# High-density QCD pairing in compact star structure

###### Key Words.:

Stars: neutron – Equation of state-Strange quark matter in a color flavor locked (CFL) state can be the true ground state of hadronic matter for a much wider range of the parameters of the model (the gap of the QCD Cooper pairs , the strange quark mass and the Bag Constant ) than the state without any pairing. We review the equation of state (EOS) of CFL strange matter and study the structure of stellar objects made up of this phase, highlighting the novel features of the latter. Although the effects of pairing on the equation of state are thought to be small, we find that CFL stars may be in fact much more compact than strange stars (SS). This feature may be relevant in view of some recent observations claiming the existence of exotic and/or deconfined phases in some nearby neutron stars (NS).

## 1 Introduction

The study of a hypothetical stability of strange quark matter (SQM), put forward in Witten’s (Witten ) seminal paper and a few important precursors (Bodmer Bodmer ; Terazawa Terazawa ; Chin and Kerman Chin ) has entered its third decade. Nothing less than the nature of the true ground state of hadronic matter is being questioned and in fact, within simple models, the hypothesis of a stable form of cold catalyzed plasma was shown to be tenable. Following the pioneering works, a general calculation of strange matter by Farhi and Jaffe (farhijaffe ) in the framework of the MIT Bag model of confinement (Degrand et al. Degrand ; Cleymans, Suhonen and Gavai Cleymans ) identified the so-called ”windows of stability”, or regions in the plane inside which the stability of SQM can be realized. Other models of confinement have been worked out to find a quite ample range of conditions for SQM to be absolutely bound (Zhang and Ru-Ken Su massdep and references therein; Hanauske et al. NJL ). However, there is a consensus that the issue of the availability of a binding energy difference for SQM to be bound is ultimately an experimental matter.

While the search for SQM in laboratory and astrophysical environments beyond their present limits continues, important theoretical developments have taken place. The most sound is probably the revival of interest in pairing interactions in dense matter, a subject already addressed in the early 1980s (Bailin and Love BL ) and revived a few years ago with new calculations of the pairing energy and related physics. It is now generally agreed (Alford, Rajagopal and Wilczek ARW ; Rapp et al. rap ; Rajagopal and Wilczeck rajagopal2 ) that (at least for very high densities) the color-flavor locked (CFL) state, with equal numbers of and quarks is likely to be the ground state of strong interactions, even if the quark masses are unequal (Alford, Berges and Rajagopal uneq ; Schäfer and Wilczek Schafer ). The equal number of flavors is enforced by the symmetry of the state, and thus electrons are absent because the mixture is automatically neutral (Rajagopal and Wilczeck rajagopal ; Steiner, Reddy and Prakash 2002).

Long awaited by theoretical physicists, the high-performance of the space X-ray missions Chandra and XMM (Weisskopf Chandra ; Becker and Aschenbach XMM ) enabled unprecedented studies of imaging and spectra of selected neutron stars with the aim of determining the masses and radii, perhaps the most simple forms of (indirectly) investigating the nature of high density matter. Adopting General Relativity as a framework, a comparison of the static models generated by integration of the Tolman-Oppenheimer-Volkof equations with observed data is expected to give information about the equation of state, and possibly other effects such as rotation, magnetic atmospheres and so on. A wonderful example of the pre-space determinations is the mass of the binary pulsar PSR 1913+16, accurate to several decimal places (van Kerkwijk vankerkwijk ). Other methods based on combinations of spectroscopic and photometric techniques have been recently proposed. At least one X-ray source is significantly above the centroid of the binary distribution ; namely Vela X-1 for which a value of has been obtained (van Kerkwijk vankerkwijk ). Recently, not only the masses have been determined, but also indications of the radii became available, suggesting a very compact structure. For instance, claims of high compactness have been made from the analysis of the binary Her X-1 (Li, Lai and Wang li ; Dey et al. Dey98 ) with and km, and of the isolated nearby RX J185635-3754 (Pons et al. Pons ) with and km. In both cases, the results have been revisited and challenged by other groups (Reynolds et al. Rey ; Kaplan, van Kerkwijk and Anderson Kaplan ) who in turn found figures around the expected for conventional neutron star models. This stresses the cautionary remarks made by several researchers about the high-compactness objects and guarantees further studies, already undertaken in most cases. Li et al. (Li99b ) have also added the source 4U 1728-34 to the candidate list, showing that conventional accretion models indicate a very compact source in the mass-radius plane. As always, the actual distance to the source is a matter of concern. Needless to say, these results have yet to be confirmed carefully. Nonetheless, it is worthwhile to consider the possibility that at least some compact stars are extremely compact, or in other words, that their radii are less than the ”canonical” favored by neutron matter models for .

