One night in early January 1634, Saint Francis Xavier appeared in a vision to Father Marcello Mastrilli, a Jesuit priest near death. The saint promised Father Mastrilli that "All those who implore my help daily for nine consecutive days, from the fourth to the twelfth of March inclusive and worthily receive the Sacraments of Penance and Holy Communion on one of the nine days will experience my protection and may hope with entire assurance to obtain from God any Grace they ask that is for the good of their souls and the glory of God."
The Novena of Grace has successfully helped many of the faithful for centuries. You can pray this novena any time during the year, not just on the dates specified by St. Francis Xavier.
O Saint Francis Xavier, well beloved and full of charity, in union with thee, I reverently adore the Majesty of God; and since I rejoice with exceeding joy in the singular gifts of grace bestowed upon thee during thy life, and thy gifts of glory after death, I give Him hearty thanks therefore; I beseech thee with all my heart's devotion to be pleased to obtain for me, by thy effectual intercession, above all things, the grace of a holy life and a happy death. Moreover, I beg of thee to obtain for me…(here mention the spiritual or temporal favour to be prayed for). But if what I ask of thee so earnestly doth not tend to the glory of God and the greater good of my soul, do thou, I pray obtain for me what is more profitable to both these ends.
Blessed be Jesus in the most holy Sacrament of the Alter.
Blessed be he glorious assumption.
Blessed be St. Joseph her most chase spouse.
When we say the Rosary we are imitating what the mysteries contain and obtain what they promise. We are praying for strength and wisdom and for the graces the Rosary can give us and others. The word "mysteries" refers to particular events in the lives of Jesus and Mary.


In this tutorial you will prepare a timetable for your school. Data about this school can be found in Appendix D. You will start by transferring this data into six files. This procedure has been done for you and are kept in form *.tSt. You will copy them to the form *.txt.
Activate the simple text editor, Edit.Com, by typing EDIT at the DOS prompt and pressing the ENTER button.
Open the file Asas.TXT. Browse through the file and compare with the instructions in Appendix A-I. Likewise do the same for the other five files. More details can be found in the chapter on INPUT.
Start the Timetable Generator. When the copyright message appears, press the OK button to see the main menu. Select INPUT (use the arrow keys and then press <Enter> or type the mnemonic letter for INPUT, i.e. the Alt key and the letter I pressed together).
How can I be sure that all data was transferred correctly into the six files ? Activate the UJI menu. Test each of the six files, there should be no errors detected. Make some errors on purpose to any one of the six files, then test the file. Did the UJI routine detect the errors you made ? Correct the mistakes, save the file and then test the file again to make sure that there are no errors.
Sometimes information (data to be arranged) given by the various heads of department can have error. Two useful routines found in the JANA menu are the table of class against subject which shows the frequency of periods allocated and the table of class against subject which shows the number of periods allocated for each lecturer. Try all of them and browse through the files created by these routines.
If you do not understand the meaning of the codes used, use the KOD menu to select the relevant code.
Press KELUAR to go to the main menu when you are finished.
Before you generate the timetable, you must make sure that the data to be arranged together with the conditions set is feasible (What makes you think that you did not put such heavy constrains that it is impossible for all data to be slotted in the timetable?). This program does not attempt to give definite proofs if a set of data is impossible to be put in or not, but, it has algorithms to indicate the likelihood of an impossible condition.
Press the SEMUA button to generate the timetable for all the classes, with the fixed data first followed by blocks, double periods and finally single. The final number of rejects you should expect after the TUNGGAL PULIH is used should be about 50 % of the rejects.
In this example we will not modify the Arahan.TXT nor Makasas.TXT. Instead we will deal with the problem of rejects later. In a real situation you would be advised to deal with the problem first as the number of rejects are significant.
Do the BLOK. You should get some rejects. Repeat the procedure until you get a feel of the minimum number of rejects possible. Do the BLOK again until you are satisfied with the number of rejects. Because Blocks are the most difficult to arrange, block problems must be solved immediately. Normally you might rethink having a particular kind of block (and thus modify Makasas.TXT) or relax conditions in Arahan.TXT or override the conditions only for the rejects. To override the conditions you need to use the BAIKPULIH routine. (Read the BAIKPULIH section below and then continue from here).
Next the double periods, mark the SAMBUNG check box and then choose KEMBAR. By marking the Sambung check box you have instructed the computer to continue with timetable generation with previous arranged data intact. You should get a number of rejects. In a real situation you would have actually quit this menu and selected the BAIK PULIH routine to deal with the double period rejects before returning to deal with the single period data. Proceed with single period data after marking the SAMBUNG check box.
You should get about quite a number of single period rejects. Since it is expected that the single period data would be the last to be dealt with, which in turn would give rise to more rejects of this type, a separate routine, the TUNGGAL PULIH is provided. This routine makes use of more rigorous search algorithms to arrange the rejected data (only single period data) and so it could take some time. Change the default tries (25) to 2 to get a feel of the amount out time it takes. Each time you hear a beep, it means a data is accepted. As long as data gets accepted, the process time shortens after each round.
This routine is quite intuitive. Try to find a slot which you would agree to (although some conditions has to be ignored). If satisfied, accept the slot(s). Repeat this procedure until slots are found for all the rejected data. Make use of the LAPORAN to get detail information of reason for rejects for each slots to select the least compromised slot. Sometimes you will find that none of the slots are satisfactory. In cases like these, use the PERUBAHAN routine.
In this routine you have less visual help, so you would be advised to use a printed general timetable (using INDUK in CETAK routine). You could check if your change is valid by pressing the TERIMA button. It will give you feedback on why a data in question is not to be allowed. You will have the option of ignoring the warning or canceling the command. Make use of the LAPORAN as described in Baik Pulih above.
Use this routine to arrange the rooms.
The rest of the routines are also quite intuitive and details can be found in the relevant chapters.


"I Sing the Body Electric" is a poem by Walt Whitman from his 1855 collection Leaves of Grass.
Its original publication, like the other poems in Leaves of Grass, did not have a title. In fact, the line "I sing the body electric" was not added until the 1867 edition. At the time, "electric" was not yet a commonly used term.
In 1969, author Ray Bradbury published I Sing the Body Electric, a science fiction anthology named after the poem and including a short story by the same title. The short story was based on a 1962 Twilight Zone episode that Bradbury had also written.
"I Sing the Body Electric" is the title of a 1972 Columbia Records album by the jazz fusion group Weather Report.
"I Sing the Body Electric" is the title and first line of a song from the 1980 musical film Fame.
"I Sing the Body Electric" was the theme song heard in the PBS fitness and aerobics series Body Electric.
"I Sing the Body Electric" was referenced by Susan Sarandon's character, Annie Savoy, in the 1988 movie Bull Durham.
"I Sing the Body Electro" was the 1998 debut solo album of Kurtis Mantronik.
"I Sing the Body Electric" is the title of a 2011 song by the trip hop band Arms and Sleepers.
The 2012 song, "Body Electric", from Lana Del Rey's third EP, Paradise, alludes to Whitman in the lyric, "Whitman is my daddy." "I sing the body electric" is the song's chorus. Part of the poem is recited in Del Rey's short film Tropico.
"The Body Electric" is a song on the 1984 Rush album Grace Under Pressure.
"Celebrate the Body Electric (It Came from an Angel)" is a song on the 2008 Ponytail album Ice Cream Spiritual.
"I Sing the Body Electric, Especially When my Power's Out" is a poem by American poet Andrea Gibson (from the album Flower Boy).
"I Sing the Body Electric" was used in an episode of the podcast Welcome to Night Vale, where it is somewhat parodied and made strange.
"I Sing the Body Electric" was performed by Klaus Schulze in Rouen in April 1976 running time 49:11 and released on his Historic Edition CD in 1995 and reissued on his Ultimate Edition in 2000.
↑ "Lana Del Rey hates personal critics". STV. STV Group plc. Retrieved 5 October 2012.
↑ Moore, Alex. "Here's Lana Del Rey's new Walt Whitman-referencing track, 'The Body Electric'". Death and Taxes. Retrieved 5 October 2012.


Bot is stuck when he trying to back to the spot after killing stone outside the "circle"
Premium: IgorSeba, now waithack doesn't work, speedhack with onehit loses hits, can't automatically sell items, can't automatically buy potions, farmbot doesn't work properly, bot stucks sometimes without any reason.
We dont need more users here.
Still attackspeed and selling items doesn't work,I've patched m2bob 3 times.


Are you needing that festive feeling but the breakfast sherry just isn't cutting it? Is your eggnog not quite noggy enough? Are your toes not feeling the mistle? Don't worry. In the most divine collaboration, Arcola Theatre residents Basement Orchestra are joining forces with The Spiritual Playground to bring you an extravaganza of yuletide fun and frolics that will knock you for six geese a-laying. There will be music. There will be singing. There will be games. There will be glitter, sleigh bells, dancing and angels, all held together by Shesus herself. This is going to stuff your stocking, baste your turkey and put a Christmas tree up your chimney. It's Christmas, but not as we know it.
One of those 'so good they played it twice' kind of affairs. Show times are 3pm and 7pm.


Well-being, authenticity, physical and spiritual health, self-confidence. Who does not feel at sympathize with these words here and there some discomfort? In private and professional life, it is to stay to your woman or your man, but too often on what costs? How often do we tear ourselves together to give us and others no nakedness?
How often do we come up against our own limits, but did not find the measure, time or requirements to bring our own ways to a common denominator.
Who am I? How do I see myself? How do others see me?
How is it possible for me to have confidence in myself and in my life? Strategies for a "less is more" (control vs. release). Esteem and forgiveness for more "Courage to Myself"!
Broader limits, build confidence, gain freedom, live and enjoy solidarity and loyalty, the Courage to Myself for a fulfilling way to "We".
Outdoor activities with reflections and transfers.
Creating the personal self-development plan (SDP) for a lifelong Courage to Myself.
The second day offers to all fans of adventure and daring an ideal opportunity to meet your own boundaries, to overcome them, to grow with obstacles, to face challenges and ultimately triumphant – and guarantees much bigger and stronger – to reach your highest point.
Obviously in the high ropes course it is not only about individual performance. While everyone has to pass its own test of courage and has to overcome fear, the entire action should also be considered being a team event.


#12 25245 tom_tyler Released SDP 2019.1.25238 (2019/03/02).
#11 21810 tom_tyler Released SDP 2017.1.21808 (2017/03/08).
#8 16581 robert_cowham Standardised processing and formatting.
#7 15976 russell_jackson Added try/except around client delete.
#5 15314 russell_jackson Added quotes around client name.
#3 11583 russell_jackson Added -Fs to client delete in case shelves can't be deleted for some reason.
groups that have a / in the name.


text:
\section{Introduction}
\label{intro}

The cosmological standard model includes dark matter (DM) as an essential component, commonly considered to be a neutral particle not part of the standard model of particle physics (SM). Assuming thermal production in the early Universe, a weakly interacting massive particle (WIMP) in the GeV-TeV mass is a strong candidate, since the Weak Interaction of the SM yields just the right annihilation cross section to predict the observed relic density of DM, a relation known as the WIMP miracle.
This default candidate is the main target of experimental DM search, and since the weak interaction couples universally to all leptons and quarks, its parameter space is successively scanned and ruled out by direct detection experiments based on WIMP-nucleon interactions~\cite{Escudero:2016gzx,Baer:2016ucr,Liu:2017kmx,Aprile:2018dbl,Aprile:2019dbj}
and indirect searches looking for the products of annihilation into hadronic channels, such as anti-protons~\cite{Evoli:2015vaa,Cuoco:2017iax} and {$\gamma$-rays}~\cite{Abdalla:2018mve}.

