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

D Dunbar, DESY 07 2/38 Plan -motivation -organising Calculations -tree amplitudes -one-loop amplitudes -unitarity based techniques -factorisation based techniques -prospects

D Dunbar, DESY 07 3/38 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

D Dunbar, DESY 07 4/38 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

D Dunbar, DESY 07 5/38 Organisation 2: Spinor Helicity Xu, Zhang,Chang 87 Gluon Momenta Reference Momenta -extremely useful technique which produces relatively compact expressions for amplitudes in terms of spinor products

D Dunbar, DESY 07 6/38 We can continue and write amplitude completely in fermionic variables For massless particle with momenta Amplitude a function of spinors now -Amplitude is a function on twistor space Witten, 03 Transforming to twistor space gives a different organisation

D Dunbar, DESY 07 7/38 Organisation 3: Supersymmetric Decomposition Supersymmetric gluon scattering amplitudes are the linear combination of QCD ones+scalar loop -this can be inverted

D Dunbar, DESY 07 8/38 Organisation 4: General Decomposition of One-loop n-point Amplitude Vertices involve loop momentum propagators p degree p in l p=n : Yang-Mills

D Dunbar, DESY 07 9/38 Passarino-Veltman reduction Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator

D Dunbar, DESY 07 10/38 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

D Dunbar, DESY 07 11/38 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

D Dunbar, DESY 07 12/38 Six-Gluon Tree Amplitude factorises

D Dunbar, DESY 07 13/38 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

D Dunbar, DESY 07 14/38 The Six Gluon one-loop amplitude Bern, Dixon, Dunbar, Kosower Bern, Bjerrum-Bohr, Dunbar, Ita Bidder, Bjerrum-Bohr, Dixon, Dunbar Bedford, Brandhuber, Travaglini, Spence Britto, Buchbinder, Cachazo, Feng Bern, Chalmers, Dixon, Kosower Mahlon Xiao,Yang, Zhu Berger, Bern, Dixon, Forde, Kosower Forde, Kosower Britto, Feng, Mastriolia 81% `B’ ~14 papers

D Dunbar, DESY 07 15/38 Methods 1: Unitarity Methods -look at the two-particle cuts -use this technique to identify the coefficients

D Dunbar, DESY 07 16/38 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 l i 2 =0

D Dunbar, DESY 07 17/38 Fermionic Unitarity -try to use analytic structure to identify terms within two-particle cuts bubbles

D Dunbar, DESY 07 18/38 Generalised Unitarity -use info beyond two-particle cuts -see also Dixon’s talk re multi-loops

D Dunbar, DESY 07 19/38 Box-Coefficients -works for massless corners (complex momenta) Britto,Cachazo,Feng or signature (--++)

D Dunbar, DESY 07 20/38 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 -key feature : work with on-shell physical amplitudes Ellis, Giele, Kunszt

D Dunbar, DESY 07 21/38 Methods 2: On-shell recursion: tree amplitudes 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 Britto,Cachazo,Feng (and Witten)

D Dunbar, DESY 07 22/38 Provided, Residues occur when amplitude factorises on multiparticle pole (including two-particles) then

D Dunbar, DESY 07 23/38 -results in recursive on-shell relation Tree Amplitudes are on-shell but continued to complex momenta (three-point amplitudes must be included) 12 (c.f. Berends-Giele off shell recursion)

D Dunbar, DESY 07 24/38 Recursion for One-Loop amplitudes? Analytically continuing the 1-loop amplitude in momenta leads to a function with both poles and cuts in z

D Dunbar, DESY 07 25/38 Expansion in terms of Integral Functions - R is rational and not cut constructible (to O(  )) cut construcible recursive? -amplitude is a mix of cut constructible pieces and rational

D Dunbar, DESY 07 26/38 Recursion for Rational terms -can we shift R and obtain it from its factorisation? 1)Function must be rational 2)Function must have simple poles 3)We must understand these poles Berger, Bern, Dixon, Forde and Kosower

D Dunbar, DESY 07 27/38 -to carry out recursion we must understand poles of coefficients -multiparticle factorisation theorems Bern,Chalmers

D Dunbar, DESY 07 28/38 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

D Dunbar, DESY 07 29/38 Potential Recursion Relation + 

D Dunbar, DESY 07 30/38 Example: Split Helicity Amplitudes Consider the colour-ordered n-gluon amplitude -two minuses gives MHV -use as a example of generating coefficients recursively

D Dunbar, DESY 07 31/38 r-r- r look at cluster on corner with “split” -shift the adjacent – and + helicity legs -criteria satisfied for a recursion -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)

D Dunbar, DESY 07 32/38 -for, special case of 3 minuses,

D Dunbar, DESY 07 33/38 For R see Berger, Bern, Dixon, Forde and Kosower

D Dunbar, DESY 07 34/38 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

D Dunbar, DESY 07 35/38 Example of Spurious singularities Collinear Singularity Multi-particle pole Co-planar singularity

D Dunbar, DESY 07 36/38 Spurious singularities link different coefficients =0 at singularity -these singularities link different coefficients together

D Dunbar, DESY 07 37/38 Six gluon, what next,

D Dunbar, DESY 07 38/38 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