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Recursive Approaches to QCD Matrix Elements

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### Recursive Approaches to QCD Matrix Elements

David Dunbar,

Swansea University, Wales

including work with

Z. Bern, S Bidder, E Bjerrum-Bohr, L. Dixon, H Ita, D Kosower W Perkins K. Risager

RADCOR 2005

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
- -One-Loop n-point on-shell amplitudes unknown for n>5
- -In this talk we discuss how recent (> Dec 03) ideas of a “weak-weak” duality might help,
- - apply to 2g 4g

2 4 talks of Binoth,Dittmaier

Duality with String Theory

- In Dec 03 Witten proposed a Weak-Weak duality between
- A) Yang-Mills theory ( N=4 )
- B) Topological String Theory

S-matrix of two theories should be identical

-True for tree level gluon scattering

Rioban, Spradlin,Volovich

proposal constructive for N=4 but… N<4..??

Duality Details

- Amplitude a function
- Spinor helicity replaces polarisation by fermionic variables
- Use fermionic coordinates and replace bosonic momenta by `twistors`
- Amplitude now function entirely of fermionic variables
- Fourier (Penrose transform) one of spinors,
- Amplitude in twistor space is closest to string theory

Xu, Zhang,Chang 87

Is the duality useful?

Theory A :

hard, interesting

Theory B:

easy

Topological

String Theory:

harder, uninteresting

Perturbative QCD,

hard, interesting

-duality may be useful indirectly

Inspired by duality – the CSW/MHV-vertex construction

Cachazo Svercek Witten 04, (Nair)

- Promotes MHV amplitude to fundamental object by
- -Off-shell continuation
- -makes YM perturbative expansion match that of B-instantons

(colour ordered amplitudes)

Parke-Taylor, Berends-Giele

MHV-vertex construction

- Works for gluon scattering tree amplitudes
- Works for (massless) quarks
- Works for Higgs and W’s
- Works for photons
- Works for gravity

Wu,Zhu; Su,Wu; Georgiou Khoze

Badger, Dixon, Glover, Forde, Khoze, Kosower Mastrolia

Ozeren+Stirling

Bjerrum-Bohr,DCD,Ita,Perkins, Risager

-Works for one-loops in SUPERSYMMETRIC theories

-In trouble for non-SUSY theories

Bedford,Brandhuber, Spence and Travaglini;

Qigley,Rozali

A(+++…++)

Inspired by duality –BCFW construction for tree amplitudes

Britto,Cachazo,Feng (and Witten)

- Return of the analytic S-matrix!
- Shift amplitude so it is a complex function of z

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)

CSW vs BCF

- Difference
- CSW asymmetric between helicity sign
- BCF chooses two special legs
- Similarities-
- both rely upon analytic structure

CSW can be derived from a type of analytic shift

Risager; Bjerrum-Bohr,Dunbar,Ita,Perkins and Risager, 05

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

n-point MHV amplitudes supersymmetric theories

Six-point N=4 amplitudes

Bern,Dixon,Dunbar and Kosower 94/95

Supersymmetric Decomposition

Supersymmetric gluon scattering amplitudes are the linear combination of QCD ones+scalar loop

-this can be inverted

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

Box-Coefficients

Britto,Cachazo,Feng

S

-works for massless corners (complex momenta)

or signature (--++)

-works for non-supersymmetric

Bjerrum-Bohr,Bidder,DCD,Perkins

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, Perkins

Britto, Buchbinder Cachazo, Feng, 04/05

recursive?

recursive?

The Final Pieces : scalar contributions-last component of QCD amplitudes

- R is rational and not cut constructible (to O())

-can we avoid direct integration?

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

Complication,

Either R or the coefficients of integral functions may containSpurious Singularitieswhich 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

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

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’)

Results: can we avoid them?

It has been demonstrated, using

A) (-++….+++), (+++++….++) and

B) (--++++) and (--++….+++)

1)

that shifts can be found which allow calculation of rational parts recursively

Bern, Dixon Kosower

Forde, Kosower

Shifts can be found which allow the integral coefficients to be computed recursively

2)

A(---..--+++…++)

Bern, Bjerrum-Bohr, Dunbar, Ita

Conclusions can we avoid them?

- -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!

-also fermions, masses, multiloop……..???.......

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