Exact results for the entanglement entropy and the energy radiated by a quark (1312.5682v4)

Abstract: We consider a spherical region with a heavy quark in the middle. We compute the extra entanglement entropy due to the presence of a heavy quark both in ${\cal N}=4 $ Super Yang Mills and in the ${\cal N}=6$ Chern-Simons matter theory (ABJM). This is done by relating the computation to the expectation value of a circular Wilson loop and a stress tensor insertion. We also give an exact expression for the Bremsstrahlung function that determines the energy radiated by a quark in the ABJM theory.

Exploration of Entanglement Entropy and Energy Radiation in Gauge Theories

The paper by Aitor Lewkowycz and Juan Maldacena offers an in-depth analysis of entanglement entropy in gauge theories, particularly focusing on configurations involving a heavy quark. The researchers present exact computations within the context of both $\mathcal{N}=4$ Super Yang-Mills and $\mathcal{N}=6$ Chern-Simons matter theories (ABJM), exploiting methodologies that relate these calculations to fundamental geometric observations and the behavior of associated Wilson loops.

Key Results and Methodologies

The analysis begins by investigating the additional entanglement entropy imparted by the presence of a heavy quark within a spherical region. The research utilizes connections between the computation of entanglement entropy and the expectation values of a circular Wilson loop augmented by stress tensor insertions. In conformal field theories (CFTs), the paper argues that this relationship simplifies to computing the entropy in a thermal field theory defined on hyperbolic spaces, particularly at an inverse temperature parameterized by $\beta = 2\pi$.

A significant result presented is the derivation of an exact expression for the Bremsstrahlung function in the ABJM theory, which describes the energy emission by the quark in response to its motion. The paper reveals that this computation spans both weak and strong coupling regimes, capturing the contributions of gauge fields in the process.

Numerical Findings and Analytical Insight

The research delivers notable quantitative results, with expressions for ${\cal N}=4$ SYM entanglement entropy and related quantities. For instance, for ${\cal N}=4$ SYM, the entanglement entropy associated with the Wilson loop is provided by the formula: $$S_W = \left( 1 - \frac{4}{3} \lambda \partial_{\lambda} \right) \log \langle W_{\circ} \rangle$$ where $\langle W_{\circ} \rangle$ denotes the expectation value of the circular Wilson loop. This equation encapsulates behaviors across various coupling strengths $\lambda$, illustrating a sound synthesis of strong and weak coupling insights.

Theoretical and Practical Implications

The theoretical implications of this work are extensive – it informs about the universal aspects of entanglement entropy in CFTs and provides comprehensive insights into the mathematical structure of gauge theories involved. Practically, as the formulation heavily relies on the paradigms such as the AdS/CFT correspondence and localization techniques, it contributes to refining computational methods useful in quantum field theories and condensed matter systems.

Future Directions in Research

The discussion alludes to several future avenues: potential improvements in understanding quantum entanglement on string worldsheets, further exploration using integrability tools in the setting of ABJM theory, and enhancements in theoretical physics to better describe the interaction between accelerated charges and their quantum fields. Comparable entropic analyses in different dimensions or through varied regularization procedures could generate deeper insights into conformal field theories' entropic characteristics.

Overall, this paper stands as a rigorous and expansive work, yielding exact results that bridge theoretical predictions with potentially observable consequences in strongly coupled gauge theories. It sets the stage for further significant explorations into the quantum aspects of gauge theories, particularly through the lens of entropic measures.