Reply on the Comment on ``New Conditions for a Total Neutrino Conversion in a Medium''
M. V. Chizhov$^1$ and S. T. Petcov$^{2}$
hep-ph0504247.tex
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\title
{ Reply on the Comment on ``New Conditions for a Total Neutrino 
Conversion in a Medium''
}
\author
{ M. V. Chizhov$^1$ and S. T. Petcov$^{2}$}
\address
{
$^1$Faculty of Physics, University of Sofia, 1164 Sofia, Bulgaria\\
$^2$SISSA, I-34014 Trieste, Italy}
% $^3$INRNE, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria}
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\begin{abstract}

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\medskip
{\bf Chizhov and Petcov Reply:} 
We have found in \cite{ChPet991}
{\it new conditions for a total 
neutrino conversion in a medium}
(see also \cite{ChPet992,SP98}).
% % In the Comment \cite{ASPRL},
It is claimed in \cite{ASPRL}
% Akhmedov and Smirnov claim in \cite{ASPRL} 
that our results are a particular case
of enhancement of neutrino oscillations, 
suggested in \cite{Param86,Akh88} 
and widely discussed 
in the literature.
% and discussed, e.g., in \cite{KS89}.
% % We refute the points in \cite{ASPRL} 
We refute 
% their points, 
these claims,
confirming the 
novelty of our results.

  We have studied in \cite{ChPet991} 
the transitions
$\nu_{e} \rightarrow \nu_{\mu (\tau)}$, etc.
of neutrinos which 
crossed $n =2~(3)$ alternating 
layers with constant 
densities $N_{1}$ and $N_{2}$.
The probability of the transitions, 
$P_{2(3)} (\nu_a \rightarrow \nu_b)\equiv P_{2(3)}$,
is given by \cite{ChPet991,ChPet992}:
$$P_2 = 1 - Y^2 - X_{3}^2,~~~~~~
P_3 = 1 - \bar{Y}^2 - \bar{X}_{3}^2,~\eqno(1)$$
\noindent where $Y$, $X_3$ are defined
in \cite{ASPRL} (eqs. (1), (2)),
$\bar{Y} = - c_2 + 2c_1Y,~\bar{X}_{3} = 
- s_2 \cos 2\theta_2 - 2s_1 \cos (2\theta_1)Y$,
$c_{j}~(s_{j})\equiv \cos\phi_{j}~(\sin\phi_{j})$,
$\theta_{j}$ and 2$\phi_{j}$, $j=1,2$, 
being the mixing angle in matter in layer $j$
% % with density $N_{j}$ 
and the neutrino state 
phase difference
after neutrinos 
crossed this layer.
The {\it new} {\it conditions} 
{\it for 
a total neutrino conversion}
follow from (1) \cite{ChPet992}:
$$n=2:~~Y = 0,~X_3 = 0;~~~~
n=3:~~\bar{Y} = 0,~ \bar{X}_3 = 0.\eqno(2)$$
\noindent The solutions 
of these conditions were  given in 
\cite{ChPet991} (eqs. IV and (22)).
% % , see also \cite{ChPet992}).
We have shown also \cite{ChPet991} 
that, e.g., for $n=2$, (2) are conditions for 
a {\it maximal constructive 
interference} between 
the amplitudes of neutrino 
transitions in the two layers.
Thus, a clear physical interpretation
of the absolute maxima of $P_2$ 
% (\nu_a \rightarrow \nu_b)
% associated with conditions (2) 
is that of {\it constructive 
interference maxima}.

(i) In connection with eq. (2) 
and the related effect 
of total neutrino conversion
the authors of \cite{ASPRL} write: 
``... the ``new effect of 
total neutrino conversion''
\cite{ChPet991} is nothing 
but a particular case
of the parametric resonance enhancement 
of neutrino oscillations, suggested in'' 
\cite{Param86,Akh88} ``and 
widely discussed in the literature...''.
We note
% would like to 
% emphasize  
that the two sets of two  
conditions in (2) 
and their solutions 
were not derived and/or
discussed in any form  
in \cite{Param86,Akh88}
or in any other article 
% on the subject of interest
published before \cite{ChPet991,ChPet992}. 
They do not follow from the conditions
of enhancement of 
$P(\nu_a \rightarrow \nu_b)$
found in \cite{Param86,Akh88} 
or \cite{KS89}.
% and thus are 
% not a particular case of the latter. 

