Jet fragmentation due to a quark/diquark pick-up in high energy heavy ion collisions
ABSTRACT We propose a model aimed at explaining jet quenching, large azimuthal asymmetry and baryon/meson ratio at large transverse momenta pt=2-10 GeV observed at RHIC. Its main point is that a QCD string can be cut by matter quarks and/or diquarks before its natural breaking. We model the early quark/diquark production via QCD sphalerons.
arXiv:hep-ph/0305160v2 2 Jun 2003
Jet Fragmentation due to a Quark/Diquark Pick-up
in High Energy Heavy Ion Collisions
J.Casalderrey-Solana and E.V. Shuryak
Department of Physics and Astronomy,
State University of New York, Stony Brook NY 11794-3800, USA
(February 1, 2008)
We propose a model aimed at explaining jet quenching, large azimuthal asymmetry and
baryon/meson ratio at large transverse momenta pt = 2 − 10GeV observed at RHIC. Its main
point is that a QCD string can be cut by matter quarks and/or diquarks before its natural breaking.
We model the early quark/diquark production via QCD sphalerons.
1.Observations. The field of heavy ion collisions en-
tered the new era with first experiments at Relativistic
Heavy Ion Collider (RHIC), which revealed a number of
new phenomena.While the totality of data for most
secondaries, with pt < 2 GeV/c, agree with a picture
of strong collective explosion of matter close to equilib-
rium, and is well described by relativistic hydrodynamics
, for pt> 2 GeV/c the regime changes. Based on pp
data it has been anticipated that this region can be de-
scribed perturbatively, by a standard parton model, with
modest modifications due to initial and final state inter-
action. (Such ideology was implemented e.g. in popular
event generator HIJING.) Instead, RHIC experiments at
pt= 2 − 10GeV have found that: (i) hadron yields, rel-
ative to pp or parton model, are smaller by the “ jet
quenching factor” Q ∼ 1/3; (ii) the azimuthal asymme-
try is unexpectedly large, with v2=< cos(2φ) >∼ .1 for
mid-peripheral collisions; (iii) the baryon/meson ratio is
unexpectedly large, ∼ 1, well above that in the usual
jet fragmentation; (iv) a clear link between the two last
points is seen from the fact that v2for baryons (p,Λ) is
larger than for mesons (π,K). In this paper we propose a
mechanism which may account for all these observations.
2.Recent literature. A simple theoretical argument
made by one of us  is that in any model of jet quenching
by absorption is limited by the regime of surface emis-
sion. Its consequence is the geometrical limit: v2(b) <
2(b) where the r.h.s. is uniquely related to the shape
of the almond-shaped nuclear overlap region at impact
parameter b. However, as emphasized in , STAR data
seem to exceed the geometric limit , which rules out all
purely absorptive models.
It was suggested by Lin and Ko  and Voloshin and
Molnar  that quark coalescence into hadrons enhances
v2, and vbaryon/vmeson≈ 3/2. The coalescence has been
further discussed in Refs [6,7]. It is concluded that the co-
alescence of 2 or 3 hard partons is very improbable. The
coalescence of soft thermalized partons produces thermal-
like distribution of hadrons in the rest frame of each mat-
ter cell, reproducing basically the Cooper-Frie formula
used in all hydro papers. How many partons participates
in coalescence seem to be nearly irrelevant. A coalescence
of hard-soft kind may enhance the baryon/meson ratio.
However, due to jet quenching the hard partons only con-
tribute to hadronic spectra if they are produced at the
surface and move outward. The soft ones, to be lifted
by flow, have to wait a significant time of ∼ 10fm/c, at
which point hard ones are too far away. Our model to
be introduced below includes a hard-soft coalescence at
very early time, < 1.5fm/c. Hard and soft partons are
not at the same place, but are connected by a string.
3.The model. Unlike models mentioned above, we
focus on the matter-induced modification of the jet frag-
mentation process. Our first new point is that although
the outgoing hard quark (or gluon) leaves the system
promptly, the QCD string (or 2 strings) is still extended
behind and is crossing the excited matter. The string
helps to explain the timing problem mentioned above,
among other ones. The second new point is that the QCD
sphalerons can provide promptly quarks or diquarks, con-
veniently concentrated on thin and expanding spheres .
