Quasiparticle interaction in nuclear matter with chiral three-nucleon forces1footnote 11footnote 1Work supported in part by BMBF, GSI and by the DFG cluster of excellence: Origin and Structure of the Universe.

# Quasiparticle interaction in nuclear matter with chiral three-nucleon forces111Work supported in part by BMBF, GSI and by the DFG cluster of excellence: Origin and Structure of the Universe.

J. W. Holt, N. Kaiser, and W. Weise Physik Department, Technische Universität München, D-85747 Garching, Germany
###### Abstract

We derive the effective interaction between two quasiparticles in symmetric nuclear matter resulting from the leading-order chiral three-nucleon force. We restrict our study to the Landau parameters of the central quasiparticle interaction computed to first order. We find that the three-nucleon force provides substantial repulsion in the isotropic spin- and isospin-independent component of the interaction. This repulsion acts to stabilize nuclear matter against isoscalar density oscillations, a feature which is absent in calculations employing low-momentum two-nucleon interactions only. We find a rather large uncertainty for the nuclear compression modulus due to a sensitive dependence on the low-energy constant . The effective nucleon mass on the Fermi surface, as well as the nuclear symmetry energy , receive only small corrections from the leading-order chiral three-body force. Both the anomalous orbital -factor and the Landau-Migdal parameter (characterizing the spin-isospin response of nuclear matter) decrease with the addition of three-nucleon correlations. In fact, remains significantly smaller than its value extracted from experimental data, whereas still compares well with empirical values. The inclusion of the three-nucleon force results in relatively small -wave () components of the central quasiparticle interaction, thus suggesting an effective interaction of short range.

## I Introduction

In a recent publication holt11 () we have studied the quasiparticle interaction in isospin-symmetric nuclear matter employing realistic chiral nucleon-nucleon (NN) interactions. The second-order calculation performed in ref. holt11 () explored the order-by-order convergence of the perturbative expansion for the quasiparticle interaction as well as the scale dependence resulting from the renormalization group evolution bogner03 (); bogner10 () of the underlying two-body interaction. Although good agreement was found for a number of nuclear observables, such as the effective mass at the Fermi surface, the nuclear symmetry energy and the spin-isospin response encoded in the parameter , both the compression modulus of isospin-symmetric nuclear matter and the anomalous orbital -factor were found to differ appreciably from empirical values extrapolated from collective excitations of finite nuclei. Most seriously, the compression modulus of nuclear matter (encoded in the Fermi liquid parameter ) was found to be negative at both first- and second-order across a wide range of cutoff scales.

Previous studies sjoberg73a (); sjoberg73b (); dickhoff83 (); backman85 (); holt07 () have suggested that interactions induced by the polarization of the medium (first studied by Babu and Brown for the case of liquid He babu73 ()) provide sufficient repulsion to stabilize nuclear matter against isoscalar density fluctuations. In fact, already the leading-order contribution to the Babu-Brown induced interaction is quite repulsive holt11 (), though not enough to achieve stability at normal nuclear matter saturation density when employing chiral or low-momentum NN interactions. For such potentials, numerous calculations fritsch05 (); bogner05 (); siu09 (); hebeler11 () of the equation of state have revealed the necessity of three-nucleon forces in driving saturation toward the empirical density  fm and energy per particle   MeV.

To improve the microscopic description of the quasiparticle interaction, we compute in the present work the first-order (perturbative) contribution to the Landau parameters resulting from the leading-order chiral three-nucleon force. Previous work kaiser06 () performed within the framework of chiral effective field theory included explicit -isobar degrees of freedom in a calculation of the isotropic central Fermi liquid parameters. In the present study we employ the high-precision Idaho NLO chiral NN interaction entem03 () together with the NLO chiral three-nucleon force epelbaum06 (), which subsumes certain processes involving virtual -isobar excitations considered in ref. kaiser06 (). The additional repulsion provided by the leading-order three-nucleon force is expected to play an important role for the Landau parameter , but the extent to which other nuclear observables respond to additional three-nucleon correlations has not been studied systematically in calculations employing high-precision NN potentials.

