Pairing Matrix Elements and Pairing Gaps with Bare, Effective and Induced Interactions

The dependence on the single-particle states of the pairing matrix elements of the Gogny force and of the bare low-momentum nucleon-nucleon potential $v_{low-k}$ is studied in the semiclassical approximation for the case of a typical finite, superfluid nucleus ($^{120}$Sn). It is found that the matrix elements of $v_{low-k}$ follow closely those of $v_{Gogny}$ on a wide range of energy values around the Fermi energy $e_F$, those associated with $v_{low-k}$ being less attractive. This result explains the fact that around $e_F$ the pairing gap $\Delta_{Gogny}$ associated with the Gogny interaction (and with a density of single-particle levels corresponding to an effective $k$-mass $m_k\approx 0.7 m$) is a factor of about 2 larger than $\Delta_{low-k}$,being in agreement with $\Delta_{exp}$= 1.4 MeV. The exchange of low-lying collective surface vibrations among pairs of nucleons moving in time-reversal states gives rise to an induced pairing interaction $v_{ind}$ peaked at $e_F$. The interaction $(v_{low-k}+ v_{ind})Z_{\omega}$ arising from the renormalization of the bare nucleon-nucleon potential and of the single-particle motion ($\omega-$mass and quasiparticle strength $Z_{\omega}$) due to the particle-vibration coupling leads to a value of the pairing gap at the Fermi energy $\Delta_{ren}$ which accounts for the experimental value.


Abstract
The dependence on the single-particle states of the pairing matrix elements of the Gogny force and of the bare low-momentum nucleonnucleon potential v low−k -designed so as to reproduce the low-energy observables avoiding the use of a repulsive core -is studied in the semiclassical approximation for the case of a typical finite, superfluid nucleus ( 120 Sn). It is found that the matrix elements of v low−k follow closely those of v Gogny on a wide range of energy values around the Fermi energy e F , those associated with v low−k being less attractive. This result explains the fact that around e F the pairing gap ∆ Gogny associated with the Gogny interaction (and with a density of single-particle levels corresponding to an effective k-mass m k ≈ 0.7m) is a factor of about 2 larger than ∆ low−k , being in agreement with ∆ exp = 1.4 MeV. The exchange of low-lying collective surface vibrations among pairs of nucleons moving in time-reversal states gives rise to an induced pairing interaction v ind peaked at e F . The interaction (v low−k + v ind )Z ω arising from the renormalization of the bare nucleon-nucleon potential and of the single-particle motion (ω−mass

Introduction
An economic description of pairing correlations in finite nuclei is provided by Hartree-Fock-Bogoliubov (HFB) theory [1,2] making use of phenomenological interactions like e.g. the finite range Gogny force [3] or density dependent zero-range forces combined with appropriate energy cut-offs (cf., e.g., [4,5]). Such a description leads to values of the pairing gap which are in overall agreement with the experimental findings. Note that in these calculations, the density of levels at the Fermi energy ρ(e F ) is controlled by the so called k−mass (i.e. ρ(e F ) ∼ m k ) which, as a rule, is smaller than the bare nucleon mass (e.g. m k ≈ 0.7m in the case of the Gogny force).
On the other hand, a number of studies have shown that the superfluid properties of nuclear systems, ranging from infinite nuclear and neutron matter to finite atomic nuclei, are strongly influenced by polarization phenomena [6]. In these calculations one starts from a bare nucleon-nucleon potential (Argonne, Bonn, Paris, etc.) adding afterwards the renormalization processes. Recently, the pairing gap, the quasiparticle spectrum and the collective modes of 120 Sn have been calculated solving the Dyson-Gor'kov equation [7], in a single-particle space characterized by m k = 0.7m, allowing the nucleons to interact in the 1 S 0 channel through a v 14 Argonne N-N potential taking into account the variety of renormalization processes (self-energy, fragmentation, induced interaction and vertex corrections) arising from the coupling of the particles with surface vibrations. While the bare N-N interaction accounts for about half of the pairing gap, overall agreement with the experimental findings is achieved by including medium polarization effects.
