We apply the method of graphical functions that was recently extended to six dimensions for scalar theories, to $phi^3$ theory and compute the $beta$ function, the wave function anomalous dimension as well as the mass anomalous dimension in the $over
line{mbox{MS}}$ scheme to five loops. From the results we derive the corresponding renormalization group functions for the Lee-Yang edge singularity problem and percolation theory. After determining the $varepsilon$ expansions of the respective critical exponents to $mathcal{O}(varepsilon^5)$ we apply recent resummation technology to obtain improved exponent estimates in 3, 4 and 5 dimensions. These compare favourably with estimates from fixed dimension numerical techniques and refine the four loop results. To assist with this comparison we collated a substantial amount of data from numerical techniques which are included in tables for each exponent.
Recently, the existence of a candidate a-function for renormalisable theories in three dimensions was demonstrated for a general theory at leading order and for a scalar-fermion theory at next-to-leading order. Here we extend this work by constructin
g the a-function at next-to-leading order for an N=2 supersymmetric Chern-Simons theory. This increase in precision for the a-function necessitated the evaluation of the underlying renormalization-group functions at four loops.
The 3-point vertices of QCD are examined at the symmetric subtraction point at one loop in the Landau gauge in the presence of the Gribov mass, gamma. They are expanded in powers of gamma^2 up to dimension four in order to determine the order of the
leading correction. As well as analysing the pure Gribov-Zwanziger Lagrangian, its extensions to include localizing ghost masses are also examined. For comparison a pure gluon mass term is also considered.
The effects of the Gribov copies on the gluon and ghost propagators are investigated in SU(2) Euclidean Yang-Mills theory quantized in the maximal Abelian gauge. By following Gribovs original approach, extended to the maximal Abelian gauge, we are ab
le to show that the diagonal component of the gluon propagator displays the characteristic Gribov type behavior. The off-diagonal component is found to be of the Yukawa type, with a dynamical mass originating from the dimension two gluon condensate, which is also taken into account. Furthermore, the off-diagonal ghost propagator exhibits infrared enhancement. Finally, we make a comparison with available lattice data.
We review the techniques used to renormalize quantum field theories at several loop orders. This includes the techniques to systematically extract the infinities in a Feynman integral and the implementation of the algorithm within computer algebra. T
o illustrate the method we discuss the renormalization of phi^4 theory and QCD including the application of the critical point large $N$ technique as a check on the anomalous dimensions. The renormalization of non-local operators in QCD is also discussed including the derivation of the two loop correction to the Gribov mass gap equation in the Landau gauge.
We review the current status of the application of the local composite operator technique to the condensation of dimension two operators in quantum chromodynamics (QCD). We pay particular attention to the renormalization group aspects of the formalis
m and the renormalization of QCD in various gauges.
The ghost condensate <epsilon^{abc} cbar^b c^c> is considered together with the gluon condensate <A^2> in SU(2) Euclidean Yang-Mills theories quantized in the Landau gauge. The vacuum polarization ceases to be transverse due to the nonvanishing conde
nsate <epsilon^{abc} cbar^b c^c>. The gluon propagator itself remains transverse. By polarization effects, this ghost condensate induces then a splitting in the gluon mass parameter, which is dynamically generated through <A^2>. The obtained effective masses are real when <A^2> is included in the analysis. In the absence of <A^2>, the already known result that the ghost condensate induces effective tachyonic masses is recovered. At the one-loop level, we find that the effective diagonal mass becomes smaller than the off-diagonal one. This might serve as an indication for some kind of Abelian dominance in the Landau gauge, similar to what happens in the maximal Abelian gauge.
A class of covariant gauges allowing one to interpolate between the Landau, the maximal Abelian, the linear covariant and the Curci-Ferrari gauges is discussed. Multiplicative renormalizability is proven to all orders by means of algebraic renormaliz
ation. All one-loop anomalous dimensions of the fields and gauge parameters are explicitly evaluated in the MSbar scheme.
We investigate a dynamical mass generation mechanism for the off-diagonal gluons and ghosts in SU(N) Yang-Mills theories, quantized in the maximal Abelian gauge. Such a mass can be seen as evidence for the Abelian dominance in that gauge. It originat
es from the condensation of a mixed gluon-ghost operator of mass dimension two, which lowers the vacuum energy. We construct an effective potential for this operator by a combined use of the local composite operators technique with algebraic renormalization and we discuss the gauge parameter independence of the results. We also show that it is possible to connect the vacuum energy, due to the mass dimension two condensate discussed here, with the non-trivial vacuum energy originating from the condensate <A^2>, which has attracted much attention in the Landau gauge.
Massive renormalizable Yang-Mills theories in three dimensions are analysed within the algebraic renormalization in the Landau gauge. In analogy with the four dimensional case, the renormalization of the mass operator A^2 turns out to be expressed in
terms of the fields and coupling constant renormalization factors. We verify the relation we obtain for the operator anomalous dimension by explicit calculations in the large N_f. The generalization to other gauges such as the nonlinear Curci-Ferrari gauge is briefly outlined.
