Origin of the gap to Jin-Martin maximal growth in the dispersive primal S-matrix bootstrap

The dispersive primal S-matrix bootstrap of de Rham-Tolley-Wang-Zhou constructs two-to-two amplitudes of identical massive scalars in four dimensions in which partial-wave unitarity, crossing symmetry, and Martin analyticity hold by construction, and in which the Jin-Martin theorem — growth at fixed $t$ in the window $-4m^2<t<4m^2$ strictly slower than $|s|^2$ — is enforced through the subtraction structure of fixed-$t$ dispersion relations. The extremal amplitudes on the boundary of the allowed low-energy-coupling region nevertheless reach a fitted Regge growth exponent of only about 1.7-1.8 near $t\to4m^2$ (around 1.5 or slightly faster under fractional subtractions), leaving a small gap to the $|s|^2$ limit. The open problem is to determine the origin of that gap: either the full constraint set of unitarity, crossing, and Martin analyticity caps the achievable growth exponent strictly below 2 — a genuine theoretical obstruction and an improvement beyond Jin-Martin — or the gap is an artifact of finite truncations and precision, in which case consistent amplitudes with growth arbitrarily close to $|s|^2$ exist and should be constructible. The question is open.

Background

The modern S-matrix bootstrap asks which two-to-two scattering amplitudes $\mathcal{M}(s,t)$ of a relativistic quantum field theory are consistent with the general principles of unitarity, analyticity, and crossing symmetry, and charts the allowed space numerically. The standard four-dimensional testbed is the scattering of identical scalar particles of mass $m$, described by the Mandelstam variables $s,t,u$ — the invariant squared energies of the three exchange channels — which obey $s+t+u=4m^2$. The amplitude is expanded in partial waves as $\mathcal{M}(s,t)=16\pi\sum_{\ell\,\mathrm{even}}(2\ell+1)a_\ell(s)P_\ell(1+2t/(s-4m^2))$, with $P_\ell$ the Legendre polynomials (only even $\ell$ appear, due to the $t\leftrightarrow u$ symmetry of identical-particle scattering), and $S$-matrix unitarity becomes the inequality $\mathrm{Im}\,a_\ell(s)\ge\tfrac{1}{2}\sqrt{(s-4m^2)/s}\,|a_\ell(s)|^2$ for $s\ge4m^2$ (equivalently $|1+i\sqrt{(s-4m^2)/s}\,a_\ell(s)|\le1$), saturated as an equality in a purely elastic scattering regime. Crossing symmetry states that the three channel amplitudes are one and the same analytic function, $\mathcal{M}(s,t)=\mathcal{M}(s,u)=\mathcal{M}(t,s)$. In the primal approach of Paulos, Penedones, Toledo, van Rees and Vieira, the whole amplitude is parameterized — after compactifying the cut complex planes into unit disks via so-called $\rho$-variables — and the constraints are imposed as a semidefinite program; in the dual approach, exclusion bounds are instead derived from the convex dual problem. A standard output is the allowed region of the leading crossing-symmetric low-energy couplings $g_0=\mathcal{M}(4m^2/3,4m^2/3)/(32\pi)$ and $g_2$ (a normalized second $s$-derivative at the symmetric point $s=t=u=4m^2/3$), whose boundary shows a characteristic kink structure.

On the analytic side, rigorous theorems bound the high-energy behavior. Froissart and Martin proved that the forward total cross-section cannot grow faster than $\log^2 s$; Jin and Martin then showed that a fixed-$t$ dispersion relation requires at most two subtractions, whence, within the Martin analyticity domain $-4m^2<t<4m^2$ — the $t$-interval in which analyticity is rigorously established from unitarity — the amplitude obeys $|\mathcal{M}(s,t)|<|s|^{1+\gamma(t)}$ for some $0<\gamma(t)<1$ that may depend on the amplitude: growth at fixed $t$ is strictly slower than $s^2$, but the theorem yields the bound amplitude by amplitude rather than a single cap shared by the whole class (the Jin-Martin bound). A subtracted dispersion relation reconstructs the amplitude on the physical cut from its discontinuity, with each subtraction (a multiplication by inverse powers of $s$ before the Cauchy integral) softening the assumed asymptotic growth.

