Revised neutrino-gallium cross section and prospects of BEST in resolving the Gallium anomaly

O(1) eV sterile neutrino can be responsible for a number of anomalous results of neutrino oscillation experiments. This hypothesis may be tested at short base-line neutrino oscillation experiments, several of which are either ongoing or under construction. Here we concentrate on the so-called Gallium anomaly, found by SAGE and GALLEX experiments, and its foreseeable future tests with BEST experiment at Baksan Neutrino Observatory. We start with a revision of the neutrino-gallium cross section, that is performed by utilizing the recent measurements of the nuclear final state spectra. We accordingly correct the parameters of Gallium anomaly and refine the BEST prospects in testing it and searching for sterile neutrinos. We further evolve the previously proposed idea to investigate the anomaly with Zn-65 artificial neutrino source as a next option available at BEST, and estimate its sensitivity to the sterile neutrino model parameters following the Bayesian approach. We show that after the two stages of operation BEST will make 5$\sigma$-discovery of the sterile neutrinos, if they are behind the Gallium anomaly.


I. INTRODUCTION
Sterile neutrinos are hypothetical massive Majorana fermions, singlets with respect to the Standard Model (SM) gauge group, which have been introduced to explain oscillations of the SM (or active) neutrinos via mixing with them. There are no direct evidences for the sterile neutrinos, unless one interprets the results of the (anti)neutrino oscillation experiments, missing a fit by three active neutrinos, as observations of O(1) eV sterile neutrinos. Though rather speculative, this interpretation encourages physicists to put forward various experimental proposals [1] to check this hypothesis and hunt the sterile neutrinos. One of such proposals, Baksan Experiment on Sterile Transitions (BEST) [1][2][3][4][5] is a short base-line oscillation experiment aimed at searches/measurements of disappearance of electron neutrino by capturing it on gallium, ν e + 71 Ga → e − + 71 Ge (1) Neutrinos come from an artificial source, which is supposed to be 51 Cr. It provides with direct testing the Gallium anomaly [6][7][8] which includes 4 measurements in total, and 3 out of 4 was performed with 51 Cr sources. In this paper we study the recently proposed idea [5] to perform after 51 Cr-based experiment the second stage of the BEST operation with the neutrino source based on the isotope 65 Zn. The main advantage of 65 Zn with respect to 51 Cr is higher availability. At the same time, neutrino spectra of 51 Cr and 65 Zn are significantly different which allows us to achieve more 'uniform' sensitivity to the sterile neutrino parameter space after BEST subsequent operation with the two artificial sources. In this case, to estimate the BEST sensitivity to sterile neutrino parameters we calculate the cross section of the process (1) at neutrino energies expected for the 65 Zn source. To this end we use the computer program "speccros" by John Bahcall which we adapt to account for recent measurements of Refs. [9,10]. We revise the estimates of the cross section of process (1) utilized by SAGE and GALLEX, and consequently refine the parameters of sterile neutrino model favored by the Gallium anomaly [4,11,12]. Then we reestimate the prospects of testing the Gallium anomaly at BEST with artificial source based on 51 Cr isotope. Finally, we find the sensitivity of BEST with 65 Zn source to the sterile neutrino model parameters. In accord with expectation, we observe that running the subsequent experiments with 51 Cr and 65 Zn neutrino sources improves considerably the BEST performance. In particular, it would allow to make 5 σ-discovery and determine the sterile neutrino model parameters with 10% accuracy.
The paper is organized in the following way. The neutrino-gallium cross section is revisited in Sec. II. In particular, here we obtain formulas valid for 37 Ar, 51 Cr and 65 Zn sources. Sec. III contains a sketch of BEST. In Sec. IV we apply the obtained in previous sections results to refine the Gallium anomaly, revise the BEST sensitivity with 51 Cr source and investigate BEST perspectives with 65 Zn source in testing the Gallium anomaly and searches for sterile neutrinos. We summarize in Sec. V.

II. NEUTRINO-GALLIUM CROSS SECTION
The general formula for neutrino absorption cross section accounting for nuclear transitions in reaction (1) can be cast in the following form [13]: where expression in brackets is the dimensionless phase space factor and σ 0 refers to the scale of the neutrino capture cross section. Scale factor σ 0 can be written as [14,15] where α is the fine-structure constant, J f is spin of the final nuclear state, J i is spin of the initial nuclear state, Z is atomic number of the final nucleus, f t 1/2 ( 71 Ge) is the product of dimensionless phase-space factor f for the kinematically allowed electron capture, the inverse process to the reaction (1), and t 1/2 ( 71 Ge) is the half-life of 71 Ge. This factor is defined as where G V , G A are the vector and axial coupling constants of nucleon, determined from the neutron decay [16], and GT are the squares of the transition matrix elements, which the vector current (Fermi transitions) and the axial-vector current (Gamow-Teller transitions) contribute to [17,18]. These allowed transitions are illustrated in Table I, and the squared transition matrix TABLE I. Types of allowed transitions. S is total spin of the leptons. ∆L is change of the total angular momentum of the system. ∆P is change of parity of the system.

