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Celestial Amplitudes&

Sabrina Gonzalez Pasterski,Princeton PCTS

8

8 Celestial Holographypurportsadual.it#ybetweengravitationa1 scatteringin asymptotically flat spacetimes and a CFT living on the celestial sphere .

8 Celestial Amplitudes are 5-matrix elements in a basis where they transform asconformal comelators in D-2 .

Bo

•:

f i

Today we will review the current status of this dictionary .

0 Part I Asymptotic Symmetries

8 Part II Celestial Amplitudes

The results presented are thanks to :

Adamo,Arkani -Hamed

,Ashtekar

,Atanasov

, Athira , Avery , Ball , Banerjee, Barnich , Bhatakar, Cachazo, Campiglia, Casali , Chang , Cheung , Coi-10, Compere ,Crawley, de la Fuente , Donnay , Dumitrescu , Fan , Fiorucci , Fotopoulos,Giribet

,Gonzalez

,Gosh

,Guevara

,Haco

,Hamada

, Hawking , He , HimWich , Huang , Kopec , Laddha , Lam, Law, Li , Lippstreu , Liu , Long , Lysov ,Magnea,Manu ,Mao

,Mason

,Melton

,Miller

,Mirzacyan, Mitra ,

Mizera,Nandan

,Nande

,

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

,Stieberger,

Strominger, sandrum , Suskind , Taylor , Trevisani ,Troessaert, Venugopalan, Verlinde,Udovich , Wald , Wen ,Yuan

,Zhiboedou

,Zhu

,Zktnikou

. . .

12:20 - 13:10 Discussion session w/ Andy f Tomasz !

The first step is to match the symmetries on both sides of the proposedduality . . .

aqg.az-•Bo•

• -

••••→-••a•

a.

g. µfBBmom

B.at•÷÷••••.•.•:•..÷!N÷mMPB ago

BB 8 .; a.

I 1

Lorentz transformations of Minkowski space act as global conformaltransformations on the celestial sphere .

For what follows we will want to keep in mind how to connect positionand momentum space descriptions of the scattering problem

17, E)

✗ w -

ly,z , E)

p2 = -m2 p2 = 0

pm =Fy ( It y?-1 2-E

,z+E

,ilE- 2-1

,I -y'- 2-E) pm = w/ It 2-E, z+E , ilE- 2-1 , I - 2-E)

For 5-matrix elements we would typically specify a set of on-shell momentafor the in and out particles .

while in position space we need to specify the field configurationson early and late Cauchy slices .

→" µf•aao•qoBB_azBBqaBq⇒→

For gravitational scattering we want to allow fluctuations of thebulk geometry and will specify the free data at the conformal boundary .

Let's look at the Penrose diagram for Minka .

The causal structure ofthe boundary will be the same for asymptotically flat spacetimes .

✗ 2- = e#tan¥

it,

Itm 1=0

r

m= 0

t.io

✓⇒+ r

É

Massless excitations enter and exit along null hypersurfaces I-1=-112×5?We call this 5h cross -section the celestial sphere .

Let's compare this to the free mode expansion of the field operators :

hmu = If ,d¥p£wg[e%a✗ei9✗ + equate- i9× ]✗ = ±

The saddle point approximation localizes the momentum direction toalign 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 )

,Lio L J

wa -

I-

v=t+r 92=0

it

He, Lysov , Mitra ,

Strominger 44

For massive fields we have a similar identification between the latetime momentum and a point on the hyperboloid resolving timelike infinity .

ok, y , -2 , E) ~ e-

*Hrm E-" aly ,z,E)e-i-m-h.c.it

.

It✗ L S

r

ly,¥ , E)

% p2 = -m2

✓⇒+ r

É

de Boer,soloclutch in 103 Campiglia ,

Laddha 'is

we'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

ytr# corrections

d§= - did -2dudr +

2r28zz-dzdzi-2-MBdu2-rczzdz2-rcz-z-dz-2-DZC.cz dudz + DECEEdude + . . .

