Nuclear surface properties in relativistic effective field theory

We perform Hartree calculations of symmetric and asymmetric semi-infinite nuclear matter in the framework of relativistic models based on effective hadronic field theories as recently proposed in the literature. In addition to the conventional cubic and quartic scalar self-interactions, the extended models incorporate a quartic vector self-interaction, scalar-vector non-linearities and tensor couplings of the vector mesons. We investigate the implications of these terms on nuclear surface properties such as the surface energy coefficient, surface thickness, surface stiffness coefficient, neutron skin thickness and the spin-orbit force.


Introduction
Quantum hadrodynamics (QHD) and the relativistic treatment of nuclear systems has been a subject of growing interest during recent years [1,2,3,4,5]. The σ−ω model of Walecka [1] and its non-linear extensions with cubic and quartic self-interactions of the scalar-meson field [6] have been widely used to this end. This model contains Dirac nucleons together with neutral scalar and vector mesons as well as isovector-vector ρ mesons. At the mean field (Hartree) level, it already includes the spin-orbit force, the finite range and the density dependence which are essential ingredients of the nuclear interaction. This simple model has become very popular in relativistic calculations and describes successfully many properties of the atomic nucleus.
From a theoretical point of view, the non-linear σ − ω model with cubic and quartic scalar self-interactions was classed within renormalizable field theories which can be characterized by a finite number of coupling constants. However, very recently, generalizations of this model that include other non-linear interactions among the meson fields and tensor couplings have been presented on the basis of effective field theories by Serot et al. [5,7,8,9].
The effective theory contains many couplings of non-renormalizable form that are consistent with the underlying symmetries of QCD. Consequently, one must find some suitable expansion parameters and develop a systematic truncation scheme. For this purpose the concept of naturalness has been employed: it means that the unknown couplings of the theory should all be of the order of unity when written in appropriate dimensionless form using naive dimensional analysis [5,7,8,9]. Then, one can estimate the contributions coming from different terms by counting powers in the expansion parameters and truncating the Lagrangian at a given level of accuracy.
One important fact is the observation that at normal nuclear densities the scalar and vector meson fields, denoted by Φ and W , are small as compared with the nucleon mass M and that they change slowly in finite nuclei. This implies that the ratios Φ/M, W/M, |∇Φ|/M 2 and |∇W |/M 2 are useful expansion parameters when the effective field theory is applied to the nuclear many-body problem. From this viewpoint, if all the terms involving scalar and meson self-interactions are retained in the Lagrangian up to fourth order, one recovers the well-known non-linear σ−ω model plus some additional terms [5,7,9]. For the truncation to be consistent, the corresponding coupling constants should exhibit naturalness and cannot be arbitrarily dropped out without an additional symmetry argument. The effective Lagrangian truncated at fourth order contains thirteen free parameters that have been fitted to reproduce twenty-nine finite nuclei observables [5,9]. Remarkably, the fitted parameters turn out to be natural and the results are not dominated by the last terms retained. This evidence confirms the utility of the principles of naive dimensional analysis and naturalness and shows that truncating the effective Lagrangian at the first lower orders is justified.
The term with a vector-meson quartic self-interaction has been considered previously in relativistic mean field (RMF) calculations from a phenomenological point of view. Bodmer [10] considered this coupling to avoid the negative coefficient of the quartic scalar selfinteraction that appears in many non-linear σ−ω parametrizations that correctly describe the atomic nucleus [11]. In some special situations this negative term can lead to a pathological behaviour of the scalar potential. On the other hand, the equation of state is softened at moderate high densities when the vector non-linearity is taken into account. The quartic vector self-interaction has also been phenomenologically used by Gmuca [12,13] in a nonlinear σ−ω model for parametrizing Dirac-Brueckner-Hartree-Fock calculations of nuclear matter. The same idea was developed by Toki et al. and applied to study finite nuclei [14] and neutron stars [15]. Recently, the properties of high-density nuclear and neutron matter have been analyzed in the RMF approach taking into account scalar and vector non-linearities [8].
The tensor couplings of the vector ω and ρ mesons to the nucleon were investigated by Reinhard et al. [3,16] as an extension of the RMF model, and more recently by Furnstahl et al. [9,17] from the point of view of relativistic effective field theory. In these works it was shown that the tensor coupling of the ω meson has an important bearing on the nuclear spin-orbit splitting.
