Electromagnetic neutrinos in laboratory experiments and astrophysics
Carlo Giunti, Konstantin A. Kouzakov, Yu-Feng Li, Alexey V. Lokhov, Alexander I. Studenikin, Shun Zhou
SSeptember 17, 2018
Electromagnetic neutrinos in laboratory experimentsand astrophysics
Carlo Giunti , Konstantin A. Kouzakov , Yu-Feng Li , Alexey V. Lokhov ,Alexander I. Studenikin ∗ , and Shun Zhou An overview of neutrino electromagnetic properties,which open a door to the new physics beyond the Stan-dard Model, is given. The effects of neutrino electro-magnetic interactions both in terrestrial experimentsand in astrophysical environments are discussed. Theexperimental bounds on neutrino electromagneticcharacteristics are summarized. Future astrophysicalprobes of electromagnetic neutrinos are outlined.
The importance of neutrino electromagnetic propertieswas first mentioned by Pauli in 1930, when he postulatedthe existence of this particle and discussed the possibilitythat the neutrino might have a magnetic moment [1]. Sys-tematic theoretical studies of neutrino electromagneticproperties started after it was shown that in the extendedStandard Model with right-handed neutrinos the mag-netic moment of a massive neutrino is, in general, nonva-nishing and that its value is determined by the neutrinomass [2–8].Neutrinos remained elusive until the detection of reac-tor neutrinos by Reines and Cowan around 1956 [9]. How-ever, there was no sign of a neutrino mass. After the discov-ery of parity violation in 1957, the two-component theoryof massless neutrinos was proposed [10–12], in which aneutrino is described by a Weyl spinor and there are onlyleft-handed neutrinos and right-handed antineutrinos. Itwas however clear [13–15] that two-component neutrinoscould be massive Majorana fermions and that the two-component theory of a massless neutrino is equivalent tothe Majorana theory in the limit of zero neutrino mass.The two-component theory of massless neutrinos waslater incorporated in the Standard Model of Glashow,Weinberg and Salam [16–18], in which neutrinos are mass-less and have only weak interactions. In the StandardModel Majorana neutrino masses are forbidden by the SU(2) L × U(1) Y symmetry. We now know that neutrinosare massive, because many experiments observed neu-trino oscillations (see the review articles [19–24]), whichare generated by neutrino masses and mixing [25–28].Therefore, the Standard Model must be extended to ac-count for the neutrino masses. There are many possibleextensions of the Standard Model which predict differentproperties for neutrinos (see [21, 29, 30]). Among them,most important is their fundamental Dirac or Majoranacharacter. In many extensions of the Standard Model neu-trinos acquire also electromagnetic properties throughquantum loops’ effects which allow interactions of neu-trinos with electromagnetic fields and electromagneticinteractions of neutrinos with charged particles.Hence, the theoretical and experimental study of neu-trino electromagnetic interactions is a powerful tool in thesearch for a more fundamental theory beyond the Stan-dard Model. Moreover, the electromagnetic interactionsof neutrinos can generate important effects, especiallyin astrophysical environments, where neutrinos propa-gate over long distances in magnetic fields in vacuum andin matter. Unfortunately, in spite of many efforts in thesearch of neutrino electromagnetic interactions, up tonow there is no positive experimental indication in favor ∗ Corresponding author E-mail: [email protected] INFN, Sezione di Torino, and Dipartimento di Fisica Teorica,Universit‘a di Torino2 Institite of High Energy Physics, Chinese Academy of Sci-ences, Beijing, China3 Department of Nuclear Physics and Quantum Theory of Col-lisions, Faculty of Physics, Lomonosov Moscow State Univer-sity, 119991 Moscow, Russia4 Department of Theoretical Physics, Faculty of Physics,Lomonosov Moscow State University, 119991 Moscow, Russia5 Joint Institute for Nuclear Research, 141980 Dubna, MoscowRegion, Russia6 Institute for Nuclear Research, Russian Academy of Sciences,117312 Moscow, Russia
Copyright line will be provided by the publisher a r X i v : . [ h e p - ph ] N ov . Giunti et al.: Electromagnetic neutrinos in laboratory experiments and astrophysics of their existence. However, it is expected that the Stan-dard Model neutrino charge radii should be measured inthe near future. This will be a test of the Standard Modeland of the physics beyond the Standard Model which con-tributes to the neutrino charge radii. Moreover, the exis-tence of neutrino masses and mixing implies that neutri-nos have (diagonal and/or transition) magnetic moments.Since their values depend on the specific theory whichextends the Standard Model in order to accommodateneutrino masses and mixing, experimentalists and theo-rists are eagerly looking for them.The paper is organized as follows. Section 2 deliversthe general form of the electromagnetic interactions ofDirac and Majorana neutrinos in the one-photon approx-imation, which are expressed in terms of electromagneticform factors. In Section 3 we discuss some basic pro-cesses which are induced by the neutrino electromagneticproperties and some important effects due to the inter-action of neutrinos with classical electromagnetic fields.In Section 4 we overview the experimental constraints onthe neutrino electric and magnetic moments, the electriccharge (millicharge), the charge radius and the anapolemoment. In Section 5 future astrophysical probes of neu-trino electromagnetic properties and interactions are out-lined. Finally, Section 6 summarizes this work. In this Section we discuss the general form of the electro-magnetic interactions of Dirac and Majorana neutrinosin the one-photon approximation. In the Standard Model,the interaction of a fermionic field f with the electromag-netic field A µ is given by the interaction Hamiltonian H ( f )em = j ( f ) µ A µ = e f f γ µ f A µ , (1)where e f is the charge of the fermion f .For neutrinos the electric charge is zero and there areno electromagnetic interactions at tree-level . At the sametime, such interactions can arise at the quantum levelfrom loop diagrams at higher order of the perturbative ex-pansion of the interaction. We know that there are at leastthree massive neutrino fields in nature, which are mixedwith the three active flavor neutrinos ν e , ν µ , ν τ . Therefore,we discuss the case of three massive neutrino fields ν i ν i ( p i ) ν f ( p f ) γ ( q )Λ fi Figure 1
Effective one-photon coupling of neutrinos with theelectromagnetic field, taking into account possible transitionsbetween two different initial and final massive neutrinos ν i and ν f . with respective masses m i ( i =
1, 2, 3). In the one-photonapproximation, the effective electromagnetic interactionHamiltonian is given by H ( ν )em = j ( ν ) µ A µ = (cid:88) i , f = ν f Λ f i µ ν i A µ , (2)where we take into account possible transitions betweendifferent massive neutrinos. The physical effect of H ( ν )em isdescribed by the effective electromagnetic vertex in Fig. 1.In momentum-space representation, this vertex dependsonly on the four-momentum q = p i − p f transferred tothe photon and can be expressed as follows: Λ µ ( q ) = (cid:161) γ µ − q µ / q / q (cid:162) (cid:163) f Q ( q ) + f A ( q ) q γ (cid:164) − i σ µν q ν (cid:163) f M ( q ) + i f E ( q ) γ (cid:164) , (3)in which Λ µ ( q ) is a 3 × × f Q = f † Q , f M = f † M , f E = f † E , f A = f † A , (4)where Q , M , E , A refer respectively to the real charge, mag-netic, electric, and anapole neutrino form factors. TheLorentz-invariant form of the vertex function (3) is alsoconsistent with electromagnetic gauge invariance thatimplies four-current conservation.For the coupling with a real photon in vacuum ( q = f f iQ (0) = e f i , f f iM (0) = µ f i , f f iE (0) = (cid:178) f i , f f iA (0) = a f i , (5)where e f i , µ f i , (cid:178) f i and a f i are, respectively, the neutrinocharge, magnetic moment, electric moment and anapolemoment of diagonal ( f = i ) and transition ( f (cid:54)= i ) types.A Majorana neutrino is a neutral spin 1/2 particlewhich coincides with its antiparticle. The four degrees Copyright line will be provided by the publishereptember 17, 2018 of freedom of a Dirac field (two helicities and two particle-antiparticle) are reduced to two (two helicities). Sincea Majorana field has half the degrees of freedom of aDirac field, its electromagnetic properties are also re-duced. Namely, in the Majorana case the charge, magneticand electric form-factor matrices are antisymmetric andthe anapole form-factor matrix is symmetric. Since f M Q , f M M and f M E are antisymmetric, a Majorana neutrino doesnot have diagonal charge and dipole magnetic and elec-tric form factors [13, 14, 31]. It can only have a diagonalanapole form factor. On the other hand, Majorana neu-trinos can have as many off-diagonal (transition) form-factors as Dirac neutrinos. Neutrino electric charge.
