Stueckelberg Fields Research Articles

On massive higher spin supermultiplets in d = 4

In this work we discuss the cubic interactions for massless spin 3/2 gravitino with massive higher spin supermultiplets using three superblocks (2, 3/2), (5/2, 2) and (3, 5/2) as the first non-trivial examples. We use gauge invariant formalism for the massive higher spin fields and, as is common in such cases, we face an ambiguity related with the possible field redefinitions due to the presence of Stueckelberg fields. From one hand, we show how this ambiguity can be used as one more way to classify possible cubic vertices. We also note that all these field redefinitions do not change the part of the Lagrangian which appears in the unitary gauge (where all Stueckelberg fields are set to zero) so we still have some important independent results. From the other hand, we show how using the so-called unfolded formalism one can fix these ambiguities and obtain consistent deformations for all massive field gauge invariant curvatures which is the most important step in the Fradkin-Vasiliev formalism. Unfortunately, this works for the massive fields only so the way to construct deformations for the massless field curvatures is still has to be found.

Conformal Yang-Mills field in (A)dS space

Ordinary-derivative (second-derivative) Lagrangian formulation of classical conformal Yang-Mills field in the (A)dS space of six, eight, and ten dimensions is developed. For such conformal field, we develop two gauge invariant Lagrangian formulations which we refer to as generic formulation and decoupled formulation. In both formulations, the usual Yang-Mills field is accompanied by additional vector and scalar fields where the scalar fields are realized as Stueckelberg fields. In the generic formulation, the usual Yang-Mills field is realized as a primary field, while the additional vector fields are realized as auxiliary fields. In the decoupled formulation, the usual Yang-Mills field is realized as massless field, while the additional vector fields together with the Stueckelberg are realized as massive fields. Some massless/massive fields appear with the wrong sign of kinetic terms, hence demonstrating explicitly that the considered models are not unitary. The use of embedding space method allows us to treat the isometry symmetries of (A)dS space manifestly and obtain conformal transformations of fields in a relatively straightforward way. By accompanying each vector field by the respective gauge parameter, we introduce an extended gauge algebra. Levy-Maltsev decomposition of such algebra is noted. Use of the extended gauge algebra setup allows us to present concise form for the Lagrangian and gauge transformations of the conformal Yang-Mills field. Higher-derivative representation of the Lagrangian is also obtained.

Pole-skipping for massive fields and the Stueckelberg formalism

Pole-skipping refers to the special phenomenon that the pole and the zero of a retarded two-point Green’s function coincide at certain points in momentum space. We study the pole-skipping phenomenon in holographic Green’s functions of boundary operators that are dual to massive p-form fields and the dRGT massive gravitational fields in the AdS black hole background. Pole-skipping points for these systems are computed using the near horizon method. The relation between the pole-skipping points of massive fields and their massless counterparts is revealed. In particular, as the field mass m is varied from zero to non-zero, the pole-skipping phenomenon undergoes an abrupt change with doubled pole-skipping points found in the massive case. This arises from the breaking of gauge invariance due to the mass term and the consequent appearance of more degrees of freedom. We recover the gauge invariance using the Stueckelberg formalism by introducing auxiliary dynamical fields. The extra pole-skipping points are identified to be associated with the Stueckelberg fields. We also observe that, as the mass varies, some pole-skipping points of the wave number q may move from a non-physical region with complex q to a physical region with real q.

Cosmological Mass of the Photon Related to Stueckelberg and Higgs Mechanisms

I consider the electro-weak (EW) masses and interactions generated by photons using vacuum expectation values of Stueckelberg and Higgs fields. I provide a prescription to relate their parametric values to a cosmological range derived from the fundamental Heisenberg uncertainty principle and the Einstein–de Sitter cosmological constant and horizon. This yields qualitative connections between microscopic ranges acquired by W± or Z0 gauge Bosons and the cosmological scale and minimal mass acquired by g-photons. I apply this procedure to an established Stueckelberg–Higgs mechanism, while I consider a similar procedure for a pair of Higgs fields that may spontaneously break all U(1) × SU(2) gauge invariances. My estimates of photon masses and their additional parity-breaking interactions with leptons and neutrinos may be detectable in suitable accelerator experiments. Their effects may also be observable astronomically through massive g-photon condensates that may contribute to dark matter and dark energy.

