The Sun: light dark matter and sterile neutrinos

The existence of such an LDM field can by identified with a dilation field of an extradimensional extension of the Standard Model or/and a CP-violating pseudo-Goldstone boson of a spontaneously broken global symmetry. For some of these models, $\phi$ couples to the Standard Model fields, and as such it induces periodic time variation in particle masses and couplings. In such theories the gauge invariance suggests that the $\phi$ should possess an identical coupling constant to charged leptons, in which case scalar interactions with the electrons provide a good opportunity for detection through atomic clocks (e.g., Arvanitaki et al., 2016), accelerometers (e.g., Arvanitaki et al., 2018), and gravitational wave detectors (e.g., Lopes and Silk, 2014; Graham et al., 2016).

Similarly, this $\phi$ field can couple to neutrinos. Once again, these types of interactions generically result in time-varying corrections to the neutrino masses, neutrino mass differences, and mixing angles, which can be searched for in the neutrino flux signals on present and future experimental neutrino detectors (Aharmim et al., 2013; Abe et al., 2016; Borexino Collaboration et al., 2018; Aalbers et al., 2020). If $\phi$ couples weakly to the neutrinos, over a large range of masses, it can significantly modify the neutrino oscillations probabilities leading to a distorted survival electron neutrino probability function (Berlin, 2016; Krnjaic et al., 2018).

The motivation for such a model comes from the possibility of this composite particle physics model resolving two observational problems:

1. 1.

   The classical cold dark matter model leads to several inconsistencies with the cosmological observational data, such as the missing satellite problem and the cusp problem (e.g., Primack, 2009). Light dark matter resolves such problems if dark matter is totally or partially made of light scalar particles with a mass of the order of $10^{-22}{\rm eV}$ (Hu et al., 2000; Peebles, 2000). In hierarchical models of structure formation, such type of dark matter is able to explain the flatness observed on the profiles of the distribution of gas and stars in halos and filaments (Mocz et al., 2019).

2. 2.

   Although the standard three neutrino flavor model produces a reasonable good global fit to all the neutrino data (Esteban et al., 2019), there are now many hints that point out to the possibility of the existence of a fourth neutrino. This one does not have any other interaction than gravity and for that reason is known as a sterile neutrino (Diaz et al., 2019). It was found that a flavor oscillation model made of the 3 active neutrinos plus a sterile neutrino could explain some of the observed anomalies found on the Short baseline neutrino oscillation experiments (Giunti et al., 2012, 2013), Liquid Scintillator Neutrino Detector (Aguilar et al., 2001), and MiniBooNE Short-Baseline Neutrino Experiment (MiniBooNE Collaboration et al., 2018), as well as the anomalies related with GALLEX and SAGE solar-neutrino detectors – the so-called Gallium anomalies (Kostensalo et al., 2019). For instance, Kostensalo et al. (2019) found that the data favour a $3+1$ neutrino flavor model with $m_{4}=\;1.1eV$ and mixing matrix element $U_{e4}=0.11$.

One possibility to resolve both problems (neutrino anomalies and structure formation) is to consider that dark matter is made of a light scalar field that couples to a sterile neutrino (e.g., Farzan, 2019). The interactions of $\phi$ and $\nu_{s}$ could impede the oscillations in the universe and thereby improve the agreement between the structure formation and cosmological observations (e.g., Dasgupta and Kopp, 2014; Hannestad et al., 2014).

If such $\phi$ and $\nu_{s}$ particles exist today, they were produced abundantly in the early universe. For instance, sterile neutrinos can be produced via mixing with active neutrinos (Dodelson and Widrow, 1994), in some scenarios such neutrino production is being enhanced by the oscillations between active and sterile neutrinos (Bezrukov et al., 2019, 2020; de Gouvêa et al., 2020) or by the lepton asymmetry (Shi and Fuller, 1999). The production of light dark matter can take many forms, such as vector bosons by parametric resonance production (Dror et al., 2019). For instance, some models predict a sterile neutrino abundance of $\Omega_{s}^{2}h^{2}=0.12\left({\sin^{2}{(2\theta_{s})}}/{3.5\times 10^{-9}}\right)\left({m_{\nu_{s}}}/{7keV}\right)$ where $m_{\nu_{s}}$ and $\theta_{s}$ is the sterile neutrino mass and mixing angle (Kusenko, 2009). For the light dark matter field some authors obtained $\Omega_{\phi}^{2}=0.1\left({a_{o}}/{10^{17}\;{\rm GeV}}\right)^{2}\left({m_{a}}/{10^{-22}\;{\rm eV}}\right)^{1/2}$, where $a_{o}$ is a parameter that relates to the initial misalignment angle of the axion, and $m_{a}$ is the axion mass (Hui et al., 2017; Niemeyer, 2019). Conveniently, we will assume that in the present-day universe the total dark matter abundance is given by

