Update to the “New Minimal Standard Model”

Bruce Hoeneisen

Universidad San Francisco de Quito, Quito, Ecuador.

DOI: 10.4236/ijaa.2025.151004                

Abstract

The Standard Model of quarks and leptons is a great achievement. However, this model is incomplete: it does not include the observed dark matter, neutrino masses and mixings, the matter-antimatter asymmetry of the universe, the current acceleration of the expansion of the universe, nor inflation. We seek a minimal extension of the Standard Model to account for these observations. We constrain its parameters and discuss potential tests for this updated “New Minimal Standard Model”.

Keywords

New Minimal Standard Model, Dark Matter, Neutrino Oscillations, Leptogenesis, Inflation

1. Introduction

In 2004 the authors of “The New Minimal Standard Model” [1] set out to “include the minimal number of new degrees of freedom to accommodate convincing (e.g., $>5\sigma$ ) evidence for physics beyond the Minimal Standard Model”, “based on the principle of minimal particle content and the most general renormalizable Lagrangian.”. To this end, the following fields were added to the Standard Model (SM):

• A real scalar field $S$ as dark matter. To make $S$ stable, the $Z_2$ symmetry $S\leftrightarrow -S$ is imposed. The most general renormalizable Lagrangian from [1] is reproduced in Section 3 below.

• The simplest way to include the observed present acceleration of the universe expansion is to add Einstein’s cosmological constant Λ.

• To account for neutrino oscillations, two right handed neutrinos are added. These neutrinos are assumed to be of the Majorana type to allow Type I baryogenesis via leptogenesis in the CP-violating and lepton number non-conserving decay of the heavy neutrino mass eigenstates. “The lighter right-handed neutrino should be heavier than 10¹⁰ GeV to have enough CP asymmetry. [1]” A Majorana neutrino has no additive conserved charges, and hence allows a “Majorana” mass term, see Section 3 below.

• Finally, to account for the observed nearly scale-invariant, adiabatic, and Gaussian density fluctuations in the early universe, a real scalar field Φ is added to drive inflation.

The virtue of the “New Minimal Standard Model” is that it is data driven, constraining, allows predictions, and hence is testable.

The present study attempts to update the “New Minimal Standard Model” (NMSM) with the evidence gathered in the intervening twenty years, namely the discovery of the top quark and Higgs boson, new measured properties of dark matter (to be discussed in Section 2), and new ideas on leptogenesis (to be discussed in Section 4). We take cosmological parameter values and notation from [2].

2. Dark Matter

Let us consider non-relativistic warm dark matter when the universe is nearly homogeneous, and let us assume that this dark matter is a gas of particles of mass $m_S$ . These particles may be collisionless, or have elastic dark matter-dark matter collisions. The root-mean-square thermal velocity of the dark matter particles is $v_{h\text{rms}}\left(a\right)$ , where $a\left(t\right)$ is the expansion parameter of the universe, normalized to $a\left(t_0\right)=1$ at the present time $t_0$ . As the universe expands, $v_{h\text{rms}}\left(a\right)$ varies in proportion to $1/a$ (assuming the dark matter particles have no excited states), so

$v_{h\text{rms}}\left(1\right)=v_{h\text{rms}}\left(a\right)a$ (1)

is an adiabatic invariant. This adiabatic invariant defines the “warmness” of the dark matter, and has been measured with dwarf galaxy rotation curves [3] [4]:

$v_{h\text{rms}}\left(1\right)=406\pm 69\text{m}/\text{s}.$ (2)

The transition from ultra-relativistic to non-relativistic dark matter occurs at an expansion parameter

$a_{h\text{NR}}=\frac{v_{h\text{rms}}\left(1\right)}{c}=\left(1.35\pm 0.23\right)\times {10}^{-6},$ (3)

when the universe density is dominated by radiation.

Let $P_{\text{CDM}}\left(k\right)$ be the comoving linear power spectrum of density fluctuations in the cold dark matter ΛCDM cosmology. Due to warm dark matter free-streaming, or to dark matter-dark matter particle elastic collisions, the linear power spectrum becomes attenuated at high $k$ by a cut-off factor ${\tau }^2\left(k\right)$ , i.e., $P\left(k\right)={\tau }^2\left(k\right)P_{\text{CDM}}\left(k\right)$ . We define the comoving cut-off wavevector $k_c$ so ${\tau }^2\left(k_c\right)=1/e$ at expansion parameter $a=0.01$ (other definitions in the literature are ${\tau }^2\left(k_c\right)=1/2$ or 1/4). We have chosen $a=0.01$ because numerical integrations to account for non-linear density evolution and baryonic feedback often start at $a\approx 0.01$ , and because free-streaming is almost complete by $a\approx 0.01$ .

For $v_{h\text{rms}}\left(1\right)=406$ m/s the comoving free-streaming cut-off wavevector is

$k_{\text{fs}}\approx 0.28\frac{2\pi }{l_{\text{fs}}}=0.32{\text{Mpc}}^{-1},$ (4)

$l_{\text{fs}}$ is the comoving dark matter free-streaming length.

The dark matter self interaction cross-section per unit mass ${\sigma }_{SS}/m_S$ is observed to be less than 0.47 cm²/g at the present time [2]. The corresponding mean distance between collisions is greater than $1/\left(\left({\sigma }_{SS}/m_S\right){\Omega }_c{\rho }_{\text{crit}}\right)=3\times {10}^5$ Mpc. In conclusion, dark matter-dark matter collisions have a negligible contribution to $k_c$ .

$k_c$ has been estimated from galaxy stellar mass distributions (1 to 3 Mpc⁻¹), galaxy ultra-violet luminosity distributions (2 to 4 Mpc⁻¹), first galaxies (1.6 to 3 Mpc⁻¹), and reionization (1 to 4 Mpc⁻¹), see [4] and references therein. For reionization see also [5]. Studies of the Lyman-α forest of quasar light place lower bounds on the “standard thermal relic” mass $m_X$ in the range 550 eV [6] up to 5.7 keV [7]. These limits, due to assumed warm dark matter free-streaming, are really limits on $k_c$ , i.e., $k_c\gtrsim 3$ Mpc⁻¹ up to $k_c\gtrsim 39$ Mpc⁻¹. These measurements of $k_c$ are in disagreement with each other, and with (2), and are sensitive to the non-linear regenerated power spectrum that is large and uncertain, see figure 4 of [8], and figure 18 of [9]. Future Cosmic Microwave Background (CMB) measurements may reach and test $k_{\text{fs}}$ of (4). A direct measurement of the linear power spectrum of density perturbations will become possible with observations of weak gravitational lensing of the CMB radiation [8].

Every measurement has its issues. When measurements disagree at most one is correct, and all need to be revised. The purpose of the present study is to explore the consequences of the assumption that $v_{h\text{rms}}\left(1\right)$ of (2) is correct and of cosmological origin [4] [10]-[12], in spite of the tensions mentioned above that need to be resolved.

Galaxy observations alone can not obtain the dark matter particle mass, see Section 5 of [12]. To make progress, we need to assume a cosmological model of dark matter. To break the degeneracy between dark matter temperature and mass, we assume that ultra-relativistic dark matter has zero chemical potential. With this assumption, for spin zero dark matter, with the measured $v_{h\text{rms}}\left(1\right)$ of (2), we obtain the dark matter particle mass [4]

$m_S=177\pm 23\text{eV},$ (5)

and the dark matter-to-photon temperature ratio, after $e^{-}{e}^{+}$ annihilation, while dark matter is still ultra-relativistic, to be

$\frac{T_h}{T_{\gamma }}=0.343\pm 0.015,$ (6)

sufficiently cold to not spoil the success of big-bang nucleosynthesis.

These measurements are consistent with spin zero dark matter in thermal and diffusive equilibrium with the early Standard Model sector, i.e. no “freeze-in”, that decouples from the Standard Model sector while still ultra-relativistic, and does not decay or annihilate when it becomes non-relativistic, i.e. no “freeze-out” [4]. Our assumption is self-consistent because early equilibrium with the Standard Model sector implies that ultra-relativistic dark matter has zero chemical potential.

To the dark matter we assign a scalar field $S$ with $Z_2$ symmetry, i.e. $S\leftrightarrow -S$ , so that dark matter is stable. Since dark matter has not been observed in direct or indirect searches, we will assume that dark matter has no $SU{\left(3\right)}_c$ , $SU{\left(2\right)}_L$ or $U{\left(1\right)}_Y$ gauge interactions. We therefore assume that the only interaction of $S$ with the Standard Model sector is a contact coupling

$-\frac{1}{2}{\lambda }_{\varphi S}\left({\varphi }^{†}\varphi \right)S^2$ with the Higgs boson $\varphi$ . The coupling $\left|{\lambda }_{\varphi S}\right|$ is sufficiently strong to bring $S$ and the Standard Model sector into thermal and diffusive equilibrium by the Electro-Weak Symmetry Breaking (EWSB) temperature $T_{\text{EWSB}}=159\pm 1$ GeV [13]. When the Higgs boson becomes non-relativistic and decays or annihilates, dark matter decouples from the Standard Model sector. In this scenario, for spin zero dark matter, the predicted dark matter mass and temperature ratio (after $e^{+}{e}^{-}$ annihilation) are $m_S=150\pm 2$ eV and $T_h/T_{\gamma }=0.345$ [4], in agreement with the measurements. In this scenario, spin $\frac{1}{2}$ Dirac or Majorana dark matter, and spin 1 vector dark matter, disagree with the observations by more than $5\sigma$ [4]. See Figure 1.

