Properties of Nuclear Matter

Kyrl Alekseevich Bugaev Bogolyubov ITP, Kiev, Ukraine

1. Equation of state (EoS) of nuclear matter: from Walecka model to thermodynamically self-consistent! models (1.5 hours). ! 2. Induced surface tension EoS for nuclear! and hadronic matter and quantum virial coefficients (1.5 hours). ! ! 3. Statistical multifragmentation model (SMM) of atomic nuclei, its exact analytical solution and nuclear liquid-gas phase transition (1 hour). ! ! 4. Critical exponents of classical and statistical EoS (1 hour). !

! ! Truskavets, October 11, 2018! Why Nuclear Matter?

1. Since it is sufficiently simple object and can be studied at low and intermediate energies of nuclear reactions

2. Nuclear Matter has a liquid-gas phase transition and hence it can be used as a realistic test site to verify the ideas on phase transition signals in finite systems 3. Still it is located at the frontier: superheavy elements; vacuum e.-m. instability against e+e- production, if total electric charge in reaction > 137; reactions with radioactive nuclei; neutron stars equation of state (EoS)

4. It has plenty of various applications in our life! NM Frontiers NN--ZZ diagramsdiagrams ofof thethe atomicatomic nucleinuclei superheavy NN--ZZ diagramsdiagrams ofof thethe atomicatomicelements nucleinuclei

The valleyThe of valley stability of stability

25 A. Titov, ILE, Osaka-U, 2010 Lecture 1. Elementary Particles of Nature 25 A. Titov, ILE, Osaka-U, 2010 Lecture 1. Elementary Particles of Nature NM Frontiers

Till now we do not know whether stability island exists there!

? ?

Till now these reactions are not well understood!

26 A. Titov, ILE, Osaka-U, 2010 Lecture 1. Elementary Particles of Nature Nuclear Liquid = Compound Nucleus EvEvaappoorraattioionn/fis/fisssioionn t>1000 fm/c mass distribution of fragments t=0 fm/c EvEvaappoorraattioionn/fis/fisssioionn t>1000 fm/c But a few MeV A more... t=0 fm/c A + B A’ + X Incliquidrease few MeVA A + B A’ + X droplet CN CN Probability (a.u.) Moderate heating Probability (a.u.) ρ∼ρ 0 Moderate heating

ρ∼ρ Intermediate Mass Fragment A’ Intermediate Mass Fragment A’ 0 Compound Nρuc0 le usis (C normalN) is an equilibra nuclearted hot nuc ledensityus whose excitation energy is distributed over many microscopic d.o.f. (introduced by Niels Bohr in 1936-39) Initial state Liquid Sequential evaporation model—Weiskopf 1937, ! Statistical fission model—Bohr-Wheeler 1939, Frenkel 1939 0 T Compound Nucleus (CN) is an equilibrated hot nucleus whose excitation energy c is distributed over many microscopic d.o.f. (introduced by Niels Bohr in 1936-39) Mixed Phase Sequential evaporation model—Weiskopf 1937, Nuclear Matter Statistical fission model—Bohr-Wheeler 1939, Frenkel 1939 Final states Phase Diagram Gas

5 10 15 20 T (MeV) 10 NNuuccleleaarr bbNuclearrreeaakk-- uuMultifragmentationNNppuuccleleaarr bbrreeaakk -- =uu pNop Liquid! t 0 fm/c = t=0 fm/c NNuuccleleaarr bbrreeaatk>100k--uupp fm/c t>100 different fm/c mass p t=0 fm/cp distribution t>100 fm/c of fragments But a few MeV A more... p A + B A’ + X Increase few MeVA A + B A’ + X But a few MeV A more... A + B A’ + X Increase few MeVA A + B A’ + X or oror Probability (a.u.) Probability (a.u.) Heating Probability (a.u.) Probability (a.u.) Peripheral ρ<ρ P~0 AA collision 0 Equilibrated source Heating slow expansion Heating ’ ’ Peripheral Perρipher<ρ alP~0 Intermediate Mass Fragmentρ<ρA P~0 Intermediate Mass Fragment A Power-low fr0agm ent mass distribution aroundEqui criticallibr poiatnted, 0Y ( A sour)~A-τ ce Equilibrated source AA collision AA collision Liquid Intermediate Mass Fragment A’ Intermediate Mass Fragment A’ Cansl beow wel expansil understoodon within an equilibrium statislstiowcalInitial appr expansi oachstate on !0 (Randrup&Koonin, D.H.E. Gross et al, Bondorf-Mishustin-Botvina, Hahn&Stoecker,...) Tc Initial state Liquid -τ -τ !0 Power-low fragment mass distrPowibutieron-l owaround fragm crentitical m asspoi dintst, Yribut(A)i~Aon aroundMixed cri tPhaseical point, Y(A)~A Nuclear Matter Can be well understood within Canan equi be lwibrelilum under statstioodstical w iapprthin oachan equilibrium statistical approach Tc Final states (Randrup&Koonin, D.H.E. PhaseGross etDiagram al, Bondorf-Mishustin-Botvina, Hahn&Stoecker,...) (Randrup&Koonin, D.H.E. Gross et al, Bondorf-Mishustin-Botvina, Hahn&Stoecker,...) Gas Mixed Phase 5 10 15 20NuclearT (MeV) Matter 10 Final states Phase Diagram Gas

5 10 15 20 T (MeV) 10 Nuclear Multifragmentation as a Phase Transition

But a few MeV A more... A + B A’ + X Increase few MeVA A + B A’ + X Probability (a.u.) Probability (a.u.)

Intermediate Mass Fragment A’ Intermediate Mass Fragment A’

Initial state Liquid !0

Mixed Phase Nuclear Matter Final states Phase Diagram Gas

5 10 15 20 T (MeV) 10 Our Major Aims

1. Study the nuclear liquid-gas PT using different approaches

2. Become familiar with mean-field approximation and statistical models of cluster type etc Outline of lecture I

1. Historical introduction

2. Formal definition of grand canonical ensemble (GCE) and L. van Hove axioms of statistical mechanics

3. Properties of heavy nuclei and nuclear matter

4. Walecka model and nuclear matter EoS

5. Phenomenological generalization of Walecka model

6. Summary Strong Interaction Discovery

DiscoveryDiscovery ofof thethe protonproton (( RutherfordRutherford 19181918 ))

shower of hydrogen nuclei - Protons atomic number of hydrogen is 1 Conclusion: proton is a constituent of a nucleus

First nuclear reaction performed in a laboratory!

21 A. Titov, ILE, Osaka-U, 2010 Lecture 1. Elementary Particles of Nature Strong Interaction Discovery DiscoveryDiscovery ofof thethe neutronneutron (( ChadwickChadwick 19321932 ))

(I&F Joliot Curie !!! p flux)

James Chadwick unknown high penetrating radiation (neutral particle) (1891-1974)

22 A. Titov, ILE, Osaka-U, 2010 Lecture 1. Elementary Particles of Nature Strong Interaction Discovery DiscoveryDiscovery ofof thethe neutronneutron (( ChadwickChadwick 19321932 ))

James Chadwick unknown neutral particle = neutron (1891-1974)

The Nobel Prize in Physics 1935

neutron is a constituent of a nucleus

23 A. Titov, ILE, Osaka-U, 2010 Lecture 1. Elementary Particles of Nature Strong Interaction Discovery

PredictionPrediction ofof pionpion ((ππππππ)) In 1935 H. Yukawa introduced interaction between nucleons (proton and neutron) in nucleus. “Nucleons (protons and neutrons) are held together by force stronger than electrostatic repulsion of protons”

Hideki Yukawa (1907-1981) Nobel Prize 1949 In fact, Yukawa predicted a new particle –quanta of strong interaction meson, with mass

41 A. Titov, ILE, Osaka-U, 2010 Lecture 1. Elementary Particles of Nature Discovery of muons came first…

Discovery of muon: J. Street and E. Stevenson 1937 They found (in cloud chamber) penetrating cosmic ray tracks with unit charge but mass in between electron and proton (Yukawa particle?). Muons were proven not to have any nuclear interactions and to be just heavier versions of electrons. H. Bethe and R. Marshak suggested that the muon might be the decay product of the particle needed in the Yukawa theory, so the search of Yukawa particle was continued. Later was known that µ meson decays to electron and two invisible neutrinos via weak interactions (β decay): µ ! 2ν+e.

Particle Electric charge Mass (x 1.6 10-19 C) (GeV=x 1.86 10-27 kg) e −1 0.0005 µµµ −1 0.106 p +1 0.938 n 0 0.940 γγγ 0 0 Why does the Nature need muons?

elementary particles by 1938 A. Titov, ILE, Osaka-U, 2010 Lecture 1. Elementary Particles of Nature Discovery of pion

DiscoveryDiscovery ofof pionpion (1947)(1947)

Cecil F. Powell (1903-1969) details on the next slide Nobel Prize 1950

•Detected in cosmic rays captured in photographic emulsion. •Unlike muons, they do interact with nuclei. •Charged pions eventually decay to muons: π ! µ + ν.

