Superfield Continuous Spin Equations of Motion

Physics Letters B 793 (2019) 445–450

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Physics Letters B

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Superﬁeld continuous spin equations of motion ∗ I.L. Buchbinder a,b, S. James Gates Jr. c, K. Koutrolikos c, a Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk 634041, Russia b National Research Tomsk State University, Tomsk 634050, Russia c Department of Physics, Brown University, Box 1843, 182 Hope Street, Barus & Holley 545, Providence, RI 02912, USA a r t i c l e i n f o a b s t r a c t

Article history: We propose a description of manifestly supersymmetric continuous spin representations in 4D, N = 1 Received 26 March 2019 Minkowski superspace at the level of equations of motions. The usual continuous spin wave function Accepted 9 May 2019 is promoted to a chiral or a complex linear superﬁeld which includes both the single-valued (span Available online 15 May 2019 integer helicities) and the double-valued (span half-integer helicities) representations thus making Editor: M. Cveticˇ their connection under supersymmetry manifest. The set of proposed superspace constraints for both Keywords: superﬁeld generate the expected Wigner’s conditions for both representations. Supersymmetry © 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license Continuous spin (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction they were never observed. The rest, fall in the category of continuous spin representations (CSR) [2–12]. This type of representations It is generally accepted that the elementary excitation in any is massless (vanishing C1) and is characterized by a non vanishing 2 fundamental theory are classified according to the symmetries of eigenvalue of the second Casimir C2 = μ , where μ is a real, con- the vacuum. It follows that the free elementary particles are as- tinuous parameter with dimensions of mass. There are two such sociated with the unitary and irreducible representations of rel- representations, the single-valued one and the double-valued. The evant spacetime symmetry groups. For this reason, special atten- size of both of them is countable infinite and their spectrum in- tion is paid to the consideration of maximally symmetric space- cludes all integer separated integer or half integer helicity states times (Minkowski, de Sitter and anti-de Sitter). In the case of 4D respectively with multiplicity one.1 Minkowski spacetime, Wigner [1], classified all such representa- The infinite number of degrees of freedom per spacetime point tions. The one particle representations are labeled by the mass was the reason why Wigner rejected the use of such representa- and spin quantum numbers, which correspond to the eigenvalues tions, claiming that the heat capacity of a gas of continuous spin m of the two invariant Casimir operators (quadratic) C1 = P Pm and particles is infinite [13]. Further attempts to relate these repre- C = m (quartic) 2 W Wm, where Pm and Wm are the momentum and sentations with physical systems have also failed. In Refs. [6,7]it Pauli-Lubanski vectors respectively. The one particle states inside was shown that the free field description of these representations the representations are labeled by the eigenvalues of the corre- breaks causality or locality thus making impossible to construct sponding Cartan subalgebra (like the spin/helicity in the direction a consistent quantum field theoretic description. Not surprising, of motion). these representations have been ignored. Yet, the same two exotic Some of these representations appear in local field theories and properties (presence of a continuous dimensionfull parameter and string theories. These are the familiar finite size representations an infinite tower of massless helicities) are very appealing from the that describe massless particles with fixed integer or half-integer point of view of higher spin (gravity) theories. Consistent interact- helicity and massive particles with integer or half-integer spin. ing higher spin theories require the presence of infinite massless Asubset of them has been observed in nature. Other represen- particles with arbitrary high helicities [14] and a dimensionfull pa- tations are the tachyonic particles which are characterized by negative eigenvalues of C1. Their presence indicates instabilities and 1 An alternative terminology for such representations is “infinite spin”. This is a more appropriate name because it captures the essence of the spectrum of these * Corresponding author. representations (spin is not bounded) and avoids the use of the misleading term E-mail addresses: [email protected] (I.L. Buchbinder), “continuous spin” (the helicity of the states is not continuous). Nevertheless, for [email protected] (S.J. Gates), [email protected] historical reasons continuous spin is the prevailed nomenclature and that is the (K. Koutrolikos). one we will use. https://doi.org/10.1016/j.physletb.2019.05.015 0370-2693/© 2019 The Author(s). 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. 446 I.L. Buchbinder et al. / Physics Letters B 793 (2019) 445–450

