Gluonic evanescent operators: two-loop anomalous dimensions

Abstract

Evanescent operators are a special class of operators that vanish in four-dimensional spacetime but are non-zero in d = 4−2ϵ dimensions. In this paper, we continue our systematic study of the evanescent operators in the pure Yang-Mills theory and focus on their two-loop renormalization. We develop an efficient strategy to compute the two-loop divergences of form factors of high-dimensional and high-length operators by combining the d-dimensional unitarity method and the improved tensor reduction techniques. Two-loop anomalous dimensions are obtained for the dimension-10 basis in the planar YM theory, for which both the $$ \overline{\textrm{MS}} $$ MS ¯ scheme and the finite-renormalization scheme are used. We verify that the two-loop anomalous dimensions are the same in these two schemes at the Wilson-Fisher conformal fixed point. Our computation shows that the evanescent operators are indispensable in order to obtain the correct two-loop anomalous dimensions. This work provides a first computation of the two-loop anomalous dimensions of the complete set of dimension-10 operators. The method we use is also expected to provide an efficient strategy for the two-loop renormalization of general high-dimensional operators.

Is Yang-Mills Theory Unitary in Fractional Spacetime Dimensions?

“…Second, for the ADs, the operator mixing matrix will enlarge in general because of the appearance of new operators containing matter fields. A preliminary study of the YM coupled to scalars shows that the complex anomalous dimensions persist; details will be presented in [18]. It would be interesting to check this for general gauge theory models.…”

“…We mention that in principle one can compute the Gram matrix elements in (2) by Wick contraction, which is however cumbersome for high dimensional operators. Instead, we develop an efficient method for computing the Gram matrix based on form factors; details will be given in [18].…”

“…This kind of match is not a general feature though; for example, the relation breaks down for the dimension-12 length-5 operators where there are 8 negative-norm states but only 14 complex ADs. More details will be given in [18].…”

Double copy for tree-level form factors. Part II. Generalizations and special topics

“…Note that tr f (−+) or tr f (−−+) are zero, which explains why the MHV case is simple 23. A systematic construction of gluonic evanescent operators in YM theory, as well as their relation to D-dimensional form factors, has been recently studied in[20,21].…”