Towards the S-matrix of massless QFTs on the Riemann sphere Piotr - PowerPoint PPT Presentation

Towards the S-matrix of massless QFTs on the Riemann sphere Piotr Tourkine, University of Cambridge IGST 2017, ENS Paris In collaboration with Eduardo Casali, Yannick Herfray, - Yvonne Geyer, Lionel Mason, Ricardo Monteiro. - 1

Outline • Motivations • Review: CHY formulae & Twistor strings • Loops • Null strings 2

Scattering equations formalism [arXiv:1307.2199] Scattering of Massless Particles in Arbitrary Dimensions F. Cachazo, S. He, E. Y. Yuan [arXiv:1311.2564] Ambitwistor strings and the scattering equations L. Mason, D. Skinner 3

Scattering equations formalism [arXiv:1307.2199] Scattering of Massless Particles in Arbitrary Dimensions F. Cachazo, S. He, E. Y. Yuan [arXiv:1311.2564] Ambitwistor strings and the scattering equations L. Mason, D. Skinner Witten; Roiban Spradlin Volovich 4

« » Quote from David Skinner, at Amplitudes 2015 (Zürich) 5

The scattering equations Fairlie, Roberts ‘72, Gross Mende, ‘87 and n null momenta k i ∈ R d , k 2 P : CP 1 → C d Let i = 0 n k i � � z , z i ∈ CP 1 P ( z ) = k i = 0 dz z − z i i = 1 Scattering equations: z 2 z 1 k i · k j � Res P 2 � … = 0 z i = � z i − z j i , j z n 6

The scattering equations Fairlie, Roberts ‘72, Gross Mende, ‘87 k i · k j � Res P 2 � = 0 z i = � z i − z j i , j 7

The scattering equations k i · k j � Res P 2 � = 0 z i = � z i − z j i , j • Solve for z i in terms of k i . k j • SL(2,C) invariance • (n-3)! solutions Arise in large α ’ limit of string theory Gross, Mende ‘87 8

Cachazo-He-Yuan formulae (2013) n � � dz i ¯ � � sc. eqns. F ( k i , � j , z i ) � tree-level amplitude = i = 1 scattering data: 
 ⟶ F = I L × I R kinematics, polarisations, 
 colour structure, … I kin I kin ∼ gravity I colour I kin ∼ gauge theory I L / R ∈ { I kin , I colour } = ⇒ I colour I colour ∼ scalar

Cachazo-He-Yuan formulae (2013) n � � dz i ¯ � � sc. eqns. F ( k i , � j , z i ) � tree-level amplitude = i = 1 I kin I kin ∼ gravity I colour I kin ∼ gauge theory I L / R ∈ { I kin , I colour } = ⇒ I colour I colour ∼ scalar just for completeness : I kin = Pf � ( M ) Tr ( T a 1 . . . T a n ) I colour = ( z 1 − z 2 ) . . . ( z n − z 1 ) + permutations

Cachazo-He-Yuan formulae (2013) n � � dz i ¯ � � sc. eqns. F ( k i , � j , z i ) � tree-level amplitude = i = 1 � F � = n-point field theory amplitude � Jac solutions

The scattering equations • Global residue theorem Dolan, Goddard 2014 • Many works on avoiding having to solve Baadsgaard, Bjerrum-Bohr, Bourjaily, Damgaard, Feng K 2 → 0 K = k 1 + · · · + k k , z 2 z 1 z k+1 z 2 z 1 … ⟶ … … 1 z k z n 12 K 2 z n

Review, continued Cachazo He Yuan 2015 13

Ambitwistor strings x' x Mason, Skinner 2013 • A chiral string in ambitwistor space ∂ X − e � P · ¯ 2 P 2 S A = X' Σ X

Ambitwistor strings x' x Mason, Skinner 2013 • A chiral string in ambitwistor space ∂ X − e � P · ¯ 2 P 2 S A = X' Σ X Ambitwistor space = space of compexified null geodesics X' x' x X space-time

Mason, Skinner 2013 Ambitwistor strings x' x • A chiral string in ambitwistor space ∂ X − e � P · ¯ 2 P 2 S A = X' Σ X • Spectrum: type II supergravity α � = 0 of string theory.

Mason, Skinner 2013 Ambitwistor strings x' x • A chiral string in ambitwistor space ∂ X − e � P · ¯ 2 P 2 S A = X' Σ NO STRINGY MODES X • Spectrum: type II supergravity α � = 0 of string theory.

Mason, Skinner 2013 Ambitwistor strings x' x • A chiral string in ambitwistor space ∂ X − e � P · ¯ 2 P 2 S A = X' Σ NO STRINGY MODES X • Spectrum: type II supergravity α � = 0 of string theory. • Can be extended to loops + + ... Adamo, Casali, Skinner ’14, Ohmori ‘15 18

Review, continued • Deep link to colour kinematics • Extended to curved space Adamo, Casali, Skinner ‘14 • Computation of form factors He, Liu ’16; Brandhuber, Hughes, Panerai, Spence, Travaglini ’16; Bork Onishchenko ‘17 • ℐ models; explain the BMS origin of soft theorems Adamo, Casali, Skinner ’14; Geyer Lipstein Mason ‘14 • Ambitwistor models for a zoo of theories Casali, Geyer, Mason, Monteiro, Roehrig 19

Ambitwistor strings : loops ! • New zero modes for P (= loop momenta) • New scattering equations fixing the moduli of the surface in terms of the loop momenta • Total number of eqns : 3g-3+n 20

