Celestial Amplitudes 8 - ICTP - SAIFR
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"" au .
&
Celestial Amplitudes
Sabrina Gonzalez Pasterski Princeton PCTS
,
88 Celestial Holograph ypurportsadual.it#ybetweengravitationa1
flat spacetimes and CFT
living
scattering
the celestial sphere
in
asymptotically a on .
8 Celestial Amplitudes are 5- matrix elements in a basis where
they transform as
conformal come lators in D- 2 .
f
:
i
Bo
•will the current status of this
Today we review
dictionary .
0 Part I
Asymptotic Symmetries
8 Part II Celestial Amplitudes
The results presented are thanks to :
Adamo Arkani Hamed Ashtekar, Atanasov Bhatakar , Cachazo Campiglia , Casali , Chang Cheung , Coi -10 Compere Crawley de la Fuente
Barn ich
Athi ra , Avery , Ball Banerjee, , Donnay ,
Dumitrescu , Fan Fiorucci , Fotopoulos
-
, , , , , , , ,,
, , ,
Giribet Gonzalez , Gosh , Haco , Hamada , Hawking He , Him Wich Ladd ha , Lam , Law Li , Lippstreu Liu , Long , Lysov , Magnea Manu , Mao , Mason , Melton Miller, Mirza cyan Mitra Mizera Nandan Nan de
,
Guevara ,
, , Huang Kopec
, ,
, , , , , , , , ,
Narayanan , Nguyen ,
Nichols, Oblak , Oliveri , Pandey S.P. , Pate , Paul Perry , Porfyriadis , Prabhu, Puhm , Radar ice , Raju , Rojas Rosso Ruzziconi , Saha Satter Samal Schreiber , Schwab , Sen Seo Seraj Shao Sharma , Shiu , Shrivastava Stieber ger
, ,
, , , , , , , , , , ,
Strom Taylor
inger, sand rum , Suskind
, ,
Trevisan i , Troessaert
, Venugopal an ,
Verlinde , Udovich , Wald , Wen Yuan Zhiboedou , Zhu , Zktnikou
, ,
. . .
w/ !
12:20 -
13:10 Discussion session
Andy f TomaszThe first step is to match the
symmetries on both sides of the proposed
duality . . .
•÷÷•• •.•.•:•. ÷
•
-•Bo• -
N÷
aqg.az mom ••••→-••a•
g.
µfBB B.at
a.
m MPB ago
BB
8 .
; a.
I 1
Lorentz transformations Minkowski space act conformal
of as
global
transformations on the celestial sphere .
?⃝For what follows we will want to keep in mind how to connect position
and momentum space descriptions of the scattering problem
17 , E)
✗ w -
ly , z , E)
m2
p2 =
p2 =
0
-
Fy ( il E- 2-1
?
pm = It
y
-1 2- E
,
z +E
, ,
I -
'
y
-
2- E) pm =
w/ It 2- E
,
z +E
,
il E- 2-1
,
I -
2- E)
For 5- matrix elements we would
typically specify a set of on
-
shell momenta
for the in and out particles .while inposition space we need to
specify the field configurations
on
early and late Cauchy slices .
→
"
qµf•a o•qoBB_a→
zBBqaB
For
gravitational scattering we want to allow fluctuations of the
bulk
geometry and will specify the free data at the conformal boundary .Let's look at the Penrose for
diagram Minka .
The causal structure of
the
boundary will be the same for
asymptotically flat spacetimes .
#
✗
2- = e tan ¥
it
,
It m 1=0
r
m = 0
t.io
✓ + r
É
Massless I-1=-112×5?
excitations enter and exit
along null hypersurfaces
We call this 5h cross section the celestial
-
sphere .Let's compare this to the free mode expansion of the field operators
:
If ,d¥p£wg[e%a✗ei9 i9×
]
✗
equate
-
hmu = +
✗= ±
The saddle point approximation localizes the momentum direction to
align with the
corresponding point on the celestial sphere .
¥1 Eh =
iaa¥-zpf°°dwg[ a- lwgni ) e- iwoi-aflwgxleih.ci]
it
,
+
I )
pig ✗ e- iwqu iwqrltcoso
-
.
