N=1 Supersymmetric Non-Abelian Compensator Mechanism for Extra Vector Multiplet

We present a variant formulation of N=1 supersymmetric compensator mechanism for an arbitrary non-Abelian group in four dimensions. This formulation resembles our previous variant supersymmetric compensator mechanism in 4D. Our field content consists of the three multiplets: (i) A Non-Abelian Yang-Mills multiplet (A_\mu^I, \lambda^I, C_{\mu\nu\rho}^I), (ii) a tensor multiplet (B_{\mu\nu}^I, \chi^I, \varphi^I) and an extra vector multiplet (K_\mu^I, \rho^I, C_{\mu\nu\rho}^I) with the index I for the adjoint representation of a non-Abelian gauge group. The C_{\mu\nu\rho}^I is originally an auxiliary field dual to the conventional auxiliary field D^I for the extra vector multiplet. The vector K_\mu^I and the tensor C_{\mu\nu\rho}^I get massive, after absorbing respectively the scalar \varphi^I and the tensor B_{\mu\nu}^I. The superpartner fermion \rho^I acquires a Dirac mass shared with \chi^I. We fix all non-trivial cubic interactions in the total lagrangian, all quadratic terms in supersymmetry transformations, and all quadratic interactions in field equations. The action invariance and the super-covariance of all field equations are confirmed up to the corresponding orders.


Introduction
Recently, there have been considerable developments [1] [2] for the supersymmetrization of the Proca-Stueckelberg compensator mechanism [3]. The supersymmetrization of non-Abelian compensator mechanism was first performed in late 1980's [4]. Abelian supersymmetric Proca-Stueckelberg mechanism [5] has a direct application to MSSM [6]. In [1], general representations of non-Abelian group are analyzed, and higher-order terms have been also fixed. Even though the original Higgs mechanism [7] has been established experimentally [8], the Proca-Stueckelberg type compensator mechanism for massive gauge fields [3] is still an important theoretical alternative.
The C µνρ I -field is dual to the conventional auxiliary field D I . The 'dilaton' ϕ I (or B µν I ) is absorbed into the longitudinal component of A µ I (or C µνρ I ), making the latter massive [2]. This compensation mechanism works even with C µνρ I in the adjoint representation.
In this present paper, we demonstrate yet another field content as a supersymmetric compensator system in which an extra vector in the adjoint representation absorbs a scalar.
After the absorption, the on-shell DOF count as A µ I (2), λ I (2); K µ I (3), ρ I (4), C µνρ I (1), as summarized in the following Table. DOF Our new system differs from our recent work [2] in terms of the three aspects: (i) Our present system has three multiplets VM, TM and EVM, while that in [2] has only VM and TM. The new multiplet is EVM (K µ I , ρ I , C µνρ I ), where K µ I (or C µνρ I ) absorbs ϕ I (or B µν I ), getting massive.
(ii) The vector field getting massive is not A µ I , but is the extra vector field K µ I .
(iii) The VM (A µ I , λ I ) has no auxiliary field, while the EVM has the auxiliary field C µνρ I .
So our VM is on-shell, while our TM and EVM are off-shell.

Field Strengths and Tensorial Transformations
The field strengths for our bosonic fields A µ I , B µνρ I , C µνρ I , K µ I and ϕ I are respectively We use m for the YM coupling constant, while D µ is the YM-covariant derivative. The G in (2.1b) instead of G is a reminder that this field strength has an extra term m −1 F ∧B.
Similarly, D µ in (2.1e) is used to be distinguished from D µ . The m C and m K -terms in the respective field strength G and Dϕ are suggestive that these field strengths can be absorbed into the field redefinitions of C and K.
The Bianchi identities for our field strengths are There should be proper tensorial transformations [1][2] associated with B µν I , C µνρ I and K µ I which are symbolized as δ β , δ γ and δ κ . The last δ κ is for the extra vector K µ I which is also a kind of 'tensor' in adjoint representation: where δ α is the standard YM gauge transformation.
Our field strengths are all covariant under δ α , while invariant under δ β , δ γ , δ γ and δ κ : The transformations (2.3c) and (2.3d) indicate that the C µνρ I and K µ I -fields respectively can absorb the compensators B µν I and ϕ I .

Lagrangian and N=1 Supersymmetry
Once the invariant field strengths F, G, H, L and Dϕ have been established, it is straightforward to construct a lagrangian, invariant also under N = 1 supersymmetry. Our 3) We also use the symbol ⌊ ⌈r⌋ ⌉ for totally antisymmetric indices ρ1···ρr to save space. Our notation is (η µν ) ≡ diag. (−, +, +, +), up to quartic terms O(φ 4 ). The kinetic terms of B and ϕ, namely, the (G µνρ I ) 2 and (D µ ϕ) 2 -terms, which respectively contain m 2 C 2 and m 2 K 2 -terms, play the role of mass terms for the C and K -fields, after the absorptions of DB by C and D ϕ by K.
Because of N = 1 supersymmetry, this compensator mechanism between TM and EVM works also for fermionic partners. Namely, the original χ I -field in TM is mixed with the ρ I -field in EVM, forming the Dirac mass term m (χ I ρ I ).
The N = 1 supersymmetry transformation rule of our multiplets is An important corollary is for the arbitrary variations of our field strengths: A special case of (3.3) is the supersymmetry transformation rule, Note the peculiar m −1 F ∧ B -term in G in (2.1b). The general variation of this term is The last term is proportional to (δA) ∧ L with the original m −1 cancelled by m in the former resulting in only a m 0 -term, interpreted as the third term in (3.3b). The first term of (3.6) with m −1 is absorbed into the second term of δ B νρ I in (3.4a). This sort of sophisticated Chern-Simons terms at order m −1 has not been presented in the past, to our knowledge. This is the result of intricate play between the TM and EVM, where the latter absorbs the former as a compensator multiplet. depending on the number of γ -matrices sandwiched by ǫ and λ. This is carried out by adding the non-trivial λρ -terms in δ Q χ, λχ -terms in δ Q ρ, and χρ -terms in δ Q λ. For the category (IV) for m 1 φ 3 -terms has four sectors: (i) mλρ 2 , (ii) mλ 3 , (iii) mλχ 2 , and (iv) mλρ 2 . The confirmation of all of these sectors are relatively easy, consistently with the λρ -terms in δ Q χ, λχ -terms in δ Q ρ, and χρ -terms in δ Q λ.

