Linearized Non-Minimal Higher Curvature Supergravity

In the framework of linearized non-minimal supergravity (20/20), we present the embedding of the $R + R^2$ model and we analyze its field spectrum. As usual, the auxiliary fields of the Einstein theory now become propagating, giving rise to additional degrees of freedom, which organize themselves into on-shell irreducible supermultiplets. By performing the analysis both in component and superspace formulations we identify the new supermultiplets. On top of the two massive chiral superfields reminiscent of the old-minimal supergravity embedding, the spectrum contains also a consistent physical, massive, vector supermultiplet and a tachyonic ghost, massive, vector supermultiplet.


Introduction
Supergravity, [1], as the low energy limit of superstring theory, offers the proper setup to study high energy gravitational phenomena. Among others, it provides an appropriate framework for the accommodation of cosmic inflation. The constraints on the latter released by the Planck collaboration [2] favor inflationary models which are characterized by plateau potentials with a tiny tensor-to-scalar ratio r [3]. Among the candidates is a higher curvature gravitational model, the Starobinsky model of inflation [4] which stands out for its simplicity in providing a microscopic description of the mechanism responsible for the quasi de Sitter phase during inflation. This is a particular higher curvature gravitational theory of the type described in [5]. It is classically equivalent to a theory of standard gravitation coupled to an additional propagating real scalar degree of freedom [6], with a sufficiently flat potential at large values, ideal to drive inflation.
To outline the procedure, we start with the free theory of nonminimal supergravity which includes a set of dynamical components that describe gravity (helicity ± 2) with its superpartener, the gravitino (helicity ± 3/2) and another set of auxiliary components just so the SUSY algerbra will close off-shell. Afterwards we introduce the higher curvature terms of the form R 2 . Due to the higher derivatives, the auxiliary fields of the free theory start propagating and organize themselves into supermultiplets. Nevertheless, these supermultiplets will have to be on-shell because only their dynamical degrees of freedom appear in the action, no auxiliary fields. The goal is to uncover these newly formed on-shell supermultiplets and their properties. In order to do that, we quickly realize that, we do not need to start with the full theory but its linearized version will do. The results of this analysis for the case of old-minimal supergravity [11,12] are two physical, massive, chiral supermultiplets and for the case of new-minimal supergravity [16] is a physical, massive, vector supermultiplet. Linearized supergravity is nothing else but the theory of massless, irreducible representation of the super-Poincaré group with superhelicity Y =3/2. The superspace and component formulation of the massless, arbitrary superhelicity, irreducible representations and their properties have been studied in detail in a series of papers [39][40][41]. For our purpose, we will use the formalism and the results of [39] and adapt them for the case of superhelicity Y =3/2.
The presentation of this work is organized in the following way. In Section 2 we briefly review the results of [39] for the case of linearized, non-minimal supergravity in both superspace and components. Then in section 3 we construct the R 2 action in superspace and project to components. In section 4 we combine the two previous results to construct the Starobinsky model (R+R 2 ) in this framework and study its spectrum. We perform the analysis at the component level for both bosons and fermions. At the end we verify our results by performing a duality in superspace that reveals exactly the same spectrum and demonstrates the classical equivalence between the R + R 2 theory of non-minimal supergravity and the standard non-minimal supergravity coupled to two massive chiral supermultiplets, a massive vector supermultiplet and a tachyonic ghost, massive vector supermultiplet. An interesting remark is that all of the massive supermultiplets turn out to have equal masses.

Superhelicity Y = 3 as Linearized Non-Minimal Supergravity
From the investigation of free, massless, higher superspin theories [39] we can extract the 4D, N = 1 superspace action for linearized non-minimal supergravity which contains the real bosonic superfield H αα and the fermionic superfield χ α as a compensator. The action is invariant under the following transformation which forces the following Bianchi Identities The superfields T αα and G α are the variations of the action (2) with respect to the unconstrained superfields H αα and χ α . Their explicit expressions are The two superfields T αα and G α in (5) have mass dimensionality [T αα ] = 2, [G α ] = 3/2 1 .
To prove that indeed this action describes the desired representation, using the equations of motion we can now show that a gauge invariant chiral superfield F αβγ exists ([F αβγ ]=5/2) and on-shell (T αα = G α = 0), it satisfies the desired constraints in order to describe a super-helicity Y =3/2 systemDα At the component level, the above superspace action describes the dynamics of the following bosons namely, in 4-component notation, of three vectors A µ (A αα ), u µ (u αα ) and v µ (v αα ), three scalars (S, P, h) and a symmetric traceless rank-2 tensor (the graviton) h µν (h αβαβ ). The corresponding gauge transformations acting on the bosons are which leave 4 degrees of freedom for each vector, 1 for each scalar and 5 for the symmetric traceless tensor, a total of 20 degrees of freedom to fill the bosonic part of the non-minimal irreducible supersymmetric multiplet. The bosonic sector of the Lagrangian density is where L h=±2 describes a massless helicity ±2 particle and [ √ −gR]| linearized is the linearized Einstein-Hilbert Lagrangian, keeping only the terms quadratic in the fields. At this linear approximation, the Ricci scalar takes the form (up to an overall normalization) 1 The highest spin component of Hαα is a propagating boson.

