Massive Abelian Gauge Research Articles

Modified Stückelberg Formalism: Free Massive Abelian 2-Form Theory in 4D

We demonstrate that the celebrated Stückelberg formalism is modified in the case of a massive four (3 + 1)-dimensional (4D) Abelian 2-form theory due to the presence of a self-duality discrete symmetry in the theory. The latter symmetry entails upon the modified 4D massive Abelian 2-form gauge theory to become a massive model of Hodge theory within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism where there is the existence of a set of (anti-)co-BRST transformations corresponding to the usual nilpotent (anti-)BRST transformations. The latter exist in any arbitrary dimension of spacetime for the usual Stückelberg-modified massive Abelian 2-form gauge theory. The modification in the Stückelberg technique is backed by the precise mathematical arguments from the differential geometry where the exterior derivative and Hodge duality operator play the decisive roles. The modified version of the Stückelberg technique remains invariant under the discrete duality transformations which also establish a precise and deep connection between the off-shell nilpotent (anti-)BRST and (anti-)co-BRST transformations. We have clarified a simple trick of using the equations of motion to remove the higher derivative terms in the appropriate Lagrangian densities so that our 4D theory can become consistent.

Noether Theorem and Nilpotency Property of the (Anti-)BRST Charges in the BRST Formalism: A Brief Review

In some of the physically interesting gauge systems, we show that the application of the Noether theorem does not lead to the deduction of the Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST charges that obey precisely the off-shell nilpotency property despite the fact that these charges are (i) derived by using the off-shell nilpotent (anti-)BRST symmetry transformations, (ii) found to be the generators of the above continuous symmetry transformations, and (iii) conserved with respect to the time-evolution due to the Euler–Lagrange equations of motion derived from the Lagrangians/Lagrangian densities (that describe the dynamics of these suitably chosen physical systems). We propose a systematic method for the derivation of the off-shell nilpotent (anti-)BRST charges from the corresponding non-nilpotent Noether (anti-)BRST charges. To corroborate the sanctity and preciseness of our proposal, we take into account the examples of (i) the one (0 + 1)-dimensional (1D) system of a massive spinning (i.e., SUSY) relativistic particle, (ii) the D-dimensional non-Abelian one-form gauge theory, and (iii) the Abelian two-form and the Stu¨ckelberg-modified version of the massive Abelian three-form gauge theories in any arbitrary D-dimension of spacetime. Our present endeavor is a brief review where some decisive proposals have been made and a few novel results have been obtained as far as the nilpotency property is concerned.

Dark Z at the International Linear Collider

A dark Z is a massive Abelian gauge boson which is coupled to the Standard Model through both kinetic and mass mixing with the electroweak sector. We study the phenomenology of the dark Z at an energy-frontier electron-positron collider, such as the proposed International Linear Collider (ILC). We show that precision electroweak constraints and the current bounds from the hadron colliders allow a dark Z that is kinematically accessible at the ILC with 250 GeV or 500 GeV center-of-mass energy. Further, the reach of the ILC searches for a dark Z significantly exceeds the expected reach of the high luminosity LHC. If a signal consistent with a dark Z is discovered, it would motivate a dedicated run of the ILC at the center-of-mass energy matching the dark Z mass. We demonstrate that a short one month run at design luminosity could measure the dark Z chiral couplings to fermions with percent precision. This measurement can be used to discriminate between competing theoretical models of the resonance: for example, the dark Z can be distinguished from a dark photon with purely kinetic mixing.

Massive 4D Abelian 2-form theory: Nilpotent symmetries from the (anti-)chiral superfield approach

The off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetries are obtained by using the (anti-)chiral superfield approach (ACSA) to Becchi–Rouet–Stora–Tyutin (BRST) formalism for the four [Formula: see text]-dimensional (4D) Stückelberg-modified massive Abelian 2-form gauge theory. We perform exactly similar kind of exercise for the derivation of the off-shell nilpotent (anti-)co-BRST symmetry transformations, too. In the above derivations, the symmetry invariant restrictions on the superfields play very important and decisive roles. To prove the sanctity of the above nilpotent symmetries, we generalize our 4D ordinary theory (defined on the 4D flat Minkowskian space–time manifold) to its counterparts [Formula: see text]-dimensional (anti-)chiral super-submanifolds of the [Formula: see text]-dimensional supermanifold which is parametrized by the superspace coordinates [Formula: see text] where [Formula: see text] [Formula: see text] are the bosonic coordinates and a pair of Grassmannian variables [Formula: see text] are fermionic: [Formula: see text], [Formula: see text] in nature. One of the novel observations of our present endeavor is the derivation of the Curci–Ferrari (CF)-type restrictions from the requirement of the symmetry invariance of the coupled (but equivalent) Lagrangian densities of our theory within the framework of ACSA to BRST formalism. We also exploit the standard techniques of ACSA to capture the off-shell nilpotency and absolute anticommutativity of the conserved (anti-)BRST as well as the conserved (anti-)co-BRST charges. In a subtle manner, the proof of the absolute anticommutativity of the above conserved charges also implies the existence of the appropriate CF-type restrictions on our theory. This proof is also a novel result.

