Uncovering Multiple CP-Nonconserving -Decay ββ...
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Uncovering Multiple CP-Nonconserving Mechanisms of (ββ ) 0ν -Decay S. T. Petcov SISSA/INFN, Trieste, Italy, Kavli IPMU, University of Tokyo, Tokyo, Japan CISNP13, University of South Carolina Columbia, U.S.A. May 19, 2013
Transcript of Uncovering Multiple CP-Nonconserving -Decay ββ...
Uncovering Multiple CP-Nonconserving
Mechanisms of (ββ)0ν-Decay
S. T. Petcov
SISSA/INFN, Trieste, Italy,
Kavli IPMU, University of Tokyo, Tokyo, Japan
CISNP13, University of South Carolina
Columbia, U.S.A.
May 19, 2013
Symposium in honor ofFrank Avignone and Ettore Fiorini
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song cycle that takes the listener on a dynamic ... guitarists Chris Clermont and Jamie
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Google: Frank Avignone; 2013: 2 850 000• Sensitivity of CUORE to Neutrinoless Double-Beta Decay F. Alessandria, E. Andreotti,
R. Ardito, C. Arnaboldi, F. T. Avignone III, M. Balata, I. Bandac, T. I. Banks, G. Bari,
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• Majorana Collaboration Passes DOE/NSF Critical Design Review Monday, June 4th,
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Bertrand, M. Boswell, V. Brudanin, M. Busch, Y. D. Chan, C. D. Christofferson, J. I.
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• On June 21, 2010, the University of Zaragoza in Spain conferred an honorary doctorate
on Professor Frank Avignone in recognition of his many contributions to the field of
particle astrophysics. The formal ceremony followed a 500-year-old format, in which he
gave his acceptance speech in Spanish. This was Avignones second honorary doctoral
degree, the first one having been bestowed by the University of Buenos Aires in 2004.
Avignone joined the USC faculty in January 1965, and although he retired as a state
employee in 1998, he has worked fulltime under contract at USC as a research professor
ever since. He continues to hold the title of Carolina Endowed Professor of Physics and
Astronomy and is the leader of the Particle Astrophysics Group. He served as chair of
the department from May 1979 until he took his new position in July 1998.
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professore ordinario. Settore scientifico disciplinare:. Settore FIS/04 - Fisica Nucle-
are E Subnucleare. Dipartimento: ... www.unimib.it/go/Home/Pagine-Speciali/Elenco-
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Current Academic Responsibilities:
Direttore dei Laboratori di Radioattivita‘ e di Criogenia.
Other Professional Responsabilities:
Responsabile nazionale degli esperimenti ”Mibeta” e ”Cuore”.
Coordinatore del Network Europeo sui Rivelatori Termici.
• Particle physicists plumb the depths for Roman lead - 13 July 1991 ...
The physicists, Gianni Fiorentini and Ettore Fiorini, want the lead for exper-
iments that are of critical importance in particle physics and cosmology. ...
www.newscientist.com/article/mg13117772.600- particle-physicists-plumb-the-depths-
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• Premio Enrico Fermi 2007 della Societa Italiana di Fisica
Motivazioni: a Ettore Fiorini per il contributo alla scoperta delle correnti deboli neutre e
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• Ortvay Kollokvium
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(Recepient of the Marx Medal): ”The neutrinoless beta decay and the nature of neutrino”
... ortvay-koll.elte.hu/Programs/Abstracts/marx-emlek2008tavasz.abs.html
• IUPAP COMMISSION C12: 1999-2002
E-mail: [email protected]. JONSON, B. (1996) Department of Physics,
Chalmers University of Technology, SE-412 96 Goeteborg, SWEDEN. ...
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Ettore Fiorini. INFN Milan. [email protected].
G. Smadja. IPN Lyon. [email protected].
Joe Silk. Nuclear and Astrophysics Laboratory ...
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• COMUNICATO STAMPA (Press release) 11 febbraio 2008
“L’occhio” nucleare rivela: Napoleone non e stato avvelenato”
“The nuclear “eye” reveals: Napoleon was not poisoned”
11 feb 2008 ... Ettore Fiorini, docente di Fisica Nucleare all’ Universita‘
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www.ricercaitaliana.it/stdoc/pdfnapoleon.pdf
• FOXNews.com - Arsenic Poisoning Ruled Out in Napoleon’s Death ...
Feb 13, 2008 ... The researchers, including Ettore Fiorini of the Italian Na-
tional Institute of Nuclear Physics and the University of Milano-Biccoca, ...
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Napoleon didn’t die from arsenic poisoning - Telegraph
Drs Ettore Fiorini and Ezio Previtali of INFN, who did the study with Angela
Santagostino of the University of Milan at a small nuclear reactor at the ...
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Ettore Fiorini, ordinario di Fisica nucleare e subnucleare all’Universita‘ Milano Bicocca,
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hunt - physicsworld.com physicsworld.com/cws/article/news/42445 Apr 23, 2010 When
nuclear physicist Ettore Fiorini at the University of Milan-Bicocca read about the find in
a newspaper he went to Cagliari to offer the ...
Ever on the look-out for ultra-low radioactive materials to shield their sensitive exper-
iments, nuclear physicists have struck gold with a consignment of lead that lay on the
floor of the Mediterranean Sea for 2000 years. The almost completely inert ancient
lead will be used to line the CUORE neutrino experiment located under the Gran Sasso
mountain in central Italy.
• Neutrinoless Double Beta Decay - Neutrino Unbound -
Infn www.nu.to.infn.it/Neutrinoless Double Beta Decay/ [1-3]: Neutrino physics with
cryogenic detectors, Fiorini, Ettore, Prog. Part. Nucl. Phys. 64 (2010) 241-248. [1-4]:
Liquid Xenon Detectors for Particle Physics and ...
• Present Experimental Techniques, Results and
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...
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by 2 - Related articles Ettore Fiorini. Abstract. After a brief introduction on the main
features of Double Beta Decay (DBD) and on its origin, its importance is stressed in
view of the ...
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in Japan on 11th March 2011, ...
• Neutrino physics: Beta test : Nature News Comment www.nature.com/news/neutrino-
physics-beta-test-1.10988 Jul 12, 2012 A definitive sighting, says Ettore Fiorini, a particle
physicist at the University of Milano-Bicocca in Italy, would be one of the most important
...
