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Recursive Approaches to QCD Matrix Elements

Recursive Approaches to QCD Matrix Elements. David Dunbar, Swansea University, Wales. including work with Z. Bern, S Bidder, E Bjerrum-Bohr, L. Dixon, H Ita, D Kosower W Perkins K. Risager. RADCOR 2005. Q C D Matrix Elements.

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Recursive Approaches to QCD Matrix Elements

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  1. Recursive Approaches to QCD Matrix Elements David Dunbar, Swansea University, Wales including work with Z. Bern, S Bidder, E Bjerrum-Bohr, L. Dixon, H Ita, D Kosower W Perkins K. Risager RADCOR 2005

  2. QCD Matrix Elements • -QCD matrix elements are an important part of calculating the QCD background for processes at LHC • -NLO calculations (at least!) are needed for precision • -One-Loop n-point on-shell amplitudes unknown for n>5 • -In this talk we discuss how recent (> Dec 03) ideas of a “weak-weak” duality might help, • - apply to 2g  4g 2  4 talks of Binoth,Dittmaier

  3. Duality with String Theory • In Dec 03 Witten proposed a Weak-Weak duality between • A) Yang-Mills theory ( N=4 ) • B) Topological String Theory S-matrix of two theories should be identical -True for tree level gluon scattering Rioban, Spradlin,Volovich proposal constructive for N=4 but… N<4..??

  4. Duality Details • Amplitude a function • Spinor helicity replaces polarisation by fermionic variables • Use fermionic coordinates and replace bosonic momenta by `twistors` • Amplitude now function entirely of fermionic variables • Fourier (Penrose transform) one of spinors, • Amplitude in twistor space is closest to string theory Xu, Zhang,Chang 87

  5. Is the duality useful? Theory A : hard, interesting Theory B: easy Topological String Theory: harder, uninteresting Perturbative QCD, hard, interesting -duality may be useful indirectly

  6. Inspired by duality – the CSW/MHV-vertex construction Cachazo Svercek Witten 04, (Nair) • Promotes MHV amplitude to fundamental object by • -Off-shell continuation • -makes YM perturbative expansion match that of B-instantons (colour ordered amplitudes) Parke-Taylor, Berends-Giele

  7. + _ _ + _ + _ _ + _ + _ + + _ -three point vertices allowed -number of vertices = (number of -) -1

  8. MHV-vertex construction • Works for gluon scattering tree amplitudes • Works for (massless) quarks • Works for Higgs and W’s • Works for photons • Works for gravity Wu,Zhu; Su,Wu; Georgiou Khoze Badger, Dixon, Glover, Forde, Khoze, Kosower Mastrolia Ozeren+Stirling Bjerrum-Bohr,DCD,Ita,Perkins, Risager -Works for one-loops in SUPERSYMMETRIC theories -In trouble for non-SUSY theories Bedford,Brandhuber, Spence and Travaglini; Qigley,Rozali A(+++…++)

  9. Inspired by duality –BCFW construction for tree amplitudes Britto,Cachazo,Feng (and Witten) • Return of the analytic S-matrix! • Shift amplitude so it is a complex function of z Amplitude becomes an analytic function of z, A(z) Full amplitude can be reconstructed from analytic properties

  10. Provided, then Residues occur when amplitude factorises on multiparticle pole (including two-particles)

  11. 1 2 -results in recursive on-shell relation (three-point amplitudes must be included)

  12. CSW vs BCF • Difference • CSW asymmetric between helicity sign • BCF chooses two special legs • Similarities- • both rely upon analytic structure CSW can be derived from a type of analytic shift Risager; Bjerrum-Bohr,Dunbar,Ita,Perkins and Risager, 05

  13. One-Loop Amplitudes • One Loop Gluon Scattering Amplitudes in QCD • -Four Point : Ellis+Sexton • -Five Point : Bern, Dixon,Kosower • -Six-Point and beyond--- present problem n-point MHV amplitudes supersymmetric theories Six-point N=4 amplitudes Bern,Dixon,Dunbar and Kosower 94/95

  14. Supersymmetric Decomposition Supersymmetric gluon scattering amplitudes are the linear combination of QCD ones+scalar loop -this can be inverted

  15. N=4 One-Loop Amplitudes –solved! • Amplitude is a a sum of scalar box functions with rational coefficients (BDDK,1994) • Coefficients are ``cut-constructable’’ (BDDK,1994) • …. • …. • Quadruple cuts turns calculus into algebra (Britto,Cachazo,Feng,2005) • Box Coefficients are actually coefficients of terms like

  16. Box-Coefficients Britto,Cachazo,Feng S -works for massless corners (complex momenta) or signature (--++) -works for non-supersymmetric Bjerrum-Bohr,Bidder,DCD,Perkins

  17. N=1 One-Loop Amplitudes -???? • Important to choose a good basis of functions • A) choose chiral multiplet • B) use D=6 boxes • Amplitude also cut constructible • -six gluon amplitudes now obtained using unitarity Bidder,Bjerrum-Bohr,Dixon, Dunbar, Perkins Britto, Buchbinder Cachazo, Feng, 04/05

  18. cut construcible recursive? recursive? The Final Pieces : scalar contributions -last component of QCD amplitudes - R is rational and not cut constructible (to O()) -can we avoid direct integration?

  19. Recursion for Rational terms -can we shift R and obtain it from its factorisation? • Function must be rational • Function must have simple poles • We must understand these poles

  20. -understanding poles -multiparticle factorisation theorems Bern,Chalmers

  21. Complication, Either R or the coefficients of integral functions may containSpurious Singularitieswhich are not present in the full amplitude It is important and non-trivial to find shift(s) which avoid these spurious singularities whilst still affecting the full R/coefficient

  22. Collinear Singularity Multi-particle pole Co-planar singularity Example of Spurious singularities

  23. Spurious singularities spoil understanding of residues – can we avoid them? Splitting Amplitude into C and R is not unique The integral functions can be defined to include rational pieces, e.g rather than avoids a spurious singularity as r1 (r=s/s’)

  24. Results: It has been demonstrated, using A) (-++….+++), (+++++….++) and B) (--++++) and (--++….+++) 1) that shifts can be found which allow calculation of rational parts recursively Bern, Dixon Kosower Forde, Kosower Shifts can be found which allow the integral coefficients to be computed recursively 2) A(---..--+++…++) Bern, Bjerrum-Bohr, Dunbar, Ita

  25. State of Play Six Gluon Scattering 2g- 4g X X X X X X X X X X X X X X X X X

  26. Conclusions • -Reasons for optimism in computing one-loop QCD matrix elements • -Recent progress uses UNITARITY and FACTORISATION as key features of on-shell amplitudes • -Inspired by Weak-Weak duality but not dependant upon it • -after much progress in highly super-symmetric theories the (harder) problem of QCD beginning to yield results • -first complete result for a partial 2g  ng amplitude! -also fermions, masses, multiloop……..???.......

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