Loop Calculations of Amplitudes with Many Legs

1. Loop Calculations of Amplitudes with Many Legs David Dunbar, Swansea University, Wales, UK DESY 2007

2. Plan • -motivation • -organising Calculations • -tree amplitudes • -one-loop amplitudes • -unitarity based techniques • -factorisation based techniques • -prospects

3. 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 • - apply to 2g  4g -this talk about one-loop n-gluon scattering n>5 -will focus on analytic methods

4. Organisation 1: Colour-Ordering • Gauge theory amplitudes depend upon colour indices of gluons. • We can split colour from kinematics by colour decomposition • The colour ordered amplitudes have cyclic symmetric rather than full crossing symmetry Colour ordering is not is field theory text-books but is in string texts -leading in colour term Generates the others

5. Gluon Momenta Reference Momenta Organisation 2: Spinor Helicity Xu, Zhang,Chang 87 -extremely useful technique which produces relatively compact expressions for amplitudes in terms of spinor products

6. We can continue and write amplitude completely in fermionic variables • For massless particle with momenta • Amplitude a function of spinors now Transforming to twistor space gives a different organisation -Amplitude is a function on twistor space Witten, 03

7. Organisation 3: Supersymmetric Decomposition Supersymmetric gluon scattering amplitudes are the linear combination of QCD ones+scalar loop -this can be inverted

8. degree p in l p Vertices involve loop momentum propagators Organisation 4: General Decomposition of One-loop n-point Amplitude p=n : Yang-Mills

9. Passarino-Veltman reduction Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator

10. Passarino-Veltman reduction • process continues until we reach four-point integral functions with (in yang-mills up to quartic numerators) In going from 4-> 3 scalar boxes are generated • similarly 3 -> 2 also gives scalar triangles. At bubbles process ends. Quadratic bubbles can be rational functions involving no logarithms. • so in general, for massless particles Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator

11. Tree Amplitudes MHV amplitudes : Very special they have no real factorisations other than collinear -promote to fundamental vertex??, nice for trees lousy for loops Cachazo, Svercek and Witten

12. Six-Gluon Tree Amplitude factorises

13. One-Loop QCD Amplitudes • One Loop Gluon Scattering Amplitudes in QCD • -Four Point : Ellis+Sexton, Feynman Diagram methods • -Five Point : Bern, Dixon,Kosower, String based rules • -Six-Point and beyond--- present problem

14. - - - 93 - - - 93 94 94 94 06 94 94 05 06 94 94 05 06 94 06 05 05 94 05 06 06 94 05 06 06 The Six Gluon one-loop amplitude ~14 papers 81% `B’ Berger, Bern, Dixon, Forde, Kosower Bern, Dixon, Dunbar, Kosower Britto, Buchbinder, Cachazo, Feng Bidder, Bjerrum-Bohr, Dixon, Dunbar Bern, Chalmers, Dixon, Kosower Bedford, Brandhuber, Travaglini, Spence Forde, Kosower Xiao,Yang, Zhu Bern, Bjerrum-Bohr, Dunbar, Ita Britto, Feng, Mastriolia Mahlon

15. Methods 1: Unitarity Methods -look at the two-particle cuts -use this technique to identify the coefficients

16. Unitarity, • Start with tree amplitudes and generate cut integrals -we can now carry our reduction on these cut integrals Benefits: -recycle compact on-shell tree expression -use li2=0

17. Fermionic Unitarity -try to use analytic structure to identify terms within two-particle cuts bubbles

18. Generalised Unitarity -use info beyond two-particle cuts -see also Dixon’s talk re multi-loops

19. Box-Coefficients Britto,Cachazo,Feng -works for massless corners (complex momenta) or signature (--++)

20. Unitarity -works well to calculate coefficients -particularly strong for supersymmetry (R=0) -can be used, in principle to evaluate R but hard -can be automated Ellis, Giele, Kunszt -key feature : work with on-shell physical amplitudes

21. Methods 2: On-shell recursion: tree amplitudes Britto,Cachazo,Feng (and Witten) • Shift amplitude so it is a complex function of z Tree amplitude becomes an analytic function of z, A(z) -Full amplitude can be reconstructed from analytic properties

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

23. 1 2 -results in recursive on-shell relation (c.f. Berends-Giele off shell recursion) Tree Amplitudes are on-shell but continued to complex momenta (three-point amplitudes must be included)

24. Recursion for One-Loop amplitudes? • Analytically continuing the 1-loop amplitude in momenta leads to a function with both poles and cuts in z

25. cut construcible recursive? recursive? Expansion in terms of Integral Functions - R is rational and not cut constructible (to O()) -amplitude is a mix of cut constructible pieces and rational

26. 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 Berger, Bern, Dixon, Forde and Kosower

27. -to carry out recursion we must understand poles of coefficients -multiparticle factorisation theorems Bern,Chalmers

28. Recursion on Integral Coefficients Consider an integral coefficient and isolate a coefficient and consider the cut. Consider shifts in the cluster. • Shift must send tree to zero as z -> 1 • Shift must not affect cut legs -such shifts will generate a recursion formulae

29. Potential Recursion Relation  +

30. Example: Split Helicity Amplitudes • Consider the colour-ordered n-gluon amplitude -two minuses gives MHV -use as a example of generating coefficients recursively

31. -look at cluster on corner with “split” -shift the adjacent – and + helicity legs -criteria satisfied for a recursion - - - - r- + r+1+ + + -we obtain formulae for integral coefficients for both the N=1 and scalar cases which together with N=4 cut give the QCD case (with, for n>6 rational pieces outstanding)

32. -for , special case of 3 minuses,

33. For R see Berger, Bern, Dixon, Forde and Kosower

34. Spurious Singularities • -spurious singularities are singularities which occur in • Coefficients but not in full amplitude • -need to understand these to do recursion • -link coefficients together

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

36. 1 1 2 2 2 2 1 1 Spurious singularities link different coefficients =0 at singularity -these singularities link different coefficients together

37. Six gluon, what next,

38. Conclusions • -new techniques for NLO gluon scattering • -lots to do • -but tool kits in place? • -automation? • -progress driven by very physical developments: unitarity and factorisation