Nuclear energy density functional from chiral pion-nucleon dynamics revisited
aa r X i v : . [ nu c l - t h ] D ec Nuclear energy density functional fromchiral pion-nucleon dynamics revisited
Physik Department, Technische Universit¨at M¨unchen, D-85747 Garching, Germany email: [email protected]
Abstract
We use a recently improved density-matrix expansion to calculate the nuclear energydensity functional in the framework of in-medium chiral perturbation theory. Our cal-culation treats systematically the effects from 1 π -exchange, iterated 1 π -exchange, andirreducible 2 π -exchange with intermediate ∆-isobar excitations, including Pauli-blockingcorrections up to three-loop order. We find that the effective nucleon mass M ∗ ( ρ ) enteringthe energy density functional is identical to the one of Fermi-liquid theory when employingthe improved density-matrix expansion. The strength F ∇ ( ρ ) of the ( ~ ∇ ρ ) surface-term asprovided by the pion-exchange dynamics is in good agreement with that of phenomenolog-ical Skyrme forces in the density region ρ / < ρ < ρ . The spin-orbit coupling strength F so ( ρ ) receives contributions from iterated 1 π -exchange (of the “wrong sign”) and fromthree-nucleon interactions mediated by 2 π -exchange with virtual ∆-excitation (of the “cor-rect sign”). In the region around ρ / ≃ .
08 fm − where the spin-orbit interaction innuclei gains most of its weight these two components tend to cancel, thus leaving all roomfor the short-range spin-orbit interaction. The strength function F J ( ρ ) multiplying thesquare of the spin-orbit density comes out much larger than in phenomenological Skyrmeforces and it has a pronounced density dependence. PACS: 12.38.Bx, 21.30.Fe, 21.60.-n, 31.15.EwKeywords: Nuclear energy density functional; Density-matrix expansion; Chiral pion-nucleondynamics
The nuclear energy density functional approach is the many-body method of choice in order tocalculate the properties of medium-mass and heavy nuclei in a systematic manner [1]. In thiscontext non-relativistic Skyrme forces [2, 3, 4, 5, 6] have gained much popularity because of theiranalytical simplicity and their ability to reproduce nuclear properties over the whole periodictable within the self-consistent Hartree-Fock approximation. Another widely and successfullyused approach to nuclear structure calculations are relativistic mean-field models [7, 8]. In thesemodels the nucleus is described as a collection of independent Dirac quasi-particles moving inself-consistently generated scalar and vector mean-fields. The footprints of relativity becomevisible through the large nuclear spin-orbit interaction which emerges in that framework fromthe interplay of the strong scalar and vector mean-fields. These counteract in producing the(attractive) central potential but act coherently to generate the strong spin-orbit potential. Work supported in part by BMBF, GSI and the DFG cluster of excellence: Origin and Structure of theUniverse. ~ ∇ ρ ) , or the spin-orbit term, ~ ∇ ρ · ~J , arise in these calculations exclusively from the long-range 1 π - and 2 π -exchange dynamics in an inhomogeneous many-nucleon system characterizedby a local density ρ ( ~r ) and a local spin-orbit density ~J ( ~r ).In a recent paper by Gebremariam, Duguet and Bogner [18] an improved density-matrixexpansion has been developed for spin-unsaturated nuclei. It has been demonstrated thatphase-space averaging techniques allow for a consistent expansion of both the spin-independent(scalar) part as well as the spin-dependent (vector) part of the density-matrix. A further keyfeature of the new method has been to take into account the deformation displayed by the localdensity distribution at the surface of most nuclei. The accuracy of the new phase-space averageddensity-matrix expansion and the original one of Negele and Vautherin has been gauged viathe Fock energy (densities) arising from (schematic finite-range) central, tensor and spin-orbitinteractions for a large set of semi-magic nuclei. For a central force the Fock energy dependsprimarily on the spin-independent (scalar) part of the density-matrix and a few percent accuracyis reached for both variants of the density-matrix expansion. On the other hand the Fock energydue to a tensor force is determined by the spin-dependent (vector) part of the density-matrix.In that case the original density-matrix expansion of Negele and Vautherin leads to an error ofabout 50%, whereas the new one based on phase-space averaging techniques reduces the errordrastically to only a few percent. This is the same level of accuracy as obtained for interactionterms involving the spin-independent (scalar) part of the density-matrix. For further detailson these extensive and instructive test studies we refer to ref.