# The parton bubble model compared to central Au Au collisions (0% to 5%) at
$\sqrt{s_{NN}}$=200 GeV.

R.S. Longacre  Brookhaven National Laboratory, Upton, New York 11973

###### Abstract

In an earlier paper we developed a Parton Bubble Model (PBM) for RHIC, high-
energy heavy-ion collisions. PBM was based on a substructure of a ring of
localized bubbles (gluonic hot spots) which initially contain 3-4 partons
composed of almost entirely gluons. The bubble ring was perpendicular to the
collider beam direction, centered on the beam, at midrapidity, and located on
the expanding fireball surface of Au Au central collisions (0-10%) at
$\sqrt{s_{NN}}$=200 GeV. The bubbles emitted correlated particles at kinetic
freezeout, leading to a lumpy fireball surface. For a selection of charged
particles (0.8 GeV/c $<$ $p_{t}$ $<$ 4.0 GeV/c), the PBM reasonably
quantitatively (within a few percent) explained high precision RHIC
experimental correlation analyses in a manner which was consistent with the
small observed HBT source size in this transverse momentum range. We
demonstrated that surface emission from a distributed set of surface sources
(as in the PBM) was necessary to obtain this consistency. In this paper we
give a review of the above comparison to central Au Au collisions. The bubble
formation can be associated with gluonic objects predicted by a Glasma Flux
Tube Model (GFTM) that formed longitudinal flux tubes in the transverse plane
of two colliding sheets of Color Glass Condensate (CGC), which pass through
one another. These sheets create boost invariant flux tubes of longitudinal
color electric and magnetic fields. A blast wave gives the tubes near the
surface transverse flow in the same way it gave transverse flow to the bubbles
in the PBM. In this paper we also consider the equivalent characteristics of
the PBM and GFTM and connect the two models. In the GFTM the longitudinal
color electric and magnetic fields have a non-zero topological charge density
$F\widetilde{F}$. These fields cause a local strong CP violation which effects
charged particle production coming from quarks and anti-quarks created in the
tube or bubble.

###### pacs:

25.75.Nq, 11.30.Er, 25.75.Gz, 12.38.Mh

##  I Parton bubble model development

Our interest in and the eventual development of the PBM goes back many years
to the early nineteen eighties. Van Hove’s workVanHove  in the early eighties
considered the bubbles as small droplets of quark-gluon plasma (QGP) assumed
to be produced in ultra-relativistic collisions of hadrons and/or nuclei. His
work recognized that experimental detectors only see what is emitted in the
final state at kinetic freezeout. Therefore final state predictions of his
string model work on plasma bubbles or any other theory required specific
convincing predictions that are observable in the final state which can be
measured by experiment. His work predicted that the final state rapidity
distribution dn/dy of hadrons would exhibit isolated maxima of width $\Delta
y\sim 1$ (single bubble) or rapidity bumps of a few units (due to those cases
producing a few bubbles). These rapidity regions would also contain
traditional signals of QGP formation. Even this early work by Van Hove needed
his bubbles near the surface of the final state fireball in order to
experimentally measure the particles emitted in the final state at kinetic
freezeout. We and other searched for these Van Hove bubbles, but no one ever
found any significant evidence for them.

Our next paper2000  was our final attempt to make a theoretical treatment of
the single bubble case (similar to the Van Hove case) which could be
experimentally verified. It is possible that with enough statistics one in
principle could find single to a few bubble events. We developed a number of
event generators which could possibly provide evidence for striking signals
resulting from these bubbles. One should note that in that paper we had
included in Sec. 11 mathematical expressions describing how charged pions are
effected by strong color electric and color magnetic fields which are present
and parallel in a CP-odd bubble of metastable vacuum. This section was
motivated by the work of Kharzeev and PisarskiPisarski . For example eq. 3-4
of the section show the boosts of the $\pi^{+}$ and the $\pi^{-}$ we
estimated. Thus in 2000 we were already interested in and were investigating
and publishing predictions for this phenomenon.

When we considered the RHIC quantum interference data in 2000Adler  it became
clear that the fireball surface was rapidly moving outward. This implied a
very large phase space region covered by the fireball, and also implied that
it would be very unlikely for there to be only one isolated bubble (small
source size) with a large amount of energy sitting on the surface. It seemed
more likely that there would be many bubbles or gluonic hot spots around the
expanding surface. This led to a paperthemodel  which was our earlier version
of Ref.PBM . In that paper we concluded that the behavior of the Hanbury-Brown
and Twiss (HBT) measurementsAdler ; HBT  should be interpreted as evidence of
a substructure of bubbles located on the surface of the final state fireball
in the central rapidity region at kinetic freezeout. The HBT radii were
decreasing almost linearly with transverse momentum and implying to us a
source size of $\sim$2 fm radii bubbles were on the surface and could be
selected for if we considered transverse momenta above 0.8 GeV/c. These
momenta would allow sufficient resolution to resolve individual bubbles of
$\sim$2 fm radii. We further concluded that these HBT quantum interference
observations were likely due to phase space focusing of the bubbles pushed by
the expanding fireball. Thus the HBT measurements of source size extrapolated
to transverse momenta above 0.8 GeV/c were images of the bubbles. The HBT
correlation has the property of focusing these images on top of each other for
a ring of bubbles transverse to and centered on the beam forming an average
HBT radius.

Thus our model was a ring of bubbles sitting on the freezeout surface with
average size $\sim$2 fm radius perpendicular to the beam at mid-rapidity. The
phase space focusing would also lead to angular correlations between particles
emitted by the bubbles and be observable. Fig. 1 shows the geometry of the
bubbles. For the background particles that account for particles in addition
to the bubble particles we used unquenched HIJINGhijing  with its jets
removed. We assumed that most of the jets were eliminated from the central
events because of the strong jet quenching observed at RHICquench1 . We
investigated the feasibility of using charged-particle-pair correlations as a
function of angles which should be observable due to the phase space focusing
of the particles coming from individual bubbles. At that point we fully
expected that these correlations would be observable in the STAR detectorstar
at RHIC if one analyzed central Au Au collisions. Correlation analyses are
powerful tools in detecting substructures. Historically substructures have
played an important role in advancing our understanding of strong (non
perturbative) interactions. Our earlier paper explained the general
characteristics of the angular correlation data, and was also consistent with
HBT measurements in a qualitative manner. This motivated us to develop a
reasonably quantitative model, the parton bubble modelPBM  which is discussed
in the following.

Figure 1: The bubble geometry is an 8 fm radius ring perpendicular to and
centered on the beam axis. It is composed of twelve adjacent 2 fm radius
spherical bubbles elongated along the beam direction by the Landau
longitudinal expansion. The upper left figure is a projection on a plane
section perpendicular to the beam axis. The lower left figure is a projection
of the bubble geometry on a plane containing the beam axis. The lower right
figure is a perspective view of the bubble geometry. Due to rotational
invariance about the beam axis the only direction that is meaningful to define
is the beam axis shown in the lower 2 figures.

##  II Parton bubble modelPBM

In this publicationPBM  we developed a QCD inspired parton bubble model (PBM)
for central (impact parameter near zero) high energy heavy ion collisions at
RHIC. The PBM is based on a substructure consisting of a single ring of a
dozen adjoining 2-fm-radius bubbles (gluonic hot spots) transverse to the
collider beam direction, centered on the beam, and located at or near mid-
rapidity. The at or near mid-rapidity refers to the boost invariant region
that exist in the RHIC heavy ion collisions which spans 2-4 units of rapidity.
Because of this spread in $\eta$, we will use the relative spread of particles
in $\eta$ ($\Delta\eta$) in order to capture the angular correlations along
the beam axis. The ring resides on the fireball blast wave surface (see Fig.
1). We assumed these bubbles are likely the final state result of quark-gluon-
plasma (QGP) formation since the energy densities produced experimentally are
greater than those estimated as necessary for formation of a quark-gluon-
plasma. Thus this is the geometry for the final state kinetic freezeout of the
QGP bubbles on the surface of the expanding fireball treated in a blast wave
model.

The average behavior of emitted final state particles coming from the surface
bubbles at kinetic freezeout is given by the ring of twelve bubbles formed by
energy density fluctuations near the surface of the expanding fireball of the
blast wave. One should note that the blast wave surface is moving at its
maximum velocity at freezeout (3c/4). For central events each of the twelve
bubbles have 3-4 partons per bubble each at a fixed $\phi$ for a given bubble.
This number of partons was determined from correlation data analyzed by the
STAR experimentPBM . The transverse momentum ($p_{t}$) distribution of the
charged particles is similar to pQCD but has a suppression at high $p_{t}$
like the data.

