# Large Mixing Angle Sterile Neutrinos and Pulsar Velocities

###### Abstract

We investigate the momentum given to a protoneutron star, the pulsar kick, during the first 10 seconds after temperature equilibrium is reached. Using a model with two sterile neutrinos obtained by fits to the MiniBoone and LSND experiments, which is consistent with a new global fit, there is a large mixing angle, and the effective volume for emission is calculated. Using formulations with neutrinos created by URCA processes in a strong magnetic field, so the lowest Landau level has a sizable probability, we find that with known parameters the asymmetric sterile neutrino emissivity might account for large pulsar kicks.

PACS Indices:97.60.Bw,97.60.Gb,97.60.JD

## I Introduction

The existence of sterile neutrinos, in addition to three active neutrinos of the standard model, is of great interest for both particle physics and astrophysics. The Los Alamos LSND experiment found evidence for antineutrino oscillationlsnd . An analysis of LSND, along with other short-baseline experiments, foundscs04 that the standard model is not consistent with the data and the best model has two sterile neutrinos in addition to three active neutrinos.

The recent experiment by the MiniBooNE Collaborationmini found that the data for electron neutrino appearance showed an excess at low energies, in comparison to what was expected in the standard model. This data, along with the LSND data, has been analyzed in a model with two light sterile neutrinosms07 , and compared to MiniBooNE datas07 . The mixing angles of two light sterile neutrinos were extracted. (See, however, Ref.mm08 , which questions the accuracy of the results of Refms07 ). An analysis of MiniBooNE, LSND, and other experimentsgsm07 , based in part on earlier work with the seesaw mechanismsgmg05 , also found that the preferred model to fit the data consists of two light sterile neutrinos, in addition to three active neutrinos.

Very recently a global analysis of updated MiniBooNE data, along with data from LSND, KARMEN, NOMAN, Bugey, CHOOZE, CCFR84, and CDHS has been carried outkdcss09 . There are two possible scenerios, both compatible with the results of Refs.ms07 ; s07 . Fits to appearance and disappearance with a 3+1 hypothesis have very good probabilities, and also find CPT violation. The CPT violation, with different mixing angles and mass differences for neutrinos vs antineutrinos, might be understood as matter effects, similar to the MSW effectw78 ; ms85 , but this has not yet been proved. Also a 3+2 model was shown to give a good probability and is compatible within the range of the parameters for the 3+2 model used in Refs.ms07 ; s07 , but without the uncertainty pointed out in Ref.mm08 .

In the present work we apply this range of fitsms07 ; s07 to MiniBooNE/LSND to the study of pulsar velocities. The gravitational collapse of a massive star often leads to the formation of a neutron star, a pulsar. It has been observed that many pulsars move with linear velocities of 1000 km/s or greater, called pulsar kicks. See Ref.hp97 for a review. Due to the high density, the active neutrinos have a small mean free path, and only escape at the surface of the neutrinosphere. During the first ten seconds after temperature equilibrium is attained during the gravitational collapse of a massive star, the main cooling is via emmision of neutrinos produced by the URCA process, and during the next ten seconds via neutrinos produced by the modified URCA process. We have investigated the pulsar kicks which arise from the modified URCA processes in the time interval 10-20 sec after the supernova collapses, when the neutrinosphere is just inside the protoneutron star. With a strong magnetic field and temperature, so that the population of the lowest Landau level is approximately 0.4 of the total occupation probability, and we find large pulsar kickshjk07 .

The largest neutrino emission after the supernova collapse takes place during the first 10 seconds, with URCA processes dominant. The possibility of pulsar kicks from anisotropic neutrino emission due to strong magnetic fields during this time was discussed more than two decades agochu84 . It has been showndor that, with the strength of the magnetic field expected during this period, the lowest Landau level has a sizable occupation probability, which produces the neutrino emission asymmetry that is needed for pulsar kicks. However, due to the high opacity for standard model neutrinos in the dense region within the neutrinosphere, few neutrinos are emitted, and the large pulsar kick is not obtainedlq98 .