[18].The purpose of the present work is to match with these new developments and to reconsiderthe nuclear energy density functional as it emerges from chiral pion-nucleon dynamics on the2asis of the improved density-matrix expansion of ref.[18]. Our paper is organized as follows. Insection 2 we recall the explicit form of the improved density-matrix expansion of Gebremariam,Duguet and Bogner [18]. Its Fourier-transform to momentum space provides the adequatetechnical tool to calculate the nuclear energy density functional in a diagrammatic framework.As a first interesting result we find that for the zero-range Skyrme force the new and the olddensity-matrix expansion lead to identical results. Differences between the two versions aretherefore to be expected for the interaction contributions arising from the long-range 1 π - and2 π -exchange between nucleons. In section 3, we present the analytical results for the density-dependent strength functions F τ ( ρ ), F so ( ρ ) and F J ( ρ ) from which the nuclear energy densityfunctional is composed. We restrict ourselves here to the isospin-symmetric case of equalproton and neutron number. These analytical expressions give individually the effects due to1 π -exchange, iterated 1 π -exchange, and irreducible 2 π -exchange with intermediate ∆-isobarexcitations, including Pauli-blocking corrections up to three-loop order. Section 4 is devoted toa discussion of our numerical results and finally section 5 ends with a summary and concludingremarks. In the appendix the three-body spin-orbit coupling strength F so ( ρ ) is presented foran alternative description of the 2 π -exchange three-nucleon interaction. The starting point for the construction of an explicit nuclear energy density functional is thedensity-matrix as given by a sum over the occupied energy eigenfunctions Ψ α of the (non-relativistic) many-fermion system. According to Gebremariam, Duguet and Bogner [18] thebilocal density-matrix can be expanded in relative and center-of-mass coordinates, ~a and ~r , asfollows: X α Ψ α ( ~r − ~a/ † α ( ~r + ~a/
2) = 3 ρak f j ( ak f ) − a k f j ( ak f ) (cid:20) τ − ρk f − ~ ∇ ρ (cid:21) + 3 i ak f j ( ak f ) ~σ · ( ~a × ~J ) + . . . , (1)where j ( x ) = (sin x − x cos x ) /x is the spherical Bessel function of index 1. The other quantitiesappearing on the right hand side of eq.(1) are the local nucleon density: ρ ( ~r ) = 2 k f ( ~r )3 π = X α Ψ † α ( ~r )Ψ α ( ~r ) , (2)written here in terms of the local Fermi-momentum k f ( ~r ), the local kinetic energy density: τ ( ~r ) = X α ~ ∇ Ψ † α ( ~r ) · ~ ∇ Ψ α ( ~r ) , (3)and the local spin-orbit density: ~J ( ~r ) = X α Ψ † α ( ~r ) i ~σ × ~ ∇ Ψ α ( ~r ) . (4) We are considering for equal proton and neutron number the spherical phase-space averaged version withoutquadrupolar deformation of the local Fermi momentum distribution. It brings about already most of theimprovements [18]. ~r in eq.(1) and will do so in thefollowing. It is important to note that a pairwise filling of time-reversed orbitals α has beenassumed in eq.(1), so that (various possible) time-reversal-odd fields do not come into play [1].The main difference of this improved density-matrix expansion to the original one of Negele andVautherin [15] lies in the index of the Bessel function multiplying the kinetic energy and spin-orbit densities in eq.(1). The Fourier-transform of the (expanded) density-matrix with respectto both coordinates ~a and ~r defines a ”medium insertion” for the inhomogeneous many-nucleonsystem characterized by the time-reversal-even fields ρ ( ~r ), τ ( ~r ) and ~J ( ~r ):Γ( ~p, ~q ) = Z d r e − i~q · ~r (cid:26) θ ( k f − | ~p | ) + π k f h k f δ ′ ( k f − | ~p | ) − δ ( k f − | ~p | ) i × (cid:18) τ − ρk f − ~ ∇ ρ (cid:19) − π k f δ ( k f − | ~p | ) ~σ · ( ~p × ~J ) (cid:27) . (5)The double line in the left picture of Fig. 1 symbolizes this medium insertion together with theassignment of the out- and in-going nucleon momenta ~p ± ~q/
