Neutrinos in the flavor-dependent $U(1)_{F}$ model

The fermion sector in the Standard Model (SM) exhibits a large hierarchical structure of masses across the three families, and the quark flavor mixings described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix Cabibbo:1963yz ; Kobayashi:1973fv are not predicted from first principles within the SM. These phenomena are known as the mass hierarchy puzzle and the flavor puzzle, which indicate the observed fermionic mass spectrum and mixings are still enigmatic in particle physics. Additionally, the observed tiny but nonzero neutrino masses and flavor mixing among the three generations of neutrinos not only make the flavor puzzle in the SM more acutely, but also provide unambiguous evidence of new physics (NP) beyond the SM.

Addressing the mass hierarchy puzzle, flavor puzzle, nonzero neutrino masses and neutrino mixings, the flavor-dependent $U(1)_{F}$ model (FDM) proposed in our previous work Yang:2024kfs ; Yang:2024znv simultaneously accounts for these phenomena. Furthermore, the analysis demonstrates that the six quark masses, the CKM matrix can be accurately fitted within the model. This study focuses on exploring whether the neutrino sector within the FDM can account for the neutrino-related observations. Experimentally, the observed neutrino mass-squared differences are ParticleDataGroup:2024cfk

	$\displaystyle\Delta m_{12}^{2}\equiv m_{\nu_{2}}^{2}-m_{\nu_{1}}^{2}=(7.53\pm 0.18)\times 10^{-5}\;{\rm eV}^{2},$
	$\displaystyle\Delta m_{23}^{2}\equiv m_{\nu_{3}}^{2}-m_{\nu_{2}}^{2}\approx(2.455\pm 0.028)\times 10^{-3}\;{\rm eV}^{2},\;({\rm NH})$
	$\displaystyle\Delta m_{32}^{2}\equiv m_{\nu_{2}}^{2}-m_{\nu_{3}}^{2}\approx(2.529\pm 0.029)\times 10^{-3}\;{\rm eV}^{2},\;({\rm IH})$		(1)

where NH, IH correspond to normal hierarchy (NH) and inverse hierarchy (IH) neutrino masses respectively. For Majorana neutrinos, the observed neutrino oscillations can be expressed as

	$\displaystyle U_{\rm PMNS}=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\ -s_{12}c_{23}-c_{12}s_{13}s_{23}e^{i\delta}&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}&c_{13}s_{23},\\ s_{12}s_{23}-c_{12}s_{13}c_{23}e^{i\delta}&-c_{12}s_{23}-s_{12}s_{13}c_{23}e^{i\delta}&c_{13}c_{23}\end{array}\right)$		(5)
	$\displaystyle\qquad\qquad\times{\rm diag}(e^{i\rho},e^{i\sigma},1),$		(6)

where $c_{ij}\equiv\cos\theta_{ij}$, $s_{ij}\equiv\sin\theta_{ij}$, $\delta$ is usually referred to as the Dirac CP phase, and $\rho,\;\sigma$ are the possible CP phases for Majorana neutrinos. Particle Data Group (PDG) ParticleDataGroup:2024cfk collects the measured results of the mixing angles $\theta_{ij}$ and the Dirac CP phase $\delta$, the results read

	$\displaystyle s_{12}^{2}=0.307\pm 0.013,\;s_{13}^{2}=(2.19\pm 0.07)\times 10^{-2},\;\delta=(1.19\pm 0.22)\pi$
	$\displaystyle s_{23}^{2}=0.558^{+0.015}_{-0.021}\;\;({\rm NH}),\;s_{23}^{2}=0.553^{+0.016}_{-0.024}\;\;({\rm IH}).$		(7)

Massive and mixing neutrinos can induce the nonzero neutrino transition magnetic dipole moments (MDM), which are predicted to be zero in the SM. Furthermore, Dirac and Majorana neutrinos exhibit different structures in their transition MDM Fujikawa:1980yx ; Pal:1981rm ; Nieves:1981zt ; Kayser:1982br ; Shrock:1982sc ; Kayser:1984ge ; Bell:2005kz ; Bell:2006wi ; Giunti:2014ixa ; Xu:2019dxe ; Ge:2022cib . Therefore, observing the neutrino transition MDM is a crucial method to explore the neutrino-related NP. The transition MDM can be tested experimentally in various ways. For instance, measurements of neutrino scattering cross sections with electron peaks in the low momentum transfer region (e.g., the reactor experiment GEMMA Beda:2013mta and the solar experiment Borexino Borexino:2017fbd ), stellar cooling observations of red giants Diaz:2019kim and white dwarfs Corsico:2014mpa ; MillerBertolami:2014oki ; Hansen:2015lqa , and future dark matter direct detection experiments such as Xenon1T XENON:2020rca and PandaX PandaX-II:2020udv ; PandaX-II:2021nsg hold the potential to observe the neutrino transition MDM. Additionally, nonzero Majorana neutrino transition MDM within the core of supernova explosions may leave a potentially observable imprint on the energy spectra of neutrinos and antineutrinos from supernovae deGouvea:2012hg ; deGouvea:2013zp . Therefore, we also present the predicted neutrino transition MDM in the FDM.

