A chargeless complex vector matter field in supersymmetric scenario

In this paper we construct and study a formulation of a chargeless complex vector matter field in a supersymmetric framework. To this aim we combine two no-chiral scalar superfields in order to take the vector component field to build the chargeless complex vector superpartner where the respective field strength transforms as matter fields by a global $U(1)$ gauge symmetry. To the aim to deal with consistent terms without breaking the global $U(1)$ symmetry it imposes a choice to the complex combination revealing a kind of symmetry between the choices and eliminate the extra degrees of freedom consistently with the supersymmetry. As the usual case the mass supersymmetric sector contributes as a complement to dynamics of the model. We obtain the equations of motion of the Proca's type field, for the chiral spinor fields and for the scalar field on the mass-shell which show the same mass as expected. This work establishes the firsts steps to extend the analysis of charged massive vector field in a supersymmetric scenario.


I. INTRODUCTION
Due to the fundamental role played by supersymmetry on strings theory putting together quantum theories of the gravitational interactions, electroweak and strong forces, the studies on supersymmetric theories are of great interest to high energy physicists [1][2][3][4][5][6][7][8][9][10][11][12][13]. Supersymmetry deals with graded Lie algebra the unique reliable algebra extension which holds to be consistent withs the S-matrix in relativistic quantum field theory [14][15][16][17]. Recalling that this special symmetry correlate fermionic and bosonic fields, called superpartners, which put them together in a superfield formulation, what stresses the important role played by the matter-like fields to construct appropriated supersymmetric models [18,19].Moreover supersymmetric models with chiral superfields and global gauge invariance for the matter fields are elegantly constructed [13,17,19]. Thus quarks, leptons and vector bosons which participate in usual gauge theories, as electroweak and chromo-dynamics, in an supersymmetric extension coexist along their superpartners: squarks, sleptons and the fermionic partner of the vector bosons. For instance the supersymmetric version of quantum electrodynamics involves a vector supermultiplet whose contents are a massless photon and its spin- 1 2 superpartner, the photino [13,17]. The theoretical formulation of supersymmetric gauge vector field has been largely studied [13,19,20]. It was shown that gauge vector field component emerges from non-chiral scalar superfields when one uses some suitable constraint (Wess-Zumino) to remove exceeding non-physical components fields [17,19]. Nevertheless, there is a lack of studies on models that describe supersymmetric vector matter fields. Such models are interesting in order to understand the supersymmetric model of electroweak theory because it contains charged vector particles. Furthermore it can improve our knowledge of the form of nuclear atomic structure and its interaction at high energy. Furthermore, charged vector fields, with matter symmetry, are relevant in the vacuum polarization theory that can be connected to Lorentz symmetry violation in high energy physics . The aim of this work is to get the supermultiplet that accommodates the charged matter vector field and their supersymmetric partners, and also to get the most appropriate supersymmetric action for this field. To this end we are going to formulate a supersymmetric Lagrangian starting from neutral non-chiral superfield which contains the vector (matter) field. The present paper is outlined as follows: in Section II we present a model that acommodates two real vector matter field; in Section III we compose the previous model in a complex form and we present the Dirac superspinor field Ψ; in section IV we present a general conclusion.

II. TWO NEUTRAL VECTOR MATTER SUPERFIELDS MODEL
We are going to present a neutral (real) formulation for vector matter field. To this aim we start from a general non-chiral scalar superfield which includes in the matter multiplet a vector field as irreducible representation of the Lorentz group. In order to build more ahead a complex extension we introduce two neutral real scalar superfields doubling the number of degrees of freedom, that are written as where the superfields Φ and Λ are particular constructions of matter vector supermultiplet which include real vector fields X µ (x) and Y µ (x) with helicity ±1; the fields ϕ a (x), λ a (x), χ a (x) and ζ a (x) are two-components Weyl fermions with helicity ± 1 2 ; and the fields C(x), D(x), A(x), S(x), M (x), N (x), ρ(x), τ (x) are real scalar fields with spin-0. It is easy to verify that to both superfields the number of bosonic and fermionic degrees of freedom are the same. We stress that we only have applied the reality condition on the superfields what does not spoil the matter structure of these multiplets. Therefore the dynamics to neutral supersymmetric vector fields can be obtained through suitable field-strengths which accommodates the real superfields Φ and Λ.
In order to construct the supersymmetric field-strengths for the real superfields Φ and Λ, whose we call neutral supersymmetric field-strengths, we are going to apply supersymmetric covariant derivatives on the above scalar superfields which results in chiral superfields, in such way that and by similarity for Λ, we have that the W a and Ω a are chiral spinor superfields. We can redefine the superfields in the chiral superspace coordinates, namely Φ(y µ , θ a ),Φ(z µ ,θȧ), Λ(y µ , θ a ) and Λ(z µ ,θȧ), such that y µ = x µ + iθσ µθ and z µ = x µ − iθσ µθ . Hence the supersymmetric covariant derivatives are defined as According to this definitions we can compute the field-strengths W a and Ω a , and we have that , and similarly for the Wȧ and Ωȧ. So we are in conditions to construct the supersymmetric model in terms of the superfields W a and Ω a where neutral vector matter field is present. The kinetic part can be written as We have adopted the usual conventions for the spinor algebra, for the superspace parametrization, and the translation invariance of the integral in the chiral coordinates [13,19]. Then we obtain that the expression (6) has the following component expansion The action (7) describes the kinetic part of supersymmetric neutral vector field. However, to write the full action that corresponds to the underlined field theory we might also consider the supersymmetric mass term, given by where α 2 is the mass parameter. As usual, the "mass" part of the action presents kinetic terms, beyond the usual mass terms, which were eliminated by spinor chirality property of superfields W a and Ω a . Furthermore, we could infer that the mass term in the action (8) arises as a dynamical complement to the supersymmetric vector matter fields. Indeed supersymmetric matter-like fields are formulated with chiral spinor superfields.

