Gauge-invariant Composite Operators Research Articles

Gauge invariant operators in the SU(2) Higgs model: Ward identities and renormalization

The renormalization properties of two local gauge invariant composite operators $(O,{R}_{\ensuremath{\mu}}^{a})$ corresponding, respectively, to the gauge invariant description of the Higgs particle and of the massive gauge vector boson, are analyzed to all orders in perturbation theory by means of the algebraic renormalization in the $SU(2)$ Higgs model, with a single scalar in the fundamental representation, when quantized in the Landau gauge in Euclidean space-time. The present analysis generalizes earlier results presented in the case of the $U(1)$ Higgs model. A powerful global Ward identity, related to an exact custodial symmetry, is derived for the first time, with deep consequences at the quantum level. In particular, the gauge invariant vector operators ${R}_{\ensuremath{\mu}}^{a}$ turn out to be the conserved Noether currents of the above-mentioned custodial symmetry. As such, these composite operators do not renormalize, as expressed by the fact that the renormalization $Z$-factors of the corresponding external sources, needed to define the operators ${R}_{\ensuremath{\mu}}^{a}$ at the quantum level, do not receive any quantum corrections. Using this Ward identity, one can also prove that the longitudinal component of the two-point correlation function $⟨{R}_{\ensuremath{\mu}}^{a}(p){R}_{\ensuremath{\nu}}^{b}(\ensuremath{-}p)⟩$ exhibits only a tree level nonvanishing contribution which, moreover, is momentum independent, thus it is not associated to any physical propagating mode. Finally, we point out that the renowned nonrenormalization theorem for the ghost-antighost-vector boson vertex in Landau gauge remains true to all orders, also in the presence of the Higgs field.

Spectral properties of local gauge invariant composite operators in the SU(2) Yang\u2013Mills\u2013Higgs model

The spectral properties of a set of local gauge (BRST) invariant composite operators are investigated in the SU(2) Yang–Mills–Higgs model with a single Higgs field in the fundamental representation, quantized in the ’t Hooft R_{xi }-gauge. These operators can be thought of as a BRST invariant version of the elementary fields of the theory, the Higgs and gauge fields, with which they share a gauge independent pole mass. The two-point correlation functions of both BRST invariant composite operators and elementary fields, as well as their spectral functions, are investigated at one-loop order. It is shown that the spectral functions of the elementary fields suffer from a strong unphysical dependence from the gauge parameter xi , and can even exhibit positivity violating behaviour. In contrast, the BRST invariant local operators exhibit a well defined positive spectral density.

Polylogarithms from the Bound-State S-matrix

Higher-point functions of gauge invariant composite operators in N=4 super Yang-Mills theory can be computed via triangulation. The elementary tile in this process is the hexagon introduced for the evaluation of structure constants. A glueing procedure welding the tiles back together is needed to return to the original object. In this note we present work in progress on n-point functions of BPS operators. In this case, quantum corrections are entirely carried by the glueing procedure. The lowest non-elementary process is the glueing of three adjacent tiles by the exchange of two single magnons. This problem has been analysed before. With a view to resolving some conceptional questions and to generalising to higher processes we are trying to develop an algorithmic approach using the representation of hypergeometric sums as integrals over Euler kernels.

Multi-particle finite-volume effects for hexagon tessellations

Correlation functions of gauge-invariant composite operators in mathcal{N} = 4 super Yang-Mills theory can be computed by integrability using triangulations. The elementary tile in this process is the hexagon, which should be glued by appropriately inserting resolutions of the identity involving virtual (“mirror”) magnons. We consider this problem for five-point functions of protected operators. At one-loop in the ’t Hooft coupling, it is necessary to glue three adjacent tiles which involves two virtual magnons scattering among each other. We show that the result can be simplified by using an adapted mirror rotation and employing appropriate summation techniques. The mirror-particle contributions then yield hyperlogarithms of weight two. Finally, we use these results to investigate braiding prescriptions introduced in earlier work on the problem.

