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The dataset generation failed
Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 34
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 37099)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 34
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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text
string | meta
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|---|---|
\section{Introduction}
\label{sec:introduction}
Quantum deformations based on the $\kappa $-Poincar\'{e}-Hopf algebra
constitute an important branch of research that enables us to address
problems in condensed matter and high energy physics through field
equations. These field equations were first presented in Ref. \cite
{PLB.1992.293.344}, where a new real quantum Poincar\'{e} algebra with
standard real structure, obtained by contraction of $U_{q}\left( O\left(
3,2\right) \right) $. The resulting algebra of this contraction is a
standard real Hopf algebra and depends on a dimension-full parameter $\kappa
$ instead of $q$. Since then, the algebraic structure of the $\kappa $
-deformed Poincar\'{e} algebra has been investigated intensively and have
become a theoretical field of increasing interest \cite
{PLB.1994.329.189,PLB.1994.334.348,
CQG.2010.27.025012,PLB.2012.711.122,NPB.2001.102-103.161,PLB.2002.529.256,PRD.2011.84.085020,JHEP.2011.1112.080,EPJC.2013.73.2472, PRD.2009.79.045012,EPJC.2006.47.531,EPJC.2008.53.295,PLB.2013.719.467,CQG.2004.21.2179,JHEP.2004.2004.28,PLB.1995.359.339, PLB.1994.339.87,PRD.2013.87.125009,PRD.2012.85.045029,PRD.2009.80.025014}
. Through the field equations from the $\kappa $-Poincar\'{e} algebra ($
\kappa $-Dirac equation \cite
{PLB.1993.302.419,PLB.1993.318.613,EPJC.2003.31.129}), we can study the
physical implications of the quantum deformation parameter $\kappa $ in
relativistic and nonrelativistic quantum systems. In this context, we
highlight the study of relativistic Landau levels \cite{PLB.1994.339.87},
the Aharonov-Bohm effect taking into account spin effects \cite
{PLB.1995.359.339}, the Dirac oscillator \cite{PLB.2014.731.327,PLB.2014.738.44} and the
integer quantum Hall effect \cite{EPL.2016.116.31002}.
When we want to study the relativistic quantum dynamics of particles with
spin, we must obviously consider the presence of external fields, which
include the vector and scalar fields. The inclusion of vector and scalar
potentials in the Dirac equation reveals interesting properties of
symmetries in nuclear theory. The first contributions in this subject
revealed the existence of $SU(2)$ symmetries, which are known in the
literature as pseudospin and spin symmetries \cite
{AoP.1971.65.352,NPB.1975.98.151}. Some investigations have been made in
this scenario in order to give a meaning to these symmetries. However, it
was only in a work by Ginocchio, that pseudospin symmetry was revealed. He
verified that pseudospin symmetry in nuclei could arise from nucleons moving
in a relativistic mean field, which has an attractive scalar and repulsive
vector potential nearly equal in magnitude \cite{PRL.1997.78.436} (for a
more detailed description see Ref. \cite{PR.2005.414.165}). Spin and
pseudo-spin symmetries in the Dirac equation have been studied under
different aspects in recent years (see Refs. \cite
{JPG.1999.25.811,PRA.2015.92.062137}). Some studies have been developed
taking into account the spin and pseudospin symmetry limits to study
relativistic dynamics of physical systems interacting with a class of
potentials \cite{CTP.2012.58.807,FBS.2013.54.1839,EPJA.2009.43.73,
AMC.2010.216.545,PRA.2012.86.032122,PRC.2012.86.052201,AoP.2015.356.83,AoP.2015.362.196}.
The present work is proposed to investigate the $\kappa $-deformed Dirac
equation derived in Refs. \cite{PLB.1993.318.613} in the
context of minimum, vector and scalar couplings under spin and pseudospin
symmetric limits.
The structure of the paper is as follows. In Sec. \ref{sec:II}, we present
the $\kappa $-deformed Dirac equation with couplings from which we derive
the $\kappa $-deformed Pauli-Dirac equation, by using the usual procedure
that consists of squaring the $\kappa $-deformed Dirac equation. In Sec. \ref
{sec:III}, we consider the equation of Pauli and establish the spin and
pseudospin symmetries limits. As an application, we consider the particle
interacting with an uniform magnetic field in the $z$-direction in two
different physical situations: (i) particle interacting with a harmonic
oscillator and (ii) particle interacting with a linear potential. We obtain
expressions for the energy eigenvalues and wave functions in both limits.
In Sec. \ref{sec:c}, we present our comments and conclusions.
\section{The $\protect\kappa$-deformed Dirac equation with couplings}
\label{sec:II}
We begin with the deformed Dirac equation invariant under the $\kappa $
-deformed Poincar\'{e} quantum algebra \cite{PLB.1993.302.419,
PLB.1993.318.613}
\begin{equation}
\left\{ \left( \gamma _{0}p_{0}-\gamma _{i}p_{i}\right) +\frac{1}{2}
\varepsilon \left[ \gamma _{0}\left( p_{0}^{2}-p_{i}p_{i}\right) -Mp_{0}
\right] \right\} \psi =M\psi . \label{e1}
\end{equation}
The interactions can be performed through the following prescriptions \cite
{greiner.rqm.wf}:
\begin{eqnarray}
p_{i} &\rightarrow &p_{i}-eA_{i}, \\
E &\rightarrow &E-\nu \left( r\right) , \\
M &\rightarrow &M+w\left( r\right) .
\end{eqnarray}
As we are interested in a planar dynamics, i.e., when the third directions
of the fields involved are zero, we choose the following representation for
the gamma matrices \cite{NPB.1988.307.909}:
\begin{eqnarray}
\gamma _{0} &=&\sigma _{3}, \label{sigma1} \\
\alpha _{1} &=&\gamma _{0}\gamma _{1}=\sigma _{1}, \label{sigma2} \\
\alpha _{2} &=&\gamma _{0}\gamma _{2}=s\sigma _{2}, \label{sigma3}
\end{eqnarray}
where the parameter $s$, which has a value of twice the spin value, can be
introduced to characterizing the two spin states, with $s=+1$ for spin "up"
and $s=-1$ for spin "down". In the above representation, the $\kappa $
-deformed Dirac including the interactions can be written as
\begin{align}
& \left[ \mathbf{\alpha }\cdot \left( \mathbf{p}-e\mathbf{A}\right) +\gamma
_{0}\left( M+w\left( r\right) \right) \right] \psi -\left[ E-\nu \left(
r\right) \right] \psi \notag \\
& +\frac{\varepsilon }{2}\left[ es\left( \mathbf{\sigma }\cdot \mathbf{B}
\right) \psi +\gamma _{0}\left( \left( \mathbf{\alpha }\cdot \mathbf{p}
\right) w\left( r\right) \right) \psi +M\gamma _{0}\left( \mathbf{\alpha }
\cdot \mathbf{p}\right) \psi \right] \notag \\
& +\frac{\varepsilon }{2}\left[ \gamma _{0}w\left( r\right) \left( \mathbf{
\alpha }\cdot \mathbf{p}\right) \psi -\gamma _{0}e\left( \mathbf{\alpha }
\cdot \mathbf{A}\right) M\psi \right] \notag \\
& -\frac{\varepsilon }{2}\left[ \gamma _{0}e\left( \mathbf{\alpha }\cdot
\mathbf{A}\right) w\left( r\right) \psi \right] =0. \label{d1}
\end{align}
Let us now determine the Dirac equation in its quadratic form. This can be
accomplished by applying the operator
\begin{align}
& \mathbf{\alpha }\cdot \left( \mathbf{p}-e\mathbf{A}\right) +\gamma
_{0}\left( M+w\left( r\right) \right) +\left[ E-\nu \left( r\right) \right]
\notag \\
& +\frac{\varepsilon }{2}\left[ es\left( \mathbf{\sigma }\cdot \mathbf{B}
\right) +\gamma _{0}\left( \left( \mathbf{\alpha }\cdot \mathbf{p}\right)
w\left( r\right) \right) +M\gamma _{0}\left( \mathbf{\alpha }\cdot \mathbf{p}
\right) \right] \notag \\
& +\frac{\varepsilon }{2}\left[ \gamma _{0}w\left( r\right) \left( \mathbf{
\alpha }\cdot \mathbf{p}\right) -\gamma _{0}e\left( \mathbf{\alpha }\cdot
\mathbf{A}\right) M-\gamma _{0}e\left( \mathbf{\alpha }\cdot \mathbf{A}
\right) w\left( r\right) \right]
\end{align}
in Eq. (\ref{d1}). The result is the $\kappa $-deformed Dirac-Pauli equation
\begin{align}
& \left( \mathbf{p}-e\mathbf{A}\right) ^{2}\psi +\mathbf{\alpha }\cdot \left[
\mathbf{p}\nu \left( r\right) \right] \psi -\gamma _{0}\mathbf{\alpha }\cdot
\left[ \mathbf{p}w\left( r\right) \right] \psi \notag \\
& +\left[ M+w\left( r\right) \right] ^{2}\psi -\left[ E-\nu \left( r\right)
\right] ^{2}\psi -es\sigma _{z}B\psi \notag \\
& -\frac{\varepsilon }{2}\left\{ \gamma _{0}\left[ \mathbf{p}^{2}w\left(
r\right) \right] +\gamma _{0}\left[ \left( \mathbf{\alpha }\cdot \mathbf{p}
\right) w\left( r\right) \right] \left[ \left( \mathbf{\alpha }\cdot \mathbf{
p}\right) \right] \right\} \psi \notag \\
& +\frac{\varepsilon }{2}\left\{ 2is\gamma _{0}\left[ \mathbf{\sigma }\cdot
\left[ \left( \mathbf{p}w\left( r\right) \right) \times \mathbf{p}\right] -e
\mathbf{\sigma }\cdot \left( \mathbf{p}w\left( r\right) \right) \times
\mathbf{A}\right] \right\} \psi \notag \\
& +\frac{\varepsilon }{2}\left\{ e\gamma _{0}\left[ \left( \mathbf{\alpha }
\cdot \mathbf{p}\right) w\left( r\right) \right] \left[ \left( \mathbf{
\alpha }\cdot \mathbf{A}\right) \right] -w\left( r\right) \left[ \left(
\mathbf{\alpha }\cdot \mathbf{p}\right) w\left( r\right) \right] \right\}
\psi \notag \\
& +\frac{\varepsilon }{2}\left\{ 2MesB+2w\left( r\right) esB-M\left[ \left(
\mathbf{\alpha }\cdot \mathbf{p}\right) w\left( r\right) \right] \right\}
\psi \notag \\
& +\frac{\varepsilon }{2}\left\{ M\gamma _{0}\left[ \left( \mathbf{\alpha }
\cdot \mathbf{p}\right) \nu \left( r\right) \right] +\gamma _{0}w\left(
r\right) \left[ \left( \mathbf{\alpha }\cdot \mathbf{p}\right) \nu \left(
r\right) \right] \right\} \psi =0. \label{dirackp}
\end{align}
In order to apply this equation to some physical system, we need to choose a
representation for the vector potential $\mathbf{A}$ and the scalar potentials $
w\left( r\right) $ and $\nu \left( r\right) $. For certain particular
choices of these quantities, we can study the physical implications of
quantum deformation on the properties of various physical systems of
interest.
For the field configuration, we consider a constant magnetic field along the
$z$-direction (in cylindrical coordinates), $\mathbf{B}=B\mathbf{\hat{z}}$,
which is obtained from the vector potential (in the Landau gauge) \cite
{Book.1981.Landau},
\begin{equation}
\mathbf{A}=\frac{Br}{2}\mathbf{\hat{\varphi}}. \label{Al}
\end{equation}
In this configuration, Eq. (\ref{dirackp}) reads
\begin{equation}
X+\frac{\varepsilon }{2}Y=0, \label{diracsp}
\end{equation}
with
\begin{align}
X& =-\frac{\partial ^{2}\psi }{\partial r^{2}}-\frac{1}{r}\frac{\partial
\psi }{\partial r}-\frac{1}{r^{2}}\frac{\partial ^{2}\psi }{\partial \varphi
^{2}}+ieB\frac{\partial \psi }{\partial \varphi } \notag \\
& +\frac{1}{4}e^{2}B^{2}r^{2}\psi +\left[ M+w\left( r\right) \right]
^{2}\psi -\left[ E-\nu \left( r\right) \right] ^{2}\psi \notag \\
& -es\sigma _{z}B\psi +i\left[ \frac{\partial w\left( r\right) }{\partial r}
\right] \gamma _{0}\alpha _{r}\psi -i\left[ \frac{\partial \nu \left(
r\right) }{\partial r}\right] \alpha _{r}\psi ,
\end{align}
and
\begin{align}
Y& =\gamma _{0}\left[ \frac{\partial ^{2}w\left( r\right) }{\partial r^{2}}+
\frac{1}{r}\frac{\partial w\left( r\right) }{\partial r}\right] \psi -\gamma
_{0}\left[ \frac{\partial w\left( r\right) }{\partial r}\right] \frac{
\partial \psi }{\partial r} \notag \\
& -is\left[ \frac{1}{r}\frac{\partial w\left( r\right) }{\partial r}\right]
\frac{\partial \psi }{\partial \varphi }-2is\left[ \frac{1}{r}\frac{\partial
w\left( r\right) }{\partial r}\right] \frac{\partial \psi }{\partial \varphi
} \notag \\
& +es\left[ \frac{\partial w\left( r\right) }{\partial r}\right] \frac{Br}{2}
\psi -2es\left[ \frac{\partial w\left( r\right) }{\partial r}\right] \frac{Br
}{2}\psi \notag \\
& +iw\left( r\right) \alpha _{r}\left[ \frac{\partial w\left( r\right) }{
\partial r}\right] \psi -iw\left( r\right) \gamma _{r}\left[ \frac{\partial
\nu \left( r\right) }{\partial r}\right] \psi \notag \\
& +iM\alpha _{r}\left[ \frac{\partial w\left( r\right) }{\partial r}\right]
\psi -iM\gamma _{r}\left[ \frac{\partial \nu \left( r\right) }{\partial r}
\right] \psi \notag \\
& +2MesB\psi +2w\left( r\right) esB\psi
\end{align}
where the matrices (\ref{sigma1})-(\ref{sigma3}) are now given in
cylindrical coordinates, $\gamma _{r}=i\sigma _{\varphi }$, $\gamma
_{\varphi }=-is\sigma _{r}$, with \cite{EPJC.2015.75.321}
\begin{eqnarray}
\alpha _{r} &=&\gamma _{0}\gamma _{r}=\left(
\begin{array}{cc}
0 & e^{-is\varphi } \\
e^{is\varphi } & 0
\end{array}
\right) , \label{alphar} \\
\alpha _{\varphi } &=&\gamma _{0}\gamma _{\varphi }=\left(
\begin{array}{cc}
0 & -ie^{-is\varphi } \\
ie^{is\varphi } & 0
\end{array}
\right) , \label{alphaphi} \\
\gamma _{0} &=&\sigma _{z}=\left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right) . \label{gammaz}
\end{eqnarray}
For convenience, we will attribute expressions to functions $\nu \left(
r\right) $ and $w\left( r\right) $ in Eq. (\ref{diracsp}) only in the next
section, when we treat analysis of spin and pseudo-spin symmetries. We will
argue after that only some particular choices for these functions will lead
to a differential equation that admits an exact solution.
\section{Symmetries limits}
\label{sec:III}
To implement the spin and pseudospin symmetries limits, we make in Eq. (\ref
{diracsp}) the requirement that $w\left( r\right) =\pm \nu \left( r\right) $
, where the plus(minus) signal refers to spin(pseudo-spin) symmetry,
respectively \cite{PRL.1997.78.436}. Next, by using $\psi =\left( \psi
_{+},\psi _{-}\right) ^{T}$, the first and second lines in Eq. (\ref{diracsp}
) can be written in a simple form, which allows us to solve them separately.
Furthermore, as mentioned above, we need to choose a representation for the
radial function $\nu \left( r\right) $. We give a representation in terms of
cylindrically symmetric scalar potentials which lead to results well-known
in the literature.
\subsection{Particle interacting with a harmonic oscillator}
\label{sec:A}
Because of applications to various physical systems, we consider the
potential of a harmonic oscillator, $\nu \left( r\right) =ar^{2}$, where $a$
is a constant. By adopting solutions of the form
\begin{equation}
\psi _{\pm }=\left(
\begin{array}{c}
\sum_{m}f_{+}\left( r\right) e^{im\varphi } \\
i\sum_{m}f_{-}\left( r\right) e^{i\left( m+s\right) \varphi }
\end{array}
\right) ,
\end{equation}
we arrive at radial equations
\begin{align}
\frac{d^{2}f_{\pm }(r)}{dr^{2}}& +\left( \frac{1}{r}+\varepsilon ar\right)
\frac{df_{\pm }(r)}{dr}-\frac{\left( m^{\pm }\right) ^{2}}{r^{2}}f_{\pm }(r)
\notag \\
& -\left( \omega ^{\pm }\right) ^{2}r^{2}f_{+}(r)+k^{\pm }f_{+}(r)=0,
\label{K1}
\end{align}
where $k^{+}=E^{2}-M^{2}+\left( m+s\right) eB-\varepsilon \left(
2a+3sma+MesB\right) $, $k^{-}=E^{2}-M^{2}+eB\left( m+s\right)
+esB-\varepsilon \left[ 2a-3sa\left( m+s\right) +MesB\right] $, $\left(
\omega ^{+}\right) ^{2}=e^{2}B^{2}/4+2\left( M+E\right) a-\varepsilon esaB/2$
, $\left( \omega ^{-}\right) ^{2}=e^{2}B^{2}/4+2\left( E-M\right)
a-\varepsilon esaB/2$, $m^{+}=m$ and $m^{-}=m+s$.
By using solutions of the form
\begin{equation}
f_{\pm }(\rho )=e^{-\frac{1}{2}\left( \kappa ^{\pm }+1\right) \rho }\rho ^{
\frac{1}{2}\left\vert m^{\pm }\right\vert }F_{\pm }\left( \rho \right)
,\;\;\rho =\omega ^{\pm }r^{2} \label{fA}
\end{equation}
where $\kappa ^{\pm }=\varepsilon a/2\omega ^{\pm }$, Eq. (\ref{K1}) becomes
\begin{align}
\rho \frac{d^{2}F_{\pm }}{d\rho ^{2}}& +\left( 1+\left\vert m^{\pm
}\right\vert -\rho \right) \frac{dF_{\pm }}{d\rho } \notag \\
& -\left[ \frac{1}{2}\left( 1+\left\vert m^{\pm }\right\vert +\kappa ^{\pm
}\right) -\frac{k^{\pm }}{4\omega ^{+}}\right] F_{\pm }=0. \label{Kummer}
\end{align}
Equation (\ref{Kummer}) is of the confluent hypergeometric equation type and
its solution is given in terms of the Kummer functions. In this manner, the
general solution for Eq. (\ref{K1}) is given by \cite{Book.2010.NIST}
\begin{align}
f_{\pm }(\rho )& =\mathit{c}_{1}\,e^{-\frac{1}{2}\left( 1+\kappa ^{\pm
}\right) \rho }\rho ^{\frac{1}{2}\left\vert m^{\pm }\right\vert } \notag \\
& \times {\mathrm{M}\left( \frac{1}{2}\left( 1+\left\vert m^{\pm
}\right\vert +\kappa ^{\pm }\right) -\frac{k^{\pm }}{4\omega ^{\pm }}
,1+\left\vert m^{\pm }\right\vert ,\rho \right) } \notag \\
& +\mathit{c}_{2}\,{e^{-\frac{1}{2}\left( 1+\kappa ^{\pm }\right) \rho }\rho
^{-\frac{1}{2}\left\vert m^{\pm }\right\vert }} \notag \\
& \times {{\mathrm{M}\left( \frac{1}{2}\left( 1-\left\vert m^{\pm
}\right\vert +\kappa ^{\pm }\right) -\frac{k^{\pm }}{4\omega ^{\pm }}
,1-\left\vert m^{\pm }\right\vert ,\rho \right) ,}}
\end{align}
where $M$\ are the Kummer functions. In particular, when $\left(
1+\left\vert m^{\pm }\right\vert +\kappa ^{\pm }\right) /2-k^{\pm }/4\omega
^{\pm }=-n$, with $n=0,1,2,...$, the function $\mathrm{M}$ becomes a
polynomial in $\rho $ of degree not exceeding $n$. From this condition, we
extract the energies for the spin and pseudospin symmetries limits, given
respectively by
\begin{align}
E^{2}-M^{2}& =2\sqrt{\frac{e^{2}B^{2}}{4}+2\left( M+E\right) a-\frac{1}{2}
\varepsilon esaB} \notag \\
& \times \left( 2n+\left\vert m\right\vert +1\right) -eB\left( m+s\right)
\notag \\
& +\varepsilon \left( 3a+3sma+MesB\right) , \label{energy1} \\
E^{2}-M^{2}& =2\sqrt{\frac{e^{2}B^{2}}{4}+2\left( E-M\right) a+\frac{1}{2}
\varepsilon esaB} \notag \\
& \times \left( 2n+1+\left\vert m+s\right\vert \right) -eB\left( m+s\right)
-esB \notag \\
& -\varepsilon \left[ a-3sa\left( m+s\right) +MesB\right] . \label{energy2}
\end{align}
These energies are a relativistic generalization of the Landau levels in the
context of quantum deformation. When $a$ and $\varepsilon $ are null, we
obtain
\begin{eqnarray}
E^{2}-M^{2} &=&eB\left[ 2n+1+\left\vert m\right\vert -m-s\right] , \\
E^{2}-M^{2} &=&eB\left[ 2n+1+\left\vert m+s\right\vert -\left( m+s\right) -s
\right] .
