A dynamical theory for linearized massive superspin 3 / 2 1

: We present a new theory of free massive superspin Y = 3 / 2 irreducible representation of the 4 D, N = 1 Super-Poincar´e group, which has linearized non-minimal supergravity (superhelicity Y = 3 / 2) as it’s massless limit. The new results will illumi-nate the underlying structure of auxiliary superﬁelds required for the description of higher massive superspin systems.


Introduction
After four decades of exploring the topic of supersymmetry (SUSY), the problem of writing a manifestly susy-invariant action that describes a free, off-shell massive arbitrary superspin irreducible representation of the Super-Poincaré group still possesses puzzles. Although the non-supersymmetric case of massive higher spin theory has been developed [1,2] and is well understood, the off-shell supersymmetric case has yet to be understood with a comparable level of clarity. There has been progress for on-shell supersymmetry [3], but these results don't capture the rich off-shell structure of supersymmetric theories. There is a need for a manifestly susy invariant theory of massive integer and half-integer superspins which includes all the auxiliary superfields a theory of this nature is expected to possess.
Progress in this direction was made with the works presented in [4][5][6] where free massive irreducible representations of superspin 1 and 3/2 were constructed. These results provided a proof of concept that constructions like these are possible, but they don't shed light to the heart of the problem which is to determine the set of auxiliary superfields required to describe an arbitrary superspin system with a proper massless limit. Specifically in [4] the focus was on massive extension of linearized old-minimal supergravity and newminimal supergravity. These theories do not generalize to the arbitrary spin case, therefore the results obtained do not provide clues about the underlying structure of the auxiliary superfields for the general case. This is not the case with the work presented in [6] where a free massive extension of linearized non-minimal supergravity is derived. Linearized non-minimal supergravity supermultiplet is a member of a tower of irreducible representations that can be extended to the arbitrary super-helicity and that makes it a good starting point. However their construction uses a lagrange multiplier technique in order to impose constraints that were not derived in a dynamical way.

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We will show that there is an alternative formulation of the theory where all the constraints required, for the description of a free massive irreducible representation of Y = 3/2, are dynamically generated from the equations of motion of a set of superfields {H αα . , χ α , u α }. Superfields H αα . , χ α in the massless limit form the free linearized nonminimal theory (superhelicity Y = 3/2) and u α is an auxiliary superfield that decouples when m → 0. Finally, the theory presented here is a free theory without interactions. The fully interacting, non-linear problem is still an open and very difficult one and it is one of the motivations for this kind of investigations. In a realistic approach we can not talk about interactions if we haven't established the free theories first. The results presented here extend our understanding for the free massive theory of the non-minimal superspin 3/2 supermultiplet, which is the first non-trivial. Furthermore we provide clues for some of the degrees of freedom that must be present in the non-linear, interacting theory. These are the superfields that have auxiliary status in the free linearized theory.
Our presentation is organized as follows: in section 2, we quickly review the representation theory of the 4D, N = 1 Super-Poincaré group for a free massive arbitrary superspin system. In section 3, we present the constraints imposed in the theory in order to have a proper massless limit. In the following section 4, we start with a warm up exercise by quickly reproducing the massive theory for superspin Y = 1/2. In the last section 5 we present the new massive theory for Y = 3/2.

Arbitrary superspin representation theory
The irreducible representations of the Super-Poincaré group are labeled by it's two Casimir operators. The first one is the mass and the other one is a supersymmetric extension of the Poincaré Spin operator. For the massive case the Super-spin casimir operator takes the form where W 2 is the ordinary spin operator (the square of the Pauli-Lubanski vector), P (o) is the projection operator P (o) = − 1 m 2 D γD2 D γ and the parameter λ satisfies the equation In order to diagonalize C 2 we want to diagonalize both W 2 , P (o) . The superfield Φ α(n)α .
(m) that does this and describes the highest possible representation (highest superspin) λ = n + m 2 ,

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has to satisfy the following: where all dotted and undotted indices are fully symmetrized and the spin content of this A superfield that describes a superspin Y system has index structure such that n+m = 2Y − 1 where n, m are integers. This Diophantine equation has a finite number of different solutions for (n, m) pairs but the corresponding superfields are all equivalent because we can use the ∂ ββ operator to convert one kind of index to another. So we can pick one of them to represent the entire class.
One last comment has to be made about the reality of the representation. The reality condition imposed on the superfield differs with the character of the superfield. The bosonic superfields, have even total number of indices therefore describe half-integer superspin systems, Y = s + 1/2. In this case we can pick to have On the other hand the fermionic superfields have odd total number of indices and describe integer superpsin systems, Y = s + 1. For that case we can pick n = s + 1, m = s (Ψ α(s+1)α . (s) ) and the reality condition is the Dirac (s) .

The massless limit
Representation theory tells us the type of superfield and constraints we need in order to describe a specific irreducible representation. We would like to have a dynamical way to derive these constraints, through an action. Very quickly we realize that, for that to happen we need a set of auxiliary superfields to help us generate these constraints, as in the non-supersymmetric free massive arbitrary spin story. To find the minimum number and type of these auxiliary superfields needed, is the heart of the problem. That sounds like an intuitive trial-and-error process, but there is a hidden clue and that is the massless limit of the theory. We demand the massless limit of our massive theory to give the corresponding massless irreducible representation.
The list of available massless highest superhelicity irreducible representations was presented in [7][8][9][10], and [11]. There is one infinite tower for theories of integer superhelicity and two different infinite towers for theories of half integer super-helicities (figure 1). However there are a few theories that don't fall in one of the three infinite towers, like the old minimal, new minimal and new-new minimal supermultiplets. These are special cases that can not be generalized to the arbitrary superhelicity. If our goal is towards the construction JHEP03(2014)030 of an arbitrary massive superspin supermultiplet, then it is obvious that we should start with massless theories that are members of an infinite tower and not a special case.
The conclusion is that the construction of massive theories must start with the corresponding massless action, the addition of some extra auxiliary superfields (if necessary) and appropriate (self)interaction terms proportional to m and m 2 , so the massless theory decouples in the massless limit.

