Charming loops in exclusive rare FCNC $B$-decays

Rare $B$-decays induced by flavour-changing neutral currents (FCNC) is one of the promising candidates for probing physics beyond the Standard model. However, for identifying potential new physics from the data, reliable control over QCD contributions is necessary. We focus on one of such QCD contributions - the charming loops - that potentially can lead to difficulties in disentangling new physics effects from the observables and discuss the possibility to gain control over theoretical predictions for charming loops.


Introduction
The interest in the contributions of charm to rare FCNC decays of the B-mesons is to a great extent motivated by the fact that virtual charm-quark loops, including charmonia states which appear in the physical region of several FCNC B-decay, have a strong impact on the B-decay observables [1] thus providing a "noise" for the extraction of possible new physics effects. FCNC decays are forbidden in the SM and proceed through loop contributions, see Figure 1.  The u-quark contribution to the CKM unitarity is strongly suppressed, leading to so the contributions of the top and the charm have approximately the same CKM strength, while the u-quark contribution in the loop may be neglected.

Top-quark contribution to FCNF B-decays
In B-decays, the characteristic energy scale µ ∼ m b is much lower than the masses of the t-quark and the W and Z bosons, therefore these heavy degrees of freedom may be integrated out [2][3][4][5]. As the result, e.g. for the top contribution to the effective Hamiltonian describing the b → sγ decay, Fig. 2, one finds the following expression: To calculate the effective Hamiltonian for the b → sl + l − transition, one takes into account the contribution generated by the electromagnetic penguin operator (2), and adds the contribution of box and penguin diagrams described by operators O 9V and O 10A (Fig. 3) yielding

Charm-quark contribution to FCNC B-decays
The c-quark is dynamical at the scale µ ≃ m b , so at most one can integrate out the W-boson contribution and come to the four-fermion effective Hamiltonian The SM Wilson coefficients at the scale µ 0 = 5 GeV have the values [corresponding to

Factorizable charm contributions
Using Eq. (4), factorizable charm contributions to the b → sll amplitude reads [13]: where g cc (m 2 c ,q 2 ),m c = m c /m b ,q 2 = q 2 /m 2 b ), is the function describing charm contribution to vacuum polarization. At leading order in α s , one has The factorizable charm contribution to the b → sl + l − l effective Hamiltonian has the same Lorentz structure as the C 9V term in the top contribution, Eq. (3). So, the relative strength of the factorizable charm/top contributions is essentially determined by the ratio of the Wilson coefficients C 2 + 3C 1 and C 9V (obvioulsly, the top contribution contains also other Lorentz structures, therefore the total top contribution is further enhanced; also the charm contribution contains also nonfactorizable pieces to be discussed later; these are proportional to C 2 ). The combination C 2 + 3C 1 depends strongly on the precise value of the low-energy scale µ ∼ m b . For instance, at µ = m b , one finds C 2 + 3C 1 = −0.3, while C 9V = −4.21. Indeed, there is some numerical (although not parametric) suppression of the factorizable charm contribution but this suppression is the subject to the precise choice of the scale µ. This indicates the importance of higher-order QCD corrections.
As the final step, the calculation of the amplitude of an exclusive FCNC B-decay requires the B-meson weak decay form factors. For instance, for B → γl + l − decay, one needs the B → γ transition form factors We have already noticed that the strong sensitivity of the combination (3C 1 + C 2 ) to the precise value of the scale µ indicates the importance of higher-order QCD corrections. The latter should be synchronously taken into account in the Wilson coefficients and in the Bdecay form factors; this is however a very subtle and difficult problem.