\end{eqnarray}
which are the usual relativistic Landau levels with the inclusion of the
element of spin.
\subsection{Particle interacting with a linear potential}
\label{sec:B}
Let us consider the case where the particle interacts with a linear
potential, $ar$. In this case, we make $w\left( r\right) =\nu \left(
r\right) =ar$ (where $a$ is a positive constant) in Eq. (\ref{diracsp}) to
the limits of spin and pseudo-spin symmetries and proceed as before.
In the case of the spin symmetry limit, the resulting equation is given by
\begin{align}
& \frac{d^{2}f\left( r\right) }{dr^{2}}+\left( \frac{1}{r}+\frac{\varepsilon
a}{2}\right) \frac{df\left( r\right) }{dr}-\frac{\left( m^{\pm }\right) ^{2}
}{r^{2}}f\left( r\right) -\omega ^{2}r^{2}f\left( r\right) \notag \\
& -\mu ^{\pm }rf\left( r\right) -\frac{k^{\pm }}{r}f\left( r\right) +l^{\pm
}f\left( r\right) =0, \label{f}
\end{align}
with $m^{+}=m$, $m^{-}=m+s$, $\omega =eB/2$, $\mu ^{+}=2\left( E+M\right)
a+3\varepsilon aesB/4$, $\mu ^{-}=2\left( E-M\right) a-3\varepsilon aesB/4$,
$k^{+}=\varepsilon a\left( 1+3sm\right) /2$, $k^{-}=\frac{1}{2}\varepsilon a
\left[ 1-3s\left( m+s\right) \right] $, $l^{+}=E^{2}-M^{2}+eB\left(
m+s\right) -\varepsilon MesB$ and $l^{-}=E^{2}-M^{2}+eB\left( m+s\right)
+esB\left( 1-\varepsilon M\right) $. In Eq. (\ref{f}), the $+(-)$ sing refer
to spin and pseudo-spin symmetries, respectively. By performing the variable
change, $x=\sqrt{\omega }r$, Eq. (\ref{f}) assumes the form
\begin{align}
& \frac{d^{2}f\left( x\right) }{dx^{2}}+\left( \frac{1}{x}+\frac{1}{2}\kappa
\right) \frac{df\left( x\right) }{dx}-\frac{m^{2}}{x^{2}}f\left( x\right)
-x^{2}f\left( x\right) \notag \\
& -a_{L}^{\pm }xf\left( x\right) -\frac{a_{C}^{\pm }}{x}f\left( x\right) +
\frac{l^{\pm }}{\omega }f\left( x\right) =0, \label{fb}
\end{align}
where we have defined the parameters $\kappa =\varepsilon a/\sqrt{\omega }$,
$a_{L}^{\pm }=\mu ^{\pm }/\omega \sqrt{\omega }$ e $a_{C}^{\pm }=k^{\pm }/
\sqrt{\omega }$. Note that the choice $w\left( r\right) =\nu \left( r\right)
=ar$ induces a Coulomb-like interaction in the resulting eigenvalue
equation. The origin of this Coulomb potential is due purely to the quantum
deformation and boundary symmetries involved.
Equation (\ref{fb}) is of the Heun equation type, which is a homogeneous,
linear, second-order, differential equation defined in the complex plane.
This equation can be put into its canonical form using the solution
\begin{equation}
f\left( x\right) ={x}^{\left\vert m^{\pm }\right\vert }e{^{-\frac{1}{2}{x}
^{2}}}e{^{-\frac{1}{2}\left( a_{L}^{\pm }+\frac{1}{2}\kappa \right) x}y}
_{\pm }\left( x\right) ,
\end{equation}
where ${y}_{+}$ satisfies the biconfluent Heun differential equation
\begin{align}
& {y}_{\pm }^{^{\prime \prime }}+\left[ \frac{\alpha ^{\pm }+1}{x}-2{x}
-\beta ^{\pm }\right] {y}_{\pm }^{\prime }+\Big\{\gamma ^{\pm }-\alpha ^{\pm
}-{2} \notag \\
& -\frac{1}{2x}\left[ {\beta ^{\pm }}\left( \alpha ^{\pm }+{1}\right)
+\delta ^{\pm }\right] \Big\}y_{\pm }=0, \label{heunb}
\end{align}
with $\alpha ^{\pm }=2{\left\vert m^{\pm }\right\vert }$, $\beta ^{\pm
}=a_{L}^{\pm }$, $\gamma ^{\pm }=\left( \beta ^{\pm }\right) ^{2}/4+l^{\pm
}/\omega $ and $~\delta ^{\pm }={\kappa /2}+2a_{C}^{\pm }$. Equation (\ref
{heunb}) has a regular singularity at $x=0$, and an irregular singularity at
$\infty $ of rank $2$. Usually, the solution of this equation is given in
terms of two linearly independent solutions as
\begin{align}
y_{+}\left( x\right) =& \mathit{N}\left( \alpha ^{\pm },\beta ^{\pm },\gamma
^{\pm },\delta ^{\pm };x\right) \notag \\
& +{x}^{-\alpha ^{\pm }}\mathit{N}\left( -\alpha ^{\pm },\beta ^{\pm
},\gamma ^{\pm },\delta ^{\pm };x\right) , \label{heunc}
\end{align}
where (assuming that $\alpha ^{\pm }$ is not a negative integer)
\begin{equation}
\mathit{N}\left( \alpha ^{\pm },\beta ^{\pm },\gamma ^{\pm },\delta ^{\pm
};x\right) =\sum\limits_{q=0}^{\infty }\frac{\mathcal{A}_{q}^{+}\left(
\alpha ^{\pm },\beta ^{\pm },\gamma ^{\pm },\delta ^{\pm }\right) }{\left(
1+\alpha ^{\pm }\right) _{q}}\frac{x^{q}}{q!} \label{frb}
\end{equation}
are the Heun functions. After the insertion of this solution into Eq. (\ref
{heunb}), we find ($q\geqslant 0$)
\begin{equation}
\mathcal{A}_{0}=1, \label{A0}
\end{equation}
\begin{equation}
\mathcal{A}_{1}^{\pm }=\frac{1}{2}\left[ \delta ^{\pm }+\beta ^{\pm }\left(
1+\alpha ^{\pm }\right) \right] , \label{A1}
\end{equation}
\begin{align}
\mathcal{A}_{q+2}^{\pm }& =\left\{ \left( q+1\right) \beta ^{\pm }+\frac{1}{2
}\left[ \delta ^{\pm }+\beta ^{\pm }\left( 1+\alpha ^{\pm }\right) \right]
\right\} \mathcal{A}_{q+1}^{\pm } \notag \\
& -\left( q+1\right) \left( q+1+\alpha ^{\pm }\right) \left[ \gamma ^{\pm
}-\alpha ^{\pm }-2-2q\right] \mathcal{A}_{q}^{\pm }, \label{recur}
\end{align}
\begin{equation}
\left( 1+\alpha ^{\pm }\right) _{q}=\frac{\Gamma \left( q+\alpha ^{\pm
}+1\right) }{\Gamma \left( \alpha ^{\pm }+1\right) },\;\;q=0,1,2,3,\ldots
\text{.}
\end{equation}
From the recursion relation (\ref{recur}), the function $N\left( \alpha
^{\pm },\beta ^{\pm },\gamma ^{\pm },\delta ^{\pm };x\right) $ becomes a
polynomial of degree $n$ if and only if the two following conditions are
imposed:
\begin{equation}
\gamma ^{\pm }-\alpha ^{\pm }-2=2n,\;\;n=0,1,2,\ldots \label{cndA}
\end{equation}
\begin{equation}
\mathcal{A}_{n+1}^{\pm }=0, \label{cndB}
\end{equation}
where $n$ is a positive integer. In this case, the $\left( n+1\right) $th
coefficient in the series expansion is a polynomial of degree $n$ in $\delta
^{\pm }$. When $\delta ^{\pm }$ is a root of this polynomial, the $n+1$th
and subsequent coefficients cancel and the series truncates, resulting in a
polynomial form of degree $n$ for $N\left( \alpha ^{\pm },\beta ^{\pm
},\gamma ^{\pm },\delta ^{\pm };x\right) $. From condition (\ref{cndA}), we
extract the energies at the spin and pseudo symmetries limit, given
respectively by:
\begin{align}
E_{nm}^{2}& -M^{2}=2\omega \left( {n}+{\left\vert m\right\vert +1}\right) -
\frac{a^{2}}{\omega ^{2}}\left( E_{nm}+M\right) ^{2} \notag \\
& -\frac{3a^{2}}{2\omega }\left( E_{nm}+M\right) \varepsilon s+2\omega \left[
\varepsilon Ms-\left( m+s\right) \right] , \label{Energy+}
\end{align}
\begin{align}
E_{nm}^{2}& -M^{2}=2\omega \left( {n}+{\left\vert m+s\right\vert +1}\right) -
\frac{a^{2}}{\omega ^{2}}\left( E_{nm}-M\right) ^{2} \notag \\
& +\frac{3a^{2}}{2\omega }\left( E_{nm}-M\right) \varepsilon s-2\omega \left[
s\left( 1-\varepsilon M\right) +\left( m+s\right) \right] . \label{Energy-}
\end{align}
The equations (\ref{Energy+}) and (\ref{Energy-}) can not represent the
spectrum for the system in question. The energy of a physical system must be
a function involving all the parameters present in the equation of motion.
As expected, from the condition (\ref{cndA}) alone one cannot a priori
derive the energy of the system. In the case of the energies above, this is
justified by the absence of the parameters $a_{C}^{\pm }$. Moreover, it also
does not provide the energy spectrum for all values of $n$. On the other
hand, by analyzing more carefully the condition (\ref{cndB}), we see that it
admits a natural application of (\ref{cndA}), so that it is a necessary and
sufficient condition for the derivation of the energy of the particle. Let
us consider the solution (\ref{frb}) up to second-order in $x$\ of the
expansion,
\begin{align}
N\left( \alpha ^{\pm },\beta ^{\pm },\gamma ^{\pm },\delta ^{\pm };x\right)
& =\frac{\mathcal{A}_{0}}{\left( 1+\alpha ^{\pm }\right) _{0}}+\frac{
\mathcal{A}_{1}^{\pm }}{\left( 1+\alpha ^{\pm }\right) _{1}}x \notag \\
& +\frac{\mathcal{A}_{2}^{\pm }}{\left( 1+\alpha ^{\pm }\right) _{2}}\frac{
x^{2}}{2!}+\ldots . \label{Heun}
\end{align}
By using the relation of recurrence (\ref{recur}) and Eqs. (\ref{A0})-(\ref
{A1}), the coefficient above $\mathcal{A}_{2}^{\pm }$ can be determined. If
we want to truncate solution (\ref{Heun}) in $x$, we must impose that $
A_{1}^{\pm }=0$ through the condition (\ref{cndB}); when we truncate in $
x^{2}$, we make $A_{2}^{\pm }=0$, and so on. For each of these cases, we
have an associated energy. Thus, for $A_{1}^{\pm }=0$, it means that we are
investigating the particular solution for $n=0$. Then, from Eq. (\ref
{A1}), we have
\begin{equation}
\frac{1}{2}\left( \delta ^{\pm }+\beta ^{\pm }\tilde{m}^{\pm }\right) =0,
\label{scdA}
\end{equation}
where $\tilde{m}^{\pm }=1+\alpha ^{\pm }$. Solving (\ref{scdA}) for $E$, we
find the energies corresponding to the spin and pseudo-spin symmetries. They
can be written explicitly as
\begin{align}
E_{0m}+M& =-\frac{\varepsilon \omega }{2\left( 1+2{\left\vert m\right\vert }
\right) }\left[ 3s\left( {\left\vert m\right\vert }+m\right) +\frac{3}{2}
\left( s+{1}\right) \right] , \label{En0A} \\
E_{0m}-M& =\frac{\varepsilon \omega }{2\left( 1+2{\left\vert m+s\right\vert }
\right) } \notag \\
& \times \left[ 3s\left( {\left\vert m+s\right\vert }+m+s\right) +\frac{3}{2}
\left( s-{1}\right) \right] . \label{En0B}
\end{align}
These energies, after imposing $\varepsilon =0$, lead to the ground state, $
E_{0m}=-M$ and $E_{0m}=M$. Analogously, for $\mathcal{A}_{2}^{\pm }=0$, we
get ($n=1$)
\begin{equation}
\frac{1}{2}\beta ^{\pm }\left( \delta ^{\pm }+\beta ^{\pm }\tilde{m}^{\pm
}\right) +\frac{1}{4}\left( \delta ^{\pm }+\beta ^{\pm }\tilde{m}^{\pm
}\right) ^{2}-2\tilde{m}^{\pm }=0. \label{ndf}
\end{equation}
In this relation, we can observe that the only parameter that depends on the
energy $E$ is the parameter $a_{L}^{\pm }$ (through the parameter $\beta
^{\pm }$). So, we only need to solve it for $a_{L}^{\pm }$. The result is
given by
\begin{align}
E_{1m}& +M=\pm \frac{\omega }{a}\sqrt{\frac{2\omega }{2{\left\vert
m\right\vert }+3}} \notag \\
& -\frac{\varepsilon \omega }{2}\left[ \frac{\left( 1+{\left\vert
m\right\vert }\right) \left[ 1+2\left( 1+3sm\right) \right] }{\left( 1+2{
\left\vert m\right\vert }\right) \left( 3+2{\left\vert m\right\vert }\right)
}+\frac{3}{2}s\right] , \label{enyA}
\end{align}
for the spin symmetry limit, and
\begin{align}
E_{1m}& -M=\pm \frac{\omega }{a}\sqrt{\frac{2\omega }{3+2{\left\vert
m+s\right\vert }}} \notag \\
& -\frac{\varepsilon \omega }{2}\left[ \frac{3\left( 1+{\left\vert
m+s\right\vert }\right) \left( 1-2s\left( m+s\right) \right) }{\left( 1+2{
\left\vert m+s\right\vert }\right) \left( 3+2{\left\vert m+s\right\vert }
\right) }-\frac{3}{2}s\right] , \label{enyB}
\end{align}
for the pseudo-spin symmetry limit. If $\varepsilon =0$ in Eqs. (\ref{enyA})-(\ref{enyB}), we find
\begin{eqnarray}
E_{1m}+M &=&\pm \frac{\omega }{a}\sqrt{\frac{2\omega }{3+2{\left\vert
m\right\vert }}}, \\
E_{1m}-M &=&\pm \frac{\omega }{a}\sqrt{\frac{2\omega }{3+2{\left\vert
m+s\right\vert }}}.
\end{eqnarray}
The energies obtained from condition (\ref{cndA}) (Eqs. (\ref{Energy+})-(\ref
{Energy-})) together with those obtained from (\ref{cndB}) (Eqs. (\ref{En0A}
)-(\ref{En0B}) and (\ref{enyA})-(\ref{enyB})) specify the energy eigenvalues
for the system governed by equation (\ref{f}). However, we can verify that
these energies are connected to each other through the parameters $\omega $
and $a$, so that we have a constraint on the energies. In particular, if we
solve Eqs. (\ref{enyA})-(\ref{enyB}) for $\omega $ and replace them in (\ref
{Energy+})-(\ref{Energy-}), we will have expressions for the energies
involving the quantities $\alpha ^{\pm }$, $\beta ^{\pm }$, $\gamma^{\pm }$
, $\delta ^{\pm }$, which contains the Coulomb potential coupling constant $
a_{C}^{\pm }$, the mass of the particle $M$, the effective angular momentum $
m^{\pm }$ and the frequency $\omega $.
For a specific physical system described by Eq. (\ref{f}), its corresponding
energy spectrum are the modified Landau levels. In the absence of magnetic
field, the energy eigenvalues are equivalent to those of a planar harmonic
oscillator being corrected only by the parameter of quantum deformation.
Because the spectrum of the system has the generalized form of the spectrum
of a relativistic oscillator is more convenient to fix the frequency the
frequency $\omega $ to give the energies corresponding to each value of $n$.
It is an immediate calculation to solve the equations Eqs. (\ref{En0A})-(\ref
{En0B}) and (\ref{enyA})-(\ref{enyB}) for $\omega $. For each specific
frequency, $\omega _{0m}$, $\omega _{01}$, we have the following energies:
\begin{align}
& E_{0m}^{2}-M^{2}=2\omega _{0m}\left( {\left\vert m\right\vert +1}\right) -
\frac{a^{2}}{\omega _{0m}^{2}}\left( E_{0m}+M\right) ^{2} \notag \\
& -\frac{3a^{2}}{2\omega _{0m}}\left( E_{0m}+M\right) \varepsilon s+2\omega
_{0m}\left[ \varepsilon Ms-\left( m+s\right) \right] , \label{Energy0fi+} \\
& E_{0m}^{2}-M^{2}=2\omega _{0m}\left( {\left\vert m+s\right\vert +1}\right)
-\frac{a^{2}}{\omega _{0m}^{2}}\left( E_{0m}-M\right) ^{2} \notag \\
& +\frac{3a^{2}}{2\omega _{0m}}\left( E_{0m}-M\right) \varepsilon s-2\omega
_{0m}\left[ s\left( 1-\varepsilon M\right) +\left( m+s\right) \right] ,
\label{Energy0fi-}
\end{align}
and
\begin{align}
& E_{1m}^{2}-M^{2}=2\omega _{1m}\left( {\left\vert m\right\vert +2}\right) -
\frac{a^{2}}{\omega _{1m}^{2}}\left( E_{1m}+M\right) ^{2} \notag \\
& -\frac{3a^{2}}{2\omega _{1m}}\left( E_{1m}+M\right) \varepsilon s+2\omega
_{1m}\left[ \varepsilon Ms-\left( m+s\right) \right] , \label{Energy1fi+} \\
& E_{1m}^{2}-M^{2}=2\omega _{1m}\left( {\left\vert m+s\right\vert +2}\right)
-\frac{a^{2}}{\omega _{1m}^{2}}\left( E_{1m}-M\right) ^{2} \notag \\
& +\frac{3a^{2}}{2\omega _{1m}}\left( E_{1m}-M\right) \varepsilon s-2\omega
_{1m}\left[ s\left( 1-\varepsilon M\right) +\left( m+s\right) \right] ,
\label{Energy1fi-}
\end{align}
To determine the energies corresponding to $n=2,3,4,\ldots $, we
must make use of the above recipe. However, the polynomials of degree $n\geq
3$ resulting from condition (\ref{cndB}), in general, only some roots are
physically acceptable.
\section{Conclusions}
\label{sec:c}
We have studied the relativistic quantum dynamics of a spin-$1/2$ charged
particle with minimal, vector and scalar couplings in the quantum deformed
framework generated by the $\kappa $-Poincar\'{e}-Hopf algebra. The problem
have been formulated using the $\kappa $-deformed Dirac equation in two
dimensions. The $\kappa $-deformed Pauli equation was derived to study the
dynamics of the system taking into account the spin and pseudospin
symmetries limits. For the $\kappa $-deformed Dirac-Pauli equation obtained
(Eq. (\ref{diracsp})), we have argued that only particular choices of radial
function $\nu \left( r\right) $ lead to exactly solvable differential
equations. We have considered the case where the particle interacts with an
uniform magnetic field, a planar harmonic oscillator and a linear potential.
We have verified that the linear potential leads to a Coulomb-type term in
the $\kappa $-deformed sector of the radial equation. The resulting equation
obtained is a Heun-type differential equation. Analytical solutions for both
spin and pseudospin symmetries limits enabled us to obtain expressions for
the energy eigenvalues (through the use of the Eqs. (\ref{cndA}) and (\ref
{cndB})) and wave functions. Because of the limitations imposed by the
condition (\ref{cndB}), we have derived expressions for the energies
corresponding only to $n=0$ (Eqs. (\ref{Energy0fi+})-(\ref{Energy0fi-})) and
$n=1$ (Eqs. (\ref{Energy1fi+})-(\ref{Energy1fi-})). We have shown that the
presence of the spin element in the equation of motion introduces a
correction in the expressions for the bound state energy and wave functions.
\section*{Acknowledgments}
This work was supported by the Brazilian agencies CAPES, CNPq and FAPEMA.
\bibliographystyle{apsrev4-1}
\input{dirac-kappa.bbl}
\end{document}
|
{
"timestamp": "2017-04-18T02:07:00",
"yymm": "1704",
"arxiv_id": "1704.04847",
"language": "en",
"url": "https://arxiv.org/abs/1704.04847"
}
|
\section{Introduction}
Convolutional neural networks have become ubiquitous in computer vision ever since AlexNet \cite{krizhevsky2012imagenet} popularized deep convolutional neural networks by winning the ImageNet Challenge: ILSVRC 2012 \cite{russakovsky2015imagenet}. The general trend has been to make deeper and more complicated networks in order to achieve higher accuracy \cite{simonyan2014very,szegedy2015rethinking,szegedy2016inception,he2015deep}. However, these advances to improve accuracy are not necessarily making networks more efficient with respect to size and speed. In many real world applications such as robotics, self-driving car and augmented reality, the recognition tasks need to be carried out in a timely fashion on a computationally limited platform.
