$\displaystyle\left\|\partial_{k}(\rho\circ
u_{\rho})_{i}-\partial_{k}(\sigma\circ
u_{\sigma})_{i}\right\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq\left\|\rho^{\prime}\circ(u_{\rho})_{i}\partial_{k}(u_{\rho})_{i}-\rho^{\prime}\circ(u_{\rho})_{i}\partial_{k}(u_{\sigma})_{i}\right\|_{L^{\infty}(\Omega)}$
$\displaystyle+\left\|\rho^{\prime}\circ(u_{\rho})_{i}\partial_{k}(u_{\sigma})_{i}-\rho^{\prime}\circ(u_{\sigma})_{i}\partial_{k}(u_{\sigma})_{i}\right\|_{L^{\infty}(\Omega)}$
$\displaystyle+\left\|\rho^{\prime}\circ(u_{\sigma})_{i}\partial_{k}(u_{\sigma})_{i}-\sigma^{\prime}\circ(u_{\sigma})_{i}\partial_{k}(u_{\sigma})_{i}\right\|_{L^{\infty}(\Omega)}.$

The first term is easily bounded using the fact that $\rho^{\prime}$ is
bounded on $\mathbb{R}$. The second term is bounded owing to $\rho^{\prime}$
being Lipschitz continuous and $\nabla u_{\sigma}$ being bounded. We bound the
last term by using the $L^{\infty}$ norm of $\rho^{\prime}-\sigma^{\prime}$
and the fact that $\nabla u_{\sigma}$ is bounded. We finally obtain the
following upper bound

$\displaystyle\left\|\partial_{k}(\rho\circ
u_{\rho})_{i}-\partial_{k}(\sigma\circ
u_{\sigma})_{i}\right\|_{L^{p}(\Omega)}$
$\displaystyle\leq\underbrace{\left[C_{2}\left\|\rho^{\prime}\right\|_{L^{\infty}(\mathbb{R})}+C_{1}C_{\boldsymbol{\vartheta}}\operatorname{Lip}{\rho^{\prime}}\left\|\sigma^{\prime}\right\|_{L^{\infty}(\mathbb{R})}^{\alpha}\right]}_{=C_{2}^{\prime}}\left\|\rho-\sigma\right\|_{L^{p}(\mathbb{R})}$
$\displaystyle+\underbrace{\left[C_{3}\left\|\rho^{\prime}\right\|_{L^{\infty}(\mathbb{R})}+C_{\boldsymbol{\vartheta}}\left\|\sigma^{\prime}\right\|_{L^{\infty}(\mathbb{R})}^{\alpha}\right]}_{=C_{3}^{\prime}}\left\|\rho^{\prime}-\sigma^{\prime}\right\|_{L^{p}(\mathbb{R})}.$

Since $\sigma^{\prime}$ is bounded on $\mathbb{R}$ it holds
$\left\|\partial_{k}(\sigma\circ
u_{\sigma})_{i}\right\|_{L^{\infty}(\Omega)}\leq\left\|\sigma^{\prime}\right\|_{L^{\infty}(\mathbb{R})}\left\|\nabla
u_{\sigma}\right\|_{L^{\infty}(\Omega,\mathbb{R}^{d\times n})}$. We conclude
by taking the maximum on $1\leq i\leq m$ and $1\leq k\leq d$: the three bounds
still hold for constants $C_{1}^{\prime}$, $C_{2}^{\prime}$ and
$C_{3}^{\prime}$ indicated above,
$C_{\boldsymbol{\vartheta}}^{\prime}=C_{\boldsymbol{\vartheta}}$ and
$\alpha^{\prime}=\alpha+1$. ∎

##  Appendix B Algorithms

###  B.1. Mesh extraction

In this section, we provide more detail on the way to construct an adapted
mesh to a neural network based on a CPWL approximation of its activation
function, as described in Alg. 1. We recall the notations introduced in Sect.
5.

Suppose that $\pi[u_{k}]$ is a CPWL map and that its restriction to
$R\subset\mathbb{R}^{d}$ is
$\pi[u_{k}]_{|R}(\boldsymbol{x})=\boldsymbol{W}_{R}\boldsymbol{x}+\boldsymbol{b}_{R}$.
For $1\leq i\leq m$, we write $\boldsymbol{w}_{i}$ and $b_{i}$ the $i$-th row
of $\boldsymbol{W}_{R}\in\mathbb{R}^{m\times d}$ and
$\boldsymbol{b}_{R}\in\mathbb{R}^{m}$. Let $\pi[\rho]$ be a CPWL function, and
$(\xi_{j})_{1\leq j\leq K}$ denote its breakpoints. We also introduce the
local coefficients $(\alpha_{j},\beta_{j})$ of $\pi[\rho]$ such that the
restriction of $\pi[\rho]$ to $\left[\xi_{j},\xi_{j+1}\right]$ has the
expression $x\mapsto\alpha_{j}x+\beta_{j}$.

To identify the regions where $\pi[\rho]\circ\pi[u_{k}]_{|R}$ is linear, we
need to consider the equations
$\boldsymbol{w}_{i}\cdot\boldsymbol{x}+b_{i}=\xi_{j}$ for all $1\leq i\leq m$
and $1\leq j\leq K$. These hyperplanes intersect in $R$ and define subregions
adapted to $\pi[\rho]\circ\pi[u_{k}]_{|R}$. This is the step that corresponds
to $\mathtt{cutRegion}$ in Alg. 1. We now present specific algorithms to
define these subregions in dimensions zero, one, and two.

####  B.1.1. Dimensions zero and one

In dimension zero, a region is reduced to one single point so nothing needs to
be done. In the one-dimensional case, $R$ is a segment that we write
$\left[p_{-},p_{+}\right]$. We first compute the range of each linear map
$h_{i}:x\mapsto\boldsymbol{w}_{i}\cdot x+b_{i}$ on $R$ by computing their
values at $p_{-}$ and $p_{+}$. We then find which breakpoints $\xi_{j}$ belong
to that range via a binary search, and we compute the corresponding
antecedents $p_{i,j}$ of $\xi_{j}$ by $h_{i}$ with normalised coordinates as
follows

$p_{i,j}=\frac{1}{2}(p_{-}+p_{+})+\frac{t_{i,j}}{2}(p_{+}-p_{-}),\qquad
t_{i,j}=\frac{2\xi_{j}-(h_{i}(p_{+})+h_{i}(p_{-}))}{h_{i}(p_{+})-h_{i}(p_{-})}\in\left[-1,+1\right].$

We found this formulation more numerically stable compared to expressing
$p_{i,j}$ in terms of $w_{i}$, $b_{i}$, and $\xi_{j}$. We sort the normalised
coordinates $t_{i,j}$ and obtain the corresponding sorted abscissae $(p_{l})$.
For each segment $\left[p_{l},p_{l+1}\right]$, we evaluate the linear maps
$h_{i}$ at $\frac{1}{2}(p_{l}+p_{l+1})$ and retrieve the coefficients
$(\alpha_{l,i},\beta_{l,i})$ of $\pi[\rho]$ at this location to update the
local expression $(\boldsymbol{W}_{R},\boldsymbol{b}_{R})$ into
$(\mathrm{diag}(\boldsymbol{\alpha}_{l})\boldsymbol{W}_{R},\mathrm{diag}(\boldsymbol{\alpha}_{l})\boldsymbol{b}_{R}+\boldsymbol{\beta}_{l})$.

####  B.1.2. Dimension two

The two-dimensional case is much more involved as the hyperplanes are lines
that can intersect with the boundary of $R$ and among one another. We consider
four steps.

(i) Intersection of the hyperplanes with $\partial R$. Since $R$ is a convex
polygon, each hyperplane
$h_{i}:\boldsymbol{x}\mapsto\boldsymbol{w}_{i}\cdot\boldsymbol{x}+b_{i}$ can
intersect with $\partial R$ at two points at most. If the number of
intersections is zero or one, then $R$ lies entirely on one side of the
hyperplane thus we only need to handle the case when there are two
intersection points. One notable corner case occurs when the hyperplane
coincides with an edge of the polygon. In this degenerate case, the region is
not cut.

The clipping of a line by a convex polygon can be done in logarithmic time,
using a method akin to binary search. We refer the reader to [25] for the full
algorithm. We write $(P_{i,j}^{-},P_{i,j}^{+})$ the clipping of $h_{i}$ by
$R$, and $(P_{l})$ the collection of these vertices.

