Long baseline neutrino experiments, mass hierarchy and delta(CP)
aa r X i v : . [ h e p - ph ] F e b Long baseline neutrino experiments, mass hierarchyand δ CP C. R. Das ∗ , Jo˜ao Pulido † CENTRO DE F´ISICA TE ´ORICA DE PART´ICULAS (CFTP)Departamento de F´ısica, Instituto Superior T´ecnicoAv. Rovisco Pais, P-1049-001 Lisboa, Portugal
Abstract
We investigate the possibilities offered by the long baseline experiments T2K, No ν a,LBNE and LAGUNA for the evaluation of the neutrino mass hierarchy and CP violat-ing phase δ CP . We consider a neutrino and antineutrino energy in the interval [0.5,12]GeV. It is found that the clearest possible distinction between the two hierarchy sig-natures is provided by LAGUNA for an (anti)neutrino energy E (¯ ν ) ν ≃ . ν µ → ¯ ν µ ) ν µ → ν µ disappearance channel. For LBNE at E ¯ ν ≃ ν µ → ¯ ν µ channel may also provide a distinction, although not so clear, and for No ν a this maybe even less clear. These results are essentially the same for the θ first and secondoctant solutions. As for δ CP determination, LAGUNA also offers the best chances at E ¯ ν ≃ . ν µ → ¯ ν e channel with θ in either octant. Regarding nonstan-dard interactions, the best possibility for their investigation resides in No ν a whosesource-far detector distance can become a magic baseline if the hierarchy is normalfor channels ν µ → ν µ , ¯ ν µ → ¯ ν e , ¯ ν µ → ¯ ν µ with energy (8.9-9.1) GeV, (11.6-12) GeV,(8.8-12) GeV. If the hierarchy is inverse, magic baselines for No ν a occur for ν µ → ν e , ν µ → ν µ , and ¯ ν µ → ¯ ν µ with energy (11.8-12) GeV, (8.9-12.3) GeV, (8.8-8.9) GeV. ForLAGUNA a magic baseline appears for the ¯ ν µ → ¯ ν µ channel only at (4.7-5.0) GeV ininverse hierarchy. We have also investigated a possible complementarity between T2Kand No ν a. ∗ E-mail: [email protected] † E-mail: [email protected]
Introduction
The solar and atmospheric neutrino anomalies [1] - [8] have since long ago motivated in-terest [9] in long baseline (LBL) neutrino experiments ( [10] - [14]), aiming at an accurateevaluation of some of the neutrino parameters ‡ . Among these, the values of the masssquared differences and mixing angles are by now reasonably well established from the dataof the solar [1] - [7], atmospheric [8] and reactor experiments [16] - [18]. Still, several issueson neutrino parameters are still open, namely the absolute mass, the mass hierarchy andthe magnitude of δ CP , the CP violating phase. LBL neutrino experiments, either fromaccelerators [11] - [14], [19], neutrino factories or beta beams [20], [21] will be essential inproviding the answer to the two most important questions, namely, whether the hierarchyis normal (NH, ∆ m >
0) or inverse (IH, ∆ m <
0) and whether or not CP violationoccurs in the leptonic sector ( sin δ CP = 0).The recent measurement of the θ mixing angle [16], [18] reduces the parameter de-generacy [22] - [25] inherent in the LBL three neutrino analysis, so that the above goalshave become within reach of the forthcoming experiments. Besides the existing T2K [12]and No ν a [13], the latter expected to start taking data soon, two other major projects arebeing considered: LBNE [19] and LAGUNA/LBNO [26]. The first, whose proposal is undercurrent evaluation, aims at sending a neutrino beam from Fermilab to Homestake ( ∼ ∼ µ − → e − ¯ ν e ν µ , µ + → e + ν e ¯ ν µ , the precise content of the neutrino and antineutrino beam can be accurately determined.There have been many studies of CP symmetries and mass hierarchy via neutrino os-cillations for present and future facilities (see e.g. [31] - [36]) and the prospects for theirdetermination have considerably improved after the measurement of θ [16], [18]. In thepresent work we take advantage of this fact and explore the possibility for extracting δ CP and the mass hierarchy from the T2K [12], No ν a [13], LBNE [19] and LAGUNA/LBNO [26] ‡ For a recent review on LBL neutrino experiments see [15]. δ CP can be overcome with the data from a single experiment. In fact to this end one de-tector located sufficiently far away from the source may be enough, as already noted in [37].Of course the possible redundancy provided by one or two extra detectors will always bewelcome for the verification and accuracy of the results § .This paper is organized as follows: we start in section 2 with the derivation of the generalexpression for the oscillation probability in matter with constant density without resortingto any