New mechanism of string breaking must act before (in
the jet center of mass frame) than both (i) the usual
spontaneous string-breaking, as described by the Lund-
model ; and (ii) the perturbative gluon radiation from
a parton. ( Due to strongly falling ptspectra, even mod-
est energy loss makes the contribution irrelevant.) An
approximate expression for the time of both phenomena
(i) and (ii) combined  can be written as a condition
τ < τstring breaking≈
σ + p2
where σ = (.44 GeV/c)2is the usual string tension.
The “string cutting” is the main idea of the model,
while the sphalerons used as a description of matter at
early time is admittedly an extreme one, and can possi-
bly be later combined/replaced by another one. It was
chosen due to its simple geometrical rules, and also to
maximize both (i) the radiative jet quenching and (ii)
quark and diquark pick-up rate.
will find it reasonable to go to the extreme in the first
exploratory study, since so many other models we and
others tried have failed.
Theory of the QCD sphalerons is discussed in detail
in . In brief, they are unstable classical soliton-like
objects with masses ∼ 3 GeV/c, excited from the part of
the vacuum wave function under the topological barrier
by high energy collisions.
For discussion of their production in hadronic collisions
in experiments see e.g.  and references therein.
We hope the reader
Like jets, sphalerons are produced in (semi)-hard gg
collisions and thus have the same distribution of the orig-
ination points in the transverse plane. Once produced,
they evolve into expanding spheres, also moving with a
speed of light. The radiative energy loss in a single cross-
ing of a parton through the sphaleron was calculated in
. Using its results we estimated that this generate
quenching of a parton by about an order of magnitude,
provided the time is shorter than τ < τsp. After that
time we stop the model.
The rules of the model thus are as follows: (i) If a jet
goes through the sphaleron sphere, it is eliminated; (ii)
If a jet escapes all the sphalerons but its string is crossed
by a sphere of one of them, the string is cut by q pickup.
(iii) With a probability Pqqa diquark instead of a quark
is picked up. The parameters we will use below are (i) the
sphaleron density dNsp/dy = 200 in central AuAu; (ii)
Pqq= 1; (iii) the lifetime of the process τsp= 1.5fm/c;
Further evolution of a ¯ q − string − q (q − string −
qq) systems is done following the Lund model. As they
are produced earlier compared with the creation of pairs
in the usual fragmentation, those have relatively small
4.Fragmentation in the Lund model (e.g.
PYTHIA [13,14] event generator) describe a gluon jet
as a pair of strings that stretches from it to the forward-
backward-moving quarks . The usual treatment only
fulfills the Lund Area Law on average which is inade-
quate for small masses dominated by the few-body de-
cays. Fragmentation of the low energy quark jets has
been studied only recently  in this framework, pass-
ing the tests provided by the BES collaboration .
Unfortunately, the corresponding fragmentation of the
low mass gluon strings have not yet been studied. Since
the production of relevant particles at RHIC is dominated
by gluon jets, we have to address the issue. Additionally,
we found that the issue of the shape of the gluon string is
very important. It in turn is related to string-string at-
traction, first introduced by Montvay . If the tension
of the double string is less than twice that of a single one,
σg/(2σq) ≡ r < 2, the minimal energy configuration of a
qg¯ q jet system have shapes shown in Fig.1. The value of
r is not yet fixed from data. T. Sj¨ ostrand  concluded
only that the this ratio should be r > 1.5 in order to
describe the 3-jet data, we use a value r = 1.8 .
The junction moves according to the forces produced
by strings along the direction of the gluon, with the ve-
locity v = r/2 ≈ 0.9, and the whole fragmentation is
simplified in its rest frame. In this frame, if no breaking
occurs, the partons will oscillate in the direction of the
strings in a yo-yo motion. The turning time of the gluon
(in this frame) is tc= Egj/σg where Egj is the energy
in the junction rest frame.
The string configuration just described can interact
with quarks located at sphaleron spheres.
can interact with several sphalerons, we assume that the
fragmentation is determined by the one that cuts the q-
strings closest to the junction, in its rest frame. (As the
If a string
gluonic piece of a string is very short and extended out-
wards, the cutting happens predominantly in the quark
part as shown in Fig.1 by the horizontal dashed lines.)
Once the pick up happens, one is left with a system
with low invariant mass M. For example, hadrons of
pt= 3 − 4 GeV/c come from jet subsystems with M ∼
3GeV , which is precisely where few-body fragmentation
starts to be important. One may work out the complete
exclusive fragmentation distribution for the low energy
gluons, which we have not yet done.
