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Predicting Outcomes of Electric Dipole and Magnetic Moment Experiments

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Predicting Outcomes of Electric Dipole and Magnetic Moment Experiments 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.
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