To study the phase space f symmetries one needs to *aµM¥ÑÑ••EEsoaÉÉ#_§¥→BÉ→§qgg.tlpick a convenient gauge I

8 specify physical falloffs µµRope

B-zoziez-s-r-IMM.at#-rgs@-zins¥1K

¥¥%EoIBondi

,van der Burg , Metzner

'62 Sachs '62

Residual diffeomorphisms that preserve the falloffs and act non - triviallyon the asymptotic data are part of the Asymptotic SymmetryGroup .

ASG =Allowed SymmetriesTrivial Symmetries

The ASG will be much larger then the group of isometries of anygiven spacetime within this class .

BMS 3 Poincaré# generators : A 10

Bondi,van der Burg , Metzner

'62 Sachs '62

8 Supertranslations induce angle - dependent shifts in the Time coordinate

} / = flz,E) Ju

It

8 Superrotations extend global conformal transformations to local ckvs

} / = YZ /z) Jz + E- DZYZ /E)Ju+ cc .It § 0

go0 o

o oo o g

o'o o

o

g0 ¥

oooo

g'

og

Bondi,van der Burg , Metzner

'62 Sachs '62 Barnich,Troessaert 41 $

These asymptotic symmetries manifest in scattering amplitudes assoft theorems

.

Lout / QS - SQ 1in) <→"m w <outlawed 51in> ✗ Coutts / in>w→o

8 The canonical charges are co-dim 2 and can be written as anintegral of data on It?

SduJul . ) <→ Io W

8 Most of the symmetries are spontaneously broken by the vacuumand the term that generates inhomogeneous shifts is linear in hmu

Q 3- Qs ~ Sdu Jufdw e-iwualwx)

Weinberg'65 Strominger 43 He

, Lysov , Mitra ,Strominger 44

Moreover,these Ward identities are naturally organized in terms

of currents in a 2D CFT.

4D ASG <→ Soft Theorem <→ 2D CFTWard Identity Ward Identity

g o

or•% ; o

O o g 8 ✗n

D o g ×

f o ¥ ×.

ooo,

-

g'

og

'

.

0

Strominger 43

In particular , the subleading soft graviton theorem gives us acandidate stress tensor

.

Tzz =g÷G fduudufdtdwfz.DE cww

However,

to construct operators with definite 2D conformal weightswe 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,Strominger 44 Kopec , Lysov , S.P. , Strominger 44 Kapec ,

Mitra,Radarin

,Strominger

' 16

In particular , the subheading soft graviton theorem gives us acandidate stress tensor

.

Tzz =g÷G fduudufdtdwfz.DE cww

However,

to construct operators with definite 2D conformal weightswe will need to change our scattering basis .

✗n

CTzz.ci#On---wim-wJwoiHalwx1S1in> ✗

r ✗.

h.EE/-w?wIsul,h-..--tzl-w?wisu ) .

.

I

Cachazo,Strominger 44 Kopec , Lysov , S.P. , Strominger 44 Kapec ,

Mitra,Radarin

,Strominger

' 16

A conformal primary wavefunction is a function of a bulk point X"

and a reference point we ① which transforms as follows

☒ Is /AT ✗"

i EI-÷ ,I = Icw + d) ☐+7 c-w-i-d-P-JD.sn/II,-lXiw.w-l

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

Taking an inner product of such a wavefunction with the field operatorgives a (quasi) - primary operator with 2D conformal dimensions and spin J .

0s ,±

☐⇒ twin)= i /①4×4 , II.⇒ 1×+1 ;with

These operators carry a ± label indicating in versus out ,selected by

taking uts ut ie .

u qm

9M = ( It WÑ , w-10 , ilño - wt , I - wñ)

~ ÷xP5. P

.

,Shao 47 Donnay,

S.P.,Puhm '

20

For m = 0 these wavefunctions are gauge equivalent to a Mellintransform of on- shell plane waves .

To" ' ☐ - l

Eu,. . . µ ,,,e±iw9X±- D

,J~ Idw w

A similar transform exists when m -40 .In either case

,we can apply

this transform directly to 5-matrix elements to land on Celestial Amplitudes .

