1. Webs of soft gluons from QCD to N=4 super-Yang-Mills theory Lance Dixon (SLAC) KEK Symposium “Towards precision QCD physics” in memory of Jiro Kodaira March 10, 2007

2. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 2 26 years ago…

3. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 3 26 years ago… I was also at SLAC in the summer of 1981 – but as an undergraduate working on the Mark III experiment at SPEAR I had no idea what “Summing Soft Emission” meant – although I did sneak into some of the SLAC Summer Institute lectures on The Strong Interactions So I could not yet appreciate the beauty of this formula:

4. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 4 Outline The two-loop soft anomalous dimension matrix in QCD – it’s all about K Multi-loop analogs of K (cusp anomalous dim.) – we now (probably) know them to all loop orders in large N c N=4 super-Yang-Mills theory, and tantalizing pieces of them in QCD as well Aybat, LD, Sterman, hep-ph/0606254, 0607309 Bern, Czakon, LD, Kosower, Smirnov, hep-th/0610248 Eden, Staudacher, hep-ph/0603157 Beisert, Eden, Staudacher, hep-th/0610251 Kotikov, Lipatov, Onishchenko, Velizhanin, hep-th/0404092 Benna, Benvenuti, Klebanov, Scardicchio, hep-th/0611135

5. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 5 IR Structure of QCD Amplitudes [Massless Gauge Theory Amplitudes] Expand multi-loop amplitudes in d=4-2  around d=4 (  =0) Overlapping soft (1/  ) + collinear (1/  ) divergences at each loop order imply leading poles are ~ 1/  2L at L loops Pole terms are predictable, due to soft/collinear factorization and exponentiation, in terms of a collection of constants (anomalous dimensions) Mueller (1979); Akhoury (1979); Collins (1980), hep-ph/0312336; Kodaira, Trentadue (1981); Sen (1981, 1983); Sterman (1987); Botts, Sterman (1989); Catani, Trentadue (1989); Korchemsky (1989); Magnea, Sterman (1990); Korchemsky, Marchesini, hep-ph/9210281; Giele, Glover (1992); Kunszt, Signer, Trócsányi, hep-ph/9401294; Kidonakis, Oderda, Sterman, hep-ph/9801268, 9803241; Catani, hep-ph/9802439; Dasgupta, Salam, hep-ph/0104277; Sterman, Tejeda-Yeomans, hep-ph/0210130; Bonciani, Catani, Mangano, Nason, hep-ph/0307035; Banfi, Salam, Zanderighi, hep-ph/0407287; Jantzen, Kühn, Penin, Smirnov, hep-ph/0509157 Same constants control resummation of large logarithms near kinematic boundaries – as Jiro Kodaira understood so well

6. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 6 Soft/Collinear Factorization Magnea, Sterman (1990) Sterman, Tejeda-Yeomans, hep-ph/0210130 S = soft function (only depends on color of i th particle; matrix in “color space”) J = jet function (color-diagonal; depends on i th spin) H = hard remainder function (finite as ; vector in color space) color: Catani, Seymour, hep-ph/9605323; Catani, hep-ph/9802439

7. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 7 For the case n=2, gg  1 or qq  1, the color structure is trivial, so the soft function S = 1 Thus the jet function is the square-root of the Sudakov form factor (up to finite terms): _ The Sudakov form factor

8. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 8 Jet function By analyzing structure of soft/collinear terms in axial gauge, find differential equation for jet function J [i] (~ Sudakov form factor): finite as    contains all Q 2  dependence Mueller (1979); Collins (1980); Sen (1981); Korchemsky, Radyushkin (1987); Korchemsky (1989); Magnea, Sterman (1990) Pure counterterm (series of 1/  poles); like  ( ,  s ), single poles in  determine completely also obey differential equations (ren. group): cusp anomalous dimension

9. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 9 can be extracted from fixed-order calculations of form factors or related objects Solution to differential equations  s = running coupling in D=4-2  _ E.g. at three loops Moch, Vermaseren, Vogt, hep-ph/0507039, hep-ph/0508055 Magnea, Sterman (1990) Jet function solution

10. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 10 Soft function For generic processes, need soft function S Much less well-studied than J Also obeys a (matrix) differential equation: Kidonakis, Oderda, Sterman, hep-ph/9803241 soft anomalous dimension matrix Solution is a path-ordered exponential: depends on massless 4-velocities ; momenta are

11. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 11 Equivalently, consider web function W or eikonal amplitude of n Wilson lines E.g. for n=4, 1 + 2  3 + 4: Computation of soft anomalous dimension matrix Remove jet function contributions by dividing by appropriate Sudakov factors Only soft gluons  couplings classical, spin-independent Take hard external partons to be scalars Expand vertices and propagators 

12. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 12 1-loop soft anomalous dim. matrix 1/  poles in 1-loop graph yield: Kidonakis, Oderda, Sterman, hep-ph/9803241 Agrees with known divergences of generic one loop amplitudes: Giele, Glover (1992); Kunszt, Signer, Trócsányi, hep-ph/9401294; Catani, hep-ph/9802439 Finite, hard parts scheme-dependent! Expansion of 1-loop amplitude

13. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 13 2-loop soft anomalous dim. matrix Classify web graphs according to number of eikonal lines (nE) 4E graphs factorize trivially into products of 1-loop graphs. 1-loop counterterms cancel all 1/  poles, leave no contribution to Two 3E graphs – each looks as if it might give a complicated color structure depending on 3 legs!

14. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 14 But: vanishes due to antisymmetry after changing to light-cone variables with respect to A, B = 0 factorizes into 1-loop factors, allowing its divergences to be completely cancelled by 1-loop counterterms and +

15. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 15 The 2E graphs All were previously analyzed for the cusp anomalous dimension Korchemsky, Radyushkin (1987); Korchemskaya, Korchemsky, hep-ph/9409446 Same analysis can be used here (although color flow is generically different)  All color factors become proportional to the one-loop ones,  Proportionality constant dictated by cusp anomalous dimension 2-loop soft anomalous dimension – it’s all about K

16. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 16 Implications for resummation To resum a generic hadronic event shape requires diagonalizing the exponentiated soft anomalous dimension matrix in color space Because of the proportionality relation, same diagonalization at one loop (NLL) still works at two loops (NNLL), and eigenvalue shift is trivial! Result foreshadowed in the bremsstrahlung (CMW) scheme Catani, Marchesini, Webber (1991) for redefining the strength of parton showering using Kidonakis, Oderda, Sterman, hep-ph/9801268, 9803241; Dasgupta, Salam, hep-ph/0104277; Bonciani, Catani, Mangano, Nason, hep-ph/0307035; Banfi, Salam, Zanderighi, hep-ph/0407287

17. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 17 Why N=4 super-Yang-Mills theory? Most supersymmetric theory possible without gravity Uniquely specified by local internal symmetry group – e.g., number of colors N c for SU(N c ) An exactly scale-invariant (conformal) field theory: for any coupling g,  (g) = 0 Connected to gravity and/or string theory by –AdS/CFT correspondence, a weak/strong duality Remarkable “transcendentality” relations with QCD

18. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 18 “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) 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

19. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 19 Moch Vermaseren, Vogt (MVV), hep-ph/0403192, hep-ph/0404111 in QCD through 3 loops: K from Kodaira, Trentadue (1981)

20. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 20 in N=4 SYM through 3 loops: KLOV prediction Finite j predictions confirmed (with assumption of integrability) Staudacher, hep-th/0412188 Confirmed at infinite j using on-shell amplitudes, unitarity Bern, LD, Smirnov, hep-th/0505205 and with all-orders asymptotic Bethe ansatz Beisert, Staudacher, hep-th/0504190 leading to an integral equation Eden, Staudacher, hep-th/0603157

21. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 21 An all-orders proposal Integrability, plus an a ll-orders asymptotic Bethe ansatz led to the following proposal for the cusp anomalous dimension in large N c N=4 SYM: where is the solution to an integral equation with Bessel-function kernel Perturbative expansion: ? Eden, Staudacher, hep-ph/0603157

22. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 22 ES proposal (cont.) Eden, Staudacher, hep-ph/0603157 Because of various assumptions made, particularly an overall dressing factor, which could affect the entire “world-sheet S-matrix”, and which was known to be non-trivial at strong-coupling, the ES proposal needed checking via another perturbative method, particularly at 4 loops.

23. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 23 AdS/CFT duality suggests that weak-coupling perturbation series for planar N=4 SYM should have very special properties: strong-coupling limit is equivalent to weakly-coupled strings in large-radius AdS 5 x S 5 background –  -model classically integrable too – world-sheet  -model coupling is Cusp anomalous dimension should be given semi-classically, by energy of a long string, a soliton in the  -model, spinning in AdS 5 First two strong-coupling terms known Cusp anomalous dimension via AdS/CFT Maldacena, hep-th/9711200; Gubser, Klebanov, Polyakov, hep-th/9802109 Gubser, Klebanov, Polyakov, hep-th/0204051 Bena, Polchinski, Roiban, hep-th/0305116 Frolov, Tseytlin, hep-th/0204226

24. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 24 Four-loop planar N=4 SYM amplitude BCDKS, hep-th/0610248 Very simple – only 8 loop integrals required!

25. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 25 Soft/collinear simplification in large N c (planar) limit 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):

26. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 26 Sudakov form factor in planar N=4 SYM Expand in terms of  =0, so running coupling in D=4-2  has only trivial (engineering) dependence on scale , simplifying differential equations

27. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 27 General amplitude in planar N=4 SYM Insert result for form factor into n-point amplitude extract cusp anomalous dimension from coefficient of pole We found a numerical result consistent with: compared with ES prediction, a single sign flip at four loops! We also argued that at order the signs of terms containing should be flipped as well, …

28. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 28 Independently… At the same time, investigating the strong-coupling properties of the dressing factor led Beisert, Eden and Staudacher [hep-th/0610251] to propose an integral equation with a new kernel: with With the “2”, the result is to flip signs of odd-zeta terms in ES prediction, to all orders (actually,  2k+1  i  2k+1 ) Arutyunov, Frolov, Staudacher, hep-th/0406256; Hernandez, Lopez, hep-th/0603204; …

29. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 29 Soon thereafter … Benna, Benvenuti, Klebanov, Scardicchio [hep-th/0611135] solved BES integral equation numerically, by expanding in basis of Bessel functions. Solution agrees stunningly well with “KLV approximate formula,” which incorporates the known strong-coupling behavior

30. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 30 Conclusions for part 2 Combining a number of approaches, an exact solution for the cusp anomalous dimension in planar N=4 SYM appears to be in hand. Result provides a very interesting test of the AdS/CFT correspondence. Through KLOV conjecture, the exact solution provides all-loop information about certain “most transcendental” terms in  K (  s ) in perturbative QCD. The multi-loop analogs of K are related to the energy of a spinning string in anti-de Sitter space! What would Jiro Kodaira make of all this?!

31. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 31 Extra Slides

32. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 32 Soft computation (cont.) Regularize collinear divergences by removing Sudakov-type factors (in eikonal approximation), from web function, defining soft function S by: Soft anomalous dimension matrix determined by single ultraviolet poles in  of S:

33. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 33 Proportionality at 3 loops? Again classify web graphs according to number of eikonal lines (nE) 6E and 5E graphs factorize trivially into products of lower-loop graphs; no contribution to thanks to 2-loop result 4E graphs also trivial ??? and then there are more 4E graphs, and the 3E and 2E graphs… use same (A,B) change of variables

34. Mar. 10, 2007 L. Dixon Webs of soft gluons KEK symposium 34 Consistency with explicit multi-parton 2-loop computations Results for Organized according to Catani, hep-ph/9802439 After making adjustments for different schemes, everything is consistent Anastasiou, Glover, Oleari, Tejeda-Yeomans (2001); Bern, De Freitas, LD (2001-2); Garland et al. (2002); Glover (2004); De Freitas, Bern (2004); Bern, LD, Kosower, hep-ph/0404293 And electroweak Sudakov logs for 2  2 also match Jantzen, Kühn, Penin, Smirnov, hep-ph/0509157