42 A. Titov, ILE, Osaka-U, 2010 Lecture 1. Elementary Particles of Nature Discovery of pion

First Pion

43 A. Titov, ILE, Osaka-U, 2010 Lecture 1. Elementary Particles of Nature Discovery of pion

Observed by Powell, Oct. 1947

44 A. Titov, ILE, Osaka-U, 2010 Lecture 1. Elementary Particles of Nature After the discovery of pion…

ElementaryElementary particlesparticles (discovered)(discovered) atat endend ofof 40th40th

Particle Electric charge Mass (x 1.6 10-19 C) (GeV=x 1.86 10-27 kg) e −1 0.0005 µµµ −1 0.106 γγγ 0 0 p +1 0.938 n 0 0.940 πππ +1,−1,(0) 0.14

45 A. Titov, ILE, Osaka-U, 2010 Lecture 1. Elementary Particles of Nature After the discovery of pion more hadrons were found! StrangeStrange particlesparticles 19471947

since them come in “V-particles” pairs with V-like tracks

46 A. Titov, ILE, Osaka-U, 2010 Lecture 1. Elementary Particles of Nature After the discovery of pion more hadrons and hyper nuclei were found! WhyWhy theythey areare ““strangestrange””

1. They are produced in pairs 2 The probability of a production is much greater, than probability of their decay

Murray Gell-Mann and Kazuhiro Nishijima introduced new quantum number: “Strangeness” …and concluded that “Strangeness is conserved in strong interactions (production) and violated in weak interactions (decay)

47 A. Titov, ILE, Osaka-U, 2010 Lecture 1. Elementary Particles of Nature hypernuclei! NB: Hagedorn Spectrum Follows from

Stat.Bootstrap Model, Veneziano Model, S.Frautschi, 1971 K.Huang,S.Weinberg, 1970 Hadrons are built from hadrons Used in string models

800 M.I.T. Bag Model,

dN/dm 700 J.Kapusta, 1981 Hadron mass spectrum 600 Hadrons are quark-gluon bags PDG 2004 500

Large Nc limit of 3+1 400 QCD 300

T. Cohen, 2009 200

! K" N", # $ % & 100

But experimentally 0 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 it is not seen... m (GeV) 20 ...Hagedorn Spectrum is seen, but not where it is supposed2 to be! to the data, itConsideris custom theary integralto form oft hexperimentale accumulate densityd of hadronic states: number of states of mass lower than m,

Nexp(m) = giΘ(m − mi), (2) !i where gi = (2Ji+1)(2Ii+1) is the spin-isospin degeneracy of the ith state, and mi is its mass. The theoretical counterpart of EqIt. ( 2is) i s exponential m Nfortheo r ( m 1) GeV<= ρ( m

Yukava interaction is not a fundamental one

The fundamental interaction of strongly interacting particles is due to colored gluons, the source of color interaction are quarks

Yukava interaction is not sufficient to construct nuclei, we need strong repulsion at very short range and moderate range attraction 3 virial coecients for the low density limit. Using these results, we discuss a few basic con- straints on the quantum EoS which are necessary to model the properties of nuclear/neutron and hadronic matter. The work is organized as follows. In Sect. 2 we analyze the quantum VdW EoS and its virial expansion, and discuss the pitfalls of this EoS. The quantum version of the IST EoS is suggested and analyzed in Sect. 3. In Sect. 4 we obtain several virial expansions of this model and discuss the Third Law of thermodynamics for the IST EoS. Some simplest applications to nuclear and hadronic matter EoS are discussed in Sect. 5, while our conclusions are formulated in Sect 6.

II. QUANTUM VIRIAL EXPANSION FOR THE VDW QUASI-PARTICLES

Necessary Apparatus Microcanonical Ensemble (MCE) of N Boltzmann particles

1 N d3x d3p 0 1 k k 2 2 Zmc(E, N, V )= E m + p U(xl,xj) N! (2⇡h¯)3 B k C Z k=1 B j q l,j C Y B X X C B C B potential energyC @ A EXACTLY conserves energy E and number of particles N (or| charge){z }

x_k and p_k are, respectively, coordinate and momentum of particle k

Canonical Ensemble (CE) of N Boltzmann particles

1 E Zce(T,N,V )= dEe T Zmc(E, N, V )= Z0 2 2 N 3 3 m + pk U(xl,xj) 1 d xkd pk j l,j = exp 2 q 3 3 P P N! Z k=1 (2⇡h¯) T Y 6 7 6 7 4 5 conserves E on average, but number of particles N (or charge) EXACTLY Grand Canonical Ensemble 4

Grandcanonical Ensemble (GCE) of Boltzmann particles

1 µN Zgc(T,µ,V )= e T Zce(E, N, V )= NX=0 µN 2 2 N 3 3 m + pk U(xl,xj) 1 e T d xkd pk j l,j = exp 2 q 3 3 P P N=0 N! Z k=1 (2⇡h¯) T X Y 6 7 6 7 4 5 conserves E and N on average only!

Pressure is deﬁned as

ln [Zce(T,N,V )] thermal p = T lim V !1 V @ ln [Zce(T,N,V )] mechanical p = T @V Apparently, in thermodynamic limit they should coincide

Similarly to the ordinary gases, in the hadronic or nuclear systems the source of hard-core repulsion is connected to the Pauli blocking e↵ect between the interacting fermionic constituents existing interior the composite particles (see, for instance, [2]). This e↵ect appears due to the requirement of antisymmetrization of the wave function of all fermionic constituents existing in the system and at very high densities it may lead to the Mott e↵ect, i.e. to a dissociation of composite particles or even the clusters of particles into their constituents [2]. Therefore, it is evident that at suciently high densities one cannot ignore the hard-core repulsion or the ﬁnite (e↵ective) size of composite particles and the success of traditional EoS used in the theory of real gases [9] based on the hard-core repulsion approach tells us that this is a fruitful framework also for quantum systems. Hence we start from the simplest case, i.e. the quantum VdW EoS [14, 15]. The typical form of EoS for quantum quasi-particles of mass mp and degeneracy factor dp is as follows

p(T,µ,n )=p (T,⌫(µ, n )) P (T,n ) , (1) id id id int id L. van Hove Axioms of Statistical Mechanics

A1: in the Grand Canonical Ensemble the pressure p ≥ 0 can be only the function of V, T and µ

A2: in thermodynamic limit along the curve T=const the pressure p can be only a monotonically decreasing function ofesponding inverse density States 1/(� p/� µ) . Exception is a phase transition region, where p=const for T=const

A3: kinks of pressure p can exist in thermodynamic limit only! For finite V they are transformed into regular behavior of isotherms. =1/(� p/� µ) Maxwell's rule eliminates the oscillating behavior of the isotherm in the phase transition zone by defining it as a certain isobar in that zone. Hence the Van der Waals EoS does not considers phase transitions correctly!

17 4

Grandcanonical Ensemble (GCE) of Boltzmann particles

1 µN Zgc(T,µ,V )= e T Zce(E, N, V )= NX=0 µN 2 2 N 3 3 m + pk U(xl,xj) 1 e T d xkd pk j l,j = exp 2 q 3 3 P P N=0 N! Z k=1 (2⇡h¯) T X Y 6 7 6 7 4 5 conserves E and N on average only!

Pressure is deﬁned as

ln [Zce(T,N,V )] thermal p = T lim V !1 V @ ln [Zce(T,N,V )] mechanical p = T Thermodynamics in Grand Canonical@V Ensemble Apparently, in thermodynamic limit they should coincide

Other thermodynamic quantities can be found from identities:

I Law p + ✏ = Ts+ µn

@p @p II Law s = ; n = @T @µ

III Law s 0, if T 0 ! !

✏ is energy density (E/V ), s is entropy density, n is baryonic charge density

F is free energy, E is internal energy ∂F F = E + T =⇒ if E = Const =⇒ F = E ∂T Full SMM 2 2 J.P. Bondorfif et Eal., =Phys.+ Rep.aT 257= (1995)⇒ F131.= −aT MoNucleardel Assum Matterptions: Properties Bethe—Weizsaecker1. Degrees of freedo mformula- fragm forent sbindingof k-nu cenergyleons of nucleus 2. Interactionofo fZt hprotonse nucle oandns w (A-Z)ithin neutrons each fragment - by Weizs¨acker mass formula

2 (A/2 − Z)2 Z2 Mo3del Parameters EW = (mN+−W0)A+a2A +a3 +a4 1 +( corrections1.1) due to shell effects A A3 In addition - binding energy of the nucleons inside fragment depends on temperature T Binding energy of nucleons at T = 0 W0 =-16 MeV T 2 T 2 Eint = EW + A =⇒ Fint = EW − A ( 1.2) Surface eneϵr0gy of spherical frϵa0gments a2 = 16 ÷ 18.5 Phenomenological Fisher’s term (correction to inﬁnite MeV matter) FF = −τ T ln(A) ( 1.3) S3y.mInmtereatcrtiyonebneetwrgeeyn fragments: a3 = 100 MeV a. background Coulomb Coulomb energy a = 0.7121 MeV small! 3Z2e2 1 4 EC = 0 , R = (3V/4π)3 ( 1.4) 0 5 R Ebx.cRiteaputlisoionn oefntehregfyragomfenfrtsagbymEexnclutded volume ϵ0 = 16 MeV

Vk = k b, Vfree = V − k b nk ( 1.5) PhenoImaginemenolo ag matterical nF withishe 50%r’s tofe rprotonsm andτF 50%= 1of. 8neutrons,4 ÷ 3.5 but protons have !no electric charge = symmetric nuclear matter Proper nucleon volume b ≈ 1 ρ0 −3 normal nuclear density ρ0 = 0.16 fm Of special interest b = b(T ) ?

