Deciphering the Majorana nature of sub-ev neutrinos by using their statistical property

Deciphering the Majorana nature of sub-ev neutrinos by using their statistical property C. S. Kim In collaboration with Dibyakrupa Sahoo Department of Physics and IPAP, Yonsei University, Seoul, South Korea FCFM-BUAP Puebla, Mexico 17 February, 2017 1 / 36

Contents 1. Introduction 2. Looking for Majorana neutrinos via L = 2 processes 2.1 0νββ 2.2 Rare meson decays 2.3 Collider searches at LHC 3. Statistical nature of neutrinos (compared with their interaction nature) 4. Effective Dalitz plot method 5. Conclusion 2 / 36

Introduction Neutrinos have mass. Neutrinos are massless in SM, m ν = 0. All neutrinos are only left-handed (ν L ). D mass = m ν (ν R ν L + ν L ν R ), m ν = Y ν v 2, where Y ν = Higgs-neutrino Yukawa coupling constant, and v = Higgs VEV. No way to generate mass without right-handed neutrinos (ν R ). But observations of neutrino oscillation imply that neutrinos have mass, m ν 0. The Nobel Prize in Physics 2015 was awarded jointly to Takaaki Kajita (Super-Kamiokande) and Arthur B. McDonald (Sudbury Neutrino Observatory) for the discovery of neutrino oscillations, which shows that neutrinos have mass. 3 / 36

Introduction How to give neutrinos mass? There are various suggestions as to how neutrinos can get mass. Dirac mass: Assumption: ν R exists. Lagrangian: D mass = md ν (ν Rν L + ν L ν R ). Disadvantage: No reason for m D to be small. ν Majorana mass: Assumption: neutrino anti-neutrino. Lagrangian: M = 1 mass 2 mm ν C ν L ν L + ν L ν C. L Disadvantage: M mass is not invariant under SU(2) L U(1) Y gauge group, so not allowed by SM. 4 / 36

Introduction How to give neutrinos mass? Dirac-Majorana mass: Assumptions: ν R exists, and neutrino anti-neutrino. Lagrangian: D+M = 1 mass 2 ml ν C ν L ν L + 1 2 mr ν C ν R ν R m D (ν ν Rν L ) + H.c. = 1 2 NC L MN L + H.c., where N L = νl ν C R m L and M = ν m D ν m D ν m R ν is the mass matrix. Note: Out of m L, ν mr and ν md two are real and positive, but the ν remaining one is complex, in general. We choose m D to be complex. ν Problem: ν L and ν R have no definite mass, due to Dirac mass m D. ν Solution: Go to a basis in which mass matrix is diagonal. N L = Un L, with ν1l n L = and U T m1 0 MU = M. ν 2L 0 m 2 D+M = 1 mass 2 m 1 ν C 1 ν 1 + m 2 ν C 2 ν 2 + H.c., with ν k = ν kl + ν C (k = 1, 2) being Majorana neutrinos and kl m m 2,1 = 1 L m 2 + 2 ν ν mr ± L ν + m R ν + 4 m D 2 ν. Disadvantages: (i) No explanation for small mass of neutrino, and (ii) one mass out of m D, ν mr, ν ml is complex, in general. ν 5 / 36

Introduction How to give neutrinos mass? See-saw mechanism: A simpler version of Dirac-Majorana mass, with a nice twist. Assumptions: m L = 0 and ν md ν mr. ν Lagrangian: D+M where N L = = 1 mass 2 mr ν νl ν C R Mass eigenvalues: m 2,1 = 1 2 1 2 mr ν = m 1 and M = ν C R ν R m D (ν ν Rν L ) + H.c. = 1 2 NC L MN L + H.c., 0 m D ν is the mass matrix. m D ν m R ν m Rν ± m Rν 2 + 4 m Dν 2 m D 2 ν 1 ± 1 ± 2 m D 2 ν. m R ν and m 2 m R ν. m R ν Advantage: m 1 m 2, so light neutrinos are possible. Challenges: To find the heavy ν 2 experimentally. To prove that both the light ν 1 and heavy ν 2 are Majorana neutrinos. ν 2 ν 1 As ν 2 gets heavier, ν 1 become lighter. 6 / 36