The surface properties of nuclei play a crucial role in certain situations. This is the case, for instance, of saddle-point configurations in nuclear fission or fragment distributions in heavy-ion collisions. Within a context related to the liquid droplet model (LDM) and the leptodermous expansion [18], the surface properties can be extracted from semi-infinite nuclear matter calculations either quantally or semiclassically (though the total curvature energy coefficient can only be computed semiclassically [19]). In the non-relativistic case most of the calculations of the surface properties have been carried out using Skyrme forces, quantally [20] or semiclassically with the help of the extended Thomas-Fermi (ETF) method [20,21]. In the relativistic case the nuclear surface has been analyzed within the σ−ω model since a long time ago. The calculations have been performed semiclassically using the relativistic Thomas-Fermi (TF) method or its extensions (RETF), for symmetric [1,6,22,23,24] and asymmetric [25,26] matter, and also in the quantal Hartree approach [27,28,29].
In the framework of the relativistic model and effective field theory, the main purpose of the present work is to carefully analyze the influence on surface properties of the quartic vector non-linearity, of the newly proposed scalar-vector self-interactions and of the tensor coupling. We shall investigate quantities such as the surface energy coefficient, surface thickness, spin-orbit strength, surface stiffness coefficient and neutron skin thickness obtained from Hartree calculations of symmetric and asymmetric semi-infinite nuclear matter.
The paper is organized as follows. Section 2 is devoted to the basic theory. The results on the surface properties of symmetric matter are discussed in Section 3. Section 4 addresses the case of asymmetric systems. The summary and conclusions are given in the last section. 4 2 Mean field equations for symmetric semi-infinite nuclear matter Following Ref. [7], to derive the mean field equations one starts from an energy functional containing Dirac baryons and classical scalar and vector mesons. The energy functional can be obtained from the effective Lagrangian in the Hartree approach using many-body techniques [5,9]. However, this energy functional can also be considered as an expansion in Φ/M, W/M, |∇Φ|/M 2 and |∇W |/M 2 of a general energy density functional that contains all the correlation effects. The theoretical basis of this functional lies on the extension of the Hohenberg-Kohn theorem [30] to QHD [31]. Using the Kohn-Sham scheme [32] with the mean fields playing the role of Kohn-Sham potentials, one finds similar mean field equations to those obtained from the Lagrangian [31], but including effects beyond the Hartree approach through the non-linear couplings [4,7,8,9].
A semi-infinite system of uncharged nucleons corresponds to a one-dimensional geometry where half the space is filled with nuclear matter at saturation and the other half is empty, so that a surface develops around the interface. The fields and densities change only along the direction perpendicular to the medium. Specifying the energy density functional considered in Refs. [5] and [9] to symmetric semi-infinite nuclear matter with the surface normal pointing into the z direction one has where the index α runs over all occupied states of the positive energy spectrum, Φ ≡ g s φ 0 and W ≡ g v V 0 (notation as in Ref. [1]). Except for the terms with α 1 and α 2 , the functional (2.1) is of fourth order in the expansion. We retain the fifth-order terms α 1 and α 2 because 5 in Refs. [5] and [9] they have been estimated to be numerically of the same magnitude as the quartic scalar term in the nuclear surface energy.
The mean field equations are obtained by minimizing with respect to ϕ † α , Φ and W : The baryon, scalar and tensor densities are respectively The expression of the four-component spinors ϕ α (z) in the semi-infinite medium was given by Hofer and Stocker in Ref. [27].
In a semi-infinite nuclear matter calculation the sum over the single-particle states is replaced by an integration over momenta: where Ω stands for the volume of the box, the factor 2 takes into account the isospin degree of freedom and λ describes the spin orientation of the nucleons. Introducing the Fermi 6 momentum k F , the integration domain is restricted to k 2 x + k 2 y + k 2 z = k 2 ⊥ + k 2 z ≤ k 2 F , with k z ≥ 0 if the bulk nuclear matter is located at z = −∞. Following the method outlined in Ref. [27] one finds two sets (λ = ±1) of first-order differential equations for the orbital part of the upper and lower components of the Dirac spinors: where a = (k z , k ⊥ , λ) and M * (z) = M − Φ(z) is the Dirac effective mass of the nucleons.