In the present paper we want to shed light into the physics of these results, by studying the magnitude of the different pairing (J π = 0 + ) matrix elements as well as their dependence on the energy of the single-particle states lying in the vicinity of the Fermi energy in Section 2, and by discussing the calculation of the pairing gaps associated with these matrix elements in Section 3.

Matrix elements
The matrix element of the induced interaction can be written as [8] where L, M and n denote the quantum numbers of the exchanged collective vibrations, f Ln (r) = β Ln R 0 (dU/dr) is the associated radial formfactor (β Ln is the deformation parameter associated with the n−th mode of multipolarity L, R 0 is the ground-state radius and U is the average potential),hω Ln is the vibrational energy of the n−th mode, while e ν and e F are the single-particle and Fermi energies. E 0 is the pairing correlation energy per Cooper pair, which is of the order of −∆. In practice we have used E 0 = -2 MeV. The pre-factor 2 in Eq. (1) comes from the two possible time orderings associated with the one-phonon exchange.
In Fig. 1 we show the value of the diagonal matrix elements νν|v ind |νν of the induced interaction for the nucleus 120 Sn as a function of the singleparticle energy. The single-particle levels entering Eq. (1) have been calculated making use of a standard Saxon-Woods potential, whose parameters are provided in the caption of Fig. 1. In most of our calculations we use this potential and an associated effective mass m k equal to the bare mass. In the last section we shall use instead an effective mass m k = 0.7m, which simulates the typical outcome of a Hartree-Fock calculation (with, e.g., the Gogny interaction). In those cases, the Woods-Saxon parameters are changed in such a way that the Fermi energy remains close to the experimental value (V , see e.g. Fig. 8). The parameters β Ln andhω Ln have been determined by diagonalizing a separable multipole-multipole interaction in the quasiparticle-random phase approximation (QRPA) adjusting the coupling constants so as to reproduce the energies and transition probabilities of the lowest-lying states of each spin and parity. Also shown in Fig. 1 are the diagonal matrix elements of the Gogny interaction. Here, and in the rest of the paper, we shall employ the D1 parametrization.
The matrix elements of v ind , which display an average value equal to -0.15 MeV, are peaked at the Fermi energy, within an energy range of essentially 5 MeV around e F . This behaviour reflects the fact that ν ′ν′ |v ind |ν,ν is controlled, through the energy denominator appearing in Eq.(1), by the energy of the low-lying collective vibrations (hω L (n) < 5 MeV). Outside this energy range, the values of ν ′ν′ |v ind |ν,ν become very small, less than 50 keV in absolute value. The matrix elements of v Gogny decrease on average smoothly in magnitude all the way from the deepest levels up to the continuum threshold, displaying a value of about -0.3 MeV at e F .
The induced interaction matrix elements depend on specific properties of finite nuclei, namely the single-particle quantum numbers and energies, and the energy, the zero-point amplitude and the transition density of the vibrational states. To study the global features of the bare, of the induced, and of the Gogny interactions as well as to provide a more transparent form for their matrix elements, we shall present, in the rest of this section, also the matrix elements obtained using the semiclassical Thomas-Fermi (TF) approximation. In fact, this approximation averages out shell effects typical of finite nuclei and provides the overall energy dependence of the matrix elements. Within this context we follow Ref. [10] where semiclassical expressions for two-body matrix elements have been presented. To obtain a smooth behaviour of the diagonal matrix elements as a function of the continuous energy variable E, we compute the Fourier transform of a two-body interaction v( r 1 , r 2 ) using plane waves of momenta p 1 and p 2 . The average of the quantal matrix elements is carried out using the normalized semiclassical density matrix, In the above relation the semiclassical level density g is R c being the classical turning point. We assume a constant value for the effective mass m k , neglecting its possible dependence on position. The expression for the semiclassical diagonal matrix element of v( r 1 , r 2 ) as a function of the single-particle energy, is consequently [10] v For an interaction that depends only on the magnitude of the relative coordinate s ≡ r 1 − r 2 , like the Gogny interaction in the S=0, T=1 channel, one can integrate first over d 3 s and then over the momenta d 3 p 1 and d 3 p 2 , obtaining the expression where R indicates the center of mass, v(k, k) is the on-shell pairing matrix element of the interaction v(s) in the 1 S 0 channel and in a plane wave basis, evaluated at the local momentum In the next section we shall calculate the pairing gap associated with the different interactions, and therefore we shall need the matrix elements between all pairs of particles moving in time reversal. For this purpose we shall employ a formula analogous to that given in Eq.
