The expectation is that the Standard Model size of CP violation in D decaysis O(10−3 ) or less. At present, the most sensitive searches involve the D0 → K + K − , D0 → π + π −and D0 → K ± π ∓ modes.
The neutral D mesons decay via a singly-Cabibbo-suppressed transition to the CP eigenstates
K + K − and π + π − . These decays are dominated by Standard-Model tree diagrams. Thus, we canwrite, for f = K + K − or π + π − ,Thi(13.53a)

−iφT
f

h

i

(13.53b)

Af = ATf e+iφf 1 + rf ei(δf +φf ) ,Āf = ATf e

1 + rf ei(δf −φf ) ,T

where ATf e±iφf is the Standard Model tree-level contribution, φTf and φf are weak, CP violatingphases, δf is a strong phase difference, and rf is the ratio between a subleading (rf  1) contribution with a weak phase different from φTf and the Standard Model tree-level contribution.
Neglecting rf , λf is universal, and we can define an observable phase φD via
λf ≡ −|q/p|eiφD .
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(13.54)

14

13. CP Violation in the Quark Sector(In the limit of CP conservation, choosing φD = 0 is equivalent to defining the mass eigenstatesby their CP eigenvalue: |D∓ i = p|D0 i ± q|D0 i, with D− (D+ ) being the CP -odd (CP -even) state;that is, the state that does not (does) decay into K + K − .)We define the time integrated CP asymmetry for a final CP eigenstate f as follows:
R∞

R∞
0
0
0 Γ (Dphys (t) → f )dt − 0 Γ (D phys (t) → f )dt
R
af ≡ R ∞.
∞000 Γ (Dphys (t) → f )dt + 0 Γ (D phys (t) → f )dt(13.55)(This expression corresponds to the D meson being tagged at production, hence the integration goesfrom 0 to +∞; measurements are also possible with ψ(3770) → D0 D0 , in which case the integration
goes from −∞ to +∞ giving slightly different results; see the discussion in Section 13.1.3.) We
take x, y, rf  1 and expand to leading order in these parameters. We can then separate thecontribution to af into three parts [64],iaf = adf + am
f + af ,(13.56)

with the following underlying mechanisms:
1. adf signals CP violation in decay (similar to Eq. (13.21)):
adf = 2rf sin φf sin δf .

(13.57)

2. am
f signals CP violation in mixing (similar to Eq. (13.30)). With our approximations, it isuniversal:


y q
p
m
a =−
−
cos φD .
(13.58)2 p
q
3. aif signals CP violation in the interference of mixing and decay (similar to Eq. (13.31)). With
our approximations, it is universal:
x
a =
2
i



q
p
+
p
q



sin φD .

(13.59)In the SM, both am and ai are O(10−5 ) or less, while ad could be up to two orders of magnitudelarger.
One can isolate the effects of direct CP violation by taking the difference between the CP
asymmetries in the K + K − and π + π − modes:∆aCP ≡ aK + K − − aπ+ π− = adK + K − − adπ+ π− ,(13.60)where we neglected a residual, experiment-dependent, contribution from indirect CP violation due
to the fact that there may be a decay time-dependent acceptance function that can be different for
the K + K − and π + π − channels. The current average gives [13, 62]:
adK + K − − adπ+ π− = (−0.161 ± 0.028) × 10−2 ,(13.61)demonstrating CP violation in charm decay. While the asymmetry is somewhat larger than thetheoretical predictions that preceded the measurement, it can in principle be explained by nonperturbative QCD effects.
One can also isolate the effects of indirect CP violation in the following way. Consider thetime-dependent decay rates in Eq. (13.17a) and Eq. (13.17b). The mixing processes modify the
31st May, 2024

13. CP Violation in the Quark Sector

15

time dependence from a pure exponential. However, given the small values of x and y, the time dependences can be recast, to a good approximation, into purely exponential form, but with modified
decay-rate parameters [65, 66] (given here for the K + K − final state):
ΓD0 →K + K − = Γ × [1 + |q/p| (y cos φD − x sin φD )] ,(13.62a)

ΓD0 →K + K − = Γ × [1 + |p/q| (y cos φD + x sin φD )] .

(13.62b)

One can define CP -conserving and CP -violating combinations of these two observables (normalized
to the true width Γ ):
ΓD0 →K + K − + ΓD0 →K + K −−12Γ
= (y/2) (|q/p| + |p/q|) cos φD − (x/2) (|q/p| − |p/q|) sin φD ,
Γ 0
+ − − Γ 0
D →K + K −AΓ ≡ D →K K
2Γ= −(am + ai ) .

yCP ≡(13.63a)(13.63b)In the limit of CP conservation (and, in particular, within the Standard Model), yCP = (Γ+ −Γ− )/2Γ = y (where Γ+ (Γ− ) is the decay width of the CP -even (-odd) mass eigenstate) and AΓ = 0.
Indeed, present measurements imply that CP violation is small [62],yCP − yCP (Kπ) = (+0.697 ± 0.028) × 10−2 ,−2AΓ = (0.009 ± 0.011) × 10

