Cogenesis of Majoron dark matter and baryon asymmetry from a high scale inverse seesaw mechanism

Abstract

The inverse seesaw mechanism provides a concise scenario to produce tiny neutrino masses. Here we investigate the Majoron dark matter arising from the $\mathrm{U}{(1)}_{\mathrm{L}}$ symmetry spontaneous breaking and matter–antimatter asymmetry in the framework of a high scale inverse seesaw mechanism. Due to the existence of a lepton-number-violating mass term, the Majoron gets its mass from radiative corrections. On the other hand, the Noether’s theorem indicates that the Majoron can obtain a non-zero initial velocity in the early universe, and then generates the relic abundance via kinetic misalignment mechanism. With the same initial Majoron velocity, the observed baryon asymmetry can also be explained through the so-called spontanous bayogenensis mechanism.

Introduction

Although the Standard Model (SM) of particle physics achieves the great success in explaining the phenomena relevant to strong and electroweak interactions. It is unable to solve the following two problems at least: Dark Matter (DM) [1, 2] and the Baryon Asymmetry of the Universe (BAU) [3]. Based on the principle of the Occam Razor, it’s of great significance to seek a unified theory of new physics which can solve both of the above two puzzles together. Over the past decades, the QCD axion arising from the Peccei–Quinn (PQ) symmetry spontaneous breaking has become one of the most popular candidates for ultra-light dark matter [4, 5, 6, 7, 8, 9, 10–11]. Additionally, the study of QCD axions also leads to the axion-like particles (ALPs) with similar properties [12], such as Majoron [13, 14, 15, 16–17], a type of particle produced after the spontaneous breaking of the $\mathrm{U}{(1)}_{\mathrm{L}}$ symmetry with $``{\mathrm{L}}^{\prime \prime }$ being the lepton number. Recently, the common origin of QCD axions (or ALPs) and BAU has attracted widespread interest [18, 19, 20–21]. The idea is: Suppose that the relevant global U(1) symmetries are explicitly broken by higher-dimensional operators, the angular mode of the complex scalar field, i.e. the dark matter candidate, obtains a non-zero initial velocity that can serve as an additional chemical potential to deviate the matter–antimatter symmetry.

In this paper, we investigate a new Majoron dark matter generation mechanism and spontanous baryogenesis within the framework of a higher scale inverse seesaw mechanism [22, 23]. With the help of a lepton-number-violating (LNV) Majorana mass term of right-handed neutrinos, the Majoron can interact directly with the heavy right-handed neutrinos and gets tiny but non-zero mass via one-loop radiative corrections. At meanwhile, the LNV term also results in a non-zero initial velocity of Majoron. However, we found that it is too small to produce enough baryon asymmetry. To alleviate this flaw, we introduce a higher dimensional potential term of Higgs singlet that explicitly breaks the lepton-number symmetry, and can then address the observed Majoron dark matter relic density as well as the baryon asymmetry of the universe at one stroke.

The paper is organized as follows: In Sect. 2 we introduce the modified inverse seesaw mechanism and study the cosmological evolution of Majoron in detail. Section 3 is devoted to the study of the BAU. In Sect. 4 we discuss the direct detection signals of the Majoron dark matter. Finally, a brief concluding remark is presented in Sect. 5.

[See PDF for image]

Fig. 1

One-loop Feynman diagrams for generating Majoron mass, where the cross represents the mass insertion and the solid dot represents the vertex of Majoron-$S_R$ interaction given in Eq. (3)

The model

At first, we bierfly review the basic structure of inverse seesaw mechanism, which extends the Standard Model by adding three heavy right-handed neutrinos $N_R$, three SM gauge-singlet neutrinos $S_R$ and a Higgs singlet $\mathrm{\Phi }$ that carries two units lepton-number. The effective potential beyond the SM is described by [21, 22–23]

1

where H is the SM Higgs doublet and ${\ell }_L$ is the left-handed lepton doublet. $Y_{\ell }$ and $Y_{\text{s}}$ are the Yukawa coupling matrices. There is a lepton-number-violating Majorana mass term for $S_R$ neutrinos, indicating that the value of ${\mu }_S$ is naturally small in light of ’t Hooft’s naturalness criterion [24]. At very high temperature, the $\mathrm{U}{(1)}_{\mathrm{L}}$ is broken spontanously and $\mathrm{\Phi }$ develops a non-zero vacuum expectation value (VEV). Then the Higgs singlet can be parameterized as