rameter to weight the higher number of derivatives required by [Jmn, Jrs]=iηmr Jns − iηmsJnr + iηnsJmr − iηnr Jms , (1) the interactions. This dimensionfull parameter is usually identified [J , P ]=iη P − iη P , with the radius of (A)dS spacetime. CSR naturally provide both mn r mr n nr m these features, hence in principle it can be seen as a good can- [Pm, Pn]=0 . didate model for interacting higher spins in flat spacetime and even possibly bypassing some of the no-go theorems [12]. For this The stabilizer subgroup (little group) is generated by the trans- m = reason, recently there has been an increased interest in the study lations generator Pm and the Pauli-Lubanski vector Wm, W 1 mnrs of CSR [15–36] investigating various kinematic (covariant on-shell, 2 ε Jnr Ps with the algebra off-shell descriptions) and dynamic (interactions, scattering ampli- [ ]= [ m n]= mnrs m = tudes) properties (for a review see [27]). Wm, Pn 0 , W , W iε Wr Ps , W Pm 0 , (2) The object of this work is to study the continuous spin rep- [Jmn, Wr ]=iηmr Wn − iηnr Wm . resentations in the presence of supersymmetry. In Ref. [8]it was shown that the supersymmetry charges are compatible (commute) It is useful to keep in mind that when we consider for realiza- with the transverse vector generators of iso(3, 1): i , i = 1, 2 tions of representations of the above algebra in terms of fields which define the CSR. Hence, it is expected that the single-valued (wavefunctions) the abstract generators Jmn , Pm are replaced by CSR can be combined with the double-valued CSR in order to as- the corresponding operators Jmn =−ix[m∂n] − iMmn, Pm =−i∂m semble supersymmetric continuous spin representation (sCSR) that that generate the group action in the space of fields. Notice that include both integer and half-integer helicities. In this letter we Wm, due to its structure, does not depend on the “orbital” part aim to find a covariant, on-shell description of sCSR that makes (x[mPn]) of Jmn and only the “intrinsic” spin (Mmn) part sur- 2 2 the supersymmetry manifest. For that we use the 4D Minkowski, vives. The Casimirs of (1)are C1 = P and C2 = W and their N = 1superspace formulation. We find that the usual Wigner eigenvalues label the representations. Additionally, Pm is one of wavefunction is elevated to a chiral or complex linear superfield. the Cartan subalgebra generators hence the one particle states Its bosonic component corresponds to two single valued CSR and have at least one additional label pm, the eigenvalue of Pm. The its fermionic component corresponds to one double-valued CSR. eigenvalue (label) pm is restricted to the finite set of representa- We propose a set of covariant superspace constraints that such a tive momenta km, since all other possible momenta are generated wavefunction must satisfy on-shell in order to describe a sCSR as by the action of various group elements (boosts). The different this is defined by the super-Poincaré algebra. By projecting into kinds of representative momenta are that of a massive particle components we show that these constraints give back Wigner’s at rest km = (−m, 0, 0, 0) (timelike), that of a massless particle equations for the single and double valued CSR as expected. During km = (−E, 0, 0, E) (lightlike) or that of a tachyonic particle km = the time of writing, new work appeared [37] which demonstrates (m, 0, 0, 0) (spacelike). the connection of the single valued CSR with the double valued For massless (C1 = 0) particles, Pm will be identified with pm = CSR under on-shell supersymmetry. The authors showed that on- (−E, 0, 0, E). The orthogonality of Wm with Pm fixes the structure shell supersymmetry transformations map one CSR to the other. to the Pauli-Lubanski vector to be: Our results generalize this for off-shell supersymmetry transformation since the superfields provide the necessary auxiliary fields to Wm = pm W + m (3) close the algebra of the transformations without the use of equa- = = tions. where m is the transverse vector m (0, 1, 2, 0) with 1 The plan of the paper is as follows. In section 2, we review E(J23 + J20), 2 =−E(J13 + J10) and W = J12. According to (2) the group theoretical description of CSR using the method of these elements satisfy the algebra the eigenvalues of the Casimirs of the Poincaré algebra. Then we review the discussion of massless representations of the super- [1, 2]=0 , i[1, W ]=2 , i[2, W ]=−1 (4) Poincaré algebra and we extend the discussion to the definition of This is the algebra of group E2 and describes the symmetries of sCSR. In section 3, we consider the superfield description of such representations and derive the superspace, covariant, equations of the two dimensional euclidean plane perpendicular to the motion 4 motion it must satisfy. In section 4, we discuss the component pro- of the massless particle. Instead of i one can use the equivalent ± = ± jection and recover the on-shell description of the single valued set of generator 1 i2, where the above algebra takes and double valued CSR. the simpler form: [ ± ∓]= [ ±]=± ± 2. Review of CSR and sCSR from the symmetry algebra viewpoint , 0 , J12, (5) The second Casimir takes the form For the definition and classification of the various representations, we are following the method of diagonalizing the Casimirs 2 2 + − C2 = (1) + (2) = (6) and the Cartan subalgebra generators of the stabilizer subgroup of the four dimensional Poincaré group and its N = 1 supersymmet- Due to the structure of (4)there are two natural sets of eigenstates ric extension. that one can use in order to describe the various representations. The first set includes the helicity states (|λ), which are the eigen- 2.1. 4D Poincaré algebra states of J12. They are labeled by a discrete integer or half-integer helicity and they are mixed under the nontrivial action of i . The 2 The Poincaré group is the group of isometries of Minkowski second set includes the angle states, which are the eigenstates of spacetime and it is generated by the set of rigid motions, i.e. trans- 5 1 and 2. They are labeled by a continuous angle parameter 3 lations (Pm) and rotations (Jmn ) which satisfy the algebra :

