1. Unitarity Methods in Quantum Field Theory
David Dunbar,
Swansea University, Wales, UK
Hidden Structures in Quantum Field Theory, Copenhagen 2009
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2. Objective Experiment Theory
precise predictions
Experiment
Theory
We want technology to calculate these predictions quickly, flexibly and accurately
-use calculations to probe theory
-despite our successes we have a long way to go

3. Feynman Diagram of One- loop n-point Amplitude (in Massless Theory)
degree p in l
Vertices involve loop momentum
p=n : Yang-Mills
p=2n Gravity
propagators

4. Passarino-Veltman reduction of 1-loop
Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator
Four dimensional cut construcible
-coefficients are rational functions of |ki§ using spinor helicity
-feature of Quantum Field Theory

5. Unitarity Techniques
Alternate to Feynman diagram techniques Discuss and demystify Some applications Most at one-loop Looking at analytic techniques

6. -cross-order relation
Unitarity of S-matrix
-cross-order relation
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7. -use unitarity to identify the coefficients
Unitarity Methods
-look at the two-particle cuts K
-use unitarity to identify the coefficients

8. Topology of Cuts
-look when K is timelike, in frame where K=(-K0,0,0,0)
3-momenta l1 and l2 are back to back on surface of sphere
imposing an extra condition

9. Generalised Unitarity
-use info beyond two-particle cuts

10. Analytic Structure K1 K2 z Forde
-triple cut reduces to problem in complex analysis
-real momenta corresponds to unit circle
-A(z) can be extended to all z
poles at z=0 are triangles functions
poles at z  0 are box coefficients

11. -works for massless corners (complex momenta)
Box-Coefficients
Britto,Cachazo,Feng
-works for massless corners (complex momenta)

12. Unitarity Techniques
-use generalised unitarity, step by step. This is a choice, since one can just use C2
-C2 most complicated/time consuming
Different ways to approach this
reduction to covariant integrals
fermionic
analytic structure

13. Unitarity using canonical Forms
-act directly on C2 with amplitudes written in Spinor helicity
-integrand is function of
-really recognising standard integrals which can be done using any method- once!

14. Reduction to covariant integrals
-convert fermionic variables
-converts integral into n-point integrals
-advantages:
connects to conventional reduction technique
everyone understands!
-organise according to order of li

15. -simplest non-trivial term
in the two-particle cut kb P
-linear triangle

16. Extend the canonical form
-algebraic identity

17. more canonical forms Use identity,
Gram determinant of three mass triangle

18. -better to recombine and rationalise

19. -another useful identity,
-leading order is blind to label on li

20. Higher Order Canonical Forms
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21. Spurious singularities,
-A singularity in the coefficient not present in the amplitude
-singularity cancels between integral functions
-can combine integral functions

22. Canonical Forms for Triple CutK1 K2
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23. Applications: 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 : lots of People, lots of techniques

24. Organisation of QCD amplitudes: divide amplitude into smaller physical pieces
-QCD gluon scattering amplitudes are the linear combination of
Contributions from supersymmetric multiplets
-use colour ordering; calculate cyclically symmetric partial amplitudes
-organise according to helicity of external gluon

25. The Six Gluon one-loop amplitude94 05 06 - 93
~13 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

26. The Six Gluon one-loop amplitude93 - - - 93 94 94 94 06 94 94 05 06 94 94 05 06 94 05 05 06 94 05 06 06 94 05 06 06
unitarity
MHV
Difficult/Complexity
recursion
feynman

27. -supersymmetric approximations
(++++++) 1
(-+++++) 6
(--++++) 12
(-+-+++)
(-++-++)
(---+++)
(--+-++)
(-+-+-+) 2
-for fixed colour structure we have 64 helicity structures
-specify colour structure, 8 independent helicities

28. QCD is almost supersymmetric….
N=4
SUSY
(--++++)
0.32
0.04
(-+-+++)
0.30
(-++-++)
0.37
(---+++)
0.16
0.06
(--+-++)
0.36
(-+-+-+)
0.13
0.02
-working at the specific kinematic point of Ellis, Giele and Zanderaghi
(looking at the finite pieces)
QCD is almost supersymmetric….

29. Approximate Universality for N=4
dcd, Ettle Perkins
(again at EGZ kinematic point)
Compare to QCD and N=1,
-comparison is renormalisation scale dependant, helicity emplitudes converge at very large renormalisation scale. Effect a IR artifact?

30. The Seven Gluon one-loop amplitude93 93 94 94 94 05 94 94 05 06 94 94 06 05 05 05 05 06 05 09 05 09 05 09
Refs at

31. Using Canonical forms for Eg. SevenPoint N=1 Contributions
-20 rational coefficients of the integral functions determine contribution

32. very similar to N=4 form, Bern Dixon Kosower
-general n-point forms can be constructed
for many boxes
very similar to N=4 form, Bern Dixon Kosower
D Dunbar,, NBA, Aug 09

33. D Dunbar,, NBA, Aug 09

35. Unitarity -works well to calculate coefficients
-particularly strong for supersymmetry (R=0)
-can be automated
Ellis, Giele, Kunszt ;Ossola, Pittau, Papadopoulos
Berger Bern Dixon Febres-Cordero Forde Ita Kosower Maitre
-extensions to massive particles progressing
Ellis, Giele, Kunzst, Melnikov
Britto, Feng Yang;
Mastrolia
Britto, Feng Mastrolia
Badger, Glover, Risager
Anastasiou, Britto, Feng, Kunszt, Mastrolia

36. How do we calculate R?
D- dimensional Unitarity
Factorisation/Recursion
Feynman Diagrams

37. D-dimensional Unitarity
-in dimensional regularisation amplitudes have an extra -2 momentum weight
-consequently rational parts of amplitudes have cuts to O()
-consistently working with D-dimensional momenta should allow us to determine rational terms
-these must be D-dimensional legs
Van Neerman
Bern Morgan
Britto Feng Mastrolia
Bern,Dixon,dcd, Kosower
Brandhuber, Macnamara, Spence Travaglini
Kilgore

38. Conclusions -new techniques for NLO gluon scattering
-progress driven by very physical developments: unitarity and factorisation
-amplitudes are over constrained
-nice to live on complex plane (or with two times)
-still much to do: extend to less specific problems