Open String Pair Production from the Interacting Dressed-Angled D-Strings

Hamidreza Daniali and Davoud Kamani

Department of Physics, Amirkabir University of Technology (Tehran
Polytechnic), Iran
P.O.Box: 15875-4413
e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

We calculate the rate of the open string pair production from the interaction
of two non-intersecting D-strings at angle. The D-strings have been equipped
with the $U(1)$ gauge potentials in the presence of the Kalb-Ramond field.
This rate will be discussed when the D-strings intersect each other. Our
computations is in the framework of the bosonic string theory.

PACS numbers: 11.25.-w; 11.25.Uv

Keywords: Dressed D-string; Amplitude; String pair production; Intersecting
D-strings.

##  1 Introduction

The D-branes are essential objects for various descriptions of the string
theory [1, 2, 3]. For computing the lowest order stringy interaction amplitude
of the D-branes in the closed string channel, one can apply the boundary state
formalism [4]-[17]. Equivalently, it is possible to calculate the same
amplitude via the one-loop annulus of open string [1, 14, 18]. By adding
dynamics, background fields and different internal gauge potentials to the
D-branes, one can reveal more properties of them [11]-[15], [19]-[23]. For the
branes with a large separation, the interaction occurs via the exchange of the
massless closed strings, which is attractive [14, 20, 23]. However, in the
small distance limit, the nature of the branes interaction (attraction or
repulsion) completely is ambiguous. As we shall see, the open string channel
provides the most accurate description for such configurations.

The electric field on a D-brane provides the necessary energy for creating the
open string pairs, analogous to the Schwinger pair production [24]. Thus, in
the presence of the electric fields, the interaction amplitude of the D-branes
possesses an imaginary part, which leads to the open string pair production.
In fact, the production rate of the open string pairs is infinitesimally
small. However, when the magnetic fields are applied to the system of the
branes this rate is extremely modified [25]-[29].

The open string pair production, similar to the other investigations of the
branes, has been usually studied for the parallel branes. However, the branes
at angles have been also applied in various subjects, e.g. in the super Yang-
Mills theories [30, 31], in the modeling of the black holes [32], in the non-
chiral supersymmetric field theories [33]-[35], and so on. Therefore, we shall
study the open string pair production via two angled-dressed D-strings
(D1-branes) which are not intersecting. The D-strings have been dressed with
the different $U(1)$ gauge potentials and live in a nonzero Kalb-Ramond
background field. The production rate will be also investigated for two
intersecting D-strings. Our calculations is in the context of the bosonic
string theory

This paper is organized as follows. In Sec. 2, we obtain the boundary state,
associated with a skew-dressed D-string in the presence of the Kalb-Ramond
field. In Sec. 3, at first, we calculate the interaction amplitude of a system
of two D-strings at angle with a nonzero distance. Then, we compute the pair
production rate of the open strings from the interaction of this system. In
the subsection 3.1, we investigate this rate for the intersecting D-strings.
The conclusions will be given in Sec. 4.

##  2 The boundary state corresponding to an oblique-dressed D-string

In order to compute the open string pair production, it is necessary to
calculate the interaction amplitude. The angled D-strings (D1-branes) of the
setup have been dressed with the $U(1)$ gauge potentials in the presence of a
constant background $B$-field. The interaction will be computed in the closed
string channel. Thus, we use the boundary state formalism. Hence, at first we
construct the boundary state, corresponding to a D-string along the
$x_{1}$-direction. So, we begin with the following string action

$\displaystyle S$ $\displaystyle=$
$\displaystyle-\dfrac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}{\rm
d}^{2}\sigma\left(\sqrt{-h}h^{ab}G_{\mu\nu}(X)+\epsilon^{ab}B_{\mu\nu}(X)\right)\partial_{a}X^{\mu}\partial_{b}X^{\nu}$
(1) $\displaystyle+$
$\displaystyle\dfrac{1}{2\pi\alpha^{\prime}}\int_{\partial\Sigma}{\rm d}\sigma
A_{\alpha}\partial_{\sigma}X^{\alpha},$

where $\mu,\nu$ are the spacetime indices and $h^{ab}$, with
$a,b\in\\{\tau,\sigma\\}$, represents the metric of the closed string
worldsheet. The indices $\alpha,\beta\in\\{0,1\\}$ indicate the worldsheet of
the D-string. For the next purposes the indices $i,j\in\\{2,\cdots,d-1\\}$
will represent transverse directions to the D-string. We shall apply the flat
metric $G_{\mu\nu}=\eta_{\mu\nu}={\rm diag}(-1,1,\cdots,1)$ for the spacetime,
a constant Kalb-Ramond field $B_{\mu\nu}$, and the Landau gauge
$A_{\alpha}=-2^{-1}F_{\alpha\beta}X^{\beta}$ with the constant field strength
$F_{\alpha\beta}$. In fact, this gauge dedicates a squared structure to the
action (1).

Setting the action variation to zero yields the equation of motion and also
the following boundary state equations

$\displaystyle\left[\partial_{\tau}X^{0}(\sigma,\tau)-\mathcal{E}\partial_{\sigma}X^{1}(\sigma,\tau)\right]_{\tau=0}|B^{\prime}\rangle=0,$
$\displaystyle\left[\partial_{\tau}X^{1}(\sigma,\tau)-\mathcal{E}\partial_{\sigma}X^{0}(\sigma,\tau)\right]_{\tau=0}|B^{\prime}\rangle=0,$
$\displaystyle\left[X^{i}(\sigma,\tau)-y^{i}\right]_{\tau=0}|B^{\prime}\rangle=0,$
(2)

where $\mathcal{E}\equiv F_{01}-B_{01}$, the parameters $y^{i}$ indicate the
position of the D-string, and $|B^{\prime}\rangle$ represents the
corresponding boundary state of it.

Now consider a D-string parallel to the $x^{1}x^{2}$-plane, which makes the
angle $\phi$ with the $x^{1}$-direction. By utilizing the well-known closed
string mode expansion, the zero-mode part of the boundary state equations for
the oblique D-string become

$\displaystyle\hat{p}^{0}|B\rangle_{0}=0,$
$\displaystyle\left(\hat{p}^{1}\cos\phi+\hat{p}^{2}\sin\phi\right)|B\rangle_{0}=0,$
$\displaystyle\left[(\hat{x}^{2}-y^{2})\cos\phi-(\hat{x}^{1}-y^{1})\sin\phi\right]|B\rangle_{0}=0,$
$\displaystyle(\hat{x}^{i^{\prime}}-y^{i^{\prime}})|B\rangle_{0}=0,$ (3)

in which $i^{\prime}\in\\{3,4,\cdots,d-1\\}$, and $|B\rangle$ shows the
boundary state of the oblique D-string. We employed the decomposition
$|B\rangle=|B\rangle_{0}\otimes|B\rangle_{\rm osc}$. Using the quantum
mechanical technics, the zero-mode portion of the boundary state finds the
following solution

$\displaystyle|B\rangle_{0}$ $\displaystyle=$
$\displaystyle\frac{T_{1}}{2}\delta\left[(\hat{x}^{2}-y^{2})\cos\phi-(\hat{x}^{1}-y^{1})\sin\phi\right]|p^{0}=0\rangle|p^{1}=0\rangle|p^{2}=0\rangle$
(4) $\displaystyle\times$
$\displaystyle\left(\prod_{i^{\prime}=3}^{d-1}\delta(\hat{x}^{i^{\prime}}-y^{i^{\prime}})|p^{i^{\prime}}=0\rangle\right),$

where
$T_{1}=g^{-1}_{s}\sqrt{\pi}\;2^{(10-d)/4}(4\pi^{2}\alpha^{\prime})^{(d-6)/4}$
is the tension of the D-string.

