# Neutrino Electromagnetic Form Factor and Oscillation Effects on Neutrino Interaction With Dense Matter

###### Abstract

The mean free path of neutrino - free electron gas interaction has been calculated by taking into account the neutrino electromagnetic form factors and the possibility of neutrino oscillation. It is shown that the form factor effect becomes significant for a neutrino magnetic moment and for a neutrino radius MeV. The mean free path is found to be sensitive to the and transition probabilities.

###### pacs:

13.15.+g, 13.40.Gp, 25.30.Pt, 97.60.Jd, 14.60.Pq^{†}

^{†}preprint: FIS-UI-TH-04-01

Neutrino interaction with dense matter plays an important role in astrophysics, e.g., in the formation of supernova and the cooling of young neutron stars sanjay ; Horo1 ; Horo66 ; niembro ; mornas ; leinson ; fabbri ; Horo2 . Earlier calculation on neutrino interactions with electrons gas, dense and hot matter, based on the standard model has been performed by Horowitz and Wehrberger Horo1 ; Horo66 . Some relativistic calculations of neutrino mean free path in hot and dense matter have been also done in Refs. niembro ; mornas ; leinson ; fabbri . Recently, due to a demand on a more realistic neutrino mean free path for supernova simulations, a mean free path calculation by taking into account the weak magnetism of nucleons has been also performed Horo2 .

However, certain phenomena such as solar neutrinos, atmospheric neutrinos problems, and some astrophysics and cosmology arguments need explanations beyond the standard model assumption of neutrino’s properties such as neutrino oscillation kuo ; pulido , the helicity flipping of neutrinos ayala ; enqvist ; gaemers ; grimus1 and neutrino electromagnetic form factors. We note that the upper bound of the neutrino magnetic moment extracted from the Super-Kamiokande solar data liu ; beacom falls in the range of , where stands for the Bohr magneton. Other experimental limits grimus ; dara give 1.0 , whereas signals from Supernova 1987A (SN1987A) require that 1.0 . These bounds have been derived by considering the helicity flipping neutrino scattering in a supernova core nuno . In the case of random magnetic fields inside the sun, one can obtain a direct constraint on the neutrino magnetic moment of 1.0 , similar to the bounds obtained from the star cooling mira . In addition, data from muon neutrino- and anti neutrino-electron scatterings allen ; Kerimov and a close examination to the data over the years from Kamiokande II and Homestake according to Mouro et al. Mourao , similarly give a neutrino average squared radius 25 MeV with = + . The definitions of and will be explained later.

Therefore, in connection with the demand on realistic neutrino mean free path in dense and hot matter, an extension of the previous study niembro ; mornas ; leinson ; fabbri ; Horo2 which takes into account the electromagnetic form factors of neutrinos and neutrino oscillations is inevitable. As a first step before that, in this report we calculate the mean free path of neutrino-free electrons gas where those effects are included. Here we assume that neutrinos are massless and the RPA correlations can be neglected. Furthermore, we use zero temperature approximation in this calculation.

In the standard model, where the momentum transfer is much less than the mass, direct and contributions to the matrix element can be written as an effective four-point coupling Horo66 ; Kerimov

(1) |

where is the coupling constant of weak interaction, and are neutrino and electron spinors, respectively, and the current is defined by

(2) |

The vector and axial vector couplings and can be written in terms of Weinberg angle (where Horo66 ; niembro ) as and (the upper sign is for , the lower sign is for and ).

The electromagnetic properties of Dirac neutrinos are described in terms of four form factors, i.e., and , which stand for the Dirac, anapole, magnetic, and electric form factors, respectively. The matrix element for the neutrino-electron interaction which contains electromagnetic form factors reads Kerimov

(3) | |||||

where , , and are neutrino and electron masses, respectively. In the static limit, the reduced Dirac form factor and the neutrino anapole form factor are related to the vector and axial vector charge radii and through Kerimov

(4) |

In the limit of , and define respectively the neutrino magnetic moment and the (CP violating) electric dipole moment nardi ; Kerimov . Here we use =+ .