=
( Z, E )
L J
Lio wa
-
, I
-
it
v=t+r 92=0
He Mitra , Strom inger 44
, Lysov ,For massive fields we have a similar identification between the late
time momentum and the hyperboloid timelike infinity
a
point on
resolving .
i-m-h.c.it
ok y E)
*
e- Hrm
"
aly E) e-
, , -2 , ~
E-
,
z
,
.
✗ L S
It
r
ly ,¥ , E)
m2
% p2 = -
✓ + r
É
de Boer solo clutch in 103
, Campi glia ,
Ladd ha
'
iswe're interested in
gravitational scattering in spacetimes with 1=0 .
The
outgoing radiation is captured by the behavior of the metric at
large r
,
fixed u .
✓ flat tr # corrections
d§= did -2 dudr y
2r28zz-dzdzi-2-MBdu2-rczzdz2-rcz-z-dz-2-DZC.cz
-
+
dudz +
DECEE dude +
. . .
To
study *aµM¥Ñ
the phase space f
symmetries one needs to
g g. ••EEsoaÉÉ#_§¥→BÉ→§q
tl
I
µ
pick convenient
gauge a
8
specify physical falloffs
Rope
B-zoziez-s-r-IMM.at#-rgs@-zins
¥1K
¥¥%EoI
Bondi
' '
Metzner 62 62
,
van der
Burg , SachsResidual diffeomorphisms that preserve the falloffs and act non trivially -
the Asymptotic
on the
asymptotic data are part of Symmetry Group .
Allowed Symmetries
ASG =
Trivial
Symmetries
ASG will be much larger then the
The
group of isometries of any
given spacetime within this class .
BMS 3 Poincaré
#
generators : A 10
Bondi
' '
Metzner 62 62
,
van der
Burg , Sachs8 Super translations shifts the Time coordinate
induce
angle dependent
- in
}/ =
flz , E) Ju
It
8 Super rotations extend global conformal transformations to local ckvs
}/ =
YZ / z) Jz +
E- DZYZ / E) Ju + cc .
0
§
It
0
go o
o o
o o
' g
o o o
o
0 ¥
g
ooo o
'
g og
Bondi ,
van der
Burg , Metzner
'
62 Sachs
'
62 Barn ich Troessaert 41
,
$manifest in
These
soft theorems
asymptotic symmetries
.
scattering amplitudes as
"m
Lout / QS -
SQ 1in) ✗
Coutts / in>
w→o
8 The canonical charges are co
-
dim 2 and can be written as an
integral of data on It?
Sdu Jul ) .
<
→
Io W
8 Most of the
symmetries are
spontaneously broken by the vacuum
and the term that shifts
generates inhomogeneous is linear in hmu
Q 3-
Qs ~
Sdu Jufdw e- iwualwx)
'
65 Strom He
Weinberg inger 43 , Lysov ,
Mitra , Strom inger 44Moreover, these Ward identities
of currents 2D CFT
are
naturally organized in terms
in a .
4 D ASGthe subleading soft theorem
In
particular , graviton gives us a
candidate stress tensor .
Tzz =g÷G fduudufdtdwfz.DE cww
However to construct operators with definite
,
2D conformal
weights
we will need to
change our
scattering basis .
✗
n
< Tzz Oi . . . Ori > =
§ / ¥zup ¥1] < Oi +
. . . Ori )
✗
.
×
h.IE/-w?wIsnl,h-..--tzl-w?wisu )
.
.
I
Cachazo 44 Kapec Mitra , Radar in , Strom
Kopec Lysov 44
'
Strom S.P. Strom 16
, inger , , , inger ,
ingertheorem
In
particular ,
the subheading soft
graviton gives us a
candidate stress tensor .
Tzz =g÷G fduudufdtdwfz.DE cww
However to construct operators with definite
,
2D conformal
weights
we will need to
change our
scattering basis .
CTzz.ci#On---wim-wJwoiHalwx1S1in>
✗
n
✗
r ✗
.
h.EE/-w?wIsul,h-..--tzl-w?wisu )
.
.