Slightly above the temperature $T_{\text{EWSB}}$ the NMSM is composed of the dark matter scalar $S$ , the Higgs boson $\varphi$ , the gauge bosons ${\overrightarrow{G}}_i$ , $\overrightarrow{W}$ and $\overrightarrow{B}$ , and mass-less fields that carry the Weyl-L or Weyl-R 2-dimensional irreducible representations of the proper Lorentz group, see Figure 2. The Weyl-L representation is also denoted “left-handed”, or $\left(\frac{1}{2},0\right)$ [14], or $\left(\frac{1}{2},-\frac{3}{2}\right)$ [15]. The Weyl-R representation is also denoted “right-handed”, or $\left(0,\frac{1}{2}\right)$ [14], or $\left(\frac{1}{2},\frac{3}{2}\right)$ [15]. Below the temperature $T_{\text{EWSB}}$ , the Higgs boson acquires a vacuum expectation value increasing to $v/\sqrt{2}\approx 246/\sqrt{2}$ GeV that ties together Weyl-L and Weyl-R fields obtaining massive 4-dimensional Dirac fields, as shown with dashed lines in Figure 2. The fields in Figure 2 have wavelengths much shorter than their mean free path.

Consider $T\gg T_{M_H}$ . The $SS\leftrightarrow hh$ cross-section, with $\varphi \to h/\sqrt{2}$ , is

$\sigma \left(SS\leftrightarrow hh\right)=\frac{{\left|{\lambda }_{\varphi S}\right|}^2}{16\pi s},$ (7)

where $s$ is the Mandelstam variable, i.e., the square of the total center of mass energy. We are interested in the mean time to the next collision $t_c=1/\left(\sigma nc\right)$ . The mean value of $s$ , obtained with the Maxwell distribution, is $\langle s\rangle ={\left(2kT\right)}^2$ . The cross-section (7) scales as $a^2$ . The number density of dark matter particles $n$ scales as $T^3\propto a^{-3}$ , so the $S$ particle mean-free-path $1/\left(\sigma n\right)$ scales as $a$ , slower than the proper distance to the horizon $\propto a^2$ . So $S$ and $\varphi$ approach equilibrium towards the future. To agree with the observed “no freeze-in and no

Figure 1. The “no freeze-in and no freeze-out” dark matter scenario is illustrated for spin zero warm dark matter particles coupled to the Higgs boson. $T$ is the photon temperature, and the $n$ ‘s are particle number densities. The abbreviations stand for “Electro-Weak Symmetry Breaking”, “Big Bang Nucleosynthesis”, “EQuivalence” of matter and radiation densities, and “DECoupling” of photons from the proton-electron plasma when it recombines to neutral hydrogen. Dark matter particles become non-relativistic at $a_{h\text{NR}}$ . Time advances towards the right. Figure from [4].

Figure 2. The elementary particles of the “New Minimal Standard Model” are shown. Before EWSB the fields carry 2-dimensional Weyl-L (filled dots) and Weyl-R (empty dots) representations of the proper Lorentz group. After EWSB, the vacuum expectation value $v$ of the Higgs boson couples Weyl-L and Weyl-R fields into 4-dimensional Dirac fields, as indicated by the dashed lines. Circles and vertical lines illustrate, respectively, $SU{\left(3\right)}_c$ and $SU{\left(2\right)}_L$ transformations. These transformations are mediated, respectively, by gluons $G_i^{\mu }$ and weak $W_j^{\mu }$ gauge bosons. $Y$ is the charge corresponding to the $U{\left(1\right)}_Y$ symmetry with gauge boson $B^{\mu }$ . $I_3$ is the iso-spin component of $SU{\left(2\right)}_L$ . The “New Minimal Standard Model” adds to the Standard Model the sterile particles $S$ , $N_1$ and $N_2$ with $Y=I=I_3=0$ , and Einstein’s cosmological constant (not shown).

freeze-out” scenario, we need $S$ and $\varphi$ to come into thermal and diffusive equilibrium by $T=M_H$ , where $M_H$ is the Higgs boson mass. This requirement implies $\left|{\lambda }_{hS}\right|\gtrsim 6\times {10}^{-7}$ . For the leptogenesis scenario to be discussed in Section 4 we require thermal and diffusive equilibrium by $T_{\text{EWSB}}$ , so $\left|{\lambda }_{\varphi S}\right|\gtrsim 7\times {10}^{-7}$ . To not exceed the Higgs boson invisible decay width 0.37 MeV [2] we need $\left|{\lambda }_{\varphi S}\right|<6\times {10}^{-3}$ .

The non-relativistic dark matter elastic scattering cross-sections in two channels are

$\sigma \left(SS\leftrightarrow h^{\text{*}}\leftrightarrow SS\right)=\frac{9}{2^8\pi s}{\left(\frac{{\lambda }_{\varphi S}v}{M_H}\right)}^4,$ (8)

$\sigma \left(SS\leftrightarrow SS\right)=\frac{{\left|{\Lambda }_S\right|}^2}{16\pi s},$ (9)

with $s={\left(2m_S\right)}^2$ . The observed limit on the dark matter elastic scattering cross-section per unit mass 0.47 cm²/g [2] requires $\left|{\text{Λ}}_{\varphi S}\right|\lesssim {10}^{-4}$ and $\left|{\text{Λ}}_S\right|\lesssim 4\times {10}^{-8}$ . From the preceeding estimates, we take

$7\times {10}^{-7}\lesssim \left|{\lambda }_{hS}\right|\lesssim {10}^{-4}.$ (10)

3. Lagrangian

To account for the observed neutrino flavor oscillations, and to allow lepton number violation needed for leptogenesis, we add to the Standard Model Lagrangian, $m\ge 2$ sterile, i.e. $SU{\left(3\right)}_c\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$ gauge singlet, Majorana neutrinos with the Weyl-R fields $N_j$ with $j=1,2,\cdots ,m$ . The case $m=2$ leaves one of the light neutrinos mass-less. We also add to the Standard Model the scalar dark matter field $S$ with $Z_2$ symmetry, i.e. $S\leftrightarrow -S$ . To simplify the notation, and without loss of generality, we work in a weak flavor basis with diagonal $SU{\left(2\right)}_L$ gauge interactions, diagonal charged lepton Yukawa coupling matrices $Y^l$ , and diagonal sterile neutrino Majorana mass matrix $M_N$ , see Figure 2. Then, the action of the “New Minimal Standard Model” is

${\mathcal{S}}_J={\displaystyle \int {\text{d}}^{4}x}\sqrt{-g}\left\{{ℒ}_{SM}+{ℒ}_N+{ℒ}_S+{ℒ}_{S\varphi }+{ℒ}_{\text{grav}}\right\},$ (11)

${ℒ}_N=+{\overline{N}}_k{\gamma }^{\mu }i{\partial }_{\mu }{N}_k$ (12)

$-\frac{1}{2}{\overline{N}}_j^c{M}_{N,jj}{N}_j-\frac{1}{2}{\overline{N}}_j{M}_{N,jj}{N}_j^c$ (13)

$-Y_{k\alpha }^{\nu \text{*}}{\overline{L}}_{\alpha }{\varphi }^c{N}_k-Y_{k\alpha }^{\nu }{\overline{N}}_k{\varphi }^{c†}{L}_{\alpha },$(14)

${ℒ}_S=+\frac{1}{2}{\partial }_{\mu }S\cdot {\partial }^{\mu }S-\frac{1}{2}{\overline{m}}_S^2{S}^2-\frac{{\lambda }_S}{4!}{S}^4,$ (15)

${ℒ}_{S\varphi }=-\frac{1}{2}{\lambda }_{\varphi S}\left({\varphi }^{†}\varphi \right)S^2,$ (16)

${ℒ}_{\text{grav}}=-{\Omega }_{\text{Λ}}{\rho }_{\text{crit}}-\left(\frac{M_P^2}{2\left(8\pi \right)}+{\xi }_{\varphi }{\varphi }^{†}\varphi +{\xi }_S{S}^2\right)R.$ (17)

Here, $\alpha =\left(e,\mu ,\tau \right)$ denotes a flavor index, and $j,k=1,2,\cdots ,m$ are indices for the singlet Majorana neutrinos. Sums over repeated indices are understood. To use Dirac notation, the Weyl-L (Weyl-R) 2-dimensional fields are embedded into 4-dimensional Dirac fields by adding null Weyl-R (Weyl-L) components.