4 i are the two generators of translations along the two perpendicular directions 2 Its part connected to the identity. and J12 is the generator of rotation along the axis of motion. 3 We follow the discussions in [3,38,18]. 5 This is the origin of the “continuous” spin terminology. I.L. Buchbinder et al. / Physics Letters B 793 (2019) 445–450 447

and the action of J12 results to a shift of this angle. The two sets 2.2. 4D, N = 1 super-Poincaré algebra of states are related through a Fourier transformation. The definition of helicity states is: For the supersymmetric extension of the Poincaré algebra with only one supersymmetry,6 we add to the list of Poincaré genera- J |λ=λ|λ , C |λ= 2|λ (7) ¯ 12 2 μ tors the four fermionic generators of supersymmetry Qα and Qα˙ . Therefore, in addition to (1)wemust consider the following: where the eigenvalue of C2 is a real positive number parametrized by the dimensionfull parameter μ. It is straightforward to show ˙ ˙ ˙ ± [ ]= β [ ¯ α]= ¯ α ¯ β that the action of on a helicity state increases (or reduces) the Jmn, Qα i(σmn)α Qβ , Jmn, Q i(σmn) β˙Q , (16) ¯ helicity by one unit [Pm, Qα]=0 , [Pm, Qα˙ ]=0 , ± ± ± { }= { ¯ ¯ }= { ¯ }=− m J12 |λ=(λ ± 1) |λ⇒ |λ=μ|λ ± 1 (8) Qα, Qβ 0 , Qα˙ , Qβ˙ 0 , Qα, Qα˙ (σ )αα˙ Pm . ± where the normalization of state |λ is fixed by the orthonor- In this case the stabilizer subgroup must preserve both Pm and mality of the helicity eigenstates and their Casimir eigenvalue. Qα and is straightforward to show that it is generated by Pm and ± Hence by a repeated action of we can construct an infinite Zm, where set of linearly independent states with integer separated helicity ˙ = 1 mnrs + αα[ ¯ ˙ ] values Zm 2 ε Jnr Ps c(σm) Qα, Qα . (17) ± ( )n|λ=μn|λ ± n , n ∈ N (9) This is the supersymmetric version of the Pauli-Lubanski vector. It is important to realize that supersymmetry not only appears in All of these states belong in the same representation because they the second term but also in the first term, through the genera- C have the same 2 eigenvalue μ tor of rotations. This can be seen by considering the superfield C ± n| = 2 ± n| = realization of the generator of rotations which takes the form 2 ( ) λ μ ( ) λ , n 0, 1, 2, ... (10) J =− + ¯β˙ ¯ α˙ ¯ − β α − M mn ix[m∂n] iθ (σmn) β˙∂α˙ iθ (σmn)β ∂α i mn. Notice Furthermore, by doing a full rotation of the state |λ we get that this is not the same as the non-supersymmetric case, because 2iπλ e |λ, thus λ is either ±N where N is a non-negative inte- the generator of rotations can also act on the fermionic directions ± ger (single-valued representation) or N/2 where N is a positive of superspace. The parameter c is a numerical coefficient and its half odd integer (double-valued representation). The conclusion is value differs between the massive and massless case. This is be- that