Yang–Mills–Stueckelberg theories, framing and local breaking of symmetries

We consider Yang–Mills theory with a compact structure group [Formula: see text] on a Lorentzian 4-manifold [Formula: see text] such that gauge transformations become identity on a submanifold [Formula: see text] of [Formula: see text] (framing over [Formula: see text]). The space [Formula: see text] is not necessarily a boundary of [Formula: see text] and can have dimension [Formula: see text]. Framing of gauge bundles over [Formula: see text] demands introduction of a [Formula: see text]-valued function [Formula: see text] with support on [Formula: see text] and modification of Yang–Mills equations along [Formula: see text]. The fields [Formula: see text] parametrize non-equivalent flat connections mapped into each other by a dynamical group [Formula: see text] changing gauge frames over [Formula: see text]. It is shown that the charged condensate [Formula: see text] is the Stueckelberg field generating an effective mass of gluons in the domain [Formula: see text] of space [Formula: see text] and keeping them massless outside [Formula: see text]. We argue that the local Stueckelberg field [Formula: see text] can be responsible for color confinement. We also briefly discuss local breaking of symmetries in gravity. It is shown that framing of the tangent bundle over a subspace of spacetime makes gravitons massive in this subspace.

Безмассовая частица Штюкельберга, решения с цилиндрической симметрией

The massless Stueckelberg field is studied in cylindrical coordinates.
 The field function consists of the scalar, 4-vector,
 and antisymmetric tensor. Physically observable components
 are the scalar and 4-vector. We apply the Stueckelberg
 tetrad-based matrix equation, generalized to arbitrary Riemannian
 space, including any curvilinear coordinates in the
 Minkowski space. We construct solutions with cylindric symmetry,
 while the operators of energy, of the third projection
 of the total angular momentum, and the third projection of
 the linear momentum are diagonalized. After separating the
 variables we derive the system of 11 first-order differential
 equations in polar coordinate. It is solved with the use of the
 Fedorov–Gronskiy method. According to this method, all 11
 functions are expressed through 3 main funcions. According
 to the known procedure we impose the differential constraints,
 which are consistent with the all 11 equations and
 allow us to transform these equations to algebraic form. This
 algebraic system is solved by standard methods. As a result,
 we obtain 5 linearly independent solutions. The problem of
 eliminating the gauge solutions will be studied in a separate
 paper.

Безмассовое поле Штюкельберга, точные решения в декартовых координатах и калибровочные степени свободы

In the paper we examine the massless Stueckelberg field. Among the eleven field function components, the antisymmetric tensor represents the gauge variables, whereas the scalar and vector correspond to physically observable quantities. It is shown that in Cartesian coordinates the Stueckelberg equations permit the existence of five independent solutions which describe the different states of the field. We have derived an expression for the energy-momentum tensor of the massless Stueckelberg field. We find its explicit form for arbitrary linear combination of five established solutions. We have found four combinations of five solutions which do not contribute to energy-momentum tensor, therefore they correspond to purely gauge states. There exists only one solution which corresponds to nonvanishing energy-momentum tensor, it relates to physically observable states of the massless Stueckelberg field.

One-loop radiative corrections in bumblebee-Stueckelberg model

This work aims to study the radiative corrections in a vector model with spontaneous Lorentz symmetry violation, known in the literature as the bumblebee model. We consider such a model with self-interaction quadratic smooth potential responsible for spontaneous Lorentz symmetry breaking. The spectrum of this model displays a transversal nonmassive mode, identified as Nambu-Goldstone, and a massive longitudinal mode. Besides the Lorentz symmetry, this model also exhibits gauge symmetry violation. To restore the gauge symmetry, we introduce the Stueckelberg field and calculate the two-point function by employing the principal-value (PV) prescription. The result is nontransversal, leading to a massive excited mode.

Nonminimal non-Abelian quantum vector fields in curved spacetime

The quantum effective action of non-minimal vector fields with Abelian or non-Abelian gauge degrees of freedom in curved spacetime is studied. The Proca or Yang-Mills fields are coupled to a local mass-like term acting in both coordinate and gauge spaces. Pathologies due to gauge invariance in the ultraviolet are avoided through the introduction of a non-Abelian version of the Stueckelberg field. It is found that the breaking of gauge invariance induced by the mass term affects only the tree level part of the effective action. The ultraviolet divergent part of the effective action to one loop is obtained using the method of covariant symbols and dimensional regularization. Formulas are given valid for any spacetime dimension and explicit results are shown for the two-dimensional case. As already happened for a single vector field, the ultraviolet divergences are local but not of polynomial type.

A note on higher rank descriptions of massless and massive spin-1 particles

The Maxwell theory can be written as a first order model with the help of a two-form auxiliary field, such master action allows the proof of duality between 1-form and D−3 forms. Here we show that the replacement of the two-form auxiliary field by an arbitrary (non symmetric) rank-2 tensor leads to a new massless spin-1 dual theory in terms of a partially antisymmetric rank-3 tensor. In the massive spin-1 case we have a non symmetric generalization of the massive two-form theory (Kalb–Ramond). The coupling of the massive non symmetric spin-1 model to matter fields is investigated via master actions.We also show that massive models with severe discontinuity in their massless limit can also be obtained from Kaluza–Klein dimensional reduction of massless higher rank tensors which become Stueckelberg fields after the reduction.