approximations. In section 3 we start by comparing the results of our numericalcalculation with the leading term approximations used in the literature [39], [40], [41]. Fromthe oscillation probability expression we obtain the biprobability plots [39] for a few relevantpairs of oscillation channels in which most of our analysis will be based. Thus in subsection3.1 we get such plots for fixed distance, varying energy and several values of δ CP , in thecases of normal and inverse hierarchies. It will be seen that within preferred energy rangesthe relative ordering of the iso- δ CP curves remains invariant as the energy changes, whilefor other energy regions at the same distance they successively intersect each other. Inthese cases, for a small change in energy any two neighbouring curves may swop theirrelative position. This entails a less accurate, if at all possible, determination of δ CP in thecorresponding energy range at that distance. In fact the biprobability curves for constantenergy as a function of δ CP are closed contours which intersect the iso- δ CP curves. Sincethe neutrino energy cannot be known with arbitrarily good accuracy, the two parameters(beam energy and distance from source) must be conveniently chosen to ensure that the iso- δ CP curves do not intersect each other within the energy uncertainty range. Moreover theregions of sudden variations of the CP phase along the closed contours should be avoidedso that δ CP could be determined with a reasonable accuracy. In subsection 3.2 we obtainthe biprobabilities for fixed energy, varying distance and several values of δ CP also for bothhierarchies. Closed contours are now obtained for constant distances as a function of the CP phase and the same rule as before applies for the requirement of its accurate determination.Thus far our analysis will only use the first octant solution for the θ mixing angle and ismainly focused on the No ν a, LBNE and LAGUNA experiments. It turns out that the latterexperiment appears to be particularly promising as regards its contribution to providingdefinite conclusions for the mass hierarchy and the δ CP range. At the end of section 3.2we provide a comparison between the θ first and second octant solutions in the threeexperiments. Section 4 is dedicated to a comparison of T2K and No ν a emphasizing theprospective distinction in these experiments between the signatures of mass hierarchies andbetween the signatures of the first and second octant solutions for θ . Finally in section 5we summarize the present work and draw our conclusions. § The idea of two detectors placed at different baselines was first proposed in [38]. However, owing to thelimited knowledge on the mass differences and mixing angles at the time, a solution to the degeneracies wasfar from reachable. The oscillation probability
The starting point for our neutrino oscillation analysis is the standard matter Hamiltonianwhich in the flavour basis reads H ( f ) = U ∆ m E
00 0 ∆ m E U † + V k (1)where ∆ m ji = m j − m i , V k = √ G F N e k , G F is the Fermi constant and N ek is the electrondensity in the k th layer. The latter can be expressed in terms of the mass density ρ k , theelectron number density Y k and the nucleon mass m N as N ek = Y k ρ k m N . (2)For the cases we study we need only to consider propagation within the Earth’s crust, forwhich ρ k = 3 g cm − and Y k = 1 /
2. In eq.(1) U is the usual leptonic mixing matrix P M N S relating the neutrino mass basis to the flavour basis ¶ | ν α > = U | ν i > (3)and in a like manner for future purpose one can define a unitary transformation relating themass matter eigenstates to the flavour ones | ν α > = U ′ | ν ′ i > . (4)Given the definition of U (eq.(3)), denoting by H ( m ) the mass basis Hamiltonian operatingbetween mass eigenstates | ν i > and < ν j | and H ( f ) the flavour basis one operating betweenflavour eigenstates | ν α > and < ν β | , we have < ν j | H ( m ) | ν i > = < ν β | U H ( m ) U † | ν α > = < ν β | H ( f ) | ν α > (5)hence H ( f ) = U H ( m ) U † . (6)Since H ( f ) and H ( m ) are related by a unitary transformation their eigenvalues are the same,therefore denoting by V f and V respectively the matrices that diagonalize H ( f ) and H ( m ) V † f H ( f ) V f = V † H ( m ) V = H D (7)where H D is the diagonalized Hamiltonian. Using (6) one obtains the relation betweenmatrices V f and V , V f = U V (8) ¶ We use the Particle Data Group notation [42] for the U matrix.
3o that
U V also diagonalizes the flavour basis Hamiltonian as can be easily checked fromeqs.(1) and (6). Moreover applying the definition of U ′ (eq.(4)) < ν β | H ( f ) | ν α > = < ν ′ j | ( U ′ ) † H ( f ) U ′ | ν ′ i > (9)which shows that U ′ also diagonalizes H f , thus U ′ = V f = U V .Since the neutrino Hamiltonian (1) satisfies a Schr¨odinger like equation i ddx ν α = H ( f ) αβ ν β ( α, β = e, µ, τ ) , (10)a state produced as flavour α at the origin becomes, after traveling a distance Lν β ( L ) = S βα ( L, ν α (0) (11)where S βα ( L,
0) satisfies the same Schr¨odinger equation. So in index notation S βα ( L,
0) = ( U ′ ) βi exp (cid:20) − i Z L ( U ′ ∗ ) γi ( H ( f ) ) γδ ( U ′ ) δj dx (cid:21) ( U ′ ∗ ) αi = ( U ′ ) βi e − iλ i L ( U ′ ∗ ) αi (12)where λ i ’s are the mass matter eigenvalues. The oscillation probability P ( ν α → ν β , L ) = | S βα | (13)is then evaluated through P ( ν α → ν β , L ) = ( U ′ ∗ ) βi e iλ i L ( U ′ ) αi ( U ′ ) βj e − iλ j L ( U ′ ∗ ) αj (no sum in α, β ) (14)with the following neutrino parameter values [43], sin θ = 0 . , sin θ = 2 . × − , sin θ = 0 . , ∆ m = 7 . × − eV , ∆ m = 2 . × − eV . (15)For the sake of comparison we will present one example with the second octant solution for θ [44] sin θ = 0 . . (16)We will use equation (14) throughout the paper as the basis for our calculations. ν a, LBNE, LAGUNA The channels to analyse in this section are the ones for which the three experiments (No ν a,LBNE and LAGUNA) are dedicated, namely the muon neutrino disappearance channel( ν µ → ν µ ), the inverse golden channel or electron neutrino appearance ( ν µ → ν e ) and theirantineutrino counterparts for normal and inverse hierarchy. We denote these channels as µµ ,4 e , their counterparts as ¯ µ ¯ µ , ¯ µ ¯ e respectively and accordingly their oscillation probabilitiesas P µµ , P µe , P ¯ µ ¯ µ , P ¯ µ ¯ e . As mentioned in the introduction our main results are presented interms of biprobability graphs.In the current literature the oscillation probabilities for long baselines are usually calcu-lated by resorting to approximations up to first or second order in ’small’ parameters suchas [27], [39], [40], [41], [46]2 √ G F N e E ∆ m , ∆ m E L, ∆ m ∆ m , θ , ∆ m ∆ m = ǫ ≃ θ . In particular, as it is known nowadays, the approximation ǫ ≃ θ is rather innacurate.Here we will use instead the exact numerical expressions from (14) and show in fig.1 thecomparison between our results and those from the approximations used in the literature.In the four top panels of fig.1 we plot the oscillation probability P µe as a function of distancefor eight values of δ CP equally spaced from 0 to 360 degrees and in the two bottom panelsthe biprobability plots for the channel pair µe, ¯ µ ¯ e . Here we consider normal hierarchy onlyand a neutrino energy E=2.3 GeV. Notice the large discrepancies in the probabilities andthe locations of the intersections from panel to panel. These intersections correspond to themagic baselines to which we will return in section 3.2. In this subsection we just consider the No ν a [13] and LBNE [19] experiments and study theconstant δ CP curves as a function of energy for fixed distance.No ν a is planned to start taking data in 2013 and consists of a 330 ton near detector atthe Fermilab site and a 14 kiloton far detector. The latter is a liquid scintillator situated 12km off-axis at 810 km in the Neutrino Main Injector (NuMI) beam produced at Fermilab.It is dedicated mainly to observe ν µ → ν e oscillation along with its antiparticle counterpart.The advantage of an off-axis location is that the neutrino energy is nearly independent fromthe parent meson energy, therefore the beam energy spread is much smaller than for anon-axis one [45]. Also using the NuMI beam, but not yet finally approved, is LBNE with a34 kton liquid Argon far detector at 1290 km distance and a neutrino energy in the interval[0.5,12] GeV.In each of the four panels of fig.2 we show the biprobability curves for the pair of channels µe , µµ for eight values of δ CP equally spaced from 0 to 360 degrees with energy running from0.5 GeV to 100 GeV. The top two panels refer to a distance of 810 km and the bottom twoare for 1290 km which are the source-far detector distances for No ν a and LBNE respectively.The left panels are for normal and the right ones are for inverse hierarchy. In each panel weplot the biprobability curves for three energies which are ellipses of large eccentricity: for810 km we take E=2.3 GeV which is the preferred No ν a energy (middle ellipses