At this energy, quark jet fragmentation is dominated
by three and four body decays . However, if in the
interaction with the sphaleron the string picks up a di-
quark, a two body decay or a single string breaking shown
in Fig.1 would be enough. So, whenever a baryon is pro-
duced, the fragmentation is described by a combination
of a two body decay plus the standard fragmentation de-
scribed by standard fragmentation functions. In the for-
mer case, the energy-momentum conservation together
with the linear potential of the string determine uniquely
the four momentum of the two particles produced. The
kinematics alone ensure that the effect disappears with
the increasing ptof the gluon.
In our simulation, with r = 1.8, the 2-body fragmen-
tations die out around pt∼ 8Gev/c, which corresponds
to the length of the gluonic string of 1 fm in the junc-
tion rest frame: longer strings should break by the usual
mechanism . The typical time and position of pick
up is such that in the junction rest frame, the turning
time of the gluon defined above is smaller than the pick
up time. This means that the string works as a slingshot,
allowing the gluon to give all its energy to the baryon or
pion. By requiring that the invariant masses of the two
subsystems after the spontaneous breaking correspond to
a nucleon and a π, we obtain two possible four momenta
(corresponding to the two possible cuts). Surprisingly the
previous remark makes the hadron that absorbs the gluon
string to move in opposite direction to the initial one be-
cause in the moment of the pick up the gluon is moving
back-wards. This characteristic is maintained when we
translate to the original frame. The outcome is that we
obtain a very energetic particle (the one that does not
absorb the gluon string) with fraction of the three mo-
mentum of the gluon close to one. In fact, as the second
hadron moves in the opposite direction, this fraction can
be even slightly bigger than one.
FIG. 1. String configuration of a qg¯ q event. The pick up
from the sphaleron produces a q and a diquark. The two
permitted spontaneous breakings are shown
The two possible ways of breaking correspond to boost-
ing either the meson or the nucleon produced. The re-
quirement of producing physical masses is more easily
fulfilled when the π does not absorb the gluon string (as
it is expected, given its small mass). So, in many cases
only one fragmentation is possible. When both are pos-
sible, we have assumed that the two cuts happen with
equal probability. In principle this does not have to be
true and phase space and dynamical (area law suppres-
sion) considerations should be taken into account. We
have also assumed that only π´s or nucleons are pro-
duced. A more realistic model should also consider other
Summarizing this point: a gluon fragmentation func-
tion at moderate pt contains a part in which only two
hadrons are generated. Those are harder than the usual
ones because one of the particles takes almost all the mo-
mentum of the initial gluon. Due to string dynamics, it
can even happen that the three momentum of one of the
particles is even bigger than the momentum of the gluon.
5.The results follow from numerical simulation of
non-central AuAu collisions at mid-rapidity, in which we
produce quark and gluon jets according to standard nu-
clear shapes and the structure functions of the parton
model, reproducing pp data. All trajectories are traced,
some jets are quenched by sphalerons; from those that
escape some have their strings cut off, with remnants
fragmenting as described above.
The pion quenching factor is a combination of absorp-
tion due to quenching and enhancement due to modi-
fied fragmentation. The results shown in Fig 2 are in
agreement with the data from PHENIX  for pt =
3 − 7GeV/c.
pt > 8 GeV/c where we are only left with the strong
absorption produced by the sphalerons.
The baryon/meson ratio close to 1 is not trivial to ob-
tain: even though the only process we consider always
generates one nucleon and one π, the boosts that these
particles obtain are different. In general it is easier to
generate high energy π´s than protons. These π´s come
from the string piece that does not include the gluon
string; so it will be easier to generate a particle with
small mass from this cut. The other cut is suppressed
by kinematic reasons. So, if we fix the ptof the gluons
we will always obtain more π than protons. However,
we observe that when the nucleon is boosted, it carries a
bigger fraction of the momentum. Finally, when we con-
volute yields with the cross section we obtain the ratio
shown in Fig.3. We found that the value decreases with
ptas it is expected, although slowly.
We have set the probability of picking up diquarks
Pqq = 1. As the mechanism of production of particles
links the nucleon and π production, the ratio of those
particles coming from this mechanism will be indepen-
dent of Pqq. Pqq determines, however, the strength of
the effect. If we reduce it, the pick up of quarks (that
we have not included in the version reported here, for
simplicity) would start to play a role.