Looking for Majorana neutrinos via L = 2 processes Neutrinos are the only elementary fermions known to us that can have Majorana nature. Majorana neutrinos: ν ν. Majorana neutrinos violate lepton flavor number (L), they mediate L = 2 processes. W ± l ± i ν k ν k d 4 p U (2π) 4 li k U lj k k m k + p p 2 m 2 k W ± l ± j L = 2 processes play crucial rule to probe Majorana nature of ν s. neutrinoless double-beta (0νββ) decay Rare meson decays with L = 2 Collider searches at LHC 7 / 36

Looking for Majorana neutrinos via L = 2 processes Decay rate of any L = 2 process with final leptons l + 1 l+ 2 : 2 m Γ L=2 U k l1 ku l2 k p 2 m 2 k + im, kγ k k where we have used the fact that (1 γ 5 ) p(1 γ 5 ) = 0. Light ν: 2 Γ L=2 U l1 ku l2 km k = ml1 2 l 2. Heavy ν: Resonant ν: k 2 U l1 ku l2 k Γ L=2. Γ L=2 k m k Γ (N i) Γ (N f) m N Γ N. 8 / 36

Looking for Majorana neutrinos via L = 2 processes Neutrinoless double-beta (oνββ) decay Process: Black-box diagram for 0νββ d d 0νββ u e e u 9 / 36

Looking for Majorana neutrinos via L = 2 processes Neutrinoless double-beta (oνββ) decay Double-beta (2νββ) decay has been observed in 10 isotopes, 48 Ca, 76 Ge, 82 Se, 96 Zr, 100 Mo, 116 Cd, 128 Te, 130 Te, 150 Nd, 238 U, with half-life T 1/2 10 18 10 24 years. Giovanni Benato (for the GERDA collaboration), arxiv:1509.07792 0νββ (forbidden in SM) is yet to be observed in any experiment. T 0ν 1/2 76 Ge 1.9 10 25 years. 10 / 36

Looking for Majorana neutrinos via L = 2 processes Neutrinoless double-beta (oνββ) decay The half-life of a nucleus decaying via 0νββ is, T 0ν 1/2 1 = G0ν M 0ν mββ 2, where G 0ν is phase space factor, M 0ν is the nuclear matrix element, (large theoretical uncertainty) 3 m ββ is effective Majorana mass. m ββ = U 2 m ek k is complex, in general, and can be zero due to possible cancellations arising from phases in U ek. k=1 11 / 36

Looking for Majorana neutrinos via L = 2 processes Neutrinoless double-beta (oνββ) decay S. M. Bilenky and C. Giunti, arxiv:1203.5250 Mod. Phys. Lett. A 27, 1230015 (2012) If m ββ is vanishingly small, there will hardly be any neutrinoless double-beta decay events. 12 / 36

Looking for Majorana neutrinos via L = 2 processes Rare meson decays: M + M l + 1 l+ 2 Processes: M + M l + 1 l+ 2, where M = K, D, D s, B, B c and M = π, K, D,... G. Cvetic, C.S. Kim, arxiv:1606.04140 (PRD 94, 053001, 2016) G. Cvetic, C. Dib, S. Kang, C. S. Kim, arxiv:1005.4282 (PRD 82, 053010, 2010) l + 1 l + 2 l + 1 Q q 1 Q q 1 M + N M M + M N q q 2 l + 2 q q 2 No nuclear matrix element unlike 0νββ, but probes Majorana nature of massive neutrino(s) N. 13 / 36

Looking for Majorana neutrinos via L = 2 processes Rare meson decays: M + M l + 1 l+ 2 14 / 36

Looking for Majorana neutrinos via L = 2 processes Collider searches at LHC Processes: W + e + e + µ ν µ, W + µ + µ + e ν e. Involves heavy neutrino N which can have Majorana nature as well. C. Dib, C.S. Kim, arxiv:1509.05981 (PRD 92, 093009, 2015); C. Dib, C.S. Kim, K. Wang, J. Zhang, arxiv:1605.01123 (PRD 94, 013005, 2016) e + e + µ e + µ e + W + N W (Lepton Number Violating) ν µ Decay widths: LNV: Γ W + e + e + µ ν µ = UNe 4 ˆΓ, W + N W + (Lepton Number Conserving) LNC: Γ W + e + e + µ ν µ = UNe U 2 Nµ ˆΓ, G 3 2 F where ˆΓ = M3 W 12 96 m5 N 1 m2 N 2π 4 Γ N M 2 1 m2 N W 2M 2. W ν e 15 / 36

