Strength of reduced two-body spin-orbit interaction from chiral three-nucleon force

# Strength of reduced two-body spin-orbit interaction from chiral three-nucleon force

M. Kohno Physics Division, Kyushu Dental College, Kitakyushu 803-8580, Japan
###### Abstract

The contribution of a chiral three-nucleon force to the strength of an effective spin-orbit coupling is estimated. We first construct a reduced two-body interaction by folding one-nucleon degrees of freedom of the three-nucleon force in nuclear matter. The spin-orbit strength is evaluated by a Scheerbaum factor obtained by the -matrix calculation in nuclear matter with the two-nucleon interaction plus the reduced two-nucleon interaction. The problem of the insufficiency of modern realistic two-nucleon interactions to account for the empirical spin-orbit strength is resolved. It is also indicated that the spin-orbit coupling is weaker in the neutron-rich environment. Because the spin-orbit component from the three-nucleon force is determined by the low-energy constants fixed in the two-nucleon sector, there is little uncertainty in the present estimation.

###### pacs:
21.30.Fe, 21.45.Ff, 21.65.-f

Spin-orbit field in atomic nuclei is essential to reproduce well-established single-particle shell structure. The empirical strength of the spin-orbit potential, however, has not been fully understood on the basis of the realistic nucleon-nucleon force. The possible role of intermediate isobar -excitation to the nuclear spin-orbit field was considered in parallel with the construction of the two-pion-exchange three-nucleon force (3NF) by Fujita and Miyazawa FM57 (). The problem was reinvestigated in the early 1980s OTT80 (); AB81 () to search for the additional spin-orbit strength. Later, the Illinois group showed PP93 () that their 3NF makes a substantial contribution to the spin-orbit splitting in N.

Kaiser and his collaborators investigated, in their several papers KFW (); NK03 (); NK04 (), the nuclear spin-orbit coupling in the framework of chiral perturbation theory. The large contributions generated by iterated one-pion exchange and the 3NF almost cancel each other KFW (); NK03 (), and the short-range spin-orbit strength in the form of the effective four-nucleon contact-coupling deduced from realistic nucleon-nucleon interactions accounts well NK04 () for the empirical one. Because the contact-interaction in the chiral perturbation, however, is still needed to be regulated for the application to low-energy nuclear structure calculations, and their arguments for various contributions seem not to be fully unified, it is worthwhile to analyze the effective strength of the spin-orbit coupling by applying the established microscopic theory, namely the lowest-order Brueckner theory, to the two-nucleon and three-nucleon interactions in the chiral effective field theory (Ch-EFT).

The Thomas form of an average single-particle spin-orbit potential has been used to describe nucleon spin-orbit coupling:

 U0ℓs1rdρ(r)dr\boldmathℓ⋅% \boldmathσ, (1)

where the radial function is a nucleon total density distribution. The relation of the strength to a two-body effective spin-orbit interaction was derived by Scheerbaum SCH76b (). By defining the constant for the triplet odd component of the effective two-body spin-orbit interaction

 BS(¯q)=−2π¯q∫∞0drr3j1(¯qr)v3Oℓs(r), (2)

with being a spherical Bessel function, the single-particle spin-orbit potential for spin-saturated nuclei may be written as

 Uℓs,τ(r)=12BS(¯q)1rd{ρ(r)+ρτ(r)}dr\boldmathℓ⋅\boldmathσ, (3)

where specifies either a proton or neutron. We refer to as a Scheerbaum factor, which is different from the original constant in Ref. SCH76b () by a factor of . Scheerbaum prescribed fm on the basis of the wavelength of the density distribution. We employ this prescription. If we assume a naive relation , we recover the Thomas form, Eq. (1), with . It has also been customary to use a -type two-body spin-orbit interaction

 iW(\boldmathσ1+\boldmathσ2)⋅(∇r×δ(\boldmathr)∇r) (4)

in nuclear Hartree-Fock calculations using -type Skyrme interactions SKY75 (); SKY80 () and even with finite range effective forces, e.g., the Gogny force GP77 (). This two-body force provides a single-particle spin-orbit potential:

 12W1rd{ρ(r)+ρτ(r)}dr\boldmathℓ⋅\boldmathσ. (5)

Therefore, the strength may be identified as the Scheerbaum factor . The empirical value of is around MeVfm in various nuclear Hartree-Fock calculations. As will be shown below, the modern nucleon-nucleon interactions underestimate the spin-orbit strength by about 25 %.

