Proton driver optimization for new generation neutrino superbeams to search for sub-leading numu->nue oscillations ($\theta_{13}$ angle)

We perform a systematic study of particle production and neutrino yields for different incident proton energies $E_p$ and baselines $L$, with the aim of optimizing the parameters of a neutrino beam for the investigation of $\theta_{13}$-driven neutrino oscillations in the $\Delta m^2$ range allowed by Superkamiokande results. We study the neutrino energy spectra in the ``relevant'' region of the first maximum of the oscillation at a given baseline $L$. We find that to each baseline $L$ corresponds an ``optimal'' proton energy $E_p$ which minimizes the required integrated proton intensity needed to observe a fixed number of oscillated events. In addition, we find that the neutrino event rate in the relevant region scales approximately linearly with the proton energy. Hence, baselines $L$ and proton energies $E_p$ can be adjusted and the performance for neutrino oscillation searches will remain approximately unchanged provided that the product of the proton energy times the number of protons on target remains constant. We apply these ideas to the specific cases of 2.2, 4.4, 20, 50 and 400 GeV protons. We simulate focusing systems that are designed to best capture the secondary pions of the ``optimal'' energy. We compute the expected sensitivities to $\sin^22\theta_{13}$ for the various configurations by assuming the existence of new generation accelerators able to deliver integrated proton intensities on target times the proton energy of the order of ${\cal O}(5\times 10^{23})\rm\ GeV\times\rm pot/year$.


Introduction
The firmly established disappearance of muon neutrinos of cosmic ray origin [1,2] strongly points toward the existence of neutrino oscillations [3].
The approved first generation long baseline (LBL) experiments -K2K [4], MI-NOS [5], ICARUS [6] and OPERA [7] -will search for a conclusive and unambiguous signature of the oscillation mechanism. They will provide the first precise measurements of the parameters governing the main muon disappearance mechanism. In particular, the CERN-NGS beam [8,9], specifically optimized for tau appearance, will allow to directly confirm the hints for neutrino flavor oscillation.
In addition to the dominant ν µ → ν τ oscillation, it is possible that a sub-leading transition involving electron-neutrinos occur as well. In the "standard interpretation" of the 3-neutrino mixing, the ν µ → ν e oscillations at the ∆m 2 ≈ 2.5 × 10 −3 eV 2 indicated by atmospheric neutrinos is driven by the so-called θ 13 angle. Indeed, given the flavor eigenstates ν α (α = e, µ, τ ) related to the mass eigenstates ν ′ i (i = 1, 2, 3) where ν α = U αi ν ′ i , the mixing matrix U is parameterized as: with s ij = sin θ ij and c ij = cos θ ij .
The best sensitivity for this oscillation is expected for ICARUS at the CERN-NGS. Limited by the CNGS beam statistics at low energy, this search should allow to improve by roughly a factor 5 (see Ref. [6]) the CHOOZ [10] limit on the θ 13 angle for ∆m 2 ≈ 3 × 10 −3 eV 2 . Beyond this program, new methods will be required in order to improve significantly the sensitivity.
At present, the only well established proposal in this direction is the JHF-Kamioka project [11]. In its first phase, 5 years of operation with the Super-K detector, it aims to a factor 20 improvement over the CHOOZ limit.
In Ref. [12], we have studied an optimization of the CNGS optics that would allow to increase the neutrino flux yield at low energy by a factor 5 compared to the baseline τ -optimization of the CNGS beam. This would yield an improvement in the sensitivity by about a factor two, or equivalently an improvement of a factor 10 compared to CHOOZ.
In this paper, FLUKA [13,14] Monte Carlo simulations are employed to perform a systematic study of particle production and neutrino yields for different beam energy and baselines. Focusing systems adapted to the low energy range are also investigated, to obtain realistic estimates of the achievable neutrino rates.
