MSU May 15, 2015 M. Horoi CMU
Nuclear structure for tests of fundamental symmetries
Mihai Horoi
Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA
Ø Support from NSF grant PHY-1404442 and DOE/SciDAC grants DE-SC0008529/SC0008641 is acknowledged
MSU May 15, 2015 M. Horoi CMU
5/12/15, 7:47 AMScientific Opportunities | frib.msu.edu
Page 2 of 3http://frib.msu.edu/content/scientific-opportunities
What is the origin of simple patterns in complex nuclei?What are the heaviest nuclei that can exist?
Source: 2006 brochure from the RIA users community
Nuclear AstrophysicsNuclear physics and astronomy are inextricably intertwined. In fact, more than ever, astronomicaldiscoveries are driving the frontiers of nuclear physics while our knowledge of nuclei is driving progress inunderstanding the universe.
Because of its powerful technical capabilities, FRIB will forge tighter linksbetween the two disciplines. Rare isotopes play a critical role in theevolution of stars and other cosmic phenomena such as novae andsupernovae, but up to now the most interesting rare isotopes have beenlargely out of the reach of terrestrial experiments. FRIB will provideaccess to most of the rare isotopes important in these astrophysicalprocesses, thus allowing scientists to address questions such as:
How are the elements from iron to uranium created?How do stars explode?What is the nature of neutron star matter?
Recent astronomical missions such as the Hubble Space Telescope, Chandra X-ray Observatory,Spitzer Space Telescope, and the Sloan Digital Sky Survey have provided new and detailed informationon element synthesis, stellar explosions, and neutron stars over a wide range of wavelengths. However,scientists attempting to interpret these observations have been constrained by the lack of information onthe physics of unstable nuclei.
FRIB and future astronomy missions such as the Joint Dark Energy Mission, and the Advanced ComptonTelescope will complement each other and provide a potent combination of tools to discover answers toimportant questions that confront the field.
Source: 2006 brochure from the RIA users community
Fundamental InteractionsNuclear and particle physicists study fundamental interactions for twobasic reasons: to clarify the nature of the most elementary pieces ofmatter and determine how they fit together and interact. Most of whathas been learned so far is embodied in the Standard Model of particlephysics, a framework that has been both repeatedly validated byexperimental results and is widely viewed as incomplete.
"[Scientists] have been stuck in that model, like birds in a gilded cage,ever since [the 1970s]," wrote Dennis Overbye in a July 2006 essayfor The New York Times. "The Standard Model agrees with everyexperiment that has been performed since. But it doesn't say anythingabout the most familiar force of all, gravity. Nor does it explain whythe universe is matter instead of antimatter, or why we believe there are such things as space and time."
Rare isotopes produced at FRIB's will provide excellent opportunities for scientists to devise experimentsthat look beyond the Standard Model and search for subtle indications of hidden interactions and minutelybroken symmetries and thereby help refine the Standard Model and search for new physics beyond it.
Sources: 2006 brochure from the RIA users community, New York Times
Applied Benefits
Nuclei, a laboratory for studying fundamental interactions and fundamental symmetries
- Double-beta decay: 76Ge, 82Se, 130Te, 136Xe
- EDM: 199Hg, 225Ra, 211Rn, etc
- PNC: 14N, 18F, 19F, 21Ne (PRL 74, 231 (1995))
- Beta decay: super-allowed, angular correlations, etc
Classical Double Beta Decay Problem
MSU May 15, 2015 M. Horoi CMU
Adapted from Avignone, Elliot, Engel, Rev. Mod. Phys. 80, 481 (2008) -> RMP08
€
Qββ
A.S. Barabash, PRC 81 (2010)
136Xe 2.23×1021 0.010 €
T1/ 2−1(2ν) =G2ν (Qββ ) MGT
2v (0+)[ ] 2
€
T1/ 2−1(0v) =G0ν (Qββ ) M
0v (0+)[ ] 2 < mββ >
me
%
& '
(
) *
2
€
mββ = mkUek2
k∑
2-neutrino double beta decay
neutrinoless double beta decay
Neutrino Masses
MSU May 15, 2015 M. Horoi CMU
- Tritium decay:
- Cosmology: CMB power spectrum, BAO, etc,
€
Δm212 ≈ 7.5 ×10−5 eV 2 (solar)
Δm322 ≈ 2.4 ×10−3 eV 2 (atmospheric)
€
c12 ≡ cosθ12 , s12 = sinθ12 , etc
Two neutrino mass hierarchies
€
3H → 3He + e− +ν e
mν e= Uei
2mi2
i∑ < 2.2eV (Mainz exp.)
