Breakdown of NRQCD Factorization ?? J. P. Ma Institute of Theoretical Physics, Beijing 北京 May Taipei Summer Institute on Strings, Particles and Fields, 台北

Content 1.NRQCD factorization 2.Recent Result at two-loop level 3.An examination at one-loop level 4.Summary and Conclusion

1. NRQCD factorization Roughly 10 years ago: A heavy quarkonium H: consist of a heavy quark pair Because the heavy quark moves with a small velocity it is a nonrelativistic bound state The state can be classified with Quantum mechanics problem if one knows the potential……… and perturbative QCD can be used because the heavy quark It leads to the Color-Singlet Model, many predictions can be made and it is successful

Success hides problems (as usual?) Coulomb singularities at one loop… Calculations with P-wave H even have infrared singularities at tree level……… ………………………………….. A fundamental problem: Heavy quarks: nonrelativistic quantum mechanics gluons light quarks: relativistic quantum field theory A state in quantum field theory: many components The heavy quark pair can have different quantun numbers as H does and can be found in color-octet states…….

Solutions for all problems came around For small v, one can derive from full QCD an effective theory – nonrelativistic QCD(NRQCD)- by expanding v. A power counting of v to determine relative importance of different components. The production of a heavy quark pair: perturbative QCD The formation of a heavy quark pair into H : NRQCD matrix elements, nonperturbative A factorization approach NRQCD factorization G.T. Bodwin, E. Braaten and G.P. Lepage: Phys. Rev. D51 (1995) …………..

For an inclusive production of H: (factorization) : The production rate : Perturbative coefficients for the production of the quark pair. (The rate at tree level) : NRQCD matrix elements for the formation nonperturbative, n: the state of the pair. If the factorization holds, the coefficients do not contain any soft divergence…….. The matrix elements are universal.

A striking success of the approach: Quarkonium production at Tevatron: A huge discrepancy between CSM and experiment. NRQCD approach can explain the data by adding color-octet Contributions. E. Braaten and S. Fleming: Phys.Rev.Lett.74: ,1995 ……………..

+ other many successes……. All progresses of quarkonium physics in the last decade are summarized in the yellow report of the quarkonium working group HEAVY QUARKONIUM PHYSICS. Dec pp. e-Print Archive: hep-ph/ Again, success combined with problems………

Phenomenologically: the puzzle of polarizations

The measured polarization do not agree with NRQCD predictions!!!

A fundamental problem is: The proof of NRQCD factorization ( A nasty task) There is no proof for that there is no soft divergence in those perturbative coefficients: In some cases at one loop level, one can prove it: + ………

A “rumor” was there two years ago…. It says that the factorization does not hold………..

2. Recent Result at two-loop level (and the motivation of our work) The key point to explain production rates at Tevatron is by adding the gluon fragmentation into a heavy quark pair in color octet. For the gluon fragmentation function one also has the factorization shown before. At parton level to v 4 the factorization for a color-octet quark pair reads:

The first line is at order v 0, it contains only S-wave states. The second line is at order v 2, they are relativistic corrections to those states in the first line. The third line is at order v 2 too, it contains all P-wave states. Some of these perturbative functions have been calculated at one loop level and are free from any soft divergence.

A 2(3)-year project shows: There are some I.R. divergence at two-loop level, and at order v 2 G.C. Nayak, J.W. Qiu and G. Sterman, Phys. Lett. B613 (2005) 45 NQS To restore the factorization, NRQCD matrix elements at order of v 2 should be modified. They suggest to add a gauge link because they are not gauge invariant. The gauge link is determined by the moving direction of the pair, hence the quarkonium.

At first look, these results are “strange”, because If the singularity appears in P-wave channel (the 3. line), one should find it in previous studies at one-loop level, because at tree level a gluon can not fragment into P-wave state. NRQCD is formulated in the rest frame of a heavy quark pair, hence the factorization is made in the rest frame. This implies that the formation of the pair into a quarkonium H is independent how H moves (naturally). With the suggested gauge link, it does!

If it appears in relativistic corrections ( in the second line), it will be not “strange”, because no one has studied them at one-loop before. This will be equivalent to the breakdown of QCD factorization at higher twist, e.g., Drell-Yan process………. A two-loop calculation is complicated ( it is why the rumor). For understanding the breakdown or to which extent the factorization fails, it would be nice to have a one- loop case, where P-wave states can be produced at tree level…….. And also to study the factorization of relativistic corrections…….

3. An examination at one-loop level We study NRQCD factorization in The NRQCD factorization reads:

The coefficients should be free from I.R, if the factorization is right. To determine these coefficients we replace H with a heavy quark pair: with the momenta in the rest frame of the pair: Velocity :

At tree level : ( two Feynman diagrams) The one-loop correction consists of a real- and virtual part. Each part receives contributions from more than 10 Feynman diagrams…….. However, we only need those containing soft divergences. Dimensional regularization: only I.R. poles are kept.

Virtual correction :

It is proportional to the tree-level result !! The double pole is from a overlap of soft- and collinear region…….. Coulomb singularity

The real correction: two gluons or a light quark pair in the final state. The two gluons can be emitted by heavy quarks : also can be generated by gluon splitting

The contribution from heavy-quark emission …….. Denote contributions from states irrelevant for the factorization, but relevant for the KLN cancellation. The contribution from the interference with gluon splitting: The double pole here will be cancelled by that from the virtual correction.

The contribution from the gluon splitting into two gluons and a light quark pair: The total of one-loop correction: Only a singularity at order of v 2 and the Coulomb singularity remain. But it is not from P-wave states, but from relativistic corrections. ……….contribution from other states.

KLN cancellation: In the above, the states of the quark pair are not summed. If one does the sum: In the sum there should be no soft divergence.

To complete the factorization we need to calculate NRQCD matrix elements at one-loop. For any operators considered here we have: It leads to all coefficients are free from Coulomb singularity

At order v 2, there are operator mixing. The relevant mixing: Then: All those coefficients are free from I.R. at one-loop level

The relativistic corrections are also factorizable ! ( all results given here can be found in a paper, which will appear in 2 or 3 days or weeks, by J.P. Ma and Z.G. Si. ) All these results are incompatible with those in NQS paper. If NQS finds the singularity is for P-wave states and modifies corresponding matrix elements with an gauge link, factorization with the modified matrix elements can be made in gluon fragmentation, but the same can NOT be made here. Universality of matrix elements is lost……

A close look at gluon fragmentation (GF): In the standard definition of GF function in any H there is a gauge link U determined by the moving direction of H. Even in the rest frame of H, U is always there. That is why the formation of a quark pair into a quarkonium will depend on the direction. Actually, the divergence found by NQS is from the gauge link U in the definition……… A deep reason for all these “incompatibilities” is…….

The incompatibility between NRQCD factorization and QCD factorization based on twist expansion. Twist expansion (roughly): If a hadron is produced with large P T, an expansion is made in At the leading order (twist-2), the concept of gluon fragmentation appears…. NRQCD factorization (roughly): an expansion in Two limits are not compatible. One is able to show this in detail ( Work under progress……)

4. Summary and Conclusion NRQCD factorization is still right in cases examined, also in the case with relativistic corrections. The factorizations for usual production rates and for gluon fragmentation are different. Universality is lost. NRQCD factorization is incompatible with the QCD factorization based on twist expansion ( will be shown in detail).

Thank the organizers of the Summer Institute Thank all of you!