Gauge-invariant spectral description of the U (1) Higgs model from local composite operators

The spectral properties of a set of local gauge-invariant composite operators are investigated in the U(1) Higgs model quantized in the ’t Hooft Rξ gauge. These operators enable us to give a gauge-invariant description of the spectrum of the theory, thereby surpassing certain incommodities when using the standard elementary fields. The corresponding two-point correlation functions are evaluated at one-loop order and their spectral functions are obtained explicitly. As expected, the above mentioned correlation functions are independent from the gauge parameter ξ, while exhibiting positive spectral densities as well as gauge-invariant pole masses corresponding to the massive photon and Higgs physical excitations.

Exploratory study of the off-shell properties of the weak vector bosons

Gauge invariance requires even in the weak interactions that physical, observable particles are described by gauge-invariant composite operators. Such operators have the same structure as those describing bound states, and consequently the physical versions of the $W^\pm$, the $Z$, and the Higgs should have some kind of substructure. To test this consequence, we use lattice gauge theory to study the physical weak vector bosons off-shell, especially their form-factor and weak radius, and compare the results to the ones for the elementary particles. We find that the physical particles show substantial deviations from the structure of a point-like particle. At the same time the gauge-dependent elementary particles exhibit unphysical behavior.

Gauge invariant composite operators of QED in the exact renormalization group formalism

Using the exact renormalization group (ERG) formalism, we study the gauge invariant composite operators in QED. Gauge invariant composite operators are introduced as infinitesimal changes of the gauge invariant Wilson action. We examine the dependence on the gauge fixing parameter of both the Wilson action and gauge invariant composite operators. After defining ‘gauge fixing parameter independence,’ we show that any gauge independent composite operators can be made ‘gauge fixing parameter independent’ by appropriate normalization. As an application, we give a concise but careful proof of the Adler–Bardeen non-renormalization theorem for the axial anomaly in an arbitrary covariant gauge by extending the original proof by A Zee.

On a discrete symmetry of the Bremsstrahlung function in $ \mathcal{N} $ = 4 SYM

We consider the quark anti-quark potential on the three sphere in planar N = 4 SYM and the associated vacuum potential in the near BPS limit with L units of R-charge. The associated Bremsstrahlung function BL has been recently computed analytically by means of the Thermodynamical Bethe Ansatz. We discuss it at strong coupling by computing it at large but nite L. We provide strong support to a special symmetry of the Bremsstrahlung function under the formal discrete Z2 symmetry L ! 1 L. In this context, it is the counterpart of the reciprocity invariance discovered in the past in the spectrum of various gauge invariant composite operators. The Z2 symmetry has remarkable consequences in the scaling limit where L is taken to be large with xed ratio to the 't Hooft coupling. This limit organizes in inverse powers of the coupling and resembles the semiclassical expansion of the dual string theory which is indeed known to capture the leading classical term. We show that the various higher-order contributions to the Bremsstrahlung function obey several constraints and, in particular, the next-to-leading term, formally associated with the string one-loop correction, is completely determined by the classical contribution. The large L limit at strong coupling is also discussed.

Study of the lightest glueball states inSU(2)Euclidean Yang-Mills theory in the maximal Abelian gauge

A qualitative study of the lightest glueball states in Euclidean $SU(2)$ Yang-Mills theory quantized in the maximal Abelian gauge is presented. The analysis is done by generalizing to the maximal Abelian gauge the so-called replica model, already successfully introduced in the Landau gauge. As it will be shown, the gluon and ghost propagators obtained from the replica model are of the same type of those already introduced in [M. A. L. Capri, V. E. R. Lemes, R. F. Sobreiro, S. P. Sorella, and R. Thibes, Phys. Rev. D 77, 105023 (2008).], whose behavior turns out to be in agreement with that available from the lattice data on the maximal Abelian gauge. The model turns out to be renormalizable to all orders, while enabling us to introduce gauge-invariant composite operators for the study of the lightest glueballs ${J}^{PC}={0}^{++}$, ${2}^{++}$ and ${0}^{\ensuremath{-}+}$. The spectral representation for the correlation functions of these operators are evaluated to the first order, and the corresponding spectral densities are shown to be positive. Under the assumption of Abelian dominance, it turns out that the hierarchy for the masses of the lightest glueballs in the maximal Abelian gauge is in agreement with that already obtained in the Landau gauge, a feature which provides evidence for the gauge independence of the spectrum of the theory.

