Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology

1

16 3 Neutrinos beyond the

S

tandard Mode

l

this interesting case Eq.

(

3.158

)

can be used to understand the relationshi

p

betwee

n

φ

and two Ma

j

orana phases ρ

a

n

d

σ

at the

q

uasi-ﬁxed

p

oint

.

(

d

)

The running behaviors of ρ an

d

σ

are re

l

ative

l

ymi

ld

in comparison

w

i

t

h

t

h

at of

φ

. A remarkable feature of ˙

ρ

and ˙

σ

is that the

y

will vanish, i

f

both ρ an

d

σ are initially vanishing. This observation indicates tha

t

ρ an

d

σ

c

annot simultaneously be generated fro

m

φ

v

ia the one-loop RGEs.

(

e

)

The unphysical phase

s

φ

μ

a

n

d

φ

τ

o

n

l

y

h

ave re

l

ative

l

ymi

ld

runnin

g

eﬀects, while the running behavior of

φ

e

m

a

y

be violent for suﬃcientl

y

small

θ

l

.

A quasi-ﬁxed point of

φ

e

i

sa

l

so expecte

d

in t

h

e

l

imit

θ

l

→

0

an

d

un

d

e

r

the circumstance given by Eq.

(

3.157

)

or Eq.

(

3.158

).

3

.

6

.

3

Evolution of Dirac Neutrino Mass Parameter

s

The Yukawa coupling matrix of Dirac neutrino

s

Y

ν

YY

o

beys the RGE given in

E

q.

(

3.138

)

. Because neutrino masses are extremely smaller than charged

-

l

epton masses, the

(

Y

ν

YY

Y

†

ν

YY

)

term can be safely neglected from Eq.

(

3.138

)

.We

a

re then left with

16

π

2

d

Y

ν

YY

d

t

=



α

ν

+

C

l

ν

(

Y

l

Y

†

l

YY

)



Y

ν

YY

,

16

π

2

d

Y

†

ν

YY

d

t

=

Y

†

ν

Y



α

ν

+

C

l

ν

(

Y

l

YY

Y

†

l

YY

)



.

(

3.159

)

D

eﬁning the Hermitian quantit

y

κ



≡

Y

ν

YY

Y

†

ν

YY

,

we obtain

16

π

2

d

κ



d

t

=

2

α

ν

κ



+

C

κ



(

Y

l

YY

Y

†

l

YY

)

κ



+

κ



(

Y

l

YY

Y

†

l

YY

)



,

(

3.160

)

whe

r

e

α

ν

≈

−

0

.

4

5

g

2

1

−

2

.

25

g

2

+3

y

2

t

(

SM

)

or

α

ν

≈−

0

.

6

g

2

1

−

3

g

2

+3

y

2

t

(

MSSM

),

a

nd

C

κ

=

C

l

ν

. This RGE is rather similar to the one given in Eq.

(

3.130

)

for

Majorana neutrinos. Without loss o

fg

enerality, we choose the

ﬂ

avor basi

s

w

h

e

r

e

Y

l

Y

=

D

l

≡

D

ia

g

{

y

e

,y

μ

,

y

τ

}

ho

l

ds

.In

t

hi

sbas

i

s

κ



c

an be dia

g

onalized

b

y

the unitar

y

trans

f

ormatio

n

V

†

κ



V

=

3

κ



≡

Dia

g

{

y

2

1

,y

2

,y

2

3

}

w

i

th

V

b

ein

g

the MNS matrix an

d

y

i

b

eing the eigenvalues o

f

Y

ν

YY

.

Let us denot

e

V

≡

PU

P



,

i

n

wh

i

ch

P

=D

iag

{

e

i

φ

e

,

e

i

φ

μ

,

e

i

φ

τ

}

,

P



≡

D

iag

{

e

i

ρ

,

e

i

σ

,

1

}

,

an

d

U

is the

C

KM-

l

ike matrix containin

g

three neutrino mixin

g

an

g

les and one

C

P-violatin

g

p

hase. Note that

P

d

oes not take

p

art in the RGEs because it can be remove

d

f

rom Eq.

(

3.160

)

in the chosen basis. The phases in

P



a

re a

l

so unp

h

ysica

l

,

b

u

t

they have their own R

G

Es. Followin

g

astrai

g

ht

f

orward procedure analo

g

ou

s

to the one from Eq.

