Direct WIMP detection rates for transitions in isomeric nuclei

The direct detection of dark matter constituents, in particular the weakly interacting massive particles (WIMPs), is central to particle physics and cosmology. In this paper we study WIMP induced transitions from isomeric nuclear states for two possible isomeric candidates: $\rm^{180}Ta$ and $\rm^{166}Ho$. The experimental setup, which can measure the possible decay of $\rm^{180}Ta$ induced by WIMPs, was proposed. The corresponding estimates of the half-life of $\rm^{180}Ta$ are given in the sense that the WIMP-nucleon interaction can be interpreted as ordinary radioactive decay.


I. INTRODUCTION
At present there are plenty of evidences of dark matter (DM) from i) cosmological observations, the combined MAXIMA-1 [1], BOOMERANG [2], DASI [3], COBE/DMR Cosmic Microwave Background (CMB) observations [4,5], as well as the recent WMAP [6] and Planck [7] data and ii) the observed rotational curves in the galactic halos, see e.g. the review [8].It is, however, essential to directly detect such matter in order to unravel the nature of its constituents.
In this paper, we will focus on the spin dependent WIMP nucleus interaction.This cross section can be sizable in a variety of models, including the lightest super-symmetric particle (LSP) [29,[32][33][34], in the co-annihilation region [35], where the ratio of the SD to to the SI nucleon cross section, depending on tan β and the WIMP mass can be large, e.g. 10 3 in the WIMP mass range 200-500 GeV.
Furthermore more recent calculations in the super-symmetric SO (10) model [36], also in the co-annihilation region, predict ratios of the order of 2 • 10 3 for a WIMP mass of about 850 GeV.Models of exotic WIMPs, like Kaluza-Klein models [20,21] and Majorana particles with spin 3/2 [37], also can lead to large nucleon spin induced cross sections, which satisfy the relic abundance constraint.This interaction is very important because it can lead to inelastic WIMP-nucleus scattering with a non a prospect proposed some time ago [38] and considered in some detail by Ejiri and collaborators [39].Indeed for a Maxwell-Boltzmann (M-B) velocity distribution the average kinetic energy of the WIMP is: So, for sufficiently heavy WIMPs, the available energy via the high velocity tail of the M-B distribution may be adequate [40] to allow scattering to low lying excited states of certain targets, e.g. of 57.7 keV for the 7/2 + excited state of 127 I, the 39.6 keV for the first excited 3/2 + of 129 Xe, 35.48 keV for the first excited 3/2 + state of 125 Te and 9.4 keV for the first excited 7/2 + state of 83 Kr.In fact calculations of the event rates for the inelastic WIMPnucleus transitions involving the above systems have been performed [41,42].However, these levels live for a very short time (much less than 1 µs), and are unsuitable for the expected long-term exposures to search for WIMPnucleus interactions.At the same time, there is a set of long-lived nuclear meta-stable states that can be artificially directly produced in reactors and accelerators and prepared in the form of samples for their long exposure under low-background experimental conditions.The possible candidates are listed in Tab.I Table I: Long-lived isomeric states with high spin differences that can be effectively produced and used for the DM search.Interest in the inelastic WIMP-nucleus scattering has recently been revived by a new proposal to search for the de-excitation of meta-stable nuclear isomers [43] after such collisions.The longevity of these isomers is related to a strong suppression of γ and β-transitions, typically inhibited by a large difference in the angular momentum for the nuclear transition.Collisional de-excitation by DM is possible since heavy DM particles can have a momentum exchange with the nucleus comparable to the inverse nuclear size, hence lifting tremendous angular momentum suppression of the nuclear transition.
In this work we consider very long-lived isomeric states that can be directly and efficiently produced, then (if necessary) chemically separated and prepared for long-term exposures.The lifetimes of these isomeric states must be preferably longer than the time of their many-years exposures to detect interactions with DM.The baseline of this research is placed on a thorough analysis of the possibility of observing the effect of the interaction of DM particles with isomeric states in 166 Ho and 180 Ta, of which the first can be obtained in a reactor, and the second is completely quasi-stable and can be separated from a mixture of tantalum isotopes in nature.To search for a signal from DM, the cryogenic microcalorimetry method can be used, which has shown its many times better efficiency and accuracy compared to the previously used semiconductor spectroscopy approach.
Given the requirement from existing work [43], this paper provides detailed calculations for the dedicated nuclear theory evaluation of WIMP induced transition.Meanwhile a new type of detection technique is proposed to perform the experiment.The article is arranged as below.In section II and III, the basic kinematics and cross section formula are laid out.In section IV, the general nuclear structure consideration is introduced.In section V and VI, the detailed cross section calculations for two different targets are shown and in section VII and VIII, an experimental proposal with sensitivity are given, followed by a discussion section in IX.

