Oscillations Between two types of Neutrinos in Vacuum

Examensarbete 15 hp

Juni 2012

Oscillations Between two types

of Neutrinos in Vacuum

Oscillations Between two types of Neutrinos in

Vacuum

Olof Svedvall

Kandidatprogrammet i Fysik

Uppsala Universitet

Handledare: Stefan Leupold

Examinator: Stefan Leupold

June 25, 2012

In this thesis the oscillation probability between two ultra rela-tivistic Dirac/Majorana neutrinos in vacuum with a narrow spread in momentum is calculated. The cause of the oscillation is in both cases given by real and symmetric mass matrices. The oscillation probabilities P (t) in SI units are in both cases given by P (t) = sin2_{(2θ) sin}2₍t(m211c4−m222c4)

4~E(~k) ), where θ is a mixing angle, m11 and m22

Contents

1 Introduction 2

2 Majorana 3

2.1 Majorana Action . . . 3 2.2 Solution to The Majorana Equation . . . 7

3 Oscillation Probability 8

3.1 Oscillations between two Dirac neutrinos in Vacuum . . . 8 3.2 Oscillations between two Majorana

neutrinos in Vacuum . . . 12

4 Conclusion 15

5 Summary 17

1

Introduction

When scientists in the middle of the 20th century proposed that the sun got its energy from nuclear fusion [1], they also prepared a very delicate problem that would take decades to solve. To test the theory about the energy pro-duction in the sun two scientists, Raymond Davis Jr and John N Bahcall, in 1964 proposed an experiment to count the number of neutrinos coming from the sun [2]. The first result came in 1968 and showed that the predicted amount of neutrinos from the solar model was three times greater than the actual amount of neutrinos found in the experiment. Other experiments fol-lowed that also came to the conclusion that the amount of solar neutrinos was low compared to predictions from the solar model [3]. This difference between theory and experiment was named ”The Solar Neutrino Problem” [2].

The solution to what was later called The Solar Neutrino Problem had already been proposed by Bruno Pontecorvo in 1957 [3]. Neutrinos come in three flavors [4, p. 165]: electron-neutrino, muon-neutrino and tau-neutrino. Furthermore neutrinos are, according to the textbook version of the standard model, massless [2], which means that they cannot oscillate between different flavor states. Pontecorvo questioned what was later called the textbook ver-sion of the standard model and proposed the idea that neutrinos had mass and hence could oscillate between these flavor states. Pontecorvo calculated, that if neutrinos could oscillate then the predicted amount of solar neutrinos by the solar model would differ from the measured amount of solar neutrinos for a detector (such as the one used by Raymond Davis Jr and John N Bah-call in 1964) that do not detect all flavors of neutrinos. Pontecorvo’s idea that neutrinos could oscillate, which solves The Solar Neutrino Problem, was confirmed in the first years of the 21st century [2].

In this thesis the neutrino oscillation will be calculated for two neutrino flavors in vacuum. It is not known if neutrinos are Dirac particles1 _or

Majo-rana particles2_{. Dirac particles are treated extensively in the standard course}

book for quantum field theory ”An Introduction to Quantum Field Theory” by Michael E. Peskin and Daniel V. Schroeder [5] but Majorana particles are not, therefore section 2 is about the Majorana action and its equation of motion. In section 3 the neutrino oscillation probability is calculated for both Dirac and Majorana neutrinos.

1_{Dirac particles are not their own anti-particles}

2

Majorana

In ”An Introduction to Quantum Field Theory” by Michael E. Peskin and Daniel V. Schroeder [5] chapter 3.2 the Weyl or chiral representation of the proper orthochronous Lorentz group is given. From the chiral representation it is possible to derive two two-dimensional representations of the proper orthochronous Lorentz group, one is called the two-dimensional left-handed representation and the other is called the two-dimensional right-handed rep-resentation. Objects that are transformed by the chiral representation are called Dirac spinors, objects transformed by the two-dimensional left-handed representation are called left-handed Majorana spinors and objects trans-formed by the two-dimensional handed representation are called right-handed Majorana spinors.

In problem 3.4 ”Introduction to Quantum Field Theory” [5] an action for a left handed Majorana spinor is presented. In subsection 2.1 we will show that this action is real and that it leads to the equation of motion known as the Majorana equation. We will also show that the Majorana equation is Lorentz covariant and that the Majorana equation leads to the Klein Gordon equation. In subsection 2.2 we will find the solution to the Majorana Equation.

