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Date: Tue, 28 Mar 2000 12:45:28 -0500
From: Stefano Rigolin
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To: Steve Geer
Subject: comments on draft1
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Dear Steve,
I read carefully the last version (n.1) of the draft, pointing
especially the attention to the theoretical and oscillation sections.
In the following find enclosed some comments and suggestions. I divided
them in two sections (1) physics and (2) notation.
I hope they could be useful for the preparation of the final version.
I found that you and the other of FNAL have had an impressive work
in collecting all this in few weeks.
Please tell me if you think that I can be useful for reading/checking
deeply some particular parts of the draft. I'm starting to read it again
(so maybe tell me if you have already a new version)
Stefano
PS Jaunjo wrote me that he hadn't time to give a detailed look
to the draft and to contribute to it, so he feels better not to
be in the author list of this paper.
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(1) PHYSICS
1) In page 5, at the end of point (2) it is addressed the possibility
of a \nu_\mu-> \nu_s oscillation for explaining atmospheric anomaly.
As explained in a following session the sterile explanation seems
almost completely ruled out. Maybe a comment should be added.
>> done
2) In section 2, when speaking about the CC rates at for example 10000 km
probably it should be added that this is for "no oscillation" scenario.
In eq. 7,8 \gamma and \beta are not defined. Their definitions is
straightforward but maybe for helping non expert readers can be added.
>> Done
3) In Table 1 and figure 6 it is not written which is the number
of effective muons in the beam that it is used. The data in the
table are slightly different from the one used in the plot. Is this
the reason (or probably it is due to some other thing as averaging
over a 1 mrad angle or something like this ...) ? Maybe should be
commented.
>> done
4) In eq. 20 and following text it will be better, for avoiding
confusion to call M_R as m_R. because M_R denote the matrix
(see eqs. 19 and 21)
>> done
5) Subsection 3.1.2. it is written "In the approximation that we neglect
oscillations driven by the small $\delta m^2$ scale ..." and after in
the eq. 27 this small scale appears. Also it is to be chosen if in this
and the following eqs. it is to be used "=" or "\approx" that now it
is mixed.
Maybe in eq. 27 and 30 could be added as in 28,29 the explicit form
of the matrix element in terms of \theta_ij angles. They are used
in section 3.1.3.
>> done
In Eq. 28 and 29 appears the matrix element U_{13} (and following)
that are not defined and should be changed with U_{e3} and
correspondent change for the others (see def in eq. 22)
>> done
It is to be defined \rho(x). I argued that x = r/R = normalized distance.
>> rho now defined
Eq. 31 and 33 are exactly the same, maybe one of the two is not
necessary.
>> Its a little pedestrian, but we think useful
6) Page 21. "This evolution equation can be solved numerically". It can
be solved also analitycally see for example
H. W. Zaglauer and K. H. Schwarzer, Z. Phys. C 40 (1988) 273.
Probably we agree that in this case is it "complicated" to have a
physical meaning. In any case approximate formulas with one mass
are also be developed for example in
O. Yasuda, hep-ph/9809205
For two masses analytical formula see for example also
A. Cervera et al., hep-ph/0002108
where expansion in \delta m^2_{12} is performed.
>> Yes you are right, but in practice everyone solves this numerically,
>> for the reason you pointed out.
7) In eq. 38 the \theta_{23}^m is not defined. So either
eq. 38 either eq. 36 should be changed. In the one mass approximation
\theta_{23} is not "shifted" by matter so it is defined as in vacuum
and in my opinion eq. 38 should be corrected deleting the "m" over
\theta_{23}.
>> done
8) Table 3. \delta_{23} in the second lines should be \delta_{12}
>> done
In pag. 25 "the 4x4 neutrino mixing matrix ... to be [28]"
Here it is cited the proceeding of Nufact99. May it will be better
to cite also the published version (to appear in Nucl. Phys.):
A. Donini et al., hep-ph/9909254
>> done
After eq. 45. the definition of R_{13}(\theta_{13},\delta) should be
a 4x4 matrix and not a 3x3. So in this case (0,0,0,1) should be added
(as column/raw)
>> done
9) In subsection 3.5.1 in showing the results of fig. 27 and 32 maybe it
should be said that here the background is not included. In these figs.
it appears that going to small L should be better for the \theta_{13}
sensitivity. This is not true if you include background estimations
as in fig. 39. This, I think, is one important point to be stressed
in supporting the "necessity" to go farer (3000 km !!!).
In the other plots the inclusion or not of the backgrounds seems
to me less "fundamental" as data at a fixed L are "parallel transported".
(For example only an overall shift in fig. 28). This do not change
substantially the conclusions, while in fig. 27 and 32 it does.
>> Good points. For Fig. 27 the backgrounds are less than 1 event for
>> L > 2800 km. Added sentence to say this and no backgrounds in calcultion.
>> Added sentence to point out backgrounds at short L.
>> Added sentence in e->tau discussion about no backgrounds included,
>> longer baselines favored for low nackgrounds.
10) In more than one point it is stressed that the dependence with the
energy is proportional E^{-1.6}. In my "naive" estimation it should
be 1/E^{3/2} as the number of event grows with E^3 while the error
only with E^{3/2} and so the ratio between the two.
Is your estimation numerical ? Could E^{-1.5} be as good as E^{-1.6} ?
