The 2-Neutrino Exchange Potential with Mixing: A Probe of Neutrino Physics and CP Violation

TL;DR

This paper introduces a new method for calculating the 2-neutrino exchange potential, incorporating neutrino mixing and CP violation, which can help probe fundamental neutrino properties and CP violation effects.

Contribution

A novel approach to compute the spin-independent 2-neutrino exchange potential including three-flavor mixing and CP violation effects.

Findings

New calculation method for 2-neutrino exchange potential

Inclusion of neutrino mixing and CP violation in potential calculations

Potential to probe neutrino properties and CP violation experimentally

Abstract

The 2-neutrino exchange potential is a Standard Model weak potential arising from the exchange of virtual neutrino-antineutrino pairs which must include all neutrino properties, including the number of flavors, their masses, fermionic nature (Dirac or Majorana), and CP violation. We describe a new approach for calculating the spin-independent 2-neutrino exchange potential, including the mixing of three neutrino mass states and CP violation.

Equations

$$V_{\nu ,\bar{\nu }}(r)=\frac{G_F^2{g}_{V,1}^f{g}_{V,2}^f{m}_{\nu }^3}{4{\pi }^3{r}^2}{K}_3(2m_{\nu }r),$$

$$U_{\alpha a}=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{\rm CP}}\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{\rm CP}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{\rm CP}}&s_{23}c_{13}\\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{\rm CP}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{\rm CP}}&c_{23}c_{13}\end{array}\right).$$

$$V_{{\mathrm{N}}_1,{\mathrm{N}}_2}(r)=\frac{G_F^2{g}_{V,1}^{{\mathrm{N}}_1}{g}_{V,2}^{{\mathrm{N}}_2}}{4{\pi }^3{r}^2}\sum _{a=1}^3{m}_a^3{K}_3(2m_{a}r),$$

$$V_{\mathrm{N}\alpha }(r)=\frac{G_F^2{g}_V^{\mathrm{N}}}{4{\pi }^3{r}^2}\sum _{a=1}^3{m}_a^3\left(g_V^{\alpha }+|U_{\alpha a}{|}^2\right)K_3(2m_{a}r).$$

$$V_{\alpha \beta }(r)$$

$$\begin{array}{rl}{V}_{\alpha \beta ,\mathrm{mix}}(r)& \simeq \frac{G_F^2}{4{\pi }^3{r}^2}[\sum _{a>b}^3\frac{\mathrm{Re}\left(U_{\alpha a}^{\ast }{U}_{\alpha b}{U}_{\beta b}^{\ast }{U}_{\beta a}\right)}{4}\\ & \left\{m_{+}^{ab}\left[{\left(m_{+}^{ab}\right)}^2+{\left(m_{-}^{ab}\right)}^2\right]K_3\left(m_{+}^{ab}r\right)-\frac{4{\left(m_{-}^{ab}\right)}^2}{r}{K}_2\left(m_{+}^{ab}r\right)\right\}].\end{array}$$

$$V_{\alpha \beta }(r)=V_{\alpha \beta }^{(0)}(r)+V_{\alpha \beta }^{(\mathrm{CP})}(r){\sin }^2\left(\frac{{\delta }_{\mathrm{CP}}}{2}\right),$$

Full text

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The neutrino sector of the Standard Model (SM) holds great potential for revealing new physics. Interestingly, the unsolved problems of neutrino physics [e.g., the masses of the three neutrino mass states, the neutrino’s fermionic nature (Dirac or Majorana), number of flavors, existence of sterile neutrinos and CP violation] all impact the 2-neutrino exchange potential (2-NEP), the weak interaction force arising from the exchange of virtual neutrino–antineutrino pairs. The formulas for the single neutrino flavor 2-NEP were first derived by Feinberg and Sucher[1] and Fischbach[2] assuming massless and massive neutrinos, respectively. A number of other authors have also investigated the 2-NEP.[3, 4, 5] Lusignoli and Petrarca[6] developed an integral formula for the 2-NEP with mixing of three neutrino flavors, but did not include all of the electroweak contributions. Here, we describe a new derivation of the 2-NEP with mixing that incorporates neutral-current (NC) and charged-current (CC) weak interactions and CP violation, and discuss the possibilities of using the 2-NEP as a probe of neutrino physics.[7]

In our approach for calculating the 2-NEP, we express the neutrino fields in the Schrödinger picture and then use time-independent perturbation theory to calculate the second-order energy shift of the neutrino-field vacuum energy due to the presence of two stationary fermions. We ignore infinite self-energy corrections, which only depend on the positions of a single fermion. The spin-independent contribution, which depends on the separation distance $r$, is finite, and for the single-flavor case involving NC weak interactions is found to be given by[7]

[TABLE]

where $G_{F}$ is the Fermi constant, $g_{V}$ is the vector coupling constant, $m_{\nu}$ is the neutrino mass, and $K_{n}(x)$ is the modified Bessel function.

