1

74. Polarization in B Decays

74. Polarization in B Decays
Revised July 2023 by A.V. Gritsan (Johns Hopkins U.).
We review the notation used in polarization measurements in particle production and decay,
with a particular emphasis on the B decays and the CP -violating observables in polarization measurements. We look at several examples of vector-vector and vector-tensor B meson decays, whilemore details about the theory and experimental results in B decays can be found in a separatemini-review [1] in this Review.

Figure 74.1: Definition of the production and helicity angles in the sequential process ab → X →P1 P2 → (p11 p12 )(p21 p22 ). The three helicity angles include θ1 and θ2 , defined in the rest frame of
the two daughters P1 and P2 , and Φ, defined in the X frame as the angle between the two decay
planes. The two production angles θ∗ and Ψ are defined in the X frame, where Ψ is the angle
between the production plane and the average of the two decay planes.
Figure 74.1 illustrates angular observables in an example of the sequential process ab → X →P1 P2 → (p11 p12 )(p21 p22 ) [2]. The angular distributions are of particular interest because they
are sensitive to spin correlations and reveal properties of particles and their interactions, such as
quantum numbers and couplings. In the case of a spin-zero particle X, such as B meson or a
Higgs boson, there are no spin correlations in the production mechanism and the decay chain is tobe analyzed. The angular distribution of decay products can be expressed as a function of threehelicity angles which describe the alignment of the particles in the decay chain. The analyzer of the
B-daughter polarization is normally chosen for two-body decays, as the direction of the daughtersin the center-of-mass of the parent (e.g., ρ → 2π) [3], and for three-body decays as the normal
to the decay plane (e.g., ω → 3π) [4]. An equivalent set of transversity angles is sometimes usedin polarization analyses [5]. The differential decay width depends on complex amplitudes Aλ1 λ2 ,S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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74. Polarization in B Decays

2corresponding to the X-daughter helicity states λi .
In the case of a spin-zero B-meson decay, its daughter helicities are constrained to λ1 = λ2 = λ.
Therefore we simplify amplitude notation as Aλ . Moreover, most B-decay polarization analyses
are limited to the case when the spin of one of the B-meson daughters is 1. In that case, there
are only three independent amplitudes corresponding to λ = 0 or ±1 [6], where the last√two canbe expressed in terms of parity-even and parity-odd amplitudes Ak,⊥ = (A+1 ± A−1 )/ 2. The
overall decay amplitude involves three complex terms proportional to the above amplitudes andthe Wigner d functions of helicity angles. The exact angular dependence would depend on thequantum numbers of the B-meson daughters and of their decay products, and can be found in
the literature [6, 7]. When both B-meson daughters are tensor mesons and the smaller of the two
daughter spins is J1 > 1, this formalism can be easily extended by√introducing the parity-evenand parity-odd amplitudes of higher order Akn,⊥n = (A+n ± A−n )/ 2, with 1 < n ≤ J1 , while
the general angular parameterization may be found in Ref. [7]. However, we limit the following
discussion to J1 = 1. The differential decay rate would involve six real quantities αi , includinginterference terms,XdΓ=
αi fi (cos θ1 , cos θ2 , Φ) ,(74.1)
Γ d cos θ1 d cos θ2 dΦ
iwhere each fi (cos θ1 , cos θ2 , Φ) has unique angular dependence specific to particle quantum numbers, and the αi parameters are defined as:|A0 |2= fL ,
Σ|Aλ |2|Ak |2 + |A⊥ |2
α2 == (1 − fL ) ,Σ|Aλ |2|Ak |2 − |A⊥ |2
α3 == (1 − fL − 2 f⊥ ) ,Σ|Aλ |2=m(A⊥ A∗k ) q
α4 == f⊥ (1−fL −f⊥ ) sin(φ⊥ −φk ) ,Σ|Aλ |2<e(Ak A∗0 ) q
α5 == fL (1 − fL − f⊥ ) cos(φk ) ,Σ|Aλ |2=m(A⊥ A∗0 ) p
α6 == f⊥ fL sin(φ⊥ ) ,Σ|Aλ |2
α1 =(74.2)(74.3)(74.4)(74.5)(74.6)(74.7)

