Higher Spin Superfield interactions with Complex linear Supermultiplet: Conserved Supercurrents and Cubic Vertices

We continue the program of constructing cubic interactions between matter and higher spin supermultiplets. In this work we consider a complex linear superfield and we find that it can have cubic interactions only with supermultiplets with propagating spins $j=s+1$, $j=s+1/2$ for any non-negative integer $s$ (half-integer superspin supermultiplets). We construct the higher spin supercurrent and supertrace, these compose the canonical supercurrent multiplet which generates the cubic interactions. We also prove that for every $s$ there exist an alternative minimal supercurrent multiplet, with vanishing supertrace. Furthermore, we perform a duality transformation in order to make contact with the corresponding chiral theory. An interesting result is that the dual chiral theory has the same coupling constant with the complex linear theory only for odd values of $s$, whereas for even values of $s$ the coupling constants for the two theories have opposite signs. Additionally we explore the component structure of the supercurrent multiplet and derive the higher spin currents. We find two bosonic currents for spins $j=s$ and $j=s+1$ and one fermionic current for spin $j=s+1/2$.


Introduction
In a recent paper [1] the cubic interactions between the chiral supermultiplet and higher spin supermultiplets were constructed via Noether's procedure. The conclusion was that both massive and massless chiral superfields couple only to half-integer superspin supermultiplets (s + 1, s + 1/2) with an interesting twist that the integer s must be odd for the case of massive chirals. Furthermore, excplicit expressions were given for the supercurrent multiplet which includes the higher spin supercurrent J α(s)α(s) 4 and the higher spin supertrace T α(s−1)α(s−1) .
This paper is the continuation of that program for a different matter supermultiplet. In this work, the role of matter will be played by the complex linear supermultiplet and the goal is to (i ) find the spin selection rules for consistent cubic interactions between the complex linear and the 4D, N = 1 super-Poincaré higher spin supermultiplets and (ii ) give explicit expressions for the higher spin supercurrent supermultiplets.
The existence of variant descriptions of supersymmetric theories has been well documented [2] and one of the most well known examples are the minimal and non-minimal formulations of 4D, N = 1 supergravity. The complex linear supermultiplet is another well known, non-minimal, description of the scalar multiplet. In a previous paper [5] we derived the supercurrent multiplets for various free and interacting (including higher derivatives) theories of a complex linear superfield (Σ) by investigating the linearized coupling to supergravity generated by linearized superdiffeomorphisms.
In this paper, we generalize the linearized superdiffeomorphism transformation of Σ in order to include higher rank parameters in a manner consistent with the linearity constraint of Σ (D 2 Σ = 0).
Using this higher spin transformation of Σ, we perform a perturbative Noether's procedure in order to construct an invariant theory, up to first order in coupling constant. This will reveal the cubic interactions of Σ with the 4D, N = 1 super-Poincaré higher spin supermultiplets. From that action we can read off the corresponding higher spin supercurrent multiplet which generates the cubic vertices.
The results we find are that a complex linear superfield can have cubic interactions only with half-integer superspin supermultiplets (s + 1, s + 1/2) exactly like a chiral superfield. Moreover, we provide explicit expressions for the both the canonical and minimal supercurrent multiplets. Additionally, we investigate the duality transformation between the complex linear and the chiral in the presence of these higher spin cubic interactions. Interestingly enough we find that the charge (the coupling constant that controls these cubic interactions) of the dual theory differs from the charge of complex linear by a sign that depends on the value of s. Specifically, for even values of s they are opposite and for odd values of s are the same. We also calculate the set of component higher spin currents. There are two bosonic conserved currents, one for spin j = s and one for spin j = s + 1 and there is a fermionic current for spin j = s + 1/2. These currents agree with the expressions derived from the chiral theory presented in [1].