II. KINEMATICS
Evaluation of the differential rate for a WIMP induced transition A i iso (E x ) for an isomeric nuclear state at excitation energy E x to another one A f iso (E ′ x ) (or to the ground state) proceeds in a fashion similar to that of the standard inelastic WIMP induced transition, except in the consideration of kinematics.We will make a judicious choice of the final nuclear state that can decay in a standard way to the ground state or to another lower excited state: with χ the DM particle (WIMP).Assuming that all particles involved are non relativistic we get: where q is the momentum transfer to the nucleus q = p χ − p ′ χ and m A is the isomer mass.So the above equation becomes where ∆ > 0, ξ is the cosine of the angle between the incident WIMP and the recoiling nucleus, υ the oncoming WIMP velocity, µ r = m A mχ m A +mχ the reduced mass of the WIMP-nucleus system and E R the nuclear recoil energy.From the above expression it is immediately apparent that Thus we find the next condition This means that for −∆ + m A µr E R > 0 the allowed range of velocities is where υ esc is the escape velocity, the maximum allowed velocity.For −∆ + m A /µ r • E R < 0 the allowed region is At this point we should mention that in the standard inelastic scattering the region is not available.Note the difference in sign between the previous equation and Eq.( 6).As a result υ min increases with E x , which explains the suppression of the expected rates in the standard process as E x increases.Based on Appendix A, in the special case of WIMP-nucleon scattering the maximum recoil energy is given by where x = m N /m χ .Using υ 0 ≈ 0.7 • 10 −3 (in natural units) and y esc = 2.84 we obtain

III. EXPRESSIONS FOR THE CROSS SECTION
The differential cross section is given by where |M E(q)| 2 is the NME of the WIMP-nucleon interaction in dimensionless units and G F the standard weak interaction strength.Integrating over ξ by making use of the δ function we get New physics is contained in elementary nucleon interaction, so we prefer to parameterize in terms of the elementary nucleon cross section.In the case of the nucleon Eq.( 11) becomes Folding the last equation with the velocity distribution and integrating over the allowed recoil energies (see the Appendix C), we obtain IV. NUCLEAR STRUCTURE The microscopic structure of atomic nuclei is described in terms of the spherical shell model [44][45][46], introduced in 1949 in order to explain the magic numbers 2, 8, 20, 28, 50, 82, 126, . . ., at which nuclei present particularly stable configurations.The shell model is obtained from the three-dimensional isotropic harmonic oscillator, to which the spin-orbit interaction is added.It offers a satisfactory description of nuclei with few valence protons and valence neutrons outside closed shells, corresponding to the magic numbers, but it fails to explain the experimentally observed large nuclear quadrupole moments away from closed shells, where it has been suggested [47] in 1950 that spheroidal instead of spherical shapes lead to greater stability.Along this line, the collective model of Bohr and Mottelson [48,49] was introduced in 1952, in which departure from the spherical shape and from axial symmetry are described by the collective variables β and γ, respectively.Furthermore in 1955 the Nilsson model [50][51][52] was introduced, in which a cylindrical harmonic oscillator is used instead of a spherical one, characterized by a deformation ϵ, reflecting the departure of the cylindrical shape from spherical one.The single particle orbitals in the Nilsson model are labeled by Ω[N n z Λ], where N is the total number of the oscillator quanta, n z is the number of quanta along the z-axis of cylindrical symmetry, while Λ (Ω) is the projection of the orbital (total) angular momentum on the z-axis.
In what follows it will be of interest to consider the expansions of the Nilsson orbitals in the spherical shell model basis |N ljΩ⟩, where N is the principal quantum number, l (j) is the orbital (total) angular momentum, and Ω is the projection of the total angular momentum on the z-axis.The necessary expansions have been obtained as described in [53] and are shown in Tab.III and IV for three different values of the deformation ϵ.
An important remark on the expansions shown in Table III is in order.One can see that there is a basic difference between intruder orbitals (orbitals pushed within the spherical shell model by the spin-orbit interaction to the oscillator shell below) and normal parity orbitals (orbitals remaining in their own oscillator shell).Intruder orbitals remain concentrated on one spherical shell model vector at all deformations, while normal parity orbitals are concentrated on one spherical shell model orbital at small deformation, but will in general be distributed onto several spherical shell model orbitals at large deformations.The implications of this difference will become clear below.It should be mentioned that the "purity" observed in the case of the intruder orbitals is due to the fact that they do not mix with their normal parity neighbors, while the normal parity orbitals, of which there are more than one, do mix among themselves.

