1

77. Determination of CKM angles from B hadrons

77. Determination of CKM angles from B hadrons
Revised February 2024 by T. Gershon (Warwick U.), M. Kenzie (Cambridge U.) and K. Trabelsi(U. Paris-Saclay, IJCLab).

77.1 Introduction
The Cabibbo–Kobayashi–Maskawa (CKM) description of quark mixing [1, 2] leads to a numberof triangle relations between pairs of CKM matrix elements. One of these,∗
Vud Vub+ Vcd Vcb∗ + Vtd Vtb∗ = 0 ,(77.1)

is of particular interest since (i) all its terms are of comparable magnitude, and (ii) its properties
can be measured through studies of oscillations and decays of B mesons. As the area of this unitary
triangle is a measure of the amount of CP violation in the Standard Model [3], it is of particularinterest to determine the values of its angles and to test the consistency of the CKM paradigm withthe experimental measurements. The angles are defined as"V V∗
α = arg − td tb∗
Vud Vub

#

"

,

V V∗
β = arg − cd cb∗
Vtd Vtb#"V V∗
γ = arg − ud ub
Vcd Vcb∗,#,(77.2)

with an alternative notation (φ2 , φ1 , φ3 ) ≡ (α, β, γ) also widely used in the literature.
In this mini-review, the most precise methods to determine the CKM angles are described, witha particular focus on nontrivial aspects of the combination of results. More detailed discussions ofthese points can be found in Ref. [4]. A similar mini-review on the side of the unitarity triangle
adjacent to the angle γ can be found in Ref. [5]. A detailed overview of the CKM quark-mixing
matrix is given in Ref. [6] while CP violation in the quark sector is discussed in Ref. [7].

77.2 βThe relative weak (i.e. CP -violating) phase between the amplitude for any CKM-favoured
B 0 meson decay to a CP eigenstate and that for the decay following B 0 –B 0 oscillation is twice
the angle β. The decay-time-dependent CP asymmetry can be expressed as0dΓ/dt B 0phys (t) → fCP − dΓ/dt Bphys(t) → fCP



AfCP (t) ≡0dΓ/dt B 0phys (t) → fCP + dΓ/dt Bphys(t) → fCP








 ,

= Sf sin(∆mt) − Cf cos(∆mt) ,(77.3a)(77.3b)0where the notation Bphys(t) (B 0phys (t)) denotes a neutral B meson that decays at time t into thefinal state fCP , and is known (“tagged”) at time t = 0 to have flavour content corresponding to
B 0 (B 0 ). In Eq. (77.3b), ∆m denotes the mass difference between the two physical eigenstates of
the B 0 –B 0 system, while the corresponding decay-width difference is assumed to be negligible [8];moreover CP T symmetry and the absence of CP violation in B 0 –B 0 mixing is assumed throughout
this mini-review.
In the general case, one can write

2 Im(λf )Sf ≡1 + |λf |2

and

1 − |λf |2
Cf ≡,
1 + |λf |2(77.4)

A

where the parameter λf = pq Aff is defined in terms of p and q, which define the flavour content of
the mass eigenstates of the B 0 –B 0 system [8], and the amplitudes Af (Af ) for a B 0 (B 0 ) decay
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:12am

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77. Determination of CKM angles from B hadrons

to the final state fCP . In the limit that the decay amplitude is dominated by a CKM-favouredtransition, as is the case for B 0 → J/ψKS0 decays, one obtains simple relations: Sf = −ηCP sin(2β)
and Cf = 0, where ηCP is the CP eigenvalue of the final state [9,10]. This method has been pursued
intensively by experiments. The current world averages, combining results for several charmoniumkaon final states but dominated by results on B 0 → J/ψKS0 (CP odd) and B 0 → J/ψKL0 (CPeven), are [4]− ηCP Sf = 0.709 ± 0.011 ,Cf = +0.004 ± 0.010 .
(77.5)
Despite the large number of signal events in the data, the dominant uncertainties are still statistical.
One important source of potential systematic correlation between results from different experiments
is that due to “tag-side interference” [11],2 which is common to measurements exploiting production
through the e+ e− → Υ (4S) → B 0 B 0 process, including the latest results from BaBar [12] andBelle [13]. It does not, however, affect the results from LHCb [14] that now have better statisticalsensitivity. Another common source of systematic uncertainty is due to knowledge of the value of∆m, but since this quantity has been measured precisely [8] the effect remains small.
The interpretation of the value of −ηCP Sf from Eq. (77.5) as sin(2β) assumes negligible contributions from subleading amplitudes with a different weak phase to that of the tree diagram (i.e. tothat of the CKM matrix elements Vcb Vcs∗ ). This potential additional contribution is often referred
to as “penguin pollution”. All existing data, including the value of Cf in Eq. (77.5), as well as
several explicit calculations [15–18], are consistent with penguin pollution in B 0 meson decays tocharmonium-kaon decays being negligible at the current level of precision. Therefore, the value of−ηCP Sf is generally converted to sin(2β) without any correction or additional uncertainty being
assigned due to this assumption. This gives [4]