where the amplitudes have been expressed with the help of polarization parameters fL , f⊥ , φk ,and φ⊥ defined in Table 74.1. Note that the terms proportional to <e(A⊥ A∗k ), =m(Ak A∗0 ), and<e(A⊥ A∗0 ) are absent in Eqs. (2-7). However, these terms may appear for some three-body decaysof a B-meson daughter, see Ref. [7].
Overall, six real parameters describe three complex amplitudes A0 , Ak , and A⊥ . These could
be chosen to be the four polarization parameters fL , f⊥ , φk , and φ⊥ , one overall size normalization,such as decay rate Γ , or branching fraction B, and one overall phase δ0 . The phase convention isarbitrary for an isolated B decay mode. However, for several B decays, the relative phase could
produce meaningful and observable effects through interference with other B decays with the same
final states, such as for B → V KJ∗ with J = 0, 1, 2, 3, 4, ... The phase could be referenced to thesingle B → V K0∗ amplitude A00 in such a case, as shown in Table 74.1. Here V stands for any
spin-one vector meson.
Moreover, CP violation can be tested in the angular distribution of the decay as the difference
between the B and B. Each of the six real parameters describing the three complex amplitudes
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74. Polarization in B Decays

3

Table 74.1: Rate, polarization, and CP -asymmetry parameters defined for the Bmeson decays to mesons with non-zero spin. Numerical examples are shown for the
average of the B 0 → ϕK ∗ (892)0 decay measurements obtained from BABAR [8],Belle [9], and LHCb [10]. The first six parameters are defined under the assumptionof no CP violation in decay, while they are averaged between the B and B parametersin general. The last six parameters involve differences between the B and B meson
decay parameters. The phase convention δ0 is chosen with respect to a single A00
amplitude from a reference B decay mode, which is B 0 → ϕK0∗ (1430)0 for numericalresults.
parameter

definition

averageBΓ/Γtotal−6(10.1+0.6−0.5 ) × 10

fL

|A0 |2 /Σ|Aλ |20.497 ± 0.017

f⊥|A⊥ |2 /Σ|Aλ |20.225 ± 0.015

φk − πarg(Ak /A0 ) − π−0.712 ± 0.058

φ⊥ − πarg(A⊥ /A0 ) − π−0.615 ± 0.056

δ0 − πarg(A00 /A0 ) − π−0.26 ± 0.10ACP(Γ̄ − Γ )/(Γ̄ + Γ )¯(fL − fL )/(f¯L + fL )(f¯⊥ − f⊥ )/(f¯⊥ + f⊥ )−0.003 ± 0.038

A0CP
A⊥
CP
∆φk

(φ¯k − φk )/2+0.051 ± 0.053∆φ⊥(φ¯⊥ − φ⊥ − π)/2(δ¯0 − δ0 )/2+0.075 ± 0.050∆δ0−0.007 ± 0.030−0.014 ± 0.057+0.13 ± 0.08

would have a counterpart CP -asymmetry term, corresponding to three direct-CP asymmetries inthree amplitudes, and three CP -violating phase differences, equivalent to the phase measurementsfrom the mixing-induced CP asymmetries in the time evolution of B-decays [1]. In Table 74.1 andRef. [11], these are chosen to be the direct-CP asymmetries in the overall decay rate ACP , in thefL fraction A0CP , and in the f⊥ fraction A⊥
CP , and three weak phase differences:1∆φk = arg(Āk A0 /Ak Ā0 ) ,2π
1∆φ⊥ = arg(Ā⊥ A0 /A⊥ Ā0 ) − ,221∆δ0 = arg(Ā00 A0 /A00 Ā0 ) .
2(74.8)(74.9)(74.10)The π2 term in Eq. (74.8) reflects the fact that A⊥ and Ā⊥ differ in phase by π if CP is conserved.
The two parameters ∆φk and ∆φ⊥ are equivalent to triple-product asymmetries constructed fromthe vectors describing the decay angular distribution [12]. The CP -violating phase difference in thereference decay mode [11] is, in the Wolfenstein CKM quark-mixing phase convention,
1∆φ00 = arg(A00 /Ā00 ) .
2(74.11)This can be measured only together with the mixing-induced phase difference for some of theneutral B-meson decays similar to other mixing-induced CP asymmetry measurements [1].
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474. Polarization in B Decays

