Possible lattice approach to B→Dπ(K) matrix elements

We present an approach for computing the real parts of the nonleptonic . B→. DP and . B→D→P (. P=. K, π) decay amplitudes by using lattice QCD methods. While it remains very challenging to calculate the imaginary parts of these matrix elements on the lattice, we stress that their real parts play a significant role in extracting the angle . γ in the . b-. d unitarity triangle of the CKM matrix. The real part on its own gives a lower bound to the absolute magnitude of the amplitude which is in itself an important constraint for determining . γ. Also the relevant phase can be obtained by using . B decays in conjunction with relevant charm decay data. Direct four-point function calculations on the lattice, while computationally demanding, do yield the real part as that is not impeded by the Maiani-Testa theorem. As an approximation, we argue that the chiral expansion of these decays is valid in a framework similar to that of hard-pion chiral perturbation theory. In addition to constructing the leading-order operators, we also discuss the features of the next-to-leading order chiral expansion. These include the contributions from the resonance states, as well as the generic forms of the chiral logarithms.