Heavy-to-light form factors in the final hadron large energy limit of QCD

Abstract

We argue that the Large Energy Effective Theory (LEET), originally proposed by Dugan and Grinstein, is applicable to exclusive semileptonic, radiative and rare heavy-to-light transitions in the region where the energy release E is large compared to the strong interaction scale and to the mass of the final hadron, i.e. for q 2 not close to the zero-recoil point. We derive the Effective Lagrangian from the QCD one, and show that in the limit of heavy mass M for the initial hadron and large energy E for the final one, the heavy and light quark fields behave as two-component spinors. Neglecting QCD short-distance corrections, this implies that there are only three form factors describing all the pseudoscalar to pseudoscalar or vector weak current matrix elements. We argue that the dependence of these form factors with respect to M and E should be factorizable, the M -dependence ( √ M ) being derived from the usual heavy quark expansion while the E-dependence is controlled by the behaviour of the light-cone distribution amplitude near the end-point u ∼ 1. The usual expectation of the ∼ (1 − u) behaviour leads to a 1/E 2 scaling law, that is a dipole form in q 2 . We also show explicitly that in the appropriate limit, the Light-Cone Sum Rule method satisfies our general relations as well as the scaling laws in M and E of the form factors, and obtain very compact and simple expressions for the latter. Finally we note that this formalism gives theoretical support to the quark model-inspired methods existing in the literature.

$$B\rightarrow K^*\mu ^+\mu ^-$$ B → K ∗ μ + μ - optimised observables in the MSSM

“…[46,47]. In this kinematical region several form-factor relations [48,49] allow one to build the various optimised observables [28,35,38]. At low recoil (large q 2 ) the computation relies on an OPE for the relevant non-local hadronic matrix element, either in full QCD [50] or within the HQET [51].…”

On the LHCb anomaly in B → K * ℓ + ℓ −

“…In the combined limit of a heavy b-quark and of an energetic K * meson, the decay amplitude factorises to leading order in Λ/m b and to all orders in α s into process-independent non-perturbative quantities like B → K * form factors and light-cone distribution amplitudes (LCDAs) of the heavy (light) mesons and perturbatively calculable quantities, which are known to O(α 1 s ) [23,24]. Further, the seven a priori independent B → K * QCD form factors reduce to two universal soft form factors ξ ⊥, [25]. The factorisation formula applies well in the range of the dilepton mass range, 1 GeV 2 < q 2 < 6 GeV 2 .…”

NNLO vertex corrections to non-leptonic B decays: Tree amplitudes

“…Hence, we treat the differenceÕ 1 − O 1 as another evanescent operator, and regard O 1 as the only physical SCET operator. Due to the absence of soft-gluon interactions between collinear fields in different directions in the leading-power SCET Lagrangian after a field redefinition [19], O 1 factorizes into a (χχ) part, whose SCET matrix element is a light-cone distribution amplitude, and a (ξh v ) part, which defines the soft part of a heavy-to-light form factor [23,24]. On the QCD side of the matching relations (6), (9) we can write the perturbative expansion of the renormalized matrix elements as…”