Modern Methods for Loop Calculations of Amplitudes with Many Legs David Dunbar, Swansea University IOP: Theory for Experimentalists, Durham, Feb 20th

The Standard Model : Two Worlds IR world UV world D Dunbar, IOP, Durham 08

Standard Model : two worlds p p p  p  D Dunbar, IOP, Durham 08

QCD Matrix Elements n>5 -this talk about one-loop -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-particle scattering n>5 -will focus on analytic methods D Dunbar, IOP, Durham 08

Divide and Conquer : the theorists solution! -split amplitude into smaller managable peices -try to work with physical subprocesses (Feynman diagrams are not physical) -particles in standard model have colour, spin, helicity… -although experiments require inclusive cross sections it turns out more practical to evaluate exclusive D Dunbar, IOP, Durham 08

Dividing 1: Colour-Ordering Gauge theory amplitudes depend upon colour of gluons. We can split colour from kinematics by colour decomposition The colour ordered amplitudes have cyclic symmetric rather than full crossing symmetry Colour ordered amplitudes physical -leading in colour term Generates the others D Dunbar, IOP, Durham 08

Dividing 2: Spinor Helicity Xu, Zhang,Chang 87 Helicity of (massless) particle is a good quantum number in ``inner world’’ Reference Momenta Gluon Momenta -extremely useful technique which produces relatively compact expressions for amplitudes in terms of spinor products D Dunbar, IOP, Durham 08

Dividing 3: Supersymmetric Decomposition (1-loop) Supersymmetric gluon scattering amplitudes are the linear combination of QCD ones+scalar loop -this can be inverted D Dunbar, IOP, Durham 08

Dividing 4: General Decomposition of One-loop n-point Amplitude degree p in l Vertices involve loop momentum propagators p=n : Yang-Mills D Dunbar, IOP, Durham 08

Passarino-Veltman reduction Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator D Dunbar, IOP, Durham 08

Tree Amplitudes (Gluons) 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, IOP, Durham 08

Six-Gluon Tree Amplitude factorises D Dunbar, IOP, Durham 08

Conquering 1: Unitarity Methods Optical Theorem for S-matrix -look at the two-particle cuts -use this technique to identify the coefficients -a single cut only probes a subset of functions D Dunbar, IOP, Durham 08

Generalised Unitarity -use info beyond two-particle cuts Britto Cachazo, Feng D Dunbar, IOP, Durham 08

-key feature : work with on-shell physical amplitudes 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 D Dunbar, IOP, Durham 08

Conquering 2: On-shell recursion (trees) 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 D Dunbar, IOP, Durham 08

Provided, then Residues occur when amplitude factorises on multiparticle pole (including two-particles) D Dunbar, IOP, Durham 08

-results in recursive on-shell relation (c.f. Berends-Giele off shell recursion) 1 2 Tree Amplitudes are on-shell but continued to complex momenta (three-point amplitudes must be included) D Dunbar, IOP, Durham 08

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, IOP, Durham 08

Expansion in terms of Integral Functions cut construcible recursive? recursive? - R is rational and not cut constructible (to O()) -amplitude is a mix of cut constructible pieces and rational D Dunbar, IOP, Durham 08

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 D Dunbar, IOP, Durham 08

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, IOP, Durham 08

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, IOP, Durham 08

The Six Gluon one-loop amplitude 94 05 06 - 93 ~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 D Dunbar, IOP, Durham 08 Britto, Feng, Mastriolia Mahlon

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 -all has to be put back together for standard model! D Dunbar, IOP, Durham 08