The dark matter particle mass is

$m_S=\sqrt{{\overline{m}}_S^2+\frac{1}{2}{\lambda }_{\varphi S}{v}^2}=150\pm 2\text{eV},$(18)

with ${\overline{m}}_S^2$ assumed positive, while ${\lambda }_{\varphi S}$ is negative. Note that there is fine-tuning because $\sqrt{\left|{\lambda }_{\varphi S}\right|}v\gg m_S$ .

Comments on notation follow. The lepton $SU{\left(2\right)}_L$ doublets of the Standard Model (SM) are

$L_{\alpha }\equiv {\left(\begin{array}{c}{\nu }_{\alpha }\\ {\alpha }^{-}\end{array}\right)}_L\equiv \left(\begin{array}{c}{\nu }_{\alpha L}\\ {\alpha }_L^{-}\end{array}\right)=\frac{1-{\gamma }_5}{2}\left(\begin{array}{c}{\nu }_{\alpha }\\ {\alpha }^{-}\end{array}\right),$ (19)

with $\alpha =e,\mu ,\tau$ . Note that we work in a flavor basis with diagonal $SU{\left(2\right)}_L$ doublets. The fields ${\nu }_{\alpha L}$ and ${\alpha }_L^{-}$ carry the Weyl-L irreducible representation of the proper Lorentz group.

$N_j$ , with $j=1,2,\cdots ,m$ , are $SU{\left(2\right)}_L$ singlets that carry the Weyl-R irreducible representation of the proper Lorentz group. The conjugate fields $N_j^c\equiv -i{\sigma }_2{N}_j^{\text{*}}$ carry the Weyl-L irreducible representation.

After EWSB, a Weyl-L field couples to a Weyl-R field obtaining a 4-dimensional Dirac field for a massive quark, a massive charged lepton or a massive neutrino. The Dirac field ${\psi }_{\alpha }$ and its charge-conjugate ${\left({\psi }_{\alpha }\right)}^c$ , in the basis of the direct sum Weyl-L ${\oplus }_{}$ Weyl-R, are

${\psi }_{\alpha }=\left(\begin{array}{c}{\psi }_{\alpha L}\\ {\psi }_{\alpha R}\end{array}\right),{\left({\psi }_{\alpha }\right)}^c\equiv \left(\begin{array}{c}{\left({\psi }_{\alpha R}\right)}^c\\ {\left({\psi }_{\alpha L}\right)}^c\end{array}\right)=-i{\gamma }^2{\psi }_{\alpha }^{\text{*}}=\left(\begin{array}{c}-i{\sigma }_2{\psi }_{\alpha R}^{\text{*}}\\ i{\sigma }_2{\psi }_{\alpha L}^{\text{*}}\end{array}\right).$ (20)

Note the convention that the Weyl-L component of a Dirac field goes on top. We work in a basis with $\gamma$ -matrices

${\gamma }^0=\left(\begin{array}{cc}0& {\sigma }_0\\ {\sigma }_0& 0\end{array}\right),{\gamma }^k=\left(\begin{array}{cc}0& {\sigma }_k\\ -{\sigma }_k& 0\end{array}\right),{\gamma }^5=\left(\begin{array}{cc}-{\sigma }_0& 0\\ 0& {\sigma }_0\end{array}\right).$ (21)

Also, $\overline{\psi }\equiv {\psi }^{†}{\gamma }^0$ . $\overline{\psi }\psi$ is a Lorentz scalar, and $\overline{\psi }{\gamma }^{\mu }\psi$ is a Lorentz 4-vector. $R$ is the Ricci curvature scalar, and $M_P$ is the Planck mass. We assume, tentatively, that the dark matter scalar $S$ is also the inflaton, hence the quartic potential $\frac{1}{4!}{\lambda }_S{S}^4$ in (15), see Section 6.

The Higgs boson field and its conjugate are

$\varphi =\left(\begin{array}{l}{\varphi }^{+}\hfill \\ {\varphi }^0\hfill \end{array}\right),{\varphi }^c\equiv i{\sigma }_2{\varphi }^{\text{*}}=\left(\begin{array}{c}{\varphi }^{0\text{*}}\\ -{\varphi }^{+\text{*}}\end{array}\right).$ (22)

After EWSB,

$\varphi =\frac{1}{\sqrt{2}}\left(\begin{array}{c}0\\ v+h\end{array}\right),{\varphi }^c=\frac{1}{\sqrt{2}}\left(\begin{array}{c}v+h\\ 0\end{array}\right).$ (23)

Let

$M_{D,k\alpha }\equiv Y_{k\alpha }^{\nu }\frac{v}{\sqrt{2}}.$ (24)

The mass terms in (13) and (14), for $T<T_{\text{EWSB}}$ , can be written as [2]

$\frac{1}{2}\left(\overline{{\overrightarrow{v}}_L^c},\overline{\overrightarrow{N}}\right)\left(\begin{array}{cc}0& M_D^{\text{T}}\\ M_D& M_N\end{array}\right)\left(\begin{array}{c}{\overrightarrow{\nu }}_L\\ {\overrightarrow{N}}^c\end{array}\right)+h.c.\equiv \overline{{\overrightarrow{\nu }}^c}{M}_{\nu }\overrightarrow{\nu }+h.c\mathrm{..}$(25)

${\overrightarrow{\nu }}_L$ has components ${\left({\nu }_{eL},{\nu }_{\mu L},{\nu }_{\tau L}\right)}^{\text{T}}$ and ${\overrightarrow{N}}^c$ has components ${\left({\overrightarrow{N}}_1^c,{\overrightarrow{N}}_2^c,\cdots ,{\overrightarrow{N}}_m^c\right)}^{\text{T}}$. ${\left(N_j\right)}^c=-i{\sigma }_2{N}_j^{\text{*}}$ carries the representation Weyl-L, while ${\left({\nu }_{L,\alpha }\right)}^c=i{\sigma }_2{\nu }_{L,\alpha }^{\text{*}}$ carries the representation Weyl-R. The matrix $M_{\nu }$ can be diagonalized with a $\left(3+m\right)\times \left(3+m\right)$ unitary matrix $V^{\nu }$ :

${\left(V^{\nu }\right)}^{\text{T}}{M}_{\nu }{V}^{\nu }=\text{diag}\left(m_1,m_2,\cdots ,m_{3+m}\right).$ (26)

The neutrino mass eigenstates are

${\overrightarrow{\nu }}_{\text{mass}}={\left(V^{\nu }\right)}^{†}\overrightarrow{\nu }.$(27)

For $M_{N,jj}$ much greater than $M_{D,k\alpha }$ , and after EWSB, the light neutrino masses are given by the “see-saw” equation [2] [16]

$M_l\equiv \text{diag}\left(m_1,m_2,m_3\right)\equiv \text{diag}\left(m_{{\nu }_e},m_{{\nu }_{\mu }},m_{{\nu }_{\tau }}\right)\approx -V_l^{\text{T}}{M}_D^{\text{T}}{M}_N^{-1}{M}_D{V}_l,$ (28)

while the heavy neutrino masses are $M^h\approx M_N$ . $M_l$ is of dimension $3\times 3$ , real and diagonal.

The charge current interaction in the mass basis now reads [2]

$-{ℒ}_{CC}=\frac{g}{\sqrt{2}}\left[\left({\overline{e}}_L,{\overline{\mu }}_L,{\overline{\tau }}_L\right){\gamma }^{\mu }U{\overrightarrow{\nu }}_{\text{mass}}{W}_{\mu }^{-}+{\overrightarrow{\overline{\nu }}}_{\text{mass}}{U}^{†}{\gamma }^{\mu }{\left(e_L,{\mu }_L,{\tau }_L\right)}^{\text{T}}{W}_{\mu }^{+}\right]$(29)

with $U$ a $3\times \left(3+m\right)$ matrix that satisfies $U{U}^{†}=1_{3\times 3}$ . The matrix $U$ is equal to the first three rows of the matrix $V^{\nu }$ (once nonphysical phases are removed by re-phasing the neutrino fields):

