Positivity of the Veneziano Amplitude in $D=4$

Abstract

The Veneziano amplitude was put forward as a solution to the axioms of the Smatrix bootstrap. However, unitarity, reflected in the positivity of the coefficients in the Gegenbauer expansion of the amplitude is not obvious. In this note we compute the generating function of these coefficients in terms of the Appell hypergeometric function. We use this to read off an exact form of this coefficient on the leading Regge trajectory in D = 4. We find that it decays with the spin but always remains positive. Since for large spin these coefficients are expected to be smaller than those on the subleading trajectories, our result indicates the positivity of the full Veneziano amplitude in D = 4.

On the positivity of Coon amplitude in D=4

“…which precisely matches (accounting for the shifts in 2.3.1) with the result for Veneziano amplitude in [17,18].…”

“…, k. Note that the poles of our amplitude s = m 2 + [k] q start from k = 0, as opposed to [17] where the poles (s = k) start from k = −1. Thus we need to set m 2 = −1 and shift k → (k + 1) in the expressions of a q k,j to compare our results to [17]. In our convention, the leading Regge trajectory is associated with coefficients of the form a q k,k .…”

“…In order to read off the partial wave coefficients, we will use a similar argument to the one used while extracting the leading Regge coefficients for the Veneziano amplitude [17]. Firstly, we note that the h-expansion of G k (h) can be represented in the following form using equations (2.14), (2.21) and the expansion of g k,r (h) in h:…”

On the unitarity and low energy expansion of the Coon amplitude

“…In the case of string theory, the no-ghost theorem [26,27] guarantees that such states decouple from all scattering amplitudes in for d ≤ 26 or 10. At the level of the four-point function (the Veneziano amplitude), a recent paper showed that residues should all be positively expandable on Gegenbauer polynomials [28] in d ≤ 6, (see also [29] for the states on the leading Regge trajectory in d = 4). It would still be desirable to be able to bridge the gap to d = 10 or 26 and maybe the q → 1 limit of the Coon amplitude could open an avenue to be combined with the techniques of these papers.…”