LANL April 15, 2009

Double beta decay theory, in particular an “anatomy” of the nuclear matrix elements

Petr Vogel Caltech

Outline:1) Introduction - 0 as a test of the total lepton number conservation.2) Mechanism of the decay and how can one tell when the 0is actually observed.3) Calculating the nuclear matrix elements and why is it difficult.4) Similarities and differences in the quasiparticle random phase (QRPA) and nuclear shell model (NSM) approaches.

Experimental discovery of neutrino oscillations opened a `window to physics beyond the Standard Model’.

In the next stage of experiments effort will be made a) to refine our knowledge of oscillation

phenomenology (`precision neutrino physics ’ ), and b) to search for qualitatively new phenomena that should help making sense out of the things that we have seen in that window.

Study of the neutrinoless double beta decay, i.e.,

tests of the total lepton number conservation,is a primary example of this effort.

How can we tell whether the total lepton number is conserved?A partial list of processes where the lepton number would be violated:

Neutrinoless decay: (Z,A) -> (Z2,A) + 2e(), T1/2 > ~1025 yMuon conversion: - + (Z,A) -> e+ + (Z-2,A), BR < 10-12

Anomalous kaon decays: K+ -> -BRFlux of e from the Sun: BR < 10-4

Flux of e from a nuclear reactor: BR < ?Production at LHC of pair of same charge leptons with no missing energy: BR <?

Observing any of these processes would mean that the leptonnumber is not conserved, and that neutrinos are massive Majorana particles.

It turns out that the study of the 0decay is by far the mostsensitive test of the total lepton number conservation, so werestrict further discussion to this process.

0e– e–

u d d u

()R L

W W

Whatever processes cause 0, its observation would imply the existence of a Majorana mass

term: Schechter and Valle,82

By adding only Standard model interactions we obtain

Hence observing the decay guaranties that are massive Majorana particles.

()R ()L Majorana mass term

What is the nature of the `black box’? In other words, what is the mechanism of the decay? All these diagrams can contribute to the decay amplitude

Light Majorana neutrino,only Standard Model

weak interactions

Heavy Majorana neutrinointeracting with WR.

Model extended to includeright-handed current

interactions.

Light or heavy Majorananeutrino. Model extended

to include right-handed WR.Mixing extended betweenthe left and right-handed

neutrinos.

Supersymmetry with R-parity violation. Many new particlesinvoked. LightMajorana neutrinos exist also.

d u

e-

e-

WL

WL

ud

d u

WR

WR

heavy

ud

e-

e-

d u

WR

WL

ud

d u

e (selectron)

(neutralino)

ud

e (selectron)

e-

e-

e-

e-

The relative size of heavy (AH) vs. light particle (AL) exchange

to the 0 decay amplitude is (a crude estimate):

AL ~ GF2 m/<k2>, AH ~ GF

2 MW4/5 ,

where is the heavy scale and k ~ 100 MeV is the virtualneutrino momentum.For ~ 1 TeV and m ~ 0.1 – 0.5 eV AL/AH ~ 1, hence bothmechanism would contribute equally. Note that if >> 1 TeV, the heavy particle exchange wouldresult in unobservably small rate.

From the observation of the decay it is, in general, impossible to decide which of the possible graphs is relevant.

A diagnostic tool in deciding which mechanism dominates might be in linking lepton number violation (LNV) with lepton flavor violation (LFV) .

Linking LNV to LFV Summary:

- SM extensions with low ( TeV) scale LNV **

- SM extensions with high (GUT) scale LNV [m2]

** In absence of fine-tuning or hierarchies in flavor couplings. Important caveat! See: V. Cirigliano et al., PRL93,231802(2004)

Left-right symmetric model,R-parity violating SUSY, etc.possibly unrelated tom

2

R ~ O(

R = Be/Be» 10-2

Be = (e)/(ee) (Z,A) e- + (Z,A))

(Z,A) + (Z,A))Be =

As long as the mass eigenstates i, which are components of the flavor

neutrinos e, and , are Majorana neutrinos, the decay will occur for sure, with the rate

1/T1/2= G(Etot,Z) (M|<m>|2,

where G(Etot,Z) is easily calculable phase space factor, M is the nuclear

matrix element, calculable with difficulties (and discussed later), and

<m> = i |Uei|2 exp(ii) mi,

where i are unknown Majorana phases (only two of them are relevant).

The <m>, the effective neutrino Majorana mass, is the quatity

that we would like to extract from experiments.

We can relate |<m>| to other observables related to the

absolute neutrino mass.

