Double charge exchange reactions as a probe for neutrinoless double beta decay nuclear matrix elements

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Published under licence by IOP Publishing Ltd XLIII Symposium on Nuclear Physics 2020 Journal of Physics: Conference Series 1610 (2020) 012013 IOP Publishing doi:10.1088/1742-6596/1610/1/012013 1 Double charge exchange reactions as a probe for neutrinoless double beta decay nuclear matrix elements E. Santopinto1, J. Ferretti2, H. Garc´ıa-Tecocoatzi3, R. Magana Vsevolodovna1and within the NUMEN project. E-mail: [email protected] 1INFN, Sezione di Genova, via Dodecaneso 33, Genova 16146, Italy 2Department of Physics, University of Jyv¨askyl¨a, P.O. Box 35 (YFL), 40014 Jyv¨askyl¨a, Finland 3Department of Physics, University of La Plata (UNLP), 49 y 115 cc. 67, 1900 La Plata, Argentina Abstract. The formalism to describe heavy-ion double charge exchange (DCE) processes in the eikonal and small-momentum transfer approximations introduced in Phys. Rev. C 98, 061601(R) (2018) is briefly discussed. It is also shown that, under the previous approximations, the heavy-ion DCE cross-section can be factorized in terms of a reaction and a nuclear part. A double charge exchange effective potential is explicitly derived in the closure approximation and also for the first time the explicit form of the DCE nuclear matrix elements, that are of the form of double Gamow-Teller and double Fermi. The recent hypothesis of a linear correlation between double Gamow-Teller neutrinoless double beta decay and DCE nuclear matrix elements is confirmed thanks to the first explicit derivation of DCE nuclear matrix elements, and by means of microscopic IBM2 calculations. 1. Introduction Neutrinoless double beta (0νββ) decays rank as one of the most interesting Beyond the Standard Model processes. Their experimental observation would imply that the conservation of the lepton number is violated and that the neutrinos are Majorana-type particles. Moreover, it may also provide a mean to measure the neutrino mass. There are several ongoing experimental searches, including EXO [1], Cuore [2], KamLAND-Zen [3] and Gerda [4]. None of them has provided indications of 0νββ decays yet. There have been strong theoretical efforts to provide guidelines to the experimentalists. However, there are strong discrepancies among the results in the different approaches [5, 6, 7, 8, 9, 10]. In particular, the Nuclear Matrix Elements (NMEs) computed within the different nuclear models disagree by more than a factor of two. Furthermore, these results may require additional renormalization or quenching [11]. A possible mean to overcome these difficulties is to use heavy-ion double-charge exchange (DCE) processes to put constraints on neutrinoless 0νββ nuclear matrix elements. Several heavy-ion DCE experiments are ongoing at RNCP Osaka [12, 13], RIBF RIKEN [14], and LNS INFN [15, 16, 17]. The first two of them make use of high-energy heavy-ion double- XLIII Symposium on Nuclear Physics 2020 Journal of Physics: Conference Series 1610 (2020) 012013 IOP Publishing doi:10.1088/1742-6596/1610/1/012013 2 charge exchange processes in order to study multi-spin-isospin flip excitation modes, such as a high-energy double Gamow-Teller giant resonance (DGT-GR) [12], that has been predicted three decades ago [18, 19]. The experiment at LNS-INFN is aiming to extract information to put constraints on some of the nuclear matrix elements relevant to 0νββ decays [15, 16, 17]. All these experiments have triggered a strong theoretical interest [20, 21, 22, 23], including the effort in the improvement of the reaction part [24, 25, 26]. In the present contribution, we briefly discuss some aspects of the heavy-ion DCE formalism of Ref. [21]. There, for the first time the formalism to describe heavy-ion DCE was explicitly developed by making use of the eikonal approximation. 