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Nuclear Physics A 750 (2005) 259–293

Chiral approach to nuclear matter: role of two-pioexchange with virtual delta-isobar excitation✩

S. Fritsch, N. Kaiser∗, W. Weise

Physik-Department, Technische Universität München, D-85747 Garching, Germany

Received 14 June 2004; received in revised form 18 November 2004; accepted 13 December 200

Available online 21 December 2004

Abstract

We extend a recent three-loop calculation of nuclear matter by including the effects frompion exchange with single and double virtual∆(1232)-isobar excitation. Regularization dependeshort-range contributions from pion-loops are encoded in a few NN-contact coupling constanempirical saturation point of isospin-symmetric nuclear matter,E0 = −16 MeV, ρ0 = 0.16 fm−3,can be well reproduced by adjusting the strength of a two-body term linear in density (andan emerging three-body term quadratic in density). The nuclear matter compressibility comesK = 304 MeV. The real single-particle potentialU(p, kf 0) is substantially improved by the inclusion of the chiralπN∆-dynamics: it grows now monotonically with the nucleon momentump. Theeffective nucleon mass at the Fermi surface takes on a realistic value ofM∗(kf 0) = 0.88M. As aconsequence of these features, the critical temperature of the liquid-gas phase transition getsto the valueTc � 15 MeV. In this work we continue the complex-valued single-particle poteU(p, kf ) + iW(p, kf ) into the region above the Fermi surfacep > kf . The effects of 2π -exchangewith virtual ∆-excitation on the nuclear energy density functional are also investigated. Thetive nucleon mass associated with the kinetic energy density isM∗(ρ0) = 0.64M. Furthermore, wefind that the isospin properties of nuclear matter get significantly improved by including theπN∆-dynamics. Instead of bending downward aboveρ0 as in previous calculations, the energy pparticle of pure neutron matterEn(kn) and the asymmetry energyA(kf ) now grow monotonically

with density. In the density regimeρ = 2ρn < 0.2 fm−3 relevant for conventional nuclear physiour results agree well with sophisticated many-body calculations and (semi)-empirical values

✩ Work supported in part by BMBF and GSI.* Corresponding author.

E-mail address: [email protected] (N. Kaiser).

0375-9474/$ – see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysa.2004.12.042

260 S. Fritsch et al. / Nuclear Physics A 750 (2005) 259–293

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2004 Elsevier B.V. All rights reserved.

PACS: 12.38.Bx; 21.65.+f; 24.10.Cn; 31.15.Ew

Keywords: Nuclear matter properties; Two-pion exchange with virtual∆(1232)-isobar excitation

1. Introduction and preparation

In recent years a novel approach to the nuclear matter problem has emerged.element is a separation of long- and short-distance dynamics and an ordering sin powers of small momenta. At nuclear matter saturation densityρ0 � 0.16 fm−3 theFermi momentumkf 0 and the pion massmπ are comparable scales (kf 0 � 2mπ ), andtherefore pions must be included as explicit degrees of freedom in the descriptthe nuclear many-body dynamics. The contributions to the energy per particleE(kf ) ofisospin-symmetric nuclear matter as they originate from chiral pion–nucleon dynhave been computed up to three-loop order in Refs. [1,2]. Both calculations are ableproduce correctly the empirical saturation point of nuclear matter by adjusting oneparameter (either a couplingg0 + g1 � 3.23 [1] or a cut-offΛ � 0.65 GeV [2]) relatedto unresolved short-distance dynamics. The novel mechanism for saturation in theproaches is a repulsive contribution to the energy per particle generated by Pauli-blin second order (iterated) one-pion exchange. As outlined in Section 2.5 of Ref. [2mechanism becomes particularly transparent by taking the chiral limitmπ = 0. In thatcase the interaction contributions to the energy per particle are completely summarian attractivek3

f -term and a repulsivek4f -term where the parameter-free prediction for

coefficient of the latter is very close to the one extracted from a realistic nuclear mequation of state.

The single-particle properties, represented by a complex-valued momentum ansity dependent nucleon selfenergyU(p, kf ) + iW(p, kf ), have been computed within oapproach [2] in Ref. [3]. The resulting potential depthU(0, kf 0) = −53.2 MeV is in goodagreement with that of the empirical nuclear shell [4] or optical model [5]. Howevermomentum dependence of the real single-particle potentialU(p, kf 0) with its up- anddownward bending (see Fig. 3 in Ref. [3]) turns out to be too strong. As a consequthe nominal value of the effective nucleon mass at the Fermi surfacep = kf 0 would bemuch too high:M∗(kf 0) � 3M . On the other hand, the single-particle properties arothe Fermi surface are decisive for the spectrum of thermal excitations and thereforcrucially influence the low temperature behavior of isospin-symmetric nuclear matterather high critical temperatureTc � 25.5 MeV for the liquid-gas phase transition obtainin Ref. [6] is a visible manifestation of this intimate relationship.

While there is obviously room and need for improvement in our approach, onealso note at the same time that the single-particle properties in the scheme of Lut[1] (where explicit short-range terms are iterated to second order with itself and withpion exchange) come out completely unrealistic [7]. The potential depth ofU(0, kf 0) =−20 MeV is by far too weakly attractive. Most seriously, the total single-particle en
Tkin(p) + U(p, kf 0) does not rise monotonically with the nucleon momentump, thus

S. Fritsch et al. / Nuclear Physics A 750 (2005) 259–293 261

f. [7]).rature

]. The

8 inneu-

und inprop-meterisplay

act in-studiedor-

multa-ns fors (andn that

ucleonar ings. 1, 2ed thatly atle sat-r doessities

ing to

angetensor

esent-ratinge com-

tentialin the

ter andg to

implying a negative effective nucleon mass at the Fermi surface (see Fig. 3 in ReThis ruins the behavior of nuclear matter at finite temperatures since a critical tempeof Tc > 40 MeV exceeds acceptable values by more than a factor of two.

The isospin properties of nuclear matter have also been investigated in Ref. [2prediction for the asymmetry energy at saturation densityA(kf 0) = 33.8 MeV is in goodagreement with its empirical value. However, one finds a downward bending ofA(kf ) atdensitiesρ > 0.2 fm−3. Such a behavior of the asymmetry energyA(kf ) is presumablynot realistic. The energy per particle of pure neutron matterEn(kn) as a function of theneutron densityρn = k3

n/3π2 shows a similar downward bending behavior (see Fig.Ref. [2]) and at lower neutron densities, there is only rough agreement with realistictron matter calculations. The mere fact that neutron matter came out to be unboRef. [2] with no further adjusted parameter was however non-trivial. The isospinerties of nuclear matter in the scheme of Lutz et al. [1] with a second free paraadjusted, are qualitatively the same. The dashed curves in Figs. 6, 7 of Ref. [7] da downward bending ofEn(kn) andA(kf ) at even lower densities,ρ > 0.15 fm−3. Anextended version of that approach with pion-exchange plus two zero-range NN-contteractions iterated to second order and in total four adjustable parameters has beenrecently in Ref. [8]. The finding of that work is that within such a complete fourthder calculation (thus exhausting all possible terms up-to-and-includingO(k4

f )) there is nooptimal set of the four short-range parameters with which one could reproduce sineously and accurately all semi-empirical properties of nuclear matter. The conditioa good neutron matter equation of state and equally good single-particle propertieconsequently a realistic finite temperature behavior) are in fact mutually exclusive iapproach.

Calculations of nuclear matter based on the universal low-momentum nucleon–npotentialVlow-k have recently been performed in Ref. [9]. The results obtained so fHartree–Fock or Brueckner–Hartree–Fock approximation are unsatisfactory (see Fiin Ref. [9]), since no saturation occurs in the equation of state. It has been concludfor the potentialVlow-k the Brueckner–Hartree–Fock approximation is applicable onvery low densities. These findings together with the identification of a comprehensiburation mechanism in the chiral approaches hint at the fact that the Brueckner laddenot generate all relevant medium modifications which set in already at very low den(typically at about one-tenth the equilibrium density of nuclear matter, correspondFermi momenta aroundkf � mπ ).

Up to this point the situation can be summarized as follows. Chiral two-pion exchrestricted to nucleon intermediate states (basically the second-order spin–spin andforce plus Pauli blocking effects), together with a single (finetuned) contact-term repring short-distance dynamics, is already surprisingly successful in binding and satunuclear matter and reproducing key properties such as the asymmetry energy and thpression modulus. However, the detailed behavior of the nucleon single-particle poand the density of states at the Fermi surface are not well described at this ordersmall-momentum expansion.

Let us also comment on the relationship between our approach to nuclear matthe effective field theory treatment of (free) NN-scattering [10–12]. We are includin
the respective order the same long-range components from (one- and) two-pion exchange.


ucingn, weultingose ofe for

nedonly

l

al one-itious

2t2

e

the

in the

-ting ofredl scheme

arers ofansion-mallir-

, sincence of

The short-range NN-contact terms are however treated differently. Instead of introdthese terms in a potential which is then iterated in a Lippmann–Schwinger equatioadjust their strengths directly to a few empirical nuclear matter properties. The resvalues of the coupling constants are therefore not immediately comparable with ththe NN-potential. We note also that the important role of chiral two-pion exchangperipheral NN-scattering is well established [11–17].

From the point of view of the driving pion–nucleon dynamics the previously mentiochiral calculations of nuclear matter [1,2,8] are indeed still incomplete. They include(S- and) P-wave Born terms but leave out the excitation of the spin–isospin-3/2 ∆(1232)-resonance, which is the prominent feature of low-energyπN -scattering. It is also welknown that the two-pion exchange between nucleons with excitation of virtual∆-isobarsgenerates most of the needed isoscalar central NN-attraction. In phenomenologicboson exchange models this part of the NN-interaction is often simulated by a fict“σ ”-meson exchange. A parameter-free calculation of the isoscalar central potentialVC(r)

generated by 2π -exchange with single and double∆-excitation in Ref. [18] (see Fig.therein) agrees almost perfectly with the phenomenological “σ ”-exchange potential adistancesr > 2 fm, but not at shorter distances. The more detailed behavior of theπ -exchange isoscalar central potential with single virtual∆-excitation is reminiscent of thvan der Waals potential. It has the form [18]:

V(N∆)C (r) = − 3g4

A

64π2f 4π∆

e−2x

r6

(6+ 12x + 10x2 + 4x3 + x4),

with x = mπr and the prefactor includes the spin–isospin (axial) polarizability ofnucleon [19],g2

A/f 2π∆ = 5.2 fm3, from the virtualN → ∆(1232) → N transition. The

familiar r−6-dependence of the non-relativistic van der Waals interaction emergeschiral limit, mπ = 0.