Sterile neutrinos with a small mixing angle have small opacities. It has been shownfkmp03 that, using the model of Ref.dor and assuming the existence of a heavy sterile neutrino (mass 1 kev) with a very small mixing angle constrained to fit dark matter, the pulsar kicks could be explained. More recently the effects of such sterile neutrinos with large masses and small mixing angles have been studied for other processesk06 . See Ref.kus09 for a review of processes that moght be associated with dark matter sterile neutrinos.

In the present paper we use the fits of Refs.ms07 ; s07 with two sterile neutrinos to investigate the possibility of obtaining the large pulsar velocities which have been observed. The values of the masses and mixing angles are within the range found in calculations based on the seesaw mechanismgsm07 . Note that our model differs from that of Reffkmp03 in that with a much larger mixing angle there is a higher probability of sterile neutrinos, but a much smaller effective volume, due to a larger opacity. However, as we shall show, since the mean free path is much larger than those of standard neutrinos, under the conditions in which standard neutrinos produce a pulsar velocity of 2-300 km/s, the MiniBoone/LSND sterile neutrinos can give a kick of more than 1000 km/s.

## Ii Asymmetric Sterile Neutrino Emissivity and Pulsar Kicks in Light Two-Sterile Neutrino Model

Within about 1 second after the gravitational collapse of a large star, the neutrinosphere is formed with a radius of about 40 km, with temperature equilibrium. Within about 10 seconds about 98% of neutrino emission occurs, with neutrinos produced mainly by URCA processes. Due to the strong magnetic field, neutrino momentum asymmetry is produced within the neutrinosphere, but with a small mean free path they are emitted only from a small surface layer of the neutrinosphere, and the pulsar kick cannot be accounted for. If a standard active neutrino, say the electron neutrino, oscillates into a sterile neutrino, it will escape from the protoneutrino star and neutrinosphere, unless it oscillates back into the active neutrino. The mixing angle plays a key role. In the work of Fuller et alfkmp03 the mixing angle is so small that the sterile neutrinos are emitted. In the present work the starting point is the analysis of MiniBooNE and LSND data, with the two or more sterile neutrinos with small masses and large mixing angles. Before we can proceed, however, it is essential to determine possible effects of the high density and temperature of the medium on the mixing angles.

### ii.1 Mixing Angle in Neutrinosphere Matter

It has long been known that dense matter can affect neutrino states. The famous MSW effectw78 ; ms85 for understanding solar neutrinos, and the study of oscillations of high energy neutrinosams05 are studies of mixing of active neutrinos in matter. There have been many other studies. In the present work we are dealing with sterile/active neutrino mixing given by the mixing angle in neutrinosphere matter

(1) | |||||

In the work Reffkmp03 it was shown that the mixing angle for sterile neutrinos that can account for dark matter as well as those produced in the neutron star core is almost the same as the vacuum value. Starting from the much larger mixing angles for the sterile neutrinos that seem to account for the MiniBoonE, LSND data, we need the value of the mixing angles in the neutrinosphere, as we discuss below. The effective mixing angle in matter, can be related to the vacuum mixing angle, byafp01

(2) |

In Eq(2) is the finite temperature potential, while the finite density potential due to asymmetries in weakly interacting particles has been dropped as it vanishes when temperature equilibrium is reachedafp01 . A convenient form for , with the background of both neutrinos and electrons included, is given in Refnr88

(3) |

with the standard weak interaction parameters and . Assuming T=20 MeV, with 1.0, p= MeV, and 1.0 evms07 ; mm08 , we find

(4) |

Therefore, the mixing angle in the neutrinosphere medium is approximately the same as the vacuum mixing angle. This agrees with Reffkmp03 .

### ii.2 Emissivity With a Light Sterile Neutrino

We now use the fits to MiniBooNe and LSND with light sterile neutrinos to estimate pulsar kicks. The MiniBooNE results are consistent with the LSND results and CPT only if there are at least two sterile neutrinos. Models with three sterile neutrinos have also been consideredsgmg05 ; mm08 . Fits to the MiniBooNE experiment and the LSND results by Ref.ms07 in Ref.s07 with two sterile neutrinos are shown in Fig. 1.

