Quark off-shell effects in B → γγ decays

Abstract

We discuss some flavour-changing effective Lagrangians, obtained from scales above 1 GeV, which vanish on the quark mass shell. Although the effects of such effective Lagrangians are zero in the limit of vanishing bound-state interactions, we have shown that they have a significant impact on the processes K → γγ

The self-penguin contribution to K → 2π

“…Still, also in strong interactions such effects are relatively small, being proportional to M 2 /Λ 2 χ ≃ 0.07 and vanishing in the free quark limit M → 0. They might, however, become non-negligible in cases where there are cancellations between the potentially largest terms, as shown in this paper for ǫ ′ /ǫ , and for K → 2γ [9] where the potentially large pole contributions cancel when the Gell-Mann-Okubo mass formula is used.…”

“…In a series of papers [6,7,8,9], it has been pointed out that the effect of such operators are not zero in general, and that they in particular will contribute to processes like K → 2π , K → 2γ , and B → 2γ. In order to be consistent, such genuinely off-shell effects should go to zero in the limit when some typical bound state parameter goes to zero, which is indeed found to be the case.…”

Quark off-shell contributions to KL → γγ in a bound-state approach

“…Let us stress that in the meantime "the anatomy" of the off-shellness has been demonstrated in detail on the example of the heavy-light B meson [22] and a comparative study of K, B → γγ decay [23]. The off-shell part of the B → γγ amplitude that can not be transformed away has been displayed on Fig.…”

“…2 of ref. [22]. However, in order to calculate the B → γγ amplitude we had to invoke some bound-state model, different from the chiral quark model applicable for K L → γγ decay.…”

“…The necessary ingredient in such an approach is the study of the mixing of the appropriate operators under the renormalization [25]. For the purposes of such future considerations, it might therefore be worth recalling the (at first sight surprising) result of ref [22], that the operator in L F (4) has zero anomalous dimension, whereas the corresponding magnetic type operator in L σ (5) is well known (Grinstein et. al.…”