ATMOSPHERIC, ACCELERATOR AND REACTOR NEUTRINO OSCILLATIONS

ATMOSPHERIC, ACCELERATOR AND REACTOR NEUTRINO OSCILLATIONS NIUS Project Submitted By: Unnati Akhouri, Delhi University and Smruti Manjunath, Madras University Under the Supervision of: Dr. D. P. Roy, HBCSE, TIFR CONTENTS CONTENTS Contents 1 Neutrino Mixing and Oscillation 2 2 Atmospheric Neutrino Oscillation 6 3 Three Neutrino Mixing and Oscillation Formalism 4 Determination of θ13 by SBL Reactor (anti)neutrino ments 4.1 Double Chooz experiment . . . . . . . . . . . . . . . . . 4.2 RENO Experiment . . . . . . . . . . . . . . . . . . . . . 4.3 Daya Bay experiment . . . . . . . . . . . . . . . . . . . 5 Implications for LBL Accelerator Neutrino 5.1 On-axis and off-axis experiments . . . . . . 5.2 MINOS Experiment . . . . . . . . . . . . . 5.3 T2K Experiment . . . . . . . . . . . . . . . 5.4 NoVA Experiment . . . . . . . . . . . . . . 5.5 LBNE Proposal . . . . . . . . . . . . . . . . 12 Experi. . . . . . . . . . . . . . . Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 18 19 20 21 22 25 26 31 33 6 Acknowledgments 34 7 References 35 1 1 NEUTRINO MIXING AND OSCILLATION 1 Neutrino Mixing and Oscillation Neutrinos have very tiny but non-zero masses. Then, there is no reason for the three neutrino interaction eigenstates to coincide with the mass eigenstates.In general, there will be mixing between them. In this section, we have considered the mixing between two neutrino states, for it is a simple and good approximation. This mixing is represented by:      νe cos θ sin θ ν1 (1) = νµ − sin θ cos θ ν2 Here, ν1 and ν2 are the mass eigenstates with eigenvalues m1 and m2 . Note that coherent mixing between these mass eigenstates is a quantum mechanical phenomenon. Consider, for e.g the mixed state: νe = ν1 cos θ + ν2 sin θ (2) If we take the rest frame of the lighter mass eigenstate say m1 , the total energy in this frame is m1 , in Natural units ( ~= c=1). On the other hand, the ν2 component will have a higher energy m2 (plus any kinetic energy) in the same frame. This mixed state breaks energy conservation. Such a coherent admixture of unequal mass eigenstates is allowed in Quantum Mechanics, where the energy non-conservation problem is taken care of by the uncertainty principle. This leads to the phenomenon of neutrino oscillation as suggested by Pontecorvo.[2] Consider a νe state produced by a nuclear beta decay. Its ν1 and ν2 components travel with different velocities, since they have different masses. Thus, their relative sizes will change with distance, implying transformation of νe into νµ . Neutrinos with definite mass m and momentum p do not travel as point particles, due to the uncertainty principle, but as a plain monochromatic wave, represented by the wave function: ψ = e−i(Et−pl) (3) Since the energies of neutrinos E(∼ M eV ) are very high and their masses are very small(< eV ), they are extreme relativistic particles, t=l/v where v = ≃ speed of light= c=1, so t ≃ l and E= p p2 + m2 (4) m2 1 )2 p2 (5) = p(1 + Thus by taking the binomial expansion of the above equation upto two terms and neglecting higher order terms we get, E ≃p+ m2 m2 ≃p+ 2p 2E (6) 2 1 NEUTRINO MIXING AND OSCILLATION by replacing p by E as it will show difference only in the third order term. Substituting equation (4) into equation (3), ψ = e−i(Et−pl) = e−i(m 2 l/2E ) Thus, we can see that the neutrino mass eigenstate propagates with a phase 2 of e−i(m l/2E) . The wave function of the produced νe of (2) after traveling a distance l becomes: 2 2 νe → ν1 cos θe−i(m1 l/2E) + ν2 sin θe−i(m2 l/2E) (7) Also, by inverting equation 1, we can write the mass eigenstates in terms of the interaction eigenstates as follows: ν1 = νe cos θ − νµ sin θ ν2 = νe sin θ + νµ cos θ Substituting these relations into (7): 2 2 νe = (νe cos θ − νµ sin θ) cos θe−i(m1 l/2E) + (νe sin θ + νµ cos θ) sin θe−i(m2 l/2E) 2 2 = (νe cos2 θ − νµ sin θ cos θ)e−i(m1 l/2E) + (νe sin2 θ + νµ sin θ cos θ)e−i(m2 l/2E) 2 2 2 2 = νe (cos2 θe−i(m1 l/2E) +sin2 θe−i(m2 l/2E) )+νµ sin θ cos θ(−e−i(m1 l/2E) +e−i(m2 l/2E) ) (8) Thus, we see that the coefficient of the νµ term does not cancel out. The probability of νe oscillating into νµ is given by the modulus square of this coefficient: 2 2 Peµ (l) = cos θ sin θ(−e−i(m1 l/2E) + e−i(m2 l/2E) ) = cos2 θ sin2 θ e 2 −i(m2 1 +m2 )l 4E [e −i∆m2 l 4E −e i∆m2 l 4E 2 2 ] ∆m2 l sin2 2θ −i(m21 +m22 )l/4E 2 4 sin2 e 4 4E 2 2 2 where ∆m = m2 −m1 . Also, the modulus of the phase factor equals 1. Thus we 2 l see that it is oscillatory in nature with an amplitude sin2 2θ , a phase sin2 ∆m 4E and transition probability as, = ∆m2 l (9) 4E Converting the quantities of phase in equation 9 from natural units to more convenient units, i.e, ∆m2 in eV 2 , l in meter and E in MeV, then, using the relations 1M eV 2 = 1012 eV 2 and 200M eV f m = ~c, we get:   ∆m2 l , in natural units 4E Peµ (l) = sin2 2θ sin2 3 1 NEUTRINO MIXING AND OSCILLATION   1 1 ∆m2 l in convenient units X 2x10−13 1012 4E    10 ∆m2 l = 8 E   2 ∆m l = 1.25 E   ∆m2 l ≈ 1.3 E  (10) Thus in convenient units transition probability becomes, Peµ (l) ∼ sin2 2θ sin2 1.3∆m2 l E (11) In (10), the first factor gives the amplitude and the second factor the phase of neutrino oscillation. The phase in terms of oscillation wavelength is given by: πl 1.3∆m2 l = E λ  π   E  2.4E ≃ ⇒λ= 1.3 ∆m2 ∆m2 (12) For a large mixing angle: sin2 2θ ∼ 1 ; and thus, Peµ (l) ≈ sin2 1.3∆m2 l E The corresponding survival probability,Pee , is given by: Pee = 1 − Peµ (13) Thus if l << λ then sin2 πl λ ≈ 0 which implies Peµ (l) ≈ 0,Pee (l) ≈ 1. If l ≈ λ 2 then sin2 πl λ ≈ sin2 (14) π 2 ≈ 1 which implies Peµ (l) ≈ 1,Pee (l) ≈ 0. (15) If l >> λ then averaging over the phase factor we get Peµ (l) ≈ 1 1 ,Pee (l) ≈ 2 2 (16) These expected pattern of neutrino oscillation probability are tabulated below 4 1 NEUTRINO MIXING AND OSCILLATION l Peµ << λ 0 ∼ λ/2 sin2 2θ ∼ 1 >> λ (1/2) sin2 2θ ∼ 1/2 Table 1: Oscillation probability as a function of l for large mixing angles In Table 1, the factor of 1/2 in the last case comes from averaging over the phase factor. To measure ∆m2 in any experiment, l ≥ λ2 , 2.4E 2∆m2   1.2E 2 ⇒ ∆m ≥ l ⇒l≥ (17) For solar and reactor neutrino experiment, the source of νe is nuclear reaction. So, their energy E ∼ M eV . In a long baseline experiment like KamLAND, the distance between the reactor and the source, l ∼ 105 m. From (17),   1.2E 2 ∆m ≥ l ⇒ ∆m2 ≥ 10−5 eV 2 For a solar neutrino experiment, l ∼ 1011 m. From (17), ∆m2 ≥ 10−11 eV 2 For atmospheric and accelerator neutrinos, E ∼GeV, but we can use the same equation as 16 as long as l ∼ km.. For a long baseline accelerator neutrino experiment like MINOS, the distance between the source and detector, l ∼ 103 km. So, from (17), ∆m2 ≥ 10−3 eV 2 In atmospheric neutrino experiment, l is given by the diameter of the earth, i.e, l ∼ 104 km. Thus,from (17), ∆m2 ≥ 10−4 eV 2 Thus we see that these neutrino experiments can measure mass to much small scale as compared to any other experiment. 