davesne, dany

Navarro, Jesus

Pastore, Alessandro

White Rose Research Online

This is a repository copy of Linear response theory for Gogny interaction.White Rose Research Online URL for this paper:https://eprints.whiterose.ac.uk/148490/Version: Published VersionArticle:davesne, dany, Navarro, Jesus and Pastore, Alessandro orcid.org/0000-0003-3354-6432 (2019) Linear response theory for Gogny interaction. Acta Physica Polonica B. ISSN 0587-4254 eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Vol. 12 (2019) Acta Physica Polonica B Proceedings Supplement No 3LINEAR RESPONSE THEORY FOR THE GOGNYINTERACTION∗D. DavesneUniversité de Lyon, Université Lyon 143 Bd. du 11 Novembre 1918, 69622 Villeurbanne cedex, FranceandCNRS-IN2P3, UMR 5822, Institut de Physique Nucléaire de Lyon, FranceJ. NavarroIFIC (CSIC-Universidad de Valencia)Apartado Postal 22085, 46.071 Valencia, SpainA. PastoreDepartment of Physics, University of YorkHeslington, York, Y010 5DD, United Kingdom(Received December 21, 2018)We present the formalism of the linear response theory in symmetricnuclear matter for a finite-range central interaction including zero-rangespin–orbit and tensor components.DOI:10.5506/APhysPolBSupp.12.5491. IntroductionSymmetric nuclear matter (SNM) is a spin-saturated system composed ofan equal amount of neutrons and protons with no finite-size effect. Althoughit may appear as a highly ideal system, the study of its excitation modesprovides us with several useful guidelines on other systems such as atomicnuclei and neutron stars (NS).In Ref. [1], we have in great detail presented the formalism of LinearResponse (LR) theory for a Skyrme functional including both spin–orbitand tensor terms [2]. By studying the response function of the system toan external probe, we have been able to identify the critical densities andmomenta at which instabilities occur in the system [3] and relate them to the∗ Presented at the XXV Nuclear Physics Workshop “Structure and Dynamics of AtomicNuclei”, Kazimierz Dolny, Poland, September 25–30, 2018.(549)550 D. Davesne, J. Navarro, A. Pastoreappearance of finite-size instabilities in nuclei [4, 5]. We have then providedin Refs. [6, 7] a simple method based on LR in SNM to avoid such instabilitiesdirectly at the level of the fitting procedure.The response functions of SNM, or more generally of asymmetric nuclearmatter [8], have also direct application to study neutrino opacities in NS[9–11] which in turn have important effects on NS cooling [12].In this article, we present an extension of the formalism of LR theory tothe case of finite-range interactions with spin–orbit and tensor terms. Thebasic formalism has been discussed in Ref. [13], for the case of a centralpart of the Gogny interaction [14]. The authors of Ref. [15] have alreadydiscussed the LR of a Gogny interaction using continued fraction (CF) ap-proximation [16]. The main inconvenience of such a method is that mostof the integrations required to obtain the response functions need to beperformed numerically via Monte Carlo samplings. Moreover, in Ref. [15],the spin–orbit term has been neglected, which is only valid for low-transfermomenta, see discussion in Ref. [3]. The current formalism, being basedon partial wave decomposition [17], offers us the opportunity of includingall terms of the interaction. Compared to Ref. [15], our formalism requiresmore detailed analytical derivations of all matrix elements in the differentchannels, but it is numerically less expensive.The article is organised as follows: in Sec. 2, we briefly introduce theLR formalism using multipolar expansion, while in Sec. 3, we illustrate ourresults for the case of a Gogny interaction. We present our conclusions inSec. 