A consideration of mass scales also suggests to include the∆(1232)-isobar as an explicit degree of freedom in nuclear matter calculations. The delta-nucleon mass split∆ = 293 MeV is comparable to the Fermi momentumkf 0 � 262 MeV at nuclear mattesaturation density. Propagation effects of virtual∆(1232)-isobars can therefore be resolvat the densities of interest. Based on these scale arguments we adopt a calculationain which we count the Fermi momentumkf , the pion massmπ and the∆N -mass split-ting ∆ simultaneously as “small scales”.1 The non-relativistic treatment of the nuclematter many-body problem naturally goes conform with such an expansion in powsmall momenta. Relativistic corrections are relegated to higher orders in this expscheme. The leading contributions from 2π -exchange with virtual∆-excitation to the energy per particle (or the single-particle potential) are generically of fifth power in the smomenta (kf ,mπ ,∆). With respect to the counting in small momenta the effects fromreducible 2π -exchange evaluated in Refs. [2,3,6] belong to the same order. HowevertheπN∆-coupling constant is about twice as large as theπNN -coupling constant one caexpect that the∆-driven 2π -exchange effects are the dominant ones. The importan

1 The “large” scales in this context are, e.g., the nucleon massM , the vector meson massmρ,ω and the chiral
symmetry breaking scale 4πfπ .


ns of

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atterwrite

∆(1232)-degrees of freedom has also been pointed out in the “ab initio” calculatiothe Illinois group [20,21].

The purpose of the present paper is to present a calculation of (isospin-symmetisospin-asymmetric) nuclear matter which includes all effects from 2π -exchange with vir-tual ∆-excitation up to three-loop order in the energy density. The contributions tenergy per particle (or the single-particle potential) can be classified as two-body termthree-body terms. Two-body terms can be directly expressed through the NN-scattematrix (i.e., the NN-potential in momentum space). Three-body terms on the othercan be interpreted as Pauli-blocking effects on the two-body terms imposed by theFermi-sea of nucleons. The notion of “three-body term” is taken here in a more gecontext, namely in the sense that three nucleons in the Fermi sea participate inactions. The NN T-matrix generated by the in general ultra-violet divergent piondiagrams requires regularization (and renormalization). We adopt here a suitably subdispersion-relation representation of the T-matrix where this procedure is accounteda few subtraction constants. The latter constants are understood to encode unresolvedistance NN-dynamics. The associatedk3

f - andk5f -terms in the energy per particle are th

adjusted to some empirical property of nuclear matter (e.g., the maximal binding ene16 MeV).

Our paper is organized as follows: in Section 2 we start with the equation of stisospin-symmetric nuclear matter. We evaluate analytically the three-loop in-mediuagrams generated by the chiralπN∆-dynamics and perform the necessary adjustmenshort-range parameters. Section 3 deals with the real single-particle potentialU(p, kf )

whose improved momentum dependence turns out to be a true prediction. In Sewe reconsider the imaginary single-particle potentialW(p,kf ) of Ref. [3], but now ex-tended into the region above the Fermi surfacep > kf . Section 5 is devoted to the effectivnucleon massM∗(ρ) and the strength function of the( �∇ρ)2-term in the nuclear energdensity functional. In Section 6 we extend our calculation of isospin-symmetric numatter to finite temperaturesT . The main interest lies there on the critical temperaTc of the first-order liquid-gas phase transition for which we find an improved valuTc � 15 MeV. Sections 7–9 deal with the equation of state of pure neutron matterEn(kn),the asymmetry energyA(kf ) and the isovector single-particle potentialUI (p, kf ). Thesethree quantities reveal the isospin properties of the underlyingπN∆-dynamics. Explicitinclusion of the∆(1232)-degrees of freedom leads to a substantial improvement: therious downward bending ofEn(kn) andA(kf ) observed in previous chiral calculatiois now eliminated. Finally, Section 10 ends with a summary and some concludinmarks.

2. Equation of state of isospin-symmetric nuclear matter

We start the discussion with the equation of state of isospin-symmetric nuclear mfor which one has a fairly good knowledge of the empirical saturation point. We firstdown the contributions to the energy per particleE(kf ) as they arise from 2π -exchange
with single and double virtual∆-isobar excitation.


r

ibutingtically

linesree orto the

en-atingriable-encesractedto the

ntribu-rett forain thatd in the

in-act

short--rangegoodbasednt of

ssed

Fig. 1. One-loop two-pion exchange diagrams with single and double∆(1232)-isobar excitation. Diagrams fowhich the role of both nucleons is interchanged are not shown.

Fig. 1 shows the relevant one-loop triangle, box and crossed box diagrams contrto the NN T-matrix. The finite parts of these diagrams have been evaluated analyin Section 3 of Ref. [18] employing the usual non-relativistic∆ ↔ πN transition ver-tices and∆-propagator (see Eq. (4) in Ref. [18]). By closing the two open nucleonto either two rings or one ring one gets (in diagrammatic representation) the HartFock contribution to the energy density of nuclear matter. The Hartree contributionenergy per particle evidently goes linear with the nucleon densityρ = 2k3

f /3π2, namely

E(kf )(2H) = −VC(0)ρ/2 with VC(0) the isoscalar central NN-amplitude at zero momtum transfer [18]. The Fock contribution on the other hand is obtained by integrthe spin- and isospin-contracted T-matrix (depending on the momentum transfer va| �p1− �p2|) over the product of two Fermi spheres| �p1,2| < kf of radiuskf . We separate regularization dependent short-range parts in the T-matrix (originating from the divergof the loop diagrams) from the unique long-range terms with the help of a twice-subtdispersion relation. The resulting subtraction constants give rise to a contributionenergy per particle of the form:

E(kf )(ct) = B3k3f

M2+ B5

k5f

M4, (1)

whereB3 andB5 are chosen for convenience as dimensionless.M = 939 MeV stands forthe (average) nucleon mass. Note that Eq. (1) is completely equivalent to the cotion of a momentum independent andp2-dependent NN-contact interaction. We interpthe parametersB3,5 to subsume all unresolved short-distance NN-dynamics relevanisospin-symmetric nuclear matter at low and moderate densities. We emphasize agin the present approach, the high-momentum (short-distance) dynamics is encodesubtraction constantsB3,5 in a way analogous to a dispersion relation treatment of themedium T-matrix (or G-matrix), andnot at the level of a potential. The equivalent continteractions related to these subtraction constants represent the full content of thedistance T-matrix. These contact terms should therefore not be iterated with longpion-exchange pieces. At this point our scheme differs from that of Ref. [1]. Theagreement of the nucleon selfenergies with results of Dirac–Brueckner calculationson realistic NN-potentials (see Fig. 6 in Ref. [22]) gives further support to our treatmethe short-range (in-medium) dynamics by density independent contact terms.

The long-range parts of the 2π -exchange (two-body) Fock diagrams can be expre
as:


c nuclearr is 1 for

ndthese

he

gral is-

cans read

odte

diaters

ors are

Fig. 2. Hartree and Fock three-body diagrams related to 2π -exchange with single virtual∆-isobar excitation.They represent interactions between three nucleons in the Fermi sea. In the case of isospin-symmetrimatter the isospin factors of these diagrams are 8, 0, and 8, in the order shown. The combinatoric factoeach diagram.

E(kf )(2F) = 1

8π3

∞∫2mπ

dµ Im(VC + 3WC + 2µ2VT + 6µ2WT

){3µkf − 4k3

f

3µ+ 8k5

f

5µ3

− µ3

2kf

− 4µ2 arctan2kf

µ+ µ3

8k3f

(12k2

f + µ2) ln

(1+ 4k2

f

µ2

)}, (2)

where ImVC , ImWC , ImVT and ImWT are the spectral functions of the isoscalar aisovector central and tensor NN-amplitudes, respectively. Explicit expressions ofimaginary parts for the contributions of the triangle diagram with single∆-excitation andthe box diagrams with single and double∆-excitation can be easily constructed from tanalytical formulas given in Section 3 of Ref. [18]. Theµ- andkf -dependent weightingfunction in Eq. (2) takes care that at low and moderate densities this spectral intedominated by low invariantππ -masses 2mπ < µ < 1 GeV. The contributions to the energy per particle from irreducible 2π -exchange (with only nucleon intermediate states)also be cast into the form Eq. (2). The corresponding non-vanishing spectral function[13]:

ImWC(iµ) =√

µ2 − 4m2π

3πµ(4fπ)4

[4m2

π

(1+ 4g2

A − 5g4A

)

+ µ2(23g4A − 10g2

A − 1) + 48g4

Am4π

µ2 − 4m2π

], (3)

ImVT (iµ) = −6g4A

√µ2 − 4m2

π

πµ(4fπ)4. (4)

The dispersion integrals∫ ∞

2mπdµ Im(· · ·) in this and all following sections are understo

to include the contributions from irreducible 2π -exchange (with only nucleon intermediastates).

Next, we come to the three-body terms which arise from Pauli blocking of intermenucleon states (i.e., from the−2πθ(kf −| �p|) terms of the in-medium nucleon propagato[2]). The corresponding closed Hartree and Fock diagrams with single virtual∆-excitationare shown in Fig. 2. In the case of isospin-symmetric nuclear matter their isospin fact
8, 0, and 8, in the order shown. For the three-loop Hartree diagram the occurring integral


he

ss.plittings of

small-tisfieder–

e the

ree--ce

-rangeholdsndent)Fock

e thatg twodified

over the product of three Fermi spheres of radiuskf can be solved in closed form and tcontribution to the energy per particle reads:

E(kf )(3H) = g4Am6

π

∆(2πfπ)4

[2

3u6(1+ ζ ) + u2 − 3u4 + 5u3 arctan2u

− 1

4

(1+ 9u2) ln

(1+ 4u2)], (5)

with the abbreviationu = kf /mπ wheremπ = 135 MeV stands for the (neutral) pion maThe delta-propagator shows up in this expression merely via the (reciprocal) mass-s∆ = 293 MeV. Additional corrections to the delta-propagator coming from differencenucleon kinetic energies etc. will make a contribution at least one order higher in themomentum expansion. In Eq. (5) we have already inserted the empirically well-sarelationgπN∆ = 3gπN/

√2 for theπN∆-coupling constant together with the Goldberg

Treiman relationgπN = gAM/fπ (see, e.g., Eq. (5) in Ref. [18] for the∆ → Nπ decaywidth). As usualfπ = 92.4 MeV denotes the weak pion decay constant and we choosvaluegA = 1.3 in order to have a pion–nucleon coupling constant ofgπN = 13.2 [23]. Viathe parameterζ we have already included in Eq. (5) the contribution of an additional thnucleon contact interaction∼ (ζg4

A/∆f 4π )(NN)3. One notices that the 2π -exchange three

body Hartree diagram shows in the chiral limitmπ = 0 the same quadratic dependenon the nucleon density. In that limit the momentum dependentπN∆-interaction verticesget canceled by the pion-propagators and thus one is effectively dealing with a zerothree-nucleon contact-interaction. It is important to point out that this equivalenceonly after taking the spin-traces but not at the level of the (spin- and momentum depe2π -exchange three-nucleon interaction. The contribution of the (right) three-bodydiagram in Fig. 2 to the energy per particle reads:

E(kf )(3F) = − 3g4Am6

πu−3

4∆(4πfπ)4

u∫0

dx[2G2

S(x,u) + G2T (x,u)

], (6)

where we have introduced the two auxiliary functions:

GS(x,u) = 4ux

3

(2u2 − 3

) + 4x[arctan(u + x) + arctan(u − x)

]+ (

x2 − u2 − 1)ln

1+ (u + x)2

1+ (u − x)2, (7)

GT (x,u) = ux

6

(8u2 + 3x2) − u

2x

(1+ u2)2

+ 1

8

[(1+ u2)3

x2− x4 + (

1− 3u2)(1+ u2 − x2)] ln1+ (u + x)2

1+ (u − x)2. (8)

Evidently, the three-body Fock term in Eq. (6) is attractive. We note as an asidthe three-body terms could also be interpreted differently. For example, by openinneighboring nucleon lines of the diagrams in Fig. 2 one encounters a “medium mo
one-pion exchange”. Further medium modifications of theπN∆-dynamics require four


nsion.y,kcon-

le.

calcu-s

ly witht-ng

Fockto ourde theclears not

c-ergyo-

norero-

of theibilitymeseeds,5]

ed byar

errties,V

or more loops and thus contribute at a higher order in the small momentum expaThe remaining contributions toE(kf ) from the (relativistically improved) kinetic energfrom the 1π -exchange Fock diagram and from the iterated 1π -exchange Hartree and Focdiagrams have been written down in Eqs. (5)–(11) of Ref. [2]. The strongly attractivetribution from iterated 1π -exchange linear in the density and the cutoffΛ (see Eq. (15) inRef. [2]) is now of course not counted extra sinceB3 in Eq. (1) collects all such possibterms. Adding all pieces we arrive at the full energy per particleE(kf ) at three-loop orderIt involves three parameters,B3 andB5 of the two-body contact term Eq. (1) andζ whichcontrols a three-body contact term in Eq. (5).

Let us first look at generic properties of the nuclear matter equation of state in ourlation. Binding and saturation occurs in a wide range of the two adjustable parameterB3,5.However, with the full strength (ζ = 0) of the repulsiveρ2-term from the 2π -exchangethree-body Hartree diagram (see Eq. (5)) the saturation curve rises much too steepincreasing density. This causes a too low saturation densityρ0 and a too high nuclear mater compressibility,K > 350 MeV. We cure this problem in a minimal way by introducian attractive three-body contact term. Withζ = −3/4 the remaining repulsiveρ2-term inEq. (5) gets canceled by an analogous attractive contribution from the three-bodydiagram. Clearly, the need for introducing an attractive three-body contact term incalculation points to some short-distance physics whose dynamical origin lies outsipresent framework of perturbative chiral pion–nucleon interactions. It will becomein the following sections that the predictive power of our calculation is neverthelesreduced by this procedure.

We fix the minimum of the saturation curveE(kf ) to the valueE0 = −16.0 MeV. WithB3 adjusted to the valueB3 = −7.99 andB5 taken to be zero,B5 = 0, the full line inFig. 3 results. The predicted value of the saturation density isρ0 = 0.157 fm−3, corre-sponding to a Fermi momentum ofkf 0 = 261.6 MeV = 1.326 fm−1. This is very close tothe semi-empirical valueρ0 = 0.158 fm−3 obtained by extrapolation from inelastic eletron scattering off heavy nuclei [24]. The decomposition of the negative binding enE0 = −16.0 MeV into contributions of second, third, fourth and fifth power in small mmenta reads:E0 = (21.9 − 145.5 + 107.8 − 0.2) MeV with the typical balance betweelarge third and fourth order terms [2]. The very small fifth order term splits furthermas(−13.8+13.6) MeV into the contribution from the three-body contact-interaction (pportional toζ = −3/4) and a remainder. Evidently, sinceE0 = −16.0 MeV is a smallnumber that needs to be finetuned in our calculation there remains the question“convergence” of the small momentum expansion. The nuclear matter compressK = k2

f 0E′′(kf 0) related to the curvature of the saturation curve at its minimum co

out asK = 304 MeV. This number is somewhat high but still acceptable, since it exce.g., the valueK = 272 MeV obtained in the relativistic mean-field model of Ref. [2only by 12%. Inspection of Fig. 3 shows that the saturation curve is well approximata shifted parabolaE(kf ) = E0(2ρ0 −ρ)ρ/ρ2

0 with a second zero-crossing at twice nuclematter density 2ρ0. This leads to a compressibility estimate ofK = −18E0 = 288 MeV,not far from the calculated value. We have also studied variations of the parametB5.Within the limits set by a stable saturation point (and realistic single-particle propesee next section), the effects on the compressibilityK are marginal (less than 10 Me
reduction). Therefore we stay with the minimal choiceB5 = 0 (together withB3 = −7.99


nsity

ees ofrs

e re-tm-g. 3tum)

s

Hove

Fig. 3. The energy per particleE(kf ) of isospin-symmetric nuclear matter as a function of the nucleon de

ρ = 2k3f/3π2. The dotted line refers to the result of Ref. [2], with only pions and nucleons as active degr

freedom. The full line includes effects from 2π -exchange with virtual∆-excitation. The short-range parameteareB3 = −7.99 andB5 = 0.

andζ = −3/4). The dotted line in Fig. 3 shows for comparison the equation of statsulting from our previous chiral calculation [2] with noπN∆-dynamics included. In thawork the saturation densityρ0 = 0.178 fm−3 came out somewhat too high, but the copressibilityK = 255 MeV had a better value. The stronger rise of the full curve in Fiwith densityρ is a consequence of including higher order terms in the (small-momenkf -expansion.

3. Real part of single-particle potential

In this section we discuss the real partU(p, kf ) of the single-particle potential. Aoutlined in Ref. [3] the contributions to the (real) nuclear mean-fieldU(p, kf ) can beclassified as two-body and three-body potentials. The parametersB3,5 introduced in Eq. (1)reappear in a contribution to the two-body potential of the form:

U2(p, kf )(ct) = 2B3k3f

M2+ B5

k3f

3M4

(3k2

f + 5p2). (9)

Its density- and momentum-dependence is completely fixed by the Hugenholtz–Vantheorem [26] and a sum rule which connects it to the energy per particleE(kf )(ct) (see
Eqs. (5), (7) in Ref. [3]). The Fock diagrams of 2π -exchange with virtual∆-excitation


cted)

) threegram

-single-a total

gle-n

tion

give rise to a contribution to the two-body potential which can be written as a (subtradispersion integral:

U2(p, kf )(F ) = 1

2π3

∞∫2mπ


)

×{µkf + 2k3

f

15µ3

(3k2

f + 5p2) − 2k3f

3µ− µ2 arctan

kf + p

µ

− µ2 arctankf − p

µ+ µ

4p

(µ2 + k2

f − p2) lnµ2 + (kf + p)2

µ2 + (kf − p)2

}.

(10)

By opening a nucleon line in the three-body diagrams of Fig. 2 one gets (per diagramdifferent contributions to the three-body potential. In the case of the (left) Hartree diathey read altogether:

U3(p, kf )(H) = g4Am6

π

∆(2πfπ)4

{2u6(1+ ζ ) + u2 − 7u4 − 1

4

(1+ 9u2) ln

(1+ 4u2)

+ 5u3[arctan2u + arctan(u + x) + arctan(u − x)]

+ u3

2x

(2x2 − 2u2 − 3

)ln

1+ (u + x)2

1+ (u − x)2

}, (11)

with the abbreviationx = p/mπ . Note that the parameterζ = −3/4 related to the threebody contact-interaction has no influence on the momentum dependence of theparticle potential. On the other hand the (right) Fock diagram in Fig. 2 generatescontribution to the three-body potential of the form:

U3(p, kf )(F ) = − g4Am6

πx−2

4∆(4πfπ)4

{2G2

S(x,u) + G2T (x,u)

+u∫

0

dξ

[4GS(ξ,u)

∂GS(ξ, x)

∂x+ 2GT (ξ,u)

∂GT (ξ, x)

∂x

]}, (12)

with GS,T (x,u) defined in Eqs. (7), (8). The real single-particle potentialU(p, kf ) iscompleted by adding to the terms Eqs. (9)–(12) the contributions from 1π -exchange anditerated 1π -exchange written down in Eqs. (8)–(13) of Ref. [3]. The slope of the real sinparticle potentialU(p, kf ) at the Fermi surfacep = kf determines the effective nucleomass (in the nomenclature of Ref. [27], the product of “k-mass” and “E-mass” divided bythe free nucleon massM = 939 MeV) via a relation:

M∗(kf ) = M

[1− k2

f

2M2+ M

kf

∂U(p, kf )

∂p

∣∣∣∣p=kf

]−1

. (13)

The second term−k2f /2M2 in the square brackets stems from the relativistic correc

−p4/8M3 to the kinetic energy.


tic

th of

n ofshal-

to

s and

rtur-

Fig. 4. Full line: real part of the single-particle potentialU(p, kf 0) at saturation densitykf 0 = 261.6 MeV as afunction of the nucleon momentump. The dotted line includes in addition the relativistically improved kineenergyTkin(p) = p2/2M −p4/8M3. Both curves are extended into the region above the Fermi surfacep > kf 0.

The full line in Fig. 4 shows the real part of the single-particle potentialU(p, kf 0)

at saturation densitykf 0 = 261.6 MeV as a function of the nucleon momentump. Thedotted line includes in addition the relativistically improved kinetic energyTkin(p) =p2/2M − p4/8M3. With the parametersB5 = 0, ζ = −3/4 fixed andB3 = −7.99 ad-justed in Section 1 to the binding energy at equilibrium we find a potential depU(0, kf 0) = −78.2 MeV. This is very close to the resultU(0, kf 0) � −80 MeV of therelativistic Dirac–Brueckner approach of Ref. [28]. For comparison, the calculatioRef. [29] based on the phenomenological Paris NN-potential finds a somewhatlower potential depth ofU(0, kf 0) � −64 MeV. One observes that with the chiralπN∆-dynamics included, the real single-particle potentialU(p, kf 0) grows monotonically withthe nucleon momentump. The downward bending abovep = 180 MeV displayed in Fig. 3of Ref. [3] is now eliminated. The slope at the Fermi surfacep = kf 0 translates into aneffective nucleon mass ofM∗(kf 0) = 0.88M . This is now a realistic value comparedM∗(kf 0) � 3M obtained in our previous calculation [3,6] without any explicit∆-isobars.Note also that the chiral approach of Ref. [8] (where both explicit short-range termpion-exchange are iterated) has found the lower boundM∗(kf 0) > 1.4M .

