Gapped identical-scalar forward-amplitude data

Use natural units $c=\mathrm{\hslash }=1$ and metric $\eta =\mathrm{diag}(-1,1,1,1)$. An external scalar datum consists of a spin-zero species assigned a mass $m>0$ and elastic two-particle threshold $s=4m^2$. This is a mathematical input package; it does not construct a one-particle scattering state or a scattering matrix. For real on-shell momenta $p_1,p_2$ incoming and $p_3,p_4$ outgoing, with $p_j^2=-m^2$ and $p_1+p_2=p_3+p_4$, define

$s=-(p_1+p_2{)}^2,t=-(p_1-p_3{)}^2,u=-(p_1-p_4{)}^2\mathrm{.}$

Then $s+t+u=4m^2$. The forward slice is the complex-valued function $A(s)=\mathcal{A}(s,0,4m^2-s)$ obtained by analytically continuing the identical-scalar $2\to 2$ amplitude to $t=0$. The crossing action on this slice is $s\mapsto 4m^2-s$, which exchanges $s$ and $u$. Assume $A(s)=A(4m^2-s)$ wherever both sides are defined.

For context, define the physical elastic kinematic region

${\mathcal{P}}_m=\{(s,t,u)\in {\mathbb{R}}^3:s>4m^2,-(s-4m^2)\le t\le 0,u=4m^2-s-t\}\mathrm{.}$

When supplied, the full physical amplitude is a map ${\mathcal{A}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:{\mathcal{P}}_m\to \mathbb{C}$; its analytic continuation used in the forward data above is a separate supplied part of the input, with upper boundary value $A(s+i0)={\mathcal{A}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(s,0,4m^2-s)$ for $s>4m^2$. For $(s,t,u)\in {\mathcal{P}}_m$, put $x=1+2t\mathrm{/}(s-4m^2)\in [-1,1]$. For each integer $\mathrm{\ell }\ge 0$, define the normalized Legendre polynomial $P_{\mathrm{\ell }}:\mathbb{C}\to \mathbb{C}$ by

$P_{\mathrm{\ell }}(x)=\frac{1}{2^{\mathrm{\ell }}\mathrm{\ell }!}\frac{d^{\mathrm{\ell }}}{d{x}^{\mathrm{\ell }}}(x^2-1{)}^{\mathrm{\ell }},$

and let $a_{\mathrm{\ell }}:(4m^2,\mathrm{\infty })\to \mathbb{C}$ be the complex partial-wave coefficient function in this channel. When a full partial-wave expansion is asserted on the physical angular interval, require the partial sums of the right-hand side below to converge uniformly in $x\in [-1,1]$ for each fixed $s>4m^2$ to the supplied physical amplitude. Use the normalization

${\mathcal{A}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(s,t,u)=16\pi \sqrt{\frac{s}{s-4m^2}}{\sum }_{\mathrm{\ell }=0}^{\mathrm{\infty }}(2\mathrm{\ell }+1)P_{\mathrm{\ell }}\text{\hspace{0.17em}⁣}(1+\frac{2t}{s-4m^2})a_{\mathrm{\ell }}(s),P_{\mathrm{\ell }}(1)=1.$

For the forward sign alone, convergence of the series at $x=1$ suffices. The downstream forward-positivity result uses the optical-theorem sign $\mathrm{Im}A(s+i0)\ge 0$ for $s>4m^2$ as input; a full partial-wave realization is not inferred from this definition. In the displayed normalization, the sign follows if the forward series converges and the supplied partial waves obey $\mathrm{Im}{a}_{\mathrm{\ell }}(s)\ge \mathrm{\mid }{a}_{\mathrm{\ell }}(s){\mathrm{\mid }}^2\ge 0$ for every $s>4m^2$ and every $\mathrm{\ell }$.