The paper is organized as follows: In Sec.II, we present the lepton sector, including the charged lepton mass matrix, the neutrino mass matrix, and the analytical calculations of the neutrino transition MDM within the FDM. In Sec.III, we first investigate whether the lepton sector within the FDM can accommodate the measured neutrino mass-squared differences and flavor mixings. Subsequently, the numerical results of the neutrino transition MDM are presented and analyzed. We conclude the paper with a summary of findings and discussions in Sec. IV.

The FDM extends the SM by a $U(1)_{F}$ local gauge group associated with the particles’ flavor, with the third-generation fermions possessing zero $U(1)_{F}$ charge. Consequently, the third generation of right-handed neutrino is trivial under $SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)_{F}$. As a result, only two right-handed neutrinos with nonzero $U(1)_{F}$ charges are introduced, and naturally realizing the so-called minimal see-saw Ma:1998zg ; Frampton:2002qc ; Xing:2020ald in this model³³3The minimal seesaw mechanism warrants consideration and study for several compelling reasons, for example, its predictive power is enhanced by the significant reduction in the number of free parameters associated with the minimal seesaw mechanism. Further analysis about the minimal see-saw can be found in Refs. Gu:2006wj ; Davidson:2006tg ; Chan:2007ng ; Guo:2006qa ; Ren:2008yi ; Xing:2007zj ; Xing:2009ce ; Hirsch:2009ra ; Deppisch:2010fr ; Xing:2011ur ; Mondal:2012jv ; Abada:2014vea ; Abada:2014zra ; Luo:2014upa ; Mohapatra:2015gwa ; Nath:2018hjx ; Huang:2018wqp ; Xing:2020ezi ; CarcamoHernandez:2019eme .. The charges of fermions corresponding to $SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)_{F}$ are

	$\displaystyle L_{1}\sim(2,Y_{L},z),L_{2}\sim(2,Y_{L},-z),L_{3}\sim(2,Y_{L},0),$
	$\displaystyle R_{1}\sim(1,Y_{R},-z),R_{2}\sim(1,Y_{R},z),R_{3}\sim(1,Y_{R},0),$
	$\displaystyle\nu_{R_{1}}\sim(1,0,-z),\nu_{R_{2}}\sim(1,0,z),$		(8)

where nonzero $z$ denotes the $U(1)_{F}$ charge, $L_{i}$ ($L=l,\;q$) denote the $i-$generation left-handed fermion doublets, $R_{i}$ ($R=u,\;d,\;e$) denotes the $i-$generation right-handed fermion singlets in the SM, $\nu_{R_{i}}\;(i=1,2)$ denote the right-handed neutrinos. The scalar sector of the model is extended by two doublets and one singlet

	$\displaystyle\Phi_{1}=\left(\begin{array}[]{c}\phi_{1}^{+}\\ \frac{1}{\sqrt{2}}(iA_{1}+S_{1}+v_{1})\end{array}\right)\sim(2,\frac{1}{2},z),$		(11)
	$\displaystyle\Phi_{2}=\left(\begin{array}[]{c}\phi_{2}^{+}\\ \frac{1}{\sqrt{2}}(iA_{2}+S_{2}+v_{2})\end{array}\right)\sim(2,\frac{1}{2},-z),$		(14)
	$\displaystyle\Phi_{3}=\left(\begin{array}[]{c}\phi_{3}^{+}\\ \frac{1}{\sqrt{2}}(iA_{3}+S_{3}+v_{3})\end{array}\right)\sim(2,\frac{1}{2},0),$		(17)
	$\displaystyle\chi=\frac{1}{\sqrt{2}}(iA_{\chi}+S_{\chi}+v_{\chi})\sim(1,0,2z),$		(18)

where $v_{i}\;(i=1,\;2,\;3),\;v_{\chi}$ are the VEVs of $\Phi_{i},\;\chi$ respectively.

Based on the local gauge symmetry and gauge charges in Eqs. (8, 18), the Yukawa couplings in the FDM can be written as