The present paper is organized as follows. In Section II we briefly review Landau’s theory of normal Fermi liquids and discuss the connection between physical observables and various Fermi liquid parameters. We then derive analytical expressions for the first-order contribution to the central Landau parameters arising from the leading-order chiral three-nucleon interaction. In section III we present numerical results for the density and scale dependence of the quasiparticle interaction supplemented by the second-order contributions of ref. holt11 (). The latter were derived from the Idaho NLO chiral two-nucleon interaction as well as from the (universal) low-momentum NN interaction . We end with a summary and an outlook.

## Ii Nuclear quasiparticle interaction

### ii.1 Landau parameters and nuclear observables

Fermi liquid theory was introduced by Landau in the 1950’s landau57 () to describe the properties of strongly interacting normal many-fermion systems at low temperatures. The low-energy excitations about the ground state are long-lived quasiparticles that retain certain features of non-interacting (independent) particles but have modified dynamical properties such an effective mass and an effective magnetic moment. Fermi liquid theory has been used to describe a wide variety of quantum many-body systems, including various types of conductors at low temperatures, liquid He, nuclear matter, neutron matter and also finite nuclei migdal67 (); baym91 (); nozieres99 ().

The quasiparticle interaction encodes bulk equilibrium and transport properties of a Fermi liquid. It arises at second order in the expansion of the energy density in terms of powers of the quasiparticle distribution function :

 δE=∑→p,σ,τϵ→pδn→pστ+12∑→p1σ1τ1→p2σ2τ2F(→p1σ1τ1;→p2σ2τ2)δn→p1σ1τ1δn→p2σ2τ2+⋯, (1)

where is the single-particle energy and higher-order quasiparticle correlations are neglected. The central part of the quasiparticle interaction in spin- and isospin-saturated nuclear matter has the form

 F(→p1,→p2)=f(→p1,→p2)+f′(→p1,→p2)→τ1⋅→τ2+[g(→p1,→p2)+g′(→p1,→p2)→τ1⋅→τ2]→σ1⋅→σ2, (2)

where and denote the spin and isospin operators of the two nucleons on the Fermi surface . More generally, the quasiparticle interaction can include (non-central) tensor components which modify the stability conditions of nuclear matter backman79 () and are necessary for understanding the magnetic susceptibility haensel82 () and the response of nuclear matter to weak probes olsson04 (); bacca09 (). For two quasiparticles on the Fermi surface, the remaining angular dependence of their interaction can be expanded in Legendre polynomials of :

 X(→p1,→p2)=∞∑L=0XL(kF)PL(cos % θ), (3)

where represents or , and the angle is related to the relative momentum through the relation . The coefficients of the expansion in eq.(3) are referred to as the Fermi liquid parameters (FLPs). It is conventional to introduce dimensionless FLPs by multiplying by the density of states at the Fermi surface, , where is the nucleon effective mass and is the Fermi momentum, leading to

 F(→p1,→p2)=1N0∞∑L=0[FL+F′L→τ1⋅→τ2+(GL+G′L→τ1⋅→τ2)→σ1⋅→σ2]PL(cosθ). (4)

For short-range interactions the expansion in (4) is typically rapidly converging such that only a few constants characterize the dynamics of low-energy excitations. Moreover, individual parameters evaluated at the equilibrium Fermi momentum  fm are related to properties of the quasiparticles and the bulk nuclear medium:

 Quasiparticleeffectivemass:M∗MN = 1+F1/3, = 1+τ32+F′1−F16(1+F1/3)τ3, Compressionmodulus:K = 3k2FM∗(1+F0), Isospinasymmetryenergy:β = k2F6M∗(1+F′0), Spin-isospinresponse:g′NN = 4M2Ng2πNN0G′0, (5)

where the anomalous orbital -factor is given by

 δgl=F′1−F16(1+F1/3) (6)

and is the strong coupling constant. Spin observables, though largely unconstrained experimentally, receive significant contributions from the (non-central) tensor Fermi liquid parameters haensel82 ().

