<figure><img src="image_20.png" /><figcaption>**Fig. 20** Sketch of the nuclear chart showing the various cluster configurations. See text for details</figcaption></figure>

(100 cm from the target), the FAZIA telescopes, with some modifications, should be also suitable for correlation studies [101, 105] allowing performances comparable to previous studies [106, 107]. Reactions relevant for this point are FraISe beams $^{15}$C, $^{14}$Be + $^{12}$C at 40 AMeV, $^{32,38}$S + $^{28}$Si at 40 AMeV.

(iv) Zero-degree configurations (i.e., modules with specific holding structure mounted at zero degree) can also be envisaged, provided that the beam currents can be properly reduced to safe values (typically below 1–2 kHz). Precise cross section measurements require a normalization to the known Rutherford cross section (by fast plastic scintillators located below the grazing angle) or a continuous monitoring of the beam current during the experiment. Relevant example reactions might be FraISe beams $^{15}$C, $^{14}$Be and $^{16,20}$O at 40 AMeV on $^{12}$C, $^{197}$Au targets (tag LNS-ND-HIC-c2 in Table 4).

## 3.2 Clustering

In light nuclei quantum mechanics plays a major role in creating special nuclear structures such as, nuclear skins and/or nuclear halos, nuclear clusters and molecules or gas condensate (see Fig. 20). The idea of clustering as an important phenomenon in describing structure of light nuclei is almost as old as the nuclear physics itself [108, 109], and the typical cluster is the $\alpha$ particle. The basic idea of clustering is that the interaction within the cluster is strong, while the clusters weakly interact with each other. Moreover, the threshold rule holds: cluster configurations in nuclei appears at excitation energies at, or above, the decay threshold into the cluster constituents. For this reason, only weakly bound nuclei, as for example $^{6,7}$Li and $^{9}$Be, possess cluster structures in their ground state, and for this reason ground state cluster configurations are more common in unstable nuclei. Among the different types of clustering, exotic or molecular clustering, may occur. In the exotic clustering the idea of a stiff particle as cluster is dropped. It can be, in fact, a soft particle easy to excite and to break. These configurations become more and more exotic when moving toward the drip line, e.g., [110].

In molecular clustering, clusters, typically $\alpha$s, are bound together by nucleons, which act as the electrons in covalent bonding of ordinary matter. Molecular linear-chain configurations are predicted to exist, as for example in $^{14}$C [111], by antisymmetrized molecular dynamics (AMD) (see Fig. 21). Associated to these configurations two bands are predicted [112], having the $\sigma$-bond or $\pi$-bond valence nucleons.

Molecular clustering could also be the result of the dynamics of the reaction. A molecule made of the two interacting nuclei may be formed during the collision; it rotates and then breaks up without the formation of a compound system. These resonances appear as broad peaks in the elastic channel excitation function having the angular distribution peaked at backward angles [113]. On the other hand, the dynamics of the reaction could be affected by the presence of clustering.
In the last decade, studies of nuclear clustering have reached the precision era. So further studies should follow two paths: (1) fine and very precise measurements of already known cluster states in a number of key nuclei; (2) searches for new cluster structures (and their decays), in previously unstudied nuclei or even new mass regions. New theoretical predictions are being regularly made on cluster structure, often suggesting the existence of states with yet a not observed configuration. Very recent examples are studies of ¹⁰Be within the newly developed “real-time evolution method” [114], or ¹²C within the new “replica exchange Monte Carlo method” [115]. Together with old results (reviewed, e.g., in [116]), obtained within AMD method (e.g., [117]), or molecular orbital model (MOM) [118], there is now a fair amount of theoretical predictions, which awaits for new experimental data. Experimental investigation of cluster states could provide results not only on clustering effects, but also on the radii of the identified states (e.g., [119]) and/or the presence of neutron halos and skins [120]. The high precision data are also needed to identify possible states showing a very special clustering, so-called α-condensate structure (e.g., [121]).

### 3.2.1 Studies with the existing Tandem beams

For what discussed above, the LNS Tandem is an ideal accelerator, providing intense beams of ⁶Li, ⁷Li and ⁹Be, which are a standard choice for cluster studies due to their above mentioned ground state cluster structure which can be used, for example, for transfer reaction of the cluster of interest. In the study of clustering in nuclei the first problem that one has to face is to find signatures of clusterization. For example, a large reduced width for α emission, comparable with the Wigner limit, corresponds to a state where the α particle is fully preformed. The same argument holds also for clusters different than αs. Another possible signature is the identification of a sequence of excited states spaced by an energy interval ΔE following the $J = J + 1$ rule, being J the angular momentum of the state. However, the moment of inertia that can be extracted from the slope of a tentative rotational band gives information only on the deformation of states; this can be a hint of clusterization but not a direct evidence. Therefore, the observation of a number of states lying on a rotational sequence is not a clear evidence of clustering. A proper rotational band is defined as a sequence of states corresponding to the same deformed structure, i.e., the wave function describing each state must correspond to the same (clustered) internal structure. Such information can be provided by the measurement of the electromagnetic transition strength, which tests the overlaps of the initial and final states structure. In fact, for states of the same rotational band there should be an enhancement of the E2 transition probability between successive band members. Therefore, the measurement of the B(E2) gives information on the degree of collectivity of the states involved in the transition. The very small γ-decay branches of these states make such measurements very difficult to perform. However, using high-efficiency γ-detector arrays and with the correct choice of the nuclear reaction to populate those states, this is possible. This is shown, for example, in [122] where the B(E2) transition strength from the 4⁺ to 2⁺ states in ⁸Be was measured, providing a description consistent with a rotational picture and GFCM calculation. A preliminary experiment was done at LNS using the CHIMERA multidetector where γ-transitions in ¹²C were measured [123]; further higher-energy resolution studies are foreseen.

Rotational bands are widely present, especially in light nuclei. Many systems may be studied, but two systems among all are of greater importance: ¹²C and ¹⁶O.

¹²C: in ¹²C two rotational bands are present, one built on the g.s. and one on the second 0⁺ state (the so-called Hoyle state). A new rotational band associated to a triangular arrangement of the three αs have been recently proposed [124, 125] on the basis of an algebraic cluster model calculation [126, 127].

¹⁶O: in ¹⁶O several states with a cluster structure have been observed and many attempts have been made to build rotational bands, but without a definitive assignation.

The ¹⁸O having two neutrons bound to the ¹⁶O core is an excellent tool to study two neutron transfer reactions on various targets as well as to examine clustering in light and medium mass neutron-rich nuclei via (¹⁸O,¹⁶O) and (¹⁸O,²⁰Ne) reactions.

Regarding studies dedicated to the search for new cluster states, or even new type of clustering, the use of radioactive beams would be essential. As an example, the availability of a ¹⁰Be beam allows to investigate the existence of a linear-chain configuration in ¹⁴C predicted by AMD calculations [111].

The ¹⁰Be beam produced in batch mode, and delivered by the LNS Tandem, has been proven to be of excellent quality and with the highest intensity worldwide (10⁹ pps) [128]. The first experimental results concerning the linear chain in ¹⁴C are very promising, and further experiments with this beam would be very welcome,(tag LNS-ND-CLU-a0 in Table 4). In addition, preliminary results of the reaction ¹⁰Be+¹²⁰Sn at 40 MeV measured at LNS have revealed not only the ⁹Be particles coming from the breakup of ¹⁰Be into ⁹Be+n, but also the breakup channel ⁶He+⁴He. The latter could be an indication of the ⁶He-⁴He configuration of ¹⁰Be. Further studies of this reaction, using large solid angle detectors in order to measure the angular and energy distributions of the charged fragments coming from breakup of ¹⁰Be, will allow to investigate both the ⁹Be+n and the ⁶He+⁴He breakup channels of ¹⁰Be and to study its cluster structure. The interaction of ¹⁰Be with stable target nuclei with excess of neutrons (⁷Li, ⁹Be, ¹¹B, ¹³C, ¹⁸O, etc.) makes it possible to study clustering in very neutron-rich systems like ¹⁶C, ¹⁸N, ²⁰O, ²²F etc. Existing data on these nuclei are very limited or completely absent, mainly obtained using low intensity radioactive beams and consequently of poor statistics and resolution. The combination of the large detector arrays and high beam intensity, both available at the LNS Tandem facility, can significantly improve understanding of clustering in the neutron-rich nuclei.
In the last decade, studies of nuclear clustering have reached the precision era. So further studies should follow two paths: (1) fine and very precise measurements of already known cluster states in a number of key nuclei; (2) searches for new cluster structures (and their decays), in previously unstudied nuclei or even new mass regions. New theoretical predictions are being regularly made on cluster structure, often suggesting the existence of states with yet a not observed configuration. Very recent examples are studies of ¹⁰Be within the newly developed “real-time evolution method” [114], or ¹²C within the new “replica exchange Monte Carlo method” [115]. Together with old results (reviewed, e.g., in [116]), obtained within AMD method (e.g., [117]), or molecular orbital model (MOM) [118], there is now a fair amount of theoretical predictions, which awaits for new experimental data. Experimental investigation of cluster states could provide results not only on clustering effects, but also on the radii of the identified states (e.g., [119]) and/or the presence of neutron halos and skins [120]. The high precision data are also needed to identify possible states showing a very special clustering, so-called α-condensate structure (e.g., [121]).

### 3.2.1 Studies with the existing Tandem beams

For what discussed above, the LNS Tandem is an ideal accelerator, providing intense beams of ⁶Li, ⁷Li and ⁹Be, which are a standard choice for cluster studies due to their above mentioned ground state cluster structure which can be used, for example, for transfer reaction of the cluster of interest. In the study of clustering in nuclei the first problem that one has to face is to find signatures of clusterization. For example, a large reduced width for α emission, comparable with the Wigner limit, corresponds to a state where the α particle is fully preformed. The same argument holds also for clusters different than αs. Another possible signature is the identification of a sequence of excited states spaced by an energy interval ΔE following the $J = J + 1$ rule, being J the angular momentum of the state. However, the moment of inertia that can be extracted from the slope of a tentative rotational band gives information only on the deformation of states; this can be a hint of clusterization but not a direct evidence. Therefore, the observation of a number of states lying on a rotational sequence is not a clear evidence of clustering. A proper rotational band is defined as a sequence of states corresponding to the same deformed structure, i.e., the wave function describing each state must correspond to the same (clustered) internal structure. Such information can be provided by the measurement of the electromagnetic transition strength, which tests the overlaps of the initial and final states structure. In fact, for states of the same rotational band there should be an enhancement of the E2 transition probability between successive band members. Therefore, the measurement of the B(E2) gives information on the degree of collectivity of the states involved in the transition. The very small γ-decay branches of these states make such measurements very difficult to perform. However, using high-efficiency γ-detector arrays and with the correct choice of the nuclear reaction to populate those states, this is possible. This is shown, for example, in [122] where the B(E2) transition strength from the 4⁺ to 2⁺ states in ⁸Be was measured, providing a description consistent with a rotational picture and GFCM calculation. A preliminary experiment was done at LNS using the CHIMERA multidetector where γ-transitions in ¹²C were measured [123]; further higher-energy resolution studies are foreseen.

Rotational bands are widely present, especially in light nuclei. Many systems may be studied, but two systems among all are of greater importance: ¹²C and ¹⁶O.

¹²C: in ¹²C two rotational bands are present, one built on the g.s. and one on the second 0⁺ state (the so-called Hoyle state). A new rotational band associated to a triangular arrangement of the three αs have been recently proposed [124, 125] on the basis of an algebraic cluster model calculation [126, 127].

¹⁶O: in ¹⁶O several states with a cluster structure have been observed and many attempts have been made to build rotational bands, but without a definitive assignation.

The ¹⁸O having two neutrons bound to the ¹⁶O core is an excellent tool to study two neutron transfer reactions on various targets as well as to examine clustering in light and medium mass neutron-rich nuclei via (¹⁸O,¹⁶O) and (¹⁸O,²⁰Ne) reactions.

Regarding studies dedicated to the search for new cluster states, or even new type of clustering, the use of radioactive beams would be essential. As an example, the availability of a ¹⁰Be beam allows to investigate the existence of a linear-chain configuration in ¹⁴C predicted by AMD calculations [111].

The ¹⁰Be beam produced in batch mode, and delivered by the LNS Tandem, has been proven to be of excellent quality and with the highest intensity worldwide (10⁹ pps) [128]. The first experimental results concerning the linear chain in ¹⁴C are very promising, and further experiments with this beam would be very welcome,(tag LNS-ND-CLU-a0 in Table 4). In addition, preliminary results of the reaction ¹⁰Be+¹²⁰Sn at 40 MeV measured at LNS have revealed not only the ⁹Be particles coming from the breakup of ¹⁰Be into ⁹Be+n, but also the breakup channel ⁶He+⁴He. The latter could be an indication of the ⁶He-⁴He configuration of ¹⁰Be. Further studies of this reaction, using large solid angle detectors in order to measure the angular and energy distributions of the charged fragments coming from breakup of ¹⁰Be, will allow to investigate both the ⁹Be+n and the ⁶He+⁴He breakup channels of ¹⁰Be and to study its cluster structure. The interaction of ¹⁰Be with stable target nuclei with excess of neutrons (⁷Li, ⁹Be, ¹¹B, ¹³C, ¹⁸O, etc.) makes it possible to study clustering in very neutron-rich systems like ¹⁶C, ¹⁸N, ²⁰O, ²²F etc. Existing data on these nuclei are very limited or completely absent, mainly obtained using low intensity radioactive beams and consequently of poor statistics and resolution. The combination of the large detector arrays and high beam intensity, both available at the LNS Tandem facility, can significantly improve understanding of clustering in the neutron-rich nuclei.
<figure><img src="image_21.png" /><figcaption>**Fig. 21** AMD prediction of $^{10}$Be-$\alpha$ linear-chain configuration in $^{14}$C [111]</figcaption></figure>

### 3.2.2 Development of new Tandem radioactive beams

Besides $^{10}$Be, an unstable beam of interest for cluster studies is $^{14}$C. The development of a $^{14}$C beam (also radioactive with a very long half-life $t_{1/2}$ = 5700y) would, in fact, be rather useful, both for studies of clustering in the A < 20 mass region and for the search of new cluster states in the medium mass nuclei. Radioprotection safety procedures to prevent, for instance, internal and external contamination, have been taken into account by the LNS dedicated service.

The scientific arguments for $^{14}$C beam development are very similar to the ones for $^{10}$Be, it is the perfect ion beam for studying structure of nuclei with excess of neutrons. In addition to the $^{10}$Be case, due to its neutron closed shell, it was proposed that $^{14}$C may be a basic building-block for a new form of clustering in medium mass neutron-rich nuclei as for example $^{14}$C-$^{14}$C configuration in $^{28}$Mg or linear-chain configurations of three $^{14}$C in $^{42}$Ar [118, 129]. These intriguing possibilities can be probed by the measurements on medium mass stable nuclei with N>Z ($^{18}$O, $^{26}$Mg, etc.).

### 3.2.3 Development of noble gas beams at the LNS Tandem

The availability of noble gas beams at the Tandem accelerator could lead to perform new interesting physics investigations on the structure of light-to-medium mass nuclei (tag LNS-ND-CLU-b0 in Table 4) The $^{4}$He beam would also be very welcome for resonant elastic and inelastic scattering (RES) studies, since $\alpha$-particles are the basic building-blocks of classical cluster structures. The method has already been used at LNS and has given promising preliminary results (e.g., [130]).

For example, with $^{4}$He beams of energies from 9 to 20 MeV, sent to the CT2000 scattering chamber, it will be possible to investigate angular distributions and excitation functions of elastic and inelastic resonant scattering on light nuclei (as $^{9}$Be, $^{10,11}$B, $^{13}$C etc.) that will populate $\alpha$-cluster resonant states in non-self-conjugate compound nuclei (as $^{13}$C, $^{14,15}$N, $^{17}$O etc.). The availability of goniometer systems in the CT2000 chamber will facilitate the measurements of angular distributions in direct kinematics experiments by using high resolution (angular and energy) particle-detection systems [131, 132].

$^{3}$He is one of the key beams for research on the proton-rich nuclei both for reaction dynamics and clustering in nuclei with excess of protons. The two-proton transfer reactions are an adequate tool to explore various possible cluster structures in proton rich light nuclei ($^{8}$B, $^{12}$N, $^{14}$O, $^{16}$F, $^{18}$Ne, etc.), which have been poorly investigated, as well as to improve the understanding of the coupling of the proton excess to the $\alpha$-core by improving spectroscopic data on the $n\alpha+2p$ systems ($^{6}$Be, $^{10}$C, $^{14}$O, $^{18}$Ne).

Resonant scattering measurements with the $^{3}$He beam can explore the role of the $^{3}$He cluster in proton-rich environment. This beam enables studies of the $n\alpha$-clustering in N = Z nuclei at very high excitation using the moderate ion beam energies accessible by the Tandem. $^{3}$He beams in the bombarding energy domain 10–30 MeV with the geometrical features of the CT2000 chamber could allow to perform new experiments on one-proton transfer reactions on medium mass stable nuclei [133]. $^{3}$He, d) transfer reactions is in fact a powerful tool to explore single-particle proton strength in ground and excited states of nuclei without having to detect neutrons [134]. It is well known from the literature that several ambiguities in the determination of the spectroscopic factors in medium mass nuclei are still persisting and in some cases prevent the fine-tuning of interactions used in shell model calculations
<figure><img src="image_21.png" /><figcaption>**Fig. 21** AMD prediction of $^{10}$Be-$\alpha$ linear-chain configuration in $^{14}$C [111]</figcaption></figure>

### 3.2.2 Development of new Tandem radioactive beams

Besides $^{10}$Be, an unstable beam of interest for cluster studies is $^{14}$C. The development of a $^{14}$C beam (also radioactive with a very long half-life $t_{1/2}$ = 5700y) would, in fact, be rather useful, both for studies of clustering in the A < 20 mass region and for the search of new cluster states in the medium mass nuclei. Radioprotection safety procedures to prevent, for instance, internal and external contamination, have been taken into account by the LNS dedicated service.

The scientific arguments for $^{14}$C beam development are very similar to the ones for $^{10}$Be, it is the perfect ion beam for studying structure of nuclei with excess of neutrons. In addition to the $^{10}$Be case, due to its neutron closed shell, it was proposed that $^{14}$C may be a basic building-block for a new form of clustering in medium mass neutron-rich nuclei as for example $^{14}$C-$^{14}$C configuration in $^{28}$Mg or linear-chain configurations of three $^{14}$C in $^{42}$Ar [118, 129]. These intriguing possibilities can be probed by the measurements on medium mass stable nuclei with N>Z ($^{18}$O, $^{26}$Mg, etc.).

### 3.2.3 Development of noble gas beams at the LNS Tandem

The availability of noble gas beams at the Tandem accelerator could lead to perform new interesting physics investigations on the structure of light-to-medium mass nuclei (tag LNS-ND-CLU-b0 in Table 4) The $^{4}$He beam would also be very welcome for resonant elastic and inelastic scattering (RES) studies, since $\alpha$-particles are the basic building-blocks of classical cluster structures. The method has already been used at LNS and has given promising preliminary results (e.g., [130]).

For example, with $^{4}$He beams of energies from 9 to 20 MeV, sent to the CT2000 scattering chamber, it will be possible to investigate angular distributions and excitation functions of elastic and inelastic resonant scattering on light nuclei (as $^{9}$Be, $^{10,11}$B, $^{13}$C etc.) that will populate $\alpha$-cluster resonant states in non-self-conjugate compound nuclei (as $^{13}$C, $^{14,15}$N, $^{17}$O etc.). The availability of goniometer systems in the CT2000 chamber will facilitate the measurements of angular distributions in direct kinematics experiments by using high resolution (angular and energy) particle-detection systems [131, 132].

$^{3}$He is one of the key beams for research on the proton-rich nuclei both for reaction dynamics and clustering in nuclei with excess of protons. The two-proton transfer reactions are an adequate tool to explore various possible cluster structures in proton rich light nuclei ($^{8}$B, $^{12}$N, $^{14}$O, $^{16}$F, $^{18}$Ne, etc.), which have been poorly investigated, as well as to improve the understanding of the coupling of the proton excess to the $\alpha$-core by improving spectroscopic data on the $n\alpha+2p$ systems ($^{6}$Be, $^{10}$C, $^{14}$O, $^{18}$Ne).

Resonant scattering measurements with the $^{3}$He beam can explore the role of the $^{3}$He cluster in proton-rich environment. This beam enables studies of the $n\alpha$-clustering in N = Z nuclei at very high excitation using the moderate ion beam energies accessible by the Tandem. $^{3}$He beams in the bombarding energy domain 10–30 MeV with the geometrical features of the CT2000 chamber could allow to perform new experiments on one-proton transfer reactions on medium mass stable nuclei [133]. $^{3}$He, d) transfer reactions is in fact a powerful tool to explore single-particle proton strength in ground and excited states of nuclei without having to detect neutrons [134]. It is well known from the literature that several ambiguities in the determination of the spectroscopic factors in medium mass nuclei are still persisting and in some cases prevent the fine-tuning of interactions used in shell model calculations
[135, 136]. The use of high-energy-resolution Si detectors able to properly identify the emitted deuterons with low thresholds, coupled with the CT2000 chamber geometrical flexibility, will permit to measure high resolution angular distribution of this transfer reaction. Some interesting examples that have an impact also on nuclear astrophysics can be found in [137, 138].

