QCD Inequalities, Large $N_C$ and $\pi\pi$ Scattering Lengths

TL;DR

This paper investigates how the sign of pion-pion scattering lengths varies in different QCD-like theories and in the large N_C limit, revealing positive attraction in four-flavor theories and repulsion in two-flavor QCD.

Contribution

It demonstrates the sign of the pion-pion scattering length in QCD-like theories with four flavors and in large N_C QCD with two flavors, providing new insights into hadronic interactions.

Findings

Positive scattering length in four-flavor QCD-like theories.

Repulsive scattering length in two-flavor QCD at large N_C.

Clarification of the relationship between flavor content and scattering behavior.

Abstract

In this short note we show that (I) in a QCD-like theory with four (rather than two) degenerate flavors $ud u'd'$, the $\pi\pi'$ scattering length is positive (attractive); and (II) in QCD with only two (u,d) degenerate flavors the I=2 (say, $\pi^+\pi^+$ hadronic) scattering length is, in the large $N_C$ limit, repulsive. $\pi(\pi')$ are the lowest physical states coupling to $J^p = \bar{u}(x)\gamma_5d(x)$ and $J^{p'} = \bar{u}'(x)\gamma_5d'(x),$ respectively.