Predicting outcomes of electric dipole and magnetic moment experiments V. G. Baryshevskya, P. I. Porshnevb aInstitute for Nuclear Problems, Belarusian State University, Minsk, Belarus bPast affiliation: Physics Department, Belarusian State University, Minsk, Belarus Abstract The anomalous magnetic and electric dipole moments in spin motion equation acquire pseudoscalar corrections if the T(CP)- noninvariance is admitted. It allows to explain the discrepancy between experimental and theoretical values of muon (g − 2) factor under assumption that the pseudoscalar correction is the dominant source of this discrepancy. Keywords: Electric dipole moment, muon g − 2 anomaly, CP violation 1. Introduction electron, ... ) with a permanent magnetic moment, for exam- ple. It normally means that there exists an electric current The long-standing discrepancy [1, 2] between experimental that creates this moment as well as a magnetic field inside and and theoretical data for muon (g−2) factor is a potential indica- around such a particle. Without symmetry-violating polariz- tion of new physics. Evenif the pure QED contributionis by far abilities, the contributions from these fields into the system en- the largest one, the experimental precision has reached a level ergy (which are defined by correspondingmoments) are cleanly where electroweak and hadronic contributions become impor- separated from each other. Now, if the particle possesses a T- tant. The discrepancy has withstood the intense scrutiny so far odd susceptibility (polarizability), this magnetic field will in- which might indicate that other mechanisms could be contribut- duce some electric field which magnitude is proportionalto this ing as well. Relaxing T(CP)-reversal symmetries can lead to polarizability; thus it effectively induces some electric dipole additional contributions into both magnetic and electric dipole moment [7]. Specifically, it can be shown as [3, 4] moments. The primaryeffect of allowing for P/T-symmetry violation is d = d0 + µ χTodd , (1) the appearance of electric dipole moment (EDM). Additionally, a composite system (atoms and molecules) could have P/T-odd where d0 is the electric dipole moment evaluated without taking polarizabilities which are originated by a variety of symmetry- account of T-odd polarizability, µ is the particle magnetic mo- violating interactions. Conventionally, it is assumed that “sim- ment, and χTodd is the T-odd susceptibility. The latter quantity pler” particles (electron, muon, ... ) might have only EDMs as is proportional to the T-odd polarizability normalized by some extensions of standard approach. It was suggested in [3] that all characteristic volume (occupied by the particle charge distribu- tion). Again, the particle magnetic moment means that there is types of particles (atom, molecule, nucleus, neutron, electron) 3 can also have an additional property similar to P/T-odd polar- a magnetic field (inside the particle) of order of µ/R where R izabilities (in addition to nonzero EDMs). However, a specific is a characteristic particle size. This field indices an additional ∼ 3 ∼ implementation of this idea for fermions was missing; this work electric dipole moment µχTodd µβodd/R dind by means of fills this gap. T-odd polarizability βodd, see[3]. arXiv:2104.12230v1 [hep-ph] 25 Apr 2021 If a system has T-odd polarizabilities and susceptibilities By reciprocity, the EDM (1) creates an electric field which, [3, 4], see also [5, 6], it can generate a magnetic field upon ap- in turn, creates a magnetic field and additional contribution into plying the electric one; vice versa, an electric field can be gen- the magnetic moment which magnitude is again proportional to erated by the magnetic one [3, 4]. As a direct consequence of the T-odd polarizability. Hence, taking into account first the in- admitting nonzero P/T-odd polarizabilities, the magnetic and duced electric field, then the magnetic field that is induced by it, electric moments that are evaluated without these polarizabili- the additional contribution into the magnetic moment must be ties do acquire additional contributions. Saying in other words, proportional to the square of T-odd polarizability or the quan- interpreting the measured values of magnetic and electric mo- tity that plays its role, see (19) below or [7]. ments might require taking account of such polarizabilities. The squared Dirac equation allows to introduce the pseu- doscalar quantity, which plays the role of T-odd polarizability, To explain how T-odd polarizabilities mix electric and mag- into the spin motion equation [8, 9]. Since such a polarizabil- netic moments, consider a particle (atom, nucleus, nucleon, ity mixes magnetic and electric moments, both problems, the search for EDM and measurements of anomalous magnetic mo- Email addresses: [email protected] (V. G. Baryshevsky), ment (AMM), are getting directly connected. [email protected] (P. I. Porshnev) In several recent studies [7–9], we developed the phe- Preprint submitted to April 27, 2021 nomenological model which introduces the pseudoscalar cor- Upon