From Weak to Strong Coupling at Maximal Supersymmetry

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From Weak to Strong Coupling at Maximal
Supersymmetry

• David A. Kosower
• with Z. Bern, M. Czakon, L. Dixon, V. Smirnov
• 2007 Itzykson Meeting
• Integrability in Gauge and String Theory
• June 22, 2007

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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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• 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
• 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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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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• Simple structure in twistor space, completely
  unexpected from the Lagrangian

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• Novel relation between different orders in
  perturbation theory
• Compute leading-twist anomalous dimension (?
  spinning-string mass)
• Eden Staudacher (spring 2006) analyzed the
  integral equation emerging from the spin chain
  and conjectured an all-orders expression for f,
• Want to check four-loop term
• Use on-shell methods not conventional Feynman
  diagrams

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Unitarity-Based Calculations

• Bern, Dixon, Dunbar, DAK (1994)

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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
• (in N 4, a smaller set of boxes)
• Combine with use of complex momenta to determine
  box coeffs directly in terms of tree amplitudes
• Britto, Cachazo, Feng (2004)

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Generalized Cuts

• Require presence of multiple propagators at
  higher loops too

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N 4 Cuts at Two Loops
one-loop scalar box

• At one loop,
• Green, Schwarz, Brink (1982)
• Two-particle cuts iterate to all orders
• Bern, Rozowsky, Yan (1997)
• Three-particle cuts give no new information for
  the four-point amplitude

?
for all helicities
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Two-Loop Four-Point Result

• Integrand known
• Bern, Rozowsky, Yan (1997)
• Integrals known by 2000 ? could have just
  evaluated
• Singular structure is an excellent guide
• Sterman Magnea (1990) Catani (1998) Sterman
  Tejeda-Yeomans (2002)

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Two-loop Double Box

• Smirnov (1999)
• Physics is 90 mental, the other half is hard
  work Yogi Berra

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Transcendentality

• Also called polylog weight
• N 4 SUSY has maximal transcendentality 2 ?
  loop order
• QCD has mixed transcendentality from 0 to maximal

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• Infrared-singular structure is an excellent guide
• Sterman Magnea (1990) Catani (1998) Sterman
  Tejeda-Yeomans (2002)
• IR-singular terms exponentiate
• Soft or cusp anomalous dimension large-spin
  limit of trace-operator anomalous dimension
  Korchemsky Marchesini (1993)
• True for all gauge theories

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Iteration Relation in N 4

• Look at corrections to MHV amplitudes
  , at leading order in Nc
• Including finite terms
• Anastasiou, Bern, Dixon, DAK (2003)
• Conjectured to all orders
• Requires non-trivial cancellations not predicted
  by pure supersymmetry or superspace arguments

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N 4 Integrand at Higher Loops

• Bern, Rozowsky, Yan (1997)

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Iteration Relation Continued

• Confirmed at three loops
• Bern, Dixon, Smirnov (2005)
• Can extract 3-loop anomalous dimension compare
  to Kotikov, Lipatov, Onishchenko and Velizhanin
  extraction from Moch, Vermaseren Vogt result

highest polylog weight
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All-Loop Form

• Bern, Dixon, Smirnov (2005)
• Connection to a kind of conformal invariance?
• Drummond, Korchemsky, Sokatchev

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Amplitudes at Strong Coupling

• Alday Maldacena (5/2007)
• Introduce D-brane as regulator
• Study open-string scattering on it fixed-angle
  high-momentum
• Use saddle-point (classical solution) approach
• Gross Mende (1988)
• Replace D-brane with dimensional regulator, take
  brane to IR
• First strong-coupling computation of G
  (collinear) anomalous dim

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Calculation

• Dick Feynman's method is this. You write
  down the problem. You think very hard. Then you
  write down the answer. Murray Gell-Mann
• Integral set
• Unitarity
• Calculating integrals
• Computation of only needs O(²-2)
  terms

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Integrals

• Start with all four-point four-loop integrals
  with no bubble or triangle subgraphs (expected to
  cancel in N4)
• 7 master topologies (only three-point vertices)
• 25 potential integrals (others obtained by
  canceling propagators)

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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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A Posteriori, Conformal Properties

• Consider candidate integrals with external legs
  taken off shell
• Require that they be conformally invariant
• Easiest to analyze using dual diagrams
• Drummond, Henn, Smirnov, Sokatchev (2006)
• Require that they be no worse than
  logarithmically divergent
• ? 10 pseudo-conformal integrals, including all 8
  that contribute to amplitude (Sokatchev two
  others divergent off shell)

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59 ints
Bern, Carrasco, Johansson, DAK (5/2007)
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MellinBarnes Technique

• Introduce Feynman parameters (best choice is
  still an art) perform loop integrals
• Use identity
• to create an m-fold representation
• Singularities are hiding in G functions
• Move contours to expose these singularities (all
  poles in ²)
• Expand G functions to obtain Laurent expansion
  with functions of invariants as coefficients
• Done automatically by Mathematica package MB
  (Czakon)
• Compute integrals numerically or analytically

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Result

• Computers are useless. They can only give you
  answers. Pablo Picasso
• Further refinement
• Cachazo, Spradlin, Volovich (12/2006)

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Extrapolation to Strong Coupling

• Use KLV approach
• Kotikov, Lipatov Velizhanin (2003)
• Constrain at large â solve
• Predict two leading strong-coupling coefficients

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• Known strong-coupling expansion
• Gubser, Klebanov, Polyakov Kruczenski Makeenko
  (2002)
• Frolov Tseytlin (2002)
• Two loops predicts leading coefficient to 10,
  subleading to factor of 2
• Four-loop value predicts leading coefficient to
  2.6, subleading to 5!

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• This suggests the exact answer should be
  obtained by flipping signs of odd-? terms in
  original EdenStaudacher formula (Bern, Czakon,
  Dixon, DAK, Smirnov)
• The same series is obtained by consistency of
  analytic continuations (Beisert, Eden,
  Staudacher) leading to a ?-dependent dressing
  factor for the spin-chain/string world-sheet
  S-matrix and a modified EdenStaudacher equation
• The equation can be integrated numerically
  (Benna, Benvenuti, Klebanov, Scardicchio) and
  reproduces known strong-coupling terms
• The equation can be studied analytically in the
  strong-coupling region (Kotikov Lipatov Alday,
  Arutyunov, Benna, Eden, Klebanov Kostov, Serban,
  Volin Beccaria, DeAngelis, Forini) and
  reproduces the leading coefficient equivalent
  equation from string Bethe ansatz (Casteill
  Kristjansen)
• Approximation reproduces exact answer to better
  than 0.2!

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Conclusions and Perspectives

• The time has come, the Walrus said, To talk of
  many things Of AdSand CFTand scaling-limits
  Of spin-chainsand N4 Whether it is truly
  summable And what this is all good for.
• First evidence for the dressing factor in
  anomalous dimensions
• Transition from weak to strong coupling is
  smooth here
• N 4 is a guide to uncovering more structure in
  gauge theories can it help us understand the
  strong-coupling regime in QCD?
• Unitarity is the method of choice for
  performing the explicit calculations needed to
  make progress and ultimately answer the questions
  
• What is the connection of the iteration
  relation to integrability or other structures?