Low-energy CP violation and leptogenesis in a minimal seesaw model

Abstract

We investigate whether CP phases in the neutrino mixing matrix can be the source of the CP violation necessary to achieve leptogenesis. In general, low-energy CP violation in the neutrino sector is not directly linked to leptogenesis, but we show that the low-energy leptonic CP violation can give rise to the CP asymmetry required for leptogenesis in a minimal seesaw model where the Dirac neutrino mass matrix contains one-zero texture. Performing numerical analyses based on the current results of the global fit to neutrino data and the measurement of baryon asymmetry, we study how the CP phases in the neutrino mixing matrix can be constrained and show how lepton asymmetry is sensitive to two heavy Majorana neutrino masses as well as CP phases. From the constraints on the neutrino parameters, the values of the effective neutrino mass contributing to the amplitude of neutrinoless double beta decay are predicted.

1 Introduction

Thanks to the enormous number of neutrino experiments, we have been able to determine three neutrino mixing angles in neutrino mixing matrix. The remaining unknown parameters in the neutrino mixing matrix are CP violating phases. Thus, establishing leptonic CP violation (LCPV) is one of the most challenging tasks for future neutrino experiments [1]. Although we do not yet have compelling evidence for LCPV, the current global fit to available neutrino data indicates nontrivial values for the Dirac-type CP phase [2]. Recent measurements from T2K and MINOS indicate a preference for CP violation (CVP) with around $1.5 \pi$ for the Dirac-type CP phase at $1\sigma$ confidence level (C.L.) [3,4,5,6]. Much attention has been paid to the prediction of the Dirac-type LCPV phase with regards to some observables [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. Recently, it has been shown [28,29,30] that Dirac-type leptonic CP phase can be particularly predictable in terms of two neutrino mixing angles in the standard parameterization of the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) mixing matrix [31,32,33,34,35]. If neutrinos are Majorana particles, the PMNS mixing matrix will contain two more CP phases, in general. Those Majorana CP phases are not measurable through neutrino oscillations, but can be probed through neutrinoless double beta decay [36].

Baryogenesis through leptogenesis demands new source of CPV in lepton sectors [37]. From the perspective of leptogenesis, examining whether the low-energy leptonic CP phases can be sources of the CPV required for successful leptogenesis in the type-I seesaw framework is worthwhile. As will be shown later, in general, the low-energy CP phases are not directly related with leptogenesis, which is obvious from the description of the Dirac neutrino mass matrix in terms of the neutrino mixing matrix, an orthogonal matrix and diagonal mass matrices of light and heavy Majorana neutrinos in the type-I seesaw mechanism, which is the so-called Casas–Ibarra parametrization [38]. From the parameterization, one can see that the CPV necessary to achieve leptogenesis depends only on the CP phases in the orthogonal matrix, which are totally unknown.

In this work, we propose a scenario in which the orthogonal matrix in the Casas–Ibarra parameterization can be presented in terms of the entries in the PMNS mixing matrix and two neutrino masses. The model we consider is the so-called minimal seesaw model with two heavy Majorana neutrinos [39,40,41,42,43]. In the minimal seesaw model, one of the light neutrino masses should be zero and only one Dirac and one Majorana phase generically survive, so the model is very predictive because of the small number of independent parameters compared with the canonical seesaw model with three heavy Majorana neutrinos [44,45,46,47,48,49,50,51]. Inspired by the idea of the calculability of the quark mixing angles in terms of the quark masses suggested by Weinberg a long time ago [52], we introduce one-zero texture in the Dirac neutrino mass matrix and then show how the CP phases in the PMNS mixing matrix can be the source of CPV in leptogenesis. Carrying out numerical analyses based on the current results of the global fit to neutrino data, we show how the unknown parameters are constrained by confronting our predictions with the measurement for baryon asymmetry. From the constraints on the parameters, we also predict the so-called effective light neutrino mass which contributes to the amplitude of neutrinoless double beta decay [36] .

