CHARACTERISTICS OF FREE SURFACE OF HOT STRANGE QUARK MATTER

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathmatical Scincs 2016, 3, p. 6 10 CHARACTERISTICS OF FREE SURFACE OF HOT STRANGE QUARK MATTER P h y s i c s A. G. ALAVERDYAN Chair of Wav Procsss Thory and Physics YSU, Armnia Within th framwork of MIT bag modl, proprtis of hot strang quark mattr at zro prssur ar invstigatd. It is shown that th slf-boundnss proprty of strang quark mattr has a tmpratur dpndnt charactr. With incrasing tmpratur, th strang quark mattr which is slf-bound at zro tmpratur, starting with a crtain critical tmpratur bcoms not slf-bound. Kywords: hot quark mattr, strang star, fr surfac. Introduction. It is known that during comprssion, th ordinary baryonic mattr turns to a continuous quark mattr through th phas transition. Such a mattr consists of u, d, s quarks and lptons which nutraliz lctric chargs of quarks. Du to th prsnc of strang quark, this mattr is calld strang quark mattr (SQM) [1]. Sinc thr is no a fully rliabl thory for th dscription of quark mattr phas, w considrd strang quark mattr within th MIT bag modl [2]. In th MIT bag modl th quarks ar assumd to b confind to a finit rgion of spac, in th so-calld bag, by a vacuum prssur B. Th prssur from th quarks insid th bag is providd by th Frmi prssur and intractions btwn quarks is dtrmind in th on-gluon xchang approximation corrsponding to th first ordr of strong intraction coupling constant α c = g 2 /4π, whr g is th QCD coupling constant [3]. Valus of modl paramtrs B, α c and masss of quarks ar chosn such a way to wll dscrib th proprtis of hadrons. Unfortunatly, only rgions of possibl valus for th mntiond paramtrs ar dtrmind. Masss of u and d quarks do not xcd 10 MV, manwhil Frmi nrgis of quarks in SQM xcd 300 MV. Thrfor, in our modl th u- and d- quarks ar considrd to b masslss. As a rgion of possibl valus for m s, 90 200 MV is accptd. Th valu of th currnt strang quark mass, as was rportd by th Particl Data Group, is 95±5 MV [4]. Manwhil, for th strang quark mattr in th stllar intrior anothr (constitunt) mass of th strang quark is obsrvd. Accordingly, in th prsnt work w conductd numrical calculations for th thr valus of th strang quark mass m s : {95;125;150}MV. Possibl valus for th paramtr B undrli in th intrval of 40 100 MV /fm 3. Rgardlss of th valus for paramtrs B, α c, and m s, th nrgy of hot strang quark mattr (HSQM) pr baryon numbr ε, always at a crtain concntration of th baryon numbr n = n 0 has a local minimum ε min. If ε min < m n c 2, whr m n is th nutron mass, and c is th spd of light, thn SQM is slf-bound stat of mattr. Othrwis, if ε min < M( 56 F)/56, thn SQM is th tru ground stat of mattr. Clstial bodis consisting of solly such mattr ar calld strang stars [1]. Strang stars hav a distinct surfac. On th E-mail: anialavrdyan@ysu.am

Alavrdyan A. G. Charactristics of Fr Surfac of Hot Strang Quark Mattr. 7 surfac of such objct, th concntration abruptly falls from th valu n 0 to zro. Th prsnt papr is dvotd to th invstigation of hot strang mattr at n = n 0. Basic Equations for Chmical Equilibrium in HSQM. In th ordinary mattr at th tmpratur kt m c 2 (k is Boltzmann constant, m is lctron mass, c is spd of light in vacuum) lctron-positron pairs ar producd. Plasma frquncy ω p in SQM is such that ħω p 20 MV [5] (ħ is Planck constant). In th thrmal spatial fild HSQM th quants with an nrgy lss than 20 MV ar absnt. On th othr hand, lctrons in HSQM ar quasi-dgnrat, thrfor, only layrs with an nrgy of ordr 10 20 MV ar filld (this is xactly th Frmi-nrgy of lctrons in SQM). Thrfor, th lctron-positron pairs ar capabl to gnrat only photons with an nrgy 20 MV and gratr. Assuming that an ordinary star mattr was turnd to HSQM whit th xplosion of suprnova star, th tmpratur HSQM may achiv valus T 10 12 K (kt 100 MV ). Duration of such a high tmpratur stag is ngligibl with rspct to th star liftim. Anyway, thr ar som issus that cannot b addrssd without invstigation of that stllar volution stag. On of thm is th maximum valu for th mass of cold strang stars. This issu is discussd in [6]. Th thrmodynamic quilibrium stat of HSQM is dtrmind by conditions: 1. β-quilibrium: µ d = µ s = µ u + µ. 2. Equilibrium of lctron-positron pairs: µ = µ +. 2 3. Elctrical nutrality: 3 n u 1 3 (n d + n s ) n + n + = 0. 4. Baryon numbr consrvation law: n = 1 3 (n u + n d + n s ). Hr µ i stands for th chmical potntial, manwhil n i is th concntration of i th typ of particl, whr i = u, d, s, +,. Elctrons and positrons in HSQM stand for an idal Frmi gas. With approximation α c = 0 quarks also stand for such particls. In th prsnt papr th numrical calculations ar prformd within th framwork of th such approximation. Rlation btwn th chmical potntial µ i, th concntration of particl n i and th tmpratur T is givn by th formula: n i = g i p 2 d p 2π 2 xp[ε i (p) µ i ]/kt + 1, (1) 0 whr i = u, d, s,, + ; ε i (p) = m i c 4 + p 2 c 2 ; g i is dgnracy factor of th particl i (2 for th lptons, 6 for th quarks). For quasi-dgnrat quarks, th thrmal corrctions to thrmodynamic quantitis ar considrd in th approximation of (kt /µ) 2 [7, 8]. Although lptons ar ultra-rlativistic, howvr, this approximation is not applicabl to thm. Thrfor, in numrical calculations for such particls, xact intgral xprssions ar usd. Th strang quark mass is considrd in th approximation of (m s c 2 /µ) 2. For a givn valu of th baryon numbr dnsity and T, th conditions (1 4) and Eq. (1) dtrmin th chmical potntials and th concntration of all particls. For solving ths quations th mthod from [9] was usd. Charactristics HSQM at Zro Prssur. If SQM is slf-bound, thn th condition P(n,T ) = 0 idntifis th concntration of baryon numbr n 0 and all th thrmodynamic charactristics on th fr surfac of th substanc at th T. Othrwis, th charactristics of HSQM ar dfind in th vicinity of th phas quilibrium point with baryonic mattr [10]. Numrical computations ar carrid out for th valus of MIT bag modl paramtrs, namly, B = {40; 60; 80} MV / f m 3, m s = {95; 125; 150} MV and α c = 0.

