#### Abstract

The full energy shift of a massive Dirac neutrino in magnetized electron-positron plasma was investigated using the Matsubara imaginary time and real time formalisms. The neutrino dispersion in the magnetized medium was analyzed as a function of the neutrino spin and mass. It was shown that in a superstrong magnetic field the CP-symmetric plasma contribution to the neutrino energy greatly exceeds the analogous correction in the field-free case. The contribution of plasma to the anomalous magnetic moment of a neutrino was obtained.

#### 1. Introduction

In recent years, investigating neutrino properties in a dense medium at finite temperatures in the presence of external electromagnetic fields has become of considerable interest in the field of neutrino physics and astrophysics. A comprehensive review of previous studies related to collective neutrino-plasma interactions, as well as their possible astrophysical applications, is presented in [1–21]. Experimental observations of neutrino oscillations (see, e.g., [22, 23]) have provided reliable evidence that neutrinos possess nonzero masses and that they mix. The energy dispersion relation for massless Dirac neutrinos in the presence of a constant magnetic field in plasma has been studied extensively in the literature [24–33].

Nevertheless, a calculation of the neutrino self-energy in a magnetized medium that also accounts for the effects of the spin and mass of the neutrino has not been presented.

In this study, we shall concentrate on studying the full energy shift and anomalous magnetic moment (AMM) for a massive Dirac neutrino in magnetized electron-positron plasma. The paper is organized as follows. In Section 2, we discuss the general properties of the neutrino self-energy in a magnetized medium and calculate the contributions from the weak charge current to the one-loop energy shift and anomalous magnetic moment of an electron neutrino using the Matsubara temperature Green’s functions. In Section 3, we determine the contribution of the neutral weak current to the neutrino dispersion and AMM in a magnetized medium, and approximate formulas are derived for each flavor of the propagating neutrino.

In Section 4, we calculate the CP-symmetric plasma contribution of the neutrino dispersion relation in a strong magnetic field using a real time representation of Green’s function of an ideal electron-positron gas in a constant magnetic field. We also estimate numerical values of the expected contributions to the energy shift and AMM of a neutrino. Finally, we summarize our results in Section 5.

#### 2. Neutrino Self-Energy in a Magnetized Medium: Imaginary Time Representation and Charged Weak Current Contribution

The electron neutrino interacts with electrons due to both the neutral weak current and the charged weak current, whereas neutrinos of other types interact with electrons only through the neutral current. In the Feynman gauge, the three Feynman diagrams contribute to the neutrino self-energy in a magnetized medium: the bubble diagram with the -boson, the bubble diagram with the charged scalar, and the tadpole diagram with the -boson [27–29]. In a neutral medium, only the bubble diagram contributes, while in a charged medium both diagrams should be considered [26–30]. In the one-loop approximation, the mass operator determines the radiative correction to the neutrino energy in the form where is the interaction time and is the neutrino bispinor in the zero approximation without taking into account the radiative corrections.

The AMM of a lepton induced by an external field is determined by the terms in the radiative energy shift that are proportional to [34]: where is the neutrino energy, , is an external electromagnetic field potential, is the dual electromagnetic field tensor, is the particle polarization 4-vector defined as [34] and is twice the mean value of the neutrino spin vector in the rest frame. Below we shall use the temperature (Matsubara) Green’s functions method, which is also known as the imaginary time technique. For further presentation, it will suffice to mention that the diagram technique in the imaginary time formalism framework is analogous to the Feynman rules in the ordinary quantum field theory. The diagram technique in the imaginary time representation was developed according to the following replacements [35, 36]:where is the chemical potential of a macroscopic system at a finite temperature , , for fermions, and for bosons. We present here analytical expressions for the temperature Green’s function of an ideal electron-positron gas in a constant magnetic field. The electron Green’s function in a homogeneous and stationary magnetic field in the proper-time representation is given by the formula [37, 38] whereand is the charge of the electron.

The temperature Green’s function is obtained from formulas (5) and (6) by replacing with

The contribution of the -boson to the mass operator of the neutrino in the one-loop approximation is determined by the expression [39, 40]where and are the electron and -boson propagators in a magnetic field specified by the potential In the Feynman gauge the -boson propagator has the form [38, 41, 42] where we have introduced the notation and in the matrix the following elements are nonzero: Accounting to formulas (1) and (5)–(12), we obtain the following exact result for the charged weak current contribution to the massive neutrino energy shift in the magnetized medium: where , , and , and we use the shortened notation for the average value of the Pauli matrices over the neutrino spinor.

