Unitarity and Amplitudes at Maximal Supersymmetry

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Unitarity and Amplitudes at Maximal Supersymmetry

• David A. Kosower
• with Z. Bern, J.J. Carrasco, M. Czakon, L. Dixon,
  D. Dunbar, H. Johansson, R. Roiban, M. Spradlin,
  V. Smirnov, C. Vergu, A. Volovich
• Jussieu FRIF Workshop
• Dec 1213, 2008

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QCD

• Natures gift a fully consistent physical theory
• Only now, thirty years after the discovery of
  asymptotic freedom, are we approaching a detailed
  and explicit understanding of how to do precision
  theory around zero coupling
• Can compute some static strong-coupling
  quantities via lattice
• Otherwise, only limited exploration of
  high-density and hot regimes
• To understand the theory quantitatively in all
  regimes, we seek additional structure
• String theory returning to its roots

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An Old Dream Planar Limit in Gauge Theories

• t Hooft (1974)
• Consider large-N gauge theories, g2N 1, use
  double-line notation
• Planar diagrams dominate
• Sum over all diagrams ? surface or string diagram

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How Can We Pursue the Dream?

• We want a story that starts out with an
  earthquake and works its way up to a climax.
  Samuel Goldwyn
• Study N 4 large-N gauge theories maximal
  supersymmetry as a laboratory for learning about
  less-symmetric theories
• Easier to perform explicit calculations
• Several representations of the theory

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Descriptions of N 4 SUSY Gauge Theory

• A Feynman path integral
• Boundary CFT of IIB string theory on AdS5 ? S5
• Maldacena (1997) Gubser, Klebanov, Polyakov
  Witten (1998)
• Spin-chain model
• Minahan Zarembo (2002) Staudacher, Beisert,
  Kristjansen, Eden, (20032006)
• Twistor-space topological string B model
• Nair (1988) Witten (2003)
• Roiban, Spradlin, Volovich (2004) Berkovits
  Motl (2004)

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• Is there any structure in the perturbation
  expansion hinting at solvability?
• Explicit higher-loop computations are hard, but
  theyre the only way to really learn something
  about the theory

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Recent Revelations

• Iteration relation four- and five-point
  amplitudes may be expressed to all orders solely
  in terms of the one-loop amplitudes
• Cusp anomalous dimension to all orders BES
  equation hints of integrability ? Bassos talk
• Role of dual conformal symmetry
• But the iteration relation doesnt hold for the
  six-point amplitude
• Structure beyond the iteration relation yet to
  be understood

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• Traditional technology Feynman Diagrams

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Feynman Diagrams Wont Get You There

• Huge number of diagrams in calculations of
  interest factorial growth
• 8 gluons (just QCD) 34300 tree diagrams, 2.5
  107 terms
• 2.9 106 1-loop diagrams, 1.9 1010 terms
• But answers often turn out to be very simple
• Vertices and propagators involve gauge-variant
  off-shell states
• Each diagram is not gauge invariant huge
  cancellations of gauge-noninvariant, redundant,
  parts in the sum over diagrams
• Simple results should have a simple derivation
  Feynman (attr)
• Is there an approach in terms of physical states
  only?

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How Can We Do Better?

• Dick Feynman's method is this. You write down
  the problem. You think very hard. Then you write
  down the answer. Murray Gell-Mann

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New Technologies On-Shell Methods

• Use only information from physical states
• Use properties of amplitudes as calculational
  tools
• Unitarity ? unitarity method
• Underlying field theory ? integral basis
• Formalism for N 4 SUSY

Integral basis
Unitarity
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Unitarity Prehistory

• General property of scattering amplitudes in
  field theories
• Understood in 60s at the level of single
  diagrams in terms of Cutkosky rules
• obtain absorptive part of a one-loop diagram by
  integrating tree diagrams over phase space
• obtain dispersive part by doing a dispersion
  integral
• In principle, could be used as a tool for
  computing 2 ? 2 processes
• No understanding
• of how to do processes with more channels
• of how to handle massless particles
• of how to combine it with field theory false
  gods of S-matrix theory

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Unitarity as a Practical Tool