The bubble ring radius of our model was estimated by blast wave, HBT and other
general considerations to be approximately 8 fm. The bubbles emit correlated
charged particles at final-state kinetic freezeout where we select a $p_{t}$
range (0.8 GeV/c $<$ $p_{t}$ $<$ 4.0 GeV/c) in order to increase signal to
background. The 0.8 GeV/c $p_{t}$ cut increases the resolution to allow
resolving individual bubbles which have a radius of $\sim$2 fm. This space
momentum correlation of the blast wave provides us with a strong angular
correlation signal. PYTHIA fragmentation functionspythia  were used for the
bubbles fragmentation that generate the final state particles emitted from the
bubbles. A single parton using PYTHIA forms a jet with the parton having a
fixed $\eta$ and $\phi$ (see Fig. 2). The 3-4 partons in the bubble which
shower using PYTHIA all have a different $\eta$ value but all have the same
$\phi$ (see Fig. 3). The PBM explained the high precision Au Au central
(0-10%) collisions at $\sqrt{s_{NN}}=$ 200 GeVcentralproduction  (the highest
RHIC energy).

Figure 2: A jet parton shower.

Figure 3: Each bubble contains 3-4 partons as shown.

###  II.1 The Correlation Function

We utilize a two particle correlation function in the two dimensional (2-D)
space of $\Delta\phi$ vs $\Delta\eta$. The azimuthal angle $\phi$ of a
particle is defined by the angle of the particle with respect to the vertical
axis which is perpendicular to the beam axis and is measured in a clock-wise
direction about the beam. $\Delta\phi$ is the difference, $\phi_{1}$ \-
$\phi_{2}$, of the $\phi$ angle of a pair of particles (1 and 2). The pseudo-
rapidity $\eta$ of a particle is measured along one of the beam directions.
$\Delta\eta$ is the difference, $\eta_{1}$ \- $\eta_{2}$, of the $\eta$ values
of a pair of particles (1 and 2).

The two dimensional (2-D) total correlation function is defined as:

$C(\Delta\phi,\Delta\eta)=S(\Delta\phi,\Delta\eta)/M(\Delta\phi,\Delta\eta).$
(1)

Where S($\Delta\phi,\Delta\eta$) is the number of pairs at the corresponding
values of $\Delta\phi,\Delta\eta$ coming from the same event, after we have
summed over all the events. M($\Delta\phi,\Delta\eta$) is the number of pairs
at the corresponding values of $\Delta\phi,\Delta\eta$ coming from the mixed
events, after we have summed over all our created mixed events. A mixed event
pair has each of the two particles chosen from a different event. We make on
the order of ten times the number of mixed events as real events. We rescale
the number of pairs in the mixed events to be equal to the number of pairs in
the real events. This procedure implies a binning in order to deal with finite
statistics. To enhance comparison we use the same binning in our simulations
as is used by the STAR high precision experimental analysescentralproduction .
The division by M($\Delta\phi,\Delta\eta$) for experimental data essentially
removes or drastically reduces acceptance and instrumental effects. If the
mixed pair distribution was the same as the real pair distribution
C($\Delta\phi,\Delta\eta$) would be one for all values of the binned
$\Delta\phi,\Delta\eta$. In the correlations used in this paper we select
particles independent of its charge. The correlation of this type is called a
Charge Independent (CI) Correlation. This difference correlation function has
the defined property that it only depends on the differences of the azimuthal
angle ($\Delta\phi$) and the beam angle ($\Delta\eta$) for the two particle
pair. Thus the two dimensional difference correlation distribution for each
bubble which is part of C($\Delta\phi,\Delta\eta$) is similar for each of our
12 bubbles and will image on top of each other. The PBM fit to the angular
correlation data was reasonably quantitative to within a few percent (see Fig.
4 and Fig. 5)).

Figure 4: The CI (sum of all charged-particle-pairs) correlation for the 0-10%
centrality bin with a charged particle $p_{t}$ selection (0.8 $<$ $p_{t}$ $<$
4.0 GeV/c), generated by the PBM. It is plotted as a two dimensional
$\Delta\phi$ vs. $\Delta\eta$ perspective plot.

Figure 5: In each of the five labeled $\Delta\eta$ bins we show the
$\Delta\phi$ total correlation for the CI as a function of $\Delta\phi$. The
STAR Au + Au central trigger analysis results from the formulae of
Ref.centralproduction  are shown as a solid line. The parton bubble model
predictions are shown by the circular points(o) which are large enough to
include the statistical errors from a 2 million event sample. The vertical
correlation scale is not offset and is correct for the largest $\Delta\eta$
bin, 1.2 $<\Delta\eta<$ 1.5, and is the lowest bin on the figure. As one
proceeds upwards to the next bin $\Delta\eta$ the correlation is offset by
+0.01. This is added to the correlation of each subsequent $\Delta\eta$ bin.
The smallest $\Delta\eta$ and top of Figure 12 has a +0.04 offset. A solid
straight horizontal line shows the offset for each $\Delta\eta$ bin. Each
solid straight horizontal line is at 1.0 in correlation strength. The 0-10%
centrality in HIJING corresponds to impact parameter (b) range 0.0 to 4.0 fm.
The agreement is very good.

The correlation functions we employed (like the HBT correlation functions)
have the property that for the difference in angles (difference of momentum
for HBT) these correlations will image all 12 bubbles on top of each other.
This leads to average observed angles of approximately $30^{\circ}$ in
$\Delta\phi$, $70^{\circ}$ in $\Delta\eta$, originating from a source size of
about $\sim$2 fm radius which is consistent with the HBT correlation. Thus the
PBM generates $\Delta\phi$ vs $\Delta\eta$ charged-particle-pair correlations
for charged particles with $p_{t}$ in the range 0.8 Gev/c to 4.0 GeV/c as
displayed in Fig. 4 and Fig. 5. We show the CI correlation in Fig. 5 a
reasonably quantitative successful comparison with data. It should be noted
that the spread in $\Delta\eta$ of the bubble is determined by the
longitudinal momentum distribution of the 3-4 partons that make up the bubble.
This distribution which is quassian in nature was adjusted to the observed
$\Delta\eta$ width of correlation data analyzed by the STAR experimentPBM .
Furthermore the model results were consistent with the Hanbury-Brown and Twiss
(HBT) observationsHBT  that the observed source radii determined by quantum
statistical interference were reducing by a considerable factor with
increasing transverse momentum ($p_{t}$). The HBT radii are interpreted to
reduce from $\sim$6 fm at $p_{t}$ $\sim$0.2 GeV/c to $\sim$2 fm at $p_{t}$
$\sim$1 GeV/c for our $p_{t}$ range. The generally accepted explanation for
this behavior is that as $p_{t}$ increases radial flow increasingly focused
the viewed region of the final state into smaller volumes. If just one small
HBT size bubble were emitting all the correlated particles, this phenomena
would lead to large spikes of particles emitted at one limited $\phi$ angular
region in individual events. This is definitely not observed in the Au Au or
other collision data at RHIC. Therefore, a distributed ring of small sources
around the beam as assumed in the PBM is necessary to explain both the HBT
resultsHBT  and the correlation datacentralproduction . The particles emitted
from the same bubble are virtually uncorrelated to particles emitted from any
other bubble except for momentum conservation requirements. An away side peak
in the total correlation is built up from momentum conservation between the
bubbles.

##  III The PBM for other centralities(PBMEPBME )

The PBM was also recently extended to PBMEPBME  which is identical to the PBM
for central collisions (0-5%). For centralities running from 30-80% jet
quenching is not strong enough to make jets negligible therefore, a jet
component was added which was based on HIJING calculations. This jet component
accounts for more of the correlation as one moves toward peripheral bins, and
explains all of it for the most peripheral collisions. The PBME explained in a
reasonably quantitative manner (within a few percent) the behavior of the
recent quantitative experimental analysis of charge pair correlations as a
function of centralitycentralitydependence . This further strengthened the
substantial evidence for bubble substructure. The agreement of the PBM and the
PBME surface emission models with experimental analyses strongly implied that
at kinetic freezeout the fireball was dense and opaque in the central region
and most centralities (except the peripheral region) in the intermediate
transverse momentum region (0.8 $<$ $p_{t}$ $<$ 4.0 GeV/c). Thus we conclude
that the observed correlated particles are both formed and emitted from or
near the surface of the fireball. In the peripheral region the path to the
surface is always small.

The CI correlation used in Fig. 4 and the PBMPBM  is also defined as the sum
of two charged-particle-pair correlations. They are the unlike-sign charge
pairs (US) and the like-sign charge pairs (LS). This Charge Independent
correlation (CI) is defined as the correlation made up of charged-particle-
pairs independent of what the sign is, which is the average of the US plus the
LS correlations. Thus CI = (US + LS)/2. Therefore the CI is the total charge
pair correlation observed in the experimental detection system within its
acceptance.

