Physics Letters B 793 (2019) 445–450 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb 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 3 Continuous spin (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 1. Introduction they were never observed. The rest, fall in the category of continu- ous 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 1 times (Minkowski, de Sitter and anti-de Sitter). In the case of 4D respectively with multiplicity one. 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 neg- ative 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 [Jmn, Pr ]=iηmr Pn − iηnr Pm , with the radius of (A)dS spacetime. CSR naturally provide both 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- i with the transverse vector generators of iso(3, 1): , 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 transforma- tion 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.