(w , in ) (Z, E )

•

ly,¥ , E)w -

p2 = -m2 p2 = 0

So°°¥fd%G☐1y , z, E ; w ,wt / . ) fiodww"- ' f)

de Boer,soloclutch in 103 Cheung ,

de la Fuente,Sundrum '

16 S.P. , Shao ,Strominger 47

By construction , 5-matrix elements in this basisn¥→¥É¥⇒¥%%.ie#ioIiI.ATsi,zi,Ei)--T/fYdwwsi-l)Alwi,z- i

,E- i .ÉÉ¥¥¥i÷÷:÷÷!÷⇒→

i =L__oo-o--→ É¥E☒BI•ÉEBg¥jµµ*←⇒ __-osoo--s---=

will transform like comelators of quasi - primaries because theexternal particles are in boost eigenstates .

V11) 1h ,ñiz ,E) = kz + d)

-2h/ c-Etd )

-5h1h

,I ;E¥-b_d

,¥I¥->

• .

④@÷:§

i. :" ie

Joos'

62 Banerjee'18

e

By construction , 5-matrix elements in this basisn¥:i.¥E⇒¥¥¥ bi - 1) A /Wi

,zi

,E ;) €K→*:÷→%É⇒•r.ae#ioIiE.ATSi,Zi,-Zi)--T/fYdww

.EE#EIIxi--l-ooo-o---*É¥E☒BI•€÷agy¥jF.am#-----oso---------

will transform like comelators of quasi - primaries because theexternal particles are in boost eigenstates .

✓J;= jth helicity

azitb a-Ei -15Ñfsi, Fid , -¥+d ;) = ;ÉKcz . + d)

☐

(c-E. + d-1b¥] ATS; , zi , E ;)J J

• .

e e

• ! ;⑦@÷:§

i. i e

Joos'

62 Banerjee'18

e ee

This transformation is easily inverted for Si on the principal serieswhich capture finite energy radiation .

D= It it,

✗ c- IR

However translations shift the conformal dimension

pie = que>☐ ←→ → ☐+1

and we will want to analytically continue to SEE .

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 .

A /Wi , Zi , Ei ) = M ✗ 8"/Ep ;) ,

4. = Sw ; /4)← fixed

gLetting S = Ewi , q. =L-1W ; integrated

ijiodwiwisi-

4.) = foods g-""i-Ii dq.q.si-18kg.

- 1) l - li = I i= ,

so that for ne s the g. are localized by the 8"'Ep ;) and 8kg.

- 1)

n >

Git

<✓

2- ;

The remaining 5- functions imply singular low -point correlations .

S.P.,Shao

,Strominger 47 Schreiber, Vukovich ,

Zlotnikov'17

The aim of our mapfrom 413 5-matrix elements to 2D correlations

< Pout, i / S / pin , ;) /→ (OtdiJi -. .

0-

Aj , Jj- - )

is to be able to use CFT techniques to learn about amplitudes .

④

⑤

④ We are already seeing that this dual CFT is exotic , both in itscomplex spectrum ,

and in the singular behavior at n±4pt .

④Meanwhile

,the quantity we want to look at in 413 is also unusual ⑥

because we are probing scattering at all energy scales .

④• ⑧ ④ ⑥

④ ④②

② ④

To understand how to effectively use this framework we mustbuild up our dictionary !

;

Y.me?i:Ei-iiiiesianYEies#aima*+sme+ry:F÷¥¥÷*÷÷÷:÷j:÷jDressings Vertex Operators

✗ "Ssi"

g.⇒¥ ,¥-→÷i

Let us now examine how celestial amplitudes encode thebehavior of scattering in various limits

0 Infrared - soft -1hm 's,currents

, dressings , null states

8 Collinear - Celestial opes,conformal block decomposition

=8 Ultraviolet - convergence<→ ultrasoft , anti-Wilsonian paradigm

Since soft theorems motivated this program we should understandwhat they translate to in the celestial basis .

f.w*dw wa- l~also

that factorizations at various orders in w- o turn intofactorizations of the residues as A→ special values

<outta-

Cwg) 51in> = ? /I '%÷ - i P"Ep?g%") Coutts / in> + ocw)

↳ ¥,↳¥

These residues correspond to celestial currents .

☐

- y

Cheung ,de la Fuente

,Sundrum ' 16 Fan

, Fotopoulos , Taylor 49 Pate ,Radarin

,Strominger 49 Adamo, Mason ,

Sharma'19 Puhm

'19 Guevara 49

The familiar universal soft theorems correspond to Sforwhich the conformal primary wavefunctions are pure gauge .