The oscillating parts of the boundary state equations can be written in a
unified form, i.e., $\left[\alpha^{\mu}_{n}+\Omega^{\mu}_{\
\nu}(\phi,\mathcal{E})\tilde{\alpha}^{\nu}_{-n}\right]|B\rangle_{\rm osc}=0$,
where $\Omega^{\mu}_{\ \nu}=\Big{(}\mathcal{M}^{\alpha^{\prime}}_{\
\beta^{\prime}}(\phi,\mathcal{E}),-\delta^{i^{\prime}}_{\ j^{\prime}}\Big{)}$
with $\alpha^{\prime},\beta^{\prime}\in\\{0,1,2\\}$, and

$\displaystyle\mathcal{M}^{\alpha^{\prime}}_{\
\beta^{\prime}}(\phi,\mathcal{E})$ $\displaystyle\equiv$
$\displaystyle\frac{1}{1-\mathcal{E}^{2}}\begin{bmatrix}1+\mathcal{E}^{2}&-2\mathcal{E}\cos\phi&-2\mathcal{E}\sin\phi\\\
-2\mathcal{E}\cos\phi&\mathcal{E}^{2}+\cos(2\phi)&\sin(2\phi)\\\
-2\mathcal{E}\sin\phi&\sin(2\phi)&\mathcal{E}^{2}-\cos(2\phi)\end{bmatrix}.$
(5)

By employing the coherent state method, we acquire the solution

$\displaystyle|B\rangle_{\rm
osc}=\sqrt{1-\mathcal{E}^{2}}\exp\left(-\sum_{n=1}^{\infty}\frac{1}{n}\alpha^{\mu}_{-n}\Omega_{\mu\nu}\tilde{\alpha}^{\nu}_{-n}\right)|0\rangle\otimes|\tilde{0}\rangle.$

The prefactor $\sqrt{1-\mathcal{E}^{2}}$ comes from disk partition function
[9]. For the future purposes, we should note that the matrix $\mathcal{M}$ is
orthogonal and hence $\Omega$ also is an orthogonal matrix.

In the bosonic string theory, the direct product $|B\rangle_{\rm
tot}=|B\rangle_{\rm osc}\otimes|B\rangle_{0}\otimes|B\rangle_{\rm g}$ exhibits
the total boundary state, associated with the D-string, where $|B\rangle_{\rm
g}$ is the well-known boundary state of the conformal ghosts.

##  3 Open string pair production from the non-intersecting-angled D-strings

In this section, we extract the rate of the open string pair creation via the
interaction amplitude of two D-strings at angles. For preserving the
generality, assume that the electric fields and angles of the D-strings with
the $x^{1}$-direction are different. Thus, the subscripts $(1)$ and $(2)$ will
be used to show these differences.

In the closed string channel, the interaction of two D-branes takes place by
exchanging closed strings between the branes. The geometry of the worldsheet
of the exchanged closed string is a cylinder with $\tau$ as the coordinate
along the length of the cylinder, $0\leq\tau\leq t$, and $\sigma$ as the
periodic coordinate, i.e. $0\leq\sigma\leq\pi$. The interaction amplitude can
be computed via the overlap of the two boundary states, through the closed
string propagator $\mathcal{A}_{\rm
closed}=\pi\alpha^{\prime}\int_{0}^{\infty}{\rm d}t\ \ _{\rm tot}\langle
B_{(1)}|\exp(-t\mathcal{H})|B_{(2)}\rangle_{\rm tot}$, where $\mathcal{H}$ is
the total Hamiltonian of the closed string. It includes the ghost and matter
parts. After long calculations one finds

$\displaystyle\mathcal{A}_{\rm closed}$ $\displaystyle=$ $\displaystyle
V_{1}\frac{\;\sqrt{\left(1-\mathcal{E}_{(1)}^{2}\right)\left(1-\mathcal{E}_{(2)}^{2}\right)}}{\sqrt{8\pi^{2}\alpha^{\prime}}g_{s}^{2}|\sin\Phi|}\int_{0}^{\infty}{\rm
d}t\ \frac{e^{(d-2)\pi
t/12}}{t^{(d-3)/2}}\exp\left(-\frac{\mathbf{Y}^{2}}{2\pi\alpha^{\prime}t}\right)$
(6) $\displaystyle\times$
$\displaystyle\prod_{n=1}^{\infty}\left[\frac{(1-e^{-2n\pi
t})^{4-d}}{1-2e^{-2n\pi
t}\left(\frac{2\left(\cos\Phi-\mathcal{E}_{(1)}\mathcal{E}_{(2)}\right)^{2}}{\left(1-\mathcal{E}_{(1)}^{2}\right)\left(1-\mathcal{E}_{(2)}^{2}\right)}-1\right)+e^{-4n\pi
t}}\right],$

with $\Phi\equiv\phi_{(2)}-\phi_{(1)}$ is the angle between the D-strings, and
$\mathbf{Y}^{2}\equiv\sum_{j^{\prime}=3}^{d-1}\left(y_{1}^{j^{\prime}}-y_{2}^{j^{\prime}}\right)^{2}$
is the square distance between them.

Let us analyze the amplitude (6). The factor $e^{(d-2)\pi t/12}$ comes from
the Hamiltonian regularization, and $V_{1}=2\pi\delta(0)$ is the length of the
D-string. The advent of $|\sin\Phi|$ in the denominator, originates from the
property $\delta(ax)=|a|^{-1}\delta(x)$. This elucidates that the amplitude
(6) is valid only for $\Phi\neq 0$, and the parallel case is different and
cannot be obtained from this by setting $\Phi=0$. In the latter configuration,
for example, the $\mathbf{Y}$ is $(d-2)$-dimensional and $V_{1}$ should be
substituted by $V_{2}=(2\pi)^{2}\delta^{2}(0)$. The position-dependent factor
clarifies that the interaction amplitude is exponentially damped by increasing
the distance between the D-strings. The infinite product is the contributions
of the oscillatory part and the conformal ghosts portion. Precisely, the
contribution of the conformal ghosts is $(1-e^{-2n\pi t})^{2}$, and the
contribution of the transverse directions to both D-strings is $(1-e^{-2n\pi
t})^{3-d}$. The expression in the denominator of the infinite product can be
rewritten in the form $(1-\gamma e^{-2n\pi t})(1-\gamma^{\prime}e^{-2n\pi
t})$, where $\gamma\gamma^{\prime}=1$. Therefore, we receive

