Positivity of the Veneziano amplitude in D = 4

Abstract

The Veneziano amplitude was put forward as a solution to the axioms of the S-matrix 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.

Properties of infinite product amplitudes: Veneziano, Virasoro, and Coon

“…In any case, the Coon amplitude is a fruitful example for the study of general scattering amplitudes [5][6][7][8][9]. While there is yet no definitive field theory or string worldsheet realization of the Coon amplitude, accumulation point spectra like those exhibited by the Coon amplitude with q < 1 have been recently found in a stringy setup involving open strings ending on a D-brane [10].…”

“…The Veneziano amplitude was discovered in 1968 [1] and describes the scattering of four open strings. More recently, the Veneziano amplitude has been revisited in the context of the modern S-matrix bootstrap program [6][7][8][17][18][19].…”

“…The coefficient c 3,0 is negative for d > 10, indicating the non-unitarity of the superstring above its critical dimension d = 10. These coefficients were recently studied in d = 4 [8] and were recently shown to be positive for all d ≤ 6 [19]. The unitarity of superstring theory in d ≤ 10 (and thus the positivity of the c N,j ) is known from the no-ghost theorem [20][21][22], but there is yet no direct proof that c N,j > 0 for all d ≤ 10.…”

Properties of infinite product amplitudes: Veneziano, Virasoro, and Coon

“…Around this time, the Coon amplitude was revisited in the broader context of string theory [14,40]. Most recently, the Coon amplitude has reappeared in the modern S-matrix bootstrap program [5][6][7][8][9][10][11].…”

“…In any case, the Coon amplitude is a fruitful example for the study of general scattering amplitudes [5][6][7][8][9]. While there is yet no definitive field theory or string worldsheet realization of the Coon amplitude, accumulation point spectra like those exhibited by the Coon amplitude with q < 1 have been recently found in a stringy setup involving open strings ending on a D-brane [10].…”