In this relation E and E ′ indicate the energies of the single-particle states of each pair. We now turn to the case of the induced interaction, associated with the matrix elements of Eq. (1). In this case the matrix elements depend separately on r 1 and r 2 . We then integrate over the momenta first, obtaining where j 0 is the Bessel function and k ≡ 2m k h 2 (E − U(R)). The index n labels the different phonons of multipolarity L (the phonons included in the present calculation are specified at the end of this Section). Note that one can obtain numerical results from Eq. (8) by avoiding the six-dimensional integration. This is achieved by performing a multipole expansion of the quantity In Ref. [10] the reliability of the TF approximation in reproducing the quantum mechanical (QM) matrix elements has been checked in the particular case of the pairing matrix elements of a δ-interaction acting among particles moving in a harmonic oscillator (HO) potential. Note that the TF approach implies an averaging over the quantum numbers associated with the different states belonging to the major shells and displaying an energy E N = (N + 3 2 )hω. The TF matrix elements have been compared with the averaged QM matrix elements (taking degeneracies into account). Overall agreement was observed (cf. A similar comparison for the pairing matrix elements of the Gogny force in the 1 S 0 channel is reported in Fig. 2. It is seen that the TF approach provides a good approximation to the average quantum matrix elements. However, the QM diagonal matrix elements, in particular those associated with s-states, are systematically larger (in absolute value) than the non-diagonal ones. One should be aware of the fact that the semiclassical density matrix can be considered to be the analogue to the one obtained from a Strutinsky smoothing. Therefore, Eq. (2), implicitly, represents a function of about 1hω width. Thus, the matrix element in Eq. (4) contains at the quantum level cross terms corresponding to at least one major shell. Taking for example the 2s-1d shell, we have to weigh the two diagonal matrix elements with the 2s and 1d wavefunctions with a factor 1 and 25, respectively. The non-diagonal 2s-1d element obtains the weight factor 10. Performing the arithmetic average yields the crosses of Fig. 2. In this way, we see that the TF expression (4) reproduces very well the quantal average. The more realistic case of the Gogny matrix elements in the Woods-Saxon potential (including spin-orbit) is illustrated in Fig. 3. The general pattern is similar to that shown in the previous figure.
Having assessed its validity, we now employ the semiclassical approximation to compare the matrix elements of the induced interaction, the effective Gogny interaction and a bare nucleon-nucleon interaction. Concerning the latter, the matrix elements of bare nucleon-nucleon interactions which contain a repulsive core display a qualitatively different momentum dependence than that displayed by the Gogny interaction. We shall instead consider here the v low−k interaction, which has been devised to reproduce the low-momentum properties of the nucleon-nucleon force, including the experimental phase shifts, without the introduction of a repulsive core [11]. In particular, we shall employ a parametrization of v low−k devised to reproduce the properties of v 14 Argonne potential. This parametrization requires a momentum cutoff k cut equal to 2.1 fm −1 . The matrix elements of v low−k in a plane wave basis have been calculated in Ref. [12] and are reported in Fig. 4. It is seen that, while v Gogny is less attractive than v low−k for small momenta, it becomes more attractive for k larger than ≈1 fm −1 .
