The full dataset viewer is not available (click to read why). Only showing a preview of the rows.
The dataset generation failed
Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 0
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 22414)
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 0
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
datasets.exceptions.DatasetGenerationError: An error occurred while generating the datasetNeed help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
text
string | meta
dict |
|---|---|
\section{Introduction}
CP violation in the Standard Model (SM) has first been measured in the Kaon sector. The CP violating parameter measured in the famous Cronin-Fitch experiment \cite{Christenson:1964fg} is $\varepsilon_K$, which describes the mixing between CP and mass eigenstates of the neutral Kaon system. The parameter $\varepsilon_K$ measures the so-called CP violation through mixing. On the other hand, Kaons can also decay through direct CP violation. This CP violating decay is parametrized by the quantity $\varepsilon'$. The ratio of the two CP violating parameters $\varepsilon'/\varepsilon$, where we suppress $K$ in $\varepsilon_K$, is also accessible experimentally, namely through a confrontation of the $K_L\to\pi^+\pi^-$ and $K_L\to\pi^0\pi^0$ decay widths. It has been measured by the NA48 \cite{Batley:2002gn} and KTeV \cite{AlaviHarati:2002ye,
Abouzaid:2010ny} collaborations and leads to an experimental world average of
\hspace{0.1cm}
\begin{equation}\label{eq:SMexp}
(\varepsilon'/\varepsilon)_\text{exp} = (16.6 \pm 2.3) \times 10^{-4}\,.
\end{equation}
\hspace{2cm}
\noindent
The SM estimates for this observable depend on the long-distance (LD) treatment used to compute the $K\to\pi\pi$ hadronic matrix elements. As can be seen from Tab.~\ref{tab:SMpred}, the SM prediction differs for the three types of LD approaches and consequently there is some controversy over which treatment to use. The results obtained with Lattice QCD (LQCD) inputs as well as the ones in the Dual QCD (DQCD) approach are in good agreement with each other and exhibit about a $2.9\sigma$ deviation from the experimental value in eq.~\eqref{eq:SMexp}. The Chiral Perturbation Theory ($\chi$PT) approach leads to a value consistent with the SM, however exhibiting large uncertainties. Moreover the lower part of the error is consistent with the values obtained using Lattice or DQCD and therefore the situation is not conclusive.
\noindent
Taking the discrepancy between the SM prediction and the experimental value for granted, it is interesting to study beyond the SM (BSM) effect that could explain such deviations. In the following section I will review the SM prediction for $\varepsilon'/\varepsilon$ based on the DQCD approach. In Sec.~\ref{sec:BSMME} the computation of the BSM matrix elements relevant for $\varepsilon'/\varepsilon$ is discussed. In Sec.~\ref{sec:Masterform} a master formula for BSM effects in $\varepsilon'/\varepsilon$ is presented and in Sec.~\ref{sec:SMEFT} the relation between $\varepsilon'/\varepsilon$ and the SM effective theory (SMEFT) is discussed, before I summarize in Sec.~\ref{sec:concl}.
\section{$\varepsilon'/\varepsilon$ in the SM}
To describe $\varepsilon'/\varepsilon$ in a model-independent way, we use the effective Hamiltonian of three quark flavours which generates a $\Delta S=1$ transition. It consists of local operators multiplied by their corresponding Wilson coefficients and can be written as follows \cite{Buras:1991jm,Buras:1993dy,Ciuchini:1992tj,Ciuchini:1993vr}:
\begin{align}\label{eq:SMham}
\mathcal{H}_{\Delta S = 1}^{(3)} &
= - \sum_i C_i({\mu}) \, O_i\,.
\end{align}
This Hamiltonian is invariant under the unbroken gauge-group $SU(3)_c\times U(1)_{\rm em}$ and contains all the fields lighter than the charm quark as dynamical degrees of freedom. The minus sign is chosen to be in accord with the SMEFT conventions.
In the SM, the sum in eq.~\eqref{eq:SMham} contains seven four-quark operators consisting of $(V\pm A)$ currents as well as the chromomagnetic operator. The four-quark operators are generated through tree-level and box diagrams containing a $W$ boson and a gluon, as well as from QCD and Electroweak (EW) penguin diagrams. The seven effective operators can be written as linear combinations of the following vector-vector operators:
\begin{table}[tbp]
\centering
\begin{tabular}{cccc}
\toprule
Long-distance & SM prediction &
Group &
Ref.
\\
\midrule
Lattice & $(1.4 \pm 6.9) \times 10^{-4}$ &
RBC-UKQCD & \cite{Blum:2015ywa,Bai:2015nea}\\
& $(1.9 \pm 4.5) \times 10^{-4}$ &
Buras/Gorbahn/Jamin/J\"ager & \cite{Buras:2015yba}\\
& $(1.1 \pm 5.1) \times 10^{-4}$ &
Kitahara/Nierste/Tremper & \cite{Kitahara:2016nld}\\
\midrule
DQCD & $<(6.0\pm 2.4) \times 10^{-4}$ &
Buras/G\'erard & \cite{Buras:2015xba}\\
&$\qquad$ if $B_6<B_8 =B_8 \,(\text{LQCD})$ & &\\
\midrule
$\chi$PT
& $(15 \pm 7) \times 10^{-4}$ &
Gisbert/Pich & \cite{Gisbert:2017vvj}\\
\bottomrule
\end{tabular}
\captionsetup{width=0.9\linewidth}
\caption{SM estimates for $\varepsilon'/\varepsilon$, using different treatments of the long-distance effects.
}\label{tab:SMpred}
\end{table}
\begin{align}\label{eq:vecops}
O_{VAB}^q &
= (\bar s^i \gamma_{\mu} P_A d^i) (\bar q^j \gamma^{\mu} P_B q^j) \,,
&
\widetilde{O}_{VAB}^q &
= (\bar s^i \gamma_{\mu} P_A d^j) (\bar q^j \gamma^{\mu} P_B q^i) \,,
\end{align}
\hspace{2cm}
\noindent
where $P_{A,B}$ $(A,B=L,R)$ denote the chirality projection operators, $i,j$ are colour indices and $q=u,d,s$. The chromomagnetic operator reads:
\begin{align}\label{eq:chromo}
O_{8g} &
= m_s(\bar s \, \sigma^{\mu\nu} T^A P_{L} d) \, G^A_{\mu\nu} \,,
\end{align}
\hspace{2cm}
\noindent
with $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$, $\,T^A$ being the $SU(3)_c$ generators and $G^A_{\mu\nu}$ the gluonic field-strength tensor.
Having the Hamiltonian of eq.~\eqref{eq:SMham} at hand allows to compute the $\varepsilon'/\varepsilon$ observable, which is given by:
\begin{align}\label{eq:epspr}
\frac{\varepsilon'}{\varepsilon} &
= -\frac{\omega}{\sqrt{2}|\varepsilon_K|}
\left[ \frac{\text{Im}A_0}{\text{Re}A_0}
- \frac{\text{Im}A_2}{\text{Re}A_2} \right]\,.
\end{align}
\newline
\noindent
Here $\omega = {\text{Re}A_2}/{\text{Re}A_0} \approx 1/22$, reflecting the $\Delta I =1/2$ rule, and $\varepsilon_K$ is the Kaon mixing parameter mentioned before. The expression is therefore determined by the isospin amplitudes $A_{0,2}$ defined by
\begin{align}
A_{0,2} &
= \Big\langle (\pi\pi)_{I=0,2}\, \Big|\; \mathcal{H}_{\Delta S = 1}^{(3)}({\mu})
\;\Big|\, K \Big\rangle \,.
\end{align}
\hspace{2cm}
\noindent
After having fixed the Wilson coefficients of $\mathcal{H}_{\Delta S = 1}^{(3)}$ by performing a matching procedure, the only remaining task is to compute the hadronic matrix elements of the local operators in eq.~\eqref{eq:SMham}. In the following subsection, we will look into this computation by employing the DQCD approach.
\subsection{Long-distance effects in the DQCD approach}
The DQCD is based on the large $N_c$ limit, first studied by t'Hooft \cite{'tHooft:1973jz,'tHooft:1974hx} and Witten \cite{Witten:1979kh,Treiman:1986ep} for strong interactions.
To study hadronic weak decays, the following truncated Chiral Lagrangian is used \cite{Bardeen:1986vp,Bardeen:1986uz,Bardeen:1986vz}:
\begin{equation}\label{eq:chiralLag}
\mathcal{L}_{tr}=\frac{F^2}{8}\left[\text{Tr}(D^\mu UD_\mu U^\dagger)+r\text{Tr}(mU^\dagger+\text{h.c.})-\frac{r}{\Lambda^2_\chi}\text{Tr}(mD^2U^\dagger+\text{h.c.})\right] {\,,}
\end{equation}
with the unitary chiral matrix and the octet of lowest-lying pseudoscalars
\begin{equation}
U=\exp(i\sqrt{2}\frac{\Pi}{F}), \qquad
\Pi=\sum_{\alpha=1}^8\lambda_\alpha\pi^\alpha{\,.}
\end{equation}
The Lagrangian depends on the quark mass matrix and the chiral enhancement factor
\begin{equation}
m=\text{diag}(m_u,m_d,m_s)\,, \qquad r=\frac{2m_K^2}{m_s^2+m_d^2}\,.
\end{equation}
\hspace{2cm}
\noindent
It contains a hadronic mass scale $\Lambda_\chi$ corresponding to higher resonances.
\noindent
Employing now the large $N_c$ limit, the Lagrangian of eq.~\eqref{eq:chiralLag} can be matched onto the regular QCD Lagrangian containing quark and gluon fields only. In the chiral limit and at order $\mathcal{O}(p^2)$ the quark currents are then given by:
\begin{equation}\label{eq:mesrep}
(\gamma^\mu P_L)^{ba}=i \frac{F^2}{4} (\partial^\mu U U^\dagger)^{ab}, \qquad (P_L)^{ba}=- \frac{F^2}{8} r (U)^{ab}\,, \qquad (\sigma^{\mu\nu}P_L)^{ab} = 0\,,
\end{equation}
\hspace{2cm}
\noindent
for the flavour indices $a,b$. The chirality flipped versions are obtained by the replacement $U\leftrightarrow U^\dag$. These relations allow to express the local operators in terms of the lowest-lying mesons and therefore to compute their corresponding matrix elements. Furthermore, this framework allows to study the renormalization group (RG) evolution of the matrix elements up to a scale of $\mathcal{O}(1\text{GeV})$ until where the theory is valid. This RG evolution is dubbed meson evolution.
\noindent
The DQCD approach was first employed in the context of $K\to\pi\pi$ matrix elements in \cite{Bardeen:1986vp,Bardeen:1986vz,Buras:2014maa}. Its validity is confirmed by results obtained within LQCD. Among them is the correctly predicted hierarchy of the bag factors for the SM operators $Q_6$ and $Q_8$ \cite{Buras:2015xba}
\begin{equation}\label{BG}
B_6^{(1/2)} \leq B_8^{(3/2)} < 1 \, .
\end{equation}
\hspace{2cm}
\noindent
Also the explicit calculations for $B_6^{(1/2)}(m_c),\,B_8^{(3/2)}(m_c)$ are in good agreement with the Lattice results \cite{Bai:2015nea,Blum:2015ywa}. Not only for the SM four-quark operators but also for the matrix element of the chromomagnetic operator of eq.~\eqref{eq:chromo}, DQCD \cite{Buras:2018evv} agrees well with LQCD \cite{Constantinou:2017sgv}. Furthermore, the impact of final state interactions has been analysed within the DQCD approach in \cite{Buras:2016fys} and has been shown to be less important for $\varepsilon'/\varepsilon$ than for the $\Delta I=1/2$ rule, and less important than meson evolution which is responsible for (\ref{BG}).
Finally DQCD also allows, with the help of meson evolution, to understand
the pattern of the BSM $K^0-\bar K^0$ mixing matrix elements \cite{Buras:2018lgu} obtained by LQCD \cite{Carrasco:2015pra,Jang:2015sla,Boyle:2017ssm}.
More information on DQCD can be found in the original papers and in the reviews in \cite{Buras:2018hze,Buras:2014maa}.
\section{BSM matrix elements for $\varepsilon'/\varepsilon$}\label{sec:BSMME}
Generalizing the SM Hamiltonian by allowing for all possible Lorentz- and gauge invariant operators, one finds that there are 13 additional four-quark operators to be added to $\mathcal{H}_{\Delta S = 1}^{(3)}$. Three of them are vector-vector operators which are independent of the seven operators generated within the SM. They can also be written as linear combinations of the operators in eq.~\eqref{eq:vecops}. The other BSM operators consist of scalar or tensor bilinears and can be written as linear combinations of the following operators:
\begin{align}
O_{SAB}^q &
= (\bar s^i P_A d^i) (\bar q^jP_B q^j) \,,
&
\widetilde{O}_{SAB}^q &
= (\bar s^i P_A d^j) (\bar q^j P_B q^i) \,, \\
O_{TA}^q &
= (\bar s^i \sigma_{\mu\nu} P_A d^i) (\bar q^j\sigma^{\mu\nu}P_A q^j) \,,
&
\widetilde{O}_{TA}^q &
= (\bar s^i \sigma_{\mu\nu} P_A d^j) (\bar q^j\sigma^{\mu\nu}P_A q^i) \,,
\end{align}
\hspace{2cm}
\noindent
for $q=u,d,s$. Two equivalent bases for the 13 BSM operators can be found in \cite{Aebischer:2018rrz}.
\noindent
The $K\to\pi\pi$ matrix elements of these BSM operators have been calculated for the first time in \cite{Aebischer:2018rrz}, using the DQCD approach. They were computed first at the factorization scale $\mu_F$ at which the meson representation of eq.~\eqref{eq:mesrep} holds. The factorization scale corresponds to very low momenta of $\mathcal{O}(p^2\approx 0)$. Since the observable $\varepsilon'/\varepsilon$ is usually computed at the charm scale $\mu_c=\mathcal{O}(m_c)$, the running of the matrix elements has to be performed from the factorization scale up to the scale $\mu_c$ via the meson evolution for scales below $1\, {\rm GeV}$ followed by the
usual QCD evolution.
The explicit expressions and numerical values of all the matrix elements at the charm scale as well as further details of the computation can be found in \cite{Aebischer:2018rrz}. Here, we summarize only quantitatively the results of the analysis. For the different types of BSM operators, one finds for their respective matrix elements at the factorization scale $\mu_F$ and at the charm scale $\mu_c$:
\newline
\begin{itemize}
\item Vector operators: small at $\mu_F$ and at $\mu_c$.
\item Scalar operators: large at $\mu_F$, moderate at $\mu_c$.
\item Tensor operators: zero at $\mu_F$, large at $\mu_c$.
\item Scalar/Tensor operators containing three $s$ quarks: zero at $\mu_F$ and at $\mu_c$.
\end{itemize}
\section{Master formula for BSM effects in $\varepsilon'/\varepsilon$}\label{sec:Masterform}
Knowing the matrix elements for the complete set of local effective operators relevant for $\varepsilon'/\varepsilon$ allows for a model-independent analysis of the BSM effects. In this section we provide the means for such an analysis in the form of a master formula for $\varepsilon'/\varepsilon$ \cite{Aebischer:2018quc}. For this purpose, we split the observable in the following way:
\begin{align}
\frac{\varepsilon'}{\varepsilon} &
= \left(\frac{\varepsilon'}{\varepsilon}\right)_\text{SM}
+ \left(\frac{\varepsilon'}{\varepsilon}\right)_\text{BSM} \,,
\end{align}
and focus on the BSM part. Since many NP scenarios contain heavy degrees of freedom with a mass scale above the EW scale, it is reasonable to provide a master formula evaluated at the EW scale $\mu_W$. Consequently, a NP analysis of a particular model only requires a simple tree-level matching at $\mu_W$. To evaluate eq.~\eqref{eq:epspr} at the EW scale, the RG evolution of the matrix elements from $\mu_c$ up to $\mu_W$ has to be taken into account \cite{Aebischer:2017gaw,Jenkins:2017dyc}. In the running up to the EW scale new operators containing $c$ and $b$ quarks will be generated through QCD and QED mixing, leading to the more general Hamiltonian of five flavours $\mathcal{H}_{\Delta S = 1}^{(5)}$. The master formula will therefore depend on the Wilson coefficients of all such effective operators. Setting the parameter $\varepsilon_K$ as well as $\text{Re}(A_0)$ and $\text{Re}(A_2)$ appearing in eq.~\eqref{eq:epspr} to their experimental values \cite{Cirigliano:2011ny} one finds the following master formula:
\begin{align}
\label{eq:master}
\left(\frac{\varepsilon'}{\varepsilon}\right)_\text{BSM} &
= \sum_i P_i(\mu_W) ~\text{Im}\left[ C_i(\mu_W) - C^\prime_i(\mu_W)\right]
\times (1\,\text{TeV})^2,
\end{align}
with
\begin{align}
\label{eq:master2}
P_i(\mu_W) & = \sum_{j} \sum_{I=0,2} p_{ij}^{(I)}(\mu_W, \mu_c)
\,\left[\frac{\langle O_j (\mu_c) \rangle_I}{\text{GeV}^3}\right].
\end{align}
Here, the $p_{ij}^{(I)}$ contain the evolution from $\mu_c$ to $\mu_W$. The matrix elements $\langle O_j (\mu_c) \rangle_I$ are taken from LQCD \cite{Blum:2015ywa,Bai:2015nea} for the SM operators and from DQCD \cite{Aebischer:2018rrz} for the BSM operators. The crucial objects determining the impact of each Wilson coefficient on $\varepsilon'/\varepsilon$ are the $P_i$ values. These were obtained using the public codes \texttt{wcxf} \cite{Aebischer:2017ugx} for the basis change, \texttt{wilson} \cite{Aebischer:2018bkb} for the RG running and \texttt{flavio} \cite{Straub:2018kue} to compute $\varepsilon'/\varepsilon$ at the EW scale. The $P_i$ values of the full set of operators contained in $\mathcal{H}_{\Delta S = 1}^{(5)}$ can be grouped into five classes $(A-E)$, which are listed in Tab.~\ref{tab:Pis}. The operators either give a direct BSM contribution to $\varepsilon'/\varepsilon$ through their matrix element (ME) or contribute to the observable indirectly through RG mixing. For further details and the explicit values of the $P_i$'s as well as their respective uncertainties we refer to \cite{Aebischer:2018quc}.
\begin{table}
\centering
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{ccccc}
\hline
Class & Type & $O_i$ & $P_i$ & Impact \\
\hline
A & SM
& $O_{VAB}^{u,d}, \widetilde O_{VAB}^{u,d}, O_{SLR}^{d}$ & can be large & ME \\
&
& $O_{VAB}^{s,c,b}, \widetilde O_{VAB}^{c,b}, O_{SLR}^{s}$ & small & Mixing \\
\hline
B & Chromomagnetic & $O_{8g}$ & small & Mixing \\
& Scalar: $s ,c, b$ & $O_{SLL}^{s,c,b}, \widetilde O_{SLL}^{c,b} $ & small & Mixing \\
& Tensor: $s ,c, b$ & $O_{TLL}^{s,c,b}$ & small & Mixing \\
\hline
C & Scalar: $u$ & $O_{SLL}^{u}, \widetilde O_{SLL}^{u} $ & small & ME \\
& Tensor: $u$ & $O_{TLL}^{u}, \widetilde O_{TLL}^{u} $ & large & ME \\
\hline
D & Scalar: $d$ & $O_{SLL}^{d} $ & small & ME \\
& Tensor: $d$ & $O_{TLL}^{d} $ & large & ME \\
\hline
E & Scalar LR: $u$ & $O_{SLR}^{u}, \widetilde O_{SLR}^{u} $ & can be large & ME \\
\hline
\end{tabular}
\captionsetup{width=0.9\linewidth}
\caption{$P_i$ values of the effective operators relevant for $\varepsilon'/\varepsilon$ at the EW scale, grouped into five classes (A-E). The operators either contribute via their matrix element (ME) or through mixing effects to the observable.}
\label{tab:Pis}
\end{table}
\section{$\varepsilon'/\varepsilon$ meets SMEFT}\label{sec:SMEFT}
Assuming that NP manifests itself at scales much higher than the EW scale, the SMEFT \cite{Buchmuller:1985jz,Grzadkowski:2010es} consists of a valid low-energy effective theory of such a NP scenario. Therefore it is reasonable to adopt the SMEFT as an intermediate theory between any NP model and the SM. This procedure allows to describe NP effects in a model independent way. The complete tree-level matching of the SMEFT onto the weak effective theory is done in \cite{Aebischer:2015fzz,Jenkins:2017jig} and in \cite{Aebischer:2018csl} all the SMEFT operators relevant for $\varepsilon'/\varepsilon$ have been identified. There are:
\begin{itemize}
\item vector four-quark operators: $\mathcal{O}_{qq}^{(1,3)}, \mathcal{O}_{qu}^{(1,8)},\mathcal{O}_{qd}^{(1,8)}, \mathcal{O}_{ud}^{(1,8)} ,\mathcal{O}_{dd}\,,$
\item scalar four-quark operators: $\mathcal{O}_{quqd}^{(1,8)}\,,$
\item modified W and Z couplings: $\mathcal{O}_{Hq}^{(1,3)},\mathcal{O}_{Hd},\mathcal{O}_{Hud}\,,$
\item chromomagnetic dipole operator: $\mathcal{O}_{dG}$.
