# Status of Dark Photons

James M. Cline  CERN, Theoretical Physics Department, Geneva, Switzerland, and
Department of Physics,
McGill University, 3600 rue University, Montréal, Québec, Canada H3A2T8

###### Abstract

I review recent developments in the field of dark photons—here taken to be
U(1) gauge bosons with mass less than the $Z$—including both kinetically mixed
vectors and those that couple to anomaly-free U(1)’s. Distinctions between
Higgs and Stueckelberg masses are highlighted, with discussion of swampland
constraints, UV completions, and new experimental search strategies.

1\. Introduction. As we have often been reminded, the possible range of viable
dark matter (DM) masses spans many orders of magnitude, and this is also true
for DM in the form of U(1) vector gauge bosons. The heavier version is usually
called a $Z^{\prime}$ in the literature, whereas lighter ones are called dark
photons. I take the the $Z^{0}$ mass to be the upper boundary in this review
of dark photons, while below the lower limit $\sim 10^{-22}$ eV, such
particles would have too long a de Broglie wavelength to function as dark
matter in galaxies.[1] Below this limit, dark photons could still play an
interesting role in a larger dark sector, such as binding dark atoms and
giving rise to millicharged dark matter (see Ref. [2] for a review). Here I
will not discuss massless dark photons except as an approximation to massive
ones in the case where the masses are negligible relative to relevant energy
scales.

Hence, the dark photon could play the role of dark matter, dark force
mediator, or possibly both. There are two ways in which it could interact with
the standard model: through kinetic mixing, which is generically expected to
be present, or by coupling to an anomaly-free U(1) such $B-L$ or a difference
between two lepton flavors.

We first recall how kinetic mixing works.[3] The Lagrangian for the SM U(1)
gauge boson (either hypercharge, above the electroweak symmetry breaking
scale, or electromagnetism below it) plus the dark one is

${\cal
L}=-{\textstyle\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}-{\textstyle\frac{1}{4}}F^{\prime}_{\mu\nu}F^{\prime\mu\nu}-{\textstyle\frac{1}{2}}\epsilon
F_{\mu\nu}F^{\prime\mu\nu}-{\textstyle\frac{1}{2}}m_{A^{\prime}}^{2}A^{\prime}_{\mu}A^{\prime\mu}\,.$
(1)

If $m_{A^{\prime}}$ is nonzero, the kinetic term is diagonalized (up to
$O(\epsilon^{2})$ corrections) by $A_{\mu}\to A_{\mu}-\epsilon
A^{\prime}_{\mu}$. Then the dark photon will couple to particles of charge
$qe$ in the SM with strength $\epsilon qe$.111 If $m_{A^{\prime}}=0$, one can
instead diagonalize the kinetic term using $A^{\prime}_{\mu}\to
A^{\prime}_{\mu}+\epsilon A_{\mu}$. In this case, dark sector particles
coupling to $A^{\prime}$ become millicharged. In the limit of small
$m_{A^{\prime}}$, where $A^{\prime}$ can be approximated as massless, for
example in a stellar environment, the dark Higgs looks millicharged, which
leads to strong constraints. It is interesting to notice that citations of
Holdom’s seminal paper started to rise steeply in 2008, after the appearance
of Refs. [4, 5], that highlighted the importance of kinetic mixing in extended
dark sectors.

Figure 1: Left: recent limits on kinetic mixing versus mass, from Ref. [7],
with updates from C. O’Hare. Right: how the constraints are grouped by
categories. Figure courtesy of C. O’Hare.

2\. Constraints on kinetic mixing. Experimental constraints on $\epsilon$
versus $m_{A^{\prime}}$ have evolved dramatically since 2008, when there were
just a handful of limits [16]. Fig. 1 shows the current bounds individually
(left) and as classes (right). The red region overlaps with laboratory limits:
$1/r^{2}$ force law tests, light-shining-through-walls, helioscopes, axion
haloscopes, dark matter direct detectors, and a reactor neutrino experiment.
The strongest current limit $\epsilon\lesssim 10^{-16}$ at
$m_{A^{\prime}}\sim$ keV comes from $A^{\prime}$ absorption by atoms in the
XENON1T experiment.[8]

The red region also includes constraints from SM photons resonantly converting
to $A^{\prime}$, due to the photon plasma mass matching $m_{A^{\prime}}$ in
the cosmic microwave background, which would distort the cosmic microwave
background, CMB (COBE/FIRAS constraint). The green region arises from
$\gamma\to A^{\prime}$ in stars, which would cause the stars to cool faster
than observed. The foregoing constraints exist independently of whether
$A^{\prime}$ is the DM. If it is the DM, then the blue regions are constrained
by $A^{\prime}\to\gamma$ causing reionization of helium in interstellar gas
clouds. [7]

Axion haloscopes are an example of experiments designed for different physics
whose limits can be reinterpreted in terms of dark photon detection.[9, 10]
When the cavity frequency matches $m_{A^{\prime}}$, there are resonant
photon-$A^{\prime}$ oscillations, without the need for an external magnetic
field. Recently it was shown that radio telescopes can be sensitive to
conversion of $A^{\prime}\to\gamma$ in the dish leading to a monochromatic
signal. Assuming $A^{\prime}$ is the DM, LOFAR data rule out $\epsilon\gtrsim
10^{-13}$ for $m_{A^{\prime}}\sim 10^{-7}\,$eV,[11] and FAST excludes
$\epsilon\gtrsim 10^{-12}$ for $m_{A^{\prime}}\sim 10^{-5}\,$eV.[12] These
limits are projected to improve, especially with future SKA data. A
conceptually similar constraint is shown to arise in the optical from the
James Webb Space Telescope,[13] leading to $\epsilon\lesssim 10^{-11}$ for
$m_{A^{\prime}}\sim(0.05-2)$ eV.

So far we discussed light dark photons, $m_{A^{\prime}}<\,$MeV. Different
kinds of constraints apply for heavier $A^{\prime}$, notably from an array of
beam-dump and collider experiments. Fig. 2 (top row) illustrates these limits
in the case where $A^{\prime}$ couples to DM $\chi$ of mass
$m_{\chi}=m_{A^{\prime}}/4$ and coupling $g_{\chi}=0.1$ or 1. The laboratory
constraints depend on $g_{\chi}$ because it increases the invisible width of
$A^{\prime}$. However they generally resemble those arising from purely
visible decays of $A^{\prime}$. (The cyan regions are disfavored by assuming
freeze-in of DM through its interactions with $A^{\prime}$.)

Model-dependent bounds have been obtained at LHC, illustrated in Fig. 2
(bottom left). If the SM Higgs has a significant branching ratio into dark
fermions, which decay to $A^{\prime}$ plus DM, the subsequent decays of
$A^{\prime}\to\mu^{+}\mu^{-}$ yield strong limits, depending on whether they
are prompt or displaced.[15] Reach of existing and proposed experiments to
further probe the parameter space is shown in the bottom right panel. These
include HPS, FASER, SHiP, SeaQuest, HE-LHC, and FCC.

Figure 2: Top row: laboratory limits on kinetic mixing versus mass, with
$A^{\prime}$ decaying to lighter DM with coupling $g_{X}=0.1$ (left) or 1
(right), from Ref. [14]. Cyan region assumes DM freeze-in via kinetic mixing.
Bottom row: (left) ATLAS constraints from Higgs decay to dark fermions, which
then decay to $A^{\prime}$ plus DM;[15] (right) anticipated sensitivity of
future experiments for the heavy-$A^{\prime}$ parameter space.[14]

3\. Higgs or Stueckelberg mass? In presenting the previous bounds, it was
implicitly assumed that the $A^{\prime}$ gets its mass by the Stueckelberg
“mechanism,” which essentially means putting in a bare mass term as in Eq. (1)
with no other degrees of freedom. One can restore gauge invariance by
introducing a fictitious field $\theta$ and rewriting the mass term as

${\textstyle\frac{1}{2}}m_{A^{\prime}}^{2}(A^{\prime\mu}-\partial^{\mu}\theta)^{2}\,,$
(2)

but this by itself has no additional physical significance. On the other hand,
if $A^{\prime}$ gets its mass from a dark scalar $h^{\prime}$ by the Higgs
mechanism, one might expect $m_{h^{\prime}}\sim m_{A^{\prime}}$. If this is
the case, then $h^{\prime}$ behaves like a light millicharged particle, if
$m_{A^{\prime}}$ is small compared to the energy scale of interest. There is a
vertex $g^{\prime}m_{A^{\prime}}h^{\prime}A^{\prime 2}$ which leads to
$2\epsilon g^{\prime}m_{A^{\prime}}A^{\prime\mu}A_{\mu}$ in conjunction with
kinetic mixing. This yields much stronger constraints on $\epsilon$, ruling
out large regions that were allowed for the Stueckelberg case, as shown in
Fig. 3. For example, stellar cooling is accelerated by Brehmsstrahlung of
$h^{\prime}$ from photons.[16] Further limits arise from production of
$h^{\prime}$ in DM direct detection experiments through ionization of atoms in
the detector by $A^{\prime}$ produced in the sun.[17] These limits however are
model-dependent, because of the assumption $m_{h^{\prime}}\sim m_{A^{\prime}}$
and $g^{\prime}\sim e$.