The CI was generated for $\sqrt{s_{NN}}$ = 200 GeV central (0-5%) Au Au
collisions in the $p_{t}$ range 0.8 GeV/c to 4.0 GeV/c at RHIC. When the CI
was compared to the corresponding experimental analysescentralproduction ;
centralitydependence  a reasonably quantitative agreement within a few percent
of the observed CI was attained. See Ref.PBM  Sec. 4.2, Ref.PBME  Sec. V, and
Ref.centralitydependence  Sec. VI B111The STAR collaboration preferred to use
the conventional definition of CI = US + LS which is larger by a factor of 2
than the CI definition used in this present paper that is directly physically
meaningful.. This analysis and our previous workPBM ; PBME  was done without
using a jet trigger. In fact jet production in the central region was
negligible due to strong jet quenchingquench1 ; quench2 ; quench3 . As one can
see in Fig. 4 the central CI correlation of $\Delta\phi$ on the near side
($\Delta\phi$ $<$ $90^{\circ}$), is sharp and approximately jetlike at all
$\Delta\eta$. In contrast the $\Delta\eta$ dependence is relatively flat. In
the central production the average observed angles are approximately
$30^{\circ}$ in $\Delta\phi$ and $70^{\circ}$ in $\Delta\eta$. It is of
interest to note that the sharp jetlike collimation in $\Delta\phi$ persists
at all centralities (0-80%). The elongation in $\Delta\eta$ persists in the
0-30% centrality range and then decreases with decreasing centrality becoming
jetlike in the peripheral bins. These characteristics are in reasonable
quantitative agreement with the fits of the PBM and the PBME to experimental
data analysesPBM ; centralproduction ; PBME ; centralitydependence .

The difference of the US and LS correlations is defined as the Charge
Dependent (CD) correlation (CD = US - LS). The 2-D experimental CD correlation
for centralities (0-80%) has a jetlike shape which is consistent with PYTHIA
jets (vacuum fragmentation) for all centralities. This clearly implies that
both particle hadronization and emission occur from the fireball surface
region as explained in Ref.PBME  Sec. II.

##  IV Glasma flux tube model

A glasma flux tube model (GFTM)Dumitru  that had been developed considers that
the wavefunctions of the incoming projectiles, form sheets of color glass
condensates (CGC)CGC  that at high energies collide, interact, and evolve into
high intensity color electric and magnetic fields. This collection of
primordial fields is the GlasmaLappi ; Gelis , and initially it is composed of
only rapidity independent longitudinal color electric and magnetic fields. An
essential feature of the Glasma is that the fields are localized in the
transverse space of the collision zone with a size of 1/$Q_{s}$. $Q_{s}$ is
the saturation momentum of partons in the nuclear wavefunction. These
longitudinal color electric and magnetic fields generate topological Chern-
Simons chargeSimons  which becomes a source for particle production.

The transverse space is filled with flux tubes of large longitudinal extent
but small transverse size $\sim$$Q^{-1}_{s}$. Particle production from a flux
tube is a Poisson process, since the flux tube is a coherent state. As the
partons emitted from these flux tubes locally equilibrate, transverse flow
builds due to the radial flow of the blast waveGavin . The flux tubes that are
near the surface of the fireball get the largest radial flow and are emitted
from the surface. As in the parton bubble model these partons shower and the
higher $p_{t}$ particles escape the surface and do not interact. These flux
tubes are in the present paper considered strongly connected to the bubbles of
the PBM. The method used to connect the PBM and the GFTM is described and
discussed in the next section V.

$Q_{s}$ is around 1 GeV/c thus the transverse size of the flux tube is about
1/4 fm. The flux tubes near the surface are initially at a radius $\sim$5 fm.
The $\phi$ angle wedge of the flux tube is $\sim$1/20 radians or
$\sim$$3^{\circ}$. Thus the flux tube initially has a narrow range in $\phi$.
The large width in the $\Delta\eta$ correlation which in the PBM depended on
the large spread in $\Delta\eta$ of the bubble partons results from the
independent longitudinal color electric and magnetic fields that created the
Glasma flux tubes. How much of these longitudinal color electric and magnetic
fields are still present in the surface flux tubes when they have been pushed
by the blast wave will be a speculation of this paper for measuring strong CP
violation?

It has been noted that significant features of the PBM that generated final
state correlations which fit the experimentally observed correlation dataPBM ;
centralproduction ; PBME ; centralitydependence  are similar to those
predicted by the GFTMDumitru . Therefore in the immediately following section
we assume a direct connection of the PBM and the GTM, give reasons to justify
it, and then discuss its consequences, predictions and successes.

##  V The connection of the PBM and the GFTM

The successes of the PBM have strongly implied that the final state surface
region bubbles of the PBM represent a significant substructure. In this
subsection we show that the characteristics of our PBM originally developed to
fit the precision STAR Au Au correlation data in a manner consistent with the
HBT data; implies that the bubble substructure we originally used to fit these
previous data is closely related to the GFTM.

A natural way to connect the PBM bubbles to the GFTM flux tubes is to assume
that the final state at kinetic freezeout of a flux tube is a PBM final state
bubble. Thus the initial transverse size of a flux tube $\sim$1/4 fm has
expanded to the size of $\sim$2 fm at kinetic freezeout. With this assumption
we find consistency with the theoretical expectations of the GFTM. We can
generate and explain the triggered ridge phenomenon and data (see Sec. VI),
thus implying the ridge is connected to the bubble substructure. We can
predict and obtain very strong evidence for the color electric field of the
glasma from comparing our multi-particle charged particle correlation
predictions, and existing experimental correlation publications (see Sec.
VII). We have also predicted correlations which can be used to search for
evidence for the glasma color magnetic field (Sec. VII).

Of course an obvious question that arises is that since a flux tube is an
isolated system does a PBM bubble, that we assume is the final state of a flux
tube at kinetic freezeout, also act as an isolated system when emitting the
final state particles? The near side correlations signals since the original
PBMPBM  have always come virtually entirely from particles emitted from the
same bubble. Thus each bubble has always acted as an isolated system similar
to the behavior of a flux tube.

##  VI The Ridge is formed by the bubbles when a jet trigger is added to the
PBM

In heavy ion collisions at RHIC there has been observed a phenomenon called
the ridge which has many different explanationsDumitru ; Armesto ; Romatschke
; Shuryak ; Nara ; Pantuev ; Mizukawa ; Wong ; Hwa . The ridge is a long range
charged particle correlation in $\Delta\eta$ (very flat), while the
$\Delta\phi$ correlation is approximately jet-like (a narrow Gaussian). There
also appears with the ridge a jet-like charged-particle-pair correlation which
is symmetric in $\Delta\eta$ and $\Delta\phi$ such that the peak on the jet-
like correlation is at $\Delta\eta$ = 0 and $\Delta\phi$ = 0. The $\Delta\phi$
correlation of the jet and the ridge are approximately the same and smoothly
blend into each other. The ridge correlation is generated when one triggers on
an intermediate $p_{t}$ range charged particle and then forms pairs between
that trigger particle and each of all other intermediate charged particles
with a smaller $p_{t}$ down to some lower limit. The first case we will study
in this paper is a trigger charged particle between 3.0 to 4.0 GeV/c
correlated with all other charged particles which have a $p_{t}$ between 1.1
GeV/c to 3.0 GeV/c.

In this paper we will investigate whether the PBM can account for the ridge
once we add a jet trigger to our PBM generatorPBM . However this trigger will
also select jets which previously could be neglected because there was such
strong quenchingquench1 ; quench2 ; quench3  of jets in central collisions. A
jet trigger had not been used in the PBM comparison to all previous data. We
use HIJINGhijing  merely to determine the expected number of jets to add for
our added jet trigger. These jet particles were added to our PBM generator.
Thus our PBM generator now had HIJING generated background particles, bubbles
of the PBM and added jet particles from HIJING. We have already shown that our
final state particles come from hadrons at or near the fireball surface. We
reduce the number of jets by 80% which corresponds to the estimate that only
the parton interactions on or near the surface are not quenched away, and thus
at kinetic freezeout form and emit hadrons which enter the detector. This 80%
reduction is consistent with single $\pi^{0}$ suppression observed in
Ref.quench3 . We find for the reduced HIJING jets that 4% of the Au Au central
events (0-5%) centrality at $\sqrt{s_{NN}}=$ 200 have a charged particle with
a $p_{t}$ between 3.0 and 4.0 GeV/c with at least one other charged particle
with its $p_{t}$ greater than 1.1 GeV/c coming from the same jet. The addition
of the jets to the PBM generator provides the appropriate particles which are
picked up by the trigger in order to form a narrow $\Delta\eta$ correlation
signal at 0 which is also a narrow signal in $\Delta\phi$ at 0 (Fig. 14). This
narrow jet signal is present in the data and is what remains of jets after 80%
are quenched away.

