Analytic S-Matrix Theory

• Analytic S-Matrix Theory is a nonperturbative framework for scattering amplitudes that derives universal constraints from principles like analyticity, crossing symmetry, and unitarity.
• It employs dispersion relations and semidefinite programming to rigorously bound low-energy couplings, revealing key numerical bounds and phase shift behaviors.
• The approach bridges classical strong-interaction analyses with modern amplitude bootstrap techniques and extends to multi-particle and effective field theories.

Analytic S-Matrix Theory is a nonperturbative framework for the paper of scattering amplitudes in quantum physics, built on the foundational principles of analyticity, crossing symmetry, and unitarity. Rather than being derived from a Lagrangian field theory, the analytic S-matrix program focuses on deriving universal constraints, bounds, and sometimes even explicit solutions for physical amplitudes solely from general consistency requirements—properties inferred from relativity, causality, probability conservation, and particle statistics. Its methods have catalyzed both classical developments in strong-interaction physics and modern advances in amplitude bootstrap, convex optimization, and rigorous numerical bounds in quantum field theory and effective field theory.

1. Fundamental Principles: Analyticity, Crossing, and Unitarity

The analytic S-matrix is founded on three pillars:

Analyticity: The $2\to2$ scattering amplitude $A(s,t)$ is, for fixed $t$, a meromorphic function of the Mandelstam variable $s$, with only two cuts: from $s=4m^2$ to $\infty$ (physical $s$-channel) and from $s=-\infty$ to $s=-t$ (crossed $u$-channel). Polynomial boundedness at large $|s|$ (Martin’s theorem) requires, in $d=4$, at most $N=2$ subtractions for fixed $|t|<28m^2$.

The twice-subtracted fixed-$t$ dispersion relation is

$$A(s,t) = \sum_{n=0}^{1} a_n(t) s^n + \frac{s^2}{\pi} \int_{4}^{\infty} ds' \frac{\Im A(s',t)}{s'^2 (s' - s)}$$

and can be recast in symmetric Roy-like form, enforcing $s\leftrightarrow u$ crossing explicitly: $0 = \mathcal{A}(s,t) = A(s,t) - A(s_0, t_0) - \frac{1}{\pi} \int_{4}^{\infty} dv \left[ \Disc_s A(v, t) K(v, s, t; t_0) + \Disc_s A(v, t_0) K(v, t, t_0 ; s_0) \right]$ with explicit kernels $K$.

Crossing Symmetry: For identical scalars, full $s \leftrightarrow t \leftrightarrow u$ symmetry holds, subject to $s+t+u=4m^2$. In practical implementation, one imposes $s \leftrightarrow u$ crossing in the dispersion relation and $s \leftrightarrow t$ crossing via symmetrization or explicit partial-wave expansion (only even $\ell$ contribute for identical bosons).

Unitarity: The partial-wave decomposition,

$$A(s, t) = 16 \pi \sum_{\ell=0}^\infty (2\ell + 1) P_\ell(\cos\theta) f_\ell(s),\qquad \cos\theta = 1 + \frac{2t}{s-4}$$

leads to S-matrix eigenvalues

$$S_\ell(s) = 1 + i \rho(s) f_\ell(s),\qquad \rho(s) = \sqrt{\frac{s-4}{s}}$$

Elastic unitarity for $s>4m^2$ yields $\Im f_\ell(s) \geq |f_\ell(s)|^2$, equivalently,

$\mathcal{U}_\ell(s) = \begin{pmatrix} 1 - \frac{1}{2} \rho^2 \Im f_\ell & \rho \Re f_\ell \[6pt] \rho \Re f_\ell & 2 \Im f_\ell \end{pmatrix} \succeq 0$

for all $s>4$ and all even $\ell$. This forms a (countably infinite) semidefinite programming (SDP) region in the space of partial-wave amplitudes.

2. Convex Optimization and the Dual S-Matrix Bootstrap

Analyticity, crossing, and partial-wave unitarity carve out a mathematical space of allowed amplitudes that is convex when translated to SDP constraints. The bootstrap seeks rigorous bounds on physical observables by maximizing or minimizing linear functionals subject to these convex restrictions.

For example, to bound the low-energy quartic coupling,

$$g_0 := \frac{1}{2 n_0} A\left(\frac{4}{3}, \frac{4}{3}\right),\qquad n_0 = 16\pi$$

one introduces a dual SDP. The Lagrangian enforces analyticity via the Roy constraint, $s\leftrightarrow t$ crossing, and separate unitarity in elastic and inelastic regions ($s\in[4,16]$, $s>16$). The dual variables are:

• $\mathcal{W}(s,t)$, expanded in partial waves $w_\ell(s)$, encoding the Roy constraint,
• positive-semidefinite matrices $\Lambda_\ell(s)$ enforcing $\mathcal{U}_\ell(s) \succeq 0$,
• optional multipliers $E_\ell(s)$ for $\det \mathcal{U}_\ell = 0$ (elastic unitarity equality).