This paper describes an efficient network architecture and a set of two hyper-parameters in order to build very small, low latency models that can be easily matched to the design requirements for mobile and embedded vision applications. Section \ref{sec:prior} reviews prior work in building small models. Section \ref{sec:mobilenet} describes the MobileNet architecture and two hyper-parameters width multiplier and resolution multiplier to define smaller and more efficient MobileNets. Section \ref{sec:exp} describes experiments on ImageNet as well a variety of different applications and use cases. Section \ref{sec:conclusion} closes with a summary and conclusion.
\begin{figure*}
\centering
\includegraphics[width=2\columnwidth]{mobilenet_pic.pdf}
\caption{MobileNet models can be applied to various recognition tasks for efficient on device intelligence.}
\label{fig:conv_layers}
\end{figure*}
\section{Prior Work} \label{sec:prior}
There has been rising interest in building small and efficient neural networks in the recent literature, e.g. \cite{jin2014flattened,wang2016factorized,iandola2016squeezenet,wu2015quantized,rastegari2016xnor}. Many different approaches can be generally categorized into either compressing pretrained networks or training small networks directly. This paper proposes a class of network architectures that allows a model developer to specifically choose a small network that matches the resource restrictions (latency, size) for their application. MobileNets primarily focus on optimizing for latency but also yield small networks. Many papers on small networks focus only on size but do not consider speed.
MobileNets are built primarily from depthwise separable convolutions initially introduced in \cite{sifre2014rigid} and subsequently used in Inception models \cite{ioffe2015batch} to reduce the computation in the first few layers. Flattened networks \cite{jin2014flattened} build a network out of fully factorized convolutions and showed the potential of extremely factorized networks. Independent of this current paper, Factorized Networks\cite{wang2016factorized} introduces a similar factorized convolution as well as the use of topological connections. Subsequently, the Xception network \cite{chollet2016deep} demonstrated how to scale up depthwise separable filters to out perform Inception V3 networks. Another small network is Squeezenet \cite{iandola2016squeezenet} which uses a bottleneck approach to design a very small network. Other reduced computation networks include structured transform networks \cite{sindhwani2015structured} and deep fried convnets \cite{yang2015deep}.
A different approach for obtaining small networks is shrinking, factorizing or compressing pretrained networks. Compression based on product quantization \cite{wu2015quantized}, hashing \cite{chen2015compressing}, and pruning, vector quantization and Huffman coding \cite{han2015deep} have been proposed in the literature. Additionally various factorizations have been proposed to speed up pretrained networks \cite{jaderberg2014speeding, lebedev2014speeding}. Another method for training small networks is distillation \cite{hinton2015distilling} which uses a larger network to teach a smaller network. It is complementary to our approach and is covered in some of our use cases in section \ref{sec:exp}. Another emerging approach is low bit networks \cite{courbariaux2014training, rastegari2016xnor, hubara2016quantized}.
\section{MobileNet Architecture} \label{sec:mobilenet}
In this section we first describe the core layers that MobileNet is built on which are depthwise separable filters. We then describe the MobileNet network structure and conclude with descriptions of the two model shrinking hyper-parameters width multiplier and resolution multiplier.
\subsection{Depthwise Separable Convolution}
The MobileNet model is based on depthwise separable convolutions which is a form of factorized convolutions which factorize a standard convolution into a depthwise convolution and a $1 \times 1$ convolution called a pointwise convolution. For MobileNets the depthwise convolution applies a single filter to each input channel. The pointwise convolution then applies a $1 \times 1$ convolution to combine the outputs the depthwise convolution. A standard convolution both filters and combines inputs into a new set of outputs in one step. The depthwise separable convolution splits this into two layers, a separate layer for filtering and a separate layer for combining. This factorization has the effect of drastically reducing computation and model size. Figure \ref{fig:dw_conv} shows how a standard convolution \ref{fig:dw_conv_a} is factorized into a depthwise convolution \ref{fig:dw_conv_b} and a $1 \times 1$ pointwise convolution \ref{fig:dw_conv_c}.
A standard convolutional layer takes as input a $D_F \times D_F \times M$ feature map $\mathbf{F}$ and produces a $D_F \times D_F \times N$ feature map $\mathbf{G}$ where $D_F$ is the spatial width and height of a square input feature map\footnote{We assume that the output feature map has the same spatial dimensions as the input and both feature maps are square. Our model shrinking results generalize to feature maps with arbitrary sizes and aspect ratios.}, $M$ is the number of input channels (input depth), $D_G$ is the spatial width and height of a square output feature map and $N$ is the number of output channel (output depth).
The standard convolutional layer is parameterized by convolution kernel $\mathbf{K}$ of size $D_K \times D_K \times M \times N$ where $D_K$ is the spatial dimension of the kernel assumed to be square and $M$ is number of input channels and $N$ is the number of output channels as defined previously.
The output feature map for standard convolution assuming stride one and padding is computed as:
\begin{equation}
\mathbf{G}_{k,l,n} = \sum_{i,j,m} \mathbf{K}_{i,j,m,n} \cdot \mathbf{F}_{k+i-1,l+j-1,m}
\end{equation}
Standard convolutions have the computational cost of:
\begin{equation}
D_K \cdot D_K \cdot M \cdot N \cdot D_F \cdot D_F
\end{equation}
where the computational cost depends multiplicatively on the number of input channels $M$, the number of output channels $N$ the kernel size $D_k \times D_k$ and the feature map size $D_F \times D_F$. MobileNet models address each of these terms and their interactions. First it uses depthwise separable convolutions to break the interaction between the number of output channels and the size of the kernel.
The standard convolution operation has the effect of filtering features based on the
convolutional kernels and combining features in order to produce a new representation.
The filtering and combination steps can be split into two steps via the use of
factorized convolutions called depthwise separable convolutions for substantial reduction in computational cost.
Depthwise separable convolution are made up of two layers: depthwise convolutions and pointwise convolutions.
We use depthwise convolutions to apply a single filter per each input channel (input depth). Pointwise convolution, a
simple $1 \times 1$ convolution, is then used to create a linear combination of the output of the depthwise layer. MobileNets use both
batchnorm and ReLU nonlinearities for both layers.
Depthwise convolution with one filter per input channel (input depth) can be written as:
\begin{equation}
\hat{\mathbf{G}}_{k,l,m} = \sum_{i,j} \hat{\mathbf{K}}_{i,j,m} \cdot \mathbf{F}_{k+i-1,l+j-1,m}
\end{equation}
where $\hat{\mathbf{K}}$ is the depthwise convolutional kernel of size $D_K \times D_K \times M$ where the $m_{th}$ filter in $\hat{\mathbf{K}}$ is applied to the $m_{th}$ channel in $\mathbf{F}$ to produce the $m_{th}$ channel of the filtered output feature map $\hat{\mathbf{G}}$.
Depthwise convolution has a computational cost of:
\begin{equation}
D_K \cdot D_K \cdot M \cdot D_F \cdot D_F
\end{equation}
Depthwise convolution is extremely efficient relative to standard convolution. However it only filters input channels, it does not combine them to create new features. So an additional layer that computes a linear combination of the output of depthwise convolution via $1 \times 1$ convolution is needed in order to generate these new features.
The combination of depthwise convolution and $1\times1$ (pointwise) convolution is called depthwise separable convolution which was originally introduced in \cite{sifre2014rigid}.
Depthwise separable convolutions cost:
\begin{equation}
D_K \cdot D_K \cdot M \cdot D_F \cdot D_F + M \cdot N \cdot D_F \cdot D_F
\end{equation}
which is the sum of the depthwise and $1 \times 1$ pointwise convolutions.
By expressing convolution as a two step process of filtering and combining we get a reduction in computation of:
\begin{eqnarray*}
&&\frac{D_K \cdot D_K \cdot M \cdot D_F \cdot D_F + M \cdot N \cdot D_F \cdot D_F}{D_K \cdot D_K \cdot M \cdot N \cdot D_F \cdot D_F} \\
&=&\frac{1}{N} + \frac{1}{D_K^2}
\end{eqnarray*}
MobileNet uses $3 \times 3$ depthwise separable convolutions which uses between 8 to 9 times less computation than standard convolutions at only a small reduction in accuracy as seen in Section \ref{sec:exp}.
\begin{figure}[t]
\centering
\subfigure[Standard Convolution Filters]{
\includegraphics[width=.45\textwidth]{convolution2_latex.pdf}
\label{fig:dw_conv_a}
}
\subfigure[Depthwise Convolutional Filters]{
\includegraphics[width=.45\textwidth]{depthwise_latex.pdf}
\label{fig:dw_conv_b}
}
\subfigure[$1 \times 1$ Convolutional Filters called Pointwise Convolution in the context of Depthwise Separable Convolution]{
\includegraphics[width=.45\textwidth]{pointwise_latex.pdf}
\label{fig:dw_conv_c}
}
\caption{The standard convolutional filters in (a) are replaced by two layers: depthwise convolution in (b) and pointwise convolution in (c) to build a depthwise separable filter.}
\label{fig:dw_conv}
\end{figure}
Additional factorization in spatial dimension such as in \cite{jin2014flattened,szegedy2015rethinking} does not save much additional computation as very little computation is spent in depthwise convolutions.
\subsection{Network Structure and Training}
The MobileNet structure is built on depthwise separable convolutions as mentioned in the previous section except for the first layer which is a full convolution. By defining the network in such simple terms we are able to easily explore network topologies to find a good network. The MobileNet architecture is defined in Table \ref{table:mobilenet}. All layers are followed by a batchnorm \cite{ioffe2015batch} and ReLU nonlinearity with the exception of the final fully connected layer which has no nonlinearity and feeds into a softmax layer for classification. Figure \ref{fig:conv_layers} contrasts a layer with regular convolutions, batchnorm and ReLU nonlinearity to the factorized layer with depthwise convolution, $1 \times 1$ pointwise convolution as well as batchnorm and ReLU after each convolutional layer. Down sampling is handled with strided convolution in the depthwise convolutions as well as in the first layer. A final average pooling reduces the spatial resolution to 1 before the fully connected layer. Counting depthwise and pointwise convolutions as separate layers, MobileNet has 28 layers.
It is not enough to simply define networks in terms of a small number of Mult-Adds. It is also important to make sure these operations can be efficiently implementable. For instance unstructured sparse matrix operations are not typically faster than dense matrix operations until a very high level of sparsity. Our model structure puts nearly all of the computation into dense $1 \times 1$ convolutions. This can be implemented with highly optimized general matrix multiply (GEMM) functions. Often convolutions are implemented by a GEMM but require an initial reordering in memory called im2col in order to map it to a GEMM. For instance, this approach is used in the popular Caffe package \cite{jia2014caffe}. $1 \times 1$ convolutions do not require this reordering in memory and can be implemented directly with GEMM which is one of the most optimized numerical linear algebra algorithms. MobileNet spends $95\%$ of it's computation time in $1 \times 1$ convolutions which also has $75\%$ of the parameters as can be seen in Table \ref{table:percent}. Nearly all of the additional parameters are in the fully connected layer.
MobileNet models were trained in TensorFlow \cite{abadi2015tensorflow} using RMSprop \cite{tieleman2012lecture} with asynchronous gradient descent similar to Inception V3 \cite{szegedy2015rethinking}. However, contrary to training large models we use less regularization and data augmentation techniques because small models have less trouble with overfitting. When training MobileNets we do not use side heads or label smoothing and additionally reduce the amount image of distortions by limiting the size of small crops that are used in large Inception training \cite{szegedy2015rethinking}. Additionally, we found that it was important to put very little or no weight decay (l2 regularization) on the depthwise filters since their are so few parameters in them. For the ImageNet benchmarks in the next section all models were trained with same training parameters regardless of the size of the model.
\begin{table}[t]
\caption{MobileNet Body Architecture}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{l | l | l}
\hline\hline
Type / Stride & Filter Shape & Input Size \\ [0.5ex]
\hline
Conv / s2 & $3\times3\times3\times32$ & $224\times224\times3$\\
\hline
Conv dw / s1& $3\times3\times32$ dw & $112\times112\times32$\\
\hline
Conv / s1& $1\times1\times32\times64$ & $112\times112\times32$\\
\hline
Conv dw / s2& $3\times3\times64$ dw & $112\times112\times64$\\
\hline
Conv / s1& $1\times1\times64\times128$ & $56\times56\times64$\\
\hline
Conv dw / s1& $3\times3\times128$ dw & $56\times56\times128$\\
\hline
Conv / s1& $1\times1\times128\times128$ & $56\times56\times128$\\
\hline
Conv dw / s2& $3\times3\times128$ dw & $56\times56\times128$\\
\hline
Conv / s1& $1\times1\times128\times256$ & $28\times28\times128$\\
\hline
Conv dw / s1& $3\times3\times256$ dw & $28\times28\times256$\\
\hline
Conv / s1& $1\times1\times256\times256$ & $28\times28\times256$\\
\hline
Conv dw / s2& $3\times3\times256$ dw & $28\times28\times256$\\
\hline
Conv / s1& $1\times1\times256\times512$ & $14\times14\times256$\\
\hline
\multirow{2}{*}{$5\times$} Conv dw / s1& $3\times3\times512$ dw & $14\times14\times512$\\
\hspace{.53cm}Conv / s1& $1\times1\times512\times512$ & $14\times14\times512$\\
\hline
Conv dw / s2& $3\times3\times512$ dw & $14\times14\times512$\\
\hline
Conv / s1& $1\times1\times512\times1024$ & $7\times7\times512$\\
\hline
Conv dw / s2& $3\times3\times1024$ dw & $7\times7\times1024$\\
\hline
Conv / s1& $1\times1\times1024\times1024$ & $7\times7\times1024$\\
\hline
Avg Pool / s1& Pool $7\times7$ & $7\times7\times1024$\\
\hline
FC / s1 & $1024 \times 1000$ & $1\times1\times1024$\\
\hline
Softmax / s1 & Classifier & $1\times1\times1000$\\
\hline
\end{tabular}
\label{table:mobilenet}
}
\end{table}
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{conv_layers.pdf}
\caption{Left: Standard convolutional layer with batchnorm and ReLU. Right: Depthwise Separable convolutions with Depthwise and Pointwise layers followed by batchnorm and ReLU.}
\label{fig:conv_layers}
\end{figure}
\begin{table}[t]
\caption{Resource Per Layer Type}
\centering
\begin{tabular}{l | l | l}
\hline\hline
Type & Mult-Adds & Parameters \\ [0.5ex]
\hline
Conv $1 \times 1$ & 94.86\% & 74.59\% \\
\hline
Conv DW $3 \times 3$ & 3.06\% & 1.06\% \\
\hline
Conv $3 \times 3$ & 1.19\% & 0.02\% \\
\hline
Fully Connected & 0.18\% & 24.33\% \\
[1ex]
\hline
\end{tabular}
\label{table:percent}
\end{table}
\subsection{Width Multiplier: Thinner Models}
Although the base MobileNet architecture is already small and low latency, many times a specific use case or application may require the model to be smaller and faster. In order to construct these smaller and less computationally expensive models we introduce a very simple parameter $\alpha$ called width multiplier. The role of the width multiplier $\alpha$ is to thin a network uniformly at each layer. For a given layer and width multiplier $\alpha$, the number of input channels $M$ becomes $\alpha M$ and the number of output channels $N$ becomes $\alpha N$.
The computational cost of a depthwise separable convolution with width multiplier $\alpha$ is:
\begin{equation}
D_K \cdot D_K \cdot \alpha M \cdot D_F \cdot D_F + \alpha M \cdot \alpha N \cdot D_F \cdot D_F
\end{equation}
where $\alpha \in (0,1]$ with typical settings of 1, 0.75, 0.5 and 0.25. $\alpha=1$ is the baseline MobileNet and $\alpha<1$
are reduced MobileNets. Width multiplier has the effect of reducing computational cost and the number of parameters quadratically by roughly $\alpha^2$. Width multiplier can be applied to any model structure to define a new smaller model with a reasonable accuracy, latency and size trade off. It is used to define a new reduced structure that needs to be trained from scratch.
\subsection{Resolution Multiplier: Reduced Representation}
The second hyper-parameter to reduce the computational cost of a neural network is a resolution multiplier $\rho$. We apply this to the input image and the internal representation of every layer is subsequently reduced by the same multiplier. In practice we implicitly set $\rho$ by setting the input resolution.
We can now express the computational cost for the core layers of our network as depthwise separable convolutions with width multiplier $\alpha$ and resolution multiplier $\rho$:
\begin{equation}
D_K \cdot D_K \cdot \alpha M \cdot \rho D_F \cdot \rho D_F + \alpha M \cdot \alpha N \cdot \rho D_F \cdot \rho D_F
\end{equation}
where $\rho \in (0,1]$ which is typically set implicitly so that the input resolution of the network is 224, 192, 160 or 128. $\rho=1$ is the baseline MobileNet and $\rho<1$
are reduced computation MobileNets. Resolution multiplier has the effect of reducing computational cost by $\rho^2$.
As an example we can look at a typical layer in MobileNet and see how depthwise separable convolutions, width multiplier and resolution multiplier reduce the cost and parameters. Table \ref{table:layer_resource} shows the computation and number of parameters for a layer as architecture shrinking methods are sequentially applied to the layer. The first row shows the Mult-Adds and parameters for a full convolutional layer with an input feature map of size $14 \times 14 \times 512$ with a kernel $K$ of size $3 \times 3 \times 512 \times 512$. We will look in detail in the next section at the trade offs between resources and accuracy.
\begin{table}[t]
\caption{Resource usage for modifications to standard convolution. Note that each row is a cumulative effect adding on top of the previous row. This example is for an internal MobileNet layer with $D_K=3$, $M=512$, $N=512$, $D_F=14$.}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{c c c c}
\hline\hline
Layer/Modification & Million & Million \\ [0.5ex]
& Mult-Adds & Parameters \\
\hline
Convolution & 462 & 2.36 \\
Depthwise Separable Conv & 52.3 & 0.27 \\
$\alpha=0.75$ & 29.6 & 0.15 \\
$\rho=0.714$ & 15.1 & 0.15 \\
[1ex]
\hline
\end{tabular}
\label{table:layer_resource}
}
\end{table}
\section{Experiments} \label{sec:exp}
In this section we first investigate the effects of depthwise convolutions as well as the choice of shrinking by reducing the width of the network rather than the number of layers. We then show the trade offs of reducing the network based on the two hyper-parameters: width multiplier and resolution multiplier and compare results to a number of popular models. We then investigate MobileNets applied to a number of different applications.
\subsection{Model Choices}
First we show results for MobileNet with depthwise separable convolutions compared to a model built with full convolutions. In Table \ref{table:dm_full} we see that using depthwise separable convolutions compared to full convolutions only reduces accuracy by $1\%$ on ImageNet was saving tremendously on mult-adds and parameters.
\begin{table}[t]
\caption{Depthwise Separable vs Full Convolution MobileNet}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{c c c c}
\hline\hline
Model & ImageNet & Million & Million \\ [0.5ex]
& Accuracy & Mult-Adds & Parameters \\
\hline
Conv MobileNet & 71.7\% & 4866 & 29.3\\
MobileNet & 70.6\% & 569 & 4.2 \\
[1ex]
\hline
\end{tabular}
\label{table:dm_full}
}
\end{table}
We next show results comparing thinner models with width multiplier to shallower models using less layers. To make MobileNet shallower, the $5$ layers of separable filters with feature size $14 \times 14 \times 512$ in Table \ref{table:mobilenet} are removed. Table \ref{table:dm_shallow} shows that at similar computation and number of parameters, that making MobileNets thinner is $3\%$ better than making them shallower.
\begin{table}[t]
\caption{Narrow vs Shallow MobileNet}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{c c c c}
\hline\hline
Model & ImageNet & Million & Million \\ [0.5ex]
& Accuracy & Mult-Adds & Parameters \\
\hline
0.75 MobileNet & 68.4\% & 325 & 2.6 \\
Shallow MobileNet & 65.3\% & 307 & 2.9\\
[1ex]
\hline
\end{tabular}
\label{table:dm_shallow}
}
\end{table}
\subsection{Model Shrinking Hyperparameters}
Table \ref{table:wm} shows the accuracy, computation and size trade offs of shrinking the MobileNet architecture with the width multiplier $\alpha$.
Accuracy drops off smoothly until the architecture is made too small at $\alpha=0.25$.
\begin{table}[t]
\caption{MobileNet Width Multiplier}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{c c c c}
\hline\hline
Width Multiplier & ImageNet & Million & Million \\ [0.5ex]
& Accuracy & Mult-Adds & Parameters \\
\hline
1.0 MobileNet-224 & 70.6\% & 569 & 4.2 \\
0.75 MobileNet-224 & 68.4\% & 325 & 2.6 \\
0.5 MobileNet-224 & 63.7\% & 149 & 1.3 \\
0.25 MobileNet-224 & 50.6\% & 41 & 0.5 \\ [1ex]
\hline
\end{tabular}
\label{table:wm}
}
\end{table}
Table \ref{table:rm} shows the accuracy, computation and size trade offs for different resolution multipliers by
training MobileNets with reduced input resolutions. Accuracy drops off smoothly across resolution.
\begin{table}[t]
\caption{MobileNet Resolution}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{c c c c}
\hline\hline
Resolution & ImageNet & Million & Million \\ [0.5ex]
& Accuracy & Mult-Adds & Parameters \\
\hline
1.0 MobileNet-224 & 70.6\% & 569 & 4.2 \\
1.0 MobileNet-192 & 69.1\% & 418 & 4.2 \\
1.0 MobileNet-160 & 67.2\% & 290 & 4.2 \\
1.0 MobileNet-128 & 64.4\% & 186 & 4.2 \\ [1ex]
\hline
\end{tabular}
\label{table:rm}
}
\end{table}
\begin{figure}
\includegraphics[width=\linewidth]{mobilenet_multadds_cr.png}
\caption{This figure shows the trade off between computation (Mult-Adds) and accuracy on the ImageNet benchmark. Note the log linear dependence between accuracy and computation.}
\label{fig:mult-add}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{mobilenet_log_parameters_cr.png}
\caption{This figure shows the trade off between the number of parameters and accuracy on the ImageNet benchmark. The colors encode input resolutions. The number of parameters do not vary based on the input resolution. }
\label{fig:parameters}
\end{figure}
Figure \ref{fig:mult-add} shows the trade off between ImageNet Accuracy and computation for the 16 models made from the cross product of width multiplier $\alpha \in \{1,0.75,0.5,0.25\}$ and resolutions $\{224, 192, 160, 128\}$. Results are log linear with a jump when models get very small at $\alpha=0.25$.
