成蹊大学理工学研究報告
J. Fac. Sci.Tech., Seikei Univ.
Vol.44 No.1 (2007)pp.83-87

ిࢠޫࢠিಥʹ͓͚Δྭ‫ ى‬Majorana χϡʔτϦϊҏ౻Ү෉ ∗Composite Majorana Neutrino in eγ Collision
Iku Ito*
Abstract
The lepton number violating/conserving process e− γ → ν/νW is discussed in a composite model
where excited Majorana neutrinos are exchanged. The diﬀerential cross section is calculated forthe polarized initial electron beam. If the polarization of the initial electrons is right-handed,ν and ν in the ﬁnal state cannot be discriminated and hence we cannot infer whether excitedneutrino is a Majorana type or not. If the polarization of the initial electrons is left-handed, on
the other hand, the process is contaminated with those by standard model process. While thecross section is the sum of those of two processes if the ﬁnal state is νW , the interference term
between standard model process is signiﬁcant if the ﬁnal state is νW .
Keywords: lepton number violation, Majorana neutrino, electron photon collider, composite
model(Received March 26, 2007)1ΔͷΈͰ͋Δ [3]ɻIntroductionߴΤωϧΪʔՃ଎‫͍͓ͯʹݧ࣮ث‬΋ɼͨͱ͑͹ిࢠཅిχϡʔτϦϊ͸ͦͷ࣭ྔ͕ඇৗʹখ͍͞ͱ͍͏఺Ͱɼଞͷࢠিಥʹ͓͍ͯɼMajorana ཻࢠΛަ‫͢׵‬Δ͜ͱʹΑͬͯɼϑΣϧϛΦϯͱେ͖͘ҟͳ͍ͬͯΔɻඪ४໛‫͍͓ͯʹܕ‬͸‫ݫ‬e− e− → W − W − ͷΑ͏ͳϨϓτϯ਺ඇอଘͷաఔ͕‫؍‬ଌ͞ີʹ࣭ྔ 0 ͷཻࢠͱͯ͠ѻΘΕΔ͕ɼۙ೥ɼ0 Ͱ͸ͳ͍࣭ྔΛΕΔՄೳੑ͕͋ΔɻKinyua Β͸ɼ͜ͷաఔ΁ͷ Majoranaཻ࣋ͭࢠͰ͋Δ͜ͱ͕࣮‫͞ূݕͰݧ‬Ε [1]ɼඪ४໛‫ܕ‬͸मਖ਼ΛχϡʔτϦϊͷ‫د‬༩Λ‫ݕ‬౼ͨ͠ [4]ɻͦΕʹΑΕ͹ɼަ‫͢׵‬༨ّͳ͘͞Ε͍ͯΔɻ͔͠͠ɼඪ४໛‫ܕ‬Λӽ͑Δ໛‫ܕ‬͸ɼ୯ʹΔ majorana χϡʔτϦϊ͕ྭ‫ى‬χϡʔτϦϊͰ͋Ε͹ɼ࣓χϡʔτϦϊʹ࣭ྔΛ༩͑Δ͚ͩͰͳ͘ɼ࣭ྔ͕͖ΘΊͯখ‫߹݁ܕؾ‬ͷͨΊʹ (ββ)0ν ͷ੍‫ݶ‬͸ΏΔ͘ɼࢄཚஅ໘ੵ͸ྭ͍͞ཧ༝΋ಉ࣌ʹࣔ͢΋ͷͰͳ͚Ε͹ͳΒͳ͍ɻχϡʔτϦ‫ى‬χϡʔτϦϊͷ࣭ྔͱ݁߹ఆ਺ͷύϥϝʔλʹΑͬͯ͸ɼ

