New Constraints on the Axion–Electron Coupling Constant for Solar Axions

The resonant excitation of the ⁸³Kr first excited nuclear level (E = 9.4 keV) by solar axions whose fluxes depend on the axion–electron coupling constant g_Ae is sought. The γ- and X-ray photons and the conversion and Auger electrons from the excited-level relaxation are detected with a gas proportional counter of a low-background detector in the underground Baksan Neutrino Observatory (Institute for Nuclear Research, Russian Academy of Sciences). As a result, a new constraint ${\text{|}}{{g}_{{Ae}}}(g_{{AN}}^{3} - g_{{AN}}^{0}){\text{|}}$ ≤ 1.50 × 10^–17 (90% C.L.) has been obtained for the axion–electron and axion–nucleon coupling constants, which corresponds to new constraints on the axion mass m_A ≤ 320 eV and m_A ≤ 4.6 eV in the KSVZ and DFSZ axion models, respectively.

1 INTRODUCTION

Light pseudoscalar particles, axions, were introduced in the theory to solve the CP problem of strong interactions [1–3]. Despite negative results of intensive experimental searches for axions, they are still well-justified candidates for the dark-matter constituents [4]. The anomalous transparency of the Universe to high-energy γ rays [5] and the overly fast cooling of some star systems compared to the predictions of theoretical models [6] can be treated as promising astrophysical indications of the existence of axions.

The interaction of axions with matter is specified by the quantity f_A at which the Peccei–Quinn symmetry [1] is broken and is determined by the effective axion–photon (g_Aγ), axion–electron (g_Ae), and axion–nucleon (g_AN) coupling constants. The axion mass m_A and the parameter f_A are related to the respective characteristics of the ${{\pi }^{0}}$ meson as ${{m}_{A}}{{f}_{A}} \approx {{m}_{\pi }}{{f}_{\pi }}$. The corresponding expression for m_A in terms of f_A has the form [7, 8]

$${{m}_{A}} = 5.69(5)\left( {\frac{{{{{10}}^{6}} {\text{GeV}}}}{{{{f}_{A}}}}} \right) {\text{eV}}.$$

(1)

The original PQWW model of the “standard” axion [1–3] implies that PQ symmetry is broken at the electroweak scale ${{f}_{A}} = (\sqrt 2 {{G}_{F}}{{)}^{{ - 1/2}}} \simeq 250$ GeV. The constant f_A in two classes of new models KSVZ [9, 10] and DFSZ [11, 12] of the “invisible” axion can be arbitrarily large up to the Planck mass m_P ~ 10¹⁹ GeV, thus reducing both the expected axion mass and the interaction of the axion with matter.

Stars should be intense sources of axions. Intense fluxes of axions can be formed in the Sun in a number of processes whose probabilities depend on the axion coupling constants g_Aγ, g_Ae, and g_AN. The constant g_Aγ specifies the probability of conversion of photons to axions in the electromagnetic field of the solar plasma (Primakoff axions). The constant g_AN determines the emission of axions in nuclear magnetic transitions that are thermally excited at high temperatures in the center of the Sun or appear in nuclear reactions of the рр chain and CNO cycle. The constant g_Ae specifies the axion fluxes from bremsstrahlung $e + Z \to Z + e + A$ and Compton process $\gamma + e \to e + A$, as well as in discharge and recombination processes in atoms $I\text{*} \to I + A$, $e + I \to I{\text{*}} + A$. The spectra and fluxes of axions appearing in the above processes were calculated in [13–17] and are shown in Fig. 1.

Fig. 1.

(Color online) Energy spectrum of solar axions produced in (1) bremsstrahlung and (2) Compton processes and (3) sum of 1 and 2 including atomic processes as calculated with g_Ae = 10^–11.

Here, we report new results for the axion–electron coupling constant g_Ae determined from the complete set of data obtained in experiments on the search for the resonant absorption of solar axions by ⁸³Kr nuclei with a large krypton proportional counter [18, 19]. The cross section for the resonant excitation of a nuclear level depends on the constant g_AN; consequently, the rate of absorptions of axions in an experiment depends on the product of the constants g_Ae and g_AN.

