Linear response theory in asymmetric nuclear matter for Skyrme functionals including spin-orbit and tensor terms II: Charge Exchange
aa r X i v : . [ nu c l - t h ] O c t Linear response theory in asymmetric nuclear matter for Skyrme functionalsincluding spin-orbit and tensor terms II: Charge Exchange
D. Davesne, ∗ A. Pastore, † and J. Navarro ‡ Universit´e de Lyon, F-69003 Lyon, France; Universit´e Lyon 1,43 Bd. du 11 Novembre 1918, F-69622 Villeurbanne cedex, FranceCNRS-IN2P3, UMR 5822, Institut de Physique des deux Infinis de Lyon Department of Physics, University of York, York Y010 5DD, UK IFIC (CSIC-Universidad de Valencia), Apartado Postal 22085, E-46.071-Valencia, Spain (Dated: October 3, 2019)We present the formalism of linear response theory both at zero and finite temperature in thecase of asymmetric nuclear matter excited by an isospin flip probe. The particle-hole interactionis derived from a general Skyrme functional that includes spin-orbit and tensor terms. Responsefunctions are obtained by solving a closed algebraic system of equations. Spin strength functionsare analysed for typical values of density, momentum transfer, asymmetry and temperature. Weevaluate the role of statistical errors related to the uncertainties of the coupling constants of theSkyrme functional and thus determine the confidence interval of the resulting response function.
PACS numbers: 21.30.Fe 21.60.Jz 21.65.-f 21.65.Mn
I. INTRODUCTION
Transport properties of neutrinos play a crucial role in understanding the realisation of several astrophysical sce-narios as supernovae-explosions, neutron star mergers or the evolution of protoneutron stars [1–7]. The neutrinomean free path (NMFP) within dense nuclear matter at finite temperature is thus a key ingredient to understand thebehaviour of these astrophysical objects. When the neutrino pass trough the various layers of nuclear matter severalprocesses may take place as elastic scattering or absorption. In their seminal article, Iwamoto and Pethick [8] showedthat the NMFP could change by a factor of 2 to 3 in a range of densities around saturation since the neutrino canexcite a collective nuclear mode and thus lose energy and momentum substantially.In a previous series of articles [9–11], we have studied the properties of NMFP using Skyrme functionals [12] forpure neutron matter (PNM) at both zero and finite temperature. However, the PNM assumption for a stellar mediumis not realistic since a non-negligible fraction of protons is usually present [13–15] as well. Considering an environmentincluding both protons and neutrons is thus crucial.In Ref. [16], the authors have studied both neutron and charged reaction rates of neutrinos in dense nuclear matter.The work included only partially the in-medium effect at single particle level only, thus neglecting possible collectivemodes. The main outcome of the article is that the NMFP is dominated by the reaction ν e − + n → p + e − . Sucha result is also confirmed by other authors using different types of approximations [17–19]. Since NMFP is used forexample in neutrino transport radiation of hydrodynamics simulations [20], the exact value of such a quantity has adirect impact on other relevant astrophysical observables as neutrino luminosities.A fully quantitative calculation of the NMFP requires the knowledge of the nuclear strength function, which isusually determined via the random phase approximation (RPA) or linear response (LR) formalism, based on theparticle-hole (ph) interaction between particles below and above the Fermi level. The LR of asymmetric nuclearmatter to isospin-flip probes has been calculated in Ref. [21] using a zero-range Skyrme interaction restricted to itscentral part, including both direct and exchange terms. We present here results based on a general Skyrme interaction,which also includes spin-orbit and tensor components. This is of fundamental importance since the latter has beenshown to play a significant role in affecting the nuclear response function [9, 11, 22–26].The article is organised as follows. In Sec. II we briefly summarise the formalism for the computation of theasymmetric nuclear matter response to probes producing isospin flip. The results for several Skyrme functionals arepresented and analysed in Sec. III, where we discuss the effect of the spin-orbit and tensor terms on the response ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: navarro@ific.uv.es function, and also consider probes related to different isospin operators. In Sec.IV, we discuss the impact of parameteruncertainties on the response functions. Finally, our main conclusions are given in Sec. V.
