Taking a Vector Supermultiplet Apart: Alternative Fayet–Iliopoulos-Type Terms

Physics Letters B 781 (2018) 723–727

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Physics Letters B

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Taking a vector supermultiplet apart: Alternative Fayet–Iliopoulos-type terms

Sergei M. Kuzenko

Department of Physics M013, The University of Western Australia, 35 Stirling Highway, Crawley, W.A. 6009, Australia a r t i c l e i n f o a b s t r a c t

Article history: Starting from an Abelian N = 1vector supermultiplet V coupled to conformal supergravity, we construct Received 20 February 2018 from it a nilpotent real scalar Goldstino superfield V of the type proposed in arXiv:1702 .02423. It Accepted 24 April 2018 contains only two independent component fields, the Goldstino and the auxiliary D-field. The important Available online 27 April 2018 properties of this Goldstino superfield are: (i) it is gauge invariant; and (ii) it is super-Weyl invariant. Editor: M. Cveticˇ As a result, the gauge prepotential can be represented as V = V + V, where V contains only one independent component field, modulo gauge degrees of freedom, which is the gauge one-form. Making use of V allows us to introduce new Fayet–Iliopoulos-type terms, which differ from the one proposed in arXiv:1712 .08601 and share with the latter the property that gauged R-symmetry is not required. © 2018 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction where f is a real parameter characterising the scale of supersymmetry breaking. As U we choose the reducible chiral Goldstino ¯ In quantum field theory with a symmetry group G sponta- superfield X, Dα˙ X = 0, proposed in [7,8]. It is subject only to the neously broken to its subgroup H, the multiplet of matter fields constraint transforming according to a linear representation of G can be split 2 into two subsets: (i) the massless Goldstone fields; and (ii) the X = 0 . (1.2) other fields that are massive in general. Each subset transforms It was shown in [4] that X can be represented in the form nonlinearly with respect to G and linearly under H. Each subset may be realised in terms of constrained fields transforming ¯ 1 ¯ 2 ¯ X linearly under G [1,2]. In the case of spontaneously broken super- X = X + Y , f X := − D (), := −4 f , (1.3) 4 D¯ 2 X¯ symmetry [3], every superfield U containing the Goldstino may be split into two supermultiplets, one of which is an irreducible Gold- where the auxiliary field F of X is the only independent com- stino superfield1 and the other contains the remaining component ponent of the chiral scalar Y. Originally, the irreducible Goldstino fields [4], in accordance with the general relation between linear superfield was introduced in [9]to be a modified complex linear − 1 ¯ 2 = and nonlinear realisations of N = 1 supersymmetry [5]. It is worth superfield, 4 D f , which is nilpotent and obeys a holomor- recalling the example worked out in [4]. Consider the irreducible phic nonlinear constraint, ¯ chiral Goldstino superfield X , Dα˙ X = 0, introduced in [5,6]. It is 2 1 ¯ 2 defined to obey the constraints [6] = 0 , fDα =− D Dα . (1.4) 4 1 X 2 = 0 , f X =− X D¯ 2X¯, (1.1) These properties follow from (1.3). 4 The approach advocated in [4]may be pursued one step fur- ther with the goal to split any unconstrained superfield U into two supermultiplets, one of which is a reducible Goldstino supermul- E-mail address: sergei .kuzenko @uwa .edu .au. tiplet. This has been implemented in [10]for the reducible chiral 1 The notion of irreducible and reducible Goldstino superfields was introduced in Goldstino superfield X. There exist two other reducible Goldstino [4]. For every irreducible Goldstino superfield, the Goldstino is its only independent component. Reducible Goldstino superfields also contain auxiliary field(s) in addi- superfields: (i) the three-form variant of X [11,12]; and (ii) the tion to the Goldstino. nilpotent real scalar superfield introduced in [13]. In the present https://doi.org/10.1016/j.physletb.2018.04.051 0370-2693/© 2018 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 724 S.M. Kuzenko / Physics Letters B 781 (2018) 723–727 paper we make use of (ii) in order to split a U(1) vector supermul- By construction, it obeys the following nilpotency conditions tiplet into two constrained superfields. Our construction makes it 2 possible to introduce new Fayet–Iliopoulos-type terms, which dif- V = 0 , (2.8a) fer from the one recently proposed in [14] and share with the VD D V = 0 , (2.8b) latter the property that gauged R-symmetry is not required. A B In this paper, we make use of the simplest formulation for VDA DB DC V = 0 , (2.8c) N = 1 conformal supergravity in terms of the superspace geome- which mean that V is the Goldstino superfield introduced in [13].2 