The neon beams would also be very useful in studying medium/heavy clustering states.

$^{20}$Ne: due to its significant $^{16}$O+$^{4}$He cluster structure, the $^{20}$Ne beam enables the study of clustering in a number of medium mass nuclei. It is particularly suitable to examine possible $^{16}$O clustering—as the second double magic nucleus it may have a role of basic cluster in medium mass nuclei like $\alpha$s in light nuclei. Reactions of the $^{20}$Ne beam with carbon, oxygen and magnesium targets are a good tool to probe this structural mode in medium mass nuclei.

$^{22}$Ne: it is a perfect ion beam for studying two-neutron transfer reactions and structure of light and medium mass nuclei with excess of neutrons.

It's worth mentioning that the number of European facilities providing the mentioned beams is getting smaller every year, therefore LNS is, in that sense, becoming very valuable for the nuclear physics community.

### 3.2.4 Studies with in-flight radioactive beams from FraISe

The new FraISe fragment separator conceived to operate with the high-intensity primary beams which will be delivered by the LNS Cyclotron by stripping extraction, is expected to produce different light radioactive ion beams. Although maximum intensities for such beams are expected to be achieved at energies around 40–60 AMeV, calculations show that most of such beams can still be produced with interesting intensities between $10^2$ and $10^5$ pps with lower energies, of the order of 10–15 AMeV. This would open the possibility to perform a variety of investigations.

#### *Study of clustering*

$^{6}$He: having two loosely bound neutrons, the $^{6}$He beam is an excellent tool to study clustering in a number of nuclei via the ($^{6}$He,$^{4}$He) and ($^{6}$He,$^{8}$Be) reactions. This beam has been extensively used for clustering studies at many radioactive beam facilities, but its availability in recent times is quite limited. Available results obtained by the $^{6}$He beam are still limited and numerous additional measurements are needed to complete the picture of clustering in the neutron-rich light nuclei, while experimental data on clustering in medium mass neutron-rich are absent which hampers the understanding of the role of clustering in this mass region.

$^{8}$Li: campaign of measurements of the ($^{8}$Li,$^{6}$Li) and ($^{8}$Li,$^{4}$He) reactions on various targets is a promising way to study clustering in neutron-rich nuclei. It is also of significant interest for reaction dynamics studies as it provides insight in the multiparticle transfer reactions. Available experimental data are very limited in both cases, structure and dynamics.

$^{7}$Be: due to its $^{4}$He+$^{3}$He cluster structure, the $^{7}$Be is an excellent beam to study clustering in proton-rich nuclei via $^{3}$He stripping and multinucleon pick-up reactions. The most likely first experiment in a $^{7}$Be campaign would be an even simpler reaction measurement, the use of one proton pick-up reaction to explore clustering in $^{8}$B and consequent astrophysical implications.

$^{11}$C: there is significant body of evidence for the $^{7}$Be+$^{4}$He clustering in $^{11}$C, so the motivation for the development of this beam is similar to the one for the $^{7}$Be beam; it can be used to explore clustering in the proton-rich light and medium mass nuclei. One example would be to study clustering in $^{10}$C by one proton stripping reactions. The $^{10}$C structure has attracted particular interest as it is likely the only super-Borromean system, one comprising four ($2p + 2\alpha$) bound components in which any subsystem of two and three components is unbound.

$^{13}$B: molecular clustering have been predicted to exist in states of $^{13}$B, above the $\alpha$ emission threshold, having the structure $^{9}$Li-$^{4}$He. Breakup reaction of $^{13}$B in $^{9}$Li+$\alpha$ could be used to perform such an investigation. A proposal has been accepted by the LNS-PAC but the experiment could not be performed due to the COVID shut-down. This experiment would highly benefit from the increase in intensity of the future fragmentation beams. (tag LNS-ND-CLU-a1 in Table 4).

$^{18}$Ne: having two protons bound to the double magic $^{16}$O core, the $^{18}$Ne beam is an excellent tool to study the dynamics of two proton transfer reactions and clustering in the proton-rich nuclei by the ($^{18}$Ne,$^{16}$O) and multinucleon pick-up reactions on various N = Z stable nuclei. Regarding the radioactive beams from FraISe, the ones with the highest intensities (e.g., $^{18}$Ne) could be used to selectively populate cluster states using transfer reactions (in this case, on the proton-rich side of the light mass part of the chart of nuclei). Neutron-rich isotopes of beryllium and carbon could also be very useful, if they could be produced with intensities higher than $10^4$ pps.

#### *Reaction dynamics with light RIBs*

Collisions induced by halo nuclei on various targets at energies from 2 to 3 times the Coulomb barrier down to sub-barrier energies have shown that the presence of the extended low-density halo may deeply affect the reaction dynamics [139–143]. Since halo nuclei are very weakly bound, coupling to continuum may dominate the dynamics generating a suppression of the elastic scattering angular distributions in the region of the Coulomb-nuclear interference peak and an enhancement of the total reaction cross sections. Such an increase in total reaction was usually ascribed to an increased yield for direct processes such as transfer and breakup (e.g., [139, 141, 142]), but effects on fusion were also observed. Most of the available data on the above topic are relative to n-halo-induced reactions. More recently, first experimental studies of collisions induced by p-halo beams seem to suggest the presence of weaker effects when compared to the ones observed with n-halo nuclei, however further investigations are surely necessary (e.g., [143, 144]).
[135, 136]. The use of high-energy-resolution Si detectors able to properly identify the emitted deuterons with low thresholds, coupled with the CT2000 chamber geometrical flexibility, will permit to measure high resolution angular distribution of this transfer reaction. Some interesting examples that have an impact also on nuclear astrophysics can be found in [137, 138].

The neon beams would also be very useful in studying medium/heavy clustering states.

$^{20}$Ne: due to its significant $^{16}$O+$^{4}$He cluster structure, the $^{20}$Ne beam enables the study of clustering in a number of medium mass nuclei. It is particularly suitable to examine possible $^{16}$O clustering—as the second double magic nucleus it may have a role of basic cluster in medium mass nuclei like $\alpha$s in light nuclei. Reactions of the $^{20}$Ne beam with carbon, oxygen and magnesium targets are a good tool to probe this structural mode in medium mass nuclei.

$^{22}$Ne: it is a perfect ion beam for studying two-neutron transfer reactions and structure of light and medium mass nuclei with excess of neutrons.

It's worth mentioning that the number of European facilities providing the mentioned beams is getting smaller every year, therefore LNS is, in that sense, becoming very valuable for the nuclear physics community.

### 3.2.4 Studies with in-flight radioactive beams from FraISe

The new FraISe fragment separator conceived to operate with the high-intensity primary beams which will be delivered by the LNS Cyclotron by stripping extraction, is expected to produce different light radioactive ion beams. Although maximum intensities for such beams are expected to be achieved at energies around 40–60 AMeV, calculations show that most of such beams can still be produced with interesting intensities between $10^2$ and $10^5$ pps with lower energies, of the order of 10–15 AMeV. This would open the possibility to perform a variety of investigations.

#### *Study of clustering*

$^{6}$He: having two loosely bound neutrons, the $^{6}$He beam is an excellent tool to study clustering in a number of nuclei via the ($^{6}$He,$^{4}$He) and ($^{6}$He,$^{8}$Be) reactions. This beam has been extensively used for clustering studies at many radioactive beam facilities, but its availability in recent times is quite limited. Available results obtained by the $^{6}$He beam are still limited and numerous additional measurements are needed to complete the picture of clustering in the neutron-rich light nuclei, while experimental data on clustering in medium mass neutron-rich are absent which hampers the understanding of the role of clustering in this mass region.

$^{8}$Li: campaign of measurements of the ($^{8}$Li,$^{6}$Li) and ($^{8}$Li,$^{4}$He) reactions on various targets is a promising way to study clustering in neutron-rich nuclei. It is also of significant interest for reaction dynamics studies as it provides insight in the multiparticle transfer reactions. Available experimental data are very limited in both cases, structure and dynamics.

$^{7}$Be: due to its $^{4}$He+$^{3}$He cluster structure, the $^{7}$Be is an excellent beam to study clustering in proton-rich nuclei via $^{3}$He stripping and multinucleon pick-up reactions. The most likely first experiment in a $^{7}$Be campaign would be an even simpler reaction measurement, the use of one proton pick-up reaction to explore clustering in $^{8}$B and consequent astrophysical implications.

$^{11}$C: there is significant body of evidence for the $^{7}$Be+$^{4}$He clustering in $^{11}$C, so the motivation for the development of this beam is similar to the one for the $^{7}$Be beam; it can be used to explore clustering in the proton-rich light and medium mass nuclei. One example would be to study clustering in $^{10}$C by one proton stripping reactions. The $^{10}$C structure has attracted particular interest as it is likely the only super-Borromean system, one comprising four ($2p + 2\alpha$) bound components in which any subsystem of two and three components is unbound.

$^{13}$B: molecular clustering have been predicted to exist in states of $^{13}$B, above the $\alpha$ emission threshold, having the structure $^{9}$Li-$^{4}$He. Breakup reaction of $^{13}$B in $^{9}$Li+$\alpha$ could be used to perform such an investigation. A proposal has been accepted by the LNS-PAC but the experiment could not be performed due to the COVID shut-down. This experiment would highly benefit from the increase in intensity of the future fragmentation beams. (tag LNS-ND-CLU-a1 in Table 4).

$^{18}$Ne: having two protons bound to the double magic $^{16}$O core, the $^{18}$Ne beam is an excellent tool to study the dynamics of two proton transfer reactions and clustering in the proton-rich nuclei by the ($^{18}$Ne,$^{16}$O) and multinucleon pick-up reactions on various N = Z stable nuclei. Regarding the radioactive beams from FraISe, the ones with the highest intensities (e.g., $^{18}$Ne) could be used to selectively populate cluster states using transfer reactions (in this case, on the proton-rich side of the light mass part of the chart of nuclei). Neutron-rich isotopes of beryllium and carbon could also be very useful, if they could be produced with intensities higher than $10^4$ pps.

#### *Reaction dynamics with light RIBs*

Collisions induced by halo nuclei on various targets at energies from 2 to 3 times the Coulomb barrier down to sub-barrier energies have shown that the presence of the extended low-density halo may deeply affect the reaction dynamics [139–143]. Since halo nuclei are very weakly bound, coupling to continuum may dominate the dynamics generating a suppression of the elastic scattering angular distributions in the region of the Coulomb-nuclear interference peak and an enhancement of the total reaction cross sections. Such an increase in total reaction was usually ascribed to an increased yield for direct processes such as transfer and breakup (e.g., [139, 141, 142]), but effects on fusion were also observed. Most of the available data on the above topic are relative to n-halo-induced reactions. More recently, first experimental studies of collisions induced by p-halo beams seem to suggest the presence of weaker effects when compared to the ones observed with n-halo nuclei, however further investigations are surely necessary (e.g., [143, 144]).
The new FraISe fragment separator is expected to produce different n-halo and p-halo beams such as, for instance: $^6$He (2n halo), $^{11}$Li (2n halo), $^{11}$Be (1n halo), $^{14}$Be (2n halo), $^8$B (1p halo), $^{15}$C (1n halo), $^{17}$Ne (2p halo). The availability of such beams at energies of the order 10–15 AMeV opens the possibility to deepen our understanding on the above topic, by measuring elastic scattering and direct reaction angular distributions in collisions induced by different n-halo and p-halo beams on a $^{208}$Pb target at about 2–3 times the Coulomb barrier (tag LNS-ND-CLU-b1 in Table 4).

On the n-halo side, an interesting case could be for instance the reaction $^{15}$C + $^{208}$Pb which could be measured at about two times the Coulomb barrier with an estimated beam intensity of the order of $10^5$ pps. A very interesting case could be the more exotic $^{14}$Be + $^{208}$Pb system; however, here the expected currents for the exotic $^{14}$Be projectile are of the order of $10^2$ pps making the experiment much more difficult.

Other possibilities on the p-halo side could concern the $^8$B + $^{208}$Pb and $^{17}$Ne + $^{208}$Pb systems, which could be measured at energies around 2.5 times the Coulomb barrier with estimated beam currents of the order of $10^4$ pps. With p-halo projectiles, the use of a segmented and high-solid-angle detection system would also allow to measure the coincidence between the core and the halo protons gaining further insight on the breakup dynamics (e.g., [143]).

### 3.3 Fission Dynamics

The known atomic nuclei are arranged in the Chart of Nuclides, a 2D map as function of the neutron (N) and proton (P) numbers. Both stable and radioactive nuclei, for a total of 261, are found on the earth crust. The rest of the so far known nuclei, about 3200, have been artificially made through nuclear reactions. The advent of heavy-ion beam accelerators allowed the generation of new nuclei by exploiting fusion and fission reactions, and paved the way to new reaction types, e.g., fragmentation and transfer.

About 3500 unknown nuclei are predicted to be bounded by accredited nuclear models and are located in the unexplored chart's region called "Terra Incognita (TI)". Their properties represent a benchmark for current nuclear structure theories and for a deep understanding of the r-process describing the origin of the elements heavier than Fe [145, 146]. For instance, the neutron shell at N = 126 is the last "waiting point" of the r-process path. Study of the structural properties of nuclei along the N = 126 neutron shell can also contribute to the open question of the quenching of shell effects in nuclei with large neutron excess.

Currently, the properties of unknown nuclei are only predicted by phenomenological models that are built on the systematics of known nuclei. However, the predictions of such models for the unknown regions are problematic because nuclei show diverging properties when moving progressively far from the stability line. Therefore, it is of great interest to investigate nuclear reactions to deal with the production of nuclei in the TI.

The “Island of Stability”, predicted in 1966 in correspondence of a double shell closure at Z = 114 and N = 184 [147], is another zone of unknown nuclei of noteworthy interest. Superheavy nuclei (SHN, proton number Z greater than 103) belong to this region. A worldwide effort is ongoing since five decades to exactly locate the island [148]. Recently synthesized SHN from Cn (Z = 112) to Og (Z = 118) seem to belong to the border of it [149–152].

Fusion of projectile and target nuclei is the solely process that has brought to the synthesis of SHN up to Og (Z = 118) [153]. The employment of fusion reactions to synthesize heavier SHN is at risk because fusion between heavier projectiles and targets is more strongly hindered by the competition with another process called Quasifission (QF) whose yield increases with the number of protons involved [154, 155].

Multinucleon transfer (MNT) is the most promising reaction mechanism to access the TI and might also constitute a path to the SHN mass region [156]. Although it is a process identified in the late 70's, MNT has a general behavior difficult to describe with existing theoretical models due to its complexity as a many-body process and to the lack of accurate and comprehensive experimental data sets [157–160].

In this framework, we discuss here three items which can be experimentally addressed with the high-intensity facility under construction at LNS: shell effects in fission and quasifission, multinucleon transfer reactions to investigate "Terra Incognita", and a possible alternative pathway to produce superheavy elements.

#### 3.3.1 Shell effects in fission and quasifission, fission modes and the Island of stability

It is well known that the fission process is strongly affected by shell effects which appear when a nucleus is represented as a set of collective variables, such as elongation and mass asymmetry, linked by a potential energy surface (PES) [161]. Shell effects appear as peaks and valleys in the PES and trajectories in this space can be linked to observables. Within the liquid drop model (LDM) approach, the nucleus is considered as a classical incompressible “macroscopic” liquid drop in which the competition between the repulsive Coulomb force and the attractive surface force creates a smooth PES with a minimum (the ground state). During the fission process, the nucleus elongates along the line of zero mass asymmetry thus initially increasing its potential energy, until at some time the maximum of the potential energy, which is called the saddle point (the top of the fission barrier), is reached. Afterward, at even further elongation, the nucleus reaches the scission point and splits in two equal fission fragments (mass asymmetry = 0). Although the LDM approach was able to qualitatively explain why fission is one of the main decay modes of heavy nuclei, it fails to describe the experimental observation that fission, spontaneous as well as low energy-induced fission, is sometimes symmetric and sometimes asymmetric, namely, multiple humped. However, following the Strutinsky method [162], the PES should be computed as
The new FraISe fragment separator is expected to produce different n-halo and p-halo beams such as, for instance: $^6$He (2n halo), $^{11}$Li (2n halo), $^{11}$Be (1n halo), $^{14}$Be (2n halo), $^8$B (1p halo), $^{15}$C (1n halo), $^{17}$Ne (2p halo). The availability of such beams at energies of the order 10–15 AMeV opens the possibility to deepen our understanding on the above topic, by measuring elastic scattering and direct reaction angular distributions in collisions induced by different n-halo and p-halo beams on a $^{208}$Pb target at about 2–3 times the Coulomb barrier (tag LNS-ND-CLU-b1 in Table 4).

On the n-halo side, an interesting case could be for instance the reaction $^{15}$C + $^{208}$Pb which could be measured at about two times the Coulomb barrier with an estimated beam intensity of the order of $10^5$ pps. A very interesting case could be the more exotic $^{14}$Be + $^{208}$Pb system; however, here the expected currents for the exotic $^{14}$Be projectile are of the order of $10^2$ pps making the experiment much more difficult.

Other possibilities on the p-halo side could concern the $^8$B + $^{208}$Pb and $^{17}$Ne + $^{208}$Pb systems, which could be measured at energies around 2.5 times the Coulomb barrier with estimated beam currents of the order of $10^4$ pps. With p-halo projectiles, the use of a segmented and high-solid-angle detection system would also allow to measure the coincidence between the core and the halo protons gaining further insight on the breakup dynamics (e.g., [143]).

### 3.3 Fission Dynamics

The known atomic nuclei are arranged in the Chart of Nuclides, a 2D map as function of the neutron (N) and proton (P) numbers. Both stable and radioactive nuclei, for a total of 261, are found on the earth crust. The rest of the so far known nuclei, about 3200, have been artificially made through nuclear reactions. The advent of heavy-ion beam accelerators allowed the generation of new nuclei by exploiting fusion and fission reactions, and paved the way to new reaction types, e.g., fragmentation and transfer.

About 3500 unknown nuclei are predicted to be bounded by accredited nuclear models and are located in the unexplored chart's region called "Terra Incognita (TI)". Their properties represent a benchmark for current nuclear structure theories and for a deep understanding of the r-process describing the origin of the elements heavier than Fe [145, 146]. For instance, the neutron shell at N = 126 is the last "waiting point" of the r-process path. Study of the structural properties of nuclei along the N = 126 neutron shell can also contribute to the open question of the quenching of shell effects in nuclei with large neutron excess.

Currently, the properties of unknown nuclei are only predicted by phenomenological models that are built on the systematics of known nuclei. However, the predictions of such models for the unknown regions are problematic because nuclei show diverging properties when moving progressively far from the stability line. Therefore, it is of great interest to investigate nuclear reactions to deal with the production of nuclei in the TI.

The "Island of Stability", predicted in 1966 in correspondence of a double shell closure at Z = 114 and N = 184 [147], is another zone of unknown nuclei of noteworthy interest. Superheavy nuclei (SHN, proton number Z greater than 103) belong to this region. A worldwide effort is ongoing since five decades to exactly locate the island [148]. Recently synthesized SHN from Cn (Z = 112) to Og (Z = 118) seem to belong to the border of it [149–152].

Fusion of projectile and target nuclei is the solely process that has brought to the synthesis of SHN up to Og (Z = 118) [153]. The employment of fusion reactions to synthesize heavier SHN is at risk because fusion between heavier projectiles and targets is more strongly hindered by the competition with another process called Quasifission (QF) whose yield increases with the number of protons involved [154, 155].

Multinucleon transfer (MNT) is the most promising reaction mechanism to access the TI and might also constitute a path to the SHN mass region [156]. Although it is a process identified in the late 70's, MNT has a general behavior difficult to describe with existing theoretical models due to its complexity as a many-body process and to the lack of accurate and comprehensive experimental data sets [157–160].

In this framework, we discuss here three items which can be experimentally addressed with the high-intensity facility under construction at LNS: shell effects in fission and quasifission, multinucleon transfer reactions to investigate "Terra Incognita", and a possible alternative pathway to produce superheavy elements.

