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QCD constraints on the equation of state for compact stars
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Nuclear Physics A 956 (2016) 813–816 www.elsevier.com/locate/nuclphysa
QCD constraints on the equation of state for compact stars E. S. Fraga Instituto de F´ısica, Universidade Federal do Rio de Janeiro, C. P. 68528, 21941-972, Rio de Janeiro, RJ, Brazil
A. Kurkela Physics Department, Theory Unit, CERN, CH-1211 Gen`eve 23, Switzerland
J. Schaffner-Bielich Institute for Theoretical Physics, Goethe University, D-6 0438 Frankfurt am Main, Germany
A. Vuorinen Department of Physics and Helsinki Institute of Physics, P. O. Box 64, FI-00014 University of Helsinki, Finland
Abstract In recent years, there have been several successful attempts to constrain the equation of state of neutron star matter using input from low-energy nuclear physics and observational data. We demonstrate that significant further restrictions can be placed by additionally requiring the pressure to approach that of deconfined quark matter at high densities. Remarkably, the new constraints turn out to be highly insensitive to the amount - or even presence - of quark matter inside the stars. In this framework, we also present a simple effective equation of state for cold quark matter that consistently incorporates the effects of interactions and furthermore includes a built-in estimate of the inherent systematic uncertainties. This goes beyond the MIT bag model description in a crucial way, yet leads to an equation of state that is equally straightforward to use. Keywords: Equations of state of neutron-star matter, Quark matter
1. Introduction and key message The inner structure of neutron stars is determined by the equation of state (EoS) of cold and dense strongly interacting matter that is described by in-medium quantum chromodynamics (QCD) [1]. Given the outstanding property of asymptotic freedom, which is in the heart of QCD, a very important and challenging question that has been considered for decades is whether there is deconfined dense quark matter in the core of compact stars [2]. Although this is clearly a phenomenologically relevant question, finding a reliable, model-independent answer may depend on observational evidence that is still far ahead in the future. Here we consider, instead, a related but totally different question [3, 4]: does the fact that at asymptotically high http://dx.doi.org/10.1016/j.nuclphysa.2016.01.037 0375-9474/© 2016 Elsevier B.V. All rights reserved.
E.S. Fraga et al. / Nuclear Physics A 956 (2016) 813–816
814
-3
Pressure (MeV fm )
SB limit 1000
pQCD matter
inner crust
1 outer crust
neutron matter
0.001
1e-06
1
10
? 100
Central μ in maximally massive stars
1000
Quark Chemical Potential μ − μiron/3 (MeV) Fig. 1. Known limits of the stellar EoS on a logarithmic scale. On the horizontal axis we have the quark chemical potential (with an offset so that the variable acquires the value 0 for pressureless nuclear matter), and on the vertical axis the pressure. The band in the region around the question mark corresponds to the interpolating polytropic EoS used in [3].
densities one must have deconfined quark matter constrain the equation of state for compact stars? This question can be answered now. As will be clear in what follows, the answer is a sound yes, so that the main message is that even if there is no deconfined quark matter in the core of neutron stars, the form of the QCD equation of state at very large densities (which is known perturbatively) affects dramatically the equation of state for compact stars. Although a full nonperturbative determination of the pressure of the theory is still out of reach due to the so-called Sign Problem of lattice QCD, methods from chiral effective field theory (EFT) of nuclear forces [5] and high-density perturbative QCD (pQCD) [6] can now provide reliable predictions for the EoS in the limits of low density nuclear matter and dense quark matter, respectively. Furthermore, by now both approaches produce results with reliable error estimates and error bars, so that the two limits are given in a controlled fashion. The picture that emerges, also taking into account that neutron stars with masses M ∼ 2M do exist [7, 8], is exhibited in Fig. 1. For a complementary phenomenological approach, see Refs. [9, 10]. 2. Method During the last few years, several articles have addressed the determination of the neutron star EoS by combining insights from low-energy chiral EFT with the requirement that the resulting EoSs support the most massive stars observed (see e.g. Ref. [11]). In particular, the discovery of neutron stars with masses around two solar masses [7, 8] has recently been seen to lead to strong constraints on the properties of stellar matter [12]. While otherwise impressive, these analyses have solely concentrated on the low density regime, and have typically applied no microphysical constraints beyond the nuclear saturation density n0 . This has resulted in EoSs that behave very differently from that of deconfined quark matter even at rather high energy densities. To implement the correct asymptotic limit for the EoS of neutron star matter, we use the state-of-the-art result of Ref. [13], where a compact expression for the three-loop pressure of unpaired quark matter, taking into account the nonzero value of the strange quark mass, was derived (see also Refs. [14, 15] for details of the original pQCD calculation). A particularly powerful outcome of the analysis is that the high density constraint significantly reduces the uncertainty band of the stellar matter EoS even at low densities, well
E.S. Fraga et al. / Nuclear Physics A 956 (2016) 813–816 1
1 0.8
Too soft to meet pQCD
2.0
0.6
pQCD γ2
γ1
Normalized pressure P/PSB
μc: Change of polytropic index Normalized pressure P/PSB
815
1.0 X=1.2
4.0
0.4
2.0 8.0 4.0
0.2 Nuclear EoS
Too hard to match pQCD
Disallowed by mass constraint 0.8 pQCD 0.6 I 0.4
II
0.2 III 0
0 1
1.2
1.4 1.6 1.8 2 2.2 2.4 2.6 Baryon chemical potential μB [GeV]
2.8
3
Fig. 2. Matching the two limiting cases using 2-tropes. Solutions exist for μc ∈ [1.08, 2.05], γ1 ∈ [2.23, 9.2] and γ2 ∈ [1.0, 1.5].
