The Ultimate Revenge Of The Analytic S-matrix
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1 The Ultimate Revenge Of The Analytic S-matrix Rutger Boels Niels Bohr International Academy, Copenhagen 24th Nordic Network Meeting, Groningen Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 1 / 8
2 Opening credits R.B., Kasper Jens Larsen, Niels Obers, Marcel Vonk arxiv: [hep-th] R.B., Daniele Marmiroli, Niels Obers arxiv:0912.xxxx, 1001.xxxx [hep-th] Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 2 / 8
3 The big picture overall goal: scattering amplitudes with many legs and loops Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 3 / 8
4 The big picture overall goal: scattering amplitudes with many legs and loops Field theory: Feynman diagrams Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 3 / 8
5 The big picture overall goal: scattering amplitudes with many legs and loops Field theory: Feynman diagrams analytic S-matrix Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 3 / 8
6 The big picture overall goal: scattering amplitudes with many legs and loops Field theory: Feynman diagrams (off-shell techniques) analytic S-matrix (on-shell techniques) Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 3 / 8
7 Main idea of our work analytic S-matrix program (sixties) construct scattering amplitudes from their singularities and symmetry properties Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 4 / 8
8 Main idea of our work analytic S-matrix program (sixties) construct scattering amplitudes from their singularities and symmetry properties superseded by Lagrangian based approaches in seventies Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 4 / 8
9 Main idea of our work analytic S-matrix program (sixties) construct scattering amplitudes from their singularities and symmetry properties superseded by Lagrangian based approaches in seventies until recently, only success: string theory Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 4 / 8
10 Main idea of our work analytic S-matrix program (sixties) construct scattering amplitudes from their singularities and symmetry properties superseded by Lagrangian based approaches in seventies until recently, only success: string theory "Construction of a crossing-symmetric, Regge behaved amplitude for linearly rising trajectories" [Veneziano, 68] Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 4 / 8
11 Main idea of our work analytic S-matrix program (sixties) construct scattering amplitudes from their singularities and symmetry properties superseded by Lagrangian based approaches in seventies until recently, only success: string theory "Construction of a crossing-symmetric, Regge behaved amplitude for linearly rising trajectories" [Veneziano, 68] revival inspired by [Witten, 03] large # loops and legs in (S)YM and (SU)GRA [Bjerrum-Bohr s lectures] Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 4 / 8
12 Main idea of our work analytic S-matrix program (sixties) construct scattering amplitudes from their singularities and symmetry properties superseded by Lagrangian based approaches in seventies until recently, only success: string theory "Construction of a crossing-symmetric, Regge behaved amplitude for linearly rising trajectories" [Veneziano, 68] revival inspired by [Witten, 03] large # loops and legs in (S)YM and (SU)GRA [Bjerrum-Bohr s lectures] field theory is more advanced than string theory amplitude-wise idea: export techniques to string theory and exploit (cf. [Stieberger-Taylor, 06-]) Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 4 / 8
13 Britto-Cachazo-Feng-(Witten) on-shell recursion relations [04,05] amplitudes are above all functions of momenta (and quantum #s) Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 5 / 8
14 Britto-Cachazo-Feng-(Witten) on-shell recursion relations [04,05] amplitudes are above all functions of momenta (and quantum #s) change D-dim momenta while staying on-shell? p µ i ˆp µ i = p µ i + z n µ p µ j ˆp µ j = p µ j z n µ (p µ i n µ ) = (p µ j n µ ) = (n µ n µ ) = 0 Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 5 / 8
15 Britto-Cachazo-Feng-(Witten) on-shell recursion relations [04,05] amplitudes are above all functions of momenta (and quantum #s) change D-dim momenta while staying on-shell? p µ i ˆp µ i = p µ i + z n µ p µ j ˆp µ j = p µ j z n µ (p µ i n µ ) = (p µ j n µ ) = (n µ n µ ) = 0 amplitude A A(z) A(z) A(0) = z z=0 { } = Resz=finite + Res z= Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 5 / 8
16 Britto-Cachazo-Feng-(Witten) on-shell recursion relations [04,05] amplitudes are above all functions of momenta (and quantum #s) change D-dim momenta while staying on-shell? p µ i ˆp µ i = p µ i + z n µ p µ j ˆp µ j = p µ j z n µ (p µ i n µ ) = (p µ j n µ ) = (n µ n µ ) = 0 amplitude A A(z) A(z) A(0) = z z=0 { } = Resz=finite + Res z= finite z residues: lower point amplitudes Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 5 / 8
17 Britto-Cachazo-Feng-(Witten) on-shell recursion relations [04,05] amplitudes are above all functions of momenta (and quantum #s) change D-dim momenta while staying on-shell? p µ i ˆp µ i = p µ i + z n µ p µ j ˆp µ j = p µ j z n µ (p µ i n µ ) = (p µ j n µ ) = (n µ n µ ) = 0 amplitude A A(z)? A(z) { } A(0) = = Resz=finite + Res z= z=0 z finite z residues: lower point amplitudes recursion! Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 5 / 8