$U\approx \left[\left(1-\frac{1}{2}{M}_D^{†}{M}_N^{\text{*}-1}{M}_N^{-1}{M}_D\right)V_l,M_D^{†}{M}_N^{\text{*}-1}\right],$ (30)

valid for ${\left[M_D^{†}{M}_N^{\text{*}-1}\right]}_{\alpha j}\ll 1$ . These equations assume the basis with diagonal charged lepton Yukawa matrices $Y^l$ , and diagonal sterile neutrino Majorana mass matrix $M_N$ . This $3\times \left(3+m\right)$ matrix has elements $U_{\alpha j}$ with indices $\alpha =e,\mu ,\tau$ and $j=1,2,\cdots ,\left(3+m\right)$ ($j=1,2,3$ correspond to the mass eigenstates ${\nu }_e,{\nu }_{\mu },{\nu }_{\tau }$ , while $j=4,5,6$ correspond to the mass eigenstates $N_1,N_2,N_3$ for the case $m=3$ ). $U$ has two physical phases for the case $m=2$ [17], or 3 physical phases for the case $m=3$ [2]. The unitary $3\times 3$ matrix $V_l$ is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix. See current values of its elements in [2]. Multiplying (28) on the left and right by ${\left(M_l\right)}^{-1/2}$ , assuming all $m_{{\nu }_{\alpha }}\ne 0$ , obtains

$1=R^{\text{T}}R,R=i{\left(M_N\right)}^{-1/2}{M}_D{V}_l{\left(M_l\right)}^{-1/2},M_D=-i\sqrt{M_N}R\sqrt{M_l}{V}_l^{†}.$ (31)

Notation: the square root refers to the diagonal elements $m_{\alpha }$ of the diagonal matrix $M_l$ . The Casas-Ibarra matrix $R$ [18], with elements $R_{j\alpha }$ with $j=1,2,\cdots ,m$ and $\alpha =e,\mu ,\tau$ , is a complex matrix satisfying $R^{\text{T}}R=1_{3\times 3}$ . In detail, for the case $m=2$ ,

$M_D^{†}{M}_N^{\text{*}-1}=i{V}_l\left(\begin{array}{cc}\sqrt{\frac{m_{{\nu }_e}}{M_{N1}^{\text{*}}}}{R}_{1e}^{\text{*}}& \sqrt{\frac{m_{{\nu }_e}}{M_{N2}^{\text{*}}}}{R}_{2e}^{\text{*}}\\ \sqrt{\frac{m_{{\nu }_{\mu }}}{M_{N1}^{\text{*}}}}{R}_{1\mu }^{\text{*}}& \sqrt{\frac{m_{{\nu }_{\mu }}}{M_{N2}^{\text{*}}}}{R}_{2\mu }^{\text{*}}\\ \sqrt{\frac{m_{{\nu }_{\tau }}}{M_{N1}^{\text{*}}}}{R}_{1\tau }^{\text{*}}& \sqrt{\frac{m_{{\nu }_{\tau }}}{M_{N2}^{\text{*}}}}{R}_{2\tau }^{\text{*}}\end{array}\right).$ (32)

An explicit form of $R^{†}$ [17] for normal neutrino mass ordering and $m_{{\nu }_e}=0$ , is

$R^{†}=\left(\begin{array}{cc}0& 0\\ \cos z& -\sin z\\ \sin z& \cos z\end{array}\right),$ (33)

while for inverted mass ordering and $m_{{\nu }_{\tau }}=0$ ,

$R^{†}=\left(\begin{array}{cc}\cos z& -\sin z\\ \sin z& \cos z\\ 0& 0\end{array}\right).$ (34)

$z$ is a complex parameter. ${\cos }^{2}z+{\sin }^{2}z=1$ holds.

4. Leptogenesis

The observed baryon-to-photon ratio of the universe, at the present time, is $n_B/n_{\gamma }=\left(6.04\pm 0.12\right)\times {10}^{-10}$ [2]. The corresponding baryon asymmetry before baryon-anti-baryon annihilation is $\left(B-\overline{B}\right)/\left(B+\overline{B}\right)\approx 6.04\times {10}^{-10}/\left(3\times {0.399}^3\right)=3.2\times {10}^{-9}$ , where, at this temperature, $B$ is the sum of protons and neutrons (the factor 3 is $\left(3/4\right)N_f/N_b$ , with $N_f=8$ and $N_b=2$ , and the factor$0.399={\left(43/\left(11\times g_{\text{dec}}\right)\right)}^{1/3}$ ). Assuming “initial” baryon-anti-baryon symmetry, how did the universe acquire this excess of baryons? We consider the following baryogenesis scenario: out of thermal equilibrium charge conjugation and parity (CP) violating and lepton number violating interactions produce a lepton asymmetry that is partially converted to a baryon asymmetry by non-perturbative sphaleron transitions [13] [19] [20]. These transitions conserve the baryon-minus-lepton number$B-L$ , and violate$B+L$ . Sphaleron transitions freeze-out at $T_{\text{sph}}=131.7\pm 2.3$ GeV [13]. At lower temperature the sphaleron rate drops below the universe expansion rate, and the lepton and baryon asymmetries become frozen while ultra-relativistic. Thus, an excess of anti-leptons is partially converted to an exess of baryons (three left-handed anti-leptons, one of each generation, are converted to nine left-handed quarks, three of each generation). The required lepton asymmetry, assuming no sphaleron transitions, at a temperature approxiately $T_{\text{sph}}$ is obtained dividing the baryon asymmetry by ≈0.343 [21]:

$\frac{L-\overline{L}}{L+\overline{L}}\approx -9.3\times {10}^{-9},$ (35)

where the lepton number is $L\equiv {\displaystyle {\sum }_{\alpha }\left(n_{{\alpha }_L^{-}}+n_{{\nu }_{\alpha L}}\right)}$ , and anti-lepton number is $\overline{L}={\displaystyle {\sum }_{\alpha }\left(n_{{\alpha }_L^{+}}+n_{{\overline{\nu }}_{\alpha L}}\right)}$ , with $\alpha =e,\mu ,\tau$ .

Current measurements and observations under-constrain baryogenesis, so that several models are viable. Example reviews are [17] [22] [23]. An often considered scenario is “Type I Leptogenesis” with 2 [17] or more Majorana sterile neutrinos with masses 10⁹ to >10¹² GeV decaying out of thermal equilibrium and violating the CP symmetry and lepton number conservation. The high masses are needed to obtain sufficient lepton asymmetry. Such models are inspired by Grand Unified Theories. The Higgs boson was discovered at the Large Hadron Collider (LHC) and, surprisingly, no new particles were observed! The measured Higgs boson mass turns out to be exactly the mass needed so that the Standard Model may be valid up to very high energies, with no Grand Unified Theory in sight. How are these high mass neutrinos created? In the spirit of the New Minimal Standard Model, we will consider, as a proof of principle, a specific viable leptogenesis scenario [24] [25] with the lepton asymmetry being generated between the temperatures $T_{\text{EWSB}}$ and $T_{\text{sph}}$ [16], knowing that nature might have chosen otherwise.

Leptogenesis requires that the rate of transitions $L\to \overline{L}$ be greater than the rate of $\overline{L}\to L$ . Therefore, out of equilibrium CP violation and lepton number violation is needed. To obtain CP violation, two amplitudes with different “weak” phases and different “strong” phases need to interfere (as in (38) below). By definition, a “weak” phase changes sign under CP-conjugation, while a “strong” phase remains invariant. “Weak” phases come from complex Yukawa couplings $Y_{k\alpha }^{\nu }$ . “Strong” phases come from interference between (nearly) on-shell intermediate massive particle propagation. This is the motivation for considering leptogenesis at temperatures below $T_{\text{EWSB}}$ at which Standard Model particles acquire mass. In “Type I Leptogenesis” the needed “strong” phases come from on-shell mass-less particles in diagrams with loops.

As an example, consider a neutrino oscillation experiment with a base-line $l$ . The weak phases are the phases in the unitary PMNS matrix $V_l$ , or its extension $U$ with added sterile neutrinos (these phases come from complex Yukawa couplings) [2], while the strong phases are the propagation phase differences $2{\chi }_{ij}=\left(m_i^2-m_j^2\right)l/\left(2p\right)$ . The active and sterile neutrinos are ultra-relativistic in the model under consideration. The probability of a lepton conserving event, e.g. $e^{-}{W}^{+}\to {\nu }_i\to {\mu }^{-}{W}^{+}$ , is [2]

$\begin{array}{l}{P}_{\alpha \beta }=P_{\alpha \beta \text{CPC}}+P_{\alpha \beta \text{CPV}},\\ P_{\alpha \beta \text{CPC}}={\delta }_{\alpha \beta }-4{\displaystyle \sum }_{i<j}^{3+m}\mathrm{Re}\left[U_{\alpha i}{U}_{\beta i}^{*}{U}_{\alpha j}^{*}{U}_{\beta j}\right]{\sin }^2{X}_{ij},\\ P_{\alpha \beta \text{CPV}}=+2{\displaystyle \sum }_{i<j}^{3+m}\mathrm{Im}\left[U_{\alpha i}{U}_{\beta i}^{*}{U}_{\alpha j}^{*}{U}_{\beta j}\right]\sin \left(2X_{ij}\right).\end{array}$ (36)

The sums in (36) are over coherent [16] [26] mass eigenstates $i<j$ with masses $m_i$ and $m_j$ that can not be discriminated in the experiment [26] [27]. Under CP conjugation, $U\to U^{\text{*}}$ . The term $P_{\alpha \beta \text{CPC}}$ is CP conserving, while $P_{\alpha \beta \text{CPV}}$ may be CP violating. Note that to obtain CP violation, at least one physical phase in $U$ is needed, in addition to the propagation phase difference $2X_{ij}$ . Note that $X_{ij}\ne 0$ only if two neutrinos acquire different masses at $T<T_{\text{EWSB}}$ . A massive neutrino needs a Weyl_R component added to the Standard Model. Equations (36) are the standard equations used to describe neutrino oscillations in vacuum [2]. There are non-trivial subtleties described in [26] and [27].