The semiclassical matrix elements of the Gogny D1 interaction (Eq. (5)) and of the v low−k interactions calculated as a function of the single-particle energy for the nucleus 120 Sn, are displayed in Fig. 5. We remark that, close to the Fermi energy, the matrix elements of the Gogny interaction calculated with the D1S parametrization are about 10 keV less attractive. It is seen that the general trend is the same as that obtained in infinite matter, with the difference that the matrix elements of the two interactions look somewhat more similar to one another. Insight into this difference can be obtained with the help of Eq.(5). This relation is essentially an average of v(k) calculated at the various local momenta in the nucleus. From Fig. 4 we see that in the interior of the nucleus (at larger local momenta) v Gogny is more attractive than v low−k , but on the surface it tends to be less attractive than v low−k .
The semiclassical matrix elements of the induced interaction are also shown in Fig. 5. These matrix elements are strongly peaked around the Fermi energy and reproduce rather well the average of the quantal results already shown in Fig. 1. In Fig. 6 we have separated the contributions arising from the exchange of phonons of multipolarities L=2, 3 and 4. We include all the phonons with energy below 30 MeV in the calculation. The importance of the contributions with L=2 and 3 can be understood in terms of the high collectivity and low energy of the associated collective modes. The energies and deformation parameters of the low-lying modes arehω 2 = 1.17 MeV , hω 3 = 2.42 MeV,hω 4 = 2.47 MeV and β 2 = 0.12, β 3 = 0.15, β 4 = 0.07.
We also note that at the Fermi energy the matrix elements of v ind are rather large, of the order of those associated with the bare nucleon-nucleon potential v low−k (in this connection, cf. the estimate in ref. [13]) .

Pairing gaps
In this section, we shall discuss the results of calculations of the pairing gap which employ the matrix elements discussed above. In the nucleus under study, namely 120 Sn, we obtain the state-dependent pairing gaps by solving the generalized BCS equations [14]: this means that we solve the HFB equations by treating the pairing sector self-consistently, while the mean field is described in terms of the Woods-Saxon potential already used in the calculation of the matrix elements. In this way, we take into account scattering processes between nucleons lying on orbits having different number of nodes/energies [4], associated with non-diagonal matrix elements of the . This is at variance with the usual BCS method in which the pairing gap receives contributions only from scattering processes between time-reversal states. We shall refer in the following either to quantal or to semiclassical calculations; in the latter case, this is done by solving the same equations as in the quantal case, but replacing the QM matrix elements with the TF ones, using Eq. (7) and the prescription The positive energy states are obtained by setting the system in a spherical box. We have checked that convergence is achieved using R box =12 fm and including all the states from the bottom of the potential up to the positive energy E cut =50 MeV.
In infinite neutron matter, the pairing gaps calculated with the interactions v low−k and v Gogny are very similar up to Fermi momenta of the order of 0.6 fm −1 , becoming increasingly different from each other at higher momenta, as can be seen from Fig. 7 (cf. Ref. [5,12], where the D1S parametrization of the Gogny interaction has been used). The pairing gap obtained with v low−k goes to zero, around saturation, as a function of k F much faster than that associated with v Gogny ; it can also be seen that the pairing gap obtained with v low−k reproduces quite accurately the result found allowing particles to interact through the Argonne v 14 bare N-N potential [15].
In the case of 120 Sn, the pairing gap with the v 14 Argonne interaction and with m k = m, was calculated previously in ref. [16], obtaining a value of about 2.2 MeV close to the Fermi energy. We have verified that essentially the same value is obtained with v low−k , in keeping with the infinite matter case (cf. Fig. 7). In the following, however, we shall adopt the value m k = 0.7m, which is the typical value associated with Hartree-Fock calculations performed with Gogny or Skyrme forces. In Fig. 8 we compare the state-dependent pairing gap obtained with the Gogny force and with the v low−k interaction. We also compare the pairing gaps obtained inserting the semiclassical pairing matrix elements in the HFB equations. We remark that the semiclassical approach takes out the scatter from shell effects which is a desirable feature when one wishes to reveal generic trends.
The Gogny interaction leads to a pairing gap of 1.4 MeV close to the Fermi energy, in good agreement with the experimental value deduced from the odd-even mass difference. This value is about two times larger than that obtained with v low−k . Note also that the pairing gap obtained with v low−k reproduces the results obtained with the bare v 14 interaction [7]. We remark that using the D1S parametrization of the Gogny force, instead of D1, one obtains a gap at the Fermi energy of about 1.2 MeV.