We note that, in the complex plane of the variable , expression (13) is only defined for a discrete set of points on the imaginary axes. The analytical continuation of this value to the upper half-plane () defines the real component of the neutrino energy shift. This analytical continuation is possible in principle, though generally speaking it is not a trivial problem to solve. In this study, it is essential that the temperature and time Green’s functions are defined in terms of the same spectral density. The Gaussian integrals over variables in (13) can be performed, but the resulting expression is too complex to be presented in this paper. Nevertheless, we shall examine some limiting cases.

Below, I consider temperatures much lower than the -boson mass. In this case, we can neglect the thermal -boson contribution to the neutrino energy shift, which will be suppressed by the factor

First, we consider the neutrino energy shift in a pure magnetic field at and , where is the electron mass. To obtain the desired result, it is necessary to replace the summation in (13) with integration over variable according to (4). Then, after performing four Gaussian integrals over variables and , the results can be written as follows:where Results (14)-(15) coincide with the radiative energy shift of the massive Dirac neutrino in an external magnetic field and were originally published in [43]. As a consequence of (14)-(15), for comparatively weak magnetic fields, when the condition is fulfilled, the neutrino mass shift in a constant magnetic field is described by the formula where the spin number describes the particle spin orientation as being either along or contrary to the magnetic field. Using formulas (2) and (17), one can obtain the static AMM of the Dirac neutrino (DN) with the mass where is the Bohr magneton. The neutrino magnetic moment in (18) agrees with the results of [44, 45].

Result (13) was further used to describe the -boson contribution to the energy shift and AMM of the massive neutrino in magnetized electron-positron plasma.

Our goal is to calculate the complete energy shift and AMM, including the effects of the medium and linear terms in the magnetic field. To begin, we considered the energy shift in the field-free case. Using the summation formula [46]the following exact expression can be obtained for the neutrino energy shift in the electron-positron plasma when :In the high-temperature region and for relativistic neutral plasma, when the conditions are fulfilled, the temperature correction to the massive polarized DN is defined by the formula where is the neutrino momentum.

For the case of cold degenerate plasma, one obtains the following result from (20): In this equation, the density of the free electron gas is related to the chemical potential by the formula Result shown in (23) is valid when the conditions are fulfilled.

In comparison with the Wolfenstein formula for the energy shift of an electron neutrino moving in a dense electron medium in the absence of an external field, the novelty of our result (23) includes the dependence from other neutrino parameters such as the neutrino spin, energy, and momentum. If , (23) coincides with the Wolfenstein formula for the energy shift of a massless electron neutrino in a dense electron medium.

Further, we shall consider the neutrino energy shift in the case of a weak magnetic field, when the condition is satisfied simultaneously with the conditions shown in (16). First, we consider neutrinos propagating through a magnetized, charged medium, where the chemical potential that shows the asymmetry between the particles and antiparticles is and conditions (25) are also fulfilled. The pure magnetic field contribution to the energy shift is obtained immediately from (13) and is accounted for by formula (19): The case of a magnetized neutral medium should also be explored when conditions (16) and (21) are satisfied. In the approximation considered, the following result can be derived from (13):Using the relation we obtain the following expressions for the full -contributions to the massive electron neutrino energy shift for neutral and charged magnetized plasma cases, respectively:By accounting for (2), (30), and (31), we find that the -contributions to the plasma correction of the neutrino AMM have the asymptotesThe separate consideration shows that the contribution of the charged scalar to the neutrino energy shift and AMM is negligibly small compared to the -contribution, and therefore we omit it here.

#### 3. Neutral Weak Current Contribution to the Energy Shift

The tadpole contribution to the mass operator of the neutrino is determined by the expression where is the vacuum propagator of the neutral -boson at zero momentum, is the lepton propagator given by (6), the quantities and are the vector and axial-vector coupling constants, which come in the neutral current interaction of the electrons, , and . Using the imaginary time technique, the tadpole contribution to the self-energy in the case of a weak magnetic field becomeswhereIn the expressions shown above, the net number density of the electrons (positrons) and parameter are defined as It follows from (37) that the neutral weak current contribution to the AMM for each type of neutrino in a magnetized medium is described by the formula Thus, for and the AMM is defined by formula (39), and in the CP-symmetric plasma, the neutral weak current does not contribute to the neutrino dispersion and AMM. We note that the neutral weak current does not contribute to the vacuum AMM of a neutrino either.

In the limiting case (29) for a completely degenerate electron gas, the field dependent contribution of the neutral weak current to the self-energy and AMM of a Dirac neutrino moving in weak magnetic field is determined by the asymptotes The plasma contribution to the AMM of an electron neutrino, determined by the summation of weak charged and neutral current contributions, is defined by formulas (32)-(33), (39), and (41). Our results for -boson contribution in (32), which were obtained using the method of a finite temperature (Matsubara) Green’s function, coincide with the corresponding results of papers [47, 48] and differ by factor of from the result published in [49]. The sum of the corrections in (33) and (41) agrees with the result from [48] for charged plasma correction to the AMM of an electron neutrino in a weak magnetic field.