KATRIN (to takedata): goal mν e< 0.3eV
€
mii=1
3
∑ < 0.23eV
Goal : 0.01eV (5 −10 y)
€
PMNS −matrix
€
m02 = ?
€
Neutrino oscillations :− NH or IH ?− δCP = ?−Unitarity of UPMNS ?− Are there m ~ 1eV sterile neutrinos?
€
− Dirac or Majorana?− Majorana CPV α i = ?− Leptogenesis?→Baryogenesis
Neutrino ββ effective mass
MSU May 15, 2015 M. Horoi CMU
€
T1/ 2−1(0v) =G0ν (Qββ ) M
0v (0+)[ ] 2 < mββ >
me
%
& '
(
) *
2€
mββ = mkUek2
k=1
3
∑
= c122 c13
2m1 + c132 s12
2m2eiφ 2 + s13
2m3eiφ 3
Cosmology constraint
€
φ2 = α2 −α1 φ3 = −α1 − 2δ
76Ge Klapdor claim 2006
The Minimal Standard Model
MSU May 15, 2015 M. Horoi CMU
?
SM fermion masses :ψiLφYijψ jR →Yij < φ >ψiLψ jR = mD( )ijψiLψ jR
€
→neutrino is sterile: Dµ = I∂µ
€
SU(2)Ldoublet
€
SU(2)Lsinglet
€
SU(2)Ldoublet
€
mν l
SM = 0 l = e, µ, τlepton flavor conserved
€
φ
€
Local Gauge invariance of Lagrangian density L :
Dµ = I∂µ − igAµa (x)T a
T a ∈GA SM group : SU(3)c × SU(2)L ×U(1)Y
€
SU(3)c ×U(1)em
€
EWSB
Too Small Yukawa Couplings?
MSU May 15, 2015 M. Horoi CMU
arXiv:1406.5503 Standard Model fermion masses
€
-L ⊃12ψ iLYijψ jRφ →
12mDijψ iLψ jR mDij =Yij v( )
€
-L ⊃12mLR # ν R
c # ν Lc
€
Majorana
€
Yν ≈10−10 −10−9Yt
arXiv:0710.4947v3
€
φ
€
φ
€
< φ >
€
< φ >
The origin of Majorana neutrino masses
MSU May 15, 2015 M. Horoi CMU
See-saw mechanisms
mLLν ≈
(100 GeV )2
1014GeV= 0.1eV
mLLν ≈
(300keV )2
1TeV= 0.1eV
"
#$$
%$$
€
< φ >
€
< φ >
€
φ
€
φ
Left-Right Symmetric model
Weinberg’s dimension-5 BSM operator contributing to Majorana neutrino mass
φ φ
νL νL
WR search at CMS arXiv:1407.3683
Low-energy LR contributions to 0vββ decay
MSU May 15, 2015 M. Horoi CMU
€
-L ⊃12
hαβT ν βL e α L( ) Δ
− − Δ0
Δ−− Δ−
(
) *
+
, -
eRc
−νRc
(
) *
+
, - + hc
No neutrino exchange
€
η
€
λ
€
H W =GF
2jL
µ JLµ+ +κJRµ
+( ) + jRµ ηJLµ
+ + λJRµ+( )[ ] + h.c.
