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Potential Flow Of The Renormalisation Group In A Simple Two Component Model

The renormalisation group (RG) flow on the space of couplings of a simple model with two couplings is examined. The model considered is that of a single component scalar field with $ϕ^4$ self interaction coupled, via Yukawa coupling, to a fermion in flat four dimensional space. The RG flow on the two dimensional space of couplings, ${\cal G}$, is shown to be derivable from a potential to sixth order in the couplings, which requires two loop calculations of the $β$-functions. The identification of a potential requires the introduction of a metric on ${\cal G}$ and it is shown that the metric defined by Zamalodchikov, in terms of two point correlation functions of composite operators, gives potential flow to this order.