the irreducible representation of E2 algebra (4)are classified cause in the massless case, half the supersymmetry generators by a dimensionfull, continuous, parameter μ and for μ = 0they become trivial (vanish) and that qualitatively changes the struc- are infinite dimensional. The single-valued representation includes ture of the algebra. To see this once again we identify P with all integer helicity states and the double-valued representation in- m p = (−E, 0, 0, E). For this case, the supersymmetry algebra takes cludes all the half odd integer helicity states. For the special case m ± the form of μ = 0, the action of generators becomes trivial and does not lead to new states. Therefore, the infinite size representation ¯ 20 {Q , Q ˙ }=− E (18) collapses to a dimension two representation with the states |λ, α α 00 | − λ. This special case corresponds to the description of massless particles with a fixed helicity. and leads to On the other hand, the definition of the angle states is: = ¯ = Q2 0, Q2˙ 0 . (19) ± ± |θ=μ e iθ |θ , C |θ=μ2|θ (11) 2 With these constraints taken into account, the value of c is c = Such a state can be expanded in the complete basis of helicity −1/8 and the algebra of the massless, supersymmetric Pauli- states, hence we can write the ansatz Lubanski vector Zm is: | = | θ fλ(θ) λ (12) [Zm, Pn]=0 , [Zm, Qα]=0 , (20) λ m n mnrs m [Z , Z ]=iε Zr Ps , Z Pm = 0 , where f (θ) are the expansion coefficients, which due to (11)must λ [ ]= − satisfy Jmn, Zr iηmr Zn iηnr Zm .

±iθ An example of a qualitative difference between the above algebra fλ(θ) = e fλ∓1(θ) (13) and the corresponding algebra for massive particles is the com- This condition fixes uniquely the expansion coefficients, up to an mutativity of Zm with Qα. For massive representations this is no overall normalization constant longer true and there is a non-trivial right hand side in [Zm, Qα]. The orthogonality of Zm with Pm fixes its structure to be the ∼ −iλθ fλ(θ) e (14) same as in the non-supersymmetric case: hence the angle states are the Fourier dual states to the helicity Z = p Z + T (21) states m m m 1 αα˙ ¯ − where Z = J12 − (σ¯3) [Qα, Qα˙ ] and Tm is the transverse to |θ∼ e iλθ |λ . (15) 8E pm vector as m in (3), Tm = (0, T1, T2, 0) with T1 = E(J23 + λ ˙ ¯ ˙ − 1 ¯ αα[ ˙ ] =− + − 1 ¯ αα[ J20) 8E (σ1) Qα, Qα , T2 E(J13 J10) 8E (σ2) Qα, −iαJ12 ¯ It is straightforward to see that under the rotation e the an- Qα˙ ]. Using (19)the above expressions can be simplified gle state |θ will result to the state |θ + α, thus giving to J12 the interpretation of translations in the θ sector, coordinate, the J12 covariant constraints. 6 We follow the discussions in [39,40]. 448 I.L. Buchbinder et al. / Physics Letters B 793 (2019) 445–450

= − 1 [ ¯ ˙ ] + = ¯ ˙ + ¯ ˙ = ¯ ˙ Z J12 8E Q1, Q1 , (22) Qα iDα 2i∂α , Qα iDα 2i∂α (27)

