Component Actions of Liberated $\mathcal{N} = 1$ Supergravity and New Fayet-Iliopoulos Terms in Superconformal Tensor Calculus

We explicitly compute the component action of certain recently discovered new $\mathcal{N} = 1$ supergravity actions which enlarge the space of scalar potentials allowed by supersymmetry and also contain fermionic interaction terms that become singular when supersymmetry is unbroken. They are the"Liberated Supergravity"introduced by Farakos, Kehagias and Riotto, and supergravities with a new Kahler-invariant Fayet-Iliopoulos term proposed by Antoniadis, Chatrabhuti, Isono, and Knoops. This paper is complementary to our previous papers [Phys. Rev. D 103 (2021), 025008 and 105006], in which new constraints on the coupling constants of those new theories were found. In this paper we spell out many details that were left out of our previous papers.


Contents
problem that seems to plagues many supergravity models of inflation (see e.g. [2] for a review) to make alternative scenarios worth considering. One easy way out of all problems posed by supersymmetry is to assume that the world is not supersymmetric at all. On the other hand the high-energy completion of any field theory containing gravity provided by perturbative superstring theory implies that at some very high energy scale the world is supersymmetric (and 10-dimensional). In superstring theory the largest and best understood set of models of low-energy physics -such as Calabi-Yau or flux compactifications-are such that extra dimensions are compactified at energy scales which are much above the supersymmetry breaking scale. It is therefore reasonable to study four-dimensional models in which supersymmetry is broken at a high scale, e.g. at a scale M Among these possibilities, some recent proposals use explicitly supersymmetric terms that look at first sight singular, because when written in superspace they contain inverse powers of superfields.
What makes possible to write such terms is that they in fact they depend at most on inverse powers of the auxiliary fields. So they are well defined when supersymmetry is broken. We will study in particular the theory called "liberated N = 1 supergravity," proposed in Ref. [9] and the new Fayet-Iliopoulos (FI) term introduced in [3,4,5,6,7,8]. These modifications of the standard supergravity action relax the highly constrained form of scalar potentials in supergravity. (recall that the scalar potential has the universal form V = V F + V D , V F = e G (G A G AB GB − 3) and V D , for some function G of the chiral multipet scalars.). The cosmological applications to inflation of such models have been recently investigated in [4,11].
Supergravity actions contain many nonrenormalizable operators that depend on the fermionic fields.
Their existence defines an ultraviolet (UV) cutoff scale of the theory. While this property is generic in locally supersymmetic theories, it is not dangerous when the UV cutoff is the Planck scale M pl , which is in any case the ultimate cutoff for supergravity, and of course also for Einstein gravity. In the models studied in Refs. [10,11] instead, the UV cutoff is connected to the supersymmetry breaking scale, so it becomes an important parameter that constraints the domain of validity of those models.
The main purpose of this work is to show a systematic analysis of how a cutoff scale of an effective F . Then, we find constraints on the expansion coefficients C (4−δ) because these coefficients must be suppressed by a cutoff scale Λ cut of a theory for this to be valid as an effective field theory in range of some energies of interest below the cutoff; that is, (1.0.1) By taking this inequality into account, we are able to determine in what range of energies a low energy supergravity theory can be considered as an effective field theory. To do so, it is essential to explore all of the fermionic nonrenormalizable terms as we have just seen above. Checking these terms may be difficult if we use the superspace formulation as we have to fully calculate them and the superspace fields contain many more auxiliary components than needed for the off-shell closure of the supersymmetry algebra. However, we point out that using the superconformal tensor calculus enables us to compute both bosonic and fermionic terms in a systematical and economical way. Hence, we take advantage of the superconformal formalism in this work.
Since this paper is rather technical, the reader chiefly interested in finding the constraints on new supergravity terms can skip ahead to Section 4. The paper is organized as follows. In Sec. 2, we review how to embed liberated N = 1 supergravity in the superconformal tensor calculus, which was firstly introduced in Ref. [10], and explicitly compute the component action of the liberated supergravity using the tensor calculus in detail 3 . In Sec. 3, we review the component action of a new FI term, which is Kähler-invariant, proposed by Antoniadis, Chatrabhuti, Isono, and Knoops (ACIK) 4 . In Sec. 4, we recall the constraints on the liberated supergravity terms derived in [10] and derive the constraints on the new FI terms used in [11]. Both sets of constraints are based on an analysis of fermionic nonrenormalizable interactions. In Sec. 5, we briefly summarize our findings.
2 Component action of liberated N = 1 supergravity in superconformal tensor calculus In this section, we compute the component action of liberated N = 1 supergravity in superconformal tensor calculus [12,13]. We find that the superconformal Lagrangian of liberated N = 1 supergravity [9] can be written by a D-term in which we define Y ≡ S 0S0 e −K/3 where S 0 is a conformal compensator with Weyl/chiral weights (1, 1) and K is a Kähler potential with the weights (2, 0). The notation for the other fields is as follows: is a field strength multiplet with respect to the Kähler potential; U(Z,Z) is a general function of matter multiplets Z, and T (w 2 ),T (w 2 ) are chiral projection ofw 2 and its conjugate respectively.

