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Isayev kipt khnu17

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Isayev kipt khnu17

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Isayev kipt khnu17

  1. 1. Constraining central magnetic field strength in strange quark stars Alexander Isayev Kharkov Institute of Physics Technology
  2. 2. XIII Conference "Problems of Contemporary Nuclear Energetics" 2October 18, 2017 The main focus: nuclear matter in the deconfined state. If nuclear matter is hot (high temperature) or dense (high baryon density) enough, quarks will be released from the hadronic bags – the deconfinement phase transition. The conditions of high temperature are realized in high energy heavy-ion collisions at RHIC and LHC, and the conditions of high density of about several times nuclear saturation density are met in heavy neutron stars. Strange quark matter: matter consisting of deconfined u, d, s quarks. At zero temperature and pressure, the energy per baryon in strange quark matter (SQM) for a certain range of the model QCD-related parameters can be less than that for the most stable 56Fe nucleus, and, hence, SQM can be the true ground state of matter. A. Bodmer, Phys. Rev. D 4, 1601 (1971). E. Witten, Phys. Rev. D 30, 272 (1984). E. Farhi and R. L. Jaffe, Phys. Rev. D 30, 2379 (1984). Introduction
  3. 3. XIII Conference "Problems of Contemporary Nuclear Energetics" 3October 18, 2017 Astrophysical implications. SQM can be encountered in: 1. Strange quark stars, self-bound by strong interactions (in contrast to neutron stars bound by gravitational forces). 2. Hybrid stars: SQM is metastable at zero pressure, but it can appear in the high-density core of a neutron star as a result of the deconfinement phase transition. In this case, the stability of SQM is provided by the gravitational pressure from the outer hadronic layers. 3. Small nuggets, called strangelets (can be found in cosmic rays). Introduction When a critical size droplet of SQM is formed in the interior of a neutron star, the conversion of a neutron star to strange quark star or hybrid star proceeds via the laminar combustion, accompanied by quite a strong neutrino signal ~ 1051-1052 erg/s. As a result, the neutrino luminosity has the quasi-plateau, corresponding to the period of laminar combustion, which can represent a firm signature for the existence of quark matter inside compact stars.
  4. 4. XIII Conference "Problems of Contemporary Nuclear Energetics" 4October 18, 2017 Another important aspect: compact stars are endowed with the magnetic field. Neutron stars observed in nature are magnetized objects with the magnetic field strength at the surface in the range 109-1013 G. R.C. Duncan, and C. Thompson (1992): even more strongly magnetized objects, called magnetars, can exist in the universe with the field strength of about 1014-1015 G at the surface. Golden candidates for magnetars: soft gamma-ray repeaters and anomalous X-ray pulsars. Recently it was suggested that a magnetized hybrid star, or a magnetized strange quark star can be a real source of the SGRs or AXPs. In the core of a magnetar the magnetic field strength may be even larger, H ~ 1018-1020 G (from a scalar virial theorem). Introduction
  5. 5. XIII Conference "Problems of Contemporary Nuclear Energetics" 5October 18, 2017 Introduction Possible mechanisms to generate such strong magnetic fields: • turbulent dynamo amplification mechanism in a compact star with fast rotating core; • spontaneous ordering of nucleon or quark spins in the dense core of a compact star To note: In such ultrastrong magnetic fields, because of the breaking of the rotational O(3) symmetry by the magnetic field, the total pressure becomes anisotropic having different values along and perpendicular to the field direction. The pressure anisotropy should be accounted for in strong magnetic fields characteristic to the cores of magnetars.
