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

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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


Loop calculations of amplitudes with many legs

Plan

  • -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

n>5

-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

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

p

Vertices involve loop momentum

propagators

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

p=n : Yang-Mills


Passarino veltman reduction

Passarino-Veltman reduction

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


Passarino veltman reduction1

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

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


Six gluon tree amplitude

Six-Gluon Tree Amplitude

factorises


One loop qcd amplitudes

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


The six gluon one loop amplitude

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93

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06

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05

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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

Methods 1: Unitarity Methods

-look at the two-particle cuts

-use this technique to identify the coefficients


Unitarity

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


Fermionic unitarity

Fermionic Unitarity

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

bubbles


Generalised unitarity

Generalised Unitarity

-use info beyond two-particle cuts

-see also Dixon’s talk re multi-loops


Box coefficients

Box-Coefficients

Britto,Cachazo,Feng

-works for massless corners (complex momenta)

or signature (--++)


Unitarity1

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

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

Provided,

then

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


Loop calculations of amplitudes with many legs

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

Recursion for One-Loop amplitudes?

  • 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

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

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


To carry out recursion we must understand poles of coefficients

-to carry out recursion we must understand poles of coefficients

-multiparticle factorisation theorems

Bern,Chalmers


Recursion on integral coefficients

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


Potential recursion relation

Potential Recursion Relation

+


Example split helicity amplitudes

Example: Split Helicity Amplitudes

  • Consider the colour-ordered n-gluon amplitude

-two minuses gives MHV

-use as a example of generating coefficients recursively


Loop calculations of amplitudes with many legs

-look at cluster on corner with “split”

-shift the adjacent – and + helicity legs

-criteria satisfied for a recursion

-

-

-

-

r-

+

r+1+

+

+

-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)


Loop calculations of amplitudes with many legs

-for , special case of 3 minuses,


Loop calculations of amplitudes with many legs

For R see Berger, Bern, Dixon, Forde and Kosower


Spurious singularities

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


Example of spurious singularities

Collinear Singularity

Multi-particle pole

Co-planar singularity

Example of Spurious singularities


Spurious singularities link different coefficients

1

1

2

2

2

2

1

1

Spurious singularities link different coefficients

=0 at singularity

-these singularities link different coefficients together


Loop calculations of amplitudes with many legs

Six gluon, what next,


Conclusions

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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