Stueckelberg and Higgs Mechanisms: Frames and Scales

We consider Yang–Mills theory with a compact gauge group G on Minkowski space R3,1 and compare the introduction of masses of gauge bosons using the Stueckelberg and Higgs mechanisms. The Stueckelberg field ϕ is identified with a G-frame on the gauge vector bundle E and the kinetic term for ϕ leads to the mass of the gauge bosons. The Stueckelberg mechanism is extended to the Higgs mechanism by adding to the game a scalar field describing rescaling of metric on fibres of E. Thus, we associate Higgs fields as well as running coupling parameters with conformal geometry on fibres of gauge bundles. In particular, a running coupling tending to zero or to infinity is equivalent to an unbounded expansion of G-fibres or its contraction to a point. We also discuss scale connection, space-time dependent Higgs vacua and compactly supported gauge and quark fields as an attribute of confinement.

Ultra-high-energy neutrino scattering in an anomalous U(1) effective field theory

A unique characteristic of exponentially growing scattering amplitude arises in an anomalous Abelian effective field theory when an extremely light Dirac neutrino mass is introduced to break the symmetry. We show that the low energy effective Lagrangian can be made explicitly gauge invariant with the help of a nonlinear representation of the Goldstone or Stueckelberg field. We study the peculiar feature of exponential growth in the ultra-high-energy neutrino-nucleon inelastic scattering. It is found that the inelastic scattering cross section is highly sensitive to the ratio of gauge coupling to the gauge boson mass, gX/mX. When the IceCube measurement of ultra-high-energy neutrinos, which is consistent with the standard model prediction up to Eν∼6 PeV, is taken into account, the inferred constraint on gX/mX is more severe than that obtained from the events of mono-lepton+missing transverse energy at the LHC. A muon collider with a collision energy of 10 TeV can be a good environment other than hadron colliders to probe the novel effect.

Conservation laws in gauge gravity theory

We analyze the conservation laws in the gauge gravity theory which are derived for the general class of gravitational models with the action invariant under the local Poincaré and the diffeomorphism group. The consistent Noether-Lagrange formalism is developed, revealing the important role of the auxiliary Goldstone and Stueckelberg fields, with the help of which we construct the composite gauge fields that have a clear geometrical and physical meaning.

Hamiltonian analysis and positivity of a new massive spin-2 model

Recently a new model has been proposed to describe free massive spin-2 particles in D dimensions in terms of a non symmetric rank-2 tensor e μν and a mixed symmetry tensor B μ[αβ]. The model is invariant under linearized diffeomorphisms without Stueckelberg fields. It resembles a spin-2 version of the topologically massive spin-1 BF model (Cremmer–Scherk model). Here we apply the Dirac–Bergmann procedure in order to identify all Hamiltonian constraints and perform a complete counting of degrees of freedom. In D = 3 + 1 we find 5 degrees of freedom corresponding to helicities ±2, ±1, 0 as expected. The positivity of the reduced Hamiltonian is proved by using spin projection operators. We have also proposed a parent action that establishes the duality between the Fierz–Pauli and the new model. The equivalence between gauge invariant correlation functions of both theories is demonstrated.

Standard Model in Weyl conformal geometry

We study the Standard Model (SM) in Weyl conformal geometry. This embedding is truly minimal with no new fields beyond the SM spectrum and Weyl geometry. The action inherits a gauged scale symmetry D(1) (known as Weyl gauge symmetry) from the underlying geometry. The associated Weyl quadratic gravity undergoes spontaneous breaking of D(1) by a geometric Stueckelberg mechanism in which the Weyl gauge field (omega _mu ) acquires mass by “absorbing” the spin-zero mode of the {tilde{R}}^2 term in the action. This mode also generates the Planck scale and the cosmological constant. The Einstein-Proca action emerges in the broken phase. In the presence of the SM, this mechanism receives corrections (from the Higgs) and it can induce electroweak (EW) symmetry breaking. The EW scale is proportional to the vev of the Stueckelberg field. The Higgs field (sigma ) has direct couplings to the Weyl gauge field (sigma ^2omega _mu omega ^mu ). The SM fermions only acquire such couplings for non-vanishing kinetic mixing of the gauge fields of D(1)times U(1)_Y. If this mixing is present, part of the mass of Z boson is not due to the usual Higgs mechanism, but to its mixing with massive omega _mu . Precision measurements of Z mass then set lower bounds on the mass of omega _mu which can be light (few TeV). In the early Universe the Higgs field can have a geometric origin, by Weyl vector fusion, and the Higgs potential can drive inflation. The dependence of the tensor-to-scalar ratio r on the spectral index n_s is similar to that in Starobinsky inflation but mildly shifted to lower r by the Higgs non-minimal coupling to Weyl geometry.

