###### Abstract

We notice that looking for at the same time as oscillations could significantly help to reduce the errors in the leptonic CP-violating phase measurement. We show how the (“golden”) and (“silver”) transitions observed at an OPERA-like 2 Kton lead-emulsion detector at Km, in combination with the transitions observed at a 40 Kton magnetized iron detector with a baseline of Km, strongly reduce the so-called ambiguity. We also show how a moderate increase in the OPERA-like detector mass (4 Kton instead of 2 Kton) completely eliminates the clone regions even for small values of .

hep-ph/0206034

ROMA-1336/02

The silver channel at the Neutrino Factory

A. Donini^{1}^{1}1,
D. Meloni^{2}^{2}2
and
P. Migliozzi^{3}^{3}3

I.N.F.N., Sezione di Roma I and Dip. Fisica, Università di Roma “La Sapienza”, P.le A. Moro 2, I-00185, Rome, Italy

Dip. Fisica, Università di Roma “La Sapienza” and I.N.F.N., Sezione di Roma I, P.le A. Moro 2, I-00185, Rome, Italy

I.N.F.N., Sezione di Napoli, Complesso Universitario di Monte Sant’Angelo, Via Cintia ed. G, I-80126 Naples, Italy

## 1 Introduction

The present atmospheric [1]-[6] and solar [7]-[14] neutrino data are strongly supporting the hypothesis of neutrino oscillations [15]-[18] and can be easily accommodated in a three family mixing scenario.

Let the Pontecorvo-Maki-Nakagawa-Sakata matrix be the leptonic analogue of the hadronic Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix in its most conventional parametrization [19]:

with the short-form notation . Oscillation experiments are sensitive to the two neutrino mass differences and to the four parameters in the mixing matrix: three angles and the Dirac CP-violating phase, .

In particular, data on atmospheric neutrinos are interpreted as oscillations of muon neutrinos into neutrinos that are not ’s, with a mass gap that we denote by . The corresponding mixing angle is close to maximal, , and is in the range to eV [20].

The recent SNO results for solar neutrinos [12]-[14] favour the LMA-MSW [21] solution of the solar neutrino deficit with oscillations into active () neutrino states. The corresponding squared mass difference, that in this parametrization should be identified with , is - eV. Comprehensive analyses of the solar neutrino data, however, do not exclude the LOW-MSW solution [22]-[25], with eV. In both cases, the corresponding mixing angle () is large (albeit not maximal).

Finally, the LSND data [26, 27] would indicate a oscillation with a third, very distinct, neutrino mass difference: . The LSND evidence in favour of neutrino oscillation has not been confirmed by other experiments so far [28]; the MiniBooNE experiment [29] will be able to do it in the near future [30]. In the absence of an independent confirmation of the LSND evidence, we restrict ourselves to the three neutrino mixing scenario (the impact of a Neutrino Factory in the case of four neutrino mixing has been discussed in full detail in [31]-[33]).

These oscillation signals will be confirmed in ongoing and planned atmospheric and solar neutrino experiments, as well as in long baseline ones, with the latter being free of model-dependent estimations of neutrino fluxes. There is a strong case for going further in the fundamental quest of the neutrino masses and mixing angles, as a necessary step to unravel the fundamental new scale(s) behind neutrino oscillations. In particular, it is possible that in ten years from now no information whatsoever will be at hand regarding the angle (the key between the atmospheric and solar neutrino realms, for which the present bound is , [34]) and the leptonic CP violating phase .

