Effects of the long-range neutrino-mediated force in atomic phenomena

It has long been known that the exchange of a pair of (nearly) massless neutrinos between particles (see diagram on Fig. 1) produces a long-range force Gamow and Teller (1937); Feynman (1996), with the resultant potential $\sim G_{F}^{2}/r^{5}$, where $G_{F}$ is Fermi constant  Feinberg and Sucher (1968); Feinberg et al. (1989); Hsu and Sikivie (1994). However, due to a rapid decay with the distance $r$, the effects of this potential are about 20 orders of magnitude smaller than the sensitivity of the macroscopic experiments Refs. Kapner et al. (2007); Adelberger et al. (2007); Chen et al. (2016); Vasilakis et al. (2009); Terrano et al. (2015); Stadnik (2018).

A recent paper by Stadnik Stadnik (2018) introduced a new approach to obtaining constraints on this potential by considering spectra of atomic systems. In the Standard Model formulas for energy shifts produced by potential $G_{F}^{2}/r^{5}$, the Fermi constant $G_{F}$ has been replaced by an effective interaction constant $G_{\rm{eff}}$. The $G_{\rm{eff}}^{2}/r^{5}$ potential produces a small energy shift to atomic energy levels, and therefore it is possible to obtain constraints on $G_{\rm{eff}}^{2}$ from differences between highly accurate QED calculations of energy levels and experimental results Meyer et al. (2000); Karshenboim (2005). The Stadnik paper has lead to a breakthrough in sensitivity, constraints on the interaction constant $G_{\rm{eff}}^{2}$ have been improved by 18 orders of magnitude in comparison with constraints from the macroscopic experiments Refs. Kapner et al. (2007); Adelberger et al. (2007); Chen et al. (2016); Vasilakis et al. (2009); Terrano et al. (2015); Stadnik (2018).

However, the highly singular potential $G_{\rm{eff}}^{2}/r^{5}$ leads to divergent integrals in the matrix elements as $r$ approaches zero. This demonstrates the requirement of the correct extension of the potential for $r\to 0$. Ref. Stadnik (2018) used the Compton wavelength of the $Z$ boson as the cut-off radius, $r_{c}=\hbar/M_{Z}c$, for positronium and muonium, and the nuclear radius $R$ for atoms with finite nuclei. As we will show below, this oversimplified treatment in Ref. Stadnik (2018) leads to limits which were overestimated by a factor of 6 in non-hadronic atoms and underestimated by 4-5 orders of magnitude in the case of deuteron binding energy. The aim of the present paper is to provide more accurate estimates and also consider results of the measurements which have not been included in Ref. Stadnik (2018). To avoid misunderstanding, we should note that present paper is not aimed to calculate all electroweak corrections to energy levels. This should be done by a different method.

We also consider fifth forces from beyond the Standard Model that are parameterised by a Yukawa-type potential. This fifth force would require the existence of a new scalar particle to mediate the interaction, thus constraints on the coupling strength of the interaction can be found for various scalar particle masses. Limits were previously obtained using precision hydrogen $1s-2s$ spectroscopy in Ref. Ubachs et al. (2013), however we improve upon them using more recent data and include additional hydrogen-like systems.

The potential of the long-range neutrino-mediated force between two particles, presented in Ref. Stadnik (2018), is

where $\sigma_{1}$ and $\sigma_{2}$ are the Pauli spin matrix vectors of the two particles, and $a_{i}$ and $b_{i}$ represent the species-dependent parameters defined below. It is worth noting that the last term of Eq. (1) is zero for $s$-orbitals which strongly dominate in the shifts of atomic energy levels.

A potential $\sim 1/r^{5}$ gives divergent integrals ($\int_{r_{c}}d^{3}r/r^{5}\approx 1/2r_{c}^{2}$) in the matrix elements for $s$-wave. Using the nuclear radius $R$ as a cut-off, $r_{c}=R$, would give incorrect results. A more accurate approach requires first to build effective potential for electron-quark interaction and then take into account nucleon distribution $\rho(r)$ inside the nucleus. To include small distances, we present this potential for the finite size $R$ of the nucleus and cut-off for large momenta (small distances $r$) produced by the $Z$ boson propagator ($1/(q^{2}+M_{Z}^{2}$) instead of $1/M_{Z}^{2}$, see Fig. 2).

To start, we replace $1/r^{5}$ in the potential Eq. (1) with

where, for $z=M_{Z}/(2m)$,

Here $m$ is the mass of the fermion in the loop on Fig. 2. The function $F(r)\propto I(r)/r$ gives us dependence of interaction between electron and quark (or electron and other point-like fermion) on distance $r$ between them. For $\hbar/(M_{Z}c)\ll r\ll\hbar/(mc)$, we obtain $F(r)=1/r^{5}$. In this area there is no change for potential Eq. (1). For large $r\gg\hbar/(mc)$, we have $F(r)\propto\exp(-2mcr/\hbar)/r^{5/2}$. At small distance $r\ll r_{c}=\hbar/(M_{Z}c)$, function $F(r)\propto(\ln r)/r$ and has no divergency integrated with $d^{3}r$. Note that behaviour of the neutrino-exchange potential at small distance has been investigated in Ref. Xu and Yu (2022). However, they do not study this potential in the Standard Model, they considered a new scalar particle instead of $Z$ boson.