ϊ͸ Dirac ࣭ྔ߲ͱಉ࣌ʹɼMajorana ࣭ྔ߲Λ࣋ͭ͜ͱ͕‫؍‬ଌՄೳͳఔ౓ͷେ͖͞ʹͳΔɻ‫ݐ‬ઃ‫ܭ‬ը͕ਐΊΒΕ͍ͯΔՄೳͰ͋Δɻ‫ऀޙ‬͸ɼDirac ࣭ྔ߲Λ༩͑Δ SUL (2) ⊗ UY (1)কདྷͷిࢠཅిࢠিಥ‫ܕ‬Ճ଎‫[ ݧ࣮ث‬5] ʹ͓͍ͯɼ͜ͷΑ͏ରশੑͷഁΕΑΓ΋େ͖ͳΤωϧΪʔεέʔϧͰͷμΠφϛͳࣄ৅͕‫؍‬ଌ͞ΕΕ͹ɼͦΕ͸௚ͪʹ Majorana χϡʔτϦΫεʹΑΔ΋ͷͱ͢Ε͹ɼ͍ΘΏΔγʔιʔ‫[ ߏػ‬2] ʹΑͬϊͷূ‫ͳͱڌ‬ΔɻͯɼχϡʔτϦϊ࣭ྔ͕ඇৗʹখ͘͞ͳΔ͜ͱΛઆ໌Ͱ͖Δɻ͜ͷ࿦จͰ͸ɼྭ‫ى‬χϡʔτϦϊ͕ radiative coupling ΛϑΣϧϛΦϯ͕ࣗ෼ࣗ਎ͷՙి‫ڞ‬໾ͱ౳͍͠ͱ͖ɼMajo-௨ͯ͡ిࢠޫࢠিಥաఔͷࢄཚஅ໘ੵʹର͢Δ‫د‬༩Λ‫ݕ‬౼͢rana ཻࢠͱ‫ݺ‬͹ΕΔɻχϡʔτϦϊ͸ QED ిՙ͕ 0 Ͱ͋Δɻిࢠిࢠিಥ‫ܕ‬Ճ଎‫ث‬͸ɼҰํͷిࢠϏʔϜ͕ϨʔβʔΔͨΊɼཧ࿦తʹ͸ Majorana ཻࢠͰ͋Δ͜ͱ͕‫͞ڐ‬ΕΔɻޫͱ‫ࢄํޙ‬ཚ͢Δ͜ͱʹΑͬͯಘΒΕΔߴΤωϧΪʔޫࢠχϡʔτϦϊ͕ Majorana ཻࢠͰ͋Ε͹ɼϨϓτϯ਺͕อଘϏʔϜΛ࢖͏ͱɼిࢠޫࢠিಥ‫ܕ‬Ճ଎‫ͯ͠ͱث‬΋ར༻Ͱ͖Δ͠ͳ͍աఔ͕ՄೳͱͳΔɻͨͱ͑͹ɼ‫֩ࢠݪ‬ͷχϡʔτϦϊ[6]ɻ͜ͷিಥͰ͸ɼ௒ରশੑཻࢠͷ୳ࡧ΍ྭ‫ࢠిى‬ͷ୳ࡧ์ग़Λ൐Θͳ͍ೋॏϕʔλ่յʢ(ββ)0ν ͱ‫͢ه‬ʣͱ‫ݺ‬͹ΕͳͲͰόοΫάϥ΢ϯυͷগͳ͍γάφϧ͕ಘΒΕΔ͜ͱ͕Δ‫ݱ‬৅ΛҾ͖‫͢͜ى‬ɻ͔͠͠ɼͦͷΑ͏ͳ (ββ)0ν ͸‫·ࡏݱ‬‫ظ‬଴͞Ε͍ͯΔ [7, 8]ɻޫࢠ͸‫ࢄํޙ‬ཚͰಘΒΕΔͨΊɼॏͰͷͱ͜Ζ‫؍‬ଌ͞Ε͓ͯΒͣɼ൒‫ظݮ‬ͷ্‫͕ݶ‬༩͑ΒΕ͍ͯ৺‫Ͱܥ‬ͷΤωϧΪʔ͸ిࢠిࢠিಥͷ৔߹ʹൺ΂ͯ 90%ఔ∗ ੒᪟େֶཧ޻ֶ෦‫ڞ‬௨‫ڭૅج‬त (ito@st.seikei.ac.jp)౓Ͱ͋Γɼ৽ཻࢠ୳ࡧʹ͓͍ͯ΋ిࢠిࢠিಥͷ৔߹ͱଝ৭

1

─83─

ͳ͍ɻ͜ͷΑ͏ͳΦϓγϣϯͰͷ e− γ → νW աఔ͸ɼྭ‫ ى‬Ma-jorana χϡʔτϦϊ͕ଘࡏ͢Ε͹ɼe− e− → W − W − աఔͱൺ΂ͯ࢝ঢ়ଶͷΤωϧΪʔ͸ 90%ఔ౓ʹͳΔ΋ͷͷɼऴ2ྭ‫ ى‬Majorana χϡʔτϦϊ

2.1

Majorana χϡʔτϦϊϑΣϧϛΦϯ৔ ψ ͷՙి‫ڞ‬໾৔ ψ c Λ࣍ͷΑ͏ʹఆٛ͢Δɿঢ়ଶͷҐ૬ۭ͕ؒେ͖͘ͳΔͨΊɼͦΕ͚ͩେ͖ͳγάφϧ͕‫ظ‬଴Ͱ͖Δɻψ c ≡ Cψ = C(ψ † γ0 )TTͳ͓ɼऴঢ়ଶʹྭ‫ى‬χϡʔτϦϊΛੜ੒͢ΔաఔΛ௚઀‫؍‬ଌ͢ΔఏҊ΋͞Ε͍ͯΔ͕ [9]ɼͦͷͱ͖͸ɼྭ‫ى‬χϡʔτ√Ϧϊͷ࣭ྔ͸ s ͷ஋ʹΑ੍ͬͯ‫͞ݶ‬ΕΔɻ͜Εʹରͯ͠ɼ√ྭ‫ى‬χϡʔτϦϊΛަ‫͢׵‬Δաఔ͸ s ʹΑΔӡಈֶత੍‫ݶ‬(1)

ͨͩ͠ɼC ͸ՙి‫ڞ‬໾ͷϢχλϦʔԋࢉࢠͰɼDirac ͷ γ ߦ
ྻͱCγμT C † = −γμ͸ड͚ͳ͍ɻC(σ μν )T C † = −σ μνిࢠޫࢠিಥͰྭ‫ى‬χϡʔτϦϊ͕ Majorana Ͱ͋ͬͨͱͯ͠΋ɼऴঢ়ଶ͸ νW ͚ͩͰͳ͘ νW ͷ৔߹͕͋Δɻྭ‫ى‬