\end{itemize}
An effect in $\varepsilon'/\varepsilon$ stemming from SMEFT operators can result in correlations with other observables. This occurs for operators containing a quark doublet after changing from the flavour to the interaction basis, or through flavour dependent RG mixing effects. In \cite{Aebischer:2018csl} correlations of $\varepsilon'/\varepsilon$ to $\Delta S =2$ and $\Delta C=1, 2$ processes, semileptonic Kaon decays, the electroweak $T$ parameter, collider constraints as well as the neutron electric dipole moment (EDM) have been analysed. Furthermore, several tree-level mediator scenarios have been studied, which are summarised in Tab.~\ref{tab:treemed}. Further details on correlations of $\varepsilon'/\varepsilon$ and the observables mentioned here can be found in \cite{Aebischer:2018csl}.
\section{Summary}\label{sec:concl}
The hadronic matrix elements for the BSM operators relevant for $\varepsilon'/\varepsilon$ have been presented for the first time in \cite{Aebischer:2018rrz}. The newly acquired matrix elements allowed for the first time to derive a master formula for $\varepsilon'/\varepsilon$, depending on SM and BSM operators. This master formula is presented in \cite{Aebischer:2018quc} and is already included in several public codes, such as \texttt{flavio} \cite{Straub:2018kue} and \texttt{smelli} \cite{Aebischer:2018iyb}. Based on this master formula, different correlations of $\varepsilon'/\varepsilon$ to other observables have been analysed in the context of the SMEFT in \cite{Aebischer:2018csl}.
\begin{table}[tbp]
\centering
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{cccc}
\hline
Mediator & SM Representation & SMEFT & Correlation \\
\hline
$Z'$ & $(1,1)_0$ & $ \mathcal{O}_{qd}^{(1)}$ & $\varepsilon_K$ \\
& & $\mathcal{O}_{qu}^{(1)}$ & $pp\rightarrow jj$ \\
& & $\mathcal{O}_{HD}$ & T parameter \\
\hline
Coloured scalar & $(8,2)_{1/2}$ & $ \mathcal{O}_{qd}^{(1)}$ & $\varepsilon_K$ \\
& & $\mathcal{O}^{(8)}_{quqd}$ & neutron EDM \\
\hline
\end{tabular}
\captionsetup{width=0.9\linewidth}
\caption{Tree-level models, which can have a sizable effect in $\varepsilon'/\varepsilon$ and their correlations to other observables.}
\label{tab:treemed}
\end{table}
\section*{Acknowledgements}
It is a pleasure to thank the organizers of this workshop for inviting me
to this interesting event. In particular I would like to thank my great collaborators: Christoph Bobeth, Andrzej Buras, Jean-Marc
G{\'e}rard and David Straub for a very nice and inspiring collaboration. A special thanks goes to Andrzej Buras for all his support and for giving me the opportunity to give this talk. This research was
supported by the DFG cluster of excellence ``Origin and Structure of the Universe''.
\bibliographystyle{JHEP}
|
{
"timestamp": "2018-12-11T02:23:35",
"yymm": "1812",
"arxiv_id": "1812.03722",
"language": "en",
"url": "https://arxiv.org/abs/1812.03722"
}
|
\section{Introduction}
The diagnosis of skin diseases is challenging. To diagnose a skin disease,
a variety of visual clues may be used such as the individual lesional
morphology, the body site distribution, color, scaling and arrangement of lesions.
When the individual elements are analyzed separately, the recognition process
can be quite complex \cite{cox2004diagnosis}. For example,
the well studied skin cancer, melanoma, has four major clinical
diagnosis methods: ABCD rules, pattern analysis, Menzies method and $7$-Point
Checklist. To use these methods and achieve a satisfactory diagnostic accuracy, a high
level of expertise is required as the differentiation of skin lesions demands
a great deal of experience and expertise \cite{whited1998does}.
Unlike the diagnosis by human experts, which depends essentially on subjective
judgment and is not always reproducible, a computer aided diagnostic system is more
objective and reliable. Traditionally, one can use human-engineered feature
extraction algorithms in combination with a classifier to complete this
task. For some skin diseases, such as melanoma and basal cell carcinoma, this
solution is feasible as their features are regular and predictable. However,
when we extend the skin diseases to a broader range, where the features are so
complex that hand-crafted feature design becomes infeasible, the traditional
approach fails.
In recent years, deep convolutional neural networks (CNN) become very popular
in feature learning and object classification. The use of high performance GPUs
makes it possible to train a network on a large-scale dataset so as to
yield a better performance. Many studies
\cite{DBLP:journals/corr/SermanetEZMFL13,DBLP:journals/corr/IoffeS15,DBLP:journals/corr/SzegedyLJSRAEVR14,DBLP:journals/corr/SimonyanZ14a}
from the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) \cite{ILSVRC15}
have shown that the state-of-art CNN architectures are able to surpass humans in many
computer vision tasks. Therefore, we propose to construct a skin disease classifier
with CNNs.
\begin{figure}[t]
\centering
\includegraphics[scale=0.25]{images/similar}
\caption{Some visually similar skin diseases. First row (left to right):
malignant melanoma, dermatofibroma, basal cell carcinoma, seborrheic keratosis.
Second row (left to right): compound nevus, intradermal nevus, benign keratosis,
bowen's disease.}
\label{fig: similar diseases}
\vskip -0.1in
\end{figure}
However, training CNNs directly using the diagnosis labels may not be viable.
\begin{inparaenum}[1)]
\item For some diseases, their lesions are so similar that they can not be
distinguished visually. Figure \ref{fig: similar diseases} shows the
dermatology images of eight different skin diseases. We can see that
the two diseases in each column have very similar visual appearances. Thus, it
is very difficult to make a judgment between the two diseases with only the visual
information.
\item Many of the skin diseases are not so common that only a few images are available
for training. Table \ref{tab: diagnosis statistics} shows the dataset statistics
of the dermatology atlantes we used in this study. We can see that there are
tens of hundreds of skin diseases. However, most of them contain very few
images.
\item Skin disease diagnosis is a complex procedure that often involves many other
modalities, such as palpation, smell, temperature changes and microscopy
examinations \cite{cox2004diagnosis}.
\end{inparaenum}
On the other hand, lesion characteristics, which inherently describe the visual aspects of
skin diseases, arguably should be considered as the ideal ground truth for training.
For example, the two images in the first column of Figure \ref{fig: similar diseases}
can both be labeled with hyperpigmented and nodular lesion tags. Compared with
using the sometimes ambiguous disease diagnosis labels for these two images, the use of the
lesion tags can give a more consistent and precise description of the
dermatology images.
In this paper, we investigate the performance of CNNs trained with disease and
lesion labels, respectively. We collected $\num{75665}$ skin disease images from six
different publicly available dermatology atlantes. We then train a multi-class
CNN for disease-targeted classification and another multi-label CNN for
lesion-targeted classification. Our experimental results show that the top-$1$
and top-$5$ accuracies for the disease-targeted classification are $27.6\%$ and
$57.9\%$ with a mean average precision (mAP) of $0.42$. While for the
lesion-targeted skin disease classification, a much higher mAP of $0.70$ is achieved.
\section{Related Work}
Much work has been proposed for computer aided skin disease classification.
However, most of them use human-engineered feature extraction algorithms and restrict
the problem to certain skin diseases, such as melanoma \cite{arroyo2014automated,xie2014dermoscopy,Fabbrocini:2014yq,Saez:2014qy,Barata:2013fj}.
Some other works \cite{cruz2014automatic,wang2014cascaded,arevalo2015unsupervised}
use CNNs for unsupervised feature learning from histopathology images and only
focus on the detection of mitosis, an indicator of cancer. Recently,
\textit{Esteva et al.} \cite{estevadeep} proposed a disease-targeted skin
disease classification method using CNN. They used the dermatology images from the
Dermnet atlas, one of the six atlantes used in this study, and reported that their
CNN achieved $60.0\%$ top-1 accuracy and $80.3\%$ top-$3$ accuracy.
However, they performed the CNN training and testing on the same dataset without cross-validation which makes their results unpersuasive. A preliminary work \cite{liao2016deep} of this study has also discovered similar performances on skin disease classification.
\section{Datasets} \label{sec: datasets}
We collect dermatology photos from the following dermatology atlas websites:
\begin{itemize}
\item \textbf{AtlasDerm} (www.atlasdermatologico.com.br)
\item \textbf{Danderm} (www.danderm-pdv.is.kkh.dk)
\item \textbf{Derma} (www.derma.pw)
\item \textbf{DermIS} (www.dermis.net)
\item \textbf{Dermnet} (www.dermnet.com)
\item \textbf{DermQuest} (www.dermquest.com)
\end{itemize}
These atlantes are maintained by professional dermatology resource providers.
They are used by dermatologists for training and teaching purpose. All of the
dermatology atlantes have diagnosis labels for their images. For each dermatology
image only one disease diagnosis label is assigned. We use these diagnosis labels
as the ground truth to train the disease-targeted skin disease classifier.
However, each of the atlas maintains its own skin disease taxonomy and naming
convention for the diagnosis labels. It means different atlas may have different
labels for the same diagnosis and some diagnosis may have several variations.
To address this problem, we adapt the skin disease taxonomy used by the
DermQuest atlas and merge the diagnosis labels from other atlantes into it. We
choose the DermQuest atlas because of the completeness and professionalism of
its dermatology resources. In most of the cases, the labels for the same diagnoses
may have similar naming conventions. Therefore, we merge them by looking at the
word or string similarity of two diagnosis labels. We use the string pattern matching
algorithm described in \cite{ratcliff1988pattern}, where the similarity ratio
is
\begin{equation}
S = \frac{2 * M}{T}.
\end{equation}
Here, $M$ is the number of matches and $T$ is the total number of characters in
both strings. The statistics of the merged atlantes is given
in Table \ref{tab: diagnosis statistics}. Note that the total number of diagnoses
in our dataset is $\num{2113}$ which is significant higher than any of the atlas.
This is because we use a conservative merging strategy such that we merge two
diagnosis labels only when their string similarity is very high ($S > 0.8$).
Thus, we can make sure no two diagnosis labels are incorrectly merged. For those
redundant diagnosis labels, they only contain a few dermatology images. We can
discard them by choosing a threshold that filters out small diagnosis labels.
For the disease-targeted skin disease classification, we choose the AtlasDerm,
Danderm, Derma, DermIS, and Dermnet datasets as the training set and the
DermQuest dataset as the test set. Due to the inconsistency of the taxonomy and
naming convention between the atlantes, most of the diagnosis labels have only a few
images. As our goal is to investigate the feasibility of using CNNs for
disease-targeted skin disease classification, we remove these noisy diagnosis
labels and only keep those labels that have more than $300$ images. As a result
of the label refinement and cleaning, we have $\num{18096}$ images in the training set and
$\num{14739}$ images in the test set. The total number of diagnosis labels is $38$.
For the skin lesions, only the DermQuest dataset contains the lesion tags.
Unlike the diagnosis, which is unique for each image, multiple lesion
tags may be associated with a dermatology image. There are a total of $134$
lesion tags for the $\num{22082}$ dermatology images from DermQuest. However, most
lesion tags have only a few images and some of the lesion tags are duplicated.
After merging and removing infrequent lesion tags, we retain $23$ lesion tags.
\begin{table}
\renewcommand{\arraystretch}{1.3}
\caption{Dataset statistics}
\centering
\begin{tabular}{l|c|c}
\textbf{Atlas} & \textbf{\# of Images} & \textbf{\# of Diagnoses} \\
\hline\hline
AtlasDerm & \num{8766} & 478 \\
Danderm & \num{1869} & 97 \\
Derma & \num{13189} & 1195 \\
DermIS & \num{6588} & 651 \\
Dermnet & \num{21861} & 488 \\
DermQuest & \num{22082} & 657 \\
\hline\hline
\textbf{Total} & \textbf{\num{75665}} & \textbf{2113} \\
\end{tabular}
\label{tab: diagnosis statistics}
\end{table}
Since only the DermQuest dataset has the lesion tags, we use images from the
DermQuest dataset to perform training and testing. The total number of
dermatology images that have lesion tags is $\num{14799}$. As the training
and test sets are sampled from the same dataset, to avoid overfitting, we use $5$-fold
cross-validation in our experiment. We first split our dataset into $5$ evenly
sized, non-overlapping ``folds''. Next, we rotate each fold as the test set and use
the remaining folds as the training set.
\section{Methodology} \label{sec: methodology}
We use CNNs for both the disease-targeted and lesion-targeted skin disease classifications.
For the disease-targeted classification, a multi-class
image classifier is trained and for the lesion-targeted classification,
we train a multi-label image classifier.
Our CNN architecture is based on the AlexNet \cite{NIPS2012_4824} and we modify
it according to our needs. The AlexNet architecture was one of the early
wining entry of the ILSVRC challenges which is considered sufficient for this study. Readers
may refer to the latest winning entry (MSRA \cite{He2015} as of ILSVRC 2015)
for better performance. Implementation details of training and testing the CNNs
are given in the following sections.
\subsection{Disease-Targeted Skin Disease Classification} \label{sec: diagnosis MC}
For the disease-targeted skin disease classification, each dermatology image
is associated with only one disease diagnosis. Hence, we train a multi-class classifier
using CNN. We fine-tune the CNN with the BVLC AlexNet model \cite{jia2014caffe}
which is pre-trained from the ImageNet dataset \cite{ILSVRC15}. Since the
number of classes we are predicting is different with the ImageNet images, we
replace the last fully-connected layer ($1000$ dimension) with a new
fully-connected layer where the number of outputs is set to the number of skin
diagnoses in our dataset. We also increase the learning rate of the weights and
bias of this layer as the parameters of the newly added layer is randomly
initialized. For the loss function, we use the softmax function
\cite[Chapter~3]{nielsen2015neural} and connect a new softmax layer to the newly
added fully-connected layer. Formally put, let $z_j^L$ be the the weighted input
of the $j$th neuron of the softmax layer, where $L$ is the total number of the
layers in the CNN (For AlexNet, $L=9$). Thus, the $j$th activation of the
softmax layer is
\begin{equation}
a_j^L = \frac{e^{z_j^L}}{\sum_ke^{z_k^L}}
\end{equation}
And the corresponding softmax loss is
\begin{equation}
E = -\frac{1}{N} \sum_{n=1}^N\log(a_{y^n}^L)
\end{equation}
where $N$ is the number of images in a mini-batch, $y^n$ is the ground truth
of the $n$th image and $a_{y^n}^L$ is the $y^n$th activation of the softmax layer.
In the test phase, we choose the label $j$ that yields the largest activation
$a_j^L$ as the prediction, i.e.
\begin{equation}
\widehat{y} = \argmax_j{a_j^L}.
\end{equation}
\subsection{Lesion-Targeted Skin Disease Classification} \label{sec: lesion MLC}
As we mentioned early, multiple lesion tags may be associated with a
dermatology image. Therefore, to classify skin lesions we need to train a
multi-label CNN. Similar to disease-targeted skin disease classification,
we fine-tune the multi-label CNN with the BVLC AlexNet model. To train a
multi-label CNN, two data layers are required. One data layer loads the dermatology
images and the other data layer loads the corresponding lesion tags. Given
an image $\mathbf{X}_n$ from the first data layer, its corresponding lesion tags
from the second data layer are represented as a binary vector
$\mathbf{Y}_n=[y_1^n, y_2^n, \dots, y_Q^n]^T$
where $Q$ is the number of lesions in our data set and $y_j^n, j \in \{1, 2, \dots, Q\}$
is given as
\begin{equation}
y_j^n = \begin{cases}
1, & \text{if the $j$th label is associated with $\mathbf{X}_n$,} \\
0, & \text{otherwise.}
\end{cases}
\end{equation}
We replace the last fully-connected layer of the AlexNet with a new
fully-connected layer to accommodate the lesion tag vector. The learning rate of
the parameters of this layer is also increased so that the CNN can learn
features of the dermatology images instead of those images from ImageNet. For
the multi-label CNN, we use the sigmoid cross-entropy \cite[Chapter~3]{nielsen2015neural}
as the loss function and replace the softmax layer with a sigmoid cross-entropy
layer. Let the $z_j^L$ be the weighted input denoted in Section \ref{sec: lesion MLC},
then the $j$th activation of the sigmoid cross-entropy layer can be written as
\begin{equation}
a_j^L = \sigma(z_j^L) = \frac{1}{1 + e^{-z_j^L}}.
\end{equation}
And the corresponding cross-entropy loss is
\begin{equation}
E = - \frac{1}{N}\sum_{n=1}^N\sum_{j=1}^Qy_j^n\log{a_j^L} + (1-y_j^n)\log{(1 - a_j^L)}.
\end{equation}
For a given image $\mathbf{X}$, the output of the multi-label CNN is a
confidence vector $\mathbf{C}=[a_1^L, a_2^L, \dots, a_Q^L]^T$. Here,
$a_j^L$ is the $j$th activation of the sigmoid cross-entropy layer. It denotes
the confidence of $\mathbf{X}$ being related to the lesion tag $j$. In the
test phase, we use a threshold function $t(\mathbf{X})$ to determine the
lesion tags of the input image $\mathbf{X}$, i.e.
$\widehat{\mathbf{Y}} = [\widehat{y}_1, \widehat{y}_2, \dots, \widehat{y}_Q]^T$
where
\begin{equation}
\widehat{y}_j = \begin{cases}
1, & a_j^L > t(\mathbf{X}), \\
0, & \text{otherwise,}
\end{cases}
j \in \{1, 2, \dots, Q\}.
\end{equation}
For the choice of the threshold function $t(\mathbf{X})$, we adapt the method
recommended in \cite{1683770} which picks a linear function of the confidence
vector by maximizing the multi-label accuracy on the training set.
\section{Experimental Results}
In this section, we investigate the performance of the CNNs trained for
the disease-targeted and lesion-targeted skin disease classifications,
respectively. For both the disease-targeted and lesion-targeted classifications,
we use transfer learning \cite{DBLP:journals/corr/YosinskiCBL14} \footnote{We
use transfer learning and fine-tuning interchangeably in this paper.} to train
the CNNs. However, note that the ImageNet pre-trained models are trained
from images containing mostly artifacts, animals, and plants. This is very
different from our skin disease cases. To investigate the features learned only
from skin diseases and avoid using useless features, we also train the CNNs
from scratch.
We conduct all the experiments using the Caffe deep learning framework \cite{jia2014caffe}
and run the programs with a GeForce GTX 970 GPU. For the hyper-parameters,
we follow the settings used by the AlexNet, i.e., batch size $= 256$,
momentum $= 0.9$ and weight decay $= 5.0e^{-4}$. We use $0.01$ and $0.001$ learning
rate for fine-tuning and training from scratch, respectively.
\subsection{Performance of Disease-Targeted Classification}
\begin{table}
\renewcommand{\arraystretch}{1.3}
\caption{Accuracies and MAP of the disease-targeted classification}
\centering
\begin{tabular}{c|c|c|c}
\textbf{Learning Type} & \textbf{Top-1 Accuracy} & \textbf{Top-5 Accuracy} & \textbf{MAP}\\
\hline\hline
Fine-tuning & 27.6\% & 57.9\% & 0.42\\
Scratch & 21.1\% & 48.9\% & 0.35\\
\end{tabular}
\label{tab: diagnosis accuracies}
\end{table}
\begin{figure}
\centering
\includegraphics[scale=0.40]{images/confusion_matrix.pdf}
\caption{The confusion matrix of the disease-targeted skin disease classifier
with the CNN trained using fine-tuning. Row: Actual diagnosis. Column: Predicted
diagnosis.}
\label{fig: confusion matrix}
\end{figure}
To evaluate the performance of the disease-targeted skin disease classifier,
we use the top-$1$ and top-$5$ accuracies, MAP score, and the confusion matrix as
the metrics. Following the notations in Section \ref{sec: methodology}, let
$\mathbf{C}_n$ be the output of the multi-class CNN when the input is $\mathbf{X}_n$
and $\mathbf{T}_n^k$ be the labels of the $k$ largest elements in $\mathbf{C}_n$.
