A New Identity for Gauge Theory Amplitudes

Transcription

1 A New Identity for Gauge Theory Amplitudes Hidden Structures workshop, NBI Sept 10, 2008 Henrik Johansson, UCLA Z. Bern, J.J. Carrasco, HJ arxive:0805 : [hep-ph[ hep-ph] Z. Bern, J.J. Carrasco, L.J. Dixon, HJ, R. Roiban arxiv: [hep-th[ hep-th]

2 Outline Motivation Simplicity in scattering amplitudes UV superfiniteness of N = 8 SUGRA A hidden structure in cuts A surprising new identity at tree level New relations between partial amplitudes Beautiful reformulation of Kawai-Lewellen-Tye relations Outlook & Summary NBI Sept H. Johansson 2

3 Motivation - simplicity vs. complexity Physical theories - gravity and gauge theories - have surprisingly simple on-shell scattering amplitudes Feynman rules are much more complex - gravity in particular Adding SUSY increases complexity of Lagrangian & Feynman rules - yet scattering amplitudes becomes simpler Simplest theories? Arkani-Hamed, Cachazo, Kaplan N = 4 SYM - solvable (in t Hooft limit)? N = 8 SUGRA - perturbatively finite? spin theory see talks by Arkani-Hamed, Benincasa, Bjerrum-Bohr, Heslop, Ricci, Skinner, Spradlin & Vanhove S U S Y simplicity? NBI Sept H. Johansson 3

4 N = 8 supergravity superfinite N = 8 supergravity amplitude manifestly has diagram-by-diagram power counting of N = 4 SYM! Bern, Carrasco, Dixon, HJ, Kosower, Roiban; hep-th/ Bern, Carrasco, Dixon, HJ, Roiban arxiv: [hep-th] UV superfinite for D = 4, L = 3 After integration D = 6 divergence demonstrated bound is saturated NBI Sept H. Johansson 4

5 Four-loop calculation in progress N = 4 super-yang-mills case is complete N = 8 supergravity still in progress Bern, Carrasco, Dixon, HJ, Roiban (2008) Some N = 4 SYM contributions 50 distinct planar and non-planar diagrammatic topologies one Feynman diagram ~ terms in pure gravity NBI Sept H. Johansson 5

6 How we do it - the Unitarity Method optical theorem T I M E unitarity method Bern, Dixon, Dunbar and Kosower (1994) generalized unitarity Bern, Dixon and Kosower see Badger s talk quadruple cut (leading singularity) Britto, Cachazo, Feng; Buchbinder, Cachazo (2004) Cachazo and Skinner Cachazo, Spradlin, Volovich (2008) see Benincasa s talk on-shell 3-vertex (Witten) maximal cut Bern, Carrasco, HJ and Kosower (2007) NBI Sept H. Johansson 6

7 Maximal cuts - a systematic approach for any theory put maximum number of propagator on-shell simplifies calculation 3pt tree amplitudes on-shell systematically release cut conditions great control of missing terms 5pt tree Preferred technique for 4-loop calculation ---- exposes a hidden structure NBI Sept H. Johansson 7

8 A hidden structure emerges maximal cuts with one 4pt blob involves 3 diagrams surprise!.. these diagrams are not independent Yang-Mills 4 loops 1 = pt blob with off-shell internal momenta Yang-Mills 3 loops What is going on? 4pt blob (tree amplitude) has a hidden structure NBI Sept H. Johansson 8

9 Gauge theory at tree-level NBI Sept H. Johansson 9

10 Color decomposition Modern decomposition Alternative decomposition, 4pt example gauge invariant color factors Map color structures kinematic structures kinematic numerators absorbs 4-pt contact terms -- but gauge dependent! NBI Sept H. Johansson 10

11 A surprising 4pt identity color factors Jacobi identity for color And a Jacobi identity for kinematics Zhu; Goebel, Halzen, Leveille noticed this 4pt property in 1980 Kinematic numerators gauge dependent - but 4pt identity is gauge invariant gauge parameter NBI Sept H. Johansson 11

