# Chiral Symmetry Restoration using the Running Coupling Constant from the
Light-Front Approach to QCD

S. D<EMAIL_ADDRESS>Applied Mathematics Laboratory-CCTS/DFQM,
Federal University of São Carlos,
Sorocaba CEP 18052780, Brazil

###### Abstract

In this work, the distance between a quark-antiquark pair is analyzed through
both the confinement potential as well as the hadronic total cross section.
Using the Helmholtz free energy, entropy is calculated near the minimum of the
total cross section through the confinement potential. A fitting procedure for
the proton-proton total cross section is performed, defining the fitting
parameters. Therefore, the only free parameter remaining in the model is the
mass scale $\kappa$ used to define the running coupling constant of the light-
front approach to QCD. The mass scale controls the distance $r$ between the
quark-antiquark pair and, under some conditions, it allows the occurrence of
free quarks even in the confinement regime of QCD.

###### pacs:

11.30.Rd, 13.85.Lg

##  I Introduction

Chiral symmetry restoration is one of the most interesting issues in the phase
diagram of Quantum Chromodynamics (QCD). As well-known, there is evidence of a
phase transition from confinement to non-confinement phase at the saturation
scale and predicted, for example, by lattice simulations [1, 2, 3, 4]. These
results were supported by recent analyses performed by the Beam Energy Scan
(BES) program [5, 6] on experimental data from Relativistic Heavy-Ion
Collisions.

As a fundamental ingredient, the internal entropy of colliding hadrons should
be taken into account to offer a complete view of the chiral symmetry
restoration in QCD. However, the proper definition of entropy is a hard task
in any physical context. For example, entropy can be related to one-quarter of
the black hole event horizon area or Shannon’s information disorder approach.
Moreover, the correct definition of entropy is a problem that permeates
several issues as, for example, the study of entropy production in rotating
black holes [7], in the so-called Ekpyrotic universe [8], in Szekeres
spacetimes [9], in quark-gluon plasma [10], in light clusters [11], and
decoherence processes [12], among others.

Recently, a maximal entropy calculation was introduced assuming saturation
effects due to the properties of the factorizable gluon density [13]. This
result has closely connected with those obtained previously by Kharzeev and
Tuchin [14], whose achievements are due to the use of the Unruh effect
(analogous to the Hawking effect) applied to hadrons in the accelerated rest
frame. They found a transition temperature $T_{s}\approx Q_{s}/2\pi$ at the
spontaneous symmetry breaking, where $Q_{s}$ is the transferred momentum in
the gluon rest frame.

On the other hand, a possible topological phase transition in the hadron was
proposed in Ref. [15], allowing a chiral symmetry breaking below the
saturation scale with a transition temperature $T_{c}\approx 0.001$ GeV, far
below the Hagedorn temperature. Thus, above $T_{c}$ but below the Hagedorn
temperature [16, 17], the appearance of a free quark state is allowed,
resulting in an effect quite similar to the emergence of the free vortex in
the XY model. It should be emphasized that was predicted previously a chiral
symmetry breaking by McLerran and Pisarski [18] as well by Glozman and
Wagenbrunn [19], whose possible explanation is given by the presence of
quarkyonic matter, introduced years ago [18, 19]. In this scenario, the chiral
symmetry is restored even in the presence of hadronic matter.

In the present work, one supposes that the hadronic total cross section,
$\sigma_{tot}(s)$, can be divided into a finite number of non-interacting
disjoint 2D cells, each one containing a quark-antiquark ($q\bar{q}$) pair
($u\bar{u}$ and $d\bar{d}$) separated by a distance $r(s)$, depending on the
energy $\sqrt{s}$ measured in the center-of-mass system [15]. This approach is
analogous to the so-called Berezinskii-Kosterlitz-Thouless (BKT) phase
transition, which occurs without spontaneous symmetry breaking [20, 21]. In
that planar model, at low temperatures, the presence of nontrivial vortex
configuration is suppressed due to the binding effect caused by the vortex-
antivortex interaction. At high temperatures above a critical value, the
binding effect can be neglected, and the nontrivial vortex state emerges.

The main goal of the present work is to take into account the physical
consequences of the confinement potential inside the cell containing the
$q\bar{q}$-pair considering a naive model with finite temperature and null
quark chemical potential. Then, the free quark state is achieved when the
spatial separation, $r(s)$, between quark and antiquark is less than $r_{0}$,
the confining scale. Using some constraints, one obtains an asymptotic bound
to the rise of $r(s)$ as the collision energy grows. Moreover, using the
running coupling from the light-front holographic QCD (see [22] and references
therein), the non-confinement regime is achieved below the saturation scale
for $\kappa\approx 0.002$ GeV. The mass scale $\kappa$ emerges in de Alfaro,
Fubini, and Furlan’s attempt to build a fully invariant conformal Lagrangian
theory [23]. In the example they gave in the original work, the mass scale
provides the necessary infrared cut-off of the theory, and once determined the
scale furnishes the rest of the unknown parameters of the problem. In the
present context presented here, $\kappa$ is used as a free parameter
representing the $e\\!f\\!f\\!ective$ mass of the cell, being straightly
connected with the distance $r(s)$ as shall be seen.

The manuscript is organized as follows. In section II, one introduces the
relation for the entropy, total cross section, and the distance between the
quark and antiquark in the cell. In section III, one shows that, under some
conditions, the possible existence of a chiral symmetry restoration in the
confinement phase of QCD. Final remarks are left for section IV.

##  II Total Cross Section and Entropy

First of all, it should be stressed that due to the Fitzgerald-Lorentz length
contraction, the hadron contracts in the direction of its motion, although its
visual appearance, which depends exclusively on its motion, is invariant in
special relativity [24, 25]. Thus, a relativistically moving sphere is not an
ellipsoid, but it would be a rotated sphere [25]. However, as the speed
approaches $c$ in its proper reference frame, the hadron tends to become a
thin disk. Then, it is reasonable to suppose that some physical properties
observed in 2D planar objects may occur in the hadron [15]. Here, we maintain
the analogy between the hadron in high-energy collisions and 2D planar objects
as done elsewhere [15]. Thus, one assumes that hadronic total cross section
can be divided into a finite number of disjoint cells containing a single
$q\bar{q}$-pair, that interacts with each other through a confinement
potential. Then, the entropy is written as [15]

$\displaystyle S(s)=\ln\left(\frac{\sigma_{tot}(s)}{\pi r^{2}}\right),$ (1)

where $r=r(s)$ is the radius of the cell containing the $q\bar{q}$-pair. The
above assumption was based on the well-known behavior of 2D planar objects
with vortex-antivortex pairs, where the presence of a BKT phase transition
allows the emergence of new degrees of freedom and the consequent emergence of
a free vortex [20, 21].

