We study substructures of the Weyl group of conformal transformations of the metric of (pseudo)Riemannian manifolds. These substructures are identified by differential constraints on the conformal factors of the transformations which are chosen such that their composition is associative. Mathematically, apart from rare exceptions, they are partial associative groupoids, not groups, so they do not have an algebra of infinitesimal transformations, but this limitation can be partially circumvented using some of their properties cleverly. We classify and discuss the substructures with two-derivatives differential constraints, the most famous of which being known as the harmonic or restricted Weyl group in the physics literature, but we also show the existence of a lightcone constraint which realizes a proper subgroup of the Weyl group. We then show the physical implications that come from invariance under the two most important substructures, concentrating on classical properties of the energy-momentum tensor and a generalization of the quantum trace anomaly. We also elaborate further on the harmonic substructure, which can be interpreted as partial gauge fixing of full Weyl invariance using BRST methods. Finally, we discuss how to construct differential constraints of arbitrary higher-derivative order and present, as examples, generalizations involving scalar constraints with four and six derivatives.

Italy

Riccardo Martini

Gregorio Paci

Dario Sauro

Gian Vacca

Omar Zanusso

Substructures of the Weyl group and their physical applications

We study multifield extensions of Reggeon Field Theory (also equivalent to Directed Percolation model) at criticality in the perturbative ϵ-expansion below the upper critical dimension D c = 4 at one loop, for the special case when all fields have the same scale (anomalous) dimensions. Analyzing all the fixed points of the renormalization group flow for N = 2 flavors and some for N = 3, we find scale invariant solutions which are characterized by specific emergent global symmetries of the interacting potential. We also study two infinite families as a function of N having ℤ N  and S  N+1 symmetries.

Germany

Jochen Bartels

Chile

Carlos Contreras

Emergent symmetries at criticality in multi field RFT/DP

Energy momentum tensors of higher-derivative free scalar conformal field theories in flat spacetime are discussed. Two algorithms for the computation of energy momentum tensors are described, which accomplish different goals: the first is brute-force and highlights the complexity of the energy momentum tensors, while the second displays some features of their geometric origin as variations of Weyl invariant curved-space actions. New compact expressions for energy momentum tensors are given and specific obstructions to defining them as conformal primary operators in some spacetime dimensions are highlighted. Our discussion is also extended to higher-derivative free spinor theories, which are based on higher-derivative generalizations of the Dirac action and provide interesting examples of conformal field theories in dimension higher than two.

United Kingdom

Andreas Stergiou

Weyl covariance and the energy momentum tensors of higher-derivative free conformal field theories

The critical behavior of infinite families of shift symmetric interacting theories with higher derivative kinetic terms (non unitary) is considered. Single scalar theories with shift symmetry are classified according to their upper critical dimensions and studied at the leading non trivial order in perturbation theory. For two infinite families, one with quartic and one with cubic interactions, beta functions, criticality conditions and universal anomalous dimensions are computed. At the order considered, the cubic theories enjoy a one loop non renormalization of the vertex, so that the beta function depends non trivially only on the anomalous dimension. The trace of the energy momentum tensor is also investigated and it is shown that these two families of QFTs are conformally invariant at the fixed point of the RG flow.

Mahmoud Safari

United States

Scale and conformal invariance in higher derivative shift symmetric theories

We discuss the possibility to define exact RG equations for a UV regulated Wilsonian action based on a proper time (PT) regulator function. We start from a functional mapping which shows how each particular flow equation (and RG scheme) is associated to infinitely many scale dependent field redefinitions, which are related to specific coarse-graining procedures. On specializing to a sub-family of one parameter PT regulators we briefly analyze few results for the Ising Universality class in three dimensions, obtained within a second order truncation in the derivative expansion of the Wilsonian action.

Alfio Bonanno

Stefan Lippoldt

Roberto Percacci

On exact proper time Wilsonian RG flows

Starting from the leading Odderon solution of the three gluon system in perturbative QCD we introduce, as a first step towards the transition to the non-perturbative region, an infrared cutoff and use the running QCD coupling constant. In our numerical analysis we find that the fixed cut solution with intercept one persists, hinting at a physical Odderon with intercept one and a small t-slope.

The Odderon in QCD with running coupling

In this paper we encode the perturbative BFKL leading logarithmic resummation, relevant for the Regge limit behavior of QCD scattering amplitudes, in the IR-regulated effective action which satisfies exact functional renormalization group equations. This is obtained using a truncation with a specific infinite set of non local vertices describing the multi-Regge kinematics (MRK). The goal is to use this framework to study, in the high energy limit and at larger transverse distances the transition to a much simpler effective local reggeon field theory, whose critical properties were recently investigated in the same framework. We perform a numerical analysis of the spectrum of the BFKL Pomeron deformed by the introduction of a Wilsonian infrared regulator to understand the properties of the leading poles (states) contributing to the high energy scattering.

A functional RG approach for the BFKL Pomeron

We consider the family of renormalizable scalar QFTs with self-interacting potentials of highest monomial ϕ m below their upper critical dimensions d c = 2 m m − 2 $$ {d}_c=\frac{2m}{m-2} $$ , and study them using a combination of CFT constraints, Schwinger-Dyson equation and the free theory behavior at the upper critical dimension. For even integers m ≥ 4 these theories coincide with the Landau-Ginzburg description of multi-critical phenomena and interpolate with the unitary minimal models in d = 2, while for odd m the theories are non-unitary and start at m = 3 with the Lee-Yang universality class. For all the even potentials and for the Lee-Yang universality class, we show how the assumption of conformal invariance is enough to compute the scaling dimensions of the local operators ϕ k and of some families of structure constants in either the coupling’s or the ϵ -expansion. For all other odd potentials we express some scaling dimensions and structure constants in the coupling’s expansion.

Denmark

Alessandro Codello

Leading CFT constraints on multi-critical models in d &gt; 2

We extend our prescription for the construction of a covariant and background-independent effective action for scalar quantum field theories to the case where momentum modes below a certain scale are suppressed by the presence of an infrared regulator. The key step is an appropriate choice of the infrared cutoff for which the Ward identity, capturing the information from single-field dependence of the ultraviolet action, continues to be exactly solvable, and therefore, in addition to covariance, manifest background independence of the effective action is guaranteed at any scale. A practical consequence is that in this framework one can adopt truncations dependent on the single total field. Furthermore we discuss the necessary and sufficient conditions for the preservation of symmetries along the renormalization group flow.