# Neutrino propagation in media and axis of complete polarization

## Abstract

We construct a spectral representation of neutrino propagator in moving matter or in external magnetic field. In both cases there exists fixed four-dimensional axis of polarization, such that the corresponding spin projectors commute with propagator. As a result, all eigenvalues of propagator and, consequently, dispersion laws for neutrino in media are classified according to spin projection onto this axis. Use of the found spin projectors simplifies essentially the eigenvalue problem and allows to build spectral representation of propagator in moving matter or external magnetic field in analogy with the vacuum propagator.

## 1 Introduction

Neutrino physics, actively developing in last decades, is related with a wide spectrum of physical problems, including the astrophysical ones. Propagation of neutrinos in a dense matter or magnetic field leads to modification of neutrinos oscillation picture and appearance of new effects. The most prominent effect in neutrinos passing through matter is related with resonance amplification of oscillations (MSW effect) [1, 2], which solves the solar neutrino problem, see reviews [3, 4, 5].

After it a huge interest was generated to further investigations of different properties of media and its influence on fine aspects of flavor oscillations. Different methods were developed for solution of corresponding equations in media with varied density [6, 7, 8, 9, 10]. The movement of matter and its polarization, which can arise in magnetic field were taken into account [11, 12, 13, 14, 15]. Interesting results were obtained in investigations of spin dynamics in matter and transitions between different spin states [13, 16, 17]. As for applications in astrophysics, there exists a variety of conditions for neutrinos propagation: in Earth, Sun, in vicinity of Supernova – see reviews [18, 19]. We mentioned here only some aspects of investigations and only small part of relevant publications.

Possibility for neutrino to have an anomalous magnetic moment and its experimental manifestations was discussed for a long time [20, 21, 22, 23, 24, 25]. In Standard Model (SM) the magnetic moment of neutrino is arised due to loop corrections and is proportional to neutrino mass. It leads to extreme smallness of neutrino magnetic moment in SM, so the present-day interest for this subject (both in theory and experiment) is related first of all with search of new physics beyond the SM [26, 27].

Most transparent way to describe mixing and oscillations phenomena in neutrinos system is to use the quantum-mechanical equations (Schrödinger or Dirac), but the most justified is the Quantum Field Theory (QFT) approach, where production, propagation and detection of neutrino looks like a macroscopic Feynman diagram [28, 29, 30, 31, 32, 33, 34]. The necessary element of such description is the neutrino propagator.

In the present paper we build a spectral representation of neutrino propagator in matter moving with constant velocity or in constant homogeneous magnetic field.^{1} In this representation based on the eigenvalue problem a propagator looks as a sum of single poles, accompanied by orthogonal matrix projectors. Such form of propagator gives the simplest and most convenient algebraic construction and means in fact a complete diagonalization. A spectral representation was discussed earlier for dressed fermion propagator in theory with parity violation [36] and for dressed matrix propagator with mixing of few fermionic fields [37]. Note that the problem of the neutrino propagator in the presence of matter, including account for possible effect of matter motion, was also discussed in [38] (see also [39]).

In solving the eigenvalue problem for neutrino propagator in media, we find a new aspect, related with polarization of neutrino: there exist spin projectors with fixed polarization 4-vector, commuting with propagator. The properties of these spin projectors (both in matter and magnetic field) allows to reduce the algebraic problem for media to the vacuum case, properties of media modify only scalar coefficients of matrix equations.

In Sect. 2 we construct the spectral representation of neutrino propagator in a matter, moving with constant velocity. The key moment is the presence of generalized spin projectors (6), commuting with propagator, which allows to simplify the eigenstate problem (16) and to get answer for inverse propagator of the most general view (1). We discuss also the particular case of propagator in framework of Standard Model, in this case it is easy to see that the spin projection on the complete polarization axis \(z^\mu \) is not conserved.

In Sect. 3 the spectral representation is build for neutrino propagator in a constant magnetic field. In this case there exists the fixed axis of complete polarization \(z^\mu \) and again the corresponding spin projector \(\varSigma (z)\) commutes with propagator. This property allows to use the same trick (reducing of number of \(\gamma \)-matrix structures “under observation” of spin projectors) to obtain an analytical expressions for eigenvalues and eigenprojectors.

In Appendix the main facts on spectral representation for matrix of general form and some details of this representation for fermion propagator in vacuum and media are collected.

