Revisiting nonfactorizable charm-loop effects in exclusive FCNC $B$-decays

We revisit the calculation of nonfactorizable corrections induced by charm-quark loops in exclusive FCNC $B$-decays. For the sake of clarity, we make use of a field theory with scalar particles: this allows us to focus on the conceptual issues and to avoid technical complications related to particle spins in QCD. We perform a straightforward calculation of the appropriate correlation function and show that it requires the knowledge of the full generic three-particle distribution amplitude with non-aligned arguments, $\langle 0|\bar s(y)G_{\mu\nu}(x)b(0)|B(p)\rangle$. Moreover, the dependence of this quantity on the variable $(x-y)^2$ is essential for a proper account of the $\left(\Lambda_{\rm QCD}m_b/m_c^2\right)^n$ terms in the amplitudes of FCNC $B$-decays.


INTRODUCTION
The interest in the contribution of virtual charm loops in rare FCNC semileptonic and radiative leptonic decays of the B-mesons is two-fold: (i) although CKM-suppressed, the effect of the virtual charm-quark loops, including the narrow charmonia states which appear in the physical region of the B-decay, has a strong impact on the B-decay observables [1] thus providing an unpleasant "noise" for the analysis of possible new physics effects; (ii) it is known that in the charmonia region, nonfactorizable gluon exchanges dominate the amplitudes posing a challenging QCD problem.
A number of theoretical analyses of nonfactorizable effects induced by charm-quark contributions has been published in the literature. We will mention here only those that are directly related to the discussion of this letter: In [2], an effective gluon-photon local operator describing the charm-quark loop has been calculated for the real photon as an expansion in inverse charm-quark mass m c and applied to inclusive B → X s γ decays; Ref. [3] obtained a nonlocal effective gluon-photon operator for the virtual photon (i.e. without expanding in inverse powers of m c ) and applied it to inclusive B → X s l + l − decays. In [4] nonfactorizable corrections in exclusive FCNC B → K * γ decays using local OPE have been studied; in [5], these corrections have been analyzed with light-cone sum rules using local OPE for the photon-gluon operator and three-particle light-cone distribution amplitudes of K * -meson.
As emphasized in [2,3,6], local OPE for the charm-quark loop leads to a power series in Λ QCD m b /m 2 c . This parameter is of order unity for the physical masses of c-and b-quarks and thus corrections of this type require resummation. The authors of [6] derived a different form of the nonlocal photon-gluon operator compared to [3] and evaluated its effect at small values of q 2 (q momentum of the lepton pair) making use of light-cone 3-particle DA (3DA) of the B-meson with the aligned arguments, 0|s(y)G µν (uy)b(0)|B s (p) .
The goal of this letter is to emphasize that the full consistent resummation of Λ QCD m b /m 2 c n terms in the nonfactorizable amplitude requires a more complicated object, 0|s(y)G µν (x)b(0)|B s (p) , i.e., a generic 3DA with non-aligned coordinates. We perform the analysis using a field theory with scalar quarks/gluons which is technically very simple and allows one to focus on the conceptual issues; the generalization of our analysis for QCD is straightforward. We calculate nonfactorizable corrections directly, keeping control over all approximations. We adopt the counting scheme in which the parameter Λ QCD m b /m 2 c is kept of order unity, and show that the full 3DA is necessary in order to resum properly the (Λ QCD m b /m 2 c ) n corrections: the dominant contribution to the B-decay amplitude are generated not only by the light-cone terms y 2 = 0 and x 2 = 0, but also by terms of order ∼ (xy) n . Therefore, the dominant contributions to the B-decay amplitude come from the configurations when both x and y lie on the light cone, but on the different axes: if x is aligned along the (+)-axis, then y is aligned along the (−)-axis.
Expressing the B-decay amplitude via the standard 3DA with the aligned arguments, one can resum only a part of the (Λ QCD m b /m 2 c ) n corrections, whereas another source of the corrections of the same order remains unaccounted.

NONFACTORIZABLE CORRECTIONS IN A FIELD THEORY WITH SCALAR PARTICLES
In order to exemplify the details of the calculation, we consider nonfactorizable effects for the case of spinless particles. We shall use the standard QCD notations for spinor fields and assume that m b ≫ m c ≫ m s ∼ Λ QCD , but the parameter Λ QCD m b /m 2 c is of order unity.
We study the amplitude which involves weak interactions. We want to study nonfactorizable corrections due to a soft-gluon exchange between the charm-quark loop and the B-meson loop. To lowest order, the corresponding amplitude is given by the diagram of Fig. 1: where the effective Lagrangian that mimics weak four-quark interaction is taken in the form and the scalar gluon field G(x) couples to the scalar c-quarks via the interaction i.e., G involves the quark-gluon coupling. First, we consider the charm-quark loop with the emission of a soft scalar gluon. We use the gluon field in momentum representation, related to the gluon field in coordinate representation as Then the effective operator describing the gluon emission from the charm quark loop may be written as where Γ cc (κ, q) stands for the contribution of two triangle diagram with the charm quark running in the loop. The momenta κ and q are outgoing from the charm-quark loop, whereas the momentum q ′ = q + κ is emitted from the b → s vertex. p ′ is the momentum of the outgoing s † s current and p is the momentum of the B-meson, p = p ′ + q. In terms of the gluon field operator in coordinate space, we can rewrite (2.6) as By virtue of (2.7), the amplitude Eq. (2.2) takes the form Here, we encounter the B-meson three-particle amplitude with three different (non-aligned) arguments, for which we may write down the following decomposition: where λ and ω are dimensionless variables. Making use of the properties of Feynman diagrams, one may prove that they should run from 0 to 1; however, if one of the constituents is heavy, the functions are peaked near small values of λ and ω, of order Λ QCD /m b , so effectively one can run ω and λ from 0 to ∞. Notice that the function Φ(λ, ω) in (2.9) coincides with the same function that appears in the "standard" 3-particle distribution amplitude with the aligned arguments, x = uy, discussed in [7].

A. Light-cone contribution
First, let us calculate the contribution to A(q, p) from the term given by Φ(λ, ω) in the 3DA (2.9), i.e. corresponding to x 2 = y 2 = (x − y) 2 = 0. After inserting (2.9) into (2.8) we can perform the x− and y−integrals (2.10) The next step is easy: the δ-functions above allow us to take integrals over k and κ, and we find (2.11) For the sum of two triangle diagrams with the charm quark running in the loop we may use the representation (2.12) Now, we must take into account that the ω-integral is peaked at ω ∼ Λ QCD /m b so the gluon is soft: κ = −ωp and κ 2 ∼ O(Λ 2 QCD ) ≪ m 2 c . The momentum transferred in the weak-vertex is q ′ = q + κ = q − ωp, such that (2.13) By virtue of the y-integration in (2.10), the s-quark propagator takes the form (2.14) Therefore, in the bulk of the λ-integration the virtuality of the s-quark propagator is large, O(M B ). Let us notice that the q 2 -dependence of the s-quark propagator is very mild and can be neglected; the main q 2 -dependence of the amplitude A(q, p) comes from the charm-quark loop.