• Bern, Dixon, Dunbar, DAK (1994)
• Compute cuts in a set of channels
• Compute required tree amplitudes
• Reconstruct corresponding Feynman integrals
• Perform algebra to identify coefficients of
  master integrals
• Assemble the answer, merging results from
  different channels

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• One-loop all-multiplicity MHV amplitude in N 4

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Generalized Unitarity

• Can sew together more than twotree amplitudes
• Corresponds to leading singularities
• Isolates contributions of a smaller setof
  integrals only integrals with propagatorscorresp
  onding to cuts will show up
• Bern, Dixon, DAK (1997)
• Example in triple cut, only boxes and triangles
  will contribute
• ? Vanhoves talk
• Combine with use of complex momenta to determine
  box coeffs directly in terms of tree amplitudes
• Britto, Cachazo, Feng (2004)
• No integral reductions needed

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Generalized Cuts

• Require presence of multiple propagators at
  higher loops too

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Cuts

• Compute a set of six cuts, including multiple
  cuts to determine which integrals are actually
  present, and with which numerator factors
• Do cuts in D dimensions

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Integrals in the Amplitude

• 8 integrals present
• 6 given by rung rule 2 are new
• UV divergent in D (vs 7, 6 for L 2, 3)

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Dual Conformal Invariance

• Amplitudes appear to have a kind of conformal
  invariance in momentum space
• Drummond, Henn, Sokatchev, Smirnov (2006)
• All integrals in four-loop four-point calculation
  turn out to be pseudo-conformal dually
  conformally invariant when taken off shell
  (require finiteness as well, and no worse than
  logarithmically divergent in on-shell limit)
• Dual variables ki xi1 xi
• Conformal invariance in xi

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• Easiest to analyze using dual diagrams
• Drummond, Henn, Smirnov Sokatchev (2006)
• All coefficients are 1 in four-point (through
  five loops) and parity-even part of five-point
  amplitude (through two loops)

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59 ints
Bern, Carrasco, Johansson, DAK (5/2007)
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A Mysterious Connection to Wilson Loops

• Motivated by AldayMaldacena strong-coupling
  calculation, look at a dual Wilson loop at weak
  coupling at one loop, amplitude is equal to the
  Wilson loop for any number of legs (up to
  addititve constant)
• Drummond, Korchemsky, Sokatchev (2007)
• Brandhuber, Heslop, Travaglini (2007)
• Equality also holds for four- and five-point
  amplitudes at two loops
• Drummond, Henn, Korchemsky, Sokatchev (20078)

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Conformal Ward Identity

• Drummond, Henn, Korchemsky, Sokatchev (2007)
• In four dimensions, Wilson loop would be
  invariant under the dual conformal invariance
• Slightly broken by dimensional regularization
• Additional terms in Ward identity are determined
  only by divergent terms, which are universal
• Four- and five-point Wilson loops determined
  completely
• Equal to corresponding amplitudes!
• Beyond that, functions of cross ratios

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Open Questions

• What happens beyond five external legs? Does the
  amplitude still exponentiate as suggested by the
  iteration relation? Suspicions of breakdown from
  AldayMaldacena investigations
• If so, at how many external legs?
• Is the connection between amplitudes and Wilson
  loops accidental, or is it maintained beyond
  the five-point case at two loops?
• Compute six-point amplitude at two loops

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Basic Integrals
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Decorated Integrals
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Result

• Take the kinematical point
• and look at the remainder (test of the iteration
  relation)

ui independent conformal cross ratios
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Comparison to Wilson Loop Calculation

• With thanks to Drummond, Henn, Korchemsky,
  Sokatchev
• Constants in M differ compare differences with
  respect to a standard kinematic point
• Wilson Loop Amplitude!

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Questions Answered

• Does the exponentiation ansatz break down? Yes
• Does the six-point amplitude still obey the dual
  conformal symmetry? Almost certainly
• Is the Wilson loop equal to the amplitude at six
  points? Very likely

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Questions Unanswered

• What is the remainder function?
• Can one show analytically that the amplitude and
  Wilson-loop remainder functions are identical?
• How does it generalize to higher-point
  amplitudes?
• Can integrability predict it?
• What is the origin of the dual conformal
  symmetry?
• What happens for non-MHV amplitudes?