Twistors and Gauge Theory

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Twistors and Gauge Theory

DESY Theory Workshop September 30, 2005
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The Storyline

• An exciting time in gauge-theory amplitude
  calculations
• Motivation for hard calculations
• Twistor-space ideas originating with Nair and
  Witten
• Explicit calculations led to seeing simple
  twistor-space structure
• Explicit calculations led to new on-shell
  recursion relations for trees
• Combined with another class of nonconventional
  techniques, the unitarity-based method for loop
  calculations, we are at the threshold of a
  revolution in loop calculations

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• D0 event

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• Guenther Dissertori (Jan 04)

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Precision Perturbative QCD

• Predictions of signals, signalsjets
• Predictions of backgrounds
• Measurement of luminosity
• Measurement of fundamental parameters (?s, mt)
• Measurement of electroweak parameters
• Extraction of parton distributions ingredients
  in any theoretical prediction

Everything at a hadron collider involves QCD
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• Campbell (Jan 04)

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A New Duality

• Topological B-model string theory (twistor
  space)? N 4 supersymmetric gauge theory
• Weakweak duality
• Computation of scattering amplitudes
• Novel differential equations
• Nair (1988) Witten (2003)
• Roiban, Spradlin, Volovich Berkovits Motl
  Vafa Neitzke Siegel (2004)
• Novel factorizations of amplitudes
• Cachazo, Svrcek, Witten (2004)
• Indirectly, new recursion relations
• Britto, Cachazo, Feng, Witten (1/2005)

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Supersymmetry

• Most often pursued in broken form as low-energy
  phenomenology
• "One day, all of these will be supersymmetric
  phenomenology papers."

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Exact Supersymmetry As a Computational Tool

• All-gluon amplitudes are the same at tree level
  in N 4 and QCD
• Fermion amplitudes obtained through Supersymmetry
  Ward Identities Grisaru, Pendleton, van
  Nieuwenhuizen (1977) Kunszt, Mangano, Parke,
  Taylor (1980s)
• At loop level, N 4 amplitudes are one
  contribution to QCD amplitudes N 1 multiplets
  give another

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• Gauge-theory amplitude
• ?
• Color-ordered amplitude function of ki and ?i
• ?
• Helicity amplitude function of spinor products
  and helicities 1
• ?
• Function of spinor variables and helicities 1
• ?
• Support on simple curves in twistor space

Color decomposition stripping
Spinor-helicity basis
Half-Fourier transform
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Spinors

• Want square root of Lorentz vector ? need spin ½
• Spinors , conjugate spinors
• Spinor product
• (½,0) ? (0, ½) vector
• Helicity ?1 ? Amplitudes as pure functions of
  spinor variables

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Complex Invariants

• These are not just formal objects, we have the
  explicit formulæ
• otherwise
• so that the identity
  always holds
• for real momenta

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Complex Momenta

• For complex momenta
• ?
  or
• but not necessarily both!

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Lets Travel to Twistor Space!

• It turns out that the natural setting for
  amplitudes is not exactly spinor space, but
  something similar. The motivation comes from
  studying the representation of the conformal
  algebra.
• Half-Fourier transform of spinors transform
  , leave alone ? Penroses original twistor
  space, real or complex
• Study amplitudes of definite helicity introduce
  homogeneous coordinates
• ? CP3 or RP3 (projective) twistor space
• Back to momentum space by Fourier-transforming ?

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Differential Operators

• Equation for a line (CP1)
• gives us a differential (line) operator in
  terms of momentum-space spinors
• Equation for a plane (CP2)
• also gives us a differential (plane) operator

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Even String Theorists Can Do Experiments

• Apply F operators to NMHV (3 )
  amplitudesproducts annihilate them! K
  annihilates them
• Apply F operators to N2MHV (4 )
  amplitudeslonger products annihilate them!
  Products of K annihilate them

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• What does this mean in field theory?

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CachazoSvrcekWitten Construction
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How Do We Know Its Right?

