Nonlinear \( \mathcal{N}=2 \) global supersymmetry

Journal of High Energy Physics, Jun 2017

We study the partial breaking of \( \mathcal{N}=2 \) global supersymmetry, using a novel formalism that allows for the off-shell nonlinear realization of the broken supersymmetry, extending previous results scattered in the literature. We focus on the Goldstone degrees of freedom of a massive \( \mathcal{N}=1 \) gravitino multiplet which are described by deformed \( \mathcal{N}=2 \) vector and single-tensor superfields satisfying nilpotent constraints. We derive the corresponding actions and study the interactions of the superfields involved, as well as constraints describing incomplete \( \mathcal{N}=2 \) matter multiplets of non-linear supersymmetry (vectors and single-tensors).

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Nonlinear \( \mathcal{N}=2 \) global supersymmetry

Received: April Published for SISSA by Springer Open Access 0 3 4 c The Authors. 0 3 4 0 Sidlerstrasse 5 , CH-3012 Bern , Switzerland 1 LPTHE, UMR CNRS 7589, Sorbonne Universit ́es 2 Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics 3 We focus on the Goldstone de- 4 UPMC Paris 6 , 75005 Paris , France We study the partial breaking of N = 2 global supersymmetry, using a novel formalism that allows for the off-shell nonlinear realization of the broken supersymmetry, extending previous results scattered in the literature. grees of freedom of a massive N = 1 gravitino multiplet which are described by deformed N = 2 vector and single-tensor superfields satisfying nilpotent constraints. We derive the corresponding actions and study the interactions of the superfields involved, as well as constraints describing incomplete N = 2 matter multiplets of non-linear supersymmetry (vectors and single-tensors). Extended Supersymmetry; Supersymmetry Breaking - 2 global supersymmetry Single-tensor multiplet formulation Alternative proof The vacuum Dual hypermultiplet formulation Several single-tensor multiplets Nonlinear deformations Deformations of the Maxwell superfield Deformations of the single-tensor superfield Constrained multiplets The infinite-mass limit Solutions of the constraints The “long” super-Maxwell superfield The chiral-antichiral N = 2 superfield The long and short super-Maxwell superfields Long superfield and nonlinear deformations 1 Introduction 2 Partial supersymmetry breaking with one hypermultiplet 3 6 Interactions 6.1 The Chern-Simons interaction 6.2 Constrained matter multiplets The Chern-Simons interaction with deformed Maxwell multiplet The Chern-Simons interaction with deformed single-tensor multiplet The goldstino in the Maxwell multiplet The goldstino in the single-tensor multiplet A Conventions and some useful identities More on the Maxwell supermultiplet More on Im V The spontaneous breaking of global symmetries is described at low energies by a nonlinIn the case of supersymmetry, the Goldstone modes are fermions, the goldstini, and the symmetries, it can be obtained (up to field redefinitions) by a chiral superfield X satisfying of the goldstino bilinear [2–5]: X = − 2F and one obtains (on-shell) the Volkov-Akulov action [2, 6]. Besides the use of nonlinear supersymmetry as an effective low-energy theory at energies below the sgoldstino mass, it can also be realized exactly in particular vacua of type I string theory, when D-branes are combined with anti-orientifold planes that break the linear supersymmetries preserved by the D-branes, while they preserve the other half that are realized nonlinearly. In such vacua of “brane supersymmetry breaking”, superpartners of brane excitations do not exist, and supersymmetry is nonlinearly realized with the presence of a massless goldstino in the open string spectrum [7–11]. 2, broken at two different scales, is a challenging and not straightforward problem. An interesting case is N supersymmetry, which can be either a vector or a chiral multiplet. In fact, both cases have to be studied, since they constitute the Goldstone degrees of freedom of a massive spin-3/2 multiplet. Indeed, a massless spin-3/2 multiplet contains a gravitino and a graviphoton, while a massive one contains, in addition, a spin-1 and a (Majorana) spinor, so that the Goldstone modes are a vector, two 2-component spinors and two scalars [12]. When the second and nonlinear supersymmetry is taken into account, the above two a Maxwell multiplet and a hypermultiplet. The latter comes with an extra complication since it has no off-shell formulation in the standard N = 2 superspace. the presence of bosonic shift symmetries associated with the would-be Goldstone bosons providing the longitudinal components of the