The Riddle of Polarization in $B \to VV$ Transitions

Measurements of polarization fractions in $B \to VV$ transitions, with $V$ a light vector meson, show that the longitudinal amplitude dominates in $B^0 \to \rho^+ \rho^-$, $B^+ \to \rho^+ \rho^0$, and $B^+ \to \rho^0 K^{*+}$ decays and not in the penguin induced decays $B^0 \to \phi K^{*0}$, $B^+ \to \phi K^{*+}$. We study the effect of rescattering mediated by charmed resonances, finding that in $B \to \phi K^*$ it can be responsible of the suppression of the longitudinal amplitude. For the decay $B \to \rho K^*$ we find that the longitudinal fraction cannot be too large without invoking new effects.


Introduction
An important result obtained by Belle and BaBar Collaborations is the measurement of the decay widths and of the polarization fractions of several B decays to two light vector mesons [1,2,3,4]. The branching fractions measured by the two Collaborations are collected in Table 1 together with the averages. Together with these data one should collect the upper bound B(B 0 → ρ 0 ρ 0 ) ≤ 2.1 × 10 −6 from BaBar [2]. Through the analysis of angular distributions, the polarization fractions of the final states have been measured as reported in Table 2. In the decay modes B 0 → ρ + ρ − and B + → ρ 0 ρ + , ρ 0 K * + the final states are essentially in longitudinal configuration, with a larger uncertainty for B + → ρ 0 K * + ; on the contrary, in both the observed B → φK * transitions the longitudinal amplitude does not dominate, providing nearly 50% of the rate.
In this note we wish to address another effect that potentially changes the result in the penguin induced B → φK * decay without affecting the observed B → ρρ transition: rescattering of intermediate charm states. Such effects, studied long ago in B → Kπ transitions [9] and investigated recently in other B → P P and V P decays [10] as well as in factorization forbidden B transitions to charmonium final states [11], can invalidate the arguments on the basis of which the dominance of the longitudinal configuration is argued. We discuss factorization and its consequences in Section 2 and the analysis of rescattering effects for B 0 → φK * 0 in Section 3. At the end we discuss a few consequences.

Polarization in factorization-based approaches
The decay B 0 → φK * 0 is described by the amplitude with ǫ(q, λ) and η(p ′ , λ) the φ and K * polarization vectors, respectively, with λ = 0, ±1 the three helicities. Since the decaying B meson is spinless, the final vector mesons share the same helicity. A 0 and A 2 are associated to the S-and D-wave decay, respectively, and A 1 to the P-wave transition. The three helicity amplitudes A L and A ± can be written in terms of A 0,1,2 : in the transversity basis, the transverse amplitudes λ the triangular function) the common φ and K * three-momentum in the rest frame of the decaying B-meson. In terms of such amplitudes the expression of the decay rate is simply: while the three polarization fractions are given by In order to compute the amplitude eq.(1), we consider the effective weak Hamiltonian inducing theb →sss transitions, which can be written as with the operators with F µν and G a µν the electromagnetic and the gluon field strength, respectively; c i,7γ,8g (µ) are the Wilson coefficients.
The amplitude A(B 0 → φK * 0 ) obtained from (6) admits a factorized form 10 (N c is the number of colours). This formula presents the drawbacks of naive factorization, namely there is not a compensation of the scale dependence between Wilson coefficients and operator matrix elements. However, it allows us to immediately write down the polarization fractions, once the B → K * matrix element has been expressed in terms of form factors 1 , and the φ meson leptonic constant has been introduced: with the form factors V, A 1 , A 2 , A 3 and A 0 functions of q 2 . From (8-10) it is easy to write down the polarization amplitudes and check that, for large values of M B , expressions which determine the behaviour of the three amplitudes once the parametric dependence on the heavy quark mass of the form factors close to the maximum recoil point has been established. In the limit M B → ∞ and for q 2 = 0 such a dependence has been investigated [12] with the result that the three form factors V, A 1 and A 2 should be equal: A 2 /A 1 = V /A 1 = 1. One therefore expects: regardless, in this scheme, of the Wilson and CKM coefficients. Assuming generalized factorization, with the substitution of the Wilson coefficients a i with effective parameters a ef f i , it is eventually possible to reconcile the branching ratio with the experimental measurement, but not to modify the polarization fractions, since the dependence on the a i cancels out in the ratios. Therefore, in order to explain the small ratio Γ L /Γ within the Standard Model one has to look either at the finite mass corrections, or at effects beyond factorization.
For finite heavy quark mass, one can compare the experimental result for the polarization fractions in B 0 → φK * 0 decays (Table 2) with the predictions of various form factor models [13,14,15,16]. As shown in fig. 1, in many models the ratios A 2 /A 1 and V /A 1 deviate from the asymptotic prediction, suggesting that the regime of finite M B does not fully coincide with the asymptotic regime. In one case there is a marginal agreement between the form factor model and data. However, the indication of effects beyond naive and generalized factorization is clear.

