Topics in string amplitudes Harold Erbin Università di Torino (Italy) In collaboration with: – Corinne de Lacroix – Ashoke Sen – Juan Maldacena – Dimitri Skliros arXiv: 1810.07197 , 1906.06051 1 / 38
Outline: 1. Introduction Introduction Two-point amplitude Crossing symmetry: QFT Crossing symmetry: string theory Conclusion 2 / 38
Properties of string theory String theory = theory of extended objects ◮ consistency? (unitarity, crossing symmetry. . . ) ◮ differences with local point-particle QFT? ◮ non-locality? 3 / 38
Properties of string theory String theory = theory of extended objects ◮ consistency? (unitarity, crossing symmetry. . . ) ◮ differences with local point-particle QFT? ◮ non-locality? Point-particle QFT ◮ consistency assessed from S -matrix ◮ locality ∼ analyticity of S -matrix 3 / 38
Properties of string theory String theory = theory of extended objects ◮ consistency? (unitarity, crossing symmetry. . . ) ◮ differences with local point-particle QFT? ◮ non-locality? Point-particle QFT ◮ consistency assessed from S -matrix ◮ locality ∼ analyticity of S -matrix String theory properties 1. if possible, direct proof 2. otherwise, prove property consequence → indirect test 3 / 38
Properties of string theory String theory = theory of extended objects ◮ consistency? (unitarity, crossing symmetry. . . ) ◮ differences with local point-particle QFT? ◮ non-locality? Point-particle QFT ◮ consistency assessed from S -matrix ◮ locality ∼ analyticity of S -matrix String theory properties 1. if possible, direct proof 2. otherwise, prove property consequence → indirect test Natural framework: string field theory (off-shell, renormalization. . . ) 3 / 38
Plan Properties of (super)string amplitudes: 1. Tree-level 2-point amplitude with: Juan Maldacena, Dimitri Skliros [ 1906.06051 ] 2. Analyticity and crossing symmetry at all loops with: Corinne de Lacroix, Ashoke Sen [ 1810.07197 ] 4 / 38
2-point amplitude ◮ QFT A 2 ( k , k ′ ) = 2 k 0 (2 π ) D − 1 δ ( D − 1) ( k − k ′ ) (1-particle state normalization, cluster decomposition) 5 / 38
2-point amplitude ◮ QFT A 2 ( k , k ′ ) = 2 k 0 (2 π ) D − 1 δ ( D − 1) ( k − k ′ ) (1-particle state normalization, cluster decomposition) ◮ string theory 1 � d 2 z d 2 z ′ � V k ( z , ¯ z ) V k ′ ( z ′ , ¯ z ′ ) � A 2 ∼ S 2 Vol SL (2 , C ) 1 ∼ � V k ( ∞ , ∞ ) V k ′ (0 , 0) � S 2 Vol R + 5 / 38
2-point amplitude ◮ QFT A 2 ( k , k ′ ) = 2 k 0 (2 π ) D − 1 δ ( D − 1) ( k − k ′ ) (1-particle state normalization, cluster decomposition) ◮ string theory (standard lore) 1 � d 2 z d 2 z ′ � V k ( z , ¯ z ) V k ′ ( z ′ , ¯ z ′ ) � A 2 ∼ S 2 Vol SL (2 , C ) 1 ∼ � V k ( ∞ , ∞ ) V k ′ (0 , 0) � S 2 = 0 Vol R + BRST point of view: need N gh = 6 but only 2 operators c ¯ cV 5 / 38
2-point amplitude ◮ QFT A 2 ( k , k ′ ) = 2 k 0 (2 π ) D − 1 δ ( D − 1) ( k − k ′ ) (1-particle state normalization, cluster decomposition) ◮ string theory (standard lore) 1 � d 2 z d 2 z ′ � V k ( z , ¯ z ) V k ′ ( z ′ , ¯ z ′ ) � A 2 ∼ S 2 Vol SL (2 , C ) 1 ∼ � V k ( ∞ , ∞ ) V k ′ (0 , 0) � S 2 = 0 Vol R + BRST point of view: need N gh = 6 but only 2 operators c ¯ cV QFT result is universal → how to resolve contradiction? 5 / 38
2-point amplitude ◮ QFT A 2 ( k , k ′ ) = 2 k 0 (2 π ) D − 1 δ ( D − 1) ( k − k ′ ) (1-particle state normalization, cluster decomposition) ◮ string theory (standard lore) 1 � d 2 z d 2 z ′ � V k ( z , ¯ z ) V k ′ ( z ′ , ¯ z ′ ) � A 2 ∼ S 2 Vol SL (2 , C ) 1 ∼ � V k ( ∞ , ∞ ) V k ′ (0 , 0) � S 2 = 0 Vol R + BRST point of view: need N gh = 6 but only 2 operators c ¯ cV QFT result is universal → how to resolve contradiction? � V k ( ∞ , ∞ ) V k ′ (0 , 0) � S 2 ∝ δ (0) δ ( D − 1) ( k − k ′ ) = ∞ from on-shell + momentum conservation → ambiguous, need regularization / better gauge fixing 5 / 38
Analyticity and crossing symmetry Analyticity of n -point amplitude A n ( k 1 , . . . , k n ) ◮ starting point for other properties (crossing symmetry, dispersion relations) ◮ related to locality and causality 6 / 38
