Transcript Breakdown of NRQCD Factorization

Breakdown of NRQCD Factorization ??
J. P. Ma
Institute of Theoretical Physics, Beijing
北京
May 30. 2005
2005 Taipei Summer Institute on Strings,
Particles and Fields, 台北
Content
1.
2.
3.
4.
NRQCD factorization
Recent Result at two-loop level
An examination at one-loop level
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 1992-1995.
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) 1125
+ …………..
For an inclusive production of H: (factorization)
: The production rate
:
NRQCD matrix elements for the formation
nonperturbative, n: the state of the pair.
: Perturbative coefficients for the production
of the quark pair. (The rate at tree level)
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.74:3327-3330,1995
……………..
+ 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 2004. 521pp.
e-Print Archive: hep-ph/0412158
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 v4
the factorization for a color-octet quark pair reads:
The first line is at order v0, it contains only S-wave
states.
The second line is at order v2, they are relativistic
corrections to those states in the first line.
The third line is at order v2 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 v2
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 v2 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 oneloop 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:
Coulomb singularity
It is proportional to the tree-level result !! The double pole
is from a overlap of soft- and collinear region……..
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 v2 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 v2
mixing:
, there are operator mixing. The relevant
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 PT , 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!