Simpliﬁed Model - Inﬁnite nuclear 13 Matter: • No Coulomb interaction

• No symmetry energy

• Simpliﬁed treatment of isotopes and excited states Nuclear Matter Properties 5

3 1. Normal nuclear density n 0.16 fm is density at the center of heavy nuclei 0 '

2. At temperature T = 0 and normal nuclear density the system pressure p is zero. p = 0 is mechanical stability condition

3. Binding energy/nucleon at T = 0 and n = n is W = 16 MeV (see prev. slide) 0 0

4. Incompressibility constant of normal nuclear matter is @p K0 9 [200; 315] MeV ⌘ @n|T =0,n=n0 2

5. Proton ﬂow constraint (p(n) dependence at high n values)

6. Hard-core radius of nucleons R [0.3; 0.35] fm (see later) n 2

Similarly to the ordinary gases, in the hadronic or nuclear systems the source of hard-core repulsion is connected to the Pauli blocking e↵ect between the interacting fermionic constituents existing interior the composite particles (see, for instance, [2]). This e↵ect appears due to the requirement of antisymmetrization of the wave function of all fermionic constituents existing in the system and at very high densities it may lead to the Mott e↵ect, i.e. to a dissociation of composite particles or even the clusters of particles into their constituents [2]. Therefore, it is evident that at suciently high densities one cannot ignore the hard-core repulsion or the ﬁnite (e↵ective) size of composite particles and the success of traditional EoS used in the theory of real gases [9] based on the hard-core repulsion approach tells us that this is a fruitful framework also for quantum systems. Hence we start from the simplest case, i.e. the quantum

VdW EoS [14, 15]. The typical form of EoS for quantum quasi-particles of mass mp and degeneracy factor dp is as follows

p(T,µ,n )=p (T,⌫(µ, n )) P (T,n ) , (1) id id id int id 5

3 1. Normal nuclear density n 0.16 fm is density at the center of heavy nuclei 0 '

2. At temperature T = 0 and normal nuclear density the system pressure p is zero. p = 0 is mechanical stability condition

3. Binding energy/nucleon at T = 0 and n = n is W = 16 MeV (see prev. slide) Proton flow constraint 0 0

4. Incompressibility constant of normal nuclear matter is @p K0 9 [200; 315] MeV ⌘ @n|T =0,n=n0 2

5. Proton ﬂow constraint (p(n) dependence at high n values)

6. Hard-core radius of nucleons R [0.3; 0.35] fm (see later) n 2

n/n0

As you can see from these examples, some EoS do not obey this constraint!

VdW EoS [14, 15]. The typical form of EoS for quantum quasi-particles of mass mp Nuclear Matter Properties at T=0 6

EoS at T =0: p(n)=µn ✏ n [µ (m + W (n))] ⌘

✏ is energy density, W (n) is binding energy per nucleon.

From the stability condition p = 0 at T = 0 and n = n 0 )

µ µ(T =0,n= n )=m + W (n = n )=923MeV 0 ⌘ 0 0

@p @µ dW (n) =[µ (m + W (n))] + n n @n @n dn

|| |@{zp}@µ @µ dW (n) = n =0, for n = n0, @µ @n @n ) dn

since [µ (m + W (n))] = 0 for n = n 0

VdW EoS [14, 15]. The typical form of EoS for quantum quasi-particles of mass mp and degeneracy factor dp is as follows

p(T,µ,n )=p (T,⌫(µ, n )) P (T,n ) , (1) id id id int id Nuclear Matter Properties at T=0 6 6 EoS at T =0: p(n)=µn ✏ n [µ (m + W (n))] EoS at T =0: p(n)=µn ⌘ ✏ n[µ (m + W (n))] ⌘ ✏ is energy density, W (n) is binding energy per nucleon. ✏ is energy density, W (n) is binding energy per nucleon.

From the stability condition p = 0 at T = 0 and n = n 0 ) From the stability condition p = 0 at T = 0 and n = n 0 ) µ µ(T =0,n= n )=m + W (n = n )=923MeV 0 ⌘ 0 0 µ µ(T =0,n= n )=m + W (n = n )=923MeV 0 ⌘ 0 0 @p @µ dW (n) =[µ (m + W (n))] + n n @n@p @n @µ dndW (n) =[µ (m + W (n))] + n n || @n @n dn |@{zp}@µ @µ dW (n) || = n =0, for n = n0, @µ|@@{zpn}@µ @n@µ) dndW (n) = n =0, for n = n0, @µ @n @n ) dn since [µ (m + W (n))] = 0 for n = n 0 since [µ (m + W (n))] = 0 for n = n 0 At normal nuclear matter W(n) has an extremum!

Similarly to the ordinary gases, in the hadronic or nuclear systems the source of hard-coreSimilarly repulsion to the is connected ordinary gases, to the in Pauli the blocking hadronic e↵ orect nuclear between systems the interacting the source of fermionic constituents existing interior the composite particles (see, for instance, [2]). hard-core repulsion is connected to the Pauli blocking e↵ect between the interacting This e↵ect appears due to the requirement of antisymmetrization of the wave function fermionic constituents existing interior the composite particles (see, for instance, [2]). of all fermionic constituents existing in the system and at very high densities it may This e↵ect appears due to the requirement of antisymmetrization of the wave function lead to the Mott e↵ect, i.e. to a dissociation of composite particles or even the clusters of all fermionic constituents existing in the system and at very high densities it may of particles into their constituents [2]. Therefore, it is evident that at suciently lead to the Mott e↵ect, i.e. to a dissociation of composite particles or even the clusters high densities one cannot ignore the hard-core repulsion or the ﬁnite (e↵ective) size of of particles into their constituents [2]. Therefore, it is evident that at suciently composite particles and the success of traditional EoS used in the theory of real gases high densities one cannot ignore the hard-core repulsion or the ﬁnite (e↵ective) size of [9] based on the hard-core repulsion approach tells us that this is a fruitful framework composite particles and the success of traditional EoS used in the theory of real gases also for quantum systems. Hence we start from the simplest case, i.e. the quantum [9] based on the hard-core repulsion approach tells us that this is a fruitful framework VdW EoS [14, 15]. The typical form of EoS for quantum quasi-particles of mass mp also for quantum systems. Hence we start from the simplest case, i.e. the quantum and degeneracy factor dp is as follows VdW EoS [14, 15]. The typical form of EoS for quantum quasi-particles of mass mp p(T,µ,nid)=pid(T,⌫(µ, nid)) Pint(T,nid) , (1) and degeneracy factor dp is as follows

p(T,µ,n )=p (T,⌫(µ, n )) P (T,n ) , (1) id id id int id 5 6 6 3 1. Normal nuclear density n0 0.16 fm is density at the center of heavy nuclei EoS at T =0: p(n)=µn ✏ n [µ (m + W (n))] 0 ⌘ 0 ' n n = = ,n

✏ is energy density, W (n) is binding energy per nucleon.n =0 0 )=923MeV

2. At temperatureT T = 0 and normal nuclear density the system pressure p is zero. 0 n n ) = n 2 ( From the stability condition= p = 0 at T = 0 and n = n0 p = 0 is mechanical stability condition n

n ) dn W ( , 2 0 d for ) n W 2 0 ))] = 0 for ) µ µ(T =0,n= n )=m + W (n = n )=923MeV 0 0 0 n = + n )

⌘ ( 0 9 0 n n m ) n 3. Binding energy/nucleon at T = 0 and n = n0 is W0 = 16 MeV (see prev. slide) W ⌘ ))] n

= 2 ( = 0 for n + ( )=

@p dn @µ dW (n) n K n 0 dn =[µ (m + W (n))] + n n ( m n dW W @n @n dn ( W n , + = 2 )

|| d erentiating it

4. Incompressibility constant of normal nuclear matter is µ 0 m )

Nuclear Matter Properties at T=0↵ and get |@{zp}@µ @µ ,n dW (n) =0 ( µ n n

= n =0, for n = n , 0 = 0 and 0 di @p @ @ ) µ n @µ @n @n ) dn , = , n K 9 [200; 315] MeV =0 = n 0 T T =0,n=n ) ( @n 0 µ ) [ µ n n T ,n ⌘ | 2 dn n 2 ( ﬁnd ( @ @ n since [ ( =0 dW µ p n dn ))] +

since [µ (m + W (n))] = 0 for n = n T dn W 0 @ ⌘ @ 2

n ⌘ = 0 at dW ✏ ( µ n d 0 n @ @ p 5.n Proton ﬂow constraint (p(n) dependence at high n values) ) µ W @p 0 + From expression for ﬁnd µ and get n

@n + )+ ) µn ) is binding energy per nucleon. n µ n n n m ( ( =9 ( ( @ dW (n) @ dn )= µ = m + W (n)+n , di↵erentiating it n W p n W n dn ) @ @ 6.dW Hard-core radius of nucleons R [0.3; 0.35] fm (see later)

( n + = µ p 9 2 µ n 2 m =2 =[ @µ dW n d W n ⌘

( ) ( ) @ @ } =

=2 + n , 0 µ n p p 2 n µ ||

@n dn dn {z ) From expression for @ @ @ @ @ @ µ K | ) =0: n/n µ

0 0 2 T @µ d W (n = n0) = n0 for n = n0 µ @n dn2 ! ! g Similarly to the ordinary gases, in the hadronic or nuclear systems the source of is energy density,

EoS at ✏ @From thep stability condition @µ d2W (n) K 9 =9n K 9 n2 0 0 0 0 2 ⌘ @n @nT=0,n=n0 ) ⌘ dn T=0,n=n0 At normal nuclear matter W(n) has a minimum! Similarly to the ordinary gases, in the hadronic or nuclear systems the source of