1
1.2
1.4 1.6 1.8 2 2.2 2.4 2.6 Baryon chemical potential μB
2.8
3
Fig. 3. Interpolated pressure of nuclear and quark matter, normalized by the pressure of a gas of free quarks and shown together with the pQCD result at high densities.
below a possible phase transition to deconfined quark matter. This implies that the mass-radius relations we obtain are more restrictive than previous ones even for pure neutron stars. In practice, our calculation proceeds as follows (see Ref. [3] for details): at densities below 1.1n0 , we employ the chiral EFT EoS of Tews et al. [16], assuming the true result to lie within the error band given in this reference. At baryon chemical potentials above 2.6 GeV, where the relative uncertainty of the quark matter EoS is as large as the nuclear matter one at n = 1.1n0 , we on the other hand use the result of Fraga et al. [13] and its respective error estimate. Between these two regions, we assume that the EoS is well approximated by an interpolating polytrope built from two ‘monotropes’ of the form P(n) = κnΓ . These functions are first matched together in a smooth way, but we can also consider the scenario of a first-order phase transition, allowing the density to jump at the matching point of the two monotropes [13]. Varying the polytropic parameters and the transition density over ranges limited only by causality, we obtain a band of EoSs that can be further constrained by the requirement that the EoS support a two solar mass star. Fig. 2 displays an illustration of this procedure. 3. Illustrative results Fig. 3 shows the interpolated pressure of nuclear and quark matter, normalized by the pressure of a gas of free quarks and shown together with the pQCD result at high densities. All generated EoSs lie within the shaded green and turquoise areas, of which only the green ones support a star of M = 2M . Three representative EoSs marked with I-III have crosses denoting the maximal chemical potential reached at the center of the star. One can see from this figure the large reduction in the EoS uncertainty due to the tension from the mass constraint, i.e. large stellar masses require a stiffer EoS. Notice also that the uncertainties are within 30% at all densities. In Fig. 4 we show two mass-radii clouds composed of the EoSs displayed in Fig. 3. It is clear from the figure that one can reach masses up to M = 2.5M . Central densities are in the interval [3.7, 14.3]n0 . Finally, Fig. 5 displays the band for the pressure as a function of the energy density showing the contribution from the different constraints and ingredients and a comparison to the EoS obtained by Hebeler et al. [11]. 4. Final remarks As mentioned previously, the main message in this work is that even if there is no deconfined quark matter in the core of neutron stars, the form of the QCD EoS at very large densities (which is known perturbatively in a robust manner) affects dramatically the EoS for compact stars. Moreover, the existence of two-solar mass stars strongly constrains the EoS; having a first-order phase transition or a crossover not so much. ESF is supported by CAPES, CNPq and FAPERJ, AV is supported by the Academy of Finland, grant nr. 266185.
E.S. Fraga et al. / Nuclear Physics A 956 (2016) 813–816
816 2.5
10000 II
III
Without mass constraint Pressure [MeV/fm ]
I
M [Msolar]
3
2
1.5
1
1000
pQCD
100
10
1
With mass constraint HLPS Neutron matter
0.5
100 8
9
10
11 12 R [km]
13
14
1000 10000 3 Energy density [MeV/fm ]
15
Fig. 4. Two M − R clouds composed of the EoSs displayed in Fig. 3. The color coding is the same as there, as is our notation for the three representative EoSs I-III.