The paper is organized in the following manner. In §2 we consider a family of first order transformations for the complex linear superfield and their compatibility with the linearity condition. In §3 we use them for Noether's procedure in order to construct an invariant action. This will force us to introduce the cubic interactions between Σ and higher spin. These interactions are generated by the canonical higher spin supercurrent multiplet which includes the higher spin supercurrent and supertrace. In §4 we prove that always exist an alternative minimal supercurrent multiplet in which the higher spin supertrace vanishes. This is possible due to (i ) the freedom of choosing appropriate improvement terms, (ii ) the freedom to absorb trivial terms by redefining Σ and (iii ) the freedom in defining the supercurrent and supertrace up to a particular equivalence relation. In §5 we discuss the superspace conservation equations for the supercurrent multiplets and in §6 we compare the results found for the complex linear with the results for chiral, by performing the duality between them. In §7 we extract the component higher spin currents and in §8 we summarize our results.
For additional developments on the topic of supersymmetric higher spin supercurrents the reader can refer to [3,4].

First order transformation for complex linear superfield
A cubic interaction between two types of fields can be written in the form jh, where j is a current constructed from matter fields φ and h is a set of gauge fields. Because the gauge field h is defined up to a gauge redundancy, the current j must be conserved. Noether's method is a systematic and perturbative method for constructing invariant theories that respect these gauge redundancies and therefore generate the appropriate interactions between matter and gauge fields. In this approach one expands the action S[φ, h] and the transformation of fields in a power series of a coupling constant g where S i [φ, h] includes the interaction terms of order i + 2 in the number of fields and δ i is the part of transformation with terms of order i in the number of fields. The requirement for invariance up to order g 1 (cubic interactions) is In this work, the role of matter will be played by the complex linear supermultiplet, described by a complex linear superfield Σ (D 2 Σ = 0). For the role of gauge fields we consider the massless, higher spin, irreducible representations of the 4D, N = 1, super-Poincaré algebra. These were first introduced in [6,7] and later given in a superspace formulation [8][9][10] and further developed in [11][12][13][14]. Higher spin supermultiplets are parametrized by a quantum number called superspin Y , which is a supersymmetric extension of spin and it takes integer s and half-integer s + 1/2 values. Massless higher spin supermultiplets contain two irreducible representations of the Poincaré algebra with spins j and j − 1/2, therefore we denote the entire supermultiplet by (j, j − 1/2). We briefly remind the reader of the various superspace descriptions of free super-Poincaré higher spin supermultiplets: 1. The integer superspin Y = s supermultiplets (s+1/2, s) 5 are described by a pair of superfields , however we are following the formulation given in [12] 3 Ψα (s)α(s−1) and V α(s−1)α(s−1) with the following zero order gauge transformations 2. The half-integer superspin Y = s+1/2 supermultiplets (s+1, s+1/2) 6 have two formulations, the transverse and the longitudinal. The transverse description uses the pair of superfields H α(s)α(s) , χ α(s)α(s−1) with the following zero order gauge transformations whereas the longitudinal description uses the superfields The starting point of our analysis is the free action for a complex linear superfield and the zeroth order transformation of Σ, δ 0 Σ = 0. Hence according to (4), the cubic interactions of the complex linear superfield with higher spin supermultiplets, described by the S 1 [Σ, A] 7 must satisfy: Therefore it is important to find the first order, higher spin transformation of Σ (δ 1 Σ). Motivated from the structure of the transformation of Σ under superdiffeomorphisms presented in [5] we write the following ansatz 8 } with independently symmetrized dotted and undotted indices. To have this transformation consistent with the linearity of Σ (D 2 Σ = 0), we must have 6 introduced in [9], however we are following the formulation given in [12] 7 Where A is the set of superfields that participate in the description of higher spin supermultiplets. 8 We use the conventions of Superspace [15], It is worth mentioning that (10) is not the most general transformation linear in Σ one can write. For example we can have D 2 ,D 2 D 2 orD 2 D α k+1 appropriately placed in the various terms. These extra terms will modify the above constraints, nevertheless we have verified that they do not introduce extra structure regarding the coupling to higher spin supermultiplets, so we will not consider them. Solving (11) we conclude that the parameters where ℓ l α(k+1)α(k) and ℓ l are arbitrary, unconstrained superfields. A useful observation is that the constraints (11) do not mix the different l-levels and all the determined parameters (12) are functions of other parameters of the same level. Therefore, the l label does not provide any further structure and we can simplify (10) by confining ourselves in the l = 0 level 9 . Therefore, the transformation we consider is Additionally, if we want to make contact with the superdiffeomorphism transformation, we choose so that the Γ α(k)α(k) with ∆ α(k)α(k) terms for k ≥ 1 combine to give spacetime derivatives resulting 9 From this point forward we drop the l label.

