Geometry of soft scalars at one loop (2504.12371v1)

Abstract: We extend the soft theorems for scattering amplitudes of scalar effective field theories to one-loop order. Our analysis requires carefully accounting for the fact that the soft limit is not guaranteed to commute with evaluating IR-divergent loop integrals; new results for the soft limit of general scalar one-loop integrals are presented. The geometric soft theorem remains unmodified for any derivatively-coupled scalar effective field theory, and we conjecture that this statement holds to all orders. In contrast, the soft theorem receives nontrivial corrections in the presence of potential interactions, analogous to the case of non-Abelian gauge theories. We derive the universal leading-order correction to the scalar soft theorem arising from potential interactions at one loop. Explicit examples are provided that illustrate the general results.

One-Loop Scalar Soft Theorems in Effective Field Theories

The paper explores the extension of scalar soft theorems in effective field theories (EFTs) to one-loop order, emphasizing derivatively-coupled theories and those with potential interactions. Soft theorems characterize universal behaviors of scattering amplitudes as external particle energies become small. These theorems hold significant predictive power in quantum field theory (QFT) by relating complex higher-order processes to more straightforward ones.

Key Contributions

The authors anchor their discussion in the geometric formulation of EFTs, where fields reside on curved field spaces endowed with specific geometry. The main contributions of the paper include:

1. Geometric Soft Theorems at One-Loop: For scalar theories devoid of potential interactions, the geometric soft theorems are shown to remain unaltered at one-loop level. Utilizing dimensional regularization, the authors argue that IR divergences are absent in derivatively-coupled theories, allowing the soft limit to commute with loop integrations. They demonstrate this by employing a basis in Riemann normal coordinates to ensure integrand covariance and facilitate analysis.
2. Soft Limit of IR-Divergent Integrals: The paper addresses the soft limit of one-loop integrals, illustrating non-commutation scenarios between integration and soft limit evaluation. This insight is crucial for understanding why derivatively-coupled theories permit unchanged soft theorems even when quantum corrections are considered.
3. Theories with Potential Interactions: The extension to theories with potential interactions reveals necessary modifications to the soft theorem at one-loop order. The authors focus on cubic ($\phi^3$) and quartic ($\phi^4$) potential interactions. The analytic structure of such loop integrals mandates considering IR effects that scale logarithmically, thus modifying the predictions of tree-level theorems. They derive and validate a universal leading-order correction to the scalar soft theorem, capturing primary quantum corrections.
4. Examples and Applications: Through concrete examples and explicit calculations of one-loop integrals, the paper corroborates its theoretical claims. These include analyses of bubble, triangle, and box-type loop integrals within different classes of theories.

Implications and Future Directions

This work amplifies the understanding of soft theorems beyond the classical field, touching upon intricate aspects of quantized fields. The methodological advancement allows for potential applications across various domains where scalar fields play central roles, from particle physics to cosmology. Moreover, understanding quantum corrections to soft limits enriches foundational aspects of QFT, including symmetries and IR behavior.

Future research could extend these results to theories including massive states, fermions, or gauge fields, examining if the derived one-loop corrections retain their uniformity under such conditions. Such studies would contribute to the ongoing pursuit of unraveling deeper symmetries and possibly new structures within quantum theories. Furthermore, bridging these results with contemporary interpretations like celestial holography could provide novel insights into the nature of effective theories in higher-dimensional frameworks.

In conclusion, this paper marks a significant step in understanding scalar effective field theories at the loop level, detailing both the challenges and solutions when extending soft theorems into the quantum regime. By doing so, it sharpens theoretical tools crucial for analyzing and predicting the behavior of elementary processes across physics.