Abstract

The Standard Model with a classical conformal invariance holds the promise to lead to a better understanding of the hierarchy problem and could pave the way beyond the Standard Model physics. Thus, we give here a mathematical treatment of a massless quartic scalar field theory with a strong self-coupling both classically and for quantum field theory. We use a set of classical solutions recently found and show that there exist an infinite set of infrared trivial scalar theories with a mass gap. Free particles have superimposed a harmonic oscillator set of states. The classical solution is displayed through a current expansion and the next-to-leading order quantum correction is provided. Application to the Standard Model would entail the existence of higher excited states of the Higgs particle and reduced decay rates to WW and ZZ that could already be measured.

Highlights

  • Scalar field theory is an essential tool to master the main techniques in quantum field theory

  • This has been recently proved by Chishtie et al [17] and Steele and Wang [18]; extending to higher orders the computation of the effective potential, the right mass for the Higgs particle is recovered, giving a boost to the idea of conformal invariance for the Standard Model

  • Interchanging the perturbation term in the equation gives perturbation series with an expansion parameter the inverse of the other. This is the essence of the duality principle in perturbation theory. This is a general property of differential equations, which we applied to the case of the scalar field theory and which can be extended to quantum field theory in the way we displayed in this paper

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Summary

Introduction

Scalar field theory is an essential tool to master the main techniques in quantum field theory (see e.g. [1,2,3]). The success of the Coleman–Weinberg mechanism, being perturbative in origin, implies that, in order to obtain the right mass, one cannot stop at the first few terms of the perturbation series This has been recently proved by Chishtie et al [17] and Steele and Wang [18]; extending to higher orders the computation of the effective potential, the right mass for the Higgs particle is recovered, giving a boost to the idea of conformal invariance for the Standard Model. This moves the test of this idea from the existence of a further Higgs particle to the experimental determination of the selfcoupling of the Higgs field.

Classical scalar field theory
Green function
Strong coupling solution
Current expansion is the strong coupling expansion
Classical n-point functions and higher order corrections
Numerical Dyson–Schwinger equations and Green
Next-to-leading order correction
Duality principle
Callan–Symanzik equation
Systematic of renormalization
Breaking of conformal symmetry
Findings
Higgs model
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