Download pdf - Nuclear structure for tests of fundamental symmetriesnuclearphysicsworkshops.github.io/.../talks/horoi.pdf · Nuclear structure for tests of fundamental symmetries Mihai Horoi Department

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


5/12/15, 7:47 AMScientific Opportunities | frib.msu.edu

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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


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


-  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


€

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


?

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?


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


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


€

-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


arXiv:1005.1241

2β0ν rhc(η)

€

< λ >

t = εe1 −εe2

Two Non-Interfering Mechanisms


€

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?


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?


€

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


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


€

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! " # # # # # # # # # # # # # $ # # # # # # # # # # # # #


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)


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


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)


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).


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).


€

CD − Bonn SRC→

€

AV18 SRC→

The effect of larger model spaces for 48Ca


€

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


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


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


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


Ideal Coherent sums :

ΔM (0ν )∑ = 80%

ΔM (2ν )∑ =130%

Random ChangesΔME /ME < 5%

Take-Away Points


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!


€

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


€

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


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


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


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


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


€

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
