Supertranslations to all orders

The transformation laws of the general linear superfield and chiral superfields under N=1 supertranslations are tabulated to all orders in the supertranslation parameters.


Introduction
Quantum field theories with exact correspondences between bosonic and fermionic helicity states are not only basic ingredients for superstring theories, but have dominated both theoretical investigations and experimental searches for particle physics beyond the current "Standard Model" of particle physics for over three decades now. The minimal version of supersymmetric extensions of the Standard Model extends the generators M µν , p µ of the Poincaré group by a set of fermionic generators Q α and Qα in the ( 1 2 , 0) and (0, 1 2 ) representations of the proper orthochronous Lorentz group in four dimensions. It has been recognized early on that this extension of the Poincaré algebra to a super-Poincaré algebra can be represented linearly (and in a reducible, but not fully reducible manner) on a set comprising 4 complex spin-0 fields, 4 Weyl spinors, and one complex spin-1 field. This set constitutes the so called general linear multiplet or general linear superfield V , and its irreducible subsets had also been identified. Indeed, it is entirely sufficient to know the action of the supertranslation generators Q α and Qα on the components of V , or equivalently the action of the supertranslation exp[i(ζ ·Q+Q·ζ)] to first order in the parameters ζ α , ζα, to construct supersymmetric action principles and the related supercurrents. Therefore the first order transformation laws for the components of V can be found in many books and review articles on supersymmetry, and with our current understanding this is all what is needed to discuss the physical implications of supersymmetry. However, from a mathematical point of view it seems desirable to have the full transformation properties of the general linear multiplet readily available for reference. To provide such a reference is the purpose of this paper. To make these results also easily accessible for beginners in supersymmetry, the super-Poincaré algebra and the basic techniques of superspace calculations, as they pertain to the derivation of the transformation properties of the component fields, are also briefly reviewed 1 . Therefore the outline of the paper is as follows. Conventions for spinor representations of the Lorentz group and the super-Poincaré algebra are discussed in section 2. Superspace is reviewed in section 3, and the full supertranslation properties of the component fields of the general linear multiplet are reported in section 4. Chiral superfields provide a particular irreducible representation within the reducible linear multiplet. Due to their practical relevance for the supersymmetrization of matter fields, the resulting full supertranslation properties of the components of chiral superfields are listed in section 5. Appendix 1 contains a translation into the conventions of Wess and Bagger [1]. The relevant spinor identities are reviewed in appendix 2. The conventions used here differ from Wess and Bagger only with regard to the definition of superderivatives and the definition of the 2nd order epsilon spinors with lower indices. Sections 2 and 3 are included to make the paper self-contained and easily accessible, and to clarify conventions. However, the results in sections 4 and 5 are not affected by the different definitions. The cognoscenti should therefore go straight to section 4 and refer to the earlier sections only if the need arises. There is a further reason besides accessibility, why the larger part of this paper is an introductory review rather than an original research contribution. While (to my best knowledge) the full supertranslation laws of the component fields have not been published before, the methodology for calculations with linear super-multiplets in four dimensions was developed some 35 years ago by Wess, Zumino, Salam, Strathdee, and Ferrara [2,3,4,5]. The full supertranslation laws should prove useful for the further development of supersymmetry and its applications, but tabulating those transformation laws primarily closes a gap in the literature.

The super-Poincaré algebra
We use η 00 = −1 for the Minkowski metric and standard notation σ µαα with for the Pauli matrices.
1 Every serious student of supersymmetry will still have to consult Wess and Bagger [1] to learn the foundations of the subject.