(13.64a),(13.64b)where the correction yCP (Kπ) is necessary at high precision since experimentally the denominatorof the relative widths in Eq. (13.63a) is measured with the D0 → K − π + mode [67, 68].
The K ± π ∓ states are not CP eigenstates, but they are still common final states for D0 and0D decays. Since D0 (D0 ) → K − π + is a Cabibbo-favored (doubly-Cabibbo-suppressed) process,these processes are particularly sensitive to x and/or y = O(λ2 ). Taking into account that−1
 1 and x, y  1, assuming that there is no direct CP violation (these are|λK − π+ | , λK
+ π−Standard Model tree-level decays dominated by a single weak phase, and there is no contribution
from penguin-like and chromomagnetic operators), and expanding the time-dependent rates for−1
xt, yt <∼ Γ , one obtains0Γ [Dphys(t) → K + π − ] =e−Γ t |AK − π+ |2"#

rd2 + rd×q
q 2 y 2 + x2
(y 0 cos φD − x0 sin φD )Γ t +(Γ t)2 ,pp
4(13.65a)Γ [D0phys (t) → K − π + ] =e−Γ t |AK − π+ |2"×

rd2 + rdp 0
p 2 y 2 + x2
(y cos φD + x0 sin φD )Γ t +(Γ t)2qq
4#, (13.65b)

where
y 0 ≡ y cos δ − x sin δ

andx0 ≡ x cos δ + y sin δ .

(13.66)

The weak phase φD is the same as that of Eq. (13.54) (a consequence of neglecting direct CP
violation) and rd = O(tan2 θc ) is the amplitude ratio, rd = AK − π+ /AK − π+ = AK + π− /AK + π− ,−i(δ+φD ) . The parameter δ is a strongthat is, λK − π+ = rd |q/p|e−i(δ−φD ) and λ−1
K + π − = rd |p/q|ephase difference for these processes, that can be obtained from measurements of quantum correlated
31st May, 2024

16

13. CP Violation in the Quark Sector

ψ(3770) → D0 D0 decays [69, 70]. By fitting to the six coefficients of the various time-dependences,
one can determine rd , |q/p|, (x2 + y 2 ), y 0 cos φD , and x0 sin φD . In particular, finding CP violation(|q/p| =6 1 and/or sin φD 6= 0) at a level much higher than 10−3 would constitute evidence for
new physics. The most stringent constraints to date on CP violation in charm mixing have beenobtained with this method [71] and from the AΓ measurement [72].
A fit to all data [62], including also results from time-dependent analyses of D0 → KS π + π −decays, from which x, y, |q/p| and φD can be determined directly, yields no evidence for indirect
CP violation:
1 − |q/p| = +0.006 +0.015−0.016 ,(13.67a)◦(13.67b)

φD = (−2.6 ± 1.1) .

With the additional assumption of no direct CP violation in doubly-Cabibbo-suppressed D decays [73–75], more stringent constraints are obtained:
1 − |q/p| = −0.005 ± 0.007 ,◦φD = (−0.2 ± 0.3) .

(13.68a)(13.68b)More details on various theoretical and experimental aspects of D0 –D0 mixing can be found in
Ref. [32].
Searches for CP violation in charged D(s) decays have been performed in many modes. Searches
in decays mediated by Cabibbo-suppressed amplitudes are particularly interesting, since in otherchannels effects are likely to be too small to be observable in current experiments. Examples of
relevant two-body modes are D+ → π + π 0 , KS K + , φπ + and Ds+ → K + π 0 , KS π + , φK + . The mostprecise results are AD+ →KS K + = +0.0011 ± 0.0017 and ADs+ →KS π+ = +0.0038 ± 0.0048 [62]. The
precision of experiments is now sufficient that the effect from CP violation in the neutral kaon
system can be seen in D+ → KS π + decays [76, 77].
Three- and four-body final states provide additional possibilities to search for CP violation,
since effects may vary over the phase-space [78]. A number of methods have been proposed to
exploit this feature and search for CP violation in ways that do not require modelling of the
decay distribution [79–82]. Such methods are useful for analysis of charm decays since they areless sensitive to biases from production asymmetries, and are well suited to address the issue of
whether or not CP violation effects are present. They can also be applied to tagged neutral D
meson as well as to charged D(s) decays (flavor tagging is typically achieved from the charge of thepion produced in D∗+ → D0 π + decays). The results of all searches to date are consistent with theabsence of CP violation, with the most significant hint at the level of 2.7σ [83].

13.6 Beauty
13.6.1 CP violation in mixing of B 0 and Bs0 mesons
The upper bound on the CP asymmetry in semileptonic B decays [84] implies that CP violationin B 0 –B 0 mixing is a small effect (we use ASL /2 ≈ 1 − |q/p|, see Eq. (13.23)):AdSL = (−2.1 ± 1.7) × 10−3 =⇒ |q/p| = 1.0010 ± 0.0008 .

(13.69)The Standard Model prediction ishi

AdSL = O (m2c /m2t ) sin β <∼ 0.001 .
An explicit calculation gives (−4.73 ± 0.42) × 10−4 [36].
31st May, 2024

(13.70)17

13. CP Violation in the Quark Sector

The experimental constraint on CP violation in Bs0 –B 0s mixing is somewhat weaker than thatin the B 0 –B 0 system [84]AsSL = (−0.6 ± 2.8) × 10−3 =⇒ |q/p| = 1.0003 ± 0.0014 .

