Regularization of zero-range effective interactions in finite nuclei
RRegularization of zero-range effective interactions in finite nuclei
Marco Brenna, Gianluca Col`o, ∗ and Xavier Roca-Maza Dipartimento di Fisica, Universit`a degli Studi di Milano and INFN, Sezione di Milano, 20133 Milano, Italy (Dated: October 19, 2018)The problem of the divergences which arise in beyond mean-field calculations, when a zero-rangeeffective interaction is employed, has not been much considered so far. Some of us have proposed,quite recently, a scheme to regularize a zero-range Skyrme-type force when it is employed to calculatethe total energy, at second-order perturbation theory level, in uniform matter. Although this schemelooked promising, the extension for finite nuclei is not straightforward. We introduce such procedurein the current paper, by proposing a regularization procedure that is similar, in spirit, to the oneemployed to extract the so-called V low − k from the bare force. Although this has been suggestedalready by B.G. Carlsson and collaborators, the novelty of our work consists in setting on equalfooting uniform matter and finite nuclei; in particular, we show how the interactions that have beenregularized in uniform matter behave when they are used in a finite nucleus with the correspondingcutoff. We also address the problem of the validity of the perturbative approach in finite nuclei forthe total energy. PACS numbers: 21.60.Jz, 21.30.Fe, 21.10.-k, 21.10.Dr, 21.65.Mn
I. INTRODUCTION
Self-consistent mean-field approaches provide a fairlygood starting approximation to describe atomic nuclei[1]. Whereas so-called ab-initio approaches are increas-ingly successful, they cannot at present describe heavysystems and/or high-lying excited states. Mean-field ap-proaches, instead, are able to reproduce both the exper-imentally observed trends of many ground-state prop-erties (masses, radii, deformations etc.) and severalfeatures of excited states (giant resonances, rotationalbands); moreover, they can be extended if needed. Eithernon relativistic Hamiltonians of the Skyrme or Gognytype, or covariant relativistic mean-field (RMF) La-grangians have been used indeed beyond the mean-fieldapproximation, for instance in second-order calculations[2, 3], in multiparticle-multihole schemes [4], in particle-vibration coupling (PVC) models [5–17], or within thegenerator coordinate method (GCM) approach [18–21].In such approaches one introduces further correlationson top of those implicit in the mean field. Within PVC,the nucleons feel the effect of the dynamical fluctuationsof the mean field, on top of its static part; within GCM,the variational space associated with a single mean-fieldconfiguration is enlarged by superimposing several mean-field configurations, each being connected with a differentvalue of some global parameter like the quadrupole defor-mation. If effective interactions are fitted at mean-fieldlevel one would imagine that a re-fit of these interactionsis mandatory if they are employed in a different frame-work. However, this is usually not done and Skyrme andGogny forces or RMF Lagrangians are used as they are.We have in mind, for the follow-up of our discussion,mainly the PVC case since we shall consider in detail thelowest-order approximation to that model. ∗ [email protected]
If single-particle (s.p.) nuclear states are calculatedusing Skyrme or RMF Lagrangians, the corrections in-duced by PVC at lowest order are typically several hun-dreds of keV, ranging from small values to ≈ Pb [6, 23]. However, the convergence of theseresults with respect to the model space is hard to assess.Normally, one assumes that the model space is limited bythe fact that only collective vibrations should be takeninto account but this does not set a cutoff in a clear-cutway.At the same time, there is another practical and yetmore general point to be kept in mind. When zero-range forces are used in a beyond mean-field approachlike PVC, divergences arise. In other words, diagramsbeyond the HF ones (like those displayed in Fig. 1 of[6]) can be shown, by simple power counting arguments,to diverge as the model space is enlarged. This is nota specific problem of Skyrme as also Gogny forces pos-sess a zero-range term. We do not dispose at present ofa reliable, fully microscopic non relativistic Hamiltonianwithout zero-range terms. As a consequence, it is neces-sary to devise a regularization technique to absorb thisdivergence and go beyond the usual PVC calculations.In the current work, we focus on the Skyrme case.To simplify the formidable problem of finding a reli-able regularization technique for nuclei, we only considerin the following the lowest-order (that is, second order)approximation for PVC in which the phonon is replacedby a particle-hole (p-h) pair, and we focus on the correc-tion to the total energy instead of the correction to thes.p. energies. This problem has been already tackled inuniform matter [24, 25], where it has been shown thatat least a cutoff regularization is possible for a generalSkyrme interaction and an arbitrary neutron/proton ra-tio. A dimensional regularization technique [26] has also a r X i v : . [ nu c l - t h ] O c t FIG. 1. Diagrammatic representation of the first-order(Hartree-Fock) and second-order