## 2 Propagator in moving matter and spin projectors

### 2.1 Axis of complete polarization and basis

*p*and matter velocity

*u*, so altogether there exist eight \(\gamma \)-matrix structures in decomposition of propagator, if parity is not conserved. Most general expression for inverse propagator can be written as

Below we will solve the eigenvalue problem for inverse propagator of general form. As a starting point it is convenient to introduce \(\gamma \)-matrix basis with simple multiplicative properties.

*p*,

*u*and has properties of fermion polarization vector:

^{2}

*b*is the normalization factor, \(b=[p^{2}((up)^{2}-p^{2})]^{-1/2}\).

^{3}

*z*, one can construct the generalized off-shell spin projectors:

^{4}

*S*(

*p*,

*u*) (1) by unit matrix

Multiplicative properties of the matrix basis (11)

\(R_{1}\) | \(R_{2}\) | \(R_{3}\) | \(R_{4}\) | \(R_{5}\) | \(R_{6}\) | \(R_{7}\) | \(R_{8}\) | |
---|---|---|---|---|---|---|---|---|

\(R_{1}\) | \(R_{1}\) | 0 | \(R_{3}\) | 0 | 0 | 0 | 0 | 0 |

\(R_{2}\) | 0 | \(R_{2}\) | 0 | \(R_{4}\) | 0 | 0 | 0 | 0 |

\(R_{3}\) | 0 | \(R_{3}\) | 0 | \(R_{1}\) | 0 | 0 | 0 | 0 |

\(R_{4}\) | \(R_{4}\) | 0 | \(R_{2}\) | 0 | 0 | 0 | 0 | 0 |

\(R_{5}\) | 0 | 0 | 0 | 0 | \(R_{5}\) | 0 | \(R_{7}\) | 0 |

\(R_{6}\) | 0 | 0 | 0 | 0 | 0 | \(R_{6}\) | 0 | \(R_{8}\) |

\(R_{7}\) | 0 | 0 | 0 | 0 | 0 | \(R_{7}\) | 0 | \(R_{5}\) |

\(R_{8}\) | 0 | 0 | 0 | 0 | \(R_{8}\) | 0 | \(R_{6}\) | 0 |

One more useful property of \(\varSigma ^{\pm }\) is that “under observation” of the spin projectors (i.e. in \(S^+, S^-\) terms) \(\gamma \)-matrix structures may be simplified. Namely:

It can be seen from Table 1, that with use of the basis (11) the eigenvalue problem for inverse propagator (12) is separated into two different problems: one for \(R_{1} \ldots R_{4}\) and another for \(R_{5} \ldots R_{8}\). Every problem has two different eigenvalues.

### 2.2 Spectral representation of propagator in matter

- 1.
\(S\varPi _{k}=\lambda _{k}\varPi _{k}\), k = 1 ...4,

- 2.
\(\varPi _{i}\varPi _{j}=\delta _{ij}\varPi _{j}\),

- 3.
\(\sum \limits _{i=1}^4\varPi _{i}=1\).

### 2.3 Standard Model propagator

*R*-basis (12):

*z*(3). If we restrict ourselves by the first order in \(G_F\), then we have from (26) for solution with positive energy

*R*with Hamiltonian

In general case, at arbitrary direction of matter velocity, in spite of \([\varSigma ^{\pm },S]=0\), the spin projection on the axis \(z^{\mu }\) is not conserved: \([\varSigma ^{\pm },H]\ne 0\). Evidently, that for propagator of more general form than the Standard Model one, the spin projection on the axis of complete polarization *z* also is not a conservative value.

#### 2.3.1 Rest matter case

Let us consider in detail a particular case of SM propagator (22), when matter is in the rest (\({\mathbf{u}}=0, u_{0}=1\)). In this case, according to Eq. (32), spin projection is conserved and polarization vector \(z^\mu \) also corresponds to helicity state (33).

Thus, for the rest matter the well-known fact [43, 44] is reproduced that neutrino with definite helicity has a definite law of dispersion in matter.

## 3 The propagation of neutrino in an external magnetic field

In previous section we found that in moving matter there exists an axis of complete polarization \(z^{\mu }\) (3), and corresponding spin projectors (6) commute with the propagator. A similar situation arises when neutrino propagates in a magnetic field.

^{5}we can construct a spin projector with the same properties as in the case of neutrino propagation in a matter:

*R*is obvious: \(\varPsi ^{\pm }=\varSigma ^{\pm } \varPsi _0\), therefore the system looks like this:

*R*are equal to \(\pm 1\), from (50) we can find the useful relation

The inverse propagator in the external field (46), (55) is non-covariant (in particular, it contains \(\gamma ^0\)), but for algebraic problem this is not so important. At solving of eigenvalue problem with the matrix (56), the momentum vector \(p^{\mu }\) may be changed by any four numbers. Therefore, if we redefine the vector \(p^{\mu }\) in \(S^{\pm }\), we can get rid of \(\gamma ^0\) and use the ready answer for eigenvalues and eigenprojectors.