Figure 3: Left: enlarged exclusion regions for a light dark Higgs from stellar
cooling, SN1987a, and other effects.[16] Right: similar constraints, also
including $h^{\prime}$ production in DM detection experiments.[17]

Typically, phenomenological studies assume that there is no analog of the
Higgs field corresponding to the Stueckelberg mechanism, but in consistent UV
completions, namely string theory, the field $\theta$ in (2) can be thought of
as an axion-like degree of freedom, that has an accompanying radial excitation
$\phi$. Unlike the Higgs mechanism, the $A^{\prime}$ mass arises without
$\phi$ getting a VEV, and the limit $\phi\to 0$ in field space is singular
because of a kinetic term of the form $(\partial\ln\phi)^{2}$.[18] In contrast
to Higgs masses, the limit $m_{A^{\prime}}\to 0$ is thus inconsistent in the
Stueckelberg case.

Light Stueckelberg masses can naturally arise in large volume string
compactifications, but there is a lower limit on the mass, which is related to
the dimensionless volume ${\cal V}$ of the compactification and thereby the
string scale $m_{s}$ and the Planck mass, through
$m_{P}^{2}=(4\pi/g_{s}^{2}){\cal V}m_{s}^{2}$. The $A^{\prime}$ mass is of
order[19] $m_{A^{\prime}}\sim g_{s}^{3/2}(m_{P}/{\cal V})\gtrsim
g_{s}^{3/2}{\rm\,eV}\gtrsim 1{\,\rm eV}$, using $m_{s}\gtrsim 1\,$TeV (from
collider limits), hence ${\cal V}\lesssim 10^{27}$.

4\. Origin of kinetic mixing. The kinetic mixing operator
$F_{\mu\nu}F^{\prime\mu\nu}$ has dimension 4, so its coefficient $\epsilon$
can be put in “by hand.” But usually one imagines that it gets generated at
some high scale by integrating out a heavy particle that carries both U(1)′
and SM U(1) charges. In that case, its natural value is of order
$eg^{\prime}/(16\pi^{2})$. To reconcile this with the XENON1T limit would
require $g^{\prime}<10^{-11}$, precluding gauge unification. Otherwise, one
could forbid matter charged under both gauge groups, but this violates a
strong version of the weak gravity conjecture[21] (WGC, see below), which is
supported by known string compactifications.

If one ignores WGC and assumes $\epsilon=0$ the Planck scale, then pure
gravity plus Higgs mediation can generate $\epsilon$ through a 6-loop diagram,
involving three graviton exchanges and one Higgs from each (SM and dark)
sector.[20] It is also possible to evade the WCG constraints by having
exchanges of several heavy states in the loops whose contributions cancel each
other. This can happen naturally in string theory compactifications,[22, 23]
where the interaction that generates the kinetic mixing is a closed string
exchange between two $D$-branes, which host the respective $U(1)$ matter
fields (Fig. 4). Small kinetic mixing can arise from large-volume
compactifications with ${\cal V}=(2\pi R)^{6}$ (in string length units
$\ell_{s}=2\pi\sqrt{\alpha^{\prime}}$), which are associated with a light
volume modulus of mass[24] $m_{\cal V}\sim m_{P}{\cal V}^{-3/2}$. The kinetic
mixing is related to $m{\cal V}$ by[23, 19] $\epsilon\gtrsim
10^{-16}\left(m_{\cal V}/2m_{h}\right)^{8/9}$, where $m_{h}$ is the SM Higgs
mass. To avoid overclosure of the Universe by such moduli, they must be heavy
enough to decay into two Higgs bosons, hence this gives a lower bound on
$\epsilon$ that is close to the XENON1T constraint.

Figure 4: String theoretic origin of kinetic mixing between two U(1)’s
localized on different $D$ branes (right) versus field theory diagram (left),
from Ref. [23].

5\. DM Relic density from dark photons. In constrast to axions or other light
spin-0 DM candidates, the misalignment mechanism (where the field is stuck at
a nontrivial value while the Hubble rate $H$ exceeds its mass, and starts
oscillating in its potential subsequently) does not easily account for a
significant dark photon relic density. This is because vector fields in a
nontrivial gravitational background couple to the Ricci scalar in such a way
as to be diluted by the expansion of the Universe. One needs to introduce a
nonminimal coupling $RA^{\prime}_{\mu}A^{\prime\mu}$ to counteract this
effect,[9, 25] but doing so leads to the longitudinal polarization becoming
ghost-like for some range of momenta.[26] This can be overcome by modifying
the kinetic term by coupling it to a scalar field (the inflaton) via
$f(\phi)F^{\prime}_{\mu\nu}F^{\prime\mu\nu}$,[27] but this involves
significantly more model building than for a spin-0 particle, which only needs
a quadratic potential for misalignment to work.

Alternatively, a generic mechanism is through inflationary fluctuations of
$A^{\prime}$, which like for any approximately massless field are of order
$H_{I}$, the Hubble rate during inflation. Ref. [28] showed that the desired
relic density is achieved if $m_{A^{\prime}}\cong 6\times
10^{-6}{\rm\,eV}\left(10^{14}\,\rm GeV/H_{I}\right)^{4}$. Another relatively
simple mechanism is to couple dark photons to an axion via
$-(\beta/f_{a})aF^{\prime\mu\nu}\widetilde{F}^{\prime\mu\nu}$.[29, 30] The
axion can be produced by misalignment; when it starts to oscillate, a
tachyonic instability quickly transfers energy from $a$ to $A^{\prime}_{\mu}$
over a wide range of axion masses and $A^{\prime}$ masses, spanning
$m_{A^{\prime}}\sim(10^{-7}-10^{7})$ eV. A similar mechanism works for dark
Higgs oscillations producing $A^{\prime}$ pairs by parametric resonance.[31]

Dark photons can also serve as the mediator that gives rise to thermal relic
dark matter through the classic processes $\chi\bar{\chi}\to
A^{\prime}A^{\prime}$ or $\chi\bar{\chi}\to A^{\prime*}\to f\bar{f}$. However
there are many other possible production pathways that were explored in Ref.
[32]. For example, kinetic mixing can be enhanced through resonant $A\to
A^{\prime}$ conversion at finite temperature, when $m_{A^{\prime}}$ coincides
with the photon plasmon mass.

6\. Swampland constraints. The weak gravity conjecture has strong evidence,
and its variants are well motivated by string theory/quantum gravity
arguments. The magnetic WCG imples a UV cutoff
$\Lambda>\sqrt{m_{P}m_{A^{\prime}}/g^{\prime}}$ for Stueckelberg masses.[18]
Since $\Lambda>H_{I}$ during inflation for a consistent description, the
inflationary production mechanism for $A^{\prime}$ becomes limited to masses
$m_{A^{\prime}}\gtrsim 0.3\,$eV, a very significant restriction. The WCG also
demands that $\Lambda<g^{\prime}m_{P}$, which combined with the generic loop
estimate for $\epsilon$ implies that $\Lambda\lesssim(16\pi^{2}/e)\epsilon
m_{P}\sim 100\,$TeV (using the XENON constraint on $\epsilon$),[33] suggesting
new states would be accessible at the Future Circular Collider. Similarly,
combining the magnetic WGC with the LHC bound $\Lambda\gtrsim 10\,$TeV and the
loop estimate for $\epsilon$ disfavors a large region of parameter space,[34]
including interesting values $m_{A^{\prime}}\sim 10^{-11}$, $\epsilon\sim
10^{-7}$ where axion decays $a\to A^{\prime}A^{\prime}$ followed by
$A^{\prime}\to A$ resonant oscillations can boost the spectrum of low-
frequency CMB photons, suppressing the 21 cm signal as seen by the EDGES
collaboration.[35, 36]

7\. Nonabelian kinetic mixing and Composite dark photons. It was realized
early on[4] that the photon could kinetically mix with a nonabelian gauge
boson through the diagram

where $\Phi$ gets a VEV and breaks the gauge symmetry. Then kinetic mixing
comes from a dimension-5 operator, giving another possibility for suppressing
$\epsilon$, without needing small $g^{\prime}$. It was pointed out[37] that
this can also work in a confining phase, where $\langle\Phi\rangle=0$; for
example $\Phi$ binds with the dark gluon to form a massive composite dark
photon $\tilde{A}$, with
$\epsilon\sim\sqrt{\alpha\alpha^{\prime}}\,y_{X}\Lambda/(4\pi m_{X})$, where
$\Lambda$ is the dark confinement scale. An interesting feature is that the
kinetic mixing $\epsilon$ can be correlated with $m_{\tilde{A}}\sim\Lambda$ by
their dependence on $\Lambda$ (note that $\alpha^{\prime}\sim 1/\ln\Lambda$ is
not independent). Dark baryons provide a natural DM candidate in such models,
and can have direct detection signals by mediation of $\tilde{A}$.