(

3.143

)

to Eq.

(

3.151

)

in Section 3.6.2, one may easily

ﬁg

ure out the constraint equations

f

or 3

κ



,

U

,

P



a

n

dthe

ir

de

ri

vat

i

ves

.In

t

h

e

τ

-dominance approximation

(

i.e.,

D

2

l

≈

D

ia

g

{

0

,

0

,y

2

τ

}

)

, we obtain

(

Xing

,

2

006

)

˙

y

i

=

y

i

1

6

π

2



α

ν

+

C

κ

y

2

τ

|

U

τi

U

|

2



,

(

3.161

)

a

nd

3

.6 Running Behaviors o

f

Neutrino Mass Parameters 117



α



U

∗

α

U

1

˙

U

α

U

2



+i



ρU

˙

∗

e

U

1

U

e

UU

2

+˙

σU

˙

∗

μ

U

1

U

μ

U

2



=

−

C

κ

y

2

τ

1

6

π

2

ξ

1

2

U

∗

τ

UU

1

U

τ

UU

2

,



α



U

∗

α

U

1

˙

U

α

U

3



+i



ρU

˙

∗

e

U

1

U

e

UU

3

+˙

σU

˙

∗

μ

U

1

U

μ

U

3



=

−

C

κ

y

2

τ

16

π

2

ξ

13

U

∗

τ

U

1

U

τ

U

3

,



α



U

∗

α

U

2

˙

U

α

U

3



+i



ρU

˙

∗

e

U

2

U

e

UU

3

+˙

σU

˙

∗

μ

U

2

U

μ

U

3



=

−

C

κ

y

2

τ

16

π

2

ξ

23

U

∗

τ

UU

2

U

τ

UU

3

,

(

3.162

)

whe

r

e

α

r

u

n

sove

r

e

,

μ

a

n

d

τ

,an

d

ξ

ij

≡



y

2

i

+

y

2

j



/



y

2

i

−

y

2

j



w

i

th

i



=



j

.

G

iven the FX form of

U

in Eq.

(

3.152

)

, the Yukawa coupling eigenvalue

s

of three Dirac neutrinos satisfy the following one-loop RGEs

:

˙

y

1

=

y

1

16

π

2



α

ν

+

C

κ

y

2

τ

s

2

ν

s

2



,

˙

y

2

=

y

2

16

π

2



α

ν

+

C

κ

y

2

τ

c

2

ν

s

2



,

˙

y

3

=

y

3

16

π

2



α

ν

+

2

C

κ

y

2

τ

c

2



.

(

3.163

)

The RGEs of three neutrino mixing angles and one

(

physical

)

CP-violatin

g

p

h

ase are

g

iven

by

˙

θ

l

=+

C

κ

y

2

τ

16

π

2

c

ν

s

ν

cc

φ

(

ξ

13

−

ξ

2

3

)

,

˙

θ

ν

=+

C

κ

y

2

τ

16

π

2

c

ν

s

ν



s

2

ξ

1

2

+

c

2

(

ξ

1

3

−

ξ

23

)



,

˙

θ

=+

C

κ

y

2

τ

16

π

2

cs



s

2

ν

ξ

1

3

+

c

2

ν

ξ

2

3



,

˙

φ

=

−

C

κ

y

2

τ

16

π

2



c

2

l

−

s

2

l



c

−

1

l

s

−

1

l

c

ν

s

ν

cs

φ

(

ξ

1

3

−

ξ

23

)

,

(

3.164

)

where

c

φ

≡

cos

φ

an

d

s

φ

≡

sin

φ

. The RGEs of unphysica

l

ρ

a

n

d

σ

r

ea

d

˙

ρ

=

−

C

κ

y

2

τ

16

π

2

c

l

s

−

1

l

c

ν

s

ν

c

s

φ

(

ξ

13

−

ξ

23

)

,

˙

σ =

+

C

κ

y

2

τ

16

π

2

c

−

1

l

s

l

c

ν

s

ν

c

s

φ

(

ξ

13

−

ξ

23

)

.

(

3.165

)

I

t

i

sobv

i

ous that

˙

φ

=˙

ρ

+˙

σ

holds. These

RG

Es are much more concise than

those obtained in the standard parametrization o

f

U

(

Lindner

et al.

,

2005

).

Some qualitative discussions about the main features of Eqs.

(

3.163

)

,

(

3.164

)

a

nd

(

3.165

)

are in order

.

