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.
Highlights
Motivated by the need to match experimental accuracy, quantum field theory calculations have had to evolve to tackle problems of increasing complexity
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
We investigated and performed the renormalization of massless scalar effective field theories at higher orders in both the number of loops and mass dimension, working at linear order in the EFT operators
Summary
Motivated by the need to match experimental accuracy, quantum field theory calculations have had to evolve to tackle problems of increasing complexity. Explore new boundaries in mass dimension and loops for EFTs of scalar fields, and the structures that appear there and ii) the development of new general techniques to enable the renormalization of EFTs at higher order. Regarding calculations that go to higher EFT dimensions, QCD gluonic operators up to mass dimensions 16 were recently computed using generalized unitarity at two loops [62] Another related problem is the renormalization of spin-N light-cone operators. This represents a mathematically singled out (up to rotations in the space of primaries) basis for the S-matrix, i.e. the number of on-shell physical measurements one can make in a theory We explore how this picture generalizes for off-shell correlation functions and the calculation of quantum corrections; here a larger basis is required at an intermediate stage in the EFT renormalization procedure.
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