spin-1 fields, implies that the chiral multiplet can be dualized to a linear multiplet having an off-shell description when promoted to a metry [13], extending known results in the literature on Maxwell multiplets [13–17] and supersymmetry is by a (constant) deformation of the supersymmetry transformations of the fermions that cannot be absorbed in expectation values of the auxiliary fields, unlike particular relations, guaranteeing the existence of one goldstino associated with a linear combination of the two supersymmetries. The goldstino superfield of one nonlinear supersymmetry can then be obtained by imposing a nilpotent (double chiral) constraint, in analogy with X2 = 0 of N = 1. The outline of this paper is the following. In section 2, we present a model of spontaneous partial breaking of N = 2 → N admits a special superpotential that allows for partial supersymmetry breaking, in analogy with the magnetic Fayet-Iliopoulos (FI) term in the Maxwell multiplet model of [13]. field is chiral under both supersymmetries (CC), while the single-tensor superfield is chiral under the first and antichiral under the second (CA). In section 3, we discuss nonlinear section 4, we consider the infinite-mass limit that freezes half of the degrees of freedom, and derive the constrained multiplets and the corresponding nilpotent constraints. We then give the solutions of the constraints (off-shell) and derive the generalizations of the goldstino Volkov-Akulov action in the presence of a linear supersymmetry, in addition to the nonlinear one. These are the supersymmetric Dirac-Born-Infeld (DBI) action and a similar action for the linear multiplet, in agreement with previous results. We then turn to for the Maxwell and single-tensor multiplets with opposite relative chiralities compared to the “short” ones, namely CA for the Maxwell and CC for the single-tensor, so that one can write a Chern-Simons type of interaction that we discuss in section 6. This interaction leads to a super-Brout-Englert-Higgs mechansim without gravity, in which the linear multiplet is absorbed by the vector which becomes massive [19]. In section 6, we non-linear supersymmetry (vectors or single-tensors), half of the components of which are projected out. Finally, section 7 contains concluding remarks and open problems, while there are three appendices with our conventions (appendix A) and the technical details of the Maxwell multiplet (appendices B and C). hatted superfields Wc, Zb . . . have 16B + 16F fields. They are chiral with respect to the first three real scalar fields and two Weyl (or massless Majorana) spinors. In the same manner that an antisymmetric tensor is dual to a pseudoscalar with axionic shift symmetry, a single-tensor multiplet is equivalent to a hypermultiplet with shift symmetry. In both cases, the symmetry implies masslessness. In analogy with the Yang-Mills or Maxwell multiplet but in contrast with the hypermultiplet, the single-tensor multiplet admits an off-shell formulation. can be viewed as the supersymmetrization either of the gauge invariant three-form field In this section we show the existence of partial supersymmetry breaking in a large class of tional) isometry allowing for a description in terms of a dual single-tensor multiplet which admits, like the Maxwell multiplet, a fully off-shell formulation. We use this formulation to obtain these theories, dualize back to the hypermultiplet formulation and then display the strong similarity between partial breaking with a Maxwell (namely the APT model [13]) and partial breaking with a single-tensor multiplet. gauge symmetry DDL = 0 is solved by where H is any real function solving the three-dimensional Laplace equation Lkin. = ∂2H = 0. Dα˙ L = iϕα˙ − (θσµ )α˙ (vµ + i∂µ C) − θθ(∂µ ϕσµ )α˙ , vµ = (in chiral coordinates), where the real scalar C is the lowest component of L. Note also that the superpartner of L (under the second supersymmetry) is Φ = z + √2θψ − θθf . Single-tensor multiplet formulation symmetry (2.1) is then with Vb1 and Vb2 real: the gauge transformation of the single-tensor multiplet in the degenerated by Vb2. It is not a derivative unless W (Φ) ∼ Φ. Since2 the variation can also be written as3 3We usually omit derivatives when comparing lagrangian terms. δ∗ L = δn∗l L − √ (ηDΦ + ηDΦ), d2θ WΦ ηDL + h.c. = −Mf2 δ∗ d2θ W (Φ) + h.c. Lnl depends on two complex numbers, the deformation parameter Mf2 and the quantity 2 Mf2 (θη + θη), 2 Mf2 ηα˙ , Consider now the function H(L, Φ, Φ) = i −L2[WΦ − W Φ] + ΦW − ΦW , which is obviously a solution of the Laplace equation, while the action corresponding to L = i d2θd2θ h−L2[WΦ − W Φ] + ΦW − ΦW To break spontaneously the