T1 = E(J23 + J20), T2 =−E(J13 + J10), which can be used to convert between Qs and Ds. The constraints m αα˙ (19)can be written covariantly in the form (σ¯ ) pmQα = and their algebra is m αα˙ ¯ 0,(σ¯ ) pmQα˙ = 0 and in superspace they are translated to: [T , T ]=0 , i[T , Z]=T , i[T , Z]=−T . (23) 1 2 1 2 2 1 m αα˙ ˙ (σ¯ ) pm∂α = 0 , ¯ m αα p Q = 0 ⇒ Usually when describing the supersymmetric, massless represen- (σ ) m α m αα˙ 2 ¯ (σ¯ ) pmDα = 0 ⇒[D , Dα˙ ]=0 tations (see e.g. [39], [40]) the case of continuous spin represen- (28) tations is considered as non-interesting or unworthy since it had no relation to supersymmetric generalization of conventional field ¯ m αα˙ ¯ = m αα˙ (σ ) pm∂α˙ 0 , ¯ p Q¯ ˙ = 0 ⇒ theories. This is a main reason why such a case has not been dis- (σ ) m α m αα˙ ¯ ¯ 2 (σ¯ ) pmDα˙ = 0 ⇒[D , Dα]=0 cussed in details. Therefore both T1 and T2 are set to zero by hand like in non-supersymmetric case. However because the above (29) algebra is identical to the non-supersymmetric (4)one, the en- The expression for the supersymmetric Pauli-Lubanski vector Zm tire discussion for continuous spin representations of the Poincaré is: algebra can be applied as is to the super-Poincaré algebra by re- ˙ placing W and i with Z and Ti respectively plus the additional m → m =−i mnrsM + 1 ¯ m αα[ ¯ ˙ ] Z Z 2 ε nr ps 8 (σ ) Dα, Dα . (30) conditions (19). Therefore the definition of supersymmetric continuous spin representations with label μ is: In the last one, it is interesting to observe how the θ-derivatives ¯ (∂α and ∂α˙ ) originating from (27) combined with their constraints 2 ± ±iϕ C2|μ, ϕ=μ |μ, ϕ , T |μ, ϕ=μ e |μ, ϕ , (24) (28), (29)cancel the theta dependent part of Jmn leaving only the | = ¯ | = usual (internal) Poincaré part. Therefore, according to the decom- Q2 μ, ϕ 0 , Q2˙ μ, ϕ 0 position (21)we find the following for Z and Ti ± + − where T = T ± iT and C = T T . 1 2 2 ˙ =−M + 1 ¯ αα[ ¯ ˙ ] Z i 12 8E (σ3) Dα, Dα , (31) 3. Superspace realization of supersymmetric continuous spin 1 αα˙ ¯ T1 =−iE(M23 + M20) + (σ¯1) [Dα, Dα˙ ] , representations 8E ˙ = M + M + 1 ¯ αα[ ¯ ˙ ] T2 iE( 13 10) 8E (σ2) Dα, Dα . The objective of this paper is to find a 4D, N = 1Minkowski, ¯ m αα˙ = = superspace realization of sCSR in order to make the connection Once again, we can use the constraints (σ ) pmDα 0 ¯ m αα˙ ¯ under supersymmetry between the single-valued and the double- (σ ) pmDα˙ to simplify the above expressions (only the D1 and ¯ valued CSR, manifest. Therefore, the use of superfields is a natural D1˙ parts survive): choice and the question is to find the appropriate superfield and =−M + 1 [ ¯ ˙ ] the necessary set of differential constraints required for the de- Z i 12 8E D1, D1 , (32) scription of sCSR. These constraints have to be covariant under T1 =−iE(M23 + M20), T2 = iE(M13 + M10). supersymmetry, so their nature does not change under supersymmetry transformations. Hence, the constraints must be formulated Notice that the Ti found above do not seem to be aware of ¯ in terms of the supersymmetry covariant derivatives Dα and Dα˙ . the presence of supersymmetry and match precisely the non- Therefore, we must express the various objects that participate in supersymmetry discussion. The only contribution of supersymme- the discussion of section 2.2 in terms of the spinorial covariant try in the definition of sCSR seem to be the D-constraints (28), derivatives and then impose the various diagonalization conditions. (29).