Matrix at slow roll: nonrelativistic and perturbative

We analyze the slow roll dynamics of the massive version of Yang–Mills-type matrix mechanics. We find that one can reproduce in this limit the one-loop non-Hermitian matrix model, describing the dilatations of the local gauge invariant composite operators in the scalar sector of super Yang–Mills theory. This possibility is explained through the fact that the dilatation operator of the last can be identified with the radial-time Hamiltonian. We also explore the finite mass corrections to the slow roll action.

INSTABILITY IN SCALAR CHANNEL OF FERMION–ANTIFERMION SCATTERING AMPLITUDE IN MASSLESS QED3 AND ANOMALOUS DIMENSIONS OF COMPOSITE OPERATORS

Instability in the scalar channel of the fermion–antifermion scattering amplitude in massless QED3 for number of flavors less than the critical value 128/3π2 is demonstrated. The anomalous dimensions of gauge-invariant composite operators are determined to O(1/N). Exponentiation of the O(1/N) infrared logarithm is explicitly demonstrated by evaluating the contribution of the ladder diagrams.

On instanton contributions to anomalous dimensions in [formula omitted] supersymmetric Yang–Mills theory

Instanton contributions to the anomalous dimensions of gauge-invariant composite operators in the N=4 supersymmetric SU( N) Yang–Mills theory are studied in the one-instanton sector. Independent sets of scalar operators of bare dimension Δ 0⩽5 are constructed in all the allowed representations of the SU(4) R-symmetry group and their two-point functions are computed in the semiclassical approximation. Analysing the moduli space integrals the sectors in which the scaling dimensions receive non-perturbative contributions are identified. The requirement that the integrations over the fermionic collective coordinates which arise in the instanton background are saturated leads to non-renormalisation properties for a large class of operators. Instanton-induced corrections to the scaling dimensions are found only for dimension 4 SU(4) singlets and for dimension 5 operators in the 6 of SU(4). In many cases the non-renormalisation results are argued to be specific to operators of small dimension, but for some special sectors it is shown that they are valid for arbitrary dimension. Comments are also made on the implications of the results on the form of the instanton contributions to the dilation operator of the theory and on the possibility of realising its action on the instanton moduli space.

Properties of the Konishi multiplet in Script N = 4 SYM theory

We study perturbative and non-perturbative properties of the Konishi multiplet in N=4 SYM theory in D=4 dimensions. We compute two-, three- and four-point Green functions with single and multiple insertions of the lowest component of the multiplet, and of the lowest component of the supercurrent multiplet. These computations require a proper definition of the renormalized operator and lead to an independent derivation of its anomalous dimension. The O(g^2) value found in this way is in agreement with previous results. We also find that instanton contributions to the above correlators vanish. From our results we are able to identify some of the lowest dimensional gauge-invariant composite operators contributing to the OPE of the correlation functions we have computed. We thus confirm the existence of an operator belonging to the representation 20', which has vanishing anomalous dimension at order g^2 and g^4 in perturbation theory as well as at the non-perturbative level, despite the fact that it does not obey any of the known shortening conditions.

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.

Hard-soft renormalization and the exact renormalization group

The Wilsonian exact renormalization group gives a natural framework in which ultraviolet and infrared divergences can be treated separately. In massless QED we introduce, as the only mass parameter, a renormalization scale Λ R 0. We prove, using the flow equation technique, that infrared convergence is a necessary consequence of any zero-momentum renormalization condition at Λ R compatible with the effective Ward identities and axial symmetry. The same formalism is applied to renormalize gauge-invariant composite operators and to prove their infrared finiteness; in particular we consider the case of the axial current operator and its anomaly.

