Second-order corrections to the Gaussian effective potential for lambda phi(4) and other theories
Stevenson, Paul M.
Doctor of Philosophy
We formulate a systematic, nonperturbative expansion for the effective potential of $\lambda\phi\sp4$ theory. At first order it gives the Gaussian effective potential (GEP), which itself contains the 1-loop and leading-order $1\over N$ results. Here, we compute the second-order terms and hence obtain the post-Gaussian effective potential (PGEP) in 1, 2, 3, and 4 spacetime dimensions. The renormalization procedure, including the calculation of the physical mass, is discussed in detail. The results in lower dimensions agree well with the GEP when the comparison is made for the same values of the bare parameters. In 4 dimensions the divergent integrals are calculated using coordinate-space methods combined with dimensional regularization. (Difficulties with other regularizations are briefly discussed). The PGEP for the "precarious" $\lambda\phi\sp4$ theory is obtained in manifestly finite form. Remarkably, the final result takes the same mathematical form as the GEP, with only some numerical co-efficients being changed. Indeed, when parametrized in terms of the physical mass and the renormalized coupling constant, only a single coefficient is changed, from 1 to 1-1/(N + 3)$\sp2$. The "autonomous" version of the 4-dimensional theory refuses to work in this approach: one obtains indeed an autonomous-like theory, but it is unbounded below for a certain range of the parameters. The influence of fermions on scalar systems is also investigated in the post-Gaussian approximation. For the simple case of an Yukawa-type coupling with no scalar self-interaction terms the results turn out to be the same as in the Gaussian approximation.
Particle physics; Elementary particles; High energy physics