(5.95) with the counter term for the physical scheme
δAphys = λ2µ2 32π2
1
−γ−ln m2
4πµ2
⇒δλphys = 3δAphys. (5.96)
5.5 Power counting
Our experience from the previous section lets us now look for the systematic proof of renor-malisability to all orders of perturbation theory. We shall again take φ4-theory as the most simple example of a renormalisable theory.
At first we look on the so called superficial degree of divergence. This is obtained by simply counting the powers of loop momenta within the Feynman integral Γ in d-dimensional space time. A diagram withLloops yields an integral over dLmomenta. Each internal line stands for a propagator which gives a power of−2I (where I is the number of internal lines). The whole integral has thus a momentum power
Ds(d,Γ) =Ld−2I. (5.97)
For the convergence of Γ it is obviouslynecessary but by no means sufficient thatDs(Γ)<0.
The diagrams in the previous section showed that of course the divergent part had the power law in the external momenta as expected by this simple power counting method but the finite
7Note that in the effective action all three diagrams in figure 5.5 dressed with mean fields appear. Thus the counter term for the coupling has an additional factor 3 compared to the single diagram.
5.5 · Power counting
part contains non-trivial logarithms. The powers in momenta where≤2 for d= 4. Now it is evident that it is necessary for a theory to be renormalisable that the degree of divergence is negative and that the infinite part can be subtracted with help of a counter term which is of the same form as monomials of fields and its derivatives already contained in the Lagrangian.
This means that necessarily a renormalisable theory can contain only interaction monomials such that the superficial degree of divergence for proper vertex-functions is positive only for a finite set of such vertex functions and exactly those which are already in the Lagrangian.
Now we want to show thatφ4 is fulfilling this necessary requirements. This is simply done by substituting E and Lin (5.97) instead of I. The conditions are fulfilled if only the 2- and 4-point 1PI vertex functions are superficially divergent. It does not matter if the divergences arise at all orders perturbation theory but the counter terms should only contain polynomials of order O(p2) for the 2-point function and only of order O(p0) for the 4-point vertex. The 3-point vertex should be finite and also all n-point vertices for n≥5. Due to the symmetry underφ→ −φthe 3-point vertex vanishes at all.
Now we have to count the number of internal lines in terms of the number of loops and external lines. From momentum conservation at each vertex we have I −V independent momenta but 1 condition is already fulfilled by conservation of the external momenta (the sum of all momenta running into or out of the vertex diagram has to be 0), thus we have
L=I −V + 1. (5.98)
While (5.97) and (5.98) are valid for any bosonic quantum field theory now we have to use the fact that each vertex ofφ4-theory contains exactly 4 legs leading toI = (4V −E)/2 (each of the 4V legs is connected to an external point linked with another leg. The external legs do not contribute to the internal lines, each of which is the linking of two fields). This leads to the following expression for the superficial degree of divergence
D(d)S (Γ) = (d−4)V +d+
1−d 2
E. (5.99)
Ford= 4 this readsD(4)S (Γ) = 4−E. This means that the superficial degree of divergence is negative for E≥5, i.e., the first condition is fulfilled. Now we have to count the naive powers of momentum for the vertex function.
An n-point vertex function has the same naive momentum power as a coupling constant in front of a (fictive or real)φn-contribution in the Lagrangian. In our system of units the action has dimensionO(p0) and thus from the kinetic term∂µφ∂µφwe read off thatφ=O(p(d−2)/2).
This shows that anE-point proper vertex function has the naive dimensionO(pE−n(E/2−1)) = O(pD(d)(Γ)). Thus for d = 4 the φ4-theory is really superficially renormalisable because for E = 2 the naive momentum power is 2 (we have a mass counter term there to absorb the infinities into the mass) and for E = 4 the power is 0 and this gives rise to the counterterm absorbing the infinities into the bare coupling.
As we have always emphasised this ideas are not complete. The so far developed power counting arguments are only necessary but not sufficient conditions for renormalisability.
Although a diagram may have a negative naive degree of divergence it need not be finite and even worse the infinities need not be polynomial in the external momenta which seems to introduce non-local interactions into the Lagrangian. It is not a difficult task to explain how
(a) (b) (c)
Figure 5.6: An example for a convergent (a) and two superficially convergent but in fact divergent diagrams
this comes about and how this problem can be solved while the mathematical proof is a hard stuff.
So let us first give a heuristic argument how to treat these problems practically.
First take a superficially finite diagram namely the 6-point vertex of which some contribu-tions are shown in fig. 5.6. The first diagram is primitive, i.e. it does not contain any 1PI subdiagrams which can diverge. It is clear how to calculate this diagram qualitatively: One introduces a Feynman-parameter and treats the diagram making use of the standard formulas of appendix C. The result is finite by naive power counting.
But now look on diagram (b) which contains a 1-loop four-point-vertex sub-diagram which we have calculated in just some paragraphs above. This diagram is divergent, but we have also to add a counter-term diagram making use of our result of the counter-term at least in the MS-scheme. This cancels the divergencies which are not local, i.e., which are not polynoms ofp2 which look awkward on the fist look because this would spoil the idea of renormalising the divergencies with local counter terms in the effective quantum action and absorb them to the bare parameters. The same is of course true for diagram (c). The diagram with the sub-divergences subtracted is finite due to our naive power counting. If it was divergent there would remain only local over-all divergences which can be renormalised by absorbing them into the bare parameters of the theory.
But this sort of divergences is not the worst one! In these examples there was only one divergent sub-diagram which could be subtracted unambiguously. The next example, depicted in fig. 5.7 is an example of so called overlapping divergences.
Any of the bold lines in the diagrams indicates a sub-divergence. Looking on diagrams (a)-(c) each pair of these divergent sub-diagrams have one line in common. This is what is known asoverlapping divergences and realizing that there are overlapping divergences also in QED threatened the field theorists of the old days this could spoil the renormalisability completely.
To keep the reader calm we mention the solution to this problem. Just don’t worry if the divergences are overlapping or not and add the three counter terms from the one-loop four-point result obtained above and magically all difficulties vanish because this procedure leaves us with an overall divergence which islocal. This can be subtracted and put to the bare wave function normalisation factor and the bare mass of the Lagrangian. This solution also shows that in this case of the setting-sun diagram there are no non-local divergences because the subdivergences are subtracted by tadpole diagrams not dependent on the external momentum.
This is also an example for the most important point of the BPHZ formalism which