Introduction to Neutrino Oscillation Physics

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Introduction to Neutrino Oscillation Physics

Carlo Giunti

INFN, Sezione di Torino, and Dipartimento di Fisica Teorica, Universit`a di Torino mailto://[email protected]

Neutrino Unbound:http://www.nu.to.infn.it

9-13 June 2008, Benasque, Spain NUFACT’08 School

C. Giunti and C.W. Kim

Fundamentals of Neutrino Physics and Astrophysics

Oxford University Press 15 March 2007 – 728 pages

C. Giunti Neutrino Oscillation Physics 9-13 June 2008, Benasque, Spain 1

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Part I: Theory of Neutrino Masses and Mixing

Dirac Neutrino Masses Majorana Neutrino Masses Dirac-Majorana Mass Term

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Part II: Neutrino Oscillations in Vacuum and in Matter

Neutrino Oscillations in Vacuum CPT, CP and T Symmetries Two-Neutrino Oscillations

Question: Do Charged Leptons Oscillate?

Neutrino Oscillations in Matter

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Part III: Phenomenology of Three-Neutrino Mixing

Phenomenology of Three-Neutrino Oscillations Absolute Scale of Neutrino Masses

Tritium Beta-Decay

Cosmological Bound on Neutrino Masses Neutrinoless Double-Beta Decay

Conclusions

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Part I

Theory of Neutrino Masses and Mixing

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Dirac Neutrino Masses

Dirac Neutrino Masses Dirac Mass

Higgs Mechanism in SM Dirac Lepton Masses

Three-Generations Dirac Neutrino Masses Massive Chiral Lepton Fields

Massive Dirac Lepton Fields Quantization

Mixing

Flavor Lepton Numbers Total Lepton Number Mixing Matrix

Standard Parameterization of Mixing Matrix CP Violation

Example: ^#₁₂= 0 Example: ^#₁₃=⁼2 Example: m2=m3

Jarlskog Invariant^{C. Giunti} Neutrino Oscillation Physics 9-13 June 2008, Benasque, Spain 6

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Dirac Mass

◮ Dirac Equation: (i/ m)(x) = 0 (/)

◮ Dirac Lagrangian: L(x) =(x) (i/ m)(x)

◮ Chiral decomposition: L

1 ⁵

2 ^; ^R

1 +⁵

2

=_L+_R

L =Li/ L+Ri/ R m(LR+RL)

◮ In SM only L =⁾ no Dirac mass

◮ Oscillation experiments have shown that neutrinos are massive

◮ Simplest extension of the SM: addR

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Higgs Mechanism in SM

◮ SM: fermion masses are generated through the Higgs mechanism

◮ Higgs Doublet: Φ(x) =

+(x)

0(x)

!

◮ Higgs Lagrangian: L_Higgs= (DΦ)^y(DΦ) V(Φ)

◮ Higgs Potential: V(Φ) =²Φ^yΦ +(Φ^yΦ)²

◮ 2

<0,^>0=⁾ V(Φ) =

Φ^yΦ ^v₂²

2

, withv

q

2

◮ Vacuum: V_min for Φ^yΦ = ^v₂² =⁾^hΦⁱ= ^p¹

2

0 v

!

◮ Spontaneous Symmetry Breaking: SU(2)LU(1)Y ^!U(1)Q

◮ Unitary Gauge: Φ(x) = ^p¹

2

0 v+H(x)

!

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Dirac Lepton Masses

L_L ^L

`L

!

`R R

Lepton-Higgs Yukawa Lagrangian L_H

;L= y^`L_LΦ^`R yL_LΦ^eR + H.c.

Unitary Gauge Φ(x) = 1

p

2

0 v+H(x)

!

Φ =e i₂Φ= 1

p

2

v+H(x) 0

!

L_H

;L= y^`

p

2

L ^`L

0 v+H(x)

!

`R

y

p

2

L ^`L

v+H(x) 0

!

R + H.c.

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L_H

;L= y^` v

p

2

`L^`R y v

p

2

LR

y^`

p

2

`L^`RH y

p

2

LRH+ H.c.

m`=y^` v

p

2 m =y v

p

2 g`H = y^`

p

2 = m`

v gH = y

p

2 = m

v

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Three-Generations Dirac Neutrino Masses

L⁰_eL

0

eL

` 0

eL e_L⁰

1

A L⁰

L

0

L

` 0

L

0

L

1

A L⁰

L

0

L

` 0

L

0

L

1

A

` 0

eR e_R⁰ ^`⁰R

0

R ^`

0

R

0

R

0

eR

0

R

0

R

Lepton-Higgs Yukawa Lagrangian L_H

;L=

X

;=e^;;

h

Y^0`

L⁰

LΦ^`⁰R +Y⁰

L⁰

LΦ^e⁰R

i

+ H.c.