(ii) Intersection of the hyperplanes among themselves. Next, we check the
intersection of pairwise combinations of the lines obtained above. It is
important to note that for a fixed row $i$, the lines
$(P_{i,j}^{-},P_{i,j}^{+})$ are parallel. Indeed, they are solutions to
$\boldsymbol{w}_{i}\cdot\boldsymbol{x}+b_{i}=\xi$ for two different values of
$\xi$. This means that we can skip the tests corresponding to lines arising
from the same row. Let $i_{1}\neq i_{2}$ be two different row indices, $j_{1}$
and $j_{2}$ be two breakpoint indices, and suppose that
$(P_{i_{1},j_{1}}^{-},P_{i_{1},j_{1}}^{+})$ and
$(P_{i_{2},j_{2}}^{-},P_{i_{2},j_{2}}^{+})$ are two non-parallel lines. We
compute their intersection $Q$ via the following parametrisation and solve for
$s$ and $t$:

$Q=\overline{P}_{i_{1},j_{1}}+s\Delta
P_{i_{1},j_{1}}=\overline{P}_{i_{2},j_{2}}+t\Delta
P_{i_{2},j_{2}},\qquad\begin{bmatrix}\Delta P_{i_{1},j_{1}}&-\Delta
P_{i_{2},j_{2}}\end{bmatrix}\begin{bmatrix}s\\\
t\end{bmatrix}=\begin{bmatrix}\overline{P}_{1}-\overline{P}_{2}\end{bmatrix},$

where $\overline{P}_{i,j}=\frac{1}{2}(P_{i,j}^{-}+P_{i,j}^{+})$ is the middle
of the segment and $\Delta P_{i,j}=\frac{1}{2}(P_{i,j}^{+}-P_{i,j}^{-})$ is
half the direction vector. We need to check that $s$ and $t$ are between $-1$
and $+1$.

(iii) Representation of these intersections as an adjacency graph. We
initialise a connectivity graph with the edges of the region $R$. During step
(i), for each line $(P_{i,j}^{-},P_{i,j}^{+})$ that cuts the polygon, we add
the points $P_{i,j}^{-}$ and $P_{i,j}^{+}$ to the edges they belong to, and
add an arc in the graph, that connects these two points. During step (ii), we
only need to add new vertices to the graph.

(iv) Extraction of the smallest cycles in this graph. Finding the partition of
$R$ where the composition by $\pi[\rho]$ is CPWL amounts to finding the
fundamental cycle basis of the undirected graph defined above. We refer to
[26] for a definition of the fundamental cycle basis of a planar graph and for
an algorithm to build it.

Algorithm 1 $\mathtt{adaptive\char 95\relax mesh}(u)$

$\tau\leftarrow[(\Omega,\boldsymbol{I}_{d},\boldsymbol{0}_{d})]$

for $\ell\in\mathtt{layers}(u)$ do

$\boldsymbol{W}\leftarrow\mathtt{weight}(\ell),\quad\boldsymbol{b}\leftarrow\mathtt{bias}(\ell),\quad\tau^{\text{new}}\leftarrow\varnothing$

for $(R,\boldsymbol{W}_{R},\boldsymbol{b}_{R})\in\tau$ do

$\boldsymbol{W}_{R}^{\text{new}}\leftarrow\boldsymbol{W}\times\boldsymbol{W}_{R},\quad\boldsymbol{b}_{R}^{\text{new}}\leftarrow\boldsymbol{W}\times\boldsymbol{b}_{R}+\boldsymbol{b}$
$\triangleright$ Compose by linear map

for $(S,\boldsymbol{W}_{S},\boldsymbol{b}_{S})\in\mathtt{cut\char 95\relax
region}(\pi[\rho],R,\boldsymbol{W}_{R}^{\text{new}},\boldsymbol{b}_{R}^{\text{new}})$
do $\triangleright$ See Apdx. B.1.1 and Apdx. B.1.2.

$(\boldsymbol{\alpha},\boldsymbol{\beta})\leftarrow\mathtt{coefficients}(\pi[\rho],S)$
$\triangleright$ Retrieve local coefficients of $\pi[\rho]$ in $S$

$\boldsymbol{W}_{S}^{\text{new}}\leftarrow\mathtt{diag}(\boldsymbol{\alpha})\times\boldsymbol{W}_{S},\quad\boldsymbol{b}_{S}^{\text{new}}\leftarrow\mathtt{diag}(\boldsymbol{\alpha})\times\boldsymbol{b}_{S}+\boldsymbol{\beta}$
$\triangleright$ Compose by $\pi[\rho]$

$\tau^{\text{new}}\leftarrow\tau^{\text{new}}\cup\\{(S,\boldsymbol{W}_{S}^{\text{new}},\boldsymbol{b}_{S}^{\text{new}})\\}$

end for

end for

$\tau\leftarrow\tau^{\text{new}}$

end for

return $\tau$

####  B.1.3. Notes on the algorithm

In Alg. 1, $\mathtt{weight}(\ell)$ and $\mathtt{bias}(\ell)$ denote the
parameters of the network at layer $\ell$. The instruction
$\mathtt{coefficients}$ is just a lookup operation to retrieve the local
coefficients of $\pi[\rho]$. Here its outputs are vectors because the local
map on a region is vector-valued. Finally if $\boldsymbol{\alpha}$ is a
vector, $\mathtt{diag}(\boldsymbol{\alpha})$ is the diagonal matrix whose
diagonal coefficients are $\boldsymbol{\alpha}$.

###  B.2. Decomposition of convex polygons

In the general two-dimensional case, the cells of the mesh adapted to $u$ are
arbitrary convex polygons. Numerical quadrature rules are known for triangles
and convex quadrangles, so we decide to split the cells into a collection of
these two reference shapes. We encode hard rules to decompose convex polygons
with up to ten vertices into triangles and convex quadrangles. For example, an
octagon is split into one triangle and two quadrangles by considering the
three cycles $(1,2,3,4)$, $(4,5,6,7)$, $(7,8,1,4)$. Larger polygons are
recursively split in half until they have fewer than ten vertices.

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  *[FEM]: Finite Element Method
  *[PDE]: partial differential equation
  *[NN]: Neural Network
  *[PINN]: Physics-Informed Neural Network
  *[PINNs]: Physics-Informed Neural Network
  *[NNs]: Neural Network
  *[PDEs]: partial differential equation
  *[MC]: Monte Carlo
  *[CPWL]: Continuous Piecewise Linear
  *[AQ]: Adaptive Quadrature


††institutetext: Institute of Theoretical Physics, University of Wroclaw,
PL-50204 Wrocław, Poland

# Polarization effects at finite temperature and magnetic field

Pok Man Lo,111Corresponding author. Michał Szymański  Krzysztof Redlich  and
Chihiro Sasaki<EMAIL_ADDRESS>
###### Abstract

We study the screening of four-quark interaction by the ring diagram in an
effective chiral quark model. This entails a medium dependent coupling which
naturally reduces the chiral transition temperature in a class of models, and
is capable of generating an inverse magnetic catalysis at finite temperature.
These results are the first coherent description of inverse magnetic
catalysis, anchored to a reliable field-theoretical basis.

##  1 Chiral quark model with dressed interaction

Understanding the properties of QCD matter under extreme conditions, such as
those created in the laboratory during the ultra-relativistic heavy-ion
collisions or fill the core of neutron stars, demands a reliable description
of how chiral symmetry is restored and how it manifests in a medium of
partially deconfined quarks and gluons. Effective models are a robust
exploratory tool to study a dynamical system. One of the advantage is the
ability to temporarily include (or suppress) a certain class of interactions
or diagrams, thereby examining the effect in isolation. One can also gain
insights on the values of phenomenological parameters used and examine their
connections to the properties of underlying constituents. These make this
approach useful to complement the more powerful numerical methods such as
lattice QCD (LQCD).

In this paper we study the in-medium dressing of the four-quark coupling by
the polarization diagram within an effective chiral quark model. Screening of
the potential by ring diagram finds the most famous application in regulating
the long-range Coulomb forces in an electron gas GellmannBrueckner ;
ElectronGas . (See Refs Mattuck ; Leeuwen_2013  for further discussions in
condensed matter theory.) When applied to the study of the QCD phase diagram,
we shall show that it provides a natural mechanism to reducing the chiral
transition temperature in a class of models, and is capable of generating an
inverse magnetic catalysis at finite temperature.