4.2. FormalismThe response function of the system is obtained by integrating, overthe momentum, the RPA propagator [1, 18]. The latter is the solutionof the Bethe–Salpeter equation in each spin (S), spin-projection (M) andisospin (I) channel, for brevity called α ≡ (S,M, I) in this paper. It readsG(α)RPA(q, ω,k1) = GHF(q, ω,k1)+GHF(q, ω,k1)∑α′∫d3k2(2π)3V(α,α′)ph (q,k1,k2)G(α′)RPA(q, ω,k2) , (1)where q is the transferred momentum and k1,k2 is the momentum of theparticle–hole pair and ω the transferred energy. V(α,α′)ph (q,k1,k2) representsthe residual particle–hole interaction. In the case of a Skyrme interaction[1, 3, 18, 19], Eq. (1) can be solved analytically using a system of symbolicequations. For the case of a finite-range interaction as Gogny, this is notpossible. We have thus adopted the technique presented in Ref. [13] andperformed a multipolar expansion asLinear Response Theory for the Gogny Interaction 551GHF(q, ω,k1) =∑lGHFl (q, ω, k1)Yl0(cos θ1) , (2)V(α,α′)ph (q,k1,k2) =∑lm,l′m′V(α,α′)lm;l′m′(k1, k2)Ylm(k̂1)Y ∗l′m′(k̂2), (3)G(α)RPA(q, ω,k1) =∑lmG(α)lm (q, ω, k1)Ylm(k̂1), (4)where Ylm is the usual spherical harmonic. We then obtain the multipolarexpansion of Eq. (1) asG(α)lm (k1) = δm,0GHFl (k1) +∑α′∑l′m′∫dk2k22(2π)3M(α,α′)lm,l′m′(k1, k2)G(α′)l′m′(k2) , (5)where, for simplicity, we have dropped the explicit dependence on q and ω.The matrix elements M(α,α′)lm,l′m′ are defined asM(α,α′)lm,l′m′(k1, k2) =∑l1l2[(2l2 + 1)(2l1 + 1)4π(2l + 1)]1/2C l0l10l20Clml2m,l10×GHFl1 (k1)V(α,α′)lm;l′m′(k1, k2) . (6)The notation used here simplifies to the one of Ref. [13] in the case of acentral interaction for which l = l′ and m = m′. Equation (5) actuallyrepresents a system of coupled integral equations. To solve them, we dis-cretise the integrals over a uniform mesh over the range of k ∈ [0, q + kF]as done in Ref. [13] and then invert the matrix of the system. In principle,the size of the system is infinite, since a finite-range interaction contains allmultipoles. However, as discussed in Ref. [13], the contribution of higherorder multipoles becomes smaller and smaller. We have thus introduced acut-off parameter Lmax to limit the number of equations we have to solve.In Sec. 3, we discuss the convergence of our results with Lmax.3. ResultsTo test the quality of our results, we start by considering the case of aSkyrme force with tensor term, namely T44 [2]. Since the Skyrme interactionis a simple combination of S- and P-waves [20], the partial wave expansionhas a natural truncation. Notice that additional caution should be takenwhen considering an explicit tensor contribution: since the tensor term tecouples S- and D-waves, we have to take Lmax = 2. Beyond L = 2, the othercontributions are zero, due to the particular form of the Skyrme force.552 D. Davesne, J. Navarro, A. PastoreIn Fig. 1, we compare the response functions obtained with the methodgiven in Eq. (5) (dashed lines) with those obtained by using the techniquedescribed in Ref. [1]. Both calculations are done at saturation density andtransferred momentum q = kF. The results stay on top of each other: thesmall deviations are only due to the discretisation on a finite grid of Eq. (5).This example clearly proves the validity of our method based on multipolarexpansion and we will now apply it to the case of finite-range interactions.0 40 80 120 160 ω [MeV]00.511.522.53-103 Im χ(α) /π [MeV-1fm-3] (S=0,M=0,I=0)(0,0,1)(1,0,0)(1,0,1)(1,1,0)(1,1,1)(S=0,M=0,I=0)Fig. 1. Response functions −Imχ(α)/π in SNM calculated at saturation density andtransferred momentum q = kF for Skyrme T44 interaction. The small