2.1

Majorana Action

The action for a left-handed Majorana field χ(x) leading to the Majorana equation is [5, p. 73] S = Z d4x  χ†(x)i¯σµ∂µχ+ im 2 {χ T_(x)σ2_χ (x) − χ†(x)σ2χ∗_(x)}  , (1) where m > 0 and we have defined ¯σµ as

¯

σµ_{= (e, −~σ).} (2)

The matrices e and σi _{are defined as}

e=1 0 0 1  , σ1 =0 1 1 0  , σ2 =0 −i i 0  , σ3 =1 0 0 −1  , (3) the matrices σi _{are the Pauli matrices.}

χ(x) to be a field which takes as values Grassmann numbers [5, p. 73]. For Grassmann numbers α and β we have the relations

αβ _{= −βα} for any α and β, (4)

(αβ)∗ = β∗α∗. (5)

We define a Grassmann field ξ(x) as ξ(x) = X

n

αnφn(x), (6)

where αn are independent Grassmann numbers and φn(x) are orthogonal

complex functions. Combining equation (5) and equation (6) we get the following relation for Grassmann fields ξ(x) and ζ(x)

{ξa(x)ζb(x)}∗ = ζb∗(x)ξa∗(x), (7)

where a and b denotes the components of the Grassmann fields ξ(x) and ζ(x). With the help of equation (7) we will now show that the action S is real, or in other words satisfies S = S∗_{. The complex conjugate of the action S}

with the fields and matrices written out in components is S∗ = Z d4x  χ∗_a(x)i¯σ_abµ∂µχb(x) + im 2 {χa(x)σ 2 abχb(x) − χ∗a(x)σ2abχ∗b(x)} ∗ , (8) where a and b denotes components of the various fields/matrices. We now rearrange and get

S∗ = Z d4x_{[ − i¯σab}∗ (χ∗_a(x)∂χb(x))∗− im 2 {σ 2∗ ab(χa(x)χb(x))∗− σab2∗(χ∗a(x)χ∗b(x))∗}]. (9)

If we use equation (7) for the complex conjugated products of the fields and the sigma matrices satisfy

¯

σ∗_ab= ¯σba, (10)

σ_ab2∗= σ_ba2 (11) then the complex conjugated action is given by

A partial integration now leads to S = S∗_{, meaning that the action S is real.}

To see what equation of motion the action S leads to, we vary S with respect to the first and second component of χ∗_{(x) [5, p. 73], we get}

∂S ∂χ∗ 1(x) = [i¯σ_{· ∂χ(x)]}1+ mi2χ∗2(x) = 0, (13) ∂S ∂χ∗₂(x) = [i¯σ· ∂χ(x)]2− mi 2_χ∗ 1(x) = 0. (14)

The equation of motion for the field χ(x) written in a more compact form is i¯σ_{· ∂χ(x) − imσ}2χ∗(x) = 0. (15) This equation is called The Majorana equation.

It is required that the laws of physics are the same in all inertial coordi-nate systems [6, p. 7]. This leads to the fact that the equations of motion are Lorentz covariant. In order to see if the Majorana equation is Lorentz covariant we note how different objects in the Majorana equation transforms under a Lorentz transformation. The transformed objects relate to the un-transformed objects as [5, Ch. 3.2] χ_{(x) → χ}′(x′) = ΛL1 2χ(x), (16) ∂µ→ ∂µ′ = (Λ−1)νµ∂ν, (17) σ2χ∗_{(x) → σ}2χ′∗(x′) = ΛR1 2 σ2χ∗(x), (18) (ΛR1 2) −1_σ¯µ_ΛL 1 2 = Λ µ νσ¯ν. (19) The transformations ΛL 1 2

are the two dimensional left-handed transforma-tions of the proper orthochronous Lorentz group, and the ΛR

1 2

are similarly the right-handed transformations. The transformations Λµ

ν are the

transfor-mations of four-vectors in Minkowski space. The Majorana equation in the new basis x′ _{is given by}

i¯σ_{· ∂χ}′(x′_{) − imσ}2χ′∗(x′) = 0. (20) Insert the fields expressed in the old coordinate system and to the left of ¯σ insert the unit matrix in the form ΛR

1 2

(ΛR

1 2

)−1_{. Move the transformation Λ}L

1 2

over the partial derivative, This is possible because ΛL

1 2

does not depend on x. Remembering relation (19) we get

ΛR1 2(i¯

We now see that we can write

i¯σ_{· ∂χ}′(x′_{) − imσ}2χ′∗(x′) = ΛR1 2

(i¯σ_{· ∂χ(x) − imσ}2χ∗(x)). (22) This shows that the Majorana equation is Lorentz covariant.