Our starting point is an effective chiral quark model motivated by the Coulomb
Gauge QCD Govaerts:1983ft ; Kocic:1985uq ; Hirata:1989qp ; Alkofer:1989vr ;
Schmidt:1995gea ; Reinhardt:2017pyr ; Quandt:2018bbu ; conf . The Lagrangian
density reads:

$\begin{split}\mathcal{L}(x)&=\bar{\psi}(x)\,(i\not{\partial}_{x}-m)\,\psi(x)\\\
&\quad-\frac{1}{2}\,\int
d^{4}y\,\rho^{a}(x)\,V^{ab}(x,y)\,\rho^{b}(y)\end{split}$ (1)

where $m$ is the current quark mass,
$\rho^{a}(x)=\bar{\psi}(x)\gamma^{0}T^{a}\psi(x)$ is the color quark current,
and $T^{a}$ is a generator of the $SU(N_{c})$ symmetry group, with
$a=1,2,\ldots,N_{c}^{2}-1$. For the class of models where the interaction
potential $V$ is instantaneous and color-diagonal, i.e.,

$V^{ab}(x,y)\rightarrow\delta^{ab}\times\delta(x^{0}-y^{0})\,V(\vec{x}-\vec{y}),$
(2)

the gap equation for the dynamical quarks has been derived conf . It can be
summarized as follows:

$S^{-1}(p)=\not{p}-m-\Sigma(p)$ (3)

where

$\Sigma(p)=C_{F}\,\int\frac{d^{4}q}{(2\pi)^{4}}\,V(\vec{p}-\vec{q})\,i\,\gamma^{0}S(q)\gamma^{0}.$
(4)

The constant $C_{F}=\frac{N_{c}^{2}-1}{2N_{c}}$ is introduced via the
quadratic Casimir operator. Note that the use of the dressed quark propagator
$S(q)$ within the integral in Eq. (4) means that the gap equation is to be
solved self-consistently. A particularly transparent case to consider is a
contact interaction, where we replace

$V(\vec{p}-\vec{q})\rightarrow V_{0}.$ (5)

The solution to the gap equation (3) becomes

$\begin{split}M&=m+{\rm Tr}\,(\Sigma)/{\rm Tr}\,(I)\\\
&=m+C_{F}\,V_{0}\,\int\frac{d^{3}q}{(2\pi)^{3}}\,\frac{M}{2E}\\\
&\quad\times(1-2\,N_{\rm th}(E)),\end{split}$ (6)

where $E=\sqrt{q^{2}+M^{2}}$ and $N_{\rm th}(E)=({e^{\beta E}+1})^{-1}$ is the
Fermi-Dirac distribution, with temperature $T=1/\beta$. Equation (6) is of the
same form as the familiar result for quark mass ($N_{f}$ flavors) in the model
of Nambu-Jona and Lasinio (NJL) Klevansky1992 . The latter has a Lagrangian
density

$\begin{split}\mathcal{L}_{\rm
NJL}(x)&=\bar{\psi}(x)\,(i\not{\partial}_{x}-m)\,\psi(x)\\\ &\quad+G_{\rm
NJL}\,\left((\bar{\psi}(x)\,{\psi}(x))^{2}+(\bar{\psi}(x)i\gamma_{5}\vec{\tau}\psi(x))^{2}\right).\end{split}$
(7)

To leading order the gap equations of the two models would agree if we
identify

$C_{F}\,V_{0}\leftrightarrow 4\,N_{c}\,N_{f}\,(2\,G_{\rm NJL}).$ (8)

There are, however, important differences in their theoretical origin: In the
current model, the quark self energy in Eq. (4) comes from the Fock (exchange)
diagram, rather than the Hartree diagram in an NJL model. In fact, the Hartree
term vanishes in the current model (1) due to ${\rm Tr}\,T^{a}=0$. This gives
the characteristic color factor in Eq. (8), instead of a simple scaling by
$N_{c}$. Furthermore the quark current here, as dictated by the Coulomb gauge,
is of (Dirac and color) vector nature, rather than a scalar current in the NJL
case. This allows us to formally identify $V_{0}$ with the longitudinal gluon
propagator, and select the relevant Dirac structure of the polarization tensor
when dressing the potential.

##  2 Polarization tensors

The in-medium dressing of the interaction potential $V_{0}$ by the
polarization tensor $\Pi_{00}$ is implemented via conf

$\tilde{V_{0}}^{-1}={V_{0}}^{-1}-\frac{1}{2}\,N_{f}\,\Pi_{00}$ (9)

where the polarization is further approximated by the ring diagram

$\begin{split}\Pi_{00}(p^{0},\vec{p})&=\frac{1}{\beta}\,\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}{\rm
Tr}\left(\gamma^{0}S(q)\gamma^{0}S(q+p)\right).\end{split}$ (10)

Here
$\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}$
denotes a Matsubara sum over the fermionic frequencies
($\omega_{n}=(2n+1)\,\pi/\beta$), and an integral over the momenta $d^{3}q$.
Equation (9) describes the screening of the gluon propagator by the Debye
mass. The factor of $\frac{1}{2}$ in Eq. (9) comes from the color structure,
i.e. ${\rm Tr}\,T^{a}T^{b}=\frac{1}{2}\,\delta^{ab}$, and is essential to
reproduce the known result Kapusta:2006pm  of the perturbative Debye mass for
QCD, rather than for QED.

To simplify the discussion, we shall consider only the screening by the
static, finite temperature part of $\Pi_{00}$. (The static, vacuum part of
$\Pi_{00}$ naturally vanishes.) One way to obtain the quantity is by a direct
numerical evaluation of Eq. (10). A more intuitive way to obtain the result is
via a formal relation to the thermal pressure of a free (single species)
fermion gas at finite temperature and vanishing chemical potential:

$\begin{split}\Pi_{00}(p^{0}=0,\vec{p}\rightarrow\vec{0})&=\frac{1}{\beta}\,\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}{\rm
Tr}\left(\gamma^{0}S(q)\gamma^{0}S(q)\right)\\\
&=-\frac{\partial}{\partial\mu}\,\frac{1}{\beta}\,\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}{\rm
Tr}\left(\gamma^{0}S\right)\\\
&=-\frac{\partial^{2}}{\partial\mu\partial\mu}\,\frac{1}{\beta}\,\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}{\rm
Tr}\,\ln S^{-1}.\end{split}$ (11)

Here $S^{-1}(q)=(i\,\omega_{n}+\mu)\,\gamma^{0}-\vec{q}\cdot\vec{\gamma}-M$.
It is particularly useful to realize that

$\begin{split}\frac{1}{\beta}\,\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}{\rm
Tr}\left(\gamma^{0}S\right)&=\int\frac{d^{3}q}{(2\pi)^{3}}\,2\times\\\
&\quad\left(\frac{1}{e^{\beta(E-\mu)}+1}-\frac{1}{e^{\beta(E+\mu)}+1}\right).\end{split}$
(12)

Taking one further $\mu$-derivative and setting $\mu\rightarrow 0$ at the end,
yields an explicit expression of the electric mass: 222 The perturbative Debye
mass is $m_{el}^{2}=g^{2}\times\hat{m}_{el}^{2}$.

Figure 1: The electric mass (Eq. (13)) for the 2-flavor case at fixed quark
mass $M=0.18$ GeV, as a function of temperature. The results in a finite
magnetic field are calculated using Eq. (16) with physical electric charges of
$(u,d)$ quarks.

$\begin{split}\hat{m}_{el}^{2}&=-\frac{1}{2}\,N_{f}\,\Pi_{00}(p^{0}=0,\vec{p}\rightarrow\vec{0})\\\
&=\frac{1}{2}\,N_{f}\times\int\frac{d^{3}q}{(2\pi)^{3}}\,4\beta N_{\rm
th}(1-N_{\rm th}).\end{split}$ (13)

In particular, we note the following interesting limits: (1) at $M\rightarrow
0$ (or large $T$),

$\hat{m}_{el}^{2}\approx\frac{1}{2}\,N_{f}\times\frac{T^{2}}{3};$ (14)

and (2) at large $M$ (or small $T$), where the Boltzmann approximation is
valid,

$\hat{m}_{el}^{2}\approx\frac{1}{2}\,N_{f}\times\frac{2}{\pi^{2}}\,M^{2}\,(K_{2}(M/T)-K_{2}(2M/T)),$
(15)

where $K_{2}$ is the modified spherical Bessel function of the second kind. In
Fig. 1 we demonstrate a numerical calculation of the electric mass Eq. (13) at
fixed value of $M=0.18$ GeV (a typical value near the transition), and
$N_{f}=2$ flavors. The analytic limits in Eqs. (13) and (15) can be readily
verified. 333Including the second order term of the Boltzmann approximation in
Eq. (15) gives a good estimate even at high temperature, i.e. $3/\pi^{2}\times
T^{2}\approx 0.304\,T^{2}$, instead of the full quantum result $1/3\times
T^{2}$. As preparation for the analysis of the inverse magnetic catalysis at
finite temperature Andersen:2014xxa ; Miransky:2015ava , we extend the
electric mass in Eq. (13) (per flavor) to that with a finite magnetic field:

$\begin{split}\hat{m}_{el}^{2}&=\frac{1}{2}\,\frac{|e_{f}|B}{2\pi}\,\sum_{n=0}^{\infty}\,\frac{1}{2}\,\alpha_{n}\times\\\
&\quad\int\,\frac{dq_{z}}{2\pi}\,4\beta N_{\rm th}(E_{q_{z},n})(1-N_{\rm
th}(E_{q_{z},n})),\end{split}$ (16)

where $\alpha_{n}=2-\delta_{n0}$ and $e_{f}$ is the electric charge of the
species. Note that the effective dispersion relation in the presence of
magnetic field $B$ becomes:

$E_{q_{z},n}=\sqrt{q_{z}^{2}+2\,n\times|e_{f}|B+M^{2}}.$ (17)

Examining in particular the contribution from the lowest Landau level (LLL)
($n=0$), we get

$\hat{m}_{el}^{2}\approx\frac{1}{2}\,\frac{|e_{f}|B}{4\pi}\,\int\,\frac{dq_{z}}{2\pi}\,\frac{4\beta
e^{\beta\sqrt{q_{z}^{2}+M^{2}}}}{(e^{\beta\sqrt{q_{z}^{2}+M^{2}}}+1)^{2}},$
(18)

which for massless quarks reduces to 444 This gives an alternative derivation
of the result in Ref. debye .