Similarly to the ordinary gases, in the hadronic or nuclear systemshard-core the source of repulsion is connected to the Pauli blocking e↵ect between the interacting hard-core repulsion is connected to the Pauli blocking e↵ect betweenfermionic the interacting constituents existing interior the composite particles (see, for instance, [2]). fermionic constituents existing interior the composite particles (see, for instance, [2]). This e↵ect appears due to the requirement of antisymmetrization of the wave function of all fermionic constituents existing in the system and at very high densities it may lead to the Mott e↵ect, i.e. to a dissociation of composite particles or even the clusters of particles into their constituents [2]. Therefore, it is evident that at suciently high densities one cannot ignore the hard-core repulsion or the ﬁnite (e↵ective) size of composite particles and the success of traditional EoS used in the theory of real gases [9] based on the hard-core repulsion approach tells us that this is a fruitful framework also for quantum systems. Hence we start from the simplest case, i.e. the quantum

VdW EoS [14, 15]. The typical form of EoS for quantum quasi-particles of mass mp Nuclear Matter within Walecka (σ-ω) model J. D. Walecka, Annals Phys. 83, (1974) 491 Also Jonson&Teller, Duerr contributed: constituents are nucleons and static σ-ω mesons Lagrangian is a Lorentz scalar, then interaction one is (x = t, x, y, z)

scalar vector

Full Lagrangian is

Euler-Lagrangian equations of motion are Coupling constants are g_σ and g_ω For point-like nucleons σ meson => Yukawa potential! ω meson Coupled system is nucleons still complicated! m is nucleon mass, m_σ is σ-meson mass, m_ω is ω-meson mass 6

EoS at T =0: p(n)=µn ✏ n [µ (m + W (n))] ⌘

✏ is energy density, W (n) is binding energy per nucleon.

From the stability condition p = 0 at T = 0 and n = n 6 0 )

EoS at T =0: p(n)=µn ✏ n [µ (m + W (n))] µ ⌘µ(T =0 ,n= n )=m + W (n = n )=923MeV 0 ⌘ 0 0

✏ is energy density, W (n) is binding energy per nucleon. @p @µ dW (n) =[µ (m + W (n))] + n n @n @n dn

From the stability|| condition p = 0 at T = 0 and n = n0 |@{zp}@µ @µ dW (n) ) = n =0, for n = n0, @µ @n @n ) dn µ µ(T =0,n= n )=m + W (n = n )=923MeV 0 ⌘ 0 0

since [µ (m + W (n))] = 0 for n = n0 @p @µ dW (n) =[µ (m + W (n))] + n n @n @p @n dn From expression for @n ﬁnd µ and get || |@{zp}@µ @µ dWdW(n)(n) µ = m=+nW (n)+n , di=0↵erentiating, for itn = n0, @µ @n @n ) dndn )

@µ dW (n) d2W (n) =2 + n since, [µ (m + W (n))] = 0 for n = n @n dn dn2 ) 0

@p 2 From expression for ﬁnd@µ µ andd W get(n = n0) @n = n for n = n Mean-field approximation to Walecka (σ-ω) model 0 2 0 dW (@nn) dn We are interested in a static uniform and isotropicµ matter= m + beingW ( nin)+ the nground state, di↵erentiating it ! dn 2) => all spatial points and all directions are equivalent! @p @µ d W (n) K 9 =9n K 9 n2 ! 0 0 0 0 2 ⌘ @n @n2T=0,n=n0 ) ⌘ dn T=0,n=n0 => σ = <σ> averaged value, @µ dW (n) d W (n) =2 + n , ω_1 = ω_2 = ω_3 =0, @n dn dn 2 ) ω_0 = < ω_0> averaged value

2 0µ @µ d W (n = n0) => for nucleons g!!µ ) = n0 for n = n0 @n dn2 There is no explicit x-dependence in Euler-Lagrange@p equation!@µ d2W (n) K 9 =9n K 9 n2 0 0 0 0 2 ⌘ @n @nT=0,n=n0 ) ⌘ dn T=0,n=n0 Effective nucleon mass is With nucleon 4-momentum Similarly to the ordinary gases, in the hadronic or nuclear systems the source of Formal solution for nucleons g ! 0µ) Kµ = kµ g ! 0µ ! µ ! µ

Similarly to the ordinary gases, in the hadronic or nuclear systems the source of 6

EoS at T =0: p(n)=µn ✏ n [µ (m + W (n))] ⌘

✏ is energy density, W (n) is binding energy per nucleon. 6

EoS at T =0: p(n)=µn ✏ n [µ (m + W (n))] From the stability condition p = 0 at T = 0 and⌘ n = n 0 ) ✏ is energy density, W (n) is binding energy per nucleon. µ µ(T =0,n= n )=m + W (n = n )=923MeV 0 ⌘ 0 0

From the stability condition p = 0 at T = 0 and n = n0 @p @µ dW (n) ) =[µ (m + W (n))] + n n @n @n dn µ0 µ(T =0,n= n0)=m + W (n = n0)=923MeV || ⌘ |@{zp}@µ @µ dW (n) = n =0, for n = n0, @µ @n @n @)p dn 6 @µ dW (n) =[µ (m + W (n))] + n n EoS at T =0: p(n)=µn ✏ n [µ (m + W (n))] @n @n dn ⌘ || since [µ (m + W (n))] = 0 for n = n0 |@{zp}@µ @µ dW (n) ✏ is energy density, W (n) is binding energy per nucleon. = n =0, for n = n0, @µ @n @n ) dn From expression for @p find µ and get From the stability condition p = 0 at T = 0 and n = n @n 0 ) dW (n) since [µ (m + W (n))] = 0 for n = n µ = m + W (n)+n , di↵erentiating it 0 µ µ(T =0,n= n )=m + W (n = n )=923MeV 0 ⌘ 0 0 dn ) @p From expression for @n find µ and get @p @µ@µ dW (dWn) (n) d2W (n) =[µ (m + W (n))] + n n=2 + n , dW (n) @n @n@n dn dn µ = mdn+2W (n)+)n , di↵erentiating it || dn ) |@{zp}@µ @µ dW (n) MostMost achievementsachievements inin mid.mid. ofof XXXX Cent.Cent. mademade onon basibasi= nss ofof =0, for n = n0, 2 @µ @n @n ) dn @µ d 2W (n = n0) NonNon--relativisticrelativistic andand RelativisticRelativistic quantumquantum mechanicsmechanics @µ dW (n=) n0 d W (n) for n = n0 =2@n + n dn,2 2 ) NRNR QuantumQuantum mechanicsmechanics since [µ (m + W (n))] = 0@n for n = ndn dn 0 @p @µ d2W (n) @µ d2W2(n = n ) From expression for @p find µKand0 get9 =9n0 K0 9 n 0 @n ⌘ ) = ⌘n0 0 2 for n = n0 @n @nT=0,n=n0 @n dn2dn T=0,n=n0 dW (n) Nielsµ Bohr= m +WernerW (n Heisenberg)+n , di↵erentiating it Waleckadn (σ-ω) model: equation) for nucleons (1885-1962) (1901-1976). @p @µ d2W (n) K 9 =9n K 9 n2 @µ dW (n) d2W (nDirac) equation: 0 0 0 0 2 =2 + n , ⌘ @n @nT=0,n=n0 ) ⌘ dn T=0,n=n0 2 0µ ) µ µ 0µ µ RelativisticRelativistic theorytheory @n dn g!dn!µ ) K = k g!!whereµ K/ = µK 2 @µEffectived nucleonW (n = massn0) is With nucleon 4-momentum is = n0 for n = n0 @n dn2 Erwin Schrödinger Paul Dirac 0µ µ µ 0µ g!!µ ) K = k g!!µ (1887-1961) (1902-1984) 2 @p @µ35 d W (n) A. Titov, ILE, Osaka-U, 2010 Lecture 1. Elementary Particles of Nature K 9 =9n K 9 n2 0 Using0 properties of γ-matrices0 one0 can2 find the eigenvalues of ⌘ @n @nT=0,n=n0 ) ⌘ dn T=0,n=n0 Dirac operator Similarly to the ordinary gases, in the hadronic or nuclear systems the source of => g ! 0µ) Kµ = kµ g ! 0µ K/ = Kµ ! µ ! µ µ The standard procedure gives theSimilarly eigen values to thefor energy ordinary of gases, in the hadronic or nuclear systems the source of

+ 2 2 for nucleons E = k +(m⇤) + g!!µ , q for antinucleons E = k2 +(m )2 g ! . ⇤ ! µ q 34

зовалось импульсное распределение идеального газа нуклонов, апривведении‘энергии сжа-

тия’ учитывалось взаимодействие между нуклонами. Но взаимодействие должно менять им-

пульсное распределение идеального газа, и возникает проблема совместности релятивистско- 34 го Ферми распределения нуклонов по импульсам ифункциональнойформы‘энергии сжатия’. зовалось импульсное распределение идеального газа нуклонов, апривведении‘энергии сжа-

Для решения этойтия’ проблемыучитывалось взаимодействие напомним, что между в нукло релятивистскойнами. Но взаимодействие модели должно среднего менять поля им- Ва- лечки [10] (см. такжепульсное[11]) распределениевзаимодействие идеальн описываетсяого газа, и возникает скалярным проблема совместностиφ ивекторным релятивистскоU µ мезон- - го Ферми распределения нуклонов по импульсам ифункциональнойформы‘энергии сжатия’. ными полями с вершинами мезон-барионного взаимодействия в функции Лагранжа: gs ψψφ¯ Для решения этой проблемы напомним, что в релятивистской модели среднего поля Ва-

