After a brief discussion on neutrino electromagnetic properties, we consider
the problem of neutrino energy spectra in different media. It is shown that in
two particular cases (i.e., neutrino propagation in a) transversally moving
with increasing speed medium and b) rotating medium) neutrino energies are
quantized. These phenomena can be important for astrophysical applications, for
instance, for physics of rotating neutron stars.Comment: 7 pagex in LaTex, to appear in Proceedings of the XXIII Recontres de
  Physique de la Vallee D'Aoste on "Results and Perspectives in Particle
  Physics" (La Thuile, Italy, March 1-7, 2009

Balantsev, I.

Popov, Yu.

Studenikin, A.

English

arXiv

After a brief discussion on neutrino electromagnetic properties, we consider the problem of neutrino energy spectra in different media. It is shown that in two particular cases (i.e., neutrino propagation in a) transversally moving with increasing speed medium and b) rotating medium) neutrino energies are quantized. These phenomena can be important for astrophysical applications, for instance, for physics of rotating neutron stars

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DOI 10.1393/ncc/i2010-10539-7Colloquia: LaThuile09IL NUOVO CIMENTO Vol. 32 C, N. 5-6 Settembre-Dicembre 2009Neutrino magnetic moment and neutrino energy quantization inrotating mediaI. Balantsev(1), Yu. Popov(2) and A. Studenikin(1)(1) Department of Theoretical Physics, Moscow State University - Moscow, Russia(2) Skobeltsyn Institute of Nuclear Physics, Moscow State University - Moscow, Russia(ricevuto il 10 Novembre 2009; pubblicato online il 18 Gennaio 2010)Summary. — After a brief discussion on neutrino electromagnetic properties, weconsider the problem of neutrino energy spectra in different media. It is shown thatin two particular cases (i.e., neutrino propagation in a) transversally moving withincreasing speed medium and b) rotating medium) neutrino energies are quantized.These phenomena can be important for astrophysical applications, for instance, forphysics of rotating neutron stars.PACS 12.20.-m – Quantum electrodynamics.PACS 13.15.+g – Neutrino interactions.1. – Introduction—neutrino electromagnetic propertiesInitially the problem considered in this paper has originated from the studies of neu-trino electromagnetic properties and related items. There is no doubt that the recentexperimental and theoretical studies of flavour conversion in solar, atmospheric, reactorand accelerator neutrino fluxes give strong evidence of nonzero neutrino mass. A massiveneutrino can have nontrivial electromagnetic properties [1]. A recent review on neutrinoelectromagnetic properties can be found in [2]. The present situation in the domain ischaracterized by the fact that in spite of reasonable efforts in studies of neutrino elec-tromagnetic properties, there is no any experimental confirmation in favour of neutrinoelectromagnetic characteristics being nonvanishing. However, it is very plausible to as-sume that a neutrino may have nonzero electromagnetic properties. In particular, itseems very reasonable that a neutrino has a nonvanishing magnetic moment [2, 3].Neutrino magnetic moment interaction effects. If a neutrino has nontrivial electro-magnetic properties, notably nonvanishing magnetic (or electric) transition dipole mo-ments or nonzero millicharge and charge radius, then a direct neutrino coupling to pho-tons becomes possible and several processes exist important for applications [4]. A setof typical and most important neutrino electromagnetic processes involving the directneutrino couplings with photons is: 1) a neutrino radiative decay ν1 → ν2 + γ, neutrinoc© Societa` Italiana di Fisica / INFN - Laboratori Nazionali di Frascati 5354 I. BALANTSEV, YU. POPOV and A. STUDENIKINCherenkov