Can one measure nuclear matrix elements of neutrinoless double beta decay?
aa r X i v : . [ nu c l - t h ] O c t Can one measure nuclear matrix elements of neutrinoless double beta decay?
Vadim Rodin ∗ and Amand Faessler Institut f¨ur Theoretische Physik der Universit¨at T¨ubingen, D-72076 T¨ubingen, Germany (Dated: October 25, 2018)By making use of the isospin conservation by strong interaction, the Fermi 0 νββ nuclear matrixelement M νF is transformed to acquire the form of an energy-weighted double Fermi transitionmatrix element. This useful representation allows reconstruction of the total M νF provided a smallisospin-breaking Fermi matrix element between the isobaric analog state in the intermediate nucleusand the ground state of the daughter nucleus could be measured, e.g., by charge-exchange reactions.Such a measurement could set a scale for the 0 νββ nuclear matrix elements and help to discriminatebetween the different nuclear structure models in which calculated M νF may differ by as much as afactor of 5 (that translates to about 20% difference in the total M ν ). PACS numbers: 23.40.-s, 23.40.Bw 23.40.Hc, 21.60.-n,Keywords: Neutrino mass; Double beta decay; Nuclear matrix element
Neutrino is the only known spin-1/2 fermion whichmay be truly neutral, i.e., identical with its own antiparti-cle. In such a case one speaks about Majorana neutrino,to be contrasted with Dirac neutrino which is differentfrom its antiparticle [1, 2]. Majorana neutrinos naturallyappear in many extensions of the standard model (see,e.g., [3]). Also, the smallness of neutrino masses (morethan five orders of magnitude smaller than the electronmass) finds an elegant explanation within the see-sawmodel which assumes neutrinos to be Majorana parti-cles [4].The fact that neutrinos have mass has firmly been es-tablished by neutrino oscillation experiments (for reviewssee, e.g., Ref. [5]). However, the observed oscillationscannot in principle pin down the absolute scale of theneutrino masses. A study of nuclear neutrinoless doublebeta (0 νββ ) decay AZ El N −→ AZ +2 El N − + 2 e − offers amean to probe the absolute neutrino masses at the levelof tens of meV.Double beta decay is a rare decay process which mayoccur in the second order of weak interaction. It offersthe only feasible way to test the charge-conjugation prop-erty of neutrinos. The existence of the 0 νββ decay wouldimmediately prove neutrino to be a Majorana particle asassured by the Schechter-Valle theorem [6]. The 0 νββ decay is strictly forbidden in the standard model of theelectroweak interaction in which the lepton number isconserved, thus its observation would be of paramountimportance for our understanding of particle physics be-yond the standard model [7, 8, 9].The next generation of 0 νββ -decay experiments(CUORE, GERDA, MAJORANA, SuperNEMO etc.,see, e.g., Ref. [9] for a recent review) has a great discoverypotential. Provided the corresponding decay lifetimes areaccurately measured, knowledge of the relevant nuclearmatrix elements (m.e.) M ν will become indispensableto reliably deduce the effective Majorana mass from theexperimental data.Two basic theoretical approaches are used to evalu- ate M ν , the quasiparticle random phase approximation(QRPA) [10, 11], including its continuum version [12],and the nuclear shell model (NSM) [13]. There has beengreat progress in the calculations over the last five years,and now the QRPA 0 νββ nuclear m.e. of different groupsseem to converge. However, the NSM M ν are system-atically and substantially (up to a factor of 2) smallerthan the corresponding QRPA ones. There is now an ac-tive discussion in literature on what could be the reasonof such a discrepancy, a too small single-particle modelspace of the NSM or a neglect of complex nuclear config-urations within the QRPA. Even more striking is the dif-ference in the Fermi contribution to the total M ν whichcan be up to a factor of 5 larger in the QRPA calculationsthan in the NSM ones.In view of this situation, it would be extremely im-portant to find a possibility to determine M ν experi-mentally. There have been attempts to reconstruct thenuclear amplitude of two-neutrino ββ decay (which ex-perimentally is very accurately known from the directcounting ββ -decay experiments [9]) from