in the toppanels), E=1 GeV and 0.5 GeV (top and bottom ellipses respectively). In order to keep thefigures as clear as possible, we choose not to continue the iso- δ CP curves for E < δ CP is the one where the iso- δ CP curves lie the furthest apart since the phase variation is then the smoothest possible alongeach ellipse. Consequently, as it is seen from all four panels of fig.2, a value near E=0.5 GeVis the best choice for δ CP evaluation. For 1290 km (LBNE source-detector distance) we takeE=7 GeV, 1 GeV and 0.5 GeV (top, middle and bottom ellipses in the bottom panels offig.2). Again we omit the iso- δ CP curves below 1 GeV and find that the best sensitivity to δ CP is for 0.5 GeV, although a double degeneracy remains in its determination due to thevery large eccentricity of the ellipses. In the limit of increasing energies the constant δ CP curves merge and eventually coincide at the point ( P µe = 0, P µµ = 1) whereas they divergefrom each other for decreasing energy. This is consistent with the fact that the oscillationlength increases for increasing energy, so for fixed distance the neutrinos oscillate less astheir energy increases. The same as in fig.2 is done in figs.3 and 4 for the pairs of channels¯ µ ¯ e , ¯ µ ¯ µ and µe , ¯ µ ¯ e respectively. In fig.3 (for ¯ µ ¯ e , ¯ µ ¯ µ ) and in the top panels, the middle ellipsesare for E=2.3 GeV, the top and bottom ones for E=1 GeV and 0.5 GeV respectively, all for810 km distance as in fig.2. In the bottom panels (1290 km) the top, middle and bottomellipses are for E=7 GeV, 1 GeV and 0.5 GeV. Once again the iso- δ CP curves are ommitedbelow 1 GeV while 0.5 GeV is the most convenient choice as regards δ CP evaluation, if notfor a double degeneracy. As for the channel pair µe , ¯ µ ¯ e (fig.4) the 0.5 GeV contours areagain the ellipses with the longest major axes. So this energy is also the most convenientchoice for δ CP evaluation. It will be seen in the next section that an improved situation canbe found with data from LAGUNA.The possibility of distinction between hierarchies appears realistic so long as the neutrinoenergy can be appropriately tuned, as can be inferred from the comparison between the twobottom panels of fig.3 (¯ µ ¯ e , ¯ µ ¯ µ channels at the LBNE source-detector distance). In fact forE=1 GeV, which corresponds to the middle ellipse, it is seen that P ¯ µ ¯ µ ≃ .
48 for inversehierarchy whereas for normal hierarchy 0 . . P ¯ µ ¯ µ . .
66. Not so clear a distinction can beprovided at the No ν a distance for the same energy, as one can conclude from the comparisonbetween the two top panels in figs.2 and 3.In a like manner as figs.2, 3, 4, fig.5 shows the biprobability curves for the µµ , ¯ µ ¯ µ channelpair in normal hierarchy for 0.5 GeV, 1 GeV and 2.3 GeV at 810 km. In this case the iso- δ CP curves are nearly superimposed, so that an unrealistically large experimental accuracywould be needed in order to obtain information on δ CP . The same situation occurs forinverse hierarchy for the same energies and distance and for normal and inverse hierarchiesat 1290 km with 0.5 GeV, 1 GeV and 7 GeV. Hence the µµ , ¯ µ ¯ µ channel combination is oflittle use, if any. In this subsection we extend our analysis to include the LAGUNA experiment and study theconstant δ CP curves as a function of distance for fixed neutrino energy from the productionpoint up to 2290 km. This is the CERN-Pyhas¨almi mine distance where the LAGUNA6etector is planned to be installed observing neutrinos produced from CERN. We considertwo neutrino energies: E=2.3 GeV and 0.5 GeV. Our results are plotted in figs.6, 7 (for2.3 GeV) and 8, 9 (for 0.5 GeV). The top panels in fig.6 contain the biprobability curvesfor the µe , µµ channel pair, the middle ones for ¯ µ ¯ e , ¯ µ ¯ µ and the bottom ones for µe , ¯ µ ¯ e .Left hand panels are for normal and right hand ones for inverse hierarchy respectively. Theiso- δ CP curves are shown for eight values of δ CP equally spaced from 0 o to 360 o as a functionof distance, all diverging from a common point at zero distance. The constant distancecontours are for 810 km (No ν a, dotted lines), 1290 km (LBNE, dashed lines) and 2290 km(LAGUNA, full lines). k . Note that the No ν a and LBNE contours correspond to the onesalready shown in fig.2 for µe , µµ now superimposed on the iso- δ CP curves for varying energyinstead of varying distance as before. In the top right and middle left panels of fig.6 ( µe , µµ channel pair, inverse hierarchy and ¯ µ ¯ e , ¯ µ ¯ µ channel pair, normal hierachy) all curves