#### 3.3.1 Shell effects in fission and quasifission, fission modes and the Island of stability

It is well known that the fission process is strongly affected by shell effects which appear when a nucleus is represented as a set of collective variables, such as elongation and mass asymmetry, linked by a potential energy surface (PES) [161]. Shell effects appear as peaks and valleys in the PES and trajectories in this space can be linked to observables. Within the liquid drop model (LDM) approach, the nucleus is considered as a classical incompressible "macroscopic" liquid drop in which the competition between the repulsive Coulomb force and the attractive surface force creates a smooth PES with a minimum (the ground state). During the fission process, the nucleus elongates along the line of zero mass asymmetry thus initially increasing its potential energy, until at some time the maximum of the potential energy, which is called the saddle point (the top of the fission barrier), is reached. Afterward, at even further elongation, the nucleus reaches the scission point and splits in two equal fission fragments (mass asymmetry = 0). Although the LDM approach was able to qualitatively explain why fission is one of the main decay modes of heavy nuclei, it fails to describe the experimental observation that fission, spontaneous as well as low energy-induced fission, is sometimes symmetric and sometimes asymmetric, namely, multiple humped. However, following the Strutinsky method [162], the PES should be computed as
<figure><img src="image_22.png" /><figcaption>**Fig. 22** Asymmetric fission modes in spontaneous [164], neutron-induced [165] and low-energy heavy-ion reactions [166]. For $^{260}$No* the excitation energy is 41 MeV</figcaption></figure>

the sum of the macroscopic (LDM) and microscopic (shell effects) energy: $V_{\text{total}} = V_{\text{macro}}(\text{LDM}) + V_{\text{micro}}$ (shells). This naturally leads to the appearance of the asymmetric fission valleys which give rise to asymmetric mass distributions. As the microscopic shell effects depend strongly on specific neutron and proton numbers, their influence on the PES is expected to differ among nuclei, often leading to an even more complex PES with several fission valleys and the so-called fission modes [154, 163], each characterized by its unique saddle and scission points and consequently multiple humped mass distributions.

From the discovery of the fission modes, few observations can be put forward: (a) the existence of a complex structure of the PES in terms of peaks and valleys induces the appearance of several fission barriers which affect the survival probability of a nucleus against fission; (b) there can be separate thresholds for the onset of the asymmetric or symmetric fission.

In the framework of the research on fission modes, of great interest is the investigation of extreme asymmetric fission modes, or superasymmetric modes because besides giving more insight on the shell effect on the PES, could be a pathway to produce exotic neutron-rich nuclei in the region of mass around 80. The existence of a superasymmetric fission mode ($A_H/A_L > 3$) can be inferred from the multimodal nature of nuclear fission. An example of such features are given in Fig. 22. In the empirical deconvolution of the mass distributions, the asymmetric and superasymmetric division are determined by the nuclear shells at $Z = 28$ and $N = 50$ and by the tail of the second standard asymmetric component and the symmetric component, whose contribution is changing with excitation energy of the fissioning nucleus (see the case of $^{260}$No* [166]).

More experimental data for fragment charge and mass distributions for $A_L < 80$ are necessary for determining the parameters describing the superasymmetric mode in models. The fission of heavy nuclei at moderate excitation energies can be used for the production of exotic nuclei, including the very rare $^{78}$Ni [167].

It is worth noticing that the shell closures at $Z = 28$ and $N = 50$ have also a clear involvement in quasifission of heavy systems (see Fig. 23) and the cross section of fragments with mass around $^{78}$Ni is orders of magnitude bigger.

### 3.3.2 Plan for investigating shell effect in supersymmetric fission and quasifission

To plan experiments for the production and investigation of neutron-rich exotic nuclei at or below $Z = 28$, it is crucially important to have high beam intensity and the capability of measuring the charge and the mass of the fragments in that mass region. Fragments in the charge region around $Z=28$ could be produced either in superasymmetric fission or more copiously in quasifission and their charge identification can be achieved with the MAGNEX spectrometer. Additionally, a time-of-flight (TOF) arm on the opposite side of the MAGNEX spectrometer can be used to clean up the two-body reactions from background and to allow the measurements of the primary mass distributions. Reactions of interest for the superasymmetric channel and for which high-intensity beams will be available are $^{22}$Ne + $^{238}$U → $^{260}$No and $^{22}$Ne + $^{232}$Th → $^{254}$Fm. Reactions that might be a possible channel for $^{78}$Ni production are $^{48}$Ca induced on $^{208}$Pb, $^{232}$Th and $^{238}$U (tag LNS-ND-FIS-b0 in Table 4).
<figure><img src="image_22.png" /><figcaption>**Fig. 22** Asymmetric fission modes in spontaneous [164], neutron-induced [165] and low-energy heavy-ion reactions [166]. For $^{260}$No* the excitation energy is 41 MeV</figcaption></figure>

the sum of the macroscopic (LDM) and microscopic (shell effects) energy: $V_{\text{total}} = V_{\text{macro}}(\text{LDM}) + V_{\text{micro}}$ (shells). This naturally leads to the appearance of the asymmetric fission valleys which give rise to asymmetric mass distributions. As the microscopic shell effects depend strongly on specific neutron and proton numbers, their influence on the PES is expected to differ among nuclei, often leading to an even more complex PES with several fission valleys and the so-called fission modes [154, 163], each characterized by its unique saddle and scission points and consequently multiple humped mass distributions.

From the discovery of the fission modes, few observations can be put forward: (a) the existence of a complex structure of the PES in terms of peaks and valleys induces the appearance of several fission barriers which affect the survival probability of a nucleus against fission; (b) there can be separate thresholds for the onset of the asymmetric or symmetric fission.

In the framework of the research on fission modes, of great interest is the investigation of extreme asymmetric fission modes, or superasymmetric modes because besides giving more insight on the shell effect on the PES, could be a pathway to produce exotic neutron-rich nuclei in the region of mass around 80. The existence of a superasymmetric fission mode ($A_H/A_L > 3$) can be inferred from the multimodal nature of nuclear fission. An example of such features are given in Fig. 22. In the empirical deconvolution of the mass distributions, the asymmetric and superasymmetric division are determined by the nuclear shells at $Z = 28$ and $N = 50$ and by the tail of the second standard asymmetric component and the symmetric component, whose contribution is changing with excitation energy of the fissioning nucleus (see the case of $^{260}$No* [166]).

More experimental data for fragment charge and mass distributions for $A_L < 80$ are necessary for determining the parameters describing the superasymmetric mode in models. The fission of heavy nuclei at moderate excitation energies can be used for the production of exotic nuclei, including the very rare $^{78}$Ni [167].

It is worth noticing that the shell closures at $Z = 28$ and $N = 50$ have also a clear involvement in quasifission of heavy systems (see Fig. 23) and the cross section of fragments with mass around $^{78}$Ni is orders of magnitude bigger.

### 3.3.2 Plan for investigating shell effect in supersymmetric fission and quasifission

To plan experiments for the production and investigation of neutron-rich exotic nuclei at or below $Z = 28$, it is crucially important to have high beam intensity and the capability of measuring the charge and the mass of the fragments in that mass region. Fragments in the charge region around $Z=28$ could be produced either in superasymmetric fission or more copiously in quasifission and their charge identification can be achieved with the MAGNEX spectrometer. Additionally, a time-of-flight (TOF) arm on the opposite side of the MAGNEX spectrometer can be used to clean up the two-body reactions from background and to allow the measurements of the primary mass distributions. Reactions of interest for the superasymmetric channel and for which high-intensity beams will be available are $^{22}$Ne + $^{238}$U → $^{260}$No and $^{22}$Ne + $^{232}$Th → $^{254}$Fm. Reactions that might be a possible channel for $^{78}$Ni production are $^{48}$Ca induced on $^{208}$Pb, $^{232}$Th and $^{238}$U (tag LNS-ND-FIS-b0 in Table 4).
<figure><img src="image_23.png" /><figcaption>**Fig. 23** Evidence of the involvement of the shell closures at $Z = 28$ $N = 50$ in quasifission. (Top row) Mass-TKE distributions from several reactions leading to superheavy elements. (Bottom row) Mass distribution only for those events located inside the red polygon in the Mass-TKE distributions above. The effect of the shell closure is to enhance the yield of fragments being single or double magic</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Fig. 24** Cross section of isotones at $N = 126$ via multinucleon transfer and fragmentation</figcaption></figure>

### 3.3.3 Multinucleon transfer reactions to investigate "Terra Incognita"

It is well known that multinucleon transfer (MNT) is the best candidate mechanism, with stable beams, to produce neutron-rich isotopes of the Terra Incognita below $^{208}$Pb. This is shown in Fig. 24. The advantage is that the production cross sections are several orders of magnitude larger than in the case of fragmentation of heavy projectiles, such as $^{208}$Pb or $^{238}$U.

It has been also recently recognized that MNT might be decisive in the production of new superheavy elements being the measured cross sections for fragments produced with mass antisymmetrization (formation of products lighter and heavier, respectively, than the projectile and target) few orders of magnitude bigger than those expected by the most advanced models [156, 168–171]. This is an important issue because MNT may solve the many problems connected with the rise of quasifission process in competition with the fusion process in reactions aimed at producing superheavy elements [154, 155].

Multinucleon transfer is however a substantially unknown process. The important degrees of freedom that drive the dynamical evolution of the process are still matter of investigation as the role played by the shell closures or Q values. The mechanism of dissipation, which connects the single particle and collective degrees of freedom, is evidently one of the most important ingredients of any model, but largely unknown.

A deeper understanding of the features of the MNT comes from the observation that $Q_{gg}$-values and total kinetic energy loss (TKEL) play a decisive role in determining the rate of the transfer channels and the survival against neutron emission of the primary decay fragments. Indeed, the reaction products (target-like fragments (TLFs) and projectile-like fragments (PLFs)) are excited
<figure><img src="image_23.png" /><figcaption>**Fig. 23** Evidence of the involvement of the shell closures at $Z = 28$ $N = 50$ in quasifission. (Top row) Mass-TKE distributions from several reactions leading to superheavy elements. (Bottom row) Mass distribution only for those events located inside the red polygon in the Mass-TKE distributions above. The effect of the shell closure is to enhance the yield of fragments being single or double magic</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Fig. 24** Cross section of isotones at $N = 126$ via multinucleon transfer and fragmentation</figcaption></figure>

### 3.3.3 Multinucleon transfer reactions to investigate "Terra Incognita"

It is well known that multinucleon transfer (MNT) is the best candidate mechanism, with stable beams, to produce neutron-rich isotopes of the Terra Incognita below $^{208}$Pb. This is shown in Fig. 24. The advantage is that the production cross sections are several orders of magnitude larger than in the case of fragmentation of heavy projectiles, such as $^{208}$Pb or $^{238}$U.

It has been also recently recognized that MNT might be decisive in the production of new superheavy elements being the measured cross sections for fragments produced with mass antisymmetrization (formation of products lighter and heavier, respectively, than the projectile and target) few orders of magnitude bigger than those expected by the most advanced models [156, 168–171]. This is an important issue because MNT may solve the many problems connected with the rise of quasifission process in competition with the fusion process in reactions aimed at producing superheavy elements [154, 155].

Multinucleon transfer is however a substantially unknown process. The important degrees of freedom that drive the dynamical evolution of the process are still matter of investigation as the role played by the shell closures or Q values. The mechanism of dissipation, which connects the single particle and collective degrees of freedom, is evidently one of the most important ingredients of any model, but largely unknown.

A deeper understanding of the features of the MNT comes from the observation that $Q_{gg}$-values and total kinetic energy loss (TKEL) play a decisive role in determining the rate of the transfer channels and the survival against neutron emission of the primary decay fragments. Indeed, the reaction products (target-like fragments (TLFs) and projectile-like fragments (PLFs)) are excited
<figure><img src="image_25.png" /><figcaption>**Fig. 25** (left) Mass-TKE distribution of the primary fragments produced in the reaction $^{122}$Sn + $^{208}$Pb at the bombarding energy of 645 MeV; (right) Z vs. N contour plot of the production cross section per each isotope. The contour lines are in units of mb</figcaption></figure>

enough to, even at the barrier energy, evaporate neutrons. Therefore, to preserve the neutron richness of the primary fragments, their excitation energy must be kept as low as possible. However, a larger mass transfer is expected for larger TKEL. As a consequence, the rate of production of neutron-rich fragments is a delicate balance between the $Q_{gg}$ value of the specific channel and rate of energy loss. This point requires further insights to identify an optimal reaction aimed at producing a specific nuclide. How the excitation energy at the scission point is shared between the two fragments is also a matter of open discussions. In [169] it is shown that TLFs heavier than the target $^{208}$Pb are favored by the negative $Q_{gg}$ values which keep the excitation energy lower, even at larger TKEL.

### 3.3.4 Plan for a multinucleon transfer study to investigate “Terra Incognita”

Given the order of magnitude of the cross sections involved in the production of nuclides in the Terra Incognita, the availability of high-intensity beams, with an intensity gain of a factor 100, makes their production feasible in a sufficient amount to allow detailed studies.

An interesting case study is shown in Fig. 25. The Mass-TKE distribution and the production cross sections for the reaction $^{122}$Sn + $^{208}$Pb at the barrier energy were computed with the Langevin type model in Ref. [170, 171]. The benefit of this particular entrance channel is the fact that very exotic W or Os $N = 126$ isotones can be produced with a cross section of the order of few µb. Such a level of cross section can be reached in a reasonable time with a high-intensity beam. However, much care should be reserved to the experimental method. It is clearly required to measure the mass and charge of the fragments in an environment where other reaction channels are very strong. Therefore, the property of high selectivity of the detectors is mandatory. For such a task, the coupling of the MAGNEX setup, which is also being upgraded to stand high intensity, with a sub-nano-second TOF arm could be sufficient to isolate such rare channels by coincidence measurements.

### 3.3.5 A possible pathway to produce superheavy elements

In the early 80's, a campaign of experiments performed at the FLNR-JINR (Dubna) was dedicated to the study of light particle emission in heavy-ion-induced reactions at energies below 10 AMeV [172–174]. Measurements were done for different target—projectile combinations and at different angles. The emitted light particles ($p, d, ^{3,4}\text{He}$) were analyzed in energy and recorded in the focal plane of the stepped pole MSP144 magnetic spectrograph. The most abundant emissions were observed for $\alpha$ particles.

A unique feature of these reactions is that the measured $\alpha$-particle spectra show an exponential decrease that ends to the so-called kinematics upper-limit (KL) for a two-body process. This upper-limit energy value is the maximum energy an $\alpha$ particle can acquire in a two-body process without excitation of the reaction partners at 0° and is calculated from energy and momentum conservation laws using the mass excess of the reaction partners. Moreover, for an energy interval of more than about 20 MeV below the KL, no other processes but two-body are energetically possible (details are given in the references [172–174]) and therefore open.

From these studies, it also resulted that the cross section at the KL for the collinear configuration (measurements at 0°) has a relatively small value, around 6 orders of magnitude below the maximum value of the cross section in the measured spectrum which is located close to the Coulomb barrier in the exit channel. Another observation is that, near the KL, the cross sections are higher for targets that are $\alpha$ emitters and if the bombarding energies are slightly higher than the Coulomb barrier.

From that series of experiments, the most relevant one is the case of $\alpha$ emission at 0° in the reaction $^{40}\text{Ar} + ^{232}\text{Th}$ at the bombarding energy of 220 MeV. Later on, another experiment performed at the same MSP144 spectrometer [175] studied the reaction $^{48}\text{Ca} + ^{238}\text{U}$ at 270 MeV of incident energy and $\alpha$ emission at 0°. The $\alpha$ spectra at 0° in these experiments are given in Fig. 26.
<figure><img src="image_25.png" /><figcaption>**Fig. 25** (left) Mass-TKE distribution of the primary fragments produced in the reaction $^{122}$Sn + $^{208}$Pb at the bombarding energy of 645 MeV; (right) Z vs. N contour plot of the production cross section per each isotope. The contour lines are in units of mb</figcaption></figure>

enough to, even at the barrier energy, evaporate neutrons. Therefore, to preserve the neutron richness of the primary fragments, their excitation energy must be kept as low as possible. However, a larger mass transfer is expected for larger TKEL. As a consequence, the rate of production of neutron-rich fragments is a delicate balance between the $Q_{gg}$ value of the specific channel and rate of energy loss. This point requires further insights to identify an optimal reaction aimed at producing a specific nuclide. How the excitation energy at the scission point is shared between the two fragments is also a matter of open discussions. In [169] it is shown that TLFs heavier than the target $^{208}$Pb are favored by the negative $Q_{gg}$ values which keep the excitation energy lower, even at larger TKEL.

### 3.3.4 Plan for a multinucleon transfer study to investigate “Terra Incognita”

Given the order of magnitude of the cross sections involved in the production of nuclides in the Terra Incognita, the availability of high-intensity beams, with an intensity gain of a factor 100, makes their production feasible in a sufficient amount to allow detailed studies.

An interesting case study is shown in Fig. 25. The Mass-TKE distribution and the production cross sections for the reaction $^{122}$Sn + $^{208}$Pb at the barrier energy were computed with the Langevin type model in Ref. [170, 171]. The benefit of this particular entrance channel is the fact that very exotic W or Os $N = 126$ isotones can be produced with a cross section of the order of few µb. Such a level of cross section can be reached in a reasonable time with a high-intensity beam. However, much care should be reserved to the experimental method. It is clearly required to measure the mass and charge of the fragments in an environment where other reaction channels are very strong. Therefore, the property of high selectivity of the detectors is mandatory. For such a task, the coupling of the MAGNEX setup, which is also being upgraded to stand high intensity, with a sub-nano-second TOF arm could be sufficient to isolate such rare channels by coincidence measurements.

### 3.3.5 A possible pathway to produce superheavy elements

In the early 80's, a campaign of experiments performed at the FLNR-JINR (Dubna) was dedicated to the study of light particle emission in heavy-ion-induced reactions at energies below 10 AMeV [172–174]. Measurements were done for different target—projectile combinations and at different angles. The emitted light particles ($p, d, ^{3,4}\text{He}$) were analyzed in energy and recorded in the focal plane of the stepped pole MSP144 magnetic spectrograph. The most abundant emissions were observed for $\alpha$ particles.

A unique feature of these reactions is that the measured $\alpha$-particle spectra show an exponential decrease that ends to the so-called kinematics upper-limit (KL) for a two-body process. This upper-limit energy value is the maximum energy an $\alpha$ particle can acquire in a two-body process without excitation of the reaction partners at 0° and is calculated from energy and momentum conservation laws using the mass excess of the reaction partners. Moreover, for an energy interval of more than about 20 MeV below the KL, no other processes but two-body are energetically possible (details are given in the references [172–174]) and therefore open.

From these studies, it also resulted that the cross section at the KL for the collinear configuration (measurements at 0°) has a relatively small value, around 6 orders of magnitude below the maximum value of the cross section in the measured spectrum which is located close to the Coulomb barrier in the exit channel. Another observation is that, near the KL, the cross sections are higher for targets that are $\alpha$ emitters and if the bombarding energies are slightly higher than the Coulomb barrier.

From that series of experiments, the most relevant one is the case of $\alpha$ emission at 0° in the reaction $^{40}\text{Ar} + ^{232}\text{Th}$ at the bombarding energy of 220 MeV. Later on, another experiment performed at the same MSP144 spectrometer [175] studied the reaction $^{48}\text{Ca} + ^{238}\text{U}$ at 270 MeV of incident energy and $\alpha$ emission at 0°. The $\alpha$ spectra at 0° in these experiments are given in Fig. 26.
**Fig. 26** (Left) Laboratory α-particle energy spectra measured at 0° in the reaction $^{22}$Ne (178 MeV) + $^{232}$Th and $^{40}$Ar (220 MeV) + $^{232}$Th. The α two-body kinematics upper-limit energies are shown by arrows and are about 125 and 85 MeV for the two reactions, respectively. (right) Laboratory α-particle energy spectra measured at 0° and 10° in the reaction $^{48}$Ca (270 MeV) + $^{238}$U with a thin target. The α two-body kinematics limit energy is around 88 MeV

<figure><img src="image_26.png" /><figcaption>**Fig. 27** Schematic view of the experimental method to detect the heavy residue by the catcher foil and the α particle by MAGNEX</figcaption></figure>

Given the emission of α particles at the two-body KL, one may wonder what happens to the heavy residue accompanying the α particle, which should exist by conservation laws, and emitted also at 0° in the lab reference frame. At the KL, its excitation energy must be zero. For α-particle energies below the KL, the remaining energy turns into excitation of the heavy residue and the α particle will be slower (the energy decrease is close to the value of the excitation energy of the residue) while the residue will be slightly faster. If the excitation energy of the residue is higher than the neutron separation energy or fission barrier, a neutron will be promptly evaporated or the heavy residue may undergo fission, respectively. Due to the exponential slope of the α-particle spectra, for energies lower than the KL the cross section for α emission increases significantly.

