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Bounds on Mini-charged Neutrinos in the

arXiv:hep-ph/9208260v1 31 Aug 1992May, 1992UM-P-92/27Revised July, 1992OZ-92/7BA-36Bounds on Mini-charged Neutrinos in theMinimal Standard ModelK. S. Babu(a) and R. R. Volkas(b)(a)Bartol Research Institute, University of Delaware, Newark, DE 19716 U.S.A.(b)Research Centre for High Energy Physics, School of Physics, University ofMelbourne, Parkville 3052 AustraliaABSTRACTIn the minimal Standard Model (MSM) with three generations of quarksand leptons, neutrinos can have tiny charges consistent with electromagneticgauge invariance.There are three types of non-standard electric charge,given by Qst + ǫ(Li −Lj), where i, j = e, µ, τ (i ̸= j), Qst is standard elec-tric charge, Li is a family-lepton–number, and ǫ is an arbitrary parameterwhich is put equal to zero in the usual incarnation of the MSM.

These threenon-standard electric charges are of considerable theoretical interest becausethey are compatible with the MSM Lagrangian and SU(3)c⊗SU(2)L⊗U(1)Ygauge anomaly cancellation. The two most conspicuous implications of suchnon-standard electric charges are the presence of two generations of mass-less charged neutrinos and a breakdown in electromagnetic universality fore, µ and τ.We use results from (i) charge conservation in β-decay, (ii)physical consequences of charged atoms in various contexts, (iii) the anoma-lous magnetic moments of charged leptons, (iv) neutrino-electron scatter-ing, (v) energy loss in red giant and white dwarf stars, and (vi) limitson a cosmologically induced thermal photon mass, to place bounds on ǫ.While the constraints derived for ǫ are rather severe in the Le −Lµ,τ cases(ǫ < 10−17 −10−21), the Lµ −Lτ case allows ǫ to be as large as about 10−14.

The study of electromagnetism is one of the most fundamental activitiesof both theoretical and experimental physics. In the relativistic quantum do-main germane to particle physics, electromagnetism is very successfully de-scribed through the direct coupling of massless photons to electrically chargedparticles via the familiar vector current interaction.

In the minimal StandardModel (MSM), one genus of fermion – the neutrino – is taken to have no di-rect coupling with photons. However, it is not actually mandatory within thestructure of the MSM for neutrinos to possess exactly zero electric charge.The purpose of this paper is to investigate the dramatic consequences of nothaving neutrinos with precisely zero electric charge in the MSM.In order to understand how charged neutrinos can arise in the MSM, it isnecessary to study the global symmetries of the theory.

The MSM exhibitsfive U(1) invariances which commute with its non-Abelian gauge symmetrygroup SU(3)c ⊗SU(2)L. One of these is the Abelian gauge symmetry U(1)Ywhere Y is the generator of weak hypercharge, while the other four are thesymmetries U(1)B and U(1)Le,µ,τ , where B and Le,µ,τ are baryon number andthe family-lepton–numbers respectively. The usual version of the MSM isconstructed so that these last four groups are automatic global symmetriesof the classical Lagrangian, having no associated gauge fields.An interesting, non-trivial constraint on gauge models is anomaly can-cellation.

This is often imposed so that the standard proof of the renor-malizability of gauge theories applies.Alternatively, one may simply de-mand as an aesthetic principle that quantum effects not spoil the naivegauge invariance of a model, leading also to gauge anomaly cancellation.However one motivates it, it is striking that in the MSM all gauge anoma-lies from SU(3)c ⊗SU(2)L ⊗U(1)Y cancel within each fermion family. Inmodel-building one usually finds that anomaly-cancellation imposes severeconstraints on the allowed U(1) charges.It is interesting to note, therefore, that U(1)Y is not the only anomaly-free Abelian invariance of the MSM.

A simple calculation demonstrates thatdifferences in family-lepton–numbers are also completely anomaly-free1 withrespect to SU(3)c ⊗SU(2)L ⊗U(1)Y . For a three family model, there arethus three of these anomaly-free combinations, given byLeµ ≡Le −LµorLeτ ≡Le −LτorLµτ ≡Lµ −Lτ.

(1)1Note that the cancellation of the mixed gauge-gravitational anomaly is required inorder to derive these invariances as the unique anomaly-free set.1

It is important to understand that although each of these differences is indi-vidually anomaly-free, no two are anomaly-free with respect to each other.2This interesting observation immediately leads one to ask whether or notthese particular subsets of the global symmetries of the MSM have associatedgauge fields. A possibility is that one of U(1)Leµ, U(1)Leτ or U(1)Lµτ is gaugedas a local symmetry that has no role to play in electroweak physics.

