Crossing Symmetric Dispersion Relations in QFTs

TL;DR

The paper develops a crossing-symmetric dispersion framework for 2-2 scattering by mapping the Mandelstam variables to a cubic z-variable, achieving manifest three-channel crossing symmetry. It derives a general, nonperturbative set of positivity constraints and null constraints on Wilson coefficients, expressed via Gegenbauer partial waves, and presents a Feynman-block (Dyson-block) representation that illustrates how exchange and contact contributions emerge from the crossing-symmetric data. The approach yields practical results, including a method to locate the first massive string pole from a low-energy dilaton amplitude in type II strings and a numerical Froissart-type bound valid at all energies, with convergence demonstrated through string-theory amplitudes and spin truncations. The framework connects to EFT-hedron ideas and higher-dimensional S-matrix bootstrap, offering a unified, crossing-symmetric avenue to constrain EFTs and relate low-energy data to high-energy spectrum.

Abstract

For 2-2 scattering in quantum field theories, the usual fixed $t$ dispersion relation exhibits only two-channel symmetry. This paper considers a crossing symmetric dispersion relation, reviving certain old ideas in the 1970s. Rather than the fixed $t$ dispersion relation, this needs a dispersion relation in a different variable $z$, which is related to the Mandelstam invariants $s,t,u$ via a parametric cubic relation making the crossing symmetry in the complex $z$ plane a geometric rotation. The resulting dispersion is manifestly three-channel crossing symmetric. We give simple derivations of certain known positivity conditions for effective field theories, including the null constraints, which lead to two sided bounds and derive a general set of new non-perturbative inequalities. We show how these inequalities enable us to locate the first massive string state from a low energy expansion of the four dilaton amplitude in type II string theory. We also show how a generalized (numerical) Froissart bound, valid for all energies, is obtained from this approach.

Figures (4)

• Figure 1: $\delta_0^{(\max)}$ vs $2(2n+m)$, the derivative order, for $m=1,2,3,4,5$.
• Figure 2: The bound on $\bar{\sigma}(s)$ using constraints \ref{['sum01']} with $L_{max}=36, 46,60$. The black dashed line is the Froissart bound on $\bar{\sigma}$. The red line is maximum $\bar{\sigma}$ obtained using the pion bootstrap in ABPHAS.
• Figure 3: All three cases of $V(a)$. In the above plot we have assumed $\mu=4$.
• Figure 4: Comparison between crossing symmetric dispersion relation and fixed-t dispersion relation. We have truncated the $k-$ sum upto $k_{\max}=100$