That a field χ(x) satisfies the Majorana equation implies that each com-ponent of the field also satisfies the Klein-Gordon equation

(∂µ∂µ+ m2)χ(x) = 0. (23)

To see this we first divide the Majorana equation with i and get ¯

σ_{· ∂χ(x) − mσ}2χ∗(x) = 0. (24) We define a new quantity σµ _as

σµ= (e, ~σ), (25)

because then we have the relation

σ_{· ∂ ¯σ · ∂ = ∂}µ∂µ. (26)

To see if equation (26) is valid, we write the left hand side as ¯

σνσµ∂ν∂µ= e∂0∂0+ σi∂0∂i− σi∂i∂0− σiσj∂i∂j. (27)

Partial derivatives commute so the middle terms cancel out and σi_σj _satisfy

[5, p. xx (sic!)]

σiσj = eδij+ iǫijkσk. (28) Inserting this in relation (27) and remembering that ǫijk _{= −ǫ}jik _{hence leads}

to relation (26).

Multiplying equation (24) with σ · ∂ leads to

∂µ∂µχ(x) − σ · ∂ mσ2χ∗(x) = 0. (29)

Passing mσ2 _{over to the left of σ · ∂ gives}

∂µ∂µχ(x) − mσ2σ¯∗· ∂ χ∗(x) = 0. (30)

The first term is good but what about the second term in equation (30)? To see what the second term is we complex conjugate equation (24),

¯

2.2

Solution to The Majorana Equation

The Majorana equation is given by equation (15) as

i¯σ_{· ∂χ(x) − imσ}2χ∗(x) = 0. (32) Because a field that satisfies the Majorana equation also satisfies the Klein-Gordon equation, as was seen in the end of the previous subsection, the most general form of a field satisfying the Majorana equation is

χ(x) = u(p)e−i(Et−~p·~x)+ v(p)ei(Et−~p·~x), (33) where u(p) and v(p) are two-component complex objects called spinors. We know that the field χ(x) transforms with respect to the left handed proper orthochronous Lorentz group, therefore we can first find a solution for the rest frame where (E = m, ~p = 0) and then use a boost transformation to get the solution for general p. The objects u(p) and v(p) are in the rest frame given by u(p) = ǫ :=a b  , (34) v(p) = η :=c d  , (35)

where a, b, c and d are complex constants. In the rest frame the field χ(x) is then given by

χ(x) = ǫ e−imt+ η eimt. (36) To determine ǫ and η we insert relation (36) into the Majorana equation which give

i21m_(−ǫe−imt+ ηeimt_{) − imσ}2(ǫ∗eimt+ η∗e−imt) = 0. (37) Equation (37) relates ǫ and η by

−mη − imσ2ǫ∗ = 0. (38) The solution for χ(x) in the rest frame is then given as

χ(x) = ǫe−imt+ ηeimt, (39)

η_{= −iσ}2ǫ∗. (40)

To find the solution for general momentum we boost χ(x) with the boost transformation ΛL

1 2

. The field χ(x) for general momentum is then given by [5, p. 45-46]

The spinors u(p) and v(p) are then

u(p) =√σ_{· p ǫ,} (42) v_{(p) = −i}√σ_{· p σ}2ǫ∗. (43) To get the general solution in position space we integrate all momentums and sum all basis spinors, hence we get the general solution in position space

χ(x) = Z d3_p (2π)3 1 √ 2E X r={↑,↓}

(ur(~p)e−i(Et−~p·~x)+ vr(~p)ei(Et−~p·~x)). (44)

We would like the field χ(x) to create particles [5, Ch. 2.3], and insert time independent operator ar_{(~p) that when acting on the vacuum state give zero}

and time independent operator ar†_{(~p) that when acting on the vacuum state}

create a particle with spin r, momentum ~p and energy E(~p). The operator χ(x) is given by χ(x) = Z d3_p (2π)3 1 √ 2E X r={↑,↓}

