Contribution of the rho meson and quark sub-structure to the nuclear spin-orbit potential
aa r X i v : . [ nu c l - t h ] A ug Contribution of the ρ meson and quark substructure to the nuclear spin-orbit potential Guy Chanfray and J´erˆome Margueron Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3,IP2I Lyon, UMR 5822, F-69622, Villeurbanne, France (Dated: August 28, 2020)The microscopic origin of the spin-orbit (SO) potential in terms of sub-baryonic degrees of freedom is ex-plored and discussed for application to nuclei and hyper-nuclei. We thus develop a chiral relativistic approachwhere the coupling to the scalar- and vector-meson fields are controlled by the quark substructure. This ap-proach suggests that the isoscalar and isovector density dependence of the SO potential can be used to test themicroscopic ingredients which are implemented in the relativistic framework: the quark substructure of the nu-cleon in its ground-state and its coupling to the rich meson sector where the ρ meson plays a crucial role. Thisis also in line with the Vector Dominance Model (VDM) phenomenology and the known magnetic properties ofthe nucleons. We explore predictions based on Hartree and Hartree-Fock mean field, as well as various scenar-ios for the ρ -nucleon coupling, ranked as weak, medium and strong, which impacts the isoscalar and isovectordensity dependence of the SO potential. We show that a medium to strong ρ coupling is essential to reproduceSkyrme phenomenology in N = Z nuclei as well as its isovector dependence. Assuming an SU(6) valence quarkmodel our approach is extended to hyperons and furnishes a microscopic understanding of the quenching of the N Λ spin-orbit potential in hyper-nuclei. It is also applied to other hyperons, such as Σ , Ξ and Ω . Despite its crucial role for the understanding of nuclearmagic numbers [1, 2] and consequently of element abun-dances in our Universe [3], the microscopic origin of the spin-orbit (SO) interaction is still a matter of discussion. For prac-tical nonrelativistic nuclear interactions and application to fi-nite nuclei, it is often introduced as a phenomenological termcorrecting the nuclear interaction [4, 5]: it is represented by ashort-range interaction describing the coupling of the particle i with spin s i to its orbital angular momentum l i = p i × r i . Asa consequence, the SO potential is characterized by a densitygradient term (boost) with isoscalar and isovector contribu-tions, see for instance Refs. [6, 7] and references therein.One major success of the relativistic hadrodynamic modelinitiated by J. Walecka and coworkers [8, 9] comes from thenatural framework it provides for the SO coupling withoutthe need to introduce an explicit interaction. The SO inter-action originates from the relativistic nature of the hadrody-namic model, often referred to in nuclear physics as the rel-ativistic mean field (RMF) where only the Hartree potentialis considered [5], since it is generated by the coupling be-tween the up and down components of the Dirac spinor [8].In particular, a nonrelativistic potential can be derived fromthe hadrodynamic model showing that the coupling constantof the SO potential is a function of the meson coupling con-stants, which are determined from the bulk properties of nu-clear matter and/or fits to the nuclear masses. The SO split-ting appears thus as a prediction of the model since it is notfit a priori. Its detailed density functional, e.g. isoscalar andisovector density dependence, may however change from oneLagrangian to another.A functional difference between Skyrme [5] and RMF [5]nuclear interaction has been suggested by Reinhard and Flo-card [6]: the Skyrme SO potential combines together isoscalarand isovector density gradient, while the RMF is purelyisoscalar. It should however be noted that the RMF La-grangian in Ref. [6] includes only the contribution of the σ and ω mesons to the SO potential. In Ref. [10] for instancethe additional effect of the ρ vector meson and of the Fock term has been discussed, but restricting the ρ to its vectorcoupling to the nucleon and neglecting the ρ -tensor coupling.In this paper, we explicitly detail the contribution of differentmesons to the SO potential and we explore three scenarios forthe ρ meson coupling: the weak coupling which neglects the ρ -tensor contribution, and the medium and strong couplingwhich include it with increasing strength. The medium cou-pling reproduces the pure Vector Dominance Model (VDM)picture [11] while the strong coupling requires an extension ofthe pure VDM picture and is compatible with the π nucleonscattering data [12]. We show that the ρ -tensor coupling andthe Fock contribution to the mean field are crucial to reconcilerelativistic approaches with Skyrme nuclear phenomenologyand more generally to adapt to the experimental data in nucleiand hypernuclei.In the pure VDM picture the ρ -tensor coupling fully con-tributes to the anomalous magnetic moment of the nucleons.In a microscopically based chiral relativistic approach, thiscoupling originates from the composite nature of baryons intothree quarks. Hence, although the SO potential can be seen asa pure relativistic effect, its precise form deeply roots into thenature of the strong (nuclear) interaction, e.g. its chiral real-ization, the contribution of the scalar and vector meson fielddynamics as well as the quark substructure. In other words,anchoring the relations between the meson coupling constantsinto a quark substructured (bag) model, the confrontation ofthe chiral relativistic predictions for the SO potential to nu-clear data can drive to a better understanding of some mi-croscopic aspects of baryons and of their mutual interaction.There are indeed important questions involving finite nucleiand hyper-nuclei, yet unresolved, which can contribute to thebetter understanding of the strong interaction and of the quarksubstructure, such as i) what is the isoscalar and isovector de-pendence of the SO interaction? and ii) how is the SO inter-action modified in hyper-nuclei? To achieve this program weinvestigate the role of the quark wave functions in governingthe SO coupling of the exchanged mesonic degrees of freedomwith the nucleons and the hyperons. We also show the effectof the strange quarks to the SO interaction, providing a deepmicroscopic understanding of SO splitting in hyper-nuclei. Inthe following, our microscopic quark-level derivation of thespin-orbit potential closely follows the one from Guichon etal. [13].Interestingly the quark substructure of the nucleon impactsalso the saturation mechanism of the energy per particle in nu-clear matter. Experimentally, the curvature of the energy perparticle is directly measured from the energy of the isoscalargiant monopole resonance (ISGMR). It has been