(13.71)−4The Standard Model prediction is AsSL = O (m2c /m2t ) sin βs <∼ 10 , with an explicit calculation
giving (2.06 ± 0.18) × 10−5 [36].
The fit to experimental data that results in the averages quoted above has a χ2 probability of
4.5% indicating some tension between the different measurements [62]. This originates in part froma result from the D0 collaboration for the inclusive same-sign dimuon asymmetry that deviates fromthe Standard Model prediction by 3.6σ [85]. As yet, this has not been confirmed by independentstudies.
In models where Γ12 /M12 is approximately real, such as the Standard Model, an upper bound on∆Γ/∆m ≈ Re(Γ12 /M12 ) provides yet another upper bound on the deviation of |q/p| from one. This
constraint does not hold if Γ12 /M12 is approximately imaginary. (An alternative parameterization
uses q/p = (1 − ˜B )/(1 + ˜B ), leading to ASL ' 4Re(˜B ).)



13.6.2 CP violation in interference of B 0 decays with and without mixing
The small deviation (less than one percent) of |q/p| from 1 implies that, at the present level ofexperimental precision, CP violation in B 0 mixing is a negligible effect. Thus, for the purpose ofanalyzing CP asymmetries in hadronic B 0 decays, we can use
λf = e−iφM (B 0 )(Af /Af ) ,(13.72)where φM (B 0 ) refers to the phase of M12 appearing in Eq. (13.28) that is appropriate for B 0 –B 0
oscillations. Within the Standard Model, the corresponding phase factor is given bye−iφM (B 0 )= (Vtb∗ Vtd )/(Vtb Vtd∗ ) .

(13.73)The class of CP violation effects in interference between mixing and decay is studied with final
states that are common to B 0 and B 0 decays [86–88]. It is convenient to rewrite Eq. (13.26) forB 0 decays as [89–91]
Af (t) = Sf sin(∆mt) − Cf cos(∆mt) ,(13.74)2 Im(λf )Sf ≡,1 + |λf |2

1 − |λf |2
Cf ≡,
1 + |λf |2(13.75)where we assume that ∆Γ = 0 and |q/p| = 1. An alternative notation in use is Af ≡ −Cf – thisAf should not be confused with the Af of Eq. (13.1), but in the limit that |q/p| = 1 is equivalent
with the Af of Eq. (13.21).
A large class of interesting processes proceed via quark transitions of the form b → qqq 0 with0
q = s or d. For q = c or u, there are contributions from both tree (t) and penguin (pqu , where
qu = u, c, t is the quark in the loop) diagrams (see Fig. 13.2) which carry different weak phases:Af = Vqb∗ Vqq0 tf +

X





Vq∗u b Vqu q0 pqfu .

(13.76)

qu =u,c,t(The distinction between tree and penguin contributions is a heuristic one; the separation by theoperator that enters is more precise. A more detailed discussion of the operator product expansion
approach, which also includes higher order QCD corrections, can be found in Ref. [92, 93] forexample.) Using CKM unitarity, the various decay amplitudes can always be written in terms of
31st May, 2024

18

13. CP Violation in the Quark Sector

just two CKM combinations. For example, for f = ππ, which proceeds via a b → uud transition,
we can write∗tAππ = (VubVud ) Tππ + (Vtb∗ Vtd ) Pππ,(13.77)t = pt − pc . CP -violating phases in Eq. (13.77) appear only
where Tππ = tππ + puππ − pcππ and Pππππππin the CKM elements, so that
∗ )T
∗
t
Aππ
(V Vudππ + (Vtb Vtd ) Pππ


= ub
.
∗V
∗
t
Vub
Aππ
ud Tππ + Vtb Vtd Pππ(13.78)For f = J/ψK, which proceeds via a b → ccs transition, we can write∗uAψK = (Vcb∗ Vcs ) TψK + (VubVus ) PψK,

(13.79)u = putwhere TψK = tψK + pcψK − ptψK and PψKψK − pψK . A subtlety arises in this decay that is
related to the fact that B 0 decays into a final J/ψK 0 state while B 0 decays into a final J/ψK 0 state.
A common final state, e.g., J/ψKS , is reached only via K 0 –K 0 mixing. Consequently, the phase∗ V )/(V V ∗ ),factor (defined in Eq. (13.28)) corresponding to neutral K mixing, e−iφM (K) = (Vcdcs
cd csplays a role:∗ )Pu∗V(Vcb Vcs∗ ) TψK + (Vub Vus
AψKS
Vcd
ψK
cs

=− ∗ 
.
(13.80)×∗Vu∗
Vcb Vcs TψK + Vub
AψKS
P
V
V
us
cd cs
ψK

For q = s or d, there are only penguin contributions to Af , that is, tf = 0 in Eq. (13.76). (Thetree b → uuq 0 transition followed by uu → qq rescattering is included below in the P u terms.)
Again, CKM unitarity allows us to write Af in terms of two CKM combinations. For example, forf = φKS , which proceeds via a b → sss transition, we can writec + (V V ∗ ) P u∗V(Vcb Vcs∗ ) PφK
AφKS
Vcd
φK
ub us
cs

