Electromagnetic Properties of Neutrinos

Abstract

In this paper, we discuss the main theoretical aspects and experimental effects of neutrino electromagnetic properties. We start with a general description of the electromagnetic form factors of Dirac and Majorana neutrinos. Then, we discuss the theory and phenomenology of the magnetic and electric dipole moments, summarizing the experimental results and the theoretical predictions. We discuss also the phenomenology of a neutrino charge radius and radiative decay. Finally, we describe the theory of neutrino spin and spin-flavor precession in a transverse magnetic field and we summarize its phenomenological applications.

1. Introduction

The investigation of neutrino properties is one of the most active fields of research in current high-energy physics. Neutrinos are special particles, because they interact very weakly and their masses are much smaller than those of the other fundamental fermions (charged leptons and quarks). In the Standard Model neutrinos are massless and have only weak interactions. However, the observation of neutrino oscillations by many experiments (see [1–4]) imply that neutrinos are massive and mixed. Therefore, the Standard Model must be extended to account for neutrino masses. In many extensions of the Standard Model neutrinos acquire also electromagnetic properties through quantum loops effects. Hence, the theoretical and experimental study of neutrino electromagnetic interactions is a powerful tool in the search for the fundamental theory beyond the Standard Model. Moreover, the electromagnetic interactions of neutrinos can generate important effects, especially in astrophysical environments, where neutrinos propagate for long distances in magnetic fields both in vacuum and in matter.

In this paper, we review the theory and phenomenology of neutrino electromagnetic interactions. After a derivation of all the possible types of electromagnetic interactions of Dirac and Majorana neutrinos, we discuss their effects in terrestrial and astrophysical environments and the corresponding experimental results. In spite of many efforts in the search of neutrino electromagnetic interactions, up to now there is no positive experimental indication in favor of their existence. However, the existence of neutrino masses and mixing implies that nontrivial neutrino electromagnetic properties are plausible and experimentalists and theorists are eagerly looking for them.

In this review, we use the notation and conventions in [1]. When we consider neutrino mixing, we have the relation: 𝜈_𝛼𝐿=³_𝑘=1𝑈_𝛼𝑘𝜈_𝑘𝐿𝛼=𝑒,𝜇,𝜏 (1.1) between the left-handed components of the three flavor neutrino fields 𝜈_𝑒, 𝜈_𝜇, 𝜈_𝜏 and the left-handed components of three massive neutrino fields 𝜈_𝑘 with masses 𝑚_𝑘 (𝑘=1,2,3). The 3×3 mixing matrix 𝑈 is unitary (𝑈^†=𝑈⁻¹).

Neutrino electromagnetic properties are discussed in [4–7], and in the previous reviews in [8–14].

The structure of this paper is as follows. In Section 2, we discuss the general form of the electromagnetic interaction of Dirac and Majorana neutrinos, which is expressed in terms of form factors, and the derivation of the form factors in gauge models. In Section 3, we discuss the phenomenology of the neutrino magnetic and electric dipole moments in laboratory experiments. These are the most studied electromagnetic properties of neutrinos, both experimentally and theoretically. In Section 4, we review the theory and experimental constraints on the neutrino charge radius. In Section 5, we discuss neutrino radiative decay and the astrophysical bounds on a neutrino magnetic moment obtained from the study of plasmon decay in stars. In Section 6 we discuss neutrino spin and spin-flavor precession. In conclusion, in Section 7, we summarize the status of our knowledge of neutrino electromagnetic properties and we discuss the prospects for future research.

2. Electromagnetic Form Factors

The importance of neutrino electromagnetic properties was first mentioned by Pauli in 1930, when he postulated the existence of this particle and discussed the possibility that the neutrino might have a magnetic moment. Systematic theoretical studies of neutrino electromagnetic properties started after it was shown that in the extended Standard Model with right-handed neutrinos the magnetic moment of a massive neutrino is, in general, nonvanishing and that its value is determined by the neutrino mass [15–23].

Neutrino electromagnetic properties are important because they are directly connected to fundamentals of particle physics. For example, neutrino electromagnetic properties can be used to distinguish Dirac and Majorana neutrinos (see [21, 22, 24–27]) and also as probes of new physics that might exist beyond the Standard Model (see [28–30]).

2.1. Dirac Neutrinos

In the Standard Model, the interaction of a fermionic field 𝔣(𝑥) with the electromagnetic field 𝐴^𝜇(𝑥) is given by the interaction Hamiltonian: ℋ_em(𝑥)=𝑗_𝜇(𝑥)𝐴^𝜇(𝑥)=𝑞_𝔣𝔣(𝑥)𝛾_𝜇𝔣(𝑥)𝐴^𝜇(𝑥), (2.1) where 𝑞_𝔣 is the charge of the fermion 𝔣. Figure 1(a) shows the corresponding tree-level Feynman diagram (the photon 𝛾 is the quantum of the electromagnetic field 𝐴^𝜇(𝑥)).

(a)

(b)

(c)

Figure 1 

Tree-level coupling of a charged fermion 𝔣 with a photon 𝛾 (a), effective coupling of a neutrino 𝜈 with a photon (b), and effective coupling of neutrinos with a photon taking into account possible transitions between two different initial and final massive neutrinos 𝜈𝑖 and 𝜈𝑓 (c).

For neutrinos, the electric charge is zero and there are no electromagnetic interactions at tree-level (however, in some theories beyond the Standard Model neutrinos can be millicharged particles (see [13])). However, such interactions can arise at the quantum level from loop diagrams at higher order of the perturbative expansion of the interaction. In the one-photon approximation, the electromagnetic interactions of a neutrino field 𝜈(𝑥) can be described by the effective interaction Hamiltonian: ℋ_eff(𝑥)=𝑗^eff_𝜇(𝑥)𝐴^𝜇(𝑥)=𝜈(𝑥)Λ_𝜇𝜈(𝑥)𝐴^𝜇(𝑥), (2.2) where 𝑗^eff_𝜇(𝑥) is the effective neutrino electromagnetic current four-vector and Λ_𝜇 is a 4×4 matrix in spinor space which can contain space-time derivatives, such that 𝑗^eff_𝜇(𝑥) transforms as a four-vector. Since radiative corrections are generated by weak interactions which are not invariant under a parity transformation, 𝑗^eff_𝜇(𝑥) can be a sum of polar and axial parts. The corresponding diagram for the interaction of a neutrino with a photon is shown in Figure 1(b), where the blob represents the quantum loop contributions.

We are interested in the neutrino part of the amplitude corresponding to the diagram in Figure 1(b), which is given by the matrix element: 𝜈𝑝_𝑓,ℎ_𝑓𝑗^eff_𝜇(𝑥)𝜈𝑝_𝑖,ℎ_𝑖, (2.3) where 𝑝_𝑖(𝑝_𝑓) and ℎ_𝑖(ℎ_𝑓) are the four-momentum and helicity of the initial (final) neutrino. Taking into account that 𝜕^𝜇𝑗^eff_𝜇(𝑥)=𝑖𝒫^𝜇,𝑗^eff_𝜇(𝑥), (2.4) where 𝒫^𝜇 is the four-momentum operator which generate translations, the effective current can be written as 𝑗^eff_𝜇(𝑥)=𝑒^{𝑖𝒫⋅𝑥}𝑗^eff_𝜇(0)𝑒^{−𝑖𝒫⋅𝑥}. (2.5) Since 𝒫^𝜇|𝜈(𝑝)⟩=𝑝^𝜇|𝜈(𝑝)⟩, we have 𝜈𝑝_𝑓𝑗^eff_𝜇(𝑥)𝜈𝑝_𝑖=𝑒^{𝑖(𝑝𝑓−p𝑖)⋅𝑥}𝜈𝑝_𝑓𝑗^eff_𝜇(0)𝜈𝑝_𝑖, (2.6) where we suppressed for simplicity the helicity labels which are not of immediate relevance. Here we see that the unknown quantity which determines the neutrino-photon interaction is ⟨𝜈(𝑝_𝑓)|𝑗^eff_𝜇(0)|𝜈(𝑝_𝑖)⟩. Considering that the incoming and outgoing neutrinos are free particles which are described by free Dirac fields with the standard Fourier expansion in (2.139) of [1], we have 𝜈𝑝_𝑓𝑗^eff_𝜇(0)𝜈𝑝_𝑖=𝑢𝑝_𝑓Λ_𝜇𝑝_𝑓,𝑝_𝑖𝑢𝑝_𝑖. (2.7) The electromagnetic properties of neutrinos are embodied by Λ_𝜇(𝑝_𝑓,𝑝_𝑖), which is a matrix in spinor space and can be decomposed in terms of linearly independent products of Dirac 𝛾 matrices and the available kinematical four-vectors 𝑝_𝑖 and 𝑝_𝑓. The most general decomposition can be written as (see [11]) Λ_𝜇𝑝_𝑓,𝑝_𝑖=𝑓₁𝑞²𝑞_𝜇+𝑓₂𝑞²𝑞_𝜇𝛾₅+𝑓₃𝑞²𝛾_𝜇+𝑓₄𝑞²𝛾_𝜇𝛾₅+𝑓₅𝑞²𝜎_𝜇𝜈𝑞^𝜈+𝑓₆𝑞²𝜖_{𝜇𝜈𝜌𝛾}𝑞^𝜈𝜎^𝜌𝛾, (2.8) where 𝑓_𝑘(𝑞²) are six Lorentz-invariant form factors (𝑘=1,…,6) and 𝑞 is the four-momentum of the photon, which is given by 𝑞=𝑝_𝑖−𝑝_𝑓, (2.9) from energy-momentum conservation. Notice that the form factors depend only on 𝑞², which is the only available Lorentz-invariant kinematical quantity, since (𝑝_𝑖+𝑝_𝑓)²=4𝑚²−𝑞². Therefore, Λ_𝜇(𝑝_𝑓,𝑝_𝑖) depends only on 𝑞 and from now on we will denote it as Λ_𝜇(𝑞).

Since the Hamiltonian and the electromagnetic field are Hermitian (ℋ^†_eff=ℋ_eff and 𝐴^𝜇†=𝐴^𝜇), the effective current must be Hermitian, 𝑗^eff†_𝜇=𝑗^eff_𝜇. Hence, we have 𝜈𝑝_𝑓𝑗^eff_𝜇(0)𝜈𝑝_𝑖=𝜈𝑝_𝑖𝑗^eff_𝜇(0)𝜈𝑝_𝑓^∗, (2.10) which leads to Λ_𝜇𝑞=𝛾⁰Λ^†_𝜇−𝑞𝛾⁰. (2.11) This constraint implies that 𝑓₂,𝑓₃,𝑓₄𝑓arereal,₁,𝑓₅,𝑓₆areimaginary. (2.12)(2.13)

The number of independent form factors can be reduced by imposing current conservation, 𝜕^𝜇𝑗^eff_𝜇(𝑥)=0, which is required by gauge invariance (i.e., invariance of ℋ_eff(𝑥) under the transformation 𝐴^𝜇(𝑥)→𝐴^𝜇(𝑥)+𝜕^𝜇𝜑(𝑥) for any 𝜑(𝑥), which leaves invariant the electromagnetic tensor 𝐹^𝜇𝜈=𝜕^𝜇𝐴^𝜈−𝜕^𝜈𝐴^𝜇). Using (2.4), current conservation implies that 𝜈𝑝_𝑓𝒫^𝜇,𝑗^eff_𝜇(0)𝜈𝑝_𝑖=0. (2.14) Hence, in momentum space, we have the constraint: 𝑞^𝜇𝑢𝑝_𝑓Λ_𝜇𝑞𝑢𝑝_𝑖=0, (2.15) which implies that 𝑓₁𝑞²𝑞²+𝑓₂𝑞²𝑞²𝛾₅+2𝑚𝑓₄𝑞²𝛾₅=0. (2.16) Since 𝛾₅ and the unity matrix are linearly independent, we obtain the constraints: 𝑓₁𝑞²=0,𝑓₄𝑞²𝑓=−²𝑞²𝑞².2𝑚 (2.17) Therefore, in the most general case, consistent with Lorentz and electromagnetic gauge invariance, the vertex function is defined in terms of four form factors [25–27]: ^Λ_𝜇^𝑞=𝑓_𝑄^𝑞2𝛾_𝜇^−𝑓_𝑀^{𝑞2𝑖𝜎}_𝜇𝜈^{𝑞𝜈+𝑓}_𝐸^𝑞2𝜎_𝜇𝜈^𝑞𝜈𝛾₅^+𝑓_𝐴^{𝑞2𝑞2𝛾}_𝜇^−𝑞_𝜇/^𝑞𝛾₅^. (2.18) where 𝑓_𝑄=𝑓₃, 𝑓_𝑀=𝑖𝑓₅, 𝑓_𝐸=−2𝑖𝑓₆, and 𝑓_𝐴=−𝑓₂/2𝑚 are the real charge, dipole magnetic and electric, and anapole neutrino form factors. For the coupling with a real photon (𝑞²=0), 𝑓_𝑄(0)=𝑞,𝑓_𝑀(0)=𝜇,𝑓_𝐸(0)=𝜖,𝑓_𝐴(0)=𝑎, (2.19) where 𝑞, 𝜇, 𝜖, and 𝑎 are, respectively, the neutrino charge, magnetic moment, electric moment and anapole moment. Although above we stated that 𝑞=0, here we did not enforce this equality because in some theories beyond the Standard Model neutrinos can be millicharged particles (see [13]).

