Unitarity and Low Energy Expansion of the Coon Amplitude

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 Letter, 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, which connects to the known $d=26$, 10 when the deformation is removed. Incidentally, 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 couplings of the theory in terms of $q$-deformed analogs 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. Finally, we comment on the length of the Coon parton.

Figure 1

Green surface: Numerical boundary of the unitary region. Blue shaded region: Unitary region determined by $q\le q_c(m^2)$.

Figure 2

Dots: Sections of the surface mapped in Fig. 1. Solid lines: Large $d$ envelope computed analytically (see Supplemental Material [4]).

Figure 3

Plot of Wilson coefficients with cutoff $M^2=1$.

Figure 4

Plots of $a_4$ and $a_6$. Gray dashed: LSD. Yellow: spin 0,1 model. Blue dots: actual points. Veneziano is in the middle of the graph ($q=1$) and the scalar theory is at the intersection with LSD.