The top-$k$ accuracy of the multi-class CNN on the test set is given as
\begin{equation}
A_{\text{top-}k} = \frac{\sum_{n=1}^NZ_n^k}{N},
\end{equation}
where $Z_n^k$ is
\begin{equation}
Z_n^k = \begin{cases}
1, & y^n \in \mathbf{T}_n^k, \\
0, & \text{otherwise.}
\end{cases}
\end{equation}
and $N$ is the total number of images in the test set. For the MAP, we adapt the
definition described in \cite{zhu2004recall}:
\begin{equation}
\text{MAP} = \frac{1}{N}\sum_{i=1}^N\sum_{j=1}^Qp_i(j)\Delta r_i(j),
\end{equation}
where $p_i(j)$ and $r_i(j)$ denote the precision and recall of the $i$th image at
fraction $j$, $\Delta r_i(j)$ denotes the change in recall from $j-1$ to $j$ and
$Q$ is the total number of possible lesions.
Finally, for the confusion matrix $\mathbf{M}$, its elements are given as
\begin{equation} \label{eq: confusion matrix}
\mathbf{M}(i, j) = \frac{\sum_{n=1}^NI(y^n = i)I(\widehat{y}^n = j)}{N_i}
\end{equation}
where $y^n$ is the ground truth, $\widehat{y}^n$ is the prediction and $N_i$ is
the number of images whose ground truth is $i$.
Table \ref{tab: diagnosis accuracies} shows the accuracies and MAP
of the disease-targeted skin disease classifiers with the CNNs trained from
scratch or using fine-tuning. It is interesting to note that the CNN trained using transfer
learning performs better than the CNN trained from scratch only on skin diseases. It suggests that the
more general features learned from the richer set of ImageNet images can still benefit the more specific classification
of the skin diseases. And training from scratch did not necessarily help the CNN learn more
useful features related to the skin diseases. However, even for the CNN trained
with fine-tuning, the accuracies and MAP are not satisfactory. Only $27.6\%$ top-$1$
accuracy, $57.9\%$ top-$5$ accuracy, and $0.42$ MAP score are achieved.
The confusion matrix computed for the fine-tuned CNN is given in Figure
\ref{fig: confusion matrix}. The row indices correspond to the actual diagnosis
labels and the column indices denote the predicted diagnosis labels. Each cell
is computed using Equation \eqref{eq: confusion matrix} which is the
percentage of the prediction $j$ among images with ground truth $i$. A good
multi-class classifier should have high diagonal values. We find in Figure
\ref{fig: confusion matrix} that there are some off-diagonal cells with relatively
high values. This is because some skin diseases are visually similar, and
the CNNs trained with diagnosis labels still cannot distinguish among them. For example,
the off-diagonal cell at row $8$ and column $22$ has a value of $0.60$. Here,
label $8$ represents ``compound nevus'' and label $22$ stands for
``malignant melanoma''. It means about $60\%$ of the ``compound nevus'' images are
incorrectly labeled as ``malignant melanoma''. If we look at the two images
in the first column of Figure \ref{fig: similar diseases}, we can see that these
two diseases look so similar in appearance that not surprisingly the disease-targeted classifier
fails to distinguish them.
\begin{figure}
\centering
\includegraphics[scale=0.24]{images/map.pdf}
\caption{Macro-average of precision, recalls, and F-measures as well as MAP.}
\label{fig: overall metrics}
\end{figure}
\subsection{Performance of Lesion-Targeted Classification}
\begin{figure*}
\centering
\subfloat[Precisions]
{\includegraphics[width=0.32\linewidth]{images/precisions.pdf}}\hfill
\subfloat[Recalls]
{\includegraphics[width=0.32\linewidth]{images/recalls.pdf}}\hfill
\subfloat[F-measures]
{\includegraphics[width=0.32\linewidth]{images/f_scores.pdf}}\hfill
\caption{Label-based precisions, recalls, and f-measures}
\label{fig: precision, recall, f-mea}
\end{figure*}
As we use a multi-label classifier for the lesion-targeted skin disease classification,
the evaluation metrics used in this experiment are different from those used
in the previous section. To evaluate the performance of the classifier on each
label, we use the label-based precision, recall and F-measure. And to evaluate
the overall performance, we use the macro-average of the precision, recall and F-measure.
In addition, the MAP is also used as an evaluation metric of the overall performance.
Let $Y_i$ be the set of images whose ground truth contains lesion $i$ and $Z_i$ be
the set of images whose prediction contains lesion $i$. Then, the label-based
and the macro-averaged precision, recall, and F-measure can be defined as
\begin{align}
\begin{split}
P_i &= \frac{|Y_i \cap Z_i|}{|Z_i|}, P_{macro} = \frac{1}{Q}\sum_{i=1}^QP_i, \\
R_i &= \frac{|Y_i \cap Z_i|}{|Y_i|}, R_{macro} = \frac{1}{Q}\sum_{i=1}^QR_i, \\
F_i &= \frac{2|Y_i||Z_i|}{|Y_i| + |Z_i|}, F_{macro} = \frac{1}{Q}\sum_{i=1}^QF_i. \\
\end{split}
\end{align}
where $Q$ is the total number of possible lesion tags.
Figure \ref{fig: overall metrics} shows the overall performance of the lesion-targeted
skin disease classifiers. The macro-average of the F-measure is around $0.55$
and the mean average precision is about $0.70$. This is quite good for
a multi-label problem. The label-based precisions, recalls, and F-measures are
given in Figure \ref{fig: precision, recall, f-mea}. We can see that for the
lesion-targeted skin disease classification, the fine-tuned CNN performs better than the
CNN trained from scratch which is consistent with our observation in Table
\ref{tab: diagnosis accuracies}. It means for the lesion-targeted skin disease classification
problem, it is still beneficial to initialize with weights from ImageNet pretrained
models. We also see that the label-based metrics are mostly above $0.5$ in
the fine-tuning case. Some exceptions are atrophy ($0$), erythemato-squamous ($4$),
excoriation ($6$), oozing ($15$), and vesicle ($22$). The failures are mostly
due to
\begin{inparaenum}[1)]
\item the lesiona not visually salient or masked by other larger lesions, or
\item sloppy labeling of the ground truth.
\end{inparaenum}
Some failure cases are shown in Figure \ref{fig: failures}. Image $A$ is labeled
as atrophy. However, the atrophic characteristic is not so obvious and it is more
like an erythematous lesion. For image $B$, the ground truth is excoriation which
is the little white scars on the back. However, the red erythematous lesion
is more apparent. So the CNN incorrectly classified it as a erythematous lesion.
Similar case can be found in image $D$. For image $C$, the ground truth is actually
incorrect.
\begin{figure}
\centering
\includegraphics[scale=0.2]{images/failures.pdf}
\caption{Failure cases. Ground truth (left to right): atrophy, excoriation,
hypopigmented, vesicle. Top prediction (left to right): erythematous,
erythematous, ulceration, edema.}
\vskip -0.1in
\label{fig: failures}
\end{figure}
Figure \ref{fig: image retrievals} shows the image retrievals using the lesion-targeted
classifier. Here, we take the output of the second to last fully-connected layer
($4096$ dimension) as the feature vector. For each query image from the test set,
we compare its features with all the images in the training set and outputs the
$5$-nearest neighbors (in euclidean distance) as the retrievals. The retrieved
images with green solid frames match at least one lesion tag of the query
image. And those images with red dashed frames have no common lesion tags with
the query image. We can see that the retrieved images are visually and semantically
similar to the query images.
\begin{figure}[t]
\centering
\includegraphics[scale=0.04]{images/lesion_retrievals}
\caption{Images retrieved by the lesion-targeted classifier. Row 1: the query
images from the test set. Row 2-6: the retrieved images from the training set. Dotted borders annotate errors.
Ground truth of the test images from column A to D: (crust, ulceration),
(hyperpigmented, tumour), (scales), (erythematous, telangiectasis),
(nail hyperpigmentation, onycholysis), (edema, erythematous).}
\label{fig: image retrievals}
\vskip -0.1in
\end{figure}
\section{Conclusion}
In this study, we have showed that, for skin disease
classification using CNNs, lesion tags rather than the diagnosis tags should be considered as the target for automated analysis. To achieve better
diagnosis results, computer aided skin disease diagnosis systems could use
lesion-targeted CNNs as the cornerstone component to facilitate the final disease diagnosis in
conjunction with other evidences. We have built a large-scale dermatology dataset from six
professional photosharing dermatology atlantes. We have trained and tested the
disease-targeted and lesion-targeted classifiers using CNNs. Both fine-tuning
and training from scratch were investigated in training the CNN models. We found that, for
skin disease images, CNNs fine-tuned from pre-trained models perform better
than those trained from scratch. For the disease-targeted classification,
it can only achieve $27.6\%$ top-$1$ accuracy and $57.9\%$ top-$5$ accuracy as
well as $0.42$ MAP. The corresponding confusion matrix contains
some high off-diagonal values which indicates that some skin diseases cannot
be distinguished using diagnosis labels. For the lesion-targeted classification,
a $0.70$ MAP score is achieved, which is remarkable
for a multi-label classification problem. Image retrieval results also confirm
that CNNs trained using lesion tags learn the dermatology features very well.
\section*{Acknowledgment}
We gratefully thank the support from the University, and New York
State through Goergen Institute for Data Science.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2019-12-02T02:04:48",
"yymm": "1812",
"arxiv_id": "1812.03520",
"language": "en",
"url": "https://arxiv.org/abs/1812.03520"
}
|
\section{Introduction}\label{sec:intro}
There is now a lot of interest to essentially non-perturbative vacuum polarization (Casimir) effects in quasi-one-dimensional QED systems caused by charged impurities. Actually, one-dimensional QED systems with impurities appear nowadays
in many situations, which fill the range from relativistic H-like atoms in a strong homogenous magnetic field ~\cite{atom, davydov2017, sveshnikov2017, voronina2017}
up to charged impurities in low-dimensional nanostructures like semiconductor quantum wires, carbon nanotubes, in conducting
polymers, etc.~\cite{giamarchi2004}, fermionic atoms in ultracold
gases~\cite{moritz2003, recati2005a, kolomeisky2008} and defects in one-dimensional fermionic quantum liquids~\cite{recati2005b, fuchs2007, romeo2016}. Impurities have a profound effect on the physical properties of these low-dimensional systems. In certain exceptionally
clean systems, impurities can be created and controlled up to the Casimir forces between them mediated by fermions. The general literature on the Casimir effect is
vast and the reader may consult Ref.~\cite{14} for some experimental
results and Refs.~\cite{15}-\cite{19} for reviews and background
work. The Casimir interaction mediated by fermions has been intensively studied from different points of view and in different geometries during the last two decades in Refs.~\cite{20}. The main result is that for Dirac fermions we have a Casimir force whose strength and sign can
be tuned by the impurity separation and their internal structure. This provides a
physical situation where the Casimir interaction could be continuously tunable from attractive through almost completely compensated to the repulsive one by variation of an internal control parameter, realizing the known
bounds for the one dimensional Casimir interaction as
two limiting cases. In the light of
proofs showing the absence of repulsive Casimir interactions
for the photonic field in vacuum, this is a quite remarkable situation.
Moreover, in Ref.~\cite{tanaka2013} it was shown that the electronic Casimir force between two impurities on a one-dimensional semiconductor quantum wire can be of a very long range, despite nonzero effective mass of the mediator.
Of special interest in the fermionic Casimir effect is the situation, when for some reasons the impurities should be modeled as $\delta} \def\D{\Delta$-like sources, since the Dirac equation (DE) is inconsistent with direct inserting of external $\delta} \def\D{\Delta$-potentials. This problem was explored in Ref.~\cite{Jaffe2004} in terms of the energy density and interaction between two ``Dirac spikes'' as a function of a single ``spike'' parameters and the distance between them. In this model each ``spike'' is represented by a square barrier, which enters the fermion dynamics as an additional mass term, and the $\delta} \def\D{\Delta$-limit is considered via transfer-matrix, which in this limit allows for a self-consistent treatment. In Refs.~\cite{nanotubes} the Casimir interaction between two square potential barriers (``scatterers''), mediated by the massless fermions, has been considered. The Casimir force between the scatterers was found for both the case of finite width and strength of the barriers and in the $\delta} \def\D{\Delta$-limit. The result of both works is that for identical $\delta} \def\D{\Delta$-scatterers, separated by a large distance $d$, the interaction force between them reveals the conventional attractive asymptotics $\sim 1/d^2$. At the same time, for a more general case of inequivalent scatterers the magnitude and sign of the force depend on their relative spinor polarizations ~\cite{nanotubes}.
In this paper within the framework of general quasi-one-dimensional QED system we consider the Casimir interaction of two short-range Coulomb sources, either extended or $\delta} \def\D{\Delta$-like, which enter the fermion dynamics as localized electrostatic potentials. In the case of the scalar coupling, considered in Refs.~\cite{Jaffe2004,nanotubes}, the scatterers affect equally the positive- and negative-frequency fermionic modes. In the case of vector coupling the behavior of electronic and positronic components is principally different and leads to a number of new effects, the most significant of which is the discrete levels diving into the lower continuum and related non-perturbative effects of vacuum reconstruction, when the positively charged sources attain the overcritical region. The main question of interest is how these non-perturbative effects, including the effects of super-criticality, manifest themselves in Casimir forces between such sources. For identical positively charged sources, by means of the original $\ln\text{[Wronskian]}$ contour integration techniques, we find that the interaction energy between sources can exceed sufficiently large negative values and simultaneously reveal the features of a long-range force in spite of nonzero fermion mass, which could significantly influence the properties of such quasi-one-dimensional QED systems. Moreover, the Casimir force shows up a highly nontrivial behavior with increasing distance between sources, which includes separate vertical jumps at finite distances, caused by the effects of discrete levels creation-annihilation at the lower threshold, as well as different exponent rates and signs in the asymptotics. The case of two $\delta} \def\D{\Delta$-like sources is also considered in detail. To the contrary, the antisymmetric source-anti-source system reveals quite different features. In particular, in this case there is no possibility for the long-range interaction between sources. The asymptotics of the Casimir force follows the standard $\exp (-2 m s)$ law. Moreover, in the symmetric case the Casimir force between sources for small separations is attractive, while in the antisymmetric one it turns into sufficiently strong repulsion. Remarkably enough, the classic electrostatic force for such Coulomb sources should be of opposite sign. There is no evident explanation for this effect. However, the set of parameters used is quite wide to consider this effect as a general one. These results may be relevant for indirect interactions between charged defects and adsorbed species in quasi-one-dimensional QED systems mentioned above.
The single short-range positively charged Coulomb source is described as a potential square well of width $2 a$ and depth $V_0$
\begin{equation}\label{v}
V(x) =-V_0\, \theta} \def\vt{\vartheta (a-|x|) \ .
\end{equation}
Actually the potential (\ref{v}) could be interpreted as created by the charged impurity considered as a spherical shell of radius $R_0$ and charge $Z$, strongly screened for $|x|>R_0$. For this case
\begin{equation}
V_0=Z \alpha} \def\tA{\tilde {A}/R_0 \ .
\end{equation}
In this work we consider the system of two such sources, separated by the distance $s$. The component of the vacuum polarization energy $\E_{vac}$, responsible for their interaction, is defined as
\begin{equation}
\E_{vac}^{int}(s)=\E_{vac,2}(s)-\E_{vac,2}(s \to \infty) \ ,
\label{casint}\end{equation}
where $\E_{vac,2}(s)$ is the total vacuum polarization energy for the system containing two potentials like (\ref{v}), located at the distance $s$ from each other, while $\E_{vac,2}(s \to \infty)=2\, \E_{vac,1}$ with the latter being the vacuum energy of a single source.
It would be worth to note that since we consider here the sources with several parameters (for a single well these are the depth $V_0$ and the half-width $a$), the subcritical and overcritical regions for a concrete level are defined by a set of pairs $(V_0,~a)$, rather than by a single quantity $Z_{cr}$, as it occurs whenever a concrete relation between the size and charge of the Coulomb source is implied. In the case of a single source (\ref{v}) in the diagram $(V_0,~a)$ the subcritical and overcritical regions are separated by a curve (see Fig. \ref{dcr}). Therefore under the notion of the ``critical charge'' $Z_{cr,i}$ for the $i$-th discrete level we'll imply the whole set of the source parameters, which separate the sub- and overcritical regions from each other, rather than one definite quantity.
As in other works on vacuum polarization in strong Coulomb fields ~\cite{wk1956}-\cite{davydov2018}, the radiative corrections from virtual photons are neglected. Henceforth, if it is not stipulated separately, the units $\hbar=m_e=c=1$ are used. Thence the coupling constant $\alpha} \def\tA{\tilde {A}=e^2$ is also dimensionless, and the numerical calculations, illustrating the general picture, are performed for $\alpha} \def\tA{\tilde {A}=1/137.036$. However, it would be worthwhile to note that for the effective electron-hole vacuum in the quasi-one-dimensional systems like nanotubes and wires, as in graphene, the actual value of the finite structure constant and hence, the magnitude of the Casimir effects could be quite different from the one in the pure QED.
\subsection*{2. Evaluation of the Casimir energy via $\ln$[Wronskian] contour integration}
The starting point for the essentially non-perturbative evaluation of the vacuum energy in QED systems is the Schwinger vacuum average ~\cite{wk1956}-\cite{21},~\cite{mohr1998}
\begin{equation}\begin{gathered}
\E_{vac}= \dfrac{1}{2}\left(\sum\limits_{\epsilon} \def\E{\hbox{$\cal E $}_n<\epsilon} \def\E{\hbox{$\cal E $}_F}\epsilon} \def\E{\hbox{$\cal E $}_n-\sum\limits_{\epsilon} \def\E{\hbox{$\cal E $}_n \geqslant \epsilon} \def\E{\hbox{$\cal E $}_F}\epsilon} \def\E{\hbox{$\cal E $}_n\right)_V \ - \\
- \ \dfrac{1}{2}\left(\sum\limits_{\epsilon} \def\E{\hbox{$\cal E $}_n<\epsilon} \def\E{\hbox{$\cal E $}_F}\epsilon} \def\E{\hbox{$\cal E $}_n-\sum\limits_{\epsilon} \def\E{\hbox{$\cal E $}_n \geqslant \epsilon} \def\E{\hbox{$\cal E $}_F}\epsilon} \def\E{\hbox{$\cal E $}_n\right)_0 \ ,
\label{eterms}
\end{gathered}\end{equation}
\normalsize
with $\epsilon} \def\E{\hbox{$\cal E $}_n$ being the eigenvalues of the corresponding DE
\begin{equation}
\[\alpha p + \beta + V(x)\] \psi(x)= \epsilon\,\psi(x) \ ,
\label{deq}
\end{equation}
while for the positively charged sources $\epsilon} \def\E{\hbox{$\cal E $}_F$ should be chosen at the lower threshold, i.e. $\epsilon} \def\E{\hbox{$\cal E $}_F=-1$. The label $V$ indicates the presence of the external potential, while the label $0$ corresponds to the free case. Throughout the paper by solving DE the representation $\alpha} \def\tA{\tilde {A}=\sigma}\def\S{\Sigma_2\, , \beta} \def\bB{\bar {B}=\sigma}\def\S{\Sigma_3$ is used.
For the subsequent analysis it is convenient to separate in (\ref{eterms}) the contributions from the discrete spectrum and both continua and apply to the latter the well-known tool, which replaces it by the scattering phase $\delta(\epsilon)$ integration (see e.g., Refs.~\cite{raja1982, sv1991, Jaffe2004} and refs. therein). After a number of almost evident steps one obtains~\cite{sveshnikov2017}
\begin{equation}
\E_{vac}=\dfrac{1}{2 \pi}\int\limits_1^{+\infty} \delta_{tot}(\epsilon)~d\epsilon +\frac{1}{2}\sum\limits_{-1 \leqslant \epsilon} \def\E{\hbox{$\cal E $}_n < 1} (1-\epsilon} \def\E{\hbox{$\cal E $}_n) \ ,
\label{evac}
\end{equation}
where $\delta_{tot}(\epsilon)$ is the total phase shift for the given $|\epsilon} \def\E{\hbox{$\cal E $}|$, including the contributions from scattering states in both continua, while in the discrete spectrum the (effective) electron rest mass is subtracted from each level in order to retain in $\E_{vac}$ the interaction effects only.
