![Page 1: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/1.jpg)
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
![Page 2: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/2.jpg)
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
![Page 3: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/3.jpg)
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
![Page 4: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/4.jpg)
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)
![Page 5: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/5.jpg)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Simple structure in twistor space, completely unexpected from the Lagrangian
![Page 6: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/6.jpg)
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
![Page 7: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/7.jpg)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Unitarity-Based Calculations
Bern, Dixon, Dunbar, & DAK (1994)
![Page 8: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/8.jpg)
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)
![Page 9: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/9.jpg)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
Generalized Cuts
• Require presence of multiple propagators at higher loops too
![Page 10: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/10.jpg)
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
![Page 11: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/11.jpg)
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)
![Page 12: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/12.jpg)
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
![Page 13: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/13.jpg)
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
![Page 14: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/14.jpg)
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
![Page 15: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/15.jpg)
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
![Page 16: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/16.jpg)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
N =4 Integrand at Higher Loops
Bern, Rozowsky, & Yan (1997)
![Page 17: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/17.jpg)
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
![Page 18: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/18.jpg)
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
![Page 19: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/19.jpg)
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
![Page 20: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/20.jpg)
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
![Page 21: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/21.jpg)
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)
![Page 22: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/22.jpg)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
![Page 23: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/23.jpg)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
![Page 24: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/24.jpg)
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
![Page 25: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/25.jpg)
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
![Page 26: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/26.jpg)
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)
![Page 27: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/27.jpg)
From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007
59 ints
Bern, Carrasco, Johansson, DAK (5/2007)
![Page 28: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/28.jpg)
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
![Page 29: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/29.jpg)
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)
![Page 30: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/30.jpg)
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
![Page 31: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/31.jpg)
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%!
![Page 32: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/32.jpg)
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%!
![Page 33: From Weak to Strong Coupling at Maximal Supersymmetry](https://reader036.vdocuments.us/reader036/viewer/2022062410/568159a1550346895dc6f099/html5/thumbnails/33.jpg)
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?