Asymptotic Symmetries of Yang-Mills Theory (1308.0589v2)

Abstract: Asymptotic symmetries at future null infinity (I+) of Minkowski space for electrodynamics with massless charged fields, as well as non-Abelian gauge theories with gauge group G, are considered at the semiclassical level. The possibility of charge/color flux through I+ suggests the symmetry group is infinite-dimensional. It is conjectured that the symmetries include a G Kac-Moody symmetry whose generators are "large" gauge transformations which approach locally holomorphic functions on the conformal two-sphere at I+ and are invariant under null translations. The Kac-Moody currents are constructed from the gauge field at the future boundary of I+. The current Ward identities include Weinberg's soft photon theorem and its colored extension.

• The paper introduces an infinite-dimensional G Kac-Moody symmetry at null infinity derived from large gauge transformations.
• It constructs Kac-Moody currents from asymptotic gauge fields, providing a framework that reformulates soft photon and gluon theorems as Ward identities.
• It connects these symmetries to Weinberg's soft theorems, suggesting new constraints on scattering amplitudes in gauge theories.

Asymptotic Symmetries of Yang-Mills Theory: A Semiclassical Perspective

In the paper "Asymptotic Symmetries of Yang-Mills Theory," asymptotic symmetries at future null infinity of Minkowski space are analyzed within the field of electrodynamics with massless charged fields and non-Abelian gauge theories with gauge group $G$. The investigation centers on the semiclassical level, proposing an extension of the symmetry group to include an infinite-dimensional $G$ Kac-Moody symmetry. This proposed symmetry emerges from "large" gauge transformations that manifest as locally holomorphic functions on the conformal two-sphere, preserving invariance under null translations.

The paper explores a potential connection between the proposed asymptotic symmetries and Weinberg's soft photon theorem and its colored extensions. These Ward identities serve as a crucial link, suggesting that the infinite-dimensional symmetries could yield important constraints on scattering amplitudes involving soft photons or gluons. The Kac-Moody currents are constructed from the asymptotic behavior of the gauge fields, providing a pathway to recovering known results like Weinberg's relations in this novel framework.

Key Findings and Results

1. Infinite-dimensional Symmetries: It is postulated that the asymptotic symmetry group at null infinity is infinite-dimensional, including a $G$ Kac-Moody symmetry. This is analogous to the original BMS symmetry for gravity, but extended to accommodate charge and color fluxes in gauge theories.
2. Large Gauge Transformations: These transformations, which asymptotically behave as holomorphic functions on the conformal sphere, generate an extended symmetry group. Such transformations have profound implications for the scattering of massless charged fields, as they encapsulate the possibility of inserting "boundary photons" or "boundary gluons."
3. Kac-Moody Current Construction: The paper elaborates on constructing the Kac-Moody currents from the asymptotic gauge fields at future null infinity. These currents are shown to transform under large gauge transformations, revealing a structure consistent with a Kac-Moody algebra.
4. Soft Theorems as Ward Identities: A pivotal correlation is drawn between the asymptotic symmetries and Weinberg's soft theorems. The frequency-space expressions of these theorems for photons and gluons can be reformulated in terms of current algebra on the conformal sphere, emphasizing their nature as Ward identities.

Theoretical and Practical Implications

The theoretical implications are significant, suggesting that the proposed asymptotic symmetries could lead to a deeper understanding of scattering processes in gauge theories. Specifically, the non-Abelian extension identifies a symmetry that might act on color fluxes. Practically, these insights could provide new methods for calculating corrections to amplitudes involving soft emissions, essential for high-energy physics and precision tests of QCD.

Future Developments

Future research could extend this framework beyond the semiclassical regime, exploring quantum corrections and their impact on the proposed symmetry structure. Additionally, investigating how these symmetries integrate with or modify string theory scattering processes might yield new insights, particularly regarding the realizations of these symmetries in specific string compactifications.

In summary, this paper presents a comprehensive examination of the asymptotic symmetries in electrodynamics and non-Abelian gauge theories, providing an advanced understanding of the potential implications of infinite-dimensional symmetries in gauge theories and their connection to established results like Weinberg's soft theorems.