suggestedthat the softening of the equation of state around saturationdensity is induced by the polarization of the quark internalstructure of the nucleons [14–16]. This sub-nucleonic polar-ization appears in the Lagrangian as a nonlinear meson cou-pling for the σ field or alternatively as a density dependenceof the scalar-meson coupling constant. It impacts also the SOpotential, but at a subleading order which is neglected in thisstudy. It is however interesting to note that, through the SOinteraction and the saturation mechanism, the quark substruc-ture appears to have, at least, two concrete realizations impact-ing the modeling of the interaction between nucleons. Whileit is possible to ignore the microscopic mechanism suggestedby the quark substructure in practical nuclear modeling by in-troducing new terms in the Lagrangian fitted to the propertiesof finite nuclei, the mechanism we refer to suggests a moreglobal picture which provides a deep understanding of the na-ture of the nuclear interaction.This paper is organized as follows: we recall the derivationof the SO potential in atomic and nuclear physics in Sec. I.We then present a derivation of the nucleonic spin-orbit po-tential from a chiral Hartree-Fock description in Sec. II. Ourapproach is by many aspect based on the one presented inRef. [13], and also opens the possibility to perform a fullyconsistent relativistic Hartree-Fock calculation. The connec-tion to nuclear physics is emphasized in our study. In Sec. IIIwe show the impact of the nucleon substructure in nuclei andwe compare our findings to widely used parametrizations usedin nuclear structure, such as Skyrme energy density function-als (EDFs) or relativistic mean field (RMF) [5, 6]. In particu-lar we discuss the isospin dependence of the spin-orbit inter-action. Since the present approach can easily be extended topredict the SO interaction for any kind of baryon, we presentan application to hyperons in Sec. IV. We therefore apply ourgeneric results to the Λ N spin-orbit, which is known to belargely quenched, and predict SO potential for the other hy-peron systems. We then conclude this study in Sec. V. I. THE SPIN-ORBIT INTERACTION IN ATOMIC ANDNUCLEAR PHYSICS
The SO interaction exists in many quantum bound sys-tems from atoms to quarkonia, see for instance Ref. [17]. Inatomic physics its origin is well known: it is generated bythe coupling of the electron magnetic moment (spin) movingin the electric field of the nucleus, to which shall be addedthe Thomas precession [18]. An atomic electron – located atposition R – having orbital l and internal spin s angular mo- menta and moving in a central mean-field potential U ( R ) feelsa spin-orbit potential of the form W so , e ( R ) = e m e R dUdR l · s − e m e R dUdR l · s . (1)The first term in Eq. (1) comes from the interaction of theelectron magnetic moment (represented by the internal spin s )with the mean magnetic field existing in its instantaneous restframe (IRF): this is a boost effect (generating the gradient)since this mean magnetic field in the IRF originates from theLorentz transformation of the mean electric field in the restframe. However even in the absence of electric and magneticfields, the rotation of the particle curvilinear orbit involvesan additional boost (perpendicular to the motion), which isknown as the Thomas precession. It is a pure relativistic effectthat is independent of the structure and that yields the secondterm in Eq. (1). The Thomas precession reduces the impact ofthe first boosted term to the total SO potential.The SO interaction in finite nuclei is quantitatively very dif-ferent. Not only it is much larger – see for instance the discus-sion in Ref. [17] – but it has also the opposite sign comparedwith the atomic physics case. While in atomic physics, thecoupling of the electromagnetic field to the particle is only ofvector type, in nuclear physics, the interaction is spread overmore coupling channels. In particular, there is also a scalarinteraction which contributes to the SO interaction. The verylarge attractive scalar and repulsive vector self-energies, typ-ically Σ S ≈ −
400 MeV and Σ V ≈ +
350 MeV in the interiorof finite nuclei, combine together to produce the mean field(sum) and the spin-orbit potential (difference). Consequentlythe atomic formula for electrons (1) is transformed for nucle-ons ( N = p , n ) as, W so , N ( R ) ≃
12 1 m N Σ V − Σ S Σ V + Σ S R dUdR l · s . (2)Note that the structure of Eq. (1) can be recovered from Eq. (2)setting Σ S =
0. In Eq. (2) the nuclear spin-orbit potential isamplified by an order of magnitude since Σ V − Σ S is muchlarger than | Σ V + Σ S | , and the negative sign is given by thesign of Σ V + Σ S .Moreover in this picture, nucleons in the mean field cou-ple to potentials – Σ V and Σ S – which are of the order of onethird of their own mass. One can thus expect that these hugescalar and vector fields probe more than just the global struc-ture factor of nucleons, represented by its mass, but that theyare also sensitive to nucleon internal degrees of freedom, suchas quarks, gluons, pion cloud, etc... Conversely, it is diffi-cult to imagine that these huge fields have no effect on theinternal structure of the nucleon. This is the motivation of thequark-meson coupling (QMC) model proposed by Guichonin Ref. [14]. The composite nature of nucleons and the hugefields produces a polarization which softens the density de-pendence of the energy per particle around saturation density.This mechanism induces a nonlinear sigma-meson couplingor alternatively a density correction to the sigma coupling con-stant [14–16].A question immediately arizes: where do these huge scalarand vector fields come from? How are they generated fromQCD and how do they couple to the quarks (and possibly tothe pion cloud) inside the nucleon? This will be partially an-swered in the next section. II. RELATIVISTIC CHIRAL APPROACH WITHCONSTRAINTS FROM NUCLEON STRUCTURE
The link between QCD in its nonperturbative regime andthe dynamical interactions among nucleons is not yet com-pletely understood. Since the spin-orbit interaction betweenbaryons is essentially short-ranged [19], a number of authorshave linked its microscopic origin to the quark degrees offreedom, see for instance the original Ref. [20]. The rela-tion between a quark model and the spin-orbit interactionhas also been investigated in the following works, e.g. seeRefs [13, 14] for nuclei and Refs [21, 22] for hyper-nuclei. Inthese models, one usually starts with an effective realizationof the low-energy QCD Lagrangian, which can be for instancethe Nambu-Jona-Lasinio (NJL) model or nucleon orbital mod-els. Recent progresses in Lattice QCD will hopefully help theunderstanding of the nucleon interaction and the role of thequark substructure.
