Unitarity and Amplitudes at Maximal Supersymmetry David A

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Unitarity and Amplitudes at Maximal Supersymmetry David A. Kosower with Z. Bern, J. J.

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 • Nature’s gift: a fully consistent physical theory • Only now, thirty years

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 theory quantitatively in all regimes, we seek additional structure • 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) • Consider large-N

An Old Dream: Planar Limit in Gauge Theories ‘t Hooft (1974) • Consider large-N gauge theories, g 2 N ~ 1, use double-line notation • Planar diagrams dominate • 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

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 theory Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008

Descriptions of N =4 SUSY Gauge Theory • A Feynman path integral • Boundary

Descriptions of N =4 SUSY Gauge Theory • A Feynman path integral • Boundary CFT of IIB string theory on Ad. S 5 Maldacena (1997); Gubser, Klebanov, & Polyakov; Witten (1998) • Spin-chain model Minahan & Zarembo (2002); Staudacher, Beisert, Kristjansen, Eden, … (2003– 2006) • 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

 • Is there any structure in the perturbation expansion hinting at ‘solvability’? •

• 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 theory Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008

Recent Revelations • Iteration relation: four- and five-point amplitudes may be expressed to all

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 Basso’s talk • Role of ‘dual’ conformal symmetry 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

 • Traditional technology: Feynman Diagrams Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu

• Traditional technology: Feynman Diagrams Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008

Feynman Diagrams Won’t Get You There Huge number of diagrams in calculations of interest

Feynman Diagrams Won’t 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? • 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

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 Use only information from physical states • Use properties of

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 and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008

Unitarity: Prehistory • General property of scattering amplitudes in field theories • Understood in

Unitarity: Prehistory • General property of scattering amplitudes in field theories • Understood in ’ 60 s 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 – Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008

Unitarity as a Practical Tool Bern, Dixon, Dunbar, & DAK (1994) • • •

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 Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008

 • One-loop all-multiplicity MHV amplitude in N = 4 Unitarity and Amplitudes at

• 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

Generalized Unitarity Can sew together more than two tree amplitudes • Corresponds to ‘leading singularities’ • • 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 • • No integral reductions needed Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008 Britto, Cachazo, & Feng (2004)

Generalized Cuts • Require presence of multiple propagators at higher loops too Unitarity and

Generalized Cuts • Require presence of multiple propagators at higher loops too Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008

Cuts • Compute a set of six cuts, including multiple cuts to determine which

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 Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008

Integrals in the Amplitude • 8 integrals present 6 given by ‘rung rule’; 2

Integrals in the Amplitude • 8 integrals present 6 given by ‘rung rule’; 2 are new • UV divergent in D = • Unitarity and Amplitudes at Maximal Supersymmetry (vs 7, 6 for L = 2, 3) , Jussieu Workshop, Dec 12, 2008

Dual Conformal Invariance • Amplitudes appear to have a kind of conformal invariance in

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 k i = xi+1 – xi • Conformal invariance in xi Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008

 • Easiest to analyze using dual diagrams Drummond, Henn, Smirnov & Sokatchev (2006)

• 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) 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 ,

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 • Motivated by Alday–Maldacena strong-coupling calculation, look at

A Mysterious Connection to Wilson Loops • 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) • 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) In four dimensions, Wilson loop would

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 • Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008

Open Questions • What happens beyond five external legs? Does the amplitude still exponentiate

Open Questions • What happens beyond five external legs? Does the amplitude still exponentiate as suggested by the iteration relation? Suspicions of breakdown from Alday–Maldacena 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 Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008

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

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

Decorated Integrals Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008

Result • Take the kinematical point • and look at the remainder (test of

Result • Take the kinematical point • and look at the remainder (test of the iteration relation) u i — 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 , &

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! Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008

Questions Answered Does the exponentiation ansatz break down? Yes • Does the six-point amplitude

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 • Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008

Questions Unanswered • • • What is the remainder function? Can one show analytically

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? Unitarity and Amplitudes at Maximal Supersymmetry , Jussieu Workshop, Dec 12, 2008