The Z′model which ensues has recently been studied in the literature [1].Another fascinating possibility, which will be the focus of this paper, isfor the definition of the weak hypercharge in the MSM to be altered in oneof three ways:Yeµ = Yst + 2ǫLeµorYeτ = Yst + 2ǫLeτorYµτ = Yst + 2ǫLµτ,(2)where Yst is the standard hypercharge of the MSM and ǫ is a free parameter.After electroweak symmetry breaking, this leads to non-standard unbrokenelectric charges given byQij = Qst + ǫLij,(3)where Qst = I3 + Yst/2 is standard electric charge and i, j = e, µ, τ (i ̸= j).Equation (3) defines the precise ways in which electric charge quantizationcan fail in the multi-family MSM3[5]. Note that electromagnetic gauge in-variance is still exact, and the photon as usual has no zero-temperature mass(thermal masses will be considered later).The electric charge generators Qij alter the physical electric charges fortwo out of the three families of leptons.

For instance under Qµτ,Qe = −1,Qµ = −1 + ǫ,Qτ = −1 −ǫ,(4)Qνe = 0,Qνµ = ǫ,Qντ = −ǫ,2Note that the quark analogues of Eq. (1) are explicitly broken in the MSM La-grangian (this is manifested through a non-diagonal Kobayashi-Maskawa matrix), so theonly anomaly-free Abelian invariance acting on quarks is U(1)Y .3If right-handed neutrinos are added to the MSM fermion spectrum, and only Diracneutrino masses are induced after electroweak symmetry breaking, then the family-lepton–numbers are in general explicitly broken and the above form of electric charge dequantiza-tion is excluded.

In this case, however, B −L generates an anomaly-free U(1) symmetry,and so charge dequantization can ensue through Q = Qst + ǫ(B −L)[2, 3] (for bounds onǫ in this model see Ref.[2]). If bare Majorana masses are included for the right-handedneutrinos, then B −L is also explicitly broken, and no electric charge dequantization atall is allowed [4].2

while the quark charges assume their standard values, of course. The twoobservable consequences of this are that e, µ and τ do not have identicalcharges, and two neutrino flavors have equal and opposite charges.

The pur-pose of this paper is to derive phenomenological bounds on ǫ for each of thethree non-standard MSM’s.4 Our phenomenological constraints come eitherfrom physics which would be sensitive to (small) violations of electromag-netic universality for e, µ and τ, or from limits connected with the existenceof mini-charged massless neutrinos.Several phenomenological analyses on mini-charged particles have re-cently been published [2, 6, 7, 8, 9, 10, 11].Reference [2] deals specifi-cally with another form of electric charge dequantization featuring electricallycharged neutrinos (see footnote 4 above), while Ref. [6] is discussed in foot-note 3 above.

The papers in Refs. [7, 8] deal with mini-charged particles inmodels where electric-charge conservation is violated, while Refs.

[9, 10, 11],on the other hand, examine constraints on completely new and exotic parti-cles of tiny electric charge. Some of the constraints derived in these papersare immediately applicable to the models of charge dequantisation consideredhere, while others are irrelevant.

It is important to determine the specificphenomenological constraints on the parameter ǫ in Eq. (3) because of thestrong theoretical underpinning it has from the structure of the MSM.The parameter ǫ for the U(1)Yeµ and U(1)Yeτ cases (see Eq.

(2)) can be di-rectly and severely constrained from a variety of experiments. By assumingelectric charge conservation (which is exact in the models under consider-ation) in β-decay, Zorn et al[12] were able to constrain the charge of theelectron-neutrino, which in our notation leads to |ǫ| < 4 × 10−17.

A boundof |ǫ| < 10−19 is obtained from the observation of electron-neutrinos from su-pernova 1987A[13]. Also, since electrons now have a charge of Qe = −1 + ǫ,atoms are no longer electrically neutral (which is a classic signature of elec-tric charge dequantization).Reference [14] provides a useful summary ofexperiments on the neutrality of matter performed to date.