2

$$\begin{array}{c}\hfill \mathrm{\Phi }=\frac{1}{\sqrt{2}}\left(\varphi ,+,f_J\right)e^{i\theta },\end{array}$$where the angular mode ($\theta \equiv J/f_J$) is the Nambu–Goldstone particle, called the Majoron with $f_J$ being the Majoron decay constant. After substituting Eq. (2) into Eq. (1), the Majoron field in the phase can be removed from the Yukawa term by the field dependent phase transformation: where $\psi$ represents the lepton, except that in the mass term of . It leads to a Majoron-$S_R$ neutrino interaction in the form of

3

$$\begin{array}{c}\hfill {\mathcal{L}}_{\mathit{JSS}}=\frac{1}{2}{e}^{-i\theta }\overline{S_R^C}{\mu }_S{S}_R.\end{array}$$Using Eq. (3), one-loop radiative corrections lead to a Majoron potential expressed as

4

$$\begin{array}{cc}\hfill V_J=& -\frac{1}{16{\pi }^2}\left[{\mu }_S,M_S^3,\left(1,-,log,\frac{M_S^2}{M_{\text{Pl}}^2}\right),cos,\theta \right)\hfill \\ \hfill & +4{\mu }_S^2{M}_S^{2}log\frac{M_S^2}{M_{\text{Pl}}^2}cos(2\theta )-{\mu }_S^3{M}_{S}cos(3\theta )\hfill \\ \hfill & \left(+,\frac{1}{3},{\mu }_S^4,cos,(4\theta )\right],\hfill \end{array}$$where $M_S$ is defined as $Y_s{f}_J/\sqrt{2}$, and $M_{\text{Pl}}=1.22\times {10}^{19}$ GeV is the Planck mass. We adopt the $\overline{\text{MS}}$ renormalization scheme in the calculation and plot the relevant Feynman diagrams in Fig. 1. Utilizing the potential in Eq. (4), one gets the squared mass of Majoron as follows:

5

$$\begin{array}{cc}\hfill m_J^2=& \frac{1}{16{\pi }^2{f}_J^2}|m{M}_S^3\left(1,-,log,\frac{M_S^2}{M_{\text{Pl}}^2}\right)\hfill \\ \hfill & +8m^2{M}_S^{2}log\frac{M_S^2}{M_{\text{Pl}}^2}-9m^3{M}_S+\frac{16}{3}{m}^4|.\hfill \end{array}$$Given that ${\mu }_S\ll M_S$, the first term above plays a dominant role in the Majoron mass.

Majoron initial velocity

After obtaining the Majoron mass, we can solve the equation of motion of the Majoron in the Friedmann–Robertson–Walker universe to get its cosmological evolution and relic abundance [25, 26–27]:

6

$$\begin{array}{c}\hfill \frac{d^2\theta }{d{t}^2}+3H(t)\frac{d\theta }{\mathit{dt}}+m_J^2\theta =0,\end{array}$$where H is the Hubble parameter. Since the mass term of explicitly violates the lepton-number symmetry, unlike the traditional QCD axion scenario, we must consider the initial velocity of Majoron field, denoted as ${\dot{\theta }}_i$. On the one hand, Noether’s theorem indicates that ${\dot{\theta }}_i\ne 0$ due to the exist of . Then the Noether current related to the initial velocity of Majoron can be obtained by calculating the Coleman–Weinberg potential

7

$$\begin{array}{c}\hfill V_{\text{CW}}=\sum _{n=1}^{\infty }{\mathrm{C}}_{2n}{\mathrm{\Phi }}^{2n}+\mathrm{h}.\mathrm{c}.,\end{array}$$which arises from the $N_R$– $S_R$– $\mathrm{\Phi }$ interaction. Explicit calculation shows that the potential in Eq. (7) is highly suppressed by the extremely small parameter ${\mu }_S$, resulting in a negligible ${\dot{\theta }}_i$. On the other hand, since the quantum gravity effects explicitly break all global symmetries, it’s resonable to introduce the following lepton-number-violating potential [18, 21],

8

The authors in Refs. [18, 21] have already demonstrated that the potential in Eq. (8) can generate a large initial velocity ${\dot{\theta }}_i$ for Majoron, making its cosmological evolution completely different from the traditional misalignment mechanism, which is known as kinetic misalignment mechanism [28, 29].

