On the Asymptotic Behavior, for Large Values of the Time, of Solutions of Exterior Boundary Value Problems for the Wave Equation With Two Space Variables

Abstract

We study the impact of explicit chiral symmetry breaking of Wilson fermions on mesonic correlators in the ǫ-regime using Wilson chiral perturbation theory (WChPT). We generalize the ǫ-expansion of continuum ChPT to nonzero lattice spacings for various quark mass regimes. It turns out that the corrections due to a nonzero lattice spacing are highly suppressed for typical quark masses of the order aΛ 2 QCD . The lattice spacing effects become more pronounced for smaller quark masses and lead to non-trivial corrections of the continuum ChPT results at next-to-leading order. We compute these corrections for the standard current and density correlation functions. A fit to lattice data shows that these corrections are small, as expected.ArXiv ePrint: 0812.2403JHEP03 (2009)006 order. This suppression is completely analogous to the suppression of terms involving the Gasser-Leutwyler coefficients in continuum ChPT in the ǫ-regime.However, the power counting in WChPT can be different, depending on the relative size of the quark mass m and the lattice spacing a. For this reason we also consider a different power counting where m ∼ aΛ 2 QCD is no longer appropriate. In this case the lattice spacing corrections enter already at NLO. Interestingly, these corrections are entirely caused by the O(a 2 ) term in the chiral effective Lagrangian that also determines the phase diagram of the lattice theory [32]. The corrections linear in the lattice spacing, stemming from the effective Lagrangian and the effective operators, are still of higher order in the epsilon expansion. This is relevant in practice: The Wilson ChPT expressions contain only one more unknown LEC at this order, and the predictive power is not spoiled by a plethora of free fit parameters.We also use our WChPT results derived here for an analysis of the data of ref.[22]. The corrections due to the nonzero lattice spacing turn out to be very small, supporting our theoretical analysis that these corrections are in general highly suppressed. This is very encouraging for lattice simulations with Wilson fermions. The impact of explicit chiral symmetry breaking for ǫ-regime simulations is much less severe than previously thought. This makes simulations with Wilson fermions a serious and efficient alternative to those with chiral fermions.

Low Frequency Expansions for Two-Dimensional Interface Scattering Problems

“…See, for instance, [3], [4], [6], [7], [8] and [12]. The application to the study of long time asymptotics for exterior problems is treated in [1], [10] and [11]. The latter results, in contrast to ours, are rigorously proved.…”

Full low‐frequency asymptotic expansion for second‐order elliptic equations in two dimensions

“…It is well known that if u is a bounded solution of Laplace's equation in a neighbourhood of 00 then for r sufficiently large, u has the form m u(x) = Co + C (a,cosncp + b,sinnq)r-" n = O (7) In particular, this representation is valid for bounded solutions of problem (5) and of problem (1) with k = 0 iff= 0 in a neighbourhood of infinity. Therefore, condition (6) is equivalent to the requirement that Co # 0 for such solutions.…”

“…The conditions embodied in Cases I and I1 are less restrictive than those used in [16]. If condition (3) required in [16] is fulfilled then together with (7) it follows that there are only constant solutions of problem (5). This means that either case I or Case I1 apply.…”

Exterior dirichlet problem for the reduced wave equation: asymptotic analysis of low frequencies

“…In the integral (14) one can replace the kernels D i ( 9 , -+,)(z, y ) by R$; because of (21). Then by (25), ( 7 ) and (16) will hold in R ( R ) .…”