Embedding Super-Weyl-Kähler transformations as an abelian gauge symmetry into the superconformal formalism
In liberated N = 1 supergravity [9], a key idea is that the Super-Weyl-Kähler transformation can be promoted to an Abelian gauge symmetry. Liberated supergravity was constructed in [9] using the superspace formalism, where a Kähler transformation is introduced to compensate the variation of the action under a super-Weyl rescaling. In this work instead, we want to construct the equivalent liberated supergravity using the superconformal tensor calculus to analyze the fermionic interactions in a systematical and economical way. To do so, we introduce a conformal compensator multiplet, called S 0 , removing the variation while maintaining the Kähler potential invariant under superconformal symmetry. Therefore, unlike the superspace formalism, it is essential to define such a gauge transformation independently of superconformal symmetry.
To find the Super-Weyl-Kähler transformations that are compatible with the superconformal formalism, we recall first the Super-Weyl-Kähler transformations that are used in the superspace formalism. A Kähler function K(z,z), whose arguments have the vanishing Weyl/chiral weights, is defined up to a chiral gauge parameter Σ. The redefinition by Σ acts on the components of the Kähaler multiplet as where E and E are the D-and F-term densities, respectively.
Next, it may be useful to recall the relation between the superspace and superconformal formalisms.
The invariant actions from the superconformal formalism are identified with those from the superspace calculus as follows [14]: where V is a superconformal real multiplet with the Weyl/chirial weights (2,0) and S is a superconformal chiral multiplet with Weyl/chiral weights (3,3). To make the action invariant under the super-Weyl-Kähler transformations we should impose that the corresponding superconformal multiplets transform as Instead of considering the Kähler transformation of the Kähler potential a superconformal compensator is introduced in the superconformal formalism to eliminate the variation of the action transformed by a super-Weyl rescaling (also called Howe-Tucker transformation [15]). Thus, the compensator must transform as where K is invariant under the superconformal symmetry (i.e. super-Weyl rescaling), so that the action can be invariant as desired.
At this point, differently from the usual story of the superconformal symmetry, we require a "Kähler transformation" of the Kähler potential in order to construct a "liberated" supergravity that is invariant under the same Super-Weyl-Kähler transformations as an abelian gauge symmetry used in [9].
Therefore, we assume that the superconformal compensators are inert under the super-Weyl-Kähler while the Kähler potential does transform under the same transformations as above, namely as K → K + 6Σ + 6Σ, so that Y → Ye −2(Σ+Σ) , (2.1.14)

List of superconformal multiplets
In this section, we present all the superconformal multiplets of the liberated N = 1 supergravity following the notations and multiplication laws used in [13,2].