8\. Gauging SM global symmetries. In addition to kinetic mixing, $A^{\prime}$
could couple directly to SM particles by gauging one of its global symmetries.
The simplest possibilities involve anomaly-free combinations, $B-L$,
$L_{e}-L_{\mu}$, $L_{e}-L_{\tau}$ and $L_{\mu}-L_{\tau}$ since then no further
matter content is required (except possibly right-handed neutrinos). If
$\epsilon=0$ at a high scale, then it is calculable at lower scales.[38] For
example $\epsilon\sim eg^{\prime}\ln(m_{\tau}/m_{\mu})/(4\pi^{2})\sim
0.036\,g^{\prime}$ for gauged $L_{\mu}-L_{\tau}$ at vanishing momentum
transfer $q^{2}=0$; hence it makes a subdominant contribution to observable
processes.

Constraints on $g^{\prime}$ versus $m_{A^{\prime}}$ for the gauged models are
shown in Fig. 5. They are most severe when $A^{\prime}$ couples to electrons,
making $L_{\mu}-L_{\tau}$ a special case. In fact, it is the only one that can
explain the muon $g-2$ anomaly (through one-loop dressing of the muon vertex
by $A^{\prime}$ exchange), illustrated by the narrow pink band. For the same
reason, kinetically mixed models are also ruled out for $(g-2)_{\mu}$ (see
Ref. [39] for a loophole).

Figure 5: Constraints on gauge coupling versus $m_{A^{\prime}}$ for the four
anomaly-free gauge groups, from Ref. [38].

From a model-building standpoint, it might seem arbitrary to gauge only
$L_{\mu}-L_{\tau}$ as opposed to other flavor combinations, but Ref. [40]
showed that such a vector boson could be just the lightest of eight such,
coming from the gauging of SU(3) lepton flavor, broken at the $(5-10)$ TeV
scale, except for one scalar in the 6 representation with VEV $(20-200)$ GeV
to explain the lightest vector mass. The model predicts heavy neutral leptons
at the GeV scale, constrained by BBN, and that the lightest neutrino has mass
$m_{\nu_{1}}\gtrsim 10^{-4}$ eV. It can also address the Cabbibo anomaly.

The $L_{\mu}-L_{\tau}$ dark photon can have interesting implication for
neutrino physics if it is the dark matter. One can treat it as an oscillating
condensate which gives a time-dependent effective mass to $\nu_{\mu}$ and
$\nu_{\tau}$, with potential impact on neutrino oscillations. This can re-open
parameter space for sterile neutrino dark matter, by making
$\nu_{s}$-$\nu_{\mu}$ oscillations resonant and thereby making $\nu_{s}$
production efficient at smaller mixing angles than ordinarily required;[41]
this is illustrated in Fig. 6 (left). Similarly, this effect can impact
oscillations between active neutrino flavors,[42, 43, 44] giving rise to
leading constraints on the coupling, as shown in Fig. 6 (right). It was
recently shown that $B-L$ dark photons can be probed through dark matter-
neutrino scattering if the DM carries $B-L$, using IceCube observations of
neutrinos from the active galaxy NGC 1068.[45] Dark matter self-interactions
mediated by dark photons might also play a role in accelerating the mergers of
supermassive black holes,[46] to overcome the long-standing “final parsec”
problem.

Figure 6: Left: re-opened region of active-sterile neutrino mixing angle
versus sterile $\nu$ mass, enabled by $L_{\mu}-L_{\tau}$ dark photon dark
matter.[41] Right: constraints on the $L_{\mu}-L_{\tau}$ gauge coupling versus
$m_{A^{\prime}}$ from distortions of long baseline and solar neutrino
oscillations by $L_{\mu}-L_{\tau}$ dark photon DM.[44]

Acknowledgments. This work was supported by the Natural Sciences and
Engineering Research Council (NSERC) of Canada. I thank J. Jaeckel, C. O’Hare,
M. Tytgat and S. Witte for helpful discussions, and the CERN theory department
for its generous hospitality.

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# Long-term optical photometric monitoring of the FUor star V900 Mon

Serb. Astron. J. ˝ 183 (2021), 1 - 5

Original Scientific Paper

E. H. Semkov1, S. P. Peneva1 and S. I. Ibryamov2

1Institute of Astronomy and National Astronomical Observatory, Bulgarian
Academy of Sciences, 72, Tsarigradsko Shose Blvd., 1784 Sofia, Bulgaria

E–mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
2Department of Physics and Astronomy, Faculty of Natural Sciences, University
of Shumen, 115, Universitetska Str., 9712 Shumen, Bulgaria

E–mail<EMAIL_ADDRESS>

(Received: .; Accepted: .)

SUMMARY: We present results from photometric monitoring of V900 Mon, one of
the newly discovered and still under-studied object from FU Orionis type. FUor
phenomenon is very rarely observed, but it is essential for stellar evolution.
Since we only know about twenty stars of this type, the study of each new
object is very important for our knowledge. Our data was obtained in the
optical spectral region with BVRI Johnson-Cousins set of filters during the
period from September 2011 to April 2021. In order to follow the photometric
history of the object, we measured its stellar magnitudes on the available
plates from the Mikulski Archive for Space Telescopes. The collected archival
data suggests that the rise in brightness of V900 Mon began after January 1989
and the outburst goes so far. In November 2009, when the outburst was
registered, the star had already reached a level of brightness close to the
current one. Our observations indicate that during the period 2011-2017 the
stellar magnitude increased gradually in each pass band. The observed
amplitude of the outburst is about 4 magnitudes (R). During the last three
years, the increase in brightness has stopped and there has even been a slight
decline. The comparison of the light curves of the known FUor objects shows
that they are very diverse and are rarely repeated. However, the photometric
data we have so far shows that the V900 Mon’s light curve is somewhat similar
to this of V1515 Cyg and V733 Cep.

Key words. stars: pre-main sequence – stars: variables: T Tauri, Herbig Ae/Be
– stars: individual: V900 Mon

##  1 1\. INTRODUCTION

One of the main characteristics of the young stellar objects is their
photometric variability. In fact, most of the pre-main sequence (PMS) stars
show variations in brightness that are associated with the evolutionary
processes. PMS stars that undergo episodic outbursts with large amplitudes can
be divided into two types FUors and EXors (Herbig 1989). The common factor
between the two types of eruptive phenomena is that they are observed on T
Tauri stars.

The outburst of FU Orionis occurs in 1936 and for several decades this star
remained the only object of this type. First Ambartsumian (1971) draws
attention to this object by linking it to the evolution of the PMS stars and
proposes the abbreviation FUor. After the discovery of two new FUor objects -
V1057 Cyg and V1515 Cyg Herbig (1977) defined them as a new class of young
eruptive stars. Another dozen FUor objects were assigned to this class of
young variables over the next four decades (Reipurth & Aspin 2010, Audard et
al. 2014, Connelley & Reipurth 2018). These objects have been classified in
terms of their wide range of available photometric and spectral properties,
but their outbursts are thought to have the same cause: an enhanced accretion
rate from the circumstellar disk onto the central star (Hartmann & Kenyon
1996, Herbig 2007). During the outbursts, FUor objects undergo significant
increase in their accretion rate from $\sim$10-7$M_{\odot}$$/$yr up to
$\sim$10-4$M_{\odot}$$/$yr.

Several reasons are suggested to explain the enhanced accretion rate. The most
popular is that the increase is caused by thermal or gravitational instability
in the circumstellar disk (Hartmann & Kenyon 1996, Zhu et al. 2009). Another
possible triggering mechanisms could be the interactions of the circumstellar
disk with a planet or nearby stellar companion on an eccentric orbit (Lodato &
Clarke 2004, Reipurth & Aspin 2004, Pfalzner 2008) and in fall of clumps of
material formed by disk fragmentation onto the central star (Vorobyov & Basu
2005, Vorobyov et al. 2021).

Our experience in studying of FUor objects shows that the outburst can last
for several decades or even a century. As a rule, it is considered that the
time to increase the brightness is much shorter than the time to decrease it.
All known FUors share the same defining characteristics: location in star-
forming regions, association with reflection nebulae, a 4-6 mag. outburst
amplitude, an F-G supergiant spectrum during the outbursts, a strong Li I 6707
Å line in absorption, and CO bands in near-infrared spectra (Herbig 1977,
Reipurth & Aspin 2010). An important feature of FUors is the massive
supersonic wind observed as a P Cyg profile most commonly for both H$\alpha$
and Na I D lines.