Cγ5T C † = γ5(2)Majorana Λ u νϟωϧͰަ‫͢׵‬Δաఔͷɼऴঢ়ଶ͕ νW ͱͷؔ܎͕͋Δɻՙి‫ڞ‬໾৔ ψ c ͷ γ0 ‫ڞ‬໾͸νW ͷৼ෯ΛͦΕͧΕ Mmaj (νW ), Mmaj (νW ) ͱ͠Α͏ɻ“”†ψ c = Cγ0T (ψ † )T γ0 = ψ T γ0T C † γ0 = −ψ T C †͞Βʹɼऴঢ়ଶ͕ νW Ͱ͋ΔΑ͏ͳࣄ৅͸ඪ४໛‫ܕ‬ͷաఔͰ΋‫͖ى‬Δɻ͜ͷৼ෯Λ Mstd (νW ) ͱॻ͘͜ͱʹ͢Δɻ͢

(3)͜͜Ͱɼࣜ (2) ͔ΒಘΒΕΔ Cγ0T C † = −γ0 ͓Αͼɼγ0 = γ0†Δͱɼೋͭͷऴঢ়ଶʹରԠ͢Δաఔͷৼ෯͸ͦΕͧΕͰ͋Δ͜ͱΛ࢖ͬͨɻࣜ (1) ͔ΒΘ͔ΔΑ͏ʹɼՙి‫ڞ‬໾৔

e− γ → νW ɿMstd (νW ) + Mmaj (νW )ψ c ͸΋ͱͷϑΣϧϛΦϯ ψ ͱ U (1) ిՙ͕‫ʹ͍ޓ‬൓ରූ߸−e γ → νW ɿMmaj (νW )ʹͳΔɻ͜ͷͨΊɼU (1) ిՙ͕ 0 Ͱͳ͍ϑΣϧϛΦϯͱͦͱͳΔɻ͜͏ͨ͠Ճ଎‫Ͱݧ࣮ث‬͸ɼν ͱ ν ͸௚઀‫؍‬ଌͰͷՙి‫ڞ‬໾৔͕ಉཻ͡ࢠΛද͢͜ͱ͸ͳ͍ɻ͔͠͠ɼిՙ͕͖ͳ͍ͷͰɼೋͭͷऴঢ়ଶ͸ͦͷ··Ͱ͸۠ผ͕Ͱ͖ͳ

0 Ͱ͋Ε͹ɼಉཻ͡ࢠʹͳΔՄೳੑ͕͋ΔɻͦͷΑ͏ͳཻࢠ͍ɻͭ·Γɼ͜ΕΒೋͭͷաఔ͕ಉ࣌ʹ‫؍‬ଌ͞ΕΔɻ͋ͷ͜ͱΛ Majorana ཻࢠͱ͍͏ɻχϡʔτϦϊ͸ U (1)em ిͱͰड़΂ΔΑ͏ʹɼ࣮ࡍʹ͸ΧΠϥϦςΟͷ੍‫ݶ‬ͷͨΊɼ

ՙ͕ 0 Ͱ͋Δ͔ΒɼMajorana ཻࢠͷՄೳੑ͕͋ΔɻͦͷΑ

Mmaj (νW ) ͔ Mmaj (νW ) ͷͲͪΒ͔Ұํ͸ 0 Ͱ͋Δɻ͔͏ͳχϡʔτϦϊͷ͜ͱΛ Majorana χϡʔτϦϊͱ͍͏ɿΓʹɼMmaj (νW ) = 0 Ͱ͋Ε͹ɼ‫؍‬ଌ͞ΕΔஅ໘ੵ͸ν = ν c = Cν Tdσ ∝ |Mstd (νW ) + Mmaj (νW )|2(4)͋ͱͰඞཁʹͳΔؔ܎ࣜΛҰͭಋ͍͓ͯ͘ɻν ͕ Majoranaͷ Α ͏ ʹ ɼৼ ෯ Ͱ ඪ ४ ໛ ‫ ͱ ܕ‬ͷ ࿨ ͱ ͳ Δ ɻ͜ Ε ʹ ର ͠ ɼͰ͋Ε͹

Mmaj (νW ) = 0 Ͱ͋Ε͹ɼ‫؍‬ଌ͞ΕΔஅ໘ੵ͸no0|T ν T (x)ν(y) |0

dσ ∝ |Mstd (νW )|2 + |Mmaj (νW )|2ͷΑ͏ʹৼ෯ͷ 2 ৐ͷ࿨ͱͳΔɻిࢠޫࢠিಥͰ͸࢝ঢ়ଶͷిࢠ͓Αͼޫࢠͷภ‫ۃ‬Λ੍‫ޚ‬=C † 0|T {ν(x)ν(y)} |0=C † SF (x − y)(5)͜͜ͰɼSF ͸ϑΣϧϛΦϯͷϓϩύήʔλʔΛද͢ɻ

͢Δ͜ͱ͕ՄೳͰ͋Δɻͱ͘ʹిࢠͷภ‫ۃ‬Λӈ‫ࢦʹר‬ఆ͢Ε͹ɼඪ४໛‫ܕ‬ͷաఔ͸͓͖ͳ͍ɻ͕ͨͬͯ͠ɼ͜ͷ৔߹ʹ͸‫؍‬ଌ͞ΕΔஅ໘ੵ͸dσ ∝ |Mmaj (νW )|2.22໛‫ܕ‬ϥάϥϯδΞϯ

