MHV Amplitudes on Self-Dual Plane Waves Tim Adamo University of Edinburgh QCD Meets Gravity 12 December 2019 Work in progress with Lionel Mason & Atul Sharma (also work with E. Casali, A. Ilderton, S. Nekovar)
Motivation Many reasons to be interested in perturbative QFT in strong (non-trivial) background fields • Practical (strong field QED/QCD, cosmology, GWs, holography, non-perturbative effects) • Theoretical (probe robustness of structures in pQFT)
Motivation Many reasons to be interested in perturbative QFT in strong (non-trivial) background fields • Practical (strong field QED/QCD, cosmology, GWs, holography, non-perturbative effects) • Theoretical (probe robustness of structures in pQFT) Challenges cut across both! Example: tree-level gauge theory and gravity • Flat background: full tree-level S-matrix • Even simplest strong backgrounds: only 3- or (sometimes) 4-points
Today Is there any hope to make all-multiplicity statements on strong backgrounds?
Today Is there any hope to make all-multiplicity statements on strong backgrounds? YES! • Parke-Taylor-like formula for MHV gluon scattering on a self-dual plane wave background + ∞ � n � � i j � 4 � d x − e i F n ( x − ) � δ 3 k i + , ⊥ � 1 2 � � 2 3 � · · · � n 1 � i =1 −∞
Today Is there any hope to make all-multiplicity statements on strong backgrounds? YES! • Parke-Taylor-like formula for MHV gluon scattering on a self-dual plane wave background + ∞ � n � � i j � 4 � d x − e i F n ( x − ) � δ 3 k i + , ⊥ � 1 2 � � 2 3 � · · · � n 1 � i =1 −∞ • Full tree-level S-matrix conjecture for Yang-Mills on such backgrounds
Plane waves Solution to vacuum equations (in d dim.) with: • covariantly constant null symmetry n , • (2 d − 4) additional symmetries, • commuting to form Heisenberg algebra w/ center n
Plane waves Solution to vacuum equations (in d dim.) with: • covariantly constant null symmetry n , • (2 d − 4) additional symmetries, • commuting to form Heisenberg algebra w/ center n For Yang-Mills theory, PWs valued in Cartan of gauge group [Trautman, Basler-Hadicke, TA-Casali-Mason-Nekovar] d s 2 = 2 d x + d x − − ( d x ⊥ ) 2 , A = x ⊥ ˙ a ⊥ ( x − ) d x −
Plane waves Solution to vacuum equations (in d dim.) with: • covariantly constant null symmetry n , • (2 d − 4) additional symmetries, • commuting to form Heisenberg algebra w/ center n For Yang-Mills theory, PWs valued in Cartan of gauge group [Trautman, Basler-Hadicke, TA-Casali-Mason-Nekovar] d s 2 = 2 d x + d x − − ( d x ⊥ ) 2 , A = x ⊥ ˙ a ⊥ ( x − ) d x − a ⊥ ( x − ) compactly supported ↔ well-defined S-matrix [Schwinger, ˙ TA-Casali-Mason-Nekovar]
Self-dual plane waves In d = 4, complexify R 1 , 3 to C 4 : d s 2 = 2 ( d x + d x − − d z d ˜ z ). Require PW & self-duality: ∗ F = i F
Self-dual plane waves In d = 4, complexify R 1 , 3 to C 4 : d s 2 = 2 ( d x + d x − − d z d ˜ z ). Require PW & self-duality: ∗ F = i F ∂ Propagation direction of wave: n = ∂ x + � 1 � Since n 2 = 0 , α = ι α ˜ α , ι α = n α ˙ ι ˙ ι ˙ α = ˜ 0 Result: f ( x − ) d x − = ˜ z ˙ z ˙ f ( x − ) ι α ˜ α d x α ˙ α A = ˜ ι ˙ z ∧ d x − = ˙ α ∧ d x α ˙ F = ˙ f ( x − ) d ˜ ˙ β f ˜ ι ˙ α ˜ ι ˙ β d x α
SDPW kinematics The spinor-helicity formalism works on all plane waves [TA-Ilderton] SDPW have chiral on-shell kinematics
SDPW kinematics The spinor-helicity formalism works on all plane waves [TA-Ilderton] SDPW have chiral on-shell kinematics α : T a E ± α = λ α ˜ α ( x − ) e i φ k Gluon with incoming momentum k α ˙ λ ˙ α ˙ � x − z f ( x − ) + k φ k = k · x + e ˜ d t e f ( t ) k + On-shell kinematics: e α ( x − ) = λ α ˜ ˜ α := ˜ α f ( x − ) Λ ˙ α , Λ ˙ λ ˙ α + ˜ ι ˙ K α ˙ � k + α = ι α ˜ α = λ α ˜ Λ ˙ ι ˙ α α E − E + , α ˙ ι ˜ α ˙ � ι λ � [˜ λ ]
Twistor theory Twistor space: Z A = ( µ ˙ α , λ α ) homog. coords. on CP 3 PT = CP 3 \ { λ α = 0 } = CP 1 ⊂ PT via µ ˙ x ∈ C 4 given by X ∼ α = x α ˙ α λ α
Twistor theory Twistor space: Z A = ( µ ˙ α , λ α ) homog. coords. on CP 3 PT = CP 3 \ { λ α = 0 } = CP 1 ⊂ PT via µ ˙ x ∈ C 4 given by X ∼ α = x α ˙ α λ α Familiar applications from flat background: • Massless free fields ↔ cohomology on PT [Penrose, Sparling, Eastwood-Penrose-Wells] • Representation for on-shell scattering kinematics [Hodges] • Full tree-level S-matrix of N = 4 SYM [Witten, Berkovits, Roiban-Spradlin-Volovich] • Full tree-level S-matrix of N = 8 SUGRA [Cachazo-Skinner]
What’s this got to do with perturbation theory on SDPWs?