$\displaystyle\frac{1}{2}\left(\gamma+\gamma^{\prime-1}\right)=\frac{2\left(\cos\Phi-\mathcal{E}_{(1)}\mathcal{E}_{(2)}\right)^{2}}{\left(1-\mathcal{E}_{(1)}^{2}\right)\left(1-\mathcal{E}_{(2)}^{2}\right)}-1.$
(7)

By writing $\gamma\equiv e^{2i\pi\nu}$, Eq. (7) takes the feature

$\displaystyle\cos\pi\nu=\frac{\cos\Phi-\mathcal{E}_{(1)}\mathcal{E}_{(2)}}{\sqrt{\left(1-\mathcal{E}_{(1)}^{2}\right)\left(1-\mathcal{E}_{(2)}^{2}\right)}}.$
(8)

In the case of parallel D-strings there is $\cos\Phi=1$. For this
configuration we have
$1-\mathcal{E}_{(1)}\mathcal{E}_{(2)}>\sqrt{\left(1-\mathcal{E}_{(1)}^{2}\right)\left(1-\mathcal{E}_{(2)}^{2}\right)}$,
e.g. see [25]-[29]. This inequality elaborates that $\nu$ is a pure imaginary
quantity. This is an essential element for calculating the production rate of
the open string pairs. Hence, we utilize $\nu=i\tilde{\nu}$ in which
$0<\tilde{\nu}<\infty$. Consequently, Eq. (8) takes the feature

$\displaystyle\cos\Phi=\mathcal{E}_{(1)}\mathcal{E}_{(2)}+\cosh\pi{\tilde{\nu}}\;\sqrt{\left(1-\mathcal{E}_{(1)}^{2}\right)\left(1-\mathcal{E}_{(2)}^{2}\right)}.$
(9)

We observe that the perpendicularity of the two D-strings is not possible
unless for the case $\mathcal{E}_{(1)}\mathcal{E}_{(2)}<0$. Since
$\cosh\pi{\tilde{\nu}}\geq 1$, we obtain
$\mathcal{E}_{(1)}^{2}+\mathcal{E}_{(2)}^{2}\geq
2\mathcal{E}_{(1)}\mathcal{E}_{(2)}\cos\Phi+\sin^{2}\Phi$. This condition
imposes some restrictions on the angle $\Phi$ and the electric fields.
Precisely, for the given electric fields $\mathcal{E}_{(1)}$ and
$\mathcal{E}_{(2)}$, for occurring the open string pair production the
possible range of the angle $\Phi$ is defined by $\cos\Phi>x_{+}$ or
$\cos\Phi<x_{-}$, in which
$x_{\pm}\equiv\mathcal{E}_{(1)}\mathcal{E}_{(2)}\pm\
\sqrt{\left(1-\mathcal{E}_{(1)}^{2}\right)\left(1-\mathcal{E}_{(2)}^{2}\right)}$.
In fact, the electric field of a D-string is parallel to that D-string. Thus,
the angle $\Phi$ represents the angle between the electric fields too. Hence,
for happening the pair creation of open strings, the electric fields should
possess special alignments with each other.

By utilizing the Jacobi $\Theta$\- and Dedekind $\eta$-functions, the
amplitude finds the following feature

$\displaystyle\mathcal{A}_{\rm closed}$ $\displaystyle=$ $\displaystyle
iV_{1}\frac{\sinh\pi{\tilde{\nu}}\sqrt{\left(1-\mathcal{E}_{(1)}^{2}\right)\left(1-\mathcal{E}_{(2)}^{2}\right)}}{\sqrt{2\pi^{2}\alpha^{\prime}}g_{s}^{2}|\sin\Phi|}\int_{0}^{\infty}\frac{{\rm
d}t}{t^{(d-3)/2}}\frac{\exp\left(-\frac{\mathbf{Y}^{2}}{2\pi\alpha^{\prime}t}\right)}{\eta^{d-5}(it)\Theta_{1}(i\tilde{\nu}|it)}$
(10)

This amplitude is real. For the large D-strings separation, which is
equivalent to $t\rightarrow\infty$, the dominant contribution to the
interaction arises from the massless closed strings. In this case we receive
$\mathcal{A}_{\rm closed}|_{t\rightarrow\infty}>0$, which represents an
attractive force.

We now consider the small “$t$”-integration. In this limit, the sign of the
denominator in the infinite product of Eq. (6) becomes negative. Hence, the
sign of the infinite product is unclear. This implies that any statement
regarding the nature of the interaction (attraction or repulsion) is
ambiguous. In fact, for the infinitesimal value of “$t$”, the open string
amplitude is more appropriate. Thus, let us employ the Jacobi transformation
$t\rightarrow 1/t$ to convert the closed string amplitude (10) to the open
string annulus amplitude. With the use of

$\displaystyle\Theta_{1}(\hat{\nu}|\hat{\tau})=i\frac{e^{-i\pi\hat{\nu}^{2}/\hat{\tau}}}{\sqrt{-i\hat{\tau}}}\
\Theta_{1}\left(\frac{\hat{\nu}}{\hat{\tau}}\right|\left.-\frac{1}{\hat{\tau}}\right),\qquad\eta(\hat{\tau})=\frac{1}{\sqrt{-i\hat{\tau}}}\
\eta\left(-\frac{1}{\hat{\tau}}\right),$ (11)

we obtain

$\displaystyle\mathcal{A}_{\rm open}$ $\displaystyle=$ $\displaystyle
V_{1}\frac{\sqrt{\left(1-\mathcal{E}_{(1)}^{2}\right)\left(1-\mathcal{E}_{(2)}^{2}\right)}}{\sqrt{2\pi^{2}\alpha^{\prime}}g_{s}^{2}|\sin\Phi|}\sinh\pi{\tilde{\nu}}\int_{0}^{\infty}\frac{{\rm
d}t}{t^{3/2}}\frac{\exp\left(-\frac{\mathbf{Y}^{2}t}{2\pi\alpha^{\prime}}\right)e^{-\pi\tilde{\nu}^{2}t}}{\eta^{d-5}(it)\Theta_{1}(\tilde{\nu}t|it)}$
(12)

For the next purposes, we write the amplitude (12) in the form

$\displaystyle\mathcal{A}_{\rm open}$ $\displaystyle=$ $\displaystyle
V_{1}\frac{\sqrt{\left(1-\mathcal{E}_{(1)}^{2}\right)\left(1-\mathcal{E}_{(2)}^{2}\right)}}{\sqrt{8\pi^{2}\alpha^{\prime}}g_{s}^{2}|\sin\Phi|}\sinh\pi{\tilde{\nu}}\int_{0}^{\infty}{\rm
d}t\frac{e^{-2\mathcal{M}^{2}t}}{t^{3/2}\sin(\tilde{\nu}\pi t)}$ (13)
$\displaystyle\times$ $\displaystyle\prod_{n=1}^{\infty}\frac{(1-e^{-2n\pi
t})^{4-d}}{1-2\cos(2\pi\tilde{\nu}t)e^{-2n\pi t}+e^{-4n\pi t}}$

where the effective mass of the open string, stretched between the D-strings,
is given by

$\displaystyle\mathcal{M}^{2}\equiv\frac{1}{4\pi\alpha^{\prime}}\left\\{\mathbf{Y}^{2}+\pi^{2}\alpha^{\prime}\left[2\tilde{\nu}^{2}-\frac{d-2}{6}\right]\right\\}.$
(14)

When
$\mathbf{Y}<\pi\sqrt{\alpha^{\prime}\left[\frac{d-2}{6}-2\tilde{\nu}^{2}\right]}$
we receive the tachyonic shift. In this case, the integrand of Eq. (13) for
$t\rightarrow\infty$ diverges, which indicates the tachyonic instability.
Therefore, a phase transition will take place through the tachyon condensation
[36]. We discard the tachyonic shift, i.e. by choosing appropriate values for
the setup parameters, we apply $\mathcal{M}^{2}\geq 0$. Hence, the formulation
of the open string pair production is permitted.