Dark photon bounds in the dark EFT

Dark photons are massive abelian gauge bosons that interact with ordinary photons via a kinetic mixing with the hypercharge field strength tensor. This theory is probed by a variety of different experiments and limits are set on a combination of the dark photon mass and kinetic mixing parameter. These limits can however be strongly modified by the presence of additional heavy degrees of freedom. Using the framework of dark effective field theory, we study how robust are the current experimental bounds when these new states are present. We focus in particular on the possible existence of a dark dipole interaction between the Standard Model leptons and the dark photon. We show that, under certain assumptions, the presence of a dark dipole modifies existing supernovæ bounds for cut-off scales up to mathcal{O} (10–100 TeV). On the other hand, terrestrial experiments, such as LSND and E137, can probe cut-off scales up to mathcal{O} (3 TeV). For the latter experiment we highlight that the bound may extend down to vanishing kinetic mixing.

A massive field-theoretic model for Hodge theory

Within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism, we show that the four (3+1)-dimensional (4D) massive Abelian 2-form gauge theory (without any interaction with matter fields) is a model for the Hodge theory because its discrete and continuous symmetry transformations (and their corresponding Noether conserved charges) provide the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level. For this purpose, we incorporate the pseudo-scalar and axial-vector fields which appear in the theory with negative kinetic terms (but with proper definition of mass). The negative kinetic terms, for the above fields, are essential so that our theory could respect the discrete symmetry transformations which provide the physical realizations of the Hodge duality operation in the domain of differential geometry. Thus, our present endeavour, not only provides the physical realizations of all the mathematical ingredients connected with the de Rham cohomological operators of differential geometry, it also sheds light on the existence and emergence of fields with negative kinetic terms. We discuss the implications and relevance of the latter fields in the context of current models of dark matter and dark energy as well as the bouncing models of Universe.

Computation of effective front form Hamiltonians for massive Abelian gauge theory

Renormalization group procedure for effective particles (RGPEP) is applied in terms of a second-order perturbative computation to an Abelian gauge theory, as an example of application worth studying on the way toward derivation of a dynamical connection between the spectroscopy of bound states and their parton-model picture in the front form of Hamiltonian dynamics. In addition to the ultraviolet transverse divergences that are handled using the RGPEP in previously known ways, the small-x divergences are handled by introducing a mass parameter and a third polarization state for gauge bosons using a mechanism analogous to spontaneous breaking of global gauge symmetry, in a special limit that simplifies the theory to Soper's front form of massive QED. The resulting orders of magnitude of scales involved in the dynamics of effective constituents or partons in the simplified theory are identified for the fermion and boson mass counter terms, effective masses and self-interactions, as well as for the Coulomb-like effective interactions in bound states of fermions. Computations in orders higher than second are mentioned but not described in this article.

Light-front QED, Stueckelberg field and infrared divergence

Stueckelberg mechanism introduces a scalar field, known as Stueckelberg field, so that gauge symmetry is preserved in the massive Abelian gauge theory. In this work, we show that the role of the Stueckelberg field is similar to the Kulish and Faddeev coherent state approach to handle infrared (IR) divergences. We expect that the light-front quantum electrodynamics (LFQED) with Stueckelberg field must be IR finite in the massless limit of the gauge boson. We have explicitly shown the cancellation of IR divergences in the relevant diagrams contributing to self-energy and vertex correction at leading order.

Massive Abelian Gauge Bosons in Front-form Hamiltonians

It is shown how gauge bosons can be supplied a mass term using the Higgs mechanism for the purpose of regulating Hamiltonians of Abelian gauge theories in the front form of quantum dynamics.

