unitarity and factorisation in quantum field theory zurich zurich 2008 david dunbar, swansea...
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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 Quantum Field Theory
D Dunbar, Gauge Theory and Strings, ETH
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-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
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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
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QFT
S-matrix theory
String Theory
Strings and QFT both have S-matrices
-can link help with QFT?
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-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
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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
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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
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-but can be understood in field theory
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-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
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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
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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
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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
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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
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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
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The Seven Gluon one-loop amplitude
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(++++++) 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
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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
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Unitarity Methods
-look at the two-particle cuts
-use unitarity to identify the coefficients
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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
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Generalised Unitarity-use info beyond two-particle cuts
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Box-Coefficients
-works for massless corners (complex momenta)
Britto,Cachazo,Feng
or signature (--++)
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Unitarity Techniques
-turn C2 into coefficients of integral functions
Different ways to approach this
• reduction to covariant integrals
• fermionic
• analytic structure
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Reduction to covariant integrals
-advantages: •connects to conventional reduction technique
-converts integral into n-point integrals
-convert fermionic variables
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-linear triangle
in the two-particle cut
kb
P
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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
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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
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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
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How do we calculate R?
• D- dimensional Unitarity
• Factorisation/Recursion
• Feynman Diagrams
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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
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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
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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
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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)
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Provided,
Residues occur when amplitude factorises on multiparticle pole (including two-
particles)
then
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-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)
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Recursion for Loops?
cut construcible
recursive?
-amplitude is a mix of cut constructible pieces and rational
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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
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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
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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
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-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
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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
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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
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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’’
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-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?
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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
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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
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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
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-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
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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