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\title{Study of Neutrino Oscillation and Dissipative Effects in LBNE}

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\author[label1,label2]{Roberto L. N. Oliveira}
\author[label2]{Andr\' e de Gouv\^ea}
\author[label1]{Marcelo M. Guzzo}

\address[label1]{Instituto de F\'\i sica Gleb Wataghin Universidade Estadual de Campinas, UNICAMP 13083-970, Campinas, S\~ao Paulo, Brasil}

\address[label2]{Northwestern University, Department of Physics \& Astronomy, 2145 Sheridan Road, Evanston, IL 60208, USA}

\begin{abstract}
In this work we make a brief phenomenological study for three neutrino oscillation taking into account the matter effect from the open quantum system approach. Open quantum system approach has a rigorous set of the statements and when it is applied on neutrino oscillation, due to dissipative phenomena, the neutrinos may exhibit new and peculiar effects, in special, decoherence and relaxation effects. We use the most effective quantum dissipators and we study how change the neutrino behavior considering the LBNE configuration.  In particular, we show how each kind of dissipative effect is linked to each mixing angle and consequently, as the new effects are described in the survival and appearance probabilities. Interesting enough, in the LBNE case we present a new behavior at matter resonance region that can be seen even when the dissipative effect is small compared with energy dependent oscillation parameters.
\end{abstract}


\begin{keyword}
%% keywords here, in the form: keyword \sep keyword
LBNE, Dissipative effects, Neutrino Oscillation
%% MSC codes here, in the form: \MSC code \sep code
%% or \MSC[2008] code \sep code (2000 is the default)

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%% main text
\section{Introduction}
\label{1}
The neutrino oscillation model is modified when we consider neutrinos like a non isolated system. We use an open quantum system approach following the Lindblad Master Equation \cite{gor, lin} to obtain neutrino oscillation probabilities with dissipation effects. 


%In general, the energy conservation was used as a constrain for the neutrinos subsystem to obtain an upper bound for the decoherence effect\cite{lis, gab, ben}. But, there are no constrain or physical requirements that compile this choice, because in open quantum systems the energy conservation is always satisfied for global system. Some subsystem, however, can violate such energy conservation \cite{ben}. Other crucial restrictions should be imposed to subsystems and their dynamics \cite{nar}.

We show some results for LBNE configuration that is a experiment where the matter effect is import \cite{kim, lbnecollab} and by treating the neutrinos like an open quantum system \cite{lin}, we can use the effective mixing angle mapping made by Ref.~\cite{freu} and the result that appears in Ref.~\cite{workneut4} to show how the neutrino probabilities change due to dissipative effects.  

\section{Overview of Formalism and Results Preliminary}
\label{2}

We consider that neutrinos can be evolved by means of the Lindblad Master Equation that is written as  
%
\begin{equation}
\frac{d\rho_{\nu}(t)}{dt}=-i[H_{osc},\rho_{\nu}(t)]+D[\rho_{\nu}(t)]\,,
\label{i}
\end{equation}  
%
where $H_{osc}$  is a piece of the total Hamiltonian that can be given by $H_{tot}=H_{osc}+H_{R}+H_{int}$, such that, 
%
\begin{equation}
H_{osc}=H_{mass}+H_{matter}\,,
\label{ii}
\end{equation}  
%
and it can be expressed as $H_{osc}=diag.\{\tilde{E_{1}},\tilde{E_{2}},\tilde{E_{3}}\}$ if we consider the effective mass basis. This becomes similar the formalism to the neutrino oscillation with dissipative effects considering its propagation being in constant matter or in vacuum \cite{workneut4}.

The dissipative part can be written as
%     
\begin{equation}
D[\rho_{\nu}] =\frac{1}{2}\sum_{k=1}^{N^{2}-1}\bigg(\Big[V_{k},\rho_{\nu} V_{k}^{\dag}\Big]+\Big[V_{k}\rho_{\nu},V_{k}^{\dag}\Big]\bigg) \,,
\label{iii}
\end{equation}
%
where the operators $V_{k}$ act on neutrinos state only and must be such that $ \sum_{k} V^{\dag}_{k} V_{k} =\mathbbm{1}$ for the trace of $\rho_{\nu}$ to be preserved.

