Confronting solutions of the Gallium Anomaly with reactor rate data

Recently, several models have been suggested to reduce the tension between Gallium and reactor antineutrino spectral ratio data which is found in the framework of 3+1 active-sterile neutrino mixing. Among these models, we consider the extensions of 3+1 mixing with a finite wavepacket size, or the decay of the heaviest neutrino $\nu_4$, or the possibility to have a broad $\nu_4$ mass distribution. We consider the reactor antineutrino rate data and we show that these models cannot liminate the tension between Gallium and reactor rate data that is found in the 3+1 neutrino mixing framework. Indeed, we show that the parameter goodness of fit remains small. We consider also a model which explains the Gallium Anomaly with non-standard decoherence in the framework of three-neutrino mixing. We find that it is compatible with the reactor rate data.


I. INTRODUCTION
The oscillations observed in solar, atmospheric and long-baseline neutrino oscillation experiments have established the standard three-neutrino mixing framework in which the three flavor neutrinos ν e , ν µ , ν τ are unitary superpositions of three light massive neutrinos ν 1 , ν 2 , ν 3 with respective masses m 1 , m 2 , m 3 .(see, e.g., Ref. [1] and the recent global analyses in Refs.[2][3][4]).The oscillations are generated by the two independent squared-mass differences ∆m 2  21 ≈ 7.5 × 10 −5 eV 2 and |∆m 2  31 | ≈ 2.5 × 10 −3 eV 2 (with ∆m 2 kj ≡ m 2 k − m 2 j ).However, anomalies observed in shortbaseline (SBL) neutrino oscillation experiments may require the existence of a larger squared-mass difference ∆m 2 SBL ≳ 1 eV 2 , which implies the extension of the standard three-neutrino mixing framework to a model with more than three light massive neutrinos (see, e.g., the recent reviews in Refs.[5][6][7]).The simplest extension is the 3+1 model with a new massive neutrino ν 4 with mass m 4 ≳ 1 eV, such that ∆m 2  41 = ∆m 2 SBL ≳ 1 eV 2 .Since from the LEP measurements of the decay of the Z-boson [8] we know that there are only three active neutrinos, in the flavor basis the new neutrino is a sterile neutrino ν s , which does not take part in weak interactions.The sterile neutrino must be mostly mixed with the new massive neutrino, in order to have a small perturbation of the three-neutrino mixing framework which can explain the short-baseline anomalies without spoiling the fit of solar, atmospheric and long-baseline neutrino oscillation data: for α = e, µ, τ, s, where U is the unitary 4 × 4 mixing matrix such that Most puzzling is the short-baseline Gallium Anomaly (GA), which is a deficit of events observed in Gallium source experiments (GALLEX [9,10], SAGE [11,12], and BEST [13,14]) with respect to the rate expected in the three-neutrino mixing framework.Since the Gallium Anomaly deficit is relatively large, it is in tension with the measurements of short-baseline reactor neutrino experiments in the framework of 3+1 neutrino mixing [15].
It has been proposed to relieve this tension by introducing new effects which damp the oscillations in short-baseline reactor neutrino experiments: a quantum mechanical wavepacket effect [16,17], the decay of the new mass state ν 4 [17], and a broad mass distribution for ν 4 [18].The oscillation damping reduces the bounds obtained from the ratios of events measured at different distances in the short-baseline reactor experiments NEOS [19,20], DANSS [21,22], PROSPECT [23,24], and STEREO [25,26], relieving the tension between the results of these experiments and the Gallium Anomaly [17,18].
In this paper we show that, however, the new damping effects cannot relieve the tension between the results of short-baseline reactor neutrino rate experiments and the Gallium Anomaly.The rate experiments, summarized in Table 4 of Ref. [27], measured the total short-baseline reactor neutrino event rates.In 2011 the comparison with the event rates expected from the theoretical calculation of the reactor electron antineutrino flux (the HM model of Huber [28] and Mueller et al [29]) generated the Reactor Antineutrino Anomaly (RAA) [30], which is a deficit of events with respect to the prediction.However, this deficit is smaller than that of the Gallium Anomaly and it is in tension with it [15].Moreover, new reactor electron antineutrino flux calculations (the EF model of Estienne, Fallot, et al [31] and its revision [32] and the KI model of Kopeikin et al. [33]) decreased the Reactor Antineutrino Anomaly [27,34] and increased the tension with the Gallium Anomaly [15].
The new damping effects do not reduce significantly the tension between the reactor rates bound and the Gallium Anomaly, because the neutrino oscillations relevant for the reactor rates are already almost completely averaged in the 3+1 model for the values of ∆m 2 41 ≳ 1 eV 2 which fit the Gallium Anomaly.
We consider also the explanation of the Gallium Anomaly proposed in Ref. [35] through nonstandard decoherence effects in the framework of three-neutrino mixing and we obtain the condition for its compatibility with the reactor rate data.
The plan of the paper is to discuss the wavepacket effect in Section II, ν 4 decay in Section III, a broad ν 4 mass distribution in Section IV, and the three-neutrino scenario with non-standard decoherence effects in Section V. Finally, in Section VI we present a summary and conclusions.