III. THE CHARGED VECTOR MATTER SUPERFIELD MODEL
We know that the supersymmetric action for two free vector matter fields might be built through non-chiral scalar superfields [19]. Moreover, we can see that the degrees of freedom of this model are compatible with the dynamical free fields in complex space C. In this section our aim is to derive the appropriated complex superfield to describe supersymmetric charged vector fields. To this aim we need strongly define two complex non-chiral scalar superfields, such that and similarly for their complex conjugated superfields. The superfields (9) and (10) presents multiplets with charged vector fields B µ (x) e Z µ (x) and spin-1; the ξ a (x),Cȧ(x), R a (x),Ḡȧ(x), T a (x),Hȧ(x), U a (x) andQȧ(x) are Weyl fermion fields with spin-1 2 ; and the k(x), d(x), a(x) and v(x) are complex scalar fields with spin-0. So, to construct the complex extension for the neutral vector matter fields we combine the superfields Φ and Λ in such way that The combinations (11) are the two possible realizations of complex extension of neutral vector matter fields. And it is clear that they have relation via complex algebra. Indeed the transformation rule that implies in an invariant choice mechanism is given by This means that one can choose freely amongst the two equivalent complex extensions in (11) with no loss of generality of dynamical structure of the fields. We observe that the transformation rule (12) guarantees to write a consistent kinetic term for the charged vector field without the breaking of the global U (1) gauge symmetry. Another advantage that came to light is that the transformations (12) eliminates the exceeding fields which does not contribute for the supersymmetric action, what allows bosons and fermions to have the same physical degrees of freedom. Indeed the constraint relation to the superfields imply to the relations of the component fields as follows So, we can adjust the complex extension of the neutral superfields Φ and Λ by assuming the equation Σ = Φ + iΛ, where we find the following relation of fields To describe the dynamics of the supersymmetric charged vector fields with matter symmetry we need to construct an appropriated charged supersymmetric field-strength model in order to accommodate the superfields Σ and K. This can be reached starting from the following definitions where Υ a and Γ a are charged spinor superfields. As a consequence of the complex extension procedure we must relate the neutral spinor superfields Ω a and W a with the charged definitions (15) which, in the simplest way, is and by assuming the spinor identities W a Ω a = Ω a W a andWȧΩȧ =ΩȧWȧ we can find the kinetic supersymmetric Lagrangian for the charged vector fields We can observe that the left-hand side of the latter equation is the complex extension of neutral Lagrangian (7) that was written in terms of charged spinor superfields. Bearing this in mind, we can then redefine the kinetic Lagrangian (17) simply by combining the charged spinor superfields Υ a and Γ a as a "Dirac superspinor" Ψ, such that and also we assume Ψ as the adjoint Dirac superspinor representation. In this case we have that Ψ = Ψ † γ 0 = Γ aῩȧ , and so the supersymmetric action from the kinetic Lagrangian (17) is now given by We can notice that the product of Dirac superspinors ΨΨ obeys matter symmetry and it presents an interesting analogy to charged scalar superfield product S † S. In this sense we verify that Ψ and Ψ represent two chiral supersymmetric extensions for the matter vector field which can be transformed under U (1) global gauge group in the follow way where Λ is a U (1) gauge parameter, q is the charge of the symmetry. So the action (19) is then invariant under the transformations (20). In order to obtain the component Lagrangian we can expand the product ΨΨ by considering that Υ a (y , θ) = R a (y) + 2θ a d(y) + (σ µν θ) a F µν (y) − iθ 2 σ µ aḃ ∂ µḠḃ (y) Γ a (y , θ) = U a (y) + 2θ a v(y) + (σ µν θ) a Z µν (y) − iθ 2 σ µ aḃ ∂ µQḃ (y), and similarly forῩȧ andΓȧ. We notice the presence of the charged matter field-strengths, namely hence the action (19) can be expand and we obtain In this format we can recognize the dynamical term that describes the matter vector field as i 2 F µν Z µν − i 2 F * µν Z * µν . It involves both F µν and Z µν matter tensors. However, it not corresponds to the conventional kinetic term for the matter vector field, and the action (19) shows more degrees of freedom than it is necessary. In order to get rid of such fields we must assume the rule of transformation (12) which is a constraint of half of the degrees and consequently the action (19) reach the correct number of component fields. Applying the condition (12) in the action (23) we can reach the usual dynamical matter field strength term, or and so the Z µν tensor field is reabsorbed in this action. Likewise, and without loss of generality, we could have chosen the inverse relation Σ = iK † what implies to reabsorb the F µν tensor field. Then by using the whole relation (13) in action (19) we find the complex supersymmetric model for the matter vector field can be written as where the expression −F * µν (x)F µν (x) represents the usual kinetic term of the vector matter field while the terms represent with the components R a and G a the fermionic sector, and the last term corresponds to the auxiliary field d term. To completeness we are going to introduce the massive action term in the model. Observing the symmetries of non-chiral fields Σ and K the massive supersymmetric term can be suitable defined as where α 2 is a massive parameter. From non-chiral superfields Σ and K we can obtain the massive vector matter field term B * µ B µ as well as their supersymmetric parters. In order to performed it we are going to compute the action (26) by employing the condition K = iΣ † where one have that 1 2 Σ † Σ + 1 2 K † K = Σ † Σ, and by applying the definition (10) the full supersymmetric matter vector field model might be then obtained from Dirac superspinor field Ψ associated to the non-chiral scalar fields Σ and K in the follow form where the mass part of action can be obtained in component fields as where we observe the mass term to B, f (x) and l(x) fields. As in the usual supersymmetric models we note that the mass action (28) also contributes to kinetic structure, namely with the terms − 1 The action also shows mixing mass scalar and fermionic terms, By verifying the presence of extra kinetic terms in (28) it can suggest that when we particularly treat supersymmetric matter vector fields the mass action contributes with a "dynamic complement" to the kinetic action (25). Furthermore, we remark that the mass action (28) is important to match the number of bosonic and fermionic degrees of freedom of the supersymmetric matter vector action (27) for the consistency of the model. We can redefine some component fields absorbing the mass parameter as follows So the action can be re-written as analogously the mass action is now given by so the complex scalar fields d(x), f (x) and l(x) have no dynamics and arises as auxiliary fields. Hence assuming the action (27) as the sum of the redefined actions (30) and (31) and rearranging 1 the (Dirac) spinor fields Θ and Π it results in an off-shell action S os written in the following form where we denote the Dirac spinors of mass α as For the action (32) we have obtained chiral spinor mass terms given by αΘ(x)γ 5 Θ(x) and αΠ(x)γ 5 Π(x). It results that the motion equations for the fields are Taking the off-shell action (32) we note that it has 16-bosonic degrees of freedom concerning to the matter fields B µ (x), k(x), d(x), f (x), l(x) and their complex conjugated ones, as well as 16-fermionic degrees of freedom for the Dirac spinor fields Θ(x) and Π(x) and their conjugated complex ones, what is consistent to the supersymmetry. From the equations of motion (34) we note that there are three auxiliary complex scalar fields d(x), f (x), l(x) and a massless dynamical complex scalar field k(x). Moreover, as expected we have obtained a matter Proca-type equation for the field B µ . In this context, it is interesting to note that from the on-shell action that we can easily extract from the action (32) the supersymmetric generalization of matter vector field it is only possible if we include two dynamical Dirac chiral spinor fields Θ(x) and Π(x) along a massless scalar field k(x). Furthermore, a peculiar aspect of the spinor fields in the present case is that their mass terms arises as a result of their chiral structure.

