Proton-neutron pairing, α -like quarteting and ground/excited states of N=Z nuclei

. Various studies have shown that the proton-neutron (pn) pairing correlations can be accurately described not by a condensate of Cooper pairs, as considered in the majority of mean-ﬁeld calculations, but by a condensate of α -like quartets. After a short review of the quartet condensate model (QCM), it is discussed the e ﬀ ect of the pn pairing on the ground states of N = Z nuclei, analysed recently in the framework of Skyrme-HF + QCM calculations. An interesting aspect pointed out by these calculations is the strong interdependence between all types of pairing correlations. In particular, when the isoscalar pn pairing channel is switched on, the pairing correlations are redistributed among all the pairing channels without changing signiﬁcantly the total pairing energy. Due to this reason, for the majority of N = Z nuclei, the binding energy is not a ﬀ ected much when the isoscalar pairing channel is switched on. Yet, in all calculations which include both the isovector and the isoscalar pairing forces, the isoscalar pairing correlations contribute signiﬁcantly to the binding energies and coexist always with the isovector pn pairing. Finally it is presented a recent extension of the QCM approach to the excited stated of pn pairing Hamiltonians. It will be shown that the low-lying excited states of these Hamiltonians can be described by breaking a quartet from the ground state condensate and replacing it with an “excited” quartet, an approach which is analogous to the one-broken-pair approximation which is commonly used for like-particle systems.


Introduction
The role of proton-neutron (pn) pairing correlations in N = Z nuclei is an issue debated for many years [1].In the N = Z nuclei are usually considered two types of pn pairing correlations, corresponding to spin-singlet isovector (S=0,T=1) and spin-triplet isoscalar (S=1,T=0) pn pairs.The isovector pn pairing is supposed to have similar properties as the neutron-neutron (nn) and proton-proton (pp) pairing.What is the role played by the isoscalar pn pairing in nuclei is not yet clear.In particular, the isoscalar pairing is commonly associated to a long standing question in nuclear structure, i.e., whether a condensate of deuteronlike proton-neutron pairs could exist in N = Z nuclei.In spite of many experimental and theoretical studies, up to now there is not a compelling evidence for a deuteron-like condensation in nuclei.
The majority of theoretical studies on pn pairing have been done in the Hartree-Fock-Bogoliubov (HFB) approach (see [2] and the references quoated therein).The great advantage of the HFB model is the unitary treatment of both isovector and isoscalar pairing in a simple approach based on the generalised Bogoliubov transformation.This advantage comes however with important drawbacks : the non-conservation of the particle number, the isospin and the angular momentum.To restore the conservation of these quantities employing projection techniques is a difficult task and some results on this line exist only * e-mail: nsandulescu@theory.nipne.rofor the trivial case of degenerate levels [3,4].Realistic beyond-HFB calculations with particle number and angular momentum projections have been done recently, but with the projection performed after the variation [5].
Many years ago it was pointed out that the isovector pairing generates, through the isospin conservation, α-like quartet structures formed by two neutrons and two protons coupled to the total isospin T = 0 [6][7][8].In one of the first quartet formalisms, Flowers and Vujici supposed that the ground state of the isovector pairing Hamiltonian can be approximated by a BCS-type condensate expressed in term of quartets.This approach, which conserves exactly the isospin but not the particle number, is rather complicated and it was never applied in realistic calculations.An exact quartet solution was found later on for the particular case of pn pairing interactions acting on degenerate singleparticle levels [3] .It was thus shown that, in this case, the ground state of N = Z systems interacting by an isovector force or by isovector and isoscalar pairing forces of equal strength, can be expressed exactly by a quartet condensate.A realistic quartet condensation model (QCM) , for non-degenerate levels and general pairing forces, was proposed later on in Refs.[9][10][11].These QCM models will be revised below and then applied for estimating the contribution of pn pairing to the binding energies of N = Z nuclei [12].
Recently the QCM model was extended to describe the excited states of the pn Hamiltonians [13].It was shown that the low-lying excites states of these Hamiltonians can be described accurately by by breaking a pair from the quartet condensate and replacing it with an excited quartet.This formalism will be presented in the second part of the article.

