Gauge Invariant Effective Action Research Articles

On the gauge invariance of the higher-derivative Yang–Mills–Chern–Simons action

In this work, we study the perturbative generation of the gauge invariant effective action for the non-Abelian gauge field in a (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-dimensional spacetime. We present a detailed analysis of the two, three and four-point functions in order to determine the non-Abelian Chern–Simons terms (parity odd) and Yang–Mills terms (parity even). Moreover, these terms are supplemented by the higher-derivative corrections which resulted in the Alekseev–Arbuzov–Baikov effective action (parity even) plus the higher-derivative (HD) corrections to the Chern–Simons terms (parity odd). In order to highlight some features about the perturbative generation of the effective action, we present a discussion based on the dimensional analysis, which allows us to establish the general structure of the permissible terms to guarantee the gauge invariance of the higher-derivative parts.

Kaluza-Klein modes of U(1) gauge vector field on brane with codimension-d

From the paper of Fu et al. (2019) [44], it is known that the effective action of a massless U(1) gauge vector field on a codimension-2 brane is gauge invariant due to the coupling between the vector Kaluza-Klein (KK) modes with two types of scalar KK modes. It is interesting to generalize this result to a brane world model with an arbitrary number of extra dimensions. In this work, we first investigate the case with three extra dimensions. After KK decomposition, there are three types of scalar KK modes. In addition to the mutual coupling between these scalar modes, there are also coupling between the scalar and the vector KK modes. The coupling constants are not all independent. The relationships between the coupling constants enable us to obtain a gauge invariant effective action, from which we can see that the masses of the vector KK modes are contributed by all the three extra dimensions. The masses of the scalar modes, however, are contributed only by two of the three extra dimensions. Then we generalize our results to branes with codimension d (d=1,2...).

On the infinite gradient-flow for the domain-wall formulation of chiral lattice gauge theories

We examine the proposal by Grabowska and Kaplan (GK) to use the infinite gradient flow in the domain-wall formulation of chiral lattice gauge theories. We consider the case of Abelian theories in detail, for which Lüscher’s exact gauge-invariant formulation is known, and we relate GK’s formulation to Lüscher’s one. The gradient flow can be formulated for the admissible U(1) link fields so that it preserves their topological charges. GK’s effective action turns out to be equal to the sum of Lüscher’s gauge-invariant effective actions for the target Weyl fermions and the mirror “fluffy” fermions, plus the so-called measure-term integrated along the infinite gradient flow. The measure-term current is originally a local(analytic) and gauge-invariant functional of the admissible link field, given as a solution to the local cohomology problem. However, with the infinite gradient flow, it gives rise to non-local(non-analytic) vertex functions which are not suppressed exponentially at large distance. The “fluffy” fermions remain as a source of non-local contribution, which couple yet to the Wilson-line and magnetic-flux degrees of freedom of the dynamical link field.

Superstring field theory with open and closed strings

We construct Lorentz invariant and gauge invariant 1PI effective action for closed and open superstrings and demonstrate that it satisfies the classical BV master equation. We also construct the quantum master action for this theory satisfying the quantum BV master equation and generalize the construction to unoriented theories. The extra free field needed for the construction of closed superstring field theory plays a crucial role in coupling the closed strings to D-branes and orientifold planes.

U(1) gauge vector field on a codimension-2 brane

In this paper, we obtain a gauge invariant effective action for a bulk massless U(1) gauge vector field on a brane with codimension two by using a general Kaluza-Klein (KK) decomposition for the field. It suggests that there exist two types of scalar KK modes to keep the gauge invariance of the action for the massive vector KK modes. Both the vector and scalar KK modes can be massive. The masses of the vector KK modes m(n) contain two parts, m1( n) and m2( n), due to the existence of the two extra dimensions. The masses of the two types of scalar KK modes mϕ( n) and mφ( n) are related to the vector ones, i.e., mϕ( n) = m1( n) and mφ( n) = m2( n). Moreover, we derive two Schrödinger-like equations for the vector KK modes, for which the effective potentials are just the functions of the warp factor.