The whole procedure is assumed to be detector and accelerator independent. Nevertheless, to give a first estimate of the needed beam intensities, the ICARUS T3000 detector size has been assumed as a reference. This is also justified by a recent neutrino detector comparison [15], following which the 2.35 kton LAr ICARUS fiducial mass is equivalent to a detector of approximately 20 kton of steel or 50 kton of water. For this reason, we concentrate on the intrinsic electron-neutrino background from the beam and do not explicitly calculate other sources of backgrounds that are strongly related to detector performances, such as backgrounds from π 0 production in neutral current events . We assume that in reality the "best" beam would be complemented by the "best" detector and that these backgrounds will only introduce a small correction to our sensitivity estimates. In addition, we are primarily interested at this stage in the comparison among proton drivers.
We consider so-called "conventional" neutrino superbeams, in which neutrinos are produced by the decay of secondary pions obtained in high-energy collisions of protons on an appropriate target and followed by a magnetic focusing system.
In this kind of beams, the neutrino beam spectrum and its flux are essentially determined by three parameters that can be optimized appropriately: • the primary proton energy E p , • the number of protons on target N pot per year, • the focusing system, which focus a fraction of the secondary charged pions and kaons (positive or both signs depending on the focusing device).
In order to simplify the problem, we consider three "classes" of proton energies: • Low energy: protons in a range of a few GeV. We take as reference the CERN-SPL proton driver design [16] with 2.2 GeV kinetic energy and an "upgraded SPL" with similar characteristics but with 4.4 GeV protons; • Medium energy: we take 20 GeV proton energy, similar to the CERN PS machine and the 50 GeV of the JHF facility [11]; • High-energy: we take the highest energies, i.e. the 400 GeV protons like in the case of the CERN SPS.
The purpose of this work is to understand the required intensities for the various proton energies, in other words, is it more favorable to employ low-energy, highintensity proton or high-energy, low-intensity proton machines?
2 Observing the oscillation -choosing L and E In order to maximize the probability of an oscillation, we must choose the energy of the neutrino E max and the baseline L such that However, in order to observe the oscillation, we must at least see the maximum preceded by a minimum, given by The mass difference indicated by Superkamiokande is given by ∆m 2 ≈ 2.5×10 −3 eV 2 and lies within the range 1 × 10 −3 < ∆m 2 < 4 × 10 −3 eV 2 [17].
Since we are considering one single detector at a given location as target, our baseline L for a given source is fixed. This implies that the neutrino beam spectrum should not be too narrow, e.g. it should be a relatively wide-band beam in order to cope with the uncertainty on the ∆m 2 . Since the knowledge on this parameter will improve with time, in particular with the running of K2K, MINOS and CNGS experiments, the range of energies will eventually be limited by the desire to observe the minimum and the maximum. In the meantime, we consider the full mass range indicated by atmospheric neutrino observation.
We do not wish to consider baselines longer than the current CNGS baseline of L = 730 km and use therefore a range of baselines between 100 km and 730 km. Table 1 shows the values (in MeV) of E min and E max defined above as a function for the baselines. The energies range from about 50 MeV to 300 MeV for L = 100 km and from 300 MeV to 2400 MeV for L = 730 km. Clearly, we wish to optimize for high intensity, low energy neutrino superbeams.
We note that at these low energies, neutrino interactions are clearly identifiable and have generally easily reconstructible final-states. This is an advantage for detectorrelated background suppression. In a detector with the granularity like ICARUS, neutral pions can be easily suppressed via energy ionization or by direct reconstruction of the two (well separated) decay photons.

The approximate scaling with proton energy
The understanding of the relationship between primary beam energy and neutrino production is the first step toward any optimization. To isolate this relationship from the other parameters, it is more convenient to work in the "perfect focusing" approximation, where all (positively charged) mesons produced within a given solid angle are supposed to be focused exactly on the detector direction 1 . In our case, we set P T = 0 for all secondary positively charged particles produced within 1 rad.
meter long, 2 mm radius; the decay tunnel is 150 m long and 3.5 m in radius. Target optimization will not change the general scaling, and the effect of the tunnel length on muon neutrino production can be accounted for in first approximation assuming that all neutrinos are produced by pions, and that they carry the maximum possible momentum.
The pion production rate at the exit of the target for various incident proton energies E p as estimated with FLUKA are shown in Figure 1 for energies ranging from 2.2 GeV up to the 400 GeV of the SPS for the CNGS beam. In order to compare pion productions at different proton energies, we divide the spectra by the proton energy E p .
