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Leptogenesis in the minimal gauged U(1)%&'%( model and

the sign of the cosmological baryon asymmetry

High Energy Physics Theory Group

Department of Physics, The Univ. of Tokyo

Shih-Yen Tseng

K. Asai, K. Hamaguchi, N. Nagata, S.-Y. Tseng, arXiv:2005.01039

Introduction

• Baryon asymmetry of the Universe !" ≡ $%& ~ 8.7×10.//

• Baryogenesis

- heavy particle decays in GUT scenarios (Sakharov 1967)

- electroweak baryogenesis (Kuzmin et al. 1985)

- leptogenesis (Fukugita and Yanagida 1986)

- supersymmetric condensate baryogenesis (Affleck and Dine 1985)

- and many others…

• In this work, we discuss the generation of !" through leptogenesis

- in the framework of minimal gauged U(1)Lµ−Lτ model

1

Minimal gauged U(1)%&'%( model model

• Extending the gauge sector of SM

• Less constrained by the experiments

2

SU(3)+×SU(2)%×U(1).×U(1)%/'%0GSM

U(1)%1'%& U(1)%1'%( U(1)%&'%(

R. Foot, Mod.Phys.Lett. A6 (1991) 527-530X.-G. He et al, Phys. Rev. D 43, R22

K. Asai, K. Hamaguchi, N. Nagata, arXiv:1705.00419, Eur. Phys. J. C77 (2017) 763

3

• Charge assignment

R8

kXj JBMBK�H ;�m;2/ lURVL↵�L� KQ/2Hb

G2i mb }`bi /Bb+mbb i?2 TQbbB#H2 b+2M�`BQb Q7 i?2 KBMBK�H ;�m;2/ lURVL↵�L� KQ/2HbX �b

r2 /Bb+mbb2/ BM i?2 BMi`Q/m+iBQM- QM2 2ti`� b+�H�` }2H/ Bb BMi`Q/m+2/ BM Q`/2` iQ #`2�F

i?2 lURVL↵�L� ;�m;2 bvKK2i`vX h?2`2 �`2 i?`22 72�bB#H2 r�vb iQ �bbB;M i?2 [m�MimK

MmK#2`b iQ i?2 b+�H�` }2H/ r?B+? +�M vB2H/ � M2mi`BMQ K�bb K�i`Bt i?�i �HH i?`22 �+iBp2

M2mi`BMQb KBt rBi? 2�+? Qi?2`,

UBV �M alUkVL bBM;H2i � rBi? ?vT2`+?�`;2 Y = 0 �M/ i?2 lURVL↵�L� +?�`;2 +1-

UBBV �M alUkVL /Qm#H2i �1 rBi? ?vT2`+?�`;2 Y = +1/2 �M/ i?2 lURVL↵�L� +?�`;2 +1-

UBBBV �M alUkVL /Qm#H2i �1 rBi? ?vT2`+?�`;2 Y = +1/2 �M/ i?2 lURVL↵�L� +?�`;2 �1X

6Q` i?2 +�b2 UBV- Bi Bb �HbQ TQbbB#H2 iQ i�F2 i?2 lURVLµ�L⌧ +?�`;2 iQ #2 �1X >Qr2p2`- Bi

im`Mb Qmi iQ #2 i?2 +QKTH2t +QMDm;�i2 Q7 UBV �M/ ?2M+2 i?2v �`2 2[mBp�H2MiX AM i?2 b�K2

r�v- i?2 +?QB+2 Q7 Y = �1/2 BM UBBV �M/ UBBBV Bb i?2 +QKTH2t +QMDm;�i2 Q7 UBBBV �M/ UBBV-

`2bT2+iBp2HvX

q2 }`bi +QMbB/2` i?2 lURVLµ�L⌧ ;�m;2 i?2Q`vX h?2 `2H�iBQM #2ir22M Bi �M/ Qi?2` lURVL↵�L�

rBHH #2 /Bb+mbb2/ H�i2` r?2M r2 �M�Hvx2 i?2 bi`m+im`2 Q7 i?2 KQ/2HX h?2 +?�`;2 �bbB;MK2Mi

Q7 �HH i?2 }2H/b M22/2/ BM i?2 KQ/2H +QMbi`m+iBQM �`2 bmKK�`Bx2 BM h�#H2 kXkX

h?2 H27i@?�M/2/ M2mi`BMQ ⌫` ?�b i?2 b�K2 lURVLµ�L⌧ +?�`;2 Q7 i?2 +?�`;2/ H2TiQM +QmM@

i2`T�`i- `L,RX h?`22 `B;?i@?�M/2/ M2mi`BMQb Ne- Nµ �M/ N⌧ ?�p2 i?2 lURVLµ�L⌧ +?�`;2b

0- +1- �M/ �1- `2bT2+iBp2HvX �HH [m�`Fb �`2 M2mi`�H mM/2` i?2 lURVLµ�L⌧ ;�m;2 bvKK2i`vX

qBi? i?Bb +?QB+2 Q7 [m�MimK MmK#2`b- i?2 i?2Q`v Bb 7`22 7`QK ;�m;2 �MQK�HB2b- �b b?QrM

BM i?2 T`2pBQmb +?�Ti2`X h?2 aJU@HBF2V >B;;b }2H/ �2- �M alUkVL /Qm#H2i b+�H�` rBi?

?vT2`+?�`;2 +1/2 �M/ x2`Q lURVLµ�L⌧ +?�`;2- �+ib i?2 b�K2 �b i?2 >B;;b }2H/ H BM i?2

+�b2 UBVX

Le,µ,⌧ eR, µR, ⌧R Ne,µ,⌧ � H �1,2

U(1)Y �12 @R y y +

12 +

12

U(1)Lµ�L⌧ y-YR-@R y-YR-@R y-YR-@R YR y ±1, 0

SU(2) k R R R k k

h�#H2 kXk, U(1) +?�`;2b �M/ SU(2) `2T`2b2Mi�iBQMb 7Q` i?2 }2H/bX

!" − !$

an SM(-like) Higgs field, i.e. an SU(2)L doublet scalar with hypercharge +1/2 and theU(1)Lµ�L⌧ charge zero; this scalar field is responsible for giving masses to the SM fields.

As we discussed in the previous section, we further introduce one extra scalar field tobreak the U(1)Lµ�L⌧ gauge symmetry. There are only three possibilities for the quantumnumbers of the scalar field that can yield a neutrino mass matrix with which all of thethree active neutrinos mix with each other:

(i) An SU(2)L singlet with hypercharge Y = 0 and the U(1)Lµ�L⌧ charge +1.

(ii) An SU(2)L doublet with hypercharge Y = 1/2 and the U(1)Lµ�L⌧ charge +1.

(iii) An SU(2)L doublet with hypercharge Y = 1/2 and the U(1)Lµ�L⌧ charge �1.

For the case (i), one may also think of the U(1)Lµ�L⌧ charge �1 case. However, this caseis just the complex conjugate of the case (i) and thus these two are equivalent. Similarly,the choice of Y = �1/2 in the cases of (ii) and (iii) is the complex conjugate of the cases(iii) and (ii), respectively.

In what follows, we discuss each case separately, showing the Lagrangian terms relevantto the neutrino mass structure.

2.1 Singlet

The interaction terms in the leptonic sector of the case (i) are given by

�L =� yeecRLeH

†� yµµ

cRLµH

†� y⌧⌧

cRL⌧H

†

� �eNce (Le ·H)� �µN

cµ(Lµ ·H)� �⌧N

c⌧ (L⌧ ·H)

�1

2MeeN

ceN

ce �Mµ⌧N

cµN

c⌧ � �eµ�N

ceN

cµ � �e⌧�

⇤N

ceN

c⌧ + h.c. , (2)

where H and � denote the SM Higgs and the U(1)Lµ�L⌧ -breaking singlet scalar, respec-tively, and L↵ are the left-handed lepton doublets. The dots indicate the contractionof the SU(2)L indices. After the Higgs field H and the singlet scalar � acquire VEVshHi = v/

p2 and h�i,3 respectively, these interaction terms lead to neutrino mass terms

L(N)mass = �(⌫e, ⌫µ, ⌫⌧ )MD

0

BB@

Nce

Ncµ

Nc⌧

1

CCA�1

2(N c

e , Ncµ, N

c⌧ )MR

0

BB@

Nce

Ncµ

Nc⌧

1

CCA+ h.c. , (3)

with

MD =vp2

0

BB@

�e 0 0

0 �µ 0

0 0 �⌧

1

CCA , MR =

0

BB@

Mee �eµh�i �e⌧ h�i

�eµh�i 0 Mµ⌧

�e⌧ h�i Mµ⌧ 0

1

CCA , (4)

3We can always take these VEVs to be real by using the gauge transformations.

4

• Neutrino mass matrices

4








2.1 Singlet


�L =� yeecRLeH

†� yµµ

cRLµH

†� y⌧⌧

cRL⌧H

†



c⌧ (L⌧ ·H)

�1

2MeeN

ceN

ce �Mµ⌧N

cµN

c⌧ � �eµ�N

ceN

cµ � �e⌧�

⇤N

ceN

c⌧ + h.c. , (2)



L(N)mass = �(⌫e, ⌫µ, ⌫⌧ )MD

0

BB@

Nce

Ncµ

Nc⌧

1

CCA�1

2(N c

e , Ncµ, N

c⌧ )MR

0

BB@

Nce

Ncµ

Nc⌧

1

CCA+ h.c. , (3)

with

MD =vp2

0

BB@

�e 0 0

0 �µ 0

0 0 �⌧

1

CCA , MR =

0

BB@



�e⌧ h�i Mµ⌧ 0

1

CCA , (4)


4!"#$ = − !'#$ (!)!'#$ ≃

∗ ∗ ∗∗ 0 ∗∗ ∗ 0

Nertrino physics : after the SSB of -

two-zero minor structure

Nertrino physics : analysis of !"