5 2 ATMOSPHERIC NEUTRINO OSCILLATION 2 Atmospheric Neutrino Oscillation The high energy cosmic rays, on passing through the earth’s atmosphere collide with the nuclei in it. This collision produces the π and K mesons. These mesons are not stable, and decay according to the equation: − π ± −→ µ± + νµ (νµ ) (18) − Although the π ± −→ e± + νe (νe ) decay is kinematically favoured, the decay process is dominated by (18). This is because, in the rest frame of the π(K) meson, the e± is an extreme relativistic particle. So, the helicity of the daughter particles is the same as their chirality. Also, only left-handed leptons and righthanded anti-leptons take part in weak interaction. Thus, we get the following final state of the daughter particles along with their helicities in the π rest frame: ⇐ ⇐ + e+ R ←− π −→ νeL (19) From equation18 it is clear that the net spin projection of the e+ νe pair in the direction of motion is S=1.But, the π meson has no spin and angular momentum projections along the direction of motion; i.e, L = r ∗ p = 0. So, the final state total angular momentum J = L + S = 1 for the π meson. On the other hand, for the initial state (π at rest), L = 0 and S = 0, so that J = L + S = 0. Thus, this decay is disallowed by angular momentum conservation. However, the corresponding µ± νµ decay is allowed, because µ is non-relativistic in the rest frame of the π meson, and hence its chirality is not the same as its helicity. Further, µ± decays according to the equation: − − µ± −→ νµ e+ νe (νµ e− νe ) (20) Thus, from (18) and (19), it can be seen that for every decay of a π meson, − − νe (νe ) and νµ (νµ ) are produced in the ratio 1:2. At higher energy, however, this ratio may be < 1/2, as the µ may not decay in the atmosphere due to time dilation. The main atmospheric neutrino experiment is the Super Kamioka (SK) neutrino detection experiment, which is located in the Kamioka mines in Japan. It consists of a 50 kiloton water Cerenkov detector, surrounded by thousands of photomultiplier tubes which catch Cerenkov radiation. The electron (muon) neutrinos that pass through the water interact with the protons and neutrons in it to produce electrons (muons), via charged current interaction, according to the equations: − νe n −→ pe− (νe p −→ ne+ ) − νµ n −→ pµ− (νµ p −→ nµ+ ) 6 2 ATMOSPHERIC NEUTRINO OSCILLATION Since an energetic electron suffers greater deflection than a muon because of its smaller mass, the Cerenkov ring produced by an electron is more diffused than the ring produced by a muon. Thus, they can be distinguished to a very good accuracy. However, it is not possible to determine the lepton charge, because there is no magnetic deflection in this experiment. − − From the measured rate of electron and muon production, the νe (νe ) and νµ (νµ ) − fluxes are estimated. It was observed that while the νe (νe ) flux matched with − the expected value, there was a clear deficit with the νµ (νµ ) flux, indicating νµ −→ ντ oscillation. Figure 1 shows the zenith angle distribution of the observed electron and muonlike events along with the corresponding theoretical predictions, with and without oscillation [3]. It can be seen that the e-like events agree with the nooscillation prediction. But, the µ-like event-rates show a deficit for both subGeV and multi-GeV neutrino energies. From (12), it can be seen that sub-GeV neutrinos have a relatively small oscillation wavelength, which accounts for the deficit being seen for neutrinos at all zenith angles. But, the multi-GeV neutrinos have relatively large oscillation wavelength, and thus, the deficit is seen only for upward going( earth-traversing) neutrinos. For such neutrinos, the relationship between the distance travelled and zenith angle is given by: l ≃ −D cos θ (21) where D ∼ 13, 000km is the diameter of earth. 7 2 ATMOSPHERIC NEUTRINO OSCILLATION Figure 1: The zenith angle distributions for fully contained 1-ring e-like and µ-like events with visible energy < 1.33 GeV (sub-GeV) and > 1.33 GeV (multiGeV). For multi-GeV µ-like events, a combined distribution with partially contained (PC) events is shown. The dotted histograms show the non-oscillated Monte Carlo events, and the solid histograms show the best-fit expectations for νµ → ντ oscillations [3]. Thus, one can measure both the Energy and the distance travelled by the neutrinos and look for the oscillatory pattern of the predicted survival probability of (7) and(9) as a function of the ratio l/E, which is depicted in figure 2. [4] 8 2 ATMOSPHERIC NEUTRINO OSCILLATION Figure 2: The SK muon like event rates relative to the theoretical predictions without oscillation(i.e, the νµ survival probability) is shown as a function of the ratio l/E along with the best νµ → ντ oscillation(black solid line). Some alternative model fits in terms of neutrino decay are also shown for comparison [4]. From figure 2, it can be seen clearly that the survival minimum (oscillation maximum) occurs at El ≃ 500km/GeV . Since l E = λ 2E , from equation 16: 1.2 l = E ∆m2 (22) This corresponds to ∆m2 ≈ 2.4 × 10−3 eV 2 , and a large oscillation amplitude implies sin2 2θ ≃ 1, which gives θ ≃ π/4. The result of the SK experiment has now been confirmed by 2 long baseline accelerator neutrino experiments in Japan (K2K) [5] and USA (MINOS)[6]. The K2K experiment uses a neutrino beam from the KEK accelerator and the SK detector, which are separated by a distance of 250km. MINOS[6] uses the neutrino beam from the Fermilab accelerator and an iron detector, which are separated by a distance of 730km. Figure 3 shows the consistency of the atmospheric neutrino oscillation parameter from all three experiments. 9 2 ATMOSPHERIC NEUTRINO OSCILLATION Figure 3: Confirmation of the atmospheric neutrino oscillation parameters of MINOS [7] results published in 2008. The 68 % and 90 % CL allowed regions are shown together with the SK-I(7) and K2K 90 % CL allowed regions [8]. From figure 3 it can be seen that the best values of atmospheric neutrino mass and mixing parameters are: ∆m2atm = ∆m232 ≃ 2.4 × 10−3 eV 2 ; sin2 2θatm = sin2 2θ23 ≃ 1 where the indices refer to the mass Eigen states in the three neutrino mixing formalism.From the absence of clear evidence of atmospheric electron neutrino oscillation, we know that an electron neutrino mixing angle in the atmospheric mass scale must be small, but the atmospheric data is not meaningful enough to give a quantitative upper bound. The first quantitative upper bound came from the CHOOZ nuclear reactor (electron neutrino) experiment in France. Finally thanks to the solar matter effect, we know the solar mass scale both in sign and magnitude along with the corresponding mixing angle from the solar and the LBL reactor (KL) experiments. These are ∆m2sol = ∆m221 ≈ 7.6X10−5 eV 2 and sin2 2θ12 ≈ 0.3 [9]. 10 2 ATMOSPHERIC NEUTRINO OSCILLATION Figure 4: Schematic diagram of the neutrino mass and mixing parameters [9]. In figure 4, the status of neutrino mass and mixing up to the stage 2010 is illustrated, where the indices refer to the neutrino mass eigen-states. The positive and negative signs of ∆m2atm correspond to the so called normal and inverted hierarchy scenarios, where the ν3 / ν1 correspond to the heaviest/ lightest mass eigen states respectively. At this stage there were three neutrino oscillation parameter yet to be determined, i.e. the third mixing angle θ13 , the sign of the atmospheric neutrino mass scale ∆m232 and CP violating phase δ . Thanks to three SBL reactor anti neutrino experiments , culminating in the Daya Bay experiment from China we now have a definite estimate of the third mixing angle which is close to its above mentioned upper limit. This will be discussed in Section 4. This has promising implications for the determination of the other two unknown quantities from the foreseeable LBL accelerator neutrino experiments. This will be discussed in Section 5. 