All normalized spectra have similar shape, with the maximum yield at low energies (p π ≈ 500 MeV). Far from the endpoint, all spectra scale approximately with the incoming proton energy. Departures from the overall scaling consist in a slightly different shape at low E p , and harder spectra at high E p .
If the detector is far away in the forward direction, the neutrino event rate as a function of neutrino energy can be derived from these spectra by considering that: • The neutrino carries a momentum that is 0.43 times the parent pion momentum • The Lorentz boost gives a factor proportional to E 2 π ∝ E 2 ν on the solid angle • The neutrino cross section grows approximately as E ν (not true at the lowest energies, where the quasi-elastic grows more rapidly, however, for the optimization of the proton energy, this approximation is adequate.) For a given E ν , these factors apply independently on the primary proton energy. Thus, we expect that the energy scaling observed on the pion production is reflected in the neutrino production. This is shown in Figure 2, where we plot the simulated muon neutrino event rate scaled with the primary proton energy.
The super-position of the curves at the lowest energies is impressive, except for one at E p = 400 GeV. Of course, by rising E p the energy-integrated event rate raises dramatically because the spectra extend to higher energies, but the event rate for a given E ν <≈ 0.15 × E p is simply proportional to E p .
This approximate scaling implies that we can define a power factor F for neutrino production as the product of the proton energy times the beam intensity: Up to now we did not take into account the neutrino oscillation probability. If the aim is that of having a neutrino beam centered on the oscillation maximum, E ν and the baseline L have to be chosen such as thus How to choose the baseline L to maximize the rate of oscillated neutrino events per proton incident on the neutrino target? The neutrino flux grows scales like 1/L 2 , like the solid angle. But L and E max are proportional, thus dividing bin by bin the simulated neutrino spectra with a factor E 2 ν one gets the shape of the event rate as a function of E max . This has been done for the perfect focusing approximation in Figure 3 for various proton energies E p .
It is evident that the most optimal situation is for E max in the range 200-600 MeV and this results holds essentially independently of the proton energy, at least as long as we discard the very high 400 GeV energy case. We can go beyond the 1/E 2 ν approximation and in the following we will use the exact expression for the oscillation probability and will compute exactly the number of oscillated events by folding the expected neutrino spectra.
Nonetheless, the situation gives us a large freedom in the choice of proton energy, provided that the intensity can be re-scaled accordingly, so that the power factor remains essentially constant 4 The optimal baseline L for each proton energy In order to study the scaling with the proton energy E p and the baseline L, we compute the number N e of oscillated ν µ → ν e events: where φ(E ν ) is the neutrino flux per proton on target. We normalize this number to a 2.35 kton (argon) detector, and assume 100% electron detection efficiency. For definiteness, we assume unless otherwise noted that ∆m 2 = 3×10 −3 eV 2 and sin 2 2θ 13 = 1 × 10 −3 .
In order to take into account the effect of focusing (realistic focusing is discussed in section 5), we focus ideally all particles with the acceptance of 1 rad and apply a constant "focusing efficiency" of 20%, i.e. scale down the rates by a factor 5. We will see that this assumption is quite realistic and at least conservative for the low energy neutrinos.
Similarly, one can also estimate the "goodness" of the neutrino flux by computing the number of muon charged current events N 0 µ,CC within the energies E min (∆m 2 = 1 × 10 −3 eV 2 ) and E max (∆m 2 = 4 × 10 −3 eV 2 ) (see Table 1) of the oscillation: We choose to normalize N 0 µ,CC to 10 19 pots and per kton of (argon) target. With these two parameters, we can easily compare all the possible options. It is fair to note that the number of oscillated events N e assumes that the search is not limited by background, which in practice is not the case, as we will show in section 6. However, we prefer to separate the issue of background (which will be discussed in section 7) and concentrate for the moment on the question of the needed proton intensity. As we will show, the background turns out to be also quite independent of the proton energy and hence it does not enter in the optimization of the proton energy.
The results as a function of the baseline L are summarized in Table 2. The configuration corresponding to the minimum I pot is shown in bold.