total energy, respectively inthe upper and lower parts of the figure. The labels outside(inside) parenthesis are those used in the text for the case ofuniform matter (finite nuclei). been proposed, whereas the general study of the renor-malizability of a Skyrme-type force has been recently ad-dressed in Ref. [27]. Also these studies concern onlyuniform matter. The extension of the techniques intro-duced in Refs. [24, 25] to finite nuclei is far from beingstraightforward and is the subject of the current paper.The total energy is depicted diagrammatically in Fig.1. We have drawn only the direct contributions but ex-change terms are properly included in our calculations.The first row of the figure depicts the mean-field orHartree-Fock (HF) total energy and the second line thesecond-order contribution to the same quantity. The la-bels outside (inside) parenthesis are momentum (generic)labels, appropriate for uniform matter (nuclei) respec-tively.In the case of uniform matter, a simple power count-ing argument dictates that the second order contributiondiverges. The simplest way to understand it is the follow-ing. The integration on momentum states is finite withrespect to the hole momentum states k , k which havethe Fermi momentum k F as an upper limit. The center-of-mass momentum conservation leaves only one furthermomentum scale, which has been chosen in Refs. [24, 25]to be the transferred momentum. If we label the parti-cle states needed for the calculation of the second-orderenergy as k , k , the matrix elements are (cid:104) k , k | V | k , k (cid:105) = (cid:104) k + q , k − q | V | k , k (cid:105) (1)(cf. Fig. 2), and the quantity q ≡ k − k − k + k δ -force) thesecond-order contribution diverges linearly, or in otherwords it scales as (cid:82) d qq , while the divergence is moresevere if momentum-dependent terms are included. InRefs. [24, 25] it has been shown that, by setting a cut-off Λ on the transferred momentum, given an interaction V that provides a reasonable total energy at mean-field k k k k k k + q k k − q FIG. 2. Representation of matrix elements in the case ofuniform matter. We compare the notation used in the presentpaper (left) with the one in which the transferred momentum q appears (right). level, it is possible to fit an interaction V Λ that repro-duces the same energy when the second-order correctionhas been included with the cutoff Λ.In finite nuclei, at variance with uniform matter, thereis not translational invariance and in dealing with thematrix elements (1) we are left with two free parame-ters or two energy scales. In the current work we havedealt with the two scales by defining in a precise fashionthe relative and center-of-mass coordinates and the asso-ciated momenta. Since the separation of center-of-massand relative motion wave functions can be done in a neatway by using an harmonic oscillator basis, the calcula-tions that we shall discuss below have been performedon that basis. We have systematically defined a cutoff λ on the relative momenta (in initial and final channel)defined as k ≡ k − k √ , k (cid:48) ≡ k − k √ . (3)Then, in our study, we will show how a simplifiedSkyrme interaction, in which only the t , t and α pa-rameters are kept and which has been regularized in uni-form matter, behaves when it is used in a finite nucleus.We restrict ourselves to the case of even-even, isospin-symmetric nuclei; in particular, we will show results for O without Coulomb and spin-orbit forces. Our ap-proach is thus self-consistent in the sense that we usethe same Skyrme interaction both at mean-field and sec-ond order level, but we compute the total energy at sec-ond order in a perturbative way, by adding the beyondmean-field contribution on top of HF solutions. We willeventually address the problem of the validity of suchperturbative approach for the total energy of a finite nu-cleus.The structure of the paper is the following. Sec. II isdevoted to a thorough explanation of the formalism wewish to introduce: in particular, in subsec. II A we dis-cuss the interaction and its regularization, whereas in thenext subsection II B we show the relationship between thenew cutoff employed in this work and the cutoff that hadbeen introduced previously in uniform matter. The spe-cific formulas that implement the regularized calculationof the total energy in a finite nucleus, on the harmonicoscillator basis, are introduced in subsec. II C. In Sec.III we describe the results obtained in the case of O.Conclusions and considerations related to the envisagedfollow-up of this work, are in Sec. IV.
II. FORMALISMA. The regularization of the interaction