The dimensionless coupling constant g_Ae in the DFSZ model is expressed in terms of the parameter f_A, which determines the axion mass m_A, and the free parameter cos²β:

$${{g}_{{Ae}}} = (1{\text{/}}3)\mathop {\cos }\nolimits^2 \beta {{m}_{e}}{\text{/}}{{f}_{A}},$$

(2)

where ${{m}_{e}}$ is the mass of the electron and β is an arbitrary angle. For the maximum value cos²β = 1 and t-aking into account Eq. (1), we obtain g_Ae = 2.99 × ${{10}^{{ - 11}}}{{m}_{A}}$, where m_A is in electronvolts.

In the KSVZ model, the axion does not interact with the electron; the effective axion–electron coupling constant calculated for the one-loop correction is [20, 21]

$${{g}_{{Ae}}} = \frac{{3{{\alpha }^{2}}{{m}_{e}}}}{{4{{\pi }^{2}}{{f}_{a}}}}\left( {\frac{E}{N}\ln \frac{{{{f}_{A}}}}{{{{m}_{e}}}} - \frac{2}{3}\frac{{4 + z + w}}{{1 + z + w}}\ln \frac{\Lambda }{{{{m}_{e}}}}} \right),$$

(3)

where $\alpha \approx 1{\text{/}}137$ is the fine-structure constant, $z = {{m}_{u}}{\text{/}}{{m}_{d}}$ = 0.56 and $w = {{m}_{u}}{\text{/}}{{m}_{s}}$ = 0.029 are the ratios of the masses of the u, d, and s quarks, Λ ≈ 1 GeV is the scale cutoff in QCD, and E/N is the model-dependent parameter about unity, which is equal to 8/3 and 0 in the DFSZ axion model and the original KSVZ axion model, respectively. The interaction of the hadron axion with the electron is suppressed by a factor of at least $\sim {\kern 1pt} {{\alpha }^{{ - 2}}}$.

According to Eqs. (2) and (3), the coupling constant g_Ae is proportional to m_A, with the coefficient of proportionality including unknown parameters cos²β and E/N for the DFSZ and KSVZ axions, respectively. The axion coupling constants g_Aγ, g_Ae, and g_AN determine both the probabilities of production of axions in various processes and cross sections for the reactions used to detect them.

The best known experiments are aimed at the search for solar axions produced through thermal-photon conversion in the field of the solar plasma. Under the assumption of the axion–photon coupling, researchers attempt to detect axions through the inverse axion–photon conversion in a laboratory magnetic field [22, 23] or in a crystal field [24, 25]. Photon count rates expected in these experiments are proportional to $g_{{A\gamma }}^{4}$. In this work, to detect solar axions appearing in reactions involving the electron, the resonant absorption of axions by ⁸³Kr nuclei is employed. In this case, the expected count rate is proportional to the dimensionless product $g_{{Ae}}^{2} \times g_{{AN}}^{2}$.

In our previous works, using a krypton proportional counter, we attempted to detect monochromatic solar axions emitted in the relaxation of the ⁸³Kr first excited nuclear level, which is excited because of the high temperature of the Sun [18], and Primakoff axions, which resonantly excite the ⁸³Kr nucleus [19]. Theoretical and experimental studies of the axion problem are reviewed in [8].

2 RESONANT EXCITATION OF THE ⁸³Kr FIRST EXCITED NUCLEAR LEVEL BY SOLAR AXIONS

Figure 1 shows the energy spectrum of axions produced in the processes caused by the axion–electron coupling that was calculated with the axion–electron coupling constant g_Ae = 10^–11 [17] and is used in further calculations. The average energy of axions is 1.6 keV and their flux almost vanishes at energies above 15 keV. The flux of axions at an energy of 9.4 keV, which co-rresponds to the ⁸³Kr first excited nuclear level, is 1.32 × 10¹¹ cm^–2 s^–1 keV^–1 [17], which is a factor of 60 lower than the maximum flux reached at an energy of 0.7 keV.