A. The nucleus 166 Ho
The even-even core of 166  67 Ho 99 is 164 66 Dy 98 , for which the experimental value of the collective deformation variable β is 0.3486 [54], thus the Nilsson deformation ϵ = 0.95β [52]  Let us now consider the formation of the above mentioned states under the light of the expansions of the Nilsson orbitals in terms of spherical shell model orbitals, shown in Tabs.III and IV.Both the proton 7/2[523] and neutron 7/2[633] orbitals are intruder ones, therefore they are mainly concentrated on the spherical shell model vectors |5 5 11/2 7/2⟩ and |6 6 13/2 7/2⟩ respectively, although other vectors with smaller coefficients also contribute, as seen in Tab.III and IV.

B. The nucleus 180 Ta
The even-even core of 180 73 Ta 107 is 178 72 Hf 106 , for which the experimental value of the collective deformation variable β is 0.2779 [54], thus the Nilsson deformation ϵ = 0.95β [52] is 0.2640 .
Several different theoretical calculations, including covariant density functional theory using the DDME2 functional [55,56], Skyrmre-Hartree-Fock-BCS2 [57], as well as a two quasi-particle plus rotor model in the mean field represented by a deformed Woods-Saxon potential [58] agree that the first neutron orbital lying above the Fermi surface of the core nucleus 178 72 Hf 106 is the 9/2[624] orbital, while the first proton orbital lying above the Fermi surface of the core nucleus 178 72 Hf 106 is the 9/2[514] orbital.Therefore it is safe to assume that these two orbitals will play a major role in the formation of the 9 − isomer state of 180 73 Ta.The question then comes from which orbitals the excited state 2 + may arise.The above mentioned covariant density functional theory using the DDME2 functional3 [55,56], calculations and Skyrmre-Hartree-Fock-BCS [57] calculations indicate that the last neutron orbital below the Fermi surface is the 5/2[512] orbital, while the last proton orbital below the Fermi surface is the 7/2[404] orbital.Then it is plausible that the 2 + excited state will come from combining the proton 9/2[514] orbital (the first orbital above the Fermi surface) with the neutron 5/2[512] orbital (the first orbital below the Fermi surface).
It is instructive to consider the formation of the aforementioned states under the light of the expansions of the Nilsson orbitals in terms of spherical shell model orbitals, shown in Tab.III and IV The orbitals participating in the formation of the 9 − isomer, proton 9/2[514] and neutron 9/2[624], are both intruder orbitals, thus the main contribution comes from the |5 5 11/2 9/2⟩ component of the former and the |6 6 13/2 9/2⟩ component of the latter.The orbitals participating in the formation of the 2 + excited state are the proton 9/2[514] (intruder) and neutron 5/2[512] (normal parity) orbitals, from which the leading contribution will come from the |5 5 11/2 9/2⟩ and |5 3 7/2 5/2⟩ vectors respectively.
V. CONSIDERATIONS OF THE 166 Ho TARGET So we will begin with the nucleus 166 Ho, which is well studied, see, e.g., [59].This is an odd-odd nucleus (Z=67, N=99), which is a deformed nucleus described in the Nilsson model by 7 2 − [523] for protons, deformed level associated with the spherical 0h level, and the 7 2 + [633] associated with the spherical level 0i for neutrons.It is convenient to express these states in a spherical basis in terms of a deformation [60].Thus for the purpose of our calculation it is sufficient to consider the expression with It is reasonable to assume that this odd-odd nucleus can be considered as a two particle system composed of one proton and one neutron in the above levels.Thus we get A. The nuclear matrix elements In spite of the fact that the coefficient C 0i 11/2 is small, the inclusion of the 0i 11/2 is mandatory to make the 0 − ground state wave function.The inclusion of the 0i 13/2 state with the large coupling is helpful but it can not lead to proton induced transitions.So one expects a suppression of NME.The operator for the transition has the structure T λ,J with rank J = 7 and orbital rank λ = J − 1, J, J + 1.The interaction cannot convert protons to neutrons or vice versa.So the λ = J = 7 is excluded by parity conservation.Thus λ = J = 6, 8, i.e.only the spin induced transitions are allowed.As a result in this case the ratio of the NME divided by the corresponding one for the nucleon can be cast in the form: The function R 2 M E(q 2 ) is independent of the scale, but it does depend on the ratio f A /f V via the coefficient The vector current does not contribute in this case, but the first line in this expression comes from the explicit dependence of the cross section on the couplings (see Eq.(B12)).In evaluating the NMEs one needs the reduced matrix element Using standard Racah techniques [61] one can obtain Eq.(D5) of the Appendix D. A detailed explicit calculation reduced matrix elements in the case of 166 Ho is given in section D 1.