In the open string channel, the small D-strings separation is equivalent to
the limit $t\to\infty$. In this case, all factors in the integrand of Eq. (13)
are positive, while the value of the factor $\sin(\tilde{\nu}\pi t)$ belongs
to the interval $(-1,1)$. This factor leads to an infinite number of simple
poles along the positive $t$-axis, $t_{m}=\frac{m}{\tilde{\nu}}$ where
$m=1,2,3,\cdots$. Each pole separately exhibits the creation of a pair of open
strings and also the decay of the system. Besides, the amplitude includes an
imaginary part. By calculating the residue, following Refs. [37, 38], and
using the well-known Schwinger formula $\mathcal{W}=-2V_{1+1}^{-1}\ {\rm Im}\
\mathcal{A}_{\rm open}$, we find the pair production rate per unit worldsheet
area $V_{1+1}$ as in the following

$\displaystyle\mathcal{W}$ $\displaystyle=$
$\displaystyle\frac{V_{1}}{V_{1+1}}\frac{\sqrt{\left(1-\mathcal{E}_{(1)}^{2}\right)\left(1-\mathcal{E}_{(2)}^{2}\right)}}{\sqrt{2\pi^{2}\alpha^{\prime}}g_{s}^{2}|\sin\Phi|}{\tilde{\nu}}^{1/2}\sinh\pi{\tilde{\nu}}\;\sum_{m=1}^{\infty}\bigg{\\{}\frac{(-1)^{m+1}}{m^{3/2}}$
(15) $\displaystyle\times$
$\displaystyle\exp\left(-\frac{2m}{\tilde{\nu}}\mathcal{M}^{2}\right)\prod_{n=1}^{\infty}\left[1-\exp\left(-\frac{2n\pi
m}{\tilde{\nu}}\right)\right]^{2-d}\bigg{\\}}.$

For the finite distance of the D-strings, especially when they are near to
each other, this quantity elaborates the creation of the open string pairs and
consequently the decay of the system. Note that the open strings production
between the D-branes has resemblances with the Casimir effect.

Eq. (9) implies that the decay rate (15) is a complicated function of the
angle $\Phi$ and the electric fields on the D-strings. We observe that by
increasing the D-strings distance $\mathbf{Y}$, the pair production rate
decreases. In other words, for the large value of the distance $\mathbf{Y}$,
the mass of each pair obviously becomes large. Consequently, the probability
of producing the heavy open string pairs is small. Besides, for the case
$d\geq 3$, as it ought to be for non-intersecting-angled D-strings, by
increasing the ${\tilde{\nu}}$ the production rate also increases.

According to Eq. (9), the electric fields cannot simultaneously vanish.
Therefore, in the decay rate (15) at least one of the electric fluxes should
be nonzero. From the physical point of view, at least an electric flux is
prominently needed to polarize the region between the D-strings. Otherwise,
the pair creation of the open strings does not occur.

For the case $\mathcal{E}_{(1)}\rightarrow 1$ or $\mathcal{E}_{(2)}\rightarrow
1$ or both, the production rate begins to diverge, and the pair production
instability occurs. In the limit $\tilde{\nu}\rightarrow 0$, which takes place
for a setup with the following relation among its parameters
$\cos\Phi\approx\mathcal{E}_{(1)}\mathcal{E}_{(2)}+\sqrt{(1-\mathcal{E}^{2}_{(1)})(1-\mathcal{E}^{2}_{(2)})}$,
we acquire the minimum rate

$\displaystyle\mathcal{W}_{\tilde{\nu}\rightarrow
0}\approx\frac{V_{1}}{V_{1+1}}\frac{\sqrt{\left(1-\mathcal{E}_{(1)}^{2}\right)\left(1-\mathcal{E}_{(2)}^{2}\right)}}{\sqrt{2\pi^{2}\alpha^{\prime}}g_{s}^{2}|\sin\Phi|}\tilde{\nu}^{3/2}\exp\left(-\frac{2}{\tilde{\nu}}\mathcal{M}^{2}\right).$
(16)

This rate prominently is infinitesimal and does not possess any physical
consequence.

###  3.1 Pair production rate of photons from the intersecting-angled
D-strings

Now let us examine the production rate for the massless open strings, i.e.
photons, from the intersecting D-strings. Thus, we should employ the critical
dimension $d=26$. With the help of Eq. (14) and $\mathbf{Y}=0$, we obtain
$\tilde{\nu}=\sqrt{2}$. The resultant rate is given by

$\displaystyle\mathcal{W}_{0}$ $\displaystyle=$ $\displaystyle\frac{V_{1}}{\pi
V_{1+1}}\frac{\sinh(\sqrt{2}\pi)}{\sqrt{\sqrt{2}\pi^{-1}\alpha^{\prime}}g_{s}^{2}}\;R\left(\mathcal{E}_{(1)},\mathcal{E}_{(2)}\right)$
(17) $\displaystyle\times$
$\displaystyle\sum_{m=1}^{\infty}\left\\{\frac{(-1)^{m+1}}{m^{3/2}}\
\prod_{n=1}^{\infty}\left[1-\exp\left(-\sqrt{2}nm\right)\right]^{-24}\right\\},$

where the dependence on the electric fields has been collected in the function
$R\left(\mathcal{E}_{(1)},\mathcal{E}_{(2)}\right)$,

$\displaystyle
R\left(\mathcal{E}_{(1)},\mathcal{E}_{(2)}\right)=\left(\frac{{\left(1-\mathcal{E}_{(1)}^{2}\right)\left(1-\mathcal{E}_{(2)}^{2}\right)}}{1-\left[\mathcal{E}_{(1)}\mathcal{E}_{(2)}+\cosh\left(\sqrt{2}\pi\right)\sqrt{{\left(1-\mathcal{E}_{(1)}^{2}\right)\left(1-\mathcal{E}_{(2)}^{2}\right)}}\;\right]^{2}}\right)^{1/2}.$
(18)

In the allowed square region $-1<\mathcal{E}_{(1)},\mathcal{E}_{(2)}<1$, this
function does not possess any smooth maximum or minimum. Besides, since
$\cosh\left(\sqrt{2}\pi\right)$ is a large number, at least one of the
electric fields should be nonzero. Otherwise, this function becomes imaginary,
which is forbidden. In addition, according to Eq. (9) (with the replacement
$\tilde{\nu}\to\sqrt{2}$), such value of $\tilde{\nu}$ leads to
$\mathcal{E}_{(1)}\to 1$, or $\mathcal{E}_{(2)}\to 1$ and or both of them.
When for example $\mathcal{E}_{(2)}$ vanishes, for any value of the angle
$\Phi$ we receive the following allowed values for $\mathcal{E}_{(1)}$, i.e.,
$\tanh\left(\sqrt{2}\pi\right)\leq|\mathcal{E}_{(1)}|<1$.

