Gapped scalar EFT forward channel and matched coefficient data

Use natural units $c=\mathrm{\hslash }=1$ on four-dimensional Minkowski space $M={\mathbb{R}}^{1,3}$ with metric $\eta =\mathrm{diag}(-1,1,1,1)$. Let $\varphi \in C^{\mathrm{\infty }}(M,\mathbb{R})$ be a real, Grassmann-even (commuting) scalar field of engineering dimension one, with a stable one-particle state of pole mass $m>0$; here stable means an isolated real mass pole with zero decay width. Stability and the existence of this state are physical input data; no construction of a state is asserted. An effective field theory (EFT) here is the selected local low-energy representative organized in powers of its cutoff; its exact amplitude is separate supplied model data. For the finite-jet local-operator convention, use the complexification for the finite-jet algebra and impose the usual reality condition on $\varphi$. Take one field species, maximum jet order $r=1$, polynomial degree at most $q=4$, weight $D=8$, and trivial internal group. Specify the spacetime-index action as follows. Let

$L_{\mathbb{R}}=\{\mathrm{\Lambda }\in {\mathrm{GL}}_4(\mathbb{R}):{\mathrm{\Lambda }}^T\eta \mathrm{\Lambda }=\eta \},L_{\mathbb{C}}=\{\mathrm{\Lambda }\in {\mathrm{GL}}_4(\mathbb{C}):{\mathrm{\Lambda }}^T\eta \mathrm{\Lambda }=\eta \}\mathrm{.}$

The real Lorentz group $L_{\mathbb{R}}$ acts on coordinates by $x^{\mathrm{\prime }\mu }={\mathrm{\Lambda }}^{\mu }{}_{\nu }{x}^{\nu }$ and on the scalar field by $(\mathrm{\Lambda }\cdot \varphi )(x)=\varphi ({\mathrm{\Lambda }}^{-1}x)$, so its field-value representation is trivial. For $0\le s\le r$, its induced action on derivative components is

${\mathrm{\partial }}_{{\mu }_1}\cdots {\mathrm{\partial }}_{{\mu }_s}(\mathrm{\Lambda }\cdot \varphi )(x)=({\mathrm{\Lambda }}^{-1}{)}^{{\nu }_1}{}_{{\mu }_1}\cdots ({\mathrm{\Lambda }}^{-1}{)}^{{\nu }_s}{}_{{\mu }_s}({\mathrm{\partial }}_{{\nu }_1}\cdots {\mathrm{\partial }}_{{\nu }_s}\varphi )({\mathrm{\Lambda }}^{-1}x),$

with the empty product for $s=0$ giving the scalar field itself. Use the complexification of these tensor representations, with $L_{\mathbb{C}}$, on the finite-jet polynomial algebra in the supplier's invariant construction; this specifies its spacetime-index representation data. Since ${\mathrm{\Lambda }}^T\eta \mathrm{\Lambda }=\eta$, the induced covector action satisfies ${\eta }^{\mu \nu }({\mathrm{\Lambda }}^{-1}{)}^{\alpha }{}_{\mu }({\mathrm{\Lambda }}^{-1}{)}^{\beta }{}_{\nu }={\eta }^{\alpha \beta }$. Thus ${\eta }^{\mu \nu }{\mathrm{\partial }}_{\mu }\varphi {\mathrm{\partial }}_{\nu }\varphi$ and its square are invariant, and the real operator ${\mathcal{O}}_8=({\mathrm{\partial }}_{\mu }\varphi {\mathrm{\partial }}^{\mu }\varphi {)}^2$ is a local Lorentz scalar of engineering dimension eight. Its coefficient in a chosen local representative is written

${\mathcal{L}}_{\mathrm{E}\mathrm{F}\mathrm{T}}=-\frac{1}{2}{\mathrm{\partial }}_{\mu }\varphi {\mathrm{\partial }}^{\mu }\varphi -\frac{1}{2}{m}^2{\varphi }^2+\frac{C_8({\mu }_{\ast })}{{\mathrm{\Lambda }}^4}{\mathcal{O}}_8+\cdots ,$

where $\mathrm{\Lambda }>2m$ is the positive energy cutoff of the EFT description, $0<{\mu }_{\ast }<\mathrm{\Lambda }$ is the chosen scale at which the coefficient and matching identity are specified, and $C_8({\mu }_{\ast })\in \mathbb{R}$ is dimensionless in the fixed coefficient convention and local representative. No basis-independence under field redefinitions, equations of motion, or integration by parts is asserted for $C_8$ itself.

