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.: AAAA

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

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

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

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

-cross-order relation Unitarity of S-matrix -cross-order relation D Dunbar, NBA, Aug 09

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

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

Generalised Unitarity -use info beyond two-particle cuts

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

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

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

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!

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

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

Extend the canonical form -algebraic identity

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

-better to recombine and rationalise

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

Higher Order Canonical Forms D Dunbar,, NBA, Aug 09

Spurious singularities, -A singularity in the coefficient not present in the amplitude -singularity cancels between integral functions -can combine integral functions

Canonical Forms for Triple Cut K1 K2 D Dunbar,, NBA, Aug 09

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

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

The Six Gluon one-loop amplitude 94 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

The Six Gluon one-loop amplitude 93 - - - 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 http://pyweb.swan.ac.uk/~dunbar/sixgluon.html

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

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….

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?

The Seven Gluon one-loop amplitude 93 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 http://pyweb.swan.ac.uk/~dunbar/sevengluon.html

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

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

D Dunbar,, NBA, Aug 09

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

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

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

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