Amplitudes in N=8 supergravity and Wilson Loops

Transcription

1 Amplitudes in N=8 supergravity and Wilson Loops Gabriele Travaglini Queen Mary, University of London Brandhuber, Heslop, GT Brandhuber, Heslop, Nasti, Spence, GT [hep-th] [hep-th] Galileo Galilei Institute, Firenze, June 2008

2 Understand why scattering amplitudes (in maximally supersymmetric theories) are simple geometry in Twistor Space (Witten) recursive structures in the perturbative S-matrix of gauge theories Simplicity hidden by Feynman diagrams diagrams not not separately gauge invariant unphysical singularities Motivations Unitarity-based & twistor-inspired methods gauge-invariant, on-shell data at each intermediate step of calculation also in non-supersymmetric theories

3 Amplitudes in N=4 super Yang-Mills are even simpler (and more mysterious...) All one-loop amplitudes expressed in terms of box functions (Bern, Dixon, Dunbar, Kosower) Iterative structures in N=4 splitting amplitudes and planar MHV amplitudes (Anastasiou, Bern, Dixon, Kosower; Bern, Dixon, Smirnov) - Splitting amplitudes: universal quantities, govern collinear limits - MHV: gluon helicities are a permutation of planar: leading in 1/N

4 Intriguing relation to Wilson loops (Drummond, Korchemsky, Sokatchev + Henn; Brandhuber, Heslop, GT) Dual conformal symmetry - integral functions in planar amplitudes (Drummond, Henn, Smirnov, Sokatchev) - Wilson loops (Drummond, Henn, Korchemsky, Sokatchev) Maximal transcendentality

5 We will consider N=8 supergravity maximally supersymmetric nonplanar Our goals: look for iterative relations in N=8 supergravity MHV amplitudes No multi-particle poles - MHV in N=4/N=8; all-plus in non-supersymmetric YM/Gravity relate Wilson loops to amplitudes idea: find more similarities between the two maximally supersymmetric theories

6 Common features N=4/N=8: Absence of triangle and bubble subgraphs in amplitudes ( no-triangle hypothesis ) (Bern, Dixon, Perelstein, Rozowsky; Bern, Bjerrum- Bohr, Dunbar; Bjerrum-Bohr, Dunbar, Ita; Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager; Bjerrum-Bohr, Vanhove) N=8 conjectured to be perturbatively finite (Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager; Chalmers; Bern, Dixon, Roiban; Green, Russo, Vanhove; Bern, Carrasco, Dixon, Johansson, Kosower, Roiban) Gauge theory/gravity: KLT relations (Kawai, Lewellen, Tye) UV behaviour under complex shifts (Bedford, Brandhuber, Spence, GT; Cachazo, Svrcek; Benincasa, Boucher-Veronneau, Cachazo; Arkany-Hamed, Kaplan)

7 In the rest of the talk: MHV amplitudes in N=4 SYM iterative structures in the perturbative expansion Gregory Korchemsky s talk this afternoon one-loop n-point amplitudes and Wilson loops (Brandhuber, Heslop, GT) 4-point MHV amplitude in N=8 Supergravity (Brandhuber, Heslop, Nasti, Spence, GT) iterative structures Wilson loops

8 N=4 Yang-Mills

9 Simplest one-loop amplitude n-point MHV amplitude in N=4 SYM at one loop: A 1 loop MHV = A tree MHV ( ) Colour-ordered partial amplitude, leading term in 1/N Sum of two-mass easy box functions, all with coefficient 1 Q Diagrammatic interpretation

10 Computed in 1994 by Bern, Dixon, Dunbar and Kosower using unitarity Rederived in 2004 with loop MHV diagrams... (Brandhuber, Spence, GT)...and, more recently, with a weakly-coupled Wilson loop calculation, with the Alday-Maldacena polygonal contour (Brandhuber, Heslop, GT)

11 Surprising regularities at higher loops n-point MHV amplitude in N=4 SYM A n,mhv = A tree n,mhvm n M n := 1 + L=1 a L M (L) n? [ = exp L=1 a L( )] f (L) (ɛ)m (1) n (Lɛ)+C (L) + O(ɛ) (Bern, Dixon, Smirnov) a g 2 N/(8π 2 ) M (1) n (ɛ) is the all-orders in ɛ one-loop amplitude, D =4 2ɛ f (L) (ɛ) =f (L) 0 + ɛf (L) 1 + ɛ 2 f (L) 2 anomalous dimension of twist-two operators at large spin, Higher-loop amplitudes expressed in terms of lower loop amplitudes γ (L) K /4