2. The momentum transfer ~q isprovided by the Fourier components of the inhomogeneous (matter) distributions ρ ( ~r ), τ ( ~r ) and ~J ( ~r ). As a check one verifies that the Fourier transform (1 / π ) R d p e − i~p · ~a of the expression inthe curly brackets in eq.(5) reproduces exactly the right hand side of the (improved) density-matrix expansion written in eq.(1). In comparison to the version of Γ( ~p, ~q ) which followed fromNegele and Vautherin’s density-matrix expansion [16] the weight function of the kinetic energydensity τ ( ~r ) has changed from 35(5 ~p − k f ) θ ( k f − | ~p | ) to 2 k f [ k f δ ′ ( k f − | ~p | ) − δ ( k f − | ~p | )] andthat of the spin-orbit density ~J ( ~r ) has changed from δ ( k f −| ~p | ) − k f δ ′ ( k f −| ~p | ) to − δ ( k f −| ~p | ).For an inhomogeneous many-nucleon system this leads to a different weighting of the momentumdependent nucleon-nucleon interactions in the vicinity of the local Fermi-surface | ~p | = k f ( ~r ),with appropriate consequences for the energy density functional.Going up to second order in spatial gradients (i.e. deviations from homogeneity) the energydensity functional relevant for N = Z even-even nuclei reads: E [ ρ, τ, ~J ] = ρ ¯ E ( ρ ) + (cid:20) τ − ρk f (cid:21)(cid:20) M − k f M + F τ ( ρ ) (cid:21) +( ~ ∇ ρ ) F ∇ ( ρ ) + ~ ∇ ρ · ~J F so ( ρ ) + ~J F J ( ρ ) . (6)Here, ¯ E ( ρ ) is the energy per particle of isospin-symmetric nuclear matter evaluated at thelocal nucleon density ρ ( ~r ). The (small) correction term − k f / M in eq.(6) stems from therelativistically improved kinetic energy and reflects in this way the relativistic increase of mass.The density-dependent functions F τ ( ρ ), F ∇ ( ρ ), F so ( ρ ) and F J ( ρ ) arising from two- and three-nucleon interactions encode new dynamical information specific for the inhomogeneous many-nucleon system. In particular, F ∇ ( ρ ) measures the energy associated with density gradients atthe nuclear surface and F ∇ ( ρ ) gives the strength of the spin-orbit coupling.Returning to eq.(5) one sees that F τ ( ρ ) emerges via a perturbation on top of the density ofstates θ ( k f − | ~p | ). The single-particle potential in nuclear matter can be obtained in the sameway by introducing a (three-dimensional) delta-function as the perturbation. Consequently, thestrength function F τ ( ρ ) can be expressed in terms of the momentum and density-dependentsingle-particle potential U ( p, k f ) as follows: F τ ( ρ ) = 12 k f ∂U ( p, k f ) ∂p (cid:12)(cid:12)(cid:12) p = k f . (7)4 Γ( ~p, ~q ) ~p − ~q/ ~p + ~q/ ~r + ~a/ ~r − ~a/ Figure 1: Left: The double line symbolizes the medium insertion defined by eq.(5). Next areshown: The one-pion exchange Fock diagram and the iterated one-pion exchange Hartree andFock diagrams. Their isospin factors for isospin-symmetric nuclear systems are 6, 12 and − τ − ρk f / − ~ ∇ ρ/
4. Performing a partial integration ofthe energy R d r E [ ρ, τ, ~J ] one is lead to the decomposition: F ∇ ( ρ ) = 14 ∂F τ ( ρ ) ∂ρ + F d ( ρ ) , (8)where F d ( ρ ) comprises all those contributions for which the ( ~ ∇ ρ ) -factor originates directlyfrom the momentum dependence of the interactions in an expansion up to order ~q . Sinceno information about the density-matrix expansion beyond its (fixed) nucleon matter piece θ ( k f − | ~p | ) goes into the derivation of the strength function F d ( ρ ) the pertinent contributionsfrom the 2 π -exchange dynamics are still given in unchanged form by eqs.(12,15,19,24,28) inref.[16] and eqs.(26,30) in ref.[17].As a first test case for the improved density-matrix expansion (summarized in eq.(5)) wehave applied it to the (zero-range) Skyrme force [2, 19] and found that it gives identical results: F τ ( ρ ) (Sk) = ρ
16 (3 t + 5 t ) , F d ( ρ ) (Sk) = 132 (3 t − t ) ,F so ( ρ ) (Sk) = 34 W , F J ( ρ ) (Sk) = 132 ( t − t ) , (9)for the energy density functional as the original density-matrix expansion of Negele and Vau-therin [15]. Obviously, for contact-interactions with their simple quadratic momentum depen-dence the different weighting of interaction strength in the vicinity of the Fermi-surface hasno visible effect. A stronger influence of the actual form of the density-matrix expansion istherefore expected for the contributions arising from the long-range 1 π - and 2 π -exchange. Thepertinent analytical expressions are collected in the next section. In this section we present analytical formulas for the three density-dependent strength functions F τ ( ρ ), F so ( ρ ) and F J ( ρ ) as derived (via the improved density-matrix expansion [18]) from 1 π -exchange, iterated 1 π -exchange, and irreducible 2 π -exchange diagrams with intermediate ∆-isobar excitations, including Pauli-blocking corrections up to three-loop order. We give foreach diagram only the final result omitting all technical details related to extensive algebraicmanipulations and solving elementary integrals. Further explanations about the organizationand performance of our diagrammatic calculation can be found in section 3 of ref.