=− ∗  c
,
×
∗V
u
∗
AφKS
V
V
Vcb Vcs PφK + Vub
P
us
cd cs
φK

(13.81)c = pctuutwhere PφK
φK − pφK and PφK = pφK − pφK .
Since in general the amplitude Af involves two different weak phases, the corresponding decays
can exhibit both CP violation in the interference of decays with and without mixing, Sf 6= 0, and CP
violation in decay, Cf 6= 0. (At the present level of experimental precision, the contribution to Cffrom CP violation in mixing is negligible, see Eq. (13.69).) If the contribution from a second weakphase is suppressed, then the interpretation of Sf in terms of Lagrangian CP -violating parameters
is clean, while Cf is small. If such a second contribution is not suppressed, Sf depends on hadronicparameters and, if the relevant strong phase difference is large, Cf is large.
A summary of b̄ → q̄q q̄ 0 modes with q 0 = s or d is given in Table 13.1. The b → ddq transitions
lead to final states that are similar to those from b → uuq transitions and have similar phasedependence. Final states that consist of two vector mesons (ψφ and φφ) are not CP eigenstates,
and angular analysis is needed to separate the CP -even from the CP -odd contributions.
The cleanliness of the theoretical interpretation of Sf can be assessed from the information in
the last column of Table 13.1. In case of small uncertainties, the expression for Sf in terms of CKM
phases can be deduced from the fourth column of Table 13.1 in combination with Eq. (13.73) (and,for b → qqs decays, the example in Eq. (13.80)). Here we consider several interesting examples.
For B 0 → J/ψKS and other b → ccs processes, we can neglect the P u contribution to Af , in
the Standard Model, to an approximation that is better than one percent, giving

λψKS = −e−2iβ ⇒ SψKS = sin(2β) ,31st May, 2024

CψKS = 0 .

(13.82)

13. CP Violation in the Quark Sector

19

B

0

d or s

or

Bs

∗

Vqb

b

q
q′
Vqq′

(a) tfB0q

d or s

or

Bs

f

qu

b
∗

Vq ub

q′
Vquq′

(b) pf quq

fq

Figure 13.2: Feynman diagrams for (a) tree and (b) penguin amplitudes contributing to B 0 → f
or Bs0 → f via a b → qqq 0 quark-level process.
Table 13.1: Summary of b̄ → q̄q q̄ 0 modes with q 0 = s or d. The second and third columns
give examples of hadronic final states (usually those which are experimentally most convenient tostudy). The fourth column gives the CKM dependence of the amplitude Af , using the notation of
Eqs. ((13.77), (13.79), (13.81)), with the dominant term first and the subdominant second. The
suppression factor of the second term compared to the first is given in the last column. “Loop”
refers to a penguin versus tree-suppression factor (it is mode-dependent and roughly O(0.2 − 0.3))
and λ ' 0.23 is the expansion parameter of Eq. (13.34).
b̄ → q̄q q̄ 0 B 0 → fb̄ → c̄cs̄
ψKSb̄ → s̄ss̄
φKSb̄ → ūus̄
π 0 KSb̄ → c̄cd¯ D+ D−b̄ → s̄sd¯ KS KSb̄ → ūud¯ π + π −b̄ → c̄ud¯ DCP π 0b̄ → c̄us̄ DCP KS

Bs0 → f
ψφ
φφ
K +K −
ψKS
φKS
ρ0 KS
DCP KSDCP φ

CKM dependence of Af∗ V )P u(Vcb∗ Vcs )T + (Vubus
∗ V )P u(Vcb∗ Vcs )P c + (Vubus
∗ V )T(Vcb∗ Vcs )P c + (Vubus
(Vcb∗ Vcd )T + (Vtb∗ Vtd )P t
(Vtb∗ Vtd )P t + (Vcb∗ Vcd )P c
∗ V )T + (V ∗ V )P t
(Vub
ud
tb td
∗ V )T 0(Vcb∗ Vud )T + (Vubcd∗
∗ V )T 0(Vcb Vus )T + (Vubcs

Suppression
loop × λ2
λ2
2
λ /loop
loop
<
∼1
loop
λ2
<
∼1

It is important to verify experimentally the level of suppression of the penguin contribution. Meth31st May, 2024

20

13. CP Violation in the Quark Sector

ods based on flavor symmetries [94–97] allow limits to be obtained. All are currently consistent
with the P u term being negligible. Explicit calculations [97–100] also support this conclusion.
In the presence of new physics, Af is still likely to be dominated by the T term, but the mixing
amplitude might be modified. Thus, model-independently, Cf ≈ 0 while Sf cleanly determines the∗ )). The experimental measurement gave the first precision test of
mixing phase (φM − 2 arg(Vcb Vcd
the Kobayashi-Maskawa mechanism. The latest world average [62] isSψK 0 = +0.709 ± 0.011 .

(13.83)(We use K 0 throughout to denote results that combine KS and KL modes, but use the sign
appropriate to KS .) The consistency of this measurement with the predictions for sin 2β makes itvery likely that this mechanism is indeed the dominant source of CP violation in the quark sector.
For B 0 → φKS and other b → sss processes (as well as some b → uus processes), we can neglect
the subdominant contributions, in the Standard Model, to an approximation that is good to theorder of a few percent:λφKS = −e−2iβ ⇒ SφKS = sin 2β ,CφKS = 0 .