The relic density of Majoron dark matter

The kinetic misalignment mechanism shows that the oscillation temperature determined by the traditinal oscillation condition $3H(T_{\text{osc}}^{\text{con}})=m_J$ from Eq. (6) is delayed due to the kinetic effect ( ${\dot{\theta }}_i\ne 0$). And the kinetic oscillation temperature $T_{\text{osc}}^{\text{k}}$ can be estimated from $1/2\dot{\theta }{(T_{\text{osc}}^{\text{k}})}^2{f}_J^2=V_{J,\text{max}}$ [28, 29], which leads to $\dot{\theta }(T_{\text{osc}}^{\text{k}})=\sqrt{2}{m}_J$ [21]. The significance of the dynamic effects in the kinetic mechanism is characterized by the critical initial velocity, namely, ${\dot{{\theta }_i}}_{\text{C}}$. In the case of $\dot{{\theta }_i}={\dot{{\theta }_i}}_{\text{C}}$, the traditioanal oscillation temperature is equal to the kinetic oscillation temperature, i.e. $T_{\text{osc}}^{\text{k}}=T_{\text{osc}}^{\text{con}}$. Provided that the initial velocity of Majoron is much smaller than the critical initial velocity ($|{\dot{\theta }}_i|\ll |{\dot{{\theta }_i}}_{\text{C}}|$), the kinetic effects are negligible and the evolution of Majoron is consistent with that in the traditional misalignment scenario. Once the oscillation temperature is obtained, the energy density of the Majoron dark matter at present reads as [21]

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$$\begin{array}{c}\hfill {\rho }_J(T_0)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}{m}_J^2{f}_J^2⟨{\theta }_{J,i}^2⟩{\displaystyle \frac{g_{\ast s}^{}(T_0)}{g_{\ast s}^{}(T_{\text{osc}}^{\text{con}})}}{\left({\displaystyle \frac{T_0}{T_{\text{osc}}^{\text{con}}}}\right)}^3,\hfill & |{\dot{\theta }}_i|\ll |{\dot{{\theta }_i}}_{\text{C}}|\hfill \\ m_J^2{f}_J^2⟨{\theta }_{J,i}^2⟩{\displaystyle \frac{g_{\ast s}^{}(T_0)}{g_{\ast s}^{}(T_{\text{osc}}^{\text{k}})}}{\left({\displaystyle \frac{T_0}{T_{\text{osc}}^{\text{k}}}}\right)}^3,\hfill & |{\dot{\theta }}_i|\ge |{\dot{{\theta }_i}}_{\text{C}}|\hfill \end{array}\right)\end{array}$$where $T_0$ is the current CMB temperature, $g_{\ast s}^{}$ is the effective relativistic degrees of the freedom, and the angle brackets represent the initial value of $\theta$ averaged over $[-\pi ,\pi )$ [30]. Here we only consider the post-inflationary case and take it to be $\theta (t_i)=\sqrt{⟨{\theta }_{J,i}^2⟩}=\pi /\sqrt{3}$ [12, 30].

In Fig. 2 we plot the relic abundance of the Majoron dark matter as the function of ${\dot{\theta }}_i$. The benchmark decay constant is selected as $f_J={10}^{13}$ GeV. The remaining mass parameters are set to be ${\mu }_S=1.15\times {10}^{-35}$ GeV and $M_S={10}^{10}$ GeV. Different from the traditional inverse seesaw mechanism, we take a larger seesaw scale ($M_S$) and a smaller LVN mass (${\mu }_S$). The horizontal dashed line shows the observed dark matter relic desity of the universe, i.e. ${\mathrm{\Omega }}_J{h}^2\simeq 0.12$ [31, 32]. Figure 2 shows that only a sufficiently large initial velocity ($|{\dot{\theta }}_i|\ge |{\dot{{\theta }_i}}_{\text{C}}|$) can significantly enhance the relic density of Majoron dark matter. Compared to the critical initial velocity, smaller initial velocities have minimal impact on ${\mathrm{\Omega }}_J{h}^2$. That is to say, the kinetic effect of Majoron only works when the initial velocity is close to or above the critical velocity [21, 28, 29]. Actually, the higher dimensional operators in Eq. (8) can not generate an initial velocity as large as $|{\dot{\theta }}_i|>{10}^{18}$ GeV as shown in Fig. 2 [28, 29]. Fortunately, we only need a relatively small initial velocity (${\dot{\theta }}_i=-3.285\times {10}^6$ GeV) to generate enough dark matter and baryon asymmetry.

[See PDF for image]

Fig. 2

The relic density of Majoron dark matter as the function of initial velocity ${\dot{\theta }}_i$. The vertical green dashed line represents the critical initial velocity $|{\dot{{\theta }_i}}_{\text{C}}|$. The red curve corresponds to $f_J={10}^{13}$ GeV. The mass parameters are taken to be ${\mu }_S=1.15\times {10}^{-35}\text{GeV}$ and $M_S={10}^{10}$ GeV. The gray region is ruled out due to the overabundance of dark mtter. The initial velocity shown with blue dashed line is relevant to the observed baryon asymmetry

Baryon asymmetry of the universe

In this section, we are ready to compute the BAU induced by Majoron dark matter. In general, the BAU is characterized by the ratio of net baryon number density ($n_B$) to entropy density (s)

