Free energy on the sphere for non-abelian gauge theories
Fabiana De Cesare, Lorenzo Di Pietro, Marco Serone
Abstract
A bstract We compute the S d partition function of the fixed point of non-abelian gauge theories in continuous d , using the ϵ -expansion around d = 4. We illustrate in detail the technical aspects of the calculation, including all the factors arising from the gauge-fixing procedure, and the method to deal with the zero-modes of the ghosts. We obtain the result up to NLO, i.e. including two-loop vacuum diagrams. Depending on the sign of the one-loop beta function, there is a fixed point with real gauge coupling in d > 4 or d < 4. In the first case we extrapolate to d = 5 to test a recently proposed construction of the UV fixed point of 5 d SU(2) Yang-Mills via a susy-breaking deformation of the E 1 SCFT. We find that the F theorem allows the proposed RG flow. In the second case we extrapolate to d = 3 to test whether QCD 3 with gauge group SU( n c ) and n f fundamental matter fields flows to a CFT or to a symmetry-breaking case. We find that within the regime with a real gauge coupling near d = 4 the CFT phase is always favored. For lower values of n f we compare the average of F between the two complex fixed points with its value at the symmetry-breaking phase to give an upper bound of the critical value $$ {n}_f^{\ast } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>n</mml:mi> <mml:mi>f</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:math> below which the symmetry-breaking phase takes over.