(ii) For $n$ alternating layers 
one has according to 
\cite{ASPRL} 
$$ P_n(\nu_a \rightarrow \nu_b) = 
\frac{X_1^2 + X_2^2}{X_1^2 + X_2^2  + X_3^2}\sin^2 \Phi_p,~~\eqno(3)
$$
\noindent where ${\bf X} = (X_1,X_2,X_3)$ 
is a real vector, ${\bf X}^2 = 1 - Y^2$;
${\bf X}$ and $\Phi_p$ 
% % $ = (n/2)\Phi$ 
are defined in \cite{ASPRL} (eqs. (1) - (5)). 
% % the number of layers $n$ is even, and  
% % $$\Phi_p = \frac{n-1}{2}\Phi+\varphi,~
% % \varphi = \arctan\left(|{\bf X}| s_1\sin
% %  2\theta_1/Z\right),
% % ~\varphi = \arcsin\left(s_1\sin 
% % 2\theta_1/\sqrt{1-X_3^2/|{\bf X}|^2}\right)
% % \eqno(6)$$ 
% % \noindent $Z = s_2\sin 2\theta_2 + Y s_1\sin 2\theta_1$,
% % for odd number of layers. 
% % In the $n =2~(3)$ case 
% % studied in \cite{ChPet991,ChPet992},
% % $\Phi_p = \Phi$ ($\Phi_p = \Phi + \phi)$,
% % eqs. (1), (4) imply that
% % actually $\sin^2 \Phi_p \sim {\bf X}^2$
% % and expression (5) reduces to (3).
According to \cite{ASPRL}, eq. (3) 
describes parametric oscillations,
and parametric resonance occurs when 
% % $X_3 = 0$,
$X_3\equiv -(s_1c_2 \cos 2\theta_1+c_1s_2 \cos 2\theta_2)=0$, 
the latter being
the ``resonance condition''.
Due to eq. (1), 
any resonance interpretation  
of the probabilities
$P_{2,3}$
% % $(\nu_a \rightarrow \nu_b)$
based solely on eq. (3) 
seems physically questionable.
Actually,
the denominator in (3) is 
always canceled by $\sin^2\Phi_p$, 
and $P_n$ 
% $(\nu_a \rightarrow \nu_b)$
is just  a polynomial without
any explicit resonance features: 
for even $n$, 
% % $\sin\Phi_p = \sin\Phi~U_{n/2-1}(\cos\Phi)$, 
$\sin^2\Phi_p = {\bf X}^2U^{2}_{n/2-1}(Y)$,
% $(\cos\Phi)$, 
$U_n(x)$ being the
well-known Chebyshev polynomial of the 
second kind, $Y = \cos\Phi =c_1c_2 - 
\cos(2\theta_2 - 2\theta_1)s_1s_2$;
% % (see I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals,
% % Series, and Products, ed. A. Jeffrey, Academic Press, Inc., 1994)
for odd 
% $n$
$n \geq 3$,
$$P_n=\left[
s_1\sin2\theta_1\cos\left(\frac{n-1}{2}\Phi\right)+
Z U_{\textstyle\frac{n-3}{2}}(Y)
\right]^2,\eqno(4)$$
\noindent 
% $n \geq 3$, 
$Z = s_2\sin 2\theta_2 + Y s_1\sin 2\theta_1$.
As should also be clear from (1), 
% the condition 
$X_{3} = 0$ 
{\it alone does not 
ensure the existence 
even of a local maximum} of 
$P_{2,3}$.
% % $(\nu_a \rightarrow \nu_b)$}. 
The same conclusion is valid 
for {\it any finite} $n$.
% number of layers}.