II. LINEAR RESPONSE FORMALISM
The LR formalism for asymmetric nuclear matter has already been presented in previous articles [11, 26] but forno charge-exchange processes. Our aim here is to extend the calculation to probes represented by external fields ofthe form X j e i q · r j Θ ( S ) j τ ± j , (1)where Θ ( S ) j refers to the operator or σ zj , respectively for total spin S = 0 ,
1, and τ ± j are the usual isospin raising andlowering operators. The response functions for this kind of probes imply ( n, p ) and ( p, n ) charge exchange reactions,where a proton converts into a neutron and vice versa. More precisely, the operators τ + and τ − create particle-holeexcitations of two different species, namely pn − and np − , respectively. Both channels are such that the total isospinand its third component are equal to 1. In the following, we will focus on the operator τ + , which requires the phinteraction between pn − states only. Whenever necessary, excitations created by τ − may be simply obtained just byexchanging proton and neutron labels. A. Residual interaction
The residual interaction or ph interaction is currently defined as the second functional derivative of the energyfunctional [27, 28] so that the matrix elements we are interested in can be obtained as: h pn − | V ph | pn − i = δ h E i δρ np δρ pn , (2)where ρ np and ρ pn represents off-diagonal matrix elements of the one-body density. Because the Skyrme functional [12]does not contain explicit charge exchange terms [30], there are no rearrangement contributions, so that the ph inter-action coincides with the particle-particle interaction h pn | V | np i . Note that only the Pauli exchange term contributesto that interaction.Hereafter, the different ph channels will be labelled ( S, M ) where S stands for the total spin and M its projection.Moreover, we will follow the standard notation [11, 29] and indicate a general matrix element as V ( SM ; S ′ M ′ ) ph ( k , k , q ),where k , k are the hole momenta, q is the transferred momentum and we omit the proton and neutron indices inorder to simplify the notation. For the general Skyrme functional defined in Ref. [12], the ph interaction can bewritten as V ( SM ; S ′ M ′ ) ph ( k , k , q ) = δ ( S, S ′ ) δ ( M, M ′ ) h W ( S, ( q ) + W ( S, ( k − k ) i + 8 C ∇ s δ SS ′ δ S δ MM ′ δ M q + 4 C F (cid:26) ( − ) M ( k ) − M ( k ) M ′ δ SS ′ δ S − δ SS ′ δ S δ M δ M ′ q (cid:27) + 4 qC ∇ J ( δ S ′ δ S M ( k ) − M + δ S ′ δ S M ′ ( k ) M ′ ) , (3)where the parameters W ( S,T =1) i are defined as14 W (0 , = 2 C ρ, + 2 C ρ,γ ρ γ − (cid:20) C ∆ ρ + 12 C τ (cid:21) q , (4)14 W (1 , = 2 C s, + 2 C s,γ ρ γ − (cid:20) C ∆ s + 12 C T (cid:21) q , (5)14 W (0 , = C τ , (6)14 W (1 , = C T . (7)The constants C X , with X = ∆ s, F, . . . , are the coupling constants of the density functional. If the functional hasbeen obtained from an effective interaction, these coupling constants can also be expressed in terms of the interactionparameters as shown in Ref. [11]. Finally, it is worth noticing that the charge-exchange process is not coupled to the nn − and pp − excitations, contrarily to the non-isospin flip processes [26]. B. The nuclear strength function
The calculation of the response function requires the prior knowledge of the RPA ph propagator, which itself satisfiesthe following Bethe-Salpeter (BS) equation G ( pn − ,SM ) RP A ( k , q, ω ) = G ( pn − ) HF ( k , q, ω )+ G ( pn − ) HF ( k , q, ω ) X S ′ M ′ Z d k (2 π ) V ( SM ; S ′ M ′ ) ph ( k , k , q ) G ( pn − ,S ′ M ′ ) RP A ( k , q, ω ) . (8)The Hartree-Fock (HF) propagator is independent of ( S, M ). At variance with the non-isospin flip case [26], it hasnow the form G ( pn − ) HF ( k , q, ω ) = n n ( k ) − n p ( k + q ) ω − [ ε p ( k + q ) − ε n ( k )] + iη , (9)where ε τ = n,p are the HF single particle energies ε τ ( k ) = k m ∗ τ + U τ . (10)As usual, m ∗ τ stands for the effective mass and U τ is the single particle potential [31], excluding the k -dependencewhich is absorbed into the effective mass. . The occupation number n τ ( k ) is either a step function θ ( k ( τ ) F − k ) at zerotemperature or a Fermi-Dirac distribution n τ ( k ) = h ( ε τ ( k ) − µ τ ) /T i − , (11)at finite temperature.Once the BS equation is solved, we calculate the response function of the system as χ ( pn − ,SM ) ( q, ω ) = 2 Z d k (2 π ) G ( pn − ,SM ) RP A ( k , q, ω ) . (12)Finally, the strength function is S ( pn − ,SM ) ( q, ω ) = − π Im χ ( pn − ,SM ) ( q, ω )1 − e − (˜ ω − µ p + µ n ) /T , (13)where ˜ ω = ω − ( U p − U n ). We observe that using the detailed balance theorem we can relate the pn − to np − strengthfunctions as S ( pn − ,SM ) ( q, ω ) = e − (˜ ω − µ p + µ n ) /T S ( np − ,SM ) ( q, − ω ) . (14)The method used to solve the BS equation has been discussed in Refs. [26, 29]. Essentially, it implies a closed linearsystem for several momentum integrals of the RPA propagator. This linear system can then be cast in a matrix formas AX = B , (15)where A is the interaction matrix containing the ph matrix elements as well as momentum integrals of the HFpropagator, X contains the unknown momentum integrals of RPA propagators, including the response function and B contains only momentum integrals of HF propagators. These matrices are explicitly given in Appendix B. III. RESULTS
We now come to the presentation and discussion of the response functions at zero and finite temperature usingdifferent functionals and for various isospin asymmetries, defined by the parameter Y = ρ n − ρ p ρ , (16)where ρ n ( p ) is the neutron (proton) density and ρ = ρ n + ρ p . The PNM case corresponds to the value Y = 1, whilethe isospin saturated symmetric nuclear matter (SNM) is obtained for Y = 0. Asymmetric nuclear matter (ANM)corresponds to intermediate values of Y . Hereafter, we will consider two representative asymmetries, namely Y = 0 . Pb and Y = 0 . β -equilibrium within a neutron star.We present results obtained with the following Skyrme functionals: SLy5 [32], T22, T44 [33], and Skxta [34]. Theformer contains central and spin-orbit terms, the other three also include a tensor term. The first three functionals havebeen derived using the same Saclay-Lyon fitting protocol while the fourth one was obtained with a different protocol.For the reasons explained below, we also present results for the central part of the Skyrme SGII interaction [35]. A. Results with central terms only
The formalism of the charge exchange linear response theory has been already presented and discussed in Ref. [21],but limited to the central part of the SGII. Because our algorithm to obtain the response function is more complexthat the one of Ref. [21], we have decided to perform the same calculations to benchmark our results. In this way wehave detected that the value of parameter x used in [21] was erroneously divided by a factor of ten . We have thuschecked that both algorithms give exactly the same results, provided the same parameters are introduced.In Fig. 1, we display the strength function obtained with the central part only of the SGII interaction at saturationdensity and Y = 0 . S = 1 (solid lines) and S = 0 (dashed lines). The different panels refer todifferent values of transfer momentum q/k F . A comparison with Fig. 3 of Ref. [21] shows that although qualitativelyin agreement, there are significant quantitative differences, in particular in the location of the collective states.As already stressed in Ref. [21], an important point is the presence at T = 0 of collective oscillations at negativeenergies, for small momenta. This is related to the absence of Coulomb effects, which certainly modify the singleparticle spectrum for protons. A possible way to simulate Coulomb effects in infinite nuclear matter would be addinga repulsive shift to the proton mean field (see e.g. [36]). In the present case, as the neutron Fermi energy lies higherthan the proton one, ( n, p ) transitions at low energy and momentum are possible in some cases. Such transitions donot exist in the case of ph excitations of the type np − , as will be shown later on in Fig. 7.The results displayed in Fig. 1 have been obtained by ignoring the spin-orbit term of interaction SGII. It wasshown in Ref. [37] that the spin-orbit term couples the spin channels S = 0 and 1, acting differently for channels withdifferent M -component. However, the effect on the response function is rather small, even at momentum transferrelatively large as compared to the Fermi momentum. Indeed, by inspecting Eq. B1, we observe that the spin-orbitterm contributes via a q -term and thus becomes negligible for small transferred momenta. This is not the case ofthe tensor which plays a role at both low and high transferred momenta [25, 26]. We thus consider now Skyrmefunctionals containing both spin-orbit and tensor terms. B. Results with the full interaction
In Fig. 2, are shown the strength functions obtained with the four considered Skyrme functionals, at density ρ/ρ = 0 .