We observe that for each baseline there is an optimal proton energy E optimal p , which minimizes the required integrated proton intensity I pot to observe a fixed number of oscillated events. This is also visible in Figure 8, where we plot the integrated beam intensity needed to obtain 5 oscillated ν e events in a 2.35 kton detector for ∆m 2 = 3 × 10 −3 eV 2 and sin 2 (2θ 13 ) = 10 −3 for different beam energies and baselines. The point of minimum pots corresponds approximately to the maximum ν µ event rate. This is because in that point, the secondary pion yield energy spectrum per proton on target, is best matched to the neutrino oscillation probability. Conversely, for  Table 2: Integrated ν µ CC events per kton and 10 19 p.o.t within the relevant energy interval N 0 µ,CC and integrated beam intensity I pot , assuming a constant 20% focusing efficiency wrt perfect focusing, needed to obtain N e = 5 events (see text) as a function of the baseline L for various proton energies. The configuration corresponding to the minimum I pot is shown in bold. each proton energy there is an optimal baseline L opt , which maximizes the integrated neutrino oscillation probability in the neutrino energy region which corresponds to the largest weighted pion yield at that proton energy.
For a 2.2 GeV proton driver, the optimal baseline L opt is approximately L opt ≈ 150 km. For a 4.4 GeV proton driver, it is approximately L opt ≈ 200 km. For 20 GeV, we find L opt ≈ 450 km. For energies above 50 GeV, the optimal baseline is around 700 km.
For actual baselines smaller than the optimal baseline, L < L opt , the neutrino oscillation maximum occurs at lower energy and the yield of corresponding pions for the given proton energy is lower than in the optimal case. Hence, we need a higher intensity to compensate for this effect.
For actual baselines greater than the optimal baseline, L > L opt , the neutrino oscillation maximum occurs at higher energy and the yield of corresponding pions for the given proton energy is lower than in the optimal case. Indeed, at some point, the optimal neutrino oscillation energy corresponds to a pion energy, which is kinematically forbidden for the given incident proton energy. Hence, we need again a higher proton intensity to compensate for the kinematical suppression.
At the optimal baselines L opt the power factors are the following: This directly confirms the approximate scaling with F , apart at the lowest and the highest proton energies. Strictly speaking, a proton energy of 20 GeV appears to be the most economical choice in terms of protons, with a proton economy of about a factor 2 compared to the 2.2 and 400 GeV cases.

Results in real focusing
The standard focusing system for all neutrino beams up to now is based on magnetic horns. However, new solutions should be envisaged for low-energy, intense beams. The angular acceptance of the system has to be large (the average transverse momentum of reaction products is of the order of 200-300 MeV, comparable with the total momentum for the shortest baselines), and the amount of material within the secondary beam cone must be as little as possible, both to preserve the flux and to avoid heating/damage at high intensities.
In principle, the focusing system should cancel the transverse momentum of all secondary particles relative to the direction of flight toward the detector and this independently of their momentum p. Indeed, this is the definition of the ideal focusing.
In practice, a horn system can be designed to focalize a given signed momentum, i.e. like p. A FODO system focalizes like |p|. A series of coils should focalize like 1, i.e. independent of p. Hence, this last solution seems to be attractive in our situation.
We have found with the help of full simulations and tracking of the particles in the focusing system that (1) for the lowest energy configuration, a series of coils does indeed provide a quite optimal focusing and (2) for the higher energy configurations, where the coil focusing becomes impracticable, the traditional horn focusing can be efficiently used.

Details on coil focusing
One possible solution is to exploit the focusing capabilities of magnetic field gradients, such as the fringe field at the ends of a solenoid, or the decreasing field far from a current loop. The fact that a particle traveling almost parallel to the axis of a solenoid suffers a change of its transverse momentum when traversing the fringe region is not intuitive but can be understood by realizing that any change of the longitudinal component of an axially symmetric field is associated through Maxwell's divergence equation to a radial component of the field.
The particle motion in axially symmetric fields can be described on the basis of the Busch's theorem, which in turn follows from the conservation of canonical angular momentum. The derivation and applications of this theorem can be found in textbooks (see for instance [18,19]), here we give a summary to better illustrate our focusing system. Suppose we have a particle having electric charge q and total energy E, moving in a static, non-uniform axially symmetric field, at a distance r from the field axis (z−axis), with an angular velocityφ. Busch's theorem states that where p φ = γmφ is the azimuthal component of the particle momentum, and Ψ is the magnetic flux linked by a circle of radius r centered on the field axis.