It is usually believed thatthe neutrino electric charge e ν = f Q (0) is zero. This isoften thought to be attributed to the gauge-invarianceand anomaly-cancellation constraints imposed in theStandard Model. In the Standard Model of SU(2) L × U(1) Y electroweak interactions it is possible to get [32] a gen-eral proof that neutrinos are electrically neutral, which isbased on the requirement of electric charges’ quantiza-tion. The direct calculations of the neutrino charge in theStandard Model for massless (see, for instance [33, 34])and massive neutrinos [35, 36] also prove that, at least atthe one-loop level, the neutrino electric charge is gauge-independent and vanishes. However, if the neutrino has amass, it still may become electrically millicharged. A briefdiscussion of different mechanisms for introducing mil-licharged particles including neutrinos can be found in[37]. In the case of millicharged massive neutrinos, electro-magnetic gauge invariance implies that the diagonal elec-tric charges e ii ( i =
1, 2, 3) are equal . It should be men-tioned that the most stringent experimental constraintson the electric charge of the neutrino can be obtainedfrom neutrality of matter. These and other constraints, in-cluding the astrophysical ones, are discussed in Section 4. Neutrino charge radius.
Even if the electric chargeof a neutrino is zero, the electric form factor f Q ( q ) canstill contain nontrivial information about neutrino staticproperties [38]. A neutral particle can be characterized bya superposition of two charge distributions of oppositesigns, so that the particle form factor f Q ( q ) can be non-zero for q (cid:54)=
0. The mean charge radius (in fact, it is thecharged radius squared) of an electrically neutral neutrinois given by 〈 r ν 〉 = d f Q ( q ) d q (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) q = , (6) which is determined by the second term in the power-series expansion of the neutrino charge form factor.Note that there is a long-standing discussion (see [38]for details) on the possibility to obtain for the neutrinocharged radius a gauge-independent and finite quantity.In one of the first studies [33], it was claimed that inthe Standard Model and in the unitary gauge the neu-trino charge radius is ultraviolet-divergent and so it isnot a physical quantity. However, it was shown [39] thatin the unitary gauge it is possible to obtain for the neu-trino charge radius a gauge-dependent but finite quantity.Later on, it was also shown [3] that considering additionalbox diagrams in combination with contributions fromthe proper diagrams it is possible to obtain a finite andgauge-independent value for the neutrino charge radius.In this way, the neutrino electroweak radius was definedin [40, 41] and an additional set of diagrams that give con-tribution to its value was discussed in [42]. Finally, in aseries of papers [43–45] the neutrino electroweak radius asa physical observable has been introduced. This, however,revived the discussion [46–49] on the definition of the neu-trino charge radius. Nevertheless, in the correspondingcalculations, performed in the one-loop approximationincluding additional terms from the γ − Z boson mixingand the box diagrams involving W and Z bosons, thefollowing gauge-invariant result for the neutrino chargeradius has been obtained [49]: 〈 r ν e 〉 = × − cm . Thistheoretical result differs at most by an order of magni-tude from the available experimental bounds on 〈 r ν 〉 (seeSection 4 for references and more detailed discussion).Therefore, one may expect that the experimental accuracywill soon reach the level needed to probe the neutrinoeffective charge radius. Neutrino electric and magnetic moments.
The mostwell studied and understood among the neutrino elec-tromagnetic characteristics are the dipole magnetic andelectric moments, which are given by the correspondingform factors at q = µ ν = f M (0), (cid:178) ν = f E (0). (7)The diagonal magnetic and electric moments of a Diracneutrino in the minimally-extended Standard Model withright-handed neutrinos, derived for the first time in [4],are respectively µ Dii = e G F m i (cid:112) π ≈ × − µ B (cid:179) m i (cid:180) , (cid:178) Dii =
0, (8)where µ B is the Bohr magneton. According to (8) the valueof the neutrino magnetic moment is very small. How-ever, in many other theoretical frameworks (beyond theminimally-extended Standard Model) the neutrino mag-netic moment can reach values that are of interest for the Copyright line will be provided by the publisher . Giunti et al.: Electromagnetic neutrinos in laboratory experiments and astrophysics next generation of terrestrial experiments and also acces-sible for astrophysical observations. Note that the bestlaboratory upper limit on a neutrino magnetic moment, µ ν ≤ × − µ B (90% CL), has been obtained by theGEMMA collaboration [50] (see Section 4), and the bestastrophysical limit is µ ν ≤ × − µ B (90% CL) [51]. Thelatter bound comes from the constraints on the possibledelay of helium ignition of a red giant star in globular clus-ters due to the cooling induced by the energy loss in theplasmon-decay process γ ∗ → ν ¯ ν (see Fig. 2(b)). Recentlythe limit has been updated in [52] using state-of-the-artastronomical observations and stellar evolution codes,with the results µ ν < (cid:40) × − µ B (68% CL),4.5 × − µ B (95% CL). (9)This astrophysical bound on a neutrino magnetic mo-ment is applicable to both Dirac and Majorana neutri-nos and constrains all diagonal and transition dipole mo-ments. Neutrino anapole moment.
The notion of an anapolemoment for a Dirac particle was introduced by Zeldovich[53] after the discovery of parity violation. In order tounderstand the physical characteristics of the anapolemoment, it is useful to consider its effect in the interac-tions with external electromagnetic fields. The neutrinoanapole moment contributes to the scattering of neutri-nos with charged particles. In order to discuss its effects,it is convenient to consider strictly neutral neutrinos with f Q (0) = f Q ( q )such that f Q ( q ) = q ˜ f Q ( q ). (10)Then, from Eq. (6), apart from a factor 1/6, the reducedcharge form factor at q = f Q (0) = 〈 r ν 〉 /6. (11)Let us now consider the charge and anapole parts of theneutrino electromagnetic vertex function, as Λ Q , A µ ( q ) = (cid:161) γ µ q − q µ / q (cid:162) (cid:163) ˜ f Q ( q ) + f A ( q ) γ (cid:164) . (12)Since for ultrarelativistic neutrinos the effect of γ is onlya sign which depends on the helicity of the neutrino, thephenomenology of neutrino anapole moments is similarto that of neutrino charge radii. Hence, the limits on theneutrino charge radii discussed in Section 4 apply also tothe neutrino anapole moments multiplied by a factor of6. Neutrino-electron elastic scattering.