ྭ‫ى‬χϡʔτϦϊ৔Λ ν ∗ Ͱද͢ɻ͜Εͱిࢠ͓Αͼ͍ܰ͋Δ͍͸χϡʔτϦϊͱͷήʔδ૬‫༻࡞ޓ‬ͷ Lagrangian Λ࣍ͷ‫ʹܗ‬dσ ∝ |Mmaj (νW )|2ॻ͜͏ [10]ɿͱͳΔɻλe ∗ν (x)σ μν (ηL ΓL + ηR ΓR ) ν(x)Fμν (x)2mν ∗` ´λ g ∗†ν (x)σ μν ηLΓ L + ηR
ΓR e(x)Wμν(x)+2mν ∗+h.c.
(6)L=ୈ 2 ষͰ͸ɼྭ‫ ى‬Majorana χϡʔτϦϊͱిࢠͳΒͼν ͳ͍͠ ν ͱͷ૬‫༻࡞ޓ‬Λ‫ن‬ఆ͢Δ໛‫ܕ‬ϥάϥϯδΞϯΛड़΂Δɻୈ 3 ষͰ͸ɼ͜ͷ໛‫ܕ‬ϥάϥϯδΞϯʹ΋ͱ͍ͮͨࢄཚஅ໘ੵͷ‫݁ࢉܭ‬ՌΛࣔ͢ɻྭ‫ى‬χϡʔτϦϊͷ࣭ྔʹԠͯ͡ɼγάφϧ͕ͲͷΑ͏ʹ‫͑ݟ‬Δ͔ɼόοΫάϥ΢ϯυͱͷ͜͜Ͱɼe, g ͸ऑి࣓૬‫༻࡞ޓ‬ͷ݁߹ఆ਺ɼmν ∗ ͸ྭ‫ى‬χ۠ผɼͳͲʹ͍ͭͯٞ࿦͢ΔɻϡʔτϦϊͷ࣭ྔɼFμν , Wμν ͸ޫࢠͱ W Ϙκϯ৔Λද͢ɻ2─84─

ΓL(R) ≡ (1 ∓ γ 5 )/2 ͸ΧΠϥϦςΟ΁ͷࣹӨԋࢉࢠΛද͢ɻ

·ͨɼλ, λ·ͨɼࣜ (9) ͷ T ੵ಺Ͱ࠷ॳͷ 3 ߲ͷੵ͸͓Αͼ ηL(R) , ηL(R)͸͍ͣΕ΋ dimensionless ͷν T (σ μν )T (ηL ΓL + ηR ΓR )Tఆ਺Ͱ͋Δɻྭ‫ى‬χϡʔτϦϊ͕ྭ‫ ͱࢠిى‬SU (2) ೋॏ߲= ν T C † · C(σ μν )T C † · C(ηL ΓL + ηR ΓR )T C † · CΛ૊Ή໛‫͍͓ͯʹܕ‬͸ ηL = ηL, η R = ηR
Ͱ͋Δɻͦͷ৔= (−ν c ) · (−σ μν ) · (ηL ΓL + ηR ΓR ) · C߹ʹ͸ɼParticle Data Group ͱಉ͡ notation ʹͳΔɻ͜͜Ͱ͸ྭ‫ࢠిى‬ͷ࣭ྔ͕େ͖͍ͱͯ͠ɼͦͷ‫د‬༩͸ߟ͑ͳͱॻ͚ΔͷͰͱΛಠཱʹͱ͓ͬͯ͘ɻ͍ͨ͜ͱʹ͢Δɻη L (R) ͱ ηL(R)Zd4 p ie−ip(x−y)(2π)4 p2 − mν ∗ 2j×ν̄(p2 )|T ν c (x)σ μν (ηL ΓL + ηR ΓR )( p + mν ∗ )ﬀ×σ ρτ (ηLΓ L + ηRΓR )e(y) |e(p1 )

ͩ͠ɼmax(|ηL |, |ηR |) = max(|ηL|, |ηR|) = 1 ͷΑ͏ʹਖ਼ 
ηR = 0 Λຬͨ‫ن‬Խ͠ɼ·ͨΧΠϥϦςΟ৚݅ ηL ηR = ηL= 1, ηR = ηR= 0 ͷͱ͢΋ͷͱ͢Δɻͱ͘ʹ ηL = ηL͖͸ Cabibbo, Maiani, Srivastava [11] ͓ΑͼɼHagiwara,Komamiya, Zeppenfeld [12] ͷ૬‫ؼʹ༻࡞ޓ‬ண͢Δɿ„«1 ∗ μν
τa a
L=L σ+ g  f  Y Bμν ΓL L + h.c. (7)gf Wμν
2Λ2ͨͩ͠1λ=(f − f  ),2mν ∗4Λ
ͱ͓͍ͨɻ

(11)
ν c (x) ͱ e(y) Ͱ͸͞·Εͨ෦෼Λ Γμνρτ ͱ͓͘ɻ√λ
2f=2mν ∗4Λ

ΓL(R) ( p + mν ∗ )ΓL(R) = mν ∗ ΓL(R)ΓL(R) ( p + mν ∗ )ΓR(L) = pΓR(L)

Takasugi ͸ɼ͜ͷ૬‫ʹ༻࡞ޓ‬΋ͱ͖ͮɼ76 Ge ͷ (ββ)0ν(12)ͱͳΔ͜ͱΛ࢖͏ͱjΓμνρτ = σ μν mν ∗ (ηL ηLΓ L + ηR η RΓR ) ɹͷण໋Λ‫[ ͨ͠ࢉܭ‬13]ɻ࣮ͦͯ͠‫ݧ‬஋ͷ্‫͔ݶ‬Βύϥϝʔλ x = (mν ∗ /MW )2 ʹ͍ͭͯͨͱ͑͹ɼྭ‫ࢠిى‬ͷ࣭ྔ͕