Avoiding hadronic interaction of DM requires the introduction of a new force and corresponding charge, which is only carried by the DM and leptons. In the initial version of this Leptophilic Dark Matter~\cite{Fox:2008kb},
all lepton generations carry the same charge, resulting in equal branching ratios in the annihilation of DM. In this case, the strongest constraints on the DM annihilation cross section come from observation of dwarf galaxies in {$\gamma$-rays}
based on the $DM + DM \rightarrow \tau^{+} + \tau^{-}$ channel, which due to its higher {$\gamma$-ray} multiplicity yields limits about half a magnitude more strict than those on  $DM + DM \rightarrow e^{+} + e^{-}$ channel and $DM + DM \rightarrow \mu^{+} + \mu^{-}$ channel~\cite{Ackermann:2015zua,Archambault:2017wyh}. These limits are subject to about one order of magnitude variation from uncertainty on the halo shape and resulting J-factors~\cite{Hiroshima:2019wvj}, which however is independent of the annihilation channel.
Most recently very strict limits on hadronic and the $DM + DM \rightarrow \tau^{+} + \tau^{-}$ channel based on the morphology of {$\gamma$-ray} flux from the galactic center have been brought forward~\cite{Abazajian:2020tww}, giving explicitly no such constraint on $DM + DM \rightarrow e^{+} + e^{-}$ channel and $DM + DM \rightarrow \mu^{+} + \mu^{-}$ channel.

On the other hand, search for DM annihilation in positron and electron cosmic rays with detectors such as AMS-02~\cite{Aguilar:2013pos,Accardo:2014pos,Aguilar:2014ele,Aguilar:2014all,Aguilar:2019ele,Aguilar:2019owu}, CALET~\cite{Adriani:2017all,Adriani:2018ktz} , DAMPE~\cite{Ambrosi:2017all} and the Cosmic Ray Subsystem on the Voyager probes is most sensitive to the electron channel, since its signature is a sharp drop in the spectrum at the mass of the DM particle which can be recognized above a smooth astrophysical background~\cite{Ibarra:2013zia,Motz:2015cua,Boudaud:2016mos}.
For GeV-TeV range DM, the target region is the local DM halo within $\sim$kpc range due to the energy loss and resulting limited propagation distance of electron cosmic rays.
This complementarity can reduce the possible impact of astrophysical uncertainties in the case of DM with universal coupling to leptons. For DM with selective coupling to the different lepton flavors, either search with {$\gamma$-rays} or charged cosmic rays may have preferential sensitivity.

Apart from DM, the other strong indication of physics beyond the SM is the neutrino mass, and many theoretical models extending the SM aim at solving both issues simultaneously, examples being radiative seesaw models at one-loop~\cite{Ma:2006km}, two-loop~\cite{Kanemura:2011mw, Kajiyama:2013zla}, and three-loop~\cite{Krauss:2002px, Aoki:2008av, Gustafsson:2012vj}.
Several models extending the SM by an additional  U(1) gauge symmetry have been proposed, which favor annihilation or decay to tau and/or muon as a possible DM-only explanation of the positron excess~\cite{Kohri:2013sva,Ko:2014lsa,Han:2019diw}, while also featuring a mechanism for giving the neutrinos mass.

In this context we investigate if a thermally produced DM candidate based on a flavor-specific $U(1)_{e-\mu}$ gauge symmetry coupling only to electron and muon is also feasible, corroborated by simultaneous explanation of the neutrino sector. This kind of DM would be a favorable target to search in electron-positron cosmic rays while being less detectable by {$\gamma$-ray} search. After establishing the particle physics model
defining the properties of the DM, we discuss its cosmic-ray signatures and implications from available CALET and AMS-02 data.
While introduction of a new flavor-specific gauge interaction lacks the elegance of the classical WIMP, studying such a model seems worthwhile as it allows to keep a thermal production mechanism and a WIMP-like DM candidate. This should be seen against the trend of DM candidate theory becoming more and more diversified to avoid constraints on the WIMP and WIMP-like particles~\cite{Baer:2014eja}.

Our extension of the SM is based on a radiatively induced neutrino mass (scotogenic model), which originally provides us with an appropriate explanation of the hierarchy among the Yukawa sector of the SM. The ratio between the top Yukawa quark coupling($\sim1$) and the electron Yukawa coupling($\sim10^{-6}$) is of the order $10^6$, which respectively are the heaviest and lightest masses in the fermion sector of the SM.
However, the ratio between the electron Yukawa coupling and the typical neutrino Yukawa coupling($\sim10^{-13}$) is of the order $10^{7}$. If we assume the neutrino mass to be of Dirac type and to be induced at tree level, which is the same as for the other matter sectors in the SM, this would suggest that there is a huge gap
between the neutrino coupling and the other three Yukawa couplings.
The scotogenic model generates neutrino mass at loop level, with newly introduced fields running inside the loop. It is found that with a 0.01 loop suppression factor and two Yukawa couplings at one-loop level in the neutrino mass formula, the order of Yukawa coupling at one-loop level is minimally $10^{-6}$, which is comparable to the electron Yukawa coupling. We fix the mass scale of one
new field to be on the order of one TeV, which allows for the new scale to be tested by current experiments.
Another advantage of this model is its predicted correlation between the DM candidate properties and the neutrino mass, since the DM field is running inside the neutrino loop. Therefore, the neutrino interacts with SM-like Higgs only though the DM field in the generation of the neutrino mass. This provides a natural explanation for the tininess of the neutrino mass, and phenomenology apart from direct and indirect DM search.
Further phenomenology arises from the not so small strength of the Yukawa couplings and their nontrivial structure to induce the neutrino mixings as well as mass eigenvalues, which might cause lepton flavor violations (LFVs) that are severely restricted by current experiments such as MEG~\cite{TheMEG:2016wtm,Renga:2018fpd}.

To realize a sufficiently high cross section yielding the observed relic density in thermal production of the DM candidate, the annihilation process should be s-channel dominated, which however is helicity-suppressed for a Majorana particle. Therefore we introduce a $Z_3$ discrete symmetry under which the DM is charged, giving it a Dirac nature and ensuring its stability.
Also, we impose a $Z_2$~discrete~symmetry to forbid tree level neutrino mass, where this symmetry is softly broken in the Higgs potential and its broken term contributes to generating the tiny neutrino mass.
The neutrino mass is induced at two-loop level, where we introduce two types of neutral fermions; Dirac type and Majorana type.
In the neutrino sector, the $U(1)_{e-\mu}$ symmetry also plays an important role in predicting the neutrino mass.
Because the nonzero charges (-1 or 1) have to be assigned to only two families, the minimal number of new fermions are two families, which is also the minimal number to explain the active neutrino oscillation data and their mass eigenvalues.
Furthermore, the two families are required to allow gauge anomaly cancellation in a minimal manner.
Thus, we predict one massless neutrino that causes the other two massive neutrinos to be uniquely determined by the experimental results, which are the squared solar mass difference and squared atmospheric mass difference, as we will discuss for both cases of normal hierarchy (NH) and inverted hierarchy (IH) in detail.

This paper is organized as follows:
In \secref{partphysmod}, we explain our particle physics scenario and formulate the lepton sector and the Higgs potential, the masses and mixings for the two new fermions and the active neutrinos, and the mass of the new gauge boson and its interactions, also discussing LFVs.
In \secref{DMprop}, we will discuss our DM candidate, in which we briefly explain why it is not subject to current bounds from direct detection searches, and
explain calculation of the DM relic density.
We also show that Breit-Wigner enhancement may lead to a significant boost factor ($B$) on the annihilation cross section, which may increase the signatures to the level detectable by current indirect DM search.
In \secref{DMsig}, the electron and positron cosmic-ray signature of the DM candidate X is explained, and after introduction of propagation and astrophysical background models, the $e^{-}+e^{+}$~flux measured by CALET~\cite{Adriani:2018ktz} and the $e^{+}$-only flux measured by AMS-02~\cite{Aguilar:2019owu} are interpreted including the DM signature. It is shown that step-like spectral structures in the CALET spectrum could be identified with the signature of the DM candidate, identifying the best-fit regions in $m_{\rm X}$ vs. $B$ space.
Finally we summarize and conclude our results in \secref{sumconcl}.

 \section{Particle Physics Model}
\label{partphysmod}
In this section, we review our model~\cite{footnote1}.
At first, we explain our motivation for introducing new symmetries and fields.
Then, we construct the Lagrangian and Higgs potential, and continue with formulating the neutral fermions, LFVs, and the additional gauge boson sector.

\subsection{Particle Contents and Lagrangian}

We introduce three families of vector-like fermions ($N_{e},N_\mu,N_\tau$), and two families of
Majorana fermions ($\nu_{R_e},\nu_{R_\mu}$) in the fermion sector, so that
we can construct a two-loop induced neutrino mass model. These fermions are minimally required to reproduce the neutrino oscillation data and cancel the anomaly for $\nu_R$.
We extend the scalar sector by introducing an isospin doublet inert boson $\eta$, an isospin singlet inert boson $S$, and a singlet boson $\varphi$ that gives nonzero
Vacuum Expectation Values (VEVs)
to spontaneously break the $U(1)_{e-\mu}$ symmetry as shown later, where the SM-like  scalar boson is symbolized by $H$. Here we denote their
VEVs
as $\langle\varphi\rangle\equiv v_\varphi/\sqrt2$ and $\langle H\rangle\equiv v_H/\sqrt2$, respectively.
In addition, we impose three additional symmetries; gauged symmetry $U(1)_{e-\mu}$ and discrete Abelian symmetries $Z_3$ and $Z_2$.
The first symmetry defines the newly introduced interaction with only the two first generations of leptons, giving the model the intended property of avoiding gauge interactions with the $\tau$-lepton,
while the second one provides stability of potential DM candidates $N,\eta,S$, and assures the Dirac feature of $N$. We associate the lightest Dirac particle $N$ with DM, since the heavier ones can decay into the lighter ones via five-dimensional terms even though the decay is forbidden within the renormalizable theory. The field contents and their assignments are summarized in Table~\ref{tab:1} for fermions and Table~\ref{tab:2} for bosons.

\begin{table*}[t
\begin{footnotesize}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|}\hline
 & \multicolumn{6}{c|}{ SM fermions} & \multicolumn{5}{c|} {New fermions} \\\hline
Fermions  ~&~ $L_{L_e}$ ~&~ $L_{L_\mu}$ ~&~ $L_{L_\tau}$ ~&~ $e_R$ ~&~ $\mu_R$ ~&~ $\tau_R$ ~&~ $N_{e}$ ~&~ $N_{\mu}$ ~&~ $N_{\tau}$ ~&~ $\nu_{R_e}$ ~&~ $\nu_{R_\mu}$ ~
\\\hline
$SU(3)_C$  & $\bm{1}$  & $\bm{1}$  & $\bm{1}$   & $\bm{1}$  & $\bm{1} $  & $\bm{1}$ & $\bm{1}$  & $\bm{1} $  & $\bm{1}$ & $\bm{1}$ & $\bm{1}$   \\\hline

