Introduction to Neutrino Oscillation Physics

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

Part I: Theory of Neutrino Masses and Mixing Dirac Neutrino Masses

Majorana Neutrino Masses Dirac-Majorana Mass Term

Part II: Neutrino Oscillations in Vacuum and in Matter Neutrino Oscillations in Vacuum

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

Part I

Theory of Neutrino Masses and Mixing

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= m^₃

Jarlskog Invariant^{C. Giunti} Neutrino Oscillation Physics 9-13 June 2008, Benasque, Spain 5

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(^L^R+^R^L)

◮ In SM only ^L =⁾ no Dirac mass

◮ Oscillation experiments have shown that neutrinos are massive

◮ Simplest extension of the SM: add^R

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(D^Φ) 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)L^U(1)Y ^!U(1)Q

◮ Unitary Gauge: Φ(x) = ^p1

2

0 v+ H(x)

!

Dirac Lepton Masses

L_L^{ L}

`L

!

`R ^R

Lepton-Higgs Yukawa Lagrangian L_H

;L= y^{`</sup>L<sub>L</sub>Φ<sup>`}R y^L_LΦ^eR + H.c.

Unitary Gauge

Φ(x) = 1

p

2

0 v+ H(x)

!

Φ = ie ₂Φ^= 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.

L_H

;L= y^` v

p

2

`L<sup>`</sup>R y^ v

p

2

L^R

y^`

p

2

`L<sup>`</sup>RH y^

p

2

L^RH+ H.c.

m`= y<sup>`</sup> v

p

2 m = y^ v

p

2 g`H = y<sup>`</sup>

p

2 = m`

v gH = y^

p

2 = m

v

Three-Generations Dirac Neutrino Masses

L⁰_eL ^

0



 0

eL

` 0

eL ^e_L⁰

1

A L⁰

L ^

0



 0

L

` 0

L^

0

L

1

A L⁰

L^

0



 0

L

` 0

L^

0

L

1

A

` 0

eR ^e_R^{0 `0}R ^

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^`0R + Y^0

L⁰

LΦ^e0R

i

+ H.c.

Unitary Gauge

Φ(x) = ^p1

2

0



0 v+ H(x)

1

A

Φ = ie ₂Φ^= ^p1

2

0



v+ H(x) 0

1

A

L_H

;L=

v+ H

p

2



X

;=e^;;

h

Y^0`

` 0

L^`

0

R + Y^0

 0

L^

0

R

i

+ H.c.

L_H

;L=

v+ H

p

2



h

ℓ⁰_LY^0`ℓ⁰_R + ν_L⁰ Y^0ν_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`</sup> Y<sub>e</sub><sup>0`}



Y_e^0`

Y^0`

e Y^0`



Y^0`



Y^{0`</sup>e Y<sup>0`} Y^0` 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

M^0`= v

p

2Y^0` M^0 = v

p

2Y^0

L_H

;L=

v+ H

p

2



h

ℓ⁰_LY^0`ℓ⁰_R + ν_L⁰ Y^0ν_R⁰

i

+ H.c.

Diagonalization of Y^{0`</sup> andY<sup>0</sup> withunitary V<sub>L</sub><sup>`},V_R^{`</sup>,V<sub>L</sub><sup></sup>,V<sub>R</sub><sup></sup> ℓ<sup>0</sup><sub>L</sub>= V<sub>L</sub><sup>`}ℓ_L ℓ⁰_R = V_R^`ℓ_R ν_L⁰ = V_L^nL ν_R⁰ = V_R^nR

Kinetic terms are invariant under unitary transformations of the fields

L_H

;L=

v+ H

p

2



h

ℓ_LV_L^{`y</sup>Y<sup>0`}V_R^`ℓ_R+ ν_LV_L^yY^0V_R^ν_R

i

+ H.c.

V_L^{`y</sup> Y<sup>0`} V_R^{`</sup> = Y<sup>`} Y^{`</sup> = y<sup>`} Æ

 (^;= e^;;) V_L^y Y^0 V_R^ = Y^ Y_kj^ = y_k^Ækj (k^;j = 1^;2^;3)

Real and Positive y^`,y_k^

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

18 9 3 9

Massive Chiral Lepton Fields

ℓ_L= V_L^`yℓ⁰_L 0

B

B

B

B



e_L

L

L

1

C

C

C

C

A

ℓ_R = V_R^`yℓ⁰_R  0

B

B

B

B



e_R

R

R

1

C

C

C

C

A

nL= V_L^yν_L⁰  0

B

B

B

B



1L

2L

3L

1

C

C

C

C

A

nR = V_R^yν_R⁰  0

B

B

B

B



1R

2R

3R

1

C

C

C

C

A

L_H

;L=

v+ H

p

2



h

ℓ_LY^`ℓ_R + nLY^n_R

i

+ H.c.

=

v+ H

p

2



"

X

=e^;;

y^`

`

L^`R +

3

X

k=1

y_k^_kL^_kR

#

+ H.c.

