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Long baseline phenomenology with an intermediate γ beta-beam Christopher Orme March 7th 2008 Based on arXiv:0802.0255 Davide Meloni, Olga Mena, Christopher Orme, Sergio Palomares-Ruiz and Silvia Pascoli. Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 1 / 23 1 Introduction 2 3-ν phenomenology 3 Determining (θ13 , δ) and the resolution of degeneracies 4 Case study: A beta-beam for the CERN-Boulby baseline (1050 km) Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 2 / 23 2-ν oscillations "solar" parameters "atmospheric" parameters -2 5 10 90% CL (2 dof) 20 global 2007 GLOBAL 4 ∆m31 [eV ] ∆m31 [10 eV ] ∆m21 [ 10 eV ] 2 MINOS 2 2 15 solar 3 SK+K2K+MINOS global -5 −3 2 5 10 2 2 2 5 CH KamLAND OO 5 1 -3 SOL+KAML Z atmospheric 10 +CHOOZ 0 0 -2 -1 0.2 0.4 0.6 0.8 0 0.25 0.5 0.75 1 10 10 2 2 sin θ12 2 sin θ23 sin θ13 Schwetz, arXiv:0710.5027 Present oscillation data and solar neutrino data points to 2 2-ν mixing schemes There are two mass squared splitings of different orders. Only a bound on θ13 presently exists. Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 3 / 23 3-ν mixing The neutrino mixing matrix (PMNS) can be parameterised by three mixing angles and a complex phase 1 0 0 c 0 s13 e−i δ c s12 0 13 12 U = 0 −s23 0 1 0 −s12 0 (1) c23 c12 iδ 0 s23 c23 s13 e 0 c13 0 0 1 If θ13 = 0 then the solar and atmospheric regimes decouple into 2 2-ν oscillation schemes. The CP phase is not physical. If θ13 = 0 then the two regimes are weakly coupled with θ13 and δ manifesting themselves through sub-dominant oscillation processes. Experimentally, θ13 is known to be small or zero. Need high luminosity experiments to observe it. Degeneracies (later) and systematics will present a major stumbling block. Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 4 / 23 Present knowledge and unknowns Parameter Best ﬁt 3σ ∆m21 (eV2 ) 2 7.9 × 10−5 7.1-8.9 ×10−5 sin θ12 2 0.30 0.24-0.40 −3 |∆m31 | (eV2 ) 2 2.5 × 10 1.9-3.2 ×10−3 sin θ23 2 0.5 0.34-0.68 sin θ13 2 unknown < 0.041 δ unknown no bound Task of current and next-generation experiments is improve accuracy of the knowns and pin down the unknowns. The ultimate goal in neutrino oscillation physics is to measure or, at least get a bound on, θ13 , δ and sign(∆m31 ). 2 Experiments running or about to will improve accuracy and make the ﬁrst attempt to determine the unknowns. Superbeams, Beta-beams and Neutrino Factories will be the experiments of choice for the long term future. Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 5 / 23 We are in the era of precision not discovery Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 6 / 23 3-ν oscillations in matter To ﬁnd neutrino oscillation probabilities one needs to solve the Schrondinger equation for neutrinos in the ﬂavour basis d i |ν(t ) = H |ν(t ) (2) dt where 0 0 0 A 0 0 1 † H U 0 2 ∆m21 0 U + 0 0 0 (3) 2E 2 0 0 ∆m31 0 0 0 Here, U is the usual PMNS mixing matrix in vacuum and A is given by √ 1 Z L A= 2GF ¯e (L) n with ¯e = n ne (L )dL L 0 Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 7 / 23 Probabilities For θ13 = 0.05 and L = 1050 km: 0.07 νe → νµ, δ=0 νe → νµ, δ=90 0.06 νe → νµ, δ=-90 0.05 Probability 0.04 0.03 0.02 0.01 0 0.5 1 1.5 2 2.5 3 Energy GeV Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 8 / 23 Analytical approach - solar oscillations as a perturbation It is desirable to have an analytical form for the probabilities, however, the complete diagonisation gives a result lacking in physical intuition. Instead, treat the solar effects as a perturbation of the 2-ν atmospheric oscillation by noting the following 2 ∆m21 1 2 ∆m31 30 2 Then to ﬁrst order in ∆m21 we may write ∆13 ±A−B 2 0 0 0 0 0 ¯† ¯† †¯ ¯ H =U 0 0 0 U + U U 0 ∆12 0 U U (4) ∆13 ±A+B 0 0 2 0 0 0 Eigenvalues and eigenvectors are then found to diagonalise the above. With the mixing matrix to ﬁrst order in ∆12 known, one can then get the probabilities. Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 9 / 23 The oscillation probability formula 2 ∆mij Introducing the abbreviation ∆ij ≡ 2E , the oscillation probability can be expressed as Peµ = Patm + Pint + Psol where 2 ∆13 (A ∆13 )L Patm = s23 sin2 2θ13 2 sin2 A ∆13 2 ∆12 ∆13 AL (A ∆13 )L ∆13 L Pint = J sin sin cos δ A A ∆13 2 2 2 2 ∆12 AL Psol = c23 sin2 2θ12 2 sin2 A 2 The result is accurate for the baseline and energies considered here. Fit is still good for longer baselines and higher energies. Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 10 / 23 The problem of degeneracies Determination of the unknown parameters (θ13 , δ, sign(∆m31 ), θ23 ) suffers from the problem of 2 degeneracies. In general, we can have (θ13 , δ ) such that Peµ (θ13 , δ, ∆m31 , θ23 ) = Peµ (θ13 , δ , ∆m31 , θ23 ) + 2 + 2 Peµ (θ13 , δ, ∆m31 , θ23 ) = Peµ (θ13 , δ , ∆m31 , θ23 ) − 2 − 2 Known as the intrinsic degeneracy The sign of ∆m31 and octant of θ23 are also unknown so one can also have 2 Peµ (θ13 , δ, ∆m31 , θ23 ) = Peµ (θ13 , δ , −∆m31 , θ23 ) 2 2 and π Peµ (θ13 , δ, ∆m31 , θ23 ) = Peµ (θ13 , δ , ∆m31 , 2 2 − θ23 ) 2 Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 11 / 23 Degenerate solutions graphically L = 130 km, E = 250 MeV 4.0 positive sign solution negative sign solution 3.5 mixed sign solution 3.0 no solution Suppose that the appearence 2.5 probability has been measured for both CP[P(ν)] (%) 2.0 a neutrino and anti-neutrino run 1.5 Set θ23 = π/4 1.0 no solution There are 4 solutions that can ﬁt the CP+, sin 2θ13 = 0.055 (upper) 2 0.5 CP+, sin 2θ13 = 0.05 (lower) 2 data - 2 for each mass hierarchy. CP−, sin 2θ13 = 0.0586 (upper) 2 CP−, sin 2θ13 = 0.0472 (lower) 2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 P(ν) (%) Minakata, Nunokawa and Parke, Phys.Rev. D66 (2002) 093012, arXiv:hep-ph/0208163 Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 12 / 23 Techniques for the removal of degeneracies The location of the intrinsic degeneracy in parameter space is energy dependent. All next generation or later experiments should therefore have good energy resolution to eliminate these clone solutions. Count rates, systematics (and correlations) are then the only barriers to (θ13 , δ). 2 In the presence of matter, the oscillation probability is dependent on sign(∆m31 ). The effect mimics that of δ so one needs a large matter effect or favourable δ to resolve the degeneracy. Baselines L ≥ 700 km are necessary to have any possibility of measurement. The θ23 octant is very hard to get a handle on - an effect relating to solar oscillations. For a LBL optimised for atmospheric oscillations, this is a subdominant process requiring small θ13 or low E to be seen. Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 13 / 23 The Durham-Roma approach The crucial difference to most studies is that we propose the use of 2 maxima as opposed to the usual neutrino and anti-neutrino runs (5+5). To see this, write the consider an experiment with ﬂux Φνe (νe ) (Eν ), the number of events in the ith bin is being given by Z Ei +∆E Ni (¯13 , ¯ = NT t θ δ) ε(Eν )σνµ (νµ ) (Eν )Peµ (Eν , ¯13 , ¯ νe (νe ) (Eν )dEν ± θ δ)Φ Ei COnsider further inﬁnite energy resolution so that ± ± N1 (E1 ) = c1 Peµ and N2 (E2 ) = c2 Peµ This type of analysis allows one to get at the features of the probability instrumental to the existence of the degeneracies. Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 14 / 23 The Durham-Roma approach Contd. For the intrinsic degeneracy, the clone solution (θ13 , δ) is found by solving N1 (¯13 , ¯ sign(∆m31 ), ¯23 ) θ δ, 2 θ θ = N1 (θ13 , δ, sign(∆m31 ), ¯23 ) 2 N2 (¯13 , ¯ sign(∆m31 ), ¯23 ) θ δ, 2 θ θ = N2 (θ13 , δ, sign(∆m31 ), ¯23 ) 2 By taking E1 as ﬁrst oscillation maximum and E2 and solving these equations, one ﬁnds no intrinsic clone solutions Similarly, for the sign degeneracy we can show A sin2 2θ13 θ sin2 2¯13 1 + 4 ∆13 sin δ sin ¯ δ The octant degeneracy persists for sin2 2θ13 > 10−3 . Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 15 / 23 The beta-beam Proposal to accelerate then store β emitting ions to source a pure neutrino beam Flux well-known theoretically No contamination of beam as in Superbeam No need for magnetised detectors 18 Ne, 6 He, 8 B and 8 Li are favoured nuclides. Need ∼ 1018 ions per year to compare with physics reach of Superbeams and Neutrino Factories. Beam collimated with opening angle 1/γ with rest frame energies E0 being boosted to 2γE0 on-axis Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 16 / 23 A beta-beam for the CERN-Boulby baseline (1050 km) 1st atmopsheric maximum at Boulby is at E ∼ 2 GeV (from ∆m31 L/4E = π/2). 2 18 Ne boosted to γ = 450 covers this and beyond. Assuming the detector has a low energy threshold, second oscillation maximum is avaliable as well. Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 17 / 23 Details of simulation Two exposures are considered 1 × 1021 ions per kton-year (low statistics) 5 × 1021 ions per kton-year (high statistics) For predicted number of events Ni and simulated number of event ni , we calculate a χ2 of the following form χ2 = ∑(ni − Ni )Ci−1 (nj − Nj ) ,j i ,j with the covariance matrix Ci−1 = δij (δni )2 ,j with δni = ni + (fsys · ni )2 We set fsys = 0.02 and include a background of 0.03 atmospheric events per kton-year (assuming 10−3 duty cycle) We bin with 200 MeV in the range [0.4, 2.0], with a ﬁnal bin [2.00, 3.06] Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 18 / 23 Fits to simulated data θ13 = 1 deg θ13 = 3 deg The black contours are the 1σ, 2σ and 3σ ﬁts. The red contours are a clone solution. Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 19 / 23 CP-violation CP discovery potential 180 Blue line - high statistics, red line - low statistics 90 δ A 1050 km beta-beam will have good sensitivity for 0 sin2 2θ13 ∼ few × 10−4 −90 For low statistics, still better than superbeam. Sensitivity for −180 −5 −4 −3 −2 −1 sin2 2θ13 ∼ few × 10−3 10 10 10 10 10 2 sin (2θ13) Test the hypothesis δ = 0. Apply a χ2 test over the (θ13 , δ) plane Contours are for 99 % conﬁdence and 2 d.o.f. Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 20 / 23 Mass hierarchy Blue line - high statistics, red line - low statistics Can determine the mass hierarchy for all δ when sin2 2θ13 < 0.03 (low statistics) sin2 2θ13 < 0.01 (high statistics) Need longer baseline for a dominant matter effect and resolution of degeneracy Calculate χ2 assuming NH and ruling out IH and vice versa. Contours are for 99 % conﬁdence and 2 d.o.f. Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 21 / 23 Octant degeneracy octant sensitivity 180 90 High statistics only δ Sensitivity is best for small θ13 as 0 this is the region where the solar term dominates −90 If θ13 is close to the present limit then sensitivity is poor. −180 −3 −2 −1 10 10 10 2 sin (2θ13) For θ13 close to the present limit, need info from low energies in this case (region of solar oscillations). Limited by low cross-sections and ﬂux. Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 22 / 23 Summary and conclusions The physics that links solar and atmospheric neutrino mixing is still undermined. This regime is to be explored by looking for sub-dominant processes in near and long-term future neutrino oscillation experiments. It has been demonstrated that the problem of degenerate solutions can be overcome with a single neutrino helicity when detectors have a good energy resolution and low energy thresholds. The information provided by second oscillation maximum compensates for not running in anti-neutrinos. The beta-beam is an experimental option that can be used on the intermediate baselines of Europe, for example the CERN-Boulby baseline. The physics reach is better than for a Superbeam, providing good sensitivities to the unknown parameters if sin2 2θ13 ∼ few × 10−3 . Christopher Orme () Long baseline phenomenology with an intermediate γ beta-beam March 7th 2008 23 / 23

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