Interpolation of partial and full supersymmetry breakings in $\cal{N} = 2$ supergravity

We discuss an $\cal{N}=2$ supergravity model that interpolates the full and the partial supersymmetry breakings. In particular, we find the conditions for an $\cal{N}=0$ Minkowski vacuum, which is continuously connected to the partial-breaking ($\cal{N}=1$ preserving) one. The model contains multiple (Abelian) vector multiplets and a single hypermultiplet, and is constructed by employing the embedding tensor technique. We compute the mass spectrum on the Minkowski vacuum, and find some non-trivial mass relations among the massive fields. Our model allows us to choose the two supersymmetry-breaking scales independently, and to discuss the cascade supersymmetry breaking for the applications to particle phenomenology and cosmology.


Introduction
Extended (N ≥ 2) supergravity naturally appears from higher dimensional supergravity and string compactifications (see [1,2] for review). Its interactions are more restricted (predictive) than N = 1 supergravity, which has been intensively investigated from the viewpoint of particle phenomenology and cosmology. Due to the restrictions, it is a non-trivial task to obtain a phenomenologically favorable supersymmetry-breaking vacuum. This fact is related to the no-go theorem for the partial breaking of extended supergravity [3,4]. For example, in N = 2 supergravity, the naive gauging (electric gauging in the frame where

Set up
In this section, we specify the model which is a generalization of Ref. [7]. Here we follow the convention of [46] and use the unit M P = 1, where M P = 2.4 × 10 18 GeV is the reduced Planck mass.

Vector sector
The vector sector is governed by the prepotential F (X Λ ), which is a holomorphic function of n v + 1 complex variables X Λ (Λ = 0, 1, · · · , n v ), and is homogeneous of degree two. In general, it can be parameterized as (2.4) where f is an arbitrary holomorphic function. In N = 2 supergravity, the theory has (onshell) Sp(2n v + 2, R) symmetry which acts on the holomorphic section, , (Λ, Σ = 0, 1, · · · , n v ) (2.5) where F Σ = ∂F/∂X Σ , and the index M specifies 2n v + 2 components of the symplectic vector.
Based on the holomorphic section Ω, the Kähler potential K is given in a manifestly symplectic invariant way as (2.6) 3 The hyperscalars b u (u = 0, · · · , 3) have the quaternionic structure. 4 Note that α is not a spinor index. The spinor indices are suppressed throughout this paper.
where C is a symplectic invariant tensor, We take a special coordinate as where z i are identified as physical scalars in vector multiplets. Then, F Λ = {F 0 , F i } becomes where the subscript i on f denotes the derivative with respect to z i . The Kähler potential is then written by This is the general form of the Kähler potential. The derivatives of the Kähler potential, the Kähler metric, and the Levi-Civita connection are computed as Furthermore, for later convenience, we list several quantities which appear in the Lagrangian: 14) Finally, the gauge kinetic functions N ΛΣ are given by (2.17)

Hyper sector
As for the hyper sector, we consider the following quaternion-Kähler metric [7], which describes a nonlinear sigma model on SO(4, 1)/SO (4). As shown recently in Ref. [15], this is the unique metric of a single hypermultiplet for the partial breaking in Minkowski space. The vielbein U αA = U αA u db u can be read off as where A = 1, 2 and α = 1, 2 represent the SU(2) and Sp (2) indices respectively (their conventions are shown in Appendix A). σ x is the standard Pauli matrices. The SU(2) connection is given by Note that Eq. (2.18) admits three commuting isometries: where c m are real constants. 5 Then, the Killing vectors k u m which generate these transformations and the moment maps corresponding to k u m are

Gauging
In order to discuss the supersymmetry breaking, we will gauge some of the isometries of the hyper sector. For this purpose, we employ the embedding tensor formalism [10,11], which is useful for discussing the general gauging of the extended supergravity (see also [47,48] for a review). This formalism formally introduces a double copy of the gauge fields, i.e., the electric gauge fields A Λ µ and the magnetic gauge fields A µΣ (Λ, Σ = 0, 1, · · · , n v ), and gauges some of the global symmetries with the gauge couplings, which is called the embedding tensor. The tensor Θ m M must satisfy several conditions for the self-consistency of the theory [10,11]. In our case where no isometry on the vector sector is gauged, the only corresponding constraint is Then the covariant derivative is defined by where T m are generators of the isometries (2.21), thus k u m = T m b u . We also define Note that the magnetic vectors A µΛ also participate in the gauging. Our interest is the N = 2 supergravity system with two breaking scales, which can realize the N = 1 (partially broken) vacuum in some limits of the gauge couplings. We need to gauge two of the isometries in Eq. (2.21) to obtain the partial breaking [7][8][9] because the N = 1 massive gravitino multiplet contains two massive vector fields, which come from the gauging of two isometries. Also, we need a magnetic entry in Eq. (2.23) for the partial breaking because it is proven in Ref. [12] that purely electric gauging cannot preserve N = 1 supersymmetry. From these observations, we take the embedding tensor as . This is a generalization of the one discussed in Ref. [15]. Our setup reduces to the model of Ref. [7] when e 2 = E i = 0 and n v = 1.

