Decoherenceeffects in the neutrinomixing mechanism: active and sterile neutrinos in the three flavor scheme
Abstract
In this work we analyse the effects of quantum decoherence upon the precession of the neutrino polarization vector in a supernovaelike environment. In order to perform the study we have determined the timedependence of the polarization vector by solving the equation of motion for different neutrinomixing schemes. The results of the calculations show that the onset of decoherence depends strongly on the parameters of the adopted mixing scheme. As examples we have considered : a) the mixing between active neutrinos and b) the mixing between active end sterile neutrinos.
a,b]M. E. Mosquera, b,1]O. Civitarese^{1}^{1}footnotetext: Corresponding author. Prepared for submission to JCAP
Decoherenceeffects in the neutrinomixing mechanism: active and sterile neutrinos in the three flavor scheme

Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata,
Paseo del Bosque, (1900) La Plata, Argentina 
Department of Physics, University of La Plata,
c.c. 67 (1900), La Plata, Argentina
Keywords: decoherence, neutrino fluxes
Contents
1 Introduction
The study of neutrinos produced in scenarios of astrophysical interest is a cross disciplinary subject which relates aspects of cosmology, particle and nuclear physics. Neutrinos produced in different processes such as relic neutrinos, supernova neutrinos, neutrinos produced in microquasar’s jets, among others, might arrive to Earth without being affected by local interactions. They exhibite oscillations in flavor, a phenomena predicted long ago by Pontecorvo. The review of Ref.[1] gives a nice and rigorous introduction to the subject. The parameters which are associated to neutrino oscillations have been determined experimentally [2, 3, 4, 5, 6], thought the absolute scale of the neutrino mass is still unknown. This value can eventually be determined if the neutrinoless double beta decay is observed [7].
From quantum mechanics one can determine the evolution of pure states into mixed states due to interactions with a background [8, 9]. Therefore, and due to the quantum nature of neutrinos, it is expected that decoherence can modify the neutrino’s wave packet in their way from the source to the detectors [8, 9, 10, 11, 12].
In this work we focus our attention in the study of the effects due to decoherence acting on neutrinos produced in supernovae, for different neutrinomixing schemes. In Ref. [13] we have studied the onset of decoherence on the spectrum of neutrinos produced in microquasar jets. The calculation was performed using two active neutrinos and in the oneactive plus onesterile neutrino ()scheme. In this work we have extended the formalism of Ref. [11] in order to include three neutrino states. As initial distribution function we have used a FermiDirac with values of temperature and mean energy adequate to the supernova environment [14, 15]. We have considered free as well as interacting neutrinos.
The work is organized as follows. In Section 2 we briefly introduce the formalism needed to calculate the time evolution of the energy spectrum for the superposition of three neutrinomass eigenstates. In Section 3 we describe the interactions and initial conditions used to calculate the time evolution of the polarization vector in presence of decoherence in the three active neutrinos scheme and scheme (two active plus one sterile neutrinos). The results of the calculations are presented and discussed in Section 4. Our conclusions are drawn in Section 5.
2 Formalism
The time evolution of the occupation number of neutrinos is governed by the equation of motion [11]
(2.1) 
where the squared brackets reads for the commutator, the dot represents the time derivative and is the masssquared matrix in the flavor basis. The density matrix in the flavor basis is , and is a matrix whose diagonal terms are the neutrino number densities, stands for the neutrino energy and is the Fermi constant.
Given an unitary mixing matrix , which relates the masseigenstates with the flavoreigenstates, one can write the masssquared matrix in the flavor basis as
where is the identity matrix, stands for the mass of the eigenstate and . This matrix can be written in terms of the GellMann matrices as
(2.2) 
By comparing the two previous equations one obtains the expression of the vector , which in matrix form reads
(2.3) 
The matrix can be expressed in terms of GellMann matrices as:
(2.4) 
In the previous expression is the polarization vector in the flavor basis. With the use of Eq. (2.1) it is transformed into
(2.5) 
where , and is the masssquared difference between mass eigenstates. The vector is the total (or global) polarization vector. The cross product is the inner product between two vectors with the structure constant of the SU(3) group. The vector fixes a certain orientation of the background (Eq. (2.3)). The parameter stands for the strength of the neutrinoneutrino interaction [11]. The initial condition for , is given by
(2.6) 
where , and are the electron, muon and tauneutrino spectral normalized functions. These distribution functions are defined in Section 3.
The quantity which measures decoherence is the order parameter which is defined as the length of the component of perpendicular to the vector normalized at . In this way
(2.7) 
For the case of three active neutrinos the mixing matrix reads
(2.8) 
where and stand for and respectively. For the scheme we have used
(2.9) 
and, in this case, the frequency is defined as .
3 Neutrino spectra
For the calculation of the factor of decoherence we have used Gaussianlike functions to model the interactions between neutrinos. We assume that the interactions depend on the frequency of the interacting neutrinos [11, 13].