The most sensitiveand widely used method for the experimental investiga-tion of the neutrino magnetic moment is provided by di-rect laboratory measurements of low-energy elastic scat-tering of neutrinos and antineutrinos with electrons inreactor, accelerator and solar experiments . Detailed de-scriptions of several experiments can be found in [58, 59].Extensive experimental studies of the neutrino mag-netic moment, performed during many years, are stimu-lated by the hope to observe a value much larger than theprediction in Eq. (8) of the minimally extended StandardModel with right-handed neutrinos. It would be a clearindication of new physics beyond the extended StandardModel. For example, the effective magnetic moment in¯ ν e - e elastic scattering in a class of extra-dimension mod-els can be as large as about 10 − µ B [60]. Future higherprecision reactor experiments can therefore be used toprovide new constraints on large extra-dimensions.The possibility for neutrino-electron elastic scatteringdue to neutrino magnetic moment was first consideredin [61] and the cross section of this process was calcu-lated in [62] (for related short historical notes see [63]).Here we would like to recall the paper by Domogatskyand Nadezhin [64], where the cross section of [62] wascorrected and the antineutrino-electron cross section wasconsidered in the context of the earlier experiments withreactor antineutrinos of [65, 66], which were aimed toreveal the effects of the neutrino magnetic moment. Dis-cussions on the derivation of the cross section and on theoptimal conditions for bounding the neutrino magneticmoment, as well as a collection of cross section formu-lae for elastic scattering of neutrinos (antineutrinos) onelectrons, nucleons, and nuclei can be found in [63, 67].Let us consider the process ν (cid:96) + e − → ν (cid:96) (cid:48) + e − , (13)where a neutrino or antineutrino with flavor (cid:96) = e , µ , τ andenergy E ν elastically scatters off a free electron (FE) at restin the laboratory frame. Due to neutrino mixing, the finalneutrino flavor (cid:96) (cid:48) can be different from (cid:96) . There are twoobservables: the kinetic energy T e of the recoil electronand the recoil angle χ with respect to the neutrino beam, Copyright line will be provided by the publishereptember 17, 2018 which are related bycos χ = E ν + m e E ν (cid:104) T e T e + m e (cid:105) . (14)The electron kinetic energy is constrained from theenergy-momentum conservation by T e ≤ E ν E ν + m e . (15)Since, in the ultrarelativistic limit, the neutrino mag-netic moment interaction changes the neutrino helicityand the Standard Model weak interaction conserves theneutrino helicity, the two contributions add incoherentlyin the cross section which can be written as [67], d σ ν (cid:96) e − dT e = (cid:181) d σ ν (cid:96) e − dT e (cid:182) FESM + (cid:181) d σ ν (cid:96) e − dT e (cid:182) FEmag . (16)The weak-interaction cross section is given by (cid:181) d σ ν (cid:96) e − dT e (cid:182) FESM = G F m e π (cid:110) ( g ν (cid:96) V + g ν (cid:96) A ) + ( g ν (cid:96) V − g ν (cid:96) A ) (cid:181) − T e E ν (cid:182) + (cid:163) ( g ν (cid:96) A ) − ( g ν (cid:96) V ) (cid:164) m e T e E ν (cid:111) , (17)with the standard coupling constants g V and g A given by g ν e V = θ W + g ν e A = g ν µ , τ V = θ W − g ν µ , τ A = − g A → − g A .The neutrino magnetic-moment contribution to thecross section is given by [67] (cid:181) d σ ν (cid:96) e − dT e (cid:182) FEmag = πα m e (cid:181) T e − E ν (cid:182) (cid:181) µ ν (cid:96) µ B (cid:182) , (20)where µ ν (cid:96) is the effective magnetic moment discussedin the following Section. It is called traditionally “mag-netic moment”, but it receives contributions from boththe electric and magnetic dipole moments (see details inSection 4).The two terms ( d σ ν (cid:96) e − / dT e ) FESM and ( d σ ν (cid:96) e − / dT e ) FEmag exhibit quite different dependencies on the experimen-tally observable electron kinetic energy T e . One can see that small values of the neutrino magnetic moment canbe probed by lowering the electron recoil energy thresh-old. In fact, considering T e (cid:191) E ν in Eq. (20) and neglectingthe coefficients due to g ν (cid:96) V and g ν (cid:96) A in Eq. (17), one canfind that ( d σ / dT e ) FEmag exceeds ( d σ / dT e ) FESM for T e (cid:46) π α G m e (cid:181) µ ν µ B (cid:182) . (21)The current experiments with reactor antineutrinoshave reached threshold values of T e as low as few keV. Asdiscussed in Section 6, these experiments are likely to fur-ther improve the sensitivity to low energy deposition inthe detector. At low energies however one can expect amodification of the free-electron formulas (17) and (20)due to the binding of electrons in the germanium atoms,where e.g. the energy of the K α line, 9.89 keV, indicatesthat at least some of the atomic binding energies are com-parable to the already relevant to the experiment valuesof T e . It was demonstrated [69–73] by means of analyti-cal and numerical calculations that the atomic bindingeffects are adequately described by the so-called steppingapproximation introduced in [74] from interpretation ofnumerical data. According to the stepping approach, (cid:181) d σ ν (cid:96) e − dT e (cid:182) SM = (cid:181) d σ ν (cid:96) e − dT e (cid:182) FESM (cid:88) j n j θ ( T e − E j ), (22) (cid:181) d σ ν (cid:96) e − dT e (cid:182) mag = (cid:181) d σ ν (cid:96) e − dT e (cid:182) FEmag (cid:88) j n j θ ( T e − E j ), (23)where the j sum runs over all occupied atomic sublevels,with n j and E j being their occupations and binding ener-gies. Neutrino-nucleus coherent scattering.
As mentionedabove, the most sensitive probe of neutrino electromag-netic properties is provided by direct laboratory measure-ments of (anti)neutrino-electron scattering at low ener-gies in solar, accelerator and reactor experiments (theirdetailed description can be found in [38, 58, 59, 75–77]).The coherent elastic neutrino-nucleus scattering [78] hasnot been experimentally observed so far, but it is expectedto be accessible in the reactor experiments when low-ering the energy threshold of the employed Ge detec-tors [79–81].Let us consider the case of electron neutrino scatter-ing off a spin-zero nucleus with even numbers of protonsand neutrons, Z and N . The matrix element of this pro-cess, taking into account the neutrino electromagnetic Copyright line will be provided by the publisher . Giunti et al.: Electromagnetic neutrinos in laboratory experiments and astrophysics properties, reads M = (cid:183) G F (cid:112) u ( k (cid:48) ) γ µ (1 − γ ) u ( k ) C V + π Z eq (cid:179) e ν e + e q 〈 r ν e 〉 (cid:180) ¯ u ( k (cid:48) ) γ µ u ( k ) − π Z e µ ν e q ¯ u ( k (cid:48) ) σ µν q ν u ( k ) (cid:184) j ( N ) µ , (24)where C V = [ Z (1 − θ W ) − N ]/2, j ( N ) µ = ( p µ + p (cid:48) µ ) F ( q ),with p and p (cid:48) being the initial and final nuclear four-momenta. For neutrinos with energies of a few MeV themaximum momentum transfer squared ( | q | max = E ν )is still small compared to 1/ R , where R , the nucleus ra-dius, is of the order of 10 − − − MeV − . Therefore, thenuclear elastic form factor F ( q ) can be set equal to one.Using (24), one obtains the differential cross section in thenuclear-recoil energy transfer T N as a sum of two compo-nents. The first component conserves the neutrino helic-ity and can be presented in the form (cid:181) d σ ν e N dT N (cid:182) Q SM = η (cid:181) d σ ν e N dT N (cid:182) SM , (25)where η = − (cid:112) π e ZG F C V (cid:104) e ν e MT − e 〈 r ν e 〉 (cid:105) ,with M being the nuclear mass, and (cid:181) d σ ν e N dT N (cid:182) SM = G F π MC V (cid:181) − T N T max N (cid:182) (26)is the Standard Model cross section due to weak interac-tion [82], with T max N = E ν E ν + M .The second, helicity-flipping component is due to themagnetic moment only and is given by [67] (cid:181) d σ ν e N dT N (cid:182) mag = παµ ν e Z T N (cid:195) − T N E ν + T N E ν (cid:33) . (27)Clearly, any deviation of the measured cross sectionof the process under discussion from the well-knownStandard Model value (26) will provide a signature of thephysics beyond the Standard Model (see also [83–86]).Formulas (25) and (27) describe such a deviation due toneutrino electromagnetic interactions. Radiative decay and related processes.