β = 22.6 +0.5
−0.4

◦

,

(77.6)

where only the solution consistent with the Standard Model is reported (methods to resolve thetrigonometric ambiguity in the result are discussed below). It is also possible to use data-driven
methods, typically based on flavour symmetries plus some additional assumptions, to constrain
the effects of penguin pollution [19–21]. In this case it is necessary to consider each charmoniumkaon final state separately, since the penguin pollution to each may differ. The most common
approach [19], which relies on experimental information on B 0 → J/ψπ 0 decays, currently gives anadditional uncertainty on sin(2β) from B 0 → J/ψKS0 of around 0.01.
It is possible to avoid the issue of penguin pollution in the measurement of β by using B 0
meson decays to a charm- and light-meson final state, such as DCP π 0 (where DCP represents a D0
meson decaying into a CP eigenstate), instead of the charmonium-kaon final states. These decays∗ instead of V V ∗ ), which can in principle bias
do have a CKM-suppressed contribution (Vub Vcd
cb ud
the determination of sin(2β) from Sf , but this can be calculated and is known to be negligible
at current precision. The requirement that the neutral D meson decays to a final state that iscommon to both D0 and D0 , such as the CP -even eigenstate K + K − , reduces the sample size thatis available for analysis. Consequently, the world average [4], sin(2β) = 0.71 ± 0.09, with thesechannels is not as precise as that from the charmonium-kaon states.
Converting experimental results on sin(2β) into constraints on β leads to a trigonometric ambiguity in the range [0◦ , 180◦ ]. This can be resolved with experimental measurements of cos(2β),
which can be obtained from decay-time-dependent analyses of B 0 meson decays to multibody (nonCP -eigenstate) final states. Among the charmonium-kaon decays, study of B 0 → J/ψK ∗ (892)0
with K ∗ (892)0 → KS0 π 0 is the most promising approach, but due to the limited sample size thathas been analysed to date the precision is not sufficient to resolve the ambiguity conclusively. The
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77. Determination of CKM angles from B hadrons

charm- and light-meson channels such as B 0 → Dπ 0 with D → KS0 π + π − have been shown to provide good statistical power for this purpose, with a joint analysis of BaBar and Belle data givingcos(2β) = 0.91 ± 0.25 [22, 23], sufficient to rule out the alternative solution for β.

77.3 αIn the limit that only tree amplitudes contribute to B 0 meson decays to light mesons, suchas B 0 → π + π − , then the observables of the decay-time-dependent CP asymmetry of Eq. (77.3)
would allow a straight-forward determination of 2α: Sf = +ηCP sin(2α) and Cf = 0. In general,
however, the determination of α is complicated by the presence of contributions from b → d(uū)
neutral-current penguin transitions, which have a similar level of CKM-suppression as the b → u(ūd)
charged-current tree amplitudes but have a different weak phase. Consequently, one obtains insteadfor B 0 → π + π −q
Sπ+ π− = 1 − Cπ2+ π− sin(2α − 2∆α),(77.7)
where ∆α is the a priori unknown penguin contribution.
This contribution from the penguin amplitude can be accounted for in an analysis relating theamplitudes for isospin partner
decays, e.g. A+− for B 0 → π + π − , A+0 for B + → π + π 0 , A00 for
B 0 → π 0 π 0 decays and A+− , A−0 , A00 for their charge conjugates. The isospin analysis relies on
the fact that there is no penguin contribution to A+0 and A−0 , because π ± π 0 is a pure isospin-2
state, and the (∆I = 21 ) QCD-penguin amplitudes only contribute to the isospin-0 final state. One
therefore obtains the following isospin triangle relations [25]1
A+0 = √ A+− + A00
2

