Finite truncated forward-amplitude bootstrap data

Let $B(v)$ and ${\nu }_0=2m^2>0$ be the pole-subtracted forward-amplitude data of the amplitude-input definition. In addition to being even and holomorphic near $v=0$, retain its real-analyticity condition $B(\bar{v})=\overline{B(v)}$. If $B(v)={\sum }_{j\ge 0}{\beta }_j{v}^j$ near $0$, uniqueness of the Taylor expansion gives ${\beta }_j=\overline{{\beta }_j}$ from this condition and ${\beta }_{2n+1}=0$ from evenness. Hence the even Taylor coefficients are real. Fix an integer truncation order $N\ge 1$. For $n\ge 1$ define the Taylor coefficient $c_n=B^{(2n)}(0)\mathrm{/}(2n)!$ and the dimensionless forward on-shell coefficient $b_k={\nu }_0^{2k+2}{c}_{k+1}\in \mathbb{R}$ for $0\le k\le N$. Since $v$ and ${\nu }_0$ have units $(\text{mass}{)}^2$, $c_n$ has units $(\text{mass}{)}^{-4n}$ and $b_k$ is dimensionless. Here “Wilson coefficient” refers only to this normalized on-shell amplitude coordinate; it does not identify a generic coefficient of an off-shell Lagrangian operator basis.

For each integer $r\ge 0$ with $2r\le N$, form the quadratic constraint

${\sum }_{i,j=0}^r{a}_i{a}_j{b}_{i+j}\ge 0\text{for every }(a_0,\dots ,a_r)\in {\mathbb{R}}^{r+1}\mathrm{.}$

For each integer $r\ge 0$ with $2r+1\le N$, form the two additional quadratic constraints

${\sum }_{i,j=0}^r{a}_i{a}_j{b}_{i+j+1}\ge 0,{\sum }_{i,j=0}^r{a}_i{a}_j(b_{i+j}-b_{i+j+1})\ge 0\text{for every }(a_0,\dots ,a_r)\in {\mathbb{R}}^{r+1}\mathrm{.}$

The finite order-$N$ forward-amplitude bootstrap feasible set ${\mathcal{K}}_N$ is the subset of ${\mathbb{R}}^{N+1}$ consisting of vectors $(b_0,\dots ,b_N)$ that satisfy every applicable constraint above. The finite truncated bootstrap problem is to determine whether a supplied coefficient vector belongs to ${\mathcal{K}}_N$. Passing this finite test is a necessary condition supplied by forward positivity; it is not defined to be sufficient for the existence of a full amplitude-data package as defined above.

Gapped identical-scalar forward-amplitude data

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

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

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