S-Matrix and Amplitude Bootstrap

An S-matrix bootstrap starts from observable analytic data—symmetry, particle spectrum, singularities, unitarity, crossing, and high-energy bounds—and asks which amplitudes satisfy all of them. Perturbative amplitude bootstrap methods often solve a finite ansatz using factorization and cuts; nonperturbative numerical bootstraps impose inequalities on functional approximations. Both are conditional: uniqueness or an exclusion bound is only as strong as the declared spectrum, analyticity domain, asymptotics, and ansatz completeness.

Required background. Causality, Growth, and Analytic Domains supplies the hypotheses behind analyticity and polynomial bounds. Fixed-t and Partial-Wave Dispersion supplies subtracted dispersion relations and partial-wave projections.

Helpful background. Constructibility, Boundary Terms, and Failure Modes supplies the perturbative distinction between factorization data and contributions at infinity.

Bootstrap data and an ansatz

For elastic scattering of identical massive scalars, a candidate amplitude $\mathcal M(s,t,u)$ begins with

$$s+t+u=4m^2, \qquad \mathcal M(s,t,u)=\mathcal M(t,s,u)=\mathcal M(u,t,s),$$

for a fully crossing-symmetric species. Its singularity data include stable-particle poles, multiparticle branch cuts, and possible subtractions. A useful decomposition is

$$\mathcal M =\mathcal M_{\mathrm{poles}} +\mathcal M_{\mathrm{cuts}} +\mathcal P,$$

where $\mathcal P$ is a subtraction or contact ambiguity constrained by growth and low-energy data.

In perturbation theory, one chooses basis functions with the allowed singularities and solves for coefficients by matching factorization residues, generalized cuts, crossing, soft behavior, gauge invariance, and power counting. A finite basis can prove uniqueness only within that basis and after possible rational or boundary terms have been included.

Partial-wave unitarity

In $D=4$, choose the elastic normalization

$$\mathcal M(s,t) =16\pi\sum_{\ell=0}^\infty (2\ell+1)a_\ell(s)P_\ell(\cos\theta),$$

and

$$S_\ell(s)=1+2i\rho(s)a_\ell(s), \qquad \rho(s)=\sqrt{1-\frac{4m^2}{s}}.$$

Here $s>4m^2$ is approached from the upper rim of the physical cut and the square-root branch is chosen so that $\rho(s)>0$. Unitarity gives

$$\operatorname{Im}a_\ell(s) \ge \rho(s)|a_\ell(s)|^2,$$

with equality between the two-particle threshold and the first inelastic threshold, provided no other channel is open. Equivalently, $S_\ell$ lies in or on the unit disk. Different amplitude conventions move factors of $16\pi$ and $\rho$; a numerical bootstrap must fix them before imposing a disk constraint.

Crossing couples partial waves in different channels, so one cannot choose each $a_\ell(s)$ independently. Analytic parametrizations—conformal maps, dispersion integrals, or basis expansions—turn these coupled conditions into constraints on finitely many coefficients plus a truncation error.

Dispersion and low-energy data

For $t$ in a domain where fixed-$t$ analyticity holds, a schematic $N$-subtracted dispersion relation is

$$\mathcal M(s,t) =P_{N-1}(s,t) +\frac{(s-s_0)^N}{\pi} \int_{s_{\mathrm{th}}}^\infty \mathrm ds' \frac{\operatorname{Im}\mathcal M(s'+i0,t)} {(s'-s_0)^N(s'-s)} +\text{left-cut and pole terms}.$$

Here $s_0$ lies in the assumed fixed-$t$ analytic domain. Moving it changes the subtraction polynomial $P_{N-1}$ but not the analytic content. Positive absorptive parts in appropriate forward combinations can constrain derivatives of the low-energy amplitude. These constraints remain conditional on the subtraction count, pole removal, crossing combination, mass gap, forward-limit regularity, and high-energy behavior. The bootstrap does not eliminate those hypotheses; it organizes their consequences.

Known low-energy coefficients, resonance locations, or asymptotic bounds can be used as inputs or objective functions. A bound such as “maximize one EFT coefficient” means maximize over the explicitly defined feasible set, not over all imaginable quantum field theories.

Perturbative and numerical bootstraps

The two common workflows solve related but distinct problems.

Perturbative amplitude bootstrap. Choose a loop or rational ansatz with a finite singularity/function basis. Impose locality, cuts, factorization, symmetries, soft limits, and ultraviolet power counting. Validate the result against independent unitarity cuts and low-point limits. Missing basis elements appear as undetermined contact, rational, or boundary terms.

Nonperturbative numerical S-matrix bootstrap. Parameterize analytic partial waves or the full amplitude, impose crossing and unitarity inequalities at sampled or functional points, truncate spin or basis size, and optimize an observable. Establish convergence by increasing all truncations and by constructing primal and dual certificates where available.

For gapped $1+1$-dimensional Lorentz-invariant QFT, the dispersion and numerical construction for identical neutral particles is worked out in Paulos et al. 2017, § 2.1, pp. 6–10. A separate construction in $3+1$ dimensions treats elastic scattering of the lightest identical real scalar, imposing partial-wave unitarity on a crossing-symmetric analytic ansatz Paulos et al. 2019, §§ 3–4, pp. 8–22. Neither construction proves uniqueness for a different dimension, spectrum, spin content, or analyticity class without repeating the analytic and convergence analysis.