Figure \ref{fig:parameters} shows the trade off between ImageNet Accuracy and number of parameters for the 16 models made from the cross product of width multiplier $\alpha \in \{1,0.75,0.5,0.25\}$ and resolutions $\{224, 192, 160, 128\}$.
Table \ref{table:mncompare} compares full MobileNet to the original GoogleNet \cite{szegedy2015going} and VGG16 \cite{simonyan2014very}. MobileNet is nearly as accurate as VGG16 while being 32 times smaller and 27 times less compute intensive. It is more accurate than GoogleNet while being smaller and more than 2.5 times less computation.
Table \ref{table:mncompare2} compares a reduced MobileNet with width multiplier $\alpha=0.5$ and reduced resolution $160\times160$. Reduced MobileNet is $4\%$ better than AlexNet \cite{krizhevsky2012imagenet} while being $45\times$ smaller and $9.4\times$ less compute than AlexNet. It is also $4\%$ better than Squeezenet \cite{iandola2016squeezenet} at about the same size and $22 \times$ less computation.
\begin{table}[t]
\caption{MobileNet Comparison to Popular Models}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{c c c c}
\hline\hline
Model & ImageNet & Million & Million \\ [0.5ex]
& Accuracy & Mult-Adds & Parameters \\
\hline
1.0 MobileNet-224 & 70.6\% & 569 & 4.2 \\
GoogleNet & 69.8\% & 1550 & 6.8 \\
VGG 16 & 71.5\% & 15300 & 138 \\ [1ex]
\hline
\end{tabular}
\label{table:mncompare}
}
\end{table}
\begin{table}[t]
\caption{Smaller MobileNet Comparison to Popular Models}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{c c c c}
\hline\hline
Model & ImageNet & Million & Million \\ [0.5ex]
& Accuracy & Mult-Adds & Parameters \\
\hline
0.50 MobileNet-160 & 60.2\% & 76 & 1.32 \\
Squeezenet & 57.5\% & 1700 & 1.25 \\
AlexNet & 57.2\% & 720 & 60 \\ [1ex]
\hline
\end{tabular}
\label{table:mncompare2}
}
\end{table}
\subsection{Fine Grained Recognition}
We train MobileNet for fine grained recognition on the Stanford Dogs dataset \cite{KhoslaYaoJayadevaprakashFeiFei_FGVC2011}.
We extend the approach of \cite{krause2015unreasonable} and collect an even larger but noisy training set than \cite{krause2015unreasonable} from the web.
We use the noisy web data to pretrain a fine grained dog recognition model and then fine tune the model on the Stanford Dogs training set.
Results on Stanford Dogs test set are in Table \ref{table:dogs}. MobileNet can almost achieve the state of the art results from
\cite{krause2015unreasonable} at greatly reduced computation and size.
\begin{table}[t]
\caption{MobileNet for Stanford Dogs}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{c c c c}
\hline\hline
Model & Top-1 & Million & Million \\ [0.5ex]
& Accuracy & Mult-Adds & Parameters \\
\hline
Inception V3 \cite{krause2015unreasonable} & 84\% & 5000 & 23.2 \\
1.0 MobileNet-224 & 83.3\% & 569 & 3.3 \\
0.75 MobileNet-224 & 81.9\% & 325 & 1.9 \\
1.0 MobileNet-192 & 81.9\% & 418 & 3.3 \\
0.75 MobileNet-192 & 80.5\% & 239 & 1.9 \\[1ex]
\hline
\end{tabular}
\label{table:dogs}
}
\end{table}
\subsection{Large Scale Geolocalizaton}
\begin{table}[t]
\setlength\tabcolsep{3pt}
\caption{Performance of PlaNet using the MobileNet architecture. Percentages are the fraction of the Im2GPS test dataset that were localized within a certain distance from the ground truth. The numbers for the original PlaNet model are based on an updated version that has an improved architecture and training dataset.}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{c c c c}
\hline\hline
Scale & Im2GPS \cite{hays2014large} & PlaNet \cite{weyand2016planet} & PlaNet \\
& & & MobileNet \\
\hline
\small{Continent (2500 km)} & 51.9\% & 77.6\% & 79.3\% \\
\small{Country (750 km)} & 35.4\% & 64.0\% & 60.3\% \\
\small{Region (200 km)} & 32.1\% & 51.1\% & 45.2\% \\
\small{City (25 km)} & 21.9\% & 31.7\% & 31.7\% \\
\small{Street (1 km)} & 2.5\% & 11.0\% & 11.4\% \\ [1ex]
\hline
\end{tabular}
\label{table:planet}
}
\end{table}
PlaNet \cite{weyand2016planet} casts the task of determining where on earth a photo was taken as a classification problem. The approach divides the earth into a grid of geographic cells that serve as the target classes and trains a convolutional neural network on millions of geo-tagged photos. PlaNet has been shown to successfully localize a large variety of photos and to outperform Im2GPS \cite{hays2008im2gps,hays2014large} that addresses the same task.
We re-train PlaNet using the MobileNet architecture on the same data. While the full PlaNet model based on the Inception V3 architecture \cite{szegedy2015rethinking} has 52 million parameters and 5.74 billion mult-adds. The MobileNet model has only 13 million parameters with the usual 3 million for the body and 10 million for the final layer and 0.58 Million mult-adds. As shown in Tab.~\ref{table:planet}, the MobileNet version delivers only slightly decreased performance compared to PlaNet despite being much more compact. Moreover, it still outperforms Im2GPS by a large margin.
\subsection{Face Attributes}
Another use-case for MobileNet is compressing large systems with unknown or esoteric training procedures. In a face attribute classification task, we demonstrate a synergistic relationship between MobileNet and distillation~\cite{hinton2015distilling}, a knowledge transfer technique for deep networks. We seek to reduce a large face attribute classifier with $75$ million parameters and $1600$ million Mult-Adds. The classifier is trained on a multi-attribute dataset similar to YFCC100M~\cite{thomee2016yfcc100m}.
We distill a face attribute classifier using the MobileNet architecture. Distillation~\cite{hinton2015distilling} works by training the classifier to emulate the outputs of a larger model\footnote{The emulation quality is measured by averaging the per-attribute cross-entropy over all attributes.} instead of the ground-truth labels, hence enabling training from large (and potentially infinite) unlabeled datasets. Marrying the scalability of distillation training and the parsimonious parameterization of MobileNet, the end system not only requires no regularization (e.g. weight-decay and early-stopping), but also demonstrates enhanced performances. It is evident from Tab.~\ref{table:faceattr} that the MobileNet-based classifier is resilient to aggressive model shrinking: it achieves a similar mean average precision across attributes (mean AP) as the in-house while consuming only $1\%$ the Multi-Adds.
\begin{table}[t]
\setlength\tabcolsep{3pt}
\caption{Face attribute classification using the MobileNet architecture. Each row corresponds to a different hyper-parameter setting (width multiplier $\alpha$ and image resolution).}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{c c c c}
\hline\hline
Width Multiplier / & Mean & Million & Million \\
Resolution & AP & Mult-Adds & Parameters \\
\hline
1.0 MobileNet-224 & 88.7\% & 568 & 3.2 \\
0.5 MobileNet-224 & 88.1\% & 149 & 0.8 \\
0.25 MobileNet-224 & 87.2\% & 45 & 0.2 \\
1.0 MobileNet-128 & 88.1\% & 185 & 3.2 \\
0.5 MobileNet-128 & 87.7\% & 48 & 0.8 \\
0.25 MobileNet-128 & 86.4\% & 15 & 0.2 \\
\hline
Baseline & 86.9\% & 1600 & 7.5 \\
\end{tabular}
\label{table:faceattr}
}
\end{table}
\subsection{Object Detection}
MobileNet can also be deployed as an effective base network in modern object detection systems.
We report results for MobileNet trained for object detection on COCO data based on the recent work that won the 2016 COCO challenge \cite{cocodetection2016}.
In table \ref{table:objectdetection}, MobileNet is compared to VGG and Inception V2 \cite{ioffe2015batch} under both Faster-RCNN \cite{ren2015faster} and SSD \cite{liu2015ssd} framework.
In our experiments, SSD is evaluated with 300 input resolution (SSD 300) and Faster-RCNN is compared with both 300 and 600 input resolution (Faster-RCNN 300, Faster-RCNN 600).
The Faster-RCNN model evaluates 300 RPN proposal boxes per image. The models are trained on COCO train+val excluding 8k minival images and evaluated on minival.
For both frameworks, MobileNet achieves comparable results to other networks with only a fraction of computational complexity and model size.
\begin{table}[t]
\setlength\tabcolsep{3pt}
\caption{COCO object detection results comparison using different frameworks and network architectures. mAP is reported with COCO primary challenge metric (AP at IoU=0.50:0.05:0.95)}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{c c c c c}
\hline\hline
Framework & Model & mAP & Billion & Million \\
Resolution & & & Mult-Adds & Parameters \\
\hline
& deeplab-VGG & 21.1\% & 34.9 & 33.1 \\
SSD 300 & Inception V2 & 22.0\% & 3.8 & 13.7 \\
& MobileNet & 19.3\% & 1.2 & 6.8 \\
\hline
Faster-RCNN & VGG & 22.9\% & 64.3 & 138.5 \\
300 & Inception V2 & 15.4\% & 118.2 & 13.3 \\
& MobileNet & 16.4\% & 25.2 & 6.1 \\
\hline
Faster-RCNN & VGG & 25.7\% & 149.6 & 138.5 \\
600 & Inception V2 & 21.9\% & 129.6 & 13.3 \\
& Mobilenet & 19.8\% & 30.5 & 6.1 \\
\hline
\end{tabular}
}
\label{table:objectdetection}
\end{table}
\begin{figure}
\includegraphics[width=\linewidth]{mobilessd.jpg}
\caption{Example objection detection results using MobileNet SSD.}
\label{fig:detection}
\end{figure}
\subsection{Face Embeddings}
The FaceNet model is a state of the art face recognition model \cite{schroff2015facenet}. It builds face embeddings based on the triplet loss. To build a mobile FaceNet model we use distillation
to train by minimizing the squared differences of the output of FaceNet and MobileNet on the training data. Results for very small MobileNet models can be found in table \ref{table:facenet}.
\begin{table}[t]
\caption{MobileNet Distilled from FaceNet}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{c c c c}
\hline\hline
Model & 1e-4 & Million & Million \\ [0.5ex]
& Accuracy & Mult-Adds & Parameters \\
\hline
FaceNet \cite{schroff2015facenet} & 83\% & 1600 & 7.5 \\
1.0 MobileNet-160 & 79.4\% & 286 & 4.9 \\
1.0 MobileNet-128 & 78.3\% & 185 & 5.5 \\
0.75 MobileNet-128 & 75.2\% & 166 & 3.4 \\
0.75 MobileNet-128 & 72.5\% & 108 & 3.8 \\
[1ex]
\hline
\end{tabular}
\label{table:facenet}
}
\end{table}
\section{Conclusion} \label{sec:conclusion}
We proposed a new model architecture called MobileNets based on depthwise separable convolutions. We investigated some of the important design decisions leading to an efficient model. We then demonstrated how to build smaller and faster MobileNets using width multiplier and resolution multiplier by trading off a reasonable amount of accuracy to reduce size and latency. We then compared different MobileNets to popular models demonstrating superior size, speed and accuracy characteristics. We concluded by demonstrating MobileNet's effectiveness when applied to a wide variety of tasks. As a next step to help adoption and exploration of MobileNets, we plan on releasing models in Tensor Flow.
{\small
\bibliographystyle{ieee}
|
{
"timestamp": "2017-04-18T02:07:24",
"yymm": "1704",
"arxiv_id": "1704.04861",
"language": "en",
"url": "https://arxiv.org/abs/1704.04861"
}
|
\section{Introduction}
There are important missing pieces in the Standard Model (SM),
for example, a candidate for the dark matter (DM), and tiny neutrino masses and their flavor mixings.
The SM should be extended so as to supplement these missing pieces.
The so-called seesaw mechanism is a natural way
to reproduce the tiny neutrino masses~\cite{Minkowski:1977sc,Yanagida:1979as,
GellMann:1980vs,Glashow:1979nm,Mohapatra:1979ia},
where heavy Majorana right-handed neutrinos (RHNs) are introduced.
The minimal gauged $B-L$ model~\cite{Mohapatra:1980qe,Marshak:1979fm,Wetterich:1981bx,
Masiero:1982fi,Mohapatra:1982xz,Buchmuller:1991ce}
is one of the simplest extensions of the SM with an extra gauge symmetry,
in which the accidentally anomaly-free global $B-L$ (baryon number minus lepton number) in the SM
is gauged.
Three RHNs play an essential roll to cancel the gauge and gravitational anomalies of the model.
Associated with the $B-L$ symmetry breaking, the RHNs acquire their Majorana masses,
and hence the seesaw mechanism is automatically implemented.
The minimal $B-L$ model can be generalized
to the so-called minimal U(1)$^\prime$ model~\cite{Appelquist:2002mw}.
Here, the U(1)$^\prime$ gauge group is defined as a linear combination of the U(1)$_{B-L}$
and the SM U(1)${_Y}$ gauge groups,
so that the U(1)$^\prime$ model is anomaly-free.
In our previous work~\cite{Oda:2015gna,Das:2016zue},
we have investigated the minimal U(1)$^\prime$ model with classically conformal invariance.\footnote{
See Refs.~\cite{Hempfling:1996ht,Dias:2006th,Espinosa:2007qk,Chang:2007ki,Foot:2007as,
Foot:2007ay,Meissner:2006zh,Foot:2007iy,Meissner:2007xv,Meissner:2008gj,Iso:2009ss,
Iso:2009nw,Holthausen:2009uc,Farzinnia:2013pga,Heikinheimo:2013fta,Farzinnia:2014xia,
Lindner:2014oea,Khoze:2014xha,Gabrielli:2013hma,Altmannshofer:2014vra,Karam:2015jta,
Haba:2015yfa,Okada:2015gia,Latosinski:2015pba,Wang:2015sxe,Goertz:2015dba,Haba:2015lka,
Haba:2015nwl,Ghorbani:2015xvz,Haba:2015qbz,Ahriche:2015loa,Ishida:2016ogu,Hatanaka:2016rek,
Karam:2016rsz,Marzola:2016xgb,Das:2015nwk,Kannike:2016wuy,Marzola:2017jzl}
for recent work on new physics models with classically conformal invariance.}
In this model, the U(1)$^\prime$ gauge symmetry is radiatively broken
through the Coleman-Weinberg (CW) mechanism~\cite{Coleman:1973jx}.
Given a negative mixing quartic coupling between the SM Higgs and the U(1)$^\prime$ Higgs fields,
once the U(1)$^\prime$ Higgs field develops a vacuum expectation value (VEV),
a negative mass squared of the SM Higgs doublet is generated,
and thus the electroweak symmetry breaking is naturally triggered.
In this model context, we have investigated the electroweak vacuum instability problem in the SM.
Employing the renormalization group (RG) equations at the two-loop level
and the central values for the world average masses
of the top quark ($m_t=173.34$ GeV~\cite{ATLAS:2014wva})
and the Higgs boson ($m_h=125.09$ GeV~\cite{Aad:2015zhl}),
we have performed parameter scans to identify
the parameter region for resolving the electroweak vacuum instability problem.
We have also investigated the ATLAS and CMS search limits at the LHC Run-2 (2015)
for the U(1)$^\prime$ gauge boson ($Z^\prime$)~\cite{TheATLAScollaboration:2015jgi,CMS:2015nhc},
and identified the allowed parameter regions in our model.
Combining the constraints from the electroweak vacuum stability and the LHC Run-2 results,
we have found a lower bound on the $Z^\prime$ boson mass.
We also have calculated self-energy corrections to the SM Higgs doublet field through the heavy states,
the right-handed neutrinos and the $Z^\prime$ boson, and have found the naturalness bound
as $m_{Z^\prime} \lesssim 6$ TeV, in order to reproduce the right electroweak scale
for the fine-tuning level better than 10\%.
The so-called weakly interacting massive particle (WIMP) is one of the most promising candidates
of the DM in our Universe,
which is in thermal equilibrium in the early Universe.
Among many possibilities, a simple way to introduce a WIMP DM
in the minimal U(1)$^\prime$ model has been proposed
in~\cite{Okada:2010wd} (see also \cite{Anisimov:2008gg}),
where $Z_2$-parity is introduced and an odd-parity is assigned to one RHN,
while all the other particles is assigned to be $Z_2$-even.
We adapt this scheme in our minimal U(1)$^\prime$ model with the classically conformal invariance,
and the $Z_2$-odd RHN is a DM candidate,
while the other two RHNs are utilized for the seesaw mechanism.
Note that only two RHNs are sufficient to reproduce the neutrino oscillation data,
and the observed baryon asymmetry in the Universe
through leptogenesis~\cite{Fukugita:1986hr}.
This system is called the minimal seesaw \cite{King:1999mb,Frampton:2002qc}.
In our model, there are two ways for the RHN DM to interact with the SM particles.
One is mediated by the $Z^\prime$ boson ($Z^\prime$-portal)
and the other is by the two Higgs bosons (Higgs portal)
which are two mass eigenstates consisting of the SM Higgs and the U(1)$^\prime$ Higgs bosons.
Recently, the $Z^\prime$-portal DM scenarios~\cite{Burell:2011wh,Basso:2012ti,Dudas:2013sia,Das:2013jca,Chu:2013jja,
Lindner:2013awa,Alves:2013tqa,Kopp:2014tsa,Agrawal:2014ufa,Hooper:2014fda,
Ma:2014qra,Alves:2015pea,Ghorbani:2015baa,Sanchez-Vega:2015qva,Duerr:2015wfa,
Alves:2015mua,Ma:2015mjd,Okada:2016gsh,Okada:2016tzi,Chao:2016avy,Biswas:2016ewm,
Accomando:2016sge,Fairbairn:2016iuf,Klasen:2016qux,Dev:2016xcp,Altmannshofer:2016jzy,
Okada:2016tci,Kaneta:2016vkq,Arcadi:2017kky}
have been intensively investigated,
while the Higgs portal RHN DM scenarios~\cite{Okada:2010wd,Okada:2012sg,Basak:2013cga}
have been analyzed in detail.
In this paper, we consider the classically conformal U(1)$^\prime$ extended SM with the RHN DM.
As we mentioned above, the allowed parameter regions in the classically conformal model
are severely constrained
in order to solve the electroweak vacuum instability problem,
and to satisfy the LHC limits from the search for $Z^\prime$ boson resonance.
In addition to these constraints, we will investigate the RHN DM physics.
Because of the nature of classical conformality,
we find the mass mixing between the SM Higgs and the U(1)$^\prime$ Higgs bosons is very small,
so that the RHN DM pair annihilation process mediated by the Higgs bosons is highly suppressed.
Therefore, we focus on the study of the $Z^\prime$-portal RHN DM~\cite{Okada:2016gsh,Okada:2016tci},
and identify allowed parameter regions to reproduce the observed DM relic density
from the Planck 2015 result~\cite{Aghanim:2015xee}.
We will show that the DM physics, LHC phenomenology,
and the electroweak vacuum stability condition complementarily work to narrow down
the allowed parameter regions.
For the identified allowed regions,
we also calculate the spin-independent cross section of the RHN DM with nucleons
and compare our results with the current upper bounds from the direct DM search experiments.
This paper is organized as follows:
In the next section, we introduce the classically conformal U(1)$^\prime$ extended SM
with $Z^\prime$-portal RHN DM.
We briefly review our previous work
on the classically conformal U(1)$^\prime$ model~\cite{Oda:2015gna, Das:2016zue}.
In Sec.~\ref{Sec_relic_density}, we calculate the relic density of the $Z^\prime$-portal RHN DM.
In Sec.~\ref{Sec_collider}, we study the $Z^\prime$ boson production at the LHC Run-2 (2016)~\cite{ATLAS:2016cyf,CMS:2016abv},
and obtained the constraints on the model parameter space from the search results of the $Z^\prime$ boson resonance by the ATLAS and the CMS Collaborations.
In Sec.~\ref{Sec_allowed_region}, we combine all the results in the previous sections
and narrow allowed regions.
In Sec.~\ref{Sec_direct_detection}, for the allowed parameter regions,
we calculate the spin-independent cross section of the RHN DM with nucleons.
The last section is devoted to conclusions.
\section{The classically conformal U(1)$^{\prime}$ extended SM with RHN DM}
\label{Sec_U1p}
In this section, we will briefly review the results in Ref.~\cite{Das:2016zue}.
Although the model is extended to incorporate the RHN DM,
the results presented here are essentially the same as those in Ref.~\cite{Das:2016zue}.