Although the experimental access to $r(s)$ is forbidden at the present-day
energies, the theoretical determination of some physical bounds is possible,
in principle, by using high-energy theorems as well as taking into account
different approaches to entropy $S(s)$. Therefore, one can rewrite relation
(1) as

$\displaystyle r^{2}=\frac{1}{\pi}\sigma_{tot}(s)e^{-S(s)}.$ (2)

A remarkable realization of Axiomatic Quantum Field Theory in the early 1960s
is the Froissart-Martin bound. However, despite its beauty, the Froissart-
Martin result is an asymptotic inequality: It does not give us a hint about
the behavior of $\sigma_{tot}(s)$ at intermediate energies or even the
”correct” functional form of $\sigma_{tot}(s)$ at high energies. That bound is
usually written as

$\displaystyle\sigma_{tot}(s)\leq\frac{\pi}{m_{\pi}^{2}}\ln^{2}(s/s_{0}),$ (3)

where $m_{\pi}$ is pion mass and $\sqrt{s_{0}}$ is some initial energy
(usually $\sqrt{s_{0}}=1.0$ GeV). Of course, for $0<e^{-S(s)}$, one can write
the inequality

$\displaystyle
r^{2}\leq\left[\frac{1}{m_{\pi}^{2}}\ln^{2}(s/s_{0})\right]e^{-S(s)},$ (4)

which acts as an upper bound for $r$. Notice, in addition, that the total
cross section is related to the imaginary part of the forward scattering
amplitude through the well-known optical theorem

$\displaystyle\sigma_{tot}(s)=\frac{\mathrm{Im}F(s)}{s},$ (5)

where $F(s)$ is the forward scattering amplitude. Thus, one can also write

$\displaystyle r^{2}=\left[\frac{\mathrm{Im}F(s)}{s}\right]e^{-S(s)}.$ (6)

The usage of specific parameterizations for $\sigma_{tot}(s)$ can help to
understand the behavior of $r$ depending on $\sqrt{s}$. In addition, some
known results can also be used: For example, for small values of transferred
momentum $\sqrt{|t|}$ in the center-of-mass system, the scattering amplitude
is mainly given by its absorptive part. Then, one can use the differential
cross section, for example, for $\sqrt{|t|}<0.1$ GeV, to infer effects
introduced by $t$ in Eq. (6).

On the other hand, considering the result given by Eq. (4), we can analyze the
distance $r$ according to $e^{-S(s)}$.

Firstly, consider the case where $S(s)$ is not a logarithmic function of its
variable. Then for an increasing entropy of the form $S(s)\sim s/s_{0}$ (for
example), the distance $r$ fastly decreases, leading to the presence of free
quarks due to chiral symmetry restoration. In contrast, for a system with
decreasing entropy of the form $S(s)\sim s_{0}/s$, the distance $r$ increases
as the collision energy grow.

Secondly, considering that $S(s)$ is a logarithmic function of its variable.
Then for a system with a slowing increasing entropy given by
$S(s)\sim\ln(s/s_{0})$, $r$ tends to zero for sufficiently high energy. Of
course, the decreasing entropy case is given by $S(s)\sim\ln(s_{0}/s)$, which
implies the increase of $r$ as $s$ grow. Both situations where entropy
decreases as the collision energy grows, although interesting, are not treated
here.

It is important to stress that the definition of $S(s)$ only makes sense here
if one takes into account the relative distance between the $q$ and $\bar{q}$.
Thus, $S(s)$ should explicitly depend on $r$. Following the approach presented
here, one introduces the distance $r$ in $S(s)$ considering the confinement
potential, as shall be seen.

##  III Finite Temperature: Far Below the Saturation Scale

###  III.1 Confinement Potential and Running Coupling in the Confinement Phase
of QCD

As a key point for the problem, the internal energy of the colliding hadron
cannot be deduced either from the first principles of QCD or thermodynamics.
Thus the use of potential energy may be a useful approach to get some insights
into the physical processes, as performed since the 1950s [26, 27, 28, 29].

There are several confinement potentials in the literature (for example, see
[30, 31, 32, 33, 34]). For the sake of simplicity, one considers only the
Cornell confinement potential [32, 33, 34], which presents a good agreement
with experimental data for the spectrum of light and heavy mesons [35, 36,
37]. That potential is usually written as

$\displaystyle V(r)=-\frac{4}{3}\frac{\alpha_{s}(r)}{r}+\sigma r,$ (7)

where $\alpha_{s}(r)$ is the Fourier sine transform of the running coupling
constant $\alpha_{s}(Q)$, and $\sqrt{\sigma}$ is the string tension. For
static-source $\sqrt{\sigma}=0.46$ GeV [38] while an average estimation
$\sqrt{\sigma}=0.405$ GeV is obtained for cold strongly interacting matter
[39]. Without loss of generality, one adopts hereafter $\sqrt{\sigma}=0.4$
GeV.

The running coupling constant of QCD is far from a unified definition in the
confinement regime of QCD. Although the perturbative sector of QCD has a well-
defined definition, the same does not occur with the non-perturbative regime,
where several approaches can be used. Two basic examples of definitions are
the ”analytic” running coupling obtained by Shirkov and Solovtsov by using the
spectral density from Källén-Lehmann relation [40] and the running coupling
from the light-front approach to QCD [41].

In this work, one uses the light-front holographic QCD approach to the running
coupling [42, 22], whose choice is based on both its simplicity as well good
description of the available experimental data [22]. In the light-front
holographic approach, the running coupling is set for all values of $Q$ as
[22]

$\displaystyle\alpha_{s}(Q)=e^{-Q^{2}/4\kappa^{2}},$ (8)

where $\kappa$ is the mass scale (also called mass parameter) determined from
the soft-wall model [43, 44], and defined for the hadron as

$\displaystyle M^{2}=4\kappa^{2}(L+S/2+n),$ (9)

where $M$ is the hadron mass, $L$, $S$, and $n$ refer to internal orbital
angular momentum, internal spin, and radial quantum number, respectively. The
soft-wall model is based on a quadratic dilaton field [44] that leads to a
harmonic confining potential whose strength is measured by $\kappa$. The
consequence of such potential is the breaking of conformal invariance.

