Threshold Resummation for TopQuark Pair Production at ILC
J.P. Ma
ITP, CAS, Beijing
2005年直线对撞机国际研讨会, Tsinghua Univ.
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…..
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!
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…………..
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……..
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???
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 gluon’s mass is zero!.
Same as in QED, known as Coulomb potential…….
Adding the correction:
At higher orders:
Each gluon-exchange
gives a factor:
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……..(?)
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……..
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:
The contribution in the limit is proportional to
In the Coulomb region:
Neglecting
( q0)2
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……. )
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
:two-component field for top quark
: Anti-top
This effective theory describe the interaction with
momenta around:
Coulomb region:
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
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!!!
3: Resummation
The total cross-section :
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!
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!
So far, the resummation is done for NNLO:
Solving the equation with the potential V, one obtains
the result. It is finite.
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 < 50 MeV