10

$$\begin{array}{c}\hfill Y_B=\frac{n_B}{s},\end{array}$$where $s=2{\pi }^{2}g_{\ast s}^{}{T}^3/45$ is the entropy density. The spontanous baryogenesis mechanism shows that the non-zero Majoron velocity can contribute an additional chemical potential to the Boltzmann transport equations associated with SM particles, leading to the breaking of matter–antimatter symmetry [19, 20]. Consequently, the Boltzmann transport equations should be modified to [19, 21]

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$$\begin{array}{c}\hfill -\frac{d\left({\mu }_i,/,T\right)}{dlnT}=\frac{-1}{g_i}\sum _{\alpha }{n}_i^{\alpha }\frac{{\gamma }_{\alpha }}{H}\left[\sum _j,n_j^{\alpha },\left(\frac{{\mu }_j}{T}\right),-,n_S^{\alpha },\frac{\dot{\theta }(T)}{T}\right],\\ \hfill \end{array}$$where the chemical potential ${\mu }_{i,j}$ of SM particles can be related to their number density $n_{i,j}$ through $n_i=g_i{\mu }_i{T}^2/6$ with $g_i$ being the degrees of freedom. It should be noted that $n_{i,j}^{\alpha }$ and $n_S^{\alpha }$ with superscript $\alpha$ represent the charge of the particle species i, j and the component of the Majoron source vector, respectively. ${\gamma }_{\alpha }$ is the interaction rate per unit time.

As discussed in Sect. 2, we need to calculate the baryon asymmetry produced around the temperature of ${10}^{13}$ GeV. Then we can only consider the chemical potentials of the following 11 species [19]: $i=(e,\mu ,\tau ,L_{12},L_3,q_{12},t,b,Q_{12}$, $Q_3,H)$ with $g_i=\left(1,,,1,,,1,,,4,,,2,,,12,,,3,,,3,,,12,,,6,,,4\right)$, and the explicit result of the charge vectors for various particles can be found in Ref. [19]. On the other hand, the source vector $n_S^{\alpha }$ can be derived from the triangle anomalies induced by the Majoron field-dependent phase transformation discussed in Sect. 2. As a result, the Majoron interacts with SM leptons in the form of [21, 33]

12

$$\begin{array}{c}\hfill {\mathcal{L}}_{\text{int}}=-\frac{J}{2f_J}{\partial }_{\mu }(\overline{{\ell }_L}{\gamma }^{\mu }{\ell }_L+\overline{E_R}{\gamma }^{\mu }{E}_R),\end{array}$$which gives the expression of source vector as $n_S^{\alpha }=\frac{1}{2}{\sum }_i{n}_i^{\mathbf{L}}{n}_i^{\alpha }=\frac{1}{2}\left(n_{\tau }^{\alpha },+,n_{L_{12}}^{\alpha },+,n_{L_3}^{\alpha }\right)$ [21], where $n_i^{\mathbf{L}}=(1,1,1,0,0,0,0$, 0, 0, 0, 0) is the lepton-number vector of the 11 particle species above. The dimension-5 Weinberg operator closely related to the LNV interaction is coupled with Majoron in the form of ${\mathcal{L}}_{\text{int}}\supset \frac{1}{2M_S}\frac{J}{f_J}\ell \ell HH$, with the decoupling temperature being $T_{\text{W}}\simeq 6\times {10}^{12}\text{GeV}$ [19] in the case of active neutrino mass $m_{\nu }=0.05$ eV.

The diagram for the Majoron–Weinberg operator interaction is depicted in Fig. 3 [34]. Then source vector reads as

13

$$\begin{array}{c}\hfill \left(n_S^{\mathit{WS}},,,n_S^{W_{12}},,,n_S^{W_3},,,n_S^{\mathit{SS}},,,n_S^{Y_{\tau }},,,n_S^{Y_t},,,n_S^{Y_b}\right)=\left(\frac{3}{2},,,1,,,1,,,0,,,0,,,0,,,0\right).\end{array}$$Asymmetries are produced sequentially: The lepton-number asymmetry is first converted to $\mathbf{B}-\mathbf{L}$ symmetry, where the $\mathbf{B}-\mathbf{L}$ asymmetry [19, 20] is defined as $n_{\mathrm{B}-\mathbf{L}}=\frac{T^2}{6}{\mu }_{\mathrm{B}-\mathbf{L}}$ with [19]

