Neutrino Oscillation in Dense Matter Speaker: Law Zhiyang, National University of Singapore.

Neutrino Oscillation in Dense Matter

Speaker: Law Zhiyang,

National University of Singapore

Outline:

1. Lagrangian & Field Equations (Review)

2. Conserved Norm (In Uniform Matter Distribution)

3. Solutions to Field Equations (Uniform Matter)

4. Neutrino Oscillation in Uniform Matter

5. Conserved Norm (In Non-Uniform Matter)

6. Conclusion

1. Lagrangian & Field Equations (Review):

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Lν l

mat = − 2GF ν L γ μ JμCC NCC + Jμ

NC N NCν L ;

JμNC = fLγ μ fL

f∑ T3 fL

− 2Q f Sin 2θw( ) ,

JμNC = 2 fLγ μ fL

f∑ ,

NCC =1 00 0 ⎛ ⎝ ⎜

⎞ ⎠ ⎟ , N NC =

1 00 1 ⎛ ⎝ ⎜

⎞ ⎠ ⎟ , ν L =

ν eL

ν μL

⎛

⎝ ⎜

⎞

⎠ ⎟ For 2 flavors( )

Neutrino Field Equations:

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iγ μ∂μν L − M +ν R − 2GFγ μ JμCC NCC + Jμ

NC N NC ν L = 0 ,

iγ μ∂μν R − Mν L = 0 ,

Eliminate right-handed field:

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−∂2ν L − M +Mν L − i 2GFγ ν ∂ν γ μ (JμCC NCC + Jμ

NC N NC )ν L = 0 *

• see Halprin, PRD 34, #11, 3462(1986)

* * Giunti et al PRD 43, #1, 164(1991)

Assume matter is isotropic & non-relativistic:

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fLγ μ fL matter≈ 1

2ρ fδμ 0 ** ; ρ f = Background fermion # density

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∂t2 −∇ 2 + M +M + i[(∂t +

r σ ⋅

r ∇)A + A(∂t +

r σ ⋅

r ∇)]ν L = 0 ;

Neutrino Field Equations:

• Notation from Kiers et al, PRD 56, # 9, 5776(1997).

where ,

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A =2(α − β ) 0

0 −2β ⎛ ⎝ ⎜

⎞ ⎠ ⎟ , α = GF

2ρ e ,

β = GF

2ρ f

f∑ T3 fL

− 2Q f Sin2θw( ) *

2. Conserved Norm (uniform matter):

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∂t2 −∇ 2 + M +M + i[(∂t +

r σ ⋅

r ∇)A + A(∂t +

r σ ⋅

r ∇)]ν L = 0 ;

- To consider oscillation probability, a conserved norm relevant to the left-handed field is needed :

For uniform, time-independent matter distribution, “A” is constant :

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(∂t +r σ ⋅

r ∇)A → 1

2(∂t +

r σ ⋅

r ∇)A

∂ 2 + M +M + i 12

(∂t +r σ ⋅

r ∇)A + A(∂t +

r σ ⋅

r ∇)

⎡ ⎣ ⎢

⎤ ⎦ ⎥

⎧ ⎨ ⎩

⎫ ⎬ ⎭

ν L = 0 ;

Conserved Norm (uniform matter):

Consider 2 arbitrary wave functions (flavor basis) :

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ν La , ν L

b

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ν +La r

∂ 2 + M +M + i 12

(∂t +r σ ⋅

r ∇)A + A(

r ∂ t +

r σ ⋅

r ∇)

⎡ ⎣ ⎢

⎤ ⎦ ⎥

⎧ ⎨ ⎩

⎫ ⎬ ⎭

ν Lb = 0 (1)

ν +Lb r

∂ 2 + M +M + i 12

(∂t +r σ ⋅

r ∇)A + A(

r ∂ t +

r σ ⋅

r ∇)

⎡ ⎣ ⎢

⎤ ⎦ ⎥

⎧ ⎨ ⎩

⎫ ⎬ ⎭

ν La = 0

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ν +La s

∂ 2 + M +M − i 12

(∂t +r σ ⋅

r ∇)A + A(

s ∂ t +

r σ ⋅

s ∇)

⎡ ⎣ ⎢

⎤ ⎦ ⎥

⎧ ⎨ ⎩

⎫ ⎬ ⎭

ν Lb = 0 (2)

Conserved Norm (uniform matter):

(1) - (2) :

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∂t ν +La t ∂ tν L

b( ) +

r ∇ ⋅ ν +

La t

∇ν Lb

( )

+ν +La

i (∂t +r σ ⋅

r ∇)A + A(

r ∂ t +

r σ ⋅

r ∇) + A(

s ∂ t +

r σ ⋅

s ∇)[ ] ν L

b = 0 ,

→ ∂t ν +La

it ∂ t − A( )ν L

b( ) +

r ∇ ⋅ ν +

La

−it

∇ −r σ A( )ν L

b( ) = 0 ,

⇒ Conserved Norm : J 0 = ν +L i

t ∂ t − A( )ν L ,

Current : r J = ν +

L −it

∇ −r σ A( )ν L

3. Solutions to Field Equations (Uniform Matter) :-Field equation needs to be solve to calculate probability current. For uniform matter:

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∂t2 −∇ 2 + M +M + i[A(∂t +

r σ ⋅

r ∇)]ν L = 0 ;

- Assume “ν” is traveling along, and with spin in the x-axis :

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⇒ ∂t2 −∂x

2 + M +M + i[A(∂t + s∂x )]ν L = 0 ; s = ±1

try Ansatz : ν L ∝ e i(kx−Et ) :

−E 2 + k 2 + M +M + [A(E − sk)]ν L = 0 ,

−E 2 + k 2 − (E − sk)2β 00 2β ⎛ ⎝ ⎜

⎞ ⎠ ⎟+ M +M + (E − sk)

2α 00 0 ⎛ ⎝ ⎜

⎞ ⎠ ⎟

1 2 4 4 4 4 3 4 4 4 4 ν L = 0 ,

Ω

Solutions to Field Equations (Uniform Matter) :

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M +M =cosθ sinθ−sinθ cosθ ⎛ ⎝ ⎜

⎞ ⎠ ⎟⋅

m12 0

0 m22

⎛

⎝ ⎜

⎞

⎠ ⎟⋅

cosθ sinθ−sinθ cosθ ⎛ ⎝ ⎜

⎞ ⎠ ⎟ ,

Ω = M +M + (E − sk)2α 00 0

⎛ ⎝ ⎜

⎞ ⎠ ⎟

= Um E,k( ) ⋅λ1 00 λ 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟⋅Um

+ E,k( ) ,

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λ1/ 2 = 12

m (2α (E − sk) − Δcos(2θ))2 + (Δsin(2θ))2 + Σ + 2α (E − sk) ,

Δ = m22 − m1

2 , Σ = m22 + m1

2 ,

Um (E,k) =cosθm sinθm

−sinθm cosθm

⎛ ⎝ ⎜

⎞ ⎠ ⎟ ; sin(2θm ) = Δsin(2θ)

(2α (E − sk) − Δcos(2θ))2 + (Δsin(2θ))2

where ;

Solutions to Field Equations (Uniform Matter) :Effective mass matrix (diagonal elements):

€

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M1/ 2

2 = 12

m (2α (E − sk) − Δcos(2θ))2 + (Δsin(2θ))2 + Σ

+ 2(E − sk)(α − 2β ) ,

E 2 = M1/ 2

2 + k 2 ,

⇒ E 2 − k 2 −1/2Σ − (E − sk)(α − 2β ) = m12

(2α (E − sk) − Δcos(2θ))2 + (Δsin(2θ))2

This distinguishes the propagation modes (k1/2) for a given “E”

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⇒ E 2 − k 2 −1/2Σ − (E − sk)(α − 2β )2 = α 2(E − sk)2 −αΔcos(2θ)(E − sk) +1/4Δ2

This quartic equation gives 4 solutions: 2 propagation modes, k1/2 , and “negative” & “positive” energy solutions *

• Similar dispersion relations is found by Kiers et al, PRD 56, # 9, 5776(1997).

Solutions to Field Equations (Uniform Matter) :Taking positive energy solution in propagation basis:

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ν Lm1 ∝ e i(k1x−Et )

0

⎛

⎝ ⎜

⎞

⎠ ⎟ , ν L

m2 ∝0

e i(k2x−Et )

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Solutions in flavor basis :

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Um (E,k1/ 2) =cosθm sinθm

−sinθm cosθm

⎛ ⎝ ⎜

⎞ ⎠ ⎟ ;

sin(2θm ) = Δsin(2θ)(2α (E − sk1/ 2) − Δcos(2θ))2 + (Δsin(2θ))2

,

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ν L1 = Um (E,k1) ⋅ν L

m1 =cosθ1

−sinθ1

⎛ ⎝ ⎜

⎞ ⎠ ⎟ e i(k1x−Et ), ν L

1 = Um (E,k2) ⋅ν Lm2 =

sinθ2

cosθ2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ e i(k2x−Et )

General solution (flavor basis) :

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ν L = A1ν L1 + A2ν L

2

4. Neutrino Oscillation in Uniform MatterAn example: Oscillation of νe :

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At x = 0 , t = 0 :

ν L =ν e

0 ⎛ ⎝ ⎜

⎞ ⎠ ⎟= A1ν L

1 + A2ν L2 =

A1 cos(θ1) + A2 sin(θ2)−A1 sin(θ1) + A2 cos(θ2) ⎛ ⎝ ⎜

⎞ ⎠ ⎟

⇒ − A1 sin(θ1) + A2 cos(θ2) = 0

⇒ A1 =1 , A2 = sin(θ1)cos(θ2)

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Current : J x = ν +L −i

t ∂ x − sA( )ν LRecall :

This is diagonal in flavor space; can identify νe-component & ν-component probability current.

