One-loop Anomalous Dimensions Research Articles

Integrable spin chain and operator mixing in [formula omitted] supersymmetric theories

We study operator mixing, due to planar one-loop corrections, for composite operators in D=4 supersymmetric theories. We present some N=1,2 Yang–Mills and Wess–Zumino models, in which the planar one-loop anomalous dimension matrix in the sector of holomorphic scalars is identified with the Hamiltonian of an integrable quantum spin chain with SU(3) or SU(2) symmetry, even if the theory is away from the conformal points. This points to a more universal origin of the integrable structure beyond superconformal symmetry. We also emphasize the role of the superpotential in the appearance of the integrable structure. The computations of operator mixing in our examples by solving Bethe ansatz equations show some new features absent in N=4 SYM.

Yang-Mills duals for semiclassical strings onAdS5×S5

We consider a semiclassical multiwrapped circular string pulsating on S_5, whose center of mass has angular momentum J on an S_3 subspace. Using the AdS/CFT correspondence we argue that the one-loop anomalous dimension of the dual operator is a simple rational function of J/L, where J is the R-charge and L is the bare dimension of the operator. We then reproduce this result directly from a super Yang-Mills computation, where we make use of the integrability of the one-loop system to set up an integral equation that we solve. We then verify the results of Frolov and Tseytlin for circular rotating strings with R-charge assignment (J',J',J). In this case we solve for an integral equation found in the O(-1) matrix model when J'< J and the O(+1) matrix model if J'> J. The latter region starts at J'=L/2 and continues down, but an apparent critical point is reached at J'=4J. We argue that the critical point is just an artifact of the Bethe ansatz and that the conserved charges of the underlying integrable model are analytic for all J' and that the results from the O(-1) model continue onto the results of the O(+1) model.

The complete one-loop dilatation operator of [formula omitted] super-Yang–Mills theory

We continue the analysis of hep-th/0303060 in the one-loop sector and present the complete psu(2,2|4) dilatation operator of N=4 super-Yang–Mills theory. This operator generates the matrix of one-loop anomalous dimensions for all local operators in the theory. Using an oscillator representation we show how to apply the dilatation generator to a generic state. By way of example, we determine the planar anomalous dimensions of all operators up to and including dimension 5.5, where we also find some evidence for integrability. Finally, we investigate a number of subsectors of N=4 SYM in which the dilatation operator simplifies. Among these we find the previously considered so(6) and su(2) subsectors, a su(2|4) subsector isomorphic to the BMN matrix model at one-loop, a su(2|3) supersymmetric subsector of nearly eighth-BPS states and, last but not least, a non-compact sl(2) subsector whose dilatation operator lifts uniquely to the full theory.

Spinning strings in AdS5× S5 and integrable systems

We show that solitonic solutions of the classical string action on the AdS 5× S 5 background that carry charges (spins) of the Cartan subalgebra of the global symmetry group can be classified in terms of periodic solutions of the Neumann integrable system. We derive equations which determine the energy of these solitons as a function of spins. In the limit of large spins J, the first subleading 1/ J coefficient in the expansion of the string energy is expected to be non-renormalized to all orders in the inverse string tension expansion and thus can be directly compared to the 1-loop anomalous dimensions of the corresponding composite operators in N=4 super-YM theory. We obtain a closed system of equations that determines this subleading coefficient and, therefore, the 1-loop anomalous dimensions of the dual SYM operators. We expect that an equivalent system of equations should follow from the thermodynamic limit of the algebraic Bethe ansatz for the SO(6) spin chain derived from SYM theory. We also identify a particular string solution whose classical energy exactly reproduces the one-loop anomalous dimension of a certain set of SYM operators with two independent R charges J 1, J 2.

Plane-wave matrix theory from [formula omitted] super-Yang–Mills on [formula omitted

Recently a mass deformation of the maximally supersymmetric Yang–Mills quantum mechanics has been constructed from the supermembrane action in eleven-dimensional plane-wave backgrounds. However, the origin of this plane-wave matrix theory in terms of a compactification of a higher-dimensional super-Yang–Mills model has remained obscure. In this paper we study the Kaluza–Klein reduction of D=4, N=4 super-Yang–Mills theory on a round three-sphere, and demonstrate that the plane-wave matrix theory arises through a consistent truncation to the lowest lying modes. We further explore the relation between the dilatation operator of the conformal field theory and the Hamiltonian of the quantum mechanics through perturbative calculations up to two-loop order. In particular, we find that the one-loop anomalous dimensions of pure scalar operators are completely captured by the plane-wave matrix theory. At two-loop level this property ceases to exist.