Rescattering effects
If one considers the possibility of rescattering effects, there are other terms in the effective weak hamiltonian that can induce the transition B 0 → φK * 0 . Processes that should be the most relevant ones areb → ccs → sss. Such processes can give sizeable contribution to the penguin amplitudes obtained from (6) . The continuous lines correspond to the (one, two and three-σ) regions of the Belle data in Table 2 (a) and of the average of Belle and BaBar data (b) for Γ L /Γ and Γ ⊥ /Γ in B 0 → φK * 0 . The points correspond to different form factor models: QCDSR [13] (dot), LCSR [14] (triangle), MS [15] (square), BSW [16] (diamond). In order to estimate the contribution of diagrams of the type in fig.2 we can use a formalism that accounts for the heavy quark spin-flavour symmetries in hadrons containing a single heavy quark [17] and for the so called hidden gauge symmetry to describe their interaction with light vector mesons [18]. As well known, in the heavy quark limit, due to the decoupling of the heavy quark spin s Q from the light degrees of freedom total angular momentum s ℓ , it is possible to classify hadrons with a single heavy quark Q in terms of where v is the meson four-velocity and a is a light quark flavour index. The field H a is defined as H a = γ 0 H † a γ 0 ; all the heavy field operators contain a factor √ M H and have dimension 3/2.
It is possible to formulate an effective Lagrange density for the low energy interactions of heavy mesons with light vector mesons [18]. The interaction term of such a Lagrangian reads as Light vector mesons are included in (14) through the fields ρ = i g V 2ρ representing the low-lying vector octet: Invoking the mixing ω 8 − ω 0 one gets the interaction term involving φ. The interactions of heavy mesons with the light vector mesons are thus governed, in the heavy quark limit, by two couplings β and λ. From light cone QCD sum rules [19] as well as from vector mesons dominance arguments [10] one estimates β ≃ 0.9 and λ ≃ 0.56 GeV −1 , while g V is fixed to g V = 5.6 by the KSRF relation [20].
Using (14) it is easy to work out the matrix elements D ( * ) s D ( * ) K * appearing in one of the vertices in fig.2: As for the weak amplitude B 0 → D ( * )+ s D ( * )− , since there is empirical evidence that factorization reproduces the main experimental findings [21], we write it as with a 1 ≃ 1. In the heavy quark limit the matrix elements in (17) involve the Isgur-Wise function [17]: v and v ′ being B and D ( * ) four-velocities, ǫ the D * polarization vector and ξ(v · v ′ ) the Isgur-Wise form factor. As for the D ( * ) current-vacuum matrix elements defined as they can be parameterized in the heavy quark limit in terms of a single quantity f Da = f D * a . Now, the estimate of the absorptive part of the rescattering diagrams in fig. 2 ImA resc = λ can be carried out. The integration variable z = cos θ is related to the angle between the three-momenta of φ and of the emitted D ( * ) s from B vertex in fig.2. We use |V cb | = 0.042, |V cs | = 0.974 (the central values reported by the Particle Data Group [22]), f D * s = f Ds = 240 MeV [23] and ξ(y) = 2 1+y 2 .
The couplings in (16) do not account for the off-shellness of the exchanged D ( * ) s mesons in fig.2. One can introduce form factors: to account for the t-dependence of the couplings (the vertices in rescattering diagrams cannot be considered point-like since they do not involve elementary particles), g i0 being the on-shell couplings. However, the form factors are unknown. We use to satisfy QCD counting rules. We could vary the value of Λ, considering the uncertainty from the form factor F (t) in the final numerical result. Instead, since the relative sign of rescattering and factorized amplitude is also unknown, as well as the role of diagrams involving excitations and the continuum, we fix Λ = 2.3 GeV and analyze the sum varying the parameter r and approximating the long distance amplitude with eq. (20). We compute the short-distance factorized amplitude using the B → K * form factors appearing in two extreme cases in fig.1, the model [13] and the model [14],  [24] for N c = 3. The result is depicted in fig.3. For the model [13], a contribution of the rescattering amplitude is in order to obtain the measured B → φK * branching fraction. Of the two possible values of the parameter r which reproduce the experimental rate, r ≃ 0.08 and r ≃ −0.3, the former allows us to simultaneously obtain a small longitudinal polarization fraction: Γ L /Γ ≃ 0.55, compatible with the measurements. The tranverse polarization fractions turn out Γ /Γ ≃ 0.30 and Γ ⊥ /Γ ≃ 0.15. They are both consistent with measurement, but with the hierarchy Γ /Γ > Γ ⊥ /Γ. If we use the form factors in [14], for r = 0 the predicted rate exceeds the experimental datum, so that the rescattering contribution should be weighted by a negative r to  [13] (left) and in [14] (right) are used in the short-distance amplitude. r = 0 corresponds to the absence of rescattering. The three curves in (b) correspond to Γ L /Γ (continuous curve), Γ ⊥ /Γ (dashed) and Γ /Γ (dot-dashed). The horizontal lines represent the experimental result in Table 1 for the branching ratio (a) and for Γ L /Γ (b).
reconcile the branching fraction; as depicted in fig.3, in such a region (r ≃ −0.05) the longitudinal fraction increases. However, this conclusion crucially depends on the value of the Wilson coefficients a 3 − a 10 used as an input in the evaluation of the short-distance amplitude. As shown in [24]