The first three, also called classical FUor objects (FU Ori, V1515 Cyg and
V1057 Cyg) are well-studied and their light curves are published in the
literature (Clarke et al. 2005, Kopatskaya et al. 2013). Due to the large-
scale optical and infrared monitoring programs carried out in several
observatories and the contributions of amateur astronomers, some new objects
have been observed to undergo outbursts with large amplitude: V733 Cep
(Reipurht et al. 2007, Peneva et al. 2010), V2493 Cyg (Semkov et al. 2010,
Semkov et al. 2012, Miller et al. 2011), V2492 Cyg (Aspin 2011, Hillenbrand et
al. 2013, Kóspál et al. 2013), V2494 Cyg (Aspin et al. 2009), V2495 Cyg
(Movsessian et al. 2006), V582 Aur (Semkov et al. 2013, Ábrahám et al. 2018),
V2775 Ori (Fischer et al. 2012), V900 Mon (Reipurth et al. 2012), V960 Mon
(Hillenbrand et al. 2014, Kóspál et al. 2015, Gaia 18bpy (Hillenbrand et al.
2018), Gaia 18dvy (Szegedi-Elek et al. 2020). Due to the very small number of
known FUor objects, each newly discovered attracts significant attention.

The detection of a new eruptive star in Monoceros (LDN 1656 cloud) has been
reported by the amateur astronomer Jim Thommes (Thommes el. 2011). According
to the General Catalog of Variable Stars, the new variable is defined by the
designation – V900 Mon (Fig. 1). Reipurth et al. (2012) performed detailed
multi-wavelength study of V900 Mon and find significant similarities with the
objects from the group of FUors. According to the authors the outburst of V900
Mon occurred between 1953 and 2009, and its luminosity is 106 $L_{\odot}$ at a
distance of 1100 pc. Reipurth et al. (2012) also suggest that V900 Mon is a
Class I source and the spectrum of the object is very similar to those of the
classical prototype FU Orionis. The spectrum of V900 Mon in the infrared
region is much later than in the optical. Also the characteristic deep
infrared CO bandhead absorption and the lithium Li I 6707 Å line, which is
typical of young stars, are observed in case of V900 Mon.

Gramajo et al. (2014) model the SED curve of V900 Mon in the 1–200 $\mu$m
spectral range and estimate the basic parameters of the disk and the envelope
of the star. The authors confirm that V900 Mon appears as a Class I source
with disk mass 0.1 $M_{\odot}$ and disk mass accretion rate 2.0 $\times$ 10-6
$M_{\odot}$ yr-1. They found considerable similarity to those of another
eruptive star V1647 Ori after its outburst. In order to study the mass and
physical parameters of the envelope around V900 Mon Kóspál et al. (2017)
measured the 12CO and 13CO lines using the FLASH+ receiver at the APEX
telescope. The results indicate that the mass of the envelope is low compared
to other FUor objects and V900 Mon is at the end of the Class I evolutionary
stage.

Varricatt et al. (2015) observed V900 Mon using the 3.8-m United Kingdom
Infrared Telescope with the $L^{\prime}$ and $M^{\prime}$ MKO filters. The
observations were made in April and September 2015 and show a slight increase
in the brightness in the infrared region. In our first paper concerning V900
Mon, we draw attention to the observed increase in brightness in the optical
range during the period 2011-2016 (Semkov et al. 2017b). No other optical
photometric data has been published since the outburst of V900 Mon was
detected in 2009 and the object joined the FUors group.

Recently, data from observations of V900 Mon using ALMA array were published
in two papers (Takami et al. 2019, Hales et al. 2020). The presence of an
extended molecular bipolar outflow and a rotating envelope is detected. The
disk around V900 Mon is resolved and its radius is set to $\sim$ 50 au.
(Kóspál et al. 2020) note the presence of the 10 $\mu$m silicate feature in
emission that is rather characteristic of Class II sources, but define it as a
geometric effect from the high viewing angle.

##  2 2\. OBSERVATIONS AND DATA REDUCTION

###  2.1 2.1 CCD photometry

The photometric observations of V900 Mon were performed with the 2 m RCC and
the 50/70 cm Schmidt telescopes of the National Astronomical Observatory
Rozhen (Bulgaria) and with the 1.3 m RC telescope of the Skinakas
Observatory111Skinakas Observatory is a collaborative project of the
University of Crete, the Foundation for Research and Technology - Hellas, and
the Max-Planck-Institut für Extraterrestrische Physik. of the Institute of
Astronomy, University of Crete (Greece). Observations were performed with four
types of CCD camera $-$ Vers Array 1300B (1340 $\times$ 1300 pixels, 20
$\times$ 20 $\mu m/$pixel size, scale 0.26 arcmin/pixel) and ANDOR iKon-L
BEX2-DD (2048 $\times$ 2048 pixels, 13.5 $\times$ 13.5 $\mu m/$pixel size,
scale 0.17 arcmin/pixel) at the 2-m RCC telescope, ANDOR DZ436-BV (2048
$\times$ 2048 pixels, 13.5 $\times$ 13.5 $\mu m/$pixel size, scale 0.33
arcmin/pixel) at the 1.3 m RC telescope and FLI PL16803 (4096 $\times$ 4096
pixels, 9 $\times$ 9 $\mu m/$pixel size, scale 1.08 arcmin/pixel) at the 50/70
cm Schmidt telescope.

Due to the negative Declination, V900 Mon can be observed from southern Europe
only in the period September-April. All frames were exposed through a set of
standard Johnson-Cousins filters. All the data was analyzed using the same
aperture, which was chosen as 2 arcsec in radius, while the background annulus
was taken from 9 arcsec to 14 arcsec in order to minimize the light from the
surrounding nebula.

In order to facilitate the process of photometric measurement of the star, we
calibrated twelve standard stars in close proximity to it. Standard stars have
a wide range of stellar magnitudes (from 13.689 to 17.286 mag in $I$-band). In
this way, we made it possible to compare current CCD observations with those
from archival photographic plates. Calibration was made during five clear
nights in 2011 and 2012 with the 1.3 m RC telescope of the Skinakas
Observatory and one night in 2011 with the 2-m RCC telescope of the Rozhen
Observatory. Standard stars from Landolt (1992) were used as a reference.

Fig. 1. Color image of V900 Mon obtained with the 2 m RCC telescope in NAO
Rozhen. The cometary nebula arising from the star is clearly visible.

Table 1 contains the $IRVB$ photometric data for the comparison sequence. The
corresponding mean errors in the mean are also listed in the table. The
measured stellar magnitudes of the standard stars in $B$-band are still
uncertain and subject to improvement. The stars are labeled from A to L in
order of their $I$-band magnitude. In regions of star formation, a great
percentage of stars can be photometric variables. Therefore, there is a
possibility that some of our standard stars are low amplitude variables, and
we advise observers to use our photometric sequence with care.

The finding chart of the comparison sequence is presented in Fig. 2. The field
is 8 $\times$ 8 arcmin, north is at the top and east is to the left. The chart
is retrieved from the STScI Digitized Sky Survey Second Generation Red.

Table 1. Photometric data for the $BVRI$ comparison sequence.

Star $I$ $\sigma_{I}$ $R$ $\sigma_{R}$ $V$ $\sigma_{V}$ $B$ $\sigma_{B}$ A
13.689 0.019 14.099 0.019 14.551 0.014 15.3 0.1 B 13.789 0.025 14.183 0.023
14.572 0.014 15.2 0.1 C 14.471 0.024 15.084 0.015 14.821 0.014 16.8 0.1 D
14.541 0.021 15.951 0.019 17.258 0.029 19.4 0.3 E 14.902 0.026 15.525 0.018
16.141 0.018 17.1 0.2 F 15.948 0.022 16.575 0.021 17.185 0.046 18.1 0.2 G
15.953 0.040 16.480 0.021 17.167 0.034 18.7 0.2 H 15.991 0.077 16.764 0.022
17.536 0.015 $-$ $-$ I 16.076 0.077 16.996 0.014 17.889 0.042 $-$ $-$ J 16.079
0.025 16.989 0.036 17.726 0.036 18.7 0.2 K 16.469 0.055 17.465 0.031 18.267
0.028 $-$ $-$ L 17.286 0.084 18.485 0.029 19.324 0.105 $-$ $-$

Fig. 2. Finding chart for the $\it BVRI$ comparison sequence around V900 Mon.

###  2.2 2.2 Data from photographic plates from the Mikulski Archive for Space
Telescopes

In order to investigate the photometric history of V900 Mon and to search for
the beginning of the outburst, we searched the photographic plate archive of
Mikulski Archive for Space Telescopes222$https://archive.stsci.edu/cgi-
bin/dss$$\\_$$plate$$\\_$$finder$. As a result, we found scanned copies from a
total of six photographic plates obtained with the Palomar Schmidt and the UK
Schmidt telescopes. The object was identified and measured on four of them. On
the other two plates, the object is below the plate limit (XO661) and it is
outside the field of the plate (S772). Aperture photometry of the scanned
plate copies were performed with DAOPHOT routines using the same aperture
radius and the background annulus as for our CCD observations. In this way we
achieve compliance of the measurements on the photographic plates with the CCD
photometry. The results of the measured magnitudes of V900 Mon from the
scanned plate copies are given in Table 2.

Table 2. Photometric data from the Mikulski Archive for Space Telescopes.