11 3 THREE NEUTRINO MIXING AND OSCILLATION FORMALISM 3 Three Neutrino Mixing and Oscillation Formalism The three neutrino flavour eigenstates are related to the three mass eigenstates through the formula: ∗ να = ΣUαi νi , α = e, µ, τ (23) where the mixing matrix U is a 3x3 unitary matrix described by the three mixing angles and the CP violating phase δ. This matrix, called the PMNS matrix is analogous to the CKM matrix for quarks [9]. The mixing matrix can be written either in the compact form or expanded as a product of three 2x2 rotation matrices as follows:   c12 c13 s12 c13 s13 e−iδ s23 c13  U = −s12 c23 − c12 s23 s13 eiδ c12 c23 − s12 s23 s13 eiδ iδ iδ s12 s23 − c12 c23 s13 e −c12 s23 − s12 c23 s13 e c23 c13      c13 0 s13 e−iδ c12 s12 0 1 0 0 1 0  −s12 c12 0 (24) = 0 c23 s23   0 0 −s23 c23 0 0 1 −s13 eiδ 0 c13 In this equation, cij and sij denote cos θij and sin θij respectively [9]. The three mixing angles are related to the flavour components of the three mass eigenstates as: |Ue2 | |Ue1 | 2 2 = tan2 θ12 , |Uµ3 | |Uτ 3 | 2 2 2 = tan2 θ23 , |Ue3 | = sin2 θ13 (25) The vacuum oscillation probability between two neutrino flavours is given by the equation: 2 P (να → νβ ) = X Uβj e −im2 jL 2Eν ∗ Uαj (26) j In (26), the last factor comes from the decomposition of να into the mass eigenstates, the phase factor in the middle from the propagation of each mass eigenstate over a distance L, and the first factor from their recomposition into the flavour eigenstate νβ at the end. From (26), P (να → νβ ) = Uβ1 e Since U is a  Ue1 Uµ1 Uτ 1 −im2 1L 2Eν ∗ Uα1 + Uβ2 e −im2 2L 2Eν ∗ Uα2 + Uβ3 e unitary matrix, U U † = I, so     1 0 0 Ue1 Uµ1 Uτ 1 Ue2 Ue3 Uµ2 Uµ3  Ue2 Uµ2 Uτ 2  = 0 1 0 Ue3 Uµ3 Uτ 3 0 0 1 Uτ 2 Uτ 3 12 −im2 3L 2Eν 2 ∗ Uα3 (27) 3 THREE NEUTRINO MIXING AND OSCILLATION FORMALISM From the above equation, we can write: ∗ ∗ ∗ Uα1 Uβ1 + Uα2 Uβ2 + Uα3 Uβ3 = δαβ So, 2 ∗ ∗ ∗ |Uα1 Uβ1 + Uα2 Uβ2 + Uα3 Uβ3 | = δαβ (28) Another property of complex numbers is as follows: 2 2 2 2 |Z1 + Z2 + Z3 | = |Z1 | + |Z2 | + |Z3 | + 2Re(Z1∗ Z2 + Z2∗ Z3 + Z1∗ Z3 ) (29) Using this, we can write (28) as: 2 2 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Uβ1 | +|Uα2 Uβ2 | +|Uα3 Uβ3 | +2Re(Uα1 Uβ1 Uα2 Uβ2 +Uα2 Uβ2 Uα3 Uβ3 +Uα1 Uβ1 Uα3 Uβ3 ) = δαβ |Uα1 (30) Thus, from (26), (28) and (30), we get: ∗ ∗ ∗ ∗ ∗ ∗ Uβ3 + Uα1 Uβ1 Uα3 Uβ3 )+ P (να → νβ ) = δαβ − 2Re(Uα1 Uβ1 Uα2 Uβ2 + Uα2 Uβ2 Uα3 ∗ ∗ Uα2 Uβ2 e 2Re(Uα1 Uβ1 2 −i(m2 3 −m2 )L 2 −i(m2 2 −m1 )L 2Eν = δαβ 2 −i(m2 3 −m1 )L ∗ ∗ ∗ ∗ 2Eν 2Eν + Uα2 Uβ2 Uα3 Uβ3 e + Uα1 Uβ1 Uα3 Uβ3 e ) )L −i(∆m2 X X ij ∗ ∗ ∗ ∗ Re(Uαj Uβj Uαi Uβi e 2Eν ), Re(Uαj Uβj Uαi Uβi ) + 2 −2 i>j i>j where ∆m2ij = m2i − m2j =⇒ P (να → νβ ) = δαβ + 2 X ∗ ∗ Re[(Uαj Uβj Uαi Uβi )(e −i(∆m2 ij )L 2Eν i>j − 1)] (31) Using the following property of complex numbers: Re(Z1 Z2 ) = Re(Z1 )Re(Z2 ) − Im(Z1 )Im(Z2 ), P (να → νβ ) = δαβ + 2 X −2 X Also,  Re e −i(∆m2 ij )L 2Eν  Im e i>j − 1 = cos −i(∆m2 ij )L 2Eν  i>j Re  ∗ ∗ ∗ (Uαi Uαj Uβi Uβj   Re e −i(∆m2 ij )L 2Eν  −1  −i(∆m2 )L   ∗ ∗  ij ∗ Im Uαi Uαj Uβi Uβj Im e 2Eν −1 ∆m2ij L 2Eν  − 1 = − sin ! − 1 = −2 sin ∆m2ij L 2Eν 13 ! 2 ∆m2ij L 4Eν = −2 sin2 ∆ij ; = − sin 2∆ij ! (32) 3 THREE NEUTRINO MIXING AND OSCILLATION FORMALISM Thus, using (32), (31) becomes: P (να → νβ ) = δαβ −4 X ∗ ∗ ∗ Re[(Uαi Uαj Uβi Uβj ] sin2 ∆ij +2 X ∗ ∗ ∗ Im[Uαi Uαj Uβi Uβj ] sin 2∆ij i>j i>j (33) where ∆ij = ∆m2ij L 4Eν (34) The last term of (33) contains the CP violating contribution and is proportional to to sin δ. We note that only the neutrino oscillation experiments measuring the appearance probability of a new flavour can measure the CP violating contribution. For the disappearance experiments, β = α, and hence the last term of the equation vanishes. Moreover, the CP violating contribution changes sign in going from P (να → νβ ) to P (να → νβ ) and P (να → νβ ) to P (νβ → να ), since P (να → νβ ) = P (νβ → να ), by CPT invariance. Using the identity: ∆m232 = m23 − m22 = ∆31 − ∆21 (35) we can express the vacuum oscillation probability (33) in terms of sinusoidal functions of the two independent mass scales. Further, we can use the observed hierarchy between the two mass scales. Further, we can use the observed hierarchy between the two mass scales: α= ∆m221 |∆21 | ∼ = = 0.03 |∆m231 | |∆31 | (36) to write this probability in terms of a single mass scale to a very good approximation. For this purpose, it is useful to rewrite (34) in terms of convenient units, i.e, ∆ij = 1.27∆m2ij L Eν (37) where ∆m2ij is in eV 2 , the distance, L is in Km(m) and the neutrino energy Eν is in GeV(MeV) . For atmospheric or LBL accelerator neutrino experiments, Eν ∼ GeV, L ∼ 103 km; −3 )∗103 km ∼1 ∴ ∆31 = 1.27∗(2.4∗10 GeV and ∆21 ∼ α Hence, the dominant contribution from the vacuum oscillation probability comes from the ∆31 scale. This is also true for the Short Base Line reactor neutrino experiments, where Eν ∼ M eV and L ∼ 103 m. 14 3 THREE NEUTRINO MIXING AND OSCILLATION FORMALISM Thus, to a very good approximation, we have the νµ survival probability [9]: P (νµ → νµ ) ∼ = 1 − (cos4 θ13 sin2 2θ23 + sin2 θ23 sin2 2θ13 )sin2 ∆31 ∼ = 1 − sin2 2θ23 sin2 ∆31 (38) where we have neglected terms of the order cos 2θ23 and sin4 θ13 in the final step, since sin2 2θ23 ≈ 1.0 [10] and sin2 2θ13 ≈ 0.1 Thus to a very good approximation, the expression reduces to the simple two neutrino mixing formula. This implies that the values of ∆m231 and sin2 2θ13 obtained from atmospheric and LBL accelerator neutrino experiments using this simple formula hold good to a very high degree of accuracy. However, we notice that θ13 doesn’t appear in this formula. Hence, these experiments are not very useful for the determination of θ13 . The corresponding expression for the νe survival probability is: P (νe → νe ) ∼ = 1 − sin2 2θ13 sin2 ∆31 (39) This is used to determine sin2 2θ13 from SBL reactor neutrino experiments. For the KamLAND LBL reactor neutrino experiment, Eν ≈ M eV , L ≈ 105 m =⇒ ∆31 ≈ 1/α, ∆21 ≈ 1 (40) so that the oscillation terms in ∆31 approach their average values over a complete cycle. Hence, the vacuum νe survival probability is again given in terms of a single scale to a good approximation [9], i,e: 1 P (νe → νe ) ∼ = 1 − sin2 2θ13 − cos4 θ13 sin2 2θ12 sin2 ∆21 2 4 ∼ cos θ13 (1 − sin2 2θ12 sin2 ∆21 ), = (41) by neglecting the sin4 θ13 terms in the final step. This formula is used in estimating θ13 from a comparison of solar and KamLAND reactor neutrino experiments. For the appearance probability, P (νµ → νe ), the leading scale (∆31 ) contribution is suppressed by a small coefficient ∼ sin2 2θ13 . Thus, one has to consider the subleading scale contributions as well. The full expression for this vacuum oscillation probability is given by [9]: P (νµ → νe ) = sin2 2θ13 sin2 θ23 sin2 ∆31 + α sin 2θ12 sin 2θ23 sin 2θ13 [cos(δ + ∆31 ) sin ∆31 ]∆31 + (42) α2 sin2 2θ12 cos2 θ23 ∆31 Here, the first term represents the leading atmospheric scale contributions. This is suppressed by sin2 2θ13 . The second term represents the CP violating and CP conserving parts of the interference terms, which are suppressed by α sin 2θ13 . The last term represents the sub-leading solar scale contribution, which is suppressed by α2 . 