• Now 2010- Program -
INFN Bari www.ba.infn.it/ now/now2010/Program/program.html Chair: Ettore Fior-
ini. Time, Speaker, Title. 8.30 - 8.40, Gianluigi Fogli, Welcome to NOW 2010 (pptx).
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• LTD-13
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Kent Irvin (NIST Boulder) Caroline Kilbourne (NASA/Goddard) ... 13th International
Workshop on Low Temperature Detectors Stanford July 20-24, 2009
• Committees - Neutrino 2012 neu2012.kek.jp/neu2012/committees.html Ettore Fiorini
(Milano) Francis Halzen (Wisconsin) Kunio Inoue (Tohoku) Takaaki Kajita (ICRR Tokyo)
Stavros Katsanevas (Lyon) Soo-Bong Kim (Seoul) ...
• Ettore Fiorini vince il Premio Bruno Pontecorvo - Universit‘a degli ... www.unimib.it
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Uncovering Multiple CP-Nonconserving Mecha-nisms of (ββ)0ν-Decay
Based on:
1. A. Faessler, A. Meroni, S.T.P., F. Simkovic, J. Vergados, arXiv:1103.2434 (Phys. Rev. D83
(2011) 113003).
2. A. Meroni, S.T.P. and F. Simkovic, arXiv:1212.1331 (JHEP 1302 (2013) 025).
If the decay (A, Z) → (A, Z +2)+e−+e− ((ββ)0ν-decay)will be observed, the question will inevitably arise:
Which mechanism is triggering the decay?
How many mechanisms are involved?
“Standard Mechanism”: light Majorana ν exchange.
Fundamental parameter - the effective Majorana mass:
<m> =∑light
j
(
Uej
)2mj , all mj ≥ 0 ,
U - the Pontecorvo, Maki, Nakagawa, Sakata (PMNS)neutrino mixing matrix, mj - the light Majorana neutrino
masses, mj ∼< 1 eV.
U - CP violating, in general: (Uej)2 = |Uej|2 eiαj1, j =
2,3, α21, α31 - Majorana CPV phases.S.M. Bilenky, J. Hosek, S.T.P.,1980
Nuclear0νββ-decaye-
e-
p
p
νn
n
A,Z A,Z+2ν−
strong in-medium modification of the basic processdd → uue−e−(νeνe)
continuum
0+ (A,Z) (A,Z+1)
(A,Z+2)
0+
0+ 1+
1-
2- virtual excitationof states of all multipolaritiesin (A,Z+1) nucleus
V. Rodin
1e-05 0.0001 0.001 0.01 0.1 1m
MIN [eV]
0.001
0.01
0.1
1|<
m>|
[e
V]
NH
IH
QD
S. Pascoli, S.T.P., 2007 (updated by S. Pascoli in 2012)
sin2 θ13 = 0.0236 ± 0.0042, δ = 0; α21, α31 - varied in the interval [0,π];
∆m221 = 7.58 × 10−5 eV2, 1σ(∆m2
21) = 3.5%;
sin2 θ21 = 0.306, 1σ(sin2 θ21) = 6%;
|∆m231| = 2.35 × 10−3 eV2, 1σ(|∆m2
31|) = 5%.
G.L. Fogli et al., Phys. Rev. D84 (2011) 053007
2σ(|<m>| ) used.
A number of different mechanisms possible.
For a given mechanism κ we have in the case of(A, Z) → (A, Z + 2) + e− + e−:
1T0ν1/2
= |ηLNVκ |2 G0ν(E0, Z)|M ′0ν
κ |2 ,
ηLNVκ - the fundamental LNV parameter characterising
the mechanism κ,
G0ν(E0, Z) - phase-space factor (includes g4A = (1.25)4,
as well as R−2(A), R(A) = r0A1/3 with r0 = 1.1 fm),
M ′0νκ = (gA/1.25)2M0ν
κ - NME(includes R(A) as a factor).
Different Mechanisms of (ββ)0ν-Decay
V + A
WR
WR
NkR
e−
e−
V + A
χjL, NkL
V − A
V − A
WL
WL
e−
e−
Light Majorana Neutrino Exchange
ην = <m>me
.
Heavy Majorana Neutrino Exchange Mechanisms
(V-A) Weak Interaction, LH Nk, Mk ∼> 10 GeV:
ηLN
=∑heavy
k U2ek
mpMk
, mp - proton mass , Uek - CPV .
(V+A) Weak Interaction, RH Nk, Mk ∼> 10 GeV:
ηRN
=(
MWMWR
)4∑heavy
k V 2ek
mpMk
; Vek: Nk − e− in the CC .
MW∼= 80 GeV; MWR ∼> 2.5 TeV; Vek - CPV, in general.
A comment.
(V-A) CC Weak Interaction:
e(1 + γ5)ec ≡ 2eL (ec)R , ec = C(e)T ,
C - the charge conjugation matrix.
(V+A) CC Weak Interaction:
e(1 − γ5)ec ≡ 2eR (ec)L .
The interference term: ∝ me, suppressed.A. Halprin, S.T.P., S.P. Rosen, 1983
SUSY Models with R-Parity Non-conservation
λ′
111
λ′
111
uL
uL
eL
eL
uL
uL
g
dR
dR
λ′
111
λ′
111
uL
uL
eL
eL
dR
dR
g
dR
dR
λ′
111
λ′
111
uL
uL
eL
eL
uL
dR
gdR
dR
LRp = λ′111
(uL dL)( ec
R
−νceR
)
dR + (eL νeL)dR
( u∗L
−d∗L
)
+ (uL dL)dR
( e∗L−ν∗eL
)
+ h.c.
The Gluino Exchange Dominance Mechanism
ηλ′ = παs6
λ′2111
G2Fm4
dR
mpmg
1 +
(
mdRmuL
)2
2
,
GF - the Fermi constant, αs = g23/(4π), g3 - the SU(3)c gauge coupling constant, muL, mdR
and
mg - the masses of the LH u-squark, RH d-squark and gluino.
The Squark-Neutrino Mechanism
ηq =∑
kλ′11kλ′
1k12√
2GFsin 2θd
(k)
1m2
d1(k)
− 1m2
d2(k)
,
d(k) = d, s, b; θd: dkL − dkR - mixing (3 light Majorana neutrinos assumed).
The 2e− current in both mechanisms:
e(1+γ5)ec ≡ 2eL (ec)R , as in the “standard” mechanism.