##  4 Conclusions

We utilized the boundary state formalism to compute the pair production rate
of the open strings from the interaction of two non-intersecting D$1$-branes
at angle. The distance between the D-strings is arbitrary. The D-strings have
been dressed with the different $U(1)$ gauge potentials. Besides, the Kalb-
Ramond field as a background was introduced. As a special case, we obtained
the production rate of the photon pairs from the intersecting D-strings.

We observed that the production rate of the heavy open strings is very small,
as expected. This is due to the fact that the required energy for the
production of the heavy string pairs is large. Note that by increasing the
distance between the D-strings and also by enhancing of the spacetime
dimension we receive the heavy open strings.

We observed that for acquiring the open string pair creation, at least one of
the D-strings should be dressed with a nonzero electric field. Precisely, at
least an electric field is obviously required to polarize the region between
the branes. Otherwise, the pair production cannot occur.

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A Hole torn by QCD Fields.
R.S. Longacrea
aBrookhaven National Laboratory, Upton, NY 11973, USA

###### Abstract

In this paper we determine that a local negative correlation(the Hole) in
$\Delta\eta$ $\Delta\phi$ space of Au + Au collisions at $\sqrt{s_{NN}}$=200
GeV is caused by strong color QCD electric and magnetic fields which are
present in the Glasma Flux Tubes that are generated by the initial conditions.

##  1 Introduction and review of the model

In this paper we discuss a Glasma Flux Tube Model(GFTM)[1] which has tubes of
parallel strong color QCD electric and magnetic fields that are generated by
the initial conditions of Au + Au central collisions at $\sqrt{s_{NN}}$=200
GeV.

The paper is organized in the following manner:

Sec. 1 is the introduction and review of the model. Sec. 2 discuss two
particle angular correlation in the model. Sec. 3 gives the properties of the
GFTM. Sec. 4 is a comparison of charge independent two particle angular
correlation between the model and the Au + Au data. Sec. 5 is a comparison of
the charge dependent correlation of the signal coming from a single flux tube
with the Au + Au data. Sec. 6 is about QCD and the strong CP violation term.
Sec. 7 considers color electric fields and data. Sec. 8 considers color
magnetic fields and data. Sec. 9 presents the summary and discussion.

###  1.1 Glasma Flux Tube Model

A glasma flux tube model (GFTM)[1] that had been developed considers that the
wave functions of the incoming projectiles, form sheets of color glass
condensates (CGC)[2] that at high energies collide, interact, and evolve into
high intensity color electric and magnetic fields. This collection of
primordial fields is the Glasma[3, 4], 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 wave function. These
longitudinal color electric and magnetic fields generate topological Chern-
Simons charge[5] 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. The flux
tubes at the center of the transverse plane interact with each other through
plasma instabilities[3, 6] and create a locally thermalized system, where
partons emitted from these flux tubes are locally equilibrated. A hydro system
with transverse flow builds causing a radially flowing blast wave[7]. The flux
tubes that are near the surface of the fireball get the largest radial flow
and are emitted from the surface.

$Q_{s}$ 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 width in $\Delta\eta$ correlation of particles results from the
independent longitudinal color electric and magnetic fields that created the
Glasma flux tubes. In this paper we relate particle production from the
surface flux tube to a related model Parton Bubble Model(PBM)[8]. It was shown
in Ref.[9] that for central Au + Au collisions at $\sqrt{s_{NN}}$=200 the PBM
is a good approximation to the GFTM surface flux tube formation.

The flux tubes on the surface turn out to be on the average 12 in number. They
form an approximate ring about the center of the collision see Figure 1. The
twelve tube ring creates the average behavior of tube survival near the
surface of the expanding fire ball of the blast wave. The final state surface
tubes that emit the final state particles at kinetic freezeout are given by
the PBM. One should note that the blast wave surface is moving at its maximum
velocity at freezeout (3c/4).

Figure 1: The tube geometry is an 8 fm radius ring perpendicular to and
centered on the beam axis. It is composed of twelve adjacent 2 fm radius
circular tubes elongated along the beam direction as part of the flux tube
geometry. We project on a plane section perpendicular to the beam axis.

Figure 2: A jet parton shower.

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

The space momentum correlation of the blast wave provides us with a strong
angular correlation signal. PYTHIA fragmentation functions[10] were used for
the tube fragmentation that generate the final state particles emitted from
the tube. The initial transverse size of a flux tube $\sim$1/4 fm has expanded
to the size of $\sim$2 fm at kinetic freezeout. Many particles that come from
the surface of the fireball will have a $p_{t}$ greater than 0.8 Gev/c. The
final state tube size and the Hanbury-Brown and Twiss (HBT) observations[11]
of pions that have a momentum range greater than 0.8 GeV/c are consistent both
being $\sim$2 fm. A single parton using PYTHIA forms a jet with the parton
having a fixed $\eta$ and $\phi$ (see Figure 2). For central events each of
the twelve tubes have 3-4 partons per tube each at a fixed $\phi$ for a given
tube. The $p_{t}$ distribution of the partons is similar to pQCD but has a
suppression at high $p_{t}$ like the data. The 3-4 partons in the tube which
shower using PYTHIA all have a different $\eta$ values but all have the same
$\phi$ (see Figure 3). The PBM explained the high precision Au + Au central
(0-10%) collisions at $\sqrt{s_{NN}}=$ 200 GeV[12](the highest RHIC energy).

##  2 The Correlation Function for Central Au + Au Data

We utilize a two particle correlation function in the two dimensional (2-D)
space111$\Delta\phi=\phi_{1}-\phi_{2}$ where $\phi$ is the azimuthal angle of
a particle measured in a clockwise direction about the beam.
$\Delta\eta=\eta_{1}-\eta_{2}$ which is the difference of the psuedorapidity
of the pair of particles of $\Delta\phi$ versus $\Delta\eta$. The 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. 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($\Delta\phi,\Delta\eta$) should have unit value for all of the
binned $\Delta\phi,\Delta\eta$. In the correlations used first 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 tube which is part of C($\Delta\phi,\Delta\eta$) is
similar for each of the objects and will image on top of each other. We
further divide the data (see Table I) into $p_{t}$ ranges (bins).

Table I. The $p_{t}$ bins and the number of charged particles per bin with
$|\eta|$ $<$ 1.0.