These authorsobtain a bound on the electron-proton charge magnitude difference, whichtranslates into |ǫ| < 1.6 × 10−21.Some interesting terrestrial effects are possible if ǫ is nonzero because the4While this paper was being written up, we came across a preprint (Reference [6])which quotes some bounds on charge dequantization in the MSM, but it is our intentionto do a much more thorough analysis here.3

earth may be charged. We will assume first of all that the number of protonsin the earth is equal to the number of electrons.

It is certainly possible forthis assumption to be wrong, and we will comment on this issue again a littlelater on.If ǫ ̸= 0, then atoms are charged, and so mutually repulsive forces willexist between our assumed charged earth and laboratory samples of ordinarymatter. If |ǫ| is large enough, then experiments should already have beensensitive to this.

Given that no evidence of such an effect exists, we deriveupper “bounds” on |ǫ| below from a couple of considerations.Note thatthese limits are not bounds in the rigorous sense of the word, because ourassumption that the number of protons equals the number of electrons in theearth need not be correct.E¨otv¨os experiments measuring the differential attraction or repulsion ofearth with samples of material A and material B (both taken to be pureelements), lead to an upper “bound” on ǫ given byǫ2 < 10−12GmNαem ZAMA−ZBMB−1(5)where G is Newton’s constant, mN is the mass of a nucleon, αem is theelectromagnetic fine-structure constant, ZA,B are atomic numbers of materialA and B respectively, while MA,B are the masses of atoms of A and B. Fortypical materials (for instance copper and lead, see Ref. [15]) this yields|ǫ| < 10−23(6)or so.Although this “bound” is a couple of orders of magnitude betterthan the limits quoted above, it should not be taken too seriously given theelectron/proton number assumption.Experiments near the earth’s surface indicate that the earth has a radialelectric field of less than about 100 V/m [8].

With equal proton and electronnumbers, we then obtain that|ǫ| < 10−27. (7)We emphasise, however, that the assumption of equal electron and protonnumbers in the earth is important, and so this limit cannot be regarded as arigorous bound.4

How different do the electron and proton numbers need to be to invalidatethese “bounds”? Let us examine the radial electric field limit in more detail.A rigorous constraint can actually be derived if the numbers of protons andelectrons are allowed to vary.

It isNp −Ne + ǫNe < 1024(8)where Np(Ne) is the number of protons(electrons) in the earth. The protonnumber of the earth is about 1051, so with Np = Ne we recover the result ofEq.

(7). We can ask what ∆≡Np −Ne needs to be to make the limit on |ǫ|as weak as the bounds of 10−21 and 10−17 from atomic neutrality and chargeconservation in β-decay, respectively.

Given that Np ∼Ne ∼1051, we seefrom Eq. (8) that|ǫ| ∼10−21 ⇒|∆| ∼1030 and |ǫ| ∼10−17 ⇒|∆| ∼1034.

(9)If we assume that nonzero ∆is due to excess electrons, then this amountsto between 1 −104 kg of electrons. If it is due to the presumably less mobileprotons, then this is a mass in the range 103−107 kg.

By way of comparison,a cubic metre of earth has a mass of about 5500 kg. Another way of looking atthis is that it corresponds to a number density of excess electrons or protonsof about 1 −104 particles per cubic millimetre.Note also that an interesting effect can occur at the level of galaxies.Naively, a limit on |ǫ| may be obtained from the observed stability of galaxiesby requiring that electrostatic repulsion not exceed the gravitational attrac-tion.

This yields |ǫ| < (Gm2N/10)1/2 = 10−20 where equal numbers of protonsand electrons are again assumed. However, the relic neutrino cloud from theBig Bang will act as a polarizable medium at the galactic level, and so anygalactic charge will be screened to some extent.

A simple order of magnitudeestimate for the screening length is (ǫeTν)−1 where Tν ≃2K is the tempera-ture of the relic neutrinos. For |ǫ| of the order of 10−20 the screening length istherefore expected to be less than typical galactic radii.

Therefore, galacticcharges for reasonable values of |ǫ| should be rendered unobservable.Since all of the bounds on the U(1)Yeµ and U(1)Yeτ models are quite se-vere, the main interest of this paper is to derive bounds on the significantlyless constrained model defined by U(1)Yµτ . We will examine several phe-nomenological constraints on ǫ for this case.The first bound is derived by comparing the anomalous magnetic mo-ments aµ and ae of the muon and electron, respectively.

