Multi-loop beta function of gauge theory (*without* Feynman diagrams)

Question

I would like to point to the beautiful section 4.3 (page 42) of these lecture notes. I think this is the most educative exposition I have ever seen anywhere about Yang-Mill's beta function. What I love best about it is that it does it without using diagramatics (and its confusing combinatorics)(..though of course these are equivalent..but I find this language most comfortable..)

I have the following questions in relation to the above,

• In the above the author has picked out the terms quadratic in the fluctuations in equation 4.40 and evaluated the determinant and that gives the 1-loop effective action.

  What would one do in this method if one had to go to 2-loops or higher? What is the relationship between how many orders one keeps in the fluctuation field and how many loop result it translates into? (..if there a reference which shows going to higher loops in this method?..)

• Can I use the method in these notes to evaluate the corrected gauge or the Fermion propagator? If someone could outline the steps..

  Here the author has chosen a constant and static background gauge (why?) and hence he has in 4.40 no term which is a derivative of the background gauge field left. I guess one would have to lift this restriction if one had to calculate the gauge field propagator correction. With this assumption about the background gauge field lifted I guess that one would have to then compute the $\Gamma^{1-loop}$ as defined in equation 4.41 and pick out the terms quadratic in the gauge field in it and invert that.

• In the above the author has picked out from $\Gamma^{1-loop}$ only the terms quartic in the gauge field and and calculated the divergent contribution to it which is the shift in the gauge coupling constant. But the gauge coupling constant is also multiplying the term cubic in the gauge field and there is a 1-loop shift even there. What about that? Is there some theorem which guarantees that the beta-function derived by tacking the cubic term would have been the same?

I guess that since even after choosing a constant and static gauge field one can track a change in the gauge coupling constant through a quartic term, it makes sense. But if one were in a theory (or in light-cone gauge!) where the gauge coupling constant were existing only in a term which has derivatives of the gauge field then I guess this choice of a background would not work.

I would like to know of a precise way of understanding the above. (...there is also the issue of whether the 1-loop effective action done this way can throw up terms that were removed by a gauge choice..and then what would one do..)

Answer 1

OK, so there are quite a few parts to your question - I hope that I address most of them to your satisfaction.

The author of the notes is essentially using the background field method (BFM) to calculate the effective action.⁰ Many of his choices follow from his focus on calculating only the one-loop effective potential. The background field method works in all theories with scalar, gauge, fermions, gravity, superfields¹ etc... It also works at all loops, however at higher loops, diagrams are still helpful for organising the calculations.

Two of the seminal papers did not use diagramatics: Schwinger's On gauge invariance and vacuum polarization and the follow-up two-loop calculation of Ritus. However, since the BFM perturbation theory still uses a propagator-interaction separation, diagrammatics are only natural.

One advantage of using BFM calculations is that you only need to calculate "vacuum" diagrams, i.e., those with no external legs. This makes the diagrammatics and combinatorics easier. The other advantage is that the calculations become gauge covariant. The trade-off is the use of more complicated propagators.

It is most often used to calculate low energy corrections (like the effective potential for scalar fields) by keeping only the lower terms in the derivative/momentum expansion. In particular, to find the effective potential, you only need constant background scalar fields. To find the kinetic terms, you need fields with at most two derivatives.

However, it can also be used to construct perturbation theories that are very similar to the standard Feynman diagram calculations, but are explicitly covariant under gauge transformations. Good examples of this are Improved Methods for Supergraphs and New, improved supergraphs.
A really nice three-loop calculation of the Yang-Mills beta-function using the covariant background field method is hep-th/0211246. They step through the set-up of the calculation quite slowly, so it is a good paper to learn from.

The BFM relies on the following idea, see, e.g., Abbott's The Background Field Method Beyond One Loop, but the result can be extended to other theories with more complicated background-quantum splittings:
Let $\Gamma[v]$ be the effective action (Legendre transform of the connected generating function $W[J]$) where $v=v(J)=\frac{\delta W[J]}{\delta J}$ is the "classical" field generated by the sources $J$. If we modify the classical action by splitting the quantum fields into quantum + background, then the resultant modified effective action $\Gamma[v(j),V]$ now depends on both $v(J)$ and the background fields $V$. You can show (under reasonable assumptions) that $\Gamma[0, V] = \Gamma[V]$. The effective action is gauge dependent,² and the previous result is true in the background field gauge.

In the notes you linked to, the beta function was found from the quartic, derivative free correction. It can also be found from the gauge kinetic term. This works, because the method is background gauge covariant, which forces the gauge coupling and field renormalisations to be related. In fact, the background gauge potential field never needs to leave the covariant derivative and the beta function could be found from the invariant $\mathrm{tr}(F_{\mu\nu}F^{\mu\nu})$ term. This was done in the original paper by Schwinger. See Abbott for more of a discussion on this point. Also read about the Schwinger-Dewitt expansion for how to control some covariant expansions in effective action calculations. The classic paper is the Physics Report by Barvinsky & Vilkovisky. See Avramindi and Kuzenko & McArthur for more info on covariant methods.

For a good 2-loop BFM calculation involving fermionic backgrounds, see Jack & Osborn. Although they don't calculate the finite parts of the propagators. Low energy calculations with fermion backgrounds can be quite tricky, which also makes some supersymmetric BFM low-energy calculations tricky.

⁰ That said, the notes do collect some nice arguments about the structure of the effective action and effective potential.
¹ Although, in some theories (e.g., N=1 supersymmetry), the quantum-background splitting has to be nonlinear.
² A gauge independent version of the effective potential does exist - but it seems to be not very practical to calculate or use - at least, it's not used much. See, e.g., Vilkovisky or Becchi...