15 3 THREE NEUTRINO MIXING AND OSCILLATION FORMALISM We know now that sin 2θ13 ∼ 1/3, while α ∼ 1/30. Thus, the interference term and the CP violating contribution is suppressed by a factor of ∼ 10, while the last term is suppressed by a factor of ∼ 100, relative to the first term. We note that the corresponding expression forP (νe → νµ ) or P (νµ → νe ) is obtained (41) simply by changing the sign of the phase δ. Finally let us consider the earth matter effect on the above νe appearance probability. It comes from the charged current interaction of νe with electrons resulting in a potential energy term [11], ??   √ ρ −14 V = 2GF Ne ∼ Ye eV (43) 7.6X10 = g/cm3 where GF is the Fermi coupling and Ne the electron number density in the terrestrial matter. For electron neutrinos passing through the earth’s crust one can write this in terms of a nearly constant matter density and electron fraction per nucleon, ρ∼ = 3g/cm3 , Ye ∼ = 0.5. (44) In order to calculate the neutrino oscillation probability in matter, one has to solve the Schrodinger equation for the neutrino state vector in the flavour basis, i d |v(t) >= H|v(t) > dt (45) with the effective Hamiltonian
1 H≈ U diag 0, ∆m221 , ∆m231 U † + diag(V, 0, 0) 2Eν (46) For antineutrinos one has to make the replacements U → U ∗ , V → −V. (47) For the case of constant matter density one can diagonalize the effective Hamiltonian perturbatively giving ′ H = U diag(E1 , E2 , E3 )U ′† (48) where [11] A α2 sin2 2θ12 A+ + + A−1 4A   2 ∆m231 α2 sin 2θ12 αc212 − E2 ∼ = 2E 4A   2 A ∆m31 2 ∼ E3 = 1 − s13 2E A−1 ∆m231 E1 ∼ = 2E  αs212 s213 16  4 DETERMINATION OF θ13 BY SBL REACTOR (ANTI)NEUTRINO EXPERIMENTS Then the resulting oscillation probability is given in terms of these quantities, i.e. ′ ′ P (να → νβ ) = Σj Uβj e−iEj L Uαj∗ 2 (49) which is analogous to the vacuum oscillation formula (26). One can expand the oscillation probability in terms of s13 and α. Upto second order terms in these parameters, we get [11]: 4s213 s223 sin2 (A − 1)∆31 + 2 (A − 1) sin A∆31 sin(A − 1)∆31 + 2αs13 sin 2θ12 sin 2θ13 cos(∆31 + δ) A (A − 1) P (νµ → νe ) = α2 sin2 2θ12 c223 where A= (50) sin2 A∆31 A2 2Eν V ∼ Eν (GeV ) VL = =± 2∆31 ∆m231 10 For A → 0, this reduces to the vacuum oscillation probability (42). The matter effect is represented by the dimensionless quantity A. The sign of A changes with the sign of ∆m231 as well as in going from the neutrino to the corresponding antineutrino experiment. The former implies that the matter effect can be used to determine the sign of ∆m231 , while the latter implies that it can fake a CP violating effect and hence complicate the extraction of δ by comparing neutrino and antineutrino data. For off-axis experiments like T2K and NOvA, the typical beam energy is Eν ∼ 1 GeV, so that one can expand (50) in powers of A. Keeping only terms up to the first order in A, we get P (νµ → νe ) = 4s213 s223 [sin2 ∆31 + A(sin2 ∆31 − ∆31 sin 2∆31 )]+ 2αs13 sin 2θ12 sin 2θ13 cos(∆31 + δ)∆31 [sin ∆31 + A(sin ∆31 − ∆31 cos ∆31 )]+ α2 sin2 2θ12 c223 ∆231 (51) For optimal νµ → νe appearance experiments, ∆31 ∼ π/2, so that cos ∆31 and sin 2∆31 ∼ 0. Thus the relative size of the matter effect in the leading terms is sin 2A. 4 Determination of θ13 by SBL Reactor (anti)neutrino Experiments The unambiguous and by now fairly precise determination of θ13 had come since 2012 from three reactor (anti)neutrino experiments. These are described below in increasing order of precision- i,e, Double Chooz, RENO and Daya Bay. 17 4.1 4.1 4 DETERMINATION OF θ13 BY SBL REACTOR (ANTI)NEUTRINO Double Chooz experiment EXPERIMENTS Double Chooz experiment In the Double Chooz experiment, antineutrinos are produced from the 2X4.25 GW Chooz Nucelar Power Plant in France. The Double Chooz far detector is located at an average distance of 1050m from the two reactor cores. The innermost region of the detector comprises of a cylindrical target containing 10m3 of Gadolinium doped liquid scintillator to detect the reactor antineutrino via its inverse beta decay process: νe + p → e+ + n, (52) by recording the prompt signal produced by the positron in the scintillator, followed by that of a ∼ 8 MeV γ ray coming from the neutron capture in Gadolinium. The target is surrounded by a 55cm thick concentric cylinder of undoped liquid scintillator ( γ catcher) to detect γ rays escaping from the edge of the target cylinder. The whole system is surrounded by 390 PMTs to measure the scintillation energy. The synchronisation between the positron and neutron detection is a signature of the inverse beta decay events and helps to reduce the background by ∼ 10% of the original size. The experiment produced its first result in 2012 [12] after 101 live days run, which reported 4121 events against the no oscillation (θ13 = 0) prediction of 4344 ± 165 events. This was in the absence of a near detector, and the flux, which was estimated from the reactor power resulted in a fairly large systematic error. The ratio R = 0.944 ± 0.016(stat) ± 0.040(syst) (53) corresponding to the νe survival probability (39) provided a 1.7 σ evidence for non-zero θ13 . The prompt positron energy measured by the scintillator (including its annihilation energy with an electron in the detector) Epesompt = Eν + mp − mn + me ≈ Eν − 0.8M eV (54) was also found to show a spectral distortion as expected from the oscillation formulae (39 and 37). Combining the two results gave sin2 2θ13 = 0.086 ± 0.041(stat) ± 0.034(syst) (55) Subsequently, the experiment reported 228 days run [13], which doubled the statistics to ∼ 8000 events. A combined analysis of rate and spectral distortion of these events gave sin2 2θ13 = 0.109 ± 0.030(stat) ± 0.025(syst) (56) with a 2.79 σ evidence for non-zero θ13 . In 2014, the experiment published improved measurements of θ13 using the data 18 4.2 4 DETERMINATION OF θ13 BY SBL REACTOR (ANTI)NEUTRINO RENO Experiment EXPERIMENTS collected in 467.90 live days [14] from the far detector. It reported 17351 inverse beta decay events against the no-oscillation prediction of 18290+370 −330 events, when at least one reactor was running. The combined analysis of rate and spectral distortion of these events gave sin2 2θ13 = 0.090+0.032 −0.029 (57) with a 3 σ evidence for non-zero θ13 . The near detector was completed in September 2014 and started taking data since January 2015. It is placed 400 m from the liquid scintillator detector. At the 51st Moriond EW Conference in Italy in March 2016, the Double Chooz collaboration presented its first θ13 measurement exploiting the combination of two years of single- detector data and nine months of double-detector data [15]. The measured value is: sin2 2θ13 = 0.111 ± 0.018 4.2 (58) RENO Experiment This experiment detects anti-neutrinos coming from an array of six 2.8 GW reactors at the Hanbit (previously Yongwang) Nuclear