The cutoff on the relative momentum components ofthe effective interaction is analogous to that discussed inRef. [28]. The underlying philosophy is the same as in thecase of the V low − k interaction and it seems quite natural,even by invoking the original argument that the Skyrmeinteraction is a polynomial expansion in the relative mo-mentum that stops at second order [29]. Therefore, weintroduce the cutoffs on the relative momenta of the ini-tial and final states and we define a regularized interac-tion through these cutoffs. In principle, this procedurecould be avoided by using a finite-range force. However,as we stressed in the Introduction, we miss at present awidely used, reliable microscopic pure finite-range force. To identify properly the relative momenta we intro-duce center-or-mass and relative coordinates. We startby writing the velocity-independent part of the Skyrmeforce in this form: V ( r (cid:48) , r (cid:48) , r , r ) = g (cid:18) r + r (cid:19) δ ( r − r ) δ ( r − r (cid:48) ) δ ( r − r (cid:48) ) . (4)Our desired change of variables reads (cid:18) r ( (cid:48) ) R ( (cid:48) ) (cid:19) = (cid:32) √ − √ √ √ (cid:33) (cid:32) r ( (cid:48) )1 r ( (cid:48) )2 (cid:33) , (5)so that the interaction (4) can be written as V ( r (cid:48) , R (cid:48) , r , R ) = √ g (cid:18) R √ (cid:19) δ ( r ) δ ( r (cid:48) ) δ ( R − R (cid:48) )= √ g (cid:18) R √ (cid:19) v ( r (cid:48) , r ) δ ( R − R (cid:48) ) , (6)where g ( R ) = t + t [ ρ ( R )] α . The Fourier transform ofthe interaction can be written in a straightforward wayas V ( k , k , k , k ) = √
24 1Ω (cid:90) d R d R (cid:48) e − i k k √ · R (cid:48) g (cid:18) R √ (cid:19) δ ( R − R (cid:48) ) e i k k √ · R ×× (cid:90) d r d r (cid:48) e − i k − k √ · r (cid:48) v ( r (cid:48) , r )e i k − k √ · r , (7)by introducing a finite quantization volume Ω. The factorappearing in the second line of this equation can be writ-ten in terms of the variables k ≡ k − k √ and k (cid:48) ≡ k − k √ that are the conjugate variables of the relative coordi-nates defined by Eq. (5): these are the relative momenta(of the initial and final state respectively) that we havealready introduced in Eq. (3). Thus we re-write the fac-tor in the second line of Eq. (7) as v ( k (cid:48) , k ) = 1Ω (cid:90) d r d r (cid:48) e − i k (cid:48) · r (cid:48) v ( r (cid:48) , r )e i k · r = 1Ω , (8) where the last equality obviously holds if v ( r (cid:48) , r ) = δ ( r ) δ ( r (cid:48) ) as we have written in Eq. (6) (we will keepthis notation in what follows).Then, we introduce the regularized interaction as theinverse Fourier transform of (8) in which two step func-tions θ ( λ − k ) θ ( λ (cid:48) − k (cid:48) ) are introduced. In this way, λ and λ (cid:48) are the cutoffs in the relative momenta k and k (cid:48) , respectively, and the regularized interaction v λλ (cid:48) isobtained as v λλ (cid:48) ( r (cid:48) , r ) = 1Ω (cid:90) d k d k (cid:48) e i k (cid:48) · r (cid:48) v ( k (cid:48) , k ) θ ( λ − k ) θ ( λ (cid:48) − k (cid:48) )e − i k · r = 14 π λ λ (cid:48) rr (cid:48) j ( rλ ) j ( r (cid:48) λ (cid:48) ) −−−−−→ λ → + ∞ λ (cid:48) → + ∞ δ ( r ) δ ( r (cid:48) ) , (9)where the usual expansion of the plane waves in sphericalcomponents is used, and the limit in the last line comes from Eq. (3.5) of Ref. [30].In what follows, we will employ the regularized inter-action v λλ (cid:48) ( r (cid:48) , r ) to evaluate the matrix elements of theinteraction (6), and at times compare with the matrixelements obtained by using the bare interaction v ( r (cid:48) , r ). B. Uniform matter and the different choices forthe cutoff
In this subsection, we wish to establish a connectionbetween the cutoff on the transferred momentum (2) [24,25] and the cutoff on the relative momenta (3). At thesame time, we deal in this subsection with that fact thatin the procedure adopted in Refs. [24, 25] there is nocutoff affecting the HF energy. In the present scheme,we introduce a cutoff consistently in the HF and second-order energies.The HF potential energy, shown diagrammatically in the upper part of Fig. 1, is E HF = 12 (cid:88) ij (cid:104) ij | ¯ V | ij (cid:105) , (10)where ¯ V is the antisymmetrized interaction. If we writethe HF energy in symmetric nuclear matter as in Ref.[24], we obtain EA = 8 dgk F ρ (2 π ) π (cid:90) d ˜ k (cid:18) −
32 ˜ k + 12 ˜ k (cid:19) θ (1 − ˜ k ) = 38 gρ, (11)where d is the level degeneracy (4 in the case of symmetricnuclear matter) and ˜ k is defined only in this subsection,for the sake of convenience, as ˜ k ≡ k / √ k F . If we nowwish to introduce the cutoff λ on k , we have to add afactor θ (cid:16) λ √ k F − ˜ k (cid:17) . Then, Eq. (11) becomes EA = 8 dgk F ρ (2 π ) π (cid:90) d ˜ k (cid:18) −
32 ˜ k + 12 ˜ k (cid:19) θ (1 − ˜ k ) θ (cid:18) λ √ k F − ˜ k (cid:19) = 38 gρ (cid:0) β − β + 2 β (cid:1) , (12) ρ [fm − ] -16-8081624324048 E / A [ M e V ] λ = ∞λ = − λ = − λ = − λ = − FIG. 3. (Color online) Energy per particle at the HF level[cf. Eqs. (10,12)] for different values of the cutoff λ on therelative momentum k . where β = min { , λ √ k F } . Clearly, if λ > √ k F , then β = 1 and we recover the result of Eq. (11). This hasbeen tested also numerically, and the result is displayedin Fig. 3. Note the similar figure and reasoning in Ref.