Looking for Majorana neutrinos via L = 2 processes Collider searches at LHC 16 / 36

Statistical Nature of Neutrinos So far only the interaction properties of Majorana neutrinos have been exploited. Majorana neutrinos are a theorists favourite, because of their simplicity and the resulting elegance in theory, with exception of the nuclear matrix element in 0νββ. In all major searches for Majorana neutrinos, for both active and heavy neutrino cases, we have exploited only their mass dependent interaction property, int = m ν νν. m ν = 0 = no 0νββ decay or other L = 2 processes. Furthermore, for active neutrinos m ν is very small, making the concerned processes extremely rare. Therefore, we want a better alternative to 0νββ decay or other L = 2 processes. These alternative processes, Should not be a rare, i.e., when the neutrino mass is zero, the process must be allowed. Must have a unique, experimentally observable signature for Majorana neutrinos. We shall explore the quantum statistical property of Majorana neutrinos which is independent of neutrino mass. 17 / 36

Statistical Nature of Neutrinos The quantum statistical property of Majorana neutrinos does not depend upon their masses. If we have a neutrino and anti-neutrino pair (of the same flavor) in the final state of some process, the amplitude must be anti-symmetrized for Majorana neutrinos, but for Dirac case there is no such anti-symmetrization. This anti-symmetrization requirement for Majorana case arises because a Majorana neutrino and its anti-neutrino are quantum mechanically indistinguishable, i.e., they are identical fermions. Fermi-Dirac statistics for identical fermions This quantum statistical property does not depend on how heavy or light the Majorana neutrinos are. We shall consider processes whose final state contains ν l ν l (in addition to other particles) to explore the quantum statistical nature of Majorana neutrinos. 18 / 36

Effective Daitz plot method We consider only such decay modes in which the 4-momentum of neutrino and anti-neutrino can be experimentally inferred. Process: X Y [l + l ] ν l ν l (an effective three-body decay) Conditions: 1. X is some suitable resonance. 2. Y is an effective particle, which must always include l + l, with some additional (not necessary) particle(s). 3. The 4-momenta of X and all particles in Y as well as those of some intermediate resonances must be experimentally measured such that 4-momenta of ν l and ν l are experimentally deducible. 4. ν l ν l do not arise from weak neutral current interaction in the process under consideration. Some examples: X B 0 π + µ + ν µ µ ν µ Y µ + µ ν µ ν µ, X B 0 π + µ + ν µ π µ ν µ Y µ + µ ν µ ν µ, X B + D 0 K + e ν e e + ν e Y K + e + e ν e ν e. 19 / 36

Effective Daitz plot method We choose to work in a frame of reference in which exchange of ν and ν is more elegant. Gottfried-Jackson frame: p ν + p ν = 0. ν(p ν ) Invariant mass squares: θ GJ m 2 νν = (p ν + p ν ) 2 = (p X p Y ) 2, X(p X ) ν(p ν ) Y(p Y ) z m 2 Yν = (p Y + p ν ) 2 a b cos θ GJ, m Yν = (p Y + p ν ) 2 a + b cos θ GJ. where a = 1 m 2 X 2 + m2 Y + 2m2 ν s, b = 1 λ m 2 X 2, m2 Y, s 1 4m 2 ν /s, with the Källén function λ(x, y, z) defined as λ (x, y, z) = x 2 + y 2 + z 2 2 (xy + yz + zx). 20 / 36

Effective Daitz plot method Our tool for investigating the Majorana neutrinos is the effective Dalitz plot. m 2 νν + m2 Yν + m2 Yν = m2 X + m2 Y + 2m2 ν M2 (say). Since m 2 Y varies from event-to-event, M 2 does so also. Define new dimensionless ratios to take care of these event-to-event variations, m 2 νν m2 νν /M2, m 2 Yν m2 Yν /M2, m 2 Yν m2 Yν /M2, such that m 2 νν + m2 Yν + m2 Yν = 1. We can always construct a ternary plot (which along with event points we shall refer to as the effective Dalitz plot) using ( m 2 Yν, m2 Yν, m2 ) as Cartesian coordinates. νν 21 / 36

