Three-body decays $B \to \phi(\rho) K \gamma$ in perturbative QCD approach

We study the three-body radiative decays $B\to \phi(\rho) K\gamma$ induced by a flavor-changing neutral current in the perturbative QCD approach. Pseudoscalar-vector ($PV$) distribution amplitudes (DAs) are introduced for the final-state $\phi K$ ($\rho K$) pair to capture important infrared dynamics in the region with a small $PV$-pair invariant mass. The dependence of these $PV$ DAs on the parton momentum fraction is parametrized in terms of the Gegenbauer polynomials, and the dependence on the meson momentum fraction is derived through their normalizations to time-like $PV$ form factors. In addition to the dominant electromagnetic penguin, the subleading chromomagnetic penguin, quark-loop and annihilation diagrams are also calculated. After determining the $PV$ DAs from relevant branching-ratio data, the direct $CP$ asymmetries and decay spectra in the $PV$-pair invariant mass are predicted for each $B\to \phi(\rho) K\gamma$ mode.

Though the factorization theorem is not yet proved rigorously, phenomenological applications have been attempted, and abundant predictions have been made. There exist other approaches, such as final state interactions [22] and heavy meson chiral perturbation theory [23,24]. The stringent confrontation of theoretical predictions with data in the whole final state phase space is likely to reveal new dynamics, signifying the importance of threebody hadronic B meson decays.
Most of PQCD studies of the above decays focus on the kinematic configuration corresponding to edges of Dalitz plots, whose formalism can be simplified by the introduction of two-hadron distribution amplitudes (DAs) [15]. In these regions two of the three final state hadrons collimate with each other in the rest frame of the B meson. At the quark level this configuration involves the hadronization of two energetic collinear quarks, produced from the b quark decay, into the two collimated hadrons. The hadron-pair system, dominated by infrared QCD dynamics, can then be factorized out of the whole process, and defines the two-hadron DA Φ h 1 h 2 [25][26][27][28]. The factorization formula for the B → h 1 h 2 h 3 decay is then expressed as where Φ B (Φ h 3 ) denotes the B meson (h 3 hadron) DA, and ⊗ means the convolution in parton momenta. The hard kernel H for the b quark decay, similar to the two-body case, starts with the diagrams of single hard gluon exchange. An advantage of the above formalism is that both resonant and nonresonant contributions to the hadron-pair system can be included into the two-hadron DA through appropriate parametrization. Although Eq. (1) has been applied to the whole three-body phase space, it should be understood that it is precise only in the region with a small hadron-pair invariant mass. A two-hadron DA loses its accuracy in the central region of a Dalitz plot, where the major contribution to three-body decays arises from two hard gluon exchanges [15]. Nevertheless, it is also the region, where a two-hadron DA decreases with certain power law of the invariant mass, and gives a minor contribution.
In this paper we will extend the PQCD approach to the three-body radiative decays B → P V γ with P (V ) representing a pseudoscalar (vector) meson. The significance of these decays has been well recognized: the involved flavor-changing neutral current b → sγ, occurring only at loop level in the Standard Model, is sensitive to new physics effects. Following the similar reasoning, the two-hadron DAs Φ P V can be introduced to collect the dominant contribution from the region with a small P V -pair invariant mass m P V . For instance, nearly 72% of the signal events appear in the low mass region with m φK ∈ [1.5, 2.0] GeV [29]. Besides, the emitted photons from the leading electromagnetic O 7γ transition are mainly left-handed (right-handed) in B − andB 0 (B + and B 0 ) meson decays. A chirality flip may be induced by local four-quark operators and the chromomagnetic penguin operator O 8g from QCD corrections, as well as by final state interactions among various resonant channels [30]. We will address the above subjects, taking the B → φ(ρ)Kγ decays as examples.
The resonant contribution to the φK system is negligible [29,31], so the parametrization for the DAs Φ φK in Ref. [32], which contain time-like form factors with certain power-law behavior, is adopted. The resonant contributions from the states K 1 (1270) and K * (1680) to the ρK system dominate [33,34]. Therefore, the parametrization of the DAs Φ ρK follows that for quasi-two-body B meson decays [17,18,35,36], namely, the Breit-Wigner model.
In Sec. II we construct the P V DAs according to the procedure proposed in [32]: the dependence on the parton momentum fraction is parametrized in terms of the Gegenbauer polynomials, and the dependence on the meson momentum fraction is derived through the normalizations to time-like P V form factors. In Sec. III we analyze the three-body radiative decays B → φ(ρ)Kγ, determine the P V DAs from relevant branching-ratio data, and then predict their direct CP asymmetries, photon polarization asymmetries, and decay spectra in the P V -pair invariant mass. Our work is more complete than [32], because the contributions from the operators O 7γ , O 8g , and O 2 are all considered, and the annihilation diagrams are calculated. The summary is given in the last Section, and the factorization formulas for the B → φ(ρ)Kγ decay amplitudes are collected in the Appendix.