The evolution with the equation (\ref{i}), where the term $D[\rho]$ is given by (\ref{iii}), only converts a density matrix into density matrix if the time evolution is complete positive. The concept of complete positivity constrains the evolution where, it is kept inside of the physical space of Hilbert and even the evolution transforming pure states into mixed states due to dissipation effects and the quantum interpretation is satisfied \cite{len}. We can make the Von Neumann entropy, $S=-Tr[\rho_{\nu}\ln \rho_{\nu}]$, increases with time (or distance in the current case) during the process of taking a pure state into a mixed state. To this end, we must ensure that $V^{\dag}_{k}=V_{k}$ \cite{nar}.

The Eq.~(\ref{i}) can be expanded in matrix basis of the $SU(3)$ such that it assumes the following form
%
\begin{equation}
\dot{\rho}_{\vartheta}\lambda_{\vartheta}= f_{ijk}H_{i}\rho_{j}\lambda_{\vartheta}\delta_{\vartheta k}
+ \rho_{\nu}D_{\vartheta\nu}\lambda_{\vartheta}\,,
\label{iv}
\end{equation}
%
and the additional matrix, $D_{\mu\nu}$, must be real, symmetric and positive definite. By complete positivity, we have $36$ phenomenological parameters bounded among each other with inequalities which guarantee the positivity. We can use some suitable physical constraint in order to realize each dissipative effect as well as reduce physically the number of the new phenomenological parameters.

One constraint is obtained supposing the decoherence is a odd phenomenon and distinguishable of the other dissipative phenomena. So, if we put the constrint $Tr[H \rho(t)]=0$, only decoherence effect will be present on our interesting subsystem. This constraint ensure that the energy is conserved inside of interest subsystem and the form of the quantum dissipation is given by 
%
\begin{equation}
D_{\mu\mu} =-diag\{0,\Gamma_{21},\Gamma_{21},0,\Gamma_{31},\Gamma_{31},\Gamma_{32},\Gamma_{32},0\}\,.
\label{i.iv}
\end{equation}
%
%\begin{aligned}
%D_{\mu\nu}&=-diag\{0,2 (\vec{a}_{3})^{2},2 (\vec{a}_{3})^{2},0,\frac{1}{2}(\vec{a}_{3}+\vec{a}_{8})^{2},\\
%&\quad \frac{1}{2} (\vec{a}_{3}+\vec{a}_{8})^{2},\frac{1}{2} (\vec{a}_{3}-\vec{a}_{8})^{2},\frac{1}{2} (\vec{a}_{3}-\vec{a}_{8})^{2},0%\}\,,
%\end{aligned}

On the other hand, we can break this symmetry putting non zero values in the entries $D_{33}$ and $D_{88}$. As by means of complete positivity, the entries in the principal main diagonal must be bigger that the off ones, therefore, this is the most effective quantum dissipator and it can be written as       
%
\begin{equation}
D_{\mu\mu} =-diag\{0,\Gamma_{21},\Gamma_{21},\Gamma_{33},\Gamma_{31},\Gamma_{31},\Gamma_{32},\Gamma_{32},\Gamma_{88}\}\,.
\label{i.v}
\end{equation}
%
The exact definition of $\Gamma_{ij}$ and the inequalities that these parameter must satisfy to keep the complete positivity is can be found in Ref.~\cite{workneut4} and we will not reproduce here. One important thing to have in mind is that the $\Gamma_{ij}$ with $i\neq j$ are responsible by include decoherence effect while $\Gamma_{ii}$ include the relaxation effect. 
%with the following constraint given by the complete positivity
%
%\begin{equation}
%\begin{array}{ l }
%\Gamma_{12}= 2 a_{1}^{2}+2 a_{3}^{2}+a_{5}^{2}+a_{7}^{2}\geq0;\\
%\Gamma_{12}=2 a_{1}^{2}+2 a_{3}^{2}+a_{5}^{2}+a_{7}^{2}\geq0;\\
%\Gamma_{33}=4 a_{1}^{2}+a_{5}^{2}+a_{7}^{2}\geq0;\\
%\Gamma_{13}=a_{1}^{2}+\frac{1}{2} \left(2 \left(2 a_{5}^{2}+a_{7}^{2}\right)+(a[3]+a[8])^2\right)\geq0;\\
%\Gamma_{23}=\frac{1}{2} \left(2 a_{1}^{2}+2 \left(a_{5}^{2}+2 a_{7}^{2}\right)+(a[3]-a[8])^2\right)\geq0;\\
%\Gamma_{23}=\frac{1}{2} \left(2 a_{1}^{2}+2 \left(a_{5}^{2}+2 a_{7}^{2}\right)+(a[3]-a[8])^2\right)\geq0;\\
%\Gamma_{88}=3 \left(a_{5}^{2}+a_{7}^{2}\right)\geq0,
%\end{array}
%\label{ai.ii}
%\end{equation}
%