14

$$\begin{array}{cc}\hfill {\mu }_{\mathbf{B}-\mathbf{L}}& =4{\mu }_{q_{12}}+{\mu }_t+{\mu }_b+4{\mu }_{Q_{12}}+2{\mu }_{Q_3}\hfill \\ \hfill & -({\mu }_{\tau }+4{\mu }_{L_{12}}+2{\mu }_{L_3}).\hfill \end{array}$$Then the ${\mu }_{\mathbf{B}-\mathbf{L}}$ is subsequently converted to the baryon asymmetry through the formula [35, 36–37]:

15

$$\begin{array}{c}\hfill Y_{\mathbf{B}}=\frac{n_{\mathbf{B}}}{s}=\frac{T^3}{6s}\frac{{\mu }_{\mathbf{B}}}{T}=\frac{T^3}{6s}\left(\frac{28}{79},\frac{{\mu }_{\mathbf{B}-\mathbf{L}}}{T}\right)\simeq \frac{1}{{10}^3}\frac{{\mu }_{\mathbf{B}-\mathbf{L}}}{T}.\\ \hfill \end{array}$$

[See PDF for image]

Fig. 3

The Feynman diagram for the Majoron–Weinberg operator interaction [34]

The explicit expression of the transport equations shown in Eq. (11) are in the form of