[ar(~p)ur(~p)e−i(Et−~p~x)+ + ar†(~p)vr(~p)ei(Et−~p~x)],

(45)

where the ar_{(~p) and a}r†_{(~p) acting on the vacuum satisfy}

ar_{(~p)|0i = 0,} (46) p2E(~p)ar†

(~p)|0i = |r, ~p, E(~p)i. (47)

3

Oscillation Probability

In this thesis it will be assumed that only two types of neutrinos exist in the universe, in reality there exists at least three types of neutrinos [4, p. 165].

The oscillation probability between two types of ultra relativistic Dirac neutrinos in vacuum will be calculated in subsection 3.1. And in 3.2 the oscillation probability between two types of ultra relativistic Majorana neu-trinos in vacuum is calculated. In both cases the mass matrices are real and symmetric.

3.1

Oscillations between two Dirac neutrinos in

Vac-uum

The Dirac Lagrangian in vacuum for one neutrino field ψ is [5, p. 43] LDirac = ¯ψ(iγµ∂µ− m · e4×4)ψ

¯

where γµ _{is defined as} γ0 =0 1 1 0  , γi =  0 σi −σi ₀  . (49)

The sigma matrices in equation (49) are defined as in equation (3) and e4×4

in equation (48) is the 4 by 4 unit matrix.

With the help of equation (48) we construct a Lagrangian for two neutrino fields, ψα and ψβ, with a real symmetric mass matrix as

L=_¯ ψα ψ¯β  iγµ∂_µ 0 · e_4×4 0 · e4×4 iγµ∂µ  −m_mαα· e4×4 mαβ · e4×4 αβ· e4×4 mββ · e4×4  ψα ψβ  , (50) where mαα, mββ, mαβ ∈ R and mαα >0, mααmββ > m2αβ. This is the Dirac

Lagrangian equation (48) reproduced twice plus mass terms mixing the two fields.

A symmetric mass matrix can be orthogonally diagonalized by an orthog-onal matrix [7, p. 451]. Define new fields ψ′

1, ψ2′ by ψα ψβ  =cos(θ) · e4×4 − sin(θ) · e4×4 sin(θ) · e4×4 cos(θ) · e4×4  ψ′ 1 ψ′ 2  . (51)

The new fields inserted in the Lagrangian equation (50) leads to the La-grangian

L= ¯ψ′

1(iγµ∂µ− m11· e4×4)ψ1′ + ¯ψ2′(iγµ∂µ− m22· e4×4)ψ′2, (52)

where m11, m22 ∈ R and m11, m22 > 0. The new fields do not mix in the

Lagrangian, hence the solutions for them in operator form is [5, p. 58] ψ_i′(x) = Z d3p (2π)3 1 p2Ei(~p) X r={↑,↓}

[ar_i(~p)ur_i(~p)e−ipx+ br†_i (~p)v_ir(~p)eipx], (53)

ψ_i′†(x) = Z _d3_p (2π)3 1 p2Ei(~p) X r={↑,↓}

[ar†_i (~p)ur†_i (~p)eipx+ bri(~p)vr†i (~p)e−ipx]. (54)

The spinors ur

i(~p)’s and vri(~p)’s are defined as

And the elements ar i, a

r†

i , bri and b r†

i are time independent operators that

satisfy

ar_{i(~p)|0i = 0,} (56) br_{i(~p)|0i = 0,} (57) p2Eiar†i (~p)|0i = |~p, r, Ei,1i, (58)

p2Eibr†i (~p)|0i = |~p, r, Ei,−1i. (59)

The states in equations (58 and 59) are named after their quantum numbers which are from left to right momentum, spin, energy and lepton number.

