The Large energy expansion for B decays: Soft collinear effective theory
Effective field theories provide a simple and elegant method for calculating processes with several relevant energy scales [1, 2, 3, 4, 5, 6]. Part of the utility of effective theories is that they dramatically simplify the summation of logarithms of ratios of mass scales, which would otherwise make perturbation theory poorly behaved. Furthermore the systematic power counting in effective theories, and the approximate symmetries of the effective field theory can greatly reduce the complexity of calculations. Consider as an example the semi-leptonic decay of a B-meson to aD-meson. In perturbation theory the one-loop corrections will typically be enhanced by log(M/ΛQCD), where M is a heavy quark mass. These logarithms are large enough so that αS log(M/ΛQCD) ∼ 1, and the perturbative expansion breaks down. Furthermore the nonperturbative physics in the decay process is parameterized in terms of two unknown form factors. The power of effective field theories is demonstrated when we consider our example within the context of heavy quark effective theory (HQET) [7]. In HQET heavy particles have been removed from QCD so that logarithms in loop integrals are of the form log(μ/ΛQCD). Furthermore the complete series of leading logarithms α s log (μ/ΛQCD) is straightforward to sum via the renormalization group. The HQET Lagrangian has a spin-flavor symmetry which reduces the number of form-factors to a single one: the Isgur-Wise function [8]. In addition HQET tells us the normalization of the Isgur-Wise function at a kinematic point. Remarkably heavy quark spin symmetry leads to additional relations among weak decay form factors. The four form factors which are required to parameterize matrix elements of vector currents and the four axial vector current form factors reduce to the single Isgur-Wise function in the heavy quark limit. This example clearly illustrates the power of effective field theories. To motivate soft collinear effective theory (SCET) I will consider another example: the decay rate for B → Xsγ. The dominant contribution to this decay arises from the magnetic penguin operator [9]