An experimental set-up with the ambitious goal of precision measurement of the whole three-neutrino mixing parameter space is under study. This experimental programme consists of the development of a “Neutrino Factory” (high-energy muons decaying in the straight section of a storage ring and producing a very pure and intense neutrino beam, [35, 36]) and of suitably optimized detectors located far away from the neutrino source. The effort to prepare such very long baseline neutrino experiments will require a time period covering this and the beginning of the following decade. It is therefore of interest to look for the optimal conceivable factory-detectors combination. One of its main goals would be the discovery of leptonic CP violation and, possibly, its study [37]-[40]. Previous analyses [41]-[43] on the foreseeable outcome of experiments at a Neutrino Factory have shown that the determination of the two still unknown parameters in the three–neutrino mixing matrix, and , will be possible (if the LMA-MSW solution of the solar neutrino deficit is confirmed). The most sensitive method to study these topics is to measure the transition probabilities involving and , in particular . This is what is called the “golden measurement at the neutrino factory”. Such a facility is indeed unique in providing high energy and intense beams. Since these beams contain no , the transitions of interest can be measured by searching for “wrong-sign” muons: negative (positive) muons appearing in a massive detector with good muon charge identification capabilities [40].

An incredible amount of work has been devoted to this topic in the last few years: we refer the interested reader to [44]-[54] and to the refs. therein for an overview of the status-of-the-art in all its different aspects; we address to [55]-[60] and refs. therein for a comparison of the physics reach of a conventional (super)beam and of a Neutrino Factory; eventually, we point out that in [61] the idea of a beam originating from -decay (the so-called “-beam”), was advanced: it appears that the physics reach of such a beam is complementary to that of a conventional superbeam [62].

In [63] it has been noticed that the probability for neutrinos at a fixed energy (and for a given baseline) computed for a given theoretical input pair defines a continuous equiprobability curve in the () plane. Therefore, for a fixed energy, a continuum of solutions reproduce the input probability. A second equiprobability curve is defined in this plane by the probability for antineutrinos at the same energy and with the same input parameters, . The two equiprobability curves have, quite generally, two intersection points: the first of them at (), the second at a different point (). It is the intersection of equiprobability curves from the neutrinos and antineutrinos that resolves the continuum degeneracy of solutions in the () plane, restricting the allowed values for and to the two regions around () and (). This second intersection, however, introduces an ambiguity in the measurement of the physical values of and . Different proposals have been suggested to solve this ambiguity: in [63] the ambiguity is solved by fitting at two different baselines at the same time; another possibility is an increase in the energy resolution of the detector, [46, 50, 51]; see also [64].

New degeneracies have later been noticed [65], resulting from our ignorance of the sign of the squared mass difference (by the time the Neutrino Factory will be operational) and from the approximate symmetry for the atmospheric angle.

In the first part of the paper we describe how the () ambiguity arises in oscillation due to the equiprobability curves in the () plane at fixed neutrino energy. We then extend our analysis showing how the same phenomenon can be observed in a real experiment: equal-number-of-events (ENE) curves for any given neutrino energy bin appears and their intersections in the () plane explain how and where do “clone” regions arise. Our analysis is afterwards compared with the results of simulations with a realistic magnetized iron detector (studied in [66]).

We then propose to reduce the continuum degeneracy and to resolve the last ambiguity using two baselines with detectors of different design. We notice that muons proceeding from decay when ’s are produced via a transition show a different correlation from those coming from (first considered in [40]). By using a near lead-emulsion detector, capable of the -decay vertex recognition, we can therefore use the complementarity of the information from and from to solve the () ambiguity. We find that the combination of the near emulsion detector and of a massive magnetized iron detector at Km could indeed help to achieve a good resolution in the plane.

In this paper we restrict ourselves to the () ambiguity, by fixing and by choosing a given sign for (in the hypothesis that more information on the three neutrino spectrum will be available by the time the Neutrino Factory will be operational). However, we found that the core of our results do not depend on the sign of : indeed, we have been carrying on simulations with the opposite sign, with similar results (i.e., we still observe how “clone” regions disappear due to the two-detector types combination; notice, however, that the location of the “clone” regions and all the details of the simulation do depend on the sign of ). We remind that, if is not extremely small, the combined measurement of and transitions could significantly help in solving the ambiguity.