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$$\begin{array}{cc}\hfill -\frac{d}{dlnT}\left(\frac{{\mu }_{\tau }}{T}\right)& =-\frac{1}{g_{\tau }}(-1)\frac{{\gamma }_{Y_{\tau }}}{H}\left(-,\frac{{\mu }_{\tau }}{T},+,\frac{{\mu }_{L_3}}{T},-,\frac{{\mu }_H}{T}\right),\hfill \\ \hfill -\frac{d}{dlnT}\left(\frac{{\mu }_{L_{12}}}{T}\right)& =-\frac{1}{g_{L_{12}}}2\frac{{\gamma }_{\mathit{WS}}}{H}\left[\frac{2{\mu }_{L_{12}}}{T},+,\frac{{\mu }_{L_3}}{T},+,\frac{6{\mu }_{Q_{12}}}{T}\right)\hfill \\ \hfill & \left(+,\frac{3{\mu }_{Q_3}}{T},-,\frac{3}{2},\frac{\dot{\theta }(T)}{T}\right]\hfill \\ \hfill & -\frac{1}{g_{L_{12}}}\frac{2{\gamma }_{W_{12}}}{H}\left[\frac{2{\mu }_{L_{12}}}{T},+,\frac{2{\mu }_H}{T},-,\frac{\dot{\theta }(T)}{T}\right],\hfill \\ \hfill -\frac{d}{dlnT}\left(\frac{{\mu }_{L_3}}{T}\right)& =-\frac{1}{g_{L_3}}\frac{{\gamma }_{\mathit{WS}}}{H}\left[\frac{2{\mu }_{L_{12}}}{T},+,\frac{{\mu }_{L_3}}{T},+,\frac{6{\mu }_{Q_{12}}}{T}\right)\hfill \\ \hfill & \left(+,\frac{3{\mu }_{Q_3}}{T},-,\frac{3}{2},\frac{\dot{\theta }(T)}{T}\right]\hfill \\ \hfill & -\frac{1}{g_{L_3}}\frac{{\gamma }_{Y_{\tau }}}{H}\left[-,\frac{{\mu }_{\tau }}{T},+,\frac{{\mu }_{L_3}}{T},-,\frac{{\mu }_H}{T}\right]\hfill \\ \hfill & -\frac{1}{g_{L_3}}\frac{2{\gamma }_{W_3}}{H}\left[\frac{2{\mu }_{L_3}}{T},+,\frac{2{\mu }_H}{T},-,\frac{\dot{\theta }(T)}{T}\right],\hfill \\ \hfill -\frac{d}{dlnT}\left(\frac{{\mu }_{q_{12}}}{T}\right)& =\frac{1}{g_{q_{12}}}\frac{4{\gamma }_{\mathit{SS}}}{H}\hfill \\ \hfill & \left[-,\frac{4{\mu }_{q_{12}}}{T},-,\frac{{\mu }_t}{T},-,\frac{{\mu }_b}{T},+,\frac{4{\mu }_{Q_{12}}}{T},+,\frac{2{\mu }_{Q_3}}{T}\right],\hfill \\ \hfill -\frac{d}{dlnT}\left(\frac{{\mu }_t}{T}\right)& =\frac{1}{g_t}\frac{{\gamma }_{\mathit{SS}}}{H}\hfill \\ \hfill & \left[-,\frac{4{\mu }_{q_{12}}}{T},-,\frac{{\mu }_t}{T},-,\frac{{\mu }_b}{T},+,\frac{4{\mu }_{Q_{12}}}{T},+,\frac{2{\mu }_{Q_3}}{T}\right]\hfill \\ \hfill & +\frac{1}{g_t}\frac{{\gamma }_{Y_t}}{H}\left[-,\frac{{\mu }_t}{T},+,\frac{{\mu }_{Q_3}}{T},+,\frac{{\mu }_H}{T}\right],\hfill \\ \hfill -\frac{d}{dlnT}\left(\frac{{\mu }_b}{T}\right)& =\frac{1}{g_b}\frac{4{\gamma }_{\mathit{SS}}}{H}\hfill \\ \hfill & \left[-,\frac{4{\mu }_{q_{12}}}{T},-,\frac{{\mu }_t}{T},-,\frac{{\mu }_b}{T},+,\frac{4{\mu }_{Q_{12}}}{T},+,\frac{2{\mu }_{Q_3}}{T}\right]\hfill \\ \hfill & +\frac{1}{g_b}\frac{{\gamma }_{Y_b}}{H}\left[-,\frac{{\mu }_b}{T},+,\frac{{\mu }_{Q_3}}{T},-,\frac{{\mu }_H}{T}\right],\hfill \\ \hfill -\frac{d}{dlnT}\left(\frac{{\mu }_{Q_{12}}}{T}\right)& =-\frac{1}{g_{Q_{12}}}\frac{4{\gamma }_{\mathit{SS}}}{H}\hfill \\ \hfill & \left[-,\frac{4{\mu }_{q_{12}}}{T},-,\frac{{\mu }_t}{T},-,\frac{{\mu }_b}{T},+,\frac{4{\mu }_{Q_{12}}}{T},+,\frac{2{\mu }_{Q_3}}{T}\right]\hfill \\ \hfill & -\frac{1}{g_{Q_{12}}}\frac{6{\gamma }_{\mathit{WS}}}{H}\left[\frac{2{\mu }_{L_{12}}}{T},+,\frac{{\mu }_{L_3}}{T}\right)\hfill \\ \hfill & \left(+,\frac{6{\mu }_{Q_{12}}}{T},+,\frac{3{\mu }_{Q_3}}{T},-,\frac{3}{2},\frac{\dot{\theta }(T)}{T}\right],\hfill \\ \hfill -\frac{d}{dlnT}\left(\frac{{\mu }_{Q_3}}{T}\right)& =-\frac{1}{g_{Q_3}}\frac{2{\gamma }_{\mathit{SS}}}{H}\hfill \\ \hfill & \left[-,\frac{4{\mu }_{q_{12}}}{T},-,\frac{{\mu }_t}{T},-,\frac{{\mu }_b}{T},+,\frac{4{\mu }_{Q_{12}}}{T},+,\frac{2{\mu }_{Q_3}}{T}\right]\hfill \\ \hfill & -\frac{1}{g_{Q_3}}\frac{3{\gamma }_{\mathit{WS}}}{H}\left[\frac{2{\mu }_{L_{12}}}{T},+,\frac{{\mu }_{L_3}}{T},+,\frac{6{\mu }_{Q_{12}}}{T}\right)\hfill \\ \hfill & \left(+,\frac{3{\mu }_{Q_3}}{T},-,\frac{3}{2},\frac{\dot{\theta }(T)}{T}\right]\hfill \\ \hfill & -\frac{1}{g_{Q_3}}\frac{{\gamma }_{Y_t}}{H}\left[-,\frac{{\mu }_t}{T},+,\frac{{\mu }_{Q_3}}{T},+,\frac{{\mu }_{Q_H}}{T}\right]\hfill \\ \hfill & -\frac{1}{g_{Q_3}}\frac{{\gamma }_{Y_b}}{H}\left[-,\frac{{\mu }_b}{T},+,\frac{{\mu }_{Q_3}}{T},-,\frac{{\mu }_{Q_H}}{T}\right],\hfill \\ \hfill -\frac{d}{dlnT}\left(\frac{{\mu }_H}{T}\right)& =-\frac{1}{g_H}\frac{{\gamma }_{Y_{\tau }}}{H}\left[-,\frac{{\mu }_{\tau }}{T},+,\frac{{\mu }_{L_3}}{T},-,\frac{{\mu }_H}{T}\right]\hfill \\ \hfill & -\frac{1}{g_H}\frac{{\gamma }_{Y_t}}{H}\left[-,\frac{{\mu }_t}{T},+,\frac{{\mu }_{Q_3}}{T},+,\frac{{\mu }_{Q_H}}{T}\right]\hfill \\ \hfill & +\frac{1}{g_H}\frac{{\gamma }_{Y_b}}{H}\left[-,\frac{{\mu }_b}{T},+,\frac{{\mu }_{Q_3}}{T},-,\frac{{\mu }_{Q_H}}{T}\right]\hfill \\ \hfill & -\frac{1}{g_H}\frac{2{\gamma }_{W_{12}}}{H}\hfill \\ \hfill & \times \left[\frac{2{\mu }_{L_{12}}}{T},+,\frac{2{\mu }_H}{T},-,\frac{\dot{\theta }(T)}{T}\right]\hfill \\ \hfill & -\frac{1}{g_H}\frac{2{\gamma }_{W_3}}{H}\left[\frac{2{\mu }_{L_3}}{T},+,\frac{2{\mu }_H}{T},-,\frac{\dot{\theta }(T)}{T}\right],\hfill \end{array}$$

