⚛️ high-energy theory

Bounds on scattering amplitudes of non-identical scalar particles in 4d

This paper initiates the S-matrix bootstrap study of two-to-two scattering for two distinct scalar particles in four dimensions, revealing that unequal masses introduce an unbounded pseudo-physical cut which qualitatively alters the bounds on observables until the equal-mass limit is reached or additional information is supplied.

Original authors: Gabriele Ferretti, Denis Karateev, Alessandro Piazza, Marco Serone

Published 2026-09-10

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CC BY 4.0

Original authors: Gabriele Ferretti, Denis Karateev, Alessandro Piazza, Marco Serone

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast, invisible landscape of particle physics, scientists are constantly trying to map the rules that govern how matter interacts. At the heart of this effort lies a concept called the scattering amplitude, which is essentially a mathematical description of what happens when two particles collide and bounce off one another. For decades, physicists have relied on a powerful method known as the S-matrix bootstrap to explore these interactions without needing to know the specific details of the forces involved. This approach treats the universe like a puzzle where the pieces are fixed by fundamental laws: the rules of symmetry, the requirement that cause must precede effect, and the principle that probability must always add up to one. By applying these strict rules, researchers can determine which collision outcomes are possible and which are forbidden, effectively carving out the shape of reality itself. While this method has been highly successful for collisions involving identical particles, like two protons hitting each other, it has struggled to handle collisions between two different types of particles, such as a pion and a kaon, which are common in the subatomic world.

A team of researchers has now taken a significant step forward by applying this bootstrap method to the scattering of two distinct, non-identical scalar particles. Scalar particles are a simple type of matter that have no internal spin, making them an ideal starting point for understanding more complex interactions. The scientists focused on a scenario where two different particles, let's call them A and B, collide and scatter. They imposed a specific set of symmetries to ensure these particles remain stable and do not decay into other forms, a condition that simplifies the mathematical landscape. Their goal was to see if they could place strict limits on the strength of the interaction between these particles, much like determining the maximum speed a car can go based on the laws of physics rather than the engine's horsepower.

The researchers discovered that when the two particles have different masses, the problem becomes surprisingly difficult. In the case of identical particles, the mathematical rules provide clear boundaries, creating a finite box within which all possible outcomes must reside. However, when the masses differ, a strange region appears in the mathematical description known as a pseudo-physical region. This is a zone where the standard rules of probability do not directly apply, creating a gap in the constraints. Because of this gap, the team found that certain properties of the collision, such as the strength of the interaction at a specific point, become unbounded. In other words, without additional information, the interaction strength could theoretically be infinitely large or infinitely small, and the mathematical tools could not rule out these extreme possibilities. This was a stark contrast to the equal-mass case, where the pseudo-physical region disappears, and the researchers successfully recovered tight, two-sided bounds on all the interaction strengths.

To address this issue, the team explored whether adding specific assumptions about the behavior in that mysterious pseudo-physical region could restore the ability to set limits. They tested a model where the interaction in this gap was dominated by a single, simple resonance, a temporary state where the particles briefly stick together before separating. When they introduced this assumption, the unbounded directions vanished. The previously wild possibilities were tamed, and the researchers were able to establish clear upper and lower limits on the interaction strengths, even for particles with different masses. This result serves as a proof of concept, demonstrating that while the bootstrap method faces new challenges with non-identical particles, it can still yield powerful results if supplemented with reasonable physical assumptions about the hidden regions of the interaction.

The findings have direct implications for understanding real-world particle collisions, particularly those involving pions and kaons, which are fundamental components of atomic nuclei and play a crucial role in the strong nuclear force. The researchers noted that their simplified model, while abstract, captures the essential difficulties of these real-world processes. By showing that the method can be adapted to handle unequal masses, they have opened the door for future studies that could incorporate experimental data to refine our understanding of how these particles interact. The work suggests that the path forward involves combining the rigorous constraints of the bootstrap with specific knowledge about the behavior of particles in the regions where standard rules fall short, offering a promising new way to probe the fundamental structure of matter.

Technical Summary: Bounds on Scattering Amplitudes of Non-Identical Scalar Particles in 4d

Problem Statement
The S-matrix bootstrap program aims to constrain scattering amplitudes in Quantum Field Theory (QFT) using only fundamental principles: Lorentz invariance, crossing symmetry, analyticity, and unitarity. While this framework has been extensively developed for the scattering of identical particles (particularly in $d=2$ and for identical scalars in $d=4$ ), the case of non-identical particles remains largely unexplored in higher dimensions.

The scattering of non-identical particles introduces significant new analytic structures. Specifically, amplitudes can develop branch cuts whose branch points lie below the physical threshold of the corresponding process. This region is termed the "pseudo-physical region." In the case of unequal masses ( $m_A \neq m_B$ ), this region ( $4m_A^2 \leq s < 4m_B^2$ for the $AA \to BB$ process) possesses a discontinuity that is not directly constrained by physical unitarity, which strictly applies only to the physical region ( $s \geq 4m_B^2$ ). This lack of constraint qualitatively alters the bootstrap problem, potentially rendering certain observables unbounded.

