On the unitarity and low energy expansion of the Coon amplitude

Abstract

The Coon amplitude is a deformation of the Veneziano amplitude with logarithmic Regge trajectories and an accumulation point in the spectrum, which interpolates between string theory and field theory. With string theory, it is the only other solution to duality constraints explicitly known and it constitutes an important data point in the modern S-matrix bootstrap. Yet, its basics properties are essentially unknown. In this paper we fill this gap and derive the conditions of positivity and the low energy expansion of the amplitude. On the positivity side, we discover that the amplitude switches from a regime where it is positive in all dimensions to a regime with critical dimensions, that connects to the known d = 26, 10 when the deformation is removed. En passant, we find that the Veneziano amplitude can be extended to massive scalars of masses up to m 2 = 1/3, where it has critical dimension 6.3. On the low-energy side, we compute the first few coupligs of the theory in terms of q-deformed analogues of the standard Riemann zeta values of the string expansion. We locate their location in the EFT-hedron, and find agreement with a recent conjecture that theories with accumulation points populate this space. We also discuss their relation to low spin dominance.

Accumulation-point amplitudes in string theory

“…The Coon amplitude [1] is a modification of the standard Veneziano amplitude for the scattering of open strings; see ref. [2] and references therein. It reads 1…”

“…We will consider the scattering of open strings attached to the D-brane, as depicted in figure 1. 2 We analyze this problem in the large-radius limit, where…”

“…In 1969, Coon [1] constructed a one-parameter deformation of the Veneziano amplitude that appears to correspond to a unitary S-matrix [2]. While the Veneziano amplitude famously describes a spectrum with linearly increasing mass-squares, the Coon amplitude exhibits an accumulation point, with an infinite number of states below a certain energy, leading to questions about its physical relevance [3].…”

UV-Complete Gravity Amplitudes and the Triple Product

“…[10] that a triple-product form with a single pole, A(s) = 1/(m 2 −s), plays a special role in spanning the space of consistent EFTs (see Ref. [11] for the role of Coon's amplitude [12,13] in such context). The existence of such accumulation-point amplitudes was also conjectured to be important in spanning the gravitational EFT [8].…”

“…The nongravitational theory thus has partial waves (11) that decay exponentially at large spin. In contrast, for the gravitational theory the F (α) I (α) (M 2 ) term enters Eq.…”