The 9.405-keV ⁸³Kr first excited nuclear level has the spin and parity 7/2⁺. The transition to the 9/2⁺ ground state is an М1 magnetic transition (a small fraction of the Е2 transition is $\delta = 0.0129$) and can be accompanied by the emission of a γ-ray photon and by the emission or absorption of a pseudoscalar particle, an axion. The electron conversion coefficient important for target–detector experiments, where conversion and Auger electrons are absorbed in the target, is e/γ = 17.1 for the transition under study [26].

The ratio of the probabilities of the axion and electromagnetic transitions ω_A/ω_γ was calculated in the long-wavelength approximation in [27, 28] in the form

$$\frac{{{{\omega }_{A}}}}{{{{\omega }_{\gamma }}}} = \frac{1}{{2\pi \alpha (1 + {{\delta }^{2}})}}{{\left[ {\frac{{g_{{AN}}^{0}\beta {\text{*}} + g_{{AN}}^{3}}}{{({{\mu }_{0}} - 0.5)\beta {\text{*}} + {{\mu }_{3}} - \eta }}} \right]}^{2}}{{\left( {\frac{{{{p}_{A}}}}{{{{p}_{\gamma }}}}} \right)}^{3}},$$

(4)

where ${{p}_{\gamma }}$ and ${{p}_{A}}$ are the momenta of the photon and axion, respectively; δ is the ratio of probabilities of the Е2 and М1 transitions; ${{\mu }_{0}} \approx 0.88$ and ${{\mu }_{3}} \approx 4.71$ are the isoscalar and isovector nuclear magnetic moments, respectively; and β* and η are the parameters determined by particular nuclear matrix elements. The parameters β* and η for the $_{{36}}^{{83}}{\text{Kr}}$ nucleus with an odd number of nucleons and an unpaired neutron are estimated in the single-particle approximation as $\beta \text{*} = - 1$ and $\eta = 0.5$ [29].

The axion–nucleon coupling constant g_AN is the sum of the isoscalar $g_{{AN}}^{0}$ and isovector $g_{{AN}}^{3}$ parts. In the KSVZ axion model, the constants $g_{{AN}}^{0}$ and $g_{{AN}}^{3}$ can be expressed in terms of the mass of the axion as [19]

$$\begin{array}{*{20}{l}} {g_{{AN}}^{0} = - 4.03 \times {{{10}}^{{ - 8}}}({{m}_{A}}{\text{/1 eV}}),} \\ {g_{{AN}}^{3} = - 2.75 \times {{{10}}^{{ - 8}}}({{m}_{A}}{\text{/1 eV}}).} \end{array}$$

(5)

To calculate Eqs. (5), we used the particular axial vector baryon coupling constants $F = 0.462$ and $D = 0.808$ and the polarization structure function of the proton $S = 0.5$ [19], as well as the commonly accepted ratios $z = {{m}_{u}}{\text{/}}{{m}_{d}}$ = 0.56 and $w = {{m}_{u}}{\text{/}}{{m}_{s}}$ = 0.029 of the masses of the u, d, and s quarks (rather than more modern values $z = 0.47$ and $w = 0.023$ [8]) to correctly compare to the previous results.

We note that the detection of the resonant absorption of axions on the М1 transition in the ⁸³Kr nucleus is complicated by a methodological problem of a negative value of the parameter $\beta {\text{*}}$ in Eq. (4). This negative value, together with the existing wide interval of possible S and z values, leads to a large uncertainty of the ratio ω_A/ω_γ given by Eq. (4). The effect of uncertainties in the parameters S and z on the final result through a factor of ${{(g_{{AN}}^{0} - g_{{AN}}^{3})}^{2}}$ in Eq. (4) was discussed in [18, 19, 30], where it was shown in particular that the ${{(g_{{AN}}^{0} - g_{{AN}}^{3})}^{2}}$ value at the relation $S \simeq 1.2{-} 1.7z$ can be more than an order of magnitude smaller than the value calculated with S = 0.5 and z = 0.56.

In the DFSZ axion model, the axion–nucleon coupling constants $g_{{AN}}^{0}$ and $g_{{AN}}^{3}$ depend on an additional unknown parameter cos²β, but they are of the same order of magnitude [20, 31]. At cos²β = 1, the parameter ${\text{|}}g_{{AN}}^{0} - g_{{AN}}^{3}{\text{|}}$ appearing in Eq. (4) for the axion emission probability ω_A/ω_γ is a factor of 2.05 larger than this parameter for the KSVZ axion.