\subsection{The model}
\begin{table}[t]
\begin{center}
\begin{tabular}{c|ccc|rcr|c}
& SU(3)$_c$ & SU(2)$_L$ & U(1)$_Y$ & \multicolumn{3}{c|}{U(1)$^\prime$} & $Z_2$ \\
\hline
&&&&&&&\\[-12pt]
$q_L^i$ & {\bf 3} & {\bf 2}& $+1/6$ & $x_q$ & = & $\frac{1}{3}x_H + \frac{1}{6}x_\Phi$ &+ \\[2pt]
$u_R^i$ & {\bf 3} & {\bf 1}& $+2/3$ & $x_u$ & = & $\frac{4}{3}x_H + \frac{1}{6}x_\Phi$ &+ \\[2pt]
$d_R^i$ & {\bf 3} & {\bf 1}& $-1/3$ & $x_d$ & = & $-\frac{2}{3}x_H + \frac{1}{6}x_\Phi$ &+ \\[2pt]
\hline
&&&&&&&\\[-12pt]
$\ell_L^i$ & {\bf 1} & {\bf 2}& $-1/2$ & $x_\ell$ & = & $- x_H - \frac{1}{2}x_\Phi$ &+ \\[2pt]
$\nu_R^{1,2}$ & {\bf 1} & {\bf 1}& $0$ & $x_\nu$ & = & $- \frac{1}{2}x_\Phi$ &+\\[2pt]
$\nu_R^3$ & {\bf 1} & {\bf 1}& $0$ & $x_\nu$ & = & $- \frac{1}{2}x_\Phi$ & $-$ \\[2pt]
$e_R^i$ & {\bf 1} & {\bf 1}& $-1$ & $x_e$ & = & $- 2x_H - \frac{1}{2}x_\Phi$ &+ \\[2pt]
\hline
&&&&&&&\\[-12pt]
$H$ & {\bf 1} & {\bf 2}& $+1/2$ & $x_H$ & = & $x_H$\hspace*{12.5mm} &+ \\
$\Phi$ & {\bf 1} & {\bf 1}& $0$ & $x_\Phi$ & = & $x_\Phi$ &+ \\
\end{tabular}
\end{center}
\caption{
Particle contents of the U(1)$^\prime$ extended SM with $Z_2$ parity.
In addition to the SM particle contents, three generations of RHNs $\nu_R^i$
($i=1,2,3$ denotes the generation index) and U(1)$^\prime$ Higgs field $\Phi$ are introduced.
Under $Z_2$ parity, the only one RHN $\nu_R^3$ is odd,
while the other particles, including $\nu_R^1$ and $\nu_R^2$, are even.
}
\label{Tab:particle_contents}
\end{table}
The model we will investigate is the anomaly-free U(1)$^\prime$ extension of the SM
with the classically conformal invariance, which is based on the gauge group
SU(3)$_C \times$SU(2)$_L \times$U(1)$_Y \times$U(1)$^\prime$.
The particle contents of the model are listed in Table~\ref{Tab:particle_contents}.
In addition to the SM particle content, three generations of RHNs $\nu_R^i$
and a U(1)$^\prime$ Higgs field $\Phi$ are introduced.
We also introduce the $Z_2$ parity~\cite{Okada:2010wd},
and assign an odd parity to one RHN $\nu_R^3$,
while the other particles, including $\nu_R^1$ and $\nu_R^2$, have even parity.
The conservation of $Z_2$ parity ensures the stability of $\nu_R^3$, which is
a unique candidate of the DM in our model.
The covariant derivative, which is relevant to U(1)$_Y \times$ U(1)$^\prime$, is defined as
\begin{equation}
D_\mu \equiv \partial_\mu
- i\left(\begin{array}{cc} Y_{1} & Y_{X} \end{array}\right )
\left(\begin{array}{cc} g_{1} & g_{1X} \\g_{X1} & g_{X} \end{array}\right)
\left(\begin{array}{c} B_{\mu} \\B_{\mu}^\prime \end{array}\right),
\label{Eq:covariant_derivative}
\end{equation}
where $Y_{1}$ ($Y_{X}$) is U(1)$_Y$ (U(1)$^\prime$ ) charge of a particle,
and the gauge coupling $g_{X1}$ and $g_{1X}$ are introduced associated with a kinetic mixing
between the two U(1) gauge bosons.
In order to reproduce observed fermion masses and flavor mixings,
we introduce the following Yukawa interactions:
\begin{eqnarray}
{\cal L}_{\rm Yukawa}
&=& - \sum_{i=1}^{3} \sum_{j=1}^{3} Y_u^{ij} \overline{q_L^i} \tilde{H} u_R^j
- \sum_{i=1}^{3} \sum_{j=1}^{3} Y_d^{ij} \overline{q_L^i} H d_R^j
- \sum_{i=1}^{3} \sum_{j=1}^{3} Y_e^{ij} \overline{\ell_L^i} H e_R^j \nonumber \\
&& - \sum_{i=1}^{3} \sum_{j=1}^{2} Y_\nu^{ij} \overline{\ell_L^i} \tilde{H} \nu_R^j
- \sum_{i=1}^{3} Y_M^i \Phi \overline{\nu_R^{ic}} \nu_R^i + {\rm h.c.},
\label{Eq:L_Yukawa}
\end{eqnarray}
where $\tilde{H} \equiv i \tau^2 H^*$, and the fourth and fifth terms in the right-handed side
are for the seesaw mechanism to generate neutrino masses.
Without loss of generality, the Majorana Yukawa couplings in the fifth term
are already diagonalized in our basis.
Because of the $Z_2$ parity, only two generation RHNs are involved
in the neutrino Dirac Yukawa couplings
and hence the neutrino Dirac mass matrix is 2 by 3.
Once the U(1)$^\prime$ Higgs field $\Phi$ develops a VEV,
the U(1)$^\prime$ symmetry is broken
and the Majorana mass terms for the RHNs are generated.
After the electroweak symmetry breaking,
the seesaw mechanism~\cite{Minkowski:1977sc,Yanagida:1979as,
GellMann:1980vs,Glashow:1979nm,Mohapatra:1979ia} is automatically implemented,
except that only two generation RHNs are relevant.
This system is the minimal seesaw~\cite{King:1999mb, Frampton:2002qc},
which possesses a number of free parameters
$Y_{\nu}^{ij}$ and $Y_M^{j}$ ($i=1,2,3$, $j=1,2$) enough
to reproduce the neutrino oscillation data with a prediction of one massless eigenstate.
In the particle contents, the two parameters ($x_H$ and $x_\Phi$) reflect the fact that the U(1)$^\prime$ gauge group can be defined
as a linear combination
of the SM U(1)$_Y$ and the U(1)$_{B-L}$ gauge groups.
Since the U(1)$^\prime$ gauge coupling $g_{X}$ is a free parameter of the model and
it always appears as a product $x_\Phi g_{X}$ or $x_H g_{X}$,
we fix $x_\Phi=2$ without loss of generality throughout this paper.
This convention excludes the case that U(1)$^\prime$ gauge group is identical with the SM U(1)$_Y$.
The choice of $(x_H, x_\Phi)=(0, 2)$ corresponds to the U(1)$_{B-L}$ model.
Another example is $(x_H, x_\Phi)=(-1, 2)$, which corresponds to the SM with the so-called U(1)$_R$ symmetry.
When we choose $(x_H, x_\Phi)=(-16/41, 2)$, the beta function of $g_{X1}$ ($g_{1X}$) at the 1-loop level
has only terms proportional to $g_{X1}$ ($g_{1X}$)~\cite{Oda:2015gna}.
This is the orthogonal condition between the U(1)$_Y$ and U(1)$^\prime$ at the 1-loop level,
under which $g_{X1}$ and $g_{1X}$ do not evolve once we have set $g_{X1}=g_{1X}=0$ at an energy scale.
Imposing the classically conformal invariance, the scalar potential is given by
\begin{equation}
V = \lambda_H \! \left( H^\dagger H \right)^2
+ \lambda_\Phi \! \left( \Phi^\dagger \Phi \right)^2
+ \lambda_{\rm mix} \! \left( H^\dagger H \right) \! \left( \Phi^\dagger \Phi \right) ,
\label{Eq:classical_potential}
\end{equation}
where the mass terms are forbidden by the conformal invariance.
If $\lambda_{\rm mix}$ is negligibly small,
we can analyze the Higgs potential separately for $\Phi$ and $H$ as a good approximation.
This will be justified in the following subsections.
When the Majorana Yukawa couplings $Y_M^i$ are negligible compared to the U(1)$^\prime$ gauge coupling,
the $\Phi$ sector is identical with the original CW model \cite{Coleman:1973jx},
so that the radiative U(1)$^\prime$ symmetry breaking will be achieved.
Once $\Phi$ develops a VEV through the CW mechanism,
the tree-level mass term for the SM Higgs doublet is effectively generated through $\lambda_{\rm mix}$
in Eq.~(\ref{Eq:classical_potential}).
Taking $\lambda_{\rm mix}$ negative, the induced mass squared for the Higgs doublet
is negative and, as a result, the electroweak symmetry breaking is driven in the same way as in the SM.
\subsection{Radiative U(1)$^\prime$ gauge symmetry breaking}
Assuming $\lambda_{\rm mix}$ is negligibly small, we first analyze the U(1)$^\prime$ Higgs sector.
Without mass terms, the Coleman-Weinbeg potential \cite{Coleman:1973jx}
at the 1-loop level is found to be
\begin{eqnarray}
V(\phi) = \frac{\lambda_\Phi}{4} \phi^4
+ \frac{\beta_\Phi}{8} \phi^4 \left( \ln \left[ \frac{\phi^2}{v_\phi^2} \right] - \frac{25}{6} \right),
\label{Eq:CW_potential}
\end{eqnarray}
where $\phi / \sqrt{2} = \Re[\Phi]$, and
we have chosen the renormalization scale to be the VEV
of $\Phi$ ($\langle \phi \rangle =v_\phi$).
Here, the coefficient of the 1-loop quantum corrections is given by
\begin{eqnarray}
\beta_\Phi &=& \frac{1}{16 \pi^2}
\left[ 20\lambda_\Phi^2
+ 6 x_\Phi^4 \left ( g_{X1}^2 + g_{X}^2 \right)^2 - 16\sum_i(Y_M^i)^4 \right] \nonumber \\
& \simeq & \frac{1}{16 \pi^2}
\left[ 6 \left(x_\Phi g_{X} \right)^4 - 16\sum_i(Y_M^i)^4 \right] ,
\end{eqnarray}
where in the last expression, we have used
$\lambda_\Phi^2 \ll (x_\Phi g_{X})^4$ as usual in the CW mechanism
and set $g_{X1}=g_{1X}= 0$ at $ \langle \phi \rangle = v_\phi$, for simplicity.
The stationary condition $\left. dV/d\phi\right|_{\phi=v_\phi} = 0$ leads to
\begin{eqnarray}
\lambda_\Phi = \frac{11}{6} \beta_\Phi,
\label{eq:stationary}
\end{eqnarray}
and this $\lambda_\Phi$ is nothing but a renormalized self-coupling at $v_\phi$ defined as
\begin{eqnarray}
\lambda_\Phi = \frac{1}{3 !}\left. \frac{d^4V(\phi)}{d \phi^4} \right|_{\phi=v_\phi}.
\end{eqnarray}
For more detailed discussion, see Ref.~\cite{Khoze:2014xha}.
Associated with this radiative U(1)$^\prime$ symmetry breaking
(as well as the electroweak symmetry breaking),
the U(1)$^\prime$ gauge boson ($Z^\prime$ boson),
the Majorana RHNs $\nu_R^{1,2}$,
and the RHN DM particle $\nu_R^3$ acquire their masses as
\begin{eqnarray}
m_{Z^\prime} = \sqrt{(x_\Phi g_{X} v_\phi)^2 + (x_H g_{X} v_h)^2} \simeq x_\Phi g_{X} v_\phi,
\; \; m_{N^{1,2}} = \sqrt{2} Y_M^{1,2} v_\phi,
\; \; m_{\rm DM} = \sqrt{2} Y_M^3 v_\phi,
\label{Eq:mass_Zp_DM}
\end{eqnarray}
where $v_h=246$ GeV is the SM Higgs VEV, and we have used $x_\Phi v_\phi \gg x_H v_h$,
which will be verified below.
In this paper, we assume degenerate masses for $\nu_R^{1,2}$,
($Y_M^1 = Y_M^2 = y_M$, equivalently, $m_{N^{1,2}}=m_N$), for simplicity.
The U(1)$^\prime$ Higgs boson mass is given by
\begin{eqnarray}
m_\phi^2 &=& \left. \frac{d^2 V}{d\phi^2}\right|_{\phi=v_\phi}
=\beta_\Phi v_\phi^2 \simeq
\frac{1}{8 \pi^2} \left( 3(x_\Phi g_{X})^4 - 16 y_M^4 - 8 y_{\rm DM}^4\right) v_\phi^2 \nonumber \\
&\simeq& \frac{1}{8 \pi^2} \frac{ 3m_{Z^\prime}^4 - 4 m_N^4 - 2 m_{\rm DM}^4}{v_\phi^2},
\label{Eq:mass_phi}
\end{eqnarray}
where $y_{\rm DM} = Y_M^3$.
When the Yukawa couplings are negligibly small, this equation reduces to the well-known relation
derived in the original paper by Coleman-Weinberg \cite{Coleman:1973jx}.
For a sizable Majorana mass, this formula indicates that
the potential minimum disappears,
so that there is an upper bound on the RHN mass
for the U(1)$^\prime$ symmetry to be broken radiatively.
This is in fact the same reason why the CW mechanism
in the SM Higgs sector fails to break the electroweak symmetry
when the top Yukawa coupling is large as observed.
In order to avoid the destabilization of the U(1)$^\prime$ Higgs potential,
we simply set $m_{Z^\prime}^4 \gg m_N^4$ in the following analysis,
while $m_{\rm DM} \simeq m_{Z^\prime}/2$ as we will find in the next section.
Note that this condition does not mean that the Majorana RHNs must be very light,
even though a factor difference between $m_{Z^\prime}$ and $m_N$ is enough to satisfy the condition.
For simplicity, we set $y_M=0$ at $v_\phi$ in the following RG analysis as an approximation.
\subsection{Electroweak symmetry breaking}
Let us now consider the SM Higgs sector.
In our model, the electroweak symmetry breaking is achieved in a very simple way.
Once the U(1)$^\prime$ symmetry is radiatively broken,
the SM Higgs doublet mass is generated through the mixing quartic term between $H$ and $\Phi$
in the scalar potential in Eq.~(\ref{Eq:classical_potential}),
\begin{equation}
V(h) = \frac{\lambda_H}{4}h^4 + \frac{\lambda_{\rm mix}}{4} v_\phi^2 h^2,
\end{equation}
where we have replaced $H$ by $H = 1/\sqrt{2}\, (0,\,h)$ in the unitary gauge.
Choosing $\lambda_{\rm mix} < 0$, the electroweak symmetry is broken in the same way
as in the SM \cite{Iso:2009ss, Iso:2009nw}.
However, we should note that a crucial difference from the SM is that, in our model,
the electroweak symmetry breaking originates from the radiative breaking
of the U(1)$^\prime$ gauge symmetry.
At the tree level, the stationary condition $V^\prime |_{h=v_h} = 0$
leads to the relation
$|\lambda_{\rm mix}|= 2 \lambda_H (v_h/v_\phi)^2$,
and the Higgs boson mass $m_h$ is given by
\begin{equation}
m_h^2 = \left. \frac{d^2 V}{dh^2} \right|_{h=v_h} = |\lambda_{\rm mix}|v_\phi^2 = 2 \lambda_H v_h^2.
\label{Eq:mass_h}
\end{equation}
In the following RG analysis,
this is used as the boundary condition for $\lambda_{\rm mix}$ at the renormalization scale $\mu=v_\phi$.
Note that since $\lambda_H \sim 0.1$ and $v_\phi \gtrsim 10$ TeV by the large electron-positron collider (LEP) constraint \cite{LEP:2003aa, Carena:2004xs, Schael:2013ita},
$|\lambda_{\rm mix}| \lesssim 10^{-5}$, which is very small.
In our discussion about the U(1)$^\prime$ symmetry breaking, we neglected $\lambda_{\rm mix}$
by assuming it to be negligibly small.
Here we justify this treatment.
In the presence of $\lambda_{\rm mix}$ and the Higgs VEV, Eq.~(\ref{eq:stationary}) is modified as
\begin{eqnarray}
\lambda_\Phi = \frac{11}{6} \beta_\Phi + \frac{|\lambda_{\rm mix}|}{2} \left( \frac{v_h}{v_\phi} \right)^2
\simeq
\frac{1}{2 v_\phi^4} \left( \frac{11}{8 \pi^2} m_{Z^\prime}^4 + m_h^2 v_h^2 \right).
\label{eq:consistency}
\end{eqnarray}
Considering the current LHC Run-2 bound from search for $Z^\prime$ boson resonances
\cite{ATLAS:2016cyf, CMS:2016abv},
$m_{Z^\prime} \gtrsim 4$ TeV, we find that the first term in the parenthesis
in the last equality is 5 orders of magnitude greater than the second term,
and therefore we can analyze the two Higgs sectors separately.
\subsection{Solving the electroweak vacuum instability}
\begin{figure}[t]
\begin{minipage}{0.5\linewidth}
\begin{center}
\includegraphics[width=0.95\linewidth]{Fig1a.eps}
\subcaption{}\label{Fig:Higgs_stability_lambdaH}
\end{center}
\end{minipage}
\begin{minipage}{0.5\linewidth}
\begin{center}
\includegraphics[width=0.95\linewidth]{Fig1b.eps}
\subcaption{}\label{Fig:Higgs_stability_3D}
\end{center}
\end{minipage}
\caption
{
\subref{Fig:Higgs_stability_lambdaH} The evolutions of the Higgs quartic coupling $\lambda_H$ (solid line)
for the inputs $m_t=173.34$ GeV and $m_h=125.09$ GeV,
along with the SM case (dashed line).
Here, we have taken $x_H =-0.575$, $m_{Z^\prime}=4$ TeV and $\alpha_{g_X}=0.01$,
which corresponds to $v_\phi = 5.64$ TeV and $g_{X}(v_\phi) = 0.354$.
\subref{Fig:Higgs_stability_3D} The result of the three-dimensional parameter scans
for $v_\phi$, $g_{X}$ and $x_H$,
shown in ($m_{Z^\prime}/{\rm GeV}, \alpha_{g_X}, x_H$) parameter space
with $m_{Z^\prime} \simeq x_\Phi g_{X} v_\phi$.
As a reference, a horizontal plane for $x_H=-16/41$ is shown, which corresponds to the orthogonal case.
}
\label{Fig:Higgs_stability}
\end{figure}
In the SM with the observed Higgs boson mass of $m_h=125.09$ GeV~\cite{Aad:2015zhl},
the RG evolution of the SM Higgs quartic coupling shows that the running coupling becomes
negative at the intermediate scale $\mu \simeq 10^{10}$ GeV \cite{Buttazzo:2013uya}
for $m_t=173.34$ GeV~\cite{ATLAS:2014wva}, and hence the electroweak vacuum is unstable.
In our U(1)$^\prime$ extended SM, however, there is a parameter region
to solve this electroweak vacuum instability problem~\cite{Oda:2015gna, Das:2016zue}.\footnote{
In the absence of the classical conformal invariance,
the electroweak vacuum instability problem has been investigated in Refs.~\cite{Coriano:2014mpa,DiChiara:2014wha,
Coriano:2015sea,Accomando:2016sge}.}
There are only three free parameters in our model, $x_H$, $v_\phi$, and $g_{X}$,
which are also interpreted as $x_H$, $m_{Z^\prime}$, and $\alpha_{g_X}=g_X^2/(4 \pi)$.
Inputs of the couplings at $v_\phi$ are determined by these three parameters.
In Fig.~\ref{Fig:Higgs_stability}\subref{Fig:Higgs_stability_lambdaH},
we show the RG evolution of the SM Higgs quartic coupling
in our model (solid line), along with the SM result (dashed line).
Here, we have taken $x_H=-0.575$, $m_{Z^\prime}=4$ TeV and $\alpha_{g_X}=0.01$,
which corresponds to $v_\phi = 5.64$ TeV and $g_{X}(v_\phi) = 0.354$, as an example.
The Higgs quartic coupling remains positive all the way up to the Planck mass scale,
so the electroweak vacuum instability problem is solved.
In order to identify a parameter region to resolve the electroweak vacuum instability,
we perform parameter scans for the free parameters $x_H$, $v_\phi$ and $g_{X}$.
In this analysis, we impose several conditions on the running couplings at $v_\phi \leq \mu \leq M_{\rm P}$
($M_{\rm P} =2.44 \times 10^{18}$ GeV is the reduced Planck mass):
stability conditions of the Higgs potential ($\lambda_H, \lambda_\Phi > 0$),
and the perturbative conditions that all the running couplings remain in the perturbative regime, namely,
$g_i^2$ ($i=1,2,3$), $g_{X}^2$, $g_{X1}^2$, $g_{1X}^2<4\pi$
and $\lambda_H$, $\lambda_\Phi$, $\lambda_{\rm mix}<4 \pi$.
For theoretical consistency, we also impose a condition that the 2-loop beta functions
are smaller than the 1-loop beta functions (see Ref.~\cite{Das:2016zue} for detail).
In Fig.~\ref{Fig:Higgs_stability}\subref{Fig:Higgs_stability_3D}, we show the result of our parameter scans
in the three-dimensional parameter space of ($m_{Z^\prime}, \alpha_{g_X}, x_H$).
As a reference, we show a horizontal plane corresponding to the orthogonal case $x_H=-16/41$.
There is no overlapping of the plane with the resultant parameter regions
to resolve the electroweak vacuum instability.
\subsection{Naturalness bounds from SM Higgs mass corrections}
\label{Sec_naturalness}
Once the classically conformal symmetry is radiatively broken by the CW mechanism,
the masses for the $Z^\prime$ boson and the Majorana RHNs are generated,
and they contribute to self-energy corrections of the SM Higgs doublet.