try of [15], which underlies the Wess–Zumino approach [16]to old minimal supergravity [17,18]. This approach requires the super- Associated with V is the covariantly chiral spinor Wα which is Weyl transformations of [19](defined in the appendix) to belong obtained from (2.2)by replacing V with V. As shown in [13], the to the supergravity gauge group. Our notation and conventions fol- constraints (2.8)imply that low [20]. W2W¯ 2 V := −4 . (2.9) 3 2. Constructing a Goldstino superfield (DW) Choosing V = V in (2.3)gives the Goldstino superfield action pro- Consider a massless vector supermultiplet in a conformal su- posed in [13]. pergravity background. It is described by a real scalar prepotential In order for our interpretation of V as a Goldstino superfield V defined modulo gauge transformations to be consistent, its D-field should be nowhere vanishing, which is equivalent to the requirement that DW be nowhere vanishing. As δ V = λ + λ,¯ D¯ ˙ λ = 0 . (2.1) λ α follows from (2.5), this condition implies that D2 W 2 is nowhere As usual, the prepotential is chosen to be super-Weyl inert, vanishing. To understand what the latter implies, let us introduce δσ V = 0. In what follows, we assume that the top component the component fields of the vector supermultiplet following [27] (D-field) of V is nowhere vanishing. In terms of the gauge- 1 α invariant field strength [16] Wα|=ψα , − D Wα|=D , 2 ˆ ab ˆ 1 ¯ 2 ¯ D W |=2iF = i(σ ) F , (2.10) Wα := − (D − 4R)Dα V , D ˙ Wα = 0 , (2.2) (α β) αβ αβ ab 4 β | α where the bar-projection, U , means switching off the superspace our assumption means that the real scalar DW := D Wα = ¯ ¯ α˙ Grassmann variables, and Dα˙ W is nowhere vanishing. It is instructive to consider a simple supersymmetric gauge the- ˆ 1 ¯ ¯ 1 ¯ ¯ Fab = Fab − ( aσbψ + ψσb a) + ( bσaψ + ψσa b), ory in which the above assumption is compatible with the equa- 2 2 c tions of motion. Within the new minimal formulation for N = 1 Fab =∇a V b −∇b Va − Tab V c , (2.11) supergravity [21,22], the dynamics of the massless vector super- with V = e m(x) V (x) the gauge one-form, and β the grav- multiplet with a Fayet–Iliopoulos (FI) term [23]is governed by the a a m a itino. The operator ∇ denotes a spacetime covariant derivative gauge invariant and super-Weyl invariant action (see, e.g., [24]) a with torsion, 4 2 2 ¯ 1 α ¯ 2 1 S[V ]= d xd θd θ E V D (D − 4R)Dα V − 2 fLV , c cd [∇a, ∇b] = Tab ∇c + Rabcd M , (2.12) 16 2 (2.3) where Rabcd is the curvature tensor and Tabc is the torsion tensor. where L is the conformal compensator for new minimal super- The latter is related to the gravitino by gravity [25] (and as such L is nowhere vanishing). It is a real covariantly linear scalar superfield, i ¯ ¯ Tabc =− ( aσc b − bσc a). (2.13) 2 (D¯ 2 − 4R)L = 0 , L¯ = L , (2.4) For more details, see [20,27]. We deduce from the above relations with the super-Weyl transformation δσ L = (σ + σ¯ )L. The second that term in the action is the FI term, with f a real parameter. The 1 2 2 2 αβ equation of motion for V is DW =−4 fL, and it implies that DW − D W |=D − 2F Fαβ + fermionic terms . (2.14) 4 is indeed nowhere vanishing. Since DW is nowhere vanishing, we can introduce (as an ex- We conclude that the electromagnetic field should be weak enough tension of the construction in section 5.2 of [13]) the following to satisfy scalar superfield 2 αβ D − 2F Fαβ = 0 , (2.15) 2 ¯ 2 W W 2 α = V := −4 , W := W Wα . (2.5) in addition to the condition D 0. The D-field of V is (DW )3 αβ = 1 F Fαβ 2 This superfield is gauge invariant, δλV 0, and super-Weyl invari- − DW|=D1 − 2 + fermionic terms . (2.16) 2 ant, 2 D Making use of the Goldstino superfield V leads to a new = δσ V 0 , (2.6) parametrisation for the gauge prepotential given by as follows from the super-Weyl transformation laws of Wα and DW : 2 The Goldstino superfield constrained by (2.8)contains only two independent fields, the Goldstino and the auxiliary D-field. This can be shown by analogy with 3 ¯ δσ Wα = σ Wα ,δσ DW = (σ + σ )DW . (2.7) the N = 2analysis in three dimensions [26]. 2 S.M. Kuzenko / Physics Letters B 781 (2018) 723–727 725

V = V + V . (2.17) for some integer n. The superfield (2.18) correspond to n = 1. It is obvious that Vn obeys the constraints (2.8). Moreover, Vn is super- It is V which varies under the gauge transformation (2.1), δλV = Weyl invariant. The superfields V and V coincide if the gauge λ + λ¯ , while V is gauge invariant by construction. Modulo purely n gauge degrees of freedom, V contains only one independent field, prepotential V is chosen to be V. New FI-type invariants are obtained by making use of Vn instead of V in (2.25), which is the gauge one-form. There exists a different way to construct a reducible Goldstino (n) = d4xd2 d2 ¯ E (2.27) superfield in terms of V , which is given by IFI θ θ ϒVn .