Effective Daitz plot method Our tool for investigating the Majorana neutrinos is the effective Dalitz plot. Any point inside the ternary plot can be described by either polar coordinates (r, θ) or rectangular coordinates (x, y). 2 y 0 1 ν m 2 νν Y IV III θ r V VI I II ν m 2 νν = 1 3 (1 + r cos θ) = 1 (1 + y), 3 m 2 Yν = 1 2π 1 + r cos 3 3 + θ = 1 2 + 3x y, 6 m 2 Yν = 1 2π 1 + r cos 3 3 θ m 2 Yν 3 0 x 3 m 2 Yν = 1 6 2 3x y. m 2 Yν m2 Yν θ GJ π θ GJ θ θ. 22 / 36

Effective Daitz plot method We analyse the distribution of events in the effective Dalitz plot to distinguish Dirac and Majorana neutrinos. The pattern of distribution of events in the effective Dalitz plot is a consequence of dynamics. The dynamics is encoded in the transition amplitude. The amplitude for all the processes under our consideration, should be anti-symmetrized for Majorana neutrinos, while for Dirac case there is no such anti-symmetrization. The distribution of events should be completely symmetric under exchange of ν and ν for Majorana neutrinos. For Dirac neutrinos the distribution must have some asymmetry under the above exchange. 23 / 36

Effective Daitz plot method Let us analyse an example process: X B 0 π + µ + ν µ π µ ν µ Y µ + µ ν µ ν µ. Amplitude: Using Fierz rearrangement theorem, Dirac neutrinos: D (p + p ν ) α (p + + p ν ) β ψ µ (p ) γ α 1 γ 5 ψ µ + (p + ) ψ ν (p ν ) γ β 1 γ 5 ψ ν (p ν ), Majorana neutrinos: M (p + p ν ) α (p + + p ν ) β + (p + p ν ) α (p + + p ν ) β ψ µ (p ) γ α 1 γ 5 ψ µ + (p + ) ψ ν (p ν ) γ β γ 5 ψ ν (p ν ), 24 / 36

Effective Daitz plot method Let us analyse an example process: X B 0 π + µ + ν µ π µ ν µ Y µ + µ ν µ ν µ. Squaring the modulus of amplitude and summing over final spins, with the simplifying assumption that m µ = 0 = m ν, we get, Dirac neutrinos: D 2 256 (p p ν ) (p ν p + ) (p ν p + ) 2, Majorana neutrinos: M 2 128 (p ν p + ) (p ν p + ) (p p + ) (p ν p ν ) + (p (p ν + p ν )) (p + (p ν + p ν )). 25 / 36

Effective Daitz plot method Let us analyse an example process: X B 0 π + µ + ν µ π µ ν µ Y µ + µ ν µ ν µ. Kinematics: x µ ν µ p φ p ν π θ GJ θ B 0 p B p ν θ GJ z y ν µ p+ θ + µ + 26 / 36

Effective Daitz plot method Let us analyse an example process: X B 0 π + µ + ν µ π µ ν µ Y µ + µ ν µ ν µ. Integrate the amplitude modulus square over φ and get, Dirac neutrinos: D 2 2E 32πm 4 2 π + 3p 2 p 2 cos(2θ GJ ) + 8 pee + E 2+ p 2 cos θ GJ + 2E 2 + p 2 p 2 + 4E 2 E 2 +. Majorana neutrinos: M 2 32m 4 π E 2 m 2 B + 4E2 2EE B + 4E + E. Here all quantities are in the Gottfried-Jackson frame, and E B, E ±, E are the energies of B meson, µ ±, ν µ (or ν µ ) respectively, p is the magnitude of the 3-momentum of ν µ (or ν µ ), p is the magnitude of the projection of 3-momentum of µ ± on the xy-plane. The term in red is proportional to cos θ GJ and renders D 2 not symmetric under ν ν exchange which gets reflected in the distribution of events in the effective Dalitz plot. 27 / 36

