The static potential in {\cal N}=4 supersymmetric Yang-Mills at weak coupling

TL;DR

The paper analyzes the weak-coupling static potential for the 1/2 BPS Wilson loop in ${\cal N}=4$ SYM by developing an ultrasoft effective theory (a version of pNRQCD) and a multipole expansion to separate soft ($\sim 1/r$) and ultrasoft ($\sim \lambda/r$) scales. It derives the static singlet energy and potential up to next-to-leading order in the multipole expansion, and resums leading logarithms via renormalization-group running, obtaining $E_s(r)= -\lambda^{1+2\lambda/\pi}/r$ at leading-log accuracy and a next-to-leading correction $E_s(r)= -\frac{2C_F\alpha_s}{r}[1+2\frac{\lambda}{\pi}(\ln(2\lambda)+\gamma_E-1)+\mathcal{O}(\lambda^2)]$. The analysis also computes the one-loop soft potential, identifies the ultrasoft contribution, and demonstrates how the soft/ultrasoft factorization yields finite pieces in the static energy; it further extends to the ordinary Wilson loop, yielding NNLL results with a known coefficient $a_1=N_c/\pi$ and an unknown $a_2$. Overall, the work provides a controlled framework to study the weak-coupling static potential in ${\cal N}=4$ SYM, clarifies the role of massless scalars in the infrared structure, and offers insights relevant to AdS/CFT and potential QCD analogies.

Abstract

We compute the static potential associated to the locally 1/2 BPS Wilson loop in ${\cal N}$=4 supersymmetric Yang-Mills theory with ${\cal O}(λ^2/r)$ accuracy. We also resum the leading logarithms, of ${\cal O}(λ^{n+1}\ln^nλ/r)$, and show the structure of the renormalization group equation at next-to-leading order in the multipole expansion. In order to obtain these results it is crucial the use of an effective theory for the ultrasoft degrees of freedom. We develop this theory up to next-to-leading order in the multipole expansion. Using the same formalism we also compute the leading logarithms, of ${\cal O}(λ^{n+3}\ln^nλ/r)$, of the static potential associated to an ordinary Wilson loop in the same theory.

Figures (1)

• Figure 1: One loop contribution to the singlet propagator. The dotted line represents the scalar field.