Action and supersymmetry transformation
Here we show the relevant parts of the Lagrangian [46,[49][50][51]. Due to the existence of the embedding tensor as well as some magnetic vector fields, we have to introduce auxiliary two-form fields B µν,m for consistency (see Refs. [10,11] for detail). The Lagrangian is given by where R is the Ricci scalar and we have omitted the four fermi interactions. L kin consists of the kinetic terms and we further decompose it for later convenience, where H Λ µν is a gauge invariant combination of the field-strength and the two-form field, L Y denotes the interactions including the fermion bilinears and it is explicitly given by In their expressions, D u k v M = ∂ u k v M + vxy ω x u k y M and D i U M j can be found in Eq. (2.16). The interaction terms L Pauli and L der are necessary in subsection 3.6 and their explicit expressions are shown in Appendix D.
L top is required for consistency of the embedding tensor formalism and it is given by [11] (2.42) Finally, the scalar potential V is given by Note that the Lagrangian (2.28) has larger gauge symmetry, in addition to the Abelian gauge symmetry of the vector fields. Indeed, it is invariant under where Ξ µm are the gauge transformation parameters of the two-forms. We need the supersymmetry transformations of the fermions to discuss supersymmetry breaking. The relevant parts are given by where A are the transformation parameters. The ellipses represent other contributions, which vanish in the Minkowski vacuum.

Spectrum
Here we derive the mass spectrum of our model.

Scalar potential and minimization
First, let us discuss conditions the vacuum satisfies. Under the gauging (2.27), the scalar potential (2.43) is explicitly given by where we have defined Note that the scalar potential (3.1) is independent of b m (m = 1, 2, 3). Thus, the minimum is obtained by solving where ∂ i = ∂/∂z i and ∂ b 0 = ∂/∂b 0 . They give n v + 1 equations in general. By using Eq. (2.16), Eq. (3.4) can be summarized as In general, it is difficult to solve these equations for a general form of f . Therefore, for simplicity, we assume that one of z i has a nonzero expectation value (we choose it the N -th direction), and the vacuum satisfies the following conditions: (3.10) Then, the derivative of the Kähler potential and the Kähler metric become Later, we will check that these assumptions for the vacuum are valid in a concrete choice of f . Furthermore, we assume that the gauging is done only for the N -th direction, i.e., which lead to Under these simplifications, Eq. (3.6) is reduced to just one equation, (3.15) and the others are trivially satisfied. Therefore, if the vacuum satisfies either In this case, we obtain In the following discussion, we take the plus branch. 6 In summary, under the assumptions (3.7)-(3.10) and (3.13), the vacuum must satisfy either of the two conditions: These conditions have been derived in Ref. [15] in the case of a single vector multiplet (n v = 1). We will investigate how many supersymmetries are (un)broken in the vacuum in the next subsection.

Supersymmetry transformation
Let us begin with the case (ii). Under the conditions (3.18) and (3.19), the supersymmetry transformations (2.47)-(2.49) in the vacuum become As can be seen, all of the matrices have a zero eigenvalue. Indeed, defining with φ = {ψ, λ, ζ, }, we can rewrite them as Therefore, one of the two supersymmetries is broken (for − direction), but the other one is still preserved (for + direction). The minus sign in Eq. (3.19) leads to the preservation of − direction. We conclude that for nonzero e 3 and M N , we always have N = 1 preserving vacuum in the case (ii). 7 In the case (i), on the other hand, the supersymmetry transformations are In this case, the value of f N is not determined by the vacuum conditions. The vacuum can be broken to N = 0 depending on f N . If we choose f N as Eq. (3.19), the situation goes back to the case (ii), and N = 1 supersymmetry is preserved. Thus, we can interpolate the full and the partial breakings in this case. 8 In order to characterize the deviation from the partial-breaking condition (3.19), we parametrize f N as In terms of the basis (3.23), these are rewritten as The full supersymmetry breaking occurs unless ∆ = 0, −2e 3 . In the following, we call the supersymmetry transformations caused by + and − as the first and the second supersymmetry transformations, respectively.

