Sp(2) renormalization

The renormalization of general gauge theories on flat and curved space–time backgrounds is considered within the Sp(2)-covariant quantization method. We assume the existence of a gauge-invariant and diffeomorphism invariant regularization. Using the Sp(2)-covariant formalism one can show that the theory possesses gauge-invariant and diffeomorphism invariant renormalizability to all orders in the loop expansion and the extended BRST-symmetry after renormalization is preserved. The advantage of the Sp(2) method compared to the standard Batalin–Vilkovisky approach is that, in reducible theories, the structure of ghosts and ghosts for ghosts and auxiliary fields is described in terms of irreducible representations of the Sp(2) group. This makes the presentation of solutions to the master equations in more simple and systematic way because they are Sp(2)-scalars.     

Implicit regularization beyond one-loop order: gauge field theories

We extend a constrained version of implicit regularization (CIR) beyond one-loop order for gauge field theories. In this framework, the ultraviolet content of the model is displayed in terms of momentum loop integrals order by order in perturbation theory for any Feynman diagram, while the Ward–Slavnov–Taylor identities are controlled by finite surface terms. To illustrate, we apply CIR to massless abelian gauge field theories (scalar and spinorial QED) to two-loop order and calculate the two-loop beta-function of spinorial QED. PACS  11.10.Gh; 11.15.Bt; 11.15.-q     

The size of finite size effects in lattice gauge theories

If a quantum field is enclosed in a spatial box of finite volume, its mass spectrum depends on the box size L. For field theories in the continuum Lüscher has shown to all orders in perturbation theory that for large L this dependence is related to certain scattering amplitudes of the infinite volume theory. We derived the corresponding relations for lattice field theories. Assuming their validity for lattice gauge theory outside the perturbative region the magnitude of finite size effects on the spectrum is determined by a glueball coupling constant. This quantity is estimated by strong coupling methods.     

A quenching scenario for lattice gauge theories

We study lattice gauge theories with complex, random and quenched couplings. Such theories are argued to have the same continuum limits as the annealed case. The first-order phase transitions are shown to be absent and the smoother cross-over behavior of the quenched theory leads to the universal scaling law.     

Plaquette formulation and the Bianchi identity for lattice gauge theories

We extend Halpern's field-strength formulation and dual potentials (for continuum gauge theories) to abelian and non-abelian lattice gauge theories. New results include: (i) plaquette formulation of all lattice gauge theories, (ii) the strong coupling expansion is seen as (a) a perturbation in dual links or (b) a gradual restoration of the lattice Bianchi identity. To leading order in the strong coupling expansion the lattice Bianchi identity is completely ignored. Geometrical interpretation of the lattice Bianchi identity is presented along with a discussion of the “abelianization” of the non-abelian identity and its connection with gauge-invariant variables. For abelian theories we also show that the dual potential is Fourier conjugate to the Bianchi identity and that the Coulomb gas representation of these theories is easily obtained in this formulation.     

Axial gauge propagators for quarks and gluons on the Polyakov-Wilson lattice

We discuss problems encountered in defining gauge-dependent propagators in a confining theory. For precision we use a finite Polyakov-Wilson lattice to define the Yang-Mills theory and to provide the ultraviolet and infrared regularization. Gauge fixing in a class of superaxial gauges is natural in this framework. A variety of approaches for defining the propagators for quarks and gluons is discussed and the propagators are evaluated explicitly in the strong coupling limit. We speculate upon the infrared behavior of these propagators in the weak coupling limit and upon the utility and validity of the Schwinger-Dyson equations for these propagators. In conclusion we propose that the leading infrared behavior is strongly gauge dependent and governed by the masses of low-lying color singlet states in the hadron spectrum. In the ultraviolet limit, however, with a properly constructed propagator, we find no reason to question the conventional wisdom derived from perturbation theory. Our conclusions should not depend in any fundamental way on the lattice formulation of the gauge theory, except insofar as that formulation serves to give precision to the continuum functional integration.     

Scalar lattice gauge theory

Scalar lattice gauge theories are models for scalar fields with local gauge symmetries. No fundamental gauge fields, or link variables in a lattice regularization, are introduced. The latter rather emerge as collective excitations composed from scalars. For suitable parameters scalar lattice gauge theories lead to confinement, with all continuum observables identical to usual lattice gauge theories. These models or their fermionic counterpart may be helpful for a realization of gauge theories by ultracold atoms. We conclude that the gauge bosons of the standard model of particle physics can arise as collective fields within models formulated for other “fundamental” degrees of freedom.     

On the energy-momentum tensor in gauge field theories

The energy-momentum tensor in spontaneously broken non-Abelian gauge field theories is studied. The motivation is to show that recent results on the finiteness and gauge independence of S-matrix elements in gauge theories extends to observable amplitudes for transitions in a gravitational field. Path integral methods and dimensional regularization are used throughout. Green's functions Γ_μν^(j)(q; p₁,…,p_j) involving the energy-momentum tensor and j particle fields are proved finite to all orders in perturbation theory to zero and first order in q, and finite to one loop order for general q. Amputated Green's functions of the energy momentum tensor are proved to be gauge independent on mass shell.     

BRST Renormalization

We consider the renormalization of general gauge theories on curved space-time background, with the main assumption being the existence of a gauge-invariant and diffeomorphism invariant regularization. Using the Batalin-Vilkovisky (BV) formalism one can show that the theory possesses gauge invariant and diffeomorphism invariant renormalizability at quantum level, up to an arbitrary order of the loop expansion.     