Now, it is interesting to study the properties of ℋ_eff(𝑥) under a CP transformation, in order to find which of the terms in (2.18) violate CP. Let us consider the active CP transformation: U_CP𝜈(𝑥)U^†_CP=𝜉^CP𝛾⁰𝒞𝜈^𝑇𝑥_P, (2.20) where 𝜉^CP is a phase, 𝒞 is the charge-conjugation matrix (such that 𝒞𝛾^𝑇_𝜇𝒞⁻¹=−𝛾_𝜇, 𝒞^†=𝒞⁻¹ and 𝒞^𝑇=−𝒞), and 𝑥^𝜇_P=𝑥_𝜇. For the Standard Model electric current 𝑗_𝜇(𝑥) in (2.1), we have 𝑗_𝜇(𝑥)−−−−^CP→U_CP𝑗_𝜇(𝑥)U^†_CP=−𝑗^𝜇𝑥_P. (2.21) Hence, the Standard Model electromagnetic interaction Hamiltonian ℋ_em(𝑥) is left invariant by (the transformation 𝑥→𝑥_P is irrelevant since all amplitudes are obtained by integrating over 𝑑⁴𝑥) 𝐴_𝜇(𝑥)−−−−^CP→−𝐴^𝜇𝑥_P. (2.22) CP is conserved in neutrino electromagnetic interactions (in the one-photon approximation) if 𝑗^eff_𝜇(𝑥) transforms as 𝑗_𝜇(𝑥): CP⟺U_CP𝑗^eff_𝜇(𝑥)U^†_CP=−𝑗^𝜇_eff𝑥_P. (2.23) For the matrix element (2.7), we obtain CP⟺Λ_𝜇𝑞−−−−^CP→−Λ^𝜇𝑞. (2.24) One can find that under a CP transformation we have Λ_𝜇𝑞−−−−^CP→𝛾⁰𝒞Λ^𝑇_𝜇𝑞_P𝒞^†𝛾⁰, (2.25) with 𝑞^𝜇_P=𝑞_𝜇. Using the form factor expansion in (2.18), we obtain ^Λ_𝜇^{𝑞CP𝑓−−−−→−}_𝑄^{𝑞2𝛾𝜇−𝑓}_𝑀^{𝑞2𝑖𝜎𝜇𝜈𝑞}_𝜈^−𝑓_𝐸^{𝑞2𝜎𝜇𝜈𝑞}_𝜈^𝛾₅^+𝑓_𝐴^{𝑞2𝑞2𝛾𝜇−𝑞𝜇}/^𝑞𝛾₅^. (2.26) Therefore, only the electric dipole form factor violates CP: CP⟺𝑓_𝐸𝑞²=0. (2.27)

So far, we have considered only one massive neutrino field 𝜈(𝑥), but according to the mixing relation (1.1), the three flavor neutrino fields 𝜈_𝑒, 𝜈_𝜇, 𝜈_𝜏 are unitary linear combinations of three massive neutrinos 𝜈_𝑘 (𝑘=1,2,3). Therefore, we must generalize the discussion to the case of more than one massive neutrino field. The effective electromagnetic interaction Hamiltonian in (2.2) is generalized to ℋ_eff(𝑥)=𝑗^eff_𝜇(𝑥)𝐴^𝜇(𝑥)=³_𝑘,𝑗=1𝜈_𝑘(𝑥)Λ_𝜇^𝑘𝑗𝜈_𝑗(𝑥)𝐴^𝜇(𝑥), (2.28) where we take into account possible transitions between different massive neutrinos. The physical effect of ℋ_eff is described by the effective electromagnetic vertex in Figure 1(c), with the neutrino matrix element: 𝜈_𝑓𝑝_𝑓𝑗^eff_𝜇(0)𝜈_𝑖𝑝_𝑖=𝑢_𝑓𝑝_𝑓Λ_𝜇^𝑓𝑖𝑝_𝑓,𝑝_𝑖u_𝑖𝑝_𝑖. (2.29) As in the case of one massive neutrino field, Λ_𝜇^𝑓𝑖(𝑝_𝑓,𝑝_𝑖) depends only on the four-momentum 𝑞 transferred to the photon and can be expressed in terms of six Lorentz-invariant form factors: Λ_𝜇^𝑓𝑖𝑞=𝑓₁^𝑓𝑖𝑞²𝑞_𝜇+𝑓^𝑓i₂𝑞²𝑞_𝜇𝛾₅+𝑓₃^𝑓𝑖𝑞²𝛾_𝜇+𝑓₄^𝑓𝑖𝑞²𝛾_𝜇𝛾₅+𝑓₅^𝑓𝑖𝑞²𝜎_𝜇𝜈𝑞^𝜈+𝑓₆^𝑓𝑖𝑞²𝜖_{𝜇𝜈𝜌𝛾}𝑞^𝜈𝜎^𝜌𝛾. (2.30) The Hermitian nature of 𝑗^eff_𝜇 implies that ⟨𝜈_𝑓(𝑝_𝑓)|𝑗^eff_𝜇(0)|𝜈_𝑖(𝑝_𝑖)⟩=⟨𝜈_𝑖(𝑝_𝑖)|𝑗^eff_𝜇(0)|𝜈_𝑓(𝑝_𝑓)⟩^∗, leading to the constraint: Λ_𝜇^𝑓𝑖𝑞=𝛾⁰Λ_𝜇^𝑖𝑓−𝑞^†𝛾⁰. (2.31) Considering the 3×3 form factor matrices 𝑓_𝑘 in the space of massive neutrinos with components 𝑓_𝑘^𝑓𝑖 for 𝑘=1,…,6, we find that 𝑓₂,𝑓₃,𝑓₄𝑓areHermitian,₁,𝑓₅,𝑓₆areanti-Hermitian. (2.32)(2.33)

Following the same method used in (2.4)–(2.16), one can find that current conservation implies the constraints: 𝑓₁^𝑓𝑖𝑞²𝑞²+𝑓₃^𝑓𝑖𝑞²𝑚_𝑓−𝑚_𝑖=0,𝑓₂^𝑓𝑖𝑞²𝑞²+𝑓₄^𝑓𝑖𝑞²𝑚_𝑓+𝑚_𝑖=0. (2.34) Therefore, we obtain _{Λ𝜇𝑓𝑖𝑞=𝛾𝜇−𝑞𝜇}^𝑞/_{𝑞2𝑓𝑄𝑓𝑖𝑞2+𝑓𝐴𝑓𝑖𝑞2𝑞2𝛾5−𝑖𝜎𝜇𝜈𝑞𝜈𝑓𝑀𝑓𝑖𝑞2+𝑖𝑓𝐸𝑓𝑖𝑞2𝛾5,} (2.35) where 𝑓_𝑄^𝑓𝑖=𝑓₃^𝑓𝑖, 𝑓_𝑀^𝑓𝑖=𝑖𝑓₅^𝑓𝑖, 𝑓_𝐸^𝑓𝑖=−2𝑖𝑓₆^𝑓𝑖 and 𝑓_𝐴^𝑓𝑖=−𝑓₂^𝑓𝑖/(𝑚_𝑓+𝑚_𝑖), with 𝑓_Ω^𝑓𝑖=𝑓_Ω^𝑖𝑓∗(Ω=𝑄,𝑀,𝐸,𝐴). (2.36) Note that since ^𝑢_𝑓^(𝑝_𝑓⁾/^𝑞𝑢_𝑖^(𝑝_𝑖^)=(𝑚_𝑓^−𝑚_𝑖^)𝑢_𝑓^(𝑝_𝑓^)𝑢_𝑖^(𝑝_𝑖⁾, if 𝑓=𝑖, (2.35) correctly reduces to (2.18).

The form factors with 𝑓=𝑖 are called “diagonal,” whereas those with 𝑓≠𝑖 are called “off-diagonal” or “transition form factors.” This terminology follows from the expression: _{Λ𝜇𝑞=𝛾𝜇−𝑞𝜇}^𝑞/_{𝑞2𝑓𝑄𝑞2+𝑓𝐴𝑞2𝑞2𝛾5−𝑖𝜎𝜇𝜈𝑞𝜈𝑓𝑀𝑞2+𝑖𝑓𝐸𝑞2𝛾5,} (2.37) in which Λ_𝜇(𝑞) is a 3×3 matrix in the space of massive neutrinos expressed in terms of the four Hermitian 3×3 matrices of form factors: 𝑓_Ω=𝑓^†_Ω(Ω=𝑄,𝑀,𝐸,𝐴). (2.38)

For the coupling with a real photon (𝑞²=0), we have 𝑓_𝑄^𝑓𝑖(0)=𝑞_𝑓𝑖,𝑓_𝑀^𝑓𝑖(0)=𝜇_𝑓𝑖,𝑓_𝐸^𝑓𝑖(0)=𝜖_𝑓𝑖,𝑓_𝐴^𝑓𝑖(0)=𝑎_𝑓𝑖, (2.39) where 𝑞_𝑓𝑖, 𝜇_𝑓𝑖, 𝜖_𝑓𝑖, and 𝑎_𝑓𝑖 are, respectively, the neutrino charge, magnetic moment, electric moment, and anapole moment of diagonal (𝑓=𝑖) and transition (𝑓≠𝑖) types.

Considering now CP invariance, the transformation (2.23) of 𝑗^eff_𝜇(𝑥) implies the constraint in (2.24) for the 𝑁×𝑁 matrix Λ_𝜇(𝑞) in the space of massive neutrinos. Using (2.20), we obtain Λ_𝜇^𝑓𝑖𝑞−−−−^CP→𝜉^CP_𝑓𝜉^CP_𝑖^∗𝛾⁰𝒞Λ_𝜇^𝑖𝑓𝑞_P^𝑇𝒞^†𝛾⁰, (2.40) where 𝜉^CP_𝑘 is the CP phase of 𝜈_𝑘. Since the three massive neutrinos take part to standard charged-current weak interactions, their CP phases are equal if CP is conserved (see [1]). Hence, we have Λ_𝜇^𝑓𝑖𝑞−−−−^CP→𝛾⁰𝒞Λ_𝜇^𝑖𝑓𝑞_P^𝑇𝒞^†𝛾⁰. (2.41) Using the form factor expansion in (2.35), we obtain _{Λ𝜇𝑓𝑖𝑞}^CP_{𝛾−−−−→−𝜇−𝑞𝜇}^𝑞/_{𝑞2𝑓𝑄𝑖𝑓𝑞2+𝑓𝐴𝑖𝑓𝑞2𝑞2𝛾5−𝑖𝜎𝜇𝜈𝑞𝜈𝑓𝑀𝑖𝑓𝑞2−𝑖𝑓𝐸𝑖𝑓𝑞2𝛾5.} (2.42) Imposing the constraint in (2.24), for the form factors, we obtain 𝑓CP⟺^Ω𝑓𝑖=𝑓_Ω^𝑖𝑓=𝑓_Ω^𝑓𝑖∗(),𝑓Ω=𝑄,𝑀,𝐴_𝐸𝑓𝑖=−𝑓_𝐸^𝑖𝑓=−𝑓_𝐸^𝑓𝑖∗, (2.43) where, in the last equalities, we took into account the constraints (2.36). For the Hermitian 3×3 form factor matrices, we obtain that if CP is conserved 𝑓_𝑄, 𝑓_𝑀 and 𝑓_𝐴, are real and symmetric and 𝑓_𝐸 is imaginary and antisymmetric: 𝑓CP⟺^Ω=𝑓^T_Ω=𝑓^∗_Ω(),𝑓Ω=𝑄,𝑀,𝐴_𝐸=−𝑓^T_𝐸=−𝑓^∗_𝐸. (2.44)

Let us now consider antineutrinos. Using for the massive neutrino fields, the Fourier expansion in (2.139) of [1], the effective antineutrino matrix element for 𝜈_𝑖(𝑝_𝑖)→𝜈_𝑓(𝑝_𝑓) transitions is given by 𝜈_𝑓𝑝_𝑓𝑗^eff_𝜇(0)𝜈_𝑖𝑝_𝑖=−𝑣_𝑖𝑝_𝑖Λ_𝜇^𝑖𝑓𝑞𝑣_𝑓𝑝_𝑓. (2.45) Using the relation: 𝑢𝑝=𝒞𝑣^𝑇𝑝, (2.46) we can write it as 𝜈_𝑓𝑝_𝑓𝑗^eff_𝜇(0)𝜈_𝑖𝑝_𝑖=𝑢_𝑓𝑝_𝑓𝒞Λ_𝜇^𝑖𝑓𝑞^𝑇𝒞^†𝑢_𝑖𝑝_𝑖, (2.47) where transposition operates in spinor space. Therefore, the effective form factor matrix in spinor space for antineutrinos is given by Λ_𝜇^𝑓𝑖𝑞=𝒞Λ_𝜇^𝑖𝑓𝑞^𝑇𝒞^†. (2.48) Using the properties of the charge-conjugation matrix and the expression (2.35) for Λ_𝜇^𝑖𝑓(𝑞), we obtain the antineutrino form factors: 𝑓_Ω^𝑓𝑖=−𝑓_Ω^𝑖𝑓(Ω=𝑄,𝑀,𝐸),𝑓_𝐴^𝑓𝑖=𝑓_𝐴^𝑖𝑓. (2.49)(2.50) Therefore, in particular the diagonal magnetic and electric moments of neutrinos and antineutrinos have the same size with opposite signs, as the charge, if it exists. On the other hand, the diagonal neutrino and antineutrino anapole moments are equal.

