Majorana and Dirac Neutrinos: an Introduction

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Majorana and Dirac Neutrinos: An Introduction

Leonardo Peres [email protected]

Physics Institute Astroparticle Journal Club

October 1, 2020

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 1 / 31 Overview

1 Motivation

2 Dirac Equation

3 Helicity and Chirality

4 Mass and Standard Model

5 Nature of Neutrinos

6 Experimental Status

7 Conclusion

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 2 / 31 Motivation

Neutrinos I Neutrinos are the lighter particles discovered and the most abundant massive particle. I One of the main characteristics is that neutrinos are neutral particles and have spin 1/2. They do not interact via Electromagnetic force. I These particles are key to the development of a physics beyond the Standard Model. Since, they have many open questions. e.g. Neutrino Mass Hierarchy, CP Violation, Neutrino Mass and Neutrino Oscillations. I These three properties combine (massive, chargeless and spin 1/2 ) opens the possibility that neutrinos could have diﬀerent properties when compare to other leptons. They can be Majorana particles.

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 3 / 31 Dirac Equation

The Dirac Equation describes the evolution of the fields that represents spin 1/2 particles (these type of fields belong to SU(2) group and are called spinors). µ i(γ ∂µ − m)ψ = 0 (1) To be consistent with the energy–momentum relation E2 = p2 + m2, the Gamma matrices need to satisfy the Clifford Algebra:

µ ν µν {γ γ } = −2η ˆI (2)

Every solution for the 4x4 Gamma matrices that obeys the equation above is called a Representation, although they have diﬀerent properties.

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 4 / 31 Solutions of Dirac Equation

Considering free-particle solutions for the Dirac Equation and the Dirac Representation for the Gamma matrices. Writing the solutions of Dirac Equation separately, we have for solutions that correspond to particles:

+i(p·x−Et) ψi(x, t) = uie

The spinors ui are:

Whereas, for anti particles, we have:

−i(p·x−Et) ψi(x, t) = νie

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 5 / 31 Solutions of Dirac Equation

The spinors νi are:

For one particle in rest (p=0), the spinors ui are eigenstates of Sz, represented by: 1 1 σz 0 Sz = Σz = 2 2 0 σz The spinors that are solutions of Dirac Equation are not eigenstates of Sz, only if the partcles/antiparticles be propagating in ±z direction. ~p = ±pzˆ.

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 6 / 31 Helicity and Chirality

Incompatibility between Spin and Dirac Spinors

1 Sz and Hˆ D, the Dirac Hamiltonian, do not commute, it is not possible to construct a simultaneous basis of eigenstates.

Better than introduce a new basis, it is more convinient to work the pseudoscalar quantity called Helicity.

S~ · ~p Σˆ · pˆ 1 σ · pˆ 0 h = = = (3) |p| 2p 2p 0 σ · pˆ

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 7 / 31 Helicity and Chirality

Some properties of Helicity

1 Helicity and Dirac’s Hamiltonian do commute! [Σˆ · p,ˆ HˆD] = 0. 2 Helicity is not invariant under Lorentz Transformations. 3 Spinors of particles/antiparticles of spin 1/2 have eigenvalues of helicity ±1/2.

Two eigenstates of Helicity are possible. With eigenvalues: +1/2 right-handed(RH) e -1/2 left-handed(LH).

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 8 / 31 Helicity and Chirality

The eigenstates of Helicity are:

For particles, and for antiparticles:

θ θ Considering c = cos 2 e s = sen 2 , with the particle propagating in an arbitrary direction ~p = p(sinθcosφ, senθsenφ, cosθ).

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 9 / 31 Helicity and Chirality

Introducing the concept of Chirality: In the limit E >> m, the eigenstates of helicity become, approximately:

In this limit, and only in this limit, the spinors also are eigenstates of the following operator:

0 0 1 0 5 0 1 2 3 0 0 0 1 γ = iγ γ γ γ =   (4) 1 0 0 0 0 1 0 0

Which is called Chirality.

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 10 / 31 Helicity and Chirality

The eigenstates of chirality, therefore, are the same of helicity in the limit E >> m:

And, all Dirac’s spinors can be decomposed into chiral components of RH and LH by the operators: 1 1 P = (1 − γ5) ,P = (1 + γ5) R 2 L 2 Therefore, any spinor that describes a fermion can be written in the form: ψ = PRψ + PLψ.

Some Properties of PL and PR

PR + PL = 1 PLPR = 0 PRPR = PR PLPL = PL

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 11 / 31 Mass and Standard Model

I How to construct the Lagrangian for Spinors? I The terms of the Lagragian should be Lorentz scalars. I How to construct a Lorentz scalars from Dirac Spinors?