A. Foundational aspects
In this section we detail one type of strategy which connectsthe low-energy realization of QCD and the SO potential. Herethe SO potential, among other things, emerges from a localcoupling of vector and scalar fields to the quarks, which arethemselves confined by a scalar (string) potential. This canbeen done in three steps.The first step is to perform a gluon averaging of the (eu-clidean) QCD partition function to generate – at the so-called Gaussian approximation level – a chiral invariant four-quark effective Lagrangian. An efficient way is to apply theField Correlator Method (FCM) elaborated by Y. Simonov ancoworkers [23]: a very important outcome is the simultaneousand automatic generation of scalar confinement and dynami-cal chiral symmetry breaking. The whole approach dependson two QCD parameters – the string tension σ and the gluoncorrelation length, or string width, T g itself related to the gluoncondensate – and yields a long range scalar confining poten-tial V C ( r ) = σ r . What plays the role of a constituent quarkmass emerges as M ≈ σ T g . A possible crude realization butnot so bad phenomenologically is the NJL model associatedwith the following Lagrangian: L = ¯ ψ (cid:0) i γ µ ∂ µ − m (cid:1) ψ + G (cid:2) ( ¯ ψψ ) + ( ¯ ψ i γ ~ τψ ) (cid:3) − G (cid:2) ( ¯ ψγ µ ~ τψ ) + ( ¯ ψγ µ γ ~ τψ ) + ( ¯ ψγ µ ψ ) (cid:3) , (3)and complemented by a confining force of a string type [16].In the second step, as explicitly worked out in Ref. [16], q ¯ q fluctuations in the Dirac sea can be integrated out and pro-jected on to mesonic degrees of freedom. This bozonisationprocedure generates a scalar field σ (with quantum numbers of the ”sigma” meson) and vector fields ω , ρ (with quantumnumbers of the ω and ρ mesons) which couple locally to theconstituents of the nucleon (the quarks and also possibly thepion cloud which is ignored here). In addition quantum fluc-tuations generate their kinetic-energy Lagrangian. The modelallows us to calculate a quark-scalar and quark-vector cou-pling constants, g qS and g qV , as well as the mass parameters m S = m σ and m V = m ω = m ρ , which are not the on-shellmesons masses but rather represent the inverse of the corre-sponding propagators taken at zero momentum. According tothe FCM approach [23] and following reference [16], the NJLmodel can be completed by adding a confining force acting onthe NJL constituent quarks whose masses are directly propor-tional to the in-medium scalar field.The third step is to evaluate the coupling of these QCDfields to the nucleon where the quarks move in a scalar confin-ing potential. It thus consists in the emergence of an effectivenucleon-meson interaction, as detailed in the next section. B. From quarks to nucleons
We now evaluate the coupling of the QCD scalar and vec-tor fields to the nucleons, where constituent quarks move in a(scalar) confining potential. We call generically this type oforbitals model approach ”bag model”. As in Ref. [13] let usconsider a nucleon at center of mass (CM) position R within-medium effective mass M ∗ N and velocity V = P / M ∗ N , em-bedded in the nuclear mesonic fields: σ , ω and ρ . The quarklocated at R + r , feels a scalar and a vector potentials: U qS ( R + r ) = g qS σ ( R + r ) U qV ( R + r ) = g qV (cid:0) ω ( R + r ) + τ q ρ ( R + r ) (cid:1) , (4)where τ q is the isospin Pauli matrix in the third direction.The coupling of these mesonic fields to the moving nucle-ons – including relativistic effects – necessitates the knowl-edge of the three quark wave functions inside the ”bag”, asdiscussed in Ref. [13]. For that purpose the InstantaneousRest Frame (IRF) of the nucleon is introduced, with rapidity ξ such that V = tanh ξ . In the spirit of the Born-Oppenheimerapproximation, one can assume that these quarks wave func-tions are known in the IRF since the quarks have time to adjusttheir motion so that they are in their lowest energy state (seediscussion in Ref. [13] for details). Performing the boost withrapidity ξ in the IRF of this orbital model, one can define thenucleon mean-field potential as ( N = p , n ): U N ( R ) = Z IRFbag d r ′ h N | ¯ q (cid:0) r ′ (cid:1) [ U qV (cid:0) R + r ′ (cid:1) ( γ cosh ξ + γ · ˆ V sinh ξ ) + U qS (cid:0) R + r ′ (cid:1) ] q (cid:0) r ′ (cid:1) | N i . (5)We now expand the fields to first and second orders in r ′ ac-cording to U qS ( R + r ′ ) = U ( R ) + r ′ · ~ ∇ U ( R ) . Working outthe quark-nucleon matrix elements, this generates to leadingorder the ordinary mean-field potential: U N ( R ) ≡ U V ( R ) + U S ( R )= g ω ω ( R ) + g ρ ρ ( R ) h τ N i + g σ σ ( R ) . (6)where the couplings of the meson fields to the nucleons aredefined as g ω = g qV , g ρ = g qV , and g σ = g qS q S , with q S = R d r (cid:0) u ( r ) − v ( r ) (cid:1) < ∼ u and v are the up and down quark ra-dial wave functions in standard notations). These relationsreflect the quark substructure of the nucleon where the