Example: (ββ)0ν-Decay and TeV Scale See-Saw Mech-anismType I see-saw mechanism, heavy Majorana neutrinos Nj at the TeV scale:
mν ≃ −MDM−1N MT
D, M = diag(M1, M2, M3), Mj ∼ (100 − 1000) GeV.
LNCC = − g
2√
2ℓ γα (RV )ℓk(1 − γ5)Nk W α + h.c. ,
LNNC = − g
2cwνℓL γα (RV )ℓk NkL Zα + h.c.
The exchange of virtual Nj gives a contribution to |<m>| :
|<m>| ∼=∣
∣
∑
i(UPMNS)
2ei mi −
∑
kf(A, Mk) (RV )2
ek(0.9 GeV)2
Mk
∣
∣ ,
f(A, Mk)∼= f(A) .
For, e.g., 48Ca, 76Ge, 82Se, 130Te and 136Xe, the function f(A) takes the values f(A) ∼= 0.033,
0.079, 0.073, 0.085 and 0.068, respectively.
• All low-energy constraints can be satisfied in a scheme with two heavy Majorana neu-
trinos N1,2, which form a pseudo-Dirac pair:
M2 = M1(1 + z), 0 < z ≪ 1.
• Only NH and IH ν mass spectra possible.
• The Predictions for |<m>| can be modified considerably.
A. Ibarra, E. Molinaro, S.T.P., 2010 and 2011
|<m>| vs |(RV )e1| for 76Ge in the cases of NH (left panel) and IH (right panel) light neutrino
mass spectrum, for M1 = 100 GeV and i) y = 0.001 (blue), ii) y = 0.01 (green). The gray
markers correspond to |<m>std| = |∑
i(UPMNS)
2ei mi|.
A. Ibarra, E. Molinaro, S.T.P., 2010 and 2011
Illustrative examples:
T0ν1/2(
76Ge), T0ν1/2(
100Mo), T0ν1/2(
130Te) used as input,
T0ν1/2(
76Ge) ≥ 1.9×1025y , T0ν1/2(
76Ge) = 2.23+0.44−0.31×1025y
(lower limit: Heidelberg-Moscow collab., 2001; value - Klapdor-Kleingrothaus et al., 2004.)
5.8 × 1023y ≤ T0ν1/2(
100Mo) ≤ 5.8 × 1024y (lower limit - NEMO3)
3.0 × 1024y ≤ T0ν1/2(
130Te) ≤ 3.0 × 1025y (lower limit-CUORICINO)
Constraints from 3H β-decay data
Light ν exchange + “nonstandard” mechanisms
Moscow, Mainz: m(νe) < 2.3 eV; |ην|2 × 1010 < 0.21 .
KATRIN: m(νe) < 0.2 eV; |ην|2 × 1010 < 1.6 × 10−3 .
Calculation of the NMEs for 76Ge, 82Se, 100Mo, 130TeThe NME: obtained within the Self-consistent Renormalized Quasiparticle Random Phase
Approximation (SRQRPA) (takes into account the Pauli exclusion principle and conserves
the mean particle number in correlated ground state).
Two choices of single-particle basis used:
i) the intermediate size model space has 12 levels (oscillator shells N=2-4) for 76Ge and82Se, 16 levels (oscillator shells N=2-4 plus the f+h orbits from N=5) for 100Mo and 18
levels (oscillator shells N=3,4 plus f+h+p orbits from N=5) for 130Te;
ii) the large size single particle space contains 21 levels (oscillator shells N=0-5) for 76Ge,82Se and 100Mo, and 23 levels for 130Te (N=1-5 and i orbits from N=6).
The single particle energies: obtained by using a Coulomb–corrected Woods–Saxon po-
tential. Two-body G-matrix elements we derived from the Argonne and the Charge
Dependent Bonn (CD-Bonn) one-boson exchange potential within the Brueckner the-
ory. The calculations: for gph = 1.0. The particle-particle strength parameter gpp of the
SRQRPA is fixed by the data on the two-neutrino double beta decays.
Table
The phase-space factor G0ν(E0, Z) and the nuclear matrix elements M ′0νν (light Majo-
rana neutrino exchange mechanism), M ′0νN (heavy Majorana neutrino exchange mecha-
nism), M ′0νλ′ (mechanism of gluino exchange dominance in SUSY with trilinear R-parity
breaking term) and M ′0νq (squark-neutrino mechanism) for the (ββ)0ν-decays of 76Ge,
100Se, 100Mo and 130Te. The nuclear matrix elements were obtained within the Self-
consistent Renormalized Quasiparticle Random Phase Approximation (SRQRPA).
Nuclear G0ν(E0, Z) |M ′0νν | |M ′0ν
N | |M ′0νλ′ | |M ′0ν
q |transition [y−1] gA = gA = gA = gA =
NN pot. m.s. 1.0 1.25 1.0 1.25 1.0 1.25 1.0 1.25
76Ge → 76Se 7.98 10−15 Argonne intm. 3.85 4.75 172.2 232.8 387.3 587.2 396.1 594.3large 4.39 5.44 196.4 264.9 461.1 699.6 476.2 717.8
CD-Bonn intm. 4.15 5.11 269.4 351.1 339.7 514.6 408.1 611.7large 4.69 5.82 317.3 411.5 392.8 595.6 482.7 727.6
82Se → 82Kr 3.53 10−14 Argonne intm. 3.59 4.54 164.8 225.7 374.5 574.2 379.3 577.9large 4.18 5.29 193.1 262.9 454.9 697.7 465.1 710.2
CD-Bonn intm. 3.86 4.88 258.7 340.4 328.7 503.7 390.4 594.5large 4.48 5.66 312.4 408.4 388.0 594.4 471.8 719.9
100Mo → 100Ru 5.73 10−14 Argonne intm. 3.62 4.39 184.9 249.8 412.0 629.4 405.1 612.1large 3.91 4.79 191.8 259.8 450.4 690.3 449.0 682.6
CD-Bonn intm. 3.96 4.81 298.6 388.4 356.3 543.7 415.9 627.9large 4.20 5.15 310.5 404.3 384.4 588.6 454.8 690.5
130Te → 130Xe 5.54 10−14 Argonne intm. 3.29 4.16 171.6 234.1 385.1 595.2 382.2 588.9large 3.34 4.18 176.5 239.7 405.5 626.0 403.1 620.4
CD-Bonn intm. 3.64 4.62 276.8 364.3 335.8 518.8 396.8 611.1large 3.74 4.70 293.8 384.5 350.1 540.3 416.3 640.7
Important feature of the NMEs
For each mechanism κ discussed, the NMEs for the nu-clei considered differ relatively little:
|M ′κi − M ′
κj| << M ′κi, M
′κj, typically
|M ′κi−M ′
κj|0.5(M ′
κi+M ′κj)
∼ 0.1, i 6= j =76 Ge,82 Se,100 Mo,130 Te.
