Monodromies revisited with twisted homology and BCJ Piotr Tourkine, - PowerPoint PPT Presentation

Monodromies revisited with twisted homology and BCJ Piotr Tourkine, CNRS & LPTHE, Sorbonne Universités QCD meets Gravity V, dec 2019, UCLA, Mani L. Bhaumik Institute for Theoretical Physics [arXiv:1910.08514] [Work in progress] + Monodromy relations from twisted homology E. Casali, S. Mizera, P. Tourkine E. Casali, S. Mizera, P. Tourkine [arXiv:1608.01665] Phys.Rev.Lett. 117 (2016) 211601 Higher-loop amplitude monodromy relations in string and gauge theory P. Tourkine, P. Vanhove [arXiv:1707.05775] JHEP 1710 (2017) 105 One-loop monodromy relations on single cuts A. Ochirov, P. Tourkine, P. Vanhove 1

previously Set of linear relationship between open string theory loop integrands [arXiv:1608.01665] Phys.Rev.Lett. 117 (2016) 211601 Higher-loop amplitude monodromy relations in string and gauge theory α n P. Tourkine, P. Vanhove d 1 1 − α n z n 1 d − n [arXiv:1702.04963] Nucl.Phys. B925 (2017) 63-134 1 z − n Monodromy Relations in Higher-Loop String Amplitudes 2 d − α 4 n S. Hohenegger, S. Stieberger d 4 3 α z 4 d 3 [arXiv:1707.05775] JHEP 1710 (2017) 105 A 0 A B z 3 1 0 ( 1 1 c � b 1 ( 1 ) b ) One-loop monodromy relations on single cuts 1 a r 1 � C A + 0 1 1 1 0 d 1 − 2 β A. Ochirov, P. Tourkine, P. Vanhove a + 2 β 2 z 1 A 1 0 i β ( d b 1 ) i d a r 1 � 1 0 B i α 1 α r + 0 1 r ˜ ( 1 b ) ( 1 b i ) C � r + 0 1 1 C A + [arXiv:1910.08514] 0 0 1 + c d 1 B r i 0 � Monodromy relations from twisted homology α 1 r ˜ i − 1 + 1 ( b i ) r E. Casali, S. Mizera, P. Tourkine + i 1 C + c � i A 0 α A 0 0 i ˜ r i i a + i β i + 1 ( b i ) 1 β g B a � d i g 2 A 0 ( b i ) g ( b g ) r + i 1 1 B g ( c + A b g ) 0 0 i r i + g 1 d 1 c � g A 0 c + B ( g Q ( b g b g ) a + ) g g 1 g r + g 1 A 0 0 A g 0 0 g 2

now Will dissect the relations in field theory 3

motivations 4

motivations • string theoretic formalisms can often teach us a lot about generic properties of perturbative QFT • because the worldsheet is nicer than the worldline (true as well for CHY) • and because of non-pertubative physics + dualities + supersymmetry (but susy also on worldline, see Green Bjornsson) 5

motivations • The KLT relations between open (=gauge) and closed (=gravity) strings is somehow at the roots of this conference. open open closed = × • Looking at the monodromies, we will try to draw some conclusions concerning two related and interconnected questions • the “labeling problem” (BCJ around the loop) • some more speculative aspects on the double copy 6

BCJ around the clock 2 3 4 1 7

BCJ around the clock - = n t n u n s 8

BCJ around the clock • Do successive BCJ moves around the loop. • First and last diagrams do not match: there is a loop momentum shifting ambiguity . • Generalises to higher loops 9

BCJ around the clock • Do successive BCJ moves around the loop. • First and last diagrams do not match: there is a loop momentum shifting ambiguity . • Generalises to higher loops 10

BCJ around the clock • Do successive BCJ moves around the loop. • First and last diagrams do not match: there is a loop momentum shifting ambiguity . • Generalises to higher loops 11