  6. 6. XIII Conference "Problems of Contemporary Nuclear Energetics" 6October 18, 2017 Introduction Main objectives: • to determine the domain in the model parameter space for which magnetized SQM is absolutely stable, i.e., its energy per baryon is less than that for the most stable 56Fe nucleus ; • to study the impact of a strong magnetic field on the thermodynamic properties of strange quark matter under the conditions relevant to the the interior of strange quark stars with account of the effects of the pressure anisotropy; Related articles: A.A. Isayev, Phys. Rev. C 91, 015208 (2015) A.A. Isayev, J. Phys. Conf. Ser. 607, 012013 (2015) A.A. Isayev, Int. J. Mod. Phys. A 29, 1450173 (2014)
  7. 7. XIII Conference "Problems of Contemporary Nuclear Energetics" 7October 18, 2017 General Formalism: Magnetized Strange Quark Matter A theoretical framework to study SQM: the MIT bag model. In the simplified version of the MIT bag model, quarks are considered as free fermions moving inside a finite region of space called a ”bag”. The effects of the confinement are accomplished by endowing the finite region with a constant energy per unit volume, the bag constant B. The bag constant B can be also interpreted as the inward pressure - the ”bag pressure”, needed to confine quarks inside the bag. The Lagrangian density for a relativistic system of noninteracting quarks (u,d,s) and leptons (e-) in an external magnetic field reads Dirac equation for the quark and electron spinors 1 2 1 16 ( =0,1,2,3) , , , [ ( ) ] , , i i i i u d s e L i q A m F F F A A A Hx μ μν μ μ μν μν μ ν ν μ μ μ ψ γ ψ π δ μ = ∂ − − − = ∂ − ∂ = ∑ 0[ ( ) ] , , , ,i i ii q A m i u d s eμ μ μγ ψ∂ − − = =
  8. 8. XIII Conference "Problems of Contemporary Nuclear Energetics" 8October 18, 2017 General Formalism: Magnetized Strange Quark Matter The energy spectrum of free relativistic fermions in an external magnetic field has the form 2 2 1 2 sgn 2 2 | | , ( )i z i i i s k m q H n qνε ν ν= + + = + − ν=0,1,2,… enumerates the Landau levels, n is principal quantum number, s=+1 corresponds to a fermion with spin up, s = - 1 to a fermion with spin down.
  9. 9. XIII Conference "Problems of Contemporary Nuclear Energetics" 9October 18, 2017 General Formalism: Magnetized Strange Quark Matter The grand thermodynamic potential density for nonmagnetized fermions of ith species at finite temperature { } ( ) 3 3 1 1 2 ( )/ ( )/ ln lni i i iT T i i d k g T e eε μ ε μ π − − − + ⎡ ⎤ ⎡ ⎤Ω = − + + +⎣ ⎦ ⎣ ⎦∫ For magnetized fermions of ith species: ( ) 23 3 2 2 2 1 1 0 1 2 2 2 2 2 22 | | ... ... | | ... ... ( ) ( ) z z i i z s n s n dk dk q Hd kd k q H dkπ π π π π ππ ∞ ∞ ∞ ⊥ =± =±−∞ −∞ → → =∑∑ ∑∑∫ ∫ ∫ ∫ ∫ { }2 1 0 1 1 2 , ,( )/ ( )/| | ln ln i i n s i n s iT Ti i i z s n q g H T dk e e ε μ ε μ π ∞ − − − + =± ⎡ ⎤ ⎡ ⎤Ω = − + + + ⎣ ⎦ ⎣ ⎦∑∑∫ Hence or { }02 0 0 2 1 1 2 ( )/ ( )/ , | | ( ) ln ln i i i iT Ti i i z q g H T dk e eν νε μ ε μ ν ν δ π ∞∞ − − − + = ⎡ ⎤ ⎡ ⎤Ω = − − + + + ⎣ ⎦ ⎣ ⎦∑ ∫