Fig. 5. A comparison of our EoS with that of Hebeler et al. [11], labeled HLPS in the Figure. The green, turquoise and blue bands correspond to our result, thus displaying the effect of the pQCD constraint.
References [1] N. K. Glendenning, Compact Stars – Nuclear Physics, Particle Physics and General Relativity, (Springer, 2000). [2] M. Buballa, V. Dexheimer, A. Drago, E. Fraga, P. Haensel, I. Mishustin, G. Pagliara and J. Schaffner-Bielich et al., J. Phys. G 41, no. 12, 123001 (2014). [3] A. Kurkela, E. S. Fraga, J. Schaffner-Bielich and A. Vuorinen, Astrophys. J. 789, 127 (2014). [4] E. S. Fraga, A. Kurkela and A. Vuorinen, arXiv:1508.05019 [nucl-th]. [5] E. Epelbaum, H. W. Hammer and U. G. Meissner, Rev. Mod. Phys. 81, 1773 (2009). [6] U. Kraemmer and A. Rebhan, Rept. Prog. Phys. 67, 351 (2004). [7] P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and J. Hessels, Nature 467, 10811083 (2010). [8] J. Antoniadis, P. C. Freire, N. Wex, T. M. Tauris, R. S. Lynch, et al., Science 340, 6131 (2013). [9] T. Kojo, P. D. Powell, Y. Song and G. Baym, Phys. Rev. D 91, no. 4, 045003 (2015). [10] T. Kojo, arXiv:1508.04408 [hep-ph]. [11] K. Hebeler, J. M. Lattimer, C. J. Pethick and A. Schwenk, Astrophys. J. 773, 11 (2013). [12] J. M. Lattimer, Ann. Rev. Nucl. Part. Sci. 62, 485 (2012). [13] E. S. Fraga, A. Kurkela and A. Vuorinen, Astrophys. J. 781, L25 (2014). [14] A. Kurkela, P. Romatschke and A. Vuorinen, Phys. Rev. D 81, 105021 (2010). [15] A. Kurkela, P. Romatschke, A. Vuorinen and B. Wu, arXiv:1006.4062 [astro-ph.HE]. [16] I. Tews, T. Krueger, K. Hebeler and A. Schwenk, Phys. Rev. Lett. 110, 032504 (2013).
Nuclear Physics A 956 (2016) 813–816 www.elsevier.com/locate/nuclphysa
QCD constraints on the equation of state for compact stars E. S. Fraga Instituto de F´ısica, Universidade Federal do Rio de Janeiro, C. P. 68528, 21941-972, Rio de Janeiro, RJ, Brazil
A. Kurkela Physics Department, Theory Unit, CERN, CH-1211 Gen`eve 23, Switzerland
J. Schaffner-Bielich Institute for Theoretical Physics, Goethe University, D-6 0438 Frankfurt am Main, Germany
A. Vuorinen Department of Physics and Helsinki Institute of Physics, P. O. Box 64, FI-00014 University of Helsinki, Finland
Abstract In recent years, there have been several successful attempts to constrain the equation of state of neutron star matter using input from low-energy nuclear physics and observational data. We demonstrate that significant further restrictions can be placed by additionally requiring the pressure to approach that of deconfined quark matter at high densities. Remarkably, the new constraints turn out to be highly insensitive to the amount - or even presence - of quark matter inside the stars. In this framework, we also present a simple effective equation of state for cold quark matter that consistently incorporates the effects of interactions and furthermore includes a built-in estimate of the inherent systematic uncertainties. This goes beyond the MIT bag model description in a crucial way, yet leads to an equation of state that is equally straightforward to use. Keywords: Equations of state of neutron-star matter, Quark matter
1. Introduction and key message The inner structure of neutron stars is determined by the equation of state (EoS) of cold and dense strongly interacting matter that is described by in-medium quantum chromodynamics (QCD) [1]. Given the outstanding property of asymptotic freedom, which is in the heart of QCD, a very important and challenging question that has been considered for decades is whether there is deconfined dense quark matter in the core of compact stars [2]. Although this is clearly a phenomenologically relevant question, finding a reliable, model-independent answer may depend on observational evidence that is still far ahead in the future. Here we consider, instead, a related but totally different question [3, 4]: does the fact that at asymptotically high http://dx.doi.org/10.1016/j.nuclphysa.2016.01.037 0375-9474/© 2016 Elsevier B.V. All rights reserved.