Power Plant in South Korea, which are roughly equi-spaced on a line spanning ∼ 1.3 km. It uses two identical detectors placed on the perpendicular bisector of the reactor array, at distances of 294 m (near) and 1383m (far) from the array centre. Each detector consists of a cylindrical target containing 16 tons (18.6 m3 ) of Gddoped liquid scintillator to detect the prompt positron coming from the inverse beta decay process (43) along with the delayed γ rays coming from the neutron capture in Gadolinium. This is surrounded by a 60cm thick concentric cylinder of undoped liquid cylinder ( γ catcher), which is, in turn, surrounded by 354 PMTs to measure the scintillation energy. The synchronisation between the positron and neutron detection reduces the background to ∼ 3%(6%) of the signal in the near (far) detector. The RENO collaboration has reported observation of 17102 (154088) νe events in the far (near) detector based on 229 days’ data [16]. In the absence of neutrino oscillation (θ13 = 0), one can predict the number of signal events in the far detector relative to those in the near detector by rescaling the latter by 2 a weighted average of the relative flux factors (Lni /Lfi ) over the six reactors, times the relative detection efficiency factor (εf /εn ). They have found a clear deficit of ∼ 8% in the number of observed events in the far detector relative to this prediction, i.e, R = 0.920 ± 0.009(stat) ± 0.014(syst) (59) Fitting this deficit factor to the spectrum averaged oscillation formulae (39) and (37), we get: sin2 2θ13 = 0.113 ± 0.013(stat) ± 0.019(syst) 19 (60) 4.3 4 DETERMINATION OF θ13 BY SBL REACTOR (ANTI)NEUTRINO Daya Bay experiment EXPERIMENTS which contributes a 4.9 σ evidence for non-zero θ13 . Moreover, the measured prompt energy distribution shows evidence of spectral distortion as expected from the oscillation formulae. RENO released updated results in December 2013 using ∼ 800 days’ data, where they obtained a total of 457176 (53632) νe events in the near (far) detector [17]. The analysis gives: sin2 2θ13 = 0.101 ± 0.008(stat) ± 0.0190(syst) (61) which contributes a 7.88 σ evidence for non-zero θ13 . 4.3 Daya Bay experiment Figure 5: Layout of Daya Bay experiment. The two new added detectors have been shown with diamond shape. The Daya Bay experiment in southern China is the most powerful of the three SBL reactor (anti)neutrino experiments, detecting the νe coming from six 2.9 GW reactors in eight identical antineutrino detectors - four near to and four far from the reactor complex (prior to Aug 2012, it had 6 detectors). It is also the most complex one in terms of the reactor and detector layouts, as shown in figure 5. The flux-weighted baseline lengths of the two near detector halls are 470 m and 576 m, while that of the far experimental hall is 1648 m. Each detector consists of a cylindrical target containing 20 tons of Gd-doped liquid scintillator, surrounded by a concentric cylinder containing 20 tons of undoped liquid scintillator (γ catcher). The latter is surrounded by 192 PMTs to measure the scintillation energy. Daya-Bay is a low background experiment. The good overburden of each underground hall, together with the synchronisation between the detections of the prompt positron coming from the Inverse beta 20 5 IMPLICATIONS FOR LBL ACCELERATOR NEUTRINO EXPERIMENTS decay process and the delayed γ -ray coming from the neutron capture in Gd helps in reducing the background to ∼ 1.9%(3.1%) of the signal in the near (far) detectors. The first results from this experiment based on only 55 days data, with only 6 detectors, reported observation of 10416 (80376) νe events in the far (near) detectors [18]. In the absence of neutrino oscillation (θ13 = 0) one can again predict the number of νe signal events in the far detectors (EH3) relative to those in the near ones (EH1 and EH2). Here the baseline length of the near detectors, LF , corresponds to the flux-weighted average of those in EH1 and EH2 with respect to the ith reactor. There was a clear deficit of 6% in the number of observed signal events in the far detectors relative to this prediction, i.e. R = 0.94 ± 0.011(stat) ± 0.004(syst). (62) Fitting this deficit factor with the spectrum averaged oscillation formulae (38) and (36) gives a value of sin2 2θ13 = 0.092 ± 0.016(stat) ± 0.005(syst), (63) which constitutes a 5.2σ signal for a nonzero θ13 . The observed distribution of the prompt energy also shows the expected spectral distortion from these oscillation formulae. The low systematic error of this experiment has been attributed mainly to ensuring the identity of the detectors from the beginning of their fabrication [19]. The Daya Bay collaboration presented the result of their 140 days’ data [20] showing a deficit factor of R = 0.944 ± 0.007(stat) ± 0.003(syst). (64) It corresponded to a 7.7 σsignal for sin2 2θ13 = 0.089 ± 0.010(stat) ± 0.005(syst). (65) Recently, the collaboration has updated their results [21]. The new results are based on the complete data set of the 6-detectors period with the addition of the 8- Detectors period from Oct 2012 to Nov 2013, a total of 621 days. The relative measurement of the νe rate and spectrum between the near and far detectors corresponds to sin2 2θ13 = 0.084 ± 0.005 (66) This corresponds to an impressive 5.95% precision on sin2 2θ13 and hence excludes the no-oscillation hypothesis at a 16.8σlevel. 5 Implications for LBL Accelerator Neutrino Experiments The (νµ → νe ) appearance data from the LBL accelerator neutrino experiments of MINOS and T2K has provided evidence for a nonzero θ13 This was ahead 21 5.1 5 IMPLICATIONS FOR LBL ACCELERATOR NEUTRINO On-axis and off-axis experiments EXPERIMENTS of the reactor neutrino data. The resulting value of θ13 was dependent on the remaining two unknown parameters - the sign of ∆m2 31 and the value of δ (CP violating phase). The chief merit of these accelerator neutrino measurements lie in their sensitivity to these two unknown parameters, as they offer a possibility of determining them from the νµ → νe appearance data of the present and proposed accelerator neutrino experiments by using the precise value of θ13 from the forthcoming reactor neutrino data as input. Before we discuss these experiments, we present a brief discussion of the accelerator neutrino beams. 