Higher Spin supercurrent multiplet
Having found the appropriate first order transformation for the complex linear superfield, we use it to perform Noether's procedure for the cubic order terms, as described in §2 and construct the higher spin supercurrent multiplet. Starting from the free theory (8) we calculate its variation under (15): At this point it is tempting to choose the B α(k)α(k+1) parameter to be a function of the ℓ α(k+1)α(k) parameter. In principle we can write all possible terms allowed by engineering dimensions and index structure: A similar step was done in [5] and led to the coupling of the complex linear theory to the various formulations of supergravity depending on the values of d 1 , d 2 , d 3 . In this analysis supergravity would correspond to k = 0. Notice that in (16) the coefficients ofB α(k+1)α(k) andD 2 ℓ α(k+1)α(k) do not match for k ≥ 1, hence we can not repeat the same arguments in [5] for higher spin coupling. There is only one viable optionB By doing this, the variation of the action can be written in the following way where for simplicity we suppress the uncontracted indices, their symmetrization and the symmetrization factors when they are appropriate. The symbol ∂ (k) denotes a string of k spacetime derivatives. Moreover, we can use the following identity which holds for arbitrary (super)functions A and B and simplify (19) to: The superfields W α(k+1)α(k) andŪ α(k)α(k+1) are improvement terms that we can add, as discussed in [1]. Also, it is important to keep in mind that the J α(k+1)α(k+1) , T α(k+1)α(k) and J are not defined uniquely but they satisfy an equivalence relation. For example for arbitrary superfields P (1) , P (2) , P (3) , P (4) . Using (20) we can express the −i k+1 ∂ k+1 ΣΣ in the following manner 7 where X α(k+1)α(k+1) and Z α(k)α(k) are real superfields given by the following expressions: Because of (24), it is straight forward to prove that there is always a choice for the improvement term W α(k+1)α(k) that makes J α(k+1)α(k) real. Specifically, by selecting we get The reality of J α(k+1)α(k+1) and the fact that T α(k+1)α(k) becomes a total spinorial derivative (27b), allows us to modify (21) in the following way: The terms inside the square brackets are exactly the zeroth order transformations that appear in (6). Hence, in order to get an invariant theory, we must add the following cubic interaction terms between (s + 1, s + 1/2) higher spin supermultiplets (6) and the complex linear superfield is the real prepotential used for the description of the vector supermul- Superfields J α(k+1)α(k+1) (27a) and T α(k)α(k) (27c) are the Σ-generated higher spin supercurrent and higher spin supertrace respectively and together they form the higher spin supercurrent multiplet J α(k+1)α(k+1) , T α(k)α(k) . The results we get are similar to the case of the chiral superfield described in [1] and [16]. Specifically, we find that a single, free, massless, complex linear supermultiplet couples only to half integer superspin supermultiplets with a preference to the transverse formulation (6) which is the only formulation of half-integer superspins that can be elevated to N = 2 theories. Furthermore, like in the chiral case the higher spin supercurrent and supertrace include higher derivative terms, as expected from [17].

Minimal higher spin supercurrent multiplet
In the previous section we constructed what is known as the canonical supercurrent multiplet [18]. In this section we will prove that for every value of the non-negative integer k there is a unique, alternative higher spin supercurrent multiplet, called the minimal supercurrent multiplet defined by T min α(k)α(k) = 0. This will be possible due to (i ) the freedom in choosing the unconstrained improvement term U α(k+1)α(k) , (ii ) the freedom in the definition of T α(k+1)α(k) (23) and (iii ) the freedom to absorb trivial terms by redefining Σ.