3.1 Gaussian spectrum
The spectra of the electron and muontype neutrinos are given by the functions
(3.1) 
which are nonoverlapping functions of the frequency.
3.2 Supernova neutrino spectrum
Neutrinos produced in a supernova corebounce obey a FermiDirac distribution function
(3.2) 
where the temperature is related to the mean energy of the neutrino specie . The values of the mean energies were extracted from Ref. [14, 15], that is , and , where the subindex represents the muon or tauneutrino.
The distribution functions of neutrinos of a given flavor, as function of the frequency, are written:
(3.3) 
These functions are normalized to the total neutrino density
(3.4) 
The positive (negative) values of in the integral of Eq.(3.4) give the contribution of neutrinos (antineutrinos) to the neutrino density. For the calculations in the scheme we have assumed that initially there is not sterile neutrinos in the supernova environment, and in Eqs.(3.2) the factor is replaced by .
4 Results
To compute the order parameter , we have considered two different scenarios: i) three active neutrinos and ii) two active neutrinos and one sterile neutrino, as said before. The neutrino mixing parameters were extracted from Ref. [16]
(4.1) 
For the scheme scenario we have used different values of the mass square difference and the mixing angle. As initial condition, for this scheme, we assume that sterile neutrinos are not produced in the supernova, as said before.
4.1 Gaussian spectrum
We have calculated the order parameter for the Gaussianlike distribution functions presented in Section 3.1. In Fig. 1 we show the results for the parameter as a function of time using two different values of the neutrino interaction . If the neutrino interaction is turned off the order parameter acquires a nonzero, almost constant value value of . If one activates the neutrinoneutrino interaction the decoherence reduces, that is acquires a higher value at large time (. Thus, the length of the vector increases with the strength of the neutrinoneutrino interaction.
The results of the inclusion of a sterile neutrino in the computation of the order parameter (scheme) are shown in Figure 2. In the left inset of Figure 2 we present the results for and for different values of the interaction . Once again, the larger the value of , the larger the value of the order parameter. In the right inset of Fig. 2 we show the effect of the variation of the mixing angle upon the length of the vector . In all the cases, the order parameter decreases its value with time, however the final value is strongly dependent on the activesterile neutrino mixing angle.
4.2 Supernova neutrino spectrum
Here we present the results for the order parameter calculated by using the FermiDirac neutrino spectral functions of Section 3.2. We show the length of the vector in Figure 3 for the three active neutrino scheme. As one can see, the order parameter is reduced to if neutrinointeractions are neglected. When the interaction is turn on, the polarization vector oscillates towards an asymptotic nonzero value which increases with (the strength of the neutrinoneutrino interactions). The values of given in the figure are scaled to the values of the corresponding (see Eq.(2.5)).
The results of the calculations of in the scheme for two different values of the mass square difference are shown in Figure 4. The left inset corresponds to , the right inset represents the results with . We have used different mixing angles and null neutrinoneutrino interactions in both calculations. If the mixing angle is , the value of the order parameter is for the lower mass square difference, and for the larger value of . If the mixing angle is smaller, the order parameter acquires a higher value.
In Fig. 5 we present the results of for two different mixing angles (left inset: , right inset: ) in the scheme, for different values of the strength of the neutrinoneutrino interaction and . When the neutrino interaction is turned off, the reduction of the length of the vector is higher than the case with .
5 Conclusions
In this work we have studied the effects of decoherence upon neutrino distribution functions for neutrinos produced in a supernovalike environment. This analysis was performed by extending the formalism presented in Ref. [11]. Knowing the initial distribution function we have calculated the evolution of the polarization vector and consequently of the order parameter as a function of time for different neutrinoschemes.
For the case of neutrino’s Gaussian distribution functions, which model the density of neutrinos in the interior of the supernova, we have found that the polarization vector is reduced and oscillates around a nonzero value. The larger the value of the strength of the neutrinoneutrino interactions the larger is the asymptotic value of the polarization vector. Naturally, the asymptotic value of the polarization vector depends on the mixingangle between neutrino mass eigenstates.
This pattern appears also in the case of neutrinos produced in a supernovalike environment. If one sterileneutrino specie is included in the calculations the effect is reduced and it becomes strongly dependent on the mixing angle and the squared mass difference. When a sterile neutrino is oscillating with a light activeneutrino the effects of the collective oscillations become noticeable at very small times, since the maximum value for the frequency is quite large due to the mass difference, as it is the realistic strength of neutrinoneutrino interactions.
The trend of the results suggests that the asymptotic value may be reached when realistic values of strength of neutrinoneutrino interactions are used.
Acknowledgments
This work was supported by a grants (PIP616) of the National Research Council of Argentina (CONICET), and by a researchgrant of the National Agency for the Promotion of Science and Technology (ANPCYT) of Argentina. The authors are members of the Scientific Research Career of the CONICET.
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