Models of Neutrino Masses Fernando Romero López 13.05.2016
1 Introduction and Motivation 3 2 Dirac and Majorana Spinors 4 3 SU(2) L U(1) Y Extensions 11 4 Neutrino masses in R-Parity Violating Supersymmetry 20
1 Introduction and Motivation 1. Neutrinos are the lightest known leptons, although its mass has been proven to be finite. Neutrinos Oscillations. 2. Standard Model predicts vanishing neutrino masses need SM extension. 3. Easiest extension in the theory SU(2) L U(1) Y with Majorana Mass Term. 4. Othe possibilities using Supersymmetric models. Fernando Romero López Universität Bonn 3/36
2 Dirac and Majorana Spinors Dirac Spinor............................ 5 Majorana Representation..................... 6 Majorana Fermions in other Representation........... 7 Masses of Majorana Fermions................... 8 Chirality of Conjugate Fields................... 9 Mass Matrix............................ 10 based on: [1] Palash B. Pal. Dirac, Majorana and Weyl fermions. Am. J. Phys., 79:485 498, 2011.. Fernando Romero López Universität Bonn 4/36
Dirac Spinor A general complex solution of the Dirac equation is called Dirac Spinor (iγ µ µ m)ψ = 0, (1) ψ(x) = a s (p)u s (p)e ipx + b s(p)v s (p)e +ipx. (2) s p the 4x4 γ µ matrices need to satisfy: {γ µ, γ ν } = 2g µν. γ 0 γ µ γ 0 = γ µ. Fernando Romero López Universität Bonn 5/36
Majorana Representation Using purely imaginary γ matrices Majorana Representation ( ) ( ) 0 σ2 iσ1 0 γ 0 =, γ 1 =, (3) σ 2 0 0 iσ 1 ( ) ( ) 0 σ2 iσ3 0 γ 2 =, γ 3 =. (4) σ 2 0 0 iσ 3 and imposing ψ = ψ in this representation, one has real solutions, i.e. Majorana Fermions. ψ(x) = a s (p)ũ s (p)e ipx + a s(p)ũ s(p)e +ipx. (5) s p Fernando Romero López Universität Bonn 6/36
Majorana Fermions in other Representation In a general representation, the condition is ψ = ˆψ γ 0 Cψ, and it is Lorentz invariant. ψ(x) = a s (p)u s (p)e ipx + a s(p)v s (p)e +ipx, (6) s p with v s (p) = γ 0 Cu s. The Charge Conjugation matrix C has the following properties: C 1 γ µ C = γµ T. Unitary, C 1 = C. Antisymmetric, C T = C. Fernando Romero López Universität Bonn 7/36
Masses of Majorana Fermions Equivalently to real scalar fields, a factor 1/2 appears in the Lagrangian: L = 1 ( ψi / ψ ψψ) m (7) 2 Since ˆψ = γ 0 Cψ = ψ, and using the properties of C: ψ = ψ T C. (8) The mass term contains two powers of the field, so the factor 1/2 is cancelled when taking the equations of motion. Fernando Romero López Universität Bonn 8/36
Chirality of Conjugate Fields The conjugate of a left handed Majorana field is right handed: (ψ L ) c = ˆψ R and (ψ R ) c = ˆψ L In Dirac Representation, C = iγ 2 γ 0. Calculate: P L (ψ L ) c = P L γ 0 C(ψ L ) = P L γ 0 C(P L ψ) = P L γ 0 (iγ 2 γ 0 )P L ψ. Using: γ µ P R = P L γ µ, a product P L P R = 0 appears. Fernando Romero López Universität Bonn 9/36
Mass Matrix Arbitrary mass term with a SYMMETRIC mass matrix. Example: L mass = 1 ) ( η ( ) ( ) m µ ˆηR L χ L + h.c. (9) 2 µ M ˆχ R A Dirac mass term for a field Ψ can be obtained by setting m = M = 0 and identifying Ψ L η L and Ψ R ˆχ R : L mass = 1 2 µ( Ψ L Ψ R + ˆΨL ˆΨR )+ h.c. = µ( Ψ L Ψ R )+ h.c. = µ ΨΨ (10) Fernando Romero López Universität Bonn 10/36
3 SU(2) L U(1) Y Extensions Standard Model of Electroweak Interactions........... 12 Yukawa mass term......................... 13 See-saw mechanism........................ 16 Neutrinoless double beta decay (0νββ).............. 19 based on: [2] Rabindra N Mohapatra and Palash B Pal. Massive Neutrinos in Physics and Astrophysics. Fernando Romero López Universität Bonn 11/36
Standard Model of Electroweak Interactions Gauge transformation in the Standard Model (SU(2) L U(1) Y ), where σ a are the Pauli matrices. For quarks: ( ) ( ) u L (x) d L (x) = e iy1β(x) e i σa 2 θa(x) ul (x), (11) d L (x) d R(x) = e iy 2β(x) d R (x), (12) u R(x) = e iy 3β(x) u R (x). (13) Equivalent for leptons, however the sterile ν R field is not included ( singlet under all interactions). Fernando Romero López Universität Bonn 12/36
