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′ → 0 keeping Fπ 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.
Highlights
C4-pt is a constant fixed by the low-energy α → 0 limit of the amplitude
In the first part of this section we show that the Adler zeroes that we found in the four and six-pion amplitudes are present in all N -pion amplitudes, while, in the second part of this section we show that the N -pion amplitude factorizes into pion amplitudes, confirming earlier studies [14, 15] within our approach
We have reconsidered N -point pion amplitudes that were known to have Adler zeroes and we have shown that they reproduce the N -point pion amplitudes of the non-linear σ-model in the field theory limit
Summary
We revisit the Lovelace-Shapiro (LS) model [2, 3]. To this end we focus on the following ‘flavour’-ordered amplitude. One can require the presence of Adler zeroes, i.e. that the amplitude vanishes when one of the momenta goes to zero This condition can be obtained forcing the Γ function in. Constraints on α0 and the space-time dimensions D can be obtained imposing that the residues at the poles be positive definite (tree-level unitarity). Using the relation: α α0, the residue can be expanded in terms of Gegenbauer polynomials.. We consider the levels n = 0 and n = 1 obtaining the following residues. Where x = cos θ, Gn(x, D) are Gegenbauer polynomials in D space-time dimensions, f abc are the structure constants of U(Nf ), dabc is the symmetric invariant tensor of U(Nf ) and the flavour indices a, b are not summed over, while there is a sum over the flavour index c.8.
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