1. Gluon Scattering in N=4 Super-Yang-Mills Theory from Weak to Strong Coupling Lance Dixon (SLAC) LPTHE Jussieu 29 January 2008 with Z. Bern, D. Kosower, R. Roiban, V. Smirnov, M. Spradlin, C. Vergu, A. Volovich

2. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 2 N=4 Super-Yang-Mills Theory N=4 SYM: most supersymmetric theory possible without gravity: Interactions uniquely specified by gauge group, say SU(N c ), 1 coupling g Exactly scale-invariant (conformal) field theory:  (g) = 0 for all g all states in adjoint representation, all linked by N=4 supersymmetry

3. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 3 Planar N=4 SYM and AdS/CFT We will consider mostly the ’t Hooft limit,, with fixed, in which planar Feynman diagrams dominate AdS/CFT duality suggests that weak-coupling perturbation series in for large-N c (planar) N=4 SYM should have special properties, because large limit  weakly-coupled gravity/string theory Maldacena; Gubser, Klebanov, Polyakov; Witten

4. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 4 AdS/CFT in one picture

5. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 5 Gluon scattering in N=4 SYM Some quantities protected by supersymmetry, so pert. series in is trivial (e.g. energies of “BPS” states) 2  2 gluon scattering amplitudes are not protected How does series organize itself into simple result, from gravity/string point of view? Anastasiou, Bern, LD, Kosower (2002) Cusp anomalous dimension  K ( ) is a new, nontrivial example, solved to all orders in using integrability Beisert, Eden, Staudacher (2006) Recent strong-coupling confirmation for 2  2 scattering. But: problems for n gluons? Alday, Maldacena, 0705.0303[th], 0710.1060 [hep-th] Proposal:  K ( ) is one of just four functions of alone, which fully specify gluon scattering to all orders in  for any scattering angle  (value of t/s). And specify n-gluon MHV amplitudes. Bern, LD, Smirnov (2005)

6. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 6 Some questions you might have “What are gluons?” AdS/CFT most simply relates “glueballs” – color-singlet, gauge-invariant local operators – to modes of gravitational fields propagating in AdS 5 x S 5. But gluons are colored states – harder to picture at very strong coupling What does scattering mean in a conformal field theory, in which the interactions never shut off? What is the cusp anomalous dimension? What are the other functions of entering the scattering amplitude? What is the evidence for this proposal, at weak and at strong coupling?

7. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 7 Take a cue from perturbative QCD Gluons (in QCD, not N=4 SYM) are the objects colliding at the LHC (most of the time). Interactions between gluons never turn off in QCD either. In fact, it’s worse, due to asymptotic freedom – the coupling grows at large distances. Use dimensional regularization, with D=4-2 , to regulate these long-distance, infrared (IR) divergences. (Actually, dimensional reduction/expansion to preserve all the supersymmetry.) Dim. reg. breaks conformal invariance. Recover it by Laurent expansion around  = 0 through O(  0 ). In string theory, gluons can be “discovered” by tying open string ends to a D-brane in the IR, and using kinematics (large s and t) to force the string to stretch deep into the UV. But there is also a dim. reg. version of AdS 5 x S 5 Alday, Maldacena

8. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 8 Dimensional Regulation in the IR One-loop IR divergences are of two types: Soft Collinear (with respect to massless emitting line) Overlapping soft + collinear divergences imply leading pole is at 1 loop at L loops

9. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 9 IR Structure in QCD and N=4 SYM Pole terms in  are predictable due to soft/collinear factorization and exponentiation – long-studied in QCD, straightforwardly applicable to N=4 SYM Akhoury (1979); Mueller (1979); Collins (1980); Sen (1981); Sterman (1987); Botts, Sterman (1989); Catani, Trentadue (1989); Korchemsky (1989) Magnea, Sterman (1990); Korchemsky, Marchesini, hep-ph/9210281 Catani, hep-ph/9802439; Sterman, Tejeda-Yeomans, hep-ph/0210130 In the planar limit, for both QCD and N=4 SYM, pole terms are given in terms of: the beta function [ = 0 in N=4 SYM ] the cusp (or soft) anomalous dimension a “collinear” anomalous dimension

10. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 10 VEV of Wilson line with kink or cusp in it obeys renormalization group equation: Cusp anomalous dimension Polyakov (1980); Ivanov, Korchemsky, Radyushkin (1986); Korchemsky, Radyushkin (1987) Cusp (soft) anomalous dimension also controls large-spin limit of anomalous dimensions  j of leading-twist operators with spin j: Korchemsky (1989); Korchemsky, Marchesini (1993) Related by Mellin transform to limit of DGLAP kernel for evolving parton distribution functions f( x,  F ):  important for soft gluon resummations

11. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 11 Soft/Collinear Factorization Magnea, Sterman (1990); Sterman, Tejeda-Yeomans, hep-ph/0210130; talk by Moch S = soft function (only depends on color of i th particle) J = jet function (color-diagonal; depends on i th spin) h n = hard remainder function (finite as )

12. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 12 Simplification at Large N c (Planar Case) Soft function only defined up to a multiple of the identity matrix in color space Planar limit is color-trivial; can absorb S into J i If all n particles are identical, say gluons, then each “wedge” is the square root of the “gg  1” process (Sudakov form factor): coefficient of

13. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 13 Sudakov form factor Pure counterterm (series of 1/  poles); like  ( ,  s ), single poles in  determine K completely Factorization  differential equation for form factor finite as    contains all Q 2  dependence Mueller (1979); Collins (1980); Sen (1981); Korchemsky, Radyushkin (1987); Korchemsky (1989); Magnea, Sterman (1990) K, G also obey differential equations (ren. group): cusp anomalous dimension

14. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 14 are l -loop coefficients of General amplitude in planar N=4 SYM Solve differential equations for. Easy because coupling doesn’t run. Insert result for Sudakov form factor into n-point amplitude looks like the one-loop amplitude, but with  shifted to (l , up to finite terms Rewrite as collects 3 series of constants: loop expansion parameter:

15. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 15 Exponentiation in planar N=4 SYM For planar N=4 SYM, propose that the finite terms also exponentiate. That is, the hard remainder function h n (l) defined by is also a series of constants, C (l) [for MHV amplitudes]: In contrast, for QCD, and non-planar N=4 SYM, two-loop amplitudes have been computed, and hard remainders are a mess of polylogarithms in t/s Evidence based on two loops (n=4,5, plus collinear limits) and three loops (for n=4) and now strong coupling (n=4,5 only?) Bern, LD, Smirnov, hep-th/0505205 Anastasiou, Bern, LD, Kosower, hep-th/0309040; Cachazo, Spradlin, Volovich, hep-th/0602228; Bern, Czakon, Kosower, Roiban, Smirnov, hep-th/0604074 Alday, Maldacena, 0705.0303 [hep-th]

16. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 16 Evidence: from amplitudes computed via perturbative unitarity Expand scattering matrix T in coupling g Very efficient – especially for N=4 SYM – due to simple structure of tree helicity amplitudes, plus manifest N=4 SUSY Bern, LD, Dunbar, Kosower (1994) Insert expansion into unitarity relation Find representations of amplitudes in terms of different loop integrals, matching all the cuts  cutting rules: Landau; Mandelstam; Cutkosky

17. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 17 Generalized unitarity At one loop, efficiently extract coefficients of triangle integrals & especially box integrals from products of trees If one cut is good, surely more must be better Multiple cut conditions connected with leading singularities Eden, Landshoff, Olive, Polkinghorne (1966) Bern, LD, Kosower (1997); Britto, Cachazo, Feng (2004);…

18. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 18 Generalized unitarity at multi-loop level Cut 5-point loop amplitude further, into (4-point tree) x (5-point tree), in all inequivalent ways: In matching loop-integral representations of amplitudes with the cuts, it is convenient to work with tree amplitudes only. For example, at 3 loops, one encounters the product of a 5-point tree and a 5-point one-loop amplitude: Bern, LD, Kosower (2000); BCDKS (2006); BCJK (2007)

19. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 19 Planar N=4 amplitudes from 1 to 3 loops Bern, Rozowsky, Yan (1997) 2 Green, Schwarz, Brink (1982)

20. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 20 Integrals for planar amplitude at 4 loops Bern, Czakon, LD, Kosower, Smirnov, hep-th/0610248 diagrams with no 2-particle cuts

21. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 21 Integrals for planar amplitude at 5 loops Bern, Carrasco, Johansson, Kosower, 0705.1864[th] only cubic vertices (22) also quartic vertices (12)

22. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 22 Subleading in 1/N c terms Bern, Carrasco, LD, Johansson, Kosower, Roiban, hep-th/0702112 2 loops Additional non-planar integrals are required Coefficients are known through 3 loops: Bern, Rozowsky, Yan (1997) 3 loops

23. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 23 Patterns in the planar case At four loops, if we assume there are no triangle sub-diagrams, then besides the 8 contributing rung-rule & non-rung-rule diagrams, there are over a dozen additional possible integral topologies: Why do none of these topologies appear? What distinguishes them from the ones that do appear?

24. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 24 No explicit allowed (so OK) Surviving diagrams all have “dual conformal invariance” Although amplitude is evaluated in D=4-2 , all non-contributing no-triangle diagrams can be eliminated by requiring D=4 “dual conformal invariance” and finiteness. Take to regulate integrals in D=4. Require inversion symmetry on dual variables : Lipatov (2d) (1999); Drummond, Henn, Smirnov, Sokatchev, hep-th/0607160 Requires 4 (net) lines out of every internal dual vertex, 1 (net) line out of every external one. Dotted lines = numerator factors Two-loop example

25. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 25 Dual diagrams at four loops Present in the amplitude 2 diagrams possess dual conformal invariance and a smooth limit, yet are not present in the amplitude. But they are not finite in D=4 Drummond, Korchemsky, Sokatchev, 0707.0243[th] Not present: Requires on shell

26. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 26 Dual conformal invariance at five loops Only 34 are present in the amplitude Bern, Carrasco, Johansson, Kosower, 0705.1864[th] 59 diagrams possess dual conformal invariance and a smooth on-shell limit ( ) Drummond, Korchemsky, Sokatchev, 0707.0243[th] The other 25 are not finite in D=4 Through 5 loops, only finite dual conformal integrals enter the planar amplitude. All such integrals do so with weight. It’s a pity, but there does not (yet) seem to be a good notion of dual conformal invariance for nonplanar integrals…

27. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 27 Back to exponentiation: the 3 loop case L-loop formula: To check exponentiation at for n=4, need to evaluate just 4 integrals: elementary Smirnov, hep-ph/0305142 Use Mellin-Barnes integration method implies at 3 loops:

28. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 28 Exponentiation at 3 loops (cont.) Inserting the values of the integrals (including those with ) into using weight 6 harmonic polylogarithm identities, etc., relation was verified, and 3 of 4 constants extracted: n-point information still required to separate Confirmed result for 3-loop cusp anomalous dimension from maximum transcendentality Kotikov, Lipatov, Onishchenko, Velizhanin, hep-th/0404092 BDS, hep-th/0505205 Agrees with Moch, Vermaseren, Vogt, hep-ph/0508055

29. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 29 Full strong-coupling expansion Basso,Korchemsky, Kotański, 0708.3933[th] to all orders Benna, Benvenuti, Klebanov, Scardicchio [hep-th/0611135] Beisert, Eden, Staudacher [hep-th/0610251] proposal based on integrability Agrees with weak-coupling data through 4 loops Bern, Czakon, LD, Kosower, Smirnov, hep-th/0610248; Cachazo, Spradlin, Volovich, hep-th/0612309 Agrees with first 3 terms of strong-coupling expansion Gubser Klebanov, Polyakov, th/0204051; Frolov, Tseytlin, th/0204226; Roiban, Tseytlin, 0709.0681[th]

30. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 30 Pinning down CSV recently computed the four-loop coefficient numerically by expanding the same integrals to one higher power in  Cachazo, Spradlin, Volovich, 0707.1903 [hep-th] [3/2] Padé approximant incorporating all data strong coupling from Alday, Maldacena So far, no proposal for an exact solution for this quantity

31. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 31 Collinear limits Evidence for n>4: Use limits as 2 momenta become collinear: Tree amplitude behavior: One-loop behavior: Two-loop behavior: Bern, LD, Dunbar, Kosower (1994) strong-coupling: Komargodski, 0801.3274 [th]

32. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 32 Two-loop splitting amplitude iteration The two-loop splitting amplitude obeys: which is consistent with the n-point amplitude ansatz In N=4 SYM, all MHV helicity configurations are equivalent, can write Anastasiou, Bern, LD, Kosower, hep-th/0309040 and fixes n-point information required to separate these two

33. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 33 Two-loop directly for n=5 Collinear limits are highly suggestive, but not a proof (!) Even terms checked numerically with aid of Czakon, hep-ph/0511200 Cachazo, Spradlin, Volovich, hep-th/0602228 Bern, Rozowsky, Yan, hep-ph/9706392 Using unitarity, first in D=4, later in D=4-2 , the two-loop n=5 amplitude was found to be: + cyclic and odd Bern, Czakon, Kosower, Roiban, Smirnov, hep-th/0604074 + cyclic

34. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 34 Regge behavior In principle, may constrain BDS ansatz. Supporting evidence for n=4,5. Some technical problems with BNST(v1) for n>5. Naculich, Schnitzer, 0708.3069 [hep-th] Brower, Nastase, Schnitzer,Tan, 0801.3891 [hep-th]

35. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 35 Scattering at strong coupling Use AdS/CFT to compute an appropriate scattering amplitude High energy scattering in string theory is semi-classical Evaluated on the classical solution, action is imaginary  exponentially suppressed tunnelling configuration Alday, Maldacena, 0705.0303 [hep-th] Gross, Mende (1987,1988) Better to use dimensional regularization instead of r

36. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 36 Dual variables and strong coupling T-dual momentum variables introduced by Alday, Maldacena Boundary values for world-sheet are light-like segments in : for gluon with momentum For example, for gg  gg 90-degree scattering, s = t = -u/2, the boundary looks like: Corners (cusps) are located at – same dual momentum variables introduced above for discussing dual conformal invariance of integrals!!

37. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 37 Cusps in the solution Near each corner, solution has a cusp Classical action divergence is regulated by Kruczenski, hep-th/0210115 Cusp in (y,r) is the strong-coupling limit of the red wedge; i.e. the Sudakov form factor. See also Buchbinder, 0706.2015 [hep-th]

38. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 38 The full solution Divergences only come from corners; can set D=4 in interior. Evaluating the action as   0 gives: combination of Alday, Maldacena, 0705.0303 [hep-th]

39. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 39 Dual variables and Wilson lines at weak coupling Inspired by Alday, Maldacena, there has been a sequence of recent computations of Wilson-line configurations with same “dual momentum” boundary conditions: One loop, n=4 Drummond, Korchemsky, Sokatchev, 0707.0243[th] One loop, any n Brandhuber, Heslop, Travaglini, 0707.1153[th]

40. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 40 Dual variables and Wilson lines at weak coupling (cont.) Two loops, n=4 In all 3 cases, Wilson-line results match the full scattering amplitude [the MHV case for n>5] (!) – up to an additive constant in the 2-loop case. Drummond, Henn, Korchemsky, Sokatchev, 0709.2368[th] + … DHKS also remark that the one-loop MHV N=4 SYM amplitudes obey an “anomalous” (due to IR divergences) dual conformal Ward identity, which totally fixes their structure for n=4,5.

41. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 41 Assuming dual conformal invariance, first possible nontrivial “discrepancy” function from BDS, for MHVamplitudes or Wilson lines, is at n=6, where “cross-ratios” appear. [Not n=4 because x 2 i,i+1 = 0.] Dual variables and Wilson lines at weak coupling (cont.) Drummond, Henn, Korchemsky, Sokatchev, 0712.4138 [th] computed the two-loop Wilson line for n=6, and found a discrepancy + … What does this mean for amplitudes?

42. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 42 Two-loop 6-point amplitude l 2 (-2  ) l 2 One loop n=6 integrals: Bern, LD, Kosower, R. Roiban, M. Spradlin, C. Vergu, A. Volovich, in progress dual conformal invariant in D=6

43. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 43 Two loop n=6 even integrals l 2 (-2  ) all with dual conformal invariant integrands (including prefactors)

44. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 44 Two loop n=6 status “Ansatz” on previous slide still needs further cross-checking, but most likely is OK, at least through O(  ) 1/  4, 1/  3, 1/  2, 1/  all OK Need better numerics on crucial  0 terms We can say, from one higher-precision kinematic point, that the BDS ansatz needs correction. But we cannot say yet whether the correction function is dual conformal invariant, or whether it agrees (numerically) with DHKS’s function. Stay tuned!

45. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 45 Conclusions & Open Questions Remarkably, finite terms in planar gg  gg amplitudes in N=4 SYM exponentiate in a very similar way to the IR divergences. Full amplitude seems to depend on just 4 functions of  alone (one already “known” to all orders, so n=4 problem (also n=5) may be at least “1/4” solved! What is the AdS/operator interpretation of the other 3 functions? Can one find integral equations for them? How is exponentiation/iteration related to AdS/CFT, integrability, [dual] conformality, and Wilson lines? What happens for MHV amplitudes with n=6? What is the n=6 Wilson line function of DHKS, which vanishes in all collinear limits? Is it the same as that appearing in the n=6 amplitudes? What happens for non-MHV amplitudes? From structure of 1-loop amplitudes, answer must be more complex.

46. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 46 Extra Slides

47. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 47 “Leading transcendentality” relation between QCD and N=4 SYM KLOV (Kotikov, Lipatov, Onishschenko, Velizhanin, hep-th/0404092) noticed (at 2 loops) a remarkable relation between kernels for BFKL evolution (strong rapidity ordering) DGLAP evolution (pdf evolution = strong collinear ordering)  includes cusp anomalous dimension in QCD and N=4 SYM: Set fermionic color factor C F = C A in the QCD result and keep only the “leading transcendentality” terms. They coincide with the full N=4 SYM result (even though theories differ by scalars) Conversely, N=4 SYM results predict pieces of the QCD result transcendentality (weight): n for  n n for  n Similar counting for HPLs and for related harmonic sums used to describe DGLAP kernels at finite j

48. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 48 Generalized cuts computed at 4 loops Graph detection table

49. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 49 Four loops Four-loop integrals evaluated semi-numerically. Programs to automate extraction of 1/  poles from Mellin-Barnes integrals, set up numerical integration over the multiple inversion contours. Anastasiou, Daleo, hep-ph/0511176; Czakon, hep-ph/0511200; AMBRE [Gluza, Kajda, Riemann], 0704.2423[ph]; Smirnov talk No explicit check of exponentiation yet ( O(  0 ) ) However, and can be extracted from 1/  2 and 1/  coefficients in four-loop amplitude. Also need lower-loop integrals. For, only need to same order as for  0 coefficient of three-loop amplitude

50. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 50 Four-loop cusp anomalous dimension Working at, we found Existing prediction based on integrability (with no dressing factor) was: Could rewrite difference as: flipped the sign of the  3 2 term in the ES prediction, leaving the  6 term alone Later, precision on r improved dramatically to: Cachazo, Spradlin, Volovich, hep-th/0612309 BCDKS, hep-th/0610248 Eden, Staudacher, hep-th/0603157

51. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 51 Interlude: Integrability for all orders For N=4 SYM, this Hamiltonian is integrable: – infinitely many conserved charges – spectrum contains quasi-particles (magnons) – magnon scattering controlled by 2  2 S-matrix obeying Yang-Baxter equation Lipatov (1993); Minahan, Zarembo (2002); Beisert, Staudacher (2003) ;… Local single-trace operators can be mapped to 1-dimensional spin systems Anomalous dimensions found by diagonalizing dilation operator  spin-chain Hamiltonian. In planar limit, Hamiltonian is local, though range increases with number of loops

52. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 52 Solve system for any coupling by Bethe ansatz: – multi-magnon states with only phase-shifts induced by repeated 2  2 scattering – periodicity of wave function leads to Bethe condition depending on length of chain L – in limit L  ∞ this becomes an integral eqn. – 2  2 S-matrix almost fixed by symmetries, but an overall phase, the dressing factor, is not so easily deduced. – in general there can be a wrapping problem when interaction range (~number of loops) is > L ; but it is argued that the cusp anomalous dimension (L = 2, j  ∞) is immune. Integrability (cont.) Bethe (1937); … Staudacher, hep-th/0412188; Beisert, Staudacher, hep-th/0504190; Beisert, hep-th/0511013, hep-th/0511082; Eden, Staudacher, hep-th/0603157

53. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 53 Integrability (cont.) Beisert, Eden, Staudacher [hep-th/0610251] investigated strong-coupling properties of dressing factor BES found a way to continue the dressing factor to weak coupling, and proposed a new integral equation, whose only effect (remarkably) at weak coupling was to flip signs of odd-zeta terms in the ES prediction (actually,  2k+1  i  2k+1 ) – in precise agreement with our [BCDKS] simultaneous 4-loop calculation and 5-loop estimate (based on weak-strong interpolation). Arutyunov, Frolov, Staudacher, hep-th/0406256; Hernández, López, hep-th/0603204; …

54. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 54 Weak/strong-coupling interpolation Kotikov, Lipatov and Velizhanin (KLV), hep-ph/0301021 proposed the formula: to interpolate between at weak coupling and at strong coupling. Strong-coupling prediction from AdS/CFT (energy of spinning folded string): Gubser Klebanov, Polyakov, hep-th/0204051 Frolov, Tseytlin, hep-th/0204226 Roiban, Tseytlin, 0709.0681[th] computed term

55. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 55 Interpolation (cont.) Using 4 weak-coupling coefficients, plus [0,1,2] strong-coupling coefficients gives very consistent results: These 3 curves predict the five-loop coefficient: ES predicted 131.22 – but flipping signs of odd-zeta terms gave 165.65 !

56. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 56 Later, full strong-coupling expansion given by Basso, Korchemsky, Kotański, 0708.3933[th] Soon thereafter … Benna, Benvenuti, Klebanov, Scardicchio [hep-th/0611135] solved BES integral equation numerically, expanding in a basis of Bessel functions. Solution agrees stunningly well with “best KLV interpolation” – to within 0.2% for all a

57. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 57 Iteration in other theories? Two classes of (large N c ) conformal gauge theories “inherit” the same large N c perturbative amplitude properties from N=4 SYM: Khoze, hep-th/0512194; Oz, Theisen, Yankielowicz, 0712.3491 [hep-th] 1.Theories obtained by orbifold projection – product groups, matter in particular bi-fundamental rep’s Bershadsky, Johansen, hep-th/9803249 Supergravity dual known for this case, deformation of AdS 5 x S 5 Lunin, Maldacena, hep-th/0502086 Breakdown of inheritance at five loops (!?) for more general (imaginary  ) marginal perturbations of N=4 SYM? Khoze, hep-th/0512194 2.The N=1 supersymmetric “beta-deformed” conformal theory – same field content as N=4 SYM, but superpotential is modified: Leigh, Strassler, hep-th/9503121 hep-th/0612309

58. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 58 Non-MHV very different even at 1 loop MHV: logarithmic power law Non-MHV: ??

59. LPTHE Jussieu 29 Jan. 2008 L. Dixon Gluon Scattering in N=4 SYM 59 Many higher-loop contributions to gg  gg scattering deduced from a simple property of the 2-particle cuts at one loop The rung rule in N=4 SYM Bern, Rozowsky, Yan (1997) Leads to “rung rule” for easily computing all contributions which can be built by iterating 2-particle cuts