We then form two-charged-particle correlations between one-charged-particle
with a $p_{t}$ between 3.0 to 4.0 GeV/c and another charged particle whose
$p_{t}$ is greater than 1.1 GeV/c. The results of these correlations are shown
in Fig. 6. Fig. 6 is the CI correlation for the 0-5% centrality bin with the
just above stated $p_{t}$ selections on charge pairs. Since we know in our
Monte Carlo which particles are emitted from bubbles and thus form the ridge,
we can predict the shape of the ridge for the above $p_{t}$ cut by plotting
only the correlation formed from pairs of particles that are emitted by the
same bubble (see Fig. 7). The charge pair correlations are virtually all
emitted from the same bubble. Those charge pair correlations formed from
particles originating from different bubbles are small contributors which
mainly have some effect on the away side correlation. Thus Fig. 7 is the ridge
signal which is the piece of the CI correlation for the 0-5% centrality of
Fig. 6, after removing all other pairs except the pairs emitted from the same
bubble.

Figure 6: The CI correlation for the 0-5% centrality bin that results from
requiring one trigger particle $p_{t}$ above 3 GeV/c and another particle
$p_{t}$ above 1.1 GeV/c. It is plotted as a two dimensional $\Delta\phi$ vs.
$\Delta\eta$ perspective plot.

Figure 7: The ridge signal is the piece of the CI correlation for the 0-5%
centrality of Fig. 6 after removing all other particle pairs except the pairs
that come from the same bubble. It is plotted as a two dimensional
$\Delta\phi$ vs. $\Delta\eta$ perspective plot. As explained in the text
charge pair correlations formed by particles emitted from different bubbles
are small contributors which mainly have some effect on the away side
correlation for $\Delta\phi$ greater than about $120^{\circ}$.

###  VI.1 Parton $p_{t}$ correlated vs. random

A very important aspect of the GFTM is the boost that the flux tubes get from
the radial flow of the blast wave. This boost is the same all along the flux
tube and only depends on how far away from the center axis of the blast wave
the flux tube is. In the PBM the partons in the bubble received a boost in
$p_{t}$ from the radial flow field in the blast wave; that after final state
fragmentation of the bubbles gave the generated particles a consistent $p_{t}$
spectrum with the data. Since the boost from the blast wave depended on the
position of the bubble in the radial flow field, there should be a correlation
between partons $p_{t}$ within a bubble. If one redistributed the partons with
their $p_{t}$ boosts uniformly among the bubbles, the over all results for the
generated correlations without a trigger would be unchanged. One should note
none of our prior work (i.e.PBMPBM  or PBMEPBME ) contained a trigger. We have
only added a trigger here to treat the “triggered ridge” which obviously
requires it. In our treatment of the GFTM (Sec. VII) we have removed the
trigger.

Once we require a trigger demanding higher $p_{t}$ particles, we start picking
out bubbles which have more radial flow (harder particles). Thus correlated
particles which pass our $p_{t}$ selection and trigger come almost entirely
from the harder bubbles with more radial flow, while the softer bubbles which
were subject to less radial flow mainly generate low enough $p_{t}$ particles
that become background particles or are lost to the correlation analysis due
to $p_{t}$ selection. Figure 6 shows the result of our trigger and $p_{t}$
selection. If one redistributed the partons with their $p_{t}$ radial flow
boost among all the bubbles, then all bubbles become equal and there is no
longer soft and hard bubbles. With this change we can generate bubble events.
We find that both the non-triggered correlation and the $\Delta\phi$
correlation are virtually unchanged in the region less than about
$120^{\circ}$ (see Fig. 8). It seems that we move from the situation where we
have a few bubbles with a lot of correlated particles above our $p_{t}$ cut to
a lot of bubbles that have few correlated particles above our cuts. However
there is a difference in the triggered correlation’s away side or $\Delta\phi$
near the $180^{\circ}$ peak.

In Ref.PBM  we discuss this away side effect and attributed this peak to
momentum conservation. This is however not the whole story. The away side peak
is generated by the geometry of the bubble ring which requires that for every
bubble there is another bubble nearly $180^{\circ}$ away that is emitting
particles. These symmetry requirements on the geometry contribute more to the
away side peak especially for the case without a trigger or redistributed
random partons.

This geometry effect is what causes the away side correlation of elliptic flow
in a peripheral heavy ion collision. In such a collision there is a higher
energy density aligned with the reaction plane. On one side of the reaction
plane there are more particles produced because of increased energy density,
and because of the geometry of the situation. On the other side of the
reaction plane there are also more particles produced for the same reasons.
Thus there is a correlation between particles that are near each other on one
side of the reaction plane and a correlation between particles that are on
both sides of the reaction plane. Momentum conservation requires particle
communication by bouncing into each other, while geometry has no such
communication. The right hand does not need to know what the left hand is
doing. However, if physics symmetry makes them do the same thing then they can
show a correlation.

###  VI.2 The away side peak

The away side peak depends on the fragmentation of the away side bubble. For
the triggered case where there is a hard bubble with a strong correlation of
parton $p_{t}$ inside the bubble (GFTM like), we will have more correlated
particles adding to the signal on the near side. On the other hand the away
side bubble will be on average a softer bubble which will have a lot less
particles passing the $p_{t}$ cuts and thus have a smaller signal. If we
consider the case of a random parton $p_{t}$ bubble the particles passing the
$p_{t}$ cuts should be very similar on the near and the away side. In Fig. 8
we compare the two correlations generated by the correlated and the random
cases. We compare the $\Delta\phi$ correlation for the two cases (solid is
correlated and open is random)for each of the five $\Delta\eta$ bins which
cover the entire $\Delta\eta$ range (0.0 to 1.5). The vertical correlation
scale is not offset and is correct for the largest $\Delta\eta$ bin, 1.2
$<\Delta\eta<$ 1.5, which is the lowest bin on the figure. As one proceeds
upwards to the next $\Delta\eta$ bin the correlation is offset by +0.05. This
is added to the correlation of each subsequent $\Delta\eta$ bin. The smallest
$\Delta\eta$ bin on top of Fig. 8 has a +0.2 offset. A solid straight
horizontal line shows the offset for each $\Delta\eta$ bin. Each solid
straight horizontal line is at 1.0 in correlation strength. We see that the
away side correlation in the $\Delta\phi$ region greater than about
$120^{\circ}$ is larger for the random case.

Figure 8: The $\Delta\phi$ CI correlation for the 0-5% centrality bin
requiring one trigger particle $p_{t}$ above 3 GeV/c and another particle
$p_{t}$ above 1.1 GeV/c. We compare the two correlations generated by the
correlated and the random cases. We compare the $\Delta\phi$ correlation for
the two cases (solid is correlated and open is random) for each of the five
$\Delta\eta$ bins which cover the entire $\Delta\eta$ range (0.0 to 1.5). The
vertical correlation scale is not offset and is correct for the largest
$\Delta\eta$ bin, 1.2 $<\Delta\eta<$ 1.5, which is the lowest bin on the
figure. As one proceeds upwards to the next $\Delta\eta$ bin the correlation
is offset by +0.05. This is added to the correlation of each subsequent
$\Delta\eta$ bin. The smallest $\Delta\eta$ bin on top of Fig. 8 has a +0.2
offset. A solid straight horizontal line shows the offset for each
$\Delta\eta$ bin. Each solid straight horizontal line is at 1.0 in correlation
strength. We see that the away side correlation for $\Delta\phi$ greater than
about $120^{\circ}$ is larger for the random case.

Figure 9: In this figure we display the $\phi$ and $\eta$ ranges considered
for producing ridge particles in the Au Au central collisions. A typical
trigger particle ($p_{t}$ between 3.0 to 4.0 GeV/c) implies two associated
regions in the considered $\phi$ and $\eta$ ranges where the ridge particles
lie. The ridge particles we consider are separated by 0.7 in $\eta$
($\Delta\eta$) so that one eliminates particles coming from jet production.
The bulk of the ridge particles lie within $20^{\circ}$ of the $\phi$ of the
trigger particle. The width of the $\phi$ spread is $\Delta\phi$ =
$40^{\circ}$.

Figure 10: On a trigger by trigger basis we accumulate particles from the
ridge region (see Fig. 9). We then form a ratio the of $p_{t}$ spectrum for
the ridge region particles divided by the $p_{t}$ spectrum for all particles
in central Au Au events. We normalize the two $p_{t}$ spectra to have the same
counts in the 2.0 GeV/c bin. This ratio or correlation for the bubbles which
have a correlated $p_{t}$ among the partons (GFTM like) are the solid points.
The random $p_{t}$ distribution among partons in the bubble are the open
points.

