Unitarity and Factorisation in Quantum Field Theory Zurich Zurich 2008 David Dunbar, Swansea...

Unitarity and Factorisation in Quantum Field Theory

Zurich Zurich 2008

David Dunbar,

Swansea University, Wales, UK

VERSUS

Unitarity and Factorisation in Quantum Field Theory

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


3/48

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


4/48

QFT

S-matrix theory

String Theory

Strings and QFT both have S-matrices

-can link help with QFT?


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


6/48

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


7/48

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


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 factorisationRisager

-works better than expected

Brandhuber, Spence Travaglini


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


10/48

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 |ki§ using spinor helicity

-feature of Quantum Field Theory

cut construcible


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


12/48

The Six Gluon one-loop amplitude

949494949494

94 94

9494

05

06

05

0505 06

0505 06

06

06

06

0606

----

--9393

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


13/48

949494949494

94 94

9494

05

06

05

0505 06

0505 06

06

06

06

0606

----

--9393

The Six Gluon one-loop amplitude

Difficult/Complexity

unitarity

recursion

feynman

MHV


14/48

The Seven Gluon one-loop amplitude


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


16/48

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


17/48

Unitarity Methods

-look at the two-particle cuts

-use unitarity to identify the coefficients


18/48

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


19/48

Generalised Unitarity-use info beyond two-particle cuts


20/48

Box-Coefficients

-works for massless corners (complex momenta)

Britto,Cachazo,Feng

or signature (--++)


21/48

Unitarity Techniques

-turn C2 into coefficients of integral functions

Different ways to approach this

• reduction to covariant integrals

• fermionic

• analytic structure


22/48

Reduction to covariant integrals

-advantages: •connects to conventional reduction technique

-converts integral into n-point integrals

-convert fermionic variables


23/48

-linear triangle

in the two-particle cut

kb

P


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


25/48

Analytic Structure

z

K1

K2

-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

Forde


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, PapadopoulosBerger Bern Dixon Febres-Cordero Forde Ita Kosower Maitre

Ellis, Giele, Kunzst, Melnikov Britto, Feng Yang;

Britto, Feng MastroliaBadger, Glover, RisagerAnastasiou, Britto, Feng, Kunszt, Mastrolia

Mastrolia


27/48

How do we calculate R?

• D- dimensional Unitarity

• Factorisation/Recursion

• Feynman Diagrams


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


29/48

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 legsVan Neerman

Britto Feng MastroliaBern,Dixon,dcd, Kosower

Bern Morgan

Brandhuber, Macnamara, Spence Travaglini

Kilgore


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


31/48

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)


32/48

Provided,

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

particles)

then


33/48

-results in recursive on-shell relation

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

included)

1 2

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


34/48

Recursion for Loops?

cut construcible

recursive?

-amplitude is a mix of cut constructible pieces and rational


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


36/48

recursive?

Recursion on Integral Coefficients

Consider an integral coefficient and isolate a

coefficient and consider the cut. Consider shifts in the

cluster. r-

r+1++

+

+

--

- -

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

Bern, Bjerrum-Bohr, dcd, Ita


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


38/48

-amplitude has sixth order pole in [12]

1

3

4

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


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


40/48

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


41/48

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’’


42/48

-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?


43/48

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


44/48

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


45/48

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


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

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


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

Unitarity and Factorisation in Quantum Field Theory Zurich Zurich 2008 David Dunbar, Swansea...

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