  10. 10. XIII Conference "Problems of Contemporary Nuclear Energetics" 10October 18, 2017 General Formalism: Magnetized Strange Quark Matter 2 02 0 2 2 2 2 2 2 4 2 means the integer part of the argument 2 3 for quarks (# o max , , , , , , , , max | | ( ) ln , | | , , , [...] | | , i i F iii i i i F i i i i i i F i i i i i i i kq g H k m m m m q H k m m I I q H g ν ν ν ν ν ν ν ν ν ν μ δ μ π ν μ μ ν = ⎧ ⎫+⎪ ⎪ Ω = − − −⎨ ⎬ ⎪ ⎪⎩ ⎭ = + = − ⎡ ⎤− = ⎢ ⎥ ⎣ ⎦ = ∑ f colors) 1 for electrons, ⎧ ⎨ ⎩ The grand thermodynamic potential density for fermions of ith species in the external magnetic field at zero temperature reads The number density of fermions of ith species i i i T ρ μ ⎛ ⎞∂Ω = −⎜ ⎟ ∂⎝ ⎠ max ,0 ,2 0 | | (2 ) 2 i ii i i F q g H k ν ν ν ν ρ δ π = = −∑ 2 02 0 2 2 2 2 2 2 4 2 means the integer part of the argument 2 3 for quarks (# o max , , , , , , , , max | | ( ) ln , | | , , , [...] | | , i i F iii i i i F i i i i i i F i i i i i i i kq g H k m m m m q H k m m I I q H g ν ν ν ν ν ν ν ν ν ν μ δ μ π ν μ μ ν = ⎧ ⎫+⎪ ⎪ Ω = − − −⎨ ⎬ ⎪ ⎪⎩ ⎭ = + = − ⎡ ⎤− = ⎢ ⎥ ⎣ ⎦ = ∑ f colors) 1 for electrons, ⎧ ⎨ ⎩ i i i T ρ μ ⎛ ⎞∂Ω = −⎜ ⎟ ∂⎝ ⎠
  11. 11. XIII Conference "Problems of Contemporary Nuclear Energetics" 11October 18, 2017 General Formalism: Magnetized Strange Quark Matter The energy density for fermions of ith species at zero temperature 2 2 2 , 8 , 8 8 i i l i t i i i H E E B H H p B p HM B π π π = + + = − Ω − − = − Ω − + − ∑ ∑ ∑ In the MIT bag model, the total energy density, longitudinal pl and transverse pt pressures in quark matter are given by [Phys. Rev. C 82, 065802 (2010)] M=Σi Mi is the total magnetization, Mi is the partial magnetization i i i iE μ ρ= Ω + 2 02 0 2 4 max , , , , , | | ( ) ln i i F iii i i i F i i kq g H E k m m ν ν ν ν ν ν ν μ δ μ π = ⎧ ⎫+⎪ ⎪ = − +⎨ ⎬ ⎪ ⎪⎩ ⎭ ∑ i i iM H μ ∂Ω⎛ ⎞ = −⎜ ⎟ ∂⎝ ⎠ The total pressure in magnetized SQM is anisotropic. There are two sources of the pressure anisotropy: • the matter part ~ M • the field part ~ H2/8π (the Maxwell term) In strong magnetic fields, for nonferromagnetic medium the field part is dominating over the matter part and represents the major source of the pressure anisotropy.
  12. 12. XIII Conference "Problems of Contemporary Nuclear Energetics" 12October 18, 2017 Absolute stability window for magnetized SQM The absolute stability window: at zero external pressure and temperature 56 ( Fe)m m H B BMSQM ud E E ε ρ ρ < < Zero external pressure conditions in the MIT bag model description of SQM: 2 0, 8 0 l H H i i H t H H p B p H H π ⎧ = −Ω = Ω = Ω + +⎪⎪ ⎨ ∂Ω⎪ = −Ω + = ⎪ ∂⎩ ∑ 0 , i H i i M H H μ ∂Ω ∂Ω⎛ ⎞ = ⇒ = − ⎜ ⎟ ∂ ∂⎝ ⎠ ∑ In order to satisfy this equation, the bag pressure should be field dependent. Otherwise, this equation would mean that the response of the system to an external magnetic field would considerably exceed the paramagnetic response, that is hard to expect without spontaneous spin ordering in the system. 0 4 H B M Hπ ∂ − − = ∂ 1 3 B f f ρ ρ= ∑,
