Gluon Scattering in N=4 and N=2 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 (see also talk by A. Volovich) ETH Zurich Workshop on Gauge and String Theory July 3, 2008

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 2 Ante exordium: N=4 SYM to a “scatterer” 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

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 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

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 4 AdS/CFT in one picture

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 5 Gluon scattering in N=4 SYM 2  2 gluon scattering amplitudes not protected by supersymmetry (unlike e.g. BPS states). 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) Proposal is wrong for n > 5. Alday, Maldacena, [th]; Drummond, Henn, Korchemsky, Sokatchev, [th]; [th]; Bartels, Lipatov, Sabio Vera, [th]; Bern, LD, Kosower, Roiban, Spradlin, Vergu, Volovich, [th] Yet, proving its failure has been a fruitful exercise! 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)

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 6 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/ Catani, hep-ph/ ; Sterman, Tejeda-Yeomans, hep-ph/ 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

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 7 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

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 8 Soft/Collinear Factorization Magnea, Sterman (1990); Sterman, Tejeda-Yeomans, hep-ph/ 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 )

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 9 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

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 10 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

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 11 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:

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 12 Exponentiation in planar N=4 SYM For planar N=4 SYM, proposed 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. However, near-threshold Drell-Yan production in QCD shows finite-term exponentiation too. Eynck, Laenen, Magnea, hep-ph/ Evidence based on two loops (n=4,5, plus collinear limits) and three loops (for n=4) [see talk by A. Volovich for recent progress at three loops, n=5] and strong coupling (n=4,5 indirect) Bern, LD, Smirnov, hep-th/ Anastasiou, Bern, LD, Kosower, hep-th/ ; Cachazo, Spradlin, Volovich, hep-th/ ; Bern, Czakon, Kosower, Roiban, Smirnov, hep-th/ Alday, Maldacena, [hep-th]

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 13 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, [th]; Cachazo,

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 14 Planar N=4 amplitudes from 1 to 3 loops Bern, Rozowsky, Yan (1997) 2 Green, Schwarz, Brink (1982)

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 15 Integrals for planar amplitude at 4 loops Bern, Czakon, LD, Kosower, Smirnov, hep-th/

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 16 Integrals for planar amplitude at 5 loops Bern, Carrasco, Johansson, Kosower, [th]

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 17 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?

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 18 No explicit allowed (so OK) Surviving diagrams all have “pseudo (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 : Broadhurst (1993); Lipatov (1999); Drummond, Henn, Smirnov, Sokatchev, hep-th/ 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

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 19 Pseudoconformal 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, [th] Not present: Requires on shell Also works at 5 loops BCJK, DKS

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 20 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/ Use Mellin-Barnes integration method implies at 3 loops:

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 21 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 analytically, and 3 of 4 constants extracted: n-point information still required to separate – see talk by Volovich Confirmed result for 3-loop cusp anomalous dimension from maximum transcendentality Kotikov, Lipatov, Onishchenko, Velizhanin, hep-th/ BDS, hep-th/ Agrees with Moch, Vermaseren, Vogt, hep-ph/

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 22 Leading transcendentality relation between QCD and N=4 SYM KLOV (Kotikov, Lipatov, Onishschenko, Velizhanin, hep-th/ ) 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

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 23 Full strong-coupling expansion Basso,Korchemsky, Kotański, [th] “known” to all orders Benna, Benvenuti, Klebanov, Scardicchio [hep-th/ ] Beisert, Eden, Staudacher [hep-th/ ] proposal based on integrability [talk by Eden] Agrees with weak-coupling data through 4 loops Bern, Czakon, LD, Kosower, Smirnov, hep-th/ ; Cachazo, Spradlin, Volovich, hep-th/ Agrees with first 3 terms of strong-coupling expansion Gubser Klebanov, Polyakov, th/ ; Frolov, Tseytlin, th/ ; Roiban, Tseytlin, [th]

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 24 Pinning down CSV computed four-loop coefficient numerically by expanding same integrals needed for to one higher power in  Cachazo, Spradlin, Volovich, [th] [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, [th]; talk by Sterman

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 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 find consistency for n>5, but BLS (looking closer) do not, at n=6 Naculich, Schnitzer, [hep-th] Brower, Nastase, Schnitzer,Tan, [hep-th]; Bartels, Lipatov, Sabio Vera, [th] Del Duca, Glover, [th]; talks by Del Duca and Sabio Vera

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 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, [hep-th] Gross, Mende (1987,1988) Better to use dimensional regularization instead of r

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 27 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!!