It is important to stress that $\kappa$ has a value that depends on the meson
Regge trajectories analyzed, and settle down, in general, in the range
$0.3\lesssim\kappa\lesssim 0.8$ GeV [43]. Notice, in addition, that the linear
Regge trajectories given by (9) were obtained due to the quadratic dilaton
field approach [44]. However, this model entangles the explicit and
spontaneous breaking of chiral symmetries [45]. It is also important to stress
that the linearity of Regge trajectories is not true everywhere, being more
evident for light baryons and mesons [46]. Thus, definition (9) may not hold
everywhere.

Generally, is necessary a value of $\kappa\approx 0.5$ GeV to describe mesons
and baryon masses as well as the assumption of linear Regge trajectories [41,
47]. This value also satisfactorily describes the mesons trajectories of the
$\rho$ and $K^{*}$ [43]. In contrast, from fitting the nucleon form factors in
the Anti-de Sitter (AdS)/QCD soft-wall model, one has $\kappa\approx 0.40$ GeV
[48]. Moreover, from form factors analysis, the values of $\kappa$ are lower
than those for Regge trajectories and mass spectrum [43]. For example,
$\kappa=0.261\pm 0.002$ GeV assuming a constant current quark mass
$m_{q}=0.005$ GeV in a model inspired in the AdS/QCD correspondence [49].

As aforementioned, one supposes that mass scale $\kappa$ represents the
$e\\!f\\!f\\!ective$ mass of the $q\bar{q}$-pair in the cell and, then, one
assumes $\kappa<\\!<1$ GeV. This constraint will be clarified later in this
work.

Returning to the running coupling constant. To avoid the use of the mnemonic
rule from Quantum Mechanics $Q\sim 1/r$, one should use the Fourier sine
transform of Eq. (8) since, as noted by Shirkov, this rule may not be true
everywhere [50]. Then, for a rigorous mathematical approach, one writes the
Fourier sine transform of Eq. (8) as [51]

$\displaystyle\alpha_{s}(r)=\int_{0}^{\infty}e^{-Q^{2}/4\kappa^{2}}\sin(Qr)dQ,$
(10)

which can be solved in terms of Dawson integral function defined as [52, 53]

$\displaystyle D(x)=e^{-x^{2}}\int_{0}^{x}e^{-y^{2}}dy^{2},$ (11)

closely related to the error function $er\\!f(x)$ by [53]

$\displaystyle D(x)=-i\frac{\pi}{2}e^{-x^{2}}er\\!f(ix).$ (12)

The function defined by Eq. (11) can be approximated by the following series
expansion depending on $x$ is close to the origin ($x<\\!\\!<1$) [53, 54]

$\displaystyle
D(x)=\sum_{j=0}^{\infty}\frac{(-1)^{j}2^{j}}{(2j+1)!!}x^{2j+1}=x-\frac{2}{3}x^{3}+\frac{4}{15}x^{5}-...,$
(13)

where $n!!$ is the double factorial defined as

$\displaystyle n!!=\prod_{j=0}^{k}(n-2j),$ (14)

for $k=\lceil n/2\rceil-1$, where $\lceil x\rceil$ is the ceiling function
(least integer greater than or equal to $x$). In contrast, for large $x$, the
Dawson function given by Eq. (11) has an asymptotic behavior given by [54]

$\displaystyle D(x)\approx\frac{1}{2x}+\frac{1}{2^{2}x^{3}}+\frac{1\cdot
3}{2^{3}x^{5}}+\frac{1\cdot 3\cdot 5}{2^{4}x^{7}}+...$ (15)

The main goal is to study a possible chiral symmetry restoration, implying a
small value for $x=\kappa r$. For the sake of simplicity, let us suppose that
$r\lesssim r_{h}$ fm, where $r_{h}$ is the hadron radius. To achieve the
confinement phase of QCD, we impose the constraint $\kappa<\\!\\!<1$ GeV, for
which one has $\kappa r<\\!\\!<1$, resulting we can use the series given by
Eq. (13) to represent the running coupling in $r$-space, and, in particular,
$(\kappa r)^{3}<\\!\\!<1$. As shall be seen, the ratio $r/r_{0}$, for which
$r/r_{0}<1$ indicates the non-confinement regime is achieved, depends strongly
on the choice of $\kappa$.

It is important to notice that the mass scale is also present in the Fourier
sine transform since $x=\kappa r$, which introduces the need for careful
analysis to obtain the correct representation of $\alpha_{s}(r)$ through the
Dawson function. Observe that the presence of the mass scale result, after the
Fourier sine transform given by Eq. (10), in a dimensionful running coupling
written as

$\displaystyle\alpha_{s}(r)=2\kappa\sqrt{\frac{2}{\pi}}D(\kappa
r)=2\kappa\sqrt{\frac{2}{\pi}}\left[\kappa r-\frac{2}{3}(\kappa
r)^{3}+\frac{4}{15}(\kappa r)^{5}-...\right].$ (16)

Then, it is necessary to change
$\bar{\alpha}_{s}(r)\rightarrow\alpha_{s}(r)/\kappa$ in the above definition
to restore the dimensionless feature of the running coupling. Now, taking into
account the stated before, one can replace $\bar{\alpha}_{s}(r)$ in the
confinement potential given by Eq. (7)

$\displaystyle V(r)=-\frac{4}{3}\frac{\bar{\alpha}_{s}(r)}{r}+\sigma
r=-\frac{8}{3}\kappa\sqrt{\frac{2}{\pi}}\left[1-\frac{2}{3}(\kappa
r)^{2}+\frac{4}{15}(\kappa r)^{4}-...\right]+\sigma r.$ (17)

Considering the above constraints on $\kappa$ and $r$, one can retain the
series expansion up to the second order, allowing to write the confinement
potential as

$\displaystyle V(r)=-\frac{4}{3}\frac{\bar{\alpha}_{s}(r)}{r}+\sigma
r\approx-\frac{8}{3}\kappa\sqrt{\frac{2}{\pi}}\left[1-\frac{2}{3}(\kappa
r)^{2}\right]+\sigma r.$ (18)

The above result indicates a confinement potential with a finite negative
range given by

$\displaystyle V(0)=-\frac{8}{3}\kappa\sqrt{\frac{2}{\pi}},$ (19)

and dependent exclusively on the mass scale.