If the field can be considered constant within the circle of radius r, we have If particles are emitted with zero angular velocity by a source located on the axis of an axially symmetric magnetic field, both the angular momentum and Ψ are null, thus const = 0.
When exiting to a region of zero field, according to Busch's theorem the particles will have zero angular momentum. This is very appealing remembering that in an uniform (solenoidal) field, all the transverse momentum is azimuthal. However, this is not the case when the field varies, and part of the momentum can also be directed along the radius. The radial motion can also be derived exploiting Busch's theorem, but the practical use is not straightforward.
However, if the variation of the magnetic field is small on the scale of the revolution time, the theorem of adiabatic invariance states that the magnetic flux encircled by the particle trajectory remains a constant of motion. The trajectory radius can be assumed constant over a revolution and given by the usual expression and from adiabatic invariance follows Under this condition, all the rotational motion is converted into the longitudinal one, and the particle transverse momentum varies as where B z0 , p 2 T 0 are the initial magnetic field and transverse momentum. Since the period of the motion is the condition of adiabatic motion means that ∆B z must be small over distances For a pion having a longitudinal momentum of 1 GeV/c, in B=1 T, ∆z is about 20 m. Is is thus difficult to focus efficiently high energy particles through fringe fields, while this method can be very effective for low energy ones. It should be stressed that this method applies equally well to positive and negative particles, even though the resulting advantage is not great since the positive component is dominant for low energy beams.   Table 3: Same as Table 2 with the focusing system included in the simulations. The configuration corresponding to the minimum I pot is shown in bold. It is computed in order to obtain a number of oscillated events N e = 5 (see text) for the various baselines L and proton energies.

Results on horn and coil focusing
We report in Table 3 our results with full detailed simulations of the focusing systems for 2.2 GeV 4.4 GeV, 20 GeV and 400 GeV proton energies.
The coil method has been applied here to the 2.2 and 4.4 GeV proton beams, where the produced pions have small energies (only positive mesons have been considered in Table 3). The target has been assumed to be a short (30 cm) mercury target, like in the SPL [16] proposal. The non-uniform magnetic field has been obtained with ten circular loops, having a radius of 1 m, positioned from 0 to 14 meters from the target, carrying decreasing currents to give a field from 20 T to zero. Examples of particle orbits and the central magnetic field intensity are shown in Figure 4. The effect on the transverse momentum distribution can be appreciated from Figure 5, where positive pions emitted from the target within 1 radian have been considered. The decay tunnel is 150 m long.
For higher energy neutrino beams of 20 GeV and 400 GeV, the traditional two-horns system has been used. The calculations presented here refer to a first horn to focus 2 GeV/c particles, followed by a reflector to focus 3 GeV/c. The horn is placed around a graphite target, has a length of 4 meters and a current of 300 kA. The reflector starts at 6 m from the target, is 4 m long with a current of 150 kA. Examples of particle trajectories can be seen in Figure 6, and the effect on the transverse momentum in the case of a 20 GeV primary beam is shown in Figure 7. The decay tunnel is 350 m long.
In the total energy range (0-2.5 GeV), the resulting focusing efficiency varies in between 0.5 and 0.2. This motivated our choice in Section 4, when comparing all possible beam/distance options, where a common energy-independent 20% focusing efficiency had been assumed. In reality, the focusing profile is not energy independent, and the real situation can be better than this, as can be derived from the comparison of Tables 2 and 3. This effect is also visible in Figure 8.
The cautious reader can wonder why we have apparently been "conservative" in assuming a 20% efficiency for the ideal focusing. The point here is that in the ideal case we assume a constant efficiency over the whole meson energy range, while in the real focusing case one effectively reaches a situation where a part of the meson energy range is focalized with an efficiency better than 20% while other parts have lower efficiencies. We have of course optimized focusing for the energy relevant to the oscillation. It would however be incorrect to assume that this higher efficiency is constant over the full energy range.
