We consider possibilities of observing CP-violation effects in neutrino
oscillation experiments with low energy ($\sim$ several hundreds MeV).Comment: 10 pages, 1 EPS figure.Talk presented at the ICFA/ECFA Workshop
  "Neutrino Factories based on Muon Storage Rings", nu-Fact'99, Lyon, France,
  5-9 July 1999,corrected some typo

Allison

Arafune

Athanassopoulos

Barger

Becker-Szendy

Bilenky

Chau

Collaboration

De Rujula

Fogli

Fukuda

Fukugita

Goodman

Hirata

Joe Sato

Koike

Krastev

Maki

Mikheev

Minakata

Tanimoto

Toshev

Wolfenstein

English

arXiv

佐藤 丈

Saitama University Cyber Repository of Academic Resources

arXiv:hep-ph/9910442v2  16 Nov 1999CP and T violation in (long) long baselineneutrino oscillation experimentsJoe SatoDepartment of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo,113-0033, JapanAbstractWe consider possibilities of observing CP-violation effects in neutrino oscillationexperiments with low energy (∼ several hundreds MeV).1 IntroductionMany experiments and observations have shown evidences for neutrino oscil-lation one after another. The solar neutrino deficit has long been observed(1;2; 3; 4; 5). The atmospheric neutrino anomaly has been found(6; 7; 8; 9) andrecently almost confirmed by SuperKamiokande(10). There is also anothersuggestion given by LSND(11). All of them can be understood by neutrino os-cillation and hence indicates that neutrinos are massive and there is a mixingin lepton sector(12).Since there is a mixing in lepton sector, it is quite natural to imagine thatthere occurs CP violation in lepton sector. Several physicists have consideredwhether we may see CP-violation effect in lepton sector through long baselineneutrino oscillation experiments. First it has been studied in the context ofcurrently planed experiments(13; 14; 15; 16; 17) and recently in the contextof neutrino factory(18; 19; 20; 21).The use of neutrinos from muon beam has great advantages compared withthose from pion beam(22). Neutrinos from µ+(µ−) beam consist of pure νe andν¯µ (ν¯e and νµ) and will contain no contamination of other kinds of neutrinos.Also their energy distribution will be determined very well.1 Current address: Research Center for Higher Education, Kyushu University,Ropponmatsu, Chuo-ku, Fukuoka, 810-8560, JapanPreprint submitted to Elsevier Preprint 1 February 2008In this proceedings, we will consider how large CP-violation effect we will seein oscillation experiments with low energy neutrino from muon beam. Suchneutrinos with high intensity will be available in near future(23). We willconsider three active neutrinos without any sterile one by attributing the solarneutrino deficit and atmospheric neutrino anomaly to the neutrino oscillation.2 CP violation in long baseline neutrino oscillation experimentsHere we consider neutrino oscillation experiments with baseline L ∼ severalhundreds km.2.1 Oscillation probability and its approximated formulaFirst we derive approximated formulas(15) of neutrino oscillation to clarifyour notation.We assume three generations of neutrinos which have mass eigenvalues mi(i =1, 2, 3) and MNS mixing matrix U(24) relating the flavor eigenstates να(α =e, µ, τ) and the mass eigenstates in the vacuum ν ′i(i = 1, 2, 3) asνα = Uαiν′i. (1)We parameterize U(25; 26; 27) asU = eiψλ7Γeiφλ5eiωλ2=1 0 00 cψ sψ0 −sψ cψ1 0 00 1 00 0 eiδcφ 0 sφ0 1 0−sφ 0 cφcω sω 0−sω cω 00 0 1=cφcω cφsω sφ−cψsω − sψsφcωeiδ cψcω − sψsφsωeiδ sψcφeiδsψsω − cψsφcωeiδ −sψcω − cψsφsωeiδ cψcφeiδ , (2)where cψ = cosψ, sφ = sinφ, etc.The evolution equation of neutrino with energy E in matter is expressed asidνdx= Hν, (3)2whereH ≡ 12EU˜diag(m˜21, m˜22, m˜23)U˜†, (4)with a unitary mixing matrix U˜ and the effective mass squared m˜2i ’s (i =1, 2, 3). The matrix U˜ and the masses m˜i’s are determined by(28; 29; 30)U˜m˜21m˜22m˜23 U˜† = U0δm221δm231U† +a00 . (5)Here δm2ij = m2i −m2j anda ≡ 2√2GFneE = 7.56× 10−5eV2 ·(ρg cm−3)(EGeV), (6)with the electron density, ne and the averaged matter density(31), ρ. Thesolution of eq.(3) is thenν(x) = S(x)ν(0) (7)S≡Te−i∫x0dsH(s) (8)(T being the symbol for time ordering), giving the oscillation probability forνα → νβ(α, β = e, µ, τ) at distance L asP (να → νβ ;E,L)= |Sβα(L)|2 . (9)Note that P (ν¯α → ν¯β) is related to P (να → νβ) through a → −a and U →U∗(i.e. δ → −δ). Similarly, we obtain P (νβ → να) from eq.(9) by replacingδ → −δ, P (ν¯β → ν¯α) by a→ −a.Attributing both solar neutrino deficit and atmospheric neutrino anomaly toneutrino oscillation, we can assume a, δm221 ≪ δm231. The oscillation probabil-ities in this case can be considered by perturbation(15). With the additionalconditionsaL2E= 1.93× 10−4 ·(ρg cm−3)(Lkm)≪ 1 (10)3andδm221L2E= 2.53(δm221/eV2)(L/km)E/GeV≪ 1, (11)the matrix S (8) is given byS(x) ≃ e−iH0x + e−iH0x(−i)x∫0dsH1(s), (12)whereH0 =12EU00δm231U† (13)H1(x)= eiH0xH1e−iH0x, (14)H1 =12EU0δm2210U† +a00. (15)Then the oscillation probabilities are calculated, e.g., asP (νµ → νe;E,L) = 4 sin2 δm231L4Ec2φs2φs2ψ{1 +aδm231· 2(1− 2s2φ)}+2δm231L2Esinδm231L2Ec2φsφsψ{− aδm231sφsψ(1− 2s2φ) +δm221δm231sω(−sφsψsω + cδcψcω)}− 4δm221L2Esin2δm231L4Esδc2φsφcψsψcωsω. (16)As stated, oscillation probabilities such as P (ν¯µ → ν¯e), P (νe → νµ) andP (ν¯e → ν¯µ) are given from the above formula by some appropriate changes ofthe sign of a and/or δ.The first condition (10) of the approximation leads to a constraint for thebaseline length of long-baseline experiments asL≪ 1.72× 103km(ρ3g cm−3)(17)4The second condition (11) gives the energy region where we can use the ap-proximation,E ≫ 76.0MeV(δm22110−4eV2)(L300km). (18)As long as these conditions, (17) and (18) are satisfied, the approximation(12) works pretty well(15; 32).2.2 Magnitude of CP violation and matter effectThe available neutrino as an initial beam is νµ and ν¯µ in the current longbaseline experiments(33; 34). The “CP violation” gives the nonzero differenceof the oscillation probabilities between, e.g., P (νµ → νe) and P (ν¯µ → ν¯e)(15).This givesP (νµ → νe;L)− P (ν¯µ → ν¯e;L)= 16 aδm231sin2δm231L4Ec2φs2φs2ψ(1− 2s2φ)− 4aL2Esinδm231L2Ec2φs2φs2ψ(1− 2s2φ)− 8δm221L2Esin2δm231L4Esδc2φsφcψsψcωsω.(19)The difference of these two, however, also includes matter effect, or the fakeCP violation, proportional to a. We must somehow distinguish these two toconclude the existence of CP violation as discussed in ref.