arXiv subjects
Publications and source records attributed to J. A. Gracey.
The anomalous dimensions of the general flavour non-singlet quark bilinear currents ψγ^{[μ_1} ... γ^{μ_n]} ψare computed at three loops in quantum chromodynamics in the minimal subtraction scheme. The dimension of the tensor current emerges for the case n = 2 and the anomalous dimension for the general flavour singlet current is also discussed.
We present the formalism to calculate d-dimensional critical exponents in QCD in the large N_f expansion where N_f is the number of quark flavours. It relies in part on demonstrating that at the d-dimensional fixed point of QCD the critical theory is equivalent to a non-abelian version of the Thirring model. We describe the techniques used to compute critical two and three loop Feynman diagrams and as an application determine the quark wave function, eta, and mass renormalization critical exponents at O(1/N_f^2) in d-dimensions. Their values when expressed in relation to four dimensional perturbation theory are in exact agreement with the known four loop MSbar results. Moreover, new coefficients in these renormalization group functions are determined to six loops and O(1/N_f^2). The computation of the exponents in the Schwinger Dyson approach is also provided and an expression for eta in arbitrary covariant gauge is given.
We renormalize to three loops a version of the Thirring model where the fermion fields not only lie in the fundamental representation of a non-abelian colour group SU(N_c) but also depend on the number of flavours, N_f. The model is not multiplicatively renormalizable in dimensional regularization due to the generation of evanescent operators which emerge at each loop order. Their effect in the construction of the true wave function, mass and coupling constant renormalization constants is handled by considering the projection technique to a new order. Having constructed the MSbar renormalization group functions we consider other massless independent renormalization schemes to ensure that the renormalization is consistent with the equivalence of the non-abelian Thirring model with other models with a four-fermi interaction. One feature to emerge from the computation is the establishment of the fact that the SU(N_f) Gross Neveu model is not multiplicatively renormalizable in dimensional regularization. An evanescent operator arises first at three loops and we determine its associated renormalization constant explicitly.
We describe the evaluation of the anomalous dimensions of twist-2 deep inelastic light cone operators to O(1/N_f) as a check on future perturbative calculations. In particular we present recent results for the singlet gluonic operator dimension in polarised and unpolarised scattering and give three loop predictions for the O(1/N_f) gluonic eigenoperator. The section of the 3-loop O(1/N_f) DGLAP splitting function proportional to the adjoint quadratic Casimir is also calculated for the singlet gluonic operators.
We compute the O(1/N_f) correction to the predominantly gluonic flavour singlet twist-2 anomalous dimension used in polarized deep inelastic scattering. It is consistent with known two loop perturbation theory and we determine the three loop contribution at O(1/N_f). The treatment of the epsilon-tensor in the large N_f d-dimensional critical point formalism is also discussed.
We solve the Schwinger Dyson equations of the O(N) symmetric Wess-Zumino model at O(1/N^3) at the non-trivial fixed point of the d-dimensional beta-function and deduce a critical exponent for the wave function renormalization at this order. By developing the epsilon-expansion of the result, which agrees with known perturbation theory, we examine the distribution of transcendental coefficients and show that only the Riemann zeta series arises at this order in 1/N. Unlike the analogous calculation at the same order in the bosonic O(N) phi^4-theory non-zeta transcendentals, associated with for example the (3,4)-torus knot, cancel.
We extend the critical point self-consistency method used to solve field theories at their d-dimensional fixed point in the large N expansion to include superfields. As an application we compute the beta-function of the Wess-Zumino model with an O(N) symmetry to O(1/N^2). This result is then used to study the effect the higher order corrections have on the radius of convergence of the 4-dimensional beta-function at this order in 1/N. The critical exponent relating to the wave function renormalization of the basic field is also computed to O(1/N^2) and is shown to be the same as that for the corresponding field in the supersymmetric O(N) sigma model in d-dimensions. We discuss how the non-renormalization theorem prevents the full critical point equivalence between both models.
Using large N_f methods we compute the anomalous dimension of the predominantly gluonic flavour singlet twist-2 composite operator which arises in the operator product expansion used in deep inelastic scattering. We obtain a d-dimensional expression for it which depends on the operator moment n. Its expansion in powers of epsilon = (4-d)/2 agrees with the explicit exact three loop MSbar results available for n less than or equal to 8 and allows us to determine some new information on the explicit n-dependence of the three and higher order coefficients. In particular the n-dependence of the three loop anomalous dimension gamma_{gg}(a) is determined in the C_2(G) sector at O(1/N_f).