ψ¯ = ψ†γ0 ψψ¯ is a Lorentz Scalar. The Dirac Hamiltonian can be written as:

¯ µ ∂ LD = ψ iγ − m ψ (5) ∂xµ

Writing in terms of chiral components ψ = ψL + ψR, the mass term of the Lagrangian couples right- and left-handed ﬁelds. For mass terms be produced, left- and right-handed neutrinos are required. ¯ ¯ LD ∝ m(ψRψL + ψLψR)

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 12 / 31 Mass and Standard Model

I As an experimental fact that only left-handed neutrinos have been observed, neutrinos in SM remain massless, as the best theoretical description. But... Neutrino Oscillations is an experimental provement that Neutrinos do have mass!

I How a charged lepton acquire his mass? And what are the options to describe a mechanism to neutrinos have mass.

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 13 / 31 Mass and Standard Model

I Charged Leptons acquire mass through Higgs Mechanism. I The expected value of a scalar neutral ﬁeld Higgs in the vacuum is 0 diﬀerent from zero h0| φ |0i = √1 . 2 v

0 Φ = √1 2 v + H(x) I Expanding near the vaccum, we have the excitation H(x) in the ﬁeld.

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 14 / 31 Mass and Standard Model

The coupling is given by the Yukawa term, considering a Left-handed doublet of the charged lepton and the the neutrino corresponded and a singlet right handed ﬁeld of the charged lepton.

X l ¯ LH,L = − YαβLαLΦlβR + H.c. α,β=e,µ,τ

I Examples: νL LH Doublet: LeL = eL RH Singlet: leR = eR

v + H X ¯ l LH,L = − √ lαL yα lαR + H.c. 2 α=e,µ,τ

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 15 / 31 Mass and Standard Model

l l X yαv ¯ X yα ¯ LH,L = − √ lαlα − √ lαlαH (6) α=e,µ,τ 2 α=e,µ,τ 2

I The ﬁrst term is the coupling with the Higgs ﬁeld that gives the term of mass, while the second term is the coupling of the fermions and the Higgs Boson. Therefore, the mass of the fermion is the coupling term with the Higgs Field: l yαv mα = √ (7) 2

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 16 / 31 Nature of Neutrinos

Keep in mind Neutrino interactions are described, with an impressive accuracy, by the Standard Model.

What about Neutrino mass?

1 Dirac Neutrinos - Neutrinos masses are produced by the same mechanism of the other leptons.

2 Majorana Neutrinos - Other unknown mechanism gives neutrinos masses and basically implies that neutrino and antineutrino are the same.

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 17 / 31 Nature of Neutrinos

Dirac Neutrino

To produce a Dirac mass term, the Lagrangian must include a coupling between left- and right-handed neutrino ﬁelds, the right-handed must not interact via Weak force, so it is called sterile ﬁeld.

X l ¯ X ν ¯ ˜ LH,L = − Yαβ LαL Φ lβR − Yαβ LαL Φ νβR + H.c. α,β=e,µ,τ α,β=e,µ,τ

Concerning to neutrinos is a little more tricky, because the weak eigenstates are not the same mass eigenstates, but after a change of basis, we can write in the same way:

l 3 ν l 3 ν X yαv ¯ X yk v X yα ¯ X yk LH,L = − √ lαlα− √ ν¯kνk− √ lαlαH− √ ν¯kνkH α=e,µ,τ 2 k=1 2 α=e,µ,τ 2 k=1 2

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 18 / 31 Nature of Neutrinos

Problem solved?

I The coupling constant y should be very small.

I There is no evidence that the same mechanism corresponds to both eﬀects.

I The SM does not predict the mass of the particles.

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 19 / 31 Nature of Neutrinos

Majorana Neutrino Considering the Dirac Lagrangian:

µ µ ¯ ¯ LD = iγ ∂µψR + iγ ∂µψL − m(ψRψL + ψLψR)

Using Euler-Lagrange equation, we have two coupled equations:

µ µ iγ ∂µψR = mψL iγ ∂µψL = mψR

If m = 0, we have the Weyl Equations:

µ µ iγ ∂µψR = 0 iγ ∂µψL = 0

The neutrino (LH field) and antineutrino (RH field) is describe with only two components for massless particles. As neutrino have mass, is there a way to describe a massive field of neutrino with only 2 components? Yes!!! Majorana Field.

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 20 / 31 Nature of Neutrinos

Majorana Neutrino

The Field must satisfy Majorana Condition.