II. THE WAVEPACKET EFFECT
In this Section we discuss the effects of a small wavepacket width.This scenario was considered in Ref. [16] analyzing spectral shape data from reactor experiments and the Gallium data.In Ref. [17] it was considered withtin a global fit to neutrino oscillation data.The effective neutrino oscillation probability at very short baselines is given by where sin ), E is the neutrino energy, and L is the source-detector distance.
The coherence length given by where σ is the width of the wavepacket in coordinate space.The size of the wavepacket has been recently estimated in Ref. [36].The result of this analysis shows that the wavepacket is very large FIG. 1: In the left panel we show the allowed regions of parameter space at 2σ from the analyses of reactor rate data using the KI (magenta) and HM (blue) fluxes and of the Gallium data (green) for some selected values for the wavepacket width σ.The contours for each case are plotted with respect to the local minimum obtained with the wavepacket widths indicated in the legend.In the right panel we show the allowed regions after marginalizing over σ.
The parameter goodness of fit obtained for the models under consideration for the combination of Gallium (GA) data with reactor rate (RR) data using the HM flux (first row) and the KI flux (second row).
(σ ≃ 200 nm for reactor neutrinos and σ ≃ 1400 nm for Gallium experiments) and does not have any effect on neutrino oscillations in reactor and Gallium experiments.Even if this calculation is not taken into account, the required wavepacket size of Ref. [17] (σ ≈ 6.7 × 10 −5 nm) is in tension with the phenomenological bounds obtained in Refs.[37,38]: σ > 2.1 × 10 −4 nm at 90% confidence level (C.L.).However, in the current analysis of this paper we do not take into account any prior information on the wavepacket size and we consider σ as an unbounded parameter.The results of our analyses of reactor and Gallium data for neutrino oscillations with finite wavepacket size are shown in Fig. 1.For simplicity we consider only the HM and KI reactor antineutrino flux models.The HM model is the original model of the Reactor Antineutrino Anomaly and the KI model is its revision taking into account the new measurements in Ref. [33].In the case of Gallium data the choice of cross section model can have an impact on the tension with reactor rate data [15].In this paper we consider for simplicity only the Bahcall cross section model [39].
In the left panel of Fig. 1 we plot the allowed regions for some selected values of σ: the bestfit value from Ref. [38], namely σ = 3.35 × 10 −4 nm (dashed-dotted lines), the best-fit value from Ref. [17], namely σ = 6.7 × 10 −5 nm (dashed lines), and the even smaller value σ = 1 × 10 −5 nm (dotted lines).Note that the best-fit wavepacket size of Ref. [38] has little effect on the allowed regions when compared to those in the standard 3+1 analysis.When allowing for smaller wavepackets, even smaller than the best-fit value from Ref. [17], the upper bound on sin 2 2ϑ ee of the reactor allowed region, which is attained at large values of ∆m 2  41 , is almost unaffected.For very small wavepacket sizes, the Gallium allowed region is extended to low values of ∆m 2  41 , but it still requires values of sin 2 2ϑ ee larger than the reactor upper bound.Therefore, the wavepacket effect cannot relieve the tension between the reactor rate data and the Gallium data.FIG.2: In the left panel we show the allowed regions of parameter space at 2σ from the analyses of reactor rate data using the KI (magenta) and HM (blue) fluxes and of the Gallium data (green) for some selected values of the decay width Γ of ν 4 .The contours are plotted with respect to the minimum obtained using the decay widths as indicated in the legend.In the right panel we show the allowed regions after marginalizing over Γ.
In the right panel of Fig. 1 we show the region obtained after marginalizing over σ.One can see that the wavepacket effect mainly affects the regions of parameter space at low values of ∆m 2  41 .It is clear from the figure that the tension between the reactor and Gallium allowed regions persists.In order to quantify the tension we compute the parameter goodness of fit [40] for the analysis including the wavepacket effect and compare it to that obtained with the 3+1 analysis.The results are shown in Tab.I, where one can see that the inclusion of the wavepacket effect has very little impact on the parameter goodness of fit in the analysis with the HM flux model.Instead, there is an improvement in the analysis with the KI flux model, but the tension remains much worse than for the HM flux model.Therefore, we conclude that the wavepacket effect cannot eliminate the tension between the reactor rate data and the Gallium data.
Note that, in principle, the wavepacket of neutrinos in Gallium experiments does not need to have the same size as that in reactor neutrino experiments.We chose the same value for both data sets for illustration, but one can confront the reactor and Gallium allowed regions with different values of σ in Fig. 1 and see that they are anyway in tension.

