Recurrence, Unitarity and Twistors

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Recurrence, Unitarity and Twistors
including work with I. Bena, Z. Bern, V. Del
Duca, D. Dunbar, L. Dixon,D. Forde, P.
Mastrolia, R. Roiban
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• Weve heard a lot about twistors and about
  amplitudes in gauge theories, N4 supersymmetric
  gauge theory in particular.
• What are the motivations for studying amplitudes?
• ? Berns talk

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Goals of Explicit Calculations of Amplitudes

• Analytic results
• Understand structure of results
• Insight into theory
• Develop new tools
• Anomalous dimensions
• Numerical values to be integrated over phase
  space
• LHC Physics background to discoveries at the
  energy frontier
• ? Berns talk
• Computational complexity of algorithm important

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Computational Complexity of Tree Amplitudes

• How many operations (multiplication, addition,
  etc.) does it take to evaluate an amplitude?
• Textbook Feynman diagram approach factorial
  complexity
• Color ordering
• ? exponential complexity
• O(2n) different helicities at least exponential
  complexity
• But what about the complexity of each helicity
  amplitude?

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Recurrence Relations
Berends Giele (1988) DAK (1989)
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Complexity of Each Helicity Amplitude

• Same j-point current appears in calculation of Jn
  as in calculation of Jmltn
• Only a polnoymial number of different currents
  needed
• O(n4) operations for generic helicity

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Twistors New Representations for Trees

• Cachazo-Svrcek-Witten construction ? Svrceks
  talk
• simple vertices rules
• Roiban-Spradlin-Volovich representation ?
  Spradlins talk
• compact representation derived from loops
• inspired by trees obtained from infrared
  consistency equations
• Bern, Dixon, DAK
• Britto-Cachazo-Feng-Witten recurrence ? Brittos
  talk
• representation in terms of lower-n on-shell
  amplitudes
• Nice analytic forms
• In special cases, better than O(n4) operations
• Probably not the last word

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Twistors Are an Experimental Subject!

• Analysis of known results revealed simple
  structure of amplitudes at tree level and one
  loop
• Cachazo, Svrcek, Witten (2004)
• Simpler than anticipated in Wittens original
  formulation of twistor string theory
• Lead to new ideas for calculational techniques
• ? Svrceks talk
• No ab initio derivation, so checks and
  independent calculations are essential

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MHV Amplitudes

• Pure gluon amplitudes
• All gluon helicities ? amplitude 0
• Gluon helicities ? amplitude 0
• Gluon helicities ? MHV amplitude
• Parke Taylor (1986)
• Holomorphic in spinor variables
• Proved via recurrence relations
• Berends Giele (1988)

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CachazoSvrcekWitten Construction

• Vertices are off-shell continuations of MHV
  amplitudes
• Connect them by propagators i/K2
• Draw all diagrams

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Recursive Formulation

• Bena, Bern, DAK (2004)
• Recursive approaches have proven powerful in QCD
  how can we implement one in the CSW approach?

Divide into two sets by cutting propagator
Cant follow a single leg
Treat as new higher-degree vertices
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• Higher degree vertices expressed in terms of
  lower-degree ones
• Compact formula when dressed with external legs

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Beyond Pure QCD

• Add Higgs ? Dixons talk
• Add Ws and Zs
• Bern, Forde, Mastrolia, DAK (2004)
• Hybrid formalism build up recursive currents
  using CSW construction
• W current (W ? electroweak process)
• along with CSW construction

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Loop Calculations Textbook Approach

• Sew together vertices and propagators into loop
  diagrams
• Obtain a sum over 2n-point 0n-tensor
  integrals, multiplied by coefficients which are
  functions of k and ?
• Reduce tensor integrals using Brown-Feynman
  Passarino-Veltman brute-force reduction, or
  perhaps Vermaseren-van Neerven method
• Reduce higher-point integrals to bubbles,
  triangles, and boxes

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• Can apply this to color-ordered amplitudes, using
  color-ordered Feynman rules
• Can use spinor-helicity method at the end to
  obtain helicity amplitudes
• BUT
• This fails to take advantage of gauge
  cancellations early in the calculation, so a lot
  of calculational effort is just wasted.

