# Neutrino dispersion relation in a magnetized multi-stream matter background

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## Abstract

We study the propagation of a neutrino in a medium that consists of two or more thermal backgrounds of electrons and nucleons moving with some relative velocity, in the presence of a static and homogeneous electromagnetic field. We calculate the neutrino self-energy and dispersion relation using the linear thermal Schwinger propagator, we give the formulas for the dispersion relation and discuss general features of the results obtained, in particular the effects of the stream contributions. As a specific example we discuss in some detail the case of a magnetized two-stream electron, i.e., two electron backgrounds with a relative velocity \({\mathbf {v}}\) in the presence of a magnetic field. For a neutrino propagating with momentum \({\mathbf {k}}\), in the presence of the stream the neutrino dispersion relation acquires an anisotropic contribution of the form \({\hat{k}}\cdot {\mathbf {v}}\) in addition to the well known term \({\hat{k}}\cdot {\mathbf {B}}\), as well as an additional contribution proportional to \({\mathbf {B}}\cdot {\mathbf {v}}\). We consider the contribution from a nucleon stream background as an example of other possible stream backgrounds, and comment on possible generalizations to take into account the effects of inhomogeneous fields. We explain why a term of the form \({\hat{k}}\cdot ({\mathbf {v}}\times {\mathbf {B}})\) does not appear in the dispersion relation in the constant field case, while a term of similar form can appear in the presence of an inhomogeneous field involving its gradient.

## 1 Introduction and Summary

Since the discovery of the MSW effect [1, 2, 3, 4], for many years a lot of attention has been given to the calculation of the properties of neutrinos in a matter background under various conditions. The matter background modifies the neutrino dispersion relations [5, 6, 7, 8], and also induces electromagnetic couplings that can lead to effects in several astrophysical and/or cosmological settings [9]. In supernova environments the presence of the neutrino background leads to neutrino collective oscillations [10, 11, 12, 13, 14, 15] that have been the subject of significant work in the context of instabilities in supernovas [14, 15, 16].

It is now well known that the presence of a magnetic field produces an angular asymmetry in the neutrino dispersion relation when it propagates in an otherwise isotropic background medium [17]. Since many of the physical environments of interest in the contexts mentioned include the presence of a magnetic field, a significant amount of work has been dedicated to study the calculation of the neutrino self-energy in the presence of a magnetic field, or a magnetized background medium [18], and the study of the properties and propagation of neutrinos in such media [19].

In the previous calculations of the neutrino dispersion relation or index of refraction in matter in the presence of a magnetic field [20, 21, 22, 23, 24], the electron and nucleon backgrounds are taken to be at rest since there is no other reference frame defined in the problem at hand. In the present work we extend those calculations by considering a medium that contains various *stream* matter backgrounds, which have a non-zero velocity relative to each other, and including the presence of a magnetic field. The effects of moving and polarized matter on neutrino spin/magnetic moment oscillations and \(\nu _L \rightarrow \nu _R\) conversions have been studied by several authors [20, 21, 22, 23, 24]. We emphasize that our focus is different. We are concerned with the calculation of the index of refraction or dispersion relation in the magnetized stream media for a chiral Standard Model neutrino state.

In the context of plasma physics the propagation of photons in magnetized or unmagnetized *two stream* plasma systems is a well studied subject [25, 26, 27, 28, 29, 30]. Here we consider the analogous problem for neutrinos. It is expected on general grounds that the presence of the streams will produce corrections to both the the anisotropic and isotropic terms in the neutrino dispersion relations, which depend on the stream relative velocities and the magnetic field. Our goal is to determine the corrections to the neutrino dispersion relation for a neutrino that propagates in such magnetized *stream* systems. Beyond the intrinsic interest, the results are of practical application in astrophysical contexts in which the asymmetric neutrino propagation is believed to produce important effects such as the dynamics of pulsars [31, 32] and supernovas [33, 34, 35].

In the previous calculations related to the propagation of neutrinos in a matter, including the presence of a magnetic field, the electron and nucleon backgrounds are taken to be at rest since there is no other reference frame defined in the problem at hand. In the present work, we consider the case in which the medium contains various *stream* matter backgrounds, which have a non-zero velocity relative to each other, and including the presence of a magnetic field.

*normal*background. We assume that in that frame there is a constant magnetic field \({\mathbf {B}} = B{\hat{b}}\), and in that frame we define

*stream*backgrounds, which are superimposed on the normal matter background having non-zero velocity relative to the normal matter background. For definiteness, we consider only the contributions from the electrons and nucleons (\(N = n,p\)) in both the backgrounds, and to refere them we use the symbols \(s = e,N\) and \(s^\prime = e^\prime ,N^\prime \) respectively. We also use \(f_e = e,e^\prime \) to refer the electrons in either background, and similarly for the nucleons \(f_N = N,N^\prime \). The symbol

*f*stands for any fermion in either background. In particular, \(u^{\mu }_f\) denotes the velocity four-vector of any of the backgrounds.

