On higher-spin supertranslations and superrotations

Journal of High Energy Physics, May 2017

We study the large gauge transformations of massless higher-spin fields in four-dimensional Minkowski space. Upon imposing suitable fall-off conditions, providing higher-spin counterparts of the Bondi gauge, we observe the existence of an infinite-dimensional asymptotic symmetry algebra. The corresponding Ward identities can be held responsible for Weinberg’s factorisation theorem for amplitudes involving soft particles of spin greater than two.

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On higher-spin supertranslations and superrotations

Received: March On higher-spin supertranslations and superrotations Andrea Campoleoni 0 1 2 4 Dario Francia 0 1 2 3 Carlo Heissenberg 0 1 2 3 0 Open Access , c The Authors 1 Piazza dei Cavalieri 7 , I-56126 Pisa , Italy 2 ULB-Campus Plaine CP231 , B-1050 Brussels , Belgium 3 Scuola Normale Superiore and INFN 4 Universite Libre de Bruxelles and International Solvay Institutes We study the large gauge transformations of massless higher-spin elds in fourdimensional Minkowski space. Upon imposing suitable fall-o conditions, providing higherspin counterparts of the Bondi gauge, we observe the existence of an in nite-dimensional asymptotic symmetry algebra. The corresponding Ward identities can be held responsible for Weinberg's factorisation theorem for amplitudes involving soft particles of spin greater than two. neous Symmetry Breaking ArXiv ePrint: 1703.01351 Gauge Symmetry; Higher Spin Symmetry; Scattering Amplitudes; Sponta- Contents 1 Introduction and outlook Soft gravitons and BMS symmetry Higher-spin supertranslations 3.1 Spin three Consistency of the Bondi gauge Soft quanta and Goldstone theorem Higher-spin superrotations: the spin-3 example Boundary conditions reloaded Higher-spin superrotations A Spin-3 linearised eld equations in any dimension More details on the residual gauge symmetry Introduction and outlook choice of fall-o conditions at null in nity, the residual gauge transformations of Fronselectromagnetism and gravity, respectively [16{19]. the subleading corrections. tions preserving our Bondi-like fallo conditions. We investigate the general form of the will be explored in future work. our work is meant as a rst step in trying to extend the analysis at the boundary. Our mysterious infrared physics of higher-spin massless quanta. Weinberg considered the S-matrix element S (q), for arbitrary asymptotic particle states , also involving an extra soft massless particle of 4 momentum q the following picture: a factorised form that, in the notation of [2, 3], can be written !l!im0+ ! S s(q) = with i being +1 or 1 according to whether the particle i is incoming or outgoing. retarded Bondi coordinates (see e.g. [5]), t = u + r ; x1 + ix2 = x3 = sphere at (null) in nity near the point q = ! xb = (z + z; i(z (!; z; z). The polarisation vectors can be chosen as follows [25] "+(q) = p (z; 1; i; z) ; " (q) = p (z; 1; i; z) = "+(q) ; its position-space counterpart zi)(1 + zizi)s 1 where Ei and (zi; zi) characterise the massless particles scattered to null in nity. 1), that is to say, assuming that the equivalence principle holds. The action for a massless Fierz-Pauli eld h S = where E is the linearised Einstein tensor @ @ h0 + phisms, h = @ j = By analogy with the non-linear, asymptotically at case (see e.g. [5]), we consider the following form of h h dx dx = 2 Uzdudz 2 Uzdudz + r Czzdz2 + r Czzdz2 ; to gauge parameters which are u-independent and with power-like dependence on r we ponents of the metric with a certain fall-o behaviour. Yet, these can be eliminated by a gauge xing that exploits the available residual ordinary (i.e. non large) gauge symmetry. on the celestial sphere, that we can write in two equivalent ways as follows:2 dx = (T + DzDzT ) du r (DzT dz + DzT dz) ; @ = T @u + DzDzT @r DzT @z + DzT @z ; non-vanishing gauge variations are huz = hzz = Dz( T + DzDzT ) ; (u + r) D Y rated by the conformal Killing vectors on the sphere Y z(z) and Y z(z). with the residual supertranslation gauge symmetry, Q+ = T (z; z) @u DzDzCzz + DzDzCzz + J (u; z; z) J (u; z; z) lim r2J rr(u; z; z) : We assume that supertranslations act on matter elds by (x) = iT (z; z)@u (x) at I + and that this action is canonically realised by gous considerations apply to I . The correlation functions therefore satisfy n(xn)i = i h0j n(xn)Q = i X fnT (zn; zn)h = @ , where the Christo el symbols for Minkowski space in Bondi coordinates are rzz = ; zzz = @z log zz ; zuz = r zz ; rzz = while zz is the metric on the two-dimensional unit sphere. jini = i fi Ei T (zi; zi)houtjSjini ; where Q denotes the counterpart of Q+ at I , and where fi depends in principle on i dary condition DzDzCzz = DzDzCzz at I does not contribute to the left-hand side of (2.17), we e ectively obtain Q+ = T (z; z)@uDzDzCzz zz d2zdu : so that, using we can rewrite Q+ = z 1 + zz T (z; z) = z 1 + zz (for more details see [19]): DwCww du + DzCzz zz d2zdu ; Thus, using crossing symmetry, we also have du@uCzz S dv@vCzz jini = (1 + zz)2 !l!im0+houtj ! ao+ut(!x^)jini ; Performing the r ! 1 limit, so as to express Czz in terms of soft graviton creation and annihilation operators, one has 4 Dzhoutj du@uCzz S dv@vCzz jini = fi Ezii 11 ++ zzizzii houtjSjini : Czz = 8 2 (1 + zz)2 du @uCzz = h!ao+ut(!x^) + !aouty(!x^)i : which implies, by comparison with (2.23), where we have used the divergence formula !l!im0+houtj ! ao+ut(!x^)jini = lim (1 + zz) X !!0+ zi)(1 + zizi) zi)(1 + zizi) Ei(1 + zzi) zi)(1 + zizi) ginning fi = constant. (...truncated)


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Andrea Campoleoni, Dario Francia, Carlo Heissenberg. On higher-spin supertranslations and superrotations, Journal of High Energy Physics, 2017, pp. 120, Volume 2017, Issue 5, DOI: 10.1007/JHEP05(2017)120