Two “Non-Interfering” Mechanisms
Example: light LH and heavy RH Majorana ν exchanges
The corresponding LNV parameters, |ην| and |ηR| -
from “data” on T0ν1/2 of two nuclei:
1T1G1
= |ην|2|M ′0ν1,ν|2 + |ηR|2|M ′0ν
1,N |2,1
T2G2= |ην|2|M ′0ν
2,ν|2 + |ηR|2|M ′0ν2,N |2.
The solutions read:
|ην|2 =|M ′0ν
2,N |2/T1G1−|M ′0ν1,N |2/T2G2
|M ′0ν1,ν|2|M ′0ν
2,N |2−|M ′0ν1,N |2|M ′0ν
2,ν|2,
|ηR|2 =|M ′0ν
1,ν|2/T2G2−|M ′0ν2,ν|2/T1G1
|M ′0ν1,ν|2|M ′0ν
2,N |2−|M ′0ν1,N |2|M ′0ν
2,ν|2.
Solutions giving |ην|2 < 0 and/or |ηR|2 < 0 are unphysical. Given a pair (A1, Z1), (A2, Z2) of the
three 76Ge, 100Mo and 130Te we will be considering, and T1, and choosing (for convenience) always
A1 < A2, positive solutions for |ην|2 and |ηR|2 - possible for the following range of values of T2:
The positivity conditions
T1G1|M ′0ν1,N |2
G2|M ′0ν2,N |2 ≤ T2 ≤ T1G1|M ′0ν
1,ν|2G2|M ′0ν
2,ν|2(|M ′0ν
1,ν|2/|M ′0ν2,ν|2 > |M ′0ν
1,N |2/|M ′0ν2,N |2 (from Table 1) used.)
Using G1,2, and M ′0νi,ν, M ′0ν
i,N , i = 1,2, (Table 1, “CD-
Bonn, large, gA = 1.25 (1.0)”), we get the positivity
conditions for the 3 ratios of pairs of T0ν1/2 :
0.15 ≤T0ν1/2(
100Mo)
T0ν1/2
(76Ge)≤ 0.18 (0.17) ,
0.17 ≤T0ν1/2(
130Te)
T0ν1/2
(76Ge)≤ 0.22 (0.23) ,
1.14 (1.16) ≤T0ν1/2(
130Te)
T0ν1/2
(100Mo)≤ 1.24 (1.30) .
Similar results with Argonne, large, gA=1.25(1.0) NMEs:
0.15 ≤T0ν1/2(
100Mo)
T0ν1/2
(76Ge)≤ 0.18 ,
0.18 ≤T0ν1/2(
130Te)
T0ν1/2
(76Ge)≤ 0.24 (0.25) ,
1.22 ≤T0ν1/2(
130Te)
T0ν1/2
(100Mo)≤ 1.36 (1.42) .
The physical solutions possible only for remarkably nar-row intervals of T2/T1. If any of the ratios is shown tolie outside the relevant intervals, the case - excluded.
Conditions for only one mechanism being active:
|ηR|2 = 0 : |M ′0ν1,ν|2 T1 G1 = |M ′0ν
2,ν|2 T2 G2 ,
|ην|2 = 0 : |M ′0ν1,N |2 T1 G1 = |M ′0ν
2,N |2 T2 G2 .
Comments.• The feature discussed above - common to all cases of two “non-interfering” mecha-
nisms considered.
• The indicated specific half-life intervals for the various isotopes, are stable with respect
to the change of the NMEs.
• The intervals of T2/T1 depend on the type of the two “non-interfering” mechanisms.
However, the differences in the cases of the exchange of heavy Majorana neutrinos
coupled to (V+A) currents and i) light Majorana neutrino exchange, or ii) the gluino ex-
change mechanism, or iii) the squark-neutrino exchange mechanism, are extremely small.
• One of the consequences - if it will be possible to rule out one of them as the cause
of (ββ)0ν-decay, most likely one will be able to rule out all three of them.
• Using the indicated difference to get information about the specific pair of “non-
interfering” mechanisms possibly operative in (ββ)0ν-decay requires, in the cases consid-
ered by us, an extremely high precision in the measurement of the (ββ)0ν-decay half-lives
of the isotopes considered. The levels of precision required seem impossible to achieve
in the foreseeable future.
• If it is experimentally established that any of the indicated intervals of half-lives lies
outside the interval of physical solutions of |ην|2 and |ηR|2, obtained taking into account
all relevant uncertainties, one would be led to conclude that the (ββ)0ν-decay is not gen-
erated by the two mechanisms considered.
• The constraints under discussion will not be valid, in general, if the (ββ)0ν-decay is
triggered by two “interfering” mechanisms with a non-negligible (destructive) interfer-
ence term, or by more than two mechanisms none of which plays a subdominat role in
(ββ)0ν-decay.
The predictions for the half-life of a third nucleus (A3, Z3), using as input in the sys-
tem of equations for |ην|2 and |ηR|2 the half-lives of two other nuclei (A1, Z1) and
(A2, Z2). The three nuclei used are 76Ge, 100Mo and 130Te. The results shown are
obtained for a fixed value of the half-life of (A1, Z1) and assuming the half-life of
(A2, Z2) to lie in a certain specific interval. The physical solutions for |ην|2 and |ηR|2are then used to derive predictions for the half-life of the third nucleus (A3, Z3). The
latter are compared with the existing experimental lower limits. The results - ob-
tained with “CD-Bonn, large, gA = 1.25” NMEs (Table 1). One star beside the iso-
tope pair whose half-lives are used as input indicates predicted ranges of half-lives
of the nucleus (A3, Z3) that are not compatible with the existing lower bounds.
