Signatures of the Giant Pairing Vibration in the 14C and 15C atomic nuclei

Giant resonances are collective excitation modes for many-body systems of fermions governed by a mean field, such as the atomic nuclei. The microscopic origin of such modes is the coherence among elementary particle-hole excitations, where a particle is promoted from an occupied state below the Fermi level (hole) to an empty one above the Fermi level (particle). The same coherence is also predicted for the particle–particle and the hole–hole excitations, because of the basic quantum symmetry between particles and holes. In nuclear physics, the giant modes have been widely reported for the particle–hole sector but, despite several attempts, there is no precedent in the particle–particle and hole–hole ones, thus making questionable the aforementioned symmetry assumption. Here we provide experimental indications of the Giant Pairing Vibration, which is the leading particle–particle giant mode. An immediate implication of it is the validation of the particle–hole symmetry.


Supplementary Methods
Models for the background underneath the GPVs. The angular distributions of the absolute cross-section were deduced for the most intense transitions, including those populating the wide resonances observed above the two-neutron emission threshold. The differential solid angle for the full spectrometer acceptance was carefully determined taking into account the overall transport efficiency, as described in ref. 1 . A dead-time coefficient of ~30% was measured. An angular bin of 1° in the laboratory reference frame was chosen for the GPV angular distributions in order to achieve a good compromise between the statistical uncertainties in the number of counts, the background subtraction and the angular resolution.
The contribution to the angular distributions due to the continuous background in the spectra was estimated at each angle by a least-squared approach with a Gaussian model shape superimposed on a linear background as shown in Figure 1 of the main text. The adopted background model is consistent with two-neutron break-up calculations, performed considering an independent removal of the two neutrons, as described the Methods section. In order to carefully look at the projectile break-up contribution to the angular distribution, a comparison between the GPV angular distribution obtained without any background subtraction, the subtracted background and the final GPV distributions is shown in Supplementary Figure  We used different models for the background subtraction and the obtained results for the centroid and width of the resonances and also the shape of the angular distributions did not change within the quoted uncertainties. As an example, a second model for the background subtraction is show in Supplementary Figure 4  The a parameter is different from zero beyond 4 standard deviations confidence level, thus the contribution of the oscillating Bessel function is necessary in the model for describing the 14 C GPV angular distribution. This analysis shows that the oscillating shape of the 14 C GPV angular distribution is confidently established. Cross section calculations within the discretized continuum scheme. In order to describe the cross section angular distribution of the 14 C resonance at 16.9 ± 0.1 MeV the discretized continuum scheme of ref. 2 was used. The extreme cluster model approximation for the two-neutron pair was adopted. Within such approximation, the two neutrons are paired anti-parallel and coupled to a zero intrinsic angular momentum (S = 0).
For the transfer to open states the cluster model corresponds to a 3-body calculation (projectile, target, neutron pair) which is an approximation of the actual 4-body problem (projectile, target, neutron, neutron). In the case of 6 He interaction with light and heavy targets it has been shown that the three-body model is a reasonable approximation 3,4 . However, the equivalence between the three-body and four-body methods has not been proven for systems with a larger two-neutron binding energy, as the 18 O projectile (S 2n ( 18 O) = 12.188 MeV, S 2n ( 6 He) = 0.971 MeV). For this reason fine details of the calculation should not be accounted for when such 4-body to 3-body projection is applied as in the present case.
A reasonable description of the full 3-body problem is that provided by the approach of ref. 2 , where the continuum bins are derived from the Jacobi coordinates of the relative motion between the valence particle (x) and the target (T). This is assumed to be a valid approach for the description of the GPV since it corresponds to a resonance of the two-neutron + target system and lives long enough compared to the crossing time of the projectile during the reaction.
Replacing in the scattering amplitude the exact wave function by the elastic wave function 5,6 and using the prior representation, it is obtained 7 : where are the asymptotic entrance channel wave functions and the final state is the exact three-body wave function with incoming boundary conditions, expanded in terms of the x + T continuum and bound (if any) states as ∑ (2) where represents the set of bin wave functions constructed as wave packets from pure scattering states 7,8,9 . This expansion goes beyond the Distorted Wave Born Approximation (DWBA) method since couplings between final states are also considered in the wave functions (2). In expression (1) V xT is the particle-target binding potential, U c is the optical potential that describes the elastic scattering of the core by the target and U pT is the projectile-target optical potential (cluster folding potential) 2 . It is important to note that the presence of the V xT potential allows the description of the x + T resonant excitations.
The parameter free double folding São Paulo real potential was used for the present calculations 10 . The imaginary part of the potential which fits the elastic scattering, when there is no relevant couplings to the elastic channel, is represented by 0.6 times its real part. Recently, it has been shown that this prescription is the most appropriate when all the important bound state couplings are explicitly taken into account and the dissipative reaction channels (like breakup channel, and/or deep inelastic channels, or any excitation of the continuum spectrum) are globally accounted for 11 . This value was assumed for the present U pT optical potential. For the V xT part, a Woods-Saxon potential was used with a reduced radius of 1.28 fm and diffuseness of 0.6 fm, the same as ref. 12 . The depth was varied in order to fit 2n binding energy as well as the position of the 16.9 ± 0.1 MeV resonance. In the coupling scheme the three 0 + bound states of the 14 C were included (E x = 6.589, 9.746, 16.9 MeV). The convergence of the calculation was obtained considering a maximum angular momentum of 100ħ and R max = 600 fm for the 16 O and 14 C relative motion and a maximum r max = 100 fm for the 2n -12 C bins integration. The bin width was set to 2 MeV to account for the experimental width value.
The resulting calculation, performed for L = 0, is superimposed to the experimental results in Supplementary Figure 7. The absolute value is consistent with the experimental one, without any scaling factor. The model space was also enlarged in order to check if the coupling to other continuum states might affect the final result for the resonant state. It was found that the inclusion of other continuum states and higher angular momenta did not considerably affect the results.