From the asymptotic behaviour at z = −∞ (bulk nuclear matter), the condition on the energy eigenvalues is ε a = k 2 ⊥ + k 2 z + M * ∞ 2 + W ∞ , with M * ∞ and W ∞ being the nuclear matter values of M * and W . The densities for each spin orientation λ = ±1 read 13) and the total densities are given by (2.14) Using the equations of motion the energy density of the semi-infinite nuclear matter system can be written as follows: Finally, the surface energy coefficient E s is obtained from the expression [18] where a v is the energy per particle in bulk nuclear matter and r 0 is the nuclear radius constant: r 0 = (3/4πρ 0 ) 1/3 , with ρ 0 the nuclear matter density.
Another important quantity in the study of the nuclear surface structure is the spin-orbit interaction. By elimination of the lower spinor in terms of the upper spinor, one obtains a Schrödinger-type equation with a term V so (z) that has the structure of the single-particle spin-orbit potential for the non-relativistic case [27,29]. In our present model the orbital part of the spin-orbit potential reads as In the non-relativistic limit, by means of a Foldy-Wouthuysen reduction, Eq. (2.17) becomes and the nucleons are then moving in a central potential of the form

Surface properties in the symmetric case
Although previous works [23,24,25,26,27,28,29] have thoroughly investigated the properties of the nuclear surface in the standard non-linear σ −ω model, for which κ 3 and κ 4 are the only non-linearities in (2.1), here we wish to enlarge this study by including the additional non-linear and tensor couplings considered in Refs. [5,7,9]. In concrete, we want to study the role of the quartic vector self-interaction ζ 0 that has been used after the work of Bodmer [10], the role of the terms with η 1 and η 2 that couple the scalar and vector fields, of the terms with α 1 and α 2 that imply the gradients of the fields, and of the tensor coupling f v of the vector ω meson to the nucleon. While ζ 0 , η 1 and η 2 can be classified as volume contributions, the couplings α 1 , α 2 and f v are genuine surface terms. On the basis of the concept of naturalness there is no reason to omit any of these terms in the energy density functional (2.1), unless there exists a symmetry principle to forbid it. However, to clarify the impact on the surface properties of the aforementioned couplings we will analyze each one separately, as it has been similarly done in Ref. [7]. In this section we shall study symmetric systems, while in Section 4 we shall address the case of asymmetric matter.

Effect of the quartic vector self-interaction
In the conventional non-linear σ −ω model the value of the coefficients g 2 s /m 2 s , g 2 v /m 2 v , κ 3 and κ 4 can be univocally obtained by imposing that for nuclear matter at saturation the density ρ 0 , energy per particle a v , effective mass M * ∞ /M and incompressibility modulus K take given values. When the vector-meson quartic self-interaction is switched on, the Dirac equation for the baryons and the Klein-Gordon equation for the vector field in infinite nuclear matter become (throughout this subsection we set The saturation density ρ 0 and the Fermi momentum k F are related as usual by ρ 0 = 2k 3 F /3π 2 . Specifying ρ 0 , a v and M * ∞ /M, from the above equations one extracts the coupling constant g v as a function of ζ 0 (the nucleon and ω masses take their empirical values: M = 939 MeV and m v = 783 MeV). The steps to calculate g 2 s /m 2 s , κ 3 and κ 4 are then the same as when ζ 0 = 0, see e.g. Refs. [10] and [11], but now these coefficients become functions of ζ 0 . The reader will find a detailed study of the implications of the vector non-linearity ζ 0 in nuclear matter in Refs. [7] and [10].
The assumption of naturalness requires that the couplings g s /4π, g v /4π, κ 3 , κ 4 and ζ 0 should all be roughly of the order of unity. Figure 1 illustrates the variation of these couplings as a function of the non-dimensional parameter We realize that g s and g v are only weakly affected by η 0 (the change is not appreciable in the scale of Figure 1). However, η 0 has a direct effect on the scalar-meson quartic selfinteraction κ 4 , moving it from a negative value when the quartic vector term is absent values also the rest of coupling constants are natural. The behaviour gleaned from Figure   1 is rather independent of the saturation density ρ 0 and energy a v , and e.g. we have found similar trends with the specific nuclear matter properties used by Bodmer in Ref. [10]. 1 Notice that in Ref. [10] the parameter η 0 was called z.