(

a

)

Like the Majorana case, the running behaviors of three Dirac neutrino

masses

m

i

(

or equivalentl

y

i

)

are nearly identical and determined by

α

ν

,

u

n

less tan

β

i

ssu

ﬃ

ciently lar

g

eintheM

SS

M. It is also worth mentionin

g

that ˙

y

i

or ˙

m

i

(

for

i

=1

,

2

,

3)

are independent of both the CP-violating phas

e

φ and the smallest mixing angle θ

l

i

n the chosen

p

arametrization of

U

.

1

18 3 Neutrinos beyond the

S

tandard Mode

l

(

b

)

˙

θ

ν

c

onsists o

f

a term proportional t

o

ξ

1

2

=

−

(

m

2

1

+

m

2

)

/

Δ

m

2

1

.

H

e

n

ce

θ

ν

i

sin

g

eneral more sensitive to radiative corrections than

θ

l

a

nd θ

,

whose

d

erivatives

d

epen

d

on ξ

13

=

−

(

m

2

1

+

m

2

3

)

/

Δ

m

2

3

1

an

d

ξ

2

3

=

−

(

m

2

+

m

2

3

)

/

Δ

m

2

3

2

.

G

ive

n

θ

ν

a

n

d

θ

at an ener

g

y sca

l

e,

θ

l

c

an

b

era

d

iative

l

y

g

enerate

d

at anot

h

er

ener

g

y scale. In this case, however, it is impossible to simultaneously

g

enerate

the CP-violating phase

φ

.

T

h

e reason is simp

l

yt

h

at

φ

c

an a

l

ways

b

e rotate

d

a

wa

y

w

h

e

n

θ

l

is vanis

h

in

g

,an

d

t

h

e proportiona

l

ity re

l

ations

h

ip

b

etwee

n

˙

φ

a

nd si

n

φ

forbid

s

φ

t

o be generated even whe

n

θ

l

b

ecomes non-vanishing

.

(

c

)

Diﬀerent from the Majorana case, there is no nontrivia

l

q

uasi-ﬁxed

p

oin

t

i

ntheR

G

Eo

f

φ

fo

rDir

ac

n

eut

rin

os

.I

f

˙

φ

i

s required to keep

ﬁ

nite when

θ

l

a

pp

roaches zero, then

φ

i

tself must a

pp

roach zero o

r

π

.

On the other hand

,

˙

θ

l

∝

cos

φ

implies that the evolution o

f

θ

l

h

as a turnin

g

point c

h

aracterize

db

y

φ

=

π

/

2(

i.e.,

˙

θ

l

ﬂips its sign at this point

)

. Hence two interesting conclusion

s

c

an be drawn:

ﬁ

rst

,

θ

l

ca

nn

eve

r

c

r

oss

z

e

r

o

i

f

θ

l



=0andsin



φ



= 0 hold at a



c

ertain energy scale; second, CP will always be a good symmetry if

θ

l

=0o

r

s

i

n

φ

=

0

h

o

ld

s at a certain energy sca

l

e

.

(

d

)

The running behavior of

ρ

is quite similar to that o

f

φ

,simp

ly b

ecause

˙

φ

=˙

ρ



1

−

t

an

2

θ

l



holds. In addition, ˙

σ

=

−

˙

ρ

t

a

n

2

θ

l

im

p

lies that σ

o

nly gets

so

m

e

mi

ld

r

ad

i

at

i

ve co

rr

ect

i

o

n

s.

As deﬁned in Eq.

(

3.116

)

, the Jarlskog parameter

J

i

s a rephasin

g

invari-

a

nt of the MN

S

matri

x

V

a

nd measures the strength of CP and T violation i

n

eutrino oscillations.

O

ne ha

s

J

=

c

l

s

l

c

ν

s

ν

cs

2

s

i

n

φ

i

nt

h

e FX parametriza

-

tion, no matter whether three neutrinos are Dirac or Ma

j

orana particles. We

obtain the one-loo

p

RG

E

˙

J

D

JJ

=

C

κ

y

2

τ

16

π

2

J

D

JJ



c

2

ν

−

s

2

ν



s

2

ξ

12

+



c

2

−

s

2

ν

s

2



ξ

13

+



c

2

−

c

2

ν

s

2



ξ

2

3



(

3.166

)

f

or Dirac neutrinos.

