We explore the possibility of obtaining heavy hybrid stars within the
framework of the two flavor Nambu--Jona-Lasinio model with 8-quark interactions
in the scalar and in the vector channel. The main impact of the 8-quark scalar
channel is to reduce the onset of quark matter, while the 8-quark vector
channel acts to stiff the equation of state at high densities. Within the
parameter space where the 4-quark vector channel is small, and the 8-quark
vector channel sizeable, stable stars with $2 M_\odot$ and above are found to
hold quark matter in their cores.Comment: to match the published version in EPJ

Benic, Sanjin

English

arXiv

We explore the possibility of obtaining heavy hybrid stars within the framework of the two flavor Nambu--Jona-Lasinio model with 8-quark interactions in the scalar and in the vector channel. The main impact of the 8-quark scalar channel is to reduce the onset of quark matter, while the 8-quark vector channel acts to stiff the equation of state at high densities. Within the parameter space where the 4-quark vector channel is small, and the 8-quark vector channel sizeable, stable stars with 2M_sun and above are found to hold quark matter in their cores

Benić, Sanjin

Repository of Faculty of Science, University of Zagreb

DOI 10.1140/epja/i2014-14111-1LetterEur. Phys. J. A (2014) 50: 111 THE EUROPEANPHYSICAL JOURNAL AHeavy hybrid stars from multi-quark interactionsSanjin Benic´aDepartment of Physics, Faculty of Science, University of Zagreb, P.O.B. 331, HR-10002 Zagreb, CroatiaReceived: 22 June 2014Published online: 14 July 2014c© The Author(s) 2014. This article is published with open access at Springerlink.comCommunicated by D. BlaschkeAbstract. We explore the possibility of obtaining heavy hybrid stars within the framework of the two ﬂavorNambu–Jona-Lasinio model that includes 8-quark interactions in the scalar and in the vector channel. Themain impact of the 8-quark scalar channel is to reduce the onset of quark matter, while the 8-quark vectorchannel acts to stiﬀen the equation of state at high densities. Within the parameter space where the 4-quark vector channel is small, and the 8-quark vector channel sizeable, stable stars with masses of 2Mand above are found to hold quark matter in their cores.1 IntroductionOne of the most severe constraints of the equation of state(EoS) of QCD at extreme densities comes from the recent2M mass determinations of PSR J1614-2230 [1] and PSRJ0348-0432 [2] having important consequences for the ex-istence of quark matter in compact stars [3–6].In this work we assume compact stars are the so-calledhybrid stars, composed of a nuclear mantle and a quarkcore. In order to obtain heavy hybrids stars, quark mat-ter EoS should be stiﬀ, with a low onset [7]. This can beachieved in the MIT bag model by perturbative correc-tions to the EoS and small bag pressures [8], while withinthe Nambu–Jona-Lasinio (NJL) model large vector chan-nel is used for stiﬀness [9], while shift in the vacuum en-ergy [10,11] or introduction of superconductivity ensureslow onset [12]. An inﬂuence on the maximum mass is ex-pected by the nature of the phase transition itself [13–15].We propose an alternative scenario by taking into ac-count the fact that NJL is a non-renormalizable eﬀectivemodel, valid up to some scale Λ. As we approach thisscale, say by increasing the chemical potential, higher-dimensional operators should become important. It hasbeen pointed out [16,17] that including higher scalar in-teractions can reduce the critical temperature in the NJLmodel, bringing it in closer agreement with lattice resultsat ﬁnite temperature [18]. For further work on higher-dimensional operators in the context of the NJL modelsee [19–25].Our aim is to make an initial study of the eﬀect ofmultiquark interactions on occurrence of quark matterin heavy stars. In this work we will use the NJL modelparametrization of ref. [18] where the critical temperaturea e-mail: sanjinb@phy.hrat zero chemical potential is ﬁtted to lattice QCD. In ad-dition, we will also introduce the 8-quark vector channel,as a natural candidate for playing a relevant role at largedensities reached in the cores of compact stars.Our main