This paper is organized as follows: In Sect. 2, we investigate whether low-energy leptonic CP phases can give rise to the CP asymmetry in leptogenesis. In Sect. 3, we briefly review the minimal seesaw model and show how two CP phases in the PMNS mixing matrix are directly related to leptogenesis. In Sect. 4, we present the results of the constraints on the parameter space from a numerical analysis and the prediction of the effective neutrino mass contributing to the amplitude of neutrinoless double beta decay. We finally conclude in Sect. 5.

2 CP phases in the PMNS mixing matrix and leptogenesis

One of the important questions concerning leptogenesis is “Are the CP phases in the PMNS mixing matrix responsible for leptogenesis?”. To get the answer to this question, we examine how leptogenesis in the type-I seesaw model depends on the neutrino parameters. The Lagrangian for the lepton sector of the type-I seesaw model is given by [39,40,41]

$$\begin{aligned} -\mathcal{L}= Y_{li}\overline{l_{L}}_{i} \phi ^0 {l_{R}}_{j} + {Y_{\nu }}_{ij} \overline{\nu _{L}}_{i} \phi ^0 {N_{R}}_{j} + \frac{1}{2}\overline{(N_{R})^c}_{i} M_{j} {N_{R}}_{j}, \end{aligned}$$

(1)

where $i,j=1,2,3,$ and $Y_{\nu }$ is a $3 \times 3$ complex matrix. ${l_{L}}_i,{ l_{R}}_i, {\nu _{L}}_i$, and ${N_R}_i$ stand for left-handed, right-handed charged leptons, left-handed neutrino and right-handed Majorana neutrino, respectively. For our purpose, we take a basis for which heavy Majorana neutrino mass matrix M is diagonal and the charged lepton Yukawa matrix $Y_l$ is real and diagonal.

From the seesaw mechanism [53,54,55,56,57], the effective light neutrino mass matrix is given by

$$\begin{aligned} m_{\text {eff}}=Y_{\nu } \frac{1}{M} Y_{\nu }^Tv^2, \end{aligned}$$

(2)

where $v=<\phi ^0>$ is the vacuum expectation value of Higgs scalar. The effective neutrino mass matrix $m_{\text {eff}}$ can be diagonalized by using the PMNS mixing matrix given as follows [34, 35]:

$$\begin{aligned} U_{\mathrm{PMNS}}= & {} U P_{\beta }\nonumber \\= & {} \left( \begin{array}{ccc} c_{12}c_{13} &{} -s_{12}c_{13} &{} -s_{13}e^{i\delta _D}\\ s_{12}c_{23}-c_{12}s_{23}s_{13}e^{-i\delta _D} &{} c_{12}c_{23}+s_{12}s_{23}s_{13}e^{-i\delta _D}&{} -s_{23}c_{13} \\ s_{12}s_{23}+c_{12}c_{23}s_{13}e^{-i\delta _D} &{} c_{12}s_{23}-s_{12}c_{23}s_{13}e^{-i\delta _D}&{} c_{23}c_{13} \\ \end{array}\right) P_{\beta } , \end{aligned}$$

(3)

where $P_{\beta }$ is $3\times 3$ phase matrix given by $\mathrm{Diag.}~[e^{i\beta _1}, e^{i\beta _2},e^{i\beta _3}]$. Among the three phases in $P_{\beta }$, one is removable, in general. Then, the following relation holds:

$$\begin{aligned} m_{\text {eff}}=U^{*}_{\mathrm{PMNS}} m^D_{\nu } U^{\dagger }_{\mathrm{PMNS}}, \end{aligned}$$

(4)

where $m^D_{\nu }$ is a diagonal neutrino mass matrix presented by $\text{ Diag. } [m_1,m_2,m_3]$.

