Conversion of experimental half-life to effective ... - APS Link Manager

PHYSICAL REVIEW C 81, 028502 (2010)

Conversion of experimental half-life to effective electron neutrino mass in 0νββ decay Anatoly Smolnikov* Joint Institute for Nuclear Research, Dubna, Russia, and Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia

Peter Grabmayr† Kepler Center for Astro and Particle Physics, Eberhard Karls Universität Tübingen, Germany (Received 18 December 2009; published 23 February 2010) The Germanium Detector Array (GERDA) collaboration will be searching for neutrinoless double β decay of Ge. As a result it will measure the half-life T1/2 of this rare process; or at least a new value for the lower limit for T1/2 will be derived. The sensitivity of the GERDA experiment on the effective electron neutrino mass mββ depends on the theoretical value for the nuclear matrix element M and the kinematical phase space factor G. In this Brief Report we focus on existing difficulties in applying the dimensionless values of M calculated by various theoretical groups, which use different methods and parametrizations. The implicit radius dependencies in M and G are discussed. Resulting values of the neutrino mass are tabulated for various representative half-lives T1/2 representing the sensitivity of the various phases of the GERDA experiment. 76

DOI: 10.1103/PhysRevC.81.028502

PACS number(s): 23.40.Hc, 23.40.Bw, 21.10.Tg, 27.50.+e

Introduction. The Germanium Detector Array (GERDA) collaboration [1] is going to start a new measurement searching for the neutrinoless double-β (0νββ) decay of the nucleus 76 Ge. After establishing the nature of the neutrino as a Majorana particle by unambiguously identifying a line at the reaction Q value Qββ = 2039 keV, the subsequent quest will be the determination of the effective electron neutrino mass mββ 2 |Uek |2 eiαk mk , (1) mββ = Uek mk = and thus possibly ascertain the neutrino hierarchy. The effective neutrino mass depends on the mixing matrix Uj k and the the masses mk of the three mass eigenstates. Only two of the Majorana phases αk are independent, the third one is chosen arbitrarily. Due to numerous improvements compared to previous attempts [2,3], we expect to deduce as the experimental result of the GERDA experiment a value for the half-life T1/2 of the 0νββ decay from the number of counts in the peak at Qββ . Otherwise a new, much higher value for the lower limit on T1/2 will be extracted [4]. In 0νββ decay, the formulation [5–7] to convert the experimentally measured half-life T1/2 to an effective neutrino mass mββ reads {Ref. [5], Eq. (3.5.10)} 2 1 0ν 0ν 2 mββ = G |M | = F 0ν |M0ν |2 m2ββ . (2) T1/2 me The nuclear matrix element M0ν describes the overlap of the nuclear wave functions of the initial and final states. Thus, the two versions of the phase space factors F 0ν and G0ν are related

* Presently at: Max-Planck-Institut für Kernphysik, Heidelberg, Germany. † [email protected]

0556-2813/2010/81(2)/028502(4)

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as F 0ν =

G0ν . m2e

(3)

Note, the Tübingen group reversed the notation of F 0ν and G0ν . In this note we focus on the presently most relevant transition, namely the 0+ → 0+ ground-state transition with the mν part, where the mass of the propagating neutrino contributes and the two electrons are in S waves [5]. We realize different notations since the first formulation of double-β decay by Goeppert-Mayer [8] and its neutrinoless version by Furry [9]. However, we keep the convention of symbols as in Ref. [5], which describes the theory of the mν part of the interaction amplitude. Other contributions to the amplitude mediated by Majoron exchange or the (V + A) part and transitions to excited final states have different phase space factors (see, e.g., Ref. [10]). One of the earlier formulations of 0νββ decay and thus many times cited is the phase space factor by Doi et al. {see Ref. [5], Eqs. (3.5.17– 3.5.21), written as G01 } a0ν 0ν (4) G = G01 = d0ν b(ε1 , ε2 ), (me RA )2 ln2 with the constant term a0ν =

(GgA )4 m9e , 64π 5

(5)

the phase space element d0ν =

p1 ε1 p2 ε2 δ(ε1 +ε2 + Mf − Mi ) dε1 dε2 d cos θ, m5e (6)

and the kinematical factor b(ε1 , ε2 ) = F0 (Z, ε1 )F0 (Z, ε2 ),

(7)