12 First few terms of BDS conjecture: (take Log of the Ansatz) ( M (2) n = 1 2 M (3) n = 1 3 M (1) n ( M (1) n ) 2 (ɛ) + f (2) (ɛ) M (1) n (2ɛ) +O(ɛ) ) 3 (2) (ɛ) + M n (ɛ)m (1) n (ɛ) +f (3) (ɛ) M (1) n (3ɛ) +O(ɛ) and so on... Signature of two-loop iteration: M (2) n 1 2 Requires knowledge of lower-loop amplitude to higher orders in ɛ Go up by one loop only ( M (1) n ) 2 (ɛ) = f (2) (ɛ) M (1) (2ɛ) + O(ɛ) n One-loop amplitude

13 IR behaviour of Yang-Mills amplitudes Motivates BDS Ansatz (Anastasiou, Bern, Dixon, Kosower) Universal resummation of IR divergences A IR = n i=1 A div (s) =exp A div (s i,i+1 ) [ 1 8ε ( 2 a L s ) Lε γ (L) K L=1 µ 2 L 2 (for colour-ordered amplitudes) 1 4ε (Catani; Magnea, Sterman; Sterman, Tejeda-Yeomans) BDS: exponentiation of finite parts ( ] a L s ) Lε g (L) L=1 µ 2 L Exponentiated finite remainders approach constants (independent of kinematics and # of particles)

14 Checks of BDS conjecture Two and three loops at four points (Anastasiou, Bern, Dixon, Kosower; Bern, Dixon, Smirnov). Confirmed result for three-loop cusp anomalous dimension obtained assuming maximal transcendentality (Kotikov, Lipatov, Onishcenko, Velizhanin) Two loops at five points (Bern, Czakon, Kosower, Roiban, Smirnov) Problems begin at six points (Bern, Dixon, Kosower, Roiban, Spradlin, Vergu, Volovich) Exponent requires an additional finite remainder

15 N=8 Supergravity

16 N=8 supergravity MHV amplitudes At four points A N 4,MHV =8 = A tree 4,MHVM N 4 =8 tree-level amplitude factors out as in N=4 thanks to supersymmetric Ward identities Write M N =8 4 = 1 + L=1 M (L) 4 = exp [ L=1 m (L) 4 ] m (1) 4 = M (1) 4, m (2) 4 = M (2) Goal: compute the quantity In YM: ( (1)) 2 M 4 ( M (2) n 1 2 M (1) n ) 2 (ɛ) ( ) 2 M (2) YM 1 2 M (1) YM (ɛ) = f (2) (ɛ) M (1) YM (2ɛ) +O(ɛ)

17 One- and two-loop MHV amplitude One loop: ( M (1) κ ) 2 [ 4 = i s t u 2 I (1) 4 ] (s, t) +I(1) 4 (s, u) +I(1) 4 (u, t) (Green, Schwarz, Brink; Dunbar, Norridge) I (1) 4 (s, t) := d D l (2π) D 1 l 2 (l p 1 ) 2 (l p 1 p 2 ) 2 (l + p 4 ) 2 zero-mass box 2 3 No colour ordering for gravity sum over permutations (1234), (1342), (1423) 1 4

18 M (2) 4 = Two loops: ( κ ) 4 [ ] stu s 2 I (2),P 4 (s, t)+s 2 I (2),P 4 (s, u)+s 2 I (2),NP 4 (s, t)+s 2 I (2),NP 4 (s, u) + cyclic 2 (Bern, Dunbar, Dixon, Perelstein, Rozowsky) I (2),P are the planar and non-planar boxes 4, I (2),NP 4 I (2),P 4 (s, t) = d D l (2π) D d D k 1 (2π) D l 2 (l p 1 ) 2 (l p 1 p 2 ) 2 (l+k) 2 k 2 (k p 4 ) 2 (k p 3 p 4 ) 2 I (2),NP 4 (s, t) = d D l (2π) D d D k (2π) D 1 l 2 (l p 2 ) 2 (l+k) 2 (l+k+p 1 ) 2 k 2 (k p 3 ) 2 (k p 3 p 4 ) 2 s := (p 1 + p 2 ) 2,t:= (p 2 + p 3 ) 2,u:= (p 1 + p 3 ) 2 Laurent expansion explicitly evaluated by Smirnov and Tausk use it to study possible iterations

19 Iterative structure Main result: ( 2 M (2) n 1 2 M (1) n (ɛ)) = finite + O(ɛ) Finite remainder has uniform transcendentality π, log have transcendentality 1; ζ n, Li n have transcendentality n... Soft anomalous dimensions in N=4 obtained as leading transcendentality contribution of QCD result (Kotikov, Lipatov, Onishcenko, Velizhanin) Planar one- and two-loop box are transcendental. Specific combination of two-loop nonplanar box functions is transcendental another property in common with N=4 SYM? higher loops?