From the Sterbenz/Maltoni-Schwetz fits one finds for the mixing angles of the two sterile neutrinos:

(5) |

and the masses are negligibly small. Note that this is in contrast to the parameters of Ref.fkmp03 , with the constraint of dark matter giving a mixing angle of , and a mass greater than 1 keV. In our present work we will use values for in the range 0.2 to 0.004 to estimate the pulsar kick, which is compatible with the recent global analysiskdcss09

The probability of asymmetric emission, giving a pulsar kick, does not depend directly on the sterile neutrino mass in our model, but is proportional to the . It is the large mixing angles found in fits to MiniBoone and LSND that lead us to carry out the investigation in the present paper.

### ii.3 General Formulation of Neutrino Emissivity With a Strong Magetic Field

The neutrino emissivity is given in general form in many papers, e.g., see Refs fm ; bw :

where is the matrix element for the URCA process and is the product of the initial and final Fermi-Dirac functions corresponding to the temperature and density of the medium. The main source of the asymetric emissivity that produces the pulsar velocity is the fact that the electron has a large probability to be in the lowest (n=0) Landau level. See Refs jl ; mo for a discussion of Landau levels. The asymetric emissivity can be seen by considering the weak axial intereaction, ,

(7) | |||||

with , the are the nucleon spinors, and the lepton wave functions are , where and are the electron and antineutrino momenta, respectively. The key to the asymmetric emission is given be the trace over the leptonic currents, ,

(8) | |||||

with the magnetic field B in the z direction. We only consider the weak axial force, which is dominant. Using the relationship given in Eq(8) one can show that the result of the traces and integrals over the axial product matrix element has the form ()

(9) |

Details are given in Ref hjk07 , where it is shown that the asymmetric neutrino emissivity, using the general formulation of Ref fm , is

(10) | |||||

where , is the neutron star momentum, is the probability of the electron produced with the antineutrino being in the lowest Landau state, f=.52 is the probability of the neutrino being at the + z neutrinosphere surface hjk07 , is the volume at the surface of the neutrinosphere from which neutrinos are emitted, and is the time interval for the emission.

We derive and , the effective volume for the emissivity, in the next two subsections.

### ii.4 = Probability for the Electron to be in the n=0 Landau Level

Just as in our previous work in which Landau levels play a crucial rolehjk07 , only the lowest Landau level, for which the helicity is -1/2 (rather than 1/2 as with the usual Dirac spinors) gives asymmetric emission. The probability that an electron in a strong magnetic field is in the lowest (n=0) Landau level, , can be calculated from the temperature, T, and the energy spectrum of Landau levelsjl ; mo . A particle with momentum p and effective mass in a magnetic field B in the nth Landau level has the energy

(11) |

with G, and is the effective mass of the electron at the high density of the protoneutron star and neutrinosphere.

From standard thermodynamics the probability of occupation of the n=0 Landau level, P(0), is given byfkmp03 :

(12) |

where F(n), with magnetic field B, temperature T, and chemical potential , is

(13) |

The electron energy is restricted to magnitudes greater than , but the integrals in Eq.(7) are insensitive to , so we take =0 as in Reffkmp03 .

We agree with the estimate of Ref.fkmp03 for P(0). Note that if we had used the free electron mass, , in the Landau energies (Eq.(II.2)) we would have obtained a much smaller value for P(0). For B= G, MeV, =4 MeV and = 20 MeV, P(0) 0.3. This is similar to our estimate of P(n=0) 0.4 at the surface of the protoneutron star at about 10 secondshjk07 . Therefore our result for asymmetric emissivity differs from that of Fuller et al fkmp03 mainly in that we have a much larger mixing angle, and a much smaller effective volume, since the sterile neutrinos oscillate back to active neutrinos within the neutrinosphere; and therefore our emission only takes place near the surface of the neutrinoshere. However, in contrast with purely active neutrino emission in which the opacity results in very small pulsar kicks lq98 , the sterile neutrinos have a much larger effective volume, and can therefore produce much larger pulsar velocities

