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Universal Bounds for SU(3) Low Energy Constants

V. Mateu

Abstract

In this paper bounds for L_1, L_2 and L_3 are obtained in Chiral Perturbation Theory with three flavours. At the same time we test the compatibility of this theory with axiomatic principles. Following a recent paper we use dispersion relations to write positivity conditions that translate into bounds for the chiral low energy constants. As a first approach we consider the exact SU(3)_V limit and notice that if a common mass of the order of that of the kaon is adopted for the octet of pseudo-Goldstone bosons the bounds have very large O(p^6) corrections. Once the positivity conditions are adapted to account for different masses, we correct the previous bounds for a physical kaon mass and find that they tighten. We observe an overlap between the experimentally determined region and the first principles forbidden region, in the space of parameters.

Universal Bounds for SU(3) Low Energy Constants

Abstract

In this paper bounds for L_1, L_2 and L_3 are obtained in Chiral Perturbation Theory with three flavours. At the same time we test the compatibility of this theory with axiomatic principles. Following a recent paper we use dispersion relations to write positivity conditions that translate into bounds for the chiral low energy constants. As a first approach we consider the exact SU(3)_V limit and notice that if a common mass of the order of that of the kaon is adopted for the octet of pseudo-Goldstone bosons the bounds have very large O(p^6) corrections. Once the positivity conditions are adapted to account for different masses, we correct the previous bounds for a physical kaon mass and find that they tighten. We observe an overlap between the experimentally determined region and the first principles forbidden region, in the space of parameters.

Paper Structure

This paper contains 6 sections, 13 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Positivity conditions in the $\mathbf{SU(3)}$ limit
  3. Bounds on $\mathbf{L_{1},\; L_{2}}$ and $\mathbf{L_{3}}$.
  4. Symmetry corrections to the bounds
  5. Results
  6. Conclusions

Figures (1)

  • Figure 1: Mandelstam plane for the $a+b\to a+b$ process, with $m_a=m$ and $m_b=M$ (the plot corresponds to $m=m_\pi$ and $M=m_K$). The small (blue) triangle in the center is the Mandelstam triangle. The big triangle (red and blue area) is the region free from singularities. The region bounded by the thick black line corresponds to the area $\mathcal{A}$ in which the positivity conditions are satisfied.