Fermi Transitions
Gamow-Teller Transitions elements read [17], where Q + n is the transformation operator of neutron into proton without a spin flip, and the sum is taken over all nucleons in the nucleus; 2×2 spin matrices σ j are related to the Pauli matrices τ i as follows Summations in (5), (6) go over the spin matrices for all possible orientations of the angular momentum of the nucleon in the final state, since the transition probability (due to invariance with respect to rotations) should not depend on the magnetic quantum number of the initial state.
Following the works of John Bahcall [13,14,19], we introduced in (2) the value of ω 2 e G(Z, ω e ) , where G(Z, ω e ) ≡ p e F (Z, ω e )/2παZω e , is dimensionless phasespace factor averaged over the electron energies. The explicit expression is given by formula where φ(q ν ) is the neutrino energy distribution function, q ν = E ν /m e c 2 is the dimensionless neutrino energy, ω e ≡ E/m e c 2 , p e = p/m e c are the dimensionless energy and momentum of the electron. The integrals in (8) are taken over the whole spectrum of electrons, which energy can be expressed as where E ν is energy of the incoming neutrino, E ex is average excitation energy of the produced nucleus, V 0 is a correction [20] for smaller average binding energy of electron inside the nucleus with respect to that outside, and term in parenthesis is the atomic mass difference between initial M (A, Z − 1) and final M (A, Z) atomic masses.
Quantity F (Z, ω e ), which enters into formula (8), accounts for the non-planewave structure of the electron wave-function and is closely related to the Fermi function [21], that is the ratio of electron squared wave functions calculated with and without the Coulomb potential, and r denotes distance from the nucleus center to the electron. According to [14] expression (10) must be averaged over the entire finite volume V of the nucleus of radius R, that reveals The resulting correction reflects the fact that electron capture can occur at any point inside the nucleus. There are also corrections [20] to F (Z, ω e ) due to shielding of the Coulomb potential inside the nucleus. We take them into account, but find them small, at the level of percent for the set of interesting neutrino energies. The review presented above in this Section concerns only the allowed nuclear transitions. The question of the contribution of the excited states of the nucleus to the total neutrino absorption cross section is discussed below.
In paper [22] Hata and Haxton have shown that the contribution of excited states to the total neutrino absorption cross section on 71 Ga can be written as Here σ g.s. is the neutrino absorption cross section associated with gallium 71 Ga transition to the ground state of germanium 71 Ge, which is given by eq. (2), the coefficients λ Ex are the phase space factors for these transitions normalized to the ground-state phase space factor [12]. These coefficients can be calculated from eq. (8) by making use of the program "speccros" written by John Bahcall, B(GT ) g.s. is the square of the Gamow-Teller transition matrix element to the ground state (see Table I transitions to excited states with excitation energies E x of 175 keV and 500 keV, relevant for artificial sources of neutrinos based on radioactive isotopes 51 Cr [6] and 37 Ar [7]. However, for artificial neutrino source 65 Zn [24], the higher energy levels get excited in the process (1) and their contribution to the total cross section is significant, ∼ 20−30%. The coefficients λ Ex for these transitions for 65 Zn are λ 175 = 0.7969, λ 500 = 0.4791, λ 708 = 0.3145, λ 808 = 0.2466, λ 1096 = 0.0934, the squared transition matrix elements B(GT ) Ex corresponding to these energies are given in [23].
While the central value of (17) is fully consistent with previous estimate [13]: the uncertainty saturated by that of (13) is significantly larger. It happened because the value (17) was obtained from analysis of the new data [10]. We utilize the new estimate of the threshold energy of the gallium transition to the ground state of germanium (13), in contrast to the old value Q = 232.69 ± 0.15 keV used previously in [13]. We use the most recent value (13) and hence (16), which are consistent with previous results, while their errors do not dominate the uncertainties of our estimates of the neutrino-capture cross sections. Further, for each spectral line of the artificial sources 51 Cr, 37 Ar and 65 Zn presented 1 in Table II, the values of σ g.s. and λ Ex entering (12) are calculated from (2) and data [10,23] by making use of the program "speccros". Subsequently, for each neutrino energy the neutrino capture cross section is obtained including contributions of the kinematically allowed excited states, see Table II. Then the total neutrino absorption cross sections for each artificial source are obtained by summing over all energies weighted with the corresponding relative fractions, The results are as follows σ( 51 Cr) = (59.10 ± 1.14) × 10 −46 cm 2 , σ( 37 Ar) = (71.38 ± 1.46) × 10 −46 cm 2 , σ( 65 Zn) = (87.76 ± 2.03) × 10 −46 cm 2 .
We use these estimates in the following Sections.