and its mass dimension is [R] = 3. The Ricci scalar is part of the completely antisymmetric
θθ term in the expansion of the T αα superfield, specifically Also, the linearized Ricci tensor is and it resides in the fully symmetric part of the θθ term of T αα while it satisfies Similarly for the fermionic sector, we have the following components The gauge transformations of the fermionic fields are The corresponding free Lagrangian is where L h=±3/2 describes a massless Rarita-Swinger field (gravitino with helicity ± 3/2) The linearized fermionic curvatures are and they are the (anti)symmetric part of theθ term of superfield T αα Finally they satisfy 3 Constructing the R 2 theory Now we turn to the construction of a gauge invariant, higher derivative superspace action, such that it will generate R 2 terms. The reason that we restrict ourselves only to R 2 terms and we do not include for example the square of the Ricci tensor, or equivalently the Weyl tensor square, is that the inclusion of the latter terms will lead to ghost and/or tachyons states in the spectrum [5,11] .
To proceed in our cosntruction, we recall that the available gauge invariant objects are the superfields T αα , G α and F αβγ . However F αβγ , due to its chiral property and its index structure, it can only couple to itself, giving a term of the form F αβγ F αβγ . But such an object will give rise to the square of the Weyl tensor, so it is rejected. The rest of the objects could be combined in many different ways. We organize them in the following manner.
The general structure of all possible terms that we are interested in, are schematically of the form which means that any possible term will include n T αα 's, k superspace covariant derivatives and l G α 's. The dimensionality of these terms is Then, if we project to components, we have to integrate over superspaceD 2 D 2 T n D k G l |, and therefore the mass dimension of the component terms that we can, in principle, construct is 2n + The finall step is the fact that the desired R 2 term has dimensionality 6 and we require to have expressions quadratic in the components (linear approximation). Therefore we must have The solutions of this Diophantine system, and the corresponding terms allowed are given in the following table n k l term 2 0 0 T αα T αα Note that we have not included the term G α D 2 G α +c.c. since it is zero due to (4b). Moreover because of equation (4a) the terms T ααDα G α + c.c. and G αD2 G α + c.c. are identical.
Hence the R 2 superspace action must be of the form where g 0 , g 1 , g 2 ∈ R. Now what remains is to project this action to components and pick the coefficients in a way such that we generate R 2 terms and canonical kinematic terms for any additional propagating fields.
The component Lagrangian we get from the above action (28) is The basic rules for projection are 1. Use the 'Leibniz' rulē where ǫ is zero for bosonic and one for fermionic superfields.
First we focus on the bosonic sector of the theory, therefore we restrict the above calculation to the bosonic part of the projection. That means, we keep only the terms with even number of D's when acting on a bosonic superfield (like T αα ) and with odd number of D's when acting on a fermionic superfield (like G α ). We get with It is evident that I 0 | B includes a term proportional to D (ρ ,D (ρ T α)α) | D (ρ ,D (ρ T α)α) , which based on (15) makes it obvious that it generates the Ricci tensor square, R αβαβ R αβαβ , a term that is not considered here (as it leads to ghost and/or tachyonic states [5,11]). On top of that, such a term can not be canceled by any of the other contributions to the bosonic Lagrangian. Therefore the only possibility out of that, is to choose 7 The rest of the terms are relevant and after putting everything together, we find that the total bosonic sector is where g R 3 and g I 3 are the real and imaginary parts of g 3 . Notice that the higher curvature terms are accompanied by kinematic terms for all the previously auxiliary fields. This is a standard property of higher curvature supergravity.
Similarly, we find that the fermionic sector is 4 The spectrum of R + R 2 non-minimal supergravity So far we have developed the superspace action for the R and R 2 theories. In this section we combine them in order to study the spectrum of the R + R 2 theory. Specifically we will analyze the propagating degrees of freedom of the Lagrangian To do this we must first bring the full Lagrangian into a diagonal form and subsequently study their field equations. Typically one can achieve that, by doing redefinitions of the various fields and a clever choice of coefficients. But, in this case due to the fact that the L R is already diagonal, we can not perform any redefinitions and the only thing left to do is to choose appropriately the coefficients of the non-diagonal terms. 8

Bosonic sector spectrum
Following the previously explained strategy, we must impose the constraints With the above coefficients (37), we find that the linearized, bosonic part of the component Lagrangian is g m 2 P P. The equations of motion for the various fields and the degrees of freedom they allow to propagate are: From the left equation we see that three of the degrees of freedom of the vector field A αα remain auxiliary and are solved in terms of the scalar ∂ αα A αα . From the right equation we see that for g > 0, ∂ αα A αα is a physical, real, propagating, massive scalar with mass µ 2 =m 2 /6g.

