Neutrino masses from Planck scale Takashi Toma Kyoto University - PowerPoint PPT Presentation

Neutrino masses from Planck scale Takashi Toma Kyoto University The 24th Regular Meeting of the New Higgs Working Group Osaka, Japan Based on arXiv:1802.09997 + work in progress In collaboration with Alejandro Ibarra and Patrick Strobl Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 1 / 20

Outline Outline 1 Introduction Neutrino mass generation mechanisms Seesaw mechanism 2 Generation of neutrino masses from Planck scale Analytic Results Numerical Results 3 Summary Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 2 / 20

Introduction Introduction Neutrinos are massive. (massless in the Standard Model) Neutrino oscillation data ⇒ O (0 . 1) eV NH IH sin 2 θ 12 0 . 306 +0 . 012 0 . 306 +0 . 012 − 0 . 012 − 0 . 012 sin 2 θ 23 0 . 441 +0 . 027 0 . 587 +0 . 020 − 0 . 021 − 0 . 024 sin 2 θ 13 0 . 02166 +0 . 00075 0 . 02179 +0 . 00076 − 0 . 00075 − 0 . 00076 7 . 50 +0 . 19 7 . 50 +0 . 19 21 [eV 2 ] − 0 . 17 × 10 − 5 − 0 . 17 × 10 − 5 ∆ m 2 − 0 . 040 × 10 − 3 − 2 . 514 +0 . 038 3 ℓ [eV 2 ] 2 . 524 +0 . 039 − 0 . 041 × 10 − 3 ∆ m 2 Esteban et al. JHEP (2017) Very small masses of neutrinos and large mixing angles. Mild hierarchy of two heaviest masses � 6 . ⇒ different mechanism of mass generation? Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 3 / 20

Introduction Introduction Neutrino mass generation mechanisms Type-I, II, III seesaw mechanism Minkowski, Yanagida, et al. (1977) Inverse seesaw, Linear seesaw mechanisms, radiative generation of neutrino masses etc A. Zee (1980), K.S. Babu (1988), M. Malinsky et al. PRL (2005) E. Ma, PRD (2006) etc   0 m 0  , Inverse : m 0 M Linear : Linear dependence of Yukawa  0 M µ ν ν Generated via gravitational anomaly ν G. Dvali and L. Funcke, PRD (2016) Analogy to chiral symmetry breaking ν of QCD ν → exp[ iγ 5 α ] ν ν ν Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 4 / 20

Introduction Seesaw mechanism Seesaw mechanism Seesaw mechanism (Type I, Type II, Type III...) In Type I seesaw (simplest), three heavy right-handed neutrinos N R are introduced. L = − φ † ℓ L y ν N R − 1 2 N c R MN R + h . c . → − ν L m D N R − 1 2 N c R MN R + h . c . m D = y ν � φ � N c Mass matrix ν L R m ν ≈ − m D M − 1 m T � � 0 m D D + · · · → m T D M ( if m D ≪ M ) Rough picture m ν ∼ y 2 ν � φ � 2 ∼ 0 . 1 eV M Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 5 / 20

Introduction Seesaw mechanism Seesaw mechanism Ex.1: y ν ∼ O (1) for M = 10 14 GeV Ex.2: y ν ∼ O (10 − 7 ) for M = 1 GeV Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 6 / 20

Introduction Seesaw mechanism Seesaw mechanism canonical seesaw T eV scale seesaw ν MSM Intermediate scale is necessary. Cannot directly correlate between neutrino mass scale and Planck scale because of perturbavity. Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 7 / 20

ν mass from Planck scale The Model The Model Add three right-handed neutrinos. L = 1 /N i − M ij i N j − ( Y ν ) ij ˜ 2 N c 2 N i ∂ HL i N j + H . c . Assumption: (almost) rank-1 mass matrix at Planck scale.     0 0 0 M 1 0 0  , M ≈ 0 0 0 M = 0 M 2 0    0 0 M 3 0 0 M 3 at Planck scale at Electroweak scale → reduce number of parameters Right-handed Majorana neutrino masses are expected to be generated via gravitational interactions. ← No flavor discrimination   1 1 1  → Mass eigenvalues = 0 , 0 , 3 M 0 M = M 0 1 1 1  1 1 1 Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 8 / 20

ν mass from Planck scale RGE Renormalization Group Equation for M M 1 and M 2 are generated by radiative effect. ⇒ Renormalization group equation (RGE) for M . All the diagrams Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 9 / 20

ν mass from Planck scale RGE Renormalization Group Equation for M At 1-loop, only one diagram contributes = dM 1 � T M + M �� �� β 1-loop Y † Y † � dt = ν Y ν ν Y ν M (4 π ) 2 At 2-loop, there are many contributions = dM 4 � T M β 2-loop Y † Y † � � � dt = ν Y ν ν Y ν + · · · M (4 π ) 4 Rank increasing diagram the other diagrams do not increase rank of M . Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 10 / 20

ν mass from Planck scale RGE Renormalization Group Equation for M Full beta function dM 1 4 �� � T M + M �� � T M Y † Y † Y † Y † � � � � dt = ν Y ν ν Y ν + ν Y ν ν Y ν (4 π ) 4 (4 π ) 2 � 17 1 − 1 ν Y ν − 1 g 2 Y + g 2 Y † 4 Y † ν Y ν Y † 4 Y † ν Y e Y † � � � � + ν Y ν e Y ν 2 (4 π ) 4 8 �� T − 3 � � � u Y u + 3 Y † Y † e Y e + Y † ν Y ν + 3 Y † Y † 2Tr d Y d ν Y ν M � 17 1 − 1 ν Y ν − 1 g 2 Y + g 2 Y † 4 Y † ν Y ν Y † 4 Y † ν Y e Y † � � � � + (4 π ) 4 M ν Y ν e Y ν 2 8 − 3 �� � � � u Y u + 3 Y † Y † e Y e + Y † ν Y ν + 3 Y † Y † 2Tr d Y d ν Y ν We include only M and Y ν . The other contributions do not increase rank of M . Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 11 / 20

