An exact formula for the radiation of a moving quark in N=4 super Yang Mills

TL;DR

The paper derives an exact, all-coupling expression for the Bremsstrahlung function B in N=4 SYM by linking the small-angle cusp of locally BPS Wilson loops to the exactly computable circular Wilson loop via supersymmetric localization. It shows B = (1/2π^2) λ ∂_λ log ⟨W_circle⟩ (with appropriate U(N)/SU(N) adjustments) and connects this to the energy radiated by a moving quark and to the two-point function of the displacement operator on the Wilson line. By analyzing near-BPS and generalized cusp configurations, the authors express the leading deviations in terms of B evaluated at a shifted coupling, establishing a unified framework for several observables in conformal line defects. The work also bridges localization results with integrability (TBA) approaches and highlights exact weak/strong coupling limits, with cross-checks and broader implications for Wilson-loop dynamics in holographic contexts.

Abstract

We derive an exact formula for the cusp anomalous dimension at small angles. This is done by relating the latter to the computation of certain 1/8 BPS Wilson loops which was performed by supersymmetric localization. This function of the coupling also determines the power emitted by a moving quark in N=4 super Yang Mills, as well as the coefficient of the two point function of the displacement operator on the Wilson loop. By a similar method we compute the near BPS expansion of the generalized cusp anomalous dimension.

Figures (5)

• Figure 1: ( a) A Wilson line that makes a turn by an angle $\phi$. ( b) Under the plane to cylinder map, the same line is mapped to a quark anti-quark configuration. The quark and antiquark are sitting at two points on $S^3$ at a relative angle of $\pi - \phi$. Of course, they are extended along the time direction.
• Figure 2: Plot of the Bremsstrahlung function $B$ in the planar limit (solid blue curve). At weak coupling, the lower and upper dashed green curves denote the two- and three-loop approximation, respectively. It is interesting to note that the radius of convergence of the weak coupling expansion is given by the first zero of $I_1$ in (\ref{['bplanar']}), which is at $\lambda \sim - 14.7$. As one can see in the plot, the perturbative formulas become unreliable in that region. At the same time, we see that the first two orders of the strong coupling result (red dotted curve) give a qualitatively good approximation starting from that region.
• Figure 3: General class of 1/8 BPS Wilson loops. The loop lies on a two sphere $S^2$. The vector that specifies the scalar couplings can be viewed as the unit vector on $R^3$ that is orthogonal to the contour and lies on the two sphere, denoted here at a point by $\vec{m}$. The two sphere is divided in two regions, one with area $A_1$ and the other with area $A_2$.
• Figure 4: ( a) Contour with two longitude lines separated by an angle $\pi -\phi$. ( b) The same contour on the cylinder.
• Figure 5: Contour we consider for the argument. We start from two longitude segments and we deformed one of them. The deformed contour goes back to the old one near the poles. The deformation is constant along most of the longitude line. In ( a) we see the diagram on the sphere. In ( b) we see the diagram on the cylinder.