The effective potential in nonconformal gauge theories

Abstract

By using the renormalization group (RG) equation it has proved possible to sum logarithmic corrections to quantities that arise due to quantum effects in field theories. In particular, the effective potential V in the Standard Model in the limit that there are no massive parameters in the classical action (the "conformal limit") has been subject to this analysis, as has the effective potential in a scalar theory with a quartic self coupling and in massless scalar electrodynamics. Having multiple coupling constants and/or mass parameters in the initial action complicates this analysis, as then several mass scales arise. We show how to address this problem by considering the effective potential in a Yukawa model when the scalar field has a tree-level mass term. In addition to summing logarithmic corrections by using the RG equation, we also consider the consequences of the condition V ′ (v) = 0 where v is the vacuum expectation value of the scalar. If V is expanded in powers of logarithms that arise, then it proves possible to show that either v is zero or that V is independent of the scalar. (That is, either there * fbrandt@usp.br † fchishtie@uwo.ca ‡ dgmckeo2@uwo.ca 1 is no spontaneous symmetry breaking or the vacuum expectation value is not determined by minimizing V as V is "flat".)

Effective potential in massless theories

“…We find that V , if it is to have a non-vanishing extremum, is independent of φ-it is "flat". This is consistent with the requirement that V be convex [19][20][21] and with previously derived results [15][16][17][18]. Together, eq.…”

“…A shift in their value can be absorbed into Λ and Φ. Integrals involving f , b c were used to this end in eqs. (16,18,20) but when there is more than one coupling, only the one loop contributions to the RG functions are RS invariant when using a mass independent RS [32,33] and so this approach cannot be generalized and we employ K Λ and K Φ .…”

“…Again, we will only consider the conformal limit in which there is no bare mass term for the scalar field in the classical Lagrangian. The non-conformal case has been considered in [17,18].…”

“…Indeed, it has also proved possible to sum all of these logarithmic contributions to the effective potential V so that V is determined by the log independent contributions and the RG functions. When this summation is combined with the condition that V has a minimum at some non-vanishing value v of the scalar field φ, it has been shown that V in fact must be independent of φ; this occurs in a simple self-interacting scalar model, scalar electrodynamics is [15,16], a massive self-interacting model [17] and a massive model which involves interactions between the scalar and other fields [18]. This result is consistent with the general result that V must be convex [19][20][21], a condition not satisfied by the classical "Mexican hat" potential.…”

Inflation in scale-invariant theories of gravity

“…Lagrangian inevitably leads to scalar spectrum, which is incompatible with observations so it must be discarded. Motivated by this no-go result, we softly break the scale invariance with a deformation of the classical R 2 Lagrangian that mimics the resummation of higher-loop logarithmic corrections (in absence of gravity, see, for example [24]), namely…”