For an arbitrary superfield Yα, the combinationDαYα is a complex linear superfield. Therefore, one can consider the redefinition of Σ Σ =Σ + gDαYα (30) which will modify the free theory in the following manner where g is the perturbative parameter. Because we are working up to order g 1 (cubic interactions) we can ignore the last term in (31) and therefore conclude that terms in the action that depend on DΣ orDΣ (these are the on-shell equation of motion for the free theory) are trivial terms and can be absorbed by an appropriate redefinition of Σ like (30). This realization simplifies the calculation of the supercurrent multiplet. For example, if we consider the transformation of Σ (13) before fixing Γ α(k)α(k) and B α(k)α(k+1) then we get: Observe that the contribution of the B α(k)α(k+1) and Γ α(k)α(k) terms is trivial in the sense explained previously, hence we can remove them by an appropriate redefinition of Σ. This is very satisfying because, unlike the chiral case, the first order transformation of Σ (13) is far from unique. It includes two infinite families of unconstrained superfields. Despite that, as we see at the level of the action, these unconstrained parameters do not give non-trivial contributions. Therefore, the variation of the action is completely fixed without any freedom left: The superfield U α(k+1)α(k) is an improvement term similar to the improvement terms appearing in (22a). We can deduce its presence by constraining the W α(k+1)α(k) superfield in (22b) to cancel (21) which means we are already in the minimal setup of the supercurrent multiplet. Also, we must keep in mind that the above definition of J min α(k+1)α(k+1) satisfies an equivalence relation for arbitrary Θ (1) and Θ (2) superfields. Furthermore, using (24) we can rewrite (34a) Thus, we can make J min α(k+1)α(k+1) real if and only if we can find a superfield U α(k+1)α(k) that satisfies the following constraint: for some superfields ζ α(k)α(k) and a term ζ (4) α(k)α(k) that depends only on DΣ and DΣ. This condition corresponds to setting the supertrace (27c) to zero, which is exactly the demand for a minimal supercurrent multiplet.
Consider the following ansatz for a solution of (37) withλα being the unconstrained prepotential of the complex linear superfield Σ (Σ =Dαλα,Σ = D α λ α ). It is straight forward to show that 10 : From the above and equation (25b), we get that condition (37) takes the form This is a system of k + 1 linear equations for the k + 1 complex variables f p , p = 0, 1, . . . , k. If we introduce a new set of variablesf p defined as then the (41) system of equations takes the form This is the same system of equations which appeared in [1] in the process of removing the supertrace.

Conservation Equation
The off-shell invariance of the action S = S 0 + S HS-Σ cubic interactions as constructed in §3, up to order g can be expressed in terms of a set of Bianchi identities. Using them together with the onshell equations of motion of Σ we can show that the canonical higher spin supercurrent multiplet satisfies the following conservation equation: Dα k+1 J α(k+1)α(k+1) = 1 (k+1)!D Finally, we focus at the component level of the supercurrent multiplet and identify the various spacetime conserved higher spin currents. There are three higher spin currents: 5. There is a bosonic, spin j = s current J min (0,0) α(s)α(s) (62,65) which corresponds to the θ independent component of J min α(k+1)α(k+1) . This current corresponds to an R-symmetry.
Notice that the bosonic currents have two independent contributions, one coming from the bosonic sector (complex scalar) and another coming from the fermionic sector (spinor). These two contributions have been discovered independently by studying non-supersymmetric theories [20][21][22]. However, the fermionic current has a single contribution that depends on both the boson and the fermion, hence the discovery of such a higher spin current appears naturally in supersymmetric theories. The first time this type of current appeared was in [1], where the complex scalar and spinor are defined as components of a chiral superfield. In this work we find the analogue expressions for the case of using a complex linear superfield to define them. The results we get are proportional to the ones found in [1]. and consistent with the duality transformation discussed in §6.