###  VI.3 Predicted transverse momentum dependence of correlation

Particles which come from bubbles that satisfy the trigger (3.0 $<$ $p_{t}$
$<$ 4.0 GeV/c) are harder and have a different $p_{t}$ distribution compared
to the case without trigger requirements. In a given triggered event if one
could select particles that come from the bubble or ridge, one would find that
the $p_{t}$ spectrum is harder than the average $p_{t}$ spectrum. In Fig. 9 we
show how one can by trigger choose particles that are rich with ridge
particles. We consider a typical trigger particle and the associated regions
in the available experimental $\phi$ and $\eta$ ranges. The ridge particles we
consider are separated from the trigger particle by 0.7 in $\eta$. This is
done to eliminate particles coming from jet production that could also be
associated with the trigger particle. The bulk of the ridge particles lie
within $20^{\circ}$ of the $\phi$ of the trigger particle. The width of the
$\phi$ spread is $\Delta\phi$ = $40^{\circ}$. Therefore on a trigger by
trigger basis we accumulate particles from the ridge region (see Fig. 9) and
form a $p_{t}$ spectrum of charged particles. We then form a ratio of the
$p_{t}$ spectrum for the ridge region particles divided by the $p_{t}$
spectrum for all particles in central Au Au events. This ratio is highly
dependent on our ridge cut area. We can virtually remove this dependency by
normalizing the two $p_{t}$ spectra to have the same counts for the 2.0 GeV/c
bin. Figure 10 is this ratio which can be considered a correlation of
particles as a function of $p_{t}$ in the ridge region compared to particles
from the event in general. In Fig. 10 we form this ratio or correlation for
the bubbles which have a correlated $p_{t}$ among the partons (GFTM like). We
also show the correlation for bubbles which have a random $p_{t}$ distribution
among partons in the bubble. For the random case the $p_{t}$ spectrum does not
differ much from the average $p_{t}$ spectrum while there is a big difference
for the flux tube like case. This correlation could be easily measured in the
RHIC data.

###  VI.4 Comparison to data

Triggered angular correlation data showing the ridge was presented at Quark
Matter 2006Putschke . Figure 11 shows the experimental $\Delta\phi$ vs.
$\Delta\eta$ CI correlation for the 0-10% central Au Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV; requiring one trigger particle $p_{t}$ between 3 and
4 GeV/c and an associated particle $p_{t}$ above 2.0 GeV/c. The yield is
corrected for the finite $\Delta\eta$ pair acceptance. For the PBM generator,
we then form a two-charged-particle correlation between one charged particle
with a $p_{t}$ between 3.0 to 4.0 GeV/c and another charged particle whose
$p_{t}$ is greater than 2.0 GeV/c. The results of this correlation is shown in
Fig. 12. Figure 11 shows the corrected pair yield determined in the central
data whereas Fig. 12 shows the correlation function generated by the PBM which
does not depend on the number of events analyzed. We can compare the two
figures, if we realize that the away side ridge has around 420,000 pairs in
Fig. 11 while in Fig. 12 the away side ridge has a correlation of around
0.995. If we multiply the correlation scale of Fig. 12 by 422,111 in order to
achieve the number of pairs seen in Fig. 11, the away side ridge would be at
420,000 and the peak would be at 465,000. This would make a good agreement
between the two figures.

The correlation formed by the ridge particles is generated almost entirely by
particles emitted by the same bubble. We have shown in all our publications
that the same side correlation signals are almost entirely formed by particles
coming from the same bubble. Thus we can predict the shape and the yield of
the ridge for the above $p_{t}$ trigger selection and lower cut, by plotting
only the correlation coming from pairs of particles that are emitted by the
same bubble (see Fig. 13).

In Ref.Putschke  it was assumed that the ridge yield was flat across the
acceptance while in Fig. 13 we see that this is not the case. Therefore our
ridge yield is approximately 35% larger than estimated in Ref.Putschke .
Finally we can plot the jet yield that we had put into our Monte Carlo. We
used HIJINGhijing  to determine the number of expected jets, and then reduced
the number of jets by 80%. This assumes that only the parton interactions on
or near the surface that form hadrons at kinetic freezeout are not quenched
away and thus enter the detector. This 80% reduction is consistent with single
$\pi^{0}$ suppression observed in Ref.quench3 . This jet yield is plotted in
Fig. 14 where we subtracted contributions from the bubbles and the background
particles from Fig. 11.

Figure 11: Raw $\Delta\phi$ vs. $\Delta\eta$ CI preliminary correlation
dataPutschke  for the 0-10% centrality bin for Au Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV requiring one trigger particle $p_{t}$ between 3 to 4
GeV/c and an associated particle $p_{t}$ above 2.0 GeV/c.

Figure 12: The PBM generated CI correlation for the 0-5% centrality bin
requiring one trigger particle $p_{t}$ above 3 GeV/c and less than 4 GeV/c and
another particle $p_{t}$ above 2.0 GeV/c plotted as a two dimensional
$\Delta\phi$ vs. $\Delta\eta$ perspective plot. The trigger requirements on
this figure are the same as those on the experimental data in Fig. 11.

Figure 13: The ridge signal is the piece of the CI correlation for the 0-5%
centrality of Fig. 12 after removing all other particle pairs except the pairs
that come from the same bubble. It is plotted as a two dimensional
$\Delta\phi$ vs. $\Delta\eta$ perspective plot.

Figure 14: The jet signal is left in the CI correlation after the
contributions from the background and all the bubble particles are removed
from the 0-5% centrality (with trigger requirements) of Fig. 12. It is plotted
as a two dimensional $\Delta\phi$ vs. $\Delta\eta$ perspective plot.

##  VII Strong CP violations or Chern-Simons topological charge

###  VII.1 The Source $F\widetilde{F}$.

The strong CP problem remains one of the most outstanding puzzles of the
Standard Model. Even though several possible solutions have been put forward
it is not clear why CP invariance is respected by the strong interaction. It
was shown however through a theorem by Vafa-WittenWitten1  that the true
ground state of QCD cannot break CP. The part of the QCD Lagrangian that
breaks CP is related to the gluon-gluon interaction term $F\widetilde{F}$. The
part of $F\widetilde{F}$ related to CP violations can be separated into a
separate term which then can be varied by multiplying this term by a parameter
called $\theta$. For the true QCD ground state $\theta$ = 0 (Vafa-Witten
theorem). In the vicinity of the deconfined QCD vacuum metastable
domainsTytgat  $\theta$ non-zero could exist and not contradict the Vafa-
Witten theorem since these are not the QCD ground state. The metastable
domains CP phenomenon would manifest itself in specific correlations of pion
momentaPisarski ; Tytgat .

###  VII.2 Pionic measures of CP violation.

The glasma flux tube model (GFTM)Dumitru  considers the wavefunctions of the
incoming projectiles, form sheets of CGCCGC  at high energies that collide,
interact, and evolve into high intensity color electric and magnetic fields.
This collection of primordial fields is the GlasmaLappi ; Gelis . Initially
the Glasma is composed of only rapidity independent longitudinal (along the
beam axis) color electric and magnetic fields. These longitudinal color
electric and magnetic fields generate topological Chern-Simons chargeSimons
through the $F\widetilde{F}$ term and becomes a source of CP violation. How
much of these longitudinal color electric and magnetic fields are still
present in the surface flux tubes when they have been pushed by the blast wave
is a speculation of this paper for measuring strong CP violation? The color
electric field which points along the flux tube axis causes an up quark to be
accelerated in one direction along the beam axis, while the anti-up quark is
accelerated in the other direction. So when a pair of quarks and anti-quarks
are formed they separate along the beam axis leading to a separated $\pi^{+}$
$\pi^{-}$ pair along this axis. The color magnetic field which also points
along the flux tube axis (which is parallel to the beam axis) causes an up
quark to rotate around the flux tube axis in one direction, while the anti-up
quark is rotated in the other direction. So when a pair of quarks and anti-
quarks are formed they will pickup or lose transverse momentum after the
bubble is boosted radial outwards due to radial flow of the blast wave. These
changes in $p_{t}$ will be transmitted to the $\pi^{+}$ $\pi^{-}$ pairs.

It is important to note that the CP violating asymmetries in $\pi^{+}$ and
$\pi^{-}$ momenta arise through the Witten-Wess-ZuminoWitten2 ; Wess  term.
The quarks and anti quarks which, later form the $\pi^{+}$ and $\pi^{-}$,
directly respond to the color electric and color magnetic fields and receive
their boosts at the quark and anti quark level before they are pions. These
boosts are transmitted to the pions at hadronization. Thus no external
magnetic field is required in the methodology followed in this paper. The
following references demonstrate this:2000 ; Pisarski ; Tytgat ; Finch .

