We revisit the construction of the renormalized trace Θ of the Energy-Momentum tensor in the four-dimensional <math><mi>λ</mi><msup><mrow><mi>ϕ</mi></mrow><mrow><mn>4</mn></mrow></msup></math> theory, using dimensional regularization in <math><mi>d</mi><mo>=</mo><mn>4</mn><mo>−</mo><mi>ε</mi></math> dimensions. We first construct several basic correlators such as <math><mo>〈</mo><msup><mrow><mi>ϕ</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>ϕ</mi><mi>ϕ</mi><mo>〉</mo></math>, <math><mo>〈</mo><msup><mrow><mi>ϕ</mi></mrow><mrow><mn>4</mn></mrow></msup><mi>ϕ</mi><mi>ϕ</mi><mo>〉</mo></math> to order <math><msup><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msup></math> and from these the correlators <math><mo>〈</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>I</mi></mrow></msub><mi>ϕ</mi><mi>ϕ</mi><mo>〉</mo></math> and <math><mo>〈</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>I</mi></mrow></msub><msub><mrow><mi>K</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>〉</mo></math> with <math><msub><mrow><mi>K</mi></mrow><mrow><mi>I</mi></mrow></msub></math> the basis of dimension d operators. We then match the limit of their expressions on the Wilson-Fisher fixed point to the corresponding expressions obtained in Conformal Field Theory. Then, using the 3-point function <math><mo>〈</mo><mi>Θ</mi><mi>ϕ</mi><mi>ϕ</mi><mo>〉</mo></math>, we construct the operator Θ as a certain linear combination of the basis operators, using the requirements that Θ should vanish on the fixed point and that it should have zero anomalous dimension. Finally, we compute the 2-point function <math><mo>〈</mo><mi>Θ</mi><mi>Θ</mi><mo>〉</mo></math> and we show that it obeys an eigenvalue equation that gives additional information about the internal structure of the Energy-Momentum tensor operator to what is already contained in its Callan-Symanzik equation.

Greece

Nikos Irges

Leonidas Karageorgos

Energy-Momentum tensor correlators in theory I: The spin-zero sector

We introduce a discrete, graph theoretical approach to conformal field theory correlators. In a certain basis, called the squid basis, the correlator of <math><mi>N</mi></math> scalar operators can be expressed as the determinant of a natural, conformally covariant metric on a weighted graph, called the squid graph. We present the construction of this metric and discuss its possible role in constraining conformal data.

Stylianos Kastrinakis

Graph theoretical approach to conformal correlators

Using arguments from holography we propose that the deviation of the cosmological spectral index  $$n_S$$  <math> <msub> <mi>n</mi> <mi>S</mi> </msub> </math>   of scalar fluctuations from unity may be controlled almost entirely by the critical exponent  $$\eta $$  <math> <mi>η</mi> </math>   of the  $$d=3$$  <math> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </math>   Ising model.

Antonis Kalogirou

Poland

Fotis Koutroulis

Ising Cosmology

We construct the zero temperature (no compact dimensions) effective action for an SU(2) Yang–Mills theory in five dimensions, with boundary conditions that reduce the symmetry on the four-dimensional boundary located at the origin to a U(1)-complex scalar system. In order to be sensitive to the Higgs phase, we need to include higher dimensional operators in the effective action, which can be naturally achieved by generating it by expanding the corresponding lattice construction in small lattice spacing, taking the naive continuum limit and then renormalizing. In addition, we build in the effective action non-perturbative information, related to a first order quantum phase transition known to exist. As a result, the effective action acquires a finite cut-off that is low and the fine tuning of the scalar mass is rather mild.

RG flows in non-perturbative gauge-Higgs unification II: effective action for the Higgs phase near the quantum phase transition

Generalized Effective Field Theory (GEFT) is the non-renormalizable extension of an Effective Field Theory where the Wilson coefficients are endowed by their own, independent scale dependence. Such an effective theory can be constructed by quantizing a Lagrangian in the presence of all internal symmetry respecting operators of any mass dimension. The resulting theory may be practically useful in regimes of its phase diagram where the perturbative expansion and a truncation of the infinite tower of Higher Dimensional Operators (HDO) are valid. The massless limit of GEFT is especially interesting as the spontaneous breaking of internal symmetry and of scale invariance are correlated.

On RG flows in generalized effective field theory

We discuss the role of higher dimensional operators in the spontaneous breaking of internal symmetry and scale invariance, in the context of the Lorentz invariant scalar field theory. Using the <math><mi>ϵ</mi></math>-expansion we determine phase diagrams and demonstrate that (un)stable renormalization group flows computed with a certain basis of dimension 6 operators in the Lagrangian map to (un)stable renormalization group flows of another basis related to the first by field redefinitions. Crucial is the presence of reparametrization ghosts if Ostrogradsky ghosts appear.

Symmetry breaking and renormalization group flows with higher dimensional operators

Corrigendum to “Renormalization of the Abelian–Higgs model in the and Unitary gauges and the physicality of its scalar potential” [Nucl. Phys. B 924 (2017) 178–278]

We initiate the continuum description of a non-perturbative 5d lattice Yang–Mills model with 4d boundaries using the ε-expansion. In its simplest version classically the bulk has an <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></math> gauge symmetry and on the boundary there is an Abelian–Higgs system with zero scalar potential. In part I we compute the Renormalization Group flows and related quantities in a limit where the boundary decouples from the bulk.

RG flows in non-perturbative gauge-Higgs unification I

We perform an old school, one-loop renormalization of the Abelian–Higgs model in the Unitary and Rξ gauges, focused on the scalar potential and the gauge boson mass. Our goal is to demonstrate in this simple context the validity of the Unitary gauge at the quantum level, which could open the way for an until now (mostly) avoided framework for loop computations. We indeed find that the Unitary gauge is consistent and equivalent to the Rξ gauge at the level of β -functions. Then we compare the renormalized, finite, one-loop Higgs potential in the two gauges and we again find equivalence. This equivalence needs not only a complete cancellation of the gauge fixing parameter ξ from the Rξ gauge potential but also requires its ξ -independent part to be equal to the Unitary gauge result. We follow the quantum behavior of the system by plotting Renormalization Group trajectories and Lines of Constant Physics, with the former the well known curves and with the latter, determined by the finite parts of the counter-terms, particularly well suited for a comparison with non-perturbative studies.

Renormalization of the Abelian–Higgs model in the R ξ and Unitary gauges and the physicality of its scalar potential

We study the viability of five-dimensional gauge theories as candidates for the origin of the Higgs field and its mechanism for spontaneous symmetry breaking. Within the framework of lattice field theory, we consider the simplest model of an SU(2) gauge theory. We construct this theory on a five-dimensional orbifold which explicitly breaks the gauge symmetry to U(1) at the fixed points of the orbifold. Using anisotropic gauge couplings, we find that this theory exhibits three distinct phases which we label as confined, Higgs and hybrid. Within the Higgs phase, close to the Higgs-hybrid phase transition, we find that the ratio of the Higgs to gauge boson masses takes Standard Model-like values. Precisely in this region of the phase diagram, we find dimensional reduction via localisation.