©2010 The American Physical Society

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which is the product of the relativistic Fermi factors F0 (Z, εi ). The variables me , Mf , and Mi are the masses of the electron, the final and initial nucleus, and gA = GA /GV is the ratio of the vector and axial-vector coupling constants. pi are the momenta of the electrons and θ their mutual angle (pˆ 1 pˆ 2 = cos θ ). The factors F0 (Z, εi ) depend on the energy ε1 or ε2 of the respective electron and the charge Z of the daughter nucleus, which attracts the two emitted electrons. Doi et al. [5] argued that the correction of taking properly into account the effect of the uniform Coulomb field of the intermediate nucleus due to (Z − 1) is in the order of α 2 Z and thus negligible ( RA ), matching them through the continuity condition at r = RA . Note, that Doi et al. suggested a parametrization of nuclear radii as RA = 3.108 × 10−3 A1/3 /me {Ref. [5], Eq. (D.7)} with mass number A. This value corresponds to the usual parametrization of charge distributions RA = r0 A1/3 with r0 = 1.2 fm. The nuclear radius is introduced in Eq. (4) to normalize the Coulomb-type potential due to neutrino propagation as (RA /r) {see Ref. [5], Eq. (3.5.21ff)}. From Eqs. (4) through (7) it becomes quite clear that the phase space factor exhibits a proportionality

TABLE I. Kinematical phase space factor G0ν and the nuclear matrix elements M0ν calculated by different authors for 76 Ge. The product s is given in the last column multiplied by 1025 for convenience. Author

Ref.

r0 G0ν × 1015 M0ν s × 1025 Comments [eV2 × y]−1 [fm] [y −1 ]

Claim Pantis Simkovic Simkovic Rodin Simkovic Simkovic Simkovic Simkovic Caurier Barea Barea Suhonen Suhonen Suhonen Suhonen Menendez Menendez Simkovic Simkovic Simkovic

[3] [11] [12] [12] [13] [14] [14] [14] [14] [15] [16] [16] [18] [18] [18] [18] [20] [20] [21] [21] [21]

1.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.1 1.1 1.1

6.31 7.93 7.93 7.93 7.93 7.93 7.93 7.93 7.93 6.31 6.31 6.31 6.31 6.31 6.31 6.31 6.31 6.31 7.93 7.93 7.93

4.22 1.34 2.80 3.60 3.92 3.33 4.68 3.92 5.73 2.22 5.47 4.64 2.78 2.28 4.11 3.23 3.00 3.52 5.44 4.07 6.64

4.30 0.55 2.38 3.94 4.67 3.37 6.65 4.67 9.97 1.19 7.23 5.20 1.87 1.26 4.08 2.52 2.17 2.99 8.99 5.03 13.39

np pairing RQRPA RQRPA RQRPA Jastrowa Jastrowb UCOMa UCOMb SM IBM2-I IBM2-II Jastrowc Jastrowd UCOMc UCOMd SM gcne SM rge RQRPAe RQRPAae RQRPAbe

a

0ν

G

∼

gA4

1 , RA2

(8)

and for consistency the same parameter used here must also be used for the nuclear matrix elements. Difficulties in comparing results from different groups arise as theoreticians like to give their M0ν without dimensions as needed in Eq. (2). For this purpose the usually calculated matrix elements are multiplied by the nuclear radius parameterized as RA = r0 A1/3 . Unfortunately, different groups use different parameters r0 , mostly for historical reasons. Note, that the two commonly used parameters r0 are 1.1 and 1.2 fm, respectively, describing globally the mass and the charge distributions. Thus, one has to use always the consistent set of M0ν and G0ν when converting T1/2 into mββ , as was already noted by Suhonen [7]. Additionally, some authors normalized the M obtained in quenched calculations to results with the standard gA = 1.254 by the relation M0ν = (gA /1.25)2 M0ν ,

(9)

(note the on the left M) in accordance with Eqs. (4) and (8) (even when the value gA = 1.25 is printed we assume that the standard value gA = 1.254 was used). In general, the matrix elements obtained in quenched calculations differ from the results with the standard gA by much larger factors than the 9% difference due to the factor (gA /1.25)2 . When comparing M0ν of different calculations, the results should be scaled according to the respective RA (or r0 ) or the value for gA , respectively. In this Brief Report, therefore, we compile a selected number of results from more recent