It may not always be possible to have a phase-reference decay mode which would define δ0 and∆δ0 parameters. In that case, it may be possible to define the phase difference directly similarly
to Eq. (74.11):1∆φ0 = arg(A0 /Ā0 ) .
(74.12)
2
One can measure the angles of the CKM unitarity triangle, assuming Standard Model contributions
to the ∆φ0 and B-mixing phases. Examples include measurements of β = φ1 with B → J/ψK ∗and α = φ2 with B → ρρ.
Most of the B decays that arise from tree-level b → c transitions have the amplitude hierarchy|A0 | > |A+ | > |A− | which is expected from analyses based on quark-helicity conservation [13]. The
larger the mass of the vector-meson daughters, the weaker the inequality. The B meson decays toheavy vector particles with charm, such as B → J/ψK ∗ , ψ(2S)K ∗ , χc1 K ∗ , D∗ ρ, D∗ K ∗ , D∗ D∗ , andD∗ Ds∗ , show a substantial fraction of the amplitudes corresponding to transverse polarization of
the vector mesons (A±1 ), in agreement with the factorization prediction. The detailed amplitude
analysis of the B → J/ψK ∗ decays has been performed by the BABAR [14], Belle [15], CDF[16], CLEO [17], D0 [18], and LHCb [19] collaborations. Most analyses are performed under theassumption of the absence of direct CP violation. The parameter values are given in the particle
listing of this Review. The difference between the strong phases φk and φ⊥ deviates significantly
from zero. The measurements [14,15] of CP -violating terms similar to those in B → ϕK ∗ [8] shownin Table 74.1 are consistent with zero.
In addition, the mixing-induced CP -violating asymmetry is measured in the B 0 → J/ψK ∗0
decay [1, 14, 15] where angular analysis allows one to separate CP -eigenstate amplitudes. This
allows one to resolve the sign ambiguity of the cos 2β (cos 2φ1 ) term that appears in the timedependent angular distribution due to interference of parity-even and parity-odd terms. This
analysis relies on the knowledge of discrete ambiguities in the strong phases φk and φ⊥ , as discussed
below. The BABAR experiment used a method based on the dependence on the Kπ invariant
mass of the interference between the S- and P -waves to resolve the discrete ambiguity in thedetermination of the strong phases (φk , φ⊥ ) in B → J/ψK ∗ decays [14]. The result is in agreementwith the amplitude hierarchy expectation [13]. The CDF [20], D0 [21], LHCb [22], and ATLAS [23]experiments have studied the Bs0 → J/ψ(K + K − ), J/ψ(π + π − ), ψ(K + π − ) decays and provided thelifetime, polarization, and phase measurements.
The amplitude hierarchy |A0 |  |A+ |  |A− | was expected in B decays to light vector particles in both penguin transitions [24, 25] and tree-level transitions [13]. There is confirmationby the BABAR and Belle experiments of predominantly longitudinal polarization in the treelevel b → u transition, such as B 0 → ρ+ ρ− [26], B + → ρ0 ρ+ [27], and B + → ωρ+ [28]; thisis consistent with the analysis of the quark helicity conservation [13]. Because the longitudinal
amplitude dominates the decay, a detailed amplitude analysis is not possible with current B samples, and limits on the transverse amplitude fraction are obtained. The small branching fractionsof B 0 → ρ0 ρ0 , ωρ0 , ωω [28–31] indicate that b → d penguin pollution is small in the charmless,strangeless vector-vector B decays. There is a measurement of large longitudinal polarization inB 0 → ρ0 ρ0 [29–31] decays. The fraction of transverse polarization is large in decays to heavier
mesons such as B 0 → a1 (1260)+ a1 (1260)− [32].
The interest in the polarization and CP -asymmetry measurements in penguin transition, suchas b → s decays B → ϕK ∗ , ρK ∗ , ωK ∗ , or Bs0 → ϕϕ, K ∗ K ∗ , and b → d decay B → K ∗ K̄ ∗ ,is motivated by their potential sensitivity to physics beyond the Standard Model. The decay
amplitudes for B → ϕK ∗ have been measured by the BABAR, Belle, and LHCb experiments[9–11, 33, 34]. The fractions of longitudinal polarization are fL = 0.50 ± 0.05 for the B + → ϕK ∗+decay and fL = 0.497 ± 0.017 for the B 0 → ϕK ∗0 decay. These indicate significant departure from13th September, 2024

