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

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

-this talk about one-loop

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

-leading in colour term

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

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

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


Six-Gluon Tree Amplitude

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,

  • 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

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

bubbles


Generalised Unitarity

-use info beyond two-particle cuts

-see also Dixon’s talk re multi-loops


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


Potential Recursion Relation

+


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-

+

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)


-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

  • -link coefficients together


Collinear Singularity

Multi-particle pole

Co-planar singularity

Example of Spurious singularities


1

1

2

2

2

2

1

1

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


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