Comment on "Breakdown of the tensor component in the Skyrme energy density functional''
aa r X i v : . [ nu c l - t h ] A ug Comment on “Breakdown of the tensor component in the Skyrme energy densityfunctional”
H. Sagawa ∗ RIKEN Nishina Center, Wako 351-0198, Japan andCenter for Mathematics and Physics, University of Aizu, Aizu-Wakamatsu, Fukushima 965-8560, Japan
G. Col`o † Dipartimento di Fisica “Aldo Pontremoli”, Universit`a degli Studi di Milano, 20133 Milano, Italy andINFN, Sezione di Milano, 20133 Milano, Italy
Ligang Cao ‡ College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China (Dated: August 4, 2020)In a recent paper [Phys. Rev. C 101, 014305 (2020)], Dong and Shang claim that the Skyrmeoriginal tensor interaction is invalid. Their conclusion is based on the misconception that theFourier transform of tensor interaction is difficult or even impossible, so that the Skrme-type tensorinteraction was introduced in an unreasonable way. We disagree on their claim. In this note, weshow that one can easily get the Skyrme force in momentum space by Fourier transformation if onestarts from a general central, spin-orbit or tensor interaction with a radial dependence.
I. INTRODUCTION
In a recent paper published in Phys. Rev. C (Ref.[1], hereafter referred to as Dong and Shang), the au-thors claim in the abstract that “the Skyrme originaltensor interaction [...] is invalid.” One has to remindthat, although the original idea by Skyrme dates backto long time ago and the Skyrme tensor force has beenwritten out in Refs. [2, 3], there have been many stud-ies on how to implement it and fit its parameters sincethen, even in recent years. A first series of papers ap-peared in the 1970s, when the first Skyrme forces hadbeen proposed as a practical tool for Hartree-Fock (HF)and Random Phase Approximation (RPA). A second se-ries, much more numerous, has appeared in the yearssince 2000. In 2014, some of us published a review pa-per on the tensor force within mean-field and DensityFunctional Theory (DFT) approaches to nuclear struc-ture [4]. In that review, we quoted ≈ ∗ [email protected] † [email protected] ‡ [email protected] be found in Sec. II of the paper. In this Section, one canread: “The Skyrme original tensor force was introducedin an unreasonable way, because the tensor-force oper-ator S in momentum space but with an r -dependentstrength, i.e., f T ( r ) S ( k ), is applied as a starting point.”We remind, for the reader’s convenience, that S ( k ) = ( σ · k )( σ · k ) − k σ · σ . (1)We cannot see any logic behind the latter sentence inthe work by Dong and Shang. In fact, it does not haveany formal ground. In physics, nothing forbids an in-teraction to be at the same time position-dependent andmomentum-dependent. Bethe [5] was one of the first toadvocate that this ought to be the case if one wishes tointroduce an effective potential for finite nuclei. Skyrmeechoed this at the beginning of his first paper on thistopic, and it is useful at this stage to quote literally thesentence from Ref. [6]: “in the case of a finite system theeffective potential must depend upon both momenta andcoordinate.”In fact, this latter sentence does not simply lie as aground for the Skyrme tensor force, but rather it is atthe basis of the whole philosophy behind Skyrme forcesand Skyrme functionals [6, 7]. Several authors have pro-posed Skyrme-type forces with terms that have bothmomentum- and density-dependence. If the Skyrme ten-sor force is unreasonable because of the reason advocatedby Dong and Shang, the whole Skyrme force will be un-reasonable. If this were the case, this would disgrace notonly the results of a few hundred papers in which Skyrmetensor terms are introduced, but also some more ≈ papers in which Skyrme forces are used, including thepaper by Dong and Shang themselves. As we said, wesee no reason to rule out a force because it looks like amomentum-dependent operator times a function of therelative coordinate r .It should also be noticed that any comparison witha realistic potential used in Br¨uckner-HF calculations,or similar cases, is immaterial in this context. We arenot discussing any realistic potential but only the case ofan effective potential to be used in HF or DFT. Skyrmehimself, in Ref. [2], employed the word “pseudopotential”to