After that one can incorporate into the reduced matrix element the form factor associated with in each orbit, obtained via the corresponding radial integrals of the spherical Bessel j λ (qr) finding this way the single particle form factor for each orbit, see Appendix E.
Let us now consider the allowed range of the momentum transfer.As we can see the minimum velocity must be smaller than the escape velocity, see Eq.B5.For a M-B distribution see Eq.(B10) with y i = υ i /υ 0 , i = 1, 2. This momentum dependence is exhibited in Fig. 1.The NME R 2 M E(q 2 ) 0 is exhibited in Fig. 2. We observe that it is greatly suppressed.This may be surprising in view of the fact that all single particle form factors involved are much larger, see Fig. 3 and take its square.These form factors, however, are much smaller the nuclear form factors encountered in the case of the standard WIMP searches involving the elastic WIMP-nucleus scattering, see Fig .4a.
This suppression appears to be mainly due to the geometric factors involved in the reduced matrix elements, i.e. nine-j symbols, Racah functions etc as well as due to smallness of the coefficient Ci 11/2 .Indeed the total NME can be written as From the above reduced ME we find The corresponding shell model single particle form factors are exhibited in Fig. 3.There appears to be some cancellation due to their size, but it is mainly a consequence of the fact that all the coefficients C ℓ,λ are small and not of the same sign.Furthermore recall that, after incorporating the statistical factor for 166 Ho, which essentially is the ratio of the NME divided by the corresponding one for the nucleon, with the factor removed.The part of the space above q max = (65.6,233, 349, 464, 579 is not allowed for the case of m χ = (0.1, 0.5, 1, 2, 5)m A respectively (see Fig. 1 and the text for details).In addition it is also likely that the single particle form factors in the shell model are suppressed, or at least much smaller than implied by the scale set by the spherical Bessel functions j λ , see Fig. 4b.This suppression is not very significant, but ideally all fours single particle form factors could be determined experimentally.
For orientation purposes we will consider Helm like single particle form factors: This is an extension of the Helm-form factor [62], normalized so that it is unity for λ = 1 at q = 0.This expression for a = 0.6F and R = 6.87F is exhibited in Fig. 6.The NME using these form factors is exhibited in Fig. 5.There is now an improvement of two orders of magnitude, but the obtained result is still quite small.In the case of the Helm-like form factors proceeding as in the case of shell model one finds with These coefficients are also small.The negative sign in Eqs.( 19) and ( 22) leads to suppression of the NME.The numerical value of in Eq.B12 using Eq. ( 17) with f A /f V = 1, is 0.063 for A=166, expressed in units of keV −1 .The plot for 1 υ0 versus the previous one multiplied with 0.063.We prefer to express it as a function of E R in units of keV, i.e Fig. 7a.
The expressions for σ N , ϕ and R can be obtained using the relevant values for the nucleon: (kinematics factor), yielding (see the Appendix C): The same result holds for the obtained differential rate 1 since the WIMP density used in obtaining the densities is the same.The situation is, however, different if one is comparing the obtained differential rate relative to the total rate for the nucleon at some fixed value of the WIMP mass.Indeed the obtained differential rate relative to total rate of the nucleon for m N /m χ = 1 is exhibited in Fig. 7b.The exhibited differential rate contains, of course, the WIMP mass dependence arising from the WIMP density in our galaxy.
With such results the WIMP detection with the 166 Ho appears very problematic.