Cosmological aspects of cubic Galileon massive gravity

Cubic Galileon massive gravity is a development of de Rham-Gabadadze-Tolly (dRGT) massive gravity theory is which the space of the Stueckelberg field is broken. We consider the cubic Galileon term as a scalar field coupled to the graviton filed. We present a detailed study of the cosmological aspects of this theory of gravity. We analyze self-accelerating solutions of the background equations of motion to explain the accelerated expansion of the Universe. Exploiting the latest Union2 Type Ia Supernovea (SNIa) dataset, which consists of 557 SNIa, we show that cubic Galileon massive gravity theory is consistent with the observations. We also examine the tensor perturbations within the framework of this model and find an expression for the dispersion relation of gravitational waves, and show that it is consistent with the observational results.

On massive spin-3/2 in the Fradkin–Vasiliev formalism

One of the possible approaches to the construction of massive higher spin interactions is to use their gauge invariant description based on the introduction of the appropriate set of Stueckelberg fields. Recently, the general properties of such approach were investigated by Boulanger et al (2018 J. High Energy Phys. JHEP07(2018)021). The main findings of this work can be formulated in two statements. At first, there always exist enough field redefinitions to bring the vertex into abelian form where there are some corrections to the gauge transformations but the gauge algebra is undeformed. At second, with the further (as a rule higher derivative) field redefinitions one can bring the vertex into trivially gauge invariant form expressed in terms of the gauge invariant objects of the free theory. Our aim in this work is to show (using a simple example) how these general properties are realised in the so-called Fradkin–Vasiliev formalism and to see the effects (if any) that the presence of massless field, and hence of some unbroken gauge symmetries, can produce. As such example we take the gravitational interaction for massive spin-3/2 field so we complete the investigation started by Buchbinder et al (2014 Eur. Phys. J. C 74 3153) relaxing all restrictions on the number of derivatives and allowed field redefinitions. We show that in spite of the presence of massless spin-2 field, the first statement is still valid, while there exist two abelian vertices which are not equivalent on-shell to the trivially gauge invariant ones. Moreover, it is one of this abelian vertices that reproduce the minimal interaction for massive spin-3/2.

Reducible Stueckelberg symmetry and dualities

We propose a general procedure for iterative inclusion of Stueckelberg fields to convert the theory into gauge system being equivalent to the original one. In so doing, we admit reducibility of the Stueckelberg gauge symmetry. In this case, no pairing exists between Stueckelberg fields and gauge parameters, unlike the irreducible Stueckelberg symmetry. The general procedure is exemplified by the case of Proca model, with the third order involutive closure chosen as the starting point. In this case, the set of Stueckelberg fields includes, besides the scalar, also the second rank antisymmetric tensor. The reducible Stueckelberg gauge symmetry is shown to admit different gauge fixing conditions. One of the gauges reproduces the original Proca theory, while another one excludes the original vector and the Stueckelberg scalar. In this gauge, the irreducible massive spin one is represented by antisymmetric second rank tensor obeying the third order field equations. Similar dual formulations are expected to exist for the fields of various spins.

General method for including Stueckelberg fields

A systematic procedure is proposed for inclusion of Stueckelberg fields. The procedure begins with the involutive closure when the original Lagrangian equations are complemented by all the lower order consequences. The Stueckelberg field is introduced for every consequence included into the closure. The generators of the Stueckelberg gauge symmetry begin with the operators generating the closure of original system. These operators are not assumed to be a generators of gauge symmetry of any part of the original action, nor are they supposed to form an on shell integrable distribution. With the most general closure generators, the consistent gauge invariant theory is iteratively constructed, without obstructions at any stage. The Batalin–Vilkovisky form of inclusion of the Stueckelberg fields is worked out and the existence theorem for the Stueckelberg action is proven.

Axiomatic derivation of coincident general relativity and its premetric extension

A theory of gravity is deduced from the axioms of the premetric program. The starting point is the conservation of energy and momenta, and the equivalence of gravitation and inertia. The latter is what leads to the framework of the so-called purified gravity. The local and linear constitutive relation has 14 components when it is assumed to be metrical, but the compatibility of the constitutive relation with an action principle fixes uniquely the theory of coincident general relativity. The premetric formalism of purified gravity has a direct analogy with massive electromagnetism, the Planck mass corresponding to the Proca mass of the gauge boson. The metric emerges as a Stueckelberg field, and the graviton as a Goldstone boson of the broken symmetry.