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Loop Calculations of Amplitudes with Many
Legs
DESY DESY 2007
David Dunbar,
Swansea University, Wales, UK
D Dunbar, DESY 07
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Plan-motivation
-organising Calculations
-tree amplitudes
-one-loop amplitudes
-unitarity based techniques
-factorisation based techniques
-prospects
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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
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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 termGenerates the others
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Organisation 2: Spinor HelicityXu, Zhang,Chang 87
Gluon Momenta
Reference Momenta
-extremely useful technique which produces relatively compact expressions for amplitudes in terms of spinor products
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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 spaceWitten, 03
Transforming to twistor space gives a different organisation
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Organisation 3: Supersymmetric Decomposition
Supersymmetric gluon scattering amplitudes are the linear combination of
QCD ones+scalar loop
-this can be inverted
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Organisation 4: General Decomposition of One-loop n-point
Amplitude
Vertices involve loop momentumpropagators
p
degree p in l
p=n : Yang-Mills
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Passarino-Veltman reduction
Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator
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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
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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
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Six-Gluon Tree Amplitude
factorises
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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
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The Six Gluon one-loop amplitude
949494949494
94 94
9494
05
06
05
0505 06
0505 06
06
06
06
0606
----
--9393
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
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Methods 1: Unitarity Methods
-look at the two-particle cuts
-use this technique to identify the coefficients
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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
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Fermionic Unitarity
-try to use analytic structure to identify terms within two-particle cuts
bubbles
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Generalised Unitarity-use info beyond two-particle cuts
-see also Dixon’s talk re multi-loops
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Box-Coefficients
-works for massless corners (complex momenta)
Britto,Cachazo,Feng
or signature (--++)
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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
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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)
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Provided,
Residues occur when amplitude factorises on multiparticle pole (including two-
particles)
then
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-results in recursive on-shell relation
Tree Amplitudes are on-shell but continued to complex momenta (three-point amplitudes must be
included)
1 2
(c.f. Berends-Giele off shell recursion)
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Recursion for One-Loop amplitudes?
Analytically continuing the 1-loop amplitude in momenta leads to a function with both poles and cuts in z
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Expansion in terms of Integral Functions
- R is rational and not cut constructible (to O())
cut construcible
recursive?recursive?
-amplitude is a mix of cut constructible pieces and rational
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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
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-to carry out recursion we must understand poles of
coefficients
-multiparticle factorisation theoremsBern,Chalmers
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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
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Potential Recursion Relation
+
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Example: Split Helicity Amplitudes
Consider the colour-ordered n-gluon amplitude
-two minuses gives MHV
-use as a example of generating coefficients recursively
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r-
r+1+++ +
-
--
--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)
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-for , special case of 3 minuses,
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For R see Berger, Bern, Dixon, Forde and Kosower
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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
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Example of Spurious singularities
Collinear Singularity Multi-particle
poleCo-planar singularity
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Spurious singularities link different coefficients
121
2
12
12
=0 at singularity
-these singularities link different coefficients together
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Six gluon, what next,
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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
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