IV. CONCLUSION
In this paper we proposed to formulate a supersymmetric model for a charged matter vector field which is important to get a clue on the possibility of Lorentz symmetry violation in supersymmetric theories, and the role played by the simplest case of high-spin field in field and string theory. We have started from real non-chiral scalar superfield in order to obtain real matter Proca-type field in a supersymmetric Lagrangian generalization [27]. In a straightforward way, the charged model was obtained extending the real scalar non-chiral superfields to the complex space [42][43][44][45][46]. The very interesting point in this enterprise is that we had to introduce new complex superfields composed by the originals. What introduces a sort of Hodge-duality symmetry, K = iΣ † , where the new superfield obtained is essential to obtain the charged Proca-Type dynamical term with global U (1) symmetry. Furthermore this relation is crucial to match the number of degrees of freedom of the bosonic and fermionic sectors. The supersymmetric mass term B * µ B µ is also important to match of dynamical degrees of freedom of the supersymmetric model.
We can conclude that the supersymmetric mass term and Hodge-type duality symmetry are interesting features to further investigate because they play an important role to build supersymmetric matter vector field Lagrangian. Another observed characteristic is that the charged dynamical Proca-type term emerges from a product of Dirac superspinors ΨΨ which has dynamical contributions similarly to the product S † S of the super-QED model. Furthermore, we can remark that the Dirac superspinor field Ψ is also chiral because it is a combination of chiral Weyl superspinors Γ a and Υ a . Finally, we conclude that the on-shell supersymmetric action obtained (32) reveals two fermionic Dirac fields Θ(x) and Π(x), and a massless scalar field k(x) as supersymmetric partners associated to the charged vector field B µ (x).