To represent the color electric field effect we assume as the first step for
the results being shown in this paper; that we generate bubbles which have an
added boost of 100 MeV/c to the quarks in the longitudinal momentum which
represents the color electric effect. The $\pi^{+}$ and the $\pi^{-}$ which
form a pair are boosted in opposite directions along the beam axis. In a given
bubble all the boosts are the same but vary in direction from bubble to
bubble. For the color magnetic field effect, we give 100 MeV/c boosts to the
transverse momentum in an opposite way to the $\pi^{+}$ and the $\pi^{-}$
which form a pair. For each pair on one side of a bubble one of the charged
pions boost is increased and the other is decreased. While on the other side
of the bubble the pion for which the boost was increased is now decreased and
the pion which was decreased is now increased by the 100 MeV/c. All pairs for
a given bubble are treated in the same way however each bubble is random on
the sign of the pion which is chosen to be boosted on a given side. This
addition to our model is used in the simulations of the following subsections.

###  VII.3 Color electric field pionic measure.

Above we saw that pairs of positive and negative pions should show a charge
separation along the beam axis due to a boost in longitudinal momentum caused
by the color electric field. A measure of this separation should be a
difference in the pseudorapidity ($\Delta\eta$) of the opposite sign pairs.
This $\Delta\eta$ measure has a well defined sign since we defined this
difference measuring from the $\pi^{-}$ to the $\pi^{+}$. In order to form a
correlation we must pick two pairs for comparison. The pairs have to come from
the same bubble (a final state of an expanded flux tube) since we have shown
by investigating the events generated by the model that pairs originating from
different bubbles will not show this correlation. Therefore we require the
$\pi^{+}$ and the $\pi^{-}$ differ by $20^{\circ}$ or less in $\phi$. Let us
call the first pair $\Delta\eta_{1}$. The next pair ($\Delta\eta_{2}$ having
the same $\phi$ requirement) has to also lie inside the same bubble to show
this correlation. Thus we require that there is only $10^{\circ}$ between the
average $\phi$ of the pairs. This implies that at most in $\phi$ no two pions
from a given bubble can differ by more than $30^{\circ}$. In Fig. 15 we show
two pairs which would fall into the above cuts. $\Delta\eta_{1}$ and
$\Delta\eta_{2}$ are positive in Fig. 15. However if we would interchange the
$\pi^{+}$ and $\pi^{-}$ on either pair the value of its $\Delta\eta$ would
change sign. Finally the mean value shown on Fig. 15 is the mid-point between
the $\pi^{+}$ and the $\pi^{-}$ where one really uses the vector sum of the
$\pi^{+}$ and the $\pi^{-}$ which moves this point toward the harder pion.

Considering the above cuts we defined a correlation function where we combine
pairs each having a $\Delta\eta$. Our variable is related to the sum of the
absolute values of the individual $\Delta\eta$’s
($|\Delta\eta_{1}|+|\Delta\eta_{2}|$). We assign a sign to this sum such that
if the sign of the individual $\Delta\eta$’s are the same it is a plus sign,
while if they are different it is a minus sign. For the flux tube the color
electric field extends over a large pseudorapidity range therefore let us
consider the separation of pairs $|\Delta\eta|$ greater than 0.9. For the
numerator of the correlation function we consider all combinations of unique
pairs (sign ($|\Delta\eta_{1}|+|\Delta\eta_{2}|$)) from a given central Au Au
event divided by a mixed event denominator created from pairs in different
events. We determine the rescale of the mixed event denominator by considering
the number of pairs of pairs for the case $|\Delta\phi|$ lying between
$50^{\circ}$ and $60^{\circ}$ for events and mixed events so that the overall
ratio of this sample numerator to denominator is 1. By picking $50^{\circ}$
$<$ $|\Delta\phi|$ $<$ $60^{\circ}$ we make sure we are not choosing pairs
from the same bubble. For a simpler notation let (sign
($|\Delta\eta_{1}|+|\Delta\eta_{2}|$)) = $\Delta\eta_{1}+\Delta\eta_{2}$ which
varies from -4 to +4 since we have an over all $\eta$ acceptance -1 to +1 (for
the STAR TPC detector for which we calculated). The value being near $\pm$ 4
can happen when one has a hard pion with $p_{t}$ of 4 GeV/c (upper cut) at
$\eta$ = 1 with a soft pion $p_{t}$ of 0.8 GeV/c (lower cut) at $\eta$ = -1
combined with another pair; a hard pion with $p_{t}$ of 4 GeV/c at $\eta$ = -1
with a soft pion with $p_{t}$ of 0.8 GeV/c at $\eta$ = 1.

Figure 15: Pairs of pairs selected for forming a correlation due to the boost
in longitudinal momentum caused by the color electric field. The largest
separation we allow in $\phi$ for plus minus pair 1 is $\Delta\phi_{1}$ =
$20^{0}$. This assures the pair is in the same bubble. The same is true for
pair 2 so that it will also be contained in this same bubble. The mid-point
for pair 1 and 2 represents the vector sum of pair 1 and 2 which moves toward
the harder particle when the momenta differ. These mid-points can not be
separated by more than $10^{0}$ in $\Delta\phi$ in order to keep all four
particles inside the same bubble since the correlation function is almost
entirely generated within the same bubble. The $\Delta\eta$ measure is the
angle between the vector sum 1 compared to the vector sum 2 along the beam
axis (for this case $|\Delta\eta|$ = 1.1). The positive sign for
$\Delta\eta_{1}$ comes from the fact that one moves in a positive $\eta$
direction from negative to positive. The same is true for $\Delta\eta_{2}$. If
we would interchange the charge of the particles of the pairs the sign would
change.

Figure 16:  Correlation function of pairs formed to exhibit the effects of
longitudinal momentum boosts by the color electric field as defined in the
text. This correlation shows that there are more aligned pairs (correlation is
larger by $\sim$0.5% between 1.0 $<$ $\Delta\eta_{1}+\Delta\eta_{2}$ $<$ 2.0
compared to between -2.0 $<$ $\Delta\eta_{1}+\Delta\eta_{2}$ $<$ -1.0).
$\Delta\eta_{1}+\Delta\eta_{2}$ is equal to
$|\Delta\eta_{1}|+|\Delta\eta_{2}|$. As explained in the text this means there
are more pairs of pairs aligned in the same direction compared to pairs not
aligned as predicted by the color electric field. The signs and more detail
are also explained in the text.

In Fig. 16 we show the correlation function of opposite sign charge-particle-
pairs paired and binned by the variable $\Delta\eta_{1}+\Delta\eta_{2}$ with a
cut $|\Delta\eta|$ greater than 0.9 between the vector sums of the two pairs.
The events are generated by the PBMPBM  and are charged particles of 0.8 $<$
$p_{t}$ $<$ 4.0 GeV/c, and $|\eta|$ $<$ 1, from Au Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV. Since we select pairs of pairs which are near each
other in $\phi$ they all together pick up the bubble correlation and thus
these pairs of pairs over all show about a 4% correlation. In the 1.0 $<$
$\Delta\eta_{1}+\Delta\eta_{2}$ $<$ 2.0 region the correlation is 0.5% larger
than the -2.0 $<$ $\Delta\eta_{1}+\Delta\eta_{2}$ $<$ -1.0 region. This means
there are more pairs of pairs aligned in the same direction compared to pairs
of pairs not aligned. This alignment is what is predicted by the color
electric field effect presented above. In fact if one has plus minus pairs all
aligned in the same direction and spread across a pseudorapidity range,
locally at any place in the pseudorapidity range one would observe an increase
of unlike sign charge pairs compared to like sign charge pairs. At small
$\Delta\eta$ unlike sign charge pairs are much larger than like sign charge
pairs in both the PBM and the data which agree. See Figs. 10, 11, and 14 of
Ref.PBM . Figure 11 of Ref.PBM  compares the total correlation for unlike-sign
charge pairs and like-sign charge pairs in the precision STAR central
production experiment for Au Au central collisions (0-10% centrality) at
$\sqrt{s_{NN}}$=200 GeV, in the transverse momentum range 0.8 $<$ $p_{t}$ $<$
2.0 GeV/ccentralproduction . The unlike-sign charge pairs are clearly larger
in the region near $\Delta\phi$ = $\Delta\eta$ = 0.0. The increased
correlation of the unlike-sign pairs is 0.8% $\pm$ 0.002%. Figure 10 of
Ref.PBM  shows that the PBM fit to these data gives the same results. Figure
14 of Ref.PBM  shows that the CD = unlike-sign charge pairs minus like-sign
charge pairs is positive for the experimental analysis. Therefore the unlike-
sign charge pairs are considerably larger than the like-sign charge pairs. In
fact this effect is so large and the alignment is so great that when one adds
the unlike and like sign charge pairs correlations together there is still a
dip at small $\Delta\eta$ and $\Delta\phi$ see Fig. 4 and Fig. 7 of the
present paper. The statistical significance of this dip in the two high
precision experiments done independently from different data sets gathered 2
years apartcentralproduction ; centralitydependence  is huge. It would require
a fluctuation of $14\sigma$ to remove the dip. This dip is also not due to any
systematic error since both of the just cited precision experiments carefully
investigated that possibility; and found no evidence to challenge the reality
of this dip. This highly significant dip ($\sim$$14\sigma$) means that like-
sign pairs are removed as one approaches the region $\Delta\eta$ =
$\Delta\phi$ = 0.0 at a rate much larger than regular jet fragmentation. Jet
fragmentation does not have a dip in its CI correlation. Thus this is very
strong evidence for the predicted effect of the color electric field.