(Since the tau5

anomalous moment is not as precisely measured as the other two we donot need to consider it.) The dominant contribution which ǫ makes to theanomalous moment of the muon comes from the 1-loop Schwinger correction,yieldinga(1−loop)µ= (ǫ −1)3αem2πe2mµ(10)compared with the electron result a(1−loop)e= (αem/2π)(e/2me).Keepingonly linear terms in ǫ we therefore find that the muon anomalous moment isshifted from its standard value by an amount δaµ given approximately byδaµ ≃−3ǫαem2πe2mµ.

(11)We obtain a bound by simply demanding that this shift be less than theexperimental uncertainty in aµ. This approach is justified because of theimpressive agreement between the measured anomalous moments and thestandard theoretical calculations.

The best measurement of aµ[16] has anerror of ±9 × 10−9(e/2mµ) yielding,|ǫ| < 10−6. (12)This bound is many orders of magnitude less than the bounds on the gaugedU(1)Yeµ and U(1)Yeτ models.

Quite apart from the specific models we areconsidering in this paper, it is also interesting to note that this is the moststringent model-independent bound on the difference in the electric chargesof the electron and muon.The second constraint we will analyse comes from the measured νµ-escattering cross-section σ(νµe). When ǫ = 0, this process is well describedby the exchange of a Zo gauge boson in the t-channel.

For nonzero ǫ thereis an additional contribution coming from t-channel photon exchange. Wewill obtain our bound by demanding that the photon contribution to thecross-section lie within experimental errors.The exact expression for σ(νµe) includes direct Zo, direct photon and in-terference terms, and is rather complicated.

The complication arises becauseof the need to keep the electron mass finite when calculating the t-channelphoton exchange diagram. However, a useful approximate expression is ob-tained by keeping only those terms which diverge in the massless electron6

limit. The result for the ǫ-dependent contribution to the cross-section is,δσ(νµe)≃h2πα2emm2e−2πα2emmeEν−2√2αemGFx(1 −4x)lnEνmeiǫ2−2√2αemGF(1 −4x) lnEνmeǫ(13)where x ≡sin2 θW, GF is the Fermi constant, me is the electron mass, andEν is the incident neutrino energy.Experiments on νµ-e scattering use incident neutrino energies Eν of a fewGeV’s[17, 18].

Therefore the ratio Eν/me is large (> 3000), which illustrateswhy the approximate cross-section of Eq. (13) is useful.By inputting thevalues of the various quantities appearing in this expression, we see that thefirst and third terms dominate over the second.

To obtain a bound on ǫ weuse the result of the BBKOPST collaboration[18]:σ(νµe)/Eν = (1.85 ± 0.25 ± 0.27) × 10−42 cm2 GeV−1withEν = 1.5 GeV. (14)By adding the statistical and systematic errors in Eq.

(14) in quadrature, wefind that|ǫ| < 10−9(15)with both the first and third terms in Eq. (13) of roughly equal importance.Note that this bound is three orders of magnitude more stringent than thatfrom using anomalous magnetic moments.Both of the above bounds on the gauged U(1)Yµτ version of the MSM werederived from considerations that were purely within the ambit of particlephysics.We will now present two bounds which also require the use ofastrophysics and cosmology, and so our faith in their veracity will be as solidor weak as our belief in the required astrophysical and cosmological models.It is well known that bounds on weakly-coupled particles can be obtainedby requiring that their production in stars be not so strong as to causepremature (and unobserved) cooling.

In our case, the decay in red giantstars of massive plasmon states into charged νµ¯νµ and ντ ¯ντ pairs can occur.These very weakly interacting neutrinos and antineutrinos can then escapefrom the star, thus cooling it. The authors of Ref.

[10] have (effectively)calculated a bound on ǫ from red giant cooling by demanding that the rateof energy loss per unit volume to mini-charged neutrino-antineutrino pairs7

not exceed the nuclear energy generation rate per unit volume. They estimatethat the former quantity is given by, d2EdV dtν¯ν ≃1034 × ǫ2 ergs cm3 sec−1(16)and requiring that this not exceed about 106 ergs cm3 sec−1 yields the bound,|ǫ| < 10−14.

(17)This result is interesting because it is five orders of magnitude more stringentthan the limit obtained from νµ-e scattering. The authors of Ref.

[11] werealso able to derive an astrophysical bound by looking at the cooling of whitedwarf stars, obtaining|ǫ| < 10−13(18)which is an order of magnitude less severe than Eq. (17).