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 28 Cusps in the solution Near each corner, solution has a cusp Classical action divergence is regulated by Kruczenski, hep-th/ Cusp in (y,r) is the strong-coupling limit of the red wedge; i.e. the Sudakov form factor. See also Buchbinder, [hep-th]

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 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, [hep-th] Matches BDS ansatz perfectly (n=4)

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 30 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, [th] One loop, any n Brandhuber, Heslop, Travaglini, [th] talks by Travaglini, Korchemsky, Sokatchev

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 31 Dual variables and Wilson lines at weak coupling (cont.) Two loops, n=4,5 In all such cases, Wilson-line results match the full scattering amplitude [the MHV case for n>5] (!) – up to additive constants, e.g. Drummond, Henn, Korchemsky, Sokatchev, [th], [th] + … Wilson lines obey an “anomalous” (due to IR divergences) dual conformal Ward identity – totally fixes their structure for n=4,5. DHKS, [th]

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 32 Assuming dual conformal invariance, first possible nontrivial “remainder” function from ABDK/BDS, for MHV amplitudes or for 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, [th] computed the two-loop Wilson line for n=6, and found a discrepancy + … What does this mean for amplitudes?

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 33  Compute (parity-even part of) two-loop 6-point amplitude Bern, LD, Kosower, R. Roiban, M. Spradlin, C. Vergu, A. Volovich, [th]; talk by Volovich l 2 (-2  ) all with dual conformal invariant integrands (including prefactors) l / (-2  ) l

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 34 Two loop n=6 status Expression on previous slide has correct 1/  4,…, 1/  poles. O(  0 ) numerical evaluation confirms that ABDK/BDS ansatz for scattering amplitudes definitely needs correction. Numerical evidence that correction is dual conformal invariant. Most importantly, correction term agrees precisely with function from Wilson loop computation Drummond, Henn, Korchemsky, Sokatchev, [th]; [th] More recently, parity-odd part of the amplitude also computed, up to terms, using leading singularities Cachazo, Spradlin, Volovich, [th]; talk by A. Volovich also consistent with “MHV amplitudes = Wilson loops” l (-2  )

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 35 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/ ; Oz, Theisen, Yankielowicz, [hep-th] 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/ Supergravity dual known for this case, deformation of AdS 5 x S 5 Lunin, Maldacena, hep-th/ Breakdown of inheritance at five loops (!?) for more general (imaginary  ) marginal perturbations of N=4 SYM? Khoze, hep-th/ The N=1 supersymmetric “beta-deformed” conformal theory – same field content as N=4 SYM, but superpotential is modified: Leigh, Strassler, hep-th/ LD, D. Kosower, C.Vergu, 0807.nnnn

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 36 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/

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 37 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, [th]

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 38 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.

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 39 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:

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 40 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)

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 41 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)

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 42 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 in progress 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. proposal by DHKS; talks by Korchemsky & Sokatchev

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 43 Extra Slides

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 44 Two-loop directly for n=5 Even terms checked numerically with aid of Czakon, hep-ph/ Cachazo, Spradlin, Volovich, hep-th/ Bern, Rozowsky, Yan, hep-ph/ 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/ cyclic

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 45 Subleading in 1/N c terms Bern, Carrasco, LD, Johansson, Kosower, Roiban, hep-th/ loops Additional non-planar integrals are required Coefficients are known through 3 loops: Bern, Rozowsky, Yan (1997) 3 loops

ETH Zurich July 3, 2008 L. Dixon Gluon Scattering in N=4 & N=2 SYM 46 Non-MHV very different even at 1 loop MHV: logarithmic power law Non-MHV: ??