and

1
A−0 = √ A+− + A00 ,2(77.8)

from which it is possible to determine ∆α, as shown in Fig. 77.1.
Since the determination of ∆α and thus also α requires construction of amplitude-level relations,
it is not appropriate to simply average results of α from different experiments. Instead, measurements of each of the observable quantities needed to determine α are input into a combination.
For the B → ππ system, the inputs are the branching fractions of B 0 → π + π − , B + → π + π 0 andB 0 → π 0 π 0 decays, the lifetimes of the B + and B 0 mesons (which relate the branching fractions
to amplitude-level quantities), and the Sπ+ π− , Cπ+ π− and Cπ0 π0 observables. Potential sourcesof correlation must be taken into account, but these are predominantly systematic in origin andthus have a small effect on the combination, since the measurements are statistically limited. An
exception is that the LHCb measurements of (Sπ+ π− , Cπ+ π− ) [26, 27] have a significant statisticalcorrelation due to the fact that the time variable of Eq. (77.3) is the difference between production
and decay, and hence is in the range [0, ∞]. This correlation is largely absent for measurementsfrom BaBar [28] and Belle [29], where the difference between the signal and tagging B meson decay
times is measured, and hence t ∈ [−∞, ∞]. The combination itself can be performed with different
statistical approaches; the procedure described in detail in Ref. [30], based on a frequentist treatment, is used here. The knowledge of Cπ0 π0 [28, 31] is currently the limiting factor in the precision
on α from the B → ππ system, and is likely to remain so for some time due to the difficulty toreconstruct this final state.
In general, the isospin triangle construction gives a four-fold ambiguity on 2∆α (each triangle
can face either up or down), leading to an eight-fold ambiguity on α in the range [0◦ , 180◦ ]. Thisis reduced if either or both of the triangles are flat, or if the two triangles have sides of identicallength. The ambiguities can also be reduced if measurement of the Sπ0 π0 (or equivalent) observable
is available, since this can be combined with the corresponding ∆α parameter from the right-handcorner of the triangle in Fig. 77.1 to provide an additional constraint. None of these possibilities
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477. Determination of CKM angles from B hadrons

Figure 77.1: Isospin triangles for B → ππ decays, reproduced from Ref. [24]. Here, the relative
phase between A+0 andA−0 has been rotated away to simplify the picture. The total relative phase+−probed by Sπ+ π− is arg pq A
= 2α − 2∆α, including contributions from B 0 –B 0 mixing, the treeA+−level amplitudes and the correction ∆α, and exploiting the unitarity requirement α + β + γ = 180◦ .
are realised in the B → ππ system; in particular a decay-time-dependent analysis of B 0 → π 0 π 0 is
extremely challenging experimentally due to the absence of any charged particle originating fromthe B decay position. Nonetheless, solutions consistent with α = 0 can be rejected on physicalgrounds [24].
The isospin analysis can also be performed with the B → ρρ system, which contains two vector
particles in the final state and so does not have a fixed CP eigenvalue. In principle the analysis
can be performed separately for each ρρ polarization state, but in practise it is found that thelongitudinal polarization fraction, fL , is close to unity, and hence the final state is approximately
CP -even. Compared to B 0 → ππ, the ρρ modes benefit experimentally from a higher branching
fraction and smaller penguin contributions, so that the isospin triangles are flatter, reducing theambiguities. (The value of ∆α in the B → ρρ system, obtained from the isospin analysis, has asingle solution in [0, π] at (3 ± 5)◦ , while for B → ππ there are two solutions at 13◦ and 27◦ with∆α ∈ [7, 33]◦ at 68.3% confidence level (CL). The isospin analysis with either final state has an
ambiguity under ∆α ⇔ −∆α.) For the BaBar [32] and Belle [33] experiments, the high branching
fraction and smaller penguin contribution compensate for the increased difficulty to reconstructthe ρρ final state relative to ππ. Moreover, in contrast to Sπ0 π0 , measurement of Sρ0 ρ0 is possible
due to the four charged pion final state, following ρ0 → π + π − decay, as has been demonstrated by
BaBar [34].
In the B → ρπ system there are more amplitudes to consider, so that the isospin relation
corresponds to a pentagon rather than a triangle and Eq. (77.8) is modified to become√√2(A+0 + A0+ ) = A+− + A−+ + 2A00 and2(A−0 + A0− ) = A+− + A−+ + 2A00 . (77.9)
As in Eq. (77.8), the left-hand sides of these expressions correspond to a pure isospin-2 final state,
and therefore the ratio of the right-hand sides gives a pure phase term that, accounting for the B 0 –B 0 mixing phase that also contributes to the measured quantities, is 2α. The relative amplitudes
for B 0 and B 0 decays to ρ+ π − , ρ− π + and ρ0 π 0 can all be determined from a decay-time-dependent
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77. Determination of CKM angles from B hadrons