To incorporate mixing, we write the three flavor fields $\nu_{\alpha}(\vec{r})$, $(\alpha=e,\mu,\tau)$ as linear combinations of the three mass fields $\nu_{a}(\vec{r})$, $(a=1,2,3)$, i.e., $\nu_{\alpha}(\vec{r})\equiv\sum_{a=1}^{3}U_{\alpha a}\,\nu_{a}(\vec{r})$, where $U_{\alpha a}$ are components of the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix,

[TABLE]

Here, $s_{ab}=\sin\theta_{ab}$, $c_{ab}=\cos\theta_{ab}$, and $\delta_{\rm CP}$ is the CP-violation phase. For the purpose of our calculation, we note that nucleons interact with the neutrino only via NC interactions, while leptons also require the inclusion of CC interactions. We will therefore need to consider three cases for the interaction potentials: nucleon–nucleon, nucleon–lepton, and lepton–lepton.

The 2-NEP between nucleons is the simplest since the NC current interaction is independent of neutrino flavor. For nucleons #1 and #2, we merely sum Eq. (1) over the three mass states, which gives[7]

[TABLE]

where N = proton or neutron, $g_{V}^{{\rm N}}=\frac{1}{2}-2\sin^{2}\theta_{W}$ for protons and $g_{V}^{{\rm N}}=-\frac{1}{2}$ for neutrons, where $\theta_{W}$ is the Weinberg angle. The magnitude of this interaction is quite small. For two neutrons, the gravitational force is larger than the 2-NEP force when $r\gtrsim 1\,$nm.

For the case of a nucleon interacting with a lepton, one finds a result similar to the nucleon–nucleon potential except for a change of the lepton vector coupling, which depends on the PMNS matrix element corresponding to the lepton flavor,

[TABLE]

The final case, the lepton–lepton 2-NEP, is the most interesting since the mixing has the greatest impact on the form of the potential and it involves both NC and CC interactions, but we were unable to obtain a closed-form expression if all the masses are nonzero. Instead, for two lepton flavors $\alpha$ and $\beta$, we found an expansion for the 2-NEP in powers of $(m_{-}^{ab}/m_{+}^{ab})$, where $m_{\pm}^{ab}\equiv m_{a}\pm m_{b}$:[7]

[TABLE]

where to order $\left[\left(\frac{m_{-}^{ab}}{m_{+}^{ab}}\right)^{2}\right]$

[TABLE]

The lepton–lepton 2-NEP is particularly interesting because of its dependence on the CP-violating phase $\delta_{\rm CP}$. Presently, there is growing evidence[8] that $\delta_{\rm CP}\neq 0$. We show that one can write[7]

[TABLE]

where $V_{\alpha\beta}^{(0)}$ and $V_{\alpha\beta}^{({\rm CP})}(r)$ are complicated functions of $r$ independent of $\delta_{\rm CP}$, and $V_{ee}^{({\rm CP})}(r)=0$ by the definition of $\delta_{\rm CP}$. Therefore, except for the interaction between two electrons, the 2-NEP potential will depend on the CP-violating phase due to the interference of the PMNS matrix elements.

While the original study of 2-NEP arose mainly out of theoretical interests, our results raise the possibility of opening new avenues for experimental explorations of basic neutrino parameters. Because the 2-NEP involves the exchange of virtual neutrinos, all neutrino properties and energies must contribute. Besides being sensitive to the mixing angles, the 2-NEP depends directly on the actual neutrino masses, not the difference in mass squared as in neutrino-oscillation experiments. If we assume the neutrino mass spectrum lies in the range $1\,$meV $\lesssim m_{\nu}\lesssim 1\,$eV, the most promising experiments need to focus on the corresponding separations $1\,$nm $\lesssim r\lesssim 1\,\mu$m. The dependence of the 2-NEP on CP violation is also novel, the only long-range SM force with this property. Muonium is the natural system to explore these effects since the 2-NEP CP violation requires interactions between leptons other than between just electrons. In addition, as pointed out by Fischbach et al., the contribution of the 2-NEP to the nuclear binding energy provides an interesting test of the WEP as applied to neutrinos.[9] Of course, the experimental challenges in realizing these possibilities are significant, requiring measuring forces of less than gravitational strength on the nanometer scale or the Weak Equivalence Principle at the level of $\sim 10^{-17}$. Recently, Stadnik[5] examined the potential of using high-precision spectroscopy, although some of the assumptions made may be overly optimistic.[10]

This work only touches on some of the interesting questions raised by the 2-NEP. We assumed neutrinos were Dirac fermions, examined only the spin-independent interaction, and assumed the simplest neutrino vacuum state. Alternatives to these assumptions and others involving theories beyond the SM will certainly impact the 2-NEP, providing new directions for theoretical and experimental exploration.

Acknowledgments

We thank Ephraim Fischbach for useful conversations and earlier papers on the 2-NEP, which provided significant motivation for our work. We also thank Sheakha Aldaihan and Mike Snow for discussions on the derivation of potentials, which influenced our approach.