Anomalous dimensions of leading twist conformal operators

Abstract

We extend and develop a method for perturbative calculations of anomalous dimensions and mixing matrices of leading twist conformal primary operators in conformal field theories. Such operators lie on the unitarity bound and hence are conserved (irreducible) in the free theory. The technique relies on the known pattern of breaking of the irreducibility conditions in the interacting theory. We relate the divergence of the conformal operators via the field equations to their descendants involving an extra field and accompanied by an extra power of the coupling constant. The ratio of the two-point functions of descendants and of their primaries determines the anomalous dimension, allowing us to gain an order of perturbation theory. We demonstrate the efficiency of the formalism on the lowest-order analysis of anomalous dimensions and mixing matrices which is required for two-loop calculations of the former. We compare these results to another method based on anomalous conformal Ward identities and constraints from the conformal algebra. It also permits to gain a perturbative order in computations of mixing matrices. We show the complete equivalence of both approaches.Comment: 21 pages, 4 figures; references adde

Classical equation of motion and anomalous dimensions at leading order

“…Since we eventually find the critical coupling g * = g( ), neglecting the -dependence above will suffice for our purpose. We can alternatively calculate the above correlation functions by using the classical equation of motion as 8) which means γ 1 = O(g 2 ). Next we will further apply the differential operator 2 y in addition to 2 x , 2φ(x)2φ(y) = c2 x 2 y |x − y|…”

“…We start with the correlation function, φ i (x)φ j (y)χ(z) . By the assumption that the fields φ i and χ are the primary fields, the coordinate dependence is completely fixed as 8) where the three-point function above is classically vanishing, then the coefficient is O(g 1 ).…”

𝜖-expansion in critical ϕ3-theory on real projective space from conformal field theory

“…III: From the equations of motion, a particular primary operator in the free theory behaves as a descendant operator at the non-trivial fixed point. The use of the equations of motion was further developed in [6] to derive critical exponents in the critical φ 3 -theory in d = 6 − ǫ dimensions by extending the earlier works [7][8] [9]. See also [10] [11][12] [13] for more applications of the similar method in various conformal field theories (with defects).…”

Duality between Wilson loops and gluon amplitudes

“…In a conformal field theory, this mixing problem can be resolved using conformal methods that significantly reduce the complexity of the calculations [108,135]. We developed these methods further in [139]. As a particular example, we were able to determine the mixing matrix c jk at order g 2 by evaluating only order g Feynman graphs, compared to a conventional calculation at order g 4 .…”