Yukawa mass term In the SM, masses of fermions are included via Yukawa terms: ( ) ( ) L Y = c d (ū, d ) φ (+) L φ (0) d R c u (ū, d ) φ (0) (x) L φ ( ) u R + h.c.. (x) ( )(14) 0 where the Higgs field acquires a v.e.v. after the SSB φ =. v In the lepton sector, only the first one is included (equivalent to the d quark) and gives masses to charged leptons. Nothing forbids the inclusion of a u quark-like term, that gives masses to neutrinos Simplest SM extension with ν R Fernando Romero López Universität Bonn 13/36
Yukawa mass term is in general not diagonal: ν l = U lα ν α Neutrino Mixing. L CC = g W µ ll γ µ U lα ν Lα + h.c. (15) 2 However: ij 1. Upper bound from kinematical considerations is: m νe < 2.2 ev. 2. At least, 10 5 smaller mass than electron, and 10 11 for the t. 3. It seem unlikely that a single mechanism creates masses that differ by at least 10 11. 4. The added ν R cannot be detected. (Only interacts via Yukawa) Fernando Romero López Universität Bonn 14/36
Figure 1: Kurie Plot from [3]. Fernando Romero López Universität Bonn 15/36
See-saw mechanism In the simplest model, only terms with ν R now include ˆν R and ˆν L, but breaking B L symmetry. L mass = 1 ) ( ν ( ) ) 0 M T (ˆνR L ˆνL + h.c. (16) 2 M B ν R Note: diagonal terms are directly gauge invariant. The mass matrix can be decomposed: M = R T M D KR, ( ) cos θ sin θ with R = and K = sin θ cos θ ( ) 1 0, for 1 family of ν. 0 1 Fernando Romero López Universität Bonn 16/36
The eigenstates are ( ) n1l n 2L = R ( ) νl ˆν L and ( ) n1r n 2R = KR ( ) ˆνR The eigenvalues when B >> M are m 1 = M 2 /B and m 2 = B. The mass term reduces to: L mass = m 1 2 n L1n 1R m 2 2 n L2n 2R + h.c. (17) Masseigenstates n 1 and n 2 are Majorana fields: n 1 = n 1R + n 1L = cos θ(ν L ˆν R ) sin θ(ˆν L ν R ) = ˆn 1 n 2 = n 2R + n 2L = sin θ(ν L + ˆν R ) + cos θ(ˆν L + ν R ) = ˆn 2 ν R Fernando Romero López Universität Bonn 17/36
Summary of the See-saw Mechanism 1. A non-trivial mass matrix can produce two Majorana eigenstates. 2. By generating a small mass for a neutrino (m 1 = M 2 /B), a huge mass appears (m 2 = B). 3. Putting some numbers: M = m τ = 1.78 GeV and B = 5 10 9 GeV gives m 1 1 ev. 4. Two widely different scales Fine tuning?. Fernando Romero López Universität Bonn 18/36
Neutrinoless double beta decay (0νββ) Majorana mass term Lepton number violating process. It would prove the Majorana nature of neutrinos. Not observed yet. Figure 2: [2] Ongoing experiment GERDA. Fernando Romero López Universität Bonn 19/36
4 Neutrino masses in R-Parity Violating Supersymmetry Brief Introduction to SUSY.................... 21 Minimal Particle Content..................... 22 R Parity.............................. 23 Consequences of R parity Conservation.............. 25 Breaking R parity......................... 26 Neutrino-Neutralino Mixing.................... 28 Summary of ν Masses in /R p SUSY................ 32 based on: [4] R. Barbier et al. R-parity violating supersymmetry Fernando Romero López Universität Bonn 20/36
Brief Introduction to SUSY Figure 3: [5] Fernando Romero López Universität Bonn 21/36
Minimal Particle Content Each particle has its superpartner. Two Higgs doublets, that give mass to up and down/charged leptons, respectively. Figure 4: [4] Notice: 7 neutral fermions! Fernando Romero López Universität Bonn 22/36
R Parity One can assign an additive quantum number to particle and the opposite to its superpartner. This would build a symmetry group Z 2. This condition is simply realised by R = ( 1) 2S+3(B L). In the Minimal Supersymmetric Standard Model (MSSM), R Parity is conserved. Fernando Romero López Universität Bonn 23/36