where the $\Omega_{\phi}^{2}h^{2}$ and $\Omega_{s}^{2}h^{2}$ are the total $\phi$ and $\nu_{s}$ densities in the present universe, respectively. For future reference, we assume that the present-day total dark matter abundance $\Omega_{\rm DM}h^{2}=0.12$ (Planck Collaboration et al., 2018), and the dark matter density in the solar neighborhood is $\rho^{\odot}_{\rm DM}=0.39\;GeV\;cm^{-3}$ (Catena and Ullio, 2010).

In this paper, we study the impact that this light dark matter field has in the 3+1 neutrino flavor model. Specifically, we discuss how the light dark matter field modifies the neutrino flavor oscillations, and by using the current sets of solar neutrino data, we also put constraints in the parameters of such models and make predictions for the future neutrino experiments.

The article is organized as follows. In section 2, we discuss how the light dark matter drives the 3+1 neutrino flavor oscillations. In section 3, we present the neutrino flavor oscillation model in the presence of a cosmic light dark matter field. In section 4, we compute the survival electron neutrino probabilities for the electron neutrinos produced in the proton-proton (PP) chain and carbon-nitrogen-oxygen (CNO) cycle solar nuclear reactions. In section 5, we discuss the results in relation to current experiments and future ones. Finally, in section 6, we present the conclusion and a summary of our results.

If not stated otherwise, we work in natural units in which $c=\hbar=1$. In these units all quantities are measured in GeV, and we make use of the conversion rules $1m=5.068\times 10^{15}GeV^{-1}$, $1kg=5.610\times 10^{26}GeV$ and $1sec=1.519\times 10^{24}GeV^{-1}$.

We assume that in the present universe, the dark matter is composed of two fundamental particles: a light scalar boson $\phi$ and sterile neutrinos $\nu_{s}$, where $m_{\phi}$ and $m_{\nu_{s}}$ are their respective masses (Hannestad et al., 2014). The LDM field $\phi$ couples with the active neutrinos and the sterile neutrino by a Yukawa interaction $g_{\phi}\phi\nu_{s}\nu_{s}$ where $g_{\phi}$ is a dimensionless coupling (Farzan, 2019). To illustrate this effect, consider an LDM scalar $\phi$ with a Yukawa coupling to active neutrinos. Then the relevant part of the Lagrangian reads

where for convenience of representation the flavor indices have been suppressed. We also assume that the dimensionless coupling is very small ($g\ll 1$). From the Euler-Lagrange equations of $\nu$ and $\phi$, it is possible to show that the effect of $\phi$ on the propagation of the neutrino is equivalent to changing the neutrino mass from $m_{\nu}$ to $m_{\nu}+\delta m_{\nu}$. As we will see later, this $\delta m_{\nu}$ perturbation will induce time variations in the mass-squared differences and mixing angles of all neutrino flavors through $\phi$. In principle the Yukawa couplings can have any structure in the neutrino flavor space. In this work, we will focus on two convenient scenarios of great interest to neutrino detectors: mass-square differences and mixing angles (e.g., Ding and Feruglio , 2020). Moreover, we will also assume that $\delta m_{\nu}=g_{\phi}\phi$ (e.g., Smirnov and Xu, 2019).

In most studies of three-neutrino flavor models, the authors are solely interested in the modulation coming from the square mass differences $\Delta m^{2}_{ij}(t)$ (by Equation 6) and mixing angles $\theta_{ij}(t)$ (by Equation 7). For that reason, all neutrinos are assumed to couple $\phi(t)$. As a consequence, their contribution to ${\cal V}$ cancels out. Hence, it is correct to neglect the contribution of $\phi(t)$ to the Wolfenstein potential (Dev et al., 2020). Nevertheless, here in this 3+1 neutrino flavor model, as we will discuss later, we include the contribution of $\phi(t)$ in ${\cal V}$.

This 3+1 neutrino flavor model with dark matter is identical to the standard (undistorted) three-neutrino flavor model (see Equation 8). However, in this model we included a sterile neutrino, and the Wolfenstein potentials in ${\cal V}$ are modified to take into account the new LDM field $\phi$ (Miranda et al., 2015).