574. Polarization in B Decays

the naive expectation of predominant longitudinal polarization, suggesting other contributions to
the decay amplitude, previously neglected, either within the Standard Model, such as penguin
annihilation [35] or QCD rescattering [36], or from physics beyond the Standard Model [37]. The
complete set of twelve amplitude parameters measured in the B 0 → ϕK ∗0 decay is given in Table
74.1. Several other parameters could be constructed from the above twelve parameters, as suggested
in Ref. [38].
The discrete ambiguity in the phase (φk , φ⊥ , ∆φk , ∆φ⊥ ) measurements has been resolved by
BABAR in favor of |A+ |  |A− | through interference between the S- and P -waves of Kπ. The(∗)
search for vector-tensor and vector-axialvector B → ϕKJ decays with J = 1, 2, 3, 4 revealed a large+0.06
fraction of longitudinal polarization in the decay B → ϕK2∗ (1430) with fL = 0.90−0.07[11, 39], but+0.13
large contribution of transverse amplitude in B → ϕK1 (1270) with fL = 0.46−0.15 [40].
Like B → ϕK ∗ , the decays B → ρK ∗ and B → ωK ∗ may be sensitive to New Physics.
Measurements of the longitudinal polarization fraction in B → ρK ∗ [41] and in both vector-vector
and vector-tensor final states of B → ωKJ∗ [28] by BABAR and Belle reveal a large fraction oftransverse polarization, indicating an anomaly similar to B → ϕK ∗ except for a different patternin vector-tensor final states. An angular analysis of the B 0 → ρ0 K ∗0 decay mode by LHCb [42]provides much higher precision and indicates remarkably small longitudinal polarization fraction
and a significant direct CP asymmetry observed in angular distributions of B → V V decays forthe first time. A large transverse polarization is also observed in the Bs0 → ϕϕ decay by CDF [43]and LHCb [44], Bs0 → K ∗0 K̄ ∗0 decays by LHCb [45], and Bs0 → ϕK ∗0 decays by LHCb [46]. At
the same time, measurement of the polarization in the b → d penguin decays B → K ∗ K̄ ∗ indicatesa large fraction of longitudinal polarization [45, 47]. The LHCb experiment has also provided thevery first polarization results on the tensor-tensor, as well as vector-tensor, decays of the Bs0 mesonin the (Kπ)(Kπ) final state [45]. The polarization pattern in penguin-dominated B-meson decays
is not fully understood [35–37].
The three-body semileptonic B-meson decays, such as B → V `1 `2 , share many features withthe two-body B → V V decays. Their differential decay width can be parameterized with thetwo helicity angles defined in the V and (`1 `2 ) frames and with the azimuthal angle, as defined inFig. 74.1. However, since the (`1 `2 ) pair does not come from an on-shell particle, the angular distribution is unique to each point in the dilepton mass m`` spectrum. The polarization measurements
as a function of m`` provide complementary information on physics beyond the Standard Model,
as discussed for B → K ∗ `+ `− and Bs → φ`+ `− decays in Ref. [48]. The data in these modes havebeen analyzed by the BABAR, Belle, CDF, CMS, ATLAS, and LHCb experiments [49–54].
The examples of the angular distributions and observables in B → K ∗ `+ `− are discussed inRef. [48]. Two angular observables have been measured in this decay in certain ranges of the dilepton
mass m`` . One parameter is the fraction of longitudinal polarization FL , which is determined by
the K ∗ angular distribution and is similar to fL defined for exclusive two-body decays. The otherparameter is the forward-backward asymmetry of the lepton pair AF B , which is the asymmetryof the decay rate with positive and negative values of cos θ1 . A complete set of observables andangular terms has been adopted by the LHCb collaboration [54] following Ref. [48] with the FL ,AF B , and S3 − S9 coefficients in the angular distributions. Additional set of optimized observablesp(0)
Pi is derived from those, for example P2 = 2AF B /(3 − 3FL ) and P50 = S5 / FL (1 − FL ). Theseobservables have the advantage that the leading form-factor uncertainties cancel. There have beenhints of deviations from SM in the measurement of P50 and lepton flavor universality [49–54].
In summary, there has been considerable interest in the polarization measurements of B-meson
decays because they reveal both weak- and strong-interaction dynamics [35–37, 55]. New measurements will further elucidate the pattern of spin alignment measurements in rare B decays, and13th September, 2024

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74. Polarization in B Decays

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13th September, 2024