QCM approach for axially-deformed pairing Hamiltonians
In this section we consider the case of isovector-isoscalar pairing forces which scatter pairs of nucleons in timereversed states.This is a typical situation for the mean field plus paiirng calculations done in the intrinsic system of axially-deformed nuclei.The QCM formalism corresponding to this case, proposed in Ref. [10], will be summarised below.
The Hamiltonian which describes the pairing correlations in the axially-deformed system has the expression [10], where ε i,τ are the single-particle energies of neutrons (τ=1/2) and protons (τ=-1/2), while N i,τ are the particle number operators.The second term is the isovector pairing interaction expressed by the isovector pair operators The third term is the isoscalar pairing interaction and √ 2 is the isoscalar pair operator.By ν † i and π † i are denoted the creation operators for neutrons and protons in the state i, while ī is the time conjugate of the state i.The states i, which correspond to the axially deformed mean-field, are characterised by the quantum numbers i ≡ {a i , Ω i }, where Ω i is the projection of the angular momentum on the symmetry axis.
By construction, in Eq. ( 1) the pairs operators have J z = 0 but not a well-defined angular momentum J.In fact, when expressed in the laboratory frame, the isovector and the isoscalar intrinsic pairs can be written as a superposition of pairs with J = 0, 2, 4, .. and, respectively, J = 1, 3, 5, ... Therefore, the Hamiltonian (1) takes into account, in an effective way, pairing correlations which are not restricted only to the standard (J=0,T=1) and (J=1,T=0) channels.
In order to find the ground state energy of the Hamiltonian (1), we employ the quartet condensation model (QCM).Thus, according to QCM, the ground state of Hamiltonian (1) for even-even N = Z systems is approximated by the trial state [10] where n q = (N + Z)/2, while |0 is the "vacuum" state represented by the nucleons which are supposed to be not affected by the pairing interaction.The operator A † is the isovector quartet built by two isovector non-collective pairs coupled to the total isospin T = 0, i.e., Assuming that the mixing coefficients are separable, i.e., x i j = x i x j , the isovector quartet takes the form where are collective pair operators for neutron-neutron pairs (t = 1), proton-proton pairs (t = −1) and proton-neutron pairs (t = 0).The isoscalar degrees of freedom are described by the collective isoscalar pair The QCM states depend on the mixing amplitudes of the collective pair operators.They are determined variationally by minimizing the average of the Hamiltonian under the normalization condition imposed to the trial state.Details about these calculations are presented in Ref. [10].

The effect of pn pairing on the ground states of N = Z nuclei
The QCM formalism presented above has been used recently to evaluate the contribution of the pn pairing to the ground state of nuclei close to the N = Z line [12].Below we present shortly the calculation scheme and the main results for the N = Z nuclei.
To calculate the binding energies, we use a selfconsistent mean-field plus pairing formalism.The calculations are done in the intrinsic system defined by an axially deformed mean-field generated by a Skyrme functional.The QCM calculations for the Hamiltonian (1) are performed iteratively with the Skyrme-HF calculations in a similar way as in the axially-deformed Skyrme+BCS calculations.Thus, at a given iteration, the QCM equations are solved for the single-particle states generated by the Skyrme functional.Then, the occupation probabilities of the single-particle states provided by QCM are employed to get new densities and a new Skyrme functional which, in turn, is generating new single-particle states.At the convergence, the binding energy is obtained by adding to the mean-field energy the contribution of the pairing energy.The latter is calculated as the average of the pairing force from which it is extracted out the contribution of self-energy terms.For the like-particle pairing these terms are while for the pn pairing the expresions are , 02024 (2023) EPJ Web of Conferences EuNPC 2022 https://doi.org/10.1051/epjconf/202329002024290 In the expressions above, v 2 i,n(p) are the occupation probabilities for neutrons (protons) corresponding to the states included in the pairing calculations.
In the present calculations, for the isovector-isoscalar pairing interaction we employ a zero range force of the form: where PT S ,S z is the projection operator on the spin of the pairs, namely, S = 0 for the isovector force and S = 1, S z = 0 for the isoscalar force.The matrix elements of the pairing interaction (11) for the single-particle states provided by the Skyrme functional are calculated as shown in the Appendix of Ref. [14].
The most representative results for the binding energies of N = Z nuclei are presented in Fig. 1.The figure shows the binding energies residuals, i.e., the difference between the theoretical and experimental binding energies.The parameters employed in the calculations are indicated in the figure .In what follows, we shall focus on the results corresponding to the pairing force of strength V 0 =350.First of all, it can be seen that the Skyrme-HF results, obtained by using the parametrisation UNE1 [15], underestimate the binding energies by about 3-4 MeV in the middle mass region, while for the nuclei with A > 90 the calculated binding energies are larger than the experimental ones.As expected, the Skyrme-HF+BCS calculations, which take into account only the neutron-neutron (nn) and proton-proton (pp) pairing, is smoothing out the fluctuations of the HF results caused by the shell effects.For the N=Z nuclei with 60 < A < 80, where the HF fluctuations are small, in BCS approximation the residuals are decreasing by about 1 MeV compared to the HF values.From Fig. 1 it can be seen that the binding energies are increasing significantly when are taken into account the isovector pn pairing correlations, treated in the QCM approach.On the other hand, except for A=24 and A=28, the effect of the isoscalar pn pairing on the binding energies is surprisingly small.This fact is caused by the competi-tion between various pairing channels and between pairing and mean field.Thus, when the isoscalar pn channel is switched on, the like-particle and isovector pn pairing energies are decreasing.This decrease of the isovector pairing is not compensated by the pairing energy gained by opening the isoscalar channel.This is due to the fact that the off-diagonal matrix elements of the isoscalar interaction, which have the dominant contribution to the pairing energy, are small.One of the important conclusions of these calculations is that the isovector pn pairing has the most significative contribution to the binding energies of N = Z nuclei.The isoscalar pairing correlations contribute less to the binding energies but they are present in all the calculated nuclei.

