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 12–13, 2008
QCD
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Nature’s gift: a fully consistent physical theory
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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
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Can compute some static strong-coupling quantities via lattice
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Otherwise, only limited exploration of high-density and hot regimes
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To understand the theory quantitatively in all regimes, we seek
additional structure
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String theory returning to its roots
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
An Old Dream: Planar Limit in Gauge Theories
‘t Hooft (1974)
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Consider large-N gauge theories, g2N ~ 1, use double-line
notation
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Planar diagrams dominate
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Sum over all diagrams  surface or string diagram
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
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
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Study N = 4 large-N gauge theories: maximal supersymmetry as a
laboratory for learning about less-symmetric theories
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Easier to perform explicit calculations
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Several representations of the theory
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Descriptions of N =4 SUSY Gauge Theory
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A Feynman path integral
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Boundary CFT of IIB string theory on AdS5  S5
Maldacena (1997); Gubser, Klebanov, & Polyakov; Witten (1998)
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Spin-chain model
Minahan & Zarembo (2002); Staudacher, Beisert, Kristjansen, Eden, … (2003–2006)
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Twistor-space topological string B model
Nair (1988); Witten (2003)
Roiban, Spradlin, & Volovich (2004); Berkovits & Motl (2004)
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
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Is there any structure in the perturbation expansion hinting at
‘solvability’?
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Explicit higher-loop computations are hard, but they’re the only
way to really learn something about the theory
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Recent Revelations
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Iteration relation: four- and five-point amplitudes may be
expressed to all orders solely in terms of the one-loop
amplitudes
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Cusp anomalous dimension to all orders: BES equation & hints
of integrability
 Basso’s talk
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Role of ‘dual’ conformal symmetry
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But the iteration relation doesn’t hold for the six-point amplitude
Structure beyond the iteration relation: yet to be understood
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
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Traditional technology: Feynman Diagrams
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Feynman Diagrams Won’t Get You There
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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?
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
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
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
New Technologies: On-Shell Methods
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Use only information from physical states
Use properties of amplitudes as calculational tools
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Unitarity → unitarity method
Underlying field theory → integral basis
Formalism for N = 4 SUSY
Integral basis:
Unitarity
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Unitarity: Prehistory
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General property of scattering amplitudes in field theories
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Understood in ’60s at the level of single diagrams in terms of
Cutkosky rules
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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
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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
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
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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Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
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One-loop all-multiplicity MHV amplitude in N = 4
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Generalized Unitarity
Can sew together more than two
tree amplitudes
• Corresponds to ‘leading singularities’
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Isolates contributions of a smaller set
of integrals: only integrals with propagators
corresponding to cuts will show up
Bern, Dixon, DAK (1997)
Example: in triple cut, only boxes and triangles will contribute
 Vanhove’s talk
• Combine with use of complex momenta to determine box coeffs
directly in terms of tree amplitudes
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No integral reductions needed
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Britto, Cachazo, & Feng (2004)
Generalized Cuts
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Require presence of multiple propagators at higher loops too
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Cuts
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Compute a set of six cuts, including multiple cuts to determine
which integrals are actually present, and with which numerator
factors
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Do cuts in D dimensions
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Integrals in the Amplitude
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8 integrals present
6 given by ‘rung rule’; 2 are new
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UV divergent in D =
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(vs 7, 6 for L = 2, 3)
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Dual Conformal Invariance
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Amplitudes appear to have a kind of conformal invariance in
momentum space
Drummond, Henn, Sokatchev, Smirnov (2006)
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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)
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Dual variables ki = xi+1 – xi
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Conformal invariance in xi
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
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Easiest to analyze using dual diagrams
Drummond, Henn, Smirnov & Sokatchev (2006)
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All coefficients are ±1 in four-point (through five loops) and
parity-even part of five-point amplitude (through two loops)
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
59 ints
Bern, Carrasco, Johansson, DAK (5/2007)
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
A Mysterious Connection to Wilson Loops
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Motivated by Alday–Maldacena 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)
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Equality also holds for four- and five-point amplitudes at two loops
Drummond, Henn, Korchemsky, Sokatchev (2007–8)
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Conformal Ward Identity
Drummond, Henn, Korchemsky, Sokatchev (2007)
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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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Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Open Questions
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What happens beyond five external legs? Does the amplitude
still exponentiate as suggested by the iteration relation?
Suspicions of breakdown from Alday–Maldacena investigations
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If so, at how many external legs?
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Is the connection between amplitudes and Wilson loops
“accidental”, or is it maintained beyond the five-point case at two
loops?
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Compute six-point amplitude at two loops
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Basic Integrals
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Decorated Integrals
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Result
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Take the kinematical point
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and look at the remainder (test of the iteration relation)
ui — independent conformal cross ratios
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Comparison to Wilson Loop Calculation
With thanks to Drummond, Henn, Korchemsky, & Sokatchev
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Constants in M differ: compare differences with respect to a
standard kinematic point
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Wilson Loop = Amplitude!
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Questions Answered
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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
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008
Questions Unanswered
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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?
Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008