Such approach to calculation of $ \E_{vac} $ turns out to be quite effective, since the total phase shift $\delta} \def\D{\Delta_{tot}(\epsilon} \def\E{\hbox{$\cal E $})$ behaves in both (IR and UV) limits much better, than each of the elastic phases separately, and is automatically an even function of the external potential. More concretely, in the Coulomb potentials with non-vanishing source size $\delta} \def\D{\Delta_{tot}(\epsilon} \def\E{\hbox{$\cal E $})$ is finite for $\epsilon \rightarrow 1$ and decreases $\sim 1/\epsilon^3$ for $\epsilon} \def\E{\hbox{$\cal E $} \rightarrow \infty$, that provides the convergence of the phase integral in (\ref{evac}) ~\cite{davydov2017}-\cite{voronina2017},~\cite{Jaffe2004, davydov2018, voronina2018}. The sum over bound energies $1-\epsilon} \def\E{\hbox{$\cal E $}_n$ of discrete levels in the case of short-range sources like (\ref{v}) is finite from the very beginning, since such potentials allow for a finite number of discrete levels. As a result, the expression (\ref{evac}) turns out to be finite without any additional renormalization.
However, the convergence of $\E_{vac}$ in the form (\ref{evac}) is completely caused by the specifics of 1+1 D and in no way means no need for a renormalization. Renormalization via fermionic loop is required on account of the analysis of the vacuum charge density $\rho_{vac}(x)$, from which there follows that without such UV-renormalization the integral induced charge will not acquire the value that follows from quite obvious physical arguments~\cite{davydov2017}-\cite{voronina2017},~\cite{gyul1975}-\cite{mohr1998}. For the system under consideration such analysis is performed in Refs.~\cite{annphys, tmf} for both cases including the singlet and doublet of sources like (\ref{v}).
Another obvious requirement is that for $V_0 \rightarrow 0$ the value of $\E_{vac}$ should coincide with $\E_{vac}^{(1)}$, obtained in the first order of the QED perturbation theory (PT). The latter is found quite similar to the unscreened case considered in Refs.~\cite{davydov2017}-\cite{voronina2017} and for a single source like (\ref{v}) equals to
\begin{equation}
\E_{vac,1}^{(1)}=\frac{2V_0^2}{\pi^2}\, \int\limits_{0}^{+\infty}dq~\frac{\sin^2{qa}}{q^2}
\left(1-2{ \mathrm{arcsinh} (q/2) \over q \sqrt{1+(q/2)^2}}\right) \ ,
\label{evacperturb}
\end{equation}
while for the configuration containing a doublet of such identical sources, separated by the distance $d$, it is given by the following expression
\begin{equation}\begin{gathered}
\E_{vac,2}^{(1)}=\frac{2V_0^2}{\pi^2}\int\limits_{0}^{+\infty}dq~\frac{\[\sin{(q(a+d))]}-
\sin{(qd)}\]^2}{q^2} \times \\ \times \left(1-2{ \mathrm{arcsinh} (q/2) \over q \sqrt{1+(q/2)^2}}\right) \ . \label{evacperturb2}
\end{gathered}\end{equation}
It is easy to verify that the non-renormalized vacuum energy (\ref{evac}) doesn't satisfy this requirement, since the introduced below renormalization coefficient (\ref{ren}), which provides also the correspondence between $\E_{vac}^R$ and $\E_{vac}^{(1)}$ for $V_0 \rightarrow 0$, in general case turns out to be non-zero with the only exception for certain parameter sets.
For these reasons, in complete analogy with the renormalization of the charge density, considered in Refs.~\cite{davydov2017}-\cite{voronina2017},~\cite{gyul1975}-\cite{voronina2018}, we should pass from $\E_{vac}$ to the renormalized vacuum energy $\E_{vac}^R$. In the practically useful form $\E_{vac}^R$ should be represented as follows~\cite{davydov2017}-\cite{voronina2017},~\cite{davydov2018, voronina2018}
\begin{equation}
\E_{vac}^R=\E_{vac}+\lambda V_0^2 \ ,
\label{evren}
\end{equation}
where the renormalization coefficient
\begin{equation}
\lambda=\lim\limits_{V_0\rightarrow 0} \dfrac{\E_{vac}^{(1)}(V_0)-\E_{vac}(V_0)}{V_0^2} \ .
\label{ren}
\end{equation}
depends solely on the shape of the external potential and in the general 1+1 D case is a sign-alternating function with zeros \cite{davydov2017}-\cite{voronina2017}, \cite{annphys}.
The evaluation of $\E_{vac}^R$ via the sum of the phase integral and discrete levels is considered in detail in Refs.~\cite{davydov2017}-\cite{voronina2017},~\cite{davydov2018, voronina2018} for the unscreened or partially screened extended Coulomb source, and in the present case will differ only by certain technical details. However, for our purposes of a detailed study of Casimir interaction between the localized Coulomb-like external sources an alternative approach for evaluation of the non-renormalized $\E_{vac}$ turns out to be more efficient. This approach is quite similar to the calculation of the vacuum density $\rho_{vac}(x)$ via integration of specially constructed function along the Wichmann-Kroll (WK) contours \cite{wk1956, gyul1975, mohr1998}, which are shown in Fig.\ref{contour}.
\begin{figure}[b]
\begin{center}
\includegraphics[width=0.5\textwidth]{TrGcontourCF.eps}
\end{center}
\caption{The WK-contours in the complex energy plane, used for the representation of the vacuum charge density and vacuum energy via contour integrals. }
\label{contour}
\end{figure}
Namely, it is easy to see that the function
\begin{equation}
F(\epsilon,V_0)=\dfrac{\epsilon \left(\mathrm{d} J(\epsilon)/ \mathrm{d} \epsilon \right)}{J(\epsilon)} \ ,
\label{F}
\end{equation}
where $J(\epsilon} \def\E{\hbox{$\cal E $})$ is the Wronskian for the spectral problem (\ref{deq}), reveals all the pole properties, which are required for the representation of the expression (\ref{eterms}) via integrals along the WK contours, since actually $J(\epsilon} \def\E{\hbox{$\cal E $})$ is nothing else, but the Jost function of the spectral problem (\ref{deq}) \cite{sveshnikov2017}. The real zeros of $J(\epsilon} \def\E{\hbox{$\cal E $})$ lie in the interval $-1 \leqslant \epsilon} \def\E{\hbox{$\cal E $}_n < 1$ and coincide with discrete energy levels, while the complex ones reside on the second sheet of the Riemann energy surface with negative imaginary part of the wavenumber $k=\sqrt{\epsilon} \def\E{\hbox{$\cal E $}^2-1}$ and for $\mathrm{Re} \ k >0 $ correspond to the elastic resonances. Moreover, both $J(\epsilon} \def\E{\hbox{$\cal E $})$ and $\hbox{Tr} G$ as functions of the wavenumber $k$ reveal the same reflection symmetry $f^{\ast}(k)=f(-k^{\ast})$ of the Jost function.
To represent $\E_{vac}$ via integration along the WK contours, it suffices to pass to the difference \begin{equation}
\mathcal{H}(\epsilon} \def\E{\hbox{$\cal E $},V_0)=F(\epsilon} \def\E{\hbox{$\cal E $},V_0)-F(\epsilon} \def\E{\hbox{$\cal E $},0) \ ,
\label{H}
\end{equation}
normalized on the free case. As a result, the non-renormalized induced vacuum energy can be represented as
\begin{equation}
\E_{vac}=-\frac{1}{4 \pi i}\lim\limits_{R\rightarrow\infty} \left( \int\limits_{P(R)}d\epsilon ~ \mathcal{H}(\epsilon,V_0) + \int\limits_{E(R)}d\epsilon~ \mathcal{H}(\epsilon,V_0)\right) \ .
\label{41}\end{equation}
In the next step one finds by means of the analysis of the asymptotics of the function $\mathcal{H}(\epsilon,V_0)$ on the large circle in Fig.\ref{contour} that the initial integration along the contours $P(R)$ and $E(R)$ for the singlet or doublet of external potentials like (\ref{v}) can be reduced to integration along the imaginary axis \cite{annphys}. Upon taking into account the (possible) existence of negative discrete levels and proceeding further in complete analogy with the corresponding treatment of the vacuum density, performed in Refs. \cite{davydov2017}-\cite{voronina2017},\cite{gyul1975}, one finds the final expression for $\E_{vac}$ in the following form
\begin{equation}
\E_{vac}=\dfrac{1}{2 \pi} \int\limits_{-\infty}^{+\infty} dy~ \mathcal{H}(i y,V_0) - \sum\limits_{-1 \leqslant \epsilon} \def\E{\hbox{$\cal E $}_n <0} \epsilon} \def\E{\hbox{$\cal E $}_n \ .
\label{econt}
\end{equation}
For the single source (\ref{v}) the integrand in (\ref{econt}) takes the form
\begin{widetext}\begin{equation}
\begin{aligned}
\mathcal{H}(i y,V_0)=
\dfrac{iV_0\, y\,(V_0[V_0+2iy]\gamma} \def\tg{\tilde {\g}}\def\G{\Gamma} \def\bG{\bar {G}(iy)+2aj^2(iy,V_0)\gamma} \def\tg{\tilde {\g}}\def\G{\Gamma} \def\bG{\bar {G}^2(iy)\sin[2aj(iy,V_0)])}{j^2(iy,V_0)\gamma} \def\tg{\tilde {\g}}\def\G{\Gamma} \def\bG{\bar {G}^3(iy)(j(iy,V_0)\gamma} \def\tg{\tilde {\g}}\def\G{\Gamma} \def\bG{\bar {G}(iy)\cos[2aj(iy,V_0)]+(1-iV_0\,y+y^2)\sin[2aj(iy,V_0)])} - \\
- \dfrac{2ia\,V_0\,y\,j(iy,V_0)\,\gamma} \def\tg{\tilde {\g}}\def\G{\Gamma} \def\bG{\bar {G}^3(iy)\,\cos[2aj(iy,V_0)]}{j^2(iy,V_0)\gamma} \def\tg{\tilde {\g}}\def\G{\Gamma} \def\bG{\bar {G}^3(iy)\,(j(iy,V_0)\gamma} \def\tg{\tilde {\g}}\def\G{\Gamma} \def\bG{\bar {G}(iy)\cos[2aj(iy,V_0)]+(1-iV_0\,y+y^2)\sin[2aj(iy,V_0)])} \ ,
\end{aligned}
\end{equation}\end{widetext}
where
\begin{equation}\label{j}
j(\epsilon,V_0)=\sqrt{(V_0+\epsilon)^2-1} \ , \quad \gamma(\epsilon)=\sqrt{1-\epsilon^2} \ .
\end{equation}
For the doublet configuration the corresponding expression will be presented below.
The direct numerical calculation shows that both approaches to the vacuum energy (\ref{evac}) and (\ref{econt}) lead with a high precision to the same results.
Besides $\E_{vac}$, in 1+1 D the renormalization term $\lambda} \def\L{\Lambda V_0^2$ in the expression (\ref{evren}) turns out to be quite important, especially in the non-perturbative region. For a single source (\ref{v}) the dependence of the renormalization term on the source parameters is determined first of all by the coefficient $\lambda(a)$, which can be represented as
\begin{equation}
\lambda} \def\L{\Lambda(a)=\lambda} \def\L{\Lambda_1(a)-\lambda} \def\L{\Lambda_2(a) \ ,
\label{renpart}
\end{equation}
where $\lambda} \def\L{\Lambda_1$ emerges from the PT contribution $\E_{vac}^{(1)}$ to the vacuum energy
\begin{multline}
\lambda} \def\L{\Lambda_1(a)=\frac{a}{\pi}-I_1(a) \ , \\ I_1(a)=\frac{4}{\pi^2}\int\limits_0^\infty dq ~ \frac{\sin^2(qa)}{q^2}\, \left(1-2{ \mathrm{arcsinh} (q/2) \over q \sqrt{1+(q/2)^2}}\right) \ ,
\label{l1}
\end{multline}
while $\lambda} \def\L{\Lambda_2$ corresponds to the first (quadratic in $V_0$) term in $\E_{vac}$, which is found from the Born series (see Refs.~\cite{davydov2017}-\cite{voronina2017},~\cite{gyul1975},~\cite{davydov2018, voronina2018})
\begin{equation}
\lambda_2(a)=\frac{a}{\pi}-\frac{1}{16}+I_2(a) \ , \quad I_2(a)=\frac{1}{4\pi} \int\limits_{0}^{\infty} dy~\frac{e^{-4a\sqrt{1+y^2}}}{(1+y^2)^2} \ .
\label{l2}
\end{equation}
By means of the relation $\lambda_1(a)+\lambda_2(a)=a/\pi$, whose derivation requires some additional considerations and so is presented separately \cite{tmf}, one obtains
\begin{equation}
\lambda(a)=\dfrac{a}{\pi}-2\lambda_2(a)=\dfrac{1}{8}-\dfrac{a}{\pi}-2I_2(a) \ .
\label{lambdaf1}
\end{equation}
The asymptotics of $\lambda} \def\L{\Lambda(a)$ for $a \ll 1$ and $a \gg 1$, which are important for the further analysis of the Casimir interaction between separate sources, are considered in detail in Ref.~\cite{annphys}. So here we present only the required results. Namely, the asymptotics of $\lambda} \def\L{\Lambda(a)$ for $a \ll 1$ reads
\begin{equation}\label{limlambda}
\lambda(a\rightarrow 0)=\dfrac{a}{\pi}-2 a^2+O(a^3) \ ,
\end{equation}
while for large $a$ neglecting the exponentially small corrections it is given by
\begin{equation}
\lambda(a \rightarrow \infty)=\dfrac{1}{8}-\dfrac{a}{\pi} \ .
\end{equation}
As a result, for small $a$ the coefficient $\lambda (a)$ grows linearly, while for large $a$ it decreases with the same modulus slope $1/\pi$. Hence, the renormalization coefficient $\lambda (a)$ should vanish at certain $a=a_{cr}$. In the case of the single well (\ref{v}) it has a unique zero when $a_{cr}\simeq 0.297 $. More details concerning the behavior of $\lambda (a)$ are given in Ref.~\cite{annphys}.
\subsection*{3. Casimir energy of two identical positively charged short-range Coulomb sources}
Now -- having dealt with the general formulation for calculation of $\E_{vac}$ this way -- let us turn to the configuration of two such identical positively charged short-range Coulomb sources, described by the potential
\begin{equation}
V_2(x) =-V_0 \, \theta} \def\vt{\vartheta\(|x|- d\)\,\theta} \def\vt{\vartheta\(d+a -|x|\) \ .
\label{v2}
\end{equation}
Let us note that now the separate sources have the total width $a$, that provides the restoration of the initial potential well (\ref{v}) with the width $2a$ for $d \to 0$.
Further procedure of calculation and renormalization of the vacuum energy repeats completely the case of the single source and so doesn't need any special discussion besides the structure of the renormalization term in $\E_{vac,2}^R$. As in the case of one potential well, the calculation of $\E_{vac,2}^R$ requires the renormalization in the second order with respect to the external potential
\begin{equation}
\E_{vac,2}^R=\E_{vac,2}+\Lambda(a,d) V_0^2 \ ,
\label{evren2}
\end{equation}
where
\begin{equation}
\Lambda(a,d)=\Lambda_1(a,d)-\Lambda_2(a,d) \ .
\end{equation}
The components of the renormalization coefficient $\Lambda_i(a,d)$, $i=1\, , 2\,$, are expressed in terms of the corresponding coefficients $\lambda_i(a)$ for the single source as follows
\begin{equation}
\Lambda_i(a,d)=\lambda_i(a+d)+\lambda_i(d)+2\lambda_i(a/2)-2\lambda_i(d+a/2) \ .
\label{lambig12}
\end{equation}
From (\ref{lambig12}) by means of the relation $\lambda_1(a)+\lambda_2(a)=a/\pi$ one finds that $\Lambda_i(a,d)$ are subject of the same relation
\begin{equation}
\Lambda_1(a,d)+\Lambda_2(a,d)=a/\pi \ .
\end{equation}
As a result, the renormalization coefficient for the two-well problem (\ref{v2}) can be represented as
\begin{multline}
\Lambda(a,d)=a/\pi-2\Lambda_2(a,d)
= \\ = a/\pi-2\lambda_2(a+d)-2\lambda_2(d)-4\lambda_2(a/2)+4\lambda_2(d+a/2) \ .
\label{lambifin}
\end{multline}
Now let us list the results for found this way $\E_{vac,2}^R$, which are necessary for the further analysis of the Casimir interaction between separate sources. The most significant here is the dependence on the parameter $d\, , \ 0 \leq d \leq \infty$, which defines the separation of the sources in such a way that the distance $s$ between the centers of the wells is given by
\begin{equation}
s=a+2\, d \ .
\end{equation}
At first let us explore the features of the discrete spectrum of DE with the potential (\ref{v2}). For $d \to \infty $ the wells become independent, while the spectrum of DE -- twice degenerate. More concretely, with growing $d$ the even levels increase, while the odd ones, in contrast, decrease, and so for $d \rightarrow \infty$ the neighboring even and odd levels seek each other. To analyze the role of $d$ in the overcritical region the equations for the critical parameters of the source (\ref{v2}) (i.e., the set $[V_0$, $a$ , $d]$, for which the discrete levels approach the threshold of the lower continuum) should be considered. For odd levels the ``critical charges'' are determined from the equation
\begin{equation}
\sin[a\sqrt{(V_0-1)^2-1}]=0 \ ,
\label{codd}\end{equation}
which coincides with the similar equation for a single potential well up to replacement $a \to 2 a$. Since (\ref{codd}) doesn't depend on $d$, any change of $d$ for fixed $(V_0\, , a)$ doesn't yield any diving of odd levels into the lower continuum. At the same time, for even levels the equation for their ``critical charges'' takes the form
\begin{multline}
\sqrt{(V_0-1)^2-1} \cos[a\sqrt{(V_0-1)^2-1}] \ + \\ + \ 2d\,V_0 \sin[a\sqrt{(V_0-1)^2-1}]=0 \ .
\label{creven}
\end{multline}
So for even levels the ``critical charges'' depend on the distance between the sources. The parameter $d$ can be easily found from (\ref{creven}), and so the dependence $d(V_0$, $a)$ together with condition $d > 0$ defines the critical values of $d$ for even levels in the potential (\ref{v2}). The regions $(V_0\, , a)$, where the even levels diving into the lower continuum takes place by certain $d>0$, are shown as shaded ones in Fig.\ref{dcr}. The non-shaded regions in Fig.\ref{dcr} correspond to those sets of $(V_0\, , a)$, when the eq.(\ref{creven}) doesn't possess any solutions with $d >0 $. The solid and dashed curves in Fig.\ref{dcr} determine the sets $(V_0\, , a)$, when $d_{cr}=0$, and so correspond to the critical charges for a single well.
\begin{figure}[h]
\begin{center}
\includegraphics[width=1.0\linewidth]{dcr.eps}
\end{center}
\caption{The shaded regions correspond to those sets of $(V_0\, , a)$, when by varying $d$ it is possible to provide the diving of the lowest even level into the continuum. The solid lines correspond to even, while the dashed ones --- to odd ``critical charges'' for a single source (\ref{v}).}
\label{dcr}
\end{figure}
This way there appear two essentially different regimes for behavior of $\E_{vac,2}^R$ as a function of the source separation. The first one corresponds to the situation, when in the whole interval $0< d \leqslant \infty$ no discrete level attains the lower continuum, nor does any one return back from the continuum (the parameters $(V_0\, , a)$ correspond to $d<0$ in the eq. (\ref{creven})). In this case the integral vacuum charge $Q_{vac,2}$ keeps its value, while the vacuum energy $\E_{vac,2}^R$, the jumps in which are entirely due to creation-annihilation of discrete levels from the lower continuum, is a continuous function of $d$ and $s$. This regime for $\E_{vac,2}^R(s)$ is shown in Fig.\ref{vac2}a, calculated for $V_0=2$, $a=1$, which correspond to the lowest unpainted region in Fig.\ref{dcr}.
Numerical integration confirms that in this case the integral induced charge $Q_{vac}$ doesn't depend on $s$ and vanishes, since the parameters $(V_0\, , a)$ lie in the subcritical region and so varying $s$ doesn't lead to appearance of new levels at the lower threshold, while the dependence of the renormalized vacuum energy $\E_{vac,2}^R(s)$, as it follows from the Fig.\ref{vac2}a, is given by a continuous curve.
\begin{figure}[ht!]
\subfigure[]{
\includegraphics[width=\columnwidth]{Evac21.eps}
}
\hfill
\subfigure[]{
\includegraphics[width=\columnwidth]{Evac22.eps}
}
\caption{ (a): $\E_{vac,2}^{R}(s)$ for (a): $a=1$, $V_0=2$; (b): for $a=1$, $V_0=4.08$.}
\label{vac2}
\end{figure}
The second regime for $\E_{vac,2}^R(s)$ is realized for $(V_0\, , a)$, which lie in the shaded regions in Fig.\ref{dcr}. For such values of $(V_0\, , a)$ with growing $s$ one (or several) discrete levels emerge from the lower continuum by passing through the corresponding $s_{cr}=a+ 2 d_{cr}$. During this process the vacuum charge $Q_{vac}$ each time grows by $+|e|$, while $\E_{vac,2}^R(s)$ acquires a specific jump upwards by $+1$.
