unitarity and factorisation in quantum field theory

Unitarity and Factorisation in Quantum Field Theory

Zurich Zurich 2008

David Dunbar, Swansea University, Wales,

VERSUSUnitarity and Factorisation in Quantum Field Theory

D Dunbar, Gauge Theory and Strings, ETH

-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

Objective

Theory Experimentprecise predictions

We want technology to calculate these predictions quickly, flexibly and accurately

-despite our successes we have a long way to go

S-matrix theory String Theory

Strings and QFT both have S-matrices

-can link help with QFT?

-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

Duality with String Theory

Witten’s proposed of Weak-Weak duality betweenA) 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

Is the duality useful?

Theory A :Theory A : hard, hard,

interestinginteresting

Theory B: Theory B: easyeasy

Perturbative QCD,Perturbative QCD,hard, interestinghard, interesting

TopologicalTopologicalString TheoryString Theory::

harder harder -duality may be useful indirectly-duality may be useful indirectly

-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 expectedBrandhuber, Spence Travaglini

Organisation of QCD amplitudes: divide amplitude into smaller physical pieces-QCD gluon scattering amplitudes are the linear

combination ofContributions from supersymmetric multiplets

-use colour ordering; calculate cyclically symmetric partial amplitudes

-organise according to helicity of external gluon

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 |ki§ using spinor helicity

-feature of Quantum Field Theory

cut construcible

One-Loop QCD AmplitudesOne 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

The Six Gluon one-loop amplitude

949494949494

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Bern, Dixon, Dunbar, Kosower

Bern, Bjerrum-Bohr, Dunbar, Ita

Bidder, Bjerrum-Bohr, Dixon, DunbarBedford, Brandhuber, Travaglini, Spence

Britto, Buchbinder, Cachazo, Feng

Bern, Chalmers, Dixon, Kosower

MahlonXiao,Yang, Zhu

Berger, Bern, Dixon, Forde, Kosower

Forde, Kosower

Britto, Feng, Mastriolia

81% `B’

~13 papers

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The Six Gluon one-loop amplitude

Difficult/Complexity

unitarity

recursionfeynman

The Seven Gluon one-loop amplitude

(++++++) 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

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

QCD is almost supersymmetric….

(looking at the finite pieces)

-working at the specific kinematic point of Ellis, Giele and

Zanderaghi

Unitarity Methods

-look at the two-particle cuts

-use unitarity to identify the coefficients

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

sphereimposing an extra condition

Generalised Unitarity-use info beyond two-particle cuts

Box-Coefficients

-works for massless corners (complex momenta)

Britto,Cachazo,Feng

or signature (--++)

Unitarity Techniques

-turn C2 into coefficients of integral functionsDifferent ways to approach this

• reduction to covariant integrals• fermionic• analytic structure

Reduction to covariant integrals

-advantages: •connects to conventional reduction technique

-converts integral into n-point integrals

-convert fermionic variables

-linear triangle

in the two-particle cut

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

Analytic Structure

-triple cut reduces to problem in complex analysis-real momenta corresponds to unit circle

poles at z=0 are triangles functionspoles at z 0 are box coefficients

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, PapadopoulosBerger Bern Dixon Febres-Cordero Forde Ita Kosower Maitre

Ellis, Giele, Kunzst, Melnikov Britto, Feng Yang;Britto, Feng MastroliaBadger, Glover, Risager

Anastasiou, Britto, Feng, Kunszt, Mastrolia

Mastrolia

How do we calculate R?

• D- dimensional Unitarity

• Factorisation/Recursion

• Feynman Diagrams

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-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 legsVan Neerman

Britto Feng MastroliaBern,Dixon,dcd, KosowerBern Morgan

Brandhuber, Macnamara, Spence TravagliniKilgore

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

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)

Provided,

Residues occur when amplitude factorises on multiparticle pole (including two-

particles)

-results in recursive on-shell relation

Tree Amplitudes are on-shell but continued to complex momenta (three-point amplitudes must be

included)

(c.f. Berends-Giele off shell recursion)

Recursion for Loops?

cut construcible

recursive?-amplitude is a mix of cut constructible

pieces and rational

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

recursive?

Recursion on Integral Coefficients

Consider an integral coefficient and isolate a

coefficient and consider the cut. Consider shifts in the

cluster. r-

r+1+++

-we obtain formulae for integral coefficients for both the N=1 and scalar cases

Bern, Bjerrum-Bohr, dcd, Ita

Spurious Singularities

-spurious singularities are singularities which occur in

Coefficients but not in full amplitude

-need to understand these to do recursion

-link coefficients togetherBern, Dixon KosowerCampbell, Glover MillerBjerrum-Bohr, dcd, Perkins

-amplitude has sixth order pole in [12]

2s=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

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

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

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’’

-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?

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 reductionExpect triangles n > 4 , bubbles n >5 , rational n > 6

No-Triangle Hypothesis-against this expectation, it might be the case that…….

Evidence?true for 4pt n-point MHV 6-7pt NMHVproof

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

Three Loops Result

SYM: K3D-18

Sugra: K3D-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

-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

Emil Bjerrum-Bohr, IAS

Harald Ita, , UCLAUCLA

Warren Perkins

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

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

unitarity and factorisation in quantum field theory

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