In Fig. 9 we show the state-dependent pairing gap obtained with the pairing matrix elements of v total = v low−k +v ind , resulting from the sum of the matrix elements of the bare interaction v low−k and of the induced interaction v ind . The average value of the resulting pairing gap ∆ total at the Fermi energy is about 2 MeV. This value is about 25% higher than that obtained in Ref. [7], where we solved the Dyson-Gor'kov equation, taking properly into account all polarization effects (induced interaction, self-energy and vertex corrections) [17]. This is consistent with the fact that according to Nuclear Field Theory (NFT) [18], if one considers the effects of the exchange of phonons between pairs of nucleons, one has to consider at the same time processes where the phonon is absorbed by the same nucleon which has virtually excited it. Such processes lead to self-energy (ω-mass and single-particle splitting) as well as to vertex correction phenomena. Of these processes and for the nucleus under discussion ( 120 Sn), ω−mass effects [9] are the most important. Taking such effects into account is equivalent to use a residual interaction v ren = v total Z ω , where Z ω = (m ω /m) −1 is the quasiparticle strength [9] at the Fermi energy (cf. Appendix A). Consequently ∆ ren = Z ω ∆ tot . Because v ind is proportional to β 2 L /DEN L , that is to the square of the deformation parameters associated with the low-lying collective vibration and inversely proportional to the energy denominator DEN L appearing in Eq.(1), a typical error of 20% on β L and DEN L implies a 60% error in v ind . Making use of this fact and of the results of Figs. 1 and 9 one obtains, as discussed in Appendix B, ∆ ren ≈ Z ω ∆ tot ≈ 1.35 ± 0.15 MeV. It is seen that the 60% error in v ind has been reduced to a 10% in ∆ ind . This is because the stronger the particle-vibration coupling is, the larger ∆ tot but the smaller the amount of single-particle content of levels around ǫ F (and thus the smaller Z ω ), and viceversa. The self consistency between collectivity of the modes, strength of the particle-vibration coupling, and quasi-particle residue at the pole, typical of NFT, allows theory to make predictions which are more accurate than the basic parameters entering the calculation, as a result of a delicate process of cancellation of errors.

Conclusions
Summing up, we have found that the small difference existing between the matrix elements of the bare v low−k interaction and those of the Gogny interaction leads to important differences between the pairing gaps associated with the two forces (∆ Gogny ≈ 2∆ low−k , for m k ≈ 0.7m). This difference is removed and eventually overwhelmed by including the attractive contribution coming from the induced interaction v ind arising from the exchange of surface vibrations, which acts only on a rather small energy range around e F . We note that the pairing gap in 120 Sn is not much changed, including also the effect of spin fluctuations [22], which instead give the dominant (repulsive) contribution to the induced interaction in neutron matter.
The 25% excess displayed by the resulting total pairing gap with respect to the experimental value is corrected by considering the corresponding quasiparticle strength at the Fermi energy. An important question which remains to be addressed quantitatively is the ability v low−k + v ind as well as v Gogny have to describe in detail the isotopic and nuclear structure dependences displayed by ∆ exp . This subject will be addressed in a future publication.
Within this context, we note that the dependence of the pairing gaps generated by v Gogny and by v low−k + v ind on temperature and rotational frequency is expected to be quite different, as v Gogny and v low−k are essentially independent of these parameters, while v ind is quite sensitive to them.
It is also worth mentioning that a recent study in symmetric nuclear matter revealed a very strong extra attraction coming from the induced interaction [23]. Nuclear matter and finite nuclei may, however, show only qualitative similarities in this case and a quantitative link may be difficult to establish.
We thank L. Coraggio for providing us with the matrix elements of the v low−k interaction.

Simple estimate of the pairing gap
In what follows we consider some of the consequences the particle-vibration coupling has on the pairing correlations of particles moving in a single j-shell interacting through a bare nucleon-nucleon pairing potential with constant matrix elements G [19].