Next, we can obtain the differential cross section per volume for scattering of neutrinos with the initial energy and final energy on the electrons gas. It consists of the contributions from weak (W) interaction, electromagnetic (EM) interaction, as well as their interference (INT) term, i.e.,

(5) |

For each contribution, the neutrino tensors are given by

(6) |

(7) | |||||

(8) |

whereas the polarizations read

(9) |

Due to the current conservation and translational invariance, the vector polarization consists of two independent components which we choose to be in the frame of , i.e.,

The axial-vector and the mixed pieces are found to be

(10) |

and

(11) |

The explicit forms of , , and are given in Ref. Horo66 . Thus the analytical form of Eq. (5) can be obtained from the contraction of every polarization and neutrino tensors couple ().

If we take into account the possibility of the transition, the cross section can be written in the form of HP ; Morgan

Here is the cross section of the scattering. If and are replaced with and , respectively, then the cross section becomes , i.e., the cross section of the scattering. is the ’s flavor survival probability as a function of the neutrino energy.

Due to the assumption of massless neutrino, the helicity flip from left- to right-handed is only possible through it’s dipole moment. Thus, the cross section after taking into account this possibility ( transition) reads grimus

(13) |

where is the scattering via neutrino dipole moment and is the probability of to be still left handed.

Finally we can compute the mean free path from Eqs. (5), (LABEL:2), and (13), by using

In this calculation we use a neutrino energy of 5 MeV.

Figure 1 shows the total mean free path compared to the mean free path of weak interaction with various neutrino effective moments , and neutrino charge radii . The total mean free path is the coherent sum of the weak, electromagnetic and the interference contributions.

There are also evidences that cm or MeV allen ; Kerimov ; Mourao . Therefore, in the left panel of Fig. 1 we use MeV and vary between 0 and . In the right panel, we use as the strongest bound on the neutrino magnetic moment while is varied between 0 and MeV.

It is evident from the left panel of Fig. 1 that for fixed , the mean free path increases rapidly only after . As we can see from Fig. 2 this increment is due to the significant difference between total and weak cross sections starting from . The summation of the longitudinal and transversal terms of the electromagnetic contribution is responsible for this. The right panel shows that for fixed , the total mean free path and the mean free path of weak interaction show significant variance for MeV. This is also due to the fact that the summation of the longitudinal and transversal terms of the electromagnetic part of the cross section increases rapidly starting at MeV.

Figure 3 shows the effects of neutrino oscillations on the neutrino mean free path. In this case we do not calculate the transition probabilities. Instead, we only study the variation of neutrino mean free path with respect to the transition probabilities of a left handed massless neutrino electron, , oscillates to a left handed massless neutrino muon, , or flips to a right handed neutrino electron, .

By comparing the possibility of transition (left panel of Fig. 3) and transition (right panel), we can clearly see that these effects lengthen the neutrino mean free path, where the rate depends on their survival probabilities. For smaller (large flipping possibility), the path increment becomes more significant. This effect can be traced back to the value of in Eq. (13) which is smaller than that of . On the other hand, for small the possibility of oscillation does not change the neutrino mean free path dramatically. This fact arises because the difference between and in Eq. (LABEL:2) is not as large as in the case of Eq. (13). Therefore different from the mean free path with flavor changing possibility, the mean free path with helicity flipping possibility depends strongly on the value of . For example we have also found that with decreasing the mean free path grows more rapidly when we use rather than .

In conclusion, we have studied the sensitivity of the neutrino mean free path to the neutrino electromagnetic form factors and neutrino oscillations. It is found that the electromagnetic form factor has a significant role if and MeV. We note that these values are larger than their largest upper bounds. It would be interesting to see whether or not such phenomenon would also appear if contributions from the neutrino-nucleon scatterings were taken into account. Future calculation should address this question. The mean free path is also found to be sensitive to the neutrino oscillations and depends on the transition probabilities of and . This result clearly indicates that realistic mean free path calculations in the future should be performed with appropriate values of the and transition probabilities.

TM and AS acknowledge the support from the QUE project.

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