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Two-Loop Anomalous Dimension Matrix for Soft Gluon Exchange in Quantum Chromodynamics

This paper discusses the two-loop soft anomalous dimension matrix relevant to soft gluon exchanges in quantum chromodynamics (QCD). It outlines the separation of QCD amplitudes into soft, collinear, and hard functions, the computation procedures via eikonal (Wilson) lines, and the consistency with one-loop poles. The implications for resummation at next-to-next-to-leading logarithmic (NNLL) accuracy are explored, and the results regarding overlapping divergences are highlighted. The work builds upon established theories and provides essential insights into multi-loop amplitude behaviors.

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Two-Loop Anomalous Dimension Matrix for Soft Gluon Exchange in Quantum Chromodynamics

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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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