II. PSEUDOSCALAR-VECTOR DISTRIBUTION AMPLITUDES
We choose the B meson momentum P B , the P V -pair momentum P , and the photon momentum P γ in the light-cone coordinates as with the B meson mass m B and the variable η = P 2 /m 2 B ≡ ω 2 /m 2 B , ω being the invariant mass of the P V pair. Define the momenta of the vector and pseudoscalar mesons by respectively, which obey P = P 1 + P 2 , with the vector meson momentum fraction ζ and the mass ratio r V = m V /m B . The smaller pseudoscalar mass has been neglected. Write the spectator momenta in the B meson and in the P V pair as respectively, x 1 and z being the momentum fractions. We also define the polarization vectors ǫ of the P V system by A two-meson DA φ(z, ζ, ω) describes the hadronization of two collinear quarks, together with other quarks popped out of the vacuum and playing no role in a hard decay process, into two collimated mesons. It can be decomposed in terms of the eigenfunctions of the ERBL evolution equation [37,38], i.e., the Gegenbauer polynomials C 3/2 n (2z − 1), and the partial waves, i.e., the Legendre polynomials P l (2ζ − 1) [26,28]. However, for the P V system, the different spins of the pseudosalar and the vector render the Legendre polynomial expansion not applicable. Hence, we extract the ζ dependence from the normalizations of the P V DAs to the associated time-like form factors [32], which depend on the P V -pair invariant mass ω, a procedure similar to deriving the two-pion DAs via the process γγ * → π + π − [39]. To be explicit, we evaluate perturbatively the matrix elements of local currents using the vector and pseudoscalar DAs up to twist 3, where the polarization vectors of the vector meson satisfy ǫ * (V ) · P 1 = 0 and ǫ * (V ) 2 = −1, and Γ represents the possible spin projectors I, γ 5 , γ µ , γ µ γ 5 , and σ µν γ 5 . The above matrix element is precisely the normalization of the P V DA associated with the spin projector Γ, and also the P V time-like form factor associated with the local currentq ′ Γq. The goal of the perturbative calculation is to reveal the kinematic structure of the matrix element in terms of P 1 , ǫ(V ), and P 2 for each Γ, which are then approximated by the momentum P and the polarization vectors ǫ of the P V system according to the power counting rules in the heavy quark limit [32]. In this way we obtain the ζ dependence of the P V DAs up to twist 3, i.e., O(ω/m B ).
The expansions of the nonlocal matrix elements for various spin projectors Γ up to twist 3 are listed below: where n − = (0, 1, 0 T ) is a light-like vector, and the convention ǫ 0123 = −1 has been employed.
To get the first term in Eq. (7), we have applied where the coefficient 2ζ − 1 is absorbed into the DA φ , giving rise to its ζ dependence. We have also made the approximation as arriving at Eq. (8). It is found, compared to [32], that the term (P 1µ P 2ν − P 1ν P 2µ ) does not generate the twist-2 contribution (ǫ T µ P ν − ǫ T ν P µ ), since a transverse momentum and a transverse polarization have different physical meanings. Equation (10) comes from the approximation of the kinematic factor We summarize the P V DAs for the longitudinal and transverse polarizations from Eq. (7)-Eq. (11) as where φ ,t are of twist 2, and φ p,3,a,v are of twist 3. The above P V DAs contain the products of the time-like form factors F (ω), which define the normalizations of the DAs, and the z-dependent and ζ-dependent functions: Different from Ref. [32], the DA φ a in the above expressions does not depend on the meson momentum fraction ζ. Note that only the DAs for the transversely polarized P V pair are relevant to the three-body radiative decays B → P V γ considered here.
We include the first Gegenbauer moment for the function f a (z), making the φK DA φ a a bit asymmetric in the parton momentum distribution, and assume the asymptotic form The φK time-like form factors, dominated by nonresonant contributions, are parametrized with the chiral scale m 0 ≃ 1.7 GeV [40] and the threshold invariant mass m l = m φ + m K .
That is, we keep the pseudoscalar mass only in the phase space allowed for the time-like form factors. The tunable parameters a 1 and m T ≃ m , expected to be few GeV, will be determined from the fit to the data of the B → φKγ branching ratios. Since φ a gives a larger contribution, as verified numerically in the next section, the data lead stronger constraint to its first Gegenbauer moment. This explains why we introduce a 1 only into φ a .