QCM approach for the spherical-symmetric pairing Hamiltonians
In this section we present briefly the the QCM approach for the isovector-isoscalar pairing Hamiltonians with spherical symmetry, introduced in Ref. [11].
The isovector-isoscalar pairing Hamiltonian consider here has the expression The first two terms are the same as in Eq. (1) while the last term is the isoscalar pairing interaction written in term of the isoscalar pair operator where J z (11) For even-even N = Z systems the QCM ansatz for the ground state has formally the same expression as in the case of isovector pairing The difference is that now the quartet operator Q + ivs , still having total isospin T = 0, is the sum of two quartets where is the collective quartet built by a linear combination of two non-collective isovector pairs coupled to the total isospin T = 0.The isoscalar quartet Q + is is formed by two isoscalar pairs coupled to total J = 0, i.e., The parameters which define the collective quartets are determine from the minimisation of the average of the Hamiltonian on the QCM state (12).In Ref. [11] it was shown that the QCM approach provides accurate ground state energies for N = Z systems interacting by realistic isovector-isoscalar pairing forces.Recently the QCM approach presented in the previous section was extended for describing the low-lying excited states of N = Z systems interacting by isovector-isoscalar pairing forces [13].Below we present briefly this extension and the most important findings.
In correspondence with the QCM ansatz (12) for the ground state, we construct a class of excited states by replacing a quartet of the condensate with an "excited" quartet.For the case of a spherically-symmetric mean field, these states take the form where are the excited collective quartets.In order to define its coefficients Y (ν) JJ z , one has now to diagonalize the Hamiltonian (10) in the basis of non-orthogonal states To illustrate the accuracy of the approximation (16) we take as example the valence nucleons in 28 Si and assume an isovector-isoscalar pairing force corresponding to the (J = 0, T = 1) and (J = 1, T = 0) channels of the the USDB interaction.Exact and approximate spectra are shown in Fig. 2. It can be seen that the overall agreement is good.For more details see Ref. [13].

Summary and Conclusions
We have shown how the iosvector and isoscalar protonneutron (pn) pairing is treated in the quartet condensation model (QCM).In the QCM approach the pairing correlations are taken into account in terms of collective quartets formed by two neutrons and two protons coupled to total isospin T = 0 and total angular momentum J = 0.In this approach the ground state of N = Z systems is described by a condensate of quartets while the excited states are obtained by breaking a quartet from the ground state condensate and replacing it with an "excited " quartet.As an application we have analysed, in the framework of Skyrme+QCM, the effect of pn pairing on the ground states of N = Z nuclei.It is concluded that the pn pairing, in particular the isovector pairing, has a significant contribution to the binding energies of these nuclei.

Figure 1 .
Figure 1.Binding energies residuals, in MeV, for even-even N=Z nuclei as a function of A=N+Z.The results correspond to the pairing forces indicated in the figure.

Figure 2 .
Figure2.The low-lying spectrum provided by the QCM approximation (16) for the valence nucleons of 28 Si interacting by an isovector-isoscalar pairing force extracted from the USDB interaction.The numbers are the overlaps between the QCM and the exact wave functions.Energies are in MeV.