The dotted curve in Fig. 4 for the total single-particle energyTkin(p) + U(p, kf 0) hitsthe valueE(kf 0) = E0 = −16 MeV at the Fermi surfacep = kf 0, as required by theHugenholtz–Van Hove theorem [26]. This important theorem holds strictly in our (pe
bative) calculation, whereas (non-perturbative) Brueckner–Hartree–Fock approaches often


bovetiontions

gesAp-p tocaleclear

fe

agreecehase-

Fig. 5. The full line shows the real part of the single-particle potentialU(0, kf ) at nucleon momentump = 0

versus the densityρ = 2k3f/3π2. The band is obtained from the universal low-momentum NN-potentialVlow-k

in linear density approximation.

fail to respect it [30]. In Fig. 4 we have also extended both curves into the region athe Fermi surfacep > kf 0. In general this extension is not just an analytical continuaof the potential from below the Fermi surface. Whereas Eqs. (9)–(12) for the contribufrom 2π -exchange with∆-excitation apply in both regions, there are non-trivial chanin the expressions from iterated 1π -exchange. These modifications are summarized inpendix A. The smooth rise ofU(p, kf 0) as it crosses the Fermi surface and proceeds up � 400 MeV is compatible with other calculations [28,29]. Beyond this momentum sone presumably exceeds the limits of validity of the present chiral calculation of numatter.

The full line in Fig. 5 shows the potential depthU(0, kf ) for a nucleon at the bottom othe Fermi sea as a function of the nucleon densityρ = 2k3

f /3π2. The band spanned by thdotted lines stems from the universal low-momentum NN-potentialVlow-k [31] in lineardensity approximation. In this approximation the potential depth simply reads:

U(0, kf ) = 3πρ

2M

[V

(1S0)low-k(0,0) + V

(3S1)low-k(0,0)

], (14)

with V(1S0)low-k(0,0) � −1.9 fm andV

(3S1)low-k(0,0) � −(2.2± 0.3) fm [31–33], the two S-wave

potentials at zero momentum. It is interesting to observe that both potential depthsfairly well at low densities,ρ � 0.07 fm−3. This agreement is by no means trivial sinVlow-k is constructed to reproduce accurately the low-energy NN-scattering data (p
shifts and mixing angles) whereas our adjustment of theB3-term (linear in density) is made


nvature

pthe re-

dotted

terdifica-d

-atedthe-

e dia-rface

tential

at saturation densityρ0 = 0.157 fm−3. It is evident from Fig. 5 that a linear extrapolatiodoes not work from zero density up to nuclear matter saturation density. Strong cureffects set in already at Fermi momenta aroundkf � mπ if (and only if) pion-dynamics istreated explicitly.2 We note also that an “improved” determination of the potential deU(0, kf ) from Vlow-k where one takes into account its momentum dependence in thpulsive Fock contribution leads to concave curves which bend below the straight

lines in Fig. 5. In the caseV (3S1)low-k(0,0) = −1.9 fm the potential depthU(0, kf 0) at satura-

tion density would increase to−132.6 MeV (compared to−113.4 MeV in linear densityapproximation). The present observations concerning the potential depthU(0, kf ) may in-dicate why calculations based onVlow-k did so far not find saturation of nuclear mat[9]. It seems that the Brueckner ladder does not generate all relevant medium motions which set in already at rather low densitieskf � mπ (if the pion-dynamics is treateexplicitly).

4. Imaginary part of single-particle potential

In this section, we reconsider the imaginary partW(p,kf ) of the single-particle potential. To the three-loop order we are working here it is still given completely by iter1π -exchange with no contribution from theπN∆-dynamics. The new aspect here isextension into the region above the Fermi surfacep > kf , which is not an analytical continuation from below the Fermi surface. As outlined in Ref. [3] the contributions toW(p,kf )

can be classified as two-body, three-body and four-body terms. From the Hartregram of iterated 1π -exchange one finds altogether (in the region above the Fermi sup > kf ):

W(p,kf )(H) = πg4AMm4

π

(4πfπ)4

{(9+ 6u2 + 4u3

x− 2x2

)ln

[1+ (u + x)2]

+(

4u3

x+ 2x2 − 9− 6u2

)ln

[1+ (x − u)2] + 4ux

(2− u2)

+ 1

x

[(7+ 15u2 − 15x2)[arctan(u + x) − arctan(x − u)

]

+ 12u5

5− 21u

2− 8u3 ln

(1+ 4u2) −

(15u2 + 7

4

)arctan2u

]

+ 3θ(√

2u − x)

u/x∫ymin

dy(x2y2 − u2)Ay

[2s2 + s4

1+ s2− 2 ln

(1+ s2)]

2 As example for the extreme inherent non-linearities, consider the formula for the three-body poU3(0, kf )(H) in Eq. (11). Its mathematical Taylor-series expansion converges only forkf < mπ/2. This cor-

responds to tiny densities,ρ < 0.0027 fm−3.


an-nsh twoar fromthell con-tions(for

-

ratioof a

+1∫

ymin

dyAy

[3s4x2(y2 − 1)

1+ s2− 9s4

2+ 10xy

(3 arctans − 3s + s3)

+ (9+ 6u2 − 6x2y2)[s2 − ln

(1+ s2)]]}

, (15)

with x = p/mπ and the auxiliary functionsymin = √1− u2/x2 and s = xy +√

u2 − x2 + x2y2. In order to keep the notation compact we have introduced thetisymmetrization prescriptionAy[f (y)] = f (y) − f (−y). Note that there is a term iEq. (15) which vanishes identically abovep = √

2kf . A geometrical explanation for thinon-smooth behavior is that an orthogonal pair of vectors connecting the origin witpoints inside a sphere ceases to exist if the center of the sphere is displaced too fthe origin (namely by more than

√2 times the sphere radius). The orthogonality of

(momentum difference) vectors is imposed here by the non-relativistic on-mass-shedition for a nucleon. The combined two-body, three-body and four-body contributo W(p,kf ) from the iterated 1π -exchange Fock diagram read on the other handp > kf ):

W(p,kf )(F )

= πg4AMm4

π

(4πfπ)4

{u3x + u5

5x+ 3

2x

(x+u)/2∫(x−u)/2

dξ[(2ξ − x)2 − u2]1+ 4ξ2

1+ 2ξ2ln

(1+ 4ξ2)

+ 3

4πθ(

√2u − x)

1∫ymin

dy

1∫ymin

dzθ(1− y2 − z2)√

1− y2 − z2Ay

[s2 − ln

(1+ s2)]

×Az

[t2 − ln

(1+ t2)] + 3

x

1∫−1

dy

u∫0

dξ ξ2[ln(1+ σ 2) − σ 2](1− 1

R

)}, (16)

with some new auxiliary functionst = xz + √u2 − x2 + x2z2 and σ = ξ y+√

u2 − ξ2 + ξ2y2 andR = √(1+ x2 − ξ2)2 + 4ξ2(1− y2). It is also interesting to con

sider the imaginary single-particle potentialW(p,kf ) in the chiral limitmπ = 0. One findsthe following closed form expressions:

W(p,kf )|mπ=0 = 3πg4AM

(4πfπ)4

{7k5f

5p− k3

f p − 2

5p

(2k2

f − p2)5/2θ(

√2kf − p)

},

p > kf , (17)

W(p,kf )|mπ=0 = 9πg4AM

4(4πfπ)4

(k2f − p2)2

, p < kf , (18)

to which the iterated 1π -exchange Hartree and Fock diagrams have contributed in the4 : −1. The analytical results in Eqs. (17), (18) agree with Galitskii’s calculation [34]
contact-interaction to second order. In the chiral limitmπ = 0 the spin-averaged product


one isgree-lved

into

1l.energy,r-der

oten-fact

e the

Fig. 6. The imaginary part of the single-particle potentialW(p,kf 0) at saturation densitykf 0 = 261.6 MeV asa function of the nucleon momentump. The quadratic behavior around the Fermi surfacep = kf 0 with a signchange of the curvature is required by Luttinger’s theorem [35].

of two πNN -interaction vertices gets canceled by the pion propagators and thuseffectively dealing with a zero-range NN-contact interaction at second order. The ament with Galitskii’s result [34] serves as an important check on the technically invocalculation behind Eqs. (15), (16).

Fig. 6 shows the imaginary part of the single-particle potentialW(p,kf 0) at satura-tion densitykf 0 = 261.6 MeV as a function of the nucleon momentump. The quantity±2W(p,kf ) determines the width of a hole-state or a particle-state of momentump < kf

or p > kf . The finite life time of these states originates from redistributing energyadditional particle-hole excitations. Our predicted valueW(0, kf 0) = 24.0 MeV atp = 0lies in between the resultsW(0, kf 0) � 20 MeV of Ref. [36] employing the Gogny Deffective interaction andW(0, kf 0) � 40 MeV of Ref. [29] using the Paris NN-potentiaAs a consequence of the decreasing phase-space for redistributing the hole-statethe curve in Fig. 6 drops with momentump andW(p,kf 0) reaches zero at the Fermi sufacep = kf 0. According to Luttinger’s theorem [35] this vanishing is of quadratic or∼ (p − kf )2, a feature which is clearly exhibited by the curve in Fig. 6.

When crossing the Fermi surface the curvature of the imaginary single-particle ptial W(p,kf ) flips the sign. From there on a rapid fall to negative values sets in. Inthe widthΓsp = −2W(p,kf ) represents the spreading of a single-particle state abovFermi surface into two-particle-one-hole states with growing phase space asp − kf in-
creases. The range of validity of the present chiral calculation is again expected to be


not

clearxima-larityrties

ar en-chiral

om-

dratictional

rmi19)

me

ergycu-

d

tic en-urierum in-

cleon

p < 400 MeV. The rapid growth ofΓsp beyond such limited momentum scales isshared by the results of Refs. [29,36].

5. Nuclear energy density functional

The energy density functional is a general starting point for (non-relativistic) nustructure calculations within the framework of the self-consistent mean-field approtion [37]. In this context effective Skyrme forces [38–40] have gained much popubecause of their analytical simplicity and their ability to reproduce nuclear propeover the whole periodic table. In a recent work [41] we have calculated the nucleergy density functional which emerges from (leading and next-to-leading order)pion–nucleon dynamics. The calculation in Ref. [41] included (only) the 1π -exchangeFock-diagram and the iterated 1π -exchange Hartree and Fock diagrams. These few cponents alone lead already to a good nuclear matter equation of stateE(kf ). Thereforethe interest here is on the additional effects from 2π -exchange with virtual∆-excitationcontributing one order higher in the small momentum expansion. Going up to quaorder in spatial gradients (i.e., deviations from homogeneity) the energy density funcrelevant forN = Z even–even nuclei reads [41]:

E[ρ, τ ] = ρE(kf ) +[τ − 3

5ρk2

f

][1

2M− 5k2

f

56M3+ Fτ (kf )

]+ ( �∇ρ)2F∇(kf ), (19)

with ρ(�r) = 2k3f (�r)/3π2 the local nucleon density (expressed in terms of a local Fe

momentumkf (�r)) andτ(�r) the local kinetic energy density. We have left out in Eq. (the spin–orbit coupling term since the corresponding results (for∆-driven 2π -exchangethree-body spin-orbit forces3) can be found in Ref. [42]. In phenomenological Skyrparameterizations, the strength functionFτ (kf ) goes linearly with the densityρ, whileF∇(kf ) is constant. The starting point for the construction of an explicit nuclear endensity functionalE[ρ, τ ] is the bilocal density-matrix as given by a sum over the ocpied energy eigenfunctions:

∑α∈occΨα(�r − �a/2)Ψ †

α (�r + �a/2). According to Negele anVautherin [43] it can be expanded in relative and center-of-mass coordinate,�a and�r , withexpansion coefficients determined by purely local quantities (nucleon density, kineergy density and spin-orbit density). As outlined in Section 2 of Ref. [41] the Fotransform of the (so-expanded) density matrix defines in momentum space a “medisertion” for the inhomogeneous many-nucleon system:

Γ ( �p, �q) =∫

d3r e−i �q·�r θ(kf − | �p|){1+ 35π2

8k7f

(5�p2 − 3k2

f

)

×[τ − 3

5ρk2

f − 1

4�∇2ρ

]}. (20)

3 Interestingly, this three-body spin-orbit coupling is not a relativistic effect but independent of the nu
massM .


sitys

y

tion:

bu-

nd

The strength functionFτ (kf ) in Eq. (19) emerges via a perturbation on top of the denof statesθ(kf − | �p|). As a consequence of that,Fτ (kf ) can be directly expressed in termof the real single-particle potentialU(p, kf ) as:

Fτ (kf ) = 35

4k7f

kf∫0

dp p2(5p2 − 3k2f

)U(p, kf ). (21)

Note that anyp-independent contribution, in particular theB3-term in Eq. (9) and theζ -term in Eq. (11), drops out. In the medium insertion Eq. (20)τ −3ρk2

f /5 is accompanied b

−�∇2ρ/4. After performing a partial integration one is lead to the following decomposi

F∇(kf ) = π2

8k2f

∂Fτ (kf )

∂kf

+ Fd(kf ), (22)

whereFd(kf ) comprises all those contributions for which the( �∇ρ)2-factor originates di-rectly from the momentum-dependence of the interactions.