$\hat{m}_{el}^{2}\rightarrow\frac{1}{2}\,\frac{|e_{f}|B}{2\pi^{2}}.$ (19)

The limit is realized by the LLL line in Fig. 1 at high temperature. Note in
particular that the full result follows the LLL at low temperature, and
eventually reaches the limit (13) at high temperature.

Figure 2: The chiral condensate (with current quark mass contribution
subtracted), normalized to the vacuum value, versus the temperature, at zero
(left) and finite (right) magnetic field. Dashed lines represent results
obtained from a PNJL model without screening effect. The model with dressed
coupling is capable of producing the inverse magnetic catalysis at finite
temperature.

##  3 Results

###  3.1 zero magnetic field

A standard NJL model lacks a description of confinement. This is usually
remedied by considering an additional coupling of quarks to the Polyakov loop
$\ell$: a Polyakov-NJL (PNJL) model Fukushima:2003fw ; pnjl ; Sasaki:2006ww ;
Fukushima:2017csk . One known issue of this class of models is that it
overestimates the chiral transition temperature: typically giving $T_{\rm
ch}\approx 220$ MeV instead of the LQCD result $T_{\rm ch}\approx 156.5$ MeV
Td1 ; Td2 . A common way to deal with this problem is by an ad hoc rescaling
of the $T_{d}$ parameter, i.e. the deconfinement transition temperature in a
pure gauge theory, in the pure gauge potential $U_{\rm glue}$: from the
physical value of $T_{d}\approx 270$ MeV, to a lower value at $T_{d}\approx
200$ MeV shift1 ; shift2 ; mediumG1 . This is far from ideal as it points to
some missing interactions in the original model. Here we propose the screening
of the interaction by polarization as a natural resolution to the problem. For
this purpose, we generalize our model to include the coupling to $\ell$ by
implementing a statistical confinement scheme, i.e., by replacing the thermal
weight $N_{\rm th}(E)$ in Eq. (6) with ($N_{c}=3$) Skokov:2010wb ;
Skokov:2010uh ; su3pot

$\begin{split}N_{\rm
th}(E,\ell)&\rightarrow\frac{1}{3}\,\sum_{j=1}^{3}\frac{\hat{\ell}_{F}^{(j)}}{e^{\beta
E}+\hat{\ell}_{F}^{(j)}}\\\ &=\frac{1}{3}\,\frac{3\ell\,e^{-\beta
E}+6\ell\,e^{-2\beta E}+3e^{-3\beta E}}{1+3\ell\,e^{-\beta
E}+3\ell\,e^{-2\beta E}+e^{-3\beta E}},\end{split}$ (20)

where $\hat{\ell}_{F}^{(j)}$ are entries of the Polyakov loop operator in the
fundamental representation

$\hat{\ell}_{F}={\rm diag}\,\left(e^{i\gamma_{1}},1,e^{-i\gamma_{1}}\right),$
(21)

related to the Polyakov loop $\ell$ via

$\ell=\frac{1}{3}\,{\rm Tr}\,\hat{\ell}_{F}=\frac{1}{3}\,(1+2\cos\gamma_{1}).$
(22)

The final set of gap equations for quarks becomes

$M=m+C_{F}\,\tilde{V}_{0}\,M\times I_{0}(T;M,\ell),$ (23)

where

$\begin{split}I_{0}&=I_{0}^{\rm vac}+I_{0}^{T},\\\ I_{0}^{\rm
vac}&=\int\frac{d^{4}q_{E}}{(2\pi)^{4}}\,\frac{1}{q_{E}^{2}+E^{2}}\,\mathcal{R}_{4D}(q),\\\
I_{0}^{T}&=-\int\frac{d^{3}q}{(2\pi)^{3}}\,\frac{1}{2E}\times 2N_{\rm
th}(E,\ell).\end{split}$ (24)

The vacuum piece $I_{0}^{\rm vac}$ requires regularization, e.g., by a 4D
regulator $\mathcal{R}_{\rm 4D}(q)$, while the finite temperature piece
$I_{0}^{T}$ requires no regularization. 555In computing $I_{0}$ under a
general regularization scheme, the finite temperature piece can be extracted
via
$I_{0}^{T}=\lim_{\Lambda\rightarrow\infty}\,\left(I_{0}(T,\Lambda)-I_{0}(T\rightarrow
0,\Lambda)\right).$ This serves as an alternative way to define the finite
temperature contribution in a regularization independent manner regulators ;
Lo:2013lca . The screened potential is obtained from

$\tilde{V}_{0}(T;M,\ell)=\frac{1}{V_{0}^{-1}+\hat{m}_{el}^{2}(T;M,\ell)}.$
(25)

Based on Eq. (12) and the statistical weight (20), we arrive at the thermal
factor appropriate for the calculation of electric mass in Eq. (13):

$N_{\rm th}\,(1-N_{\rm
th})\rightarrow\frac{1}{3}\,\sum_{j=1}^{3}\frac{e^{\beta
E}\,\hat{\ell}_{F}^{(j)}}{(e^{\beta E}+\hat{\ell}_{F}^{(j)})^{2}}.$ (26)

The expectation value of the Polyakov loop needs to be determined from another
gap equation,

$\frac{\partial}{\partial\ell}\,(U_{\rm glue}(\ell)+U_{Q}(M,\ell))=0,$ (27)

for a given $U_{\rm glue}(\ell)$ and the quark potential $U_{Q}(M,\ell)$, the
latter describes the coupling of the Polyakov loop with quarks. These
potentials have been studied extensively Fukushima:2003fw ; Sasaki:2006ww ;
Lo:2013hla  and will not be repeated here. In this work we employ the pure
gauge potential in Ref. Lo:2013hla . This potential is unique in the sense
that the locations of minima and the curvatures around them are constrained
with the SU(3) pure gauge LQCD data sus .

The screening of $V_{0}$ in Eq. (25) introduces further field dependence in
$M$ and $\ell$, which, depending on the model parameters chosen, can induce a
first order phase transition instead of a crossover conf ; Lo:2013lca . This
is an interesting point and we shall defer its detailed study to a separate
paper. As a first numerical demonstration of the effect of polarization, we
consider a further approximation of
$\tilde{V}_{0}(T;M,\ell)\approx\tilde{V}_{0}(T;\langle M\rangle,\ell)$, and
fix $\langle M\rangle=0.136$ GeV. The set of model parameters is given by:
$\Lambda=1.076$ GeV, $G_{\rm NJL}\,\Lambda^{2}=4.232$,
$\mathcal{R}_{4D}(q)=e^{-q^{8}/\Lambda^{8}}$, and a current quark mass $m=5$
MeV. In the vacuum, this gives: $f_{\pi}=92.9$ MeV, $m_{\pi}=137.8$ MeV, and
$\langle\bar{\psi}\psi\rangle=-(250\,{\rm MeV})^{3}$ (per flavor). Solving
Eqs. (23), (25) and (27) consistently, we obtain the results in Fig. 2 (left).
The screening of the effective four-quark coupling by the electric mass leads
to a substantial drop in the chiral transition temperature.

###  3.2 finite magnetic field

While most effective chiral models can capture the magnetic catalysis in
vacuum, i.e. chiral condensate increases in magnitude at $T=0$ on increasing
$B$, they predict the opposite trend from LQCD calculations Bali:2011qj ;
Bruckmann:2013oba  on the magnetic field dependence of the chiral transition
temperature Boomsma:2009yk ; Lo:2020ptj . It is therefore of interest to
explore whether introducing the in-medium coupling via Eq. (9) could
accommodate a decreasing transition temperature with $B$. With the same model
setup we perform a numerical study of the chiral condensate, now at finite
magnetic field $B$. The result is shown in Fig. 2 (right). Without the
polarization screening in Eq. (9), the PNJL model predicts an increasing
chiral transition temperature with $B$, evident from the fact that the
condensate remains large for a wider range of temperatures. The dressed
interaction is capable of reversing this trend. Note that the electric mass
increases substantially at intermediate temperature with increasing $B$, which
can drive the phenomenon of inverse magnetic catalysis. The need for a
$B$-dependent coupling has been motivated in Refs. mediumG1 ; mediumG2 , and
Eq. (9) is a theoretical justification for introducing such a medium
dependence.

