Threshold Resummation for Top- Quark 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/ 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!

Because it is heavy: 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 Comparing with the hadronization scale ~ 150 MeV and the time for forming a toponium: 1/m t !!  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 ( q 0 ) 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

This effective theory describe the interaction with momenta around: :two-component field for top quark : Anti-top 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 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! Including the width: Regularizing the soft physics.

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