¯ µ µ и gv ψγ ψ Uµ. Длялечки ядерной[10] (см материи. также [11]) ввзаимод термодинамическомействие описывается скалярным равновесииφ ивекторным эти мезонныеU мезон- поля, ¯ как обычно полагаютными, являются полями с вершинами постоянны мезон-мибарионно классическимиго взаимодействия величинами в функции Лагранжа. Скалярное: gs ψψφ поле ¯ µ и gv ψγ ψ Uµ. Для ядерной материи в термодинамическом равновесии эти мезонные поля, описывает притяжение между нуклонами и изменяет нуклонную массу M M = M gs φ. как обычно полагают, являются постоянными классическими величинами.→Скалярное∗ поле−

Нуклонное отталкиваниеописывает описы притяжениевается между нулевой нуклонами компонентой и изменяет нуклонную(пространственные массу M M = компонентыM gs φ. → ∗ − должны исчезнутьНуклонное из-за трансляционной отталкивание описы инваетсявариантности нулевой компонентой) векторного(пространственные поля, которая компоненты добав- должны исчезнуть из-за трансляционной инвариантности) векторного поля, которая добав- 2 ляет U(ρ)=Cv ρ (Cv = const2) кэнергиинуклона( U(ρ) для антинуклона). В [67] было ляет U(ρ)=C ρ (Cv = const) кэнергиинуклона( U(ρ) для антинуклона). В [67] было v − − сформулировано обобщенноесформулировано УС обобщенное ядерной УС матери ядерной, материкоторое, которое включает включает и и модель модель среднего среднего по- по- 7 ля и чисто феноменологические модели какчастныеслучаи. Ограничиваясь нуклонными ля и чисто феноменологические модели какчастныеслучаи. Ограничиваясь нуклонными 7 степенями свободы, рассмотримWalecka следующее (σ ядерное-ω) model: УС достаточно pressure общего вида в большом 7 7 степенямиm! 2 m свободы2 , рассмотрим следующее ядерное УС достаточно общего вида в большом + ! Nканоническом=(2SN +1)(2 ансамблеIN +1)=4: , 2 0 2 pressure ρ 7 3 2 g! 2 2 g 2 2 каноническом ансамбле: γN d k k + ! m! 2 m 2 N =(2Smean-fieldN +1)(2IN +1)=4, 0 ∗ p(T,µ)= 3 (f+ + f 27)++ ρ U(!ρ0) 2 dn U(n)+N =(2Pg!(MSN) ,+1)(2(1.4)g IN +1)=4, 3 (2π) 2 2 − 2 +− 2 2 2 Degeneracy factor N =(2SN +1)(2IN +1)=4!,g!!k0 + mM⇤∗ ρ !0 + contributions!0 N =(2SN +1)(2IN +1)=4, 3 2 γ d k k " 2 2 nucleon spin S = 1 and isospin IN = 1 N p2(T,µ)= N где2 f+ и f idealфункции gas распределения with(f m*+ + massf ну)+клоновρ иU антинуклонов(ρ) dn U(n)+P (M ∗) , (1.4) 3 − − DegeneracyDegeneracy factor factorN =(2N =(2SNSN+1)(2+1)(2IINN+1)=4+1)=4, , m! 2 m 23 ! (2π) k2 + M 2 − ! ∗ 1 0 1 1 m! 2+ m! 2 N =(2SN +1)(2IN nucleon+1)=42 spin2 1S, = and isospin− I =1 Degeneracy factor N =(2SN +1)(2IN +1)=4, 0 nucleon spink + SM ∗ = µNandU2 isospin(ρ) I =N 2 + !0 2 N2=(2SN +1)(2I"N +1)=4f , exp N ∓ 2 ± +1N , 2 (1.5) 2 2 ± ≡ ⎡ ⎛" T ⎞ ⎤ 1 1 где f+ и f функции распределения нуклонов и антинуклоновnucleon spin SN = 2 and isospin IN = 2 − distribution function⎣ ⎝ ⎠ ⎦ µ барионный химический потенциал, и γN число нуклонных спин-изоспиновых1 состояний, 2 2 − DegeneracySimilarlyDegeneracy factor toN the=(2 ordinary factorSN +1)(2 gases,NI inN=(2 the+1)=4 hadronicSN +1)(2,g or nuclearI!N!k0 systems+1)=4+ mM the⇤∗ source,gµ of U!(!ρ)0 µ baryonic chemical γNf=4для симметричнойexp ядерной материи∓ . Зависи± мость эффективно+1 йнуклонноймассы, M ∗ (1.5) hard-core repulsion is connected to the Pauli± blocking⎡ e↵ect⎛" between the interactingT ⎞ ⎤ potential 1 1 1 от T и1 µ≡определяется экстремумомтермодинамическогопотенциалаSimilarly to the ordinary gases,( inмаксимум the hadronic давления or): nuclear systems the source of nucleonnucleon spin SN spin= 2 andS isospin= andIN = isospin2 I = fermionicN constituents2 existing interiorN the composite2 ⎣ particles⎝ (see, for instance, [2]). ⎠ ⎦ Similarlyhard-core to the repulsion ordinary3 is connected gases, in to the the Pauli hadronic blocking or e nuclear↵ect between systems the interacting the source of µ барионный химический потенциалδ p(T,µ) , и dPγN(Mчисло∗) нуклонныхd k M спин∗ -изоспиновых состояний, This e↵ect appears due to the requirement of antisymmetrization of the wavefermionic functionγN constituents3 existingSimilarly interior(f+ to+ thef the)=0 ordinarycomposite. (1.6) particles gases, in (see, the for hadronic instance, or[2]). nuclear systems the source of + δ Mpressure∗ , ≡ is dMknown,∗ − but how(2π )to find2 the 2values −of σ and ω_0 fields? Formally, T,hard-core µ repulsion! is connectedk + M to∗ the Pauli blocking e↵ect between the interacting of all fermionic constituents existing in the system and at very high densitiesThis e↵ itect may appears due to the requirement of antisymmetrization of the wave function γN =4для симметричной ядерной материи. Зависимость эффективно"hard-core repulsionйнуклонноймассы is connected toM the∗ Pauli blocking e↵ect between the interacting Плотность барионногоfermionic числа и плотность constituents энергиимогутбытьнайденыиз existing interior the composite(1.4) - (1.6), particlesис- (see, for instance, [2]). lead to the Mott e↵ect, i.e. to a dissociation of composite particles or evenof the all fermionic clusters constituentsfermionic existing constituents in the system existing and at interior very high the densities composite it may particles (see, for instance, [2]). of particles into theirот T constituentsи µ определяется [2]. Therefore, экстремумо it is evidentмтермодинамическогопотенциалаThis that e↵ect atlead appearssu tociently the Mott due e to↵ect, the i.e. requirement to a dissociation(максимум of antisymmetrization of composite давления particles): or of even the the wave clusters function This e↵ect appears due to the requirement of antisymmetrization of the wave function high densities one cannot ignore the hard-core repulsion or theof finite all (efermionic↵ective)of particles size constituents of into their constituents existing in [2]. the Therefore, system and it is at evident very highthat at densities suciently it may Similarly to the ordinary gases, in the hadronicδ p(T,µ) or nucleardP systems(M ) the sourced3k of Mof all fermionic constituents existing in the system and at very high densities it may composite particles and the success of traditional EoS used inlead the∗ theory toγ thehigh of Mott real densities gases e↵ect, one i.e. cannot to∗ a ignore dissociation(f the+ hard-coref )=0 of composite repulsion. or particles the finite or (e↵ evenective) the size clusters of N 3 lead to the Mott+ e↵ect, i.e. to a dissociation(1.6) of composite particles or even the clusters + δ M ∗ , ≡ dM∗ − composite(2π) particles2 and the success2 of traditional− EoS used in the theory of real gases hard-coreSimilarly repulsion[9] is to based connected the on the ordinary hard-core to the repulsion gases, Pauli blocking approach in theT, tells µ e hadronic↵ect us that between thisof or is aparticles the fruitful nuclear interacting! framework into systems theirk constituents+ theM ∗ source [2]. Therefore, of it is evident that at suciently [9] based on the" hard-coreof particles repulsion into approach their tells constituents us that this [2]. is a fruitful Therefore, framework it is evident that at suciently fermionichard-core constituents repulsionalso for existing quantum is interior connected systems. the Hence composite to we start the from Pauli particles the simplest blocking (see,high case, for edensities i.e. instance,↵ect the quantum between one [2]). cannot the ignore interacting the hard-core repulsion or the finite (e↵ective) size of VdW EoS [14, 15].