2.2. Majorana Neutrinos

A massive Majorana neutrino is a neutral spin 1/2 particle which coincides with its antiparticle. The four degrees of freedom of a massive Dirac field (two helicities and two particle-antiparticle) are reduced to two (two helicities) by the Majorana constraint: 𝜈_𝑘=𝜈^𝑐_𝑘=𝒞𝜈_𝑘^𝑇. (2.51) Since a Majorana field has half the degrees of freedom of a Dirac field, it is possible that its electromagnetic properties are reduced. From the relations (2.49) and (2.50) between neutrino and antineutrino form factors in the Dirac case, we can infer that in the Majorana case the charge, magnetic, and electric form factor matrices are antisymmetric and the anapole form factor matrix is symmetric. In order to confirm this deduction, let us calculate the neutrino matrix element corresponding to the effective electromagnetic vertex in Figure 1(c), with the effective interaction Hamiltonian in (2.28), which takes into account possible transitions between two different initial and final massive Majorana neutrinos 𝜈_𝑖 and 𝜈_𝑓. Using the neutrino Majorana fields, the Fourier expansion in (6.99) of [1], we obtain 𝜈_𝑓𝑝_𝑓𝑗^eff_𝜇(0)𝜈_𝑖𝑝_𝑖=𝑢_𝑓𝑝_𝑓Λ_𝜇^𝑓𝑖𝑝_𝑓,𝑝_𝑖𝑢_𝑖𝑝_𝑖−𝑣_𝑖𝑝_𝑖Λ_𝜇^𝑖𝑓𝑝_𝑓,𝑝_𝑖𝑣_𝑓𝑝_𝑓. (2.52) Using (2.46), we can write it as 𝑢_𝑓𝑝_𝑓Λ_𝜇^𝑓𝑖𝑝_𝑓,𝑝_𝑖+𝒞Λ_𝜇^𝑖𝑓𝑝_𝑓,𝑝_𝑖^𝑇𝒞^†𝑢_𝑖𝑝_𝑖, (2.53) where transposition operates in spinor space. Therefore, the effective form factor matrix in spinor space for Majorana neutrinos is given by Λ_𝜇^𝑓𝑖𝑝_𝑓,𝑝_𝑖=Λ_𝜇^𝑓𝑖𝑝_𝑓,𝑝_𝑖+𝒞Λ_𝜇^𝑖𝑓𝑝_𝑓,𝑝_𝑖^𝑇𝒞^†. (2.54) As in the case of Dirac neutrinos, Λ_𝜇^𝑓𝑖(𝑝_𝑓,𝑝_𝑖) depends only on 𝑞=𝑝_𝑓−𝑝_𝑖 and can be expressed in terms of six Lorentz-invariant form factors according to (2.30). Hence, we can write the 3×3 matrix Λ_𝜇(𝑝_𝑓,𝑝_𝑖) in the space of massive Majorana neutrinos as Λ_𝜇𝑞=𝑓₁𝑞²𝑞_𝜇+𝑓₂𝑞²𝑞_𝜇𝛾₅+𝑓₃𝑞²𝛾_𝜇+𝑓₄𝑞²𝛾_𝜇𝛾₅+𝑓₅𝑞²𝜎_𝜇𝜈𝑞^𝜈+𝑓₆𝑞²𝜖_{𝜇𝜈𝜌𝛾}𝑞^𝜈𝜎^𝜌𝛾, (2.55) with 𝑓_𝑘=𝑓_𝑘+𝑓^𝑇_𝑘⇒𝑓_𝑘=𝑓^𝑇_𝑘𝑓for𝑘=1,2,4,_𝑘=𝑓_𝑘−𝑓^𝑇_𝑘⇒𝑓_𝑘=−𝑓^𝑇_𝑘for𝑘=3,5,6. (2.56) Now, we can follow the discussion in Section 2.1 for Dirac neutrinos taking into account the additional constraints (2.56) for Majorana neutrinos. The hermiticity of 𝑗^eff_𝜇 and current conservation lead to an expression similar to that in (2.37): _{Λ𝜇𝑞=𝛾𝜇−𝑞𝜇}^𝑞/_{𝑞2𝑓𝑄𝑞2+𝑓𝐴𝑞2𝑞2𝛾5−𝑖𝜎𝜇𝜈𝑞𝜈𝑓𝑀𝑞2𝑓+𝑖𝐸𝑞2𝛾5,} (2.57) with 𝑓_𝑄^𝑓𝑖=𝑓₃^𝑓𝑖, 𝑓_𝑀^𝑓𝑖=𝑖𝑓₅^𝑓𝑖, 𝑓_𝐸^𝑓𝑖=−2𝑖𝑓₆^𝑓𝑖, and 𝑓_𝐴^𝑓𝑖=−𝑓₂^𝑓𝑖/(𝑚_𝑓+𝑚_𝑖). For the Hermitian 3×3 form factor matrices in the space of massive neutrinos, 𝑓_Ω=𝑓^†_Ω(Ω=𝑄,𝑀,𝐸,𝐴), (2.58) the Majorana constraints (2.56) imply that 𝑓_Ω=−𝑓^𝑇_Ω(),𝑓Ω=𝑄,𝑀,𝐸_𝐴=𝑓^𝑇_𝐴. (2.59)(2.60) These relations confirm the expectation discussed above that for Majorana neutrinos the charge, magnetic and electric form factor matrices are antisymmetric and the anapole form factor matrix is symmetric.

Since 𝑓_𝑄, 𝑓_𝑀, and 𝑓_𝐸 are antisymmetric, a Majorana neutrino does not have diagonal charge and dipole magnetic, and electric form factors. It can only have a diagonal anapole form factor. On the other hand, Majorana neutrinos can have as many off-diagonal (transition) form factors as Dirac neutrinos.

Since the form factor matrices are Hermitian as in the Dirac case, 𝑓_𝑄, 𝑓_𝑀, and 𝑓_𝐸 are imaginary, whereas 𝑓_𝐴 is real: 𝑓_Ω=−𝑓^∗_Ω(),𝑓Ω=𝑄,𝑀,𝐸_𝐴=𝑓^∗_𝐴. (2.61)(2.62)

Considering now CP invariance, the case of Majorana neutrinos is rather different from that of Dirac neutrinos, because the CP phases of the massive Majorana fields 𝜈_𝑘 are constrained by the CP invariance of the Lagrangian Majorana mass term: ℒ_𝑀=12_𝑘𝑚_𝑘𝜈^𝑇_𝑘𝒞^†𝜈_𝑘. (2.63) In order to prove this statement, let us first notice that since a massive Majorana neutrino field 𝜈_𝑘 is constrained by the Majorana relation in (2.51), only the parity transformation part is effective in a CP transformation: U_CP𝜈_𝑘(𝑥)U^†_CP=𝜉^CP_𝑘𝛾⁰𝜈_𝑘𝑥_P. (2.64) Considering the mass term in (2.63), we have U_CP𝜈^𝑇_𝑘𝒞^†𝜈_𝑘U^†_CP=−𝜉^CP_𝑘²𝜈^𝑇_𝑘𝒞^†𝜈_𝑘. (2.65) Therefore, CP⟺𝜉^CP_𝑘=𝜂_𝑘𝑖, (2.66) with 𝜂_𝑘=±1. These CP signs can be different for the different massive neutrinos, even if they all take part to the standard charged-current weak interactions through neutrino mixing, because they can be compensated by the Majorana CP phases in the mixing matrix (see [1]). Therefore, from (2.40), we have Λ_𝜇^𝑓𝑖𝑞−−−−^CP→𝜂_𝑓𝜂_𝑖𝛾⁰𝒞Λ_𝜇^𝑖𝑓𝑞_P^𝑇𝒞^†𝛾⁰. (2.67) Imposing a CP constraint analogous to that in (2.24), we obtain 𝑓CP⟺^Ω𝑓𝑖=𝜂_𝑓𝜂_𝑖𝑓_Ω^𝑖𝑓=𝜂_𝑓𝜂_𝑖𝑓_Ω^𝑓𝑖∗,𝑓_𝐸^𝑓𝑖=−𝜂_𝑓𝜂_𝑖𝑓_𝐸^𝑖𝑓=−𝜂_𝑓𝜂_𝑖𝑓_𝐸^𝑓𝑖∗, (2.68) with Ω=𝑄,𝑀,𝐴. Taking into account the constraints (2.61) and (2.62), we have two cases: CP,𝜂_𝑓=𝜂_𝑖⟺𝑓_𝑄^𝑓𝑖=𝑓_𝑀^𝑓𝑖=0,CP,𝜂_𝑓=−𝜂_𝑖⟺𝑓_𝐸^𝑓𝑖=𝑓_𝐴^𝑓𝑖=0. (2.69)(2.70) Therefore, if CP is conserved, two massive Majorana neutrinos can have either a transition electric form factor or a transition magnetic form factor, but not both, and the transition electric form factor can exist only together with a transition anapole form factor, whereas the transition magnetic form factor can exist only together with a transition charge form factor. In the diagonal case 𝑓=𝑖, (2.69) does not give any constraint, because only diagonal anapole form factors are allowed for Majorana neutrinos.

2.3. Form Factors in Gauge Models

From the demand that the form factors at zero momentum transfer, 𝑞²=0, are elements of the scattering matrix, it follows that in any consistent theoretical model the form factors in the matrix element (2.7) should be gauge independent and finite. Then, the form factors values at 𝑞²=0 determine the static electromagnetic properties of the neutrino that can be probed or measured in the direct interaction with external electromagnetic fields. This is the case for charge, dipole magnetic, and electric neutrino form factors in the minimally extended Standard Model.

In non-Abelian gauge theories, the form factors in the matrix element (2.7) at nonzero momentum transfer, 𝑞²≠0, can be noninvariant under gauge transformations. This happens because in general the off-shell photon propagator is gauge dependent. Therefore, the one-photon approximation is not enough to get physical quantities. In this case, the form factors in the matrix element (2.7) cannot be directly measured in an experiment with an external electromagnetic field. However, they can contribute to higher-order diagrams describing some processes that are accessible for experimental observation (see [31]).

Note that there is an important difference between the electromagnetic vertex function of massive and massless neutrinos [32, 33]. For the case of a massless neutrino, the matrix element (2.7) of the electromagnetic current can be expressed in terms of only one Dirac form factor 𝑓_𝐷(𝑞²) (see also [30]): 𝑢𝑝^′Λ_𝜇𝑞𝑢𝑝=𝑓_𝐷𝑞²𝑢𝑝^′𝛾_𝜇1+𝛾₅𝑢𝑝. (2.71) It follows that the electric charge and anapole form factors for a massless neutrino are related to the Dirac form factor 𝑓_𝐷(𝑞²), and hence to each other: 𝑓_𝑄𝑞²=𝑓_𝐷𝑞²,𝑓_𝐴𝑞²=𝑓_𝐷𝑞²𝑞². (2.72)

In the case of a massive neutrino, there is no such simple relation between electric charge and anapole form factors since the ^𝑞_𝜇/^𝑞𝛾₅ term in the anapole part of the vertex function (2.18) cannot be neglected.

Moreover, a direct calculation of the massive neutrino electromagnetic vertex function, taking into account all the diagrams in Figures 15–17 of [13], reveals that each of the Feynman diagrams gives nonzero contribution to the term proportional to 𝛾_𝜇𝛾₅ [32, 33]. These contributions are not vanishing even at 𝑞²=0. Therefore, in addition to the usual four terms in (2.18) an extra term proportional to 𝛾_𝜇𝛾₅ appears and a corresponding additional form factor 𝑓₅(𝑞²) must be introduced. This problem is related to the decomposition of the massive neutrino electromagnetic vertex function. The calculation of the contributions of the proper vertex diagrams (Figure 15 of [13]) and 𝛾−𝑍 self-energy diagrams (Figures 16 and 17 of [13]) for arbitrary gauge fixing parameter 𝛼=1/𝜉 in the general 𝑅_𝜉 gauge and arbitrary mass parameter 𝑎=𝑚²_𝑙/𝑚²_𝑊 shows that at least in the zeroth and first orders of the expansion over the small neutrino mass parameter 𝑏=(𝑚_𝜈/𝑚_𝑊)², the corresponding “charge” 𝑓₅(𝑞²=0) is zero. The cancellation of contributions from the proper vertex and self-energy diagrams to the form factor 𝑓₅(𝑞²) at 𝑞²≠0, 𝑓₅𝑞²=𝑓₅^{(𝛾−𝑍)}𝑞²+𝑓^(prop.vert₅^.)𝑞²=0, (2.73) was also shown [32, 33] for arbitrary mass parameters 𝑎 and 𝑏 in the ′t Hooft-Feynman gauge 𝛼=1.

Hence, in the minimally extended Standard Model one can perform a direct calculation of the neutrino vertex function leading to the four terms in (2.18) with gauge-invariant electric charge, magnetic, electric, and anapole moments.

3. Magnetic and Electric Dipole Moments

The neutrino dipole magnetic and electric form factors (and the corresponding magnetic and electric dipole moments) are theoretically the most well-studied and understood among the form factors. They also attract a reasonable attention from experimentalists, although the neutrino magnetic moment predicted in the extended Standard Model with right-handed neutrinos is proportional to the neutrino mass and, therefore, it is many orders of magnitude smaller than the present experimental limits obtained in terrestrial experiments.

3.1. Theoretical Predictions

The first calculations of the neutrino dipole moments within the minimal extension of the Standard Model with right-handed neutrinos were performed in [15–22]. The explicit evaluation of the one-loop contributions to the neutrino dipole moments in the leading approximation over the small parameters 𝑏_𝑖=𝑚²_𝑖/𝑚²_𝑊 (where 𝑚_𝑖 are the neutrino masses, 𝑖=1,2,3), that in addition exactly accounts for the dependence on the small parameters 𝑎_𝑙=𝑚²_𝑙/𝑚²_𝑊 (with 𝑙=𝑒,𝜇,𝜏), yields, for Dirac neutrinos [7, 18–23], 𝜇^D_𝑖𝑗𝜖^D_𝑖𝑗=𝑒𝐺_𝐹8√2𝜋²𝑚_𝑖±𝑚_{𝑗𝑙=𝑒,𝜇,𝜏}𝑓𝑎_𝑙𝑈^∗_𝑙𝑖𝑈_𝑙𝑗, (3.1) where 𝑓𝑎_𝑙=3411+1−𝑎_𝑙−2𝑎_𝑙1−𝑎_𝑙²−2𝑎²_𝑙ln𝑎_𝑙1−𝑎_𝑙³. (3.2) All the charged lepton parameters 𝑎_𝑙 are small. In the limit 𝑎_𝑙≪1, one has 𝑓𝑎_𝑙≃32𝑎1−^𝑙2. (3.3) From (3.1) and (3.3), the diagonal magnetic moments of Dirac neutrinos are given by 𝜇^D_𝑖𝑖≃3𝑒𝐺_𝐹𝑚_𝑖8√2𝜋²11−2_{𝑙=𝑒,𝜇,𝜏}𝑎_𝑙𝑈_𝑙𝑖². (3.4) This result exhibits the following important features. The magnetic moment of a Dirac neutrino is proportional to the neutrino mass and for a massless Dirac neutrino in the Standard Model (in the absence of right-handed charged currents), the magnetic moment is zero. The magnetic moment of a massive Dirac neutrino, at the leading order in 𝑎_𝑙, is independent of the neutrino mixing matrix and of the values of the charged lepton masses. The numerical value of the Dirac neutrino magnetic moment is 𝜇^D_𝑖𝑖≃3.2×10⁻¹⁹𝑚_𝑖𝜇eV^𝐵. (3.5) Taking into account the existing constraint of the order of 1 eV on the neutrino masses (see [1–4]), this value is several orders of magnitude smaller than the present experimental limits, which are discussed in Section 3.4.