	$\displaystyle\mathcal{L}_{Y}=Y_{u}^{33}\bar{q}_{3}\tilde{\Phi}_{3}u_{R_{3}}+Y_{d}^{33}\bar{q}_{3}\Phi_{3}d_{R_{3}}+Y_{u}^{32}\bar{q}_{3}\tilde{\Phi}_{1}u_{R_{2}}+Y_{u}^{23}\bar{q}_{2}\tilde{\Phi}_{1}u_{R_{3}}+Y_{d}^{32}\bar{q}_{3}\Phi_{2}d_{R_{2}}$
	$\displaystyle\qquad\;+Y_{d}^{23}\bar{q}_{2}\Phi_{2}d_{R_{3}}+Y_{u}^{21}\bar{q}_{2}\tilde{\Phi}_{3}u_{R_{1}}+Y_{u}^{12}\bar{q}_{1}\tilde{\Phi}_{3}u_{R_{2}}+Y_{d}^{21}\bar{q}_{2}\Phi_{3}d_{R_{1}}+Y_{d}^{12}\bar{q}_{1}\Phi_{3}d_{R_{2}}$
	$\displaystyle\qquad\;+Y_{u}^{31}\bar{q}_{3}\tilde{\Phi}_{2}u_{R_{1}}+Y_{u}^{13}\bar{q}_{1}\tilde{\Phi}_{2}u_{R_{3}}+Y_{d}^{31}\bar{q}_{3}\Phi_{1}d_{R_{1}}+Y_{d}^{13}\bar{q}_{1}\Phi_{1}d_{R_{3}}+Y_{e}^{33}\bar{l}_{3}\Phi_{3}e_{R_{3}}$
	$\displaystyle\qquad\;+Y_{e}^{32}\bar{l}_{3}\Phi_{2}e_{R_{2}}+Y_{e}^{23}\bar{l}_{2}\Phi_{2}e_{R_{3}}+Y_{e}^{21}\bar{l}_{2}\Phi_{3}e_{R_{1}}+Y_{e}^{12}\bar{l}_{1}\Phi_{3}e_{R_{2}}+Y_{e}^{31}\bar{l}_{3}\Phi_{1}e_{R_{1}}$
	$\displaystyle\qquad\;+Y_{e}^{13}\bar{l}_{1}\Phi_{1}e_{R_{3}}+Y_{R}^{11}\bar{\nu}^{c}_{R_{1}}\nu_{R_{1}}\chi+Y_{R}^{22}\bar{\nu}^{c}_{R_{2}}\nu_{R_{2}}\chi^{*}+Y_{D}^{21}\bar{l}_{2}\tilde{\Phi}_{3}\nu_{R_{1}}+Y_{D}^{12}\bar{l}_{1}\tilde{\Phi}_{3}\nu_{R_{2}}$
	$\displaystyle\qquad\;+Y_{D}^{31}\bar{l}_{3}\tilde{\Phi}_{2}\nu_{R_{1}}+Y_{D}^{32}\bar{l}_{3}\tilde{\Phi}_{1}\nu_{R_{2}}+h.c..$		(19)

As mentioned above, the quark sector in this model have been analyzed in our previous work Yang:2024kfs ; Yang:2024znv , we focus on the lepton sector in this work. After $\Phi_{1},\;\Phi_{2},\;\Phi_{3},\;\chi$ receive nonzero VEVs, the mass matrices of charged leptons and neutrinos can be written as

$\displaystyle m_{e}=\left(\begin{array}[]{ccc}0&m_{e,12}&m_{e,13}\\ m_{e,21}&0&m_{e,23}\\ m_{e,31}&m_{e,32}&m_{e,33}\end{array}\right),\;m_{\nu}=\left(\begin{array}[]{cc}0&M_{D}^{T}\\ M_{D}&M_{R}\end{array}\right),$

(25)

where $M_{D}$ is $2\times 3$ Dirac mass matrix and $M_{R}$ is $2\times 2$ Majorana mass matrix, and

	$\displaystyle m_{e,11}=m_{e,22}=0,\;m_{e,33}=\frac{1}{\sqrt{2}}Y_{e}^{33}v_{3},\;m_{e,12}=\frac{1}{\sqrt{2}}Y_{e}^{12}v_{3},\;m_{e,21}=\frac{1}{\sqrt{2}}Y_{e}^{21}v_{3},$
	$\displaystyle m_{e,13}=\frac{1}{\sqrt{2}}Y_{e}^{13}v_{1},\;m_{e,31}=\frac{1}{\sqrt{2}}Y_{e}^{31}v_{1},\;m_{e,23}=\frac{1}{\sqrt{2}}Y_{e}^{23}v_{2},\;m_{e,32}=\frac{1}{\sqrt{2}}Y_{e}^{32}v_{2},$
	$\displaystyle M_{D,11}=M_{D,22}=0,\;M_{D,12}=\frac{1}{\sqrt{2}}Y_{D}^{12}v_{3},\;M_{D,21}=\frac{1}{\sqrt{2}}Y_{D}^{21}v_{3},\;M_{D,31}=\frac{1}{\sqrt{2}}Y_{D}^{31}v_{1},$
	$\displaystyle M_{D,32}=\frac{1}{\sqrt{2}}Y_{D}^{32}v_{2},\;M_{R,12}=M_{R,21}=0,\;M_{R,11}=\frac{1}{\sqrt{2}}Y_{R}^{11}v_{\chi},\;M_{R,22}=\frac{1}{\sqrt{2}}Y_{R}^{22}v_{\chi}.$		(26)

The Hermitian charged lepton mass matrix⁴⁴4The Hermitian property ensures that the eigenvalues of charged lepton mass matrix are real, which represent the observed masses of the charged leptons. Additionally, the invariance under charge conjugation in the lepton sector further supports the requirement for the mass matrix to be Hermitian. requires real $m_{e,33}$ and