[16].5 .1 One pion exchange Fock diagram with two medium insertions The non-vanishing contributions from the 1 π -exchange Fock diagram shown in Fig. 1, includingthe relativistic 1 /M -corrections, read: F τ ( ρ ) = 3 g A m π (8 πf π ) u (cid:20)(cid:16) u + 12 (cid:17) ln(1 + 4 u ) − u (cid:21) + 3 g A m π (8 πf π M ) (cid:20) u − u − u − u ln(1 + 4 u ) (cid:21) , (10) F J ( ρ ) = 9 g A (32 m π f π ) u h u − u + ln(1 + 4 u ) i , (11)where we have introduced the convenient dimensionless variable u = k f /m π . The two-body contributions from the iterated 1 π -exchange Hartree diagram in Fig. 1 read: F τ ( ρ ) = g A M m π (8 π ) f π (cid:26) u u ln(1 + 4 u ) − u −
16 arctan 2 u (cid:27) , (12) F so ( ρ ) = 3 g A Mπm π (4 f π u ) (cid:26) u arctan 2 u − u −
54 ln(1 + 4 u ) (cid:27) . (13)The expression for F so ( ρ ) in eq.(13) gives (part of) the ”wrong-sign” spin-orbit interaction in-duced by the pion-exchange tensor force in second order. It is completed by the Fock (exchange)contribution and the respective Pauli-blocking corrections (see eqs.(15,18,22,25)). We find the following contributions from the right diagram in Fig. 1 with two medium insertionson non-neighboring nucleon propagators: F τ ( ρ ) = 3 g A M m π (4 f π ) ( πu ) Z u dx x − u x × h (1 + 8 x + 8 x ) arctan x − (1 + 4 x ) arctan 2 x i , (14) F so ( ρ ) = 3 g A M πm π (4 f π u ) (cid:26) u + Z u dx
11 + 2 x × h x (1 + x ) arctan x − (1 + 4 x ) arctan 2 x i(cid:27) , (15) F J ( ρ ) = 9 g A Mπm π (8 f π u ) (cid:26) u Z u dx
11 + 2 x h ( u − x )(1 + 4 x ) arctan 2 x +2( u − x + 2 u x − x + 2 u x − x ) arctan x i − u (cid:27) . (16)6 .4 Iterated one-pion exchange Hartree diagram with three mediuminsertions In our way of organizing the many-body calculation, the Pauli-blocking corrections are repre-sented by diagrams with three medium insertions. The corresponding contributions from theiterated 1 π -exchange Hartree diagram read: F τ ( ρ ) = 3 g A M m π (4 πf π ) ( u − ln(1 + 4 u ) + 2 u u + 2 Z dy y ln 1 + y − y × (cid:20) u y (1 + 4 u y ) (6 u y + y − u ) − u y + ln(1 + 4 u y ) (cid:21) + Z u dx x u Z − dy (cid:20) uxyu − x y + ln u + xyu − xy (cid:21)(cid:20) s ) − s + s s + ( u − x y ) s u (1 + s ) (cid:16) (5 + s ) s ′ + ( s + s )( s ′′ − s ′ ) (cid:17)(cid:21)) , (17)with the auxiliary function s = xy + √ u − x + x y , and its partial derivatives s ′ = u∂s/∂u and s ′′ = u ∂ s/∂u . F so ( ρ ) = 3 g A Mπ m π (4 f π ) Z u dx x u Z − dy ((cid:20) xy ln u + xyu − xy + u (5 x y − u ) u − x y (cid:21) × (cid:20) s + s (1 + s ) − s (cid:21) − us ( u + x y )(1 + s ) ( u − x y )+ 2 s s ′ ( s − xy )(1 + s ) ln u + xyu − xy ) , (18) F J ( ρ ) = 9 g A M u π m π f π Z dy y (1 + 4 u y ) (cid:20) y + (1 − y ) ln 1 + y − y (cid:21) . (19) The evaluation of this diagram is most tedious. It is advisable to split the contributions to thestrength functions F τ ( ρ ), F so ( ρ ) and F J ( ρ ) into ”factorizable” and ”non-factorizable” parts.These two pieces are distinguished by the feature of whether the nucleon propagator in thedenominator can be canceled or not by terms from the product of πN -interaction vertices inthe numerator. We find the following ”factorizable” contributions: F τ ( ρ ) = 3 g A M m π (4 πf π u ) (
18 (1 + 4 u + 2 u ) ln(1 + 4 u ) − u + 8 u u ln (1 + 4 u ) − u − u u Z u dx h u (1 + u + x ) − (cid:16) u + x ) (cid:17)(cid:16) u − x ) (cid:17) L i × (cid:20) (1 − u − x ) L + u − u u + x ) − u u − x ) (cid:21)) , (20)with the auxiliary function: L ( x, u ) = 14 x ln 1 + ( u + x ) u − x ) . (21)7 so ( ρ ) = 3 g A Mπ m π (8 f π u ) (