Unitary Gauge

Φ(x) = ^p¹

2

0

0 v+H(x)

1

A

Φ =e i₂Φ= ^p¹

2

0

v+H(x) 0

1

A

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L_H

;L=

v+H

p

2

X

;=e^;;

h

Y^0`

` 0

L^`

0

R +Y⁰

0

L

0

R

i

+ H.c.

L_H

;L=

v+H

p

2

h

ℓ⁰_LY^0`ℓ⁰_R +ν_L⁰ Y⁰ν_R⁰

i

+ H.c.

ℓ⁰_L 0

B

e_L⁰

0

L

0

L

1

C

A ℓ⁰_R 0

B

e_R⁰

0

R

0

R

1

C

A ν_L⁰ 0

B

0

eL

0

L

0

L

1

C

A ν_R⁰ 0

B

0

eR

0

R

0

R

1

C

A

Y^0`

0

B

Y_ee^0` Y_e^0`

Y_e^0`

Y^0`

e Y^0`

Y^0`

Y^0`e Y^0` Y^0`

1

C

A Y⁰

0

B

Y_ee⁰ Y_e⁰

Y_e⁰

Y⁰

e Y⁰

Y⁰

e Y⁰

Y⁰

1

C

A

M^0`= v

p

2Y^0` M⁰ = v

p

2Y⁰

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L_H

;L=

v+H

p

2

h

ℓ⁰_LY^0`ℓ⁰_R +ν_L⁰ Y⁰ν_R⁰

i

+ H.c.

Diagonalization of Y^0` andY⁰ withunitary V_L^`,V_R^`,V_L,V_R ℓ⁰_L=V_L^`ℓ_L ℓ⁰_R =V_R^`ℓ_R ν_L⁰ =V_LnL ν_R⁰ =V_RnR

Kinetic terms are invariant under unitary transformations of the fields

L_H

;L=

v+H

p

2

h

ℓ_LV_L^`yY^0`V_R^`ℓ_R+ν_LV_L^yY⁰V_Rν_R

i

+ H.c.

V_L^`y Y^0` V_R^` = Y^` Y^` =y^` Æ

(^;=e^;^;) V_L^y Y⁰ V_R = Y Y_kj =y_k^Æ_kj (k^;j = 1^;2^;3)

Real and Positive y^`,y_k

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V_L^yY⁰V_R = Y ⁽⁾ Y⁰ = V_R^y Y V_L 2N² N² N N²

18 9 3 9

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Massive Chiral Lepton Fields

ℓ_L=V_L^`yℓ⁰_L 0

B

e_L

L

1

C

A

ℓ_R =V_R^`yℓ⁰_R 0

B

e_R

R

1

C

A

nL=V_L^yν_L⁰ 0

B

1L

2L

3L

1

C

A

nR =V_R^yν_R⁰ 0

B

1R

2R

3R

1

C

A

L_H

;L=

v+H

p

2

h

ℓ_LY^`ℓ_R +nLYn_R

i

+ H.c.

=

v+H

p

2

"

X

=e^;;

y^`

`

L^`R +

3

X

k=1

y_k_kL_kR

#

+ H.c.

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Massive Dirac Lepton Fields

`

L+^`R (=e^;^;)

k =kL+kR (k = 1^;2^;3) L_H

;L=

X

=e^;;

y^`v

p

2

`

3

X

k=1

y_kv

p

2

kk Mass Terms

X

=e^;;

y^` p

2

`

H

3

X

k=1

y_k

p

2

kkH Lepton-Higgs Couplings

Charged Lepton and Neutrino Masses m = y^`

v

p

2 (=e^;^;) mk = y_kv

p

2 (k = 1^;2^;3) Lepton-Higgs coupling ^/Lepton Mass

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Quantization

k(x) =

Z d³p (2)³2E_k

X

h=1

a^(h)_k (p)u_k^(h)(p)e ^ip^x+b^(h)_k

y

(p)v_k^(h)(p)e^ip^x

p⁰ =E_k =

q

~p²+m²_k (/p m_k)u^(h)_k (p) = 0 (/p+m_k)v_k^(h)(p) = 0

~p^~Σ

j~p^j u^(h)_k (p) =h u_k^(h)(p)