$\displaystyle Y_{e}^{12}=Y_{e}^{21*},\;Y_{e}^{13}=Y_{e}^{31*},\;Y_{e}^{23}=Y_{e}^{32*}.$

(27)

Then the Hermitian mass matrix for charged lepton can be written as

$\displaystyle m_{e}=\left(\begin{array}[]{ccc}0&m_{e,12}&m_{e,13}\\ m_{e,12}^{*}&0&m_{e,23}\\ m_{e,31}^{*}&m_{e,32}^{*}&m_{e,33}\end{array}\right).$

(31)

Taking the measured charged lepton masses as inputs, one can obtain $m_{e,33}$, $|m_{e,23}|$, $|m_{e,12}|$ under the approximation $|m_{e,12}|,\;|m_{e,13}|,\;|m_{e,23}|\ll m_{e,33}$ as

	$\displaystyle m_{e,33}=m_{e_{1}}+m_{e_{2}}+m_{e_{3}},$
	$\displaystyle|m_{e,23}|=[(m_{e_{1}}+m_{e_{2}})m_{e,33}-|m_{e,13}|^{2}]^{1/2},$
	$\displaystyle|m_{e,12}|=\frac{|m_{e,13}||m_{e,23}|}{|m_{e,33}|}\cos(\theta_{e,12}+\theta_{e,23}-\theta_{e,13})$
	$\displaystyle\qquad\qquad+\Big{\{}[\frac{|m_{e,13}||m_{e,23}|}{|m_{e,33}|^{2}}\cos(\theta_{e,12}+\theta_{e,23}-\theta_{e,13})]^{2}+\frac{m_{e_{1}}m_{e_{2}}}{|m_{e,33}|^{2}}\Big{\}}^{1/2}|m_{e,33}|,$		(32)

where $\theta_{e,ij}\;(ij=12,\;13,\;23)$ are defined as $m_{e,ij}=|m_{e,ij}|e^{i\theta_{e,ij}}$, and $m_{e_{k}}\;(k=1,\;2,\;3)$ is $k-$generation charged lepton mass. For the $5\times 5$ neutrino mass matrix $m_{\nu}$ defined in Eq. (25), we can derive the $3\times 3$ light Majorana neutrino mass matrix $\hat{m}_{\nu}$ by taking $M_{D}^{T}M_{R}^{-1}\ll 1$

$\displaystyle\hat{m}_{\nu}\approx-M_{D}^{T}M_{R}^{-1}M_{D}=-\left(\begin{array}[]{ccc}\frac{M_{D,12}^{2}}{M_{R,22}}&0&\frac{M_{D,12}M_{D,32}}{M_{R,22}}\\ 0&\frac{M_{D,21}^{2}}{M_{R,11}}&\frac{M_{D,21}M_{D,31}}{M_{R,11}}\\ \frac{M_{D,12}M_{D,32}}{M_{R,22}}&\frac{M_{D,21}M_{D,31}}{M_{R,11}}&\frac{M_{D,31}^{2}}{M_{R,11}}+\frac{M_{D,32}^{2}}{M_{R,22}}\end{array}\right).$

(36)

$\hat{m}_{\nu}$ can be simplified by simple calculation and redefined as

$\displaystyle\hat{m}_{\nu}=\left(\begin{array}[]{ccc}\hat{m}_{\nu,11}&0&\hat{m}_{\nu,13}\\ 0&\hat{m}_{\nu,22}&\hat{m}_{\nu,23}\\ \hat{m}_{\nu,13}&\hat{m}_{\nu,23}&\frac{\hat{m}_{\nu,13}^{2}}{\hat{m}_{\nu,11}}+\frac{\hat{m}_{\nu,23}^{2}}{\hat{m}_{\nu,22}}\end{array}\right),$

(40)

where

	$\displaystyle\hat{m}_{\nu,11}=-\frac{M_{D,12}^{2}}{M_{R,22}},\;\hat{m}_{\nu,22}=-\frac{M_{D,21}^{2}}{M_{R,11}},$
	$\displaystyle\hat{m}_{\nu,13}=-\frac{M_{D,12}M_{D,32}}{M_{R,22}},\;\hat{m}_{\nu,23}=-\frac{M_{D,21}M_{D,31}}{M_{R,11}}.$		(41)

It is straightforward to verify that the determinant of the mass matrix defined in Eq. (40) is zero, indicating that one of the three light neutrinos in the FDM must be massless. In this scenario, the masses of the other two light neutrinos can be fixed by the measured neutrino mass-squared differences in Eq. (1), i.e. for the NH and IH neutrino masses we have