partial one-legtransition amplitudes to the intermediate 1 + states mea-sured in charge-exchange reactions [14]. However, sucha procedure can consistently determine M ν only if atransition via a single intermediate 1 + state dominates M ν (the so-called single-state dominance). In the caseof comparable contributions of several intermediate 1 + states the results from charge-exchange reactions cannotdirectly provide M ν , since relative phases of the contri-butions cannot be measured. Pursuing the same way toreconstruct M ν seems even more hopeless, since manyintermediate states of all multipolarities (with a rathermoderate contribution of the 1 + states) are virtually pop-ulated in the 0 νββ decay due to a large momentum ofthe virtual neutrino.The aim of this Rapid Communication is to suggest away by which at least the Fermi component of M ν candirectly be measured, e.g., in charge-exchange reactions.For the derivation of the master expressions (4),(5) belowthe well-known property of the Coulomb interaction tobe the leading source of the isospin breaking in nuclei isexploited [15, 16]. Such a measurement of M νF could seta scale for the 0 νββ nuclear m.e. and help to discrimi-nate between different nuclear structure models in whichcalculated M νF may differ by as much as a factor of 5.We start our derivation by writing down the 0 νββ nu-clear m.e. in the closure approximation in which it ac-quires the form M ν = h f | ˆ W ν | i i of the m.e. of atwo-body scalar operator ˆ W ν between the parent anddaughter ground states | i i and | f i , respectively. Thetotal 0 νββ -decay operator ˆ W ν ≡ g A ˆ W νGT − g V ˆ W νF isthe sum of the Gamow-Teller and Fermi transition oper-ators [7]:ˆ W ν = X ab P ν ( r ab ) (cid:0) g A σ a · σ b − g V (cid:1) τ − a τ − b . (1)Here, the vector and axial vector coupling constants are g V = 1 and g A = 1 .
25, respectively, and P ν ( r ab ≡| ~r a − ~r b | ) is the neutrino potential which in the simplestCoulomb approximation is just reciprocal of the distancebetween the nucleons: P ν ( r ab ) = r ab (for the sake of sim-plicity we have taken out the nuclear radius R from theusual definition of P ν [7]). In this approximationˆ W νF = X ab P ν ( r ab ) τ − a τ − b = 1 e h ˆ T − , [ ˆ T − , ˆ V C ] i , (2)where ˆ T − = P a τ − a is the isospin lowering operator,and ˆ V C = e X a = b (1 − τ (3) a )(1 − τ (3) b ) r ab is the operator ofCoulomb interaction between protons. Actually, only theisotensor component of the Coulomb interaction ˆ V tC = e X a = b T (2) ab r ab , with T (2) ab ≡ τ (3) a τ (3) b − τ a τ b , survives in thedouble commutator (2). This isotensor Coulomb inter-action does contribute to the mean Coulomb field in thenucleus, but it is easy to see that any mean-field single-particle operator drops out of the double commutator (2).Thus, the expression (2) is essentially determined by theresidual (after separating out the mean-field contribu-tion) two-body isotensor Coulomb interaction.The total nonrelativistic nuclear Hamiltonian ˆ H tot consists of the total kinetic energy of nucleons andthe strong and Coulomb two-body interactions between Using closure of the states of the intermediate nucleus AZ +1 El N − which are virtually excited in ββ -decay would be an exact proce-dure if there were no energy dependence in the 0 νββ transitionoperator. A weak energy dependence of the operator leads inreality to a “beyond-closure” correction to the total M ν of lessthan 10%. them: ˆ H tot = ˆ T + ˆ H str + ˆ V C . Assuming ˆ H str to be ex-actly isospin-symmetric h ˆ T − , ˆ H str i = 0 (we shall quan-tify later the accuracy of this assertion but it is wellknown that the isospin-breaking terms in ˆ H str are in factfairly small [15, 16]), one hasˆ W νF = 1 e h ˆ T − , [ ˆ T − , ˆ H tot ] i , (3)and, correspondingly [17], M νF = − e X s ¯ ω s h f | ˆ T − | + s ih + s | ˆ T − | i i . (4)Here, the sum runs over all 0 + states of the intermedi-ate ( N − , Z + 1) isobaric nucleus, ¯ ω s = E s − ( E i + E f ) / s ’th in-termediate 0 + state relative to the mean energy of theground states of the initial and final nuclei. To ac-count for the isospin-breaking part of ˆ H str , δM νF =1 e h f | h ˆ T − , [ ˆ T − , ˆ H str ] i | i i should be subtracted fromr.h.s. of Eq. (4).Among all the intermediate 