ν mass from Planck scale RGE RGE for mass eigenvalues → dM 1 4 � T M + M � T M �� �� Y † Y † Y † Y † � � � � dt = ν Y ν ν Y ν + ν Y ν ν Y ν (4 π ) 2 (4 π ) 4 1 = P T M + MP + 4 P T MP (4 π ) 2 Y † where P ≡ ν Y ν The RGE can be reexpressed in terms of M i and U : � � dt = 2 M i Re ˆ ˆ dM i P 2 P ii + 4 � Re diag: k M k Re , ki � � ˆ U † dU � � ii = 4 � P 2 Im diag: − 2 M i Im k M k Im , ki dt � � ij = ( M i + M j ) Re ˆ P ki ˆ ˆ U † dU � � P ij + 4 � Re non-diag: ( M j − M i ) Re k M k Re P kj , dt � � ij = ( M i − M j ) Re ˆ P ki ˆ ˆ U † dU � � P ij + 4 � Im non-diag: − ( M j + M i ) Im k M k Im P kj , dt where ˆ P = U † PU and U T MU = diag ( M 1 , M 2 , M 3 ) . Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 12 / 20

ν mass from Planck scale Fixed points Fixed points for rank 1 mass matrix If M 1 = M 2 = 0 at Planck scale → rank M = 1 � P 2 � � P 2 � � P 2 � Im 0 , Im = 0 , Im = 0 , P 31 P 32 = 0 , 21 31 32 Namely, P 32 = | P 32 | e in 1 π/ 2 , P 21 = | P 21 | e in 2 π/ 2 , solution [1] P 31 = 0 , P 31 = | P 31 | e in 1 π/ 2 , P 21 = | P 21 | e in 2 π/ 2 , solution [2] P 32 = 0 , where n 1 , n 2 = 0 , 1 , 2 , 3 . We take [1] option (but no physical difference) should get | P 21 | e in 2 π/ 2   | P 11 | 0 ⇒ mild hierarchy � 6  , | P 21 | e − in 2 π/ 2 | P 32 | e in 1 π/ 2 P = | P 22 |  | P 32 | e − in 1 π/ 2 0 | P 33 | at Planck scale (initial condition) at low scale Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 13 / 20

ν mass from Planck scale Analytic solutions Analytic solutions dM dt = P T M + MP + 4 P T MP Iterative integration: P T M ( M P ) + M ( M P ) P + 4 P T M ( M P ) P � � → M ( µ ) ≈ M ( M P ) + � µ � × log M P Diagnalize M ( µ ) For Rank M = 1 case ( M = diag(0 , 0 , M 3 ) at Planck scale) M 3 ( µ ) ≈ M 3 , � M P � → 10 14 GeV for Y ν = O (1) M 2 ( µ ) ≈ − 4 M 3 P 2 32 log µ � M P � → 10 9 GeV for Y ν = O (1) M 1 ( µ ) ≈ 8 M 3 P 2 21 P 2 32 log 2 µ → M 1 ( µ ) is comparable to four-loop order Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 14 / 20

ν mass from Planck scale Analytic solutions Analytic solutions For Rank M = 3 case ( M = diag( M 1 , M 2 , M 3 ) at Planck scale) M 3 ( µ ) ≈ M 3 , � M P � P 2 31 + P 2 � � M 2 ( µ ) ≈ − 4 M 3 log , 32 µ �� 2 � P 2 31 − P 2 � P 31 P 32 ( P 11 − P 22 ) − P 21 � M P � 32 log 2 M 1 ( µ ) ≈ 8 M 3 P 2 31 + P 2 µ 32 assumption: M 1 ∼ M 2 ≪ M 3 If tree contribution M 1 , M 2 is larger than loop induced mass, M 1 ( µ ) ≈ M 2 P 2 31 + M 1 P 2 32 P 2 31 + P 2 32 M 1 = M 2 = 0 ⇒ Rank M = 1 case is recovered. Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 15 / 20

ν mass from Planck scale Numerical results Numerical analysis ( Rank M = 1 , Rank Y ν = 2 ) Parametrization Y ν = V L Y D V † R → Y D V † R 1 D V † (4 π ) 2 V R Y 2 P = R At Planck scale M = diag (0 , 0 , M P ) Y D = diag (0 , y 2 , 1) Run RGE � � 0 Y ν v Diagonalize 6 × 6 mass matix at low energy scale Y T ν v M Note: if Rank Y ν = 1 at Planck scale, N 1 mass is not generated. Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 16 / 20

ν mass from Planck scale Numerical results Numerical analysis ( Rank M = 1 , Rank Y ν = 2 ) Heaviest: M 3 | µ ∼ M P 2nd heaviest: M 2 | µ ∼ y 4 3 M P (4 π ) 4 × mixing 4 ∼ 10 14 GeV Lightest: m 1 | µ = 0 because Y D M − 1 Y T � � Rank = 2 D 2nd lightest state (red) is always ∼ 0 . 1 eV . The other two states: y 2 v ± M 1 | µ Pseudo Dirac state is constructed by ( ν 1 , N 1 ) if y 2 � 10 − 2 . cannot generate mild hierarchy Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 17 / 20