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The Two-Loop Anomalous Dimension Matrix for Soft Gluon Exchange

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 Z ü rich September 6-8, 2006. Outline. Separation of Q C D amplitudes into soft , collinear (jet) and hard functions

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The Two-Loop Anomalous Dimension Matrix for Soft Gluon Exchange

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  1. The Two-Loop Anomalous Dimension Matrix for Soft Gluon Exchange S. M. Aybat, L.D., G. Sterman hep-ph/0606254, 0607309 Workshop HP2, ETH Zürich September 6-8, 2006

  2. Outline • Separation of QCDamplitudes 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/e poles in • explicit two-loop QCDamplitudes • Proportionality at three loops? • Implications for resummation at NNLL L. Dixon Two-Loop Soft Anomalous-Dim. Matrix

  3. IR Structure of QCD Amplitudes[Massless Gauge Theory Amplitudes] • Expand multi-loop amplitudes ind=4-2e around d=4 (e=0) • Overlapping soft (1/e) + collinear (1/e) divergences at each • loop order imply leading poles are ~1/e2Lat Lloops • Pole terms are predictable,due to soft/collinear factorization and exponentiation, in terms of a • collection of constants (anomalous dimensions) • Same constants control resummation of large logarithms • near kinematic boundaries 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 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix

  4. S = soft function (only depends on color of ith particle; • matrix in “color space”) • J = jet function (color-diagonal; depends on ith spin) • H= hard remainder function (finite as ; • vector in color space) • color: Catani, Seymour, hep-ph/9605323; Catani, hep-ph/9802439 Soft/Collinear Factorization Magnea, Sterman (1990) Sterman, Tejeda-Yeomans, hep-ph/0210130 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 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix

  6. finite as e  0; contains all Q2dependence Pure counterterm (series of 1/e poles); like b(e,as), single poles in e determine K completely K, G also obey differential equations (ren. group): cusp anomalous dimension Jet function Mueller (1979); Collins (1980); Sen (1981); Korchemsky, Radyushkin (1987); Korchemsky (1989); Magnea, Sterman (1990) • By analyzing structure of soft/collinear terms • in axial gauge, find differential equation • for jet function J[i] (~ Sudakov form factor): L. Dixon Two-Loop Soft Anomalous-Dim. Matrix

  7. _ as = running coupling in D=4-2e Jet function solution Magnea, Sterman (1990) • Solution to differential equations can be extracted from fixed-order calculations of form factors or related objects E.g. at three loops Moch, Vermaseren, Vogt, hep-ph/0507039, hep-ph/0508055 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix

  8. 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] Soft function • For generic processes, need soft functionS • Less well-studied than J • Also obeys a (matrix) differential equation: Kidonakis, Oderda, Sterman, hep-ph/9803241 soft anomalous dimension matrix L. Dixon Two-Loop Soft Anomalous-Dim. Matrix

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

  10. Soft anomalous dimension matrix determined • by single ultraviolet poles in e of S: Soft computation (cont.) • Regularize collinear divergences by removing Sudakov-type factors (in eikonal approximation), from web function, defining soft function S by: L. Dixon Two-Loop Soft Anomalous-Dim. Matrix

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

  12. 4E graphs factorize trivially into • products of 1-loop graphs. • 1-loop counterterms cancel all 1/e poles, leave no contribution to 3E graphs are of two types 2-loop soft anomalous dim. matrix • Classify web graphs according to number of eikonal lines (nE) 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 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix

  14. Same change of variables and transformation takes this factor to the one for the flipped graph: • 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 Other 3E graph factorizes contains L. Dixon Two-Loop Soft Anomalous-Dim. Matrix

  15. Same analysis can be used here, although color flow is generically different, thanks to the identity – for non-color1/e pole part of graphs: • All color factors become proportional to the one-loop ones,  Proportionality constant dictated by cusp anomalous dimension The 2E graphs Korchemsky, Radyushkin (1987); Korchemskaya, Korchemsky, hep-ph/9409446 All were previously analyzed for the cusp anomalous dimension L. Dixon Two-Loop Soft Anomalous-Dim. Matrix

  16. looks like ??? Consistency with explicit 2-loop computations • Results for • Organized according toCatani, 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 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/e poles • Color-nontrivial matrices are included • in finite part • To compare to a minimal organization • we have to commute two matrices: Electroweak Sudakov logs agree with 2  2 results Jantzen, Kühn, Penin, Smirnov, hep-ph/0509157v3 • Then everything agrees L. Dixon Two-Loop Soft Anomalous-Dim. Matrix

  18. 6E and 5E graphs factorize trivially into products of lower-loop graphs; no contribution to thanks to 2-loop result 4E graphs use same (A,B) change of variables ??? also trivial Proportionality at 3 loops? Again classify web graphs according to number of eikonal lines (nE) and then there are more 4E graphs, and the 3E and 2E graphs… 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 theproportionality 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) schemeCatani, 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 L. Dixon Two-Loop Soft Anomalous-Dim. Matrix

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