From (3.1), it can be clearly seen that in the extended Standard Model with right-handed neutrinos the static (diagonal) electric dipole moment of a Dirac neutrino vanishes, 𝜖^D_𝑖𝑖=0, in spite of possible CP violations generated by the Dirac phase in the mixing matrix (as shown in (2.27), Dirac neutrinos may have nonzero diagonal electric moments only in theories where CP invariance is violated). For a Majorana neutrino, both the diagonal magnetic and electric moments are zero, 𝜇^𝑀_𝑖𝑖=𝜖^𝑀_𝑖𝑖=0, as shown in Section 2.2.

Let us consider now the neutrino transition moments, which are given by (3.1) for 𝑖≠𝑗. Considering only the leading term 𝑓(𝑎_𝑙)≃3/2 in the expansion (3.3), one gets vanishing transition moments, because of the unitarity relation: _𝑙𝑈^∗_𝑙𝑖𝑈_𝑙𝑗=𝛿_𝑖𝑗. (3.6) Therefore, the first nonvanishing contribution comes from the second term in the expansion (3.3) of 𝑓(𝑎_𝑙), which contains the additional small factor 𝑎_𝑙=𝑚²_𝑙/𝑚²_𝑊: 𝜇^D_𝑖𝑗𝜖^D_𝑖𝑗≃−3𝑒𝐺_𝐹32√2𝜋²𝑚_𝑖±𝑚_{𝑗𝑙=𝑒,𝜇,𝜏}𝑚_𝑙𝑚_𝑊²𝑈^∗_𝑙𝑖𝑈_𝑙𝑗, (3.7) for 𝑖≠𝑗. Thus, the transition moments are suppressed with respect to the diagonal magnetic moments in (3.4). This suppression is called “GIM mechanism,” in analogy with the suppression of flavor-changing neutral currents in hadronic processes discovered in [34]. Numerically, the Dirac transition moments are given by 𝜇^D_𝑖𝑗𝜖^D_𝑖𝑗≃−4×10⁻²³𝑚_𝑖±𝑚_𝑗𝑓eV^𝑖𝑗𝜇_𝐵, (3.8) with 𝑓_𝑖𝑗=_{𝑙=𝑒,𝜇,𝜏}𝑚_𝑙𝑚_𝜏²𝑈^∗_𝑙𝑖𝑈_𝑙𝑗. (3.9)

Also, Majorana neutrinos can have nonvanishing transition magnetic and electric moments, as discussed in Section 2.2. Assuming CP conservation and neglecting model-dependent Feynman diagrams depending on the details of the scalar sector [7, 21, 22, 24], if 𝜈_𝑖 and 𝜈_𝑗 have the same CP phase, 𝜇^𝑀_𝑖𝑗=0,𝜖^𝑀_𝑖𝑗=2𝜖^D_𝑖𝑗, (3.10) whereas if 𝜈_𝑖 and 𝜈_𝑗 have opposite CP phases, 𝜇^𝑀_𝑖𝑗=2𝜇^D_𝑖𝑗,𝜖^D_𝑖𝑗=0, (3.11) with 𝜖^D_𝑖𝑗 and 𝜇^D_𝑖𝑗 given by (3.1). Hence, although the nonvanishing Majorana transition moments are twice the Dirac ones, they are equally suppressed by the GIM mechanism. However, the model-dependent contributions of the scalar sector can enhance the Majorana transition moments (see [21, 35, 36]).

In recent studies, the value of the diagonal magnetic moment of a massive Dirac neutrino was calculated in the one-loop approximation in the extended Standard Model with right-handed neutrinos, accounting for the dependence on the neutrino mass parameter 𝑏_𝑖=𝑚²_𝑖/𝑚²_𝑊 [37] and accounting for the exact dependence on both mass parameters 𝑏_𝑖 and 𝑎_𝑙=𝑚²_𝑙/𝑚²_𝑊 [32, 33]. The calculations of the neutrino magnetic moment which take into account exactly the dependence on the masses of all particles can be useful in the case of a heavy neutrino with a mass comparable or even exceeding the values of the masses of other known particles. Note that the LEP data require that the number of light neutrinos coupled to the 𝑍 boson is three [38]. Therefore, any additional active neutrino must be heavier than 𝑚_𝑍/2. This possibility is not excluded by current data (see [39]).

For a heavy neutrino with mass m_𝑖, much larger than the charged lepton masses but smaller than the 𝑊-boson mass (2GeV≪𝑚_𝑖≪80GeV), the authors of [32, 33] obtained the diagonal magnetic moment: 𝜇_𝑖𝑖≃3𝑒𝐺_𝐹8𝜋²√2𝑚_𝑖51+𝑏18^𝑖, (3.12) whereas for a heavy neutrino with mass 𝑚_𝑖 much larger than the 𝑊-boson mass, they got 𝜇_𝑖𝑖≃𝑒𝐺_𝐹8𝜋²√2𝑚_𝑖. (3.13) Note that in both cases the Dirac neutrino magnetic moment is proportional to the neutrino mass. This is an expected result, because the calculations have been performed within the extended Standard Model with right-handed neutrinos.

At this point, a question arises: “Is a neutrino magnetic moment always proportional to the neutrino mass?.” The answer is “No.” For example, much larger values of the Dirac neutrino magnetic moment can be obtained in 𝑆𝑈(2)_𝐿×𝑆𝑈(2)_𝑅×𝑈(1) left-right symmetric models with direct right-handed neutrino interactions (see [15, 40–42]). The massive gauge bosons states 𝑊₁ and 𝑊₂ have, respectively, predominant left-handed and right-handed coupling, since 𝑊₁=𝑊_𝐿cos𝜉−𝑊_𝑅sin𝜉,𝑊₂=𝑊_𝐿sin𝜉+𝑊_𝑅cos𝜉, (3.14) where 𝜉 is a small mixing angle and the fields 𝑊_𝐿 and 𝑊_𝑅 have pure 𝑉±𝐴 interactions. The magnetic moment of a neutrino 𝜈_𝑙 calculated in this model, neglecting neutrino mixing, is 𝜇_𝜈𝑙=𝑒𝐺_𝐹2√2𝜋²𝑚_𝑙𝑚1−^2𝑊1𝑚²_𝑊23sin2𝜉+4𝑚_𝜈𝑙𝑚1+^2𝑊1𝑚²_𝑊2, (3.15) where the term proportional to the charged lepton mass 𝑚_𝑙 is due to the left-right mixing. This term can exceed the second term in (3.15), which is proportional to the neutrino mass 𝑚_𝜈𝑙.

3.2. Neutrino-Electron Elastic Scattering

The most sensitive and widely used method for the experimental investigation of the neutrino magnetic moment is provided by direct laboratory measurements of low-energy elastic scattering of neutrinos and antineutrinos with electrons in reactor, accelerator and solar experiments. Detailed descriptions of several experiments can be found in [12, 43].

Extensive experimental studies of the neutrino magnetic moment, performed during many years, are stimulated by the hope to observe a value much larger than the prediction in (3.5) of the minimally extended Standard Model with right-handed neutrinos. It would be a clear indication of new physics beyond the extended Standard Model. For example, the effective magnetic moment in 𝜈_𝑒-𝑒 elastic scattering in a class of extradimension models can be as large as ~10⁻¹⁰𝜇_𝐵 [44]. Future higher precision reactor experiments can, therefore, be used to provide new constraints on large extradimensions.

The possibility for neutrino-electron elastic scattering due to neutrino magnetic moment was first considered in [45] and the cross-section of this process was calculated in [46, 47]. Discussions on the derivation of the cross-section and on the optimal conditions for bounding the neutrino magnetic moment, as well as a collection of cross-section formulas for elastic scattering of neutrinos (antineutrinos) on electrons, nucleons, and nuclei can be found in [48, 49].

Let us consider the elastic scattering: 𝜈+𝑒⁻→𝜈+𝑒⁻ (3.16) of a neutrino with energy 𝐸_𝜈 with an electron at rest in the laboratory frame. There are two observables: the kinetic energy 𝑇 of the recoil electron and the recoil angle 𝜒 with respect to the neutrino beam, which are related by 𝐸cos𝜒=^𝜈+𝑚_𝑒𝐸_𝜈𝑇𝑇+2𝑚_𝑒^1/2. (3.17) The electron kinetic energy is constrained from the energy-momentum conservation by 𝑇≤2𝐸²_𝜈2𝐸_𝜈+𝑚_𝑒. (3.18)

Since, in the ultrarelativistic limit, the neutrino magnetic moment interaction changes the neutrino helicity and the Standard Model weak interaction conserves the neutrino helicity, the two contributions add incoherently in the cross-section which can be written as [49] 𝑑𝜎=𝑑𝑇𝑑𝜎𝑑𝑇_SM+𝑑𝜎𝑑𝑇_𝜇. (3.19) The small interference term due to neutrino masses has been derived in [50].

The weak-interaction cross-section is given by 𝑑𝜎𝑑𝑇_SM=𝐺²_𝐹𝑚_𝑒𝑔2𝜋^𝑉+𝑔_𝐴²+𝑔_𝑉−𝑔_𝐴²𝑇1−𝐸_𝜈²+𝑔²_A−𝑔²_𝑉𝑚_𝑒𝑇𝐸²_𝜈, (3.20) with the standard coupling constants 𝑔_𝑉 and 𝑔_𝐴 given by 𝑔_𝑉=2sin²𝜃_𝑊+12for𝜈_𝑒,2sin²𝜃_𝑊−12for𝜈_𝜇,𝜈_𝜏,𝑔_𝐴=12for𝜈_𝑒,−12for𝜈_𝜇,𝜈_𝜏. (3.21)(3.22) In antineutrino-electron elastic scattering, one must substitute 𝑔_𝐴→−𝑔_𝐴.

The neutrino magnetic-moment contribution to the cross section is given by [49] 𝑑𝜎𝑑𝑇_𝜇=𝜋𝛼²𝑚²_𝑒1𝑇−1𝐸_𝜈𝜇_𝜈𝜇_𝐵², (3.23) where 𝜇_𝜈 is the effective magnetic moment discussed in Section 3.3.

The two terms (𝑑𝜎/𝑑𝑇)_SM and (𝑑𝜎/𝑑𝑇)_𝜇 exhibit quite different dependences on the experimentally observable electron kinetic energy 𝑇, as illustrated in Figure 2, where the values of the two terms averaged over the typical antineutrino reactor spectrum are plotted for six values of the neutrino magnetic moment, 𝜇_𝜈^(𝑁)=𝑁×10⁻¹¹𝜇_𝐵, with 𝑁=1,2,3,4,5,6 [43] (see also [49]). One can see that small values of the neutrino magnetic moment can be probed by lowering the electron recoil energy threshold. In fact, from (3.20) and (3.23), one can find that (𝑑𝜎/𝑑𝑇)_𝜇 exceeds (𝑑𝜎/𝑑𝑇)_SM for 𝜋𝑇≲²𝛼²𝐺²_𝐹𝑚³_𝑒𝜇_𝜈𝜇_𝐵². (3.24)

Figure 2 

Standard Model weak (W) and magnetic moment electromagnetic (EM) contributions to the cross section for several values of the neutrino magnetic moment [43].

It was proposed in [51] that electron binding in atoms (the “atomic ionization” effect in neutrino interactions on Ge target) can significantly increase the electromagnetic contribution to the differential cross-section with respect to the free electron approximation. However, detailed considerations of the atomic ionization effect in (anti)neutrino atomic electron scattering experiments presented in [52–57] show that the effect is by far too small to have measurable consequences even in the case of the low energy threshold of 2.8 keV reached in the GEMMA experiment [58].

3.3. Effective Dipole Moments

In scattering experiments, the neutrino is created at some distance from the detector as a flavor neutrino, which is a superposition of massive neutrinos. Therefore, the magnetic moment that is measured in these experiment is not that of a massive neutrino, but it is an effective magnetic moment which takes into account neutrino mixing and the oscillations during the propagation between source and detector [50, 59]: 𝜇²_𝜈𝜈_𝛼,𝐿,𝐸=_𝑗𝑘𝑈^∗_𝛼𝑘𝑒^{−𝑖𝑚2𝑘𝐿/2𝐸}𝜇_𝑗𝑘−𝑖𝜖_𝑗𝑘², (3.25) where we have written explicitly the dependence from the initial neutrino flavor 𝜈_𝛼, the distance 𝐿, and the energy 𝐸. In this expression of the effective 𝜇_𝜈 one can see that in general both the magnetic and electric dipole moments contribute to the elastic scattering. Note that 𝜇_𝜈(𝜈_𝛼,𝐿,𝐸) depends only on the neutrino squared-mass differences: considering for simplicity only the magnetic moment contribution, we have 𝜇²_𝜈𝜈_𝛼,𝐿,𝐸=_{𝑗𝑘𝑘′}𝑈^∗_𝛼𝑘𝑈_𝛼𝑘′𝑒^{−𝑖Δ𝑚2′𝑘𝑘𝐿/2𝐸}𝜇_𝑗𝑘𝜇_𝑗𝑘′. (3.26)

In the case of Majorana neutrinos, there are no diagonal magnetic and electric dipole moments and 𝜇²_𝜈 receives contributions only from the transition dipole moments. Furthermore, if CP is conserved, there are either only magnetic or electric transition dipole moments (see Section 2.2).

The general expression for 𝜇²_𝜈(𝜈_𝛼,𝐿,𝐸) can be simplified in some cases [59]. For instance, for Dirac neutrinos with only diagonal magnetic moments 𝜇_𝑖𝑗=𝜇_𝑖𝛿_𝑖𝑗, we have the effective flavor magnetic moment: 𝜇²_𝜈𝜈_𝛼,𝐿,𝐸→𝜇^D_𝛼²=_𝑖𝑈_𝛼𝑖²𝜇²_𝑖. (3.27) Since in this case there is no dependence on the distance 𝐿 and the neutrino energy, the magnetic cross-section is characterized by the initial neutrino flavor rather than by the composition of mass states in the detector. In this case, measurements of all flavor magnetic moments and mixing parameters can allow the extraction of all the fundamental moments 𝜇_𝑖.

