Computational Methods in Particle Physics: OnShell Methods in Field Theory

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Computational Methods in Particle
PhysicsOn-Shell Methods in Field Theory

• David A. Kosower
• University of Zurich, January 31February 14,
  2007
• Lecture III

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Review of Lecture II

• All-n forms for tree-level amplitudes
• Parke-Taylor equations
• Mangano, Xu, Parke (1986)

Maximally helicity-violating or MHV
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• MHV Rules

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Factorization Properties of Amplitudes

• As sums of external momenta approach poles,
• amplitudes factorize
• More generally as

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Factorization in Gauge Theories

• Tree level
• As but
• Sum over helicities of intermediate leg
• In massless theories beyond tree level, the
  situation is more complicated but at tree level
  it works in a standard way

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What Happens in the Two-Particle Case?

• We would get a three-gluon amplitude on the
  left-hand side
• But
• so all invariants vanish,
• hence all spinor products vanish
• hence the three-point amplitude vanishes

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• In gauge theories, it holds (at tree level) for
  n?3 but breaks down for n 2 A3 0 so we get
  0/0
• However A3 only vanishes linearly, so the
  amplitude is not finite in this limit, but should
  1/k, that is
• This is a collinear limit
• Combine amplitude with propagator to get a
  non-vanishing object

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Two-Particle Case

• Collinear limit splitting amplitude

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Universal Factorization

• Amplitudes have a universal behavior in this
  limit
• Depend on a collinear momentum fraction z

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• In this form, a powerful tool for checking
  calculations
• As expressed in on-shell recursion relations, a
  powerful tool for computing amplitudes

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Example Three-Particle Factorization

• Consider

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• As , its finite expected
  because
• As , pick up the first term with
  

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Splitting Amplitudes

• Compute it from the three-point vertex

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Explicit Values
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Collinear Factorization at One Loop
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Soft Factorization

• Universal factorization limits correspond to
  infrared-singular limits
• Recall that in addition to collinear
  singularities, amplitudes are also singular when
  a gluon momentum becomes soft
• The full amplitude does not have a simple
  factorization property in the soft limit, but
  color-ordered amplitudes do

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• The Soft factors are eikonal factors, and depend
  only on the helicity of the soft gluon,
• There are two singular invariants this
  characterizes soft singularities and
  distinguishes them from collinear ones

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Combining Factorization Limits

• In performing real integrations over singular
  regions, we have two different kinds of singular
  regions and their overlap and different
  functions to integrate in each
• Gets worse at higher loops
• Desirable to combine the two limits
• Original method Catani Seymour combine squared
  splitting and eikonal functions to give
  factorization of squared amplitude
• Also possible at the amplitude level

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Dipole Factorization

• Catani Seymour (1996)
• Unify soft collinear limits of squared matrix
  element
• Add soft limits to collinear factorization

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Dipole Factorization

• Most widely-used subtraction method in numerical
  calculations at NLO

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Antenna Factorization

• DAK (1998)
• Combine soft collinear limits at amplitude
  level
• Add collinear wings to soft core
• Use tree-level current J, computed
  recursively Berends Giele (1988),
  Dixon (1995)
• Remap three momenta to two massless momenta

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Collinear Wings

• Merge a ??1, 1 soft, and 1 ??b

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Antenna Factorization

• In any singular limit (?(a,1,b)/sab3 ? 0),

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Reconstruction Functions

• Map three momenta to two massless momenta
• Conserve momentum
• Ensure no subleading terms contribute to singular
  limits
• Generalizes to m singular emissions m ? 2

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Limiting Values

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Derivation

• Extract all poles in sa1 and s1b
• Insert complete set of states
• Put reconstructed momenta on shell, momentum
  flows between currents

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Gluon Examples

• A particular helicity
• Helicity-summed square (factorization of squared
  matrix element)

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Multicollinear Limits Using CSW
Birthwright, Glover, Khoze, Marquard (2005)

• Consider limits where
• so that
• Different helicity configurations

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• Power-counting and phase weight ? any term in
  the splitting amplitude must be of the form
• Imagine factorizing an amplitude to extract the
  splitting amplitude cannot come from
  vertices (q-dependent ones cancel), must come
  from propagators
• ? need j nearly on-shell propagators

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• In the required diagrams, use the substitutions
  (c collinear leg, x non-collinear, r off-shell
  sum of some collinear momenta, z momentum
  fractions, P sum of all collinear momenta)

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Anomalous Dimensions Amplitudes

• In QCD, one-loop anomalous dimensions of twist-2
  operators in the OPE are related to the
  tree-level Altarelli-Parisi function,
• Relation understood between two-loop anomalous
  dimensions one-loop splitting amplitudes
• DAK Uwer (2003)

Mellin Transform
Helicity-summed splitting amplitude
Twist-2 Anomalous Dimension
Altarelli-Parisi function
?

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Recursion Relations

• Considered color-ordered amplitude with one leg
  off-shell, amputate its polarization vector
• This is the BerendsGiele current
• Given by the sum of all (n1)-point color-ordered
  diagrams with legs 1 n on shell
• Follow the off-shell line into the sum of
  diagrams. It is attached to either a three- or
  four-point vertex.
• Other lines attaching to that vertex are also
  sums of diagrams with one leg off-shell and other
  on shell, that is currents

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BerendsGiele Recursion Relations
Berends Giele (1988) DAK (1989) ?
Polynomial complexity per helicity
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Properties of the Current

• Decoupling identity
• Reflection identity
• Conservation

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Explicit Solutions

• Strategy solve by induction
• Compute explicitly
• shorthand
• Compute five-point explicitly ? ansatz

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Complex Momenta

• For real momenta,
• but we can choose these two spinors independently
  and still have k2 0
• Recall the polarization vector
• but
• Now when two momenta are collinear
• only one of the spinors has to be collinear
• 
  but not necessarily both