Плотность The typical барионного form of EoS for числа quantum и плотность quasi-particles энергиalso of for massимогутбытьнайденыиз quantumm systems.high densities Hence we one start cannot(1.4) from the - ignore (1.6), simplest theис case, hard-core- i.e. the repulsion quantum or the finite (e↵ective) size of composite particlesp and the success of traditional EoS used in the theory of real gases This e↵ect appears due to the requirement of antisymmetrization of the waveVdW function EoS [14, 15]. The typical form of EoS for quantum quasi-particles of mass mp fermionic constituentsand degeneracy factor existingdp is as interiorfollows the composite particles (see, forcomposite instance, particles [2]). and the success of traditional EoS used in the theory of real gases of all fermionic constituents existing in the system and at very high[9] densities basedand on degeneracythe it hard-coremay factor repulsiondp is as follows approach tells us that this is a fruitful framework p(T,µ,n )=p (T,⌫(µ, n )) P (T,n ) , (1) [9] based on the hard-core repulsion approach tells us that this is a fruitful framework This e↵ect appears due to theid requirementid id of antisymmetrizationint idalso for quantum systems. of the wave Hence function we start from the simplest case, i.e. the quantum lead to the Mott e↵ect, i.e. to a dissociation of compositedk k2 particles1 or even the clusters palso(T,µ,n forid quantum)=pid(T,⌫ systems.(µ, nid)) P Henceint(T,n weid) , start from the simplest(1) case, i.e. the quantum 2 pid(T,⌫)=dp E(k) ⌫ , (2) dk k 1 3 (2⇡ ) 3 E(k) ( T ) VdW EoS [14, 15]. The typical form of EoS for quantum quasi-particles of mass m Z e + ⇣ pid(T,⌫)=dp E(k) ⌫ , (2) p of all fermionic constituents existing in the system and at very high densitiesVdW EoS it [14, may 15].3 The typical form of EoS for quantum quasi-particles of mass m of particles into their constituents [2]. Therefore, it is evident that at suciently (2⇡ ) 3 E(k) ( T ) p ⌫(µ, n )=µ bp+ U(T,n ) , (3) Z e + ⇣ id id and degeneracy factor dp is as follows ⌫and(µ, n degeneracyid)=µ bp+ factorU(T,ndidp )is, as follows (3) high densitieslead to one the cannot Mott ignoree↵ect, the i.e. hard-core to a16 dissociation⇡ 3 repulsion or of the composite finite (e↵ective) particles size of or even the clusters where the constant b 4V0 = 3 Rp is the excluded volume of particles with the hard- 16⇡ 3 ⌘ where the constantp(T,µ,nb 4idV)=0 = pidR(T,is⌫ the(µ, excluded nid)) volumePint(T,n of particlesid) , with the hard- (1) ⌘ 3 p p(T,µ,n id)=pid(T,⌫(µ, nid)) Pint(T,nid) , (1) compositeof particles particlescore and into radius the theirR successp (here constituentsV of0 is traditional their proper [2]. volume), EoS used Therefore, the relativistic in the theory energy it is of of evident particle real gases with that at suciently2 core radius Rp (here V0 is their properdk volume),k the relativistic1 energydk ofk particle2 with1 ~ ~ 2 2 pid(T,⌫)=dp E(k) ⌫ , (2) momentum k is E(k) k + mp and the density of point-like particles is defined~ ~ 2 2 3 pid(T,⌫)= dp E(k) ⌫ , (2) [9] based on the hard-core repulsion approach tells us that this is a fruitfulmomentum frameworkk is E(k) k + m(2p ⇡and) 3 theE(k density) ( T of point-like) particles3 is defined ⌘ Z e + ⇣ (2⇡ ) 3 E(k) ( T ) high densities one cannot@pid(T,⌫) ignoreq the hard-core repulsion or the finite (e↵ective)⌘ size of Z e + ⇣ as nid(T,⌫) . The parameter ⇣ switches between the Fermi (⇣ = 1), the @pid(T,⌫) q @⌫ as nid(T,⌫) ⌫(µ, nid)=. Theµ parameterbp+ U ⇣(T,nswitchesid) , between the Fermi (⇣ = 1), the (3) also for quantum systems. Hence⌘ we start from the simplest case, i.e. the quantum⌘ @⌫ ⌫(µ, nid)=µ bp+ U(T,nid) , (3) Bose (⇣ = 1) and the Boltzmann (⇣ = 0) statistics. The interaction part of pressure composite particles and the success of traditional EoS usedBose in ( the⇣ = 1) theory and the Boltzmann of16⇡ real ( gases⇣ = 0) statistics. The interaction part of pressure where the constant b 4V = R3 is the excluded volume16⇡ of3 particles with the hard- VdW EoS [14, 15].P The(T,n typical) and the form mean-field of EoSU(T,n for quantum) will be specified quasi-particles later. of mass mp where0 3 thep constant b 4V0 = R is the excluded volume of particles with the hard- int id id Pint(T,nid) and⌘ the mean-field U(T,nid) will⌘ be specified3 later.p [9] based on the hard-core repulsion approach tells uscore that radius thisRp (here is a fruitfulV0 is their framework proper volume), the relativistic energy of particle with and degeneracy factor dp is as follows core radius Rp (here V0 is their proper volume), the relativistic energy of particle with ~ ~ 2 2 momentum k is E(k) momentumk + mp and~k is theE(k density) ~k2 of+ point-likem 2 and the particles density is of defined point-like particles is defined also for quantum systems. Hence we start from the simplest case, i.e.⌘ the quantum ⌘ p p(T,µ,nid)=pid(T,⌫(µ, nid)) Pint(T,nid) , @pid(1)(T,⌫) q @p (T,⌫) q as nid(T,⌫) . Theas n parameter(T,⌫) ⇣ idswitches. The between parameter the Fermi⇣ switches (⇣ = between 1), the the Fermi (⇣ = 1), the 2 ⌘ @⌫ id @⌫ VdW EoS [14, 15]. The typicaldk k form of EoS1 for quantum quasi-particles of mass⌘ mp Bose (⇣ = 1) and the Boltzmann (⇣ = 0) statistics. The interaction part of pressure pid(T,⌫)=dp E(k) ⌫ , (2) Bose (⇣ = 1) and the Boltzmann (⇣ = 0) statistics. The interaction part of pressure 3 Z (2⇡ ) 3 E(k) e( T ) + ⇣ and degeneracy factor dp is as follows Pint(T,nid) and the mean-fieldPint(T,nU(T,nid) andid) will the be mean-field specifiedU later.(T,nid) will be specified later. ⌫(µ, n )=µ bp+ U(T,n ) , (3) id id p16(⇡T,µ,n3 id)=pid(T,⌫(µ, nid)) Pint(T,nid) , (1) where the constant b 4V0 = R is the excluded volume of particles with the hard- ⌘ 3 p dk k2 1 core radius Rp (here V0 is theirpid proper(T,⌫)= volume),dp the relativistic energyE(k) of⌫ particle, with (2) 3 (2⇡ ) 3 E(k) ( T ) momentum ~k is E(k) ~k2 + m 2 and the densityZ of point-likee particles+ is⇣ defined ⌘ p @pid(T,⌫) q ⌫(µ, nid)=µ bp+ U(T,nid) , (3) as nid(T,⌫) . The parameter ⇣ switches between the Fermi (⇣ = 1), the ⌘ @⌫ Bose (where⇣ = 1) the and constant the Boltzmannb 4V (⇣ == 0)16 statistics.⇡ R3 is the The excluded interaction volume part of of pressure particles with the hard- ⌘ 0 3 p Pint(T,nid) and the mean-field U(T,nid) will be specified later. core radius Rp (here V0 is their proper volume), the relativistic energy of particle with momentum ~k is E(k) ~k2 + m 2 and the density of point-like particles is defined ⌘ p q as n (T,⌫) @pid(T,⌫) . The parameter ⇣ switches between the Fermi (⇣ = 1), the id ⌘ @⌫ Bose (⇣ = 1) and the Boltzmann (⇣ = 0) statistics. The interaction part of pressure Pint(T,nid) and the mean-field U(T,nid) will be specified later. 34