If actually present in these sources, compactness would be
extremely difficult (perhaps impossible) to model using underlying
equations of state based on ordinary hadrons alone, and a natural
alternative would be to consider deconfined matter (or other
exotic components, like kaon matter or hyperons). Stars suspected
to be made of deconfined matter also include the X-ray bursters
GRO J1744-28 (Cheng et al. Cheng98 ) and SAX J1808.4-3658
(Li at al. Li99a ). This increasing availability of data on
compact neutron star structure coupled with the new developments
on the QCD ground sate leads to investigate the properties of
compact stars in the light of the CFL state ^{1}^{1}1 Alford
and Reddy (2002) have also independently analyzed the role of CFL
quark matter in compact stars in a very recent work. Their work is
rather general and focused on stars with a normal matter envelope,
while our work deals with absolutely stable CFL stars only.
However, it is worth noting that Alford and Reddy (2002) also
include some results on pure CFL stars in their Figs. 5 and 6,
which are in quantitative agreement with ours (see Section 3)..

## 2 Color flavor locked strange matter

### 2.1 The equation of state

The very complex structure of the QCD phase diagram emerging from detailed calculations by several groups is still being explored. Therefore, only schematic models are available to explore stellar structure questions. It is, however, widely agreed that if the quark mass is small enough (possibly the actual case in nature), the CFL state would be the minimum energy configuration at high densities. To proceed we need the thermodynamical potential to derive the relevant quantities. Fortunately, this potential can be found quite simply (Alford et al. alfordcfl ) to the lowest order in the gap . The procedure is to start with the of a fictional state of unpaired quark matter in which all quarks which are ”going to pair” have a common Fermi momentum , with chosen to minimize . The binding energy of the diquark condensate is included by subtracting the condensation term while the vacuum energy is introduced by means of the phenomenological bag constant allowing the mixture to confine. In this work we assume that the CFL state has been reached by deconfined matter, although the precise way in which this is done (involving weak decays, etc) merits a closer look. The thermodynamic potential of this model reads (Alford et al. alfordcfl )

(1) | |||||

where , being the chemical potential of the i-species. The common Fermi momentum is given by

(2) |

The thermodynamic quantities, such as the pressure P, the baryon number density , the particle number densities , and the energy density can be easily derived at and read (Lugones and Horvath cflprd )

(3) |

(4) |

(5) |

In this approach we shall treat the values of B, and (possibly as high as ) as free constant parameters. The full dependence of on has been known for years and certainly complicates the evaluation of the equation of state, which must be treated numerically. However, we have recently worked out an approximation to the order which has the main advantage of keeping the equation of state very simple, yet useful for most calculations, while at the same time highlights the effect of each parameter of the model (see Lugones and Horvath cflprd for details).

The stiffness of the EOS is relevant for stellar models since it has a direct impact on the compactness of the stellar configurations. As discussed by Lugones and Horvath (cflprd ), whenever is higher than , the EOS is stiffer than the unpaired SQM (that is, produces more pressure for a given energy density). Since the actual value of is not well known, we may expect either case, a stiffer or a softer EOS (for a given ).

### 2.2 Stability of the CFL phase

Self-bound matter, in which strong forces produce a zero-pressure point at finite density, is potentially interesting for astrophysics because it produces stellar sequences having in the Newtonian limit. For a given EOS the energy per baryon of the deconfined phase (at and ) must be lower than the neutron mass if matter is to be absolutely stable. Another condition to be considered results from the empirically known stability of normal nuclear matter against deconfinement at and (Farhi and Jaffe farhijaffe ), which means that the energy per baryon of deconfined matter (a pure gas of quarks and ) at zero pressure and temperature must be higher than (neglecting refinements of order to the latter statement). Work using the MIT-based EOS has shown that the latter condition imposes that the MIT bag constant B must be larger than 57 MeV fm (Farhi and Jaffe farhijaffe ).