###  III.2 Entropy

For the region near the minimum of the total cross section, the hadron volume
is almost constant, implying the entropy can be obtained by using the
Helmholtz free energy [15]

$\displaystyle S_{H}(s)=\frac{1}{T_{c}}V(r),$ (20)

where $r=r(s)$, and the transition temperature $T_{c}$ depends on the
potential approach used and $T_{c}\approx 0.001$ GeV for the Cornell potential
[15]. It is important to point out that (20) should be used only close to the
minimum of $\sigma_{tot}(s)$ since the Helmholtz free energy does not
necessarily vanish elsewhere.

The main assumption here is that the entropy given by Eq. (1) can be defined
through the Helmholtz free energy near the minimum of $\sigma_{tot}(s)$. Thus,
one can write

$\displaystyle S(s)=\gamma S_{H}(s)$ (21)

where $\gamma$ is a real parameter. The justification of such a choice is
quite difficult and is based on the fact that both definitions depend on $r$
at the minimum of $\sigma_{tot}(s)$. For the sake of simplicity, one assumes
$\gamma=1$ since it acts only as a scale to the problem.

Using potential given by Eq. (18) and entropy in Eq. (20), one can introduce a
cut-off in the series expansion of the exponential in Eq. (2) if
$V(r)/T_{c}<\\!\\!<1$. This cut-off can be obtained for $\kappa\sim
T_{c}<\\!\\!<1$ GeV. If satisfied, these constraints can indicate the
unambiguous emergence of free quarks, as shall be seen. Then, one writes the
following approximation for $r$

$\displaystyle
r^{2}\approx\left\\{1-\frac{1}{T_{c}}\left[-\frac{8}{3}\kappa\sqrt{\frac{2}{\pi}}\left(1-\frac{2}{3}(\kappa
r)^{2}\right)+\sigma r\right]\right\\}\sigma_{tot}(s),$ (22)

according to the constraints on $\kappa$, $r$, and confinement potential
adopted. Of course, different approaches for $V(r)$ and $\alpha_{s}(r)$ will
produce distinct expressions for $r$. Moreover, there are definitions for
$V(r)$ that are independent of the running coupling (see, for example, [30]).

###  III.3 Total Cross Section

Now, it is necessary to introduce a functional form for the total cross
section. In the intermediate energy range, one can use the following simple
parameterization for the total cross section [55]

$\displaystyle\sigma_{tot}(s)=a_{1}(s/s_{0})^{a_{2}}+a_{3}\ln^{\alpha_{\mathbb{P}}(0)}(s/s_{0}),$
(23)

where $a_{1}$, $a_{2}$, $a_{3}$, and $\alpha_{\mathbb{P}}(0)$ are free
parameters. In the fitting process, one uses only the experimental data
available for $pp$ total cross section above $\sqrt{s}=3.0$ GeV (including
cosmic-ray data), obtained from Particle Data Group [56]. From the fitting
procedure, one obtains the parameters shown in Table 1, where
$\alpha_{\mathbb{P}}(0)$ is typical of the hard pomeron picture [57, 58].

$a_{1}$ (mb) |  $a_{2}$ |  $a_{3}$ (mb) |  $\alpha_{\mathbb{P}}(0)$ |  $\chi^{2}/ndf$
---|---|---|---|---
$52.52\pm 0.38$ |  $0.14\pm 0.01$ |  $0.91\pm 0.08$ |  $1.61\pm 0.03$ |  1.77
Table 1: Parameters obtained by using (23) in the fitting procedure and
assuming $\sqrt{s_{0}}=1.0$ GeV. Figure 1: Behavior of $r/r_{0}$ near the
minimum of the total cross section. Solid line is for $\kappa=0.1$ GeV and
dashed line is for $\kappa=0.002$ GeV. Inner panel shows only the curve for
$\kappa=0.002$ GeV where $r/r_{0}\approx 0.066$, indicating the presence of
free quarks in the confinement phase of QCD.

As well-known, the hadronic total cross section decreases as the collision
energy grows, from the Coulomb-Nuclear region up to some minimum located in
the energy range $\sqrt{s}\approx 10\sim 30$ GeV. This behavior is attributed
to the odderon exchange, also known as the odd-$C$ Regge exchange, a three-
gluon state [59], recently observed [60]. This leading particle exchange
differentiates particle-particle from particle-antiparticle scattering and, as
$\sqrt{s}$ grows, this exchange becomes less important, implying that, at
sufficiently high energy, both total cross sections tend to the same value
(according to the Pomeranchuk theorem). In fact, this behavior for both
$\sigma_{tot}(s)$ at very high energies is attributed to the exchange of a
leading particle called pomeron (not yet observed), which does not
differentiate particle from antiparticle.

###  III.4 Results Depending on $\kappa$

Taking into account the approaches above performed as well as the fitting
result, the only remaining free parameter of the approach is the mass scale
$\kappa$.

Figure 1 shows the evolution of $r/r_{0}$ near the minimum of the total cross
section according to $\sqrt{s}$. As expected, it presents an almost flat
behavior since there is (almost) no variation on the hadronic total cross
section in this energy range. This result indicates the distance between the
$q\bar{q}$-pairs in the cell remains almost unchanged near the minimum of
$\sigma_{tot}(s)$. The solid line describes the evolution of $r/r_{0}$ for
$\kappa=0.1$ GeV (solid line) and for $\kappa=0.002$ GeV (dashed line) in the
energy range $\sqrt{s}=10\sim 30$ GeV. For $\kappa=0.1$ GeV there is no
emergence of a non-confinement phase since $r/r_{0}\approx 2.6$. In this
situation, $r>1$ fm may break the approximations performed here. In contrast,
for $\kappa=0.002$ GeV, the ratio $r/r_{0}\approx 0.066$, indicating the
unequivocal emergence of chiral restoration in the confinement phase of QCD.
It is important to point out that $\kappa\sim m_{q}$, where $m_{q}$ is the
current quark mass (also $T_{c}\sim m_{q}/2$).