Pair T0ν1/2
(A1, Z1)[yr] T0ν1/2
[A2, Z2][yr] Prediction on [A3, Z3][yr]76Ge −100 Mo T(Ge) = 2.23 · 10253.23 · 1024 ≤ T (Mo) ≤ 3.97 · 1024 3.68 · 1024 ≤ T (Te) ≤ 4.93 · 1024
76Ge −130 Te T(Ge) = 2.23 · 1025 3.68 · 1024 ≤ T (Te) ≤ 4.93 · 1024 3.23 · 1024 ≤ T (Mo) ≤ 3.97 · 1024
76Ge −100 Mo T(Ge) = 1026 1.45 · 1025 ≤ T (Mo) ≤ 1.78 · 1025 1.65 · 1025 ≤ T (Te) ≤ 2.21 · 1025
76Ge −130 Te T(Ge) = 1026 1.65 · 1025 ≤ T (Te) ≤ 2.21 · 1025 1.45 · 1025 ≤ T (Mo) ≤ 1.78 · 1025
100Mo −130 Te ⋆T(Mo) = 5.8 · 1023 6.61 · 1023 ≤ T (Te) ≤ 7.20 · 1023 3.26 · 1024 ≤ T (Ge) ≤ 4.00 · 1024
100Mo −130 Te T(Mo) = 4 · 1024 4.56 · 1024 ≤ T (Te) ≤ 4.97 · 1024 2.25 · 1025 ≤ T (Ge) ≤ 2.76 · 1025
100Mo −130 Te T(Mo) = 5.8 · 1024 6.61 · 1024 ≤ T (Te) ≤ 7.20 · 1024 3.26 · 1025 ≤ T (Ge) ≤ 4.00 · 1025
100Mo −130 Te ⋆T(Te) = 3 · 1024 2.42 · 1024 ≤ T (Mo) ≤ 2.63 · 1024 1.36 · 1025 ≤ T (Ge) ≤ 1.82 · 1025
100Mo −130 Te T(Te) = 1.65 · 10251.33 · 1025 ≤ T (Mo) ≤ 1.45 · 1025 7.47 · 1025 ≤ T (Ge) ≤ 1.00 · 1026
100Mo −130 Te T(Te) = 3 · 1025 2.42 · 1025 ≤ T (Mo) ≤ 2.63 · 1025 1.36 · 1026 ≤ T (Ge) ≤ 1.82 · 1026
“CD-Bonn, large, gA = 1.0” NMEs: intervals change by ±5%;
“Argonne, large, gA = 1.25 (1.0)” NMEs: intervals change by ±10% (±14)%.
KATRIN
3.3´1024 3.6´1024 3.9´102410-4
10-3
10-2
10-1
1
10
T1�2H100MoL @yD
ÈΗR
2 �10
16,ÈΗΝ
2 �10
12
76Ge HT1�2=2.23�1025L and 100Mo
KATRIN
2.4´1024 2.5´1024 2.6´102410-4
10-3
10-2
10-1
1
10
TH100MoL @yD
ÈΗR
2 �10
16,ÈΗΝ
2 �10
12
130Te HT1�2=3�1024L and 100Mo
KATRIN
1.4´1025 1.5´1025 1.6´1025 1.7´1025 1.8´102510-4
10-3
10-2
10-1
1
10
T1�2H100MoL @yD
ÈΗR
2 �10
16,ÈΗΝ
2 �10
12
76Ge HT1�2=1026L and 100Mo
|ην|2: solid lines; |ηR|2: dashed lines.
Physical solutions - between the two vertical lines;
the solutions in the grey area excluded by the lower limit T 0ν1/2
(76Ge) ≥ 1.9 × 1025 y.
3.7´1024 4.1´1024 4.5´1024 4.9´102410-4
10-3
10-2
10-1
1
10
T1�2H130TeL @yD
ÈΗR
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2 �10
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76GeHT1�2= 2.23 1025L and 130Te
3.7´1024 4.2´1024 4.7´1024 5.2´102410-4
10-3
10-2
10-1
1
10
T1�2H130TeL @yD
ÈΗR
2 �10
16,ÈΗΝ
2 �10
12
76GeHT1�2= 2.23 1025L and 130Te
3.7´1024 4.2´1024 4.7´1024 5.2´102410-4
10-3
10-2
10-1
1
10
T1�2H130TeL @yD
ÈΗR
2 �10
16,ÈΗΝ
2 �10
12
76GeHT1�2= 2.23 1025L and 130Te
3.7´1024 4.2´1024 4.7´1024 5.2´102410-4
10-3
10-2
10-1
1
10
T1�2H130TeL @yD
ÈΗR
2 �10
16,ÈΗΝ
2 �10
12
76GeHT1�2= 2.23 1025L and 130Te
Solutions for |ην|2 (black lines) and |ηR|2 (red lines), for given T1 = T 0ν1/2
(76Ge) = 2.23×1025 yr and
T2 = T 0ν1/2
(130Te) and the “large basis” NMEs corresponding to: i) CD-Bonn p., gA = 1.25 (solid
lines), gA = 1 (dashed lines) (u.l. panel); ii) CD-Bonn (solid lines) and Argonne (dashed lines)
p. with gA = 1.25 (u.r. panel); iii) CD-Bonn (solid lines) and Argonne (dashed lines) p. with
gA = 1.0 (l.l. panel); iv) Argonne p. with gA = 1.25 (solid lines), gA = 1 (dashed lines) (l.r. panel).
The physical (positive) solutions shown with solid (dashed) lines - between the two vertical solid
(dashed) lines. The horizontal dashed line - the prospective KATRIN limit |<m>| < 0.2 eV.
The degeneracy between the intervals of allowed valuesof T2/T1 (determined by the positivity conditions) cor-responding to different pairs of “non-interfering” mech-anisms can be lifted by using in the analysis isotopeswith largely different NMEs, e.g., 136Xe and and any of76Ge,82Se,100Mo,130Te.
A. Meroni, S.T.P. and F. Simkovic, arXiv:1212.1331.
T0ν1/2(
136Xe) > 1.6 × 1025y , 90% C.L. , EXO;
T0ν1/2(
136Xe) > 1.9 × 1025y , 90% C.L. , KamLAND − Zen.
