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Unitarity and Factorisation in Quantum Field Theory

VERSUS. Unitarity and Factorisation in Quantum Field Theory. Unitarity and Factorisation in Quantum Field Theory . David Dunbar, Swansea University, Wales, UK. Zurich 2008.

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Unitarity and Factorisation in Quantum Field Theory

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  1. VERSUS Unitarity and Factorisation in Quantum Field Theory Unitarity and Factorisation in Quantum Field Theory David Dunbar, Swansea University, Wales, UK Zurich 2008

  2. -conjectured weak-weak duality between Yang-Mills and Topological string theory in 2003 inspired flurry of activity in perturbative field theory • -look at what has transpired • -much progress in perturbation theory at both many legs and many loops (See Lance Dixon tommorow) • -unitarity • -factorisation • -QCD • -gravity

  3. Objective precise predictions Experiment Theory We want technology to calculate these predictions quickly, flexibly and accurately -despite our successes we have a long way to go

  4. QFT S-matrix theory Strings and QFT both have S-matrices String Theory -can link help with QFT?

  5. -not first time string theory inspired field theory -Parke-Taylor MHV formulae string inspired -Bern-Kosower Rules for one-loop amplitudes a’  0 -symmetry is important: embedding your theory in one with more symmetry might help understanding

  6. Duality with String Theory • Witten’s proposed of Weak-Weak duality between • A) Yang-Mills theory ( N=4 ) • B) Topological String Theory with twistor target space -Since this is a `weak-weak` duality perturbative S-matrix of two theories should be identical -True for tree level gluon scattering Rioban, Spradlin,Volovich

  7. Is the duality useful? Theory A : hard, interesting Theory B: easy Topological String Theory: harder Perturbative QCD, hard, interesting -duality may be useful indirectly

  8. + _ _ + _ + _ _ + _ + _ + + _ -eg MHV vertex construction of tree amplitudes Cachazo, Svercek, Witten -promote MHV amplitude to a fundamental vertex -works better than expected Brandhuber, Spence Travaglini -inspired by scattering of instantons in topological strings Rioban, Spradlin, Volovich -but can be understood in field theory Mansfield, Ettle, Morris, Gorsky -and by factorisation Risager

  9. 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

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

  11. 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

  12. - - - 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

  13. 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 recursion Difficult/Complexity feynman

  14. The Seven Gluon one-loop amplitude

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

  16. -working at the specific kinematic point of Ellis, Giele and Zanderaghi (looking at the finite pieces) QCD is almost supersymmetric….

  17. Unitarity Methods -look at the two-particle cuts -use unitarity to identify the coefficients

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

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

  20. Box-Coefficients Britto,Cachazo,Feng -works for massless corners (complex momenta) or signature (--++)

  21. Unitarity Techniques -turn C2 into coefficients of integral functions Different ways to approach this • reduction to covariant integrals • fermionic • analytic structure

  22. Reduction to covariant integrals -convert fermionic variables -converts integral into n-point integrals • -advantages: • connects to conventional reduction technique

  23. P kb in the two-particle cut -linear triangle

  24. Fermionic Unitarity Britto, Buchbinder,Cachazo, Feng, Mastrolia -use analytic structure to identify terms within two-particle cuts -advantages: two-dimensional rather than four dimensional, merges nicely with amplitudes written in terms of spinor variables bubbles

  25. z Analytic Structure Forde K1 K2 -triple cut reduces to problem in complex analysis -real momenta corresponds to unit circle poles at z=0 are triangles functions poles at z  0 are box coefficients

  26. 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

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

  28. Feynman Diagrams? -in general F a polynomial of degree n in l -only the maximal power of l contributes to rational terms -extracting rational might be feasible using specialised reduction Binoth, Guillet, Heinrich

  29. 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

  30. Factorisation 1) Amplitude will be singular at special Kinematic points, with well understood factorisation e.g. one-loop factorisation theorem Bern, Chalmers K is multiparticle momentum invariant 2) Amplitude does not have singularities elsewhere : at spurious singular points

  31. On-shell Recursion: tree amplitudes Britto,Cachazo,Feng (and Witten) • Shift amplitude so it is a complex function of z Tree amplitude becomes an analytic function of z, A(z) -Full amplitude can be reconstructed from analytic properties