The amplitude data consist of an identical-scalar $2\to 2$ amplitude $\mathcal{A}(s,t,u)$ and real on-shell momenta $p_j$ with $p_j^2=-m^2$, $p_1+p_2=p_3+p_4$, and

$s=-(p_1+p_2{)}^2,t=-(p_1-p_3{)}^2,u=-(p_1-p_4{)}^2,s+t+u=4m^2\mathrm{.}$

The forward slice is $A(s)=\mathcal{A}(s,0,4m^2-s)$ and assume $A(s)=A(4m^2-s)$ wherever both sides are defined. Use the amplitude-input definition's physical 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 ${\mathcal{A}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:{\mathcal{P}}_m\to \mathbb{C}$; the analytic continuation defining $A$ is separately supplied, with $A(s+i0)={\mathcal{A}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(s,0,4m^2-s)$ for $s>4m^2$. For each integer $\mathrm{\ell }\ge 0$, the coefficient function is $a_{\mathrm{\ell }}:(4m^2,\mathrm{\infty })\to \mathbb{C}$, and the normalized Legendre polynomial $P_{\mathrm{\ell }}:\mathbb{C}\to \mathbb{C}$ is

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

When a full partial-wave expansion is asserted, require its partial sums to converge uniformly in $x=1+2t\mathrm{/}(s-4m^2)\in [-1,1]$ for each fixed $s>4m^2$ to the supplied physical amplitude, with 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 an inference of the forward sign from partial waves, convergence at $x=1$ suffices together with $\mathrm{Im}{a}_{\mathrm{\ell }}(s)\ge \mathrm{\mid }{a}_{\mathrm{\ell }}(s){\mathrm{\mid }}^2\ge 0$ for every $\mathrm{\ell }$ and $s>4m^2$: each finite forward sum has nonnegative imaginary part and so does its limit. These are additional conditions on supplied partial-wave data; the normalization alone does not establish them or construct a unitary scattering matrix.

The $S$-matrix convention is $S=1+i(2\pi {)}^4{\delta }^4(p_{\mathrm{o}\mathrm{u}\mathrm{t}}-p_{\mathrm{i}\mathrm{n}})\mathcal{A}+\cdots$. The channel adopts the forward optical-theorem input $\mathrm{Im}A(s+i0)\ge 0$ for $s>4m^2$ in this normalization. This sign is an assumption on the supplied amplitude data, not a deduction from the EFT truncation.

Put ${\nu }_0=2m^2$ and $v=s-2m^2$, so $u-2m^2=-v$. Specify a finite list $(p_j,r_j{)}_{j=1}^N$, with $N\ge 0$, distinct real locations $p_j\in (0,2m^2)$, and nonzero real residues $r_j={\mathrm{Res}}_{s=p_j}A(s)$. The isolated variable simple poles of $A$ in $(0,4m^2)$ are exactly the disjoint pairs $\{p_j,4m^2-p_j\}$; thus the list contains one representative per crossing pair, always its smaller location. Crossing gives residue $-r_j$ at $s=4m^2-p_j$. For the dimensionless amplitude convention above, $r_j$ has mass dimension two. Define the crossing-symmetric pole subtraction, with the empty sum equal to zero,

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

The exact amplitude data are required to be crossing even and real analytic, with $B(-v)=B(v)$, $B(\bar{v})=\overline{B(v)}$, and $B$ holomorphic on

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

There is no massless pole or cut at the forward point $t=0$. Require finite upper and lower cut-boundary values except at $\pm {\nu }_0$. Also require $B$ to extend 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. Require the uniform high-energy condition $B(w)=o(\mathrm{\mid }w{\mathrm{\mid }}^2)$ as $\mathrm{\mid }w\mathrm{\mid }\to \mathrm{\infty }$ in the closed upper half-plane including upper cut-bank limits, and vanishing endpoint arcs for the cut integrals. For $\nu >{\nu }_0$, let $\rho (\nu )=\mathrm{Im}B(\nu +i0)$ and assign $\rho ({\nu }_0)=0$ by convention, without asserting a threshold boundary limit; changing this one endpoint value changes no Riemann integral; require $\rho \ge 0$, local Riemann integrability on every compact subinterval of $[{\nu }_0,\mathrm{\infty })$, and finite convergence of ${\int }_{{\nu }_0}^{\mathrm{\infty }}\rho (\nu ){\nu }^{-3}d\nu$. For every real $\mathrm{\mid }v\mathrm{\mid }<{\nu }_0$, the upper- and lower-bank integrals of $[B(x)-B(0)]\mathrm{/}[x^2(x-v)]$ are required to exist as finite improper Riemann limits. The positive gap $m>0$ and these analytic and spectral assumptions are IR and UV input conditions; they are not derived from a finite regulator.

Define the first nonconstant even forward coefficient by $c_1=B^{\mathrm{\prime }\mathrm{\prime }}(0)\mathrm{/}2!$, of mass dimension $-4$. Exact matching at ${\mu }_{\ast }$ is a supplied datum of the form

$c_1=\frac{4C_8({\mu }_{\ast })}{{\mathrm{\Lambda }}^4}+{\mathrm{\Delta }}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}},$

where ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}\in \mathbb{R}$ has mass dimension $-4$ and contains every other contribution to this exact $v^2$ coefficient, including other operators, loops, and matching or truncation effects. The identity is an assumption to be checked for a selected model, not a conclusion of this definition. Setting ${\mathrm{\Delta }}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}=0$ is an additional exact retained-order matching condition; power counting alone does not set it to zero.

Gapped identical-scalar forward-amplitude data

Finite-jet local operators and engineering dimension

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