5

!"#$ = − !'#$ (!)!'

#$ = *"(!",)#$*"(

PMNS matrix

./0 = cos 4/0 , 6/0 = sin 4/0 , 4/0: mixing angles,9: Dirac CP phase, :;, :<: Majorana phases

two-zero minor structure diag @$,@;,@<

Solve the equations corresponding to the zero entries in !"#$

Nertrino physics : results

6

Minimal gauged U(1)%&'%( model, normal mass ordering

safe

excluded

1)3)

1)

3)

1)3)

∑, -, < 0.146 eVat 95% C.L.

arXiv:1907.12598

take 345 = 51° in the following analysis

global-fitting group

NuFIT 4.1 (2019)withoutSK

atmospheric

data

Normal Ordering (best fit) Inverted Ordering (��2= 6.2)

bfp ±1� 3� range bfp ±1� 3� range

sin2 ✓12 0.310+0.013

�0.012 0.275 ! 0.350 0.310+0.013�0.012 0.275 ! 0.350

✓12/�

33.82+0.78�0.76 31.61 ! 36.27 33.82+0.78

�0.76 31.61 ! 36.27

sin2 ✓23 0.558+0.020

�0.033 0.427 ! 0.609 0.563+0.019�0.026 0.430 ! 0.612

✓23/�

48.3+1.1�1.9 40.8 ! 51.3 48.6+1.1

�1.5 41.0 ! 51.5

sin2 ✓13 0.02241+0.00066

�0.00065 0.02046 ! 0.02440 0.02261+0.00067�0.00064 0.02066 ! 0.02461

✓13/�

8.61+0.13�0.13 8.22 ! 8.99 8.65+0.13

�0.12 8.26 ! 9.02

�CP/�

222+38�28 141 ! 370 285

+24�26 205 ! 354

�m221

10�5 eV2 7.39+0.21

�0.20 6.79 ! 8.01 7.39+0.21�0.20 6.79 ! 8.01

�m23`

10�3 eV2 +2.523+0.032

�0.030 +2.432 ! +2.618 �2.509+0.032�0.030 �2.603 ! �2.416

withSK

atmospheric

data



sin2 ✓12 0.310+0.013

�0.012 0.275 ! 0.350 0.310+0.013�0.012 0.275 ! 0.350

✓12/�

33.82+0.78�0.76 31.61 ! 36.27 33.82+0.78

�0.75 31.61 ! 36.27

sin2 ✓23 0.563+0.018

�0.024 0.433 ! 0.609 0.565+0.017�0.022 0.436 ! 0.610

✓23/�

48.6+1.0�1.4 41.1 ! 51.3 48.8+1.0

�1.2 41.4 ! 51.3

sin2 ✓13 0.02237+0.00066

�0.00065 0.02044 ! 0.02435 0.02259+0.00065�0.00065 0.02064 ! 0.02457

✓13/�

8.60+0.13�0.13 8.22 ! 8.98 8.64+0.12

�0.13 8.26 ! 9.02

�CP/�

221+39�28 144 ! 357 282

+23�25 205 ! 348

�m221

10�5 eV2 7.39+0.21

�0.20 6.79 ! 8.01 7.39+0.21�0.20 6.79 ! 8.01

�m23`

10�3 eV2 +2.528+0.029

�0.031 +2.436 ! +2.618 �2.510+0.030�0.031 �2.601 ! �2.419

3,9, Δ-,94 from

Leptogenesis

7

• !" ≡ $"/& ~ 8.7×10.//

• Sakharov conditions- nonconserved baryon number- C and CP violations- departure from thermal equilibrium

• Nonthermal leptogenesis- 01s are produced nonthermally in inflaton decays

M. Fukugita, T. Yanagida, Phys. Lett. B174, 45 (1986)

we consider 23 < 5/

V. A. Kuzmin et al., Phys. Lett. 155B (1985) 36

sphaleron

T. Asaka et al., Phys. Lett. B 464 (1999) 12

!"

A.D. Sakharov, JETP Lett. 5 (1967) 24-27

U(1)9:.9; breaking field <

Nonthermal leptogenesis

8

• Take ! as inflaton

• Inflaton mass

(a) ✏1 (b) ✏2 (c) ✏3

Figure 2: The sign of the asymmetry parameters ✏i. In the shaded region, ✏i < 0. Here, wetake � > ⇡, as favored by the experiments. For� < ⇡, the signs of ✏i are flipped.

Figure 3: The magnitude of the asymmetry parameters ✏i as a function of � with fixed valuesof ✓.

the correctness of the prediction of baryon asymmetry. In order to successfully reproducethe baryon asymmetry in the leptogenesis scenario, nB/nL < 0 is required by the sphaleronprocesses [26]. This means that in the decays of right-handed neutrinos there must bemore anti-leptons than leptons generated so that the lepton number nL becomes negative asdesired.

4 Non-thermal leptogenesis in the minimal gaugedU(1)Lµ�L⌧ model

In this section, we investigate the non-thermal leptogenesis by the inflaton decay in theminimal gauged U(1)Lµ�L⌧ model. For minimality, we identify the U(1)Lµ�L⌧ breaking fieldas the inflaton, and consider the following Lagrangian.

L� =|@µ�|2

(1� |�|2/⇤2)2� (|�|2 � h�i2)2, (18)

7

where ⇤ > h�i. The real part of the � field, ' ⌘p2Re(�), plays the role of the inflaton.

The pole of the kinetic term guarantees that the e↵ective potential becomes very flat at largefield values [27, 28]. The inflaton potential can be canonically normalized with a change ofvariable,

L� =1

2(@ e')2 � ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

(19)

where

'p2⇤

⌘ tanh

✓e'p2⇤

◆. (20)

(| follow the discussion around eq.47-51 of 1612.05492 and rewrite them in the presentcontext.)

Using the given potential

V = ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

, (21)

we can obtain the e-folding number Ne

Ne =

Z e'N

e'f

✓V

M2PV

0

◆de'

=1

8M2P

(�⇤2 � h�i2

�cosh

✓2e'p2⇤

◆� 4 h�i2 ln

sinh

✓e'p2⇤

◆�)��e'N

e'f

, (22)

where e'N and e'f are the values of the field at the starting and ending points of the inflationrespectively. Ne is a number set to be around 50 - 60. e'f can be determined by the conditionsof the slow-roll parameters,

✏ ⌘ 1

2

✓V

0

V

◆2

M2P ⇠ 1 or |⌘| ⌘

��V

00

VM

2P

�� ⇠ 1 , (23)

and is in general much smaller than e'N . The natural logarithm in the expression of Ne canbe neglected when e' is very large. With these considerations, the initial field value can bewritten approximately as

e'N ' ⇤p2ln

✓16NeM

2P

⇤2 � h�i2

◆. (24)

8



L� =1

2(@ e')2 � ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

(19)

where

'p2⇤

⌘ tanh

✓e'p2⇤

◆. (20)



V = ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

, (21)


Ne =

Z e'N

e'f

✓V

M2PV

0

◆de'

=1

8M2P

(�⇤2 � h�i2

�cosh

✓2e'p2⇤

◆� 4 h�i2 ln

sinh

✓e'p2⇤

◆�)��e'N

e'f

, (22)


✏ ⌘ 1

2

✓V

0

V

◆2

M2P ⇠ 1 or |⌘| ⌘

��V

00

VM

2P

�� ⇠ 1 , (23)


e'N ' ⇤p2ln

✓16NeM

2P

⇤2 � h�i2

◆. (24)

8



L� =1

2(@ e')2 � ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

(19)

where

'p2⇤

⌘ tanh

✓e'p2⇤

◆. (20)



V = ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

, (21)


Ne =

Z e'N

e'f

✓V

M2PV

0

◆de'

=1

8M2P

(�⇤2 � h�i2

�cosh

✓2e'p2⇤

◆� 4 h�i2 ln

sinh

✓e'p2⇤

◆�)��e'N

e'f

, (22)


✏ ⌘ 1

2

✓V

0

V

◆2

M2P ⇠ 1 or |⌘| ⌘

��V

00

VM

2P

�� ⇠ 1 , (23)


e'N ' ⇤p2ln

✓16NeM

2P

⇤2 � h�i2

◆. (24)

8

CMB normalization

ensure the flatness of inflaton potential at large !

Baryon asymmetry

9

!" =$"% = $&

% ×$($&× $)$(

×$"$)

34 ,

-./&

2∑2 3245 6 → 8282 + ∑:;< 3: + 3< 45 6 → 8:8< ≡ 2×3>??

−2879

Nonthermal leptogenesis

!"## < 0

10

• & = 0.01• * = 10+, GeV• Λ = 10+. GeV• NuFIT inputs

depend on /

!"## = ∑1 !123 4 → 6161 + (1/2)∑<=> !< + !> 23 4 → 6<6>

Baryon asymmetry : result

11

It works!

!" ≃ 8.7×10*++cosmic string

Summary

• We have investigated the baryon asymmetry in the Universe- nonthermal leptogenesis with minimal gauged U(1)%&−%( model

• It is shown that )* can be reproduced in this framework

• Future prospect- thermal leptogenesis, etc

12

13

Backup

Introduction

• Light neutrino mass → adding RH neutrinos

! mass terms tightly restricted by the U(1)&'(&) symmetry

14

∗ 0 00 0 ∗0 ∗ 0

≃∗ 0 00 ∗ 00 0 ∗

∗ 0 00 0 ∗0 ∗ 0

∗ 0 00 ∗ 00 0 ∗

-. = −-1-2(3-1

4

this simple structure fails to explain the sizable neutrino mixing…

Seesaw mechanism

block-diagonal

P. Minkowski, Phys. Lett. B67 (1977) 421–428 T. Yanagida, Conf. Proc. C7902131 (1979) 95–99

Introduction

• Spontaneous breaking of U(1)%&'%( symmetry

introduce only one additional scalar field

)*'+ ≃∗ ∗ ∗∗ 0 ∗∗ ∗ 0

Two-zero structure leads into strong predictive power!

