Relating the Komargodski-Seiberg and Akulov-Volkov actions: Exact nonlinear field redefinition
aa r X i v : . [ h e p - t h ] O c t September, 2010
Relating the Komargodski-Seiberg and Akulov-Volkov actions:Exact nonlinear field redefinition
Sergei M. Kuzenko and Simon J. Tyler School of Physics M013, The University of Western Australia35 Stirling Highway, Crawley W.A. 6009, Australia
Abstract
This paper constructs an exact field redefinition that maps the Akulov-Volkovaction to that recently studied by Komargodski and Seiberg in arXiv:0907.2441. It isalso shown that the approach advocated in arXiv:1003.4143v2 and arXiv:1009.2166for deriving such a relationship is inconsistent. [email protected] [email protected] Introduction
The Akulov-Volkov (AV) action [1] is the second oldest supersymmetric theory in fourspace-time dimensions. It describes the low-energy dynamics of a massless Nambu-Goldstone fermionic particle which is associated with the spontaneous breaking of rigidsupersymmetry and is called the Goldstino (see [2] for a nice review of the AV model andrelated concepts). According to the general theory of the nonlinear realization of N = 1supersymmetry [1, 3, 4, 5, 6], the AV action is universal in the sense that any Goldstinomodel should be related to the AV action by a nonlinear field redefinition. Various Gold-stino models can be interesting in their own right, in particular, those models which arerealized in terms of constrained superfields. Over the years, there have appeared a numberof such superfield actions with spontaneously broken supersymmetry [11, 12, 13, 14].Recently there has been renewed interest in Goldstino couplings inspired by the work ofKomargodski and Seiberg [15]. They put forward the Goldstino model that had actuallyappeared in the literature twenty years earlier [14]. The novelty of the Komargodski-Seiberg (KS) approach is that they related the Goldstino dynamics to the superconformalanomaly multiplet X corresponding to the Ferrara-Zumino supercurrent [16]. Under therenormalization group flow, the multiplet of anomalies X defined in the UV turns out toflow in the IR to a chiral superfield X NL (obeying the constraint X NL = 0, of the typefirst introduced by Roˇcek [11]) which contains the Goldstino as a component field. For a N = 2 generalization of the KS formalism, see [17].The action derived in [14, 15] has a particularly simple form both in superspace andwhen reduced to components. However, its direct relation to the AV action and thus thestructure of its nonlinearly realized supersymmetry have not yet been studied.In [18] it was shown, using the general method of [4, 5], that the Goldstino actionintroduced by Samuel and Wess [13] can be derived from the constrained superfield for-malism of Komargodski and Seiberg. The former model is known to be equivalent to theAV theory [13].What we provide in this paper is a direct relation between the AV action and thatof Komargodski-Seiberg. Unlike [18], we do not make use of the techniques developedin [4, 5]. Instead we follow the approach pursued in [10] which can also be applied tostudy the fermionic sector of supersymmetric Euler-Heisenberg-type actions. We use thetwo-component notation and conventions adopted in [2, 19]. For instance, the fermionic sector of the N = 1 supersymmetric Born-Infeld action [7, 8] is a newGoldstino model. It has been shown to be related to the AV action by a nonlinear field redefinition [9, 10]. Setup and results