Next we turn our attention to the surface properties. In Figures 2 and 3 we have plotted, respectively, the surface energy coefficient E s and the surface thickness t of the semi-infinite density profile (the 90%-10% fall-off distance) against the η 0 parameter. The selected values of η 0 are those employed in Table 1  in all cases it stays almost equal to its value at η 0 = ∞ and goes down slightly for η 0 ≤ 2.
Noticeable departures of the surface energy coefficient and thickness from the η 0 = ∞ values can be found only if η 0 is decreased beyond the natural limit, a situation where the interaction is mainly ruled by the vector-meson quartic self-interaction.
Even though the impact of the vector-meson quartic self-interaction is small, it can help to find parameter sets for which both the surface energy coefficient and the surface thickness lie in the empirical region. This fact is illustrated in Figure 4 For η 0 ≥ 2 the dependence of E s and t upon the nucleon effective mass is similar to that found in the usual σ−ω model without a quartic vector self-interaction [23,28]. The general tendencies start to change when η 0 is lowered and leaves the natural region. As η 0 becomes smaller Figure 4 shows that the slope of the curves of E s and t as a function of M * ∞ /M changes, and that the curves for the different m s come closer together.
To get more insight about the influence of the vector-meson quartic self-interaction, we display in Figure 5 the profiles of the baryon density ρ(z) and of the surface tension density It can be seen that the local quantities depicted in Figures 5 and 6 oscillate as functions of z (Friedel oscillations). Both the surface tension density and the single-particle spinorbit potential are confined to the surface and average to zero inwards as the bulk matter is approached. The local profiles for η 0 = 2, which somehow marks the limit of naturalness as we have commented, are almost equal to those obtained in the absence of the quartic vector self-interaction. In agreement with Figures 2 and 3, only when η 0 is decreased to non-natural values one can notice some changes in the profiles, which are more visible for M * ∞ /M = 0.6 than for M * ∞ /M = 0.7. Decreasing η 0 makes the surface steeper and the thickness t smaller, produces an enhancement in the surface region of the density ρ(z) and of the mean field V c (z), and builds up Friedel oscillations in σ(z) and in the spin-orbit potential V so (z).
It is well known that the experimental spin-orbit splittings require within narrow bounds a Dirac effective mass M * ∞ /M around 0.6 in the conventional relativistic model [11,7]. Figure  6 shows that introducing a quartic vector self-interaction makes the spin-orbit well deeper.
However, it is only a minor effect: with M * ∞ /M = 0.7 it is not possible to reproduce the spinorbit interaction of the case M * ∞ /M = 0.6 at η 0 = ∞, not even if one sets η 0 = 0.5 (which in addition brings about an unreallistically small t, see Figure 3). A similar conclusion was drawn in Ref. [10] from an analysis in nuclear matter. We also have computed the non-relativistic limit V FW so (z) of the spin-orbit potential given by the expression (2.18). In agreement with Ref. [29] we have found that while V FW so (z) qualitatively reproduces the behaviour of V so (z), it strongly underestimates the quantitative depth of the fully relativistic

Influence of the volume cubic and quartic scalar-vector interactions
The next bulk terms in the energy density (2.1) that contain non-linear meson interac- In analogy to Figure 1 for η 0 , in Figure 7 we study the change of the couplings g s /4π, g v /4π, As it can be inferred from the nearly vertical lines in the η 1 -η 2 plane, the surface energy coefficient and thickness depend mostly on η 1 and are rather independent of η 2 . The consequence of increasing η 1 is a reduction of the values of E s and t. The lines of constant t turn out to be, roughly speaking, parallel to the lines of constant E s . This means that from the interplay of the parameters η 1 and η 2 it is not possible to change the value of t relative to that of E s (for example, we see in Figure 8 that t ∼ 2.2 fm if E s = 18 MeV). We have calculated the spin-orbit potential V so (z) at the Fermi surface for several values of η 1 and η 2 . We have found that these couplings have a marginal effect on the spin-orbit strength, as it happened to be the case with the other bulk non-linearity η 0 .
To get some information about the incidence on the surface energy and thickness of all the volume non-linear meson interactions together, we have repeated the calculations in the η 1 -η 2 plane setting η 0 = 2 for the quartic vector self-interaction. One finds similar features to those of Figure 8. The effect of η 0 = 2 is just shifting E s and t towards smaller values as compared with the case η 0 = ∞ (ζ 0 = 0), which is in accordance with what was found in

Influence of the non-linear terms with gradients
Now we discuss the non-linear interactions that vanish in infinite nuclear matter. We recall that these terms are actually of order 5 in the expansion of the effective Lagrangian but, following Refs. [5] and [9], we include them because they can be relevant in the surface due to their gradient structure.