(13.84)

A review of explicit calculations of the effects of subleading amplitudes can be found in Ref. [101].
In the presence of new physics, both Af and M12 can have contributions that are comparable insize to those of the Standard Model and carry new weak phases. Such a situation gives several
interesting consequences for penguin-dominated b → q̄qs decays (q = u, d, s) to a final state f :1. The value of −ηf Sf may be different from SψKS by more than a few percent, where ηf is theCP eigenvalue of the final state.
2. The values of ηf Sf for different final states f may be different from each other by more thana few percent (for example, SφKS 6= Sη0 KS ).
3. The value of Cf may be different from zero by more than a few percent.
While a clear interpretation of such signals in terms of Lagrangian parameters will be difficult
because, under these circumstances, hadronic parameters play a role, any of the above three options
will clearly signal new physics. In addition, flavor symmetry relations, such as those that relate
observables in B → Kπ decays [102, 103] can be used to provide further tests of the Standard
Model. Fig. 13.3 summarizes the present experimental results: none of the possible signatureslisted above is unambiguously established, but there is definitely still room for new physics.
For the b → uud process B → ππ and other related channels, the penguin-to-tree ratio can be
estimated using SU(3) relations and experimental data on related B → Kπ decays. The result (forππ) is that the suppression is at the level of 0.2 − 0.3 and so cannot be neglected. The expressionst )/(|V V | T ) are:for Sππ and Cππ to leading order in RP T ≡ (|Vtb Vtd | Pππ
ub ud ππhλππ = e2iα (1 − RP T e−iα )/(1 − RP T e+iα )Sππ ≈ sin 2α + 2 Re(RP T ) cos 2α sin α ,i⇒(13.85)Cππ ≈ 2 Im(RP T ) sin α .

(13.86)Note that RP T is mode-dependent and, in particular, could be different for π + π − and π 0 π 0 . If strong
phases can be neglected, then RP T is real, resulting in Cππ = 0. The size of Cππ is an indicator of
how large the strong phase is. The present experimental average is Cπ+ π− = −0.311 ± 0.030 [62].
As concerns Sππ , it is clear from Eq. (13.86) that the relative size or strong phase of the penguin
contribution must be known to extract α. The theoretical uncertainty stemming from |RP T | 6 1
is referred to in the literature as penguin pollution.
31st May, 2024

13. CP Violation in the Quark Sector

21

eff

eff

sin(2β ) ≡ sin(2φ1 ) vs CCP ≡ -ACP HFLAVMoriond 2021
PRELIMINARY

CCP ≡ -ACP

0.4

0.2
b→ccs
φK

0

0

η′ K

0

KS KS KS
0

π KS

-0.2

0

ρ KS
ω KS
f0 K

-0.4

0

0

0

π π KS
+

-

K K K

-0

0.2

0

0.4

0.6

2

0.8
1
sin(2βeff) ≡ sin(2φe1ff)

Contours give -2∆(ln L) = ∆χ = 1, corresponding to 39.3% CL for 2 dof

Figure 13.3: Summary of the results [62] of time-dependent analyses of b → qqs decays, which
are potentially sensitive to new physics.
The cleanest solution involves isospin relations among the B → ππ amplitudes [104]:1√ Aπ+ π− + Aπ0 π0 = Aπ+ π0 .
2(13.87)t is pure ∆I = 1/2 (this is not true
The method exploits the fact that the penguin contribution to Pππ
for the electroweak penguins which, however, are expected to be small), while the tree contribution
to Tππ contains amplitudes that are both ∆I = 1/2 and ∆I = 3/2. A simple geometric construction
then allows one to find RP T and extract α cleanly from Sπ+ π− . The key experimental difficulty isthat one must measure accurately the separate rates for B 0 and B 0 → π 0 π 0 .
31st May, 2024