radiation in external environment (plasma and/or electromagnetic fields), 2)photon (plasmon) decay to a neutrino-antineutrino pair in plasma γ → νν¯, 3) neutrinoscattering off electrons (or nuclei), 4) neutrino spin (spin-flavor) precession in magneticfield. Another very important phenomenon is the resonant amplification of the neutrinospin-flavour oscillations in matter that was first considered in [5].Note a new mechanism of electromagnetic radiation produced by a neutrino movingin matter and originated due to neutrino magnetic moment [6-8]. It was termed thespin light of neutrino in matter (SLν) [6]. Although the SLν was considered first withinquasiclassical approach, it was clear that this is a quantum phenomenon by its nature.The quantum theory of this radiation has been elaborated [7] (see also [8]) within devel-opment [9, 10] of a quite powerful method that implies the use of the exact solutions ofthe modified Dirac equation for the neutrino wave function in matter. For elaborationof the quantum theory of the SLν one has to find the solution of the quantum equationfor the neutrino wave function and for the neutrino energy spectrum in medium.2. – Quantum equation for neutrino in mediumThe modified Dirac equation for the neutrino wave function exactly accounting forthe neutrino interaction with matter [7]:(1){iγμ∂μ − 12γμ(1 + γ5)fμ −m}Ψ(x) = 0.This is the most general form of the equation for the neutrino wave function in whichthe effective potential Vμ = 12 (1 + γ5)fμ includes both the neutral and charged currentinteractions of neutrino with the background particles and which can also account for ef-fects of matter motion and polarization. It should be mentioned that other modificationsof the Dirac equation were previously used in [11] for studies of the neutrino dispersionrelations, neutrino mass generation and neutrino oscillations in the presence of matter.In the case of matter composed of electrons, neutrons, and protons and for the neu-trino interaction with background particles given by the standard model supplied withthe singlet right-handed neutrino one has(2) fμ =√2GF∑f=e,p,njμf q(1)f + λμf q(2)f ,whereq(1)f =(I(f)3L − 2Q(f) sin2 θW + δef), q(2)f = −(I(f)3L + δef),(3)δef ={1 for f = e,0 for f = n, p.Here I(f)3L and Q(f) are, respectively, the values of the isospin third components and theelectric charges of matter particles (f = e, n, p). The corresponding currents jμf andNEUTRINO MAGNETIC MOMENT AND NEUTRINO ENERGY QUANTIZATION ETC. 55polarization vectors λμf are(4) jμf = (nf , nfvf ), λμf =⎛⎝nf (ζfvf ), nfζf√1− v2f + nfvf (ζfvf )1 +√1− v2f⎞⎠ ,where θW is the Weinberg angle. In the above formulas (4), nf , vf and ζf (0 ≤ |ζf |2 ≤ 1)stand, respectively, for the invariant number densities, average speeds and polarizationvectors of the matter components.In the case of matter at rest it is possible to solve the modified Dirac equation fordifferent types of neutrinos moving in matter of different composition, as is shown in [7].The energy spectrum of different neutrinos moving in matter is given by(5) Eε = εη√p2(1− sαmp)2+ m2 + αm.In the general case of matter composed of electrons, neutrons and protons the matterdensity parameter α for different neutrino species is(6) ανe,νμ,ντ =12√2GFm(ne(4 sin2 θW + ) + np(1− 4 sin2 θW )− nn),where  = 1 for the electron neutrino and  = −1 for the muon and tau neutrinos.The value η = sign(1− sαmp ) in (5) provides a proper behavior of the wave functionin the hypothetical massless case. The values s = ±1 specify the two neutrino helicitystates, ν+ and ν−. The quantity ε = ±1 splits the solutions into the two branchesthat in the limit of the vanishing matter density, α → 0, reproduce the positive- andnegative-frequency solutions, respectively.In the next two sections we apply the developed method