Nuclear pairing from microscopic forces: singlet channels and higher-partial waves
NNuclear pairing from microscopic forces: singlet channels andhigher-partial waves
Stefano Maurizio
Department of Physics, University of Bologna, I-40126 Bologna, Italy
Jeremy W. Holt
Department of Physics, University of Washington, Seattle, WA 98195-1560
Paolo Finelli
Department of Physics, University of Bologna, I-40126 Bologna, ItalyINFN, Bologna section, I-40127 Bologna, Italy
Abstract
Background:
An accurate description of nuclear pairing gaps is extremely important for un-derstanding static and dynamic properties of the inner crusts of neutron stars and to explain theircooling process.
Purpose:
We plan to study the behavior of the pairing gaps ∆ F as a function of the Fermimomentum k F for neutron and nuclear matter in all relevant angular momentum channels wheresuperfluidity is believed to naturally emerge. The calculations will employ realistic chiral nucleon-nucleon potentials with the inclusion of three-body forces and self-energy effects. Methods:
The superfluid states of neutron and nuclear matter are studied by solving theBCS gap equation for chiral nuclear potentials using the method suggested by Khodel et al. ,where the original gap equation is replaced by a coupled set of equations for the dimensionless gapfunction χ ( p ) defined by ∆( p ) = ∆ F χ ( p ) and a non-linear algebraic equation for the gap magnitude∆ F = ∆( p F ) at the Fermi surface. This method is numerically stable even for small pairing gaps,such as that encountered in the coupled P F partial wave. Results:
We have successfully applied Khodel’s method to singlet ( S ) and coupled channel( SD and P F ) cases in neutron and nuclear matter. Our calculations agree with other ab-initio ap-proaches, where available, and provide crucial inputs for future applications in superfluid systems.
PACS numbers: 21.30.-x; 21.65.-f; 26.60.-c a r X i v : . [ nu c l - t h ] A ug . INTRODUCTION Superfluidity in neutron matter is connected to different aspects of neutron star physics.At the surface of the star [1, 2], where a neutron gas moves in a lattice structure of neutron-rich nuclei and a sea of relativistic electrons, a S neutron pairing gap naturally emerges,while at larger densities a (possibly anisotropic) P F gap plays a more important role (inparticular for neutron star cooling [3, 4]). At the same time, the nuclear matter case couldbe interesting for finite nuclear systems where neutron-proton pairing is relevant [5], even ifthe appearance of pairing in ordinary uniform matter is probably questionable because ofknown instabilities [6] which could hide superfluidity in a broad range of densities.The goal of this article is to solve the BCS equations starting from modern nucleon-nucleon (NN) forces based on chiral effective field theory [7–9]. In this approach one identi-fies the appropriate low-energy degrees of freedom and derives the most general Lagrangiancompatible with the symmetries and symmetry-breaking pattern of the underlying funda-mental theory (i.e., QCD). The first steps towards a realistic NN potential from first prin-ciples started almost twenty years ago within the framework of Chiral Perturbation Theory(ChPT) [10, 11]. In ChPT the nuclear potential emerges naturally as a hierarchy of termscontrolled by a power expansion in Q/ Λ χ , where Q is a soft scale (pion mass, nucleon mo-mentum) and Λ χ is a hard scale (the nucleon mass M N or the chiral symmetry breakingscale 4 πf π ). Two-nucleon forces appear at leading order ( Q/ Λ χ ) , while three-nucleon forcesappear first at order ( Q/ Λ χ ) , or next-to-next-to-leading order (N2LO).We employ primarily the high-precision NN potential developed in Ref. [7] at next-to-next-to-next-to-leading order (N3LO) in the chiral expansion, but to asses theoretical un-certainties associated with the choice of cutoff scale and regulating functions, we employin addition the chiral nuclear potentials developed in Ref. [8] in selected cases. To imple-ment the leading three-nucleon force, we include a two-body density-dependent potential[12, 13] (see also Refs. [14–17] for other approaches and relevant details). To improve con-vergence in many-body perturbation theory, it is desirable to employ nuclear interactionswith a cutoff scale below Λ ∼
