On the two-loop contributions to the pion mass

Abstract

We derive a simplified representation for the pion mass to two loops in three-flavour chiral perturbation theory. For this purpose, we first determine the reduced expressions for the tensorial two-loop 2-point sunset integrals arising in chiral perturbation theory calculations. Making use of those relations, we obtain the expression for the pion mass in terms of the minimal set of master integrals. On the basis of known results for these, we arrive at an explicit analytic representation, up to the contribution from K-K-eta intermediate states where a closed-form expression for the corresponding sunset integral is missing. However, the expansion of this function for a small pion mass leads to a simple representation which yields a very accurate approximation of this contribution. Finally, we also give a discussion of the numerical implications of our results.

Typos corrected and minor changes in Table 2. Published version. 19 pages, 1 figure, 2 tables

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• Results from the MILC collaboration's SU(3) chiral perturbation theory analysis
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• An Analytic Analysis of the Pion Decay Constant in Three-Flavoured Chiral Perturbation Theory
• Analytic representations of $m_K$, $F_K$, $m_{\eta }$ and $F_{\eta }$ in two loop $SU(3)$ chiral perturbation theory
• Quark Mass dependence at Two Loops for Meson Properties
• Strangeness content of the pion in the U(3) Nambu-Jona-Lasinio model
• Theoretical aspects of Chiral Dynamics
• An Analytic Approach to Sunset Diagrams in Chiral Perturbation Theory: Theory and Practice
• Status of Strong ChPT
• On the low energy end of the QCD spectrum
• Chiral perturbation theory in the meson sector