Using the same saturation properties and scalar mass of Figure 8, in Figure 9 we have To assess the importance of the bulk non-linear meson interactions on our study on α 1 and α 2 , we have performed calculations as in Figure 9 but setting η 0 = 2 with η 1 = η 2 = 0, and setting η 1 = 1 with η 0 = ∞ and η 2 = 0 (as indicated, the effect of η 2 is much smaller than that of η 1 ). The results show a completely similar behaviour to Figure 9. Even the slope of the contour lines of E s and t in the α 1 -α 2 plane changes only slightly. Comparing with Figure 9, when η 0 = 2 one finds that E s is shifted by approximately −1 MeV, and that when η 1 = 1 then E s is shifted by around −3 MeV. The shifts of the surface thickness t are less regular and their magnitude depends on the value of α 1 and α 2 .
In order to investigate the impact of the gradient interactions α 1 and α 2 on the spinorbit potential, in Figure 10 we have plotted V so (z) at the Fermi surface for a few selected values of α 1 and α 2 . The nuclear matter properties and the scalar mass are the same as in Figure 6, where we studied the dependence of V so (z) on η 0 . One can see that the meson interaction with α 1 = 1 and α 2 = 0 reduces the strength of V so (z) and shifts the position of the minimum slightly to the exterior. On the contrary, the interaction with α 1 = 0 and α 2 = 1 makes the potential well deeper. The combined effect is probed in the case α 1 = α 2 = 1. Since in the relativistic model the spin-orbit force is strongly correlated with the Dirac effective mass, we compare in Figure 10

Role of the tensor coupling of the omega meson
To conclude this section we investigate the influence of the ω tensor coupling which adds some momentum and spin dependence to the interaction. The natural combination for this coupling is f v /4.
Well known from one-boson-exchange potentials (where f v above is commonly written as f v /g v ), the tensor coupling was included in the fits to nuclear properties of Refs. [3,16] (conventional QHD) and [5,9] (effective field theory), and in the study of the nuclear spinorbit force in chiral effective field theories carried out in Ref. [17]. These works noticed the existence of a trade-off between the size of the ω tensor coupling and the size of the scalar field. In other words, the tensor coupling breaks the tight connection existing in relativistic models between the value of the nucleon effective mass at saturation and the empirical spin-orbit splitting in finite nuclei (which constrains M * ∞ /M to lie between 0.58 and 0.64 [7]). Including a tensor coupling the authors of Refs. [5,9,17] were able to obtain natural parameter sets that provide excellent fits to nuclear properties and spin-orbit splittings with an equilibrium effective mass remarkably higher (M * ∞ /M ∼ 0.7) than in models that ignore such coupling. We want to analyze the nature of this effect in the simpler but more transparent framework of semi-infinite nuclear matter.
In Figure 11  exemplify the incidence of f v on the spin-orbit potential, Figure 12 displays V so (z) at the Fermi surface for a few of the cases of Figure 11. We also performed the calculations for m s = 525 MeV: E s and t are shifted downwards with respect to Figure 11 and V so (z) is deeper than in Figure 12, but the global trends with f v are the same. Figure 11 shows the strong reduction of E s and t as f v increases (the slope of the curves is milder for M * ∞ /M = 0.7 than for M * ∞ /M = 0.6). Figure 12 reveals that this fact is associated with a deeper and wider spin-orbit potential. This agrees with the results of Hofer and Stocker [27] who showed in the standard RMF model that the spin-orbit coupling reduces the surface energy and thickness. At variance with the individual values of E s and t, the ratio E s /t stays to a certain extent constant with f v . The spin-orbit effect has to do with the explicit dependence of the nucleon orbital wave functions on the spin orientation λ. As described in Ref. [27] nucleons with λ = +1 feel 17 an attractive spin-orbit potential and are pushed to the exterior of the surface, whereas the spin-orbit force is repulsive for nucleons with λ = −1 which are pushed to the interior.