[See PDF for image]

Fig. 4

The red curve shows the baryon asymmetry as the function of the temperature. The gray band represents the observed baryon asymmetry from Planck Collaboration. The relevant input parameters are the same as those in Fig. 2

where the transport rate per unit time ${\gamma }_{\alpha }\equiv 6{\mathrm{\Gamma }}_{\alpha }/T^3$ are given in the Table 1 of the Ref. [19]. Next, we set the initial value of the chemical potential for 11 particle species as ${\mu }_i(f_J)=0$, and the remainning parameters are selected as [38, 39]

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$$\begin{array}{cc}& \boxed{f_J={10}^{13}\text{GeV}}:g_2=0.55,g_3=0.59,\hfill \\ \hfill & y_{\tau }=0.01,y_t=0.50,y_b=8.1\times {10}^{-3},m_{\nu }=0.05\text{eV}.\hfill \end{array}$$We show in Fig. 4 the evolution of BAU as the function of the temperature. The numerical parameters selected here are the same as those in the Fig. 2. The Horizontal gray band represents the observed baryon asymmetry: $Y_B=(8.65\pm 0.09)\times {10}^{-11}$ [32]. Note that from Eq. (13) only the the electroweak spaleron and the Weinberg operator contribute non-zero sorce terms. However, the Weinberg operator is the only source that violates the $\mathbf{B}-\mathbf{L}$ symmetry, it is ten times more effective than the electroweak spaleron [19]. As can be seen in the plot, we use red curve to represent the total BAU associated with $f_J={10}^{13}$ GeV. Near the temperature at which the $\mathrm{U}{(1)}_{\mathrm{L}}$ symmetry is spontaneously broken, the Weinberg operator is highly effective and the BAU increases at the beginning. Then the BAU drops subsequently caused by the wash-out effects since the electroweak spaleron produces a negative BAU. Once the universe cools down to the temperature of $T_W$, the interaction associated with Weinberg operator is out of thermal equilibrium and the total BAU becomes conserved. Due to the exponentially suppressed of the interaction rate of the electroweak sphaleron process after the SM electroweak phase transition, the baryon asymmetry is frozen around $T_{\text{sp,L}}\equiv 130$ GeV [40, 41], which is consistent with the observed result at present. Figures 2 and 4 indicate that both observed dark matter relic density and baryon asymmetry of the universe can be explained in the case of ${\dot{\theta }}_i=-3.285\times {10}^6$ GeV.

Direct detection of Majoron dark matter

In this section we turn to investigate the directly detectable signals for this type of Majoron dark matter on earth. As mentioned in our previous work of Ref. [33], due to the chiral anomalies cancellation, the Majoron is not coupled with electrons and di-photons but only with neutrinos and electroweak gauge bosons [21, 33, 42, 43–44]:

[See PDF for image]

Fig. 5

Inverse Primakoff process

[See PDF for image]

Fig. 6

The excluded parameter constraints on Majoron-photon-Z boson coupling (a) and decay constant (b) versus Majoron mass. The results from direct detection experiments including PandaX-4T, XENONnT and PandaX-30T are depicted with blue solid lines, cyan solid lines and blue dashed lines, respectively. The purple and orange regions in a show the constraints from LHC and LEP experiments

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$$\begin{array}{cc}& {\mathcal{L}}_{\text{int}}=\frac{m_{\nu }}{2f_J}J\overline{\nu }i{\gamma }^5\nu -N_f\frac{{\alpha }_{\text{em}}}{16\pi f_J{sin}^2{\theta }_W}J{W}_{\mu \nu }^{+}{\tilde{W}}^{-\mu \nu }\hfill \\ \hfill & -N_f\frac{{\alpha }_{\text{em}}}{32\pi f_{J}sin{\theta }_{W}cos{\theta }_W}J\left(Z_{\mu \nu },{\tilde{F}}^{\mu \nu },+,F_{\mu \nu },{\tilde{Z}}^{\mu \nu }\right)\hfill \\ \hfill & -N_f\frac{{\alpha }_{\text{em}}}{8\pi f_J{sin}^2{\theta }_W{cos}^2{\theta }_W}\left(\frac{1}{2},-,{sin}^2,{\theta }_W\right)J{Z}_{\mu \nu }{\tilde{Z}}^{\mu \nu },\hfill \end{array}$$where $N_f=3$ is the number of generation of SM leptons, ${\alpha }_{\text{em}}$ is the fine-structure constant, and ${\theta }_W$ is the Weinberg angle. However, since the Majoron-neutrino interaction is double suppressed by the factors of tiny neutrino mass $m_{\nu }$ and the extremely high decay constant $f_J$, it is almost impossible to detect the Majoron in terrestrial experiments with this scenario. Due to the interactions between Majoron-Z boson-photon ($JZ\gamma$) induced by the chiral anomalies, there exists an Inverse Primakoff process mediated by Z boson in the form of [45, 46–47]