8 Proc. of th Yrvan Stat Univ., Phys. and Math. Sci., 2016, 3, p. 6 10. Th rsults of th calculations ar shown in Fig. 1 5. Fig. 1. Tmpratur dpndnc of baryon numbr dnsity n 0 at surfac of bar strang star in units of normal nuclar numbr dnsity n N = 0.15 fm 3 for diffrnt valus of modl paramtrs B and m s. Fig. 2. Enrgy pr baryon ε(n 0, T) for SQM at zro prssur as a function of tmpratur for diffrnt valus of modl paramtrs B and m s. Fig. 3. Elctron fraction paramtr log( n / n ) : Solid lins. For SQM at zro prssur as a function of tmpratur for B=60 MV/fm 3 and diffrnt valus of m s. Dashd lins. For lctron-positron fraction diffrnc paramtr log( n / / ). n n n Fig. 4. Tmpratur dpndnc of positron to lctron ratio n / n n / n. n / n at zro prssur for diffrnt valus of modl paramtrs B and m s. Fig. 1 shows th tmpratur dpndnc of baryon numbr dnsity n 0 /n N at surfac of bar strang star for diffrnt valus of modl paramtrs B and m s (n N = 0.15 fm 3 is th normal nuclar dnsity). As xpctd, th highr th tmpratur HSQM, th lowr th dnsity of th quarks. With incrasing tmpratur, th thrmal corrctions to th prssur incras. Thrfor, at a constant prssur, th highr th tmpratur, th lowr th quark concntration. With incrasing tmpratur, th valu n 0 at th fr surfac approachs th valu of normal nuclar dnsity n N. This should b tratd with caution. Indd, as in Fig. 2 is shown th valu ε = ε(n 0,T ) m n c 2 incrass with incrasing tmpratur and bcoms positiv. That is, slf-boundnss of HSQM is violatd. Thrfor, if th cold SQM cannot b in thrmodynamic quilibrium with th ordinary baryonic mattr [11], thn at a crtain tmpratur, it may b possibl. Fig. 2 shows that th smallr th possibl valu of B, th highr th critical tmpratur valu corrsponding to th violation of slf-boundnss. Calculations show that with incrasing tmpratur of HSQM, th amount of lptons incrass dramatically. Fig. 3 shows th tmpratur dpndncis of n /n 0 and (n n +)/n 0 at P = 0 for B = 60 MV / f m 3 and m s = {95; 125; 150} MV.

