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

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

3. Standard Model : two worldsp p p  p 
D Dunbar, IOP, Durham 08

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

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

13. 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

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

15. -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

16. 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

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

18. -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

19. 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

20. 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

21. 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

22. 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

23. 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

24. The Six Gluon one-loop amplitude94 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

25. 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