(15).On the other hand, a muon ring enables to extract νe and ν¯e beam. It enablesdirect measurement of pure CP violation through “T violation”, e.g., P (νµ →νe)− P (νe → νµ) asP (νµ → νe)− P (νe → νµ) = −8δm221L2Esin2δm231L4Esδc2φsφcψsψcωsω. (20)Note that this difference gives pure CP violation.By measuring “CPT violation”, e.g. the difference between P (νµ → νe) andP (ν¯e → ν¯µ), we can check the matter effect.P (νµ → νe;L)− P (ν¯e → ν¯µ;L)= 16 aδm231sin2δm231L4Ec2φs2φs2ψ(1− 2s2φ)5-0.20-0.15-0.10-0.050.000.05Probability0.80.70.60.50.40.30.20.1Neutrino Energy / [GeV]δm231 = 3 x 10-3 eV2δm221 = 3 x 10-5 eV2sin2φ = 0.1-0.20-0.15-0.10-0.050.000.05Probability0.80.70.60.50.40.30.20.1Neutrino Energy / [GeV]δm231 = 3 x 10-3 eV2δm221 = 3 x 10-5 eV2sin2φ = 0.05-0.20-0.15-0.10-0.050.000.05Probability0.80.70.60.50.40.30.20.1Neutrino Energy / [GeV]δm231 = 3 x 10-3 eV2δm221 = 1 x 10-4 eV2sin2φ = 0.1-0.20-0.15-0.10-0.050.000.05Probability0.80.70.60.50.40.30.20.1Neutrino Energy / [GeV]δm231 = 3 x 10-3 eV2δm221 = 1 x 10-4 eV2sin2φ = 0.05-0.20-0.15-0.10-0.050.000.05Probability0.80.70.60.50.40.30.20.1Neutrino Energy / [GeV]δm231 = 1 x 10-3 eV2δm221 = 3 x 10-5 eV2sin2φ = 0.1-0.20-0.15-0.10-0.050.000.05Probability0.80.70.60.50.40.30.20.1Neutrino Energy / [GeV]δm231 = 1 x 10-3 eV2δm221 = 3 x 10-5 eV2sin2φ = 0.05-0.20-0.15-0.10-0.050.000.05Probability0.80.70.60.50.40.30.20.1Neutrino Energy / [GeV]δm231 = 1 x 10-3 eV2δm221 = 1 x 10-4 eV2sin2φ = 0.1-0.20-0.15-0.10-0.050.000.05Probability0.80.70.60.50.40.30.20.1Neutrino Energy / [GeV]δm231 = 1 x 10-3 eV2δm221 = 1 x 10-4 eV2sin2φ = 0.05(a) (b)(c) (d)(e) (f)(g) (h)Fig. 1. Graphs of P (νµ → νe) − P (νe → νµ) (solid lines; pure CP-violation ef-fects) and P (νµ → νe) − P (ν¯e → ν¯µ) (dashed lines; matter effects) as functionsof neutrino energy. Parameters not shown in the graphs are taken as follows.sin2 ω = 1/2, sin2 ψ = 1/2, sin δ = 1; ρ = g/cm3 and L = 300km.− 4aL2Esinδm231L2Ec2φs2φs2ψ(1− 2s2φ)(21)6We present in Fig.1 “T-violation” part (20) and “CPT-violation” part (21) forsome parameters allowed by the present experiments(35) 2 with sin2 ω = 1/2,sin2 ψ = 1/2, sin δ = 1 fixed. The matter density is also fixed to the constantvalue ρ = 2.5g/cm3(31). Other parameters are taken as δm231 = 3 × 10−3eV2and 1× 10−3eV2, δm221 = 1× 10−4eV2 and 3× 10−5eV2.“T-violation” effect is proportional to δm221/δm231 and, for φ≪ 1, also to sinφas seen in eq.(20) and Fig.1. Recalling that the energy of neutrino beam is ofseveral hundreds MeV, we see in Fig.1 that the “T-violation” effect amountsto at least about 5%, hopefully 10∼20%. This result gives hope to detect thepure leptonic CP violation directly with the neutrino oscillation experiments.3 CP violation in long long baseline experimentsHere we consider neutrino experiments with baseline L ∼ 10000km and seethat “T violation” will be amplified(37; 38).Since the baseline length L does not satisfy the condition (17,) we cannotmake use of the previous approximation.However, as a, δm221 ≪ δm231 is satisfied, we have approximation formulae ofthe mixing matrix in matter U˜ for neutrino,U˜ =eiψλ7Γeiφλ5eiω˜λ2 (22)tan 2ω˜=δm221s2ω−ac2φ + δm221c2ωand of “masses” in matter m˜2i