Alavrdyan A. G. Charactristics of Fr Surfac of Hot Strang Quark Mattr. 9 Fig. 5. Tmpratur dpndncis for th rlativ dviations of numbr dnsitis of u, d, s quarks from th symmtric valu 1/3. If in cold SQM th numbr of lctrons by 4 5 ordrs of magnitud is smallr than th numbr of quarks, thn at tmpratur kt = 80 MV, w hav n 0.1n 0. At first glanc it may sm that such an abrupt incras in th numbr of lctrons must b accompanid by a similar incras in th asymmtry in th numbr of diffrnt typs of quarks. Howvr, this dos not happn, bcaus th chmical potntial of th lctrons with incrasing tmpratur changs slightly, though thir numbr incrass so strongly. Th rason for this is vry simpl. Firstly, only a portion of ths lctrons nutraliz th charg of th quarks. Scondly, with incrasing tmpratur, lctron gas cass to b dgnratd. On on hand, incrasing th numbr of lctrons should lad to an incrasd dgnration and, on th othr hand, th chang of lctron gas in th dirction of Boltzmann gas prvnts an incras of chmical potntial of th lctrons. Th last factor is dominant, and th chmical potntial of th lctrons dos not xcd 10 20 MV. Evn with kt = 80 MV, th numbr of lctrons is an ordr of magnitud smallr than th numbr of quarks. Fig. 4 shows th tmpratur dpndnc of th ratio n +/n. It is clar that in cas of kt = 80 MV, th numbr of lctrons n n + which provid lctro nutrality, constitut {19; 29; 38} prcnts of th total numbr of lctrons for B = 80 MV / f m 3 and m s = {95; 125; 150} MV, rspctivly. Fig. 5 shows tmpratur dpndncis for th rlativ dviations of concntrations of u, d, s quarks from th symmtric valu 1/3 ( n i /n q 1/3). Dviation from symmtry in cas of T = 0 is du to th prsnc of non zro strang quark mass. This dviation incrass with incrasing tmpratur; manwhil th dviation for s quarks is mor than for d quarks. Numbr of u quarks changs rlativly littl. Conclusion. Numrical calculations within th framwork of MIT bag modl show that on th fr surfac of HSQM (P = 0), th following assrtions hold: with incrasing tmpratur, th rlativ numbrs of u and d quarks ar incrasd, manwhil, th rlativ numbrs of s quarks ar dcrasd; th mor th tmpratur of HSQM, th smallr th concntration of quarks n 0. With highr tmpratur kt 60 80 MV, th baryon numbr dnsity approachs th valu of normal nuclar dnsity n N ; nrgy pr baryon of HSQM sufficintly strong dpnds on th tmpratur. Bing slf-boundd at tmpratur T = 0 SQM bcoms non slf-boundd whn tmpratur incrass. Thrfor, for dfinit possibl valus of paramtrs for MIT bag modl, non slf-

10 Proc. of th Yrvan Stat Univ., Phys. and Math. Sci., 2016, 3, p. 6 10. bound hot strang quark mattr during cooling may b slf-bound. Thrfor, th hot hybrid nutron star with a cooling bcoms a strang quark star with th outr shll. Th substanc of this shll will consist of atomic nucli and a dgnrat lctron gas [12]; th numbr of lctrons in HSQM can rach 10 prcnt from th numbr of quarks. Dspit this, th chmical potntial of th lctrons dos not xcd th valu of 10 20 MV ; dspit th abrupt incras in th numbr of lctrons, with incrasing tmpratur, rlativ changs in th concntration of quarks from th valus at T = 0, ar ngligibl. Author xprsss his gratitud to profssor G.S. Hajyan for his valuabl support and consult during laboration of th papr. This work was supportd by Stat Committ of Scinc MES RA, in fram of rsarch projct SCS Extrm stats of mattr and thir astrophysical manifstations. Rcivd 06.07.2016 R E F E R E N C E S 1. Wittn E. Cosmic Sparation of Phass. // Phys. Rv. D, 1984, v. 30, 2, p. 272 285. 2. Chodos A., Jaff R.L., Johnson K., Thorn C.B., Wisskopf V.F. Nw Extndd Modl of Hadrons. // Phys. Rv. D, 1974, v. 9, 2, p. 3471 3495. 3. Farhi E., Jaff R.L. Strang Mattr. // Phys. Rv. D, 1984, v. 30, 11, p. 2379 2390. 4. Oliv K.A. t al. Th Rviw of Particl Physics (Particl Data Group). // Chin. Phys. C, 2014, v. 38, 9, p. 090001 [URL:http://pdg.lbl.gov]. 5. Alcock C., Farhi E., Olinto A. Strang Stars. // Astrophys. J., 1986, v. 310, 1, p. 261 272. 6. Hajyan G.S., Alavrdyan A.G. Hot Strang Stars. II: Numrical Rsults and Discussion. // Astrophysics, 2015, v. 58, 1, p. 77 88. 7. Landau L.D., Lifshitz E.M. Statistical Physics. M.: Fizmatlit, 2002 (in Russian). 8. Bisnovatyi-Kogan G.S. Physical Problms of th Thory of Stllar Evolution. M.: Nauka, 1989 (in Russian). 9. Hajyan G.S., Alavrdyan A.G. Hot Strang Stars. I: Equation of Stat of Hot Strang Quark Mattr. // Astrophysics, 2014, v. 57, 4, p. 559 569. 10. Glndnning N.K. Compact Stars: Nuclar Physics, Particl Physics and Gnral Rlativity. Hidlbrg, Springr, 2000. 11. Alavrdyan G.B., Harutyunyan A.R., Vartanyan Yu.L. Nutron Stars with a Quark Cor. I: Equations of Stat. // Astrophysics, 2003, v. 46, 3, p. 361 367. 12. Vartanyan Yu.L., Grigoryan A.K., Sargsyan T.R. Modls of Strang Stars with a Crust and Strang Dwarfs. // Astrophysics, 2004, v. 47, 2, p. 189 200.