1. The Two-Loop Anomalous Dimension Matrix for Soft Gluon Exchange S. M. Aybat, L.D., G. Sterman hep-ph/0606254, 0607309 Workshop HP 2, ETH Zürich September 6-8, 2006

2. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 2 Outline Separation of QCD amplitudes into soft, collinear (jet) and hard functions Computation of soft anomalous dimension matrix via eikonal (Wilson) lines to two loops (NNLL) Proportionality of one- and two-loop matrices Consistency of result with 1/  poles in explicit two-loop QCD amplitudes Proportionality at three loops? Implications for resummation at NNLL

3. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 3 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; 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

4. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 4 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

5. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 5 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

6. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 6 Jet function Pure counterterm (series of 1/  poles); like  ( ,  s ), single poles in  determine K completely 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) K, G also obey differential equations (ren. group): cusp anomalous dimension

7. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 7 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

8. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 8 Soft function For generic processes, need soft function S 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: For fixed-angle scattering with hard-scale let momenta with massless 4-velocities [ matches eikonal computation to partonic one]

9. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 9 Only soft gluons involved Couplings classical, spin-independent Take hard external partons to be scalars Expand vertices and propagators  More formally, consider web function W or eikonal amplitude of n Wilson lines E.g. for n=4, 1 + 2  3 + 4: Computation of soft anomalous dimensions

10. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 10 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:

11. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 11 1-loop soft anomalous dim. matrix 1/  poles in 1-loop graph yield: Kidonakis, Oderda, Sterman, hep-ph/9803241 Agrees with known divergences: Giele, Glover (1992); Kunszt, Signer, Trócsányi, hep-ph/9401294; Catani, hep-ph/9802439 finite, hard parts scheme-dependent! Expansion of 1-loop amplitude

12. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 12 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 3E graphs are of two types

13. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 13 Triple-gluon-vertex 3E graph vanishes Change variables to “light-cone” ones for A, B: vanishes due to antisymmetry under

14. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 14 Other 3E graph factorizes contains  The sum is color-symmetric, and factorizes into a product of 1-loop factors, which allows its divergences to be completely cancelled by 1-loop counterterms Same change of variables and transformation takes this factor to the one for the flipped graph:

15. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 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, thanks to the identity – for non-color 1/  pole part of graphs:  All color factors become proportional to the one-loop ones,  Proportionality constant dictated by cusp anomalous dimension

16. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 16 Consistency with explicit 2-loop computations Results for Organized according to Catani, hep-ph/9802439 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 looks like ???

17. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 17 2-loop consistency (cont.) Resolution is that scheme of Catani, hep-ph/9802439 is non-minimal in terms of 1/  poles Color-nontrivial matrices are included in finite part To compare to a minimal organization we have to commute two matrices: Then everything agrees Electroweak Sudakov logs agree with 2  2 results Jantzen, Kühn, Penin, Smirnov, hep-ph/0509157v3

18. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 18 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

19. Sept. 7, 2006 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix 19 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! This result was 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