Field Equations
The field equations in our system are highly non-trivial. This is due to the extra Chern-Simon-type terms in various field strengths. Even the simplest field strength D µ ϕ I has an extra term mK µ I . The explicit forms of our field equations are where the symbol .
= is for an equality by the use of field equation(s). Also, these equations The m G -term in the C -field equation (4.1c) plays the role of the mass term for the C -field after a field redefinition of C absorbing the 3DB -term in G. So does the m Dϕ -term in the K -field equation (4.1d).
The result (4.1) is based on an important lemma about the general variation of our lagrangian up to a total divergence: The symbol (δL ψD / ψ /δA µ I ) in the first line is for the contributions from the minimal couplings in the fermionic kinetic terms of λ, χ and ρ. Use is also made of the generalvariation formulae in (3.3) for arranging the whole terms.
In getting the expression (4.2), there are many non-trivial cancellations. For example, the two terms: cancel up to O(φ 3 ) upon the use of the C -field equation (4.1c). Similarly, the two terms: also cancel upon the K -field equation (4.1d) up to O(φ 3 ).
By straightforward computations, we can confirm that the supersymmetric variation of each of the field equations in (4.1) vanishes up to O(φ 3 ). This gives an independent confirmation of the consistency of our total system.
As an additional confirmation, we can show that the divergence of the A, B, C and K -field equations all vanish. For example, the divergence of the A -field equation is Here we have used other field equations, such as D / λ I .
Similarly for the case of C -field equation: where we used the B -field equation for the D G -term.

Parity-Odd Terms
We can add certain parity-odd terms to our original lagrangian L. The total invariant action is I tot ≡ I + I α, β with I α, β ≡ d 4 x L α, β with arbitrary real constants α and β: The O(φ 2 ) -term of the αL ∧ L -term is a total divergence. The α G ∧ D ϕ -term has both O(φ 2 ) and O(φ 3 ) -terms, the former of which cancels the like terms from imα(χ I γ 5 ρ I ) under the variation δ Q . The original transformation rule (3.2) is not modified by α or β. The αL ∧ L -term is an analog of the θF ∧ F -term associated with the θ -vacuum in QCD [10], or the U(1) A problem [11]. However, our αL ∧ L -term is more involved, because of the non-trivial Chern-Simons term F ϕ in L.
The invariance δ Q I tot = 0, and in particular δ Q I α, β = 0 up to O(φ 4 ) is easily confirmed. Even the non-trivial looking (fermions) 3 -terms in the variation turn out to be simple, because of the algebra,

Concluding Remarks
In this paper, we have presented a very peculiar supersymmetric system that realizes the Proca-Stueckelberg compensator mechanism [3] for an extra vector multiplet. Our present model has resemblance to our recent model [2], which had only two multiplets VM and TM.
The peculiar features of our model are summarized as (i) We have three multiplets VM, TM and EVM, where the EVM will be eventually massive.
(ii) Our peculiar field strength G = 3DB + mC − 3m −1 F ∧ B has the last term with m −1 .
(iii) Our model provides yet another mechanism of absorbing the dilaton-type scalar field ϕ I into the extra vector K µ I , different from the conventional YM gauge field A µ I .
(iv) Even the tensor C µνρ I in the EVM gets a mass absorbing B µν I in the TM.
(v) Our system accommodates also parity-odd terms, analogous to the θF ∧F -term [10] [11]. Even though our system is less economical than [2] with an additional multiplet EVM, it has its own advantage. First, we provide a mechanism for giving a mass to the extra vector K µ I in the EVM, which may be not needed as a massless particle at low energy. Second, we have a new compensator mechanism for an extra vector in the adjoint representation, which is not the YM gauge field. The derivative D µ ϕ I is simpler than exponentiations [1] [2].
General formulations for different representations (not necessarily adjoint representations) for supersymmetric compensator mechanism have been given in [1]. However, we emphasize here that the fixing of supersymmetric couplings for our system with a different field content is a highly non-trivial task. Even superspace formulation does not help so much, as described in [2]. The main reason is that the usual unconstrained formalism in terms of the singlet superfield L [12] can not describe a tensor multiplet in the adjoint representation.
Our results can be applied to diverse dimensions and also to extended supersymmetric systems.
This work is supported in part by Department of Energy grant # DE-FG02-10ER41693.