Figure 5: [4] Fernando Romero López Universität Bonn 24/36
Consequences of R parity Conservation Supersymmetric particles come in pairs. Lightest supersymmetric particle is stable. Figure 6: [6] Fernando Romero López Universität Bonn 25/36
Breaking R parity R Parity violation can have mix between particles and superparticles. The neutral electroweakly interacting spin 1/2 particles are now 7: with i = 1, 2, 3. (ψ 0 ) T = ( iλ γ, iλ Z0, h 0 u, h 0 d, ν i ), (18) However, h 0 d, ν i mix ν α, with α = 0, 1, 2, 3. Fernando Romero López Universität Bonn 26/36
In SUSY, the interactions are described by the Superpotential and the Supersymmetry breaking by the Soft Potential. They contain terms such as L µ α H u L α that create a mass term for ψ 0 (neutral and s=1/2), L mass = 1 2 (ψ0 ) T M N ψ 0. Figure 7: [4] Fernando Romero López Universität Bonn 27/36
Neutrino-Neutralino Mixing We can work in a basis for the ν α where µ α = (µ cos ξ, 0, 0, µ sin ξ) and the v.e.v is v α = (v d, 0, 0, 0). Other parameters: M 1, M 2 are masses of U(1) Y and SU(2) L gauginos; g is the SU(2) L coupling and tan β = v u /v d. Figure 8: [4] Fernando Romero López Universität Bonn 28/36
ν 1 and ν 2 remain massless at tree level. 4 heavy eigenstates (Neutralinos) and one light eigenstate appear (All Majorana fermions). In the limit sin ξ << 1, the light eigenstate coincides almost with ν 3 with: m ν3 = MZ 2 cos2 β(m 1 c 2 W + M 2s 2 W )µ cos ξ M 1 M 2 µ cos ξ MZ 2 sin 2β(M 1c 2 W + M 2s 2 W ) tan2 ξ. (19) ξ is called the misalignment angle : sin ξ 3 10 6 1 + tan 2 β m ν /1 ev. (20) Fernando Romero López Universität Bonn 29/36
In the basis of the chargino eigenstates (e, µ, τ are the lightest), the massive neutrino has the following composition: ν 3 = 1 3i=1 (µ 1 ν e + µ 2 ν µ + µ 3 ν τ ). (21) µ 2 i This also defines the neutrino mixing angles for oscillations: sin θ 13 = µ 1 3i=1, (22) µ 2 i sin θ 23 = µ 2 µ 2 2 +. (23) µ2 3 Fernando Romero López Universität Bonn 30/36
The other neutrino masses and the last mixing angle (sin 2 θ 12 ) appear only in radiative NLO corrections. Example of radiative correction induced by trilinear /R P coupling: Figure 9: [4] Fernando Romero López Universität Bonn 31/36
Summary of ν Masses in /R p SUSY. 1. Particles have R p = 1 and superparticles R p = 1. 2. Allowing R Parity violation neutrino masses! 3. Mass through neutrino-neutralino mixing: (ψ 0 ) T = ( iλ γ, iλ Z0, h 0 u, h 0 d, ν i). 4. Neutrino masses/mixings at tree level but also need loop level. Fernando Romero López Universität Bonn 32/36
Last Comments 1. There is a certain consensus regarding the fact that neutrinos have a different mechanism for mass generation. 2. The explained models are only a selection of the existent models. Additional models include extended Higgs sector (triplet, singlet), for example. 3. Finite neutrino masses are already proved and have phenomenological implications next talk. 4. The predicted Majorana nature of neutrinos can be experimentally tested. Fernando Romero López Universität Bonn 33/36
References [1] Palash B. Pal. Dirac, Majorana and Weyl fermions. Am. J. Phys., 79:485 498, 2011. [2] Rabindra N Mohapatra and Palash B Pal. Massive Neutrinos in Physics and Astrophysics. World Scientific Lecture Notes in Physics. World Scientific, Singapore, 1991. [3] Rastislav Dvornicky, Kazuo Muto, Fedor Simkovic, and Amand Faessler. The absolute mass of neutrino and the first unique forbidden beta-decay of 187Re. Phys. Rev., C83:045502, 2011.
[4] R. Barbier et al. R-parity violating supersymmetry. Phys. Rept., 420:1 202, 2005. [5] https://www.quantamagazine.org/. [6] Stephen P. Martin. A Supersymmetry primer. 1997. [Adv. Ser. Direct. High Energy Phys.18,1(1998)]. Fernando Romero López Universität Bonn 35/36
Thanks for your attention!