second supersymmetry, we first add the generic superpoLnl = = i d2θd2θ h−L2(WΦ − W Φ) + ΦW − ΦW as well as under the nonlinearly deformed second supersymmetry transformations Alternative proof Let us consider the N the partial breaking, we deform the second supersymmetry transformations of the singlea goldstino; the transformations take then the form (2.18). The deformation induces a new term in the variation of the lagrangian under the second supersymmetry: 2 Mf2 components must transform as derivatives under the first, unbroken supersymmetry. This is the case if the highest component of HL is zero or a derivative, whose solution is H = since terms linear in L do not contribute to the integral of (2.24), it becomes (since terms proportional to L0 do not contribute): Now let us consider again the derformation (2.20) of the lagrangian. With the use Consequently, the deformed lagrangian = M√f2 Z Ldef,kin. = is invariant under the first (linearly-realized) supersymmetry as well as under the second nonlinearly-realized one. It is also obvious that the lagrangians corresponding to (2.15) The vacuum superfield with the use of a prepotential function G(Z). Let us define4 Z = Φ + √2i θeDL − 41 θeθeh4iMf2 + DD Φi . = 2 We then obtain 2 Z 2 GΦΦ (DL)(DL) − 4 GΦ DD Φ − i Mf2 GΦ + h.c. (2.28) Partial supersymmetry breaking is achieved if theory (2.17) has a vacuum state invariant under the first (linear) supersymmetry. We then analyze the scalar potential, which, field lagrangian is5 − 2 It generates the scalar potential W ΦΦ[f ψψ − f ϕϕ] = −V + Lferm.. Lferm. = − 2 f − 2 e ϕϕ + h.c. Three situations can occur. and Mf2 6= 0 with V = The term depending on L in theory (2.17) does not contribute to the potential. Fermion mass terms read The analogy with partial supersymmetry breaking in a N = 2 Maxwell multiplet and X, with deformed supersymmetry variations 4 ηαDD X + i(σµ η)α ∂µ X , the invariant lagrangian is written as LMax. = 2 FXX W W − 4 FX DD X + m2X − iM 2FX + h.c. + LF.I., 2 Mf2ηα˙ + √2i 4 ηα˙ DDΦ − i(ησµ )α˙ ∂µ Φ , namely the absence of auxiliary fields in L as well as the consequent inexistence of a Dual hypermultiplet formulation The duality transformation from the single-tensor to the hypermultiplet formulation is a Lkin. = d2θd2θ hH(V, Φ, Φ) − (S + S)V i. hf i = 0. HV = S + S (Legendre transformation) In our case, the Legendre transformation is simply KV = 0 =⇒ S + S = HV = −2iV (WΦ − W Φ) HS = 0 =⇒ KS = −V . are consistent using the field equations for L and S, HL = S + S , KS = −L , DDKS = 0 , as integrability conditions. K¨ahler potential K generates a hyper-Ka¨hler metric, the first term certainly has (on-shell) The dual hypermultiplet theory reads − 2 The D-term in the first expression is the K¨ahler potential of a hyper-Ka¨hler space, fS of S does not contribute to the potential. Its field equation (WΦ − W Φ)fS − (S + S)WΦΦfΦ = 0 relations (2.42) and (2.43) with L replacing V , derivative) under the variations where KS = ∂∂S K = − 2 WΦ−W Φ i S+S . These variations are simply obtained by inserting the presence of the superpotential Mf2W is then second duality relation (2.46) in the single-tensor off-shell variations (2.3). The field equa 2 Mf2 ηα˙ , in agreement with eqs. (2.18) and (2.46). 2 Gab(DLa)(DLb) − 4 GaDDΦa − i(Mfa)2 Ga + me 2a Φa + h.c. , Ga = Gab = is invariant under the nonlinear second supersymmetry variations i δ∗La = √2(Mfa)2(θη + θη) − √ (ηDΦ + ηDΦ), fields f a: − ihGabi(Mfb)2 + me2a = 0. In this vacuum, the kinetic metric 2hRe Gabi must be invertible and the mass matrix of the The extension to a theory with several single-tensor multiplets is straigthforward. Consider the deformed N = 2 chiral superfields The lagrangian L = Mab = − 2 hRe Ga−c1ihGbcdi(Mfd)2, controlled by the third derivatives of G. Nonlinear deformations 2 single-tensor and Maxwell multiplets to engineer theories with partial supersymmetry breaking. As illustrated by eq. (2.27), a nonlinear deformation of the single-tensor multiplet can be introduced as a spurious constant component inserted in a N = 2 superfield. In this section, we study general nonlinear deformations of these multiplets, using their Deformations of the Maxwell superfield 1 W(y, θ, θe) = X + √2i θeW − 4 θeθeDD X, Wα = −iλα + θαD − 2i (σµ σνθ)αFµν − θθ (σµ ∂µ λ)α, X = x + √2 θκ − θθ F, 1 DD X = F + √2i θσµ ∂µ κ + θθ ✷x. superfield W. Defining fermion doublets and the vector Y~ is in general a complex SU(2)R triplet. But in W, the auxiliary fields correspond to Y11 = F, Y22 = F , Y12 = − √ Y~ = Im F, Re F, √ and the SU(2)R-invariant “reality” condition ≡ Yi∗j = ǫikǫjlYkl is verified: a complex value of Y~ violating this condition cannot be seen as a