Leptogenesis realized in the type-I seesaw model requires CPV in the decay of a heavy Majorana neutrino into a left-handed lepton and the Higgs scalar, which originates from the nontrivial CP phases in the Yukawa matrix $Y_{\nu }$. The lepton asymmetry $\varepsilon _i$ is given explicitly as [58,59,60]

$$\begin{aligned} \varepsilon _i= \frac{\sum _j[\Gamma (N_i\rightarrow i_j \phi ^{*})-\Gamma (N_i\rightarrow \bar{i}_j \phi )]}{\sum _j[\Gamma (N_i\rightarrow i_j \phi ^{*})+\Gamma (N_i\rightarrow \bar{i}_j \phi )]} =\frac{1}{8\pi }\frac{\sum _{j\ne i}\text{ Im }[({Y_{\nu }}^{\dagger }{Y_{\nu }})_{ij}]^2}{({Y_{\nu }}^{\dagger } {Y_{\nu }})_{ii}} g(x), \end{aligned}$$

(5)

where $g(x)=\sqrt{x}[1/(1-x)+1-(1+x) \ln ((1+x)/x)]$, with $x=|M^2_i/M^2_j|$. It is well known that the Dirac neutrino Yukawa matrix can be presented using the Casas–Ibarra parameterization as follows [38]:

$$\begin{aligned} Y_{\nu }=\frac{1}{v}U^{*}_{\mathrm{PMNS}} \sqrt{m^D_{\nu }}O \sqrt{M}, \end{aligned}$$

(6)

where $\sqrt{m^D_{\nu }}=\text{ Diag. } [\sqrt{m_1},\sqrt{m_2},\sqrt{m_3}]$, $\sqrt{M}=\text{ Diag. } [\sqrt{M_1},\sqrt{M_2},\sqrt{M_3}]$ and O is a $3\times 3$ complex orthogonal matrix that is totally unknown. Putting Eq. (6) into Eq. (5), one can easily see that $\varepsilon _i$ depends not on $U_\mathrm{PMNS}$ but on the complex orthogonal matrix O. The result shows that the CPV required for canonical leptogenesis originates not from the CP phases in $U_\mathrm{PMNS}$, but from new CP phases contained in O. Thus, the measurement of CPV through low-energy experiments, such as neutrino oscillations and neutrinoless double beta decay, may not be direct hint for canonical leptogenesis. However, that is not always true. Rather, leptogenesis can be realized with CP phases in $U_\mathrm{PMNS}$ in a specific model. In this work, we will show that the CP phases in $U_\mathrm{PMNS}$ are the source of the CPV necessary to achieve leptogenesis in a minimal seesaw model with one-zero texture in the Dirac neutrino mass matrix.

3 Minimal seesaw model

The minimal seesaw model (MSM) is an extension of the standard model (SM) with two heavy right-handed Majorana neutrinos [39,40,41,42,43]. The Lagrangian for the lepton sector of the MSM after electroweak symmetry breaking is given by [39,40,41,42,43]

$$\begin{aligned} -\mathcal{L}= \overline{l_{iL}} {m_l}_i l_{iR} + \overline{\nu _{Li}} {m_D}_{ij} N_{Rj} + \frac{1}{2} \overline{(N_{Rj})^c} M_{j} N_{Rj}, \end{aligned}$$

(7)

where $i=1,2,3,~ j=1,2$ and the Dirac neutrino mass term $m_D$ is a $3 \times 2$ complex matrix. Here, we take a basis for which heavy Majorana neutrino mass matrix M is diagonal, and the charged lepton mass matrix $m_{l}$ is real and diagonal. Since the generic $n\times n$ Dirac neutrino mass matrix contains $3n-3$ unremovable CP phases if n singlet heavy Majorana neutrinos are introduced in this basis, at least two singlet heavy Majorana neutrinos are necessary to break CP symmetry [42, 43]. That is why we call this the MSM.

The effective light neutrino mass matrix obtained from the seesaw mechanism is given by

$$\begin{aligned} m_{\text {eff}}=m_D \frac{1}{M} m_D^T, \end{aligned}$$

(8)