[28].As for the second order contribution to the total en-ergy, the relation between momenta used in the present work and those employed previously [24, 25] can be writ-ten, generalizing Eq. (3), as kk (cid:48) k (cid:48)(cid:48) = √ − √ √ − √ √ √ √ k k q . (13)The determinant of the Jacobian matrix of this transfor-mation is 1. One can note that k (cid:48) = k + √ q . (14)The second order contribution to the total energy, dis-played in the lower part of Fig. 1 with diagrams, is∆ E = 12 (cid:88) ijmn (cid:104) mn | V | ij (cid:105)(cid:104) ij | ¯ V | mn (cid:105) ε i + ε j − ε m − ε n , (15)where ε are HF s.p. energies. We evaluate this expressionin symmetric matter and we keep the notation of thissubsection, that is, ˜ k ≡ k / √ k F and ˜ k (cid:48) ≡ k (cid:48) / √ k F ,˜ k (cid:48)(cid:48) ≡ k (cid:48)(cid:48) / √ k F in an analogous way. Thus, we obtain∆ EA = χ ( ρ ) √ π (cid:90) D (˜ k , ˜ k (cid:48) , ˜ k (cid:48)(cid:48) ) d ˜ k d ˜ k (cid:48) d ˜ k (cid:48)(cid:48) k (cid:48) − ˜ k , (16)where χ ( ρ ) has been defined in Ref. [24], and the domainof integration is D (˜ k , ˜ k (cid:48) , ˜ k (cid:48)(cid:48) ) ≡ (cid:26) ˜ k , ˜ k (cid:48) , ˜ k (cid:48)(cid:48) ∈ R : ˜ k ≤ , ˜ k (cid:48)(cid:48) ≤ , (cid:16)(cid:12)(cid:12)(cid:12) ˜ k (cid:48)(cid:48) + ˜ k (cid:12)(cid:12)(cid:12) < ∩ (cid:12)(cid:12)(cid:12) ˜ k (cid:48)(cid:48) − ˜ k (cid:12)(cid:12)(cid:12) < (cid:17) ∪ (cid:16)(cid:12)(cid:12)(cid:12) ˜ k (cid:48)(cid:48) + ˜ k (cid:48) (cid:12)(cid:12)(cid:12) > ∩ (cid:12)(cid:12)(cid:12) ˜ k (cid:48)(cid:48) − ˜ k (cid:48) (cid:12)(cid:12)(cid:12) > (cid:17)(cid:27) . Our purpose is now to compare with the results ofRef. [24] and convince ourselves that we can use the in-teractions that have been fitted therein. To this aim, wemust consider the case in which the cutoff λ on relativemomenta is larger than √ k F , since otherwise the HF en-ergy should be also modified compared to the calculation with the bare force performed in Ref. [24] (cf. above).On top of this, the calculation of the integral appearingin Eq. (16) is rather cumbersome, and can be slightlysimplified if λ is larger than 2 √ k F . In this case a de-tailed analytic evaluation has been carried out [31]. Theresult can be written as∆ EA = χ ( ρ ) (cid:26) − λ − λ −
221 ˜ λ − (cid:32) λ − λ (cid:33) ln (cid:16) ˜ λ (cid:17) + (cid:32) − ˜ λ λ − λ (cid:33) ln (cid:16) ˜ λ − (cid:17) − (cid:32) − ˜ λ − λ λ (cid:33) ln (cid:16) ˜ λ + 1 (cid:17)(cid:27) , (17)where ˜ λ ≡ λ √ k F . We have checked that the part thatdoes not depend on λ is equal to the one already writtenin Ref. [24], as it should, and that the divergence is linear.To obtain a better understanding, we have evaluatednumerically the two results given by Eq. (17) and Eq. (8)of Ref. [24]. The two calculations are almost indistin-guishable when λ = √ . (18)We will then use this latter equation in the following way:when we perform a calculation of a finite system withcutoff λ we will adopt the interaction fitted in uniformsymmetric matter with the value of λ given by Eq. (18).Ultimately, we would envisage to cast uniform matterand finite nuclei in a single scheme, so to be able to fitan effective force in the same spirit of the original Skyrme force (at second order and then beyond). C. The formalism for finite nuclei using theharmonic oscillator basis
In finite nuclei the second order energy is still givenby Eq. (15) but is more conveniently written in angularmomentum-coupled representation as∆ E = 14 (cid:88) pp (cid:48) hh (cid:48) J (2 J + 1) |(cid:104) ( pp (cid:48) ) J | ¯ V | ( hh (cid:48) ) J (cid:105)| ε h + ε h (cid:48) − ε p − ε p (cid:48) , (19)where the particle-particle (pp) coupled matrix elementshave been introduced, namely (cid:104) ( αβ ) JM J | ¯ V | ( γδ ) JM J (cid:105) = (cid:88) m α m β m γ m δ (cid:104) j α m α j β m β | JM J (cid:105)(cid:104) j γ m γ j δ m δ | JM J (cid:105)(cid:104) αβ | ¯ V | γδ (cid:105) = (cid:88) m α m β m γ m δ ( − ) j α − j β + j γ − j δ ˆ J (cid:18) j α j β Jm α m β − M J (cid:19) (cid:18) j γ j δ Jm γ m δ − M J (cid:19) (cid:104) αβ | ¯ V | γδ (cid:105) , (20)where we have introduced the common shorthand nota-tion ˆ J = 2 J + 1. Actually, these latter matrix elementsdo not depend on M J because of rotational invariance.Therefore, in Eq. (19) they appear without this label;in that equation, the trivial sum over M J has been per-formed.In our calculation, we expand the single-particle wavefunctions in harmonic oscillator basis. Then, the corre-sponding matrix elements are evaluated by performingthe transformation of the initial and final two-particlestates to the center of mass and relative motion coordi-nates. As is well known, this can be done in the HO case by exploiting the Brody-Moshinsky transformations[32–34]. In this subsection we will collect only the mainequations related to the matrix elements entering our cal-culations; we shall provide some more details about themain steps of their derivation, together with other usefulformulas, in the Appendix.We shall indicate with the label I = 0, σ , τ and στ the terms of the pp-coupled matrix elements (20)that are proportional, respectively, to the identity inspin-isospin space, to σ (1) σ (2), to τ (1) τ (2) and to σ (1) σ (2) τ (1) τ (2). The final expression for these termsreads (cid:104) ( n a l a j a τ a , n b l b j b τ b ) JM J | ¯ V | ( n c l c j c τ c , n d l d j d τ d ) JM J (cid:105) I == N I F I (cid:88) Σ L i − l a − l b + l c + l d ˆ L ˆΣ ˆ j a ˆ j b ˆ j c ˆ j d G I l a l b L
12 12 Σ j a j b J l c l d L