The direct calculation of $\E_{vac,2}^R(s)$ (see Fig.\ref{vac2}b) for the parameters $V_0=4.08$ and $a=1$, which reside in the first from below shaded region in Fig.\ref{dcr}, confirms these effects completely. In particular, the numerical check shows that due to emergence at $s_{cr}\simeq 4.0709$ ($d_{cr} \simeq 1.5354$) of one even level from the lower continuum the integral vacuum charge grows from $Q_{vac}=-|e|$ for $s<s_{cr}$ up to $Q_{vac}=0$ for $s>s_{cr}$. Simultaneously for $s=s_{cr}$ there takes place a jump in the vacuum energy $\E_{vac,2}^R(s)$ by $+1$, as it follows from Fig.\ref{vac2}b. More details concerning the behavior of the charge density for these two regimes are considered in Refs.~\cite{annphys, tmf}.
\subsection*{4. Casimir forces between two identical positively charged short-range Coulomb sources}
The Casimir interaction energy $\E_{vac}^{int}(d)$ for the system of two identical short-range Coulomb sources (\ref{v2}) is determined through the relation (\ref {casint}) with subsequent renormalization. In what follows we'll use the parametrization of the source separation in terms of $d$ instead of $s$ as the most pertinent one. Indeed here the efficiency of the method (\ref{41}), based on the integration of the logarithmic derivative of the Wronskian along the WK contours (Fig.\ref{contour}) compared to evaluation of $\E_{vac}^R$ via the sum of the phase integral and discrete levels (\ref{evac}),(\ref{evren}), shows up most clearly, since it provides for $\E_{vac}^{int}(d)$ more analytic details, at least for $d \gg 1$.
Upon subtraction of $2\E_{vac}^R(V_0,\,a/2)$ from the expression (\ref{evren2}) the general structure of $\E_{vac}^{int}(d)$ takes the form
\begin{equation}\label{Evacint}
\E_{vac}^{int}(d)=I_{int}(d)-S_{int}(d)+\Lambda_{int}(d)\,V_0^2 \ ,
\end{equation}
with $I_{int}(d)$ being the contribution from the integral term, $S_{int}(d)$ -- from the negative discrete levels, while $\Lambda_{int}(d)\,V_0^2$ -- from the renormalization term, respectively.
It would be pertinent to start with the renormalization term, the asymptotics of which for large $d$ can be explored most simply and in the general form. By means of (\ref{lambdaf1}) and (\ref{lambifin}) the renormalization coefficient $\Lambda_{int}(d)=\Lambda(a,d)-2\lambda(a/2)$ can be represented as
\begin{equation}
\begin{aligned}
&\Lambda_{int}(d)=a/\pi-2\lambda_2(a+d)-2\lambda_2(d)-4\lambda_2(a/2) \ + \\ &
+ \ 4\lambda_2(d+a/2)-2(a/(2\pi)-2\lambda_2(a/2))= \\
&=4\lambda_2(d+a/2)-2\lambda_2(a+d)-2\lambda_2(d) \ .
\label{renint}
\end{aligned}
\end{equation}
\normalsize
In the next step, inserting the definition of $\lambda_2$ (\ref{l2}) into (\ref{renint}), one finds
\small
\begin{equation}\label{lambda-int}
\begin{gathered}
\Lambda_{int}(d)=4I_2(d+a/2)-2I_2(a+d)-2I_2(d)=\\ =-\dfrac{1}{2\pi}\int\limits_0^\infty \dfrac{(1-e^{-2a\sqrt{1+y^2}})^2 e^{-4d\sqrt{1+y^2}}}{(1+y^2)^2}\,dy \leqslant 0 \ .
\end{gathered}
\end{equation}
\normalsize
So the contribution to the interaction energy $\E_{vac}^{int}(d)$ from the renormalization term turns out to be strictly negative and exponentially decreasing for $d \gg 1$.
The exact form of the asymptotics of $\Lambda_{int}(d \gg 1)$ can be found from the expression (\ref{renint}) via triple integration of the MacDonald function asymptotics in the way, quite similar to the evaluation of the asymptotics of $\lambda(a\rightarrow \infty)$, considered in Ref.~\cite{annphys}, and takes the form
\begin{multline}
\Lambda_{int}(d \gg 1)=-\dfrac{e^{-4 d}}{\sqrt{2 \pi d}}\,e^{-2 a}\,\Big(\sinh^2a \ + \\ + \ \dfrac{\sinh a\,(8 a e^{-a}-13 \sinh a)}{32 d} + O\left(\dfrac{1}{d^2}\Big)\right) \ .
\label{aslam}
\end{multline}
Now let us consider the behavior of the integral term in (\ref{Evacint}) for $d \gg 1$, at first without subtracting the contribution from infinitely separated wells. Upon integration by parts it can be written as follows
\begin{equation}\label{IntWronskReg}
I(d)=-{1\over \pi}\int\limits_0^{\infty} dy~ \mathrm{Re}\left[\ln \(J_{red}(d,i y)\)\right] \ ,
\end{equation}
where the ``reduced'' Wronskian
\begin{equation}\label{Jred}
J_{red}(d,\epsilon} \def\E{\hbox{$\cal E $})=J(d,\epsilon} \def\E{\hbox{$\cal E $})/J_0(\epsilon} \def\E{\hbox{$\cal E $})
\end{equation}
contains in the nominator the Wronskian $J(d,\epsilon} \def\E{\hbox{$\cal E $})$ for the double-well potential (\ref{v2})
\begin{equation}
J(d,\epsilon} \def\E{\hbox{$\cal E $})={2\, e^{-2 a \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2}}\over \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2}}\left[ f_1^2(\epsilon} \def\E{\hbox{$\cal E $})-e^{-4 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2}} f_2^2(\epsilon} \def\E{\hbox{$\cal E $}) \right] \ ,
\end{equation}
in which
\begin{equation}
\begin{gathered}
f_1(\epsilon} \def\E{\hbox{$\cal E $})=\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2}\, \cos(a\sqrt{(V_0+\epsilon} \def\E{\hbox{$\cal E $})^2-1})- \\ - (\epsilon} \def\E{\hbox{$\cal E $}^2-1+ \epsilon} \def\E{\hbox{$\cal E $}\,V_0)\, \sin(a\sqrt{(V_0+\epsilon} \def\E{\hbox{$\cal E $})^2-1})/ \sqrt{(V_0+\epsilon} \def\E{\hbox{$\cal E $})^2-1} \ , \\
f_2(\epsilon} \def\E{\hbox{$\cal E $})=V_0 \sin(a\sqrt{(V_0+\epsilon} \def\E{\hbox{$\cal E $})^2-1})/ \sqrt{(V_0+\epsilon} \def\E{\hbox{$\cal E $})^2-1} \ ,
\end{gathered}
\end{equation}
while in the denominator the Wronskian $J_0(\epsilon} \def\E{\hbox{$\cal E $})=2\, \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2}$, corresponding to the free case $V_0=0$.
The behavior of the integral (\ref{IntWronskReg}) for large $d$ is found via the following expansion of the integrand
\begin{equation}\label{74}\begin{gathered}
\ln \[J_{red}(d,i y)\]=\ln\left( f^2_1(i y) {e^{-2 a \sqrt{1+y^2}}\over 1+y^2}\right) \ - \\ - \ e^{-4 d \sqrt{1+y^2}}\,\({f_2(i y) \over f_1(i y)}\)^2+O\(e^{-8 d \sqrt{1+y^2}}\) \ .
\end{gathered}\end{equation}
Upon substituting the expansion (\ref{74}) into the integral (\ref{IntWronskReg}) one obtains two first leading terms in the asymptotics of $I(d)$ for $d \gg 1$
\begin{equation}\label{75}\begin{gathered}
I(d) \simeq -{1\over \pi}\int\limits_0^{\infty} dy~ \mathrm{Re}\left[\ln \(f^2_1(i y) {e^{-2 a \sqrt{1+y^2}}\over 1+y^2}\)\right] \ + \\ + \ {1\over \pi}\int\limits_0^{\infty} dy~ \mathrm{Re}\left[ e^{-4 d \sqrt{1+y^2}}\,\({f_2(i y) \over f_1(i y)}\)^2\right] \ .
\end{gathered}\end{equation}
Since the first term in (\ref{75}) doesn't depend on $d$, the leading term in the asymptotics of the integral term in $\E_{vac}^{int}(d)$ for $d \gg 1$ takes the form
\begin{equation}\label{76}
\begin{gathered}
I_{int}(d)=I(d)-I(d \to \infty)= \\ = -{1\over \pi}\int\limits_0^{\infty} dy~ \mathrm{Re}\left[ \ln \((1+y^2){e^{2 a \sqrt{1+y^2}}\, \over f^2_1(i y) }\,J_{red}(d,i y)\) \right] \\ \simeq {1\over \pi}\int\limits_0^{\infty} dy~ \mathrm{Re}\left[ e^{-4 d \sqrt{1+y^2}}\,\({f_2(i y) \over f_1(i y)}\)^2\right] \ .
\end{gathered}
\end{equation}
For large $d$ the integrand in (\ref{76}) decreases rapidly with growing $y$, hence, the main contribution to the integral is provided by small $y$. Therefore it turns out to be efficient to rewrite the expression (\ref{76}) in the form
\begin{equation}\label{77}\begin{gathered}
I_{int}(d)\simeq \\ {e^{-4 d}\over \pi}\int\limits_0^{\infty} dy~\mathrm{Re}\left[ e^{-4 d (\sqrt{1+y^2}-1-y^2/2)} \({f_2(i y) \over f_1(i y)}\)^2\right]e^{-2 d y^2} \ ,
\end{gathered}\end{equation}
and thereafter to expand the square brackets in the integrand in the power series in $y$. All the integrals, emerging this way, can be calculated analytically. The final expansion of $I_{int}(d)$ for $d \gg 1$ reads
\begin{equation}\label{78}\begin{gathered}
I_{int}(d)= V^2_0\, {e^{-4 d}\over \sqrt{2 \pi d}} \ \times \\ \times \ \Bigg( {A^2 \over 2} + {1\over 8 d} \left( {3 A^2 \over 8} + B \right) + O\({1\over d^2} \) \Bigg) \ ,
\end{gathered}\end{equation}
where
\begin{equation}\label{79}
\begin{gathered}
z_0=\sqrt{V^2_0 - 1} \ , \quad A={1\over 1+ z_0 \hbox{ctg} (a z_0)} \ , \\
B=A^3\Bigg[ -3 V^2_0 \left( 1 - {\hbox{ctg}(a z_0)\over z_0} + {a \over \sin^2(a z_0)} \right)^2 A \ - \\ - \ 2-{(1+z^4_0)\over z^3_0}\,\hbox{ctg}(a z_0) \ + \\
+ \ {a\over z^2_0 \sin^2(a z_0)} \left( 1- 2 V_0^2(1- a z_0 \hbox{ctg}(a z_0))\right) \Bigg] \ .
\end{gathered}
\end{equation}
It should be specially noted that the formulae (\ref{79}) work equally well both for $V_0 > 1$ and $V_0 <1$. For $V_0=1$ upon taking in (\ref{79}) the limit $z_0 \to 0$ the expressions for $A$ and $B$ are replaced by
\begin{equation}\label{80}
\begin{gathered}
A={a\over 1+ a} \ , \quad
B=-{a^2\over 45(1+ a)^4} \times \\ \times (45+135 a +255 a^2+210 a^3 + 68 a^4+ 8 a^5) \ .
\end{gathered}
\end{equation}
So the asymptotics of the integral term in $\E_{vac}^{int}(d)$ for $d \gg 1$ turns out to be $\sim e^{-4 d}/\sqrt{d}$, which is quite similar to the behavior of the renormalization term (\ref{aslam}). It should be mentioned that the expansion (\ref{78}) can be used also for finite $d$ in the case, when the each next term in the expansion (\ref{74}) is much less than the previous one. At the same time, there might occur an alternative situation, similar to the case $a=1$, $V_0=8$, considered below, when the coefficients $A$ and $B$ turn out to be quite large. The reason is that the zero denominator in $A$ is nothing else, but the condition for existence of the level with $\epsilon} \def\E{\hbox{$\cal E $}_0=0$ in the single well. For $a=1$, $V_0=8$ the lowest level is $\epsilon} \def\E{\hbox{$\cal E $}_0 \simeq 0.02085$, and so by sufficiently small variation of the well parameters this level can be made strictly zero. It follows whence that in the general case the expansion given above doesn't hold for the case, when there exists in the well the level close to $\epsilon} \def\E{\hbox{$\cal E $}_0=0$, since in this case the expansion coefficients $A$ and $B$ become large.
In the latter case it should be taken into account by expanding the square bracket in (\ref{77}) in the power series in $y$ that the expansion of the function $f_1(i y)$ starts now from the linear in $y$ term, since the first term of the series $\cos(a z_0)+\sin(a z_0)/z_0$ vanishes. As a result, for the case $\epsilon} \def\E{\hbox{$\cal E $}_0=0$ one obtains
\begin{equation}\label{e0=0}
I_{int}(d)=- {V^2_0-1 \over (1+a) V^2_0 }\, e^{-2 d}+ O\(e^{-4 d}\) \ ,
\end{equation}
whence it follows that for this special case the rate of decrease of the integral term in (\ref{Evacint}) for $d \gg 1$ becomes sufficiently less. It should be mentioned in addition that the multiplier before the leading exponent in (\ref{e0=0}) is strictly negative, since the zero level might appear in the single well only for $V_0>1$.
Now let us consider the (possible) contribution to (\ref{Evacint}) from negative discrete levels for $d \gg 1$. In the general case, the discrete levels are determined by the corresponding zeros of the Wronskian $J(d,\epsilon} \def\E{\hbox{$\cal E $})$ and satisfy the equation
\begin{equation}\label{DiscrWronsk}
f_1^2(\epsilon} \def\E{\hbox{$\cal E $})-e^{-4 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2}} f_2^2(\epsilon} \def\E{\hbox{$\cal E $}) =0 \ .
\end{equation}
For $d \to \infty$ the eq.(\ref{DiscrWronsk}) transforms into $f_1(\epsilon} \def\E{\hbox{$\cal E $})=0$, which is obviously the equation for degenerate by parity levels in the system with two infinitely separated wells, or, equivalently, for the levels of the single well.
Let us consider one of the levels $\epsilon} \def\E{\hbox{$\cal E $}_0$ in the single well, for which $f_1(\epsilon} \def\E{\hbox{$\cal E $}_0)=0$. In the limit $d \to \infty$ the value $\epsilon} \def\E{\hbox{$\cal E $}_0$ serves as the zero approximation for corresponding even and odd levels in the double-well potential (\ref{v2}). To find the splitting of $\epsilon} \def\E{\hbox{$\cal E $}_0$ into the even and odd components for finite $d \gg 1$, let us seek the solution of (\ref{DiscrWronsk}) in the form $\epsilon} \def\E{\hbox{$\cal E $}=\epsilon} \def\E{\hbox{$\cal E $}_0+\delta\epsilon} \def\E{\hbox{$\cal E $}$, where $\delta\epsilon} \def\E{\hbox{$\cal E $}$ is a small correction to $\epsilon} \def\E{\hbox{$\cal E $}_0$. Inserting this expansion into (\ref{DiscrWronsk}) and decomposing the l.h.s. in $\delta\epsilon} \def\E{\hbox{$\cal E $}$ including the third order with account of $f_1(\epsilon} \def\E{\hbox{$\cal E $}_0)=0$, one obtains a cubic equation
\begin{equation}\label{levels}
- A_1 e^{-4 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}} + B_1 e^{-4 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}} \delta\epsilon} \def\E{\hbox{$\cal E $} + C_1 \delta\epsilon} \def\E{\hbox{$\cal E $}^2 + D_1 \delta\epsilon} \def\E{\hbox{$\cal E $}^3= 0 \ ,
\end{equation}
where
\begin{equation}\label{levels1}\begin{gathered}
A_1=f^2_2(\epsilon} \def\E{\hbox{$\cal E $}_0) \ , \\ B_1=-{2 f_2(\epsilon} \def\E{\hbox{$\cal E $}_0) \over \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}}\[2 d \epsilon} \def\E{\hbox{$\cal E $}_0 f_2(\epsilon} \def\E{\hbox{$\cal E $}_0) + \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2} f'_2(\epsilon} \def\E{\hbox{$\cal E $}_0)\] \ , \\
C_1=\[f'_1(\epsilon} \def\E{\hbox{$\cal E $}_0)\]^2 \ , \quad
D_1=f'_1(\epsilon} \def\E{\hbox{$\cal E $}_0)f''_1(\epsilon} \def\E{\hbox{$\cal E $}_0) \ .
\end{gathered}\end{equation}
Solving further the eq. (\ref{levels}) by means of successive iterations, one finds the following splitting of the unperturbed level $\epsilon} \def\E{\hbox{$\cal E $}_0$
\begin{equation}\label{corr}\begin{gathered}
\delta\epsilon} \def\E{\hbox{$\cal E $} = \pm |K_1(a)| e^{-2 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}} + K_2(a,d) e^{-4 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}} + \\ + O\(e^{-6 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}}\) \ ,
\end{gathered}\end{equation}
where
\begin{equation}\label{corr1}
q_0=V_0+\epsilon} \def\E{\hbox{$\cal E $}_0 \ , \quad K_1(a)={(1-\epsilon} \def\E{\hbox{$\cal E $}^2_0)\,(1-q^2_0)\over V_0(\epsilon} \def\E{\hbox{$\cal E $}_0+q_0+a q_0\,\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2})} \ ,
\end{equation}
\begin{widetext}\begin{equation}
\label{corr2}
K_2(a,d)={(1-\epsilon} \def\E{\hbox{$\cal E $}_0^2)^{3/2}(q^2_0-1)^2 \over 2\, V^2_0 (\epsilon} \def\E{\hbox{$\cal E $}_0+q_0+a q_0\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2})^2} \times
\end{equation}
$$
\times \Big[4 d \epsilon} \def\E{\hbox{$\cal E $}_0 \ +
{2 a^2 q^2_0\, (1-\epsilon} \def\E{\hbox{$\cal E $}_0^2)\,(\epsilon} \def\E{\hbox{$\cal E $}_0 q_0-1)+(2-\epsilon} \def\E{\hbox{$\cal E $}^2_0-q^2_0)\,(\epsilon} \def\E{\hbox{$\cal E $}_0 q_0+1) + a \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}\,(2\epsilon} \def\E{\hbox{$\cal E $}_0 q_0 \, (q^2_0-1)+(\epsilon} \def\E{\hbox{$\cal E $}_0^2-1)\,(2 q^2_0-1))\over \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}\,(q^2_0-1)\,(\epsilon} \def\E{\hbox{$\cal E $}_0+q_0+a q_0\,\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2})}\Big] \ ,
$$
\end{widetext}
whereby the upper sign in (\ref{corr}) corresponds to the odd level, while the lower -- to the even one. Here is worth to note that for discrete levels in the single well like (\ref{v}) there always holds the relation $q_0 >1$ (for details see Ref.~\cite{greiner2000}). So both $K_{1,2}$ are always well-defined, since their denominators are strictly positive.
In the case of $\epsilon} \def\E{\hbox{$\cal E $}_0<0$ for sufficiently large $d$ both levels $\epsilon} \def\E{\hbox{$\cal E $}_{odd}$ and $\epsilon} \def\E{\hbox{$\cal E $}_{even}$ become also negative, therefore their total contribution to $\E_{vac}^{int}(d)$ equals to
\begin{equation}\label{odd+even}\begin{gathered}
\epsilon} \def\E{\hbox{$\cal E $}_{odd}+\epsilon} \def\E{\hbox{$\cal E $}_{even}= \\ = 2\, \epsilon} \def\E{\hbox{$\cal E $}_0 + 2\, K_2(a,d)\, e^{-4 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}} + O\(e^{-6 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}}\) \ .
\end{gathered}\end{equation}
So in this case the contribution to $\E_{vac}^{int}(d)$ for large $d$, caused by negative discrete level $\epsilon} \def\E{\hbox{$\cal E $}_0< 0$ in the single well, takes the form
\begin{equation}\label{levels2}\begin{gathered}
S_{int}(d)=\epsilon} \def\E{\hbox{$\cal E $}_{odd}+\epsilon} \def\E{\hbox{$\cal E $}_{even}-2 \epsilon} \def\E{\hbox{$\cal E $}_0 = \\ = 2\, K_2(a,d)\, e^{-4 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}}+ O\(e^{-6 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}}\) \ .