We enumerate now the contributions to the strength functionsFτ,d(kf ) generated by2π -exchange with virtual∆-excitation. We start with (regularization dependent) contritions encoded in subtraction constants:

Fτ (kf )(ct) = B55k3

f

3M4, Fd(kf )(ct) = Bd

M4, (23)

where the new parameterBd = −M4V ′′C(0)/4 stems from two-body Hartree diagrams a

the momentum transfer dependence of the isoscalar central NN-amplitudeVC(q). Two-body Fock diagrams contribute only toFτ (kf ) via a (subtracted) dispersion integral:

Fτ (kf )(2F) = 35

24π3k4f

∞∫2mπ


)

×{ 8k7

f

35µ3− µk3

f

3− 6µ3kf + µ5

4kf

+ 5µ4 arctan2kf

µ

+ µ3

16k3f

(24k4

f − 18k2f µ2 − µ4) ln

(1+ 4k2

f

µ2

)}. (24)

The evaluation of the (left) three-body Hartree diagram in Fig. 2 leads to the results:

Fτ (kf )(3H) = 35g4Am4

π

∆(2πfπ)4

{13

4− 5

24u2+ u2

9− 35

12uarctan2u

+(

5

96u4+ 3

4u2− 3

4

)ln

(1+ 4u2)}, (25)

Fd(kf )(3H) = g4Amπ

128π2∆f 4π

{23 arctan2u − 7

uln

(1+ 4u2) − 16u − 2u(3+ 16u2)

3(1+ 4u2)2

}.

(26)


hich

(28)

y

-

Somewhat more involved is the evaluation of the (right) Fock diagram in Fig. 2 for wwe find:

Fτ (kf )(3F) = 35g4Am4

π

∆(8πfπ)4u7

u∫0

dx[2GS(x,u)GS(x,u) + GT (x,u)GT (x,u)

], (27)

GS(x,u) = 4ux

3

(6u4 − 22u2 − 45+ 30x2 − 10u2x2)

+ 4x(10+ 9u2 − 15x2)[arctan(u + x) + arctan(u − x)

]+ (

35x2 + 14u2x2 − 10x4 − 5− 9u2 − 4u4) ln1+ (u + x)2

1+ (u − x)2, (28)

GT (x,u) = ux

12

(69u4 + 70u2 − 15

) − ux3

12

(45+ 31u2) − 15ux5

4

− u

4x

(1+ u2)2(5+ 3u2) + [1+ (u + x)2][1+ (u − x)2]

16x2

× (5+ 8u2 + 3u4 − 18u2x2 + 15x4) ln

1+ (u + x)2

1+ (u − x)2, (29)

Fd(kf )(3F) = g4Amπ

π2∆(8fπ)4

{−3+ 12u2 + 26u4 + 40u6

u5(1+ 4u2)ln

(1+ 4u2)

+ 3

8u7

(1+ 2u2 + 8u4) ln2(1+ 4u2) + 2(3+ 6u2 + 16u4)

u3(1+ 4u2)

}, (30)

with GS,T (x,u) defined in Eqs. (7), (8). The strength functionsFτ,d(kf ) are completed byadding to the terms in Eqs. (23)–(30) the contributions from 1π -exchange and iterated 1π -exchange written down in Eqs. (9), (11), (12), (14) ,(15), (18), (19), (22), (24), (27),of Ref. [41]. In order to be consistent with the calculation of the energy per particleE(kf )

and the single-particle potentialU(p, kf ) we complete the 1π -exchange contribution bits relativistic 1/M2-correction:

Fτ (kf )(1π) = g2Am3

πu−5

(32πfπM)2

{280

3u6 − 15

2+ 2u

(525− 700u2 − 96u4)arctan2u

− 64u8 + 744u4 − 1777u2 +(

1050u2 − 77+ 15

8u2

)ln

(1+ 4u2)}.

(31)

The expression in Eq. (19) multiplying the kinetic energy densityτ(�r) has the interpretation of a reciprocal density dependent effective nucleon mass:

∗[ 5k2

f]−1

M (ρ) = M 1−28M2

+ 2MFτ (kf ) . (32)


nhe full

c-

ce on-to the

41]ef-nmassoursele po-ation

g and

Fig. 7. The effective nucleon massM∗(ρ) divided by the free nucleon massM as a function of the nucleodensityρ. The dotted line shows the result of Ref. [41] based on single and iterated pion-exchange only. Tline includes in addition the effects from 2π -exchange with virtual∆-excitation.

We note that this effective nucleon massM∗(ρ) (entering the nuclear energy density funtional) is conceptually different from the so-called “Landau”-massM∗(kf ) defined inEq. (13). Only if the real single-particle potential has a simple quadratic dependenthe nucleon momentum,U(p, kf ) = U0(kf ) + p2U1(kf ), do these two variants of effective nucleon mass agree with each other (modulo very small differences relatedrelativistic(kf /2M)2-correction).

In Fig. 7 we show the ratio effective-over-free nucleon massM∗(ρ)/M as a function ofthe nucleon densityρ = 2k3

f /3π2. The dotted line corresponds to the result of Ref. [based on 1π - and iterated 1π -exchange only. The full line includes in addition thefects from 2π -exchange with virtual∆-excitation. It is clearly visible that the inclusioof the πN∆-dynamics leads to substantial improvement of the effective nucleonM∗(ρ) since now it decreases monotonically with the density. This behavior is of ca direct reflection of the improved momentum dependence of the real single-partictential U(p, kf ) (see Fig. 4). Our prediction for the effective nucleon mass at saturdensity,M∗(ρ0) = 0.64M , is comparable to the typical valueM∗(ρ0) � 0.7M of phenom-enological Skyrme forces [38,39]. The full curve in Fig. 7 displays another interestinimportant feature, namely strong curvature effects at low densitiesρ < 0.05 fm−3. Theyoriginate from the explicit presence of the small mass scalemπ = 135 MeV in our calcu-
lation.


us

rated

r

s-

to zero.ntkyrme

f the

3)f the

Fig. 8. The strength functionF∇ (kf ) multiplying the( �∇ρ)2-term in the nuclear energy density functional vers

the nucleon densityρ = 2k3f/3π2. The dotted line shows the result of Ref. [41] based on single and ite

pion-exchange only. The full line includes in addition the effects from 2π -exchange with virtual∆-excitation.

Fig. 8 shows the strength functionF∇(kf ) belonging to the( �∇ρ)2-term in the nucleaenergy density functional versus the nucleon densityρ = 2k3

f /3π2. The dotted line givesthe result of Ref. [41] based on 1π - and iterated 1π -exchange only and the full line includein addition the effects from 2π -exchange with virtual∆-excitation. The subtraction constantBd (representing density independent short-range contributions) has been setIn the region around saturation densityρ0 � 0.16 fm−3 one observes a clear improvemesince there the full line meets the band spanned by the three phenomenological Sforces SIII [38], Sly [39] and MSk [40]. The strong rise ofF∇(kf ) towards low densitiesremains however. As explained in Ref. [41] this has to do with chiral singularities (oform m−2

π andm−1π ) in the contributions from 1π -exchange and iterated 1π -exchange.

The knowledge of the strength functionF∇(kf ) and the equation of stateE(kf ) allowsone to calculate the surface energy of semi-infinite nuclear matter [44] via:

as = 2(36πρ−2

0

)1/3

ρ0∫0

dρ

√ρF∇(kf )

[E(kf ) − E0

]. (33)

Here, we have inserted into the formula for the surface energyas (see Eq. (5.32) inRef. [44]) that density profileρ(z) which minimizes it. Numerical evaluation of Eq. (3gives as = 24.2 MeV. This number overestimates semi-empirical determinations o
surface energy, such asas = 20.7 MeV of Ref. [45] oras = 18.2 MeV of Ref. [44], by


ftrix

rate at8 be-

etric-

finitel

umf

n

l

to the.

tery

l

17% or more.4 The reason for our high valueas = 24.2 MeV is of course the strong rise othe strength functionF∇(kf ) at low densities. Its derivation is based on the density-maexpansion of Negele and Vautherin [43] which has been found to become inacculow and non-uniform densities [46]. Therefore one should not trust the curves in Fig.low ρ = 0.05 fm−3. Getting the right order of magnitude forF∇(kf ) in the density region0.1 fm−3 < ρ < 0.2 fm−3 is already a satisfactory result.