>> 1.6 is the result of a numerical calculation ... however the precision
>> of the calculation does not distinguish 1.6 from 1.5 ... so I have changed
>> to 1.5.
11) Table 9, in my opinion, could be "misleading". It seems that going up
with the energy the "signal/back" ratio becomes worst. Probably
this is due to the reason that no bin by bin analysis is done.
So back is constant (10^-4) for every neutrino energy. In our analisys
we had different back estimation depending from the neutrino energy.
So that in our opinion there nothing is really "missing" in going
to higher energy.
Moreover the back is not the only thing relevant. If you "sum"
back+statistics you see easily that higher energy are not so "bad".
>> Good point. I've added to the table caption a sentence.
12) Fig 33 and 4 families comments.
As is reported in the conclusion there is not much work on
the 4-families scenario (sol+atm+LNSD). Anyway there is some
results already available in the context of a neutrino factory.
Maybe this could be mentioned. See for example
A. Donini et al., hep-ph/9909254 and hep-ph/9910516
A. Kalliomaki, J. Maalampi and M. Tanimoto, hep-ph/9909301
In A. Donini et al. we studied the sensitivities to the 3+1
angles (in some approximation i.e. al the angles crossing
the LNSD gap are set to be small) and to CP violation.
We also gave the order of magnitude of the detector and the distance
for observing a typical signal (1 Ton and L = 1 km for mixing angle
determination and 1 Kton and L 10-100 km for CP violation).
We used a beam with 2x10^{20} effective muons/year.
Another thing that could be said it that probably comments
on fig. 33 could be extended also to some of the 4 neutrino
scenarios. In that case an enhancement of the \nu_e -> \nu_\tau
channel is observed. We discussed explicitally the CP violation
contribution only to \nu_\mu-> \nu_tau, but the results
were very similar and we commented this point.
>> Have added "Finally we note that the sensitivity of short and medium baseline
>> experiments to CP violation in a three--active
>> plus one sterile neutrino scenario has been considered in Ref.~\cite{donini}.
>> They concluded that a 1~kt detector and a 100~km baseline could provide a
>> clean test of CP violation, particularly in the $\tau--lepton$ appearance
>> channel."
13) In fig. 35 wrong labels appear. Full line should be 10^{19} while
dotted line 10^{21} (as explained in the text).
By the way the two plots seems to me EXACTLY the same (and this is a
little bit strange). I was expecting a (maybe) small but sizeable
difference going from 2 to 3 free parameters in the fit.
Could be an error of the authors (I think they come form Mario) that
erroneously presented two times the same data. Maybe we could ask Mario
to check them. My feeling is that if you let \rho completely free the
precision in the determination of \theta_{13} and \theta_{23} should be
worst. The other possibility is that in fig 35 (upper) they used a
fixed \rho value but including an error (say 10% or more). If is it
maybe should be commented which error they take.
>> Figure checked and updated. We are assured that the figures are indeed
>> supposed to be almost identical.
Moreover I'm not completely in agreement with the following statement
"If the basement is decreased from 7400 to 2900 ... with comparable
(although slightly worse) precision (Fig. 36)." I would not like
to seem pedant but in plot (37) you can read that for fixed \theta_{23}
the reconstruction of \theta_{13} and \rho gives you an indetermination
on \theta_{13} that at 7300 km is almost double to the one obtained
at 3500. So I'm expecting in fig. 36 for a fixed \theta_{23} that the
error in \theta_{13} is growing by at least a factor 2 going from
3000 to 7400 km. This, in our case, is entirely due to a statistical
effect. So also with different detectors the conclusion should be
the same unless they have a MUCH bigger background at 3500 respect
to 7400. Now in our opinion 3500 km is enough for having background
under control in such a way that it is not so relevant comparable with
7400 km (as is not the case for 732 km !!!)
Moreover, also in fig. 36, it should be said if and which error in
the matter effect is included.
So it seem to me that there is a "discrepancy", between the above
statement, fig. 36 and fig.37, that should be understood. Or maybe
commented why this disagreement is there.
>> Im not clear about this comment (SG). The error ellipses in Fig.37
>> seem very similar for the 2 baselines. If you can clarify a bit
>> more we can try to fix the problem for the final document.
14) Maybe in point b) of page 77 (conclusion) the results already
known for 4 families at nufact could be addressed.
>> If you propose a short addition (5 lines max) I would be happy
>> to try to include it in the final document.
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(2) NOTATION and etc.
I noticed that different notations are used throughout the paper.
1) In eq.(4,5) P_{\nu_e \leftarrow \nu_mu} while in all the other eqs.
(se for example 23 and following) it is used P (\nu_e \rightarrow \nu_mu)
Also the 1.27 \delta m^2* L/E factor is used while in many others
it is used \delta m^2* L/ (4 E). In the first case, then it will be
better to define from the beginning the units (i.e. m in eV, E in GeV
and L in km)
>> done
2) Also eqs. 10-18 probably will look much more compact defining
once for ever the units
>> I'm worried about equation being taken out of context
3) In Table 1 the comma should be avoided (i.e. 3,000 -> 3000)
>> done
4) In page 24; "Note that the Jarlskog factor 2J = 2 c_{12}", probably
the factor 2 can be suppressed.
>> done
5) Eq. 47 appear the factor 1.27 while in eq. 51 the 1/4 convention.
See also 1)
>> done
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