*I*, \({\hat{p}}_{\pm } \) and \(\gamma ^{5}\) matrix, and which is algebraically similar to the vacuum propagator. Therefore, we can use the formulas (20) for eigenvalues and eigenprojectors:

*z*(49). It turns out that, as in the case of moving medium, the projection on this axis, in general, is not a conserved quantity.

## 4 Conclusions

In the present paper we have built the spectral representation of neutrino propagator both in a moving matter and in a constant external magnetic field. In this form (17), (18), which is based on the eigenvalue problem for inverse propagator (16), the propagator looks like a sum of poles accompanied by own \(\gamma \)-matrix orthogonal projectors. The advantage of this representation is that a single term in this sum is related only with one dispersion law for particle in media. More exactly, relation of energy and momentum appears as a result of vanishing of one of eigenvalues \(\lambda _i=0\) in (18).

It turned out that both in matter and in magnetic field there exists the fixed 4-axis of complete polarization \(z^\mu \), such that all eigenvalues of propagator (and, consequently, dispersion laws) are classified according to spin projection on this axis. The found generalized spin projectors (6), (48) on the axis of complete polarization play a special role in the eigenvalue problem, simplifying essentially algebraic calculations.

In the case of moving matter the states with the definite spin projection on the found axis (3) have a definite dispersion law. In particular case of rest matter the operators \(\varSigma ^{\pm }\) are projectors on the helicity states in correspondence with known earlier results [43, 44]. Let us emphasize that for moving matter or magnetic field the vanishing of commutator with inverse propagator *S* \([S,\varSigma ^{\pm }]=0\) does not lead to conservation of spin projection on this axis, since the spin projectors \(\varSigma ^{\pm }\) do not commute, generally speaking, with Hamiltonian.

Let us note that the propagator in external magnetic field (after use of the \(\varSigma ^{\pm }\) properties) is not covariant one, it contains also \(\gamma ^0\) matrix besides the unit matrix, \({\hat{p}}\) and \(\gamma ^5\). But for the eigenvalue problem the covariance is not essential, so after transfer to non-covariant “momentum” \(p^\mu _\pm \) one can use an algebraic construction for vacuum propagator.

We considered here the cases of moving non-polarized matter or external magnetic field. In our approach one can take into account the matter polarization: it leads to simple substitution of four-velocity of matter \(u^{\mu }\) by some combination of velocity and matter polarization (see [13]), after it the same algebraic construction is repeated.

The most evident development of this approach is related with neutrino oscillation in matter, in particular, in astrophysical problems, for instance, in propagation of neutrinos through supernova envelope, see e.g. [18, 48]. The presence of the fixed axis of complete polarization and reducing the number of gamma-matrix structures should make this problem algebraically similar to the mixing problem in vacuum – see corresponding spectral representation in [37]. We suppose that dynamics of the neutrino spin in media in the presence of the off-shell axis of complete polarization also may be interesting.

## Footnotes

- 1.
Short version of this paper without discussion of propagation in external magnetic field was published in [35].

- 2.
This vector was used earlier [40] for some algebraic simplification of propagator in matter.

- 3.
Note that for space-like momentum \(p^{2}<0\) the polarization vector \(z^{\mu }\) becomes imaginary. But the product \({\hat{z}}{\hat{n}}={\hat{z}}{\hat{p}}/W\) in spin projector (6) remains real.

- 4.
We call them as generalized because of appearance of additional factor \({\hat{n}}\). But in fact the Eq. (6) is the most general form of spin projectors at dressing of fermion propagator in theories with parity violation – see details in [37]. The same modification (6) of a naive spin projector arises in matter – may be not accidently.

- 5.
This vector arises in consideration of motion of a charged relativistic fermion in a constant and homogeneous magnetic field, see 4th edition of textbook [45], §1.6 . We consider another situation: neutral fermion with an anomalous magnetic moment in a magnetic field, but it turns out that in this case the constructed spin projector also commutes with the propagator.

- 6.
Case of degenerate eigenvalues – see below an example of spectral representation of propagator.

## Notes

### Acknowledgements

We are grateful to V.A. Naumov, S.E. Korenblit, S.I. Sinegovsky and A.V.Sinitskaya for useful discussions and comments.

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