Recently, we have carried out systematic calculations holt11 () of the quasiparticle interaction in nuclear matter to second order in many-body perturbation theory employing chiral and low-momentum NN interactions. The first- and second-order contributions, shown diagrammatically in Fig. 1, have the form

 F(1)(→p1s1t1;→p2s2t2)=⟨→p1s1t1;→p2s2t2|¯V|→p1s1t1;→p2s2t2⟩≡⟨12|¯V|12⟩ (7)

and

 F(2)(→p1s1t1;→p2s2t2) = 12∑34|⟨12|¯V|34⟩|2(1−n3)(1−n4)ϵ1+ϵ2−ϵ3−ϵ4 (8) + 12∑34|⟨12|¯V|34⟩|2n3n4ϵ3+ϵ4−ϵ1−ϵ2−2∑34|⟨13|¯V|24⟩|2n3(1−n4)ϵ1+ϵ3−ϵ2−ϵ4,

where the quantity denotes the antisymmetrized two-body potential and is the usual zero-temperature Fermi distribution. These calculations revealed the importance of second-order diagrams in raising the quasiparticle effective mass from to (both lying in the phenomenological range jeukenne76 (); zuo99 ()) as well as increasing the symmetry energy from MeV to MeV (where only the second-order result lies within the range of empirical values   MeV danielewicz (); steiner ()). The isotropic spin-isospin Landau parameter increases at second order, with the effect that changes from a value of around to about , the latter being within the range favored by fits to giant Gamow-Teller resonances in heavy nuclei gaarde83 (); ericson88 (); suzuki99 (). Despite these encouraging results, the Fermi liquid parameter remained well below , giving rise to a negative compression modulus and a corresponding instability of nuclear matter against density fluctuations in the vicinity of saturation density  fm. Additionally, the anomalous orbital -factor decreased from to , which is significantly less than the empirical value extracted from giant dipole resonances nolte (). This feature followed almost entirely from the dramatic increase in the effective mass at second order. An improved microscopic description of the quasiparticle interaction may require the consistent implementation of chiral three-nucleon forces. As a first step in this program, we compute here the first-order contribution to the quasiparticle interaction from the NLO chiral three-body force.

According to eq.(1) the quasiparticle interaction is obtained by functionally differentiating the energy density twice with respect to the quasiparticle distribution functions. For a general three-nucleon force, the Hartree-Fock contribution to the energy density is given by

 E(1)3N=16trσ1τ1trσ2τ2trσ3τ3∫d3k1(2π)3d3k2(2π)3d3k3(2π)3n→k1n→k2n→k3⟨123|¯V3N|123⟩, (9)

where denotes the fully antisymmetrized three-nucleon interaction and . Functionally differentiating twice with respect to the two quasiparticle distribution functions then leaves an effective two-body interaction containing a single (loop) integral over the filled Fermi sea of nucleons.

The three-nucleon force employed in the present work is the NLO chiral three-nucleon interaction epelbaum06 (), which consists of three components. First, there is a two-pion exchange component

 V(2π)3N=∑i≠j≠kg2A8f4π→σi⋅→qi→σj⋅→qj(→qi2+m2π)(→qj2+m2π)Fαβijkταiτβj, (10)

where , MeV, MeV (average pion mass) and denotes difference between the final and initial momenta of nucleon . The quantity

 Fαβijk=δαβ(−4c1m2π+2c3→qi⋅→qj)+c4ϵαβγτγk→σk⋅(→qi×→qj) (11)

involves three terms proportional to the low-energy constants and , respectively. The summation runs over the six permutations of three nucleons. The one-pion exchange component of the three-nucleon interaction is proportional to the low-energy constant :

 V(1π)3N=−∑i≠j≠kgAcD8f4πΛχ→σj⋅→qj→qj2+m2π→σi⋅→qj→τi⋅→τj, (12)

and finally the three-nucleon contact interaction introduces the low-energy constant :

 V(ct)3N=∑i≠j≠kcE2f4πΛχ→τi⋅→τj, (13)

where MeV sets a natural scale.