5.1 On-axis and off-axis experiments The secondary particles i.e. π mesons were produced by the collisions of 12 GeV proton beams hitting a solid target like aluminium or graphite.The positively charged particles produced, primarily pions are magnetically focused in the forward direction along the proton beam axis by a system of two pulsed horn magnets,which produce a toroidal magnetic field. Then the decay of pion into a muon and a muon neutrino, given by the equation π + → µ+ + ν µ (67) This decay produces the desired neutrino beam. Let us take that the neutrino emerges from a small angle θ relative to the beam axis. In the x rest frame, considering energy and momentum conservation we get Eν CM + Eµ CM = mπ And pν CM + pµ CM = 0 ⇒ pν CM = −pµ CM = p p p This implies, p2 + mµ 2 + p2 + mν 2 = mπ As mν 2 ≈ 0 we have, (p2 + mµ 2 ) = (mπ − p)2 p + mµ 2 = p2 + mπ 2 − 2pmπ ⇒ mµ 2 = mπ 2 − 2pmπ 2 ⇒p= (mπ 2 −mµ 2 ) 2mπ √ For the neutrino four momentum vector: (p, px, p 1 − x2 , 0) where x = cos θ Eπ Lorentz boost with respect to lab frame is given by γ = m and γ = √ π 1 2 β = (1 − γ12 ) ≈ (1 − 2γ1 2 ) Thus Lorentz transformation gives,   Eν = γp(1 + βx) = γp 1 + x − 2γx2   Pν L = γp(x + β) = γp 1 + x − 2γ1 2   2 But Pν L = pν cos θ = Eν 1 − θ2 22 1 1−β 2 or 5.1 5 IMPLICATIONS FOR LBL ACCELERATOR NEUTRINO On-axis and off-axis experiments EXPERIMENTS Therefore we get,  γp 1 + x − 1 2γ 2   = γp 1 + x − Neglecting higher order terms we get, x= x 2γ 2  1− θ2 2  (1−γ 2 θ2 ) (1+γ 2 θ 2 ) And putting the value of x in Eν we get, Eν = 2γp = (1 + γ 2 θ2 )  1− mµ 2 mπ 2  Eπ (1 + γ 2 θ2 ) (68) The corresponding neutrino flux per unit area of a detector placed at a distance r from the π + decay point is given by [22] Φ=  2γ 1 + γ 2 θ2 2 1 4πr2 (69) We see from (59) and (60) that one gets the largest neutrino energy and flux for on-axis (θ = 0) neutrino experiments. All the first generation accelerator neutrino experiments, including K2K and MINOS, were on-axis experiments. However, these experiments are not well suited for νµ → νe appearance because of two serious backgrounds. Firstly, the on-axis neutrino beam has a large Eν ≈ E2π tail from that of Eπ . It results in a serious neutral current background from νµ + p → νµ + p + π0 , π0 → γγ (70) followed by the γ → e+ e− pair creation which causes the unwanted background. Secondly, the on-axis νµ beam has a νe contamination at the 1 − 2% level from the decay of the accompanying µ µ+ → e+ ν e ν µ (71) These two problems are overcome in off-axis experiments. Due to this reason, the K2K experiment has been succeeded by the off axis T2K experiments and the MINOS by NOvA. It follows from equation (59) and (60) that at θ = 0, the neutrino energy is proportional to the pion energy, which results in a broad range of neutrino beam energies when the pion energy spectrum also has a broad energy range. But after we differentiate (59) with respect to Eπ , we obtain,   mµ 2 (1 − γ 2 θ2 ) dEν (72) = 1− dEπ mπ 2 (1 + γ 2 θ2 )2 23 5.1 5 IMPLICATIONS FOR LBL ACCELERATOR NEUTRINO On-axis and off-axis experiments EXPERIMENTS which means that the neutrino energy becomes practically independent of the π pion energy at θ = γ1 = m Eπ , resulting in a quasi-monochromatic neutrino beam with   mµ 2 mπ ∼ 30M eV Eν ∼ (73) = 1− = mπ 2 2θ θ Figure 6: The energy spectrum of the T2K neutrino beam for the on-axis (θ = 0) configuration, along with several off-axis configurations [22]. We can see that the on-axis spectrum shows a broad peak at Eν ≈ 2GeV , corresponding to a broad peak at Eπ ≈ 4GeV . This would correspond to an off-axis angle and energy given by 0.140GeV mπ = = 0.035 (74) θ= Eπ 4GeV i.e θ = 0.035 = 2◦ , Ev ≈ 0.85GeV ) as shown in the figure. We note, from the figure, that the off-axis beam is quasi-monochromatic with very little spread in Eν , which effectively suppresses the neutral current background (61). The νe contamination from the secondary decay process (62) is also suppressed as it does not carry enough transverse momentum to reach the off-axis angle. This is compensated by a higher intensity of the proton beam to compensate for the decreased flux of the off-axis neutrino beam. Figure also shows that one can tune the neutrino energy to still lower values by operating at a little larger off-axis angle of θ = 2.5 or 3◦ for a closer to the maximal oscillation phase, ∆31 = 90◦ . The T2K experiment operates at θ = 2.5◦ ≥ Eν ≈ 0.68GeV (75) We shall now discuss the νµ → νe appearance measurements by the MINOS, T2K and the forthcoming NOvA experiments. 24 5 IMPLICATIONS FOR LBL ACCELERATOR NEUTRINO MINOS Experiment EXPERIMENTS 5.2 5.2 MINOS Experiment This is an on-axis experiment, designed for the measurement of νµ disappearance, which it has completed successfully. It uses the νµ beam from Fermi lab with a broad peak at Eν = 3GeV . The 5.4 kt far detector is placed at a baseline length L = 725 km, while a 1 kt near detector is placed 1 km from the target to measure the νµ flux and energy spectrum. Thus, 725km 240km L = ≈ Eν 3GeV GeV (76) And we know previously that, ∆m31 2 = ±2.4 x 10−3 eV 2   L ∆31 = 1.3∆m31 2 = 42◦ Eν (77) This is only half way to the oscillation maximum as the experiment was not designed for measuring the ν + e appearance signal . Both the detectors are tracking sampling calorimeters with alternate layers of passive (steel) and active (plastic scintillator) materials embedded in a magnetic field. The scintillation light is collected with the help of wavelength shifting fibres and is measured by PMTs [23]. It detects the ν + e appearance signal via its charged current interaction in the iron layers ν e N → e− X (78) This produces the EM shower [26]. By their distinguished shape, the EM and hadron showers can be measured using scintillator strips (The points marked by hadron showers will be slightly different from that of EM shower). But note that there is a large uncertainty in the electron identification by this detector resulting in a large neutral current background apart from that of the νe contamination in the νµ beam as we discussed earlier. The collaboration has previously published results based on 8.2X1020 Protons on Target (POT) data in 2011 reporting 62 events [24]. It provided a 1.5 σ signal for a non-zero θ13 with large estimated background of 50 ± 8. Assuming δ = 0, the central value of this angle that results from this signal is sin 2θ13 2 = 0.04 for positive ∆m31 2 and sin 2θ13 2 = 0.08 for negative ∆m31 2 (79) We can see that this result is dependent on the sign of ∆m31 2 or ∆31 . And more recently, the collaboration has updated their result with 1.07X1020 POT data [25]. It reported 88 events against a background of 69 ± 9. This provides a 2 σ signal for a non-zero θ13 . Assuming δ = 0, the central value is sin 2θ13 2 = 0.06 for positive ∆m31 2 and sin 2θ13 2 = 0.10 for negative ∆m31 2 (80) 25 5.3 5.3 5 IMPLICATIONS FOR LBL ACCELERATOR NEUTRINO T2K Experiment EXPERIMENTS T2K Experiment T2K is an off-axis neutrino experiment and it is optimised for Proton Accelerator Complex (J-PARC) produces high intensity proton beam with power of 0.7 MW. This experiment operates at an off-axis angle of θ = 2.50◦ , which corresponds to a quasi-monochromatic beam with peak of 30M eV Eν ∼ = 0.044(rad) ≈ 0.687GeV = θ (81) The contamination from the secondary decay process is reduced to the level of 0.4% and the neutral current background is also suprressed. For the far and this detector in T2K, the baseline length is L=295 Km, so ELν = 450Km GeV 2 31 L ≈ 80◦ (1.40rad). So we can see that value corresponds to |∆m31 | = 1.3∆m Eν of |∆31 | is close to the maximal oscillation phase. The far detector is the SK detector, which consists of 50 Kt water Cherenkov detector surrounded by many thousands of PMTs to measure the Cherenkov radiation energy. Two versatile multi-component near detector are placed at a distance of L= 286 m from the target. One is placed on-axis (θ=0◦ ) and the other is placed off-axis (θ=2.50◦ ). These near detectors measure the initial neutrino beam spectrum and continuously monitor its properties. They also provide accurate measurements of the differential cross- sections for