Usual representation of that relation. It shows that the <m>axis can be divided into three distinct regions. However, it createsthe impression (false) that determining <m> would decide betweenthe two competing hierarchies.

inverted

normal

degenerate

from decayfrom observationalcosmology,M = m1+m2+m3

blue shading:normal hierarchy,m2

31 > 0.red shading:inverted hierarchym2

31 < 0

shading:best fitparameters, lines95% CL errors.

minimum mass,not observable

<m> vs. theabsolute mass scale

Thanks to A. Piepke

In double beta decay two neutrons bound in the ground state of an initial even-even nucleus are simultaneously transformed into two protons that again are bound in the ground state of the final nucleus.

The nuclear structure problem is therefore to evaluate, with a sufficient accuracy, the ground state wave functions of both nuclei, and evaluate the matrix element of the 0-decay operator connecting them.

This cannot be done exactly; some approximation and/or truncation is always necessary. Moreover, there is no other analogous observable that can be used to judge the quality of the result.

Nuclear Matrix Elements:

Can one use the 2-decay matris elements for that?What are the similarities and differences?

Both 2and 0operators connect the same states.Both change two neutrons into two protons.

However, in 2the momentum transfer q < few MeV;thus eiqr ~ 1, long wavelength approximation is valid, only the GT operator need to be considered.

In 0q ~ 100-200 MeV, eiqr = 1 + many terms, thereis no natural cutoff in that expansion.

Explaining 2-decay rate is necessary but not sufficient

Basic procedures:

1) Define the valence space

2) Derive the effective hamiltonian Heff using the

nucleon-nucleon interaction

plus some empirical nuclear

data.3) Solve the equations of motion to obtain the

ground state wave functions

Two complementary procedures are commonly used:a) Nuclear shell model (NSM)b) Quasiparticle random phase approximation (QRPA)

In NSM a limited valence space is used but all configurations of valence nucleons are included.Describes well properties of low-lying nuclear states.Technically difficult, thus only few 0 calculations.

In QRPA a large valence space is used, but only a classof configurations is included. Describes collectivestates, but not details of dominantly few-particle states.Rather simple, thus many 0 calculations.

QRPA proceeds in two steps. 1) First pairing between like nucleons is included in a simple fashion:

particles quasiparticles

Bogoliubov transformation,proton and neutron Fermilevels are smeared.However, particle numbersare conserved only inaverage.

2) Then the proton-neutron interaction is included

two quasiparticlecreation operator

two quasiparticleannihilation operator

correlated groundstate, includes zero-point motion

Evaluation of M0 involves transformation to the relative coordinatesof the nucleons (the operators OK depend on rij)

unsymmetrized two-bodyradial integral involves`neutrino potentials’

From QRPA forfinal nucleus

From QRPA forinitial nucleus

overlap

Note the two separate multipole decompositions. J refers to the virtual state in odd-odd nucleus, while J refers to the angular momentum of the neutron pair transformed into proton pair.

82Se

130Te

Why it is difficult to calculatethe matrix elements accurately?

Contributions of differentangular momenta J of theneutron pair that is transformed in the decay into the proton pair with the same J.

Note the opposite signs, and thus tendency to cancel, between the J = 0 (pairing) and the J 0(ground state correlations) parts.

The same restricted s.p. space is used for QRPA and NSM. There is a reasonable agreement between the two methods

Sorry, this should be the script J

The opposite signs, and similar magnitudes of the J = 0 and J 0 parts is universal in QRPA. Here for three nuclei with coupling constant gpp adjusted so that the rate is correctly reproduced. Now two oscillator shells are included.

Dependence on the relative distance, nucleon structure, short range repulsion, higher order currents, etc.

The neutrino propagator connecting the two participating nucleonsintroduces dependence on the relative distance r ( or equivalentlymomentum transfer q ) between them.

If small values of r (or large values of q) are important, we have to worry about induced weak currents, nucleon finite size, andthe short range nucleon-nucleon repulsion.

neutrino propagator

Graphs representing theelementary amplitude.The neutrino propagatorcauses dependence of thecorresponding transitionoperator on the momentumtransfer q and, in theFourier transform on thedistance r between theparticipating nucleons.

The ``neutrino potential” isH(r) = R/r (r), where (r) is rather slowly varying function. Thus,naively, one expects thatthe typical distance is r ~ R.

(n)

(n)

(n)

(n)

(p)

(p)

(p)

(p)

(Z,A)

(Z,A) (Z+2,A)

(Z+2,A)

Full matrix element

The radial dependence of M for the three indicatednuclei. The contributionssummed over all componentsss shown in the upper panel.The `pairing’ J = 0 and`broken pairs’ J 0 partsare shown separately below.Note that these two partsessentially cancel each otherfor r > 2-3 fm. This is ageneric behavior. Hencethe treatment of small values of r and large valuesof q are quite important.

C(r)

CJ(r)

M = C(r)dr

pairing part

broken pairs part

total

The radial dependence of M for the indicated nuclei, evaluated in the nuclear shell model. (Menendes et al, arXiv:0801.3760).Note the similarity to the QRPA evaluation of the same function.

