VERSUS Unitarity and Factorisation in Quantum Field Theory

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

-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 D Dunbar, Gauge Theory and Strings, ETH 2

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

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

-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 D Dunbar, Gauge Theory and Strings, ETH 5

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 D Dunbar, Gauge Theory and Strings, ETH 6

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

-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 -and by factorisation D Dunbar, Gauge Theory and Strings, ETH Mansfield, Ettle, Morris, Gorsky Risager 8

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 D Dunbar, Gauge Theory and Strings, ETH 9

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 D Dunbar, Gauge Theory and Strings, ETH 10

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 D Dunbar, Gauge Theory and Strings, ETH 11

The Six Gluon one-loop amplitude 94 94 94 Bern, Dixon, Dunbar, Kosower 94 94 94 05 05 05 06 06 93 93 06 06 06 ~13 papers 81% `B’ Berger, Bern, Dixon, Forde, 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 D Dunbar, Gauge Theory and Strings, ETH Britto, Feng, Mastriolia Mahlon 12

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

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

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

N=4 SUSY (--++++) 0. 32 0. 04 (-+-+++) 0. 30 0. 04 (-++-++) 0. 37 0. 04 (---+++) 0. 16 0. 06 (--+-++) 0. 36 0. 04 (-+-+-+) 0. 13 0. 02 -working at the specific kinematic point of Ellis, Giele and Zanderaghi (looking at the finite pieces) QCD is almost supersymmetric…. D Dunbar, Gauge Theory and Strings, ETH 16

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

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 D Dunbar, Gauge Theory and Strings, ETH 18

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

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

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

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

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

Fermionic Unitarity Britto, Buchbinder, Cachazo, Feng, Mastrolia -use analytic structure to identify terms within twoparticle cuts -advantages: two-dimensional rather than four dimensional, merges nicely with amplitudes written in terms of spinor variables bubbles D Dunbar, Gauge Theory and Strings, ETH 24

Analytic Structure Forde K 1 z K 2 -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 D Dunbar, Gauge Theory and Strings, ETH 25

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 D Dunbar, Gauge Theory and Strings, ETH 26

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

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 D Dunbar, Gauge Theory and Strings, ETH 28

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 Ddimensional momenta should allow us to determine rational terms -these must be D-dimensional legs Bern Morgan Van Neerman Bern, Dixon, dcd, Kosower Britto Feng Mastrolia Brandhuber, Macnamara, Spence Travaglini D Dunbar, Gauge Theory and Strings, ETH Kilgore 29

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 D Dunbar, Gauge Theory and Strings, ETH 30

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 D Dunbar, Gauge Theory and Strings, ETH 31

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

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

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

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 D Dunbar, Gauge Theory and Strings, ETH 35

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

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 D Dunbar, Gauge Theory and Strings, ETH 37

-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 3 2 1 38 4 -spurious which only appears if we use complex momentum D Dunbar, Gauge Theory and Strings, ETH

u/t =-1 -s/t, expand in s 3 2 1 4 -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 D Dunbar, Gauge Theory and Strings, ETH 39

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 theory have ultra-violet singularities or is it a ``finite’’ field theory D Dunbar, Gauge Theory and Strings, ETH 40

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

-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 4 pt 3 -loop calculation • Is N=8 finite like N=4 SYM? D Dunbar, Gauge Theory and Strings, ETH 42

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

No-Triangle Hypothesis -against this expectation, it might be the case that……. Evidence? true for 4 pt n-point MHV 6 -7 pt 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 D Dunbar, Gauge Theory and Strings, ETH 44

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

-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 D Dunbar, Gauge Theory and Strings, ETH 46

Kasper. Risager, NBI Emil Bjerrum-Bohr, IAS Harald. Ita, UCLA Warren Perkins 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

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 D Dunbar, Gauge Theory and Strings, ETH 48