II-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 55 (2022)

RIKEN Accel. Prog. Rep. 55 (2022)

Comment on “Breakdown of the tensor component in the Skyrmeenergy density functional”†
H. Sagawa,∗1 G. Colò,∗2,∗3 and L. Cao∗4
In a recent paper published in Phys. Rev. C1) , hereafter referred to as Dong and Shang, the authors claimin the abstract that “the Skyrme original tensor interaction is invalid.” One should note that, although theoriginal idea by Skyrme dates back to the 1950s andthe Skyrme tensor force has been written out in Ref. 2),
there have been many studies on how to implement it
and ﬁt its parameters since then, even in recent years.
In 2014, some of us published a review paper on thetensor force within mean-ﬁeld and density functional
theory (DFT) approaches to the nuclear structure.3)
In that review, we quoted ≈30–40 papers where theSkyrme tensor is implemented and studied, authoredby diﬀerent colleagues. The conclusion of that review
paper is that evidences for a strong neutron-proton
tensor force exist, even in mean-ﬁeld and DFT studies. Therefore, if the tensor force proposed by Skyrme
were “invalid,” this would impact on a considerable
number of published works and conclusions drawn sofar. Accordingly, it would be appropriate to have astrong argument regarding this “invalidity.”
The argument against the Skyrme tensor force inRef. 1) was given as: “The Skyrme original tensor
force was introduced in an unreasonable way, because
the tensor-force operator S12 in momentum space butwith an r-dependent strength, i.e., fT (r)S12 (k), is applied as a starting point.” We note that S12 (k) can be
expressed as, S12 (k) = (σ1 · k)(σ2 · k) − k 2 σ13·σ2 . We
do not see any logic behind the sentence by Dong andShang. In physics, nothing forbids an interaction to beposition-dependent and momentum-dependent at thesame time. Bethe4) was one of the ﬁrst to advocatethat this ought to be the case if one wishes to introduce
an eﬀective potential for ﬁnite nuclei. Skyrme echoed
this at the beginning of his ﬁrst paper on this topic,and it is useful at this stage to quote literally the sentence from Ref. 5): “in the case of a ﬁnite system theeﬀective potential must depend upon both momentaand coordinate.”
Several authors have proposed Skyrme-type forces
with terms that have both momentum- and densitydependence. If the Skyrme tensor force is unreasonablebecause of the reason advocated by Dong and Shang,
the whole Skyrme force will be unreasonable. If this†
∗1
∗2
∗3
∗4

were the case, this would discredit not only the results of a few hundred papers in which Skyrme tensorterms are introduced but also some more ≈103 papers
in which Skyrme forces are used.
In the paper by Dong and Shang, it is also claimed
that the Skrme-type tensor interaction is introduced
in an unreasonable way since the Fourier transform ofsuch tensor interaction is diﬃcult or even impossible.
We disagree on this claim based on the procedure that
we discuss explicitly as follows. We show that if westart from a general central, spin-orbit or tensor interaction with a radial dependence such that the range isvery short, the Fourier transform produces the Skyrme
force in momentum space. Let us write the Fouriertransform of an interaction V (r) asV (q) = eiq·r V (r)dr,(1)where q = k − k is the momentum transfer (k andk are the initial and ﬁnal relative momenta, respectively). The tensor interaction in the coordinate spaceis expressed as
VT (r) = fT (r)S12 (r),where S12 (r) = (σ1 · r)(σ2 · r) − r2 σ13·σ2 =(2)8π3 [(σ1 ·σ2 )(2) × r2 Y2 (r̂)](0) with the spherical harmonics Y2μ .
The Fourier transform can be evaluated asVT (q) ∝ r4 drfT (r)j2 (qr)[(σ1 ·σ2 )(2) ×Y2 (q̂)](0) . (3)using the multipole expansion of a plane wave. Thespherical Bessel function j2 (qr) is proportional to q 2 ,as j2 (qr) ∼ (qr)2 /5!! in the lowest order of expansion,and VT (q) becomes
4π 8π(2)2(0)
r6 fT (r)dr.
VT (q)  −[(σ1 ·σ2 ) ×q Y2 (q̂)]
153(4)In this way, we obtain the tensor interaction in momentum space. This is a further, more detailed, andmathematically rigorous way to show that the arguments in the paper by Dong and Shang are invalid.
References

Condensed from the article in Phys. Rev. C 104, 039801(2021)RIKEN Nishina Center
Dipartimento di Fisica “Aldo Pontremoli,” Università degliStudi di Milano
INFN, Sezione di Milano
College of Nuclear Science and Technology, Beijing Normal
University

1) J. M. Dong, X. L. Shang, Phys. Rev. C 101, 014305(2020).
2) T. H. R. Skyrme, Nucl. Phys. 9, 615 (1959).
3) H. Sagawa, G. Colò, Prog. Part. Nucl. Phys. 76, 76(2014).
4) H. Bethe, Phys. Rev. 103, 1353 (1956).
5) T. H. R. Skyrme, Philos. Mag. 1, 1043 (1956).

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