Hidden zeros for particle/string amplitudes and the unity of colored scalars, pions and gluons (2312.16282v3)

Abstract: Recent years have seen the emergence of a new understanding of scattering amplitudes in the simplest theory of colored scalar particles - the Tr$(\phi^3)$ theory - based on combinatorial and geometric ideas in the kinematic space of scattering data. In this paper we report a surprise: far from the toy model it appears to be, the ''stringy'' Tr$(\phi^3)$ amplitudes secretly contain the scattering amplitudes for pions, as well as non-supersymmetric gluons, in any number of dimensions. The amplitudes for the different theories are given by one and the same function, related by a simple shift of the kinematics. This discovery was spurred by another fundamental observation: the tree-level Tr$(\phi^3)$ field theory amplitudes have a hidden pattern of zeros when a special set of non-planar Mandelstam invariants is set to zero. Furthermore, near these zeros, the amplitudes simplify, by factoring into a non-trivial product of smaller amplitudes. Remarkably the amplitudes for pions and gluons are observed to also vanish in the same kinematical locus. These properties further generalize to the ''stringy'' Tr$(\phi^3)$ amplitudes. There is a unique shift of the kinematic data that preserves the zeros, and this shift is precisely the one that unifies colored scalars, pions, and gluons into a single object. We will focus in this paper on explaining the hidden zeros and factorization properties and the connection between all the colored theories, working for simplicity at tree-level. Subsequent works will describe this new formulation for the Non-linear Sigma Model and non-supersymmetric Yang-Mills theory, at all loop orders.

• The paper demonstrates that hidden zero patterns in tree-level Tr(ϕ^3) amplitudes unify scattering processes for colored scalars, pions, and gluons.
• It employs kinematic transformations and the geometry of associahedra to reveal factorization and vanishing amplitude properties under specific Mandelstam invariant conditions.
• The research extends these findings to stringy amplitudes via continuous deformation, suggesting a universal kinematic symmetry across diverse particle theories.

An Exploration of Hidden Zero Patterns in Scattering Amplitudes

The paper of scattering amplitudes reveals a surprising unity among different theories of colored particles, guided by insights from combinatorial and geometric ideas applied to kinematic spaces. Recently, research has uncovered a striking revelation: the amplitudes of the Tr$(\phi^3)$ theory, previously considered a rudimentary model, incorporate the scattering amplitudes for both pions and non-supersymmetric gluons across any dimensional framework. A central component of this discovery involves hidden zero patterns within tree-level Tr$(\phi^3)$ field theory amplitudes, harnessed by intricate connections to associahedra within the kinematic space.

This paper explores the emergence and implications of these hidden zero patterns in scattering amplitudes for colored particles. At its core, the Tr$(\phi^3)$ theory, describing colored scalar particles, unexpectedly aligns with the amplitudes of more intricate theories such as pions and gluons. This alignment is depicted through a kinematic transformation that preserves these hidden zeros. The focal point of this paper lies in exploring the zeros hidden within tree-level Tr$(\phi^3)$ amplitudes and demonstrating their broader relevance across related theories.

Zero Patterns and Factorization in Scattering Amplitudes

Numerous key insights emerge when investigating the hidden zero patterns within Tr$(\phi^3)$ amplitudes. Notably, the amplitudes vanish when particular non-planar Mandelstam invariants are nullified. These invariant zeros, not directly visible in Feynman diagrams, become distinctly apparent when viewed through the lens of the associahedron concept derived from causal diamonds in kinematic space. This geometric viewpoint not only reveals the zeros but also elucidates a factorization property near these zero loci, leading to the decomposition of the amplitude into smaller parts. This occurs due to a unique degeneracy in the associahedron structure as one approaches these zeros.

Remarkably, the zero patterns identified within the Tr$(\phi^3)$ theory expand and are observed in the amplitude structures of pions and gluons. This universality across theories signals a profound unity between them, suggestive of a shared underlying structure or symmetry in their kinematic representations that transcends individual theoretical frameworks.

Generalization to Stringy Amplitudes

Expanding upon these foundational insights, the zeros and factorization properties extend to stringy interpretations of Tr$(\phi^3)$ amplitudes. By utilizing a positive parametrization and examining the integration of canonical forms over an associahedron space, the authors illustrate the preservation of these zeros and their associated factorizations even within the string framework. This is achieved via a continuous deformation, represented by a parameter shift in the kinematic data, which intriguingly unifies the colored scalars, pions, and gluons into a singular, cohesive functional form, hence depicting a profound kinematic symmetry encompassing a broader spectrum of particle theories.

Practical and Theoretical Implications

The existence and generalization of the hidden zeros in particle amplitudes open new avenues for theoretical physics and potential practical applications. The scalability and universality of these zeros allow a refined understanding of scattering phenomena across different physical systems, facilitating the derivation of more accurate models in high-energy physics and possibly providing insights into new symmetry structures within theoretical frameworks. The revealed connection between traditionally distinct particle amplitudes emphasizes the importance of geometric and combinatorial methods in advancing our understanding of particle interactions at both finite and infinite energy scales.

Conclusion

The exploration into the hidden zero patterns uncover novel insights into the structure of scattering amplitudes for colored particles. Through a distinct kinematic transformation and a deep understanding of associahedra, this paper establishes a unifying framework for Tr$(\phi^3)$, Non-linear Sigma Model (NLSM), and Yang-Mills (YM) theories, highlighting shared geometric features and fostering new interpretations of amplitude behaviors at both theoretical and practical levels. This work paves the way for further developments in exploring the geometric properties of scattering processes and enhancing the predictive power of theoretical models in particle physics.