###  VII.4 Color magnetic pionic measure.

After considering the color electric effect we turn to the color magnetic
effect which causes up quarks to rotate around the flux tube axis in one
direction, while the anti-up quarks rotate in the other direction. So when a
pair of quarks and anti-quarks are formed they will pickup or lose transverse
momentum. These changes in $p_{t}$ will be transmitted to the $\pi^{+}$ and
$\pi^{-}$ pairs. It was previously shown in Sec. VII A that the CP violating
asymmetries in the $\pi^{+}$ and the $\pi^{-}$ momenta arise through the
Witten-Wess-Zumino term. The quarks and anti quarks which, later form the
$\pi^{+}$ and the $\pi^{-}$, directly respond to the color electric and color
magnetic fields and receive their boosts at the quark and anti quark level
before they hadronize into pions. These boosts are transmitted to the
$\pi^{+}$ and the $\pi^{-}$.

In order to observe these differential $p_{t}$ changes one must select pairs
on one side of the bubble in $\phi$ and compare to other pairs on the other
side of the same bubble which would lie around $40^{\circ}$ to $48^{\circ}$
away in $\phi$. We defined a pair as a plus particle and minus particle with
an opening angle ($\theta$) of $16^{\circ}$ or less. We are also interested in
pairs that are directly on the other side of the bubble. We require they are
near in pseudorapidity ($\Delta\eta$ $<$ 0.2). The above requirements
constrain the four charged particles comprising both pairs to be contained in
the same bubble and be close to being directly on opposite sides of the bubble
(a final state expanded flux tube). The difference in $p_{t}$ changes due to
and predicted by the color magnetic effect should give the $\pi^{+}$ on one
side of the bubble an increased $p_{t}$ and a decreased $p_{t}$ on the other
side, while for the $\pi^{-}$ it will be the other way around. This will lead
to an anti-alignment between pairs. In Fig. 17 we show two pairs which would
fall into the above cuts. Both pairs are at the limit of the opening angle cut
$\theta_{1}$ and $\theta_{2}$ equal $16^{\circ}$. The $p_{t}$ of the plus
particle for pair number 1 is 1.14 GeV/c, while the minus particle is 1.39
GeV/c. Thus $\Delta P_{t1}$ is equal to -0.25 GeV/c. The $p_{t}$ of the plus
particle for pair number 2 is 1.31 GeV/c, the minus particle is 0.91 GeV/c and
$\Delta P_{t2}$ is equal to 0.40 GeV/c. Finally the mean value shown on Fig.
17 is the mid-point between the $\pi^{+}$ and the $\pi^{-}$ where one really
uses the vector sum of the $\pi^{+}$ and the $\pi^{-}$ which moves this point
toward the harder pion.

Figure 17: Pairs of pairs selected for forming a correlation exhibiting the
effect of changes in $p_{t}$ due to the color magnetic field. The largest
opening angle $\theta$ for plus minus pairs is $16^{\circ}$ or less. This
opening angle assures that each pair has a high probability that it arises
from a quark anti-quark pair. In this figure we have picked two pairs at this
limit ($\theta_{1}$ = $16^{\circ}$ and $\theta_{2}$ = $16^{\circ}$). The mid-
point for pair 1 and 2 represents the vector sum of pair 1 and 2 which moves
toward the harder particle when the momenta differ. These mid-points are
chosen to have $40^{\circ}$ $<$ $|\Delta\phi|$ $<$ $48^{\circ}$ in order for
the pairs to be on opposite sides of the bubble. Since we are interested in
pairs directly across the bubble we make the $\Delta\eta$ separation be no
more than 0.2. The difference in $p_{t}$ for pair 1 is $\Delta P_{t1}$ = -0.25
GeV/c, while the difference in $p_{t}$ for pair 2 is $\Delta P_{t2}$ = 0.40
GeV/c. The minus sign for 1 follows from the fact that the plus particle has
1.14 GeV/c and the minus particle has 1.39 GeV/c. The plus sign for 2 follows
from the fact that the plus particle has 1.31 GeV/c and the minus particle has
0.91 GeV/c.

Considering the above cuts we defined a correlation function where we combine
pairs each having a $\Delta P_{t}$. Our variable is related to the sum of the
absolute values of the individual $\Delta P_{t}$’s ($|\Delta P_{t1}|+|\Delta
P_{t2}|$). We assign a sign to this sum such that if the sign of the
individual $\Delta P_{t}$’s are the same it is a plus sign, while if they are
different it is a minus sign. For the flux tube the color magnetic field
extends over a large pseudorapidity range where quarks and anti-quarks rotate
around the flux tube axis, therefore we want to sample pairs at the different
sides of the tube making a separation in $\phi$ ($\Delta\phi$) between
$40^{\circ}$ to $48^{\circ}$. We are interested in sampling the pairs on the
other side so we require the separation in $\eta$ ($\Delta\eta$) be 0.2 or
less. For the numerator of the correlation function we consider all
combinations of unique pairs (sign ($|\Delta P_{t1}|+|\Delta P_{t2}|$)) from a
given central Au Au event divided by a mixed event denominator created from
pairs in different events. We determine the rescale of the mixed event
denominator by considering the number of pairs of pairs for the case
$|\Delta\eta|$ lying between 1.2 and 1.5 plus any value of $|\Delta\phi|$ for
events and mixed events so that the overall ratio of this sample numerator to
denominator is 1. By picking this $\Delta\eta$ bin for all $|\Delta\phi|$ we
have around the same pair count as the signal cut with the $\Delta\phi$
correlation of the bubbles being washed out. For a simpler notation let (sign
($|\Delta P_{t1}|+|\Delta P_{t2}|$)) = $\Delta P_{t1}+\Delta P_{t2}$ which we
plot in the range from -4 to +4 since we have an over all $p_{t}$ range 0.8 to
4.0 GeV/c. Thus the maximum magnitude of $\Delta P_{t}$’s is 3.2 GeV/c which
makes $\Delta P_{t1}+\Delta P_{t2}$ have a range of $\pm$6.4. However the
larger values near these range limits occur very rarely.

Figure 18:  Correlation function of pairs of pairs formed for exhibiting the
effect of changes in $p_{t}$ due to the color magnetic field as defined in the
text. This correlation shows that there are more anti-aligned pairs. The
correlation is larger by $\sim$1.2% for $\Delta P_{t1}+\Delta P_{t2}$ = -4.0
compared to $\Delta P_{t1}+\Delta P_{t2}$ = 4.0. $\Delta P_{t1}+\Delta P_{t2}$
is equal to $|\Delta P_{t1}|+|\Delta P_{t2}|$ where the sign is also explained
in the text.

In Fig. 18 we show the correlation function of opposite sign charged-particle-
pairs paired and binned by the variable $\Delta P_{t1}+\Delta P_{t2}$ with a
cut $|\Delta\eta|$ less than 0.2 between the vector sums of the two pairs, and
with $40^{\circ}$ $<$ $|\Delta\phi|$ $<$ $48^{\circ}$. The events are
generated by the PBMPBM  and are charged particles of 0.8 $<$ $p_{t}$ $<$ 4.0
GeV/c, with $|\eta|$ $<$ 1, from Au Au collisions at $\sqrt{s_{NN}}=$ 200 GeV.
Since we select pairs of pairs which are near each other in $\phi$
($40^{\circ}$ to $48^{\circ}$) they all together pick up the bubble
correlation and thus these pairs of pairs over all show about a 0.4%
correlation. In the -4.0 $<$ $\Delta P_{t1}+\Delta P_{t2}$ $<$ -1.0 region the
correlation increases from 0.4% to 1%, while in the 1.0 $<$ $\Delta
P_{t1}+\Delta P_{t2}$ $<$ 4.0 region the correlation decreases from 0.4% to
-0.2%. This means the pairs of pairs are anti-aligned at a higher rate than
aligned. This anti-alignment is what is predicted by the color magnetic field
effect presented above. In fact the anti-alignment increases with $p_{t}$ as
the ratio bubble particle to background increases. However these predicted
color magnetic effects have not been searched for yet and therefore there is
no experimental evidence for them. Experimental investigations of STAR TPC
detector charged particle correlations provide a method of obtaining
experimental confirmation of the anti-alignment caused by the color magnetic
field.

