Lattice Studies of the Higgs System

Abstract

Investigation of the coupled Higgs and gauge fields on the lattice has elucidated the gauge invariant formulation and several non-perturbative aspects of the Higgs mechanism. in particular its properties for strong gauge coupling and its relationship to confinement. However. until now no indication has been found for the gauge field to inhibit the vanishing of the ~4 coupling in the limit of infinite cut-off. The scalar sector dominates the properties of the Higgs mechanism, and the cut-off cannot be removed.With the gauge sector treated therefore only perturbatively. extensive analytic and numerical calculations in the pure scalar sector of the Standard Model have been performed recently on the lattice. The results indicate that the cut-off parameter in this regularization can be substantially greater than the Higgs bo.:-on mass only if this rna.."~ is not much bigger than 640 GeV, and the scalar sector is not strongly interacting.Lattice studies of the Higgs system including fermions have been initiated. Modifications of the phase diagram due to the spontaneous chiral symmetry breaking for fermions with vectorial gauge coupling have been observed. In models with strong Yukawa coupling the fermion masses increase as the expectation value of the Higgs field decreases. Therefore a method of putting chiral fermions on the lattice in a gauge invariant way and removing the unwanted fermion doublers by means of a strong coupling of the Wilson-Yukawa type to the Higgs field is very promising and currently under investigation.Review tnlk given n.t the 8th INFN Eloi:-at.ron Pro.ic-d Workshop

Gauge-independent Brout–Englert–Higgs mechanism and Yang–Mills theory with a gauge-invariant gluon mass term

“…If the fields A and φ are a set of solutions of the field equations for the U (1) gauge-scalar model with a radially fixed scalar field, they automatically satisfy the reduction condition (31) for pure U (1) gauge theory.…”

“…See [15][16][17] for a fundamental scalar and [18][19][20][21][22][23] for an adjoint scalar case. (The Fradkin-Shenker continuity does not hold even for the fundamental scalar model if the scalar field has the variable length with a sufficiently small 0 < λ 1 self-interaction coupling λ for the potential term V = λ(||φ(x)|| 2 − v 2 ) 2 [24][25][26][27][28][29][30][31][32]. Higgs phase and Confinement phase are not analytically continued in the phase diagram and separated by the phase transition.)…”

Introduction to quantum fields on a lattice: a 'robust mate'

“…The eigenvalues µ ± of M can be expressed as µ ± = e ±ω , cosh ω = 1 + 1 2 ω 2 , (2. 35) and the linear combinations sought are given by â = ν[sinh(ω q) + ip],…”

“…Recalling that ω is really aω, and similarly for ω, we see by expanding (2. 35) in powers of a, i.e. cosh(aω) = 1 + a 2 ω2 /2 + a 4 ω4 /24…”

“…However, lattice formulations of gauge-Higgs systems and to a certain extent Yukawa models are possible and have been much studied over the years. The lattice formulation of gauge-Higgs systems has interesting aspects having to do with the gauge-invariant formulation of the Higgs phenomenon, presentation and discussion of which here would lead too far [35].…”