500 GeV ͷͱ͖ λ/x < 4 TeV−1 ͱ͍͏Α͏ͳ੍‫ݶ‬Λಘͯﬀ
+ p(ηL ηR
ΓR + ηR ηLΓL ) σ ρτ (13)͍Δɻ͕ͨͬͯ͠ eγ → νW ͷࢄཚৼ෯͸3ిࢠޫࢠিಥͱྭ‫ى‬χϡʔτϦϊ

3.1

ෆมৼ෯

M=

iegλλ ck1μ 1ν k2ρ 2τ
v (p2 )Γμνρτ u(p1 )4m2ν ∗(p1 − k2 )2 − m2ν ∗ͱ ॻ ͚ Δ ɻͳ ͓ ɼCMS-HKZ ͷ ໛ ‫ Ͱ ܕ‬͸ Γμνρτ(14)=mν ∗ σ μν ΓL σ ρτ ͱ؆୯ʹͳΔɻS ԋࢉࢠͷ͏ͪɼ࢝ঢ়ଶ͕ eγ(ͦΕͧΕͷӡಈྔ p1 , k1 )ɼ

͜Εͱಉ͘͡ྭ‫ى‬χϡʔτϦϊΛަ‫ͯ͠׵‬ɼऴঢ়ଶ͕ νWऴঢ়ଶ͕ νW (ಉ͘͡ p2 , k2 ) ͷߦྻཁૉΛͱͬͨͱ͖ʹ࢒ΔͰͳ͘ νW ͷաఔ͕ՄೳͰ͋ΔɻMajorana ͕͜ͷաఔʹ‫د‬ͷ͸༩͢Δ૬‫߲༻࡞ޓ‬͸ಉ༷ʹͯ͠‫ٻ‬ΊΒΕͯjZ(−i)2 egλλ
S=Tdx4 dy 44m2ν ∗`´ɹɹ × ν ∗ (x)σ μν (ηL ΓL + ηR ΓR )ν(x)Fμν (x)“”o†ɹɹ × ν ∗ (y)σ ρτ (ηLΓ L + ηRΓR )e(y)Wρτ(y)M=(8)ߦྻཁૉͷ͏ͪɼlepton ʹ͔͔ΘΔ෦෼ΛऔΓग़͢ͱjν̄(p2 )|T ν ∗ (x)σ μν (ηL ΓL + ηR ΓR )ν(x)ﬀρτ∗×ν (y)σ (ηL ΓL + ηR ΓR )e(y) |e(p1 )
jT= ν̄(p2 )|T ν T (x)(σ μν )T (ηL ΓL + ηR ΓR )T ν ∗ (x)ﬀ×ν ∗ (y)σ ρτ (ηLΓ L + ηR
ΓR )e(y) |e(p1 )

iegλλ
k1μ 1ν k2ρ 2τ
ū(p2 )Σμνρτ u(p1 )4m2ν ∗(p1 − k2 )2 − m∗2(15)ͨͩ͠Σμνρτ=σμνj∗ ∗ ηL Γ L + ηL
ηR ΓR )mν ∗ (ηRﬀ∗ ∗ + p(ηRηR Γ R + ηL
ηL ΓL ) σ ρτ (16)∗Ͱ͋Δ͜ͱ͕Θ͔νW Ϟʔυͷৼ෯ͱൺ΂ΔͱɼηL ↔ ηRΔɻ͜ͷ৔߹΋ɼCMS-HKZ ໛‫Ͱܕ‬͸ Σμνρτ = σ μν pΓL σ ρτͱ؆୯ʹͳΔɻ(9)͜͜Ͱ ν ∗ ͕ Majorana Ͱ͋Ε͹ɼࣜ (5) ͔Βon˘¯T
0|T ν ∗ (x)ν ∗ (y) |0 = 0|T C † ν ∗ (x)ν ∗ (y) |0

3.2

ภ‫ࢄۃ‬ཚஅ໘ੵલઅͰ‫ٻ‬Ίͨৼ෯͔Βஅ໘ੵΛ‫͢ࢉܭ‬Δɻ·ͣɼe− γ →= C † SF (x − y)ν̄W ͷաఔʹ͍ͭͯɼ࢝ঢ়ଶͷޫࢠͷ Stokes ύϥϝʔλΛ

(10)

3─85─

ξ2 = 1 ͱ͠ɼిࢠͷภ‫ۃ‬Λӈ‫( ͖ר‬eR )ɼ·ͨ͸ࠨ‫( ͖ר‬eL )ͷͱ͖ʹ͸ऴঢ়ଶ͸ νW ͳͷͰɼ͜ͷ৔߹͸ඪ४໛‫ʹܕ‬Αͱͯ͠‫݁ͨ͠ࢉܭ‬Ռ͸࣍ͷΑ͏ʹͳΔɿΔաఔͱ͸‫ׯ‬ব͠ͳ͍ɻ͕ͨͬͯ͠ɼશମͷஅ໘ੵ͸ඪ४໛

