Forward positivity for a matched scalar EFT coefficient

Let $m>0$, $\mathrm{\Lambda }>2m$, and $0<{\mu }_{\ast }<\mathrm{\Lambda }$. Use natural units and the four-dimensional Minkowski metric $\eta =\mathrm{diag}(-1,1,1,1)$. Let $\varphi$ be a real scalar of engineering dimension one with a stable one-particle pole of mass $m$, where stable means an isolated real mass pole with zero decay width. In the convention $S=1+i(2\pi {)}^4{\delta }^4(p_{\mathrm{o}\mathrm{u}\mathrm{t}}-p_{\mathrm{i}\mathrm{n}})\mathcal{A}+\cdots$, let $A(s)=\mathcal{A}(s,0,4m^2-s)$ be the forward amplitude for identical on-shell external momenta $p_j^2=-m^2$, $p_1+p_2=p_3+p_4$, with $s=-(p_1+p_2{)}^2$, $t=-(p_1-p_3{)}^2$, $u=-(p_1-p_4{)}^2$, $s+t+u=4m^2$, and crossing $A(s)=A(4m^2-s)$. For a finite crossing-paired sum containing every stable variable simple pole in the open subthreshold interval

$P(s)={\sum }_j(\frac{r_j}{s-p_j}+\frac{r_j}{4m^2-p_j-s}),r_j\in \mathbb{R},p_j\in (0,4m^2)\setminus \{2m^2\},$

Put ${\nu }_0=2m^2$, $v=s-2m^2$, and $B(v)=A(2m^2+v)-P(2m^2+v)$. Require $B(-v)=B(v)$, $B(\bar{v})=\overline{B(v)}$, and holomorphy on $\mathbb{C}\setminus ((-\mathrm{\infty },-{\nu }_0]\cup [{\nu }_0,\mathrm{\infty }))$. Require finite upper and lower cut-boundary values except at $\pm {\nu }_0$, and require $B$ to extend continuously from each open half-plane onto every compact cut-bank segment avoiding $\pm {\nu }_0$. This compact-bank continuity is an explicit admissibility input. Require $B(w)=o(\mathrm{\mid }w{\mathrm{\mid }}^2)$ uniformly as $\mathrm{\mid }w\mathrm{\mid }\to \mathrm{\infty }$ in the closed upper half-plane, including upper cut-bank limits. Let $\rho (\nu )=\mathrm{Im}B(\nu +i0)$ for $\nu >{\nu }_0$, extended by $\rho ({\nu }_0)=0$ without asserting a threshold boundary limit. Require $\rho$ to be nonnegative and Riemann integrable on every compact subinterval of $[{\nu }_0,\mathrm{\infty })$, with finite ${\int }_{{\nu }_0}^{\mathrm{\infty }}\rho (\nu ){\nu }^{-3}d\nu$. For every real $\mathrm{\mid }v\mathrm{\mid }<{\nu }_0$, require the upper/lower boundary-bank integrals of $[B(x)-B(0)]\mathrm{/}[x^2(x-v)]$ to exist as finite improper Riemann limits and the small threshold-arc integrals of this same integrand to vanish as their radii tend to zero. Assume no massless pole or cut reaches $t=0$. Define $c_1=B^{\mathrm{\prime }\mathrm{\prime }}(0)\mathrm{/}2!$.

At the fixed matching scale ${\mu }_{\ast }$, suppose the selected local representative contains the real dimension-eight interaction $C_8({\mu }_{\ast })({\mathrm{\partial }}_{\mu }\varphi {\mathrm{\partial }}^{\mu }\varphi {)}^2\mathrm{/}{\mathrm{\Lambda }}^4$, with $C_8({\mu }_{\ast })$ dimensionless, and the exact matched coefficient identity is $c_1=4C_8({\mu }_{\ast })\mathrm{/}{\mathrm{\Lambda }}^4+{\mathrm{\Delta }}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}$, where ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}$ has mass dimension $-4$ and includes every other contribution to the exact $v^2$ coefficient. Then

$C_8({\mu }_{\ast })+\frac{{\mathrm{\Lambda }}^4}{4}{\mathrm{\Delta }}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}\ge 0.$

If the exact retained-order matching condition additionally gives ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}=0$, then $C_8({\mu }_{\ast })\ge 0$. The result is only a necessary bound on this specified matched channel combination; it asserts neither that a full amplitude exists for every coefficient passing the bound nor that every off-shell operator coefficient has a universal sign.

Gapped scalar EFT forward channel and matched coefficient data

Forward positivity for a gapped identical scalar

Quartic scalar contact jets descend through verified operator relations

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

Burgess, Introduction to Effective Field Theory, §§ 2.2.1, 2.2.3–2.2.4, and opening § 2.2.5

The positivity input concerns the exact pole-subtracted on-shell amplitude coefficient. The local contact vertex contributes $4C_8\mathrm{/}{\mathrm{\Lambda }}^4$ in the stated convention, while other terms at the same $v^2$ order remain in ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}$. A finite-order EFT calculation sets that remainder to zero only if the exact matching data explicitly say so. Massless forward exchange, gravitational poles, non-polynomial high-energy growth, or an unproved matching identity fall outside this theorem's hypotheses.

The matching supplier also defines the full normalized contact-jet map on its verified-relation quotient image. For a supplied exact amplitude, the remaining Taylor coordinates belong to the remainder $R(v)=B(v)-C_8{t}_{O_8}(v)\mathrm{/}{\mathrm{\Lambda }}^4$; in particular $[v^2]R={\mathrm{\Delta }}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}$. No polynomial contact term is separately required to satisfy the exact amplitude's high-energy bound.