On-Shell Recursion Relations for QCD Loop Amplitudes (Part I)

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On-Shell Recursion Relationsfor QCD Loop
Amplitudes(Part I)

• Lance Dixon, SLAC
• From Twistors to Amplitudes Workshop
• Queen Mary U. of London
• November 3, 2005

Z. Bern, LD, D. Kosower, hep-th/0501240,
hep-ph/0505055, hep-ph/0507005
C. Berger, Z. Bern, LD, D. Forde, D. Kosower, to
appear
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Motivation

• Need a flexible, efficient method to extend the
  range of known tree, and particularly 1-loop QCD
  amplitudes, for use in NLO corrections to LHC
  processes, etc.
• Unitarity is an efficient method for determining
  imaginary parts of loop amplitudes
• Efficient because it recycles
• trees into loops

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Motivation (cont.)

• Unitarity can miss rational functions that have
  no cut.
• These functions can be recovered using
  dimensional analysis, computing cuts to higher
  order in e, in dim. reg. with D4-2e
• 
  
  
  Bern, LD, Kosower, hep-ph/9602280
• Brandhuber, McNamara, Spence, Travaglini,
  hep-th/0506068
• But tree amplitudes are more complicated in
  D4-2e than in D4.
• Seems too much information is used
• n-point loop amplitudes also factorize onto
  lower-point amplitudes.
• At tree-level these data have been systematized
  into
• on-shell recursion relations Britto,
  Cachazo, Feng, hep-th/0412308
• 
  Britto, Cachazo, Feng, Witten,
  hep-th/0501052
• Efficient recycles trees into trees
• Can also do the same for loops

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On-shell tree recursion

• BCFW consider a family of on-shell amplitudes
  An(z) depending on a complex parameter z which
  shifts the momenta, described using spinor
  variables.
• For example, the (n,1) shift
• Maintains on-shell condition,
• and momentum conservation,

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MHV example

• Apply this shift to the Parke-Taylor (MHV)
  amplitudes
• Under the (n,1) shift
• So
• Consider
• 2 poles, opposite residues

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MHV example (cont.)

• MHV amplitude obeys
• Compute residue using factorization
• At
• kinematics are complex collinear
• so

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The general case

Britto, Cachazo, Feng, hep-th/0412308
Ak1 and An-k1 are on-shell tree amplitudes
with fewer legs, evaluated with 2 momenta shifted
by a complex amount.
In kth term

which solves
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Proof of on-shell recursion relations
Britto, Cachazo, Feng, Witten, hep-th/0501052
Same analysis as above Cauchys theorem
amplitude factorization
Let complex momentum shift depend on z. Use
analyticity in z.
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On-shell recursion at one loop
Bern, LD, Kosower, hep-th/0501240, hep-th/0505055

• Same techniques can be used to compute one-loop
  amplitudes
• which are much harder to obtain by other
  methods than are trees.

• Again we are trying to determine a rational
  function of z.

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A one-loop pole analysis
Bern, LD, Kosower (1993)
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The double pole diagram
Want to produce
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Unreal pole underneath the double pole
nonsingular in real Minkowski kinematics
Want to produce
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A one-loop all-n recursion relation
Same suppression factor works in the case of n
external legs!
shift
leads to
Know it works because results agree with Mahlon,
hep-ph/9312276, though much shorter formulae are
obtained from this relation
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Solution to recursion relation
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External fermions too
Can similarly write down recursion relations for
the finite, cut-free amplitudes with 2 external
fermions
and the solutions are just as compact
Gives the complete set of finite, cut-free, QCD
loop amplitudes (at 2 loops or more, all helicity
amplitudes have cuts, diverge)
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Fermionic solutions
and
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Loop amplitudes with cuts

• Can also extend same recursive technique
  (combined with unitarity) to loop amplitudes with
  cuts (hep-ph/0507005)
• Here rational-function terms contain
• spurious singularities, e.g.
• accounting for them properly yields simple
• overlap diagrams in addition to recursive
  diagrams
• No loop integrals required to bootstrap the
  rational functions from the cuts and lower-point
  amplitudes
• Tested method on 5-point amplitudes, used it to
  compute
• and more recently

Forde, Kosower, hep-ph/0509358
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Loop amplitudes with cuts (cont.)

• See also Part II (David Kosowers talk Saturday)

Generic analytic properties of shifted 1-loop
amplitude,
Cuts and poles in z-plane
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Loop amplitudes with cuts (cont.)
However, we know how to complete the cuts at
z0 to cancel the spurious pole terms, using
Li(r) functions
So we do a different subtraction

full amplitude
completed-cut part
modified rational part
New shifted rational function
has no cuts, and no spurious poles. But the
residues of the physical poles are not given by
naïve factorization onto (the rational parts of)
lower-point amplitudes, due to the rational parts
of the completed-cut terms,
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Loop amplitudes with cuts (cont.)
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Subtleties at Infinity

• It is possible that does not vanish
  at .
• It could even blow up there.
• Even if it is well-behaved, might not
  be.
• In that case,
  also wont be.

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An NMHV QCD Loop Amplitude

• David Kosower will discuss the application of
  this formalism
• to the MHV QCD loop amplitudes,

• Here I describe the application to an NMHV QCD
  loop amplitude,
• The general split-helicity case
• works identically to this case

Berger, Bern, LD, Forde, Kosower, to appear
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NMHV QCD Loop Amplitude (cont.)
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What shift to use?
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Behavior at Infinity

• Inspecting

for n5
Bern, LD, Kosower (1993)
we find that the rational terms diverge but in a
very simple way
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Behavior at Infinity (cont.)

• For general n

we assume that the rational terms diverge in the
analogous way
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Result for n6
Bern, Bjerrum-Bohr, Dunbar, Ita hep-ph/0507019

• Extra rational terms, beyond L2 terms from

where flip1 permutes
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Conclusions

• On-shell recursion relations can be extended
  fruitfully to determine rational parts of loop
  amplitudes with a bit of guesswork, but there
  are lots of consistency checks.
• Method still very efficient compact solutions
  found for
• all finite, cut-free loop amplitudes in QCD
• Recently extended same technique (combined with
  D4 unitarity) to some of the more general loop
  amplitudes with cuts, MHV and NMHV, which are
  needed for NLO corrections to LHC processes
• Prospects look very good for attacking a wide
  range of multi-parton processes in this way

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Why does it all work?

• In mathematics you don't understand things.
• You just get used to them.

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Extra Slides
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Revenge of the Analytic S-matrix?
Reconstruct scattering amplitudes directly from
analytic properties
Chew, Mandelstam Eden, Landshoff, Olive,
Polkinghorne (1960s)
Analyticity fell out of favor in 1970s with rise
of QCD to resurrect it for computing
perturbative QCD amplitudes seems deliciously
ironic!
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March of the n-gluon helicity amplitudes
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March of the tree amplitudes
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March of the 1-loop amplitudes
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MHV check

• Using
• one confirms

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To show
Britto, Cachazo, Feng, Witten, hep-th/0501052
Propagators
3-point vertices
Polarization vectors
Total
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Initial data