##  4 Conclusion

The promising results motivate further research to explore how polarization
effects modify the chiral quarks at finite baryon density. A systematic
examination of the many consequences of chiral symmetry restoration is
mandatory to construct more stringent predictions to probe possible signals of
criticality and to guide the search of the enigmatic critical endpoint by the
heavy-ion collision programme. To make progress it is essential to incorporate
other important physical effects absent in the current model. Chief among them
is to restore the momentum dependence of various quark dressing functions,
i.e. constituent mass function and the wavefunction renormalization, based on
better modeling of the confining potential conf . The external magnetic field
thus provides an additional handle to probe the relations between the quark
propagator and the gluon potential. Another important aspect is the explicit
treatment of dynamical gluons (and ghosts) in the confinement model
eric_cgauge ; Lo:2020ptj . Unlike the QED case, the electric mass of gluons
contains a pure glue contribution, which will further modify the in-medium
four-quark coupling. Due to its connection to the Cartan curvature masses
Lo:2020ptj , this needs to be determined consistently with the Polyakov loop
potential. 666Beside the statistical confinement models discussed here, there
are also confinement models which suppress the quark energy spectrum via an
infrared divergent mass (and wave function renormalization) conf . Finally, a
lattice computation of the gluon Debye mass at low temperature (with a finite
magnetic field) could validate the relevance of ring diagram screening to
inverse magnetic catalysis.

###### Acknowledgements.

This study receives supports from the Polish National Science Center (NCN)
under the Opus grant no. 2018/31/B/ST2/01663. M.S. acknowledges the support
from Preludium grant no. 2020/37/N/ST2/00367. K.R. also acknowledges the
support by the Polish Ministry of Science and Higher Education.

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# Gravity models with nonlinear symmetry realization

S.O. Alexeyev<EMAIL_ADDRESS>Sternberg Astronomical Institute, Lomonosov
Moscow State University, 119991, Moscow, Russia  Department of Quantum Theory
and High Energy Physics, Faculty of Physics, Lomonosov Moscow State
University, 119991, Moscow, Russia  D.P. Krichevskiy
<EMAIL_ADDRESS>Sternberg Astronomical Institute, Lomonosov Moscow
State University, 119991, Moscow, Russia  Physics Department, Bauman Moscow
State Technical University, Moscow, 105005, Russia  Faculty of Science,
University of Bern, CH-3012, Bern, Switzerland  B.N. Latosh
<EMAIL_ADDRESS>Bogoliubov Laboratory of Theoretical Physics, JINR,
141980, Dubna, Russia  Dubna State University, 141982, Dubna, Russia

###### Abstract

Three models with nonlinear realizations of conformal symmetry are discussed.
The simplest model can only describe a universe expanding with a deceleration
and does not include inflation. The other models are equivalent up to a
variables reparametrization. All these models contain ghost degrees of freedom
which may be excluded with an additional symmetry of the target space.

##  1 Introduction

Conformal symmetry plays a particular role in gravity. Its role is defined by
the well-known Ogievetsky theorem [1]. The theorem states that any generator
of the infinitely-dimensional coordinate transformation group is presented as
a series of commutators of generators from the conformal group $C(1,3)$ and
the affine group $A(4)$. This shows that the conformal group is strongly
related with coordinate transformations and shall have a certain influence of
gravity theory.

Particular implementations of conformal symmetry for gravity models were
studied in multiple papers, so we only mention a few key results. In paper [2]
in was shown that any metric gravity theory can be viewed as a theory with a
combined nonlinear realization of conformal and affine symmetry (see also [3,
4, 5, 6]). Explicit nonlinear realization of the conformal symmetry within
AdS/CFT approach was constructed in [7]. It also should be mentioned that
there are models with a linear realization of the conformal symmetry [8, 9],
but their applicability is debatable [10, 11, 12, 13].

In this paper we extend the discussion started in [14] of models with
nonlinear realizations of conformal symmetry proposed in [15]. In the last
paper three models were derived. The first one is defined by the Lagrangian:

$\displaystyle\begin{split}\mathcal{L}_{I}=&\cfrac{1}{2}\left[1+\cfrac{\sigma^{2}}{\varepsilon^{2}}\,\left\\{f_{2}\left(\cfrac{\psi}{\varepsilon}\right)\right\\}^{2}\right]\leavevmode\nobreak\
\eta^{\mu\nu}\,\partial_{\mu}\psi\,\partial_{\nu}\psi+\cfrac{1}{2}\,\left[f_{1}\left(\cfrac{\psi}{\varepsilon}\right)\right]^{2}\leavevmode\nobreak\
\eta^{\mu\nu}\,\eta_{(\alpha)(\beta)}\,\partial_{\mu}\sigma^{(\alpha)}\,\partial_{\nu}\sigma^{(\beta)}\\\
&-\cfrac{1}{\varepsilon}\,f_{1}\left(\cfrac{\psi}{\varepsilon}\right)\,f_{2}\left(\cfrac{\psi}{\varepsilon}\right)\leavevmode\nobreak\
\eta^{\mu\nu}\,\partial_{\mu}\psi\leavevmode\nobreak\
\sigma^{(\alpha)}\partial_{\nu}\sigma^{(\beta)}\,\eta_{(\alpha)(\beta)}.\end{split}$
(1)

Here $\psi$ and $\sigma^{(\alpha)}$ ($(\alpha)=0,\cdots,4$) are scalar fields
associated with target space coordinates on which the conformal symmetry acts
nonlinearly. Field indices $\sigma^{(\alpha)}$ taken in brackets should not be
confused with Lorentz indices. All fields have the canonical mass dimension
and $\varepsilon$ is a mass parameter corresponding to the conformal symmetry
breaking scale. Finally, functions $f_{i}(x)$ are

$\displaystyle\begin{split}f_{1}(x)&=\cfrac{e^{x}-1}{x}\,,\\\
f_{2}(x)&=\cfrac{e^{x}-x-1}{x^{2}}\,.\end{split}$ (2)

Therefore the model (1) describes five scalar fields propagating in a flat
space-time. Fields $\sigma^{(\alpha)}$ have the Minkowski space as the target
space so their indices are contracted with the Minkowski target metric
$\eta_{(\alpha)(\beta)}$.

The second model is

$\displaystyle\mathcal{L}=\eta^{\mu\nu}\nabla_{\mu}h^{(\alpha)(\beta)}\,\nabla_{\nu}h^{(\rho)(\sigma)}\,\eta_{(\alpha)(\rho)}\,\eta_{(\beta)(\sigma)},$
(3)

where the covariant derivatives are

$\displaystyle\cfrac{i}{2}\,\nabla_{\mu}h^{(\alpha)(\beta)}=$
$\displaystyle\cfrac{i}{2}\,\partial_{\mu}h^{(\alpha)(\beta)}+\sum\limits_{n=1}^{\infty}\cfrac{i}{(2n+1)!}\,(\operatorname{ad}_{h}^{2n-1}h\partial_{\mu}h)^{(\alpha)(\beta)}$
(4) $\displaystyle=$
$\displaystyle\cfrac{i}{2}\left[\partial_{\mu}h^{(\alpha)(\beta)}-\eta_{(\nu)(\sigma)}\eta_{(\mu)(\lambda)}\left(\cfrac{1}{3}\,h^{(\alpha)(\nu)}\,h^{(\beta)(\lambda)}\partial_{\mu}h^{(\sigma)(\mu)}-\cfrac{1}{3}\,h^{(\alpha)(\nu)}\,h^{(\sigma)(\mu)}\partial_{\mu}h^{(\lambda)(\beta)}\right)+\mathcal{O}(h^{5})\right]\,.$

Here all repeated bracket indices are contracted with the Minkowski metric of
the target space and $\operatorname{ad}$ operator is defined as:

$\displaystyle\left(\operatorname{ad}_{h}\partial_{\mu}h\right)^{(\mu)(\nu)}=[h,\partial_{\mu}h]^{(\mu)(\nu)}=h^{(\mu)(\rho)}\partial_{\mu}h^{(\sigma)(\nu)}\eta_{(\rho)(\sigma)}-\partial_{\mu}h^{(\mu)(\rho)}h^{(\sigma)(\nu)}\eta_{(\rho)(\sigma)}.$
(5)

The third model, in some sense, is similar to the second one:

$\displaystyle\mathcal{L}=\eta^{\mu\nu}\nabla_{\mu}h^{(\alpha)(\beta)}\,\nabla_{\nu}h^{(\rho)(\sigma)}\,\eta_{(\alpha)(\rho)}\,\eta_{(\beta)(\sigma)}+\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi.$
(6)

The same definition of covariant derivatives is used, $\phi$ is a sterile
scalar (i.e. a scalar that does not interact with any other degrees of
freedom). Further we demonstrate that the second and third models are actually
equivalent due to a certain property of the nonlinear symmetry realization
used.