 $SU(2)_L$  & $\bm{2}$  & $\bm{2}$  & $\bm{2}$   & $\bm{1}$  & $\bm{1}$   & $\bm{1}$  & $\bm{1}$  & $\bm{1}$   & $\bm{1}$& $\bm{1}$& $\bm{1}$   \\\hline

$U(1)_Y$  & $-\frac{1}{2}$ & $-\frac12$ & $-\frac12$  & $-1$ &  $-1$  &  $-1$  & $0$  & $0$ &  $0$  &  $0$   &  $0$ \\\hline

 $U(1)_{e-\mu}$ & $1$  & $-1$ & $0$ & $1$  & $-1$   & $0$ & $1$  & $-1$& $0$  & $1$  & $-1$  \\\hline

$Z_3$  & $1$  & $1$ & $1$ & $1$ & $1$ & $1$& $\omega$ & $\omega$& $\omega$ & $1$ & $1$ \\\hline

$Z_2$  & $+$  & $+$ & $+$ & $+$ & $+$ & $+$& $+$ & $+$ & $+$ & $-$ & $-$ \\\hline
\end{tabular}
\caption{Field contents of fermions
and their charge assignments under {$SU(2)_L\times U(1)_Y\times  U(1)_{e-\mu}\times Z_3\times Z_2$}, where $Z_2$ is softly broken and all the fields are singlet under $SU(3)_C$.\label{tab:1}}
 \end{footnotesize}
\vspace*{\floatsep}
\centering {\fontsize{10}{12}
\begin{tabular}{|c||c|c||c|c|c|}\hline
&\multicolumn{2}{c||}{VEV$\neq 0$} & \multicolumn{2}{c|}{Inert } \\\hline
  Bosons  &~ $H$  ~ &~ $\varphi$ ~ &~ $\eta$   ~ &~ $S$ ~ \\\hline
$SU(2)_L$ & $\bm{2}$ & $\bm{1}$ & $\bm{2}$  & $\bm{1}$    \\\hline
$U(1)_Y$ & $\frac12$ & $0$ & $\frac12$  & $0$    \\\hline
 $U(1)_{e-\mu}$ & $0$  & $1$ & $0$ & $0$   \\\hline
$Z_3$ & $1$ & $1$  & $\omega$ & $\omega$ \\\hline
$Z_2$ & $+$ & $+$  & $+$ & $-$ \\\hline
\end{tabular}%
}
\caption{Field contents of bosons
and their charge assignments under {$SU(2)_L\times U(1)_Y\times U(1)_{e-\mu}\times Z_3\times Z_2$}, where $SU(3)_C$ is singlet for all bosons, where $Z_2$ is softly broken, and all the fields are singlet under $SU(3)_C$. \label{tab:2}}
\end{table*}

{\it Anomaly cancellations}:
Since $U(1)_{e-\mu}$ gauge symmetry is anomaly free among the SM fermions, all we need to take into account is the new fermions. Furthermore, since all our fermions are neutral under $U(1)_Y$, we should consider the following two conditions: $U(1)_{e-\mu}$ and $[U(1)_{e-\mu}]^3$. Thus, one straightforwardly finds that these conditions are anomaly free in our model, since each of the fermions has opposite sign under the $U(1)_{e-\mu}$ charge.

{\it Yukawa Interactions}:
Under these fields and symmetries, the renormalizable Lagrangian for quark and lepton sector is given by
\begin{align}
-{\cal L}_{L} = & \sum_{\ell=e,\mu,\tau} \sum_{\ell'=e,\mu}
 \left[y_{\ell} \bar L_{L_\ell}  H \ell_R+y_{\eta_\ell}\bar L_{L_\ell} \tilde\eta N_{R_\ell}
  + y_{S_{\ell'}} \bar N_{L_{\ell'}} \nu_{R_{\nu_{\ell'}}} S   +M_\ell \bar N_{R_\ell} N_{L_\ell} \right]
 \nonumber\\&
+\sum_{(\alpha,\beta)}^{(e,\tau),(\tau,\mu)}{f_\varphi}_{\alpha\beta}   \bar N_{R_\alpha} N_{L_\beta}\varphi
+\sum_{(\alpha,\beta)}^{(\mu,\tau),(\tau,e)}{f'_\varphi}_{\alpha\beta}\bar  N_{R_\alpha} N_{L_\beta}\varphi^*
+{M_N}_{e\mu} \bar \nu_{R_e} \nu^C_{R_\mu}
+ {\rm h.c.},
\label{eq:lag-quark}
\end{align}
where  $\tilde \eta$ is defined by $i\sigma_2 \eta^*$, $\sigma_2$ being the second Pauli matrix,
and $\bar N_R^C\nu_R S^*$ is also allowed by our symmetries but it does not contribute to any phenomenologies. Thus we neglect this term.
$Z_2$~symmetry forbids the Dirac term $\bar L_L\tilde H \nu_R$ at tree level, where $Z_2$ is softly broken at the Higgs potential below.

{\it Scalar potential}:
The renormalizable scalar potential is given by
\begin{align}
&V=-\mu_H^2 |H|^2 + m_\eta^2 |\eta|^2 - m_\varphi^2 |\varphi|^2 + \mu_S^2 |S|^2 \nonumber\\
&+(\mu H^\dag\eta S^* +{\rm h.c.}) + \lambda_{H}|H|^4 + \lambda_{\eta}|\eta|^4+ \lambda_{\varphi}|\varphi|^4
+ \lambda_{S}|S|^4 + \lambda_{H\eta}|H|^2|\eta|^2+ \lambda'_{H\eta}|H^\dag\eta|^2\nonumber\\
&+ \lambda'_{H\varphi}|H|^2|\varphi|^2+ \lambda'_{HS}|H|^2|S|^2 + \lambda_{\eta\varphi}|\eta|^2|\varphi|^2
+ \lambda_{\eta S}|\eta|^2|S|^2 + \lambda_{\varphi S}|\varphi|^2|S|^2,
\end{align}
where the $\mu H^\dag\eta S^*$ term is softly broken under $Z_2$ symmetry, and we expect $\mu$ to be of a rather small scale compared to the electroweak scale.
We parametrize the scalar fields as
\begin{align}
&H =\left[
\begin{array}{c}
w^+\\
\frac{v_H+h+iz}{\sqrt2}
\end{array}\right],\
{\eta =\left[
\begin{array}{c}
\eta^+\\
{\eta_0}
\end{array}\right]}
,\
\varphi=\frac{v'+\rho+i z'}{\sqrt2},\
\label{component}
\end{align}
where $\eta_0$ and $S$ are complex scalars,
$v_H~\simeq 246$ GeV is VEV of the SM Higgs, and $w^\pm$, $z$, and $z'$ are respectively Nambu-Goldstone bosons (NGB)
which are absorbed by the longitudinal component of gauge bosons, denoted by $W$, $Z$, and $Z'$.
$Z'$ arises from the gauged $U(1)_{e-\mu}$ symmetry.
Then we have two neutral boson mass matrices $m^2_{h\rho}$,  $m^2_{\eta S}$ in the basis of $[h,\rho]^T$ and $[\eta_0,S]^T$, which are respectively diagonalized by $O_{a}^T m^2_{h\rho}O_{a}\equiv$Diag[$m_{h_1},m_{h_2}$] and   $O_{\alpha}^T m^2_R O_{\alpha}\equiv$Diag[$m_{H_1},m_{H_2}$],
where $m_{h_1}\approx125$ GeV is the mass of the SM Higgs.
Here we define the mixing matrices as
\begin{align}
O_{a(\alpha)} =
\left[\begin{array}{cc} c_{a(\alpha)} & s_{a(\alpha)} \\ -s_{a(\alpha)} & c_{a(\alpha)} \end{array}\right],\quad
s_{2a}=-\frac{2\lambda_{H\varphi} v v' }{m_{h_2}^2-m_{h_1}^2},\quad
s_{2\alpha}=- \frac{\sqrt2 \mu v_H }{m_{H_2}^2-m_{H_1}^2},
\label{eq:mixing}
\end{align}
where $c(s)_{a(\alpha)}$ is the short-hand notation of $\cos(\sin)_{a(\alpha)}$.
While values $s_a>0$ could be chosen within experimental limits, we take $s_a=0$ in our numerical analysis for convenience as shown later.

{\it Neutral Dirac Fermions}:
After the $e-\mu$ spontaneous breaking, the Dirac fermion mass matrix in basis of $[N_e,N_\mu,N_\tau]^T$ is found as:
\begin{align}
M_{N}\equiv \left[\begin{array}{ccc}  M_e &0 & M_{e\tau}  \\0 & M_\mu & M_{\mu\tau} \\ M_{\tau e} &  M_{\tau\mu} & M_\tau \end{array}\right]_{RL},\label{eq:ML'}
\end{align}
where $M_{e\tau}\equiv f_{\varphi_{e\tau}} v_\varphi/\sqrt2$, $M_{\tau\mu}\equiv f_{\varphi_{\tau\mu}} v_\varphi/\sqrt2$,
$M_{\mu\tau}\equiv f'_{\varphi_{\mu\tau}} v_\varphi/\sqrt2$, and $M_{\tau e}\equiv f'_{\varphi_{\tau e}} v_\varphi/\sqrt2$.
  $M_N$ is diagonalized by a bi-unitary mixing matrix as $D_N=V^\dag_R M_{N} V_L $:
\begin{align}
&V^\dag_R M_{N} M_N^\dag V_R =V^\dag_L M_{N}^\dag M_N V_L  \equiv {\text{Diag.} }\left[|M_1|^2,|M_2|^2,|M_3|^2\right],\\ &N_{L(R)_{e,\mu,\tau}}=V_{L(R)} \psi_{L(R)_{1,2,3}},\label{eq:N-mix}
\end{align}
where $M_{1,2,3}$ is the mass eigenstate, and $\psi$ is the mass eigenvector of $N$.

{\it Neutral Majorana Heavier Fermions}:
In a way similar to the Dirac fermion, the Majorana fermion mass matrix in the basis of $[\nu_{R_e},\nu_{R_\mu}]^T$ is found as:
\begin{align}
M_R \equiv \left[\begin{array}{cc}  0 & M_{N_{e\mu}}  \\ M_{N_{e\mu}}  & 0  \end{array}\right]_{}.\label{eq:MR'}
\end{align}
  $M_N$ is diagonalized by a unitary mixing matrix as $D_R=U^T M_{R}U$:
\begin{align}
&U^T M_{R} M_R^\dag U^*  \equiv {\text{Diag.} }\left[|M_{R_{1}} |^2,|M_{R_{2}}|^2\right],\\ &\nu_{R_{e,\mu}}= U \Psi_{R_{1,2}},\quad
U=\frac1{\sqrt2}\left[\begin{array}{cc}  1 & -1  \\1  & 1  \end{array}\right],
\label{eq:R-mix}
\end{align}
where $M_{R_{1,2}} = M_{N_{e\mu}}$ is the mass eigenstate, and $\Psi$ is the mass eigenvector  of $\nu_R$.

