Effective potential at three loops

I present the effective potential at three-loop order for a general renormalizable theory, using the $\overline{\mathrm{MS}}$ 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 nontrivial 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.

Figure 1

The topologies for the 2-loop and 3-loop basis vacuum integral functions used in this paper. The large dot in $F(u,z,y,v)$ corresponds to a derivative with respect to the $u$ squared mass argument. The function $\overline{F}(u,z,y,v)$ is the same as $F(u,z,y,v)$ but with a subtraction to render it infrared finite in the limit $u\to 0$. In each case, counterterms have been included to make the integrals finite as the ultraviolet regulator $\epsilon \to 0$. See Ref. [51] for the precise definitions and more information.

Figure 2

The Feynman diagram topologies contributing to the 3-loop effective potential, with numerals indicating the ordering of subscripts denoting propagator types ($S$, $F$, $\overline{F}$, $V$, or $g$) as well as the ordering of the corresponding squared mass arguments.

Figure 3

Examples of the Feynman diagram labeling scheme used in this paper, for the diagrams with loop integral functions denoted $H_{F\overline{F}SVF\overline{F}}(u,v,w,x,y,z)$ and $K_{VVSSFF}(x,w,u,z,y,v)$. Solid lines with arrows represent helicity-preserving fermion propagators. Solid lines with a dot and clashing arrows represent a helicity-violating fermion propagator. Dashed lines indicate a real scalar propagator, and wavy lines stand for real vector propagators. The squared masses are denoted by $u,v,w,x,y,z$ as labeled.