500 MeV. One approach is to employ renormalization group(RG) methods that decouple the low- and high-momentum components of the potential.Two different methods for evolving nuclear potentials to block- and band-diagonal form in amomentum-space representation, V lowk and V srg respectively, have been developed (see Refs.218–20] for detailed reviews) and used in the present study. An alternative approach wouldbe to construct from the beginning chiral nuclear potentials at lower cutoff scales [21–23].The paper is organized as follows. Section II introduces the BCS theory that is thestandard framework for a microscopic description of nucleonic pairing. In particular, thenumerical implementation first introduced by Khodel et al. [24] will be reviewed. SectionsIII A and III B describe, respectively, our predictions for pairing gaps in the singlet and inthe coupled channel cases. The role of the two-body NN interaction will be discussed alongwith the influence of three-body forces and self-energy effects. Finally, Section IV presentsour conclusions. II. THE BCS EQUATION
In this section we explain the method employed to solve the BCS equations [25] bypartial-wave decomposition [5, 24, 26, 27]. For simplicity we largely neglect spin and isospindegrees of freedom in the derivation. The BCS equation reads in terms of the NN potential V ( k , k (cid:48) ) = (cid:104) k | V | k (cid:48) (cid:105) as follows∆ ( k ) = − (cid:88) k (cid:48) (cid:104) k | V | k (cid:48) (cid:105) ∆ ( k (cid:48) )2 E ( k (cid:48) ) , (1)with E ( k ) = ξ ( k ) + | ∆( k ) | and where ξ ( k ) = ε ( k ) − µ , ε ( k ) denotes the single-particleenergy and µ is the chemical potential. As in [5], we decompose the interaction (cid:104) k | V | k (cid:48) (cid:105) = 4 π (cid:88) l (2 l + 1) P l (ˆ k · ˆ k (cid:48) ) V l ( k, k (cid:48) ) (2)and the gap function ∆( k ) = (cid:88) lm (cid:114) π l + 1 Y lm (ˆ k )∆ lm ( k ) , (3)where Y lm (ˆ k ) denotes the spherical harmonics, l is the orbital angular momentum, m is itsprojection along the z axis and P l (ˆ k · ˆ k (cid:48) ) refers to the Legendre polynomials. After performingan angle-average approximation (we do not retain the m -dependence, anisotropic pairinggaps [27] will be discussed in a forthcoming paper) we have the following equation for anyvalue of l ∆ jl ( k ) = (cid:88) l (cid:48) ( − Λ π (cid:90) dk (cid:48) V jll (cid:48) ( k, k (cid:48) ) ∆ jl (cid:48) ( k (cid:48) ) E ( k (cid:48) ) k (cid:48) , (4)3here Λ = 1 + ( l − l (cid:48) ) / j refers to the total angular momentum ( (cid:126)J = (cid:126)l + (cid:126)S ) quantumnumber including spin (cid:126)S and now E ( k ) = ξ ( k ) + (cid:80) jl ∆ jl ( k ) . Gaps with different l and j are coupled due to the energy denominator but, for the sake of simplicity, we assume thatdifferent components of the interaction mainly act on non-overlapping intervals in density.This assumption will turn out to be correct in the neutron matter case while only partiallyjustified when treating gaps for symmetric nuclear matter. To solve Eq. (4), we follow theapproach suggested by Khodel et al. [24] that has been proven to be stable even for smallvalues of the gap and to require only the initial assumption of a scale factor δ (results, ofcourse, will be δ -independent). We define an auxiliary potential W according to W ll (cid:48) ( k, k (cid:48) ) = V ll (cid:48) ( k, k (cid:48) ) − v ll (cid:48) φ ll (cid:48) ( k ) φ ll (cid:48) ( k (cid:48) ) , (5)where φ ll (cid:48) ( k ) = V ll (cid:48) ( k, k F ) /V ll (cid:48) ( k F , k F ) and v ll (cid:48) = V ll (cid:48) ( k F , k F ) so that W ll (cid:48) ( k, k (cid:48) ) vanishes onthe Fermi surface. The coupled gap equations can be rewritten as∆ l ( k ) − (cid:88) l (cid:48) ( − Λ (cid:90) dτ (cid:48) W ll (cid:48) ( k, k (cid:48) ) ∆ l (cid:48) ( k (cid:48) ) E ( k (cid:48) ) = (cid:88) l (cid:48) D ll (cid:48) φ ll (cid:48) ( k ) , (6)where dτ = k dk/π and the coefficients D ll (cid:48) satisfy D ll (cid:48) = ( − Λ v ll (cid:48) (cid:90) dτ φ ll (cid:48) ( k ) ∆ l (cid:48) ( k ) E ( k ) . (7)The gap is defined as follows ∆ l ( k ) = (cid:88) l l D l l χ l l l ( k ) , (8)where χ l l l ( k ) − (cid:88) l (cid:48) ( − Λ (cid:90) dτ (cid:48) W ll (cid:48) ( k, k (cid:48) ) χ l l l (cid:48) ( k (cid:48) ) E ( k (cid:48) ) = δ ll φ l l ( k ) , (9)and δ ll (cid:48) is the scale factor. The property that W ll (cid:48) ( k, k (cid:48) ) vanishes on the Fermi surfaceensures a very weak dependence of χ l l l ( k ) on the exact value of the gap so that, in firstapproximation, it is possible to rewrite the previous equation (9) as χ l l l ( k ) − (cid:88) l (cid:48) ( − Λ (cid:90) dτ (cid:48) W ll (cid:48) ( k, k (cid:48) ) χ l l l (cid:48) ( k (cid:48) ) (cid:112) ξ ( k (cid:48) ) + δ = δ ll φ l l ( k ) . (10)We use this equation to evaluate χ l l l ( k ) initially by matrix inversion, then we use thisfunction to self-consistently evaluate D ll (cid:48) . Finally, we solve the system given by Eqs. (7)–(9)4n a self-consistent procedure. We always assume µ = ε F and adopt the relativistic versionof the single-particle energy ε ( k ) = (cid:112) k + M N , where M N is the nucleon mass. In principle,the effective force to be included in Eq. (1) should be generated by the sum of all particle-particle irreducible Feynman diagrams [28], but in most applications to nuclear systems onlythe bare nucleon-nucleon interaction is kept [5]. Corrections to the bare force, caused bymedium polarization effects (see Refs. [29–31] and references therein) will be neglected in thepresent analysis and postponed to a forthcoming paper. As a consequence, for the pairingpotential V ( p, k ) we introduce the following ansatz: V ( p, k ) = V B ( p, k ) + V B ( p, k, m ) (cid:39) V B ( p, k ) + V eff B ( k F , p, k ) , (11)where V B is the Idaho [7] NN potential at N3LO in the chiral expansion or the Juelichversion [8], and the three-body potential is approximated by an effective two-body density-dependent potential V eff B derived by Holt et al. in Refs. [12, 13]. We employ in our cal-culations the evolved two-body potentials V lowk (with a smooth cut-off in momentum space[33]) and V srg [19] using two different evolution operators (see Sect. III B for more details).When considering self-energy effects, we simply perform the transformation M N → M ∗ N us-ing the effective mass obtained by Holt et al. in Ref. [34] using a density matrix expansiontechnique. In Ref. [34] the two-body interaction was comprised of long-range one- and two-pion exchange contributions and a set of contact terms contributing up to fourth power inmomenta as well as the leading order chiral three-nucleon interaction. The explicit formulais given by M ∗ ( ρ ) = M (cid:18) M F τ ( ρ ) − k F M (cid:19) − , (12)where the strength function F τ ( ρ ) is defined as follows F τ ( ρ ) = 12 k F (cid:18) ∂U ( p, k F ) ∂p (cid:19) p = k F (13)with U ( p, k F ) the single particle potential and − k F M a relativistic correction. In Fig. 1we plot the effective masses for nuclear and neutron matter as functions of density. Fromthe effective mass behavior we can expect that the self-energy effects will play a centralrole in the high-density components of the gap, while at low densities the effects will berather negligible. Second-order perturbative contributions to the single-particle energies areexpected to increase the effective mass [35]. Among the different versions, we employ the chiral potential in which the regulator function f ( p (cid:48) , p ) =exp[ − ( p (cid:48) / Λ) n − ( p/ Λ) n ] has the cutoff Λ = 500 MeV, and n = 2 for the 2 π exchange contributions [32]. .00 0.05 0.10 0.15 0.20 density ( fm − ) M ∗ / M Nuclear MatterNeutron Matter
FIG. 1. (color online) The effective mass in the case of nuclear (red line) and neutron (blue dashedline) matter as a function, respectively, of the total nucleon density ρ or the neutron density ρ n (see Ref. [34]) . III. PAIRING GAPSA. Singlet channel ( S ) In the singlet channel, the only difference between the nn and np potential is the chargeindependence breaking and the charge symmetry breaking terms, which are treated as smallperturbations in ChPT. In Fig. 2 we test our solution to the gap equation, in the nuclearmatter case, against previously published results [36] with the low-momentum interaction V lowk . In addition, we compute the S gap from the bare chiral NN interaction and find aqualitatively very similar behavior. The gap ∆ F reaches a maximum value of approximately3.5 MeV at k F (cid:39) .
