A stringy dispersion relation for field theory

TL;DR

This work develops a local, crossing-symmetric dispersion framework with a one-parameter stringy ambiguity λ that unifies fixed-$t$ and fixed-$s$ representations for 2-2 scattering. It derives explicit two-channel and three-channel parametric CSDRs, including higher-subtraction schemes, and demonstrates their utility through Veneziano and Virasoro–Shapiro amplitudes, yielding series that converge with poles in all channels. The formalism is extended to dispersive partial-wave expansions and applied to bound Wilson coefficients in weakly-coupled gravitational EFTs, resolving forward-limit issues via λ as an IR regulator. The authors outline a path toward n-particle dispersion relations, proposing symmetric kernels for multi-variable amplitudes and detailing the steps required to generalize the approach. Collectively, the paper provides a versatile, convergent dispersive toolkit for string-inspired S-matrix analysis and EFT constraints, with implications for bootstrap programs and beyond.”

Abstract

We derive a local, crossing symmetric dispersion relation (CSDR) for 2-2 scattering amplitudes with a parametric ambiguity motivated by string theory. Various limits of the parameter lead to the fixed-t, fixed-s, and other known CSDRs. We also present formulae for higher-subtracted cases. Several examples are discussed for illustration. In particular, for the Veneziano and the Virasoro-Shapiro amplitudes, we derive parametric series representations which manifest poles in all channels and converge everywhere. We then discuss applications of our formalism for bootstrapping weakly-coupled gravitational EFTs. We demonstrate that even in the presence of the graviton pole, one can derive bounds on the Wilson coefficients while working in the forward limit, with the parameter acting as the IR regulator instead. Finally, we derive series representations for multi-variable, totally symmetric generalisations of the Veneziano and Virasoro-Shapiro amplitudes that manifest poles in all the variables. This is a first step towards dispersion relations for n-particle scattering amplitudes.

Figures (10)

• Figure 1: The open string worldsheet picture on the left, Feynman diagrams in the middle and the product of disc representations of the channels (black crosses depict vertex operators).
• Figure 2: A typical depiction of the singularities in the integrand in the complex-$\sigma$ plane. Here $s=1+i, \: t=2+i, \: \lambda=1$ and there is an s-channel cut for $s>2$ and its t-channel counterpart. In the integrand, the t-channel cut counterpart gets mapped to the line joining $(1/3,7/3)$ and $(-1,0)$.
• Figure 3: Thick lines denote massive states in the loop and massless legs are marked with thin lines
• Figure 4: Ratio of the series to actual answer vs $\lambda$ in the fixed angle case. The black dashed line is 1, the blue dashed line is the fixed-$t$ series. The red line is the stringy representation.
• Figure 5: Ratio of the dispersive representation of the dilaton scattering amplitude and the exact answer for (a) $s_2 < -1$, (b) $s_2 > -1$ and various $\lambda$. We observe that the fixed-$t$ representation $(\lambda = -s_2)$ doesn't converge for case (b), while the parametric CSDR does, for $\lambda > 1$. We also see that the parametric CSDR is independent of $\lambda$, but dialing $\lambda$ improves the convergence rate.