Since all three flavors have the same common Fermi momentum, the absolute stability condition takes a very simple form

(6) |

Therefore, at the zero pressure point we have (without any approximation)

(7) |

where MeV. A family of curves in the plane follow, and on each one the energy per baryon is exactly for a given (Lugones and Horvath 2002).

Fig. 2 displays the stability window for the CFL phase (i.e. the region in the versus B plane where is lower than at zero pressure. Eq. (7) sets the right side boundary of the window while the left side boundary is imposed by the minimum value B = 57 MeV. The window is greatly enlarged for increasing values of with respect to the original SQM calculations which do not include any pairing (see for example, Fig. 1 of Farhi and Jaffe farhijaffe ). We have emphasized elsewhere that the pairing gap may be important for a ”CFL strange matter” state (Lugones and Horvath cflprd ; see also Madsen Madsen for a similar view).

## 3 Structure of color flavor locked stars

The study of the mass-radius relation for compact stars is a widely known tool for testing the existence of different phases of matter inside NSs. It is also well-known that simultaneous observational data on masses and radii can impose important constraints on the high density equations of state. Up to now, almost all of the measured masses of ”neutron stars” clustered within a narrow range around 1.4 . It has been conjectured that this mass scale may be due to the fact that neutron stars are formed in the gravitational collapse of supernovae and come just from the iron cores of the pre-supernovae. However, nothing fundamental precludes smaller neutron stars from existing provided some mechanism capable of creating them operates. Because small masses are not expected in core collapse models, they hold the potential for discriminating among possible equations of state. This is particularly significant given the recent claims of very low-mass/low-radii objects. As already mentioned, at least two sources, Her X-1 and RX J1856-37, are candidates for high compactness. They have been claimed to have radii and masses around (Li et al. li ; Pons et al. Pons ). Fragmentation of a larger star (perhaps due to high rotation) may be a possible way to produce sub-solar mass compact objects (Nakamura tukulito ). Nevertheless, it is clear that models and data are still subject to detailed analysis.

CFL strange matter offers the double bonus of enhancing the odds for a self-bound state, and also of allowing the existence of stable stars made up entirely of this phase. To pinpoint the effects of such a composition we have solved the Tolman-Oppenheimer-Volkoff equations of stellar structure, assuming a parameter set inside the windows of absolute stability discussed above (Lugones and Horvath cflprd ). Fig. 3 shows sequences of compact star models thus calculated for particular values of the parameters , and (see caption). These values of the latter quantities have been selected to highlight a potentially important feature of pairing in dense quark matter, namely the possibility of obtaining very compact stellar models. As expected, the mass-radius relation of stars made up of CFL strange matter presents the same qualitative shape as the curves found for strange stars . This is a direct consequence of the existence of a zero pressure point at finite density (of the order of the nuclear saturation density).

To assess the robustness of the results, and to understand the
outcome of masses and radii for a given set of parameters, we have
explored the effect of each parameters of the EOS on the static
structure of the stellar models. As shown in Figs. 4-7, and quite
analogously to the case of strange stars, the effect of increasing
while holding and fixed is to lower both the
maximum radius and the maximum mass of the
stellar configurations. The mass of the strange quark works
in the same direction although the effect is much smaller. The
effect of the pairing gap is qualitatively different and
much more interesting. An increase of increases the value
of both and . However, since the increase of
strongly enlarges the stability window, the combination
of a large with a large allows the construction of
very compact models made up of absolutely stable CFL strange
matter. This is particularly illustrative when we set . In this case the EOS adopts, to order , the same
simple form as unpaired SQM. However,
since the stability window is considerably enlarged, large values
of are allowed and very compact stars are found. This shows
how, although the effects in the EOS are only a few percent
( ), the enlargement of the stability
window allows a very rich set of stellar configurations, ranging
from very compact to quite extended objects. As is seen from the
Figures, for the smallest values of the models tend to be less
compact and show radii up to 13 km and masses as high as 2.4
. The most extended configurations are obtained for a
combination of small and high . ^{2}^{2}2 It can be
checked that when the two particular values of B used by Alford
and Reddy (2002) are selected (B = 165 MeV and B=
185 MeV; corresponding to B = 97 MeV fm and B = 153 MeV
fm respectively) the values obtained for and
from Figs. 4-7 are quantitatively in agreement (even if
the latter value has to be extrapolated if needed in these Figs.).
Moreover, a variety of CFL strange star models can be explored by
varying the parameters, although, for example, a large variation
in does not result in substantial modifications of
and . It is not possible to select the most
likely range of the parameters, which should be limited by
observational arguments much in the same way as done for strange
matter models.