In Section 2 we present our analysis of the equiprobability curves in the () plane; in Section 3 we introduce the corresponding equal-number-of-events (ENE) curves; in Section 4 we present a similar analysis for the oscillation probability and study the impact of “silver” wrong-sign muon events; in Section 5 we show our results for the combination of a near ( Km) OPERA-like detector and of a Km magnetized iron detector; in Section 6 we eventually draw our conclusions. In Appendix A a perturbative expansion in of the formulae of Sect. 2 is presented; in Appendix B we report some useful formulae for CC-interaction and decay.

## 2 equiprobability curves in the plane

We consider the transition, first. This channel has been shown to be the optimal one to measure simultaneously and at the Neutrino Factory in the context of three-family mixing, through the appearance of “wrong-sign” muons in the detector, [40]. It therefore deserves the nickname of “golden channel”.

Following eq. (1) of [63] we get for the transition probability at second order in perturbation theory in , , and (see also [67]-[69]),

(1) |

where refers to neutrinos and antineutrinos, respectively, and

with . In these formulae, (expressed in eV/GeV) and (with referring to neutrinos and antineutrinos, respectively). Finally, and .

The parameters and are the physical
parameters that must be reconstructed by fitting the experimental data
with the theoretical formula for oscillations in matter. In what
follows, the other parameters have been considered as fixed
quantities, supposed to be known by the time when the Neutrino Factory
will be operational. In particular, in the solar sector we fixed
and
eV [22]-[25], corresponding to
the LMA region of the solar neutrino problem (accordingly to the
recent SNO results [13, 14]); in the
atmospheric sector, and eV [20]
(notice that for the ambiguity [65] is absent).
Finally, we considered a fixed value for the matter parameter, eV/GeV for km and eV/GeV for km, obtained by using the average
matter density alongside the path for the chosen distance computed
with the Preliminary Earth Model [70].
For simplicity, we have not included errors on these
parameters^{4}^{4}4 It has been shown in
[63] that the inclusion of the foreseeable
uncertainties on these parameters does not modify the results on the
and measurements in a relevant manner..

Eq. (1) leads to an equiprobability curve in the plane () for neutrinos and antineutrinos of a given energy:

(2) |

We can solve eq. (2) for :

(3) |

It is useful to introduce the following functions:

(4) |

with the obvious limit and .

Eq. (3) can then be written as:

(5) |

Eq. (5) is particularly illuminating: it describes a family of two branches curves in the plane () for the neutrinos and a second family of two branches curves for the antineutrinos. The dependence on the neutrino energy resides in and in the ratio , whereas the dependence on the angle is factorized in the two -dependent functions and .

It is helpful to introduce the parameter :

constrained by the bound

(6) |

The allowed region for depend on the input parameters (), on the neutrino energy and on the baseline. In Fig. 1, we compute the allowed values of as a function of () at three different baselines, and Km, for and GeV, by numerically solving eq. (6).

It may be noticed that, for (almost) every value of in the considered range, two different regions of allowed values for exist. The first region corresponds to , whereas the second corresponds to large negative values for . In this region, is negative: this region is therefore unphysical when the sign of the mass differences and of the various angles are defined in an appropriate way [52]. We concentrate hereafter on the tiny region around .

In Fig. 2 we present the equiprobability curves for (the upper row) and (the lower row) in the () plane, at and Km for different values of the neutrino energy in the range GeV. The input values are and . In the upper row (neutrinos), it can be seen that all the equiprobability curves intersect in (namely, and ). However, notice that the equiprobability curve for a given neutrino energy intersects the curves corresponding to a different neutrino energy in a second point, at positive and negative . This second intersection depends on the energies of the two curves. In the upper branch of the neutrino equiprobability curves, no second intersection is observed, for these particular values of the input parameters. The results of Fig. 2 may be understood with the help of a perturbative expansion of eqs. (4) and (5) in terms of powers of (always possible in the allowed region, for large enough). Details on this expansion can be found in App. A.