The low-energy constants of appear already in the two-pion exchange contribution to the nucleon-nucleon interaction and can therefore be fit to NN scattering phase shifts. The analysis of the Nijmegen group rentmeester03 () resulted in the values GeV, GeV, and GeV, while Entem and Machleidt entem03 () obtain GeV, GeV, and GeV. The low-energy constants and of the mid- and short-range chiral three-nucleon interaction can be constrained by properties of nuclear few-body systems. In the present work we employ two versions of and obtained by fitting the binding energies of nuclei together with the half-life of H with the result gazit ():

 cD=−0.20,cE=−0.205, (14)

or by fitting the binding energies of H and He bogner05 ():

 cD=−2.062,cE=−0.625. (15)

In the first set the leading-order three-body force (with coefficients of Entem and Machleidt) was used together with the Idaho NLO chiral two-nucleon interaction, while in the second set the authors employed the Nijmegen low-energy constants together with the low-momentum NN interaction at the resolution scale fm. Employing these two versions of the chiral three-nucleon force (combined with two-nucleon interactions at different scales) provides a means for assessing theoretical errors at this order in the perturbative expansion.

### ii.2 Diagrammatic calculation

We begin by considering the isotropic () Fermi liquid parameters of the central quasiparticle interaction. They are obtained by angle-averaging the (in-medium) effective interaction holt10 ():

 F0(kf) = 1(4π)2∫dΩ1dΩ2⟨→p1,→p2|VmedNN|→p1,→p2⟩ (16) = f0(kf)+g0(kf)→σ1⋅→σ2+f′0(kf)→τ1⋅→τ2+g′0(kf)→σ1⋅→σ2→τ1⋅→τ2,

where denotes an antisymmetrized two-nucleon state. Both quasiparticle momenta and lie on the Fermi surface so that .

There are six topologically distinct one-loop diagrams, shown in Figs. 2 and 3, contributing to the effective interaction . Using the abbreviations and , we find for the crossed term of diagram (1) in Fig. 2 the following contribution

 F0(kf)(med,1)=(3−σ)(3−τ)g2Am3π(6π)2f4π{(2c1−c3)u31+4u2−c3u3+(c3−c1)u2ln(1+4u2)}, (17)

with . In this diagram, the two pions carry equal momenta and therefore the term in eq.(11) proportional to does not contribute. Moreover, the direct term (with zero momentum transfer) of this pion self-energy correction vanishes trivially. Analogously, only the crossed term from the one-pion exchange vertex correction (diagram (2) in Fig. 2) is nonzero with the analytical result:

 F0(kf)(med,2) = (3−σ)(3−τ)g2Am3π(24π)2f4π{3c18u5[4u2−ln(1+4u2)] (18) × [8u4+4u2−(1+4u2)ln(1+4u2)] + c3[2u2(4u2−ln(1+4u2))arctan2u+48u4+16u2+364u7ln2(1+4u2) + 12u4−16u6−30u2−924u5ln(1+4u2)+20u33−11u+1u+34u3] + c4[4u2(ln(1+4u2)−4u2)arctan2u+3+16u2−48u464u7ln2(1+4u2) + 80u6+12u4−30u2−924u5ln(1+4u2)−28u33+13u+1u+34u3]}.