Explicit model
Let us specify our model. We assume that f has the following form, 9 where c 0 , c i , c ij and c ijk are complex constants and totally symmetric for their indices. This satisfies Eqs. (3.17) and (3.18) if Also, from the assumptions (3.7)-(3.10), we obtain where we have divided the indices i = {â, N } withâ = 1, · · · , n v − 1. 10 9 In Ref. [7], the case f = z is considered. 10 The convention of the indices is summarized in Appendix A.

Fermion mass
Let us check the spectrum of the fermion sector. In the vacuum, fermion mass terms from the interactions in Eq. (2.35) are evaluated as In the second line, we have defined Also, we have assumed that where C and G are complex and real constants, for simplicity. 11 11 These can be achieved by choosing câb, câb N ∝ δâb in Eq. (3.41).
From the kinetic terms in Eq. (2.33), we obtain In terms of χ(χ) and η(η) defined above, we can rewrite them as In Eqs. (3.42) and (3.49), χ andχ are the goldstinos which correspond to the first and the second supersymmetry breaking, respectively. It can be checked directly by Thus, in the unitary gauge, χ =χ = 0, they disappear from the spectrum. Then, we obtain This gives the free parts of the fermion sector. Note that we have two massive pairs, ψ + , η and ψ − ,η, whose masses are given by |m 1 | and |m 2 | respectively. In addition, there are 2n v − 2 massive gauginos. The n v − 1 gauginos (λâ + ) have the mass |m + |, and the others (λâ − ) have |m − |. 12 Their mass scales are splitting due to the existence of e 3 .
2η • lead to the canonical normalization.

Boson mass
First, let us focus on the following terms (3.52) Each term can be decomposed explicitly as and (3.56) Note that there seems n v + 2 vectors A 0 µ , Aâ µ , A N µ , and A µN at a glance. However, we can gauge away A N µν by and eliminate its degree of freedom. Therefore, we obtain n v + 1 vectors as usual. Also, A µN and B µν,2 do not have kinetic terms, and should be integrated out. 13 As shown in Appendix B, eliminating B µν,2 gives a kinetic term for A µN . We only show the result here. After integrating out the two-form field, Eq. (3.52) at the vacuum is given by 14 where Î 00 = − e −K 4 (e 3 + Re∆) 2 + (e 2 + Im∆) 2 (e 3 + Re∆) 2 , (3.59) (3.62) For notational simplicity, here we have omitted the spacetime indices of the field-strength F µν . It can be found that two hyperscalars, b 2 and b 3 , can be eliminated by where we have redefined A µN as Then, the Lagrangian becomes It contains n v − 1 massless vector fields and 2 massive ones. There are also two massless hyperscalars, b 0 and b 1 . For the massive modes, we can diagonalize their kinetic and mass matrices by where F (B) and F (B ) denote the field-strength of B µ and B µ . Their masses are given by Remarkably, m B and m B are related to m 1 and m 2 by We see some implications of their hierarchical structures. Finally, let us check the mass of z i . Defining the fluctuation around the vacuum, we can expand the scalar potential as where the ellipsis denotes higher order couplings ofz i . The first and the second terms vanish due to the (Minkowski) vacuum conditions. 15 The third and fourth terms are expressed as by assuming Eq. (3.47). Note thatz N is massless. Taking into account the canonical normalization (zâ →zâ/G), and diagonalizing the mass matrix, we obtain 80) 15 In order to avoid the runaway of b 0 , the vacuum should be the Minkowski one. where  (3.85) and the gauginos have different masses depending on câb and câb N . The boson masses ofzâ are also similar.

Summary of mass spectrum
In table 1, we summarize the spectrum (at the tree level) obtained in the previous subsections, where ∆ ≡ ∆ + 2e 3 .
Toward a phenomenological application, let us consider a case of hierarchical supersymmetry breaking |∆ | |∆|. Then, the mass spectrum of the massive vectors {B µ , B µ } and the fermions Ψ − ≡ {ψ µ− ,η • } and Ψ + ≡ {ψ µ+ , η • } satisfies From the results shown in Appendix D, we find that all the interactions among these fields schematically take the following forms: (3.88) Through these interactions, the possible decay processes of the heavy fields B µ and Ψ − are (3.90) However, due to the mass relations (3.73) and (3.74), their decay rates vanish at least at the tree level. As for the other massive fields xâ, yâ and λâ ± , there are no specific mass relations since their masses depend on the free parameter C (or câb N in general). Thus, their decays to the light states in Ψ + can have nonvanishing rates.
In summary, we found that the direct decays from the heavy particles B, B , Ψ − to the light ones in Ψ + are not allowed (at the tree level), but those from xâ, yâ, λâ ± to Ψ + can be allowed depending on C. We schematically show these allowed/forbidden decay processes in Fig. 1. These restrictions may become important when we discuss the cosmological history based on models of extended supergravity.