3.4. Experimental Limits

The constraints on the neutrino magnetic moment in direct laboratory experiments have been obtained so far from the lack of any observable distortion of the recoil electron energy spectrum. Experiments of this type have started about 40 years ago. The strategy applied during all these years in reactor experiments is rather simple: minimize the threshold on the recoil energy for the detection of the scattered electron, keeping at the same time a reasonable background level. Since the region of interest coincides with the energy region dominated by radioactivity, low intrinsic radioactivity detectors have been employed together with active and passive shields. In addition, all the experiments were running inside laboratories with significant overburden of concrete, corresponding to several meters of water. This was enough to suppress the soft component of cosmic rays.

In all the experiments, with one exception only, the signal due to the antineutrinos from the reactor (about 2×10²⁰s⁻¹GWthermal⁻¹) is obtained from the difference between the reactor-on and reactor-off rate. Clearly, this requires the same background with the reactor on as with the reactor off.

𝜈_𝑒-𝑒 elastic scattering was first observed in the pioneering experiment [60] at the Savannah River Laboratory. The setup was made of a 15.9 kg plastic scintillator target divided into 16 optically isolated elements, totally enclosed inside a 300 kg NaI crystal shielded with lead and cadmium. Finally, the entire setup was immersed into 2200 liters of liquid scintillator. Both the NaI and the liquid scintillator detectors were working as veto against cosmics and gamma-rays from the laboratory. In the electron kinetic energy range from 1.5 MeV to 4.5 MeV, a reactor-on rate of 47.5±1 events/day was measured, to be compared with a reactor-off rate of 40.4±0.9 events/day. A revised analysis of the Savannah River Laboratory data [49] with an improved reactor neutrino spectrum and a more precise value of sin²𝜃_𝑊 gave hints for a neutrino magnetic moment on the order of (2–4)×10⁻¹⁰𝜇_𝐵.

However, lower limits were then obtained by two experiments performed at nuclear reactors in Russia. The Krasnoyarsk experiment [61] had a 103 kg target of liquid organofluoric scintillator contained into seven scintillation chambers. The absence of hydrogen in the target was a significant improvement, since the 𝜈_𝑒-𝑝 charged-current reaction, which has a much larger cross-section than 𝜈_𝑒-𝑒, had been an important background source in the Savannah River experiment. The active target was then surrounded by a passive shield of steel, copper, lead, and borated polyethylene. Finally, two layers of plastic scintillators were vetoing the cosmic muons. With a count rate of 8.3±0.3 (reactor-on) and 7.1±0.4 (reactor-off) in the electron energy range 3.15–5.18 MeV, the Krasnoyarsk collaboration obtained the limit: 𝜇_𝜈𝑒≤2.4×10⁻¹⁰𝜇_𝐵(90%C.L.). (3.28)

A completely different detector was built for the Rovno [62] experiment: 600 silicon detectors, for a total mass of 75 kg, with a passive shield of mercury, copper, cadmium absorber, and graphite. Finally, the setup was enclosed inside a veto made of plastic scintillators. With a rate of 4963±12 events/day (reactor-on) and 4921±16 events/day (reactor-off) in the electron recoil energy range 0.6–2 MeV, it has been possible to obtain a limit on the neutrino magnetic moment of 𝜇_𝜈𝑒≤1.9×10⁻¹⁰𝜇_𝐵(90%C.L.). (3.29)

Finally, more stringent limits have been obtained in the two most recent experiments at rectors. TEXONO [63] has been performed at the Kuo-Sheng nuclear power station. The detector, a 1.06 kg high-purity germanium, was completely surrounded by NaI (Tl) and CsI (Tl) crystals working as anti-Compton. The whole setup was contained inside a shield made of copper, boron-loaded polyethylene, stainless steel, lead, and plastic scintillators. A background of about 1 event/keV⋅kg⋅day could be achieved above the threshold of 12 keV, giving the limit: 𝜇_𝜈𝑒≤7.4×10⁻¹¹𝜇_𝐵(90%C.L.). (3.30)

At the moment, the world best limit is coming from the GEMMA experiment at the Kalinin nuclear power plant. A 1.5 kg high-purity germanium detector is placed inside a cup-shaped NaI crystal and surrounded by copper, lead, and plastic scintillators. With an energy threshold as low as 2.8 keV, the GEMMA collaboration obtained [58] 𝜇_𝜈𝑒≤2.9×10⁻¹¹𝜇_𝐵(90%C.L.). (3.31)

The experiment which followed a strategy different from the study of the reactor-on and reactor-off rate was MUNU. As a matter of fact, the detector [64] was able to provide not only the energy but also the topology of events. As a consequence, the initial direction of an electron track could be measured and the electron scattering angle reconstructed. This allowed to look for the reactor signal by comparing forward electrons, having as reference the reactor to detector axis, with the backward ones. In this way, the background is measured online, which eliminates problems from detector instabilities, as well as from a possible time dependence of the background itself. The central component of the detector consisted of an acrylic vessel time-projected chamber (a cylinder 90 cm in diameter and 162 cm long) filled with CF₄ at 3 bar pressure and immersed in a steel tank (2 m diameter and 3.8 m long) filled with 10 m³ liquid scintillator viewed by 48 photomultipliers. The total target mass of CF₄ was 11.4 kg. Finally, the setup was surrounded by boron-loaded polyethylene and lead. With a total rate of 6.8±0.3 events/day in the forward direction and a background of 5.8±0.17 events/day, the following upper bounds have been obtained [65]: 𝜇_𝜈𝑒≤9×10⁻¹¹𝜇_𝐵(90%C.L.). (3.32)

Several experiments at accelerators have searched for an effect due to the magnetic moment of 𝜈_𝜇 in 𝜈_𝜇-𝑒 and 𝜈_𝜇-𝑒 elastic scattering (see [66]). The current best limit has been obtained in the LSND experiment [67]: 𝜇_𝜈𝜇≤6.8×10⁻¹⁰𝜇_𝐵(90%C.L.). (3.33)

The DONUT collaboration has investigated 𝜈_𝜏-𝑒 and 𝜈_𝜏-𝑒 elastic scattering, finding the limit [68]: 𝜇_𝜈𝜏≤3.9×10⁻⁷𝜇_𝐵(90%C.L.). (3.34)

Solar neutrino experiments as Super-Kamiokande and Borexino can also search for a neutrino magnetic moment signal by studying the shape of the electron spectrum. Since the neutrino magnetic moment depends both on the mixing and on the propagation properties of the neutrino, then oscillations are here relevant.

The analysis of the recoil electron spectrum generated by solar neutrinos in the Super-Kamiokande experiment experiment gave [69] 𝜇_𝜈≤1.1×10⁻¹⁰𝜇_𝐵(90%C.L.), (3.35) where 𝜇_𝜈 is not the same as 𝜇_𝜈𝑒 since it is given by a different combination of the magnetic moment of the neutrino mass eigenstates (see Section 3.3).

The limit 𝜇_𝜈≤5.4×10⁻¹¹𝜇_𝐵(90%C.L.) (3.36) has been recently obtained in the Borexino solar neutrino scattering experiment [70]. An upper limit on the neutrino magnetic moment 𝜇_𝜈≤8.4×10⁻¹¹𝜇_𝐵 has been found in an independent analysis of the first release of the Borexino experiment data performed in [71]. It was also shown that with reasonable assumptions on the oscillation probability this limit translates into the conservative upper limits on the magnetic moments of 𝜈_𝜇 and 𝜈_𝜏: 𝜇_𝜈𝜇≤1.5×10⁻¹⁰𝜇_𝐵,𝜇_𝜈𝜏≤1.9×10⁻¹⁰𝜇_𝐵(90%C.L.). (3.37) The limit on 𝜇_𝜈𝜏 is three order of magnitude stronger than the direct limit (3.34).

The global fit [72, 73] of the magnetic moment data from the reactor and solar neutrino experiments for the Majorana neutrinos produces limits on the neutrino transition moments: 𝜇₂₃,𝜇₃₁,𝜇₁₂<1.8×10⁻¹⁰𝜇_𝐵(90%C.L.). (3.38)

Finally, an interesting new possibility for providing more stringent constraints on the neutrino magnetic moment from 𝜈_𝑒-𝑒 scattering experiments was discussed in [74] on the basis of an observation [75] that “dynamical zeros" appear in the Standard Model contribution to the scattering cross-section.

3.5. Theoretical Considerations

As it was already mentioned before, there is a gap of many orders of magnitude between the present experimental limits ~10⁻¹¹𝜇_𝐵 on neutrino magnetic moments (discussed in Section 3.4) and the prediction (3.5) of the minimal extension of the Standard Model with right-handed neutrinos. At the same time, the experimental sensitivity of reactor 𝜈_𝑒-𝑒 elastic scattering experiments has improved by only one order of magnitude during a period of about twenty years (see [49], where a sensitivity of ~10⁻¹⁰𝜇_𝐵 is discussed). However, the experimental studies of neutrino magnetic moments are stimulated by the hope that new physics beyond the minimally extended Standard Model with right-handed neutrinos might give much stronger contributions. One of the examples in which it is possible to avoid the neutrino magnetic moment being proportional to a (small) neutrino mass, that would in principle make a neutrino magnetic moment accessible for experimental observations, is realized in the left-right symmetric models considered at the end of Section 3.1.

Other interesting possibilities of obtaining neutrino magnetic moments lager than the prediction (3.5) of the minimal extension the Standard Model with right-handed neutrinos have been considered recently. In this concern, we note that it was proposed in [44] to probe a class of large extradimensions models with future reactors searches for neutrino magnetic moments. The results obtained within the Minimal Supersymmetric Standard Model with 𝑅-parity violating interactions [76, 77] show that the Majorana transition magnetic moment might be significantly above the scale of (3.5).

Considering the problem of large neutrino magnetic moments, one can write down a generic relation between the size of a neutrino magnetic moment 𝜇_𝜈 and the corresponding neutrino mass 𝑚_𝜈 [28, 29, 35, 78–80]. Suppose that a large neutrino magnetic moment is generated by physics beyond a minimal extension of the Standard Model at an energy scale characterized by Λ. For a generic diagram corresponding to this contribution to 𝜇_𝜈, one can again use the Feynman graph in Figure 1(b); the shaded circle in this case denotes effects of new physics beyond the Standard Model. The contribution of this diagram to the magnetic moment is 𝜇_𝜈∼𝑒𝐺Λ, (3.39) where 𝑒 is the electric charge and 𝐺 is a combination of coupling constants and loop factors. The same diagram of Figure 1(b) but without the photon line gives a new physics contribution to the neutrino mass: 𝛿𝑚_𝜈∼𝐺Λ. (3.40) Combining the estimates (3.39) and (3.40), one can get the relation: 𝛿𝑚_𝜈∼Λ²2𝑚_𝑒𝜇_𝜈𝜇_𝐵=𝜇_𝜈10⁻¹⁸𝜇_𝐵Λ1TeV²eV (3.41) between the one-loop contribution to the neutrino mass and the neutrino magnetic moment.

It follows that, generally, in theoretical models that predict large values for the neutrino magnetic moment, simultaneously large contributions to the neutrino mass arise. Therefore, a particular fine tuning is needed to get a large value for the neutrino magnetic moment while keeping the neutrino mass within experimental bounds. One of the possibilities [78] is based on the idea of suppressing the ratio 𝑚_𝜈/𝜇_𝜈 with a symmetry: if a 𝑆𝑈(2)_𝜈 symmetry is an exact symmetry of the Lagrangian of a model, because of different symmetry properties of the mass and magnetic moment even a massless neutrino can have a nonzero magnetic moment. If, as it happens in a realistic model, the 𝑆𝑈(2)_𝜈 symmetry is broken and if this breaking is small, the ratio 𝑚_𝜈/𝜇_𝜈 is also small, giving a natural way to obtain a magnetic moment of the order of ~10⁻¹¹𝜇_𝐵 without contradictions with the neutrino mass experimental constraints. Several possibilities based on the general idea of [78] were considered in [81–86].

Another idea of neutrino mass suppression without suppression of the neutrino magnetic moment was discussed in [35] within the Zee model [87], which is based on the Standard Model gauge group 𝑆𝑈(2)_𝐿×𝑈(1)_𝑌 and contains at least three Higgs doublets and a charged field which is a singlet of 𝑆𝑈(2)_𝐿. For this kind of models, there is a suppression of the neutrino mass diagram, while the magnetic moment diagram is not suppressed.

It is possible to show with more general and rigorous considerations [28, 29, 80] that the Λ² dependence in (3.41) arises from the quadratic divergence in the renormalization of the dimension-four neutrino mass operator. A general and model-independent upper bound on the Dirac neutrino magnetic moment, which can be generated by an effective theory beyond the Standard Model, has been derived [28, 29, 80] from the demand of absence of fine-tuning of effective operator coefficients and from the current experimental information on neutrino masses. A model with Dirac fermions, scalars, and gauge bosons that is valid below the scale Λ and respects the Standard Model 𝑆𝑈(2)_𝐿×𝑈(1)_𝑌 symmetry was considered. Integrating out the physics above the scale Λ, the following effective Lagrangian that involves right-handed neutrinos 𝜈_𝑅, lepton isodoublets and the Higgs doublet can be obtained: ℒ_eff=_𝑛,𝑗𝒞^𝑛_𝑗𝜇Λ^𝑛−4𝒪_𝑗^(𝑛)𝜇+H.c., (3.42) where 𝜇 is the renormalization scale, 𝑛≥4 denotes the operator dimension and 𝑗 runs over independent operators of a given dimension. For 𝑛=4, a neutrino mass arises from the operator 𝒪₁⁽⁴⁾=𝐿Φ𝜈_𝑅, where Φ=𝑖𝜎₂Φ^∗. In addition, if the scale Λ is not extremely large with respect to the electroweak scale, an important contribution to the neutrino mass can arise also from higher dimension operators. At this point, it is important to note that the combination of the 𝑛=6 operators appearing in the Lagrangian (3.42) contains the magnetic moment operator 𝜈𝜎_𝜇𝜈𝜈𝐹^𝜇𝜈 and also generates a contribution 𝛿𝑚_𝜈 to the neutrino mass [28, 29, 80]. Solving the renormalization group equation from the scale Λ to the electroweak scale, one finds that the contributions to the neutrino magnetic moment and to the neutrino mass are connected to each other by 𝜇^D_𝜈=16√2𝐺_𝐹𝑚_𝑒𝛿𝑚_𝜈sin⁴𝜃_𝑊9𝛼²𝑓()𝜇lnΛ/𝑣^𝐵, (3.43) where 𝛼 is the fine structure constant, 𝑣 is the vacuum expectation value of the Higgs doublet, 2𝑓=1−𝑟−3tan²𝜃_𝑊−13(1+𝑟)tan⁴𝜃_𝑊, (3.44) and 𝑟 is a ratio of effective operator coefficients defined at the scale Λ which is of order unity without fine tuning. If the neutrino magnetic moment is generated by new physics at a scale Λ∼1 TeV and the corresponding contribution to the neutrino mass is 𝛿𝑚_𝜈≲1 eV, then the bound 𝜇_𝜈≲10⁻¹⁴𝜇_𝐵 can be obtained. This bound is some orders of magnitude stronger than the constraints from reactor and solar neutrino scattering experiments discussed before.