We compute the anomalous dimensions of a set of composite operators which involve derivatives at four loops in MSbar in phi^4 theory as a function of the operator moment n. These operators are similar to the twist-2 operators which arise in QCD in the operator product expansion in deep inelastic scattering. By regarding their inverse Mellin transform as being equivalent to the DGLAP splitting functions we explore to what extent taking a restricted set of operator moments can give a good approximation to the exact four loop result.
We review the application of the critical point large N_f self-consistency method to QCD. In particular we derive the O(1/N_f) d-dimensional critical exponents whose epsilon-expansion determines the perturbative coefficients in MSbar of the field dimensions, beta-function and various twist-2 operators which occur in the operator product expansion of deep inelastic scattering.
We report on the progress in the computation of the beta-functions of phi^4 theory and QCD in the large N expansion. For the former we give an analytic formula for the critical exponent which encodes higher order coefficients in the series beyond those currently known in the MSbar scheme at O(1/N^2) in a large N expansion. This allows us to deduce new coefficients in the field anomalous dimension which relate to Kreimer's knot theory of the divergences of a quantum field theory. A similar exponent is computed for the beta-function of QCD at O(1/N_f).
Critical exponents are computed for a variety of twist-2 composite operators, which occur in polarized and unpolarized deep inelastic scattering, at leading order in the 1/N_f expansion. The resulting d-dimensional expressions, which depend on the moment of the operator, are in agreement with recent explicit two and three loop perturbative calculations. An interesting aspect of the critical point approach which is used, is that the anomalous dimensions of the flavour singlet eigenoperators, which diagonalize the perturbative mixing matrix, are computed directly. We also elucidate the treatment of gamma^5 at the fixed point which is important in simplifying the calculation for polarized operators. Finally, the anomalous dimension of the singlet axial current is determined at O(1/N_f) by considering the renormalization of the anomaly in operator form.
Counterterms that are not reducible to $ζ_{n}$ are generated by ${}_3F_2$ hypergeometric series arising from diagrams for which triangle and uniqueness relations furnish insufficient data. Irreducible double sums, corresponding to the torus knots $(4,3)=8_{19}$ and $(5,3)=10_{124}$, are found in anomalous dimensions at ${\rm O}(1/N^3)$ in the large-$N$ limit, which we compute analytically up to terms of level 11, corresponding to 11 loops for 4-dimensional field theories and 12 loops for 2-dimensional theories. High-precision numerical results are obtained up to 24 loops and used in Padé resummations of $\varepsilon$-expansions, which are compared with analytical results in 3 dimensions. The ${\rm O}(1/N^3)$ results entail knots generated by three dressed propagators in the master two-loop two-point diagram. At higher orders in $1/N$ one encounters the uniquely positive hyperbolic 11-crossing knot, associated with an irreducible triple sum. At 12 crossings, a pair of 3-braid knots is generated, corresponding to a pair of irreducible double sums with alternating signs. The hyperbolic positive knots $10_{139}$ and $10_{152}$ are not generated by such self-energy insertions.
The large N_f self-consistency programme is reviewed. As an application the QCD beta-function is computed at O(1/N_f) and the anomalous dimensions of polarized twist-2 singlet operators are determined at the same order.
The leading order coefficients of the beta-function of QCD are computed in a large N_f expansion. They are in agreement with the three loop MSbar calculation. The method involves computing the anomalous dimension of the operator (G^2_{mu nu})^2 at the d-dimensional fixed point in the non-abelian Thirring model to which QCD is equaivalent in this limit. The effect the O(1/N_f) corrections have on the location of the infrared stable fixed point for a range of N_f is also examined.
The large N_f self consistency method is applied to the computation of perturbative information in the operator product expansion used in deep inelastic scattering. The O(1/N_f) critical exponents corresponding to the anomalous dimensions of the twist 2 non-singlet and singlet operators are computed analytically as well as the non-singlet structure functions. The results are in agreement with recent explicit perturbative calculations.
We use the large $N$ self consistency method to compute the critical exponents of the fields and coupling of the supersymmetric CP(N) sigma model at leading order in $1/N$ in various dimensions. We verify that the correction to the critical beta-function slope vanishes at O(1/N) which is consistent with supersymmetry. The three dimensional model is investigated explicitly when a Chern Simons term is included supersymmetrically. We determine the modification that this has on the gauge independent quantity β^\prime(g_c) as a function of the Chern Simons coupling, \vartheta. For an N = 2 supersymmetric Chern Simons term the exponent is independent of \vartheta at O(1/N), whilst it is invariant under \vartheta \rightarrow 1/\vartheta when an N = 1 supersymmetric Chern Simons term is included.
Using the full conformal bootstrap method an analytic expression is given in d-dimensions for the anomalous dimension of the fermion at O(1/N^3) in a large N expansion of the Nambu--Jona-Lasinio model with SU(2) x SU(2) continuous chiral symmetry.