Moreover, since the survival probabilities $P_{ee,i}(E,\phi)$ (Equation 16) are time dependent through $\phi$, these quantities also vary with time. Therefore, the oscillation probability $\langle P_{ee}(E)\rangle$ (Equation 15) is generalized for each specific nuclear reaction $i$:

The LDM field $\phi$ can lead to different temporal imprints on the neutrino oscillation measurements. The specific impact depends on the mass of the LDM particle. In the following, we compute the spectra of neutrinos from any specific nuclear reaction that we know to be essentially independent of the properties surrounding solar plasma. Since in the 3+1 neutrino flavor model new processes exist to change the survival probability of electron neutrinos, this will modify the solar neutrino spectra measured on Earth. These new processes will alter the conversion rates of $\nu_{e}$ to other flavors ($\nu_{\mu}$, $\nu_{\tau}$ and $\nu_{s}$) and vice versa. Accordingly, the electron neutrino spectrum of the nuclear reaction $i$ inside the core is defined as $\Phi_{i}$, and $\Phi_{i\odot}$ is the electron neutrino spectrum arriving on Earth (Lopes, 2018b) such that:

where $P_{ee,i}(E)$ is the average survival probability of electron neutrinos for reactions in the solar interior as given by equation 16. Equally if we take the time average of equation 18, we obtain the following averaged spectrum for each nuclear reaction $i$:

where $\langle P_{ee,i}(E)\rangle$ is the average survival probability of electron neutrinos as given by equation 17.

In the parameterization for the 3+1 neutrino flavor oscillation model, we opt to adopt the recent values obtained in the data analysis of the standard three-neutrino flavor oscillation model obtained by de Salas et al. (2020), and for the sterile neutrino additional fiducial parameters we used the values obtained by Gariazzo et al. (2015). Accordingly, for a parameterisation with a normal ordering of neutrino masses, the mass-square difference and the mixing angles have the following values: $\Delta m^{2}_{21}=7.50^{+0.22}_{-0.20}\times 10^{-5}{\rm eV^{2}}$, $\sin^{2}{\theta_{12}}=0.318\pm 0.016$, and $\sin^{2}{\theta_{13}}=0.02250^{+0.00055}_{-0.00078}$. Although, $\Delta m^{2}_{31}=2.56^{+0.0003}_{-0.0004}\times 10^{-3}{\rm eV^{2}}$ and $\sin^{2}{\theta_{23}}=0.56^{+0.016}_{-0.022}$, we mention them here for reference (de Salas et al., 2020). These new parameters are consistent with previous estimations (Esteban et al., 2019; Gonzalez-Garcia et al., 2016). For the sterile neutrino, we choose the following fiducial values for the mass-square difference and mixing angles  (Gariazzo et al., 2015, 2016; Capozzi et al., 2017): $\Delta m^{2}_{41}=1.6\ {\rm eV^{2}}$, $\sin^{2}{\theta_{14}}=0.027$, $\sin^{2}{\theta_{42}}=0.014$ and the other mixing angle for the sterile neutrinos are fixed to zero. Moreover, we assume that all phases ($\delta_{13,14,34}$) and other angles related to the sterile neutrino are equal to zero.

The present-day internal structure of the Sun corresponds to an up-to-date standard solar model (SSM) that has a better agreement with neutrino fluxes and helioseismic data sets. This solar model was obtained from a one-dimensional stellar evolution code allowed to evolve in time until the present-day solar age, $4.57$ Gyr, having been calibrated to the values of luminosity and effective temperature of the present Sun, of $3.8418\times 10^{33}$ erg s^-1 and $5777$ K, respectively, as well as the observed abundance ratio at the Sun’s surface: ($Z_{\text{s}}/X_{\text{s}}$)${}_{\odot}=0.0181$, where $Z_{s}$ and $X_{s}$ are the metal and hydrogen abundances at the surface of the star (Turck-Chieze and Lopes, 1993; Bahcall et al., 1995, 2006). This stellar model was computed with the release version 12115 of the stellar evolution code MESA (Paxton et al., 2011, 2019). The details about the physics of this standard solar model in which we use the AGSS09 (low-Z) solar abundances (Asplund et al., 2009) are described in Lopes and Silk (2013) and Capelo and Lopes (2020).