Put ${\nu }_0=2m^2$ and $v=s-2m^2$, so $u-2m^2=-v$. Choose finitely many real numbers $p_j\in (0,4m^2)\setminus \{2m^2\}$ and real residues $r_j$, and set

$P(s)={\sum }_j(\frac{r_j}{s-p_j}+\frac{r_j}{4m^2-p_j-s})\mathrm{.}$

Each summand pair is the real simple-pole principal part at $s=p_j$ and its $s\leftrightarrow u$ crossing partner; hence $P(4m^2-s)=P(s)$. These poles lie off the two-particle cuts. Define the pole-subtracted forward function $B(v)=A(2m^2+v)-P(2m^2+v)$. Assume $B$ is holomorphic on

${\mathrm{\Omega }}_{{\nu }_0}=\mathbb{C}\setminus ((-\mathrm{\infty },-{\nu }_0]\cup [{\nu }_0,\mathrm{\infty })),$

real analytic there, meaning $B(\bar{v})=\overline{B(v)}$, and has finite upper and lower boundary values on both cuts except at their endpoints. Here the two-particle gap is the open interval $(-{\nu }_0,{\nu }_0)$ in the $v$ variable; no massless pole or cut reaches the forward point $t=0$.

Require additionally that $B$ extends continuously from each open half-plane onto every compact segment of either real cut bank avoiding $\pm {\nu }_0$. This is an extra admissibility assumption on the supplied data, not a consequence of finite pointwise boundary values or a claim about every quantum field theory.

Assume the exact high-energy condition $B(w)=o(\mathrm{\mid }w{\mathrm{\mid }}^2)$ uniformly as $\mathrm{\mid }w\mathrm{\mid }\to \mathrm{\infty }$ in the closed upper half-plane, including upper cut-bank limits. In particular this holds on the large upper semicircles after any fixed bounded horizontal translation. For $\nu \ge {\nu }_0$, define the absorptive density $\rho (\nu )=\mathrm{Im}B(\nu +i0)$. Require $\rho$ to be nonnegative and Riemann integrable on every compact subinterval of $[{\nu }_0,\mathrm{\infty })$, and require ${\int }_{{\nu }_0}^{\mathrm{\infty }}\rho (\nu ){\nu }^{-3}d\nu$ to converge as a finite improper Riemann integral. For every real $v$ with $\mathrm{\mid }v\mathrm{\mid }<{\nu }_0$, require the integrals along both sides of the two cuts of $[B(x)-B(0)]\mathrm{/}[x^2(x-v)]$, with boundary values taken from the upper and lower half-planes, to exist as finite improper Riemann limits. Require also that the small endpoint arcs at $\pm {\nu }_0$ for this integrand vanish in the contour limit. Since $P$ is real off its poles, $\rho (\nu )=\mathrm{Im}A(2m^2+\nu +i0)$ for $\nu >{\nu }_0$ on the right cut.

The normalized forward Taylor coefficients are $c_n=B^{(2n)}(0)\mathrm{/}(2n)!$ for integers $n\ge 1$. They have units $(\text{mass}{)}^{-4n}$; ${\nu }_0$ has units $(\text{mass}{)}^2$. No sign is assigned here to a generic off-shell Lagrangian coefficient.

Adams et al., Causality, Analyticity and an IR Obstruction to UV Completion, §4

de Rham et al., Positivity Bounds for Scalar Theories, §§2.1–2.2

Basile et al., Lectures in Quantum Gravity, §§3.2.1–3.3.1

This definition treats analyticity, crossing, growth, pole subtraction, and the forward optical-theorem sign as supplied amplitude data. It proves no physical model exists and no full unitary $S$-matrix realizes arbitrary such data. It excludes massless forward poles/cuts, massless gauge or gravitational exchange, and nonlocal amplitudes with different high-energy growth.

For completeness, changing a finite value at one endpoint preserves Riemann integrability and the integral: for any partition isolate that endpoint in an interval of length $h$; the possible change in upper and lower sums is bounded by a fixed finite bound times $h$, which tends to zero. The same equality holds on every finite truncation, and hence on each existing improper limit. This convention does not require a limit of $B$ or of its absorptive part at threshold.