1. Clustered Quark Model Calculation of Strange Quark Matter Xuesen Na Department of Astronomy, School of Physics, PKU CSQCD II

2. Motivation  Motivation: Many believe that neutron stars might be hybrid star with quark matter phase or even entirely quark star with 2 or 3 flavor quark matter. Our group propose that a strange quark star might appear in normal solid state, possibly with quark clusters on crystal lattice. It is therefore intriguing to perform some rough and naive calculation of clustering in quark matter using some simple models.

3. Most Naive Calculation  A most naive calculation: Start from Fermi gas, consider 3 flavor quark matter and color singlet, spin 1/2, 3-quark cluster (hitherto represented by subscript 'c'). Let's first try to fix cluster mass M c at certain value while using quark mass density dependent model (QMDD) to incorporate asymptotic freedom, where the parametrization follow from Peng et al. (1999)

4. Most Naive Calculation Impose chemical equilibrium of clustering- dissolving reaction (without e - ) Then we can self-consistently solve quark chemical potential μ q from the equation with given baryon number density and temperature T with degeneracy factors

5. Most Naive Calculation where is the Fermion occupation number

6. Most Naive Calculation  Result and explanation Keep Mc=900MeV and for a relatively low temperature T=10MeV and taking D 0 =(80MeV) 2, M s0 =120MeV μ q rise first rapidly then gradually with increasing baryon number density (overall value is about 300MeV), while the cluster fraction (among total baryon number) rised rapidly after μ q has reached near 300MeV and gradually saturate.

7. Most Naive Calculation  Discussion This naive method has several shortcomings (1)The clustering-dissolving reaction which is always favored at all densities while in reality, clustering should be favored only at moderate densities and at high density we should give way to pure quark gas. (2)Constant cluster mass is an oversimplification which should in fact somehow change with increasing density. (3)Volume of clusters has not been taken into account, clusters are taken as point particle.

8. Second Naive Calculation  A revised method: cluster mass density dependence To remedy for the second shortcoming, we can put some density dependence into cluster mass within some parametrization, the main feature should be that cluster mass increase with increasing density. We use the following parametrization: where ξ is a dimensionless constant of O(1)

9. Second Naive Calculation  Result and discussion Calculate the same quantities with ξ=1,2,5,8 Cluster fraction shows the expected density behavior, maybe a reasonable choice of ξ is somewhere between 5~8, while the cluster fraction still gradually cease to decrease at large densities which is also understandable since if we ignore mass, μ q ~ ν 1/3 which is the same power-law as our cluster mass parametrization

10. Excluded Volume Method  Excluded Volume Method One way to remedy the third shortcoming is to introduced an excluded volume method i.e. consider influence of cluster's finite volume on momentum space integral. Similar method was first used by J. W. Clark et al (1984) at zero temperature and latter by Bi and Shi (1987) for finite temperature as one of the possible explanation to EMC effect. (The difference of our work being that we do not introduce running coupling constant to account for interaction among deconfined quarks)

11. Excluded Volume Method  Excluded volume method Still consider color singlet spin-1/2 3-quark clusters, Interaction between free quarks and clusters and amoung clusters comes in the form of hard-ball: as a factor of available volume multiplied to every momentum integral

12. Excluded Volume Method  To self-consistently solve cluster mass, we need a relation between volume and mass for which we use the bag model The MIT bag total energy of massless quarks can be written as where terms are volume energy, zero point energy of the bag, quark kinetic energy (T. DeGrand et al 1975 (3.6) dropping gluon interaction term) Bag constant B=161.5MeV · fm -3 =(187.7MeV) 4 adopted from K. Saito & A. W. Thomas (1995) from a fit to baryon octet and then c can be determined by requireing M(R) to reach minimum at M Λ =1115MeV:

13. Excluded Volume Method  Since hadronic vacuum has been moved out of the entire region of quark matter, compared to the environment, the cluster (as MIT bag) would have an energy of which gives M-V relation and in addition the pressure (which is in fact the pressure of the entire quark matter)

14. Excluded Volume Method  Another assumption: the relativistic EoS where energy density do not include vacuum energy B and pressure is only gas pressure without vacuum term  Then quark chemical potential μ q and cluster mass M c can be solved from integral expressions of baryon number density and energy density (as 2 equations)  For QMDD model parameters we take D 0 1/2 =50,80,110MeV for m s0 =150,120,90MeV respectively

15. Excluded Volume Method  Thus baryon number density and energy density can be written where the index i run over free quark flavors (u,d,s) and cluster (c) and η is the portion of volume available to free quarks and clusters

16. Three Flavor System The parameter c in M-V relation can be extracted from bag constant and hyperon mass, we choose where B=161.5MeV · fm -3 =(187.7MeV) 4 adopted from K. Saito & A. W. Thomas (1995) from a fit to baryon octet and Lambda hyperon mass 1115MeV which gives

17. Three Flavor System  Finally, imposing the relativistic EoS P=(1/3)ε we have cluster mass M c and quark chemical potential μ q can be solved once T and is given, and through these two quantities, we can calculate portion of available volume and baryon number portion of clusters, radius of the cluster.

