Anomalous dimensions at large charge for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>×</mml:mo><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> theory in three and four dimensions
I. Jack, D.R.T. Jones
Abstract
Recently it was shown that the scaling dimension of the operator ${\ensuremath{\phi}}^{n}$ in $\ensuremath{\lambda}(\overline{\ensuremath{\phi}}\ensuremath{\phi}{)}^{2}$ theory may be computed semiclassically at the Wilson-Fisher fixed point in $d=4\ensuremath{-}\ensuremath{\epsilon}$, for generic values of $\ensuremath{\lambda}n$, and this was verified to two loop order in perturbation theory at leading and subleading $n$. This result was subsequently generalized to operators of fixed charge $Q$ in $O(N)$ theory and verified up to four loops in perturbation theory at leading and subleading $Q$. More recently, similar semiclassical calculations have been performed for the classically scale-invariant $U(N)\ifmmode\times\else\texttimes\fi{}U(N)$ theory in four dimensions, and verified up to two loops, once again at leading and subleading $Q$. Here we extend this verification to four loops. We also consider the corresponding classically scale-invariant theory in three dimensions, similarly verifying the leading and subleading semiclassical results up to four loops in perturbation theory.