The neutrino fields ψα(x), ψα†(x) are then given by

ψα(x) = Z d3p (2π)3 X r={↑,↓} [ cos(θ) p2E1(~p)

(ar₁(~p)ur₁(~p)e−ipx+ br†₁ (~p)v₁r(~p)eipx₎₋ sin(θ)

p2E2(~p)

(ar

2(~p)ur2(~p)e−ipx+ b r† 2 (~p)v2r(~p)eipx)]. (60) ψ_α†(x) = Z d3p (2π)3 X r={↑,↓} [ cos(θ) p2E1(~p)

(br₁(~p)vr†(~p)e−ipx+ ar†₁ (~p)ur†(~p)eipx₎₋ sin(θ)

p2E2(~p)

(br₂(~p)vr†(~p)e−ipx+ ar†₂ (~p)ur†(~p)eipx)]. (61) Even though we are interested in the probability that a neutrino change flavor we do not need to find the ψβ field, because the probability that a

neutrino will change flavor can be calculated as one minus the probability that a neutrino will not change flavor.

The eigenstate to the momentum operator given by the neutrino field ψ† α is |~q, r, ψα†, ti = Z d3xq 1 cos2_(θ) 2E1(~q) + sin2_(θ) 2E2(~q) 1 2E1(~q) ei~q~xur_{1, a}(~q)ψ†_{α, a(x)|0i,} (62)

where a is a spinor indices (sum over a), ~q is momentum, r is spin, ψ†

α is

and we get |~q, r, ψα†, ti = 1 q cos2_(θ) 2E1(~q) + sin2_(θ) 2E2(~q)  cos(θ) 2E1(~q) eiE1(~q)t_{|~q, r, E} 1,1i −_2Esin(θ) 2(~q) eiE2(~q)t_{|~q, r, E} 2,1i  . (63)

The normalization for these states are h~q, s, ψ†

α, t|~p, r, ψ†α, ti = (2π)3δ(3)(~p − ~q)δsr. (64)

And the identity operator expressed in these states (including the momentum states created by ψβ) is 1= X r={↑,↓} X λ={α,β} Z d3q (2π)3|~q, r, ψ † λ, tih~q, r, ψ † λ, t|. (65)

For an initial state |Ψi we can project it onto the momentum states with the use of the unit operator equation (65). This is done because we then get the probability density P′_{(t) to find a momentum state created by ψ}

α given by

P′(t) = X

r={↑,↓}

|h~q, r, ψα†, t|Ψi|2. (66)

We create an initial momentum state with spin r as |Ψi = Z d3p 1 p(2π)3 e−(~p−~k)2_/(4σ2₎ (2πσ2₎3/4 |~p, r, ψ † α, t= 0i, (67)

σ is the spread in momentum, ~k is the momentum expectation value. The neutrino oscillation probability P average over initial spin r is then given by

integral which we look up in a table, and get P_{(t) = 1 −} 1  cos2_(θ) 2E1(~k) + sin2_(θ) 2E2(~k) 2      cos2_(θ) 2E1(~k) e−iE1(~k)t ₊ sin 2_(θ) 2E2(~k) e−iE2(~q)t      2 . (69)

We can now go to the case when the momentum ~k is much larger than the masses m11, m22 which is the case for neutrinos. Then we can do the

approximations Ei(~k) = |~k| + m2

ii

2|~k|, i= 1, 2 in the exponents and Ei(~k) = |~k|

otherwise, and get

P_{(t) = 1 − | cos}2(θ)e−i

m2 11

2|~k|t+ sin2(θ)e−i m2

22 2|~k|t|2

= 1 − (cos4(θ) + sin4(θ) + 2 cos2(θ) sin2(θ) cos(t(m

2

11− m222)

2|~k| ) = 1 − (1 + 2 cos2(θ) sin2(θ)(cos(t(m

2 11− m222) 2|~k| ) − 1)) = sin2(2θ) sin2(t(m 2 11− m222) 4|~k| ). (70)

Here the following trigonometric identities have been used [8, p. 410] cos2(θ) + sin2(θ) = 1,

sin2(θ 2) =

1

2(1 − cos(θ)), sin(2θ) = 2 sin(θ) cos(θ).

(71)

In equation (70) the oscillation probability is given in natural units (i.e ~ = c = 1). Another unit system is the SI unit system. We restore ~ and c and get the oscillation probability expressed, in SI units, as

P(t) = sin2(2θ) sin2(t(m

2

11c4− m222c4)

4~E(~k) ), (72) where E(~k) is the neutrino energy.

