Crossing Symmetric Dispersion Relations in Quantum Field Theories

For 2-2 scattering in quantum field theories, the usual fixed $t$ dispersion relation exhibits only two-channel symmetry. This Letter considers a crossing symmetric dispersion relation, reviving certain old ideas from 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 nonperturbative 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.

Figure 1

${\delta }_0^{(\max )}$ vs $2(2n+m)$, the derivative order, for $m=1$, 2, 3, 4, 5.

Figure 2

The bound on $\overline{\sigma }(s)$ using constraints (13) with $L_{\max }=36$, 46, 60. The black dashed line is the Froissart bound on $\overline{\sigma }$. The red line is maximum $\overline{\sigma }$ obtained using the pion bootstrap in Ref. [35].