Dispersion Relation Bounds for pi pi Scattering

TL;DR

This work derives positivity bounds on the SU(2) chiral Lagrangian low-energy constants $\bar{l}_{1}$ and $\bar{l}_{2}$ by converting fixed-$t$ dispersion relations for $\pi\pi$ scattering into positivity conditions on the second s-derivative of the amplitude within a Mandelstam-domain region $\mathcal{A}$ that overlaps with $\chi$PT. By evaluating the $\mathcal{O}(p^{4})$ $\chi$PT amplitude at the most stringent kinematic points, the authors obtain concrete inequalities such as $\bar{l}_{1}+2\bar{l}_{2} \ge 3.925\pm0.4$, $\bar{l}_{2} \ge 1.350\pm0.4$, and $\bar{l}_{1}+3\bar{l}_{2} \ge 5.604\pm0.4$, which are stronger than previous bounds, and find that experimental fits lie within these limits. They compare with prior analyses, clarifying domain-related discrepancies, and address apparent contradictions with the linear sigma model by showing the full one-loop MS-bar LSM amplitudes satisfy the bounds for $m_{\sigma}\ge m$, while truncations can falsely suggest violations. The results reinforce how fundamental principles constrain low-energy QCD and motivate extending the positivity program to $SU(3)$ and higher-order effective theories.

Abstract

Axiomatic principles such as analyticity, unitarity and crossing symmetry constrain the second derivative of the pi pi scattering amplitudes in some channels to be positive in a region of the Mandelstam plane. Since this region lies in the domain of validity of chiral perturbation theory, we can use these positivity conditions to bound linear combinations of \bar{l}_1 and \bar{l}_2. We compare our predictions with those derived previously in the literature using similar methods. We compute the one-loop pi pi scattering amplitude in the linear sigma model (LSM) using the MS-bar scheme, a result hitherto absent in the literature. The LSM values for \bar{l}_1 and \bar{l}_2 violate the bounds for small values of m_sigma/m_pi. We show how this can occur, while still being consistent with the axiomatic principles.

Figures (8)

• Figure 1: Mandelstam plane for $\pi\,\pi$ scattering. 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 outer regions (yellow) denote the physical regions for the three crossed channels. The region bounded by the thick black line corresponds to the area $\mathcal{A}$ in which the positivity conditions are satisfied.
• Figure 2: Contour integrals leading to the fixed-$t$ dispersion relations.
• Figure 3: The $\bar{l}_{1}-\bar{l}_{2}$ region allowed by the positivity conditions is shown. The three lines correspond to the three bounds in Table \ref{['tab:bounds']}. We also show the experimentally fitted values of Ref. Colangelo with their error.
• Figure 4: Plot of $16\,\pi^{2} F_\pi^{4}\,\mathrm{d}^{2}T(s,4\,m^{2})/\mathrm{d}s^{2}\bigr|_{s=0}$ in the linear sigma model for the $\pi^{+}\pi^{0}\to\pi^{+}\pi^{0}$ process as a function of $m_{\sigma}/m$. The exact amplitude (blue, continuous line) is positive for $m_{\sigma}>m$. The amplitude up to and including $1/m_\sigma^4$ terms (red, dashed line), is positive for $m_{\sigma}>5\,m$. The $\mathcal{O}(m_\sigma^{-2})$ amplitude (green, dot-dashed line) remains negative for $m_\sigma<4.9\,m$.
• Figure 5: One-point $\sigma$ function. Double and dashed lines denote $\sigma$ particles and pions, respectively. The sum of all these tadpole graphs must vanish to ensure that perturbation theory is done around the minimum of the potential, including quantum corrections.