Discussion
The conclusion of this analysis is that FSI effects can modify the helicity amplitudes in penguin dominated processes. The numerical result depends on the interplay between Wilson coefficients, form factors and rescattering amplitude, and we have shown that the experimental observation can be reproduced. At the same time, the rescattering effects we have considered are too small to affect the observed B → ρρ decays. As a matter of fact, while the CKM factors in the tree diagram in B 0 → ρ + ρ − transition (V * ub V ud ) have similar size to the CKM factor in the FSI diagram in fig.2 (V * cb V cd ), the Wilson coefficient in current-current transition is O(1). We can expect to observe FSI effects in colour-suppressed and other penguin induced B → V V decays, such as B 0 → ρ 0 K * 0 , Let us consider B + → ρ 0 K * + . On the basis of general arguments, we cannot assess the role of FSI without an explicit calculation, due to the CKM suppression of the factorized amplitude. The determination of the rescattering amplitude, similar to that in fig.2, can be done following the same method discussed above, obtaining Γ L /Γ ≃ 0.7, i.e. smaller (even though compatible within 2-σ ) than the measurement in Table 2. Therefore, in our approach we can accomodate a small Γ L for B → φK * at the prize of having a smaller value of Γ L for B → ρK * , which is not currently excluded due to the uncertainty in the data for this mode. It is interesting to notice that an analogous prediction is done in QCD improved factorization [6], where one gets Γ L Γ (B → ρK * ) < Γ L Γ (B → φK * ). More precise measurements are in order to suggest a solution to this polarization riddle. If further measurements of polarization fractions will confirm the present situation of a small longitudinal fraction in φK * and a large longitudinal fraction in ρK * , in that case we cannot identify uniquely the rescattering mechanism for explaining the data, envisaging the exciting necessity of new effects.
Acnowledgments. One of us (PC) would like to thank CPhT,École Polytechnique, where this work was completed. We acknowledge partial support from the EC Contract No. HPRN-CT-2002-00311 (EURIDICE).