Threshold Resummation for Top-Quark Pair Production at ILC

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Threshold Resummation for Top-Quark Pair
Production at ILC

• J.P. Ma
• ITP, CAS, Beijing

2005???????????, Tsinghua Univ.
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• Content
• 1 Why Is Resummation Needed?
• 2 QCD Interaction in Coulomb Region
• and Effective Theory
• 3 Resummation
• 4 Brief Summary
• Ref A.H. Hoang, M. Beneke,
• hep-ph/0001286 and..

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• 1 Why Is Resummation Needed?
• About top
• 1995 Found at Fermilab,
• Mass 180GeV
• One of 6 quarks predicted by SM, the most heavy
  of fundamental particles!

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Heaviest known elementary particle 180GeV ?
measure properties of least known quark ? top
quark mass constrains Higgs mass ? sensitive to
new physics ? short life time probe bare quark

• Because it is heavy

• Comparing with the hadronization scale 150 MeV
• and the time for forming a toponium 1/mt !!
• a top, once produced, it will decay quickly so
  that there
• is no time to form any hadron containing the
  top!!!

And the information from the production to the
decay can be studied with perturbative theory!!
But..
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• A clean way to study top-quark is at an e e-
  collider(ILC)

Measuring the dependence
Scanning a range of s, one can determine the
mass. Interesting is At the threshold..
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• At tree-level
• In the limit

The amplitude is finite. But the phase space is
proportional to v !
Therefore, at tree-level the cross section is
zero! Determining the zero, the mass can be
determined! Is it true???
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• One-loop QCD correction

The correction is singular as
The singularity is well-known as Coulomb
singularity, it reflects that the force
between the top and anti-top is a long range
force because the gluons mass is zero!. Same as
in QED, known as Coulomb potential.
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• Adding the correction

At higher orders Each gluon-exchange gives a
factor
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The cross-section takes the general form in
perturbative QCD

The cross-section is divergent in the limit
If we resum these divergent terms, we may have
finite answer..(?)
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• A hint from QED-study in old days A resum of
  those Coulomb singularities is equivalent to
  solve the Schroedinger equation, it leads to that
  a e e- pair is formed into a bound state, i.e.,
  finite!
• But now, we must do it in a better, modern way.
• A. Identify which gluons are responsible for
  these
• these singularities,
  momentum region
• B. Use an effective theory for the interaction
  with
• those gluons, NRQCD.
• C. Resum..

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• 2 QCD Interaction in Coulomb Region and
  Effective Theory

In the limit
We neglect all terms proportional to v in
numerators. Momentum Region
Hard Soft Coulomb
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• The contribution in the limit is proportional to

In the Coulomb region
Neglecting ( q0)2
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The singularity is generated by the interaction
between a quark with a small v and a gluon with
momenta in the Coulomb region. (that is why
the name. )
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• Now, there is a standard way to describe
  interactions in different momentum region, or at
  different energy scales.

From the path integral of full QCD, one can
integrate out those dynamical freedoms with
momenta larger than
Hard
The integration can be done perturbatively. It
leads to an effective theory Non-Relativistic
QCD, NRQCD
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two-component field for top quark
Anti-top

• This effective theory describe the interaction
  with momenta around

Coulomb region
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• To obtain an effective theory for the Coulomb
  region,
• the dynamical freedoms with momenta are
  integrated

Quarks, gluons
Gluons
One obtains from NRQCD another effective theory,
called potential NRQCD, PNRQCD
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It describes the interaction in the Coulomb
region with the Coulomb potential. If one
solves PNRQCD perturbatively, one has all those
Coulomb singularities. If solved in a
nonperturbative way, they are resummed! It
really likes Quantum Mechanics!!!
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• 3 Resummation
• The total cross-section

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• In the limit one can define the Coulomb Green
  function

With a rather complicated manipulation one has
The correction is suppressed by powers of v!
Also suppressed by powers of v!
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• The properties of the Green function

NLO Correction
Including the width Regularizing the soft
physics.
Again, if one solves the equation with
perturbative theory, one has all those Coulomb
singularities. But one can solve the equation
exactly and obtain finite results for the Green
function!
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• So far, the resummation is done for NNLO

Solving the equation with the potential V, one
obtains the result. It is finite.
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• 4 Brief Summary
• The production rate near the threshold can be
  predicted precisely with the resummation.
• Naïve perturbative theory gives divergent
  results.
• Detailed study shows
• 100/fb, including experimental uncertainties,
• the top mass can be determined with error lt 50
  MeV