A study of 't Hooft and Wilson loops

An investigation is undertaken for 't Hooft loop operators in four-dimensional gauge theories. For the first time, a perimeter law is shown to be their behavior in weak coupling Wilson lattice (and continuum) non-abelian SU(N) gauge theories for all N. However, it is also argued that this perimeter law is poor criterion for quark confinement. Rather, it is suggested that non-leading long-distance behavior is what is crucial and relevant in distinguishing non-abelian from abelian (and hence confining from non-confining) theories. A new object, “the 't Hooft line”, is introduced to measure this non-leading behavior and is computed in strong coupling on the lattice. There, one finds magnetic screening characterized by a magnetic screening mass, m_s. It is shown to all orders in strong coupling that m_s is the glueball mass, a result which is expected to persist in weak coupling and in the continuum. Two further consequences of this work are that pure non-abelian gauge theories cannot be in a Higgs phase and that in such models that absence of massless physical particles implies confinement.Finally, non-leading behavior in Wilson loops is examined. The present picture of confinement suggests the absence of van der Waals forces in Yang-Mills theories.     

Isospin breaking of the narrow charmonium state of Belle at 3872 MeV as a deuson

We investigate the continuum limit of a compact formulation of the lattice U(1) gauge theory in 4 dimensions using a nonperturbative gauge-fixed regularization. We find clear evidence of a continuous phase transition in the pure gauge theory for all values of the gauge coupling (with gauge symmetry restored). When probed with quenched staggered fermions with U(1) charge, the theory clearly has a chiral transition for large gauge couplings. We identify the only possible region in the parameter space where a continuum limit with nonperturbative physics may appear.     

Higgs phenomenon without symmetry breaking order parameter

We discuss symmetry-breaking order parameters, e.g. 〈?〉, in gauge theories with Higgs scalars, ?, in suitable gauges. We show that, typically, 〈?〉 = 0. A complete set of gauge-invariant, observable composite fields for such theories, local ones and ones localized near strings (paths) is constructed. We then examine the validity of standard perturbation theory, based on assuming that 〈?〉 ≠ 0, and reformulate it in terms of our gauge-invariant fields and without assuming that 〈?〉 ≠ 0. Finally, we classify classical field configurations with non-trivial topology (“defects”) in such theories and propose a defect-gas approach to predict their effects.     

Renormalization constants and asymptotic behaviour in quantum electrodynamics

Using dimensional regularization, a field theory contains at least one parameter less than usual with the dimension of mass. The Callan-Symanzik equations for the renormalization constants then become solvable entirely in terms of the coefficient functions. Explicit expressions are obtained for all the renormalization constants in quantum electrodynamics. At non-exceptional momenta the infrared behaviour and the three leading terms in the asymptotic expansion of any Green function are controlled by the Callan-Symanzik equations. For the propagators the three leading terms are computed explicitly. The gauge dependence of the asymptotic electron propagator in momentum space is calculated in all orders of perturbation theory.     

Equivalence of stochastic and canonical quantization in perturbation theory

It is shown that the stochastic quantization method introduced by Parisi and Wu reproduces, order by order, the ordinary perturbation expansion. The proof is valid for any field theory, including gauge theories, provided one considers gauge-invariant quantities.     

Theories of quark confinement

It is conjectured that a non-Abelian gauge theory based on the color SU(3) group will confine quarks. Various techniques that have been applied to this question are reviewed. These include approximate methods based on strong coupling expansions of Hamiltonian and Euclidian lattice theories, instanton improvements on perturbation theory, and solutions of truncated Dyson-Schwinger equations for the gauge field propagator. Formal results based on electric-magnetic duality arguments and on the study of loop field theories are presented. Deconfinement at high temperatures, the inclusion of light quarks, and a possible reconciliation with a hypothetical discovery of free quarks are discussed.     

Factorisation in large-N limit of lattice gauge theories revisited

We prove using the Schwinger-Dyson equations that the factorisation property holds for all gauge-invariant Green’s function in the large-N limit of a Wilson-Polyakov lattice gauge theory.     

Quantum gravity on the lattice

I review the lattice approach to quantum gravity, and how it relates to the non-trivial ultraviolet fixed point scenario of the continuum theory. After a brief introduction covering the general problem of ultraviolet divergences in gravity and other non-renormalizable theories, I discuss the general methods and goals of the lattice approach. An underlying theme is the attempt at establishing connections between the continuum renormalization group results, which are mainly based on diagrammatic perturbation theory, and the recent lattice results, which apply to the strong gravity regime and are inherently non-perturbative. A second theme in this review is the ever-present natural correspondence between infrared methods of strongly coupled non-abelian gauge theories on the one hand, and the low energy approach to quantum gravity based on the renormalization group and universality of critical behavior on the other. Towards the end of the review I discuss possible observational consequences of path integral quantum gravity, as derived from the non-trivial ultraviolet fixed point scenario. I argue that the theoretical framework naturally leads to considering a weakly scale-dependent Newton’s constant, with a scaling violation parameter related to the observed scaled cosmological constant (and not, as naively expected, to the Planck length). Invited lecture presented at the conference “Quantum Gravity: Challenges and Perspectives”, Bad Honnef, 14–16 April 2008. To appear in the proceedings edited by Hermann Nicolai.     

Anomaly matching conditions and the moduli space of supersymmetric gauge theories

The structure of the moduli space of N = 1 supersymmetric gauge theories is analyzed from an algebraic geometric viewpoint. The connection between the fundamental fields of the ultraviolet theory, and the gauge-invariant composite fields of the infrared theory is explained in detail. The results are then used to prove an anomaly matching theorem. The theorem is used to study anomaly matching for supersymmetric QCD, and can explain all the known anomaly matching results for this case.