2. DCE potential In DCE reactions, two pairs of nucleons – one from the target and one from the projectile – interact. In particular, two protons (neutrons) are converted into two neutrons (protons) in the target, and two neutrons (protons) are converted into two protons (neutrons) in the projectile, with the mass number of the target, A, and the projectile, a, both remaining unchanged. A derivation of a DCE effective potential, describing both longand short-range interactions, was carried out in Ref. [21] by considering the one-pion-exchange and short-range-interaction diagrams depicted in Fig. 1. Figure 1. Leading diagrams in a double-charge-exchange process. From left to right, they represent a double-pion-exchange interaction, a double contact term and a mixed one-pionexchange plus contact term. Figure taken from Ref. [21]; APS CopyRight. By making use of the closure approximation, which consists in averaging over the intermediate nuclear states [27, 28], one obtains [21] VDCE(~q1, ~q2) = 4 3fπ mπ4(~σP1·~q1)(~σT1·~q1) ω1(ω1+¯ EP)~τP1 ·~τT1 (~σP2·~q2)(~σT2·~q2) ω2(ω2+¯ EP)(ω2+¯ ET)~τP2 ·~τT2 + 2 c2 T ¯ EF P+¯ EF T +c2 GT(~σP1·~σT1)(~σP2·~σT2) ¯ EGT P+¯ EGT T +cTcGT(~σP2·~σT2) ¯ EGT P+¯ EF T +cTcGT(~σP1·~σT1) ¯ EF P+¯ EGT T ×(~τP1 ·~τT1)(~τP2 ·~τT2) + fπ mπ2(~σP1·~q1)(~σT1·~q1) ω1(ω1+¯ EP)(ω1+¯ ET)~τP1 ·~τT1 ×cT(~τP2 ·~τT2) + cGT(~σP2 ·~σ2)(~τP2 ·~τT2)+ 1 ↔2i, (1) where fπ mπ2≃400 MeV·fm3[29], the values of the parameters cGT = 217 MeV fm3and cT= 151 MeV fm3are taken from the literature [29], and ωi=q~q2 i+m2 π. The labels P1, P2, T1 and T2 stand for the nucleons within the projectile (P1 and P2) and the target (T1 and T2) involved in the DCE process. The projectile and target closure energies are given by ¯ Eα p=hEa n−Ea iiαand ¯ Eα t=hEA n−EA iiα, respectively, and the superscript α=GT or F, indicates the type of energy excitation. The first line of Eq. (1) corresponds to the double-pion-exchange contribution (first diagram of Fig. 1), the second to the double-contact term (second diagram of Fig. 1), finally the third line to the mixed pion-exchange contact-term (third diagram of Fig. 1). XLIII Symposium on Nuclear Physics 2020 Journal of Physics: Conference Series 1610 (2020) 012013 IOP Publishing doi:10.1088/1742-6596/1610/1/012013 3 3. DCE cross-section and microscopic IBM2 nuclear matrix elements The previous formalism can be used to compute the DCE nuclear matrix elements within the microscopic Interacting Boson Model (IBM2) [30] and the DCE cross-sections in the lowmomentum transfer limit by making use of the distorted-wave Born approximation (DWBA). If one considers transitions between 0+and 0+ground states, the differential cross section is given by [21] dσ dΩ=k k0µ 4π2¯h22 |Tif|2,(2) where µis the reduced mass of the target-projectile system, kand k0are the incoming and outgoing momenta, and Tif is the T-matrix of the reaction. Tif can be calculated by means of the Distorted Wave Born Approximation (DWBA), Tif =DΨ− ~ k0ΦfVΨ+ ~ kΦiE=1 (2π)3/2Zd~ Rei(χ(b)−~ Q·~ R)Mif(~m),(3) where one also uses the eikonal approximation for the c.m. scattering. In the previous equation, Ψ+ ~ k,~ k0are the wave functions which describe the c.m. motion of the target and projectile ions, Φi,fthe intrinsic wave functions of the nuclei before and after the interaction, which can be written as the product of projectile and target nucleon wave functions. In the particular case of 0+ i→0+ ftransitions of the target, one gets [21] Mif(m)→2" MDGT T→T0MDGT P→P0 ¯ EGT P+¯ EGT T!+ MDF T→T0MDF P→P0 ¯ EF P+¯ EF T!