VI. CONSIDERATIONS OF THE 180 Ta TARGET
We begin by considering the transition of the isomeric 9 − state to the 2 + state.The momentum dependence of the cross section arising from the velocity distribution is different from that of 166 Ho, since the transition energy is ∆ = 37 keV.Thus the analog of Fig. 1 is given in Fig. 8.To proceed further we need to determine the structure of the target 180 Ta.As explained in section IV B in the context of the Nilsson model we can consider the proton orbital 9  2 [514] both in the initial state 9 − and the final 2 + .Furthermore for the neutrons we use 9  2 [624] for the 9 − and the 5  2 [512] for the 2 + .To proceed further we use the expansion of the Nilsson orbitals into shell model states found in Tables III and IV for deformation parameter 0.30.Note that in this case only the neutrons can undergo transitions, while the protons are just spectators.In the case of the shell model one encounters 8 transition types with odd multi-polarities.

A. Shell model form factors
The vector and axial vector reduced NMEs can be obtained using Eq.(D4) where the quantities with subscript 1 indicate neutrons and those with 2 are associated with protons.Thus we find: In the above expressions F (ℓ, ℓ ′ , λ) are the single particle form factors.The first two integers indicate orbital angular momentum quantum numbers ℓ, ℓ ′ , while the last integer λ gives the multipolarity of the transition.The quantity u corresponds to b N q, where b N is the harmonic oscillator length parameter.The NMEs have been normalized as above in the case 166 Ho, see Eq.( 17), with a compensating factor of f A appearing explicitly in the cross-section, Eq.B12.The relevant form factors are exhibited in Fig. 9.
The relevant nuclear ME is given by: Its momentum dependence is exhibited in Fig. 10a.We should note that the large value of the matrix element in the case of large f V is due to the normalization adopted to make the matrix element independent of the scale.Recall that the corresponding factor appears in the cross section.In the present work we will adopt f V = f A .

B. Phenomenological form factors
It is generally believed that the shell model single particle factors lead to large suppression, so phenomenological form factors may be preferred.One example is the the Helm form factor, see Eq.( 20).This has already been employed in the case of even transitions.We will employ here for odd (parity changing) transitions.Our treatment means that the radial integrals are independent of the angular momentum quantum numbers ℓ, ℓ ′ .The obtained results are exhibited in Fig. 10b.The reduced matrix elements for the vector and the axial vector are: RM EH A = 3.58938F 11 (a, q, R) + 6.46292F 7 + 7.80066F 9 (a, q, R) where F λ are the Helm single particle form factors.The NME is: The momentum dependence of this ME is exhibited in Fig. 10c C. Some results for 180 Ta The numerical value of in Eq.(B12) using Eq.( 17) with f A /f V = 1, is 0.068 for A=180, expressed in units of keV −1 .The plot for 1 υ0 versus the previous one multiplied with 0.063.The preference is to express it as a function of E R in units of keV, i.e Fig. 11a.It can be shown that a similar expression holds for the rate see Fig. 11b.The expressions for σ N and R can be obtained using the relevant values for the nucleon: in units of keV −1 for the Ta target.The black curve in the drawing has been reduced by a factor of 5, so the related rate must be multiplied by 5.The Helm type form factor has been employed.It also contains the WIMP mass dependence arising from the WIMP density in our galaxy.
One can integrate the differential cross section over the recoil energy E R and multiply with the total nucleus mass to obtain the WIMP-Nucleus cross section as a function of the WIMP mass m χ this is exhibited in Fig. 12.The dominant source of uncertainties for the cross section is the NME.The multipolarities of high order and the momentum transfer introduce around a 30% error.The escape velocity contributes 10%.In addition, there is an uncertainty in the model parameters given in Tabs.III, IV of about 10%.This, in turn, gives the total uncertainty as the square root of the sum of quadratures, approximately 33.2%.
The same result holds for the obtained differential rate since the WIMP density used in obtaining the densities is the same.The situation is, however, changed if one is comparing the obtained differential rate relative to the total rate for the nucleon at some fixed value of the WIMP mass.Indeed the obtained differential rate relative to total rate of the nucleon for m N /m χ = 1 is exhibited in Fig. 11b.The exhibited differential rate contains, of course, the WIMP mass dependence arising from the WIMP density in our galaxy.
One can integrate the differential rate over the recoil energy E R and multiply with the total number of nucleons to obtain the the total WIMP-Nucleus event rate as a function of the WIMP mass m χ .