\end{gathered}\end{equation}
At the same time, the zero level $\epsilon} \def\E{\hbox{$\cal E $}_0=0$ splits for finite $d$ into a pair, where only the even one is negative, which gives the following term in $\E_{vac}^{int}$
\begin{equation}\label{levels3}
S_{int}(d)=\epsilon} \def\E{\hbox{$\cal E $}_{even} = -{V^2_0-1 \over(1+a)\, V^2_0} e^{-2 d} + O\(e^{-4 d}\) \ .
\end{equation}
It should be mentioned that the analysis performed above for the discrete levels contribution to the interaction energy has the correct status only subject to condition $d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2} \gg 1$. The latter means that whenever the single well parameters are such that the level $\epsilon} \def\E{\hbox{$\cal E $}_0$ lies arbitrary close to the lower threshold, the expressions (\ref{levels2})-(\ref{levels3}) could be valid only for such separations, which provide the fulfillment of this condition.
So the resulting behavior of $\E_{vac}^{int}(d)$ for $d \gg 1$ to a high degree turns out to be subject of the single well configuration. If there are only positive levels in the single well, the asymptotics of $\E_{vac}^{int}(d)$ should be $O\(e^{-4 d}/\sqrt{d}\)$ due to the integral and renormalization terms. The strictly zero level $\epsilon} \def\E{\hbox{$\cal E $}_0=0$ yields the contributions to $I_{int}(d)$ and $S_{int}(d)$ with twice less exponent rates (\ref{e0=0}) and (\ref{levels3}), but in $\E_{vac}^{int}(d)$ these terms exactly cancel each other, hence, there remains the same exponential law of decrease $\sim e^{-4 d}$.
In presence of negative levels in the spectrum of the single well the leading term in the asymptotics of $\E_{vac}^{int}(d)$ becomes different, namely, the main contribution to the asymptotics of $\E_{vac}^{int}(d)$ will be given by the lowest $\epsilon} \def\E{\hbox{$\cal E $}_0$
\begin{equation}\label{e0<0}
\E_{vac}^{int}(d) =- 2\, K_2(a,d)\, e^{-4 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}}+ O\(e^{-6 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}}\) \ .
\end{equation}
It should be mentioned that $\epsilon} \def\E{\hbox{$\cal E $}_0$ can be arbitrarily close to $\epsilon} \def\E{\hbox{$\cal E $}_F=-1$, hence $\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}$ -- arbitrarily small (but nonzero). In this case the exponential fall-down of $\E_{vac}^{int}(d)$ takes place only at extremely large $d$ subject to condition $d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2} \gg 1$ and so the Casimir interaction between such wells acquires the features of a long-range force. It is noteworthy that this effect arises due to the lowest discrete level, rather than due to replacement of the exponential asymptotics by a power-like, what could happen only for a massless mediator similar to considered in Refs.~\cite{Jaffe2004,nanotubes}.
\begin{figure*}[ht!]
\subfigure[]{
\includegraphics[width=\columnwidth]{V0=408_a=1v1.eps}
}
\hfill
\subfigure[]{
\includegraphics[width=\columnwidth]{V0=408_a=1v2.eps}
}
\vfill
\subfigure[]{
\includegraphics[width=\columnwidth]{V0=10_a=1v1.eps}
}
\hfill
\subfigure[]{
\includegraphics[width=\columnwidth]{V0=10_a=1v2.eps}
}
\caption{The dependence of the Casimir interaction energy between two wells on the distance $d$ between them for $a=1$ and (a,b): $V_0=4.08$; (c,d): $V_0=10$. }
\label{Eint1}
\end{figure*}
The same effect is found in the work~\cite{tanaka2013}, where it was shown that the electronic Casimir force between two impurities on a one-dimensional semiconductor quantum wire can be of a very long range, despite nonzero effective mass of the mediator. It should be emphasized that in this work the electronic Casimir-Polder effect is interpreted in terms of the radiation
reaction field, where one of the two sources creates a virtual
cloud of the field around itself, and the interaction of this field
with the other atom induces the Casimir-Polder force. So in contrast to our approach based on the QED vacuum polarization there is no need to
utilize the idea of vacuum fluctuations of the field as a cause
of the electronic Casimir-Polder effect. Although these two interpretations look
qualitatively different, Milonni et al. revealed that they are two
sides of the same coin about the Casimir effect~\cite{milonni1994}-\cite{milonni}. Moreover, in the present case the analogy between these two approaches can be illustrated by means of the similarity in the answers for the origin of the long-distance behavior of Casimir force. In our case it is the negative discrete level in the single well, which lies close to the lower threshold, while in Ref.~\cite{tanaka2013} it is the single-impurity ground-state energy, which could be very small
as one of the striking features of the Van Hove singularity, which causes the appearance of the bound state just below the band edge
regardless of the bare impurity energy~\cite{tanaka2006}. And in both cases we deal with the effect, which cannot be
described by means of the perturbative methods.
The concrete type of interaction between the wells can be quite different subject to the single well parameters $V_0$ and $a$, both in the asymptotics and for finite distances between the wells. In Figs.\ref{Eint1}-\ref{Eint2} $\E_{vac}^{int}(d)$ is presented for $a=1$ and $V_0=4.08,\ 7.4,\ 8,\ 10$. It follows that for $d \gg 1$ and $V_0=4.08,\ 8, 10$ the interaction energy is positive (reflecting wells), whereas for $V_0=7.4$ the energy at large distances becomes negative (attracting wells).
Such behavior can be easily understood by means of the analysis presented above. Actually, for $V_0=4.08,\ 10$ (Fig.\ref{Eint1}) in the corresponding single well the lowest discrete level is negative ($-0.9648$ and $-0.90811$, respectively). As a result, for growing $d$ in $\E_{vac}^{int}(d)$ there takes place firstly the jump by $+1$, provided by emergence of the discrete level from the lower continuum by passing through the corresponding $d_{cr}$ (quite similar to the picture shown in Fig.\ref{vac2}b), while for $d \gg 1$ the behavior of $\E_{vac}^{int}(d)$ is defined primarily by the contribution from the discrete spectrum, which in this case has the form
\begin{equation}\label{e0=-1}\begin{gathered}
\E_{vac}^{int}(d) \simeq -2 K_2(a,d) e^{-4 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}} \to \\ \to -4 d \epsilon} \def\E{\hbox{$\cal E $}_0\, {(1-\epsilon} \def\E{\hbox{$\cal E $}_0^2)^{3/2}\,(q_0^2-1)^2 \over V^2_0\,(\epsilon} \def\E{\hbox{$\cal E $}_0+q_0+a q_0\,\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2_0})^2} e^{-4 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2}}\, >0 \ ,
\end{gathered}\end{equation}
since in the coefficient $K_2(a,d)$ under the condition $d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}_0^2} \gg 1$ the main term in the square bracket in (\ref{corr2}) will be $4 d \epsilon} \def\E{\hbox{$\cal E $}_0$.
So in presence of a negative level $\epsilon} \def\E{\hbox{$\cal E $}_0<0$ in the ``initial'' single well the interaction energy becomes positive for sufficiently large distances between wells.
\begin{figure*}[ht!]
\subfigure[]{
\includegraphics[width=\columnwidth]{V0=74_a=1v1.eps}
}
\hfill
\subfigure[]{
\includegraphics[width=\columnwidth]{V0=74_a=1v2.eps}
}
\vfill
\subfigure[]{
\includegraphics[width=\columnwidth]{V0=8_a=1v1.eps}
}
\hfill
\subfigure[]{
\includegraphics[width=\columnwidth]{V0=8_a=1v2.eps}
}
\caption{ The dependence of the Casimir interaction energy between two wells on the distance $d$ between them for $a=1$ and (a,b): $V_0=7.4$; (c,d): $V_0=8$. }
\label{Eint2}
\end{figure*}
For $V_0=7.4,\ 8$ (Fig.\ref{Eint2}) the negative levels in the single well are absent, therefore the behavior of $\E_{vac}^{int}(d)$ for $d \gg 1$ is defined by the following expression
\begin{equation}\label{e0>0}\begin{gathered}
\E_{vac}^{int}(d \gg 1) = I_{int}(d)+V_0^2\Lambda_{int}(d) \simeq \\ V^2_0\,{e^{-4 d} \over \sqrt {2\pi d}} \left[ {1\over 2}\left({1\over 1+ z_0 \hbox{ctg}(a z_0)}\right)^2-e^{-2 a}\sinh^2(a)\right] \ .
\end{gathered}\end{equation}
The sign of $\E_{vac}^{int}(d \gg 1)$ depends on the sign of the square bracket in (\ref{e0>0}). For $V_0=7.4$ the square bracket in (\ref{e0>0}) is negative, and hence, for $d \gg 1$ the wells attract each other (Fig.\ref{Eint2}b). For $V_0=8$ it is positive, since for these values of $(V_0\, , a)$ the expression $1+ z_0 \hbox{ctg}(a z_0)$ is close to zero, as it was already mentioned above, and so the asymptotics of the Casimir force is repulsive, but at the same time takes place for sufficiently larger $d$ (see Fig.\ref{Eint2}d).
\subsection*{5. Casimir forces between two $\delta} \def\D{\Delta$-wells}
Now let us explore separately the Casimir interaction between two $\delta} \def\D{\Delta$-wells, for which the width and depth are related via $a=C/V_0$ with $V_0\to \infty\, , a \to 0$ and $C>0$ being some constant, proportional to the charge of the source. It is well-known that the direct inserting of $\delta} \def\D{\Delta$-potentials into DE leads to contradictions, since DE is first order (see e.g., Ref.~\cite{Jaffe2004}). More concretely, the terms involving a $\delta} \def\D{\Delta$-function are only well defined if $\psi} \def\P{\Psi$ is continuous at the points, where the $\delta} \def\D{\Delta$-peaks are located. However, the first equation in (\ref{deq}) implies a jump in the lower component of the Dirac WF $\psi} \def\P{\Psi_2$ for continuous upper one $\psi} \def\P{\Psi_1$, while the second
requires a jump in $\psi} \def\P{\Psi_1$ for continuous $\psi} \def\P{\Psi_2$. Thus the equations are not consistent. In Ref.~\cite{Jaffe2004} this problem was solved in terms of the transfer-matrix, which in the $\delta} \def\D{\Delta$-limit remains well-defined. Here we present another approach for dealing with $\delta} \def\D{\Delta$-potentials, based on the $\ln\text{[Wronskian]}$ contour integration, described in the previous Sections.
First we consider the case of a single $\delta} \def\D{\Delta$-well, where in order to keep the correspondence with the case of finite wells, considered above, it is implied that this $\delta} \def\D{\Delta$-well is twice ``wider''. Direct evaluation of the corresponding limits for separate components in (\ref{econt}) yields the following contributions to the renormalized vacuum energy of a single $\delta} \def\D{\Delta$-well. The integral term in (\ref{econt}) gives
\begin{equation}\label{delta-int}\begin{gathered}
I\to -{1\over \pi}\int\limits_0^{\infty} dy~ \hbox{Re}\left[\ln \left( \cos(2 C)-{i y \over \sqrt{ 1 + y^2 }}\sin(2 C)\right)\right] = \\ = {1-|\cos(2 C)| \over 2} \ .
\end{gathered}\end{equation}
The equation for the discrete spectrum takes the form
\begin{equation}
\label{eq:discr1delta}
\cos(2 C)-{\epsilon} \def\E{\hbox{$\cal E $} \over \sqrt{ 1 - \epsilon} \def\E{\hbox{$\cal E $}^2 }}\sin(2 C)=0 \ ,
\end{equation}
which possesses a single root
\begin{equation}\label{eq:root}
\epsilon} \def\E{\hbox{$\cal E $}_0=\hbox{sign}(\sin(4 C))\,|\cos(2 C)| \ .
\end{equation}
Depending on the sign of $\sin(4 C)$ this root can be either positive or negative, and hence, doesn't contribute or contribute to the vacuum energy of the single $\delta} \def\D{\Delta$-well. So in the general case the non-renormalized vacuum energy of a single $\delta} \def\D{\Delta$-well can be represented as follows
\begin{equation}
\label{eq:nonrezdelta}
\E_{vac}=I-S=I-\theta(-\epsilon} \def\E{\hbox{$\cal E $}_0)\epsilon} \def\E{\hbox{$\cal E $}_0={1-\hbox{sign}(\sin(4 C))\,|\cos(2 C)| \over 2} \ .
\end{equation}
Proceeding further, on account of the asymptotics for the renormalization coefficients $\lambda} \def\L{\Lambda_1(a)$ and $\lambda} \def\L{\Lambda_2(a)$ for infinitely small width of the well, which can be easily derived from formulae (\ref{renpart})-(\ref{limlambda}), one finds
\begin{equation}
\label{eq:limits}
V^2_0\lambda_1(a)\to {V_0 C \over \pi}-C^2\to \infty , \quad V^2_0\lambda_2(a)\to C^2 \ .
\end{equation}
So in contrast to all the others terms, the PT contribution to the renormalization term doesn't possess any finite $\delta} \def\D{\Delta$-limit, and hence, $\E^R_{vac}$ for the single $\delta} \def\D{\Delta$-well is divergent:
\begin{equation}
\label{eq:endelta}\begin{gathered}
\E^R_{vac}=I-S+\lambda} \def\L{\Lambda\, V^2_0 \to \\ \to {1-\hbox{sign}(\sin(4 C))\,|\cos(2 C)| \over 2}-2\, C^2+{V_0 C \over \pi}\to\infty \ .
\end{gathered}\end{equation}
Actually, this result should be expected from general considerations, since for discontinuous potentials the Fouriet-transform $\text{\~{A}}_0(q)$ of the external potential $A_0^{ext}(x)$ decreases in the momentum space too slow and so the one-loop perturbative energy diverges. The same in essence effect appears also in more spatial dimensions by screening of the Coulomb asymptotics through the simple vertical cutoff, and it is necessary to introduce additional smoothing in order to maintain the convergence of the perturbative contribution to the energy \cite{voronina2018}. It should be clear that in the considered case of a $\delta} \def\D{\Delta$-well such smoothing would also lead to the finite answer.
However, the Casimir interaction energy between two $\delta} \def\D{\Delta$-wells turns out to be well-defined quantity without any additional smoothing, since the divergent parts doesn't depend on the distance between wells. Namely, the integral component in (\ref{Evacint}) will give in this case the following contribution to $\E_{vac}^{int}(d)$
\begin{equation}
\begin{gathered}\label{delta-int1}
I_{int}(d)=I(d)-I(d \to \infty)
\to \\ \to -{1\over 2 \pi}\,\int\limits_0^{\infty} dy~ \ln \left[ 1+(\sin C)^4\, \hbox{e}^{-4 d \sqrt{1+y^2}} \times \right. \\ \left. \times {\hbox{e}^{-4 d \sqrt{1+y^2}} - \ 2 ((1+y^2)(\hbox{ctg} C)^2 -y^2)\over ((\cos C)^2 + y^2)^2} \right] \ .
\end{gathered}
\end{equation}
Here it should be mentioned that in this case each $\delta} \def\D{\Delta$-well should be twice ``narrower'' compared to the single $\delta} \def\D{\Delta$-well, considered in (\ref{delta-int})-(\ref{eq:endelta}), what implies $C \to C/2$ in all the subsequent expressions, defining separate components in (\ref{econt}) for the two $\delta} \def\D{\Delta$-wells configuration.
In particular, the eq. for the discrete spectrum (\ref{DiscrWronsk}) splits now into two equations for two levels $\epsilon} \def\E{\hbox{$\cal E $}_\pm$
\begin{equation}\label{95}
\hbox{ctg}\,C\, \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2_\pm}\, -\epsilon} \def\E{\hbox{$\cal E $}_\pm=\mp \hbox{e}^{-2 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2_\pm}} \ ,
\end{equation}
whence there follows the next contribution to $\E_{vac}^{int}(d)$ from the negative part of the discrete spectrum
$$ S_{int}\to \theta(-\epsilon} \def\E{\hbox{$\cal E $}_+)\epsilon} \def\E{\hbox{$\cal E $}_+ + \theta(-\epsilon} \def\E{\hbox{$\cal E $}_-)\epsilon} \def\E{\hbox{$\cal E $}_- -2 \theta(-\epsilon} \def\E{\hbox{$\cal E $}_0)\epsilon} \def\E{\hbox{$\cal E $}_0 \ , $$
with $\epsilon} \def\E{\hbox{$\cal E $}_0$ being now the single level of a separated $\delta} \def\D{\Delta$-well, which differs from (\ref{eq:root}) by $C \to C/2$, namely
\begin{equation}\label{eq:root1}
\epsilon} \def\E{\hbox{$\cal E $}_0=\hbox{sign}(\sin(2 C))\,|\cos(C)| \ .
\end{equation}
Proceeding further, from (\ref{lambda-int}) one finds the following limit for the renormalization coefficient in $\E_{vac}^{int}(d)$
\begin{equation}\label{eq:lambda1delta}
\Lambda_{int}(d) V^2_0 \to -{2\, C^2\over \pi}\int\limits_0^{\infty} dy~ {\hbox{e}^{-4 d \sqrt{1+y^2}} \over 1+y^2} \ .
\end{equation}
As a result, within the $\ln\text{[Wronskian]}$ contour integration the renormalized Casimir interaction energy between two $\delta} \def\D{\Delta$-wells turns out to be a well-defined quantity.
Compared to the case of finite wells, the Casimir interaction between two $\delta} \def\D{\Delta$-sources turns out to be no less rich in the variability of the Casimir force both at finite distances and in asymptotic behavior. Namely, for $d \gg 1$ the components of $\E^{int}_{vac}(d)$ behave as follows. The integral part
(\ref{delta-int1}) turns out to be
\begin{equation}
\label{eq:deltaIintBigd}\begin{gathered}
I_{int}\simeq e^{-4 d}\, {\hbox{tg}^2 C \over 2\sqrt{2 \pi d}} \times \\ \times \left(1 + {1\over 4}\left({19\over 8}-{3\over \cos^2 C}\right){1\over d} + O\({1\over d^2}\)\right) \ ,
\end{gathered}\end{equation}
the renormalization term (\ref{eq:lambda1delta}) equals to
\begin{equation}
\label{eq:deltaLambdaBigd}
\Lambda_{int}(d) V^2_0 \simeq -e^{-4 d}\, {C^2 \over \sqrt{2\pi d}}\left(1 - {5\over 32}{1\over d} + O\({1\over d^2}\)\right) \ ,
\end{equation}
while the asymptotics of discrete levels is given by
\begin{equation}
\label{eq:deltaDiscrBigd}\begin{gathered}
\epsilon} \def\E{\hbox{$\cal E $}_\pm \simeq \epsilon} \def\E{\hbox{$\cal E $}_0 \pm e^{-2 d\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2_0}}\,(1-\epsilon} \def\E{\hbox{$\cal E $}^2_0)- e^{-4 d\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2_0}}\,\epsilon} \def\E{\hbox{$\cal E $}_0\, (1-\epsilon} \def\E{\hbox{$\cal E $}^2_0) \ \times \\ \times \ \left(1-4 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2_0}\right)/2 + O\left(e^{-6 d\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2_0}}\right) \ ,
\end{gathered}\end{equation}
approaching the level in the single $\delta} \def\D{\Delta$-well (\ref{eq:root1}) from above and from below, respectively.
If $\epsilon} \def\E{\hbox{$\cal E $}_0<0$, the contribution from the discrete spectrum for $d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2_0} \gg 1$ equals to
\begin{equation}\label{delta-discr}\begin{gathered}
-S_{int}=-(\epsilon} \def\E{\hbox{$\cal E $}_+ + \epsilon} \def\E{\hbox{$\cal E $}_- -2\epsilon} \def\E{\hbox{$\cal E $}_0)\simeq \\ \simeq e^{-4 d\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2_0}}\,\epsilon} \def\E{\hbox{$\cal E $}_0\, (1-\epsilon} \def\E{\hbox{$\cal E $}^2_0)\,\left(1-4 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2_0}\right)>0 \ ,
\end{gathered}\end{equation}
and due to the exponent $e^{-4 d\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2_0}}$ turns out to be the leading term in $\E^{int}_{vac}(d)$, implying for $\epsilon} \def\E{\hbox{$\cal E $}_0$ close to $\epsilon} \def\E{\hbox{$\cal E $}_F$ the existence of long-range forces between such $\delta} \def\D{\Delta$-wells quite similar to the case of finite wells. In turn, this is the reason of the behavior of interaction energy between wells for $C=3$ and $C=5$ for large separation (see Figs. \ref{EintD}d,f below).
At the same time, if $\epsilon} \def\E{\hbox{$\cal E $}_0>0$, then $S_{int}=0$, and the interaction energy $\E^{int}_{vac}(d)=I_{int}(d)+\Lambda_{int}(d) V^2_0$ decreases with growing $d$ much faster, namely as $O\(e^{-4 d}\)$.