	$\displaystyle m_{\nu_{1}}^{\rm NH}=0\;{\rm eV},\;m_{\nu_{2}}^{\rm NH}=(0.753\pm 0.018)^{1/2}\times 10^{-2}\;{\rm eV},$
	$\displaystyle m_{\nu_{3}}^{\rm NH}=(25.303\pm 0.298)^{1/2}\times 10^{-2}\;{\rm eV},$		(42)
	$\displaystyle m_{\nu_{3}}^{\rm IH}=0\;{\rm eV},\;m_{\nu_{1}}^{\rm IH}=(24.537\pm 0.308)^{1/2}\times 10^{-2}\;{\rm eV},$
	$\displaystyle m_{\nu_{2}}^{\rm IH}=(25.29\pm 0.29)^{1/2}\times 10^{-2}\;{\rm eV}.$		(43)

Taking the neutrino masses above as inputs, we can fix two of the free parameters in Eq. (40) by solving the two following equations

	$\displaystyle\Big{[}|\hat{m}_{\nu,22}|^{2}(|\hat{m}_{\nu,11}|^{2}+|\hat{m}_{\nu,13}|^{2}-|\hat{m}_{\nu,11}||\hat{m}_{\nu,22}|)^{2}+2|\hat{m}_{\nu,11}|^{2}|\hat{m}_{\nu,22}||\hat{m}_{\nu,23}|^{2}(|\hat{m}_{\nu,22}|$
	$\displaystyle-|\hat{m}_{\nu,11}|)+|\hat{m}_{\nu,11}|^{2}\times|\hat{m}_{\nu,23}|^{4}+2|\hat{m}_{\nu,11}||\hat{m}_{\nu,22}||\hat{m}_{\nu,13}|^{2}|\hat{m}_{\nu,23}|^{2}\cos\theta\Big{]}$
	$\displaystyle\times\Big{[}|\hat{m}_{\nu,22}|^{2}(|\hat{m}_{\nu,13}|^{2}+|\hat{m}_{\nu,11}|^{2}+|\hat{m}_{\nu,11}||\hat{m}_{\nu,22}|)^{2}+2|\hat{m}_{\nu,11}|^{2}|\hat{m}_{\nu,23}|^{2}|\hat{m}_{\nu,22}|(|\hat{m}_{\nu,11}|$
	$\displaystyle+|\hat{m}_{\nu,22}|)+|\hat{m}_{\nu,11}|^{2}|\hat{m}_{\nu,23}|^{4}+2|\hat{m}_{\nu,13}|^{2}|\hat{m}_{\nu,23}|^{2}|\hat{m}_{\nu,11}||\hat{m}_{\nu,22}|\cos\theta\Big{]}$
	$\displaystyle=|m_{\nu,11}|^{4}|m_{\nu,22}|^{4}(m_{\nu_{a}}^{2}-m_{\nu_{b}}^{2})^{2},$		(44)
	$\displaystyle|\hat{m}_{\nu,11}|^{4}|\hat{m}_{\nu,22}|^{2}+|\hat{m}_{\nu,11}|^{2}|\hat{m}_{\nu,22}|^{4}+2|\hat{m}_{\nu,13}|^{2}|\hat{m}_{\nu,11}|^{2}|\hat{m}_{\nu,22}|^{2}+2|\hat{m}_{\nu,23}|^{2}|\hat{m}_{\nu,11}|^{2}|\hat{m}_{\nu,22}|^{2}$
	$\displaystyle+|\hat{m}_{\nu,13}|^{4}|\hat{m}_{\nu,22}|^{2}+|\hat{m}_{\nu,23}|^{4}|\hat{m}_{\nu,11}|^{2}+2|\hat{m}_{\nu,13}|^{2}|\hat{m}_{\nu,23}|^{2}|\hat{m}_{\nu,11}||\hat{m}_{\nu,2}|\cos\theta$
	$\displaystyle=|\hat{m}_{\nu,11}|^{2}|\hat{m}_{\nu,22}|^{2}(m_{\nu_{a}}^{2}+m_{\nu_{b}}^{2}),$		(45)

where $\theta=2\theta_{\nu,13}+\theta_{\nu,22}-\theta_{\nu,11}-2\theta_{\nu,23}$, and $\theta_{\nu,\alpha\beta}$ ($\alpha\beta=11,\;22,\;13,\;23$) is the phase of $\hat{m}_{\nu,\alpha\beta}$, i.e. $\hat{m}_{\nu,\alpha\beta}=|\hat{m}_{\nu,\alpha\beta}|e^{i\theta_{\nu,\alpha\beta}}$.