### 3.3.6 Plan for investigating the production of superheavy elements via two-body reactions

A schematic view of a possible experimental method is shown in Fig. 27. The essential idea is to stop the reaction products onto a thin catcher foil and measure their decay each time a high-energy α particle is detected in a collinear configuration (zero degree). It is important to remark that in the two-body reaction the α particles take away an important part of momentum brought by the projectile and the resulting heavy residue will gain an energy smaller than the energy of the compound nucleus, if it occurs in the same reaction. Therefore, the heavy residue can be caught on a thinner catcher foil while the other products, in particular α particles, will go through. The silicon detectors surrounding the catcher foil serve as detectors of the heavy residue decay products (α particles or fission fragments). The detection of a high-energy α particle in the focal plane of MAGNEX could also be accompanied by the stop of the beam (beam-off method) and the wait for the decay of the heavy residue caught on the foil. The spectrometer MAGNEX can also be used to detect the α particle at 0° regardless of the presence of the heavy residue to check on the occurrence of α particles at the KL and their cross section. This preliminary step would help in planning the experiment with the catcher foil or even attempt the measurement of the coincidence between the α particle and the heavy residue. Considering the magnitude of the cross sections (of the order of few μb), this type of study can profit of the high-intensity beam at LNS. A typical reaction could be $^{32}$S(193 MeV) + $^{238}$U → α + $^{266}$Sg or $^{66}$Zn (363 MeV) + $^{232}$Th → α + $^{294}$Og (tag LNS-ND-FIS-b2 in Table 4).
**Fig. 26** (Left) Laboratory α-particle energy spectra measured at 0° in the reaction $^{22}$Ne (178 MeV) + $^{232}$Th and $^{40}$Ar (220 MeV) + $^{232}$Th. The α two-body kinematics upper-limit energies are shown by arrows and are about 125 and 85 MeV for the two reactions, respectively. (right) Laboratory α-particle energy spectra measured at 0° and 10° in the reaction $^{48}$Ca (270 MeV) + $^{238}$U with a thin target. The α two-body kinematics limit energy is around 88 MeV

<figure><img src="image_26.png" /><figcaption>**Fig. 27** Schematic view of the experimental method to detect the heavy residue by the catcher foil and the α particle by MAGNEX</figcaption></figure>

Given the emission of α particles at the two-body KL, one may wonder what happens to the heavy residue accompanying the α particle, which should exist by conservation laws, and emitted also at 0° in the lab reference frame. At the KL, its excitation energy must be zero. For α-particle energies below the KL, the remaining energy turns into excitation of the heavy residue and the α particle will be slower (the energy decrease is close to the value of the excitation energy of the residue) while the residue will be slightly faster. If the excitation energy of the residue is higher than the neutron separation energy or fission barrier, a neutron will be promptly evaporated or the heavy residue may undergo fission, respectively. Due to the exponential slope of the α-particle spectra, for energies lower than the KL the cross section for α emission increases significantly.

### 3.3.6 Plan for investigating the production of superheavy elements via two-body reactions

A schematic view of a possible experimental method is shown in Fig. 27. The essential idea is to stop the reaction products onto a thin catcher foil and measure their decay each time a high-energy α particle is detected in a collinear configuration (zero degree). It is important to remark that in the two-body reaction the α particles take away an important part of momentum brought by the projectile and the resulting heavy residue will gain an energy smaller than the energy of the compound nucleus, if it occurs in the same reaction. Therefore, the heavy residue can be caught on a thinner catcher foil while the other products, in particular α particles, will go through. The silicon detectors surrounding the catcher foil serve as detectors of the heavy residue decay products (α particles or fission fragments). The detection of a high-energy α particle in the focal plane of MAGNEX could also be accompanied by the stop of the beam (beam-off method) and the wait for the decay of the heavy residue caught on the foil. The spectrometer MAGNEX can also be used to detect the α particle at 0° regardless of the presence of the heavy residue to check on the occurrence of α particles at the KL and their cross section. This preliminary step would help in planning the experiment with the catcher foil or even attempt the measurement of the coincidence between the α particle and the heavy residue. Considering the magnitude of the cross sections (of the order of few μb), this type of study can profit of the high-intensity beam at LNS. A typical reaction could be $^{32}$S(193 MeV) + $^{238}$U → α + $^{266}$Sg or $^{66}$Zn (363 MeV) + $^{232}$Th → α + $^{294}$Og (tag LNS-ND-FIS-b2 in Table 4).
<table><caption>**Table 5** Tables with priorities and feasibility as addressed in the working group</caption><thead><tr><th>Tag</th><th>Description</th><th>Reaction</th><th>Phase</th></tr></thead><tbody><tr><td>LNS-NS-COL-a0</td><td>Isoscalar and isovector PDR excitation of <sup>38</sup>S, and other nuclei of the cocktail beam</td><td><sup>38</sup>S + <sup>12</sup>C, <sup>179</sup>Au</td><td>A</td></tr><tr><td>LNS-NS-COL-b0</td><td>Time evolution of the dynamical dipole mode in Ar+Ca/Ni systems</td><td>36,40Ar + <sup>58,64</sup>Ni, 40,48Ca</td><td>B</td></tr><tr><td>LNS-NS-COL-b1</td><td>ISGMR excited in <sup>38</sup>S and others nuclei of the cocktail beam</td><td><sup>38</sup>S + <sup>2</sup>H</td><td>B</td></tr><tr><td>LNS-NS-COL-c0</td><td>Precision measurement of IVGDR with a new detection system</td><td>Stable and FraISe beams</td><td>C</td></tr><tr><td>LNS-NS-NMETheo-a0</td><td>Studies to be performed with the present models to provide theoretical input for experiments</td><td>Theoretical model development</td><td>A</td></tr><tr><td>LNS-NS-NMETheo-b0</td><td>Evaluation of the NME models of the ββ decay with and without neutrinos</td><td>Theoretical model development</td><td>B</td></tr><tr><td>LNS-NS-NMETheo-b1</td><td>Investigations of the SCE/DEC HI reactions models</td><td>Theoretical model development</td><td>B</td></tr><tr><td>LNS-NS-NMETheo-c0</td><td>Comparison of SM/IBM/QRPA description of beta decay NME and SCE/DEC form factors</td><td>Theoretical model development</td><td>C</td></tr><tr><td>LNS-NS-SelAdSpec-a0</td><td>Study of DCE differential cross sections on selected systems at high beam intensity @15AMeV with multichannel approach</td><td>(<sup>20</sup>Ne, <sup>20</sup>O), (<sup>18</sup>O, <sup>18</sup>Ne)</td><td>A</td></tr><tr><td>LNS-NS-SelAdSpec-a1</td><td>Study of the energy dependence of DCE cross sections at high beam intensity and different energies</td><td>(<sup>20</sup>Ne, <sup>20</sup>O), (<sup>18</sup>O, <sup>18</sup>Ne)</td><td>A</td></tr><tr><td>LNS-NS-SelAdSpec-b0</td><td>Study of the <sup>100</sup>Mo response to DCE for the first time</td><td><sup>100</sup>Mo(<sup>20</sup>Ne, <sup>20</sup>O)<sup>100</sup>Ru</td><td>B</td></tr><tr><td>LNS-NS-SelAdSpec-c0</td><td>Test study of Double Isobaric Analog State gamma-gamma decay for selected case</td><td>Double Isobaric Analog State γγ decay</td><td>C</td></tr><tr><td>LNS-NS-SelAdSpec-c1</td><td>Search of Double Gamow Teller Giant Resonance</td><td>DGT GR</td><td>C</td></tr></tbody></table>

## 4 Nuclear structure

The nuclear structure working group focuses on two main topics.

The first topic is the selective study of nuclear structure response to describe the 0νββ NME, both experimentally and theoretically. As it is known, this decay can only exist if neutrinos are Majorana particles. Moreover, its observation would show that the lepton number is not conserved, providing a possible way to understanding the asymmetry between matter and antimatter present in the Universe. In this frame, the NME is a crucial ingredient in the expression of the 0νββ half-life, that expresses the transition probability of a nuclear process. To date, results produced by different models to evaluate NME show a large spread, a factor of about three, that leads to significant uncertainties both on the amount of material required in the experiments and on the neutrino mass, in case 0νββ will be observed. Reducing the uncertainty on the calculations of the NME will be essential if we wish to fully exploit the potential 0νββ measurement. Nuclear structure models must be used for simplifying the computational problem by reducing the number of active degrees of freedom (nuclear many-body problem). In this perspective, SCE and DCE reactions induced by heavy ions as well as double γ-decay process may represent key tools. Within this context, we plan to use the present state-of-the-art nuclear structure models (realistic shell model, Skyrme-QRPA, IBMs) to provide the input for direct and transfer heavy-ion SCE and DCE reactions. Comparison between results from different models will be performed (tag LNS-NS-NMETheo-a0 in Table 5, Sect. 4.1.1). In a second step, calculations with the same models will be carried out for the evaluation of the NME of the double β decay with and without neutrinos (tag LNS-NS-NMETheo-b0 in Table 5, Sect. 4.1.2). Subsequently, results from SM/IBM/QRPA descriptions will be employed to investigate possible correlation between NMEs involved in the weak and strong processes as well as in the double γ decay (tag LNS-NS-NMETheo-c0 in Table 5, Sect. 4.1.6).

To supply experimental information on 0νββ NME, the NUMEN project has proposed the idea to use heavy-ion DCE reactions as a surrogate processes of 0νββ, given the similarities with that decay such as, among the others, the same initial and final state. In the experimental approach to DCE reactions a key tool is the MAGNEX magnetic spectrometer, with its upgrade to sustain
<table><caption>**Table 5** Tables with priorities and feasibility as addressed in the working group</caption><thead><tr><th>Tag</th><th>Description</th><th>Reaction</th><th>Phase</th></tr></thead><tbody><tr><td>LNS-NS-COL-a0</td><td>Isoscalar and isovector PDR excitation of <sup>38</sup>S, and other nuclei of the cocktail beam</td><td><sup>38</sup>S + <sup>12</sup>C, <sup>179</sup>Au</td><td>A</td></tr><tr><td>LNS-NS-COL-b0</td><td>Time evolution of the dynamical dipole mode in Ar+Ca/Ni systems</td><td>36,40Ar + <sup>58,64</sup>Ni, 40,48Ca</td><td>B</td></tr><tr><td>LNS-NS-COL-b1</td><td>ISGMR excited in <sup>38</sup>S and others nuclei of the cocktail beam</td><td><sup>38</sup>S + <sup>2</sup>H</td><td>B</td></tr><tr><td>LNS-NS-COL-c0</td><td>Precision measurement of IVGDR with a new detection system</td><td>Stable and FraISe beams</td><td>C</td></tr><tr><td>LNS-NS-NMETheo-a0</td><td>Studies to be performed with the present models to provide theoretical input for experiments</td><td>Theoretical model development</td><td>A</td></tr><tr><td>LNS-NS-NMETheo-b0</td><td>Evaluation of the NME models of the ββ decay with and without neutrinos</td><td>Theoretical model development</td><td>B</td></tr><tr><td>LNS-NS-NMETheo-b1</td><td>Investigations of the SCE/DEC HI reactions models</td><td>Theoretical model development</td><td>B</td></tr><tr><td>LNS-NS-NMETheo-c0</td><td>Comparison of SM/IBM/QRPA description of beta decay NME and SCE/DEC form factors</td><td>Theoretical model development</td><td>C</td></tr><tr><td>LNS-NS-SelAdSpec-a0</td><td>Study of DCE differential cross sections on selected systems at high beam intensity @15A MeV with multichannel approach</td><td>(<sup>20</sup>Ne, <sup>20</sup>O), (<sup>18</sup>O, <sup>18</sup>Ne)</td><td>A</td></tr><tr><td>LNS-NS-SelAdSpec-a1</td><td>Study of the energy dependence of DCE cross sections at high beam intensity and different energies</td><td>(<sup>20</sup>Ne, <sup>20</sup>O), (<sup>18</sup>O, <sup>18</sup>Ne)</td><td>A</td></tr><tr><td>LNS-NS-SelAdSpec-b0</td><td>Study of the <sup>100</sup>Mo response to DCE for the first time</td><td><sup>100</sup>Mo(<sup>20</sup>Ne, <sup>20</sup>O)<sup>100</sup>Ru</td><td>B</td></tr><tr><td>LNS-NS-SelAdSpec-c0</td><td>Test study of Double Isobaric Analog State gamma-gamma decay for selected case</td><td>Double Isobaric Analog State γγ decay</td><td>C</td></tr><tr><td>LNS-NS-SelAdSpec-c1</td><td>Search of Double Gamow Teller Giant Resonance</td><td>DGT GR</td><td>C</td></tr></tbody></table>

## 4 Nuclear structure

The nuclear structure working group focuses on two main topics.

The first topic is the selective study of nuclear structure response to describe the 0νββ NME, both experimentally and theoretically. As it is known, this decay can only exist if neutrinos are Majorana particles. Moreover, its observation would show that the lepton number is not conserved, providing a possible way to understanding the asymmetry between matter and antimatter present in the Universe. In this frame, the NME is a crucial ingredient in the expression of the 0νββ half-life, that expresses the transition probability of a nuclear process. To date, results produced by different models to evaluate NME show a large spread, a factor of about three, that leads to significant uncertainties both on the amount of material required in the experiments and on the neutrino mass, in case 0νββ will be observed. Reducing the uncertainty on the calculations of the NME will be essential if we wish to fully exploit the potential 0νββ measurement. Nuclear structure models must be used for simplifying the computational problem by reducing the number of active degrees of freedom (nuclear many-body problem). In this perspective, SCE and DCE reactions induced by heavy ions as well as double γ-decay process may represent key tools. Within this context, we plan to use the present state-of-the-art nuclear structure models (realistic shell model, Skyrme-QRPA, IBMs) to provide the input for direct and transfer heavy-ion SCE and DCE reactions. Comparison between results from different models will be performed (tag LNS-NS-NMETheo-a0 in Table 5, Sect. 4.1.1). In a second step, calculations with the same models will be carried out for the evaluation of the NME of the double β decay with and without neutrinos (tag LNS-NS-NMETheo-b0 in Table 5, Sect. 4.1.2). Subsequently, results from SM/IBM/QRPA descriptions will be employed to investigate possible correlation between NMEs involved in the weak and strong processes as well as in the double γ decay (tag LNS-NS-NMETheo-c0 in Table 5, Sect. 4.1.6).

To supply experimental information on 0νββ NME, the NUMEN project has proposed the idea to use heavy-ion DCE reactions as a surrogate processes of 0νββ, given the similarities with that decay such as, among the others, the same initial and final state. In the experimental approach to DCE reactions a key tool is the MAGNEX magnetic spectrometer, with its upgrade to sustain
<figure><img src="image_27.png" /><figcaption>Fig. 28 Highlights of the nuclear structure topics</figcaption></figure>

high rates and, at the same time, to maintain the current resolution and sensitivity. The full understanding of the DCE reaction mechanism implies the study of a wide network of nuclear reactions: the new methodology proposed is the multichannel approach (tag LNS-NS-SelAdSpec-a0 in Table 5, Sect. 4.1.4). Synergies with complementary studies, such as double $\gamma$ decay, will also be discussed (tag LNS-NS-SelAdSpec-c0 in Table 5, Sect. 4.1.5). Recent experimental results on DCE give significant input to the next phase of the NUMEN project in which deeper investigations of different systems are planned with improved statistics using high-intensity beams (tag LNS-NS-SelAdSpec-a1 in Table 5 Sect. 4.1.3). Indeed, only few systems have been studied in the present conditions, due to the low cross sections (tag LNS-NS-SelAdSpec-b0 in Table 5, Sect. 4.1.5). Second-order isospin excitations of nuclei are key information bridging the gap between nuclear and neutrino physics and heavy-ion DCE reactions are promising tools in this research field, provided that nuclear structure and reaction aspects are accurately and consistently addressed. The goal is the systematic study of all the cases of nuclei candidate to undergo $0v\beta\beta$ using high-intensity beams, at different energies, and advanced spectrometry.

The second topic is the study of collectivity in nuclei and uses the CHIMERA multidetector as main tool, which could be coupled with ancillary detectors depending on the physics case. Some of the proposed items are also related to the dynamics studies, more deeply developed in the WG1 working group. In particular, the search of the Pigmy dipole resonance (PDR) will represent FraISe day-one experiments (tag LNS-NS-COL-a0 in Table 5, Sect. 4.2.1). The aim is to study the link with both the r-process and the nuclear equation of state (EOS) on one hand and on the other to follow the evolution of PDR in small nuclei, studying the possible dependence on neutron number with CHIMERA coupled with FARCOS telescopes and a new neutron detector. In a second step, the study of the dynamical dipole will be addressed in reactions induced by Ar isotopes on Ca or Ni isotopes (tag LNS-NS-COL-b0 in Table 5, Sect. 4.2.2) with the CHIMERA detector and in strong synergy with the nuclear dynamics studies, also studying the density evolution of the system. In this same timescale, we also propose the study of the Isoscalar Giant Monopole Resonance (IGMR) excited in $^{38}$S and other nuclei of the cocktail beam (tag LNS-NS-COL-b1 in Table 5, Sect. 4.2.3), using the MAGNEX magnetic spectrometer and the FraISe beams. Measuring the GMR in exotic nuclei will open the possibility to study the isospin dependence of the incompressibility. In a next step, we will also consider some experiments to be performed after important detector upgrade. Those experiments will be dedicated to the solution of some contradicting results about the saturation of the GDR width around a temperature of $T = 2.5$ MeV; to explain the sudden quenching of the GDR $\gamma$-ray yield at high excitation energy; to link the GDR disappearance with the liquid-gas phase transition; to study GDR features of light nuclei with cluster structure and the role of collectivity toward the p-drip line (tag LNS-NS-COL-c0 in Table 5, Sect. 4.2.4).

The highlights of all the nuclear structure (NS) topics are summarized in Fig. 28, where at glance it is shown the NS midterm program at LNS, where new physics scenarios will open thanks to the new, high-intensity stable and unstable beams.

## 4.1 Nuclear matrix elements toward $0v\beta\beta$

### 4.1.1 Introduction

The $0v\beta\beta$ decay is a nuclear transition in which two nucleons undergo $\beta$ decay simultaneously without the emission of neutrinos. The search for evidence of this very rare second-order electroweak process is at present one of the major goal in particle physics since nowadays it represents the most promising way to probe neutrino properties and search for deviations from the Standard Model.
<figure><img src="image_27.png" /><figcaption>**Fig. 28** Highlights of the nuclear structure topics</figcaption></figure>

high rates and, at the same time, to maintain the current resolution and sensitivity. The full understanding of the DCE reaction mechanism implies the study of a wide network of nuclear reactions: the new methodology proposed is the multichannel approach (tag LNS-NS-SelAdSpec-a0 in Table 5, Sect. 4.1.4). Synergies with complementary studies, such as double γ decay, will also be discussed (tag LNS-NS-SelAdSpec-c0 in Table 5, Sect. 4.1.5). Recent experimental results on DCE give significant input to the next phase of the NUMEN project in which deeper investigations of different systems are planned with improved statistics using high-intensity beams (tag LNS-NS-SelAdSpec-a1 in Table 5 Sect. 4.1.3). Indeed, only few systems have been studied in the present conditions, due to the low cross sections (tag LNS-NS-SelAdSpec-b0 in Table 5, Sect. 4.1.5). Second-order isospin excitations of nuclei are key information bridging the gap between nuclear and neutrino physics and heavy-ion DCE reactions are promising tools in this research field, provided that nuclear structure and reaction aspects are accurately and consistently addressed. The goal is the systematic study of all the cases of nuclei candidate to undergo $0vββ$ using high-intensity beams, at different energies, and advanced spectrometry.

The second topic is the study of collectivity in nuclei and uses the CHIMERA multidetector as main tool, which could be coupled with ancillary detectors depending on the physics case. Some of the proposed items are also related to the dynamics studies, more deeply developed in the WG1 working group. In particular, the search of the Pigmy dipole resonance (PDR) will represent FraISe day-one experiments (tag LNS-NS-COL-a0 in Table 5, Sect. 4.2.1). The aim is to study the link with both the r-process and the nuclear equation of state (EOS) on one hand and on the other to follow the evolution of PDR in small nuclei, studying the possible dependence on neutron number with CHIMERA coupled with FARCOS telescopes and a new neutron detector. In a second step, the study of the dynamical dipole will be addressed in reactions induced by Ar isotopes on Ca or Ni isotopes (tag LNS-NS-COL-b0 in Table 5, Sect. 4.2.2) with the CHIMERA detector and in strong synergy with the nuclear dynamics studies, also studying the density evolution of the system. In this same timescale, we also propose the study of the Isoscalar Giant Monopole Resonance (IGMR) excited in $^{38}$S and other nuclei of the cocktail beam (tag LNS-NS-COL-b1 in Table 5, Sect. 4.2.3), using the MAGNEX magnetic spectrometer and the FraISe beams. Measuring the GMR in exotic nuclei will open the possibility to study the isospin dependence of the incompressibility. In a next step, we will also consider some experiments to be performed after important detector upgrade. Those experiments will be dedicated to the solution of some contradicting results about the saturation of the GDR width around a temperature of $T = 2.5$ MeV; to explain the sudden quenching of the GDR γ-ray yield at high excitation energy; to link the GDR disappearance with the liquid-gas phase transition; to study GDR features of light nuclei with cluster structure and the role of collectivity toward the p-drip line (tag LNS-NS-COL-c0 in Table 5, Sect. 4.2.4).

The highlights of all the nuclear structure (NS) topics are summarized in Fig. 28, where at glance it is shown the NS midterm program at LNS, where new physics scenarios will open thanks to the new, high-intensity stable and unstable beams.