  32. Provided, then Residues occur when amplitude factorises on multiparticle pole (including two-particles)

  33. 1 2 -results in recursive on-shell relation (c.f. Berends-Giele off shell recursion) Tree Amplitudes are on-shell but continued to complex momenta (three-point amplitudes must be included)

  34. cut construcible recursive? Recursion for Loops? -amplitude is a mix of cut constructible pieces and rational

  35. 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 Berger, Bern, Dixon, Forde and Kosower -requires auxiliary recusion limits for large-z terms

  36. Recursion on Integral Coefficients - - - + - r- r+1+ + + Bern, Bjerrum-Bohr, dcd, Ita recursive? Consider an integral coefficient and isolate a coefficient and consider the cut. Consider shifts in the cluster. -we obtain formulae for integral coefficients for both the N=1 and scalar cases

  37. Spurious Singularities • -spurious singularities are singularities which occur in • Coefficients but not in full amplitude • -need to understand these to do recursion • -link coefficients together Bern, Dixon Kosower Campbell, Glover Miller Bjerrum-Bohr, dcd, Perkins

  38. 3 2 1 4 -just how powerful is factorisation? -unusual example : four graviton, one loop scattering dcd, Norridge -amplitude has sixth order pole in [12] s=0, h 1 2 i 0 -spurious which only appears if we use complex momentum

  39. 3 2 1 4 u/t =-1 -s/t, expand in s -together with symmetry of amplitude, demanding poles vanish completely determines the entire amplitude dcd, H Ita -so the, very easy to compute, box coefficient determines rest of amplitude

  40. UV structure of N=8 Supergravity -is N=8 Supergravity a self-consistent QFT -progress in methods allows us to examine the perturbative S-matrix -Does the theory have ultra-violet singularities or is it a ``finite’’ field theory

  41. ``Finite for 8 loops but not beyond’’ 2) Look at supergravity embedded within string theory 3) Find a dual theory which is solvable SuperstringTheory Dual Theory N=8 Supergravity Green, Russo, Van Hove, Berkovitz, Chalmers 1) Approach problem within the theory Abou-Zeid, Hull, Mason

  42. -results/suggestions • -the S-matrix is UV softer than one would expect. Has same behaviour as N=4 SYM • True at one-loop ``No-triangle Hypothesis’’ • True for 4pt 3-loop calculation • Is N=8 finite like N=4 SYM?

  43. N=8 Supergravity • Loop polynomial of n-point amplitude of degree 2n. • Leading eight-powers of loop momentum cancel (in well chosen gauges..) leaving (2n-8) or (2r-8) • Beyond 4-point amplitude contains triangles and bubbles but • only after reduction • Expect triangles n > 4 , bubbles n >5 , rational n > 6 r

  44. No-Triangle Hypothesis -against this expectation, it might be the case that……. Evidence? true for 4pt n-point MHV 6-7pt NMHV proof Green,Schwarz,Brink Bern,Dixon,Perelstein,Rozowsky Bjerrum-Bohr, dcd,Ita, Perkins, Risager; Bern, Carrasco, Forde, Ita, Johansson, Bjerrum-Bohr Van Hove -extra n-4 cancelations

  45. Three Loops Result Bern, Carrasco, Dixon, Johansson, Kosower and Roiban, 07 S -actual for Sugra SYM: K3D-18 Finite for D=4,5 , Infinite D=6 Sugra: K3D-16 -again N=8 Sugra looks like N=4 SYM

  46. Rockall versus Tahiti • -the finiteness or otherwise of N=8 Supergravity is still unresolved although all explicit results favour finiteness • -does it mean anything? Possible to quantise gravity with only finite degrees of freedom. • -is N=8 supergravity the only finite field theory containing gravity? ….seems unlikely….N=6/gauged….

  47. Kasper Risager, NBI Harald Ita, UCLA Warren Perkins Emil Bjerrum-Bohr, IAS Bjerrum-Bohr, Dunbar, Ita, Perkins and Risager, ``The no-triangle hypothesis for N = 8 supergravity,'‘ JHEP 0612 (2006) 072 , hep-th/0610043. May 2006 to present: all became fathers 5 real +2 virtual children TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

  48. 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 • -important to finish some process • -is N=8 supergravity finite

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