Renormalization of a model for spin-1 matter fields

In this work, the one-loop renormalization of a theory for fields transforming in the (1,0)⊕(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1,0)\oplus (0,1)$$\end{document} representation of the Homogeneous Lorentz Group is studied. The model includes an arbitrary gyromagnetic factor and self-interactions of the spin 1 field, which has mass dimension one. The model is shown to be renormalizable for any value of the gyromagnetic factor.


Introduction
In the Standard Model of particle physics, only fields transforming in the (0, 0), (1/2, 0), (0, 1/2) and (1/2, 1/2) representations of the Homogeneous Lorentz Group (HLG) are needed. There is however no guiding principle restricting the possible representations, and indeed high spin fields naturally appear in Hadron physics and in Beyond the Standard Model (BSM) scenarios like supergravity and superstrings.
In an attempt to better understand the physics of fields transforming in different representations of the HLG, a series of works have been carried [1][2][3][4][5][6][7][8] based on the projection onto subspaces of the Poincaré group. In this formalism, it has been shown that the gyromagnetic factor of spin 3/2 fields is connected with their causal propagation in an electromagnetic background [1], and with the unitarity of the Compton scattering amplitude in the forward direction [2]. The formalism can also be applied to lower spins, for example, in the spin 1 case, a similar connection between the unitarity of Compton scattering in the forward direction and the gyromagnetic factor of the field exists, which is also related to the electric quadrupole moment [3].
When the Poincaré projector method is applied to spin 1/2 fields transforming in the (1/2, 0) ⊕ (0, 1/2) representation a e-mail: ailierrivero@gmail.com b e-mail: vaquera@fisica.ugto.mx (corresponding author) [6,7], the resulting Lagrangian is a generalized version of the original second order Feynman-Gell-Man formalism [9], enhanced with an arbitrary gyromagnetic factor and fermion self interactions. The second order fermions studied in these works are conceptually different to Dirac ones, as the former propagate 8 dynamical degrees of freedom instead of 4. As shown in [6,7], there is a consistent reduction of dynamical degrees of freedom and a direct connection between the renormalization group equations for the second order fermions and the Dirac formalism if the gyromagnetic (or chromomagnetic) factor is set to the fixed value g = 2.
The goal of the present work is to study the renormalization properties of spin-1 matter fields 1 transforming in the (1, 0) ⊕ (0, 1) representation of the HLG in a model based on the Poincaré projector formalism, as a direct generalization of the spin 1/2 case [6,7].
The difference between the pure spin 1 representation (1, 0) ⊕ (0, 1), described by an antisymmetric tensor field of second rank, and the more familiar (1/2, 1/2) vector field is more dramatic in the massless case, as the Kalb-Ramond antisymmetric gauge field contains only one physical longitudinal degree of freedom [10], whereas the massless vector gauge field is characterized by 2 transverse ones. Switching to massive spin-1 particles, one must distinguish between gauge invariant and non-gauge invariant theories. It can be shown that a massive Stueckelberg compensated Kalb-Ramond gauge field is dual to a compensated massive gauge vector field [11]. However, for non-gauge invariant massive spin-1 theories, the properties of four-vector and antisymmetric tensor particles can differ significantly. In [12] the difference between spin-1 antisymmetric tensor mesons and the four-vector mesons has been studied in detail for composite hadrons. In the present work, we focus instead on pointlike massive spin-1 bosons, with emphasis on their electromagnetic properties and their possible self-interactions.
The model studied here is based on [4], where the complex antisymmetric tensor field has 6 complex degrees of freedom, making the (1, 0)⊕(0, 1) theory explicitly different to any of a massive gauge vector field. In [4] the Compton scattering of spin-1 particles described by both a massive four-vector and an antisymmetric tensor was analyzed for arbitrary values of the gyromagnetic factor, finding that the Compton scattering cross section off the parity degrees of freedom in (1, 0) ⊕ (0, 1) is finite in the forward direction, though it is still divergent elsewhere. Interestingly, for the antisymmetric tensor this result is independent of the gyromagnetic factor, while Compton scattering off the four-vector is only well behaved in all directions provided the gyromagnetic ratio is set to g = 2. Given the non-finiteness of Compton scattering in this theory, it is unclear if the renormalizable theory described here corresponds to a perturbation theory about a sensible zeroth-order Hamiltonian. However, it constitutes a unique theoretical laboratory from the point of view of the renormalization group, in the same spirit as scalar λφ 3 theory.
The structure of the paper is the following: In Sect. 2 we describe the model and the Feynman rules. The renormalization procedure is presented in Sect. 3 together with the cancellation of all the potentially divergent contributions to the one-loop vertices of the theory. Finally, the conclusions of the work are discussed in Sect. 4.