The reasons to study these models are the following. Firstly, a nonlinear
realization of a symmetry is (almost always) related with a spontaneous
symmetry breaking. In the context of the conformal symmetry this means the
following. A physical system admits the conformal symmetry in the high energy
phase. In the low energy phase the system develops a conformal symmetry
breaking ground state and perturbations propagating about that background are
subjected to a nonlinear realization of the symmetry. Due to the Goldstone
theorem after the spontaneous symmetry breaking massless modes appear and some
of them could be associated with massless gravitons [2]. The key idea is that
additional massless modes (for instance, the scalar modes of (1)) may be
associated with the inflaton. Hence it may be conjectured that the inflation
(and the inflaton driving it) is a manifestation of the spontaneously broken
conformal symmetry. Secondly, models (3) and (6) describe degrees of freedom
$h_{\mu\nu}$ that may play a role of tetrads similar to [2]. Finally
$h_{\mu\nu}$ could contain additional scalar and vector degrees of freedom.
Therefore it is interesting to study the content of models (3) and (6).

The paper is organized as follows. In Section 2 we discuss cosmological
regimes described by the model (1) and show that scalar degrees of freedom are
decoupled. Therefore they do not influence the cosmological behavior in a
meaningful way. We discuss a possible relation between this phenomenon and the
conformal symmetry together with implications for realistic cosmological
scenarios. In section 3 we discuss the field content of models (3) and (6). It
is shown that these models actually equivalent up to a coordinate redefinition
on the target space. At the same time these models contain vector ghost
degrees of freedom. We argue that these degrees of freedom can be excluded via
an introduction of an additional symmetry to the target space. However, this
symmetry should be agreed with the nonlinear realization of the conformal
symmetry which may influence the used nonlinear realization in a meaningful
way. Section 4 contains our conclusions which extend previous results [14].
Despite the reasonable motivation proposed in [15] they could hardly be
realistic without an additional development. The key reason is the presence of
ghost degrees of freedom. If ghosts are excluded these models may lose the
desirable features justifying their applicability.

##  2 Cosmological behavior

To study the cosmological behavior of the model (1) one has to introduce
gravitational degrees of freedom because their number in (1) is not enough to
describe gravitons. Therefore the standard gravitational degrees of freedom
(originating from general relativity (GR)) are required. Such procedure does
not contradict the existence of a non-linearly realized conformal symmetry
because the used nonlinear realization also generates a linear action of the
conformal group on the Lorentz group [15]. Therefore to study the cosmological
expansion we use the following model:

$\displaystyle\begin{split}\mathcal{S}=&\int
d^{4}x\sqrt{-g}\Bigg{\\{}-\cfrac{2}{\kappa^{2}}\,R+\cfrac{1}{2}\left[1+\cfrac{\sigma^{2}}{\varepsilon^{2}}\,\left\\{f_{2}\left(\cfrac{\psi}{\varepsilon}\right)\right\\}^{2}\right]\leavevmode\nobreak\
g^{\mu\nu}\,\partial_{\mu}\psi\,\partial_{\nu}\psi\\\
&+\cfrac{1}{2}\,\left[f_{1}\left(\cfrac{\psi}{\varepsilon}\right)\right]^{2}\leavevmode\nobreak\
g^{\mu\nu}\,\eta_{(\alpha)(\beta)}\,\partial_{\mu}\sigma^{(\alpha)}\,\partial_{\nu}\sigma^{(\beta)}-\cfrac{1}{\varepsilon}\,f_{1}\left(\cfrac{\psi}{\varepsilon}\right)\,f_{2}\left(\cfrac{\psi}{\varepsilon}\right)\leavevmode\nobreak\
g^{\mu\nu}\,\partial_{\mu}\psi\leavevmode\nobreak\
\sigma^{(\alpha)}\partial_{\nu}\sigma^{(\beta)}\,\eta_{(\alpha)(\beta)}\Bigg{\\}}\,,\end{split}$
(7)

where $\kappa$ is related with the Newton constant $G_{N}$ as
$\kappa^{2}=32\pi G_{N}$. Fields $\psi$ and $\sigma^{(\alpha)}$ transform
under the nonlinear conformal group action with the target space remaining
flat.

Let us consider the cosmological behavior of (7) with the open Friedmann
space-time:

$\displaystyle
ds^{2}=g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}=dt^{2}-a^{2}(t)\left(dx^{2}+dy^{2}+dz^{2}\right).$
(8)

Here $a(t)$ is the scale factor. The corresponding Einstein equations read

$\displaystyle\begin{split}G_{\mu\nu}=\cfrac{\kappa^{2}}{4}\,C_{\mu\nu}{}^{\alpha\beta}&\left[\cfrac{1}{2}\,\left[1+\cfrac{\sigma^{2}}{\varepsilon^{2}}\,\left\\{f_{2}\left(\cfrac{\psi}{\varepsilon}\right)\right\\}^{2}\right]\,\partial_{\alpha}\psi\,\partial_{\beta}\psi+\cfrac{1}{2}\,\left[f_{1}\left(\cfrac{\psi}{\varepsilon}\right)\right]^{2}\leavevmode\nobreak\
\eta_{(\mu)(\nu)}\,\partial_{\alpha}\sigma^{(\mu)}\,\partial_{\beta}\sigma^{(\nu)}\right.\\\
&\hskip
15.0pt\left.-\cfrac{1}{\varepsilon}\,f_{1}\left(\cfrac{\psi}{\varepsilon}\right)\,f_{2}\left(\cfrac{\psi}{\varepsilon}\right)\leavevmode\nobreak\
\partial_{\alpha}\psi\leavevmode\nobreak\
\sigma^{(\mu)}\partial_{\beta}\sigma^{(\nu)}\,\eta_{(\mu)(\nu)}\right]\end{split}$
(9)

where

$\displaystyle
C_{\mu\nu}{}^{\alpha\beta}\overset{\text{def}}{=}\delta_{\mu}^{\alpha}\delta_{\nu}^{\beta}+\delta_{\mu}^{\beta}\delta_{\nu}^{\alpha}-g_{\mu\nu}g^{\alpha\beta}\,.$
(10)

These equations have two non-vanishing components which can be reduced to:

$\displaystyle\begin{split}&-3\cfrac{\ddot{a}}{a}=\cfrac{1}{4}\,\kappa^{2}\,\left(\cfrac{1}{2}\,\left[1+\cfrac{\sigma^{2}}{\varepsilon^{2}}\,\left\\{f_{2}\left(\cfrac{\psi}{\varepsilon}\right)\right\\}^{2}\right]\dot{\psi}^{2}+\left[f_{1}\left(\cfrac{\psi}{\varepsilon}\right)\right]^{2}\dot{\sigma}^{(\alpha)}\dot{\sigma}_{(\alpha)}-\frac{2}{\varepsilon}f_{1}\left(\frac{\psi}{\varepsilon}\right)f_{2}\left(\frac{\psi}{\varepsilon}\right)\dot{\psi}\,\dot{\sigma}_{(\alpha)}\sigma^{(\alpha)}\right)\,,\\\
&2\dot{a}^{2}+a\ddot{a}=0\,.\end{split}$ (11)

The second equation from (11) does not contain scalar fields and describes the
behavior of the scalar factor by itself. It can be solved analytically:

$a(t)=c_{2}\left(c_{1}+3\,t\right)^{\frac{1}{3}}\,,$ (12)

where $c_{1}$ and $c_{2}$ are the integration constants defined by the
boundary conditions.

Now we analyze the result (12). First of all we note that here the universe
has only a decelerated expansion hence the model has no room for an
inflationary phase.

Secondly, one can assume that the matter content of the model (i.e. the scalar
fields) is described by the standard equation of state (EoS) $p=w\rho$. Here
$p$ is a pressure of scalar fields, $\rho$ is an energy density of the matter,
and $w$ is the EoS parameter. Solutions with such EoS are well known [16, 17],
so one can easily restore $w$ from the form of asymptotic of (12). At large
values of time the scale factor is proportional to $t^{\frac{1}{3}}$
corresponding to EoS parameter $w=1$. That is why despite the fact that scalar
fields in this model admit vanishing masses their behavior do not correspond
to a relativistic matter one with EoS parameter $w=1/3$. The reason is that
the discussed scalar fields have a non-trivial interaction sector which
influence their EoS.