forneutrino(m˜21, m˜22, m˜23)= (λ−, λ+, δm231 + ac2φ) (23)λ±=ac2φ + δm2212± 12√(−ac2ψ + δm221c2ω)2 + (δm221s2ω)2.Thus the “T violation” is given byP (νe → νµ;L)− P (νµ → νe;L)= 4sδc2φsφcψsψcω˜sω˜(sinδm˜221L2E+ sinδm˜232L2E+ sinδm˜213L2E)(24)2 Although the Chooz reactor experiment have almost excluded sin2 φ = 0.1(36),there remains still small chance to take this value.7∼ sδc2φsφcψsψcω˜sω˜(sinδm˜221L2E),here in the last equation we dropped the terms sinδm˜232L2E+sinδm˜213L2E, since theyoscillate very rapidly and will no contribution to the actual measurement.As is seen in eq. (22), due to MSW effect(28; 29) “T violation” may be am-plified very much even if the mixing angle ω is very small and hence we cantest whether there is a CP phase δ(38).4 Summary and conclusionWe considered how large CP/T violation effects can be observed making useof low-energy neutrino beam, inspired by PRISM(23).First we consider the baseline with several hundreds km. In this case more than10%, hopefully 20% of the pure CP-violation effects may be observed withinthe allowed region of present experiments. To see CP-violation effect thosebaseline length and the neutrino energy are most preferable statistically(32).Then we consider the baseline with ∼ 10000 km. We see that in this case, dueto MSW effect, “T violation” will be amplified and we can test whether theCP phase δ is large or not.References[1] GALLEX Collaboration, W. Hampel et al., Phys. Lett. B 447, 127(1999) .[2] SAGE Collaboration, J. N. Abdurashitov et al., astro-ph/9907113.[3] Kamiokande Collaboration, Y. Suzuki, Nucl. 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Kuno and Y. Mori, Talk at the ICFA/ECFA Workshop “NeutrinoFactories based on Muon Storage Ring”, July 1999; Y. Kuno, in Pro-ceedings of the Workshop on High Intensity Secondary Beam with PhaseRotation.[24] Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28, 870 (1962).[25] L. -L. Chau and W. -Y. Keung, Phys. Rev. Lett. 59, 671 (1987).[26] T. K. Kuo and J. Pantaleone, Phys. Lett. B198, 406 (1987).[27] S. Toshev, Phys. Lett. B226, 335 (1989).[28] L. Wolfenstein, Phys. Rev. D17, 2369 (1978).[29] S. P. Mikheev and A. Yu. Smirnov, Sov. J. Nucl. Phys. 42, 913 (1985).[30] V. Barger, S. Pakvasa, R.J.N. Phillips and K. Whisnant, Phys. Rev.D22, 2718 (1980).[31] M. Koike and J.Sato, Mod. Phys. Lett. A14,1297 (1999).[32] M. Koike and J.Sato, hep-ph/9909469.[33] K. Nishikawa, INS-Rep-924 (1992).[34] S. Parke, Fermilab-Conf-93/056-T (1993), hep-ph/9304271.[35] G. L. Fogli, E. Lisi, D. Montanino and G. Scioscia, Phys. Rev. D59,9033001 (1999).[36] M. Apollonio et al., Phys. Lett. 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CP and T violation in (long)long baseline neutrino oscillation experiments

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CP and T violation in (long) long baseline neutrino oscillation experiments

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CP and T violation in (long)long baseline neutrino oscillation
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CP and T violation in (long)long baseline neutrino oscillation experiments

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