æ
æ
æ
æ
à àà
à
ì ìì
ì
76Ge 100Mo82Se 130Te
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
HMj'0ΝL-HM
Xe'0ΝL�HM
Xe'0ΝL
NMEs relative difference using136Xe and gA=1.25
æ
æ
æ
æ
àà à
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76Ge 100Mo82Se 130Te
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
HMj'0ΝL-HM
Xe'0ΝL�HM
Xe'0ΝL
NMEs relative difference using136Xe and gA=1
ææ
æ
æ
à à à
à
ì ì ì
ì
76Ge 100Mo82Se 130Te0.0
0.5
1.0
1.5
HMj'0ΝL-HM
Xe'0ΝL�HM
Xe'0ΝL
NMEs relative difference using136Xe and gA=1.25
æ
æ
æ
æ
àà à
à
ì ì ì
ì
76Ge 100Mo82Se 130Te0.0
0.5
1.0
1.5
HMj'0ΝL-HM
Xe'0ΝL�HM
Xe'0ΝL
NMEs relative difference using136Xe and gA=1
The relative differences between the Argonne NMEs (upper panels) and CD-Bonn NMEs
(lower panels) (M ′0νj −M ′0ν
i )/M ′0νi , where i=136Xe and j =76Ge,82Se,100Mo,130Te, for gA = 1.25
(left panel) and gA = 1 (right panel) and for three different non-interfering mecha-
nisms: light Majorana neutrino exchange (circles), RH heavy Majorana neutrino exchange
(squares) and gluino exchange (diamonds).
The conditions |ην|2 > 0 and |ηR|2 > 0 imply for theNMEs corresponding to gA = 1.25 (1.0):
1.90 (1.85) ≤T0ν1/2(
76Ge)
T0ν1/2(
136Xe)≤ 2.70 (2.64) (Argonne NMEs) ;
1.30 (1.16) ≤T0ν1/2(
76Ge)
T0ν1/2(
136Xe)≤ 2.47 (2.30) (CD − Bonn NMEs) .
Similarly, |ηλ|2 > 0 and |ηR|2 > 0 imply:
2.70 (2.64) ≤T0ν1/2(
76Ge)
T0ν1/2(
136Xe)≤ 2.78 (2.67) (Argonne NMEs) ;
1.30 (1.16) ≤T0ν1/2(
76Ge)
T0ν1/2(
136Xe)≤ 4.43 (4.25) (CD − Bonn NMEs) .
Using the EXO limit we obtain:
T0ν1/2(
76Ge) ≥ 3.03 (2.95) × 1025 y , Argonne NMEs;
T0ν1/2(
76Ge) ≥ 2.08 (1.85) × 1025 y , CD − Bonn NMEs .
Two “Interfering” Mechanisms
Example: light Majorana ν and gluino exchanges
In this case for a given (ββ)0ν decaying (A, Z),
1T0ν1/2,i
G0νi (E,Z)
=
|ην|2|M ′0νi,ν|2 + |ηλ′|2|M ′0ν
i,λ′|2 + 2cosα|M ′0νi,λ′||M ′0ν
i,ν||ην||ηλ′|
α = arg(ηνη∗λ′) (NMEs - real).
The LNV parameters |ην|, |ηλ′| and cosα -
from “data” on T0ν1/2 of three nuclei.
The solutions read:
|ην|2 = D1D , |ηλ′|2 = D2
D , z ≡ 2cosα|ην||ηλ′| = D3D ,
D =
∣
∣
∣
∣
∣
(M ′0ν1,ν)
2 (M ′0ν1,λ′)2 M ′0ν
1,λ′M ′0ν1,ν
(M ′0ν2,ν)
2 (M ′0ν2,λ′)2 M ′0ν
2,λ′M ′0ν2,ν
(M ′0ν3,ν)
2 (M ′0ν3,λ′)2 M ′0ν
3,λ′M ′0ν3,ν
∣
∣
∣
∣
∣
, D1 =
∣
∣
∣
∣
∣
1/T1G1 (M ′0ν1,λ′)2 M ′0ν
1,λ′M ′0ν1,ν
1/T2G2 (M ′0ν2,λ′)2 M ′0ν
2,λ′M ′0ν2,ν
1/T3G3 (M ′0ν3,λ′)2 M ′0ν
3,λ′M ′0ν3,ν
∣
∣
∣
∣
∣
,
D2 =
∣
∣
∣
∣
∣
(M ′0ν1,ν)
2 1/T1G1 M ′0ν1,λ′M ′0ν
1,ν
(M ′0ν2,ν)
2 1/T2G2 M ′0ν2,λ′M ′0ν
2,ν
(M ′0ν3,ν)
2 1/T3G3 M ′0ν3,λ′M ′0ν
3,ν
∣
∣
∣
∣
∣
, D3 =
∣
∣
∣
∣
∣
(M ′0ν1,ν)
2 (M ′0ν1,λ′)2 1/T1G1
(M ′0ν2,ν)
2 (M ′0ν2,λ′)2 1/T2G2
(M ′0ν3,ν)
2 (M ′0ν3,λ′)2 1/T3G3
∣
∣
∣
∣
∣
.
Physical solutions (“positivity conditions”):
|ην|2 ≥0, |ηλ′|2 ≥0, −|ην||ηλ′| ≤ cosα|ην||ηλ′| ≤ |ην||ηλ′|.Given (i.e. having data on) T1, T2 + using the conditionon the interference term z = 2cosα|ην||ηλ′|, determinesan interval of allowed values of T3.
Ranges of half-lives T3 in the case of two interfering mechanisms:
the light Majorana neutrino exchange and gluino exchange dominance.
