Calculations of two-loop virtual corrections tob→sl+l−

B → Kℓ + ℓ − decay at large hadronic recoil

“…To accurately analyze this particular contribution, a separation of Q s − Q b component from the Q c -one is desirable in all three nonlocal hadronic amplitudes H (BK * ) 1,2,3 corresponding to the three polarization states of the K * meson. This demands a dedicated "flavour splitting" of the NLO diagrams calculated in [16,17] on one hand and a separate dispersion relation withss vector-meson states on the other hand. Also the NLO two-loop diagrams induced by the penguin operators should be included to achieve an adequate accuracy.…”

“…It has been shown in [16,17] that keeping the terms up to the third power ofŝ andm 2 c provides a sufficient numerical accuracy in the region 0.05 ≤ŝ ≤ 0.25. Here we use these expansions at q 2 < 0, restricting ourselves to |ŝ| < 0.25.…”

“…The reason why the s, c, b-quark contributions are not separated from each other is that in certain terms included in (5.1) e.g., in H (BK) f act,N LO (q 2 ), the virtual photon emissions from b and s quarks are interconnected by gauge invariance. Also the results of [17] used to calculate these contributions do not allow one the separation of the c-quark component. In future, a more detailed flavour separation can be achieved, performing a dedicated analysis of the NLO two-loop diagrams.…”

“…The definitions and nomenclature of the indices of the functions F (7,9) 1,2,8 are the same as in [16,17], where F (7,9) 1,2 are expressed as a double expansion inŝ = q 2 /m 2 b andm 2 c = m 2 c /m 2 b . It has been shown in [16,17] that keeping the terms up to the third power ofŝ andm 2 c provides a sufficient numerical accuracy in the region 0.05 ≤ŝ ≤ 0.25.…”

“…The characteristic feature of all these diagrams [8] 1 is that the corresponding hadronic matrix elements are factorizable in the same form as the LO contributions considered in the previous section. Following [8], we extract the hard-scattering coefficient functions multiplying the B → K form factor from the NLO diagrams for the b → sℓ + ℓ − decay rates calculated in [16,17].…”