It should be emphasized that if the tested neutrino moves in a medium containing a neutrino background from the same family, there is a diagram similar to tadpole diagram, but with a loop, that also contributes to the neutrino dispersion. The corresponding contribution to the energy shift may be obtained from (22) using the following replacements [24]:For the CP-symmetric medium,where is the -boson mass. As can be observed from a comparison, in the case of neutral plasma, the -contribution (43) is on the same order of magnitude as the -contribution (22). By accounting for results (30) and (43) in a comparatively weak magnetic field, the full energy shift of an electron neutrino in the neutral plasma is determined by the formulaThe corresponding result in the case of charged-asymmetric electron-positron plasma is determined by summing the charged weak current and neutral weak current contributions defined by formulas (31) and (36)-(37), respectively: In the limiting case, when , results (44) and (45) coincide with the energy shift of a massless left-handed neutrino in the magnetized electron-positron plasma obtained in [28, 29].

#### 4. Neutrino Dispersion in a CP-Symmetric Magnetized Medium: Real Time Method and Strong Magnetic Field

The presence of strong magnetic fields would be expected to influence the propagation of neutrinos in the early universe producing an imprint in neutrino oscillations at those epochs. The only tool to distinguish between different neutrino flavors is the analysis of bubble impact. Therefore, with regard to magnetic field effects on neutrino oscillations, the bubble contribution is the crucial factor.

As it was already mentioned, in the CP-symmetric plasma, the field dependent contribution on the energy shift of an electron neutrino is the -contribution. The interaction of a neutrino with a neutrally magnetized medium in this section is described based on the mass operator (1) constructed from the real time Green’s function of an electron-positron plasma at finite temperature and density in a constant magnetic field [39, 50, 51]. We also neglect thermal corrections to the -propagator and use the vacuum propagator defined by formulas (10)–(12). Using the explicit form of the propagators and (1), we represent the neutrino energy shift in the formwhere is the radiative energy shift of the neutrino in a magnetic field at zero temperature and zero density of the medium. This energy shift, as mentioned above, was investigated in detail in [43] and is omitted in this paper. The second term in (46) accounts for the effects of interest that arise from the finite temperature and density of the medium and is determined by the expressionIn this formula, is a Laguerre function [52, 53] with argument is the -component of the momentum of the intermediate -boson, is the four-momentum of the neutrino, and coefficients and , which account for the polarization state of the neutrino, arewhere and the spin number describes the neutrino spin orientation along or contrary to the magnetic field In (47), a summation is carried out over the positive and negative frequency states of the Dirac equation solution in a constant magnetic field, and the electron energy levels in a stationary magnetic field are determined by the formula [54]where is the principle quantum number and is the electron momentum projection on the magnetic field direction Number is known as the Landau level number [55]. We chose the transverse-polarization operator as the operator determining the polarization state of the Dirac neutrino and plasma particles [52, 53]:Hereand are the Pauli matrices.

Next we perform the summation in (47) over the principle quantum number using the formula [52, 53]where is a Hermit function.

As a result, the -contribution to the electron neutrino energy shift in the electron-positron plasma at temperature and chemical potential in an external field pointing in the direction is written in the form where and . In the following section, results (47) and (54) are investigated in various limiting cases.

A further simplification of (54) occurs if the neutrino moves in the direction perpendicular to the magnetic field, when . In this case, the plasma contribution to the radiative energy shift of the neutrino can be represented as the sum of two parts: where the part of the energy shift, , that depends explicitly on the orientation of the neutrino spin and the part, , of the energy shift that does not depend on the orientation of the neutrino spin are, respectively, It should be emphasized that the full dependence of the radiative energy shift from the neutrino spin number in the case when the neutrino is moving perpendicular to the magnetic field is determined only by the part of the energy shift that is explicitly dependent on the neutrino spin orientation [49]; that is, where is the neutrino AMM in the magnetized electron-positron plasma.

Let us first find the asymptotes for the values and in (56) for comparatively weak magnetic fields, when the following conditionsare fulfilled.

In a CP-symmetric plasma, when the conditionsare also fulfilled, the contribution on the electron neutrino energy shift in the field-free case and the AMM asymptotes are described by the formulas In the limiting case of a degenerate electron gas, when the following conditions are fulfilled it follows from (56) that Notice that result (63) coincides with the Wolfenstein formula for the energy shift of an electron neutrino moving in a dense electron medium in the absence of an external field, and it has no dependence on the spin orientation.