Excursions through KK modes

In this article we study Kaluza-Klein (KK) dimensional reduction of massive Abelian gauge theories with charged matter fields on a circle. Since local gauge transformations change position dependence of the charged fields, the decomposition of the charged matter fields into KK modes is gauge dependent. While whole KK mass spectrum is independent of the gauge choice, the mode number depends on the gauge. The masses of the KK modes also depend on the field value of the zero-mode of the extra dimensional component of the gauge field. In particular, one of the KK modes in the KK tower of each massless 5D charged field becomes massless at particular values of the extra-dimensional component of the gauge field. When the extra-dimensional component of the gauge field is identified with the inflaton, this structure leads to recursive cosmological particle productions.

Dark matter relic from muon anomalies

We show that the recently reported anomalies in $b\to s\mu^+\mu^-$ transitions, as well as the long-standing $g_\mu-2$ discrepancy, can be addressed simultaneously by a new massive abelian gauge boson with loop-induced coupling to muons. Such a scenario typically leads to a stable dark matter candidate with a thermal relic density close to the observed value. Dark matter in our model couples dominantly to leptons, hence signals in direct detection experiments lie well below the current sensitivity. The LHC, in combination with indirect detection searches, can test this scenario through distinctive signatures with muon pairs and missing energy.

Massive And Massless Gauge Fields Formed by Flat Connections

The Yang - Mills type massive and massless gauge theories are interpreted in the geometrical frame of holomorphic principal bundles on a complex 2 - manifold. It is seen in this formalism that, the component (1,1) of the curvature of this connection appears because of flat connections generated by holomorphic structure although connection is flat. Thus it is possible to write a Lagrangian for a Yang - Mills theory including massive and massless gauge fields. However, the mass matrix of a massive gauge field on such a bundle isn’t nilpotent and this field is generated by a noncommutative flat connection on the same bundle, then the structure group of this bundle is non - Abelian complex Lie group. However, if the gauge field is massless, then this is generated by commutative flat connection, and so the structure group of the bundle is Abelian complex Lie group. Also one sees that the second Chern number or topological charge is proportional to the total volume of the base manifold for each massless and massive gauge theories and Abelian (massless) gauge theories are indeed the theories of the Kahler potential on the complex projective space $\mathbb {C}P^{2}$ .

Bianchi attractors: a classification of extremal black brane geometries

AbstractExtremal black branes are of interest because they correspond to the ground states of field theories at finite charge density in gauge/gravity duality. The geometry of such a brane need not be translationally invariant in the spatial directions along which it extends. A less restrictive requirement is that of homogeneity, which still allows points along the spatial directions to be related to each other by symmetries. In this paper, we find large new classes of homogeneous but anisotropic extremal black brane horizons, which could naturally arise in gauge/gravity dual pairs. In 4+1 dimensional spacetime, we show that such homogeneous black brane solutions are classified by the Bianchi classification, which is well known in the study of cosmology, and fall into nine classes. In a system of Einstein gravity with negative cosmological term coupled to one or two massive Abelian gauge fields, we find solutions with an additional scaling symmetry, which could correspond to the near-horizon geometries of such extremal black branes. These solutions realize many of the Bianchi classes. In one case, we construct the complete extremal solution which asymptotes to AdS space.

Lifshitz solitons

We numerically obtain a class of soliton solutions for Einstein gravity in ($n+1$) dimensions coupled to massive Abelian gauge fields and with a negative cosmological constant with Lifshitz asymptotic behavior. We find that for all $n\ensuremath{\ge}3$, a discrete set of magic values for the charge density at the origin (guaranteeing an asymptotically Lifshitz geometry) exists when the critical exponent associated with the Lifshitz scaling is $z=2$; moreover, in all cases, a single magic value is obtained for essentially every $1<z<2$, yet none when $z>2$ sufficiently.

Massive Abelian gauge symmetries and fluxes in F-theory

F-theory compactified on a Calabi-Yau fourfold naturally describes non-Abelian gauge symmetries through the singularity structure of the elliptic fibration. In contrast Abelian symmetries are more difficult to study because of their inherently global nature. We argue that in general F-theory compactifications there are massive Abelian symmetries, such as the uplift of the Abelian part of the U(N) gauge group on D7-branes, that arise from non-Kahler resolutions of the dual M-theory setup. The four-dimensional F-theory vacuum with vanishing expectation values for the gauge fields corresponds to the Calabi-Yau limit. We propose that fluxes that are turned on along these U(1)s are uplifted to non-harmonic four-form fluxes. We derive the effective four-dimensional gauged supergravity resulting from F-theory compactifications in the presence of the Abelian gauge factors including the effects of possible fluxes on the gauging, tadpoles and matter spectrum.