results are that with scalar 8-quark inter-actions provided by ref. [18] we are able to obtain stablehybrid stars with small vector coupling in the 4-quark andzero vector coupling in the 8-quark channel. Second, wedemonstrate that the mass of the star can be increasedup to and above 2M with the 8-quark vector interaction,while still keeping the 4-quark vector interaction low.2 NJL model with 8-quark interactionsWe work within the framework of a Nf = 2 NJL modeldeﬁned as follows:L = q¯(i ∂ −m)q + μq q¯γ0q + L4 + L8, (1)where μq is the quark chemical potential and m is thecurrent mass. The interaction terms areL4 = g20Λ2[(q¯q)2 + (q¯iγ5τ q)2]− g02Λ2(q¯γμq)2, (2)L8 = g40Λ8[(q¯q)2 + (q¯iγ5τ q)2]2− g04Λ8(q¯γμq)4−g22Λ8(q¯γμq)2[(q¯q)2 + (q¯iγ5τ q)2], (3)where Λ is the model cutoﬀ. The Lagrangian (1) may bemotivated by the large Nc counting: while the 4-quarkcouplings scale as 1/Nc, the 8-quark couplings scale as1/N3c [26].Page 2 of 4 Eur. Phys. J. A (2014) 50: 111In the mean-ﬁeld approximation the Lagrangian be-comesLMF = q¯(i ∂ −M)q + μ˜q q¯γ0q − U, (4)whereM = m + 2g20Λ2〈q¯q〉+ 4g40Λ8〈q¯q〉3 − 2g22Λ8〈q¯q〉〈q†q〉2, (5)μ˜q = μq − 2g02Λ2〈q†q〉 − 4g04Λ8〈q†q〉3 − 2g22Λ8〈q¯q〉2〈q†q〉, (6)and the classical potentialU =g20Λ2〈q¯q〉2 + 3g40Λ8〈q¯q〉4 − 3g22Λ8〈q¯q〉2〈q†q〉2−g02Λ2〈q†q〉2 − 3g04Λ8〈q†q〉4. (7)Integrating out the quark degrees of freedom, the fullthermodynamic potential takes the following form:Ω = U − 2NfNc∫d3p(2π)3{E + T log[1 + e−β(E−μ˜q)]+T log[1 + e−β(E+μ˜q)]}+ Ω0, (8)where E =√p2 + M2 and β = 1/T and where Ω0 en-sures zero pressure in the vacuum. The model is solved byminimizing the thermodynamic potential with respect tothe mean ﬁelds X = 〈q¯q〉, 〈q†q〉, i.e.∂Ω∂X= 0. (9)In this work we use the parameter set of ref. [18] g20 =1.864, g40 = 11.435, m = 5.5MeV Λ = 631.5MeV. Thevector channel strengths are treated as free parameters,quantiﬁed byη2 =g02g20, η4 =g04g40. (10)We are interested in a particular region of the vector chan-nel couplings where the g02 coupling is kept small, whileg04 is increased. The reason behind our choice is as fol-lowing. Since heavy hybrid stars require a stiﬀ EoS, a re-pulsive vector coupling should be present. As the vectorcoupling renormalizes the chemical potential, it delays theonset of quark matter. This leads to a scenario where thehadronic mantle becomes too large for the pressure in thequark core to be able to hold the star against gravita-tional collapse. Therefore, the appearance of quark mat-ter in such a scenario usually makes the star unstable. Anattractive channel like e.g., superconductivity needs to beinvoked in order to lower the onset [12]. We point outthat an alternative microscopic picture is possible withmultiquark interactions: whereas stiﬀening of the EoS isprovided by the 8-quark vector channel interaction, lower-ing of the onset is accomplished by introducing a sizeable8-quark scalar channel interaction and keeping the 4-quarkvector channel interaction small.Our choice of the phenomenologically interesting pa-rameter space will have an eﬀect of stiﬀening the quarkmatter at higher densities, while at the same time keepingthe transition density low. Due to this restriction we willalso put g22 = 0 by hand. Namely, the operator controlledby the size of g22 can be important at moderate μq onlyif the vector mean ﬁeld is sizeable, which will not be thecase for the parameter region of small g02 in which we areinterested in. In addition, it will not be important at highμq since there the scalar mean ﬁeld is zero.3 Equation of stateBy evaluating the full thermodynamical potential (8) atthe minimum, we obtain the quark matter EoS as pq =−Ω. The quark number density nq and energy density 	qare deﬁned as follows:nq = − ∂pq∂μq, 	q = −pq + nqμq. (11)Beta equilibrium is taken into account by the weak pro-cesses d → u + e− + ν¯e, μ− → e− + νμ + ν¯e, implyingμu = μd + μe, μμ = μe, where the neutrino chemicalpotential is set to zero. Finally, the baryon chemical po-tential, and the baryon density are μB = 3μq = 2μd + μuand nB = nq/3, respectively.For the nuclear matter we choose the DD2 EoS [27,28].The transition from nuclear to quark matter is providedby the traditional Maxwell construction. Therefore, theEoS