If $\epsilon} \def\E{\hbox{$\cal E $}_0=0$, i.e. for $C=\pi/2+\pi n$, the expression (\ref{eq:deltaIintBigd}) isn't valid, since an essential circumstance here is that $\cos C$ entering the denominators in (\ref{delta-int1}) and (\ref{eq:deltaIintBigd}) should be non-zero. In this case the integral term transforms into
\begin{equation}
I_{int} \simeq -e^{-2 d}+ O\(e^{-4 d}\) \ ,
\end{equation}
while the contribution from the discrete spectrum contains now the level $\epsilon} \def\E{\hbox{$\cal E $}_-<0$ only and gives \begin{equation}
-S_{int} \simeq e^{-2 d}+ O\(e^{-6 d}\) \ .
\end{equation}
Therefore for $\epsilon} \def\E{\hbox{$\cal E $}_0=0$ the interaction energy between two $\delta} \def\D{\Delta$-wells decreases also as $O\(e^{-4 d}\)$.
In Figs.\ref{EintD} the dependence of the interaction energy between two $\delta} \def\D{\Delta$-wells on the distance $d$ between them for a set of different values of the parameter $C$ is shown. As it follows from Figs.\ref{EintD}c,d and e,f, depending on the concrete value of $C$ the nature of the Casimir force between wells may change from attraction to repulsion with growing $d$. In the present case this effect takes place for $C=3$ and $C=5$. For other values of $C$, shown in Figs.\ref{EintD}, the interaction energy is strictly negative and grows with increasing $d$, so the wells attract each other.
\begin{figure*}[ht!]
\subfigure[]{
\includegraphics[width=\columnwidth]{const=1.eps}
}
\hfill
\subfigure[]{
\includegraphics[width=\columnwidth]{const=10.eps}
}
\vfill
\subfigure[]{
\includegraphics[width=\columnwidth]{const=3.eps}
}
\hfill
\subfigure[]{
\includegraphics[width=\columnwidth]{const=3_v1.eps}
}
\vfill
\subfigure[]{
\includegraphics[width =\columnwidth]{const=5.eps}
}
\hfill
\subfigure[]{
\includegraphics[width=\columnwidth]{const=5_v1.eps}
}
\caption{Different types of the Casimir interaction energy between two $\delta} \def\D{\Delta$-wells as functions of the distance $d$ between them for: (a) $C=1$, (b) $C=10$, (c,d) $C=3$, (e,f) $C=5$.}
\label{EintD}
\end{figure*}
The jump-like behavior of energy at $d=3.5076$ for $C=3$ (Figs.\ref{EintD}c) and at $d=0.1479$ for $C=5$ (Fig.\ref{EintD}e) is caused by emergence of a new level at the lower threshold, provided the condition
\begin{equation}
d=-\hbox{ctg}(C)/2>0 \ ,
\end{equation}
which follows from (\ref{95}) in the limit $\epsilon} \def\E{\hbox{$\cal E $}_- \to -1$, is fulfilled. Another way to achieve this condition is to use the eq. (\ref{creven}) in the $\delta} \def\D{\Delta$-limit.
With further removal of the wells from each other this level goes up, approaching from below the unique level $\epsilon} \def\E{\hbox{$\cal E $}_0$ in the single $\delta} \def\D{\Delta$-well (\ref{eq:root1}) (for $C=3$ and $C=5$ the latter is negative). Meanwhile the second level goes down, approaching the value $\epsilon} \def\E{\hbox{$\cal E $}_0$ from above. For $C=1$ and $C=10$ there are no negative $\epsilon} \def\E{\hbox{$\cal E $}_0$, and so starting from sufficiently large $d$ the contribution from the discrete spectrum to $\E_{vac}^{int}(d)$ disappears.
\subsection*{6. Casimir forces in the source-anti-source system}
There exists only one exception, when the effect of long-range Casimir force in quasi-one-dimensional QED system with short-range Coulomb sources of the type considered above, cannot be able in principle. It is the anti-symmetric configuration of the type source-anti-source, where one of the wells is replaced by a barrier with the same width and height. For our purposes it would be pertinent to consider an even more general situation, described by the external potential of the form
\begin{equation}
W_2(x) =-\[V_1 \, \theta} \def\vt{\vartheta\(x -d \) + V_2 \, \theta} \def\vt{\vartheta\(-x -d \)\]\,\theta} \def\vt{\vartheta\(d+a -|x|\) \ ,
\label{w2}
\end{equation}
although in what follows we'll be interested first of all in the antisymmetric case with $V_1=-V_2=V_0>0$.
In the first step, for such a configuration of external short-range Coulomb sources, the calculation of corresponding vacuum charge density will be useful. For these purposes one needs to consider the trace of the Green function
\begin{equation}\label{trG}
\hbox{Tr} G(x,x;\epsilon} \def\E{\hbox{$\cal E $})={1\over J(\epsilon} \def\E{\hbox{$\cal E $})}\psi^T_L(x)\psi_R(x) \ ,
\end{equation}
with $\psi_{R,L}(x)$ being the solutions of DE (\ref{deq}) with $V(x)$ replaced by $W_2(x)$, which are regular at $\pm \infty$, respectively,
while $J(\epsilon} \def\E{\hbox{$\cal E $})$ is their Wronskian. As in eq. (\ref{F}), we use here the denotation
\begin{equation}
[f,g]_a=f_2(a)g_1(a)-g_2(a)f_1(a)
\end{equation}
for the Wronskian of functions $f(x)$ and $g(x)$, calculated at the point $x=a$.
In terms of the latter definition the Wronskian $J(\epsilon} \def\E{\hbox{$\cal E $})$ in eq. (\ref{trG}) equals to
\begin{equation}
J(\epsilon} \def\E{\hbox{$\cal E $})=[\psi_L,\psi_R] \ .
\end{equation}
For the external potential $W_2(x)$ the pertinent solutions of DE are represented in the following form
\begin{equation}\label{solsRL}
\psi_L(x)=\left\{
\begin{array}{l}
\Phi(x) \ ,\quad x\leq -d-a \ ,\\
A_L u(V_2, x) + B_L v(V_2, x) \ , \ -d-a\leq x\leq -d \ ,\\
C_L \Phi(x) + D_L \Psi(x) \ , \quad |x|\leq d \ ,\\
E_L u(V_1, x) + F_L v(V_1, x) \ , \quad d\leq x\leq d+a \ ,\\
G_L \Phi(x) + H_L \Psi(x) \ ,\quad a+d \leq x \ ,
\end{array}
\right.
\end{equation}
\begin{equation}
\psi_R(x)=\left\{
\begin{array}{l}
G_R \Psi(x) + H_R \Phi(x) \ , \quad x\leq -d-a \ ,\\
E_R u(V_2, x) - F_R v(V_2, x) \ , \ -d-a\leq x\leq -d \ ,\\
C_R \Psi(x) + D_R \Phi(x) \ , \quad |x|\leq d \ ,\\
A_R u(V_1, x) - B_R v(V_1, x) \ , \quad d\leq x\leq d+a \ ,\\
\Psi(x) \ , \quad a+d \leq x \ ,
\end{array}
\right.
\end{equation}
with the coefficients $A_{R,L}\, , B_{R,L}\, , C_{R,L}\, , D_{R,L}\, , E_{R,L}\, , F_{R,L}\, , $ $ G_{R,L}\, , H_{R,L}$ being obtained via the requirement of continuity of solutions $\psi_{R,L}(x)$ at the points $x=\pm d\, , \pm (d+a)$, while $\Phi(x)\, , \Psi(x)\, , u(V_i, x)\, , v(V_i, x)\, ,\ i=1, 2,$ are the linearly independent solutions of DE in the corresponding regions of constant potential $W_2(x)$
\begin{equation}\begin{aligned}
&\Phi(x)=\begin{pmatrix}
\sqrt{1+\epsilon} \def\E{\hbox{$\cal E $}}\ e^{x\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2}}\\
\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}}\ e^{x\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2}}
\end{pmatrix} \ , \\
&\Psi(x)=\begin{pmatrix}
\sqrt{1+\epsilon} \def\E{\hbox{$\cal E $}}\ e^{-x\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2}}\\
-\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}}\ e^{-x\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2}}
\end{pmatrix} \ , \\
\end{aligned}
\label{tsols1}
\end{equation}
\begin{equation}\begin{aligned}
&u(V_i, x)=\\
&=\begin{pmatrix}
\cos\(x\sqrt{(\epsilon} \def\E{\hbox{$\cal E $}+V_i)^2-1}\) \\
-\sqrt{\epsilon} \def\E{\hbox{$\cal E $}+V_i-1}\, \sin\(x\sqrt{(\epsilon} \def\E{\hbox{$\cal E $}+V_i)^2-1}\) / \(\epsilon} \def\E{\hbox{$\cal E $}+V_i+1\) \\
\end{pmatrix} \ , \\
&v(V_i, x)=\\
&=\begin{pmatrix}
\sqrt{\epsilon} \def\E{\hbox{$\cal E $}+V_i +1}\, \sin\(x\sqrt{(\epsilon} \def\E{\hbox{$\cal E $}+V_i)^2-1}\) / \(\epsilon} \def\E{\hbox{$\cal E $}+V_i-1\) \\
\cos\(x\sqrt{(\epsilon} \def\E{\hbox{$\cal E $}+V_i)^2-1}\)
\end{pmatrix} \ . \\
\end{aligned}
\label{tsols2}
\end{equation}
\normalsize
The cross-linking coefficients with label $R$ take the form
\begin{equation}
\begin{gathered}
A_R={[\Psi,v(V_1)]_{a+d}\over [u(V_1),v(V_1)]_{a+d}}\ ,\\ B_R={[\Psi,u(V_1)]_{a+d}\over [u(V_1),v(V_1)]_{a+d}} \ ,\\
D_R={A_R[u(V_1),\Psi]_{d}-B_R[v(V_1),\Psi]_{d}\over [\Phi,\Psi]_{d}} \ , \\ C_R={A_R[\Phi,u(V_1)]_{d}-B_R[\Phi,v(V_1)]_{d}\over [\Phi,\Psi]_{d}} \ ,\\
F_R={C_R[\Phi,u(V_2)]_{d}+D_R[\Psi,u(V_2)]_{d}\over[v(V_2),u(V_2)]_{d}} \ , \\ E_R={C_R[v(V_2),\Phi]_{d}+D_R[v(V_2),\Psi]_{d}\over[v(V_2),u(V_2)]_{d}} \ ,
\end{gathered}\label{CoefR}
\end{equation}
$$ H_R={E_R[u(V_2),\Phi]_{a+d}+F_R[v(V_2),\Phi]_{a+d}\over[\Psi,\Phi]_{a+d}} \ , $$
$$ G_R={E_R[\Psi,u(V_2)]_{a+d}+F_R[\Psi,v(V_2)]_{a+d}\over[\Psi,\Phi]_{a+d}} \ . $$
The corresponding coefficients with label $L$ are obtained from (\ref{CoefR}) by means of replacement $R \to L$ and $V_1 \leftrightarrow V_2$.
By means of (\ref{solsRL}-\ref{CoefR}) for the explicit form of $J(\epsilon} \def\E{\hbox{$\cal E $})$ one finds
\begin{equation}\begin{gathered}
J(d,\epsilon} \def\E{\hbox{$\cal E $})=2\,{e^{-2 a \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2}}\over \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2}}\, \left[ f_1(V_1,\epsilon} \def\E{\hbox{$\cal E $}) f_1(V_2,\epsilon} \def\E{\hbox{$\cal E $}) - \right. \\ \left. - e^{-4 d \sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2}} f_2(V_1,\epsilon} \def\E{\hbox{$\cal E $}) f_2(V_2,\epsilon} \def\E{\hbox{$\cal E $}) \right] \ , \label{Wronskian}
\end{gathered}\end{equation}
where
\begin{equation}\begin{gathered}\label{f12}
f_1(V_i,\epsilon} \def\E{\hbox{$\cal E $})=\sqrt{1-\epsilon} \def\E{\hbox{$\cal E $}^2}\,\cos\(a\sqrt{(V_i+\epsilon} \def\E{\hbox{$\cal E $})^2-1}\) - \\ \(\epsilon} \def\E{\hbox{$\cal E $}^2-1+V_i \epsilon} \def\E{\hbox{$\cal E $}\) \,\sin\(a\sqrt{(V_i+\epsilon} \def\E{\hbox{$\cal E $})^2-1}\)/\sqrt{(V_i+\epsilon} \def\E{\hbox{$\cal E $})^2-1} \ , \\
f_2(V_i,\epsilon} \def\E{\hbox{$\cal E $})=V_i\,\sin\(a\sqrt{(V_i+\epsilon} \def\E{\hbox{$\cal E $})^2-1}\) / \sqrt{(V_i+\epsilon} \def\E{\hbox{$\cal E $})^2-1} \ .
\end{gathered}\
\end{equation}
\begin{figure*}[ht!]
\subfigure[]{
\includegraphics[width=\columnwidth]{Ediscr_a=1_V0=408.eps}
}
\hfill
\subfigure[]{
\includegraphics[width=\columnwidth]{Ediscr_a=1_V0=408_single_well.eps}
}
\caption{(Color online). Energy levels in the case $a=1$, $V_0=4.08$ for the configurations: (a) the antisymmetric one; (b) single well (black and red lines) and barrier (orange solid and dashed lines). }
\label{Ediscr}
\end{figure*}
Let us consider now more thoroughly the antisymmetric case of the configuration barrier-well, when $V_1=-V_2=V_0>0$. As it follows from the expressions (\ref{Wronskian})-(\ref{f12}), in this case $J(d,\epsilon} \def\E{\hbox{$\cal E $})$ turns out to be an even function of the energy
\begin{equation}\label{J}
J(d,\epsilon} \def\E{\hbox{$\cal E $})=J(d,-\epsilon} \def\E{\hbox{$\cal E $}) \ .
\end{equation}
Therefore, the discrete spectrum of the problem should be sign-symmetric, i.e. the levels appear only in pairs with $\pm \epsilon} \def\E{\hbox{$\cal E $}$. Actually, the latter circumstance is the general feature of the source-anti-source system, including both the discrete spectrum and continua. Namely, all the energy eigenstates in such system are related via (up to a phase factor)
\begin{equation}\label{oddness}
\psi} \def\P{\Psi_{-\epsilon} \def\E{\hbox{$\cal E $}}(x)=\alpha} \def\tA{\tilde {A}\, \psi} \def\P{\Psi_\epsilon} \def\E{\hbox{$\cal E $} (-x) \ .
\end{equation}
The typical behavior of levels for the antisymmetric case is shown in Fig.\ref{Ediscr}a in dependence on the distance $d$ between sources for $a=1\, , V_0=4.08$. The set $(V_0\, , a)$ is taken with the same values as for the symmetric case containing two wells, considered in Sect.4. The symmetry of levels relative to the zero energy line is apparent. Note also that the highest and lowest levels appear only starting from certain $d>0$. With increasing $d$ all the levels tend to constant values, coinciding with those of the single well and barrier of the same width $a$ and depth/hight $V_0$. Such behavior follows directly from the eq. $J(d,\epsilon} \def\E{\hbox{$\cal E $})=0$. For $d\gg 1$ the second term in the expression (\ref{Wronskian}) can be neglected, hence, the resulting equation for the levels transforms into
\begin{equation}
f_1(V_0,\epsilon} \def\E{\hbox{$\cal E $})f_1(-V_0,\epsilon} \def\E{\hbox{$\cal E $})=0 \ .
\end{equation}
In turn, the latter splits into two independent equations for the levels in the single well and barrier, namely, $f_1(V_0,\epsilon} \def\E{\hbox{$\cal E $})=0$ for the well and $f_1(-V_0,\epsilon} \def\E{\hbox{$\cal E $})=0$ for the barrier, which are related by reflection $\epsilon} \def\E{\hbox{$\cal E $} \to -\epsilon} \def\E{\hbox{$\cal E $}$. So the discrete spectra of the well and barrier differ only by the sign, as
expected. For more clarity, in Fig.\ref{Ediscr}b the levels in the single well and barrier with the same parameters $a=1$, $V_0=4.08$, are shown. There exist two levels with values $\epsilon} \def\E{\hbox{$\cal E $}_1=-0.965$, $\epsilon} \def\E{\hbox{$\cal E $}_2=0.466$ in the well, while for the barrier one finds two levels with opposite signs.
The sign symmetry of the energy spectrum in the source-anti-source systems leads to significant changes in the definition and properties of vacuum polarization density and energy. The most important point here is that due to sign symmetry of the levels the whole spectrum splits into two non-intersecting parts with positive and negative energies, respectively, since the levels cannot intersect, and hence, cannot cross the zero line (see Figs.\ref{EdiscrV0}).
\begin{figure*}[ht!]
\subfigure[]{
\includegraphics[width=\columnwidth]{Ediscr_V0_a=1_d=2.eps}
}
\hfill
\subfigure[]{
\includegraphics[width=\columnwidth]{Ediscr_V0_a=2_d=2.eps}
}
\caption{The behavior of positive energy levels in the antisymmetric case of the type barrier-well in dependence on $V_0$: (a) for $d=2$, $a=1$; (b) for $d=2$, $a=2$. }
\label{EdiscrV0}
\end{figure*}
Therefore, in this case the Fermi level, dividing the electronic and positronic (electron-hole) eigenstates in the initial expressions for the vacuum averages similar to (\ref{eterms}), should be chosen equal to zero, i.e. $\epsilon} \def\E{\hbox{$\cal E $}_F=0$.
So the starting expression for the induced density should be written as
\begin{equation} \label{rhoVP}
\rho} \def\vr{\varrho_{vac}(x)=-\frac{|e|}{2}\(\sum\limits_{\epsilon} \def\E{\hbox{$\cal E $}_{n}<0} \psi} \def\P{\Psi_{n}(x)^{\dagger}\psi} \def\P{\Psi_{n}(x)-\sum\limits_{\epsilon} \def\E{\hbox{$\cal E $}_{n}>0} \psi} \def\P{\Psi_{n}(x)^{\dagger}\psi} \def\P{\Psi_{n}(x) \),
\end{equation}
where $\epsilon} \def\E{\hbox{$\cal E $}_{n}$ and $\psi} \def\P{\Psi_n(x)$ are the eigenvalues and the eigenfunctions of the corresponding DE for the antisymmetric case. Proceeding further, one finds that due to sign symmetry of the spectrum the WK-contour, shown in Fig.\ref{contour}, transforms now into the symmetric one with respect to reflection $\epsilon} \def\E{\hbox{$\cal E $} \to -\epsilon} \def\E{\hbox{$\cal E $}^{\ast}$, while its separate parts $P(R)$ and $E(R)$ each lie in their half-planes $\mathrm{Re}\, \epsilon} \def\E{\hbox{$\cal E $} <0$ for $P(R)$ and $\mathrm{Re}\, \epsilon} \def\E{\hbox{$\cal E $} >0$ for $E(R)$ and don't intersect with the imaginary axis.
As it should be expected from general grounds, there follows from eq. (\ref{rhoVP}) combined with relation (\ref{oddness}) that the vacuum density is an odd function
\begin{equation}
\rho_{vac}(x)=-\rho_{vac}(-x) \ ,
\end{equation}
reproducing this way the similar property of the external potential (\ref{w2}) in the antisymmetric case.
Applying further the same technique as in Refs.\cite{davydov2017}-\cite{voronina2017}, \cite{wk1956}-\cite{21} for the expression of the induced density in terms of $\hbox{Tr} G$, one finds
\begin{equation}\label{rhoVP1}
\rho_{vac}(x)={|e|\over 2 \pi}\int\limits_{-\infty}^{\infty} dy\, \hbox{Tr} G(x,x;i y) \ .
\end{equation}
Note that in the expression (\ref{rhoVP1}) there is no separate contribution from negative discrete levels, since the latter appears only in the case when the part $E(R)$ of the WK-contour captures a piece of negative real axis containing these discrete levels.
Since $\rho_{vac}(x)$ is odd from the very beginning, in contrast to symmetric case \cite{annphys} and all the more to the one-dimensional QED systems with long-range external Coulomb sources considered in Refs.\cite{davydov2017}-\cite{voronina2017}, the total induced charge vanishes now without any additional renormalization
\begin{equation}
Q_{vac}=\int\limits_{-\infty}^{\infty} dx\, \rho_{vac}(x)=0 \ .
\end{equation}
Nevertheless, a finite renormalization is needed due to condition that in the perturbative region $V_0 \to 0$ the renormalized vacuum density $\rho^R_{vac}(x)$ should reproduce the perturbative density $\rho^{(1)}_{vac}(x)$, calculated within the standard perturbation theory (PT) to the leading (one-loop) order \cite{davydov2017}-\cite{voronina2017}, \cite{annphys,tmf}. Actually, this procedure is equivalent to a finite renormalization and normalization conditions as known from perturbative QED (see, e.g., Ref.\cite{itzykson2012}). The explicit expression for $\rho^{(1)}_{vac}(x)$ reads
\begin{widetext}
\begin{equation}\begin{gathered}
\rho^{(1)}_{vac}(x)=\\ -{|e| \over \pi^2}\int\limits_0^{\infty} {dq \over q}\, \left(1-2{ \mathrm{arcsinh} (q/2) \over q \sqrt{1+(q/2)^2}}\right) \Big(V_1\[\sin (q(d-x)) - \sin (q(a+d-x))\] + V_2 \[\sin (q(d+x)) - \sin (q(a+d+x))\]\Big) \ .