(iii) % Contrary to what 
% is stated in \cite{ASPRL}, 
The two conditions $c_{1,2} = 0$
were not given in  
\cite{Param86,Akh88,KS89};
they were discussed first in \cite{SP98}.
What one finds in \cite{Param86,Akh88,KS89}
{\it at most} is 
$2\phi_1+2\phi_2=2\pi+2\pi k$. 
% which is not equivalent to 
% $c_{1,2} = 0$.
In addition, $c_{1,2} = 0$
are conditions of maxima of 
$P_{3,2}$ 
{\it only if fulfilled  
in a certain region
of the space of parameters}
\cite{SP98,ChPet991,ChPet992}.
% , as is discussed in detail 
% in \cite{SP98} and in
% \cite{ChPet991,ChPet992}.
Solution III is 
more than 
just $c_{1,2} = 0$ \cite{ChPet991}: 
{\it it includes as an 
% % indispensable 
integral part the region
$\cos(2\theta_2 - 2\theta_1) \leq 0$, 
where $c_{1,2} = 0$ have to hold}.
% {be satisfied}. 
Thus, solution III in \cite{ChPet991} 
is not ``reproducing''
any of the solutions 
% discussed
in \cite{Param86,Akh88,KS89}.
Moreover \cite{ChPet992},
% , as was shown in \cite{ChPet992},
$c_{1,2} = 0$
{\it lead to a maximum  
of $P_{2(3)}$
% % $(\nu_a \rightarrow \nu_b)$
in the neutrino energy 
only when they hold
on the line}
$\cos(2\theta_2 - 2\theta_1) = 0$ 
($\cos(2\theta_2 - 4\theta_1) = 0$). 
% {\it in the plane of neutrino parameters}.
The {\it three} conditions represent
% one possible 
a solution of (2) 
\cite{ChPet991,ChPet992}. 
% The enhancement conditions
% obtained in \cite{Param86,Akh88,KS89} 
% are not valid for $P_{3,2}$.

(iv) {\it There is infinite  
number of irrelevant 
solutions of 
$X_3 =0$, $c_{1,2} \neq 0$, 
for $n=2,3$}. 
% in the cases of interest}.
The existence 
of solution IV (or (22)),
% and 
its explicit form, found in \cite{ChPet991,ChPet992},
could not be, and were not, 
inferred from 
the solutions of 
$X_3 =0$, $c_{1,2} \neq 0$.
% % although the condition
% % $X_3 = 0$ was known, e.g., 
% % to the authors of \cite{ASPRL} 
% % before the publication of
% % \cite{ChPet991,ChPet992}.

(v) % Finally, 
The case of  neutrino oscillations 
in a medium with 3 layers, 
studied in  
\cite{SP98,ChPet991,ChPet992},
corresponds \cite{SP98} 
to the transitions 
in the Earth of the Earth-core-crossing
solar and atmospheric neutrinos.
Most importantly,
the {\it new conditions for
a total neutrino conversion} (2), 
% $n=3$,
found in \cite{ChPet991,ChPet992},
are fulfilled in this case 
and lead to observable effects 
% in the transitions 
\cite{ChPet991,ChPet992,SP98}.
The conditions of enhancement
of $P(\nu_a \rightarrow \nu_b)$
obtained in \cite{Param86,Akh88,KS89} 
are not valid
% realized for, i.e., 
% do not lead to enhancement of, 
for the indicated 
% neutrino 
$\nu-$transitions in the Earth. 

 Further comments on \cite{ASPRL} can be found in \cite{ChPet00}.

% \newpage
% \pagebreak[1]
\begin{thebibliography}{99}

\bibitem{ChPet991} M. Chizhov and S.T. Petcov, Phys. Rev. Lett. 
{\bf 83}, 1096 (1999) (hep-ph/9903399).

\bibitem{ChPet992} M. Chizhov and S.T. Petcov, hep-ph/9903424.

\bibitem{SP98} S.T. Petcov, Phys. Lett. {\bf B434}, 321 (1998). 
% % (E) {\bf B444}, 584 (1998). 

\bibitem{ASPRL} A.Kh. Akhmedov and A.Yu. Smirnov,
preceding {\it Comment}.

  
\bibitem{Param86} V.K. Ermilova et al., Short Notices of
the Lebedev Institute {\bf 5}, 26 (1986). 

\bibitem{Akh88} E.Kh. Akhmedov, Yad. Fiz. {\bf 47}, 475 (1988). 

\bibitem{KS89} P.I. Krastev and A.Yu. Smirnov, 
Phys. Lett. {\bf B226}, 341 (1989).

\bibitem{ChPet00} M. Chizhov and S.T. Petcov, hep-ph/0003110.

