Loop Calculations of Amplitudes with Many Legs. David Dunbar, Swansea University, Wales, UK. DESY 2007. Plan. motivation organising Calculations tree amplitudes oneloop amplitudes unitarity based techniques factorisation based techniques prospects. Q C D Matrix Elements.
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David Dunbar,
Swansea University, Wales, UK
DESY 2007
this talk about oneloop
ngluon scattering
n>5
will focus on analytic methods
Colour ordering is not is field theory textbooks but is in string texts
leading in colour term
Generates the others
Reference Momenta
Organisation 2: Spinor HelicityXu, Zhang,Chang 87
extremely useful technique which produces relatively compact expressions for amplitudes in terms of spinor products
Transforming to twistor space gives a different organisation
Amplitude is a function on twistor space
Witten, 03
Supersymmetric gluon scattering amplitudes are the linear combination of QCD ones+scalar loop
this can be inverted
degree p in l variables
p
Vertices involve loop momentum
propagators
Organisation 4: General Decomposition of Oneloop npoint Amplitudep=n : YangMills
Decomposes a npoint integral into a sum of (n1) integral functions obtained by collapsing a propagator
Decomposes a npoint integral into a sum of (n1) integral functions obtained by collapsing a propagator
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
 variables


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The Six Gluon oneloop amplitude~14 papers
81% `B’
Berger, Bern, Dixon, Forde, Kosower
Bern, Dixon, Dunbar, Kosower
Britto, Buchbinder, Cachazo, Feng
Bidder, BjerrumBohr, Dixon, Dunbar
Bern, Chalmers, Dixon, Kosower
Bedford, Brandhuber, Travaglini, Spence
Forde, Kosower
Xiao,Yang, Zhu
Bern, BjerrumBohr, Dunbar, Ita
Britto, Feng, Mastriolia
Mahlon
look at the twoparticle cuts
use this technique to identify the coefficients
we can now carry our reduction on these cut integrals
Benefits:
recycle compact onshell tree expression
use li2=0
try to use analytic structure to identify terms within twoparticle cuts
bubbles
use info beyond twoparticle cuts
see also Dixon’s talk re multiloops
Britto,Cachazo,Feng
works for massless corners (complex momenta)
or signature (++)
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 onshell physical amplitudes
Britto,Cachazo,Feng (and Witten)
Tree amplitude becomes an analytic function of z, A(z)
Full amplitude can be reconstructed from analytic properties
then
Residues occur when amplitude factorises on multiparticle pole (including twoparticles)
1 variables
2
results in recursive onshell relation
(c.f. BerendsGiele off shell recursion)
Tree Amplitudes are onshell but continued to complex momenta (threepoint amplitudes must be included)
cut construcible variables
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
can we shift R and obtain it from its factorisation?
Berger, Bern, Dixon, Forde and Kosower
multiparticle factorisation theorems
Bern,Chalmers
Consider an integral coefficient and isolate a coefficient and consider the cut. Consider shifts in the cluster.
such shifts will generate a recursion formulae
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




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, coefficients
For R see Berger, Bern, Dixon, Forde and Kosower coefficients
Collinear Singularity coefficients
Multiparticle pole
Coplanar singularity
Example of Spurious singularities1 coefficients
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, coefficients