Finally, the fact that the model admits a decelerating solution deserves a
special attention as it has ghost degrees of freedom. These ghost degrees of
freedom appear due to the metric on scalar field target space
$\eta_{(\mu)(\nu)}$:

$\displaystyle
g^{\mu\nu}\,\eta_{(\alpha)(\beta)}\,\partial_{\mu}\sigma^{(\alpha)}\,\partial_{\nu}\sigma^{(\beta)}=\partial^{\mu}\sigma^{(0)}\,\partial_{\mu}\sigma^{(0)}-\sum\limits_{i=1}^{3}\partial^{\mu}\sigma^{(i)}\,\partial_{\mu}\sigma^{(i)}\,.$
(13)

Note that these ghost degrees of freedom do not manifest themselves at the
level of cosmological solutions.

Despite the fact the an analytical solution (12) can be obtained, there are no
reasons to believe that it is stable. Equations (11) take a simple form due to
a cancellation of the stress-energy tensor of scalar fields. This
cancellation, in turn, is possible because both metric and scalar fields
depend only on the time variable. As soon as one considers metric and scalar
field perturbations propagating around the background a similar cancellation
becomes impossible.

The existence of ghost degrees of freedom makes possible implementations of
(7) unlikely. Let us discuss possible opportunities to exclude ghosts. The
first simple way is to use the inverse Higgs mechanism [18]. Unfortunately
this procedure is not applicable here because the discussed nonlinear symmetry
realization does no satisfy the necessary criteria. Another opportunity to
exclude ghosts is to introduce an additional symmetry at the scalar field
target space which would make $\sigma^{(1)}=\sigma^{(2)}=\sigma^{(3)}=0$.
Consequently, only a single massless sterile scalar field (i.e. it has neither
self-interaction nor potential) $\psi$ remains and such a field can hardly be
applied in realistic scenarios. The best opportunity wpuld be to find a
mechanism excluding ghost degrees of freedom from the model’s physical
spectrum, but allowing them to propagate only in loops. As it was shown in
[19] for such a case the scalar field $\psi$ develops non-trivial interaction
at the loop level. However, such a mechanism has yet to be found.

Summarizing, we conclude that the model (1) can hardly be applied for
realistic scenarios. The main issue is the existence of ghost degrees of
freedom that expectidly lead to an instability. Even if one finds a way to
negate the ghost problem the known cosmological solutions (12) show that the
scalar fields act as matter with EoS parameter $w=1$ and cannot drive an
inflation. Therefore one shall consider the model (1) unrealistic.

##  3 Field content

Now we switch to a discussion of the field content of second (3) and third (6)
models. Firstly, they have at least ten degrees of freedom from the symmetric
matrix $h^{(\alpha)(\beta)}$, therefore they can describe spin-$2$ massless
degrees of freedom associated with gravitons. Secondly, $h^{(\alpha)(\beta)}$
may also contain two scalar degrees of freedom (associated with its
determinant and trace) which could serve as inflatons. Finally, the third
model (6) contains a sterile scalar appearing as a consequence of the
properties of operator $D$ of the conformal group [15]. These issues are
clarified further.

First and foremost we shall address the issue related with the sterile scalar
of model (6). In the original article [15] it was missed that the sterile
scalar is related with the trace of $h^{(\alpha)(\beta)}$. The reason behind
this is due to the relation between the conformal group generators. Namely, as
degrees of freedom $h^{(\alpha)(\beta)}$ are associated with the coset
coordinates along $R_{(\alpha)(\beta)}$ direction the sterile scalar $\phi$ is
associated with the coset coordinates along $D$ direction also. However
operators $R_{(\alpha)(\beta)}$ are dependent and coupled as [1, 2, 15]:

$\displaystyle\eta^{(\alpha)(\beta)}\,R_{(\alpha)(\beta)}=2D\,.$ (14)

Therefore the corresponding coset coordinates are also dependent and the trace
$\eta_{(\alpha)(\beta)}h^{(\alpha)(\beta)}$ should be associated with $\phi$.
Hence the trace component of $h^{(\alpha)(\beta)}$ acts as a sterile massless
scalar and therefore cannot drive the inflation. Moreover, one can treat
$h^{(\alpha)(\beta)}$ as a traceless matrix with $9$ independent components
reducing the number of valuable degrees of freedom in the model. On the other
hand it guarantees the traceless of $h^{(\alpha)(\beta)}$ similar to GR
degrees of freedom.

Finally, note that this result can be obtained independently via the direct
verification. The definition
$h^{(\alpha)(\beta)}=\cfrac{1}{4}\,\eta^{(\alpha)(\beta)}\,h$ with
$h=\eta_{(\alpha)(\beta)}h^{(\alpha)(\beta)}$ causes the covariant derivative
(4) to be completely reduced to the regular ones hence the Lagrangian (3)
describes only a single massless sterile scalar.

Now we switch to $\overline{h}^{(\alpha)(\beta)}$ as the traceless part of
$h^{(\alpha)(\beta)}$ and start to study its Lagrangian. The corresponding
covariant derivative (4) is:

$\displaystyle\begin{split}&\cfrac{i}{2}\,\nabla_{\mu}\overline{h}^{(\alpha)(\beta)}\\\
&=\cfrac{i}{2}\left[\partial_{\mu}\overline{h}^{(\alpha)(\beta)}-\eta_{(\nu)(\sigma)}\eta_{(\mu)(\lambda)}\left(\cfrac{1}{3}\,\overline{h}^{(\alpha)(\nu)}\,\overline{h}^{(\beta)(\lambda)}\partial_{\mu}\overline{h}^{(\sigma)(\mu)}-\cfrac{1}{3}\,\overline{h}^{(\alpha)(\nu)}\,\overline{h}^{(\sigma)(\mu)}\partial_{\mu}\overline{h}^{(\lambda)(\beta)}\right)+\mathcal{O}(h^{5})\right]\,.\end{split}$
(15)

This derivative matches the expression (4) because the trace component is
contained only in $\partial_{\mu}h^{(\alpha)(\beta)}$. Therefore the
expression (3) determines the traceless part of $h^{(\alpha)(\beta)}$ without
changing its form.

Further as the target space metric is not positively defined the model may
contain ghosts. Let us demonstrate this explicitly. The Lagrangian density
$\mathcal{L}$ of (3) given up to $\mathcal{O}(h^{3})$ reads:

$\displaystyle\begin{split}\mathcal{L}=&\cfrac{1}{2}\,\eta_{(\alpha)(\rho)}\eta_{(\beta)(\sigma)}\,\partial_{\mu}\overline{h}^{(\alpha)(\beta)}\,\partial^{\mu}\overline{h}^{(\rho)(\sigma)}\\\
=&\cfrac{1}{2}\,\partial_{\mu}\overline{h}^{(0)(0)}\partial^{\mu}\overline{h}^{(0)(0)}+\cfrac{1}{2}\,\sum\limits_{a,b=1}^{3}\,\partial_{\mu}\overline{h}^{(a)(b)}\,\partial^{\mu}\overline{h}^{(a)(b)}-\sum\limits_{s=1}^{3}\,\partial_{\mu}\overline{h}^{(0)(s)}\,\partial^{\mu}\overline{h}^{(0)(s)}\,.\end{split}$
(16)

It is clearly seen that $\overline{h}^{(0)(s)}$ with $s=1,2,3$ have the wrong
sign kinetic term and describe ghost degrees of freedom. Because of the off-
diagonal elements $h^{(0)(i)}$ the model cannot be considered realistic until
these degrees of freedom are excluded. These ghosts cannot be excluded via the
inverse Higgs mechanism as the corresponding generators do not satisfy the
required conditions [18].

One can introduce an additional symmetry in the target space. The choice of
the target space coordinates as $\zeta^{(\alpha)}$ causes the following form
of the target space metric:

$\displaystyle g=\eta_{(\alpha)(\beta)}\,d\zeta^{(\alpha)}\otimes
d\zeta^{(\beta)}\,.$ (17)

If one demands a symmetry with respect to $\zeta^{(0)}$ inversion
($\zeta^{(0)}\to-\zeta^{(0)}$) then the field $h^{(\alpha)(\beta)}$ must have
the vanishing component $h^{(0)(i)}$ to be consistent with the considering
symmetry. This symmetry condition should be considered independently from the
particular choice of the nonlinear conformal symmetry realization. However, it
is unclear what physical reason can justify the introduction of such a
symmetry. Therefore for the time being we consider it as an ad hoc
prescription.