5, transfer momentum q/k F = 0 . Y , for spin channels ( S, M ).We observe that for SLy5 interaction, the only one with no tensor terms, the response functions in the channels (1,0)and (1,1) are fully degenerate. As discussed before, the spin-orbit term that should lift such an M -degeneracy istoo small for the selected transferred momentum and thus no effect is visible. Since the tensor term in the residualinteraction does not scale with q but with q , it is active at lower transferred momentum. We thus notice that theother three Skyrme functionals (T22, T44, Skxta) break that degeneracy in the S = 1 channel. Incidentally we have also detected a misprint in Eq. (24) of [21], the factor ( k F ( − τ ) /k F ( τ )) has to be replaced by k F ( − τ ) /k F ( τ ). -12-10 -8 -6 -4 -2 0 20.020.04 [ M e V f m - ] -20-16-12 -8 -4 0 4 8 0.51-30 -20 -10 0 1012 [ M e V f m - ] -30 -15 0 15 3012-20 0 20 40 60 [MeV] [ M e V f m - ] -20 0 20 40 60 80 [Mev] a)c)e) f)d)b) FIG. 1: (Color online). The imaginary part of the dynamical susceptibility − χ S,T ( q, ω ) /π of asymmetric nuclearmatter as a function of energy, for the Skyrme interaction SGII at saturation density of symmetric nuclear matter, Y = 0 . S = 1 and 0 channels, respectively. The location of the collectivestates is indicated by vertical lines. The transferred momentum q/k F is (a) 0.01; (b) 0.09; (c) 0.14; (d) 0.22; (e) 0.45 and (f)0.65. S ( α ) ( q , ω ) [ M e V - f m - ] Y=0.21
HF(0,0)(1,0)(1,1) -20 -10 0 ω [MeV]
123 -20 -10 0 10 ω [MeV] SLy5 T22SkxtaT44 S ( α ) ( q , ω ) [ M e V - f m - ] Y=0.5
HF (0,0)(1,0)(1,1) -45 -30 -15 0 ω [MeV]
123 -45 -30 -15 0 ω [MeV] SLy5 T22SkxtaT44
FIG. 2: (Color online). Strength function S ( α ) ( q, ω ) in asymmetric nuclear matter for the spin channels α = ( S, M ), usingthe four ph interactions considered here. Two different values of the asymmetry parameter are considered, ρ/ρ = 0 . q/k F = 0 .
2. The Hartree-Fock strength function is also displayed (dotted line) to exhibit the effects ofthe ph interaction.
Tensor effects can be small at small transferred momenta and small asymmetry (see Fig. 2, left panel), but theyare significantly enhanced with increasing asymmetry (see Fig. 2, right panel) and/or transferred momentum (Fig. 3).This confirms the intrinsic importance of the tensor on the response functions also for charge-exchange processes. Onecould naively expect that the tensor interaction affects S = 1 channels only. However, as the spin-orbit term couplesboth spin channels, it turns out that the tensor term actually acts also in the S = 0 channel. The ph interactionshould thus include both spin-orbit and tensor contributions.To show the global effect of the ph interaction itself, we have also displayed the HF strength function. We observeon Fig. 2 that the interaction has indeed a strong effect: the strength function is shifted towards the high-energyregion and a collective mode appears. For the same asymmetries, but for a higher value of the transferred momentum(see Fig. 3), the zero-sound mode is re-absorbed in the continuum part of the response. This is clearly seen in rightpanel at Y = 0 . S = 0 channel present a strong peak at the edge of the allowed region.If we now come to the tensor, we see that the effect is small for small transferred momenta ( q coupling) whateverthe asymmetry is (see Fig. 2) but becomes more pronounced at higher transferred momentum (see Fig. 3) where itincreases the accumulation of strength at high energy. S ( α ) ( q , ω ) [ M e V - f m - ] Y=0.21
HF(0,0)(1,0)(1,1) -30 0 30 60 ω [MeV]
012 -30 0 30 60 ω [MeV] SLy5 T22SkxtaT44 S ( α ) ( q , ω ) [ M e V - f m - ] Y=0.5
HF(0,0)(1,0)(1,1) -40 -20 0 20 40 60 ω [MeV]
012 -40 -20 0 20 40 60 80 ω [MeV] SLy5 T22SkxtaT44
FIG. 3: (Color online). Same as Fig.2 but for q/k F = 1. The density is also an important parameter. To quantify its effect, we reported in Fig. 4 the strength functionsat saturation density. In this case the effect of the tensor in the two spin-projection channels is opposite for allfunctionals: in the M = 0 channel the tensor leads to a strong attraction and thus an accumulation of strength atlow energy, while the opposite is true in the M = 1 channel. As expected, such an effect is absent in SLy5 since itdoes not contain an explicit tensor term, and the two curves lie on top of each other. S ( α ) ( q , ω ) [ M e V - f m - ] Y=0.21