Lower limit. Upper limit. c gA = 1.25. d gA = 1.20. e Coupled-cluster method using the N N potential CD Bonn. b

calculations for the neutrinoless double-β decay in 76 Ge – the nucleus where we expect results soon. The values of the radius parameters and the respective kinematical phase space factors given in the original articles are discussed first. In Table I the values for G0ν , M 0ν , and for ease of comparison, the new quantity s are given. Table II uses some of these results for estimates of the effective electron neutrino mass mββ . Concluding, we summarize the relevant upper limits for mββ in Table III. TABLE II. Compilation of effective electron neutrino mass mββ in units of [meV] for different T1/2 related to experimental milestones given by the claim in Ref. [3] and GERDA [1]. Author

Ref.

s × 1025 [eV2 × y]−1

T1/2 × 10−25 [y] 1.2 Claim

Menendez Suhonen Rodin Barea Simkovic

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[20] [18] [13] [16] [21]

2.99 4.08 4.67 7.23 8.99

528 452 422 340 304

2.2

3

Phase I 390 334 312 251 225

334 286 267 215 193

15

20

Phase II 149 128 119 96 86

129 111 103 83 75

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TABLE III. Range of upper limits on the effective electron neutrino mass mββ obtained from the expected sensitivity of the GERDA experiment in Phases I and II compared to the claim. Experimental sensitivity Claim GERDA GERDA

Phase I Phase II

Ref.

T1/2 [10−25 y]

mββ [eV]

[3] [1] [1]

1.2 2.2 15.0

0.30–0.53 0.23–0.39 0.09–0.15

Compilation of G0ν and M0ν . The basic formulation of the kinematical phase space factor goes back to that by Doi et al. [5]. As different authors use different values for the radius parameter and to clarify the situation, we feel it is necessary to compile the values used by the authors. Not always both values are explicitly given in the article itself, sometimes there is a reference to an original article. In some instances we had to use the published results on the half-life to recalculate (or clarify) the respective G0ν . Suhonen and his collaborators [7] used the approximation of relativistic Fermi function, but Doi and Kotani [5] as well as Boehm and Vogel [6] used the solution of the Dirac equation, which should be more accurate. All start with r0 = 1.2 fm and the resulting values scatter around G0ν = (6.3 − 6.4)10−15 /y for 76 Ge; nowadays the value G0ν = 6.31 × 10−15 /y is used. As the Tübingen group settled for r0 = 1.1 fm in their calculations of M0ν , the respective value amounts to G0ν = 7.93 × 10−15 /y. In older calculations they employed the value for G0ν calculated by Pantis et al. [11], a recalculation by Simkovic et al. [12] arrives at the same numerical value. The calculated M0ν have the dimension [fm−1 ]; but multiplied by the nuclear radius RA they become dimensionless. The ratio fr = (1.2/1.1)2 amounts to 1.19. The ratio of phase space factors fG = (7.93/6.31) = 1.26. The discrepancy of ∼6% is, apart from rounding errors in quoting r0 and other parameters, due to in the applied models and formalism for the lepton wave functions. For the entries of Table I we selected some results from older calculations (Refs. [3,11–15]) and the single work on the IBM model [16]. However, after the experimental determination of the shell-model occupancies for 76 Ge and 76 Se by Schiffer et al. [17] many new calculations [18–21] were started that take this experimental constraint into account. Effectively the spread of the theoretical values for M0ν was reduced if comparable potentials and correlations are chosen. The radius parameters r0 and the kinematical factors G0ν were checked against the respective published value for T1/2 . They are quoted explicitly in Table I to smoothen some possibly confusing statements in the original articles. T1/2 was not calculated in Ref. [16], thus we inserted G0ν compatible with the parameter r0 . The product of the kinematical factor G0ν and the absolute square of the nuclear matrix element M 0ν is divided by the electron mass squared, given in units of [eV2 ] s = G0ν |M0ν |2 m−2 e

[eV2 y]−1 .