I
Cachazo 44 Kapec Mitra , Radar in , Strom
Kopec Lysov 44
'
Strom S.P. Strom 16
, inger , , , inger ,
ingerA conformal wavefunction
"
is function of bulk
point X
primary a a
and a reference point we ① which transforms as follows
Is / AT ✗ EI-÷ I +7 c- w-i-d-P-JD.sn/II,-lXiw.w-l
" ☐
☒ i ,
=
Icw + d)
Where Ds is the
spin s rep of the Lorentz group Here we will consider
-
.
.
☒ § with 5- 1J 1 that solve the
, appropriate source free linearized eom .
5. P
.
Shao 47
,product of such a wavefunction with the field operator
Taking an inner
(quasi ) with 2D conformal dimensions and spin J
gives primary operator
-
a .
twin)= i / ① 4×4 1×+1 ; with
±
0s☐ II.
,
,
out selected
These operators carry a ± label
indicating in versus
, by
taking ut ie uts .
u
qm
9M =
( It WÑ ,
w -10
,
ilño wt , I
-
-
wñ )
~
÷xP
5. P 47
.
,
Shao
Donnay ,
S.P. , Puhm '20these wavefunctions to Mellin
For m 0 are
equivalent
=
gauge a
transform of on -
shell plane waves .
l
,e±iw9X±
☐
'
Idw
-
"
To ~
w Eu µ
D ,J ,,
. .
.
,
-
A similar transform exists when m -40 In either case , we can
apply .
to 5- matrix elements to land on Celestial Amplitudes
this transform
directly .
( Z, E )
( w , in )
•
w -
ly ,¥ , E)
m2 p2 0
p2
=
= -
So°°¥fd%G☐1y wt / ) fiodww " '
f)
-
z E; w
.
, , ,
de Boer solo clutch in 103 de la Fuente 16 S.P. Shao Strom
inger 47
'
,
Cheung , ,
Sun drum , ,elements this basis
By construction ,
5- matrix in
¥→¥É¥ ¥%%
n
.ie#ioIiI.ATsi,zi,Ei)--T/fYdwwsi-l)Alwi,z- i =L
i
,
E- i .É ¥¥¥i÷÷:÷÷!÷ →
__oo-o--→
É¥E☒BI•ÉEBg¥jµµ*← __-osoo--s---=
will transform like come lators of because the
quasi primaries
-
external boost
particles are in
eigenstates .
-2h -5h
V11) 1h ,ñiz ,
E) =
kz + d) / c- Etd ) 1h I , ;E¥-b_d ¥I¥- > ,
• ④@÷:§
.
" i
'
Joos 62 Banerjee
'
18 i. :
e
eelements this basis
By construction ,
5- matrix in
¥:i.¥E ¥¥¥
n
bi
1) A / ;) €K→*:÷→%É •
r.ae#ioIiE.ATSi,Zi,-Zi)--T/fYdww
-
Wi zi , E
.EE#EIIxi--l-ooo-o---*
,
É¥E☒BI•€÷agy¥j
F.am#-----oso---------
will transform like come lators of because the
quasi primaries
-
external boost
particles are in
eigenstates .
J; =
jth helicity
✓
;) ;ÉKcz ATS
a-Ei -15
Ñfsi azitb
(c- E. + d- 1b¥] ;)
☐
Fid , -¥+d
=
.
+ d) ; , zi
,
E
, J J
• ⑦@÷:§
e
.
e
•
! ;
'
Joos 62 Banerjee
'
18
i. i e
e e
eThis transformation is inverted for the
easily radiation
Si on
principal series
which capture finite
energy .
D= It it
,
✗ c- IR
However translations shift the conformal dimension
>☐
pie que ←→ ☐ +1
= →
and will want to continue to SEE
we
analytically .
s
'
5. P
.
Shao 47 Donnay ,
Puhm , Strominger 18 stieberger Taylor
,
48
Donnay ,
S.P. , Rehm 20
,It is
straightforward to do this transform for low point amplitudes .
fixed
/ 4)
←
"
A / Wi , Zi ,
Ei ) =
M ✗
8 / Ep ;) ,
4. = Sw ;
g
Letting S =
Ewi , q =L-1W ;
.
integrated
ijiodwiwisi 4.) foods g- ""
i-Ii dq.q.si -18kg l
-
= .