6 A superbeam to Gran Sasso (BNGS 2 ) ? 6.1 Finding the location for the source As a working hypothesis, we take for granted the existence of LNGS as an underground laboratory that can host large neutrino detectors. In particular, we have assumed for  If we take the location of the detector as fixed, the baseline L is determined by the location of the neutrino source. We have investigated various potential locations within Italy where large ENEA 3 or INFN infrastructures are already existing, as shown in Table 4. In these locations, the required conditions to host a high intensity proton driver could be met and the machine would find other applications in addition to neutrino physics.
In Table 4, the column called "angle" describes the space angle relative to the orientation of the LNGS Halls. Inclination is the incoming neutrino angle relative to the horizontal plane at LNGS. We note that due to the fortunate orientation of the LNGS Halls according to the geographical axis of the Italian peninsula, it is possible to find various laboratories within the orientation of the LNGS Halls.
Indeed, ENEA Aquilone, ENEA Brasimone, ENEA Trisaia, INFN Pavia and ENEA Ispra appear with an angle less than 15 o with respect to the LNGS Hall direction. This is an advantage for the acceptance of higher energy events given the natural longitudinal orientation of the detectors. On the other hand, in the case of ENEA Casaccia situated near Rome, the angle is 63 o . However, for the shortest baseline, we expect the relevant neutrinos to have an energy similar to that of most atmospheric neutrinos, so that we can argue that the acceptance will not be a problem given the isotropical nature of a detector like ICARUS.   Intrinsic ν e (ν e ) beam contaminations are in the range from 1% to 2% with respect to ν µ (ν µ ).
The 2.2 and 4.4 GeV proton energies are the closest to the optimal energies and only these two cases are considered in the following. We will consider 20 GeV in section 6.3 in the context of the CERN-LNGS baseline.
To appreciate the matching between the neutrino beams and the oscillation probability, we show in Figures 9 and 10 the charged current event rate at 120 km as a function of the neutrino energy. The dotted lines correspond to the oscillation probability (arb. norm.) for a ∆m 2 = 3 × 10 −3 eV 2 .
Clearly, these neutrino beams offer optimal condition to study ν µ disappearance. Indeed, the maximum of the oscillation is very well covered by the neutrino beam and hence it is quite obvious that a very precise determination of the main oscillation parameters will be accomplished. We here do not consider this any further.
We, however, concentrate instead on the ν e appearance measurement. Since we are in the presence of intrinsic ν e background from the beam at the level of 1%-2% of the ν µ component, we can improve our sensitivity by studying the energy spectrum of the ν e charged current events. This method is more sensitive than simple event counting.
In order to estimate the sensitivity, we adopt our standard fitting procedure of the various reconstructed event classes (See Ref. [20]). We assume that the neutrino and antineutrino interactions cannot be distinguished on an event-by-event basis, and hence add the ν e andν e contributions from the beam. Similarly, the oscillated spectrum is calculated by summing both ν µ → ν e andν µ →ν e oscillations, assuming the same oscillation probability for neutrino and antineutrinos.
In the present study, we considered only the energy distribution of electron events and computed the χ 2 as a function of the sin 2 2θ 13 mixing angle, scanning in ∆m 2 . The 90% C.L. sensivity region is defined by the condition χ 2 > χ 2 min + 4.6, defined by the condition that the actually observed events in the experiment coincide with the expected background. In an actual experiment, a simultaneous fit of the muon disappearance and electron appearance spectra will constrain the ∆m 2 , sin 2 θ 23 parameters and in case of negative result will limit the sin 2 2θ 13 within the allowed ∆m 2 region.
The results of the ∆m 2 scans are shown in Figure 11 for 2.2 GeV and Figure 12 for 4.4 GeV proton energy for the three assumed baselines. The curves correspond to 5 years running with 2 × 10 23 pots/year equivalent to a continous proton current of 1 mA and a fiducial mass of 2.35 kton. The assumed protons on target is compatible with an accelerator with performances similar to those of the planned CERN-SPL [16].

The CERN-GS baseline
The baseline between CERN and GS is 730 km. At this distance, the neutrino energy range relevant for ν µ → ν e search is 0.3-2.5 GeV.
