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

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Presentation on theme: "Unitarity and Factorisation in Quantum Field Theory Zurich Zurich 2008 David Dunbar, Swansea University, Wales, UK VERSUS Unitarity and Factorisation in."— Presentation transcript:

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

2 D Dunbar, Gauge Theory and Strings, ETH 2/48 -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 D Dunbar, Gauge Theory and Strings, ETH 3/48 Objective Theory Experiment precise predictions We want technology to calculate these predictions quickly, flexibly and accurately -despite our successes we have a long way to go

4 D Dunbar, Gauge Theory and Strings, ETH 4/48 QFT S-matrix theory String Theory Strings and QFT both have S-matrices -can link help with QFT?

5 D Dunbar, Gauge Theory and Strings, ETH 5/48 -not first time string theory inspired field theory -symmetry is important: embedding your theory in one with more symmetry might help understanding - Parke-Taylor MHV formulae string inspired -Bern-Kosower Rules for one-loop amplitudes  ’  0

6 D Dunbar, Gauge Theory and Strings, ETH 6/48 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 D Dunbar, Gauge Theory and Strings, ETH 7/48 Is the duality useful? Theory A : hard, interesting hard, interesting Theory B: easy Perturbative QCD, hard, interesting Topological String Theory : harder -duality may be useful indirectly

8 D Dunbar, Gauge Theory and Strings, ETH 8/48 -but can be understood in field theory + _ _ _ _ _ + + + + + + _ _ _ -eg MHV vertex construction of tree amplitudes -promote MHV amplitude to a fundamental vertex -inspired by scattering of instantons in topological strings Cachazo, Svercek, Witten Rioban, Spradlin, Volovich Mansfield, Ettle, Morris, Gorsky - and by factorisation Risager -works better than expected Brandhuber, Spence Travaglini

9 D Dunbar, Gauge Theory and Strings, ETH 9/48 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 D Dunbar, Gauge Theory and Strings, ETH 10/48 Passarino-Veltman reduction of 1-loop Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator -coefficients are rational functions of |k i § using spinor helicity -feature of Quantum Field Theory cut construcible

11 D Dunbar, Gauge Theory and Strings, ETH 11/48 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 D Dunbar, Gauge Theory and Strings, ETH 12/48 The Six Gluon one-loop amplitude 94 05 06 05 06 05 06 - --- - - 93 Bern, Dixon, Dunbar, Kosower Bern, Bjerrum-Bohr, Dunbar, Ita Bidder, Bjerrum-Bohr, Dixon, Dunbar Bedford, Brandhuber, Travaglini, Spence Britto, Buchbinder, Cachazo, Feng Bern, Chalmers, Dixon, Kosower Mahlon Xiao,Yang, Zhu Berger, Bern, Dixon, Forde, Kosower Forde, Kosower Britto, Feng, Mastriolia 81% `B’ ~13 papers

13 D Dunbar, Gauge Theory and Strings, ETH 13/48 94 05 06 05 06 05 06 - --- - - 93 The Six Gluon one-loop amplitude Difficult/Complexity unitarity recursion feynman MHV

14 D Dunbar, Gauge Theory and Strings, ETH 14/48 The Seven Gluon one-loop amplitude

15 D Dunbar, Gauge Theory and Strings, ETH 15/48 (++++++)1 (-+++++)6 (--++++)12 (-+-+++)12 (-++-++)6 (---+++)6 (--+-++)12 (-+-+-+)2 -specify colour structure, 8 independent helicities - supersymmetric approximations -for fixed colour structure we have 64 helicity structures

16 D Dunbar, Gauge Theory and Strings, ETH 16/48 N=4SUSY (--++++)0.320.04 (-+-+++)0.300.04 (-++-++)0.370.04 (---+++)0.160.06 (--+-++)0.360.04 (-+-+-+)0.130.02 QCD is almost supersymmetric…. (looking at the finite pieces) -working at the specific kinematic point of Ellis, Giele and Zanderaghi

17 D Dunbar, Gauge Theory and Strings, ETH 17/48 Unitarity Methods -look at the two-particle cuts -use unitarity to identify the coefficients

18 D Dunbar, Gauge Theory and Strings, ETH 18/48 Topology of Cuts -look when K is timelike, in frame where K=(K 0,0,0,0) l 1 and l 2 are back to back on surface of sphere imposing an extra condition

19 D Dunbar, Gauge Theory and Strings, ETH 19/48 Generalised Unitarity -use info beyond two-particle cuts

20 D Dunbar, Gauge Theory and Strings, ETH 20/48 Box-Coefficients -works for massless corners (complex momenta) Britto,Cachazo,Feng or signature (--++)

21 D Dunbar, Gauge Theory and Strings, ETH 21/48 Unitarity Techniques -turn C 2 into coefficients of integral functions Different ways to approach this reduction to covariant integrals fermionic analytic structure

22 D Dunbar, Gauge Theory and Strings, ETH 22/48 Reduction to covariant integrals -advantages: connects to conventional reduction technique -converts integral into n-point integrals -convert fermionic variables

23 D Dunbar, Gauge Theory and Strings, ETH 23/48 -linear triangle in the two-particle cut kbkb P

24 D Dunbar, Gauge Theory and Strings, ETH 24/48 Fermionic Unitarity -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 Britto, Buchbinder,Cachazo, Feng, Mastrolia