22

13. CP Violation in the Quark Sector

CP asymmetries in B → ρπ and B → ρρ can also be used to determine α. In particular, theB → ρρ measurements are presently very significant in constraining α. The extraction proceeds via
isospin analysis similar to that of B → ππ. There are, however, several important differences. First,
due to the finite width of the ρ mesons, a final (ρρ)I=1 state is possible [105]. The effect is, however,of the order of (Γρ /mρ )2 ∼ 0.04. Second, due to the presence of three helicity states for the two
vector mesons, angular analysis is needed to separate the CP -even and CP -odd components. The
theoretical expectation is that the CP -odd component is small. This is supported by experiments
which find that the ρ+ ρ− and ρ± ρ0 modes are dominantly longitudinally polarized. Third, an
important advantage of the ρρ modes is that the penguin contribution is expected to be small due
to different hadronic dynamics. This expectation is confirmed by the smallness of B(B 0 → ρ0 ρ0 ) =(0.95 ± 0.16) × 10−6 compared to B(B 0 → ρ+ ρ− ) = (24.2 ± 3.1) × 10−6 [62]. Thus, Sρ+ ρ− is not far
from sin 2α. Finally, both Sρ0 ρ0 and Cρ0 ρ0 are experimentally accessible, which may allow a precise
determination of α. However, a full isospin analysis should allow that the fractions of longitudinalpolarization in B and B decays may differ, which has not yet been done by the experiments.
Detailed discussion of the determination of α with these methods can be found in Refs. [106,107].
The latest world average is◦
α = 84.1 +4.5.
(13.88)−3.8
The consistency between the range of α determined by the B → ππ, ρπ and ρρ measurements andthe range allowed by CKM fits (excluding these direct determinations) provides further support to
the Kobayashi-Maskawa mechanism.
All modes discussed in this Section so far have possible contributions from penguin amplitudes.
As shown in Table 13.1, CP violation can also be studied with final states, typically containing
charmed mesons, where no such contribution is possible. The neutral charmed meson must be reconstructed in a final state, such as a CP eigenstate, common to D0 and D0 so that the amplitudes
for the B and B meson decays interfere. Although there is a second tree amplitude with a different
weak phase, the contributions of the different diagrams can in many cases be separated experimentally (for example by exploiting different decays of the neutral D mesons) making these channels
very clean theoretically. The first determination of sin(2β), with significance of CP violation over
5σ, with this method has recently been reported [108]. Moreover, the interference between the twotree diagrams gives sensitivity to γ, as will be discussed in Section 13.6.4.
13.6.3 CP violation in interference of Bs0 decays with and without mixing
As discussed in Section 13.6.1, the world average for |q/p| in the Bs0 system currently deviatesfrom the Standard Model expectation due to an anomalous value of the dimuon asymmetry. Attributing the dimuon asymmetry result to a fluctuation, we again neglect the deviation of |q/p|from 1, and use
0
λf = e−iφM (Bs ) (Af /Af ) .
(13.89)Within the Standard Model,−iφM (B 0 )es= (Vtb∗ Vts )/(Vtb Vts∗ ) .

(13.90)Note that in the Bs0 system, and with the sign convention of Eq. (13.8b), ∆Γ/Γ = −0.126±0.007 [62]and therefore y should not be put to zero in Eqs. (13.17a) and (13.17b). However, |q/p| = 1 is
expected to hold to an even better approximation than for B 0 mesons. One therefore obtains
Af (t) =Sf sin(∆mt) − Cf cos(∆mt),
cosh (∆Γ t/2) − A∆Γ
f sinh (∆Γ t/2)

(13.91)−2 Re(λf ).
1 + |λf |2(13.92)A∆Γ≡f

31st May, 2024

23

13. CP Violation in the Quark Sector

The presence of the A∆Γterm implies that information on λf can be obtained from analyses thatf
do not use tagging of the initial flavor, through so-called effective lifetime measurements [109].
The Bs0 → J/ψφ decay proceeds via the b → ccs transition. The CP asymmetry in this modethus determines (with angular analysis to disentangle the CP -even and CP -odd components of the
final state) sin 2βs , where βs is defined in Eq. (13.38) [110]. The Bs0 → J/ψπ + π − decay, which has
a large contribution from J/ψf0 (980) and is assumed to also proceed dominantly via the b → ccs
transition, has also been used to determine βs . In this case no angular analysis is necessary, since
the final state has been shown to be dominated by the CP -odd component [111]. The combinationof measurements yields [62]2βs = 0.049 ± 0.019 ,(13.93)consistent with the Standard Model prediction, assuming negligible penguin contributions, βs =0.0184 ± 0.0004 [20].
A time-dependent CP asymmetry was established in Bs0 → K + K − decay, which proceeds via
the b → uus transition [112]:
CKK = +0.172 ± 0.031 ,SKK = +0.139 ± 0.032 .

(13.94)For both CKK and SKK , the hadronic ratio (T /P c ) plays an important role (see Table 13.1), making
a clean theoretical interpretation challenging. First results on the b → qqs decays Bs0 → φφ andK ∗0 K ∗0 have also been reported. Parameters of CP violation have also been determined from thedecay-time distributions of Bs0 → Ds∓ K ± and Ds∓ K ± π + π − decays, involving interference between
b → cus and b → ucs amplitudes.
13.6.4 Direct CP violation in the B system
An interesting class of decay modes is that of the tree-level decays B ± → D(∗) K ± , whichallow a theoretically pristine determination of the angle γ [113–118]. The method uses the decays
B + → D0 K + , which proceeds via the quark transition b → ucs, and B + → D0 K + , which proceeds
via the quark transition b → cus, with the D0 and D0 decaying into a common final state. The
decays into common final states, such (π 0 KS )D K + , involve interference effects between the two
amplitudes, with sensitivity to the relative phase, δ + γ (δ is the relevant strong phase difference).
The CP -conjugate processes are sensitive to δ − γ. Measurements of branching ratios and CP
asymmetries allow the determination of γ and δ from amplitude triangle relations. The method
suffers from discrete ambiguities but, since all hadronic parameters can be determined from thedata, has negligible theoretical uncertainty [119].
Unfortunately, the smallness of the CKM-suppressed b → u transitions makes it difficult to
use the simplest methods alone [113–115] to determine γ. These difficulties are overcome (andthe discrete ambiguities are removed) by performing a Dalitz plot analysis for multi-body D decays [116–118]. Detailed discussion of the determination of γ with these methods can be found in
Ref. [107].
Constraints on γ from combinations of results on various B → D(∗) K (∗) processes have beenobtained by experiments [120, 121]. The latest world average is [62, 107]γ = 65.9 +2.9
−3.1

◦

.