The magneticand electric (transition) dipole moments of neutrinos, aswell as possible very small electric charges (millicharges),describe direct couplings of neutrinos with photons ν i ν f γ (a) γ ν k ¯ ν k (b) Figure 2
Feynman diagrams for neutrino radiative decay andCherenkov radiation (a) and plasmon decay (b). The depictedelectromagnetic interaction vertices are supposed to be effec-tive (such as the one-photon coupling in Fig. 1). which induce several observable decay processes. In thisSection we discuss the decay processes generated bythe diagrams in Fig. 2: the diagram in Fig. 2(a) gener-ates neutrino radiative decay ν i → ν f + γ and the pro-cesses of neutrino Cherenkov radiation and spin light( SL ν ) of a neutrino propagating in a medium; the dia-gram in Fig. 2(b) generates photon (plasmon) decay to anneutrino-antineutrino pair in a plasma ( γ ∗ → ν ¯ ν ).If the masses of neutrinos are nondegenerate, the ra-diative decay of a heavier neutrino ν i into a lighter neu-trino ν f (with m i > m f ) with emission of a photon, ν i → ν f + γ , (28)may proceed in vacuum [2,3,5–7,87–89]. Early discussionsof the possible role of neutrino radiative decay in differ-ent astrophysical and cosmological settings can be foundin [90–95]. The first estimates for the process of massiveneutrino decay were presented in [89]. They consideredvarious processes of neutrino decay, for instance, the de-cay into three neutrinos ν → ν + ν + ¯ ν and the radiativedecay ν → ν + γ .In [95] the possible existence of relic slow massive neu-trinos was considered. The radiative decay of the neutrinointo an ultraviolet photon and a light neutrino becomesthen an indicator of these relic particles. The first one-loop calculation of the neutrino radiative decay was per-formed in [5, 6] and yielded the decay rate as Γ = α G F π (cid:195) m − m m (cid:33) ( m + m ) (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) (cid:88) (cid:96) = e , µ , τ U ∗ (cid:96) U (cid:96) r (cid:96) (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) , (29) Copyright line will be provided by the publishereptember 17, 2018 where r (cid:96) (cid:39) m (cid:96) /2 m W ) ( (cid:96) = e , µ , τ ), m W is the mass ofW-boson, and U (cid:96) i are the mixing-matrix elements.The rate of neutrino radiative decay in relativistic andnon-relativistic media consisting of electrons (withoutmuons and taus) was calculated in Ref. [96] in the frame-work of finite-temperature quantum field theory. Thepresence of the medium prevents the Glashow-Iliopoulos-Maiani [97] suppression of the decay, which is strongly en-hanced in high-density matter (neutron star, supernova,etc.). In Ref. [98] the influence of dissipation and disper-sion in the medium, that can be important for the phe-nomenological studies of the early Universe, was takeninto account. As shown in Ref. [99], one can also calculatethe rate of neutrino radiative decay in matter avoiding theformalism of finite-temperature field theory by consider-ing the effective averaged interaction with the medium. Spin light of neutrino in matter.
The recent studiesof neutrino electromagnetic properties revealed a newmechanism of electromagnetic radiation by a neutrinopropagating in dense matter that has been proposed in[100]. This type of electromagnetic radiation was calledthe spin light of neutrino in matter ( SL ν ). In a quasi-classical treatment this radiation is due to neutrino mag-netic moment precession in dense background matter.The quantum theory of this phenomena has been devel-oped in [101–103].The SL ν is a process of photon emission in neutrinotransition between different helicity states in matter. As ithas been shown in Ref. [101,103], in the relativistic regimethe SL ν mechanism could provide up to one half of theinitial neutrino energy transition to the emitted radia-tion. It was also shown that the SL ν provides the spinpolarization effect of neutrino beam moving in matter(similarly to the well-known effect of the electron spinself-polarization in synchrotron radiation [104]).The characteristics of the SL ν depend on the compo-nents of the medium. The SL ν is radiated by a neutrinowith negative helicity while propagating in matter consist-ing of electrons. In the medium consisting of neutrons the SL ν is produced by an antineutrino with positive helicity.For the relativistic neutrinos the radiation is focusedinto a narrow cone in the direction of the initial neutrino.The radiation of ultra-relativistic neutrinos in matter hascircular polarization which in some cases (high density)reaches 100%. The average energy of the radiated pho-tons depends on the energy of the initial neutrinos andin dense medium reaches one half of the initial neutrinoenergy (see also [105, 106]).Along with studying the conventional spin light of neu-trino in matter with the mass for the initial and final neu-trino states one can consider the spin light process inneutrino transition between different mass states with masses m and m , m > m . The emitted photon is cou-pled to the neutrinos by the transition magnetic moment µ f i . To avoid cumbersome formulae, the effects of oscilla-tions were neglected and the matter with only a neutroncomponent was considered ( n n (cid:192) n e ≈ n p ). It was shown[107, 108] that the rate of SL ν in the neutrino radiative de-cay acquires additional terms that are proportional to thedifference of the initial and final neutrino masses squared: δ = m − m p . As opposed to the SL ν with the same massof the initial and final state, the process is kinematicallyopen for the quasi-vacuum case, when the density of thebackground medium is small. In addition, the expressionfor the rate of the process can be reduced to the results ofprevious neutrino radiative decay calculations. The influ-ence of external fields and matter on a massive neutrinodecay was further considered in [109] Neutrino interaction with electromagnetic fields.
Ifneutrinos have nontrivial electromagnetic properties,they can interact with classical electromagnetic fields. Sig-nificant effects can occur, in particular, in neutrino as-trophysics, since neutrinos can propagate over very longdistances in astrophysical environments with magneticfields. In this case even a very weak interaction can havelarge cumulative effects.A classical electromagnetic field produces spin andspin-flavor neutrino transitions [4, 110–114]. This kind ofinteraction can yield observable effects, for instance, inthe solar neutrino data [115–122]. The neutrino effectivemagnetic moment is also modified in very strong externalmagnetic fields [123]. It has been recently shown that dueto the nontrivial electromagnetic properties the produc-tion of neutrino-antineutrino pairs becomes possible invery strong magnetic fields [124]. It can be also importantto account for the external electromagnetic fields and forthe background matter simultaneously. In various astro-physical situations the effects of fields and matter caneither cancel or enhance each other.For instance, an approach based on the general-ized Bargmann-Michel-Telegdi equation can be used forderivation of an impact of matter motion and polarizationon the neutrino spin (and spin-flavor) evolution. Consider,as an example, an electron neutrino spin precession in thecase when neutrinos with the Standard Model interactionare propagating through moving and polarized mattercomposed of electrons (electron gas) in the presence ofan electromagnetic field given by the electromagnetic-field tensor F µν = ( E , B ). As discussed in [125] (see also[126, 127]) the evolution of the three-dimensional neu-trino spin vector S is given by d Sd t = µγ (cid:104) S × ( B + M ) (cid:105) , (30) Copyright line will be provided by the publisher . Giunti et al.: Electromagnetic neutrinos in laboratory experiments and astrophysics where the magnetic field B in the neutrino rest frameis determined by the transversal and longitudinal (withrespect to the neutrino motion) magnetic and electricfield components in the laboratory frame, B = γ (cid:179) B ⊥ + γ B ∥ + (cid:113) − γ − (cid:104) E ⊥ × ββ (cid:105)(cid:180) . (31)The matter term M in Eq. (30) is also composed of thetransversal M ∥ and longitudinal M ⊥ parts, M = M ∥ + M ⊥ , (32) M ∥ = γ β n (cid:113) − v e (cid:40) ρ (1) e (cid:195) − v e β − γ − (cid:33) − ρ (2) e ζ e β (cid:113) − v e + ( ζ e v e )( β v e )1 + (cid:113) − v e − γ − , (33) M ⊥ = − n (cid:113) − v e (cid:189) v e ⊥ (cid:179) ρ (1) e + ρ (2) e ( ζ e v e )1 + (cid:113) − v e (cid:180) + ζ e ⊥ ρ (2) e (cid:113) − v e (cid:190) . (34)Here n = n e (cid:113) − v e is the invariant number density ofmatter given in the reference frame for which the totalspeed of matter is zero. The vectors v e , and ζ e (0 (cid:201)| ζ e | (cid:201)
1) denote, respectively, the speed of the referenceframe in which the mean momentum of matter (electrons)is zero, and the mean value of the polarization vector ofthe background electrons in the above mentioned refer-ence frame. The coefficients ρ (1,2) e are calculated if theneutrino Lagrangian is given, and within the extendedStandard Model supplied with SU (2)-singlet right-handedneutrino ν R , ρ (1) e = ˜ G F (cid:112) µ , ρ (2) e = − G F (cid:112) µ , (35)where ˜ G F = G F (1 + θ W ). For the probability of theneutrino spin oscillations in the adiabatic approximationwe get from Eqs. (33) and (34) P ν L → ν R ( x ) = sin θ eff sin π xL eff , (36)sin θ eff = E E + ∆ , L eff = π (cid:113) E + ∆ , (37) where E eff = µ (cid:175)(cid:175)(cid:175) B ⊥ + γ M ⊥ (cid:175)(cid:175)(cid:175) , (38) ∆ = µγ (cid:175)(cid:175)(cid:175) M ∥ + B ∥ (cid:175)(cid:175)(cid:175) . (39)It follows that even without presence of an electromag-netic field, B ⊥ = B ∥ =
0, neutrino spin (or spin-flavor) os-cillations can be induced in the presence of matter whenthe transverse matter term M ⊥ is not zero. This possibil-ity is realized in the case of nonzero transversal mattervelocity or polarization. A detailed discussion of this phe-nomenon can be found in [125, 128]. Effective magnetic moment.