\end{thebibliography}

\end{document}
























   The authors of \cite{Param86,KS89} studied
the effects of {\it small} density perturbations
on the neutrino oscillations, while  
we have investigated in \cite{SP98,ChPet991,ChPet992} 
the different physical problem
of {\it large} ``perturbations'' of density.
In \cite{Akh88} neutrino oscillations in a medium 
consisting of {\it even} number $n$
of alternating layers with densities 
$N_1$ and $N_2$ have been considered.
However,  i) the two layers were assumed to have 
equal widths, $L_1 = L_2$, and i) an enhancement of 
the neutrino transitions was 
found to take place for small vacuum mixing angles
at densities $N_{1,2}$ 
much smaller than the MSW resonance density.
The authors of \cite{Param86,Akh88} 
were interested in and found the conditions
for the classical parametric resonance 
in neutrino oscillations, which can take place 
after many periods of 
density modulation of the oscillations. 
From reading the articles \cite{Param86,Akh88,KS89}
it becomes clear that their authors had in mind
astrophysical applications of their results
\footnote{In all the concrete examples 
and corresponding plots 
in \cite{KS89}, for instance, the physical 
quantities having a dimension of length are 
given in units of the solar radius 
and are comparable with it; the values 
of the densities used 
in the examples differ substantially from 
those met in the Earth - they are noticeably 
smaller than the Earth mantle and core densities.}, and 
the results obtained in \cite{Param86,Akh88,KS89} 
may indeed have astrophysical applications.
Our studies \cite{SP98,ChPet991,ChPet992} 
were concerned primarily with the 
neutrino oscillations in the Earth. 
As a consequence,
the conditions of the enhancement 
of $P(\nu_a \rightarrow \nu_b)$
obtained in \cite{Param86,Akh88,KS89}
differ from those found 
by us in \cite{SP98,ChPet991,ChPet992}.
In the two- and three-layer medium cases 
i) and ii) discussed by us 
(e.g., in the case of the Earth)
the enhancement found 
in \cite{Akh88}, for example, is not realized.


*****************************************

% Konetz kuska.
*****************************************







 Analogous results valid 
in the case of medium with three layers 
(case ii)) considered in \cite{SP98,ChPet991,ChPet992}.
Using the correct expression for $\varphi$,
eq. (4), one finds again that the 
% ``parametric resonance'' form in which  
% the authors of \cite{ASNPB} 
% write the 
probability of interest
$P_3(\nu_a \rightarrow \nu_b)$
has the same form as in eq. (6) \cite{ChPet992}
$$ 
P_3(\nu_a \rightarrow \nu_b)  
= 1 - \bar{Y}^2 - \bar{X}_{3}^2,\eqno(8)$$
\noindent where 
$$\bar{Y} = - c_2 + 2c_1Y,~\bar{X}_{3} = 
- s_2 \cos 2\theta_2 - 2s_1 \cos (2\theta_1)~Y.\eqno(9)$$ 
\noindent The conditions for 
a total neutrino conversion in this case
read \cite{ChPet992}:
$$\bar{Y} = 0,~~~~~ \bar{X}_3 = 0. \eqno(10).$$
\noindent Their solutions 
were given in \cite{ChPet991,ChPet992}. 


That $X_{3} = 0$ {\it alone} 
is a condition for {\it local maxima}
of the probability 
$P_n(\nu_a \rightarrow \nu_b)$
was suggested in E. Kh. Akhmedov,
Nucl. Phys. B538 (1999) 25 (to be quoted 
further as NP B538, 25),
on the basis of the form of the probability
in eq. (3).
Of the three solutions
for the extrema of 
$P_3(\nu_a \rightarrow \nu_b)$
found in NP B538, 25,
the solution $c_1 = 0,~c_2 = 0$ 
was already discussed 
in detail in ref. \cite{SP98} 
(compare  eqs. (11) - (16) and (24) in \cite{SP98} with
conditions (1) on page 37 of NP B538, 25), while 
the other two correspond to MSW transitions
(they were also briefly discussed  
in \cite{SP98}).









hep-ph0504247.extracted_bib
\begin{thebibliography}{99}

\bibitem{ChPet991} M. Chizhov and S.T. Petcov, Phys. Rev. Lett. 
{\bf 83}, 1096 (1999) (hep-ph/9903399).

\bibitem{ChPet992} M. Chizhov and S.T. Petcov, hep-ph/9903424.

\bibitem{SP98} S.T. Petcov, Phys. Lett. {\bf B434}, 321 (1998). 
% % (E) {\bf B444}, 584 (1998). 

\bibitem{ASPRL} A.Kh. Akhmedov and A.Yu. Smirnov,
preceding {\it Comment}.

  
\bibitem{Param86} V.K. Ermilova et al., Short Notices of
the Lebedev Institute {\bf 5}, 26 (1986). 

\bibitem{Akh88} E.Kh. Akhmedov, Yad. Fiz. {\bf 47}, 475 (1988). 

\bibitem{KS89} P.I. Krastev and A.Yu. Smirnov, 
Phys. Lett. {\bf B226}, 341 (1989).

\bibitem{ChPet00} M. Chizhov and S.T. Petcov, hep-ph/0003110.

\end{thebibliography}