where $1/M=\text{ Diag. } [1/M_1,1/M_2]$. We can diagonalize $m_{\text {eff}}$ using $U_\mathrm{PMNS}$. Obviously one of the three light neutrino masses is zero in the MSM. For the normal hierarchy (NH) of the neutrino mass spectrum, $m_1=0$ whereas $m_3=0$ for the inverted hierarchy (IH). Thus, the lighter neutrino mass is a solar neutrino oscillation scale $\sqrt{\Delta m^2_\mathrm{sol}}$ whereas the heavier neutrino mass is an atmospheric neutrino oscillation scale $\sqrt{\Delta m^2_\mathrm{atm}}$. Thanks to this fact, only one phase in $P_{\beta }$, which is denoted as $\delta _M$, is unremovable. Explicitly, $(m_{\text {eff}})_{ij}$ can be written in terms of the entries of $U_{\mathrm{PMNS}}$ and the light neutrino masses as follows:

$$\begin{aligned} (m_{\text {eff}})_{ij}= & {} \left\{ \begin{array}{cc} (U^{*}_{\mathrm{PMNS}})_{i2}(U^{*}_{\mathrm{PMNS}})_{j2} m_2 + (U^{*}_{\mathrm{PMNS}})_{i3}(U^{*}_{\mathrm{PMNS}})_{j3} m_3, &{} \text{ for } \text{ NH } \\ (U^{*}_{\mathrm{PMNS}})_{i1}(U^{*}_{\mathrm{PMNS}})_{j1} m_1 + (U^{*}_{\mathrm{PMNS}})_{i2}(U^{*}_{\mathrm{PMNS}})_{j2} m_2, &{} \text{ for } \text{ IH }. \end{array} \right. \end{aligned}$$

(9)

From Eqs. (4) and (8), one can obtain the relation [38]

$$\begin{aligned} m_D \frac{1}{\sqrt{M}} O^{T}=U^{*}_{\mathrm{PMNS}} \sqrt{m^D_{\nu }}, \end{aligned}$$

(10)

where O is a $2\times 2$ complex orthogonal matrix that is totally unknown, and the zero column on the right-hand side of Eq. (10) due to a zero mass eigenvalue of $m^D_{\nu }$ is ignored. We parameterize O in terms of two complex variables x and y as follows:

$$\begin{aligned} O=\left( \begin{array}{cc} x &{} y \\ -y &{} x \end{array} \right) . \end{aligned}$$

(11)

The construction of the neutrino mass matrix is necessary for model building, which, in turn, may unravel the underlying dynamics of the generation of neutrino masses, mixing and CPV. A long time ago, Weinberg proposed that the so-called “Cabibbo mixing angle”, the mixing angle between the d-quark and the s-quark, could be predicted in terms of the ratio of two quark masses, $m_d$ and $m_s$, at leading order by assuming that the mixing mass term $\bar{d}_L d_R$ in a weak basis is zero whereas the other mixing mass terms are nonzero [52]. Taking the zeros in the fermion mass matrix or fermion Yukawa matrix has been well known to make the mixing parameters predictive. In this context, the zero texture approach has been widely followed in the literature. In particular, the two-zero texture has been relatively more successful in both the flavor, as well as the non-flavor, basis [61,62,63,64,65,66,67,68,69]. Following this idea, we propose an ansatz that one entry of the Dirac neutrino mass matrix $m_D$ is zero. As will be shown, the ansatz causes the CP phases in $U_\mathrm{PMNS}$ to become the source of the CPV required for leptogenesis. Without loss of generality, among the six possible entries in the Dirac neutrino mass matrix, we study only the case in which the (1,1) entry of $m_D$ is zero. Zeros for the other entries do not alter the main conclusion in this work. Then, from Eq. (10), the requirement of $(m_D)_{11}=0$ leads to the following relation,

$$\begin{aligned} y=\frac{ (U_{\mathrm{PMNS}})_{12}\sqrt{m_2}}{ (U_{\mathrm{PMNS}})_{13}\sqrt{m_3}} x. \end{aligned}$$

(12)

Applying the orthogonality of the matrix O, we finally obtain

$$\begin{aligned} x^2= & {} \frac{ (U_{\mathrm{PMNS}})_{13}^2 m_3}{ (U_{\mathrm{PMNS}})_{12}^2 m_2+ (U_{\mathrm{PMNS}})_{13}^2 m_3}, \end{aligned}$$