15

)* ≃∗ 0 ∗0 0 ∗∗ ∗ ∗

or

What are the contents?

• The SM

• Three RH neutrinos

• One extra scalar field16

k

L =

⌫L

eL

!eR Q =

uL

dL

!uR dR H

SU(3)C R R j j j RSU(2)L k R k R R kU(1)Y �

12 @R 1

623 �

13

12

h�#H2 RXR, GSM ⌘ SU(3)C⇥SU(2)L⇥U(1)Y `2T`2b2Mi�iBQM 7Q` 72`KBQMb �M/ i?2 >B;;b#QbQM BM i?2 aJX PMHv QM2 ;2M2`�iBQM Bb b?QrM 7Q` bBKTHB+BivX

h?2 >B;;b K2+?�MBbK Bb BMi`Q/m+2/ BM Q`/2` iQ ;2M2`�i2 i?2 K�bb2b Q7 H2TiQMb �M/ ;�m;2

#QbQMbX �M SU(2) b+�H�` /Qm#H2i }2H/ Bb �//2/ rBi? MQM@x2`Q p�+mmK 2tT2+i�iBQM p�Hm2

�M/ i?Bb #`2�Fb i?2 Q`B;BM�H ;�m;2 ;`QmT BMiQ SU(3)C ⇥ U(1)EM X �7i2` �TTHvBM; i?2

>B;;b K2+?�MBbK- i?2 K�bb2b Q7 i?2 +?�`;2/ 72`KBQMb- i?2 q± #QbQMb- i?2 w #QbQM �M/

i?2 >B;;b #QbQM �`2 ;2M2`�i2/- r?BH2 i?2 T?QiQMb +Q``2bTQM/BM; iQ i?2 `2K�BMBM; U(1)EM

bvKK2i`v �`2 K�bbH2bb �b 2tT2+i2/X AM i?2 2M/- r2 ?�p2 K�ii2`b +QMbBbiBM; Q7 H2TiQMb �M/

[m�`Fb �M/ BMi2`�+iBQM K2/B�iQ`b BM+Hm/BM; T?QiQMb- q± �M/ w #QbQMb �M/ QM2 M2mi`�H

>B;;b #QbQM `2M/2`BM; i?2 K�bb Q7 2H2K2Mi�`v T�`iB+H2bX

h?2 T?vbB+�H TB+im`2 Q7 i?2 aJ ?�b #22M T`Qp2/ i?�i Bi rQ`Fb r2HH mT iQ i?2 2M2`;v b+�H2

Q7 O(10) h2o �i i?2 G>*X >Qr2p2`- i?2`2 �`2 biBHH TH2Miv Q7 +?�HH2M;2b �M/ Kvbi2`B2b BM

T?vbB+b i?�i i?2 aJ +�MMQi 2tTH�BMX 6Q` BMbi�M+2-

Ç bQH�` �M/ �iKQbT?2`B+ M2mi`BMQ ~mt2b UM2mi`BMQ Qb+BHH�iBQMV-

Ç K�ii2`@/QKBM�i2/ mMBp2`b2 U#�`vQ;2M2bBbV-

Ç ;�H�tv `Qi�iBQM +m`p2b �M/ ;`�pBi�iBQM�H H2MbBM; Q7 ;�H�tv +Hmbi2`b U/�`F K�ii2`VX

"2bB/2b i?2 iQTB+b K2MiBQM2/ �#Qp2- i?2`2 �`2 Qi?2` ?BMib Q7 T?vbB+b #2vQM/ i?2 bi�M@

/�`/ KQ/2H U"aJV- 2X;X i?2 ?B2`�`+?v T`Q#H2K- i?2 BM~�iBQM Q7 i?2 mMBp2`b2- i?2 /�`F

2M2`;v- �tBQMb- 2i+X h?Bb i?2bBb Bb K�BMHv `2H�i2/ iQ i?2 M2mi`BMQ T?vbB+b �M/ i?2 K�ii2`@

�MiBK�ii2` �bvKK2i`v Q7 i?2 mMBp2`b2X AM i?Bb BMi`Q/m+iBQM- r2 rBHH }`bi #`B2~v /2b+`B#2

i?2 bi�M/�`/ TB+im`2 Q7 i?2 M2mi`BMQ Qb+BHH�iBQMb �M/ i?2 #�`vQM �bvKK2i`v Q7 i?2 mMB@

p2`b2X L2ti r2 rBHH /Bb+mbb i?2 KQiBp�iBQM #2?BM/ i?2 KQ/2Hb mb2/ BM i?Bb i?2bBb �M/ i?2

�MQK�HB2b BM Z6h r?B+? 7Q`Kb �M BKTQ`i�Mi BM;`2/B2Mi Q7 i?2 KQ/2H #mBH/BM; BM i?2 "aJ

T?vbB+bX

Input parameters

17

global-fitting groupNuFIT 4.1 (2019)

withoutSK

atmospheric

data



sin2 ✓12 0.310+0.013

�0.012 0.275 ! 0.350 0.310+0.013�0.012 0.275 ! 0.350

✓12/�

33.82+0.78�0.76 31.61 ! 36.27 33.82+0.78

�0.76 31.61 ! 36.27

sin2 ✓23 0.558+0.020

�0.033 0.427 ! 0.609 0.563+0.019�0.026 0.430 ! 0.612

✓23/�

48.3+1.1�1.9 40.8 ! 51.3 48.6+1.1

�1.5 41.0 ! 51.5

sin2 ✓13 0.02241+0.00066

�0.00065 0.02046 ! 0.02440 0.02261+0.00067�0.00064 0.02066 ! 0.02461

✓13/�

8.61+0.13�0.13 8.22 ! 8.99 8.65+0.13

�0.12 8.26 ! 9.02

�CP/�

222+38�28 141 ! 370 285

+24�26 205 ! 354

�m221

10�5 eV2 7.39+0.21

�0.20 6.79 ! 8.01 7.39+0.21�0.20 6.79 ! 8.01

�m23`

10�3 eV2 +2.523+0.032

�0.030 +2.432 ! +2.618 �2.509+0.032�0.030 �2.603 ! �2.416

withSK

atmospheric

data



sin2 ✓12 0.310+0.013

�0.012 0.275 ! 0.350 0.310+0.013�0.012 0.275 ! 0.350

✓12/�

33.82+0.78�0.76 31.61 ! 36.27 33.82+0.78

�0.75 31.61 ! 36.27

sin2 ✓23 0.563+0.018

�0.024 0.433 ! 0.609 0.565+0.017�0.022 0.436 ! 0.610

✓23/�

48.6+1.0�1.4 41.1 ! 51.3 48.8+1.0

�1.2 41.4 ! 51.3

sin2 ✓13 0.02237+0.00066

�0.00065 0.02044 ! 0.02435 0.02259+0.00065�0.00065 0.02064 ! 0.02457

✓13/�

8.60+0.13�0.13 8.22 ! 8.98 8.64+0.12

�0.13 8.26 ! 9.02

�CP/�

221+39�28 144 ! 357 282

+23�25 205 ! 348

�m221

10�5 eV2 7.39+0.21

�0.20 6.79 ! 8.01 7.39+0.21�0.20 6.79 ! 8.01

�m23`

10�3 eV2 +2.528+0.029

�0.031 +2.436 ! +2.618 �2.510+0.030�0.031 �2.601 ! �2.419

NuFIT 4.1 (2019)

withoutSK

atmospheric

data



sin2 ✓12 0.310+0.013

�0.012 0.275 ! 0.350 0.310+0.013�0.012 0.275 ! 0.350

✓12/�

33.82+0.78�0.76 31.61 ! 36.27 33.82+0.78

�0.76 31.61 ! 36.27

sin2 ✓23 0.558+0.020

�0.033 0.427 ! 0.609 0.563+0.019�0.026 0.430 ! 0.612

✓23/�

48.3+1.1�1.9 40.8 ! 51.3 48.6+1.1

�1.5 41.0 ! 51.5

sin2 ✓13 0.02241+0.00066

�0.00065 0.02046 ! 0.02440 0.02261+0.00067�0.00064 0.02066 ! 0.02461

✓13/�

8.61+0.13�0.13 8.22 ! 8.99 8.65+0.13

�0.12 8.26 ! 9.02

�CP/�

222+38�28 141 ! 370 285

+24�26 205 ! 354

�m221

10�5 eV2 7.39+0.21

�0.20 6.79 ! 8.01 7.39+0.21�0.20 6.79 ! 8.01

�m23`

10�3 eV2 +2.523+0.032

�0.030 +2.432 ! +2.618 �2.509+0.032�0.030 �2.603 ! �2.416

withSK

atmospheric

data



sin2 ✓12 0.310+0.013

�0.012 0.275 ! 0.350 0.310+0.013�0.012 0.275 ! 0.350

✓12/�

33.82+0.78�0.76 31.61 ! 36.27 33.82+0.78

�0.75 31.61 ! 36.27

sin2 ✓23 0.563+0.018

�0.024 0.433 ! 0.609 0.565+0.017�0.022 0.436 ! 0.610

✓23/�

48.6+1.0�1.4 41.1 ! 51.3 48.8+1.0

�1.2 41.4 ! 51.3

sin2 ✓13 0.02237+0.00066

�0.00065 0.02044 ! 0.02435 0.02259+0.00065�0.00065 0.02064 ! 0.02457

✓13/�

8.60+0.13�0.13 8.22 ! 8.98 8.64+0.12

�0.13 8.26 ! 9.02

�CP/�

221+39�28 144 ! 357 282

+23�25 205 ! 348

�m221

10�5 eV2 7.39+0.21

�0.20 6.79 ! 8.01 7.39+0.21�0.20 6.79 ! 8.01

�m23`

10�3 eV2 +2.528+0.029

�0.031 +2.436 ! +2.618 �2.510+0.030�0.031 �2.601 ! �2.419

Inflaton physics

18

The power spectrum of the curvature perturbation P⇣ is

P⇣ =V

3

12⇡2M6PV

02

=⇤6

96⇡2M6P

(1�

⇤coth

✓e'p2⇤

◆�2)4

sinh4

✓e'p2⇤

◆tanh2

✓e'p2⇤

◆

'

