All Loop Scattering As A Sampling Problem

Abstract: How to turn the flip of a coin into a random variable whose expected value equals a scattering amplitude? We answer this question by constructing a numerical algorithm to evaluate curve integrals - a novel formulation of scattering amplitudes - by a Monte Carlo strategy. To achieve a satisfactory accuracy we take advantage of tropical importance sampling. The crucial result is that the sampling procedure can be realized as a stochastic process on surfaces which can be simulated efficiently on a computer. The key insight is to let go of the Feynman-bias that amplitudes should be presented as a sum over diagrams, and instead re-arrange the sum as suggested by a dual triangulation of curve integrals. We attach an implementation of this algorithm as an ancillary file, which we have used to evaluate amplitudes for the massive $\mathrm{Tr}(\phi)^3$ theory in $D=2$ space-time dimensions, up to 10-loops. Interestingly, we observe experimentally that the number of sample points required to achieve a fixed accuracy remains significantly smaller than what the number of diagrams would suggest. Finally we propose an extension of our method which is inspired by ideas from artificial intelligence. We use the stochastic process to define a parametrization for a space of distributions, where we formulate importance sampling for an arbitrary curve integrand as a convex optimization problem.

All Loop Scattering as a Sampling Problem: A Technical Overview

The paper "All Loop Scattering As A Sampling Problem" by Giulio Salvatori presents a novel approach to evaluating scattering amplitudes via a Monte Carlo integration strategy. The core objective is transitioning from traditional Feynman diagrams to a dual framework that re-arranges these amplitudes as a single curve integral. This shift encapsulates amplitudes not as a sum over diagrams but rather under a unified integral structure known as "curve integrals," taking advantage of tropical importance sampling to achieve efficient numerical calculation.

The novel framework, referred to as surfaceology, utilizes a tropicalized representation of Symanzik polynomials, converting the problem into one suitable for Monte Carlo evaluation. This formulation is particularly useful for the massive $\mathrm{Tr}(\phi)^3$ theory in two dimensions, where the curve integral formalism elegantly circumvents the issues typically associated with exponentially growing diagram counts. Salvatori introduces a dual sampling method that draws upon the geometric properties of a fan structure formed by allowable curves on a surface.

The paper is structured into several key sections:

1. Introduction and Background:
   The paper introduces surfaceology, emphasizing its foundation in modern scattering amplitude calculations. This section serves to prioritize curves over diagrams, offering a new dissected view on how to represent such amplitudes in a refined, computational manner.

2. Graphs and Surfaces:
   It provides a detailed comparison between traditional Feynman integrals and the proposed curve integrals, highlighting conceptual similarities and mathematical formulations. A necessary background on tropical geometry also helps in understanding the curve integrals' parametric space.

3. Tropicalization of Symanzik Polynomials:
   Insight is given into translating Symanzik polynomials into their tropical analogs, producing the Hepp bounds—a numeric tool key to assessing integrand contributions through the project's Monte Carlo integration method.

4. Monte Carlo Strategy and Dual Sampling:
   Crucially, this section tackles the Monte Carlo sampling setup, where a dual barycentric decomposition of the Feynman fan simplifies the computation. The powerful stochastic process is detailed step by step, presenting a framework divorced from the cumbersome need to compute every component of a traditional Feynman series.

5. Applications and Implications:
   The discussion extends to potential applications beyond $\mathrm{Tr}(\phi)^3$ theory. The implications span improved sampling strategies inspired by machine learning to generalizing the variable space to more complex gauge theories.

6. Conclusion and Future Directions:
   Salvatori suggests multiple potential research directions, such as divergences inclusion and upper bound polynomial exploration, broadening the future utility of surfaceology.

Central to this innovative approach is the recurrence relation on surfaces, introducing recursive algorithms that capture the essence of curves through computational reduction. The work presents numerical results that underscore the efficiency of the dual algorithm in reducing computational overhead. What is observed is a significant reduction in the number of samples needed for high accuracy compared to what naive diagram enumeration would predict.

The implications of this approach are significant. The transition from Feynman diagrams to a curve integral framework marks a pivotal step in efficiently handling large-loop and high-multiplicity scattering amplitudes in theoretical physics. This paper's methods open avenues for tackling other complex amplitude calculations in theoretical frameworks, potentially reformulating gauge theories and exploring quantization at higher dimensions.

In summary, Salvatori's work is a crucial addition to computational physics, providing an alternate route to evaluating scattering amplitudes steeped in the modern language of tropical geometry and combinatorial structures. The dual sampling method, combined with surfaceology's deeper insights into geometric and algebraic aspects of Feynman integrals, positions this research at the forefront of numerical strategies in high-energy theoretical physics. Future developments and applications could revolutionize how researchers approach amplitude calculations, making high-loop computations accessible and efficient.