Abstract
We perform a careful study of the infrared sector of massless non-abelian gauge theories in four-dimensional Minkowski spacetime using the covariant phase space formalism, taking into account the boundary contributions arising from the gauge sector of the theory. Upon quantization, we show that the boundary contributions lead to an infinite degeneracy of the vacua. The Hilbert space of the vacuum sector is not only shown to be remarkably simple, but also universal. We derive a Ward identity that relates the n-point amplitude between two generic in- and out-vacuum states to the one computed in standard QFT. In addition, we demonstrate that the familiar single soft gluon theorem and multiple consecutive soft gluon theorem are consequences of the Ward identity.
Highlights
Infrared (IR) divergences present in the scattering matrix elements of gauge and gravitational theories have long been known to physicists [1, 2], and numerous attempts in the 1970s and 1980s have been made to render the scattering matrix elements in such theories IR finite.1 A key idea in these approaches is to use modified asymptotic states to define the scattering matrix, wherein the charged external states are dressed with a coherent state of soft photons [3,4,5,6,7,8,9]
We perform a careful study of the infrared sector of massless non-abelian gauge theories in four-dimensional Minkowski spacetime using the covariant phase space formalism, taking into account the boundary contributions arising from the gauge sector of the theory
Many may argue that there is no need for an IR finite S-matrix when the inclusive cross-section is IR finite, advances in our understanding of how the soft theorems of quantum field theories (QFTs) are related to asymptotic symmetries have brought newfound appreciation for what the IR divergences in the S-matrix elements signify
Summary
Infrared (IR) divergences present in the scattering matrix elements of gauge and gravitational theories have long been known to physicists [1, 2], and numerous attempts in the 1970s and 1980s have been made to render the scattering matrix elements in such theories IR finite. A key idea in these approaches is to use modified asymptotic states to define the scattering matrix, wherein the charged external states are dressed with a coherent state of soft (low energy) photons [3,4,5,6,7,8,9]. As was shown in [19, 33], one can obtain an IR finite S-matrix after incorporating this infinite degeneracy and the corresponding in- and out-states are the coherent states constructed in [3, 5,6,7,8,9] This infinite degeneracy in gauge (and gravitational) theories may seem surprising at first, its existence can be deduced from a careful but straightforward application of the covariant phase space formalism [34,35,36,37,38] to gauge theories. We will perform an analysis of the phase space of gauge theories, study the corresponding Hilbert space, including the infinite-dimensional vacuum degeneracy, and derive a factorization formula for the scattering matrix element between any two vacuum states in the Hilbert space. We show how the leading soft gluon theorem involving a single soft gluon as well as multiple consecutive soft gluons are consequences of the Ward identity
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