Five loop renormalization of the Wess-Zumino model

We renormalize the Wess-Zumino model at five loops in both the minimal subtraction (MSbar) and momentum subtraction (MOM) schemes. The calculation is carried out automatically using a routine that performs the D-algebra. Generalizations of the model to include $O(N)$ symmetry as well as the case with real and complex tensor couplings are also considered. We confirm that the emergent SU(3) symmetry of six dimensional O(N) phi^3 theory is also a property of the tensor O(N) model. With the new loop order precision we compute critical exponents in the epsilon expansion for several of these generalizations as well as the XYZ model in order to compare with conformal bootstrap estimates in three dimensions. For example at five loops our estimate for the correction to scaling exponent is in very good agreement for the Wess-Zumino model which equates to the emergent supersymmetric fixed point of the Gross-Neveu-Yukawa model. We also compute the rational number that is part of the six loop MSbar beta-function.

Critical exponent $eta$ at $O(1/N^3)$ in the chiral XY model using the large $N$ conformal bootstrap

We compute the $O(1/N^3)$ correction to the critical exponent $eta$ in the chiral XY or chiral Gross-Neveu model in $d$-dimensions. As the leading order vertex anomalous dimension vanishes, the direct application of the large $N$ conformal bootstrap formalism is not immediately possible. To circumvent this we consider the more general Nambu-Jona-Lasinio model for a general non-abelian Lie group. Taking the abelian limit of the exponents of this model produces those of the chiral XY model. Subsequently we provide improved estimates for $eta$ in the three dimensional chiral XY model for various values of $N$.

Asymptotic freedom from the two loop term of the $beta$-function in a cubic theory

We renormalize a six dimensional cubic theory to four loops in the MSbar scheme where the scalar is in a bi-adjoint representation. The underlying model was originally derived in a problem relating to gravity being a double copy of Yang-Mills theory. As a field theory in its own right we find that it has a curious property in that while unexpectedly there is no one loop contribution to the $beta$-function the two loop coefficient is negative. It therefore represents an example where asymptotic freedom is determined by the two loop term of the $beta$-function. We also examine a multi-adjoint cubic theory in order to see whether this is a more universal property of these models.

Six dimensional ultraviolet completion of the $CP(N)$ $sigma$ model at two loops

We extend the recent one loop analysis of the ultraviolet completion of the $CP(N)$ nonlinear $sigma$ model in six dimensions to two loop order in the MSbar scheme for an arbitrary covariant gauge. In particular we compute the anomalous dimensions of the fields and $beta$-functions of the four coupling constants. We note that like Quantum Electrodynamics (QED) in four dimensions the matter field anomalous dimension only depends on the gauge parameter at one loop. As a non-trivial check we verify that the critical exponents derived from these renormalization group functions at the Wilson-Fisher fixed point are consistent with the $epsilon$ expansion of the respective large $N$ exponents of the underlying universal theory. Using the Ward-Takahashi identity we deduce the three loop MSbar renormalization group functions for the six dimensional ultraviolet completeness of scalar QED.

Symmetric point flavour singlet axial vector current renormalization at two loops

We calculate the two loop correction to the quark 2-point function with the non-zero momentum insertion of the flavour singlet axial vector current at the fully symmetric subtraction point for massless quarks in the modified minimal subtraction (MSbar) scheme. The Larin method is used to handle $gamma^5$ within dimensional regularization at this loop order ensuring that the effect of the chiral anomaly is properly included within the construction.

On the self-consistency of off-shell Slavnov-Taylor identities in QCD

Using Hopf-algebraic structures as well as diagrammatic techniques for determining the Slavnov-Taylor identities for QCD we construct the relations for the triple and quartic gluon vertices at one loop. By making the longitudinal projection on an external gluon of a Greens function we show that the gluon self-energy of that leg is consistently replaced by a ghost self-energy. The resulting identities are then studied by evaluating all the graphs for an off-shell non-exceptional momentum configuration. In the case of the 3-point function this is for the most general momentum case and for the 4-point function we consider the fully symmetric point.

Off-shell quark bilinear operator Greens functions at two loops

We construct the two loop Greens functions for a quark bilinear operator inserted at non-zero momentum in a quark 2-point function for the most general off-shell configuration. In particular we consider the quark mass operator, vector and tensor currents as well as the second moment of the flavour non-singlet Wilson operator.

Large $N_f$ quantum field theory

We review the development of the large $N$ method, where $N$ indicates the number of flavours, used to study perturbative and nonperturbative properties of quantum field theories. The relevant historical background is summarized as a prelude to the introduction of the large $N$ critical point formalism. This is used to compute large $N$ corrections to $d$-dimensional critical exponents of the universal quantum field theory present at the Wilson-Fisher fixed point. While pedagogical in part the application to gauge theories is also covered and the use of the large $N$ method to complement explicit high order perturbative computations in gauge theories is also highlighted. The usefulness of the technique in relation to other methods currently used to study quantum field theories in $d$-dimensions is also summarized.

Fermion bilinear operator critical exponents at $O(1/N^2)$ in the QED-Gross-Neveu universality class

We use the critical point large $N$ formalism to calculate the critical exponents corresponding to the fermion mass operator and flavour non-singlet fermion bilinear operator in the universality class of Quantum Electrodynamics (QED) coupled to the Gross-Neveu model for an $SU(N)$ flavour symmetry in $d$-dimensions. The $epsilon$ expansion of the exponents in $d$ $=$ $4$ $-$ $2epsilon$ dimensions are in agreement with recent three and four loop perturbative evaluations of both renormalization group functions of these operators. Estimates of the value of the non-singlet operator exponent in three dimensions are provided.

Four loop renormalization of the Gross-Neveu model

We renormalize the SU(N) Gross-Neveu model in the modified minimal subtraction (MSbar) scheme at four loops and determine the beta-function at this order. The theory ceases to be multiplicatively renormalizable when dimensionally regularized due to the generation of evanescent 4-fermi operators. The first of these appears at three loops and we correctly take their effect into account in deriving the renormalization group functions. We use the results to provide estimates of critical exponents relevant to phase transitions in graphene.