One-loop scattering equations Adamo, Casali, Skinner 2014 q = exp ( 2i πτ ) � � � 1 ( z ij , � ) = 0 , i = 2 , . . . , n − 1 � · k i + k i · k j � 1 j � = i n n � 2 + 2 θ � θ � θ 1 ( z 0i , � ) θ � � � θ 1 ( z 0i , � ) + θ 1 ( z 0j , � ) = 0 � · k i k i · k j 1 1 1 i = 1 i � = j Integrands are easy to write, string theory-like computation 21

One-loop scattering equations Adamo, Casali, Skinner 2014 • Difficulty: hard to solve. • number of solutions ? • modular invariance ? after all this is just field theory. • Only solution in special regime Casali, Tourkine 2015 22

Outline • Motivations. • Review: CHY formulae & Twistor strings Geyer Mason Monteiro Tourkine • Loops reloaded PRL 2015, JHEP 2016, PRD 2016 • Null strings 23

From loops to trees q = exp ( 2i πτ ) 24

From loops to trees 1-loop Ambitwistor string amplitudes have one additional scattering equation d τδ ( P 2 ) 25

From loops to trees q = exp ( 2i πτ ) 26

From loops to trees ∂ 1 q = exp ( 2i πτ ) ¯ δ ( f ( z )) := ¯ f ( z ) � 1 � dq � � = − 1 = ... = − 1 d τδ ( P 2 ) → ¯ � � � P 2 P 2 � 2 q � q = 0 Integration by parts: sets q=0 27

From loops to trees q = exp ( 2i πτ ) 28

From loops to trees + � 1 = � 2 × − � Integration by parts loop momentum � d D � 1 tree-level amplitude 1-loop amplitude = � 2 × with two more points 29

From loops to trees • (n-1)!-2(n-2)! solutions • n-point one-loop amplitudes in (super)Yang-Mills, (super)gravity • Maps to ‘Q-cut’ representation Baadsgaard, Bjerrum-Bohr, Bourjaily, Caron-Huot, Damgaard, Feng 30

Two loops Geyer Monteiro Mason Tourkine, PRD 2016 1 “ “ = 1 � 2 2 � 2 1 � 1 � 2 Prescription for max SUSY, 
 gravity and gauge theories, 
 at four points 31

the dream New formulation of the perturbative expansion based on this procedure to all loops +… + + + 1. whole new formalism 2. compact formulae 3. long term perspectives; resummation ? 32

the dream vs reality • Beyond 2 loops there are new unexpected issues • These models are really too unusual after all • So we looked for a string theory origin Siegel ‘15 33

Outline • Motivations. • Review: CHY formulae & Twistor strings • Loops Casali, Tourkine, JHEP 2016 • Null strings Casali, Herfray, Tourkine work in progress Bandos, 1404.1299 Bagchi, Chakrabortty, Parekh 1507.04361 34

A clue: the scattering equations • Also appear in the high energy limit of string theory ! 35

Paradox T = ( 2 πα � ) � 1 M 2 ~ T J M 2 J 36

Paradox T = ( 2 πα � ) � 1 tension to zero M 2 ~ T J M 2 Theory of higher spins J 37

Paradox T = ( 2 πα � ) � 1 tension to zero M 2 ~ T J M 2 Theory of higher spins vs Ambitwistor strings = 
 just normal QFTs J 38

LST action Lindstrom Sundborg Theodoris ’91 Nambu-Goto 39

LST action ↳ Hamiltonian action H = λ ( P 2 + T 2 X � 2 ) + µ P · X � Virasoro constraints 40

LST action 41

LST action Casali, Tourkine, 2016 Gauge λ =0 and special quantization reduces to ambitwistor string ∂ X − e � P · ¯ 2 P 2 S A = Σ � V α ∂ α X · V β ∂ β X Integrate P out: S LST = 42

LST action � V α ∂ α X · V β ∂ β X S LST = Lindstrom Sundborg Theodoris ’91 Bagchi, Chakrabortty, Parekh 1507.04361 equations of motion : 43

LST action � V α ∂ α X · V β ∂ β X S LST = equations of motion : D h αβ Open question : for Polyakov’s strings, gives integration over the moduli space. What does give here ? D V α 44

Classical and quantum Null = tensionless tension to zero ???

1+1+1+1+1+… = -1/2 In string theory, this Casimir effect 
 gives the tachyon.

1+1+1+1+1+… = -1/2 In string theory, this Casimir effect 
 gives the tachyon.

Classical and quantum ^ L 0 = + 2 � + + Quantum effect in the angular momentum suppresses wild fluctuations 48

Classical and quantum • Constraints : P 2 and P.X’ • Constraints form a GCA 2 (=BMS 3 ) algebra Bagchi, Gopakumar, Mandal, Miwa 0912.1090  [ L n , L m ] = ( n − m ) L n + m + d 6 m ( m 2 − 1 ) δ m + n , d = η µ  µ  [ L n , M m ] = ( n − m ) M n + m ,  [ M n , M m ] = 0 ,  49

Classical and quantum � � L 0 = nx n p − n → L 0 = n : x n p − n : L 0 → L 0 − 2  L m | phys � = 0 , m > 0   ( L 0 � 2 ) | phys � = 0 ,  M m | phys � = 0 , m � 0  50

A mysterious quantization ambiguity J. Gamboa, C. Ramirez, M. Ruiz-Altaba, 1989 “Weyl ordering” Normal ordering