So

J

D

JJ

= 0 will be a stable result independent o

f

th

e

renormalization scales,

p

rovided

θ

l

o

r

si

n

φ

i

nitially vanishes at a

g

iven scale

.

In com

p

arison, we have

˙

J

M

JJ

=

C

κ

y

2

τ

16

π

2

"

J

M

JJ





c

2

ν

−

s

2

ν



s

2



ζ

−

1

12

c

2

(

σ

−

ρ

)

+

ζ

12

s

2

(

σ

−

ρ

)



+



c

2

−

s

2

ν

s

2



ζ

−

1

13

c

2

ρ

+

ζ

1

3

s

2

ρ



+



c

2

−

c

2

ν

s

2



ζ

−

1

23

c

2

σ

+

ζ

2

3

s

2

σ



+

c

ν

s

ν

cs

2



C

1

2

3

ζ

1

2

+

C

13

3

ζ

1

3

+

C

23

CC

3

ζ

2

3

#

(

3.167

)

f

or Ma

j

orana neutrinos, where

C

12

=

c

l

s

l

s

2

c

φ

c

(

σ

−

ρ

)

s

(

σ

−

ρ

)

,

C

1

3

=



c

l

s

l

c

φ



s

2

ν

−

c

2

ν

c

2



+



c

2

l

−

s

2

l



c

ν

s

ν

c



c

ρ

s

ρ

,

C

23

C

=



c

l

s

l

c

φ



c

2

ν

−

s

2

ν

c

2



−



c

2

l

−

s

2

l



c

ν

s

ν

c



c

σ

s

σ

.

(

3.168

)

Thi

s

r

esu

l

ts

h

ows t

h

at

J

M

JJ

c

an be radiatively

g

enerated

f

rom two nontrivia

l

Majorana phase

s

ρ

a

n

d

σ

, even if it is initially vanishing at a given scale

.

R

e

f

erences 11

9

Ta

k

in

g

ρ

=

σ

=0,wearr

i

ve a

t

C

12

=

C

1

3

=

C

23

C

= 0 as well as ˙

ρ

=˙

σ

=0.

But it is impossible to obtain the equalit

y

˙

J

M

J

(

ρ

=

σ

=

0

)=

˙

J

D

J

,

because

ζ

−

1

12

=

ξ

12

,

ζ

−

1

3

=

ξ

13

an

d

ζ

−

1

23

=

ξ

2

3

(

or equivalently

m

1

m

2

=

m

1

m

3

=

m

2

m

3

=

0

)

cannot simultaneously hold. This observation demonstrates tha

t

the runnin

g

behavior of

J

M

JJ

is essentiall

y

diﬀerent from that of

J

D

JJ

e

ven if

one switches oﬀ two Majorana phases

(

Xing and Zhang, 2007

).

S

o

f

ar a lot o

f

numerical exercises have been done to illustrate the runnin

g

behaviors of neutrino mass and mixing parameters from the seesaw scale t

o

the electroweak scale, or vice versa. Some particular attention has been pai

d

to the possibility o

fg

eneratin

g

the smallest neutrino mixin

g

an

g

le

θ

13

(

or

θ

l

)

via the eﬀect of its RGE evolution

,

and to the radiative corrections to a fe

w

typica

l

neutrino mixin

g

patterns inc

l

u

d

in

g

t

h

e“tri-

b

imaxima

l”

an

d

“

d

emo-

c

ratic” patterns

(

see, e.g., Antusc

h

et al.

,

2002c;

A

ntusch and Ratz, 2002; Me

i

a

nd Xing, 2004; Plentinger and Rodejohann, 2005; Mei and Xing, 2005; Lu

o

a

n

d

Xin

g

, 2006a; Li

n

et al

.

,

2010

)

. To achieve appreciable running eﬀects,

h

owever, one usually has to make at least one o

f

the

f

ollowin

g

assumptions

:

(

a

)

three neutrino masses are nearly degenerate to enhance the magnitudes of

ζ

−

1

ij

ζ

o

r

ξ

i

j

,

(

b

)

the value of tan

β

i

ssu

ﬃ

ciently lar

g

e to enhance the ma

g

nitud

e

of

y

2

τ

=

m

2

τ

(

1+tan

2

β

)

/

v

2

i

n the MSSM case, and

(

c

)

signiﬁcant threshold

eﬀects of heavy particles above the seesaw scale of Majorana neutrinos. W

e

do

n

ot elabo

r

ate those

n

u

m

e

ri

cal

r

esults

in

th

i

sbook.