(13)

$$\begin{aligned} y^2= & {} \frac{ (U_{\mathrm{PMNS}})_{12}^2 m_2}{ (U_{\mathrm{PMNS}})_{12}^2 m_2+ (U_{\mathrm{PMNS}})_{13}^2 m_3}. \end{aligned}$$

(14)

We see from Eqs. (13) and (14) that the orthogonal matrix can be parameterized using the PMNS mixing matrix and light neutrino masses, so leptogenesis in this scenario depends only on the CP phases measurable in low-energy experiments such as neutrino oscillations and neutrinoless double beta decay. In the end, the Dirac neutrino mass matrix in the weak basis can be written as

$$\begin{aligned} m_D= \left( \begin{array}{cc} (W_{12} x -W_{13} y )\sqrt{M_1} &{} (W_{12} y+W_{13} x )\sqrt{M_2} \\ (W_{22} x -W_{23} y)\sqrt{M_1} &{} (W_{22} y+W_{23} x)\sqrt{M_2} \\ (W_{32} x -W_{33} y)\sqrt{M_1}&{} (W_{32} y+W_{33} x)\sqrt{M_2} \end{array} \right) , \end{aligned}$$

(15)

where $W_{i2}= (U_{\mathrm{PMNS}})_{i2}\sqrt{m_2}$ and $W_{i3}= (U_{\mathrm{PMNS}})_{13}\sqrt{m_3}$. Inserting Eq. (15) into Eq. (5), we can predict the CP asymmetry $\varepsilon _i$ in terms of three neutrino mixing angles; two CP phases in $U_\mathrm{PMNS}$; and two light neutrino masses $m_1, m_2$ and two heavy neutrino masses $M_1, M_2$. Among them, the two heavy neutrino masses and a Majorana type CP phase $\delta _M$ are totally unknown. For the Dirac type CP phase $\delta _D$, although it has not yet been measured, a possible range derived from the recent result of the global fit analysis, which will be used to calculate $\varepsilon _i$ in our numerical analysis, is known to exist. Since we consider that leptogenesis is generated by the decay of the lightest Majorana neutrino ${N_R}_1$ under the assumption $M_1<M_2$, taking into account $\varepsilon _1$ in our study is sufficient.

On the other hand, the light neutrino masses, the mixing angles and the CP phases in $U_\mathrm{PMNS}$ may affect the amplitude of $\nu 0 \beta \beta$, which is proportional to $|\sum _{i}U_{ei}^2 m_i|$ explicitly presented as follows [36]:

$$\begin{aligned} |\sum _{i}U_{ei}^2 m_i| \equiv M_{ee}=\left\{ \begin{array}{cc} \left| m_2 s^2_{12}c^2_{13}+m_3 s^2_{13} e^{-2 i (\delta _D+\delta _M)}\right| , &{}~~~~ \text{ for } \text{ NH } \\ c_{13}^2\left| m_1 c_{12}^2+m_2 s^2_{12} e^{2 i \delta _M}\right| , &{} ~~~~~\text{ for } \text{ IH }. \end{array} \right. \end{aligned}$$

(16)

For NH, $M_{ee}$ depends on both $\delta _D$ and $\delta _M$ whereas only $\delta _M$ contributes to $M_{ee}$ for IH. Since $s^2_{13}$ is quite small, we expect the amplitude for IH to be much larger than that for NH.

In the next section, we numerically study how the unknown parameters in the MSM can be constrained by the measurement of the baryon asymmetry and the recent results of the global fit to neutrino data. Using constraints on the CP phases, we also show how the values of $M_{ee}$ can be predicted.

4 Numerical results

We numerically analyze how the independent parameters discussed above are constrained. For our numerical analysis, we adopt the latest result for the three neutrino mixing angles and the Dirac-type CP phase $\delta _D$ as inputs taken from the global fit to neutrino data at the $1\sigma$ C.L. [2], which are explicitly presented in Table 1. For two light neutrino masses, we use the best fit results, $\Delta m^2_{21}=7.4\times 10^{-5} \text{ eV}^2$ for the solar mass scale and $|\Delta m^2_{31(2)}|=2.415\times 10^{-3}~\text{ eV}^2$ for the atmospheric mass scale.

