1. Gluon Scattering in N=4 Super-Yang-Mills Theory from Weak to Strong Coupling Lance Dixon (SLAC) work with Z. Bern, M. Czakon, D. Kosower, R. Roiban, V. Smirnov, M. Spradlin, C. Vergu, A. Volovich London Triangle String Seminar, King’s College 15 October 2008

2. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 2 N=4 super-Yang-Mills theory 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. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 3 Planar N=4 SYM and AdS/CFT In the ’t Hooft limit, fixed, planar 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 on AdS 5 x S 5 Maldacena; Gubser, Klebanov, Polyakov; Witten

4. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 4 AdS/CFT in one picture

5. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 5 Gluon scattering in N=4 SYM gg  gg 4-gluon scattering amplitude not protected by supersymmetry (unlike e.g. anomalous dimensions of BPS states). How does series organize itself into simple result, from gravity/string point of view? Anastasiou, Bern, LD, Kosower, hep-th/0309040 Can we propose an exact form for the scattering amplitude, which interpolates between weak and strong coupling? Cusp anomalous dimension  K ( ) is a new, nontrivial example, solved to all orders in using integrability Beisert, Eden, Staudacher, hep-th/0610251

6. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 6 A proposal Proposal is wrong for n > 5. Alday, Maldacena, 0710.1060; Drummond, Henn, Korchemsky, Sokatchev, 0712.4138; Bartels, Lipatov, Sabio Vera, 0802.2065; Bern, LD, Kosower, Roiban, Spradlin, Vergu, Volovich, 0803.1465 Yet, demonstrating its failure has also been fruitful…  K ( ) one of just four functions of alone, which fully specify gg  gg scattering to all orders in  for any scattering angle  (value of t/s). Also n-gluon (2  n-2) MHV amplitudes g - g -  g + g - g + g - g + g + ? Bern, LD, Smirnov (2005) Good evidence in favor of proposal for n = 4,5, due to symmetry of AdS 5 x S 5 : (fermionic) T-duality  dual (super)conformal invariance Kallosh, Tseytlin, hep-th/9808088; Alday, Maldacena, 0705.0303; 0710.1060; Berkovits, Maldacena, 0807.3196; Beisert, Ricci, Tseytlin, Wolf, 0807.3228; Ricci, next talk

7. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 7 IR Structure in QCD and N=4 SYM Regularize IR with D = 4-2 . Pole terms in  predicted by soft/collinear factorization + exponentiation – long-studied in QCD, straightforward to apply 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 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

8. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 8 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 controls large-spin limit of anomalous dimensions  j of leading-twist operators with spin j: Korchemsky (1989); Korchemsky, Marchesini (1993)

9. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 9 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

10. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 10 Soft/Collinear Factorization Magnea, Sterman (1990); Sterman, Tejeda-Yeomans, hep-ph/0210130 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 )

11. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 11 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

12. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 12 Sudakov form factor Factorization  differential equation for form factor 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

13. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 13 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:

14. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 14 Exponentiation in planar N=4 SYM For planar N=4 SYM, finite terms also exponentiate. That is, the hard remainder function h n (l) defined by is 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. However, near-threshold Drell-Yan production in QCD shows finite-term exponentiation too. Eynck, Laenen, Magnea, hep-ph/0305179 Evidence based originally on two loops (n=4,5, plus collinear limits) and three loops (for n=4) later strong coupling (n=4, 5 indirect) 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

15. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 15 Perturbative Evidence gathered by 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); Bern, Czakon, LD, Kosower, Smirnov (2006); Bern, Carrasco, Dixon, Johansson, Kosower, Roiban (2007); BCJK (2007); Cachazo, Skinner, 0801.4574; Cachazo, 0803.1988

16. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 16 Planar N=4 amplitudes from 1 to 3 loops Bern, Rozowsky, Yan (1997) 2 Green, Schwarz, Brink (1982)

17. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 17 Integrals for planar amplitude at 4 loops Bern, Czakon, LD, Kosower, Smirnov, hep-th/0610248

18. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 18 Integrals for planar amplitude at 5 loops Bern, Carrasco, Johansson, Kosower, 0705.1864

19. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 19 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?

20. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 20 Dual Conformal Invariance A conformal symmetry acting in momentum space, on dual or sector variables x i First seen in N=4 SYM planar amplitudes in the loop integrals Broadhurst (1993); Lipatov (1999); Drummond, Henn, Smirnov, Sokatchev, hep-th/0607160 x5x5 x1x1 x2x2 x3x3 x4x4 k invariant under inversion:

21. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 21 Pseudoconformal diagrams at four loops Present in the amplitude Not present: Requires on shell Also works at 5 loops BCJK, DKS Require 4 (net) lines out of every internal dual vertex, 1 (net) line out of every external one. Dotted lines = numerator factors

22. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 22 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:

23. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 23 Exponentiation at 3 loops (cont.) Inserting values of 4-point integrals (including those with ) into using weight 6 harmonic polylogarithm identities, etc., relation was verified analytically, and 3 of 4 constants extracted: Need 5-point information to separate. See Spradlin, Volovich, Wen, 0808.1054 Confirmed 3-loop cusp anomalous dimension result from maximum transcendentality Kotikov, Lipatov, Onishchenko, Velizhanin, hep-th/0404092 BDS, hep-th/0505205 Agrees with Moch, Vermaseren, Vogt, hep-ph/0508055

24. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 24 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

25. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 25 Regge / high-energy behavior Study limits with large rapidity separations between final-state gluons Everything consistent with Regge/BFKL factorization for n=4,5. BNST, DDG find consistency for n>5, but BLS-V (looking closer) do not, at n=6 Drummond, Korchemsky, Sokatchev, 0707.0243; Naculich, Schnitzer, 0708.3069; Brower, Nastase, Schnitzer,Tan, 0801.3891; 0809.1632; Bartels, Lipatov, Sabio Vera, 0802.2065, 0807.0894 Del Duca, Glover, 0802.4445; Del Duca, Duhr, Glover, 0809.1822

26. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 26 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 Gross, Mende (1987,1988) Better to use dimensional regularization instead of r

27. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 27 Dual variables and strong coupling T-dual momentum variables introduced 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!!

28. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 28 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. Buchbinder, 0706.2015

29. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 29 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 Matches BDS ansatz perfectly (n=4)

30. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 30 Dual variables and Wilson lines at weak coupling Inspired by Alday, Maldacena, a sequence of computations of Wilson-line configurations with same “dual momentum” boundary conditions: One loop, n=4 Drummond, Korchemsky, Sokatchev, 0707.0243 One loop, any n Brandhuber, Heslop, Travaglini, 0707.1153

31. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 31 Dual variables and Wilson lines at weak coupling (cont.) Two loops, n=4,5 In every case, Wilson-line matches full scattering amplitude [MHV amplitude for n>5] (!) – up to additive constants, e.g. Drummond, Henn, Korchemsky, Sokatchev, 0709.2368, 0712.1223 + … Wilson lines obey an “anomalous” (due to IR divergences) dual conformal Ward identity – totally fixes their structure for n=4,5. DHKS, 0712.1223

32. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 32 Assuming dual conformal invariance (possessed also by ABDK/BDS ansatz), first possible nontrivial “remainder” function, for MHV amplitudes or for Wilson lines, is at n=6, where “cross-ratios” appear. [Not n=4,5 because x 2 i,i+1 = 0.] Dual variables and Wilson lines at weak coupling (cont.) Drummond, Henn, Korchemsky, Sokatchev, 0712.4138 computed two-loop Wilson line for n=6, found a discrepancy R 6 (2) + … What does this mean for amplitudes?

33. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 33  Compute (parity-even part of) two-loop 6-point amplitude Bern, LD, Kosower, R. Roiban, M. Spradlin, C. Vergu, A. Volovich, 0803.1465 l 2 (-2  ) all with dual conformal invariant integrands (including prefactors) l / (-2  ) l

34. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 34 Two loop n=6 amplitude Find all the correct 1/  4,…, 1/  poles. O(  0 ) numerical evaluation confirms that ABDK/BDS ansatz for scattering amplitudes needs correction. A 6 (2) = A 6 (2) BDS + R 6 (2) Correction term R 6 (2) agrees precisely (numerically) with dual conformal invariant Wilson loop function Drummond, Henn, Korchemsky, Sokatchev, 0712.4138; 0803.1466 More recently, parity-odd part of the amplitude also computed, up to terms, using leading singularities Cachazo, Spradlin, Volovich, 0805.4832 also consistent with “MHV amplitudes = Wilson loops” l (-2  )

35. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 35 What is R 6 (2) ? Vanishes in 2-particle collinear limits: because ansatz was built to be consistent with two-loop collinear behavior: a = 1 b = 2 Totally symmetric function of 3 dual conformal ratios: Should be weight 4 transcendentality function: Li 4 (…) + …

36. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 36 What is R 6 (2) ? (cont.) Nontrivial in 3-particle collinear limits: Because all 3 u i are generic, we would know R 6 (2) if we knew this splitting amplitude

37. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 37 What is R 6 (2) ? (cont.) Nontrivial in Minkowski multi-Regge kinematics Bartels, Lipatov, Sabio Vera, 0802.2065, 0807.0894 Limiting behavior found by BLS-V can be put into dual conformal invariant form: But what is R 6 (2) for generic u i ?

38. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 38 All orders 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 1.Theories obtained by orbifold projection [AdS 5 x S 5 /Z n ] – 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 LD, D. Kosower, C.Vergu, 08mm.nnnn

39. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 39 The simplest Z 2 orbifold of SU(2N c ) N=4 SYM replaces 2 of the 3 adjoint chiral multiplets by bi-fundamental representations under the unbroken gauge group SU(N c ) x SU(N c ) with N=2 SUSY. Each gauge group has the same gauge coupling This theory, whose supergravity dual is AdS 5 x S 5 /Z 2, inherits its amplitudes from N=4 SYM at large N c However, there is a marginal perturbation which makes the two couplings unequal, The supergravity dual for this case does not seem to be known, at least not for general For the theory reduces to N=2 SU(N c ) gauge theory with N f = N c flavors of N=2 hypermultiplets [N=2 SCFT]. N=2 SCFT Iteration? Bershadsky, Johansen, hep-th/9803249

40. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 40 Consider the scattering of four gluons in the theory. At one loop, there are only terms. Four-gluon scattering in N=2 SCFT For N f = N c, the 4-gluon planar amplitude is identical to that in N=4 SYM, because factor of N f makes up for a missing color index. Glover, Khoze, Williams, 0805.4190

41. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 41 Four gluons at two loops in N=2 SCFT At two loops, there are and terms, which for sum back up to the N=4 SYM amplitude. 3 of 4 types of color contributions are identical to N=4 SYM at large N c, for same reason as at 1 loop. But the 4 th is only leading-color as a term.

42. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 42 N=2 SCFT four-gluon two-loop poles We find that the N=2 amplitude is equal to the N=4 amplitude, plus a finite, but nonvanishing, remainder: where The fact that the remainder is finite means that the cusp and collinear anomalous dimension are same as in N=4 through 2 loops:

43. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 43 N=2 SCFT four-gluon two-loop finite terms The finite remainder is different for the two independent, nonvanishing helicity configurations, and. It has some interesting properties: violates dual conformal invariance (not constant) violates uniform, maximal transcendentality (but not by much)

44. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 44 N=2 SCFT implications? Violation of maximal transcendentality is not too surprising. Interesting that the violation is “small” and does not occur (through two loops) in cusp or collinear anomalous dimensions. What does violation of dual conformal invariance mean? If there is a supergravity dual, perhaps it does not have a T duality (approximate) symmetry. Another possibility could be that there is a dual (super) conformal symmetry, but it acts on N=2 MHV amplitudes differently from how it acts on N=4 MHV amplitudes, c.f. situation for N=4 NMHV amplitudes. Berkovits, Maldacena (2008); Drummond, Henn, Korchemsky, Sokatchev (2008)

45. King's College 15 Oct. 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 are exponentiation/iteration, AdS/CFT, integrability, [dual] conformality & Wilson lines related? N=2 SCFT as a “control” sample? Why are MHV amplitudes = Wilson lines, (at least through n=6)? Berkovits, Maldacena, 0807.3196 What is the n=6 “remainder” function in N=4 SYM? What happens for non-MHV amplitudes? From form of 1-loop amplitudes, answer must be more complex. dual superconformal proposal by DHKS

46. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 46 Extra Slides

47. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 47 Two-loop directly for n=5 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

48. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 48 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

49. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 49 Full strong-coupling expansion Basso,Korchemsky, Kotański, 0708.3933 “known” 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

50. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 50 Pinning down CSV computed four-loop coefficient numerically by expanding same integrals needed for to one higher power in  Cachazo, Spradlin, Volovich, 0707.1903 [3/2] Padé approximant incorporating all data strong coupling from Alday, Maldacena So far, no proposal for an exact solution for this quantity. Hope that new equation: may help in this regard LD, Magnea, Sterman, 0805.3515

51. King's College 15 Oct. 2008 L. Dixon Gluon Scattering in N=4 SYM 51 Non-MHV very different even at 1 loop MHV: logarithmic power law Non-MHV: ??