Computational Methods in Particle Physics: OnShell Methods in Field Theory

1
Computational Methods in Particle
PhysicsOn-Shell Methods in Field Theory

• David A. Kosower
• University of Zurich, January 31February 19,
  2007
• Lecture VII

2
Computing QCD Amplitudes

• N 4 pure QCD 4 fermions 3 complex
  scalars
• QCD N 4 dN 1 dN 0

chiral multiplet
scalar
cuts rational
cuts rational
cuts
rational
D4 unitarity
D4 unitarity
D4 unitarity
bootstrap or on-shell recursion relations
3

• At tree level,
• True in supersymmetric theories at all loop
  orders
• Non-vanishing at one loop in QCD
• but finite no possible UV or IR singularities
• Separate V and F terms

4
Factorization at One Loop
5

• Cut terms have spurious singularities ? rational
  terms do too
• Such terms emerge for example from two-mass
  triangles, along with
• which removes the singularity as

6
Completing the Cut

• To solve this problem, define a modified
  completed cut, adding in rational functions to
  cancel spurious singularities
• We know these have to be there, because they are
  generated together by integral reductions
• Spurious singularity is unique
• Rational term is not, but difference is free of
  spurious singularities

7
On-Shell Recursion at Loop Level

• Bern, Dixon, DAK (17/2005)
• Complex shift of momenta
• Behavior as z ? ? require A(z) ? 0
• Basic complex analysis treat branch cuts
• Knowledge of complex factorization
• at tree level, tracks known factorization for
  real momenta
• at loop level, same for multiparticle channels
  and - ?-
• Avoid ?

8
Derivation

• Consider the contour integral
• Vanishing of integral gives us equation for A(0)
  in terms of other poles and branch cuts
• First problem poles include spurious
  singularities

Rational terms
Cut terms
9

• Replace cut terms C by completed cut terms C
  eliminates spurious poles
• C entirely known from four-dimensional unitarity
  method
• Assume as
• Modified separation
• so

10

• Perform integral residue sum for
• so

Unitarity Method
Recursion Relation
11
Separating Rational Terms in Loop Factorization

• Only single poles in splitting amplitudes with
  cuts (like tree)
• Cut terms ? cut terms
• Rational terms ? rational terms
• Define new vertices to be the complete rational
  parts of amplitudes

12

• Recursion on rational pieces would build up
  complete rational terms R, not modified terms
• Recursion gives

Potentially double-counted overlap
13

• Subtract off overlap terms

Compute explicitly from known C also have a
diagrammatic expression
14
One-Loop Recursion Relations
15
Five-Point Example

• Look at
• Cut terms
• have required large-z behavior under
  shift
• Free of spurious singularities

16
Five-Point Example

• Look at
• Recursive diagrams use shift

17

• (a) Tree vertex
  vanishes

18

• (b) Loop vertex
  vanishes scalar contribution to
  vanishes

19

• (c) Loop vertex
  vanishes

20

• (d) Tree vertex
  vanishes

21

• (e) Loop vertex
  vanishes

22

• (f) Diagram doesnt vanish

23
(No Transcript)
24
Five-Point Example (cont.)

• Overlap contributions
• Take rational terms in C

25

• Apply shift, extract residues in each channel

26

• Overlap contributions

27
General Helicities

• Those are my principles, and if you don't like
  them... well, I have others Groucho Marx
• Can we relax the requirements
  as ?
• Yes! Use an auxiliary recursion relation
• Large-z behavior using auxiliary shift
  

28

• shifts appear to have good
  behavior
• but may involve ? channels
• so avoid them for the primary recursion
• use them for the auxiliary recursion where bad
  channels disappear as

29
Unitarity
On-shell recursion
Auxiliary on-shell recursion
30
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

31
Generalized Cuts

• In practice, replace loop amplitudes by their
  cuts too

32
At a Frontier

• Integral basis not known a priori
• Collinear factorization known to two loops
• Recursion relations for rational parts not
  derived potential subtleties
• Work out cuts to derive possible integrals as
  well as their coefficients
• Some QCD calculations at two loops
• Most work in N 4 supersymmetric theory

33
(No Transcript)
34
(No Transcript)
35
Cuts

• Compute a set of six cuts, including multiple
  cuts to determine which integrals are actually
  present, and with which numerator factors

36
On-Shell Methods

• Physical states
• Use of properties of amplitudes as calculational
  tools
• Kinematics Spinor Helicity Basis ? Twistor space
• Tree Amplitudes On-shell Recursion Relations ?
  Factorization
• Loop Amplitudes Unitarity (SUSY)
  Unitarity On-shell Recursion QCD