Nonfactorizable charm contribution
In addition to factorizable charm contributions, one needs to take into account nonfactorizable charm effects induced by soft gluons emitted from the charm-quark loop and absorbed in the B-meson loop. The latter are not reduced to the product of charm contribution to the vacuum polarization and the meson transition form factors, but have a more complicated structure. Nonfactorizable charm contributions to the effective Hamiltonian H b→sl + l − eff should be obtained as the convolution of the octet-octet part of the effective b → scc effective Hamiltonian with the electromagnetic ieQ c (cγ α c)A e.m. α (10) and the strong vertices of the charm-quark field. Here A e.m. α and A N β are the photon and the gluon field, respectively; T L = λ L /2, with λ L the Gell-Mann matrices, the indices L, N = 1, . . . , 8.
Nonfactorizable charm contributions are governed by a large Wilson coefficient C 2 . So, nonfactorizabe and factorizable charm-loop effects are expected to be of the same order of magnitude. Comparing the patterns of broad charmonia measured in l + l − collisions and in B → (K, K * )l + l − decays, one concludes that in the charmonia region, nonfactorizable gluon exchanges are indeed at least equivalently important as factorizable charm effects.
A number of theoretical analyses of nonfactorizable effects induced by charm-quark contributions has been published in the literature: an effective gluon-photon local operator describing the charm-quark loop has been calculated in [6] for the real photon as an expansion in inverse charm-quark mass m c and applied to inclusive B → X s γ decays; Ref. [7] obtained a nonlocal effective gluon-photon operator for the virtual photon and applied it to inclusive B → X s l + l − decays. In [8] nonfactorizable corrections in exclusive FCNC B → K * γ decays using local OPE have been studied; in [9,10], 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 [6,7,[11][12][13], local OPE for the charm-quark loop leads to a power se- . 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 [13] derived a different form of the nonlocal photon-gluon operator compared to [7] 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)⟩. In [14] it was shown that the consistent resummation of In the next sections we discuss in detail the contributions of charm to the amplitudes of excusive FCNC B-decays. To avoid technical complication related to a Lorentz/spinorial structure of the amplitudes, we illustrate the calcuation using a field theory with scalar quarks/gluons; the generalization to QCD is straightforward. We nevertheless keep the QCD notations for quark fields and assume the following hierarchy of quark masses m b ≫ m c ≫ m s ∼ Λ QCD , and adopt the counting scheme in which the parameter Λ QCD m b /m 2 c is kept of order unity. far below the charm quark threshold, q 2 ≪ 4m 2 c . Above the charm threshold, perturbative QCD calculations are not applicable, and approaches based on hadron degrees of freedom should be used. However, vacuum polarization can be measured experimentally in a broad range of q 2 , including the region above the charm threshold.

Factorizable charm-loop effects
The form factor describing a FCNC transition B s → γ * (q)γ * (p ′ ) is defined as follows: Here p = q + p ′ and p 2 = M 2 B , and in this formula the quark fields are Heisenberg field operators in QCD. By expanding these operators in powers of α s , one generates the corresponding expansion of the form factor.

Leading-order contribution
At leading order in α s , the form factor is given by the diagram of Figure 6. This contribution to the form factor may be represented as follows: Here the B-meson Bethe-Salpeter wave function, ⟨0|s(x)b(0)|B s (p)⟩, depends on two variables x 2 and xp. One can parametrize it by a Fourier transform in variable xp and perform the Taylor expansion in variable x 2 : Because of the general properties of Feynman diagrams, the integration in the variable ξ runs from 0 to 1. We now insert this expansion in Eq. (12) and study the relative size of the different contributions generated by the expansion (14).
• The term proportinal to ϕ 0 in (14) corresponds to x 2 = 0 and thus describes the light-cone configuration of the quarks inside the B s -meson. Its contribution to the form factor (12) is easy to calculate: Taking into account that (p − q) 2 = p ′2 , and thus 2qp = p 2 − q 2 − p ′2 , we obtain We now have to take into account the crucial property of the light-cone two-particle distribution amplitude (2DA) ϕ 0 (ξ) of the heavy meson B s : since the b-quark is heavy, it carries the major part of the B-meson momentum, such that ϕ 0 (ξ) is peaked near ξ ∼ Λ QCD /m b . We then find that k 2 ∼ −Λ QCD m b , i.e., the propagating s-quark is highly virtual, and therefore the perturbative expression for its propagator is well justified.
• We now turn to the x 2 terms in the expansion (14): these terms describe the deviations from the light-cone configuration. To calculate these contributions, it is convenient to substitute in Eq. [15) x α = −i ∂ ∂k α e ikx . By performing the parts integration, the k α -derivative acts on the s-quark propagator. Taking into account that k 2 ∼ −Λ QCD m b , we find that the contribution of a term (x 2 ) n → (Λ QCD /m b ) n compared to LC term. So the integral for the form factor (12) is indeed dominated by the light-cone quark configuration, whereas the deviations from the LC in the BS wave function of the B-meson are suppressed by powers of a small parameter Λ QCD /m b .