T0ν1/2[y](fixed) T0ν
1/2[y](fixed) Allowed
T(Ge) = 2.23 · 1025 T(Mo) = 5.8 · 1024 5.99 · 1024 ≤ T(Te) ≤ 7.35 · 1024
T(Ge) = 2.23 · 1025 T(Te) = 3 · 1024 2.46 · 1024 ≤ T(Mo) ≤ 2.82 · 1024
T(Ge) = 1026 T(Mo) = 5.8 · 1024 6.30 · 1024 ≤ T(Te) ≤ 6.94 · 1024
T(Ge) = 1026 T(Te) = 3 · 1024 2.55 · 1024 ≤ T(Mo) ≤ 2.72 · 1024
T(Ge) = 2.23 · 1025 T(Te) = 3 · 1025 2.14 · 1025 ≤ T(Mo) ≤ 3.31 · 1025
T(Ge) = 1026 T(Te) = 3 · 1025 2.38 · 1025 ≤ T(Mo) ≤ 2.92 · 1025
“CD-Bonn potential, large, gA = 1” NMEs
T0ν1/2[y](fixed) T0ν
1/2[y](fixed) Allowed
T(Ge) = 2.23 · 1025 T(Mo) = 5.8 · 1024 3 · 1024 ≤ T(Te) ≤ 8.62 · 1024
T(Ge) = 2.23 · 1025 T(Te) = 3 · 1024 2.55 · 1024 ≤ T(Mo) ≤ 6.18 · 1024
T(Ge) = 2.23 · 1025 T(Te) = 3 · 1025 1.33 · 1025 ≤ T(Mo) ≤ 3.88 · 1026
T(Ge) = 1026 T(Mo) = 5.8 · 1024 3.62 · 1024 ≤ T(Te) ≤ 6.04 · 1024
T(Ge) = 1026 T(Te) = 3 · 1024 3.11 · 1024 ≤ T(Mo) ≤ 4.70 · 1024
T(Ge) = 1026 T(Te) = 3 · 1025 2.15 · 1025 ≤ T(Mo) ≤ 8.29 · 1025
“Argonne potential, large, gA = 1.25” NMEs
T0ν1/2[y](fixed) T0ν
1/2[y](fixed) Allowed
T(Ge) = 2.23 · 1025 T(Mo) = 5.8 · 1024 3 · 1024 ≤ T(Te) ≤ 9.22 · 1024
T(Ge) = 2.23 · 1025 T(Te) = 3 · 1024 2.55 · 1024 ≤ T(Mo) ≤ 7.92 · 1024
T(Ge) = 2.23 · 1025 T(Te) = 3 · 1025 1.19 · 1025 ≤ T(Mo) ≤ 2.55 · 1027
T(Ge) = 1026 T(Mo) = 5.8 · 1024 3.15 · 1024 ≤ T(Te) ≤ 5.85 · 1024
T(Ge) = 1026 T(Te) = 3 · 1024 3.25 · 1024 ≤ T(Mo) ≤ 5.49 · 1024
T(Ge) = 1026 T(Te) = 3 · 1025 2.08 · 1025 ≤ T(Mo) ≤ 1.20 · 1026
“Argonne Potential, large, gA = 1” NME
T0ν1/2[y](fixed) T0ν
1/2[y](fixed) Allowed
T(Ge) = 2.23 · 1025 T(Mo) = 5.8 · 1024 3 · 1024 ≤ T(Te) ≤ 1.11 · 1025
T(Ge) = 2.23 · 1025 T(Te) = 3 · 1024 2.63 · 1024 ≤ T(Mo) ≤ 2.04 · 1025
T(Ge) = 2.23 · 1025 T(Te) = 3 · 1025 9.19 · 1024 ≤ T(Mo) ≤ 2.36 · 1026
T(Ge) = 1026 T(Mo) = 5.8 · 1024 3 · 1024 ≤ T(Te) ≤ 5.07 · 1024
T(Ge) = 1026 T(Te) = 3 · 1024 3.82 · 1024 ≤ T(Mo) ≤ 9.44 · 1024
T(Ge) = 1026 T(Te) = 3 · 1025 1.96 · 1025 ≤ T(Mo) ≤ 6.54 · 1026
KATRIN
Mainz �Moscow
6.´1024 6.5´1024 7.´1024 7.5´102410-4
10-3
10-2
10-1
1
T1�2H130TeL @yD
ÈΗΛ’
2 �10
14,ÈΗΝ
2 �10
10
76GeH2.23 1025L,100MoH5.8 1024L and 130Te HfreeL
KATRIN
Mainz �Moscow
2.5´1024 2.6´1024 2.7´1024 2.8´102410-4
10-3
10-2
10-1
1
T1�2H100MoL @yD
ÈΗΛ’
2 �10
14,ÈΗΝ
2 �10
10
76GeH2.23 1025L,130Te H3´1024L and 100Mo HfreeL
KATRIN
Mainz �Moscow
6.3´1024 6.5´1024 6.7´1024 6.9´102410-4
10-3
10-2
10-1
1
T1�2H130TeL @yD
ÈΗΛ’
2 �10
14,ÈΗΝ
2 �10
10
76GeH1026L, 100MoH5.8 1024L and 130Te HfreeL
KATRIN
Mainz �Moscow
2.5´1024 2.6´1024 2.7´102410-4
10-3
10-2
10-1
1
T1�2H100MoL @yD
ÈΗΛ’
2 �10
14,ÈΗΝ
2 �10
10
76GeH1026L,130Te H3´1024L and 100Mo HfreeL
KATRIN
Mainz �Moscow
2.´1025 2.5´1025 3.´1025 3.5´102510-4
10-3
10-2
10-1
1
T1�2H100MoL @yD
ÈΗΛ’
2 �10
14,ÈΗΝ
2 �10
10
76GeH2.23 1025L,130TeH3´1025L and 100Mo HfreeL
KATRIN
Mainz �Moscow
2.4´1025 2.6´1025 2.8´1025 3.´102510-4
10-3
10-2
10-1
1
T1�2H100MoL @yD
ÈΗΛ’
2 �10
14,ÈΗΝ
2 �10
10
GeH1026L, 130TeH3´1025L and 100Mo HfreeL
|ην|2 × 1010 : solid lines; |ηλ′|2 × 1014: dashed lines. All solutions: cosα ∼= −1.
Allowed regions (physical solutions) - white areas;
the solutions in the grey (blue) areas - excluded.
The horizontal solid (dashed) line - the Moscow-Mainz limit |<m>| < 2.3 eV
(the prospective KATRIN limit |<m>| < 0.2 eV).
KATRIN
Mainz �Moscow
4.5765 1024 4.58´1024 4.5835 102410-5
10-4
10-3
10-2
10-1
1
T1�2H130TeL@yD
ÈΗΛ’
2 �10
14,ÈΗΝ
2 �10
10,z�10
12
GeH2.23�1025L,100MoH3.7�1024L and 130Te HfreeL
4.577 1024 4.58´1024 4.583 10240
Π
2
Π
T1�2H130TeL@yD
Α@r
adD
PhaseΑ
A case of constructive interference: cosα > 0.
|ην|2 × 1010 (solid line), |ηλ′|2 × 1014 (dashed line) and
z × 1012 = 2cosα|ην||ηλ′| × 1012 (dot-dashed line).
Conditions for |ην|2 > 0, |ηλ′|2 > 0 and z = 0 (no int.),or z > 0 (constructive int.), or z < 0 (distructive int.).