Considering the nonzero neutrino masses and flavor mixings, we can obtain the nonzero neutrino transition MDM, which may have significant astrophysical consequences. Generally, the MDM of Dirac fermions can be written as

$\displaystyle\mathcal{L}_{\rm MDM}=\frac{1}{2}\mu_{ij}^{D}\bar{\psi}_{i}\sigma^{\mu\nu}\psi_{j}F_{\mu\nu},$

(46)

where $i,j$ are the indices of generation, $\psi_{i,j}$ are the four component fermions, $F_{\mu\nu}$ is the electromagnetic field strength, $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$, and $\mu_{ij}^{D}$ ($i\neq j$) is the transition MDM between $\psi_{i}$ and $\psi_{j}$. After matching between the effective theory and the full theory, one can get all high dimension operators together with their coefficients. And it is enough to retain the dimension 6 operators Feng:2008cn ; Feng:2008nm ; Feng:2009gn

	$\displaystyle{\mathcal{O}}_{1}^{L,R}=e\bar{\psi}_{i}(i/\!\!\!\!\mathcal{D})^{3}P_{L,R}\psi_{j},$
	$\displaystyle{\mathcal{O}}_{2}^{L,R}=e\overline{(i{\mathcal{D}}_{\mu}\psi_{i})}\gamma^{\mu}F\cdot\sigma P_{L,R}\psi_{j},$
	$\displaystyle{\mathcal{O}}_{3}^{L,R}=e\bar{\psi}_{i}F\cdot\sigma\gamma^{\mu}P_{L,R}(i{\mathcal{D}}_{\mu}\psi_{j}),$
	$\displaystyle{\mathcal{O}}_{4}^{L,R}=e\bar{\psi}_{i}(\partial^{\mu}F_{\mu\nu})\gamma^{\nu}P_{L,R}\psi_{j},$
	$\displaystyle{\mathcal{O}}_{5}^{L,R}=em_{\psi_{i}}\bar{\psi}_{i}(i/\!\!\!\!\mathcal{D})^{2}P_{L,R}\psi_{j},$
	$\displaystyle{\mathcal{O}}_{6}^{L,R}=em_{\psi_{i}}\bar{\psi}_{i}F\cdot\sigma P_{L,R}\psi_{j},$		(47)

in the following calculations. In Eq. (47), ${\mathcal{D}}_{\mu}=\partial_{\mu}+ieA_{\mu}$ and $A_{\mu}$ denotes the photon field, $P_{L,R}=(1\mp\gamma_{5})/2$, $m_{\psi_{i}}$ is the mass of fermion $\psi_{i}$. It is obvious that, all dimension $6$ operators in Eq. (47) induce the effective couplings among photons and fermions, and only the operators ${\mathcal{O}}_{2}^{L,R}$, ${\mathcal{O}}_{3}^{L,R}$, ${\mathcal{O}}_{6}^{L,R}$ can make contributions to the MDM of fermions. In addition, if the full theory is invariant under the combined transformation of charge conjugation, parity and time reversal (CPT), the Wilson coefficients $C_{2}^{L,R}$, $C_{3}^{L,R}$, $C_{6}^{L,R}$ of the corresponding operators ${\mathcal{O}}_{2}^{L,R}$, ${\mathcal{O}}_{3}^{L,R}$, ${\mathcal{O}}_{6}^{L,R}$ satisfy the relations

$\displaystyle C_{3}^{L,R}=(C_{2}^{R,L})^{*},\;\;\;C_{6}^{L}=(C_{6}^{R})^{*}.$

(48)

After applying the equations of motion to the external fermions, the transition MDM $\mu_{ij}^{D}$ can be written in the form of $C_{2}^{L}$, $C_{2}^{R}$, $C_{6}^{R}$ as

$\displaystyle\mu_{ij}^{D}=4m_{e}\Re(m_{\psi_{i}}C_{2}^{R}+m_{\psi_{j}}C_{2}^{L*}+m_{\psi_{i}}C_{6}^{R})\mu_{B},$

(49)

where $\Re(\cdot\cdot\cdot)$ denote the operation to take the real part of a complex number, $m_{e}$ is the electron mass, the Bohr magneton $\mu_{B}$ is defined as $\mu_{B}\equiv e/(2m_{e})$.

In the FDM, the Feynman diagrams contributing to neutrino transition MDM are plotted in Fig. 1. Then the nonzero coefficients can be written by neglecting the tiny Yukawa couplings of neutrinos as

	$\displaystyle C_{2}^{L(a)}=\frac{1}{2M_{W}^{2}}C_{L}^{We_{k}\nu_{i}}C_{L}^{We_{k}\nu_{j}}\Big{[}I_{1}(x_{e_{k}},x_{W})-I_{3}(x_{e_{k}},x_{W})\Big{]},$
	$\displaystyle C_{2}^{L(b)}=\frac{1}{2M_{W}^{2}}C_{L}^{We_{k}\nu_{i}}C_{L}^{We_{k}\nu_{j}}\Big{[}I_{2}(x_{e_{k}},x_{W})+I_{3}(x_{e_{k}},x_{W})\Big{]},$
	$\displaystyle C_{2}^{L(c)}=\frac{1}{4M_{W}^{2}}C_{L}^{H_{k}\bar{e}_{l}\nu_{i}}C_{L}^{H_{k}\bar{e}_{l}\nu_{j}}\Big{[}I_{3}(x_{e_{l}},x_{H_{k}})-I_{2}(x_{e_{l}},x_{H_{k}})\Big{]},$
	$\displaystyle C_{2}^{L(d)}=\frac{1}{4M_{W}^{2}}C_{L}^{H_{k}\bar{e}_{l}\nu_{i}}C_{L}^{H_{k}\bar{e}_{l}\nu_{j}}\Big{[}2I_{2}(x_{e_{l}},x_{H_{k}})-I_{1}(x_{e_{l}},x_{H_{k}})-I_{3}(x_{e_{l}},x_{H_{k}})\Big{]},$		(50)

where $C_{abc}^{L}$ is the coupling constant for left-hand part of particles $a,\;b,\;c$, $x_{i}=m_{i}^{2}/M_{W}^{2}$, $M_{W}$ is the W boson mass and Zhang:2014iva ; Yang:2021duj