Kähler potential multiplet
Let us consider n physical chiral multiplets of matter z I ≡ {z I , P L χ I , F I } where I = 1, 2, 3, · · · , n and their anti-chiral multipletszĪ ≡ {zĪ , P R χĪ ,FĪ }. 5 Then, according to the superconformal tensor calculus, the Kähler potential multiplet can be written as follows: The complex conjugates arezĪ ≡ (z I ) * , χĪ ≡ (χ I ) C ,χĪ ≡ (χ I ) C , andFĪ ≡ (F I ) * (The barred index is the complex conjugate index, so that the handedness of fermion becomes opposite, i.e. (P L/R χ) The chiralities of fermion are specified as χ I ≡ PLχ I and χĪ ≡ PRχĪ . The Majorana conjugates are (P L/R χ) =χP L/R (The handedness is preserved.). where Here the covariant derivatives 6 of chiral multiplets of matter with the weights (0, 0) are given by (2.2.12) Note that the only bosonic contribution to D K is given by and we especially denote this byF. We also note that theF is positive definite up to terms containing spatial gradients; so, for small spatial gradients, (2.2.14) 6 From Eq. (16.34) in Ref. [2] we find that for a general superconformal chiral multiplet (z I , PLχ I , F I ) with Weyl/chiral weights (w, c = w) and gauge symmetries with Killing vector fields k I A , the full superconformal covariant derivatives Da are given by

Compensator multiplet
Chiral compensators S 0 ,S 0 with the Weyl/chiral weights (1,1) are defined as follows. The chiral supermultiplets S 0 = {s 0 , P L χ 0 , F 0 } andS 0 = {s * 0 , P R χ 0 , F * 0 } can be embedded into the superconformal formalism as Then, the composite real compensator S 0S0 is where the covariant derivatives of the conformal compensator (s 0 , P L χ 0 , F 0 ) with Weyl/chiral weights (1, 1) are given by The composite real compensator Υ ≡ S 0S0 e −K/3 (2.2.34) The conventional superconformal gauge is defined by the choice Let us define We also defined: Next, we shall consider a gauge fixing that is equivalent to the gauge condition given by η = 0 in Ref. [9]. In fact, this can be obtained by imposing Λ K = 0. We will call this gauge the "Liberated SUGRA gauge". Then, the only non-vanishing superconformal components of the multiplets W 2 (K),W 2 (K) in this gauge are given by Another representation of the chiral multiplet is We will also need the following definitions: and Note that we have to insert K W = Λ W = D W = 0 and HW = ΛW = DW = 0 as given in Eqs.
(2.2.47) and (2.2.54). In addition, the complex conjugate multipletw 2 can be obtained by taking a replacement Q →Q in the above expressions.
In the Liberated SUGRA gauge (Λ K = 0), the non-vanishing components are given by Furthermore, with the superconformal gauge choice Z Υ = 0, the non-vanishing components are The superconformal setting for this and its complex conjugate are then given by In the Liberated SUGRA gauge, the non-vanishing components are given by With the superconformal gauge choice Z Υ = 0, we have Final form of the composite multiplet of liberated N = 1 supergravity To construct the final composite multiplet which will give the new term in the action of liberated supergravity, we first collect the following chiral multiplets: Then, the lowest component of the final composite real multiplet is where i, j = 0, I, W, T andj,k,l =0,Ī,W ,T and

Bosonic Lagrangians
The bosonic contribution to the scalar potential can be found from the term D N . Using the above results and Eq. (2.2.112), we find the equivalent bosonic contribution to the scalar potential as follows: In the superconformal gauge Υ = 1, we get