If the U(1)$^\prime$ gauge symmetry breaking scale is very large,
the self-energy corrections may exceed the electroweak scale
and require us to fine-tune the model parameters in reproducing the correct electroweak scale.
See \cite{Casas:2004gh} for related discussions.
As heavy states, we have the RHNs and $Z^\prime$ boson,
whose masses are generated by the U(1)$^\prime$ gauge symmetry breaking.
Since the original theory is classically conformal and defined as a massless theory,
the self-energy corrections to the SM Higgs doublet originate
from corrections to the mixing quartic coupling $\lambda_{\rm mix}$.
Thus, what we calculate to derive the naturalness bounds are quantum corrections to
the term $\lambda_{\rm mix} h^2 \phi^2$ in the effective Higgs potential
\begin{eqnarray}
V_{\rm eff} \supset
\frac{\lambda_{\rm mix}}{4} h^2 \phi^2
+ \frac{\beta_{\lambda_{\rm mix}}}{8} h^2 \phi^2 \left( \ln\left[\phi^2\right] + C \right),
\end{eqnarray}
where the logarithmic divergence and the terms independent of $\phi$ are all encoded in $C$.
Here, the major contributions to quantum corrections are from the $Z^\prime$ boson loops:
\begin{eqnarray}
\beta_{\lambda_{\rm mix}} &\supset& \frac{12 x_H^2 x_\Phi^2 g_X^4}{16 \pi^2}
- \frac{4 \left( 19 x_H^2 + 10 x_H x_\Phi + x_\Phi^2 \right) x_\Phi^2 y_t^2 g_X^4}{\left(16 \pi^2\right)^2},
\end{eqnarray}
where the first term is from the one-loop diagram,
and the second one is from the two-loop diagram \cite{Iso:2009ss,Iso:2009nw}
involving the $Z^\prime$ boson and the top quark.
By adding a counter-term, we renormalize the coupling $\lambda_{\rm mix}$ with the renormalization condition,
\begin{eqnarray}
\frac{\partial^4 V_{\rm eff}}{\partial h^2 \partial \phi^2} \Big|_{h=0, \phi=v_\phi} = \lambda_{\rm mix},
\end{eqnarray}
where $\lambda_{\rm mix}$ is the renormalized coupling.
As a result, we obtain
\begin{eqnarray}
V_{\rm eff} \supset
\frac{\lambda_{\rm mix}}{4} h^2 \phi^2
+ \frac{\beta_{\lambda_{\rm mix}}}{8} h^2 \phi^2 \left( \ln\left[\frac{\phi^2}{v_\phi}\right] - 3 \right).
\end{eqnarray}
Substituting $\phi=v_\phi$, we obtain the SM Higgs self-energy correction as
\begin{eqnarray}
\Delta m_h^2 &=& - \frac{3}{4} \beta_{\lambda_{\rm mix}} v_\phi^2 \nonumber \\
&\sim& - \frac{9}{4 \pi} x_H^2 \alpha_{g_X} m_{Z^\prime}^2
+ \frac{3 m_t^2} {32 \pi^3 v_h^2} \left(19x_H^2 + 20x_H + 4\right) \alpha_{g_X} m_{Z^\prime}^2.
\label{Eq:Delta_mh2}
\end{eqnarray}
For the stability of the electroweak vacuum, we impose $\Delta m_h^2 \lesssim m_h^2$ as the naturalness.
The most important contribution to $\Delta m_h^2$ is the first term of Eq.~(\ref{Eq:Delta_mh2})
generated through the one-loop diagram with the $Z^\prime$ gauge boson,
and the second term becomes important in the case of the U(1)$_{B-L}$ model,
where $x_H=0$.
If $\Delta m_h^2$ is much larger than the electroweak scale,
we need a fine-tuning of the tree-level Higgs mass ($|\lambda_{\rm mix}| v_\phi^2/2$)
to reproduce the correct SM Higgs VEV, $v_h=246$ GeV.
We simply evaluate a fine-tuning level as
\begin{eqnarray}
\delta = \frac{m_h^2}{2 |\Delta m_h^2|}.
\end{eqnarray}
Here, $\delta =0.1$, for example, indicates that we need to fine-tune the tree-level Higgs mass squared
at the 10\% accuracy level.
\section{Relic density of the RHN DM}
\label{Sec_relic_density}
In this section, we calculate the thermal relic density of the RHN DM
and identify the model parameter region to be consistent
with the Planck 2015 measurement~\cite{Aghanim:2015xee}
(68\% confidence level):
\begin{eqnarray}
\Omega_{\rm DM} h^2 &=& 0.1198 \pm 0.0015.
\end{eqnarray}
In our model, the RHN DM particles mainly annihilate into the SM particles
through the $s$-channel process mediated by the U(1)$^\prime$ gauge boson $Z^\prime$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.5\linewidth]{Fig2.eps}
\caption
{
The relic density of the RHN DM
as a function of its mass ($m_{\rm DM}$).
We have fixed $x_H=-0.575$ and $m_{Z^\prime}=4$ TeV,
and have shown the relic densities for various values of the gauge coupling,
$\alpha_{g_X}=0.002$, $0.00235$, $0.003$, $0.004$ and $0.005$
(solid lines from top to bottom).
The two horizontal lines denote the range of the observed DM relic density,
$0.1183 \leq \Omega_{\rm DM} h^2 \leq 0.1213$ in the Planck 2015 results~\cite{Aghanim:2015xee}.
}
\label{Fig:Omega_h2}
\end{center}
\end{figure}
The Boltzmann equation of the RHN DM is given by
\begin{eqnarray}
\frac{dY}{dx} &=& -\frac{xs \langle \sigma v \rangle}{H(m_{\rm DM})} (Y^2 - Y_{\rm EQ}^2),
\label{Eq:Boltzmann}
\end{eqnarray}
where temperature of the Universe is normalized by the mass of the RHN DM
$x=m_{\rm DM}/T$,
$H(m_{\rm DM})$ is the Hubble parameter at $T=m_{\rm DM}$,
$s$ is the entropy density,
$Y=n/s$ is the yield of the RHN DM
which is defined by the ratio of the number density $n$ to $s$,
$Y_{\rm EQ}$ is the yield in the thermal equilibrium,
and $\langle \sigma v \rangle$ is the thermal averaged product of the RHN DM
annihilation cross section $\sigma$ and relative velocity $v$.
Explicit formulas of these are summarized as follows:
\begin{eqnarray}
s &=& \frac{2 \pi^2}{45} g_* \frac{m_{\rm DM}^3}{x^3}, \nonumber \\
H(m_{\rm DM}) &=& \sqrt{\frac{\pi^2}{90} g_*} \frac{m_{\rm DM}^2}{M_{\rm P}}, \nonumber \\
sY_{\rm EQ} &=& \frac{g_{\rm DM}}{2 \pi^2} \frac{m_{\rm DM}^3}{x} K_2(x),
\end{eqnarray}
where
$g_{\rm DM}=2$ is the number of degrees of the freedom for the RHN DM,
$g_*$ is the effective total number of degrees of freedom for particles in thermal equilibrium
(in this paper, we set $g_*=106.75$ for the SM particles),
and $K_2$ is the modified Bessel function of the second kind.
The thermally-averaged annihilation cross section times velocity is given by
\begin{eqnarray}
\langle \sigma v \rangle
&=& (sY_{\rm EQ})^{-2} g_{\rm DM}^2 \frac{m_{\rm DM}}{64 \pi^4 x}
\int_{4m_{\rm DM}^2}^{\infty} ds \hat{\sigma}(s) \sqrt{s}
K_1\left( \frac{x \sqrt{s}}{m_{\rm DM}} \right),
\end{eqnarray}
where the reduced cross section is defined as $\hat{\sigma}(s) = 2 (s-4m_{\rm DM}^2) \sigma(s)$
with the total cross section $\sigma(s)$,
$K_1$ is the modified Bessel function of the first kind.
The total cross section of the RHN DM annihilation process
$\nu_R^3 \nu_R^3 \to Z^\prime \to f \bar{f}$
($f$ denotes the SM fermion)\footnote{
Although there are also other annihilation processes,
such as $\nu_R^3 \nu_R^3 \to \phi \phi$,
$\nu_R^3 \nu_R^3 \to \phi Z^\prime$
and $\nu_R^3 \nu_R^3 \to Z^\prime Z^\prime$
(see, for example, Ref.~\cite{Bell:2016uhg}),
all these cross sections are estimated to be much less than 1 pb,
which is a typical cross section to reproduce
$\Omega_{\rm DM} h^2 \simeq 0.1$,
for $\alpha_{g_X} \sim 0.01$
(see Figs.~\ref{Fig:mZp_alpha} and \ref{Fig:xH_alpha}),
$y_{\rm DM} \sim g_X$,
and $m_{\rm DM} \sim 1$ TeV.
}
is calculated as
\begin{eqnarray}
\sigma(s) &=& \frac{\pi}{3} \alpha_{g_X}^2
\frac{\sqrt{s (s-4m_{\rm DM}^2)}}{(s-m_{Z^\prime}^2)^2+m_{Z^\prime}^2 \Gamma_{Z^\prime}^2} \nonumber \\
&\times& \left[ \frac{103x_H^2 + 86x_H + 37}{3}
+ \frac{17x_H^2 + 10x_H + 2 + (7x_H^2 + 20x_H + 4)\frac{m_t^2}{s}}{3}
\sqrt{1-\frac{4m_t^2}{s}} \right. \nonumber \\
&& \hspace{20mm}
\left. + 18 x_H^2 \frac{(s-m_{Z^\prime}^2)^2}{s (s-4m_{\rm DM}^2)}
\frac{m_{\rm DM}^2 m_t^2}{m_{Z^\prime}^4}
\sqrt{1-\frac{4m_t^2}{s}} \right],
\end{eqnarray}
where the total decay width of $Z^\prime$ boson is given by
\begin{eqnarray}
\Gamma_{Z^\prime} &=&
\frac{\alpha_{g_X} m_{Z^\prime}}{6} \!\!
\left[ \frac{103x_H^2 + 86x_H + 37}{3}
+ \frac{17x_H^2 + 10x_H + 2 + (7x_H^2 + 20x_H + 4)\frac{m_t^2}{m_{Z^\prime}^2}}{3}
\sqrt{1-\frac{4m_t^2}{m_{Z^\prime}^2}} \right. \nonumber \\
&& \hspace{20mm}
\left. + 2 \left( 1 - \frac{4m_N^2}{m_{Z^\prime}^2} \right)^{\frac{3}{2}}
\theta \left( \frac{m_{Z^\prime}^2}{m_N^2} -4\right)
+ \left( 1 - \frac{4m_{\rm DM}^2}{m_{Z^\prime}^2} \right)^{\frac{3}{2}}
\theta \left( \frac{m_{Z^\prime}^2}{m_{\rm DM}^2} -4\right) \right].
\label{DecayWidthZp}
\end{eqnarray}
Here, we have neglected all SM fermion masses except for the top quark mass $m_t$.
By solving the Boltzmann Eq.~(\ref{Eq:Boltzmann}) numerically,
we find the asymptotic value of the yield $Y(\infty)$,
and the present DM relic density is given by
\begin{eqnarray}
\Omega_{\rm DM} h^2 &=& \frac{m_{\rm DM} s_0 Y(\infty)}{\rho_c / h^2},
\end{eqnarray}
where $s_0=2890$ cm$^{-3}$ is the entropy density of the present Universe,
and $\rho_c/h^2=1.05 \times 10^{-5}$ GeV/cm$^3$ is the critical density.
Our analysis involves four parameters,
namely $\alpha_{g_X}$, $m_{Z^\prime}$, $m_{\rm DM}$ and $x_H$.
For $m_{Z^\prime}=4$ TeV and $x_H=-0.575$,
we show in Fig.~\ref{Fig:Omega_h2} the resultant RHN DM relic density
as a function of the RHN DM mass $m_{\rm DM}$,
along with the range of the observed DM relic density,
$0.1183 \leq \Omega_{\rm DM} h^2 \leq 0.1213$~\cite{Aghanim:2015xee}
(two horizontal dashed lines).
The solid lines from top to bottom show the resultant RHN DM relic densities
for various values of the gauge coupling, $\alpha_{g_X}=0.002$, $0.00235$, $0.003$, $0.004$ and $0.005$.
The plots indicate the lower bound on $\alpha_{g_X} \geq 0.00235$
for $m_{Z^\prime}=4$ TeV and $x_H=-0.575$
in order to reproduce the observed relic density.
In addition, we can see that the enhancement of the RHN DM annihilation cross section
via the $Z^\prime$ boson resonance is necessary to satisfy the cosmological constraint
and hence, $m_{\rm DM} \simeq m_{Z^\prime}/2$.
\section{Collider constraints on the U(1)$^\prime$ $Z^\prime$ boson}
\label{Sec_collider}
\begin{figure}[t]
\begin{minipage}{0.5\linewidth}
\begin{center}
\includegraphics[width=0.95\linewidth]{Fig3a.eps}
\subcaption{}\label{Fig:LHC2016_ATLAS}
\end{center}
\end{minipage}
\begin{minipage}{0.5\linewidth}
\begin{center}
\includegraphics[width=0.95\linewidth]{Fig3b.eps}
\subcaption{}\label{Fig:LHC2016_CMS}
\end{center}
\end{minipage}
\caption
{
\subref{Fig:LHC2016_ATLAS} The cross section as a function of the $Z^\prime_{SSM}$ mass (solid line)
with $k=1.16$, along with the LHC Run-2 ATLAS result
from the combined dielectron and dimuon channels in Ref.~\cite{ATLAS:2016cyf}.
(Here we have also shown the ALTAS 2015 result \cite{TheATLAScollaboration:2015jgi}
for comparison.)
\subref{Fig:LHC2016_CMS} The cross section ratio as a function of the $Z^\prime_{SSM}$ mass (solid line)
with $k=1.42$, along with the LHC Run-2 CMS result
from the combined dielectron and dimuon channels in Ref.~\cite{CMS:2016abv}.
(Here we have also shown the CMS 2015 result \cite{CMS:2015nhc} for comparison.)
}
\label{Fig:LHC2016}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.5\linewidth]{Fig4.eps}
\caption
{
The lower bound on $m_{Z^\prime}/g_X$ as a function of $x_H$,
obtained by the limits from the final LEP 2 data~\cite{Schael:2013ita}
at 95\% confidence level.
}
\label{Fig:LEP}
\end{center}
\end{figure}
The ATLAS and the CMS Collaborations have searched for $Z^\prime$ boson resonance
at the LHC Run-1 with $\sqrt{s}=8$ TeV,
and continued the search at the LHC Run-2 with $\sqrt{s}=13$ TeV.
The most stringent bounds on the $Z^\prime$ boson production cross section times branching ratio
have been obtained by using the dilepton final state.
For the so-called sequential SM $Z^\prime$ ($Z^\prime_{SSM}$) model \cite{Barger:1980ti},
where the $Z^\prime_{SSM}$ boson has exactly
the same couplings with the SM fermions as those of the SM $Z$ boson,
the latest cross section bounds from the LHC Run-2 results lead to lower bounds
on the $Z^\prime_{SSM}$ boson mass
as $m_{Z^\prime_{SSM}} \geq 4.05$ TeV in the ATLAS 2016 results \cite{ATLAS:2016cyf} and
$m_{Z^\prime_{SSM}} \geq 4.0$ TeV in the CMS 2016 results \cite{CMS:2016abv}, respectively.
We interpret these ATLAS and CMS results into the U(1)$^\prime$ $Z^\prime$ boson case
and derive constraints on $x_H$, $\alpha_{g_X}$ and $m_{Z^\prime}$.
We calculate the dilepton production cross section
for the process $pp \to Z^\prime +X \to \ell^{+} \ell^{-} +X$.
The differential cross section with respect to the invariant mass $M_{\ell \ell}$ of the final state dilepton
is described as
\begin{eqnarray}
\frac{d \sigma}{d M_{\ell \ell}}
= \sum_{a,b}
\int^{1}_{\frac{M^2_{\ell \ell}}{E^2_{\rm CM}}} d x_1 \frac{2M_{\ell \ell}}{x_1 E^2_{\rm CM}}
f_{a}(x_{1}, M^2_{\ell \ell}) f_{b}\left(\frac{M^2_{\ell \ell}}{x_{1} E^2_{\rm CM}}, M^2_{\ell \ell} \right)
\hat{\sigma} (\bar{q} q \to Z^\prime \to \ell^+ \ell^-),
\label{CrossLHC}
\end{eqnarray}
where $f_a$ is the parton distribution function for a parton $a$,
and $E_{\rm CM} =13$ TeV is the center-of-mass energy of the LHC Run-2.
In our numerical analysis, we employ CTEQ5M~\cite{Pumplin:2002vw} for the parton distribution functions.
In the case of the U(1)$^\prime$ model,
the cross sections for the colliding partons are given by
\begin{eqnarray}
\hat{\sigma} (\bar{u} u \rightarrow Z^\prime \to \ell^+ \ell^-)
&=& \frac{\pi \alpha_{g_X}^2}{81}
\frac{M_{\ell \ell}^2}{(M_{\ell \ell}^2-m_{Z^\prime}^2)^2 + m_{Z^\prime}^2 \Gamma_{Z^\prime}^2}
(85x_H^4 + 152x_H^3 + 104x_H^2 + 32x_H + 4),
\nonumber\\
\hat{\sigma} (\bar{d} d \rightarrow Z^\prime \to \ell^+ \ell^-)
&=& \frac{\pi \alpha_{g_X}^2}{81}
\frac{M_{\ell \ell}^2}{(M_{\ell \ell}^2-m_{Z^\prime}^2)^2 + m_{Z^\prime}^2 \Gamma_{Z^\prime}^2}
(25x_H^4 + 20x_H^3 + 8x_H^2 + 8x_H + 4),
\label{CrossLHC2}
\end{eqnarray}
where the total decay width of the $Z^\prime$ boson is given in Eq.~(\ref{DecayWidthZp}).
By integrating the differential cross section over a range of $M_{\ell \ell}$ set by the ATLAS
and CMS analyses, respectively, we obtain the cross section
as a function of $x_H$, $\alpha_{g_X}$ and $m_{Z^\prime}$,
which are compared with the lower bounds obtained by the ATLAS and CMS Collaborations.
In interpreting the ATLAS and the CMS results for the U(1)$^\prime$ $Z^\prime$ boson,
we follow the strategy in \cite{Okada:2016gsh}.
We first analyze the sequential SM $Z^\prime$ model to check the consistency of our analysis
with the one by the ATLAS and the CMS Collaborations.
With the same couplings as the SM, we calculate the differential cross section of the process
$pp \to Z^\prime_{SSM}+X \to \ell^+ \ell^- +X$ like Eq.~(\ref{CrossLHC}).
According to the analysis by the ATLAS Collaboration at the LHC Run-2,
we integrate the differential cross section for the range of
120 GeV$\leq M_{\ell \ell} \leq 6000$ GeV \cite{ATLAS:2016cyf}
and obtain the cross section of the dilepton production process
as a function of the $Z^\prime_{SSM}$ boson mass.
Our result is shown as a solid line in Fig.~\ref{Fig:LHC2016}\subref{Fig:LHC2016_ATLAS},
along with the plots presented by the ATLAS Collaboration \cite{ATLAS:2016cyf}
(Here we have also shown the ALTAS 2015 result \cite{TheATLAScollaboration:2015jgi} for comparison.
We can see that the ATLAS 2016 result has dramatically improved
the bound obtained by the ATLAS 2015 result.).
In Fig.~\ref{Fig:LHC2016}\subref{Fig:LHC2016_ATLAS},
the experimental upper bounds on the $Z^\prime$ boson production
cross section are depicted as the horizontal solid (red) curves.
The theoretical $Z^\prime$ boson production cross section presented in \cite{ATLAS:2016cyf}
is shown as the diagonal dashed line,
and the lower limit of the $Z^\prime_{SSM}$ boson mass is found to be $4.05$ TeV,
which can be read off from the intersection point of the theoretical prediction (diagonal dashed line)
and the experimental cross section bound (horizontal lower solid (red) curve).
In order to take into account the difference of the parton distribution functions
used in the ATLAS analysis and our analysis, and QCD corrections of the process,
we have scaled our resultant cross section by a factor $k=1.16$
in Fig.~\ref{Fig:LHC2016}\subref{Fig:LHC2016_ATLAS},
with which we can obtain the same lower limit of the $Z^\prime_{SSM}$ boson mass
as $4.05$ TeV.
We can see that our result (solid line) in Fig.~\ref{Fig:LHC2016}\subref{Fig:LHC2016_ATLAS}
with the factor of $k=1.16$
is very consistent with the theoretical prediction (diagonal dashed line) presented
by the ATLAS Collaboration.
We use this factor in the following analysis for the U(1)$^\prime$ $Z^\prime$ production process,
when we interpret the ATLAS 2016 result.
We apply the same strategy and compare our results for the $Z^\prime_{SSM}$ model
with those in the CMS 2016 results~\cite{CMS:2016abv}.
According to the analysis by the CMS Collaboration,
we integrate the differential cross section for the range of
$0.95 \; m_{Z^\prime_{SSM}} \leq M_{\ell \ell} \leq 1.05 \; m_{Z^\prime_{SSM}}$ \cite{CMS:2016abv}
and obtain the cross section.
In the CMS analysis, the limits are set on the ratio of the $Z^\prime_{SSM}$ boson cross section
to the $Z/\gamma^*$ cross section:
\begin{eqnarray}
R_{\sigma} &=&
\frac{\sigma (pp \to Z^\prime+X \to \ell \ell +X)}
{\sigma (pp \to Z+X \to \ell \ell +X)},
\end{eqnarray}
where the $Z/\gamma^*$ production cross sections
in the mass window of $60$ GeV$\leq M_{\ell \ell} \leq 120$ GeV
are predicted to be $1928$ pb at the LHC Run-2 \cite{CMS:2016abv}.