Left − right symmetric model
€
H W =GF
2jL
µJLµ+ + h.c.
€
jL /Rµ = e γ µ 1∓ γ 5( )ν e
Low-energy effective Hamiltonian
DBD signals from different mechanisms
MSU May 15, 2015 M. Horoi CMU
arXiv:1005.1241
2β0ν rhc(η)
€
< λ >
t = εe1 −εe2
Two Non-Interfering Mechanisms
MSU May 15, 2015 M. Horoi CMU
€
ην =mββ
me
≈10−6
€
ηNR =MWL
MWR
#
$ %
&
' (
4
Vek2 mp
Mkk
heavy
∑ ≈10−8
Assume T1/2(76Ge)=22.3x1024 y
€
ην , ηNR ⇐GGe0νT1/ 2Ge
0ν[ ]−1
= MGe(0ν ) 2ην
2+ MGe
(0N ) 2ηNR2
GXe0νT1/ 2Xe
0ν[ ]−1
= MXe(0ν ) 2ην
2+ MXe
(0N ) 2ηNR2
&
' (
) (
€
See also PRD 83,113003 (2011)
T1/20ν!" #$
−1≈G0ν M (0ν ) 2 ηνL
2+ M (0N ) 2 ηNR
2!"'
#$( No interference terms!
Is there a more general description?
MSU May 15, 2015 M. Horoi CMU
Long-range terms: (a) - (c )
€
Aββ ∝ T[L (t1)L (t2)]∝ jV −AJV −A+( ) jαJβ+( )
α, β :V − A, V + A, S + P, S − P, TL , TR
€
G010ν , G06
0ν , G090ν
Doi, Kotani, Takasugi 1983
Short-range terms: (d)
€
Jµν = u i2γ µ ,γν[ ] 1± γ 5( )d
€
Aββ ∝ L
More long-range contributions?
MSU May 15, 2015 M. Horoi CMU
€
SUSY&LRSM :Prezeau, Ramsey −Musolf ,Vogel, PRC 68, 034016 (2003)
Hadronization /w R-parity v. and heavy neutrino €
SUSY /wR − parity v. : e.g. Rep.Prog.Phys. 75,106301(2012)
Summary of 0vDBD mechanisms
• The mass mechanism (a.k.a. light-neutrino exchange) is likely, and the simplest BSM scenario.
• Low mass sterile neutrino would complicate analysis • Right-handed heavy-neutrino exchange is possible, and
requires knowledge of half-lives for more isotopes. • η- and λ- mechanisms are possible, but could be ruled
in/out by energy and angular distributions. • Left-right symmetric model may be also (un)validated
at LHC/colliders. • SUSY/R-parity, KK, GUT, etc, scenarios need to be
checked, but validated by additional means. MSU May 15, 2015 M. Horoi CMU
MSU May 15, 2015 M. Horoi CMU
2v Double Beta Decay (DBD) of 48Ca
€
f7 / 2€
p1/ 2
€
f5 / 2
€
p3 / 2
€
G
€
T1/ 2−1 =G2v (Qββ ) MGT
2v (0+)[ ] 2
€
48Ca 2v ββ# → # # 48Ti
€
Ikeda sum rule(ISR) = B(GT;Z →Z +1)∑ − B(GT;Z →Z −1)∑ = 3(N − Z)
Ikeda satisfied in pf !
€
E0 =12Qββ + ΔM Z +1
AT−ZAX( )
The choice of valence space is important!