Methodology
The authors initiate the study of two-to-two scattering amplitudes involving two distinct real scalar particles, $A$ and $B$ , in $d=4$ spacetime dimensions. To simplify the setup while retaining the essential features of non-identical scattering, they impose a $Z_2 \times Z_2$ global symmetry. Under this symmetry, $A$ and $B$ carry charges $(-,+)$ and $(+, -)$ respectively. This symmetry ensures:

Absolute stability of $A$ and $B$ (as they are the lightest states in their charged sectors).
The absence of cubic couplings and one-particle exchange poles.
The absence of triangular anomalous thresholds.
A reduction of the full set of processes to a minimal closed system under crossing symmetry: $AB \to AB$ and $AA \to BB$ .

The study employs the primal S-matrix bootstrap method. The amplitude is parametrized as a power series expansion in conformal variables ( $\rho$ -variables) that map the cut complex plane to the unit disk. The ansatz explicitly accommodates the branch cut structure, including the pseudo-physical region. The coefficients of this expansion are treated as optimization variables.

The constraints imposed are:

Analyticity: The amplitude is analytic in the complex plane except for physical and pseudo-physical cuts.
Crossing Symmetry: Relations between $AB \to AB$ and $AA \to BB$ amplitudes are enforced.
Unitarity: The partial wave amplitudes must satisfy positivity conditions derived from the Gram matrix of asymptotic states. For the $AB \to AB$ channel, this is a single-channel constraint. For the $AA \to BB$ channel, the constraint applies only in the physical region ( $s \geq 4M^2$ ), leaving the pseudo-physical region ( $4m^2 \leq s < 4M^2$ ) unconstrained by unitarity in the minimal setup.

The optimization problem is formulated as a Semi-Definite Program (SDP) and solved numerically using the solver SDPB. The authors investigate convergence with respect to the truncation parameters: the maximum degree of the polynomial ansatz ( $N_{max}$ ), the maximum spin ( $L_{max}$ ), and the number of sampling points in the physical region ( $n_{pts}$ ).

Key Contributions and Results

Unboundedness in the Unequal Mass Case ( $\mu > 1$ ):
For unequal masses, the authors find that the absence of unitarity constraints in the pseudo-physical region leads to several unbounded observables:
- The coupling $\lambda_{0,0}$ (the amplitude value at a symmetric point) is unbounded from both above and below.
- The coupling $\lambda_{2,0}$ has a lower bound ( $\geq 0$ ) but no upper bound.
- The coupling $\lambda_{2,1}$ has a non-trivial absolute lower bound but no upper bound.
- The scalar scattering length $a_0$ is unbounded from both sides.
- However, fixing one observable (e.g., $\lambda_{2,0}$ ) can restore boundedness for others (e.g., a convergent lower bound for $\lambda_{2,1}$ and $a_0$ ).
Boundedness in the Equal Mass Limit ( $\mu = 1$ ):
In the limit where $m_A = m_B$ , the pseudo-physical region collapses to a point and disappears. Consequently, the bootstrap problem recovers the standard behavior seen in identical particle scattering: all couplings ( $\lambda_{0,0}, \lambda_{2,0}, \lambda_{2,1}$ ) admit both upper and lower bounds. The allowed regions in the coupling space are found to be substantially larger than those for identical scalars.
Restoring Boundedness via Pseudo-Physical Constraints:
As a proof of concept, the authors demonstrate that introducing additional assumptions about the pseudo-physical region can restore boundedness. By parametrizing the discontinuity in the pseudo-physical region using a Muskhelishvili-Omnès function (assuming dominance by a scalar resonance), they recover two-sided bounds for all couplings and a convergent lower bound for the scattering length. This suggests that physical input regarding the pseudo-physical region is crucial for obtaining finite bounds in the unequal mass case.
Comparison with Linearized Unitarity:
The numerical bounds obtained for $\lambda_{2,1}$ are found to be very close to (within a few percent of) the analytic lower bounds derived from linearized unitarity and positivity, providing a consistency check for the numerical implementation.

Significance and Scope
The paper claims to provide a starting point for future studies of physical processes such as pion-kaon ( $\pi K \to \pi K$ ) and pion-nucleon ( $\pi N \to \pi N$ ) scattering, which involve non-identical particles. The authors emphasize that their work is a theoretical exploration of the minimal system closed under crossing symmetry.

The primary significance lies in identifying the "pseudo-physical region" as the source of unboundedness in the bootstrap for non-identical particles. The results highlight that without additional information about the amplitude's behavior in this region (which is not fixed by standard unitarity), the space of allowed amplitudes is significantly less constrained than in the identical particle case. The paper does not claim to have solved the full phenomenological problem for QCD but rather establishes the theoretical framework and identifies the specific analytic challenges that must be addressed to apply the S-matrix bootstrap to realistic hadronic scattering.

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