Probing Dark Matter Axions using the Hyperfine Structure Splitting of Hydrogen Atoms

QCD axions can be a substantial part of dark matter if their mass $m_a\sim10^{-5}$eV. Since the axions were created by the misalignment mechanism, their local energy spectrum density is large. Consequently, the axion-induced atomic transition rate is enhanced if the atomic energy gap matches the axion mass. The hyperfine splitting between the spin 0 singlet ground state and the spin 1 triplet state of hydrogen is $0.59\times10^{-5}$eV, which is close to the preferred mass of dark matter axions. With an energy gap adjustment by applying a weak Zeeman magnetic field, dark matter axions can induce atomic hydrogen transitions. Furthermore, because the total spins of the hydrogen triplet and singlet differ, the axion-induced transitions are detectable by a Stern--Gerlach apparatus or a sensitive magnetic field detector. A potential realization of the proposed scheme can be similar to existing hydrogen masers.


I. INTRODUCTION
The existence of cold dark matter in our universe is generally accepted due to abundant astrophysical and cosmological evidence. Observations indicate that the energy density in our universe is dominated by 73% of dark energy followed by approximately 23% of dark matter. The ordinary matter made of the standard model particles accounts for less than 4%. Although the existence of dark matter is well established, its particular properties remain unknown.
The strong CP problem is one of the most important puzzles in particle physics today.
By assuming that the PQ symmetry broke after inflation and the resulting axions constituted the majority of dark matter (DM), an important DM axion window is the PQ scale f a ∼ 10 11 GeV and the axion mass m ∼ 10 −5 eV [24][25][26][27]40]. This is often called the classical window. There is an additional anthropic window for QCD DM axions if the PQ symmetry was broken before cosmological inflation. In the latter scenario, the anthropic selection resulted in a small initial misalignment angle; consequently, the PQ scale can be much larger than the prediction in the classical window. This scenario is constrained by cosmic microwave background (CMB) observations and is consistent with the low-scale inflation model [26]. Typically, if H I < 10 10 GeV, f a ≳ 10 14 GeV and m a ≲ 10 −7 eV. Some recent studies, such as [53,59,61], relax these constraints. If one also considers axion-like particle dark matter, the theoretical mass range is even larger.
There are many proposed and ongoing experimental studies searching for axions due to the uncertainties of axion mass [18-20, 22, 28-33, 35-39, 41-50, 52, 54-58] etc. In this paper, we propose to use splitting of the 1S state of hydrogen atoms to probe the DM axions.
Axions couple to fermions. Because of the high energy spectrum density of the DM axions, resonant atomic transitions are greatly enhanced compared to the nonresonant effects. This phenomenon has been considered in [33] in a general way, and [38] later suggested using molecular oxygen to search axions with a mass of approximately 10 −3 eV. Interestingly, the hyperfine splitting of the hydrogen 1S state is 0.59 × 10 −5 eV (see FIG.1); therefore, it could be matched with the preferred dark matter QCD axion mass by applying a small external magnetic field. The magnetic moment of the hydrogen atoms can be detected as the sign of atomic transitions. Because a flip of either the proton or the electron causes the transition of the states, this scheme works for the KSVZ axions as well. Thus, using the hyperfine splitting of atomic hydrogen allows us to simultaneously probe both the axion-electron and axion-proton couplings with a single transition compared to just the axion-electron couplings in earlier proposals. The induced quantum transitions could be counted with a Stern-Gerlach apparatus or a maser-like device (see FIG. 3,4,5). The anthropic window of QCD axions or axion-like particles can also be explored by splitting the 1S triplet state (see FIG. 2).

II. QCD AXION MASS CONSIDERATIONS
If the QCD axions were created by the misalignment mechanism in the early universe and compose the majority of dark matter, the preferred mass range can be theoretically constrained. Certainly, some mechanisms, such as [59], could result in a larger mass window.
After the Peccei-Quinn symmetry breaking, the equation of motion of the axion field a in the Friedmann-Robertson-Walker (FRW) universe is where R is the scale factor, H =Ṙ/R is the Hubble parameter, and V (a) is the potential of the axion field. The potential depends on the temperature T of the background and can be written as follows: where Λ Q ∼ 200MeV, and b ∼ O(0.01) depending on the particular axion models. When T ≲ Λ Q , the axion mass is Because the Hubble parameter H was large in the early universe, the field potential V was negligible. Consequently, the initial misalignment angle θ 0 = a 0 /f a was frozen. The axion field started to oscillate when the Hubble parameter dropped to H ≈ m a (T osc )/3, at which point the field potential was no longer negligible. Thus, the axion energy density at that time wasρ where < θ 2 0 > is averaged over horizon so it gives rise to an order of one number. Subsequently, the energy density decreased as matter. Assuming the axions are the major part of dark matter, the implied parameter window is If PQ symmetry breaking occurred before inflation, due to anthropic selection, f a could be significantly larger than in the previous scenario. The presence of the axion field during inflation generated isocurvature perturbations, which are constrained by the CMB [26], but recent studies, [53], etc., showed some relaxations. A possible DM QCD axion window for the low-scale inflation scenario, H I ≲ 10 10 GeV, is Other production mechanisms and axion-like particles generally could have a much more relaxed mass constraint, but these two windows draw great interest due to their elegance and simplicity.