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Traditional Methods in the N4 One-LoopSeven-Poin
t Amplitude

• 227,585 diagrams
• _at_ 1 in2/diagram three bound volumes of Phys.
  Rev. D just to draw them
• _at_ 1 min/diagram 22 months full-time just to draw
  them
• So of course one doesnt do it that way

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Can We Take Advantage

• Of tree-level recurrence relations?
• Of new twistor-based ideas for reducing
  computational effort for analytic forms?

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Unitarity

• Basic property of any quantum field theory
  conservation of probability. In terms of the
  scattering matrix,
• In terms of the transition matrix
  we get,
• or
• with the Feynman i?

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• This has a direct translation into Feynman
  diagrams, using the Cutkosky rules. If we have a
  Feynman integral,
• and we want the discontinuity in the K2 channel,
  we should replace

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• When we do this, we obtain a phase-space integral

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In the Bad Old Days of Dispersion Relations

• To recover the full integral, we could perform a
  dispersion integral
• in which so long as
  when
• If this condition isnt satisfied, there are
  subtraction ambiguities corresponding to terms
  in the full amplitude which have no
  discontinuities

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• But its better to obtain the full integral by
  identifying which Feynman integral(s) the cut
  came from.
• Allows us to take advantage of sophisticated
  techniques for evaluating Feynman integrals
  identities, modern reduction techniques,
  differential equations, reduction to master
  integrals, etc.

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Computing Amplitudes Not Diagrams

• The cutting relation can also be applied to sums
  of diagrams, in addition to single diagrams
• Looking at the cut in a given channel s of the
  sum of all diagrams for a given process throws
  away diagrams with no cut that is diagrams with
  one or both of the required propagators missing
  and yields the sum of all diagrams on each side
  of the cut.
• Each of those sums is an on-shell tree amplitude,
  so we can take advantage of all the advanced
  techniques weve seen for computing them.

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Unitarity Method for Higher-Order Calculations

• Bern, Dixon, Dunbar, DAK (1994)
• Proven utility as a tool for explicit one- and
  two-loop calculations
• Fixed number of external legs
• All-n equations
• Tool for formal proofs all-orders collinear
  factorization
• Yields explicit formulae for factorization
  functions two-loop splitting amplitude
• Recent work also by Bedford, Brandhuber, Spence,
  Travaglini Britto, Cachazo, Feng Bidder,
  Bjerrum-Bohr, Dixon, Dunbar, Perkins

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Unitarity-Based Method at One Loop

• Compute cuts in a set of channels
• Compute required tree amplitudes
• Form the phase-space integrals
• Reconstruct corresponding Feynman integrals
• Perform integral reductions to a set of master
  integrals
• Assemble the answer

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Unitarity-Based Calculations

• Bern, Dixon, Dunbar, DAK (1994)
• In general, work in D4-2? ? full answer
• van Neerven (1986) dispersion relations converge
• At one loop in D4 for SUSY ? full answer
• Merge channels rather than blindly summing find
  function w/given cuts in all channels

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The Three Roles of Dimensional Regularization

• Ultraviolet regulator
• Infrared regulator
• Handle on rational terms.
• Dimensional regularization effectively removes
  the ultraviolet divergence, rendering integrals
  convergent, and so removing the need for a
  subtraction in the dispersion relation
• Pedestrian viewpoint dimensionally, there is
  always a factor of (s)?, so at higher order in
  ?, even rational terms will have a factor of
  ln(s), which has a discontinuity

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Integral Reductions

• At one loop, all n?5-point amplitudes in a
  massless theory can be written in terms of nine
  different types of scalar integrals
• boxes (one-mass, easy two-mass, hard
  two-mass, three-mass, and four-mass)
• triangles (one-mass, two-mass, and three-mass)
• bubbles
• In an N4 supersymmetric theory, only boxes are
  needed.