differencesbetween solid and dashed lines are the consequence of the method (discretisation)used to solve Bethe–Salpeter equations. See the text for details.In Figs. 2–3, we present the response functions in the different spin–isospin channels for different values of the maximum angular momentumLmax in the case of Gogny D1S interaction [14]. The response functionsare calculated at ρ = 0.16 fm−3 and at two different values of transferred0 20 40 60 80 ω [MeV]0.40.81.21.62-103 Im χ(α) /π [MeV-1fm-3](0,0,0) LMax=1(1,0,0)(1,1,0)LMax=2LMax=30 20 40 60 80 ω [MeV]0.511.522.53-103 Im χ(α) /π [MeV-1fm-3](0,0,1) LMax=1(1,0,1)(1,1,1)LMax=2LMax=3Fig. 2. Response function −Imχ(α)/π in SNM calculated at ρ = 0.16 fm−3 andtransferred momentum q = 0.5kF for Gogny D1S interaction as a function of cut-off in angular momentum expansion Lmax.Linear Response Theory for the Gogny Interaction 553momentum q = 0.1kF, q = kF. Gogny D1S interaction is not equippedwith a tensor term, thus the spin–orbit term only induces a splitting of thedifferent projections of spin M = 0,±1. Since the transferred momentum isstill quite small, such a splitting is fully negligible as discussed in Ref. [15].0 50 100 150 200 ω [MeV]0.20.40.60.811.2-103 Im χ(α) /π [MeV-1fm-3](0,0,0) LMax=1(1,0,0)(1,1,0)LMax=2LMax=30 50 100 150 200 ω [MeV]0.511.52-103 Im χ(α) /π [MeV-1fm-3](0,0,1) LMax=1(1,0,1)(1,1,1)LMax=2LMax=3Fig. 3. The same as Fig. 2, but for q = kF.We clearly observe that Lmax = 2 is sufficient to obtain a reasonabledescription of the response function, confirming the results of Ref. [13].Finally, we compare in Fig. 4 our calculations obtained using Eq. (1)with a cut-off of Lmax = 2 with the ones of Ref. [15], based on the ContinuedFraction (CF) approximation [16]. Following Ref. [15], the terms included inthe calculations are the central part of Gogny D1S interaction. The densityis ρ = 0.174 fm−3 and the transferred momentum q = kF. Without tensor orspin–orbit terms, there is no longer a splitting in the different M projectionsof the total spin S. For such a reason, we present only results with M = 0.We can see that both methods are in excellent agreement, thus proving thevalidity of our results.40 80 120 160ω[MeV]00.511.5-103 Imχ(α) /π [MeV-1fm-3](0,0,0) CF(0,0,1)(1,0,0)(1,0,1)(0,0,0) Eq.5Fig. 4. Response function for the central term of Gogny D1S interaction. Thecalculations are done at ρ = 0.174 fm−3 and q = kF. See the text for details.554 D. Davesne, J. Navarro, A. PastoreIt is also interesting to compare the S = 0 channels from Fig. 3 and Fig. 4since we can observe a non-negligible difference of the low-energy part of theisoscalar response function. To understand the role of spin–orbit in thischannel, it is useful to observe the expression of the response function givenin Ref. [3]: due to the spin–orbit term, the interaction terms of the S = 1do contribute in the S = 0 channel and the term mixing the two channelsis typically proportional to the transferred momentum to the power of 4.As a consequence, when the transferred momentum is small, we can neglectsuch a term, but for large transferred momenta, this term starts playing animportant role.A very important aspect of the LR formalism is the detection of poles inthe response functions. In Fig. 5, we considered the example of Gogny D1Sinteraction and showed the evolution of the response function in the channel(0,0,1) for different values of transferred momentum. The calculations havebeen performed at ρ = 1.5ρ0, ρ0 being the saturation density of Gogny D1Sinteraction because it is known (see Ref. [15]) that at this particular value ofdensity, D1S exhibits a pole at qc = 2.53 fm−1. Effectively, we