6⇡2

"Ne

�⇤2 � h�i2

�

MP⇤

#2

. (25)

The constraint on the power spectrum from the Planck experiment is P⇣ ' (2.1 ± 0.03) ⇥10�9 [13] (denoted as a parameter As in the paper). This leads to a prediction of the size ofthe perturbative inflaton coupling constant

' 2.9⇥ 10�2 ⇥✓50

Ne

◆2

⇥✓1014 GeV

⇤

◆2

⇥✓1� h�i2

⇤2

◆�2

. (26)

The Hubble parameter during the inflation is roughly

Hinf =

sV

3M2P

' 4.1⇥ 108 ⇥✓

⇤

1014 GeV

◆. (27)

The scalar spectral index ns and the tensor-to-scalar ratio r are

ns = 1� 6✏+ 2⌘

= 1�24M2

P⇤2 sech4

⇣e'p2⇤

⌘tan2

⇣e'p2⇤

⌘

h⇤2 tan2

⇣e'p2⇤

⌘� h�i2

i2

+4M2

P sech2⇣

e'p2⇤

⌘ h2�h�i2 � ⇤2

�+�7⇤2 � 3 h�i2

�sech2

⇣e'p2⇤

⌘� 5⇤2 sech4

⇣e'p2⇤

⌘i

h⇤2 tan2

⇣e'p2⇤

⌘� h�i2

i2

' 1� 3⇤2

2M2P

1

N2e

+

(� 2

Ne+

✓7⇤2 � 3 h�i2

4M2P

◆1

N2e

�"5⇤2

�⇤2 � h�i2

�

16M4P

#1

N3e

), (28)

r = 16✏ =

2

48MP⇤ sech2

⇣e'p2⇤

⌘tan

⇣e'p2⇤

⌘

⇤2 tan2⇣

e'p2⇤

⌘� h�i2

3

5

2

'✓

2⇤

MPNe

◆2

.

The Planck best-fit value of the scalar spectral index is ns = 0.9649 ± 0.0042 [13]. We seethat to the lowest order of Ne, ns ' 1�2/Ne with Ne ⇠ 50, it is compatible with the Planckresult and the prediction of the tensor-to-scalar ratio is vanishingly small and unable to bedetected.

9

!" ≃ 2.1×10)*

(| Identify the inflaton mass.) The inflaton mass squared can be extracted from the secondderivative of the potential. We have

m2� =

d2V

de'2

��e'=e'min

= 4 h�i2✓1� h�i2

⇤2

◆2

. (29)

The inflaton mass is then m� ' 3.4 ⇥ 1013 ⇥ (h�i /⇤) GeV and we see that it depends onthe ratio h�i /⇤.The rate of inflatons decaying into heavy right-handed neutrinos is given by (| need cross-check)

�(� ! NN) =X

i,j

|hij|2m�

8⇡

1�

✓Mi +Mj

m�

◆2�s

1� 2

✓M

2i +M

2j

m2�

◆+

✓M

2i �M

2j

m2�

◆2

,

(30)

where hij = h↵�⌦↵i⌦�j and is symmetric under the exchange of i and j. The repeatedindices are not summed here. Finally, let us calculate the baryon asymmetry of the presentscenario. The final baryon asymmetry is given by [29, 30, 31, 32]

YB ⌘ nB

s=

X n�

s· nN

n�· nL

nN· nB

nL(31)

=3

4

TR

m�·2

3X

i=1

✏iBr(� ! NiNi) +X

m 6=n

(✏m + ✏n)Br(� ! NmNn)

�·✓�28

79

◆,

(32)

where the sums are over all possible decay channels. In the first line of the equation, we haveseparate the baryon asymmetry into four factors. In order from left to right they indicatethe amount of inflaton depending on the inflaton mass and the reheating temperature, thebranching ratio of di↵erent decay channels, the CP asymmetry in each decay channel andthe lepton-baryon conversion factor from the sphaleron processes, respectively.(| Reheating temperature and the condition for the non-thermal leptogenesis.) In order torealize the non-thermal leptogenesis scenario, the heavy right-handed neutrinos are requiredto depart from thermal equilibrium. This condition is satisfied when the scale of the reheatingafter the inflation epoch is lower than the mass scale of the right-handed neutrinos, whichmeans the reheating temperature must be smaller than the mass of the lightest right-handedneutrino, TR < M1. The reheating temperature is given by

TR =

✓90

⇡2g⇤

◆ 14 p

��MP , (33)

where g⇤ is the relativistic degrees of freedom.Let us briefly summarize the parameters in the model. In the neutrino sector, we use thefollowing parameters:

10

aBKBH�`Hv i?2 TQr2` bT2+i`mK Q7 i?2 +m`p�im`2 T2ìm`#�iBQMb P⇣ Bb /2}M2/ �b

h⇣(~k) ⇣(~k0)i = (2⇡)3�(~k + ~k0)2⇡2

k3P⇣(k) . UjXRd8V

6`QK 1[bUjXk8V- UjXkdV �M/ UjXRe3V r2 Q#i�BM

P⇣(k) =H4

BM74⇡2(�)2

=V 3

12⇡2(V 0)2M6G

. UjXRdeV

AM 1[bX UjXRd9V �M/ UjXRdeV i?2 p�Hm2b Q7 i?2 BM~�iQM TQi2MiB�H V �M/ Bib /2`Bp�iBp2 V 0 �`22p�Hm�i2/ �i k = aHX h?2 +m`p�im`2 �M/ /2MbBiv T2ìm`#�iBQMb T`Q/m+2 �MBbQi`QTB2b Q7i?2 +QbKB+ KB+`Qr�p2 #�+F;`QmM/ U*J"VX h?2 Q#b2`p�iBQM Q7 *J" #v i?2 `2+2Mi SH�MFb�i2HHBi2 T`QpB/2 mb i?2 T`2+Bb2 p�Hm2 Q7 i?2 �KTHBim/2 Q7 i?2 +m`p�im`2 T2ìm`#�iBQMb-

[P⇣ ]1/2 = 4.93⇥ 10�5 , UjXRddV

�i k⇤ = 0.002 JT+�1Xh?2 bT2+i`mK BM/2t ns Q7 i?2 +m`p�im`2 T2ìm`#�iBQMb Bb /2}M2/ �b

ns � 1 =d lnP⇣(k)

d ln k, UjXRd3V

r?B+? K2�Mb P⇣ / kns�1X lbBM; 1[X UjXRdeV i?2 bT2+i`�H BM/2t Bb r`Bii2M �b

ns � 1 = 2d ln(V 3/2/V 0)

d ln k. UjXRdNV

h?2 `2H�iBQM #2ir22M i?2 +QbKQHQ;B+�H b+�H2 L = k�1 �M/ i?2 27QH/ N Bb ;Bp2M #v

N ⇠ 50 + ln

✓k�1

1000JT+

◆, UjXR3yV

r?B+? H2�/b iQd ln k = �dN = �

V

V 0M2G

d� . UjXR3RV

h?mb-

ns � 1 = �2M2G

d ln(V 3/2/V 0)

(V/V 0)d�

= �2M2G

V 0

V

✓3

2

V 0

V�

V 00

V 0

◆

= �3M2G

✓V 0

V

◆2

+ 2M2G

V 00

V. UjXR3kV

qBi? mb2 Q7 i?2 bHQr@`QHH T�`�K2i2`b ✏ �M/ ⌘ (UkXNy �M/ UkXNdV) i?2 bT2+i`�H BM/2t ns +�M#2 2tT`2bb2/ �b

ns = 1� 6✏+ 2⌘ . UjXR3jV

dd

aBKBH�`Hv i?2 TQr2` bT2+i`mK Q7 i?2 +m`p�im`2 T2ìm`#�iBQMb P⇣ Bb /2}M2/ �b

h⇣(~k) ⇣(~k0)i = (2⇡)3�(~k + ~k0)2⇡2

k3P⇣(k) . UjXRd8V

6`QK 1[bUjXk8V- UjXkdV �M/ UjXRe3V r2 Q#i�BM

P⇣(k) =H4

BM74⇡2(�)2

=V 3

12⇡2(V 0)2M6G

. UjXRdeV

AM 1[bX UjXRd9V �M/ UjXRdeV i?2 p�Hm2b Q7 i?2 BM~�iQM TQi2MiB�H V �M/ Bib /2`Bp�iBp2 V 0 �`22p�Hm�i2/ �i k = aHX h?2 +m`p�im`2 �M/ /2MbBiv T2ìm`#�iBQMb T`Q/m+2 �MBbQi`QTB2b Q7i?2 +QbKB+ KB+`Qr�p2 #�+F;`QmM/ U*J"VX h?2 Q#b2`p�iBQM Q7 *J" #v i?2 `2+2Mi SH�MFb�i2HHBi2 T`QpB/2 mb i?2 T`2+Bb2 p�Hm2 Q7 i?2 �KTHBim/2 Q7 i?2 +m`p�im`2 T2ìm`#�iBQMb-