  13. 13. XIII Conference "Problems of Contemporary Nuclear Energetics" 13October 18, 2017 Absolute stability window for magnetized SQM In order to determine the absolute stability window: 1. To find the respective chemical potentials of all fermion species: a) Charge neutrality 2 3 0u d s e ρ ρ ρ ρ −− − − = b) Chemical equilibrium with respect to weak processes , , e e e e d u e u e d s u e u e s s u d u ν ν ν ν − − − − → + + + → + → + + + → + + ↔ + ,d u d se μ μ μ μ μ−= + = c) From the stability constraint: 56 ( Fe) 930 MeVm H B MSQM E ε ρ = ≈ 2. The upper (lower) bound on the bag pressure from the absolute stability window: 2 ( ) , , , ( , , ) ( ) 8 u l i i u d s e i u d e H B H π= = = − Ω −∑ 1) Upper bound on B: 56 19 ( Fe) 930 MeV (for 10 G)m H B ud E Hε ρ = ≈ <2) Lower bound on B:
  14. 14. XIII Conference "Problems of Contemporary Nuclear Energetics" 14October 18, 2017 Absolute stability window for magnetized SQM The maximum value of the upper bound Bu: Bmax=Bu(H=0)≈85 MeV/fm3 for ms=90 MeV, Bmax≈80 MeV/fm3 for ms=120 MeV, Bmax≈75 MeV/fm3 for ms=150 MeV Bu vanishes at Hmax≈ 3.3·1018 G for ms=90 MeV; at Hmax≈ 3.2·1018 G for ms=120 MeV at Hmax≈ 3.1·1018 G for ms=150 MeV Thus, in order magnetized SQM would be absolutely stable, the magnetic field strength should satisfy the constraint H<Hmax. In fact, Hmax represents the upper bound on the magnetic field strength which could be reached in a magnetized strange quark star mu = md = 5 MeV, and ms = 90; 120; 150 MeV; For the lower bound Bl: Bmax=Bl(H=0)≈57 MeV/fm3; Bl vanishes at Hl,0≈ 2.7·1018 G
  15. 15. XIII Conference "Problems of Contemporary Nuclear Energetics" 15October 18, 2017 Absolute stability window for magnetized SQM The maximum values of the bounds ρB u , ρB l : ρB u≈5.6ρ0, ρB l≈4.4ρ0 for ms=90 MeV, ρB u≈5.4ρ0, ρB l≈4.3ρ0 for ms=120 MeV, ρB u≈5.2ρ0 , ρB l≈4.1ρ0 for ms=150 MeV The increase of the current mass ms leads to the decrease of the upper bound on ρB u: ρB u(H=0) ≈ 2.4ρ0 for ms=90 MeV; ρB u(H=0) ≈ 2.3ρ0 for ms=120 MeV; ρB u(H=0) ≈ 2.2ρ0 for ms=150 MeV Magnetic fields H>~ 1018 G strongly affect the upper and lower bounds on the baryon number density from the absolute stability window. mu = md = 5 MeV, and ms = 90; 120; 150 MeV;
  16. 16. XIII Conference "Problems of Contemporary Nuclear Energetics" 16October 18, 2017 Magnetized strange quark stars For the parameters within the absolute stability window, magnetized strange quark stars made up entirely of SQM and self-bound by strong interactions can be formed. Aim: to study thermodynamic properties of magnetized SQM under conditions relevant to the interior of strange quark stars with account of the effects of the pressure anisotropy.
  17. 17. XIII Conference "Problems of Contemporary Nuclear Energetics" 17October 18, 2017 SQM in inhomogeneous magnetic field Magnetic field varies from the core to the surface. This variation can be modeled by the dependence H(μB): 0 0 ( ) 1 B B B B s cenH H H e γ μ μ β μ μ ⎛ ⎞− − ⎜ ⎟ ⎝ ⎠ ⎡ ⎤ ⎢ ⎥= + − ⎢ ⎥ ⎣ ⎦ In numerical calculations: Hs = 1015 G, the exponential parametrization with β=45, γ=3. μB0 is the baryon chemical potential at the surface of a strange quark star, Hcen and Hs– central and surface magnetic field strengths The current quark masses: mu = md = 5 MeV, and ms = 150 MeV. The bag pressure: B = 74 MeV/fm3 - smaller than the upper bound B ≈ 75 MeV/fm3 from the absolute stability window for these current quark masses.