Table I
---
$p_{t}$ range | amount
$4.0GeV/c-1.1GeV/c$ | 149
$1.1GeV/c-0.8GeV/c$ | 171
$0.8GeV/c-0.65GeV/c$ | 152
$0.65GeV/c-0.5GeV/c$ | 230
$0.5GeV/c-0.4GeV/c$ | 208
$0.4GeV/c-0.3GeV/c$ | 260
$0.3GeV/c-0.2GeV/c$ | 291

Since we are choosing particle pairs, we choose for the first particle
$p_{t1}$ which could be in one bin and for the second particle $p_{t2}$ which
could be in another bin. Therefore binning implies a matrix of $p_{t1}$ vs
$p_{t2}$. We have 7 bins thus there are 28 independent combination. Each
combination will have a different number of enters. In order to take out this
difference one uses multiplicity scaling[13, 14]. The diagonal bins one scales
event average of Table I. For the off diagonal combination one uses the
product of square root of corresponding diagonal event averages. In Figure 4
we show the correlation function equation 1 for the highest diagonal bin
$p_{t}$ 4.0 to 1.1 GeV/c. Figure 5 is the smallest diagonal bin $p_{t}$ 0.3 to
0.2 GeV/c.

Figure 4: $\Delta\phi$ vs.$\Delta\eta$ CI correlation for the 0-5% centrality
bin for Au + Au collisions at $\sqrt{s_{NN}}=$ 200 GeV requiring both
particles to be in bin 7 $p_{t}$ greater than 1.1 GeV/c and $p_{t}$ less than
4.0 GeV/c.

Figure 5: $\Delta\phi$ vs.$\Delta\eta$ CI correlation for the 0-5% centrality
bin for Au + Au collisions at $\sqrt{s_{NN}}=$ 200 GeV requiring both
particles to be in bin 1 $p_{t}$ greater than 0.2 GeV/c and $p_{t}$ less than
0.3 GeV/c.

Figure 6: $\Delta\phi$ vs.$\Delta\eta$ CI correlation for the 0-5% centrality
bin for Au + Au collisions at $\sqrt{s_{NN}}=$ 200 GeV for all 28 combination
of $p_{t1}$ vs $p_{t2}$ (see text).

Figure 7: $\Delta\phi$ vs.$\Delta\eta$ CI correlation for the 0-5% centrality
bin for Au + Au collisions at $\sqrt{s_{NN}}=$ 200 GeV for all 28 combination
of $p_{t1}$ vs $p_{t2}$ not rescaled (see text).

Once we use multiplicity scaling we can compare all 28 combination. In Figure
9 we show all 28 plots, but the z-axis has been rescaled so that each appear
the same size. In Figure 10 we again show the 28 plots all having the same
scale. This make it easier to see how fast the correlation signals drop off
with lowering the momentum. These plots show the properties of parton
fragmentation. P1P7 has the same signal as P2P6, P3P5, and P4P4.

##  3 The properties of the Glasma Flux Tube Model

For the above correlations on average there was 12 final state tubes on the
surface of the fireball per central Au + Au collision. Each tube on the
average showered into 49 charged particles. The soft uncorrelated particles
accounted for 873 or 60% of the charged particles. Since the tubes are sitting
on the surface of the fireball and being push outward by radial flow, not all
particles emitted from the tube will escape. Approximately one half of the
particles that are on the outward surface leave the fireball and the other
half are absorbed by the fireball (see Figure 8). This means that 20% of the
charged particles come from tube emission, and 294 particles are added to the
soft particles increasing the number to 1167 (see Table III).

Table III. Parameters of the GFTM for charged particles.

Table III
---
variable | amount | fluctuations
$tubes$ | 12 | 0
$particles$ | 24.5 | 5
$soft$ | 1167 | 34

The particles that emitted outward are boosted in momentum, while the inward
particles are absorbed by the fireball. Out of the initial 49 particles per
tube the lower $p_{t}$ particles have larger losses. In Table IV we give a
detailed account of these percentage losses and give the average number of
charged particles coming from each tube for each $p_{t}$ bin.

Table IV. Parameters of the GFTM for $p_{t}$ of the charged particles .

Table IV
---
$p_{t}$ | amount | %survive
$4.0GeV/c-1.1GeV/c$ | 4.2 | 100
$1.1GeV/c-0.8GeV/c$ | 3.8 | 76
$0.8GeV/c-0.65GeV/c$ | 3.2 | 65
$0.65GeV/c-0.5GeV/c$ | 4.2 | 54
$0.5GeV/c-0.4GeV/c$ | 3.2 | 43
$0.4GeV/c-0.3GeV/c$ | 3.3 | 35
$0.3GeV/c-0.2GeV/c$ | 2.6 | 25

In the surface GFTM we have thermalization and hydro flow for the soft
particles, while all the two particle angular correlations come from the tubes
on the surface. The charge particle spectrum of the GFTM is given by a blast
wave model and the direct tube fragmentation is only 20% of this spectrum. The
initial anisotropic azimuthal distribution of flux tubes is transported to the
final state leaving its pattern on the ring of final state flux tubes on the
surface. This final state anisotropic flow pattern can be decomposed in a
Fourier series ($v_{1}$, $v_{2}$, $v_{3}$, …). These coefficients have been
measure[15] and have been found to be important in central Au + Au collisions.

Figure 8: Each tube sits on the surface.

##  4 Charge Independent Correlation compared with the Au + Au Data

How well does the glasma flux tube model (GFTM)[1][8] reproduce the Au + Au
central collision data at $\sqrt{s_{NN}}=$ 200 GeV. A complete analysis for Au
+ Au central data 0-10% within a $p_{t}$ range of 0.8 $<$ $p_{t}$ $<$ 2.0
GeV/c can be found in Ref.[12]. Here we compare two of our $p_{t}$ ranges in a
$\Delta\eta$ range 0.6 $<$ $\Delta\eta$ $<$ 0.9 for Au + Au central collision
data 0-5% at $\sqrt{s_{NN}}=$ 200 GeV. The first $p_{t}$ range bin is P7P7
where P7 is 1.1 $<$ $p_{t}$ $<$ 4.0 Gev/c(see Figure 9). The surface flux tube
correlations are very strong in this bin. When we let one of the particles
move down in $p_{t}$ to P4 0.5 $<$ $p_{t}$ $<$ 0.65 Gev/c(see Figure 10) the
correlation weakens a lot. We see that the GFTM does a good job reproducing
the charge independent data.

Figure 9: $\Delta\phi$ for GFTM vs Au + Au data for $\Delta\eta$ range 0.6 $<$
$\Delta\eta$ $<$ 0.9 Au + Au central collision data 0-5% at $\sqrt{s_{NN}}=$
200 GeV requiring both particles to be in bin 7 $p_{t}$ greater than 1.1 GeV/c
and $p_{t}$ less than 4.0 GeV/c.

Figure 10: $\Delta\phi$ for GFTM vs Au + Au data for $\Delta\eta$ range 0.6
$<$ $\Delta\eta$ $<$ 0.9 Au + Au central collision data 0-5% at
$\sqrt{s_{NN}}=$ 200 GeV requiring one particle to be in bin 7 $p_{t}$ greater
than 1.1 GeV/c and $p_{t}$ less than 4.0 GeV/c and the second to be in bin 4
$p_{t}$ greater than 0.65 eV/c and $p_{t}$ less than 0.8 GeV/c.