Conclusion
In this paper, we have investigated the N = 2 supergravity model that interpolates the full and the partial breakings of supersymmetries, which is a generalization of Ref. [7]. We Figure 1: Decay modes to Ψ + extend the model by introducing additional vector multiplets. As can be inferred from the studies of the partial breaking of extended supergravity [7][8][9]12], the magnetic gauging is important and we chose the embedding tensor as Eq. (2.27) with Eq. (3.13). Then we found the conditions for an N = 0 Minkowski vacuum, which is continuously connected to N = 1 preserving ones.
The breaking scales of the two supersymmetries can be chosen independently in our model (see Eqs. (3.35)-(3.37)). Thus, we can discuss the case of the cascade supersymmetry breakings in which the two supersymmetry breaking scales are separated hierarchically. This is phenomenologically interesting when we consider the string compactifications. In such a case, an approximate N = 1 supersymmetry appears at intermediate energies between the two scales. In contrast to other phenomenological supersymmetric models, we can quantitatively discuss the effects of the second supersymmetry breaking on light modes in the approximate N = 1 sector.
We computed the mass spectrum (see table 1), which is indeed characterized by two different scales, |∆| and |∆ |. We found that there are non-trivial relations in the spectrum, (3.73) and (3.74). Even for |∆ | |∆|, there are no direct decay processes from the heavy fields (B, B , Ψ − ) whose masses are of O(|∆ |) to the first gravitino (Ψ + ) in our model (see figure. 1). This property may be important when models based on extended supergravity 16 are applied to cosmological scenarios.
There are several issues to be addressed: The first one is the generality of the gauging. In this work, we have not discussed the most general gauging, taking the embedding tensor as Eq. (2.27) with the simplification (3.13). While our model includes that of Ref. [7] as a special case, it is worth investigating the other gauging and checking how general our result is. Also, we may allow other vacuum solutions without assuming the ansatz (3.7)-(3.10), and include higher-order terms in z i of the prepotential f . It would be also interesting to extend this analysis to the system with multiple hypermultiplets and non-Abelian gauge symmetries as in Ref. [39]. We leave them for future works.

A.1 Index of vector fields
The index of the vector fields Λ is decomposed as in table 2.
The SU(2) and Sp(2) invariant tensors satisfy AB BC = −δ A C , 12 = 12 = 1, (A.1) and the indices of SU (2) and Sp (2) vectors are raised and lowered by The Pauli matrices are (σ x ) B A (x = 1, 2, 3) are Their indices are raised and lowered by AB and AB defined above. We denote the chirality of the spinors as (A.7)

B Integrating out two-form field
Here we show the process to integrate out the auxiliary two-form field, which is needed for the embedding tensor formalism [10,11]. The two-form field appears in Eqs. (2.32), (2.42). Also, it is contained in Eq. (D.6) which is necessary for deriving the interactions in Appendix D. We summarize them as, where Q µν Λ comes from Eq. (D.6), and is defined by Here * denotes a Hodge dual defined by * T µν = − i 2 ε µνρσ T ρσ . In the gauge (3.57), the E.O.M of B µν,2 yields where we have introduced an indexΛ = {0,â}. This can be solved as By substituting the solution into the Lagrangian (B.1), we obtain (B.14) The first and the second line in Eq. (B.8) describe the kinetic terms and theta couplings of physical vector fields. The third line shows the interactions including the vector and the fermion biliner couplings (we have neglected four fermi interactions), which are used in Appendix D.

C Expectation values of gauge kinetic function
Here we list the explicit expressions of the vacuum expectation values of the gauge kinetic function and the couplings appearing in Eq. (B.8).

D Interaction
Here, we calculate the interaction terms up to three point couplings. We focus on the interactions of the massive fermions including two gravitinos. Before specifying the interactions on the vacuum, here we enumerate the undefined quantities in subsection 2.3 : The covariant derivatives of the fermions, L Pauli , and L der .
From Eq. (D.6), we can obtain the vector and fermion bilinear couplings. It also contains the two-form field B µν,2 and affects the process of integrating out B µν,2 (see Appendix B).