G)Od,s=i(O , -1-00%-52c. pure gauge if

1J / A soft -1hm .Current Asymisym .

1 1 w"

J large v11 )

3/2 42 W-1/2 S large SUSY

2 1 w"

P supertranslations

0 WO T superrotations /Diff/54

Cheung ,de la Fuente

,Sundrum ' 16 5. P

.Shao / 17 Donnay, Puhm , Strominger

/ 18 Donnay , S.P. , Puhm'

205.1? ,Puhm ,Trevisani'21

These IS,J ) are also special from the point of view of the global

conformal multiple-1s . Primary states

41h15> =L , thin> = 0

have primary descendants when

41L - 1)"

1h,5) = - K / 2h-1k- 1) 1L - , )"-114,57=0

and ditto for -4 .

When both conditions are met we get nested primaries .

1¥ .

':-)

II.Ñ 1L- ,)"

I ¥4> I '¥)

I "¥¥>

These correspond to an infinite tower of 'conformallysoft' theorems

at de l -27=0. 01¥

a-

so out

0✓

> 0

>✓

While the soft theorems of the previous table descend to operatorsgenerating asymptotic symmetry transformations

OD, J,

> ⑤2-☐ ,-J

Q[y]=Jd%Osoft

- Y05ftc-

Banerjee , Pandey, Paul 49 Guevara , Himwich , Pate ,Strominger

' 21 S.P.,Puhm

,Trevisani

'21

These correspond to an infinite tower of 'conformallysoft' theorems

at de l -27=0. 01¥

a-

so out

0✓

to

>✓

Whose partners can themselves be expressed as primary descendantsof dressingmodes . ?

⑤D, J,

> ⑤2-☐ ,-J

>✓

Banerjee , Pandey, Paul 49 Guevara , Himwich , Pate ,Strominger

' 21 S.P.,Puhm

,Trevisani

'21

Soft factors relate amplitudes with and without an extra singleparticle emission /absorption .

Exchanges between the charged external legs exponential so thatamplitudes without soft radiation vanish .

This vanishing can be interpreted as non - conservation for the chargesgenerating asymptotic symmetry transformations .

One can avoid this vanishing by dressing the operators . m~g.gr

=

Chung' 65 Faddeev

,Kulish ' 70 Kapec , Perry ,

Radar ice,Strominger

' 17

These dressings can be adapted to the conformal basis .For ESM

W ;= exp [ - eQ ; %¥_pÉo ( Etna - E✗ µ at )] = ei QiIott ; it ;)

c- D= I TSLl 2 ,E) ascendant

and we see that the dressing takes the form of a vertex operator .Moreover the celestial amplitudes factorize

A =A soft 1- hard

where

1- soft = (eiQ.IO/ZiEil...eiQnEk-niEn1 )

while 1-hard equals the amplitude for dressed operators .

Arkani -Hamed,Pate

,Radariu

,Strominger 120

The spontaneous symmetry breaking dynamics for the 413asymptotic symmetries is captured by simple 2D models .

Large- V11 ) ←→ free boson

with the important observation that the levels of the 2Dcurrent algebra are set by the cusp anomalous dimension in 4D .

( Iolz,E) Idw

,inD= 4%2 1nA ,=p In / z -wt

Nande,Pate

,Strominger 47 HimWich , Narayanan, Pate, Paul , Strominger 120 Nguyen, Salzer '

20 -

'21

Currents arise from tuning si to special values .

Rather than tuningi,we can tune Zij

= Zi -Zj .

Collinear limits in 4D should be captured by a celestial OPE .

Consider

taking two gluons collinear

Osa, ,+ ,/Z

, ,E) O☐bµ ,

(Zz,Ea) ~ - if£÷ CCS , ,

Aa) O'

Dis z- l , -11( Zz 12=2)

A very interesting observation is that we can use symmetries tofind CCA

,S2)

.