III. ν 4 DECAY
We turn our attention now to the second scenario considered in Ref. [17], the possible decay of ν 4 .As in the wavepacket scenario this model improves the fit of neutrino oscillation data according to Ref. [17].In this case the neutrino oscillation probability at short baselines is given by where Γ = 1/τ is the decay width and τ is the lifetime of ν 4 .Since ν 4 has a much larger mass than the other neutrinos we approximate m 4 ≃ ∆m 2 41 .We fit Eq. 4 assuming that Γ is a completely free parameter, not taking into account any possible bounds.The results are shown in Fig. 2. In the left panel we show the results for some selected values of Γ, while in the right panel we show the allowed regions after marginalization over the decay width.As can be seen when fixing Γ = 0.35 eV (which is the best-fit value from Ref. [17]),

FIG. 3:
In the left panel we show the allowed regions of parameter space at 2σ from the analyses of reactor rate data using the KI (magenta) and HM (blue) fluxes and of the Gallium data (green) for some selected values of the breadth b of a broad ν 4 mass distribution.The contours are plotted with respect to the minimum obtained using the breadth values indicated in the legend.In the right panel we show the allowed regions after marginalizing over b.
the allowed region of the reactor rate analysis with the HM fluxes opens up towards lower masses, while in the case of the KI flux the bound at small masses becomes less stringent.However, the upper limit on sin 2 2ϑ ee , which is reached for large values of ∆m 2  41 , is unaffected by the decay.This remains true when allowing for larger values of Γ and also when marginalizing over it, as shown in the right panel of the figure.
The effect of the decay on the Gallium region is also not helping in reducing the tension with the reactor rate data.The allowed parameter space is larger than in the standard 3+1 case, but the new region requires even larger mixing angles.Therefore, the tension between the reactor rate data and the Gallium data is not eliminated by the decay.This can be seen from the values of the parameter goodness of fit in Tab.I, where one can see that considering the HM flux model the inclusion of the decay does not change the parameter goodness of fit with respect to that obtained with the 3+1 analysis.There is instead an improvement in the analysis with the KI flux model, but the tension remains much worse than for the HM flux model.Hence, we conclude that the addition of ν 4 decay to the 3+1 model cannot eliminate the tension between the reactor rate data and the Gallium data.

IV. BROAD ν 4 MASS DISTRIBUTION
In this Section we consider a model with a broad ν 4 mass distribution, which was developed in Ref. [41] and applied to Gallium and reactor spectral ratio data in Ref. [18].In this model it is assumed that the fourth mass-eigenstate can be modeled as a state with a central mass squared m 2 4 and finite breadth b.In this scenario the effective neutrino oscillation probability at short baselines is given by where sinc(x) = sin(x) x .Note that sinc(0) = 1 and therefore for b = 0 the standard 3+1 neutrino oscillation probability is recovered.FIG.4: Constraints on the decoherence length λ 21 from reactor rate data using the KI flux model.The shaded region is the approximate preferred region of Ref. [35] for the explanation of the Gallium Anomaly.The horizontal dashed line indicates the 2σ confidence level.
The results of the fit of Eq. ( 5) are shown in Fig. 3.As in the previous cases, we show the contours for selected values of b in the left panel and the contours obtained after marginalizing over b in the right panel.As can be seen from the left panel of Fig. 3, the best-fit value of b from Ref. [18] (b = 0.055 eV2 ) has little effect on the allowed regions.For the much larger value b = 0.5 eV 2 the reactor rate data give upper bounds for sin 2 2ϑ ee which are independent of ∆m 2   41   and have the same values as those of the 3+1 analysis for large values of ∆m 2  41 in both analyses with the HM and KI flux models.The Gallium region is slightly extended to lower values of ∆m 2   41   without changing significantly the lower bound for sin 2 2ϑ ee , which remains in tension with the reactor upper bound.
The right panel in Fig. 3 shows that also after marginalizing over b the tension between the reactor and Gallium allowed regions persists1 .This is confirmed by the values of the parameter goodness of fit in Tab.I, which are the same as those obtained in the ν 4 decay analysis.Therefore, we conclude that also a broad ν 4 mass distribution cannot eliminate the tension between the reactor rate data and the Gallium data.