Figure 11: a)2-D perspective plot of the unlike sign pairs for a single tube
GFTM signal in Au + Au central collision data 0-5% at $\sqrt{s_{NN}}=$ 200 GeV
requiring both particles to be in bin 7 $p_{t}$ greater than 1.1 GeV/c and
$p_{t}$ less than 4.0 GeV/c. b)2-D perspective plot of the unlike sign pairs
single tube data signal(see text) in Au + Au central collision data 0-5% at
$\sqrt{s_{NN}}=$ 200 GeV requiring both particles to be in bin 7 $p_{t}$
greater than 1.1 GeV/c and $p_{t}$ less than 4.0 GeV/c.

Figure 12: a)2-D perspective plot of the like sign pairs for a single tube
GFTM signal in Au + Au central collision data 0-5% at $\sqrt{s_{NN}}=$ 200 GeV
requiring both particles to be in bin 7 $p_{t}$ greater than 1.1 GeV/c and
$p_{t}$ less than 4.0 GeV/c. b)2-D perspective plot of the like sign pairs
single tube data signal(see text) in Au + Au central collision data 0-5% at
$\sqrt{s_{NN}}=$ 200 GeV requiring both particles to be in bin 7 $p_{t}$
greater than 1.1 GeV/c and $p_{t}$ less than 4.0 GeV/c.

Figure 13: Inside each flux tube a large color electric field causes pairs of
color and anti-color up quarks to be aligned in this field. The up quark ends
up in a $\pi^{+}$ and the anti-up quark ends up in a $\pi^{-}$.

##  5 Charge dependent correlations of the signal coming from a single flux
tube

In this section we are going to explore the detailed shape of the correlations
coming from a single flux tube. We will follow the method used in Refs.[8, 12,
13, 14]. We can display the correlations coming from a single flux tube if we
form S of equation 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
flux tube in each event. We then sum S over all the events.
M($\Delta\phi,\Delta\eta$) remains 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. Another way to form is same
correlation is to form S of equation 1 coming from all pairs except those from
the same flux tube in each event(we call this the background). We then
subtract this S(background) from S made out of all the pairs. If we want to
display the single flux tube correlation of the data, we subtract
S(background) from the S generated by the data.

In order to explore the charge dependence, we separate pairs into like sign
pairs and unlike sign pairs. We will show how well GFTM describes surface flux
tubes by looking at P7P7 charge dependent correlation. In Figure 11 we show
the unlike sign pair correlation of the single flux tube and the background
subtracted data. In Figure 12 we show the same for like sign pairs. It should
be noted that when we compare the data there are some pair wise track cuts.
For Figure 11 unlike sign pairs with a $\Delta\eta$ $<$ 0.1 there is a
$\Delta\phi$ $<$ $20^{0}$ cut and 0.1 $<$ $\Delta\eta$ $<$ 0.2 there is a
$\Delta\phi$ $<$ $10^{0}$ cut. These cuts remove track merging and the Coulomb
effect. For Figure 12 like sign pairs with a $\Delta\eta$ $<$ 0.2 there is a
$\Delta\phi$ $<$ $5^{0}$ cut. These cuts remove track merging and the HBT
effect.

We see in both Figure 11 and Figure 12, that the correlation from each pair
choice is well described by GFTM. We see at small $\Delta\phi,\Delta\eta$
there is a local bump for unlike sign pairs, while there is negative bump or
dip for like sign pairs. This dip or hole torn is caused by QCD field shown in
Figure 13. This field focus unlike charge pairs into the same
$\Delta\phi,\Delta\eta$ and defocus like charge pairs out of this same region.

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

###  6.1 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-Witten[16] 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 domains[17] with $\theta$ non-zero could exist and not
contradict the Vafa-Witten theorem since it is not the QCD ground state. The
metastable domains CP phenomenon would manifest itself in specific
correlations of pion momenta[17, 18].

###  6.2 Pionic measures of CP violation.

The glasma flux tube model (GFTM)[1] considers the wave functions of the
incoming projectiles, form sheets of CGC[2] at high energies that collide,
interact, and evolve into high intensity color electric and magnetic fields.
This collection of primordial fields is the Glasma[3, 4]. 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 charge[5] 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(see Figure 13). 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 pick up or lose transverse momentum(see
Figure 14). These changes in $p_{t}$ will be transmitted to the $\pi^{+}$
$\pi^{-}$ pairs.

Figure 14: Inside each flux tube a large color magnetic field causes pairs of
color and anti-color up quarks to be pushed away from this field causing a
difference in $P_{t}$ between quark and anti quark. The up quark ends up in a
$\pi^{+}$ and the anti-up quark ends up in a $\pi^{-}$.

To represent the color electric field effect we assume as the first step for
the results being shown in this paper; that we generate tubes 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
tube all the boosts are the same but vary in direction from tube to tube. 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 tube one of the charged pions boost is
increased and the other is decreased. While on the other side of the tube 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 tube are
treated in the same way however each tube 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 sections.

##  7 Color electric field pionic measure and data.

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 define sign if 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
tube (a final state of an expanded flux tube) since pairs originating from
different tubes 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 tube to show this
correlation. Thus we require that there is only $10^{\circ}$ between the
average $\phi$ of each pair. This implies that at most in $\phi$ no two pions
can differ by more than $30^{\circ}$. In Figure 15 we show two pairs which
would fall into the above cuts. $\Delta\eta_{1}$ and $\Delta\eta_{2}$ are
positive in Figure 15 but 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 Figure 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 combination 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 tube. 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 overall $\eta$ 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.

In Figure 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
PBM[8] and are charged particles of 0.8 $<$ $p_{t}$ $<$ 4.0 GeV/c, and
$|\eta|$ $<$ 1, from Au + Au central (0-10%) 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 tube correlation and thus these pairs of pairs
overall 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 Figures. 10, 11, and 14 of
Ref.[8]. Figure 11 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 (0-10%) collisions at $\sqrt{s_{NN}}$=200 GeV, in the
transverse momentum range 0.8 $<$ $p_{t}$ $<$ 2.0 GeV/c[12]. 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 shows that the PBM fit to these data gives the same results.
Figure 14 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 Figure 4 and Figure 5 of this
paper. The statistical significance of this dip in the two high precision
experiments done independently from different data sets gathered 2 years
apart[12, 14] 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. Thus this is very
strong evidence for the predicted effect of the color electric field.

Let us now apply the above analysis to 0-10% central Au + Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV. In figure 17 we show that aligned pairs positive
$\Delta\eta_{1}+\Delta\eta_{2}$ is 0.2% greater than not aligned pairs
negative $\Delta\eta_{1}+\Delta\eta_{2}$. This effect is about half the value
we had simulated, but shows that this alignment effect is present in the data.
We can obtain a measure of how significant this difference is by forming a
ratio of not aligned(unlike) over aligned(like)(see Figure 18). If there was
no effect the ratio would be a straight line at 1.0. We see that there is a
dip of 0.2% at a $\Delta\eta_{1}+\Delta\eta_{2}$ = 1.4 with an average value
shift of 0.14% over the expected range.

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 tube. The same is true for pair
2 so that it will also be contained in this same tube. 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 cannot be separated
by more than $10^{0}$ in $\Delta\phi$ in order to keep all four particles
inside the same tube since the correlation function is entirely generated
within the same tube. 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.