6. Nuclear matter at finite temperatures

In this section, we discuss nuclear matter at finite temperaturesT � 30 MeV. We areparticularly interested in the first-order liquid-gas phase transition of isospin-symmnuclear matter and its associated critical point(ρc, Tc). As outlined in Ref. [6] a thermodynamically consistent extension of the present (perturbative) calculational scheme totemperatures is to relate it directly to the free energy per particleF (ρ,T ), whose naturathermodynamical variables are the nucleon densityρ and the temperatureT . In that casethe free energy densityρF (ρ,T ) of isospin-symmetric nuclear matter consists of a sof convolution integrals over interaction kernelsKj multiplied by powers of the density onucleon states in momentum space:5

d(pj ) = pj

2π2

[1+ exp

p2j − 2Mµ

2MT

]−1

. (34)

The effective one-body “chemical potential”µ(ρ,T ) entering the Fermi–Dirac distributioin Eq. (34) is determined by the relation to the particle densityρ = 4

∫ ∞0 dp1 p1d(p1). We

summarize now the additional interaction kernels arising from 2π -exchange with virtua∆-excitation. The two-body kernels read:

K(ct)2 = 24π2B3

p1p2

M2+ 20π2B5

p1p2

M4

(p2

1 + p22

), (35)

K(F )2 = 1

π

∞∫2mπ


)

×{µ ln

µ2 + (p1 + p2)2

µ2 + (p1 − p2)2− 4p1p2

µ+ 4p1p2

µ3

(p2

1 + p22

)}. (36)

Note that theB3-term in Eq. (35) generates a temperature independent contributionfree energy per particle,F(ρ,T )(B3) = 3π2B3ρ/2M2, which drops linearly with density

4 One could reproduce the surface energyas = 20.7 MeV of Ref. [45] by adjusting the short-range parameBd in Eq. (32) to the valueBd = −75. The full curve forF∇ (kf ) in Fig. 8 would then be shifted downward b

29 MeV fm5. Compared toB3 = −8 the fitted numberBd = −75 seems to be rather large.5 Since the temperatureT is comparable to an average kinetic energy we countT of quadratic order in smal

momenta.


three-ree-

ntribu-(7),lcula-

t of thehich hasre-

col-nlearically

real-

al ex-clearityserva-

Temperature and density dependent Pauli blocking effects are incorporated in thebody kernelK3. The contributions of the Hartree and Fock diagrams in Fig. 2 to the thbody kernel read:

K(H)3 = 3g4

Ap3

∆f 4π

{2p1p2(1+ ζ ) + 2m4

πp1p2

[m2π + (p1 + p2)2][m2

π + (p1 − p2)2]− m2

π lnm2

π + (p1 + p2)2

m2π + (p1 − p2)2

}, (37)

K(F )3 = − g4

A

4∆f 4πp3

[2X(p1)X(p2) + Y(p1)Y (p2)

], (38)

X(p1) = 2p1p3 − m2π

2ln

m2π + (p1 + p3)

2

m2π + (p1 − p3)2

, (39)

Y(p1) = p1

4p3

(5p2

3 − 3m2π − 3p2

1

)

+ 3(p21 − p2

3 + m2π )2 + 4m2

πp23

16p23

lnm2

π + (p1 + p3)2

m2π + (p1 − p3)2

. (40)

The remaining kernels building up the free nucleon gas part and the interaction cotions from 1π -exchange and iterated 1π -exchange have been written down in Eqs. (4)–(11), (12) of Ref. [6]. It is needless to say that the extension of our nuclear matter cation to finite temperaturesT does not introduce any new adjustable parameter.

Fig. 9 shows the calculated pressure isothermsP(ρ,T ) = ρ2∂F (ρ,T )/∂ρ of isospin-symmetric nuclear matter at six selected temperaturesT = 0,5,10,15,20,25 MeV. As itshould be these curves display a first-order liquid-gas phase transition similar to thavan der Waals gas. The coexistence region between the liquid and the gas phase (wto be determined by the Maxwell construction) terminates at the critical temperatuTc.From there on the pressure isothermsP(ρ,T ) grow monotonically with the nucleon density ρ. We find here a critical temperature ofTc � 15 MeV and a critical density ofρc �0.053 fm−3 � ρ0/3. This critical temperature is close to the valueTc = (16.6± 0.9) MeVextracted in Ref. [47] from experimentally observed limiting temperatures in heavy ionlisions. In comparison, a critical temperature ofTc = (20± 3) MeV has been extracted iRef. [48] from multi-fragmentation data in proton-on-gold collisions. Most other nucmatter calculations find a critical temperature somewhat higher than our value, typTc � 18MeV [49–51]. The reduction ofTc in comparison toTc � 25.5 MeV obtained pre-viously in Ref. [6] results from the much improved momentum dependence of thesingle-particle potentialU(p, kf 0) near the Fermi surfacep = kf 0 (see Fig. 4). As a general rule the critical temperatureTc grows with the effective nucleon massM∗(kf 0) at theFermi surface.

The single-particle properties around the Fermi surface are decisive for the thermcitations and therefore they crucially influence the low temperature behavior of numatter. The inclusion of the chiralπN∆-dynamics leads to a realistic value of the densof (thermally excitable) nucleon states at the Fermi surface. This is an important ob
tion.


ra-

arisonctorshes

. (2).

Fig. 9. Pressure isothermsP(ρ,T ) = ρ2∂F (ρ,T )/∂ρ of isospin-symmetric nuclear matter at finite tempeture T . The coexistence region of the liquid and the gas phase ends at the critical point:ρc � 0.053 fm−3,Tc � 15 MeV.

7. Equation of state of pure neutron matter

This section is devoted to the equation of state of pure neutron matter. In compto the calculation of isospin-symmetric nuclear matter in Section 2 only the isospin faof the 2π -exchange diagrams with virtual∆-excitation change. We summarize now tcontributions to the energy per particleEn(kn) of pure neutron matter. The two-body termread:

En(kn)(ct) = Bn,3

k3n

M2+ Bn,5

k5n

M4, (41)

En(kn)(2F) = 1

8π3

∞∫2mπ

dµ Im(VC + WC + 2µ2VT + 2µ2WT

){3µkn − 4k3

n

3µ+ 8k5

n

5µ3

− µ3

2kn

− 4µ2 arctan2kn

µ+ µ3

8k3n

(12k2

n + µ2) ln

(1+ 4k2

n

µ2

)}, (42)

with Bn,3 andBn,5 two new subtraction constants. The relative weights of isoscalar (VC,T )and isovector (WC,T ) NN-amplitudes have changed by a factor 3 in comparison to Eq
The three-body terms generated by the Hartree and Fock diagrams in Fig. 2 read:


disti-energy

ningree-bids

m) of

ofour

Fig. 10. The energy per particleEn(kn) of pure neutron matter as a function of the neutron densityρn = k3n/3π2.

The dashed curve gives the result of Ref. [2]. The full curve includes theπN∆-dynamics and two adjusteshort-range parametersBn,3 = −0.95 andBn,5 = −3.58. The dashed-dotted curve stems from the sophcated many-body calculation of the Urbana group [20]. The dotted curve gives one half of the kineticEkin(kn)/2 = 3k2

n/20M .

En(kn)(3H) = g4

Am6π

6∆(2πfπ)4

[2

3u6 + u2 − 3u4 + 5u3 arctan2u

− 1

4

(1+ 9u2) ln

(1+ 4u2)], (43)

En(kn)(3F) = − g4

Am6πu−3

4∆(4πfπ)4

u∫0

dx[G2

S(x,u) + 2G2T (x,u)

], (44)

with GS,T (x,u) defined in Eqs. (7), (8). We emphasize that in this section the meaof u changes tou = kn/mπ , wherekn denotes the neutron Fermi momentum. All ththree-body diagrams in Fig. 2 have now the same isospin factor 2/3 since between neutrons only the 2π0-exchange is possible. Note that the Pauli-exclusion principle fora three-neutron contact-interaction and therefore Eq. (43) is free of anyζ -term. The ad-ditional contributions toEn(kn) from the (relativistically improved) kinetic energy, fro1π -exchange and from iterated 1π -exchange have been written down in Eqs. (32)–(37Ref. [2].

Fig. 10 shows the energy per particleEn(kn) of pure neutron matter as a functionthe neutron densityρn = k3

n/3π2. The dashed (concave) curve gives the result of
previous calculation in Ref. [2]. The full curve includes the chiralπN∆-dynamics. The


r-

ulationmatter

neu-in

n den-t upny-on ofer thanof

ndentn of

proton

asym-

wave)t also

ration

short-range parametersBn,3 andBn,5 (controlling the contribution of a nn-contact inteaction to En(kn)) have been adjusted to the valuesBn,3 = −0.95 andBn,5 = −3.58.6

The dashed-dotted curve in Fig. 10 stems from the sophisticated many-body calcof the Urbana group [20], to be considered as representative of realistic neutroncalculations. Moreover, the dotted curve gives one half of the kinetic energy of a freetron gas,Ekin(kn)/2 = 3k2

n/20M . Results of recent quantum Monte Carlo calculationRef. [52] have demonstrated that the neutron matter equation of state at low neutrosities ρn < 0.05 fm−3 is well approximated by this simple form. One observes thato ρn = 0.16 fm−3 our result forEn(kn) is very close to that of the sophisticated mabody calculation [20,52]. At higher densities we find a stiffer neutron matter equatistate. Again, one should not expect that our approach works at Fermi momenta largkn = 350 MeV (corresponding toρn = 0.19 fm−3). One of the most important resultsthe present calculation is that the unrealistic downward bending ofEn(kn) (as shown bythe dashed curve in Fig. 10) disappears after the inclusion of the chiralπN∆-dynamics.This is a manifestation of improved isospin properties.

8. Asymmetry energy

As a further test of isospin properties we consider in this section the density depeasymmetry energyA(kf ). The asymmetry energy is generally defined by the expansiothe energy per particle of isospin-asymmetric nuclear matter (described by differentand neutron Fermi momentakp,n = kf (1∓ δ)1/3) around the symmetry line:

Eas(kp, kn) = E(kf ) + δ2A(kf ) +O(δ4). (45)

Following the scheme in the previous sections we summarize the contributions to themetry energyA(kf ). The two-body terms read:

A(kf )(ct) = (2Bn,3 − B3)k3f

M2+ (3Bn,5 − B5)

10k5f

9M4, (46)

A(kf )(2F) = 1

12π3

∞∫2mπ

dµ

{Im

(VC + 2µ2VT

)[µkf − 2k3

f

µ+ 16k5

f

3µ3

− µ3

4kf

ln

(1+ 4k2

f

µ2

)]+ Im

(WC + 2µ2WT

)

×[3µkf + 2k3

f

µ− µ

4kf

(8k2

f + 3µ2) ln

(1+ 4k2

f

µ2

)]}. (47)

In Eq. (46) we have taken care of the fact that there are only two independent (S-NN-contact couplings which can produce terms linear in density. It is surprising tha

6 The short-range parametersBn,3 andBn,5 have been adjusted such that the asymmetry energy at satu
density takes on the valueA(kf 0) = 34 MeV.


s

sram in

e-t it-give

tro-

6)–

fixed

n-. Theetryshfourthg

scopic

the other coefficient 10(3Bn,5−B5)/9 in front of thek5f /M4-term is completely fixed. Thi

fact can be shown on the basis of the most general order-p2 NN-contact interaction writtendown in Eq. (6) of Ref. [53]. Out of the seven low-energy constantsC1, . . . ,C7 only twoindependent linear combinations,C2 andC1 +3C3 +C6, come into play for homogeneouand spin-saturated nuclear matter. The contribution of the three-body Hartree diagFig. 2 to the asymmetry energyA(kf ) has the following analytical form:

A(kf )(3H) = g4Am6

πu2

9∆(2πfπ)4

[(9

4+ 4u2

)ln

(1+ 4u2) − 2u4(1+ 3ζ )

− 8u2 − u2

1+ 4u2

], (48)

with the abbreviationu = kf /mπ . The parameterζ = −3/4 is again related to the threnucleon contact interaction∼ (ζg4

A/∆f 4π )(NN)3 which has the interesting property tha

contributes equally but with opposite sign to the energy per particleE(kf ) and the asymmetry energyA(kf ). Furthermore, both three-body Fock diagrams in Fig. 2 add up torise a contribution to the asymmetryA(kf ) which can be represented as:

A(kf )(3F) = g4Am6

πu−3

36∆(4πfπ)4

u∫0

dx{4GS01GS10 − 2G2

S01 − 6G2S10

+ 2GS(3GS + 8GS01 − 3GS02 + 2GS11 − 3GS20) + 2GT 01GT 10

− 7G2T 01 − 3G2

T 10 + GT (3GT + 8GT 01

− 3GT 02 + 2GT 11 − 3GT 20)}. (49)

The auxiliary functionsGS,T (x,u) have been defined in Eqs. (7), (8) and we have induced a double-index notation for their partial derivatives:

GIjk(x,u) = xjuk ∂j+kGI (x,u)

∂xj ∂uk, I = S,T , 1� j + k � 2. (50)

For notational simplicity we have omitted the argumentsx and u in the integrand ofEq. (49). The asymmetry energyA(kf ) is completed by adding to the terms in Eqs. (4(49) the contributions from the (relativistically improved) kinetic energy, 1π -exchange anditerated 1π -exchange written down in Eqs. (20)–(26) of Ref. [2].

