# Constraining exotic spin dependent interactions of muons and electrons

## Abstract

Many experiments have been performed to search for the exotic spin-dependent interactions in ranges from \(\sim \upmu \)m to astrophysical range which corresponds to the energy scale of less than \(\sim \)10 eV. At present, nearly all known experiments searching for these new interactions at the macroscopic range are for protons, neutrons, and electrons. Constraints at this range for other fermions such as muons are scarce, though muons might be the most suspicious particles which might take part in new interactions, considering their involvement of several well-known puzzles of modern physics. We use the anomalous magnetic moment and electric dipole moment (EDM) to study the exotic spin-dependent interactions for muons and electrons. The muon’s magnetic moment might indicate existing of the pseudo-scalar–pseudo-scalar (PP) type interaction. We set up a constraint for the scalar–pseudo-scalar (SP) type interaction at the interested range for muons. For the PP type interaction of electrons, we obtained a new constraint at the range of \(\sim \) nm to \(\sim \) 1 mm. Since all the present experiments searching for the new forces give zero results, it is reasonable to consider that these new interactions might only couple to muons. We propose to further search for the new interactions using the muon spin rotation techniques.

## 1 Introduction

New interactions beyond the Standard Model are possible. Suggested solutions for several important problems of modern physics have led to new interactions mediated by new particles [1]. Exotic macroscopic forces mediated by WISPs (weakly-interacting slim particles) is an example. Reference [2] classified the new interactions into 16 different types and most of them are spin dependent. These new interactions have ranges from nanometers to astronomical distance which corresponds to the mediating-boson mass of \(\sim 10^{-18}\) eV to \(\sim 100\) eV. Methods of precision measurement such as the torsion pendulum [3], co-magnetometer of polarized noble gases [4, 5], SERF magnetometer [6], polarized \(^3\)He atom beam [7], etc are convenient to probe these new interactions at a much lower energy scale than the high energy physics.

*r*the distance between the interacting particles. The spin–spin dependent potential induced by the PP interaction is:

In this work, we discuss the possibility that the anomalous magnetic moment of the muon might indicate existing of the new SS or PP interaction mediated by ALPs lighter than \(\sim 100\) eV. By using the muon’s EDM (Electric Dipole Moment), we could establish a constraint for the parity-violating monopole–dipole interaction mediated by ALPs lighter than \(\sim 100\) eV for muons. When applying the method to the electrons, constraints of \(g_S^{e}g_S^{e}\) and \(g_P^{e}g_P^{e}\) are obtained for the range between \(\sim \)1 and \(\sim \)1 mm.

## 2 The anomalous magnetic moment and EDM induced by new interactions

*SS*(

*x*) and

*PP*(

*x*) are defined as:

## 3 Applications to the muon and electron

By using the anomalous magnetic moment, constraints of \(|g_S^{e}g_S^{e}|\) and \(|g_P^eg_P^e|\) for electrons can be established as shown in Figs. 3 and 4, respectively. One can see that this method could derive a new constraint for \(V_{PP}\) in the range between \(\sim \)nm to \(\sim \)mm. When using the electron EDM, the derived constraint for \(g_S^eg_P^e\) at the interested force range is at a level of \(\sim 10^{-17}\). It gives a constraint \(\sim \)3 orders more stringent than a recent experimental work [36] in the range near \(\sim \mu \)m. At long interaction ranges(\(m_\phi <\sim \) keV), using data from atomic and molecular EDM experiments [37], Ref. [38] gives a constraint on \(g_S^eg_P^e\) \(\sim \)3 times more stringent than this work. At very short ranges(\(m_\phi >\sim \)MeV), the loop induced EDM dominates thus the method presented in this work might work better [39]. For the new boson heavier than \(\sim \)MeV, it is more convenient to be detected by the method of high energy physics and obviously out of the scope of this work.

## 4 Conclusion and discussion

The 3.7\(\sigma \) difference between the theoretical prediction and the experimental measurement of the anomalous magnetic moment of muons might indicate nonzero new interactions of SS or PP type with a range larger than \(\sim \)1 nm. We constrain the monopole–dipole type interaction (originated from the SP) for muons, as shown in Fig. 2. Previously, many experiments and studies have been performed for spin-dependent new interactions associated with electrons, protons, and neutrons only. Although short-range muonic forces with an energy scale larger than \(\sim \) MeV have been proposed to solve the proton radius puzzle, studies considering macroscopic range muonic interactions mediated by ALPs have not been conducted yet, according to our best knowledge. The charge radius puzzles of the muonic hydrogen and deuteron and the anomalous magnetic moment of muons suggest that new physics or new interactions might exist for muons.

It would be fascinating to include muons into the searching business for the macroscopic new spin dependent interactions which are mediated by ALPs or other new light bosons. Low energy muon beams with 100% polarization are frequently used in condensed matter physics [40] and fundamental physics [29]. If a nonmagnetic mass-source can be put in the region close to muon beams as in Fig. 5 [7], constraints on \(g_Sg_P^\mu \) at long distances can be obtained by measuring changes of the muon polarization. Furthermore, if polarized electron spin-density sources, as the \(\mu \)-metal shielded \(\hbox {SmCo}_5\) [3, 41], are used in experiments, then constraints on \(g_P^eg_P^\mu \) can be established. It is not hard to imagine that the muonic new interactions mediated by light vector particles can also be searched for using experimental schemes as in Ref. [7].

Using the same method, we can also give constraints of \(g_S^eg_S^e\) and \(g_P^eg_P^e\) for electrons. This method works at small distances from \(\sim \) nm to \(\sim \) mm. Moreover, it can give pure constraints only between the electrons, while many other methods cannot easily isolate the contributions from other fermions like protons or neutrons. Our results for the electron are consistent with zero.

The fact that no new interactions have been detected for electrons, neutrons and protons indicates the muon might be an interesting target for these new forces at long ranges. It is possible that these new interactions might be muonic which means they only couple to the muons. Though studies have performed for muonic new interactions at very short ranges (new boson mass heavier than \(\sim \) MeV), to the best of our knowledge, no experiments have ever been done to search for these spin-dependent new interactions at long ranges (new boson mass lighter than \(\sim \)100 eV).

## Notes

### Acknowledgements

We acknowledge support from the National Natural Science Foundation of China, under Grant 91636103, 11675152, 11875238. This work was also supported by National Key Program for Research and Development (Grant 2016YFA0401504). We thank Dr. Queiroz and Dr. Stadnik for providing us useful references. We thank Dr. Stadnik for helpful discussions.

## Supplementary material

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