background value of N = 1 superfields X or Wα. expected for goldstino fermions, should be introduced with δκα = √2(A2ǫα + Γηα) + . . . δλα = √2(B2ηα + Γǫα) + . . . breaking is in any case incompatible with the reality condition (3.7): the auxiliary fields F and D are not able to induce partial breaking with their background values; in other words, the deformation parameters cannot be absorbed in the background values of the While a chiral-chiral (CC) superfield is relevant to study deformations of the Maxwell multiplet, the single-tensor multiplet is conveniently described using a chiral-antichiral (CA) N = 2 superfield Z, with the expansion in the appropriate coordinates (y, θ, θe), Dα˙ yeµ = Deα yeµ = 0. A particular deformation e with partial supersymmetry breaking has been earlier described [eq. (2.28)] and we wish to generalize it. Since fermion fields are in the components7 of Z, the deformation parameters will add vector and the deformations are encoded in two complex numbers Ae2 and Be2 only. The nonlinear variations of the spinors are and generic values of Ae2 and Be2 break both supersymmetries. Partial breaking occurs if generate partial breaking on its own. cludes a tensor with gauge symmetry. A generic lagrangian generated by the CA superfield L = d2θeG(Z) + me2 Φ + h.c. = Llin. + Lnl . where Lnl includes all terms generated by the deformations with parameters Ae2 and Be2. In the function G(Z), a term linear in Z is irrelevant (it contributes with a derivative) and the component expansion of the lagrangian depends on the second and higher derivatives 1 G′′′(z)[f ψψ + f ϕϕ] − me2 f + h.c. Lnl = −G′′(z) Be f + Ae2 f + Ae2Be2i h 2 − 21 G′′′(z)hBe2 ψψ + Ae2 ϕϕi + h.c. and the scalar potential and the fermion bilinear terms read respectively V (z, z) = Lferm. = 2 Re G′′ Be2 G′′ + Ae2 G′′ + me 2 Re G′′ (Be G′′ + Ae2 G′′ + me 2) − Be G′′′ + h.c. G′′′ 2 2 G′′′ 2 2 2 Re G′′ (Be G′′ + Ae2 G′′ + me2) − Ae G′′′ + h.c. The kinetic metric of the multiplet is 2 Re G′′(z). Notice that these formulas do not depend on the real scalar C in L, which always leads to a flat direction. 2hRe G′′i MΦMΦ zz − 2 MΦψψ − 2 MΦ ψψ , 2hRe G′′i MΦMΦ zz − 2 MΦϕϕ − 2 MΦ ϕϕ , 2 [Re G′′(z)] f = G′′(z)Be + G′′(z)Ae + me2 − 2 1 G′′′(z)ϕϕ − 2 signs) breaks both supersymmetries, assuming that V has a ground state hzi. Constrained multiplets When supersymmetry is partially broken in the Maxwell or single-tensor (hypermultiplet) the field equation of this superfield is a constraint which allows for the elimination of the massive chiral superfield. One is then left with a nonlinear realization of N = 2 in terms of which the lagrangian is L = 1 Z 1 Z + h.c. + LF.I. + h.c. + LF.I. deformed superfield W = X + √2i θeW + θeθe B2 − 4 DD X , − 4 hFXXX i χχ − 4 hF XXX i χχ MX = 2 RehFXX i Since the auxiliary fields f and D vanish in the ground state, the mass terms of the fermion and, since the kinetic metric is RehFXX i, the mass of X is the metric of the scalar manifold), thus disproving the claim made in [25]. Expanding the field equation of X and retaining only the term in hFXXX i leads to the constraint and the constraint (4.4) is then equivalent to [16] W W − 2 XDDX + 2 B2 X = 0 , 2 = 0 . We now turn to the partial breaking in a single-tensor theory. Again, the two options chiral-antichiral superfield which induces the nonlinear deformation L = − 4 2 GΦΦ(Φ)(DL)(DL) + me2Φ + h.c. , The lowest component is the field equation for the auxiliary field f , Gzz(z) (f − Be ) = me 2 kinetic metric normalization 2 Re Gzz(hzi). can be sent to infinity keeping Gzz(hzi) finite as in the Maxwell case. In this limit, and the field equation becomes8 which does not depend on the function G and which was first given in [18]. This equation =⇒ n = 0 (n ≥ 2). The second supersymmetry variation of the constraint (4.9) is The invariance of the constraint then follows from the results (4.10). Moreover, since = −2 2 ∂µ (ησµ DLΦ) . 2 = 0 . Solutions of the constraints The solution of (4.4), and thus of (4.5), was first given in [14]. In our conventions, it is The bosonic part of lagrangian (4.2) then takes the form X = − 2B2 1 − D 4B4 + a + 4B4 a = (D2W 2 + D2W 2) , b = − D2W 2) . L|bos = 8m2B2 1 − 1 − B14 (−Fµν F µν + 2D2) − 4B1 8 (Fµν F˜µν )2 The equation of motion for D is then and, substituting back into (4.16), one arrives at [14, 16] L|bos = 8m2B2 1 − = 8m2B2 1 − D = 0 , It is also possible to add the FI term to the lagrangian (4.16). Solving the equation of motion for D then gives − B4 D2 = − ξ2 + 2 · 162m4 and substituting back to (4.16), we find that the latter takes the form L|bos = 8m2B2 1 − = 8m2B2 1 − 1 − B4 (−Fµν F µν + 2D2) − 4B8 (Fµν F˜µν )2 which means