~p^~Σ

j~p^j v_k^(h)(p) = h v_k^(h)(p)

fa^(h)_k (p)^;a^(h

0)^y

k (p⁰)^g=^fb^(h)_k (p)^;b^(h

0)^y

k (p⁰)^g= (2)³2E_k^Æ³(^~p ^~p⁰)^Æ_hh⁰

fa^(h)_k (p)^;a^(h

0)

k (p⁰)^g=^fa^(h)_k ^y(p)^;a^(h

0)^y

k (p⁰)^g= 0

fb^(h)_k (p)^;b^(h

0)

k (p⁰)^g=^fb_k^(h)^y(p)^;b^(h

0)^y

k (p⁰)^g= 0

fa^(h)_k (p)^;b^(h

0)

k (p⁰)^g=^fa^(h)_k ^y(p)^;b^(h

0)^y

k (p⁰)^g= 0

fa^(h)_k (p)^;b^(h

0)^y

k (p⁰)^g=^fa^(h)_k ^y(p)^;b^(h

0)

k (p⁰)^g= 0

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Mixing

Charged-Current Weak Interaction Lagrangian L^(CC)

I = g

2

p

2j_W W+ H.c.

Weak Charged Current: j_W =j_W

;L+j_W

;Q

Leptonic Weak Charged Current j_W

;L =

X

=e^;;

0

1 ⁵

` 0

= 2

X

=e^;;

0

L

` 0

L= 2ν_L⁰ ℓ⁰_L

ℓ⁰

L=V_L^`ℓ_L ν⁰

L=V_LnL

j_W

;L= 2nLV_L^yV_L^`ℓ_L= 2nLV_L^yV_L^`ℓ_L= 2nLU^yℓ_L Mixing Matrix

U^y=V_L^yV_L^` U =V_L^`yV_L

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◮ Definition: Left-Handed Flavor Neutrino Fields ν_L=UnL=V_L^`yν_L⁰ =

0

B

eL

L

1

C

A

◮ They allow us to write theLeptonic Weak Charged Currentas in the SM:

j_W

;L= 2ν_L

ℓ_L= 2

X

=e^;;

L

`

L

◮ Each left-handed flavor neutrino fieldis associated with the

corresponding charged lepton fieldwhich describes a massive charged lepton (e,,).

◮ In practiceleft-handed flavor neutrino fields are useful for calculations in the SM approximation of massless neutrinos (interactions).

◮ If neutrino masses must be taken into account, it is necessary to use j_W

;L= 2n_LU^yℓ_L= 2

3

X

k=1

X

=e^;;

UkkL

`

L

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Flavor Lepton Numbers

Flavor Neutrino Fields are useful for defining Flavor Lepton Numbers

as in the SM L_e L L

(e^;e ) +1 0 0 (

; ) 0 +1 0 (

; ) 0 0 +1

Le L L

(_e^c^;e⁺) 1 0 0

c

;

+

0 1 0

(^c

;

+) 0 0 1

L=L_e+L+L

Standard Model: Lepton numbers are conserved

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◮ Leptonic Weak Charged Current is invariant under the globalU(1) gauge transformations

`

L^!eⁱ^'^`L L^!eⁱ^'L (=e^;^;)

◮ If neutrinos are massless (SM), Noether’s theorem implies that there is, for each flavor, a conserved current:

j

=L

L+^`

`

j

= 0 and a conserved charge:

L =

Z

d³x j⁰

(x) ₀L = 0

: L: =

Z d³p (2)³2E

h

a^{( )} ^y(p)a^{( )} (p) b⁽⁺⁾ ^y(p)b⁽⁺⁾ (p)

i

+

Z d³p (2)³2E

X

h=1

h

a^(h)^y

`

(p)a^(h)

`

(p) b^(h)^y

`

(p)b^(h)

`

(p)

i

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◮ Lepton-Higgs Yukawa Lagrangian:

L_H

;L=

v+H

p

2

"

X

=e^;;

y^`

`

L^`R +

3

X

k=1

y_k_kL_kR

#

+ H.c.

◮ Mixing: L=

3

X

k=1

UkkL ⁽⁾ kL =

X

=e^;;

U

kL

L_H

;L=

v+H

p

2

X

=e^;;

"

y^`

`

L^`R+L 3

X

k=1

Uky_k_kR

#

+ H.c.