HF(0,0)(1,0)(1,1) ω [MeV] ω [MeV] SLy5 T22SkxtaT44 S ( α ) ( q , ω ) [ M e V - f m - ] Y=0.5
HF(0,0)(1,0)(1,1) -60 0 60 120 ω [MeV] ω [MeV] SLy5 T22SkxtaT44
FIG. 4: (Color online). Same as Fig.2 but for q/k F = 1 and ρ/ρ = 1. One can also assess the role of the tensor contribution by comparing SLy5, T22 and T44 results in Fig. 4. Asdiscussed in Ref [33] the T ij interactions have been fitted with very similar protocol as SLy5, but including an explicittensor term. In order to get some insight of the role played by the tensor, we have thus repeated the calculations withT44 only, turning off (on) the tensor coupling constants C F and C ∇ s The result is illustrated in Fig. 5 for Y=0.21 atsaturation density and transferred momentum q/k F = 1. -40 0 40 80 120 160 ω [MeV] S ( α ) ( q , ω ) [ M e V - f m - ] HF(0,0)(1,0)(1,1)
FIG. 5: (Color online). Response function for T44 using q/k F = 1 and ρ/ρ = 1 and Y=0.21. The solid lines refer to thecalculation without tensor while open symbols refer to full interaction. We clearly observe that the tensor interaction acts in all channels and that its effect is quite remarkable especiallyin the (1,0) channel. In this case we notice that the calculations without tensor (solid line) exhibit an accumulationof strength at high energy, thus meaning a repulsive residual interaction.As previously discussed, in this case the(1,0) and (1,1) are essentially degenerate since the spin-orbit contribution is too small at low transferred momenta toprovide a visible effect. When the tensor is taken into account the response function (dashed lines) accumulates atlow energy thus meaning a strongly attractive interaction. As already discussed previously in Refs [9, 22, 23, 25, 26]the tensor has a very important role in determining the excited states of a nuclear system.
C. Thermal effects S ( α ) ( q , ω ) [ M e V - f m - ] T=0.1 e F HF(0,0)(1,0)(1,1) -20 -10 0 10 ω [MeV]
123 -20 -10 0 10 ω [MeV] SLy5 T22SkxtaT44 S ( α ) ( q , ω ) [ M e V - f m - ] T=0.5 e F HF(0,0)(1,0)(1,1) -30 -20 -10 0 10 ω [MeV]
123 -30 -20 -10 0 10 ω [MeV] SLy5 T22SkxtaT44
FIG. 6: (Color online). Same as Fig.2 at Y=0.21, but at T = 0 . e F (left panel) and T = 0 . e F (right panel). In Fig. 6, we illustrate the impact of temperature on the response function. For simplicity, we consider the samevalues of density and transferred momentum as in Fig. 2, but now considering a finite temperature of T = 0 . e F and T = 0 . e F , where e F is the Fermi energy. We consider here only the case Y=0.21 since the other asymmetry leadsto very similar results. As is well-known [11, 21, 31, 38–40], the effect of temperature is to wash out the structure ofthe response function and spread its strength. Moreover, a system at finite temperature can deexcite by giving someenergy to the probe. These effects are clearly visible when comparing with Fig. 2. The strength becomes broader,and in some cases the collective mode is absorbed into the continuum.Going from T = 0 . e F to T = 0 . e F we observe that the limits of the strength function are increased and the peaksacquire a larger width. By further increasing the temperature, we may thus observe the complete disappearance ofthe peaks observed here and obtaining a smooth strength function over a large energy domain. D. Role of the isospin operator
Finally, we investigate the role of the isospin operator τ in the probe defined in Eq. (1). In the case of asymmetricnuclear matter two probes are possible: τ z for non-isospin flip process and τ ± for charge exchange excitations. In thecase of non-isospin flip probes, the relevant quantum numbers of each ph pair are the total spin ( S ), spin projection( M ) and isospin ( T ). In the case of isospin-flip the ph-pairs are either pn − (corresponding to the τ + operator) or np − (for τ − ). Also in this case the pairs are coupled to S and M , the value of T being equal to 1.In Fig. 7, we illustrate the difference between the two probes for the T44 functional and asymmetry Y = 0 .