Fermionic Lagrangians
In this section, we investigate fermionic terms in the liberated N = 1 supergravity Lagrangian. We focus in particular on the matter-coupling and on the most divergent fermionic terms, in order to explore interesting interactions and check the limits of validity of the liberated N = 1 supergravity as an effective theory.
First of all, let us recall that the final composite multiplet N in terms of the superconformal chiral We we also denote their lowest components as W ≡Λ K P L Λ K ,W ≡Λ K P R Λ K , T ≡ C T , andT ≡ CT .
Note that the final composite multiplet consists of the four superconformal chiral multiplets only.
Remember that C T ∼FW Generically, the matter couplings are found from the following contributions: Note that the matter couplings of fermions can be classified into seven types: The derivatives of the N are given in general by with respect to z I ,z I , which are the lowest component of the matter chiral multiplets; q = q 1 + q 2 where q 1 (q 2 ) is the order of the derivative with respect to the compensator scalar s 0 (s * 0 ); p = p 1 + p 2 where p 1 (p 2 ) is the order of the derivative with respect to the field strength multiplet scalar W (W ); m = m 1 + m 2 where m 1 (m 2 ) is the order of the derivative with respect to the chiral projection multiplet scalar T (w 2 ) (T (w 2 )); k is the order of the derivative with respect to the matter multiplet scalar z I ; n is the order of the derivative acting on the new term U with respect to the matter multiplet; q is the total order of derivative with respect to the compensator scalars s 0 and s * 0 . An explicit form of the derivatives of N is given in Appendix A.
The mass dimension 8  Now, let us focus on the case such that q = 0 and k = n which gives the most singular fermionic terms in the limit that the D-term vanishes. The most singular terms are those that contain the highest power of the auxiliary field D in the denominator and therefore are the nonrenormalizable operators associated with the smallest UV cutoff mass scale. That is, we consider that the matter scalar derivatives act only on the new term U and there are no the derivatives with respect to the compensator scalar. Then, we have In particular, since r = p + m + n (0 ≤ r ≤ 4), it reduces to Remember that we called r the total order of the derivatives acting on the N . Here n is the number of derivatives acting on the matter scalars in the new term U; they produce U (n) . Finally, p = p 1 + p 2 is the sum of the number of derivatives w.r.t the multiplets (W,W ) acting on U.

Structure of the fermionic components
Next, let us explore the detailed structure of the chiral fermions of the superconformal multiplets.
First of all, the compensator and matter fermions are given by Note that χ 0 and χ I are fundamental fermions, and later in the S-gauge, the compensator chiral fermions will be replaced by the matter ones according to: P L χ 0 = 1 3 s 0 K I P L χ I and P R χ 0 = 1 3 s * 0 KĪ P R χĪ .
On the contrary, the other fermions χ W and χ T are composite. Hence, we need to find their specific structure.
First, the W(K) 2 -multiplet fermion, say χ W , is found to be where Note that the composite fermion χ W contains powers of the matter fermions ranging from one to seven, and it is nonvanishing on-shell. Second, the chiral projection multiplet T (w 2 ) fermions are found to be as follows: 1f + · · · + 9 fermions, whereF ≡ 2K IJ −∂ µ z I ∂ µzJ + F IFJ ; "|F 1 " denotes the terms proportional toF 1 , and Note that χ T is also nonvanishing on-shell.
In the above calculation we have used the formula: (2.4.21) Here ≈ means equality up to terms proportional to the equations of motion of free massless matter fields. Such terms produce only terms that contain additional factors of the matter field masses in the numerator and therefore give rise to either renormalizable operators or nonrenormalizable operators weighted by a mass scale higher than that associated to those terms that do not vanish on shell. The details of the above calculation is given in Appendix A.1.
Finally, we present here the chiral fermions of the superconformal multiplets up to multiple fermion terms.
Notice that none of the chiral fermions χ i vanish on-shell, and that only χ T dependens onF and includes the factor of Υ −2 . All of these properties affect the mass dimension of the expansion coefficients of the nonrenormalizable Lagrangians.
We finally expand W andW as follows: Notice that W andW can be represented by products of two, four, and six fundamental fermions.