Figures 1 and 2 show the impact of the time-dependent mass-square difference (Equation 6) and mixing angles (Equation 7) on the averaged electron survival probability $\langle P_{ee}(E)\rangle$ (Equation 15) for which the LDM field $\phi$ has a fixed amplitude (Equation 5): $\epsilon_{\phi}=0$ or $\epsilon_{\phi}=1.5\%$. We also show LDM models for which the sterile neutrino couples to $\phi$ with strength $G_{\phi}$.

The overall shape of the curve $\langle P_{ee}(E)\rangle$ depends on $G_{\phi}$ times $n_{\phi}$ in the potential $V_{\nu_{s}\phi}$ or the ratio $G_{\phi}/m_{\phi}$ as previously mentioned. For instance, in a LDM model in which we fix $m_{\phi}=10^{-9}\;{\rm eV}$ (or $n_{\phi}$), an increase of $G_{\phi}$ from $10^{20}$ to $10^{22}\;G_{\rm F}$ leads $\langle P_{ee}(E)\rangle$ to vary significantly, as shown in Figure 1. As expected this change in $\langle P_{ee}(E)\rangle$ is more pronounced for high energy neutrinos where the MSW effect is more significant. If we choose higher values of $m_{\phi}$ the results will somehow be similar (see Figure 1).

Evidently, for an LDM model in which $m_{\phi}$ decreases by a certain amount ($n_{\phi}=\rho_{\phi}/m_{\phi}$), the constancy of $G_{\phi}/m_{\phi}$ in $V_{\nu_{s}\phi}$ implies that $G_{\phi}$ can increase by the same order of magnitude to obtain the same MSW effect on the $\langle P_{ee}(E)\rangle$ curve (see Figure 1). For instance, an LDM model with $m_{\phi}=10^{-23}\;{\rm eV}$ and $G_{\phi}=10^{7}G_{\rm F}$ will have $\langle P_{ee}(E)\rangle$ identical to an LDM model with $m_{\phi}=10^{-9}\;{\rm eV}$ and $G_{\phi}=10^{21}G_{\rm F}$ , since in both LDM models we have the same ratio: $G_{\phi}/m_{\phi}\approx 10^{30}$. The same argument explains the reason why the coupling constant between sterile neutrinos and more massive dark matter particles is much smaller in those models than in the present study. For instance, this is the case for particles captured from the dark matter halo by the Sun. Since over time, the star accreted a significant amount of dark matter (e.g., Lopes , 2018a), for these models $G_{\phi}$ is significantly smaller than the value found for the present study.

The most important feature of such a class of LDM models is the time dependence of the dark matter field $\phi(t)$ and its imprint in the flavor oscillation parameters’ mass-square differences (Equation 6) and mixing angles (Equation 7). As predicted by equation 9, there are many $P_{ee}(E,\phi)$ with near similar behavior. Figure 1 shows an ensemble of time-dependent $P_{ee}(E,\phi)$ as a pink band. The difference between curves relates to the dependence of the oscillation parameters on time. In this LDM model it is assumed there is a very negligible interaction between sterile neutrinos and $\phi$ (for which $G_{\phi}\approx 0$). The figure also shows $\langle P_{ee}(E)\rangle$ (red curve) the time-averaged of the ensemble of $P_{ee}(E,\phi)$ curves that we compute using equation 15.

Although there are several parameters that contribute for the time variability of $P_{ee}(E,\phi)$ (Equation 9), the main contributions come from $\theta_{21}$ and $\theta_{13}$. The variability related $\theta_{24}$ is relevant for the high energies. We notice that the contributions coming from $\Delta m^{2}_{21}$ and $\theta_{32}$ are much smaller than all the parameters mentioned above. The amplitude of the $P_{ee}(E,\phi)$ pink band is defined by the value of $\epsilon_{\phi}$ for which we adopt the fiducial value of $\epsilon_{\phi}=1.5\%$. It is worth pointing out that the $P_{ee}(E,\phi)$ band is much larger for low-energy than for higher-energy values. Moreover, the averaged value of this ensemble given by $\langle P_{ee}(E)\rangle$ is identical to $\langle P_{ee}(E)\rangle$ with $\epsilon_{\phi}\approx 0$. Figure 2 shows the variability of $P_{ee}(E,\phi)$ for a LDM with different $G_{\phi}$ values. These results are identical to the model in figure 1. Nevertheless, the LDM model with the largest $G_{\phi}$ has (an orange) band with a smaller amplitude around $\langle P_{ee}(E)\rangle$. Once again, the band thickness decreases for neutrinos with higher energy for all these models.