de Rham, Tolley, Wang and Zhou introduced a primal bootstrap that parameterizes, instead of the full amplitude, the imaginary parts of the partial waves on the physical cut, and reconstructs the amplitude through the twice-subtracted fixed-$t$ dispersion relation, so that crossing, unitarity and the Jin-Martin behavior are built in by construction. They also introduced fractionally subtracted dispersion relations of real subtraction order $2r$, which allow one to impose any assumed high-energy growth $|\mathcal{M}(s,t)|\lesssim|s|^{2r}$, $0\le r\le1$, and to probe how the allowed region depends on it. The method reproduces known dual-bootstrap results (for example the spin-2 glueball coupling bound of Guerrieri, Hebbar and van Rees) and gives access to the Regge behavior — the large-$|s|$ growth exponent $\alpha(t)$ at fixed $t$, interpretable in Regge theory as a $t$-channel trajectory of resonances, whose value at $t=0$ is the intercept — of the extremal amplitudes on the boundary of the allowed $(g_0,g_2)$ region. The fastest growth they find is 'at least $s^{1.7\sim1.8}$' near $t\to4m^2$ (and around $s^{1.5}$ or slightly faster in the fractionally-subtracted analysis), close to but below the $s^2$ Jin-Martin limit. Since Jin-Martin itself already forbids growth at or above $s^2$, the substantive question is whether consistent amplitudes can approach that limit arbitrarily closely, or whether the full constraint set caps the exponent strictly below it.

Problem Statement

For the two-to-two scattering of identical scalars of mass $m$ in four dimensions ($s+t+u=4m^2$), consider amplitudes $\mathcal{M}(s,t)$ satisfying: (i) full partial-wave unitarity, $\mathrm{Im}\,a_\ell(s)\ge\tfrac{1}{2}\sqrt{(s-4m^2)/s}\,|a_\ell(s)|^2$ for the partial-wave coefficients $a_\ell(s)$ of the expansion defined in the Background, for all even $\ell\ge0$ and all $s\ge4m^2$; (ii) crossing symmetry, $\mathcal{M}(s,t)=\mathcal{M}(s,u)=\mathcal{M}(t,s)$; and (iii) analyticity in the Martin domain $-4m^2<t<4m^2$ with fixed-$t$ dispersion relations, so that the Jin-Martin bound — growth strictly slower than $|s|^2$ at fixed $t$ in that window — holds. The dispersive primal bootstrap constructs this space at finite truncation (spin cutoff $\ell_{\max}$, energy cutoff $s_{\max}$ of the dispersive reconstruction on the physical cut, and polynomial degree $k_{\max}$ of the ansatz for each partial wave), and its extremal boundary amplitudes attain a maximal fitted growth exponent of only $\alpha\approx1.7$-$1.8$ near the upper edge $t\to4m^2$. Determine the origin of the gap between this maximal constructed growth and the $|s|^2$ Jin-Martin limit, settling the dichotomy posed in the source: (a) show that the gap is not fundamental, by exhibiting — through improved numerical convergence at larger truncations and/or analytic or constructive amplitudes — that amplitudes satisfying (i)-(iii) exist whose fixed-$t$ growth in the gap region near $t\to4m^2$ approaches $|s|^2$ arbitrarily closely (exponents arbitrarily close to 2) — what establishes (a) is a family, whether of explicit analytic or constructive amplitudes or obtained through truncation-controlled convergence as the truncation parameters are increased, whose fixed-$t$ exponents in the gap region have supremum 2; or (b) show that the gap is fundamental, by proving that (i)-(iii) together imply a strictly stronger growth bound than Jin-Martin — a single cap $\alpha_*<2$ on the growth exponent, uniform both over all amplitudes satisfying (i)-(iii) and over $t$ in the gap region near $t\to4m^2$ (that is, some $\delta>0$ with $\alpha(t)\le2-\delta$ for every such amplitude and every $t$ in that region) — so that no consistent amplitude can approach $|s|^2$ growth anywhere in the gap region.

References

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