Analyticity and crossing symmetry Analyticity of n -point amplitude A n ( k 1 , . . . , k n ) ◮ starting point for other properties (crossing symmetry, dispersion relations) ◮ related to locality and causality Crossing symmetry: ◮ relations between amplitudes with exchange of particles/anti-particles in initial/final states ◮ often assumed or observed (scattering amplitude program. . . ) 6 / 38
Analyticity and crossing symmetry Analyticity of n -point amplitude A n ( k 1 , . . . , k n ) ◮ starting point for other properties (crossing symmetry, dispersion relations) ◮ related to locality and causality Crossing symmetry: ◮ relations between amplitudes with exchange of particles/anti-particles in initial/final states ◮ often assumed or observed (scattering amplitude program. . . ) Why a general proof? ◮ ensure observed examples not accident of simple amplitudes ◮ learn about fundamental properties of QFT 6 / 38
Method Proof idea in QFT [Bros-Epstein-Glaser, ’64-65] : 1. prove analyticity of S-matrix in “primitive domain” ∆ 2. analytic extension H (∆) 3. show that 2) ⇒ crossing symmetry 7 / 38
Method Proof idea in QFT [Bros-Epstein-Glaser, ’64-65] : 1. prove analyticity of S-matrix in “primitive domain” ∆ 2. analytic extension H (∆) 3. show that 2) ⇒ crossing symmetry Remarks: ◮ 1) is non-perturbative (full S-matrix) ◮ 2) and 3) are general statements from theory of several complex variables 7 / 38
Method Proof idea in QFT [Bros-Epstein-Glaser, ’64-65] : 1. prove analyticity of S-matrix in “primitive domain” ∆ 2. analytic extension H (∆) 3. show that 2) ⇒ crossing symmetry Remarks: ◮ 1) is non-perturbative (full S-matrix) ◮ 2) and 3) are general statements from theory of several complex variables String theory: ◮ interactions non-locality → no position space Green functions ◮ prove 1) perturbatively from Feynman diagrams 7 / 38
Outline: 2. Two-point amplitude Introduction Two-point amplitude Crossing symmetry: QFT Crossing symmetry: string theory Conclusion 8 / 38
Gauge-fixed amplitude ◮ 2-point amplitude A 0 , 2 ( k , k ′ ) = 8 πα ′− 1 � d 2 z d 2 z ′ � V k ( z , ¯ z ) V k ′ ( z ′ , ¯ z ′ ) � S 2 Vol K 0 K 0 := PSL (2 , C ) 9 / 38
Gauge-fixed amplitude ◮ 2-point amplitude A 0 , 2 ( k , k ′ ) = 8 πα ′− 1 � d 2 z d 2 z ′ � V k ( z , ¯ z ) V k ′ ( z ′ , ¯ z ′ ) � S 2 Vol K 0 K 0 := PSL (2 , C ) ◮ simple gauge-fixing A 0 , 2 ( k , k ′ ) = 8 πα ′− 1 � V k ( ∞ , ∞ ) V k ′ (0 , 0) � S 2 Vol K 2 K 2 := U (1) × R + = dilatation × rotation 9 / 38
Gauge-fixed amplitude ◮ 2-point amplitude A 0 , 2 ( k , k ′ ) = 8 πα ′− 1 � d 2 z d 2 z ′ � V k ( z , ¯ z ) V k ′ ( z ′ , ¯ z ′ ) � S 2 Vol K 0 K 0 := PSL (2 , C ) ◮ simple gauge-fixing A 0 , 2 ( k , k ′ ) = 8 πα ′− 1 � V k ( ∞ , ∞ ) V k ′ (0 , 0) � S 2 Vol K 2 K 2 := U (1) × R + = dilatation × rotation ◮ evaluate CFT correlation function + regularize zero-modes κ 0 → 0 (2 π ) D − 1 δ ( D − 1) ( k + k ′ ) 16 π 2 i δ ( κ 0 ) A 2 ( k , k ′ ) = lim α ′ Vol K 2 z ′ ) � S 2 = i (2 π ) D δ ( D ) ( k + k ′ ) Normalization: � V k ( z , ¯ z ) V k ′ ( z ′ , ¯ . | z − z ′ | 4 numerator = zero-modes e i ( k + k ′ ) · x for Lorentzian target spacetime 9 / 38
Compute CKV volume (1) ◮ Volume regularization � d 2 z � 2 π � ∞ d r Vol K 2 = | z | 2 = 2 d θ r 0 0 � ∞ � ∞ d τ e i ετ = 4 π d τ = 4 π lim ε → 0 −∞ −∞ Vol ε K 2 = 8 π 2 δ ( ε ) ◮ ( τ, ε ) Euclidean worldsheet (time, energy) on the cylinder (dimensionless) ◮ problem: Lorentzian spacetime, dimensionful energy → need Wick rotation and rescaling 10 / 38
Compute CKV volume (2) Jacobian from mode expansions without oscillators: 1. worldsheet Wick rotation τ = i t , ε = − i E 2. Lorentzian regularized volume Vol M , E K 2 = 8 π 2 i δ ( E ) 3. Lorentzian mode expansion X 0 = x 0 + α ′ k 0 t 4. scale between spacetime and worldsheet times / energies ξ 0 E = α ′ k 0 κ 0 t = = ⇒ α ′ k 0 ( ξ 0 , κ 0 ) dimensionful worldsheet variables 5. regularized Lorentzian volume Vol M ,κ 0 K 2 = 8 π 2 i δ ( κ 0 ) α ′ k 0 11 / 38
Result κ 0 → 0 (2 π ) D − 1 δ ( D − 1) ( k + k ′ ) 16 π 2 i δ ( κ 0 ) A 2 ( k , k ′ ) = lim α ′ Vol M ,κ 0 K 2 Vol M ,κ 0 K 2 = 8 π 2 i δ ( κ 0 ) α ′ k 0 Recover QFT result: A 2 ( k , k ′ ) = 2 k 0 (2 π ) D − 1 δ ( D − 1) ( k + k ′ ) 12 / 38
Recommend
More recommend