5

PDG 2024

1 − CL

1.0

HFLAV

B 0 → (ρπ)0Combination

B → ππB → ρρ

0.8
0.6
0.4
0.2
0.0

0

25

50

75

100

125

150

175
α [◦ ]

Figure 77.2: World average of α, as well as contributions from individual modes, in terms of
1−CL.
analysis of the π + π − π 0 Dalitz plot, so that study of this channel alone allows determination ofα [35]. This analysis in principle leads to a single solution for α in [0◦ , 180◦ ], but the precision ofcurrent measurements [36–38] is limited.
The isospin analysis used to determine α is believed to be valid to high precision, and theoreticaluncertainties in the procedure are usually neglected. Nonetheless, it should be noted that theanalysis assumes the absence of electroweak penguin amplitudes, which can contribute to ∆I = 32
transitions with a different weak phase to that of the tree amplitudes [39, 40]. Moreover, isospinbreaking effects such as π 0 , η, η 0 mixing would impact on the relations of Eq. (77.8). A further
complication in the B → ρρ system is the effect of the non-zero ρ meson width [41]. Estimates ofthe size of these effects on the determined value of α are typically at the 1◦ level or less [30]. By
contrast, methods to determine α using SU(3) or other flavour symmetries are generally considered
to have larger theoretical uncertainties and are not included here.
The world average obtained for the angle α from isospin analysis of B → ππ, ρπ and ρρ decays
is [4]◦
α = (84.1 +4.5
(77.10)−3.8 ) ,where the quoted uncertainty is at the 68.3% CL and does not include effects due to isospinbreaking. This world average, together with results split by decay mode, is shown in Fig. 77.2.
The combination has a total of 57 experimental inputs from which 24 parameters are determined,
and an overall χ2 of 22.4, which corresponds to a p-value of 91%. Thus, there is excellent overall
consistency between the inputs, despite the tension apparent in Fig. 77.2 between the results from31st May, 2024

77. Determination of CKM angles from B hadrons

6

B 0 → (ρπ)0 and the others. The combination gives a single best-fit for α in [0◦ , 180◦ ], but anambiguous solution exists at α ⇔ α + 180◦ . A secondary minimum close to zero is disfavoured [30].

77.4 γThe angle γ is the weak phase between Cabibbo-favoured b → c and suppressed b → u quarktransitions and can be determined by exploiting interference between them. Explicitly, the ratio of
suppressed to favoured amplitudes is parameterized byrB ei(δB ±γ) =

Asup
,
Afav

(77.11)

where rB is the ratio of amplitude magnitudes, δB the strong phase difference and the + or − signdepends on whether the transition involves a b̄ or b quark, respectively. Measurement of γ in thisway has negligible theoretical uncertainty in the Standard Model [42], and therefore this approachprovides a benchmark against which determinations from other methods, typically involving loop
diagrams, can be compared.
Interference between these amplitudes is realised in B + → DK + decays, where D represents anadmixture of D0 and D0 mesons. The simplest case is that of D decays to CP -eigenstates (GLW
method [43, 44]), either CP -even such as K + K − (CP +) or CP -odd such as KS0 π 0 (CP −). The
normalized decay rate and CP asymmetry are given byΓ (B − → DCP ± K − ) + Γ (B + → DCP ± K + )2= 1 + rB
± 2rB cos(δB ) cos(γ) ,
−
0
−
+
0
+
Γ (B → D K ) + Γ (B → D K )±2rB sin(δB ) sin(γ)Γ (B − → DCP ± K − ) − Γ (B + → DCP ± K + )=ACP ± =
2 ± 2r cos(δ ) cos(γ) .
−−++Γ (B → DCP ± K ) + Γ (B → DCP ± K )1 + rB
B
B