Double-copy • works amazingly well • up to 4 loops • 5 loops : needs to be modified. Why ? [arXiv:1708.06807] Phys.Rev. D96 (2017) 126012 Five-loop four-point integrand of N =8 supergravity as a generalized double copy Z. Bern, J. J. M. Carrasco, W. Chen, H. Johansson, R. Roiban, M. Zeng 12

• In this talk, we will see how much we can extract from the monodromy relations and twisted homology to attack these two questions • Will require a closer look at the field theory limit of the monodromy relations 13

Outline • Loop momentum in string theory • Monodromy relations • Tree-level : sum and di ff erence • Loop-level : same 14

The field theory limit • One topology in string theory (genus = number of loops) descends to all possible graphs in the field theory limit • Scherk 70’s : tree-level • Bern-Kosower string based rules 90’s : one-loop 15

The loop momentum in string theory • String theory has a uniform definition of the loop momentum P μ = ∂ t X μ ℓ μ = ∮ ∂ t X μ d σ • zero-mode of momentum field : • Descends to all the topologies of graphs generated in the field theory limit. Example : 16

field theory limit of the monodromy relations 17

warm-up : tree-level α n − 1 α n α 4 C − α 3 z n − 1 z 4 z n d 2 d 3 d 4 d n − 2 d n − 1 d 1 d 1 z 2 z 3 C + 18

warm-up : tree-level z 1 1 0 ∞ z 2 z 3 z 4 e i πα ′ � s A ( s , u ) + A ( s , t ) + e − i πα ′ � t A ( t , u ) = 0 e − i πα ′ � s A ( s , u ) + A ( s , t ) + e + i πα ′ � t A ( t , u ) = 0 valid for complex kinematics so not quite just complex conjugation 19

warm-up : tree-level e i πα ′ � s A ( s , u ) + A ( s , t ) + e − i πα ′ � t A ( t , u ) = 0 e − i πα ′ � s A ( s , u ) + A ( s , t ) + e + i πα ′ � t A ( t , u ) = 0 Sum and di ff erence gives cosines and sines : A ( s , u ) = sin( α ′ � π t ) sin( α ′ � π s ) A ( t , u ) cos( πα ′ � s ) A ( s , u ) + A ( s , t ) + cos( πα ′ � t ) A ( t , u ) = 0 20

warm-up : tree-level e i πα ′ � s A ( s , u ) + A ( s , t ) + e − i πα ′ � t A ( t , u ) = 0 e − i πα ′ � s A ( s , u ) + A ( s , t ) + e + i πα ′ � t A ( t , u ) = 0 As , one is left with α ′ � → 0 A ( s , u ) + A ( s , t ) + A ( t , u ) = 0 sA ( s , u ) + tA ( t , u ) = 0 21

loop-level • generic mechanism • boundary e ff ects 22

One-loop A 00 A 0 z m a − a + b 1 d m − 1 b 1 β C + z m +1 z m +1 z m − 1 C − b 2 d m − 2 b 2 α m − 1 z m +2 z m +2 z m − 2 α m − 2 α m +1 B B α m +2 α 3 z n − 1 z n − 1 z 3 α n − 1 b n − m d 2 b n − m α 2 z n z n α n z 2 d 1 b n − m +1 b n − m +1 c − c + A 0 A 00 z m 23

One-loop A 00 A 0 z m a − a + b 1 d m − 1 b 1 β C + z m +1 z m +1 z m − 1 C − b 2 d m − 2 b 2 α m − 1 z m +2 z m +2 z m − 2 α m − 2 α m +1 B B α m +2 α 3 z n − 1 z n − 1 z 3 α n − 1 b n − m d 2 b n − m α 2 z n z n α n z 2 d 1 b n − m +1 b n − m +1 c − c + A 0 A 00 z m 24

