Gluon scattering in ${\cal N}=4$ Super Yang-Mills at finite temperature

TL;DR

This work extends the Alday-Maldacena framework to finite temperature, defining gluon scattering amplitudes in N=4 SYM via a lightlike Wilson loop at the horizon of the T-dual AdS black hole. It shows that at T>0 the gluon amplitude and the Wilson loop are no longer equivalent observables and provides a concrete forward 4-point amplitude calculation using both cutoff and generalized dimensional regularization, with the result depending on $L_-/T$ and $L_{2-}/T$ rather than Mandelstam variables. The analysis reveals IR divergences governed by the horizon and highlights a lack of a smooth $T\to 0$ limit, suggesting a link to transport properties like viscosity while noting the need for broader kinematic coverage. The appendix consolidates the standard Wilson loop treatments and the Penrose diagram structure to underpin the geometric interpretation of the finite-temperature observables.

Abstract

We extend the AdS/CFT prescription of Alday and Maldacena to finite temperature $T$, defining the amplitude for gluon scattering in ${\cal N}=4$ Super Yang-Mills at strong coupling from string theory. It is defined by a lightlike ''Wilson loop'' living at the horizon of the T-dual to the black hole in AdS space. Unlike the zero temperature case, this is different from the Wilson loop contour defined at the boundary of the AdS black hole metric, thus at finite $T$ there is no relation between gluon scattering amplitudes and the Wilson loop. We calculate the amplitude at strong coupling for forward scattering of a low energy gluon ($E<T$) off a high energy gluon ($E\gg T$) in both cut-off and generalized dimensional regularization. The generalized dimensional regularization is defined in string theory as an IR modified dimensional reduction. For this calculation, the corresponding usual Wilson loop is related to the jet quenching parameter of the finite temperature ${\cal N}=4$ SYM plasma, while the gluon scattering amplitude is related to the viscosity coefficient.

Figures (9)

• Figure 1: Contours for endpoints of worldsheets. a) Contour for the Wilson loop giving $V_{q\bar{q}}(L)$, with $T\gg L$. b) Contour for the calculation of the Wilson loop giving the jet-quenching parameter, with $L_-\gg L$. c) Contour for the calculation of the gluon amplitude, with $L_-\gg L_{2-}$. It is a lightlike polygon with rectangular projection on the spatial $(y_1,y_2)$ plane
• Figure 2: $L_{2-}/r_0$ as a function of $a$
• Figure 3: $L_{2-}/r_0$ as a function of $a$
• Figure 4: Integral in S, divided by $\sqrt{a^4+1}$, as a function of $a$, for $\epsilon=0.01$
• Figure 5: Integral in S, divided by $\sqrt{a^4+1}$, as a function of $a$, for $\epsilon=0.01$