Pion-pion scattering at low energy

TL;DR

This work provides a complete two-loop evaluation of elastic ππ scattering in CHPT, including a thorough treatment of renormalization, scheme dependence, and EOM ambiguities. By first illustrating the loop expansion in an $N$-component φ^4 theory and then applying the methods to CHPT, the authors extract the pion mass, decay constant, and scattering amplitude up to $O(p^6)$, with explicit dependence on both $O(p^4)$ and $O(p^6)$ low-energy constants. A resonance-saturation analysis estimates the $O(p^6)$ LECs (dominantly from vector mesons) and yields threshold parameters and phase shifts that are compared with experimental data, highlighting robustness in some channels and sensitivity to $O(p^4)$ inputs in others. The results offer a framework for precision tests of QCD at low energies and stress the importance of Roy-equation based analyses to refine the $O(p^6)$ sector and to robustly extract threshold parameters from data.

Abstract

We present technical details of the evaluation of the elastic pi-pi scattering amplitude to two loops in chiral perturbation theory. In particular, we elaborate on the renormalization procedure at the two-loop order and on the evaluation of the relevant Feynman diagrams that can all be expressed in terms of elementary functions. For the sake of clarity, we discuss these matters both in the N-component $φ^4$ theory (in its symmetric phase) and in chiral perturbation theory. Estimates for the relevant low-energy constants of $O(p^6)$ are presented. Threshold parameters and phase shifts are then calculated for two sets of $O(p^4)$ coupling constants and compared with experiment. We comment on the extraction of threshold parameters from phase shift data.

Figures (9)

• Figure 1: The two--loop graphs in $\phi^4$ theory. Graph a corresponds to the contribution (\ref{['eq_za']}), and analogously for the diagrams b--e. The solid--dashed lines stand for the propagator (\ref{['eqpropg']}). The black circles denote a vertex generated by the lagrangian $\left.{\cal L}\right|_{C_i=0}$ in (\ref{['eqlagrangian']}), and the filled squares stand for the contributions from the counterterm $C_1$. Finally, the contributions from the counterterm $C_2$ are indicated with an open square in graph e.
• Figure 2: Diagrams that contribute to the two--point function at two--loop order. A filled square (open square) stands for the contributions from the counterterm $C_1 (C_2).$
• Figure 3: Summing up the graphs d$_1\cdots$ d$_9$ in Fig. \ref{['f2']}. See text after Eq. (\ref{['eqmassg2']}).
• Figure 4: The elastic $\phi\phi\rightarrow\phi\phi$ scattering amplitude to two loops. The numbers on the external lines denote the momenta. Crossed diagrams and insertions on external lines are not displayed. A filled box (open box) denotes contributions from the counterterm $C_1 (C_2)$.
• Figure 5: Summing up the diagrams c$_1\cdots$ c$_5$ in Fig. \ref{['f4']}. See text after Eq. (\ref{['eqscatta']}).