The pick up mechanism disappears for
3 3.54 4.55 5.566.57
0 200 GeV
3 3.54 4.55 5.56 6.57
p/ 200 GeV
FIG. 2. (a) Quenching factor for pions and (b) p/π ra-
tio versus the transverse momentum at impact parameter
b = 6fm. Experimental values are PHENIX results in the
10-20 % cetrality bin,√s = 200GeV/c, from (a)  and (b)
As already explained, the production of only two π
would not be dominant because the invariant mass is too
big. Other channels could be important, for example the
production of ρ instead of π. The introduction of ρ does
not lead directly to the reduction of the p/π ratio. We
have assumed in the simulation that all strings produce
a π, but this is not necessarily true. So by introducing
any other channel we will reduce the number of π even
though all of them will generate high energy nucleons.
Azimuthal asymmetry follows from underlying geome-
try in a non-trivial way. The absorption reduces the con-
tribution of jets produced in the center of the almond,
pushing the production of particles that escape toward
the surface halo of the nuclei (see Fig.3(b)).This is pre-
cisely the problem of all absorptive models.
However, the density of sphalerons is smaller in the
halo and the (di)quark pick up (which boosts the frag-
mentation) is more effective for more central jets. The
outcome of both processes is that the emission is dom-
inated by a relatively thin layer, see Fig.3(a). As it is
shaped approximately as the surface of the overlap re-
gion of two hard spheres, the model produces values of
v2close to the geometric limit.
-8 -6 -4-202468
Nent = 9045
Mean x = -0.01322
Mean y = -0.08997
RMS x = 2.708
RMS y = 3.408RMS y = 3.408
x y distribution for pi
Nent = 9045
Mean x = -0.01322
Mean y = -0.08997
RMS x = 2.708
Nent = 9045
Mean x = -0.06487
Mean y = 0.003184
RMS x = 3.105RMS x = 3.105
RMS y = 3.674 RMS y = 3.674
x y distribution no cut
Nent = 9045
Mean x = -0.06487
Mean y = 0.003184
FIG. 3. Transverse plane distribution of the jets that es-
cape the nuclei (b = 6fm along the x axis) from pure absorp-
tion (right) and after the pick up (left). We can observe the
smaller contribution of the halo.
Fig 3(b) shows the distribution of jets in the transverse
plane (at b = 6fm) that escape the absorption. The big
contribution from the halo reduces the value of v2 to
0.049 ± 0.001 . Fig 3(a) shows the same distribution for
π´s when the pick up is considered. The contribution of
the halo is reduced, yielding a v2at b = 6fm of 0.076±
0.004 and 0.080 ± 0.004 for pions and nucleons. These
values are found to be approximately independent of pt
The experimental values at the same impact parameter
reported by STAR  are 0.12±0.02 and 0.13±0.01,(all
charged) for√s = 130 and 200AGeV/c.
Although after the pick up our results are lower than
the experimental values, they are still larger than the re-
sults of pure absorption (in spite of having a quenching
factor ten times bigger). What is important, with the
pick-up mechanism we have been able to reduce the con-
tribution of the halo to the final production. Note also,
that the nucleon asymmetry is somewhat larger than the
pion one, although not much.
6.Conclusions. We have presented a model in which
the medium interacts not only with the partons produced
in the collision, but also with the strings (color fields)
that the partons stretch. This interaction modifies the
whole process of fragmentation and leads to fragmenta-
tion functions that are harder than the vacuum ones.
We have presented a very simplified model in which we
have assumed that in the relevant energy the fragmen-
tation process is dominated by two body decay, and we
have given an explanation for large p/π ratio and small
quenching factor. We have also significantly improved
the value of v2 from pure absorption models. Further
work is needed to tune the parameters better and make
the model more realistic, hopefully bringing it in even
better agreement with the experimental data.
This work is partly supported by the US DOE grant.
We thank B.Kopeliovich, C.M.Ko, R.J.Fries and T.
Sj¨ ostrand for helpful comments.
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ies obtained for the double qq string tension has been
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th/0106185; hep-th/0111090. It is not however the same
as the gluonic string.
 The f2(2010) meson is a candidate for the 2++glue-ball.
It could be represented as a gluonic string spinning with
angular momentum 2. Its width is ∼ 200MeV , which
means that the corresponding gluonic string lives around
1 fm/c before breaking. It allows us to estimate the
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