4[ ln(1 + 4 u ) − u ] arctan 2 u + 28 u + 8 u + 3 u − u + 10 u u ln(1 + 4 u ) + 3 + 20 u + 16 u u ln (1 + 4 u )+4 Z u dx (cid:26) L h x − (1 + u ) + 3 + 2 u − u − (3 + 7 u ) x + 5 x i − ux − (1 + u ) L + 3 u x − (1 + u ) (cid:27)) , (22) F J ( ρ ) = 9 g A Mπ m π (8 f π u ) ( u + 4 u − u − u u ln (1 + 4 u ) − u − u − u u ln(1 + 4 u ) + Z u dx (cid:26) L u (cid:20) x (1 + u ) +2(1 − u )(1 + u ) + (5 + 2 u + 5 u ) x − (6 + 10 u ) x + 11 x (cid:21) + Lu (cid:20) u + 2 u − − x (1 + u ) (cid:21) + 32 x (1 + u ) (cid:27)) . (23)The ”non-factorizable” contributions (stemming from nine-dimensional principal value integralsover the product of three Fermi-spheres of radius k f ) read on the other hand: F τ ( ρ ) = 3 g A M m π (4 πf π ) Z − dy Z − dz yz θ ( y + z − | yz |√ y + z − (cid:26) u z (2 z − u z × h ln(1 + 4 u y ) − u y i θ ( y ) θ ( z ) + Z u dx x s u (1 + s ) × h t − ln(1 + t ) ih ( s + s )(2 s ′ − s ′′ ) − (3 + s ) s ′ i(cid:27) , (24) F so ( ρ ) = 3 g A Mπ m π (4 f π ) Z − dy Z − dz yz θ ( y + z − | yz |√ y + z − (cid:26) y z θ ( y ) θ ( z )1 + 4 u y × h arctan(2 uz ) − uz i + Z u dx x s s ′ t t ′ u (1 + s )(1 + t ) ( txy − sxz − st ) (cid:27) , (25) F J ( ρ ) = 9 g A Mπ m π (4 f π ) Z − dy Z − dz yz θ ( y + z − | yz |√ y + z − (cid:26) y θ ( y ) θ ( z ) u (1 + 4 u y ) × h ln(1 + 4 u z ) − u z i + Z u dx x s s ′ t t ′ (1 − y − z )4 u (1 + s )(1 + t ) (cid:27) , (26)with the auxiliary function t = xz + √ u − x + x z and its partial derivative t ′ = u∂t/∂u .For the numerical evaluation of the dy dz -double integrals in eqs.(24,25,26) it is advantageousto first antisymmetrize the integrands in y and z and then to substitute z = √ y ζ + 1 − y .This way the integration region becomes equal to the unit-square 0 < y, ζ < At next order in the small momentum expansion comes the irreducible 2 π -exchange including(also) intermediate ∆-isobar excitations. We employ a (subtracted) spectral-function represen-8ation of the πN ∆-loops and find the following (two-body) contributions: F τ ( ρ ) = 18 π Z ∞ m π dµ Im( V C + 3 W C + 2 µ V T + 6 µ W T ) × (cid:20) µk f + 8 k f µ − µ k f ( µ + 2 k f ) ln (cid:18) k f µ (cid:19)(cid:21) , (27) F J ( ρ ) = 316 π Z ∞ m π dµ ( Im( V C + 3 W C ) (cid:20) µ k f ( µ + 2 k f ) ln (cid:18) k f µ (cid:19) − µk f − µ (cid:21) +Im( V T + 3 W T ) (cid:20) µk f − µ + µ k f − µ k f ( µ + 4 k f ) ln (cid:18) k f µ (cid:19)(cid:21)) . (28)The imaginary parts Im V C , Im W C , Im V T and Im W T of the isoscalar and isovector central andtensor NN-amplitudes due to 2 π -exchange with single and double ∆-excitation can be found insection 3 of ref.[20]. The additional contributions from the irreducible 2 π -exchange with onlynucleon intermediate states are accounted for by inserting into eqs.(27,28) the imaginary parts:Im W C = q µ − m π πµ (4 f π ) (cid:20) m π (1 + 4 g A − g A ) + µ (23 g A − g A −
1) + 48 g A m π µ − m π (cid:21) , (29)Im V T = − g A q µ − m π πµ (4 f π ) . (30)At leading order the irreducible 2 π -exchange generates no spin-orbit NN-interaction. It emergesfirst as a relativistic 1 /M -correction. In order to see the size of such relativistic effects we haveevaluated the energy density functional with a two-body interaction composed of the (isoscalarand isovector) spin-orbit NN-amplitudes V SO and W SO written in eqs.(22,23) of ref.[21]. Wefind with it the following contribution to the spin-orbit coupling strength: F so ( ρ ) = g A m π πM (4 f π ) (cid:26)(cid:18) g A − (cid:19)(cid:20) u ln(1 + u ) − u (cid:21) + 185 − g A (cid:18) g A − u − u (cid:19) arctan u (cid:27) , (31)which has been subtracted at ρ = 0 in order to eliminate (regularization dependent) short-distance components. As a consequence of that subtraction only the Fock terms are includedin the expressions in eqs.(27,28,31). ∆ -excitation The Pauli-blocking correction to the 2 π -exchange with single ∆-excitation is equivalent to thecontribution of a (genuine) three-nucleon force. In fact, one is dealing here with the same three-nucleon interaction as originally introduced by Fujita and Miyazawa [22]. Moreover, it has beenshown in ref.[13] that the inclusion of this long-range 2 π -exchange three-nucleon interaction isessential in order to reproduce the empirical saturation point of nuclear matter when usingthe low-momentum NN-potential V low − k in Hartree-Fock calculations. It is therefore equallyinteresting to see its effects on the nuclear energy functional E [ ρ, τ, ~J ].The pertinent Hartree and Fock three-body diagrams related to 2 π -exchange with virtual∆-excitation are shown in Fig. 2. The central diagram with parallel pion-lines vanishes for9igure 2: Hartree and Fock three-body diagrams related to 2 π -exchange with virtual ∆-isobarexcitation. Their isospin factors for isospin-symmetric nuclear systems are 8, 0, and 8, respec-tively.isospin-symmetric nuclear systems. Returning to the medium insertion written in eq.