of exact solutions to twoparticular cases when neutrino is propagating a medium transversally moving with in-creasing speed [12, 9] and in a rotating medium of constant density. In both cases theobtained energy spectrum of the neutrino is quantized like the energy spectrum of anelectron is quantized in a constant magnetic field.3. – Neutrino quantum states in a medium transversally moving withincreasing speedFirst we consider a neutrino propagating in a medium composed of neutrons thatmove perpendicular to the neutrino path with linearly increasing speed. This can beregarded as the first approach to modelling of neutrino propagation inside a rotatingneutron star [12, 9]. The corresponding modified Dirac equation for the neutrino wavefunction is given by (1) with the matter potential accounting for rotation,(7) fμ = −G(n, nv), v = (ωy, 0, 0),where G = GF√2. Here ω is the angular frequency of matter rotation around the OZ axis,it also is supposed that the neutrino propagates along the OY axis. For the neutrino56 I. BALANTSEV, YU. POPOV and A. STUDENIKINwave function components ψ(x) we get from the modified Dirac equation (1), a set ofequations(1),[i(∂0 − ∂3) + Gn]ψ1 + [−(i∂1 + ∂2) + Gnωy]ψ2 = mψ3,(8)[(−i∂1 + ∂2) + Gnωy]ψ1 + [i(∂0 + ∂3) + Gn]ψ2 = mψ4,i(∂0 + ∂3)ψ3 + (i∂1 + ∂2)ψ4 = mψ1,(i∂1 − ∂2)ψ3 + i(∂0 − ∂3)ψ4 = mψ2.In a general case, it is not a trivial task to find solutions of this set of equations.The problem is reasonably simplified in the limit of a very small neutrino mass, i.e.when the neutrino mass can be ignored in the left-hand side of (8) with respect to thekinetic and interaction terms in the right-hand sides of these equations. In this case twopairs of the neutrino wave function components decouple one from each other and fourequations (8) split into the two independent sets of two equations, that couple togetherthe neutrino wave function components in pairs, (ψ1, ψ2) and (ψ3, ψ4).The second pair of eqs. (8) does not contain a matter term and is attributed to thesterile right-handed chiral neutrino state, ψR. The corresponding solution can be takenin the plain-wave form(9) ψR ∼ L− 32 exp[i(−p0t + p1x + p2y + p3z)]ψ,where pμ is the neutrino momentum. Then for the components ψ3 and ψ4 we obtainfrom (8) the following equations:(10)(p0 − p3)ψ3 − (p1 − ip2)ψ4 = 0,−(p1 + ip2)ψ3 + (p0 + p3)ψ4 = 0.Finally, from (10) for the sterile right-handed neutrino we get(11) ψR =e−ipxL3/2√2p0(p0 − p3)⎛⎜⎜⎝00−p1 + ip2p3 − p0⎞⎟⎟⎠ ,where px = pμxμ, pμ = (p0, p1, p2, p3) and xμ = (t, x, y, z). This solution, as it shoulddo, has the vacuum dispersion relation.In the neutrino mass vanishing limit the first pair of eqs. (8) corresponds to the activeleft-handed neutrino. The form of these equations is similar to the corresponding equa-tions for a charged particle (e.g., an electron) moving in a constant magnetic field B givenby the potential A = (By, 0, 0) (see, for instance, [13]). To display the analogy, we notethat in our case the matter current component nv plays the role of the vector potentialA. The existed analogy between an electron dynamics in an external electromagneticfield and a neutrino dynamics in background matter is discussed in [9].(1) The chiral representation for Dirac matrices is used.NEUTRINO MAGNETIC MOMENT AND NEUTRINO ENERGY QUANTIZATION ETC. 57The solution of the first pair of eqs. (8) can be taken in the form(12) ψL ∼ 1Lexp[i(−p0t + p1x + p3z)]ψ(y),and for the components ψ1 and ψ2 of the neutrino wave function we obtain from (8) thefollowing equations:(13)(p0 + p3 + Gn)ψ1 −√ρ(∂∂η− η)ψ2 = 0,√ρ(∂∂η+ η)ψ1 + (p0 − p3 + Gn)ψ2 = 0,whereη =√ρ(x2 +p1ρ), ρ = Gnω.