€
B(GT) =f ||σ⋅ τ || i
2
(2Ji +1)
Horoi, Stoica, Brown,
PRC 75, 034303 (2007) €
gAστquenched$ → $ $ $ 0.77gAστ
ISR 48Ca 48Ti pf 24.0 12.0
f7 p3 10.3 5.2
Closure Approximation and Beyond in Shell Model
MSU May 15, 2015 M. Horoi CMU
€
MS0v = ˜ Γ ( ) 0 f
+ ap+ ˜ a n( )
JJk Jk a # p
+ ˜ a # n ( )J
0i+
p # p n # n J k J
∑ p # p ;J q2dq ˆ S h(q) jκ (qr)GFS
2 fSRC2
q q + EkJ( )
τ1−τ2−
(
) * *
+
, - -
∫ n # n ;J − beyond
Challenge: there are about 100,000 Jk states in the sum for 48Ca
Much more intermediate states for heavier nuclei, such as 76Ge!!!
No-closure may need states out of the model space (not considered).
€
MS0v = Γ( ) 0 f
+ ap+a # p
+( )J
˜ a # n ˜ a n( )J$ % &
' ( )
0
0i+ p # p ;J q2dq ˆ S
h(q) jκ (qr)GFS2 fSRC
2
q q+ < E >( )τ1−τ2−
$
% &
'
( ) ∫ n # n ;J
as
− closureJ, p< # p n< # n p< n
∑
Minimal model spaces 82Se : 10M states 130Te : 22M states 76Ge : 150M states
€
M 0v = MGT0v − gV /gA( )2
MF0v + MT
0v
ˆ S =σ1τ1σ2τ 2 Gamow −Teller (GT)τ1τ 2 Fermi (F)
3( ! σ 1⋅ ˆ n )(! σ 2 ⋅ ˆ n ) − (! σ 1⋅! σ 2)[ ]τ1τ 2 Tensor (T)
&
' (
) (
€
many − body! " # # # # # # # $ # # # # # # #
€
two − body! " # # # # # # # # # # # # # $ # # # # # # # # # # # # #
MSU May 15, 2015 M. Horoi CMU
1 2 3 4 5 6 7 8 9 10 11 12 13 14Closure energy <E> [MeV]
2.8
3
3.2
3.4
3.6
3.8pure closure, CD-Bonn SRCmixed, CD-Bonn SRCpure closure, AV18 SRCmixed, AV18 SRC
0 50 100 150 200 250N, number-of-states cutoff parameter
3.2
3.3
3.4
3.5
3.6mixed, <E>=1 MeVmixed, <E>=3.4 MeVmixed, <E>=7 MeVmixed, <E>=10 MeV
50 100 150 200 2500
0.5
1
1.5
2Er
ror [
%]
error in mixed NME
J=0 J=1 J=2 J=3 J=4 J=5 J=6 J=7 J=8 J=9Spin of the intermediate states
-0.2
0
0.2
0.4
0.6
0.8
1 GT, positiveGT, negativeFM, positiveFM, negative
82Se: PRC 89, 054304 (2014)
€
Mmixed (N) = Mno−closure (N) + Mclosure (N = ∞) −Mclosure (N)[ ]
GXPF1A FPD6 KB3G JUN450
0.5
1
1.5
2
2.5
3
3.5
4
Opti
mal
clo
sure
ener
gy [
MeV
]
48Ca
46Ca
44Ca
76Ge
82Se
New Approach to calculate NME: New Tests of Nuclear Structure
MSU May 15, 2015 M. Horoi CMU I=0 I=1 I=2 I=3 I=4 I=5 I=6 I=7 I=8 I=9Spin of the neutron-neutron (proton-proton) pairs-3
-2
-1
0
1
2
3
4
5
6
7
GT, positiveGT, negativeFM, positiveFM, negative
Brown, Horoi, Senkov
PRL 113, 262501 (2014)
MSU May 15, 2015 M. Horoi CMU