RCP ± =

(77.12a)(77.12b)

These relations assume the absence of direct CP violation in the charm system; experimentallyallowed deviations from this assumption are too small to cause a significant bias on γ [7, 45]. It isconvenient to determine the RCP ± quantities through a double ratio, normalizing to B + → Dπ +decays involving the same final states, since this cancels potential sources of systematic uncertainty
due to the branching fractions of the D decays that are used; small possible effects of CP violation in
B + → Dπ + decays are a source of systematic uncertainty in this procedure. The GLW method canbe extended to include final states that are almost CP -eigenstates [46], as is the case in D → π + π − π 0
and D → K + K − π 0 decays, via inclusion of a factor encoding the fraction of CP -even (or CP -odd)
content, F± , which dilutes the sensitivity to γ by reducing the size of the interference terms (theterms linear with rB ) in Eq. (77.12).
For other D decays, the ratio of amplitudes for the D0 and D0 decays to the final state ofinterest has to be accounted for in the formalism. The ADS method [47, 48] uses D decays to
final states such as K ∓ π ± , which involve interference between Cabibbo-favoured (CF) and doublyCabibbo-suppressed (DCS) transitions. The observables in this case are
AADS =

Γ (B − → [K + π − ]D K − ) − Γ (B + → [K − π + ]D K + )2rB rD sin(δB + δD ) sin(γ)= 2
2 + 2r r cos(δ + δ ) cos(γ) ,
−
+
−
−
+
−
+
+
Γ (B → [K π ]D K ) + Γ (B → [K π ]D K )rB + rD
B D
B
D(77.13a)RADS =

Γ (B − → [K + π − ]D K − ) + Γ (B + → [K − π + ]D K + )
2
2
= rB
+ rD
+ 2rB rD cos(δB + δD ) cos(γ) ,Γ (B − → [K − π + ]D K − ) + Γ (B + → [K + π − ]D K + )(77.13b)where rD and δD are the amplitude magnitude ratio and strong phase difference between the CF andDCS D decay. An alternative pair of observables, (R− , R+ ), is also sometimes used, where R− (R+ )31st May, 2024

777. Determination of CKM angles from B hadrons

is the ratio of decay rates between the suppressed and favoured transitions for B − (B + ) decays. The
R− and R+ observables are statistically independent, while AADS and RADS are not (in particular,
the uncertainty on AADS depends on the central value of RADS ). However, the pair (R− , R+ ) has
more correlated sources of systematic uncertainty compared to (AADS , RADS ). The observablesof Eq. (77.13) are therefore usually preferred once a significant signal is established. The ADS
method can also be extended to include decays to multibody final states, such as D → K ± π ∓ π 0
and D → K ± π ∓ π + π − , by addition of a coherence factor [49] which appears in the interferenceterms of Eq. (77.13) and accounts for dilution of the sensitivity due to variation of the decay
amplitude across the phase space of the final state. A similar method can be used for singly
Cabibbo-suppressed D decays to non-CP eigenstates such as K ∗ K [50].
For D decays to multibody self-conjugate final states (BPGGSZ method [51, 52]), such as D →KS0 π + π − , one can write the partial decay rate as a function of the position in the phase space interms of the “Cartesian parameters” x± + iy± = rB ei(δB ±γ) :2 2dΓ (B ± → [KS0 π + π − ]D K ± ) = A2(∓,±) + rB
A(±,∓) + 2A(±,∓) A(∓,±) [x± cD(±,∓) + y± sD(±,∓) ],(77.14)

