Introduction to neutrino mass models Lecture 1: Weyl, Majorana, Dirac, SM Igor Ivanov CFTP, Instituto Superior T´ecnico,Lisbon University of Warsaw January 8-11, 2018 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Content of the lecture set 1 Basics: Weyl vs Majorana vs Dirac fermions; various mass terms 2 Seesaw and radiative mass models 3 Tribimaximal mixing from A4 symmetry group Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 2/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM 1 Weyl spinors 2 Dirac vs Majorana fermions 3 Charge conjugation 4 Left and right-handed projectors 5 Mass terms 6 Neutrinos in the SM Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 3/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Weyl fermions Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 3/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Weyl spinors Isotropy laws are SO(3) symmetric basic objects (fields) transform under ! ! SO(3) irreducible representations: scalar φ(x), vector Ai (x), tensor Tij (x), etc. Relativistic theory: laws are Lorentz invariant fields transform as irreducible representations of SO(1; 3). But since ! so(1; 3) su(2)L su(2)R ; ' × we get more representations: scalar φ(x), left-handed Weyl spinor a(x) y (a = 1; 2), right-handed Weyl spinor a_ (_a = 1; 2), 4-vector Aµ(x), tensors etc. Chiral spinors exist because our space-time can accommodate them! Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 4/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Weyl spinors Isotropy laws are SO(3) symmetric basic objects (fields) transform under ! ! SO(3) irreducible representations: scalar φ(x), vector Ai (x), tensor Tij (x), etc. Relativistic theory: laws are Lorentz invariant fields transform as irreducible representations of SO(1; 3). But since ! so(1; 3) su(2)L su(2)R ; ' × we get more representations: scalar φ(x), left-handed Weyl spinor a(x) y (a = 1; 2), right-handed Weyl spinor a_ (_a = 1; 2), 4-vector Aµ(x), tensors etc. Chiral spinors exist because our space-time can accommodate them! Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 4/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Weyl spinors Notation: left-handed = LH = \left"; right-handed = RH = \right"; LH and RH spinors are different fields! They belong to different spaces; they transform differently under boosts and rotations. Do not think of them of two \polarizations" of a single field! Helicity and chirality are different notions! Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 5/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Weyl spinors Scalars fields can be real or complex. But Weyl spinors must be complex: y = . This is because the generators of su(2)L su(2)R are complex. 6 × They live in different spaces! y y conjugation ( a) = ( )a_ ; LH RH ; −−−−−−! because the generators of su(2)L su(2)R are swapped under conjugation. × To avoid confusion, we adopt for now the following convention: Weyl spinors without (e.g. ) are always LH and carry undotted indices a y Weyl spinors with (e.g. χy) are always RH and carry dotted indicesa _ y Later we will switch to the traditional notation with νL and νR . Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 6/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Weyl spinors Scalars fields can be real or complex. But Weyl spinors must be complex: y = . This is because the generators of su(2)L su(2)R are complex. 6 × They live in different spaces! y y conjugation ( a) = ( )a_ ; LH RH ; −−−−−−! because the generators of su(2)L su(2)R are swapped under conjugation. × To avoid confusion, we adopt for now the following convention: Weyl spinors without (e.g. ) are always LH and carry undotted indices a y Weyl spinors with (e.g. χy) are always RH and carry dotted indicesa _ y Later we will switch to the traditional notation with νL and νR . Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 6/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Lorentz invariant combinations ij ij 3D rotations: invariant symbol is δ : δ ai bj is SO(3) invariant. Lorentz symmetry: several invariant symbols possible. LH LH : combining a and χb × ab 0 1 ab = aχb is Lorentz invariant. 1 0 ! − Conventional shorthand notation: ab ab ba χ aχb = ( χb a)= χb a χψ : ≡ − − − − ≡ ab Majorana mass term for Weyl spinors contains νν νaνb. ≡ − Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 7/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Lorentz invariant combinations ij ij 3D rotations: invariant symbol is δ : δ ai bj is SO(3) invariant. Lorentz symmetry: several invariant symbols possible. LH LH : combining a and χb × ab 0 1 ab = aχb is Lorentz invariant. 1 0 ! − Conventional shorthand notation: ab ab ba χ aχb = ( χb a)= χb a χψ : ≡ − − − − ≡ ab Majorana mass term for Weyl spinors contains νν νaνb. ≡ − Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 7/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Lorentz invariant combinations RH RH : combining y and χy × a_ b_ _ 0 1 _ a_b = a_b yχy is Lorentz invariant. 1 0 ! a_ b_ − _ Majorana mass term contains νyνy +a_bνyνy. ≡ a_ b_ This convention is consistent with the rule ( χ)y = χy y: y ab y ab ∗ y y a_b_ y y b_a_ y y y y ( χ) ( aχb) = ( ) (χb) ( a) = χ = + χ χ : ≡ − − − b_ a_ b_ a_ ≡ Full Majorana mass term contains νν + νyνy. Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 8/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Lorentz invariant combinations RH RH : combining y and χy × a_ b_ _ 0 1 _ a_b = a_b yχy is Lorentz invariant. 1 0 ! a_ b_ − _ Majorana mass term contains νyνy +a_bνyνy. ≡ a_ b_ This convention is consistent with the rule ( χ)y = χy y: y ab y ab ∗ y y a_b_ y y b_a_ y y y y ( χ) ( aχb) = ( ) (χb) ( a) = χ = + χ χ : ≡ − − − b_ a_ b_ a_ ≡ Full Majorana mass term contains νν + νyνy. Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 8/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Lorentz invariant combinations Equivalent notation via raising and lowering indices: a ab b ab b ; a ab ; ab = ; ≡ ≡ − and the same for RH spinors. The convenient index-free notation is ab b b χ aχb = χb = bχ : ≡ − − In short, there are many ways to write the same expression. Just remember that the Lorentz invariant quantity is χ = 2χ1 1χ2 : − Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 9/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Lorentz invariant combinations y LH RH: a and χ do not match we need something extra! × b_ ! ! Correct combination: LH vector RH usingσ ¯µ = (I; σk ) × × − y µ y µ ab_ χ σ¯ Vµ χ (¯σ ) bVµ is Lorentz invariant. ≡ a_ Again, intuitive rule works: (χyσ¯µ )y = yσ¯µχ : _ In addition to up-index tensors ab, a_b, (¯σµ)ab_ , we define lower-indices tensors µ ab, a_b_ ,(σ )ab_ , which follow several consistency rules: µ aa_ ab a_b_ µ µ (¯σ ) = (σ ) _ ; (σ )aa_(¯σµ) _ = 2ab _ ; etc. bb bb − a_b Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 10/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Dirac vs Majorana Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 10/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Lagrangian For real scalar field φ(x), we construct 1 µ 1 2 = @µφ @ φ m φ φ ; L 2 − 2 We need something similar for the Weyl LH field a(x). mass term: y y ab a_b_ y y + a b + : ≡ − a_ b_ µ kinetic term @µ @ is bad because it is not positively definite. y µ But single-derivative term i (¯σ )@µ is OK. y µ y y µ y µ y µ i (¯σ )@µ = i(@µ ) (¯σ ) = i @µ (¯σ ) +i (¯σ )@µ : − − | {z } =0 Igor Ivanov (CFTP, IST) Neutrino mass models 1 UW, January 2018 11/46 Weyl spinors Dirac vs Majorana Charge conjugation LH/RH projectors Mass terms ν in the SM Lagrangian y µ 1 ∗ y y = i (¯σ )@µ m + m : L − 2 Mass parameter m can be complex m = m eiα, but the phase is not a physical parameter and can be removed via a globalj j field redefinition iα/2 e (x) = new(x): y µ 1 y y = i (¯σ )@µ m( + ) : L − 2 After that, we are no longer allowed to rephase ! We can only flip its sign.