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T Todd ihapraetmgei oet o exam- for moment, magnetic permanent a with ) odpolarizability, -odd y is,Belarus Minsk, ty, b sthe is is,Belarus Minsk, , β T odd T d odplrzblt omlzdb some by normalized polarizability -odd ff µχ odssetblt.Telte quantity latter The susceptibility. -odd e [ see , = cieyidcssm lcrcdipole electric some induces ectively Todd 19 d 0 1 eo r[ or below ) T + rae neeti edwhich, field electric an creates ) 3 7 ∼ odplrzblt rtequan- the or polarizability -odd 8 ]. χ µ – , µ experiments 9 µβ 9 stepril antcmo- magnetic particle the is ,w eeoe h phe- the developed we ], .Snesc polarizabil- a such Since ]. Todd odd / , R 7 3 T ]. ∼ odpolarizability, -odd 3 d , ind 4 µ/ ] g ymasof means by R pi 7 2021 27, April − 3 )factor 2) where T men- em ( l in- ill tional into n tribu- CP it, y (1) his T z- )- is R o y e - - 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 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 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 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 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 and 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 [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. doscalar density into the spin equation. In turn, it allows to To improve the accuracy of regular Dirac equation, it is com- derive the contribution of T-odd polarizability into both EDM monly upgraded [29–31] by adding two field-dependent terms and AMM that we discussed above directly within the formal- µν ism of squared Dirac equation. Following our previous work ae 5 σ i∂/ − eA/ − m − ( + idγ ) Fµν ψ = 0 , (3) [8], it is convenient to define the new parameter " 2m 2 # q iψγ¯ 5ψ where a and d are the anomalous magnetic and electric dipole β = 0 = (2) ¯ moments. It is convenient to eliminate γ5 from (3) by introduc- r0 ψψ free ing the single tensor coefficient where both quantities q0 and r0 are pseudodensity and density of free fermions. This parameter is both P- and T-odd and can 1 µν i∂/ − eA/ − m − bµνσ ψ = 0 , (4) play a role similar to the P/T-odd polarizability. 2 ! 2 which is defined by p. 119], the squared Dirac equation is obtained by applying the adjoint operator to (4) ae ˜ bµν = Fµν − dFµν , (5) 1 1 2m (i∂/ − eA/ + m + b0 σµν)(i∂/ − eA/ − m − b0 σµν)ψ = 0 , (10) µ 2 µν 2 µν µν µνρσ here the dual field tensor F˜ is defined as ε Fρσ/2. Hence, 0 where bµν is defined with (5) by means of d0 and a0. The step- the is extended by adding two terms stem- by-step WKB-based derivation of spin equation which follows ming from the additional fermion form-factors. from (10) is given in [7, 9]. The key difference from the previ- The step-by-step WKB-based derivation of spin equation ous section is allowing a nonzero pseudoscalar q , 0 or, equiv- with AMM from equation (3) is given in [17], and with both alently, admitting β , 0. Remarkably, we obtain exactly the AMM and EDM terms in [7, 9]. It is given by same BMT-like equation which we repeat here as dsµ ge ae dsµ ge ae = Fµν s + sρF uνuµ − 2d F˜µν s + sρF˜ uνuµ (6) = Fµν s + sρF uνuµ −2d F˜µν s + sρF˜ uνuµ (11) dτ 2m ν m ρν ν ρν dτ 2m ν m ρν ν ρν     = − ffi where the spin and vectors are defined as sµ = where again 2a g 2. However, the moments (or coe cients (ψγ¯ 5γµψ)/(ψψ¯ ) and uµ = (ψγ¯ µψ)/(ψψ¯ ) respectively, τ is proper in front of field-dependentterms in the above equation) are now time, and 2a = g − 2. given as There exists the well known connection between the BMT- 2m e a = a + d β, d = d − a β. (12) like equation (6) and the Thomas equation (7) which describes 0 0 e 0 0 2m the spin in the laboratory frame [10, 32–34]. These As we discussed in the introduction, particles are typically as- equations are self-consistent within this approximation order, if sumed to have no T-odd polarizability in addition to EDM. As- the fermion motion is driven by the regular Lorentz-Maxwell suming instead that such additional polarizabilities are