Signs of analyticity in fermion scattering

TL;DR

This work uses S-matrix analyticity and fixed-s dispersion relations to bound leading 4-fermion operators in effective field theories of fermions. It shows that, under Regge behavior, the sign of these operators is constrained and that the presence of higher-spin bound states, such as the deuteron in NN scattering, necessitates subtractions that can invalidate naive positivity bounds. The analysis connects non-perturbative bound-state physics to observable quantities like scattering lengths and Sommerfeld enhancement, and it highlights cases (e.g., Weyl fermions with no Regge behavior) where these constraints can be evaded by infrared modifications. The results illuminate the limitations of perturbative approaches to quark-mass dependence in nuclear forces and suggest potential applications to electroweak current-current operators in chiral Lagrangians.

Abstract

We show that the signs of the leading irrelevant interactions for Dirac fermions are constrained by the analytic structure of the S-matrix. If Regge behavior obtains, negative signs indicate the presence of higher-spin bound states that spoil the convergence of the dispersion integrals and drive the corresponding operators relevant. For nucleon-nucleon scattering, the negativity of some of the low-energy interactions signals the presence of a spin-1 bound state: the deuteron. We connect the divergence of the dispersion integral to the "Sommerfeld enhancement" of the cross-section for low-energy scattering. We also discuss how this illuminates potential pitfalls in using perturbative methods to understand the dependence of the low-energy nuclear interaction on the masses of the light quarks. Finally, we suggest the possibility of applying similar reasoning to the current-current operators in the electroweak effective lagrangian, where no bound states spoil convergence of the dispersion relations.

Figures (5)

• Figure 1: Diagrams representing the $s$ channel and $t$ channel scattering processes respectively, related by crossing of the particles labeled $1'$ and $2$. (In this paper, Feynman diagrams are always drawn with time flowing from left to right.)
• Figure 2: Elastic unitarity can be visualized as the cutting across two internal fermion lines, yielding Eq. (\ref{['elastic']}).
• Figure 3: Non-perturbative contribution to the spectral function in Eq. (\ref{['disp']}) that persists as $t \to \infty$ for small fixed $s$. The solid lines represent nucleons (protons and neutrons) and the dashed lines represent pions.
• Figure 4: Cartoon of the behavior of the scattering length $a$ as a function of $x \equiv g m / \mu$ for particles of mass $m$ interacting via the Yukawa potential given in Eq. (\ref{['yukawaV']}).
• Figure 5: Plots of $u(r) \equiv r \psi (r)$ (in gray) for solutions to the Schrödinger equation in three dimensions with $E = l =0$, for a square well potential (plotted in black) with depth $V_0$ and range $L$. Only $V_0$ varies between the plots shown in (a), (b), and (c). The value of the $r$-intercept is equal to the scattering length $a$.