is obtained by requiring local charge neutrality. Thefull pressure in the quark phase takes into account thecontribution of electrons and muonsp(μq) = pu(μu) + pd(μd) + pe(μe) + pμ(μμ), (12)where pe,μ is the pressure of a free electron (muon) gas.We emphasize that the Maxwell construction is merely thelimit of large surface tension at the quark-hadron inter-face in a more elaborate approach that takes into accountﬁnite-size eﬀects, see [29]. The following results ultimatelydepend also on the choice of the construction of the phasetransition.The hybrid EoS are given in ﬁg. 1 for η2 = 0.05 anda range of η4. Owing to 8-quark scalar interactions, andsmall 4-quark vector interactions, onset of quark matteris rather low: for η2 = 0.05 and η4 = 0.0 it is around	  500MeV/fm3. Even a drastic increase of η4 has barelyany inﬂuence on the onset: this is only natural, since thereis extra suppression due to high dimensionality of the cor-responding operator.The inﬂuence of η4 is best seen by inspecting the speedof sound: while small or almost vanishing vector interac-tions yield the relativistic value c2s  1/3, the speed ofsound signiﬁcantly increases with 	 already for η4 = 1.0,see right panels of ﬁg. 1. This gradual stiﬀening of the EoScan also be seen as a microscopic mechanism of a postu-lated scenario of a medium-dependent parameter η2 [30],which is able to provide a very stiﬀ quark EoS [7]. Assum-ing that the vector mean ﬁeld is given by the density ofEur. Phys. J. A (2014) 50: 111 Page 3 of 4500 1000 1500 [MeV/fm3]101001000p [MeV/fm3 ]2=0.05, 4=0.02=0.05, 4=1.02=0.05, 4=2.02=0.05, 4=5.0DD2500 1000 1500 [MeV/fm3]0.40.60.81c s22=0.05, 4=0.02=0.05, 4=1.02=0.05, 4=2.02=0.05, 4=5.0ε εη ηη ηη ηη ηη ηη ηη ηη ηFig. 1. (Color online) On the left panel we show the EoS ofhybrid matter. Thick black curve is the hadronic contribution.Flat region corresponds to Maxwell construction. We ﬁxη2 = 0.05. Quark matter curves are denoted as following: theorange, dashed line accounts for η4 = 0.0, dash-dotted, greenline for η4 = 1.0, dash-double-dotted, indigo line for η4 = 2.0and double-dash-dotted, magenta line for η4 = 5.0. Rightpanel shows the speed of sound of quark matter with the samelabeling.massless fermions, an approximative expansion in g02 andg04 for the speed of sound c2s = ∂p/∂	 as a function of thequark chemical potential can be shown to holdc2s 13+32g026π2μ2qΛ2+512g2024π4μ4qΛ4+75776g30224π6μ6qΛ6+3768320g40248π8μ8qΛ8+8192g0448π8μ8qΛ8, (13)illustrating that the g04 term starts to be important onlyat higher chemical potentials. While the 4-quark vectorinteraction respects causality, strong 8-quark vector inter-action may violate the causal limit, see ref. [31] for anexplicit example in the nucleonic NJL model. This is thereason why the EoS with η4 = 5.0 turns acausal alreadyat 	 ∼ 2000MeV/fm3. In addition, for such high densities,the quark chemical potential is approaching Λ where themodel should not be trusted anymore. Therefore, resultsobtained for this extreme scenario are given for illustrativepurposes.4 Hybrid starsThe static, spherically symmetric stars are obtained as so-lutions of the Tolman-Oppenheimer-Volkoﬀ (TOV) equa-tions for the EoS shown in the previous section. In ﬁg. 2 wedisplay the resulting mass-radius and mass-central energydensity diagrams for η2 = 0.05.Due to the early onset of quark matter, in ﬁg. 2 we areable to obtain stable stars with pure quark matter in theircores even for very small vector coupling. Such a scenariois not easy to achieve in the NJL model with only fourth-order scalar and vector operators [12,32], see however [11].The strong vector interaction increases the speed of soundand makes quark matter stiﬀ, but at the same time it ap-pears at too high energy densities. The higher-dimensional10 12 14R [km]11.522.5M [Msun]500 1000 1500 2000c [MeV/fm3]2=0.05, 4=0.02=0.05, 4=1.02=0.05, 4=2.02=0.05, 4=5.0DD2PSR J1614-2230PSR J0348+0432causalityεη ηη ηη ηη ηFig. 2. (Color online) Left panel shows the M -R and the rightpanel the M -c diagram of compact star solutions within themodel. We ﬁx η2 = 0.05. The detachment from the hadronicbranch given by the thick, black curve and the maximum massdepends on η4. Hybrid star