nonzero force. The evolution of spin three-vector si is given by leads to corrections to magnetic and electric moments that are evaluated without T-odd polarizabilities. Physical reasons be- ds = s × Ω , (7) hind the appearance of T-odd polarizabilities in a phenomeno- dt logical models are T-odd interactions of fermions with some where the precession frequency is given by other background field that might be of pseudoscalar nature. A specific mechanism behind the appearance of T-odd polar- e 1 aγ 1 izability of quasiclassical fermions is outside the scope of this Ω= a + B − (v · B) v − a + v × E work; we can however point to the analogy with atomic sys- m" γ ! γ + 1 γ + 1! # tems where certain types of hypothetical interactions between γ electrons and nucleus lead to appearance of such po- + 2d E − (v · E) v + v × B . (8) " γ + 1 # larizabilities [4]. The form of the additional contributions to magnetic and electric moments (evaluated without T-odd po- The spin three-vector s is given in the rest frame, while the larizabilities) could be guessed from the ; fields are in the laboratory frame, and γ is the Lorentz factor. however, it would not be possible to derive numerical factors In a non-relativistic frame the precession frequency becomes in front of these contributions. The formalism developed here allows to derive the exact form of additional T-odd contribu- e 1 Ωnr = a + 1 B − a + v × E + 2d E + v × B . (9) tions and additionally discover that the Dirac pseudoscalar can m 2 be viewed as an effective T-odd polarizability of fermion. h   i  In the semiclassical approximation, we assume that the space Inspecting the coefficients (12), we see that the pseudoscalar part of wave packet moves along an orbit that is a solution mixes the magnetic and electric moments in them. However, of corresponding Hamilton-Jacobi equation. More accurately, since |d0|≪|a0e/m| and |β| ≪ 1, the correction to a is of second such orbits can be described by eigenfunctions of scalar part of order of smallness. It is similar to T-odd and P-odd atomic po- Hamiltonian, since particle trajectories are only weakly influ- larizabilities that are originated by symmetry-violating interac- enced by spin in this approximation. The spin evolution along tions of atomic electrons with nucleus nucleons [4]. The scalar such trajectories is given by the spin precession (7) with fre- parts of these polarizabilities are pseudoscalar quantities which quency (8)or(9). colorblue The applicability conditions of this mix electric and magnetic contributions into the system en- approximation are discussed in [10, 28]. ergy. It is quite a typical situation in systems where symmetry- violating interactions are allowed, as we discussed in [7]. In the scenario where the EDM is screened (d0 ∼ β), see below, the 3. Squared Dirac equation pseudoscalar correction to the magnetic moment is of order of β2 which physical origin was discussed in the introduction. Consider now the equation (3) where we re-label coefficients The remaining task for this section is to re-write the spin d and a as d0 and a0 respectively; they are quantities that are equation for the laboratory frame. Since (11) has the same func- used in conventional theories which do not take account of T- tional form as the original BMT equation (6), the known con- odd polarizability. Next, following the traditional approach[10, nections and relationships [10, 35] between two equations can 3 be directly used. The spin precession equation has the same on many assumptions. While the new T(CP)- symmetry violat- form (7) ing effects are extensively studied in the field theory [38], the ds = s × Ωβ , (13) contributions into EDM and AMM that are derived there do not dt take account of T-odd polarizability. where three-vectors of spin s and velocity v keep their conven- Several scenarios are possible depending on the magnitudes tional definitions. The precession frequency is again given by of dipole moments and ratio of β, see the table.