The cross section $\sigma ({{E}_{A}})$ for axion absorption at the energy E_A is given by an expression similar to the cross section for the resonant absorption of γ-ray photons with the correction to the ratio ω_A/ω_γ. The total cross section for axion absorption can be obtained by integrating $\sigma ({{E}_{A}})$ over the solar-axion spectrum $(d{{\Phi }_{A}}{\text{/}}d{{E}_{A}})$ [30]. As a result, the rate of absorption of solar axions R_A by the ⁸³Kr nucleus is given by the expression

$${{R}_{A}} = \pi {{\sigma }_{{0\gamma }}}\Gamma (d{{\Phi }_{A}}{\text{/}}d{{E}_{A}})({{\omega }_{A}}{\text{/}}{{\omega }_{\gamma }}),$$

(6)

where ${{\sigma }_{{0\gamma }}} = 1.22 \times {{10}^{{ - 18}}}$ cm² is the maximum γ-ray absorption cross section and $\Gamma = 2.95 \times {{10}^{{ - 12}}}$ keV is the width of the ⁸³Kr first excited nuclear level.

The flux of these axions is proportional to $g_{{Ae}}^{2}$, and the ratio ω_A/ω_γ depends on the parameter ${{(g_{{AN}}^{3} - g_{{AN}}^{0})}^{2}}$. The resulting expression for the rate of axion absorption R_A by the ⁸³Kr nucleus in units of inverse seconds per atom in terms of the coupling constant has the model-independent form

$${{R}_{A}} = 2.15 \times {{10}^{4}}g_{{Ae}}^{2}{{(g_{{AN}}^{3} - g_{{AN}}^{0})}^{2}}{{({{p}_{A}}{\text{/}}{{p}_{\gamma }})}^{3}}.$$

(7)

Substituting Eqs. (5) for the constants $g_{{AN}}^{0}$ and $g_{{AN}}^{3}$ in terms of the axion mass m_A obtained in the KSVZ model, the axion absorption rate can be expressed in terms of g_Ae and m_A, the latter being measured in electronvolts,

$${{R}_{A}} = 3.53 \times {{10}^{{ - 12}}}g_{{Ae}}^{2}m_{A}^{2}{{({{p}_{A}}{\text{/}}{{p}_{\gamma }})}^{3}}.$$

(8)

The total number of detected axions depends on the number of ⁸³Kr nuclei in the target, measurement time, and detector efficiency. The probability of observing the 9.4-keV peak depends on the background level of the experimental setup.

3 EXPERIMENTAL SETUP

The experimental setup is described in detail in our previous works [18, 19]. Here, we present only the basic characteristics. A detector with a gas proportional counter is located in the low-background underground Baksan Neutrino Observatory (Institute for Nuclear Research, Russian Academy of Sciences) at a depth of 4900 mwe, where the muon flux is (2.6 ± 0.09) m^–2 day^–1, which is lower than that on the ground by a factor of 5 × 10⁶ [32].

The cylindrical copper counter has a total volume of 10.8 L. A gold-plated tungsten wire running along the counter axis serves as an anode. To exclude the influence of edge effects on charge collection, the anode diameter is increased, which limits the working volume of the chamber to 8.8 L. The counter is filled with krypton enriched to 99.9% in the ⁸³Kr isotope at a pressure of 1.8 bar. The mass of the ⁸³Kr isotope in the working volume of the counter is 58 g. The passive shield of the counter consists of sequential copper, lead, and polyethylene layers. A 12.5-MHz digitizer is used to measure the amplitude of the pulse, the duration of its rising edge, and the secondary photoemission pulse. The procedure of analysis of the pulse shape is described in [33, 34]. The detection efficiency for the γ-and X-ray photons, as well as Auger and conversion electrons, arising from the relaxation of the ⁸³Kr 9.4-MeV excited level as estimated through a Monte Carlo simulation with the Geant4 package is $\varepsilon = 0.825$ [19].