III. SKETCH OF BEST
The BEST experiment is described in detail in Ref. [4]. Here we merely recall the general idea of this experiment.
The experimental setup consists of two concentric zones filled with liquid gallium. The first zone is a sphere of radius R 1 = 0.66 m, in the center of which there is an artificial neutrino source about 0.1 m in size. Such a size makes it possible to place in the center of the first zone a source of neutrinos 51 Cr with activity 3 MCi. The second zone is a cylinder of radius R 2 = 1.096 m and height 2 × R 2 . The image of the experimental setup is shown in Fig. 2. The liquid gallium is irradiated by a neutrino flux from an artificial source. As a result of reaction (1), germanium atoms are formed, which are then chemically extracted from the zones. Possible transitions to sterile neutrinos would affect the neutrino flux. Hence the numbers of extracted atoms are sensitive to the presence of light sterile neutrinos.
The total mass of gallium is 50 tons. The original proposal [1,2] suggests to exploit the isotope 51 Cr as the artificial neutrino source with radioactivity of about 3 MCi. At the same time other candidates may be considered, and one of the most promising is 65 Zn [5]. It provides different neutrino spectrum giving the opportunity to test somewhat different region of sterile neutrino parameter space. Also the half-life of 65 Zn is longer (244 d compared to 27 d for 51 Cr), thus giving more time to make longer measurements with the sufficient activity of the source. However, the artificial source 65 Zn of the same activity has a noticeably larger size than the source 51 Cr, which reduces the oscillation signal after averaging over the source volume. This must be avoided, and a special investigation is required to find the reliable technical solution and optimize the source volume. For the present study we take as a realistic option to adopt the smaller 65 Zn source with activity of about 1 MCi, which will be acceptably compact. The volume occupied by the source within the first zone will increase slightly, but this will not negatively affect the isotropy of target irradiation. Likewise, with such activity it will be possible to keep sufficiently high homogeneity of the zinc source. Finally, the lower power of the source is partly compensated by larger cross section (21). Although the predicted production rate from the 65 Zn source with activity of about 1 MCi is about two times smaller compared to the 3MCi 51 Cr source, nevertheless the expected number of germanium atoms to be extracted from the vessels are still sufficiently large with respect to the solar background. The statistical errors grow insufficiently and the total uncertainty of the extraction is dominated by systematics, which we expect to be the same as in case of 51 Cr source.

