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
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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
• -One-Loop n-point on-shell amplitudes unknown for
  ngt5
• -In this talk we discuss how recent (gt Dec 03)
  ideas of a weak-weak duality might help,
• - apply to 2g ? 4g

2 ? 4 talks of Binoth,Dittmaier
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Duality with String Theory

• In Dec 03 Witten proposed a Weak-Weak duality
  between
• A) Yang-Mills theory ( N4 )
• B) Topological String Theory

S-matrix of two theories should be identical
-True for tree level gluon scattering
Rioban, Spradlin,Volovich
proposal constructive for N4 but Nlt4..??
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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
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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
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Inspired by duality the CSW/MHV-vertex
construction
Cachazo Svercek Witten 04, (Nair)
Parke-Taylor, Berends-Giele
(colour ordered amplitudes)

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

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-three point vertices allowed
-number of vertices (number of -) -1
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MHV-vertex construction

• Works for gluon scattering tree amplitudes
• Works for (massless) quarks
• Works for Higgs and Ws
• Works for photons
• Works for gravity

Wu,Zhu Su,Wu Georgiou Khoze
Badger, Dixon, Glover, Forde, Khoze, Kosower
Mastrolia
OzerenStirling
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()
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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
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Provided,
then
Residues occur when amplitude factorises on
multiparticle pole (including two-particles)
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-results in recursive on-shell relation
(three-point amplitudes must be included)
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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
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One-Loop Amplitudes

• One Loop Gluon Scattering Amplitudes in QCD
• -Four Point EllisSexton
• -Five Point Bern, Dixon,Kosower
• -Six-Point and beyond--- present problem

n-point MHV amplitudes supersymmetric theories
Six-point N4 amplitudes
Bern,Dixon,Dunbar and Kosower 94/95
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Supersymmetric Decomposition
Supersymmetric gluon scattering amplitudes are
the linear combination of QCD onesscalar loop
-this can be inverted
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N4 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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Box-Coefficients
Britto,Cachazo,Feng
S
-works for massless corners (complex momenta)

or signature (--)
-works for non-supersymmetric
Bjerrum-Bohr,Bidder,DCD,Perkins
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N1 One-Loop Amplitudes -????

• Important to choose a good basis of functions
• A) choose chiral multiplet
• B) use D6 boxes
• Amplitude also cut constructible
• -six gluon amplitudes now obtained using unitarity

Bidder,Bjerrum-Bohr,Dixon, Dunbar, Perkins
Britto, 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(?))
-can we avoid direct integration?
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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

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-understanding poles
-multiparticle factorisation theorems

Bern,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
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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 r?1 (rs/s)
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Results

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
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State of Play Six Gluon Scattering
2g-? 4g
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
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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!

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