intersectat one point. This corresponds to one of the channel probabilities being independent of δ CP at one particular distance, the so called magic baseline. Such distance can be evaluated byexpanding the appearance probability P µe = | ( U ′ ) ei e − iλ i L ( U ′ ∗ ) µi | (17)and separating the δ CP dependent and independent terms. We obtain P µe = 2 s c s c s c ( As δ + Bc δ ) + δ CP independent terms (18)with A = sin ( λ − λ ) L − sin ( λ − λ ) L + sin ( λ − λ ) L (19) B = c θ [1 − cos ( λ − λ ) L ] − cos ( λ − λ ) L + cos ( λ − λ ) L where c = cos θ , ... and L is the distance from the source to the detector. In order toensure the independence of P µe on δ CP one must require the vanishing of the quantity inbrackets in (18). To this end we note that the three arguments of the sines and cosines inthe expressions for A and B cannot simultaneously vanish for any possible baseline distanceand moreover such condition would only be a sufficient one for P µe to become independentof δ CP . So one must impose tan δ CP = − BA (20)for any δ CP , which requires A = B = 0. In this way for E=2.3 GeV one gets for the firstthree magic baselines for the µe channel in inverse hierarchy L magic ≃ km, ≃ km and ≃ km (see fig.7). Hence the common point of all curves in the top right panelof fig.6 corresponds to the shortest one at 1980 km . Analogously, for the ¯ µ ¯ e , ¯ µ ¯ µ channelpair in normal hierarchy, the magic baselines occur for ¯ µ ¯ e , the shortest one being located at k The preferred neutrino energy for LAGUNA has not been decided yet. Here we consider the sameenergies for all three experiments so that the corresponding closed contours appear in the same panel. magic ≃ km , and the following ones at approximately 4020 km and 6030 km . Theycan be easily determined in the same way as in the previous case with the replacements U ′ → U ′ ∗ V → − V. (21)Magic baselines are particularly useful for the investigation of nonstandard interactions [46],[47], [48].Fig.8 shows the biprobability contours for a neutrino energy E=0.5 GeV and the samechannels as fig.6 at the No ν a (dotted), LBNE (dashed) and LAGUNA (full) distances. Forsimplicity the iso- δ CP curves are ommited from fig.8.From the inspection of figs.6 and 8 (neutrino energies 2.3 GeV and 0.5 GeV respectively)one can infer the prospects for distinguishing between normal and inverse hierarchies. Tothis end one must compare the left and right panels of these figures. Starting with fig.6(middle panels) it is seen that for LBNE alone (bottom ellipse appearing as a line segment)normal hierarchy gives 0 . . P ¯ µ ¯ e . . P ¯ µ ¯ µ ∼ .
06 while inverse hierarchy gives0 . . P ¯ µ ¯ e . . P ¯ µ ¯ µ ∼ .
09. Also for LBNE and appearance probabilities only, thecomparison of the bottom panels if fig.6 shows for normal hierarchy 0 . . P µe . . . . P ¯ µ ¯ e . .
034 while for inverse hierarchy such probabilities are interchanged. Anothertypical case can be observed from the top panels of fig.6 with the uppermost ellipses whichcorrespond to the LAGUNA experiment. Here it is seen that 0 . . P µe . .
060 and P µe . .
020 for normal and inverse hierarchy respectively, hence some overlap exists in thetwo probabilities. As for the µµ channel one has 0 . . P µµ ≃ .
95 and 0 . . P µµ . . . P µe . .
14, 0 . . P µµ . .
40, while for inverse hierarchy 0 . . P µe . . . . P µµ . . . . P ¯ µ ¯ e . .
21, 0 . . P ¯ µ ¯ µ . .
50 while in inverse hierarchy 0 . . P ¯ µ ¯ e . .
16, 0 . . P ¯ µ ¯ µ . . µµ and ¯ µ ¯ µ at an energy near 0.5 GeV in the LAGUNA experiment. In fact inall cases a dedicated observation of fig.8 shows that the difference between the probabilitiesfor normal and inverse hierarchies is substantial P µµ ( N H ) − P µµ ( IH ) ≥ . P ¯ µ ¯ µ ( N H ) − P ¯ µ ¯ µ ( IH ) ≥ .
32 (22)and moreover it is also seen that the other two experiments ( No ν a and LBNE) cannotoffer such a possibility. Since this is the most favourable case for hierarchy determination,we have also evaluated this probability difference using the second octant solution for θ (eq.(16)). The result is 8 µµ ( N H ) − P µµ ( IH ) ≥ . P ¯ µ ¯ µ ( N H ) − P ¯ µ ¯ µ ( IH ) ≥ .
29 (23)Furthermore if the hierarchy is normal, inspection of the bottom left panel of fig.8 showsthat there are realistic prospects for δ CP evaluation. In fact for LAGUNA, whose contourappears in this scale similar to a circle, and for the µe channel, we have 0 . . P µe . . . . P µe . .
13, the two possible values of P ¯ µ ¯ e arereasonably apart from each other: either P ¯ µ ¯ e . .
04 or 0 . . P ¯ µ ¯ e . .