The finding that the relative distances r < 2- 3 fm, and correspondinglythat the momentum transfer q > ~100 MeV means that one needs toconsider a number of effects that typically play a minor role in thestructure of nuclear ground states:a) Short range repulsionb) Nucleon finite sizec) Induced weak currents (Pseudoscalar and weak magnetism)

Each of these, with the present treatment, causes correction(or uncertainty) of ~20% in the 0 matrix element.

There is a consensus now that these effects must be included butno consensus how to treat them, in particular a).

(5.3)(4.0)

(4.1)

(5.0)

Dependence on the distance between the two transformed nucleons and the effect of different treatments of short range correlations. This causes changes of M by ~ 20%.

Graph by F. Simkovic

C(r)

Contributions of different parts of the nucleon current.Note that the AP (axial-pseudoscalar interference) contains q2/(q2 + m

2), and MM contains q2/4Mp2.

76Ge76Se

Full estimated range of M within QRPA framework and comparison with NSM (higher order currents now included in NSM)

The 2 matrix elements, unlike the 0 ones, exhibit pronounced shelleffects. They vary fast as a function of Z or A.

0 nuclear matrix elements calculated very recently with the Interacting Boson Model-2, see Barea and Iachello, Phys. Rev. C79, 044301(2009).

Why are the QRPA and NSM matrix elements different?

Various possible explanations:a) Assumed occupancies of individual valence orbits might be differentb) In QRPA more single particle states are includedc) In NSM all configurations (seniorities) are includedd) In NSM the deformation effects are includede) All of the above

p

f5/2

g9/2

Pf5/2

g9/2

P 0.5f5/2 0.8g9/2

0.7

Neutron orbit occupancies, original Woods-Saxon vs.adjusted effective mean field. For 76Ge -> 76Se

experiment

Assumed occupancies of individual valence orbits might be different

Experiment from J.P.Schiffer et al, Phys.Rev.Lett. 100, 1120501(2008),used (d,p),(p,d),(3He,),(,3He) to derive occupancies of neutron orbits

P 1.8 0.15 f5/2 2.0 0.25g9/2 0.2 0.25P 2.1 0.15f5/2 3.2 0.25

g9/2 0.8 0.25

P 0.3f5/2 1.2g9/2 0.6

Proton orbit occupancies, original Woods-Saxon vs.adjusted effective mean field.

experiment

Experiment from B.P.Kay et al, Phys.Rev.C79,021301(2009), based on (d,3He)

Full estimated range of M within QRPA framework and comparison with NSM (higher order currents now included in NSM)

New QRPA value with adjusted mean field so that experimentaloccupancies are reproduced

New NSM value with adjusted mean field (monopole) where experimentaloccupancies are better reproduced

In QRPA more single particle states are included

Contribution of initial neutron orbit pairs against the final proton pairs.The nonvalence orbits are labeled as r. Adding all parts with r-type orbits gives +2.83 - 3.22 = -0.39 which is only ~12% of the total matrix element 3.27In the figure all entries are, however, normalized so that their sum is unity..

(from the PhD thesis of J. Menendez)

In NSM the 0 matrix elements converge slowly as higher and higherseniorities (more complicated configurations) are included. In QRPAthe sm = 4 should be well described, but higher sm might not be.

From the PhD thesis of J. Menendez

NSM evaluation of the (hypothetical) mirror decay 66Ge -> 66Se.The deformation of 66Se is artificially changed, while 66Ge is notchanged. The matrix element is reduced significantly if the deformations of the initial and final nuclei are different.

0 matrix element

It appears, therefore, that all of these effects, possible differencesin the assumed occupancies of valence orbits, additional single particle states included in QRPA but not in NSM, inclusion ofcomplicated configurations (higher seniority and/or deformation)in NSM but only crudely in QRPA, can and probably do affectthe resulting nuclear matrix elements, and might explain thedifferent outcomes of the two methods. In particular, the difference in deformation of the initial andfinal nuclei makes the evaluation of the matrix element for150Nd -> 150Sm very difficult.

Summary

1) There is, as of now, agreement of all practitioners on what needs to be included in the evaluation of the 0 nuclear matrix elements, even though there is no complete agreement how to do it (e.g. for

the short range correlations).2) The NSM and QRPA have both many basic features in common, in particular the (sometimes severe) cancellation between the

effect of pairing and `broken pairs’ configurations and in the radial

distance dependence.3) There are still noticeable differences between the two methods,

and several possible causes have been identified.4) Both methods predict that the 0 nuclear matrix elements

should vary slowly and rather smoothly with A and Z, unlike the 2

matrix elements. That makes the comparison of experiments with different

sources easier.