Unitarity and Amplitudes at Maximal Supersymmetry

Unitarity and Amplitudes at Maximal Supersymmetry

David A. KosowerDavid A. Kosowerwith 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 WorkshopDec 12–13, 2008

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

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


An Old Dream: Planar Limit in Gauge Theories

‘t Hooft (1974)

• Consider large-N gauge theories, g2N ~ 1, use double-line notation

• Planar diagrams dominate

• Sum over all diagrams surface or string diagram


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 the theory


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)


• 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


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


• Traditional technology: Feynman Diagrams


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?


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


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


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

• Understood in ’60s 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 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


• One-loop all-multiplicity MHV amplitude in N = 4


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

Vanhove’s talk• Combine with use of complex momenta to determine

box coeffs directly in terms of tree amplitudesBritto, Cachazo, & Feng (2004)

• No integral reductions needed


Generalized Cuts

• Require presence of multiple propagators at higher loops too


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


• 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


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 ki = xi+1 – xi

• Conformal invariance in xi


• Easiest to analyze using dual diagramsDrummond, 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)


59 ints

Bern, Carrasco, Johansson, DAK (5/2007)


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)


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


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



Basic Integrals


Decorated Integrals


Result

• Take the kinematical point

• and look at the remainder (test of the iteration relation)

ui — independent conformal cross ratios


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!


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


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?

Documents

Unitarity and Amplitudes at Maximal Supersymmetry