On the N-pion extension of the Lovelace-Shapiro model

TL;DR

This work extends the Lovelace-Shapiro four-pion amplitude to an N-pion framework by shifting the NS model tachyon mass to zero, preserving super-projective invariance and producing massless pions that reproduce Adler zeroes. The authors show that 4- and 6-pion flavour-ordered amplitudes are unitary by embedding NS tachyons in D=10 with specific internal KK momenta, and they analyze higher-point ghosts that arise beyond six pions. The N-pion amplitude factorises consistently into lower-point pion amplitudes at pion poles, and in the field-theory limit $\alpha'\to 0$ (with $F_\pi$ fixed) it reduces to the non-linear $\sigma$-model; the six-pion case explicitly matches the NL$\sigma$-model structure. The work also provides a detailed analysis of resonances (rho, sigma) and related amplitudes, exposing both the potential for phenomenological relevance and the intrinsic limitations due to ghosts at higher $N$, while offering avenues toward holographic and bootstrap connections.

Abstract

We reconsider a modification of the N-point amplitude of the Neveu-Schwarz (NS) model in which the tachyon becomes a pion by shifting its mass to zero and keeping the super-projective invariance of the integrand of the amplitude. For the scattering of four particles it reduces to the amplitude written by Lovelace and Shapiro that has Adler zeroes. We confirm that also the N-pion amplitude has Adler zeroes and show that it reduces to that of the non-linear $σ$-model for $α' \rightarrow 0$ keeping $F_π$ fixed. The four- and six-point flavour-ordered amplitudes satisfy tree-level unitarity since they can be derived from the correspondent amplitudes of the NS model in ten dimensions by suitably choosing the components of the momenta of the external mesons in the six extra dimensions. Negative norm states (ghosts) are shown to appear instead in higher-point amplitudes. We also discuss several amplitudes involving different external mesons.

Figures (1)

• Figure 1: Spectrum of NS (left) and LS (right) model in four dimensions, Regge trajectories in blue (red) have G-parity $+1$ ($-1$). Bullets represent 'physical' states, open circles represent 'missing' states.