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Hidden Structures in Quantum Field Theory, Copenhagen 2009

Unitarity Methods in Quantum Field Theory. David Dunbar, Swansea University, Wales, UK. Hidden Structures in Quantum Field Theory, Copenhagen 2009. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A. Objective. precise predictions. Experiment.

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Hidden Structures in Quantum Field Theory, Copenhagen 2009

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  1. UnitarityMethodsin Quantum Field Theory David Dunbar, Swansea University, Wales, UK Hidden Structures in Quantum Field Theory, Copenhagen 2009 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

  2. Objective 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. degree p in l p Vertices involve loop momentum Feynman Diagram of One- loop n-point Amplitude (in Massless Theory) 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. Unitarity of S-matrix -cross-order relation

  7. 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 imposingan extra condition

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

  10. z Analytic Structure Forde K1 K2 -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. 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. P kb -simplest non-trivial term in the two-particle cut -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

  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 Cut K1 K2

  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. - - - 93 - - - 93 94 94 94 06 94 94 05 06 94 94 05 06 94 06 05 05 94 05 06 06 94 05 06 06 The Six Gluon one-loop amplitude ~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 amplitude - - - 93 - - - 93 94 94 94 06 94 94 05 06 94 94 05 06 06 94 05 05 94 05 06 06 94 05 06 06 unitarity MHV Difficult/Complexity recursion feynman http://pyweb.swan.ac.uk/~dunbar/sixgluon.html

  27. -supersymmetric approximations -for fixed colour structure we have 64 helicity structures -specify colour structure, 8 independent helicities

  28. -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 amplitude 93 93 94 94 94 05 94 94 06 05 94 06 94 05 05 05 05 06 05 09 09 05 05 09 Refs at http://pyweb.swan.ac.uk/~dunbar/sevengluon.html

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

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

  33. 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 Britto, Feng Yang; Ellis, Giele, Kunzst, Melnikov Mastrolia Britto, Feng Mastrolia Badger, Glover, Risager Anastasiou, Britto, Feng, Kunszt, Mastrolia

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

  35. 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 Bern Morgan Van Neerman Britto Feng Mastrolia Bern,Dixon,dcd, Kosower Brandhuber, Macnamara, Spence Travaglini Kilgore

  36. 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

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