The present CNGS design [8] is optimized for ν τ appearance, thus for a relatively high-energy neutrino beam. The 400 GeV/c SPS beam will deliver 4.5 10 19 protons per year on a graphite target, made of spaced rods to reduce the re-interaction rate. The two magnetic horns (horn and reflector) are tuned to focus 35 and 50 GeV/c mesons, with an acceptance of the order of 30 mrad. The decay tunnel length is 1 km. With the standard CNGS parameters, the low-energy neutrino flux is low, as can be seen from the entries flagged by † in Table 6.
In Ref. [12], we have studied a L.E. optimization of the 400 GeV protons of the CNGS in order to improve the sensitivity to θ 13 . This yielded an improvement of a factor 5 in flux at low energy compared to the τ optimization.
Here, we study the 20 GeV proton energy (we call this the PS++) and compare it to 400 GeV. Expected neutrino fluxes and rates obtained with real focusing systems are reported in Table 6.
The results of the ∆m 2 − sin 2 2θ 13 sensitivity scans are shown in Figure 14. The curves correspond to 5 years running with 2 × 10 21 or 2 × 10 22 pots/year and a fiducial mass of 2.35 kton.

The intrinsic ν e background
As well known, the intrinsic electron (anti)neutrinos in the beam are produced either in the decay of muons coming from kaons or pions via the chain π + /K + → µ + + ν µ , µ + → ν e + e + +ν µ , or directly in the three-body K e3 kaon decays.
When looking for ν µ → ν e oscillations, this contamination will eventually be the limiting factor. It is therefore essential to understand its level and it is also worth understanding if the beam design can be optimized to minimize this background.
Rather than the ν e /ν µ ratio, we decide to consider the ratio ν µ / √ ν e in order to better estimate the effect of the backrgound on the ν µ → ν e oscillation sensitivity.
Which source of ν e is relevant to our study?
Due to the large difference between the π and µ decay lengths, the electron neutrino background depends on the length of the decay tunnel l. This dependence should be more evident for low beam energies.
Kaon production strongly depends on the proton energy, as shown in Figure 15, where a threshold effect is clearly visible. For the lowest proton energies, 2.2 and 4.4 GeV, the fraction of kaon relative to pions is a few per mille. Above 20 GeV, it is on the order of a little less than 10%.
At low ν e energies, the production is shared by kaon and muon decays, while kaon decays alone are responsible for the high energy tail. For low proton beam energies, kaon production is much lower and practically all the intrinsic electron contamination comes from muon decays.
A first order estimate of the effect of the decay tunnel length l on the muon-induced background can be derived assuming forward decay at each step, and counting for each neutrino energy the fraction of parent particles that decay within a path l. The fraction D π of π decayed after a length l is simply given by Muons have to be generated by a pion first, thus The decay lengths λ ′ π , λ µ depend on the energy of the meson. We can fix a neutrino energy E ν , the same for ν µ and ν e . For ν µ production, we can then assume p π = E ν /0.43. For the ν e produced in the decay of a muon, there is no fixed relation, but we can take the average of the ν e energy in the muon rest frame, getting p µ ≈ E ν /0.6. The grand-parent pion had therefore a momentum equal to p ′ π ≈ p µ ≈ E ν /0.6. With these approximations, D π and D µ can be expressed as a function of l/E ν . In these units, one π decay length corresponds to l/E ν = 130 m/GeV.
It is a priori obvious that short decay tunnels reduce the relative probability of muon production and decay. However, the pion decay yield is also affected.
To study the background from muons, we consider the statistically significant ratio πdecays/ √ µdecays as the correct estimator for ν µ / √ ν e as a function of the decay tunnel length. This ratio is shown in Figure 16, where the assumption of a fixed ν e energy in µ decay has been relieved: the hatched band corresponds to 20%-80% of the maximum ν e energy in the µ rest frame.
We find that the ratio does not show dramatic variations between 0 and 4 pion decay lengths. We therefore conclude that not much is to be gained by reducing the length of the decay tunnel.