¯T ψR = CψL (8)

The charge conjugation matrix depends on the Representation. Writing a Majorana Field: ¯T ψ = ψL + ψR = ψL + CψL

But, the condition is exactly the Charge Conjugation operation for ψL field. Cˆ ¯T c c ¯T c ψL −→CψL = ψL ψR = ψL ψ = Cψ ψ = ψ The Majorana condition implies that the charged conjugated field is the same field. Particle and antiparticle fields are the same. The charged current of a Majorana Field is always zero, by definition. (Ok! Neutrinos are chargeless particles).

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 21 / 31 Nature of Neutrinos

Majorana Mass Term

I Given the Dirac Lagrangian below, it is necessary to find a right- handed function of the left-handed field which transforms as left- handed field under Lorentz transformations and can be replaced by the right-handed field. D Lmass = −mνν¯ = −m(¯νRνL +ν ¯LνR) Using Majorana Condition:

c T νR = νL = Cν¯L 1 LM = [¯ν iγµ∂ ν +ν ¯c iγµ∂ νc − m(¯νc nu +ν ¯ νc )] 2 L µ L L µ L L L L L

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 22 / 31 Nature of Neutrinos

Majorana vs. Dirac Neutrino CPT Theorem CPT operation is a fundamental symmetry of the Universe. It converts particles into antiparticles with opposite eigenstate of Chirality.

Figure 1: Dirac Neutrino Scheme

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 23 / 31 Nature of Neutrinos

Figure 2: Majorana Condition

Figure 3: Dirac Neutrino and Majorana Neutrino

T Particle??? Antiparticle??? ν = νL + Cν¯L

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 24 / 31 Experimental Status

Neutrinoless Double Beta Decay

I The most promising experimental approach is the search for Neutri- noless Double Beta Decay (2β0ν ). I The Beta Decay of some nucleus are energetically forbidden, or highly suppressed. If the parent nucleus is also stable against alpha and gamma decays, the Double Beta Decay occurs.

The SM predicts:

− (Z,A) −→ (Z + 2,A) + 2e + 2¯νe

If neutrinos are Majorana particles, the following process is also possible:

(Z,A) −→ (Z + 2,A) + 2e−

Majorana particles implies Lepton Number Violation! ∆L = 2

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 25 / 31 Experimental Status

Why the SM does not allow a 2β0ν?

The particle–antiparticle mismatch

A ν¯e emitted in the upper leptonic vertex cannot be absorbed in the lower leptonic vertex, which is capable only of absorbing a νe.

The helicity mismatch The helicity of the neutral lepton emitted in the upper leptonic vertex is positive and the lower leptonic vertex can absorb only a neutral lepton with negative helicity.

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 26 / 31 Experimental Status

If neutrino is a Majorana Particle ...

Particle–antiparticle matching

ν¯e = νe. The electron neutrino must be a Majorana particle. In this case, the total lepton number is not conserved.

Helicity matching

mνe 6= 0. In this case, the upper leptonic vertex can emit a neutrino with negative

helicity with relative amplitude mνe /Eνe , which is absorbed by the lower leptonic vertex with relative amplitude equal to unity.

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 27 / 31 Experimental Status

I 2β0ν has a unique signature in the spec- trum since there are no neutrinos that could ”take the energy away” available in the decay.

I 11 isotopes were identify capable of double beta decay.

I The ﬁrst detection a 2β2ν event was made in 1987.

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 28 / 31 Experimental Status

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 29 / 31 Conclusion

I This is an introduction to the subject, there is the possibility of neutrinos be a combination of a Dirac and Majorana particle which com- plicates the scenario a little. I There are others experimental approaches to this question, as search for neutrino decays, and measurements of very low energy neutrino’s interactions. I The Standard Model describes, in good agreement with measurements, all neutrino interactions with others SM particles, even considering massless neutrinos. I The fact that neutrinos have mass and do not have charge allows the possibility of neutrinos be Majorana particles. I It is unlikely that Higgs Mechanism produces neutrino masses. I The importance of determinate if neutrinos are Dirac or Majorana particles lays in the possibility of exclude proposed models for physics beyond the SM. Thank you! Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 30 / 31 Bibliography

1 Symmetry and the Standard Model, Matthew Robinson, Springer, 2011.

2 Neutrino Physics, Kai Zuber, Third Edition.

3 Modern Particle Physics, Mark Thomson, Cambridge University Press, India, 2015.

4 Fundamentals of Neutrino Physics and Astrophysics, Carlo Giunti and Chung W. Kim, Oxford University Press.

5 Particle Physics, B.R. Martin and G. Shaw, Fourth Edition, 2017, John Wiley Sons.

6 P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020).

Leonardo Peres Majorana and Dirac Neutrinos October 1, 2020 31 / 31