Warming up with Y = 1/2
So if we want to construct the theory of superspin 1/2 we start with the theory of superhelicity 1/2, add terms proportional to m and m 2 and check if we can generate the desired constraints. If not then we add extra auxiliary fields until we do. The starting action is: To describe Y = 1 2 , H must satisfy D 2 H = 0 and H = m 2 H. The equation of motion is which gives

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There is also another way to obtain these results and that isà la Stückelberg. The observation is that at least on-shell the massive superspin 1 2 can be seen as the result of the combination of the massless superhelicity 1 2 plus the massless superhelicity 0. So we start with the actions for superhelicity 1 2 and 0 and we introduce (self)interaction terms proportional to m and m 2 The equations of motion are: If we manage to show that on-shell Φ = 0 then E (Φ) = 0 D 2 H = 0 H = m 2 H (a 3 = a 1 ). With that goal in mind we attempt to eliminate H from the equation of Φ and choose coefficients in such a way to find Φ = 0. We begin by defining I =D 2 E (H) + mE (Φ) and then notice Due to this symmetry, the chiral superfield Φ can be gauged away completely and therefore it's equation of motion (or the Bianchi identity) will give the desired constraint of D 2 H = 0. The action for this case is S = d 8 z c HDγD 2Dγ H + cm 2 H 2 + 2cm H Φ +Φ + 2c ΦΦ , (4.11)

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and the gauge fixed action is identical with the action obtained from the first derivation. We would like to know if similar 'Stückelberg' constructions can occur for the higher superspin theories, like it is the case for the higher spin theories 5 New massive Y = 3/2 theory Now we will follow a similar strategy to build a theory of superspin 3 2 . The starting point is the theory of superhelicity 3 2 , in specific we choose the theory of non-minimal supergravity (s = 1 in [9]). Linear Non-minimal supergravity is formulated in terms of a real bosonic superfield H αα . and a fermionic superfield χ α . We will add mass corrections to that action and check if 1) we can make χ α vanish on-shell (auxiliary status) and 2) we can generate the constraints on H αα . demanded by representation theory D α H αα . = 0, . The starting action is given by H αα . , (5.1) and the equations of motion are: Now we may use these equations and attempt to remove any H αα -dependence to derive one equation that depends solely on χ α . That will tell us if we can pick coefficients in a way that χ α vanishes on-shell. Consider the following linear combination of equations of motion where each such equation of motion is obtained by the variation of the action with regard to the respective superfields indicated by the subscripts in the first equation below: The following choice of coefficients will remove any H αα . dependence from the equation above: A + B = 0 , Aa 4 − 1 = 0 , a 1 = 0 , (5.5)

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and imposing these leads to the form of I α to be given by From this we see there is no choice of coefficients that will make χ α vanish on-shell. Therefore we must introduce an auxiliary superfield. Its purpose will be to impose a constraint on χ α when it vanishes. That constraint will be used to simplify the above expression for I α and set χ α to zero. But a more careful examination of I α will convince us that there is no unique constraint on χ α that will make all terms (except the last one) vanish. The inescapable conclusion is that we have to treat χ α = 0 as the desired constraint. This suggests that we must introduce a spinorial superfield u α that couples with χ α through only a mass term ∼ mu α χ α . Hence when u α = 0 then immediately we see χ α = 0.
We must update the action with the introduction of a few new terms: the interaction term mu α χ α and the kinetic energy terms for u α (the most general quadratic action). The new action is and the updated equations of motion are Now we repeat the process of eliminating H αα . , but since u α doesn't couple to H αα .
nothing will be changed regarding the H αα . -dependent terms. The same choice of coefficients as in (24) must be made to remove H αα . . So the updated expression for I α is we denote by J α whose explicit form is given by . D αū α . .

(5.20)
This is the superspace action that describes a superspin Y = 3 2 system with the minimum number of auxiliary superfields and has a massless limit that gives the free linearized non-minimal supergravity. This action is a representative of a family of actions that are all equivalent and connected through superfields redefinitions of the form H αα . , where z i , w i are complex (5.22)

Summary and conclusions
We started with the 3 2 superhelicity theory of free linearized non-minimal supergravity, formulated in terms of a real vector superfield H αα . and a fermionic compensator χ α . We then added mass terms to it in an attempt to discover a theory for massive superspin 3 2 system, only to find that it is not possible and we need the help of an extra fermionic auxiliary superfield u α which must couple only to χ α through a mass term. Finally using the equations of motion we manage to show that on-shell u α = 0 We have managed to derive yet another formulation of free massive supergravity supermultiplet and most importantly probe the set of auxiliary superfields required for the construction of higher superspin theories. The fermionic superfield u α is the first nontrivial auxiliary superfield needed beyond the massless theory. As we go to even higher JHEP03(2014)030 superspin values we should discover more and more of these objects. The hope is that after the study of some non-trivial low superspin examples, such as the one demonstrated here, we will have a deeper understanding on the number, type and role of these auxiliary objects. When that happens we might be in a position to construct the arbitrary massive superspin irreducible representation in an inductive manner.