^{1}

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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),

Of all the parts of the Standard Model, the neutrino sector arguably holds the greatest promise for revealing new physics beyond the Standard Model. Since the discovery of the electron-neutrino over sixty years ago

The observation of neutrino oscillations

To accomplish this, we will first quickly review previous derivations of the 2NEP. Then we will introduce an alternative method for calculating the single flavor 2NEP based on expressing the neutrino fields in the Schrödinger picture and using time-independent perturbation theory to evaluate the shift in neutrino field vacuum energy due to the presence of two fermions. This approach has been used previously to calculate the Yukawa interaction between fermions exchanging scalar bosons

The possibility that a long-range force could arise from the exchange of virtual neutrinos has been known since the 1930s

Feynman diagram of neutrino-antineutrino exchange between fermions which leads to Eq.

Shortly after Feinberg and Sucher’s publication

Before continuing, a comment should be made on referring to the 2NEP as being “long-ranged.” For light neutrinos,

All of the above derivations of the 2NEP assume a single flavor of neutrinos, but mixing of neutrino flavors has now been observed experimentally

In order to establish our notation and approach for the mixing case, in this section we derive the 2NEP by calculating the change in the single neutrino field vacuum energy due to the presence of two fermions which couple to the field. The potential energy of these two particles is defined as the shift in the vacuum energy that depends on the separation of the particles. In general, the field interaction will also lead to self-energy corrections which are independent of the particle separation.

In our calculation, we will assume the two fermions are located at positions

We begin by writing the total Hamiltonian describing the two fermions interacting with the neutrino field as

To facilitate a comparison of our results with previous derivations of the 2NEP, in this section we will assume the interaction Hamiltonian

Values for the vector coupling

We will now use ordinary time-independent perturbation theory to calculate the energy of the system due to the interaction. The first-order energy correction of the ground state does not lead to a potential energy between the particles because it is the sum of two terms, each of which only depends on the position of one particle:

The 2NEP must come from the second-order ground state energy correction which can be written as

To evaluate Eq.

Observations reveal there are three flavors of neutrinos,

It is important to note that the vacuum that we will use subsequently in perturbation theory calculations is the eigenstate of the Hamiltonian in Eq.

According to the Standard Model, the interaction of neutrinos with other fermions is through the flavor fields

In general, the low-energy effective Hamiltonian density describing the interaction of neutrinos with fermions is the sum of two contributions

Feynman diagrams showing how the NC (top) and CC (bottom) processes involving vector bosons

Since we are interested in only neutrino interactions with nonrelativistic fermions, we focus on only the spin-independent interaction Hamiltonians involving nucleons and charged leptons. Nucleons only experience the NC interaction so their interaction Hamiltonian is the generalization of Eq.

While the interaction between neutrinos and other fermions is most naturally expressed in terms of the flavor fields

The interaction of the neutrino fields with the charged lepton is a bit more complicated due to the additional contribution from the charged current interaction. Transforming the flavor neutrino field in the neutrino current into mass fields gives

In this section, we will use the Hamiltonians describing the interactions of fermions with the neutrino fields given in the previous section to derive the 2NEPs for nucleon-nucleon, nucleon-lepton, and lepton-lepton interactions including neutrino mixing. As in the single neutrino case, the derivation of the 2NEP with mixing will start with the second-order energy shift of the vacuum due to two fermions #1 and #2 which depends on the fermions separation, Eq.

Since the interaction of neutrinos with nucleons is flavor independent, the derivation of the 2NEP is a straightforward extension of the single neutrino case. For two nucleons, the second-order vacuum energy shift is obtained from Eq.

Now let us consider the interaction energy for a nucleon (particle #1) and a charged lepton (particle #2). Then the second-order vacuum shift is

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.

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

Now let us consider the interaction energy between two electrons. The three diagrams contributing to the general lepton-lepton 2NEP are shown in Fig.

Neutrino mass and PMNS matrix parameters used in numerical calculations (Table 14.1, Ref.

It is important to note that the asymptotic expansion used to obtain Eq.

Let us now compare our results for the 2NEP between two electrons with mixing with the integral expression derived by Lusignoli and Petrarca

A straightforward generalization of the calculation carried out in Eq.

Similarly, if the lightest neutrino is massless, the mixing term analogous to Eq.

While the mixing of neutrino mass states does modify the 2NEPs involving leptons and nucleons, the most important effects are seen in interactions involving two leptons. The consequences of the 2NEPs involving two leptons derived in the previous section are explored in Figs.

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

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

The absolute value of the ratio

The relative ratios of the mixing portion

There is growing evidence that

To summarize, we have developed an alternative approach to derive the 2NEP between two fermions based on expressing the neutrino fields in terms of the Schrödinger picture and finding the change in vacuum energy using time-independent perturbation theory. We incorporated mixing of three neutrino flavors using the PMNS matrix

While there was recent work which suggested the observation of the effects of the 2NEP might be within reach of spectroscopy experiments

In this paper we have focused our attention on the spin-independent 2NEP, but the work by Stadnik

We thank Ephraim Fischbach for insightful discussions and for his earlier work which motivated this paper, and the anonymous referee for comments which greatly improved our presentation. We also thank Wabash College for providing summer stipends which supported some of this work.

In this Appendix, we calculate the second-order energy shift of the single neutrino vacuum by two fermions which depends on the fermion separation, Eq.

To evaluate the sums in Eq.