# Loop Calculations of Amplitudes with Many Legs - PowerPoint PPT Presentation

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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

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

### 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

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

factorises

### 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

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

-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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-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)

-for , special case of 3 minuses,

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

### 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

1

1

2

2

2

2

1

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### Spurious singularities link different coefficients

=0 at singularity

-these singularities link different coefficients together

Six gluon, what next,

### 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