With the discussed prescription the traceless part of $h^{(\alpha)(\beta)}$
includes six independent components which can fit one spin-$2$ degree of
freedom and one spin-$0$ degree of freedom. To obtain the vanishing trace we
set

$\displaystyle\overline{h}^{(0)(0)}=\sum\limits_{a=1}^{3}\overline{h}^{(a)(a)}.$
(18)

In such a presentation the only non-vanishing components are $h^{(a)(b)}$
where $a,b=1,2,3$. Within such a setup the original Lagrangian (3) up to
$\mathcal{O}(h^{4})$ order looks as:

$\displaystyle\begin{split}\mathcal{L}=&\partial_{\mu}h^{(1)(1)}\,\partial^{\mu}h^{(1)(1)}+\partial_{\mu}h^{(2)(2)}\,\partial^{\mu}h^{(2)(2)}+\partial_{\mu}h^{(3)(3)}\,\partial^{\mu}h^{(3)(3)}\\\
&+\partial_{\mu}h^{(1)(2)}\,\partial^{\mu}h^{(1)(2)}+\partial_{\mu}h^{(2)(3)}\,\partial^{\mu}h^{(2)(3)}+\partial_{\mu}h^{(3)(1)}\,\partial^{\mu}h^{(3)(1)}\\\
&\partial_{\mu}h^{(1)(1)}\,\partial^{\mu}h^{(2)(2)}+\partial_{\mu}h^{(2)(2)}\,\partial^{\mu}h^{(3)(3)}+\partial_{\mu}h^{(3)(3)}\,\partial^{\mu}h^{(1)(1)}+\mathcal{O}(h^{4})\,.\end{split}$
(19)

Generically this Lagrangian is non-diagonal. It could be made diagonal in the
following representation:

$\displaystyle\begin{split}\zeta_{1}=&\cfrac{1}{\sqrt{6}}\,\left[h^{(1)(1)}+h^{(2)(2)}+h^{(3)(3)}\right]\,,\\\
\zeta_{2}=&h^{(1)(1)}-h^{(3)(3)}\,,\\\
\zeta_{3}=&h^{(1)(1)}-h^{(2)(2)}\,,\end{split}$
$\displaystyle\begin{split}\zeta_{4}=&h^{(1)(2)}\,,\\\
\zeta_{5}=&h^{(2)(3)}\,,\\\ \zeta_{6}=&h^{(3)(1)}\,.\end{split}$ (20)

The quadratic part of the diagonal Lagrangian reads

$\displaystyle\mathcal{L}=\sum\limits_{i=1}^{6}\partial_{\mu}\zeta_{i}\,\partial^{\mu}\zeta_{i}$
(21)

so the corresponding field equations are reduced to the Klein-Gordon equation

$\displaystyle\square\,\zeta_{i}=0\,.$ (22)

As a result the model describes massless degrees of freedom. However one lacks
a condition which can fix their chirality.

Mass and chirality of a particle are fixed by eigenvalues of the Poincare
group Casimir operators [20, 21]. The D’Alamber operator $\square$ defined to
the mass operator which is one of the two Poincare group Casimir operators.
The second Casimir operator is the Pauli-Lubanski vector

$\displaystyle
W^{\mu}=\cfrac{1}{2}\,\varepsilon^{\mu\nu\alpha\beta}\,P_{\nu}\,M_{\alpha\beta}\,$
(23)

where $M_{\alpha\beta}$ are Lorentz generators building Lorentz group action
on a given degree of freedom. The degrees of freedom $h^{(\alpha)(\beta)}$
should be mixed to diagonalize the Lagrangian therefore they can be only
scalars of the trivial Lorenz group action. Hence the model has no room for
spin-$2$ degrees of freedom.

This section could be summarized as follows. Both discussed models (3) and (6)
really are equal because of the relations between the conformal group
generators with the influence on the coset coordinates and the corresponding
nonlinear symmetry realization. Secondly, the model (3) contains ghost degrees
of freedom that can be excluded from the model only with an ad hoc
prescription of auxiliary symmetry of the model target space. Such a
prescription makes the model to be not natural. Nonetheless, the model with
the additional symmetry is healthy and describes massless degrees of freedom
appearing to be scalar ones. All these factors make the considered model not
realistic.

##  4 Discussion and conclusions

Three models with particular nonlinear conformal symmetry realizations [15]
were studied. We extend the consideration started in [14] and demonstrate that
the discussed models could become realistic only after significant
modifications.

The first model (1) seems not to be realistic. Firstly, the original model
contains five degrees of freedom propagating in a flat space-time. In [15] it
was argued that an extension to a curved space-time case may realize a viable
inflationary scenario. Following [14] we show that in this model a universe
expands with a deceleration (EoS parameter $w=1$) and, therefore, there is no
inflation in it. Secondly, the model contains ghost degrees of freedom.
Although they do not appear at the cosmological solution it is reasonable to
expect that they make the solution unstable. Therefore the model is not
applicable unless ghosts are excluded from the perturbation spectrum.

Secondly, (3) and (6) were analyzed and found out to be equivalent. This
feature was missed both in [15] and [14]. The models are equivalent up to a
parametrization of dynamical variables. As it is pointed in the previous
section the trace component of coset coordinates $h^{(\alpha)(\beta)}$ present
in model (3) is the coset coordinate $\phi$ present in model (6). The coset
coordinate $\phi$ is associated with operator $D$ which is related with the
trace of operators $R_{(\alpha)(\beta)}$ which, in turn, are associated with
coordinated $h_{(\alpha)(\beta)}$ (see (14)). Therefore models (3) and (6)
merely provide different parametrizations of the same model.

Finally, we analyzed the field content of model (3). The trace component of
$h_{(\alpha)(\beta)}$ acts as a sterile massless scalar which excludes its
practical application. The traceless part $\overline{h}_{(\alpha)(\beta)}$
contains nine degrees of freedom. They are three ghosts
$\overline{h}^{(0)(s)}$ with $s=1,2,3$ which cannot be excluded with the
inverse Higgs mechanism[18]. An additional symmetry of the target space may
exclude the ghosts. Therefore an opportunity to introduce such a symmetry
should be studied.

## Acknowledgment

The work (B.L.) was supported by the Foundation for the Advancement of
Theoretical Physics and Mathematics “BASIS”. The work (S.A.) was also
supported by the Interdisciplinary Scientific and Educational School of Moscow
University “Fundamental and Applied Space Research”.

## Appendix A

Here it is necessary to demonstrate some features of an interaction between
$h^{(\mu)(\nu)}$ and matter degrees of freedom within model (3). The issue
provides an additional reason to believe that the model can hardly be
considered realistic.

For the sake of simplicity we consider a single massless vector field (which
can be associated with the electromagnetic field). Accordingly to [15] such a
vector field is described by the standard Lagrangian

$\displaystyle\mathcal{L}_{\text{vec}}=-\cfrac{1}{4}\,F_{\mu\nu}F^{\mu\nu}\,.$
(24)

However, a definition of the field tensor contains covariant derivatives:

$\displaystyle F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}\,.$ (25)

The covariant derivative, in turn, reads

$\displaystyle\begin{split}\nabla_{\mu}A^{\nu}=\partial_{\mu}A^{\nu}+\sum^{\infty}_{n=0}\frac{i}{(2n+2)!}(ad^{2n}_{h}h\partial_{\mu}h)^{(\alpha)(\beta)}(M_{(\alpha)(\beta)})^{\nu}_{\sigma}A^{\sigma}\\\
=\partial_{\mu}A^{\nu}+\frac{i}{2}\eta_{(\gamma)(\sigma)}h^{(\alpha)(\gamma)}\partial_{\mu}h^{(\beta)(\sigma)}(M_{(\alpha)(\beta)})^{\nu}_{\sigma}A^{\sigma}+\mathcal{O}(h^{2})\,.\end{split}$
(26)

Here
$\left(M_{(\alpha)(\beta)}\right)_{\mu\nu}=i(\eta_{(\alpha)\mu}\eta_{(\beta)\nu}-\eta_{(\alpha)\nu}\eta_{(\beta)\mu})$
is the vector representation of Lorentz group generators.

The corresponding Lagrangian has a few peculiar features. First and foremost,
at the leading order the model describes interaction between two modes
$h^{(\mu)(\nu)}$ and two vectors $A^{\mu}$. Within GR, in contrast, the
leading order interaction between gravity and a massless vector field is a
cubic interaction describing an interaction between two vectors and a
graviton. Secondly, in contrast with GR the interaction can contain only odd
powers of $h^{(\mu)(\nu)}$.

Implications of these features lead to significant differences between GR and
(3). The most interesting one is related with $2\to 2$ scattering. Within GR
there are tree-level amplitudes describing gravitational scattering of
massless vectors. Within model (3) such a scattering appears only at the one-
loop level.

It is necessary to point out that to pinpoint the exact difference in such
scattering processes a more detailed analysis is required. However, even at
the current level the model (3) appears to be inconsistent with GR.

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