Applying Scheerbaum’s formulation to the momentum-space -matrix calculation in nuclear matter with the Fermi momentum , we obtain the corresponding spin-orbit strength as follows FK00 ():

 BS(¯q)=1k3F∑JT(2J+1)(2T+1)∫qmax0dq ×W(¯q,q){(J+2)GJT1J+1,1J+1(q)+GJT1J,1J(q) −(J−1)GJT1J−1,1J−1(q)}. (6)

Here, and the weight factor is

 W(¯q,q)=⎧⎪⎨⎪⎩θ(kF−¯q)for0≤q≤|kF−¯q|2k2F−(¯q−2q)28¯qqfor|kF−¯q|2≤q≤kF+¯q2, (7)

where is a step function. In Eq. (6), is the abbreviation of the momentum-space diagonal -matrix element in the spin-triplet channel with the total isospin , total spin , and orbital momenta and .

Calculating in the lowest-order Brueckner theory with the continuous prescription for intermediate spectra, as presented below explicitly in Table 1, modern two-body nucleon-nucleon potentials are found to give smaller values of around 90 Mevfm compared with the empirical one. As has been well known that LOBT calculations in symmetric nuclear matter with realistic two-nucleon force do not reproduce correct saturation property. However, in most case, calculated energies at the empirical saturation point fm are close to the empirical energy of about MeV. This suggests that matrices provide basic information on the effective nucleon-nucleon interaction in the nuclear medium, by incorporating important short-range correlations, Pauli effects and dispersion effects.

Now we consider the contribution of the 3NF. In this article, we estimate it in a two-step procedure. First, the 3NF defined in momentum space is reduced to an effective two-nucleon interaction by folding one-nucleon degrees of freedom:

 ⟨\boldmathk′1σ′1τ′1,\boldmathk′2σ′2τ′2|v12(3)|\boldmathk1σ1τ1,\boldmathk2σ2τ2⟩A=13∑\boldmathk3σ3τ3⟨\boldmathk′1σ′1τ′1,\boldmathk′2σ′2τ′2,% \boldmathk3σ3τ3|v123|\boldmathk1σ1τ1,\boldmathk2σ2τ2,\boldmathk3σ3τ3⟩A. (8)

Here, we have to assume that remaining two nucleons are in the center-of-mass frame, namely . The density-dependent effective two-nucleon interaction as the effect of the 3NF has been commonly introduced in the literature KAT74 (); FP81 (); HKW10 (). Note that the suffix means an antisymmetrized matrix element; namely and , and the factor in Eq. (8) is an additional statistical one. This statistical factor has been often slipped in the literature. The recent derivation of the effective two-body interaction from the Ch-EFT 3NF by Holt, Kaiser and Weise HKW10 () also seems not to be an exception. If an adjustable strength is introduced, the statistical factor may be hidden in the fitting procedure. In our case of using the Ch-EFT 3NF, the low-energy constants except for and are fixed. Although there may be a room to adjust and , the contributions to the energy from these terms are rather small, if they are in a reasonable range. In addition, and do not contribute to the reduce two-nucleon spin-orbit interaction. By comparing the nuclear matter energy directly calculated from and that by the reduced , the error due to this approximation can be checked to be less than 10 %, if we calculate Born energy without including a form factor.