[P⇣ ]1/2 = 4.93⇥ 10�5 , UjXRddV

�i k⇤ = 0.002 JT+�1Xh?2 bT2+i`mK BM/2t ns Q7 i?2 +m`p�im`2 T2ìm`#�iBQMb Bb /2}M2/ �b

ns � 1 =d lnP⇣(k)

d ln k, UjXRd3V

r?B+? K2�Mb P⇣ / kns�1X lbBM; 1[X UjXRdeV i?2 bT2+i`�H BM/2t Bb r`Bii2M �b

ns � 1 = 2d ln(V 3/2/V 0)

d ln k. UjXRdNV

h?2 `2H�iBQM #2ir22M i?2 +QbKQHQ;B+�H b+�H2 L = k�1 �M/ i?2 27QH/ N Bb ;Bp2M #v

N ⇠ 50 + ln

✓k�1

1000JT+

◆, UjXR3yV

r?B+? H2�/b iQd ln k = �dN = �

V

V 0M2G

d� . UjXR3RV

h?mb-

ns � 1 = �2M2G

d ln(V 3/2/V 0)

(V/V 0)d�

= �2M2G

V 0

V

✓3

2

V 0

V�

V 00

V 0

◆

= �3M2G

✓V 0

V

◆2

+ 2M2G

V 00

V. UjXR3kV

qBi? mb2 Q7 i?2 bHQr@`QHH T�`�K2i2`b ✏ �M/ ⌘ (UkXNy �M/ UkXNdV) i?2 bT2+i`�H BM/2t ns +�M#2 2tT`2bb2/ �b

ns = 1� 6✏+ 2⌘ . UjXR3jV

dd

jXjXjX *m`p�im`2 T2ìm`#�iBQMb T`Q/m+2/ #v BM~�iBQMG2i mb 2biBK�i2 i?2 +m`p�im`2 T2ìm`#�iBQMb T`Q/m+2/ #v i?2 BM~�iQM ~m+im�iBQMbX q2i�F2 i?2 ~�i bHB+2 �i ti r?2M i?2 `2H2p�Mi b+�H2 #2+QK2b bmT2`@?Q`BxQM Uk/a ' HV �M/ i?2mMB7Q`K /2MbBiv bHB+2 �i tf �7i2` `2?2�iBM;X 6`QK 1[bX UjXR8eV �M/ UkXRRyV i?2 +m`p�im`2T2ìm`#�iBQM �i tf Bb ;Bp2M #v

⇣ = �N =V

V 0MG

�� = �HBM7

�� . UjXRe3V

lbBM; 1[X UjXRedV r2 Q#i�BM

� =

8><

>:

23HBM7

��

�

��k=aH

U_.V

35HBM7

��

�

��k=aH

UJ.V. UjXReNV

PM+2 � Bb Q#i�BM2/- i?2 /2MbBiv T2ìm`#�iBQM Bb +�H+mH�i2/ mbBM; i?2 SQBbbQM 2[m�@iBQM UjXRy9V-

k2� = �4⇡Ga2⇢

� + 3H(1 + w)

V

k

�

= �4⇡Ga2⇢� , UjXRdyV

r?2`2 � Bb i?2 ;�m;2 BMp�`B�Mi /2MbBiv T2ìm`#�iBQMX lbBM; i?2 6`B2/K�MM 2[m�iBQM i?2/2MbBiv T2ìm`#�iBQM � BM i?2 K�ii2` /QKBM�i2/ 2`� Bb r`Bii2M �b

� = �2

3

k2

a2H2� = �

2

5

k2

a2H2HBM7

��

�UJ.V . UjXRdRV

h?2 TQr2` bT2+i`mK Q7 i?2 /2MbBiv T2ìm`#�iBQMb P(k) Bb i?2M ;Bp2M #v

h�(~k)�(~k0)i = (2⇡)3�(~k + ~k0)2⇡2

k3P(k) . UjXRdkV

lbBM; 1[bUjXk8V- UjXkdV �M/ UjXRdRV i?2 TQr2` bT2+i`mK Bb r`Bii2M �b

P(k) =4

25

✓k2

a2H2

◆2 H2BM7

(�)2P��(k) . UjXRdjV

h?mb- r2 Q#i�BM

P(k) =1

25⇡2

✓k2

a2H2

◆2 H4BM7

(�)2=

1

25⇡2

✓k2

a2H2

◆2 V 3

3(V 0)2M6G

. UjXRd9V

de

jXjXjX *m`p�im`2 T2ìm`#�iBQMb T`Q/m+2/ #v BM~�iBQMG2i mb 2biBK�i2 i?2 +m`p�im`2 T2ìm`#�iBQMb T`Q/m+2/ #v i?2 BM~�iQM ~m+im�iBQMbX q2i�F2 i?2 ~�i bHB+2 �i ti r?2M i?2 `2H2p�Mi b+�H2 #2+QK2b bmT2`@?Q`BxQM Uk/a ' HV �M/ i?2mMB7Q`K /2MbBiv bHB+2 �i tf �7i2` `2?2�iBM;X 6`QK 1[bX UjXR8eV �M/ UkXRRyV i?2 +m`p�im`2T2ìm`#�iBQM �i tf Bb ;Bp2M #v

⇣ = �N =V

V 0MG

�� = �HBM7

�� . UjXRe3V

lbBM; 1[X UjXRedV r2 Q#i�BM

� =

8><

>:

23HBM7

��

�

��k=aH

U_.V

35HBM7

��

�

��k=aH

UJ.V. UjXReNV

PM+2 � Bb Q#i�BM2/- i?2 /2MbBiv T2ìm`#�iBQM Bb +�H+mH�i2/ mbBM; i?2 SQBbbQM 2[m�@iBQM UjXRy9V-

k2� = �4⇡Ga2⇢

� + 3H(1 + w)

V

k

�

= �4⇡Ga2⇢� , UjXRdyV

r?2`2 � Bb i?2 ;�m;2 BMp�`B�Mi /2MbBiv T2ìm`#�iBQMX lbBM; i?2 6`B2/K�MM 2[m�iBQM i?2/2MbBiv T2ìm`#�iBQM � BM i?2 K�ii2` /QKBM�i2/ 2`� Bb r`Bii2M �b

� = �2

3

k2

a2H2� = �

2

5

k2

a2H2HBM7

��

�UJ.V . UjXRdRV

h?2 TQr2` bT2+i`mK Q7 i?2 /2MbBiv T2ìm`#�iBQMb P(k) Bb i?2M ;Bp2M #v

h�(~k)�(~k0)i = (2⇡)3�(~k + ~k0)2⇡2

k3P(k) . UjXRdkV

lbBM; 1[bUjXk8V- UjXkdV �M/ UjXRdRV i?2 TQr2` bT2+i`mK Bb r`Bii2M �b

P(k) =4

25

✓k2

a2H2

◆2 H2BM7

(�)2P��(k) . UjXRdjV

h?mb- r2 Q#i�BM

P(k) =1

25⇡2

✓k2

a2H2

◆2 H4BM7

(�)2=

1

25⇡2

✓k2

a2H2

◆2 V 3

3(V 0)2M6G

. UjXRd9V

de

Non-thermal leptogenesis

19

• Taking ! as inflaton



L� =1

2(@ e')2 � ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

(19)

where

'p2⇤

⌘ tanh

✓e'p2⇤

◆. (20)



V = ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

, (21)


Ne =

Z e'N

e'f

✓V

M2PV

0

◆de'

=1

8M2P

(�⇤2 � h�i2

�cosh

✓2e'p2⇤

◆� 4 h�i2 ln

sinh

✓e'p2⇤

◆�)��e'N

e'f

, (22)


✏ ⌘ 1

2

✓V

0

V

◆2

M2P ⇠ 1 or |⌘| ⌘

��V

00

VM

2P

�� ⇠ 1 , (23)


e'N ' ⇤p2ln

✓16NeM

2P

⇤2 � h�i2

◆. (24)

8



L� =1

2(@ e')2 � ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

(19)

where

'p2⇤

⌘ tanh

✓e'p2⇤

◆. (20)



V = ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

, (21)


Ne =

Z e'N

e'f

✓V

M2PV

0

◆de'

=1

8M2P

(�⇤2 � h�i2

�cosh

✓2e'p2⇤

◆� 4 h�i2 ln

sinh

✓e'p2⇤

◆�)��e'N

e'f

, (22)


✏ ⌘ 1

2

✓V

0

V

◆2

M2P ⇠ 1 or |⌘| ⌘

��V

00

VM

2P

�� ⇠ 1 , (23)


e'N ' ⇤p2ln

✓16NeM

2P

⇤2 � h�i2

◆. (24)

8

-20 -10 10 20

20

40

60

80 "( $%)

-20 -10 10 20

50

100

150

200

250 "( $%)

Inflaton potential

20

!( #$)

-3×1018 -2×1018 -1×1018 1×1018 2×1018 3×1018

5.0×1069

1.0×1070

1.5×1070

2.0×1070

2.5×1070

3.0×1070

3.5×1070

-1.5×1015 -1.0×1015 -5.0×1014 0 5.0×1014 1.0×1015 1.5×1015

5.0×1058

1.0×1059

1.5×1059

2.0×1059 !( #$)

Cosmic string

21

7.3 Cosmic Strings �221

gauge field, Am, and a complex Higgs field, fk, which carries U(1) charge e,

.0 = DAD"41,1 — 1 Ft,,,Pa' — A(40.(1) — cr212)2 , �(7.46)

where

Fill, = 49,,A, — ovAm, DIA, = 804) — ieilm4). � (7.47)

We immediately recognize that the theory is spontaneously broken as V(4)) is minimized for (01)2 = cr 2/2. The physical states after SSB are a scalar boson of mass M3 = 2Ao-2 and a massive vector boson of mass /14 = e2 o-2.