20 Remainder is simpler compared to full M (2) 4 M (2) 4 1 ( κ ) 4 2 (M(1) 4 )2 = [u 2[ k(y)+k(1/y) ] + s 2[ k(1 y)+k(1/(1 y) ] 8π where + t 2[ k(y/(y 1)) + k(1 1/y) ]] + O(ɛ) k(y) := L4 6 + π2 L 2 2 +i 4S 1,2 (y)l log4 (1 y)+4s 2,2 (y) 19π4 90 ( 2 3 π log3 (1 y) 4 3 π3 log(1 y) 4Lπ Li 2 (y)+4πli 3 (y) 4πζ(3) ) y = s/t, L := log(s/t)

21 IR behaviour of (super)gravity amplitudes Exponentiation of one-loop divergences (Weinberg) Similar to QED Soft and collinear amplitudes unrenormalised (Bern, Dunbar, Dixon, Perelstein, Rozowsky) No colour ordering: M IR = i<j M div (s ij ) 4 pts, one loop, M (1) IR = c Γ ( κ 2 ) 2 2 ɛ [ ] s log( s)+t log( t)+u log( u) ɛ 1 IR divergence softer than in YM

22 Our result is in agreement with the expected IR singularities Cancellation of leading and subleading singularities in the difference M (2) (M(1) 4 )2

23 Steven Weinberg, Phys. Rev. 140: B516 (1965)

24 Beyond four points One-loop amplitude no longer proportional to the tree-level amplitude Requires more thinking!

25 Wilson Loops Gregory Korchemsky s talk this afternoon

26 Amplitudes in N=4 and Wilson Loops (Drummond, Korchemsky, Sokatchev; Brandhuber, Heslop, GT; Drummond, Henn, Korchemsky, Sokatchev) MHV amplitudes in N=4 super Yang-Mills appear in a completely different calculation: < W[C] > Contour C is determined by the momenta of the scattered particles Strong coupling calculation of Alday and Maldacena

27 The contour of the Wilson loop: this contour corresponds to a seven-point amplitude colour ordering Tr(T 1 T 2 T 7 ) at strong coupling, boundary of worldsheet tends to boundary of dual AdS space as IR cutoff is removed p $ p %! $! %! & p i = k i k i+1 lightlike momenta p #! # p &! ' k s are T-dual (region) momenta p "! " p! momentum conservation!! p ' n p i = 0 i=1 closed contour dual conformal symmetry acts on the T-dual momenta

28 Result: < W[C] > is the n-point MHV amplitude in N=4 SYM (modulo tree-level prefactor) Completely unexpected! Eikonal approximation usually only reproduces IR behaviour; we also get finite parts Conjecture: (Log) < W[C] > = (Log) M persists at higher loops Recently checked at two loops by Drummond, Henn, Korchemsky, Sokatchev for the four-, five-, and six-point case Discussed earlier today by Gregory Korchemsky

29 < W[C] > at one loop, n points (Brandhuber, Heslop, GT) Calculation done (almost) instantly. Two classes of diagrams: p 4 k 5 p 5 k 6 p 4 k 5 q = p 5 k 6 k 4 p 6 k 4 p 6 p 3 k 7 p 3 k 7 k 3 p 7 k 3 p 7 p 2 p 1 k 1 k 2 Gluon stretched between two segments meeting at a cusp p = p 2 p 1 k 1 k 2 Gluon stretched between two non-adjacent segments A. Infrared divergent B. Infrared finite

30 Clean separation between infrared-divergent and infrared-finite terms Important advantage, as ε can be set to zero in the finite parts from the start From diagrams in class A : M (1) n IR = 1 ε 2 n i=1 ( ) ε si,i+1 s i,i+1 =(p i + p i+1 ) 2 is the invariant formed with the momenta meeting at the cusp µ 2

31 Diagram in class B, with gluon stretched between p and q gives a result proportional to F ε (s,t,p,q)= Z 1 0 P 2 + Q 2 s t dτ p dτ q [ ( ) P 2 +(s P 2 )τ p +(t P 2 )τ q +( s t + P 2 + Q 2 )τ p τ q ] 1+ε Explicit evaluation shows that this is the finite part of a 2-mass easy box function Two-dimensional representation of a four-dimensional integral function

32 s := (p + P) 2 p 4 k 5 q = p 5 k 6 k 4 p 6 p 3 k 7 t := (q + P) 2 k 3 p 7 p = p 2 p 1 k 1 k 2 ( ) Q In the example: p = p 2 q = p 5 P = p 3 + p 4, Q = p 6 + p 7 + p 1 One-to-one correspondence between Wilson loop diagrams and finite parts of 2-mass easy box functions Explains why each box function appears with coefficient equal to 1 in the expression of the one-loop N=4 MHV amplitude