12 12 Σ j c j d J λ λ (cid:48) π (cid:88) n i N i n f N f M L ( N f Ln f n a l a n b l b ) M L ( N i Ln i n c l c n d l d ) (21) (cid:90) d RR R N f L ( √ βR ) g ( R ) R N i L ( √ βR ) (cid:90) d r rR n i ( βr ) j ( rλ ) (cid:90) d r (cid:48) r (cid:48) R n f ( βr (cid:48) ) j ( r (cid:48) λ (cid:48) ) . Here the single-particle states are labelled by the usualquantum numbers in spherical symmetry, n, l, j , togetherwith the third component of the isospin τ . These single-particle states are expanded in the harmonic oscilla-tor basis, and β is the harmonic oscillator parameter, β ≡ (cid:112) mω/ (cid:126) . The harmonic oscillator single-particlestates and their radial wave functions R are defined inEq. (A1). The symbol M L corresponds to the Brody-Moshinsky coefficients. The quantities N I , F I , G I are de-fined in the Appendix. Although the structure of theformula should look clear, as it includes the transforma-tions (i) to LS -coupling, (ii) to the harmonic oscillatorbasis, (ii) to the center-of-mass and relative coordinates,reading the Appendix may shed some further light on it.The two-body matrix elements (21) constitute thebackbone of our calculation. Nonetheless, since as ex-plained above we will also use the renormalized interac-tion at the mean field level, it is useful to provide in thissubsection the final form of the one-body matrix elements of the HF Hamiltonian, that are h ( α ) ab = t ab (22)+ (cid:88) βε β ≤ ε F (cid:88) cd c ∗ β,c (cid:104) ac | ¯ V | bd (cid:105) c β,d (23)+ 12 (cid:88) βγε β,γ ≤ ε F (cid:88) cdef c ∗ β,c c ∗ γ,d (cid:104) cd | ∂ ¯ V∂c ∗ α,a | ef (cid:105) c β,e c γ,f , (24)where c denotes the expansion coefficients of the s.p.states on the harmonic oscillator basis, the Greek let-ters represent the set of quantum number which identifya s.p. state and the Latin letters indicate the harmonicoscillator basis quantum number.The explicit expression for the term (22) can be easilyfound in Ref. [35]. The second term (23) can be writ-ten using Eq. (21). Nevertheless, the expression can befurther simplified because of two simple considerations[36]: • we are dealing with even-even nuclei, thus the ma-trix elements of the operator σ (1) σ (2) vanishes; • there is no charge mixing of the HF states, so theisospin exchange operator P τ reduces to a Kro-necker delta.With these simplifications and by using the orthogonalityrelations for the 9 − j symbol, we get (cid:88) βε β ≤ ε F (cid:88) cd c ∗ β,c (cid:104) ac | ¯ V | bd (cid:105) c β,d = (cid:88) βε β ≤ ε F (cid:88) cd (cid:88) J ˆ J ˆ j α c ∗ β,c (cid:104) ( ac ) JM | ¯ V | ( bd ) JM (cid:105) c β,d = (cid:88) βε β ≤ ε F (cid:88) cd c ∗ β,c c β,d (cid:88) L (cid:88) n i N i n f N f ˆ L ˆ j β ˆ l α ˆ l β M L ( N f Ln f al α cl β ) M L ( N i Ln i bl α dl β ) × (cid:18) − δ q α ,q β (cid:19) (cid:90) d RR R N f L ( √ βR ) g ( R ) R N i L ( √ βR ) (25) × λ λ (cid:48) π (cid:90) d r rR n i ( βr ) j ( rλ ) (cid:90) d r (cid:48) r (cid:48) R n f ( βr (cid:48) ) j ( r (cid:48) λ (cid:48) ) . Following the same strategy, the last term (24), which isthe rearrangement term, can be written as12 (cid:88) βγε β,γ ≤ ε F (cid:88) cdef c ∗ β,c c ∗ γ,d (cid:104) cd | ∂ ¯ V∂c ∗ α,a | ef (cid:105) c β,e c γ,f = 12 (cid:88) βγε β,γ ≤ ε F (cid:88) cdef c ∗ β,c c ∗ γ,d c β,e c γ,f × (cid:88) L (cid:88) n i N i n f N f ˆ L ˆ j γ ˆ j β ˆ l γ ˆ l β M L ( N f Ln f cl β dl γ ) M L ( N i Ln i el β f l γ ) × (cid:18) − δ q γ ,q β (cid:19) (cid:90) d RR R N f L ( √ βR ) g (cid:48) ( R ) R N i L ( √ βR ) (26) × λ λ (cid:48) π (cid:90) d r rR n i ( βr ) j ( rλ ) (cid:90) d r (cid:48) r (cid:48) R n f ( βr (cid:48) ) j ( r (cid:48) λ (cid:48) ) , where g (cid:48) ( R ) = t α π R al α ( βR ) R bl α ( βR ) ρ α − ( R ). III. RESULTS
In our work we have focused on the calculation of thetotal energy (19) in O. As explained in the previousSections, we aim at using interactions fitted with a cut-off regularization in uniform matter and check that thisstrategy is enough to prevent the divergence of the totalenergy in the finite system. The relation between thecutoffs that are used throughout our procedure has beengiven in Eq. (18) above, and reads λ = √ . The interactions V Λ associated with the re-fit of sym-metric matter, when the second-order contribution has an associated cutoff Λ, are provided in Table I. As al-ready mentioned, we employ an harmonic oscillator ba-sis. The oscillator parameter β ≡ (cid:112) mω/ (cid:126) is 0.5 fm − and the number of oscillator shells is n max = 10. Theradial wave functions are calculated up to a maximumvalue of r given by R = 12 fm.In Fig. 4 we display the mean-field energy obtainedwith the renormalized interactions, as a function of λ , bymeans of the line labeled with SkP Λ . As a reference weprovide the same quantity calculated with the bare inter-action SkP (line labeled with SkP), and the mean fieldvalue (without cutoff) which is associated with the hor-izontal line and is − . TABLE I. Parameter sets (named SkP Λ ) obtained in the fits associated with different values of the cutoff Λ compared with theoriginal set SkP, labeled with SkP (first line) [37]. t t α t t α SkP − / . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − λ [fm − ]-250-200-150-100-50050 E H F [ M e V ] SKPSKP Λ FIG. 4. (Color online) Total HF energy as a function of acutoff λ . The dashed (dot-dashed) curve corresponds to theresult obtained with SkP with only t , t terms (SkP Λ ). Thethin dotted line corresponds to the result without any cutoff,obtained with SkP with only t , t term, and is meant toguide the eye for the convergence of the dashed line. See thetext for a more detailed discussion. tal value of the binding energy of O which is known tobe − .