(10)

The intermediate quantity s is introduced for ease of comparison. Its inverse square root gives directly the value of

the neutrino mass for a half-life of 1025 years. Actually, this factor s can serve as the unique number for the comparison of different calculations. Finally, it is used to convert the half-life T1/2 into values for the effective neutrino mass mββ in units of [eV] mββ = (sT1/2 )−1/2 .

(11)

For various predictions of M0ν the resulting values for the neutrino mass are shown in Table II; actually the values for mββ are given in units of [meV] in this table. The halflives were chosen with respect to the germanium experiments [1,3]. T1/2 = 1.2 × 1025 y (90% C.L. confidence level) resulted from an analysis of 71.7 kg × y of the Heidelberg-Moscow (HdM) experiment [3], the other four values come from the GERDA proposal [1]. The experimental program foresees two phases. The first phase will use the eight detectors available from the previous experiments, HdM [3] and IGEX [2]. A total of 17.9 kg of enriched material is available for Phase I; the collaboration aims for an exposure of 15 kg × y. A background level of b ∼ 0.01 cts/(keV × kg × y) is expected, which sets the limit at T1/2 = 2.2 × 1025 y. If no background is present, a stronger limit on T1/2 = 3 × 1025 y can be reached. For Phase II about 37.5 kg of enriched material was bought from which about 20 kg sensitive mass (i.e., detectors) will be produced. Additionally, the background level will be lowered further by an order of magnitude. The two factors will increase the limit on the half-life as given in the last two columns of Table II. Correspondingly, the estimated upper limits on the effective neutrino mass will be reduced. Generally, the results for gA = 1.25 were chosen. We selected one result for each group among the most recent calculations accounting for the experimental occupancies, however, not excluding Refs. [13,16]. Nucleon-nucleon correlations of the Jastrow-type are rather general, however, the unitary correlation operator method (UCOM) correlations are more sophisticated and thus more relevant for the present discussion. More recently, the two-nucleon wave functions were calculated by the coupled-cluster method (CCM) employing consistently the realistic nucleon-nucleon potentials (as, e.g., the CD-Bonn). The entries are sorted for ascending values of s. Concluding from Table II one can obtain, for example with the expected statistics in Phase I, a lower limit on T1/2 = 2.2 × 1025 y, which can be translated based on the most recent calculations for M0ν of 76 Ge, into upper limits on the neutrino mass in the range of 225 < mββ < 390 meV. Summarizing the estimates of Table II for the various experimental sensitivities of GERDA, we find effective neutrino masses mββ for the range of M0ν from 5.94 to 3.52 (with r0 = 1.2 fm) as presented in Table III. This time the effective electron neutrino mass mββ is tabulated in units of [eV]. The details of the background contributions are discussed in Refs. [1,4]. Conclusion. Different ranges of the upper limits on mββ were presented in many articles or conference contributions even when the same limits on the half-life T1/2 for 76 Ge were given. This can be connected with a certain freedom in selection of M 0ν values as well as with difficulties in

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using results of theoretical calculations from different groups as it was pointed out in this Brief Report. Thus, for future presentations (at least until new values of M are published) we can recommend the estimates of the GERDA sensitivity in terms of mββ as summarized in Table III for the range of M 0ν values from 5.94 to 3.52 (with r0 = 1.2 fm) according to the set of the most recent calculations (published in 2009, see Table II). We hope that the suggested unification by introducing the intermediate parameter s will help practical comparisons and will avoid confusion. It is clear that the problem of the correct use of the kinematical factor G0ν and the nuclear matrix elements M 0ν discussed here for the 76 Ge case pertains also for the other

[1]

[2] [3]

[4] [5] [6] [7] [8] [9] [10] [11]

GERDA,

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isotopes considered as candidates in the search for 0νββ decay. Starting from the parameters discussed here (i.e., r0 and gA ) the proper phase space factor can be found also for the desired isotope. The wave functions of the electrons were calculated in a homogeneous charge distribution. It might be desirable to investigate the influence of realistic charge distributions on the relativistic Fermi factors and the resulting kinematical factors G0ν . The authors appreciated helpful discussions with J. Suhonen, V. Rodin, and F. Simkovic. This work was supported in part by the German BMBF (05A08VT1) and the Russian RFBR (07/02/01050).

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