## 4.1 Nuclear matrix elements toward $0vββ$

### 4.1.1 Introduction

The $0vββ$ decay is a nuclear transition in which two nucleons undergo β decay simultaneously without the emission of neutrinos. The search for evidence of this very rare second-order electroweak process is at present one of the major goal in particle physics since nowadays it represents the most promising way to probe neutrino properties and search for deviations from the Standard Model.
**Fig. 29** NMEs for $0\nu\beta\beta$ candidates as a function of mass number A. Results are from shell model, interacting boson model, quasi-particle random-phase approximation, and energy density functional theory, and are obtained by using the free value of the axial vector coupling constant $g_A$. Further explanations can be found in [176], from which the figure was taken

<figure><img src="image_28.png" /></figure>

At present, the strongest limits on the $0\nu\beta\beta$ decay have been set as > $10^{26}$ yr by the GERDA and KamLAND-ZEN experiments using $^{76}$Ge and $^{136}$Xe nuclei, respectively, and as > $10^{24}$ yr by the CUORE experiment using $^{130}$Te. The goal of future experiments, as for instance ton-scale CUPID (CUORE Upgrade with Particle Identification), is to push the sensitivity to the point at which the decay can be observed if neutrinos are indeed Majorana particles and their masses are arranged in a pattern known as the “inverted hierarchy”. In this context, the NMEs of the $0\nu\beta\beta$ decay related to the wave functions of the parent and granddaughter nuclei play a fundamental role. In fact, the half-life of this process, when considering only the light neutrino exchange, may be expressed as

$$[T_{1/2}^{0\nu}]^{-1} = G^{0\nu}|M^{0\nu}|^2 \left(\frac{\langle m_{\nu} \rangle}{m_e}\right)^2 \quad (2)$$

where $G^{0\nu}$ is the so-called phase-space factor which has recently been re-evaluated with improved precision, $M^{0\nu}$ is the nuclear matrix element, and $\langle m_{\nu} \rangle$ and $m_e$ are, respectively, the effective neutrino mass and the electron mass.

The knowledge of the NMEs when combined with the present limit on $\langle m_{\nu} \rangle$ is a key point in the choice of the material and the required amount to use in experiments. However, as mentioned above, the calculated matrix elements for nuclei of experimental interest are currently characterized by large uncertainties, predictions from different models differing by a factor of two or three (see Fig. 29). Large efforts have therefore been devoted by the community to better constrain and improve the available nuclear structure models.

It is important to note that, in addition to weak processes, nuclear charge exchange (CE) transitions also include excitations induced by the strong interaction, like the Isobaric Analog (IA) or the Gamow-Teller (GT) Resonances [177–181]. In most cases, these transitions share the same initial and final states of corresponding weak processes and their NMEs can be evaluated within the same nuclear structure models which are usually adopted for the investigation of $0\nu\beta\beta$ NMEs. A comparison with a more comprehensive set of experimental observables than it has been done so far, including, for instance, M1 polarizability and spin dipole transitions, would be highly desirable to further constrain and improve the nuclear structure models.

The connection between nuclear single CE excitations and $\beta$ decay has been widely explored in the past. As far as double excitations are concerned, the analogy between strong and weak processes was recently pointed out in Ref. [182], where double Gamow-Teller (DGT) DCE matrix elements are evaluated within the shell-model framework, showing the existence of a linear correlation between DGT DCE and $0\nu\beta\beta$ NMEs. This linear correlation between the DGT DCE matrix elements and the $0\nu\beta\beta$ NMEs was also demonstrated in Ref. [183]. In this paper, the explicit form of DGT DCE matrix elements is derived from chiral effective field theory, and it is shown that, under certain approximations, the DCE cross section can be factorized into a reaction and nuclear structure part, showing the possibility to extract DCE matrix elements. These combined analyses of strong and weak processes are expected to lead to a more comprehensive understanding of the spin-isospin phenomenology, including mutual relationships between the NMEs.

Another interesting possibility to extract information about $0\nu\beta\beta$ NMEs is the study of second-order electromagnetic transitions emitting two photons ($\gamma\gamma$), in particular the double-magnetic dipole decays [184]. Shell-model calculations show a good linear correlation between the $A_{Z-2}X_{N+2} \rightarrow A_ZY_N + 2e^-$ and $A_ZY_N^* \rightarrow A_ZY_N + 2\gamma$ NMEs, the $Y^*$ referring to the double Isobaric Analog State (DIAS). This means that future $\gamma\gamma$ measurements, as planned at LNS, can be employed to extract information on the $0\nu\beta\beta$ NMEs.

Nuclear CE modes can be excited by nuclear reactions, which offer a quite appealing opportunity to explore the physics we have discussed so far. Indeed, already in the past, SCE reactions, mainly induced by light projectiles, have been the major source of information on the isospin and spin-isospin modes of excitation in nuclei. On one hand, these reaction mechanisms provide important clues on the corresponding terms in the effective nucleon-nucleon interaction in the medium; on the other, their study includes the possibility to isolate, out of the reaction cross section, NMEs presenting formal analogies with those of single $\beta$ decay.
**Fig. 29** NMEs for $0\nu\beta\beta$ candidates as a function of mass number A. Results are from shell model, interacting boson model, quasi-particle random-phase approximation, and energy density functional theory, and are obtained by using the free value of the axial vector coupling constant $g_A$. Further explanations can be found in [176], from which the figure was taken

<figure><img src="image_28.png" /></figure>

At present, the strongest limits on the $0\nu\beta\beta$ decay have been set as > $10^{26}$ yr by the GERDA and KamLAND-ZEN experiments using $^{76}$Ge and $^{136}$Xe nuclei, respectively, and as > $10^{24}$ yr by the CUORE experiment using $^{130}$Te. The goal of future experiments, as for instance ton-scale CUPID (CUORE Upgrade with Particle Identification), is to push the sensitivity to the point at which the decay can be observed if neutrinos are indeed Majorana particles and their masses are arranged in a pattern known as the “inverted hierarchy”. In this context, the NMEs of the $0\nu\beta\beta$ decay related to the wave functions of the parent and granddaughter nuclei play a fundamental role. In fact, the half-life of this process, when considering only the light neutrino exchange, may be expressed as

$$[T_{1/2}^{0\nu}]^{-1} = G^{0\nu}|M^{0\nu}|^2 \left(\frac{\langle m_{\nu} \rangle}{m_e}\right)^2 \quad (2)$$

where $G^{0\nu}$ is the so-called phase-space factor which has recently been re-evaluated with improved precision, $M^{0\nu}$ is the nuclear matrix element, and $\langle m_{\nu} \rangle$ and $m_e$ are, respectively, the effective neutrino mass and the electron mass.

The knowledge of the NMEs when combined with the present limit on $\langle m_{\nu} \rangle$ is a key point in the choice of the material and the required amount to use in experiments. However, as mentioned above, the calculated matrix elements for nuclei of experimental interest are currently characterized by large uncertainties, predictions from different models differing by a factor of two or three (see Fig. 29). Large efforts have therefore been devoted by the community to better constrain and improve the available nuclear structure models.

It is important to note that, in addition to weak processes, nuclear charge exchange (CE) transitions also include excitations induced by the strong interaction, like the Isobaric Analog (IA) or the Gamow-Teller (GT) Resonances [177–181]. In most cases, these transitions share the same initial and final states of corresponding weak processes and their NMEs can be evaluated within the same nuclear structure models which are usually adopted for the investigation of $0\nu\beta\beta$ NMEs. A comparison with a more comprehensive set of experimental observables than it has been done so far, including, for instance, M1 polarizability and spin dipole transitions, would be highly desirable to further constrain and improve the nuclear structure models.

The connection between nuclear single CE excitations and $\beta$ decay has been widely explored in the past. As far as double excitations are concerned, the analogy between strong and weak processes was recently pointed out in Ref. [182], where double Gamow-Teller (DGT) DCE matrix elements are evaluated within the shell-model framework, showing the existence of a linear correlation between DGT DCE and $0\nu\beta\beta$ NMEs. This linear correlation between the DGT DCE matrix elements and the $0\nu\beta\beta$ NMEs was also demonstrated in Ref. [183]. In this paper, the explicit form of DGT DCE matrix elements is derived from chiral effective field theory, and it is shown that, under certain approximations, the DCE cross section can be factorized into a reaction and nuclear structure part, showing the possibility to extract DCE matrix elements. These combined analyses of strong and weak processes are expected to lead to a more comprehensive understanding of the spin-isospin phenomenology, including mutual relationships between the NMEs.

Another interesting possibility to extract information about $0\nu\beta\beta$ NMEs is the study of second-order electromagnetic transitions emitting two photons ($\gamma\gamma$), in particular the double-magnetic dipole decays [184]. Shell-model calculations show a good linear correlation between the $A_{Z-2}X_{N+2} \rightarrow A_ZY_N + 2e^-$ and $A_ZY_N^* \rightarrow A_ZY_N + 2\gamma$ NMEs, the $Y^*$ referring to the double Isobaric Analog State (DIAS). This means that future $\gamma\gamma$ measurements, as planned at LNS, can be employed to extract information on the $0\nu\beta\beta$ NMEs.

Nuclear CE modes can be excited by nuclear reactions, which offer a quite appealing opportunity to explore the physics we have discussed so far. Indeed, already in the past, SCE reactions, mainly induced by light projectiles, have been the major source of information on the isospin and spin-isospin modes of excitation in nuclei. On one hand, these reaction mechanisms provide important clues on the corresponding terms in the effective nucleon-nucleon interaction in the medium; on the other, their study includes the possibility to isolate, out of the reaction cross section, NMEs presenting formal analogies with those of single $\beta$ decay.
In the case of heavy-ion reactions, an especially interesting aspect is the broad range of projectile-target combinations which, for example, allow to project out selectively specific features, e.g., spin flip and non-spin flip transitions [185]. SCE reactions with heavy ions are expected to also populate states with higher multiplicities, as those corresponding to the intermediate states of the $0\nu\beta\beta$ decay [181]. Nuclear structure inputs are crucial for the calculation of the associated reaction cross sections, which can be tested against experimental results.

DCE reactions with heavy ions [20, 186], whose investigation is pursued within the NUMEN project, are a new exciting frontier, also because of their possible analogy with double $\beta$ decay. The key aspects are that the two (weak and strong) processes, beyond involving the same nuclear configurations, are both represented by transition operators that are a superposition of short-range isospin, spin-isospin and rank-two tensor components with a relevant available momentum (100 MeV/c or so) [20]. Thus DCE experimental cross sections can be considered as important benchmarks for theoretical models; tuning the models on experimental data will increase their predictive power for the evaluation of $0\nu\beta\beta$ NMEs, possibly reducing the undesired spread presently existing among the various calculations [176]. It is also interesting to note that, besides the intimate connection to double-$\beta$ decay, heavy-ion DCE reactions are of genuine interest for nuclear reaction and structure physics since they give access to the systematic studies of the hitherto unexplored territory of multiple excitations of elementary isovector modes of both GT spin flip and Fermi non-spin flip character. Requests for combined stable ejectiles and high resolution spectroscopy are satisfied only by heavy-ion beams because neither light ion nor $(\pi^{\pm}, \pi^{\mp})$ DCE reactions are simultaneously fulfilling these demands, see, e.g., [181].

### 4.1.2 Heavy-ion-induced direct reactions to access observables correlated to $0\nu\beta\beta$ decay

#### Nucleon transfer and charge exchange reactions

The competition of direct and transfer DCE mechanisms is an important issue requiring systematic experimental and accompanying theoretical studies. Experience with heavy-ion SCE reactions shows that the transfer channels in general strongly depend on the kinematic conditions and especially on the spectroscopic properties of the involved ions, so transfer processes can be controlled by an appropriate choice of projectile–target combinations and incident energy, as widely discussed in Ref. [181]. Direct SCE and DCE processes, however, are hard collisional processes on momentum scales defined by the masses of isovector mesons. As such, their yields are only little affected by kinematic conditions or single particle structures. As a rule of thumb, direct SCE and DCE mechanisms dominate provided that the reaction take place at energies well above the mutual Coulomb barrier and projectile–target combinations of incompatible single-particle structures are used, that is easily realized by choosing largely different masses for projectile and target nuclei. This result was confirmed in the recently published study of Ref. [187], where it was demonstrated that the multinucleon transfer contribution to the $^{116}$Cd($^{20}$Ne,$^{20}$O)$^{116}$Sn DCE reaction cross section at 15 AMeV is negligible.

At the start of the NUMEN project, neither reaction nor structure theory had at hand approved methods for second-order processes of this kind. In the last years, significant progresses were made in developing the appropriate theoretical formalism and the numerical methods and computer codes, enabling now the exploration of all reaction channels by using the same experimental conditions and a unique theoretical framework in the so-called multichannel approach described in detail in Sect. 4.1.4. These studies are a selective tool to acquire spectroscopic information and at the same time they allow to pinpoint and constrain the intermediate channels relevant for the population of SCE and DCE channels.

A necessary step forward in the attempt to extract the DCE cross sections and get information on the $0\nu\beta\beta$ NMEs, is to coherently combine the reaction amplitudes of transfer and direct SCE and DCE processes in order to properly account for the quantal interference effects between these competing reaction mechanisms. A pending problem, yet to be solved, is to describe the transfer states and the configurations reached by direct SCE/DCE transitions through compatible structure models. The task is to make commensurable the shell model results, best suited to describe single-particle and pair structures as discussed in other parts of this report, to the QRPA results used for SCE and DCE transitions.

#### Reaction mechanism in direct double charge exchange reactions

Heavy-ion direct DCE reactions can be described in terms of a combination of two single direct CE processes, where mechanisms reflecting the presence of short-range correlations can also be included. The treatment of uncorrelated two-step process leads to an expression of projectile and target NMEs close to the ones involved in $2\nu\beta\beta$ decay of the corresponding nuclei [188, 189]. On the other hand, mechanisms underlying the role of correlations (Majorana-like mechanisms) lead to NMEs exhibiting analogies with the ones involved in the elusive $0\nu\beta\beta$ decay [186, 190]. However, generally speaking, direct DCE NMEs contain a considerably more complex multipole and spin structure than the corresponding weak counterparts.

Within the sequential (uncorrelated) reaction mechanism, heavy-ion direct DCE reactions are described as a double-SCE process (dSCE), i.e., as a sequence of two independent direct SCE reactions. Second-order DWBA represents the natural framework for developing the theory of the dSCE reaction mechanism. The propagation of the two nuclei generated by the first SCE reaction (the intermediate channel) is accounted for by a Green function. Expanding the latter in the basis of nuclear states and distorted waves associated with the intermediate channel, the dSCE transition matrix element becomes a superposition of two SCE transition matrix elements. Initial and final state interactions (distortion effects) of the two SCE processes constituting the dSCE reaction, are very important and have been studied in detail. Whereas distortion effects of the intermediate channel compensate each other to a large
In the case of heavy-ion reactions, an especially interesting aspect is the broad range of projectile-target combinations which, for example, allow to project out selectively specific features, e.g., spin flip and non-spin flip transitions [185]. SCE reactions with heavy ions are expected to also populate states with higher multiplicities, as those corresponding to the intermediate states of the $0\nu\beta\beta$ decay [181]. Nuclear structure inputs are crucial for the calculation of the associated reaction cross sections, which can be tested against experimental results.

DCE reactions with heavy ions [20, 186], whose investigation is pursued within the NUMEN project, are a new exciting frontier, also because of their possible analogy with double $\beta$ decay. The key aspects are that the two (weak and strong) processes, beyond involving the same nuclear configurations, are both represented by transition operators that are a superposition of short-range isospin, spin-isospin and rank-two tensor components with a relevant available momentum (100 MeV/c or so) [20]. Thus DCE experimental cross sections can be considered as important benchmarks for theoretical models; tuning the models on experimental data will increase their predictive power for the evaluation of $0\nu\beta\beta$ NMEs, possibly reducing the undesired spread presently existing among the various calculations [176]. It is also interesting to note that, besides the intimate connection to double-$\beta$ decay, heavy-ion DCE reactions are of genuine interest for nuclear reaction and structure physics since they give access to the systematic studies of the hitherto unexplored territory of multiple excitations of elementary isovector modes of both GT spin flip and Fermi non-spin flip character. Requests for combined stable ejectiles and high resolution spectroscopy are satisfied only by heavy-ion beams because neither light ion nor $(\pi^{\pm}, \pi^{\mp})$ DCE reactions are simultaneously fulfilling these demands, see, e.g., [181].

### 4.1.2 Heavy-ion-induced direct reactions to access observables correlated to $0\nu\beta\beta$ decay

#### Nucleon transfer and charge exchange reactions

The competition of direct and transfer DCE mechanisms is an important issue requiring systematic experimental and accompanying theoretical studies. Experience with heavy-ion SCE reactions shows that the transfer channels in general strongly depend on the kinematic conditions and especially on the spectroscopic properties of the involved ions, so transfer processes can be controlled by an appropriate choice of projectile–target combinations and incident energy, as widely discussed in Ref. [181]. Direct SCE and DCE processes, however, are hard collisional processes on momentum scales defined by the masses of isovector mesons. As such, their yields are only little affected by kinematic conditions or single particle structures. As a rule of thumb, direct SCE and DCE mechanisms dominate provided that the reaction take place at energies well above the mutual Coulomb barrier and projectile–target combinations of incompatible single-particle structures are used, that is easily realized by choosing largely different masses for projectile and target nuclei. This result was confirmed in the recently published study of Ref. [187], where it was demonstrated that the multinucleon transfer contribution to the $^{116}$Cd($^{20}$Ne,$^{20}$O)$^{116}$Sn DCE reaction cross section at 15 AMeV is negligible.

At the start of the NUMEN project, neither reaction nor structure theory had at hand approved methods for second-order processes of this kind. In the last years, significant progresses were made in developing the appropriate theoretical formalism and the numerical methods and computer codes, enabling now the exploration of all reaction channels by using the same experimental conditions and a unique theoretical framework in the so-called multichannel approach described in detail in Sect. 4.1.4. These studies are a selective tool to acquire spectroscopic information and at the same time they allow to pinpoint and constrain the intermediate channels relevant for the population of SCE and DCE channels.

A necessary step forward in the attempt to extract the DCE cross sections and get information on the $0\nu\beta\beta$ NMEs, is to coherently combine the reaction amplitudes of transfer and direct SCE and DCE processes in order to properly account for the quantal interference effects between these competing reaction mechanisms. A pending problem, yet to be solved, is to describe the transfer states and the configurations reached by direct SCE/DCE transitions through compatible structure models. The task is to make commensurable the shell model results, best suited to describe single-particle and pair structures as discussed in other parts of this report, to the QRPA results used for SCE and DCE transitions.

#### Reaction mechanism in direct double charge exchange reactions

Heavy-ion direct DCE reactions can be described in terms of a combination of two single direct CE processes, where mechanisms reflecting the presence of short-range correlations can also be included. The treatment of uncorrelated two-step process leads to an expression of projectile and target NMEs close to the ones involved in $2\nu\beta\beta$ decay of the corresponding nuclei [188, 189]. On the other hand, mechanisms underlying the role of correlations (Majorana-like mechanisms) lead to NMEs exhibiting analogies with the ones involved in the elusive $0\nu\beta\beta$ decay [186, 190]. However, generally speaking, direct DCE NMEs contain a considerably more complex multipole and spin structure than the corresponding weak counterparts.

Within the sequential (uncorrelated) reaction mechanism, heavy-ion direct DCE reactions are described as a double-SCE process (dSCE), i.e., as a sequence of two independent direct SCE reactions. Second-order DWBA represents the natural framework for developing the theory of the dSCE reaction mechanism. The propagation of the two nuclei generated by the first SCE reaction (the intermediate channel) is accounted for by a Green function. Expanding the latter in the basis of nuclear states and distorted waves associated with the intermediate channel, the dSCE transition matrix element becomes a superposition of two SCE transition matrix elements. Initial and final state interactions (distortion effects) of the two SCE processes constituting the dSCE reaction, are very important and have been studied in detail. Whereas distortion effects of the intermediate channel compensate each other to a large
extent, distortion effects related to the entrance and exit channels play a significant role both in the order of magnitude and in the diffraction pattern of the angular distributions, as observed in the case of the single processes.

To single out the information on both projectile and target dSCE NMEs from the analysis of the DCE cross section, some approximations are needed in the treatment of the intermediate channel. In analogy with the weak decays, both the single state dominance hypothesis and the closure approximations were tested. It was found that the order of magnitude of the experimental cross sections can be reproduced only when considering several nuclear states in the intermediate channel, thus supporting the validity of the closure approximation. Within such a scheme and with a proper composition of the nuclear structure elements of the two steps, the structure kernel can be decoupled from the reaction dynamics [188]. Finally, a proper rearrangement of the angular momenta couplings allows to get separate information on projectile and target NMEs [189]. This will give the opportunity to access direct information on NMEs from DCE cross section measurements.

dSCE calculations have been already applied to the study of DCE reactions involving the systems $^{40}$Ca + $^{18}$O, $^{76}$Se + $^{18}$O, $^{76}$Ge + $^{20}$Ne and $^{116}$Cd + $^{20}$Ne. For the first, the lightest one of these nuclear systems, a good reproduction of the experimental DCE angular distribution was found, while the cross section of the reactions involving heavier systems is presently underestimated, pointing to a possible contribution of other DCE reaction mechanisms. In particular, the Majorana-like reaction mechanism, which is currently under investigation, describes the DCE process as a sequence of two correlated SCE reactions. In each of the two interacting nuclei, the correlation is provided by the exchange of neutral mesons between the two nucleons involved in the DCE process. The exchange of charged (responsible for the CE) and neutral mesons (yielding correlations) can occur in different ways, which contribute coherently to the DCE process [186, 190]. Finally, the total DCE reaction can proceed via sequential as well as Majorana-like mechanisms, so that all possible reaction paths coherently contribute to the whole process. Further investigations of the different DCE reaction mechanisms will allow a more complete analysis of the DCE data and a deeper understanding of the underlying interactions.