To obtain lepton number violation, and also to account for the observed neutrino flavor oscillations, we add $m=2$ (or more) sterile Majorana neutrinos $N_1$ and $N_2$ to the Standard Model. Let us consider the universe in the temperature range of interest between $T_{\text{EWSB}}$ and $T_{\text{sph}}$ . The probability to observe a lepton violating event, e.g. $e^{-}{W}^{+}\to {\nu }_i\to {\mu }^{+}{W}^{-}$ , is [16]

$\begin{array}{l}{P}_{\alpha \overline{\beta }}=P_{\alpha \overline{\beta }\text{CPC}}+P_{\alpha \overline{\beta }\text{CPV}}\ll 1,\\ P_{\alpha \overline{\beta }\text{CPC}}=\frac{1}{2E^2}\left\{{\left|{\displaystyle \sum }_i^{3+m}{m}_i{U}_{\alpha i}^{*}{U}_{\beta i}^{*}\right|}^2-4{\displaystyle \sum }_{i<j}^{3+m}{m}_i{m}_j\mathrm{Re}\left[U_{\alpha i}{U}_{\beta i}{U}_{\alpha j}^{*}{U}_{\beta j}^{*}\right]{\sin }^2{X}_{ij}\right\},\\ P_{\alpha \overline{\beta }\text{CPV}}=\frac{1}{2E^2}\left\{2{\displaystyle \sum }_{i<j}^{3+m}{m}_i{m}_j\mathrm{Im}\left[U_{\alpha i}{U}_{\beta i}{U}_{\alpha j}^{*}{U}_{\beta j}^{*}\right]\sin \left(2X_{ij}\right)\right\},\end{array}$(37)

where we have included the angle averaged polarization miss-match factors $\sqrt{2\left[\left(E-p\right)/\left(E+p\right)\right]}\approx m/\left(\sqrt{2}E\right)$ for an ultra-relativistic neutrino [16]. The probability $P_{\overline{\alpha }\beta }$ for the CP-conjugate event, e.g. $e^{+}{W}^{-}\to {\overline{\nu }}_i\to {\mu }^{-}{W}^{+}$ , is obtained by $U\to U^{\text{*}}$ . Note that the term $P_{\alpha \overline{\beta }\text{CPC}}$ violates lepton number but conserves the CP symmetry. Note that the term $P_{\alpha \overline{\beta }\text{CPV}}$ is lepton number violating and may be CP violating, and is the source of the leptogenesis studied in this article. Note that CPT-invariance holds.

Let us outline the derivation of (37) in order to discuss coherence issues. The cross section ${\sigma }_{\alpha \overline{\beta }}$ includes the interference of two amplitudes

$\propto {\left|U_{\beta i}^{\text{*}}\frac{m_i}{\sqrt{2}E}{\text{e}}^{-i{A}_i}{U}_{\alpha i}^{\text{*}}+U_{\beta j}^{\text{*}}\frac{m_j}{\sqrt{2}E}{\text{e}}^{-i{A}_j}{U}_{\alpha j}^{\text{*}}\right|}^2.$ (38)

with

$A_{i\left(j\right)}=\left(E\pm \text{Δ}E\right)t-\sqrt{{\left(E\pm \text{Δ}E\right)}^2-{\left(m\mp \text{Δ}m\right)}^2}\left(l\pm \text{Δ}l\right),$ (39)

where $\text{Δ}E$ is the neutrino energy uncertainty, $\text{Δ}l$ is the base-line uncertainty, and $m\mp \text{Δ}m$ are the masses of ${\nu }_i$ and ${\nu }_j$ . We consider the case of ultra-relativistic neutrinos, so $E\gg m_{i\left(j\right)}$ . Equation (38) obtains (37). Consider the universe at the temperature $T_{\text{sph}}=132$ GeV. The age of the universe is $t_u=1.4\times {10}^{-11}$ s. The mean time of an electron to the next collision $e^{-}{W}^{+}\to {\nu }_e\to e^{-}{W}^{+}$ , or of an anti-neutrino to the next collision ${\overline{\nu }}_e{W}^{+}\to e^{+}\to {\overline{\nu }}_e{W}^{+}$ , is $t_c\approx 6\times {10}^{-23}$ s. The corresponding electron energy quantum uncertainty in this case is $\text{Δ}E\approx \hslash /t_c\approx 0.01$ GeV.

The build-up of the lepton asymmetry is described by these equations, where $j$ is the time step of duration $t_c$ , i.e., the lifetime of the neutrinos:

$n_{\beta ,j+1}=n_{\beta ,j}+{\displaystyle \sum }_{\alpha }\left[n_{\alpha ,j}\left(P_{\alpha \beta \text{CPC}}+P_{\alpha \beta \text{CPV}}\right)+n_{\overline{\alpha },j}\left(P_{\alpha \overline{\beta }\text{CPC}}-P_{\alpha \overline{\beta }\text{CPV}}\right)\right],$(40)

$n_{\overline{\beta },j+1}=n_{\overline{\beta },j}+{\displaystyle \sum }_{\alpha }\left[n_{\alpha ,j}\left(P_{\alpha \overline{\beta }\text{CPC}}+P_{\alpha \overline{\beta }\text{CPV}}\right)+n_{\overline{\alpha },j}\left(P_{\alpha \beta \text{CPC}}-P_{\alpha \beta \text{CPV}}\right)\right].$(41)

Now subtract (41) from (40), sum over $\beta$ , and divide by $L+\overline{L}$ . The terms with $P_{\alpha \beta \text{CPC}}$ and $P_{\alpha \beta \text{CPV}}$ do not change $L-\overline{L}$ and are neglected. The terms with $P_{\alpha \overline{\beta }\text{CPV}}>0$ build the lepton asymmetry $L-\overline{L}<0$ , until the terms with $P_{\alpha \overline{\beta }\text{CPC}}>0$ saturate the build up. The lepton asymmetry (without sphaleron transitions) becomes [16]

$\frac{L-\overline{L}}{L+\overline{L}}=-t_u{\displaystyle \sum }_{\alpha \overline{\beta }}\frac{P_{\alpha \overline{\beta }\text{CPV}}}{t_c}.$(42)

This asymmetry needs to reach the value (35). Equation (42) is valid while wash-out from $P_{\alpha \overline{\beta }\text{CPC}}$ does not dominate, i.e. while

$\left(n_{\alpha }-n_{\overline{\alpha }}\right)P_{\alpha \overline{\beta }\text{CPC}}+\left(n_{\alpha }+n_{\overline{\alpha }}\right)P_{\alpha \overline{\beta }\text{CPV}}>0.$(43)

$P_{\alpha \overline{\beta }\text{CPV}}$ includes a factor $\text{sin}\left(2{\chi }_{ij}\right)$ with $2{\chi }_{ij}\approx \left(m_i^2-m_j^2\right)t_c/\left(2E\right)$ . We only include coherent events with small $t_c$ with $2{\chi }_{ij}\lesssim \pi /2$ . Events with $2{\chi }_{ij}\gtrsim \pi /2$ are incoherent and cancel on average. The coherent events have $P_{\alpha \overline{\beta }CPV}\propto t_c$ , so $t_c$ drops out of (42).

