Abstract
Using Chiral Perturbation Theory at one-loop we analyze the consequences of twisted boundary conditions. We point out that due to the broken Lorentz and reflection symmetry a number of unexpected terms show up in the expressions. We explicitly discuss the pseudo-scalar octet masses, axial-vector and pseudo-scalar decay constants and electromagnetic form-factors. We show how the Ward identities are satisfied using the momentum dependent masses and the non-zero vacuum-expectation-values values for the electromagnetic (vector) currents. Explicit expressions at one-loop are provided and an appendix discusses the needed one-loop twisted finite volume integrals.
Highlights
We study in detail the finite volume corrections from the isospin current matrix element π0(p )|dγμu|π+(p) which is used in lattice QCD to obtain information on the pion radius
We have worked out the expressions in one-loop Chiral Perturbation Theory (ChPT) for masses, axial-vector and pseudo-scalar decay constants as well as the vacuum expectation value and the two-point function for the electromagnetic current
In particular we showed how one needs more form-factors than in the infinite volume limit and obtained expressions for those at one-loop order
Summary
ChPT is the effective field theory describing low energy QCD as an expansion in masses and momenta [7,8,9]. We use the external field method [8, 9] to incorporate electromagnetism, quark masses as well as couplings to other quark-antiquark operators. To do this we introduce the field χ and the covariant derivative χ = 2B0(s + ip), DμU = ∂μU − irμU + iU lμ. We assume the temporal direction to be infinite in extent and use the LSZ theorem to obtain the needed wave function renormalization by keeping the spatial momentum constant and taking the limit in (p0) to p2 = m2.
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