Effective Daitz plot method Let us analyse an example process: X B 0 π + µ + ν µ π µ ν µ Y µ + µ ν µ ν µ. Signature of Majorana neutrinos: The distribution of events in the effective Dalitz plot is completely symmetric under m 2 Yν m2 Yν for Majorana neutrinos. Dirac neutrinos lead to an asymmetry in the effective Dalitz plot under the same exchange. 28 / 36

Effective Daitz plot method The distribution of events in the effective Dalitz plot can be described by a Fourier decomposition. We can construct the full effective Dalitz plot, 0 θ 2π. The distribution of events inside it has no discontinuity, or infiniteness. Let (r, θ) denote the distribution of events inside the effective Dalitz plot. Then, D (r, θ) = S Dn (r) sin(nθ) + CDn (r) cos(nθ) (Dirac neutrinos) n=0 M (r, θ) = C M (r) cos(nθ) n n=0 (Majorana neutrinos) where S D n (r) and CD,M n (r) are the Fourier coefficients which are some functions of masses and energies of the particles involved. 29 / 36

Effective Daitz plot method The Dirac and Majorana neutrinos leave two distinct signatures in the effective Dalitz plot. dr M (r, θ) = dr M (r, θ), (Majorana neutrinos) dr D (r, θ) dr D (r, θ), (Dirac neutrinos) where we have carried out integrations radially, i.e. we add all the events inside the effective Dalitz plot along the radial direction at any chosen polar angle. It must be emphasized that this distinction between Dirac and Majorana neutrinos is always present in our effective Dalitz plot irrespective of neutrino mass. 30 / 36

Effective Daitz plot method The signature of Majorana neutrinos can be quantified in terms of some easily observable asymmetries. Sextant asymmetry: 2 m 2 νν Y y IV III 0 V II 1 ν VI I ν m 2 Yν m 2 Yν 3 0 x 3 = N I N VI N I + N + N II N V VI N II + N + N III N IV V N III + N, IV where N i denotes the number of events in the ith sextant. 31 / 36

Effective Daitz plot method The signature of Majorana neutrinos can be quantified in terms of some easily observable asymmetries. Binned asymmetry: 2 2 θ m 2 νν θ m Y θ m 2 θ y 0 Left Right 1 ν ν m 2 Yν 3 0 x 3 = N(θ m ) N( θ m ) N(θ m ) + N( θ m ), θ m m 2 Yν where N(θ m ) is the number of events in an angular bin θ m ± θ. 32 / 36

Effective Daitz plot method There exist a plethora of processes which can be looked at using our approach. X intermediate resonances final state (Y)ν l ν l B 0 D l + ν l K 0 l + l ν l ν l B + D 0 l + ν l (K l + l ) ν l ν l B 0, D 0, K 0 π ± l ν ( ) l (l + l ) ν l ν l π + π (l + l ) ν l ν l B 0, D 0 π + K (l + l )ν l ν l K 0 S π + π γ (l + l γ)ν l ν l B 0, D 0, K 0 L, J/ψ(1S) π + π π 0 (l + l π 0 )ν l ν l 33 / 36

Effective Daitz plot method There exist a plethora of processes which can be looked at using our approach. X intermediate resonances π + π K 0 S π + K π 0 K + K K 0 S π + π 2π 0 K l + ν l D 0 final state (Y)ν l ν l (l + l K 0 S )ν lν l (l + l π 0 )ν l ν l (l + l K 0 S )ν lν l (l + l 2π 0 )ν l ν l (l + l ) ν l ν l D + K 0 l + ν l (π + l l + ) ν l ν l π + π ω (l + l ω)ν l ν l J/ψ(1S) π + π η π + π φ (l + l η)ν l ν l (l + l φ)ν l ν l π + π ωπ 0 (l + l ωπ 0 )ν l ν l Υ (2S) π + π Υ (1S) (l + l Υ (1S))ν l ν l 34 / 36

Conclusion Processes are not rare for our case. These are not dependent on neutrino mass, and we are using the statistical property of neutrinos. Majorana and Dirac neutrinos have completely distinct signatures, which survive even when one considers neutrinos to be massless. The signatures are quantifiable by easily observable asymmetries defined on effective Dalitz plots. For m ν 0, L = 2 processes 0, but our asymmetries 0. 35 / 36

Conclusion By using our methodology of considering effective Dalitz plots for suitably well choosen processes one can look for the Majorana nature of active neutrinos. Thank You 36 / 36