(5) wefind from the left three-body Hartree diagram in Fig. 2 the following contributions: F τ ( ρ ) = g A m π ∆(2 πf π ) (cid:26)(cid:16) u + 34 (cid:17) ln(1 + 4 u ) − u (3 + 10 u )1 + 4 u (cid:27) , (32) F so ( ρ ) = 3 g A m π π ∆(4 f π ) (cid:26) u + 8 u − u u ln(1 + 4 u ) (cid:27) , (33) F J ( ρ ) = 3 g A m π π ∆(4 f π ) (cid:26) u ln(1 + 4 u ) + 4 u − u + 8 u u (cid:27) , (34)with ∆ = 293 MeV the delta-nucleon mass splitting. We have used the value 3 / √ πN ∆- and πN N -coupling constants. Note that the expression for F so ( ρ )in eq.(33) gives the (dominant part of the) three-body spin-orbit coupling strength suggestedoriginally by Fujita and Miyazawa [22]. The three-body effects on the energy density functionalare completed by the contributions from the right Fock diagram in Fig. 2 which read: F τ ( ρ ) = g A m π ∆(4 πf π ) ( u h u − (1 + 2 u ) ln(1 + 4 u ) i arctan 2 u − u
3+ 31 u −
314 + 58 u + 3 + 22 u + 176 u + 288 u u ln (1 + 4 u )+ 316 u + 196 u (248 u − u − u − u −
9) ln(1 + 4 u )+ 1 u Z u dx (cid:26) G S (cid:20) u ( u + x )1 + ( u + x ) + 2 u ( x − u )1 + ( u − x ) − xL (cid:21) + G T (cid:20) u x (3 u − − ux u ( u + x )1 + ( u + x ) + u ( x − u )1 + ( u − x ) + L x (3 x + 6 u x − x − u − u + 3) (cid:21)(cid:27)) , (35)with the auxiliary functions: G S ( x, u ) = 4 ux u −
3) + 4 x h arctan( u + x ) + arctan( u − x ) i +( x − u −
1) ln 1 + ( u + x ) u − x ) , (36)10 T ( x, u ) = ux u + 3 x ) − u x (1 + u ) + 18 (cid:20) (1 + u ) x − x + (1 − u )(1 + u − x ) (cid:21) ln 1 + ( u + x ) u − x ) . (37) F so ( ρ ) = g A m π π ∆(8 f π ) ( u − u − u − u − u + (cid:18) u + 394 u + 13 u + 6 u − u (cid:19) ln(1 + 4 u ) − u (64 u + 80 u + 36 u + 5) ln (1 + 4 u ) ) , (38) F J ( ρ ) = g A m π π ∆(8 f π u ) ( h − u − u ln(1 + 4 u ) i arctan 2 u − u + 3 u +272 u − u + 94 u + (cid:18) u − u − u (cid:19) ln(1 + 4 u ) + 364 u × (3 + 16 u + 144 u ) ln (1 + 4 u ) + 34 (58 u + 31 u −
63) arctan 2 u + 663 u
16 + 495 u − u u (29 − u + 52 u ) ln(1 + 4 u )+ Z u dx (cid:26) L u (cid:20) x (1 + u ) (3 u − − x (1 + u ) + (1 + u ) × (50 u − − u ) + 4 x (35 u + 5 u − u − − x +13 x (1 + 2 u − u ) + 2 x (65 u − (cid:21) + 3 L u (cid:20) x (1 + u ) + 1 x (1 + u ) (3 − u ) + 6(25 u + 5 u − u + 19) (cid:21) − x (1 + u ) + 32 x (1 + u ) (3 + 11 u ) (cid:27)) . (39)In the appendix we present the three-body spin-orbit coupling strength F so ( ρ ) for an alternativedescription of the 2 π -exchange three-nucleon interaction (using ππN N -contact vertices insteadof propagating ∆-isobars). A good check of all formulas collected in this section is providedby their Taylor series expansion in k f . Despite the superficial opposite appearance the leadingterm in the k f -expansion is always a non-negative power of k f (which is higher for three-bodycontributions than for two-body contributions). In this section we present and discuss our numerical results obtained by summing the series ofcontributions given in section 3. The physical input parameters are: g A = 1 . f π = 92 . m π = 135 MeV (neutral pionmass) and M = 939 MeV (nucleon mass). We recall that with these physical parameters anda few adjustable short-distance couplings the nuclear matter equation of state ¯ E ( ρ ) and manyother nuclear matter properties [17] can be well described by the chiral pion-nucleon dynamicstreated to three-loop order. 11 .02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 ρ [fm -3 ] M * ( ρ ) / M Effective nucleon mass
Figure 3: The effective nucleon mass M ∗ ( ρ ) divided by the free nucleon mass M as a functionof the nuclear density ρ .Returning to the energy density functional E [ ρ, τ, ~J ] in eq.(6) one observes that the ex-pression multiplying the kinetic energy density τ ( ~r ) has the meaning of a reciprocal density-dependent effective nucleon mass: M ∗ ( ρ ) = M (cid:20) − k f M + 2 M F τ ( ρ ) (cid:21) − . (40)It is identical to the so-called ”Landau” mass introduced in Fermi-liquid theory, since it derivesin the same way from the slope of the single-particle potential U ( p, k f ) at the Fermi-surface p = k f . This consistency of effective nucleon masses follows from the improved density-matrixexpansion of Gebremariam, Duguet and Bogner [18], but it did not hold for the original density-matrix expansion of Negele and Vautherin [15, 16].Fig. 3 shows the ratio of effective to free nucleon mass M ∗ ( ρ ) /M as a function of the nucleardensity ρ = 2 k f / π . One observes a reduced effective nucleon mass which reaches the value M ∗ ( ρ ) = 0 . M at nuclear matter saturation density ρ = 0 .