3.1. From Hilbert space to superspace 3.2. Superﬁeld description of sCSR

As mentioned previously, once we consider the (super)field de- Looking back to the wavefunction description of CSR, there are scription of the various representations the abstract generators will two clues that provide some guidance. The first one is the infinite size of the representations with all integer separated helicities be replaced by the familiar differential operators that describe the participating in the spectrum of the theory. That means that we group action in the space of (super)fields: can not describe CSR with a finite collection of tensor fields and ¯β˙ α˙ ¯ β α one should consider the countable infinite set of increasing rank Jmn → Jmn =−ix[m∂n] + iθ (σ¯mn) ˙∂α˙ − iθ (σmn)β ∂α (25) β bosonic (or fermionic) tensor fields. The second clue is that the − iMmn , action of rotations on the angle states gives a shift in the angle pa- 1 ¯α˙ m rameter. This indicates that the intrinsic spin generator Mmn can Qα → Qα = i∂α + θ (σ )αα˙ ∂m , 2 be interpreted as the derivation with respect to an appropriate “in- ¯ ˙ → ¯ ˙ = ¯ ˙ + 1 α m ˙ Qα Qα i∂α 2 θ (σ )αα∂m , ternal” coordinate not related to spacetime. Both of these features suggest that one should introduce an auxiliary coordinate ξ and P → P =−i∂ = p . m m m m m consider the generating “functions” φ(ξ, x). Morally, an expansion The sets of spinorial covariant derivatives with respect to super- in terms of ξm will generate an infinite list of spacetime fields with symmetry are all possible ranks and the ξ -orbital angular momentum generator 7 ξ[mπn] will correspond to the intrinsic spin generator (−iMmn ) ˙ = + i ¯α m ˙ ¯ ˙ = ¯ ˙ + i α m ˙ and thus giving to all these fields the appropriate helicity value. Dα ∂α 2 θ (σ )αα∂m , Dα ∂α 2 θ (σ )αα∂m . (26)

From the above, trivially the relation between Qα and Da can be 7 [ m ] = m written as: πm is the conjugate variable to ξm such that ξ , πn iδn . I.L. Buchbinder et al. / Physics Letters B 793 (2019) 445–450 449