IV. REVISION OF THE GALLIUM ANOMALY AND SEARCHES AT BEST
For the revision of the results for neutrino absorption cross sections we begin with discussion of uncertainties.
The main contribution to the uncertainty of the neutrino absorption cross section is associated with corrections from the excited states. To calculate the uncertainty of neutrino cross section, the results of [10,23], as well as the known uncertainty of σ 0 are accounted for. Assuming the measurements of B(GT ) for different energy levels to be independent, we calculate the overall error for each spectral line of the artificial sources as the square root of the sum of the squared standard deviations of all values entering (12).
The obtained values of the cross sections for 51 Cr and 37 Ar and their relative uncertainties deviate insignificantly from the previous study in [12]. However, we take different value of the energy of the gallium transition to the ground state of germanium [10], as well as another value of the transition matrix element to the ground state (16). We find the uncertainty of the cross sections to be about two percent, while earlier for the BEST experiment the uncertainty of +3.6 %/-2.8 % [13] has been adopted.
It is worth noting that the measurement of the threshold energy of the gallium-germanium transition does not contain unknown uncertainties in the nuclear structure, which could explain the anomalous results of the SAGE [6,7] and GALLEX [8] experiments. This result was further discussed in Ref. [10].
The results obtained in Section II imply that despite the fact that we applied new value of the threshold energy of the gallium transition into the ground state of germanium, than previously done, and despite the utilization of the recent measurements of the transitions matrix elements [23], the central values and their uncertainties have not changed much, in comparison with the values presented in [12]. The refined values of the ratios of observed-to-expected number of events R in gallium experiments (gallium anomaly), which we represent in this paper, see Table III, almost completely coincide with the values presented in [12].
TABLE III. Values of the magnitudes of the gallium anomaly, obtained on the basis of refined data on the neutrino absorption cross section, using the value of Q = 233.5 ± 1.2 keV, the transition matrix element to the ground state BGTg.s. = 0.086 ± 0.001 and the transition matrix elements to excited states taken from Ref. [23]. Thus, taking into account the refined value of the neutrino absorption cross section on gallium found in this paper, the resulting error of the experiment BEST [3] for the source 51 Cr is 4.9 % for each of the zones and 4.2 % for the total target, instead of 5.5 % and 4.8 %, respectively. For the artificial neutrino source 65 Zn with activity of 1 MCi in the BEST experiment, the resulting errors will be the same as for the 3 MCi 51 Cr source if the irradiation plan with the 65 Zn source is identical to that presented in Ref. [3].
The anomalous lack of neutrinos presented in Table III can be explained by oscillations of electron neutrinos into sterile partners [27]. The combined results of SAGE and GALLEX, obtained on the basis of refined data, are presented in Fig. 3. The result shown in Fig. 3 shows that the best fit values ∆m 2 = 2.5 eV 2 and sin 2 (2ϑ) = 0.3 are slightly different (by about 10%) from those presented in [4]. The refined regions of the neutrino oscillation parameters to be tested at the BEST [3]  found by applying the formulas from [4]. Assuming the BEST with source 51 Cr fully confirms the anomaly, the most favorable regions (all data of the three experiments are included) of sterile neutrino model parameter space are presented in Fig. 6. Comparing these plots with similar ones in Ref. [4] one can conclude that after revision of the neutrino capture cross section all signal regions become more compact, hence the sensitivity of BEST to the sterile neutrino model certainly increases.
To illustrate the power of the source 65 Zn in further testing the sterile neutrino hypothesis, we present in Fig. 7 the anomaly-favored region after the second run of BEST operating with the source 65 Zn. The sensitivity of the second run is estimated in exactly the same way as has been done in [4] for the source 51 Cr. For the favored by gallium anomaly best fit values of the sterile neutrino model the expected signal rates in the two vessels of BEST correspond to ratios R=(0.827,0.781). One clearly observes from Figs. 6 and 7 the significant improvement in the sensitivity after the combined analysis of the two runs (assuming both confirm the Gallium anomaly). Finally, if both runs find no hint of sterile neutrinos, the exclusion region will expand with respect to that in Fig. 4, and it is presented in Fig. 8.

V. SUMMARY
In this work updated data [10,23] on neutrino absorption cross section on gallium and the program "speccros" are used to refine the neutrino absorption cross section, which is done for 71 Ga and neutrino sources 51 Cr, 37 Ar and 65 Zn.
The results obtained for the sources 51 Cr and 37 Ar agree with the estimates presented in [12]. This suggests that the leading uncertainties in the cross section for neu-FIG. 8. Allowed regions of oscillation parameters in case the BEST experiment does not find any anomalies after two runs: the ratios R of observed-to-expected without sterile neutrinos germanium atoms in both vessels for both sources are consistent with unity, (1, 1).
trino capture are the uncertainties of the matrix elements of nuclear transitions to excited states. The analysis of the capture cross sections for all three types of neutrino sources considered in this paper reveals that taking into account all the uncertainties in the determination of the threshold energy of the gallium transition to the ground state of germanium and taking into account the uncertainties of the matrix elements of the transitions to excited states give an uncertainty of the cross sections of about 2 %. This result shows that the central values and errors of the cross sections (19)-(21) cannot explain the anomalous results of SAGE [6,7] and GALLEX [8]: the anomalous results remain intact.
Thus, the main results published in [4] where the data [23], [10] have not used, remain true, and the experiment BEST [3] has high potential in testing the hypothesis of electron neutrino oscillations into sterile neutrinos.
To summarize, we present the refined estimates of BEST sensitivity to models with light sterile neutrinos mixed with electron neutrinos. The obtained results strongly suggest to use the new artificial source based on the isotope 65 Zn at the second stage of BEST operation, which allow us to reduce the degeneracy in sensitivity to the sterile neutrino model parameters. To illustrate this point we present in Fig. 9 the sensitivity contours in case of both stages exploiting the 51 Cr sources. One can conclude by comparing the plots in Figs. 7, 9 that while 5-σ discovery of the sterile neutrinos is mostly due to double statistics (one stage is not enough to achieve this goal), the second source with different neutrino energies definitely provides with better cornering the signal regions with respect to the case of identical sources. We study possible impact of the future BEST results on status of the Gallium anomaly.
We thank S. Kulagin, F. Simkovic and O. Smirnov for valuable discussions. The work was supported by the RSF grant 17-12-01547.