-
1) l -
i= I
i= ,
"'
so that for ne s the g. are localized by the 8 Ep ;) and 8kg .
-
1)
n
>
Git
✓
<
2- ;
The 5- functions low point correlations
remaining imply singular
-
.
'
S.P. Shao Strom Schreiber , Vukovich , Zlotnikov 17
, , inger 47The aim of our from 413 5- matrix elements to 2D correlations
map
< Pout i / ;) ( Ot )
-
S / pin /→ 0
diJi Aj Jj
- -
- .
.
, , ,
⑤
is to be able to use CFT techniques to learn about amplitudes .
④
that this dual CFT
We are
already seeing is exotic , both in its
④
complex spectrum ,
and in the singular behavior at n±4pt .
④
Meanwhile , the want to look at unusual
quantity we in 413 is also ⑥
because we are
probing scattering at all energy scales .
•
⑧ ②
④
④
⑥
④ ④
② ④To understand how to effectively use this framework we must
build up our
dictionary !
✗:FY.÷¥¥÷*÷÷÷: ies#aima*+sme+÷ry j:÷j
me?i:Ei-i esianYE-→÷i
;
Dressings
Vertex Operators
"
Ssi "
g. ¥ ¥ ,
Let us now examine how celestial amplitudes encode the
various limits
behavior of
scattering in
0 Infrared soft -1hm 's currents states
dressings null
-
, , ,
8 Collinear -
Celestial opes conformal block decomposition
,
8 Ultraviolet -
convergence
<
→ ultrasoft , anti Wilsonian-
paradigmSince soft theorems motivated this program we should understand
what the
they translate to in celestial basis .
also
l
f.w*dw wa
-
~
that factorizations at various orders in w -
o turn into
factorizations of the residues as A→ special values
< outta Cw -
g) 51in > =
? /I '%÷ P"Ep?g%" ) Coutts / -
i in > + ocw)
↳
¥ ,
↳
¥
☐
These residues correspond to celestial currents .
- y
rominger 49 19 Guevara 49
' '
de la Fuente , Sun drum '
Fan Fotopoulos Taylor 49 Pate Radar in St Adamo , Mason , Sharma
Cheung ,
16 , , , ,
19 PuhmThe familiar universal soft theorems correspond to Sfor
which the conformal wavefunctions
primary are
pure gauge .
Od G)
,s=i(O -1-00%-52 ,
if
c.
pure gauge
1J / A soft -1hm .
Current Asymisym .
" v11 )
1 1 w J large
-1/2
3/2 42 W S large SUSY
"
2 1 w P translations
super
0 WO T superrotations /Diff / 54
'
Cheung de la Fuente , Sundrum Puhm Puhm 205.1? Trevisan i'21
16
Donnay Donnay ,Puhm
/ /
Strominger
'
,
5. P Shao 17 ,
18 S.P.
, , ,
.
,of the
These IS ,
J ) are also special from the point of view
global
conformal multiple-1s .
Primary states
41h15 > =L , thin> = 0
have descendants when
primary
1) ) " -114,57=0
"
41L 1) 1h 5)
-
,
= -
K / 2h -1k -
1L -
,
and ditto for -4 When both conditions
.
are met we
get nested primaries .
1¥ :-).
'
II. Ñ "
1L -
,
)
I ¥4
> I '
¥)
I "¥¥>correspond infinite tower of ' conform ally soft theorems
'
These to an
at de l -27=0 .
01¥
a-
so
out
✓
0
>
0
>✓
While the soft theorems of the
previous table descend to
operators
transformations
generating asymptotic symmetry
OD ,
J
,
> ⑤2- ☐ -
J
,
Q[y]=Jd%Osoft Y -
'
05ft c-
Banerjee Pandey , Paul 49 Guevara , Himwich , Pate 21 S.P. Puhm Trevisan i 21
'
Strom
, , inger , ,correspond infinite tower of ' conform ally soft theorems
'
These to an
at de l -27=0 .
01¥
a-
so
out
✓
0
to
>✓
Whose
partners can themselves be
expressed as
primary descendants
of
dressing modes .