(13.95)The consistency between the range of γ determined by the B → DK measurements and therange allowed by CKM fits (excluding these direct determinations) provides further support to the
Kobayashi-Maskawa mechanism. As more data become available, determinations of γ from Bs0 →Ds∓ K ± [122, 123] and B 0 → DK ∗0 [124–128] are expected to also give competitive measurements.
31st May, 2024

13. CP Violation in the Quark Sector

24

Decays to the final state K ∓ π ± provided the first observations of direct CP violation in both B 0
and Bs0 systems. The asymmetry arises due to interference between tree and penguin diagrams [129],
similar to the effect discussed in Section 13.6.2. In principle, measurements of AB 0 →K + π− andABs0 →K − π+ could be used to determine the weak phase difference γ, but lack of knowledge of therelative magnitude and strong phase of the contributing amplitudes limits the achievable precision.
The uncertainties on these hadronic parameters can be reduced by exploiting flavor symmetries,
which predict a number of relations between asymmetries in different modes. One such relation isthat the partial rate differences for B 0 and Bs0 decays to K ∓ π ± are expected to be approximately
equal and opposite [130], which is consistent with current data. It is also expected that the partial
rate asymmetries for B 0 → K + π − and B + → K + π 0 should be approximately equal; however, theexperimental results currently show a significant discrepancy [62]:AB 0 →K + π− = −0.0836 ± 0.0032 ,AB + →K + π0 = +0.029 ± 0.012 .

(13.96)

It is therefore of great interest to understand whether this originates from Standard Model QCD
corrections, or whether it is a signature of new dynamics. Improved tests of a more precise relation
between the partial rate differences of all four Kπ final states [131–134], currently limited byknowledge of the CP asymmetry in B 0 → KS π 0 decays, may help to resolve the situation.
It is also of interest to investigate whether similar patterns appear among the CP violating
asymmetries in B meson decays to final states containing one pseudoscalar and one vector meson.
Since the vector resonance decays to two particles, such channels can be studied through Dalitz
plot analysis of the three-body final state. Model-independent analyses of B + → K + K − K + ,π + π − K + , π + π − π + and K + K − π + decays have revealed large CP violation effects in certain regionsof phase space [135]. For the B + → K + K − π + decay, an amplitude analysis has established a large
CP violation effect associated with ππ ↔ KK S-wave rescattering [136]. In B + → π + π − π +decays, amplitude analysis has established CP violation effects in the decay amplitude involving
the f2 (1270) resonance, in the π + π − S-wave at low invariant mass, and in the interference between
the π + π − S-wave and the P-wave B + → ρ(770)0 π + amplitude [137, 138]. For the other channels it
remains to be seen whether the CP violation effects are associated to particular resonances or tointerference effects, which will be necessary to understand the underlying dynamics.

13.7 Summary and Outlook
CP violation has been experimentally established in K, D and B meson decays. A full list of
CP asymmetries that have been measured at a level higher than 5σ is given in Section 13.8. In
Section 13.1.4 we introduced three types of CP -violating effects. Examples of these three types
include the following:
1. All three types of CP violation have been observed in K → ππ decays:!

1 Aπ 0 π 0
A + −Re(0 ) =− π π= (2.5 ± 0.4) × 10−6 ,(I)(13.97a)6 Aπ 0 π 0Aπ + π −1qRe() =1−
= (1.66 ± 0.02) × 10−3 , (II)(13.97b)2p1
Im() = − Im(λ(ππ)I=0 )= (1.57 ± 0.02) × 10−3 . (III)(13.97c)2
2. For D mesons, CP violation in decay has been established in the difference of asymmetries
for D0 → K + K − and D0 → π + π − decays.
∆aCP =|AK + K − /AK + K − |2 − 1 |Aπ+ π− /Aπ+ π− |2 − 1−= (−0.161 ± 0.028) × 10−2 , (I)|AK + K − /AK + K − |2 + 1 |Aπ+ π− /Aπ+ π− |2 + 1(13.98)31st May, 2024

13. CP Violation in the Quark Sector25

3. In the B meson system, CP violation in decay has been observed in, for example, B 0 → K + π −transitions, while CP violation in interference of decays with and without mixing has been
observed in, for example, the B 0 → J/ψKS channel:|AK − π+ /AK + π− |2 − 1= −0.0836 ± 0.0032 ,|AK − π+ /AK + π− |2 + 1= Im(λψK 0 )= +0.709 ± 0.011 .

AK + π − =(I)(13.99a)

SψK 0

(III)

(13.99b)

