From Weak to Strong Coupling at Maximal
Supersymmetry
David A. KosowerDavid A. Kosower
with Z. Bern, M. Czakon, L. Dixon, & V. Smirnov
2007 Itzykson Meeting: Integrability in Gauge and String Theory
June 22, 2007
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
QCD
• Nature’s 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
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
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 they’re the only way to really learn something about the theory
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
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 modelMinahan & Zarembo (2002); Staudacher, Beisert, Kristjansen,
Eden, … (2003–2006)
• Twistor-space topological string B modelNair (1988); Witten (2003)
Roiban, Spradlin, & Volovich (2004); Berkovits & Motl (2004)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Simple structure in twistor space, completely unexpected from the Lagrangian
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
• 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
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Unitarity-Based Calculations
Bern, Dixon, Dunbar, & DAK (1994)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Generalized Unitarity• Can sew together more than two
tree amplitudes• Corresponds to ‘leading singularities’
• Isolates contributions of a smaller setof integrals: only integrals with propagatorscorresponding 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)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Generalized Cuts
• Require presence of multiple propagators at higher loops too
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
• At one loop,
Green, Schwarz, & Brink (1982)
• Two-particle cuts iterate to all ordersBern, Rozowsky, & Yan (1997)
• Three-particle cuts give no new information for the four-point amplitude
N =4 Cuts at Two Loops
== = =
for all helicitiesfor all helicities
one-loop scalar boxone-loop scalar box
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
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)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Two-loop Double Box
Smirnov (1999)
Physics is 90% mental, the other half is hard work — Yogi Berra
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Transcendentality
• Also called ‘polylog weight’
• N = 4 SUSY has maximal transcendentality = 2 loop order
• QCD has mixed transcendentality: from 0 to maximal
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
• Infrared-singular structure is an excellent guideSterman & 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
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Iteration Relation in N = 4
• Look at corrections to MHV amplitudes , at leading order in Nc
• Including finite termsAnastasiou, Bern, Dixon, & DAK (2003)
• Conjectured to all orders• Requires non-trivial cancellations not predicted
by pure supersymmetry or superspace arguments
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
N =4 Integrand at Higher Loops
Bern, Rozowsky, & Yan (1997)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Iteration Relation Continued
• Confirmed at three loopsBern, 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
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
All-Loop Form
Bern, Dixon, & Smirnov (2005)
Connection to a kind of conformal invariance? Drummond, Korchemsky, & Sokatchev
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Amplitudes at Strong CouplingAlday & Maldacena (5/2007)
• Introduce D-brane as regulator• Study open-string scattering on it: fixed-angle high-
momentum
• Use saddle-point (classical solution) approachGross & Mende (1988)
• Replace D-brane with dimensional regulator, take brane to IR
• First strong-coupling computation of G (“collinear”) anomalous dim
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
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
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Integrals
• Start with all four-point four-loop integrals with no bubble or triangle subgraphs (expected to cancel in N=4)
7 master topologies (only three-point vertices)
25 potential integrals (others obtained by canceling propagators)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
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
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
• 8 integrals present• 6 given by ‘rung rule’; 2 are new
• UV divergent in D = (vs 7, 6 for L = 2, 3)
Integrals in the Amplitude
211
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
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)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
59 ints
Bern, Carrasco, Johansson, DAK (5/2007)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Mellin–Barnes 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 Γ functions• Move contours to expose these singularities (all
poles in ²)• Expand Γ functions to obtain Laurent expansion
with functions of invariants as coefficients• Done automatically by Mathematica package MB
(Czakon)• Compute integrals numerically or analytically
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
ResultComputers are useless. They can only give you answers. — Pablo Picasso
• Further refinement
Cachazo, Spradlin, & Volovich (12/2006)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Extrapolation to Strong Coupling
• Use KLV approachKotikov, Lipatov & Velizhanin (2003)
• Constrain at large â: solve
• Predict two leading strong-coupling coefficients
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
• 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%!
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
• This suggests the ‘exact’ answer should be obtained by flipping signs of odd-ζ terms in original Eden–Staudacher formula (Bern, Czakon, Dixon, DAK, Smirnov)
• The same series is obtained by consistency of analytic continuations (Beisert, Eden, & Staudacher) leading to a ζ-dependent ‘dressing’ factor for the spin-chain/string world-sheet S-matrix and a modified Eden–Staudacher 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%!
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Conclusions and Perspectives“The time has come,” the Walrus said, “To talk of many things: Of AdS–and CFT–and scaling-limits– Of spin-chains–and N=4– 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?