Most astrophysicistsare confident that the stellar structure and evolution of red giants and whitedwarfs are sufficiently well understood that these bounds are to be takenvery seriously. It may nevertheless be wise to caution that, due to the verynature of the subject matter, one cannot ascribe as much confidence on thesebounds as one can on bounds of purely particle physics origin.We now turn to an interesting cosmological consequence of having mini-charged neutrinos.

The standard hot Big Bang model of cosmology predictsthe existence of a thermal background of each flavor of neutrino. The tem-perature of this bath of thermal neutrinos is found to be slightly less thanthe 3K temperature of the microwave photon background, Tν ≃2K.

Becausemuon- and tau-neutrinos are charged in the U(1)Yµτ model, they form a back-ground “cosmic plasma” which permeates the entire universe. All particles,and in particular photons, have to propagate through this thermal heat bathof neutrinos.

Photons will therefore acquire a nonzero “electric mass” frominteracting with this medium (in a similar manner to the aforementioned ac-quisition by photons of a nonzero plasmon mass in stellar interiors). Knownbounds on photon electric masses5 will therefore constrain ǫ, since it is im-5In principle, a photon can also have a “magnetic mass.” However, a nonzero magneticmass cannot arise from thermal effects[19], so it is irrelevant to the present discussion.Note that the most stringent bounds on the photon mass are derived from knowledge ofmagnetic fields, and are thus constraints on the magnetic rather than the electric mass ofthe photon.8

possible for photons to avoid propagating through the neutrino backgroundplasma.The thermal electric mass of the photon is calculated through the 1-loopcontribution of the charged neutrinos to the photon vacuum polarization ten-sor, where the internal neutrino propagators are taken at finite temperature.Since a similar calculation is performed in Ref. [20], we will omit technicaldetails of how this computation is done.

The result is[melγ ]2 = Nν2π3 ǫ2αem(kTν)2,(19)where Nν = 2 is the number of charged neutrino flavors and k is Boltzmann’sconstant. The best bound on the photon electric mass comes from a test ofGauss’s Law (or, equivalently, Coulomb’s Law), and is[21]melγ < 10−25GeV.

(20)The resulting bound for ǫ is thereforeǫ < 10−12. (21)It is interesting that this limit is stronger than those obtained from particlephysics measurements, but less severe than those obtained from energy lossin stellar objects.We should remark here that the derivation of the astrophysical bound[Eqs.

(17)-(18)] and the cosmological bound [Eq. (21)] on ǫ assumes that νµand ντ have masses less than about 10 keV.

Otherwise, (a) the plasmon decayinto νν will be forbidden kinematically inside red giants and white dwarfswhere the typical temperature is of order 10 keV and (b) the cosmologicalmass density constraint requires that the keV neutrinos decay or annihiliatein the early stages of the evolution of the universe, so that they will notbe around today to give a thermal mass to the photon. While it is truethat in the MSM the neutrinos have no zero temperature masses (as thephoton, the neutrinos also acquire a thermal mass of order ǫ2 T from thebackground photons), by slightly modifying the Higgs sector (e.g.

adding atriplet Higgs), it is possible to give a small ‘Dirac’ mass for νµ and ντ withoutviolating charge conservation. The present experimental limits on the massesof νµ and ντ are 270 keV and 35 MeV respectively, so it is not impossible toinvalidate the bounds in Eqs.

(17)-(18) and in Eq. (21).9

As noted above, by some minor modifications to the Higgs sector neu-trinos can acquire tiny masses without violating electromagnetic gauge in-variance. However, there will be no neutrino mixing and hence no neutrinooscillations in this case.

Therefore, the MSM with mini-charged neutrinoscannot account for the apparent deficit in the flux of neutrinos coming fromthe sun through any form of neutrino oscillation mechanism. On the otherhand, by utilising another class of extensions to the basic model, the neutrinodeficit may be explained by endowing νe with a transition magnetic momentwith either (νµ)c or (ντ)c, depending on whether the U(1)Yeµ or U(1)Yeτ caseis considered.

This mechansim would also have the advantage of explainingthe possible anticorrelation of the solar neutrino flux with sunspot activity.Another important cosmological question to consider is whether chargedrelic neutrinos can induce an overall charge for the universe. If they can thenelectrostatic repulsion will contribute to the expansion of the universe.