\subsection{Active Neutrino Mass}
The dominant contribution to the active neutrino mass matrix arises from the canonical seesaw model, but the Dirac mass matrix $m_D$ is given at one-loop level. Thus the neutrino mass is induced at two-loop level.
Before formulating the neutrino sector, we evaluate the number of complex parameters.
First of all, three components of $y_\eta$ can be real by phase redefinition for $L_{L_{e,\mu,\tau}}$, which implies that the phases of $N_{R_{e,\mu,\tau}}$ and $e_R,\mu_R,\tau_R$ are fixed.
Second, the two components of $y_S$ can also be real by the redefinition for $\nu_{R_{e,\mu}}$, which suggests that the phases of $N_{L_{e,\mu}}$ are fixed.
Finally, one phase in $M_N$ can be real by the phase redefinition for $N_{L_{\tau}}$. Here we identify $M_\tau$ to be real.
Thus, we have six phases in $M_N$.
The canonical seesaw is given by the following form:
\begin{align}
m_\nu\approx -m_D M_R^{-1} m_D^T,
\end{align}
where $m_D$ is found as follows~\cite{Kanemura:2014rpa, Han:2018zcn}:
\begin{align}
m_D=\frac{y_{\eta_{a}}V_{R_{ai}} M_i V^\dag_{L_{ib}} y_{S_{ib}} s_\alpha c_\alpha}{(4\pi)^2}
\left[\frac{m_{H_1}^2}{M^2_i-m_{H_1}^2}\ln\left[\frac{m_{H_1}^2}{M^2_i}\right]
-
\frac{m_{H_2}^2}{M^2_i-m_{H_2}^2}\ln\left[\frac{m_{H_2}^2}{M^2_i}\right]
\right],
\end{align}
The neutrino mass matrix is then diagonalized by a unitary matrix $U_\nu$ as \linebreak {$U_\nu^T m_\nu U_\nu ={\rm diag}(m_1,m_2,m_3)\equiv D_\nu$}.
Here we can identify $U_\nu$ as the PMNS matrix~\cite{Maki:1962mu}
because of the diagonal mass matrix of the charged leptons, which is achieved by the $U(1)_{e-\mu}$ gauge symmetry.
Each of the mixings is then given by:
\begin{align}
\sin^2\theta_{13}=|U_{\nu_{13}}|^2,\quad
\sin^2\theta_{23}=\frac{|U_{\nu_{23}}|^2}{1-|U_{\nu_{13}}|^2},\quad
\sin^2\theta_{12}=\frac{|U_{\nu_{12}}|^2}{1-|U_{\nu_{13}}|^2}.
\end{align}

In case of NH, we find that
the neutrino mass eigenvalues and the effective neutrinoless double beta decay $\langle m_{ee}\rangle$ are respectively given in terms of observables and phases as
\begin{align}
&{m}_1^2  = 0,\quad {m}_2^2  = \Delta m^2_{\rm sol},\quad
 {m}_3^2 \simeq  \Delta m^2_{\rm atm}
 ,\\
 %
& \langle m_{ee}\rangle \simeq
\left|\Delta m_{\rm sol} \sin^2\theta_{12} \cos^2\theta_{13}e^{i\alpha_{21}}
+ \Delta m^2_{\rm atm}  \sin^2\theta_{13}e^{i(-2\delta_{CP})}\right| ,
 \label{eq:NH-nu}
\end{align}
where $\Delta m^2_{\rm atm}$ and $\Delta m^2_{\rm sol}$ are respectively atmospheric mass difference square and solar mass difference square which are observables~\cite{Esteban:2018azc}; therefore these three neutrino mass eigenvalues are uniquely determined.
Here, we redefine the neutrino mass eigenstate as {$|D_\nu|^2\equiv\frac{s^4_\alpha c^4_\alpha}{(4\pi)^8}{\rm diag}(0,|\tilde m_2|^2,|\tilde m_3|^2)$}. Then, $s_\alpha(c_\alpha)$ can be rewritten by
\begin{align}
s^4_\alpha c^4_\alpha =(4\pi)^8 \frac{\Delta m^2_{\rm atm}}{|\tilde m_3|^2},
\end{align}
which implies that $s_\alpha$ is determined by the two parameters $\Delta m^2_{\rm atm}$ and $|\tilde m_3|^2$.
Also, $\Delta m^2_{\rm atm}$ is fixed by
\begin{align}
\Delta m^2_{\rm sol} =  \frac{|\tilde m_2|^2}{|\tilde m_3|^2}  \Delta m^2_{\rm atm}.
\end{align}

Similar to the case of NH,  we also find
the neutrino mass eigenvalues and $\langle m_{ee}\rangle$ in case of IH to be
\begin{align}
&{m}_3^2  = 0,\quad {m}_2^2  = \Delta m^2_{\rm atm},\quad
 {m}_1^2 =  \Delta m^2_{\rm atm}-\Delta m^2_{\rm sol},\\
 %
 & \langle m_{ee}\rangle =
\left|\sqrt{\Delta m^2_{\rm atm}-\Delta m^2_{\rm sol}} \cos^2\theta_{12} \cos^2\theta_{13}
+\Delta m_{\rm atm} \sin^2\theta_{12} \cos^2\theta_{13}e^{i\alpha_{21}}\right|.
 \label{eq:IH-nu}
\end{align}
And $s_\alpha$ and $\Delta m^2_{\rm sol}$ are found by
\begin{align}
s^4_\alpha c^4_\alpha =(4\pi)^8 \frac{\Delta m^2_{\rm atm}}{|\tilde m_2|^2},\quad
\Delta m^2_{\rm sol} = \left(1- \frac{|\tilde m_1|^2}{|\tilde m_2|^2}\right)\Delta m^2_{\rm atm}.
\end{align}
Here, we redefine the neutrino mass eigenstate as $|D_\nu|^2\equiv\frac{s^4_\alpha c^4_\alpha}{(4\pi)^8}{\rm diag}(|\tilde m_1|^2,|\tilde m_2|^2,0)$.

}

\subsection{Lepton Flavor Violations}
Lepton Flavor Violations (LFVs) arise from the term $y_\eta$ at one-loop level, and
their branching ratios are given by
\begin{align}
& {\rm BR}(\ell_a\to\ell_b\gamma)= \frac{48\pi^3\alpha_{\rm em} C_{ab} }{(4\pi)^4{\rm G_F^2}}
\left|\sum_{i=1-3}Y_{bi} Y^\dag_{ia} F(M_i,m_{\eta^-})\right|^2,\\

& F_{2}(m_a,m_b)=\frac{2 m_a^6 +3 m_a^4 m_b^2 -6 m_a^2 m_b^4 + m_b^6+12 m_a^4 m_b^2 \ln(m_b/m_a)}{12(m_a^2-m_b^2)^4},
\end{align}
where $Y \equiv  y_\eta V_R$, ${\rm G_F}\approx 1.17\times10^{-5}$[GeV]$^{-2}$ is the Fermi constant, $\alpha_{\rm em}\approx1/128$ is the fine structure constant at the Z-boson scale, $C_{21}\approx1$, $C_{31}\approx 0.1784$, and $C_{32}\approx 0.1736$.
Experimental upper bounds are respectively given by Refs.~\cite{TheMEG:2016wtm, Aubert:2009ag,Renga:2018fpd} as
\begin{equation}
{\rm BR}(\mu\to e\gamma)\lesssim 4.2\times10^{-13},\quad {\rm BR}(\tau\to e\gamma)\lesssim 3.3\times10^{-8},\quad {\rm BR}(\tau\to \mu\gamma)\lesssim 4.4\times10^{-8}
\label{LFVbounds}
\end{equation}
and these bounds give constraints on the related Yukawa couplings and masses in the loop.
It is worthwhile to mention the muon anomalous magnetic moment $\Delta a_\mu$.
Although we have a new contribution to $\Delta a_\mu$ from the same term as LFVs, its sign is negative, which is opposite to the experimental result. Thus, we assume a different effect to dominantly cause the anomaly and do not discuss it further.

\subsection{$Z_{e-\mu}$ Gauge Boson}
After the $U(1)_{e-\mu}$ symmetry breaking, we find the massive $Z_{e-\mu}$ gauge boson that is denoted by $Z'$ hereafter, and its mass $m_{Z'}$ is given by
\begin{align}
m_{Z'}=g' v_\varphi,
\end{align}
where $g'$ is the gauge coupling of the $U(1)_{e-\mu}$ symmetry and we neglect kinetic mixing for simplicity.
Gauge interactions among $Z'$ are given by
\begin{align}
&T =g' Z'_\mu(\bar e\gamma^\mu e - \bar\mu \gamma^\mu\mu+ \bar \nu_e \gamma^\mu P_L\nu_e- \bar \nu_\mu \gamma^\mu P_L\nu_\mu ) \label{eq:int-dm}
\\
&+ g' Z'_\mu
\sum_{i,j=1-3}\left(\frac12 \bar \psi_i (W_{N_L}+W_{N_R})_{ij} \gamma^\mu \psi_j
+\frac12\bar \psi_i (W_{N_L}-W_{N_R})_{ij} \gamma^\mu\gamma_5 \psi_j
+ \bar \Psi_i W_{R_{ij}} \gamma^\mu P_R\Psi_j \right),\nonumber
 \end{align}
 where $W_{N_{L(R)}}\equiv V_{L(R)}^\dag {\rm Diag}[1,-1,0] V_{L(R)}$, and  $W_R\equiv U^\dag {\rm Diag}[1,-1,0] U$.
Then each of the decay rates of $Z'$ is given by
 \begin{align}
& \Gamma(Z'\to e\bar e)\approx   \Gamma(Z'\to \mu\bar \mu)\approx  \Gamma(Z'\to \nu_{e,\mu}\bar \nu_{e,\mu})  \approx \frac{g'^2 }{12\pi} m_{Z'}, \\
& \Gamma(Z'\to X\bar X)\approx \frac{|(W_{N_L}+W_{N_R})_{11}|^2 g'^2 }{12\pi} m_{Z'} \left(1+\frac{m_X^2}{m_{Z'}^2}\right)\sqrt{\left(1-\frac{4m_X^2}{m_{Z'}^2}\right)},
\label{eq:dr1}
 \end{align}
 where we have
assumed $2 M_1< m_{Z'} <M_{R_{1,2}}, M_2,M_3$ and $M_1$ is considered to be the DM in the next section.
When the decay rate of $ \Gamma(Z'\to X\bar X)$ can be negligible, the branching ratios are respectively found as
 \begin{align}
& {\rm BR}(Z'\to e\bar e)\approx   {\rm BR}(Z'\to \mu\bar \mu)\approx  {\rm BR}(Z'\to \nu_{e,\mu}\bar \nu_{e,\mu})  \approx \frac13.
\label{eq:dr2}
 \end{align}

 Since $Z'$ couples to an electron and positron pair, we have to impose the following constraint which comes from LEP~\cite{Schael:2013ita}:
 \begin{align}
 4950\ {\rm GeV} \lesssim \frac{m_{Z'}}{g'},\label{eq:lep}
 \end{align}
 where we have adopted a conservative bound.
Here, we briefly mention other possibilities to detect signatures at colliders in the future, for the case of the $Z'$-mass being of the order of 100~GeV.
First is the Large Hadron Collider (LHC), which can observe the mode {$q\bar q\to Z/\gamma\to e^{+}e^-Z'(\mu^+\mu^-Z') \to \{2e^{+}2e^-,2\mu^{+}2\mu^-,e^{+}\mu^-e^{-}\mu^+\}$~\cite{Sirunyan:2018nnz}}. Second is the future International Linear Collider (ILC) which could measure modes {$e^+ e^-\to Z'\to \{e^+ e^-,\mu^+\mu^-\}$}~\cite{Baer:2013cma, Fujii:2017vwa}.
So far there is no analysis of LHC data for above channels, thus LHC provides no constraint on the model parameters.