Figure 2: For panels (a) and (c), solid line ($\kappa=0.002$ GeV), dashed line
($\kappa=0.1$ GeV), and dotted line ($\kappa=0.500$ GeV). Panels (b) and (d)
shows the behavior of the running coupling for $\kappa=0.002$ GeV. The fast
decreasing of $\alpha_{s}(Q)$ in panel (b) represents a very slow increase in
the position space (d).

One observes that when $\kappa$ diminishes, the distance $r$ between quark and
antiquark in the cell also decreases. Then, the mass scale measures the
$e\\!f\\!f\\!ective$ mass of the cell, and when $\kappa\lesssim 0.002$ GeV,
the quark and antiquark are free to achieve new degrees of freedom. It does
not mean that all pairs inside the hadron will achieve the non-confinement
phase at this energy range: Only the pairs where $\kappa\lesssim 0.002$ GeV
$\sim m_{q}$. The unbinding pairs are shielded by binding pairs that have not
achieved the non-confinement phase, and, probably, these initial free quarks
are created near the center of the hadron, explaining why we do not see these
free states in the confinement regime of QCD.

It is important to stress that the chiral restoration can also be achieved
assuming, for example, $\kappa\lesssim 0.035$ GeV, resulting in
$r/r_{0}\lesssim 0.94$. However, high values for $\kappa$ cannot ensure the
validity of the truncation of the exponential series performed to obtain Eq.
(22).

The appearance of free quarks in the confinement phase of QCD at the center of
the colliding hadron may explain the hollowness effect [61, 62], marked by the
appearance of a gray area near the center of the hadron (please, see [63], and
references therein). This effect, roughly speaking, indicates that the black
disk limit cannot be achieved even for asymptotic energies.

In Figure 2, panels (a) and (b) show the behavior of $\alpha_{s}(Q)$ using (8)
and depending on $\kappa$. Solid line is for $\kappa=0.002$ GeV (almost
invisible in panel (a)), dashed line is for $\kappa=0.1$ GeV, and for
$\kappa=0.500$ GeV one has the dotted line. Observe that adopting $\kappa$ as
a free parameter introduces the possibility of a non-confinement phase even
for small $Q$. Panels (c) and (d) are for $\bar{\alpha}_{s}(r)$ using the same
values for $\kappa$. In panel (c), the solid line for $\kappa=0.002$ GeV is
almost flat compared to the other curves, while in panel (a), considering the
same $\kappa$, the solid line presents a fast decrease.

##  IV Final Remarks

In the BKT phase transition, the vortex-antivortex pair are unbinding at the
critical temperature. An analogy for this phase transition can be performed,
treating $q\bar{q}$-pairs as the vortex-antivortex pairs [15]. As obtained in
this work, under some physical conditions, the $q\bar{q}$-pairs can no longer
exist even in the confinement phase of QCD. The possible chiral symmetry
restoration in the confinement phase should be understood as the co-existence
of free quarks and colorless states during the collision.

It is interesting to point out that a few years ago was predicted both the
existence of quarkyonic matter (which is not a quark) and the possibility that
it suffers a chiral symmetry restoration in the confinement phase of QCD [18,
19]. According to McLerran and Pisarski [18], for a large number of colors
$N$, it is reasonable that for a large chemical potential $\mu_{q}$ and low
temperature, a confining regime exists as well as the preservation of chiral
symmetry.

Using the running coupling constant defined in the light-front approach for
QCD, we obtain a confinement potential depending on the mass scale $\kappa$,
considered here as the effective mass of the $q\bar{q}$-pair. The Helmholtz
free energy near the minimum of the total cross section allows connecting
entropy with the confinement potential. A fitting procedure is implemented to
$\sigma_{tot}(s)$ using $pp$ experimental data. Then, the ratio $r/r_{0}$ is
analyzed depending only on $\kappa$.

Considering the mass scale as a free parameter, one assumes that $\kappa$
represents the effective mass contained in the cell. For $\kappa=0.002$ GeV,
the ratio $r/r_{0}\approx 0.066$ indicates the emergence of chiral symmetry
restoration in the confinement phase of QCD. It is important to stress that
$\kappa$ is close to the current quark mass, i.e. at the chiral symmetry
restoration, one may expect $\kappa\sim m_{q}$. On the other hand, for
$\kappa\gtrsim 0.035$ GeV, there is no emergence of free quarks in the
confinement phase of QCD. Notice, furthermore, that in the range $0.002$
GeV$\lesssim\kappa\lesssim 0.035$ GeV the chiral symmetry restoration is
allowed if the approximations performed could be ensured.

As a final conjecture, the decreasing of $\kappa$, resulting in the decreasing
of $r$, may be a consequence of the odderon exchange occurring from the
Coulomb-Nuclear region up to the minimum of $\sigma_{tot}(s)$. As the
collision energy smoothly grows above the minimum of $\sigma_{tot}(s)$, the
odderon is no longer the leading particle exchange, which could imply a
decrease in the emergence of free quarks. However, above the minimum of
$\sigma_{tot}(s)$ the creation of free quark through odderon exchange,
although less effective due to the diminishing (not the vanishing) in the
odderon exchange, possibly continues up to the hadron dissociation (at the
saturation scale). Since the emergence of free quarks leads to the increasing
of entropy inside the hadron, then the growth of $\sigma_{tot}(s)$ above the
minimum may be caused (in thermodynamics language) by the increase of entropy,
which leads to the hadron dissociation at the saturation scale [64].

## Acknowledge

SDC thanks to UFSCar for the financial support.