Convergence of the integral in the matrix elements on the distance $r\sim r_{c}=\hbar/M_{Z}c$ indicates that this interaction in atoms may be treated as a contact interaction (see Fig. 1). We can replace $F(r)$ by its contact limit, $F(r)\to C\delta({\bf r})$

where we assume $z=M_{Z}/(2m)\gg 1$. Note that if we would assume potential $1/r^{5}$ with the cut-off $r_{c}=\hbar/M_{Z}c$, the result would be 6 times bigger:

Using Eq. (4), the potential in Eq. (1) in the contact limit may be presented as, using natural units $\hbar=c=1$,

In Ref. Grifols et al. (1996), the potential was obtained for a Majorana neutrino loop instead of a Dirac neutrino loop. Using these results, we conclude that the neutrino-exchange potential for Majorana neutrinos requires the adjustment to $I(r)$ as follows

This indicates that the nature of neutrinos may, in principle, be detected from the difference in Dirac and Majorana potentials. At small distance, the Dirac neutrino and Majorana neutrino potentials are practically the same, the difference is proportional to $(m_{\nu}cr/\hbar)^{2}$ and is very small. In the contact interaction limit, the relative difference is $\sim(m_{\nu}/M_{Z})^{2}$. However, the asymptotic expression at large distance changes: for Majorana neutrinos we have $I^{(M)}_{2}(r)/r\propto\exp(-2mcr/\hbar)/r^{7/2}$, whereas $I(r)/r\propto\exp(-2mcr/\hbar)/r^{5/2}$ for Dirac neutrinos. Therefore, the ratio of Dirac potential to Majorana potential $\sim m_{\nu}cr/\hbar$ Grifols et al. (1996). Thus, the difference is negligible at small distances and only becomes significant at large distances $r\gtrsim\hbar/m_{\nu}c$. Unfortunately, effects of the neutrino-exchange potential are many orders of magnitude smaller than sensitivity of current macroscopic experiments Refs. Kapner et al. (2007); Adelberger et al. (2007); Chen et al. (2016); Vasilakis et al. (2009); Terrano et al. (2015), motivating future experimental work.

At large distance a dominating contribution to the vacuum polarization by the $Z$ boson field is given by the lightest particles which are neutrinos. However, at distance $r$ all particles with the Compton wavelength $\hbar/mc>r$ give a significant contribution. Following Ref. Stadnik (2018) we present interaction constants for potentials (1,6) in the following form:

where $N_{\rm{eff}}$ is the effective number of particles (normalised to one neutrino contribution) mediating the interaction on Fig. 2. Contribution, which is not proportional to $N_{\rm{eff}}$, appears due the diagrams with $W$ boson. For example, for interaction between electron and quark, such diagrams involve electron neutrino - see Ref. Hsu and Sikivie (1994).

In atoms dominating contribution comes from the distance $r\sim\hbar/M_{Z}c$. Summation of the contributions from $\nu,\,e,\,\mu,\,\tau,\,u,\,d,\,s,\,c,\,b$ (all with mass $m\ll M_{Z}$) gives $N_{\rm{eff}}=14.5$ Stadnik (2018). Consider an interaction between electron and nucleon with an exchange by electron neutrino, electron has values $a_{e}^{(1)}=1/2+2\sin^{2}(\theta_{W})$ and $b_{e}^{(1)}=1/2$, while nucleons have values $a_{n}^{(1)}=-1/2$, $a_{p}^{(1)}=1/2-2\sin^{2}(\theta_{W})$, $b_{n}^{(1)}=-g_{A}/2$, and $b_{p}^{(1)}=g_{A}/2$, where $g_{A}\approx 1.27$. For the contributions from the other neutrino species, there is no $W$ boson contribution and we have values for charged leptons $a_{l}^{(2)}=2\sin^{2}(\theta_{W})-1/2$, $b_{l}^{(2)}=-1/2$, $a_{N}^{(2)}=a_{N}^{(1)}$, and $b_{N}^{(2)}=b_{N}^{(1)}$. Value of the $\sin^{2}(\theta_{W})=0.239$ for a small momentum transfer Tanabashi et al. (2018), where $\theta_{W}$ is the Weinberg angle.

Simple two-body systems provide the most accurate values of the difference between experimental result and result of QED calculation of the transition energies. Following Ref. Stadnik (2018), we use these differences to obtain limits on the effective interaction constant $G_{\rm{eff}}$. We consider hydrogen, muonium and positronium spectra and deuteron binding energy. A summary of our calculations is presented in Table 1.