Gauge-invariant fields and flow equations for Yang–Mills theories

We discuss the concept of gauge-invariant fields for non-abelian gauge theories. Infinitesimal fluctuations around a given gauge field can be split into physical and gauge fluctuations. Starting from some reference field the gauge-invariant fields are constructed by consecutively adding physical fluctuations. An arbitrary gauge field can be mapped to an associated gauge invariant field. An effective action that depends on gauge-invariant fields becomes a gauge-invariant functional of arbitrary gauge fields by associating to every gauge field the corresponding gauge-invariant field. The gauge-invariant effective action can be obtained from an implicit functional integral with a suitable “physical gauge fixing”. We generalize this concept to the gauge-invariant effective average action or flowing action, which involves an infrared cutoff. It obeys a gauge-invariant functional flow equation. We demonstrate the use of this flow equation by a simple computation of the running gauge coupling and propagator in pure SU(N)-Yang–Mills theory.

Probing the string winding sector

We probe a slice of the massive winding sector of bosonic string theory from toroidal compactifications of Double Field Theory (DFT). This string subsector corresponds to states containing one left and one right moving oscillators. We perform a generalized Kaluza Klein compactification of DFT on generic 2n-dimensional toroidal constant backgrounds and show that, up to third order in fluctuations, the theory coincides with the corresponding effective theory of the bosonic string compactified on n-dimensional toroidal constant backgrounds, obtained from three-point amplitudes. The comparison between both theories is facilitated by noticing that generalized diffeomorphisms in DFT allow to fix generalized harmonic gauge conditions that help in identifying the physical degrees of freedom. These conditions manifest as conformal anomaly cancellation requirements on the string theory side. The explicit expression for the gauge invariant effective action containing the physical massless sector (gravity+antisymmetric+gauge+ scalar fields) coupled to towers of generalized Kaluza Klein massive states (corresponding to compact momentum and winding modes) is found. The action acquires a very compact form when written in terms of fields carrying O(n, n) indices, and is explicitly T-duality invariant. The global algebra associated to the generalized Kaluza Klein compactification is discussed.

Transgression Field Theory at the Interface of Topological Insulators

Topological phases of matter can be classified by using Clifford algebras through Bott periodicity. We consider effective topological field theories of quantum Hall systems and topological insulators that are Chern-Simons and BF field theories. The edge states of these systems are related to the gauge invariance of the effective actions. For the edge states at the interface of two topological insulators, transgression field theory is proposed as a gauge invariant effective action. Transgression actions of Chern-Simons theories for (2+1)D and (4+1)D and BF theories for (3+1)D are constructed. By using transgression actions, the edge states are written in terms of the bulk connections of effective Chern-Simons and BF theories.

Gauge-invariant implementation of the Abelian-Higgs model on optical lattices

We present a gauge-invariant effective action for the Abelian Higgs model (scalar electrodynamics) with a chemical potential $\mu$ on a 1+1 dimensional lattice. This formulation provides an expansion in the hopping parameter $\kappa$ which we test with Monte Carlo simulations for a broad range of the inverse gauge coupling $\beta_{pl}$ and small values of the scalar self-coupling $\lambda$. In the opposite limit of infinitely large $\lambda$, the partition function can be written as a traced product of local tensors which allows us to write exact blocking formulas. Their numerical implementation requires truncations but there is no sign problem for arbitrary values of $\mu$. We show that the time continuum limit of the blocked transfer matrix can be obtained numerically and, in the limit of infinite $\beta_{pl}$ and with a spin-1 truncation, the small volume energy spectrum is identical to the low energy spectrum of a two-species Bose-Hubbard model in the limit of large onsite repulsion. We extend this procedure for finite $\beta_{pl}$ and derive a spin-1 approximation of the Hamiltonian. It involves new terms corresponding to transitions among the two species in the Bose-Hubbard model. We propose an optical lattice implementation involving a ladder structure.

