Modern methods in scattering amplitudes -- a bird’s eye view

Radu Roiban Pennsylvania State University Modern methods in scattering amplitudes -- a bird’s eye view of the last ~6 months

Radu Roiban Pennsylvania State University New New physics mathemacs

General principles String theory

AdS/CFT Color/kinemacs duality & & integrability QFT Double copy S matrix

Scaering Twistor and ambitwistor equaons string(s) Coffee Duality Color/kinemacs Bern, Carrasco, Johansson In perturbave field theory: L D d p 1 n C ni = ni(p↵ p, ✏ p↵,...) L loop L m 2+2L l i i · · m = i g D 2 A (2⇡) Si p a1bc ca2d i l=1 ↵i ↵i Ci = ...f f ... X2G3 Z Y Q are not ni Ci + Cj + Ck =0 ni + nj + nk =0 gauge-invariants $ • Kinemac Jacobi relaons color relaons required by gauge invariance 3 • Present in many theories: YM+maer, QCD, Coulomb branch, , Z-theory, BLG,… - 5-point 2-loop all-plus amplitude Mogull, O’Connell remarkably-complicated expression; remarkably bad powercounng - Explicit color/kinemacs-sasfying numerators for NLSM Du, Fu - Suggeson for a(nother) symmetry behind BCJ amplitudes relaons Brown, Naculich momentum-dependent shi of color factors - Generalizaon of BCJ amplitude relaons at higher loop Vanhove, Tourkine also He, Schloerer; earlier Boels, Isermann - Can be defined for form factors of certain operators; Boels, Kniehl, Tarasov, Yang first 5-loop computaon – the form factor of the 20’ operator in N=4 sYM Yang Can be defined for correlaon funcons of certain operators cf. Engelund, RR Color/kinemacs Bern, Carrasco, Johansson

- Generalizaon of BCJ amp. relaons at higher loop Vanhove, Tourkine also He, Schloerer; earlier Boels, Isermann - Tree amplitudes relaons rel’s between cuts w/ extra linear numerator factors expect rel’s between loop amp’s w/ extra linear numerator factors - From loop-level monodromy relaons in string theory (issues w/ moduli space integraon?) - Loop momentum-dependent relaons between amplitudes’ integrands up to total derivaves

- Examples in field theory limit at 1 loop: p 1 n k k A(2,...,i,1,i+1,...pp +1,...,n)+ k k A(2,...pp +1,...,i,1,i+1,...,n) 1 · 2...i | 1 · p+1...i | i=2 i=p X X n = A(2,...pp +1,...,i,1,i+1,...,n)[l k ] | · 1 i=p X A(1, 2 ...n)[l k1]+A(2, 1 ...n)[(l + k2) k1]+ + A(1 ...n 1, 1,n)[(l + k23...n 1) k1]=0 · · ··· · - Not restricted to 1-loop; 2-loop examples are available Using this result one may be able to argue for: - loop-level color/kinemacs duality w/o explicit construcon of integrand Color/kinemacs and the double copy Bern, Carrasco, Johansson In perturbave field theory: m 2+2L L D L loop L+1  d pl 1 nin˜i m = i D 2 M 2 (2⇡) Si p i l=1 ↵i ↵i ⇣ ⌘ X2G3 Z Y Q - Property of many pure & YM/Maxwell-Einstein SGs w/ further maer, open string theory, self-dual gravity, , EYM+SSB,… R + R3

- 5-loop double copy of N=4 sYM Sudakov form factor Gang Yang - first 5-loop N=8 supergravity expression - physical interpretaon is under debate; not necessarily a form factor of local op. - integraon remains an open problem see talk by Mao Zeng - on general grounds one expects it to have slightly worse UV properes than the 4-point 5-loop amplitude (effecvely less supersymmetry)

- Progress in the idenficaon of SG symmetries i.t.o. YM operaons Anastasiou, Borsten, Duff, Hughes, Marrani, - Double-copy structure of twin supergravies Nagy, Zoccali different susy compleon of idencal bosonic sectors

