Abstract
We study the fixed point that controls the IR dynamics of QED in d = 4 − 2ϵ dimensions. We derive the scaling dimensions of four-fermion and bilinear operators beyond leading order in the ϵ-expansion. For the four-fermion operators, this requires the computation of a two-loop mixing that was not known before. We then extrapolate these scaling dimensions to d = 3 to estimate their value at the IR fixed point of QED3 as function of the number of fermions Nf . The next-to-leading order result for the four-fermion operators corrects significantly the leading one. Our best estimate at this order indicates that they do not cross marginality for any value of Nf , which would imply that they cannot trigger a departure from the conformal phase. For the scaling dimensions of bilinear operators, we observe better convergence as we increase the order. In particular, the ϵ-expansion provides a convincing estimate for the dimension of the flavor-singlet scalar in the full range of Nf .
Highlights
The conformal bootstrap approach to derive bounds on the scaling dimensions of some monopole operators
In order to assess the reliability of the -expansion in QED, and to improve the estimates from the one-loop extrapolations, it is desirable to extend the calculation of these anomalous dimensions beyond leading order in
The rest of the paper is organized as follows: in section 2 we set up our notation and describe the fixed point of QED in d = 4 − 2 ; in section 3 we present the computation of the two-loop anomalous dimension matrix (ADM) of the four-fermion operators, and the result for their scaling dimension at the IR fixed point in d = 4 − 2 ; in section 4 we present the same result for the bilinear operators; in section 5 we extrapolate the scaling dimensions to d = 3, and plot the resulting dimensions as a function of Nf for the various operators we consider; in section 6 we present our conclusions and discuss possible future directions
Summary
To compute the anomalous dimension of local operators Oi, we add these operators to the Lagrangian. We first present the computation of the two-loop anomalous dimension of the four-fermion operators and use it to obtain the O( 2) IR scaling dimension at the fixed point. The additional contribution that cancels this basis-dependence originates from the O( ) term γ(1,−1) in the one-loop ADM. Such O( ) terms are induced in every scheme that contains finite renormalizations, such as the flavor scheme. There are a few non-trivial ways of partially testing the correctness of the two-loop results: i) We performed all computations in general Rξ gauge This allowed us to explicitly check that the mixing of gauge-invariant operators does not depend on ξ. A reader more interested in the results for the scaling dimensions may proceed directly to section 3.4
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