The amplitude analysis on the resonant structure of the final state in the B + → K + π − π + γ decay [34] provides a useful guideline for parametrizing the resonant contribution to the B → ρKγ mode, for which K 1 (1270) and K * (1680) are the major intermediate resonances.
The resonance K 1 (1270) is a mixture of the K 1A (1 3 P 1 ) and K 1B (1 1 P 1 ) states, θ K being the mixing angle. With the insertion of Eq. (20), the quasi-two-body B → K 1 (1270)(→ ρK)γ decay amplitude can be expressed as the combination of the B → K 1A (→ ρK)γ amplitude and the B → K 1B (→ ρK)γ amplitude, such that the K 1A and K 1B meson DAs with the specific symmetry in the z dependence can be employed. The ρK DAs for the B → K 1 (1270)(→ ρK)γ modes are then parametrized as where the mixing factor c K and the Gegenbauer moments associated with the K 1A (K 1B ) state take the values In principle, the Gegenbauer moments for the ρK DAs are free parameters. Here we adopt those for the K 1A and K 1B DAs [41,42] as their typical values in the numerical study below.
In this paper we have proposed different parametrizations of the nonresonant and resonant contributions: for the former, the final state interaction is ignored, and their form factors are real and normalized to F (m l ) = 1. Because it arises from a wider range of the invariant mass, the different power-law behaviors of the form factors, F φK T ∼ 1/ω 2 and F φK a,v (ω) ∼ m 0 /ω 3 at large ω [43][44][45], have been taken into account. For the latter, the major piece comes from the region around the resonance mass, so it is reasonable not to differentiate the power-law behaviors of the form factors in φ t,a,v , but parametrize them in terms of the same Breit-Wigner model.
Since the Gegenbauer moments of the K * (1680) DAs are unknown, we simply assume the asymptotic form z(1 − z) for the z dependence of the corresponding ρK DAs. The standard Breit-Wigner model is aslo used for the associated form factor where the parameter c, characterizing the strength relative to the amplitude from the resonance K 1 (1270), will be determined from the fit to the data of the B → ρKγ branching ratios. There is no interference between the K 1 and K * states due to different quantum numbers. Denoting the amplitude from the K 1 (1270) (K * (1680)) channel by A 1 (A 2 ), we compute the total amplitude squared for the B → ρKγ decays by [34] in which the factor 1/(1 + c 2 ) plays the role of an overall normalization.

III. NUMERICAL RESULTS
As stated before, the evaluation of the three-body radiative B meson decay amplitudes reduces to that of two-body ones [46][47][48] with the introduction of the P V DAs. We consider the O 7γ , O 8g , and O 2 operators, and the annihilation contributions, performing an analysis more complete than in [32], where only the emission diagrams from O 7γ were taken into account. The explicit factorization formulas for various contributions are collected in the Appendix. The B → P V γ differential decay rate, i.e. the decay spectrum in the P V invariant mass, is then derived from where the vector meson momentum fraction ζ is bounded by m 2 l /ω 2 ≤ ζ ≤ 1, and A R(L) denotes the amplitude for the right-handed (left-handed) photon emission.
The inputs for the masses (in units of GeV) [49], the widths of the K 1 and K * mesons (in units of GeV) [34], and the mean lifetimes of the B mesons (in units of ps) are listed below: In addition, we take the B meson decay constant f B = 0.190 GeV, which is in agreement with the lattice results f B = 0.186 ± 0.004 GeV [57] and f B = 0.186 ± 0.013 GeV [58], and with those from the recent theoretical studies [59,60].
We consider the following theoretical errors.
which match the observed values well. The negative a 1 implies that the light spectator quark intends to carry, as expected, a smaller fraction of the φK pair momentum. It has been examined that the above results are stable against the variation of m T, around few GeV.
The central value of B(B + → φK + γ) in Eq. (29) is lower than the prediction (2.9 +0.7 −0.5 ) ×10 −6 in [32] because of the combined effects of the following changes: retaining the kaon mass here suppresses the phase space, and the inclusion of the parton k T renders hard propagators more off-shell, but the asymmetric DA φ a compensates the above reduction a bit.