factorthree refers to the quark number. Let us also mention that thescalar piece of the mean-field potential and the decrease ofthe nucleon mass in the medium are related to the decrease ofthe chiral condensate associated with partial chiral symmetryrestoration at finite density [15, 16].Moreover, after performing exactly the inverse boost, thisprocedure allows us to build a nucleon located at point R withenergy in the laboratory frame [13], E ( R ) = q M ∗ N ( R ) + P ∗ ( R ) (7)with M ∗ N ( R ) = M N + Σ S ( R ) , and P ∗ ( R ) = M ∗ N ( R ) ˆ V sinh ξ . (8)At this level, one comment concerning the coupling to thescalar field is in order. In the detailed approach described inRefs. [16, 24], the treatment of the nucleon coupling to thescalar field – seen as a fluctuation of the chiral field – is alittle more involved since it leads to the concept of the nu-cleon response to the scalar field as was originally introducedby Guichon [14]. One net effect is the density dependence ofthe scalar coupling constant corresponding to the progressivereduction of the scalar field, which thus generates the repul-sion needed for the saturation mechanism. This mechanismis precisely what was proposed by P. Guichon in his pioneer-ing paper [14] at the origin of the QMC model. Although thisscalar field decoupling mechanism is essential for the satura-tion properties, here we disregard this effect for the SO poten-tial since it is a subleading effect. Another consequence of thescalar nature of the coupling would be to replace in the scalarpotential the baryonic density ρ by the scalar density ρ S . Thisis again a subleading correction which goes beyond the scopeof the present study.For practical applications to nuclear physics our relativis-tic chiral approach is very similar to the original QMCmodel [25] but they are at least two important differences atthe principle level. First in the QMC model, the nucleon (MITbag) model only insists on confinement whereas, in our ap-proach, chiral symmetry breaking (and its partial restorationat finite density) is present by construction ; in particular thesigma field has a perfectly-well-defined chiral status; it is chi-ral invariant and reflects part of the evolution of the quark con-densate associated with partial chiral restoration at finite den-sity [15, 16]. Second, in the QMC model the local couplingof the three meson fields to the quarks is introduced by handwith six parameters (three coupling constants and three massparameters), whereas in our approach, they are generated bythe underlying bosonization of the effective QCD Lagrangian. C. Spin orbit potential
From Eq. (5), the SO potential is defined as the second orderin the gradient expansion of the quark position r ′ , W so , N ( R ) = g qV M ∗ N ( R ) Z IRFbag d r ′ h N | ¯ q ( r ′ ) h ~ γ · P r ′ · (cid:16) ~ ∇ω ( R )+ τ q ~ ∇ρ ( R ) (cid:17)i q ( r ′ ) | N i . (9)The ω contribution involves the nucleon matrix element of aone-body quark operator, which can be calculated knowingthe up and down quark wave functions: * g qV ∑ i r i γ i · P + = − g qV ∑ i h σ i × P i Z d r r u i ( r ) v i ( r )= − g ω h σ N × P i Z d r r u ( r ) v ( r ) ≡ − g ω µ S M N h σ N × P i , (10)and the ρ contribution gives * g qV ∑ i r i γ i · P τ i + = − g qV ∑ i h σ i τ i × P i Z d r r u i ( r ) v i ( r )= − g ρ h σ N × P τ N i Z d r r u ( r ) v ( r ) ≡ − g ρ µ V M N h σ N × P τ N i , (11)where in the above equations we used the octet matrix ele-ments, h N | ∑ u , d σ i | N i = h N | σ N | N i , (12) h N | ∑ u , d σ i τ i | N i = h N | σ N τ N | N i . (13)The ω and ρ contributions to the SO potential associatedwith the boost are directly proportional to the isoscalar andisovector nucleon magnetic moments µ S and µ V , defined inEqs. (10) and (11), respectively, and calculated in this typeof bag model. For the SO potential we have implicitly usedin the matrix elements the vacuum quark wave functions, ig-noring the quark polarization (see above discussion). To beconsistent the nucleon effective mass is replaced by its freevalue. From Eqs. (10) and (11), we see that the magnetic mo-ments satisfy the SU(6) bag model ratio µ V / µ S = µ V = µ p − µ n = . µ S = µ p + µ n = .
88 [26]. Moreover it is also possibleto explicitly evaluate the radial integral R d r r u ( r ) v ( r ) to getthe absolute value of µ S and consequently µ p = µ S , µ n = − µ S , µ V = µ S . In bag model with a scalar confining inter-action the above radial integral can be obtained independentlyof the precise shape of the confining interaction by using theDirac equation. The result is µ S = M N ε (cid:16) + q S (cid:17) ≡ M N ε (cid:18) g A + (cid:19) (14)where ε is the eigenenergy of the lowest orbital, q S is theintegrated one-quark scalar density and g A is the weak axial-vector coupling constant calculated in the model. Consider-ing the experimental value g A = .
26 and taking M N ≈ ε ,one automatically obtains the correct order of magnitude, µ S ≈ .