### ii.5 Estimate of = Effective Volume for Emission

To estimate we make use of the early study of opacity in about the first 20s of the creation of a neutron star via a supernova collapse ip82 ; bl86 ; bml81 , and a recent detailed study of neutrino mean free pathsss07 . Since the mean free path of the sterile neutrino is determined by that of the standard neutrino to which it oscillates, , we make use of studies of active neutrino mean free paths. First note that the neutrino mean free path is given by

(14) | |||||

where is the weak matrix element and is the Fermi distribution. For the calculation of the sterile neutrino, for which , one can use the value of with a factor of from the matrix element and another such factor from the occupation probability. From the results of previous authors, for T in the 10 to 20 Mev range and in the 20 to 40MeV range, we estimate that This gives a range for the effective sterile neutrino mean free path

(15) |

For a neutrinosphere radius of 40 km, with this gives us .

From Eq.(10), = 40 km, and =1.0 cm,

(16) |

with . Taking the mass of the neutron star to equal the mass of our sun, gm, we obtain for the velocity of the neutron star

(17) |

.

For example, for T=10 Mev = kK,

(18) |

which means that sterile neutrino emission could account for the large pulsar kick with the parameters extracted from Refs ms07 ; s07 . If we use the physical parameters that give Eq.(18) for electron neutrinos, we obtain a pulsar velocity of = 95 km/s, which is consistent with previous predictions by several authors.

It should be noted that the study of the MiniBooNE and LSND results are in progress, and the mixing angles that result could be different from those obtained in Refs.ms07 ; s07 . For this reason we have used a range of parameters. Preliminary data from the MiniBooNE/Minos experimentminos08 , however, is consistent with the MiniBooNE resultsmini . We also once more point out that although Ref mm08 questions the accuracy of the peramaters extracted by Ref ms07 , our model is compatible with the recent global analysiskdcss09 , with very good probabilities.

## Iii Conclusions

Because of the strong magnetic fields in protoneutron stars and the associated neutrinosphere, the electrons produced in the URCA processes that dominate neutrino production in the first 10 seconds have a sizable probability, P(0), to be in the lowest (n=0) Landau level. This leads to asymmetric neutrino momentum. With the mixing angles found in Refsms07 ; s07 , we find that the sterile neutrinos produced during this period for high luminosity pulsars can give the pulsars velocities of greater than 1000 km/s, as observed. We emphasize that our results are consistent sterile neutrino parameters based on a global analysis of data from MiniBoonE, LSND, KARMEN, NOMAN, Bugey, CHOOZE, CCFR84, and CDHSkdcss09 , rather than a model. There is a high probability that light sterile neutrinos with a large mixing angle exist, which is the basis of our work.

There is a strong correlation of the pulsar velocity with temperature, T. Since it is difficult to determine T accurately, it is difficult for us to predict the velocity of a pulsar whose kick arises from sterile neutrino emission. On the other hand, if the pulsar kick arises from the asymmetric emission of active neutrinos produced by the modified URCA processes after 10 seconds, also proportional to P(0)hjk07 , then T can be determined by an accurate measurement of the neutrinos from the supernova. Therefore, in future years, with much more accurate neutrino detectors, one could predict the velocity of the resulting pulsar. Unfortunately, the energy of emitted sterile neutrinos cannot be measured. From our results in the present paper and those in Refhjk07 , high luminosity pulsars receive a large kick both from sterile neutrinos in the first ten seconds and standard neutrinos in the second ten seconds.

pulskicksterile8-5-09 Acknowledgements

This work was supported in part by DOE contracts W-7405-ENG-36 and DE-FG02-97ER41014. The authors thank Terry Goldman; and William Louis, Gerald Garvey and other LANL members of the MiniBooNE Collaboration for helpful discussions.

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