зовалось импульсное распределение идеального газа нуклонов, апривведении‘энергии сжа-

тия’ учитывалось взаимодействие между нуклонами. Но взаимодействие должно менять им-

Нуклонное отталкиваниеописывает описы притяжениевается между нулевой нуклонами компонентой и изменяет нуклонную(пространственные массу M M = компонентыM gs φ. → ∗ − должны исчезнутьНуклонное из-за трансляционной отталкивание описы инваетсявариантности нулевой компонентой) векторного(пространственные поля, которая компоненты добав- должны исчезнуть из-за трансляционной инвариантности) векторного поля, которая добав- 2 ляет U(ρ)=Cv ρ (Cv = const2) кэнергиинуклона( U(ρ) для антинуклона). В [67] было ляет U(ρ)=C ρ (Cv = const) кэнергиинуклона( U(ρ) для антинуклона). В [67] было v − − сформулировано обобщенноесформулировано УС обобщенное ядерной УС матери ядерной, материкоторое, которое включает включает и и модель модель среднего среднего по- по- ля и чисто феноменологические модели какчастныеслучаи7 . Ограничиваясь нуклонными ля и чисто феноменологические модели какчастныеслучаи. Ограничиваясь нуклонными 7 степенями свободы, рассмотримWalecka следующее (σ ядерное-ω) model: УС достаточно pressure общего вида в большом 7 степенями свободы, рассмотрим следующее ядерное УС достаточно общего вида в большом 7 m! 2 m 2 каноническом ансамбле: + ! N =(2SN +1)(2IN +1)=4, 2 0 2 pressure ρ 7 3 2 g! 2 2 g 2 2 каноническом ансамбле: γN d k k + ! m! 2 m 2 N =(2Smean-fieldN +1)(2IN +1)=4, p(T,µ)= (f + f )+0ρ U(ρ) dn U(n)+P (M ∗) , 3 + 27 + !0 2+ N =(2g! SN +1)(2(1.4)g IN +1)=4, 3 (2π) 2 2 − 2 − 2 contributions2 2 Degeneracy factor N =(2SN +1)(2IN +1)=4!,g!!k0 + mM⇤∗ ρ !0 + !0 N =(2SN +1)(2IN +1)=4, 3 2 γ d k k " 2 2 1 N 1 nucleon spin SN =p(T,µand isospin)= IN = где f+ и f idealфункции gas распределения with(f m*+ + massf ну)+клоновρ иU антинуклонов(ρ) dn U(n)+P (M ∗) , (1.4) 2 2 3 − − DegeneracyDegeneracy factor factorN =(2N =(2SNSN+1)(2+1)(2IINN+1)=4+1)=4, , m! 2 m 23 ! (2π) k2 + M 2 − ! ∗ 1 0 1 1 m! 2+ m! 2 N =(2SN +1)(2IN nucleon+1)=42 spin2 1S, = and isospin− I =1 Degeneracy factor N =(2SN +1)(2IN +1)=4, 0 nucleon spink + SM ∗ = µNandU2 isospin(ρ) I =N 2 + !0 2 N2=(2SN +1)(2I"N +1)=4f , exp N ∓ 2 ± +1N , 2 (1.5) 2 2 ± ≡ ⎡ ⎛" T ⎞ ⎤ 1 1 где f+ и f функции распределения нуклонов и антинуклоновnucleon spin SN = 2 and isospin IN = 2 − distribution function⎣ ⎝ ⎠ ⎦ µ барионный химический потенциал, и γN число нуклонных спин-изоспиновых1 состояний, 2 2 − DegeneracyDegeneracy factor N =(2 factorSN +1)(2NIN=(2+1)=4SN +1)(2,gI!N!k0+1)=4+ mM⇤∗ ,gµ U!(!ρ)0 µ baryonic chemical Similarly to the ordinary gases, in theγN hadronicf=4для симметричной orexp nuclear systems ядерной the материи source∓ . Зависи of± мость эффективно+1 йнуклонноймассы, M ∗ (1.5) ± ≡ ⎡ ⎛" T ⎞ ⎤ potential nucleon spin S hard-core= 1 and repulsion isospin1 isI connected= 1 to the Pauliот T и1 blockingµ определяется e↵ect between экстремумо theмтермодинамическогопотенциалаSimilarly interacting to the ordinary gases,( inмаксимум the hadronic давления or): nuclear systems the source of nucleonN spin2 SN = andN isospin2 IN = fermionic constituents2 existing interior the composite2 ⎣ particles⎝ (see, for instance, [2]). ⎠ ⎦ Similarlyhard-core to the repulsion ordinary3 is connected gases, in to the the Pauli hadronic blocking or e nuclear↵ect between systems the interacting the source of µ барионный химический потенциалδ p(T,µ) , и dPγN(Mчисло∗) нуклонныхd k M спин∗ -изоспиновых состояний, This e↵ect appears due to the requirement of antisymmetrization of the wavefermionic functionγN constituents3 existingSimilarly interior(f+ to+ thef the)=0 ordinarycomposite. (1.6) particles gases, in (see, the for hadronic instance, or[2]). nuclear systems the source of + δ Mpressure∗ , ≡ is dMknown,∗ − but how(2π )to find2 the 2values −of σ and ω_0 fields? Formally, T,hard-core µ repulsion! is connectedk + M to∗ the Pauli blocking e↵ect between the interacting of all fermionic constituents existing in the system and at very high densitiesThis e↵ itect may appears due to the requirement of antisymmetrization of the wave function γN =4для симметричной ядерной материи. Зависимость эффективно"hard-core repulsionйнуклонноймассы is connected toM the∗ Pauli blocking e↵ect between the interacting Плотность барионногоfermionic числа и плотность constituents энергиимогутбытьнайденыиз existing interior the composite(1.4) - (1.6), particlesис- (see, for instance, [2]). lead to the Mott e↵ect, i.e. to a dissociation of composite particles or evenof the all fermionic clusters constituentsfermionic existing constituents in the system existing and at interior very high the densities composite it may particles (see, for instance, [2]). от T и µ определяетсяThermodynamics экстремумомтермодинамическогопотенциалаThis e↵ ectrequireslead appears to the Mott extremum due e to↵ect, the i.e. requirement to values a dissociation(максимум of of antisymmetrization ofcorresponding composite давления particles): or of even the the wave clusters function of particles into their constituents [2]. Therefore, it is evident that at suciently This e↵ect appears due to the requirement of antisymmetrization of the wave function high densities one cannot ignore the hard-core repulsionpotential or theof finite all for (efermionic↵ective)of a particles given size constituents of intoset their of variables constituents existing in [2]. the(ensemble)! Therefore, system and it is at evident very highthat at densities suciently it may 3 of all fermionic constituents existing in the system and at very high densities it may Similarly to the ordinary gases, in the hadronicδ p(T,µ) or nucleardP systems(M ∗) thehigh source densitiesd k of one cannotM ∗ ignore the hard-core repulsion or the finite (e↵ective) size of composite particles and the success of traditional EoS used inlead the theory toγ theN of Mott real gases e↵ect, i.e. to a dissociation(f+ + f )=0 of composite. particles(1.6) or even the clusters 3 lead to2 the Mott− e↵ect, i.e. to a dissociation of composite particles or even the clusters hard-coreSimilarly repulsion is to connected the ordinary to the gases, Pauli+ δ blockingM in∗ the,T, µ e hadronic↵≡ect betweendM∗ or− the nuclearcomposite interacting(2 systemsπ) particles2 and the the success source of traditional of EoS used in the theory of real gases [9] based on the hard-core repulsion approachFor tells the us GC that thisensembleof is aparticles fruitful !one framework into has their tok constituents require+ M ∗ a [2]. maximum Therefore, of it pressure! is evident that at suciently [9] based on the" hard-coreof particles repulsion into approach their tells constituents us that this [2]. is a fruitful Therefore, framework it is evident that at suciently fermionic constituentsalso for existing quantum interior systems. the Hence composite we start from particles the simplest (see,high case, for densities i.e. instance, the quantum one [2]). cannot ignore the hard-core repulsion or the finite (e↵ective) size of hard-core repulsion is connectedПлотность барионного to the Pauli числа blocking и плотность e↵ect энергиalso for betweenимогутбытьнайденыиз quantum systems. thehigh densitiesinteracting Hence we one start cannot(1.4) from the - ignore (1.6), simplest theис case, hard-core- i.e. the repulsion quantum or the finite (e↵ective) size of This e↵ect appearsVdW due to EoS the [14, requirement 15]. The typical of form antisymmetrization of EoS for quantum quasi-particles ofcomposite the wave of particles function mass mp and the success of traditional EoS used in the theory of real gases VdW EoS [14, 15]. Thecomposite typical form particles of EoS and for the quantum success quasi-particles of traditional of mass EoSm usedp in the theory of real gases fermionic constituentsand degeneracy factor existingdp is as interiorfollows the composite particles (see, for instance, [2]). [9] basedand on degeneracythe hard-core factor repulsiond is as follows approach tells us that this is a fruitful framework of all fermionic constituents existing in the system and at very high densities it may [9]p based on the hard-core repulsion approach tells us that this is a fruitful framework p(T,µ,n )=p (T,⌫(µ, n )) P (T,n ) , (1) This e↵ect appears due to theid requirementid id of antisymmetrizationint idalso for quantum systems. of the wave Hence function we start from the simplest case, i.e. the quantum lead to the Mott e↵ect, i.e. to a dissociation of composite2 particles or even the clusters palso(T,µ,n forid quantum)=pid(T,⌫ systems.(µ, nid)) P Henceint(T,n weid) , start from the simplest(1) case, i.e. the quantum dk k 1 2 pid(T,⌫)=dp E(k) ⌫ , (2) dk k 1 3 VdW EoS [14, 15]. The typical form of EoS for quantum quasi-particles of mass m (2⇡ ) 3 E(k) ( T ) pid(T,⌫)=dp E(k) ⌫ , (2) p of all fermionic constituents existingZ in thee system+ ⇣ and at very high densitiesVdW EoS it [14, may 15].3 The typical form of EoS for quantum quasi-particles of mass mp of particles into their constituents [2]. Therefore, it is evident that at suciently Z (2⇡ ) 3 E(k) e( T ) + ⇣ ⌫(µ, nid)=µ bp+ U(T,nid) , and degeneracy factor(3) dp is as follows ⌫and(µ, n degeneracyid)=µ bp+ factorU(T,ndid)is, as follows (3) high densitieslead to one the cannot Mott ignoree↵ect, the i.e. hard-core to a dissociation repulsion or of the composite finite (e↵ective) particles size of or even the clusters p 16⇡ 3 where the constant b 4V0 = 3 Rp is the excluded volume of particles with the hard- 16⇡ 3 ⌘ where the constantp(T,µ,nb 4idV)=0 = pidR(T,p is⌫ the(µ, excluded nid)) volumePint(T,n of particlesid) , with the hard- (1) composite particles and the success of traditional EoS used in the theory of real gases ⌘ 3 p(T,µ,n id)=pid(T,⌫(µ, nid)) Pint(T,nid) , (1) of particlescore into radius theirRp (here constituentsV0 is their proper [2]. volume), Therefore, the relativistic energy it is of evident particle with that at suciently2 core radius Rp (here V0 is their properdk volume),k the relativistic1 energydk ofk particle2 with1 2 2 pid(T,⌫)=dp E(k) ⌫ , (2) momentum ~k is E(k) ~k + m and the density of point-like particles is defined~ ~ 2 2 3 pid(T,⌫)= dp E(k) ⌫ , (2) [9] based on the hard-core repulsion approachp tells us that this is a fruitfulmomentum frameworkk is E(k) k + m(2p ⇡and) 3 theE(k density) ( T of point-like) particles3 is defined ⌘ Z e + ⇣ (2⇡ ) 3 E(k) ( T ) high densities one cannot@p (T,⌫) ignoreq the hard-core repulsion or the finite (e↵ective)⌘ size of Z e + ⇣ id @pid(T,⌫) q as nid(T,⌫) @⌫ . The parameter ⇣ switches between the Fermias (⇣ n=id( 1),T,⌫ the) ⌫(µ, nid)=. Theµ parameterbp+ U ⇣(T,nswitchesid) , between the Fermi (⇣ = 1), the (3) also for quantum systems. Hence⌘ we start from the simplest case, i.e. the quantum⌘ @⌫ ⌫(µ, nid)=µ bp+ U(T,nid) , (3) Bose (⇣ = 1) and the Boltzmann (⇣ = 0) statistics. The interaction part of pressure composite particles and the success of traditional EoS usedBose in ( the⇣ = 1) theory and the Boltzmann of16⇡ real ( gases⇣ = 0) statistics. The interaction part of pressure where the constant b 4V = R3 is the excluded volume16⇡ of3 particles with the hard- VdW EoS [14, 15]. The typical form of EoS for quantum quasi-particles of mass mp where0 3 thep constant b 4V0 = R is the excluded volume of particles with the hard- Pint(T,nid) and the mean-field U(T,nid) will be specified later. Pint(T,nid) and⌘ the mean-field U(T,nid) will⌘ be specified3 later.p [9] based on the hard-core repulsion approach tells uscore that radius thisRp (here is a fruitfulV0 is their framework proper volume), the relativistic energy of particle with and degeneracy factor dp is as follows core radius Rp (here V0 is their proper volume), the relativistic energy of particle with ~ ~ 2 2 momentum k is E(k) momentumk + mp and~k is theE(k density) ~k2 of+ point-likem 2 and the particles density is of defined point-like particles is defined also for quantum systems. Hence we start from the simplest case, i.e.⌘ the quantum ⌘ p p(T,µ,nid)=pid(T,⌫(µ, nid)) Pint(T,nid) , @pid(1)(T,⌫) q @p (T,⌫) q as nid(T,⌫) . Theas n parameter(T,⌫) ⇣ idswitches. The between parameter the Fermi⇣ switches (⇣ = between 1), the the Fermi (⇣ = 1), the 2 ⌘ @⌫ id @⌫ VdW EoS [14, 15]. The typicaldk k form of EoS1 for quantum quasi-particles of mass⌘ mp Bose (⇣ = 1) and the Boltzmann (⇣ = 0) statistics. The interaction part of pressure pid(T,⌫)=dp E(k) ⌫ , (2) Bose (⇣ = 1) and the Boltzmann (⇣ = 0) statistics. The interaction part of pressure 3 Z (2⇡ ) 3 E(k) e( T ) + ⇣ and degeneracy factor dp is as follows Pint(T,nid) and the mean-fieldPint(T,nU(T,nid) andid) will the be mean-field specifiedU later.(T,nid) will be specified later. ⌫(µ, n )=µ bp+ U(T,n ) , (3) id id p16(⇡T,µ,n3 id)=pid(T,⌫(µ, nid)) Pint(T,nid) , (1) where the constant b 4V0 = R is the excluded volume of particles with the hard- ⌘ 3 p dk k2 1 core radius Rp (here V0 is theirpid proper(T,⌫)= volume),dp the relativistic energyE(k) of⌫ particle, with (2) 3 (2⇡ ) 3 E(k) ( T ) momentum ~k is E(k) ~k2 + m 2 and the densityZ of point-likee particles+ is⇣ defined ⌘ p @pid(T,⌫) q ⌫(µ, nid)=µ bp+ U(T,nid) , (3) as nid(T,⌫) . The parameter ⇣ switches between the Fermi (⇣ = 1), the ⌘ @⌫ Bose (where⇣ = 1) the and constant the Boltzmannb 4V (⇣ == 0)16 statistics.⇡ R3 is the The excluded interaction volume part of of pressure particles with the hard- ⌘ 0 3 p Pint(T,nid) and the mean-field U(T,nid) will be specified later. core radius Rp (here V0 is their proper volume), the relativistic energy of particle with momentum ~k is E(k) ~k2 + m 2 and the density of point-like particles is defined ⌘ p q as n (T,⌫) @pid(T,⌫) . The parameter ⇣ switches between the Fermi (⇣ = 1), the id ⌘ @⌫ Bose (⇣ = 1) and the Boltzmann (⇣ = 0) statistics. The interaction part of pressure Pint(T,nid) and the mean-field U(T,nid) will be specified later. 34