The leptogenesis problem at hand is under-constrained. As a proof-of-principle, let us consider examples with $m=2$ sterile neutrinos and normal neutrino mass ordering, with $m_{{\nu }_e}=0$ , $m_{{\nu }_{\mu }}=0.009$ eV and $m_{{\nu }_{\tau }}=0.050$ eV. We write $z$ in (33) in the form $z=\alpha +iB$ with real $\alpha$ and $B$ . Leptogenesis requires $e^B\gg 1$ ,

so $\cos z\approx -i\sin z\approx \frac{1}{2}{\text{e}}^B{\text{e}}^{-i\alpha }$ . Let us estimate the cross-section of $W^{+}{e}^{-}\to {\nu }_e\to W^{+}{e}^{-}$ scattering at $T_{\text{sph}}$ [28]:

$\sigma \approx \frac{1}{16\pi s}\frac{g^4}{4}{\left|V_{le{\nu }_e}\right|}^{4}K$ (44)

with

$K\equiv \frac{1}{4\pi }{\displaystyle \int \text{d}\Omega }\cdot 2\left(-\frac{u}{s}-\frac{s}{u}\right)$ (45)

$={\displaystyle {\int }_{-1+ϵ}^1\text{d}\cos \theta \left(\frac{1+\cos \theta }{2}+\frac{2}{1+\cos \theta }\right)}$ (46)

$\approx 1+2\ln \left(\frac{2}{ϵ}\right)\approx 1+2\ln \left(\frac{16\cdot {\left(2T_{\text{sph}}\right)}^4}{M_W^4}\right)\approx \mathrm{8.5.}$ (47)

$\theta$ is the angle between the outgoing and incoming electron. The value of $m_i^2-m_j^2$ fixes the maximum coherent neutrino mean free path $l_c$ corresponding to $2{\chi }_{ij}=\pi /2$ . We present calculations at $T_{\text{sph}}$ . The age of the universe at $T_{\text{sph}}$ is $1.4\times {10}^{-11}$ s. The mean free path and lifetime of ${\nu }_e$ due to the reaction $W^{+}{e}^{-}\to {\nu }_e\to W^{+}{e}^{-}$ is $l_c\approx 1/\left(\sigma n_{eL}\right)\approx 2\times {10}^{-14}$ m and $t_c=l_c/c\approx 6\times {10}^{-23}$ s, respectively, and similarly for ${\nu }_{\mu }$ and ${\nu }_{\tau }$ . $n_{eL}$ is the density of left-handed electrons. $2{\chi }_{ij}=\pi /2$ at this $l_c$ if $m_{{\nu }_i}^2-m_{{\nu }_j}^2\approx 9$ GeV².

• Example 1: $m=2$ . Active coherent neutrinos, e.g. ${\tau }^{-}{W}^{+}\to {\nu }_{\tau }{\nu }_{\mu }{\nu }_e\to {\mu }^{+}{W}^{-}$ . In this example $2{\chi }_{\tau \mu }\approx {10}^{-21}$ is too small to have a viable leptogenesis solution.

• Example 2: $m=2$ . Three active and one sterile coherent neutrinos, e.g. ${\tau }^{-}{W}^{+}\to {\nu }_e{\nu }_{\mu }{\nu }_{\tau }$ ,$N_1\to {\mu }^{+}{W}^{-}$ . For $2{\chi }_{e4}\approx \pi /2$ the needed coherent sterile neutrino mass is $M_{N_1}\approx 3$ GeV. We choose the second sterile neutrino to be incoherent, e.g. $M_{N_2}=20$ GeV. We obtain

$\frac{P_{\tau \overline{\mu }\text{CPV}}}{t_c}=\frac{M_{N_1}}{E^2}\mathrm{Im}\{{\displaystyle \sum }_{i=1}^3\left(m_{{\nu }_i}{V}_{l\tau i}{V}_{l\mu i}\right)$(48)

$\cdot {\left(i{V}_{l\tau {\nu }_{\mu }}\sqrt{\frac{m_{{\nu }_{\mu }}}{M_{N_1}}}\cos z+i{V}_{l\tau {\nu }_{\tau }}\sqrt{\frac{m_{{\nu }_{\tau }}}{M_{N_1}}}\sin z\right)}^{*}$

$\cdot {\left(i{V}_{l\mu {\nu }_{\mu }}\sqrt{\frac{m_{{\nu }_{\mu }}}{M_{N_1}}}\cos z+i{V}_{l\mu {\nu }_{\tau }}\sqrt{\frac{m_{{\nu }_{\tau }}}{M_{N_1}}}\sin z\right)}^{*}\}\frac{M_{N_1}^2}{2E}$ (49)

$\approx 0.06\cdot \frac{m_{{\nu }_{\tau }}^2}{E^2}{\text{e}}^{i2\alpha }{\text{e}}^{2B}\frac{M_{N_1}^2}{2E}.$ (50)

It is possible to choose $\alpha$ so that the product of the four $U$ ‘s in (37) is purely imaginary. This choice maximizes $P_{\tau \overline{\mu }\text{CPV}}$ while obtaining zero wash-out from $P_{\tau \overline{\mu }\text{CPC}}$. From (35), (42) and (50) we obtain ${\text{e}}^B\approx 3500$ . We veryfy that the largest element of $M_D^{†}{M}_N^{-1}$ is ≈0.003 ≪ 1, leaving some room for larger baryon-to-photon ratio than observed. In conclusion, with just two sterile Majorana neutrinos it is possible to obtain successful leptogenesis in this channel.

• Example 3: $m=2$ . Two sterile coherent neutrinos, e.g. ${\tau }^{-}{W}^{+}\to N_1$ ,$N_2\to {\mu }^{+}{W}^{-}$ . There is a single pair of coherent neutrinos: the two heavy sterile neutrinos $N_1$ and $N_2$ , which in this example have masses $M_{N_1}=10-0.2$ GeV and $M_{N_2}=10+0.2$ GeV to still be coherent with $2{\chi }_{ij}\lesssim \pi /2$ , and much less than $E$ to be ultra-relativistic. We obtain

$\begin{array}{c}{P}_{\tau \overline{\mu }\text{CPV}}=\frac{M_{N_1}{M}_{N_2}}{E^2}\mathrm{Im}\{\left(i{V}_{l\tau {\nu }_{\mu }}\sqrt{\frac{m_{{\nu }_{\mu }}}{M_{N_1}}}\cos z+i{V}_{l\tau {\nu }_{\tau }}\sqrt{\frac{m_{{\nu }_{\tau }}}{M_{N_1}}}\sin z\right)\\ \cdot \left(i{V}_{l\mu {\nu }_{\mu }}\sqrt{\frac{m_{{\nu }_{\mu }}}{M_{N_1}}}\cos z+i{V}_{l\mu {\nu }_{\tau }}\sqrt{\frac{m_{{\nu }_{\tau }}}{M_{N_1}}}\sin z\right)\\ \cdot {\left(-i{V}_{l\tau {\nu }_{\mu }}\sqrt{\frac{m_{{\nu }_{\mu }}}{M_{N_2}}}\sin z+i{V}_{l\tau {\nu }_{\tau }}\sqrt{\frac{m_{{\nu }_{\tau }}}{M_{N_2}}}\cos z\right)}^{*}\\ \cdot {\left(-i{V}_{l\mu {\nu }_{\mu }}\sqrt{\frac{m_{{\nu }_{\mu }}}{M_{N_2}}}\sin z+i{V}_{l\mu {\nu }_{\tau }}\sqrt{\frac{m_{{\nu }_{\tau }}}{M_{N_2}}}\cos z\right)}^{*}\}2{\chi }_{ij}\\ =0,\end{array}$(51)

because the product $U_{\tau 4}{U}_{\mu 4}{U}_{\tau 5}^{\text{*}}{U}_{\mu 5}^{\text{*}}$ is real, so there is no lepton asymmetry in this channel.

Therefore, instead of adding two sterile neutrinos, let us consider three.

• Example 4: $m=3$ . Two sterile coherent neutrinos, e.g. ${\tau }^{-}{W}^{+}\to N_1$ , $N_2\to {\mu }^{+}{W}^{-}$ . The Casas-Ibarra matrix ${\left(R^{†}\right)}_{\alpha j}$ , satisfying $R^{\text{T}}R=1_{3\times 3}$ , can be written as

$R^{†}=\left(\begin{array}{ccc}1& 0& 0\\ 0& c_{23}& s_{23}\\ 0& -s_{23}& c_{23}\end{array}\right)\cdot \left(\begin{array}{ccc}{c}_{13}& 0& s_{13}\\ 0& 1& 0\\ -s_{13}& 0& c_{13}\end{array}\right)\cdot \left(\begin{array}{ccc}{c}_{12}& s_{12}& 0\\ -s_{12}& c_{12}& 0\\ 0& 0& 1\end{array}\right),$ (52)

where $c_{ij}\equiv \cos z_{ij}$ , $s_{ij}\equiv \sin z_{ij}$ , and the $z$ ’s have the form $z_{ij}\equiv {\alpha }_{ij}+i{B}_{ij}$ , with ${\alpha }_{ij}$ and $B_{ij}$ real. The problem is under-constrained, so, as a proof of principle, we may set $z_{12}=0$ , and $B_{13}=1$ . Then, for $\text{exp}\left(B_{23}\right)\gg 1$ ,

$U_{\tau 4}{U}_{\mu 4}{U}_{\tau 5}^{*}{U}_{\mu 5}^{*}\approx \left(-i{V}_{l\tau {\nu }_{\mu }}\sqrt{\frac{m_{{\nu }_{\mu }}}{M_{N_1}}}{s}_{13}{s}_{23}-i{V}_{l\tau {\nu }_{\tau }}\sqrt{\frac{m_{{\nu }_{\tau }}}{M_{N_1}}}{s}_{13}{c}_{23}\right)$