9. However, the MIT bag model predicts g A = . ε = . / R , which tends to give too low a magnetic mo-ment for a reasonable value of the bag radius R ≤ . V C ( r ) = γ σ r where σ is the string tension, giving good results for both g A and the magnetic moments [27].In the following we introduce the anomalous isoscalar κ ω and isovector κ ρ magnetic moments through the followingdefinitions: µ S ≡ + κ ω and µ V ≡ + κ ρ , in units of the nu-clear magneton µ N .Injecting Eqs. (10) and (11) into (9), the boost piece of theSO potential takes the following form: W boostso , N ( R ) = − g ω ( + κ ω ) M N M ∗ N ( R ) ~ ∇ω ( R ) · h σ N × P i− g ρ ( + κ ρ ) M N M ∗ N ( R ) ~ ∇ρ ( R ) · h σ N × P τ N i . (15)This contribution has to be supplemented by the Thomasprecession (TP) piece, also derivable in the above approach[13]: W TPso , N ( R ) = − g σ M N M ∗ N ( R ) ~ ∇σ ( R ) · h σ N × P i + g ω M N M ∗ N ( R ) ~ ∇ω ( R ) · h σ N × P i + g ρ M N M ∗ N ( R ) ~ ∇ρ ( R ) · h σ N × P τ N i . (16)Again, for consistency, we also replace the nucleon effectivemass coming from the boost by the bare nucleon mass in thefollowing.Finally, the SO potential is given by the sum of the boostand TP contributions, W so , N ( R ) = W boostso , N ( R ) + W TPso , N ( R ) . (17)As a side remark, let us mention that the above results canalso be derived from a relativistic theory such as that utilizedin Ref. [24] where the nucleon-vector meson coupling La-grangian written with standard notation reads: L ω = − g ω ω µ ¯ Ψγ µ Ψ − g ω κ ω M N ∂ ν ω µ Ψ ¯ σ µν Ψ , L ρ = − g ρ ρ a µ ¯ Ψγ µ τ a Ψ − g ρ κ ρ M N ∂ ν ρ a µ Ψ ¯ σ µν τ a Ψ . (18)The origin of the tensor ( ρ and ω ) couplings in such a La-grangian is not resolved but instead given as an input. In ourapproach instead, these couplings are derived from the quarksubstructure of the baryons. Meson Boost Thomas Total Associatedprecession gradient w σ N g σ m σ g σ m σ ∇ n w ω N ( + κ ω ) g ω m ω − g ω m ω ( + κ ω ) g ω m ω ∇ n w ρ N ( + κ ρ ) g ρ m ρ − g ρ m ρ ( + κ ρ ) g ρ m ρ ∇ n TABLE I. Meson decomposition of the nucleon direct (Hartree) spin-orbit potential multiplying the term ( l · s ) τ / ( RM N ) . The densitiesare n ≡ n n + n p and n ≡ n p − n n . III. SPIN-ORBIT POTENTIAL IN NUCLEI
The equations of motion for the meson fields can be usedto express the SO potential in terms of the nucleon densi-ties. Starting from Eqs. (15) and (16) and assuming largevector-meson masses (i.e., neglecting Darwin terms), e.g. ω ( R ) = g ω m ω n ( r ) , we obtain, after elementary manipulations(namely, R · h σ N × P i = − ( l · s ) ), an expression for spher-ical nuclei involving the radial derivative of the total nu-cleon density n ( r ) ≡ n p ( r ) + n n ( r ) and the isovector density n ( r ) ≡ n p ( r ) − n n ( r ) (note the convention for n which is op-posite to the usual nuclear one) as, W boostso , N ( R ) = R M N h g ω m ω ( + κ ω ) dn dR ± g ρ m ρ ( + κ ρ ) dn dR i ( l · s ) τ . (19)and W TPso , N ( R ) = R M N h g σ m σ dn dR − g ω m ω dn dR ∓ g ρ m ρ dn dR i ( l · s ) τ . (20)where the ± and ∓ signs in the previous equations refer re-spectively to the proton and neutron cases.In practice, due to the small value of κ ω ∼ − .
13, the ω -tensor coupling can be safely neglected. The case of the ρ meson is less simple. The pure Vector Dominance Model(VDM) picture [11], i.e., the strict proportionality between theelectromagnetic current and the vector-meson fields, impliesthe identification of κ ρ with the anomalous part of the isovec-tor magnetic moment of the nucleon, i.e. , κ ρ = .
7, hereaftercalled the medium coupling for the ρ meson. For instancethe effective Lagrangian PKA1 [28] assumes a value of about3.2, which is comparable to the one suggested by the VDMpicture. However pion-nucleon scattering data [12] suggesta larger value κ ρ = .