In scattering experimentsthe neutrino is created at some distance from the de-tector as a flavor neutrino, which is a superposition ofmassive neutrinos. Therefore, the magnetic moment thatis measured in these experiments is not that of a singlemassive neutrino, but it is an effective magnetic momentwhich takes into account neutrino mixing and the oscilla-tions during the propagation between source and detector[68, 129]. In the following, when we refer to an effectivemagnetic moment of a flavor neutrino without indicationof a source-detector distance L it is implicitly understoodthat L is small, such that the effective magnetic moment isindependent of the neutrino energy and from the source-detector distance. In such a case, the effective magneticmoment is given by [38] µ ν (cid:96) (cid:39) µ ν (cid:96) (cid:39) (cid:88) f = (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) (cid:88) i = U ∗ (cid:96) i (cid:161) µ f i − i (cid:178) f i (cid:162)(cid:175)(cid:175)(cid:175)(cid:175)(cid:175) . (40)Another situation where the effective magnetic mo-ment does not depend on the neutrino energy and onthe source-detector distance is when the source-detectordistance is much larger than all the oscillation lengths L f i = π E ν / | ∆ m f i | . The effective magnetic moment inthis case is evaluated as [38] µ ν (cid:96) (cid:39) µ ν (cid:96) (cid:39) (cid:88) i = | U (cid:96) i | (cid:88) f = (cid:175)(cid:175) µ f i − i (cid:178) f i (cid:175)(cid:175) . (41)Note that in the case of solar neutrinos, which have beenused by the Super-Kamiokande [130] and Borexino [131]experiments to search for neutrino magnetic moments, Copyright line will be provided by the publishereptember 17, 2018 one must take into account the matter effects. The lattercan be done by replacing the neutrino mixing matrix inEq. (41) with the effective mixing matrix in matter at thepoint of neutrino production inside the Sun (see [38] andreferences therein).It is also interesting to note that flavor neutrinos canhave effective magnetic moments even if massive neu-trinos are Majorana particles. In this case, since massiveMajorana neutrinos do not have diagonal magnetic andelectric dipole moments, the effective magnetic momentsof flavor neutrinos receive contributions only from thetransition dipole moments.The constraints on the neutrino magnetic momentsin direct laboratory experiments have been obtained sofar from the lack of any observable distortion of the recoilelectron energy spectrum. Experiments of this type havestarted in the 50’s at the Savannah River Laboratory wherethe ¯ ν e - e − elastic scattering process was studied [65, 66,140] with somewhat controversial results, as discussedby [67]. The most significant experimental limits on theeffective magnetic moment µ ν e which have been obtainedin measurements of reactor ¯ ν e - e − elastic scattering afterabout 1990 are listed in Tab. 1 (some details of the differentexperimental setups are reviewed in [77]).An attempt to improve the experimental bound on µ ν e in reactor experiments was undertaken in [141], where itwas suggested that in ¯ ν e interactions on an atomic targetthe atomic electron binding (“atomic-ionization effect”)can significantly increase the electromagnetic contribu-tion to the differential cross section with respect to thefree-electron approximation. However, the dipole approx-imation used to derive the atomic-ionization effect is notvalid for the electron antineutrino cross section in reactorneutrino magnetic moment experiments. Instead, the freeelectron approximation is appropriate for the interpreta-tion of the data of reactor neutrino experiments and thecurrent constraints in Tab. 1 cannot be improved by con-sidering the atomic electron binding [69–73,142,143]. Thehistory and present status of the theory of neutrino-atomcollisions is reviewed in [144].The current best limit on µ ν e has been obtained in2012 in the GEMMA experiment at the Kalinin NuclearPower Plant (Russia) with a 1.5 kg highly pure germaniumdetector exposed at a ¯ ν e flux of 2.7 × cm − s − at adistance of 13.9 m from the core of a 3 GW th commercialwater-moderated reactor [50]. The competitive TEXONOexperiment is based at the Kuo-Sheng Reactor NeutrinoLaboratory (Taiwan), where a 1.06 kg highly pure germa-nium detector was exposed to the flux of ¯ ν e at a distance of 28 m from the core of a 2.9 GW th commercial reactor[135] .Searches for effects of neutrino magnetic momentshave been performed also in accelerator experiments. TheLAMPF bounds on µ ν e in Tab. 1 have been obtained with ν e from µ + decay [136]. The LAMPF and LSND bounds on µ ν µ in Tab. 1 have been obtained with ν µ and ¯ ν µ from π + and µ + decay [136, 138]. The DONUT collaboration [139]investigated ν τ - e − and ¯ ν τ - e − elastic scattering, findingthe limit on µ ν τ in Tab. 1.Solar neutrino experiments can also search for a neu-trino magnetic moment signal by studying the shape ofthe electron spectrum [129]. Table 1 gives the limits ob-tained in the Super-Kamiokande experiment [130] for µ ( E ν (cid:38) (cid:39) cos ϑ
13 3 (cid:88) i = (cid:175)(cid:175) µ i − i (cid:178) i (cid:175)(cid:175) + sin ϑ
13 3 (cid:88) i = (cid:175)(cid:175) µ i − i (cid:178) i (cid:175)(cid:175) , (42)where ϑ is the mixing angle, and that obtained in theBorexino experiment [131] for µ S ( E ν (cid:46) (cid:39) µ ν e , (43)where µ ν e is given by Eq. (41).Information on neutrino magnetic moments has beenobtained also with global fits of solar neutrino data [147–149]. Considering Majorana three-neutrino mixing, theauthors of [149] obtained, at 90% CL, (cid:113) | µ | + | µ | + | µ | < × − µ B , (44)from the analysis of solar and KamLAND, and (cid:113) | µ | + | µ | + | µ | < × − µ B , (45)adding the Rovno [133], TEXONO [150] and MUNU [151]constraints.The neutrino magnetic moment contribution to the(anti)neutrino-electron elastic scattering process flips theneutrino helicity. If neutrinos are Dirac particles, this pro-cess transforms active left-handed neutrinos into sterileright-handed neutrinos, leading to dramatic effects on theexplosion of a core-collapse supernova [152–161], wherethere are also contributions from the (anti)neutrino-proton and (anti)neutrino-neutron elastic scattering. Re-quiring that the entire energy in a supernova collapse Copyright line will be provided by the publisher . Giunti et al.: Electromagnetic neutrinos in laboratory experiments and astrophysics Method Experiment Limit CL ReferenceReactor ¯ ν e - e − Krasnoyarsk µ ν e < × − µ B
90% [132]Rovno µ ν e < × − µ B
95% [133]MUNU µ ν e < × − µ B
90% [134]TEXONO µ ν e < × − µ B
90% [135]GEMMA µ ν e < × − µ B
90% [50]Accelerator ν e - e − LAMPF µ ν e < × − µ B
90% [136]Accelerator ( ν µ , ¯ ν µ )- e − BNL-E734 µ ν µ < × − µ B
90% [137]LAMPF µ ν µ < × − µ B
90% [136]LSND µ ν µ < × − µ B
90% [138]Accelerator ( ν τ , ¯ ν τ )- e − DONUT µ ν τ < × − µ B
90% [139]Solar ν e - e − Super-Kamiokande µ S ( E ν (cid:38) < × − µ B
90% [130]Borexino µ S ( E ν (cid:46) < × − µ B
90% [131]
Table 1
Experimental limits for different neutrino effective magnetic moments. is not carried away by the escaping sterile right-handedneutrinos created in the supernova core, the authors of[159, 160] obtained the following upper limit on a genericneutrino magnetic moment: µ ν (cid:46) (0.1 − × − µ B , (46)which is slightly more stringent than the bound µ ν (cid:46) (0.2 − × − µ B obtained in [156].There is a gap of many orders of magnitude betweenthe present experimental limits on neutrino magnetic mo-ments of the order of 10 − µ B and the prediction smallerthan about 10 − µ B in Eq. (8) of the minimal extensionof the Standard Model with right-handed neutrinos. Thehope to reach in the near future an experimental sensi-tivity of this order of magnitude is very small, taking intoaccount that the experimental sensitivity of reactor ¯ ν e - e elastic scattering experiments has improved by only oneorder of magnitude during a period of about twenty years(see [67], where a sensitivity of the order of 10 − µ B is dis-cussed). However, the experimental studies of neutrinomagnetic moments are stimulated by the hope that newphysics beyond the minimally extended Standard Modelwith right-handed neutrinos might give much strongercontributions. Neutrino millicharge.