that the addition of the FI term only changes the prefactor of the Born-Infeld lagrangian included in L. or equivalently of (4.13). In our conventions, it is (DL)2(DL)2 where we have assumed that Be is real for simplicity and a = Due to the constraint (4.13), only if G has linear dependence on Z will it contribute to (3.15). However, e − 4 + h.c. = derivative . Consequently, (3.15) takes the form L = me2 = − e 4Be4 + a + 4Be4 + h.c. (4.25) The “long” super-Maxwell superfield In section 6 we will construct supersymmetric interactions of deformed or constrained single-tensor and Maxwell supermultiplets. We will find it useful to describe the Maxwell multiplet in terms of a chiral-antichiral superfield, with 16B + 16F components, as an alternative to the 8B +8F chiral-chiral superfield (3.1). In the present and technical section, To begin with, both types of superfields exist for the single-tensor multiplet. In particular, the latter can be described either by the “short” (8B + 8F ) chiral-antichiral (CA) superfield (3.11), (and its AC conjugate), or by a “long” chiral-chiral (CC) superfield [19] Zb = Y + √ are related by9 = me2Be2 1 − 1 − B4 6 Hµνρ Hµνρ + ∂µ C∂µ C − 9Be8 (ǫµνρσ Hνρσ∂µ C)2 Z = − 2 De De Zb + Moreover, using (2.10), we find L|bos = me2Be2 1 − (DL)2|bos = θ2 (υµ υµ + 2iυµ ∂µ C − ∂µ C∂µ C) , ea|bos = 4 υ2 − (∂C)2 , − B8 and the real linear superfield L is There is a gauge invariance acting on the long CC superfield. According to eqs. (5.1) Hence, Z is invariant under 9Identities in Apprendix A may help. Ωα˙ = ωα˙ + (θσµ )α˙ Vµ − θθ λα˙ . L(x, θ, θ) = Φ(x, θ, θ) − θω − θσµ θ Vµ + θθθλ − 2 θθ θσµ ∂µ ω + θθθθ ∂µ Vµ Upon defining the chiral-chiral superfield Wc = U + √ W = − 2 DeDe Wc + W = X + √2i θeαDα˙ DαΩα˙ + where W is a Maxwell (chiral-chiral) superfield (3.1). This gauge invariance eliminates superfield for the super-Maxwell theory. Wc = U + √ V = 2(L + L) other words, instead of V being simply a real superfield. This new condition follows from with a real linear L. In other words, W is invariant under 1 Y = U + √2i θeDL − θeθe 4 DD U . Eq. (5.1) indicates that this gauge variation is induced by a single-tensor supermultiplet in a “short” chiral-antichiral superfield. The long and short super-Maxwell superfields To summarize, to describe the single-tensor and the Maxwell multiplet, we have obtained and one short (8B + 8F ) superfield: Long, 16B + 16F Short, 8B + 8F Gauge variation, 8B + 8F of L. This is discussed in appendix B. Counting off-shell degrees of freedom in the “long” Maxwell multiplet is interesting. Firstly, X and U include 8B + 8F fields while the complex linear L has 12B + 12F components.10 The variation (5.16) is not the gauge transformation of the super-Maxwell theory: it which is a symmetry of Wc.11 A comparison of 2(L + L) with the standard expansion of the are respectively Aµ = −4 Re Vµ , D = −4 ∂µ Im Vµ . Replacing the scalar D by the divergence of a vector field has nontrivial consequences which integration constant appearing when solving the field equation of Im Vµ and a well-defined procedure for the elimination of Im Vµ shows that the theories formulated with either D or Im Vµ are physically equivalent. 11See appendix B. According to relation (5.12), the nonlinear deformation Wnl can be transferred to a deforterm would be Wcnl = − 2 This is the case if the deformation can be viewed as a background value of the auxiliary F in X, which never leads to partial breaking. A similar argument holds for the single-tensor superfield with relation (5.3). Then, to consider a general deformation and in particular if the interest is in partial supersymmetry breaking, the deformed short version of the superfields must be used. Since these short superfields have different chiralities, writing an interaction of two deformed supermultiplets is problematic. The Chern-Simons interaction to its counterpart with linear N = 2. Maxwell multiplets respectively, the Chern-Simons interaction with (real) coupling g can be written as a N = 1 D-term [16, 19]: LCS = −g It is invariant under the second supersymmetry variations (2.3) and (B.1) and it is also and some partial integrations: LCS = g X = The expressions (6.1) and (6.2) differ by a derivative term. The chiral form can be extended the Maxwell and single-tensor multiplets respectively [19]:12 LCS = ig 2 Z integral). This expression is also invariant under the gauge transformation (5.6) of Zb, since, for any pair of (short) Maxwell multiplets W1 and W2, 2 Z are derivative terms. Finally, one can also write the Chern-Simons lagrangian using the chiral-antichiral superfields Z (short) and Wc (long) for the single-tensor and the Maxwell multiplet respecLCS = ig 2 Z This can be verified either by direct calculation or by using relation (5.12) and partial up to a derivative term under the gauge transformation (B.13) of Wc, since, for any pair of (short) single-tensor multiplets Z1, Z2, 2 Z are derivative terms. LCS = g In components, using expansions (2.10) and (5.10), we find that (under a spacetime − g ∂µ ϕσµ ω − g ωσµ ∂µ ϕ , The Chern-Simons interaction with deformed Maxwell multiplet The nonlinearly-deformed Maxwell multiplet is described by the CC superfield W, including the deformation terms (3.8). This leads to the Chern-Simons interaction Lnl = ig 2 Z = LCS + ig 12See eqs. (3.1) and (5.2). 