21. Wealso fix the density of the system to ρ/ρ = 1 and the transferred momentum to q/k F = 1. In panel (a) of Fig. 7, wereport the isospin-flip case for the two operators τ + (solid lines) and τ − (dashed lines). We observe that the domainof energy where the response function exists is quite different in the case np − and pn − . This shift in the energydomain is due to the difference of chemical potential for the two species. Given the current asymmetry, the resultsimply shows that is more favourable to a pn − pair due to the large neutron excess of the system than the oppositeprocess. -30 0 30 60 90 120 150 180 ω [MeV] S ( α ) ( q , ω ) [ M e V - f m - ] (0,0) pn -1 (1,0) pn -1 (1,1) pn -1 (0,0) np -1 (1,0) np -1 (1,1) np -1 a) -30 0 30 60 90 120 150 180 ω [MeV] S ( α ) ( q , ω ) [ M e V - f m - ] (0,0,0)(0,0,1)(1,0,0)(1,0,1)(1,1,0)(1,1,1) b) FIG. 7: (Color online). Response function for T44 using q/k F = 1 and ρ/ρ = 1 and Y=0.21. Panel a) shows the charge-exchange cases, namely with operators τ + (solid lines) and τ − (open symbols), for the different ( S, M ) channels. Panel b)shows the strength functions for the case with no charge exchange, for the different (
S, M, T ) channels.
For completeness, we compare in panel (b) of Fig. 7 the strength functions for the operators τ + (solid lines) and τ z (dashed lines), under the same conditions as panel (a). Apart from the fact that there are now more channels,due to the degeneracy breaking of M , the main difference is that the strength is considerably reduced in the chargeexchange channels. We may thus expect this to have an impact on astrophysical observables such as NMFP in densestellar matter [8]. IV. ERROR ANALYSIS
In this section, we provide the first quantitative analysis of statistical uncertainties on the nuclear response function.Following Refs [41, 42], the error V y on a given observable y ( x , a ) depending on some independent variable x and aset of parameters a is obtained as V y ( x ) = X ij ∂y ( x , a ) ∂a i C ij ∂y ( x , a ) ∂a j (17)where C is the covariance matrix [43]. The partial derivative respect to parameter space in Eq.17 are done using afinite-difference method as discussed in Ref. [44]. To perform such calculations a critical ingredient is thus representedby C . Unfortunately very few Skyrme functionals provide published values for the covariance matrix. In the followingwe will restrict to the UNEDF0 [45] and UNEDF1 [46] functionals, since all relevant statistical informations requiredto perform error propagations are available.The only limitation of this functionals is that they have been explicitly fitted taking into account only time-eventerms of the functional [12]. This means that essentially the S = 1 channel of the response function is not determined,we thus consider the response function of the system only in the S=0 channel.In Fig.8, we show the response function S (0 , ( q, ω ) (dashed line) obtained with UNEDF0 and UNEDF1 for thecharge exchange operator τ − (left panel) and τ + (right panel). By performing the full error propagation as definedin Eq.17, we have obtained the coloured bands appearing in the figure. The band has been drawn to represent onestandard deviation. FIG. 8: (Colour online). Response function for UNEDF0 and UNEDF1 in the S=0 channel (dashed line). The colour bandsrepresent the 1 σ error due to the statistical uncertainties in the parameters of the functional. See text for details We observe that the main features of the response function are not much impacted by the statistical uncertainties, i.e the attractive/repulsive structure of the response function is not affected by error. The latter have a strong impacton the actual height of the peaks. This analysis is relevant for any model taking the response functions as input.For example, the calculations on NMFP may be strongly impacted by these error bars, but further investigations arerequired.