Let us make two ﬁnal remarks. First, the chosen ﬂavor basis

(

i.e.,

Y

l

YY

=

D

l

)

i

s stable against radiative corrections. This point can be seen from Eq.

(

3.131

)

or from Eq.

(

3.138

)

in the neglect of the

(

Y

ν

YY

Y

†

ν

YY

)

term:

16

π

2

d

D

l

d

t

=



α

l

+

C

l

C

D

2

l



D

l

,

(

3.169

)

whose le

f

t- and ri

g

ht-hand sides are both dia

g

onal. Henc

e

D

l

i

tsel

f

keeps di

-

a

gonal between the seesaw scale and the electroweak scale, although its three

eigenvalues get modiﬁed at diﬀerent energy scales. I

f

D

l

were unsta

bl

e against

q

uantum corrections, the R

G

Es o

f

neutrino mass and mixin

g

parameters ob

-

tained above would no more make sense. Second, the running behaviors o

f

l

eptons are more or less related to those of quarks because their RGEs ar

e

essentia

ll

yentan

gl

e

d

wit

h

eac

h

ot

h

er. Even t

h

ou

gh

t

h

ree

l

i

gh

t neutrinos im-

pose little inﬂuence on the evolution of nine charged fermions, the opposite

i

s not true. That is why one has to use a

f

ull set o

f

R

G

Es o

f

lepton Yukawa

c

ouplin

g

s, quark Yukawa couplin

g

sand

g

au

g

e couplin

g

stodoacomplete

n

umerical analysis of the evolution of neutrino masses, ﬂavor mixing angles

a

nd

C

P-violatin

g

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g

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4

S

eesaw Mechanisms o

f

Neutrino Masses

The ori

g

in of nonzero but tiny neutrino masses has been a bi

g

puzzle in par-

ticle physics. Among various scenarios of neutrino mass generation which ar

e

s

urvivin

g

today, the seesaw mechanisms are most popular

f

or model buildin

g.

In this chapter we shall ﬁrst outline a few possible ways to

g

o beyond th

e

s

tandard model

(

SM

)

and generate ﬁnite neutrino masses, and then focus o

n

s

evera

l

typica

l

seesaw mec

h

anisms an

dd

iscuss

h

ow to

g

et a

b

a

l

ance

b

etween

their theoretical naturalness and experimental testabilit

y

. Possible collider

s

ignatures of TeV seesaw mechanisms and their consequences on non-unitary

n

eutrino mixin

g

and

C

P violation will also be discussed.

4.1 How to

G

enerate Tiny Neutrino Masse

s

N

eutrinos are massless in the

S

M,

j

ust because the structure o

f

the

S

M itsel

f

i

stoosim

p

le to accommodate massive neutrinos.

•

T

wo fundamentals of the

S

Marethe

SU

(

2

)

L

×

U

(

1

)

Y

g

auge symmetr

y

and the Lorentz invariance. Both o

f

them are mandatory to

g

uarantee

t

hat the SM is a consistent quantum ﬁeld theor

y.

•

T

he particle content of the SM is economical. There are no right-hande

d

neutrinos, so a Dirac neutrino mass term is not a

ll

owe

d

.T

h

ereison

ly

on

e

Higgs doublet, so a gauge-invariant Majorana mass term is forbidden.

•

T

he

S

M is a renormalizable theory, in which there does not exist a

dimension-5 operator that may

g

ive each neutrino a Majorana mass.

In other words, the SM accidentl

y

possesses the

B

−

L

s

y

mmetr

y

which as

-

s

ures three neutrinos to be exactly massless. Unlike the masslessness of the

photon, which is naturally

g

uaranteed by the electroma

g

neti

c

U

(

1

)

Q

g

au

g

e

s

ymmetry, the masslessness of three neutrinos in the SM is not an immedi

-

a

te consequence o

f

any

f

undamental law or symmetry principle. But today’

s

n

eutrino experiments have convincin

g

ly indicated the existence o

f

neutrino

oscillations

(

Nakamura

e

tal

.

,

2010

)

. This quantum phenomenon can appea

r

Z.-Z. Xing et al., Neutrinos in Particle Physics, Astronomy and Cosmology

Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology

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