The main objective of the present work is the calculation of the neutrino dispersion relations with the simultaneous presence of the stream background and the magnetic field. Our work is based on the calculation of the thermal self-energy diagrams shown in Fig. 1, using the thermal Schwinger propagator, linearized in *B*, including only the electrons in both backgrounds, and to the leading order \(O(1/m^2_W)\) terms. The results of the calculation are summarized in Eqs. (60)–(63) for the self-energy, and in Eqs. (68)–(72) for the corresponding dispersion relations. The main result is that for a neutrino propagating with momentum \({\mathbf {k}}\) in the presence of a stream, the neutrino dispersion relation acquires an anisotropic contribution of the form \({\hat{k}}\cdot {\mathbf {u}}_{s^\prime }\) in addition to the well known term \({\hat{k}}\cdot {\mathbf {B}}\), and the standard isotropic term receives an additional contribution proportional to \({\mathbf {B}}\cdot {\mathbf {u}}_{s^\prime }\). The term involving \({\hat{k}}\cdot ({\mathbf {u}}_{s^\prime }\times {\mathbf {B}})\) does not appear in the dispersion relation, due to time-reversal invariance.

In Sect. 2 we summarize the general parametrization of the self-energy, review the relevant formulas for the electron thermal propagator and the main ingredients involved in the calculation are given in Sect. 3. The formulas for the parameter coefficients that appear in the neutrino thermal self-energy are obtained and summarized in Sect. 4. The calculation of the contribution of a nucleon stream is also given there as an illustration of possible generalizations. In Sect. 5 we discuss and summarize the main features of the results obtained for the neutrino dispersion relation, and comment on related work, in particular the calculation in the case of an inhomogeneous external field.

## 2 General considerations

*W*propagator, and we do not consider the momentum dependent terms nor its dependence on the magnetic field. To this order in \(1/m^2_W\) the \(a_f, g_f, {\tilde{g}}_f\) terms in Eq. (10) vanish and we do not consider them any further. Regarding the other terms, for our particular case in which the field is a pure

*B*field in the rest frame of the normal background, Eq. (5) implies that \(\varSigma _s\), \((s=e,n,p)\), is reduced towhich is the form used in Ref. [17]. However, for the stream backgrounds, using Eq. (4),where

## 3 Thermal propagators

### 3.1 Electron propagator

*B*-dependent part of the Schwinger propagator for the electron [38, 39, 40, 41, 42, 43, 44, 45]

*f*the function \(\eta _f(p\cdot u_f)\) is given by

*B*-independent, partand

*B*-dependent part of the

*vacuum*Schwinger propagator. It is useful to note that

### 3.2 Nucleon propagator

*B*-dependent part of the neutrino thermal self-energy. The formula analogous to Eq. (28) for the thermal Schwinger propagator for a nucleon including the anomalous magnetic moment coupling, was obtained in Ref. [37]. Adapting that result to our case, the thermal Schwinger propagator for a nucleon in either the normal or stream background (\(f_N = N,N^\prime )\) is

## 4 Calculation

### 4.1 *W*-diagram

*W*diagram in Fig. 1 gives a contribution to the neutrino thermal self-energy

*B*-independent and

*B*-dependent contribution to the neutrino thermal self-energy, respectively. By simple Dirac algebra they can be expressed in the formwhere we have used Eq. (31), and

*f*. Denoting the energy and momentum of the background particles in that reference frame by \(\mathcal{E}_f\) and \(\mathbf {P}\), a straightforward evaluation yields

### 4.2 *Z*-diagram

*Z*diagram we need the following neutral current couplings,

*Z*diagram contribution is

#### 4.2.1 Electron background contribution

#### 4.2.2 Nucleon background contribution

*B*-independent and

*B*-dependent contributions of each background (\(f_N = N,N^\prime \)). Denoting them by \(\left( \varSigma ^{(Z)}_{f_N}\right) _{T}\) and \(\left( \varSigma ^{(Z)}_{f_N}\right) _{TB}\), respectively. From Eq. (53),

*X*stands for either subscript,

*T*or

*TB*. The calculation involving \(S^{(f_N)}_{T}(p)\) and the \(G_N(p)\) term of \(S^{(f_N)}_{TB}(p)\) follows the steps that lead to Eq. (54). On the other hand, using Eq. (35) it follows that

### 4.3 Summary

## 5 Discussion and conclusions

### 5.1 Dispersion relations

*k*(to the order \(1/m^2_W\) that we are considering in this work), the solutions are \(k^0 = \omega _{\pm }({\mathbf {k}})\), where

It has been suggested repeatedly in the literature that the anisotropic terms in the neutrino dispersion relations can have effects in several astrophysical environments including pulsars [31] and the dynamics of supernovas [33, 34, 35]. The resonance condition for neutrino oscillations in a magnetic field depends on \({\hat{k}}\cdot {\mathbf {B}}\), and therefore is satisfied at different depths, corresponding to different densities and temperatures. This difference results in an asymmetry in the momentum distributions of the neutrinos. In the presence of a stream background, the neutrino asymmetry will depend on the relative orientation of the three vectors \({\mathbf {k}},{\mathbf {B}},{\mathbf {u}}_f\).

