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Majorana Neutrinos Lecture III: Majorana neutrinos Petr Vogel, Caltech NLDBD school, October 31, 2017 Whatever processes cause 0νββ, its observation would imply the existence of a Majorana mass term and thus would represent ``New Physics’’: Schechter and Valle,82 – – (ν) e e R 0νββ νL W u d d u W By adding only Standard model interactions we obtain (ν)R → (ν)L Majorana mass term Hence observing the 0νββ decay guaranties that ν are massive Majorana particles. But the relation between the decay rate and neutrino mass might be complicated, not just as in the see-saw type I. The Black Box in the multiloop graph is an effective operator for neutrinoless double beta decay which arises from some underlying New Physics. It implies that neutrinoless double beta decay induces a non-zero effective Majorana mass for the electron neutrino, no matter which is the mechanism of the decay. However, the diagram is almost certainly not the only one that generates a non-zero effective Majorana mass for the electron neutrino. Duerr, Lindner and Merle in arXiv:1105.0901 have shown that 25 evaluation of the graph, using T1/2 > 10 years implies that -28 δmν = 5x10 eV. This is clearly much too small given what we know from oscillation data. Therefore, other operators must give leading contribution to the neutrino masses. Petr Vogel: QM of Majorana particles Weyl, Dirac and Majorana relativistic equations: µ Free fermions obey the Dirac equation: (γ pµ m) =0 − µ 01 0 ~ 10 Lets use the following(γ p representationµ γm0 =) 0 =01 of~ the= 0γ matrices:− 1 γ5 = 0 1 − B 10C B ~ 0 C B 0 1 C B C B C B C µ @ A @ A @ − A 01(γ pµ m0) =0~ 10 γ0 = 0 1 ~ −= 0 − 1 γ5 = 0 1 B 10C B ~ 0 C B 0 1 C B C B C B C 01 @ A 0 @ ~ A 10@ − A γ0 = 0 The 1Dirac~ equations= 0 − can 1be thenγ5 rewritten= 0 as 1two coupled B 10C B ~ 0 C B 0 1 C B two-componentC B equationsC B C @ A @ A @ − A m +(E ~p ~ ) + =0 − − − (E + ~p ~ ) m + =0 − − Here ψ- = (1 - γ5)/2 Ψ = ψL, and ψ+ = (1 + γ5)/2 Ψ = ψR are the chiral projections. 1 1 1 (γµp m) =0 µ − (γµp m) =0 01 0 ~ 10 µ − γ0 = 0 1 ~ = 0 − 1 γ5 = 0 01 1 0 ~ 10 10 ~ 0 γ0 =0 0 11 ~ = 0 − 1 γ5 = 0 1 B C B C B B 10C C B ~ 0 C B 0 1 C B C B C B B C C B C B C @ A @ A @ @ − A A @ A @ − A m +(E ~p ~ ) + =0 m +(E ~p ~ ) + =0 − − − In the −limit of m -> 0 these two equations decouple(E + ~ andp ~ ) we obtainm + =0 − − − − two(E states+ ~p ~ with) definitem chirality+ =0 and helicity: − ~ pˆ = − · ± ± ± Thus, massless~ pˆ fermions= obey the two-component(E Weyl~ equationsp~) + =0 − · · ± ± ± E + ~ p~) =0 · − (E ~ p~) + =0 − · ( E + ~ p~) =0 · − The states ψ+ = ψR and ψ- = ψL , so-called Weyl spinors, (also Called vad der Waerden spinors) transform independently under the two nonequivalent simplest representations of the Lorentz group. 1 1 µ (γ pµ (mγµ) p =0m) =0 − µ − µ 01 0 ~ 10 (γ pµ m) γ=0= 0 1 01~ = 0 1 0 γ~ = 0 1 10 0 γ0 = 0 1 ~−= 0 5 1 γ5 = 0 1 − B 10C B ~ 0 C − B 0 1 C B C B 10CB BC ~ 0 C B CB 0 1 C @ A B C@ BA C @ − AB C 01 0 ~ 10@ A @ A @ − A 0 1 0 1 0 m 1 +(E ~p ~ ) =0 γ0 = ~ = − γ5 = m +(E+ ~p ~ ) + =0 B 10C B ~ 0 C B 0− 1 −C −− B C B C B (E +C~p ~−) m −=0 @ A @ A @ − A (E + ~p ~ ) + m + =0 − − − − For massivem fermions+(E there~p ~ ) +are=0 two possible~ p ˆrelativistic = − − − ± ~ pˆ ± = equations(E of+ motion.~p ~ ) m + =0 · ·± ± ± ± − − (E ~ (p~E) +~ =0p~) + =0 − · − · 1) The four component~ pˆ = Dirac equation, orE its+ ~equivalentp~E) + ~ =0 p~two) =0 · ± ± ± · − · − (E ~ p~) + =0 + + coupled two-component− · equations, with = 0 1= 0 1 B C B C E + ~ p~) =0 B − C B C · − @ A @ − A (E ~p ~ ) m✏ =0 2) Alternatively, as suggested + by Majorana, there canR be twoR⇤ = 0 1 − ~ − nonequivalent, relativistic two-component(E equations0 + ~ p0) L m0✏ L⇤ =0 B C B C − @ − A 01 (E ~p ~ ) R m✏ R⇤ =0 where ✏ = iσy = 0 1 B 10C − ~ − B C (E0 + ~ p0) L + m0✏ L⇤ =0 @ − A 01 These✏ two= i σMajoranay = 0 equations1 are totally independent, as B 10C B C indicated by different@ − energies,A momenta and masses. 