Renormalization and non-renormalization of scalar EFTs at higher orders

Abstract

We renormalize massless scalar effective field theories (EFTs) to higher loop orders and higher orders in the EFT expansion. To facilitate EFT calculations with the R* renormalization method, we construct suitable operator bases using Hilbert series and related ideas in commutative algebra and conformal representation theory, including their novel application to off-shell correlation functions. We obtain new results ranging from full one loop at mass dimension twelve to five loops at mass dimension six. We explore the structure of the anomalous dimension matrix with an emphasis on its zeros, and investigate the effects of conformal and orthonormal operators. For the real scalar, the zeros can be explained by a ‘non-renormalization’ rule recently derived by Bern et al. For the complex scalar we find two new selection rules for mixing n- and (n− 2)-field operators, with n the maximal number of fields at a fixed mass dimension. The first appears only when the (n− 2)-field operator is conformal primary, and is valid at one loop. The second appears in more generic bases, and is valid at three loops. Finally, we comment on how the Hilbert series we construct may be used to provide a systematic enumeration of a class of evanescent operators that appear at a particular mass dimension in the scalar EFT.

Five-loop anomalous dimensions of $ϕ^Q$ operators in a scalar theory with $O(N)$ symmetry

“…The same methods may also be applied to other operators, like the ∂ 2m φ n -type operators considered in [34], and the operators in more complicated O(N ) representations considered in [35]. Q is that the superficial degree of UV divergence is 0.…”

Theoretical developments in the SMEFT at dimension-8 and beyond

“…Another manifestation of the above mentioned role that conformal symmetry plays in the structure of EFT operator bases can be seen through the appearance of selection rules in the anomalous dimension matrix of the theory: certain entries are zero only for operator bases consisting of conformal primary operators [26].…”

“…Ref [26] also exemplified the use of Hilbert series in conjunction with off-shell techniques to calculate quantum corrections in EFT. Indeed, state-of-the-art calculations of quantum corrections (e.g.…”

“…Another way in which the above is pertinent to loop calculations is down to the fact that the Hilbert series and operator construction technologies introduced in Ref [8] work in d spacetime dimensions (where a spinor-helicity approach does not in general exist), which is relevant to calculations that employ dimensional regularization. In this direction, evanescent operators (that vanish in integer dimensions, but can nevertheless leave an imprint through loop corrections) can be studied using the above techniques; systematic enumeration of a particular class of evanescent operators using Hilbert series has been presented [26]. This particular class of operators appears at very high mass dimension in the SMEFT; it would be interesting to see if other classes of evanescent operators in the SMEFT, such as those that appear at mass dimension 6 and beyond, can be analyzed systematically with Hilbert series.…”

Renormalization of gluonic leading-twist Operators in covariant Gauges

“…To this end, we apply the R * operation [63][64][65][66], using a formulation that is valid for a general Feynman rule of the inserted operator [67][68][69][70] Z δΓ…”