Uncovering Hidden Connections in Particle Physics

by

Author: Denis Avetisyan

A new data-driven approach is revealing fundamental relationships within scattering amplitudes, potentially simplifying complex calculations and leading to new discoveries.

Researchers used symbolic regression to rediscover known analytical structures like KLT relations and CPQR, demonstrating a pathway for data-driven discovery in gauge theory and gravity.

Despite longstanding challenges in analytically determining relationships within quantum field theory, this paper, ‘Learning the S-matrix from data: Rediscovering gravity from gauge theory via symbolic regression’, demonstrates that modern machine-learning techniques can autonomously rediscover key structures in scattering amplitudes directly from numerical data. Specifically, we show that the Kawai-Lewellen-Tye (KLT) relations, alongside the Kleiss-Kuijf and Bern-Carrasco-Johansson (BCJ) relations, can be recovered using symbolic regression and linear algebra applied to colour-ordered Yang-Mills amplitudes. This data-driven approach, achieving high accuracy up to five external legs, establishes symbolic regression as a powerful tool for exploring the analytic structure of scattering amplitudes and begs the question: can this method uncover entirely new relationships hidden within the complex landscape of quantum field theories?

The Inevitable Bottleneck: Counting Interactions

The determination of scattering amplitudes – mathematical expressions describing the probabilities of particle interactions – lies at the heart of modern particle physics, yet presents a rapidly escalating computational challenge. As the number of particles involved in a given interaction increases, the complexity of calculating these amplitudes grows at a startling rate, often exhibiting factorial growth. This means that even seemingly simple processes, involving just a few particles, can require immense computational resources, quickly exceeding the capabilities of even the most powerful supercomputers. The issue doesn’t stem from a lack of theoretical understanding, but rather from the sheer number of possible interactions and permutations that must be accounted for when transitioning from theoretical prediction to precise numerical result. Consequently, the ability to accurately predict and test the Standard Model, and to search for physics beyond it, is directly constrained by this computational bottleneck, motivating ongoing research into more efficient algorithms and alternative computational frameworks.

The standard method for calculating the probabilities of particle interactions – through Feynman diagrams – faces a fundamental limitation: computational complexity increases factorially with the number of particles involved. This means that even seemingly straightforward processes, with just a few interacting particles, rapidly become intractable for even the most powerful supercomputers. Each additional particle and interaction adds exponentially to the number of diagrams that must be calculated and integrated, quickly overwhelming available resources. For instance, calculating a process involving ten particles requires considering a vastly larger number of diagrams than one involving only five, hindering precision and limiting the ability to test theoretical predictions against experimental results. This factorial growth represents a significant bottleneck in modern particle physics, pushing researchers to explore alternative computational techniques to overcome these limitations and unlock a more complete understanding of the universe at its most fundamental level.

The precision with which physicists can test and refine the Standard Model, and explore physics beyond it, is fundamentally constrained by the computational demands of calculating particle interactions. These calculations rely on scattering amplitudes, which describe the probabilities of particles colliding and transforming. However, as the energy of collisions increases, or the number of particles involved grows, the complexity of these calculations escalates dramatically. This isn’t merely a matter of needing faster computers; the computational cost grows factorially, quickly exceeding the capacity of even the most powerful supercomputers. Consequently, the ability to make precise predictions – and thus rigorously compare theory with experimental results from facilities like the Large Hadron Collider – is severely hampered, potentially obscuring subtle signals of new physics or preventing a complete understanding of known interactions. This limitation necessitates the development of innovative computational techniques to overcome the factorial growth and unlock a more complete picture of the fundamental forces governing the universe.

Finding the Hidden Order: Color, Kinematics, and Relations

Scattering amplitudes, representing the probabilities of particle interactions, exhibit underlying structural properties that have enabled significant advancements in their calculation. Traditional approaches treated these amplitudes as monolithic expressions; however, recognizing that they can be decomposed into color-ordered amplitudes – categorized by the color factors associated with the interacting particles – has proven highly effective. This decomposition leverages the fact that color factors are conserved throughout the calculation, allowing for separate treatment and simplification. By isolating these color-ordered components, calculations become more manageable and reveal relationships between different processes, ultimately reducing the computational complexity required to determine scattering outcomes. \mathcal{A} = \sum_{c} \mathcal{A}_c \mathcal{C}_c , where \mathcal{A} is the full amplitude, \mathcal{A}_c represents the color-ordered amplitude, and \mathcal{C}_c denotes the corresponding color factor.