The model-independent limit on a Majorana neutrino transition magnetic moment 𝜇^M_𝜈 was also discussed in [28, 29, 80]. However, the limit in the Majorana case is much weaker than that in the Dirac case, because for a Majorana neutrino, the magnetic moment contribution to the mass is Yukawa suppressed. The limit on 𝜇^M_𝜈 is also weaker than the present experimental limits if 𝜇^M_𝜈 is generated by new physics at the scale Λ∼1TeV. An important conclusion of [28, 29, 80], based on model-independent considerations of the contributions to 𝜇_𝜈, is that if a neutrino magnetic moment of order 𝜇_𝜈≥10⁻¹⁵𝜇_𝐵 were observed in an experiment, it would give a proof that neutrinos are Majorana rather than Dirac particles.

4. Neutrino Charge Radius

Even if the electric charge of a neutrino is vanishing, the electric form factor 𝑓_𝑄(𝑞²) can still contain nontrivial information about neutrino electromagnetic properties. Considering 𝑓_𝑄(0)=0, in the static limit (𝑞²→0), the electric form factor is given by 𝑓_𝑄𝑞²=𝑞²𝑑𝑓_𝑄𝑞²𝑑𝑞²_𝑞2=0+⋯. (4.1) The leading contribution can be expressed in terms of a neutrino charge radius considering a static spherically symmetric charge distribution of density 𝜌(𝑟) (with 𝑟=|^→𝑥|) in the so-called “Breit frame,” where 𝑞₀=0. In this approximation, we have 𝑓_𝑄𝑞²=𝜌(𝑟)𝑒^{𝑖→𝑞⋅→𝑥}𝑑³𝑥=4𝜋𝜌(𝑟)sin𝑞𝑟𝑟𝑞𝑟²𝑑𝑟, (4.2) where 𝑞=|^→𝑞|. Since 𝑑𝑓_𝑄/𝑑𝑞²|_𝑞2=0=−⟨𝑟²⟩/6, with ⟨𝑟²⟩=∫𝑟²𝜌(𝑟)𝑑𝑟, the neutrino charge radius is defined by 𝑟²_𝜈=−6𝑑𝑓_𝑄𝑞²𝑑𝑞²_𝑞2=0. (4.3)

Note that ⟨𝑟²_𝜈⟩ can be negative, because the charge density 𝜌(𝑟) is not a positively defined function of 𝑟.

In one of the first studies [31], it was claimed that in the Standard Model and in the unitary gauge, the neutrino charge radius is ultraviolet-divergent and so it is not a physical quantity. A direct one-loop calculation [32, 33] of proper vertices and 𝛾−𝑍 self-energy (Figures 15 and 16 of [13]) contributions to the neutrino charge radius performed in a general 𝑅_𝜉 gauge for a massive Dirac neutrino gave also a divergent result. However, it was shown [88], using the unitary gauge, that, by including in addition to the usual terms also contributions from diagrams of the neutrino-lepton neutral current scattering (𝑍 boson diagrams), it is possible to obtain for the neutrino charge radius a gauge-dependent but finite quantity. Later on, it was also shown [16] that in order to define the neutrino charge radius as a physical quantity, one has also to consider box diagrams (see Figure 18 of [13]), which contribute to the scattering process 𝜈+ℓ→𝜈+ℓ, and that in combination with contributions from the proper diagrams it is possible to obtain a finite and gauge-independent value for the neutrino charge radius. In this way, the neutrino electroweak radius was defined [89, 90] and an additional set of diagrams that give contribution to its value was discussed in [91]. Finally, in a series of papers [92–94] the neutrino electroweak radius as a physical observable has been introduced. In the corresponding calculations, performed in the one-loop approximation including additional terms from the 𝛾−𝑍 boson mixing and the box diagrams involving 𝑊 and 𝑍 bosons, the following gauge-invariant result for the neutrino charge radius has been obtained: 𝑟²_𝜈𝛼=𝐺_𝐹4√2𝜋²𝑚3−2log^2𝛼𝑚²_𝑊, (4.4) where 𝑚_𝑊 and 𝑚_𝛼 are the 𝑊 boson and lepton masses (𝛼=𝑒,𝜇,𝜏). This result, however, revived the discussion [95–98] on the definition of the neutrino charge radius. Numerically, for the electron neutrino electroweak radius, it yields [92–94] 𝑟²_𝜈𝑒=4×10⁻³³cm², (4.5) which is very close to the numerical estimations obtained much earlier in [89, 90].

Note that the neutrino charge radius can be considered as an effective scale of the particle’s “size,” which should influence physical processes such as, for instance, neutrino scattering off electron. To incorporate the neutrino charge radius contribution in the cross-section (3.20), the following substitution [49, 99, 100] can be used: 𝑔_𝑉→12+2sin²𝜃_𝑊+23𝑚²_𝑊𝑟²_𝜈𝑒sin²𝜃_𝑊. (4.6) Using this method, the TEXONO collaboration obtained [101] −2.1×10⁻³²cm²<𝑟²_𝜈𝑒<3.3×10⁻³²cm²(90%C.L.). (4.7)

Other available bounds on the electron neutrino charge radius are from primordial nucleosynthesis [102] 𝑟²_𝜈𝑒≲7×10⁻³³cm², (4.8) from SN 1987A [103] 𝑟²_𝜈𝑒≲2×10⁻³³cm², (4.9) from neutrino neutral-current reactions [104] −2.74×10⁻³²cm²<𝑟²_𝜈𝑒<4.88×10⁻³²cm²(90%C.L.), (4.10) from solar experiments (Kamiokande II and Homestake) [105, 106] 𝑟²_𝜈𝑒<2.3×10⁻³²cm²(95%C.L.), (4.11) from an evaluation of the weak mixing angle sin²𝜃_𝑊 by a combined fit of all electron neutrino elastic scattering data [107] −0.13×10⁻³²cm²<𝑟²_𝜈𝑒<3.32×10⁻³²cm²(90%C.L.). (4.12)

Comparing the theoretical value in (4.5) with the experimental limits in (4.7)–(4.12), one can see that they differ at most by one order of magnitude. Therefore, one may expect that the experimental accuracy will soon reach the value needed to probe the theoretical predictions for the neutrino effective charge radius.

The effects of new physics beyond the Standard Model can also contribute to the neutrino charge radius. Let us only mention that the anomalous 𝑊𝑊𝛾 vertex contribution to the neutrino effective charge radius has been studied in [30], and shown to correspond to a contribution ≲10⁻³⁴cm² to |⟨𝑟²_𝜈𝑒⟩|. Note that this is only one order of magnitude lower than the expected value of the charge radius in the Standard Model.

A detailed discussion on the possibilities to constrain the 𝜈_𝜏 and 𝜈_𝜇 charge radii from astrophysical and cosmological observations and from terrestrial experiments can be found in [108].

5. Radiative Decay and Plasmon Decay

If the masses of neutrinos are nondegenerate, the radiative decay of a heavier neutrino 𝜈_𝑖 into a lighter neutrino 𝜈_𝑓(with𝑚_𝑖>𝑚_𝑓) with emission of a photon, 𝜈_𝑖→𝜈_𝑓+𝛾, (5.1) may proceed in vacuum [15, 16, 18–21, 23, 109, 110]. Early discussions of the possible role of neutrino radiative decay in different astrophysical and cosmological settings can be found in [111–116].

For the case of a Dirac neutrino, the decay rate in the minimal extension of the Standard Model with right-handed neutrinos is [15, 16, 18–21, 23, 109, 110] Γ_𝜈^𝐷_𝑖→𝜈^𝐷_𝑗+𝛾=𝛼𝐺²_𝐹128𝜋⁴𝑚²_𝑖−𝑚²_𝑗𝑚_𝑗³𝑚²_𝑖+𝑚²_{𝑗𝑙=𝑒,𝜇,𝜏}𝑓𝑎_𝑙𝑈_𝑙𝑗𝑈^∗_𝑙𝑖², (5.2) where 𝑓(𝑎_𝑙) is given by (3.2). Recalling the results for the Dirac neutrino magnetic and electric transition moments 𝜇_𝑖𝑗 and 𝜖_𝑖𝑗, given in (3.1), one can rewrite (5.2) as (see [117, 118]) Γ_{𝜈𝑖→𝜈𝑗+𝛾}=𝜇_𝑖𝑗²+𝜖²_𝑖𝑗𝑚8𝜋^2𝑖−𝑚²_𝑗𝑚_𝑗³. (5.3) For degenerate neutrino masses (𝑚_𝑖=𝑚_𝑗), the process is kinematically forbidden in vacuum.

Note that there are models (see for instance [119]) in which the neutrino radiative decay rates (as well as the magnetic moment discussed above) of a nonstandard Dirac neutrinos are much larger than those predicted in the minimally extended Standard Model.

For Majorana neutrinos, if CP is violated, the decay rate is given by (5.3). If CP is conserved, we have two cases: if the Majorana neutrinos 𝜈_𝑖 and 𝜈_𝑗 have the same CP eigenvalues, (2.69) and (3.10) imply that the decay process is induced purely by the neutrino electric transition dipole moment, because 𝜇_𝑖𝑗=0; on the other hand, if the two Majorana neutrinos have opposite CP eigenvalues, from (2.70) and (3.11), one can see that the transition is purely of magnetic dipole type (𝜖_𝑖𝑗=0).

For numerical estimations, it is convenient to express (5.3) in the following form: Γ_{𝜈𝑖→𝜈𝑗+𝛾}𝜇=5.3^eff𝜇_𝐵²𝑚²_𝑖−𝑚²_𝑗𝑚²_𝑗³𝑚_𝑖1eV³𝑠⁻¹, (5.4) with the effective neutrino magnetic moment 𝜇_eff=|𝜇_𝑖𝑗|²+|𝜖²_𝑖𝑗|.

The neutrino radiative decay can be constrained by the absence of decay photons in reactor 𝜈_𝑒 and solar 𝜈_𝑒 fluxes. The limits on 𝜇_eff that have been obtained from these considerations are much weaker than those obtained from neutrino scattering terrestrial experiments. Stronger constraints on 𝜇_eff (though still weaker than the terrestrial ones) have been obtained from the neutrino decay limit set by SN 1987A and from the limits on the distortion of the Cosmic Microwave Background Radiation (CMBR). These limits can be expressed as (see [117, 118] and references therein) 𝜇_eff𝜇_B<0.9×10⁻¹eV𝑚_𝜈²Reactor𝜈_e,0.5×10⁻⁵eV𝑚_𝜈²Sun𝜈_e,1.5×10⁻⁸eV𝑚_𝜈²SN1987A(allflavors),1.0×10⁻¹¹eV𝑚_𝜈^9/4CMBR(allflavors). (5.5) Detailed discussions (and corresponding references) on the astrophysical constraints on the neutrino magnetic and electric transition moments, summarized in Figure 3, can be found in [117, 118].

Figure 3 

Astrophysical limits on neutrino transition moments [117, 118].

For completeness, we would like to mention that other processes characterized by the same signature of (5.1) have been considered (for a review of the literature see [120–124]).(i)The photon radiation by a massless neutrino (𝜈_𝑖→𝜈_𝑗+𝛾,𝑖=𝑗) due to the vacuum polarization loop diagram in the presence of an external magnetic field [125].(ii)The photon radiation by a massive neutrino with nonvanishing magnetic moment in constant magnetic and electromagnetic wave fields [126–129].(iii)The Cherenkov radiation due to the nonvanishing neutrino magnetic moment in an homogeneous and infinitely extended medium, which is only possible if the speed of the neutrino is larger than the speed of light in the medium [130, 131].(iv)The transition radiation due to a nonvanishing neutrino magnetic moment which would be produced when the neutrino crosses the interface of two media with different refractive indices [132–134].(v)The Cherenkov radiation of a massless neutrino due to its induced charge in a medium [135, 136]. (Note that the neutrino electromagnetic properties are in general affected by the external environment. In particular, a neutrino can acquire an electric charge in magnetized matter [135, 136] and the neutrino magnetic moment depends on the strength of external electromagnetic fields [127, 137, 138]. A recent study of the neutrino electromagnetic vertex in magnetized matter can be found in [139]. See also [122, 123] for a review of neutrino interactions in external electromagnetic fields.)(vi)The Cherenkov radiation of massive and massless neutrinos in a magnetized medium [120, 140].(vii)The neutrino radiative decay (𝜈_𝑖→𝜈_𝑗+𝛾,𝑖≠𝑗) in external fields and media (see [141–146] and references therein).(viii)The spin light of neutrino in matter (𝑆𝐿𝜈) that is a mechanism of electromagnetic radiation due to the precession or transition of magnetic or electric (transition) moments of massive neutrinos when they propagate in background matter [121, 147–151].

A very interesting process, for the purpose of constraining neutrino electromagnetic properties, is the photon (plasmon) decay into a neutrino-antineutrino pair: 𝛾^∗→𝜈+𝜈. (5.6) This process becomes kinematically allowed in media, because a photon with the dispersion relation 𝜔²_𝛾+^→𝑘²_𝛾>0 roughly behaves as a particle with an effective mass.