18 Britto-Cachazo-Feng-(Witten) on-shell recursion relations [04,05] amplitudes are above all functions of momenta (and quantum #s) change D-dim momenta while staying on-shell? p µ i ˆp µ i = p µ i + z n µ p µ j ˆp µ j = p µ j z n µ (p µ i n µ ) = (p µ j n µ ) = (n µ n µ ) = 0 amplitude A A(z)? A(z) { } A(0) = = Resz=finite + Res z= z=0 z finite z residues: lower point amplitudes recursion! string amplitudes as infinite sums over three point amplitudes different possible shifts related by crossing symmetry z related to Regge behavior Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 5 / 8
19 On-shell recursion relations in string theory announcement [us, 09] all disc-level amplitudes in a flat background obey on-shell recursion in any string theory (and probably the sphere too) Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 6 / 8
20 On-shell recursion relations in string theory announcement [us, 09] all disc-level amplitudes in a flat background obey on-shell recursion in any string theory (and probably the sphere too) example: Koba-Nielsen amplitudes large-z adjacent shifts non-adjacent shifts ( lim A(z) z α (p i +p i+1 ) 2 +1 z lim A(z) e ±(α )z z α (p i +p j ) 2 +1 z G 0 + O ( ) 1 z ( G 0 + O ( 1 z ) Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 6 / 8
21 On-shell recursion relations in string theory announcement [us, 09] all disc-level amplitudes in a flat background obey on-shell recursion in any string theory (and probably the sphere too) example: Koba-Nielsen amplitudes large-z adjacent shifts non-adjacent shifts ( lim A(z) z α (p i +p i+1 ) 2 +1 z lim A(z) e ±(α )z z α (p i +p j ) 2 +1 z G 0 + O ( ) 1 z ( G 0 + O ( 1 z analysis adjacent shifts adapted from [Bardakci-Ruegg, 68] non-adjacent shifts through monodromy relations [Plahte, 70] (see Søndergaard s talk) ) Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 6 / 8
22 On-shell recursion relations in string theory announcement [us, 09] all disc-level amplitudes in a flat background obey on-shell recursion in any string theory (and probably the sphere too) example: Koba-Nielsen amplitudes large-z adjacent shifts non-adjacent shifts lim z A(z) z α (p i +p i+1 ) 2 +? ( G 0 + O ( ) 1 z lim A(z) e ±(α )z z α (p i +p j ) 2 +? ( G z 0 + O ( ) 1 z analysis adjacent shifts adapted from [Bardakci-Ruegg, 68] non-adjacent shifts through monodromy relations [Plahte, 70] (see Søndergaard s talk) other states polarization dependent, probably universal smooth field theory limit for gluons, gravitons Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 6 / 8
23 CFT point of view how generic are recursion relations in string theory? curved backgrounds? loops? study CFT Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 7 / 8
24 CFT point of view how generic are recursion relations in string theory? curved backgrounds? loops? study CFT adjacent shifts from CFT argument (based on [Brower et.al., 06]) : e ip1x(y) :: e ip2x(0) : = ( ) 1 2πα p 1 p 2 : e i(p 1X(y)+p 2 X(0))+izn µ (X µ(y) X µ(0)) y assume yz 1 and integrate Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 7 / 8
25 CFT point of view how generic are recursion relations in string theory? curved backgrounds? loops? study CFT adjacent shifts from CFT argument (based on [Brower et.al., 06]) : e ip1x(y) :: e ip2x(0) : = ( ) 1 2πα p 1 p 2 : e i(p 1X(y)+p 2 X(0))+izn µ (X µ(y) X µ(0)) y assume yz 1 and integrate CFT derivation of monodromy relations [us, 09] based on :V 1 (z 1 ): :V 2 (z 2 ): :V 2 (z 2 ): :V 1 (z 1 ): R 12 in flat backgrounds R 12 e ±2πiα (p 1 p 2 )Heavy(z 1,z 2 ) Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 7 / 8
26 CFT point of view how generic are recursion relations in string theory? curved backgrounds? loops? study CFT adjacent shifts from CFT argument (based on [Brower et.al., 06]) : e ip1x(y) :: e ip2x(0) : = ( ) 1 2πα p 1 p 2 : e i(p 1X(y)+p 2 X(0))+izn µ (X µ(y) X µ(0)) y assume yz 1 and integrate CFT derivation of monodromy relations [us, 09] based on :V 1 (z 1 ): :V 2 (z 2 ): :V 2 (z 2 ): :V 1 (z 1 ): R 12 in flat backgrounds R 12 e ±2πiα (p 1 p 2 )Heavy(z 1,z 2 ) general: R 12 obeys the Yang-Baxter equation... Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 7 / 8
27 Conclusions and outlook proven on-shell recursion for open strings in flat backgrounds Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 8 / 8
28 Conclusions and outlook proven on-shell recursion for open strings in flat backgrounds not shown: B-fields, parallel separated branes, closed string, pure spinor super-shifts,... Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 8 / 8
29 Conclusions and outlook proven on-shell recursion for open strings in flat backgrounds not shown: B-fields, parallel separated branes, closed string, pure spinor super-shifts,... join the fun! calculate actual amplitudes extract information (e.g. non-abelian DBI action) study curved backgrounds study loops your question? Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 8 / 8
30 Conclusions and outlook proven on-shell recursion for open strings in flat backgrounds not shown: B-fields, parallel separated branes, closed string, pure spinor super-shifts,... join the fun! calculate actual amplitudes extract information (e.g. non-abelian DBI action) study curved backgrounds study loops your question? string theory: the ultimate revenge of the analytic S-matrix Rutger Boels (NBIA) Ultimate Revenge Of The Analytic S-matrix 24th Nordic Network Meeting 8 / 8
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