◮ Invariant for

`

L^!eⁱ^'^`L^; L^!eⁱ^'L

`

R ^!eⁱ^'^`R^; 3

X

k=1

Uky_k_kR ^!eⁱ^'

3

X

k=1

Uky_k_kR

◮ But kinetic part of neutrino Lagrangian is not invariant L⁽⁾

kinetic =

X

=e^;;

Li/ L+

3

X

k=1

kRi/ _kR

because ^P³_k=1Uky_kkR is not a unitary combination of the kR’s

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L^D

mass=

eL L L

0

B

m^D_ee m^D_e m_e^D

m^D

e m^D

m^D

m^De m^D m^D 1

C

A 0

B

eR

R

1

C

A+ H.c.

Le,L,L are not conserved

Lis conserved: L(R) =L(L) ⁾ ^j∆L^j= 0

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Total Lepton Number

◮ Dirac neutrino masses violateconservation of Flavor Lepton Numbers

◮ Total Lepton Number is conserved, because Lagrangian is invariant under the global U(1)gauge transformations

kL^!eⁱ^'kL^; kR ^!eⁱ^'kR (k = 1^;2^;3)

`

L^!eⁱ^'^`L^; ^`R ^!eⁱ^'^`R (=e^;^;)

◮ From Noether’s theorem:

j =

3

X

k=1

k

k +

X

=e^;;

`

j = 0 Conserved charge: L =

Z

d³x j⁰

(x) ₀L = 0 : L : =

3

X

k=1

Z d³p (2)³2E

X

h=1

h

a^(h)_k^y(p)a^(h)_k (p) b^(h)_k^y(p)b^(h)_k (p)

i

+

X

=e^;;

Z d³p (2)³2E

X

h=1

h

a^(h)^y

`

(p)a^(h)

`

(p) b^(h)^y

`

(p)b^(h)

`

(p)

i

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Mixing Matrix

◮ Leptonic Weak Charged Current: j_W

;L= 2nLU^yℓ_L

◮ U =V_L^`yV_L =

0

B

U₁₁ U₁₂ U₁₃ U₂₁ U₂₂ U₂₃ U₃₁ U₃₂ U₃₃

1

C

A

0

B

U_e1 U_e2 U_e3 U1 U2 U3

U1 U2 U3

1

C

A

◮ UnitaryNN matrix depends on N² independent real parameters

N = 3 =⁾

N(N 1)

2 = 3 Mixing Angles N(N+ 1)

2 = 6 Phases

◮ Not all phases are physical observables

◮ Only physical effect of mixing matrix occurs through its presence in the Leptonic Weak Charged Current

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◮ Weak Charged Current: j_W

;L = 2

3

X

k=1

X

=e^;;

kLUk

`

L

◮ Apart from theWeak Charged Current, the Lagrangian is invariant under the global phase transformations

k ^!eⁱ^'^k_k (k = 1^;2^;3)^; ^` ^!eⁱ^'^` (=e^;^;)

◮ Performing this transformation, theCharged Currentbecomes j_W

;L= 2

3

X

k=1

X

=e^;;

kLe ⁱ^'^kU

keⁱ^'^`L

j_W

;L= 2 e ⁱ⁽^'¹ ^'^e⁾

| {z }

1 3

X

k=1

X

=e^;;

kL e ⁱ⁽^'^k ^'¹⁾

| {z }

N 1=2

Uk eⁱ⁽^' ^'^e⁾

| {z }

N 1=2

`

L

◮ There are 1 + (N 1) + (N 1) = 2N 1 = 5 arbitrary phases of the fields that can be chosen to eliminate 5 of the 6 phases of the mixing matrix

◮ 2N 1 and not2N phases of the mixing matrix can be eliminated because a common rephasing of all the fields leaves the Charged Current invariant⁽⁾ conservation of Total Lepton Number.