V. CONCLUSIONS
In this paper we have generalised the LR formalism presented in Ref. [21] so that Skyrme spin-orbit and tensorterms can be included. Moreover, the response functions were calculated for a general Skyrme functional [12], thusproviding more flexibility to study the behaviour of some particular coupling constants. We have investigated theevolution of the strength function as a function of the density of the system and the transferred momentum for somerepresentative Skyrme functionals. We have observed that the presence of an explicit tensor term induces majoreffects on the strength function.Finally, we have generalised the formalism to the case of arbitrary isospin asymmetry and temperature so that thecurrent methodology may be easily adopted to perform calculations of astrophysical interest as neutrino mean freepath. In a recent work [9], we have illustrated the role of the tensor on NMFP for the pure neutron matter case.From the results presented in the current article, we have shown that the tensor play a crucial role on the strengthfunctions also in the case of isospin-flip probes. We thus may expect to observe a noticeable impact also in NMFPcalculated in asymmetric matter. We leave this aspect for a future investigation.0
Acknowledgements
The work of JN was supported by grant FIS2017-84038-C2-1P, Mineco (Spain). The work of AP was funded by agrant from the UK Science and Technology Facilities Council (STFC): Consolidated Grant ST/P003885/1.
Appendix A: β pn functions To transform the Bethe-Salpeter equation into a closed linear system of algebraic equations, the propagators haveto be integrated over the momentum with some weights. The HF propagator appears in the following integrals: β pni ( q, ω, T ) = Z d k (2 π ) G pn − HF ( k , q , ω, T ) F i ( k , q ) (A1) F i ( k , q ) = 1 , k · q q , k q , (cid:20) k · q q (cid:21) , ( k · q ) k q , k q , (cid:20) k · q q (cid:21) , (cid:20) k · q q (cid:21) , ( k · q ) k q . The imaginary part of these integrals has been already presented in Ref. [11]. The real part is obtained numericallyusing the dispersion relation Re β pni ( q, ω, T ) = − π Z + ∞−∞ dω ′ Im β pni ( q, ω ′ , T ) ω − ω ′ . (A2) Appendix B: Matrix elements
We give here the explicit expressions of the matrices required to determine the strength functions.
1. Channel S=0
The interaction matrix reads A = − β pn ˜ W − q β pn W − β pn − W qβ pn − W − q β pn ˜ W − q β pn W − q β pn W q β pn W − qβ pn ˜ W − q β pn W − qβ pn W q β pn W where ˜ W = W + 16 q (cid:0) C ∇ J (cid:1) ( β np − β np )1 + q ( β np − β np ) (cid:0) W − C F (cid:1) (B1)The other two matrices read X = h G pn − ,pn − , RP A ih k G pn − ,pn − , RP A i q π h kY G pn − ,pn − , RP A i B = β pn q β pn qβ pn
2. Channel S=1 M= ± A = − β pn ˜ W − q β pn W − β pn W qβ pn W − C F β pn − C F q ( β pn − β pn ) − q [ C F ] βpn βpn − βpn − βpn βpn − βpn zpn, +8[ C F ] q β pn βpn − βpn zpn, − q β pn ˜ W − q β pn W − q β pn W q β pn W + 8 q [ C F ] βpn βpn − βpn βpn zpn, − q C F β pn − C F q ( β pn − β pn ) − qβ pn ˜ W − q β pn W − qβ pn W q β pn W − qC F β pn − C F q ( β pn − β pn ) − q [ C F ] βpn βpn − βpn − βpn βpn − βpn zpn, +8[ C F ] q β pn βpn − βpn zpn, − q ( β pn − β pn ) ˜ W − q ( β pn − β pn ) W − q ( β pn − β pn ) W q ( β pn − β pn ) W − q C F ( β pn − β pn ) − C F q ( β pn − β pn + β pn )+8[ C F ] q βpn − βpn zpn, and X = h G pn − ,pn − , RP A ih k G pn − ,pn − , RP A i q π h kY G pn − ,pn − , RP A i π h k | Y | G pn − ,pn − , RP A i ; B = β pn q β pn qβ pn q ( β pn − β pn ) ˜ W = W + 8 q [ C ∇ J ] z pn, ( β pn − β pn ) + q [ C F ] (cid:20) β pn − β pn ) − W q ( β pn − β pn ) z pn, (cid:21) (B2) z pn,S = W S q ( β pn − β pn ) (B3)
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