6, hereafter called the strong ρ cou-pling. Many approaches in finite nuclei while including the ρ -vector coupling neglect the ρ -tensor coupling; see, for in-stance, Ref. [10, 29]. In the following, we also explore thiscase, defined as the weak ρ coupling. Finally, the decompo-sition of the SO potential for the various meson channels areshown in Table I.For the nucleonic sector, the SO potential is usually ex- ρ contribution no ρ weak ρ medium ρ strong ρ w ρ N (fm ) 0 ≃ . ≃ . ≃ . W H / W H ≃ . < ∼ > ∼ W HF / W HF ≃ . < ∼ . > ∼ . ( W + W ) H ≃ . ≃ . ≃ . ≃ . ) ( W + W ) HF ≃ . ≃ . ≃ . ≃ . )TABLE II. Summary of the results showing the various scenarios forthe ρ . First row, ρ coupling w ρ N for the weak, medium and strongscenarios. For the other mesons, we have w ω N ≃ and w σ N ≃ . . Then we show the predictions for the ratio W / W , seeEqs. (22), (23), (24), and for the half sum ( W + W ) /
2; see Eq. (27),for Hartree (RMF) and Hartree-Fock (RHF) cases. pressed as, W so , N ( R ) = R ( W ∇ n τ + W ∇ n − τ ) ( l · s ) τ , (21)hence directly exhibiting its isospin dependence. Forthe Skyrme interaction, the ratio ( W / W ) Skyrme = ρ ) we have ( W / W ) RMF , no ρ = W H and W H for thedirect (Hartree) contribution in terms of the quantities w iN ( i = σ , ω , ρ ) defined in Table I, W H ≡ M N (cid:2) w σ N + w ω N + w ρ N (cid:3) , (22) W H ≡ M N (cid:2) w σ N + w ω N − w ρ N (cid:3) . (23)For the orientation of the following discussion let us rea-sonably consider that the σ and the ω contributions are sim-ilar, as suggested from most phenomenological studies [5,9]. For instance if we choose the omega coupling adjustedfrom standard VDM phenomenology ( g qV = . m ω =
780 MeV) [11], one obtains g ω / m ω = ( × . × / ) ≃ g ω / m ω = . g σ / m σ shall beslightly larger, leading to w σ N ∼ . w ω N at maximum. The ρ coupling constant is one third of the ω coupling constant. Soin the absence of ρ -tensor coupling (weak ρ ), we also expectthat the ρ contribution will be w weak ρ N ∼ . w ω N . However inthe medium- and strong- ρ cases, w med ρ N and w strong ρ N will bearound ten times as large; see Table II for typical values.Hence for a typical RMF approach without the ρ field,Eqs. (22) and (23) predict W H / W H =
1, as expected from thesimplest ” σ - ω ” Walecka model; see, for instance Ref. [6]. Itis interesting to note that, for the medium (strong) ρ coupling, one has W H / W H < ∼ W H / W H > ∼
3) significantly larger than W H / W H ≃ . ρ coupling. In Ref. [10] this ra-tio calculated at the Hartree level (RMF) remains very closeto 1.1 (see Fig. 1 of this paper) for various nuclei ( O, Si,
Pb), which is consistent with the weak- ρ hypothesis.We see the considerable effect of the ρ -tensor coupling tothe isovector density dependence of the SO potential, whichin our approach is interpreted as a purely quark substructureeffect. The question then arises of its survival when exchangeterms are included. From the approach of Ref. [24], it is pos-sible to construct an energy density functional [30] as in theQMC model [25, 31], from which one can deduce the ex-change (Fock) contribution to the spin-orbit potential. Theresults for the direct (Hartree) are given in Eqs. (22) and (23),and we give hereafter the exchange (Fock) and total (Hartree-Fock) contribution to W and W as, W F = M N (cid:0) w σ N + w ω N + w ρ N (cid:1) , W F = M N w ρ N , W HF = M N (cid:0) w σ N + w ω N + w ρ N (cid:1) , W HF = M N ( w σ N + w ω N ) . (24)Completely ignoring the contribution of the ρ meson, onegets (cid:2) W HF / W HF (cid:3) no ρ = .
5, which can be seen as the ba-sic Walecka model result with exchange correction included.Introducing the ρ contribution in the weak scenario, i.e., ig-noring the tensor coupling, one obtains (cid:2) W HF / W HF (cid:3) weak ρ ∼ .
7, not far from the ratio 1 . − . O, Si,
Pb, in a RHFcalculation which also incorporates density dependent cou-plings. For the medium ρ coupling, the ratio becomes (cid:2) W HF / W HF (cid:3) med ρ < ∼ .
25, and for the strong ρ coupling, itbecomes (cid:2) W HF / W HF (cid:3) strong ρ > ∼ .
25. One can also comparewith the conventional Skyrme EDF parametrization for which W / W =
2, see for instance Refs. [5, 6], which turns out tobe close to our estimate for medium and strong ρ scenarios.These results are summarized in Table II.One can observe from the results given in Table II that theratio W / W is clearly influenced by the contribution of the ρ meson, as well as by the Fock term in the mean field. Forall cases, the Fock term contributes to shift the ratio W / W towards the phenomenological Skyrme value ( ≃ ρ meson coupling.It is also interesting to look at the influence of the ρ mesonon the absolute value of the SO potential. For this purposeone can look at its isoscalar component, i.e., the SO potentialfelt by one nucleon in a N = Z nucleus. In the SLy5 SkyrmeEDF approach this potential is parametrized with the W ≃
120 MeV fm ≃ . parameter [33] according to: [ W so ] Skyrme N = Z ( R ) = W R dn dR l · s . (25)In our microscopic approach, the same quantity is given by: [ W so ] Micro N = Z ( R ) = W + W R dn dR l · s . (26)We see that we have to compare 3 W / ≃ .