Plate Survey Name Epoch Emulsion+Filter Exp. Plate Scale Magnitude ID yyyy-mm-
dd hh:mm min. arc. sec. ER700 Equatorial Red 1989-01-08 14:19 IIIaF + OG590
63.0 1.01 R=18.60$\pm$0.04 IS700 SERC-I Survey 1985-02-10 10:56 IVN + RG715
90.0 1.01 I=17.22$\pm$0.03 S772 SERC-J Survey 1983-01-17 12:25 IIIaJ + GG395
60.0 1.70 out of plate S700 SERC-J Survey 1983-02-07 12:27 IIIaJ + GG395 65.0
1.70 V=20.7$\pm$0.3 XO661 POSS-I O 1953-01-17 06:42 xx103aO 10.0 1.01 B$<$21
XE661 POSS-E Red 1953-01-17 07:52 xx103aE + plexi 40.0 1.70 R=18.01$\pm$0.05

##  3 3\. RESULTS AND DISCUSSION

The results of our photometric CCD observations of V900 Mon are summarized in
Table 3. The columns provide the Julian Date of observation, $\it IRVB$
magnitudes of V900 Mon, the telescope and CCD camera used. The typical
instrumental errors in the reported magnitudes are $0.01$-$0.02$ for $I$, $R$
and $V$-band, and $0.03-0.05$ for $B$-band.

The $BVRI$ light curves of V900 Cyg during the period of our observations are
plotted in Fig. 3. The filled diamonds represent the CCD observations from the
50/70 cm Schmidt telescope, the filled squares CCD observations from the 2 m
RCC telescope and the filled triangles the CCD observations from the 1.3 m RC
telescope.

Table 3. Photometric $IRVB$ observations of V900 Mon during the period
September 2011 - April 2021.

J.D. (24…) I R V B Tel CCD J.D. (24…) I R V B Tel CCD 55815.612 13.30 14.94
16.11 1.3RC ANDOR 57369.520 13.02 14.55 15.54 16.74 2RCC VA 55816.627 13.32
14.95 1.3RC ANDOR 57370.481 13.06 14.63 15.66 16.95 2RCC VA 55824.604 13.33
14.99 16.14 1.3RC ANDOR 57371.481 13.03 14.58 15.58 16.85 2RCC VA 55842.593
13.26 14.89 15.92 1.3RC ANDOR 57372.489 13.05 14.58 15.42 Sch FLI 55847.578
13.32 14.97 16.15 1.3RC ANDOR 57374.444 13.09 14.63 15.56 16.65 Sch FLI
55865.569 13.31 14.92 15.98 17.34 2RCC VA 57425.352 13.09 14.53 15.51 Sch FLI
55866.594 13.32 14.93 16.02 17.33 2RCC VA 57426.341 13.12 14.60 15.77 Sch FLI
55892.555 13.26 14.85 15.92 17.25 2RCC VA 57485.288 13.00 14.61 15.56 16.88
Sch FLI 55893.481 13.24 14.82 Sch FLI 57756.433 13.01 14.66 15.70 Sch FLI
55895.547 13.33 14.80 Sch FLI 57781.348 12.96 14.53 15.52 16.54 Sch FLI
55896.502 13.33 14.79 Sch FLI 57781.429 12.99 14.38 15.71 16.91 2RCC VA
55925.505 13.16 14.65 Sch FLI 57785.361 12.97 14.39 15.63 16.99 2RCC VA
55928.422 13.20 14.74 15.90 Sch FLI 57786.380 12.97 14.37 15.67 16.95 2RCC VA
56003.303 13.29 14.85 Sch FLI 57800.329 12.91 14.44 15.43 16.49 Sch FLI
56013.301 13.22 14.80 15.85 17.01 2RCC VA 57801.356 12.90 14.58 15.62 16.60
Sch FLI 56173.625 13.24 14.91 1.3RC ANDOR 57813.305 12.90 14.52 15.53 Sch FLI
56180.622 13.28 14.92 1.3RC ANDOR 57817.312 12.80 14.15 15.20 Sch FLI
56183.627 13.29 14.92 16.08 1.3RC ANDOR 57845.236 13.01 14.62 15.60 Sch FLI
56193.613 13.31 14.93 16.13 1.3RC ANDOR 57846.298 13.00 14.48 15.61 Sch FLI
56209.626 13.28 14.86 15.94 Sch FLI 58013.588 13.10 14.62 15.65 Sch FLI
56249.563 13.36 14.97 16.04 Sch FLI 58043.588 13.02 14.60 15.60 Sch FLI
56275.512 13.14 14.66 15.65 2RCC VA 58080.536 12.98 14.57 15.52 16.66 Sch FLI
56276.505 13.26 14.82 15.89 17.24 2RCC VA 58081.540 13.01 14.59 15.63 16.66
Sch FLI 56328.352 13.20 14.66 15.71 Sch FLI 58113.469 13.02 14.60 15.63 16.82
Sch FLI 56329.358 13.26 14.78 15.75 Sch FLI 58114.452 12.91 14.40 15.30 16.41
Sch FLI 56371.338 13.21 14.74 15.71 16.96 2RCC VA 58217.268 12.95 14.52 15.50
Sch FLI 56553.604 13.21 14.81 15.92 1.3RC ANDOR 58218.260 12.98 14.56 15.72
Sch FLI 56655.480 13.20 14.65 15.63 Sch FLI 58220.262 12.98 14.66 15.64 Sch
FLI 56656.445 13.23 14.86 15.81 Sch FLI 58220.281 13.13 14.46 15.72 17.08 2RCC
ANDOR 56681.395 13.18 14.75 15.66 Sch FLI 58543.302 13.04 14.55 15.56 16.70
Sch FLI 56694.362 13.19 14.78 15.83 17.21 2RCC VA 58547.316 13.15 14.63 15.70
16.80 Sch FLI 56715.289 13.19 14.65 15.50 Sch FLI 58864.428 13.19 14.61 15.62
Sch FLI 56738.253 13.13 14.62 15.53 Sch FLI 58865.437 13.01 14.36 15.59 Sch
FLI 57005.439 13.14 14.66 15.77 Sch FLI 58867.444 13.26 14.65 15.92 17.24 2RCC
ANDOR 57006.523 13.16 14.69 15.75 Sch FLI 58869.420 13.11 14.46 15.61 16.71
2RCC ANDOR 57016.486 13.16 14.69 15.64 16.91 2RCC VA 58870.429 13.24 14.74
15.81 Sch FLI 57072.314 13.05 14.42 15.23 Sch FLI 58901.330 13.05 14.44 15.44
Sch FLI 57074.328 13.18 14.69 15.67 Sch FLI 58930.281 13.15 14.60 15.75 Sch
FLI 57268.618 13.12 14.70 15.80 1.3RC ANDOR 59176.505 13.19 14.54 15.61 16.69
Sch FLI 57269.622 13.10 14.69 15.74 1.3RC ANDOR 59177.516 13.09 14.43 15.42
16.58 Sch FLI 57330.548 13.07 14.60 15.51 Sch FLI 59220.388 12.96 14.35 15.50
16.87 2RCC ANDOR 57331.561 13.08 14.56 15.49 Sch FLI 59250.392 13.10 14.43
15.53 2RCC ANDOR 57333.594 13.11 14.65 15.61 Sch FLI 59251.337 13.12 14.45
15.64 16.86 2RCC ANDOR 57714.551 13.02 14.47 15.70 16.99 2RCC VA 59314.253
12.94 14.30 15.40 16.45 2RCC ANDOR 57715.544 13.06 14.47 15.72 17.05 2RCC VA
59316.258 12.93 14.32 15.33 16.54 Sch FLI 57716.536 13.06 14.48 15.70 17.02
2RCC VA

Fig. 3. BVRI light curves of V900 Mon for the period of our observations 2011
September $-$ 2021 April.

The historical $VRI$ light curves of V900 Mon from all available observations
are plotted in Fig. 4. Our CCD observations are indicated by the same symbols
as on Fig. 3. The filled circles on Fig. 4 represent data from Mikulski
Archive for Space Telescopes, listed in this paper. The empty circles
represent data from DSS reported in Table 1 of the paper of Reipurth et al.
(2012).

Archival data shows that before the outburst V900 Mon was variable, which is
especially noticeable in the $R$-light. The amplitude of variability is about
1 magnitude, which is a typical variability of T Tauri stars. There are
several FUor objects for which the data for photometric variability before the
outburst is available (FU Ori, V1057 Cyg, V2493 Cyg, V582 Aur, Gaia 18dvy).
They all show non-periodic changes in brightness, that proves the affiliation
of the prototypes of FUors to the class of T Tauri stars.

The exact time of the beginning of the outburst of V900 Mon cannot be
determined now. But it can certainly be concluded that the outburst occurred
after January 1989. Due to the lack of data for the period 1989-2011, we
cannot determine the rate of increase in brightness and the nature of the
light curve before reaching the maximum brightness. The first photometric data
for V900 Mon from 2009 and 2010 reported by Thommes el. (2011) and Reipurth et
al. (2012) were obtained in the $g^{\prime}r^{\prime}i^{\prime}z^{\prime}$
system and are not compatible with our results. Also, this data was obtained
with a smaller aperture (1 arcsec), and as noted by Reipurth et al. (2012),
the aperture radius is of great importance for the measured value of the
stellar magnitudes.