The most severe experimentalconstraint on neutrino electric charges is that on the effec-tive electron neutrino charge e ν e , which can be obtainedfrom electric charge conservation in neutron beta decay n → p + e − + ¯ ν e , from the experimental limits on the non-neutrality of matter which constrain the sum of the proton and electron charges, e p + e e , and from the experimentallimits on the neutron charge e n [162, 163]. Several experi-ments which measured the neutrality of matter give theirresults in terms of e mat = Z ( e p + e e ) + Ne n A , (47)where A = Z + N is the atomic mass of the substanceunder study, Z is its atomic number and N is its neutronnumber. From electric charge conservation in neutronbeta decay, we have e ν e = e n − ( e p + e e ) = AZ ( e n − e mat ) . (48)The best recent bound on the non-neutrality of matter[164], e mat = ( − ± × − e , (49)has been obtained with SF , which has A = Z =
70. Using the independent measurement of the charge ofthe free neutron [165] e n = ( − ± × − e , (50)we obtain e ν e = ( − ± × − e . (51)This value is compatible with the neutrality of matter limitin Tab. 2, which has been derived [162,163] from the valueof e n in Eq. (50) and e mat = (0.8 ± × − e [166]. Copyright line will be provided by the publishereptember 17, 2018
It is also interesting that the effective charge of ¯ ν e canbe constrained by the SN 1987A neutrino measurementstaking into account that galactic and extragalactic mag-netic field can lengthen the path of millicharged neutrinosand requiring that neutrinos with different energies ar-rive on Earth within the observed time interval of a fewseconds [167]: | e ν e | (cid:46) × − ( E ν /10 MeV)( d /10 kpc)( B /1 µ G) (cid:115) ∆ t / t ∆ E ν / E ν , (52)considering a magnetic field B acting over a distance d and the corresponding time t = d / c . E ν ≈
15 MeV is theaverage neutrino energy, ∆ E ν ≈ E ν /2 is the energy spread,and ∆ t ≈ B ≈ − µ G acting over thewhole path d (cid:39)
50 kpc, which corresponds to t (cid:39) × s, gives | e ν e | (cid:46) × − e . (53)2. A galactic field B ≈ µ G acting over a distance d (cid:39)
10 kpc, which corresponds to t (cid:39) × s, gives | e ν e | (cid:46) × − e . (54)The last two limits in Tab. 2 have been obtained [170,171] considering the results of reactor neutrino magneticmoment experiments. The differential cross section of the¯ ν e – e − elastic scattering process due to a neutrino effectivecharge e ν e is given by (see [172]) (cid:181) d σ dT e (cid:182) charge (cid:39) πα m e T e e ν e . (55)In reactor experiments the neutrino magnetic moment issearched by considering data with T e (cid:191) E ν . The ratio ofthe charge cross section (55) and the magnetic momentcross section in Eq. (20), for which we consider only thedominant part proportional to 1/ T e , is given by R = ( d σ / dT e ) charge ( d σ / dT e ) mag (cid:39) m e T e (cid:161) e ν e / e (cid:162) (cid:161) µ ν e / µ B (cid:162) (56)Considering an experiment which does not observe anyeffect of µ ν e and obtains a limit on µ ν e , it is possible toobtain, following [171], a bound on e ν e by demanding thatthe effect of e ν e is smaller than that of µ ν e , i.e. that R (cid:46) e ν e (cid:46) T e m e (cid:181) µ ν e µ B (cid:182) e . (57)The last limit in Tab. 2 has been obtained from the 2012results [50] of the GEMMA experiment, considering T e atthe experimental threshold of 2.8 keV. Let us finally note that a strong limit on a generic neu-trino electric charge e ν can be obtained by consideringthe influence of millicharged neutrinos on the rotation ofa magnetized star which is undergoing a core-collapse su-pernova explosion (the neutrino star turning mechanism, ν ST) [173]. During the supernova explosion, the escapingmillicharged neutrinos move along curved orbits insidethe rotating magnetized star and slow down the rotationof the star. This mechanism could prevent the generationof a rapidly rotating pulsar in the supernova explosion. Im-posing that the frequency shift of a forming pulsar due tothe neutrino star turning mechanism is less than a typicalobserved frequency of 0.1 s − and assuming a magneticfield of the order of 10 G, the author of [173] obtained | e ν | (cid:46) × − e . (58)Note that this limit is much stronger than the astrophysi-cal limits in Tab. 2. Neutrino charge radius.