III. DARK MATTER AXION-INDUCED QUANTUM TRANSITIONS
At the laboratory scale, the dark matter axions can be considered free steaming. Therefore, they satisfy the Klein-Gordon equation: In addition, because the cold dark matter particles are nonrelativistic, the axion field can be written as [51]: where α j is a random number of the Rayleigh distribution P (α j ) = α j e −α j /2 , f (v j ) is the local DM speed distribution, v j ≪ c is the local velocity relative to the laboratory, and ϕ j is a phase factor. f (v j ) is different depending on the particular halo models but is generally very sharp, so the DM axions can effectively be considered a mono-frequency field with a small frequency spread in experimental searches.
The averaged field strength:ā 0 ≈ √ 2ρ CDM /m a , where ρ CDM ≈ 1GeV/cm 3 is the local dark-matter energy density. The local axion velocity v ≈v j depends on the dark matter halo structure and the relative position of Earth. Assuming that the dark matter particles did not lose energy during the formation of galactic halos, their speed can be assumed to be approximately equal to the speed of the Sun, i.e., v ∼ 10 −3 c.
The local dark matter energy spectrum density is where δv is the velocity spread. The typical estimation is T a is the effective dark matter temperature. Some authors suggest a lower dark matter temperature, which leads to δv ∼ 10 −7 c [34].
The axions couple to electrons and protons via: where ψ is the fermion field. f = e, p refers to electrons and protons, respectively. In atoms, electrons and protons can be regarded as nonrelativistic. Therefore, where p f and σ f are the momentum operator and the spin operator of the fermions, respectively. For the atomic transitions, the first term is subdominated compared to the second term. One could find this by exploring the commutator ⃗ where H is the atomic Hamiltonian; Please see [35] Eq.(11)- (14) for details. Because r · ⃗ σ|B⟩m f , the first term in the bracket of Eq. (12) is proportional to m 2 a a 0r , wherer ∼ 10 −11 m is the Bohr radius. Thus, m 2 a a 0r < 10 −10 m a a 0 when m a < 10 −5 eV. The second term is proportional to vm a a 0 > 10 −3 m a a 0 . Therefore, the first term can be dropped.
In addition, the wavelength of the axions is λ = 2πm −1 a , which is much larger than the Bohr radiusr of atoms. Consequently, Eq.(12) becomes where ω a = m a (1 + v 2 /2) is the energy of the axions. When the energy gap between the atomic states matches the energy of the axions, the induced transition rate is Eq. (14) can only be applied when the initial atomic state |i > has a lifetime longer than the axion oscillation time 2π/m a , which is true for the hydrogen 1S state. The resonant transitions also require a match between the transferred energy and the atomic energy gap, which can be realized by using the Zeeman effect (see Figs. 1, 2). For the classical window, let us consider the energy splitting between the singlet state and the triplet state: = (B/Tesla) × 11.6 × 10 −5 eV + 5.9 × 10 −6 eV , A recent work estimated that f a ∼ 10 15 GeV [60]. Then, the external field should be approximately 0.001 T. Note that when the Zeeman field is very weak, the |1, 1 > and |1, −1 > splitting is almost equal in energy; thus, |1, 0 > transitions to both states due to axion DM energy spectrum spreading. The transition rate Eq. (14) is then doubled for the anthropic window searches.
The mass range that can be scanned is limited by the available strength of the Zeeman field. For current technology, the classical window can be fully covered, and the anthropic window is limited by nT, which is approximately 10 −14 eV∼ 10 −7 eV.
The axion models predict that at least one of g e and g p is on the order of one. Therefore, The event rate is then For δv ∼ 10 −3 c, m a ∼ 10 −5 eV, f a ∼ 10 11 GeV, and N ∼ 1mole, the event rate is 2.0 s −1 . If δv ∼ 10 −7 c [34], only 10 −8 mole atoms are required to achieve a similar event rate. We see that a smaller mass can partially nullify the increase in f a .
The major source of noise using this approach is the thermal excitation of atomic states; then, the optimal temperature T o satisfies where ∆E ≈ m a . If m a ≈ 10 −6 eV, we have T o ≈ 20mK. When the temperature is higher than the optimal temperature, the detection requires a longer integration time that satisfies: where R n is the thermal-induced transition rate.