A minimal scalar example

A contact interaction already shows why crossing and exact unitarity are independent constraints. Consider one stable real scalar of mass $m>0$ with $\mathbb Z_2$ symmetry and

$$\mathcal L_{\mathrm{int}}=-\frac{\lambda}{4!}\phi^4, \qquad \mathcal M_{\mathrm{tree}}=-\lambda.$$

In the elastic window $4m^2<s<16m^2$, use the identical-particle convention

$$\begin{aligned} \mathcal M(s,z) &=32\pi\sum_{\substack{\ell\ge0\\ \ell\ \mathrm{even}}} (2\ell+1)a_\ell(s)P_\ell(z),\\ a_\ell(s) &=\frac{1}{64\pi}\int_{-1}^{1}\mathrm dz\, P_\ell(z)\mathcal M(s,z),\\ S_\ell&=1+2i\rho a_\ell. \end{aligned}$$

The constant tree amplitude gives

$$a_0^{\mathrm{tree}}=-\frac{\lambda}{32\pi}, \qquad a_{\ell\ge2}^{\mathrm{tree}}=0.$$

It is entire, crossing symmetric, and polynomially bounded, yet

$$|S_0^{\mathrm{tree}}|^2 =1+\frac{\rho^2\lambda^2}{256\pi^2}>1 \qquad(\lambda\ne0).$$

This does not invalidate perturbation theory: the excess begins at $O(\lambda^2)$, where one-loop absorption is required. Expanding elastic unitarity fixes

$$\operatorname{Im}a_0^{\mathrm{1\text{-}loop}} =\rho|a_0^{\mathrm{tree}}|^2 =\frac{\rho\lambda^2}{1024\pi^2},$$

and hence

$$\operatorname{Im}\mathcal M^{\mathrm{1\text{-}loop}}(s,0) =\frac{\rho\lambda^2}{32\pi}.$$

The independent optical-theorem check uses the identical-final-state tree cross section

$$\sigma_{\mathrm{tree}}=\frac{\lambda^2}{32\pi s},$$

so $s\rho\,\sigma_{\mathrm{tree}}$ reproduces the same imaginary part. The calculation is convention sensitive: a $16\pi$ partial-wave expansion must be accompanied by consistently redefined $a_\ell$ and $S_\ell$. Its lesson is that a crossing-symmetric analytic seed can satisfy perturbative unitarity order by order while failing the exact unit-disk condition when truncated.

At tree level, suppose the only exchanged stable scalar has mass $M$ and cubic coupling $g$. With a pole-sign convention in which the residue at $s=M^2$ is $-g^2$, crossing suggests the illustrative basis

$$\mathcal M_{\mathrm{ansatz}}(s,t,u) =g^2\left( \frac1{M^2-s} +\frac1{M^2-t} +\frac1{M^2-u} \right) +c_0+c_1(s^2+t^2+u^2)+\cdots.$$

Because $s+t+u=4m^2$, a crossing-symmetric term linear in the Mandelstam variables is already absorbed into $c_0$. Factorization fixes the pole residues but leaves contact coefficients. Growth assumptions, derivative counting, low-energy measurements, or positivity constraints can restrict them; factorization alone cannot. This elementary example is the same warning encountered in on-shell recursion: correct poles do not determine a polynomial boundary term.

Validation and convergence

A bootstrap result should report:

1. the spectrum and every assumed pole or cut;
2. the analyticity domain and crossing equations;
3. amplitude and partial-wave normalization;
4. the high-energy or subtraction assumption;
5. basis, spin, grid, and precision truncations;
6. monotonicity or stability of bounds under systematic enlargement;
7. a reconstructed amplitude or certificate that satisfies independent points and physical limits;
8. sensitivity to removing or adding an allowed state or contact term.

The frontier evidence table labels exact conditional constraints as established and keeps numerical completeness and uniqueness tied to their stated ansatz. This evidence statement was checked against the cited primary constructions on 9 August 2026; it is not a claim that any finite numerical ansatz is complete.

Common pitfalls

Claiming uniqueness from poles and crossing alone. Contact and subtraction terms can share all factorization poles. State what fixes or bounds them.

Imposing a unit disk with mismatched normalization. Derive $S_\ell$ from the amplitude convention before coding inequalities.

Showing one stable truncation. Vary spin, basis size, grid, precision, and asymptotic assumptions independently. Accidental plateaus are possible.

Calling a perturbative ansatz nonperturbative. Loop-order power counting and a chosen function basis are additional assumptions even when no diagrams are used.

Exercises

Add a crossing-symmetric constant $c_0$ to the scalar pole ansatz. Verify that every pole residue and crossing equation remains unchanged. What can constrain $c_0$?

Solution

A constant is entire, so it changes no factorization residue, and it is invariant under every permutation of $s,t,u$. Factorization and crossing therefore leave $c_0$ undetermined. Low-energy data, a dispersion relation with specified subtraction constants, derivative power counting, positivity, or a sufficiently strong asymptotic condition can constrain it; each is additional input.

Where to continue

• Forward-Limit Positivity Bounds develops one important family of conditional low-energy constraints.
• Scattering Analyticity, Crossing, and Rigorous Bounds gives the theorem-first treatment under locality, spectrum, stability, and temperedness hypotheses.
• Function Spaces, Symbols, and Coaction Patterns supplies finite function ansätze for perturbative bootstraps.
• Color–Kinematics Duality contains the shared evidence table.

References

• Paulos, Miguel F., Joao Penedones, Jonathan Toledo, Balt C. van Rees, and Pedro Vieira. “The S-Matrix Bootstrap II: Two Dimensional Amplitudes.” Journal of High Energy Physics 11 (2017): 143. doi:10.1007/JHEP11(2017)143. Open PDF.
• Paulos, Miguel F., Joao Penedones, Jonathan Toledo, Balt C. van Rees, and Pedro Vieira. “The S-Matrix Bootstrap. Part III: Higher Dimensional Amplitudes.” Journal of High Energy Physics 12 (2019): 040. doi:10.1007/JHEP12(2019)040. Open PDF.