The neutrino charge radiushas an effect in the scattering of neutrinos with chargedparticles. The most useful process is the elastic scat-tering with electrons. Since in the ultrarelativistic limitthe charge form factor conserves the neutrino helicity, aneutrino charge radius contributes to the weak interac-tion cross section ( d σ / dT e ) SM of ν (cid:96) – e − elastic scatteringthrough the following shift of the vector coupling constant g ν (cid:96) V [42, 67, 177, 178]: g ν (cid:96) V → g ν (cid:96) V + m W 〈 r ν (cid:96) 〉 sin θ W . (59)Using this method, experiments which measure neutrino-electron elastic scattering can probe the neutrino chargeradius. Some experimental results are listed in Tab. 3. Inaddition, the authors of [176] obtained the following 90%CL bounds on 〈 r ν µ 〉 from a reanalysis of CHARM-II [175]and CCFR [179] data: − × − < 〈 r ν µ 〉 < × − cm . (60)More recently, the authors of [180] obtained the follow-ing 90% CL bounds on 〈 r ν e 〉 from a combined fit of allavailable ν e – e − and ¯ ν e – e − data: − × − < 〈 r ν e 〉 < × − cm . (61)The single photon production process e + + e − → ν + ¯ ν + γ has been used to get bounds on the effective ν τ chargeradius, assuming a negligible contribution of the ν e and ν µ charge radii [176, 181, 182]. For Dirac neutrinos, theauthors of [176] obtained − × − < 〈 r ν τ 〉 < × − cm . (62) Copyright line will be provided by the publisher . Giunti et al.: Electromagnetic neutrinos in laboratory experiments and astrophysics Limit Method Reference | e ν τ | (cid:46) × − e SLAC e − beam dump [168] | e ν τ | (cid:46) × − e BEBC beam dump [169] | e ν | (cid:46) × − e Solar cooling (plasmon decay) [163] | e ν | (cid:46) × − e Red giant cooling (plasmon decay) [163] | e ν e | (cid:46) × − e Neutrality of matter [163] | e ν e | (cid:46) × − e Nuclear reactor [170] | e ν e | (cid:46) × − e Nuclear reactor [171]
Table 2
Approximate limits for different neutrino effective charges. The limits on e ν apply to all flavors. Method Experiment Limit [cm ] CL ReferenceReactor ¯ ν e - e − Krasnoyarsk |〈 r ν e 〉| < × −
90% [132]TEXONO − × − < 〈 r ν e 〉 < × −
90% [174]Accelerator ν e - e − LAMPF − × − < 〈 r ν e 〉 < × −
90% [136]LSND − × − < 〈 r ν e 〉 < × −
90% [138]Accelerator ν µ - e − BNL-E734 − × − < 〈 r ν µ 〉 < × −
90% [137]CHARM-II |〈 r ν µ 〉| < × −
90% [175]
Table 3
Experimental limits for the electron neutrino charge radius. In the TEXONO, LAMPF, LSND, BNL-E734, and CHARM-IIcases, the published limits are half, because they use a convention which differs by a factor of 2 (see also Ref. [176]).
Comparing the theoretical Standard Model values withthe experimental limits in Tab. 3 and those in Eqs. (60)–(62), one can see that they differ at most by one orderof magnitude. Therefore, one may expect that the exper-imental accuracy will soon reach the value needed toprobe the Standard Model predictions for the neutrinocharge radii. This will be an important test of the Stan-dard Model calculation of the neutrino charge radii. If theexperimental value of a neutrino charge radius is foundto be different from the Standard Model prediction it willbe necessary to clarify the precision of the theoretical cal-culation in order to understand if the difference is due tonew physics beyond the Standard Model.The neutrino charge radius has also some impact onastrophysical phenomena and on cosmology. The limitson the cooling of the Sun and white dwarfs due to theplasmon decay process discussed in the previous Sectioninduced by a neutrino charge radius led the authors of[183] to estimate the respective limits |〈 r ν 〉| (cid:46) − cm and |〈 r ν 〉| (cid:46) − cm for all neutrino flavors. From the cooling of red giants the authors of [181] inferred the limit |〈 r ν 〉| (cid:46) × − cm .If neutrinos are Dirac particles, e + – e − annihilationscan produce right-handed neutrino-antineutrino pairsthrough the coupling induced by a neutrino charge radius.This process would affect primordial Big-Bang Nucleosyn-thesis and the energy release of a core-collapse supernova.From the measured He yield in primordial Big-Bang Nu-cleosynthesis the authors of [184] obtained |〈 r ν 〉| (cid:46) × − cm , (63)and from SN 1987A data the authors of [185] obtained 〈 r ν 〉 (cid:46) × − cm , (64)for all neutrino flavors. Copyright line will be provided by the publishereptember 17, 2018
Solar neutrinos.
The precision measurements of low-energy neutrinos from the Sun in the ongoing and forth-coming solar neutrino experiments will not only pro-vide us with more accurate values of neutrino oscilla-tion parameters [186], but also offer a precious oppor-tunity to test the Mikheyev-Smirnov-Wolfenstein (MSW)matter effect [187, 188] and to probe the solar properties,such as the core metallicity (by measuring the CNO neu-trino flux) and the total luminosity (through determiningthe pp neutrino flux). Furthermore, the observations ofsolar neutrinos in the future water-Cherenkov detectorHyper-Kamiokande [189] and liquid-scintillator detectorsSNO+ [190], JUNO [191], RENO50 [192] and LENA [193]will greatly improve current knowledge about the electro-magnetic properties of neutrinos.Besides reactor antineutrino experiments, the neutrino-electron elastic scattering of low-energy solar neutrinoscan also be used to measure the neutrino magnetic prop-erties. The contribution of the neutrino magnetic dipolemoment to the elastic ν e - e − cross section becomes morepredominant as the electron kinetic energy T e decreasessince it is inversely proportional to T e at low energy. There-fore, the measurements of solar neutrinos in the B [130], Be [131] and pp [194] processes may provide us excellentopportunities to constrain the neutrino magnetic dipolemoment. As already reported in Table 1, the current upperlimits at 90% C.L. obtained from the measurements of Band Be neutrinos are 1.1 × − µ B [130] and 5.4 × − µ B [131], respectively. Note that these are limits on effectivemagnetic moments which are different combinations ofthe magnetic dipole moments of massive neutrinos, asdiscussed at the beginning of Section 4.In future, large liquid-scintillator detectors will im-prove the precision of low-energy solar neutrino measure-ments, and can give better limits on the magnetic dipolemoment. There will be a liquid-scintillator detector witha 20 kiloton target mass and a high energy resolution of3%/ (cid:112) E /MeV at JUNO, and the LENA detector will be 2.5times larger. In consequence, JUNO [191] (LENA [193])will register about four thousand (ten thousand) Be elas-tic ν e - e − events per day in its detectable window above250 keV, which means that the statistical uncertaintiescan be negligible after years of data-taking. Therefore, theachievable limit on the neutrino magnetic dipole momentmainly depends on the systematics, and in particular onthe radioactive and cosmogenic backgrounds.Another interesting solar neutrino process due to neu-trino magnetic properties is the spin-flavor precession mechanism [38]. As discussed in Section 3, besides thestandard MSW resonant transition, there might be in-teresting transitions between the left-handed and right-handed components of solar neutrinos in the presenceof the solar magnetic field. In the case of Dirac neutrinos,the additional transition happens between the active andsterile neutrino states and can be a sub-leading effect inneutrino oscillation probabilities. More interestingly, inthe case of Majorana neutrinos, right-handed neutrinosof the electron flavor produced in the spin-flavor preces-sion can be detected with the inverse beta decay reaction,which can significantly reduce the singles background us-ing the coincidence of prompt and delayed signals of thereaction. The recent measurement from Borexino [195]constrains the transition probability to be smaller than1.3 × − (90% C.L.), which corresponds to an upper limitof 10 − µ B − − µ B for the neutrino magnetic dipolemoment. Future liquid-scintillator detectors (e.g. JUNO,RENO50 and LENA) are 1-2 orders of magnitude largerthan Borexino and may improve the transition probabilitylimits by one order of magnitude. This observation couldbe free of the reactor antineutrino background when oneconcentrates on the energy region larger than 10 MeV. Supernova neutrinos.