IV. POSSIBLE EXPERIMENTAL SET-UPS
The setup of the experiment can be very similar to the original experiment demonstrating the Zeeman splitting transitions (see Fig.3), where in addition to the Zeeman field, a small varying magnetic field that varies at GHz was applied. A major difference is that in the proposed experiment, the dark matter axions stimulate the transitions; therefore, a varying field is not needed. On the other hand, a hydrogen maser-like device with some modifications could be ideal for the experiment (see Fig.4). The molecular hydrogen is dissociated in the wherev H = 3kT /m is the rms velocity of atoms, l is the mean distance between collisions, P is a factor counting the reaction percentage of collisions, which is usually taken as O(0.1), and T is the storage bulb temperature. The activation energy E a depends on the chemical reactions of the atom and the bulb wall [1], which is typically much higher than the thermal energy of atoms at the proposed experimental temperature.
Atomic State Relaxations: Ideally, if the magnetic field in the storage bulb can be perfectly uniform, it will not induce atomic transitions. However, if there is a nonuniform part of the magnetic field, atomic motion induces transitions. Therefore, some counts should be taken if the atom leakage is large. The |0, 0 > state is much less perturbed by this phenomenon, while the |1, 0 > state contributions dominate. The relaxation rate depends on the nonuniformity of the field, the storage bulb geometry, and the atomic speedv H , which is small after reaching the proposed atomic temperature. In the limit of a low strength Zeeman magnetic field, which is the case in the proposed setup, this relaxation rate is typically much smaller than the axion-induced transition rate ∼ O(Hz).
In addition, if there are different spin states in the storage bulb, hydrogen-hydrogen spin-exchange collisions can contribute to atomic state relaxation. The experimental and theoretical analyses predict that this effect is density related [1]. This decay rate is ∝ 10 −10 N sec −1 , so several stages of the state selection filter are preferable to filter the different spin states before they enter the bulb. and ω is the center of the transition frequency. The value of Q H depends on the particular experimental setup but remains constant during the integration time. As long as it is higher than the axion DM quality factor Q a = 1/δv 2 a ∼ 10 6 , the sensitivity will not be reduced.

V. SCANNING AND SENSITIVITY
Because the exact value of the axion mass is unknown, one needs to scan the interested mass range. This can be done by turning the Zeeman magnetic field B. Assuming to scan ∆f ≡ ∆m/2π in a single working year, the magnetic field is tuned as To cover approximately O(10 −5 )eV mass range, one needs to scan approximately several GHz. To cover a portion of the anthropic axion mass window, e.g., m a ∼ 10 −8 eV, the scanning bandwidth is approximately several MHz. Since the bandwidth of the DM axion wave is ∆f a = 1 2 m a δv 2 , the effective integration time t int = ∆f a /R scaning is t int = 3.8 × 10 7 s( m a 10 −5 eV )δv 2 ( GHz ∆f ) .  Subsequently, the atoms enter a storage bulb surrounded by a tuneable Zeeman effect magnet.
Once inside the bulb, the atomic energy gaps are matched with the axion mass. Some atoms are resonantly excited to the m j = 1 state, which is detectable by the magnetic field detector.
Assuming operating at the optimal temperature and high confidence event detection, 3σ detection requires N Rt int ≥ 3, which leads to:

VI. CONCLUSION
The QCD axion is a well-motivated dark matter candidate. It could provide information about the ultrahigh-energy new physics, f a ≳ 10 11 GeV. It may also generate rich cosmological phenomena, for example, implications for the inflation Hubble scale H I . Thus, it is crucial and promising to pursue direct laboratory searches of axions or axion-like particles.
Nevertheless, the construction of experimental probes is rather challenging because the axions only weakly couple to the standard model particles, and the energy carried by each axion is very small. Therefore, the axion interactions are weak in terms of energy. spin instead of the transferred energy might be more straightforward.