To explain the procedure of obtaining more explicitly, we write the reduced spin-orbit component originating from the term of the Ch-EFT 3NF:

 −c1g2Am2πf4π∑1≤i

where , MeV, is a pion mass, and is a momentum transfer of the -th nucleon. The momentum transfer of the third nucleon is dictated by the relation . The folding of the 3NF by one nucleon is carried out without incorporating a three-body form factor. A form factor is later introduced on the two-body level. The folding in symmetric nuclear matter with the Fermi momentum gives, besides the central and tensor components, the following spin-orbit term:

 c1g2Am2πf4π1(2π)3∭|\boldmathk3|≤kFd\boldmathk3 ×i(\boldmathσ1+\boldmathσ2)⋅(−\boldmathk′1×\boldmathk1+(\boldmathk′1−\boldmathk1)×\boldmath% k3)((\boldmathk′1−\boldmathk3)2+m2π)((\boldmathk1−\boldmathk3)2+m2π). (10)

When carrying out the folding in pure neutron matter, the restriction of the isotopic spin brings about an additional factor of .

The partial-wave decomposition of the above spin-orbit term becomes

 −δS1c1g2Am2πf4πℓ(ℓ+1)+2−J(J+1)2ℓ+1 {Qℓ−1W,0(k′1,k1)−Qℓ+1W,0(k′1,k1)−Wℓℓs,0(k′1,k1)} (11)

for the orbital and total angular momenta and . The functions and are defined by

 QℓW,0(k′1,k1) ≡ 2π(2π)312∫kF0dk3Qℓ(x′)Qℓ(x), (12) Wℓℓs,0(k′1,k1) ≡ 2π(2π)312k′1k1∫kF0dk3k3 (13) ×{k′1Qℓ(x)(Qℓ−1(x′)−Qℓ+1(x′)) +k1Qℓ(x′)(Qℓ−1(x)−Qℓ+1(x))},

where is a Legendre function of the second kind, and and , respectively.

The spin-orbit component arises also from the term of the Ch-EFT 3NF. This case, in addition to the replacement of the coupling constant, an additional factor appears in the denominator in Eq. (10). The partial-wave decomposition reads

 δS1c3g2A2f4πℓ(ℓ+1)+2−J(J+1)2ℓ+1[(m2π+12(k′21+k21)){Qℓ−1W,0(k′1,k1)−Qℓ+1W,0(k′1,k1)−Wℓℓs,0(k′1,k1)} +3k′1k1{QℓW,0(k′1,k1)−(ℓ−1)Qℓ−2W,0(k′1,k1)+(ℓ+2)Qℓ+2W,0(k′1,k1)+ℓ−12ℓ−1Wℓ−1ℓs,0(k′1,k1)+ℓ+22ℓ+3Wℓ+1ℓs,0(k′1,k1)} −δℓ1k′1k12(F0(k′1)+F0(k1)−F1(k′1)−F1(k1))], (14)

where the new functions and are defined by

 F0(k)≡1(2π)3∭|\boldmathk3|≤kFd\boldmathk31(\boldmathk−% \boldmathk3)2+m2π, (15) F1(k)≡1(2π)31k2∭|% \boldmathk3|≤kFd\boldmathk3\boldmathk⋅\boldmathk3(\boldmathk−\boldmathk3)2+m2π. (16)

Adding the reduced two-nucleon interaction to the Ch-EFT two-nucleon interaction, we repeat the LOBT -matrix calculation. Although explicit expressions are not shown in this Letter except for the spin-orbit part, we include all central, tensor and spin-orbit components of the reduced interaction . The form factor in a functional form of is introduced for with the cut-off mass 550 MeV. We use the low-energy constants fixed for the Jülich Ch-EFT potential by Hebeler et al. HEB (); , and . Other constants are GeV, GeV, and GeV. Because the reduction of the 3NF to the two-nucleon force was carried out in nuclear matter, may not be directly applied to very light nuclei, such as H and He.

First, we comment on calculated saturation curves, which are given in Fig. 1. Without the contribution of the 3NF, the saturation curve attains its minimum at larger as a function of the Fermi momentum than the empirical saturation momentum, as has been known. Nucleon-nucleon interactions, AV18 AV18 (), NSC97 NSC (), and Jülich NLO with the cutoff mass of 550 MeV EHM09 () give similar saturation curves, and the CD-Bonn potential CDB () predicts somewhat deeper binding. For the reference of what saturation curve is preferable for nuclear mean filed calculations, we also show the result with the Gogny D1S interaction GP77 ().