For this simple model, the value of the occupation numbers U ν and V ν must be the same for all the 2j + 1 orbitals. In particular, the occupation probability for the case when the system is occupied with N = Ω particles (half filled shell), where Consequently, the pairing gap is given by the relation Because the density of levels is proportional to the ω−mass (m ω ), the effective (dressed) degeneracy can be written in terms of the quasiparticle strength at the Fermi energy Z ω = (m ω /m) −1 where m ω = m(1 + λ), λ being the mass enhancement factor. In the case of nuclei this dimensionless quantity (which measures the strength with which nucleons couple to low-lying collective vibrations) is of the order of 0.5, closer to the strong than to the weak coupling situation of BCS (in which case ∆ ∼ λ) [21].
Because of their coupling to vibrations, nucleons spend part of the time in more complicated configurations than pure single-particle states. The quantity (quasiparticle strength [9]) Z ω = (1 + λ) −1 ≈ 0.7 measures the content of single-particle strength present in levels around the Fermi energy available for nucleons to interact through a two-body interaction, in particular through through a pairing force. Consequently, due to the self-energy processes arising from the particle-vibration coupling phenomenon, the pairing strength becomes GZ ω 2 . The exchange of vibrations between pairs of nucleons moving in time reversal states close to the Fermi energy gives rise to an effective pairing interaction of strength g p−v Z ω 2 , where g p−v stands for the induced pairing interaction controlled by the particle-vibration coupling vertices. Taking these effects into account, one can write Identifying G with v low−k and g p−v with v ind , one finds from Fig. 9 that As seen from Eq. (A.6), the number to be compared with experimental findings is ∆ ren = Z ω × 2MeV ≈ 1.4MeV. (A.8) Note that in the above discussion we have not considered the errors to be expected in estimating both g p−v and Z ω . This subject is taken up in Appendix B. , displayed as a function of the single-particle energy e ν of the state ν calculated using the bare nucleon mass and the single-particle wavefunctions of a Woods-Saxon potential with standard parameters (depth V 0 = -49 MeV, diffusivity a =0.65 fm, and radius R 0 = 6.16 fm), including the spin-orbit term, parametrized according to ref. [20]. Also shown by means of vertical lines is the position of the Fermi energy, e F = −9.1 MeV. Note the different scale in the two figures.  The nucleus 120 Sn. The same as Fig. 2 (i.e. also with m k = m) but for the fact that the single-particle wavefunctions have been calculated making use of the Woods-Saxon potential including the spin-orbit term already used for Fig. 1, where the diagonal matrix elements have already been shown. The non-diagonal matrix elements are plotted here at an energy e ν which is the average between the energies of the initial and final states.   e F Figure 6: The nucleus 120 Sn. The semiclassical matrix elements of the induced interaction, as a function of the energy of the single-particle levels (calculated with m k = m) associated with the multipolarities L = 2, 3, and 4 of the formfactor (see Eq.(1)), are shown by a dashed, dotted and solid line respectively. The sum of the three contributions is displayed by means of a dash-dotted line (cf. Fig. 5).    Figure 8: State-dependent pairing gaps of 120 Sn calculated with a Woods-Saxon potential (with depth V 0 = -64 MeV, diffusivity a = 0.65 fm and radius R 0 = 6.17 fm), as a function of the single-particle energy. The k−mass m k was set equal to 0.7m. The Fermi energy is e F =-8.6 MeV. Solid triangles (open squares) display the results of a HFB calculation with the Gogny interaction, with quantal (semiclassical) matrix elements. The solid diamonds refer instead to a HFB calculation using the semiclassical matrix elements of the v low−k potential.  Fig. 8 (i.e. with m k = 0.7m) and different interactions. Diamonds and triangles show again the gaps obtained with the v low−k and v Gogny interactions, respectively. Stars show the gap associated with the matrix elements obtained summing the matrix elements of v low−k and v ind . The latter have been evaluated using Eqs. (9) and (10) .