The [formula omitted] SYM integrable super spin chain

Recently it was established that the one-loop planar dilatation generator of N=4 super-Yang–Mills theory may be identified, in some restricted cases, with the Hamiltonians of various integrable quantum spin chains. In particular Minahan and Zarembo established that the restriction to scalar operators leads to an integrable vector so(6) chain, while recent work in QCD suggested that restricting to twist operators, containing mostly covariant derivatives, yields certain integrable Heisenberg XXX chains with non-compact spin symmetry sl(2) . Here we unify and generalize these insights and argue that the complete one-loop planar dilatation generator of N=4 is described by an integrable su(2,2|4) super spin chain. We also write down various forms of the associated Bethe ansatz equations, whose solutions are in one-to-one correspondence with the complete set of all one-loop planar anomalous dimensions in the N=4 gauge theory. We finally speculate on the non-perturbative extension of these integrable structures, which appears to involve non-local deformations of the conserved charges.

Stringing spins and spinning strings

We apply recently developed integrable spin chain and dilatation operator techniques in order to compute the planar one-loop anomalous dimensions for certain operators containing a large number of scalar fields in N =4 Super Yang-Mills. The first set of operators, belonging to the SO(6) representations [J,L-2J,J], interpolate smoothly between the BMN case of two impurities (J=2) and the extreme case where the number of impurities equals half the total number of fields (J=L/2). The result for this particular [J,0,J] operator is smaller than the anomalous dimension derived by Frolov and Tseytlin [hep-th/0304255] for a semiclassical string configuration which is the dual of a gauge invariant operator in the same representation. We then identify a second set of operators which also belong to [J,L-2J,J] representations, but which do not have a BMN limit. In this case the anomalous dimension of the [J,0,J] operator does match the Frolov-Tseytlin prediction. We also show that the fluctuation spectra for this [J,0,J] operator is consistent with the string prediction.

Nonperturbative propagators, running coupling, and the dynamical quark mass of Landau gauge QCD

The coupled system of renormalized Dyson-Schwinger equations for the quark, gluon, and ghost propagators of the Landau gauge QCD is solved within truncation schemes. These employ bare as well as nonperturbative Ans\"atze for the vertices such that the running coupling as well as the quark mass function are independent of the renormalization point. The one-loop anomalous dimensions of all propagators are reproduced. Dynamical chiral symmetry breaking is found; the dynamically generated quark mass agrees well with the phenomenological values, and the corresponding results from lattice calculations. The effects of unquenching the system are small. In particular the infrared behavior of the ghost and gluon dressing functions found in previous studies is almost unchanged as long as the number of light flavors is smaller than four.

The Bethe-ansatz for Script N = 4 super Yang-Mills

We derive the one loop mixing matrix for anomalous dimensions in N=4 Super Yang-Mills. We show that this matrix can be identified with the Hamiltonian of an integrable SO(6) spin chain with vector sites. We then use the Bethe ansatz to find a recipe for computing anomalous dimensions for a wide range of operators. We give exact results for BMN operators with two impurities and results up to and including first order 1/J corrections for BMN operators with many impurities. We then use a result of Reshetikhin's to find the exact one-loop anomalous dimension for an SO(6) singlet in the limit of large bare dimension. We also show that this last anomalous dimension is proportional to the square root of the string level in the weak coupling limit.

BMN gauge theory as a quantum mechanical system

We rigorously derive an effective quantum mechanical Hamiltonian from N=4 gauge theory in the BMN limit. Its eigenvalues yield the exact one-loop anomalous dimensions of scalar two-impurity BMN operators for all genera. It is demonstrated that this reformulation vastly simplifies computations. E.g., the known anomalous dimension formula for genus one is reproduced through a one-line calculation. We also efficiently evaluate the genus two correction, finding a non-vanishing result. We comment on multi-trace two-impurity operators and we conjecture that our quantum-mechanical reformulation could be extended to higher quantum loops and more impurities.

A new double-scaling limit of [formula omitted] super-Yang–Mills theory and pp-wave strings

The metric of a spacetime with a parallel plane (pp)-wave can be obtained in a certain limit of the space AdS 5 × S 5. According to the AdS/CFT correspondence, the holographic dual of superstring theory on that background should be the analogous limit of N=4 supersymmetric Yang–Mills theory. In this paper we shall show that, contrary to widespread expectation, non-planar diagrams survive this limiting procedure in the gauge theory. Using matrix model techniques as well as combinatorial reasoning it is demonstrated that a subset of diagrams of arbitrary genus survives and that a non-trivial double scaling limit may be defined. We exactly compute two- and three-point functions of chiral primaries in this limit. We also carefully study certain operators conjectured to correspond to string excitations on the pp-wave background. We find non-planar linear mixing of these proposed operators, requiring their redefinition. Finally, we show that the redefined operators receive non-planar corrections to the planar one-loop anomalous dimension.

On the logarithmic behaviour in Script N = 4 SYM theory

We show that the logarithmic behaviour seen in perturbative and non perturbative contributions to Green functions of gauge-invariant composite operators in N=4 SYM with SU(N) gauge group can be consistently interpreted in terms of anomalous dimensions of unprotected operators in long multiplets of the superconformal group SU(2,2|4). In order to illustrate the point we analyse the short-distance behaviour of a particularly simple four-point Green function of the lowest scalar components of the N=4 supercurrent multiplet. Assuming the validity of the Operator Product Expansion, we are able to reproduce the known value of the one-loop anomalous dimension of the single-trace operators in the Konishi supermultiplet. We also show that it does not receive any non-perturbative contribution from the one-instanton sector. We briefly comment on double- and multi-trace operators and on the bearing of our results on the AdS/SCFT correspondence.