Since the beginning of our observations in 2011, we have been recording a
gradual rise in brightness. But during the last three observation seasons
(from the end of 2018 to the beginning of 2021), the increase in brightness
has stopped and there has even been a slight decline. The increase in
brightness is noticeable in all filters, as for the R-band it is about 0.5
mag. for the eight-year period from 2011 to 2017. Therefore, V900 Mon has
reached its maximum brightness during observation season 2017$-$2018 and we
should expect a decrease in brightness in the coming years. But occasionally
it is possible to observe two peaks at maximum brightness, as is the case of
V2493 Cyg outburst (Semkov et al. 2012). Whether we have really registered the
maximum of the FUor type outburst should be confirmed by subsequent
observations.

Long-term photometric data from observations of known FUor objects show a wide
variety of the light curves. Even the first three classical FUor objects show
different rates of increase and decrease in brightness (Clarke et al. 2005).
As the number of known FUor objects increases, the diversity of the light
curves increases even more. Some objects show a very rapid rise in brightness
within a few months or a year (FU Ori, V1057 Cyg, V2493 Cyg) (Clarke et al.
2005, Semkov et al. 2012). In others, the rise in brightness can take several
years and even achieves to 20-30 years period (V1515 Cyg, V1735 Cyg, V733 Cep)
(Clarke et al. 2005, Peneva et al. 2009, Peneva et al. 2010).

Such diversity is observed for the periods of decline in brightness. Usually
the decline in brightness lasts for decades or even a century. But there are
objects where a comparatively rapid decline in brightness has been observed.
For example, V960 Mon in which the brightness decreases by 2 mag (V) over a
period of five years (Takagi et al. 2020), and V582 Aur for which two deep
decreases in brightens by about 3 mag (R) have been observed separated by a
five-year period (Semkov et al. 2013, Ábrahám et al. 2018). There are also
objects that for a period of several decades do not practically change their
brightness as in the case of V1735 Cyg (Peneva et al. 2009). Also, V2493 Cyg
remains for several years with an almost constant magnitude at the maximum
level of brightness (Semkov et al. 2017a).

This variety of photometric behavior strongly supports the idea that FUor
objects are not a homogeneous group and that the causes of this phenomenon can
be several different mechanisms (Vorobyov et al. 2021). Trying to compare the
available photometric data for V900 Mon with the other FUors, we register the
following similarities. The slow rise in brightness during the recent years is
similar to that of objects V1515 Cyg and V733 Cep. We do not yet have data on
how the decline in brightness will continue, but the beginning of a gradual
decline is also similar to the above two objects.

Fig. 4. Historical VRI light curves of V900 Mon for the period 1953
January$-$2021 April. In the figure, the data from the $I$ pass band are
colored in black, from the $R$ pass band in red and from the $V$ pass band in
green.

##  4 4\. CONCLUSIONS

During the last twenty years, we have undertaken intensive photometric
monitoring of several unexplored FUor objects. This data can also be used to
detect new FUor or EXor events and to determine the type of the outburst.
FUors are the most noticeable young variable objects due to the large
amplitudes and eruptions lasting for years. Due to the small number of such
objects, we do not know enough about their photometric history.

The data collected so far for V900 Mon shows that it can be classified with
confidence as a $bonafide$ FUor. An outburst with an amplitude of 4 magnitudes
was registered, which began after 1989. Maximum brightness was observed in the
period 2017-2018, after which a decrease in brightness began.

Our results show the need for systematical photometric monitoring of the
objects of special interest. Collecting more data from observations as well as
from astronomical archives can help clarify theories about the early stages of
stellar evolution. We support the idea that the type of light curve of a FUor
object can be evidence of the specific cause of its outburst.

Acknowledgements – This work was partly supported by the Bulgarian Scientific
Research Fund of the Ministry of Education and Science under the grant DN
18-13/2017 and DN 18-10/2017. The authors thank the Director of Skinakas
Observatory Prof. I. Papamastorakis and Prof. I. Papadakis for the award of
telescope time. This research has made use of the NASA’s Astrophysics Data
System Abstract Service, the SIMBAD database and the VizieR catalogue access
tool, operated at CDS, Strasbourg, France. Some of the data presented in this
paper was obtained from the Mikulski Archive for Space Telescopes (MAST).
STScI is operated by the Association of Universities for Research in
Astronomy, Inc., under NASA contract NAS5-26555.

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Also at ]Department of Physics, Faculty of Science, Universiti Teknologi
Malaysia, 81310 Skudai, Johor Bahru, Malaysia.

# Muon capture reaction on 100Mo to study nuclear responses for double beta
decays and astro-neutrinos

I.H.Hashim1,2,Ejiri2, T.Shima2,A.Sato3,Y.Kuno3,N.Kawamura4, S.Miyake4,
K.Ninomiya4 1Department of Physics, Faculty of Science, Universiti Teknologi
Malaysia, Johor Bahru, Malaysia.  [<EMAIL_ADDRESS>2Research Center for Nuclear
Physics, Osaka University, Osaka, Japan.  3Department of Physics, Osaka
University, Osaka, Japan.  4Material Life Science Facility, J-PARC, Tokai,
Japan.

###### Abstract

The negative-muon capture reaction (MCR) on the enriched 100Mo isotope was
studied for the first time to investigate neutrino nuclear responses for
neutrino-less double beta decays and supernova neutrino nuclear interactions.
MCR on 100Mo proceeds mainly as 100Mo($\mu$,xn)100-xNb with $x$ being the
number of neutrons emitted from MCR. The Nb isotope mass distribution was
obtained by measuring delayed $\gamma$-rays from radioactive 100-xNb. By using
the neutron emission model after MCR, the neutrino response (the strength
distribution) for MCR was derived. Giant resonance (GR)-like distribution at
the peak energy around 11-14 MeV, suggests concentration of the MCR strength
at the muon capture GR region.

DOI

PACS number(s)

23.40.-s, 13.35.Hb, 14.60.St.

###### pacs:

Valid PACS appear here

##  I Introduction

Neutrino nuclear responses (square of nuclear matrix element, NME) are crucial
for neutrino studies in nuclei. Neutrino properties such as the Majorana
nature, the absolute mass scale and others beyond the standard electro-weak
model are studied by investigating neutrino-less double beta decays (DBD) ref1
; ref2 ; ref3 ; ref4 ; ref5 . Here the $\nu$ responses for DBD ($\beta^{-}$
and $\beta^{+}$ responses) are necessary to get the neutrino properties beyond
the standard model. The DBD matrix element is expressed in terms of the
product of $\beta^{-}$ and $\beta^{+}$ matrix elements of $M(\beta^{-}$) and
$M(\beta^{+}$). Astro-neutrino nucleo-syntheses and astro-neutrino nuclear
reactions are studied by investigating astro neutrino nuclear interactions.
Then one needs nuclear responses for astro neutrinos and astro anti-neutrinos
ref1 ; ref2 . So far, the single $\beta^{-}$ matrix element $M(\beta^{-})$ and
the neutrino response have been extensively studied by charge exchange
reactions ref1 ; ref5 .

The present report aims to show that negative muon ($\mu$) capture reaction
(MCR) is used to study the neutrino nuclear responses relevant to the
$\beta^{+}$ response involved in DBD and anti-neutrino response associated
with astro anti-neutrino reactions. The MCR response for 100Mo shows a giant
resonance (GR) distribution at the peak energy around 11-14 MeV.

Figure 1: Reaction and decay schemes for astro neutrinos and double beta
decays.

MCR is a kind of muon charge exchange reaction (MCER) via the weak boson W±,
where muon becomes muon neutrino and proton in the target nucleus transform
into neutron. The response represents the square of the nuclear matrix element
$M(\mu)$. The reaction and the NME are expressed as

$\mu+^{A}_{Z}X\rightarrow^{A}_{Z-1}Y+\nu_{\mu},~{}~{}~{}M(\mu),$ (1)

where ${}^{A}_{Z}X$ with $A$ and $Z$ being the mass number and the atomic
number of the target nucleus and ${}^{A}_{Z-1}Y$ is the nucleus after MCR. The
corresponding astro anti-neutrino reaction and the NME are given as

$\bar{\nu}_{e}+^{A}_{Z}X\rightarrow^{A}_{Z-1}Y+e^{+},~{}~{}~{}M(\bar{\nu}).$
(2)

DBD via the light Majorana $\nu$ exchange process with the $\beta^{+}$ and
$\beta^{-}$ responses is written as the neutrino emission and re-absorption
process as

${}^{A}_{Z}X\rightarrow^{A}_{Z-1}Y+\nu_{e}+e^{+},~{}~{}~{}M(\beta^{+}).$ (3)
${}^{A}_{Z-2}X\rightarrow^{A}_{Z-1}Y+\bar{\nu}_{e}+e^{-},~{}~{}~{}M(\beta^{-}),$
(4)

where ${}^{A}_{Z-1}$Y is the intermediate nucleus. Note that the
$\mu,\bar{\nu}$ and $\beta^{+}$ NMEs are associated with $\tau^{+}$
p$\rightarrow$n transition, while $\nu$ and $\beta^{-}$ NMEs are with
$\tau^{-}$ n$\rightarrow$ p one. These reaction and decay schemes are shown in
Fig. 1. Here the DBD NME $M(\beta\beta)$ is given by the sum of NMEs
$M_{i}(\beta\beta)$ over all relevant states in the intermediate nucleus
${}^{A}_{Z-1}Y$. The NMEs $M_{i}(\beta\beta)$ are associated indirectly with
the single $\beta$ NMEs of $M_{i}(\beta^{+})$ and $M_{i}(\beta^{-})$ via the
intermediate state i. Thus information of $M(\beta^{\pm})$ is used to help
evaluate $M(\beta\beta)$.