25 D Dunbar, Gauge Theory and Strings, ETH 25/48 Analytic Structure z K1K1 K2K2 -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 Forde

26 D Dunbar, Gauge Theory and Strings, ETH 26/48 Unitarity -works well to calculate coefficients -particularly strong for supersymmetry (R=0) -can be automated -extensions to massive particles progressing Ellis, Giele, Kunszt ;Ossola, Pittau, Papadopoulos Berger Bern Dixon Febres-Cordero Forde Ita Kosower Maitre Ellis, Giele, Kunzst, Melnikov Britto, Feng Yang; Britto, Feng Mastrolia Badger, Glover, Risager Anastasiou, Britto, Feng, Kunszt, Mastrolia Mastrolia

27 D Dunbar, Gauge Theory and Strings, ETH 27/48 How do we calculate R? D- dimensional Unitarity Factorisation/Recursion Feynman Diagrams

28 D Dunbar, Gauge Theory and Strings, ETH 28/48 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 Dunbar, Gauge Theory and Strings, ETH 29/48 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 Britto Feng Mastrolia Bern,Dixon,dcd, Kosower Bern Morgan Brandhuber, Macnamara, Spence Travaglini Kilgore

30 D Dunbar, Gauge Theory and Strings, ETH 30/48 Factorisation 1) Amplitude will be singular at special Kinematic points, with well understood factorisation Bern, Chalmers e.g. one-loop factorisation theorem K is multiparticle momentum invariant 2) Amplitude does not have singularities elsewhere : at spurious singular points

31 D Dunbar, Gauge Theory and Strings, ETH 31/48 On-shell Recursion: tree amplitudes 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 Britto,Cachazo,Feng (and Witten)

32 D Dunbar, Gauge Theory and Strings, ETH 32/48 Provided, Residues occur when amplitude factorises on multiparticle pole (including two-particles) then

33 D Dunbar, Gauge Theory and Strings, ETH 33/48 -results in recursive on-shell relation Tree Amplitudes are on-shell but continued to complex momenta (three-point amplitudes must be included) 12 (c.f. Berends-Giele off shell recursion)

34 D Dunbar, Gauge Theory and Strings, ETH 34/48 Recursion for Loops? cut construcible recursive? -amplitude is a mix of cut constructible pieces and rational

35 D Dunbar, Gauge Theory and Strings, ETH 35/48 Recursion for Rational terms -can we shift R and obtain it from its factorisation? 1)Function must be rational 2)Function must have simple poles 3)We must understand these poles Berger, Bern, Dixon, Forde and Kosower -requires auxiliary recusion limits for large-z terms

36 D Dunbar, Gauge Theory and Strings, ETH 36/48 recursive? Recursion on Integral Coefficients Consider an integral coefficient and isolate a coefficient and consider the cut. Consider shifts in the cluster. r-r- r+1 + + + + - - - - -we obtain formulae for integral coefficients for both the N=1 and scalar cases Bern, Bjerrum-Bohr, dcd, Ita

37 D Dunbar, Gauge Theory and Strings, ETH 37/48 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 D Dunbar, Gauge Theory and Strings, ETH 38/48 -amplitude has sixth order pole in [12] 1 3 4 2 s=0, h 1 2 i  0 -spurious which only appears if we use complex momentum -just how powerful is factorisation? -unusual example : four graviton, one loop scattering dcd, Norridge

39 D Dunbar, Gauge Theory and Strings, ETH 39/48 u/t =-1 -s/t, expand in s 1 3 4 2 - 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 D Dunbar, Gauge Theory and Strings, ETH 40/48 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 D Dunbar, Gauge Theory and Strings, ETH 41/48 Superstring Theory 2) Look at supergravity embedded within string theory N=8 Supergravity 1) Approach problem within the theory Dual Theory 3) Find a dual theory which is solvable Green, Russo, Van Hove, Berkovitz, Chalmers Abou-Zeid, Hull, Mason ``Finite for 8 loops but not beyond’’

42 D Dunbar, Gauge Theory and Strings, ETH 42/48 -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 D Dunbar, Gauge Theory and Strings, ETH 43/48 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 D Dunbar, Gauge Theory and Strings, ETH 44/48 No-Triangle Hypothesis -against this expectation, it might be the case that……. Evidence? true for 4pt n-point MHV 6-7pt NMHV proof Bern,Dixon,Perelstein,Rozowsky Bjerrum-Bohr, dcd,Ita, Perkins, Risager; Bern, Carrasco, Forde, Ita, Johansson, Green,Schwarz,Brink Bjerrum-Bohr Van Hove -extra n-4 cancelations

45 D Dunbar, Gauge Theory and Strings, ETH 45/48 Three Loops Result  SYM: K 3D-18 Sugra: K 3D-16 Finite for D=4,5, Infinite D=6 -actual for Sugra -again N=8 Sugra looks like N=4 SYM Bern, Carrasco, Dixon, Johansson, Kosower and Roiban, 07

46 D Dunbar, Gauge Theory and Strings, ETH 46/48 -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…. Rockall versus Tahiti

47 Emil Bjerrum-Bohr, IAS, UCLA Harald Ita, UCLA Warren Perkins TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A Kasper Risager, NBI 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

48 D Dunbar, Gauge Theory and Strings, ETH 48/48 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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