Loop monodromies 4 3 2 25

Loop monodromies e i πα ′ � k 1 ⋅ ( k 2 + k 3 ) 4 1 4 3 2 e i πα ′ � k 1 ⋅ ℓ 3 3 3 4 1 4 1 2 2 e i πα ′ � k 1 ⋅ k 2 2 3 2 4 1 26

4 1 3 2 1 4 4 3 2 3 3 3 4 1 4 2 1 2 2 1 3 2 4 4 1 3 2 27

Relations I (1234) + e i πα ′ � k 1 ⋅ k 2 I (2134) + e i πα ′ � k 1 ⋅ ( k 2 + k 3 ) I (2314) − e i πα ′ � k 1 ⋅ ℓ I (234 | 1) + J up − J down = 0 Sum and di ff erence, plus field theory limit gives KK and fundamental BCJ Let’s look at the ’s in the formula above I 28

I (1234) + e i πα ′ � k 1 ⋅ k 2 I (2134) + e i πα ′ � k 1 ⋅ ( k 2 + k 3 ) I (2314) − e i πα ′ � k 1 ⋅ ℓ I (234 | 1) 29

I (1234) + e i πα ′ � k 1 ⋅ k 2 I (2134) + e i πα ′ � k 1 ⋅ ( k 2 + k 3 ) I (2314) − e i πα ′ � k 1 ⋅ ℓ I (234 | 1) 2 3 1 3 3 1 + ( ℓ + k 2 ) · k 1 + ( ℓ + k 23 ) · k 1 ℓ · k 1 � 1 4 2 4 2 4 30

2 3 1 3 3 1 + ( ℓ + k 2 ) · k 1 + ( ℓ + k 23 ) · k 1 ℓ · k 1 � 1 4 2 4 2 4 2 � · k 1 = ( � + k 1 ) 2 − � 2 2 ( � + k 2 ) · k 1 = ( � + k 1 + k 2 ) 2 − ( � + k 2 ) 2 2 ( � + k 2 + k 3 ) · k 1 = ( � + k 1 + k 2 + k 3 ) 2 − ( � + k 2 + k 3 ) 2 31

- 2 3 1 + 3 3 1 + + + + - - 1 4 2 4 2 4 � 2 � · k 1 = ( � + k 1 ) 2 − � 2 2 ( � + k 2 ) · k 1 = ( � + k 1 + k 2 ) 2 − ( � + k 2 ) 2 2 ( � + k 2 + k 3 ) · k 1 = ( � + k 1 + k 2 + k 3 ) 2 − ( � + k 2 + k 3 ) 2 32

• The graphs that appear in the monodromy relations are grouped by BCJ triplets. [arXiv:1608.01665] Phys.Rev.Lett. 117 (2016) 211601 Higher-loop amplitude monodromy relations in string and gauge theory P. Tourkine, P. Vanhove [arXiv:1707.05775] JHEP 1710 (2017) 105 One-loop monodromy relations on single cuts A. Ochirov, P. Tourkine, P. Vanhove • What happens at the boundary ? 33

I (1234) + e i πα ′ � k 1 ⋅ k 2 I (2134) + e i πα ′ � k 1 ⋅ ( k 2 + k 3 ) I (2314) − e i πα ′ � k 1 ⋅ ℓ I (234 | 1) + J up − J down = 0 Now let’s look at the J’s 34

Now let’s look at the J’s [arXiv:1707.05775] JHEP 1710 (2017) 105 One-loop monodromy relations on single cuts A. Ochirov, P. Tourkine, P. Vanhove 35

“Triangles” 2 + + . . . 2 2 1 n − 1 n − 1 n 1 n 1 n − 1 n • Bern-Kosower rules (90’s) : explain which integrands generate these triangles when z 1 → z n 36

Now let’s look at the J’s [arXiv:1707.05775] JHEP 1710 (2017) 105 One-loop monodromy relations on single cuts A. Ochirov, P. Tourkine, P. Vanhove New graphs : “ exotic triangles” [work in progress] z 1 z m = it z m = it z 1 ` ` m 1 m 1 37

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