$\cdot \left(-i{V}_{l\mu {\nu }_{\mu }}\sqrt{\frac{m_{{\nu }_{\mu }}}{M_{N_1}}}{s}_{13}{s}_{23}-i{V}_{l\mu {\nu }_{\tau }}\sqrt{\frac{m_{{\nu }_{\tau }}}{M_{N_1}}}{s}_{13}{c}_{23}\right)$

$\cdot {\left(i{V}_{l\tau {\nu }_{\mu }}\sqrt{\frac{m_{{\nu }_{\mu }}}{M_{N_2}}}{c}_{23}-i{V}_{l\tau {\nu }_{\tau }}\sqrt{\frac{m_{{\nu }_{\tau }}}{M_{N_2}}}{s}_{23}\right)}^{*}$

$\cdot {\left(i{V}_{l\mu {\nu }_{\mu }}\sqrt{\frac{m_{{\nu }_{\mu }}}{M_{N_2}}}{c}_{23}-i{V}_{l\mu {\nu }_{\tau }}\sqrt{\frac{m_{{\nu }_{\tau }}}{M_{N_2}}}{s}_{23}\right)}^{\text{*}}$ (53)

$\approx 0.03\cdot \frac{m_{{\nu }_{\tau }}^2}{M_{N_1}{M}_{N_2}}{\text{e}}^{-i2{\alpha }_{13}}{\text{e}}^{4B}.$ (54)

It is possible to choose ${\alpha }_{13}$ so $U_{\tau 4}{U}_{\mu 4}{U}_{\tau 5}^{\text{*}}{U}_{\mu 5}^{\text{*}}$ is purely imaginary. This choice maximizes $P_{\tau \overline{\mu }\text{CPV}}$, while the wash-out from $P_{\tau \overline{\mu }\text{CPC}}$ is zero. As an example, we take $M_{N_1}=10-0.2$ GeV and $M_{N_2}=10+0.2$ GeV, corresponding to $2{\chi }_{45}\approx \pi /2$ . $N_3$ needs to be incoherent, so we choose, for example, $M_{N_3}=20$ GeV. From (35), (37) and (42) we obtain ${\text{e}}^B\approx 72$ . We verify that the largest element of $M_D^{†}{M}_N^{-1}$ is $\approx 7\times {10}^{-5}\ll 1$ , leaving room for larger baryon-to-photon ratio than observed.

In summary, successful baryogenesis can be achieved by coherent flavor-violating and CP-violating oscillations of Majorana neutrinos in the temperature range from $T_{\text{EWSB}}$ to $T_{\text{sph}}$ [24] [25]. For coherent interference between one sterile neutrino and the active neutrinos, successful leptogenesis is obtained with $m\ge 2$ sterile neutrinos added to the Standard Model. One of these sterile neutrinos has a mass of order 3 GeV. For coherent interference between two sterile neutrinos, at least $m=3$ sterile neutrinos need to be added to the Standard Model. The $m\ge 3$ cases are under-constrained. We note that the dark matter field $S$ plays no role in these leptogenesis scenarios.

5. Sterile Neutrino Production and Decay

Let us list some sterile neutrino production cross-sections and decy rates (assumed to be above threshold):

$\sigma \left({\nu }_{\tau }{W}^{-}\leftrightarrow {\tau }^{-}\leftrightarrow N_1{W}^{-}\right)\approx \frac{K}{16\cdot \pi s}{\left|V_{l\tau {\nu }_{\tau }}\right|}^2\frac{g^4}{4}{\left|V_l\sqrt{M_l}{R}^{†}{M}_N^{-1/2}\right|}_{\tau 4}^2,$ (55)

$\sigma \left({\tau }^{-}{W}^{+}\leftrightarrow {\nu }_{\tau }\leftrightarrow N_{1}h\right)\approx \frac{g^2}{32\cdot \pi s{v}^2}{\left|V_{l\tau {\nu }_{\tau }}\right|}^2{\left|\sqrt{M_N}R\sqrt{M_l}{V}_l^{†}\right|}_{4\tau }^2.$ (56)

Once $N_1$ becomes non-relativistic at $T\approx M_{N_1}$ :

$\sigma \left(N_1{N}_1\to {\nu }_{\tau }{\nu }_{\tau }\right)\approx \frac{M_{N_1}^4}{12\pi s{v}^4{m}_H^4}{\left|\sqrt{M_N}R\sqrt{M_l}{V}_l^{†}\right|}_{\tau 4}^4.$ (57)

As an example, consider sterile neutrino production in the channel of (55) at the temperature $T_{\text{sph}}$ . The age of the universe is $1.4\times {10}^{-11}$ s, while the mean time for a ${\nu }_{\tau }$ to convert to a $N_1$ is ≈10⁻¹⁷ s (with the parameters of Example 2, e.g. ${\text{e}}^B=3500$ ), or ≈10⁻¹⁴ s (with the parameters of Example 4, e.g. ${\text{e}}^B=72$ ). Note that $N_1$ comes into thermal and diffusive equilibrium with the Standard Model sector as soon as the Higgs boson comes into equilibrium with $S$ , and $N_1$ becomes depleted from the universe as soon as the temperature drops below the threshold $M_W+M_{N_1}$ .

On the other hand, the depletion of $N_1$ in the channel (57), when $N_1$ becomes non-relativistic, is negligible.

6. Inflation

The original “New Minimal Standard Model” [1] adds a real scalar field Φ that drives inflation. The Lagrangian of Section 3 includes two scalar fields: the warm dark matter field $S$ , and the Higgs boson field $\varphi$ . $S$ or $\varphi$ may drive inflation [29]-[33] so we tentatively do not add the field Φ to the updated “New Minimal Standard Model”. The field Φ allows, and requires, more interacting terms with the Standard Model sector [1], which may become necessary when pre-heating and re-heating are better understood.

Higgs inflation is nearly critical: the Higgs quartic coupling ${\lambda }_{\varphi }$ may run negative at energies approaching the Planck scale (within the experimental uncertainties of the top quark mass, the Higgs boson mass, and the $SU{\left(3\right)}_c$ coupling constant ${\alpha }_3$ ) [33] [34]. Inflation driven by the dark matter field $S$ is not critical. We will assume that the dark matter field $S$ drives inflation.

A detailed study of inflation can be found in [33], see Figure 3. Here we present a few comments on the Lagrangian (11):

• For ${\lambda }_S=0$ the dark matter particles have a large mass ≈10¹³ GeV corresponding to cold dark matter. For the observed warm dark matter we need the quartic potential with ${\lambda }_S>0$ .

• The non-minimal couplings ${\xi }_{\varphi }$ and ${\xi }_S$ to the curvature Ricci scalar $R$ are needed for inflation and also for renormalization [31].

Figure 3. Past and future history of the universe. Vertical axis: natural logarithm of the expansion parameter $a$ , i.e., “e-folds”. Horizontal axis: natural logarithm of comoving distances ${\text{M}}_{\text{P}}r/a$ in Planck units. The red line corresponds to a fixed point in space at a reference comoving distance $1/k=20$ Mpc. The blue line is the comoving distance to the horizon ${\text{M}}_{\text{P}}/\left(aH\right)$ in Planck units. The thin green box (with a width too small to be resolved on this scale) is the prediction $\left({\text{M}}_{\text{P}}/\left(a_f{H}_f\right),a_f\right)$ for the end of inflation with the observed values of the comoving power spectrum amplitude $P_{\zeta }\left(k\right)$ and tilt $n_s$ , at 68% confidence, with no free parameter! The Cosmic Microwave Background radiation (CMB) that we see today propagates to us from point “X”. Symbols “-” and “×” illustrate segments of “light” rays inside and outside of the horizon. Figure from [33].

• The measured comoving power spectrum $P_{\zeta }\left(k_{\text{ref}}\right)$ at the reference wave-vector $k_{\text{ref}}=0.05{\text{Mpc}}^{-1}$ , and its tilt $n_s$ [2], pass a closure test with no free parameters [33], i.e., the end of inflation coincides, within observational uncertainties, with the beginning of pre-heating, followed by re-heating and the hot Big Bang cosmology. This coincidence is quite amazing!

Some numerical results are summarized in Table 1.

7. A Brief History of the Universe

The problem at hand is under-constrained, so several alternatives are possible. One viable sequence of events follows [4] [16] [33], see Figure 1 and Figure 3. Note that some of the numbers are model-dependent.

1) The dark matter field $S$ is the only field in the early universe. $S$ drives inflation. The reference comoving distance $1/k_{\text{ref}}=20$ Mpc enters the horizon at expansion parameter $a_i=\text{exp}\left(-121.28\pm 0.12\right)$ .

2) Inflation ends at $a_f=\text{exp}\left(-64.1\pm 6.8\right)$ .

3) Pre-heating is followed by re-heating, and the hot Big Bang cosmology follows.