• Physicists proof
• Correct factorization properties (collinear,
  multiparticle)
• Compare numerically with conventional recurrence
  relations through n11 to 20 digits
• Derive using new on-shell recursion relations
• Risager (8/2005)

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Recursion Relations
Berends Giele (1988) DAK
(1989) ? Polynomial complexity per helicity
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On-Shell Recurrence Relations

• Britto, Cachazo, Feng (2004)
• Amplitudes written as sum over factorizations
  into on-shell amplitudes but evaluated for
  complex momenta

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• Massless momenta

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Proof Ingredients

• Less is more. My architecture is almost nothing
  Mies van der Rohe
• Britto, Cachazo, Feng, Witten (2004)
• Complex shift of momenta
• Behavior as z ? ? need A(z) ? 0
• Basic complex analysis
• Knowledge of factorization at tree level, tracks
  known multiparticle-pole and collinear
  factorization

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C
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Proof

• Consider the contour integral
• Determine A(0) in terms of other poles
• Poles determined by knowledge of factorization in
  invariants
• At tree level

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• Very general relies only on complex analysis
  factorization
• Applied to gravity
• Bedford, Brandhuber, Spence, Travaglini
  (2/2005)
• Cachazo Svrcek (2/2005)
• Massive amplitudes
• Badger, Glover, Khoze, Svrcek (4/2005, 7/2005)
• Forde DAK (7/2005)
• Integral coefficients
• Bern, Bjerrum-Bohr, Dunbar, Ita (7/2005)

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Unitarity Method for Higher-Order Calculations

• Bern, Dixon, Dunbar, DAK (1994)
• Proven utility as a tool for explicit
  calculations
• Fixed number of external legs
• All-n equations
• Tool for formal proofs
• Yields explicit formulae for factorization
  functions
• Color ordering
• Key idea sew amplitudes not diagrams

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Unitarity-Based Calculations

• Bern, Dixon, Dunbar, DAK (1994)
• At one loop in D4 for SUSY ? full answer(also
  for N 4 two-particle cuts at two loops)
• In general, work in D4-2? ? full answer
• van Neerven (1986) dispersion relations converge
• Merge channels find function w/given cuts in all
  channels
• Generalized cuts require more than two
  propagators to be present

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Unitarity-Based Method at Higher Loops

• Loop amplitudes on either side of the cut
• Multi-particle cuts in addition to two-particle
  cuts
• Find integrand/integral with given cuts in all
  channels
• In practice, replace loop amplitudes by their
  cuts too

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On-Shell Recursion at Loop Level

• Bern, Dixon, DAK (2005)
• Subtleties in factorization factorization in
  complex momenta is not exactly the same as for
  real momenta
• For finite amplitudes, obtain recurrence
  relations which agree with known results
  (Chalmers, Bern, Dixon, DAK Mahlon)
• and yield simpler forms
• Simpler forms involve spurious singularities

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• Amplitudes contain factors like
  known from collinear limits
• Expect also as subleading
  contributions, seen in explicit results
• Double poles with vertex
• Non-conventional single pole one finds the
  double-pole, multiplied by

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Eikonal Function
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Rational Parts of QCD Amplitudes

• Start with cut-containing parts obtained from
  unitarity method, consider same contour integral

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• Start with same contour integral
• Cut terms have spurious singularities, absorb
  them into but that means there is a
  double-counting subtract off those residues

Rational terms
Cut terms
Cut terms
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A 2?4 QCD Amplitude

• Bern, Dixon, Dunbar, DAK (1994)

Only rational terms missing
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A 2?4 QCD Amplitude

• Rational terms
• and check this mornings mailing for more!

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Road Ahead

• Opens door to many new calculations time to do
  them!
• Approach already includes external massive
  particles (H, W, Z)
• Reduce one-loop calculations to purely algebraic
  ones in an analytic context, with polynomial
  complexity
• Massive internal particles
• Lots of excitement to come!

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