13Eqs. (3.11) and (5.11). Lnl = g tent. One can get around this problem by using l > 1 deformed Maxwell multiplets (namely one “long” single-tensor and at least two “short” and deformed Maxwell multiplets), as then the relevant equation of motion would take the form of a tadpole-like condition gaBa2 = 0 , a = 1, . . . , l , where ga would be the coupling of each Chern-Simons interaction. This is in agreement with the claim made in [26] and [27], namely that one cannot couple hypermultiplets to a single Maxwell multiplet in a theory with partial breaking induced by the latter. The Chern-Simons interaction (6.8) can be combined with the kinetic lagrangian Lkin. = 2 Z for the two multiplets, as well as with an FI contribution LFI = ξ d2θd2θ V2 + The theory depends then on a function H solving the Laplace equation and on an arbitrary holomorphic function F . Imposing the constraint W 2 = 0 (where W is deformed) eliminates X, which becomes a function X(W W ) of W W and its derivatives. Moreover, due to the constraint, the lagrangian no longer depends on F and it reduces to has been analyzed in [19]. The Chern-Simons interaction with deformed single-tensor multiplet In the analogous procedure for the nonlinear single-tensor multiplet, the CA superfield (3.11) with deformation (3.13) is coupled to the long Maxwell CA superfield (5.11): Lnl = ig 2 Z = LCS + ig Lnl = g L = Lnl + 2 Z 2 Z where Z is deformed and we have added an FI term for V2. Upon imposing the constraint (4.13), G does not contribute to (6.15), since 2 Z 2 d2θ D Φ + h.c. = deriv. term 1 Z and the bosonic part of (6.15) becomes Lbos = d2θeF (W)|bos + h.c. − 2ξ ∂µ Im Vµ 2 2 δ∗ iBe U = √2i Be ηD L , ever, the equation of motion of U is inconsistent as that of Y of the previous subsection — this problem can be solved by coupling the “long” Maxwell multiplet(s) to at least two “short” and deformed single-tensor multiplets.15 The complete theory has then lagrangian − 26 ǫµνρσ HνρσAµ + C ∂µ Im Vµ − Be2 Im FU · 1 − 1 − B4 6 Hµνρ Hµνρ + ∂µ C∂µ C − 9Be8 (ǫµνρσ Hνρσ∂µ C)2 ′′ is a real superfield. 15Note that there is no reason to identify the imaginary part of the auxiliary field of U with a fourform field as was done for Y in [19]. In particular, the variation of Y under the gauge transformation whose solution is 14See appendix B. analogue, the Born-Infeld lagrangian, does in ref. [19]. where Be has been assumed to be real and FU is the auxiliary field of U . Notice that the lagrangian (4.27) has acquired a field-dependent coefficient (g Re x + 2 me2)Be2 as its the equation of motion for the auxiliary field Im Vµ is ∂µ 16 Re Fxx ∂ν Im Vν + 2g C = 0 , 16 Re Fxx ∂ν Im Vν + 2g C = −λ , make the identification In this vacuum, x corresponds to a flat direction of the potential and is massless. The chiral multiplet X; the Chern-Simons coupling results in the vector multiplet W absorbing the goldstino multiplet, while X remains massless. Consequently, we observe a mechanism analogous to the super-Brout-Englert-Higgs effect without gravity [19], which is induced by the Chern-Simons coupling of the previous subsection (6.1.1). Constrained matter multiplets that couples a Maxwell to a single-tensor multiplet, where one of the two contains the goldstino. In both cases, upon imposing a nilpotent constraint on the goldstino multiplet, the Chern-Simons interaction generates a super-Brout-Englert-Higgs phenomenon without massless chiral multiplet remains in the spectrum. Here, we discuss generalisations of the nilpotent constraint in order to describe, besides the goldstino, incomplete matter multiplets of non-linear supersymmetry in which half of the degrees of freedom are integrated out of the spectrum, giving rise to constraints. The scalar potential of the theory is then whose supersymmetric vacuum is at V = < C >= 2 canonically normalized mass MC,can that C aquires is then 4 Re Fxx 2g Re x + 4m2 2 = 0. with the goldstino being part of either a nilpotent (deformed) Maxwell multiplet W with The goldstino in the Maxwell multiplet