Anomalous Dimensions Of Composite Operators Research Articles

Bounds on multiscalar CFTs in the \u03b5 expansion

We study fixed points with N scalar fields in 4 − ε dimensions to leading order in ε using a bottom-up approach. We do so by analyzing O(N) invariants of the quartic coupling λijkl that describes such CFTs. In particular, we show that λiijj and {lambda}_{ijkl}^2 are restricted to a specific domain, refining a result by Rychkov and Stergiou. We also study averages of one-loop anomalous dimensions of composite operators without gradients. In many cases, we are able to show that the O(N) fixed point maximizes such averages. In the final part of this work, we generalize our results to theories with N complex scalars and to bosonic QED. In particular we show that to leading order in ε, there are no bosonic QED fixed points with N < 183 flavors.

N=4 super-Yang-Mills correlators without anticommuting variables

Quantum correlators of pure supersymmetric Yang-Mills theories in D=3,4,6 and 10 dimensions can be reformulated via the non-linear and non-local transformation (`Nicolai map') that maps the full functional measure of the interacting theory to that of a free bosonic theory. As a special application we show that for the maximally extended N=4 theory in four dimensions, and up to order O(g^2), all known results for scalar correlators can be recovered in this way without any use of anti-commuting variables, in terms of a purely bosonic and ghost free functional measure for the gauge fields. This includes in particular the dilatation operator yielding the anomalous dimensions of composite operators. The formalism is thus competitive with more standard perturbative techniques.

Exact results from the geometry of couplings and the effective action

We invent a method that exploits the geometry in the space of couplings and the known all-loop effective action, in order to calculate the exact in the couplings anomalous dimensions of composite operators for a wide class of integrable σ-models. These involve both self and mutually interacting current algebra theories. We work out the details for important classes of such operators. In particular, we consider the operators built solely from an arbitrary number of currents of the same chirality, the composite operators which factorize into a chiral and an anti-chiral part, as well as those made up of three currents of mixed chirality. Remarkably enough, the anomalous dimensions of the former two sets of operators turn out to vanish. In our approach, loop computations are completely avoided.

The local Callan-Symanzik equation: structure and applications

The local Callan-Symanzik equation describes the response of a quantum field theory to local scale transformations in the presence of background sources. The consistency conditions associated with this anomalous equation imply non-trivial relations among the $\beta$-function, the anomalous dimensions of composite operators and the short distance singularities of correlators. In this paper we discuss various aspects of the local Callan-Symanzik equation and present new results regarding the structure of its anomaly. We then use the equation to systematically write the n-point correlators involving the trace of the energy-momentum tensor. We use the latter result to give a fully detailed proof that the UV and IR asymptotics in a neighbourhood of a 4D CFT must also correspond to CFTs. We also clarify the relation between the matrix entering the gradient flow formula for the $\beta$-function and a manifestly positive metric in coupling space associated with matrix elements of the trace of the energy momentum tensor.

Discontinuity relations for the AdS4/CFT3 correspondence

We study in detail the analytic properties of the Thermodynamic Bethe Ansatz (TBA) equations for the anomalous dimensions of composite operators in the planar limit of the 3D N=6 superconformal Chern–Simons gauge theory and derive functional relations for the jump discontinuities across the branch cuts in the complex rapidity plane. These relations encode the analytic structure of the Y functions and are extremely similar to the ones obtained for the previously-studied AdS5/CFT4 case. Together with the Y-system and more basic analyticity conditions, they are completely equivalent to the TBA equations. We expect these results to be useful to derive alternative nonlinear integral equations for the AdS4/CFT3 spectrum.

Anomalous dimensions of composite operators in scalar field theories

The problem of calculation of one-loop anomalous dimensions of composite operators can be reformulated as a spectral problem for the Hamiltonian of an N-particle quantum-mechanical system with pairwise interaction. We give a brief review of the corresponding technique using as examples the scalar field theories φ3 and φ4.As an application we calculate the exact anomalous dimensions for operators of a certain type. Bibliography: 27 titles.

Baxter equation for long-range [formula omitted] magnet

We construct a long-range Baxter equation encoding anomalous dimensions of composite operators in the SL(2|1) sector of N=4 supersymmetric Yang–Mills theory. The formalism is based on the analytical Bethe ansatz. We compare predictions of the Baxter equations for short operators with available multiloop perturbative calculations.

Statistical mechanics for dilatations in [formula omitted] super-Yang–Mills theory

Matrix model describing the anomalous dimensions of composite operators in N = 4 super-Yang–Mills theory up to one-loop level is considered at finite temperature. We compute the thermal effective action for this model, which we define as the log of the partition function restricted to the states of given fixed length and spin. The result is obtained in the limit of high as well as low temperature.