The complex field 4) can be written in terms of two real fields: 4) = (0 + i,01)/. If the VEV is chosen to lie in the real direction, then the potential becomes V(0) = (A/4)(02 — u2)2, where (OD = (0)/-ii. However, energetics do not determine the phase of (4)) since the vacuum energy depends only upon III; this fact follows from the U(1) gauge symmetry. Defining the phase of the VEV by ((b) = (o-14 exp(i0), we see that 0 = 8(50 can be position dependent. However, 4) must be single valued; i.e., the total change in 0, AO, around any closed path must be an integer multiple of 27r. Imagine a closed path with 6.0 = 27r. As the path is shrunk to a point (assuming no singularity is encountered), AO cannot change continuously from AO = 27r to AO = 0. There must, therefore, be one point contained within the path where the phase 8 is undefined, i.e., (4)) = 0. The region of false vacuum within the path is part of a tube of false vacuum. Such tubes of false vacuum must either be closed or infinite in length, otherwise it would be possible to deform the path around the tube and contract it to a point without encountering the tube of false vacuum. In most instances, these tubes of false vacuum have a characteristic transverse dimension far smaller than their length, so they can be treated as one-dimensional objects and are called "strings."'

The string solution to the equations of motion for the Lagrangian in (7.46) was first found by Nielsen and Olesen [161. At large distances from an infinite string in the z-direction, their solution is

(b �-- (o 1 Vi)exp(iN0), 15There should be no confusion between superstrings and the (cosmic) strings con-

sidered here.

222 �Phase Transitions

A, � [1n(-44./cr)] , � (7.48)

where 9 is the polar angle in the x-y plane, and N is the "winding number" of the string.16 Note this choice for A. and 4. results in a finite-energy solution, since at large distances from the string, Fo,, -> 0, and Do4. -) 0 (the 8o4. and Ao contributions to DA exactly cancel). There is no general solution to the coupled equations of motion for 4, and Ao. However, the Higgs field 4. and the 0-component of the gauge field for a string solution can be well approximated by

- exp( -r/ri)] exp(-iO),

[1 - exp(-r/r2 )]2 AB = �(7.49) er where r1 and r2 are proportional to cr-1 and depend upon the coupling constants e and ) [17]. The results of a variational calculation for �and AB are shown in Fig. 7.9.

The Hamiltonian (or energy) per unit length for an infinite string in the z-direction is given by

d H r dra = - f rdrd6 ,C dz

= �1°° 121rrdrde �2

82 + �- ie o �o �r 8B 2

17(4)) + B 2

1(7.50) 2

where 1:1 = x A is the "magnetic" field associated with the U(1) gauge field. The total magnetic flux within the string (or vortex, as it is often referred to) is N(27r/e).17 Because there are no closed-form solutions for A„ and 4,, A cannot be calculated in closed form, except for the case My = Ms

"In general we will be interested only in N = 1; for Mv/Ms < 1, it is energetically favorable for a string with N > 1 to decay into N strings with N = 1. Even if N > 1 strings are stable, the number of N > 1 strings formed in the phase transition is much less than the number of N = 1 strings [18].

17The strings discussed here are called gauge strings. There are also strings associated with the SSB of a global U(1) symmetry (i.e., e = 0). Such strings are called global (or axionic) strings. Other than the fact that their energy per length is logarithimically divergent (cut off by the distance to the next string), their cosmological properties are very similar to gauge strings.

! ≃ #$%

tan�=0.5tan�=1tan�=1.7tan�=15

0 10 ° 20 ° 30 ° 40 ° 50 ° 60 ° 70 ° 80 ° 90 °

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

�

-�×105

/�2

Leptogenesis

22

Baryon asymmetry and asymmetry parameter

Observed baryon asymmetry of the universe

Relation between baryon and lepton generated

Sign of asymmetry parameter is negative

22Kento Asai, Univ. of Tokyo The minimal gauged U(1)%& model and leptogenesis PPAP, 8/3/2018

4, Implications for leptogenesis

!"/!$ !%/!$

('(, '*, '+) ≡ ' cos 1, sin 1 cos4, sin 1 sin4

Leptogenesis

23

2

Leptogenesis

24

• !" ≡ $"/& ~ 8.7×10.//

• Decays of RH neutrinos generate lepton asymmetryNeutrino Physics Leptogenesis in the Universe

+ +NiNj

NjNi

Ni

H

`↵ `↵

H H

`↵

`, `

H

`

H

Figure 1.1: The CP asymmetry in type-I seesaw leptogenesis results from the interference between treeand 1-loop wave and vertex diagrams. For the 1-loop wave diagram, there is an additional contributionfrom L-conserving diagram to the CP asymmetry which vanishes when summing over lepton flavours.

1.2.3 Classical Boltzmann equations

We work in the one flavour regime and consider only the decays and inverse decays ofN1. If leptogenesisoccurs at T >

⇠ 1012GeV, then the charged lepton Yukawa interactions are out of equilibrium, and thisdefines the one flavour regime. The assumption that only the dynamics of N1 is relevant can berealized if for example the reheating temperature after inflation is TRH ⌧ M2,3 such that N2,3 arenot produced. In order to scale out the e↵ect of the expansion of the Universe, we will introduce theabundances, i.e. the ratios of the particle densities ni =

Rd3pfi to the entropy density s = 2⇡2

45 g⇤T 3:

Yi ⌘ni

s. (1.18)

The evolution of the N1 density and the lepton asymmetry Y�L = 2Y�` ⌘ 2(Y`�Y¯) 4 can be describedby the following classical Boltzmann equations (BE)[53]

dYN1

dz= �D1(YN1 � Y eq

N1), (1.19)

dY�L

dz= ✏1D1(YN1 � Y eq

N1)�W1Y�L, (1.20)

where z ⌘ M1/T and the decay and washout terms are respectively given by

D1(z) =�N1z

sH(M1)= K1z

K1(z)

K2(z), W1(z) =

1

2D1(z)

Y eq

N1(z)

Y eq

`

, (1.21)

with Kn the n-th order modified Bessel function of second kind. Y eq

N1and Y eq

`read:5

Y eq

N(z) =

45

2⇡4g⇤z2K2(z), Y eq

`=

15

4⇡2g⇤. (1.22)

From eq. (1.19) and eq. (1.20), the solution for Y�L can be written down as follows

Y�L(z) = Y�L(zi)e�R

z

zi

dz0W1(z0)

�

Zz

zi

dz0✏1(z0)dYN1

dz0e�

Rz

z0 dz00W1(z00) (1.23)

where zi is some initial temperature when N1 leptogenesis begins, and we have assumed that anypreexisting lepton asymmetry vanishes Y 0

�L(zi) = 0. Notice that ignoring thermal e↵ects, the CP

asymmetry is independent of the temperature ✏1(z) = ✏1 (c.f. eq. (1.17)).

4The factor of 2 comes from the SU(2)L degrees of freedoms.

5To write down a simple analytic expression for the equilibrium density of N1, we assume Maxwell-Boltzmann distri-

bution. However, follwing [54], the normalization factor Yeq

`is obtained from a Fermi-Dirac distribution.

8

0 = Γ 3 → 56 − Γ 3 → 569

Γ 3 → 56 + Γ 3 → 569

M. Fukugita, T. Yanagida, Phys. Lett. B174, 45 (1986)

Leptogenesis

25

• Baryon asymmetry in the universe

• Decays of RH neutrinos generate lepton asymmetry

S. Davidson et al. / Physics Reports 466 (2008) 105–177 123

• Including a fourth lepton generation allows thermal leptogenesis at T ⇠ TeV [172–174].• Consider the case that the number of heavy singlet neutrinos is larger than three. Their contribution to the effective

operator (`�)(`�) has no effect on the bounds on the flavoured asymmetries ✏↵↵ . In contrast, the contribution ofmanysinglets in weak washout can enhance the final baryon asymmetry, allowing thermal leptogenesis at Treheat that is afactor ⇠30 lower than in the case of three singlet neutrinos [175]. In the case where leptogenesis is unflavoured, extrasinglets weaken the upper bound on |✏| from 3M1(m3 � m1)/(16⇡v2) to 3M1m3/(16⇡v2) [175,176].

4. For degenerate light neutrinos,mmax > matm, so the individual flavour asymmetries can be larger. However, the efficiencyfactor is smaller, so formmax . eV (a conservative interpretation of the cosmological bound [177–179]) the lower boundonM1 is similar to Eq. (5.12) [62,180,181].

One can go beyond the effective theory and incorporate the N2,3 states as dynamical degrees of freedom. For a not-toodegenerate Ni spectrum,Mi � Mj � �D (the case ofMi � Mj ⇠ �D is discussed in Section 10.2.2), one obtains

✏↵↵ =1

(8⇡)

1[�Ñ�]11

X

j

Im�(�⇤

↵1)(�Ñ�)1j�↵j

g�xj�+

1(8⇡)

1[�Ñ�]11

X

j

Im�(�⇤

↵1)(�Ñ�)j1�↵j

11 � xj

, (5.13)

where

xj ⌘ M2j /M

21 ,

and, within the SM [56],

g(x) =px

11 � x

+ 1 � (1 + x) ln✓1 + xx

◆�x�1�! �

32px

�5

6x3/2+ · · · . (5.14)

In the MSSM, N1 decays to a slepton+Higgsino, as well as to lepton+Higgs. The sum of the asymmetries to leptons and tosleptons is about twice larger than the SM asymmetry [56]:

g(x) = �px✓

2x � 1

+ ln [1 + 1/x]◆

x�1�! �

3px

�3

2x3/2+ · · · . (5.15)

The first term of Eq. (5.13) corresponds to the first two loop diagrams in Fig. 5.1 while the second line [56,63,66,95]corresponds to the last diagram. This contribution violates the single lepton flavours but conserves the total lepton number,and thus it vanishes when summed over ↵:

✏ ⌘

X

↵

✏↵↵ =1

(8⇡)

1[�Ñ�]11

X

j

Im�[(�Ñ�)1j]

2 g�xj�

. (5.16)

As discussed in Sections 5.2 and 9.4.3, the upper bound on the flavoured CP asymmetries ✏↵↵ can be used to obtain alower bound on the reheat temperature. Here we discuss the upper bound on the total CP asymmetry ✏ =

P↵ ✏↵↵ of Eq.