	$\displaystyle I_{1}(x_{1},x_{2})=\frac{1}{16\pi^{2}}\Big{[}\frac{1+\log x_{2}}{x_{2}-x_{1}}-\frac{x_{1}\log x_{1}-x_{2}\log x_{2}}{(x_{2}-x_{1})^{2}}\Big{]},$
	$\displaystyle I_{2}(x_{1},x_{2})=\frac{1}{32\pi^{2}}\Big{[}\frac{3+2\log x_{2}}{x_{2}-x_{1}}-\frac{2x_{2}+4x_{2}\log x_{2}}{(x_{2}-x_{1})^{2}}-\frac{2x_{1}^{2}\log x_{1}-2x_{2}^{2}\log x_{2}}{(x_{2}-x_{1})^{3}}\Big{]},$
	$\displaystyle I_{3}(x_{1},x_{2})=\frac{1}{96\pi^{2}}\Big{[}\frac{11+6\log x_{2}}{x_{2}-x_{1}}-\frac{15x_{2}+18x_{2}\log x_{2}}{(x_{2}-x_{1})^{2}}+\frac{6x_{2}^{2}+18x_{2}^{2}\log x_{2}}{(x_{2}-x_{1})^{3}}$
	$\displaystyle\qquad\qquad\quad+\frac{6x_{1}^{3}\log x_{1}-6x_{2}^{3}\log x_{2}}{(x_{2}-x_{1})^{4}}\Big{]}.$		(51)

Eq. (50) shows that the contributions from the charged Higgs are proportional to the couplings $C_{L}^{H_{k}\bar{e}_{l}\nu_{i}}$, which are suppressed by the small charged lepton masses. As a result, the contributions to the neutrino transition MDM in the FDM are dominated by those from the $W$ boson-related interactions. Moreover, only $C_{2}^{L}=C_{2}^{L(a)}+C_{2}^{L(b)}+C_{2}^{L(c)}+C_{2}^{L(d)}$ is predicted to be nonzero in the FDM as shown in Eq. (50), hence the transition MDM of Majorana neutrinos in the FDM can be written as

$\displaystyle\mu_{ij}=\mu_{ij}^{D}-\mu_{ji}^{D},$

(52)

with

$\displaystyle\mu_{ij}^{D}=4m_{e}\Re(m_{\nu_{j}}C_{2}^{L*})\mu_{B},$

(53)

and $m_{\nu_{j}}$ denotes the neutrino mass.

XENONnT XENON:2022ltv presented an upper bound on the MDM of solar neutrinos, which reads $\mu_{\rm solar}<6.3\times 10^{-12}\mu_{B}$. In general, $\mu_{\rm solar}$ can be approximated by the transition MDM of neutrinos approximately as Akhmedov:2022txm

$\displaystyle\mu_{\rm solar}=|\mu_{12}|^{2}c_{13}^{2}+|\mu_{13}|^{2}(c_{13}^{2}c_{12}^{2}+s_{13}^{2})+|\mu_{23}|^{2}(c_{13}^{2}s_{12}^{2}+s_{13}^{2}).$

(54)

The measured neutrino mass-squared differences in Eq. (1), flavor mixings in Eq. (6) and charged lepton masses will impose strict constraints on the parameters in Eq. (26). Generally, $m_{e}$ in Eq. (25) and $\hat{m}_{\nu}$ in Eq. (40) can be diagonalized by

$\displaystyle m_{e}^{\rm diag}=U_{L}^{e,\dagger}m_{e}U_{R}^{e},\;m_{\nu}^{\rm diag}=U^{\nu,T}\hat{m}_{\nu}U^{\nu},$

(55)

where $U_{L}^{e},\;U_{R}^{e},\;U^{\nu}$ are the $3\times 3$ unitary matrices. Then $U_{\rm PMNS}$ in the FDM can be defined theoretically as

$\displaystyle U_{\rm PMNS}=U_{L}^{e,\dagger}\cdot U^{\nu}.$

(56)

PDG collects the measured results of $|U_{\rm PMNS}|$ and Dirac CP phase $\delta$ in $3\sigma$ ranges, and we consider ParticleDataGroup:2024cfk