136Xe ββ Experimental Results Publication Experiment T2ν
1/2 T0ν1/2(lim) T0ν
1/2(Sens)
PRL 110, 062502 KamLAND-Zen > 1.9x1025 y
1.1x1025 y
PRC 89, 015502 EXO-200 (2.11 0.04 0.21)x1021 y Nature 510, 229 EXO-200 >1.1x1025 y 1.9x1025 y
PRC 85, 045504 KamLAND-Zen (2.38 0.02 0.14)x1021 y
€
±
€
±
€
±
€
±
€
Mexp2ν = 0.0191− 0.0215 MeV −1
EXO-200
arXiv:1402.6956, Nature 510, 229
MSU May 15, 2015 M. Horoi CMU
136Xe 2νββ Results
€
στ →0.74στ quenching
0g7/2 1d5/2 1d3/2 2s5/2 0h11/2 model space
0h11/2
2s5/2
1d3/2
1d5/2
0g7/2
0h9/2
0g9/2
0h11/2
2s5/2
1d3/2
1d5/2
0g7/2
0h9/2
0g9/2
0h11/2
2s5/2
1d3/2
1d5/2
0g7/2
0h9/2
0g9/2
€
136Xe(0+)
€
136Cs(1+)
€
136Ba(0+)
€
M 2ν = 0.064 MeV −1
€
Mexp2ν = 0.019 MeV −1
€
B(GT;Z →Z +1)∑ − B(GT;Z →Z −1)∑ = 52
Ikeda: 3(N − Z) = 84
0g9/2 0g7/21d5/2 1d3/2 2s5/2 0h11/2 0h9/2
€
B(GT;Z →Z +1)∑ − B(GT;Z →Z −1)∑ = 84
Ikeda: 3(N − Z) = 84
New effective interaction,
0h11/2
2s5/2
1d3/2
1d5/2
0g7/2
0h9/2
0g9/2
np - nh
n (0+) n (1+) M(2v)
0 0 0.062
0 1 0.091
1 1 0.037
1 2 0.020 Horoi, Brown,
PRL 111, (2013)
MSU May 15, 2015 M. Horoi CMU
S. Vigdor talk at LRP Town Meeting, Chicago, Sep 28-29, 2014
€
T1/ 2 >1×1026 y, after ? years
€
T1/ 2 > 2.4 ×1026 y, after 3 years
€
T1/ 2 >1×1026 y, after 5 years€
T1/ 2 >1×1026 y, after 5 years
€
T1/ 2 > 2 ×1026 y, after ? years€
T1/ 2 > 6 ×1027 y, after 5 years! (nEXO)
€
Goals (DNP14 DBD workshop) :
IBA-2 J. Barea, J. Kotila, and F. Iachello, Phys. Rev. C 87, 014315 (2013).
QRPA-En M. T. Mustonen and J. Engel, Phys. Rev. C 87, 064302 (2013).
QRPA-Jy J. Suhonen, O. Civitarese, Phys. NPA 847 207–232 (2010).
QRPA-Tu A. Faessler, M. Gonzalez, S. Kovalenko, and F. Simkovic, arXiv:1408.6077
ISM-Men J. Menéndez, A. Poves, E. Caurier, F. Nowacki, NPA 818 139–151 (2009). SM M. Horoi et. al. PRC 88, 064312 (2013), PRC 89, 045502 (2014), PRC 89, 054304 (2014), PRC 90, 051301(R) (2014), PRC 91, 024309 (2015), PRL 110, 222502 (2013), PRL 113, 262501(2014).
MSU May 15, 2015 M. Horoi CMU
IBM-2 PRC 91, 034304 (2015)
IBA-2 J. Barea, J. Kotila, and F. Iachello, Phys. Rev. C 87, 014315 (2013).
QRPA-Tu A. Faessler, M. Gonzalez, S. Kovalenko, and F. Simkovic, arXiv:1408.6077
SM M. Horoi et. al. PRC 88, 064312 (2013), PRC 90, PRC 89, 054304 (2014), PRC 91, 024309 (2015), PRL 110, 222502 (2013).