Theoretical calculations of NMEs for neutrino nuclear responses are very hard
since they are very sensitive to nuclear correlations, nuclear medium effects
and nuclear models, as discussed in review articles ref4 ; ref5 . Thus
experimental studies of them are very valuable. Experimental studies of the
neutrino nuclear responses for DBDs and astro neutrinos are described in
review articles ref1 ; ref2 ; ref3 ; ref4 ; ref5 .

The NMEs $M(\nu)$ for astro neutrino and the $M(\beta^{-})$ for DBD (the left
hand side of Fig. 1) have been studied extensively by using high energy-
resolution (3He,t) experiments at RCNP ref1 ; ref4 ; ref5 . On the other hand,
there are no appropriate high energy-resolution nuclear probes for the astro
anti-neutrino $M(\bar{\nu})$ and DBD $M(\beta^{+})$ (the right hand side of
Fig.1).

Figure 2: Reaction and decay schemes for MCR on 100Mo. Highly excited states
in 100Nb around 30 MeV de-excite by emitting 3 neutrons, populating finally
the 97Nb RI, while those around 12 MeV de-excite by emitting 1 neutron,
populating finally the 99Nb. The residual nuclei are identified by observing
$\gamma$-rays characteristic of the RIs.

The present MCR provides useful information relevant to $M(\bar{\nu})$ and
$M(\beta^{+})$. Unique features of MCR is the large energy and momentum
regions of $E$=0-50 MeV and $p$= 20-100 MeV/c, which are the regions involved
in neutrino-less DBD neutrinos and medium energy supernova neutrinos.

In MCR, a well-bound proton in the target nucleus ${}^{A}_{Z}X$ is shifted up
to a vacant neutron shell, and one gets mostly the excited nucleus
${}^{A}_{Z-1}$Y∗ with the excitation energy $E$. If particle bound, it decays
by emitting $\gamma$-rays to the ground state of ${}^{A}_{Z-1}$Y. On the other
hand, if ${}^{A}_{Z-1}$Y∗ is unbound, it de-excites by emitting a number ($x$)
of neutrons in case of medium heavy nucleus since proton emission is
suppressed much by the Coulomb barrier.

Finally one residual nucleus ${}^{A-x}_{Z-1}Y$ is obtained. If it is
radioactive, we identify it by measuring characteristic $\gamma$-rays of the
residual nucleus. The number of the emitted neutrons reflects the excitation
energy $E$, larger $x$ corresponds to the higher $E$ region. Thus one can
derive the MCR response, i.e. the strength distribution, as a function of $E$,
from the mass ($A-x$) distribution of the residual isotopes, as suggested in
1972 ref6 , and also in 2001’s ref2 ; ref7 ; ref8 . The MCER and the neutron
emission schemes are illustrated in Fig.2.

So far nuclear MCR $\gamma$-rays were measured to study the nuclear reaction
mechanisms ref9 ; ref10 . Prompt $\gamma$-rays from bound states excited by
$\mu$ capture AX($\mu$,$\gamma$)AY reactions were investigated to study
$\beta^{+}$ responses for low lying bound states ref11 . Here it is hard to
extract the $\beta^{+}$ strengths to individual states because the states are
populated not only directly by MCR but also indirectly from so many higher
states populated by MCR via $\gamma$ transitions.

In the present work, we focus on gross structure of the MCER strength
distribution by measuring the mass distribution of residual isotope (RI)
${}^{A-x}Y$. The RI yield is well obtained by measuring the yield of
$\gamma$-rays from ${}^{A-x}Y$ and the known $\gamma$-branching ratio. A
neutron emission code was developed to link the mass number $A-x$ of the
residual nucleus to the initial excitation energy $E$ of ${}^{A}_{Z-1}Y^{*}$.

Figure 3: Schematic plane view of (a) the target and the coaxial-type Ge
detectors for prompt X and prompt/short-lived $\gamma$-rays and (b) the Ge
detectors for the off-line measurements for the long-lived RIs.

##  II Muon Irradiation Experiment

The present MCR experiment is made on ${}^{100}_{42}$Mo as a typical medium
heavy nucleus for astro-physics and DBD interests ref12 ; ref13 . The target
used is a thick 100Mo with 40 mg/cm2. Low energy $\mu^{-}$ beams with p = 28
MeV/c from the D-beam line at J-PARC MLF were used to irradiate the target for
7 hours. Here most muons were stopped and captured into the target nucleus to
form excited states in ${}^{100}_{41}$Nb. Then $\gamma$-rays from short-lived
Nb isotopes with $A$=100 and 99 were measured on-line using planar and
coaxial-end type HPGe detectors as shown in the left hand side of Fig.3, while
$\gamma$-rays from long-lived Nb isotopes and other isomers were measured off-
line at a separate room by using 2 coaxial Ge detectors as shown in the right
hand side of Fig.3.

##  III Gamma rays from MCR on enriched Mo and Nb mass distribution

The prominent $\gamma$-ray peaks from short-lived and long-lived Nb isotopes
were clearly observed as shown in Fig.4. The typical $\gamma$-rays from the
MCR products are listed in Table1. The 535.0 keV $\gamma$-ray from the short-
lived 100Nb and 137 keV $\gamma$-ray from the short-lived 99Nb were measured
by the on-line Ge detector setting, while others from the long-lived 99Mo,
${}^{99}Tc^{m}$, 98Nb, 97Nb, 96Nb, and 95Nb were measured by the off-line Ge
detector setting. Among them, the 66 hr 99Mo is the $\beta^{-}$ decay product
from 99Nb and the 6 hr 99Tcm is the isomeric state produced by the $\beta^{-}$
decay from 99Nb.

Figure 4:  Gamma ray spectra from MCR on 100Mo. (a) On-line spectrum for the
prompt $\gamma$-rays and delayed ones from short-lived RIs. (b) Off-line
spectrum for the delayed $\gamma$-rays from long-lived RIs.

The number of the Nb RIs 100-xNb produced by MCR on 100Mo was evaluated from
the observed $\gamma$-ray yields corrected for the Ge detector efficiency, the
$\gamma$-ray branching ratio, and their decays during the muon irradiation and
the $\gamma$-ray measurement. The obtained RI mass ($A-x$) distribution is
shown in Fig.5. The 100Nb yield at $x$=0 is quite small, but jumps up
drastically at $x$=1, and decreases gradually as $x$ increases down to the
mass $A$=95 and $x$=5. This is similar to the distributions in other target
nuclei ref7 ; ref9 ; ref10 ; ref17 .

Table 1: Nuclear isotopes observed by MCR ($\mu$, xn) reactions. Half lives are given by s: second, m: minutes, h: hour or d: day. Isotope |  lifetime |  Energy(keV)
---|---|---
100Nb |  2.99 s |  535.0
99Nb |  2.6 m |  137.0
99Mo |  66.0 |  181.1, 739.5
99Tc |  6 h |  140.5
98Nb |  1.23 h |  722.0, 787.0
97Nb |  0.85 h |  657.9
96Nb |  23.3 h |  459.0, 568.7, 778.0
95Nb |  34.9 d |  765.0
94Nb |  51.8 m |  75.5, 366.9, 891.7
93Nb |  6.85 h |  2424.9

Let us evaluate the RI mass distribution on the basis of the MCR strength
distribution and the statistical neutron emission model. MCR excitations are
expressed in terms of the vector excitations with the spin transfers of
$\Delta J^{\pi}=0^{+},1^{-},2^{+}$ and the axial-vector ones with $\Delta
J^{\pi}=1^{+},2^{-}$. Among them the 0+ Fermi and the 1+ GT excitations are
reduced much since the 0$\hbar\omega$ Fermi and GT excitations for the
$\beta^{+}$ and the anti-neutrino responses are blocked by the neutron excess
in medium heavy nuclei of the present interest. The 1- excitation with the
1$\hbar\omega$ jump may show the giant resonance (GR) like the E1 GR in case
of the photon capture reaction (PCR).