16 fm − . This is compatiblewith the range 0 . < M ∗ ( ρ ) /M < . ρ ) has to be reached until the subleading πN ∆-dynamics canrevert the tendency of the iterated 1 π -exchange to increase the effective nucleon mass. Thesame feature has also been observed for the p -wave Landau parameter f ( k f ) in ref.[23] (seeFig. 3 therein), a quantity which is intimately related to the effective nucleon mass M ∗ ( ρ ).Next, we show in Fig. 4 the strength function F ∇ ( ρ ) belonging to the ( ~ ∇ ρ ) surface-term.The dashed line corresponds to the truncation to 1 π - and iterated 1 π -exchange, whereas thefull line includes in addition the 2 π -exchange and the associated three-body contributions.Taking the band spanned by phenomenological Skyrme forces [3, 4, 5, 6] as a benchmark onemay conclude that the subleading 2 π -exchange dynamics leads to some improvement. The12 .02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 ρ [fm -3 ] F g r a d ( ρ ) [ M e V f m ] (density-gradient) term Skyrme forces
Figure 4: The strength function F ∇ ( ρ ) of the surface-term ( ~ ∇ ρ ) in the nuclear energy densityfunctional versus the nuclear density ρ . The dashed line corresponds to the truncation to 1 π -and iterated 1 π -exchange. The full line includes also 2 π -exchange and associated three-bodycontributions.improved density-matrix expansion [18] has furthermore flattened and shifted downward thecurve for F ∇ ( ρ ) in comparison to our previous calculation (see Fig. 8 in ref.[17]) based on theNegele-Vautherin density-matrix expansion. We also note that in the relevant density region ρ / < ρ < ρ the main contribution to the strength function F ∇ ( ρ ) comes from the component F d ( ρ ) (see eq.(8)) which is insensitive to the density-matrix expansion beyond its fixed nuclearmatter part θ ( k f − | ~p | ).Of particular interest is the strength F so ( ρ ) of the spin-orbit coupling provided by the explicitpion-exchange dynamics. The dashed curve in Fig. 5 shows the ”wrong-sign” spin-orbit couplingstrength arising from iterated 1 π -exchange (i.e. the pion-exchange tensor force in second order).Its value at half nuclear matter density ρ / .
08 fm − decomposes as (( − . .
7) +(76 . − . = − . into Hartree and Fock pieces supplemented by therespective Pauli-blocking corrections. This net negative result amounts to about −
40% ofthe empirical spin-orbit coupling strength F (emp) so ≃
90 MeVfm . In comparison to our previouscalculation [16] based on the Negele-Vautherin density-matrix expansion which gave at ρ / − . the ”wrong-sign” spin-orbit coupling strength has substantially decreasedin magnitude. The full line in Fig. 5 shows the spin-orbit coupling strength after including thesubleading 2 π -exchange, in particular the three-body contributions eqs.(33,38). One finds nowa pronounced cancellation in the density region around ρ / .
08 fm − , where the spin-orbitinteraction in nuclei gains actually most of its weight. Such an almost complete cancellationleaves then all room for the short-distance NN-dynamics (not treated explicitly in this work)to account for the strong spin-orbit coupling in nuclei. In fact, it has been shown in ref.[24]that the empirical value F (emp) so ≃
90 MeVfm of the spin-orbit coupling strength in nuclei isin perfect agreement with the one extracted from realistic nucleon-nucleon potentials. The13 .02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 ρ [fm -3 ] -60-40-2002040 F s o ( ρ ) [ M e V f m ] spin-orbit coupling Figure 5: The strength function F so ( ρ ) of the spin-orbit coupling term ~ ∇ ρ · ~J in the nuclearenergy density functional versus the nuclear density ρ . Dashed line: 1 π - and iterated 1 π -exchange only. Full line: 2 π -exchange and three-body contributions added.intimate connection between the strong Lorentz scalar and vector mean-fields and the (short-range) spin-orbit part of the NN-potential has been elucidated in ref.[25] via (relativistic) Dirac-Brueckner calculations of the in-medium nucleon self-energy.Moreover, we note that the spin-orbit coupling strength generated by the irreducible 2 π -exchange as a relativistic 1 /M -correction (see eq.(31) in section 3.6) contributes little to the can-cellation between ”wrong-sign” and ”correct-sign” parts shown in Fig. 5. At ρ / .
08 fm − this piece amounts to just about − . . Furthermore, we have convinced ourselves thatthe spin-orbit NN-amplitudes from 2 π -exchange with ∆-excitation ( V SO and W SO collectedin the appendix of ref.[20]) lead to an even smaller effect. These NN-amplitudes make up atwo-body contribution to F so ( ρ ) that scales again with 1 /M .Finally, we show in Fig. 6 the strength function F J ( ρ ) belonging the squared spin-orbitdensity ~J in the nuclear energy density functional as a function of the nuclear density ρ . Oneobserves that the inclusion of the subleading 2 π -exchange strongly reduces the values of F J ( ρ ).In comparison to the (narrow) band spanned by phenomenological Skyrme forces [3, 4, 5, 6] ourprediction for the strength function F J ( ρ ) is much larger in the whole density region 0 < ρ < ρ .In addition, the density dependence of F J ( ρ ) comes out markedly different, due to the long-rangecharacter of the pion-exchange interactions. For orientation, we reproduce by the dashed-dottedline in Fig. 6 the leading contribution from the 1 π -exchange Fock diagram (see eq.(11)). Wealso note that in comparison to the calculation based on the Negele-Vautherin density-matrixexpansion the magnitude of the strength function F J ( ρ ) has substantially increased (see Fig. 5in ref.