where the notation (+, −) is shorthand for the dependence on the Dalitz-plot position — thesquared invariant masses of KS0 π + and KS0 π − combinations, respectively. The quantities A(+,−)
and A(−,+) represent the magnitudes of the D0 and D0 decay amplitudes at the position (+, −)and are interchangable with their CP conjugate amplitudes because CP conservation is assumed
in the D decay (i.e. A(−,+) = A(+,−) ). The quantities cD(±,∓) and sD(±,∓) are the cosine andsine of the strong phase difference, δD(+,−) = arg(A(+,−) ) − arg(A(+,−) ), between the D0 and
D0 amplitudes. These quantities can be determined from an amplitude model, although thisleads to a hard-to-quantify systematic uncertainty associated to the composition of the model. An
alternative, “model-independent”, approach involves dividing the phase space into appropriate bins.
In this case, the analysis benefits from external input on the values of cD and sD integrated overeach bin. Measurements of these external parameters have been performed for the D → KS0 π + π −decay by the CLEO-c and BES-III collaborations [53–56]. The use of common input values forthese parameters in model-independent determinations of γ with the BPGGSZ method by different
experiments is a source of correlation between experiments that is currently negligible but will
become more significant as the available B meson data samples increase in size.
The discussion above refers to B + → DK + decays, but analogous measurements can be madealso for additional channels such as B + → D∗ K + (with D∗ → Dπ 0 , Dγ) and B + → DK ∗+ (withK ∗+ → KS0 π + , K + π 0 ). In the limit that these can be treated purely as two-body decays, theexpressions for B + → DK + are modified only by ensuring the rB and δB parameters are specificto each B decay. Moreover, for B + → D∗ K + decays an effective shift of the strong phase by π
between D∗ → Dπ 0 and Dγ decays [57] has to be taken into account. In case the finite width ofthe decaying resonance is non-negligible, as is the case for the K ∗ (892) state, additional amplitudes
can contribute leading to a dilution of the sensitivity, which can be accounted for in the formalism
through the introduction of a relevant coherence factor. For the B 0 → DK ∗0 decay, full amplitude
analysis of the B 0 → DK + π − Dalitz plot provides additional sensitivity compared to the quasitwo-body approach [58, 59].
It is also possible to measure γ using decay-time-dependent analysis of the Bs0 meson [60]. The
weak phase arising in the interference between direct decay of Bs0 → Ds∓ K ± and decay via mixing
is (γ − 2βs ), where βs is the angle associated with Bs0 → J/ψφ decays in a similar way to therelation between β and B 0 → J/ψKS0 decays described in Sec. 77.2. Sufficient information canbe obtained from the tagged, decay-time-dependent rates of Bs0 → Ds∓ K ± decays that this weak
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877. Determination of CKM angles from B hadrons