We have verified these results directly by the full simulation of the neutrino beams for various decay tunnel lengths. Answers are reported in Table 7 for various proton energies and decay tunnel lengths. The 6th column shows the expected ν e contamination relative to the ν µ and the last column lists the statistically relevant ratio ν µ / √ ν e . For the 2.2 GeV proton energy, the ratio ν e /ν µ varies from 0.3% for l = 20 m up to 1.7% for l = 150 m, but this happens at a high cost of genuine ν µ 's. The statistically relevant ratio ν µ / √ ν e varies from 0.67 for l = 20 m down to 0.47 for l = 150 m. This is a modest loss. We also stress that the naive √ N scaling is not adequate for an appearance experiment where we are looking for few events and hence we conclude that genuine ν µ rate is more important than a slightly better ν e /ν µ ratio.
Remains the issue of the proton energy. Naively, one would expect that the higher the proton energy, the higher is the background. We find however that the intrinsic electron neutrino background does not strongly depend on the proton energy.
This was verified directly for various proton energies and baselines. Results of the calculation are shown in Table 8, all normalized to 10 19 pots. We observe that (1) for the shortest baselines, the ratio ν e /ν µ is increasing dramatically with proton energy. Accordingly, (2) the ratio ν µ / √ ν e decreases. However, we must rescale this ratio to take into account the approximate scaling of the number of events with the proton energy. Since we expect we consider the rescaled ratio ν µ / ν e · E p in the last columns of the Table 8. These are also plotted in Figure 17 as a function of the baseline L. Numerically, we find that the rescaled ratios ν µ / ν e · E p are almost the identical at the optimal baselines of each proton energy, so not much is too be gained by varying the proton energy.
Summarizing, we find that the electron neutrino background in the relevant region is not dependent on the proton energy and only determined by the decay tunnel length, but that its optimisation is very limited. For the maximum neutrino muon flux, it is at the level of the 1% for any of the considered setups.

Summary and Conclusions
In this document, we have performed a two-dimensional scan, varying the beam energy and baseline parameters to optimize the conditions for the investigation of θ 13 driven neutrino oscillations in the whole Superkamiokande allowed ∆m 2 range. We find that: • The optimal baselines for θ 13 searches are in the range 100-700 km for proton energies varying from 2.2 to 400 GeV.
• The needed beam intensity scales approximately with the inverse of the beam energy.
• In terms of proton economics, the optimum beam energy is around 20 GeV, but lower beam energies are appealing for the shortest baselines.
Realistic focusing system for low and medium baselines have also been studied. In this case:  • Focusing efficiencies of 30-50% can be achieved in the energy range of interest.
The whole procedure is detector and accelerator independent. Nevertheless, to give a first estimate of the needed beam intensities, the ICARUS detector has been assumed as a reference.
We can draw the following observations: • a 2.2 GeV or 4.4 GeV high-intensity proton machine (i.e.à la CERN-SPL) is well matched to a baseline in the range 100-300 km. It is not matched to a baseline of 730 km (i.e. CERN-LNGS).
• a 20 GeV machine is best matched to a baseline of 730 km (i.e. CERN-LNGS). However, an integrated intensity in the range of 10 23 pots are required, which is about two orders of magnitude higher than the intensity deliverable by the current CERN-PS in a reasonable amount of time.
• a 400 GeV energy is reasonably matched to a baseline of 730 km (i.e. CERN-LNGS). For the 400 GeV, the required intensity is in the range of 10 22 pots, which is about one order of magnitude higher than the intensity deliverable by the current CERN-SPS in a reasonable amount of time.
Finally, we stress that the present study is essentially a theoretical one. All the "real" work has still to be accomplished in order for one of these options to become reality.    GeV proton beam on a Hg target. Dotted tracks: p π < 0.5 GeV, dot-dashed tracks 0.5 < p π < 1 GeV, continuous tracks 1 < p π < 2 GeV, dashed tracks p π > 2 GeV. The thick curve is the on-axis magnetic field value (scale on the right). Loop positions are marked on top and bottom. R(cm) Figure 6: Particle trajectories in the horn+reflector focusing, from a 20 GeV proton beam on a C target. Dotted tracks: p π < 0.5 GeV, dot-dashed tracks 0.5 < p π < 1 GeV, continuous tracks 1 < p π < 6 GeV, dashed tracks p π > 6 GeV.