- Double-copy structure for (derivave) correcons to DBI ↵0 Carrasco, Mafra, Schloerer deformaons of NLSM; Z theory Color/kinemacs and the double-copy Bern, Carrasco, Johansson Classical soluons/resumed tree-level perturbaon theory: - me-dependent Kerr-Schild-type soluons Luna, Monteiro, Nicholson, O'Connell, White  µ ⌫ g =¯g + h h = k k g¯µ⌫ k k =0 (k D)k =0 µ⌫ µ⌫ µ⌫ µ⌫ 2 µ ⌫ · 1 M 1 Aµ = g kµ hµ⌫ = kµk⌫ 4⇡r ! 2 4⇡r - Perturbave soluon for several color charges asymptoc radiaon field for gravity+dilaton+massive scalar Goldberger, Ridgeway - Algorithm for perturbave construcon of general Luna, Monteiro, Nicholson classical soluons of gµ⌫ Bµ⌫ O'Connell, Ochirov, Westerberg, White Color/kinemacs and the double-copy Bern, Carrasco, Johansson - Technical issues: + can have unexpectedly high powers of loop mom. Mogull, O’Connell + frustrangly difficult to find manifest c/k-sasfying representaons - Many open quesons: - Can we double copy at loop level in the absence of a manifest c/k representaon? See talks by Chen & Carrasco - Develop 4-, 5- and higher-loop integraon technology See talk by Mao Zeng generalize 2-loop integral reducon strategy of Johansson, Kosower, Larsen; Zhang - All double copies of gauge th’s are supergravies, but See talk by Chiodaroli are all supergravies double copies? - Is there a criterion for when a q can/cannot be a double-copy? - Color/kinemacs vs. fundamental principle? - Complete explicit soluon for the tree-level S matrices More in other talks - Complete id. of (duality) symm’s and of their physical consequences See talk by Duff - Understand the kinemac algebra and its off-shell realizaon - Is there a direct link between c/k and the UV properes of double copy? - Are all classical soluons of (super)gravity double-copies? - Find explicit of (tree-level) S-matrix of perturbave a QFT in curved space - … Scaering equaons/CHY n k k i · j =0 ( )i =1,...,n i j 8 j=i=1 6X - Originally: dominant worldsheet config’s for high energy fixed angle scaering in s.t. Gross, Mende - In four dimensions: describe the curves in the connected prescripon (RSV) of Wien’s twistor string theory describe (s)YM and CSG Wien

- In all dim’s: Govern YM and gravity ( ) scaering amplitudes +Bµ⌫ , Cachazo, He, Yuan Also stability condion for a system of charges in two dimensions Scaering equaons n k k i · j =0 ( )i =1,...,n i j 8 j=i=1 6X - Originally: dominant worldsheet config’s for high energy fixed angle scaering in s.t. Gross, Mende - In four dimensions: describe the curves in the connected prescripon (RSV) of Wien’s twistor string theory describe (s)YM and CSG Wien

- In all dim’s: Govern YM and gravity ( ) scaering amplitudes +Bµ⌫ , Cachazo, He, Yuan

2 H Double-copy flavor: n =Pf0 n( k, ✏, )Pf0 n( k, ✏˜, ) ; = () I { } { } In C ! X Scaering equaons n k k i · j =0 ( )i =1,...,n i j 8 j=i=1 6X - Originally: dominant worldsheet config’s for high energy fixed angle scaering in s.t. Gross, Mende - In four dimensions: describe the curves in the connected prescripon (RSV) of Wien’s twistor string theory describe (s)YM and CSG Wien

- In all dim’s: Govern YM and gravity ( ) scaering amplitudes +Bµ⌫ , Cachazo, He, Yuan Extended to a myriad of other theories Cachazo, He, Yuan, Zhang, Mizera, Liu, … “compactify” Gravity: BI compacfy = dimensional reducon compactify compactify “compactify” squeeze EM DBI generalize = relink denominators & add traces generalize single trace “compactify” “compacfy” = compacfy + generalize EYM YM NLSM compactify squeeze = convert a graviton into a gluon

4 YMS squeeze by removing ½ of pol. tensor corollary generalize gen. YMS

Figure 1. Theories studied by CHY and operations relating them. null momenta of the massless particles in the scattering process, and

n k P ()= i , i=1 i X and G = SL(2, C) C3 is the residual gauge symmetry of the ambitwistor string ⇥ fixed according to the standard Fadeev-Popov procedure. The integrand naturally l r decomposes into factors I and I that depend on the i,ki and the polarization and/or colour data of the particles whose scattering is being computed and depends on the theory. The delta functions 1 ¯(z)=@¯ = ( z)( z)dz¯ 2⇡iz < = impose the scattering equations: k P ( )=0.Theintegralsessentiallyreducetoa i · i sum over solutions to the scattering equations of IlIr multiplied by a Jacobian factor. The Il and Ir can be chosen from five di↵erent choices and the various theories arise from the di↵erent possible combinations. Ambitwistor strings are chiral infinite tension analogues of RNS strings that can be interpreted, after reduction of constraints, as strings whose target space is the space

–3– Scaering equaons n k k i · j =0 ( )i =1,...,n i j 8 j=i=1 6X - More recently: - A lot of effort for solving scaering equaons and evaluang amp’s in this formalism - Organizaon of the expanded Pfaffian; -integraon Lam, Yao - Algorithm for obtaining covariant expressions for any Bjerrum-Bohr, Bourjaily, mulplicity and in any dimension Damgaard, Feng