E.S. Fraga et al. / Nuclear Physics A 956 (2016) 813–816
814
-3
Pressure (MeV fm )
SB limit 1000
pQCD matter
inner crust
1 outer crust
neutron matter
0.001
1e-06
1
10
? 100
Central μ in maximally massive stars
1000
Quark Chemical Potential μ − μiron/3 (MeV) Fig. 1. Known limits of the stellar EoS on a logarithmic scale. On the horizontal axis we have the quark chemical potential (with an offset so that the variable acquires the value 0 for pressureless nuclear matter), and on the vertical axis the pressure. The band in the region around the question mark corresponds to the interpolating polytropic EoS used in [3].
densities one must have deconfined quark matter constrain the equation of state for compact stars? This question can be answered now. As will be clear in what follows, the answer is a sound yes, so that the main message is that even if there is no deconfined quark matter in the core of neutron stars, the form of the QCD equation of state at very large densities (which is known perturbatively) affects dramatically the equation of state for compact stars. Although a full nonperturbative determination of the pressure of the theory is still out of reach due to the so-called Sign Problem of lattice QCD, methods from chiral effective field theory (EFT) of nuclear forces [5] and high-density perturbative QCD (pQCD) [6] can now provide reliable predictions for the EoS in the limits of low density nuclear matter and dense quark matter, respectively. Furthermore, by now both approaches produce results with reliable error estimates and error bars, so that the two limits are given in a controlled fashion. The picture that emerges, also taking into account that neutron stars with masses M ∼ 2M do exist [7, 8], is exhibited in Fig. 1. For a complementary phenomenological approach, see Refs. [9, 10]. 2. Method During the last few years, several articles have addressed the determination of the neutron star EoS by combining insights from low-energy chiral EFT with the requirement that the resulting EoSs support the most massive stars observed (see e.g. Ref. [11]). In particular, the discovery of neutron stars with masses around two solar masses [7, 8] has recently been seen to lead to strong constraints on the properties of stellar matter [12]. While otherwise impressive, these analyses have solely concentrated on the low density regime, and have typically applied no microphysical constraints beyond the nuclear saturation density n0 . This has resulted in EoSs that behave very differently from that of deconfined quark matter even at rather high energy densities. To implement the correct asymptotic limit for the EoS of neutron star matter, we use the state-of-the-art result of Ref. [13], where a compact expression for the three-loop pressure of unpaired quark matter, taking into account the nonzero value of the strange quark mass, was derived (see also Refs. [14, 15] for details of the original pQCD calculation). A particularly powerful outcome of the analysis is that the high density constraint significantly reduces the uncertainty band of the stellar matter EoS even at low densities, well
E.S. Fraga et al. / Nuclear Physics A 956 (2016) 813–816 1
1 0.8
Too soft to meet pQCD
2.0
0.6
pQCD γ2
γ1
Normalized pressure P/PSB
μc: Change of polytropic index Normalized pressure P/PSB
815
1.0 X=1.2
4.0
0.4
2.0 8.0 4.0
0.2 Nuclear EoS
Too hard to match pQCD
Disallowed by mass constraint 0.8 pQCD 0.6 I 0.4
II
0.2 III 0
0 1
1.2
1.4 1.6 1.8 2 2.2 2.4 2.6 Baryon chemical potential μB [GeV]
2.8
3
Fig. 2. Matching the two limiting cases using 2-tropes. Solutions exist for μc ∈ [1.08, 2.05], γ1 ∈ [2.23, 9.2] and γ2 ∈ [1.0, 1.5].
1
1.2
1.4 1.6 1.8 2 2.2 2.4 2.6 Baryon chemical potential μB
2.8
3
Fig. 3. Interpolated pressure of nuclear and quark matter, normalized by the pressure of a gas of free quarks and shown together with the pQCD result at high densities.