Consider the case in which the goldstino is in a deformed Maxwell multiplet W0, given W0 = X0 + √2i θeW0 + θeθe B2 − 4 X0 = −2 − DDX0 W0W1 = 0 , We now use (6.28) and the identity to solve the second of equations (6.31), which yields where h is a chiral superfield. This expression verifies the first eq. (6.31) for all h and the third eq. (6.31) if where W1 is an undeformed (and short) Maxwell multiplet given by (3.1): The constraint (6.29) then yields the following set of equations X0X1 = 0 , − 4 DD(X0X1 + X1X0) + W0W1 = 0 . (W0W1)W0α = − 2 (W0W0)W1α X1 = −4 − DDX0 + h W0W0 , h = −2 − DDX0)2 X1 = −4 − DDX0 − 2 − DDX0)2 W0W0 . One may further use the solution (4.14) for X0 and solve (6.35) to obtain X1 as a function of W0, W1 and their derivatives; the constraint (6.29) eliminates X1. dabcWbWc = 0 ; a, b, c = 1, . . . , l introduced in [28, 29] to obtain coupled DBI (Dirac-Born-Infeld) actions. In eqs. (6.36), all Wa are in general deformed with different deformation parameters Ba and the constants W0Zb = 0 , where Zb is a “long”16 single-tensor multiplet given by (5.2). Equation (6.37) then leads to X0Y = 0 , − 2 DD(X0Y + Y X0) − i W0χ = 0 , which, following the same steps as before, yield W0Wg = 0 , Y = 4i − DDX0 − 2 − DDX0)2 W0W0 , Zb + Wg , One can also check if the expression (6.39) is covariant under the gauge variation (5.6) Under (6.41), the expression (6.39) becomes Xg = −4 − DDX0 − 2 − DDX0)2 W0W0 , which, as was previously shown, is actually the consequence of leads to an overconstrained system of equations. that is the variation of (6.37) under (6.40). The expression (6.39) is thus invariant only under the reduced gauge transformations (6.40) subject to the constraint (6.43). These are not sufficient to eliminate all unphysical components of Zb. In the physically-relevant linear superfield L however, Wg disappears: − DDX0) Z0, given by which satisfies (4.13) or equivalently eq. (4.10) [18]: since Wg verifies the Bianchi identity. The goldstino in the single-tensor multiplet Now let us consider the case in which the goldstino is in a deformed single-tensor multiplet Z02 = 0 , Z0Z1 = 0 , where Z1 is an undeformed (and short) single-tensor multiplet given by (3.11) Following the same steps as before, as well as the identity (DL0DL1)Dα˙ L0 = − 2 DL0DL1 − 2 DL0DL0 , deabcZbZc = 0 ; a, b, c = 1, . . . , l , in analogy with the system (6.36), where deabc are totally symmetric constants, in order to obtain a coupled action of non-linear (deformed) single-tensor multiplets. Finally, we consider the constraint Z0Wc = 0 , as before, we obtain U = 4i − 2 2iX + DDU DL0DL0 , which eliminates U . Using the same reasoning as before, one can show that the solution (6.54) is invariant under the reduced gauge variation (5.16) Following the same procedure as for the solution of the constraint (6.37), one can use the terms of the N = 1 chiral superfield X:17 Wc + Zg , Z0Zg = 0 . This result defines L up to the addition of an arbitrary chiral field: as expected, the constraint equation (6.57) is invariant under the Maxwell gauge transformation gauge ambiguity (6.55). 17Since L is real linear, SL is complex linear for any chiral S. vanishing expectation value for the F-component of the goldstino superfield arising a priori through a non-trivial deformation that cannot be obtained by an expectation value of the auxiliary fields. supersymmetry (vectors and single-tensors that have off-shell descriptions), as well as generalisations of the nilpotent constraints describing incomplete matter multiplets. The interactions are of the Chern-Simons type and describe a super-Brout-Englert-Higgs phemultiplet. The constraints describe, in the case of a goldstino in a Maxwell multiplet, a goldstino in a linear multiplet, the constraints describe either incomplete single-tensors It would be interesting to study the interactions of the Goldstone degrees of freedom of It is not clear whether our results are sufficient to provide a description of such a system. Another open but related question is the coupling to supergravity realising partial breaking of N = 2 supersymmetry and its rigid limit. Acknowledgments J.-P. D. wishes to thank the LPTHE at UPMC, Paris and CNRS for hospitality and support. The work of J.