celestial amplitudes
TRANSCRIPT
""⇒au.
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>
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