VII. EXPERIMENTAL APPROACH FOR THE DARK MATTER SEARCH
Calculations performed in previous chapters have been dedicated to the most prominent candidates, which were taken from the list given in Tab.I.The DM collision with the isomeric state has an exceedingly small cross section.Therefore, the isomeric states with longer half-lives are favored due to the smaller contribution of natural decay to their total decay rates.The 180m Ta is an outstanding candidate for investigation.As mentioned above this nuclide has been proposed [43] and treated by gamma-ray spectrometry with HPGe-detectors [63].The expected gamma-lines of the direct isomer decay and the further 180 Ta ground state decay (half-life 8.1 h) are shown in Tab.II.The decay energies are precisely determined thanks to the recent accurate measurements of excitation energy of the isomeric state by the Penning trap mass spectrometry: 76.80 (33) keV [64].The energy of transitions expected as a result of the decay of the isomer to the ground state and the daughter nuclides of 180 W and 180 Hf are shown in Tab.II.Additionally, the values of the internal electron conversion coefficients indicated in the fifth column of Tab.II.Only the gamma-transition with energy 103.5 keV has been used to search for possible response to DM [63].The 93.3 keV line from the electron capture is too similar to the background lines from 234 Th.The 103.5 keV gamma-line with   the branching ratio 15% registered with the detector efficiency < 0.3% belongs to de-excitation of daughter nuclide 180 W.This results in a non-observation of any signal from the DM interaction obtained with HPGe-detector with total photon registration efficiency of < 4 • 10 −4 .Meanwhile, Low Temperature Detectors (LTD) are widely used to search for rare events in nuclear and particle physics.Great success was achieved with Magnetic Micro Calorimeters (MMC) [65,66] that can measure particle and photon energy with detection angle coverage close to 4π.The use of such detectors will increase the sensitivity of recording rare events by many orders of magnitude in comparison with the conventional germanium detector used in [63].In addition to the angular advantages in such detectors, it is possible to efficiently register the transition energies transmitted by internal conversion electrons, which considerably prevail in the decays of isomeric states with low transition energies.Another very important advantage is the energy resolution, which exceeds the extreme limits of any semiconductor detectors.The MMC-cryogenic detector consists of a metallic absorber that stores the released energy and has a very good connection to the temperature sensor -a paramagnetic alloy, which resides in a low magnetic field.The sensor is weakly connected to a thermal bath.The energy released in the absorber from a radioactive source completely enclosed in the absorber raises the temperature of the detector.This leads to a change of the sensor magnetization which is read out as a change of flux by SQUID magnetometer.Such type of an MMC has been used in the spectra measurement of 163 Ho within the ECHo project [65].The energy resolution on the level of 1.2 eV was achieved at X-ray energy of 6 keV.The same detector can be used to search for DM-particles with interaction to isomeric state of 166m Ho (see Tab. I).This isotope can plentifully be produced in reactors by neutron irradiation of 165 Ho.However, the small cross section for the DM-scattering obtained in our calculations for 166m Ho makes this detection unrealistic due to the strong background contribution of events from natural 166m Ho radioactive decay.
We emphasize again that 180m Ta is a promising candidate because its extremely long lifetime has very small background from its natural radioactive decay.Unlike the germanium detector, the MMC-technique allows measurement of all decay channels: characteristic X-ray and gamma-radiation and equally importantly, peaks from internal con-version electrons, which prevail in the spectrum of low-energy transitions, as can be seen in the fourth column of Tab.II.The last column of the same table shows intensive expected electron energies of these transitions.The most intensive electron spectrum belongs to the energy region below 100 keV.The most prominent characteristic K X-rays for Ta and Hf nuclides that follow the electron conversion process in the interval from 64.4 to 67.0 keV are beyond the electron conversion spectra and can be also detected by the MMC-detector with high precision of approximately 100 eV.Thus, in comparison with the germanium detector used in the paper [63], the MMC-method can provide plenty of indicators of DM interactions with the Ta-isomeric state.As a matter of fact some careful elaboration concerning the long-term stability in vacuum and maintenance at low temperature, as well as the physicochemical properties of Ta can be taken into account.
The 180m Ta isomeric state is the only one in nature, having the abundance of 0.012% in natural tantalum.Isotopic extraction of tantalum is very difficult.However 3 mg of isomer diluted in 30 mg of 181 Ta was already used in [67].Hopefully, a similar target is achieved by our proposed experiment instead of the method of [63] in which 180 mg of 180m Ta in 1.5 kg of natural tantalum is used.The total achievable mass of 3 mg for 180m Ta corresponds to N N ≈ 10 19 and this value is later used for the half-life calculation in VIII.
Another possibility to produce 180m Ta is the reaction (p, pn) on 181 Ta.Readout system can use SQUID microwave multiplexing [65].The natural background will be approximately 1 event per year if the expected total lifetime of this isomer is 10 19 year [68].As can be seen from these estimates, the observation of the DM-effect is very challenging.It becomes more realistic if the cross section attributable to DM de-excitation and/or the natural lifetime of the isomeric state deviate for two-three orders of magnitude from used one.Such deviations are entirely possible.As long as no signature is observed in the long term experiment, limits can be constrain the DM de-excitation.However, if the 180m Ta decay is observed, then measurements with increased exposure time in deeper underground locations may be indicative of a DM response.