What’s this got to do with perturbation theory on SDPWs? Theorem [Ward, 1977] There is a 1:1 correspondence between: • SD SU( N ) Yang-Mills fields on C 4 , and • rank N holomorphic vector bundles E → PT trivial on every X ⊂ PT (+ technical conditions)
What’s this got to do with perturbation theory on SDPWs? Theorem [Ward, 1977] There is a 1:1 correspondence between: • SD SU( N ) Yang-Mills fields on C 4 , and • rank N holomorphic vector bundles E → PT trivial on every X ⊂ PT (+ technical conditions) Upshot: twistor theory trivializes the SD sector
SDPWs in Twistor Space Can construct E → PT explicitly; holomorphicity encoded by partial connection on E : � s [˜ ι µ ] � d s D = ¯ ¯ ¯ δ 2 ( ι − s λ ) ∂ + A , A = d t f ( t ) s C ∗ D 2 = 0 Easy to show that ¯
SDPWs in Twistor Space Can construct E → PT explicitly; holomorphicity encoded by partial connection on E : � s [˜ ι µ ] � d s D = ¯ ¯ ¯ δ 2 ( ι − s λ ) ∂ + A , A = d t f ( t ) s C ∗ D 2 = 0 Easy to show that ¯ Penrose transform: gluons encoded by E -twisted cohomology on PT - helicity ↔ H 0 , 1 D ( PT , O ( − 4) ⊗ E ) ¯ + helicity ↔ H 0 , 1 D ( PT , O ⊗ E ) ¯
So what?
So what? Can give perturbative formulation of SDPW background field Yang-Mills in PT [Mason, Boels, TA-Mason-Sharma] Properties: • Perturbative around SD sector • Generating functional for MHV interactions localized on lines X ⊂ PT • Explicit expansion using holomorphic trivialization of E | X
MHV amplitude Evaluating this expansion gives: + ∞ � � n � i j � 4 � d x − e i F n ( x − ) � δ 3 k i + , ⊥ � 1 2 � � 2 3 � · · · � n 1 � i =1 −∞ for Volkov exponent � x − n − 1 1 � K α ˙ α ( x − ) := K α ˙ α ( x − ) , F n ( x − ) := d t K 2 ( t ) i K + i =1
Twistor action proves formula is correct, but...
Twistor action proves formula is correct, but... Red flag: only one residual lightfront integral! Expect n − 2 for n -point tree amplitude
Twistor action proves formula is correct, but... Red flag: only one residual lightfront integral! Expect n − 2 for n -point tree amplitude Resolution: field redefinition recasts Yang-Mills action such that all MHV vertices have single lightfront integral [Mansfield]
Twistor action proves formula is correct, but... Red flag: only one residual lightfront integral! Expect n − 2 for n -point tree amplitude Resolution: field redefinition recasts Yang-Mills action such that all MHV vertices have single lightfront integral [Mansfield] Other sanity checks & features: • Explicit checks at 3- and 4-points • Perturbative limit (MHV n + background → MHV n +1 ) • Flat background limit • Generalization to N = 4 SYM
Full tree-level S-matrix? Easy guess for N k MHV, based on holomorphic maps Z : CP 1 → PT � n � � k +1 a =0 d 4 | 4 U a γ − 1 � A i γ i d σ i � i vol GL (2 , C ) tr σ i − σ i +1 i =1 where: a =0 U a σ a is a degree k + 1 holomorphic map • Z ( σ ) = � k +1 • { σ i } ⊂ CP 1 punctures on CP 1 • A ∈ H 0 , 1 D ( PT , O ⊗ E ) twistor wavefunctions ¯ • γ holomorphic frame trivializing E over image of Z
Full tree-level S-matrix? Easy guess for N k MHV, based on holomorphic maps Z : CP 1 → PT � n � � k +1 a =0 d 4 | 4 U a γ − 1 � A i γ i d σ i � i vol GL (2 , C ) tr σ i − σ i +1 i =1 where: a =0 U a σ a is a degree k + 1 holomorphic map • Z ( σ ) = � k +1 • { σ i } ⊂ CP 1 punctures on CP 1 • A ∈ H 0 , 1 D ( PT , O ⊗ E ) twistor wavefunctions ¯ • γ holomorphic frame trivializing E over image of Z Currently just a conjecture...
Summary Upshot: it is possible to make all-multiplicity statements in strong backgrounds! Many exciting things to do: • Prove/correct N k MHV conjecture • Gravitational SDPWs • Double copy for full tree-level SDPW S-matrix • Generalize to generic PW backgrounds
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