4 RESULTS

The measurements were carried out. The energy spectrum of the proportional gas counter signals detected over a live time of 776.6 day is shown in Fig. 2. The most intense peak observed in the spectrum is due to the X-ray K Cu lines (${{K}_{{\alpha 1}}} = 8.048$ keV, ${{K}_{{\alpha 2}}} = 8.028$ keV, and ${{K}_{\beta }} = 8.905$ keV) from the copper frame of the counter.

Fig. 2.

(Color online) Spectrum of the proportional counter measured in 776.6 days and its theoretical approximations: (1) the Cu X-ray emission peak, (2) the peak from the ⁸¹Kr decay and the Kr and Br X-ray emission, and (3) the 9.4-keV “axion” peak with the number of events 3S_lim.

The second peak observed at an energy of ~13.5 keV consists of several peaks with close energies. The long-lived ⁸¹Kr isotope ($\tau = 3.3 \times {{10}^{5}}$ yr) is formed from the stable ⁸²Kr and ⁸⁰Kr isotopes under the action of neutrons and decays through electron capture into the ⁸¹Br ground state with a probability of 99.7%. The absorption of the characteristic X rays and Auger electrons from bromide in the sensitive volume of the detector is responsible for the 13.47-keV peak corresponding to the binding energy of the electron in the K shell of the Br atom. X-ray photons emitted by krypton (K_α12 = 12.65 keV) and bromine (K_α12 = 11.92 keV) beyond the sensitive volume of the gas chamber make an additional contribution to the broadened 13.5-keV peak.

The measured spectrum is approximated in the range of 4–20 keV by the sum of a continuous background and four Gaussian peaks. The function describing the continuous background is the sum of a constant component and an exponential function of the energy E:

$${{S}_{{bkg}}}(E) = a + b\exp ( - cE) + \sum\limits_1^4 {{S}_{i}}G(E,{{E}_{i}},{{\sigma }_{i}}),$$

(9)

where a, b, and c are varying parameters. Three Gaussians describe the known 8.04-keV K_α12 Cu and 8.905-keV K_β Cu peaks and the wide 13.5-keV peak. The fourth Gaussian describes the 9.405-keV axion peak whose position and width are set to the parameters of the E₁ peak K_α12 Cu.

The approximation of the energy spectrum in the range of 4–20 keV corresponding to χ² = 156.3/147 and P = 0.28 is shown by the solid line in Fig. 2. The 9.4-keV “axion” peak is not manifested statistically. To establish an upper bound on the number of counts in the 9.4-keV peak, we used the standard method to determine the profile ${{\chi }^{2}}({{S}_{4}})$ and the probability f-unction $P({{\chi }^{2}}({{S}_{4}}))$. The upper bound on the number of peak events thus determined is ${{S}_{{\lim }}} = 140$ for 90% C.L.

The determined upper bound on the number of events in the 9.4-keV peak provides bounds on the axion–electron coupling constant g_Ae, $g_{{AN}}^{3}$ – $g_{{AN}}^{0}$, and the axion mass m_A according to Eqs. (7) and (8). The expected number of detected axions is

$${{S}_{A}} = {{R}_{A}} {{N}_{{83{\text{Kr}}}}} T \epsilon \leqslant {{S}_{{\lim }}},$$

(10)

where ${{N}_{{{\text{83Kr}}}}} = 5.24 \times {{10}^{{23}}}$ is the number of ⁸³Kr nuclei in the target, $T = 6.71 \times {{10}^{7}}$ s is the measurement time, and $\epsilon = 0.825$ is the detection efficiency.

According to Eqs. (7) and (10), under the condition ${{({{p}_{A}}{\text{/}}{{p}_{\gamma }})}^{3}} \cong 1$, which is valid for the masses of the axion m_A < 2 keV, we obtain the upper bound

$${\text{|}}{{g}_{{Ae}}}(g_{{AN}}^{3} - g_{{AN}}^{0}){\text{|}} \leqslant 1.50 \times {{10}^{{ - 17}}}$$

(11)

for 90% C.L. Bound (11) is a model-independent constraint on the coupling constants of the axion or any pseudoscalar axion-like particle to the electron and nucleon.