## 4 Discussion

We have addressed in this work stellar models constructed with the simplest version of a ”CFL strange matter” phase. The CFL phase at zero temperature has been modelled as an electrically neutral and colorless gas of quark Cooper pairs, in which quarks are paired in such a way that all the flavors have the same Fermi momentum and hence the same number density (Rajagopal and Wilczeck rajagopal ). Since the strange quark is massive, some energy must be spent in order to keep a common Fermi momentum for all three flavors. However, more than that is gained from the energy gap of the pairing. The model allows CFL strange matter to be the true ground state of strong interactions for a wide range of the parameters , and . It is remarkable that the strange matter hypothesis may be quite boosted by pairing interactions. We have explored the mass-radius relation for all the values of the parameter space of the EOS that give absolutely stable CFL strange matter. Very compact configurations are found that could help to explain the recently claimed compactness of a few neutron stars. Also, for some parameter choices the models are massive and extended.

A feature that has been largely explored and debated in the context of strange star models is the existence of a normal matter crust (Zdunik crust ). Bare quark surfaces may alter drastically the ability of radiating photons, and thus they may produce interesting signatures for their identification (Page and Usov PU ). Conversely, a normal matter crust (held in mechanical equilibrium by the electrostatic potential at the surface) may hide most of the features of exotic matter. By its very construction, it is clear that (in striking contrast with the well-studied SS) no crust could be present in the case of CFL strange stars. This is directly related to the absence of electrons in the mixture (Rajagopal and Wilczek rajagopal ), and thus to the absence of an electrostatic potential to prevent normal matter from being converted to the stable CFL state. CFL strange stars must have bare surfaces within these models. This in turn means that the transport properties of the state, which are still being explored (see, for example Shovkovy and Ellis ISPE ; Jaikumar, Prakash and Schafer 2002; Reddy, Sadzikowski and Tachibana 2003), hold the clue to the identification of these compact stars. We expect the photon emission properties of CFL strange stars to resemble those of bare SS. Briefly, pairing effects should affect the plasma frequency through the baryon number density as a small correction of order so that will not be very different from the expected for bare SS . The equilibrium photon radiation will show a very hard spectrum and a tiny luminosity, making CFL strange stars very difficult to detect. On the other hand, while the thermal emission of photons from the bare quark surface of a hot strange star (mainly by electron-positron pair production) has been shown to be much higher than the Eddington limit (Page and Usov PU ), this would not be the case for CFL SS since no electrons will be present at the surface.

As already mentioned, the fact of considering CFL SQM to be absolutely stable together with the absence of electrons in the mixture precludes the existence of a crust of normal matter, then CFL strange stars are bare by construction. In a more general approach, Alford and Reddy (2002b) found mass-radius relations with essentially the same shape as that for neutron stars made up of normal nuclear matter when considering a wider range of parameters in which ”hybrid” stars are allowed. This corresponds to a large part of the parameter space and not only do they find a normal matter envelope for all these models, but also the composition of the less massive objects becomes pure nuclear matter, as expected. In addition, as pointed out in Section 3, a subset of the models of Alford and Reddy (2002) correspond to our ”CFL strange stars”, and the full range of these self-bound models can be obtained with the aid of Figs. 4-7. The self-bound CFL state may be compared to stable diquark states, which have been found to produce very similar stellar structural properties than CFL strange matter within completely different models (Horvath SDM ; Horvath and de Freitas Pacheco HP ; Lugones and Horvath vigilantes ). We finally note that, due to the presence of and in the EOS, a linear form is obtained in particular cases only, therefore the scaling behavior of the Tolman-Oppenheimer-Volkov equations that allows the construction of simple expressions for the maximum masses and radii is lost for these CFL stars. The relevance of this modelling to understand actual observed stars is not clear as yet, but certainly the latter will continue to advance to the point at which this and other questions will be answered with confidence.

Acknowledgements: G. Lugones acknowledges the Instituto de Astronomia, Geofísica e Ciências Atmosféricas de São Paulo and the financial support received from the Fundação de Amparo à Pesquisa do Estado de São Paulo. J.E. Horvath wishes to acknowledge the CNPq Agency (Brazil) for partial financial support.

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