Effective potential at three loops

Abstract

I present the effective potential at three-loop order for a general renormalizable theory, using the \MSbar renormalization scheme and Landau gauge fixing. As applications and illustrative points of reference, the results are specialized to the supersymmetric Wess-Zumino model and to the Standard Model. In each case, renormalization group scale invariance provides a consistency check. In the Wess-Zumino model, the required vanishing of the minimum vacuum energy yields an additional check. For the Standard Model, I carry out the resummation of Goldstone boson contributions, which provides yet more opportunities for non-trivial checks, and obtain the minimization condition for the Higgs vacuum expectation value at full three-loop order. An infrared divergence due to doubled photon propagators appears in the three-loop Standard Model effective potential, but it does not affect the minimization condition or physical observables and can be eliminated by resummation.Comment: 48 pages, 8 ancillary file

Standard model parameters in the tadpole-free pure MS¯ scheme

“…The quantities ∆ n have been given up to 3-loop order in ref. [14] and the 4-loop order contribution at leading order in QCD is found in ref. [15].…”

“…We choose instead to expand the Higgs field around a loop-corrected VEV v, which is defined to be the minimum of the full effective potential [8][9][10] in Landau gauge. For the Standard Model (and indeed for a general renormalizable field theory), the effective potential has now been obtained at 2-loop [11,12] and 3-loop [13,14] orders, with the 4-loop contributions known [15] at leading order in QCD. The choice of Landau gauge is made because other gauge-fixing choices lead to unpleasant technical problems including kinetic mixing between the longitudinal components of the vector and the Goldstone scalar degrees of freedom.…”

“…with n-loop order contributions ∆ n that are free of spurious imaginary parts and infrared divergences and do not depend at all on the Goldstone boson squared mass. (The 1/λ in this equation is the source of the tadpole effects noted above if one chooses to expand in terms of v tree rather than v.) The full 3-loop contributions were given in [14] in terms of 2-loop and 3-loop basis integrals that can be efficiently evaluated numerically using the computer code 3VIL [26], ‡ and the 4-loop contribution was obtained at leading order in QCD in [15]. However, a numerical illustration of these effects was deferred.…”

The light CP-even MSSM Higgs mass including N$$^\mathbf {3}$$LO+N$$^\mathbf {3}$$LL QCD corrections

“…However, if (some of) the SUSY particles are very heavy, then the perturbative coefficients receive large logarithmic contributions, which spoil the perturbative series. Currently, loop corrections up to the two-loop level are known in the on-shell scheme [5][6][7][8][9][10][11][12][13][14][15][16] and up to three-loop level in the DR scheme [10][11][12][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. The corresponding FO Higgs pole mass results are available through implementations into publicly available spectrum generators [8,[32][33][34][35][36][37][38][39][40][41].…”

Minimal SU(3)×SU(3) symmetry breaking patterns

“…(3.3). 4 The computation of the one-loop effective potential is a mature subject (see, e.g., [24]), and even the three-loop effective potential has been recently computed for a general renormalizable theory [25].…”