Complex conjugation turns undotted indices into dotted indices and vice versa,
and hermiticity of the Pauli matrices implies for the complex conjugate matrices We pull spinor indices with the two-dimensional epsilon spinors The equations (1) then imply that the conjugate Pauli matrices with upper spinor indices are Numerically we have with the upper index positions for the barred matrices and lower index positions for the unbarred matrices, Although not formally required, use of upper indices for barred Pauli matrices and lower indices for unbarred Pauli matrices is a useful and very common convention.
Relations for Pauli matrices are meticulously compiled in Ref. [1]. For convenience, we recall those relations which are directly relevant for the derivation of supertranslations to all orders, and The factor ǫ 0123 = ±1 was included to allow for ready use of both conventions for the four-dimensional ǫ-tensor.
We will briefly recall below that pulling spinor indices with the 2nd order epsilon spinors is motivated by the fact that this yields Lorentz invariant spinor products where the anti-commutation property of spinors was used. Conjugation also implies re-ordering of spinor quantities, such that conjugation of equation (10) yields The vector representation matrices of the Lorentz algebra appear as structure constants in the Poincaré algebra. The spinor representations of a proper orthochronous Lorentz transformation The following relations hold, These relations are used in the derivation of supersymmetric Maxwell or Yang-Mills actions.
We can now write down the super-Poincaré algebra in the form implies that equation (15) can also be written as The super-Poincaré algebra satisfies all the pertinent super-Jacobi identities as a consequence of the representation properties of the vector and spinor representations of the Lorentz algebra. The particular super-Jacobi identity holds as a consequence of the fact that the Pauli matrices have the same form in every inertial frame, This can be verified from equations (7) by commuting the σ λ matrices into the middle positions in the products on the right hand side. It can also be verified as a direct consequence of equation (9).

N = 1 Superspace
The Poincaré algebra is realized on spacetime coordinates x µ through derivative operators In a nutshell, superspace is based on the observation that this construction can be extended to the super-Poincaré algebra by supplementing Minkowski spacetime with fermionic coordinates coordinates θ α and θα and corresponding fermionic derivatives The super-Poincaré algebra is then realized on the superspace coordinates (x µ , θ α , θα) by amending the representations (17) of the bosonic operators with the realizations for the fermionic operators, and complementing the Lorentz generators to include the action on Q α and Qα, The commutation relations and therefore We can calculate the transformation properties of the component fields by comparing with the expansion of the right hand side of equation (22) with respect to the fermionic variables θ α and θα.

Supertranslations of the general linear multiplet
Supertranslations shift the argument x of component fields to We can calulate the transformation properties of the components of the supermultiplet V to all orders in the transformation parameters ζ, ζ, by expanding the right hand side of equation (22) in all orders in θ and θ. The first step requires the expansion of component fields with respect to the coordinate shift ∆x µ = i ζ · σ µ · θ − θ · σ µ · ζ , e.g.
and corresponding expansions for combinations of the other eight component fields with various factors, which are different in each case due to the presence of fermionic variables in the extra factors. Altogether, this includes 35 more relations, e.g.
Substitution of all the expansions in terms of standard words in the Grassmann variables into equation (22) For conversion of the last equation into standard "words" in the Grassmann variables ζ and ζ, note that from equation (9) The remaining transformation equations are These transformation laws are compatible with the reality constraints V (x, θ, θ) = V + (x, θ, θ) which define the vector multiplet,

Supertranslations of the chiral multiplet
Besides the superderivatives (19,20) one can also define the supercovariant derivatives [4,5] such that and The condition for chiral superfields is therefore invariant under supertranslations. The basic solutions The equation The chiral superfield corresponds to the following substitutions in the general superfield V , It is clear from the construction, but can also be checked explicitly that these constraints are compatible with the transformation properties (25-33) of the full linear multiplet. We find the following supertranslations of the components of the chiral multiplet, ψ ′ (x) = 2iσ µ · ζ∂ µ φ(x) + ζζ 2 ∂ 2 φ(x) + ψ(x) Please note that this presentation does not involve the usual rescaling of the spinor component in the chiral superfield, which is required for canonically normalized kinetic terms.

Conclusions
The supertranslation properties of the component fields of a general linear supermultiplet and of a chiral multiplet were reported to all orders in the translation parameters ζ and ζ in equations (25-33) and equations (36-38), respectively. On the one hand, one can think of these results as explicit parametrizations of orbits of supertranslations in the space of component fields of a supersymmetric theory. On the other hand, one can consider the transformed fields again as superfields in variables (x, ζ, ζ), because e.g.