Matrix elements of unstable states

V Bernard,U.-G Meißner,A Rusetsky,D Hoja

doi:10.1007/jhep09(2012)023

Abstract

Abstract Using the language of non-relativistic effective Lagrangians, we formulate a systematic framework for the calculation of resonance matrix elements in lattice QCD. The generalization of the Lüscher-Lellouch formula for these matrix elements is derived. We further discuss in detail the procedure of the analytic continuation of the resonance matrix elements into the complex energy plane and investigate the infinite-volume limit.

Highlights

In this paper, we address these questions in detail
Note that the matrix element displayed in eq (1.4) should be understood as a mere notation: in the spectrum, there exists no isolated resonance state with a definite momentum. As it is clear from eq (1.4), this definition of the resonance matrix elements necessarily implies an analytic continuation into the complex plane
7 Conclusions i) In this paper, by using the technique of the non-relativistic effective Lagrangians in a finite volume, we were able to formulate a procedure for extracting the resonance matrix elements on the lattice

Summary

Extraction of the resonance poles in moving frames

The initial and final states in a form factor have non-zero momenta. For this reason, one has to formulate a procedure for extracting resonance pole positions in moving frames. The terms that vanish as P0 → Q0 can be omitted on The justification for this is the fact that the parameters in the potential are determined by matching to the physical S-matrix elements (on shell), order by order in the low-energy expansion. Where the derivative is taken with respect to the variable p∗ Substituting this expression into eq (3.6), performing the integral over P0 and taking into account the fact that the “free” poles at P0 = i(w1(k) + w2(k)) cancel in the integrand, the final expression for the matrix element in eq (3.6) for x0 − y0 > 0 reads: 0|O(x0; P, k)O†(y0; P, k)|0. This expression allows one to extract the absolute value of a scalar form factor in the timelike region from the measured matrix element En(P)|j(0)|0 in a finite volume. It can be shown that, by using our method, one exactly reproduces the Luscher-Lellouch formula in moving frames [27, 44, 45]

Extraction of resonance matrix elements in a finite volume

Analytic continuation and fixed points

Matrix elements at nonzero momentum transfer