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

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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

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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

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Introduction

Neutrinos are massive. (massless in the Standard Model) Neutrino oscillation data ⇒ O(0.1) eV

NH IH sin2 θ12 0.306+0.012

−0.012

0.306+0.012

−0.012

sin2 θ23 0.441+0.027

−0.021

0.587+0.020

−0.024

sin2 θ13 0.02166+0.00075

−0.00075

0.02179+0.00076

−0.00076

∆m2

21 [eV2]

7.50+0.19

−0.17 × 10−5

7.50+0.19

−0.17 × 10−5

∆m2

3ℓ [eV2] 2.524+0.039 −0.040 × 10−3 −2.514+0.038 −0.041 × 10−3 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

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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

Inverse :   0 m 0 m 0 M 0 M µ   , Linear : Linear dependence of Yukawa Generated via gravitational anomaly

G. Dvali and L. Funcke, PRD (2016)

Analogy to chiral symmetry breaking

f QCD ν → exp[iγ5α]ν

ν ν ν ν ν ν

Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 4 / 20

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Introduction Seesaw mechanism

Seesaw mechanism

Seesaw mechanism (Type I, Type II, Type III...) In Type I seesaw (simplest), three heavy right-handed neutrinos NR are introduced. L = −φ†ℓLyνNR − 1 2N c

RMNR + h.c.

→ −νLmDNR − 1 2N c

RMNR + h.c.

mD = yνφ Mass matrix νL Nc

R

mD

mT

D M

→

mν ≈ −mDM−1mT

D + · · ·

(if mD ≪ M) Rough picture mν ∼ y2

νφ2

M ∼ 0.1 eV

Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 5 / 20

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Introduction Seesaw mechanism

Seesaw mechanism

Ex.1: yν ∼ O(1) for M = 1014 GeV Ex.2: yν ∼ O(10−7) for M = 1 GeV

Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 6 / 20

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Introduction Seesaw mechanism

Seesaw mechanism

νMSM canonical seesaw T eV scale seesaw

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

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ν mass from Planck scale The Model

The Model

Add three right-handed neutrinos. L = 1 2Ni∂ /Ni − Mij 2 Nc

i Nj − (Yν)ij ˜

HLiNj + H.c. Assumption: (almost) rank-1 mass matrix at Planck scale. M ≈   0 0 0 0 0 0 0 0 M3   , M =   M1 0 0 M2 0 0 M3   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 M = M0   1 1 1 1 1 1 1 1 1   → Mass eigenvalues = 0, 0, 3M0

Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 8 / 20

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ν mass from Planck scale RGE

Renormalization Group Equation for M

M1 and M2 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

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ν mass from Planck scale RGE

Renormalization Group Equation for M

At 1-loop, only one diagram contributes β1-loop

M

= dM dt = 1 (4π)2

Y †

ν Yν

T M + M

Y †

ν Yν

At 2-loop, there are many contributions

β2-loop

M

= dM dt = 4 (4π)4

Y †

ν Yν

T M

Y †

ν Yν

+ · · ·

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

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ν mass from Planck scale RGE

Renormalization Group Equation for M

Full beta function dM dt = 1 (4π)2

Y †

ν Yν

TM + M

Y †

ν Yν

+

4 (4π)4

Y †

ν Yν

TM

Y †

ν Yν

+

1 (4π)4 17 8

g2

Y + g2 2

Y †

ν Yν

− 1

4Y †

ν YνY † ν Yν − 1

4Y †

ν YeY † e Yν

−3 2Tr

Y †

e Ye + Y † ν Yν + 3Y † u Yu + 3Y † d Yd

Y †

ν Yν

T M + 1 (4π)4M 17 8

g2

Y + g2 2

Y †

ν Yν

− 1

4Y †

ν YνY † ν Yν − 1

4Y †

ν YeY † e Yν

−3 2Tr

Y †

e Ye + Y † ν Yν + 3Y † u Yu + 3Y † d Yd

Y †

ν Yν

We include only M and Yν.

The other contributions do not increase rank of M.

Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 11 / 20

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ν mass from Planck scale RGE

RGE for mass eigenvalues

→ dM dt = 1 (4π)2

Y †

ν Yν

T M + M

Y †

ν Yν

+

4 (4π)4

Y †

ν Yν

T M

Y †

ν Yν

= P TM + MP + 4P TMP

where P ≡ 1 (4π)2Y †

ν Yν

The RGE can be reexpressed in terms of Mi and U:

Re diag:

dMi dt = 2MiRe ˆ

Pii + 4

k MkRe

ˆ

P 2

ki

,

Im diag: −2MiIm

U †dU

dt

ii = 4

k MkIm

ˆ

P 2

ki

,

Re non-diag: (Mj − Mi) Re

U †dU

dt

ij = (Mi + Mj) Re ˆ

Pij + 4

k MkRe

ˆ

Pki ˆ Pkj

,

Im non-diag: − (Mj + Mi) Im

U †dU

dt

ij = (Mi − Mj) Re ˆ

Pij + 4

k MkIm

ˆ

Pki ˆ Pkj

,

where ˆ P = U †PU and U TMU = diag(M1, M2, M3).

Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 12 / 20

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ν mass from Planck scale Fixed points

Fixed points for rank 1 mass matrix

If M1 = M2 = 0 at Planck scale → rankM = 1 Im

P 2

21

0, Im
P 2

31

= 0, Im
P 2

32

= 0, P31P32 = 0,

Namely, solution [1] P31 = 0, P32 = |P32|ein1π/2, P21 = |P21|ein2π/2, solution [2] P32 = 0, P31 = |P31|ein1π/2, P21 = |P21|ein2π/2, where n1, n2 = 0, 1, 2, 3. We take [1] option (but no physical difference)

P =   |P11| |P21|ein2π/2 |P21|e−in2π/2 |P22| |P32|ein1π/2 |P32|e−in1π/2 |P33|   ,

should get ⇒ mild hierarchy 6 at Planck scale (initial condition) at low scale

Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 13 / 20

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ν mass from Planck scale Analytic solutions

Analytic solutions

Iterative integration: dM dt = P TM + MP + 4P TMP → M(µ) ≈ M(MP) +

P TM(MP) + M(MP)P + 4P TM(MP)P
× log

µ MP

Diagnalize M(µ)

For Rank M = 1 case (M = diag(0, 0, M3) at Planck scale) M3(µ) ≈ M3, M2(µ) ≈ −4M3P 2

32 log

MP µ

→ 1014 GeV for Yν = O(1)

M1(µ) ≈ 8M3P 2

21P 2 32 log2

MP µ

→ 109 GeV for Yν = O(1)

→ M1(µ) is comparable to four-loop order

Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 14 / 20

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ν mass from Planck scale Analytic solutions

Analytic solutions

For Rank M = 3 case (M = diag(M1, M2, M3) at Planck scale) M3(µ) ≈ M3, M2(µ) ≈ −4M3

P 2

31 + P 2 32

log

MP µ

,

M1(µ) ≈ 8M3

P31P32 (P11 − P22) − P21
P 2

31 − P 2 32

2 P 2

31 + P 2 32

log2 MP µ

assumption: M1 ∼ M2 ≪ M3

If tree contribution M1, M2 is larger than loop induced mass, M1(µ) ≈ M2P 2

31 + M1P 2 32

P 2

31 + P 2 32

M1 = M2 = 0 ⇒ Rank M = 1 case is recovered.

Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 15 / 20

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ν mass from Planck scale Numerical results

Numerical analysis (Rank M = 1, Rank Yν = 2)

Parametrization Yν = VLYDV †

R → YDV † R

P = 1 (4π)2VRY 2

DV † R

At Planck scale M = diag (0, 0, MP) YD = diag (0, y2, 1) Run RGE Diagonalize 6 × 6 mass matix

Yνv

Y T

ν v M

at low energy scale

Note: if Rank Yν = 1 at Planck scale, N1 mass is not generated.

Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 16 / 20

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ν mass from Planck scale Numerical results

Numerical analysis (Rank M = 1, Rank Yν = 2)

Heaviest: M3|µ ∼ MP 2nd heaviest: M2|µ ∼ y4

3MP

(4π)4 × mixing4 ∼ 1014 GeV Lightest: m1|µ = 0 because Rank

YDM−1Y T

D

= 2

2nd lightest state (red) is always ∼ 0.1 eV. The other two states: y2v ± M1|µ Pseudo Dirac state is constructed by (ν1, N1) if y2 10−2. cannot generate mild hierarchy

Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 17 / 20

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ν mass from Planck scale Numerical results

Numerical analysis (Rank M = 2, Rank Yν = 2)

At Planck scale M = diag (0, M2, MP) YD = diag (0, y2, 1) M2 = 109 GeV Heaviest, 2nd heaviest, lightest are same with Rank M = 1 case. Mild hierarchy of small neutrino masses can be obtained if 10−4 y2 10−2

Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 18 / 20

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ν mass from Planck scale Number of parameters

Number of Parameters

Parametrization: Yν = VLYDV †

R

⇒ Yν = YDV †

R

For rank Yν = 3 and rank M = 3 (usual case) eigenvalues y1,2,3, M1,2,3 and 6 mixing angles and 6 CP phases ⇒ yi, Mi and 3 mixings + 3 CP phases in Y †

ν Yν

For rank Yν = 2 and rank M = 2 (realistic case) Number of parameters are reduced. Relevant CP phases: 3 ⇒ 1 Predictive Leptogenesis is expected N1 → ℓH and N1 → ℓH∗ Asymmetry parameter ǫ ∝

i=1 Im(Y †

ν Yν) 2 1i

(Y †

ν Yν)11

N masses also dynamically change.

Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 19 / 20

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Summary

1 If right-handed neutrino masses are highly hierarchical at the Planck

scale, radiative corrections dominate right-handed neutrino masses at low scale.

2 Without intermediate scale, one of small neutrino masses is naturally

generated via seesaw mechanism.

3 Number of parameters are reduced and this framework leads predictive

phenomenology.

Future Works

1 More detailed analysis. 2 Application to leptogenesis, sterile neutrino dark matter? 3 Application to the other scenarios with hierarchical mass spectrum.

Takashi Toma (Kyoto University) The 24th Meeting of NHWG@Osaka 21st December 2018 20 / 20