21. In this way theambiguity in δ CP can possibly be lifted. If otherwise P µe is found to lie close to either endof the interval, namely 0.05 or 0.14, the value of δ CP is also unambigously determined, asthere is only one possibility in each case: δ CP ∼ o or δ CP ∼ o respectively. So farall results obtained are for the θ first octant solution (eq.(15)). Still for the energy valuebeing considered now (E=0.5 GeV) we show in fig.9 the same as fig.8 for the second octantsolution ( θ > ◦ , eq.(16)) whose results are similar. For larger energies the distinctionbetween the first and second octant solutions is even less clear as all contours pertaining tothe three experiments become closer. We will return to this point in the next section.810 kmChannel E (GeV) P µµ − − µ ¯ e − . − . × − ¯ µ ¯ µ − − The energy and oscillation probability for a magic baseline in normal hierarchy atNo ν a (810 km): for each oscillation channel and the neutrino energy range the probabilityshown is nearly independent of δ CP . No magic baseline is found to exist for LBNE andLAGUNA in the energy range [0.5,12] GeV in normal hierarchy.
810 km 2290 kmChannel E (GeV) P E (GeV) P µe − . − . × − — — µµ − − µ ¯ µ − − − . − . × − Table 2:
The same as table 1 in inverse hierarchy for No ν a (810 km) and LAGUNA (2290km). No magic baseline is found to exist for LBNE in the energy range [0.5,12] GeV ininverse hierarchy. Finally given the source-detector distances for No ν a, LBNE and LAGUNA, we haveestimated the necessary neutrino energy for each distance to become a magic baseline andthe corresponding oscillation probability. We analysed the energy interval [0.5,12] GeV forwhich all three detectors are designed. Our results are shown in tables 1 and 2. For eachoscillation channel we show the energy range where the probability is nearly constant with9etectors located at 810 km from the neutrino source (No ν a) and 2290 km (LAGUNA).For LBNE no magic baseline is found to exist in this energy range. We note that for the¯ µ ¯ µ channel both distances can become magic baselines, whereas for µe with normal and¯ µ ¯ e with inverse hierarchy none of the distances is suitable. Since at the magic baselinethe oscillation probability becomes δ CP independent, any significant deviation from theoscillation prediction will be the signature of nonstandard interactions [46], [47], [48]. ν a The main objective of the T2K experiment [12] is to discover ν e appearance from ν µ . Thecollaboration reported their first results in June 2011 [49] which were later improved [50]and also reported evidence for ν µ disappearance [51]. The experiment is a long baselineoff-axis one with a 295 km source-detector distance from Tokai (J-PARC) to Kamioka (Su-perKamiokande) and its peak neutrino energy is around 600 MeV.Our aim in this section is twofold: to obtain the predictions for the oscillation proba-bilities P µe and P µµ at T2K displaying them in terms of biprobability plots and to explorethe possible complementarity between T2K and No ν a. Our results are shown in the fourpanels of fig.10. We consider normal and inverse hierarchies (left and right panels) and firstand second octant solutions for θ (top and bottom panels).We recall that in the previous section the prospective data from one experiment werecompared at the same energy for different hierarchies and/or different octants. In no case thecontours from two different experiments were ever directly compared. Thus it was ensuredthat the comparisons were made at the same point in the oscillation phase. Exploring thecomplementarity between T2K and No ν a, since the two experiments operate at differentphases, requires a reduction of the probability to the same phase. Given the fact that T2Kruns with a peak neutrino energy E=0.6 GeV at 295 km distance one must operate No ν a,whose source-detector distance is 810 km, at a neutrino energy value satisfying (cid:18) LE (cid:19) Noνa = (cid:18) LE (cid:19) T K (24)which gives E Noνa ≃ ν a (E=1.65 GeV). Hence observing the left and right panels of fig.10on each row, it is seen that( P µe ) Noνa > ( P µe ) T K (normal hierarchy) ( P µe ) Noνa < ( P µe ) T K (inverse hierarchy) . On the other hand observing the top and bottom panels on each column,( P µµ ) Noνa > ( P µµ ) T K ( θ < ◦ ) ( P µµ ) Noνa < ( P µµ ) T K ( θ > ◦ ) . Although these inequalities apply for any value of δ CP , the differences in the probabilitiesare so small that the possibility of ever detecting them in this way is slim.10 Summary and conclusions
We have investigated the prospects for distinguishing normal from inverse neutrino masshierarchies, δ CP determination and first vs. second octant θ solutions with T2K, No ν a,LBNE and LAGUNA long baseline experiments. We examined the oscillation channelswhich are possibly relevant for these, namely µe , µµ and their antineutrino counterparts.Owing to the baseline distances involved, the neutrinos are assumed to pass through theEarth’s mantle only. The starting point for our numerical analysis is the general formula forthe matter oscillation probability derived in section 2. The discrepancies between the resultsfrom the leading term approximations existing in the literature based on the smallness ofthe mass square differences ratio or the θ mixing angle were also examined.The main results of our paper are described in section 3.2 and displayed in fig.8. They aresuggestive of the importance of LAGUNA and its operation at a neutrino energy around 0.5GeV. Otherwise we have found that there are also good possibilities to distinguish betweenhierarchies with other experiments provided the neutrino energy can be tuned to reasonableaccuracy. To this end the oscillation channels which can provide the best information arethe muon and antimuon disappearance ones, namely ν µ → ν µ and ¯ ν µ → ¯ ν µ . For a neutrinoenergy E ≃ P ¯ µ ¯ µ ≃ . . . P ¯ µ ¯ µ . .