The thin dotted curve shows the result in which the plane wave expectation value of the 3NF is added to the result of the two-nucleon NLO. The thick dotted curve alongside the thin dotted curve is the result with the plane wave expectation value of the reduced two-nucleon interaction . The difference between the thin and thick curves is due to the difference of the form-factors and the necessary approximation in Eq. (8).

The solid curve is the result of the -matrix calculation with including the reduced two-nucleon interaction, . Although the energy is seen to be underestimated by a few MeV, the saturation property is largely improved by the repulsive contribution from the three-nucleon force. It is not necessary at present to expect a perfect agreement with the empirical properties in the LOBT calculation in nuclear matter.

Now we examine the spin-orbit strength. We tabulate values for of Eq. (6) at fm calculated in the LOBT with modern nucleon-nucleon interactions: AV18 AV18 (), NSC97 NSC (), CD-Bonn CDB (), and Jülich NLO EHM09 (). The Scheerbaum factors obtained by realistic two-nucleon forces are seen to be similar but insufficient to explain the strength needed in nuclear mean field calculations. Namely only about three-fourths of the empirically needed strength is accounted for. The two-body part of the Ch-EFT, NLO, shows little difference with other realistic two-nucleon force. It is also noticed that values at fm, namely at the half of the normal density, change little from those at the normal density with fm. It turns out, as the last column of Table I shows, that the addition of the reduced two-body interaction from the Ch-EFT 3NF bring about a good effect to fill the gap, though the 3NF contribution is smaller at fm. This is in accord with the important role of the 3NF to the spin-orbit splitting demonstrated in quantum Monte Carlo calculations of low-energy neutron-alpha scattering NOL (). Although there are ambiguities from the form factor and uncertainties inherent in the folding procedure without taking into account nucleon-nucleon correlations, no additional adjustable parameter exists, because low-energy constants and which contribute solely to the spin-orbit strength are determined on the two-nucleon sector.

As noted after Eq. (10), the reduced two-body spin-orbit term in neutron matter is one-third of that in symmetric nuclear matter. Actual -matrix calculations using the Ch-EFT NLO plus in pure neutron matter with fm tell that values at fm are 84.7 and 93.5 MeVfm without and with the reduced two-nucleon interaction , respectively. If fm is assumed, the corresponding values are 87.0 and 94.6 MeVfm, respectively. Again, the -dependence is weak. While the spin-orbit strength from the two-nucleon force is scarcely different from that in symmetric nuclear matter, the additional contribution from the three-nucleon force is in fact almost one-third of that in symmetric nuclear matter. Thus, the spin-orbit strength is expected to be smaller in the neutron-rich environment. This seems to be consistent with the trend observed in the shell structure near the neutron drip line SCH04 () that a decreasing spin-orbit interaction is preferable with increasing neutron excess.

In summary, we have estimated quantitatively the contribution of the three-nucleon force of the chiral effective field theory to the single-particle spin-orbit strength, using the formulation by Scheerbaum SCH76b (). We first introduced the reduced two-body interaction by folding one-nucleon degrees of freedom of the 3NF in nuclear matter. Making partial-wave expansion of the resulting two-body interaction and adding it to the genuine two-nucleon interaction with including the necessary statistical factor of , we carried out LOBT -matrix calculations in infinite matter and evaluated the Scheerbaum factor corresponding to the spin-orbit strength. Because the spin-orbit field in the atomic nuclei is fundamentally important as the nuclear magic numbers exhibit, it is important to learn that the inclusion of the 3NF in the chiral effective field theory can account for the spin-orbit strength empirically required for nuclear mean filed calculations. Because the relevant low-energy constants and are determined in the two-nucleon interaction sector, there should be little uncertainty for the additional spin-orbit strength except for the treatment of the two-body form factor. We have also noted that the additional spin-orbit strength from the 3NF should be weaker in neutron-excess nuclei.

###### Acknowledgements.
This work is supported by Grant-in-Aid for Scientific Research (C) from the Japan Society for the Promotion of Science (Grant No. 22540288). The author thanks H. Kamada for valuable comments concerning the Ch-EFT interaction. He is also grateful to M. Yahiro for his interest in this work.

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