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# 2D to 1D magnetic interactions evolution in Cu2F5-x through electron doping
by fluoride non-stoichiometry

Dmitry M. Korotin<EMAIL_ADDRESS>M.N. Mikheev Institute of Metal Physics
of Ural Branch of Russian Academy of Sciences, 18 S. Kovalevskaya St.,
Yekaterinburg, 620137, Russia.  Dmitry Y. Novoselov  M.N. Mikheev Institute of
Metal Physics of Ural Branch of Russian Academy of Sciences, 18 S.
Kovalevskaya St., Yekaterinburg, 620137, Russia.  Department of Theoretical
Physics and Applied Mathematics, Ural Federal University, 19 Mira St.,
Yekaterinburg 620002, Russia  Vladimir I. Anisimov  M.N. Mikheev Institute of
Metal Physics of Ural Branch of Russian Academy of Sciences, 18 S.
Kovalevskaya St., Yekaterinburg, 620137, Russia.  Department of Theoretical
Physics and Applied Mathematics, Ural Federal University, 19 Mira St.,
Yekaterinburg 620002, Russia

###### Abstract

The copper fluoride Cu2F5 is a compound with 2D-magnetic exchange interactions
between the Cu ions in the $S=1$ and $S=\frac{1}{2}$ spin-states. Using ab-
initio calculations, we predict that the existence of 5% vacancies in the
fluoride sublattice of Cu2F5 results in the drastic transformation of the
spin-state of all copper ions and the final spin-states are $S=\frac{1}{2}$
and $S=0$. Consequently, the anisotropy of magnetic interactions increases,
and the 1D linear chains of the Cu $d^{9}$, $S=\frac{1}{2}$ ions appear. We
also propose a microscopic mechanism of such exchange interaction
transformation via CuO6 octahedra elongation.

##  I Introduction

Cuprates are well-known objects for the low-dimensional magnetism appearance.
With such structural building blocks as CuO6 octahedra and CuO4 plaquettes,
and due to the presence of copper ion in $d^{9}$ electronic configuration,
there is a variety of magnetic structures. In perovskite-like KCuF3 [1, 2],
which structure is formed with corner-sharing CuF6 octahedra, there is the
G-type antiferromagnetic ordering of moments with existence of 1D magnetic
chains of Cu ions. 2D antiferromagnetic ladders are realized in Srn-1Cun+1O2n
[3] and SrCu2O3 [4] with edge-sharing CuO4 plaquettes. And the corner-sharing
plaquettes in Sr2CuO4 [5] and AgCuVO4 [6] led to appearance of the one-
dimensional chain of $S=\frac{1}{2}$ Cu ions.

The plethora of spin lattices mentioned above exist in the copper-oxygen
complexes. The stability of the copper-fluoride complex, Cu2F5, was predicted
recently [7]. Structurally it is formed by both blocks: the octahedra (CuF6)
and the plaquettes (CuF4). Consequently, one can expect that the magnetic
anisotropy, similar to the one seen in the cuprates, could appear in the
fluoride with or without additional doping with carriers. In our previous work
[8], within the DFT+U calculations, we show that in stoichiometric Cu2F5, the
Cu ions are in the $S=1$ and $S=\frac{1}{2}$ spin-states ($d^{8}$ and $d^{9}$
electronic configuration). Additionally, we obtained that the Heisenberg
exchange interaction along the $a$ crystal axis is tiny. In the (100)-plane
there is antiferromagnetic superexchange between the half-filled $d$-orbitals
of the nearest Cu ions through fluoride $p$-states located in between. As a
consequence, the 2D spin-lattice exists in stoichiometric Cu2F5.

One can assume that if an extra electron occupies the half-filled $d$-orbital
of Cu ion in Cu2F5, the corresponding Cu-$d$ $\leftrightarrow$ F-$p$
$\leftrightarrow$ Cu-$d$ superexchange interaction will be destroyed. More
extra electrons will cause a significant modification of the exchange
interactions. Our purpose was to find conditions under which the one-
dimensional magnetic interactions prevail in copper fluoride.

We have produced additional electrons in the cell considering the doped
compound – Cu2F5-x. Each fluoride vacancy results in an extra electron that
occupies one of the copper $d$-orbitals. As is shown in the following
sections, not only electronic and magnetic, but the crystal structure of the
compound also evolves following the doping.

##  II Methods

In this paper, we follow the methodology defined in the previous one,
describing Cu2F5 [8]. All calculations were performed using Quantum-ESPRESSO
[9] package with pseudopotentials from pslibrary set [10]. The exchange-
correlation functional was chosen to be in Perdew-Burke-Ernzerhof [11] form.
The energy cutoff for plane wave wave functions and charge density expansion
was set to 50 Ry and 400 Ry, respectively. Integration in the reciprocal space
was done on a regular $8\times 8\times 8$ $k$-points mesh in the irreducible
part of the Brillouin zone.

Electronic correlations were treated within the DFT+U method [12] with the
Hubbard $U$ value equals 6 eV. As it was shown for the parent compound Cu2F5,
even the $U$ value of 4 eV is enough for the band gap to appear. The variation
of the $U$ from 4 to 8 eV doesn’t change the electronic structure
qualitatively and affects the Heisenberg exchange interaction parameters only
slightly. The Hund parameter $J$ = 0.9 eV was set to its typical value for
cuprates [13, 14].

The convergence criteria used for crystal cell relaxation within DFT+U are:
total energy $<10^{-6}$ Ry, total force $<10^{-3}$ Ry/Bohr, pressure $<0.1$
kbar.

##  III  Results

Figure 1: Crystal structure of Cu2F5 and the three types of fluoride ions that
were removed to obtain the structure of Cu2F5-x. Blue spheres denote Cu ions
inside the ligand’s octahedron, red spheres denote Cu ions in the center of
plaquettes, and gray spheres – F ions. Visualized using VESTA [15].

There are three different structural sites for the fluoride ion in Cu2F5 that
could be substituted by a vacancy: (a) the fluoride shared by CuF4 plaquette
and CuF6 octahedron that are placed along the _a_ axis of the cell; (b) the
ion shared by CuF4 plaquette and CuF6 octahedron in the direction of the _b_
axis; (c) the fluoride belonging to the two CuF6 octahedra along the _c_ axis.
The Cu-F-Cu bond angle is 180 for the (b) and (c) ions and only 129 degrees
for the case-(a) fluoride ion, which results in much weaker magnetic exchange
interaction in the direction of the _a_ axis. The absence of one of these
F-ions would naturally lead to a change in the electronic configuration of the
nearest Cu ions. With the destruction of the Cu-F-Cu superexchange path, the
disappearance of one fluoride ion probably will change the magnetic exchange
interaction pattern in Cu2F5.