We can draw some conclusion from what observed in Fig. 2 and from the previous considerations on the energy dependence of the equiprobability curves. In particular, it is to be expected that by fitting experimental data for neutrinos only it should be quite difficult to determine the physical parameters () with good accuracy. We expect, instead, that the fitting procedure will identify a low region whenever the family of equiprobability curves are not well separated, within the experimental energy resolution. In particular, at short distance ( Km) it is to be expected a good determination of (notice that is generally less than ) and no determination whatsoever of the CP-violating phase . At the intermediate distance, Km, the equiprobability curves for neutrinos (for this particular set of input parameters) do not depend strongly on the energy in the upper branch, whereas a larger separation can be seen in the lower branch. Therefore, we expect a low region alongside the upper branch of the equiprobability curves spanning from around the single point corresponding to the physical parameters (at ) to the (diluted) region where the curves show the second intersection (that by periodicity in the axis happens to be in the lower branch).

Finally, at large distance ( Km) we expect a low region in the region and (notice that the small spread in the variable for this baseline is in agreement with Fig. 1).

A great improvement in the reconstruction of the physical parameters from the experimental data is achievable using at the same time neutrino and antineutrino data. This can be seen in Fig. 3, where the equiprobability curves for neutrinos and antineutrinos (for the same input parameters as in Fig. 2) have been superimposed. At short distance, the two family of equiprobability curves overlap for any value of , and no improvement is to be expected. However, at the intermediate distance the equiprobability curves for neutrinos and antineutrinos overlap only in the vicinity of the physical point and in the region of the second intersection, whereas in the intermediate region they are quite well separated, both in the upper and lower branch. We expect, in this case, that the fitting procedure of the whole set of neutrino and antineutrino data will identify two separate low region, around the physical point and around the region where all the curves show the second intersection. This second allowed region in the parameter space was first observed in [63] and subsequently confirmed in [46, 50, 51]. Finally, at large distance we expect no significant improvement with respect to the previous case.

Notice that these considerations can be drawn by looking at the equiprobability curves for neutrinos and antineutrinos, only. We will see in the following section how the theoretical expectation is indeed reproduced in the “experimental data”.

## 3 Number of “wrong-sign” muons in the detector

The experimental information is not the transition probability but the number of muons with charge opposite to that of the muons circulating in the storage ring, that in the following will be often called “golden” muons. The events are then grouped in bins of energy, with the size of the energy bin depending on the energy resolution of the considered detector. In general,

(7) |

is the number of wrong-sign muons in the i-th energy bin for the
input pair (); is the neutrino
(antineutrino) energy^{5}^{5}5The neutrino energy can be reconstructed
if the considered detector has a hadronic calorimeter capable to measure
the energy of the hadronic shower () in the CC
interactions with good precision..
The charged current neutrino and antineutrino
interaction rates can be computed using the approximate expressions
for the neutrino-nucleon cross sections on an isoscalar target,

(8) |

In the laboratory frame the neutrino fluxes, boosted along the muon momentum vector, are given by:

(9) | |||||

Here, , is the parent muon energy, , is the number of useful muons per year obtained from the storage ring and is the angle between the beam axis and the direction pointing towards the detector. In what follows, the fluxes have been integrated in the forward direction with an angular divergence (taken to be constant) mr. The effects of the beam divergence and the QED one-loop radiative corrections to the neutrino fluxes have been properly taken into account in [71]. The overall correction to the neutrino flux has been shown to be of .

In the same approximations as for eq. (1), we get for the number of events per bin:

(10) |

where we introduced a short-form notation for the following integrals:

(11) |

For a fixed energy bin and fixed input parameters (), we can draw a continuous curve of equal number of events in the () plane,

(12) |

as it was the case for the transition probability, eq. (2).

We therefore get an implicit equation in ,

(13) |

where

(14) |

and and are the -dependent functions introduced in eq. (4). Solving for ,

(15) |

we get equal-number-of-events curves (ENE) in the () plane, see Fig. 4.