66 for normal hierarchy (see fig.3, bottom panels),whereas for No ν a these are 0 . . P ¯ µ ¯ µ . .
68 (inverse) and 0 . . P ¯ µ ¯ µ . .
56 (normal) ascan also be seen from fig.3 (top panels). On the other hand for a neutrino energy E ≃ µµ channel in the LAGUNA far detector one expects 0 . . P µµ . . . . P µµ . .
40 (fig.8, top panels). Moreover for the ¯ µ ¯ µ channeland for LAGUNA with inverse hierarchy we obtain 0 . . P ¯ µ ¯ µ . .
06 and with normalhierarchy 0 . . P ¯ µ ¯ µ . .
50 (fig.8, middle panels). The difference between the oscillationprobabilities P ¯ µ ¯ µ for normal and inverse hierarchy is thus 0.32 or larger and for P µµ it is0.30 or larger (see eq.(22)). Hence the µµ and ¯ µ ¯ µ channels at LAGUNA with a neutrinoenergy E ≃ ν a, presentedin fig.10, does not offer a clear perspective for the mass hierarchy determination.For the θ solution in the second octant, we have also checked the possible distinctionbetween hierarchies. The chances look almost identical as for the first octant solution,although slightly disfavoured in the ¯ µ ¯ µ channel (see eq.(23)). Distinguishing between firstand second octant solutions on the other hand looks difficult as can be seen from thecomparison between figs.8 and 9 and from fig.10.As for δ CP determination the LAGUNA far detector offers good chances for its feasabilityon the basis of observing the µe and ¯ µ ¯ e channels. Again the neutrino energy must be tunedto E ≃ P ¯ µ ¯ e . .
04 and 0 . . P ¯ µ ¯ e . .
21, a relatively narrow interval for δ CP can be determined(see fig.8, bottom left panel). Again, for θ in the second octant the prospects are practically11he same.Finally, regarding nonstandard interactions, since the baseline distances are a priorifixed, these will become magic for a conveniently chosen value of the neutrino energy. Thisenergy was evaluated in section 3 along with the corresponding oscillation probability. Giventhis probability, any deviation from such a value is a signature of nonstandard interactions.In the appropriate neutrino energy interval for the three experiments, namely [0.5,12] GeV,the range to search for magic baselines is E ν ≥ ν a and (4.7 − δ CP phase range, the mass hierarchy and may be nonstandard interactionsif they really exist. Acknowledgments
We acknowledge discussions with Evgeni Akhmedov and Luis Lavoura. C. R. Das grate-fully acknowledges a scholarship from Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT, Por-tugal) ref. SFRH/BPD/41091/2007. This work was partially supported by FCT throughthe projects CERN/FP/123580/2011 PTDC/FIS/ 098188/2008 and CFTP-FCT Unit 777which are partially funded through POCTI (FEDER).