45 fm in theSLy5 Skyrme EDF with ( W + W ) / (cid:20) W + W (cid:21) H no ρ = M N ( w σ N + w ω N ) , (cid:20) W + W (cid:21) H with ρ = M N ( w σ N + w ω N ) , (cid:20) W + W (cid:21) HF no ρ = M N ( w σ N + w ω N ) , (cid:20) W + W (cid:21) HF with ρ = M N ( w σ N + w ω N ) + M N w ρ N . (27)The two last rows of Table II provide estimates for theSO potential under various scenarios for the ρ coupling. Wesee that the contribution of the ρ meson, including its ten-sor piece, is of utmost importance to reproduce the Skyrmephenomenology otherwise the SO potential would be stronglyunderestimated by almost a factor of two. Also note that thequantitative agreement of the microscopic approach with theSkyrme interaction has been discussed within the QMC modelin Ref. [31]. Moreover the strength of ρ -tensor coupling,which is still under discussion, can possibly be determinedfrom the isoscalar and isovector density dependence of the SOinteraction extracted from finite nuclei data.Let us make a further comment: in the present case we haveneglected the influence of the nucleon effective mass, namelyits lowering with respect to the bare nucleon mass which is po-sition dependent and varies from about 0.6-0.7 M N in the bulkup to about M N at the surface. At leading order, the SO po-tential is increased by about 10%-30% up, see Eqs. (15) and(16), depending on the coordinate position R ; to be consistentone should nevertheless take into account the decrease of thein-medium scalar coupling constant g σ , which reduces the SOpotential but to a lower extent. As a result, the SO potentialwill increase by about 10%-20% while the ratio W / W willbe almost unchanged. The Skyrme phenomenology will thusbe recovered more consistently for the isoscalar and isovec-tor density dependence by considering the medium-to-strong-coupling cases. These results certainly deserve a more de-tailed calculation but a firm conclusion is nevertheless that arealistic relativistic calculation (RHF) certainly requires theinclusion of the ρ -tensor coupling, which can ultimately belinked to the quark substructure of the nucleon. The symme-try (SLS) and the antisymmetric (ALS) spin-orbit terms to theenergy splitting are discussed in Refs. [34, 35]. IV. SPIN-ORBIT POTENTIAL IN HYPER-NUCLEI
Let us now come to the question of the SO potential inhyper-nuclei. Recent precision measurements of E p − to s − shell orbitals of a Λ hyperon in Λ C give a
Baryon composition S B T B L B I B p uud 1 5 / n udd 1 − / − Λ uds 0 0 2 / Σ + uus 4 / / / Σ uds 4 / / Σ − dds 4 / − / / − Ξ uss − / − / / Ξ − dss − / / / − Ω − sss 0 0 0 0TABLE III. Matrix elements for nucleons and hyperons. p / - p / SO splitting of only (152 ±
65) keV [36] to be com-pared with about 6 MeV in ordinary p − shell nuclei (differentby a factor ≈ Λ SO potential therefore appears to beweaker by at least an order of magnitude than the nucleonicSO potential. This effect was originally suggested from phe-nomenological analyses indicating a strong suppression of the Λ spin-orbit potential [37, 38].Since the seminal work by Brockmann and Weise [39]where the reduction of the SO potential in Λ hypernucleiwas obtained in a relativistic Hartree approach, this effecthas been investigated within several models: From one-bosonexchange N Λ potentials [34, 40–42], which tend to overesti-mate the N Λ spin-orbit potential; from SU(3) generalizationof standard nuclear RMF models [39, 43–45]; from the naiveSU(6) quark model with flavor symmetry breaking, which nat-urally explains the small spin-orbit coupling of the Λ hyperon;from a quark model picture combined to Dirac phenomenol-ogy [21, 46]; or from combining the quark model with scalar-and vector-meson exchange (QMC, quark-meson couplingmodel) [47], where Pauli blocking in the Λ N - Σ N coupledchannels is incorporated phenomenologically. We finallymention the flavor-SU(3) in-medium chiral effective-field the-ory approaches, where strangeness is being included. Analmost complete cancellation is found between short-rangecontributions and long-range terms [48, 49], even includ-ing the three-body spin-orbit interaction of Fujita-Miyazawatype [50]. This scenario has been tested over a large set ofhyper-nuclei [51, 52]. This list is only partial and many otherapproaches have been developed.Our chiral relativistic approach can be extended to the fulloctet including hyperons. In case of a single hyperon hypernu-cleus, the mesonic mean field originating from the ensembleof baryons with a large majority of nucleons is not modified.When considering the spin-orbit felt by an hyperon, the sum-mation on the quarks appearing in Eqs. (10) and (11) has to belimited to the u and d quarks since the strange quark does notcouple to the σ , ω and ρ fields. We neglect here interaction ofhyperons mediated by strange mesons. Assuming SU(3) fla-vor symmetry, one can derive a general expression for the SOpotential experienced by any baryon B = N or Y where Y = Λ , Ξ , Σ or Ω [47]: W so , B ( R ) = W Boostso , B ( R ) + W TPso , B ( R ) , (28)where W Boostso , B ( R ) = RM N (cid:20) g ω m ω ( + κ ω ) S B dn dR + g ρ m ρ (cid:0) + κ ρ (cid:1) T B dn dR ( l · s ) B (29) W TPso , B ( R ) = RM N "(cid:18) g σ m σ − g ω m ω (cid:19) L B dn dR − g ρ m ρ I B dn dR ( l · s ) B (30)with S B = (cid:10) ∑ i = u , d σ i (cid:11) B h σ B i B , T B = (cid:10) ∑ i = u , d σ i τ i (cid:11) B h σ B i B , (31) L B = * ∑ i = u , d i + B , I B = * ∑ i = u , d τ i + B , (32)where only the Hartree term is considered since we treat thesingle hyperon case. The relevant SU(6) matrix elements S B , T B , L B and I B are given in Table III.A first general remark is that the Thomas precession –which was already small for the nucleon SO potential due tothe compensation between the scalar and the vector terms, andthe small contribution of the ρ term – is also small