β e 1 aγ 1 exp exp Ω = a + B − (v · B) v − a + v × E) d ∆a = a − a0 β Comment m γ γ + 1 γ + 1 h   i 1 0 0 0 NoNewPhysics(NP) γ exp + 2d E − (v · E) v + v × B . (14) 2 d 0 0 NP,conventionalmodel " γ + 1 # 3 dexp , 0 , 0 NP, mixedcase, new model exp where the coefficients are nowdefinedas in (12). We see that all 4 0 |a | > |a0| , 0 NP, screened EDM, new model terms keep their original physical meaning and form (including Thomas precession), while both magnetic and electric moments In this context, the coefficient a0 is associated with the theo- are adjusted with pseudoscalar corrections. retical value ath, since the existing theoretical evaluations of The nonrelativistic precession frequency follows from (14) both magnetic and electric moments do not take into account as the possibility of nonzero β. The conventional models are still β e 1 adequate in describing experiments at the current level of accu- Ω = a + 1 B − a + v × E) + 2d E + v × B (15) nr m 2 racy (Case 1 and 2). If the g − 2 discrepancy persists, then it h i    signals that β , 0. It means the pseudoscalar corrections must where we dropped terms with powers of velocity equal or be taken into account according to (16). The corrections offset higher than two. Here the moments are given by both magnetic and electric moments that are evaluated without 2m e taking into account the β-correction. Case 3 favors experiments a = a + d β, d = d − a β, (16) 0 0 e 0 0 2m with the heaviest fermions, since the relative strength of correc- tions scales as m−2, see [7]. which are the same expressions as (12); they are given here The most restrictive scenario (Case 4), which is also the most again to emphasize that the same expressions are valid for the predictive, is the assumption that the EDM is screened by the nonrelativistic case in this approximation order. vacuum polarization cloud. It follows then from (16) that As a final comment for this section, we would like to men- tion that we obtained the additional contribution into the EDM, exp ae d ≈ 0 → d0 ≈ β. (18) see the second equation in (16), in the form a0 µB β, where 2m µB = e/2m is the . A similar form was discussed before [3], see also the discussion around equation (1) regard- Substituting it into the first equation in (16), for γ → 1 for ex- ing the influence of magnetic field on induced EDM. Later, this ample, we see that the model yields the additional contribution 2 form was also used in [36, 37], however in the very different β a0 into the magnetic moment context. The electron EDM was parameterized there as exp 2 a ≈ a0 1 + β . (19) de = a0 µB λ, (17)  Hence, the new model predicts that the experimental data will where the dimensionless coefficient λ combines model-specific exceed the existing theoretical values mass ratios, mixing angles, and couplings. This ad hoc form 1 exp was proposed based on purely dimensional grounds in [37] . a > |a0| . (20) The form (17) was known since [3], it is derived here strictly from the squared Dirac equation; more, the dual contribution The inequality holds independently of signs of magnetic into the AMM, see the first equation in (16), has not been anomaly a0, the static pseudoscalar q0 (or β since r0 > 0), known so far. Comparing the ad hoc form (17) with our equa- and dipole moment d0. Clearly, the underlying assumption is tions (16), we can match the parameters as λa0 ∼ βa0 ∼ χ that both experiment and theoretical evaluation achieved a level which is the T-odd susceptibility from (1). where the pseudoscalar correction becomes the primary source exp of difference (a − a0). 4. Evaluating contribution of T-odd pseudoscalar into Physically, the prediction (18)-(20), can be explained as fol- AMM and EDM lows. The vacuum polarization cloud around the bare charge screens the particle EDM, however the electric screening leads Let us label experimentally measured moments as aexp and to the increased magnetic anomaly. Saying otherwise, the un- dexp respectively. The treatment of experimental data is based balanced distribution gets converted into the ad- ditional magnetic moment by means of pseudoscalar density 1 In parameterized forms, the anomalous moment a0 is often replaced by the which plays a role similar to P- and T-odd polarizabilities, see leading (Schwinger) term α/2π. [7]. 4 These considerations allowed to suggest the new explanation electrons and muons. 