As is well known, the electro-magnetic dipole interaction of massive neutrinos couplesleft-handed neutrinos to the right-handed ones. If neu-trinos are Dirac particles, right-handed neutrino statesare sterile and can be copiously produced in the super-nova core, where large magnetic fields may exist. Whilethe left-handed neutrinos are trapped inside the super-nova core and come out by diffusion, the sterile ones canfreely escape from the core immediately after production.Since the energy loss caused by right-handed neutrinosshould not shorten significantly the duration of the neu-trino signal, which has been observed by the Kamiokande-II, IMB and Baksan experiments to be about ten seconds,one can obtain the restrictive limit on the neutrino mag-netic dipole moment µ ν (cid:46) × − µ B [163]. However,this bound applies only to massive Dirac neutrinos, sincethe right-handed states of Majorana neutrinos interactas Standard Model antineutrinos and do not induce anyextra energy loss because they are trapped in the core.Although it was pointed out long time ago that theneutrino-neutrino refraction in the supernova environ-ment may be very important for neutrino flavor conver-sions, the nonlinear evolution of neutrino flavors has re-cently been found to dramatically change neutrino energyspectra [196]. Depending on the initial neutrino fluxes andenergy spectra, a complete swap between neutrino spec-tra of electron and non-electron flavors can take placein the whole or a finite energy range, as a direct conse-quence of collective neutrino oscillations. The impact of Copyright line will be provided by the publisher . Giunti et al.: Electromagnetic neutrinos in laboratory experiments and astrophysics nonzero transition magnetic moments for massive Ma-jorana neutrinos on collective neutrino oscillations hasbeen explored in Ref. [197, 198]. For a magnetic field of10 G and µ ν ≈ − µ B , which is just two orders of mag-nitude larger than the Standard-Model prediction corre-sponding to neutrino masses of the order of 0.1 eV, thepattern of spectral splits of supernova neutrinos may beobserved in future experiments.For a future galactic supernova, a number of largewater-Cherenkov (Super-Kamiokande [199] and Hyper-Kamiokande), scintillator (JUNO, RENO50 and LENA) andliquid-argon (DUNE [200]) detectors will be able to per-form a high-statistics measurement of galactic supernovaneutrinos. In the case of a galactic supernova at a typicaldistance of 10 kpc, the JUNO detector will record about5000 inverse beta-decay events, implying a precise deter-mination of ν e energy spectrum. In addition, the charged-current interaction ν e + C → e − + N contributes to afew hundred events, which together with the elastic ν e - e − scattering leads to a possible measurement of ν e energyspectrum. Finally, the number of elastic neutrino-protonscattering events reaches two thousand, since JUNO isexpected to achieve a threshold around 0.1 or 0.2 MeVfor the proton recoil energy. Combining these measure-ments with the information from the water-Cherenkovand liquid-argon detectors, we hope to pin down the neu-trino energy spectra with reasonable accuracy. The identi-fication of the spectral splits will allow us to probe valuesof the neutrino magnetic moments which are extremelysmall and impossible to detect in other terrestrial exper-iments. Unfortunately, the experimental determinationof the neutrino magnetic moments will be complicatedby the distorsions of the neutrino spectra induced by theordinary Mikheyev-Smirnov-Wolfenstein effects in thesupernova envelope and by the Earth matter effects. Cosmological observations.
The early Universe is an-other place where neutrinos can be in thermal equilib-rium and play a very important role. The phase transi-tions in the early Universe can have generated primor-dial magnetic fields, which populated the right-handedneutrinos if neutrinos are Dirac particles and have finitemagnetic dipole moments. If the magnetic dipole interac-tion rate of neutrinos is larger than the Hubble expansionrate during the epoch of primordial nucleosynthesis, theright-handed neutrinos are in thermal equilibrium andcontribute to the effective number of neutrino speciesby ∆ N ν =
3, which will modify the correct predictionsof the standard BBN theory for the abundance of lightnuclear elements. As shown in Refs. [201,202], the require-ment for the magnetic dipole interaction rate to be smallerthan the Hubble expansion rate at T =
200 MeV, when theQCD phase transition occurs, leads to an upper bound on the neutrino magnetic dipole moment. For a primor-dial magnetic field B = − G and the size of magneticfield domain λ = µ ν < − µ B , which is several orders of magnitude belowcurrent experimental limits [202]. In this review we outlined some aspects of the physics ofelectromagnetic neutrinos. No experimental evidence infavor of neutrino electromagnetic interactions has beenobtained so far. All the neutrino electromagnetic charac-teristics have rather stringent upper bounds, which aredue to laboratory experiments or from astrophysical ob-servations.The most accessible neutrino electromagnetic prop-erty may be the charge radius, for which the StandardModel gives a value which is only about one order of mag-nitude smaller than the experimental upper bounds. Ameasurement of a neutrino charge radius at the level pre-dicted by the Standard Model would be another spectacu-lar confirmation of the Standard Model, after the recentdiscovery of the Higgs boson (see [203]). However, such ameasurement would not give information on new physicsbeyond the Standard Model unless the measured value isshown to be incompatible with the Standard Model valuein a high-precision experiment.The strongest current efforts to probe the physics be-yond the Standard Model by measuring neutrino electro-magnetic properties is the search for a neutrino magneticmoment effect in reactor ¯ ν e - e − scattering experiments.The current upper bounds reviewed in Section 4 are morethan eight orders of magnitude larger than the predictiondiscussed in Section 2 of the Dirac neutrino magnetic mo-ments in the minimal extension of the Standard Modelwith right-handed neutrinos. Hence, a discovery of a neu-trino magnetic moment effect in reactor ¯ ν e - e − scatteringexperiments would be a very exciting evidence of non-minimal new physics beyond the Standard Model.In particular, the GEMMA-II collaboration expectsto reach around the year 2017 a sensitivity to µ ν e ≈ × − µ B in a new series of measurements at the KalininNuclear Power Plant with a doubled neutrino flux ob-tained by reducing the distance between the reactor andthe detector from 13.9 m to 10 m and by lowering theenergy threshold from 2.8 keV to 1.5 keV [50,204]. The cor-responding sensitivity to the neutrino electric millichargewill reach the level of | e ν e | ≈ × − e [171]. Copyright line will be provided by the publishereptember 17, 2018
There is also a GEMMA-III project to further lowerthe energy threshold to about 350 eV, which may allow theexperimental collaboration to reach a sensitivity of µ ν e ≈ × − µ B . The corresponding sensitivity to neutrinomillicharge will be | e ν e | ≈ × − e [171].An interesting possibility for exploring very small val-ues of µ ν e in ¯ ν e - e − scattering experiments has been pro-posed in Ref. [205] on the basis of the observation [206]that “dynamical zeros” induced by a destructive interfer-ence between the left-handed and right-handed chiralcouplings of the electron in the charged and neutral cur-rent amplitudes appear in the Standard Model contribu-tion to the scattering cross section. It may be possible toenhance the sensitivity of an experiment to µ ν e by select-ing recoil electrons contained in a forward narrow conecorresponding to a dynamical zero (see Eq. (14)).In the future, experimental searches of neutrino elec-tromagnetic properties may be performed also with newneutrino sources, as a tritium source [207], a low-energybeta-beam [207, 208], a stopped-pion neutrino source[83], or a neutrino factory [208]. Recently the authors ofRef. [209] proposed to improve the existing limit on theelectron neutrino magnetic moment with a megacurie Cr neutrino source and a large liquid Xe detector.Neutrino electromagnetic interactions could have im-portant effects in astrophysical environments and in theevolution of the Universe and the current rapid advancesof astrophysical and cosmological observations may leadsoon to the very exciting discovery of nonstandard neu-trino electromagnetic properties. In particular, futurehigh-precision observations of supernova neutrino fluxesmay reveal the effects of collective spin-flavor oscillationsdue to Majorana transition magnetic moments as smallas 10 − µ B [197, 198].Let us finally emphasize the importance to pursue theexperimental and theoretical studies of electromagneticneutrinos, which could open a door to new physics be-yond the Standard Model. Acknowledgements.
This work was supported in part by thejoint project of the Russian Foundation for Basic Research(RFBR) under Grant No. 15-52-53112 and National Natu-ral Science Foundation of China (NSFC) under Grant No.11511130016. The work of C. Giunti was partially supportedby the PRIN 2012 research grant 2012CPPYP7. The work ofK. A. Kouzakov, A. V. Lokhov, and A. I. Studenikin was alsosupported in part by the RFBR under Grant No. 14-22-03043- ofi-m, and that of Yu-Feng Li and Shun Zhou by the NSFCunder Grant Nos. 11135009, 11305193, by the Innovation Pro-gram of the Institute of High Energy Physics under Grant No.Y4515570U1, and by the CAS Center for Excellence in ParticlePhysics (CCEPP). K. A. Kouzakov also acknowledges supportfrom the RFBR under Grant No. 14-01-00420-a.
Key words.
Neutrino, Beyond Standard Model, Neutrino elec-tromagnetic properties.
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