The general conditions were derived. Below -the conditions for “CD-Bonn, large, ga = 1.24” NMEs.
Given T1, |ην|2 > 0, |ηλ′|2 > 0, z > 0:
z > 0 :
0.14T1 < T2 ≤ 0.16T1 ,4.44T1 T2
3.74T1 − 0.93T2
≤ T3 ≤ 2.10T1 T2
1.78T1 − 0.47T2
;
0.16T1 < T2 < 0.18T1 ,4.44T1 T2
3.74T1 − 0.93T2
≤ T3 ≤ 4.10T1 T2
3.44T1 − 0.81T2
.
Given T1, z > 0 only if T2 lies in a relatively narrow interval and T3 has a value in extremely
narrow intervals; a consequence of the values of Gi and of the NMEs used: for the
3 nuclei considered, |Mi − Mj| << Mi, Mj, |Λi − Λj| << Λi,Λj, i 6= j = 1,2,3, and typically
|Mi − Mj|/(0.5(Mi + Mj)) ∼ 10−1, |Λi − Λj|/(0.5(Λi + Λj)) ∼ (10−2 − 10−1),
Mi ≡ M ′0νi,ν, Λi ≡ M ′0ν
i,λ′, i =76 Ge,100 Mo,130 Te.
Given T1, |ην|2 > 0, |ηλ′|2 > 0, z < 0:
z < 0 :
T2 ≤ 0.14T1 , T3 ≤ 2.10T1 T2
1.78T1 − 0.47T2
;
0.14T1 < T2 ≤ 0.18T1 , T3 ≤ 4.44T1 T2
3.74T1 − 0.93T2
;
0.18T1 < T2 < 4.23T1 , T3 ≤ 4.10T1 T2
3.44T1 − 0.81T2
;
T2 ≥ 4.23T1 T3 > 0 .
The intervals of values of T2 and T3 - very different from those corresponding to the cases
of two “non-interfering” mechanisms (the only exception - the second set of intervals
which partially overlap with the latter ).
This difference can allow to discriminate experimentally between the two possibilities of
(ββ)0ν-decay being triggered by two “non-interfering” mechanisms or by two “destruc-
tively interfering” mechanisms.
Given T1, |ην|2 = 0, |ηλ′|2 > 0 (z = 0):
T2 = 0.14T1 , T3 =2.10T1T2
1.78T1 − 0.47T2
∼= 0.18T1 .
Given T1, |ην|2 > 0, |ηλ′|2 = 0 (z = 0):
T2 = 0.18T1 , T3 =4.10T1T2
3.44T1 − 0.81T2
∼= 0.22T1 .
Additional consequence of “positivity” and “interfer-ence” conditions.
Given T0ν1/2 of one isotope, say of 76Ge (T1) + an exper-
imental lower bound on the T0ν1/2 of a 2nd isotope, e.g.,
of 130Te (T3), the conditions imply a constraint on theT0ν1/2 of any 3rd isotope, say of 100Mo (T2).
The constraint depends noticeably on the type of the two“interfering” mechanisms generating the (ββ)0ν-decayand can be used, in principle, to discriminate betweenthe different possible pairs of “interfering” mechanisms.
Example: T1 = 2.23 × 1025 y (76Ge), T3 > 3.0 × 1024
y (130Te), constraint on T2 (100Mo); “CD-bonn (Ar-gonne), large, gA = 1.25” NMEs used.
Light Neutrino and gluino exchange mechanisms:
T2 ≡ T0ν1/2(
100Mo) > 2.46 (2.47) × 1024 y.
(Increasing the value of T 0ν1/2
(76Ge) leads to the increasing of the value of the lower limit.)
Light Neutrino and LH Heavy neutrino exchanges:
T0ν1/2(
100Mo) > 2.78 (2.68) × 1024 y.
(The value of the lower limit increases with the increasing of the value of T 0ν1/2
(76Ge).)
Squarks-neutrino and gluino exchange mechanisms:
T0ν1/2(
100Mo) > 7.92 (22.1) × 1023 y.
( For larger values of T 0ν1/2
(76Ge), this lower bound assumes larger values.)
LH Heavy neutrino and gluino exchange mechanisms:
1.36 × 1024 y < T0ν1/2(
100Mo) < 3.42 × 1024 y .Increasing the value of T 0ν
1/2(76Ge) leads to a shift of the interval to larger values; for a
sufficiently large T 0ν1/2
(76Ge) > 1026 y - only a lower bound on T 0ν1/2
(100Mo). Using the NMEs
derived with the Argonne potential - only a lower bound: T 0ν1/2
(100Mo) > 5.97×1023 y. The
difference between the results obtained with the two sets of NMEs can be traced to fact
that the determinant D, calculated with the second set of NMEs, has opposite sign to
that, calculated with the first set of NMEs. As a consequence, the dependence of the
physical solutions for |ηLN |2 and |ηλ′|2 on T1, T2 and T3 in the two cases of NMEs is very
different.
The constraints thus obtained can be used, e.g., to exclude some of the possible cases
of two “interfering” mechanisms inducing the (ββ)0ν-decay: if, e.g., it is confirmed that
T 0ν1/2
(76Ge) = 2.23 × 1025 y, and in addition it is established that T 0ν1/2
(100Mo) ≤ 1024 y,
that combined with the experimental lower limit on T 0ν1/2
(130Te) would rule out i) the
light neutrino and gluino exchanges, and ii) the light neutrino and LH heavy neutrino
exchanges, as possible mechanisms generating the (ββ)0ν-decay.
Conclusions.
If the decay (A, Z) → (A, Z +2)+e−+e− ((ββ)0ν-decay)will be observed, the questions will inevitably arise:
Which mechanism is triggering the decay?
How many mechanisms are involved?
Discussed how one possibly can answer these questions.• The measurements of the (ββ)0ν-decay half-lives withrather high precision and the knowledge of the relevantNMEs with relatively small uncertainties is crucial for es-tablishing that more than one mechanisms are operativein (ββ)0ν-decay.• The method considered can be generalised to the caseof more than two (ββ)0ν-decay mechanisms.• It allows to treat the cases of CP conserving and CPnonconserving couplings generating the (ββ)0ν-decay ina unique way.
To Frank and Ettore:
Happy Anniversary!
Thank you for continuing to be the driving force in the field of (ββ)0ν-decay reserach and
inspiration for the younger generation of researchers working in this field.
My best wishes for personal happiness and many professional successes!
