Loop Calculations of Amplitudes with Many

Legs

DESY DESY 2007

David Dunbar,

Swansea University, Wales, UK

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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