# ,(4) where MDGT and MDF are Double-Gamow-Teller (DGT) and Double-Fermi (DF) nuclear matrix elements, respectively, of the projectile/target (A = P, T). The previous matrix elements are defined as [21] MDGT A→A0=cGT DΦ(A0) J0X n,n0 [~σn×~σn0](0)~τn~τn0Φ(A) JE(5) and MDF A→A0=ctDΦ(A0) J0X n,n0 ~τn~τn0Φ(A) JE,(6) where the sum runs over the nucleons (n, n0) involved in the process. They are calculated in the microscopic Interacting Boson Model (IBM2) [21, 30]. Finally, the cross section of Eq. (2) can be written in the eikonal approximation and low-momentum transfer limit as dσ dΩ→k k0µ 4π2¯h22 2F(θ) MDGT T→T0MDGT P→P0 ¯ EGT P+¯ EGT T +MDF T→T0MDF P→P0 ¯ EF P+¯ EF T! 2 ,(7) where F(θ) is an angular distribution [21, Eq. (14)]. 4. Linear correlation between DCE and 0νββ nuclear matrix elements In Ref. [20], the authors discussed the possible emergence of a linear correlation between DGT DCE and 0νββ nuclear matrix elements by means of a large-scale shell-model calculation. In Ref. [21], the previous hypothesis was confirmed by making use of a different nuclear model, the microscopic IBM2. Moreover, as also discussed in the previous sections, the existence of a procedure to factorize the DCE cross sections in terms of reaction and nuclear parts was explicitly XLIII Symposium on Nuclear Physics 2020 Journal of Physics: Conference Series 1610 (2020) 012013 IOP Publishing doi:10.1088/1742-6596/1610/1/012013 4 demonstrated in the eikonal approximation. Most important, a microscopic description of DCE processes was developed with the derivation of a DCE potential in the closure approximation [21]. Thanks to this, one may think to use the present and forthcoming experimental data of heavy-ion DCE cross-sections to place an upper limit on 0νββ NMEs in terms of the DCE experimental data at very forward angles, whereas in the case of larger scattering angles, the nuclear part is expected to be a convolution of beam and target NMEs. The emergence of the linear correlation between DCE and 0νββ nuclear matrix elements can be shown by calculating the previous DCE and 0νββ NMEs within the same nuclear model and for a sufficiently large set of nuclei. Then, a simple linear regression analysis can be conducted and a regression line can be drawn. If one plots the microscopic IBM2 results [21, 31] for the 116Cd →116Sn, 128Te →128Xe, 82Se →82Kr, and 76Ge →76Se DGT DCE and 0νββ NMEs, one obtains the clear regression line shown in Fig. 2. Figure 2. Correlation between calculated DCE-DGT NMEs [21] and 0νββ-DGT NMEs [31]. The orange square, green triangle, red star, and blue circle stand for 116Cd →116Sn, 128Te → 128Xe, 82Se→82Kr and 76Ge →76Se data, respectively. Figure from Ref. [21]; APS CopyRight. Finally, it is worth to note that the slopes of our curves in Fig. 2 and [21, Figs. 3] and of those reported in [20, Figs. 4] are quite different. The main reason for this mismatch resides in the procedures used in Ref. [20] and [21] to extract the form of the DCE potential, which is needed to calculate the DCE NMEs. In the first case, the authors did not derive the DCE matrix elements explicitly, but they made a guess on the form only of the DGT DCE nuclear matrix elements ; see [20, Eq. (6)]. In the second case, the authors derived the DCE potential of Eq. (1) and [21, Eq. (3)] explicitly from the diagrams of Figs. 1. Because of this, in the results of [20, Figs. 4] one can appreciate the emergence of a linear correlation between DGT DCE and 0νββ nuclear matrix elements, but the slopes of the curves are “random”. On the contrary, the slopes of the curves in Fig. 2 and [21, Figs. 3] are the “physical” ones. 5. Conclusion The formalism to calculate the cross-sections and Nuclear Matrix Elements (NMEs) of heavy-ion Double Charge Exchange (DCE) processes of Ref. [21] was briefly described. It was also shown that the heavy-ion DCE cross-section can be factorized in terms of a reaction and a nuclear part and that there is a linear correlation between DCE NME’s and neutrinoless NME’s [21]. This will make it possible to extract the 0νββ NMEs from experimental measurements of DCE cross-sections. 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