[16]).Besides representing the non-local Fock contributions from tensor forces etc. in the energydensity functional the ~J -term leads to another interesting side effect. Namely, it gives riseto an extra spin-orbit single-particle mean-field 2 F J ( ρ ) ~J in addition to the ”normal” one,14 .02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 ρ [fm -3 ] F J ( ρ ) [ M e V f m ] J - term Skyrme forces 1 π -exchange Figure 6: The strength function F J ( ρ ) accompanying the squared spin-orbit density ~J in thenuclear energy density functional versus the nuclear density ρ . Dashed line: 1 π - and iterated1 π -exchange only. Full line: 2 π -exchange and three-body contributions added. F so ( ρ ) ~ ∇ ρ . It would be interesting to investigate the role of this additional (nucleus-dependent)spin-orbit mean-field together with the large values and the strong density dependence of F J ( ρ )as predicted by in-medium chiral perturbation theory. In this work we have used the recently improved density-matrix expansion of Gebremariam,Duguet and Bogner [18] to calculate the nuclear energy density functional E [ ρ, τ, ~J ] relevant for N = Z even-even nuclei in the framework of in-medium chiral perturbation theory. Our calcu-lation treats systematically the effects from 1 π -exchange, iterated 1 π -exchange, and irreducible2 π -exchange with intermediate ∆-isobar excitations, including Pauli-blocking corrections up tothree-loop order.We find that the effective nucleon mass M ∗ ( ρ ) entering the energy density functional be-comes identical to the one of Fermi-liquid theory when employing the improved density-matrixexpansion. The strength F ∇ ( ρ ) of the ( ~ ∇ ρ ) surface-term as provided by the pion-exchangedynamics is in good agreement with that of phenomenological Skyrme forces in the densityregion ρ / < ρ < ρ .The spin-orbit coupling strength F so ( ρ ) receives contributions from iterated 1 π -exchange (ofthe “wrong sign”) and from three-nucleon interactions mediated by 2 π -exchange with virtual∆-excitation (of the “correct sign”). In the region around ρ / ≃ .
08 fm − where the spin-orbitinteraction in nuclei gains most of its weight these two components tend to cancel, thus leavingall room for the short-range spin-orbit interaction. The empirical value F (emp) so ≃
90 MeVfm ofthe spin-orbit coupling strength in nuclei agrees perfectly with the one extracted from the short-15 ρ [fm -3 ] F s o ( ρ ) [ M e V f m ] three-body spin-orbit Figure 7: Three-body spin-orbit coupling strength F so ( ρ ) as a function of the nuclear density ρ .range spin-orbit component of realistic NN-potentials [24]. This part of the NN-interactiondrives at the same time the strong Lorentz scalar and vector mean-fields on which the wholesuccess of the relativistic Dirac phenomenology rests.The strength function F J ( ρ ) multiplying the squared spin-orbit density comes out muchlarger than from phenomenological Skyrme forces and it has a pronounced density dependencedue to the long-range character of the pion-exchange interaction. The interplay between the twocomponents of the total nuclear spin-orbit mean-field 2 F J ( ρ ) ~J + F so ( ρ ) ~ ∇ ρ should be furtherexplored together with the large values and strong density dependence of F J ( ρ ) as predictedby in-medium chiral perturbation theory.In comparison to refs.[16, 17] where the density-matrix expansion of Negele and Vautherinhas been employed, we find an improved description of the nuclear energy density functional E [ ρ, τ, ~J ] on the basis of the improved density-matrix expansion [18]. In view of the factthat short-range contributions do not change (as exemplified here for the Skyrme force), acancellation of the net two-pion exchange spin-orbit coupling strength around half nuclearmatter density ρ / .
08 fm − is more satisfactory than having this cancellation around ρ as discussed in ref.[26]. In any case, the effective field theory formulation of nuclear forcesprovides short-range contributions to all four strength functions F τ ( ρ ), F ∇ ( ρ ), F so ( ρ ) and F J ( ρ )and these can be fine-tuned in nuclear structure calculations. Appendix: Three-body spin-orbit coupling strength
In this appendix we present and discuss the result for the three-body spin-orbit couplingstrength F so ( ρ ) one obtains from an alternative description of the 2 π -exchange three-nucleoninteraction. Instead of the sequential πN → ∆ → πN transition with intermediate ∆-isobar16xcitation one can employ the second order chiral ππN N -contact vertex [27]: if π n δ ab ( c ~q a · ~q b − c m π ) + c ǫ abc τ c ~σ · ( ~q a × ~q b ) o , (41)to built up the 2 π -exchange three-nucleon interaction. Here, ~q a,b denote out-going pion momentaand we have already dropped the c term proportional to the product of two pion energies. Inthe present application these (off-shell) pion energies are equal to differences of nucleon kineticenergies, thus producing a relativistic 1 /M -correction. The pertinent in-medium diagrams arethose shown in Fig. 2 with the ∆-propagator shrunk to a point. Returning to the mediuminsertion written in eq.(5) we find from the corresponding three-body Hartree diagram (withtwo closed nucleon rings) the following contribution to the spin-orbit coupling strength: F so ( ρ ) = 3 g A m π (8 π ) f π (cid:26) u (4 c − c ) − c u + (cid:20) u ( c − c ) + 3 c − c u (cid:21) ln(1 + 4 u ) (cid:27) , (42)where u = k f /m π . It is completed by the contribution of the three-body Fock diagram (with asingle closed nucleon ring) which reads: F so ( ρ ) = g A m π π (4 f π u ) ( c (cid:20) u − u + 32 u − u u ln(1 + 4 u )+ 3 + 16 u + 16 u u ln (1 + 4 u ) (cid:21) + ( c + c ) (cid:20) u − u u
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