1 1 1 (γµp m) =0 µ − 01 0 ~ 10 γ0 = 0 1 ~ = 0 − 1 γ5 = 0 1 B 10C B ~ 0 C B 0 1 C B C B C B C @ A @ A @ − A µ m (γ +(p E m)~ p ~ )=0 + =0 − − µ − − (E + ~p ~ ) m + =0 01 0 − −~ 10 γ0 = 0 1 ~ = 0 − 1 γ5 = 0 1 B 10C ~ Bpˆ~ =0 C B 0 1 C B C B C B C @ A ·@ ± ±A ± @ − A m( E +(~ E p~) ~+p ~ )=0 + =0 − − − · − (E +E ~+p ~ ~) p~) m =0+ =0 ·− − − ~ pˆ = + · =±0 ±1 ± B C (E ~ Bp~) +C=0 − ·@ − A (E E~+p ~ )~ Rp~) m✏=0 R⇤ =0 − · − − (E + ~ p~ ) + m ✏ =0 0 0 L + 0 L⇤ = 0 1 B 01C B C ✏ = iσy =@ 0 − A 1 B 10C (E ~p ~ ) R B m✏ ⇤ C=0 @ − R A − ~ − (E0 + ✏~ pL⇤0) L + m0✏ L⇤ =0 R Lets L compare= 0 − once1 , more theR = Dirac0 and1 Majorana equations B L C B ✏ ⇤ C B C 01B R C @✏ = iσyA= 0 1@ A B 10C D: (E ~ p~) RB m LC=0 − · @−− A M: (E ~ p~) m✏ =0 ✏ L⇤ R R⇤ R L =(E0−+−~ · p~)1 , − mR = =00 1 B L C L R B ✏ ⇤ C B C B R C @ · A − @ A It is( Eeasy+ ~to seep~) thatL + mthey✏ L⇤become=0 identical if m = 0 as well (E ~ · p~) R m L =0 if ψ = ε− ψ* . ·Similarly− for the other pair and ψ = -ε ψ* (LE ~ R p~) m✏ ⇤ =0 R L − · R1 − R D: (E + ~ p~) m =0 · L − R M: (E + ~ p~) + m✏ ⇤ =0 · L L 1 The four-component Dirac field is therefore equivalent to two degenerate, m = m’, two-component Majorana fields, with * the corresponding relation between ψL and ε ψ R Charge conjugation trasformation: Dirac bispinor transforms as * Charge conjugation matrix C in Weyl representation is Therefore the Dirac bispinor ψD cannot be an eigenstate of charge conjugation unless m = 0. 1 γ5 c 1+γ5 c c L = −2 :( L) = 2 =( )R 1+γ5 c 1 γ5 c c In contrast,R = 2 Majorana:( R) bispinor= −2 =( ) L with only two independent components transforms as i↵i In other words,li it ltransformsi0 e li to itself under charge conjugation. ! ! c This is so-called Majorana condition, ψM is identical with ψM . In general, the Majorana field can be defined as c χ(x)=[ (x)+⌘C (x)]/p2 By appropriate choice of the phase we obtain a field that is an eigenstate of charge conjugation with λc = +-1. 1 Neutrinos interact with chiral projection eigenstates. The chirality, i.e. the eigenvalues of operators (1 +- γ5)/2 is a conserved, Lorentz invariant quantity for massive or massles particles. On the other hand, the helicity, the projection of spin on momentum, is not conserved. For massive particle there is always a frame in which the momentum points in the opposite direction. For massles particles chirality and helicity are identical. But the chiral projections ψL and ψR do not obey the Dirac eq. unless m=0. If ψ is a chirality eigenstate, i.e. γ5ψ = λψ then the charge conjugation state ψc is also an eigenstate, but with λc = -λ. Also, if the Majorana state χ has the charge conjugation c c eigenvalue λ , the state γ5χ has the opposite eigenvalue –λ . Therefore, the charge conjugation eigenstates (Majorana states) cannot be simultaneously eigenstates of chirality. c The Majorana mass term is mL νL νL . c However, the objects νL and νL are not the mass eigenstates, i.e. the particles with definite mass. They are just the neutrinos in terms of which the model is constructed. The mass term c c mL νL νL induces mixing of νL and νL . As a result of the mixing, the neutrino mass eigenstates are These mass eigenstates are explicitly charge conjugation eigenstates. However, they do not have fixed chirality. Four distinct states of a massive Dirac neutrino and the transformation among them. νL can be converted into the opposite helicity state by a Lorentz transformation, or by the torque exerted be an external B or E field. Dirac neutrino ν Lorentz, B, E D ν L νR νR νL CPT CPT Lorentz There are only two distinct states of a Majorana neutrino νM. Under the Lorentz transformation νL νL νR is transformed into the same state νR as by the Lorentz transformation. The dipole magnetic and electric moments must vanish. CPT Number of parameters in the mixing matrix. Why there are more CP phases for Majorana neutrinos? CKM matrix for quarks: In the quark mass eigenstate basis one can make a phase rotation of the u-type and d-type quarks, thus iΦ(u) -iΦ(d) V -> e V e , where Φ(u) = diag(φu , φc , φt) , etc. The N x N unitary matrix V has N 2 parameters.
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