phase can be determined, up to an ambiguity, together with the strong phase difference between,
and the ratio of the magnitudes of, the suppressed and favoured amplitudes. Since βs is known
to good precision [8], measurements of the decay-time-dependent CP -asymmetry observables in
Bs0 → Ds∓ K ± decays can be used to infer constraints on γ. Alternatively, if effects of penguin
pollution in Bs0 → J/ψφ decays [17, 18] are a concern, as they will become in the future, resultsfrom the Bs0 → Ds∓ K ± mode can be combined with an independent precise measurement of γ toprovide a penguin-free determination of βs .
The average for γ requires a non-trivial combination due the complicated relations between theobservables and the physics parameters of interest, such as in Eqs. (77.12), (77.13) and (77.14).
Moreover, hadronic parameters such as rB and δB defined in Eq. (77.11) are common to all different D decay modes (but differ for each B decay mode). Thus, it is not correct to simply average
results for γ obtained by different experiments or in different channels. Instead, measurements ofrate asymmetries, rate ratios and the Cartesian parameters are taken as inputs to the combination,from which results are obtained not only for γ but also for the hadronic parameters. Independentmeasurements of auxiliary parameters such as rD and δD are also treated as inputs to the combination. In some cases the B decay data can help to reduce uncertainties on these auxiliary parameters
and therefore a simultaneous fit of charm and beauty data can provide stronger constraints [61];this approach however is not currently used for the world average.
The precision to which γ can be measured with a particular B decay is approximately inversely
proportional to the value of rB . Thus, results from channels with smaller yields but larger valuesof rB , such as B 0 → DK ∗0 and Bs0 → Ds∓ K ± (rB ≈ 0.3–0.4), can have a significant impact on
the world average and are included in the combination. By contrast the B + → Dπ + mode, forwhich large samples are available but rB ≈ 0.005, has little impact and is also more sensitive topotential systematic biases; hence it is not included. The sensitivity of the world average at present
is dominated by results from B + → DK + , where rB ≈ 0.1, in particular results with the GLW [62],
ADS [62] and BPGGSZ [63] methods.
The world average obtained for the angle γ, obtained by combining results from B + → DK + ,D∗ K + , DK ∗+ , DK + π + π − , B 0 → DK + π − , Bs0 → Ds∓ K ± and Bs0 → Ds∓ K ± π + π − decays, is [4]γ = (65.7 ± 3.0)◦ ,(77.15)where the quoted uncertainty is at the 68.3% CL.
Effects related to charm and kaon mixing and CP violation are generally negligible at thecurrent level of precision, in particular for modes with rB & 0.1. An exception is that a dependenceof the selection efficiency on the charm decay time can induce a dependence of the observables on
charm mixing parameters [64]. Such effects can be important at hadron collider experiments suchas LHCb, but can be and are corrected for. Interactions of neutral kaons with detector material canalso cause a bias in determination of γ from modes with low values of rB [65], such as the BPGGSZ
method applied to B + → Dπ + , but are negligible in modes with larger rB values. A further
subtlety is that the identification of the weak phase between suppressed and favoured amplitudes
in B → DK decays with γ, as defined in Eq. (77.2), assumes that the 2 × 2 submatrix of the CKM∗ / (V V ∗ )] = 0. This is true to an excellent approximation in thematrix is real, i.e. that arg [Vud Vus
cd cs
Standard Model, and is known experimentally from independent studies of the charm system [45]to contribute negligible bias to current measurements. Nonetheless, in future it will be possible totest directly this assumption by comparing the value of γ obtained from the B → DK and B → Dπsystems.
Effects from correlated uncertainties between amplitude models and strong phase differences incharm decays are negligible and are not explicitly accounted for in the combination, nor are effects
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77. Determination of CKM angles from B hadrons

9

1 − CL

1.0

HFLAV

PDG 2024

0.8

Bs0 → Ds− K +Bs0 → Ds− K + π + π −B + → D0 K ∗+B + → D∗0 K +B 0 → D0 K ∗0
B + → D0 K +Combined

0.6

0.4

68.3%

0.2
95.4%
0.0

0

50

100

150

200

γ [◦ ]

η̄

Figure 77.3: World average of γ ≡ φ3 , as well as contributions from individual modes, in termsof 1−CL.

0.7
0.6

HFLAV

α
β
γ

PDG 2024

0.5
0.4
0.3
0.2
0.1
0.0
0.0

contours hold 68.3%, 95.4% CL

0.2

0.4

0.6

0.8

1.0

ρ̄
Figure 77.4: Constraints from the measurements of the angles of the CKM unitarity triangle inthe (ρ, η) plane.

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77. Determination of CKM angles from B hadrons

related to charm and kaon mixing and CP violation. This world average, together with results split
by decay mode, is shown in Fig. 77.3. The combination has a total of 173 experimental inputs fromwhich 36 parameters are determined, an an overall χ2 of 159.5, which corresponds to a p-value of
9% indicating reasonable agreement between the inputs. The combination gives a single solution
for γ in [0◦ , 180◦ ], but an ambiguous solution exists at γ ⇔ γ + 180◦ .

77.5 Summary
Experimental progress has resulted in all three angles of the CKM unitarity triangle being
measured with good accuracy, with β known to subdegree precision and both α and γ known tobetter than 5◦ . The constraints from these three measurements in the (ρ, η) plane are shown in
Fig. 77.4; further discussion and comparison with constraints from independent measurements canbe found in Ref. [6]. The determinations of all three angles remain statistically limited, but it
will be a challenge for experiments to ensure that this remains the case as the precision improves.
Consequently, the correct treatment of sources of correlation between the measurements that go
into the world average combinations is becoming increasingly important.
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