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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

Loop Calculations of Amplitudes with Many Legs

David Dunbar,

Swansea University, Wales, UK

DESY 2007


  • -motivation

  • -organising Calculations

  • -tree amplitudes

  • -one-loop amplitudes

  • -unitarity based techniques

  • -factorisation based techniques

  • -prospects

Q c d matrix elements
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


-will focus on analytic methods

Organisation 1 colour ordering
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

Organisation 2 spinor helicity

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

We can continue and write amplitude completely in fermionic variables
We can continue and write amplitude completely in fermionic variables

  • 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
Organisation 3: Supersymmetric Decomposition variables

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

-this can be inverted

Organisation 4 general decomposition of one loop n point amplitude

degree p in l variables


Vertices involve loop momentum


Organisation 4: General Decomposition of One-loop n-point Amplitude

p=n : Yang-Mills

Passarino veltman reduction
Passarino-Veltman reduction variables

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

Passarino veltman reduction1
Passarino-Veltman reduction variables

  • 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
Tree Amplitudes variables

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

Six gluon tree amplitude
Six-Gluon Tree Amplitude variables


One loop qcd amplitudes
One-Loop QCD Amplitudes variables

  • 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

The six gluon one loop amplitude

- variables
































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


Methods 1 unitarity methods
Methods 1: Unitarity Methods variables

-look at the two-particle cuts

-use this technique to identify the coefficients

Unitarity, variables

  • Start with tree amplitudes and generate cut integrals

-we can now carry our reduction on these cut integrals


-recycle compact on-shell tree expression

-use li2=0

Fermionic unitarity
Fermionic Unitarity variables

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


Generalised unitarity
Generalised Unitarity variables

-use info beyond two-particle cuts

-see also Dixon’s talk re multi-loops

Box coefficients
Box-Coefficients variables


-works for massless corners (complex momenta)

or signature (--++)

Unitarity variables

-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
Methods 2: On-shell recursion: tree amplitudes variables

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, variables


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

1 variables


-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
Recursion for One-Loop amplitudes? variables

  • Analytically continuing the 1-loop amplitude in momenta leads to a function with both poles and cuts in z

Expansion in terms of integral functions

cut construcible variables



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
Recursion for Rational terms variables

-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

To carry out recursion we must understand poles of coefficients
-to carry out recursion we must understand poles of coefficients

-multiparticle factorisation theorems


Recursion on integral coefficients
Recursion on Integral Coefficients 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

Potential recursion relation
Potential Recursion Relation coefficients


Example split helicity amplitudes
Example: Split Helicity Amplitudes coefficients

  • 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” coefficients

-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)

Spurious singularities
Spurious Singularities coefficients

  • -spurious singularities are singularities which occur in

  • Coefficients but not in full amplitude

  • -need to understand these to do recursion

  • -link coefficients together

Example of spurious singularities

Collinear Singularity coefficients

Multi-particle pole

Co-planar singularity

Example of Spurious singularities

Spurious singularities link different coefficients

1 coefficients








Spurious singularities link different coefficients

=0 at singularity

-these singularities link different coefficients together

Conclusions coefficients

  • -new techniques for NLO gluon scattering

  • -lots to do

  • -but tool kits in place?

  • -automation?

  • -progress driven by very physical developments: unitarity and factorisation