We also examine the constraints obtained from molecular systems for the neutrino-exchange interaction. On the molecular scale, it is sufficient to use Eq. (1) (only $a_{1}a_{2}$ contribute) as the nuclei are separated by a distance at least $a_{B}$. Additionally, this also means that only neutrinos contribute to the interaction, so we use $N_{\rm{eff}}=3$. We consider molecular hydrogen systems and thus present the potential

where the interacting particles are nucleons. Ref. Salumbides et al. (2015) used precision molecular spectroscopy to obtain constraints with regards to gravity in extra dimensions, including a potential with $1/r^{5}$ dependence. This was done similarly to our method in Section III, where the difference between theoretical and experimental results was used to obtain constraints on the interaction strength.

We utilise the findings of Ref. Salumbides et al. (2015) to obtain constraints from the systems H₂, D₂, and HD⁺. Values for the chosen systems are $a_{p}^{(1)}a_{p}^{(2)}=1.5\times 10^{-3}$, $a_{n}^{(1)}a_{n}^{(2)}=0.75$, and $a_{p}^{(1)}a_{n}^{(2)}=-3.3\times 10^{-2}$. Using the results of Ref. Salumbides et al. (2015), we find constraints on the interaction strength of the neutrino-exchange interaction presented in Table 2.

In addition to constraints on the neutrino-exchange interaction, one can also use spectroscopy to place constraints on a fifth force from beyond the Standard Model. The force can be phenomenologically parameterised by a Yukawa-type potential

where $\beta$ is the coupling strength and $m$ is the mass of a scalar particle mediating Yukawa-type interaction. Constraints were obtained using molecular hydrogen spectroscopy in Ref. Salumbides et al. (2013) and atomic hydrogen $1s-2s$ spectroscopy in Ref. Ubachs et al. (2013). It was found that atomic systems provide stronger limits than molecular systems. We improve upon Ref. Ubachs et al. (2013) by using more recent data and provide additional constraints from muonium $1s-2s$, positronium $1s-2s$, hydrogen $1s-3s$, and deuteron.

The energy shifts for $1s-2s$ and $1s-3s$ transitions found from the expectation values of the Yukawa potential for hydrogen-like atoms (in units $\hbar=c=1$)

where $\tilde{a}_{B}$ is the reduced Bohr radius. Note that in both transitions, in the large mass limit we observe $\delta E\propto\beta/m^{2}$. The energy shift for deuteron, using the wave function in Eq. 27, is given by

where we have the exponential integral ${\rm Ei}(x)=-\int_{-x}^{\infty}e^{-t}/t\ dt$. To obtain constraints we use limits on energy shifts for hydrogen, muonium, and positronium transitions and deuteron binding energy presented in Section III. For comparison, we include the previous constraint from Ref. Ubachs et al. (2013) which used $|\delta E|=2.1\times 10^{-10}$ eV. Our results are presented in Figure 3. Hydrogen $1s-3s$ gives the strongest constraints at small masses $|\beta/\alpha|<9.2\times 10^{-13}$, while at large masses this constraint is $|\beta/\alpha|<1.5\times 10^{-20}\ (m/\rm{eV})^{2}$.

Our work is motivated by the Stadnik paper Stadnik (2018), which demonstrated that the sensitivity of atomic spectral data to the neutrino mediated potential, introduced in Refs. Gamow and Teller (1937); Feynman (1996); Feinberg and Sucher (1968); Feinberg et al. (1989); Hsu and Sikivie (1994), is up to 18 orders of magnitude better than the sensitivity of macroscopic experiments to this potential. However, an oversimplified cut-off treatment of this potential at small distance has led to inaccurate results in Ref. Stadnik (2018), especially in systems with finite size of the particles. For example, cut-off at the nuclear radius gave limits on the interaction strength which are five orders of magnitude weaker than the limits obtained in the present work. We argue that one should firstly build effective interaction between point-like particles, like electrons and quarks. On the second step this interaction should be integrated over the nuclear volume.

In this paper, we calculated energy shifts, produced by neutrino potentials, and extracted limits on the strength of this potential from hydrogen-like systems, namely hydrogen, muonium, positronium, and deuteron. We also extracted constraints from spectra of H₂, D₂ and HD⁺ molecules. Following Ref. Stadnik (2018), we presented our results as constraints on the ratio of the effective strength of the neutrino-mediated potential $G_{\rm{eff}}^{2}$ to the squared Fermi constant $G_{F}^{2}$. The best limit was obtained from the muonium hyperfine structure, $G_{\rm{eff}}^{2}/G_{F}^{2}<10^{3}$. Our constraints are expected to be significantly enhanced in the near future, for example, with the muonium $1s-2s$ measurement predicted to be improved by three orders of magnitude by the currently ongoing experiment MuMASS Crivelli (2018). Constraints from atomic spectroscopy on the coupling strength and mass of a new scalar particle mediating a fifth force are also obtained and are an order of magnitude stronger compared to Ref. Ubachs et al. (2013).

This work was supported by the Australian Research Council Grants No. DP190100974 and DP200100150.