Plasmon decay generated by the neutrino coupling to photons due to a magnetic moment 𝜇_𝜈 (and/or to a neutrino electric millicharge 𝑞_𝜈) was first considered in [152] as a possible source of energy loss of the Sun. The requirement that the energy loss does not exceed the solar luminosity gave [117, 118] 𝜇_𝜈≲4×10⁻¹⁰𝜇_𝐵, (5.7) and 𝑞_𝜈≲6×10⁻¹⁴𝑒.

The tightest astrophysical bound on a neutrino magnetic moment is provided by the observed properties of globular cluster stars. The plasmon decay (5.6) inside the star liberates the energy 𝜔_𝛾 in the form of neutrinos that freely escape the stellar environment. This nonstandard energy loss cools a red giant star so fast that it can delay helium ignition. From the lack of observational evidence of this effect, the following limit has been found [153]: 𝜇_𝜈≤3×10⁻¹²𝜇_𝐵, (5.8) and 𝑞_𝜈≲2×10⁻¹⁴𝑒. This is the most stringent astrophysical constraint on a neutrino magnetic moment, applicable to both Dirac and Majorana neutrinos. The same limit applies for the neutrino magnetic transition moments as well as for the electric (transition) moments.

Recently, it has been shown that the additional cooling due to neutrino magnetic moments generates qualitative changes to the structure and evolution of stars with masses between 7 and 18 solar masses, rather than simply changing the time scales of their burning [154]. The resulting sensitivity to the neutrino magnetic moment has been estimated to be at the level of (2−4)×10⁻¹¹𝜇_𝐵.

6. Spin-Flavor Precession

If neutrinos have magnetic moments, the spin can precess in a transverse magnetic field [155–157].

Let us first consider the spin precession of a Dirac neutrino generated by its diagonal magnetic moment 𝜇. The spatial evolution of the left-handed and right-handed helicity amplitudes 𝜑_𝐿(𝑥) and 𝜑_𝑅(𝑥) in a transverse magnetic field 𝐵_⟂(𝑥) is given by 𝑖d𝜑d𝑥^𝐿(𝑥)𝜑_𝑅(𝑥)=0𝜇𝐵_⟂(𝑥)𝜇𝐵_⟂(𝑥)0𝜑_𝐿(𝑥)𝜑_𝑅(𝑥). (6.1) The differential equation (6.1) can be solved through the transformation: 𝜑_𝐿(𝑥)𝜑_𝑅(𝑥)=1√211𝜑−11⁻(𝑥)𝜑₊(𝑥). (6.2) The new amplitudes 𝜑₋(𝑥) and 𝜑₊(𝑥) satisfy decoupled differential equations, whose solutions are 𝜑_∓(𝑥)=exp±𝑖^𝑥₀d𝑥^′𝜇𝐵_⟂𝑥^′𝜑_∓(0). (6.3) If we consider an initial left-handed neutrino, we have 𝜑_𝐿(0)𝜑_𝑅(0)=10⇒𝜑₋(0)𝜑₊(0)=1√211. (6.4) Then, the probability of 𝜈_𝐿→𝜈_𝑅 transitions is given by 𝑃_{𝜈𝐿→𝜈𝑅}(𝑥)=𝜑_𝑅(𝑥)²=sin^2𝑥₀d𝑥′𝜇𝐵_⟂𝑥^′. (6.5) Note that the transition probability is independent from the neutrino energy (contrary to the case of flavor oscillations) and the amplitude of the oscillating probability is unity. Hence, when the argument of the sine is equal to 𝜋/2, there is complete 𝜈_𝐿→𝜈_𝑅 conversion.

The precession 𝜈_𝑒𝐿→𝜈_𝑒𝑅 in the magnetic field of the Sun was considered in 1971 [155] as a possible solution of the solar neutrino problem. If neutrinos are Dirac particles, right-handed neutrinos are sterile and a 𝜈_𝑒𝐿→𝜈_𝑒𝑅 conversion could explain the disappearance of active solar 𝜈_𝑒𝐿’s.

In 1986, it was realized [156, 157] that the matter effect during neutrino propagation inside of the Sun suppresses 𝜈_𝑒𝐿→𝜈_𝑒𝑅 transition by lifting the degeneracy of 𝜈_𝑒𝐿 and 𝜈_𝑒𝑅. Indeed, taking into account matter effects, the evolution equation (6.1) for a Dirac neutrino becomes 𝑖d𝜑d𝑥^𝐿(𝑥)𝜑_𝑅(𝑥)=𝑉(𝑥)𝜇𝐵_⟂(𝑥)𝜇𝐵_⟂(𝑥)0𝜑_𝐿(𝑥)𝜑_𝑅(𝑥), (6.6) with the appropriate potential 𝑉(𝑥) which depends on the neutrino flavor: 𝑉_𝛼(𝑥)=𝑉_CC(𝑥)𝛿_𝛼𝑒+𝑉_NC(𝑥). (6.7) Here, 𝑉_CC and 𝑉_NC are the charged-current and neutral-current potentials given by 𝑉_CC(𝑥)=√2𝐺_𝐹𝑁_𝑒(𝑥),𝑉_NC(𝑥)1=−2√2𝐺_𝐹𝑁_𝑛(𝑥), (6.8) where 𝑁_𝑒(𝑥) and 𝑁_𝑛(𝑥) are the electron and neutron number densities in the medium. For antineutrinos, 𝑉_𝛼(𝑥)=−𝑉_𝛼(𝑥).

In the case of a constant matter density, the differential equation (6.6) can be solved analytically with the orthogonal transformation: 𝜑_𝐿(𝑥)𝜑_𝑅(𝑥)=cos𝜉sin𝜉𝜑−sin𝜉cos𝜉⁻(𝑥)𝜑₊(𝑥). (6.9) The angle 𝜉 is chosen in order to diagonalize the matrix operator in (6.6): sin2𝜉=2𝜇𝐵_⟂Δ𝐸_𝑀, (6.10) with the effective energy splitting in matter Δ𝐸_M=𝑉²+2𝜇𝐵_⟂². (6.11) The decoupled evolution of 𝜑_∓(𝑥) is given by: 𝜑_∓(𝑥)−𝑖=exp2𝑉∓Δ𝐸_𝑀𝜑_∓(0). (6.12) For an initial left-handed neutrino, 𝜑₋(0)𝜑₊(0)=cos𝜉sin𝜉, (6.13) leading to the oscillatory transition probability: 𝑃_{𝜈𝐿→𝜈𝑅}(𝑥)=𝜑_𝑅(𝑥)²=sin²2𝜉sin²12Δ𝐸_𝑀𝑥. (6.14) Since in matter Δ𝐸_𝑀>2𝜇𝐵_⟂, the matter effect suppresses the amplitude of 𝜈_𝐿→𝜈_𝑅 transitions. However, these transitions are still independent from the neutrino energy, which does not enter in the evolution equation (6.6).

When it was known, in 1986 [156, 157], that the matter potential has the effect of suppressing 𝜈_𝐿→𝜈_𝑅 transitions because it breaks the degeneracy of left-handed and right-handed states, it did not take long to realize, in 1988 [158, 159], that the matter potentials can cause resonant spin-flavor precession if different flavor neutrinos have transition magnetic moments (spin-flavor precession in vacuum was previously discussed in [24]).

Let us denote with 𝜓_𝛼ℎ(𝑥) the flavor and helicity amplitudes (with 𝛼=𝑒,𝜇,𝜏 and ℎ=±1), that is, 𝜑_𝛼−1(𝑥)≡𝜑_𝛼𝐿(𝑥) and 𝜑_𝛼+1(𝑥)≡𝜑_𝛼𝑅(𝑥). Considering neutrino mixing, the evolution of the flavor and helicity amplitudes is given by 𝑖d𝜓_𝛼ℎ(𝑥)d𝑥=_{𝛽ℎ′=±1𝑘}𝑈_𝛼𝑘𝑚²_𝑘𝑈2𝐸^∗𝛽𝑘+𝑉_𝛼(𝑥)𝛿_𝛼𝛽𝛿_ℎℎ′+𝜇_𝛼𝛽𝐵_⟂(𝑥)𝛿_−ℎℎ′𝜓_𝛽ℎ′(𝑥), (6.15) with the effective magnetic moments in the flavor basis: 𝜇_𝛼𝛽=_𝑘,𝑗𝑈_𝛼𝑘𝜇_𝑘𝑗𝑈^∗_𝛽𝑗. (6.16)

For a Dirac neutrino, from (2.36), we have 𝜇_𝑗𝑘=𝜇^∗_𝑘𝑗⇒𝜇_𝛽𝛼=𝜇^∗_𝛼𝛽. (6.17) If CP is conserved, from (2.43), and the reality of the mixing matrix, for a Dirac neutrino, we obtain CP⇒𝜇_𝑗𝑘=𝜇_𝑘𝑗⇒𝜇_𝛽𝛼=𝜇_𝛼𝛽. (6.18)

For a Majorana neutrino, from (2.59) and (2.61), we have 𝜇_𝑗𝑘=−𝜇_k𝑗,𝜇_𝑘𝑗=−𝜇^∗_𝑘𝑗. (6.19) Hence, in the mass basis of Majorana neutrinos, there are no diagonal magnetic moments and the transition magnetic moments are imaginary. If CP is not conserved, the mixing matrix is not real and the constraints (6.19) do not imply similar relations between the effective magnetic moments in the flavor basis, for which we have only the relation in (6.17) as for Dirac neutrinos. In particular, Majorana neutrinos can have diagonal effective magnetic moments in the flavor basis if CP is not conserved. Let us emphasize that both Dirac and Majorana phases contribute to this effect. Therefore, it occurs also in the case of two-neutrino mixing, in which there is one Majorana phase.

On the other hand, if CP is conserved, there is no additional constraint on the magnetic moments of Majorana neutrinos in the mass basis, as we have seen in Section 2.2. However, in this case, the mixing matrix is real and we have CP⇒𝜇_𝛽𝛼=−𝜇_𝛼𝛽,𝜇_𝛼𝛽=−𝜇^∗_𝛼𝛽. (6.20) Hence, only if CP is conserved, there are no diagonal magnetic moments of Majorana neutrinos in the flavor basis as in the mass basis.

In the following, we discuss the spin-flavor evolution equation in the two-neutrino mixing approximation, which is interesting for understanding the relevant features of neutrino spin-flavor precession.

Considering Dirac neutrinos, from (6.15), it follows that the generalization of (6.1) to two-neutrino 𝜈_𝑒−𝜈_𝜇 mixing is, using an analogous notation, 𝑖d𝜑d𝑥^𝑒𝐿(𝑥)𝜑_𝜇𝐿(𝑥)𝜑_𝑒𝑅(𝑥)𝜑_𝜇𝑅(𝑥)𝜑=H^𝑒𝐿(𝑥)𝜑_𝜇𝐿(𝑥)𝜑_𝑒𝑅(𝑥)𝜑_𝜇𝑅(𝑥), (6.21) with the effective Hamiltonian matrix: −H=Δ𝑚²4𝐸cos2𝜗+𝑉_𝑒Δ𝑚²4𝐸sin2𝜗𝜇_𝑒𝑒𝐵_⟂(𝑥)𝜇_𝑒𝜇𝐵_⟂(𝑥)Δ𝑚²4𝐸sin2𝜗Δ𝑚²4𝐸cos2𝜗+𝑉_𝜇𝜇^∗_𝑒𝜇𝐵_⟂(𝑥)𝜇_𝜇𝜇𝐵_⟂(𝑥)𝜇_𝑒𝑒𝐵_⟂(𝑥)𝜇_𝑒𝜇𝐵_⟂(𝑥)−Δ𝑚²4𝐸cos2𝜗Δ𝑚²𝜇4𝐸sin2𝜗_∗𝑒𝜇𝐵_⟂(𝑥)𝜇_𝜇𝜇𝐵_⟂(𝑥)Δ𝑚²4𝐸sin2𝜗Δ𝑚²4𝐸cos2𝜗, (6.22) where we have used the constraint (6.17) for the transition magnetic moments. The matter potential can generate resonances, which occur when two diagonal elements of H become equal. Besides the standard MSW resonance in the 𝜈_𝑒𝐿⇆𝜈_𝜇𝐿 channel for 𝑉_CC=Δ𝑚²cos2𝜗/2𝐸 (see [1–4]), there are two possibilities (1)There is a resonance in the 𝜈_𝑒𝐿⇆𝜈_𝜇𝑅 channel for 𝑉_𝑒=Δ𝑚²2𝐸cos2𝜗. (6.23) The density at which this resonance occurs is not the same as that of the MSW resonance, because of the neutral-current contribution to 𝑉_𝑒=𝑉_CC+𝑉_NC. The location of this resonance depends on both 𝑁_𝑒 and 𝑁_𝑛. (2)There is a resonance in the 𝜈_𝜇𝐿⇆𝜈_𝑒𝑅 channel for 𝑉_𝜇=−Δ𝑚²2𝐸cos2𝜗. (6.24) If cos2𝜗>0, this resonance is possible in normal matter, since the sign of 𝑉_𝜇=𝑉_NC is negative, as one can see from (6.8).

In practice, the effect of these resonances could be the disappearance of active 𝜈_𝑒𝐿 or 𝜈_𝜇𝐿 into sterile right-handed states.

Let us consider now the more interesting case of Majorana neutrinos, which presents two fundamental differences with respect to the Dirac case. (A)If CP is conserved, Majorana neutrinos can have only a transition magnetic moment 𝜇_𝑒𝜇=−𝜇_𝜇𝑒=−𝜇^∗_𝑒𝜇 in the flavor basis. (B)The right-handed states are not sterile, but they interact as right-handed Dirac antineutrinos.