### 4.1.3 The experimental approach to double charge exchange reactions

The DCE reactions induced by heavy ions present some distinctive characteristics which need to be properly taken into account for a successful experimental study. The involved tiny cross sections (~10 nbarn) call for very precise measurements, maintaining sufficient energy and angular resolution to distinguish among the different possible transitions. Indeed, angular distributions at forward scattering angles, including zero degrees, are necessary to extract meaningful spectroscopic information and provide insight into the reaction mechanism. High performances in terms of medium-/heavy-ion identification is also required. Moreover, a careful control of the signal-to-background ratio is needed due to the large yields generated in the collisions by other reaction channels different from DCE.

In such a context, the magnetic spectrometry presents important advantages compared to other detection techniques. In particular, the MAGNEX magnetic spectrometer [25] plays the central role in the NUMEN experimental campaign. Indeed, its large acceptance in angle (50 msr) and momentum (−14%, +10%) allows to explore a wide angular range and a broad momentum transfer window. Moreover, its focal plane detector (FPD) ensures good identification and tracking performances [191]. However, it was not designed to work with high-intensity beams such as those that will be available at the INFN-LNS facility once the accelerator upgrade is completed [192]. Therefore, a new FPD has been designed with the aim of fulfilling specific requirements. The new FPD consists of two sections: a gas tracker allowing the ions to be tracked downstream of the magnetic elements and a particle identification (PID) wall.

The gas tracker is a proportional drift chamber filled with isobutane gas continuously flowing at a pressure that ranges from a few up to tens of mbar. The multiplication stage is based on multiple thick GEM (M-THGEM) technology [193] which is robust and can withstand the high rate expected at the focal plane (about $3 \times 10^4$ pps/cm). When a reaction ejectile crosses the active volume of the gas tracker the primary electrons drift toward the multiplication stage, are multiplied inside the M-THGEM holes and then drift to a segmented anode. From the charges induced on the anodic pads, it is possible to extract the projection of the track on the horizontal plane, while from the drift time of the electrons it is possible to get information on the vertical coordinates, thus obtaining a full 3D reconstruction of the track. The gas tracker was designed to provide a resolution of 5 mrad FWHM on the horizontal and vertical angles $\theta$ and $\phi$, and a position resolution better than 0.5 mm FWHM.

The identification of the heavy ions is performed by the PID wall. It is an array of 720 telescopes, each with a surface area of $15 \times 15 \text{ mm}^2$. The $\Delta E$ stage is a 100-µm-thick silicon carbide (SiC) detector, while the residual energy $E_r$ stage is a 5-mm-thick CsI(Tl) coupled to a photodiode. The choice of SiC and CsI(Tl) detectors was motivated by the following requirements:

* High radiation hardness, since the detectors must withstand a rate of $10^{11}$ions/(cm$^2 \times$ yr).

* Good energy resolution (about 2–3 %).

* Good time resolution ($\approx 2–3$ ns).

* Operation in a low-pressure gas environment.

The degree of segmentation of the PID wall is a compromise between the requirement of a high efficiency and a low number of electronic channels on the one hand and a low double hit probability on the other hand.
extent, distortion effects related to the entrance and exit channels play a significant role both in the order of magnitude and in the diffraction pattern of the angular distributions, as observed in the case of the single processes.

To single out the information on both projectile and target dSCE NMEs from the analysis of the DCE cross section, some approximations are needed in the treatment of the intermediate channel. In analogy with the weak decays, both the single state dominance hypothesis and the closure approximations were tested. It was found that the order of magnitude of the experimental cross sections can be reproduced only when considering several nuclear states in the intermediate channel, thus supporting the validity of the closure approximation. Within such a scheme and with a proper composition of the nuclear structure elements of the two steps, the structure kernel can be decoupled from the reaction dynamics [188]. Finally, a proper rearrangement of the angular momenta couplings allows to get separate information on projectile and target NMEs [189]. This will give the opportunity to access direct information on NMEs from DCE cross section measurements.

dSCE calculations have been already applied to the study of DCE reactions involving the systems $^{40}$Ca + $^{18}$O, $^{76}$Se + $^{18}$O, $^{76}$Ge + $^{20}$Ne and $^{116}$Cd + $^{20}$Ne. For the first, the lightest one of these nuclear systems, a good reproduction of the experimental DCE angular distribution was found, while the cross section of the reactions involving heavier systems is presently underestimated, pointing to a possible contribution of other DCE reaction mechanisms. In particular, the Majorana-like reaction mechanism, which is currently under investigation, describes the DCE process as a sequence of two correlated SCE reactions. In each of the two interacting nuclei, the correlation is provided by the exchange of neutral mesons between the two nucleons involved in the DCE process. The exchange of charged (responsible for the CE) and neutral mesons (yielding correlations) can occur in different ways, which contribute coherently to the DCE process [186, 190]. Finally, the total DCE reaction can proceed via sequential as well as Majorana-like mechanisms, so that all possible reaction paths coherently contribute to the whole process. Further investigations of the different DCE reaction mechanisms will allow a more complete analysis of the DCE data and a deeper understanding of the underlying interactions.

### 4.1.3 The experimental approach to double charge exchange reactions

The DCE reactions induced by heavy ions present some distinctive characteristics which need to be properly taken into account for a successful experimental study. The involved tiny cross sections (~10 nbarn) call for very precise measurements, maintaining sufficient energy and angular resolution to distinguish among the different possible transitions. Indeed, angular distributions at forward scattering angles, including zero degrees, are necessary to extract meaningful spectroscopic information and provide insight into the reaction mechanism. High performances in terms of medium-/heavy-ion identification is also required. Moreover, a careful control of the signal-to-background ratio is needed due to the large yields generated in the collisions by other reaction channels different from DCE.

In such a context, the magnetic spectrometry presents important advantages compared to other detection techniques. In particular, the MAGNEX magnetic spectrometer [25] plays the central role in the NUMEN experimental campaign. Indeed, its large acceptance in angle (50 msr) and momentum (−14%, +10%) allows to explore a wide angular range and a broad momentum transfer window. Moreover, its focal plane detector (FPD) ensures good identification and tracking performances [191]. However, it was not designed to work with high-intensity beams such as those that will be available at the INFN-LNS facility once the accelerator upgrade is completed [192]. Therefore, a new FPD has been designed with the aim of fulfilling specific requirements. The new FPD consists of two sections: a gas tracker allowing the ions to be tracked downstream of the magnetic elements and a particle identification (PID) wall.

The gas tracker is a proportional drift chamber filled with isobutane gas continuously flowing at a pressure that ranges from a few up to tens of mbar. The multiplication stage is based on multiple thick GEM (M-THGEM) technology [193] which is robust and can withstand the high rate expected at the focal plane (about $3 \times 10^4$ pps/cm). When a reaction ejectile crosses the active volume of the gas tracker the primary electrons drift toward the multiplication stage, are multiplied inside the M-THGEM holes and then drift to a segmented anode. From the charges induced on the anodic pads, it is possible to extract the projection of the track on the horizontal plane, while from the drift time of the electrons it is possible to get information on the vertical coordinates, thus obtaining a full 3D reconstruction of the track. The gas tracker was designed to provide a resolution of 5 mrad FWHM on the horizontal and vertical angles $\theta$ and $\phi$, and a position resolution better than 0.5 mm FWHM.

The identification of the heavy ions is performed by the PID wall. It is an array of 720 telescopes, each with a surface area of $15 \times 15 \text{ mm}^2$. The $\Delta E$ stage is a 100-µm-thick silicon carbide (SiC) detector, while the residual energy $E_r$ stage is a 5-mm-thick CsI(Tl) coupled to a photodiode. The choice of SiC and CsI(Tl) detectors was motivated by the following requirements:

* High radiation hardness, since the detectors must withstand a rate of $10^{11}$ions/(cm$^2 \times$ yr).

* Good energy resolution (about 2–3 %).

* Good time resolution ($\approx 2–3$ ns).

* Operation in a low-pressure gas environment.

The degree of segmentation of the PID wall is a compromise between the requirement of a high efficiency and a low number of electronic channels on the one hand and a low double hit probability on the other hand.
**Fig. 30** Three-dimensional view of the MAGNEX spectrometer, the scattering chamber, the quadrupole and dipole magnets, the focal plane detector and the exit beam lines

<figure><img src="image_29.png" /></figure>

In cases where the energy resolution of MAGNEX ($\Delta E/E \approx 1/1000$) is not sufficient to separate transitions to the states of interest in the residual nuclei, measurements will be performed in coincidence between MAGNEX and the G-NUMEN $\gamma$-ray calorimeter. G-NUMEN will be an array of $\approx 110$ cerium-activated lanthanum-bromide (LaBr$_3$(Ce)) scintillation detectors [26] that will be positioned around the reaction target to detect the prompt $\gamma$-rays from reaction products.

The new detection system is specifically designed to suit the different experimental runs of the NUMEN campaign. Important differences exist in the experimental settings depending on the explored reaction channels, as detailed below. $\beta^{-}\beta^{-}$-like transitions in the target nuclei are probed via the ($^{20}$Ne,$^{20}$O) and ($^{12}$C,$^{12}$Be) DCE reactions, induced by $^{20}$Ne$^{10+}$ and $^{12}$C$^{6+}$ beams, respectively, while $\beta^{+}\beta^{+}$-like transitions are investigated via the ($^{18}$O,$^{18}$Ne) DCE reaction, induced by $^{18}$O$^{8+}$ beams. In this context it is worth noting that, thanks to the spectrometer large momentum acceptance, many reaction channels can be measured simultaneously, i.e., within a unique magnetic setting. This condition guarantees the optimal exploitation of the beam time as well as stable working conditions.

The DCE channel is characterized by the lowest cross section if compared to competing quasi-elastic reactions. Therefore, specific experimental strategies in terms of beam intensities and FPD acceptances have been defined in order to optimize the data taking.

In measurements focused on the DCE g.s. to g.s. transitions that can be energetically resolved by MAGNEX, the beam intensity will be of the order of $10^{13}$ pps, while the FPD momentum acceptance will be reduced to 10%, thus excluding other reaction channels that would produce a prohibitively high count rate. The cases where the MAGNEX inclusive measurements cannot separate the DCE g.s. transition from the low-lying ones will require the coincident detection of the emitted $\gamma$-rays with the G-NUMEN array. Then, a reduction of the beam intensity to $\approx 10^{12}$ pps is foreseen in order to limit the average number of reactions per beam bunch to about 1, thus keeping the best observational limit for the DCE g.s. transitions. Finally, the reaction channels different from the DCE will be explored with a further reduced beam intensity ($\approx 10^{11}$ pps) but exploiting the full FPD acceptance. The use of the G-NUMEN array in such runs could also be considered since with such a beam intensity the total reaction rate will be reasonable. These sets of conditions allow to maintain the data throughput from the digital electronics to manageable levels with affordable solutions for transfer and storage.

Stringent constraints are put on the reaction targets by the above experimental conditions. These targets will typically be prepared by evaporation of the isotope of interest, with a high level of enrichment, onto a substrate. The targets should be thin ($\approx 10^{18}$ atoms/cm$^2$) and uniform to preserve a good energy resolution for the ejectiles detected by MAGNEX FPD. Given the high beam intensities, the power deposited in these fragile targets will be significant ($\approx 1$ W), thus requiring an active cooling system. The proposed solution is to use a 2-$\mu$m-thick backing of Highly Oriented Pyrolytic Graphite (HOPG), whose high thermal conductivity promotes the heat dissipation in a copper sample holder, in contact with the cold head of a cryocooler. [194, 195]. The HOPG will also serve as a post-stripper, which will shift the charge distribution of the ejectiles closer to a full-stripping condition. This is particularly relevant for reaction studies induced by $^{20}$Ne where $8^+$ and $9^+$ charge states could enter the active FPD region giving rise to a substantial background, possibly limiting the tolerable beam intensity [196].

A view of the MAGNEX spectrometer after the upgrade is illustrated in Fig. 30.

### 4.1.4 The multichannel approach

The full understanding of the complete DCE reaction mechanism implies the necessity of studying a wide network of nuclear reactions.

The analysis of each reaction channel allows to access the many features of the dynamical process and the structure of the colliding nuclei involved in the DCE reaction:

* The initial and final state interactions, such as the ones competing to the intermediate partitions, are responsible for the distortion of the incoming and outgoing wave functions involved in nuclear reactions and play a central role in all the reaction mechanisms. Studies of *elastic and inelastic scattering* are mandatory to investigate nucleus-nucleus potential and nuclear deformation, respectively.
**Fig. 30** Three-dimensional view of the MAGNEX spectrometer, the scattering chamber, the quadrupole and dipole magnets, the focal plane detector and the exit beam lines

<figure><img src="image_29.png" /></figure>

In cases where the energy resolution of MAGNEX ($\Delta E/E \approx 1/1000$) is not sufficient to separate transitions to the states of interest in the residual nuclei, measurements will be performed in coincidence between MAGNEX and the G-NUMEN $\gamma$-ray calorimeter. G-NUMEN will be an array of $\approx 110$ cerium-activated lanthanum-bromide (LaBr$_3$(Ce)) scintillation detectors [26] that will be positioned around the reaction target to detect the prompt $\gamma$-rays from reaction products.

The new detection system is specifically designed to suit the different experimental runs of the NUMEN campaign. Important differences exist in the experimental settings depending on the explored reaction channels, as detailed below. $\beta^{-}\beta^{-}$-like transitions in the target nuclei are probed via the ($^{20}$Ne,$^{20}$O) and ($^{12}$C,$^{12}$Be) DCE reactions, induced by $^{20}$Ne$^{10+}$ and $^{12}$C$^{6+}$ beams, respectively, while $\beta^{+}\beta^{+}$-like transitions are investigated via the ($^{18}$O,$^{18}$Ne) DCE reaction, induced by $^{18}$O$^{8+}$ beams. In this context it is worth noting that, thanks to the spectrometer large momentum acceptance, many reaction channels can be measured simultaneously, i.e., within a unique magnetic setting. This condition guarantees the optimal exploitation of the beam time as well as stable working conditions.

The DCE channel is characterized by the lowest cross section if compared to competing quasi-elastic reactions. Therefore, specific experimental strategies in terms of beam intensities and FPD acceptances have been defined in order to optimize the data taking.

In measurements focused on the DCE g.s. to g.s. transitions that can be energetically resolved by MAGNEX, the beam intensity will be of the order of $10^{13}$ pps, while the FPD momentum acceptance will be reduced to 10%, thus excluding other reaction channels that would produce a prohibitively high count rate. The cases where the MAGNEX inclusive measurements cannot separate the DCE g.s. transition from the low-lying ones will require the coincident detection of the emitted $\gamma$-rays with the G-NUMEN array. Then, a reduction of the beam intensity to $\approx 10^{12}$ pps is foreseen in order to limit the average number of reactions per beam bunch to about 1, thus keeping the best observational limit for the DCE g.s. transitions. Finally, the reaction channels different from the DCE will be explored with a further reduced beam intensity ($\approx 10^{11}$ pps) but exploiting the full FPD acceptance. The use of the G-NUMEN array in such runs could also be considered since with such a beam intensity the total reaction rate will be reasonable. These sets of conditions allow to maintain the data throughput from the digital electronics to manageable levels with affordable solutions for transfer and storage.

Stringent constraints are put on the reaction targets by the above experimental conditions. These targets will typically be prepared by evaporation of the isotope of interest, with a high level of enrichment, onto a substrate. The targets should be thin ($\approx 10^{18}$ atoms/cm$^2$) and uniform to preserve a good energy resolution for the ejectiles detected by MAGNEX FPD. Given the high beam intensities, the power deposited in these fragile targets will be significant ($\approx 1$ W), thus requiring an active cooling system. The proposed solution is to use a 2-$\mu$m-thick backing of Highly Oriented Pyrolytic Graphite (HOPG), whose high thermal conductivity promotes the heat dissipation in a copper sample holder, in contact with the cold head of a cryocooler. [194, 195]. The HOPG will also serve as a post-stripper, which will shift the charge distribution of the ejectiles closer to a full-stripping condition. This is particularly relevant for reaction studies induced by $^{20}$Ne where $8^+$ and $9^+$ charge states could enter the active FPD region giving rise to a substantial background, possibly limiting the tolerable beam intensity [196].

A view of the MAGNEX spectrometer after the upgrade is illustrated in Fig. 30.

### 4.1.4 The multichannel approach

The full understanding of the complete DCE reaction mechanism implies the necessity of studying a wide network of nuclear reactions.

The analysis of each reaction channel allows to access the many features of the dynamical process and the structure of the colliding nuclei involved in the DCE reaction:

* The initial and final state interactions, such as the ones competing to the intermediate partitions, are responsible for the distortion of the incoming and outgoing wave functions involved in nuclear reactions and play a central role in all the reaction mechanisms. Studies of *elastic and inelastic scattering* are mandatory to investigate nucleus-nucleus potential and nuclear deformation, respectively.
<figure><img src="image_30.png" /><figcaption>**Fig. 31** Networks of nuclear reactions populated in: a $^{18}$O+$^{130}$Xe and b $^{20}$Ne+$^{130}$Te systems. Arrows represent all the possible nuclear reactions connecting the initial and final partitions. Color labels are indicated in the legend</figcaption></figure>

* The occupation probabilities of valence orbits active in the decay and reaction dynamics is one of the most relevant features of the nuclear wave functions. In this framework, the study of one-nucleon transfer reactions is an important tool to access the single-particle configurations in nuclear states.

* Nuclear many-body properties, such as the pairing interaction, play a crucial role in the nuclear structure of the ββ-candidates nuclei. Two-nucleon transfer reactions are very sensitive to those nuclear features and their study is one the most powerful tool to investigate them.

* The double-SCE contribution to the total DCE can be estimated considering a folding of two SCE reaction amplitudes [188]. For that reason, it is particularly important to improve the description of the single charge exchange experimental data.

All the studies present in literature on heavy-ion-induced direct reactions are focused on few (often one) reaction channels at the time. In this way the information extracted from data analysis cannot be fully constrained and important parameters need to be taken either from other experimental studies performed in similar conditions, or by model calculations. The comprehensive study of a wide ensemble of reaction channels explored in the same experimental conditions and consistently described by a unique theoretical framework would be desirable to make significant step forward in the field. These limitations can be overcome by means of multichannel approach, recently proposed within the NUMEN project.

Thanks to the large momentum acceptance of the MAGNEX spectrometer, it is possible to detect the ejectiles coming from the mentioned nuclear reaction channels in only few sets of the quadrupole and dipole magnetic fields. This simultaneous measurement of different reaction channels helps to highlight possible systematic uncertainties in the evaluation of the integrated beam charge and target thickness, in the measurement of the scattering ion tracks at the FPD and in the reconstruction of the scattering parameters.

Two examples of these networks are shown in Fig. 31a, b in the case of the $^{18}$O and $^{20}$Ne-induced reactions on $^{130}$Xe and $^{130}$Te target nuclei, respectively. The components of the complete DCE reaction mechanism are represented by arrows. In addition to the direct DCE and the two-step double-SCE diagonal paths, many multinucleon transfer routes lead to the same final DCE partition. All the processes shown in Fig. 31a, b coexist during the reaction and compete each other. While the gray arrows in Fig. 31 refer to the transitions not directly measurable in the case of $^{18}$O and $^{20}$Ne-induced reactions, the other colored arrows refer to measured reaction channels: elastic and inelastic scattering (el. and inel.), one-neutron (n), two-neutron (n–n), one-proton (p), and two-proton (p–p) transfers, SCE and DCE.

The multichannel approach has been recently applied to the $^{20}$Ne+$^{116}$Cd [187, 197, 198], $^{18}$O+$^{40}$Ca [199–201], $^{18}$O+$^{48}$Ti [202–204] and $^{18}$O+$^{12}$C [205] systems at 15.3 AMeV incident energy.

### 4.1.5 Recent experimental results on DCE and future plans with the upgraded facility

Two recent experimental studies of DCE reactions have been performed at INFN-LNS, the $^{40}$Ca ($^{18}$O, $^{18}$Ne) $^{40}$Ar [4] and the $^{130}$Te ($^{20}$Ne, $^{20}$O) $^{130}$Xe reaction [206]. The beams have been delivered by K-800 superconducting cyclotron and the ejectiles produced in the reactions were momentum-analyzed by the MAGNEX spectrometer [25] and detected by its focal plane detector [191]. For both systems, not only the DCE reactions but also the complete net of competing processes (elastic scattering, single charge exchange and transfer reactions) has been measured at very forward angles including zero degrees. Concerning the first reaction, the $^{18}$O beam was accelerated with an energy of 270 MeV and impinged on a thin $^{40}$C target. In the second experiment, the $^{20}$Ne beam
<figure><img src="image_30.png" /><figcaption>**Fig. 31** Networks of nuclear reactions populated in: a $^{18}$O+$^{130}$Xe and b $^{20}$Ne+$^{130}$Te systems. Arrows represent all the possible nuclear reactions connecting the initial and final partitions. Color labels are indicated in the legend</figcaption></figure>

* The occupation probabilities of valence orbits active in the decay and reaction dynamics is one of the most relevant features of the nuclear wave functions. In this framework, the study of one-nucleon transfer reactions is an important tool to access the single-particle configurations in nuclear states.