below a possible phase transition to deconfined quark matter. This implies that the mass-radius relations we obtain are more restrictive than previous ones even for pure neutron stars. In practice, our calculation proceeds as follows (see Ref. [3] for details): at densities below 1.1n0 , we employ the chiral EFT EoS of Tews et al. [16], assuming the true result to lie within the error band given in this reference. At baryon chemical potentials above 2.6 GeV, where the relative uncertainty of the quark matter EoS is as large as the nuclear matter one at n = 1.1n0 , we on the other hand use the result of Fraga et al. [13] and its respective error estimate. Between these two regions, we assume that the EoS is well approximated by an interpolating polytrope built from two ‘monotropes’ of the form P(n) = κnΓ . These functions are first matched together in a smooth way, but we can also consider the scenario of a first-order phase transition, allowing the density to jump at the matching point of the two monotropes [13]. Varying the polytropic parameters and the transition density over ranges limited only by causality, we obtain a band of EoSs that can be further constrained by the requirement that the EoS support a two solar mass star. Fig. 2 displays an illustration of this procedure. 3. Illustrative results Fig. 3 shows the interpolated pressure of nuclear and quark matter, normalized by the pressure of a gas of free quarks and shown together with the pQCD result at high densities. All generated EoSs lie within the shaded green and turquoise areas, of which only the green ones support a star of M = 2M . Three representative EoSs marked with I-III have crosses denoting the maximal chemical potential reached at the center of the star. One can see from this figure the large reduction in the EoS uncertainty due to the tension from the mass constraint, i.e. large stellar masses require a stiffer EoS. Notice also that the uncertainties are within 30% at all densities. In Fig. 4 we show two mass-radii clouds composed of the EoSs displayed in Fig. 3. It is clear from the figure that one can reach masses up to M = 2.5M . Central densities are in the interval [3.7, 14.3]n0 . Finally, Fig. 5 displays the band for the pressure as a function of the energy density showing the contribution from the different constraints and ingredients and a comparison to the EoS obtained by Hebeler et al. [11]. 4. Final remarks As mentioned previously, the main message in this work is that even if there is no deconfined quark matter in the core of neutron stars, the form of the QCD EoS at very large densities (which is known perturbatively in a robust manner) affects dramatically the EoS for compact stars. Moreover, the existence of two-solar mass stars strongly constrains the EoS; having a first-order phase transition or a crossover not so much. ESF is supported by CAPES, CNPq and FAPERJ, AV is supported by the Academy of Finland, grant nr. 266185.
E.S. Fraga et al. / Nuclear Physics A 956 (2016) 813–816
816 2.5
10000 II
III
Without mass constraint Pressure [MeV/fm ]
I
M [Msolar]
3
2
1.5
1
1000
pQCD
100
10
1
With mass constraint HLPS Neutron matter
0.5
100 8
9
10
11 12 R [km]
13
14
1000 10000 3 Energy density [MeV/fm ]
15
Fig. 4. Two M − R clouds composed of the EoSs displayed in Fig. 3. The color coding is the same as there, as is our notation for the three representative EoSs I-III.
Fig. 5. A comparison of our EoS with that of Hebeler et al. [11], labeled HLPS in the Figure. The green, turquoise and blue bands correspond to our result, thus displaying the effect of the pQCD constraint.
References [1] N. K. Glendenning, Compact Stars – Nuclear Physics, Particle Physics and General Relativity, (Springer, 2000). [2] M. Buballa, V. Dexheimer, A. Drago, E. Fraga, P. Haensel, I. Mishustin, G. Pagliara and J. Schaffner-Bielich et al., J. Phys. G 41, no. 12, 123001 (2014). [3] A. Kurkela, E. S. Fraga, J. Schaffner-Bielich and A. Vuorinen, Astrophys. J. 789, 127 (2014). [4] E. S. Fraga, A. Kurkela and A. Vuorinen, arXiv:1508.05019 [nucl-th]. [5] E. Epelbaum, H. W. Hammer and U. G. Meissner, Rev. Mod. Phys. 81, 1773 (2009). [6] U. Kraemmer and A. Rebhan, Rept. Prog. Phys. 67, 351 (2004). [7] P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and J. Hessels, Nature 467, 10811083 (2010). [8] J. Antoniadis, P. C. Freire, N. Wex, T. M. Tauris, R. S. Lynch, et al., Science 340, 6131 (2013). [9] T. Kojo, P. D. Powell, Y. Song and G. Baym, Phys. Rev. D 91, no. 4, 045003 (2015). [10] T. Kojo, arXiv:1508.04408 [hep-ph]. [11] K. Hebeler, J. M. Lattimer, C. J. Pethick and A. Schwenk, Astrophys. J. 773, 11 (2013). [12] J. M. Lattimer, Ann. Rev. Nucl. Part. Sci. 62, 485 (2012). [13] E. S. Fraga, A. Kurkela and A. Vuorinen, Astrophys. J. 781, L25 (2014). [14] A. Kurkela, P. Romatschke and A. Vuorinen, Phys. Rev. D 81, 105021 (2010). [15] A. Kurkela, P. Romatschke, A. Vuorinen and B. Wu, arXiv:1006.4062 [astro-ph.HE]. [16] I. Tews, T. Krueger, K. Hebeler and A. Schwenk, Phys. Rev. Lett. 110, 032504 (2013).