The AV action [1] is S AV [ λ, ¯ λ ] = 12 κ Z d x (cid:16) − det Ξ (cid:17) , (1)where κ denotes the dimensionful coupling constant andΞ ab = δ ab + κ ( v + ¯ v ) a b , v ab = i λσ b ∂ a ¯ λ , ¯ v ab = − i ∂ a λσ b ¯ λ . (2)By construction, S AV is invariant under the nonlinear supersymmetry transformations δ ξ λ α = 1 κ ξ α − i κ (cid:0) λσ a ¯ ξ − ξσ a ¯ λ (cid:1) ∂ a λ α . (3)Expanding out the determinant in (1) and denoting the trace of a matrix M = ( M ab )with Lorentz indices as h M i = tr( M ) = M aa yields S AV [ λ, ¯ λ ] = − Z d x h v + ¯ v i + 2 κ (cid:16) h v i h ¯ v i − h v ¯ v i (cid:17) + κ (cid:16) (cid:10) v ¯ v (cid:11) − h v i h v ¯ v i − (cid:10) v (cid:11) h ¯ v i + 12 h v i h ¯ v i + c . c . (cid:17)! . (4)As demonstrated in [10], the 8th-order terms vanish.The Goldstino action constructed in [14, 15] is S KS [ ψ, ¯ ψ ] = − Z d x (cid:16) h u + ¯ u i + 12 f ∂ a ¯ ψ ∂ a ψ + 18 f ψ ¯ ψ ∂ ψ ∂ ¯ ψ (cid:17) , (5)where we defined u ab = i ψσ b ∂ a ¯ ψ and its complex conjugate. In the following section, wefind that the constant f is related to κ via 2 f = κ − .Below, we find that the nonlinear field redefinition which maps the action (1) to theaction (5), i.e. S AV [ λ α ( ψ, ¯ ψ ) , λ α ( ψ, ¯ ψ )] = S KS [ ψ, ¯ ψ ], can be chosen to be λ α ( ψ, ¯ ψ ) = ψ α − i κ σ a ¯ ψ ) α ( ∂ a ψ ) − κ ψ α (cid:16) h u ¯ u i − h u i h ¯ u i + 12 (cid:10) ¯ u (cid:11) − ∂ a ψ ∂ a ¯ ψ + 14 ¯ ψ (cid:3) ψ (cid:17) + κ ψ α (cid:0) (cid:10) u ¯ u (cid:11) + 32 h u ¯ u i h ¯ u i + 34 h u i (cid:10) ¯ u (cid:11) (cid:1) . (6)The inverse field redefinition is ψ α ( λ, ¯ λ ) = λ α + i κ σ a ¯ λ ) α ( ∂ a λ ) (cid:16) κ h ¯ v i (cid:17) + κ λ α (cid:0) h v ¯ v i − (cid:10) ¯ v (cid:11) − h ¯ v i + 12 ∂ a λ ∂ a ¯ λ + 34 ¯ λ (cid:3) λ (cid:1) (7) − κ λ α (cid:0) (cid:10) v ¯ v (cid:11) + 12 h v ¯ v i h ¯ v i − h v i (cid:10) ¯ v (cid:11) − h v i h ¯ v i + 34 h ¯ v i ∂ a λ ∂ a ¯ λ (cid:1) . Deriving the nonlinear field redefinition
In this section, we sketch the derivation of (6). A more detailed presentation of ourmethod will be given in a separate publication.Our goal is to find a nonlinear field redefinition λ α → λ α ( ψ, ¯ ψ ) = ψ α + O ( κ ) thatsatisfies S AV [ λ ( ψ, ¯ ψ ) , ¯ λ ( ψ, ¯ ψ )] ≡ ˜ S AV [ ψ, ¯ ψ ] = S KS [ ψ, ¯ ψ ] . (8)Since both actions S AV [ λ, ¯ λ ] and S KS [ ψ, ¯ ψ ] are invariant under R -symmetry, the nonlineartransformation we are looking for must be covariant under R -symmetry. The most generalfield transformation of this type is λ α ( ψ, ¯ ψ ) = ψ α + κ ψ α h α u + α ¯ u i + i κ ( σ a ¯ ψ ) α ( ∂ a ψ ) (cid:16) α + κ h β u + β ¯ u i (cid:17) (9)+ κ ψ α (cid:0) β h u ¯ u i + β h u i h ¯ u i + β (cid:10) ¯ u (cid:11) + β h ¯ u i + β ∂ a ψ ∂ a ¯ ψ + β ¯ ψ (cid:3) ψ (cid:1) + κ ψ α (cid:0) γ (cid:10) u ¯ u (cid:11) + γ h u ¯ u i h ¯ u i + γ h u i (cid:10) ¯ u (cid:11) + γ h u i h ¯ u i + γ h ¯ u i ∂ a ψ ∂ a ¯ ψ (cid:1) . This is equivalent to the field redefinition used in [10] up to some 7-fermion identities.The general field redefinition at O ( κ ) acts on the AV action to give˜ S AV = − Z d x n h u + ¯ u i + κ (cid:16) (cid:0) h u i − (cid:10) u (cid:11) + c . c . (cid:1) + (cid:0) ( α + α ) h u i + c . c . (cid:1) − (cid:0) α (cid:10) u (cid:11) + c . c . (cid:1) + 2Re( α ) h u i h ¯ u i − Re( α ) ∂ a ψ ∂ a ¯ ψ (cid:17) + O ( κ ) o , (10)where we have rewritten all terms in the minimal basis (cid:10) u (cid:11) , (cid:10) ¯ u (cid:11) , h u i h ¯ u i , h u i , h ¯ u i , ∂ a ψ ∂ a ¯ ψ . (11)Obviously, if we are to match S KS to this order we need α = 0 , Re( α ) = 0 , α = − , f = κ − . (12)The imaginary part of α , which we will denote as α i2 , is not fixed at this order.The effect of (9) with (12) on the AV action at O ( κ ) can be similarly analysed. If wesplit all coefficients into their real and imaginary parts, β j = β r j + i β i j , then the restrictionson the β j can be written as β r1 = 4 β r6 + 2 β r8 , β i1 = 2 α i2 + 4 β i6 − β i8 , β r3 = − (cid:0) β r6 (cid:1) , β i3 = − α i2 + β i6 ) ,β r2 = 32 − β r4 + 4 β r6 − β r7 − β r8 , β i2 = − α i2 + β i4 − β i7 + β i8 , (13) β r5 = 12 + 2 β r6 + β r8 , β i5 = α i2 + 2 β i6 − β i8 , β r6 = − (cid:0) α i2 ) (cid:1) . β r4 , β