Figure 17:  Correlation function of pairs formed to exhibit the effects of
longitudinal momentum boosts by the color electric field as defined in the
text. For this correlation we are using real 0-10% central Au + Au collisions
at $\sqrt{s_{NN}}=$ 200 GeV data. The aligned pairs positive values are larger
by $\sim$0.2% between 1.0 $<$ $\Delta\eta_{1}+\Delta\eta_{2}$ $<$ 2.0 than
between -2.0 $<$ $\Delta\eta_{1}+\Delta\eta_{2}$ $<$ -1.0). 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.

Figure 18: We can obtain a measure of how significant this difference is by
forming a ratio of not aligned(unlike) over aligned(like). If there was no
effect the ratio would be a straight line at 1.0. We see that there is a dip
of 0.2% at a $\Delta\eta_{1}+\Delta\eta_{2}$ = 1.4 with an average value shift
of 0.14% over the expected range.

##  8 Color magnetic pionic measure and data.

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 pick up or lose transverse
momentum. These changes in $p_{t}$ will be transmitted to the $\pi^{+}$ and
$\pi^{-}$ pairs. In order to observe these differential $p_{t}$ changes one
must select pairs on one side of the tube in $\phi$ and compare to other pairs
on the other side of the same tube 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 tube. 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 tube and be close to being directly on opposite sides of
the tube (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 tube 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 Figure 19 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 Figure
19 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 $\Delta P_{t}$. 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
combination 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 tubes 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 overall $p_{t}$ 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 19: 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 tube. Since we are interested in
pairs directly across the tube 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.

In Figure 20 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 PBM[8] and are charged particles of 0.8 $<$ $p_{t}$ $<$
4.0 GeV/c, with $|\eta|$ $<$ 1, from Au + Au central (0-10%) 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
tube correlation and thus these pairs of pairs overall 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 tube particle to background increases.

We apply the magnetic analysis to 0-10% central Au + Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV. The error bars on the correlation is such that we
need increase our $\Delta\phi$ cut to $32^{\circ}$ $<$ $|\Delta\phi|$
$<$$48^{\circ}$. The STAR TPC efficiency varied with azimuthal angle because
of sector boundaries in the TPC. 12 boundaries generate effects every
$30^{\circ}$. Pairs of pairs have a lower efficiency if separated by
$30^{\circ}$. Pairs that lie with a $\phi$ separation between $25^{\circ}$ and
$35^{\circ}$ have both pairs going into sector boundaries in twelve
orientations in the TPC. This efficiency is less than other angles where one
of its pairs will go into sector boundaries at 24 orientations in the TPC.
Thus the numerator of the correlation is reduced more in the TPC data. In
figure 21 we show that aligned pairs positive $\Delta P_{t1}+\Delta P_{t2}$ is
0.1% less than not aligned pairs negative $\Delta P_{t1}+\Delta P_{t2}$. This
is about half the value that we had simulated, and shows that this alignment
effect is present in the data. The average value of the correlation has been
reduced because of the TPC efficiency. We can obtain a measure of how
significant this difference is by forming a ratio of not aligned(unlike) over
aligned(like)(see Figure 22). If there was no effect the ratio would be a
straight line at 1.0. We see that there is a rise of 0.1% in a parabolic shape
which is a good fit at a $\Delta P_{t1}+\Delta P_{t2}$ = 2.0.

Figure 20:  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.

Figure 21:  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 For this
correlation we are using real 0-10% central Au + Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV data. The not aligned pairs negative values are
larger by $\sim$0.1% at -2.0 $\Delta P_{t1}+\Delta P_{t2}$ . This means there
are more pairs of pairs aligned in the opposite direction compared to pairs
aligned as predicted by the color magnetic field.

Figure 22: We can obtain a measure of how significant this difference is by
forming a ratio of not aligned(unlike) over aligned(like). If there was no
effect the ratio would be a straight line at 1.0. We see that there is a rise
of 0.1% in a parabolic shape which is a good fit at a $\Delta P_{t1}+\Delta
P_{t2}$ = 2.0.

##  9 Summary and Discussion

In this paper we discuss a Glasma Flux Tube Model(GFTM)[1] which has tubes of
parallel strong color QCD electric and magnetic fields that are generated by
the initial conditions of Au + Au central collisions at $\sqrt{s_{NN}}$=200
GeV. We summarize the assumptions made and the reasoning that led to the
development and construction of GFTM, which successfully explained the charge-
particle-pair correlations in the central (0-10% centrality) $\sqrt{s_{NN}}=$
200 GeV Au Au data[12].

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.
Initially the transverse space is filled with flux tubes of large longitudinal
extent but small transverse size $\sim$$Q^{-1}_{s}$. The flux tubes that are
near the surface of the fireball get the largest radial flow and are emitted
from the surface. These partons shower and the higher $p_{t}$ particles escape
the surface and do not interact. $Q_{s}$ is around 1 GeV/c thus the size of
the flux tube is about 1/4 fm initially. The flux tubes near the surface are
initially at a radius $\sim$5 fm. The $\phi$ angle wedge of the flux tube
$\sim$ 1/20 radians or $\sim$$3^{\circ}$. The flux tubes on the surface turn
out to be on the average 12 in number. They form an approximate ring about the
center of the collision see Figure 1. The twelve tube ring creates the average
behavior of tube survival near the surface of the expanding fire ball of the
blast wave. The final state surface tubes that emit the final state particles
at kinetic freezeout for this paper are given by the PBM[8]. One should note
that the blast wave surface is moving at its maximum velocity at freezeout
(3c/4).

We explored the charge dependence correlation where we separate pairs into
like sign pairs and unlike sign pairs. In Figure 11 we show the unlike sign
pair correlation of the single flux tube and the background subtracted data.
In Figure 12 we show the same for like sign pairs. We see in both Figure 11
and Figure 12, that the correlation from each pair choice is well described by
GFTM. We see at small $\Delta\phi,\Delta\eta$ there is a local bump for unlike
sign pairs, while there is negative bump or dip for like sign pairs. This dip
or hole torn is caused by QCD field shown in Figure 13. This field focus
unlike charge pairs into the same $\Delta\phi,\Delta\eta$ and defocus like
charge pairs out of this same region.

Strong CP violation (Chern-Simons topological charge) is treated in Sec. 6.

We develop 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 charge[5] 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 pick up 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 tube to tube 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. We observe in the STAR TPC $\sqrt{s_{NN}}$
= 200 GeV central Au Au collision data that these correlations are as expected
and is a strong confirmation of this theory(see Sec. 7 and Sec. 8). The much
larger unlike-sign pairs than like-sign pairs in the data and the strong dip
of the CI correlation at small $\Delta\eta$ (see Sec. 5) shows very strong
evidence supporting the color electric alignment prediction in the
$\sqrt{s_{NN}}$ = 200 GeV central Au Au collision data analyses at RHIC[8,
12]. 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 the strongest evidence for the predicted effect of the color
electric field.

##  10 Acknowledgments

This research was supported by the U.S. Department of Energy under Contract
No. DE-AC02-98CH10886. The author thanks Sam Lindenbaum and William Love for
valuable discussion and Bill for assistance in production of figures. It is
sad that both are now gone.

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