Assuming CP conservation, the evolution equation of the amplitudes is given by (6.21) with the effective Hamiltonian matrix: −H=Δ𝑚²4𝐸cos2𝜗+𝑉_𝑒Δ𝑚²4𝐸sin2𝜗0𝜇_𝑒𝜇𝐵_⟂(𝑥)Δ𝑚²4𝐸sin2𝜗Δ𝑚²4𝐸cos2𝜗+𝑉_𝜇−𝜇_𝑒𝜇𝐵_⟂(𝑥)00𝜇_𝑒𝜇𝐵_⟂(𝑥)−Δ𝑚²4𝐸cos2𝜗−𝑉_𝑒Δ𝑚²4𝐸sin2𝜗−𝜇_𝑒𝜇𝐵_⟂(𝑥)0Δ𝑚²4𝐸sin2𝜗Δ𝑚²4𝐸cos2𝜗−𝑉_𝜇. (6.25) Again, besides the standard MSW resonance in the 𝜈_𝑒𝐿⇆𝜈_𝜇𝐿 channel, there are two possible resonances(1)There is a resonance in the 𝜈_𝑒𝐿⇆𝜈_𝜇𝑅 channel for 𝑉_CC+2𝑉_NC=Δ𝑚²2𝐸cos2𝜗. (6.26)(2)There is a resonance in the 𝜈_𝜇𝐿⇆𝜈_𝑒𝑅 channel for 𝑉_CC+2𝑉_NC=−Δ𝑚²2𝐸cos2𝜗. (6.27)The location of both resonances depend on both 𝑁_𝑒 and 𝑁_𝑛. If cos2𝜗>0, only the first resonance can occur in normal matter, where 𝑁_𝑛≃𝑁_𝑒/6. A realization of the second resonance requires a large neutron number density, as that in a neutron star.

The neutrino spin oscillations in a transverse magnetic field with a possible rotation of the field-strength vector in a plane orthogonal to the neutrino-propagation direction (such rotating fields may exist in the convective zone of the Sun) have been considered in [160–163]. The effect of the magnetic-field rotation may substantially shift the resonance point of neutrino oscillations. Neutrino spin oscillations in electromagnetic fields with other different configurations, including a longitudinal magnetic field and the field of an electromagnetic wave, were examined in [164–169].

It is possible to formulate a criterion [163] for finding out if the neutrino spin and spin-flavor precession is significant for given neutrino and background medium properties. The probability of oscillatory transitions between two neutrino states 𝜈_𝛼𝐿⇆𝜈_𝛽𝑅 can be expressed in terms of the elements of the effective Hamiltonian matrices (6.22) and (6.25) as 𝑃_{𝜈𝛼𝐿⇆𝜈𝛽𝑅}=sin²𝜗_effsin²𝑥𝜋𝐿_eff, (6.28) where sin²𝜗_eff=4H²_𝛼𝛽4H²_𝛼𝛽+H_𝛽𝛽−H_𝛼𝛼²,𝐿_eff=2𝜋4H²_𝛼𝛽+H_𝛽𝛽−H_𝛼𝛼². (6.29) The transition probability can be of order unity if the following two conditions hold simultaneously: (1) the amplitude of the transition probability must be sizable (at least sin²𝜗_eff≳1/2); (2) the neutrino path length in a medium with a magnetic field should be longer than half the effective length of oscillations 𝐿_eff. In accordance with this criterion, it is possible to introduce the critical strength of a magnetic field 𝐵_cr which determines the region of field values 𝐵_⟂>𝐵_cr at which the probability amplitude is not small (sin²𝜗_eff>1/2): 𝐵_cr=12𝜇H_𝛽𝛽−H_𝛼𝛼², (6.30) where 𝜇 is 𝜇_𝑒𝑒, 𝜇_𝜇𝜇, 𝜇_𝑒𝜇, or 𝜇_𝜇𝑒 depending on the type of neutrino transition process in question.

Consider, for instance, the case of 𝜈_𝑒𝐿⇆𝜈_𝜇𝑅 transitions of Majorana neutrinos. From (6.25) and (6.30), it follows [163] that 𝐵_cr=12𝜇Δ𝑚²cos2𝜗−√2𝐸2𝐺_F𝑁_eff, (6.31) where 𝑁_eff=𝑁_𝑒−𝑁_𝑛. For getting numerical estimates of 𝐵_cr, it is convenient to rewrite (6.31) in the following form: 𝐵_cr𝜇≈43^𝐵𝜇cos2𝜗Δ𝑚²eV²MeV𝐸−2.5×10⁻³¹𝑁_effcm⁻³Gauss. (6.32)

An interesting feature of the evolution equation (6.21) in the case of Majorana neutrinos is that the interplay of spin precession and flavor oscillations can generate 𝜈_𝑒𝐿→𝜈_𝑒𝑅 transitions [170]. Since 𝜈_𝑒𝑅 interacts as right-handed Dirac antineutrinos, it is often denoted by 𝜈_𝑒𝑅, or only 𝜈_𝑒, and called “electron antineutrino.” This state can be detected through the inverse 𝛽-decay reaction: 𝜈_𝑒+𝑝→𝑛+𝑒⁺, (6.33) having a threshold 𝐸_th=1.8MeV.

The possibility of 𝜈_𝑒→𝜈_𝑒 transitions generated by spin-flavor precession of Majorana neutrinos is particularly interesting for solar neutrinos, which experience matter effects in the interior of the Sun in the presence of the solar magnetic field (see [10, 171]). Taking into account the dominant 𝜈_𝑒→𝜈_𝑎 transitions due to neutrino oscillations (see [1–4]), with 𝜈_𝑎=cos𝜗₂₃𝜈_𝜇−sin𝜗₂₃𝜈_𝜏 and sin²2𝜗₂₃>0.95 (90% C.L.) [66], the probability of solar 𝜈_𝑒→𝜈_𝑒 transitions is given by [172] 𝑃_{𝜈𝑒→𝜈𝑒}≃1.8×10⁻¹⁰sin²2𝜗₁₂𝜇_𝑒𝑎10⁻¹²𝜇_𝐵𝐵_⟂0.05𝑅_⊙10kG², (6.34) where 𝜇_𝑒𝑎 is the transition magnetic moment between 𝜈_𝑒 and 𝜈_𝑎, sin²2𝜗₁₂=0.857^+0.023_−0.025 [66] and 𝑅_⊙ is the radius of the Sun.

It is also possible that spin-flavor precession occurs in the convective zone of the Sun, where there can be random turbulent magnetic fields [173–175]. In this case [176], 𝑃_{𝜈𝑒→𝜈𝑒}≈10⁻⁷𝑆²𝜇_𝑒𝑎10⁻¹²𝜇_𝐵²𝐵20kG²3×10⁴km𝐿_max^𝑝−1×8×10⁻⁵eV²Δ𝑚²₂₁^𝑝𝐸10MeV^𝑝cos²𝜗₁₂0.7^𝑝, (6.35) where 𝑆 is a factor of order unity depending on the spatial configuration of the magnetic field, 𝐵 is the average strength of the magnetic field at the spatial scale 𝐿_max, which is the largest scale of the turbulence, 𝑝 is the power of the turbulence scaling, Δ𝑚²₂₁=7.50^+0.19_−0.20×10⁻⁵eV² [66], and 𝐸 is the neutrino energy. A possible value of 𝑝 is 5/3 [173–175], corresponding to Kolmogorov turbulence. Conservative values for the other parameters are 𝐵=20kG and 𝐿_max=3×10⁴km.

In 2002, the Super-Kamiokande Collaboration established for the flux of solar 𝜈_𝑒’s a 90% C.L. an upper limit of 0.8% of the Standard Solar Model (SSM) neutrino flux in the range of energy from 8 to 20 MeV [177] by taking as a reference the BP00 SSM prediction 𝜙^BP00_8B=5.05×10⁶cm⁻²s⁻¹ for the solar ⁸B flux [178] and assuming an undistorted ⁸B spectrum for the 𝜈_𝑒’s. This limit was improved in 2003 by the KamLAND Collaboration [179] to 2.8×10⁻⁴ of the BP00 SSM prediction at 90% C.L. by measuring 𝜙_𝜈𝑒<370cm⁻²s⁻¹ (90% C.L.) in the energy range 8.3–14.8 MeV, which corresponds to 𝜙_𝜈𝑒<1250cm⁻²s⁻¹ (90% C.L.) in the entire ⁸B energy range assuming an undistorted spectrum.

Recently, the Borexino collaboration established the best limit on the probability of solar 𝜈_𝑒→𝜈_𝑒 transitions [180]: 𝑃_{𝜈𝑒→𝜈𝑒}<1.3×10⁻⁴(90%C.L.), (6.36) by taking as a reference 𝜙^SSM_8B=5.88×10⁶cm⁻²s⁻¹ [181] and assuming an undistorted ⁸B spectrum for the 𝜈_𝑒’s. They measured 𝜙_𝜈𝑒<320cm⁻²s⁻¹ (90% C.L.) for 𝐸_𝜈𝑒>7.3MeV, which corresponds to 𝜙_𝜈𝑒<760cm⁻²s⁻¹ (90% C.L.) in the entire ⁸B energy range assuming an undistorted spectrum.

The implications of the limits on the flux of solar 𝜈_𝑒’s on Earth for the spin-flavor precession of solar neutrinos have been studied in several papers [172–175, 182–185], taking into account the dominant 𝜈_𝑒→𝜈_𝜇,𝜈_𝜏 transitions due to neutrino oscillations (see [1–4]). Using (6.34) and (6.36), we obtain 𝜇_𝑒𝑎≲1.3×10⁻¹²7MG𝐵_⟂0.05𝑅_⊙𝜇_B, (6.37) with 600G≲𝐵_⟂(0.05𝑅_⊙)≲7MG [180]. In the case of spin-flavor precession in the convective zone of the Sun with random turbulent magnetic fields, (6.35) and (6.36) give, assuming 𝑝=5/3, 𝜇_𝑒𝑎≲4×10⁻¹¹S⁻¹20kG𝐵𝐿_max3×10⁴km^1/3𝜇_B. (6.38)

The spin-flavor mechanism was also considered [186] in order to describe time variations of solar-neutrino fluxes in Gallium experiments. The effect of a nonzero neutrino magnetic moment is also of interest in connection with the analysis of helioseismological observations [187].

The idea that the neutrino magnetic moment may solve the supernova problem, that is, that the neutrino spin-flip transitions in a magnetic field provide an efficient mechanism of energy transfer from a protoneutron star, was first discussed in [188] and then investigated in some detail in [189–191]. The possibility of a loss of up to half of the active left-handed neutrinos because of their transition to sterile right-handed neutrinos in strong magnetic fields at the boundary of the neutron star (the so-called boundary effect) was considered in [163].

7. Summary and Perspectives

In this review, we have considered the electromagnetic properties of neutrinos with focus on the most important issues related to the problem. The main results discussed in the paper can be summed up as follows.

In the most general case, the neutrino electromagnetic vertex function is defined in terms of four form factors: the charge, dipole magnetic, and electric and anapole form factors. This decomposition is consistent with Lorentz and electromagnetic gauge invariance. The four form factors at zero momentum transfer 𝑞²=0 are, respectively, the neutrino charge, magnetic moment, electric moment, and anapole moment. These quantities contribute to elements of the scattering matrix and describe neutrino interactions with real photons.

An important characteristic of neutrino electromagnetic properties is that they are different for Dirac and Majorana neutrinos. In particular, Majorana neutrinos cannot have diagonal magnetic or electric moments. Thus, studies of neutrino electromagnetic interactions can be used as a procedure to distinguish whether a neutrino is a Dirac or Majorana particle.

Moreover, CP invariance in the lepton sector puts additional constraints on the neutrino form factors and can be tested with experimental probes of neutrino electromagnetic interactions.

Up to now, no effect of neutrino electromagnetic properties has been found in terrestrial laboratory experiments and in the analyses of astrophysical and cosmological data. However, massive neutrinos have nontrivial electromagnetic properties in a wide set of theoretical frameworks, including the simplest extension of the Standard Model with inclusion of singlet right-handed neutrinos. Therefore, the search for nonvanishing neutrino electromagnetic properties is of great interest for experimentalist and theorists.

The neutrino dipole magnetic (and also electric) moment is theoretically the most well studied and understood among the neutrino electromagnetic moments. In models which extend the Standard Model with the addition of singlet right-handed neutrinos, a Dirac neutrino has a nonzero magnetic moment proportional to the neutrino mass, which yields a very small value for the magnetic moment, less than about 3×10⁻¹⁹𝜇_𝐵 for a neutrino mass smaller than 1 eV (3.5). Extra terms contributing to the magnetic moment of the neutrino that are not proportional to the neutrino mass may exist, for example, in the framework of left-right symmetric models. In this type of models, as well as in other generalizations of the Standard Model, as, for instance, in supersymmetric and extradimension models, values of the neutrino magnetic moment much larger than 10⁻¹⁹𝜇_𝐵 can be obtained.

The most severe terrestrial experimental upper bound for the effective electron antineutrino magnetic moment, 𝜇_𝜈𝑒≤2.9×10⁻¹¹𝜇_𝐵 at 90% C.L. (3.31), has been recently obtained in the direct antineutrino-electron scattering experiment performed by the GEMMA collaboration [58]. This value is still about an order of magnitude weaker than the upper bound 𝜇_𝜈≲3×10⁻¹²𝜇_𝐵 (5.8) obtained from astrophysics (considering the cooling of red giant stars) [153].

There is a gap of some orders of magnitude between the present experimental limits ~10⁻¹¹÷10⁻¹²𝜇_𝐵 on neutrino magnetic moments and the predictions of different extensions of the Standard Model which hint at a range ~10⁻¹⁴÷10⁻¹⁵𝜇_𝐵 [28, 29, 80] (see Section (3.5)). The terrestrial experimental constraints have been improved by only one order of magnitude during a period of about twenty years. Further improvements are very important, but unfortunately at the moment there is no new idea which could lead to fast improvements in the near future. On the other hand, astrophysical studies could allow significant improvements of the sensitivity to nontrivial neutrino electromagnetic properties and maybe find a positive indication in their favor. In particular, neutrino flows in extreme astrophysical environments with very strong magnetic fields are sensitive to small values of the neutrino electromagnetic moments. An example is the modelling of neutrinos propagation during core-collapse supernovae (see [192] and references therein) where very strong magnetic fields are believed to exist and in which the influence of neutrino electromagnetic properties has not yet been taken into account.

Acknowledgments

The work of A. Studenikin on this paper has been partially supported by the Russian Foundation for Basic Research (Grants no. 11-02-01509 and no. 12-02-06833) and the Ministry of Education and Science of Russia (State Contract no. 12.741.11.0180 and Projects no. 2012-1.2.1-12-000-1012-1958 and no. 2012-1.1-12-000-1011-6097). C. Giunti would like to thank the Department of Physics of the University of Torino for hospitality and support.