Based on Standard Model predictions, further observations of CP violation in B 0 , B + and Bs0
decays seem likely in the near future, at both LHCb and its upgrades [139–141] as well as the BelleII experiment [142]. The first observation of CP violation in b baryons is also likely to be withinreach of LHCb. Further improvements in the sensitivity to CP violation effects in the charm sector
can also be anticipated, though uncertainty in the Standard Model predictions makes it difficult
to forecast whether or not additional discoveries will be forthcoming. A number of upcoming
experiments have potential to make significant progress on rare kaon decays. Observables that are
subject to clean theoretical interpretation, such as β from SψKS , βs from Bs0 → J/ψφ, B(KL →π 0 νν) and γ from CP violation in B → DK decays, are of particular value for constraining thevalues of the CKM parameters and probing the flavor sector of extensions to the Standard Model.
Progress in lattice QCD calculations is also needed to complement the anticipated experimental
results. Other probes of CP violation now being pursued experimentally include the electric dipole
moments of the neutron and electron, and the decays of tau leptons. Additional processes thatare likely to play an important role in future CP studies include top-quark production and decay,
Higgs boson decays and neutrino oscillations.
All measurements of CP violation to date are consistent with the predictions of the KobayashiMaskawa mechanism of the Standard Model. In fact, it is now established that the KM mechanism
plays a dominant role in the CP violation measured in the quark sector. However, a dynamicallygenerated matter-antimatter asymmetry of the universe requires additional sources of CP violation,and such sources are naturally generated by extensions to the Standard Model. New sources might
eventually reveal themselves as small deviations from the predictions of the KM mechanism, orelse might not be observable in the quark sector at all, but observable with future probes such as
neutrino oscillations or electric dipole moments. The fundamental nature of CP violation demands
a vigorous search.
A number of excellent reviews of CP violation are available [143–150], where the interested
reader may find a detailed discussion of the various topics that are briefly reviewed here.
We thank David Kirkby for significant contributions to earlier versions of this review.

13.8 Observed CP violation effects
We conclude by listing the observables where CP violation has been observed at a level above
5σ [44, 62, 84]:
• Indirect CP violation in K → ππ and K → π`ν decays, and in the KL → π + π − e+ e− decay,
is given by|| = (2.228 ± 0.011) × 10−3 .
(13.100)• Direct CP violation in K → ππ decays is given byRe(0 /) = (1.65 ± 0.26) × 10−3 .

(13.101)• CP violation in the interference of mixing and decay in the tree-dominated b → cc̄s transitions,such as B 0 → ψK 0 , is given by:
SψK 0 = +0.709 ± 0.011 .
31st May, 2024

(13.102)

26

13. CP Violation in the Quark Sector• CP violation in the interference of mixing and decay in modes governed by the tree-dominatedb → cūd transitions is given bySD(∗) h0 = +0.71 ± 0.09 ,(13.103)• CP violation in the interference of mixing and decay in various modes related to b → cc̄dtransitions is given bySψπ0 = −0.86 ± 0.14 ,SD+ D− = −0.84 ± 0.12 ,SD∗± D∓ = −0.81 ± 0.06 ,SD∗+ D∗− = −0.71 ± 0.09 .

(13.104)• CP violation in the interference of mixing and decay in various modes related to b → q q̄s(penguin) transitions is given by
SφK 0 = +0.74 +0.11−0.13 ,
Sη0 K 0 = +0.63 ± 0.06 ,Sf0 K 0 = +0.69 +0.10−0.12 ,
+0.09
SK + K − KS = +0.68 −0.10 .

(13.105)• CP violation in the interference of mixing and decay in the B 0 → π + π − mode is given by
Sπ+ π− = −0.666 ± 0.029 .

(13.106)• Direct CP violation in the B 0 → π + π − mode is given by
Cπ+ π− = −0.311 ± 0.030 .

(13.107)• Direct CP violation in the Bs0 → K + K − mode is given by
CK + K − = 0.172 ± 0.031 .
(∗)(13.108)(∗)• Direct CP violation in B + → D+ K + decays (D+ is the CP -even neutral D(∗) state) are
given by
AB + →D+ K + = +0.139 ± 0.009

andAB + →D+∗ K + = −0.109 ± 0.019 ,(13.109)

while the corresponding quantity in the case that the neutral D meson is reconstructed in
the suppressed K − π + final state is
AB + →DK − π+ K + = −0.453 ± 0.026 ,(13.110)• Direct CP violation has also been observed in B + → DK + decays through differences between
the phase-space distributions of subsequent D → KS π + π − and D → K − π + π + π − decays.
• Direct CP violation in the B 0 → K + π − mode is given by
AB 0 →K + π− = −0.0836 ± 0.0032 .

(13.111)• Direct CP violation in the Bs0 → K − π + mode is given by
ABs0 →K − π+ = +0.225 ± 0.012 .
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(13.112)

27

13. CP Violation in the Quark Sector• Direct CP violation in B + → K + K − π + , B + → K + K − K + and B + → π + π − π + decays isgiven by
AB + →K + K − π+ = −0.115 ± 0.008 ,AB + →K + K − K + = −0.0365 ± 0.0036 ,AB + →π+ π− π+ = 0.076 ± 0.005 ,(13.113)• Large CP violation effects have been observed model-independently in certain regions of thephase space of B + → K + K − K + , K + K − π + , π + π − K + and π + π − π + decays. An amplitude analysis has established a large CP violation effect associated with ππ ↔ KK S-wave
rescattering in B + → K + K − π + decays. In B + → π + π − π + decays, amplitude analysis has
established CP violation effects in the decay amplitude involving the f2 (1270) resonance, inthe π + π − S-wave at low invariant mass, and in the interference between the π + π − S-wave
and the P-wave B + → ρ(770)0 π + amplitude.
• Direct CP violation has been established in the difference of asymmetries for D0 → K + K −and D0 → π + π − decays∆aCP = (−0.161 ± 0.028) × 10−2 .
(13.114)
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