619 MeV [38]. The HF result obtained withSkP can be understood, since when the cutoff increasesthe calculation tends obviously to the exact one, whilereducing the momentum components amounts to tryingto minimize the energy in a Hilbert space which is not complete: although the variational principle cannot berigorously invoked, it is plausible that the energy doesnot attain its minimum and is instead smaller in abso-lute value. It is instructive to note that the convergenceto the exact result is obtained only for λ of the order of2.5 fm − : in fact, inside this light nucleus the densitycan rise up to 0.25 fm − (cf. Fig. 5) and the associatedmaximum effective (local) k F is therefore ≈ − ,so that the maximum value for the momenta defined inEq. (3) can be as high as 2.2 fm − . The result asso-ciated with the renormalized interaction is more subtleto understand. For low values of the cutoff energy therenormalization of the interaction is not significant (asone can notice from the values of the parameters in Ta-ble I). This is understandable, since for small values of λ the second-order contribution in infinite matter is small,and one needs to weakly renormalize the interaction inorder to obtain in the same system the HF energy asso-ciated with the bare interaction. As a consequence, forsmall values of λ the curves associated with the bare andrenormalized interactions overlap. However, for large val-ues of λ the total energy still decreases in absolute valuewhen it is calculated with the renormalized interaction.In this case, in fact, most of the momentum componentsare retained, but the interaction is strongly renormalized(again, this can be seen from the values of the parametersin Table I) and, as a consequence, the mean-field totalenergy is small. As a conclusion, we infer from Fig. 4that for either too small or too large values of λ the sys-tem calculated at mean-field level with the renormalizedinteraction is far away from the system we would like toreproduce by adding the second-order contribution: inother words, perturbation theory is doomed to fail forthose values of λ , especially if we start from a situationin which the total energy is positive at mean-field butnot only in that case. In practice, we restrict our follow-ing discussion to, and draw conclusions from, values of λ between ≈ − (although we will show inthe figures some results associated with a broader rangeof values for λ ). r [fm]10 -3 -2 -1 ρ ( r ) [f m − ] full HF full HF λ = 0.85 fm − λ = 1.41 fm − λ = 1.13 fm − λ = 2.26 fm − λ = 2.83 fm − λ = 3.11 fm − λ = 0.57 fm − λ = 1.98 fm − λ = 1.70 fm − λ = 0.28 fm − FIG. 5. (Color online) Total density profiles obtained with therenormalized SkP Λ interactions. The thick black line refersto the bare interaction. The inset shows some detail of theregion in which the density attains its largest values. This view is in part confirmed by the results shown inFig. 5: here we display the different profiles for the totaldensity emerging from the HF calculations when differentrenormalized interactions are employed. Along the sameline of the discussion in the previous paragraph, if thecutoff is small a large fraction of the high momentumor small distance components of the relative motion arecut, and the system becomes very dilute, almost like auniform unbound nucleon gas.We now discuss our main results, that are summarizedin the three panels of Figs. 6 and 7. In the left panel ofFig. 6, the total energy calculated at second order withthe renormalized interactions is shown, for various val-ues of λ , as a function of the maximum particle energy ε max p . For the sake of clarity only a selection of the resultsobtained with the interactions associated with differentvalues of Λ are displayed. For values of λ between ≈ − , the results are close to one another. Evenmore importantly, these results do not depend on thevalue of ε max p , at least if this is larger than ≈
80 MeV.This can be understood on the basis of a simple semi-classical argument: particles having energies larger than ≈
80 MeV, and having thus very large kinetic energies,would contribute to the total energy through matrix el-ements associated with momentum components that areactually eliminated by our choice of the cutoff. There-fore, the most important conclusion is that our proposed renormalization procedure can work, and the extra scaleassociated with the maximum value of the particle en-ergy, or with the total momentum, does not spoil thatprocedure.The stability of the renormalized results for the secondorder energy, is also visible in the right panel of Fig. 6.The curves associated with values of λ between ≈ − , for a broad range of values of ε max p , lie in theshaded box that corresponds to ≈
10% error in the totalenergy. A quick glance of the behavior of our resultsis allowed by the plot of Fig. 7, that collects the sameinformation already provided in the two panels of Fig. 6by means of a more intuitive three-dimensional globalrepresentation.