- Studies of double-so lim’s, collinear limits at NLO Nandan, Plea, Wormsbecher; He,Liu, Wu; etc - Deformaons of YM by Tr[F 3] He, Zhang n m n m Pf = ( ) P = ( ) (N + c)P n i1...im ! Pn i>1 i1...im 1 i1...im n 1 i1...im n i + X+i =n i + X+i =n 1 ··· m 1 ··· m - Tree-level form factors in 4 dimensions He, Liu Brandhuber, Hughes, Panerai, Spence, Travaglini

- Explicit 1-trace n-gluon + 1, 2 & 3-graviton amplitudes and 2-trace all-mulplicity gluons in EYM i.t.o. pure-YM amplitudes Nandan, Plea, Schloerer, Wen (1-graviton from monodromy rel’s by Seberger, Taylor EYM from Scaering equaons

- CHY integrand for n gluons and r gravitons

- Idea: reorganize rather than integrate straight out; expose YM paral amplitudes

Nandan, Plea, Schloerer, Wen - 1-graviton amplitude:

- 2-graviton amplitude:

- 3-graviton amplitude: impressive but somewhat complicated

Alternave expressions from double copy -- more on Chiodaroli’s talk Scaering equaons – some open quesons

- Why scaering eqs describe both high-energy fixed angle scaering and low energy? Disnct and disconnected string theory regimes

- Are correcons to low energy limit of std. string theory governed by scaering ↵0 eqs?

- Is there a scaering eq presentaon of amplitudes for massive parcles/string states?

- Is there a modificaon of the CHY formalism that describes correlaon funcons? Form fcts are described by modified CHY & unitarity relates them to correlaon fcts

- …

- Is the ambitwistor string the only algorithmic way to generalize CHY to loop level? Twistor and ambitwistor and physical strings

Ambitwistor strings: - describe holomorphic maps to spaces of cpx null geodesics - natural framework for scaering equaons Mason, Skinner

+ Acons:

+ BRST closure + absence of : X()X(0) finite number of states h i closed: + Integrated v.ops.: open:

+ Correlaon funcons: Coordinate is Lagrange mulplier integrate out on any ⌃ Constraint on P : n kiµ Genus-dependent soluons: g =0 Pµ()=d i=1 i X funcons in vertex operators enforce the scaering equaons Twistor and ambitwistor and physical strings

Ambitwistor string at higher loops Adamo, Casali, Skinner Moduli space integral contains further constraints on P 2

One-loop amplitudes Geyer, Monteiro, Mason, Tourkine - Supersymmetric and non-supersymmetric theories He, Yuan - Complex structure parameter integraon Baadsgaard, Bjerrum-Bohr, Bourjaily, localizes on degeneraon loci Damgaard, Feng - General mulplicity formal expressions - Correct properes - Some explicit examples at low mulplicity - Relaon to the Q-cut construcon of field theory S matrix

Two-loop amplitudes Geyer, Monteiro, Mason, Tourkine - Complex structure parameter integraon localizes on degeneraon loci - Some four-point examples See later talks for details - Relaon to the Q-cut construcon of field theory S matrix Some open quesons:

- Relaon between ambitwistor string and physical string in presence of interacons

Various aempts Mason, Skinner; Casali, Tourkine Huang, Siegel, Yuan; Siegel

- Formulate and use ambitwistor string in curved target space Adamo Recent results for AdS 5 Adamo, Skinner, Williams

- Is there a criterion idenfying theories which admit an ambitwistor formulaon? EYM? Massive parcles?

- …. More excing results - Universality in string interacons - Evidence + conjecture that leading transcendental coefficient at each order in is universal in all string theories ↵0 Huang, Schloerer - Older conjecture that leading small terms are not renormalized between ↵0 weak and strong coupling (smallest number of derivaves) Chepelev, Tseytlin - Evidence + conjecture of linear relaon between heteroc string tree-level S matrix and single-trace S matrix of gauge mulplet, to all orders in ↵0 Schloerer - For , the 4-point amp. of any weakly-coupled ↵0s, ↵0t 1 Caron-Huot, Komargodski, th. of massive higher spins coincides with the Veneziano amp. Sever, Zhiboedov - Developments in/from the Q-cut representaon of S matrix - An aempt at a systemac classificaon of EFTs Cheung, Kampf, Novotny, Shen, Trnka - Properes of mul-loop integrated amplitudes - Integrand posivity amplitude posivity Dixon, Hippel, McLeod, Trnka - Bootstrap w/ Input from Steinmann relaons Caron-Huot, Dixon, McLeod, von Hippel - EYM from YM via collinear limits/graviton = 2 collinear gluons Seberger, Taylor - Developments in nonplanar on-shell diagrams Herrmann, Trnka Bourjaily, Franco, Galloni, Wen - … Many excing developments and we should be looking forward to the talks and discussions of the next week for more