It should be noticed that the early universe was almost charge symmetric, with a particle-antiparticle asymmetry of only [17, 56].

Let us first investigate the neutrino dispersion in the neutral medium with a strong magnetic field under the conditions Such a strong magnetic field can be expected to exist in the early universe before the period of neutrino decoupling era, which is believed to occur around the energy scale of 1 MeV when the age of the universe was approximately 1 second [30, 32].

Because in the strong-field limit (65) the gap between the electron Landau levels is larger than the temperature, only the lowest Landau level contributions in formula (47) of the plasma contribution to the neutrino energy shift should be used to be consistent.

From (47) in the LLL approximation [30], the asymptote for the electron neutrino energy shift in a CP-symmetric electron-positron plasma in a strong magnetic field takes the form It follows from (66) that under the conditions shown in (65) the plasma contribution to the neutrino AMM is determined by formula (61) obtained in the weak field approximation.

We also note that in a superstrong magnetic field, the CP-symmetric plasma contribution (66) to the neutrino energy greatly exceeds the similar correction (22) in the field-free case:To estimate the energy shift of the neutrino, we set , , and MeV. Then, from (22) and (66) we have eV and eV. For , , and MeV, one can obtain that eV and eV, respectively. As follows from the estimates given in case (65), the energy shift of the neutrino can be substantially greater than its value in the magnetic field free case.

In the limiting case of a degenerate electron gas and relatively strong magnetic fields, when the first term of the expansion of (49) in the parameter corresponds to replacing the Fermi distribution function by a -function and the energy shift and AMM of the neutrino are determined by (47). In this case, it is only necessary to take the term in the sum. It follows from (68) that for at cm^{−3} only the ground level can be occupied with electrons. For example, in modern models of neutron stars, the star core (with its thickness ~0.1 of its radius) appears as a crystal lattice of the ions immersed in a highly degenerate gas of relativistic electrons with an electron density cm^{−3}, temperature K, and magnetic field strength G [57–59]. Fields of the order of 10^{16} G and larger could exist in the magnetars [30, 60]. The typical energy of neutrinos emitted via URCA processes and at the conditions of a supernova core is about 1 MeV and MeV, respectively.

If we require also that the condition be fulfilled, and taking into account the relationship between the chemical potential and the density of the electron gas [57, 59] we find the following asymptotes in the first approximation: To estimate numerical values of the obtained correction (70) to the static AMM of a neutrino in the limiting case (68) we set which corresponds to a chemical potential value given by . Then, from (70), we have . For , MeV, and MeV one can find and , respectively. Assuming that , MeV, and MeV, we obtain that and respectively.

Now let us suppose that the neutrino moves parallel (or antiparallel) to the magnetic field. From (54) it follows that in this case the neutrino energy shift can be determined by the following exact formula: The structure of (72) is in agreement with the results obtained when a neutrino moves along the magnetic field; its chirality is conserved and it coincides with the spin polarization along the field, given by the operator .

In a comparatively weak magnetic field and for neutral relativistic plasma, when conditions (58) and (59) are fulfilled, the energy shift (68) in the field-free case takes the form The pure magnetic field contribution to the energy shift is determined by the asymptotesFinally, we shall discuss the connection between the neutrino dispersion and AMM if the neutrino moves parallel (or antiparallel) to the magnetic field. To give the correct interpretation for result (74), we notice that it can be written in the form where is the neutrino polarization 4-vector defined by the formula (3). The first terms in (75) in the limiting case, when , coincide with the energy shift of a massless left-handed neutrino in the magnetized electron-positron plasma obtained in [28, 29]. The AMM of a neutrino is determined by the term in the energy shift that is proportional to and coincides with the results in (61) obtained for a neutrino moving perpendicular to a magnetic field.

#### 5. Conclusions

We argue that the results of this work represent an advance on previous studies [24–33] examining the neutrino dispersion in the magnetized medium. Our calculations may also have practical applications in the modern studies of cosmology and supernova physics, which investigate the possible existence of massive sterile neutrinos. At present, the nonrelativistic sterile neutrinos with masses on the order of 2 to 50 keV are treated as the most popular candidates for particles that form dark matter [7, 8, 12–15]. It is clear, however, that the presented calculations could not be simply applied to massive sterile neutrinos. This is a separate problem that depends on the mixing scheme, type, and magnetic moment (transition or diagonal) of the neutrino and will be considered elsewhere.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The author is grateful to V. Ch. Zhukovsкii, A. E. Shabad, A. V. Borisov, and A. I. Ternov for useful discussions.