The physics of neutrinos~Dirac vs. Majorana~
C.S. Lim (林 青司)(Editor-in-Chief, PTEP, The Physical Society of Japan)

TCHoU Symposium
at the Semi-Centennial Anniversary of the University of Tsukuba(Sept. 28, ’23)

１．Key roles played by neutrinos
・P violation in weak interactions :× in SM (Standard Model)

・definitive evidence of BSM (beyond standard model) :neutrino oscillations ⇒ massive neutrinos
・ unsolved problem in SM:←matter-antimatter asymmetry in the universe

heavy Majorana neutrino ``leptogenesis” (Fukugita & Yanagida )・ cosmic neutrino background (CνB) of 1.9 Kvery good probe of very early universe (origin of our universe)
Cosmic Background Neutrino Decay Search (COBAND) exp.

()

2. Dirac vs. Majorana
・Charged fermion : Dirac fermion
described by 2×2 = 4 component complex Dirac spinor
・Is there fermion without discrimination between particle –antiparticle ?
Possible only for neutral fermions, like neutrino (photino, ..) : Majorana fermion
described by Majorana spinor:・Take scalar particle (s = 0) . Charged scalar is described by a complex field(gauge transformation)⇒: complex：charge conjugation = complex conjugation.

・A scalar without discrimination between particle –antiparticle (e.g. Higgs):

・Similarly Majorana spinor is 4-component real spinor.
Majorara is more fundamental than Dirac.

・Similarly, a Dirac spinor is equivalent to 2 Majorana spinors with identical masses:

・If neutrinos are Majorana, lepton number L ： indefinite``Majorana mass term” causes L violation.
In general, mass term causes chirality-flip and necessitates ``chiral partner”

Dirac (active → ``sterile”)

Majorana(active → active,)3. Neutrino oscillation
A sort of lepton-flavor changing process :e.g.
Assume 2 generation scheme

Transition probability at time t :The physics is basically the same as ``beat”

For neutrino oscillation,・⇒・massive neutrinos

4. Three types of neutrino
Neutrino oscillation ⇒ massive neutrinos ⇒ BSM
We minimally extend SM by introducing
The most general mass term for neutrino (1 generation):(a) Dirac ()mass eigenvalues:(degenerated)mass eigenstates:,: Majorana1 Dirac = 2 Majorana with mass degeneracy
mixing angle:(between active & sterile)(N.B.) Though mixing is maximal, we cannot expect neutrino oscillation (active ⇔ sterile)
because of mass degeneracy.

(b) Seesaw ()・・mass-eigenstates:
・mass-eigenvalues:⇒smallness of neutrino masses is naturally explained@ low energies, only 1 light Majorana neutrino

message: no neutrino oscillation in 1 generation scheme for Dirac or Seesaw

(c) pseudo-Dirac()(Wolfenstein, …)・ almost Dirac &: almost maximal

・ But, the degeneracy of mass eigenvalues is slightly lifted:
two independent light Majorana neutrinos for each generation⇒neutrino oscillation (active ⇔ sterile) with maximal mixing,
even in 1 generation scheme !!
( M. Kobayashi & C.S.L., PRD 64 (’01)013003;
see also M. Kobayashi, C.S.L. & M.M. Nojiri, PRL 67(’91)1685 )5. Neutrino oscillations and Majorana nature of neutrino
How can we reveal the Majorana nature ?
・Promising terrestrial experiment: 0νββ()(KamLAND-ZEN, GERDA, PIKACHU, …)
challenging, the decay width being suppressed by small neutrino mas (- squared).
・How about neutrino oscillation ?
Unfortunately, Dirac and Seesaw provide identical predictions (cannot be discriminated)since ,
(i) chirality-flipping oscillation,
are strongly suppressed by

(ii) chirality-preserving oscillation,(Dirac), cannot discriminate two cases:(Seesaw)

Majorana nature cannot be revealed, as long as we take seesaw as correct scenario
for Majorana neutrino.
In both of Dirac & seesaw scenarios, the unitarity among active states holds:

６．What we have learned from the neutrino oscillation(Chirality preserving) ordinary flavor oscillations do not discriminate Dirac & Seesaw⇒ described by 3×3 matrix (PMNS) with 3 mixing angles
and CP violating phase, together with 2 mass-squared differences,【remaining issues】
・ mass ordering (normal or inverted) : the sign of・ CP violation in leptonic sector:
T2HK (T2KK), LBNF/DUNE(NB) Majorana CP phase, important in leptogenesis, is not attainable.