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◮ The mixing matrix contains N(N+ 1)

2 (2N 1) = (N 1) (N 2)

2 = 1 Physical Phase

◮ It is convenient to express the33 unitary mixing matrix only in terms of the four physical parameters:

3 Mixing Anglesand1 Phase

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Standard Parameterization of Mixing Matrix

0

B

eL

L

1

C

A=

0

B

U_e1 U_e2 U_e3 U1 U2 U3

U1 U2 U3

1

C

A 0

B

1L

2L

3L

1

C

A

U = R₂₃W₁₃R₁₂

=

0

B

1 0 0

0 c₂₃ s₂₃ 0 s₂₃ c₂₃

1

C

A 0

B

c₁₃ 0 s₁₃e ⁱ^Æ¹³

0 1 0

s₁₃eⁱ^Æ¹³ 0 c₁₃

1

C

A 0

B

c₁₂ s₁₂ 0 s₁₂ c₁₂ 0 0 0 1

1

C

A

=

0

B

c12c13 s12c13 s13e ⁱ^Æ¹³ s12c23 c12s23s13eⁱ^Æ¹³ c12c23 s12s23s13eⁱ^Æ¹³ s23c13

s₁₂s₂₃ c₁₂c₂₃s₁₃eⁱ^Æ13 c₁₂s₂₃ s₁₂c₂₃s₁₃eⁱ^Æ13 c₂₃c₁₃

1

C

A

c_ab cos^#ab s_absin^#ab 0^#ab

2 0^Æ132 3 Mixing Angles^#₁₂,^#₂₃,^#₁₃ and1 Phase ^Æ₁₃

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CP Violation

◮ U =U ⁽⁾ CP symmetry

◮ General conditions for CP violation (14 conditions):

1. No two charged leptons or two neutrinos are degenerate in mass (6 conditions)

2. No mixing angle is equal to0or ⁼2 (6 conditions) 3. The physical phase is different from 0or(2 conditions)

◮ These 14 conditions are combined into the single condition detC ⁶= 0 C = i[M⁰M⁰^y^;M^0`M^0`^y]

detC = 2J

m²

2 m²

1

m²

3 m²

1

m²

3 m²

2

m²

m²_e

m²

m²_e

m²

◮ Jarlskog invariant: J =⁼m

h

U3U_e2U2U_e3

i

[C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039, Z. Phys. C 29 (1985) 491]

[O. W. Greenberg, Phys. Rev. D 32 (1985) 1841]

[I. Dunietz, O. W. Greenberg, Dan-di Wu, Phys. Rev. Lett. 55 (1985) 2935]

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Example:

^#12

= 0

U =R₂₃R₁₃W₁₂

W₁₂=

0

B

cos^#₁₂ sin^#₁₂e ⁱ^Æ¹² 0 sin^#₁₂e ⁱ^Æ¹² cos^#₁₂ 0

0 0 1

1

C

A

#12= 0 =⁾ W₁₂=

0

B

1 0 0 0 1 0 0 0 1

1

C

A=1 real mixing matrix U =R₂₃R₁₃

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Example:

^#13

=

⁼

2

U =R₂₃W₁₃R₁₂

W₁₃=

0

B

cos^#₁₃ 0 sin^#₁₃e ⁱ^Æ¹³

0 1 0

sin^#₁₃eⁱ^Æ¹³ 0 cos^#₁₃

1

C

A

#13=⁼2 =⁾ W₁₃=

0

B

0 0 e ⁱ^Æ¹³

0 1 0

eⁱ^Æ¹³ 0 0

1

C

A

U =

0

B

0 0 e ⁱ^Æ¹³

s₁₂c₂₃ c₁₂s₂₃eⁱ^Æ¹³ c₁₂c₂₃ s₁₂s₂₃eⁱ^Æ¹³ 0 s12s23 c12c23eⁱ^Æ¹³ c12s23 s12c23eⁱ^Æ¹³ 0

1

C

A

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U =^B

0 0 e ⁱ^Æ¹³

jU1^jeⁱ¹ ^jU2^jeⁱ² 0

C

A

1 2=1 2 1 1 =2 2

k ^!eⁱ^'^kk (k = 1^;2^;3)^; ^`

!eⁱ^'^` (=e^;^;) U ^!

e ⁱ^'^e 0 0 0 e ⁱ^' 0 0 0 e ⁱ^'

0 0 e ⁱ^Æ¹³

jU1^jeⁱ¹ ^jU2^jeⁱ² 0

!

eⁱ^'¹ 0 0 0 eⁱ^'² 0 0 0 eⁱ^'³

U = ⁰ ⁰

eⁱ⁽ ^Æ¹³ ^'^e⁺^'³⁾

jU1^jeⁱ⁽¹ ^'⁺^'¹⁾^jU2^jeⁱ⁽² ^'⁺^'²⁾ 0

jU1^jeⁱ⁽¹ ^'⁺^'¹⁾ ^jU2^jeⁱ⁽² ^'⁺^'²⁾ 0

!

'1= 0 ^'=1 ^' =