The problem arises in the reconstruction of the physical parameters from a data set consisting of some given number of events per bin, for a given number of bins (depending on the specific detector energy resolution). As it was the case for the equiprobability curves in the previous section, all the ENE curves intersect in the physical point () and any given couple of curves intersect in a second point in the same region as in Fig. 2. As it can be seen in Fig. 4, the second intersection differs when considering different couples of curves, but lies always in a restricted area of the () plane, the specific location of this region depending on the input parameters (), see eqs. (23) and (24). The analysis of the data will therefore identify two allowed regions: the “physical” one (around the physical value, ) and the “clone” solution, spanning all the area where a second intersection between any two ENE curves occurs. This is the source of the ambiguity pointed out in [63].

In the remaining of this section, we apply the analysis in energy bins of [40, 63]. Let be the total number of wrong-sign muons detected when the factory is run in polarity , grouped in energy bins specified by the index , and three possible distances, 1, 2, 3 (corresponding to Km, Km and Km, respectively). In order to simulate a typical experimental situation we generate a set of “data” as follows: for a given value of the oscillation parameters, the expected number of events, , is computed; taking into account backgrounds and detection efficiencies per bin, and , we then perform a gaussian (or poissonian for events) smearing to mimic the statistical uncertainty:

(16) |

Finally, “data” are fitted to the theoretical expectation as a function of the neutrino parameters under study, using a gaussian minimization:

(17) |

where is the statistical error for (errors on background and efficiencies are neglected) or a poissonian minimization:

(18) |

whenever events are Poisson-distributed around the theoretical values (see [60] and refs. therein). We verified that the fitting of theoretical numbers to the smeared (“experimental”) ones is able to reproduce the values of the input parameters (the best fit always lies within a restricted region around ).

The following “reference set-up” has been considered: neutrino beams resulting from the decay of ’s and ’s per year in a straight section of an GeV muon accumulator. An experiment with a realistic 40 Kton detector of magnetized iron and five years of data taking for each polarity is envisaged. Detailed estimates of the corresponding expected backgrounds and efficiencies have been included in the analysis, following [66]. Notice that this set-up is exactly the same of [40, 63].

In the first row of Figs. 5-6, we present the results of the fit to five bins of data for decaying muons of one single polarity, . The energy resolution of the detector is GeV. In all cases we observe the pattern depicted in the previous section, with a good determination of and an extremely poor determination of . In the second row we fit to five bins of data for decaying muons of both polarities. The results follow again the theoretical analysis of this and of the previous section and are in perfect agreement with what presented in [63]. In particular, notice how at the intermediate distance it is possible now to reconstruct with an error of tens of degrees in two separate regions of the parameter space.

## 4 The equiprobability and ENE curves

We present in this section the possibility to use a different channel, namely the oscillation probability, to improve the reconstruction of the physical parameters () in combination with the results for the transition described in [40, 63] and in the previous section.

The oscillation probability at second order in perturbation theory in , , and is:

(19) |

where refers to neutrinos and antineutrinos, respectively, and

with . Notice that and differs from the corresponding coefficients for the transition for the exchange, only. The term is identical for the two channels, but it appears with an opposite sign. This sign difference in the -term is crucial, as it determines a different shape in the plane for the two sets of equiprobability curves.

In Fig. 7, we superimposed the equiprobability curves for the and oscillations at a fixed distance, Km, with input parameters and , for different values of the energy, GeV. The effect of the different sign in front of the -term in eqs. (1) and (19) can be seen in the opposite shape in the () plane of the curves with respect to the ones. Notice that all the curves of both families met in the “physical” point, , , and that now three would-be “clone” regions (i.e., the spread regions where the intersections of any given couple of equiprobability curves lie) can be seen.

As a final comment we signal that, if is not extremely small (in such a way that the and terms dominate over the terms in eqs. (1) and (19)), the combined measurement of and transitions could in principle solve the ambiguity.

