Loop Calculations of Amplitudes with Many Legs

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Loop Calculations of Amplitudes with Many Legs. David Dunbar, Swansea University, Wales, UK. DESY 2007. Plan. -motivation -organising Calculations -tree amplitudes -one-loop amplitudes -unitarity based techniques -factorisation based techniques -prospects. Q C D Matrix Elements.

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### Loop Calculations of Amplitudes with Many Legs

David Dunbar,

Swansea University, Wales, UK

DESY 2007

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

n-gluon scattering

n>5

-will focus on analytic methods

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

Generates the others

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

• 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

Organisation 3: Supersymmetric Decomposition

Supersymmetric gluon scattering amplitudes are the linear combination of QCD ones+scalar loop

-this can be inverted

degree p in l

p

Vertices involve loop momentum

propagators

p=n : Yang-Mills

Passarino-Veltman reduction

Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator

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

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

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

~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

Methods 1: Unitarity Methods

-look at the two-particle cuts

-use this technique to identify the coefficients

Unitarity,

-we can now carry our reduction on these cut integrals

Benefits:

-recycle compact on-shell tree expression

-use li2=0

Fermionic Unitarity

-try to use analytic structure to identify terms within two-particle cuts

bubbles

Generalised Unitarity

-use info beyond two-particle cuts

Box-Coefficients

Britto,Cachazo,Feng

-works for massless corners (complex momenta)

or signature (--++)

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

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

Provided,

then

Residues occur when amplitude factorises on multiparticle pole (including two-particles)

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)

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

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

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

-multiparticle factorisation theorems

Bern,Chalmers

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

Example: Split Helicity Amplitudes
• Consider the colour-ordered n-gluon amplitude

-two minuses gives MHV

-use as a example of generating coefficients recursively

-look at cluster on corner with “split”

-shift the adjacent – and + helicity legs

-criteria satisfied for a recursion

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r-

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-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)

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

Collinear Singularity

Multi-particle pole

Co-planar singularity

Example of Spurious singularities

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1