Abstract
The inter-quark potential is dominated by anti-screening effects which underly asymptotic freedom. We calculate the order g6 anti-screening contribution from light fermions and demonstrate that these effects introduce a non-local divergence. These divergences are shown to make it impossible to define a coupling renormalisation scheme that renormalises this minimal, anti-screening potential. Hence the beta function cannot be divided into screening and anti-screening parts beyond lowest order. However, we then demonstrate that renormalisation can be carried out in terms of the anti-screening potential.
Highlights
The inter-quark potential is dominated by anti-screening effects which underly asymptotic freedom
We have seen that the decomposition of the beta function into screening and antiscreening structures breaks down beyond one loop
This we saw by calculating the light quark contributions to the anti-screening potential: non-local divergences arose in fermion loops in the minimal potential at order g6 which are not cancelled by an anti-screening beta function
Summary
We will calculate the quark contribution, i.e., the nf dependent terms, to the minimal dressing. We will show that at next to next to leading order quarks produce an anti-screening effect. This contribution is needed to ensure gauge invariance at higher orders. They arise by higher order expansions of the dressings and will involve Green’s functions such as g5 0|AAA|0 and g6 0|AAAA|0 These Green’s functions will only depend on the nf light fermions through loops and it is easy to see that they will first introduce quark contributions beyond order g6 in the coupling.
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