?
⑤D J
,
,
> ⑤2- ☐ -
J
,
>✓
'
Banerjee Pandey , Paul 49 Guevara , Himwich , Pate 21 S.P. Puhm Trevisan i 21
'
Strom
, , inger , ,Soft factors relate amplitudes with and without an extra
single
particle emission / absorption .
between the that
Exchanges charged external
legs exponential so
without soft radiation vanish
amplitudes .
This interpreted as
vanishing conservation
can be non for the
charges
-
generating asymptotic symmetry transformations .
One can avoid this
vanishing by dressing the operators .
m~g.gr
=
Faddeev Kulish
Perry Radar ice , Strom
'
65 ' '
Chung ,
70
Kapec , , inger
17conformal basis
These
dressings can be adapted to the .
For ESM
Iott ; it ;)
W ;= exp [ -
e Q
; %¥_pÉo ( Etna -
E✗ µ at ) ] = ei Qi
TS
c- D= I Ll 2 , E) ascendant
and we see that the
takes the form of dressing a vertex
operator .
Moreover the celestial amplitudes factorize
A A soft 1- hard
=
where
1- soft =
( eiQ.IO/ZiEil...eiQnEk-niEn1 )
1- hard
while
equals the amplitude for dressed
operators .
Arkani Hamed , Pate Radariu Strom 120
inger
-
, ,The
spontaneous symmetry breaking dynamics for the 413
asymptotic symmetries is captured by simple 2D models .
V11 ) ←→ free boson
Large
-
with the important observation that the levels of the 2D
anomalous dimension in 4D
current algebra are set
by
the
cusp .
( Iolz E) Idw , in D=
,
4%2 1nA ,=p In / z -
wt
Nan de , Pate Strom 47 Him Wich Narayanan , Pate
inger Paul Strom 120 Nguyen , Salzer ' '
,
inger 20 21
-
,
, ,Currents arise from
tuning si to special values .
Rather than
tuning
tune Zi Zj
Zij
=
i we can
-
, .
Collinear limits in 4D should be captured by a celestial OPE .
Consider
taking two
gluons collinear
/ Z E)
O☐bµ if£÷
'
Osa ( Zz Ea) ~ -
CCS , Aa) O l -11
( Zz 12=2 )
,+ ,
, ,
, , , Dis
-
, z ,
A observation is that symmetries to
very interesting
we can use
find CCA , S2 ) .
:* ,
i
✗
-
,
/
Fan , Fotopoulos Taylor
,
49 Pate , Radar in ,
Strom Inger , Yuan 49From translation invariance
CIA , ,
A a) =
CIA , -11 Az) ,
+
CIA ,
,
dat 1)
while from the
leading soft gluon theorem
¥1 Is 1) Cls Sal
1 -
=
.
, ,
, ,
Moreover the kernel of the descend relation for the sub
,
any that imposes
leading
soft gluon gives an additional global symmetry
IS , 2) CCS
-
, -1,121=14+12 -
3) CCS , ,
Sa)
This recursion solved
giving
can be ,
CCS Sa) BCS -1 Da 1) Blx g)
= -
=
, , ,
, , ,
collinear limit
which matches what we
get from transforming the .
Pate , Radar in ,
Strom Inger , Yuan 49Despite the behavior of low point functions , collinear limits let
singular
us extract the Ciju .
With this data and of the spectrum
our
understanding ,
we can
apply
CFT
machinery to celestial amplitudes .
For example
:
8 the of
Using symmetries to
go beyond leading singular terms
the OPE and constrain correlations .
8 Examining conformal block
decompositions to interpret intermediate
and radial quantization in CCFT
exchanges .
pi
Q Q
pi
g ,
V5 .
s
pie
>
pi, 02 04
Lam , Shao
'
17 Nandan Schreiber Vukovich , Zlotnikov '
Banerjee
'
Gosh , Gonzo 20 Law Zlotnikou 120 Fan , Fotopoulos , Stieber Zhu 121 Atanasov , Melton Radar in Strom '
, , 19 , , ger , Taylor , , , inger 21Let's take a closer look at 2 → 2
scattering Starting from . the
momentum space amplitude
Mls -4×8 / Ep ;)
" '
1- =
,
the
corresponding celestial amplitude can be written as
Ñ= # A / B ,
Z ) ,
D= Hsi 1) -
,
z= -2122-22
2- 132-24
where for massless external states
out
14-1 Zij 43 hi hi 5/3 Tri ñi
)
✗
-
-
- -
# ✗
-
Zij
glitz -
z)
is ;
.