85 fm − when the bare interaction is used in the two-body sector, whilea somewhat reduced gap (by almost 0 . V lowk .In the neutron matter case, at the two-body level, there is good agreement with the gapcomputed from well known realistic potentials like the CD-Bonn or Nijmegen interactions6 .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 k F ( fm − ) ∆ F ( M e V ) N LON LO ( V lowk ) Hebeler [37]
FIG. 2. (color online) The S gap for nuclear matter computed with the realistic chiral potential ofRef. [7, 32] at N3LO (red line) and the corresponding V lowk potential (blue dashed line). With thegreen dashed-dotted line we include, as a benchmark, a similar calculation performed by Hebeler et al. [36]. [5, 37, 38], but at larger densities the N3LO gap exhibits a higher value. This can beexplained by observing that the phase shifts from the chiral N3LO potential exhibit moreattraction than the CD-Bonn potential for high momenta, as already observed by Hebeler et al. [36]. In Fig. 3 we compare our full calculation for the gap, i.e., with the completepotential in Eq. (11) and the density-dependent effective mass in Eq. (12), with recentresults by Hebeler et al. [36], where the authors started from a chiral N3LO interaction andevolved to a sharp low-momentum interaction . Also presented for comparison are ab-initio results obtained in the last several years: Auxiliary Field Diffusion Monte Carlo (AFDMC)[39] with AV8’ [40] + UIX [41] potentials, Quantum Monte Carlo (QMC) [42], where theauthors have retained the S -wave part of the AV18 [43] interaction, and Correlated BasisFunctions (CBF) [44] still with AV8’ plus UIX. We observe that at low densities the gap We used a rather different approach to construct our V lowk . The RG procedure has been performed withdifferent cutoffs and regulating functions, in particular a Fermi-Dirac function f Λ ( k ) = 1 / (1 + e ( k − Λ ) /ε )and an exponential cutoff f Λ ( k ) = e − ( k / Λ ) n [33]. The results show a very weak cutoff-dependence. .0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 k F ( fm − ) ∆ F ( M e V ) N LO +3 B + M ∗ Hebeler [37]
AFDMC [40]
QMC [43]
CBF [45]
FIG. 3. (color online) The S gap for neutron matter computed with the realistic chiral potentialof [7, 32] at N3LO plus the three-body contribution of Eq. (11) and the inclusion of the effectivemass in Eq. (12). As a comparison, we include a similar calculation by Hebeler [36] with a greendashed line and a set of ab-initio simulations with different many-body techniques: AFDMC (bluecircles) [39], QMC Green Functions (green squares) [42] and CBF (yellow diamonds) [44]. See thetext for additional details. behaviors are very similar, but beyond Fermi momenta of k F ≈ . − the gaps computedwith the Argonne potentials decrease rapidly in contrast to those from chiral interactions.At the present time, it is hard to assess if disagreement is due to different choices in thenuclear Hamiltonian or different many-body methods.It is useful to consider separately the different physical effects governing the S pairinggap. In Fig. 4 we plot the gaps obtained with two-body interactions alone (the dotted linesrepresent the bare and the renormalized N3LO potentials), with the inclusion of effectivethree-body forces (dashed lines) and considering also self-energy effects (solid lines). Byconstruction, we expect that at low densities the three-body effects are rather small, whileonly at higher densities do they become appreciable. The main role of both three-body forcesand the effective mass is to substantially reduce the attractive strength in the S channel (for8 .4 0.6 0.8 1.0 1.2 k F ( fm − ) ∆ F ( M e V ) N LON LO ( V lowk ) N LO +3 BN LO ( V lowk ) +3 BN LO +3 B + M ∗ N LO ( V lowk ) +3 B + M ∗ FIG. 4. (color online) The S gap for neutron matter. In this figure we show all the contributionsto the pairing gap ∆ F starting from the inclusion of the bare two-body potential (red dotted line)or V lowk (blue dotted line) and then including effective three-body forces (dashed lines) and adensity-dependent effective mass (solid lines). higher partial waves the situation is more involved, see Sect. III B). B. Higher partial waves ( SD and P F ) In addition to the S channel, in the nuclear matter case a non-vanishing gap appearsin the SD channel. The presence of a bound state in this channel and the very high phaseshifts in the S channel indicate that the interaction is more attractive than in the otherchannels. As a consequence the gap has a magnitude of about 10 MeV, as can be seen inFig. 5, with conventional realistic potentials. There is no agreement on the details of thegap in this channel, but both Elgarøy et al. [45] and Takatsuka et al. [26] suggest thepossibility of a gap of such magnitude (see curves labeled, respectively, by BONN-A andOPEG in Fig. 5). While BONN-A [46] is a complete one-boson exchange potential, OPEG[26] contains only the one-pion exchange tail and a Gaussian repulsive core. The combined9 .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 k F ( fm − ) ∆ F ( M e V ) N LON LO +3 B + M ∗ BONN − A [46] OPEG [27]