\section{Properties of Dark Matter Particles}
\label{DMprop}
The DM candidate in this model is the lightest Dirac fermion $X\equiv \psi_1$, and its mass given by $m_X\equiv M_1$.
In this section we study with which model parameters DM consisting of X and $\mathrm{\bar X}$ is viable, with the goal of showing the existence of an allowed region,
leaving a complete scan of the whole possible parameter space for future work.
First, we briefly discuss detectability by direct detection searches and the reason why we take $s_a=0$.
Then, we
explain the calculation of the DM relic density which is determined by gauge interaction via s-channel, and perform a numerical analysis to
explore the region around the pole $m_X=m_{Z'}/2$ which satisfies all discussed constraints.
Finally, we discuss that by applying Breit-Wigner enhancement to our model, the annihilation cross section in the current Universe can be increased by a boost factor $B$
compared to a generic thermally produced DM with velocity independent annihilation cross section.

\subsection{Direct Detection}
The latest bound on spin-independent scattering is reported by the XENON1T experiment, which gives an upper limit on the spin independent elastic DM-nucleon cross section \linebreak $\sigma$:  $\sigma<4.1\times 10^{-47}$ cm$^2$ at $m_X=30$ GeV with 90\% confidence level~\cite{Aprile:2018dbl}.
Our DM dominantly interacts with nuclei only via the mixing of $s_a$ at tree level arising from the terms $f_\varphi^{(')}$.  Then, our scattering cross section is given by
\begin{align}
\sigma\approx \frac{\mu^2_{nX}}{2\pi v^2} m_n^2 C^2|(V^\dag_R f'_\varphi V_L)_{11}|^2 (c_a s_a)^2\left(-\frac{1}{m_{h_1}^2} + \frac{1}{m_{h_2}^2} \right)^2,
\end{align}
where $\mu_{nX}\equiv m_n m_X/(m_n+m_X)$, $m_n$ is the mass of neutron, and $C\approx 0.3$ is determined by lattice simulation among DM and nucleon.
The easiest way to evade this constraint is to assume $s_a$ to be zero.
LHC results also favor $s_a$ to be small with an upper bound of $s_a\lesssim 0.2$~\cite{Heinemeyer:2013tqa}.
Thus, with the choice of $s_a=0$, no direct detection bounds need to be considered for our DM candidate. We leave exploring possible bounds for the case $s_a>0$ for future work.

\subsection{Relic Density
In the following we discuss the relic density of DM.
The relevant processes arise from Yukawa interaction via $y_\eta$ and kinetic interaction via $g'$.
In case of Yukawa interaction, the coupling is mainly restricted by $\mu\to e\gamma$, which is typically of the order 0.01, although it depends on its flavor structure. Then the cross section via $y_\eta$ is $10^{-17}$ GeV$^{-2}$ at most.
Thus, Yukawa contribution is negligibly small compared to the cross section $\sim10^{-9}$~GeV$^{-2}$ required to explain the relic density.
As a result, the dominant cross section to the relic density comes from kinetic interaction.

We make use of the micrOMEGAs package~\cite{Belanger:2013oya} to calculate the speed averaged cross section $\langle\sigma v_{\rm rel}\rangle$, and the relic density.
micrOMEGAs is adapted to this model by defining the properties of the interaction mediated by $Z'$ in the form of a kinetic term simplified from Eq.~\eqref{eq:int-dm} as follows:
\begin{align}
&T =g' Z'_\mu(\bar e\gamma^\mu e - \bar\mu \gamma^\mu\mu+ \bar \nu_e \gamma^\mu P_L\nu_e- \bar \nu_\mu \gamma^\mu P_L\nu_\mu + a_x \bar \psi_1 \gamma^\mu \psi_1)  \label{eq:int-dm-mm}
\end{align}
where
\begin{align}
 a_x = \frac12 |(W_{N_L}+W_{N_R})_{11}| .
\end{align}
The model parameter space is thus effectively given by $m_X$ , $m_{\rm Z'}$ , $g'$  and  $a_x$ , with  $a_x$ taking values in the interval~$[0,1]$.

The evolution of the DM abundance is given through the Boltzmann equation
\begin{align}
\frac{dY}{dx}=-
\frac{x s[x] }{H}\langle\sigma v_{\rm rel}\rangle (Y^2-Y_{\rm EQ}^2), \label{eq:boltz}
\end{align}
where $s[x]$ is the entropy density and $H$ is the Hubble parameter, which are respectively given by
\begin{align}
s[x]=\frac{2\pi^2 g_\star}{45}\frac{M_X^3}{x^3},\quad
H= \sqrt{\frac{4\pi^3 g_\star}{45}} \frac{M_X^2}{M_{\rm PL}}.
\end{align}
Here $g_\star\approx$ 107 is the total number of effective relativistic degrees of freedom,
and the Planck mass \linebreak{$M_{\rm PL}\approx 1.22\times 10^{19}$~GeV}.
Finally, the DM relic density is given by
\begin{align}
\Omega h^2\approx 2.74\times 10^8  \left[ \frac{M_X}{\rm GeV}\right] Y_\infty,
\end{align}
where $Y_\infty$ is the final DM abundance~\cite{Srednicki:1988ce, Edsjo:1997bg, Guo:2009aj, Ibe:2008ye}.
Observed relic density at $2 \sigma$ is given by Ref.~\cite{Ade:2013zuv} as
\begin{align}
\Omega h^2=0.1199\pm0.0054.
\end{align}
In the numerical analysis, we adopt a rather relaxed value range, $0.11\lesssim \Omega h^2\lesssim 0.13$, and the LEP constraint expressed in Eq.~\eqref{eq:lep} is imposed.

The Breit-Wigner effect causes a higher DM annihilation rate than for a thermally produced DM with velocity independent annihilation cross section where average velocity $v$ is low, notably in the galactic DM halo near the position of the Solar system ($v \sim 10^{-3}$), and in the era of CMB formation ($v \sim 10^{-6}$).
To express this enhancement, we define the boost factor as the ratio of the speed averaged cross section of our model at a given value of $v$ under the condition of obtaining the correct relic density by solving Eq.~\eqref{eq:boltz}, and the speed averaged cross section required to obtain the correct relic density for a thermally produced DM with velocity independent annihilation cross section, given by
\begin{align}
\langle\sigma v_{\rm rel}\rangle_{\rm th}\approx 3\times 10^{-26} {\rm cm^{3} s^{-1}}= 2.573\ 10^{-9}\ {\rm GeV}^{-2}.
\end{align}
The boost factor for annihilation in the galactic halo near the Solar System, in the current epoch is then given by
\begin{align}
B = \frac{\langle\sigma v_{\rm rel}\rangle_{\rm c}}{\langle\sigma v_{\rm rel}\rangle_{\rm th}},
\end{align}
and the boost factor for annihilation in the CMB formation era by
\begin{align}
B_{\rm CMB} = \frac{\langle\sigma v_{\rm rel}\rangle_{\rm CMB}}{\langle\sigma v_{\rm rel}\rangle_{\rm th}},
\end{align}
where $\langle\sigma v_{\rm rel}\rangle_{\rm c}$ corresponds to {$x = v^{-2} \approx 10^6$}, while $\langle\sigma v_{\rm rel}\rangle_{\rm CMB}$ corresponds to $x\approx 10^{12}$~\cite{footnote2}.

\subsection{Numerical Analysis}
We have performed a numerical analysis to find the allowed region for obtaining the correct relic density of DM, where neutrino oscillation data is implicitly reproduced and the LFV constraint is satisfied.
We have analyzed parameter sets with fixed values of  {$m_X=[20,100,400,2000]$ GeV}, while $m_{Z'}$ and $g'$ are determined by randomly selected values of $\delta\equiv 1-\frac{m_{Z'}^2}{4m_X^2}$ and $\gamma\equiv \frac{\Gamma_{Z'}}{m_{Z'}}$, with the mixing matrix $M$ also being chosen randomly under the condition of $m_X$ being the smallest eigenvalue, from which we calculate the effective input parameter $a_x$. To cover this parameter space, O$(10^7)$ parameter sets are calculated per value of $m_X$ with a flat distribution in the range {[-8,-3]} of both  $\log_{10}(-\delta)$ and $\log_{10}(\gamma)$.
The properties of the parameter sets satisfying $0.11\lesssim \Omega h^2\lesssim 0.13$ and $4950\ {\rm GeV} \lesssim \frac{m_{Z'}}{g'}$ are further analyzed.

We also conduct a numerical analysis on LFVs for these parameter sets, finding that if the Yukawa couplings are below $\sim 10^{-2}$,
the experimental limits given in Eq.~\eqref{LFVbounds} are not exceeded for any of the parameter sets.
Given that the Yukawa couplings are independent from the parameters defining the DM properties, there is no constraint from LFV on the studied parameter space.

With micrOMEGAs, we calculate the speed averaged cross section for {$v = 10^{-3}$} and {$v = 10^{-6}$} to obtain $B$ and $B_{\rm CMB}$ for each parameter set respectively, taking  {$\langle\sigma v_{\rm rel}\rangle_{\rm th} = \frac{0.12}{\Omega h^2}\times  3\times 10^{-26} {\rm cm^{3} s^{-1}}$} to compare with the generic model yielding the same relic density.
Figure~\ref{g-d} shows boost factor $B$ for the parameter sets which satisfy $0.11\lesssim \Omega h^2\lesssim 0.13$ and $ 4950\ {\rm GeV} \lesssim \frac{m_{Z'}}{g'}$ in the $\log_{10}(-\delta)$ vs. $\log_{10}(\gamma)$ plane, indicating that these two parameters determine the value of $B$. The left-top plot is for $m_X=20$ GeV, the right-top plot for $m_X=100$ GeV, the left-bottom plot for $m_X=400$ GeV, and the right-bottom plot for $m_X=2000$ GeV.

\begin{figure*}
\begin{center}
\includegraphics[width=0.49\linewidth]{scatter-bflog-parametersMX20-atx.png}
\includegraphics[width=0.49\linewidth]{scatter-bflog-parametersMX100-atx.png}
\includegraphics[width=0.49\linewidth]{scatter-bflog-parametersMX400-atx.png}
\includegraphics[width=0.49\linewidth]{scatter-bflog-parametersMX2000-atx.png}

\caption{Scatter plots in terms of $-\delta$ and $\gamma$ of the parameter sets fulfilling \mbox{$0.11 < \Omega h^2 < 0.13$} and \mbox{$4950\ {\rm GeV} \lesssim \frac{m_{Z'}}{g'}$}, where the left-top figure is the one for $M_X=20$~GeV, the right-top one for $M_X=100$~GeV, the left-bottom one for $M_X=400$~GeV, and the right-bottom one for~$M_X=2000$ GeV. Boost factor $B$ is encoded by the color of the dots.
The gray line indicates the limit from CMB anisotropy based on $B_{\rm CMB}$ using Eq.~\eqref{CMBconstr}, the purple lines the limit on $B$ from analysis of CALET and AMS-02 data in \secref{DMsig}, and the green lines in the plot for $m_X = 400$~GeV are the boundaries of the $2 \sigma$ region given in Figure~\ref{limits} for explaining a structure in the CALET spectrum as a signature of this DM candidate.
The region excluded by the LEP bound $ 4950\ {\rm GeV} \lesssim \frac{m_{Z'}}{g'}$ is colored red, while in the white region, $\Omega h^2 > 0.13$ for all parameter sets.
}
  \label{g-d}
\end{center}\end{figure*}