:* ,

✗i

,

-

/

Fan, Fotopoulos , Taylor 49 Pate ,

Radarin,StromInger , Yuan 49

From 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 descendany relation for the subleading

soft gluon gives an additional global symmetry that imposesIS

,-2) CCS

,-1,121=14+12 - 3) CCS , ,Sa)

This recursion can be solved, giving

CCS, ,Sa) = BCS ,

-1,Da - 1)

,Blx

, g) =

which matches what we get from transforming the collinear limit .Pate ,

Radarin,StromInger , Yuan 49

Despite the singular behavior of low point functions ,collinear limits let

us extract the Ciju .

With this data and our understanding of the spectrum ,we can apply

CFT machinery to celestial amplitudes .For example :

8 Using symmetries to go beyond the leading singular terms ofthe OPE and constrain correlations .

8 Examining conformal block decompositions to interpret intermediateexchanges and radial quantization in CCFT.

pig

,

pi Q Q

V5.

s

pie>

pi, 02 04

Lam, Shao

'17 Nandan

,Schreiber

,Vukovich

, Zlotnikov'

19 Banerjee, Gosh , Gonzo'20 Law

,Zlotnikou 120 Fan

, Fotopoulos, Stieberger , Taylor , Zhu 121 Atanasov ,Melton

,Radar in

,Strominger

'21

Let's take a closer look at 2→ 2 scattering . Starting from themomentum space amplitude

1- = Mls,-4×8 " ' /Ep;)

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 43 - hi - hi-

5/3-Tri -ñi glitz - z )).

✗

is ;Zij Zij

• -in

-2 = E

✗

Arkani -Hamed,Pate

,Radariu

,Strominger 120

The stripped amplitude is probed at all energy scales

ftp.z/--fi0dwwB-1Mlw3-zwY8 The convergence of this integral for external for external dimensionson the principal series is tied to the UV behavior .

8 Poles at BE 223 present in field theory are absent in quantum gravity.

s s

us imprint of UV completion, . , . . .

M~EAI.co-2m if e- ✗'w'

ArKani -Hamed,Pate

,Radarin

,Strominger 120

Meanwhile, expanding M around w=O 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 Positivity constraints from amplitudes translate to positivity constraintson the residues of the simple poles at D= 222 .

I imprint of causality

Lam, Shao

'17 Arkani -Hamed

,Pate

,Radarice

,Strominger 120 Chang, Huang, Huang, Li

'21

We have a framework that

0 makes symmetry enhancements manifest ,

T.mn#Et..::::q-Lorentz Invariance Conformal Symmetry8 reorganizes soft and collinear limits , qq.snsinagrsi.im#s

Vertex operators

•

• Celestial OPES✗ I.;••.É¥e8 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 steps forward are to

8 Expand our dictionary8 Connect to adjacent subfields ①8 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

Relativity"""

cfyhim.

Ashtekar i_wm. Banerjee Himwich Miller

Wald Memory Effects Gosh Currents Paulcheung

Crawley Narayanan

Ruzziconi

one,

Oblak

VenugopalanBa"

SpectrumTnessaert

poss, Horizons gushing Guevarade " """

Saha

gna,FanCanonical Charges Donna

,zniooedou Conformal/ysoft Theorems

seraj Barnick Giribet Mirza'yanSP

'

Adamo' '

Fotopoulos ①①£ Yuan

Fiorucci Laddha compere Hamada Schreiber .'

Law ZhuPate

ToyModels✓

Conformal Block Decomposition nsusen↳ Asymptotic symmetries . '

Nandan

Holography"""

CampigliaWen Shiu Magnea

Mao Zktnik"Atanasov

Udovich Melton

gayer

Porfyriadis Kopec IR Divergences stromincserShao

He Lippstreu Dressings AshtekarLam

MitraMan" Miter" Bhatakar Chang Radariu Strings Mason Prabhu

CachaoLi Sen UV behavior Taylor Raju ¥_ sundrum

Soft Theorems

Black Hole Info - ParadoxAmira

""° Amplitudes Armani -Hamedstieoerser shrivastava

spence yelleshpur Huang Rojas Loop EffectsSupersymmetry Brandner Gonzalez Sharma Hawking Hao

Avery JiangTravaglini Rihm Collinear Limits Perry Firewalls

Lesson Double Copy Adamo .-3 verlinde

Dumitrescuschwab Casali