As in LBNE the matter potential has an important role, we will use the same analytical approximation scheme that was used in the Ref. \cite{freu}. In this scheme the matter parameters are given in terms of the usual effective parameters and this way, it is easy to apply this formalism to arrive the same expressions for survival and appearance probabilities as was given in Ref. \cite{workneut2} to the vacuum.
\begin{figure}[!t]
%\hspace{-1.5 cm}
\includegraphics[width=2.9in, height=1.5in]{g12s.jpg}
\includegraphics[width=2.9in, height=1.5in]{g13s.jpg}
\includegraphics[width=2.9in, height=1.5in]{g23s.jpg}
\label{chi}
%\vspace{-0.8 cm}
\textbf{\small Figure 1:}{\small Survival and appearance probabilities to decoherence effects in LBNE.}
\end{figure}


In this situation,  we can write together the complete probabilities for appearance and survival consiredering this muon source as   
%
%\begin{eqnarray}
%P_{\nu_{\alpha}\rightarrow \nu_{\alpha'}}&=& \sum_{ii} U_{\alpha' i}U^{*}_{\alpha i} U_{\alpha i}U^{*}_{\alpha' i}e^{-\gamma_{ii} x}\nonumber\\ && -2 \sum_{i>j} Re\big(U_{\alpha' i}U^{*}_{\alpha i} U_{\alpha j}U^{*}_{\alpha' j}\big)e^{-i \Gamma_{ij}}\cos \Delta_{ij} x\nonumber\\ &&
%\label{i.xii}
%\end{eqnarray}
%
\begin{equation}
\begin{aligned}
P_{\nu_{\alpha}\rightarrow \nu_{\alpha'}}=& \frac{1}{3}+\frac{1}{2}(|U_{\alpha 1}|^{2}-|U_{\alpha 1}|^{2})(|U_{\alpha' 1}|^{2}-|U_{\alpha' 1}|^{2})\\ &\times e^{-i \Gamma_{33}x}+\frac{1}{6}e^{-i \Gamma_{88}x} \times \\&\sum_{k}\big(|U_{\alpha k}|^{2}-3|U_{\alpha 3}|^{2}\big)\big(|U_{\alpha' k}|^{2}-3|U_{\alpha' 3}|^{2}\big) \\ & +2 \sum_{j>k} Re\big(U_{\alpha' j}U^{*}_{\alpha j} U_{\alpha k}U^{*}_{\alpha' k}\big)e^{-i \Gamma_{jk}x}\cos \Delta_{jk} x\\ & +2 \sum_{j>k} Im\big(U_{\alpha' j}U^{*}_{\alpha j} U_{\alpha k}U^{*}_{\alpha' k}\big)e^{-i \Gamma_{jk}x}\sin \Delta_{jk} x\,,
\end{aligned}
\label{i.xii}
\end{equation}
%
where $\Delta_{jk}=\Delta m^{2}_{jk}/2E$ to $j>k$ and $\Gamma_{jk}$ are given in Eq.~(\ref{i.v}). The $U_{\alpha i}$ are expression that can be obtained using the map results given in Ref.~\cite{freu}.

So, considering the LBNE, the distance between the source and the detector is $1300$ km and to oscillation parameters we have used the following values: $\Delta m^{2}_{12}=8\times 10^{-23}$ GeV, $\Delta  m^{2}_{31}=2.5\times 10^{-23}$ GeV, $\theta _{23}=0.74$, $\theta _{12}=0.58$, $\theta _{13}=0.14$, $\delta =0$ and the usual Earth density~\cite{kim}. For a specific $\Gamma_{ji}$ that we are interested, we vary its value about three different values as we can see in the figures 1 and 2. All of plots were made considering normal hierarchy.