Our result for the $Z^\prime_{SSM}$ model is shown as the solid line
in Fig.~\ref{Fig:LHC2016}\subref{Fig:LHC2016_CMS},
along with the plot presented in \cite{CMS:2016abv}
(Here we have also shown the CMS 2015 result \cite{CMS:2015nhc} for comparison.
We can see that the CMS 2016 result has dramatically improved
the bound obtained by the CMS 2015 result.).
The analyses in this CMS paper leads to the lower limits of the $Z^\prime_{SSM}$ boson mass
as $4.0$ TeV,
which is read off from the intersection point of the theoretical prediction (diagonal dashed line)
and the experimental cross section bound (horizontal lower solid (red) curve).
In order to obtain the same lower mass limits,
we have scaled our resultant cross section by a factor $k=1.42$
in Fig.~\ref{Fig:LHC2016}\subref{Fig:LHC2016_CMS}.
With this $k$ factor, our result (solid line) is very consistent
with the theoretical prediction (diagonal dashed line) presented
in Ref.~\cite{CMS:2016abv}.
We use this $k$ factor in our analysis to interpret the CMS result
for the U(1)$^\prime$ $Z^\prime$ boson case.
The search for effective 4-Fermi interactions mediated by the $Z^\prime$ boson at the LEP
leads to a lower bound on $m_{Z^\prime}/g_X$~\cite{LEP:2003aa,Carena:2004xs,Schael:2013ita}.
Employing the limits from the final LEP 2 data~\cite{Schael:2013ita} at 95\% confidence level,
we follow Ref.~\cite{Carena:2004xs} and derive a lower bound on $m_{Z^\prime}/g_X$
as a function $x_H$.
Our result is shown in Fig.~\ref{Fig:LEP}.
\section{Combined results}
\label{Sec_allowed_region}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.5\linewidth]{Fig5.eps}
\caption
{
The allowed regions to solve the electroweak instability problem
for $m_{Z^\prime}$ and $\alpha_{g_{X}}$ with a fixed $x_H=-0.575$
at the TeV scale,
along with the dark matter lower bound ((blue) right solid line) on $\alpha_{g_{X}}$,
the LHC Run-2 (2016) CMS upper bound ((red) solid line) on $\alpha_{g_{X}}$
and the LHC Run-2 ATLAS (2016) upper bound ((red) dashed line) on $\alpha_{g_{X}}$
from direct search for $Z^\prime$ boson resonance.
The (green) shaded region in between two solid lines satisfies all constraints.
Here, the naturalness bounds for 10\% (right dotted line) and 30\% (left dotted line) fine-tuning levels
are also depicted.
}
\label{Fig:mZp_alpha}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{minipage}{0.5\linewidth}
\begin{center}
\includegraphics[width=0.95\linewidth]{Fig6a.eps}
\subcaption{}\label{Fig:xH_alpha_mzp=4000}
\vspace{5mm}
\end{center}
\end{minipage}
\begin{minipage}{0.5\linewidth}
\begin{center}
\includegraphics[width=0.95\linewidth]{Fig6b.eps}
\subcaption{}\label{Fig:xH_alpha_mzp=3750}
\vspace{5mm}
\end{center}
\end{minipage}
\begin{minipage}{0.5\linewidth}
\begin{center}
\includegraphics[width=0.95\linewidth]{Fig6c.eps}
\subcaption{}\label{Fig:xH_alpha_mzp=3500}
\end{center}
\end{minipage}
\begin{minipage}{0.5\linewidth}
\begin{center}
\includegraphics[width=0.95\linewidth]{Fig6d.eps}
\subcaption{}\label{Fig:xH_alpha_mzp=3000}
\end{center}
\end{minipage}
\caption{
Allowed parameter regions in the ($x_H$, $\alpha_{g_X}$)-plain
for various $m_{Z^\prime}$ values.
\subref{Fig:xH_alpha_mzp=4000} is for $m_{Z^\prime}=4$ TeV.
The shaded region indicates the parameter space for solving the electroweak vacuum instability.
The (blue) convex-downward solid line shows the cosmological lower bound on $\alpha_{g_X}$ as a function of $x_H$.
The (red) convex-upward solid (dashed) line shows the upper bound on $\alpha_{g_X}$
obtained from the $Z^\prime$ boson search
by the CMS~\cite{CMS:2016abv} (ATLAS~\cite{ATLAS:2016cyf}) Collaboration,
and the (red) dashed-dotted lines show the LEP bounds.
The (green) shaded region in between two solid lines satisfies all constraints.
Here, the naturalness bounds for 10\% (dashed line) and 30\% (dotted line) fine-tuning levels
are also depicted.
\subref{Fig:xH_alpha_mzp=3750}, \subref{Fig:xH_alpha_mzp=3500}
and \subref{Fig:xH_alpha_mzp=3000} are the same as \subref{Fig:xH_alpha_mzp=4000},
but $m_{Z^\prime}=3.75$ TeV, 3.5 TeV and 3 TeV, respectively.
}
\label{Fig:xH_alpha}
\end{figure}
Now let us combine all the constraints that we have obtained in the previous sections
from the RHN DM physics, collider phenomenology, and the electroweak vacuum stability.
In Fig.~\ref{Fig:mZp_alpha}, we show the allowed region in the ($m_{Z^\prime}$, $\alpha_{g_X}$)-plain
for fixed $x_H=-0.575$, as an example.
The shaded region indicates the parameter space for solving the electroweak vacuum instability.
The (blue) right solid line shows the lower bound on $\alpha_{g_X}$ as a function of $m_{Z^\prime}$
to reproduce the observed DM relic density of the Planck result~\cite{Aghanim:2015xee}.
The (red) left solid (dashed) line shows the upper bound on $\alpha_{g_X}$
obtained from the search results for $Z^\prime$ boson resonance
by the CMS~\cite{CMS:2016abv} (ATLAS~\cite{ATLAS:2016cyf}) Collaboration.
The (green) shaded region in between two solid lines satisfies all constraints.
These three constraints are complementary to narrow down the allowed region
to be 4 TeV $\lesssim m_{Z^\prime} \lesssim 8$ TeV and
$0.009 \lesssim \alpha_{g_X} \lesssim 0.017$.
We also show the naturalness bounds for 10\% (right dotted line)
and 30\% (left dotted line) fine-tuning levels.
In Fig.~\ref{Fig:xH_alpha}, we show allowed parameter regions in the ($x_H$, $\alpha_{g_X}$)-plain
for various $m_{Z^\prime}$ values.
Fig.~\ref{Fig:xH_alpha}\subref{Fig:xH_alpha_mzp=4000} is for $m_{Z^\prime}=4$ TeV.
The shaded region indicates the parameter space for solving the electroweak vacuum instability.
The (blue) convex-downward solid line shows the lower bound on $\alpha_{g_X}$ as a function of $x_H$
to reproduce the observed DM relic density.
The (red) convex-upward solid (dashed) line shows the upper bound on $\alpha_{g_X}$
obtained from the search results for $Z^\prime$ boson resonance
by the CMS~\cite{CMS:2016abv} (ATLAS~\cite{ATLAS:2016cyf}) Collaboration,
and the (red) dashed-dotted lines also show the LEP bounds.
The (green) shaded region in between two solid lines satisfies all constraints.
These three constraints are complementary to narrow down the allowed region
to be $-1.1 \lesssim x_H \lesssim -0.4$ and
$0.002 \lesssim \alpha_{g_X} \lesssim 0.02$.
We also show the naturalness bounds for 10\% (dashed line) and 30\% (dotted line) fine-tuning levels.
Figs.~\ref{Fig:xH_alpha}\subref{Fig:xH_alpha_mzp=3750},
\ref{Fig:xH_alpha}\subref{Fig:xH_alpha_mzp=3500}
and \ref{Fig:xH_alpha}\subref{Fig:xH_alpha_mzp=3000}
are the same as Fig.~\ref{Fig:xH_alpha}\subref{Fig:xH_alpha_mzp=4000},
but $m_{Z^\prime}=3.75$ TeV, 3.5 TeV and 3 TeV, respectively.
From Fig.~\ref{Fig:xH_alpha}\subref{Fig:xH_alpha_mzp=3750},
the allowed region to satisfy these three constraints indicates
$-0.9 \lesssim x_H \lesssim -0.5$ and $0.003 \lesssim \alpha_{g_X} \lesssim 0.015$
for fixed $m_{Z^\prime}=3.75$ TeV.
As $m_{Z^\prime}$ decreases, the LHC upper bound lines are shifted downward,
while the DM lower bound line remains almost the same (it slightly moves to downward).
Therefore, the allowed region between the LHC upper bounds and the DM lower bound narrows.
On the other hand, the shaded region remains almost the same,
so that the (green) shaded region disappears for $m_{Z^\prime} \lesssim 3.5$ TeV.
\section{Direct detection of RHN DM}
\label{Sec_direct_detection}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.5\linewidth]{Fig7.eps}
\caption
{
For a fixed $x_H=-0.575$,
the resultant spin-independent cross section $\sigma_{\rm SI}$
as a function of $m_{Z^\prime}$.
Here, for a fixed $m_{Z^\prime}$ value,
$\alpha_{g_X}$ is taken from the shaded region in Fig.~\ref{Fig:mZp_alpha}
to solve the electroweak vacuum instability problem.
The (green) shaded region in between around 3.5 TeV and 9 TeV
corresponds to the (green) shaded parameter region in Fig.~\ref{Fig:mZp_alpha},
which satisfies all three constraints,
the electroweak vacuum stability condition, the LHC Run-2 bound,
and the cosmological constraint from the observed RHN DM relic density.
The (red) upper solid (dashed) line shows the XENON1T~\cite{Aprile:2017iyp}
(LUX 2016~\cite{Akerib:2016vxi}) upper bound on $\sigma_{\rm SI}$
as a function of $m_{Z^\prime} \simeq 2 m_{\rm DM}$,
and the (red) dotted line shows the prospective reach for the upper bound on $\sigma_{\rm SI}$
in the next-generation successor of the LUX experiment,
the LUX-ZEPLIN (LZ) DM experiment~\cite{Szydagis:2016few}.
}
\label{Fig:mZp_sigma}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{minipage}{0.5\linewidth}
\begin{center}
\includegraphics[width=0.95\linewidth]{Fig8a.eps}
\subcaption{}\label{Fig:xH_sigma_mzp=4000}
\vspace{5mm}
\end{center}
\end{minipage}
\begin{minipage}{0.5\linewidth}
\begin{center}
\includegraphics[width=0.95\linewidth]{Fig8b.eps}
\subcaption{}\label{Fig:xH_sigma_mzp=3750}
\vspace{5mm}
\end{center}
\end{minipage}
\begin{minipage}{0.5\linewidth}
\begin{center}
\includegraphics[width=0.95\linewidth]{Fig8c.eps}
\subcaption{}\label{Fig:xH_sigma_mzp=3500}
\end{center}
\end{minipage}
\begin{minipage}{0.5\linewidth}
\begin{center}
\includegraphics[width=0.95\linewidth]{Fig8d.eps}
\subcaption{}\label{Fig:xH_sigma_mzp=3000}
\end{center}
\end{minipage}
\caption{
The resultant $\sigma_{\rm SI}$ in the ($x_H$, $\sigma_{\rm SI}$)-plain
for various $m_{Z^\prime}$ values,
corresponding to the parameter regions shown in Fig.~\ref{Fig:xH_alpha}.
\subref{Fig:xH_sigma_mzp=4000} shows our results for $m_{Z^\prime}=4$ TeV.
The shaded regions indicate the parameter space for solving the electroweak vacuum instability.
The (green) shaded region in the range of $-1.1 \lesssim x_H \lesssim -0.4$
corresponds to the (green) shaded region
in Fig.~\ref{Fig:xH_alpha}\subref{Fig:xH_alpha_mzp=4000},
which satisfies all three constraints,
the electroweak vacuum stability condition, the LHC Run-2 bound,
and the cosmological constraint from the observed RHN DM relic density.
The (red) upper solid (dashed) line shows the XENON1T~\cite{Aprile:2017iyp}
(LUX 2016~\cite{Akerib:2016vxi}) upper bound on $\sigma_{\rm SI}$,
and the (red) dotted line shows the prospective reach for the upper bound on $\sigma_{\rm SI}$
in the LZ DM experiment~\cite{Szydagis:2016few}.
Figs.~\subref{Fig:xH_sigma_mzp=3750}, \subref{Fig:xH_sigma_mzp=3500}
and \subref{Fig:xH_sigma_mzp=3000}
are the same as \subref{Fig:xH_sigma_mzp=4000},
but for $m_{Z^\prime}=3.75$ TeV, 3.5 TeV and 3 TeV
corresponding to Fig.~\ref{Fig:xH_alpha}\subref{Fig:xH_alpha_mzp=3750},
\ref{Fig:xH_alpha}\subref{Fig:xH_alpha_mzp=3500}
and \ref{Fig:xH_alpha}\subref{Fig:xH_alpha_mzp=3000}, respectively.
}
\label{Fig:xH_sigma}
\end{figure}
A variety of experiments are underway and also planned for directly detecting
a DM particle through its elastic scattering off with nuclei.\footnote{
We can also consider an indirect detection
of the RHN DM through cosmic rays from
a pair annihilation of the RHN DMs.
However, using the parameters in the allowed regions
shown in Sec.~\ref{Sec_allowed_region},
we have found that the pair annihilation cross section
is much smaller than the current upper bounds
obtained from, for example,
the Fermi-LAT experiments \cite{Charles:2016pgz}.
}
In this section, we calculate the spin-independent elastic scattering cross section
of the RHN DM particle via the Higgs bosons exchange,\footnote{
There is another process for the RHN DM to scatter off with nuclei
via $Z^\prime$-boson exchange.
Since the RHN DM is a Majorana particle,
only its interaction with nuclei is spin-dependent.
We have calculated this spin-dependent cross section
to be $\sigma_{\rm SD} \sim 10^{-9}$ pb,
which is far below the current upper bounds,
$\sigma_{\rm SD} \lesssim 10^{-4}$ pb
obtained from the LUX~\cite{daSilva:2017swg}
and the IceCube~\cite{Aartsen:2016zhm} experiments.
}
and compare our results with the current experimental results
and a prospective reach by future experiments.
From Eq.~(\ref{Eq:mass_Zp_DM}),
the U(1)$^\prime$ Higgs VEV $v_\phi$ is expressed
as a function of $m_{Z^\prime}$, $ \alpha_{g_{X}}$ and $x_H$:
\begin{eqnarray}
v_\phi^2 &=& \frac{m_{Z^\prime}^2}{16 \pi \alpha_{g_{X}}}
\left[1 - 4 \pi \alpha_{g_{X}} \left( \frac{x_H v_h}{m_{Z^\prime}} \right)^2 \right]
\; \simeq \; \frac{m_{Z^\prime}^2}{16 \pi \alpha_{g_{X}}}.
\label{Eq:v_phi2}
\end{eqnarray}
In Sec.~\ref{Sec_relic_density}, we have also shown that $m_{\rm DM} \simeq m_{Z^\prime}/2$
to satisfy the experimental relic density of the $Z^\prime$-portal RHN DM,
which means $y_{\rm DM} \simeq m_{Z^\prime}/2\sqrt{2} v_\phi \simeq \sqrt{2 \pi \alpha_{g_X}}$,
and then Eq.~(\ref{Eq:mass_phi}) is approximately expressed as
\begin{eqnarray}
m_\phi^2 &\simeq& \frac{1}{8 \pi^2} \frac{23}{8} \frac{m_{Z^\prime}^4}{v_\phi^2}
\; \simeq \; \frac{23}{4 \pi} \alpha_{g_X} m_{Z^\prime}^2.
\label{Eq:m_phi2_sim}
\end{eqnarray}
Using the SM Higgs boson mass in Eq.~(\ref{Eq:mass_h}),
the scalar mass matrix is found to be
\begin{eqnarray}
{\cal M} &=&
\left(
\begin{array}{cc}
m_h^2 & -m_h^2 \left( \frac{v_h}{v_\phi} \right) \\
-m_h^2 \left( \frac{v_h}{v_\phi} \right) & m_\phi^2
\end{array}\right).
\end{eqnarray}
The mass eigenstates $h^\prime$ and $\phi^\prime$ are defined as
\begin{eqnarray}
\left( \begin{array}{c} h^\prime \\ \phi^\prime \end{array} \right)
&=& \left(
\begin{array}{cc}
\cos \theta & - \sin \theta \\
\sin \theta & \cos \theta
\end{array}\right)
\left( \begin{array}{c} h \\ \phi \end{array} \right),
\end{eqnarray}
with the mixing angle $\theta$ given by
\begin{eqnarray}
\tan 2 \theta &=&
\frac{2m_h^2(v_h/v_\phi)}{m_h^2-m_\phi^2},
\label{Eq:tan_2theta}
\end{eqnarray}
and their mass eigenvalues are given by
\begin{eqnarray}
m_{h^\prime}^2
&=& m_h^2 \cos^2 \theta + m_\phi^2 \sin^2 \theta
+ 2m_h^2 \frac{v_h}{v_\phi} \sin \theta \cos \theta
\; \simeq \; m_h^2, \nonumber \\
m_{\phi^\prime}^2
&=& m_h^2 \sin^2 \theta + m_\phi^2 \cos^2 \theta
- 2m_h^2 \frac{v_h}{v_\phi} \sin \theta \cos \theta
\; \simeq \; m_\phi^2.
\label{Eq:m_scalar2}
\end{eqnarray}
Here, we have used the fact that except for the special case,
$m_h^2 \simeq m_\phi^2$,
the mixing angle is always small because of the suppression
by $v_h/v_\phi$ with $v_h=246$ GeV and $v_\phi \gtrsim 10$ TeV.
Thus, the mass eigenstate $h^\prime$ is the SM-like Higgs boson,
while $\phi^\prime$ is the U(1)$^\prime$-like Higgs boson.
The spin-independent elastic scattering cross section with nucleon is given by
\begin{eqnarray}
\sigma_{\rm SI}
&=& \frac{1}{\pi} \left( \sqrt{2} y_{\rm DM} \sin \theta \cos \theta \right)^2
\left( \frac{\mu_{\rm DM,N}}{v_h} \right)^2 f_N^2
\left( \frac{1}{m_{h^\prime}^2} - \frac{1}{m_{\phi^\prime}^2} \right)^2 \nonumber \\
&\simeq& 4 \theta^2 \alpha_{g_X}
\left( \frac{\mu_{\rm DM,N}}{v_h} \right)^2 f_N^2
\left( \frac{1}{m_h^2} - \frac{1}{m_\phi^2} \right)^2 ,
\label{DD}
\end{eqnarray}
where $\mu_{\rm DM,N} = m_N m_{{\rm DM}}/(m_N+m_{{\rm DM}})$ is
the reduced mass of the RHN DM-nucleon system with the nucleon mass $m_N=0.939$ GeV, and
\begin{eqnarray}
f_N &=& \left( \sum_{q=u,d,s} f_{T_q} + \frac{2}{9}f_{TG} \right) m_N
\end{eqnarray}
is the nuclear matrix element accounting for the quark and gluon contents of the nucleon.
In evaluating $f_{T_q}$, we use the results from the lattice QCD simulation \cite{Ohki:2008ff}:
$f_{T_u} +f_{T_d} \simeq 0.056$ and $|f_{T_s}|\leq 0.08$.
For conservative analysis, we take $f_{T_s}=0$ in the following.
Using the trace anomaly formula, $\sum_{q=u,d,s} f_{T_q} + f_{TG}=1$
\cite{Crewther:1972kn,Chanowitz:1972vd,Chanowitz:1972da,Collins:1976yq,Shifman:1978zn},
we obtain $f_N^2 \simeq 0.0706 \; m_N^2$.
Using Eqs.~(\ref{Eq:v_phi2}), (\ref{Eq:m_phi2_sim}) and (\ref{Eq:tan_2theta}),
$\sigma_{\rm SI}$ is expressed as a function of only two free parameters:
$\alpha_{g_X}$ and $m_{Z^\prime}$.
For a fixed $x_H=-0.575$,
the resultant spin-independent cross section $\sigma_{\rm SI}$
as a function of $m_{Z^\prime}$
is depicted in Fig.~\ref{Fig:mZp_sigma}.
Here, for a fixed $m_{Z^\prime}$ value,
$\alpha_{g_X}$ is taken from the shaded region in Fig.~\ref{Fig:mZp_alpha}
to solve the electroweak vacuum instability problem.
The (green) shaded region in between around 3.5 TeV and 9 TeV
corresponds to the (green) shaded parameter region in Fig.~\ref{Fig:mZp_alpha},
which satisfies all three constraints,
the electroweak vacuum stability condition, the LHC Run-2 bound,
and the cosmological constraint from the observed RHN DM relic density.
The (red) upper solid (dashed) line shows the XENON1T~\cite{Aprile:2017iyp}
(LUX 2016~\cite{Akerib:2016vxi}) upper bound on $\sigma_{\rm SI}$
as a function of $m_{Z^\prime} \simeq 2 m_{\rm DM}$,
and the (red) dotted line shows the prospective reach for the upper bound on $\sigma_{\rm SI}$
in the next-generation successor of the LUX experiment,
the LUX-ZEPLIN (LZ) DM experiment~\cite{Szydagis:2016few}.
Our resultant spin-independent cross section appears below the future reach.