* Nuclear many-body properties, such as the pairing interaction, play a crucial role in the nuclear structure of the ββ-candidates nuclei. Two-nucleon transfer reactions are very sensitive to those nuclear features and their study is one the most powerful tool to investigate them.

* The double-SCE contribution to the total DCE can be estimated considering a folding of two SCE reaction amplitudes [188]. For that reason, it is particularly important to improve the description of the single charge exchange experimental data.

All the studies present in literature on heavy-ion-induced direct reactions are focused on few (often one) reaction channels at the time. In this way the information extracted from data analysis cannot be fully constrained and important parameters need to be taken either from other experimental studies performed in similar conditions, or by model calculations. The comprehensive study of a wide ensemble of reaction channels explored in the same experimental conditions and consistently described by a unique theoretical framework would be desirable to make significant step forward in the field. These limitations can be overcome by means of multichannel approach, recently proposed within the NUMEN project.

Thanks to the large momentum acceptance of the MAGNEX spectrometer, it is possible to detect the ejectiles coming from the mentioned nuclear reaction channels in only few sets of the quadrupole and dipole magnetic fields. This simultaneous measurement of different reaction channels helps to highlight possible systematic uncertainties in the evaluation of the integrated beam charge and target thickness, in the measurement of the scattering ion tracks at the FPD and in the reconstruction of the scattering parameters.

Two examples of these networks are shown in Fig. 31a, b in the case of the $^{18}$O and $^{20}$Ne-induced reactions on $^{130}$Xe and $^{130}$Te target nuclei, respectively. The components of the complete DCE reaction mechanism are represented by arrows. In addition to the direct DCE and the two-step double-SCE diagonal paths, many multinucleon transfer routes lead to the same final DCE partition. All the processes shown in Fig. 31a, b coexist during the reaction and compete each other. While the gray arrows in Fig. 31 refer to the transitions not directly measurable in the case of $^{18}$O and $^{20}$Ne-induced reactions, the other colored arrows refer to measured reaction channels: elastic and inelastic scattering (el. and inel.), one-neutron (n), two-neutron (n–n), one-proton (p), and two-proton (p–p) transfers, SCE and DCE.

The multichannel approach has been recently applied to the $^{20}$Ne+$^{116}$Cd [187, 197, 198], $^{18}$O+$^{40}$Ca [199–201], $^{18}$O+$^{48}$Ti [202–204] and $^{18}$O+$^{12}$C [205] systems at 15.3 AMeV incident energy.

### 4.1.5 Recent experimental results on DCE and future plans with the upgraded facility

Two recent experimental studies of DCE reactions have been performed at INFN-LNS, the $^{40}$Ca ($^{18}$O, $^{18}$Ne) $^{40}$Ar [4] and the $^{130}$Te ($^{20}$Ne, $^{20}$O) $^{130}$Xe reaction [206]. The beams have been delivered by K-800 superconducting cyclotron and the ejectiles produced in the reactions were momentum-analyzed by the MAGNEX spectrometer [25] and detected by its focal plane detector [191]. For both systems, not only the DCE reactions but also the complete net of competing processes (elastic scattering, single charge exchange and transfer reactions) has been measured at very forward angles including zero degrees. Concerning the first reaction, the $^{18}$O beam was accelerated with an energy of 270 MeV and impinged on a thin $^{40}$C target. In the second experiment, the $^{20}$Ne beam
<figure><img src="image_31.png" /><figcaption>**Fig. 32** a Excitation energy spectrum from $^{40}$Ca ($^{18}$O, $^{18}$Ne) $^{40}$Ar DCE. The symbols g.s.△ and 1.46△ indicate the $^{40}$Ca ($^{18}$O, $^{18}$Ne$_{1.87}$MeV) $^{40}$Ar$_{g.s.}$ and $^{40}$Ca ($^{18}$O, $^{18}$Ne$_{1.87}$MeV) $^{40}$Ar$_{1.47}$MeV transitions, respectively. Figure from Ref. [4]. **b** Excitation energy spectrum for the $^{130}$Te ($^{20}$Ne, $^{20}$O) $^{130}$Xe DCE reaction. Inset: zoomed view for $E_x$ < 10 MeV. Figure from Ref. [206]</figcaption></figure>

was produced with an energy of 306 MeV and impinged on a thin $^{130}$Te target. Adopting the identification technique described in [207] the ions of interest were selected. A 10$^{th}$-order trajectory reconstruction software was used to determine the excitation energy spectra, shown in Fig. 32.

In the DCE energy spectrum of Fig. 32a, the $^{40}$Ar$_{g.s.}$ is clearly separated from the not resolved doublet of states $^{40}$Ar 2+ at 1.460 MeV and $^{18}$Ne 2+ at 1.887 MeV. At higher excitation energy the measured yield is spread over many overlapping states. The cross section around zero-degree is ≈ 11 µb/sr for the $^{40}$Ca ($^{18}$O, $^{18}$Ne) $^{40}$Ar$_{g.s.}$. Multistep transfer reactions are expected to be strongly suppressed in the population of the mismatched ($L = 0, Q = -2.9$ MeV) $^{40}$Ar$_{g.s.}$. Indeed, the cross section around zero-degree is ≈ 3 µb/sr for $^{40}$Ca ($^{18}$O, $^{20}$Ne) $^{38}$Ar$_{g.s.}$ two-proton transfer [200].

Regarding the $^{130}$Te ($^{20}$Ne, $^{20}$O) $^{130}$Xe DCE reaction, as it can be seen in Fig. 32b, only a few events were detected for the $^{130}$Te$_{g.s.}$ → $^{130}$Xe$_{g.s.}$ transition, leading to an integrated cross section of 13 nb in the angular range $0° < \theta_{lab} < 9.5°$ and in the energy range -1 MeV < $E_x$ < 1 MeV. A contribution due to the first excited state of $^{130}$Xe 2+ at 0.536 MeV may be included in the given cross section. The high density of the $^{130}$Xe excited states and the finite experimental energy resolution lead to a rather structureless shape of the excitation energy spectrum above the g.s. region.

These results give a significant input to the next phase of the NUMEN project in which a deeper investigation of the system is planned with improved statistics by using high-intensity beams.

With the upgraded facility, it is planned to start the measurements of DCE reactions in both $\beta^+\beta^+$ and $\beta^-\beta^-$ directions on the partner nuclei: $^{76}$Ge ⇐⇒ $^{76}$Se and $^{116}$Cd ⇐⇒ $^{116}$Sn. These systems were already explored with the low beam intensity at 15 AMeV incident energy during the recent NUMEN experimental activity. The study of such systems at different incident beam energies (from 5 to 60 AMeV) are foreseen at the beginning of time schedule in order to investigate the energy dependence of DCE cross sections. As one of the first systems, the exploration the $^{100}$Mo target nucleus, which is candidate for the $0\nu\beta\beta$ decay, is also planned, because of the great interest of the scientific community to this isotope nucleus.

### 4.1.6 Double $\gamma$ decay

Other measurements that may represent an important tool to acquire information on the $0\nu\beta\beta$ NMEs are related to the $\gamma\gamma$ decay. The two-photon decay process is a second-order process in quantum electrodynamics. In the nuclear case, this means that an excited nuclear state simultaneously emits two $\gamma$-ray energy-quanta with a continuous $\gamma$-energy spectrum, with the sum energy of the two $\gamma$ rays equaling to the energy difference between the initial and the final state. Until very recently, the double $\gamma$ decay mode was only observed in $0^+ \rightarrow 0^+$ transitions in nuclei which have ground and first excited $0^+$ states so that the decay of the first excited state by a single photon is strictly forbidden [208]. In 2015, Walz et al. published a Nature letter [209] where, for the first time, they observed the competing double-$\gamma$ nuclear decay ($\gamma\gamma/\gamma$-decay) from a $11/2^-$ isomeric state in $^{137}$Ba to its $3/2^+$ ground state following the $\beta$ decay of $^{137}$Cs. The branching ratio of this process was measured to be $(2.05 \pm 0.37) \times 10^{-6}$. In that experiment, the authors were able also to determine the important multiplicities involved in the $\gamma\gamma/\gamma$ decay. The $\gamma\gamma/\gamma$-decay from the $11/2^-$ isomeric state in $^{137}$Ba has been confirmed in a recent experiment performed at Extreme Light Infrastructure-Nuclear Physics facilities [210]. However, in contrast to the conclusions of [209], the new experiment points to the dominant role of the octupole-dipole term in comparison to the quadrupole-quadrupole term.

The decay rate of the competitive $\gamma\gamma/\gamma$-decay, which is more than five orders of magnitude smaller than that of the allowed $\gamma$ decay, makes its observation very challenging. However, these measurements may represent an important tool to acquire information
<figure><img src="image_31.png" /><figcaption>**Fig. 32** a Excitation energy spectrum from $^{40}$Ca ($^{18}$O, $^{18}$Ne) $^{40}$Ar DCE. The symbols g.s.△ and 1.46△ indicate the $^{40}$Ca ($^{18}$O, $^{18}$Ne$_{1.87}$MeV) $^{40}$Ar$_{g.s.}$ and $^{40}$Ca ($^{18}$O, $^{18}$Ne$_{1.87}$MeV) $^{40}$Ar$_{1.47}$MeV transitions, respectively. Figure from Ref. [4]. **b** Excitation energy spectrum for the $^{130}$Te ($^{20}$Ne, $^{20}$O) $^{130}$Xe DCE reaction. Inset: zoomed view for $E_x$ < 10 MeV. Figure from Ref. [206]</figcaption></figure>

was produced with an energy of 306 MeV and impinged on a thin $^{130}$Te target. Adopting the identification technique described in [207] the ions of interest were selected. A 10$^{th}$-order trajectory reconstruction software was used to determine the excitation energy spectra, shown in Fig. 32.

In the DCE energy spectrum of Fig. 32a, the $^{40}$Ar$_{g.s.}$ is clearly separated from the not resolved doublet of states $^{40}$Ar 2+ at 1.460 MeV and $^{18}$Ne 2+ at 1.887 MeV. At higher excitation energy the measured yield is spread over many overlapping states. The cross section around zero-degree is ≈ 11 µb/sr for the $^{40}$Ca ($^{18}$O, $^{18}$Ne) $^{40}$Ar$_{g.s.}$. Multistep transfer reactions are expected to be strongly suppressed in the population of the mismatched ($L = 0, Q = -2.9$ MeV) $^{40}$Ar$_{g.s.}$. Indeed, the cross section around zero-degree is ≈ 3 µb/sr for $^{40}$Ca ($^{18}$O, $^{20}$Ne) $^{38}$Ar$_{g.s.}$ two-proton transfer [200].

Regarding the $^{130}$Te ($^{20}$Ne, $^{20}$O) $^{130}$Xe DCE reaction, as it can be seen in Fig. 32b, only a few events were detected for the $^{130}$Te$_{g.s.}$ → $^{130}$Xe$_{g.s.}$ transition, leading to an integrated cross section of 13 nb in the angular range $0° < \theta_{lab} < 9.5°$ and in the energy range -1 MeV < $E_x$ < 1 MeV. A contribution due to the first excited state of $^{130}$Xe 2+ at 0.536 MeV may be included in the given cross section. The high density of the $^{130}$Xe excited states and the finite experimental energy resolution lead to a rather structureless shape of the excitation energy spectrum above the g.s. region.

These results give a significant input to the next phase of the NUMEN project in which a deeper investigation of the system is planned with improved statistics by using high-intensity beams.

With the upgraded facility, it is planned to start the measurements of DCE reactions in both $\beta^+\beta^+$ and $\beta^-\beta^-$ directions on the partner nuclei: $^{76}$Ge ⇐⇒ $^{76}$Se and $^{116}$Cd ⇐⇒ $^{116}$Sn. These systems were already explored with the low beam intensity at 15 AMeV incident energy during the recent NUMEN experimental activity. The study of such systems at different incident beam energies (from 5 to 60 AMeV) are foreseen at the beginning of time schedule in order to investigate the energy dependence of DCE cross sections. As one of the first systems, the exploration the $^{100}$Mo target nucleus, which is candidate for the $0\nu\beta\beta$ decay, is also planned, because of the great interest of the scientific community to this isotope nucleus.

### 4.1.6 Double $\gamma$ decay

Other measurements that may represent an important tool to acquire information on the $0\nu\beta\beta$ NMEs are related to the $\gamma\gamma$ decay. The two-photon decay process is a second-order process in quantum electrodynamics. In the nuclear case, this means that an excited nuclear state simultaneously emits two $\gamma$-ray energy-quanta with a continuous $\gamma$-energy spectrum, with the sum energy of the two $\gamma$ rays equaling to the energy difference between the initial and the final state. Until very recently, the double $\gamma$ decay mode was only observed in $0^+ \rightarrow 0^+$ transitions in nuclei which have ground and first excited $0^+$ states so that the decay of the first excited state by a single photon is strictly forbidden [208]. In 2015, Walz et al. published a Nature letter [209] where, for the first time, they observed the competing double-$\gamma$ nuclear decay ($\gamma\gamma/\gamma$-decay) from a $11/2^-$ isomeric state in $^{137}$Ba to its $3/2^+$ ground state following the $\beta$ decay of $^{137}$Cs. The branching ratio of this process was measured to be $(2.05 \pm 0.37) \times 10^{-6}$. In that experiment, the authors were able also to determine the important multiplicities involved in the $\gamma\gamma/\gamma$ decay. The $\gamma\gamma/\gamma$-decay from the $11/2^-$ isomeric state in $^{137}$Ba has been confirmed in a recent experiment performed at Extreme Light Infrastructure-Nuclear Physics facilities [210]. However, in contrast to the conclusions of [209], the new experiment points to the dominant role of the octupole-dipole term in comparison to the quadrupole-quadrupole term.

The decay rate of the competitive $\gamma\gamma/\gamma$-decay, which is more than five orders of magnitude smaller than that of the allowed $\gamma$ decay, makes its observation very challenging. However, these measurements may represent an important tool to acquire information
on the $0\nu\beta\beta$ NMEs. In fact, as mentioned in the Introduction, a good linear correlation between the $\gamma\gamma$ and $0\nu\beta\beta$ NMEs has been demonstrated in [184] by using large-scale shell model calculations. This work reveals that the correlation holds for $\gamma\gamma$ transitions driven by the spin or orbital angular momentum due to the dominance of zero-coupled nucleon pairs, a feature common to $0\nu\beta\beta$ decay. Experimental studies of $\gamma\gamma$ decay for some selected DIAS relevant for the $0\nu\beta\beta$ decay are planned.

### 4.1.7 Nuclear structure input: comparison between different theoretical approaches

The three models mostly employed to describe spectroscopic properties and provide the needed input for reaction studies are: the shell model, the quasi-particle random-phase approximation and the interacting boson model. We shall briefly describe the main features of these models and discuss their present state-of-the-art as well as their future perspectives concerning the calculation of matrix elements involved in $\gamma\gamma$ decay and CE transitions induced by weak and strong interactions. In order to develop a comprehensive understanding of these phenomena and gain more insight on possible correlations between the corresponding NMEs, it is of great relevance to assess to which extent the results are sensitive to the structure inputs provided by the different approaches.

#### The shell model

The shell model (SM) is considered as the basic scheme for a microscopic description of the nucleus. Within the SM, the complexity of the many-body system is simplified by considering the nucleus as a closed-shell core with additional valence nucleons that interact in a subspace of the Hilbert space (model space) through an effective Hamiltonian, $H_{\text{eff}}$, accounting for contributions from configurations external to the model space. The SM $H_{\text{eff}}$ can be constructed by following a phenomenological approach, i.e., by adjusting its matrix elements to reproduce a set of experimental data, or can be derived microscopically by means of many-body perturbation theory starting from realistic free nuclear potentials. Non-perturbative methods are recently become available. The use of effective operators for the calculation of observables is also required to account for the missing correlations in the wave functions.

During seventy years of SM calculations, phenomenological interactions have shown to be very successful in reproducing a huge amount of nuclear structure properties. The microscopic approach to both the Hamiltonian and transition/decay operators, we called realistic SM, has largely progressed in the two decades or so, and it is by now also well set [211].

A number of papers have been devoted to the $0\nu\beta\beta$ NME calculation within the SM framework. However, several problems are still open and further developments are certainly required in the next years [212]. We have learned that (i) SM calculations adopting equally reasonable $H_{\text{eff}}$s lead to a rather narrow spread among the predictions of the NMEs ranging between 10 and 30%; (ii) the closure approximation usually employed in the calculation of the NMEs is good within 10% [213]; and (iii) the results are slightly sensitive to the variations of correlations induced by the strong short-range two-nucleon interaction. Single-$\beta$ strengths and $2\nu\beta\beta$ matrix elements predicted by realistic SM calculations are in good agreement with experiment without resorting to quenching factors of the free value of the axial vector coupling $g_A$. The quenching is, in fact, provided by the renormalization procedure employed in the derivation of the effective operators. In this connection, it is worth mentioning that renormalization effects, which are found to be relevant in the $2\nu\beta\beta$ decay, play a minor role in the calculated $0\nu\beta\beta$ NME [214].

Further investigations will be focused on: (i) the extension of the SM model space with the aim to include spin-orbit partners which may be particularly relevant in $\beta$ decay processes; (ii) the contribution of the electroweak two-body currents, derived within the framework of the chiral perturbation theory, on the $0\nu\beta\beta$ operator. Finally, it would be desirable to overcome the limits of the closure approximation in the derivation of the $0\nu\beta\beta$ operator within many-body perturbative theory.

In order to gain a more comprehensive understanding of spin-isospin phenomenology, including possible mutual relationships between the transition matrix elements involved in the $\beta$ decays and CE reactions, one needs a consistent description of the nuclear structure input required in both processes. Some work along this line has been performed in [182]. However, a more extensive investigation is needed using state-of-the-art SM calculations, already employed in $\beta$ decay studies.

Within the NUMEN project, large efforts have been devoted so far, as discussed in Sect. 4.1.2, to the analyses of the nucleon transfer channels. As next step, SM calculations should be employed to provide target and projectile form factors of the SCE reactions which contain the transition matrix elements induced by one-body operators, and then focus will be placed on the same quantities needed in DCE reaction calculations. Studies within the same framework are planned for the $\gamma\gamma$ decay of some selected DIAS.

#### The quasi-particle random-phase approximation

The random-phase approximation (RPA) and the quasi-particle RPA (QRPA) are well-known [215] and largely applied microscopic methods to describe nuclear excitations [216, 217], both in the charge conserving and charge exchanging case. Their main advantage, in comparison to other nuclear structure models, is that large single-particle spaces can be used, allowing thus for a full description of the total excitation strength and the corresponding energy weighted sum rules. In the context of CE excitations of relevance in the present contribution, these approximations have been applied in a number of works (see [215] and references therein). Energy density functionals, specially devised for the description of spin-isospin resonances at the RPA/QRPA level are available in the literature [218, 219].

In the RPA and QRPA, excited states are approximated by a linear superposition of one particle-one hole and two-quasi-particle excitations, respectively, neglecting thus high order configurations of the many particle—many hole type. This limit of the RPA and QRPA has been partially overcome by new extensions, such as the particle vibration coupling (PVC) [220, 221], the second RPA (SRPA) [222], the quasi-particle time blocking approximation [223] or the quasi-particle phonon model [224]. These extensions
on the $0\nu\beta\beta$ NMEs. In fact, as mentioned in the Introduction, a good linear correlation between the $\gamma\gamma$ and $0\nu\beta\beta$ NMEs has been demonstrated in [184] by using large-scale shell model calculations. This work reveals that the correlation holds for $\gamma\gamma$ transitions driven by the spin or orbital angular momentum due to the dominance of zero-coupled nucleon pairs, a feature common to $0\nu\beta\beta$ decay. Experimental studies of $\gamma\gamma$ decay for some selected DIAS relevant for the $0\nu\beta\beta$ decay are planned.

### 4.1.7 Nuclear structure input: comparison between different theoretical approaches

The three models mostly employed to describe spectroscopic properties and provide the needed input for reaction studies are: the shell model, the quasi-particle random-phase approximation and the interacting boson model. We shall briefly describe the main features of these models and discuss their present state-of-the-art as well as their future perspectives concerning the calculation of matrix elements involved in $\gamma\gamma$ decay and CE transitions induced by weak and strong interactions. In order to develop a comprehensive understanding of these phenomena and gain more insight on possible correlations between the corresponding NMEs, it is of great relevance to assess to which extent the results are sensitive to the structure inputs provided by the different approaches.

#### The shell model

The shell model (SM) is considered as the basic scheme for a microscopic description of the nucleus. Within the SM, the complexity of the many-body system is simplified by considering the nucleus as a closed-shell core with additional valence nucleons that interact in a subspace of the Hilbert space (model space) through an effective Hamiltonian, $H_{\text{eff}}$, accounting for contributions from configurations external to the model space. The SM $H_{\text{eff}}$ can be constructed by following a phenomenological approach, i.e., by adjusting its matrix elements to reproduce a set of experimental data, or can be derived microscopically by means of many-body perturbation theory starting from realistic free nuclear potentials. Non-perturbative methods are recently become available. The use of effective operators for the calculation of observables is also required to account for the missing correlations in the wave functions.