To follow the line of reasoning adopted for the channel, we should now discuss how the channel can be used in a realistic experiment. A number of modifications with respect to the case of the “golden” channel should be taken into account.

First, the approximate expressions for the neutrino-nucleon cross section on an isoscalar target, eq. (8), are no longer appropriate in the case of a CC interaction inside the detector. In this case we used the reported values for the cross-section [72] that have been applied in the CHORUS and OPERA experiment to compute the expected number of CC events. The considered cross-section includes mass effects in the DIS region following [73] (see App. B for details), as well as the elastic and quasi-elastic contributions to the cross section.

Second, the will decay in flight into a muon of the same charge
and two neutrinos, with a branching ratio , [19]. This “silver” wrong-sign muon is the experimental
signal we are looking for, to be identified and to be separated from
the “golden” wrong-sign muons originated from CC
interactions^{6}^{6}6We adopt the nick-name of “silver” muon events
due to the lesser statistical significance with respect to “golden”
ones.. The first tool to distinguish the two sets of wrong-sign
muons is their different energy distribution (see App. B
for details on the differential decay rate). It has been shown in
[66] that in the magnetized iron detector considered
in the previous section, muons from decay cannot be
distinguished from the main background represented by muons from
charmed mesons decay by means of kinematical cuts.
In order to take advantage of this channel, we
should therefore use a different kind of detector: for this reason we
concentrate in the remaining of the paper on a lead-emulsion detector,
where the observation of the decay vertex allows to
distinguish “golden” and “silver” wrong-sign muons, and the
latter from the charmed mesons decay background.
We must mention that the oscillations were previously
considered in [48] for a liquid argon detector, using
kinematical cuts to identify ’s. It could be of interest to explore
further the possibility of using “silver” muon events in such a detector
to reduce or eliminate the () ambiguity.

In what follows, we consider an OPERA-like detector with a mass of 2 Kton and spectrometers capable of muon charge identification (see the OPERA proposal for details, [74]) located at Km down the neutrino source (obviously, in the back of our mind we are thinking of the CNGS set-up). The results for both the “golden” and the “silver” channel at the near emulsion detector will be combined with results for the “golden” channel obtained with the magnetized iron detector located at the optimal distance for the measurement of leptonic CP violation, Km.

In this paper, we will first restrict ourselves to an ideal OPERA-like
detector with perfect efficiency and no background. Afterwards, we
take into account the realistic estimates of the energy-dependent
reconstruction efficiency and of the most relevant
backgrounds^{7}^{7}7 A dedicated careful analysis of the “silver”
muons reconstruction efficiency and of the background as a function of
the neutrino energy for this specific detector is currently under
progress, [75]. A key issue is the maximum affordable
amount of charge discrimination, both for the emulsion and the
magnetized iron detector. Estimates can be found in
[66] for the magnetized iron detector and in Fig. 86
of [74] for the emulsion detector.. Eventually, we
will consider how an increase in the detector mass or an improvement
on the signal/noise ratio affects our results.

Schematically, starting from a positive charged muon in the storage ring, “silver” muons are obtained by the following chain:

whereas “golden” muons come from:

If we group the events in bins of the final muon energy , with the size of the energy bin depending on the energy resolution of the considered detector, the number of “golden” muons in the i-th energy bin for the input pair () and for a parent muon energy is:

(20) |

(remember that is the oscillation probability for neutrinos and antineutrinos, respectively, see Sect. 2), whereas the number of “silver” muons in the i-th energy bin is:

(21) | |||||

In both equations, stands for a convolution integral on the intermediate energy: for example,

gives the number of muons in the i-th bin in the final muon energy , for a given neutrino energy (see App. B for details). In Fig. 8 we present a direct comparison of “golden” and “silver” muons grouped in five energy bins with GeV, for a parent muon energy GeV with input parameters , . We consider here a near detector ( Km) with a mass of 2 Kton and perfect reconstruction efficiency for both channels. Notice that in Fig. 8 we have not included quasi-elastic and resonance contributions to the cross-section that could enhance production at low neutrino energy.