In the Cu2F5 crystal cell containing 8 Cu ions (see Fig. 1), we removed the F
ion in (a), (b), or (c) position sequentially, as described above, and
performed full cell relaxation within the DFT+U approach to obtain a ground
state crystal structure. The used cell size corresponds to a 5% concentration
of fluoride vacancies (Cu2F4.75), and we are focused on this simplest and the
most visual case.

After the relaxation, we compared the enthalpies of the obtained structures.
The lowest enthalpy has the cell where the vacancy is placed instead of the F
ion in the (b) position. If F is removed from the (c) position, the enthalpy
is +45 meV / formula unit higher and it is +46 meV/formula unit higher for the
vacancy in the (a) position. We conclude that the favorable vacancy
localization site in Cu2F5 is the F ion shared by CuF4 plaquette and CuF6
octahedron in the direction of the _b_ axis. Below under the Cu2F4.75 or
Cu2F5-x, we mean the corresponding crystal structure.

Parameter | Cu2F5 | Cu2F4.75
---|---|---
Cell volume (Å3) | 369.3 | 393.6
Cuocta \- Cuocta distance along $a$ (Å) | 6.98 | 6.96
Cuocta \- Cuocta distance along $b$ (Å) | 7.61 | 8.08
Cuocta \- Cuocta distance along $c$ (Å) | 7.60 | 7.52
Average Cuplaq-F bond length (Å) | 1.90 | 1.80
Average Cuocta-F bond length (Å) | 1.93 | 2.04
Average Cuocta-F distance along $b$ (Å) | 1.93 | 2.26
Table 1: Transformation of crystal structure of Cu2F5 with electrons doping
via vacancies.

The vacancy creation results in the crystal structure distortions presented in
Table 1. The unit cell volume increase of 6.6% happens from the elongation of
the cell along the $b$ lattice vector. At the same time, there is an expansion
of the CuF6 octahedra and a decrease of the average Cu-F bond length within
the CuF4 plaquette in Cu2F5-x.

The two Cu ions, that had the F ion in between in Cu2F5, and have the vacancy
site instead in Cu2F4.75, are slightly shifted to each other along the _b_
crystal axis. Hereafter, the Cu-F bonds of the _ex_ -CuF4 plaquette and the
_ex_ -CuF6 octahedron are distorted when they lose the shared fluorine ion.
For the next-nearest to the vacancy Cu-F octahedra and plaquettes, one can
still say that the local environment for the Cu ion remains an octahedron and
a plaquette. The structure files for Cu2F5 and Cu2F4.75 could be found here
[16].

The Cu2F4.75 is an antiferromagnetic insulator with a 0.91 eV band gap. The
calculated partial densities of states (pDOS) are presented in Fig. 2. A
plausible assumption would be to say that all copper ions inside the fluoride
octahedra have very similar pDOSes despite the various distortions of CuF6
structures regarding the distance from the vacancy. The same is true for the
CuF4 plaquettes. The corresponding pDOSes are marked in Fig. 2 as Cuocta and
Cuplaq respectively. Here and below, we refer to the Cu $3z^{2}-r^{2}$,
$x^{2}-y^{2}$, etc. orbitals in terms of the local coordinate system for each
Cu ion, with the $z$ direction is perpendicular to the plaquette plane for the
Cuplaq ion and the $z$ direction is along with the crystal $b$ vector for the
Cuocta ion.

Figure 2: Partial densities of states for Cu2F5 (upper panel) and Cu2F5-x
(middle and lower panels). Positive/negative pDOSes correspond to spin-up/down
states respectively.

From the analysis of pDOS, one can see that Cu ions in the octahedral
environment obtained an additional electron and became Cu2+ ions in $d^{9}$,
$S=\frac{1}{2}$ electronic configuration: all $t_{2g}$ and $3z^{2}-r^{2}$
orbitals are filled, and the $x^{2}-y^{2}$ orbital is half-filled. At the same
time, the plaquette-surrounded Cu ions are now in $d^{8}$, $S=0$ configuration
and have negligible magnetic moments. Both Cuex-plaq and Cuex-octa ions are in
$d^{9}$, $S=\frac{1}{2}$ electronic configuration with the hole at the
$x^{2}-y^{2}$ orbital. We started our calculation from the cell containing 4
Cuocta ions in $d^{8}$, $S=1$ configuration + 4 Cuplaq ions in $d^{9}$,
$S=\frac{1}{2}$ configuration + an electron from the vacancy. At the end we’ve
obtained 3 Cuplaq ions in $d^{8}$, $S=0$ \+ 5 Cu ions (3 Cuocta, Cuex-plaq,
Cuex-octa) in $d^{9}$, $S=\frac{1}{2}$ configuration.

In the Cu2F5 [8] the lower unoccupied state is the $3z^{2}-r^{2}$ orbital of
the Cuocta ions (Fig. 2, upper panel, right graph). We assumed that as a
consequence of the doping, the additional electron will occupy some of these
orbitals. Consequently, metallization of the Cu2F5 with electron doping was
expected due to the appearance of the partially filled Cuocta $3z^{2}-r^{2}$
states. Surprisingly we observe an electron transfer from the Cuplaq ions to
the Cuocta ions in Cu2F5 with doping. We interpret it in the following way.
Due to the partial occupation of the Cuocta $3z^{2}-r^{2}$ orbital, the
corresponding octahedron elongates in the $c$ direction, that results in the
expansion of the cell along the $c$ axis. Consequently, all the CuF6 octahedra
are elongated. Since the Cu-F distances in the $b$ direction within the
octahedron become large, the Cuocta $3z^{2}-r^{2}$ orbital turns energetically
more favorable than even the Cuplaq $x^{2}-y^{2}$ orbital. Electrons that
occupied the Cuplaq $x^{2}-y^{2}$ state in the stoichiometric Cu2F5 leave it
in Cu2F5-x and fill the Cuocta ions d-shell. As a result, Cuplaq ions that
have the $d^{9}$ configuration in Cu2F5, become $d^{8}$ in the doped
structure, and Cuocta ions change their configuration in reverse way.

To confirm such evolution of crystal and electronic structure arises from the
electronic degrees of freedom, not from interactions between the vacancy
states, we modeled the electrons doping in a stoichiometric Cu2F5. The
fluoride ion wasn’t removed. We just added one extra electron in the cell and
relaxed the crystal structure.