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_k^v

p

2

k^k Mass Terms

X

=e^;;

y^` p

2

`

`

H

3

X

k=1

y_k^

p

2

k^kH Lepton-Higgs Couplings Charged Lepton and Neutrino Masses

m = y^`

v

p

2 (= e^;;) mk = y_k^v

p

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

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 = Ek =

q

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

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= 2 ν_L^{0 }ℓ⁰_L ℓ⁰

L= V_L^`ℓ_L ν⁰

L= V_L^nL

j_W^

;L= 2 nLV_L^{y }V_L^{`</sup>ℓ<sub>L</sub>= 2 nLV<sub>L</sub><sup>y</sup>V<sub>L</sub><sup>` }ℓ_L= 2 nLU^{y }ℓ_L Mixing Matrix

U^y= V_L^yV_L^{`</sup> U = V<sub>L</sub><sup>`y}V_L^

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

0

B



eL



L



L

1

C

A

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

j_W^

;L= 2 νL

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= 2 n_LU^{y }ℓ_L= 2

3

X

k=1

X

=e^;;

U^k^kL



`

L

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

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

`

L^!e^i'`L ^L^!e^i'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

◮ 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

Uk^kL ^{() }kL =

X

=e^;;

U^

k^L

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^!e^i'`L^{; }L^!e^i'L

`

R ^!e^i'`R^; 3

X

k=1

Uky_k^_kR ^!e^i'

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_k^kR is not a unitary combination of the ^kR’s

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



R

1

C

A+ H.c.

Le,L,L are not conserved

Lis conserved: L(^R) = L(^L) ^{) j}∆L^j= 0

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^!e^i'kL^{; }kR ^!e^i'kR (k = 1^;2^;3)

`

L^!e^{i'`</sup>L<sup>; `}R ^!e^i'`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

Mixing Matrix

◮ Leptonic Weak Charged Current: j_W^

;L= 2 nLU^{y }ℓ_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

◮ UnitaryN^N 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

◮ Weak Charged Current: j_W^

;L = 2

3

X

k=1

X

=e^;;

kLU^k



`

L

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

k ^!e^i'k_k (k = 1^;2^;3)^{; ` !</sup>e<sup>i'`} (= e^;;)

◮ Performing this transformation, theCharged Currentbecomes j_W^

;L= 2

3

X

k=1

X

=e^;;

kLe ^i'kU^

ke^{i' `}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

U^k e^{i(' 'e)}

| {z }

N 1=2



`

L

◮ There are 1 + (N 1) + (N 1) = 2 N 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.

◮ The mixing matrix contains N(N + 1)

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

2 = 1 Physical Phase

◮ It is convenient to express the3^3 unitary mixing matrix only in terms of the four physical parameters:

3 Mixing Anglesand1 Phase

Standard Parameterization of Mixing Matrix

0

B



eL



L



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 ^iÆ13

0 1 0

s₁₃e^iÆ13 0 c₁₃

1

C

A 0

B



c₁₂ s₁₂ 0 s₁₂ c₁₂ 0 0 0 1

1

C

A

=

0

B



c12c13 s12c13 s13e ^iÆ13 s12c23 c12s23s13e^iÆ13 c12c23 s12s23s13e^iÆ13 s23c13

s₁₂s₂₃ c₁₂c₂₃s₁₃e^iÆ13 c₁₂s₂₃ s₁₂c₂₃s₁₃e^iÆ13 c₂₃c₁₃

1

C

A

c_ab ^cos^#ab s_ab^sin^#ab 0^#ab ^



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

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 det C ⁶= 0 C = i[M^0M^0y;M^{0`</sup>M<sup>0`y}]

det C = 2 J



m²

2 m²

1



m²

3 m²

1



m²

3 m²

2





m²



m²_e



m²

m²_e



m²

m²





◮ Jarlskog invariant: J =⁼m

h

U3U_e2U^2U_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]

Example:

= 0

U = R₂₃R₁₃W₁₂

W₁₂=

0

B



cos^#₁₂ sin^#₁₂e ^iÆ12 0 sin^#₁₂e ^iÆ12 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

Example:

=

2

U = R₂₃W₁₃R₁₂

W₁₃=

0

B



cos^#₁₃ 0 sin^#₁₃e ^iÆ13

0 1 0

sin^#₁₃e^iÆ13 0 cos^#₁₃

1

C

A

#13=^=2 =⁾ W₁₃=

0

B



0 0 e ^iÆ13

0 1 0

e^iÆ13 0 0

1

C

A

U =

0

B



0 0 e ^iÆ13

s₁₂c₂₃ c₁₂s₂₃e^iÆ13 c₁₂c₂₃ s₁₂s₂₃e^iÆ13 0 s12s23 c12c23e^iÆ13 c12s23 s12c23e^iÆ13 0

1

C

A

U =^B

0 0 e ^iÆ13

jU1^je^{i1 j}U2^je^i2 0

jU1^je^{i1 j}U2^je^i2 0

C

A



1 ^2=^1 ^2^{ }1 ^1 =^2 ^2^

k ^!e^i'kk (k = 1^;2^;3)^{; `}

!e^i'` (= e^;;) U ^!



e ^i'e 0 0 0 e ^i' 0 0 0 e ^i'

 0 0 e ^iÆ13

jU1^je^{i1 j}U2^je^i2 0

jU1^je^{i1 j}U2^je^i2 0

!



e^i'1 0 0 0 e^i'2 0 0 0 e^i'3



U = ^{0 0}

e^{i( Æ13 'e +'3)}

jU1^je^{i(1 '+'1)j}U2^je^{i(2 '+'2)} 0

jU1^je^{i(1 '+'1) j}U2^je^{i(2 '+'2)} 0

!

'1= 0 ^'=^1 ^' =^1 ^'2=^{' }2=^1 ^2

'2=^{' }2^=^1 ^2^=^1 ^2 OK!

U =

 0 0 ^1

jU1^{j j}U2^j 0

jU1^{j j}U2^j 0