##  VIII Summary and Discussion

In this article we summarize the assumptions made and the reasoning that led
to the development and construction of our parton bubble model (PBM)PBM ,
which successfully explained the charge-particle-pair correlations in the
central (0-10% centrality) $\sqrt{s_{NN}}=$ 200 GeV Au Au
datacentralproduction . The PBM is also consistent with the Au Au central
collision HBT results. This is presented and discussed in Sec. 4 of Ref.PBM
and Sec. I and II of this paper. In Sec. III we discussed extending our model,
which was a central collision model, to be able to treat the geomery of bubble
production for 0-80% collision centralities (PBMEPBME ) such as measured and
analyzed in RHIC datacentralitydependence .

The bubbles represent a significant substructure of gluonic hot spots formed
on the surface of a dense opaque fireball at kinetic freezeout. The origins of
these hot spots come from a direct connection between our Parton Bubble Model
(PBMPBM ) and the Glasma Flux Tube Model (GFTM)Dumitru ). In the GFTM a flux
tube is formed right after the initial collision of the Au Au system. This
flux tube extends over many units of pseudorapidity ($\Delta\eta$). A blast
wave gives the tubes near the surface transverse flow in the same way it gave
flow to the bubbles in the PBM. This means the transverse momentum ($p_{t}$)
distribution of the flux tube is directly translated to the $p_{t}$ spectrum.
In the PBM this flux tube is approximated by a sum of partons which are
distributed over this same large $\Delta\eta$ region. Initially the transverse
space is filled with flux tubes of large longitudinal extent but small
transverse size $\sim$$Q^{-1}_{s}$. The flux tubes that are near the surface
of the fireball get the largest radial flow and are emitted from the surface.
As in the parton bubble model these partons shower and the higher $p_{t}$
particles escape the surface and do not interact. $Q_{s}$ is around 1 GeV/c
thus the size of the flux tube is about 1/4 fm initially. The flux tubes near
the surface are initially at a radius $\sim$5 fm. The $\phi$ angle wedge of
the flux tube $\sim$ 1/20 radians or $\sim$$3^{\circ}$. In Sec. V (also see
Sec. IV) we connect the GFTM to the PBM: by assuming the bubbles are the final
state of a flux tube at kinetic freezeout, and discussing evidence for this
connection which is further developed in Sec. VI and Sec. VII.

With the connection of the PBM to the GFTM two new predictions become possible
for our PBM. The first is related to the fact that the blast wave radial flow
given to the flux tube depends on where the tube is initially in the
transverse plane of the colliding Au Au system. The tube gets the same radial
boost all along its longitudinal length. This means that there is correlated
$p_{t}$ among the partons of the bubble. In this paper we consider predictions
we can make in regard to interesting topics and comparisons with relevant data
which exist by utilizing the parton bubble model (PBMPBM ), and its features
related to the glasma flux tube model (GFTM)Dumitru ).

Topic 1: The ridge is treated in Sec. VI.

Topic 2: Strong CP violation (Chern-Simons topological charge) is treated in
Sec. VII.

We show in Sec. VI that if we trigger on particles with 3 to 4 GeV/c and
correlate this trigger particle with an other charged particle of greater than
1.1 GeV/c, the PBM can produce a phenomenon very similar to the ridgeDumitru ;
Armesto ; Romatschke ; Shuryak ; Nara ; Pantuev ; Mizukawa ; Wong ; Hwa . (See
Figs. 6-12). We then selected charged particles inside the ridge and predicted
the correlation that one should observe when compared to the average charged
particles of the central Au Au collisions at $\sqrt{s_{NN}}$ = 200
GeVcentralproduction ; centralitydependence .

In Sec. VI D (Comparison to data): Triggered experimental angular correlations
showing the ridge were presented at Quark Matter 2006Putschke . Figure 7 shows
the experimental $\Delta\phi$ vs. $\Delta\eta$ CI correlation for 0-10%
central Au Au collisions at $\sqrt{s_{NN}}$ = 200; requiring one trigger
particle $p_{t}$ between 3 to 4 GeV/c and an associated particle $p_{t}$ above
2.0 GeV/c. The yield is corrected for the finite $\Delta\eta$ pair acceptance.

For the PBM generator, we then form a two charged particle correlation between
one charged particle with a $p_{t}$ between 3.0 to 4.0 GeV/c and another
charged particle whose $p_{t}$ is greater than 2.0 GeV/c. These are the same
trigger conditions as in Ref.Putschke  which is shown in Fig. 11, that shows
the corrected pair yield in the central data. Fig. 12 shows the correlation
function generated by the PBM which does not depend on the number of events
analyzed. The two figures were shown to be in reasonable agreement when
compared as explained previously in Sec. VI D. In Fig. 13 we show the ridge
signal predicted by the PBM for very similar data but with 0-5% centrality.
Figure 14 shows the extraction of the jet signal. Explanations are given in
the text.

The second prediction is a development of a predictive pionic measure of the
strong CP Violation. The GFTM flux tubes are made up of longitudinal color
electric and magnetic fields which generate topological Chern-Simons
chargeSimons  through the $F\widetilde{F}$ term that becomes a source of CP
violation. The color electric field which points along the flux tube axis
causes an up quark to be accelerated in one direction along the beam axis,
while the anti-up quark is accelerated in the other direction. So when a pair
of quarks and anti-quarks are formed they separate along the beam axis leading
to a separated $\pi^{+}$ and $\pi^{-}$ pair along this axis. The color
magnetic field which also points along the flux tube axis (which is parallel
to the beam axis) causes an up quark to rotate around the flux tube axis in
one direction, while the anti-up quark rotates in the other direction. So when
a pair of quarks and anti-quarks are formed they will pickup or lose
transverse momentum. These changes in $p_{t}$ will be transmitted to the
$\pi^{+}$ and $\pi^{-}$ pairs.

The above pionic measures of strong CP violation are used to form correlation
functions based on four particles composed of two pairs which are opposite
sign charge-particle-pairs that are paired and binned. These four particle
correlations accumulate from bubble to bubble by particles that are pushed or
pulled (by the color electric field) and rotated (by the color magnetic field)
in a right or left handed direction. The longitudinal color electric field
predicts aligned pairs in a pseudorapidity or $\Delta\eta$ measure. The
longitudinal color magnetic field predicts anti-aligned pairs in a transverse
momentum or $\Delta P_{t}$ measure. The observations of these correlations
would be a strong confirmation of this theory. The much larger unlike-sign
pairs than like-sign pairs in the PBM and the data; and the strong dip of the
CI correlation at small $\Delta\eta$ (see Fig. 4 and Sec. VII C for full
details) shows very strong evidence supporting the color electric alignment
prediction in the $\sqrt{s_{NN}}$ = 200 GeV central Au Au collision data
analyses at RHICPBM ; centralproduction . This highly significant dip
($\sim$$14\sigma$) means that like-sign pairs are removed as one approaches
the region $\Delta\eta$ = $\Delta\phi$ = 0.0. Thus this is very strong
evidence for the predicted effect of the color electric field. The color
magnetic anti-aligned pairs in the transverse momentum prediction, treated in
Sec. VII D as of now has not been observed or looked for. However our
predicted specific four charged particle correlations can be used to search
for experimental evidence for the color magnetic fields.

Our success in demonstrating strong experimental evidence for the expected
color electric field effects from previously published data suggests that the
unique detailed correlations we have presented for searching for evidence for
the predicted color magnetic field effects should be urgently investigated. If
we are lucky and the predicted color magnetic effects can be confirmed
experimentally we would have strong evidence for the following:

1) CP is violated in the strong interaction in isolated local space time
regions where topological chargeSimons  is generated.

2) The glasma flux tube model (GFTM) which was evolved from the color glass
condensate (CGC) would be found to be consistent with a very significant
experimental check.

3) The parton bubble model event generator (PBM) is clearly closely connected
to the GFTM. The bubble substructure strongly supported by the PBM is likely
due to the final state of the flux tube at kinetic freezeout.

##  IX Acknowledgments

This research was supported by the U.S. Department of Energy under Contract
No. DE-AC02-98CH10886. The author thank William Love for valuable discussion
and assistance in production of figures. Finally this manuscript is in memory
of my fellow collaborator Prof. Sam Lindenbaum which help form many of the
ideas of bubbles in heavy ion collisions.

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