(5.16) [73,74]:

|✏| <3

16⇡(mmax � mmin)M1

v2u

⇥ �(m,mmax,mmin) (5.17)

wheremmax (mmin) is the largest (smallest) light neutrino mass, and � ⇠ 1 can be found in [162].The interesting feature of the bound (5.17), is that it decreases for degenerate light neutrinos. This was used to obtain

an upper bound on the light neutrino mass scale from unflavoured leptogenesis [135,182] (discussed in Section 9.4.1), andexplains the interest in the form of the function � . However, themaximumCP asymmetry in a given flavour is unsuppressedfor degenerate light neutrinos [62], so flavoured leptogenesis can be tuned to work for degenerate light neutrinos, asdiscussed in Section 9.4.1.

5.3. Implications of CPT and unitarity for CP violation in decays

S-matrix unitarity and CPT invariance give useful constraints on CP violation (see e.g. [24,31,183]). This section is a briefreview of some relevant results for CP violation in decays. CP transforms a particle `↵ into its antiparticlewhichwe representas `↵ .

Useful relations, between matrix elements and their CP conjugates, can be obtained from the unitarity of the S-matrixS = 1 + iT:

1 = SSÑ = (1 + iT)(1 � iTÑ) (5.18)

which implies that iTab = iT⇤

ba � [TTÑ]ab. Assuming that the transition matrix T can be perturbatively expanded in somecoupling constant �, it follows from

|Tab|2� |Tba|

2= �2 Im

�[TTÑ]abT⇤

ba

+��[TTÑ]ab

��2 (5.19)

S. Davidson et al. / Physics Reports 466 (2008) 105–177 123

• Including a fourth lepton generation allows thermal leptogenesis at T ⇠ TeV [172–174].• Consider the case that the number of heavy singlet neutrinos is larger than three. Their contribution to the effective

operator (`�)(`�) has no effect on the bounds on the flavoured asymmetries ✏↵↵ . In contrast, the contribution ofmanysinglets in weak washout can enhance the final baryon asymmetry, allowing thermal leptogenesis at Treheat that is afactor ⇠30 lower than in the case of three singlet neutrinos [175]. In the case where leptogenesis is unflavoured, extrasinglets weaken the upper bound on |✏| from 3M1(m3 � m1)/(16⇡v2) to 3M1m3/(16⇡v2) [175,176].

4. For degenerate light neutrinos,mmax > matm, so the individual flavour asymmetries can be larger. However, the efficiencyfactor is smaller, so formmax . eV (a conservative interpretation of the cosmological bound [177–179]) the lower boundonM1 is similar to Eq. (5.12) [62,180,181].

One can go beyond the effective theory and incorporate the N2,3 states as dynamical degrees of freedom. For a not-toodegenerate Ni spectrum,Mi � Mj � �D (the case ofMi � Mj ⇠ �D is discussed in Section 10.2.2), one obtains

✏↵↵ =1

(8⇡)

1[�Ñ�]11

X

j

Im�(�⇤

↵1)(�Ñ�)1j�↵j

g�xj�+

1(8⇡)

1[�Ñ�]11

X

j

Im�(�⇤

↵1)(�Ñ�)j1�↵j

11 � xj

, (5.13)

where

xj ⌘ M2j /M

21 ,

and, within the SM [56],

g(x) =px

11 � x

+ 1 � (1 + x) ln✓1 + xx

◆�x�1�! �

32px

�5

6x3/2+ · · · . (5.14)

In the MSSM, N1 decays to a slepton+Higgsino, as well as to lepton+Higgs. The sum of the asymmetries to leptons and tosleptons is about twice larger than the SM asymmetry [56]:

g(x) = �px✓

2x � 1

+ ln [1 + 1/x]◆

x�1�! �

3px

�3

2x3/2+ · · · . (5.15)

The first term of Eq. (5.13) corresponds to the first two loop diagrams in Fig. 5.1 while the second line [56,63,66,95]corresponds to the last diagram. This contribution violates the single lepton flavours but conserves the total lepton number,and thus it vanishes when summed over ↵:

✏ ⌘

X

↵

✏↵↵ =1

(8⇡)

1[�Ñ�]11

X

j

Im�[(�Ñ�)1j]

2 g�xj�

. (5.16)

As discussed in Sections 5.2 and 9.4.3, the upper bound on the flavoured CP asymmetries ✏↵↵ can be used to obtain alower bound on the reheat temperature. Here we discuss the upper bound on the total CP asymmetry ✏ =

P↵ ✏↵↵ of Eq.

(5.16) [73,74]:

|✏| <3

16⇡(mmax � mmin)M1

v2u

⇥ �(m,mmax,mmin) (5.17)

wheremmax (mmin) is the largest (smallest) light neutrino mass, and � ⇠ 1 can be found in [162].The interesting feature of the bound (5.17), is that it decreases for degenerate light neutrinos. This was used to obtain

an upper bound on the light neutrino mass scale from unflavoured leptogenesis [135,182] (discussed in Section 9.4.1), andexplains the interest in the form of the function � . However, themaximumCP asymmetry in a given flavour is unsuppressedfor degenerate light neutrinos [62], so flavoured leptogenesis can be tuned to work for degenerate light neutrinos, asdiscussed in Section 9.4.1.

5.3. Implications of CPT and unitarity for CP violation in decays

S-matrix unitarity and CPT invariance give useful constraints on CP violation (see e.g. [24,31,183]). This section is a briefreview of some relevant results for CP violation in decays. CP transforms a particle `↵ into its antiparticlewhichwe representas `↵ .

Useful relations, between matrix elements and their CP conjugates, can be obtained from the unitarity of the S-matrixS = 1 + iT:

1 = SSÑ = (1 + iT)(1 � iTÑ) (5.18)

which implies that iTab = iT⇤

ba � [TTÑ]ab. Assuming that the transition matrix T can be perturbatively expanded in somecoupling constant �, it follows from

|Tab|2� |Tba|

2= �2 Im

�[TTÑ]abT⇤

ba

+��[TTÑ]ab

��2 (5.19)

!" =$"%$&%

': Yukawa coupling

Leptogenesis

26

• Baryon asymmetry in the universe

• Decays of RH neutrinos generate lepton asymmetry

• Sphaleron processes

Baryon asymmetry and asymmetry parameter

Observed baryon asymmetry of the universe

Relation between baryon and lepton generated

Sign of asymmetry parameter is negative

22Kento Asai, Univ. of Tokyo The minimal gauged U(1)%& model and leptogenesis PPAP, 8/3/2018

4, Implications for leptogenesis

! = 2879× ! − )

Leptogenesis 3

Sphaleron bL

bL

tL

sL

sL

cL

dL

dL

uL

!e

!µ

!!

Figure 1. One of the 12-fermion processes which are in thermal equilibrium in the

high-temperature phase of the standard model.

topological charge, i.e. a transition from one vacuum to another one, corresponds to a

change in baryon and lepton numbers,

!B = !L = nf!NCS , (2)

where nf is the number of generations of quarks and leptons, i.e. nf = 3 in the SM.

Note that, although both baryon and lepton number are violated, the linear combination

B ! L is still conserved at the quantum level.

At low temperatures, when the electroweak symmetry is broken, the di"erent

vacua are separated by a potential barrier, whose height is determined by the vacuum

expectation value (VEV) of the Higgs field, v = "!#, i.e. the scale of electroweaksymmetry breaking. Hence, processes changing the topological charge are tunneling

processes whose rate is unobservably small, due to the smallness of the electroweak

coupling constant. In the low temperature regime being probed at accelerator

experiments, B and L are therefore conserved to a very good approximation, in accord

with experimental observations (cf. [10]).

When the standard model particles form a heat bath of temperature T the situationchanges. At high temperatures, T $ TEW % 100 GeV, the Higgs VEV ‘evaporates’,

leading to a restoration of the electroweak symmetry and the disappearance of the

potential barriers separating the di"erent vacua. B and L violating transitions are then

no longer suppressed [11].

The rate at which these processes occur is related to the free energy of field

configurations which carry topological charge. In the electroweak part of the SM theseso-called sphaleron processes lead to an e"ective interaction of all left-handed fermions

[7] (cf. Fig. 1),

OB+L =!

i

(qLiqLiqLilLi) , (3)

V. A. Kuzmin et al., Phys. Lett. 155B (1985) 36

Right-handed neutrino sector

27

• RH neutrino mass matrix

• Parameters

ed

h?Bb Bb i?2 umF�r� +QmTHBM; K�i`Bt i?�i r2 ?�p2 iQ BMb2ì BMiQ i?2 7Q`KmH� Q7 i?2 *S

�bvKK2i`vX h?2 `2H�i2/ T�ì Q7 i?2 H�;`�M;B�M BM i?2 K�bb 2B;2M#�bBb Q7 `B;?i@?�M/2/

M2mi`BMQb Bb

�L = �

X

↵=e,µ,⌧

3X

i=1

�↵i

�L↵ ·H

�N

c

i �1

2

3X

i=1

M/B�;Ri

Nc

i Nc

i + ?X+X , U9X3dV

r?2`2

MR = ⌦⇤M

/B�;R

⌦†, N

c

i =

X

↵

⌦†i↵N

c

↵ , �↵i =

X

�

�↵�⌦�i , ⌦†⌦ = I . U9X33V

LQr i?2 T`2T�`�iBQMb �`2 /QM2X h?2 7QHHQrBM; `2bmHib �`2 Q#i�BM2/ #v +?QQbBM; i?2 #2bi@

}i p�Hm2b Q7 i?2 BMTmi T�`�K2i2`b- ✓12- ✓13- �m221 �M/ �m

231- T2`7Q`K2/ #v i?2 Lm6Ah

;HQ#�H@}i ;`QmT (Rk) HBbi2/ BM h�#H2 RXk �M/ i?2 KBtBM; �M;H2 ✓23 Bb b2i iQ #2 52� BM Q`/2`

iQ #2 +QMbBbi2Mi rBi? i?2 +QMbi`�BMi 7`QK i?2 SH�M+F 2tT2`BK2MiX �b r2 ?�p2 �H`2�/v