As a consequence of this a depletion of particles with λ = −1 occurs at the surface. This behaviour is contrasted in Figure 13 for f v = 0 and f v = 0.6 in the case M * ∞ /M = 0.7. The figure depicts the profiles of the total baryon and tensor densities as well as those of their spin components ρ λ (z) and ρ λ T (z) for λ = ±1, Eqs. (2.11)-(2.14). When the spin-orbit strength is large, attraction dominates over repulsion and more particles accumulate at the surface than particles are removed from it. Then the total baryon density is enhanced at the surface region and it falls down more steeply.

Surface properties in the asymmetric case
We briefly recall some basic definitions concerning nuclear surface symmetry properties (further details on the relativistic treatment of asymmetric infinite and semi-infinite nuclear matter can be found in Refs. [25,26,29]). For a bulk neutron excess δ 0 = (ρ n0 −ρ p0 )/(ρ n0 +ρ p0 ) (i.e., the asymptotic asymmetry far from the surface), a surface energy coefficient can be computed as where E(z) is the total energy density of the system of neutrons and protons, a v (δ 0 ) denotes the energy per particle in nuclear matter of asymmetry δ 0 , and ρ(z) = ρ n (z) + ρ p (z) with ρ n and ρ p referring to the neutron and proton densities, respectively. According to the liquid droplet model (LDM) [18], for small values of the neutron excess E s (δ 0 ) can be expanded as follows: In this equation J stands for the bulk symmetry energy coefficient, L reads for the LDM coefficient that expresses the density dependence of the symmetry energy, and Q is the socalled surface stiffness coefficient that measures the resistance of the system against pulling the neutron and proton surfaces apart. All of these macroscopic coefficients are familiar from semi-empirical LDM mass formulae.
Another quantity of interest is the neutron skin thickness Θ, namely the separation between the neutron and proton surface locations: In finite nuclei Θ would correspond to the difference between the equivalent sharp radii of the neutron and proton distributions. In the small asymmetry limit the LDM predicts a linear behaviour of Θ with δ 0 : For calculations of finite nuclei of small overall asymmetry I = (N − Z)/A, the LDM expansion of the energy can be written as where a C is the Coulomb energy coefficient. Notice that I = δ 0 in finite nuclei.
To describe asymmetric matter in the relativistic approach we need to generalize the energy density (2.1) by including the isovector ρ meson. In terms of the mean field R = g ρ b 0 , with b 0 the time-like neutral component of the ρ-meson field, the additional contributions to Eq. (2.1) read The symmetry energy coefficient turns out to be In the conventional model one has f ρ = η ρ = 0. The isovector tensor coupling f ρ was included in the calculations of Refs. [3,5,9,16]. The new non-linear coupling η ρ between the ρ-and σ-meson fields is of order 3 in the expansion and it has been introduced in Refs. [5,9]. We will not consider higher-order non-linear couplings involving the ρ meson since the expectation value of the ρ field is typically an order of magnitude smaller than that of the ω field [5,9]. For example, in calculations of the high-density nuclear equation of state, Müller and Serot [8] found the effects of a quartic ρ meson coupling (R 4 ) to be only appreciable in stars made of pure neutron matter. On the other hand, in analogy to the couplings α 1 and α 2 for the σ and ω fields, we also tested a surface contribution −α 3 Φ (∇R) 2 /(2g 2 ρ M) and found that the impact it has on the properties we will study in this section is absolutely negligible.
As we have seen, the quantity that governs the surface properties in the regime of low asymmetries is the surface stiffness Q. Table 1 Table 1 shows that the influence on the surface stiffness of the volume self-interactions η 0 , η 1 and η 2 is not very large for natural values of these couplings. In the present case Q is slightly increased by decreasing η 0 (i.e., by increasing the quartic vector coupling ζ 0 ). For η 1 = 1 we find a non-negligible increase of Q, which signals a larger rigidity of the nuclear system against the separation of the neutron and proton surfaces. The effect of η 2 is again moderate as compared to that of η 1 . Q is augmented by a positive η ρ coupling, while a negative η ρ induces a lower value of Q. Some visible changes in Q take place when the α 1 and α 2 gradient interactions are taken into account. Due to the opposite behaviour of Q with α 1 and α 2 , the tendencies compensate in a case like α 1 = α 2 = 1, but the net effect is reinforced e.g. if α 1 = −α 2 = 1.