33 Gravity Wilson Loops (Brandhuber, Heslop, Nasti, Spence, GT) Requirements for candidate Wilson loop: invariance under coordinate transformations contour dictated by particle momenta has the same symmetries as the scattering amplitude

34 Obvious choice: where U α β(c) := P exp Tr U(C) Γ is the Christoffel connection [ ] iκ dy µ Γ α µβ(y) C invariant under coordinate transformations already studied in perturbation theory by Modanese Result has nothing to do with amplitude! κ 2 C dx µ dy ν Γ α µβ(x)γ β να(y) κ 2 divergent expression, reminiscent of the loop equation... C dx µ dy µ δ (D) (x y)

35 Try again work in linearised approximation g µν (x) =η µν + κh µν (x) W [C] := [ ] exp iκ dτ h µν (x(τ))ẋ µ (τ)ẋ ν (τ) C Same expression used in gravity eikonal approximation (Kabat & Ortiz; Fabbrichesi, Pettorino, Veneziano, Vilkovisky ) For cusped contours, gauge invariance violated at cusps Exponent can be rewritten as where T µν (x) := energy-momentum tensor of free particle d D x T µν (x)h µν (x) dτ ẋ µ (τ)ẋ ν (τ) δ (D) (x x(τ))

36 Try anyway in order to have correct symmetries, we consider W := W [C 1234 ] W [C 1243 ] W [C 1324 ] C ijkl is a contour obtained by joining p i,p j,p k,p l in this order At one loop, W (1) = W (1) [C 1234 ]+W (1) [C 1243 ]+W (1) [C 1324 ]

37 Results Tree-level prefactor missing (as in YM) Relative normalisation between IR singular and finite parts incorrect by a factor of from overcounting cusp contributions in W; minus sign more difficult to explain Result gauge dependent (but very close to correct one...)

38 < W > at one loop Diagrammatics identical to YM case. 2 classes of diagrams: x 3 x 3 p 2 p 2 p 3 p3 x 2 x 2 p 1 x 1 x 4 p 4 x 1 p 1 p 4 x 4 Graviton stretched between two edges meeting at a cusp Graviton stretched between two non-adjacent edges A. Infrared divergent B. Infrared finite

39 From diagrams in class A (after summing over permutations): κ 2 c(ɛ) ɛ 2 [ ( s) 1 ɛ +( t) 1 ɛ +( u) 1 ɛ] leading divergence cancels due to s + t + u = 0 subleading term proportional to expected 1/ɛ term: M (1) IR = c Γ ( κ 2 ) 2 2 ɛ [ ] s log( s)+t log( t)+u log( u)

40 From diagrams in class B: κ 2 c(ɛ) u 2 1 [ 4 ( log 2 s ) + π 2] t finite part of zero-mass box function sum over all permutations reproduces finite part of amplitude ( M (1) κ ) 2 [ 4 = i s t u 2 I (1) 4 ] (s, t) +I(1) 4 (s, u) +I(1) 4 (u, t)

41 Conformal gauge (Brandhuber, Heslop, Nasti, Spence, GT) Defined as the gauge where cusp diagrams vanish used in Yang-Mills, where Wilson loop is gauge invariant In this gauge, we obtain the correct N=8 supergravity amplitude Special case of a de Donder gauge fixing: L = 1 2 ( µh νρ ) 2 +( ν h ν µ) ( µh λ λ) 2 + h λ λ µ ν h µν Free Lagrangian of linearised gravity L (gf) = α ( ν h ν µ 1 ) α-gauge fixing µh α α α = -2 usual de Donder gauge conformal gauge α = 2ɛ 1+ɛ

42 Graviton propagator in configuration space: [ conf µν,µ ν (x) ɛ+1 ɛ ( 1 ( x 2 ) η 1+ɛ µ (µη ν)ν + ) ] ɛ 1 2(ɛ+1) η 2 µν η µ ν +2( x 2 ) x 2+ɛ (µ η ν)(ν x µ ) Cfr. gluon propagator in configuration space: conf µν (x) ɛ+1 ɛ [ ] 1 ( x 2 +iε) η 1+ɛ µν 2 x µx ν x 2 J µν (x) := η µν 2 x µx ν x 2 Inversion tensor

43 Summary Not quite same iterative structure as in N=4 uniform transcendentality of the result finite remainder, relatively simple expression Wilson loop almost reproduces amplitude Gauge-dependent expression Result closely related to correct answer Conformal gauge Can we do better?