Twistor Inspired techniques in Perturbative Gauge Theories-II

Twistor Inspired techniques in Perturbative Gauge Theories-II

including work with Z. Bern, S Bidder, E

Bjerrum-Bohr, L. Dixon, H Ita, W Perkins K. Risager

KIAS-KIAST KIAS-KIAST 2005

David Dunbar,

Swansea University, Wales

D Dunbar, KIAS-KIAST 05

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Seminar II

Hadron Colliders, LHC

Need for NLO computations

Pieces of NLO computations

QCD calculations


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Hadron Colliders LHC


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5


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LHC Physics

-hadron machines are DISCOVERY machines (SPS:W+Z,Tevatron: t)

-LHC will “hunt the higgs”

-hunt SUSY

-hunt new physics


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Higgs Production and Decay

H

g

g

W/Z

W/Z

-four final state particles end-point of Higgs production-very often four jets

eg


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10


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-most decay end-points of new physics can be simulated by background standard model processes

-very important to have robust accurate predictions for

background decay rates/event shapes/angular distribution

based upon known physics

-jets are “inclusive processess” : experimentally we cannot

distinguish colour, helicity, spin.


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Pieces of Theoretical Prediction

Probability of producing final state =

Structure FunctionsMatrix Elements Hadronisation

-piece that twistors may help with


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Need for NLO Matrix Elements calculations for jets

Consider 2g -> 2g

+g4

2

g2

2

2

g3


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g2+g4 +g6

2

2

2

g3

g4

+g5

NNLO


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Wny is NLO neccessary

-accurancy, QCD is strong(ish)

-scale dependance

-cone-size dependance


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One-Loop Amplitudes

One Loop Gluon Scattering Amplitudes in QCD

-Four Point : Ellis+Sexton

-Five Point : Bern, Dixon,Kosower

-Six-Point and beyond--- present problem

-Five and Six-Point mixed procecess

n-point MHV amplitudes supersymmetric theories

Bern,Dixon,Dunbar and Kosower 94/95

Six-point N=4 amplitudes


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General Decomposition of One-loop Amplitude

Linear in loop momentum propagators

n

degree n in l


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Passerino-Veltman reduction

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

k

l

l-k


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-process continues until we reach four-point integral functions with (in yang-mills up to quartic numerators)

-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

Functions of a single kinematic invariant, ln(s)


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Supersymmetric Decomposition

Supersymmetric gluon scattering amplitudes are the linear combination of

QCD ones+scalar loop

-this can be inverted


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N=4 One-Loop Amplitudes –solved!

Amplitude is a a sum of scalar box functions with rational coefficients (BDDK,1994)

Coefficients are ``cut-constructable’’ (BDDK,1994)

Quadruple cuts turns calculus into algebra (Britto,Cachazo,Feng,2005)

Box Coefficients are actually coefficients of terms like


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N=4 Susy

In N=4 susy there are cancelations between the states of different spin circulating in the loop.

Leading four-powers of loop momentum cancel (in well chosen gauges..)

N=4 lie in a small subspace of the allowed possible amplitudes


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Basis in N=4 Theory

‘‘easy’ two-mass easy’ two-mass boxbox

‘‘hard’ two-mass hard’ two-mass boxbox


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Box-Coefficients

-works for massless corners (complex momenta)

S

Britto,Cachazo,Feng

or signature (--++)-works for non-supersymmetric

Bjerrum-Bohr,Bidder,DCD,Perkins


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Box Coefficients-Twistor Structure

Box coefficients has coplanar support for NMHV 1-loop

amplitudes

-true for both N=4 and QCD!!!


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N=1 One-Loop Amplitudes -????

Important to choose a good basis of functions

A) choose chiral multiplet

B) use D=6 boxes

Amplitude also cut constructible

-six gluon amplitudes now obtained using unitarity

Bidder,Bjerrum-Bohr,Dixon, Dunbar, PerkinsBritto, Buchbinder Cachazo, Feng, 04/05


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The Final Pieces : scalar contributions

-last component of QCD amplitudes- R is rational and not cut constructible (to

O())

cut construcible

recursive?recursive?

-can we avoid direct integration?


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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


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-understanding poles

-multiparticle factorisation

theoremsBern,Chalmers


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Complication,

Either R or the coefficients of integral functions may contain Spurious Singularities which are not present in the full amplitude

It is important and non-trivial to find shift(s) which avoid these spurious singularities whilst still affecting the full R/coefficient


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Example of Spurious singularities

Collinear Singularity Multi-particle

poleCo-planar singularity


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Splitting Amplitude into C and R is not unique

The integral functions can be defined to include rational pieces, e.g

rather than

avoids a spurious singularity as r1 (r=s/s’)

Spurious singularities spoil understanding of residues – can we avoid them?


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Results:

It has been demonstrated, using

A) (-++….+++), (+++++….++) and

B) (--++++) and (--++….+++)

that shifts can be found which allow calculation of rational parts recursivelyBern, Dixon Kosower

1)

2) Shifts can be found which allow the integral coefficients to be computed

recursivelyA(---..--+++…++)

Bern, Bjerrum-Bohr, Dunbar, Ita

Forde, Kosower


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State of Play Six Gluon Scattering

X

X X

XX

X

X

X

X

X X

X X

X

X

XX

2g- 4g


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Conclusions-Reasons for optimism in computing one-loop QCD matrix

elements

-Recent progress uses UNITARITY and FACTORISATION as key features of on-shell amplitudes

-Inspired by Weak-Weak duality but not dependant upon it

-after much progress in highly super-symmetric theories the (harder) problem of QCD beginning to yield results

-first complete result for a partial 2g ng amplitude!

-NNLO is the goal for LHC

-analytic vs. numerical?

-fermions, masses, multi-loops,…..

Twistor Inspired techniques in Perturbative Gauge Theories-II

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