Electroweak Sudakov logarithms and real gauge-boson radiation in the TeV region

Electroweak radiative corrections give rise to large negative, double-logarithmically enhanced corrections in the TeV region. These are partly compensated by real radiation and, moreover, affected by selecting isospin-noninvariant external states. We investigate the impact of real gauge boson radiation more quantitatively by considering different restricted final state configurations. We consider successively a massive abelian gauge theory, a spontaneously broken SU(2) theory and the electroweak Standard Model. We find that details of the choice of the phase space cuts, in particular whether a fraction of collinear and soft radiation is included, have a strong impact on the relative amount of real and virtual corrections.

Absolutely anticommuting (anti-)BRST symmetry transformations for topologically massive Abelian gauge theory

We demonstrate the existence of the nilpotent and absolutely anticommuting Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetry transformations for the four (3+1)-dimensional (4D) topologically massive Abelian U(1) gauge theory that is described by the coupled Lagrangian densities (which incorporate the celebrated (B∧F) term). The absolute anticommutativity of the (anti-) BRST symmetry transformations is ensured by the existence of a Curci–Ferrari type restriction that emerges from the superfield formalism as well as from the equations of motion which are derived from the above coupled Lagrangian densities. We show the invariance of the action from the point of view of the symmetry considerations as well as superfield formulation. We discuss, furthermore, the topological term within the framework of superfield formalism and provide the geometrical meaning of its invariance under the (anti-)BRST symmetry transformations.

Trkalian fields and Radon transformation

We write the spherical curl transformation for Trkalian fields using differential forms. Then we consider Radon transform of these fields. The Radon transform of a Trkalian field satisfies a corresponding eigenvalue equation on a sphere in transform space. The field can be reconstructed using knowledge of the Radon transform on a canonical hemisphere. We consider relation of the Radon transformation with Biot–Savart integral operator and discuss its transform introducing Radon–Biot–Savart operator. The Radon transform of a Trkalian field is an eigenvector of this operator. We also present an Ampere-law type relation for these fields. We apply these to Lundquist solution. We present a Chandrasekhar–Kendall-type solution of the corresponding equation in the transform space. Lastly, we focus on the Euclidean topologically massive Abelian gauge theory. The Radon transform of an anti-self-dual field is related by antipodal map on this sphere to the transform of the self-dual field obtained by inverting space coordinates. The Lundquist solution provides an example of quantization of topological mass in this context.

TOPOLOGICALLY MASSIVE ABELIAN GAUGE THEORY

We discuss three mathematical structures which arise in topologically massive Abelian gauge theory. First, the Euclidean topologically massive Abelian gauge theory defines a contact structure on a manifold. We briefly discuss three solutions and the related contact structures on the flat 3-torus, the AdS space, the 3-sphere which respectively correspond to Bianchi type I, VIII, IX spaces. We also present solutions on Bianchi type II, VI and VII spaces. Secondly, we discuss a family of complex (anti-)self-dual solutions of the Euclidean theory in Cartesian coordinates on [Formula: see text] which are given by (anti)holomorpic functions. The orthogonality relation of contact structures which are determined by the real parts of these complex solutions separates them into two classes: the self-dual and the anti-self-dual solutions. Thirdly, we apply the curl transformation to this theory. An arbitrary solution is given by a vector tangent to a sphere whose radius is determined by the topological mass in transform space. Meanwhile a gauge transformation corresponds to a vector normal to this sphere. We discuss the quantization of topological mass in an example.

THE STUECKELBERG FIELD

In 1938, Stueckelberg introduced a scalar field which makes an Abelian gauge theory massive but preserves gauge invariance. The Stueckelberg mechanism is the introduction of new fields to reveal a symmetry of a gauge-fixed theory. We first review the Stueckelberg mechanism in the massive Abelian gauge theory. We then extend this idea to the standard model, stueckelberging the hypercharge U(1) and thus giving a mass to the physical photon. This introduces an infrared regulator for the photon in the standard electroweak theory, along with a modification of the weak mixing angle accompanied by a plethora of new effects. Notably, neutrinos couple to the photon and charged leptons have also a pseudo-vector coupling. Finally, we review the historical influence of Stueckelberg's 1938 idea, which led to applications in many areas not anticipated by the author, such as strings. We describe the numerous proposals to generalize the Stueckelberg trick to the non-Abelian case with the aim to find alternatives to the standard model. Nevertheless, the Higgs mechanism in spontaneous symmetry breaking remains the only presently known way to give masses to non-Abelian vector fields in a renormalizable and unitary theory.