The vector 2+ and the axial-vector 2- excitations may show a broad GR-like
distributions as the 2$\hbar\omega$ and spin dipole GRs. Accordingly, we
assume MCR strength distribution of $B(\mu,E)$ given by the sum of the two GR
strengths of $B_{1}(\mu,E)$ and $B_{2}(\mu,E)$,

$B(\mu,E)=B_{1}(\mu,E)+B_{2}(\mu,E),$ (5)
$B_{i}(\mu,E)=\frac{B_{i}(\mu)}{(E-E_{Gi})^{2}+(\Gamma_{i}/2)^{2}},$ (6)

where $E_{Gi}$ and $\Gamma_{i}$ with $i$=1 and 2 are the GR energy and the
width for the $i$th GR, and the constant $B_{i}(\mu)$ is expressed as
$B_{i}(\mu)=\sigma_{i}\Gamma_{i}/(2\pi)$ with $\sigma_{i}$ being the total
strength integrated over the excitation energy.

Figure 5:  Nb RI mass distribution for MCRs on 100Mo. The black and light grey
histograms are the experimental and calculated yields.

Excited states ${}^{A}_{Z-1}$Y∗ populated by MCR de-excite mostly by emitting
neutrons at the pre-equilibrium (PEQ) and equilibrium (EQ) stages ref18 . If
the states are neutron unbound, and do by emitting $\gamma$-rays to the ground
state if they are bound. Here we ignore proton emission which is prohibited by
the Coulomb barrier in case of the medium and heavy nuclei. The energy
spectrum of the first neutron E${}_{n}^{1}$ is given by ref14 .

$S(E_{n}^{1})=k[E_{n}^{1}exp(-\frac{E_{n}^{1}}{T_{EQ}(E)}+pE_{n}^{1}exp(-\frac{E_{n}^{1}}{T_{PEQ}(E)})],$
(7)

where $E_{n}^{1}$ is the first neutron kinetic energy, $T_{EQ}(E)$ and
$T_{PEQ}(E)$ are the EQ and PEQ nuclear temperatures and $p$ is the fraction
of the PEQ neutron emission. The neutron emission from the EQ stage is a kind
of neutron evaporation from thermal equilibrium phase.

The EQ temperature is expressed as $T_{EQ}(E)=\surd(E/a)$ with $a$ being the
level density parameter ref14 . The parameter $a$ is expressed as $a=A$/8 MeV
for the nucleus with mass number $A$. $T_{PEQ}(E)$ is given by $b\times
T_{EQ}(E)$ with $b\approx$3 for MCR with low-momentum ($\approx$ 50-90 MeV/c)
and low-excitation ($E\approx$ 10-50 MeV). The PEQ contribution for the first
neutron emission depends on the nuclear size, getting smaller as the nuclear
size becomes larger. It is estimated to be around $p\approx 0.6A^{-1/3}$ for
the present MCR case by referring to the observed neutron energy spectra ref7
.

The residual nucleus ${}^{A-1}_{Z-1}$Y after the first neutron emission de-
excites by emitting the second neutron or $\gamma$-rays depending on the
excitation energy above or below the neutron threshold energy. The ground
state of ${}^{A-1}_{Z-1}$Y is populated after the $\gamma$ emission. The $2$nd
neutron n2 is the EQ evaporation neutron, and then the $3$rd neutron is
emitted if the residual nucleus after the $2$nd neutron emission is neutron-
unbound, and so on. Then, one gets finally the residual isotopes of
${}^{A-x}_{Z-1}$Y with $x$ = 0,1,2,3,… depending on the excitation energy $E$
and the number $x$ of the emitted neutrons. Some of them are $\beta$-unstable
RIs.

The neutron number $x$ and the mass number $A-x$ distributions reflect the
strength distribution $B(\mu,E)$ of the nucleus ${}^{A}_{Z-1}$Y∗ after MCR,
the highly excited states around 30-40 MeV emits 3-4 neutrons while the low
excited states around 11-14 MeV emit one neutron as illustrated in Fig.6. In
other words, the GR-like strength around 11-14 MeV leads preferentially to
population of ${}^{A-1}_{Z-1}$Y after 1 neutron emission, and the population
of ${}^{A-x}_{Z-1}$y decrease as $x$ increases. These features are just what
have been observed ref8 ; ref7 ; ref12 .

Figure 6:  The MCR strength distribution suggested from the experimental RI
distribution. $E_{G1}$ and $E_{G2}$ are the MCR GRs at around 12 MeV and 30
MeV.

We compare the observed RI mass distribution for MCR on 100Mo with the
calculation based on the strength distribution and the EQ/PEQ neutron emission
model. The obtained RI mass distribution is compared with the observed one in
Fig.6. The agreement with the observed data is quite good where $\chi^{2}$ is
0.06. The parameters used for the calculation are $E_{G1}$=12 MeV with
$\Gamma_{1}$=8 MeV, $E_{G2}$=30 MeV with $\Gamma_{2}$=8 MeV, and the cross
section ratio is $\sigma_{1}/\sigma_{2}$=1/6. The first GR corresponds to the
large population of the mass $A-1$ with $x$=1 neutron emission, while the
second GR match with the population of the RIs with the mass around $A-3$ and
$A-4$ with $x$=3-4 neutron emission.

The MCR GR may be compared with the photon capture reaction giant resonance
(PCR GR). The energy of 12 MeV is a bit smaller than the PCR GR energy of 14
MeV, but the width of 8 MeV is much larger than the width of 5 MeV for PCR GR.
MCR GR consists of mixed components of $J^{\pi}$=1-, 1+, 2-,… while PCR GR is
only one component of $J^{\pi}$=1-.

MCRs in other nuclei have been studies as discussed in the review paper ref7 ;
ref9 ; ref10 ; ref16 ; ref17 . The one neutron emission is dominant in most
MCRs, being consistent with the present observation on 100Mo, and with the GR
strength corresponding to the one neutron emission. Neutron energy spectra
were observed for MCRs on 32S, 40Ca, 207Pb and 209Bi. They reported low-energy
EQ and medium energy PEQ components from neutron time-of-flight (TOF)
measurement. They are reproduced by the EQ/PEQ neutron emission model with the
GR1 and GR2 strength distribution given in equation (6).

##  IV Concluding remarks and Perspectives

MCR, as the lepton-sector charge exchange reaction (CER) via the weak boson,
is shown to be used to study neutrino nuclear responses relevant to DBD and
astro-neutrino reactions. It provides unique information on $\beta^{+}$ side
DBD NMEs and astro anti-neutrino NMEs in the energy and momentum regions of
$E\approx$5-50 MeV and $p\approx$95-50 MeV/c.

These are just the regions associated with the neutrino-less DBD and supernova
neutrinos. Nb RIs produced by MCR on 100Mo are identified by measuring delayed
$\gamma$-rays characteristic of the RIs. The Nb mass distribution shows large
yield at $A$=99 after $x$=1 neutron emission, and decreases gradually as $A$
decreases till $A$=95 ($x$=5). The neutron emission is analyzed in terms of
the EQ/PEQ neutron emission model.

The observed RI distribution reflects the MCR strength distribution with a
broad peak at $E\approx$ 11-14 MeV and the small bump at $E\approx$ 30-40 MeV.
The broad peak is a kind of $\mu$-capture GR analogous to the E1 photon-
capture GR. It is noted that similar GRs are observed for the DBD
$\beta^{-}$-side responses and the astro neutrino responses ref1 ; ref4 ; ref5
.

The RCNP MuSIC DC muon beam and the J-PARC MLF pulsed muon beam are promising
for further studies of neutrino nuclear responses. The lifetime measurement is
under progress to study the absolute strength (square of absolute NME). The
absolute response, together with the strength distribution, help theories to
evaluate the DBD NMEs and astro-neutrino synthesis/interaction NMEs. In other
words, nuclear models to be used for DBD NME calculations should reproduce the
MCR strength distribution as observed. Experimental studies of MCRs on all DBD
nuclei are under progress ref18 .

The EQ/PEQ neutron emission code was developed for the neutron emission
following MCR on the present 100Mo with the large neutron excess and $Z$=42.
One has to include proton emissions as well in medium heavy nuclei with less
neutron excess since the proton binding energy gets lower relative to the
neutron one, and also in light nuclei where the Coulomb barrier gets lower.
The EQ/PEQ code with both neutrons and protons are being developed at UTM and
RCNP.

The present experiment is made on delayed $\gamma$-rays from RIs produced by
MCR to get the yields of the RIs since the $\gamma$-ray branching ratios are
well known. Then we get the gross structure of the strength distribution up to
$E\approx$ 60 MeV. The strength (response) to individual states below the
neutron threshold energy can be studied by measuring prompt $\gamma$-rays from
MCR.

The population of the $i$th state is in principle obtained from the difference
between the summed yield of all the $\gamma$-rays from the $i$th state and
that of all the $\gamma$-rays feeding the $i$th state from higher states.
However, in medium heavy nuclei with rather high level density, it is a
challenge to measure accurately all the $\gamma$-rays from and to the
individual states.

## References

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