The dual program extremizes over $w_\ell(s)$ and $X_\ell(s) = \Lambda_\ell^{11}(s)$: $$\min_{\Lambda_\ell\succeq 0,\, \pm\,\bar w_\ell(s)\geq 0} \pm D^\pm = \sum_{\ell=0}^L \int_4^{\mu_e^2} ds\, X_\ell(s)$$ with normalization $\int_4^{\mu_e^2} ds\, w_0(s) = 1/2$ and $\bar{w}_\ell(s)$ known linear functionals of $w_k$ via the dispersion-relation kernel.

The resulting bounds $g_0 \leq D^+$ (upper), $g_0 \geq D^-$ (lower) are rigorous for any finite $L$ and discretization, and converge monotonically as $L$ and resolution are increased.

3. Numerical Implementation and Results in Four Dimensions

Implementation in $d=4$ ($m=1$) consists of:

• Truncation at a maximal even spin $L$ (for the SDP block),
• Energy domain cutoffs: $\mu_e^2 \leq 12$ for even, $\mu_o^2 \leq 8$ for odd $\ell$,
• Chebyshev polynomial expansions for $w_\ell(s)$ and $X_\ell(s)$ on $[4, \mu_e^2]$,
• Fine $s$-grid for semidefinite and positivity constraints.

The SDP (e.g., solved using SDPB) yields: $-8.08 \lesssim g_0 \lesssim 2.7272$ This substantially improves longstanding bounds in $\pi\pi$ scattering and nearly matches primal S-matrix bootstrap results. The extremal solutions saturating the upper bound correspond to strong attractive interaction (positive scattering length), while the lower-bound solution exhibits maximal inelasticity. In both cases, spin-$\ell$ phase shifts extracted from the dual solution agree with primal numerics and converge to exact answers as $L\to\infty$.

This procedure provides direct, computer-assisted, and fully rigorous nonperturbative bounds on low-energy couplings, independent of detailed QFT Lagrangian input.

4. Physical Interpretation and Systematic Rigorousness

The analytic S-matrix reformulation as a convex optimization problem (SDP) endows the $S$-matrix bootstrap with the following robust features:

• For any finite truncation, the computed bounds on low-energy couplings are strict,
• Systematic improvement—raising $L$, refining grids, enlarging analyticity domain—can only tighten these bounds,
• There is no duality gap: dual and primal formulations converge in the limit.

This achieves a rigor comparable to that of the conformal bootstrap. The method is not limited to scalar theories, but directly generalizes to amplitudes with global symmetries, multiple species, and enables the exploration of spaces of allowed EFT parameters, subject to fundamental physical principles only.

5. Broader Implications and Extension to General Quantum Theories

The convex-analytic S-matrix approach demonstrates that nonperturbative physical constraints—analyticity, crossing, and unitarity—are both necessary and sufficient to bound couplings of low-energy effective theories. This establishes a rigorous paradigm wherein:

• All consistent QFTs and quantum gravity EFTs appear as points (sometimes extremal) in the allowed region carved out by the SDP,
• Observed scattering amplitudes in Nature must reside inside the convex region determined by these general principles,
• Absence of an underlying Lagrangian is no obstruction: entire spaces of consistent quantum theories can be mapped by convex optimization.

The method is directly extensible to more elaborate scattering problems, including higher-point functions, theories with multiple particles or global symmetries, and to the paper of resonance phenomena via imposing zeros/poles at unphysical locations.

6. Comparison and Integration with Other Analytic S-Matrix Frameworks

In comparison to traditional dispersion-relation approaches, the dual/convex SDP framework has the following distinguishing advantages:

• Explicit, rigorous, and systematically improvable computer bounds,
• Algorithmic tractability for high-precision bounding in multiple dimensions,
• Direct analogy to, and occasionally integration with, conformal bootstrap, spectral-function positivity bounds, and convex geometric analyses of QFT observables.

Related modern approaches, such as the geometric-function-theory (GFT) bounds on Wilson coefficients (Haldar, 2022) or analytic inversion techniques combining threshold and large-spin expansions (Correia et al., 2020), can be incorporated with convex-SDP-based S-matrix bootstrapping to further sharpen or generalize rigorous bounds on observable parameters in quantum field theory.

7. Outlook

Analytic S-matrix theory, resurrected in the form of convex optimization and dual functionals, provides not only foundational understanding of nonperturbative amplitude structure but also practical tools for bounding EFT coefficients, constraining theory space, and potentially solving for scattering amplitudes without appeal to a Lagrangian. Ongoing developments in higher-dimensional generalization, algorithmic and functional innovations, and the integration with amplitude geometry and numerical conformal bootstrap techniques are expected to further the reach and power of the analytic S-matrix program (Kruczenski et al., 2022).