Weak Transition Matrix Elements¶from Finite-Volume Correlation Functions

Abstract

The two-body decay rate of a weakly decaying particle (such as the kaon) is shown to be proportional to the square of a well-defined transition matrix element in finite volume. Contrary to the physical amplitude, the latter can be extracted from finite-volume correlation functions in euclidean space without analytic continuation. The K → ππ transitions and other non-leptonic decays thus become accessible to established numerical techniques in lattice QCD.

Computing the hadronic vacuum polarization function by analytic continuation

“…Besides this, both the energy and the amplitude extracted from C µν ( k, t) are affected by the finite lattice volume. Such effects should be treated properly using the Lellouch−Lüscher method [20] and Meyer's proposal in Ref [21], which is, however, beyond the scope of this work.…”

Lattice 2001: Reflections

“…Interesting possibilities were discussed recently [2,3] to avoid the Maiani-Testa problem [4] due to three external particles in the K → ππ amplitude. Preliminary results using three-point functions have been presented in Martinelli's plenary talk [5,6].…”

Matrix elements of light quarks

“…[14,15]. For ∆I = 3/2 transitions, an extended study of the chiral behaviour of the matrix elements of the operatorsQ 4,7,8 , renormalized non-perturbatively, was performed.…”

“…ii) Compute directly ππ|Q i (µ)|K I=0,2 [14,15,32,33]. The main difficulty in the latter case is due to the relation between K → ππ matrix elements computed in a finite Euclidean space-time volume and the corresponding physical amplitudes (the infrared problem).…”

“…The main topics in this review are: i) the determination of the ∆I = 1/2 and ∆I = 3/2 K → ππ amplitudes [3,4,7,8]; ii) the main contributions to ǫ ′ /ǫ due to the electropenguin operators (Q 7 and Q 8 in the standard notation) and to the strong penguin operator Q 6 [3,4,7,8]; iii) matrix elements of SUSY operators which can be computed on the lattice [9]; iv) studies of the chiral expansion of the relevant amplitudes in finite and infinite volumes, in the quenched and unquenched cases [10]- [13]; v) studies of Final State Interactions (FSI) in lattice simulations [8]; vi) the extraction of the physical amplitudes from matrix elements computed in the Euclidean space-time on a finite volume [14,15]. The problem of the renormalization and improvement of the relevant operators will be mentioned and discussed along with the presentation of the different arguments.…”