0 + states, the isobaricanalog state (IAS) dominates the sum (4). In fact, h IAS | ˆ T − | i i ≈ √ N − Z is the largest first-leg transitionm.e. (a few percents of the total Fermi strength N − Z may go to the highly-excited isovector monopole reso-nance (IVMR) since the IAS and IVMR get mixed mainlyby the Coulomb mean field). Similarly, the second-legFermi transition dominantly populates the double IAS(DIAS) in the final nucleus. Due to the isotensor part ofthe Coulomb interaction (which also gives the only con-tribution to the double commutator (2)), the final g.s.gets an admixture of the DIAS where the correspond-ing mixing m.e. is h f | DIAS i = − h f | ˆ V tC | DIAS i E DIAS , with E DIAS ≈ ω IAS . Thereby, one gets h f | ˆ T − | IAS i 6 = 0.Other quantitative arguments for the dominance of theIAS in the sum (4) follow from the representation of thedouble commutator: h ˆ T − , [ ˆ T − , ˆ V tC ] i = ˆ V tC (cid:16) ˆ T − (cid:17) + (cid:16) ˆ T − (cid:17) ˆ V tC − T − ˆ V tC ˆ T − . It is clear that the first term V tC ( T − ) dominates them.e. h f | h ˆ T − , [ ˆ T − , ˆ V tC ] i | i i , since the other m.e., be-cause of ˆ T + | f i ≈ h f | ˆ V tC (cid:16) ˆ T − (cid:17) | i i = h f | ˆ V tC | DIAS ih DIAS | (cid:16) ˆ T − (cid:17) | i i .Thus, M νF is determined by the amplitude of the dou-ble Fermi transition via the IAS in the intermediate nu-cleus into the ground state of the final nucleus wherethe second Fermi transition amplitude is due to an ad-mixture of the DIAS in the final nucleus to the groundstate of the parent nucleus: h f | ˆ T − | IAS ih IAS | ˆ T − | i i = h f | DIAS ih DIAS | ˆ T − | IAS ih IAS | ˆ T − | i i . Finally, onecan write M νF ≈ − e ¯ ω IAS h f | ˆ T − | IAS ih IAS | ˆ T − | i i . (5)Therefore, the total M νF can be reconstructed accord-ing to Eq. (5), if one is able to measure the ∆ T = 2isospin-forbidden m.e. h f | ˆ T − | IAS i , for instance incharge-exchange reactions of the ( n, p )-type (also thesame m.e. determines M νF , but it would be much moredifficult to extract it). Using the QRPA calculation re-sults for M νF [10, 11], this m.e. can roughly be esti-mated as h f | ˆ T − | IAS i ∼ . h IAS | ˆ T − | i i ≈√ N − Z . This strong suppression of h f | ˆ T − | IAS i re-flects the smallness of the isospin violation in nuclei. TheIAS has been observed as a prominent and extremely nar-row resonance and its various features have well beenstudied by means of (p,n), ( He,t) and other charge-exchange reactions, see, e.g. [18]. This gives us hopethat a measurement of h f | ˆ T − | IAS i in the ( n, p ) charge-exchange channel might be possible. More generally,a measurement by whichever experimental mean of the∆ T = 2 admixture of the DIAS in the final ground statewould be enough to determine M νF .A qualitative analysis of the physics involved in cal-culations of M νF can be conducted further. One candefine an operator ˆ V tC = e R X ab T (2) ab which is ob-tained by the substitution of 1 r ab by a constant 1¯ R in the definition of the isotensor Coulomb interaction.Such an operator ˆ V tC is diagonal in the basis of isospineigenstates and does not mix in the first order theDIAS and the final ground state, h DIAS | ˆ V tC | + f i =0. The matrix element 1 e h f | h ˆ T − , [ ˆ T − , ˆ V tC ] i | i i =12 ¯ R X s h f | ˆ T − | + s ih + s | ˆ T − | i i is by a large factor e R ¯ ω IAS ≪ V tC from ˆ V tC in Eq. (2)only a small change in M νF is introduced. Such a subtrac-tion with an appropriate choice of ¯ R ∼ R allows to cutoff the contribution to M νF from the long internucleondistances, where r ab has a smooth behavior and whichare relevant for the Coulomb mean field. Therefore, themajor contribution to M νF should come from the shortdistances where the gradient of r ab is the largest. Thisprovides a natural qualitative explanation of the numer-ical results of both the QRPA and NSM [10, 13] whichconsistently show the short-range character of the partial r -dependent contribution to M ν .Of course, by measuring only M νF one does not get the total m.e. M ν but rather its subleading contribu-tion. However, knowledge of M νF itself brings a veryimportant piece of information. For instance, it willallow to investigate the A dependence of M νF . Also,it can help a lot to discriminate