Thesimple answer to this question is that no overall charge for the universe willbe generated because electric charge conservation is still exact in our models.This will follow provided, of course, that a neutral universe is posited as aninitial condition for the Big Bang. Charged neutrinos are therefore no moreproblematic in this regard than any other stable charged particles.In summary then, we have discovered that experiments on the neutralityof atoms places a bound given by |ǫ| < 10−21 on the allowed non-standardelectric charges Qst + ǫ(Le −Lµ) and Qst + ǫ(Le −Lτ).A direct boundon the electron-neutrino charge of |ǫ| < 4 × 10−17 is obtained from similarexperiments where charge conservation in β-decay is assumed.

However, oneof the main points of our paper is that the other allowed non-standard chargeQst + ǫ(Lµ −Lτ) is constrained far less profoundly. Upper bounds on ǫ of10−14, 10−13, 10−12, 10−9 and 10−6 were derived from, respectively, energyloss in red gaint stars, energy loss in white dwarf stars, the thermal electricmass of the photon, νµ-e scattering and the anomalous magentic moment ofthe muon.AcknowledgementsK.S.B.

would like to acknowledge discussions with D. Seckel, and bothauthors would like to thank him for raising the issue of a thermal photonmass. K.S.B.

is supported by the U.S. Department of Energy. R.R.V.

wouldlike to thank M. J. Thomson for a helpful correspondence, and X.-G. He,K. C. Hines and K. Liffman for discussions.R.R.V.

is supported by theAustralian Research Council through a Queen Elizabeth II Fellowship.10

References[1] X.-G. He, G. C. Joshi, H. Lew and R. R. Volkas, Phys.

Rev. D43, R22(1991); ibid.

D44, 2118 (1991). [2] R. Foot, G. C. Joshi, H. Lew and R. R. Volkas, Mod.

Phys. Lett.

A5,95 (1990). For reviews on electric-charge quantization see R. Foot, G.C.

Joshi, H. Lew and R. R. Volkas, Mod. Phys.

Lett. A5, 2721 (1990);R. Foot, H. Lew and R. R. Volkas, University of Melbourne report No.UM-P-92/52.

[3] N. G. Deshpande, Oregon Report OITS-107 (1979) (unpublished). [4] K. S. Babu and R. N. Mohapatra, Phys.

Rev. Lett.

63, 938 (1989); Phys.Rev. D41, 271 (1990).

[5] R. Foot, Mod. Phys.

Lett. A6, 527 (1991).

As far as we are aware, thisis the first place in which the anomly-freedom of family-lepton–numberdifferences was pointed out. [6] E. Takasugi and M. Tanaka, Osaka University preprint OS-GE 21-91(unpublished).

[7] E. Takasugi and M. Tanaka, Phys. Rev.

D44, 3706 (1991); M. Maruno,E. Takasugi and M. Tanaka, Prog.

Theor. Phys.

86, 907 (1991). [8] K. S. Babu and R. N. Mohapatra, Phys.

Rev. D42, 3866 (1990).

[9] R. N. Mohapatra and I. Z. Rothstein, Phys.

Lett. B247, 593 (1990); R.N.Mohapatra and S. Nussinov, University of Maryland Report UMD-PP-008 (1991).

[10] S. Davidson, B. Campbell and D. Bailey, Phys. Rev.

D43, 2314 (1991). [11] M. I. Dobroliubov and A. Yu.

Ignatiev, Phys. Rev.

Lett. 65, 679 (1990).

[12] J. C. Zorn, G. E. Chamberlain and V. W. Hughes, Phys. Rev.

129, 2566(1963). [13] G. Barbiellini and G. Cocconi, Nature (London) 329, 21 (1987).11

[14] M. Marinelli and G. Morpugo, Phys. Lett.

B137, 439 (1984). See alsoH.

F. Dylla and J. G. King, Phys. Rev.

A7, 1224 (1973). [15] L. B. Okun, Leptons and Quarks, pp.

166-167 (North-Holland, Amster-dam, 1984). [16] J. Bailey et al., Nucl.

Phys. B150, 1 (1979).

[17] For a summary of these experiments see U. Amaldi et al., Phys. Rev.D36, 1385 (1987) and G. Costa et al., Nucl.

Phys. B297, 245 (1988).

[18] K. Abe et al., Phys. Rev.

Lett. 58, 636 (1987).

[19] E. S. Fradkin, 1965 Proceedings of the Lebedev Physical Institute 297-138 (English translation 1967 by Consultants Bureau, New York). [20] J. F. Nieves, P. B. Pal and D. G. Unger, Phys.

Rev. D28, 908 (1983)[21] E. R. Williams, J. E. Faller and H. A. Hill, Phys.

Rev. Lett.

26, 721(1971).12

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