2 In this connection, an experiment to for the (g − 2) muon discrepancy between the best experimental measure the magnetic anomaly of τ- is predicted to the exp (aµ) and latest theoretical aµ values [1] find the similar gap and hence has an additional interest since it could confirm the universality of (20). It is explained by the exp −9 (aµ) − aµ = 2.8 × 10 . (21) presence of vacuum polarization around the bare charges that screens the intrinsic EDM which in turn leads to the increased The efforts to address remaining theoretical uncertainties in magnetic anomaly. Saying differently, the nonspherical electric (21) have been relentless; the most significant uncertainties charge distribution gets effectively converted into the additional are contributions from higher-order hadronic light-by-light dia- magnetic moment. grams [39–41] and hadronic vacuum polarization [22–25, 42], for which the teams seem to be making good progress recently. Still, the discrepancy remains and currently stands at 4.2σ level Acknowledgments [2]. Assuming now that the main factor behind the discrepancy (21) is the β-correction (19) allows to determine both dµ and βµ, We thank B. Malaescu for pointing the reference [43] to us, see their estimates in [8]. and R. Talman for valuable discussion of our results which However, the electron AMM data that were available to us by helped to clarify the presented material. exp the time of preparing [8] showed that (ae) − ae < 0; clearly, it contradicts to the model prediction (20). We commented in [8] that the only way to resolve this conflict within the devel- References oped model and under the assumption that electron EDM is screened is to accept that the currently available electron data [1] T. Aoyama, N. Asmussen, M. Benayoun, J. Bijnens et al, The had not reached the accuracy level where the β-correction be- anomalous magnetic moment of the muon in the Standard comes dominant. Our model does not include any other param- Model, Physics Reports 887 (2020) 1–166, arXiv: 2006.04822. doi:10.1016/j.physrep.2020.07.006. eters that can be fitted to reverse the inequality (20); it is quite [2] B. Abi et al, Measurement of the Positive Muon Anoma- restrictive in this case. Since that time, the new value of fine lous Magnetic Moment to 0.46 ppm, Physical Review Let- structure constant has become available [43]. It has allowed to ters 126 (14) (2021) 141801, arXiv:2104.03281 [hep-ex]. reevaluate the discrepancy between experimental and theoreti- doi:10.1103/PhysRevLett.126.141801. [3] V. G. Baryshevsky, Time-reversal-violating generation of static mag- cal values of electron magnetic anomaly [44] as netic and electric fields and a problem of electric dipole mo- ment measurement, Physical Review Letters 93 (4) (2004) 043003. exp −13 (ae) − ae = 4.8 × 10 . (22) doi:10.1103/PhysRevLett.93.043003. [4] V. G. Baryshevsky, High- nuclear optics of polarized particles, Now, the difference in electron AMM values between the ex- World Scientific Publishing Company, Singapore; Hackensack, NJ, 2012. [5] B. Ravaine, M. G. Kozlov, A. Derevianko, Atomic CP-violating polariz- periment and theory agrees with our model prediction. ability, Physical Review A 72 (1) (2005) 012101, arXiv:hep-ex/0504017. The standard model predicts the very small EDM values of doi:10.1103/PhysRevA.72.012101. around 10−39e · cm which was recently re-estimated by taking [6] M. G. Kozlov, A. Derevianko, Proposal for a Sensitive Search for the of the Electron with Matrix-Isolated Radicals, into account the one-loop vector meson contributions [45, 46]. Physical Review Letters 97 (6) (2006) 063001, arXiv:physics/0602111 Assuming that the latest estimate (22) withstands the scrutiny, [physics.atom-ph]. doi:10.1103/PhysRevLett.97.063001. our phenomenological model is able to explain this difference [7] V. G. Baryshevsky, P. I. Porshnev, Search for electric dipole moment, arXiv:2010.14218 [hep-ph] (Oct. 2020). and extract both de and βe. A much higher value of intrinsic [8] V. G. Baryshevsky, P. I. 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These gaps are predicted to have conventional phenomenology that does not include pseudoscalar corrections, the clear signature, |aexp| > ath , and are now observed for both hence the gaps.

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