MSU May 15, 2015 M. Horoi CMU
€
CD − Bonn SRC→
€
AV18 SRC→
The effect of larger model spaces for 48Ca
MSU May 15, 2015 M. Horoi CMU
€
f7 / 2€
p1/ 2
€
f5 / 2
€
p3 / 2
€
d5 / 2
€
d3 / 2
€
s1/ 2
€
sd − pf
€
N = 2€
N = 3
M(0v) SDPFU SDPFMUP 0 0.941 0.623 0+2 1.182 (26%) 1.004 (61%)
€
!ω
€
!ω
SDPFU: PRC 79, 014310 (2009)
SDPFMUP: PRC 86, 051301(R) (2012)
arXiv:1308.3815, PRC 89, 045502 (2014)
M(0v) 0 / GXPF1A 0.733 0 +2nd ord./GXPF1A 1.301 (77%)
€
!ω
€
!ω
PRC 87, 064315 (2013)
Experimental info needed
MSU May 15, 2015 M. Horoi CMU
Σ B(
GT)
0
0.5
1
Energy (KeV)0 500 1,000 1,500 2,000 2,500
B(GT
)
0
0.05
0.1
0.15
Energy (KeV)0 500 1,000 1,500 2,000 2,500
ExperimentalTheoretical
MSU May 15, 2015 M. Horoi CMU
48Ca: M0v vs the Effective Interaction and SRC
€
M 0v
0
0.2
0.4
0.6
0.8
1
1.2
GXPF1 GXPF1A KB3 KB3G FPD6
No SRC M-S SRC CD-Bonn SRC AV18 SRC
M. Horoi, S. Stoica, arXiv:0911.3807, Phys. Rev. C 81, 024321 (2010)
€
Prediction : M 0v = 0.85 ± 0.15 T1/2 (0v )≥1026 y# → # # # # mββ ≤ 0.230± 0.045eV
Effects of Changing Matrix Elements of Hamiltonian: 82Se
MSU May 15, 2015 M. Horoi CMU
T=0
T=1
ΔME /ME = 5%
ΔM ≡ΔNMENME
×100
Ideal Coherent sums :
ΔM (0ν )∑ = 80%
ΔM (2ν )∑ =130%
Effects of Changing Matrix Elements of Hamiltonian: 82Se
MSU May 15, 2015 M. Horoi CMU
Ideal Coherent sums :
ΔM (0ν )∑ = 80%
ΔM (2ν )∑ =130%
Random ChangesΔME /ME < 5%
Take-Away Points
MSU May 15, 2015 M. Horoi CMU
Black box theorem (all flavors + oscillations)
Observation of 0νββ will signal New Physics Beyond the Standard Model.
0νββ observed ó
at some level
(i) Neutrinos are Majorana fermions.
(ii) Lepton number conservation is violated by 2 units
€
(iii) mββ = mkUek2
k=1
3
∑ = c122 c13
2m1 + c132 s12
2m2eiφ 2 + s13
2m3eiφ 3 > 0
Regardless of the dominant 0νββ mechanism!
MSU May 15, 2015 M. Horoi CMU
€
T1/ 2−1(0v) =G0ν (Qββ ) M
0v (0+)[ ] 2 < mββ >
me
%
& '
(
) *
2
€
φ2 = α2 −α1 φ3 = −α1 − 2δ
Take-Away Points The analysis and guidance of the experimental efforts need accurate Nuclear Matrix Elements.
€
mββ ≡ mv = c122 c13
2m1 + c132 s12
2m2eiφ 2 + s13
2m3eiφ 3
MSU May 15, 2015 M. Horoi CMU
€
Σ = m1 +m2 +m3 from cosmology€
mββ = c122 c13
2m1 + c132 s12
2m2eiφ 2 + s13
2m3eiφ 3
Take-Away Points Extracting information about Majorana CP-violation phases may require the mass hierarchy from LBNE(DUNE), cosmology, etc, but also accurate Nuclear Matrix Elements. €
φ2 = α2 −α1 φ3 = −α1 − 2δ
Recent Constraints from Cosmology
MSU May 15, 2015 M. Horoi CMU
2
parameter, Supernovae and Baryonic Acoustic Oscilla-tions (BAOs).