Superspace approach to anomalous dimensions in [formula omitted] SYM

In a N=1 superspace setup and using dimensional regularization, we give a general and simple prescription to compute anomalous dimensions of composite operators in N=4 , SU( N) supersymmetric Yang–Mills theory, perturbatively in the coupling constant g. We show in general that anomalous dimensions are responsible for the appearance of higher order poles in the perturbative expansion of the two-point function and that their lowest contribution can be read directly from the coefficient of the 1/ ϵ 2 pole. As a check of our procedure we rederive the anomalous dimension of the Konishi superfield at order g 2. We then apply this procedure to the case of the double trace, dimension 4, superfield in the 20 of SU(4) recently considered in the literature. We find that its anomalous dimension vanishes for all N in agreement with previous results.

NLO evolution for large scale distances, positivity constraints and the low-energy model of the nucleon

We present a computationally reliable and accurate method for solving the Gribov-Lipatov-Altarelli-Parisi equations at next to leading order, both in the non-singlet and in the singlet case. It requires solving numerically the renormalization group equations for the anomalous dimensions of composite operators in the complex plane, and finally performing an inverse Mellin transformation. In this way the group property of renormalization is exactly preserved, i.e. performing two successive scale transformations coincides exactly with a direct one making parton distributions independent of the integration path used to connect two different scales. This is relevant when large scale differences are involved and makes upward or downward evolution fully equivalent. Thus, it becomes possible to evolve the known parton distributions and leading twist contributions to the structure functions from Q 2 = m b 2 to the lowest possible scale imposed by positivity and unitarity.

QED logarithms in the electroweak corrections to the muon anomalous magnetic moment

We employ an effective Lagrangian approach to derive the leading-logarithm two-loop electroweak contributions to the muon anomalous magnetic moment, a_mu. We show that these corrections can be obtained using known results on the anomalous dimensions of composite operators. We confirm the result of Czarnecki et al. for the bosonic part and present the complete sin^2 \theta_W dependence of the fermionic contribution. The approach is then used to compute the leading-logarithm three-loop electroweak contribution to a_mu. Finally we derive, in a fairly model-independent way, the QED improvement of new-physics contributions to a_mu and to the electric dipole moment (EDM) of the electron. We find that the QED corrections reduce the effect of new physics at the electroweak scale by 6% (for a_mu) and by 11% (for the electron EDM).

A simple scheme for the calculation of the anomalous dimensions of composite operators in the 1/ N expansion

A simple method for the calculation of the anomalous dimensions of composite operators up to order 1/ N 2 is developed. We demonstrate the effectiveness of this approach by computing the critical exponents of the (⊗ φ) s and ( φ ⊗ (⊗ ∂) n φ) operators to order 1/ N 2 in the non-linear σ-model. The special simplifications due to the conformal invariance of the model are discussed.

Generic scaling relation in the scalar model

The results of an analysis of the one-loop spectrum of the anomalous dimensions of composite operators in the scalar model are presented. We give a rigorous constructive proof of the hypothesis on the hierarchical structure of the spectrum of anomalous dimensions - the naive sum of any two anomalous dimensions generates a limit point in the spectrum. Arguments in favour of the non-perturbative character of this result and the possible ways of generalizing it to other field theories are briefly discussed.

The spectrum of the anomalous dimensions of composite operators in the ge-expansion in the N-vector models

The spectrum of the anomalous dimensions of the composite operators (with an arbitrary number of fields n and derivatives l) in the N-vector model in first order of the ge-expansion is investigated. The exact solution for the operators with the number of fields ⩽ 4 is presented. The behaviour of the anomalous dimensions in the large- l limit has been analyzed. The qualitative description of the spectrum structure for the operators with an arbitrary number of fields is given.

Conformal symmetry and the spectrum of anomalous dimensions in the N-vector model in 4−ϵ dimensions

The subject of this paper is to study the critical N-vector model in 4− ϵ dimensions in one-loop order. We analyse the spectrum of anomalous dimensions of composite operators with an arbitrary number of fields and gradients. For composite operators with three elementary fields and gradients we work out the complete spectrum of anomalous dimensions, thus extending the old solution of Wilson and Kogut for two fields and gradients. In the general case we prove some properties of the spectrum, in particular a lower limit 0 + O( ϵ 2). Thus one-loop contributions generally improve the stability of the nontrivial fixed point in contrast to some 2 + ϵ expansions. Furthermore we explicitly find conformal invariance at the nontrivial fixed point.

Finite fixed points - rule rather than exception

We calculate anomalous dimensions of composite operators in 2+1 dimensional scalar, Yukawa and QED theories. In the strong coupling regime the anomalous dimensions are large and some composite operators entering the lagrangian become irrelevant. Theories flow to nonperturbative fixed points. The renormalizable theories describing this limit are correspondingly the nonlinear σ-model, the four-Fermi interaction and the “chargeless” QED.

Anomalous Dimensions of Composite Operators in the Gross-Neveu Model in 2+ Dimensions

Anomalous Dimensions of Composite Operators in the Gross-Neveu Model in 2+ Dimensions