VIII. HALF-LIFE TIME ESTIMATION
The 180 Ta isomer is predicted to decay to the ground state, but the half-life has not yet been determined.There are only experimental and theoretical upper limits available for this isotope, since decay has never been observed in a real experiment.The expected half-life time should exceed 10 19 years [68].And the lower experimental bound for the total half-life time is 4.5 • 10 16 years (90% C.L.) [69].The interaction with the WIMP and subsequent decay can be interpreted as normal decay, and from this the half-life time can be estimated.
As shown in the previous section, the overall background is one event per year.Since all DM searches do not observe any events (signal is zero), statistical evaluation can be simplified using the conventional Feldman-Cousins approach [70].The expected number of signal events n exp can be written as where Φ WIMP flux, σ tot total cross-section of WIMP isomer interaction, T exposure time, N N number of isomer atoms, ε detection efficiency.With zero signal this n exp can be associated with a particular upper limit n up .For instance in our case the 95% C.L. n up equals 2.33 (signal zero, background one for 1 year exposure [70]).
From another side, signal can be interpreted as normal radioactive decay that follows the exponential decay law.In this case n exp can be expressed as with expected half-life time T 1/2 .Combining Eq. ( 26) and Eq. ( 27), we can express T 1/2 as a function of T As can be seen, the T 1/2 does not depend on the amount of isomer and detection efficiency, because in our case the cross-section is already determined from theory.Based on that, we can estimate the sensitivity of the proposed experimental setup to the WIMP-nucleon interaction, where the last one can be interpreted as normal radioactive decay.In Fig. 13 the half-life time of 180m Ta as function of WIMP mass is depicted.Several levels of the experimental sensitivity with different exposure time are also shown.We can conclude that the WIMP-nucleon interaction can be measured with 95% C.L. for masses ≤ 130 GeV (10 years exposure time is assumed).The conservative value of the detector efficiency was chosen as 1%, and the result depends linearly from ε, hence it can be easily scaled for higher values of ε.

IX. DISCUSSION
We have seen that, not unexpectedly, the nuclear ME encountered in the inelastic WIMP-nucleus scattering involving isomeric nuclei is much smaller than that involved in the elastic process considered in the standard WIMP searches.This occurs for two reasons: a) the form factor in the elastic being favorable, see Fig. 4a, and b) in the elastic case the cross section is proportional to the mass number A 2 .In the present case the NME for 180 Ta, as indicated by the coefficients appearing in Eq. (24), is not unusually small compared to other typical inelastic processes.The Nilsson model is expected to work well in the case of 180 Ta, but the obtained event rate is quite small.At the same time, our calculations demonstrated a significant suppression in the matrix element for 166 Ho, that allowed us to rule out it from the list of candidates for experimental proposal.
Following the estimated WIMP-nucleon cross section and the current mass of 180 Ta isomer (N N ≈ 10 19 ), the estimated half-life time is between 10 15 and 10 18 years, covering current experimental limit 4.5 • 10 16 years (90% C.L.) [69].However it is still not enough to reach the expected half-life time for 180m Ta.That is why various experimental approaches are required to disentangle this puzzle.Further improvement can be achieved using larger mass of isomer with better detection efficiency.At the same time, the experiment can exploit the signal provided by the subsequent standard decay of the 2 + state to the ground state, that is not available in the conventional WIMP searches.
The above integrals can by computed analytically where The functions ψ i (y i , y esc ) depend on the momentum transfer.This depends on the specific nuclear target and will be discussed below.The expression contained in the last square bracket is momentum dependent and provides a restriction in the range and distribution of momentum.It is given by Figs. 8 and 1 for the nuclei of interest in this work.
Sometimes is useful to modify the above formulas using dimensionless variables.let us define η = qR, where is the nuclear radius.Then where the functions ψ 1 and ψ 2 are given by Eq.(B10) via Eq.(B11).
In shell model calculations instead of the nuclear radius R one may use the harmonic oscillator size parameter b A (see Appendix E).There remains the crucial part of the calculation in involving the NME and the associated nuclear form factor.