Neutrino Oscillation in Uniform Matter

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J x (ν e ) = 2(k1 − s(α − β ))cos2(θ1) + (k2 − s(α − β ))tan2(θ2)sin2(θ1) + (k1 + k2 − 2s(α − β ))tan(θ2)sin(θ1)cos(θ1)cos[(k2 − k1)x]

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J x (ν μ ) = 2(k1 + k2 + 2sβ )sin2(θ1)1− cos[(k1 − k2)x]

νe-component :

ν-component :

From:

€

Conserved Norm : J 0 = ν +L i

t ∂ t − A( )ν L , (this is time independent)

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J x (total) = Constan t in x


This implies that the currents are positive when (k1-(-)) , (k2-(-))

& (k1+)) , (k2+)) are positive;

Because :

J(ν) = positive

J(total) = constant = positive

J(ν) = positive

Therefore a consistent oscillation probability can be calculated.


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J x (total)

= 2 (k1 − s(α − β ))A12 + (k2 − s(α − β ))A2

2

+ (k1 + k2 − 2s(α − β ))A1A2 sin(θ1 + θ2) ,

Prob ν e →ν e( ) = J x (ν e )J x (total)

,

After some algebra :

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Pr ob ν e → ν e( ) =1− Ampsin2[(k2 − k1)x /2] ,

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2(k1 + k2 − 2s(α − β )) sin(θ1)sin(θ2)cos(θ1)cos(θ2)

(k1 − s(α − β ))[cos2(θ1) + sin(θ1)sin(θ2)cos(θ1)cos(θ2)

] + (k2 − s(α − β ))[sin2(θ1)sin2(θ2)cos2(θ2)

+ sin(θ1)sin(θ2)cos(θ1)cos(θ2)

]

Amp. =


Check vacuum oscillation in ultra-relativistic limit :

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=0 , β = 0 , k2 = k + Δk , k1 = k :

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Amp. = 2(2k + Δk)sin2(θ)cos2(θ)k + Δk cos2(θ)

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Δk ≈ 0

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Amp. = sin2(2θ)

This is the standard result for oscillation amplitude (vacuum).


Example 2 :

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≈0 , compared to β and E

Maybe in a dense astrophysical medium :

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sin(2θm ) = Δsin(2θ)(2α (E − sk1/ 2) − Δcos(2θ))2 + (Δsin(2θ))2

≈ sin(2θ)

Similarly for the dispersion relations :

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E 2 − k 2 −1/2Σ − (E − sk)(α − 2β ) = m12

(2α (E − sk) − Δcos(2θ))2 + (Δsin(2θ))2

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k 2 + 2sβk + (1/2(Σ mΔ) − E 2 − 2βE) = 0


Solving for k :

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k1 = −sβ ± (β + E)2 − m12 ,

k2 = −sβ ± (β + E)2 − m22 ,

left propagating solution

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k1 = −sβ + (β + E)2 − m12 ,

k2 = −sβ + (β + E)2 − m22 ,


Oscillation length :

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k1 + sβ ≈ (β + E) − m12

2(β + E)

k2 + sβ ≈ (β + E) − m22

2(β + E) ⇓

1losc

∝ Δ2(β + E)

modification to standard value


Amplitude :

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= ( (β + E)2 − m1

2 + (β + E)2 − m22 )

2 ( (β + E)2 − m12 )cos2(θ) + ( (β + E)2 − m2

2 )sin2(θ)[ ]× sin2(2θ)

Resonance :

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( (β + E)2 − m12 + (β + E)2 − m2

2 )

2 ( (β + E)2 − m12 )cos2(θ) + ( (β + E)2 − m2

2 )sin2(θ)[ ]= 1

sin2(2θ)

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(β + E ) = m12 − m2

2 tan2(θ)1− tan2(θ)

But if “” , “E” >> “m12” , “m2

2” just vacuum amplitude

5. Conserved Norm (In Non-Uniform Matter)

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∂t2 −∇ 2 + M +M + i[(∂t +

r σ ⋅

r ∇)A + A(∂t +

r σ ⋅

r ∇)]ν L = 0 ;

Recall field equation :

We replace :

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(∂t +r σ ⋅

r ∇)A → 1

2(∂t +

r σ ⋅

r ∇)A

Due to the short wave length of neutrinos :

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(∂t +r σ ⋅

r ∇)A ≈ 0

Hence probability current derived is also applicable to non-uniform matter.

6. Conclusion :

1. A quantum mechanical treatment of neutrino oscillation in matter is discussed.

2. A relevant conserved current is derived

3. Oscillation length & amplitude is studied for special cases together with resonance.

4. Conserved current is also applicable to non-uniform matter.

The End

Neutrino Oscillation in Dense Matter Speaker: Law Zhiyang, National University of Singapore.

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Transcript of Neutrino Oscillation in Dense Matter Speaker: Law Zhiyang, National University of Singapore.