As one could expect the isoscalar tensor coupling f v has a notable effect on Q, even for the relatively small value f v = 0.3 that we have used in Table 1. On the contrary, Q is virtually insensitive to the isovector tensor coupling f ρ . The reason is that the derivative of the R(z) field is much smaller than that of the W (z) field. In the least-square fits to ground-state properties of Refs. [3,16] nothing was gained by the ρ tensor coupling. In any case, the best fits of Refs. [5,9] have f ρ ≈ 4.
From Table 1 we recognize that the main changes in the coefficient L arise from the η ρ coupling. As a rule of thumb, increasing values of Q are associated with decreasing values of L for the bulk couplings η 0 , η 1 , η 2 and η ρ . Since L is a bulk quantity, it is not modified by the surface interactions. Figures 14 and 15 illustrate the dependence on asymmetry of the neutron skin thickness, surface energy and surface thickness for some of the cases considered in Table 1. The figures   21 extend up to δ 0 = 0.3, which widely covers the range relevant for laboratory nuclei (δ 0 ≤ 0.2).
As the system becomes neutron rich we can appreciate how a neutron skin develops and Θ grows from its vanishing value at δ 0 = 0. For small asymmetries the growth is linear in δ 0 , as predicted by the LDM. The surface energy coefficient grows quadratically with increasing neutron excess and the LDM equation (4.2) is clearly a good approximation. In general, the interactions having thicker neutron skins (smaller values of Q) also have larger surface energies.
The parameter η ρ can be used for the fine tuning of the symmetry properties of the interaction without spoiling the predictions for symmetric systems. If in the conventional ansatz g ρ is fixed by the value of the symmetry energy J, in the extended model J depends on a combination of g ρ and η ρ , Eq. (4.7). Therefore, η ρ provides in practice a mechanism that can help to simultaneously adjust Q (to get the required neutron skin Θ) and J (to keep the fit to the masses) preserving the symmetric surface properties.

22
Within relativistic mean field theory, we have investigated the influence on nuclear surface properties of the non-linear meson interactions and tensor couplings recently considered in the literature. These interactions, beyond standard QHD, are based on effective field theories. The effective field theory approach allows one to expand the non-renormalizable couplings, which are consistent with the underlying QCD symmetries, using naive dimensional analysis and the naturalness assumption [5,7,8,9].
The quartic vector self-interaction ζ 0 makes it possible to obtain a desirable positive value of the coupling constant κ 4 of the quartic scalar self-interaction, for realistic nuclear matter properties and within the bounds of naturalness. This ζ 0 coupling has only a slight impact on the surface properties. Nevertheless, it helps to find parametrizations where both the surface energy coefficient E s ant the surface thickness t lie in the empirical region. The ζ 0 vector non-linearity makes the spin-orbit potential well deeper, although the effect is almost negligible. Concerning the volume non-linear couplings η 1 and η 2 , they also allow one to obtain positive values of κ 4 in the region of naturalness, depending somewhat on the saturation properties (incompressibility and effective mass). The surface properties are not much affected by these bulk terms either, and it turns out that η 2 has a marginal effect as compared to that of η 1 .
The equilibrium properties do not depend on the couplings α 1 and α 2 that involve the gradients of the fields. Thus, these couplings serve to improve the quality of the surface properties without changing the bulk matter. In the conventional σ − ω model the only parameter not fixed by the saturation conditions is the mass of the scalar meson. In the α 1 -α 2 plane the lines of constant E s have a different slope than those of constant t. It is then possible to keep a fixed value of E s and to modify the value of t by choosing α 1 and α 2 appropriately. The range of variation of E s and t with α 1 and α 2 is wider than with the volume couplings. This justifies including these gradient terms in the energy functional in spite of being of order 5 in the expansion. The α 1 and α 2 surface meson interactions also influence the spin-orbit potential, but the effect is not extremely significant.
The effective model is augmented with a tensor coupling of the ω meson to the nucleon. 23 An outstanding feature is the drastic consequences it has for the spin-orbit force. We have emphasized how inclusion of f v permits to obtain a spin-orbit strentgh similar to that of M * ∞ /M ∼ 0.6 with larger values of the equilibrium nucleon effective mass, contrary to the phenomenology known from models without such a coupling.
We have discussed the implications of the extra couplings of the extended model on various surface symmetry properties. We have restricted ourselves to the regime of low asymmetries, where the liquid droplet model can be applied and the surface stiffness coefficient Q is the key quantity. In particular we have pointed out the role that the non-linearity η ρ of the isovector ρ-meson field may play in the details of the symmetry properties.