W 2 W¯ 2 Vˆ := −4 DW . (2.18) 3. U(1) duality invariant models and BI-type terms D2 W 2D¯ 2 W¯ 2 Unlike V defined by (2.5), this gauge-invariant superfield is not Ref. [27]presented a general family of U(1) duality invariant manifestly super-Weyl invariant. Nevertheless, it proves to be in- models for a massless vector supermultiplet coupled to off-shell variant under the super-Weyl transformations, supergravity, old minimal or new minimal. Such a theory is described by a super-Weyl invariant action of the form ˆ δσ V = 0 , (2.19) 1 4 2 2 SSDVM[V ; ϒ]= d xd θ E W as follows from the observation [28](see also [27]) that 2 2 2 ¯ 2 ¯ W 1 4 2 2 ¯ W W ω ω D2 − 4R¯ (2.20) + d xd θd θ E , . ϒ2 4 ϒ2 ϒ2 ϒ2 is super-Weyl invariant for any compensating (nowhere vanishing) (3.1) real scalar ϒ with the super-Weyl transformation law E := 1 D2 2 ¯ Here is the chiral density, ω 8 W , and (ω, ω) is areal = + ¯ analytic function satisfying the differential equation [29,30] δσ ϒ (σ σ )ϒ . (2.21) ∂(ω ) In the new minimal supergravity, we can identify Im − ω¯ 2 = 0 , := . (3.2) ∂ω ϒ = L . (2.22a) These self-dual dynamical systems are curved-superspace exten- sions of the globally supersymmetric systems introduced in [29, In the case of the old minimal formulation for N = 1 supergravity 30]. The curved superspace extension [28], SSBI[V ], of the super- [16–18], we choose symmetric Born–Infeld action [28,31] corresponds to the choice ¯ ¯ ϒ = , Dα˙ = 0 , (2.22b) g2 (ω, ω¯ ) = , A = g2(ω + ω¯ ), where the chiral compensator has the super-Weyl transforma- + 1 + + + 1 2 1 2 A 1 A 4 B tion law δσ = σ . By construction, the superfield (2.18)obeys the nilpotency con- B = g2(ω − ω¯ ), (3.3) ditions (2.8). It may be shown that the composites (2.5) and (2.18) with g a coupling constant. coincide if V is chosen to be V or Vˆ . Thus the two Goldstino su- In flat superspace (which, in particular, corresponds to ϒ = 1), perfields V or Vˆ differ only in the presence of a gauge field. It the fermionic sector of (3.1)was shown [27]to possess quite re- follows from the definition (2.18) that Vˆ is well defined provided markable properties. Specifically, only under the additional restric- the condition (2.15) holds. The same definition tells us that the tion D-field of Vˆ is equal to = 3 1 uu¯ (0, 0) 3 (0, 0), (3.4) − DWˆ |=D + fermionic terms . (2.23) 2 the component fermionic action proves to coincide, modulo a non- The composite (2.18)was introduced in a recent paper [14]. The linear field redefinition, with the Volkov–Akulov action [3]. This authors of [14]put forward the supersymmetric invariant ubiquitous appearance of the Volkov–Akulov action in such models was explained in [32]. If the FI term is added to the flat-superspace ˆ 4 2 2 ¯ ˆ counterpart of (3.1), then the auxiliary scalar D develops a non- IFI = d xd θd θ E ϒV (2.24) vanishing expectation value, in general, for its algebraic equation of motion has a non-zero solution. As a result, the supersymme- as a novel FI term that does not require gauged R-symmetry. The try becomes spontaneously broken, and thus the photino action compensating superfield ϒ was chosen in [14]to be the old mini- should be related to the Goldstino action, due to the uniqueness of mal expression (2.22b). the latter. We propose an alternative FI-type invariant In supergravity, the situation is analogous to the rigid supersymmetric case. Let us add a standard FI term to the vector multi- I = d4xd2θd2θ¯ E ϒV . (2.25) FI plet action (3.1) coupled to new minimal supergravity, It also does not require gauged R-symmetry. In addition, it does 3 S = d4xd2θd2θ¯ ELlnL + S [V ; L] not require (2.15). 2 SDVM κ Actually, the above constructions can be generalised by intro- 4 2 2 ¯ ducing a gauge-invariant Goldstino superfield of the form − 2 f d xd θd θ ELV . (3.5)

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