reviewing the available experimental data for both rections into the spin motion of fermions. The approach is AMM and EDM problems within the same phenomenologi- based on the squared Dirac equation with both AMM and EDM cal approach, the following prediction can be made. Higher included. The underlying physical hypothesis is the assump- precision measurements of EDMs will continue yielding null tion that free fermions possess a small nonzero pseudoscalar results, while similar higher precision measurements of AMMs 5 q0 = iψγ¯ ψ. This real-valued quantity is both P- and T-odd will continue confirming the gap against corresponding theoret- [10, 11]. Bare charges are reduced by vacuum polarization ical evaluations. The physical reason is the conversion of non- which is one of the radiative corrections that is predicted by spherical electric moment into the additional magnetic anomaly the field theory. Effectively, a charged fermion is viewed as the by means of pseudoscalar density which plays a role P- and T- core charge surroundedby the coat of virtual pairs, including its odd polarizability. This prediction is the extreme case among antifermions. The idea that the polarization of vacuum screen- other possible scenarios that were reviewed in [8]. ing cloud around a bare charge should be taken into account in The (g − 2) discrepancy has withstood the intense scrutiny phenomenological models is not new [12, 13], see also selected for quite some time already; however it is not a foregone works on nucleon polarizability [4, 14]. A magnetic field inside conclusion that it could not be resolved without new physics particle (estimated for a dipole with muon Compton length) one day [21]. The evaluation of hadronic vacuum polarization could be enormous, be an order of 1016G. A virtual electron is still evolving; the estimates differ substantially between or muon, from the muon polarization coat, would have the cy- different teams [22–25]. Anyway, following the latest review clotron frequency up to several dozens of MeV. Therefore, ef- [1] and new experimental data [2], the discrepancy currently fects related to the vacuum polarization coat could be signif- stands at (3.7 − 4.2)σ. We have shown here that there exists icant. However, a specific implementation of how to include the T-odd contribution into AMM (that is proportional to β2) such effects into phenomenological models was missing. As in addition to the conventional contributions that have been we discussed in [8], the inclusion of static pseudoscalar density considered in the literature so far. Hence the magnetic anomaly into the phenomenologicalmodel is our attempt to take account is given as aTeven + aTodd where the second term depends on of the vacuum polarization cloud within the single-particle rep- the pseudoscalar β from (2). This additional contribution can resentation. possibly explain the muon g − 2 discrepancy. The squared Dirac equation has been viewed as an alternative to the regular first-order one [10, 15–19]. It properly recovers The paper is organized as follows. In the next section, we the term with g-factor and is frequently used in various applica- briefly outline the derivation of conventional spin motion equa- tions [10]. Since it also recovers the Bargmann-Michel-Telegdi tion in weak external fields. Next, we recap the formalism (BMT) equation in the quasiclassical limit [17, 20] and the based on the squared Dirac equation with both AMM and EDM Pauli term for the nonrelativistic motion [10, p. 120], it can be terms included. Lastly, we discuss contributions of T(CP)-odd seen as equally confirmed by atomic phenomenology as the pseudoscalar into both magnetic and electric dipole moments. original Dirac equation. To ensure the mathematical equiva- The analysis of latest (g − 2) data for muons and electrons does lency between two types of Dirac equations, only solutions of not contradict the potentially significant contributions of pseu- second-order equation that also satisfy the first-order equation doscalar corrections into the magnetic anomaly. are conventionally chosen [10, p. 120]. However, in doing so, we will end up with exactly the same situation as with the 2. Conventional approach regular Dirac equation. The solutions of squared equation that are filtered in such a way will have zero pseudoscalar for the Spin motion equationscan be derived or justified in many dif- field-free case; we do not gain anything new then. Therefore, ferent ways. A large literature has been dedicated to this topic we will not impose this additional requirement on the solutions since the BMT equation was published 60+years ago [26]; the of squared Dirac equation; it admits new types of solutions topic itself goes back to the Frenkel’s work [27]. For homo- compared to the regular approach. geneous weak fields and the quasiclassical motion [28], a con- sistent derivation leads to the BMT equation which can be up- Main hypothesis and new physics: Next, we consider the graded with both anomalous magnetic and electric dipole mo- squared Dirac equation which allows to introduce the pseu- ments.