\label{rho1}
\end{gathered}\end{equation}\end{widetext}
It should be quite clear without any additional comments that in the antisymmetric case the perturbative density is an odd function by construction, hence, in this case the total induced charge $Q^{(1)}_{vac}$, calculated to the leading order of PT by means of $\rho^{(1)}_{vac}(x)$, vanishes (actually this statement holds also for the non-symmetric case, for details see, e.g., Ref.\cite{davydov2018}).
Thus, by means of the standard renormalization procedure for the vacuum density considered in Refs. \cite{davydov2017}-\cite{voronina2017}, \cite{wk1956}-\cite{21}, one obtains
\begin{equation}\label{rhoR}
\rho^R_{vac}(x) = \rho^{(1)}_{vac}(x) + \rho^{(3+)}_{vac}(x) \ ,
\end{equation}
where
\begin{equation}\label{rho3+}
\rho^{(3+)}_{vac}(x)= {|e|\over 2 \pi}\int\limits_{-\infty}^{\infty} \! dy\, \[\hbox{Tr} G(x,x; i y)- \hbox{Tr} G^{(1)}(x,x; i y)\] \ .
\end{equation}
In the expression (\ref{rho3+}) the function $\hbox{Tr} G^{(1)}(x,x; \epsilon} \def\E{\hbox{$\cal E $})$ is the first-order term in the expansion of the Green function in the Born series in powers of $V_0$ (for the antisymmetric case).
\begin{figure*}[ht!]
\subfigure[]{
\includegraphics[width=\columnwidth]{a=1_V0=8_d=2.eps}
}
\hfill
\subfigure[]{
\includegraphics[width=\columnwidth]{a=2_V0=2_d=2.eps}
}
\caption{The renormalized vacuum charge density in the antisymmetric case for the following sets of the system parameters: (a) $d=2$, $a=1$, $V_0=8$; (b) $d=2$, $a=2$, $V_0=2$. }
\label{rho}
\end{figure*}
In Fig.\ref{rho} the renormalized vacuum charge density is shown for the following sets of the system parameters: (a) $d=2$, $a=1$, $V_0=8$; (b) $d=2$, $a=2$, $V_0=2$. In general, the behavior of density is quite similar to those achieved for the case of two wells in Ref.\cite{annphys} with the main exception that now the density is odd.
After these preliminary considerations let us turn to calculation of the Casimir energy for the antisymmetric configuration. Repeating the procedure of passing from the initial definition of the vacuum energy by means of the Schwinger average (\ref{eterms}) to the integration over the imaginary axis in (\ref{IntWronskReg}), considered in detail for the symmetric case, for the non-renormalized vacuum energy one obtains
\begin{equation}\label{IntWronskReg1}
\E_{vac}(d)=-{1\over \pi}\int\limits_0^{\infty} dy\, \mathrm{Re}\left[\ln J_{red}(d,i y)\right] \ ,
\end{equation}
with the same definition of the ``reduced'' Wronskian $J_{red}(d,i y)$ as in (\ref{Jred}).
For the antisymmetric case one obtains
\begin{equation}\begin{gathered}
J_{red}(d, i y)={e^{-2 a \sqrt{1+y^2}}\over 1+y^2} \times \\ \times \left[ |f_1(V_0, i y))|^2 -e^{-4 d \sqrt{1+y^2}} |f_2(V_0, i y)|^2 \right] \ ,
\end{gathered}\end{equation}
with $f_i(V_0, iy)$ being defined in (\ref{f12}). Note also that in contrast to (\ref{econt}), due to the same reasons as in (\ref{rhoVP1}), in the expression (\ref{IntWronskReg1}) there is no separate contribution from the negative discrete levels.
The renormalized vacuum energy is represented as
\begin{equation}
\E^R_{vac}(d)=\E_{vac}(d)+\lambda(d)\, V^2_0 \ ,
\end{equation}
where the renormalization coefficient $\lambda(d)=\lambda_1(d)-\lambda_2(d)$ contains two terms of the following form
\begin{widetext}\begin{equation}
\lambda_1(d)=\lim_{V_0\to 0}\, \E^{(1)}_{vac}(d)/V^2_0 \ , \quad
\lambda_2(d)=\lim_{V_0\to 0}\, \E_{vac}(d)/V^2_0 = {a\over \pi}-{1\over 8}+{1\over \pi}\,\int\limits_0^{\infty} dy\, {1-2 e^{-4 d\sqrt{1+y^2}}\sinh^2(a\sqrt{1+y^2}) \over 2 (1+y^2)^2}\,e^{-2 a\sqrt{1+y^2}} \ ,
\end{equation}
where the first-order perturbative vacuum energy $\E^{(1)}_{vac}(d)$ is given by the following expression, calculated within PT in the one-loop approximation for the antisymmetric case
\begin{multline}
\E^{(1)}_{vac}(d)={1\over 2}\int\limits_{-\infty}^{\infty} dx\, \rho^{(1)}_{vac}(x) A^{ext}_0(x)=-{1\over \pi^2}\,\int\limits_0^{\infty}\,{dq\over q^2}\, \left(1-2{ \mathrm{arcsinh} (q/2) \over q \sqrt{1+(q/2)^2}}\right) \times \\
\times \[-2+2 \cos(a q)-\cos(2 d q)-\cos(2 (a+d)q) + 2\cos((a+2 d)q)\] \ ,
\label{EvacV0}
\end{multline}\end{widetext}
where $A^{ext}_0(x)$ is related to the external potential $W_2(x)$ in DE via $W_2(x)=-|e|A^{ext}_0(x)$.
In the antisymmetric case there holds also the relation $\lambda_1+\lambda_2=a/\pi$, which allows to represent the renormalization coefficient in a more convenient form, namely
\begin{equation}
\lambda(d)={a\over \pi}-2\lambda_2(d) \ .
\end{equation}
To explore the Casimir force in the source-anti-source system let us start with the behavior of non-renormalized vacuum energy $\E_{vac}(d)$ for large $d \gg 1$. In this case the expression (\ref{IntWronskReg1}) simplifies up to
\begin{equation}
\E_{vac}(d \gg 1)=-{1\over \pi}\,\int\limits_0^{\infty} dy\, \ln \left[ {e^{-2 a \sqrt{1+y^2}}\over 1+y^2} |f_1(V_0, i y))|^2\right] \ ,
\end{equation}
and coincides with the non-renormalized total energy of the system, containing infinitely separated barrier and well with the same width $a$ and depth/height $V_0$, but preserving the antisymmetry property (\ref{oddness}) of the whole configuration. Otherwise, considering the limiting configuration as a direct sum of the single barrier and single well without antisymmetry property, we should deal with their contributions according to (\ref{econt}) for the well and to similar expression for the barrier, where the additional sum includes now positive discrete levels and enters with opposite sign. As a result, in this case the limiting vacuum energy will contain twice the sum over discrete levels, entering the expression (\ref{econt}). However, such a configuration cannot be considered as a physically correct limit for $\E_{vac}(d \gg 1)$, since the antisymmetry property is lost.
So the non-renormalized interaction energy in the coupled barrier-well system equals to
\begin{equation}\label{Eint}\begin{gathered}
\E_{int}(d)=\E_{vac}(d)-\E_{vac}(d \to \infty)=\\ -{1\over \pi}\,\int\limits_0^{\infty} dy\, \ln \[J_{red}(d,i y)\, (1+y^2)\, {e^{2 a \sqrt{1+y^2}} \over |f_1(V_0,i y)|^2 }\] \ .
\end{gathered}\end{equation}
Expanding the integrand in the r.h.s. of (\ref{Eint}) for $d \gg 1 $ up to $O\(e^{-8 d \sqrt{1+y^2}}\)$, one obtains
\begin{equation}\label{EintApp}
\E_{int}(d) \simeq - {1\over \pi}\,\int\limits_0^{\infty} dy\, e^{-4 d \sqrt{1+y^2}} \ \Bigg| {f_2(V_0,i y) \over f_1(V_0,i y)}\Bigg|^2 \ .
\end{equation}
Further expansion of the expression (\ref{EintApp}) for large $d$ proceeds quite similar to the symmetric case and leads to the next answer
\begin{equation}
\E_{int}(d) \simeq -V^2_0\, {e^{-4 d} \over \sqrt{2 \pi d}}\,\left( {A^2 \over 2} + {1\over 8 d} \left( {3 A^2 \over 8} + B \right)\right) + O\({1\over d^2}\) \ ,
\end{equation}
where $z_0$ and $A$ are defined as in (\ref{79}), while
\begin{multline}
B=A^3\Bigg( -V^2_0 \left( 1 - {\hbox{ctg}(a z_0)\over z_0} + {a \over \sin^2(a z_0)} \right)^2 A \ - \\ - \ 2-{(1+z^4_0)\over z^3_0}\hbox{ctg}(a z_0) \ +\\
+ \ {a\over z^2_0 \sin^2(a z_0)} \left( 1- 2 V^2_0(1- a z_0 \hbox{ctg}(a z_0))\right) \Bigg) \ .
\end{multline}
As in the symmetric case, these expressions are valid both for $V_0<1$ and $V_0>1$, while for $V_0=1$, when $z_0=0$, they should be replaced by
\begin{equation}\begin{gathered}
A={a\over 1+ a} \ , \\
B=-a^2\, {45+135 a +165 a^2+90 a^3 + 28 a^4+ 8 a^5 \over 45(1+ a)^4} \ .
\end{gathered}\end{equation}
Moreover, the expansion of $\E_{int}(d)$ for large $d$, presented above, becomes invalid, when in the single well (or barrier) there exists the level with zero energy, since in this case the denominator in $A$ vanishes, i.e. $ \sin(a z_0)+ z_0 \cos(a z_0)=0$. Therefore this case requires for a separate analysis, similar to considered for two wells in Sect.4.
\begin{figure*}[ht!]
\subfigure[]{
\includegraphics[width=\columnwidth]{a=1_V0=408.eps}
}
\hfill
\subfigure[]{
\includegraphics[width=\columnwidth]{a=1_V0=74.eps}
}
\vfill
\subfigure[]{
\includegraphics[width=\columnwidth]{a=1_V0=10.eps}
}
\vfill
\subfigure[]{
\includegraphics[width=\columnwidth]{a=1_V0=8v1.eps}
}
\hfill
\subfigure[]{
\includegraphics[width=\columnwidth]{a=1_V0=8v2.eps}
}
\caption{The behavior of $\E_{int}^R(d)$ for $a=1$ and
(a) $V_0=4.08$; (b) $V_0=7.4$; (c) $V_0=10$; (d,e) $V_0=8$. }
\label{Eint3}
\end{figure*}
The renormalization coefficient for $\E_{int}^R(d)$ coincides with the corresponding one in the two-wells configuration up to the sign, namely
\begin{equation}
\L_{int}(d) ={1 \over 2\pi}\, \int\limits_0^{\infty} dy\, e^{-4 d \sqrt{1+y^2}}\, {\left(1- e^{-2 a \sqrt{1+y^2}}\right)^2 \over (1+y^2)^2} \geqslant 0 \ ,
\end{equation}
and so for large $d$ reveals the same asymptotics as in (\ref{aslam}) with different sign.
\begin{figure*}[ht!]
\subfigure[]{
\includegraphics[width=\columnwidth]{Ediscr_V1_V2=-2_a=1_d=2.eps}
}
\hfill
\subfigure[]{
\includegraphics[width=\columnwidth]{Ediscr_V1_V2=-2_a=1_d=2v1.eps}
}
\vfill
\subfigure[]{
\includegraphics[width=\columnwidth]{Ediscr_V1_V2=-2_a=2_d=2.eps}
}
\hfill
\subfigure[]{
\includegraphics[width=\columnwidth]{Ediscr_V1_V2=-2_a=2_d=2v1.eps}
}
\caption{(Color online). The behavior of the levels in the barrier-well system without antisymmetry of the potential in dependence on the well depth $V_1$ with fixed height of the barrier $V_2=-2$: (a,b) for $d=2$, $a=1$; (c,d) for $d=2$, $a=2$. }
\label{EdiscrV1}
\end{figure*}
As a result, the leading term in the renormalized Casimir energy $\E_{int}^R(d)$ for the antisymmetric case turns out to be the following
\begin{equation}\label{Casimir}\begin{gathered}
\E^R_{int}(d)=\E_{int}(d) + \L_{int}(d) V_0^2 \simeq \\ \simeq V^2_0\, {e^{-4 d} \over \sqrt{2 \pi d}}\, \left[ e^{-2 a} \sinh^2 a -{A^2\over 2} \right] \ .
\end{gathered}\end{equation}
In (\ref{Casimir}) the multiplier in square brackets is sign-alternating in dependence on the single source parameters $(V_0\, , a)$. In particular, for the set $a=1$ and $V_0=4.08, 7.4, 10$, considered in Sect.4, this multiplier is positive, hence, the sources reflect at large separations, whereas for $a=1$ and $V_0=8$ it is negative and so the sources attract. Note that in the last case the Casimir force changes from reflection to attraction by increasing $d$. The behavior of $\E_{int}^R(d)$ starting from sufficiently small separations up to large $d$-asymptotics is shown in Figs.\ref{Eint3}.
Apart from these peculiar features, the general answer for the Casimir force in the antisymmetric case is substantially different from the symmetric one, since now the asymptotics of the Casimir force for large separations between sources is subject of the standard $\exp (-2 m s)$ law. Moreover, it is the unique specifics of the source-anti-source system, since it is the only case, when the symmetry between the positive and negative energy eigenstates according to (\ref{oddness}) takes place. The direct consequence of this symmetry is that the separate contributions from negative discrete levels are absent both in final expressions (\ref{rhoVP1}) and (\ref{IntWronskReg1}) for $\rho} \def\vr{\varrho_{vac}(x)$ and $\E_{vac}(d)$. Indeed this circumstance underlies the standard $\exp (-2 m s)$ fall-down of the Casimir force for large separations between sources, since in the symmetric case the breakdown of the latter is caused by the contribution from the negative discrete levels. Namely, the main contribution to the asymptotics of $\E_{vac}^{int}(d)$ will be given by the lowest $\epsilon} \def\E{\hbox{$\cal E $}_0<0$ according to eq. (\ref{e0<0}). As soon as the strict antisymmetry of the external potential (\ref{w2}) is broken, the spectrum immediately transforms into the standard non-symmetric form, where the levels are able to approach the threshold of the lower continuum, for instance, with growing depth of the well. This circumstance reminds the well-known quantum-mechanical effect, when in the one-dimensional potential well with arbitrary small depth and size, but with equal height of both walls, there exists always at least one discrete level, which can be very shallow, but disappears as soon as the height of the walls becomes different. As an illustration of this property of the antisymmetric case in Figs.\ref{EdiscrV1} the behavior of the levels in the barrier-well system without antisymmetry of the potential $W_2(x)$ in dependence on the well depth $V_1$ with fixed height of the barrier $V_2$ is shown.
\subsection*{7. Conclusion}
To conclude, in this work by means of the $\ln\text{[Wronskian]}$ contour integration techniques for calculating the Casimir effect we have shown the magnitude of the Casimir force variability for two short-range Coulomb sources, embedded in the background of one dimensional massive Dirac fermions. The main result is that essentially non-perturbative vacuum QED-effects, including the effects of super-criticality, are able to add a set of new properties to Casimir forces between such sources, which turn out to be more diverse compared to the case of scattering potentials with scalar coupling to fermions, considered in Refs.~\cite{Jaffe2004, nanotubes}. In particular, we have shown that the interaction energy between two identical positively charged short-range Coulomb sources can exceed sufficiently large negative values and simultaneously reveal some features similar to a long-range force, like the electronic Casimir force between two impurities on a one-dimensional semiconductor quantum wire despite nonzero effective mass of the mediator ~\cite{tanaka2013}, which could significantly alter the properties of such quasi-one-dimensional QED-systems.
The most intriguing circumstance here is that in the symmetric case their mutual interaction is governed first of all by the structure of the discrete spectrum of the single source, in dependence on which it can be tuned to give an attractive, a repulsive, or an (almost) compensated Casimir force with various rates of the exponential fall-down, quite different from the standard $\exp (-2 m s)$ law. Let us mention once more that the essence of the long-range interaction between sources, which appears whenever the single well contains a level $\epsilon} \def\E{\hbox{$\cal E $}_0$ close to the lower threshold, is that under these conditions the exponential fall-down starts at extremely large distances $d \gg \(1-\epsilon} \def\E{\hbox{$\cal E $}_0^2\)^{-1/2}$ between sources, rather than by replacement of the exponential asymptotics by a power-like behavior, what could happen only for a massless mediator. No less interesting is the pattern of Casimir interaction observed in the $\delta} \def\D{\Delta$-limit with sources of negligible width, which can also be explored in detail within the presented $\ln\text{[Wronskian]}$ contour integration approach. The latter circumstance could be quite important, since in some reasonable cases the best description for impurities is achieved indeed in the $\delta} \def\D{\Delta$-limit.
A special attention should be paid to the antisymmetric source-anti-source system, which reveals quite different features. In particular, in this case there is no possibility for the long-range interaction between sources. The asymptotics of the Casimir force follows the standard $\exp (-2 m s)$ law. Moreover, the symmetric and antisymmetric cases are substantially different for small separations between sources. Namely, there follows from Figs.\ref{Eint1}-\ref{EintD},\ref{Eint3} which are calculated for the same sets of single source parameters up to replacement well-barrier, that in the symmetric case the Casimir interaction between sources is attractive, while in the antisymmetric one it turns into sufficiently strong repulsion. Remarkably enough, the classic electrostatic force for such Coulomb sources should be of opposite sign. There is no evident explanation for this effect. However, the set of parameters used is quite wide to consider this effect as a general one.
These results may be relevant for indirect interactions between charged defects and adsorbed species in quasi-one-dimensional QED systems mentioned above.
\subsection*{7. Acknowledgments}
The authors are very indebted to Prof. P.K.Silaev and Dr. O.V.Pavlovsky from MSU Department of Physics for interest and helpful discussions. KS is especially grateful to Ms. A.Kondakova, MSU Department of Physics, Solid State Division for information on the current situation in the research of quantum wires. This work has been supported in part by the RF Ministry of Sc. $\&$ Ed. Scientific Research Program, projects No. 01-2014-63889, A16-116021760047-5, and by RFBR grant No. 14-02-01261.
|
{
"timestamp": "2019-03-05T02:24:51",
"yymm": "1812",
"arxiv_id": "1812.03416",
"language": "en",
"url": "https://arxiv.org/abs/1812.03416"
}
|
"\\section{Introduction} \n\\label{sect:intro}\n\nA driving question in diophantine geometry is to p(...TRUNCATED)
| {"timestamp":"2019-11-15T02:07:20","yymm":"1812","arxiv_id":"1812.03423","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\\label{sec:introduction}\nLet $(M,g)$ be a (pseudo-)Riemannian manifold of (...TRUNCATED)
| {"timestamp":"2019-08-06T02:18:32","yymm":"1812","arxiv_id":"1812.03591","language":"en","url":"http(...TRUNCATED)
|
"\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nOptional type systems must be designed with clo(...TRUNCATED)
| {"timestamp":"2018-12-11T02:17:42","yymm":"1812","arxiv_id":"1812.03571","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\nPeople rely on word-of-mouth learning when deciding\nwhether to vaccinat(...TRUNCATED)
| {"timestamp":"2020-06-30T02:04:26","yymm":"1812","arxiv_id":"1812.03354","language":"en","url":"http(...TRUNCATED)
|
"\\section{\\label{Intro}Introduction}\n\nIn physics, topological equivalence classes refer to Hamil(...TRUNCATED)
| {"timestamp":"2019-03-27T01:06:31","yymm":"1812","arxiv_id":"1812.03738","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\r\n\r\n\r\n\r\n\r\nIt is well known that the Virasoro algebra plays signi(...TRUNCATED)
| {"timestamp":"2019-06-18T02:21:00","yymm":"1812","arxiv_id":"1812.03435","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\nAmong the three CKM \\cite{C,KM} angles, the uncertainty on $\\phi_3$ is(...TRUNCATED)
| {"timestamp":"2018-12-11T02:11:01","yymm":"1812","arxiv_id":"1812.03440","language":"en","url":"http(...TRUNCATED)
|
End of preview.