branches appear in the same linestyles as the corresponding EoS in ﬁg. 1. Hatched regions markexperimental constraints from two heaviest sources, lower, redis [1], while higher, blue is [2].vector operator stiﬀens the EoS and without inﬂuencingthe onset signiﬁcantly, gives a mechanism for 2M hybridstars, see ﬁg. 2. In order to cover the present experimentalwindow provided by PSR J1614-2230 and PSR J0348-0432we have found that values η4 up to η4 ∼ 2.0 are suﬃcient.The extreme case with η4 = 5.0 leads to a signiﬁcant in-crease in the mass, yielding M ∼ 2.25M.5 ConclusionsObserving heavy compact stars oﬀers a promising perspec-tive on constraining the cold, dense EoS beyond satura-tion density. A compelling discrimination of many possi-ble scenarios of dense matter will be possible once precisemeasurements of also star radii become available [33]. Thedata of both masses and radii will enable Bayesian inver-sion of the TOV equation leading to the EoS [34].We have studied one possible scenario where multi-quark interactions coming from higher-dimensional oper-ators in the NJL model might play an important role atlarge densities. A sizeable 8-quark scalar channel is in-troduced in order to achieve a low onset. With a small4-quark vector coupling onset is still low, while the rela-tively large 8-quark coupling is used to gradually stiﬀenthe EoS at high densities. Within this parameter space wewere able to fulﬁll and go beyond the 2M constraint.The scenario of small 4-quark vector coupling is inaccordance with the results of ref. [35,36], but a morethorough study is needed to reveal how would the in-troduction of the 8-quark vector coupling inﬂuence theirresults. In addition, we stress that the consensus con-cerning the status of vector interactions in quark mat-ter is yet to be reached in the community, as some otherstudies [18,37–39] advocate a diﬀerent scenario where thePage 4 of 4 Eur. Phys. J. A (2014) 50: 1114-quark vector coupling is sizeable. Using the results fromthe lattice QCD measurements at ﬁnite T [40–42] and atimaginary chemical potential [43] a more thorough studyof the model presented here is needed to achieve betterconstraints on the three vector channel couplings, and toinvestigate the consequences at large densities. Finally, itremains to be explored what can such a setup say aboutthe existence of strangeness in compact stars.The author acknowledges illuminating discussions withD. Blaschke and thanks D.E. Alvarez-Castillo for help withthe TOV code. The author also thanks the Yukawa Institutefor Theoretical Physics, Kyoto University, where part of thiswork was performed during the YITP-T-13-05 workshop on“New Frontiers in QCD”. This work is supported by the MIAUproject of the Croatian Science Foundation and by the theCOST Action MP1304 NewCompStar.Open Access This is an open access article distributedunder the terms of the Creative Commons AttributionLicense (http://creativecommons.org/licenses/by/4.0), whichpermits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.References1. P. Demorest et al., Nature 467, 1081 (2010).2. J. Antoniadis et al., Science 340, 6131 (2013).3. F. Ozel, Nature 441, 1115 (2006).4. T. Klahn, D. Blaschke, F. Sandin, C. Fuchs, A. Faessler,H. Grigorian, G. Ropke, J. Trumper, Phys. Lett. B 654,170 (2007).5. M. Alford, D. Blaschke, A. Drago, T. Klahn, G. Pagliara,J. Schaﬀner-Bielich, Nature 445, E7 (2007).6. J.M. Lattimer, M. 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Heavy hybrid stars from multi-quark interactions

We explore the possibility of obtaining heavy hybrid stars within the framework of the two flavor Nambu-Jona-Lasinio model that includes 8-quark interactions in the scalar and in the vector channel. The main impact of the 8-quark scalar channel is to reduce the onset of quark matter, while the 8-quark vector channel acts to stiffen the equation of state at high densities. Within the parameter space where the 4-quark vector channel is small, and the 8-quark vector channel sizeable, stable stars with masses of 2 M
⊙ and above are found to hold quark matter in their cores

Heavy hybrid stars from multi-quark interactions

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