The measured value of the ratio of the baryon asymmetry to the photon number density, $\eta _B$, is given by [70]

$$\begin{aligned} \eta _B= \frac{n_B - n_{\bar{B}}}{n_\gamma } = (8.65\pm 0.085) \times 10^{-11}, \end{aligned}$$

(17)

where $n_{B(\bar{B})}$ and $n_\gamma$ are the baryon (anti-baryon) number density and the photon number density, respectively. Following Ref. [18], the asymmetry $\eta _B$ can be explicitly given as

$$\begin{aligned} \eta _B =\frac{3}{4}\frac{a_{sph}}{f}\varepsilon _i \kappa _f(K_i) \quad K_i = \frac{\Gamma _D}{H(M_i)}~~(i=1,2), \end{aligned}$$

(18)

where we have taken the values of $a_{sph}$ and f presented in Ref. [18] and

$$\begin{aligned} \Gamma _D= & {} \frac{(Y^{\dagger }_{\nu }Y_{\nu })_{11}}{16 \pi }M_1, \nonumber \\ H(T)= & {} \sqrt{\frac{4\pi ^3g_*}{45}}\frac{T^2}{M_{pl}}, \nonumber \\ z_B(K_i)= & {} 1 + \frac{1}{2}\ln \left( 1 + \frac{\pi K_i^2}{1024}\left[ \ln \left( \frac{3125\pi K_i}{1024}\right) \right] ^5\right) , \nonumber \\ \kappa _f(K_i)= & {} \frac{2}{z_B(K_i)K_i}\left( 1-\exp \left\{ -\frac{1}{2}z_B(K_i)K_i\right\} \right) . \end{aligned}$$

(19)

The parameter $\kappa _f$ is the so-called efficiency factor, which is studied in Refs. [21, 22], and $g_{*}$ is the number of relativistic particles after the decay of ${N_R}_i$ is decoupled [71]. In our numerical analysis, we use the analytic fits for $\kappa _f$ explicitly presented in Ref. [18] without solving the complicated Boltzmann equations which are in good approximations to the actual solutions.

Table 1 Three neutrino mixing angles and the Dirac-type CP phase from the latest global fit to neutrino data at $1\sigma$ [2]

In this work, we first calculate $\eta _B$ by scanning three neutrino mixing angles and $\delta _D$ for the range of $1\sigma$ C.L. from the results of the global fit and $\delta _M$ from 0 to $2\pi$ for fixed values of $M_1$ and $M_2$. Then, comparing the numerical results for $\eta _B$ with the experimental results given by Eq. (17), we obtain allowed regions of parameter space. From our numerical analysis, we have found that leptogenesis works only for $M_1>5 \times 10^{9}$ GeV and that the results are almost independent of $M_2$ for $M_2 \gtrsim 100 M_1$.¹ Thus, for the sake of simplicity, we keep $M_2 = 100 M_1$ in our analysis.

4.1 Case of NH

Fig. 1

Allowed region of parameter space $(\delta _D, \delta _M)$. Left (Right) panel corresponds to $M_1=10^{10}$ ($M_2=10^{13}$) GeV. The purple, green and light blue regions correspond to $M_2=10^{12}$ GeV, $10^{13}$ GeV and $10^{14}$ GeV in the left panel and to $M_1=5\times 10^{10}$ GeV, $10^{11}$ GeV and $10^{12}$ GeV in the right panel, respectively