The computational complexity of scattering amplitudes in quantum field theory is significantly lessened by employing relations such as the Kleiss-Kuijf and BCJ (Bern-Carrasco-Johansson) relations. The Kleiss-Kuijf relation demonstrates that amplitudes can be expressed as sums over color-ordered partial amplitudes, effectively decoupling the color factors from the kinematic factors. The BCJ relation, extending this principle, reveals a recursive structure within amplitudes, allowing for the computation of any n-point amplitude using only a minimal set of independent amplitudes – specifically, n-2 independent amplitudes are sufficient. This reduction in the number of required calculations arises because these relations demonstrate that seemingly independent amplitudes are, in fact, related by specific on-shell recursion rules and color decomposition, avoiding redundant computations and simplifying the overall calculation process.

Scattering amplitudes, while seemingly complex, are demonstrably not arbitrary mathematical functions; they adhere to specific constraints revealed by relations such as the Kleiss-Kuijf and BCJ relations. These relations demonstrate that amplitudes satisfy certain identities, meaning a subset of amplitudes can be expressed in terms of others, effectively reducing the number of independent calculations required. Specifically, the BCJ relation A(1,2,3,...,n) = \sum_{\sigma \in S_n} \frac{1}{2} \text{Ber}(\sigma) A(1, \sigma(2), ..., \sigma(n)) shows that amplitudes can be rewritten by summing over permutations, highlighting a hidden symmetry. This indicates that the space of possible amplitudes is highly structured and constrained, rather than consisting of all possible functions satisfying basic physical requirements.

Reverse Engineering Reality: Symbolic Regression and Amplitude Discovery

Symbolic regression addresses the problem of determining the mathematical expression that best fits a given dataset of scattering amplitudes. Unlike traditional fitting procedures which assume a functional form, symbolic regression automatically searches for the underlying equation directly from the data, effectively performing “reverse engineering” of the amplitude’s analytical form. This is achieved by exploring a large space of potential functions – typically constructed from elementary mathematical operations and variables representing kinematic quantities – and selecting the expression that minimizes a defined error metric. The process is particularly valuable in high-energy physics where analytical expressions for scattering amplitudes can be complex and difficult to derive through conventional methods, and the ability to automatically rediscover or discover new formulas from numerical data is highly desirable.

Symbolic regression algorithms utilize techniques such as Column-Pivoted QR decomposition (CPQR) to efficiently identify and retain only the statistically significant terms within a proposed functional form. CPQR operates by systematically decomposing a matrix representing the candidate terms, prioritizing columns based on their contribution to the overall variance explained. This process effectively ranks terms by relevance, allowing for the discarding of those with negligible impact on the regression. More recently, Neural Networks are being integrated to perform similar filtering and selection of relevant terms, offering increased scalability and the potential to handle more complex functional spaces beyond those readily addressed by traditional linear regression methods like CPQR. Both approaches minimize model complexity and improve the interpretability of the resulting amplitude formulas.

Symbolic regression techniques have demonstrated the capability to reproduce established scattering amplitude formulas from data, notably including the Parke-Taylor formula, which expresses tree-level amplitudes for gluons. Successful reconstruction using these methods has been achieved for amplitudes involving up to five kinematic points, representing a significant milestone in automated formula discovery. Beyond reproducing known results, the approach also holds the potential to identify previously unknown amplitude structures, offering a pathway to expand our understanding of high-energy particle interactions; however, current limitations restrict accurate reconstruction beyond five points.

