arXiv:1202.1368v1 [hep-ph] 7 Feb 2012
Do we need Feynman diagrams for higher orders perturbation theory?
aNational Institute of Physics and Nuclear Engineering, PO Box MG-6, Bucharest-Magurele, Romania.
(Dated: February 8, 2012)
We compute the two loop correction to the beta function for Yang Mills theories in the background
gauge field method and using the background gauge field as the only source. The calculations are
based on the separation of the one loop effective potential into zero and positive modes contributions
and are entirely analytical. No two loop Feynman diagrams are considered in the process.
PACS numbers: 11.10 Ef, 11.10 Gh, 11.10 Hi, 11.15 Bt.
The instanton approach for SU(N) gauge theories with or without fermions has been initiated by ’t Hooft  and
further developed in  and . In this method the separation of quantum degrees of freedom into zero modes (spin
dependent) and positive modes (spin independent) is crucial. Moreoverthe so called zero modes have an ” antiscreening
” effect which is ultimately responsible for asymptotic freedom. The presence of fermions has an opposite effect. In
 we suggest that in essence the magnetic properties of the QCD vacuum play a decisive role in the chiral symmetry
breaking. Furthermore we show in  that in the process of gluino decoupling from supersymmetric QCD separation
into zero and positive modes is very important.
A very useful method for computing beta functions for the gauge coupling constant is the background gauge field
method  which is based on the decomposition of the gauge field into a background gauge field and a fluctuating
field, the quantum gauge field. Even from the dawn of this method the background gauge field was regarded as an
alternate source. However the regular sources J(x) and η(x), η′(x) (corresponding to the quantum gauge fields and
ghost respectively) are introduced and one uses the conventional functional formalism to derive beta function or other
loop corrections. The reason is simple; the background gauge field does not couple linearly to the other fields (as
linear terms are canceled) and it is not obvious how one can compute simply Green functions with the background
gauge field as a source.
In the present work we determine the two loop contribution to the beta function for Yang Mills theories using the
background gauge field as the only source present in the functional formalism. Of course the beta function is known
up the fourth order  in the MS scheme so our main interest lies in the method that we introduce and the possibility
for that to be developed for higher orders. We rely on the well-known result of the one loop effective potential (derived
either in the perturbative or in the instanton approach) and on the decomposition of the one loop operators into spin
dependent and spin independent operators corresponding to each field. Our derivation is entirely based on an analytic
functional approach (see the Appendix) that does not involve the computation of any two loop Feynman diagram.
II. THE METHOD
The Yang Mills Lagrangian in the background gauge field method (where the gauge field is separated into Ba
µis the background gauge field) has the expression:
L = −
µ)2+ ¯ ca[(−D2)ac− DµfabcAb
This lagrangian contains quantum gauge fields Aa
bution and a higher order one. The procedure for extracting the quadratic terms in this lagrangian is standard and
after integration leads to the one loop effective potential .
µand ghosts ca, ¯ caand can be separated into a quadratic contri-
from which we can deduce,
It is useful to give in what follows some results regarding the functional derivatives of the square of the gauge tensor
mνta= Fµν, where tais the generator in the adjoint representation).
8i[(Fρσ)ae(x)δcdδ(x − y) − (Fρσ)dcδaeδ(x − y)].
Furthermore from this one can deduce:
d4xd4yd4ud4vδ(x − u)δ(y − v)facmfden×
ρσ(u)δ(δ(u − v))exp[k2
µν)2× exp[(k1+ k2)
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