IV. CONCLUSIONS
The problem associated with the fact that zero-rangeforces produce divergent results when employed beyondmean-field has become object of a renewed interest.Skyrme forces have zero-range character but also Gognyforces possess a zero-range terms; pure finite-range forceshave not been so widely developed and systematically ap-plied in non-relativistic approaches.The problem of the renormalization of these diver-gences has been tackled first in a simple system like uni-form nuclear matter, restricting to the case of the second-order correction to the energy. The purpose of this workis to make a significant first step in the direction of afull renormalization scheme for the Skyrme force in finitenuclei. Our main idea is to work in a harmonic oscilla-tor basis, so that the center-of mass and relative motioncoordinates and associated momenta can be neatly sep-arated. The cutoff we set on the momentum associatedwith the relative motion can be then related to the oneused in the previous calculations of nuclear matter. Wehave illustrated such formal scheme in full detail in thispaper.As a numerical application, we have limited ourselvesto O calculated with a simple, momentum-independentSkyrme force. Our main results are listed as follows: • under certain conditions, it is possible to relate thecutoff λ on the relative momentum to the cutoff Λthat has been previously introduced in [24, 25] [cf.Eq. (18) and the related discussion]; • if calculations of the total energy at second orderare envisaged, with a cutoff λ , the interactions in-troduced in uniform matter using the associatedvalue of Λ can be employed; • the practical way to do these calculations is to workin harmonic oscillator basis and change the form ofthe interaction so that relative-momentum compo-nents larger than λ are cut; • at least for a reasonable range of values, λ ≈ − . ε p max [MeV]-200-150-100-50050 E + ∆ E [ M e V ] λ = 0.28 fm − λ = 0.85 fm − λ = 1.70 fm − λ = 1.98 fm − λ = 2.26 fm − λ = 2.55 fm − λ = 2.83 fm − λ = 3.11fm − λ [fm − ]-160-140-120-100-80-60 E + ∆ E [ M e V ] ε p max = 20 MeV ε p max = 40 MeV ε p max = 60 MeV ε p max = 80 MeV ε p max = 100 MeV ε p max = 120 MeV ε p max = 140 MeV ε p max = 160 MeV ε p max = 180 MeV ε p max = 200 MeV FIG. 6. (Color online) Total energy at second order as a function of the maximum particle energy (left panel) or as a functionof the cutoff (right panel). All curves are obtained with renormalized interactions. See the text for a discussion. Λ (cid:64) fm (cid:45) (cid:68) (cid:182) p max (cid:64) MeV (cid:68) (cid:45) (cid:45) (cid:45) (cid:69)(cid:43)(cid:68) E (cid:64) MeV (cid:68) (cid:69)(cid:43)(cid:68) E (cid:64) MeV (cid:68) (cid:45) (cid:45) (cid:45)
FIG. 7. (Color online) The same as Fig. 6 in a three-dimensional representation, that is, total energy at secondorder as a function of both the maximum particle energy andthe cutoff. fm − , the results turn out to be stable, namely thedivergence does not show up; • in particular, the second energy scale associatedwith the total momentum, that in finite systems isassociated with the maximum value of the particleenergy ε max p , does not seem to spoil this stability.This can be justified by semiclassical arguments.In terms of perspectives, several issues remain to beconsidered. Our results look promising only in a limitedwindow for values of the cutoff. We are inclined to thinkthat this is due to the choice of a perturbative scheme tocalculate the second-order energies. At variance with thecase of infinite matter, a consistent second-order calcu-lation that employs the proper equations (Dyson equa-tion for the s.p. energies, Koltun sum rule for the total energy) is probably called for. On top of this, we aredealing with a simple Skyrme force which is not very re-alistic and the extension to the full force must be alsoenvisaged. Only after this, one could judge if the plan ofdevising a zero-range force that is fitted and consistentlyused beyond mean-field, is feasible. In this respect, ourresults can be seen as promising but we can draw onlyqualitative, and not too much quantitative, conclusionsat the present stage. ACKNOWLEDGMENTS
M.B. thanks IPN Orsay for the warm hospitality dur-ing the time when part of this work was carried out. Theauthors are grateful to P. F. Bortignon, B. G. Carlsson,M. Grasso, E. Khan, K. Moghrabi, J. Piekarewicz, P.Ring and N. Van Giai for useful discussions and com-ments.
Appendix A: Derivation of the pp-coupled matrixelements
We discuss here, in some detail, the steps that are nec-essary to evaluate the particle-particle coupled matrixelements of Eq. (20) and we give some intermediate for-mula which can be useful for the reader.Let us consider a two-particle state | n a l a j a m a , n b l b j b m b (cid:105) in a harmonic oscillator potential.The single particle wave functions are | nljmτ (cid:105) = ψ τnljm ( r ) = i l R nl ( βr ) (cid:104) Y l (ˆ r ) ⊗ χ (cid:105) jm ξ τ , (A1)where β = mω (cid:126) . If the two states are coupled to to-tal angular momentum JM as in Eq. (20), we need to1switch from the j – j coupling scheme to the L – S one be- fore making the Brody-Moshinsky transformation. Then,the two-particle states read | n a l a j a m a τ a , n b l b j b m b τ b (cid:105) = (cid:88) JM J ΛΣ ˆΛ ˆΣˆ j a ˆ j b (cid:104) j a m a j b m b | JM J (cid:105) l a l b Λ