We mention that in the discussion above, in particular in writing Eq. (72), we have considered a two-stream system without explaining its physical origin, therefore in this sense the stream velocity \({\mathbf {v}}\) is not specified. However, the results can be used in specific applications or situations in which the stream velocity is determined and/or restricted by the particular physical conditions of the problem, for example if the stream velocity is due to the drift of electrons in the *B* field. In such a case, since the Lorentz forces makes charged particles drift only along the *B* axis but not in the perpendicular plane, the results can be applied to that case as well by taking \({\mathbf {v}}\) to be on the \({\mathbf {B}}\) axis.

### 5.2 Comment on the \(F^{\mu \nu }u_{f\nu }\gamma _\mu \) term

*T*violation in the context of our calculation, the \(O^\prime _E\) term is not generated. On the other hand \(O^\prime _M\) is even under time-reversal but odd under

*CP*, and therefore it can be generated if the background is

*CP*asymmetric.

*CP*and even under time-reversal. Therefore, it can be present in the effective Lagrangian without implying time-reversal violation and even if the background and the interactions are

*CP*-symmetric. This contrasts with \(O^\prime _{M}\) which is

*CP*-odd and therefore does not exist if the background is

*CP*-symmetric (neglecting the

*CP*violating effects of the weak interactions). \(O^{\prime \prime }_E\) can give additional anisotropic contributions to the neutrino dispersion relation [e.g., Eq. (72)] that are not present otherwise, with different kinematic properties from the constant field case. For example, in the presence of a static but inhomogeneous field, it gives a term involving the gradient of \({\hat{k}}\cdot ({\mathbf {v}}\times {\mathbf {B}})\).

Of course this type of term (with derivatives of the electromagnetic field) do not appear in the approach we are using in the present work based on the electron thermal propagator in a constant *B* field. Instead we have to resort to the type of approach employed in Ref. [17], which is based on calculating the electromagnetic vertex first, and then taking the static limit in a suitable way to obtain the self-energy in the (inhomogeneous) external field. We have performed this calculation and the results are presented separately [48].

### 5.3 Conclusions

To summarize, in this work we have studied the propagation of a neutrino in a *magnetized two stream plasma system*. Specifically, we considered a medium that consists of a *normal* electron background plus another electron *stream* background that is moving as a whole relative to the normal background. In addition, we assume that in the rest frame of the normal background there is a constant magnetic field.

Using the thermal Schwinger propagator for the electrons in the medium we have calculated the neutrino self-energy in such environment, linearized in *B* and to the leading order \(O(1/m^2_W)\) terms. The results of the calculation are summarized in Eqs. (60)–(63). From the self-energy the dispersion relations were obtained in the standard way, and the corresponding formulas are summarized in Eqs. (68)–(72).

In the presence of the stream (with velocity \({\mathbf {v}}\) relative to the normal background), the dispersion relation acquires an anisotropic term of the form \({\hat{k}}\cdot {\mathbf {v}}\) in addition to the well known term of the form \({\hat{k}}\cdot {\mathbf {B}}\), and the standard isotropic term receives an additional contribution proportional to \({\mathbf {B}}\cdot {\mathbf {v}}\) that involves the stream velocity and the magnetic field. We explained why a term of the form \({\hat{k}}\cdot ({\mathbf {v}}\times {\mathbf {B}})\) does not appear in the dispersion relation, due to time-reversal invariance, and why a term of similar kinematic form can appear in the presence of an inhomogeneous magnetic field, involving the derivative of the field. We have given the explicit formulas for the dispersion relations and outlined possible generalizations, for example to include the nucleon contribution or the case of non-homogeneous fields. We have made simple estimates of the magnitude of the asymmetric terms proportional to \({\hat{k}}\cdot {\mathbf {v}}\) and \({\hat{k}}\cdot {\mathbf {B}}\), and found that they can be comparable for acceptable values of the parameters involved.

In the context of plasma physics the propagation of photons in *two stream plasma* systems is a well studied subject. Here we have started to carry out an analogous study for the case of neutrinos. The present work is limited in several ways, for example by restricting ourselves to an electron background and stream, the linear approximation in the *B* field, and the calculation of only the leading \(O(1/m^2_W)\) terms. However, the results reveal interesting effects that are potentially important in several physical contexts, such as supernova dynamics and gamma-ray bursts physics where the effects of such systems are a major focus of current research, and in this sense our work motivates and paves the way for further calculations without these simplifications.

S.S is thankful to Japan Society for the promotion of science (JSPS) for the invitational fellowship. The work of S.S. is partially supported by DGAPA-UNAM (México) Project No. IN110815 and PASPA-DGAPA, UNAM.

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