Spin-independent two-neutrino exchange potential with mixing and $CP$ violation

We develop a new approach for calculating the spin-independent two-neutrino exchange potential (2NEP) between nonrelativistic fermions which places emphasis on the neutrino vacuum state, an area of theoretical interest in recent years. The 2NEP is a natural probe of fundamental issues of neutrino physics such as neutrino masses, flavor mixing, the number of neutrino flavors, neutrino nature (Dirac or Majorana), $CP$ violation, and the neutrino vacuum state. We explore the dependence of the 2NEP on the mixing of neutrino mass states assuming normal and inverted mass ordering for nucleon-nucleon, nucleon-lepton, and lepton-lepton interactions, and the $CP$-violation phase in the neutrino mixing matrix.

Figure 1

Feynman diagram of neutrino-antineutrino exchange between fermions which leads to Eq. (1). The thicker (thinner) lines denote fermion (neutrino) propagators.

Figure 2

Feynman diagrams showing how the NC (top) and CC (bottom) processes involving vector bosons $Z^0$ and $W^{\pm }$ and neutrinos lead to the effective process described by Eqs. (26) and (27) when the energies involved satisfy $E\ll M_Z,M_W$. Note that nucleons and charged leptons interact via NC processes while only leptons participate in the CC processes.

Figure 3

Feynman diagrams contributing to the 2NEP. Here (a) represents the NC-NC interaction, (b) one of the two NC-CC diagrams contributing to the lepton weak form factor, and (c) the CC-CC diagram. For the nucleon-nucleon 2NEP, only the diagram (a) contributes, while diagrams (a) and (b) contribute to the nucleon-lepton 2NEP. Finally, (a), (b), and (c) contribute to the lepton-lepton 2NEP.

Figure 4

Plots of the total 2NEP between nucleons and electrons with NO of neutrino masses: the black lines represent the neutron-electron 2NEP, while the gray lines represent the proton-electron 2NEP. We assumed the smallest neutrino mass state $m_{\min }=0\mathrm{eV}$ (short-dashed), $m_{\min }=0.05\mathrm{eV}$ (long-dashed) and $m_{\min }=0.25\mathrm{eV}$ (solid). Values for all other parameters used were obtained from the Particle Data Group [44] given in Table 2.

Figure 5

Plots of the total 2NEP between (a) 2 electrons and (b) electron and muon. Black lines represent NO of neutrino masses while the gray lines represent IO. It is assumed respectively that $m_{\min }=0\mathrm{eV}$ (short-dashed), $m_{\min }=0.05\mathrm{eV}$ (long-dashed), $m_{\min }=0.25\mathrm{eV}$ (solid), and that neutrinos are massless without mixing (black, dot-dashed). Values for all other parameters were obtained from the Particle Data Group [44] given in Table 2.

Figure 6

The ratios of the contribution from neutrino mixing to the overall 2NEP between: (a) Two electrons, (b) two muons, (c) two taus, (d) electron and muon, (e) electron and tau and (f) muon and tau. Black lines represent NO of neutrino masses while the gray lines represent IO. It is assumed respectively that $m_{\min }=0\mathrm{eV}$ (short-dashed), $m_{\min }=0.05\mathrm{eV}$ (long-dashed), $m_{\min }=0.25\mathrm{eV}$ (solid). Values for all other relevant parameters were obtained from the Particle Data Group [44] given in Table 2.

Figure 7

The absolute value of the ratio $V_{e\mu }^{(CP)}(r)/V_{e\mu }^{(0)}(r)$ from Eq. (61) for several possibilities of minimum neutrino masses. Black lines represent NO of neutrino masses while the gray lines represent IO. It is assumed respectively that $m_{\min }=0\mathrm{eV}$ (short-dashed), $m_{\min }=0.05\mathrm{eV}$ (long-dashed), $m_{\min }=0.25\mathrm{eV}$ (solid). Values of all other parameters were obtained from the Particle Data Group [44] given in Table 2. Note: For $r>0$, each cusp in the IO curves corresponds to where the contribution of the $CP$-violating term changes its sign.