Calculation of the continuum-lattice HQET matching for the complete basis of four-fermion operators: reanalysis of the [formula omitted] mixing

In this work, we find the expressions of continuum HQET four-fermion operators in terms of lattice operators in perturbation theory. To do so, we calculate the one-loop continuum-lattice HQET matching for the complete basis of ΔB = 2 and ΔB = 0 operators (excluding penguin diagrams), extending and completing previous studies. We have also corrected some errors in previous evaluations of the matching for the operator O LL . Our results are relevant to the lattice computation of the values of unknown hadronic matrix elements which enter in many very important theoretical predictions in B-meson phenomenology: B 0− B 0 mixing, Γ B and τ B s lifetimes, SUSY effects in ΔB = 2 transitions and the B s width difference Δγ B s . We have reanlyzed our lattice data for the B B parameter of the B 0− B 0 mixing on 600 lattices of size 24 3 × 40 at β = 6.0 computed with the SW-Clover and HQET lattice actions. We have used the correct lattice-comtinuum matching factors and boosted perturbation theory with tadpole improved heavy-light operators to reduce the systematic error in the evaluation of the renormalization constants. Our best estimate of the renormalization scale independent B-parameter is B ̂ B = 1.29 ± 0.08 ± 0.06 , where the first error is statistical and the second is systematic coming from the uncertainty in the determination of the renormalization constants. Our result is in good agreement with previous results obtained by extrapolating Wilson data. As a byproduct, we also obtain the complete one-loop anomalous dimension matrix for four-fermion operators in the HQET.

Evolution of the light-cone distribution function for a heavy quark

We compute the one-loop anomalous dimension for the light-cone distribution function of a heavy quark and solve the corresponding evolution equation analytically. Some implications of the results for inclusive B decays are discussed.

Renormalization of chiral-even twist-3 light-cone wave functions for vector mesons in QCD

We present the one-loop anomalous dimension matrices for the chiral-even twist-3 (nonsinglet) conformal operators, which govern the scale-dependence of the vector meson light-cone wave functions through the conformal expansion. It is clarified that the constraints from the charge-conjugation invariance and the chirality conservation allow only one independent anomalous dimension matrix for each conformal spin.

Next-to-leading orderQ2evolution of the transversity distributionh1(x,Q2)

We present a calculation of the two-loop anomalous dimension for the transversity distribution h_1(x,Q^2), $\gamma^{h(1)}_n$, in the MS scheme of the dimensional regularization. Due to the chiral-odd nature, h_1 does not mix with the gluon distributions, and thus our result is the same for the flavor-singlet and nonsinglet distributions. At small n (moment of h_1), $\gamma^{h(1)}_n$ is significantly larger than $\gamma^{f(1)}_n$ (the anomalous dimension for the nonsinglet f_1), but approaches $\gamma^{f(1)}_n$ very quickly at large n, keeping the relation $\gamma^{h(1)}_n > \gamma^{f(1)}_n$. This feature is in parallel to the relation between the one-loop anomalous dimension for f_1 and h_1.

Renormalization of Twist-Four Operators in QCD Bjorken and Ellis–Jaffe Sum Rules

The QCD effects of twist-four operators on the first moment of nucleon spin-dependent structure function g1(x,Q2) are studied in the framework of operator product expansion and renormalization group method. We investigate the operator mixing through renormalization of the twist-four operators including those proportional to the equation of motion by evaluating off-shell Green's functions in the usual covariant gauge as well as in the background gauge. Through this procedure we extract the one-loop anomalous dimension of the spin-1 and twist-four operator which determines the logarithmic correction to the 1/Q2 behavior of the contribution from the twist-four operators to the first moment of g1(x,Q2).

Baryonic currents and their correlators in the heavy quark effective theory

HQET currents with the quantum numbers of the ground-state baryons are discussed. One-loop anomalous dimensions are calculated, and exact one-loop matching to QCD currents is found. Two-point correlators of these currents are calculated taking into account d⩽9 terms of the OPE. Sum rules for heavy baryons Λ Q and Σ Q (∗) are analysed.

Ultraviolet divergences in1Nexpansions of asymptotically free theories

We calculate the ultraviolet divergences of the nonlinear sigma and Gross-Neveu models in the 1/N expansion near two dimensions (d = 2-epsilon). Beyond the leading order the theories develop logarithmic as well as pole singularities in epsilon. Identical divergences occur when the renormalization constants calculated in perturbation theory are expanded in powers of 1/N rather than in powers of the coupling constants. The necessary infinite renormalizations of the 1/N expansions are completely determined by the two-loop ..beta.. functions and one-loop anomalous dimensions of perturbation theory. These results can be extended to non-Abelian gauge theories in four dimensions which admit 1/N expansions.