  18. 18. XIII Conference "Problems of Contemporary Nuclear Energetics" 18October 18, 2017 SQM in inhomogeneous magnetic field From the set of four equations one can find the chemical potentials of quarks and electrons and then the baryon chemical potential at the surface of a strange quark star: In order to determine the baryon chemical potential μB0 at the surface of a strange quark star for this set of parameters: 2 3 0u d s e ρ ρ ρ ρ −− − − = 1) Charge neutrality 2) Chemical equilibrium with respect to weak processes ,d u d se μ μ μ μ μ−= + = 3) Further we assume a spherically symmetric radial distribution of the magnetic field inside a star. Then the longitudinal pressure pL should vanish at the surface of a star: 2 0 8 l i i H p B π = − Ω − − =∑ 0 0 0 0 927.4 MeVB u d sμ μ μ μ= + + ≈
  19. 19. XIII Conference "Problems of Contemporary Nuclear Energetics" 19October 18, 2017 Anisotropic pressure of SQM in inhomogeneous H Transverse (the upper 3 curves) and longitudinal (the lower 3 curves) pressures in magnetized SQM vs. μB at T=0 and variable central field Hcen. The full dots correspond to the points where pL'(μB)=0. States of MSQM with the central magnetic field, characterized by the appearance pL'(μB)<0, are unstable: instability is developed along the magnetic field
  20. 20. XIII Conference "Problems of Contemporary Nuclear Energetics" 20October 18, 2017 Energy density of magnetized SQM vs. μB The energy density E (upper curves except the lower curve) and its matter part contribution Em=E-H2/8π (the lower curve) vs. μB. The relative role of the matter Em and magnetic field Ef contributions to E=Em+ Ef: the matter part dominates over the field part at such μB and H.
  21. 21. XIII Conference "Problems of Contemporary Nuclear Energetics" 21October 18, 2017 Anisotropic EoS of magnetized SQM Because of the pressure anisotropy in strongly magnetized SQM, the equation of state of the system is also highly anisotropic.
  22. 22. XIII Conference "Problems of Contemporary Nuclear Energetics" 22October 18, 2017 Conclusions In summary, we have considered the behavior of SQM at zero temperature under additional constraints of charge neutrality and beta equilibrium within the framework of the MIT bag model in strong magnetic fields up to 1019 G. It has been determined the absolute stability window of magnetized SQM in the planes “H – B” and “H – ρB”, for which at zero external pressure: 1) magnetized SQM has the energy per baryon less than that of the most stable 56Fe nucleus, 2) the energy per baryon of magnetized two-flavor quark matter is larger than that of 56Fe nucleus. It has been clarified that the upper bound on the bag pressure from the absolute stability window vanishes at H~3· 1018 G. The magnitude of this field, in fact, plays the role of the upper bound on the magnetic field strength which can be reached in a strongly magnetized strange quark star.
  23. 23. XIII Conference "Problems of Contemporary Nuclear Energetics" 23October 18, 2017 Conclusions For the parameters within the domain of absolute stability, magnetized strange quark stars made up entirely of SQM and self-bound by strong interactions can be formed. It has been studied the impact of a strong magnetic field, varying with the baryon chemical potential, on thermodynamic properties of SQM in the interior of a magnetized strange quark star: • It is clarified that the central magnetic field strength is bound from above by the value at which the derivative pL'(μB) vanishes first somewhere in the interior of a strange quark star under varying the central field. Above this upper bound, the instability along the magnetic field is developed in magnetized SQM. • The total energy density E, transverse pT and longitudinal pL pressures in magnetized SQM have been calculated as functions of the baryon chemical potential. Also, the highly anisotropic EoS has been determined in the form of pT(E) and pL(E) dependences.
  24. 24. XIII Conference "Problems of Contemporary Nuclear Energetics" 24October 18, 2017 Thank you for attention

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