3.2

Oscillations between two Majorana

neutrinos in Vacuum

fields χα and χβ with a real symmetric mass matrix as L=χ † α χ†_β T  i¯σµ_∂ µ 0 · e2×2 0 · e2×2 i¯σµ∂µ  χα χβ  + i 2 χT α χT_β T mαα· σ2 mαβ · σ2 mαβ· σ2 mββ· σ2  χα χβ  −χ † α χ†_β T mαα· σ2 mαβ· σ2 mαβ· σ2 mββ· σ2  χ∗ α χ∗_β ! , (73) where mαα, mββ, mαβ ∈ R and mαα >0, mααmββ > m2αβ.

As in the Dirac case where we could orthogonally diagonalize the mass matrix, the mass matrix in equation (73) can also be orthogonally diagonal-ized. We define new fields χ′

1 and χ′2 as χα χβ  =cos(θ) − sin(θ) sin(θ) cos(θ)  χ′ 1 χ′ 2  . (74)

The Lagrangian for the new fields is L= " χ′₁† χ′₂† #T  i¯σµ_∂ µ 0 · e2×2 0 · e2×2 i¯σµ∂µ  χ′ 1 χ′₂  + i 2   χ′_T 1 χ′₂T T m11· σ2 0 · e2×2 0 · e2×2 m22· σ2  χ′ 1 χ′ 2  − " χ′₁† χ′₂† #T m11· σ2 0 · e2×2 0 · e2×2 m22· σ2  χ′_∗ 1 χ′₂∗   , (75) where m11, m22 ∈ R and m11, m22>0. Because the new fields do not mix in

the Lagrangian, the solutions to the new fields can be found from equation (45).

The neutrino field χα is then given by

χα = Z d3p (2π)3 X r={↑,↓} ( cos(θ) p2E1(~p)

[ar₁(~p)ur₁(~p)e−ipx+ ar†₁ (~p)vr₁(~p)eipx_]− sin(θ)

p2E2(~p)

[ar₂(~p)ur₂(~p)e−ipx+ ar†₂ (~p)v₂r(~p)eipx]),

(76)

where the operators ar i and a

r†

i acting on the vacuum satisfy the relations

The field χβ is not needed because the neutrino oscillation probability

will be calculated as one minus the probability that a neutrino do not change flavor.

In order to simplify calculations regarding momentum states we specify the momentum to be in z-direction and the spins to be ǫ↑ _{= (1, 0) and}

ǫ↓ = (0, 1). In the Dirac case we calculated the oscillation probability for ultra relativistic neutrinos, and we will do this also here. Therefore we approximate the energy in the spinors ur_{(~p)’s and v}r_{(~p)’s to depend only on momentum}

~

p. We then get the following relations between Majorana spinors −u↑†1 (~p)v ↑ i(~p) = 0, −u↑†1 (~p)v ↓ i(~p) = 2p3, −v1↓∗†(~p)u ↑∗ i (~p) = 2p3, −v1↓∗†(~p)u ↓∗ i (~p) = 0. (79)

The eigenstates with momentum ~q, spin r and flavor χα are then

|~q, ↓, χα, ti = Z d3xq 1 cos2_(θ) 2E1(~q) + sin2_(θ) 2E2(~q) ei~q~x(−1)u ↑† 1 (~q) 2q3 χα|0i = cos(θ) 2E1(~q)e iE1(~q)t|~q, ↓, χ′ 1i − sin(θ) 2E2(~q)e iE2(~q)t|~q, ↓, χ′ 2i q cos2_(θ) 2E1(~q) + sin2_(θ) 2E2(~q) , (80) |~q, ↑, χα, ti = Z d3xq 1 cos2_(θ) 2E1(~q) + sin2_(θ) 2E2(~q) ei~q~x(−1)v ↓∗† 1 (~q) 2q3 χ ∗ α|0i = cos(θ) 2E1(~q)e iE1(~q)t|~q, ↑, χ′ 1i − sin(θ) 2E2(~q)e iE2(~q)t|~q, ↑, χ′ 2i q cos2_(θ) 2E1(~q) + sin2_(θ) 2E2(~q) , (81)

where {↓} is spin down, {↑} is spin up and t is time. The normalization for these states is

h~p, r, χα, t|~q, s, χα, ti = (2π)3δrsδ(3)(~p − ~q). (82)

To calculate the neutrino oscillation probability we do exactly as we did in the Dirac case 3.1, therefore only the major parts will be outlined in the following.