Additional gauge fixing for physical theory in the liberated supergravity
First of all, we consider the conventional superconformal gauge which is chosen by Note that the first condition is the D-gauge fixing which gets us to the Einstein frame; the second one is the improved S-gauge fixing; the third one is the A-gauge fixing; the last one is the K-gauge fixing.
To compare our results with the formulation of liberated supergravity in [9], we choose a gauge given by In both the conventional superconformal gauge (2.5.2,2.5.3,2.5.4) and in (2.5.5), the relevant multiplets are

5.6)
z I = z I , P L χ I , F I ,zĪ = zĪ , P R χĪ ,FĪ , (2.5.7) (2.5.11) Let us further observe that all the terms coupled to N and its derivatives with respect to 0, I, T vanish because N contains the product of WW , which is zero in the liberated gauge, i.e. W =W = 0. We see that all the fermionic terms coupled to the derivatives of N with respect to W or T vanish as well, because these terms always couple to the fermions χ W and χ T which vanish in the gauge. The only non-vanishing contribution is given by only the bosonic term, N WW F WFW which gives us the new term U to the scalar potential. Therefore, in both gauges, the liberated supergravity Lagrangian is specified by where L SU GRA is the standard supergravity Lagrangian which contains the auxiliary fields F 0 , F I and where the last term coupled to the parameter ξ corresponds to the new FI terms.

Component action of the new FI term
We consider matter chiral multiplets Z i , the chiral compensator S 0 , a real multiplet V , and another real multiplet (V ) D , whose lowest component is the auxiliary D term of the real multiplet V . Their superconformal multiplets are given as follows: where Notice that when finding the multipletw 2 , we can just replace h by its complex conjugate h * .
3.1.2 T (w 2 ),T (w 2 ) chiral projection multiplets: (Weyl/Chiral) weights = (0, 0) The second superconformal multiplets that we need are the composite chiral projection multiplets T (w 2 ) andT (w 2 ) with Weyl/chiral weights (0, 0). From their component supermultiplets defined by we find the corresponding superconformal multiplets and their complex conjugates as follows: (3.1.26) whose superconformal components are given by where (3.1.37) We present here a superconformal composite real multiplet R with Weyl/chiral weights (2, 0). Defining we represent the composite one R as whose lowest component is and ∆ ≡ s 0s0 e −K/3 . Then, the superconformal multiplet of the new Fayet-Iliopoulos term can be written by using R · (V ) D = {C,Z,H,K,B µ ,Λ,D}, (3.1.44) whose superconformal multiplet components are as follows:

Bosonic term of the new FI term
The new FI term is obtained from the term D 2 f WW F W FW inside the D-termD. Thus, we get This section studies the fermionic nonrenormalizable terms in the Lagrangians containing either liberated N = 1 terms or new FI terms. We will make full use of the superconformal tensor calculus to find the limits of validity of those effective field theories.

Constraining liberated N = 1 supergravity
We investigate first the nonrenormalizable fermionic terms in "liberated" supergravity, to find possible constraints on the liberated term U. To begin with, we rewrite the coefficients where we used Υ = s 0 s * 0 e −K/3 .
The most powerful constraints on U come from the most singular fermionic terms in the Lagrangian.
Such singular terms are those containing the highest power ofF in the denominator. These powers are linearly proportional to m because they are due to taking derivatives of N with respect to the lowest component of the multiplet T (w 2 ). Therefore, we must investigate the fermionic interactions containing only derivatives with respect to the chiral projection and matter scalar indices, i.e. T and I, to obtain the terms that are coupled to U (n) and contain the maximal inverse powers ofF. Such terms are those with q = p = 0 and k = n.
In particular, we find by direct inspection of the fermionic interactions that the most singular terms are given by the couplings to the derivatives proportional to N TT , N W TT , N WW TT if our theory has a single chiral matter multiplet, while they are given by N T TTT for two or more chiral matter multiplets.
The latter terms vanish identically for a single multiplet because of Fermi statistics. This has been investigated in detail in Ref. [10]. Here we merely recall the results from Ref. [10]: • For the single chiral matter multiplet case, (4.1.2) • For two or more chiral matter multiplets, From this, by calling Λ cut the cutoff UV scale of liberated supergravity and using the inequality 1 Λ δ−4 cut (δ = 2(6 − n) for N mat = 1 and δ = 2(8 − n) for N mat ≥ 2) we obtain where 0 ≤ n ≤ 2 for N mat = 1, where 0 ≤ n ≤ 4 for N mat ≥ 2.