Appendix C: The nucleon cross section and event rate
We have seen that Eq.( 12) is correct, but it does not indicate the range of q involved.To find it we return to the basic expression q 2 2mu2 − qυξ = 0.This leads to This implies that Folding expression of Eq.( 12) with the velocity distribution, we obtain We must now integrate over the recoil energy from zero to (E R ) max given by Eq.( 9) ⟨σ N (υ)⟩ = υ 0 (4m N υ 2 0 ) Spϕ(x), ϕ(x) = (E R )max 0 dE R ψ(y min , y esc ). (C7) The function can only be obtained numerically.It is exhibited in Fig. 14.It is clear that for values of m χ much larger than m N the cross section becomes independent of m χ .Adopting the value ϕ(x) ≈ 1 we obtain: The WIMP-nucleon interaction is not known.Let us assume that it is of V − A type in dimensionless units.Then U [ℓ i , 1/2, j i , λ, s, J, ℓ ′ i , 1/2, j ′ i ]⟨ℓ i , 0; λ, 0|ℓ ′ i , 0⟩⟨n ′ i ℓ ′ i |j λ (kr)|n i ℓ i ⟩, i = 1, 2 (D6) A detailed explicit calculation reduced matrix elements in the case of 166 Ho is given in section D 1.
1.The explicit calculation reduced matrix elements in the case of 166 Ho Using standard Racah techniques one finds [61]:

Figure 1 :
Figure 1: The allowed momentum distribution (defined in Appendix B) arising from the maximum allowed velocity (escape velocity) of the distribution for 166 Ho.Different line colors correspond to the WIMP masses m χ = (0.1, 0.5, 1, 2, 5)m A

Figure 3 :
Figure 3: The shell model single particle form factors encountered in the case of a WIMP induced nuclear transition with ∆J = 7 for 166 Ho, with λ is the orbital rank of the operator (multi-polarity).

Figure 4 :
Figure 4: (a) The shell model form factor encountered in the case of a WIMP induced elastic nuclear transition for 166 Ho and the same with the Helm form factor.The NME in this case is obtained by multiplying this form factor with the number of nucleons in the nucleus, here A = 166.(b) The function j λ (qR) for an A=166 nucleus in the range of q of interest in the present work for the appropriate values of λ.

Figure 5 :
Figure 5: The same as in Fig. 2 obtained with Helm type single particle form factors.The restrictions on the allowed momenta are the same as in Fig. 2.

Figure 6 :
Figure 6: The Helm form factors F F H λ (q) for λ = 6, red line, and for λ = 8, black line, in the case of 166 Ho and momentum transfer of interest in the present work.

Figure 9 :
Figure 9: The form factors for different F are exhibited.

Figure 10 :
Figure 10: (a) The momentum dependence of the expression R 2 M E (q 2 ) is exhibited as a function of q for different values of f V .(b) The Helm type form factors for λ = 7, λ = 9 and λ = 11.(c) The momentum dependence of R 2M EH (q 2 ) as a function of q for different values of f V .Helm form factors have been used.

Figure 12 :
Figure 12: The total WIMP-Nucleus cross section as a function of WIMP mass for the target nucleus 180m Ta.

Figure 13 :
Figure 13: The solid red line is the half-life time as a function of the WIMP mass for 180m Ta.Dashed lines indicate 95% C.L. sensitivity for the proposed experimental setup with different exposures.Assumed parameters for calculating sensitivity ε = 0.01, N N = 10 19 .

Figure 14 :
Figure 14: The dependence on the function ϕ(x) on the mass m χ appearing in the case of the nucleon-WIMP scattering

Table II :
Transition energies that can arise from the decay of the 9 − isomeric state of 180m Ta.

Table IV :
Expansions of proton Nilsson orbitals Ω[N n z Λ] in the shell model basis |N ljΩ⟩ for three different values of the deformation ϵ.