Table 1. Inflation parameters obtained from the measured comoving power spectrum $P_{\zeta }\left(k_{\text{ref}}\right)\equiv {\text{Δ}}_R^2=\text{exp}\left(3.044\pm 0.014\right)\times {10}^{-10}$ at the reference wave-vector $k_{\text{ref}}=0.05{\text{Mpc}}^{-1}$ , and its tilt $n_s=0.965\pm 0.004$ [2]. $a_i$ and $S_i$ are the expansion parameter $a$ and amplitude $S$ when the comoving distance $1/k_{\text{ref}}$ enters the horizon. $N=\text{ln}\left(a_f/a_i\right)$ is the number of e-folds of inflation, $a_f$ is the expansion parameter at the end of inflation, and $r$ is the tensor-to-scalar ratio. Here we assume that the dark matter field $S$ drives inflation. For $\varphi$ driving inflation replace $S\to \varphi$ . For details see [33].

Parameter	Value
$N\equiv \text{ln}\left(a_f/a_i\right)$	$57.1\pm 6.6$
$\text{ln}\left(a_i\right)$	$-121.28\pm 0.12$
$\text{ln}\left(a_f\right)$	$-64.1\pm 6.8$
$r$	$\left(3.7\pm 0.9\right)\times {10}^{-3}$
$\sqrt{{\xi }_S}{S}_i$	$\left(1.74\pm 0.10\right){\text{ M}}_{\text{P}}$
${\lambda }_S^{1/4}{S}_i$	$\left(8.05\pm 0.03\right)\times {10}^{-3}{\text{ M}}_{\text{P}}$
${\lambda }_S/{\xi }_S^2$	$\left(4.6\pm 1.1\right)\times {10}^{-10}$

4) $S$ and the Higgs boson $\varphi$ , and hence the gauge bosons ${\overrightarrow{G}}_i$ , $\overrightarrow{W}$ and $\overrightarrow{B}$ , and Weyl-L and Weyl-R fields, come into thermal and diffusive equilibrium before $T=T_{\text{EWSB}}$ if $\left|{\lambda }_{\varphi S}\right|\gtrsim 7\times {10}^{-7}$ . This is the “Weyl” epoch.

5) The Higgs boson $\varphi$ acquires a vaccuum expectation value at the temperature $T_{\text{EWSB}}=159\pm 1$ GeV. Weyl_L and Weyl_R fields pair up into massive Dirac fields. This is the “Dirac” epoch. Linear superpositions of ${\overrightarrow{W}}_{\mu }$ and $B_{\mu }$ become massive $W^{+}$ , $W^{-}$ and $Z$ bosons, leaving the photon $\gamma$ mass-less.

6) All particles of the NMSM (including the sterile neutrinos) reach equilibrium.

7) Coherent CP-violating and lepton number-violating oscillations between a sterile neutrino (with mass of order 3 GeV) and the three active neutrinos (or between two sterile neutrinos if $m\ge 3$ ) produce a lepton asymmetry. Due to sphaleron transitions [13] [19] [20], part of the lepton asymmetry is converted to a baryon asymmetry.

8) At $T_{\text{sph}}=131.7\pm 2.3$ GeV, sphalerons freeze-out, i.e., the conversion rate of anti-leptons to three quarks (for each generation), become less than the universe expansion rate. The lepton and baryon asymmetries freeze-out.

9) At $T=m_H=125.20\pm 0.11$ GeV the Higgs boson becomes non-relativistic and decays. $S$ decouples from the remaining fields of the Standard Model sector.

10) Sterile neutrinos $N_i$ are depleted from the universe as soon at the temperature drops below $M_W+M_{N_i}$ .

11) Particles and anti-particles annihilate as they become non-relativistic. Baryon-anti-baryon annihilations result in the present day baryon-to-photon ratio $n_B/n_{\gamma }=6.12\pm 0.04\times {10}^{-10}$ .

12) The first three minutes: Big Bang Nucleosynthesis forms helium, and other light, nuclei.

13) The light neutrinos decouple. Electrons and positrons annihilate.

14) The warm dark matter particles $S$ , of mass $m_S=150\pm 2$ eV, become non-relativistic at expansion parameter $a_{h\text{NR}}=\left(1.35\pm 0.23\right)\times {10}^{-6}$ .

15) Matter density begins to dominate at $a\approx 1/3403$ , and the universe becomes transparent to photons as neutral hydrogen atoms form at $a\approx 1/1091$ .

8. Tests

We present a list of observations that may test the “New Minimal Standard Model” or constrain its parameters:

• High resolution weak lensing measurements of the Cosmic Microwave Background (CMB) radiation can measure precisely the linear power spectrum of density fluctuations $P\left(k\right)$ up to a wave-vector $\approx 40h{\text{Mpc}}^{-1}$ [8]. Such a measurement would definitely rule out or confirm the interpretation that $v_{h\text{rms}}\left(1\right)$ of (2) is of cosmological origin.

• Next generation measurements of the CMB power spectrum may reach and constrain $k_{\text{fs}}$ of (4).

• Observations and simulations of the Lyman-α forest of quasar light set limits to the warm dark matter free-streaming cut-off wavevector $k_{\text{fs}}$ , see, e.g. [6]. Measurements of $v_{h\text{rms}}\left(1\right)$ and of $k_{\text{fs}}$ , and perhaps also Figure 2 of [8], are in disagreement with these Lyman-α limits (see Section 10 of [4]). Resolving this discrepancy with new observations and studies will constrain possible extensions of the Standard Model.

• The cores of some dwarf galaxies are dominated by dark matter. Dedicated studies of the cores of these galaxies can further constrain the dark matter adiabatic invariant $v_{h\text{rms}}\left(1\right)$ , which is of cosmological origin [4] [12]. An example is [35] (that obtains $v_{h\text{rms}}\left(1\right)=515\pm 15$ (stat) m/s [4]).

• If nature has chosen only two sterile neutrinos, then one of them has a mass of order 3 GeV. Neutrino oscillation experiments, focusing on lepton number violating events, can constrain the parameters in Example 2 of Section 4.

• Constrain or measure the dark matter-dark matter elastic scattering cross-section per unit mass $\sigma \left(SS\to SS\right)/m_S$ . The current upper limit is 0.47 cm²/g. From the limit on ${\lambda }_{\varphi S}\gtrsim 7\times {10}^{-7}$ obtained in Section 2, the lower limit is ≈10⁻⁹ cm²/g. Galaxy properties imply that dark matter has self-elastic interactions [12].

• Constrain or measure the power spectrum tensor-to-scalar ratio. The predicted value is $r=\left(3.7\pm 0.9\right)\times {10}^{-3}$ at $k_{\text{ref}}=0.05{\text{Mpc}}^{-1}$ (see Table 1), while the current limit is $r<0.036$ at 95% confidence [2].

• The Higgs boson quartic coupling ${\lambda }_{\varphi }$ is critical. Within the experimental uncertainties of the top quark mass, the Higgs boson mass, and the $SU{\left(3\right)}_c$ coupling constant ${\alpha }_3$ , ${\lambda }_{\varphi }$ may run negative at high energy [34]. Understanding this near criticality, and tighter experimental constraints, may shed light on the early universe.

• Measurements or tighter limits on the $0\nu 2\beta$ effective Majorana electron-neutrino mass ${\langle m\rangle }_{ee}=\left|m_1{V}_{le{\nu }_e}^2+m_2{V}_{le{\nu }_{\mu }}^2+m_3{V}_{le{\nu }_{\tau }}^2\right|$ may shed light on the Dirac or Majorana nature of neutrinos [2] [17].

• Re-heating following inflation needs new theoretical ideas.

9. Conclusions

The Standard Model of quarks and leptons is a great achievement. However, this model is incomplete: it does not include the observed dark matter, neutrino masses and mixings, the matter-antimatter asymmetry of the universe, the current acceleration of the expansion of the universe, nor inflation. In 2004, Hooman Davoudiasl, Ryuichiro Kitano, Tianjun Li, and Hitoshi Murayama proposed the “New Minimal Standard Model” to account for these observations [1]. This minimal extension of the Standard Model is valid to this day! The minimal extension of the Standard Model to account for all current confirmed observations is the addition of a scalar warm dark matter field $S$ with $Z_2$ symmetry $S\leftrightarrow -S$ , two sterile neutrinos with both Yukawa and Majorana mass terms to give mass to two of the active neutrinos, and allow lepton number violation needed for baryogenesis via leptogenesis, and Einstein’s cosmological constant. See Figure 2.

In the intervening twenty years, the top quark and Higgs boson were discovered, and great advances in neutrino physics, cosmology, dark matter, baryogenesis via leptogenesis, and inflation have been achieved. As a result, we have been able to explore details of the “New Minimal Standard Model”, constrain some of the new parameters, and propose tests.

Acknowledgements

I thank Karsten Müller for many useful discussions, and an anonymous reviewer for detailed and constructive feedback that has improved this article.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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