More recently, by using a new sample of quasar spec-tra from SDSS-III and Baryon Oscillation SpectroscopicSurvey searches and a novel theoretical framework whichincorporates neutrino non-linearities self consistently,Palanque-Delabrouille et al. [8] have obtained a new tightlimit on ⌃. This constraint was derived both in frequen-tist and bayesian statistics by combining the Planck 2013results [5] with the one-dimensional flux power spectrummeasurement of the Lyman-↵ forest of Ref. [7]. In partic-ular, from the frequentist interpretation (which is in ex-cellent agreement with the bayesian results), the authorscompute a probability for ⌃ that can be summarized ina very a good approximation by:
��2(⌃) =(⌃� 22meV)2
(62meV)2. (5)
Starting from the likelihood function L / exp�(��2/2)with��2 as derived from Fig. 7 of Ref. [8], one can obtainthe following limits:
⌃ < 84meV (1�C.L.)
⌃ < 146meV (2�C.L.)
⌃ < 208meV (3�C.L.)
(6)
which are very close to those predicted by the Gaussian��2 of Eq. 5.
It is worth noting that, even if this measurement iscompatible with zero at less than 1�, the best fit value isdi↵erent from zero, as expected from the oscillation dataand as evidenced by Eq. 5.
Furthermore, the (atmospheric) mass splitting � ⌘p�m2 ' 49meV [2] becomes the dominant term of Eqs.
3 and 4 in the limit m ! 0. Under this assumption,in the case of NH (IH) ⌃ reduces approximately to �(2�). This explains why this result favors, for the firsttime, the NH mass spectrum, as pointed out in Ref. [8]and as advocated in older theoretical works [9].
It is the first time that some data indicate a prefer-ence for one specific mass hierarchy. Nonetheless, theseresults on ⌃ have to be taken with due caution. In fact,claims for a non-zero value for the cosmological mass(from a few eV to hundreds of meV) are already presentin the literature (see e. g. Refs. [10, 11]). In particular,it has been recently suggested that a total non-zero neu-trino mass, around 0.3 eV, could alleviate some tensionspresent between cluster number counts (selected both inX-ray and by Sunyaev-Zeldovich e↵ect) and weak lensingdata [12, 13]. In some cases, a sterile neutrino particlewith mass in a similar range is also advocated [14, 15].However, these possible solutions are not supported byCMB data or BAOs for either the active or sterile sectors.In fact, a combination of those data sets strongly disfa-vors total masses above (0.2-0.3) eV [4]. More precisemeasurements from cosmological surveys are expected in
the near future (among the others, DESI1 and the Euclidsatellite2) and they will probably allow more accuratestatements on neutrino masses.