In the calculation of the asymmetry energy we use consistently the previouslyshort-distance parametersB3 = −7.99 andBn,3 = −0.95, as well asB5 = 0 andBn,5 =−3.58. Fig. 11 shows the asymmetry energyA(kf ) as a function of the nucleon desity ρ = 2k3

f /3π2. The dashed (concave) curve corresponds to the result of Ref. [2]full curve includes the chiralπN∆-dynamics. The corresponding value of the asymmenergy at saturation densityρ0 = 0.157 fm−3 is A(kf 0) = 34.0 MeV. It decomposes aA(kf 0) = (12.1 + 119.3 − 109.9 + 12.5) MeV into contributions of second, third, fourtand fifth power of small momenta, again with a balance between large third andorder terms [2]. The valueA(kf 0) = 34.0 MeV is consistent with most of the existinempirical determinations of the asymmetry energy. For example, a recent micro
estimate in a relativistic mean-field model (constrained by some specific properties of


-

en-of

e-

ourty

hiral

nce

spin-rotons

Fig. 11. The asymmetry energyA(kf ) as a function of the nucleon densityρ = 2k3f/3π2. The dashed curve

shows the result of Ref. [2]. The full curve includes the chiralπN∆-dynamics.

certain nuclei) gave the valueA(kf 0) = (34± 2) MeV [54]. For comparison, other empirical values obtained from extensive fits of nuclide masses areA(kf 0) = 36.8 MeV[45] or A(kf 0) = 33.2 MeV [55]. The slope of the asymmetry energy at saturation dsity, L = kf 0A

′(kf 0), is likewise an interesting quantity. As demonstrated in Fig. 11Ref. [56] the neutron skin thickness of208Pb is linearly correlated with the slope paramter L. We extract from the full curve in Fig. 11 the valueL = 90.8 MeV. This predictionis not far fromL � 100 MeV quoted in Ref. [45] andL = 119.2MeV obtained from the“standard” relativistic force NL3 [57]. Furthermore, we extract from the curvature ofasymmetry energyA(kf ) at saturation densityρ0 the positive asymmetry compressibiliKas= k2

f 0A′′(kf 0) − 2L = 160.5 MeV.

Again, the most important feature visible in Fig. 11 is that the inclusion of the cπN∆-dynamics eliminates the (unrealistic) downward bending of the asymmetryA(kf )

at higher densitiesρ > 0.2 fm−3 (as displayed by the dashed curve in Fig. 11). This is omore a manifestation of improved isospin properties.

9. Isovector single-particle potential

In this section we generalize the calculation of the single-particle potential to isoasymmetric (homogeneous) nuclear matter. Any relative excess of neutrons over p
in the nuclear medium leads to a different “mean-field” for a proton and a neutron. This


tial in

term)-

ontact

ented

d-55))–

fact is expressed by the following decomposition of the (real) single-particle potenisospin-asymmetric nuclear matter:

U(p, kf ) − UI (p, kf )τ3δ +O(δ2), δ = ρn − ρp

ρn + ρp

, (51)

with U(p, kf ) the isoscalar (real) single-particle potential discussed in Section 3. Thelinear in the isospin-asymmetry parameterδ = (ρn − ρp)/(ρn + ρp) defines the (realisovector single-particle potentialUI (p, kf ), andτ3 → ±1 for a proton or a neutron. Without going into further technical details we summarize now the contributions toUI (p, kf ).The two-body terms read:

UI (p, kf )(ct) = (2Bn,3 − B3)2k3

f

M2+ (2Bn,5 − B5)

5k3f

3M4

(k2f + p2), (52)

UI (p, kf )(2F) = k2f

12π3

∞∫2mπ

dµ Im(VC − WC + 2µ2VT − 2µ2WT

)

×{

4kf

µ3

(k2f + p2 − µ2) + µ

pln

µ2 + (kf + p)2

µ2 + (kf − p)2

}. (53)

The contribution of the three-body Hartree diagram in Fig. 2 and the three-body cterm can be written in analytical form:

UI (p, kf )(3H) = 2g4Am6

πu5

9∆(2πfπ)4

{1

xln

1+ (u + x)2

1+ (u − x)2

− 2u

[1+ (u + x)2][1+ (u − x)2] − 2u(1+ 3ζ )

}, (54)

and the total contribution of both three-body Fock diagrams in Fig. 2 can be represas:

UI (p, kf )(3F) = g4Am6

πux−2

18∆(4πfπ)4

{2GS(x,u)

∂GS(x,u)

∂u+ GT (x,u)

∂GT (x,u)

∂u

+ 2GS(u,u)∂GS(u, x)

∂x+ GT (u,u)

∂GT (u, x)

∂x

−u∫

0

dξ

[2∂GS(ξ,u)

∂u

∂GS(ξ, x)

∂x+ 7

∂GT (ξ,u)

∂u

∂GT (ξ, x)

∂x

]},

(55)

wherex = p/mπ andu = kf /mπ . The auxiliary functionsGS,T (x,u) have been definein Eqs. (7), (8). The (real) isovector single-particle potentialUI (p, kf ) (restricted to the region below the Fermi surfacep � kf ) is completed by adding to the terms in Eqs. (52)–(the contributions from 1π -exchange and iterated 1π -exchange written down in Eqs. (28(36) of Ref. [58]. The imaginary isovector single-particleWI(p, kf ) (below the Fermi
surfacep � kf ) has been discussed in Section 4.2 of Ref. [58].


etricpoten-

inetic)

the. (56)

vesty.le-

tential

Fig. 12. The real isovector single-particle potentialUI (p, kf ) as a function of the nucleon densityρ = 2k3f

/3π2.The dashed and full curves correspond to the sectionsp = 0 (bottom of the Fermi sea) andp = kf (at the Fermisurface), respectively.

The generalization of the Hugenholtz–Van Hove theorem [26] to isospin-asymmnuclear–matter gives a model-independent relation for the isovector single-particletial UI (p, kf ) at the Fermi surface (p = kf ):

UI (kf , kf ) = 2A(kf ) − k2f

3M+ k4

f

6M3− kf

3

∂U(p, kf )

∂p

∣∣∣∣p=kf

. (56)

The second and third term on the right hand side just subtract non-interacting (kcontributions from the asymmetry energyA(kf ). We find at saturation densitykf 0 =261.6 MeV an isovector single-particle potential ofUI (kf 0, kf 0) = 40.4 MeV. This is con-sistent with the valueU1 � 40 MeV [5] deduced from nucleon–nucleus scattering inframework of the optical model. The generalized Hugenholtz–Van Hove theorem Eqserves also as an excellent check on our analytical and numerical calculations.

Fig. 12 shows the real isovector single-particle potentialUI (p, kf ) as a function of thenucleon densityρ = 2k3

f /3π2. The dashed and full curves correspond to the sectionsp = 0(bottom of the Fermi sea) andp = kf (at the Fermi surface), respectively. One obsera splitting of both curves which sets in atρ � 0.10 fm−3 and increases with the densiAt saturation densityρ0 = 0.157 fm−3 the difference between the (real) isovector singparticle potential atp = kf 0 andp = 0 is UI (kf 0, kf 0) − UI (0, kf 0) = 7.1 MeV. This ismuch less than the analogous difference for the (real) isoscalar single-particle po
U(kf 0, kf 0) − U(0, kf 0) = 26.4 MeV (see also Fig. 4).


ker

orem

by in-

ntumThese

readys has

etricg-body

Fig. 13. The full line shows the real isovector single-particle potentialUI (p, kf 0) at saturation densitykf 0 = 261.6 MeV as a function of the nucleon momentump.

The full line in Fig. 13 shows the (real) isovector single-particle potentialUI (p, kf 0)

at saturation densitykf 0 = 261.6 MeV as a function of the nucleon momentump. In theregion below the Fermi surfacep � kf 0 the momentum dependence of this curve is weathan that of the (real) isoscalar single-particle potentialU(p, kf 0) shown by the left half ofthe full line in Fig. 4. Finally, we note that the generalized Hugenholtz–Van Hove theEq. (56) (which allows for an alternative determination ofUI (kf , kf ) shown in Fig. 12)holds with high numerical precision in our calculation.

10. Summary and concluding remarks

In this work we have extended a recent three-loop calculation of nuclear mattercluding the effects from two-pion exchange with single and double virtual∆(1232)-isobarexcitation. We have encoded all short-distance contributions (from the high-momeparts of the pion-loops integrals etc.) in a few adjustable contact-coupling constants.constants are different from those of the chiral NN-potential since they include aliterations to all orders in the nuclear medium. A wide variety of nuclear propertiebeen investigated in this framework. The empirical saturation point of isospin-symmnuclear matter,E0 = −16 MeV, ρ0 = 0.157 fm−3, can be well reproduced by adjustinthe strength of a two-body term linear in density and weakening an emerging three
term quadratic in density. The various constraints set by empirical nuclear matter properties


real

on getsgetti-

lns and

erepul-y and

in theer ex-

e pion

-scaleses

smalloatura-

auli-ignoregy dy-

re-hange

ion ofpat-

ential

ny

(saturation point and potential depth) lead to the (minimal) parameter choiceB5 = 0 andζ = −3/4, with little freedom for further variation. The momentum dependence of thesingle-particle potentialU(p, kf ) is improved significantly by including the chiralπN∆-dynamics. As a consequence the critical temperature of the liquid-gas phase transitilowered to the realistic valueTc � 15 MeV. The isospin properties of nuclear matteralso substantially improved by including the chiralπN∆-dynamics. The energy per parcle of pure neutron matterEn(kn) and the asymmetry energyA(kf ) now show a monotonicgrowth with density. In the density regimeρ = 2ρn < 0.2 fm−3 relevant for conventionanuclear physics, we find good agreement with sophisticated many-body calculatio(semi-)empirical values.