Renormalization of gauge-invariant composite operators in the light-cone gauge.

We generalize to composite operators concepts and techniques which have been successful in proving renormalization of the effective Action in light-cone gauge. Gauge invariant operators can be grouped into classes, closed under renormalization, which is matrix-wise. In spite of the presence of non-local counterterms, an ``effective" dimensional hierarchy still guarantees that any class is endowed with a finite number of elements. The main result we find is that gauge invariant operators under renormalization mix only among themselves, thanks to the very simple structure of Lee-Ward identities in this gauge, contrary to their behaviour in covariant gauges.

Driving operators relevant: A feature of the Chern-Simons interaction.

By computing anomalous dimensions of gauge invariant composite operators $(\bar\psi\psi)^n$ and $(\phi^*\phi)^n$ in Chern-Simons fermion and boson models, we address that Chern-Simons interactions make these operators more relevant or less irrelevant in the low energy region. We obtain a critical Chern-Simons fermion coupling, ${1\over \kappa_c^2} = {6\over 19}$, for a phase transition at which the leading irrelevant four-fermion operator $(\bar\psi\psi)^2$ becomes marginal, and a critical Chern-Simons boson coupling, ${1\over \kappa_c^2} = {6\over 34}$, for a similar phase transition for the leading irrelevant operator $(\phi^*\phi)^4$. We see this phenomenon also in the $1/N$ expansion.

Renormalization of gauge-invariant operators and the axial anomaly

The renormalization properties of gauge-invariant composite operators that vanish when the classical equations of motion are used (class ${\mathrm{II}}^{\mathrm{a}}$ operators) and which lead to diagrams where the Adler-Bell-Jackiw anomaly occurs are discussed. It is shown that gauge-invariant operators of this kind do need, in general, nonvanishing gauge-invariant (class I) counterterms.

Higher dimensional renormalization group invariant vacuum condensates in quantum chromodynamics

The renormalization of the five- and six-dimensional scalar gauge invariant composite operators is performed. The corresponding renormalization group invariant vacuum condensates are studied. Our main results are that the six-dimensional gluon condensate does not mix with the quartic quark condensates nor the other way round. We also show that the factorization hypothesis of the four-quark operator does not lead to renormalization group invariant quark vacuum condensates. So, one has to be careful when keeping the six-dimensional operators in the QCD sum rule analysis.

Green's Functions for Composite Particles in Quantum Chromodynamics

We discuss the bound state problem of quantum chromodynamics. We give the reduc­ tion formula for nonlocal gauge-invariant composite operators which describe composite particles using the method of Haag, Nishijima and Zimmermann. We develop the method of auxiliary fields for nonlocal gauge-invariant composite operators. § 1. Introduction Different models of strong interactions have been proposed corresponding to different energy regions considered; that is, current algebra and chiral dynamics for the low energy region; the dual resonance model and the dual string model for the Regge energy region; and the parton model for the deep inelastic region. Recently, quantum chromodynamics (QCD) has attracted much interest of particle physicists as a model which may combine properties of the models of three energy regions. Quarks are probably confined; at least, they have not appeared in the experi­ ments up to the present. Therefore the ordinary perturbative expansion with quarks and gluons as intermediate states may not reveal the real physical world, because the ordinary perturbation theory restricts asymptotic conditions. The ex­ pansions must probably be carried out using bound states of quarks and gluons as intermediate states. Moreover, the Regge pole theory and the dual model indicate that not only stable hadrons but also unstable resonances should be included in the intermediate states, while the dual string model suggests that the intermediate states have semiclassical string configurations. Further, only the states which can be described by color singlet states have been observed and no massless particles participating in strong interactions seem to exist. These facts suggest that the color gauge invariance remains unbroken even by the state vectors and the asymptotic states are gauge-invarianel, 2J (see also Ref.