i4 , β i6 , β r7 , β i7 , β r8 and β i8 are not fixed at this order.A similar analysis is performed at O ( κ ) and we find that to match ˜ S AV to S KS weneed γ = 1 , γ r2 = − α i2 (cid:0) α i2 + 2 β i6 − β i8 (cid:1) + 2( β r7 + β r8 + γ r5 ) ,γ r3 = − α i2 (cid:0) α i2 + 2 β i6 + β i7 − β i8 (cid:1) + 2 β r4 + 3 β r8 ,γ i3 = − α i2 (cid:0) ( α i2 ) + − β r7 − β r8 (cid:1) − β i4 − β i6 − β i8 ,γ r4 = − α i2 (cid:0) α i2 + β i4 + 2 β i6 + β i7 + β i8 (cid:1) + (cid:0) β r4 − β r7 + β r8 (cid:1) . (14)The free parameters at this order are γ i2 , γ i4 , γ i5 and γ r5 , the first three of which havecompletely dropped out the calculation.It can be shown that all the free parameters can be accounted for by the symmetriesof either one of the two actions. In particular γ i2 , γ i4 and γ i5 correspond to single termtrivial symmetries of any Goldstino action.From the above results we see that out of the original 32 real parameters in thenonlinear field redefinition, 12 remain unfixed by the requirement that ˜ S AV = S KS . Sincethese freedoms may be recovered by a symmetry transformation of either action, we maysimply set all free parameters to zero and get the field redefinition (6).Some results of this section were obtained with computer assistance [20]. The core ofthe computer program is the generation of a canonical form for expressions involvingspinors, which is necessary for comparing expressions. All Fierz-type identities wereautomatically satisfied by choosing a representation for the Pauli matrices and defininga definite ordering for spinors and their derivatives. Total derivatives, where relevant,were removed from expressions by generating a set of replacement rules that performedthe appropriate integration by parts to yield a unique form for the expression. Furtherdetails of the algorithm will be given in a separate publication. It has been pointed out, e.g. [14, 18], that S KS does not have definite transformationproperties under the supersymmetry transformation (3). But now that we have an explicitmapping from S AV to S KS we can use it to find the supersymmetry transformation under4hich S KS is invariant. We get δ ξ ψ α = δ ξ ψ α ( λ, ¯ λ ) = δ ξ λ β · δδλ β ψ α ( λ, ¯ λ ) + δ ξ ¯ λ ˙ β · δδ ¯ λ ˙ β ψ α ( λ, ¯ λ ) (cid:12)(cid:12)(cid:12) λ = λ ( ψ, ¯ ψ ) (15)= 1 κ ξ α − i κ (cid:18)(cid:0) ψσ a ¯ ξ − ξσ a ¯ ψ (cid:1) ∂ a ψ α − ( σ a ¯ ψ ) α ∂ a ( ξψ ) −
12 ( σ a ¯ ξ ) α ∂ a ψ (cid:19) + O ( κ ) . Finally, we would like to comment on the field redefinition found by Zheltukhin [21,22]. In these papers, written in the four-component spinor notation, the required fieldredefinition was sought in the form ψ ( λ ) = λ + κ χ ( λ ) + O ( κ ). By requiring that S KS [ ψ ( λ )] = S AV [ λ ] a solution was found for χ . The key step in the O ( κ ) calculationreported in [21] is the factorisation( ∂ m ¯ λ ) (cid:0) γ m χ + ζ m ( λ ) (cid:1) = 0 , (16)where we have introduced ζ m ( λ ) = i2 (cid:0) λ ( ¯ λ ,m λ ) + γ λ ( ¯ λ ,m γ λ ) (cid:1) − i4 (cid:0) γ m λ ( ¯ λ ,n γ n λ ) − γ n λ ( ¯ λ ,n γ m λ ) (cid:1) (17)and denoted by ¯ λ ,m the derivative of ¯ λ with respect to x m . In [21, 22], it was then inferredfrom (16) that γ m χ + ζ m ( λ ) = 0 . (18)Unfortunately, the 16 equations (18) for the 4 components of χ are inconsistent. Thiscan be seen by taking a time- or space-like vector p m and contracting both sides of (18)with ( p n γ n ) − p m = − p − ( p n γ n ) p m . Then, the first term in the relation obtained will be p -independent, while the second remains p -dependent. Acknowledgements:
SJT is grateful to Ian McArthur, Paul Abbott and Joseph Novak for useful discussions.SMK acknowledges email correspondence with Alexander Zheltukhin. The work of SMKis supported in part by the Australian Research Council.
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