Using Eqs. (8) and (11), one can establish the following upper bound on the product of the constant g_Ae and mass m_A for the KSVZ axion:

$${\text{|}}{{g}_{{Ae}}} \times {{m}_{A}}{\text{|}} \leqslant 1.17 \times {{10}^{{ - 9}}} {\text{eV}}.$$

(12)

The upper bound (12) for the DFSZ axion at cos²β = 1 is almost two times lower ${\text{|}}{{g}_{{Ae}}} \times {{m}_{A}}{\text{|}} \leqslant 5.72 \times {{10}^{{ - 10}}}$ eV. The upper bound (12) on the allowed ${\text{|}}{{g}_{{Ae}}} \times {{m}_{A}}{\text{|}}$ values makes it possible to compare the results with the results of other experiments on the search for solar axions, in particular, with the results of the search for the axio-electric effect in atoms [36, 37] (Fig. 3).

Fig. 3.

(Color online) Upper bounds on the coupling constant g_Ae (line 1) obtained in this work compared to the experimental results (2, 3) on the resonant absorption of axions by the ¹⁶⁹Tm nucleus [16, 35], (4, 6) on search for the axio-electric effect in (4) Si(Li) [36] and (6) LUX [37], (5) on the neutrino luminosity of the Sun [14], and (7) on astrophysical bounds [39]. The regions of excluded values lie above the corresponding lines.

The upper bound (12) excludes a new region of the coupling constant g_Ae at relatively large masses of the axion m_A and is a factor of almost 4 lower than the experimental result on the detection of resonant excitation of the ¹⁶⁹Tm first excited nuclear level [16, 35]. Since the 8.4-keV М1 transition in the ¹⁶⁹Tm nucleus is primarily a proton transition ($\beta \simeq 1$), the ratio ω_A/ω_γ does not include uncertainty.

Figure 3 shows the band of possible g_Ae and m_A values in the KSVZ and DFSZ axion models. Using Eqs. (2) and (3) for the axion–electron coupling constant, one can obtain upper bounds on the axion mass in these two models from Eq. (12). The upper bound (12) on the product g_Ae and m_A excludes the axion masses m_A > 320 eV in the KSVZ model axion (E/N = 8/3). The upper bound on the DFSZ axion at cos²β = 1 and Eq. (2) is much lower: m_A ≤ 4.6 eV.

The upper bounds established in this work on the coupling constant of solar axions to the electrons for masses ${{m}_{A}} \geqslant 0.3$ keV are the most stringent laboratory constraints and are close to astrophysical constraints. It is interesting that the analysis of the luminosity of white dwarfs indicates a nonzero |g_Ae| value in the in-terval of (0.7–2.2) × 10^–13 [8, 38], although the c-orresponding upper bound is above the bound ${\text{|}}{{g}_{{Ae}}}{\text{|}} \leqslant 1.3 \times {{10}^{{ - 13}}}$ obtained for red giants in some globular clusters [39] (Fig. 3).

5 CONCLUSIONS

The resonant absorption of 9.4-keV solar axions by ⁸³Kr nuclei with the excitation of the first excited nuclear level has been sought. Gamma- and X-ray photons, as well as conversion and Auger electrons, have been detected with a large gas proportional counter that is filled with the ⁸³Kr isotope and is placed in a low-background detector in the underground Baksan Neutrino Observatory (Institute for Nuclear Research, Russian Academy of Sciences). As a result, a new upper bound ${\text{|}}{{g}_{{Ae}}}(g_{{AN}}^{3} - g_{{AN}}^{0}){\text{|}} \leqslant$ 1.50 × 10^–17 (90% C.L.) has been obtained for the axion–electron and axion–nucleon coupling constants. This bound corresponds to constraints on the axion–electron coupling constant g_Ae and the axion mass m_A ${\text{|}}{{g}_{{Ae}}} \times {{m}_{A}}{\text{|}} \leqslant 1.17 \times {{10}^{{ - 9}}}$ eV and ${\text{|}}{{g}_{{Ae}}} \times {{m}_{A}}{\text{|}} \leqslant 5.72 \times$ 10^‒10 eV and on the axion mass m_A ≤ 320 eV and m_A ≤ 4.6 eV in the KSVZ and DFSZ (cos²β = 1) axion models, respectively.