can observethe shape evolution of the response function when q approaches the criticalvalue qc, showing the stability of our numerical approach and the ability ofthe formalism to detect instabilities.50 100 150 200 250 300ω [MeV0246810-103 χ(0,0,1) /π[MeV-1fm-3] q=1.84 fm-1q=1.99 fm-1q=2.15 fm-1q=2.30fm-1Fig. 5. Response functions for Gogny D1S interaction for the channel (0,0,1) anddifferent values of the transferred momentum. See the text for details.4. ConclusionsIn this paper, we have presented the formalism of the LR theory forfinite-range interactions using multipolar expansion of the Bethe–Salpeterequation. We have tested our formalism against the results of the LR for-malism given in Ref. [1] for the case of Skyrme T44 [2]. In particular, thanksto the multipolar expansion, we clearly observe the tensor coupling betweenS- and D-waves. As a further benchmark of the formalism, we have comparedLinear Response Theory for the Gogny Interaction 555the result for the case of a Gogny interaction with Ref. [15]. In this case, theresults are in excellent agreement as well. The advantage of our technique(compared to Ref. [15]) is the explicit inclusion of spin–orbit term in theresidual interaction. As explicitly shown, this may lead to non-negligibleeffects on the response function since it induces an extra coupling betweenthe S = 0 and S = 1 channels.Finally, we have explored how the current formalism describes the pres-ence of poles in the response function. The goal is to continue in that direc-tion so that we can include directly a test to detect instabilities in the fittingprocedure itself. This can be important in the context of a future Gogny-likeparametrisation devoted to nuclear astrophysics for instance [21, 22].The work of J.N. has been supported by grant FIS2017-84038-C2-1-P,Mineco (Spain). The work of A.P. is supported by the UK Science and Tech-nology Facilities Council under grants No. ST/L005727 and ST/M006433.REFERENCES[1] A. Pastore, D. Davesne, J. Navarro, Phys. Rep. 563, 1 (2015).[2] T. Lesinski et al., Phys. Rev. C 76, 014312 (2007).[3] A. Pastore et al., Phys. Rev. C 85, 054317 (2012).[4] V. Hellemans et al., Phys. Rev. C 88, 064323 (2013).[5] A. Pastore, D. Tarpanov, D. Davesne, J. Navarro, Phys. Rev. C 92, 024305(2015).[6] A. Pastore et al., Phys. Scr. T154, 014014 (2013).[7] P. Becker et al., Phys. Rev. C 96, 044330 (2017).[8] D. Davesne, A. Pastore, J. Navarro, Phys. Rev. C 89, 044302 (2014).[9] N. Iwamoto, C.J. Pethick, Phys. Rev. D 25, 313 (1982).[10] A. Pastore et al., Phys. Rev. C 86, 044308 (2012).[11] A. Pastore et al., Phys. Rev. C 90, 025804 (2014).[12] P.S. Shternin et al., Mon. Not. R. Astron. Soc. Lett. 412, L108 (2011).[13] J. Margueron, Nguyen Van Giai, J. Navarro, Phys. Rev. C 72, 034311 (2005).[14] J. Dechargé, D. Gogny, Phys. Rev. C 21, 1568 (1980).[15] A. De Pace, M. Martini, Phys. Rev. C 94, 024342 (2016).[16] J. Margueron, J. Navarro, Nguyen Van Giai, P. Schuck, Phys. Rev. C 77,064306 (2008).[17] D. Davesne, J. Meyer, A. Pastore, J. Navarro, Phys. Scr. 90, 114002 (2015).[18] C. García-Recio, J. Navarro, L.L. Salcedo, Nguyen Van Giai, Ann. Phys.214, 293 (1992).556 D. Davesne, J. Navarro, A. Pastore[19] P. Becker et al., J. Phys. G: Nucl. Part. Phys. 42, 034001 (2015).[20] D. Davesne, P. Becker, A. Pastore, J. Navarro, Ann. Phys. 375, 288 (2016).[21] X. Viñas et al., arXiv:1810.07469 [nucl-th].[22] M. Martini, A. De Pace, K. Bennaceur, arXiv:1806.02080 [nucl-th].

Linear response theory for Gogny interaction

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Linear response theory for Gogny interaction

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