The Pauli-blocked two-pion exchange component (diagram (3) in Fig. 2) has both a nonvanishing direct and crossed term. Their sum takes the form

 F0(kf)(med,3) = g2Am3π(4π)2f4π{24(c3−c1)u−8c3u3+(3c3−4c1)3uln(1+4u2) (19) + 6(6c1−5c3)arctan2u+(3−σ)(3−τ)c49∫u0dx(Y2−X2) + (1+σ)(1+τ)∫u0dx[3c1Z2+c32(X2+2Y2)]},

where the auxiliary functions and arising from Fermi sphere integrals over a pion propagator read

 X(u,x) = 2x−12uln1+(u+x)21+(u−x)2, (20) Y(u,x) = 5x2−3u2−34x+4x2+3(1+u2−x2)216ux2ln1+(u+x)21+(u−x)2, (21) Z(u,x) = 1+x2−u2−14uxln1+(u+x)21+(u−x)2. (22)

Note that the direct term contributes only to the spin- and isospin-independent Landau parameter .

There are two diagrammatic contributions from the mid-range one-pion exchange chiral three-nucleon force, labeled as (4) and (5) in Fig. 3. The crossed term from the -exchange vertex correction (diagram (4)) leads to the contribution

 F0(kf)(med,4)=(3−σ)(3−τ)gAcDm3π(24π)2f4πΛχ[4u3−uln(1+4u2)], (23)

and the sum of direct and crossed terms from diagram (5) yields

 F0(kf)(med,5)=(3−σ−τ−στ)gAcDm3π(4π)2f4πΛχ{2u33−u+arctan2u−14uln(1+4u2)}. (24)

Finally, the three-nucleon contact term generates a contribution proportional to the nuclear density . The sum of direct and crossed terms reads

 F0(kf)(med,6)=(σ+τ+στ−3)cEk3f4π2f4πΛχ. (25)

We note that for all three-body contributions the spin-spin and isospin-isospin components of the quasiparticle interaction are equal, .

The -wave () Fermi liquid parameters are obtained by weighting the angular integrals by the first Legendre polynomial :

 F1(kf) = 3(4π)2∫dΩ1dΩ2(^p1⋅^p2)⟨→p1,→p2|VmedNN|→p1,→p2⟩ (26) = f1(kf)+g1(kf)→σ1⋅→σ2+f′1(kf)→τ1⋅→τ2+g′1(kf)→σ1⋅→σ2→τ1⋅→τ2.

After this weighting, the two short-range contributions (diagrams (5) and (6) in Fig. 3) to the quasiparticle interaction vanish. We provide the expressions for the four remaining pieces below. The crossed term from the pion self-energy correction takes the form

 F1(kf)(med,1) = (3−σ)(3−τ)g2Am3π48π2f4π{(2c1−c3)u1+4u2+(6c1−5c3)u (27) + [2(c3−c1)u+3c3−4c12u]ln(1+4u2)}.

The crossed term from the one-pion exchange vertex correction reads

 F1(kf)(med,2) = (3−σ)(3−τ)g2Am3π(16π)2f4π{c14u7[4u2−(1+2u2)ln(1+4u2)] (28) × [8u4+4u2−(1+4u2)ln(1+4u2)] + c33u4[4(4u2−(1+2u2)ln(1+4u2))arctan2u + 96u6+80u4+22u2+332u5ln2(1+4u2) + 56u6−32u8−60u4−48u2−912u3ln(1+4u2) + 4u53(7−4u2)−14u3+5u+32u] + c43u4[8((1+2u2)ln(1+4u2)−4u2)arctan2u + 3+22u2−16u4−96u632u5ln2(1+4u2) + 160u8−136u6−60u4−48u2−912u3ln(1+4u2) −