In particular, the Fig. 2 shows the relaxation effects. They allow a bigger appearance neutrinos than the standard pattern to all of energy bins.  This happens because the relation effects change the constant terms in the probabilities and they are responsible for becoming the probabilities a perfect mixing of the state, i. e., in the rate $1/3:1/3:1/3$. This rate is to when the propagation achieves its asymptotic limit $x\rightarrow \infty$. However, because they eliminate the constant terms in the probabilities, the maximum and minimum values of the probabilities change as well.

The decoherence effects, $\Gamma_{21}$, $\Gamma_{31}$, $\Gamma_{32}$, are represented in the Fig. 1. On the survival plots, we can see that the decoherence effect from $\Gamma_{32}$ will be most effective than $\Gamma_{21}$, $\Gamma_{31}$, and this is expected because the main oscillation term comes from the usual parameters that describe transitions between the families $\nu_{\mu}$ and $\nu_{\tau}$. However, from energy above of $E=10$ Gev the behavior due to $\Gamma_{21}$, $\Gamma_{31}$ parameters can change the probabilities in an exotic way because on the resonance region, the behavior depend on the $\nu_{e}$ parameters.

The appearance probabilities, we have the most interesting behavior due to the dissipative effects. Essentially, for understanding the strange behavior that was brought by $\Gamma_{21}$ and $\Gamma_{32}$ in the appearance probability, we need to keep in mind that decoherence effect acts in the decoupling of the oscillation between the families that are related by these parameters. So, in the Fig. 1 we can see that before $E=2$ GeV $\Gamma_{21}$ increases the appearance probability, while after $E=2$ Gev $\Gamma_{21}$  decreases it. Still in the Fig. 1, the $\Gamma_{32}$ always decreases the maximum probability, but it increases all the minimum.

$\Gamma_{31}$ is the most intriguing decoherence parameter because it acts under the effective potential mass terms. So, around of $4$ Gev and before, its behavior is similar of the other decorence terms, except that $\Gamma_{31}$ brings the little phase that is caused if the decoherence value is high. This happens because it eliminates the oscillation where the effective potential is most effective leading the oscillation for a vacuum regime.

 The other interesting thing is a new peak that occurs around of $10$ GeV that can be completely explained by means of the appearance probability in two families, where it is clear that the diference between constant and the amplitude term at the ressonace region results in this peak.
%
\begin{figure}
%\hspace{-1.5 cm}
\includegraphics[width=2.9in, height=1.5in]{g33s.jpg}
\includegraphics[width=2.9in, height=1.5in]{g88s.jpg}
\label{chii}
%\vspace{-0.8 cm}
\textbf{\small Figure2:}{\small Survival and appearance probabilities to relaxation parameters in LBNE.}
\end{figure}
%



\section{Comments and Conclusion}
\label{4}

We investigated dissipative effects in neutrinos oscillations concerning on the proposed LBNE set-up. We find that nontrivial effects can be observed at LBNE. Particularly interesting effects are expected in the ``resonance region'', which is just outside the energy range of LBNE neutrinos. However, a considerable neutrino appearance is hoped in the end of spectrum that LBNE surely will have access. This will put a limit on $\Gamma_{31}$ parameter.

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\bibitem{kim}
C.~Giunti, C.~W. Kim, Fundamentals of neutrino physics and astrophysics, Oxford
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\bibitem{lbnecollab}
C.~Adams, et~al., Exploring fundamental symmetries of the universe,
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\bibitem{freu}
M.~Freund, Analytic approximations for three neutrino oscillation parameters
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\bibitem{workneut4}
R.~L.~N. Oliveira, M.~M. Guzzo, Quantum dissipation in a neutrino system
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\bibitem{len}
R.~Alicki, K.~Lendi, Quantum dynamical semigrou-ps and applications, Lect.
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\bibitem{nar}
F.~Benatti, H.~Narnhofer, Entropy behaviour under completely positive maps,
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\bibitem{workneut2}
R.~L.~N. Oliveira, M.~M. Guzzo, Dissipation and $\theta_{13}$ in neutrino
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