12 12 Σ j a j b J | [ n a n b , ( l a , l b )Λ , ( 12 ,
12 )Σ , τ a τ b ] JM J (cid:105) = (cid:88) J ΛΣ (cid:88) M Λ M Σ M J M l M L (cid:88) σ a σ b i l a + l b ( − ) j a − j b +Λ+ M Λ +Σ+ M Σ + L + l ˆ J ˆΛ ˆΣ ˆ j a ˆ j b l a l b Λ
12 12 Σ j a j b J (cid:32) j a j b Jm a m b − M J (cid:33) (cid:32) Λ Σ JM Λ M Σ − M J (cid:33) (cid:32)
12 12 Σ σ a σ b − M Σ (cid:33) (cid:32) L l Λ M L M l − M Λ (cid:33)(cid:88) nlNL M Λ ( N Lnl ; n a l a n b l b ) R nl ( βr ) R NL ( βR ) Y lM l (ˆ r ) Y LM L ( ˆ R ) χ σ a (1) χ σ b (2) ξ τ a (1) ξ τ b (2) , (A2)where the center of mass and relative motion coordinateshave been introduced as in Eq. (5) and the correspondingBrody-Moshinsky coefficients are denoted by M Λ [32–34].In addition, several intermediate quantum numbers areintroduced.We want to compute the matrix elements of theSkyrme interaction, written as in Eq. (4), between the two-particle states (A2). We are interested in the anti-symmetrized interaction ¯ V = V (1 − P M P σ P τ ), where P M is the Majorana exchange operator while P σ and P τ arethe spin and isospin exchange operators. In the case ofthe center of mass and relative motion coordinate systemthe Majorana operator is non-trivial. The exchange op-erator acts on the two-particle state (A2) in the followingway: | n b l b j b m b τ b , n a l a j a m a τ a (cid:105) = P M P σ P τ | n a l a j a m a τ a , n b l b j b m b τ b (cid:105) = i l a + l b ˆ j a ˆ j b (cid:88) JM J (cid:88) Λ M Λ (cid:88) Σ M Σ (cid:88) LM L (cid:88) lM l (cid:88) Nn (cid:88) σ a σ b ( − l a + l b − L ˆ J ˆΛ ˆΣ l a l b Λ
12 12 Σ j a j b J ( − j a − j b +Λ+ M Λ +Σ+ M Σ + L + l M Λ ( N Lnl ; n a l a n b l b ) R nl ( βr ) R NL ( βR ) Y lM l (ˆ r ) Y LM L ( ˆ R ) (cid:32) j a j b Jm a m b − M J (cid:33) (cid:32) Λ Σ JM Λ M Σ − M J (cid:33) (cid:32)
12 12 Σ σ a σ b − M Σ (cid:33) (cid:32) L l Λ M L M l − M Λ (cid:33) P σ [ χ σ a (1) χ σ b (2)] P τ [ ξ τ a (1) ξ τ b (2)] , that is equal to Eq. (A2) except for the P σ , P τ operatorsand the phase factor ( − l a + l b − L . This can be checkedby direct calculation.As mentioned in the main text (cf. Subsec. II C) weprovide separately the expressions for the different spin and isospin terms of the pp-coupled matrix elements (20),and we shall use for them, respectively, the label I thatcan assume the values I = 0 , σ, τ, στ . We also introducethe following quantities:2 F I = δ τ a τ c δ τ b τ d if I = 0 , σ (cid:80) µ ( − ) τ a + τ b + µ (cid:32) τ c µ − τ a (cid:33) (cid:32) τ d − µ − τ b (cid:33) if I = τ, στ G I = I = 0 , τ ( − ) (cid:40)
12 12 Σ
12 12 (cid:41) if I = σ, τ N I = if I = 0 − if I = σ, τ − I = στ M I = (cid:40)(cid:0) − ( − l c + l d − L (cid:1) if I = 0( − l c + l d − L if I = στ. Then, the general expression for the four terms of the matrix elements reads (cid:104) ( n a l a j a τ a , n b l b j b τ b ) JM J | ¯ V | ( n c l c j c τ c , n d l d j d τ d ) JM J (cid:105) I == N I F I (cid:88) ΛΣ Ll i − l a − l b + l c + l d ( − ) l M I G I l a l b Λ
12 12 Σ j a j b J l c l d Λ
12 12 Σ j c j d J ˆΛ ˆΣ ˆ j a ˆ j b ˆ j c ˆ j d ˆ l (cid:88) n i N i n f N f M Λ ( N f Ln f l ; n a l a n b l b ) M Λ ( N i Ln i l ; n c l c n d l d ) (cid:90) d RR R N f L ( √ βR ) g ( R ) R N i L ( √ βR ) (cid:90) d r d r (cid:48) r r (cid:48) R n f l ( βr (cid:48) ) v lm ( r (cid:48) , r ) R n i l ( βr ) . (A3)It can be a useful exercise to insert in this expressionthe standard coefficients of the multipole expansion ofthe velocity-independent part of the Skyrme interaction, that are v lm ( r (cid:48) , r ) = ( − ) l ˆ l π δ ( r ) r δ ( r (cid:48) ) r (cid:48) . (A4)Then the matrix element reads (cid:104) ( n a l a j a τ a , n b l b j b τ b ) JM J | ¯ V | ( n c l c j c τ c , n d l d j d τ d ) JM J (cid:105) I == N I F I (cid:88) Σ L i − l a − l b + l c + l d ˆ L ˆΣ ˆ j a ˆ j b ˆ j c ˆ j d π G I l a l b L
12 12 Σ j a j b J l c l d L
12 12 Σ j c j d J (cid:88) n i N i n f N f M L ( N f Ln f n a l a n b l b ) M L ( N i Ln i n c l c n d l d ) (A5) R n i (0) R n f (0) (cid:90) d RR R N f L ( √ βR ) g ( R ) R N i L ( √ βR ) . This expression does not include isospin coupling (whenthis is considered, cf. the analogous expression in e.g. Ref. [32]).For the renormalized interaction the multipole expan-3sion coefficients can be found, instead, as v λλ (cid:48) lm ( r (cid:48) , r ) = 14 π λ λ (cid:48) rr (cid:48) j ( λr ) j ( λ (cid:48) r (cid:48) ) ( − ) l ˆ l (cid:88) m (cid:90) dˆ r (cid:48) dˆ r Y ∗ lm (ˆ r ) Y lm (ˆ r (cid:48) )= 14 π λ λ (cid:48) rr (cid:48) j ( λr ) j ( λ (cid:48) r (cid:48) ) ( − ) l ˆ l (cid:88) m πδ l, δ m, = 14 π λ λ (cid:48) rr (cid:48) j ( λr ) j ( λ (cid:48) r (cid:48) ) δ l, ; (A6)then, the corresponding matrix element reads (cid:104) ( n a l a j a τ a , n b l b j b τ b ) JM J | ¯ V | ( n c l c j c τ c , n d l d j d τ d ) JM J (cid:105) I == N I F I (cid:88) Σ L i − l a − l b + l c + l d ˆ L ˆΣ ˆ j a ˆ j b ˆ j c ˆ j d G I l a l b L
12 12 Σ j a j b J l c l d L
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