Springer Lecture Notes in Physics 716

310 U. C. Täuber
phase transition; but u becomes irrelevant for d>4: one then expects meanfield
(Gaussian) critical exponents. At the upper critical dimension d c =4,
the nonlinear coupling u is marginally relevant: this will induce logarithmic
corrections to the mean-field scaling laws, see Table 7.1.
It is obviously not a simple task to treat the IR-singular perturbation
expansion in a meaningful, well-defined manner, and thus allow nonanalytic
modifications of the critical power laws (note that mean-field scaling is completely
determined by dimensional analysis or power counting). The key of
the success of the RG approach is to focus on the very specific symmetry
that emerges near critical points, namely scale invariance. There are several
(largely equivalent) versions of the RG method; we shall here formulate
and employ the field-theoretic variant [1–6, 8, 13]. In order to proceed,
it is convenient to evaluate the loop integrals in momentum space by means
of dimensional regularisation, whereby one assigns finite values even to UVdivergent
expressions, namely the analytically continued values from the UVfinite
range. For example, even for noninteger dimensions d and σ, weset
∫
d d k
(2π) d
k 2σ Γ (σ + d/2) Γ (s − σ − d/2)
(τ + k 2 ) s =
2 d π d/2 Γ (d/2) Γ (s)
The renormalisation program then consists of the following steps:
τ σ−s+d/2 . (7.53)
1. We aim to carefully keep track of formal, unphysical UV divergences. In
dimensionally regularised integrals (7.53), these appear as poles in ɛ =
d c − d; their residues characterise the asymptotic UV behaviour of the field
theory under consideration.
2. Therefrom we may infer the (UV) scaling properties of the control parameters
of the model under a RG transformation, namely essentially a change
of the momentum scale µ, while keeping the form of the action invariant.
This will allow us to define suitable running couplings.
3. We seek fixed points in parameter space where certain marginal couplings
(u here) do not change anymore under RG transformations. This describes
a scale-invariant regime for the model under consideration, where the UV
and IR scaling properties become intimately linked. Studying the parameter
flows near a stable RG fixed point then allows us to extract the
asymptotic IR power laws.
As a preliminary step, we need to take into account that the fluctuations
will also shift the critical point downwards from the mean-field phase transition
temperature T 0 c ; i.e., we expect the transition to occur at T c