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Basis in N4 Theory
easy two-mass box
hard two-mass box
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Example MHV at One Loop
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• Start with the cut
• Use the known expressions for the MHV amplitudes

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• Most factors are independent of the integration
  momentum

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• We can use the Schouten identity to rewrite the
  remaining parts of the integrand,
• Two propagators cancel, so were left with a box
  the ?5 leads to a Levi-Civita tensor which
  vanishes
• Whats left over is the same function which
  appears in the denominator of the box

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• We obtain the result,

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• Knowledge of basis opens door to new methods of
  computing amplitudes
• Need to compute only the coefficients
• Algebraic approach by Cachazo based on
  holomorphic anomaly ? Brittos talk
• Britto, Cachazo, Feng (2004)
• Knowledge of basis not required for the
  unitarity-based method

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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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• Cuts require two propagators to be present
  corresponding to a massive channel
• Can require more than two propagators to be
  present generalized cuts
• Break up amplitude into yet smaller and simpler
  pieces more effective recycling of tree
  amplitudes
• Triple cuts
  Bern, Dixon, DAK (1996)
• all-n next-to-MHV
  amplitude Bern, Dixon, DAK (2004)

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Generalized Cuts

• Isolate different contributions at higher loops
  as well

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An Amazing Result Planar Iteration Relation

• Bern, Rozowsky, Yan (1997)
• Anastasiou, Bern, Dixon, DAK (2003)
• This should generalize

Ratio to tree
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• With knowledge of the integral basis, quadruple
  cut gives general numerical solution for N4
  one-loop coefficients (four equations for
  four-vector specify it)
• Use complex loop momenta to obtain solution even
  with three-point vertices (which vanish on-shell
  for real momenta)
• Britto, Cachazo, Feng (2004)

? Brittos talk
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Loops From MHV Vertices

• Sew together two MHV vertices
• Brandhuber, Spence, Travaglini (2004)

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• Simplest off-shell continuation lacks i?
  prescription
• Use alternate form of CSW continuation
• ?
• DAK (2004)
• to map the calculation on to the cut
• Brandhuber, Spence, Travaglini (2004)

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One-Loop Seven-Point NMHV Amplitude

• Bern, Del Duca, Dixon
• Results remarkably simple (6 pages, each
  coefficient essentially a one-liner) to draw all
  the Feynman diagrams in 6 pages, each would have
  to fit in 1 mm2
• Structure
• 3-mass, easy hard 2-mass, 1-mass boxes

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Seven-Point Coefficients

3-mass
cubic
collinear
Easy 2-mass
planar
multiparticle
Hard 2-mass
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The All-n NMHV Amplitude in N4

• Quadruple cuts show that four-mass boxes are
  absent
• Triple cuts lead to simple expression for
  three-mass box coefficient
• Triple cuts or soft limits lead to expression for
  hard two-mass box coefficient as a sum of
  three-mass box coefficients
• Infrared equations lead to expressions for easy
  two-mass (and one-mass) box coefficients as sums
  of three-mass box coefficients

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Triple Cuts

• Write down the three vertices, pull out
  cut-independent factor
• Use Schouten identity to partial fraction second
  factor

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• Use another partial fractioning identity with
  cubic denominators
• Isolate box with three cut momenta and

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All-n Results Structure

• ? Dixons talk

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Infrared Consistency Equations

• N4 SUSY amplitudes are UV-finite, but still have
  infrared divergences due to soft gluons
• Leading divergences are universal to a gauge
  theory independent of matter content same as QCD
• Would cancel in a physical cross-section
• General structure of one-loop infrared
  divergences
• Giele Glover (1992) Kunszt, Signer, Trocsanyi
  (1994)

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• Examine coefficients of
• Gives linear relations between coefficients of
  different boxes
• n (n3)/2 equations enough to solve for easy
  two-mass and one-mass coefficients in terms of
  three-mass and hard two-mass coefficients odd n
• Alternatively, gives new representation of trees
• also Roiban, Spradlin, Volovich (2004)
• Britto, Cachazo, Feng, Witten (2004/5)

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Infrared Divergences

• Bena, Bern, Roiban, DAK (2004)
• BST calculation MHV diagrams map to cuts
• Generic diagrams have no infrared divergences
• Only diagrams with a four-point vertex have
  infrared divergences
• One can define a twistor-space regulator for
  those
• Separates the issue of infrared divergences from
  the formulation of the string theory

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Summary

• Unitarity is the natural tool for loop
  calculations with twistor methods
• Large body of explicit results useful for both
  phenomenology and twistor investigations