6B;m`2 9X8, h?2 H2TiQM �bvKK2i`v �b � 7mM+iBQM Q7 i?2 T�`�K2i2` � �i bQK2 }t2/p�Hm2b Q7 ✓X

/Bb+mbb2/ �i i?2 2M/ Q7 b2+iBQM 9XR- i?2 #�`vQM �bvKK2i`v +QMM2+ib rBi? i?2 B � L

[m�MimK MmK#2`X AM i?2 H2TiQ;2M2bBb i?2 H2TiQM �bvKK2i`v Bb }`bi ;2M2`�i2/ �M/ #v 1[X

U9XjjV r2 b22 i?�i BM i?Bb +�b2 i?2 bB;M Q7 #�`vQM �M/ H2TiQM �bvKK2i`B2b �`2 QTTQbBi2X

h?Bb K2�Mb i?�i 7Q` � bm++2bb7mH H2TiQ;2M2bBb- �M �KQmMi Q7 M2;�iBp2 H2TiQM MmK#2` Kmbi

#2 T`Q/m+2/ BM i?2 *S@pBQH�iBM; /2+�vb Q7 i?2 `B;?i@?�M/2/ M2mi`BMQb (e9)X >2M+2- r2

M22/ iQ +?2+F r?2i?2` � M2;�iBp2 H2TiQM MmK#2` +�M #2 `2�HBx2/ BM i?2 7`�K2rQ`F Q7 i?2

KBMBK�H ;�m;2/ lURVLµ�L⌧ KQ/2HX 6B;X 9X8 b?Qrb i?2 H2TiQM �bvKK2i`v �b � 7mM+iBQM

ed

h?Bb Bb i?2 umF�r� +QmTHBM; K�i`Bt i?�i r2 ?�p2 iQ BMb2ì BMiQ i?2 7Q`KmH� Q7 i?2 *S

�bvKK2i`vX h?2 `2H�i2/ T�ì Q7 i?2 H�;`�M;B�M BM i?2 K�bb 2B;2M#�bBb Q7 `B;?i@?�M/2/

M2mi`BMQb Bb

�L = �

X

↵=e,µ,⌧

3X

i=1

�↵i

�L↵ ·H

�N

c

i �1

2

3X

i=1

M/B�;Ri

Nc

i Nc

i + ?X+X , U9X3dV

r?2`2

MR = ⌦⇤M

/B�;R

⌦†, N

c

i =

X

↵

⌦†i↵N

c

↵ , �↵i =

X

�

�↵�⌦�i , ⌦†⌦ = I . U9X33V

LQr i?2 T`2T�`�iBQMb �`2 /QM2X h?2 7QHHQrBM; `2bmHib �`2 Q#i�BM2/ #v +?QQbBM; i?2 #2bi@

}i p�Hm2b Q7 i?2 BMTmi T�`�K2i2`b- ✓12- ✓13- �m221 �M/ �m

231- T2`7Q`K2/ #v i?2 Lm6Ah

;HQ#�H@}i ;`QmT (Rk) HBbi2/ BM h�#H2 RXk �M/ i?2 KBtBM; �M;H2 ✓23 Bb b2i iQ #2 52� BM Q`/2`

iQ #2 +QMbBbi2Mi rBi? i?2 +QMbi`�BMi 7`QK i?2 SH�M+F 2tT2`BK2MiX �b r2 ?�p2 �H`2�/v

6B;m`2 9X8, h?2 H2TiQM �bvKK2i`v �b � 7mM+iBQM Q7 i?2 T�`�K2i2` � �i bQK2 }t2/p�Hm2b Q7 ✓X

/Bb+mbb2/ �i i?2 2M/ Q7 b2+iBQM 9XR- i?2 #�`vQM �bvKK2i`v +QMM2+ib rBi? i?2 B � L

[m�MimK MmK#2`X AM i?2 H2TiQ;2M2bBb i?2 H2TiQM �bvKK2i`v Bb }`bi ;2M2`�i2/ �M/ #v 1[X

U9XjjV r2 b22 i?�i BM i?Bb +�b2 i?2 bB;M Q7 #�`vQM �M/ H2TiQM �bvKK2i`B2b �`2 QTTQbBi2X

h?Bb K2�Mb i?�i 7Q` � bm++2bb7mH H2TiQ;2M2bBb- �M �KQmMi Q7 M2;�iBp2 H2TiQM MmK#2` Kmbi

#2 T`Q/m+2/ BM i?2 *S@pBQH�iBM; /2+�vb Q7 i?2 `B;?i@?�M/2/ M2mi`BMQb (e9)X >2M+2- r2

M22/ iQ +?2+F r?2i?2` � M2;�iBp2 H2TiQM MmK#2` +�M #2 `2�HBx2/ BM i?2 7`�K2rQ`F Q7 i?2

KBMBK�H ;�m;2/ lURVLµ�L⌧ KQ/2HX 6B;X 9X8 b?Qrb i?2 H2TiQM �bvKK2i`v �b � 7mM+iBQM

!" = −!%&!'()!%

(+,, +., +/) ≡ + cos 5, sin 5 cos8, sin 5 sin8


28

• As a concrete example, we first consider this simple scenario

• Pairs of RH neutrinos generated from inflaton decays

• NR out-of-equilibrium decay : < "#

• Taking $ as inflaton








2.1 Singlet


�L =� yeecRLeH

†� yµµ

cRLµH

†� y⌧⌧

cRL⌧H

†



c⌧ (L⌧ ·H)

�1

2MeeN

ceN

ce �Mµ⌧N

cµN

c⌧ � �eµ�N

ceN

cµ � �e⌧�

⇤N

ceN

c⌧ + h.c. , (2)



L(N)mass = �(⌫e, ⌫µ, ⌫⌧ )MD

0

BB@

Nce

Ncµ

Nc⌧

1

CCA�1

2(N c

e , Ncµ, N

c⌧ )MR

0

BB@

Nce

Ncµ

Nc⌧

1

CCA+ h.c. , (3)

with

MD =vp2

0

BB@

�e 0 0

0 �µ 0

0 0 �⌧

1

CCA , MR =

0

BB@



�e⌧ h�i Mµ⌧ 0

1

CCA , (4)


4


29




(a) ✏1 (b) ✏2 (c) ✏3

Figure 2: The sign of the asymmetry parameters ✏i. In the shaded region, ✏i < 0. Here, wetake � > ⇡, as favored by the experiments. For� < ⇡, the signs of ✏i are flipped.

Figure 3: The magnitude of the asymmetry parameters ✏i as a function of � with fixed valuesof ✓.

the correctness of the prediction of baryon asymmetry. In order to successfully reproducethe baryon asymmetry in the leptogenesis scenario, nB/nL < 0 is required by the sphaleronprocesses [26]. This means that in the decays of right-handed neutrinos there must bemore anti-leptons than leptons generated so that the lepton number nL becomes negative asdesired.

4 Non-thermal leptogenesis in the minimal gaugedU(1)Lµ�L⌧ model

In this section, we investigate the non-thermal leptogenesis by the inflaton decay in theminimal gauged U(1)Lµ�L⌧ model. For minimality, we identify the U(1)Lµ�L⌧ breaking fieldas the inflaton, and consider the following Lagrangian.

L� =|@µ�|2

(1� |�|2/⇤2)2� (|�|2 � h�i2)2, (18)

7


30






L� =1

2(@ e')2 � ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

(19)

where

'p2⇤

⌘ tanh

✓e'p2⇤

◆. (20)



V = ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

, (21)


Ne =

Z e'N

e'f

✓V

M2PV

0

◆de'

=1

8M2P

(�⇤2 � h�i2

�cosh

✓2e'p2⇤

◆� 4 h�i2 ln

sinh

✓e'p2⇤

◆�)��e'N

e'f

, (22)


✏ ⌘ 1

2

✓V

0

V

◆2

M2P ⇠ 1 or |⌘| ⌘

��V

00

VM

2P

�� ⇠ 1 , (23)


e'N ' ⇤p2ln

✓16NeM

2P

⇤2 � h�i2

◆. (24)

8



L� =1

2(@ e')2 � ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

(19)

where

'p2⇤

⌘ tanh

✓e'p2⇤

◆. (20)



V = ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

, (21)


Ne =

Z e'N

e'f

✓V

M2PV

0

◆de'

=1

8M2P

(�⇤2 � h�i2

�cosh

✓2e'p2⇤

◆� 4 h�i2 ln

sinh

✓e'p2⇤

◆�)��e'N

e'f

, (22)


✏ ⌘ 1

2

✓V

0

V

◆2

M2P ⇠ 1 or |⌘| ⌘

��V

00

VM

2P

�� ⇠ 1 , (23)


e'N ' ⇤p2ln

✓16NeM

2P

⇤2 � h�i2

◆. (24)

8



L� =1

2(@ e')2 � ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

(19)

where

'p2⇤

⌘ tanh

✓e'p2⇤

◆. (20)



V = ⇤4

"tanh2

✓e'p2⇤

◆�✓h�i⇤

◆2#2

, (21)


Ne =

Z e'N

e'f

✓V

M2PV

0

◆de'

=1

8M2P

(�⇤2 � h�i2

�cosh

✓2e'p2⇤

◆� 4 h�i2 ln

sinh

✓e'p2⇤

◆�)��e'N

e'f

, (22)


✏ ⌘ 1

2

✓V

0

V

◆2

M2P ⇠ 1 or |⌘| ⌘

��V

00

VM

2P

�� ⇠ 1 , (23)


e'N ' ⇤p2ln

✓16NeM

2P

⇤2 � h�i2

◆. (24)

8


31




from CMB normalization


32

• Inflaton decay

leptogenesisin the minimal gauged u(1) '% model and the ... › ugap › ws › am2020 ›...

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