distinguish clearly his approach from any one basedon a fundamental interaction. In modern language, wewould say that we deal with an effective feld theory inwhich the coupling terms depend on δ ( r − r ) timesderivatives. An analogous case is that of the pionlessEFT.In the paper by Dong and Shang, it is also claimed thatthe Skrme-type tensor interaction is introduced in an un-reasonable way since the Fourier transform of such tensorinteraction is difficult or even impossible. We disagreeon this claim based on the procedure that we discuss ex-plicitly in the next Section. In the next Section we willshow that if we start from a general central, spin-orbitor tensor interaction, with a radial dependence such thatthe range is very short, the Fourier transform producesthe Skyrme force in momentum space. This is a further,more detailed and mathematically rigorous, way to showthat the arguments in the paper by Dong and Shang areinvalid. II. FOURIER TRANSFORM OF THE TERMSOF THE SKYRME FORCE
Let us write the Fourier transform of an interaction V ( r ) as V ( q ) = Z e i q · r V ( r ) d r , (2)where q = k − k ′ is the momentum transfer ( k and k ′ are the initial and final relative momenta). For a centralinteraction V C ( r ), the integral in (2) can be performedby using the multipole expansion of a plane wave e i q · r = 4 π X λµ i λ j λ ( qr ) Y λµ (ˆ q ) Y ∗ λµ (ˆ r ) , (3)where j λ ( qr ) is the spherical Bessel function of rank λ ,and Y λµ ( q ) is a spherical harmonic. Thus, the integralfor a central interaction becomes V ( q ) ∝ Z j ( qr ) V C ( r ) r dr. (4)The constant and the lowest-order momentum-dependentterms of the Skyrme interaction are obtained by meansof the Taylor expansion j ( qr ) = 1 − ( qr ) / · · · . Theconstant term of this expansion provides the t term inthe Skyrme force. The second term is written as V ( q ) ∝ q Z V C ( r ) r dr, (5) with q = k + k ′ − k · k ′ . (6)The first two terms in Eq. (6) provide the t term of theSkyrme force, while the third term gives the t term.These two terms mimic the finite range in the effec-tive two-body interaction. The radial integrals in Eqs.(4) and (5) can be easily performed for any commonlyadopted Yukawa-type or Gaussian-type finite-range in-teraction.The spin-orbit interaction V LS , V LS = f LS ( r )( σ + σ ) · ( r − r ) × ( p − p ) , (7)can be also expanded, after being Fourier-transformedthrough Eq. (2), in the momentum operator q . Theradial dependence of V LS ( r ) is expressed by the sphericalharmonics Y µ , so that the expansion of Eq. (3) producesthe spherical Bessel function j ( qr ). We do not repeatthese steps here as they can be found in Ref. [7].The tensor interaction in the coordinate space is ex-pressed as V T ( r ) = f T ( r ) S ( r ) , (8)where S ( r ) = ( σ · r )( σ · r ) − r σ · σ . (9)The tensor operator of (9) is rewritten using the sphericalharmonic Y µ and becomes S ( r ) = r π σ · σ ) (2) × r Y (ˆ r )] (0) . (10)The Fourier transform can be evaluated as V T ( q ) = Z e i q · r V T ( r ) d r ∝ Z r drj ( qr )[( σ · σ ) (2) × Y (ˆ q )] (0) . (11)The spherical Bessel function j ( qr ) is proportional to q , as j ( qr ) ∼ ( qr ) / V T ( q ) becomes V T ( q ) ≃ − π r π σ · σ ) (2) × q Y (ˆ q )] (0) Z r f T ( r ) dr. (12)In this way, we obtain the tensor interactions in momen-tum space in the form S ( q ) = r π σ · σ ) (2) × q Y (ˆ q )] (0) (13)= ( σ · q )( σ · q ) − q σ · σ { [( σ · k ′ )( σ · k ′ ) − ( σ · σ ) k ′ ]+ [( σ · k )( σ · k ) − ( σ · σ ) k ] }− { ( σ · k ′ ) ( σ · k ) + ( σ · k ′ ) ( σ · k ) −
23 [( σ · σ ) k ′ · k ] } . (15)In the above expression, the operator k = ( ∇ − ∇ ) / i acts on the right and k ′ = − ( ∇ ′ − ∇ ′ ) / i acts on theleft. The first two terms in Eq. (15) are the so-calledtriplet-even tensor term (T-term in Skyrme), whereas the third and fourth terms correspond to a triplet-oddterm (U-term). In this way we can validate the Skyrmetype tensor interaction as the lowest-order expansion ofa finite-range tensor force.acts on theleft. The first two terms in Eq. (15) are the so-calledtriplet-even tensor term (T-term in Skyrme), whereas the third and fourth terms correspond to a triplet-oddterm (U-term). In this way we can validate the Skyrmetype tensor interaction as the lowest-order expansion ofa finite-range tensor force.