2 2‫ʹܕ‬ΑΔաఔͷஅ໘ੵ ((22) ͷୈೋ߲) ͱ (18) ͷ࿨ͱͳͬͯ

2λ λ (eg)dσ 2|ηR ηR| f (s, t, u)(eR ) =dt
8πm4ν ∗(17)‫؍‬ଌ͞ΕΔɻλ2 λ2 (eg)2dσ
 2|ηR ηL| g(s, t, u)(eL ) =dt
8πm4ν ∗(18)͸ νW Ͱ͋Δ͕ɼඪ४໛‫ͱܕ‬ͷ࿨ͱͯ͠‫؍‬ଌ͞ΕΔͱ͖ʹҎ্Λ·ͱΊΔͱɼஅ໘ੵ͕‫ׯ‬বͷಛ௃Λࣔͤ͹ऴঢ়ଶ͸ऴঢ়ଶ͸ νW ͷࣄ৅Ͱ͋Γɼͦͷͱ͖ͷྭ‫ى‬χϡʔτϦͨͩ͠
2um2ν ∗ MWs + 2tu
f (s, t, u) = −2s2 (u − m2ν ∗ )2

ϊ͸ Majorana ͱ͍͏͜ͱʹͳΔɻೋͭͷ৔߹Ͱɼ࣮ࡍͲ
ͷ͘Β͍ͷ͕ࠩ͋Δͷ͔ఆྔతʹࣔ͢ɻਤ 1 ͱ 2 ͸ɼ͍ͣ

(19)Ε΋࢝ঢ়ଶͷిࢠͷภ‫ۃ‬Λࠨ‫͠ͱ͖ר‬ɼ|ηL | = 1 ͷ৔߹ͱ22)(u − MW)u2 su + (s − MW(20)g(s, t, u) = − 22 2s(u − mν ∗ )|ηR | = 1 ͷ৔߹ʹ͍ͭͯɼॏ৺‫Ͱܥ‬ͷ W ͷ֯౓෼෍Λඪ√s=४໛‫ܕ‬աఔͱͷൺΛͱͬͯදͨ͠΋ͷͰ͋Δɻ‫ࢉܭ‬͸

s, t, u ͸ Mandelstam ม਺Ͱɼిࢠ͓ΑͼχϡʔτϦϊͷ࣭

1 TeVɼλ = λ = 1 ͱͯ͠ߦͬͨɻ·ͨྭ‫ى‬χϡʔτϦϊ√ͷ࣭ྔ͸ਤ 1 Ͱ͸ mν ∗ = 272 GeV( x = 3.4)ɼਤ 2 Ͱ͸

2ྔ͕ແࢹͰ͖ΔΤωϧΪʔྖҬͰ s + t + u = MWͷؔ܎͕͋Δɻmν ∗ = 500 GeV ͱͨ͠ɻ|ηR | = 1 ͷ৔߹͸ ηR ͷҐ૬ʹ͸࣍ʹɼeγ → νW ͷաఔʹ͍ͭͯɼిࢠͷภ‫͕ۃ‬ӈ‫ͱ͖ר‬ΑΒͳ͍͕ɼ|ηL | = 1 ͷ৔߹͸ ηL ͷҐ૬ʹԠͯ͡‫ׯ‬ব߲ͷࠨ‫Ͱͱ͖ר‬அ໘ੵ͸ͦΕͧΕλ2 λ2 (eg)2dσ 2|ηL ηR| f (s, t, u)(eR ) =dt
8πm4ν ∗େ͖͕͞มΘΔͷͰɼਤʹ͸ ηL = ±1 ͷ৔߹Λࣔͨ͠ɻ(21)
3(dσ/dt)/(dσ/dt)std

λ2 λ2 (eg)2dσ 2ɹɹ (eL ) =|ηL ηL| g(s, t, u)
dt8πm4ν ∗22 2(eg) u(u − MW )ɹɹɹɹɹɹ −2 2
8πs3 (t − MW)√2u2 (u − MW2λλ (eg)2 ∗ )∗
ɹɹ −(ηL ηL + ηL η  L )22222πs mν ∗(u − mν ∗ )(t − MW)(22)

ηL= 1

2.521
ηL=10.50ͱॻ͚Δɻ͜͜Ͱɼిࢠ͕ࠨ‫͖ר‬ͷͱ͖ʹ͸ɼඪ४໛‫ʹܕ‬ΑΔ|ηR|=11.5-60

աఔ͔Βͷ‫د‬༩͕͋Γɼৼ෯͸ඪ४໛‫ܕ‬ͷ‫د‬༩ͱྭ‫ى‬χϡʔ

-40-200204060

θ of W boson (deg)τϦϊͷ‫د‬༩ͷ࿨ͱͳΔɻ(22) Ͱ͸ӈลͷୈೋ߲͕ඪ४໛‫ܕ‬ʹΑΔաఔͷஅ໘ੵɼୈࡾ߲͸ඪ४໛‫ܕ‬ͷաఔͱ ν ∗ ‫ܦ‬༝ͷਤ 1:աఔͷ‫ׯ‬ব߲ʹ͋ͨΔɻ