19

$$\begin{array}{c}\hfill J+\text{T}\to \gamma +\text{T},\end{array}$$where $\text{T}$ represents the various targets, and the diagram of this detection channel is depicted in Fig. 5.

The expected number of events for Majoron signals from the Inverse Primakoff is written as [48, 49–50]

20

$$\begin{array}{c}\hfill N_{\mathrm{event}}=N_T·{\mathrm{\Phi }}_J·{\sigma }^{\mathrm{IP}},\end{array}$$where $N_T$ is the number of targets, ${\mathrm{\Phi }}_J$ is the Majoron flux in the initial state and ${\sigma }^{\mathrm{IP}}$ denotes the scattering cross section of Inverse Primakoff process. Due to the lack of Majoron-electron interaction, the Majoron flux can be only produced through the Primakoff process [51, 52, 53, 54, 55–56]. Then the total differential scattering cross section of the Inverse Primakoff process is

21

$$\begin{array}{c}\hfill \frac{d{\sigma }_{\text{total}}^{\mathrm{IP}}}{d\mathrm{\Omega }}=\frac{d{\sigma }_e^{\mathrm{IP}}}{d\mathrm{\Omega }}+\frac{d{\sigma }_p^{\mathrm{IP}}}{d\mathrm{\Omega }}+\frac{d{\sigma }_n^{\mathrm{IP}}}{d\mathrm{\Omega }},\end{array}$$where the subscripts e, p and n represent the case where the target is an electron, proton and neutron, respectively. Their analytic expressions are given in the appendix. We show the Constraints on Majoron-photon-Z boson coupling $g_{JZ\gamma }$ and Majoron mass $m_J$ in Fig. 6. The green, dashed blue and red curves represent the Constraints from XENONnT, PandaX-4T and PandaX-30T direct detection experiments, respectively. The results from LHC and LEP experiments [57, 58–59] are shown in the colored regions for comparison.

Conclusion

In this paper, we have investigated the common origin of Majoron dark matter and baryon asymmetry of the universe from a high scale inverse seesaw mechanism. A lepton-number-violating mass term of the Majorana neutrino gives rise to a non-zero mass and initial velocity for Majoron dark matter, resulting in its relic abundance through the kinetic misalignment mechanism. In addition, the same initial velocity converts the original lepton asymmetry into baryon asymmetry. Then two new physics phenomena can be explained naturally in a concise model. Furthermore, we would like to make a brief comment on the neutrino mass of $m_{\nu }=0.05$ eV selected in the numerial analysis. As we can see, the traditonal inverse seesaw mechanism gives a neutrino mass in the form of $m_{\nu }=M_D{\mu }_S{M}_D^T/(M_S^T{M}_S)$ [37], which implies $m_{\nu }\ll 0.05$ eV in this model. In other words, this modified inverse seesaw model aims to account for both dark matter and baryon asymmetry at the expense of explaining the active neutrino mass. It is unable to provide a sufficiently large neutrino mass, which may come from other mechanisms such as radiation generation mechanisms [60, 61, 62, 63–64].

Acknowledgements

The authors thank Jing-Jing Feng and Xin Wang for valuable discussions.

Funding

Ying-Quan Peng is supported by the funding support from Sichuan University of Science and Engineering (No.2024RC027). Xiu-Fei Li is supported by Inner Mongolia Autonomous Region Natural Science Foundation under grant No.2025QN01030, Scientific Research Foundation of Inner Mongolia University under grant No.10000-22311201/036 and Scientific Research Foundation of Inner Mongolia Autonomous Region under grant No.12000-15042261.

Data Availability Statement

My manuscript has no associated data. [Authors’ comment: Data sharing not applicable to this article as no datasets were generated or analysed during the current study.]

Code Availability Statement

My manuscript has no associated code/software. [Authors’ comment: Code/Software sharing not applicable to this article as no code/software was generated or analysed during the current study.]

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