12 Similar identity at higher points Decomposing 5pt amplitude in terms of 15 cubic diagrams kinematic numerator color factor propagators Equivalent to partial amplitudes etc... Kinematic Jacobi identity holds but is no longer gauge invariant! NBI Sept H. Johansson 12

13 not gauge invariant yet physical In a general theory we can solve the 15 n i s at 5 pts 9 independent kinematic Jacobi identities plus 2 constraints: 4 undetermined n i s (pure gauge transformations) etc Any 5pt tree is a linear combination of two basis amplitudes Α 5 (.....) = α Α 5 (1,2,3,4,5) + β Α 5 (1,4,3,2,5) true for any gauge theory and in D-dimensions NBI Sept H. Johansson 13

14 Tree level n-points _ a conjecture A gauge theory tree amplitude can be expanded in purely cubic diagrams full amplitude partial amplitude color factors Jacobi identity true for both color and kinematics as long as gauge invariance is enforced for (n - 3)! partial amplitudes only (n - 3)! linearly independent partial amplitudes Checked through 8 pts! NBI Sept H. Johansson 14

15 All-n formula -- partial amplitude relations General relations for gauge theory partial amplitudes where and and and t k is the position of leg k in the set {ρ} Α n ( ) = α 1 Α n (1,2,..,n) + α 2 Α n (2,1,..,n) + + α (n-3)! Α n (3,2,..,n) NBI Sept H. Johansson 15

16 Examples 4pts 5pts 6pts NBI Sept H. Johansson 16

17 Gravity & Gauge theory relations NBI Sept H. Johansson 17

18 KLT relations Kawai, Lewellen and Tye Originally a string theory tree level identity closed string (left-mover open string) (right-mover open string) Field theory limit gravity theory ~ (gauge theory) (gauge theory) gravity states are direct products of gauge theory states 1 grav = 1 gauge 1 gauge NBI Sept H. Johansson 18

19 New identity + KLT Feeding the new identity through KLT gives + = a beautiful numerator squaring relationship Compare to gauge theory gauge theory color gravity = 2 Unlike KLT this squaring relationship is between local objects n i and is manifestly crossing (Bose) symmetric NBI Sept H. Johansson 19

20 Holds at all-n tree level At 5 points true given that n i and ñ i satisfy kinematic Jacobi identities gauge theory color gravity = 2 At n points Checked trough 8 points! NBI Sept H. Johansson 20

21 Gauge theory Outlook - beyond on-shell & tree-level? Generalization to loop-level kinematic Jacobi identity? Find special gauge where Feynman rules manifestly obeys the identity Lagrangian understanding highly desirable Gravity Gravity Feynman rules = (gauge theory Feynman rules) 2 possible? Might clarify the KLT relations in terms of the Lagrangians Possible matching off-shell and non-perturbative physics between gravity and gauge theory NBI Sept H. Johansson 21

22 Summary We uncovered a new Jacobi-like identity for tree diagrams in gauge theory amplitudes The identity is gauge dependent - presumably there is a special gauge where it is manifest - nonetheless the identity relates and constrains physical information The identity imply new relations for gauge invariant partial amplitudes Combine with KLT to uncover a new local and manifestly crossing (Bose) symmetric squaring relationship between gravity and gauge theories It is possible that this identity holds on the Lagrangian level - can deepen our understanding of gauge theory and gravity NBI Sept H. Johansson 22

23 Extra slides NBI Sept H. Johansson 23

24 Pure Yang-Mills Feynman gauge propagator : vertices: NBI Sept H. Johansson 24

25 Pure Einstein gravity de Donder gauge propagator : cubic vertex: After symmetrization ~ 100 terms! NBI Sept H. Johansson 25

26 Gauge theory amplitude properties Tree level, adjoint representation Well-known partial amplitude properties gauge invariant cyclic symmetry reflection symmetry (n - 1)! photon -decoupling identity Kleiss-Kuijf relations (n - 2)! New relations reduce independent basis to (n - 3)! NBI Sept H. Johansson 26

27 Identity holds in unitarity cuts A peek at loop level Some known examples point at loop-level identity but not checked in general NBI Sept H. Johansson 27