eγ→√
ν(ν)W Ͱ ͷ W ͷ ֯ ౓ ෼ ෍ ɻ s=1 TeV, mν ∗ = 272 GeVɻηL = ±1(dashed line) ͱ |ηR | = ηR = 0 ͳͷͰɼྭ‫ى‬ΧΠϥϦςΟอଘ͔Β ηL ηR = ηL1(dotted line)ɻχϡʔτϦϊͷγάφϧ͸ɼ࢝ঢ়ଶͷిࢠͷภ‫͕ۃ‬ӈ‫͔͖ר‬ࠨ‫͖ר‬ͷ͍ͣΕ͔ɼͦͯ͠ऴঢ়ଶ͕ νW ͔ νW ͷ͍ͣΕ͔ͷνϟϯωϧͰ͔͠‫؍‬ଌ͞Εͳ͍ɻ
·ͣɼ|ηR| = 1 Ͱ͋Ε͹ɼిࢠͷภ‫ۃ‬Λӈ‫ͯ͠ʹ͖ר‬িಥ͠2(dσ/dt)/(dσ/dt)stdͨͱ͖ɼྭ‫ى‬χϡʔτϦϊ‫ܦ‬༝ͷࣄ৅͕‫؍‬ଌ͞ΕΔɻ|ηR | = 1Ͱ͋Ε͹ऴঢ়ଶ͸ νW ɼ|ηL | = 1 Ͱ͋Ε͹ νW ͱͳΔɻඪ४໛‫Ͱܕ‬͸ eR γ → νW ͷஅ໘ੵ͸ 0 ͳͷͰɼ͍ͣΕͷ৔߹ʹ΋όοΫάϥ΢ϯυ͸ͳ͍ɻ͔͠͠ɼऴঢ়ଶͷ ν ͱ ν ͸‫؍‬ଌʹΑͬͯ۠ผͰ͖ͳ͍ͷͰɼ|ηR | = 1 ͳͷ͔ɼͦΕͱ΋|ηL | = 1 ͳͷ͔ɼ͕ͨͬͯ͠ɼྭ‫ى‬χϡʔτϦϊ͕ Majorana͔Ͳ͏͔͸͜Ε͚ͩͰ͸൑அͰ͖ͳ͍ɻ

ηL= 11.5|ηR|=110.5

ηL=10-60
| = 1 Ͱ͋Ε͹ɼిࢠͷภ‫ۃ‬Λࠨ‫ͯ͠ʹ͖ר‬িಥҰํɼ|ηL-40-200204060

θ of W boson (deg)ͨ͠ͱ͖ɼྭ‫ى‬χϡʔτϦϊ‫ܦ‬༝ͷࣄ৅͕‫؍‬ଌ͞ΕΔɻ͜ͷͱ͖͸ɼ͍ͣΕͷऴঢ়ଶͰ΋ඞͣඪ४໛‫ʹܕ‬ΑΔաఔ eγ → νW

√ν(ν)W Ͱ ͷ W ͷ ֯ ౓ ෼ ෍ ɻ s͕όοΫάϥ΢ϯυͱͳΔɻ|ηL | = 1 Ͱ͋Ε͹ɼν ∗ ʹΑΔऴ

ਤ 2:

ঢ়ଶ͕ νW Ͱ͋Δ͔Βɼ͜Ε͸ඪ४໛‫ʹܕ‬ΑΔ eγ → νW ͷ1 TeV, mν ∗ = 500 GeVɻηL = ±1(dashed line) ͱ |ηR | =աఔͱՄ‫ׯ‬বͰ͋Γɼ(22) ͷΑ͏ʹɼೋͭͷաఔͷৼ෯ͷ࿨

1(dotted line)ɻ

Λͱͬͨ΋ͷ͕அ໘ੵʹ‫د‬༩͢Δɻ͜Εʹରͯ͠ɼ|ηR | = 14─86─eγ→=ඪ४໛‫ܕ‬աఔͰ͸֯౓෼෍͕લํʹूத͢Δɻ͜Ε͸அ໘ͨ͠ͱ͖ඪ४໛‫ʹܕ‬ΑΔ νW աఔͷஅ໘ੵ 80 pb ʹൺ΂Ε2 2) ʹൺྫ͢ΔͨΊͰ͋Δɻ͜Εʹൺ΂Δͱɼੵ͕ 1/(t − MW2͹খ͍͞΋ͷͷɼϧϛϊγςΟ͕ 1033 s−1 cm−2 ఔ౓͋Ε͹ྭ‫ى‬χϡʔτϦϊͷ‫د‬༩͸ɼg(s, t, u) ͕ u ʹൺྫ͢ΔͨΊɼ‫ݕ‬ग़Մೳͳൣғʹ͋Δ͜ͱ͕Θ͔Δɻ΍͸ΓલํʹϐʔΫ͕͋Δɻ͔͠͠ɼඪ४໛‫ܕ‬աఔʹΑΔ෼
෍ͱൺ΂Ε͹෯͕޿͘ɼ|ηR | = 1 ͷͱ͖͸֯౓͕େ͖͘ͳΔʹͭΕͯඪ४໛‫ܕ‬աఔͱͷ͕ࠩஶ͘͠ͳΔɻࢀߟจ‫ݙ‬Ұํɼ|ηL | = 1 ͷͱ͖͸ͦͷූ߸ʹΑͬͯඪ४໛‫ͱܕ‬ͷࠩ͸ҟͳΔɻηL = 1 ͷͱ͖͸ɼθ = 0 ͔Β཭ΕΔͱ͍ͬͨΜ[1] S. Fukuda et al. [Super-Kamiokande Collaboration],