-P. D. has been supported by the Germaine de Sta¨el franco-swiss bilateral program (project no. 2015-17). C.M. would like to thank the Albert Einstein Center for Fundamental Physics of the Institute for Theoretical Physics of the University of Bern for very warm hospitality and for financially supporting her stay there. Conventions and some useful identities The notation [. . .] in (2.1) is used for antisymmetrization with weight one. Specifically, 6 ∂µ Bνρ ± 5 permutations . {Dα, Dα˙ } = −2i(σµ )αα˙ ∂µ : As a consequence, [Dα, DD] = −4i(σµ D)α∂µ , The Maxwell field-strength chiral superfields are defined as where V is a real superfield. In addition, DD DD Y = −✷Y. DD DαL = −2 Dα˙ DαDα˙ L = 4i(σµ D)α∂µ L , where L is a complex linear superfield. More on the Maxwell supermultiplet superfields V1 and V2 with second supersymmetry variations δ∗V2 = √2i(ηD + ηD)V1. δ∗Λc = √2i ηDΛℓ. (B.3) X = Variations (B.1) imply: 4 ηαDD X + i(σµ η)α ∂µ X , 4 ηα˙ DD X − i(ησµ )α˙ ∂µ X . These are the second supersymmetry variations of the components of the “short” chiralchiral superfield (3.1): To go to the “long” Maxwell multiplet, one introduces the complex linear L with and variations (B.1) suggest to write V2 = 2(L + L), close18 and the number of off-shell fields is not a multiple of 8B + 8F . To find the complete multiplet, we rely upon the chiral-antichiral superfield written in its two forms (5.7) and (5.11): Wc = U + √ Wc = U + √ 2 θeΩ − θeθe 2i X + 41 DD U , 2 θeD L − θeθe 2i X + 41 DD U . Since the first expression is a chiral-antichiral superfield with 16B + 16F components,19 the second supersymmetry variations 18See below. supplementary components which are actually invisible in Wc: the gauge variation (B.6) a supplementary condition on the chiral X. This is where X = helps by firstly adding 4B + 4F fields to reach 24B + 24F with U and L and secondly by turning the second supersymmetry variations (B.9) into δ∗L = √ ηD V1 + √ (ηD U + ηD U ) , δ∗L = √ ηD V1 − √ (ηD U + ηD U ) , 24B + 24F fields. Since δ∗ V2 = √2i (ηD + ηD)V1 , the 16B + 16F multiplet with superfields V1 and V2 is included in the long representation. The long multiplet has two gauge variations generated by two independent single-tensor which is a short chiral-antichiral multiplet similar to eq. (5.1). This is the gauge trans[δ1∗, δ2∗] L = 2i (η1σµ η2 − η2σµ η1) ∂µ L − iΛℓ 20Verifying explicitly the closure of the algebra is relatively easy. and the multiplet is not complete without U .21 The two sets of gauge variations (B.12) remove 16B + 16F components in the long supermultiplet, to obtain the 8B + 8F physically relevant components of the super-Maxwell (1B), the two complex scalars in X (4B) and two Majorana gauginos (8F ). More on Im V as is usually the case, a component of a real superfield V , but it appears in the expansion auxiliary scalar field D in the expansion of V is replaced by the divergence of a vector field. Comparing expansion (5.10) of L with with the auxiliary scalar D, its lagrangian is quadratic in D: LD = A D2 + (B + ξ)D, In particular, A would be the gauge kinetic metric in super-Maxwell theory (hence the positivity condition). To integrate over D, it is legitimate to solve the field equation This theory does not have any symmetry and the (supersymmetric) ground state is at variables of A and B is of course given by LD = − = −V. ∂zLD = −∂z = −∂zV. quadratic lagrangian for the divergence of a vector field, L = A(∂µ Vµ )2 + (B + ξ) ∂µ Vµ , 21In this gauge, variations (B.7) hold. 23They do not depend on derivatives of fields. These scalar fields are collectively denoted by z. instead of expression (C.1). Now, the FI term is a derivative which does not contribute to the dynamical equations and the field equation for Vµ is Its solution right-hand side of the solution is not a derivative of off-shell fields. This situation is not new in the literature. Redefine Vµ = the lagrangian (C.4) becomes ∂µ Vµ = (∂µ Vµ )2 = − 24 1 F µνρσ Fµνρσ , LF = − 48 A F µνρσ Fµνρσ + (B + ξ) ǫµνρσ Fµνρσ . been studied as a potential source for a cosmological constant [30]. Another example is the massive Schwinger model [31]24 where the Maxwell lagrangian 24As also explained in ref. [30]. L = Returning to our lagrangian (C.4) and solution (C.6), if we substitute the solution field equation of z. We obtain L = − = −V Comparing with expression (C.3), equivalence is obtained if we identify the integration except if A is constant (the super-Maxwell theory has then canonical kinetic terms), in which case the second constant term in the potential is irrelevant. With this procedure, of the super-Maxwell theory. Notice that a derivative term may in general contribute to currents. The canonical Tµν = ξ [∂ν Vµ − ηµν ∂ρVρ] energy-momentum is zero, assuming the absence of boundary contributions): Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [INSPIRE]. 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Ignatios Antoniadis, Jean-Pierre Derendinger, Chrysoula Markou. Nonlinear \( \mathcal{N}=2 \) global supersymmetry, Journal of High Energy Physics, 2017, 1-38, DOI: 10.1007/JHEP06(2017)052