The exclusive B meson decays into the resonances K 1 (1270) and K * (1680) have been reported by BaBar with the branching ratios [34], The branching factions of the K 1 (1270) and K * (1680) transitions into the ρK final state [49] B exp (K 1 (1270) → ρK) = (42 ± 6)%, then lead to Note that the total branching ratio [34] deviates from the sum of Eqs. (32) and (33), since the nonresonant contribution and the other minor K 1 (1400), K * (1410), and K * 2 (1430) resonant contributions, which may cause destructive interference, have been neglected. The free parameter c in Eq. (24), characterizing the magnitude of the K * (1680) resonant contribution relative to the K 1 (1270) one, will be determined by the fit to Eqs. (32) and (33). Note that the parameter c has reflected the strength of the K * (1680) → ρK transition.
and the branching ratios of the B 0 meson decays are predicted to be For θ K = 58 • , we choose the best fit value c = 1.8, obtaining We mention that an upper bound ω ≤ 1.8 GeV for the ρK invariant mass, the same as the experimental cutoff [34], has been applied to the calculation. To test the sensitivity of the branching ratios to the mixing angle θ K , we fix c = 2.0, and display the θ K dependence of the B + → K 1 (1270 + )(→ ρ 0 K + )γ branching ratio in Fig. 1. The coordinates θ K ∼ 33 • and 58 • happen to locate on the two sides of a peak, explaining why the results in Eqs.
whose theoretical uncertainties contain only those associated with the considered resonances.
The isospin symmetry then yields the estimate The direct CP asymmetry in the B → P V γ decay is define by Since the difference of the weak phases between V * tb V ts and V * cb V cs is negligible, the dominant O 7γ contribution can induce an appreciable CP asymmetry only through its interference with the amplitudes proportional to V * ub V us . We predict the direct CP asymmetries (in units of percentage) whose errors are smaller than those of the branching fractions, due to the cancellation of partial theoretical uncertainties in the ratio in Eq. (42). Both the Belle and BaBar Collaborations have measured the direct CP asymmetries (in units of perventage) [29,31] A which are consistent with our prediction in Eq. (43).
The photon polarization parameter is defined by [61], whose measurements provide a crucial test for the Standard Model [62,63]. We find λ γ ≃ 1 in our framework, implying that the left-handed contribution is tiny in both the B → φKγ Fitting the PQCD factorization formulas to the branching-ratio data, we have fixed the free parameters in the P V DAs, which were then employed to predict the direct CP asymmetries, the decay spectra and the photon polarization parameter of the B → φ(ρ)Kγ modes.
The O 7γ , O 8g , and O 2 operators, and the annihilation contributions have been taken into account, so this work is more complete than in the literature [32], where only the emission diagrams from O 7γ were considered. It has been shown that our results are in good agreement with all the existing data. More precise data from future experiments will help testing our predictions, including other minor resonant contributions which have been ignored here, and improving the application of the PQCD formalism to more three-body B meson decays.
The analysis of the B → K * πγ decays is similar, but requires the inclusion of all the K 1 (1270), K 1 (1400), K * (1410), and K * (1680) intermediate resonances [34]. Five parameters are then needed to describe the interference among the resonances with the same spin parity, three of them accounting for the magnitudes and two for the phases. The present data are not sufficient to determine these parameters, so we will leave the B → K * πγ modes to a future investigation.  The effective Hamiltonian relevant to the b → s transition is given by [65] with the Wilson coefficients C and the local operators where the terms associated with the strange quark mass in the O 7γ and O 8g operators have been dropped.
The dominant contributions to the three-body radiative decays B → P V γ comes from O 7γ , whose diagrams are displayed in Fig. 4 with the photon being emitted from the operator.
The factorization formulas for the emissions of the right-handed and left-handed photons are written as respectively, where the left-helicity amplitude M L 7γ vanishes because of the neglect of the strange quark mass.
for the first two diagrams, where Q b(s) labels the charge of the b (s) quark in units of the electron charge e, and as The diagrams with the photon being emitted by an external quark are shown in Fig. 6.
The effective vertexb →sg resulting from the loop integration in the MS scheme is given by [66] where k is the virtual gluon momentum and m i , i = u, c, are the masses of the quarks in for the first two diagrams, and as for the last two diagrams.
In the case where the photon is emitted from the quark loop, as displayed in Fig. 7, the sum of the effective vertexb →sγg * produces [67,68] where with q (k) being the photon (gluon) momentum. The amplitudes for the two diagrams are expressed as in which the function h ′ e is defined by with The annihilation diagrams are exhibited in Fig. 8 a 1 = C 2 + C 1 /3, a 4 = C 4 + C 3 /3, a 6 = C 6 + C 5 /3, The factorization formulas for the annihilation contributions are given by M L(a,LR) Finally, we sum the squared amplitudes for the B + meson decays in the helicity basis, |A(B + )| 2 = i=R,L |A i (B + )| 2 , deriving We have the similar sum for the B 0 meson decay amplitudes with It is seen that the O 7γ contributions to the B + and B 0 meson decays are identical.
The hard functions are written as The Sudakov factor S t (x) from the threshold resummation follows the parametrization in [69] S t (x) = 2 1+2a Γ(3/2 + a) with the Sudakov exponents γ q = −α s /π being the quark anomalous dimension. The function s(Q, b) is expressed as n f being the number of the quark flavor, and γ E the Eular constant.