7． Two possible ways to reveal Majorana nature(relying on neutrino oscillation)(1) Neutrino oscillation of pseudo-Dirac neutrinos
Oscillation without chirality flip :, which makes difference:If oscillations of this type are found, it will be a clear signal of the pseudo-Dirac neutrinos, and therefore Majorana nature of neutrinos
This scenario is disfavored as the solution of solar or atmospheric neutrino deficit.
(e.g.
→from Super-Kamiokande & SNO for solar ν)(from solar ν)

However、we have a chance to detect the oscillation of pseudo Diracneutrinos by use of neutrinos with astrophysical or cosmic origin
utilizing their very long baseline.
・ Flavor composition of ultra-high energy astrophysical neutrino flux
(detected e.g. by Ice-Cube)

is distorted by this pseudo-Dirac type oscillation(J.F. Beacon, N.F. Bell, D. Hooper, J.G. Learned, S. Pakvasa, T.J. Weiler, PRL 92(’04)011101)
Ordinary flavor oscillation:
Taking into account pseudo-Dirac type oscillation, it can be e.g.
(if largestis large enough )This is based on a formula【 Formalism 】( M. Kobayashi & C.S.L., PRD 64 (’01)013003 )We have 2 × 3 = 6 light neutrinos ⇒ mass matrix: 6×6 : terrible !
However, invoking
・ diagonalize the Dirac mass matrix by use of PMNS matrix
・ there appear degeneracies in the mass eigenvalues for each generation
・ perform exact diagonalization in each of 2×2 sub-matrix
to get(Recent discussion in the literature)
・A possible sign in SN1987A data (KⅡ, IMB, Baksan) ?
(I. Martinez-Soler, Y.F. Perez-Gonzalez, M. Sen, Phys.Rev.D 105 (2022) 9, 095019)

mild preference for・Diffuse supernova neutrino background ( Y.F. Perez-Gonzalez, CERN Neutrino Platform ’23 )← SK-Gd, HK, DUNE・Capture rate of CνB (PTOLEMY exp. ) : sensitive to( Y.F. Perez-Gonzalez, M. Sen, 2308.05147[hep-ph])(2) RSFP (Resonant Spin-Flavor Precession)( C.S.L. & W.J. Marciano, PRD37 (’88)1368; E.Kh. Akhmedov, PL B213(’88)64 )Neutrino oscillation with chirality-flip, but without suppression factor,(thus making discrimination between Dirac and Majorana possible):spin precession under strong external magnetic field (the interior of the Sun,supernovae etc.), whose probability(in the vacuum): nothing to do with neutrino masses
・ Originally, RSFP was proposed to solve the solar neutrino problem.

Though ordinary precession
is suppressed by ``matter effect” ,if we consider spin-flavor precession such as
the matter effect leads to resonant enhancement (a la MSW) of the precession.
・For Majorana neutrino, RSFP is L-violating (active →active) oscillation, such as
Interaction term relevant for this process:where⇒ for Majorana neutrino the transition magnetic moment
is inevitable

・Radiative decay of neutrinos

utilized in COBAND exp.
to search for CνB is exactly due to the transition magnetic moment⇒ RSFP & COBAND both need BSM to realize ``sizable” neutrino
magnetic moment, while keeping neutrino masses small(Recent discussion in the literature)
・For Majorana neutrinos, RSFP & flavor mixing will imply( C.S.L., M. Mori, Y. Oyama, A. Suzuki, Phys.Lett.B 243 (’90) 389 ;E. Akhmedov, P. Martinez-Mirave, JHEP10 (’22)144 )it would be clear signal of Majorana nature (L violating・from the Sun)due to RSFP is also expected for supernova neutrino produced byneutronization burst, and may be tested by HK & DUNE
(S. Jana, Y.P. Porto-Silva, M. Sen, ICHEP 2022)