αα˙ αα˙ ∂ αα˙ 1 The role of this auxiliary coordinate is to provide a mechanism in ξ ∂ ˙ φ = 0 ,∂ φ =− φ,ξ ξ ˙ φ = φ, (43) αα αα˙ μ αα 2 order to group the infinite set of components in to a multiplet with ∂ξ the correct book keeping for their helicities in order to match the 2φ = 0 spectrum of CSR. This approach turned out successful and provides = | the correct (Wigner’s) covariant conditions for the field description and the lowest fermionic component ψα Dα θ=0=θ¯ describes of CSR. the double-valued CSR: All these features remain present in the case of sCSR, thus it is αα˙ αα˙ ∂ ξ ∂αα˙ ψβ = 0 ,∂ ψβ =−μψβ , (44) natural to follow a similar path. For these reasons we consider an ∂ξαα˙ expanded version of superspace by introducing the auxiliary coor- ˙ αα ˙ = 1 α ˙ = ξ ξααψβ 2 ψβ ,∂ αψα 0 . dinate ξm such that the action of internal spin generator −iMmn on standard superspace superfield tensors is reproduced by the ac- Similarly, the complex linear superfield description of sCSR − ∂ tion of iξ[m ∂ξn] on the extended superspace, rank zero (scalar) takes the form: superfield (ξ, x, θ, θ)¯ . Notice that the extension of superspace αα˙ ¯ = takes place only in the bosonic sector, in order to keep the same ξ DαDα˙ 0 , (45) number of superchargers. Therefore T can be written as: ˙ ∂ i DαD¯ α =−iμ , (46) ∂ξαα˙ ∂ ∂ m m αα˙ 1 T1 =−iξ2 p + i p ξm , (33) ξ ξ ˙ = , (47) ∂ξm ∂ξ2 αα 2 ¯ 2 ∂ ∂ D = 0 , (48) T = iξ pm − i pmξ 2 1 m 1 m ∂ξ ∂ξ Dα = 0 , (49) The definition of sCSR is: and it is straightforward to show that the components ϕ = | = ¯ | θ=0=θ¯ and λ Dα θ=0=θ¯ satisfy the same (43), (44) condi- T (x,ξ,θ,θ)¯ ∼ μ(x,ξ,θ,θ)¯ (34) tions and thus provide a description of integer and half-integer i CSR. Of course this alternative description of sCSR exists due to pmξ (x,ξ,θ,θ)¯ = 0 , ⇒ m the well-known duality between chiral and complex linear super- m ∂ ¯ = ¯ p ∂ξm (x,ξ,θ,θ) iμ(x,ξ,θ,θ) fields which flips eq. (41), (42)with (48), (49). An interesting observation is that due to (40), (47)the solutions C (x,ξ,θ,θ)¯ = μ2(x,ξ,θ,θ)¯ (35) 2 must be searched in the space of distributions. This is a charac- m ¯ ¯ ⇒ ξ ξm(x,ξ,θ,θ)= (x,ξ,θ,θ) teristic property of CSR and sCSR. In addition, the coordinate ξαα˙ ¯ 2 ¯ ¯ can not be written as the product of two twistors because that Q2(x,ξ,θ,θ)= 0 ⇒[D , Dα˙ ](x,ξ,θ,θ)= 0 (36) will make it a lightlike coordinate (ξαα˙ = ωαω¯ α˙ ). However one can ¯ ¯ = ⇒[¯ 2 ] ¯ = introduce two sets of twistors ωI with I = 1, 2. Then one can de- Q2˙ (x,ξ,θ,θ) 0 D , Dα (x,ξ,θ,θ) 0 (37) α compose ξαα˙ in the following manner: The first three equations are Wigner’s conditions for CSR. The last two are the additional supersymmetric constraints in order to de- = I ¯ J ξαα˙ ωαωα˙ εIJ (50) scribe sCSR. Equations (36), (37)are solved by either a chiral su- ¯ 2 This decomposition makes contact with the description in [36,37] perfield (Dα˙ = 0) with the equation of motion D = 0or by a complex linear superfield (D¯ 2 = 0) with the equation of mo- where two twistors πα and ρa and their conjugates were used for the description of CSR. The correspondence is ωI ={π , ρ } and tion Dα = 0. α α α thus it will relate the component fields found here with the ones used in the BRST description done in [36]. 4. Components discussion and the recovery of the single and double valued CSR 5. Summary and discussion

For the case of the chiral superfield description, eqs. (34), (35), In this work we define the supersymmetric continuous spin (36), (37) take the form8: representation (sCSR) (eq. (24)). To find a superfield description of N = αα˙ ¯ it we extended standard 4D, 1superspace with the addition ξ Dα˙ Dα = 0 , (38) of an auxiliary, commuting, coordinate ξαα˙ in order to construct ¯ α˙ α ∂ =− generating superfunction that group together the countable infinite D D ˙ iμ , (39) ∂ξαα number of supersymmetric multiplets of increasing superhelicity αα˙ 1 that appear in the spectrum of sCSR. These are the supersymmet- ξ ξαα˙ = , (40) 2 ric extension of Wigner’s wavefunctions used for the description ¯ Dα˙ = 0 , (41) of CSR. We find two descriptions. The first is based on a chiral su- ¯ ¯ = D2 = 0 . (42) perfield (ξ, x, τ , θ) (Dα˙ 0) and the proposed set of covariant equations of motion it must satisfy is: By projecting these superspace equations into equations for the αα˙ ¯ = component fields of we find that the lowest component φ = ξ Dα˙ Dα 0 , | ¯ describes the single-valued CSR ˙ ∂ θ=0=θ D¯ αDα =−iμ , ∂ξαα˙ αα˙ = 1 8 αα˙ 1 m αα˙ ∂ ξ ξαα˙ , Convert the vector index to spinorial indices ξ = (σ¯ ) ξ and ˙ = 2 2 m ∂ξαα m ∂ 2 = (σ )αα˙ ∂ξm . D 0 . 450 I.L. Buchbinder et al. / Physics Letters B 793 (2019) 445–450

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