Physics Letters B 781 (2018) 723–727 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb 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 supersym- metry 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 sub- set 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 VDA DB V = 0 , (2.8b) latter the property that gauged R-symmetry is not required. 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) D 3 2. Constructing a Goldstino superfield ( W) 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. 1 1 It is instructive to consider a simple supersymmetric gauge the- ˆ ¯ ¯ ¯ ¯ 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.