V. NON-STANDARD DECOHERENCE EFFECTS
An alternative solution, which does not require a fourth neutrino, has been proposed in Ref. [35].In this reference the authors explain the Gallium Anomaly in the framework of three neutrino mixing with a non-standard loss of coherence of the low-energy neutrinos in Gallium source experiments.The neutrino oscillation probability in this scenario is given by where is the decoherence parameter.Following Ref. [35], we set E 0 = 0.75 MeV.In Eq. ( 6) it is assumed that there is full decoherence in the 13-sector, according to the fit of Gallium data in Ref. [35].The authors of Ref. [35] show that the Gallium Anomaly could be explained with energy dependencies n ≲ −2 and argue that extreme dependencies n ≲ −10 do not affect the observed oscillations of higher-energy solar and reactor neutrinos.Such extreme energy dependencies have never been considered before [42][43][44][45][46][47][48][49].
We calculate bounds on the decoherence length λ 21 using reactor rate data.For simplicity, this time we consider only the KI flux model (the results with the HM model are similar).The marginal ∆χ2 = χ 2 − χ 2 min as a function of λ 21 is shown in Fig. 4 for several choices of n.The shaded region in the figure is the approximate preferred region of Ref. [35] for the explanation of the Gallium Anomaly with n ≲ −2.One can see that at 2σ this explanation of the Gallium Anomaly is not in conflict with reactor rate data as long as n ≲ −7.More extreme energy dependencies, such as n = −12 considered in Ref. [35], cannot be tested with reactor rate data.Therefore, we conclude that the explanation of the Gallium Anomaly proposed in Ref. [35] is allowed by the reactor rate data.

VI. CONCLUSIONS
In this paper we have examined three models which have been proposed [16][17][18] to relieve the tension between Gallium and reactor experiments that is found in the 3+1 neutrino mixing framework [15].While this is true when considering reactor spectral ratio data [16][17][18], we have shown that these models are in strong tension with reactor rate data.In Tab.I we summarize the parameter goodness of fit [40] of the analyses of Gallium data and reactor rate data for each model and we compare it with the one obtained from the standard 3+1 analysis.As can be seen, the goodness of fit obtained using the HM reactor antineutrino flux is practically the same in the three models as in the 3+1 analysis.When considering the KI flux instead, the goodness of fit is better than in the standard 3+1 case.Unfortunately, it has still a very low value.Therefore, we conclude that the three models that we have considered cannot eliminate the tension between Gallium and reactor rate data that is found in the 3+1 neutrino mixing framework.
We considered also, in Section V, a model with non-standard decoherence proposed in Ref. [35] to explain the Gallium Anomaly in the three-neutrino mixing framework.We have shown that it is compatible with the reactor rate data if the power n of the energy dependence of the decoherence parameter is n ≲ −7.Therefore, the explanation of the Gallium Anomaly with n ≲ −10 proposed in Ref. [35] is allowed by the reactor rate data 2 .
Reactor rate data are often discarded in analyses arguing that the results are model dependent.We disagree with this argument.While it is true that there is a model dependence, the differences among the results with different antineutrino flux models are not huge and they give upper bounds on the 3+1 oscillation parameter sin 2 2ϑ ee [27] which must be taken into account.A good indication that the reactor flux models are more reliable now than some years ago is that there is good agreement [27] between the predictions of the EF [31] and KI [33] flux models, which have been obtained using two completely different techniques.Many results in particle physics are obtained under certain assumptions and are not criticized or discarded for being model-dependent.Neither should reactor rate data be ignored.
Regarding the Gallium Anomaly, several oscillation explanations are in tension with the reactor rate data [15][16][17][18]50].Other explanations, with Standard Model and beyond the Standard Model physics, have been suggested in Refs.[51,52], but some of the Standard Model explanations are already excluded by the new measurements of 71 Ge decay presented in Ref. [53].As of now, the