arXiv subjects
Publications and source records attributed to J. A. Gracey.
The critcal exponent $ω$ is evaluated at $O(1/N)$ in $d$-dimensions in the Gross-Neveu model using the large $N$ critical point formalism. It is shown to be in agreement with the recently determined three loop $β$-functions of the Gross-Neveu-Yukawa model in four dimensions. The same exponent is computed for the chiral Gross-Neveu and non-abelian Nambu-Jona-Lasinio universality classes.
We construct several towers of scalar quantum field theories with an $O(N)$ symmetry which have higher derivative kinetic terms. The Lagrangians in each tower are connected by lying in the same universality class at the $d$-dimensional Wilson-Fisher fixed point. Moreover the universal theory is studied using the large $N$ expansion and we determine $d$-dimensional critical exponents to $O(1/N^2)$. We show that these new universality classes emerge naturally as solutions to the linear relation of the dimensions of the fields deduced from the underlying force-matter interaction of the universal critical theory. To substantiate the equivalence of the Lagrangians in each tower we renormalize each to several loop orders and show that the renormalization group functions are consistent with the large $N$ critical exponents. While we focus on the first two new towers of theories and renormalize the respective Lagrangians to $16$ and $18$ dimensions there are an infinite number of such towers. We also briefly discuss the conformal windows and the extension of the ideas to theories with spin-$\frac{1}{2}$ and spin-$1$ fields as well as the idea of lower dimension completeness.
The renormalization group functions for six dimensional scalar $ϕ^3$ theory with an $F_4$ symmetry are provided at four loops in the modified minimal subtraction (MSbar) scheme. Aside from the anomalous dimension of $ϕ$ and the $β$-function this includes the mass operator and a $ϕ^2$-type operator. The anomalous dimension of the latter is computed explicitly at four loops for the $\mathbf{26}$ and $\mathbf{324}$ representations of $F_4$. The $ε$ expansion of all the related critical exponents are determined to $O(ε^4)$. For instance the value for $Δ_ϕ$ agrees with recent conformal bootstrap estimates in $5$ and $5.95$ dimensions. The renormalization group functions are also provided at four loops for the group $E_6$.
We evaluate the quartic ghost and quark Green's functions as well as the gluon-ghost, gluon-quark and ghost-quark 4-point functions of Quantum Chromodynamics at one loop at the fully symmetric point in a linear covariant gauge. Similar expressions for the analogous Green's functions in Quantum Electrodynamics are also provided.
We renormalize the six dimensional cubic theory with an $O(N)$ $\times$ $O(m)$ symmetry at three loops in the modified minimal subtraction (MSbar) scheme. The theory lies in the same universality class as the four dimensional Landau-Ginzburg-Wilson model. As a check we show that the critical exponents derived from the three loop renormalization group functions at the Wilson-Fisher fixed point are in agreement with the large $N$ $d$-dimensional critical exponents of the underlying universal theory. Having established this connection we analyse the fixed point structure of the perturbative renormalization group functions to estimate the location of the conformal window of the $O(N)$ $\times$ $O(2)$ model.
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 constructing 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.
We review recent activity in the construction of the renormalization group functions for O(N) scalar and gauge theories in six and higher dimensions. The theories lie in their respective universality classes at the Wilson-Fisher fixed point. The critical exponents at this fixed point in the various dimensions are all in agreement with the known exponents determined in the large Nexpansion.
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.
We compute Green's functions with a bilinear quark operator inserted at non-zero momentum for a generalized momentum configuration to two loops. These are required to assist lattice gauge theory measurements of the same quantity in matching to the high energy behaviour. The flavour non-singlet operators considered are the scalar, vector and tensor currents as well as the second moment of the twist-2 Wilson operator used in deep inelastic scattering for the measurement of nucleon structure functions.
The a-function is a proposed quantity defined in even dimensions which has a monotonic behaviour along RG flows, related to the beta-functions via a gradient flow equation. We study the a-function for a general scalar theory in six dimensions, using the beta-functions up to three-loop order for both the MSbar and MOM schemes (the latter presented here for the first time at three loops).
We construct the six dimensional Quantum Chromodynamics (QCD) Lagrangian in a linear covariant gauge and subsequently renormalize it at two loops in the MSbar scheme. The coupling constant corresponding to the gauge interaction is asymptotically free for all numbers of quark fields, Nf. Analysing the beta-functions yields a rich spectrum of fixed points. For instance, the conformal window in the six dimensional theory is at Nf = 16 for the SU(3) colour group. The critical theory structure is similar to that of an O(N) scalar theory in eight dimensions. Using the large N expansion the latter is shown to be in the same universality class as the Heisenberg ferromagnet. Similarly using the large Nf expansion, six dimensional QCD is shown to be in the same class as the two dimensional non-abelian Thirring model and four dimensional QCD. Abelian gauge theories are also renormalized at high loops in six and eight dimensions. It is shown that the gauge parameter only appears in the electron anomalous dimension at one loop, similar to four dimensions.
The vertex structure of QCD fixed in the maximal abelian gauge (MAG) and Curci-Ferrari gauge is analysed at two loops at the fully symmetric point for the 3-point functions corresponding to the three momentum subtraction (MOM) renormalization schemes. Consequently the three loop renormalization group functions are determined for each of these three schemes in each gauge using properties of the renormalization group equation.
We renormalize six dimensional phi^3 theory in the modified minimal subtraction (MSbar) scheme at four loops. From the resulting beta-function, anomalous dimension and mass anomalous dimension we compute four loop critical exponents relevant to the Lee-Yang edge singularity and percolation problems. Using resummation methods and information on the exponents of the relevant two dimensional conformal field theory we obtain estimates for exponents in dimensions 3, 4 and 5 which are in reasonable agreement with other techniques for these two problems. The renormalization group functions for the more general theory with an O(N) symmetry are also computed in order to obtain estimates of exponents at various fixed points in five dimensions. Included in this O(N) analysis is the full evaluation of the mass operator mixing matrix of anomalous dimensions at four loops. We show that its eigen-exponents are in agreement with the mass exponents computed at O(1/N^2) in the non-perturbative large N expansion.
We analyse the critical exponents relating to the quark mass anomalous dimension and beta-function at the Banks-Zaks fixed point in Quantum Chromodynamics (QCD) in a variety of representations for the quark in the momentum subtraction (MOM) schemes of Celmaster and Gonsalves. For a specific range of values of the number of quark flavours, estimates of the exponents appear to be scheme independent. Using the recent five loop modified minimal subtraction (MSbar) scheme quark mass anomalous dimension and estimates of the fixed point location we estimate the associated exponent as 0.263-0.268 for the SU(3) colour group and 12 flavours when the quarks are in the fundamental representation.
We provide the full set of renormalization group functions for the renormalization of QCD in the minimal MOM scheme to four loops for the colour group SU(N_c).
We determine the R-ratio for massless quarks in several renormalization schemes to various loop orders. These are the momentum subtraction schemes of Celmaster and Gonsalves as well as the minimal momentum subtraction scheme. The dependence of the R-ratio on the schemes is analysed.
We compute the two loop corrections to the gluon propagator for low momentum in a gluon mass model. This model has recently been proposed as an alternative to the Gribov construction in the way it handles Gribov copies in the gauge fixing. The corrections provide improvements for estimating the point where the gluon propagator freezes in relation to lattice data.
We compute the full one loop correction to the quartic vertex of QCD at the fully symmetric point. This allows us to define a new momentum subtraction (MOM) scheme in the class of schemes introduced by Celmaster and Gonsalves. Hence using properties of the renormalization group equation, the two loop renormalization group functions for this scheme are given.