Next-to-leading-order contribution
We now calculate α s -corrections generated by the expansion of the T -product in Eq. (12). The diagram of Fig. 7(a) corresponds to the radiative correction to the s-quark propagator. As we have shown above, the s-quark is highly virtual, so the calculation of the radiative correction is straightforward. We do not give any detail of this calculation but just notice that the corresponding contribution is again given in terms of the two-particle BS amplitude of Eq. (14). Similar to the LO contribution discussed above, the LC term in the expansion of Eq. (14) dominates the B s -decay amplitude, whereas the terms containing powers of (x 2 ) n lead to power-suppressed terms ∼ (Λ QCD /m b ) n in the B s -decay amplitude.
Another correction to the B s decay amplitude shown in Fig. 7(b) has a different structure and needs a more complicated three-particle quark-antiquark-gluon distribution amplitude (3DA) ⟨0|s(y)G(x ′ )b(0)|B s (p)⟩ for its calculation. A detailed discussion of 3Da will be given  in the next Section. Here we just emphasize, that the LC term in 3DA leads to the dominant part of the B s decay amplitude, whereas all the terms in 3DA containing powers of (x ′2 ) n , (y 2 ) n , and ((x ′ − y) 2 ) n , lead to the contributions to the B s decay amplitude, that are suppressed by powers of (Λ QCD /m b ) n . So the situation is very similar to the case of the diagram of Fig. (5. This is due to the fact that the factorizable contributions discussed in this Section contain only one fermion loop containing the heavy quark b. We shall see in the next Section that the situation is different in the case of nonfactorizabe contributions. In the latter case, the amplitude involves two quark loops with different masses, m b and m c , and this fact changes the suppression of the contributions to the B s amplitude, generated by the off-LC terms in the 3DA of B s .

Nonfactorizable charm-loop effects
The charm contribution to the amplitude of an FCNC B-decay reads where quark fields are understood as Heisenberg field operators in a theory that involves weak and strong interactions. Our goal is to study nonfactorizable corrections due to a soft-gluon exchange between the charm-quark loop and the B-meson loop. To the lowest order, the corresponding amplitude is given by the diagram of Fig. 5: The effective Lagrangian that mimics weak four-quark interaction is chosen in a simple form while the interaction of the scalar gluon field G(x) and the scalar c-quarks is taken to be

Quarks inside the charm-quark loop with soft gluon emission are perturbative
The charm contribution is described by the three-point function. For our analysis it is important, that in the region q 2 ≪ 4m 2 c , q ′2 and κ 2 are also far below the charm thresholds, i.e. the charm loop is perturbative. To show that, we use the gluon field in momentum representation, which is related to the gluon field in coordinate representation as follows 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 diagrams 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 outgoingss 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 (23) as

Three-particle antiquark-quark-gluon distribution amplitude of B-meson
By virtue of (24), the amplitude Eq. (19) takes the form Here, we encounter the B-meson three-particle amplitude with three (non-aligned) arguments, for which one may write down the following decomposition: Here λ and ω are dimensionless variables. Making use of the properties of Feynman diagrams, one may show that they should run from 0 to 1. However, if one of the meson constituents is heavy, it carries the major fraction of the meson momentum and as the result the function Φ(λ, ω) is strongly peaked in the region So, effectively one can run the ω and λ integrals from 0 to ∞; such integration limits in fact emerge in the DAs within heavy-quark effective theory [13,15]. We emphasize that for the results presented below only the peaking of the DAs in the region (27) is essential. Notice also that the function Φ(λ, ω) in (26) coincides with the same function that appears in the "standard" 3-particle distribution amplitude with the aligned arguments, x = uy, discussed in [15].

Light-cone contribution
First, let us calculate the contribution to A(q, p) from the term that corresponds to x 2 = y 2 = (x − y) 2 = 0 in the 3DA (26). This is very easily calculable: by inserting (26) into (25) we can perform the x− and y−integrals Next, the δ-functions above kill the integrals over k and κ, and we find As the result one obtains for the sum of two triangle diagrams (with the charm quark running in the loop in two opposite directions) the following expression for the amplitude: Now, we must take into account that the ω-integral is peaked at ω ∼ Λ QCD /m b so the gluon is soft:

s-quark produced in a weak b → s transition is highly virtual
The momentum transferred in the weak-vertex is q ′ = q + κ = q − ωp, such that By virtue of the y-integration in (28), the s-quark propagator takes the form Therefore, in the bulk of the λ-integration the virtuality of the s-quark propagator is large, O(M B ). Therefore, the use of the Feynman expression for the s-quark propagator is well justified.