In Fig.~\ref{Fig:xH_sigma}, we show the resultant $\sigma_{\rm SI}$
in the ($x_H$, $\sigma_{\rm SI}$)-plain for various $m_{Z^\prime}$ values,
corresponding to the parameter regions shown in Fig.~\ref{Fig:xH_alpha}.
Fig.~\ref{Fig:xH_sigma}\subref{Fig:xH_sigma_mzp=4000} shows our results for $m_{Z^\prime}=4$ TeV.
The shaded regions indicate the parameter space for solving the electroweak vacuum instability.
The (green) shaded region in the range of $-1.1 \lesssim x_H \lesssim -0.4$
corresponds to the (green) shaded region
in Fig.~\ref{Fig:xH_alpha}\subref{Fig:xH_alpha_mzp=4000},
which satisfies all three constraints,
the electroweak vacuum stability condition, the LHC Run-2 bound,
and the cosmological constraint from the observed RHN DM relic density.
The (red) upper solid (dashed) line shows the XENON1T~\cite{Aprile:2017iyp}
(LUX 2016~\cite{Akerib:2016vxi}) upper bound on $\sigma_{\rm SI}$,
and the (red) dotted line shows the prospective reach for the upper bound on $\sigma_{\rm SI}$
in the LZ DM experiment~\cite{Szydagis:2016few}.
Figs.~\ref{Fig:xH_sigma}\subref{Fig:xH_sigma_mzp=3750},
\ref{Fig:xH_sigma}\subref{Fig:xH_sigma_mzp=3500}
and \ref{Fig:xH_sigma}\subref{Fig:xH_sigma_mzp=3000}
are the same as Fig.~\ref{Fig:xH_sigma}\subref{Fig:xH_sigma_mzp=4000},
but for $m_{Z^\prime}=3.75$ TeV, 3.5 TeV and 3 TeV
corresponding to Fig.~\ref{Fig:xH_alpha}\subref{Fig:xH_alpha_mzp=3750},
\ref{Fig:xH_alpha}\subref{Fig:xH_alpha_mzp=3500}
and \ref{Fig:xH_alpha}\subref{Fig:xH_alpha_mzp=3000}, respectively.
Fig.~\ref{Fig:xH_sigma}\subref{Fig:xH_sigma_mzp=3750} has
a (green) shaded region in the range of $-0.9 \lesssim x_H \lesssim -0.5$
to satisfy the three constraints,
while Figs.~\ref{Fig:xH_sigma}\subref{Fig:xH_sigma_mzp=3500}
and \ref{Fig:xH_sigma}\subref{Fig:xH_sigma_mzp=3000}
have no such region.
\section{Conclusions}
\label{Sec_conclusion}
We have considered the DM scenario in the context
of the classically conformal U(1)$^\prime$ extended SM,
with three RHNs and the U(1)$^\prime$ Higgs field.
The model is free from all the U(1)$^\prime$ gauge and gravitational anomalies
in the presence of the three RHNs.
We have introduced a $Z_2$-parity in the model,
under which an odd-parity is assigned to one RHN,
while all the other particles are assigned to be $Z_2$-even.
In our model, the $Z_2$-odd RHN serves as a stable DM candidate,
while the other two RHNs are utilized for the the minimal seesaw mechanism
in order to reproduce the neutrino oscillation data
and the observed baryon asymmetry in the Universe through leptogenesis.
In this model, the U(1)$^\prime$ gauge symmetry is radiatively broken
through the CW mechanism,
by which the electroweak symmetry breaking is triggered.
There are three free parameters in our model,
the U(1)$^\prime$ charge of the SM Higgs doublet ($x_H$),
the new U(1)$^\prime$ gauge coupling ($\alpha_{g_X}$),
and the U(1)$^\prime$ gauge boson ($Z^\prime$) mass ($m_{Z^\prime}$).
In this model context,
we have first investigated a possibility to resolve the electroweak vacuum instability
with the current world average of the experimental data,
$m_t =173.34$ GeV and $m_h=125.09$ GeV.
By analyzing the RG evolutions
of the couplings of the model at the two-loop level,
we have performed a parameter scan
for the three parameters, $m_{Z^\prime}$, $\alpha_{g_X}$ and $x_H$,
and have identified parameter regions which can solve the electroweak instability problem
and keep all coupling values in the perturbative regime up to the Planck mass scale.
We have found that the resultant parameter regions are very severely constrained.
Next, we have calculated the thermal relic density of the RHN DM
and identified the model parameter region to reproduce the observed DM relic density
of the Planck 2015 measurement.
In our model, the RHN DM particles mainly annihilate into the SM particles
through the $s$-channel process mediated by the $Z^\prime$ boson.
We have obtained the lower bound
on $\alpha_{g_X}$ as a function of $m_{Z^\prime}$ and $x_H$ from the observed DM relic density.
We have also considered the LHC Run-2 bounds
from the search for the $Z^\prime$ boson resonance
by the recent ATLAS and CMS analysis,
which lead to the upper bounds on $\alpha_{g_X}$ as a function of $m_{Z^\prime}$ and $x_H$.
The LEP results from the search for effective 4-Fermi interactions mediated by the $Z^\prime$ boson
can also constrain the model parameter space, but the LEP constraints are found to be weaker than
those obtained from the LHC Run-2 results.
Finally, we have combined all the constraints.
The cosmological constraint on the RHN DM yields the lower bound on $\alpha_{g_X}$ as a function of $m_{Z^\prime}$ and $x_H$,
while the upper bound on $\alpha_{g_X}$ is obtained from the LHC Run-2 results,
so that these constraints are complementary to narrow the allowed parameter regions.
We have found that only small portions in these allowed parameter regions
can solve the electroweak vacuum instability problem.
In particular, no allowed region to satisfy all constraints exists for $m_{Z^\prime} \lesssim 3.5$ TeV.
For the obtained allowed regions,
we have calculated the spin-independent cross section of the RHN DM with nucleons.
We have found that the resultant cross section well below the current experimental upper bounds.
\section*{Acknowledgements}
The work of D.-s.T. and S.O. was supported by Advanced Medical Instrumentation unit [Sugawara unit]
and Mathematical and Theoretical Physics unit [Hikami unit], respectively,
of the Okinawa Institute of Science and Technology Graduate University.
The work of N.O. was supported in part by the United States Department of Energy (DE-SC0013680).
\bibliographystyle{utphys}
|
{
"timestamp": "2017-11-13T02:02:06",
"yymm": "1704",
"arxiv_id": "1704.05023",
"language": "en",
"url": "https://arxiv.org/abs/1704.05023"
}
|
\section{Introduction}
Interest in local measurements of non-equilibrium conductors at the nanoscale motivated various kinds of spatially resolved sensors.
The energy averaged approaches characterize the effective local temperature of the electronic system, via the resistive measurements \cite{Wu2013}, the measurements of tiny heat fluxes \cite{Prokudina2014}, \cite{Menges2016} and the nearfield imaging of a terahertz emission \cite{Weng_Arxiv2016}. Raman thermography permits a local evaluation for the lattice temperature \cite{Doerk2010}. Energy resolved electronic measurements are conventionally based on the energy selective tunneling, which involves a superconducting tunnel probe \cite{Pothier1997} or a quantum dot \cite{Altimiras2010}. In both cases, the energy resolution is naturally limited to the excitations below, respectively, the superconducting gap or dot level spacing.
An alternative approach to gain energy selectivity, without an obviously limited excitation energy, was suggested in 1999 by Gramespacher and B\"{u}ttiker \cite{Gramespacher1999}. They derived a relation between the local electronic energy distribution and the shot noise of a tunneling contact, which served as a bias controlled energy selective probe, see \cite{Meair2011,Kühne2015651} for later developments. Recently, a local noise thermometry was demonstrated by means of diffusive semiconductor nanowires with a resistance much higher than the conductor under test \cite{Tikhonov2016.SciRep}. Here, we extend this approach and perform energy resolved local noise measurement in a metallic interconnect.
\section{Sensing electron distribution function}
It has been shown recently, that an InAs nanowire (NW) can be used as a miniature noise probe, capable of non-invasive local shot noise measurements in a non-equilibrium conductor \cite{Tikhonov2016.SciRep}. In a system of interest, one end of such nanowire (a test end) contacts a non-equilibrium conductor, while the other (a cold end) is kept at the base temperature $T_0$, thus having an equilibrium electronic energy distribution (EED) at $T_0$. In the case of elastic diffusion, the spatially dependent EED in the NW can be represented as a linear combination of distributions at its ends: $f(x,\varepsilon)= (1-\frac{x}{L})f_{\mathrm{test}}(\varepsilon) + \frac{x}{L}f_{\mathrm{cold}}(\varepsilon)$, where $x$ is the coordinate along the NW \cite{Nagaev1992}. Thus the spectral density of the NW current fluctuations $S_I=\frac{4k_B}{R}T_S$ depends on the EEDs at both NW ends, since the sensor measures the average noise temperature along the NW: $T_S=\int{T_N(x)\frac{dx}{L}}$, $T_N(x)\equiv\int{f(x,\varepsilon)(1-f(x,\varepsilon))d\varepsilon/k_B}$. For $T_0 \ll T_N(0)$ this results in $T_S=\alpha T_N(0)$, where $\alpha$ slightly depends on the shape of $f_{\mathrm{test}}(\varepsilon)$ \cite{Sukhorukov1999}.
\begin{figure}[t]
\begin{center}
\vspace{0mm}
\includegraphics[width=0.9\columnwidth]{sample.pdf}
\end{center}
\caption{A sketch of the sample, featuring the golden strip (red) and NW (green). A strip has two current leads, one connected to the current source $I_H$, and other connected to the ground. One end of the NW, denoted as a test end is connected to the center of the golden strip, while another, denoted as a cold end is used to drive current $I_\mathrm{NW}$ through the NW and to measure current fluctuations $S_\mathrm{I}$.
}\label{fig_sample}
\end{figure}
\par
Here we consider the case, when the electrochemical potential at the cold end can be modified by applying external voltage bias $V_\mathrm{b}$. In this case, the general equation \cite{Nagaev1992} for current fluctuations $S_I=\frac{4}{RL}\int dx\int d\varepsilon f(x,\varepsilon)(1-f(x,\varepsilon))$ leads to following result:
\begin{equation}
\begin{split}
S_I = \frac{4}{3R} \Big[ &\int d\varepsilon f_{\mathrm{cold}}(\varepsilon)(1-f_{\mathrm{cold}}(\varepsilon)) + \\
&\int d\varepsilon f_{\mathrm{test}}(\varepsilon)(1-f_{\mathrm{test}}(\varepsilon)) + \\
&\frac{1}{2}\int d\varepsilon f_{\mathrm{test}}(\varepsilon)(1-f_{\mathrm{cold}}(\varepsilon)) + \\ &f_{\mathrm{cold}}(\varepsilon)(1-f_{\mathrm{test}}(\varepsilon)) \Big].
\end{split}
\end{equation}
The first two terms represent $T_N(0)$ and $T_N(L)=T_0$ respectively and are independent of $V_\mathrm{b}$. The other terms, however, contain the product $f_{\mathrm{cold}}(\varepsilon)f_{\mathrm{test}}(\varepsilon)$, which enables the energy selectivity of the noise measurement \cite{Gramespacher1999}. Assuming that $f_{cold}(\varepsilon)=(\exp(\frac{\varepsilon - eV_\mathrm{b}}{k_B T_0})+1)^{-1}$, we obtain the derivative $dS_I/dV_\mathrm{b}$, which simplifies in the limit $T_0 \ll T_N(0)$:
\begin{equation}
\begin{split}
\frac{dS_I}{dV_b} = &\frac{2e}{3R}\int d\varepsilon (1-2 f_{\mathrm{test}}(\varepsilon))\frac{df_{\mathrm{cold}}(\varepsilon)}{d\varepsilon} \\
\approx &\frac{2e}{3R} (1-2 f_{\mathrm{test}}(\varepsilon=eV_\mathrm{b})).
\end{split}
\end{equation}
Thus, the measured noise and the EED under test are related as:
\begin{equation}
f_{\mathrm{test}}(\varepsilon=eV_\mathrm{b}) = 1/2 - \frac{3R}{4e}\frac{dS_\mathrm{I}}{dV_\mathrm{b}}
\end{equation}
Here, to prove this concept of the EED measurement, we consider a device (figure \ref{fig_sample}) consisting of a short metal strip with the test end of an InAs NW connected to strip's center. The EED $f_{\mathrm{test}}(\varepsilon)$ in the center of the strip is controlled by the external bias current $I_H$. The opposite, cold end of the NW, which is connected to noise measurement circuit is used to apply the bias $V_b$ (hence, the current $I_{\mathrm{NW}}=V_b/R_{NW}$). Other contacts and side gates were not used in the present experiment. This sample has been previously used in \cite{Tikhonov2016.SciRep}, more details on the fabrication can be found in \cite{Tikhonov2016.SST}.
\begin{figure}[t]
\begin{center}
\vspace{0mm}
\includegraphics[width=0.9\columnwidth]{calibration.pdf}
\end{center}
\caption{Power detector response dependence on total noise circuit resistance. Colors correspond to different temperatures. The detector response changes due to change in thermal noise $4k_BT/R$. The resistance-dependent gain coefficient is determined via calibration procedure (see text).
}\label{fig1}
\end{figure}
\par
\section{Experimental technique}
To measure shot noise, we used resonant amplification of voltage fluctuations from the cold end, loaded on a $\rm10\,k\Omega$ resistor. Signal is amplified by $\mathrm{\sim75\,dB}$ with an amplifier chain. All measurements were performed in a $\rm ^3He/^4He$ dilution refrigerator with a $\rm30\,mK$ base temperature. A homebuilt low-T amplifier at $\rm\sim800\,mK$ was utilized as a first stage. Noise spectral power density was measured at $\rm\sim8\,MHz$ in a $\rm\sim500\,kHz$ band.
To precisely obtain the unknown full gain coefficient of amplification circuit the Johnson-Nyquist noise calibration procedure was performed. At a given equilibrium $T$, we measured the thermal noise $S_I = 4 k_B T /R_{\mathrm{par}}$ of the sample, the load resistor and the RF transistor, all connected in parallel. The total load resistance was varied between $R_{\mathrm{par}}\approx40\,\rm\Omega$ and $R_{\mathrm{par}}\approx5\,\rm k\Omega$ with the help of transistor gate voltage.
In Fig \ref{fig1}, we plot the output signal of noise amplification circuit $P_{\mathrm{det}}$ as a function of $R_{par}$ at different $T$. The shape of resulting curves is determined by resistance-dependent full conversion coefficient $G(R_{\mathrm{par}})$, $P_{\mathrm{det}} = G(R_{\mathrm{par}}) S_\mathrm{I}$. This calibration allows to determine both $G(R_{\mathrm{par}})$ and the input current noise of the first stage ($2.5\times10^{-27}\,\rm A^2/Hz$). In addition, we verified that the lowest achievable electronic temperature in our setup is $\rm\approx100\pm20\,$mK.
Throughout the shot-noise measurements the transistor was pinched off.
\section{Results and discussion}
\subsection{Local noise thermometry}
\begin{figure}[t]
\begin{center}
\vspace{0mm}
\includegraphics[width=0.9\columnwidth]{noise_tikhonov_style.pdf}
\end{center}
\caption{Three-terminal local noise measurements in the center of a metal strip as a function of the strip current $I_H$. Black line is a shot-noise fit, see text.
}\label{fig2}
\end{figure}
\par
To test the device operation a local noise measurement with an unbiased NW ($I_{\mathrm{NW}}=0$), similar to \cite{Tikhonov2016.SciRep} was performed at the base temperature. The resulting $T_S$ dependence is shown in figure \ref{fig2} together with a shot noise fit for $T_S$, assuming local equilibrium with temperature:
\begin{equation}
T_N(0)=\sqrt{T_0^2 + \frac{3}{4\pi^2}(e r I_H/k_B)^2},
\label{eq_balance}
\end{equation}
determined by balance between Joule heating and Wiedemann-Franz heat conductance \cite{Nagaev1995}. The strip resistance fit parameter equals $r = \rm5.2\,\Omega$, substantially higher that the value $r \sim \rm3\,\Omega$ previously obtained in \cite{Tikhonov2016.SciRep}. Most probably, this is explained by the fact, that in previous experiments the sample was immersed into liquid helium, which resulted in better thermalization of the current leads.
\begin{figure}[t]
\begin{center}
\vspace{0mm}
\includegraphics[width=0.9\columnwidth]{noise_th_and_exp.pdf}
\end{center}
\caption{(a) The experimentally measured current noise as a function of NW current. Solid lines represent equilibrium configuration, when no current flows through metal the strip at different temperatures. Symbols represent non-equilibrium configurations with different strip currents. Dashed line shows Fano factor $\rm1/3$ slope.
(b) Theoretical prediction of the current noise for equilibrium $\rm300\,mK$ case (red dashed line) and non-equilibrium $\rm100\,mK/27\,\mu A$ case (dark blue solid line). Inset: a sketch of EEDs at test and cold end for equilibrium and non-equilibrium cases at an equal $I_{\mathrm{NW}}$. Solid blue lines for non-equilibrium case are Fermi-Dirac distributions with different temperatures and chemical potential offset, while red dashed lines for equilibrium case are Fermi-Dirac distributions with the same temperature differing only by offset.
}\label{fig3}
\end{figure}
\par
\subsection{Electronic energy distribution}
To verify the concept of the EED measurement the shot noise dependence on $I_{NW}$ was measured in two configurations: equilibrium strip ($I_H=0$ and $T_0$ above the base) and non-equilibrium strip ($I_H\neq0$ and base $T_0$). The $T_0$ in the first case and the $I_H$ in the second case were adjusted such that in the absence of the NW bias ($I_{\mathrm{NW}}=0$) the $T_S$ is the same in two experiments.
The green curve in figure \ref{fig3}a corresponds to the standard shot noise measurement at the base temperature and $I_H=0$. The Fano factor $F\approx1/3$ proves the elastic diffusive transport in the NW \cite{Beenakker_Buettiker_1992}\cite{Nagaev1992}, which is crucial for operating the noise sensor.
\begin{figure}[t]
\begin{center}
\vspace{0mm}
\includegraphics[width=0.9\columnwidth]{noise_Rdiff.pdf}
\end{center}
\caption{NW differential resistance $R_{\mathrm{diff}}=dV/dI$ as a function of NW current I at different overheat regimes.
}\label{fig_Rdiff}
\end{figure}
\par
At increasing $I_H$ (symbols) or $T_0$ (lines) the measured noise increases, as expected. However, the noise in the non-equilibrium case is indistinguishable from equilibrium case with higher corresponding $T_0$ (red line with dark blue symbols, and dark red line with blue symbols). To verify if this observation is consistent with the EED sensing we plot the theoretical predictions for current noise in corresponding configurations, see figure \ref{fig3}b. For the non-equilibrium case, the electron temperature at the test end was calculated using eq. (\ref{eq_balance}). Similar to the experimental data, the results for the equilibrium and non-equilibrium cases are also almost indistinguishable in figure \ref{fig3}b. We conclude that the EED in the middle of the current biased strip is very well captured by the Fermi-Dirac distribution with the local temperature given by eq. (\ref{eq_balance}). A direct comparison to the experimental data is complicated because of a slightly nonlinear current-voltage response of the NW, which gives rise to $I_{\mathrm{NW}}$ dependent differential resistance (see figure \ref{fig_Rdiff}). Note that in spite of this similarity, the case of non-equilibrium strip is characterized by a strong temperature gradient along the NW, which manifests itself in thermoelectric measurements \cite{Tikhonov2016.SST}. The EEDs on the two ends of the NW are sketched in the inset of figure \ref{fig3}b.
\section{Conclusion}
In summary, we experimentally realized the concept of the energy selective local noise measurement. The nonlinear current-voltage response of the NW complicates accurate extraction of the local EED under test. Yet, comparison with the theoretical calculations is consistent with the Fermi-Dirac shaped EED.
\section*{Acknowledgment}
We thank D.V.\,Shovkun and E.S\, Tikhonov for fruitful discussions. We gratefully acknowledge S. Roddaro and L. Sorba for providing us with the NW devices. This work was supported by the Russian Science Foundation under the grant RSF-DFG: 16-42-01050 and the bilateral CNR-RFBR project 15-52-78023.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2017-04-21T02:05:10",
"yymm": "1704",
"arxiv_id": "1704.04899",
"language": "en",
"url": "https://arxiv.org/abs/1704.04899"
}
|
"\\section*{Introduction}\nGiven a \\emph{bipartite quiver} $Q$ which is just a directed bipartite g(...TRUNCATED)
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"\\section{Introduction}\n\nVanadium dioxide VO$_2$ is a material of long standing interest and is o(...TRUNCATED)
| {"timestamp":"2017-04-18T02:08:32","yymm":"1704","arxiv_id":"1704.04917","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\\label{intro}\n\nThere are many types of graph tournaments such as round (...TRUNCATED)
| {"timestamp":"2017-04-18T02:07:45","yymm":"1704","arxiv_id":"1704.04879","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\n\nCurrent astrophysical and cosmological observational data indicate that (...TRUNCATED)
| {"timestamp":"2017-04-18T02:09:10","yymm":"1704","arxiv_id":"1704.04961","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\n\nSignal propagation in optical fibers can be modeled by the stochastic no(...TRUNCATED)
| {"timestamp":"2018-09-25T02:12:29","yymm":"1704","arxiv_id":"1704.04904","language":"en","url":"http(...TRUNCATED)
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"\\section{Introduction}\n\nIn this paper we prove a Bombieri-Vinogradov type theorem for general mu(...TRUNCATED)
| {"timestamp":"2017-06-19T02:01:11","yymm":"1704","arxiv_id":"1704.04831","language":"en","url":"http(...TRUNCATED)
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