all charged and neutral current interaction which is not possible with SK detector. These measurements are used to estimate the backgrounds to the appearance signal. The signal is detected in the SK detector via the charged current interaction, which is dominated at this beam energy by the quasi-elastic process νe + p → e− + n; νµ + p → µ− + n (82) These and produce single Cherenkov ring events. As we have discussed previously, the electron ring can be distinguished from the muon ring by its diffused nature at 1% level for such events. Selecting single Cherenkov ring events with electron like ring reduces the estimated background from the near detector data to the level of appearance signal for . With some further cuts on the event topology and the reconstructed neutrino energy reduces the latter to about 1/3rd of the signal size. The first result of T2K is based on its 1.51020 POT data which reported 6 events against an estimated background of 1.5 ± 0.3(syst) [26], which provides a 2.5σ signal for nonzero θ13 . Assuming δ = 0 the resulting central value of the angle is sin2 2θ13 = 0.11 for positive ∆m231 and sin2 2θ13 = 0.14 for negative ∆m231 (83) Updated result from T2K shows 3X1020 POT data reporting 11 events against background of 3.2 ± 0.4(syst) [27], which comes mainly from νe contamination 26 5.3 5 IMPLICATIONS FOR LBL ACCELERATOR NEUTRINO T2K Experiment EXPERIMENTS and neutral current events. This constitutes a 3.2 σ signal for nonzero θ13 . Here again assuming δ = 0, we have the value 2 sin2 2θ13 = 0.11+0.053 −0.040 for positive ∆m31 and 2 sin2 2θ13 0.14+0.063 −0.049 for negative ∆m31 (84) From the result we can observe that there is a relative insensitivity to the mass hierarchy. This is due to the small baseline length or equivalently the small optimised beam energy of Eν ≈ 0.68GeV , corresponding to A ≈ ±6.8%. This is small enough for using the first order formula (51). One can clearly see from this formula a matter effect of ≈ ±10% for the above values of A and |∆31 |. It accounts for the relative size of the two central values in equation 84. Similarly for small A one can understand the ±20% variation of the central value over the full range of δ from the relative size of the first and second term in (42). The experiment published results based on data taken from January 2010 to May 2013 [28]. The total neutrino beam exposure at SK corresponded to 6.57X102 0 POT. A total of 28 νe events were detected with an energy distribution consistent with an appearance signal, against an expected background of 4.92 ± 0.55 events. This constitutes a 7.3σ signal for non-zero θ13 . The variation of sin2 2θ13 over the full δ cycle has been shown in figure 7. For δ = 0, the best fit value for confidence level 68% was: sin2 2θ13 = 0.140+0.038 −0.032 f orN ormalHierarchy sin2 2θ13 = 0.170+0.045 −0.037 f orInvertedHierarchy 27 (85) 5.3 5 IMPLICATIONS FOR LBL ACCELERATOR NEUTRINO T2K Experiment EXPERIMENTS Figure 7: The 68% and 90% CL allowed regions for sin2 2θ13 , as a function of δ, assuming normal hierarchy (top) and inverted hierarchy (bottom). The solid line represents the best fit sin2 2θ13 value for given δ values. The shaded region shows the average sin2 2θ13 value from the PDG2012 [28] The point estimates for the oscillation parameters and the predicted number of events when the reactor measurement are included in the likelihood function suggest that sin2 θ13 ’s estimate is smaller than the result obtained with T2K data only. The likelihood is maximum for Normal Mass Hierarchy and for δcp = −π/2, where the appearance probability is largest. (χ2 is the likelihood function, described in [29]) The 68% and 90% Confidence Level Regions for the two mass hierarchies constructed using ∆χ2 with respect to the best-fit point, the one for the normal hierarchy, are presented in figures 8 and 9. Figure 10 gives the following excluded regions for δcp at the 90% Confidence Level : δcp = [0.15, 0.83]π for Normal Hierarchy and δcp = [−0.08, 1.09]π for Inverted Hierarchy. 28 5.3 5 IMPLICATIONS FOR LBL ACCELERATOR NEUTRINO T2K Experiment EXPERIMENTS Figure 8: 68% (dashed) and 90% (solid) CL regions from the analysis that includes results from reactor experiments with different mass hierarchy assumptions using ∆χ2 with respect to the best-fit point, the one from the fit with normal hierarchy. The parameter |∆m2 | represents ∆m232 or ∆m213 for normal and inverted mass hierarchy assumptions respectively [29]. 29 5.3 5 IMPLICATIONS FOR LBL ACCELERATOR NEUTRINO T2K Experiment EXPERIMENTS Figure 9: Comparison of 68% (dashed) and 90% (solid) CL regions combined with the results from reactor experiments with different mass hierarchy assumptions using ∆χ2 with respect to the best-fit point, the one from the fit with normal hierarchy. The parameter |∆m2 | represents ∆m232 or ∆m213 for normal and inverted mass hierarchy assumptions respectively [29]. Figure 10: Profiled ∆χ2 as a function of δcp with the results of the critical ∆χ2 values for the normal and inverted hierarchies for the joint fit with reactor constraint, with the excluded regions found overlaid [29]. The T2K experiment plans to achieve a 26 fold increase in data to 7.8X1021 POT over the next few years[29]. By that time the sin2 2θ13 measurement from 30 5.4 5 IMPLICATIONS FOR LBL ACCELERATOR NEUTRINO NoVA Experiment EXPERIMENTS the reactor neutrino experiments would have reached an accuracy of 5%. The projected νµ → νe signal from this T2K data is shown over the full cycle of δ in figure along with the sin 2θ13 2 measured in reactor neutrino experiments. As one sees from this figure, a comparison of the two results would be able to find a nonzero δ signalling CP violation at the 90% CL level over typically half the δ cycle. There are also plans to achieve a large increase in data in the future by increasing the intensity of the beam and/or the size of the detector to a ∼ 1 Mt Hyper-Kamiokande water Cherenkov detector [30]. This will make it possible to select only a few regions of the δ parameter space depending on the mass hierarchy (sign of ∆m31 2 ). The mass hierarchy itself can be determined at 3σ level by the atmospheric neutrino data at the Hyper-Kamiokande detector, as discussed in the next section. Besides, there are proposals to extend the experiment by adding another far detector at Okinoshima, at a baseline length of 658 km and off-axis angle of 0.78 degrees. And we can see from (73), this would triple the beam energy to Eν ≈ 2GeV and the resulting sensitivity to the mass hierarchy by a similar factor. Thus one can combine the two lots of far detector data to simultaneously determine the mass hierarchy and the value of δ. 5.4 NoVA Experiment This is the off-axis sequel to the MINOS experiment, which has started operation in 2013 [31]. This experiment is optimized for appearance measurements using the high intensity proton beam from Fermilab [32] which has a beam power of 0.7 MW like J-PARC. It operates at an off-axis angle of θ ≈ 0.8, corresponding to a quasi-monochromatic beam with peak Eν ≈ 30Mθ eV = 0.015(rad) ≈ 2GeV . The far detector is located at a baseline length of L = 810 Km which correKm and|∆31 | ≈ 70◦ (1.33rad) . This value is also close to sponds to ELν ≈ 