Deviations from the light-cone
We now turn to the calculation of the contributions to A(q, p) generated by terms ∼ x 2 , y 2 , (x− y) 2 in the 3DA (26). The terms containing powers of 4-vectors y and x in the integral (25) can be calculated by parts integration. Taking into account the results (28), we find the following relative contributions of the terms containing different powers of the coordinate variables: Clearly, all terms containing powers of x 2 and/or y 2 in the 3DA lead to the suppressed contributions to A(q, p) and may be neglected within the considered accuracy. However, the terms containing powers of xy lead to the contributions containing powers of Λ QCD m b /m 2 c , i. e., of order unity within the adopted counting rules. The kinematics of the process is thus rather simple: the vectors x and y are directed along the light-cone [e.g., x along the (+) axis, and y along the (−) axis], but the 4-vector x − y is obviously not directed along the light cone. Therefore, the full dependence of 3DA on the variable (x − y) 2 is needed in order to properly resum corrections of order

Conclusions and outlook
We reviewed the general framework for the theoretical description of semileptonic and radiative leptonic B-decays induced by flavour-changing neutral currents with an emphasis on gaining control over QCD effects induced by charming loops, including both factorizable and nonfactorizable contributions. The charm-loop effects are known to be small compared to the effects of the t-quark in the loop at small values of the dilepton pair (although there is no parametric suppression of charm compared to that of top), but dominate the amplitudes of FCNC rare B-decays in the region of charmonium resonances. As an illustration of the typical magnitude of the contribution of charm, we show the predicted differential distribution in B → l + l − decays from [19] where charm is clearly seen to provide the main contribution to the branching ratios in the charmonia region. Charm contributions in the charmonia resonance region also strongly influence various observables; Fig. 9 shows the prediction for the lepton forward-backward asymmetry in B s → γl + l − from [19]. The contribution of charm away from the charmonia region is expected at the level of a few percent thus providing an unpleasant "noise" for the search for the potential new physics effects. This calls for a better understanding of the charm contributions in FCNC B-decays and gaining a theoretical control over these contributions.
To put our emphasis on the conceptual aspects, we discussed the case of all scalar particles, avoiding in this way conceptually unimportant technical details.  Figure 9. The predicted differential branching ratios dBr/dq 2 in B → γl + l − from [19]. Figure 10. The predicted forward-backward asymmetry A FB (q 2 ) in B → γl + l − decays from [19].
Our main conclusions may be formulated as follows: • The relevant object that arises in the calculation of the nonfactorizable corrections is the three-particle DA (or the vertex function) with independent (non-aligned) coordinates: This object may be viewed as a four-point function and thus depends on five independent variables (the external momentum p is on the mass shell, p 2 = M 2 B ). It is convenient to choose the variables xp, yp, x 2 , y 2 , and (x − y) 2 as these five independent variables. One may parametrize the function by its Fourier transform in the variables xp and yp, and to write the Taylor series in the variables x 2 , y 2 , and (x − y) 2 . It should be emphasized that the function Φ(ω, λ) here is precisely the same function that parametrizes the standard 3DA with the aligned arguments, x = uy, discussed in [15].
At small q 2 ≤ m 2 c , terms of order x 2 and y 2 in the 3DA of B s meson yield small corrections to the nonfactorizable amplitude of B-decay compared to the LC term in the 3DA: for terms O(x 2 ) the suppression parameter is Λ 2 QCD /m 2 c , and for terms O(y 2 ) the suppression parameter is Λ QCD /m b . However, terms ∼ (xy) n in the 3DA yield the contributions of order in the B s decay amplitude. These contributions are of O(1) in our counting scheme and should be properly resummed. (Recall that the difference between the local OPE [6] and the light-cone OPE [13] is of the same order). We emphasize that the knowledge of merely the light-cone 3DA corresponding to x 2 = 0, y 2 = 0 and (x − y) 2 = 0, is not sufficient: While the LC 3DA allows one to resum a part of the large corrections of the order Λ QCD m b /m 2 c , some terms of the same order remain unaccounted for. A consistent resummation of these corrections requires the knowledge of the full dependence of ⟨0|s(y)G(x)b(0)|B s (p)⟩ on the variable (x − y) 2 . 1 The kinematics of the process looks simple: the 4-vectors x and y are directed along the light-cone [e.g., x along the (+)-axis, and y along the (−)-axis], but the 4-vector x − y is obviously not directed along the LC; therefore, the full dependence of the 3DA (34) on the variable (x − y) 2 is needed in order to properly resum corrections of order ( Λ QCD m b /m 2 c ) n . 1 As emphasized to us by Danny van Dyk, the actual suppression parameter of those contributions to the amplitude of FCNC B s decay induced by the (x−y) 2 terms in 3DA vs the LC terms in 3DA is Λ QCD m b /(4m 2 c −q 2 ). For |q 2 | ∼ m 2 c , this agrees with our claim. If one considers the region of q 2 ∼ −m 2 b , then the off-LC effects turn out to be suppressed. However, whether or not for small timelike q 2 these terms may be neglected, requires further detailed analysis.