The crossed term from Pauli-blocked two-pion exchange is given by

 F1(kf)(med,3) = g2Am3π(4π)2f4π∫u0dx{(3−σ)(3−τ)c49(3X2b+2X2c−3X2a) (29) + (1+σ)(1+τ)[3c1(Z2a+2Z2b)+c32(3X2a+6X2b+4X2c)]},

with auxiliary functions

 Za(u,x) = xu+u2−x2−14u2ln1+(u+x)21+(u−x)2, (30) Zb(u,x) = x2−3u2−34ux+4(1+u2+u4)−(u2+x2−1)216u2x2ln1+(u+x)21+(u−x)2, (31) Xa(u,x) = 1u−u2+x2+14u2xln1+(u+x)21+(u−x)2, (32) Xb(u,x) = 18ux2[2x2(1+u2)−3x4−3(1+u2)2] (33) + 1+u2+x232u2x3[3(1+u2−x2)2+4x2]ln1+(u+x)21+(u−x)2, Xc(u,x) = 18ux2[3(1+u2)2−8u2x2−3x4] (34) + 332u2x3(x2−1−u2)[(1+u2+x2)2−4u2x2]ln1+(u+x)21+(u−x)2.

Finally, the only nonvanishing term from the mid-range three-nucleon force is the crossed term from diagram (4) in Fig. 3, which reads

 (35)

Again, one observes that for all contributions, .

This completes the first-order calculation of the Landau parameters arising from the NLO chiral three-nucleon force. A good check of the formulas in eqs.(17)–(35) is given by their Taylor expansions in . The leading terms are of the form: . Note that full consistency with the second-order calculation in ref. holt11 () would require the inclusion of the subleading NLO chiral three-body force (recently derived in ref. bernard08 ()). Investigations along these lines are in progress.

## Iii Results

In this section we study the density-dependence of the Landau parameters derived in the previous section from the leading-order chiral three-nucleon interaction. The values of the five low-energy constants occuring at this order have significant uncertainties, and we employ two different sets which have been fit to reproduce properties of nuclei. These contributions are then combined with the results of ref. holt11 () for the second-order quasiparticle interaction computed with the corresponding two-nucleon interactions. The impact on various nuclear observables is discussed.

In Fig. 4 we plot the Fermi liquid parameters (in units of fm) as a function of the nuclear density (normalized to that of saturated nuclear matter fm) employing the set of low-energy constants

 (36)

which have been used with the Idaho NLO chiral two-nucleon interaction to reproduce the binding energies of nuclei and the triton half-life. We observe that for densities greater than , both the and Fermi liquid parameters depend approximately linearly on the density, which is not immediately obvious from the analytical expressions given in the previous section. The largest effect on the quasiparticle interaction is a dramatic increase in the isotropic spin- and isospin-independent component, , which at nuclear matter saturation density is about four times larger than any of the other contributions to the parameters. In fact, both and remain negligibly small for all densities considered here, while the parameter decreases monotonically with the density and reaches the value  fm at nuclear matter saturation density . The Fermi liquid parameters all decrease with increasing density. The relatively small change in provides only a small downward correction to the quasiparticle effective mass . This observation, combined with the small change in discussed above, indicate that three nucleon forces have a relatively small effect on the nuclear symmetry energy . The significant reduction in the parameter decreases the nucleon anomalous orbital -factor considerably below its empirical value.

In Table 1 we show the various contributions to the Fermi liquid parameters arising from the six topologically-distinct diagrams of Figs. 2 and 3. These are labeled as for . Contributions from the long-range two-pion exchange component of the three-body force are significantly larger than those from the mid- and short-range three-body forces. However, since both and are medium modifications to one-pion exchange but enter with opposite sign, to a large extent they cancel in all Fermi liquid parameters. This leaves the Pauli-blocked two-pion exchange diagram as the dominant contribution, except in the Fermi liquid parameters and where its effects are surprisingly small.

It is instructive to compare the final results to what would be obtained in the chiral limit ():

 (f0)χ=1.287fm2,(g0)χ=(f′0)χ=0.136fm2,(g′0)χ=−0.424fm2 (f1