(14)For the wave function we finally get(15) ψL =ρ14 e−ip0t+ip1x+ip3zL√(p0 − p3 + Gn)2 + 2ρN⎛⎜⎜⎜⎝(p0 − p3 + Gn)uN (η)−√2ρNuN−1(η)00⎞⎟⎟⎟⎠ ,where uN (η) are Hermite functions of order N . For the energy of the active left-handedneutrino we get(16) p0 =√p23 + 2ρN −Gn, N = 0, 1, 2, . . . .The energy depends on the neutrino momentum component p3 along the rotation axisof matter and the quantum number N that determines the magnitude of the neutrinomomentum in the orthogonal plane. For description of antineutrinos one has to considerthe “negative sign” energy eigeinvalues (see similar discussion in sect. 2). Thus, theenergy of an electron antineutrino in the rotating matter composed of neutrons is given by(17) p˜0 =√p23 + 2ρN + Gn, N = 0, 1, 2, . . . .Obviously, generalization for different other neutrino flavours and matter composition isjust straightforward (see (4) and (6)).Thus, it is shown [12] that the transversal motion of an active neutrino and antineu-trino is quantized in moving matter very much like an electron energy is quantized in aconstant magnetic field that corresponds to the relativistic form of the Landau energylevels (see [13]).58 I. BALANTSEV, YU. POPOV and A. STUDENIKIN4. – Neutrino energy in a rotating mediumNow we consider a more consistent model of a neutrino motion in a rotating matter.For this case we choose the effective matter potential in (1) in the following form:(18) fμ = −G(n, nv), v = (−ωy, ωx, 0).Contrary to the case considered in the previous section, eq. (1) with the potential (18)describes the case when a neutrino is moving in a rotating medium. It is shown belowhow eq. (1) with (18) can be solved and the corresponding neutrino energy spectrum isobtained.The solution of eq. (1) with (18) can be sought in the form(19) ψ(t, x, y, z) = e−ip0t+ip3z⎛⎜⎜⎜⎝ψ1(x, y)ψ2(x, y)ψ3(x, y)ψ4(x, y)⎞⎟⎟⎟⎠ .Substituting (19) into (1) with (18) and using the explicit form of the Dirac matrices inthe chiral representation, we arrive at a system of linear equations for the neutrino wavefunction components:−(p0 + p3 + Gn)ψ1 + i{(∂∂x− i ∂∂y)+ Gnω(x− iy)}ψ2 = −mψ3,(20)i{(∂∂x+ i∂∂y)−Gnω(x + iy)}ψ1 + (p3 − p0 −Gn)ψ2 = −mψ4,(p0 − p3)ψ3 + i(∂∂x− i ∂∂y)ψ4 = mψ1,+i(∂∂x+ i∂∂y)ψ3 + (p0 + p3)ψ4 = mψ2.In the polar coordinates x + iy = reiφ, x− iy = re−iφ one has(21)∂∂x+ i∂∂y= eiφ(∂∂r+ir∂∂φ),∂∂x− i ∂∂y= e−iφ(∂∂r− ir∂∂φ),and the system of eqs. (20) transforms to−(p0 + p3 + Gn)ψ1 + ie−iφ{∂∂r− ir∂∂φ+ ρr}ψ2 = −mψ3,(22)ieiφ{∂∂r+ir∂∂φ− ρr}ψ1 + (p3 − p0 −Gn)ψ2 = −mψ4,(p0 − p3)ψ3 + ie−iφ(∂∂r− ir∂∂φ)ψ4 = mψ1,+ieiφ(∂∂r+ir∂∂φ)ψ3 + (p0 + p3)ψ4 = mψ2.NEUTRINO MAGNETIC MOMENT AND NEUTRINO ENERGY QUANTIZATION ETC. 59It is possible to show that the operator of the total momentum Jz = Lz + Sz, whereLz = −i ∂∂φ , Sz = 12σ3, commutes with the corresponding Hamiltonian of the consideredsystem. Therefore the solutions can be taken in the form(23)⎛⎜⎜⎝ψ1ψ2ψ3ψ4⎞⎟⎟⎠ =⎛⎜⎜⎜⎝iχ1(r)ei(l−1)φχ2(r)eilφiχ3(r)ei(l−1)φχ4(r)eilφ⎞⎟⎟⎟⎠ ,that are the eigenvectors for the total momentum operator Jz with the correspondingeigenvalues l − 12 . After substitution of (23) the system (22) can be rewritten in thefollowing form:−(p0 + p3 + Gn)χ1 +{ddr+lr+ ρr}χ2 = −mχ3,(24) {ddr− l − 1r− ρr}χ1 + (p0 − p3 + Gn)χ2 = mχ4,(p0 − p3)χ3 +(ddr+lr)χ4 = mχ1,(ddr− l − 1r)χ3 − (p0 + p3)χ4 = −mχ2.For further consideration it is convenient to introduce the rising and decreasing operatorsR+ =ddr− l − 1r− ρr, R− = ddr+lr+ ρr.(25)After application of the decreasing R− and increasing R+ operators to the second andfirst equations of (22) correspondingly, one gets(26)R−R+χ1 + ((p0 + Gn)2 − p23 −m2)χ1 = m(Gnχ3 + ρrχ4),R+R−χ2 + ((p0 + Gn)2 − p23 −m2)χ2 = m(Gnχ4 + ρrχ3).Note that the system (24), as well as the system (8), can be solved exactly in the limitof vanishing neutrino mass m → 0. In order to find the nonzero-mass correction to theenergy spectrum of a neutrino in a bound state in matter, the neutrino square integrablewave function should be found. Therefore, in analogy with the zero-mass