Among the cosmological constraints to the model, CMB anisotropy provides the strictest bound, since in principle the annihilation rate increases with decreasing relative velocity~\cite{Hisano:2011dc}. The limit calculated from 2015 Planck CMB anisotropy measurement~\cite{Kawasaki:2015peu} excludes
\begin{align}
\langle\sigma v_{rel}\rangle_{th} > \frac{m_X}{GeV} \times 1.4 \times 10^{-27} {\rm cm}^3 {\rm s}^{-1}
\end{align}
 for annihilation to $e^{-}+e^{+}$, and
\begin{align}
\langle\sigma v_{rel}\rangle_{th} > \frac{m_X}{GeV} \times 3.6 \times 10^{-27}{\rm cm}^3 {\rm s}^{-1}
\end{align}
 for annihilation to $\mu^{-} + \mu^{+}$. While velocity dependence of the annihilation cross section was not considered for these results, it can be assumed that the annihilation cross section in the CMB formation era is decisive. With the limits in principle being inversely proportional to the energy injected into the thermal bath, the limit for the annihilation of $X$ can be calculated as
\begin{align}
B_{\rm CMB} > (\frac{1}{3}\times 1.4^{-1} + \frac{1}{3}\times 3.6^{-1})^{-1} \times \frac{10^{-27}}{3 \times 10^{-26 }} \times \frac{m_{\rm X}}{\rm GeV} . \label{CMBconstr}
\end{align}
For example, at $m_X = 400$~GeV, $B_{\rm CMB} > 40.32$ would be excluded, with the part of the parameter space excluded by this and corresponding limits for other values of $m_X$ indicated in Figure~\ref{g-d}.

\begin{figure*}
\begin{center}
\includegraphics[width=0.49\linewidth]{scatter-brlog-parametersMX20-atx.png}
\includegraphics[width=0.49\linewidth]{scatter-brlog-parametersMX100-atx.png}

\caption{Scatter plots showing the ratio of the boost factor in the galactic halo $B$ to the boost factor at the time of CMB formation $B_{\rm CMB}$  encoded by the color of the dots. The left figure is the one for $m_X=20$~GeV, the right one for $m_X=100$~GeV. We omit the figures for $m_X=400$~GeV and $m_X=2000$~GeV, since there is no allowed region where $B \not\approx B_{\rm CMB}$ for these values of $m_X$.
}
  \label{br}

\end{center}\end{figure*}

We find $B_{\rm CMB} \approx B$ for most of the studied parameter space, except for two regions at $\gamma  \lesssim 10^{-7}$ as shown in Figure~\ref{br}, matching the results shown in Ref.~\cite{Xiang:2017jou}. While there is a region in which $B$ is up to three orders of magnitude larger than $B_{\rm CMB}$, it is ruled out by the constraint on $B_{\rm CMB}$.

\section{Dark Matter Signatures in Cosmic-Ray Spectra}
\label{DMsig}
Electron and positron cosmic rays from annihilation of the DM candidate particle X are a potential signature of the proposed model.
In this section we evaluate measured cosmic-ray spectra by CALET and AMS-02 to derive limits on the annihilation cross section and to identify potential correlations of spectral structures with the DM signature.  The results of CALET and AMS-02 agree well for the $e^{-}+e^{+}$~spectrum, which is a prerequisite for the combined fitting of the CALET $e^{-}+e^{+}$~spectrum and the AMS-02 $e^{+}$-only spectrum without assuming an inherent systematic offset. Due to the systematic difference of the DAMPE $e^{-}+e^{+}$~spectrum results~\cite{Ambrosi:2017all} from both AMS-02 and CALET spectra, we chose not to consider them in our study.

\subsection{Electron and Positron Flux from Annihilation in the Galactic Halo}

To predict the shape of the spectral component from annihilation of X and $\mathrm{\bar X}$ , the positron spectra (identical to electron spectrum due to the symmetry of the process) per annihilation in the electron and muon channels have been calculated with PYTHIA 8.2~\cite{Sjostrand:2014zea}, which in turn were used as input for the propagation calculation with {DRAGON~\cite{Gaggero:2013rya}} to obtain the flux at Earth. For the local DM density, $\rho_0 = 0.3 $~GeV/cm$^3$ is assumed, and the speed averaged annihilation cross section normalized to the value predicted for a thermal relic DM,  $\langle\sigma v_{rel}\rangle_{th} = 3 \times 10^{-26}$~cm$^3$~s$^{-1}$. The choice of the DM halo shape model has no strong impact on the spectrum as the propagation range of electrons is limited and discussed models agree around the position of the solar system~\cite{Weber:2009pt}. A NFW parametrization~\cite{Navarro:1996gj} is used for the calculation. The flux for annihilation of X and $\mathrm{\bar X}$ is composed according to the branching ratio from Eq.~\eqref{eq:dr2} as the sum of one third of the normalized flux for electron channel and one third of the flux for muon channel, with the annihilation to neutrinos not contributing.

\pagebreak
For the propagation calculation, we consider two propagation models strongly distinct in diffusion zone height $L$ and diffusion coefficient normalization $D_0$, denoted Model~A and Model~B.

Model A comprises a gradual change in the slope of the diffusion coefficient with rigidity~\cite{Genolini:2017dfb} according to \begin{equation} D(R) = D_0 \left(\frac{R}{R_0}\right)^{\delta_l} \left(1 + \left(\frac{R}{R_b}\right)^{\frac{{\delta_l-\delta_h } }{s}} \right)^{-s} \ , \label{DReq}\end{equation} with $\delta_l = 0.62 $ , $\delta_h = 0.33 $, $ R_0 = 4 $~GV , $R_b = 350 $~GV , $D_0 = 1.1 \times 10^{28}\ \mathrm{cm^2/s}$, and a softness parameter $s = 0.15$. These propagation parameters are derived from calculation of the nuclei spectra with DRAGON. Setting the diffusion zone half-height $L = 3$~kpc and the width of the spiral arm thickness to 0.65~kpc, this model reproduces the AMS-02 B/C ratio~\cite{Aguilar:2016vqr} and proton spectrum~\cite{Aguilar:2015ooa} measurements if assuming a common injection index {$\gamma_{i}$ = -2.32} for all nuclei. This model predicts the hardening in the proton spectrum matching the index change as recently measured by CALET~\cite{Adriani:2019aft} as a pure propagation effect, without any break in the injection index.

Model B is designed as an alternative with high diffusion coefficient already at low energy, choosing $D_0 = 3.7 \times 10^{28}\ \mathrm{cm^2/s}$ which implies a much larger diffusion zone half-height of $L = 15$. With a constant diffusion coefficient index $\delta_h = 0.5$, the slope changes in B/C ratio and proton spectrum are explained as the effects of diffusive acceleration (Alfven speed $v_A = 12$~km/s) and two smooth breaks in the nuclei injection spectrum at 12~GV and 500~GV, changing the power law index from 2.0 to 2.36 and from 2.36 to 2.1 respectively. Here, the spiral arm width is taken as the default value of 0.3~kpc.

Figure~\ref{propmods} shows the comparison of the calculated nuclei spectra for both propagation models with experimental data.

\begin{figure*}

\centering
	\resizebox{0.95\linewidth}{!}{\includegraphics{ProBCSCIfluxes-D1d1-d0d62-I2d9-L3-R0d0-CSM-SW0d65-2002041547.png}}
\begin {center}
\caption{Proton flux and B/C ratio as reproduced by propagation Model A and Model B compared to data from AMS-02 and CALET, with charge independent solar modulation potential {$\Phi_0= \Phi = 500$~MV}.  $\Phi_1$ represents an additional potential for positive charge only at low energy following Ref.~\cite{Cholis:2015gna}. \label{propmods}}
\end {center}
\end{figure*}

\subsection{Astrophysical Background Flux Model}

Potential signatures of DM in the electron and positron spectra need to be distinguished from the background spectrum from astrophysical sources. The three main components comprising the background spectra are primary electrons accelerated by supernova remnants (SNR), secondary electrons and positrons from interaction of nuclei cosmic rays with the interstellar medium, and an extra source of electrons and positrons as an explanation of the positron excess, for which nearby pulsars are the prime candidate~\cite{Hooper:2008kg,Malyshev:2009tw,Kawanaka:2009dk,Feng:2015uta}. The pulsar scenario is supported by the discovery of {$\gamma$-ray} emission around nearby pulsars~\cite{Hooper:2017gtd,Fang:2018qco} and thus chosen over other discussed explanations of the positron excess such as secondary production in dense clouds around SNRs~\cite{Blasi:2009hv,Fujita:2009wk,Blasi:2009bd,Mertsch:2009ph,Blum:2013zsa,Cholis:2013lwa,Kohri:2015mga}. While a DM-only explanation of the positron excess is also not ruled out, it requires specific conditions such as decaying dark matter yielding softer spectra than the electron and muon channel annihilation of our DM candidate~\cite{Buch:2016jjp,Bhattacharyya:2017kfq,Farzan:2019qdm}.

The model used for describing the background spectra and fitted to electron and positron cosmic-ray data is the sum of the above mentioned components, with the electron spectrum written as
\begin{equation} \Phi_e^{-} = C_e E^{-(\gamma_e - \Delta\gamma_e)} \left(1+ \left( \frac{E}{E_b}\right)^{\frac{\Delta\gamma_e}{s}} \right)^s e^{-\left(\frac{E}{E_{cut_d}}\right)} +\frac{C_s}{C_{norm}} \Phi_{s(e^{-})} +  \Phi_{ex}  \ , \end{equation}
and the positron spectrum as
\begin{equation} \Phi_e^{+} = \frac{C_s}{C_{norm}} \Phi_{s(e^{+})} +  \Phi_{ex} \ . \end{equation}

Due to their large energy loss in propagation, the spectrum of primary electrons depends on the distribution of individual SNR in the galactic neighborhood of the solar system, which is yet mostly unknown. As an effective model of the local (after propagation) primary electron spectrum from all contributing SNR, it is parametrized by a power law with a soft spectral break (normalization $C_e$, index $\gamma_e$, break position $E_b$ and index change $\Delta\gamma_e$ are free fit parameters, softness $s = 0.05$ is fixed) at low energy, and a high-energy exponential cut-off at $E_{cut_d}$ representing radiative energy loss of high energy electrons. $E_{cut_d}$ is not well constrained by the measurement and therefore treated as a fixed nuisance parameter for which we consider values of 2~TeV, 4 TeV and 10~TeV.

The secondary positron ($\Phi_{s(e^{+})}$) and electron ($\Phi_{s(e^{-})}$) fluxes are taken from the output of numerical propagation calculation with DRAGON for the nuclei spectra from which the propagation conditions are derived. The propagation conditions are used consistently for calculation of fluxes from secondaries, pulsars and Dark Matter. With an initial scale factor $C_{norm}$ obtained from normalizing the proton flux to measurements of AMS-02~\cite{Aguilar:2015ooa}, a common rescaling factor ($C_s/C_{norm}$) is included in the fit as free parameter to account for remaining uncertainties in secondary particle production.