for the hy-peron potential for the same reason; see Eq. (30).Let us give explicitly the SO potential for the neutral hy-perons, namely Λ , Σ and Ξ . To simplify the writing weomit the prefactor 1 / RM N and we define G σ = g σ / m σ , G ω = g ω / m ω , G ρ = g ρ / m ρ , also keeping in mind that G σ ≈ G ω ≈ ( + κ ρ ) G ρ ≈ G ρ in the case of medium and strong ρ couplings. W so , Λ = ( G σ − G ω ) dn dR ( l · s ) Λ (33) W so , Σ = (cid:20) ( G σ − G ω ) + G ω ( + κ ω ) (cid:21) dn dR ( l · s ) Σ (34) W so , Ξ = (cid:26)(cid:20) ( G σ − G ω ) − G ω ( + κ ω ) (cid:21) dn dR − (cid:20) G ρ + G ρ (cid:0) + κ ρ (cid:1)(cid:21) dn dR (cid:27) ( l · s ) Ξ (35)to be compared with the SO potential for neutrons, includinghere the Fock contribution, W so , n = (cid:20) W + W dn dR − W − W dn R (cid:21) ( l · s ) n = (cid:26)(cid:20) ( G σ + G ω ( + κ ω )) + G ρ (cid:0) + κ ρ (cid:1)(cid:21) dn dR − (cid:20) G ρ (cid:0) + κ ρ (cid:1) + ( G σ + G ω ( + κ ω )) (cid:21) dn dR (cid:27) ( l · s ) n (36) In the particular case of the Λ hyperon, Eq. (34), Thomasprecession is however the only term which survives, leadingto a strong reduction of the SO potential (by a factor of about50, as in the experimental data) with respect to the neutroncase. Similar conclusions have also been obtained in variousanalyses, see Refs. [21, 37, 38, 47].Concerning the Σ case, Brockmann [53] has predicted asmall spin-orbit splitting, in contrast with quark-model pre-dictions suggesting a strong spin-orbit splitting [20]. In ourcase we predict for symmetric nuclei an increase of the Σ SOpotential by about thirty percent with respect to the neutroncase taken at the Hartree level, for N=Z, i.e., Σ Ca as a typicalexample. However, if the Fock term is taken into account forthe neutron case the Hartree term is increased by a factor of5 / ρ contribution has to be added. Hence, for theweak- ρ coupling, the Σ SO potential is expected to be veryclose to the neutron case whereas, for the medium- or strong- ρ coupling the Σ SO potential is expected to be twenty percentsmaller than the neutron SO potential. The QMC model (seeFig. 3 of Ref. [47]) predicts a slight decrease in the case of Σ Ca of this order of magnitude. We also observe that the Σ SO potential is dominated by the ω meson, which induces agreat stability in our results, almost independent of the ρ sce-nario.For the cascade case we predict, again for symmetric nuclei,a significant reduction by a factor of one fifth with respect tothe neutron case with in addition a change of sign (also ob-served in Ref. [47]) . One peculiarity of the Ξ SO potentialis that the contribution of the ω meson is quenched. In asym-metric nuclei such as Y Pb, the reduction of the hyperon spin-orbit potential compared with the neutron spin-orbit potentialis even accentuated.Finally, for the Ω hyperon there is no SO potential since itsis composed only of strange quarks.So, in conclusion, we predict very different SO potentialsfor hyperons based on different meson coupling mechanisms:the cancellation of the boost contribution strongly quenchesthe Λ SO potential, the dominance of the ω coupling for the Σ SO potential induces a very stable prediction, which is 20%smaller than for nucleons, and finally, the quenching of the ω contribution for the Ξ SO potential makes smaller by a fac-tor of order five depending on the ρ scenario compared withnucleons with in addition a change of sign. V. CONCLUSIONS
In this paper, the predictions of a chiral relativistic approachfor the SO potential in nuclei and hyper-nuclei are analyzed.The basic inputs are introduced at the quark substructure levelin such a way that the ω and ρ coupling constants are com-patible with the standard VDM phenomenology and that the σ coupling allows a plausible saturation mechanism. Thestrength of the anomalous magnetic moment generated fromthe ρ -tensor coupling is also predicted from the VDM picture,and we explore some departure from it. Specifically, we studythree distinct scenarios: the weak-coupling case (no ρ tensor),the medium coupling case (suggested from quark substructureand VDM), and the strong-coupling case (deduced from pion-nucleon scattering data). In finite nuclei, the important role ofthe ρ meson is underlined and we compare our results to theusual approximations, where either the ρ meson is neglectedor the Fock term is not calculated (as in the RMF approach).We show that the systematics of SO splitting in finite nucleicould be used to better determine the strength of the ρ mesoncoupling. An important result is that the Skyrme phenomenol-ogy can be recovered only in the case of the medium to strong ρ coupling.The present model is based on the quark substructure of thenucleon, sensitive both to the confinement mechanism and toSU(3) symmetry for the values or relations between the σ , ω and ρ mesons coupling constants. The strong sensitivity of theresults on the ρ meson strength – and in particular on its ten- sor piece affecting the nucleon anomalous magnetic moment –suggests that the confrontation of the present phenomenologi-cal analysis for systematics in finite nuclei could shed light onthe quark substructure of nucleons.The same chiral relativistic model is applied to hypernucleiwhere is shows that the SO potential in these cases can be verydifferent. It is quenched for Λ , decreased by 20% for the Σ ,and reduced by one fifth for the Ξ case. For each case, themechanism is different and analyzed in the present approach.Extending the present model to the description of finite nu-clei, it will be interesting to analyze the isotope shifts in thePb region since it is expected to be closely related to the spin-orbit interaction as well [6]. In the future, we should includeother contributions, such as the π and δ mesons missing in thepresent approach.mesons missing in thepresent approach.