The total number of events for these parameters are and . The strong reduction in the number of “silver” muons with respect to the “golden” muons with the same input parameters depends on the suppression due to the branching ratio and to the different DIS cross-section for muons and taus.

Following the same procedure used to get the “golden” muons ENE curves presented in Fig. 4 we can compute ENE curves for “silver” muons. These curves are reported in Fig. 9 in the case of Km.

In Fig. 10 we superimpose “golden” and “silver” ENE curves for Km and . Notice how, as it was expected from the equiprobability curves analysis, the two sets of curves have opposite concavity in the () plane. As for the equiprobability curves, all lines met in the “physical” point. Therefore, a combined analysis of the two sets of data should present a well defined global minimum around the “physical” region, whereas the local minima situated in the three “clone” regions are considerably raised with respect to what presented in the previous section where only the “golden” muon signal was considered.

## 5 Combination of “golden” and “silver” muon events

We follow the analysis in energy bins outlined in the previous section and in [40, 63]: we produce a theoretical data set () for fixed input parameters and then we smear it as in eq. (16) to obtain an “experimental” data set (). Finally, “experimental” data are fitted and 68.5, 90 and 99 % C.L. contours in the () plane are drawn.

In Fig. 11 we present the results of this analysis comparing two different possibilities: in the upper row we combine two realistic magnetized iron detectors at and Km; in the lower row we combine an ideal OPERA-like detector at Km and a realistic magnetized iron detector at Km.

First, we present our results for the combination of the two iron
detectors (where only “golden” muons, , can be
used). We draw in each figure the contours for different input parameters:
three values for and and
three values for the phase and .
In each figure, therefore, fits to nine input parameter pairs
() are shown: this has to be compared with
Figs. 5 and 6 where in each plot
the results of a fit to one single input parameter pair was presented.

On the left, only five years of data taking for
circulating in the storage ring are considered.
Notice that for any given input pair is always reconstructed within a
error; on the contrary, roughly any value for the CP-violating phase is
allowed. The situation is drastically improved on the right, where
five years of data taking for each muon polarization are
considered. In this case, the phase is reconstructed with a
precision of tens of degrees for all values of the input parameters.
Notice, however, how some “clone” region is still present at 90 %
C.L. (e.g., for the
small region around ;
for the
small region around ;
see also Fig. 5).
These results can be easily understood in terms of the
theoretical analysis of the equiprobability and equal-number-of-events
curves for the channel of Sects. 2
and 3.

We present now results for the combination of a near emulsion detector (with
both “golden” and “silver” muons, and )
and a not-so-far iron detector (with “golden” muons, only). On the left, again only
five years of data taking for circulating in the storage ring are considered.
Notice that a significant reduction in the reconstruction errors on the phase
is already achieved.
On the right, we simply add to the first five years of data taking for
the polarity further five years for the opposite polarity in the iron
detector, only. We have not included a further five year operational time
for the emulsion detector to take into account the mass decrease due to the
brick removal in the first five year period^{8}^{8}8This is a quite conservative assumption,
being the expected number of bricks to be removed looking for “silver” muons
smaller than in the case of oscillations
considered in the OPERA proposal, [74]..
Notice, however, that a quite relevant improvement with respect to the one-polarity
two-detector types case (lower left) is achieved in the reconstruction
error. More important, an improvement with respect to the two-polarities
one-detector type (upper right) can also be observed. In particular, the “clone”
regions have completely disappeared, due to the combination of “silver” and “golden”
muons with a different -dependent oscillation probability.
The effect of the inclusion of “silver” muons can be seen in Fig. 12,
where we present the two-polarities two-detector types combination
(Fig. 11, lower right) compared with the same combination but
with only a “golden” muon signal. In this figure we can clearly see that “clone”
regions are still present and that the near emulsion detector is too small to compete
with a 40 Kton iron detector located at the same distance down the neutrino source.