Double-logarithmic asymptotics of scattering amplitudes in gravity and supergravity

We review the Balitsky-Fadin-Kuraev-Lipatov approach to high-energy scattering in QCD and supersymmetric gauge theories. At a large number of colors, the equations for the gluon composite states in the t-channel have remarkable mathematical properties including their Mobius invariance, holomorphic separability, duality symmetry, and integrability. We formulate a theory of Reggeized gluon interactions in the form of a gauge-invariant effective action local in particle rapidities. In the maximally extended N=4 supersymmetry, the Pomeron is dual to the Reggeized graviton in the ten-dimensional anti-de Sitter space. As a result, the Gribov Pomeron calculus should be reformulated here as a generally covariant effective field theory for the Reggeized gravitons. We construct the corresponding effective action, which allows calculating the graviton Regge trajectory and its couplings. We sum the double-logarithmic contributions for amplitudes with graviton quantum numbers in the t-channel in the Einstein-Hilbert gravity and its supersymmetric generalizations. As the supergravity rank N increases, the double-logarithmic amplitudes begin to decrease rapidly compared with their Born contributions.

Gauge theory of topological phases of matter

We study the response of quantum many-body systems to coupling some of their degrees of freedom to external gauge fields. This serves to understand the current Green functions and transport properties of interacting many-body systems. Our analysis leads to a “gauge theory of states of matter” complementary to the well-known Landau theory of order parameters. We illustrate the power of our approach by deriving and interpreting the gauge-invariant effective actions of (topological) superconductors, 2D electron gases exhibiting the quantized Hall- and spin-Hall effect, 3D topological insulators, as well as axion electrodynamics. We also use the theory to elucidate the structure of surface modes in these systems.

Parton-distribution functions for the pion and kaon in the gauge-invariant nonlocal chiral-quark model

We investigate the parton-distribution functions (PDFs) for the positively charged pion and kaon at a low-renormalization scale ~ 1 GeV. To this end, we employ the gauge-invariant effective chiral action from the nonlocal chiral-quark model, resulting in that the vector currents are conserved. All the model parameters are determined phenomenologically with the normalization condition for PDF and the empirical values for the pseudoscalar-meson weak-decay constants. We consider the momentum dependence of the effective quark mass properly within the model calculations. It turns out that the leading local contribution provides about 70 % of the total strength for PDF, whereas the nonlocal one that is newly taken into account in this work for the gauge invariance does the rest. High-Q^2 evolution to 27 GeV^2 is performed for valance-quark distribution function (VQDF), using the DGALP equation. The moments for the pion and kaon VQDFs are also computed. The numerical results are compared with the empirical data and theoretical estimations, and show qualitatively agreement with them.

Effective action in a higher-spin background

We consider a free massless scalar field coupled to an infinite tower of background higher-spin gauge fields via minimal coupling to the traceless conserved currents. The set of Abelian gauge transformations is deformed to the non-Abelian group of unitary operators acting on the scalar field. The gauge invariant effective action is computed perturbatively in the external fields. The structure of the various (divergent or finite) terms is determined. In particular, the quadratic part of the logarithmically divergent (or of the finite) term is expressed in terms of curvatures and related to conformal higher-spin gravity. The generalized higher-spin Weyl anomalies are also determined. The relation with the theory of interacting higher-spin gauge fields on anti de Sitter spacetime via the holographic correspondence is discussed.

The effective action and the triple Pomeron vertex

We study integrations over light-cone momenta in the gauge invariant effective action of high energy QCD. A regularization mechanism which allows for the evaluation of the longitudinal integrations is presented. After a rederivation of the reggeized gluon and the BFKL-equation from the effective action, we study the 1-3 and 2-4 reggeized gluon transition vertex of QCD Reggeon field theory and discuss their connection with the usual triple Pomeron vertex of perturbative QCD.