Figures 3 and 4 compare our predictions with current solar neutrino data (e.g., Aalbers et al., 2020). These figures show that LDM models with relatively low values of $\epsilon_{\phi}$ and $G_{\phi}$ are compatible overall with current solar neutrino data coming from Borexino, Super-Kamiokande, and SNO. Clearly, this analysis has also shown that the precision of our current solar neutrino experiments is not able to distinguish between some of these LDM models. Nevertheless, it is already possible to put some constraints on these LDM models. For instance, we found that LDM models with $G_{\phi}\approx 0$ must have a $\epsilon_{\phi}$ smaller than $3\%$ to be consistent with all data, including $pp$ measurements of the Borexino detector (Borexino Collaboration et al. , 2020) (see Figure 3); and any LDM models must have a ratio $G_{\phi}/m_{\phi}$ smaller than $10^{30}$, otherwise they become inconsistent with $pp$ and ⁷Be measurements for several solar detectors (Borexino Collaboration et al. , 2020; Agostini et al., 2019; Borexino Collaboration et al., 2018; Bellini et al., 2010) (see figure 4). Figure 4 shows a LDM model with a $m_{\phi}=10^{-9}\;{\rm eV}$ and $G_{\phi}=10^{22}\;G_{\rm F}$ with a ratio $G_{\phi}/m_{\phi}$ of the order of $10^{31}\;G_{\rm F}$ (see Figure 4); This ratio is one order of magnitude larger than the critical $G_{\phi}/m_{\phi}$ value of $10^{30}$ discussed in the previous section. Figure 4 also shows that the variability related with time dependence on $P_{ee}(E,\phi)$ decreases for large values of $G_{\phi}$.

There is another important effect that also contributes to the time variability of $P_{ee}(E,\phi)$. The PP chain and CNO cycle, nuclear reactions occur at different distances from the center of the Sun and each nuclear reaction emits neutrinos in a well-defined energy range. As a consequence, the electron neutrinos produced in each specific nuclear reaction will be affected differently by the MSW effect. As such, this effect will also contribute to the overall variability of electron neutrinos $P_{ee,i}(E,\phi)$ (see Equation 16) and their time-averaged $\langle P_{ee,i}(E,\phi)\rangle$ (see Equation 17).

The time-dependent electron neutrino survival probability will have a significant impact on the neutrino spectra of the different nuclear reactions. Accordingly, figures 5 and 6 show the spectra correspond to two neutrino types: $pp$ and ${}^{8}B$ neutrinos. An essential difference between these two spectra relates to the thickness of the $\epsilon_{\phi}$ band for a fixed value since thickness decreases with neutrino energy. Therefore the $\epsilon_{\phi}$ band is more significant for a $pp$ spectrum than for a ${}^{8}B$ neutrino spectrum. This is an effect identical to the one discussed previously for the $P_{ee}(E,\phi)$ functions. Therefore, the measurement of solar neutrino fluxes and solar neutrino spectrum in the energy range below $0.2\;MeV$ will provide the strongest constraint for such a class of dark matter models. Figure 5 and 6 show the spectra of ⁸B and $pp$, if we assume the precision expected to be attained by the Darwin experiment (Aalbers et al., 2020). Figure 6 also shows the precision expected for the Darwin experiment.

By studying a large range of dark matter particle masses (from $10^{-9}$ to $10^{-23}\;{\rm eV}$) we found that depending on the mass of these LDM particles and the value of the generalized Fermi constant, the shape of electron neutrino survival probability and their spectra can vary with time. We establish that for LDM particles with low masses (low-frequency regime), the solar neutrino detectors can observe the electron neutrino survival probability changing with time. Conversely, for dark matter particles with higher masses (high-frequency regime), this impact can be determined by measuring the time-averaged electron neutrino survival probability.

It was possible to establish, using data of current solar neutrino measurements, that those models with a $G_{\phi}/m_{\phi}$ ratio smaller than $10^{30}\;G_{\rm F}eV^{-1}$ agree with current solar neutrino data from the Borexino, SNO and Super-Kamiokande detectors. We also found that for models with a near-zero constant, the time-variability amplitude must be smaller than 3%. Such a constraint is equivalent to the condition ${g_{\phi}\sqrt{2\rho^{\odot}_{\rm DM}}}/{(m_{\phi}m_{\nu})}\;\left({\Omega_{\phi}}/{\Omega_{\rm DM}}\right)^{1/2}\leq 0.03$.

Finally, we also found that the precision expected in the measurements to be made by the Darwin detector will allow us to put powerful constraints to this class of models.