• -
in
-2 = E
✗
Arkani Hamed , Pate Radariu Strom 120
inger
-
, ,The stripped amplitude is probed at all scales
energy
ftp.z/--fi0dwwB-1Mlw3-zwY
8 The
convergence of this
integral for external for external dimensions
on the principal series is tied to the UV behavior .
8 Poles field quantum gravity
at BE 223 present in
theory are absent in .
s s
us
imprint of UV completion
, . , . . .
' '
✗ w
M~EAI.co if e-
-2m
Ar Kani Hamed Pate Radar in Strom 120
inger
-
, , ,Meanwhile , M around w=O
expanding gives
M =
San ,m , rlz-IWZYGNWYmlog.TW/1uv)
8 The logs coming from running couplings give higher order poles in B.
f.
↳
dww-watogbw-Ya-bfidww-wa~E.is
8 constraints from amplitudes translate
Positivity to
positivity constraints
on the residues of the
simple poles at D= 222 .
I
imprint of causality
Chang , Huang , Huang ,
'
Lam , Shao Arkani Hamed , Pate Rada rice Strom 21
'
17 120 Li
inger
-
, ,We have a framework that
0
T. m n#Et. . : : q -
enhancements manifest
makes
symmetry ,
.;••.É¥e I
Conformal
Lorentz Invariance Symmetry
✗
8 reorganizes soft and collinear limits Vertex operators
,
• qq.snsinagrsi.im#s Celestial OPES
•
8 and is sensitive to the deep UV .
We have
given examples of
8 how to translate features of amplitudes into objects within the celestial CFT
8 and what kind of
properties we can demand of CCFTS .The next forward
steps are to
8 Expand dictionary
①
our
8 Connect to adjacent subfields
8 Look for an intrinsic construction
Relativity Conformal
Bootstrap
Asymptotic Symmetries
>
11
>
Celestial Amplitudes String Theory
^
Soft Theorems
Amplitudes AISKFT"" "
Null states Samal Trevisan :
Pandey state /operator
him
Relativity cfy
.
Ashtekar i_wm .
Banerjee Himwich Miller
Currents Crawley
Memory Effects Gosh Paul
Narayanan
Wald
cheung
Ruzziconi Oblak Ba "
Spectrum
Tnessaert
Horizons
Venugopal an de " " " "
one,
gushing Guevara
poss, Saha
Canonical
Charges
Fan
Conformal / ysoft Theorems
gna,
Donna zniooedou
,
SP
①① £
'
Barnick Giribet Mirza
'yan
'
seraj
'
Adamo Yuan
Fotopoulos
Fiorucci
Laddha Hamada Schreiber ' Pate
Toy Models
compere .
Law Zhu
Asymptotic symmetries Conformal Block Decomposition
✓
↳ nsusen
. '
Mao Udovich Melton
Zktnik" "" "
Atanasov Nandan
Holography
Wen
Campiglia
Shiu
Magnea gayer
Porfyriadis Kopec
IR Divergences
stromincser
Shao
Soft Theorems Lam
He Lippstreu
Dressings Ashtekar
Strings
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Man " Miter" Mason Prabhu
Mitra Bhatakar Chang Radariu
Cachao
Amplitudes
Li Raju sundrum
Sen UV behavior Taylor
""°
stieoerser shrivastava
Amira Armani Hamed
-
spence yelleshpur Huang
Rojas
Loop Effects Black Hole Info Paradox -
Supersymmetry Brandner
Gonzalez Sharma
Hawking Hao
Avery
Jiang
Travaglini Rihm
Collinear Limits Perry
Firewalls
Double
Copy -3
Lesson Adamo . verlinde
Dumitrescu
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