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)