In passing we note that the inclusion of the chiralπN∆-dynamics guarantees thspin-stability of nuclear matter [59]. These improvements can be traced back tosive two-body Fock terms as well as three-body terms with a very specific densitmomentum dependence. Open questions concerning the role of yet higher orderssmall momentum expansion (and its “convergence”) remain and should be furthplored.

Our calculation takes seriously the fact that there exist two hadronic scales, thmassmπ = 135 MeV and the delta-nucleon mass splitting∆ = 293 MeV, which aresmaller than or comparable to the Fermi momentumkf 0 � 262 MeV of equilibrated nuclear matter. Propagation effects of quasi-particles associated with these “light”are resolvable. Therefore pions and∆-isobars must be included as explicit degreof freedom in the nuclear many-body problem. Controlled by an expansion inscales (kf ,mπ,∆), the dynamics of the interactingπN∆-system is worked out up tthree-loop order. In this approach the basic mechanism for nuclear binding and stion are attractive 2π -exchange interactions of the van der Waals type on which Pblocking acts in the nuclear medium. Most other phenomenological approachesthese “light” physical degrees of freedom and parameterize the relevant low-enernamics in terms of strongly coupled heavy scalar and vector bosons (σ,ω,ρ, δ, etc.). Theirpropagation takes place on length scales of 0.5 fm or less and can therefore not besolved in the domain relevant to nuclear physics. We are guided instead by a cof paradigm, namely that the nuclear many-body problem involves the separatscales that is characteristic for low-energy QCD and its (chiral) symmetry breakingtern.

Appendix A. Real single-particle potential above the Fermi surface

In this appendix we summarize the continuation of the real single-particle potU(p, kf ) calculated in Section 3 of Ref. [3] into the region above the Fermi surfacep > kf .The expressions Eqs. (8), (9) in Ref. [3] for the two-body potentials from the 1π -exchangeFock diagram and the iterated 1π -exchange Hartree diagram remain valid without achanges. Eq. (10) in Ref. [3] for the two-body potential from the iterated 1π -exchange
Fock diagram gets replaced by:


y

U2(p, kf ) = g4AMm4

π

(4π)3f 4π

{u3 + 3

4x

(u+x)/2∫(x−u)/2

dξu2 − (2ξ − x)2

1+ 2ξ2

× [(1+ 8ξ2 + 8ξ4)arctanξ − (

1+ 4ξ2)arctan2ξ]}

, (A.1)

with the abbreviationsu = kf /mπ andx = p/mπ . Eq. (11) in Ref. [3] for the three-bodpotential from the iterated 1π -exchange Hartree diagram gets modified to:

U3(p, kf )

= 6g4AMm4

π

(4πfπ)4

{ 1∫ymin

dy

{[2uxy + (

u2 − x2y2) lnu + xy

|u − xy|]

×Ay

[2s2 + s4

2(1+ s2)− ln

(1+ s2)]

+s−xy∫

xy−s

dξ

[2uξ + (

u2 − ξ2) lnu + ξ

u − ξ

](xy + ξ)5

[1+ (xy + ξ)2]2}

+1∫

−1

dy

u∫0

dξξ2

x

[2σ 2 + σ 4

1+ σ 2− 2 ln

(1+ σ 2)] ln

x + ξy

x − ξy

}. (A.2)

Finally, Eq. (13) in Ref. [3] for the three-body potential from the iterated 1π -exchangeFock diagram is replaced by:

U3(p, kf )

= 3g4AMm4

π

(4πfπ)4

{G2(x)

8x2+

u∫0

dξ G(ξ)

[1+ ξ2 − x2 − 1

4xξln

1+ (x + ξ)2

1+ (x − ξ)2

]

+1∫

ymin

dy

1∫ymin

dzθ(y2 + z2 − 1)

4√

y2 + z2 − 1Ay

[s2 − ln

(1+ s2)]Az

[ln

(1+ t2) − t2]

+1∫

−1

dy

u∫0

dξξ2

x

[ln

(1+ σ 2) − σ 2]

×(

lnx + ξy

x − ξy+ 1

Rln

xR + (x2 − ξ2 − 1)yξ

xR + (1− x2 + ξ2)yξ

)}, (A.3)

where we have again introduced the auxiliary function:

( 2 2) 1 [ 2][ 2] 1+ (u + x)2

G(x) = u 1+ u + x −4x

1+ (u + x) 1+ (u − x) ln1+ (u − x)2

. (A.4)


p-

2.

in.ferences

ferences

For the definition of the quantitiesymin, s, σ , t andR and the antisymmetrization prescrition Ay we refer to Section 4.

References

[1] M. Lutz, B. Friman, Ch. Appel, Phys. Lett. B 474 (2000) 7.[2] N. Kaiser, S. Fritsch, W. Weise, Nucl. Phys. A 697 (2002) 255.[3] N. Kaiser, S. Fritsch, W. Weise, Nucl. Phys. A 700 (2002) 343.[4] A. Bohr, B.R. Mottelson, Nuclear Structure, vol. I, Benjamin, New York, 1969, Chapter 2.4.[5] P.E. Hodgson, Growth Points in Nuclear Physics, vol. 3, Pergamon, Elmsford, NY, 1981, Chapter 2.[6] S. Fritsch, N. Kaiser, W. Weise, Phys. Lett. B 545 (2002) 73.[7] S. Fritsch, N. Kaiser, Eur. Phys. J. A 17 (2003) 11.[8] S. Fritsch, N. Kaiser, Eur. Phys. J. A 21 (2004) 117.[9] J. Kuckei, F. Montani, H. Müther, A. Sedrakian, Nucl. Phys. A 723 (2003) 32.

[10] C. Ordonez, L. Ray, U. van Kolck, Phys. Rev. C 53 (1996) 2086.[11] D.R. Entem, R. Machleidt, Phys. Rev. C 68 (2003) 041003, and references therein.[12] E. Epelbaum, W. Glöckle, U.-G. Meißner, nucl-th/0405048, and references therein.[13] N. Kaiser, R. Brockmann, W. Weise, Nucl. Phys. A 625 (1997) 758.[14] M.C.M. Reentmeester, R.G.E. Timmermans, J.L. Friar, J.J. de Swart, Phys. Rev. Lett. 82 (1999) 499[15] M.R. Robilotta, C.A. da Rocha, Nucl. Phys. A 615 (1997) 391.[16] D.B. Kaplan, M.J. Savage, M.B. Wise, Phys. Lett. B 424 (1998) 390.[17] S. Fleming, T. Mehen, I.W. Stewart, Nucl. Phys. A 677 (2000) 313.[18] N. Kaiser, S. Gerstendörfer, W. Weise, Nucl. Phys. A 637 (1998) 395.[19] M. Ericson, A. Figureau, J. Phys. G 7 (1981) 1197.[20] A. Akmal, V.R. Pandharipande, D.G. Ravenhall, Phys. Rev. C 58 (1998) 1804, and references there[21] S.C. Pieper, V.R. Pandharipande, R.B. Wiringa, J. Carlson, Phys. Rev. C 64 (2001) 014001, and re

therein.[22] P. Finelli, N. Kaiser, D. Vretenar, W. Weise, Nucl. Phys. A 735 (2004) 449.[23] M.M. Pavan, et al., Phys. Scr., T 87 (2000) 65.[24] I. Sick, private communication;

See also: D.B. Day et al., Phys. Rev. C 40 (1989) 1011.[25] D. Vretenar, G.A. Lalazissis, R. Behnsch, W. Pöschl, P. Ring, Nucl. Phys. A 621 (1997) 853.[26] N.M. Hugenholtz, L. Van Hove, Physica 24 (1958) 363.[27] C. Mahaux, R. Sartor, Adv. Nucl. Phys. 22 (1991) 1, and references therein.[28] G.Q. Li, R. Machleidt, R. Brockmann, Phys. Rev. C 45 (1992) 2782.[29] P. Grange, J.P. Cugnon, A. Lejeune, Nucl. Phys. A 473 (1987) 365.[30] P. Bozek, P. Czerski, Eur. Phys. J. A 11 (2001) 271, and references therein.[31] S.K. Bogner, T.T.S. Kuo, A. Schwenk, Phys. Rep. 386 (2003) 1, and references therein.[32] A. Schwenk, G.E. Brown, B. Friman, Nucl. Phys. A 703 (2002) 745.[33] H. Feldmeier, T. Neff, R. Roth, J. Schnack, Nucl. Phys. A 632 (1998) 61.[34] V.M. Galitskii, Sov. Phys. JEPT 7 (1958) 104.[35] J.M. Luttinger, Phys. Rev. 121 (1961) 942.[36] R.W. Hasse, P. Schuck, Nucl. Phys. A 445 (1985) 205.[37] M. Bender, P.-H. Heenen, P.-G. Reinhard, Rev. Mod. Phys. 75 (2003) 121, and references therein.[38] M. Beiner, H. Flocard, N. Van Giai, P. Quentin, Nucl. Phys. A 238 (1975) 29.[39] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Schaeffer, Nucl. Phys. A 635 (1998) 231, and re

therein.[40] F. Tondour, S. Goriely, J.M. Pearson, M. Onsi, Phys. Rev. C 62 (2000) 024308.[41] N. Kaiser, S. Fritsch, W. Weise, Nucl. Phys. A 724 (2003) 47.[42] N. Kaiser, Phys. Rev. C 68 (2003) 054001.[43] J.W. Negele, D. Vautherin, Phys. Rev. C 5 (1972) 1472.
[44] M. Brack, C. Guet, H.B. Haakansson, Phys. Rep. 123 (1985) 275.


0.

[45] J.P. Blaizot, Phys. Rep. 64 (1980) 171.[46] J. Dobaczewski, work in progress and private communications.[47] J.B. Natowitz, et al., Phys. Rev. Lett. 89 (2002) 212701.[48] V.A. Karnaukhov, et al., Phys. Rev. C 67 (2003) 011601.[49] B. Friedman, V.R. Pandharipande, Nucl. Phys. A 361 (1981) 502.[50] G. Sauer, H. Chandra, U. Mosel, Nucl. Phys. A 264 (1976) 221.[51] J.I. Kapusta, Finite-Temperature Field Theory, Cambridge Univ. Press, Cambridge, 1989, Chapter 1[52] J. Carlson, J. Morales, V.R. Pandharipande, D.G. Ravenhall, Phys. Rev. C 68 (2003) 025802.[53] C. Ordonez, U. van Kolck, Phys. Lett. B 291 (1992) 459.[54] D. Vretenar, T. Niksic, P. Ring, Phys. Rev. C 68 (2003) 024310.[55] P.A. Seeger, W.M. Howard, Nucl. Phys. A 238 (1975) 491.[56] R.J. Furnstahl, Nucl. Phys. A 706 (2002) 85.[57] T. Niksic, D. Vretenar, P. Finelli, P. Ring, Phys. Rev. C 66 (2002) 024306.[58] N. Kaiser, Nucl. Phys. A 709 (2002) 251.
[59] N. Kaiser, Phys. Rev. C 70 (2004) 054001.

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