During seventy years of SM calculations, phenomenological interactions have shown to be very successful in reproducing a huge amount of nuclear structure properties. The microscopic approach to both the Hamiltonian and transition/decay operators, we called realistic SM, has largely progressed in the two decades or so, and it is by now also well set [211].

A number of papers have been devoted to the $0\nu\beta\beta$ NME calculation within the SM framework. However, several problems are still open and further developments are certainly required in the next years [212]. We have learned that (i) SM calculations adopting equally reasonable $H_{\text{eff}}$s lead to a rather narrow spread among the predictions of the NMEs ranging between 10 and 30%; (ii) the closure approximation usually employed in the calculation of the NMEs is good within 10% [213]; and (iii) the results are slightly sensitive to the variations of correlations induced by the strong short-range two-nucleon interaction. Single-$\beta$ strengths and $2\nu\beta\beta$ matrix elements predicted by realistic SM calculations are in good agreement with experiment without resorting to quenching factors of the free value of the axial vector coupling $g_A$. The quenching is, in fact, provided by the renormalization procedure employed in the derivation of the effective operators. In this connection, it is worth mentioning that renormalization effects, which are found to be relevant in the $2\nu\beta\beta$ decay, play a minor role in the calculated $0\nu\beta\beta$ NME [214].

Further investigations will be focused on: (i) the extension of the SM model space with the aim to include spin-orbit partners which may be particularly relevant in $\beta$ decay processes; (ii) the contribution of the electroweak two-body currents, derived within the framework of the chiral perturbation theory, on the $0\nu\beta\beta$ operator. Finally, it would be desirable to overcome the limits of the closure approximation in the derivation of the $0\nu\beta\beta$ operator within many-body perturbative theory.

In order to gain a more comprehensive understanding of spin-isospin phenomenology, including possible mutual relationships between the transition matrix elements involved in the $\beta$ decays and CE reactions, one needs a consistent description of the nuclear structure input required in both processes. Some work along this line has been performed in [182]. However, a more extensive investigation is needed using state-of-the-art SM calculations, already employed in $\beta$ decay studies.

Within the NUMEN project, large efforts have been devoted so far, as discussed in Sect. 4.1.2, to the analyses of the nucleon transfer channels. As next step, SM calculations should be employed to provide target and projectile form factors of the SCE reactions which contain the transition matrix elements induced by one-body operators, and then focus will be placed on the same quantities needed in DCE reaction calculations. Studies within the same framework are planned for the $\gamma\gamma$ decay of some selected DIAS.

#### The quasi-particle random-phase approximation

The random-phase approximation (RPA) and the quasi-particle RPA (QRPA) are well-known [215] and largely applied microscopic methods to describe nuclear excitations [216, 217], both in the charge conserving and charge exchanging case. Their main advantage, in comparison to other nuclear structure models, is that large single-particle spaces can be used, allowing thus for a full description of the total excitation strength and the corresponding energy weighted sum rules. In the context of CE excitations of relevance in the present contribution, these approximations have been applied in a number of works (see [215] and references therein). Energy density functionals, specially devised for the description of spin-isospin resonances at the RPA/QRPA level are available in the literature [218, 219].

In the RPA and QRPA, excited states are approximated by a linear superposition of one particle-one hole and two-quasi-particle excitations, respectively, neglecting thus high order configurations of the many particle—many hole type. This limit of the RPA and QRPA has been partially overcome by new extensions, such as the particle vibration coupling (PVC) [220, 221], the second RPA (SRPA) [222], the quasi-particle time blocking approximation [223] or the quasi-particle phonon model [224]. These extensions
aim at describing the spreading widths of collective excitations and their fragmentation, due to the coupling with more complicated states, like 2p-2h or 4-quasi-particle configurations. In the PVC model, this is achieved in an effective way by taking into account the coupling of single-nucleon states to the collective low-lying nuclear vibrations described by the RPA/QRPA phonons. In the SRPA model, the phonon operators contain explicitly the 2p-2h configurations, besides the 1p-1h ones already introduced in RPA. Some of these extensions have been applied also in the context of CE excitations and single $\beta$ decay, improving significantly on the RPA/QRPA results [225–227].

The task to be accomplished in the near future are in the following. (1) The RPA/QRPA models based on the Skyrme interaction will be used to evaluate: (1.i) the $2\nu\beta\beta$ and $0\nu\beta\beta$ NMEs; (1.ii) the form factors needed for the DCE reaction calculations; and (1.iii) the off-diagonal polarizabilities to account for the $\gamma\gamma$ decay probability. (2) The same quantities described in (1) will be also investigated by using the PVC and SRPA models, in order to study the effect of higher order configurations.

### The interacting boson model

The interacting boson model (IBM) is an algebraic model for even–even nuclei that describes the collective excitations in the even–even nuclei in terms of a system of interacting bosons with angular momentum $L = 0$ (s-boson) and 2 (d-boson) [228], and in the proton–neutron version (IBM-2) [228, 229] includes a distinction between proton and neutron bosons. For odd–even nuclei, the interacting boson-fermion model (IBFM-2) [230, 231] is employed, where the bosons are coupled to the single-particle degrees of freedom of the odd nucleon, while odd–odd nuclei are described by coupling the single-particle degrees of freedom of the odd neutron and the odd proton to the bosons (IBFFM) [232–234]. All together these models are called the interacting boson models (IBMs).

The formalism to compute the spectroscopic amplitudes from an even–even to an odd–odd nucleus in the IBFFM scheme has been developed for the first time in [234]. In future, one- and two-particle spectroscopic amplitudes of all competing transfer channels will be calculated within the interacting boson schemes, as already done in previous articles within the NUMEN collaboration [197, 235].

Then, calculations of the DCE nuclear matrix elements in closure approximation, useful to put constraints on the $0\nu\beta\beta$ NMEs [183], will be completed [183]. Finally, developments for other nuclei of interest for the NUMEN project that can be described by means of the IBMs are planned in order to calculate the radial transition densities entering the form factors for the SCE reaction code.

## 4.2 Collectivity in nuclei

One of the main expressions of collectivity in nuclei are the nuclear GR that were discovered more than 75 years ago [236]. GRs were described as modes of a system having the characteristics of a two component fluid (protons and neutrons) [237, 238] that can move either in phase (isoscalar modes) or out of phase (isovector modes). In particular, GDRs were investigated at LNS with the MEDEA and TRASMA arrays [239, 240]. The experimental work was strongly supported by an excellent theoretical activity [241, 242]. Further investigations at LNS and LNL have been devoted to other pseudo-collective phenomena like the so-called dynamical dipole mode [243–246] or the PDR (Pygmy dipole resonance) [247]. The research activity in these fields will be pursued in the next future, taking the opportunities given by the LNS upgrades, e.g., using the intense radioactive beams produced with FraISe (see Sect. 2.2.2) [17, 18].

By following the order from the experiments ready for the FraISe day one to those that need further tests and resources, in Sect. 4.2.1 we will shortly describe measurements investigating the excitation and decay of the PDR. In Sect. 4.2.2 we will present new experiments planned on the dynamical dipole mode. In Sect. 4.2.3 we will describe the study on the ISGMR on exotic nuclei, to be performed with MAGNEX spectrometer. Finally in Sect. 4.2.4 we will present the concept of a new detection system replacing the dismissed MEDEA array, coupled to some telescope array of the FAZIA type, for more accurate GDR investigations.

### 4.2.1 Search for Pygmy resonances

The study of the low-lying E1 strength below and above the neutron emission threshold, known as PDR, is of paramount interest because it has an important link with both r-process and the EOS parameters, as discussed in Sect. 3.1. Some of its features, as the degree of collectivity, are still under debate. Investigations on the possible single-particle nature of PDR are presently carried out at INFN-LNS. The low-lying E1 transitions in $^{96}$Mo were populated with (d,p) and (p,d) reactions. The outgoing particles are detected by the MAGNEX spectrometer. The measured angular distributions are compared with DWBA calculations to extract the corresponding spectroscopic factors [248]. Experimental investigations and theoretical predictions show that the PDR is connected to the neutron excess in nuclei and its strength is larger in nuclei far from the stability valley. One of the most important feature of the PDR is the isospin mixing effect [249, 250]. Due to this, as shown in a large number of stable nuclei and in few unstable ones, the PDR excitation can be achieved with both isoscalar and isovector probes [249–251]. In stable nuclei and at excitation energies below the neutron separation threshold, the PDR states are separated in two groups: one group, sitting at low energy, refers to states excited by both probes, while the higher energy group is populated only by isovector probes. This effect is known as PDR (or isospin) splitting. The excitation of the PDR at energies above the neutron emission threshold has been observed, until now, only
aim at describing the spreading widths of collective excitations and their fragmentation, due to the coupling with more complicated states, like 2p-2h or 4-quasi-particle configurations. In the PVC model, this is achieved in an effective way by taking into account the coupling of single-nucleon states to the collective low-lying nuclear vibrations described by the RPA/QRPA phonons. In the SRPA model, the phonon operators contain explicitly the 2p-2h configurations, besides the 1p-1h ones already introduced in RPA. Some of these extensions have been applied also in the context of CE excitations and single $\beta$ decay, improving significantly on the RPA/QRPA results [225–227].

The task to be accomplished in the near future are in the following. (1) The RPA/QRPA models based on the Skyrme interaction will be used to evaluate: (1.i) the $2\nu\beta\beta$ and $0\nu\beta\beta$ NMEs; (1.ii) the form factors needed for the DCE reaction calculations; and (1.iii) the off-diagonal polarizabilities to account for the $\gamma\gamma$ decay probability. (2) The same quantities described in (1) will be also investigated by using the PVC and SRPA models, in order to study the effect of higher order configurations.

### The interacting boson model

The interacting boson model (IBM) is an algebraic model for even–even nuclei that describes the collective excitations in the even–even nuclei in terms of a system of interacting bosons with angular momentum $L = 0$ (s-boson) and 2 (d-boson) [228], and in the proton–neutron version (IBM-2) [228, 229] includes a distinction between proton and neutron bosons. For odd–even nuclei, the interacting boson-fermion model (IBFM-2) [230, 231] is employed, where the bosons are coupled to the single-particle degrees of freedom of the odd nucleon, while odd–odd nuclei are described by coupling the single-particle degrees of freedom of the odd neutron and the odd proton to the bosons (IBFFM) [232–234]. All together these models are called the interacting boson models (IBMs).

The formalism to compute the spectroscopic amplitudes from an even–even to an odd–odd nucleus in the IBFFM scheme has been developed for the first time in [234]. In future, one- and two-particle spectroscopic amplitudes of all competing transfer channels will be calculated within the interacting boson schemes, as already done in previous articles within the NUMEN collaboration [197, 235].

Then, calculations of the DCE nuclear matrix elements in closure approximation, useful to put constraints on the $0\nu\beta\beta$ NMEs [183], will be completed [183]. Finally, developments for other nuclei of interest for the NUMEN project that can be described by means of the IBMs are planned in order to calculate the radial transition densities entering the form factors for the SCE reaction code.

## 4.2 Collectivity in nuclei

One of the main expressions of collectivity in nuclei are the nuclear GR that were discovered more than 75 years ago [236]. GRs were described as modes of a system having the characteristics of a two component fluid (protons and neutrons) [237, 238] that can move either in phase (isoscalar modes) or out of phase (isovector modes). In particular, GDRs were investigated at LNS with the MEDEA and TRASMA arrays [239, 240]. The experimental work was strongly supported by an excellent theoretical activity [241, 242]. Further investigations at LNS and LNL have been devoted to other pseudo-collective phenomena like the so-called dynamical dipole mode [243–246] or the PDR (Pygmy dipole resonance) [247]. The research activity in these fields will be pursued in the next future, taking the opportunities given by the LNS upgrades, e.g., using the intense radioactive beams produced with FraISe (see Sect. 2.2.2) [17, 18].

By following the order from the experiments ready for the FraISe day one to those that need further tests and resources, in Sect. 4.2.1 we will shortly describe measurements investigating the excitation and decay of the PDR. In Sect. 4.2.2 we will present new experiments planned on the dynamical dipole mode. In Sect. 4.2.3 we will describe the study on the ISGMR on exotic nuclei, to be performed with MAGNEX spectrometer. Finally in Sect. 4.2.4 we will present the concept of a new detection system replacing the dismissed MEDEA array, coupled to some telescope array of the FAZIA type, for more accurate GDR investigations.

### 4.2.1 Search for Pygmy resonances

The study of the low-lying E1 strength below and above the neutron emission threshold, known as PDR, is of paramount interest because it has an important link with both r-process and the EOS parameters, as discussed in Sect. 3.1. Some of its features, as the degree of collectivity, are still under debate. Investigations on the possible single-particle nature of PDR are presently carried out at INFN-LNS. The low-lying E1 transitions in $^{96}$Mo were populated with (d,p) and (p,d) reactions. The outgoing particles are detected by the MAGNEX spectrometer. The measured angular distributions are compared with DWBA calculations to extract the corresponding spectroscopic factors [248]. Experimental investigations and theoretical predictions show that the PDR is connected to the neutron excess in nuclei and its strength is larger in nuclei far from the stability valley. One of the most important feature of the PDR is the isospin mixing effect [249, 250]. Due to this, as shown in a large number of stable nuclei and in few unstable ones, the PDR excitation can be achieved with both isoscalar and isovector probes [249–251]. In stable nuclei and at excitation energies below the neutron separation threshold, the PDR states are separated in two groups: one group, sitting at low energy, refers to states excited by both probes, while the higher energy group is populated only by isovector probes. This effect is known as PDR (or isospin) splitting. The excitation of the PDR at energies above the neutron emission threshold has been observed, until now, only
**Fig. 33 a** The γ-ray decay of pygmy resonance excited in $^{68}$Ni with a isoscalar excitation measured at GSI, adapted from ref. [252], and b isoscalar excitation measured at INFN-LNS adapted from Ref. [247]. The similarity of the two spectral shapes suggest that no isospin splitting emerges

<figure><img src="image_32.png" /></figure>

in unstable nuclei [247, 252, 253]. At INFN-LNS the γ-decay of the PDR in the $^{68}$Ni above the neutron emission threshold was measured using a $^{12}$C target as isoscalar probe [247]. The $^{68}$Ni beam was delivered via the FRIBs@LNS facility [16]. The PDR was excited, via inelastic scattering, mostly with nuclear interaction due to the low Coulomb field of the carbon target. The γ-decay channel of the PDR was studied using the CHIMERA multidetector [2, 254], while the FARCOS array provided the full isotopic identification of $^{68}$Ni and other reaction products [24]. The shape of the observed resonance, Fig. 33b, is compared with previous relativistic Coulomb excitation measurements [252, 253], Fig. 33a. Even if the strength is different, the similar shape indicates that the isospin splitting does not seem to be present at the energy above the neutron emission threshold.

However, due to statistics limitations and by the modest energy resolution, it is not possible to draw definite conclusions on the absence of isospin splitting.

Aiming at a better understanding of the nature of the PDR, with the help of the new facility FraISe [18] we plan to continue the study of the PDR in $^{68}$Ni, using both isoscalar and isoscalar probes. Moreover we want to extend the study to other systems, in the region of light and medium nuclei such as $^{20}$O, $^{34}$Si, $^{38}$S, $^{48}$Ar. The aim is also to measure the neutron shell occupancy dependence of the PDR both below and above the particle emission threshold. A nucleus of particular interest could be the $^{38}$S, one of the first nuclei that will be produced by FraISe. Moreover, it will be important also to search for the PDR above the particle emission threshold in stable nuclei, such as $^{48}$Ca and $^{64}$Ni. The PDR will be excited in Coulex experiments using Pb/Au targets in order to have a strong Coulomb field. A carbon target will be used as an almost pure isoscalar probe due to the low intensity of the Coulomb field. It may be considered the use of a proton target ($CH_2$) as a further messenger to study the interplay between isoscalar and isoscalar excitation.

We plan to carry on experiments with a more powerful and efficient setup. Indeed, we will use the CHIMERA multidetector and FARCOS array in its complete configuration. The use of a new prototype of neutron detector (see Sect. 3.1.6) is also envisaged in order to fully characterize the PDR.

### 4.2.2 Dynamical dipole

Dynamical dipole pre-equilibrium γ-ray emission was observed in fusion and binary reactions around 10–20 MeV/nucleon for different systems in various experiments [243, 245, 246, 255]. It was observed in heavy-ion collisions characterized by strong initial dipole moment connected to the charge/mass asymmetry of the reaction partners. In the framework of a semi-classical approximation, this pre-equilibrium emission is a γ-ray E1 emission. The strength of such γ-emission, as a function of its energy, is determined by the second-order derivative of the total electrical dipole moment (averaged over the selected events) through its Fourier transform. Semi-classical calculations, based on transport models (BNV) [244] and on the CoMD [255], show that the fast variation of the dipole moment, and the consequent γ-ray emission, takes place during the early stage of the interaction below 120fm/c, when the largest changes of the density are produced, as shown in the example of Fig. 34b. Interestingly, according to model predictions (due to conservation rules) statistical decay after the collision does not affect the values of $\bar{D}$, i.e., the time derivative of the electrical dipole (see also Appendix of Ref. [243]). Therefore, the measured variable, if well reconstructed, should retain the asymptotic D value (see Fig. 34a, dashed line).
**Fig. 33 a** The γ-ray decay of pygmy resonance excited in $^{68}$Ni with a isovector excitation measured at GSI, adapted from ref. [252], and b isoscalar excitation measured at INFN-LNS adapted from Ref. [247]. The similarity of the two spectral shapes suggest that no isospin splitting emerges

<figure><img src="image_32.png" /></figure>

in unstable nuclei [247, 252, 253]. At INFN-LNS the γ-decay of the PDR in the $^{68}$Ni above the neutron emission threshold was measured using a $^{12}$C target as isoscalar probe [247]. The $^{68}$Ni beam was delivered via the FRIBs@LNS facility [16]. The PDR was excited, via inelastic scattering, mostly with nuclear interaction due to the low Coulomb field of the carbon target. The γ-decay channel of the PDR was studied using the CHIMERA multidetector [2, 254], while the FARCOS array provided the full isotopic identification of $^{68}$Ni and other reaction products [24]. The shape of the observed resonance, Fig. 33b, is compared with previous relativistic Coulomb excitation measurements [252, 253], Fig. 33a. Even if the strength is different, the similar shape indicates that the isospin splitting does not seem to be present at the energy above the neutron emission threshold.

However, due to statistics limitations and by the modest energy resolution, it is not possible to draw definite conclusions on the absence of isospin splitting.

Aiming at a better understanding of the nature of the PDR, with the help of the new facility FraISe [18] we plan to continue the study of the PDR in $^{68}$Ni, using both isoscalar and isovector probes. Moreover we want to extend the study to other systems, in the region of light and medium nuclei such as $^{20}$O, $^{34}$Si, $^{38}$S, $^{48}$Ar. The aim is also to measure the neutron shell occupancy dependence of the PDR both below and above the particle emission threshold. A nucleus of particular interest could be the $^{38}$S, one of the first nuclei that will be produced by FraISe. Moreover, it will be important also to search for the PDR above the particle emission threshold in stable nuclei, such as $^{48}$Ca and $^{64}$Ni. The PDR will be excited in Coulex experiments using Pb/Au targets in order to have a strong Coulomb field. A carbon target will be used as an almost pure isoscalar probe due to the low intensity of the Coulomb field. It may be considered the use of a proton target (CH$_2$) as a further messenger to study the interplay between isoscalar and isovector excitation.

We plan to carry on experiments with a more powerful and efficient setup. Indeed, we will use the CHIMERA multidetector and FARCOS array in its complete configuration. The use of a new prototype of neutron detector (see Sect. 3.1.6) is also envisaged in order to fully characterize the PDR.

### 4.2.2 Dynamical dipole

Dynamical dipole pre-equilibrium γ-ray emission was observed in fusion and binary reactions around 10–20 MeV/nucleon for different systems in various experiments [243, 245, 246, 255]. It was observed in heavy-ion collisions characterized by strong initial dipole moment connected to the charge/mass asymmetry of the reaction partners. In the framework of a semi-classical approximation, this pre-equilibrium emission is a γ-ray E1 emission. The strength of such γ-emission, as a function of its energy, is determined by the second-order derivative of the total electrical dipole moment (averaged over the selected events) through its Fourier transform. Semi-classical calculations, based on transport models (BNV) [244] and on the CoMD [255], show that the fast variation of the dipole moment, and the consequent γ-ray emission, takes place during the early stage of the interaction below 120fm/c, when the largest changes of the density are produced, as shown in the example of Fig. 34b. Interestingly, according to model predictions (due to conservation rules) statistical decay after the collision does not affect the values of $\bar{D}$, i.e., the time derivative of the electrical dipole (see also Appendix of Ref. [243]). Therefore, the measured variable, if well reconstructed, should retain the asymptotic D value (see Fig. 34a, dashed line).