A Glimmer of Unity: KLT Relations and Data-Driven Discovery

Symbolic regression, a technique employing machine learning to discover mathematical expressions from data, proves instrumental in verifying and extending foundational relationships within theoretical physics, notably the Kawaii-Lewellen-Tye (KLT) relations. These relations, which elegantly connect the amplitudes describing gluon interactions – the carriers of the strong force – with those of gravitons – the hypothetical carriers of gravity – suggest a deep underlying unity between these fundamental forces. By training algorithms on ‘on-shell’ data – data representing physically realizable particle interactions – researchers can effectively rediscover known relationships like the KLT formulas and, crucially, explore potential generalizations beyond existing theoretical frameworks. This data-driven approach offers a powerful complement to traditional analytical methods, potentially revealing hidden structures and accelerating progress towards a more complete understanding of the universe’s fundamental laws.

The Kawai-Lewellen-Tye (KLT) relations represent a profound connection between the amplitudes describing the interactions of gluons – the force carriers of the strong nuclear force – and gravitons, the hypothetical force carriers of gravity. These relations mathematically demonstrate that graviton amplitudes can be expressed in terms of products of gluon amplitudes, suggesting a deep underlying unity between these seemingly disparate forces. This connection isn’t merely a mathematical curiosity; it provides a potential pathway towards constructing a consistent quantum theory of gravity, a long-sought goal in physics. By expressing gravity in terms of gauge theories like the strong force, the KLT relations offer a framework for tackling the complexities of quantum gravity and potentially unifying all fundamental forces within a single, coherent theoretical structure.

A novel computational pipeline has successfully re-established fundamental relationships within high-energy physics using a purely data-driven methodology. This approach, leveraging symbolic regression techniques, not only confirmed the validity of established formulas like the Parke-Taylor formula-a cornerstone of scattering amplitude calculations-but achieved this rediscovery with remarkable efficiency, completing the process in approximately 102 seconds. Critically, the pipeline demonstrated exceptional accuracy, exhibiting a maximum relative error of only 10-16 when tested on previously unseen data, highlighting its potential for uncovering and validating complex relationships in physics beyond current theoretical frameworks. This precision suggests a powerful new tool for exploring the mathematical structure underlying fundamental forces and particles.

The pursuit of simplifying complex expressions through data-driven methods, as demonstrated in this work on scattering amplitudes, feels predictably human. One anticipates the inevitable complications arising when elegant theories meet the realities of production systems – or, in this case, the sheer volume of data. It reminds one of Leonardo da Vinci’s observation: “Simplicity is the ultimate sophistication.” This paper attempts to rediscover known analytical structures – a clever way of phrasing the act of rebuilding something that, while theoretically sound, proves fragile in practice. Better one well-understood, albeit complex, relation than a hundred approximations that break under scrutiny. The application of symbolic regression offers a path to navigate this complexity, but it won’t eliminate it.

What’s Next?

The exercise of rediscovering known physics with algorithmic tools feels, predictably, like building a very elaborate telescope to confirm the sun still rises. The real test lies not in replicating established results – the KLT relations, the CPQR representation – but in the inevitable encounter with structures that resist neat analytical forms. The current framework, while elegant in its automation of symbolic manipulation, skirts the fundamental issue: data, however plentiful, remains a projection of reality, not reality itself. Expect the next generation of these approaches to wrestle with the ambiguities inherent in the training sets, the subtle biases baked into the choice of basis, and the lurking possibility that the ‘new’ relations discovered are simply artifacts of the search algorithm.

The promise of a ‘data-driven’ approach to fundamental physics often implies a reduction in human intuition, a displacement of theoretical insight. But the selection of relevant observables, the design of the initial data sets, and the interpretation of the algorithmic output all remain profoundly human endeavors. One anticipates a future where these methods become less about ‘discovery’ and more about efficient systematization – a powerful tool for navigating the landscape of known physics, but unlikely to conjure truly novel structures from the void. Tests, after all, are a form of faith, not certainty.

The inevitable scaling challenges also loom. Scattering amplitudes grow in complexity with alarming speed. Even with streamlined representations, the computational cost of exploring higher-order corrections, or even moderately complex processes, will quickly become prohibitive. The true measure of success may not be the elegance of the discovered relations, but the pragmatic ability to extract useful predictions from intractable calculations-to make something that doesn’t crash on Mondays.

Original article: https://arxiv.org/pdf/2602.15169.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

See also:

2026-02-18 19:37