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Four top panels (clockwise): P µe as a function of distance from 0 to 7500km evaluated with our numerical approach, and with the approximate approximations givenin [41], [40], [39] for eight values of δ CP equally spaced from 0 to 360 . In our case (topleft panel) constant δ CP lines for 225 , 270 , 315 coincide with those for 135 , 90 , 45 .Two bottom panels: biprobability plots P µe , P ¯ µ ¯ e at 810 km (left) and 2290 km (right) fromthe neutrino source. Full, dotted, dashed and dot-dashed contours are obtained from ourapproach, and from the approximate expressions in refs. [41], [40], [39]. All results are for θ in the first octant (eq.(15)), normal hierarchy and neutrino energy 2.3 GeV. P µµ P µ e µ e P µµ Figure 2:
Biprobability plots for the ( µe, µµ ) channel pair with energy in the range [0.5,100]GeV and θ < ◦ (eq.(15)) showing the constant δ CP curves (which merge at coordinates(0,1) for large energy) and the constant energy contours. Top panels: 810 km distance with(from top to bottom) 1 GeV, 2.3 GeV, 0.5 GeV contours. Bottom panels: 1290 km distancewith (from top to bottom) 7 GeV, 1 GeV. 0.5 GeV contours. Left and right panels: normaland inverse hierarchy respectively. Points marked as ⊡ , (cid:4) , ⊙ , • , △ , N , ▽ , H are for δ CP = ,45 , 90 , 135 , 180 , 225 , 270 , 315 . P µ µ P µ e µ e
0 0.02 0.04 0.06 0.08 P µ µ
0 0.02 0.04 0.06 0.08 0.1
Figure 3:
The same as fig.2 for the ( ¯ µ ¯ e, ¯ µ ¯ µ ) channel pair. P µ e P µ e µ e P µ e Figure 4:
Biprobability plot for the ( µe, ¯ µ ¯ e ) channel pair with energy in the range [0.5,100]GeV, δ CP < ◦ . Top panels: the contours for 810 km with 2.3 GeV (dotted), 1 GeV (thinfull) 0.5 GeV (thick full). Bottom panels: the contours for 1290 km with 7 GeV (dotted),1 GeV (thin full) 0.5 GeV (thick full). Due to the figure scale, the contour for 7 GeV inthe bottom panel can hardly be seen. The constant δ CP curves merge for large energy atcoordinates (0,0) and the values of δ CP are marked as in figs.2 and 3. P µ µ P µµ Figure 5:
Biprobability curves for constant δ CP for the ( µµ, ¯ µ ¯ µ ) channel pair at 810 kmsource/detector distance in normal hierarchy with θ < ◦ . Their intersections with theconstant energy contours are marked. Their closeness prevents determination of δ CP . Theneutrino energy interval is [0.5,12] GeV. P µ e P µ e µ e P µ µ P µ e µ e P µµ P µ e µ e Figure 6:
Biprobability graphs for neutrino energy 2.3 GeV ( θ < ◦ , eq.(15)): the curvesfor constant δ CP for varying distance and the contours for No ν a (dotted), LBNE (thin full)and LAGUNA (thick full). Left and right panels are for normal and inverse hierarchy. Top,middle and bottom panels: ( µe, µµ ), ( ¯ µ ¯ e, ¯ µ ¯ µ ) and ( µe, ¯ µ ¯ e ) channel pairs. Values of δ CP oneach contour are marked as in fig.2: ⊡ , (cid:4) , ⊙ , • , △ , N , ▽ , H are for δ CP = , 45 , 90 , 135 ,180 , 225 , 270 , 315 . Merging of δ CP curves occurs for 0 km distance. km P µ e Figure 7: P µe (inverse hierarchy, θ < ◦ , eq.(15))) as a function of distance from 0 to6000 km for eight values of δ CP equally spaced from 0 to 360 . Full, dotted, dashed, dot-dashed and dot-double dashed lines are for δ CP =0 , 45 , 90 , 135 , 180 respectively. δ CP =225 , 270 , 315 lines coincide with 135 , 90 , 45 . The neutrino energy is 2.3 GeV. Theanalogue of this graph for normal hierarchy is shown in the top left panel of fig.1. P µ e P µ e µ e P µ µ P µ e µ e P µµ P µ e µ e Figure 8:
Biprobability graphs for neutrino energy 0.5 GeV ( θ < ◦ , eq.(15)): the con-tours for No ν a (dotted), LBNE (dashed) and LAGUNA (full). The values of δ CP on eachcontour are marked as in fig.2: ⊡ , (cid:4) , ⊙ , • , △ , N , ▽ , H are for δ CP = , 45 , 90 , 135 , 180 ,225 , 270 , 315 . Left and right panels: normal and inverse hierarchy. Top, middle andbottom panels: ( µe, µµ ), ( ¯ µ ¯ e, ¯ µ ¯ µ ) and ( µe, ¯ µ ¯ e ) channel pairs. The determination of the masshierarchy and δ CP range appear more favourable in this case than for 2.3 GeV (fig.6). P µ e P µ e µ e P µ µ P µ e µ e P µµ P µ e µ e Figure 9:
The same as fig.8 for the second octant solution ( θ > ◦ , eq.(16)). Fromcomparison with fig.8 it is seen that hardly any distinction can be made between the twosolutions. P µµ P µ e µ e P µµ Figure 10:
Comparison between T2K and No ν a: left and right panels are for normaland inverse hierarchy, top and bottom ones for θ in the first (eq.(15)) and second octant(eq.(16)). Dotted contours are for T2K and full contours for No ν a. Values of δ CP aremarked in each contour and follow the conventions of the previous figures.aremarked in each contour and follow the conventions of the previous figures.