For the flux of the primary positron source causing the positron excess $\Phi_{ex}$, the least complex solution of a single young pulsar is assumed in the base model, for which the Monogem pulsar (PSR J0659+1414) is chosen.
The power-law with cut-off injection spectrum (defined by spectral index $\gamma_{ex}$ and cut-off energy $E_{cut_{ex}}$) of the pulsar is propagated using the analytic solution of the propagation equation for a point source as explained e.g. in Ref.~\cite{Feng:2015uta}, adapted to include the gradual change in the diffusion coefficient of propagation Model A, yielding the propagated flux from the pulsar as
\begin{equation} \Phi_{pulsar} = \frac{Q_0 \eta}{\pi^{3/2} r^3_{dif}} E^{-\gamma_{ex}} \left(1-\frac{E}{E_{max}} \right)^{(\gamma_{ex}-2)} e^{-\frac{E/E_{cut_{ex}}}{1-E/E_{max}}-\frac{r^2}{r^2_{dif}}} \ ,  \end{equation}
in which the characteristic diffusion distance $r_{dif}$ is expressed as
\begin{equation} r_{dif} = 2 \sqrt{\frac{D(E) t_{dif}}{1-\delta(E)} \frac{E_{max}}{E} \left[1-\left(1-\frac{E}{E_{max}} \right)^{(1-\delta(E))} \right]}\ , \end{equation}
with $E_{max}= (b_0\ t_{dif})^{-1}$, $b_0=1.4\ 10^{-16}$~GeV~s$^{-1}$, $D(E)$ given by Eq.~\eqref{DReq} and $\delta(E)$ approximated as the local index of $D(E)$ at $E$.

The distance to the Monogem pulsar $r = 0.28$~kpc is taken from the ATNF catalog~\cite{Manchester:2004bp}, as well as its age $T = 1.11 \times 10^2$~kyr and energy loss rate $\dot E = 3.81 \times 10^{34}$~erg~s$^{-1}$. The initial rotation energy of the pulsar $Q_0 = 1.48 \times 10^{48}$~erg is calculated as $Q_0 = \dot E\ T^2 / \tau$, where $\tau = 10$~kyr is the assumed spin-down timescale~\cite{Feng:2015uta}, so that the spectrum scales with the acceleration efficiency $\eta$, which is a free parameter in the fitting. A common assumption is that the accelerated particles are trapped for some time in the pulsar wind nebula (PWN) forming around the pulsar, and released with the dissolution of the PWN.  The release delay $T_r$ is thus subtracted from the age $T$ of the pulsar to determine the diffusion time $t_{dif}$, with $T_r$ scanned in steps of 1~kyr considering the range up to 100~kyr~\cite{Malyshev:2009tw}.

\pagebreak
\subsection{Fit of the Background Model to CALET and AMS-02 Data}
The model is fitted to the data of CALET~\cite{Adriani:2018ktz} based on total flux $\Phi_e^{-}+\Phi_e^{+}$ and data of \mbox{AMS-02}~\cite{Aguilar:2019owu} for $E>10$~GeV based on $\Phi_e^{+}$ by minimizing the sum of $\chi^2$ of both comparisons, with systematic uncertainties of both measurements taken into account. For the CALET measurement, the $1 \sigma$ deviation $\Delta_{(k,i)}$ as a function of each data point's energy ($E_i$) is listed in the supplemental material of Ref.~\cite{Adriani:2018ktz} for the systematic uncertainty associated with the following parts of the analysis:  Normalization, tracking, charge selection, electron identification, Monte Carlo model dependence.
A systematic shift of the data-points is performed as part of the fit function with weights $w_k$ as free parameters and the squared weight of each uncertainty is added to the total $\chi^2$ of the fit as given by
\begin{equation} \chi^2_{CALET} = \left( \sum\limits_{i} \frac{(\Phi_i) + \sum_k \Delta_{(k,i)} w_k - J_{i})}{\sigma_i^2} \right) + \sum\limits_k w^2_k \ , \end{equation}
where $i$ iterates over the data points and $k$ over the different systematic uncertainty types. Systematic errors associated with the trigger and the boosted decision tree proton rejection are added quadratically to the statistical error.
For the AMS-02 measurement, the error on mean energy $\sigma_E$ in each bin is translated into an error on flux $\sigma_{J(E)}$ using the power law index $\gamma_{e^+}$ also shown in Ref.~\cite{Aguilar:2019owu} via the relation $\sigma_{J(E)} = J(E) (\sigma_E/E) (\gamma_{e^+}-1)$.

The lower boundary of $E>10$~GeV for the data points used in the fitting is chosen due to charge and time dependent solar modulation effects expected below this energy~\cite{Cholis:2015gna}. Solar modulation effects above this energy are calculated using the force field approximation with a modulation potential of $\Phi = 500$~MV, common for both charge signs. To check the potential influence of the parameter on our results, $\Phi = 300$~MV and $\Phi = 700$~MV are used as alternative fixed values.

\begin{figure*}[!htb]
\centering
	\resizebox{\linewidth}{!}{\includegraphics{totfluxplot-puls-J0659+1414-XCALEAMS-NP-SEC-SX10P0d500E0d500-binavg-FPS6d2B350d0H0d33S0d15-DC1d1-LESB0d05-FNUI-SF-SRT.png}}
	\resizebox{\linewidth}{!}{\includegraphics{totfluxplot-puls-J0659+1414-XCALEAMS-NP-SEC-SX10P0d500E0d500-binavg-FPS5d0-DC3d7-LESB0d05-FNUI-SF-SRT.png}}
\begin {center}
\caption{The base model as fitted to CALET and AMS-02 data using propagation Model A (top) and Model B (bottom) in the default case of $E_{cut_{d}} = 10$~TeV and $\Phi = 500$~MV. See legends for explanation of markers and lines, the values for the parameters introduced in the text are given in the box to the right of the graphs. \label{basemod}}
\end {center}
\end{figure*}

The results of the base-model fit are shown in Figure~\ref{basemod} for the two propagation models. For Model A, the best fit is obtained with $T_r = 9 $~kyr, while for Model B, $T_r = 56$~kyr yields lowest $\chi^2$. The best fit for both propagation models uses $E_{cut_{d}} = 10$~TeV, which is thus taken as the default case. The reduced $\chi^2$ is in either case $\chi^2/ndof \approx 0.5$, indicating that the base model already more than adequately describes the data.  With the Geminga pulsar as source of the positron excess, the fit quality is significantly worse unless $T_r>100$~kyr, which is the reason why we chose the Monogem pulsar.

\FloatBarrier

\subsection{Limit on Annihilation cross section from CALET and AMS-02 Data}

The predicted flux from DM annihilation is added to the base model as an additional component of $\Phi_{ex}$ with varied boost factor $B$, and the change of $\chi^2$ studied.
To derive a limit on $B$, or equivalently the speed averaged annihilation cross section $\langle\sigma v_{rel}\rangle_{c}$, $B$ is increased in steps until $\chi^2$ exceeds the 95\%~CL threshold for the respective number of degrees of freedom~\cite{footnote3}.
To determine the precise value of $B$ for which the 95\%~CL threshold is crossed, the scan is repeated from the last allowed value with a factor 10 smaller step size, down to a step size of 0.01. To avoid reporting a too stringent limit due to the fitting function having no unique minimum, the "Migrad" and "Simplex" minimizers of Minuit~\cite{James:1994vla} are used in alternation with different starting points as explained in Ref.~\cite{Motz:2015cua}. Multiplying the normalization cross section by the scale factor at which the 95\%~CL threshold is crossed yields the limit on cross section $\langle\sigma v_{rel}\rangle_{c}$.
By performing this procedure with $m_X$ scanned in steps of 5~GeV up to 500~GeV, 25~GeV from 500 to 1~TeV, 50 GeV from 1~TeV to 2~TeV and 100~GeV above 2~TeV, limits depending on $m_X$ are calculated, which are shown in Figure~\ref{limits}. It is found that the limit varies only slightly under change of the nuisance parameters $\Phi$ and $E_{cut_{d}}$.
In principle, these limits are subject to the modeling of the astrophysical background flux being a good representation of the actual spectrum, disregarding whether it is an correct interpretation e.g. whether or not the Monogem pulsar is indeed the dominating source of the positron excess. However to judge the conservativeness of the limits, it should be considered that for the peaked DM signal to be hidden by structures of the background from multiple astrophysical sources, these structures would have to form a deficit in a rather narrow energy range which can be considered an implausible coincidence given the smoothness of the spectrum in general.
Due to these reasons, we consider the limits rather conservative, however to estimate the utmost possible influence of the background variability, also limits without any assumption on the background were calculated, using the method described above, but with only excess of the flux from DM annihilation over the flux measured by CALET and AMS-02 contributing to $\chi^2$. They are also shown in Figure~\ref{limits} for comparison.

\begin{figure*}[!hptb]
\centering
	\resizebox{\linewidth}{!}{\includegraphics{PAdrpyann-emu-XCALEAMS-NP-SX10P0d500E0d500-binavg-FPS6d2B350d0H0d33S0d15-DC1d1-LESB0d05-FNUI-J0659+1414-SF-RT9000-.png}}
	\resizebox{\linewidth}{!}{\includegraphics{PAdrpyann-emu-XCALEAMS-NP-SX10P0d500E0d500-binavg-FPS5d0-DC3d7-DR12d0-LESB0d05-FNUI-J0659+1414-SF-RT56000-.png}}

\begin {center}
\caption{Limit on the annihilation cross section as a function of $m_X$, compared to limits for $e^\pm$ and $\mu^\pm$ channels from {$\gamma$-ray} observation of dwarf galaxies with Fermi-LAT from the supplemental material of Ref.~\cite{Ackermann:2015zua} multiplied with three to account for the branching fraction. The purple area shows the variation of the limit among all cases with nuisance parameters $\Phi$ and $E_{cut_{d}}$ changed between 300~MV,500~MV,700~MV and 2~TeV,4~TeV,10~TeV respectively. The dashed gray line indicates the limit without any background modeling, with only excess over measured flux contributing to $\chi^2$. Also, the shown limit is the worst in the range from variation of $\Phi$. The dark green line shows the cross section as a function of $m_X$ of the best fit where $\Delta \chi^2 > $2.2977 ($1 \sigma$). The green and yellow areas show the areas with $1 \sigma$ fit improvement for the default case and all cases respectively. The dark yellow and dark green areas show the areas with $2 \sigma$ fit improvement. The top panel is for propagation Model A, the bottom panel for Model B. \label{limits} }
\end {center}
\end{figure*}

\subsection{Structures in the CALET Spectrum as Possible Dark Matter Signatures}

The addition of the predicted DM flux and increase of $B$ for limit calculation causes eventually an increase in $\chi^2$ of the fit. However, it is found that the addition of the DM flux with $B$ smaller than the limit value improves the fit compared to the base model with a pulsar extra source in two ranges of $m_X$, corresponding to step-like structures in the CALET spectrum.
Given the excellent energy resolution combined with detailed energy calibration~\cite{Asaoka:2017clb} over the wide dynamic range~\cite{Asaoka:2018ope} of CALET,
it is permissible to assume that the measured structures are features of the physical spectrum and not measurement artifacts, thus warranting an interpretation.
To quantify the significance of interpreting the spectral structures as a signature of the proposed DM candidate, the optimal value of $B$ and associated maximal $\chi^2$ reduction are determined depending on $m_X$. Using an approach similar to the limit calculation, $B$ is initially scanned in 20 steps between zero and the 95\%~CL limit value, and then the interval around the value with best $\chi^2$ scanned in nested intervals.