18. Results and Discussion  We calculated: cluster mass M c, quark chemical potential μ q, cluster radius R, available volume η and cluster fraction for T=10,50 MeV and ν=1~20ν 0, where ν 0 is normal nuclear density 0.159fm -3

19. Results and Discussion  Cluster mass M c vs baryon number density for T=10,50MeV D 0 1/2 =50,80,110MeV M c for different settings are almost the same, and if we fit this with our previous naive parametrization, we would find ξ~5.2  M c difference from the first setting: T=10MeV D 0 1/2 =50MeV

20. Results and Discussion  Quark chemical potential vs baryon number density for T=10,50MeV D 0 1/2 =50,80,110MeV  μ q difference from the first setting: T=10MeV D 0 1/2 =50MeV

21. Results and Discussion  Cluster volume R vs baryon number density for T=10,50MeV D 0 1/2 =50,80,110MeV  R difference from the first setting: T=10MeV D 0 1/2 =50MeV

22. Results and Discussion  Cluster fraction vs baryon number density for T=10,50MeV D 0 1/2 =50,80,110MeV

23. Results and Discussion  Available portion η vs baryon number density for T=10,50MeV D 0 1/2 =50,80,110MeV

24. Results  For baryon number density in the range of about 2~20 times normal nuclear density and temperature not too high (10~50MeV), our treatment gets the result that 1.Cluster mass M c is in the range ~800-1200 MeV (increasing) 2.Quark chemical potential in the range 300~400MeV (increasing) 3.Cluster radius R in the range 0.9~0.5fm (decreasing) 4.Available volume is about 15%~30% while 80%~90% of baryon number is in cluster, in other words, most volume and baryon number is occupied by clusters at moderate densities similar to Clark's two flavor system, the behavior of available portion vary with temperature

25. Some Discussion  This is still only a naive calculation and values should not be taken seriously  Strictly speaking this method is not fully self- consistent: excluded volume effective thermodynamical potential is not statistically representable (Yukalov & Yukalova 1997), i.e. not coming in the form  Modification from what is presented in the abstract: we dropped 2-quark clusters because we are mainly interested in low temperature behavior which is realistic in astrophysical context, hence antiquarks and gluons are rare so do mesonlike clusters on the other hand, diquarks are never colorless

26. Some Discussion  Also, electrons are not considered otherwise they would complicate the chemical equilibrium  In this treatment, interaction between quarks, quarks and clusters and clusters are taken into account only through (1)chemical reaction of cluster forming and dissolving (2)hard-ball repulsion which only scale phase space entirely without influencing structure, which are surely unrealistic and oversimplifying.  Of course density range from 1~20v 0 is too large for clustered phase (if it exists) to be realized, we expect deconfinement transition to take place at lower densities and at some higher densities 2SC and CFL superconducting phases but we make no attempt to study phase transition itself since this is only a very naive and rough method which would give nonsensical estimates of transition density, latent heat, etc.

27. Some Discussion  In our treatment (and also Clark's two flavor system) EoS is not an output but an input due to the use of MIT bag model relations.  It is hard to both have simple and reasonable M-V relation and EoS input while on the other hand have realistic cluster fraction, which should decrease from almost 1 to negligible value at high density to connect with other method. The reason behind this is perhaps that quarks are not dynamically confined, which is a future task.

28. Summary We studied the possible phase of three flavor quark matter mainly characterised by clustering of deconfined quarks into colorless 3-quark spin-1/2 clusters with QMDD model in Fermi gas picture guided by astrophysical interests. In the most naivest consideration (M c ~v 1/3 parametrization of cluster mass density dependence) cluster fraction is highly sensitive to model constant We then adopted excluded volume method with cluster M-V relation and relativistic EoS from MIT bag model and find that for moderate densities most baryon number and volume are taken by clusters

29. Thank you !