Noncommutative induced gauge theories on Moyal spaces

Noncommutative field theories on Moyal spaces can be conveniently handled within a framework of noncommutative geometry. Several renormalisable matter field theories that are now identified are briefly reviewed. The construction of renormalisable gauge theories on these noncommutative Moyal spaces, which remains so far a challenging problem, is then closely examined. The computation in 4-D of the one-loop effective gauge theory generated from the integration over a scalar field appearing in a renormalisable theory minimally coupled to an external gauge potential is presented. The gauge invariant effective action is found to involve, beyond the expected noncommutative version of the pure Yang-Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic term, which for the noncommutative φ4-theory on Moyal space ensures renormalisability. A class of possible candidates for renormalisable gauge theory actions defined on Moyal space is presented and discussed.

Noncommutative induced gauge theory

We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4-dimensional Moyal space and compute in position space the one-loop Yang–Mills-type effective theory generated from the integration over the scalar field. We find that the gauge-invariant effective action involves, beyond the expected noncommutative version of the pure Yang–Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic oscillator term, which for the noncommutative ϕ4-theory on Moyal space ensures renormalisability. The expression of a possible candidate for a renormalisable action for a gauge theory defined on Moyal space is conjectured and discussed.

Feynman rules for effective Regge action

Starting from the gauge invariant effective action in the quasi-multi-Regge kinematics (QMRK), we obtain the effective reggeized gluon (R)–particle (P) vertices of the following types: R P P , R R P , R R P P , R P P P , R R P P P , and R P P P P , where the on-mass-shell particles are gluons, or sets of gluons with small invariant masses. The explicit expressions satisfying the Bose-symmetry and gauge invariance conditions are obtained. As a comment to the Feynman rules for derivation of the amplitudes in terms of effective vertices we present a “vocabulary” for practitioners.

Gauge invariant effective action for the Polyakov line in theSU(N)Yang-Mills theory at high temperatures

We integrate out fast varying quantum fluctuations around static A_4 and A_i fields for the SU(N) gauge group. By assuming that the gluon fields are slowly varying but allowing for an arbitrary amplitude of A_4 we obtain two variants of the effective high-temperature theory for the Polyakov line. One is the effective action for the gauge-invariant eigenvalues of the Polyakov line, and it is explicitly Z(N) symmetric. The other is the effective action for the Polyakov line itself as an element of the SU(N). In this case the theory necessarily includes the spatial components A_i to ensure its gauge invariance under spatial gauge transformations. We derive the 1-loop effective action in the `electric' and `magnetic' sectors, summing up all powers of A_4.

Effective actions, Wilson lines and the IR/UV mixing in noncommutative supersymmetric gauge theories

We study IR/UV mixing effects in noncommutative supersymmetric Yang-Mills theories with gauge group U(N) using background field perturbation theory. We compute three- and four-point functions of background fields, and show that the IR/UV mixed contributions to these correlators can be reproduced from an explicitly gauge-invariant effective action, which is expressed in terms of open Wilson lines.

Leading logarithmic approximation in QCD

The leading logarithmic approximation (LLA) for the scattering amplitudes in QCD is reviewed. The double-logarithmic asymptotics of scattering amplitudes is obtained as a solution to nonlinear evolution equations in the infrared cutoff. The DGLAP equation describes an evolution of parton distributions with increasing parton virtuality. The evolution of the amplitudes with respect to the scale in the longitudinal subspace is given by the BFKL equation. The gluon and quarks in QCD lie on the Regge trajectories calculable in perturbation theory. Mesons and baryons are composite states of Reggeized quarks. Similarly the Pomeron and Odderon are colorless ground states of Reggeized gluons. In the case of multicolor QCD, the Reggeon field theory in LLA is completely integrable. The Reggeon interactions in QCD are derived from a gauge-invariant effective action. In particular, next-to-leading corrections to the BFKL equation in QCD and in supersymmetric gauge models are obtained in this way.