FIG. 5. The gap in the SD channel. We plot our calculations with the N3LO interaction (red line)in comparison with results obtained employing BONN-A potential [45] (blue curve) and OPEG[26] (yellow line). All results suggest a very large pairing gap (around 10 MeV), but completecalculations including three-body forces and effective masses (see Eqs. (11) and (12)), shown in thedashed red curve, indicate a substantial reduction and a sizable modification of the gap’s shape. effect of three-body forces and self-energy effects leads not only to a sizable reduction of thegap itself but also to a shift of the gap’s maximum at k F ≈ − and a rapid decrease athigher Fermi momenta.Due to the large densities over which the pairing gap remains finite, it is questionablewhether low-momentum interactions, V lowk , with a block-diagonal momentum-space cutoffon the order of Λ ∼ . − are appropriate. A better approach is provided by the SimilarityRenormalization Group (SRG), where off-diagonal momentum-space matrix elements aresupressed. In this case, we study nuclear Hamiltonians H = T rel + V evolved through theSRG procedure [19], where we define a class of Hamiltonians H s = U s HU † s ≡ T rel + V s (14)10 .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 k F ( fm − ) ∆ F ( M e V ) G s = TG s = PHP + QHQ . . . . FIG. 6. (color online) The evolution of the pairing gap in the SD channel with SRG-evolvedinteractions. We employed two different evolution operators G s : T rel (red band) and P HP + QHQ (blue band). The arrows denote the flow variable λ (related to s through λ ≡ s − / ) which is variedfrom 4.7 fm − down to 1.1 fm − . with a generator η s = dU s ds U † s = − η † s . (15)If we choose η s = [ G s , H s ] the flow equation takes the form dH s ds = [[ G s , H s ] , H s ] . (16)As shown in [19], results obtained from SRG-evolved interactions are very similar to thoseobtained from V lowk if an appropriate G s is chosen. Moreover, the SRG interaction has manysalient features of low-momentum interactions, such as independence of the physical observ-ables from the operator G s , perturbativeness and universality. In the literature it is commonto encounter also the dimensional parameter λ = s − / fm − . A very interesting feature ofthe SRG procedure is that the tensor interaction strength is reduced as s increases, and thismodification to the interaction can strongly modify the SD gap. Since all physical observ-ables should remain unchanged under an SRG transformation, this variation represents an11 .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 k F ( fm − ) ∆ F ( M e V ) N LOCD − BONNAV AV FIG. 7. (color online) The gap in the P F channel obtained from the N3LO [32] (red line)interaction in comparison with several realistic NN potentials taken from Ref. [5]. Chiral potentials,by definition, can be trusted only up to momenta close to the cutoff (beyond the cutoff, the pairinggap is symbolized with a dashed line). uncertainty estimate in the pairing strength. A common choice for G s is T rel , and in thiscase as s increases, V s approaches the diagonal form. We tested one more generator G s = P Λ H s P Λ + Q Λ H s Q Λ , (17)where P Λ and Q Λ are, respectively, the projector and the exclusion operators in the subspace { k < Λ } (see Sect. 3.4 in Ref. [19]). From Eq. (16) it is easy to see that, if H is a twobody Hamiltonian expressed in the second quantization formalism, ( dH s /ds ) s =0 will alsoinclude three body interactions. In this way, the evolution over the flow will naturallyinduce many-body interactions. The errors arising from omitting the induced many-bodyforces can be estimated by analyzing the dependence of the physical observables on the flowparameter λ . Our results are shown in Fig. 6, where we tested the two evolution operators.For G s = T rel (red color) we found that the gap becomes quite stable for λ < . − ,where the maximum is reduced to approximately 5 MeV (a factor of 2 smaller compared12 .5 1.0 1.5 2.0 k F ( fm − ) ∆ F ( M e V ) N LON LO +3 BN LO +3 B + M ∗ BHF [48]