Figure 1 shows the allowed region of parameter space $(\delta _D, \delta _M$) for a fixed $M_1=10^{10}$ GeV (left panel) and for a fixed $M_2=10^{13}$ GeV (right panel) in the case of NH of the neutrino mass spectrum. The purple, green and light blue regions correspond to $M_2=10^{12}$ GeV, $10^{13}$ GeV and $10^{14}$ GeV in the left panel and to $M_1=5\times 10^{10}$ GeV, $10^{11}$ GeV and $10^{12}$ GeV in the right panel, respectively. The results show that small values of $\delta _M (<\pi /4)$ are allowed for $\delta _D <\pi$ and that $\delta _M$ decreases from $2\pi$ as $\delta _D$ increases in the range $\pi<\delta _D<1.4\pi$. Comparing both panels, we see that leptogenesis in this scenario is more sensitive to $M_1$ than $M_2$. For a given $\delta _D$, a smaller $\delta _M$ for $\delta _M>\pi$ and a larger one for $\delta _M <\pi$ are allowed as $M_1$ increases. We also see from Fig. 1 that some regions of $\delta _M$ are excluded, for instance, $\pi /4<\delta _M<\pi /2$ for the cases of $M_2=10^{14}$ GeV (left panel) and $M_1=10^{11}$ GeV (right panel).

Fig. 2

Prediction of $M_{ee}$ (eV) in terms of $\delta _D$ (left panel) and $\delta _M$ (right panel) for the allowed regions of parameters obtained in Fig. 1. The meanings of the colors are the same as in Fig. 1

Figure 2 shows how the values of $M_{ee}$ are predicted in terms of $\delta _D$ (left panel) and $\delta _M$ (right panel) for the allowed regions of the parameters obtained in Fig. 1. As expected, $M_{ee}$ is insensitive to $M_1$ and $M_2$. Since $M_{ee}$ depends on a combination of $\delta _D$ and $\delta _M$, as can be seen in Eq. (16), the shapes of the predicted regions in both panels are the same. Smaller values of $\delta _D (\delta _M$) lead to larger values of $M_{ee}$.

4.2 Case of IH

In the case of IH, because $m_3=0$, $\delta _D$ affects neither leptogenesis nor the amplitude of neutrinoless double beta decay. In Fig. 3, we plot the allowed region of parameter space $(\delta _D, \delta _M)$ for fixed $M_1=10^{10}$ GeV (left panel) and $M_2=10^{13}$ GeV (right panel). The meanings of the colors in the plots are the same as in Fig. 1. Figure 4 shows how the values of $M_{ee}$ are predicted in terms of $\delta _D$ (left panel) and $\delta _M$ (right panel) for the allowed regions of parameters obtained in Fig. 3. In Figs. 3 and 4, we see that the allowed regions obtained from the constraint on $\varepsilon _1$ are independent of $\delta _D$. Only the regions near $\delta _M=0, 2\pi$ are allowed. In addition, the results are insensitive to the heavy Majorana neutrino masses. As expected, the predicted values of $M_{ee}$ for IH are much larger than those for NH.

Fig. 3

Allowed region of parameter space $(\delta _D, \delta _M)$. Left (Right) panel corresponds to $M_1=10^{10}$ ($M_2=10^{13}$) GeV. The purple, green and light blue regions correspond to $M_2=10^{12}$ GeV, $10^{13}$ GeV and $10^{14}$ GeV in the left panel and to $M_1=5\times 10^{10}$ GeV, $10^{11}$ GeV and $10^{12}$ GeV in the right panel, respectively

Fig. 4

Prediction of $M_{ee}$ (eV) in terms of $\delta _D$ (left panel) and $\delta _M$ (right panel) for the allowed regions of parameters obtained in Fig. 3

5 Conclusion

In this work, we have addressed whether the LCPV measurable through low-energy experiments such as neutrino oscillations and neutrinoless double beta decay can be responsible for the CP asymmetry necessary to achieve leptogenesis. In general, low-energy CPV is not directly linked to leptogenesis, but we have shown that the low-energy CP phases in $U_\mathrm{PMNS}$ can be the source of the CP asymmetry required for leptogenesis in the minimal seesaw model where the Dirac neutrino mass matrix contains one-zero texture. Performing numerical analyses based on the current results of the global fit to neutrino data at $1\sigma$, we have obtained allowed regions of the two CP phases in $U_\mathrm{PMNS}$ by comparing our calculations with the measurement of baryon asymmetry, and we discuss how they depend on two heavy Majorana neutrino masses. From the constraints on the neutrino parameters, we have studied how the values of $M_{ee}$, which contributes to the amplitude of neutrinoless double beta decay, are predicted in terms of CP phases.