	$\displaystyle 0.801<|U_{\rm PMNS}|_{11}<0.842,\;0.518<|U_{\rm PMNS}|_{12}<0.580,\;0.143<|U_{\rm PMNS}|_{13}<0.155,$
	$\displaystyle 0.498<|U_{\rm PMNS}|_{22}<0.690,\;0.634<|U_{\rm PMNS}|_{23}<0.770,\;0.53\pi<\delta<1.85\pi$		(57)

in the numerical calculations. To confront the predictions of $U_{\rm PMNS}$ with the experimental data, we extract the standard-parametrization parameters according to the formulas Xing:2020ald

	$\displaystyle s_{13}=|U_{\rm PMNS}|_{13},\;s_{12}=\frac{|U_{\rm PMNS}|_{12}}{\sqrt{1-(|U_{\rm PMNS}|_{13})^{2}}},\;s_{23}=\frac{|U_{\rm PMNS}|_{23}}{\sqrt{1-(|U_{\rm PMNS}|_{13})^{2}}},$
	$\displaystyle\delta={\rm arg}\Big{(}\frac{U_{\rm PMNS,12}U_{\rm PMNS,23}U_{\rm PMNS,13}^{*}U_{\rm PMNS,22}^{*}+s_{12}^{2}c_{13}^{2}s_{13}^{2}s_{23}^{2}}{c_{12}s_{12}c_{13}^{2}s_{13}c_{23}s_{23}}\Big{)},$
	$\displaystyle\rho={\rm arg}(U_{\rm PMNS,11}U_{\rm PMNS,13}^{*})-\delta,\;\sigma={\rm arg}(U_{\rm PMNS,12}U_{\rm PMNS,13}^{*})-\delta.$		(58)

In the context of the minimal seesaw mechanism, the Majorana CP phase associated with the massless neutrino has no physical impact. Consequently, only a single Majorana CP phase is physically relevant Xing:2020ald , i.e. $\sigma$ in the NH case ($m_{\nu_{1}}=0$) and $\sigma-\rho$ which can be redefined as $\sigma$ Mei:2003gn in the IH case ($m_{\nu_{3}}=0$). In addition, to determine the best fit for the mass-squared differences in Eq. (1) and the flavor mixings, Dirac CP phase $\delta$ in Eq. (7), we perform a $\chi^{2}$ test, which can be constructed as

$\displaystyle\chi^{2}=\sum_{1}\Big{(}\frac{O_{i}^{\rm th}-O_{i}^{\rm exp}}{\sigma_{i}^{\rm exp}}\Big{)}^{2},$

(59)

where $O_{i}^{\rm th}$ denotes the $i-$th observable computed theoretically, $O_{i}^{\rm exp}$ is the corresponding experimental value and $\sigma_{i}^{\rm exp}$ is the uncertainty in $O_{i}^{\rm exp}$.

Taking the charged lepton masses $m_{e}=0.511\;{\rm MeV},\;m_{\mu}=105.658\;{\rm MeV},\;m_{\tau}=1.777\;{\rm GeV}$ as inputs, we scan the parameter space

	$\displaystyle|\hat{m}_{\nu,11}|\sim(10^{-6},\;10^{-1})\;{\rm eV},\;|\hat{m}_{\nu,22}|\sim(10^{-6},\;10^{-1})\;{\rm eV},\;|m_{e,13}|\sim(0,\;0.3)\;{\rm GeV},$
	$\displaystyle\theta_{\nu,\alpha\beta}\sim(-\pi,\;\pi),\;\theta_{e,ij}\sim(-\pi,\;\pi),$		(60)

where $\alpha\beta=11,22,13,23$, $ij=12,13,23$, $m_{e,33}$, $|m_{e,12}|$, $|m_{e,23}|$, $|\hat{m}_{\nu,13}|$, $|\hat{m}_{\nu,23}|$ can be obtained by Eqs. (32, 44, 45). Then keeping the mass-squared differences in Eq. (1) in the experimental $3\sigma$ ranges and Eq. (57) in the scanning, we plot the results of $|\hat{m}_{\nu,13}|-|\hat{m}_{\nu,11}|$, $|\hat{m}_{\nu,23}|-|\hat{m}_{\nu,22}|$, $|m_{e,13}|-|\hat{m}_{\nu,13}|$, $|\hat{m}_{\nu,23}|-\cos\theta$ for NH neutrino masses in Fig. 2 ($a_{1}$), Fig. 2 ($b_{1}$), Fig. 2 ($c_{1}$), Fig. 2 ($d_{1}$) respectively. Similarly, the results of $|\hat{m}_{\nu,13}|-|\hat{m}_{\nu,11}|$ ($a_{2}$), $|\hat{m}_{\nu,23}|-|\hat{m}_{\nu,22}|$ ($b_{2}$), $|m_{e,13}|-|\hat{m}_{\nu,13}|$ ($c_{2}$), $|\hat{m}_{\nu,13}|-\cos\theta$ ($d_{2}$) for IH neutrino masses are also plotted in Fig. 2. The red stars denote the best fit corresponding to $\chi^{2}=\mathbf{0.169}$ for NH neutrino masses and $\chi^{2}=\mathbf{0.312}$ for IH neutrino masses. The results of the best fit are listed in Tab. 1,