Constraining new Fayet-Iliopoulos terms
In this section 9 , we explore the possible constraints on the new FI parameter ξ. First of all, let us find the fermionic terms coming from the new FI terms. From its Lagrangian, we observe that the most singular terms come from the derivatives of the function f , defined in Eq. (3.1.43) and below, with respect to the multiplet T since they produce the inverse powers of the D-term 'D'. The function f was defined by Then, the most singular terms in the Lagrangians has the generic form The most singular terms appear in the Lagrangian as follows: where we omitted −(m 1 !m 2 !c) since this is O(1).
Next, we explore the constraints coming from the new FI term. First of all, we recall that the new FI term proposed by Antoniadis et al. can be added to the standard FI term, when the theory contains also a gauged U(1) R-symmetry U (1) R . Then, we have the following auxiliary D-term Lagrangian where ξ is the new FI constant, ξ ≡ i(G i k i − Gīkī). Here k = k i ∂ i is the Killing vector of a standard R-symmetry U R (1) and G is the supergravity G-function. If we restore explicit powers of the gauge coupling g, then we get Plugging this result into the Lagrangian again, we obtain This is a constraint on the new FI term constant ξ, written in terms of the UV cutoff Λ cut .

Summary
In Sec.  [10,11]. specifically, we relate the "liberated" scalar potential U to the UV cutoff and we derive the constraints on the ACIK-FI term used in [11]. What makes our constraints powerful is that differently from standard supergravity, both liberated supergravity and the ACIK-FI terms introduce nonrenormalizable interactions proportional to inverse powers of the supersymmetry breaking scale M S . This makes it impossible to send M S to zero while keeping the UV cutoff of the theory finite. The most singular nonrenormalizable interactions in the limit M S → 0 are cumbersome, multi-fermion operators, but they can be found and studies using the superconformal tensor calculus in a systematic and economical way. Superconformal calculus techniques are general so can be applied to any supergravity theory. Our analysis is based on standard Effective Field Theory methods and it is thus complementary to swampland [16] constraints.

A.2 Derivatives of N
In this appendix we give explicit formulas for the derivatives of N with respect to the multiplets i = 0, I, W, T . The first derivative of N is and the fourth derivative of N is l ∂0∂T ∂ I ∂ J + δ I i δ J j δK k δT l ∂K∂T ∂ I ∂ J + δ I i δ J j δT k δL l ∂L∂T ∂ I ∂ J +2δ I i δ J j δ (0 k δW ) l ∂0∂W ∂ I ∂ J + δ I i δ J j δK k δW l ∂K∂W ∂ I ∂ J + δ I i δ J j δW k δL l ∂L∂W ∂ I ∂ J +2δ I i δ J j δ (W k δT ) l ∂T ∂W ∂ I ∂ J + δ I i δ T j δ0 k δ0 l ∂ 2 0 ∂ I ∂ T + δ I i δ T j δT k δT l ∂ 2 T ∂ I ∂ T +δ I i δ T j δK k δ0 l ∂0∂K∂ I ∂ T + δ I i δ T j δ0 k δL l ∂0∂L∂ I ∂ T + δ I i δ T j δK k δL l ∂K∂L∂ I ∂ T +2δ I i δ T j δ (0 k δT ) l ∂0∂T ∂ I ∂ T + δ I i δ T j δK k δT l ∂K∂T ∂ I ∂ T + δ I i δ T j δT k δL l ∂L∂T ∂ I ∂ T +2δ