between different nu-clear structure models in which calculated M νF may dif-fer by as much as a factor of 5. In addition, the ratio M νF /M νGT may be more reliably calculable in differentmodels than M νF and M νGT separately. Let us put for-ward here some simple arguments in support of the lat-ter statement. Since only small internucleon distancesdetermine M ν , then only nucleon pairs in the spatialrelative s -wave must dominantly contribute to the m.e..The isotensor Coulomb interaction only couples T = 1pairs which must then be in the state with the totalspin S = 0 to assure antisymmetry of the total two-body wave function. Because of this and the fact that σ · σ | S = 0 , T = 1 i = − | S = 0 , T = 1 i , a naturalestimate for the Gamow-Teller m.e. is M νGT = − M νF provided the neutrino potential is the same in both Fand GT cases. The high-order terms of the nucleon weakcurrent which are present in the case of the GT m.e., butabsent in the F m.e., change a bit this simple estimateto M νGT /M νF ≈ − .
5. Also, an uncertainty of few per-cent may come from the difference in the mean nuclearexcitation energies in the F and GT cases. It is worthnoting that the recent QRPA results [10, 11, 12] are ingood correspondence with these simple estimates.Here, we want to estimate possible corrections to thesimplest closure approximation discussed above. Due touniversality and conservation of the vector current, allthe corrections of the vector current vertices should bethe same independently of which virtual particle, neu-trino or photon, is exchanged between them. This is truefor the effects of short-range correlations and the finitenucleon size. A small difference of a few procent in therealistic potentials may arise from different mean nuclearexcitation energies while exchanging the neutrino or pho-ton but this effect seems to be rather reliably calculable.Another difference can arise from those corrections to thepropagator of the virtual photon, as for instance the vac-uum polarization correction, that are missing in the caseof the virtual neutrino. The effect of the the vacuum po-larization is about 0.5% and can simply be accounted forby a proper renormalization of the electron charge.The effect of isospin nonconservation in the strong two-body interaction can be estimated to be at the level of2%–3% [15, 16]. One can then directly compare the ra-dial dependencies of the isospin-breaking part of the two-body strong interaction in the S = 0 , T = 1 channel andthe Coulomb interaction within the relevant short rangeof 1–2 fm to find the dominating source of the isospinbreaking. Following Ref. [19], one can approximate theradial dependence of the isospin-breaking strong two-body central potential as (0.02–0.03) × f π π e − mπr r (¯ h = c =1). With f π π ≈ .
08 one arrives at the conclusion that thissource of the isospin non-conservation must be about 20–30 % of that caused by the Coulomb interaction. Thoughthere are rather large relative uncertainties in calculatingthe isospin-breaking part of the two-body strong interac-tion, by assuming that this correction could in principlebe evaluated with a moderate accuracy of 30 %, a resid-ual uncertainty of only 10 % in M νF is thereby induced.Thus, the main message of this Rapid Communicationthat, at least in principle, M νF is measurable remainsintact in the most realistic situation (though minor cor-rections may be needed).To conclude, we have shown in this Rapid Communi-cation that the Fermi 0 νββ nuclear m.e. can be recon-structed if one is able to measure the isospin-forbiddenFermi m.e. between the ground state of the final nu-cleus and the isobaric analog state in the intermediatenucleus, for instance by means of charge-exchange reac-tions of the ( n, p )-type. Knowledge of M νF would bringa quite important piece of information on the total 0 νββ nuclear m.e.. Simple arguments show that the estimate M νGT /M νF ≈ − . M νF may differ byas much as a factor of 5.The authors acknowledge support of both the DeutscheForschungsgemeinschaft within the SFB TR27 “Neutri-nos and Beyond” and the EU ILIAS project under Con-tract No. RII3-CT-2004-506222. ∗ Electronic address: [email protected][1] B. Kayser, F. Gibrat-Debu, and F. Perrier,
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