case we takeχ3 = χ4 = 0 in the lowest order of perturbation series expansion. Thus we arrive at thesystem(27)R−R+χ1 + ((p0 + Gn)2 − p23 −m2)χ1 = 0,R+R−χ2 + ((p0 + Gn)2 − p23 −m2)χ2 = 0.The solution of (27) can be written in the form(28)(χ1χ2)=(C1Ll−1s (ρr2)C2Lls(ρr2)),60 I. BALANTSEV, YU. POPOV and A. STUDENIKINwhere Lls are the Laguerre functions [13]. After substitution of (28) into (27) and takinginto account the properties of the increasing and decreasing rising operators,(29)R+Ll−1s (ρr2) = −2√ρ(s + l)Lls(ρr2),R−Lls(ρr2) = 2√ρ(s + l)Ll−1s (ρr2),we get from (27) the equation for the neutrino energy spectrum in matter:(30) m2 + p23 + 4(s + l)ρ− (p0 + Gn)2 = 0.Solving this equation we get for the neutrino energies(31) p0 = ±√m2 + p23 + 4Nρ−Gn, N = 0, 1, 2, . . . ,where the quantum number N = s+ l is introduced. As usually, two signs in the solutioncorrespond to the neutrino and antineutrino energies, correspondingly,(32) p0 =√m2 + p23 + 4Nρ−Gn, p˜0 =√m2 + p23 + 4Nρ + Gn.From the obtained energy spectrum it is just straightforward that the transversalmotion momentum of an antineutrino is given by(33) p˜⊥ = 2√NGω.The quantum number N determines also the radius of the quasiclassical orbit in matter(it is supposed that N  1 and p3 = 0),(34) R =√NGnω.It follows that antineutrinos can have bound orbits inside a rotating star. To make anestimation of magnitudes, let us consider a model of a rotating neutron star with radiusRNS = 10 km, matter density n = 1037 cm−3 and angular frequency ω = 2π × 103 s−1.For this set of parameters, the radius of an orbit is less than the typical star radius RNSif the quantum number N ≤ Nmax = 1010. Therefore, antineutrinos that occupy orbitswith N ≤ 1010 can be bounded inside the star. The scale of the bounded antineutrinosenergy estimated by (32) is of the order p˜0 ∼ 1 eV. It should be underlined that withinthe quasiclassical approach the neutrino binding on circular orbits is due to an effectiveforce that is orthogonal to the particle speed. Note that there is another mechanism ofneutrinos binding inside a neutron star when the effect is produced by a gradient of thematter density (see the last paper in [11]). A discussion on the “matter-induced Lorentzforce” that can be introduced in order to explain a neutrino motion on quasiclassicalcircular orbits can be found in [9].We argue that the effect of a neutrino energy quantization can have important con-sequences for physics of rotating neutron stars.∗ ∗ ∗One of the authors (AS) thanks G. Bellettini, G. Chiarelli, M. Greco andG. Isidori for the kind invitation to participate in the XXIII Rencontres de PhysiqueNEUTRINO MAGNETIC MOMENT AND NEUTRINO ENERGY QUANTIZATION ETC. 61de La Valle´e D’Aoste on Results and Perspectives in Particle Physics and also thanksall the organizers for their hospitality in La Thuile. The authors are also thankful toA. Grigoriev for fruitful discussions.REFERENCES[1] Marciano W. J. and Sanda A. I., Phys. Lett. B, 67 (1977) 303; Lee B. W. and ShrockR. E., Phys. Rev. D, 16 (1977) 1444; Fujikawa K. and Shrock R. E., Phys. Rev. Lett.,45 (1980) 963.[2] Giunti C. and Studenikin A., hep-ph/0812.3646 preprint (2008).[3] Studenikin A., Nucl. Phys. B (Proc. Suppl.), 188 (2009) 220.[4] Raffelt G., Phys. Rev. Lett., 64 (1990) 2856; Stars as Laboratories for FundamentalPhysics, in Univ. of Chicago Press, edited by Schellinger P., Thomas G. and TryneskiJ. (Chicago, USA) 1996; Raffelt G., Phys. Rep., 320 (1999) 319; Haft M., RaffeltG. and Wiess A., Astrophys. 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Neutrino magnetic moment and neutrino energy quantization in

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Neutrino magnetic moment and neutrino energy quantization in rotating media

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