= −1) ͱൺ͕খ͘͞ͳΓɼ΄͔ͷ 2 ͭͷ৔߹ (ηR = 1, ηL

Phys. Rev. Lett. 81 (1998) 1562; S.N. Ahmed et͸ࡍཱͬͨ෼෍Λࣔ͢͜ͱ͕Θ͔Δɻ

al. [SNO Collaboration], Phys. Rev. Lett. 87 (2001)
071301; K. Eguchi et al. [KamLAND Collaboration],Phys. Rev. Lett. 90 (2003) 021802

3.3

શஅ໘ੵ

[2] M. Gell-Mann, P. Ramond and R. Slansky, in Supergravity, eds. P.van Neuwenhuizen and D.Z. Freedman

|ηR| = 1 ͷ৔߹ɼ࢝ঢ়ଶͷిࢠͷภ‫ۃ‬Λӈ‫͢ͱ͖ר‬Ε͹ɼ

(North Holland 1979); T. Yanagida, in Proceedingsྭ‫ى‬χϡʔτϦϊ‫ܦ‬༝Ͱऴঢ়ଶ͕ νW ·ͨ͸ νW ͕‫؍‬ଌ͞

of Workshop on Uniﬁed Theory and Baryon NumberΕΔɻ͜ΕΒͷ۠ผ͸Ͱ͖ͳ͍ͷͰɼ͜Ε͚ͩͷ͜ͱ͔Βྭ

of the Universe, eds. O. Sawada and A. Sugamoto‫ى‬χϡʔτϦϊ͕ Majorana ͔Ͳ͏͔͸Θ͔Βͳ͍ɻ͔͠͠ɼ(KEK, 1979)ඪ४໛‫Ͱܕ‬͸‫ࣄ͍ͳ͖ى‬৅Ͱ͋ΓɼόοΫάϥ΢ϯυͳ͠ʹ

[3] For a recent review, K. Zuber, Acta Phys. Polon.

‫؍‬ଌͰ͖ΔͨΊɼྭ‫ى‬χϡʔτϦϊͷγάφϧͱͳΔɻ࣮ࡍ

B37 (2006) 1905ʹશஅ໘ੵ͕Ͳͷ͘Β͍ͷେ͖͞ʹͳΔͷ͔Λ਺஋Ͱࣔͯ͠
͓͜͏ɻ

[4] R. Kinyua and E. Takasugi, Prog. Theor. Phys. 100(17) ͋Δ͍͸ (21) Λ‫͞ڐ‬ΕΔӡಈֶతྖҬʹΘͨͬͯੵ෼͢Ε͹

σ=λ2 λ2 (eg)2|ηL(R) |2 F (x, y)216πMW(1998) 607[5] ACFA, JHEPC and KEK, KEK Report 2003-7(23)ͱ ॻ ͚ Δ ɻ͜ ͜ Ͱ ɼF (x, y) ͸ ແ ࣍ ‫ ݩ‬ύ ϥ ϝ ʔ λ x(2003)≡[6] C. Akerlof, University of Michigan preprint, UM-81-22m2ν ∗ /MWͱ y ≡ MW/s ͷؔ਺Ͱ

59 (1981); I.F. Ginzburg, G.L. Kotkin, V.G. Serbo˛˛˛˛y(1 − 4x + 4xy − 6x2 y)xy˛F (x, y) =log ˛˛x
1 − y + xy ˛ɹ+and V.I. Telnov, Nucl. Instrum. Methods 205 (1983)
47: I.F. Ginzburg, G.L. Kotkin, S.L. Panﬁl, V.G.

(1 − y)(1 − 2y + 6xy)y 2 (1 − x)(1 − y)−xx(1 − y + xy)(24)Serbo and V.I. Telnov, Nucl. Instrum. Methods 219(1984) 5[7] I. Watanabe et al., KEK Report 97-17 (1997)√Ͱ͋Δɻ s = 1 TeV(y = 0.0064) ͱͨ͠ͱ͖ɼF ͷ஋͸࣍[8] T. Kon, I. Ito and Y. Chikashige, Phys. Lett. B287ͷΑ͏ʹͳΔɿ

(1992) 277xɹ F (x, 0.0064)

50.14610

5.98 × 10−250
100[9] J. Peressutti, O.A. Sampayo and J.I. Aranda, Phys.
Rev. D64 (2001) 073007; S. Bray, J.S. Lee and A.
Piaftsis, Phys. Lett. B628 (2006) 250−34.71 × 10

[10] Particle Data Group, J. Phys. G33 (2006) 1−3

1.18 × 10[11] N. Cabbibo, L. Maiani and Y. Srivastava, Phys. Lett.
139B (1984) 459(23) Ͱ[12] K. Hagiwara, S. Komamiya and D. Zeppenfeld, Z.

(eg)2= 0.13 × 10−6 GeV−2 = 51 pb
216πMW

Phys. C29 (1985) 115[13] E. Takasugi, Prog. Theor. Phys. 98 (1997) 977

ͳͷͰɼͨͱ͑͹ x = 10(mν ∗ = 253 GeVɼҎԼಉ͡) ͷͱ͖ͷஅ໘ੵ͸ |ηL(R) |2 = 1 ͱͯ͠ɼ3.0λ2 λ2 pbɼx =50(566 GeV) ͷͱ͖͸ 0.24λ2 λ2 pb ͱͳΔɻλ, λ ͕͍ͣΕ΋ 1 ఔ౓ͷେ͖͞ͱ͢Δͱɼ͜Ε͸ిࢠͷภ‫ۃ‬Λࠨ‫ʹ͖ר‬

5─87─