\begin{figure*}[t]

\centering
	\resizebox{0.495\linewidth}{!}{\includegraphics{Kdrpyann-emu-XCALEAMS-NP-SX10P0d500E0d500-binavg-FPS6d2B350d0H0d33S0d15-DC1d1-LESB0d05-FNUI-J0659+1414-SF-RT9000.png}}
        \resizebox{0.495\linewidth}{!}{\includegraphics{Kdrpyann-emu-XCALEAMS-NP-SX10P0d500E0d500-binavg-FPS5d0-DC3d7-DR12d0-LESB0d05-FNUI-J0659+1414-SF-RT56000.png}}

\begin {center}
\caption{Fit improvement ($\chi^2$ reduction) by addition of flux from Dark Matter to the base model for propagation Model A (left) and Model B (right) as a function of $m_X$ (red), together with the values of $B$ giving the best fit (green). The shaded regions indicate the change from variation of the nuisance parameters.  \label{fitimp}}

\end {center}
\end{figure*}

The best-fit $B$ and $\chi^2$ improvement as a function of DM mass are shown in Figure~\ref{fitimp}. The largest $\chi^2$ improvement compared to the single pulsar case is $\Delta\chi^2$ = 6.84 at $m_X = 390$~GeV with
$B = 40.1$
for propagation Model A. The significance exceeds the $2\sigma$ significance level for the two additional free parameters ($m_X$ and $B$ or $\langle\sigma v_{rel}\rangle_{c}$) independent of the chosen values for the nuisance parameters. For propagation Model B, the best fit for the default case of $E_{cut_{d}} = 10$~TeV improves by $\Delta\chi^2$ = 3.80 also at $m_X = 390$~GeV, with larger improvement if choosing smaller $E_{cut_{d}}$.
The necessary Breit-Wigner enhancement of $B \approx 40$ is predicted within the theoretical framework of the DM candidate as shown in Figure~\ref{g-d}, with part of the $2\sigma$-region being below the constraint from CMB anisotropy. It is also not ruled out by overproduction of {$\gamma$-rays} as shown by the comparison with limits from dwarf-galaxy observation by Fermi-LAT in Figure~\ref{limits}.

With both propagation models, the fit also improves with addition of the DM signal above 1~TeV where another a step-like structure exists in the CALET spectrum. For the default case, the fit improvement is maximal at $m_X = 1350$~GeV for both Model A and Model B, but it is much less significant than the improvement for $m_X = 390$~GeV. Apart from the larger errors, the significance is low, since even for the direct annihilation to $e^{-}+e^{+}$, the DM signal spectrum is not localized (hard) enough to match the structure well, as the best-fit graphs in Figure~\ref{bestfitsTeV} demonstrate.
The structure at $m_X = 390$~GeV is thus a better candidate for being a DM signature, despite the 1~TeV structure being visually more prominent.

\begin{figure*}

\centering
	\resizebox{\linewidth}{!}{\includegraphics{Ktotfluxplot-DM-drpyann-emu-XCALEAMS-NP-SX10P0d500E0d500-binavg-FPS6d2B350d0H0d33S0d15-DC1d1-LESB0d05-FNUI-J0659+1414-SF-RT9000-390.png}}
	\resizebox{\linewidth}{!}{\includegraphics{Ktotfluxplot-DM-drpyann-emu-XCALEAMS-NP-SX10P0d500E0d500-binavg-FPS5d0-DC3d7-DR12d0-LESB0d05-FNUI-J0659+1414-SF-RT56000-390.png}}
\begin {center}
\caption{The best fits for the default case with $M_{\rm DM} = 390$~GeV for propagation Model A (top) and Model B (bottom). See legends for explanation of markers and lines, the values for the parameters introduced in the text are given in the box to the right of the graphs. \label{bestfits}}
\end {center}
\end{figure*}

\begin{figure*}

\centering
	\resizebox{\linewidth}{!}{\includegraphics{Ktotfluxplot-DM-drpyann-emu-XCALEAMS-NP-SX10P0d500E0d500-binavg-FPS6d2B350d0H0d33S0d15-DC1d1-LESB0d05-FNUI-J0659+1414-SF-RT9000-1350.png}}
	\resizebox{\linewidth}{!}{\includegraphics{Ktotfluxplot-DM-drpyann-emu-XCALEAMS-NP-SX10P0d500E0d500-binavg-FPS5d0-DC3d7-DR12d0-LESB0d05-FNUI-J0659+1414-SF-RT56000-1350.png}}
\begin {center}
\caption{The best fits for the default case in the TeV region with $M_{\rm DM}= 1350$~GeV for propagation Model A (top) and Model B (bottom). See legends for explanation of markers and lines, the values for the parameters introduced in the text are given in the box to the right of the graphs. \label{bestfitsTeV}}

\end {center}
\end{figure*}

\clearpage
\section{Summary and Conclusions}
\label{sumconcl}

We have shown the viability of a GeV-TeV range WIMP-like DM candidate featuring flavor-dependent interaction only with electron and muon, allowed by current constraints from direct and indirect detection. The DM candidate is predicted in the framework of a scotogenic model at two-loop level, where we have accommodated two families of Dirac neutral fermions and Majorana fermions under gauge $U(1)_{e-\mu}\times Z_3\times Z_2$. The Dirac fermion with lightest mass is our DM candidate and it runs inside the neutrino loop, which is a typical feature of the scotogenic scenario.
The two families are the minimal extension to understand the neutrino oscillation data, its mass eigenstate and the gauge anomaly cancellations at the same time.
Due to introducing two families, we have predicted that the two mass eigenstates of active neutrinos are uniquely given by the two observed mass difference squares, $\Delta m^2_{sol}$ and $\Delta m^2_{atm}$ depending on the hierarchy order, with the lightest mass eigenvalue being always zero.
Next, we have shown the allowed region yielding the correct relic density of DM in terms of $m_{X}$ and $m_{Z'}$, while imposing the constraint from LEP. Furthermore we have shown that the annihilation cross section may be increased by a boost factor $B$ from Breit-Wigner enhancement, depending on the two factors $\delta$ and $\gamma$, finding that $(|\delta|,\gamma)\lesssim {\cal O}(10^{-4}-10^{-3})$
can give ${\cal O}(10-100)\ B$.

After calculating the expected signature of the DM candidate in electron and positron cosmic rays for two largely distinct propagation models, we performed a combined search in the measured $e^{-}+e^{+}$~CALET and $e^+$ AMS-02 spectra on top of an astrophysical background model assuming a single young pulsar as the source of the positron excess.
As outcome we presented limits on the annihilation cross section close to those from {$\gamma$-ray} observation with Fermi-LAT, as well as a possible association of structures in the CALET spectrum with a DM signature.
The significance of the fit improvement by adding the DM signature to the base model exceeds the $2\sigma$-level depending on the propagation model, with the best fit for $M_{\rm DM}$ at 390~GeV with a value of $B$ which is well within the range predicted by the Breit-Wigner enhancement.  These results demonstrate the significance of the step-like structure itself, and while other interpretations are possible, for example by overlapping spectra from individual astrophysical sources~\cite{Motz:ICRC2019}, it is shown that the annihilation of the DM candidate from the model presented herein also provides a suitable explanation.

\pagebreak
\section*{Acknowledgments}
The research performed by H.M. leading to the results shown in this paper was supported by a Waseda Grant for Special Research Projects (Project number: 2020C-539).
The research performed by H.O. was supported by an appointment to the JRG Program at the APCTP through the Science and Technology Promotion Fund and Lottery Fund of the Korean Government. It was also supported by the Korean Local Governments - Gyeongsangbuk-do Province and Pohang City (H.O.). H. O. is sincerely grateful for being a KIAS member, and to Log Cabin at POSTECH for providing a nice space to
come up with ideas for this project. K.K. was supported by JSPS KAKENHI Grants No. JP17H01131, MEXT Grant-in-Aid for Scientific Research on Innovative Areas JP15H05889, JP18H04594, JP19H05114. This work was also supported in part by KAKENHI 19H04617 (Y.A.)

redpajama_set_name: RedPajamaArXiv


In 22 last "dives" executed by Cassini, the mission took a risk and went where no spacecraft had dared venture: the region between Saturn and its distinctive rings. These bold, final months — dubbed the Grand Finale — provided researchers with a trove of discoveries, some of which are covered in the Oct. 5 issue of the journal Science and in companion articles posted online by Geophysical Review Letters (GRL).
For 40 years, calculations based on remote observations indicated that Saturn's magnetic field carried ions and charged dust grains from the rings to the planet's mid latitudes. But the Grand Finale produced the first direct measurements of the narrow region below the rings and led to a startling discovery: fast-moving molecules and particles smaller than two nanometers streaming along the plane of the rings into Saturn's atmosphere by another process known as atmospheric drag.
To add more intrigue to the finding, Mitchell determined the nanometer-sized dust was coming from Saturn's innermost ring — the D-ring — something that experts initially thought highly unlikely.
Cassini's mass spectrometer, INMS, measured the composition and structure of Saturn's atmosphere, but also complemented MIMI's observations of the particle influx. In a GRL paper, Mark Perry, a member of the INMS science team, reports that most of the mass pulled from the rings is composed of molecules and particles even smaller than those measured by MIMI. Unlike the rings, which are 95 percent water, hydrocarbons comprise most of the material falling from the rings into Saturn. "We now can explain the startling amount of material falling into Saturn, but the hydrocarbons are still mostly a mystery to solve," Perry said.
At the current rate, the inner rings would last only tens of thousands of years, much less than the age of the rings, which are at least 30 million years old. INMS measurements show a variation in the amount of material dragged from the rings. There is a tentative link between the amount of material and clumps that appeared in 2015 on the inner edge of the D-ring, suggesting that the current, possibly unsustainable flux may be temporary.
These newly discovered particles and processes alter the evolutionary landscape of the rings and provide an exciting, rich field for research aimed at understanding the origin and history of the rings. Where did the material originally come from? Was it from a one-time event? How long ago? Has it always occurred, or is this something unique to our time period? The discovery and subsequent broader questions can tell us more about the solar system, said Mitchell. "If it happens on Saturn, it could potentially also occur at other planets and in other stellar systems," he said.
Peter Kollmann, a member of the MIMI science team, shared first-authorship of the Science paper reporting the discovery of an inner radiation belt passing through Saturn's D-ring. The belt contains highly energetic protons — several giga-electron volts worth — much higher than what had previously been observed in the well-known outer radiation belt that extends around the planet. "It wasn't clear that we would find a radiation belt there or if the radiation would be absorbed by the D-ring," Kollmann said, adding that the fact the inner radiation belt is able to exist with the inner ring under unique conditions makes it a fascinating find.
The Cassini-Huygens mission is a cooperative project of NASA, the European Space Agency and the Italian Space Agency. JPL, a division of the California Institute of Technology in Pasadena, manages the Cassini-Huygens mission for NASA's Science Mission Directorate, Washington. The Cassini orbiter was designed, developed and assembled at JPL. Learn more about the Cassini mission and its Grand Finale at https://saturn.jpl.nasa.gov/.