The identity operator given in momentum states (including the momen-tum states created by χβ) is given by

An initial momentum state with spin r and average momentum ~k is con-structed at t = 0 as |Ψi = Z d3q 1 p(2π)3 e−(~q−~k)2/(4σ2) (2πσ2₎3/4 |~q, r, χα, t= 0i, (84)

σ is here the spread in momentum. The neutrino oscillation probability averaged over spin states r is then given by

P_{(t) = 1 −} 1 2 X s,r={↑,↓} Z d3p 1 (2π)3|h~p, s, χα, t|Ψi| 2 = 1 − Z d3qe −(~q−~k)2_/(2σ2₎ (2πσ2₎3/2    cos2_(θ) 2E1(~q)e −iE1(~q)t+sin2(θ) 2E2(~q)e −iE2(~q)t    2  cos2_(θ) 2E1(~q) + sin2_(θ) 2E2(~q) 2 . (85)

We here see that the probability is exactly the same as in the Dirac case 3.1. So by doing the same assumptions and approximations we get the neutrino oscillation probability as

P(t) = sin2(2θ) sin2(t(m

2

11− m222)

4|~k| ). (86)

And given in SI units it is

P(t) = sin2_{(2θ) sin}2₍t(m112 c4− m222c4)

4~E(~k) ). (87)

4

Conclusion

The oscillation probability calculated in section 3.1 have amplitude 1 2sin

2_(2θ)

and angular speed 2(m211c4−m222c4)

4~E(~k) . The appendix contains a graph with two

curves to illustrate how the angular speed depends on the mass difference. The curve with amplitude 1 have the lowest mass difference.

The neutrino oscillation probability calculated in section 3.2 is equal to the neutrino oscillation probability calculated in section 3.1. We conclude from this that in a neutrino oscillation experiment it is not possible to distin-guish between Dirac and Majorana neutrinos if the neutrinos are subjected to the following constraints

• Two neutrinos in the Lagrangian.

• Neutrinos are ultra relativistic. • Neutrinos in vacuum.

The constraints that was used in this thesis could be altered in the fol-lowing ways

• One or more neutrinos could be included in the Lagrangian.

• Neutrinos could have narrow spread in position space instead of mo-mentum space.

• Non ultra relativistic neutrinos could be used instead of ultra relativistic ones.

• Neutrinos could interact with other particles. • Mass matrix could be complex.

This could lead to different oscillation probability for Dirac and Majorana neutrinos.

5

Summary

In the sixties during the 20th century the neutrino flux from the sun was measured to be one third of the neutrino flux as predicted from known theory at the time [2]. This difference in neutrino flux between experiment and theory was named ”The Solar Neutrino Problem”.

The solution to The Solar Neutrino Problem was found in the first years of the 21st century when experiments confirmed that neutrinos could oscillate between flavor states [2].

In this thesis the oscillation probability between two ultra relativistic Dirac/Majorana neutrinos with a narrow spread in momentum in vacuum was calculated, in both cases the mass matrix was real and symmetric. It was found that the oscillation probability was the same for both Dirac and Majorana neutrinos with those conditions and it is given in SI units as P(t) = sin2(2θ) sin2(t(m112 c4−m222c4)

4~E(~k) ), where θ is a mixing angle, m11 and m22

are the neutrino masses, t is time and E(~k) is the kinetic energy of one of the neutrinos.

References

[1] Bahcall J N. How the Sun Shines.

http://www.nobelprize.org/nobel prizes/physics/articles/fusion/. 19 Apr 2012.

[2] Bahcall J N. Solving the Mystery of the Missing Neutrinos.

http://www.nobelprize.org/nobel prizes/physics/articles/bahcall/. 19 Apr 2012.

[3] Goswami S. Solar Neutrino Experiments: An Overview. arXiv:hep-ph/0303075v1. 19 Apr 2012.

[4] Povh B, Scholz C, Rith K, Zetsche F. Particles And Nuclei. sixth edition. :Springer; 2008.

[5] Peskin M E, Schroeder D V. An introduction to quantum field theory. : Westview Press; 1995.

[6] Rindler W. Introduction to Special Relativity. : Oxford University Press Inc; 1991.

[7] Lay C L. Linear algebra and its applications. third edition. : Greg Tobin; 2003.

A

Graph

Neutrino Oscillation probability

Time (s)

Probability