III. CONTRIBUTION OF THE THREE LIGHTNEUTRINOS TO NEUTRINOLESS DOUBLE
BETA DECAY
The close connection between the neutrino mass mea-surements obtained in the laboratory and those probedby cosmological observations was outlined long ago [16].In the case of 0⌫��, a bound on ⌃ allows the derivationof a bound on m
��
. This can be done by computing mas a function of ⌃ and by solving the quartic equationthus obtained.It appears therefore useful to adopt the representation
originally introduced in Ref. [17], where m��
is expressedas a function of ⌃.The resulting plot, according to the values of the os-
cillation parameters of Ref. [2], is shown in the left panelof Fig. 1. The extreme values for m
��
after variationof the Majorana phases can be easily calculated, see e. g.Refs. [3, 18]. This variation, together with the uncertain-ties on the oscillation parameters, results in a wideningof the allowed regions. It is also worth noting that theerror on ⌃ contributes to the total uncertainty. Its e↵ectis a broadening of the light shaded area on the left sideof the minimum allowed value ⌃(m = 0) for each hierar-chy. In order to compute this uncertainty, we consideredGaussian errors on the oscillation parameters, namely
�⌃ =
s✓@ ⌃
@ �m2�(�m2)
◆2
+
✓@ ⌃
@�m2�(�m2)
◆2
. (7)
The following inequality allows the inclusion of the newcosmological constraints on ⌃ from Ref. [8]:
(y �m��
(⌃))2
(n�[m��
(⌃)])2+
(⌃� ⌃(0))2
(⌃n
� ⌃(0))2< 1 (8)
where m��
(⌃) is the Majorana E↵ective Mass as a func-tion of ⌃ and �[m
��
(⌃)] is the 1� associated error, com-puted as discussed in Ref. [3]. ⌃
n
is the limit on ⌃ derivedfrom Eq. 5 for the C. L. n = 1, 2, 3, . . . By solving the in-equality for y, it is thus possible to get the allowed con-tour for m
��
considering both the constraints from oscil-lations and from cosmology. In particular, the Majoranaphases are taken into account by computing y along thetwo extremes ofm
��
(⌃), namelymmax
��
(⌃) andmmin
��
(⌃),and then connecting the two contours. The resulting plotis shown in the right panel of Fig. 1.The most evident feature of Fig. 1 is the clear di↵er-
ence in terms of expectations for both m��
and ⌃ in
1http://desi.lbl.gov/cdr
2http://www.euclid-ec.org
Σ =m1 +m2 +m3
arXiV:1505.02722
MSU May 15, 2015 M. Horoi CMU
Take-Away Points Alternative mechanisms to 0νββ need to be carefully tested: many isotopes, energy and angular correlations.
These analyses also require accurate Nuclear Matrix Elements.
€
T1/ 20ν[ ]−1
= G0ν M jη jj∑
2
= G0ν M (0ν )ηνL + M (0 N ) ηNL +ηNR( ) + ˜ X λ < λ > + ˜ X η <η > +M (0 ' λ )η ' λ + M (0 ˜ q )η ˜ q +2
€
ην , ηNR ⇐GGe0νT1/ 2Ge
0ν[ ]−1
= MGe(0ν ) 2ην
2+ MGe
(0N ) 2ηNR2
GXe0νT1/ 2Xe
0ν[ ]−1
= MXe(0ν ) 2ην
2+ MXe
(0N ) 2ηNR2
&
' (
) (
SuperNEMO; 82Se
MSU May 15, 2015 M. Horoi CMU
2 3 4 5 6 7 8 9 10Closure energy <E> [MeV]
3
3.2
3.4
3.6
3.8
4pure closure, CD-Bonn SRCmixed, CD-Bonn SRCpure closure, AV18 SRCmixed, AV18 SRC
76Ge
J=0 J=1 J=2 J=3 J=4 J=5 J=6 J=7 J=8 J=9Spin of the intermediate states
-0.2
0
0.2
0.4
0.6
0.8
1
GT, positiveGT, negativeFM, positiveFM, negative
I=0 I=1 I=2 I=3 I=4 I=5 I=6 I=7 I=8 I=9Spin of the neutron-neutron (proton-proton) pairs-3
-2
-1
0
1
2
3
4
5
6
7
GT, positiveGT, negativeFM, positiveFM, negative
€
Mmixed (N) = Mno−closure (N) + Mclosure (N = ∞) −Mclosure (N)[ ]
Take-Away Points Accurate shell model NME for different decay mechanisms were recently calculated.
The method provides optimal closure energies for the mass mechanism.
Decomposition of the matrix elements can be used for selective quenching of classes of states, and for testing nuclear structure.
Collaborators:
• Alex Brown, NSCL@MSU • Roman Senkov, CMU and CUNY • Andrei Neacsu, CMU • Jonathan Engel, UNC • Jason Holt, TRIUMF
MSU May 15, 2015 M. Horoi CMU