FIG. 8. (color online) The gap in the P F channel with only N3LO potential (dotted red line),with three-body forces (dashed red line) and including also self-energy effects (solid red line). Incomparison we plot the results of recent BHF calculations [47] (blue curve). to the bare potential). In the range 1 . − ≤ λ ≤ . − the variation in the size ofthe gap is on the order of 0.5 MeV or less. When using G s given by Eq. (17) we obtainedvery similar results, confirming the approximate independence of the physical results on thechoice of G s , but with a reduced cutoff-dependence.In the neutron matter case, while at low density the dominant channel is the S partialwave, at higher densities the high-momentum components (which are repulsive) become moreimportant, suppressing the gap, and this happens at k F ≈ . − . At these densities,the only channel which substantially contributes to the neutron matter gap is the coupled P F , where the coupling is due to the tensor interaction. As can be seen in Fig. 7, thereis a significant dependence of the gap on the potential model, though the peak in the gapconsistently occurs between 2 . ≤ k F ≤ . − . At the high densities and associatedmomentum scales relevant for pairing in this channel, realistic NN interactions are not aswell constained by fits to phase shifts, which partially explains the differences in the observedgaps. As explained in [7], in this channel one expects a crucial contribution from the three-13ion-exchange topology at N LO and from the contact term at N LO, which should reducethe attraction in this channel. All reasonable interactions give a gap of magnitude ≈ Q/ Λ χ .In Fig. 8 we plot predictions for the P F gap including three-body forces (dashed red line)and self-energy effects (solid red line) in comparison with a very recent Brueckner-Hartree-Fock calculation by Dong et al. [47], where the authors employed the Bonn B potential [48]and a microscopic three-body force constructed by Li et al. [49]. Our complete calculationnicely agrees with [47], in particular for small momenta, and suggests a sizeable reductionof the gap if many-body forces are taken into account.In Fig. 9 we show also the P F gap we have computed from the Juelich ChPT potentials[8]. Because the P F gap extends towards very large densities (even beyond the reasonablelimits of applicability of a ChPT approach) is very interesting to test the robustness of pre-vious calculations (see Figs. 7 and 8) against a different theoretical approach. In fact, in thelast years Epelbaum et al. developed a new scheme in the construction of a realistic chiralpotential where, instead of a Dimensional Regularisation scheme for chiral-loop integrals, afinite cutoff Λ is kept in the range of 500 −
800 MeV which appears to be physically reason-able and matches well with the cutoff used in the Lippmann-Schwinger (LS) equation. As aconsequence, in our calculations we employed two different cutoffs: Λ LS for the LS equation(with non relativistic kinematics) and Λ π for the spectral-function regulator (SFR) of thetwo-pion exchange potential (varied between 500 and 700 MeV). For Fermi momenta upto nearly k F = 1 . − , the predictions from the different potentials are nearly universaland agree reasonably well with the predictions from the Entem and Machleidt chiral N3LOpotential. However, beyond this density there is a significant scale dependence in the the-oretical predictions, in particular to Λ π . This uncertainty has to be taken into account ifmicroscopic P F gaps are used to describe the cooling process of neutron stars [50]. IV. CONCLUSIONS
We have presented calculations of the pairing gaps in infinite nuclear and neutron matteremploying realistic two- and three-body nuclear forces derived within the framework of chiraleffective field theory. The BCS gap equation is solved employing Khodel’s method, which isfound to be stable even for small values of the pairing gap. Three-nucleon forces help reduce14 .6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 k F ( fm − ) ∆ F ( M e V ) Λ LS =450 / Λ π =500Λ LS =600 / Λ π =600Λ LS =550 / Λ π =600Λ LS =450 / Λ π =700Λ LS =600 / Λ π =700 FIG. 9. The gap in the P F channel as a function of the resolution scale in the Juelichnucleon-nucleon interactions [8]. The scales refer respectively to the cutoffs (units of MeV) inthe Lippmann-Schwinger equation (Λ LS ) and the spectral function regulator in multi-pion ex-change loop diagrams(Λ π ). It appears that the magnitude of the gap’s maximum is very sensitiveto Λ π and, to a lesser content, to Λ LS . the strength of pairing in the S and coupled SD channels, while for the coupled P F channel the three-nucleon forces enhances the gap. In all cases considered in the presentwork, consistent nucleon effective masses reduce pairing correlations. Of particular interestis the scale dependence of the P F pairing gap, which exhibits a nearly universal behaviorat low densities in all chiral potentials considered. This works sets the stage for futureapplications to pairing gaps in finite-temperature neutron matter [50]. V. ACKNOWLEDGEMENTS
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