2.
For u αα we find This describes the propagation of a real, massive, scalar ∂ αα u αα with equations of motion ∂ αα u αα = m 2 6g ∂ αα u αα and mass µ 2 =m 2 /6g, and the propagation of a real, massive vector with the same mass described by the divergent-less field defined aŝ u αα =u αα − 3g m 2 ∂ αα ∂ ββ u ββ , with equations of motion û αα = m 2 6gû αα . Both of them are tachyonic ghosts (for g > 0) since they appear in the Lagrangian with an opposite overall sign.
3. For v αα we have As before this equation includes both the spin zero part, described by ∂ αα v αα and the spin one part , described by thev αα =v αα − 3g m 2 ∂ αα ∂ ββ v ββ . Both of them have the same mass µ 2 =m 2 /6g and are physical for g > 0.

The gravitational sector of the action is
which can be re-expressed with the help of a Lagrange multiplier φ in the following form where [φ] = 1. Now we perform the following redefinition of h The change of L h=±2 is and the change of R is δR = −6c φ.
Therefore we get for S ′ | B We choose c such that the cross term vanish hence we get which describes a helicity ±2 and a physical (for g > 0), real, massive, scalar φ with mass µ 2 = m 2 /(6g).

Fermionic sector spectrum
In order to verify the fermionic spectrum, we start with equation (35) and make the same choice of coefficients as in (37), which give The equations of motion for the various fields are 1. From β α andβα we find which describe a pair of massive Weyl spinors with Dirac mass µ 2 = m 2 /(6g).

From ρ α andρα we find
which again describe a pair of massive Weyl spinors with Dirac mass µ 2 = m 2 /(6g). Note that, in order to reveal the fermions that belong into the tachyonic -ghost vector multiplet, we have to diagonalize the Lagrangian (52). Once we do that, we will get one positive and one negative eigenvalue, which signals the propagation of one physical and one tachyonic -ghost fermion.
3. The rest of the action includes L h=±3/2 and R α and can be expressed in the following way The change of L h=±3/2 is and the change of R α is So we get that Finally we choose d in order to cancel the interaction term with R α and we get The equations of motion from Lagrangian (61) on top of the massless gravitino, give two massive Weyl spinors with Dirac mass µ 2 = m 2 /(6g).
Therefore the spectrum of fermions gives, as expected, the same structure.

Superspace Duality
From our previous considerations, we find that this higher curvature theory has additional propagating degrees of freedom. Since this is a supersymmeric theory it should be possible to identify the multiplet structure of these new degrees of freedom directly from superspace manipulations. In other words we expect to find that our higher curvature theory is classically equivalent to a particular set of matter fields coupled to standard non-minimal supergravity (i.e. a supergravity with no higher curvature terms). The Superspace action for the above choice of coefficients is of the form for the chiral Φ=DαḠα and the real vector V αα =i(D αḠα +DαG α ). The action (62) can be re-written as We now choose coefficients c and d to eliminate the cross terms involving superfields Φ and V αα respectively, which gives c = −k and d = −l, leading to It is obvious that, the above action contains linearized non-minimal supergravity with no higher curvature terms and an independent additional matter sector. Before we conclude let us study the on-shell superfield content of the matter sector, and compare to our findings from the component discussion.
The equations of motion for superfields F αα , T , S are Looking for the solution of the above equations, we do the following ansatz: where V and W are on-shell, real, superfields which they satisfy the constraints D 2 V = D 2 V = 0, D 2 W =D 2 W = 0 and we have for their equations of motion By doing that, we realize that there are two on-shell chiral supermultiplets, described by the chiral superfieldsD 2 T and S and they satisfy the following equations of motion 13 The above equations (69) and (70) solve the system (67) if we set From (69) and (70) we see that indeed we get two vector supermultiplets and two chiral supermultiplets with equal masses µ 2 = m 2 6g . The final expression for the superspace action is where g and l are free, non-zero parameters. Furthermore, due to the different integration by parts properties of the two operators ∂ αα and [D α ,Dα], we immediately conclude that there will be an overall minus in front of the terms quadratic to W , illustrating that, the W massive vector supermultiplet will be a tachyonic ghost one. The above performed superspace duality demonstrated the classical equivalence between the higher curvature non-minimal supergravity theory and the non-minimal supergravity with the addition of a specific spectrum that we are expecting from the previous component discussions.

Conclusions
In this work we have studied the spectrum of the Starobinsky model R + R 2 , embedded in the framework of non-minimal supergravity. We have utilized the linearized theory since it is sufficient for the understanding of the field content. As expected from a supergravity theory, on top of the scalaron degree of freedom, there are previously auxiliary fields which now pick up kinematic terms due to to the new action. We have identified these fields and the way they organize inside supermultiplets. Our findings show that the 20/20 higher curvature supergravity is classically equivalent to a 20/20 supergravity coupled to two vector supermultiplets (one of which is a tachyonic ghost multiplet) and two chiral supermultiplets with equal masses. Therefore, the embedding of the R + R 2 theory in non-minimal supergravity is reminiscent of the corresponding embedding of the general quadratic gravity (with R 2 and Weyl square terms) in minimal supergravity, as in both cases unphysical states appear in the spectrum.