As a result, the same effect was qualitatively reproduced. The cell volume
increase by 17.5% with its significant elongation along $b$ direction (Cuocta-
Cuocta distance grows up to 8.08 Å). The CuF6 octahedra are stretched along
$b$ direction too, with the corresponding average Cuocta-F bond length equals
2.25 Å. The spin-state of the copper ions was also changed as a result of the
existence of additional electrons: the Cuocta ions became $d^{9}$,
$S=\frac{1}{2}$.

Figure 3: Two half-filled $d$-orbitals of Cu ions that provide the
superexchange interaction via F $p$-orbital along the $b$ axis in
stoichiometric Cu2F5. In Cu2F5-x the $d_{3z^{2}-r^{2}}$ become fully filled
and the superexchange is suppressed.

We concluded that the electronic configuration and spin state of the copper
ions in Cu2F5 evolve in the same manner if an extra electron appears
regardless of the origin of the extra electron (vacancy or manual increase in
the number of carriers within the cell). We continue our presentation and
reasoning below for Cu2F5-x with the vacancy.

The half-filled $d_{3z^{2}-r^{2}}$-orbital of Cuocta ions in Cu2F5 gives the
superexchange interaction path between Cuocta and Cuplaq ions along the $b$
crystal axis as it is shown in Figure 3. The electrons hoping between Cuocta
$d_{3z^{2}-r^{2}}$ and Cuplaq $d_{x^{2}-^{y}2}$ via the F-$p$ orbital in
between led to the antiferromagnetic exchange. In Cu2F5-x the possibility of
such a hoping of electrons is suppressed, since Cuocta $d_{3z^{2}-r^{2}}$
becomes filled. Consequently, the corresponding superexchange interaction
along the $b$ crystal axis vanishes.

The described evolution of electronic structure has an outcome that the 1D
chains of Cuocta ions with a hole on the $d_{x^{2}-y^{2}}$-orbital appear.
Taking into account the existence of the fluoride ion between such copper ions
and following the Goodenough-Kanamori rule [17], the antiferromagnetic
superexchange interaction will emerge along the $c$ crystal axis.

Figure 4: Evolution of spin-states and patterns of the exchange interaction
between the Cu ion from Cu2F5 (left) to Cu2F5-x (right). Blue elements denote
Cu ions in the octahedral surrounding, red elements denote Cu ions in the
center of fluoride plaquettes. The blue and red half-filled triangles
correspond to Cuex-octa, $S=\frac{1}{2}$ and Cuex-plaq, $S=\frac{1}{2}$ ions
respectively. The filled, half-filled and empty circles denote $S=1$,
$S=\frac{1}{2}$, and $S=0$ spin-states of the ions. The (100) lattice planes
are shown with a light green color for an eye guide. The vacancy position is
shown with a black cross. The strongest exchange interactions are $J_{2D}$ in
Cu2F5 (green line) and $J_{1D}$ in Cu2F5-x (violet line). Other exchange
interactions are negligible. Fluorine ions are not shown for clarity.

Using Green’s function method based on magnetic-force linear response theory
[2], we computed the Heisenberg exchange interaction between Cu ions up to the
9th nearest neighbor. The model Hamiltonian has the form: $H=-\sum_{\langle
ij\rangle}J_{ij}{\bf e}_{i}{\bf e}_{j},$ where ${\bf e}_{i}$ are the unit
vectors pointing in the direction of the $i$th site magnetization, and the
summation runs once over each ion pair.

Only one exchange interaction survived under doping. The antiferromagnetic
exchange between the Cuocta ions along the $c$-axis is $J_{1D}$ = -29.5 meV.
The second largest interaction $\approx$ -3.3 meV is between Cuocta and Cuex-
plaq ions along the [101]-direction. The pattern of the 1D magnetic chains
stems from such exchanges with the interchain interaction being an order of
magnitude smaller than the intrachain one. It is shown in Fig. 4, right panel.
The figure illustrates also the evolution of the strongest exchange
interactions in Cu2F5 that arose from electron doping. The absolute value of
the exchange remains almost the same:$J_{2D}\approx-33~{}\textrm{meV}$ in
Cu2F5, but the dimension of the interaction decreases from 2D to 1D as a
result of doping.

We point here to an analogy to the exchange interaction pattern that exists in
KCuF$3$. The potassium copper fluoride is formed with the CuF6 octahedra, and
all copper ions have a hole on the $d_{x^{2}-y^{2}}$ orbital [14]. Since there
is the Jahn-Teller distortions of the octahedra, the lobes of the half-empty
$d$-orbitals of the nearest Cu ions are perpendicular to each other in the
[001]-plane. Consequently, the superexchange via the F $p$-orbitals in the
[001]-plane is negligible in KCuF3 and the 1D chains of antiferromagnetically
ordered moments appear along the $c$ crystal axis. Therefore, depsite the
different building blocks of the structure in Cu2F5-x (octahedra and
plaquettes) and KCuF3 (octahedra only), there is a similarity in the magnetic
interactions picture between these two compounds.

##  IV Conclusion

Using the DFT+U calculations, we explored the influence of the fluoride
vacancy appearance on the crystal, electronic and magnetic structure of
Cu2F5-x. Extra electrons, which resulted from the absence of the fluorine ion,
are shown to result in the elongation of the CuF6 octahedra along the $b$
crystal axis and then the $3z^{2}-r^{2}$ orbital of all Cuocta ions becomes
occupied. As a result, all the Cu ions in the center of CuF6 octahedra get
$S=\frac{1}{2}$ spin configuration, and the Cu ions inside the CuF4 plaquettes
become non-magnetic ($S=0$). Such significant effect become apparent even when
one extra electron per 4x formula unit is added. The antiferromagnetic linear
chains of copper ions appear along the $c$-axis of the crystal. The interchain
exchange interaction is ten times smaller than the largest intrachain one. Our
calculations show consistently that Cu2F5-x can be described as a quasi-one-
dimensional $S=\frac{1}{2}$ Heisenberg chain in a good approximation.

## Acknowledgments

Calculation of the ground state crystal structure for the doped Cu2F5-x was
carried out within the state assignment of Ministry of Science and Higher
Education of the Russian Federation (theme “Electron” No. 122021000039-4).
Results on the spin-lattice evolution with doping were obtained with the
support of the Russian Science Foundation (project No. 19-12-00012).

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