450 GeV the maximum oscillation phase. The far and near detectors are fully active segmented scintillation detectors of similar design and weigh about 14 kt and 0.3 kt respectively [33]. These detectors mainly consist of long and narrow plastic cells filled with liquid scintillator, which are arranged vertically and horizontally in layers. Each cell is connected by wavelength shifting fiber to a photodetector for measuring the scintillation energy. The νe and νµ events are detected through the electron and muon produced via charged current interactions. The diffused profile of the electron track is well distinguished from the straight muon track in the scintillation detectors. The beam energy of Eν ≈ 2GeV corresponds to A ≈ ±0.2 from, which is three times larger than that of the T2K experiment. Therefore one expects a matter effect of ≈ ±30% for the νµ → νe appearance probability of for positive (negative) ∆m31 2 . However, a modulation of similar size can also come from the variation of the CP violating phase δ, which means that the two effects cannot be disentangled from the νµ → νe data alone. Therefore the NOvA experiment plans to complete 3+3 years of νµ → νe + νµ → νe appearance measurements.The νµ → νe oscillation probability is obtained from that of νµ → νe by changing A → −A and δ → −δ 31 5.4 5 IMPLICATIONS FOR LBL ACCELERATOR NEUTRINO NoVA Experiment EXPERIMENTS Figure 11: The blue and red ellipses show possible values of the oscillation probabilities for a fixed value of sin2 2θ13 = 0.095, the blue for the normal hierarchy and the red for the inverted hierarchy. On each coloured ellipse, the choice of the δ phase varies as one moves around the ellipse as indicated by the symbols. The mass hierarchy may be determined depending on where the NOvA measurements (data, black stars) lie on the coloured ellipses (prediction) [33]. Thus one can in principle determine δ and the sign of A by measuring both these oscillation probabilities. In practice, however this is not easy, as also seen from figure. It shows the two predicted contours of νe and νe appearance probabilities corresponding to the full cycle of δ for the two signs of ∆m31 2 . The two contours appear to have only a small overlap. However, the typical 2σ error bars of these probabilities are expected to be ∼ 0.015 after 3 + 3 years’ run. This corresponds to an effective overlap region of the two contours covering a little over half of each[35]. This means that one can resolve the mass hierarchy only over a little less than half the δ cycle, centred around δ ≈ π2 (3π/2) if the actual sign of ∆m31 2 is negative (positive). And, it will not be possible to determine a nonzero value of δ at the 2σ level[33] . It shows that combining the projected Nova and T2K data does not enhance the δ range of the 2σ resolution of mass hierarchy; but it provides a 1σ resolution over the entire cycle of δ [34]. Moreover the combined data can observe a nonzero value of δ (signalling CP violation) at the 1.5σ(90% CL) level over most of the δ cycle, although still not be able to reach the 2σ level]. NOvA announced the first results of νe appearance analysis in August 2015 [35] with 2.74X1020 POT equivalent exposure (This is one-thirteenth of the overall 32 5.5 5 IMPLICATIONS FOR LBL ACCELERATOR NEUTRINO LBNE Proposal EXPERIMENTS planned exposure). In the first of νe appearance measurement, a of νe event is identified by charged current (CC) interactions where the electron-neutrino is converted into an electron. The of νe analysis at NOvA makes use of two electron event identification algorithms (EID). Result from the primary EID shows that the range of 0 < δ < 0.65π in the inverted hierarchy is disfavoured at the 90% C.L., while the secondary EID result shows that inverted hierarchy is disfavoured at > 2.2σ, while normal hierarchy 0< δ < π is mildly disfavoured (> 1σ) [35]. 5.5 LBNE Proposal Finally, there is a proposal for a new on-axis long baseline neutrino experiment (LBNE) using a new beam line from Fermilab with an initial beam power of 0.7 MW, which can be upgraded up to 2.2 MW [35]. It will have a 10 kt liquid Argonne time projection chamber as the far detector located at the Homestake mines, at a baseline length of 1300 km. It will be able to resolve the mass hierarchy at≥ 2σ level over the entire δ cycle on its own, and at ≥4 σ level in combination with NOvA and T2K data [33]. Moreover it will be able to determine a nonzero δ, signalling CP violation, over most of its parameter space (0.2π < |δ| < 0.8π) at a 2 σ level on its own, and at ≥ 3 σ level in combination with the NOvA and T2K data [33]. As an alternative to the LBNE experiment there is a proposal to upgrade NOvA by installing a 30 kt liquid Argonne detector at its far site. The combined data from this experiment along with those of the original NOvA and T2K experiments will be able to resolve the mass hierarchy at ≥ 2σ level over the full δ cycle and also observe a nonzero δ signal at ≥ 2σ level over the range 0.2π < |δ| < 0.8π [33]. In summary, we hope to determine the mass hierarchy (sign of ∆m31 2 ) and the CP violating phase with the data from T2K and NOvA experiments along with their proposed upgrades/extensions over the next decade or two. Thanks to the fairly sizeable value of the third mixing angle (sin 2θ13 2 ≈ 0.1), it will be possible to achieve this with the conventional superbeam experiments in the foreseeable future instead of waiting for the beta beam or neutrino factory experiments. However, precision measurements of δ and other neutrino oscillation parameters and resolution of the remaining degeneracies will require these latter experiments of the more distant future. 33 6 ACKNOWLEDGMENTS 6 Acknowledgments We would like to express our deepest acknowledgment to all those who helped us in completing this report successfully. We owe our profound gratitude to our mentor, Prof. D. P. Roy, eminent physicist, TIFR, HBCSE for his persistent guidance and support all along the duration of our NIUS project work. His intense knowledge on the topic was an inspiration to us towards working in the project sincerely. Without his teaching and guidance, the knowledge we built up would not have been possible. We also extend our thanks to him for arranging such a wonderful experience at BARC, Mumbai. We also thank Homi Bhabha Centre for Science Education (HBCSE) for giving us the huge opportunity and support through the NIUS program. Finally, We wish to express our sincere thanks to our parents, friends and fellow NIUS camp mates for their constant support and encouragement throughout the completion of this project as well as our stay at TIFR. Thank You. 34 7 REFERENCES 7 References (1) D. P. Roy, Physics News 39 (3), 51 (2009); arXiv:0809.1767. (2) B. Pontecorvo, Sov. Phys. JETP 26 (1968) 984 . (3) Nakamura K et al (Particle Data Group) 2010 J. Phys. G: Nucl. Part. Phys. 37 075021. (4) SK Collaboration: Y. Ashie et al., Phys. Rev. Lett. 93 (2004) 101801 [hep- ex/0404034]. (5) K2K Collaboration: M. H. Ahn et al., Phys. Rev. D74 (2006) 072003 [hepex/0606032]. (6) MINOS Collaboration: arXiv:0708.1495[hep-ex]. (7) Y. Ashie et al., Phys. Rev. D71, 112005 (2005). (8) P. 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