The model
Our model comprises a massive complex spin-1 antisymmetric tensor field B αβ in the (1, 0) ⊕ (0, 1) representation of the HLG, minimally coupled to U (1) EM with arbitrary gyromagnetic factor and mass dimension one, allowing for self interaction terms. The Lagrangian of the model is given by where D μ = ∂ μ + ie A μ is the covariant derivative, and the tensors used are given by The kinetic part of the Lagrangian is of Klein-Gordon type and spin-1 information is encoded by a Pauli-like term modulated by an arbitrary gyromagnetic factor g and the four independent quartic self-interactions that can be built from the covariant basis for the (1, 0)⊕(0, 1) representation space, given by the complete set of tensors presented in [8], namely In our analysis, the gauge freedom is fixed by the R ξ contribution with arbitrary gauge fixing parameter ξ , rendering the complete Lagrangian of the model as The Feynman rules corresponding to the above Lagrangian are presented in Fig. 1, where all momenta are incoming.
The gauge invariance of the theory imposes two important Ward-Takahashi identities (see [5] for their derivation in the analogous spin 1/2 case). The first one relates the tensortensor-photon (TTγ ) vertex function −ie μ (q, p, − p − q), where q is the momentum of the photon, with the tensor selfenergy −i ( p) according to The second one involves the tensor-tensor-photon-photon (TTγ γ ) vertex ie 2 μν (q, q , p, p ), with photon momenta q and q , and the TTγ vertex, and reads (2.7)

Renormalization
In this section, we analyze the renormalization properties of the model at one-loop level, studying the UV divergent parts of all the potentially divergent vertex functions. In this work, we use dimensional regularization with d = 4 − 2 and the naive prescription for the chirality operator χ This approach does not lead to inconsistencies as χ appears in pairs for all the processes involved. The subtraction scheme used in the study is the minimal subtraction (MS) one.

Counterterms
Taking Eq. (2.5) as the bare Lagrangian, with all bare quantities denoted by a 0 subscript, its parameters are the tensor mass m 0 , the tensor charge e 0 and the gyromagnetic factor g 0 . The renormalized fields are defined in terms of the bare ones through It is convenient to split the Lagrangian as the sum of two terms where the first piece is the renormalized Lagrangian, and has the same structure as Eq. (2.5) , (3.4) and the second one contains the relevant counterterms with the following definitions In d = 4−2 dimensions, the renormalized parameters must be scaled according to e r → μ e r , g r → g r , λ r i → μ 2 λ r i , m r → m r , where μ is the arbitrary scale introduced by dimensional regularization. In what follows, we will omit the r subscript for the renormalized parameters. In this notation, the Feynman rules for counterterms are given in Fig. 2.
In the following subsections, we will compile the results obtained for the calculation of all the divergent processes showing that all the divergencies can be absorbed successfully into the given set of counterterms provided by the theory.

Vacuum polarization
There are two diagrams contributing to the vacuum polarization, depicted in Fig. 3. The divergent piece, denoted by −i μν (q) * is given by (3.9) and can be removed in the MS scheme by fixing the counterterm δ 1 as (3.10)

Tensor self-energy
In Fig. 4 are shown the three diagrams contributing to the Tensor self-energy. The divergent part of this amplitude is (3.11) and the counterterms that cancel the UV divergence are then given by     As expected, the contribution to the γ γ γ vertex from the diagrams in Fig. 5 vanishes identically from the charge conjugation invariance of the theory.

TTγ vertex
The one-loop contribution to the TTγ vertex comes from the four diagrams in Fig. 6. Its divergent piece can be written as (3.14) and is canceled by the corresponding counterterm with Notice that this result is consistent with the Ward identity * μ (0, p, − p) = − ∂ * ( p) ∂ p μ . (3.17) as δ e = δ 2 . Gauge invariance also fixes the counterterm involved in the finiteness of the TTγ γ vertex, as Eq. (2.7) dictates that

γ γ γ γ vertex
The one-loop correction to the γ γ γ γ vertex involves 21 diagrams, shown in Fig. 8, and there is no counterterm available to cancel a potential divergence in this case. By an explicit calculation, we have found that the divergent piece of the total amplitude vanishes exactly.

Summary and conclusions
In this work, we have studied the one-loop renormalization of the electrodynamics of fields transforming under the (1, 0) ⊕ (1, 0) representation of the HLG in the Poincaré projector formalism. The analysis has been done in an arbitrary covariant gauge, with arbitrary gyromagnetic factor and including all the independent parity conserving selfinteractions. The main conclusion of the work is that the theory is renormalizable for any value of the gyromagnetic factor, displaying a rich set of renormalization group equations. In contrast to the analogous spin 1/2 case studied in [6], there is no non-trivial finite value for the gyromagnetic factor that allows the existence of a pure electrodynamics without the inclusion of self interactions.