зовалось импульсное распределение идеального газа нуклонов, апривведении‘энергии сжа-

тия’ учитывалось взаимодействие между нуклонами. Но взаимодействие должно менять им-

пульсное распределение идеального газа, и возникает проблема совместности релятивистско- го Ферми распределения нуклонов по импульсам ифункциональнойформы‘энергии сжатия’.

Для решения этой проблемы напомним, что в релятивистской модели среднего поля Ва-

лечки [10] (см. также [11]) взаимодействие описывается скалярным φ ивекторнымU µ мезон-

ными полями с вершинами мезон-барионного взаимодействия в функции Лагранжа: gs ψψφ¯ ¯ µ и gv ψγ ψ Uµ. Для ядерной материи в термодинамическом равновесии эти мезонные поля, как обычно полагают, являются постоянными классическими величинами. Скалярное поле 7

описывает притяжение между нуклонами и изменяет нуклонную массу M M = M gs φ. → ∗ − Нуклонное отталкивание описывается нулевой компонентой (пространственные компоненты m! m 7 должны+ исчезнуть!2 из-за трансляционной2 =(2 инвариантностиS +1)(2) векторногоI +1)=4 поля, ,которая добав- 0 N N N 2 2 2 ляет U(ρ)=C ρ (Cv = const) кэнергиинуклона( U(ρ) для антинуклона). В [67] было v 4 − сформулировано обобщенное УС ядерной матери, которое включает и модель среднего по- m! 2 m 2 7 + ! DegeneracyN =(2SN +1)(2 factorIN +1)=4=(2S , +1)(2I +1)=4,g! m⇤ Grandcanonical Ensemble (GCE)2 of Boltzmann0 2 particles ля и чисто феноменологическиеN моделиN какчастныеслучаиN . Ограничиваясь! 0 нуклонными 7 1 1 1 µN nucleon spin степенямиSN = свободыandm isospin, рассмотримm IN = следующее ядерное УС достаточно общего вида в большом T 2 ! 2 2 2 7 Zgc(T,µ,V )= e Zce(E, N, V )= + ! N =(2SN +1)(2IN +1)=4, 2 0 2 DegeneracyNX=0 factor N =(2SN +1)(2каноническомIN +1)=4 ансамбле,g: !!0 m⇤ µN 2 2 ρ N 3 3 m + pk U(xl,xj3) 2 7 m1 !e T2 1 md xkd2 pk j 1 l,j γN d k k nucleon spin S = and isospin I = p(T,µ)= (f+ + f )+ρ U(ρ) dn U(n)+P (M ∗) , (1.4) =+ !N 2 expNN2 =(2P 2q SN +1)(2DegeneracyP IN factor+1)=4N =(23 SN +1)(2, IN +1)=4,g!!0 m⇤ 0 3 3 (2π) 2 p 2 −p − N=02 N! Zk=1 2(2⇡h¯) m! T m! k + M ∗ !0 X Y 6 2 27 =0 =0 Walecka6 + (σ!-0ω) 1 model: 7 " maximum1 N =(2SN +1)(2 of pressureIN +1)=4, 4 nucleon spin SN = and isospin5IN != 0 m где2f+ и fm функции2 2 распределения2 нуклонов и антинуклонов conserves E and N on average only! ! 2 − 2 + !0 N =(2SN +1)(2IN +1)=4, 1 2 2 32 2 − Degeneracy factor N =(2SN +1)(2p IN +1)=4p ,g!!0 dkkm+⇤ M ∗ µ U(ρ) =0 =0 f exp ∓ ± 2 +1 , (1.5) g±!≡N⎡ ⎛" [f+ Tf ] m!!⎞0 =0⎤ 7 7 !Degeneracy0 factor)N =(2pSN(2+1)(2⇡)p3 IN+1)=4 ,g!!0 m⇤ Pressure is defined as1 1 Z7=0 =0 nucleon spin SN = and isospin IN = !⎣ ⎝ ⎠ ⎦ 7 2 Degeneracy2 factorµ N =(2SN +1)(20 IN +1)=4γ ,g!!0 m⇤ 1барионный химический потенциал1 , и N число нуклонных спин-изоспиновых состояний, ln [Zce(T,N,V )] nucleon spin SN = and isospin2 INm!=2 m 2 thermal p = T lim m! 2 m m + ! 3 2 N =(2SN +1)(2g! IN +1)=4, 1 γ2N =4для симметричной2 1 ядернойdk0 материи. Зависимость эффективнойнуклонноймассыM ∗ nucleonV spin S +=V and! isospin I =N(=(2m2 SNm+1)(2⇤)2 IN +1)=4!0 ,= n !1 N 2 0 N 2g!2N [f+ f ] m!!0 =0 2 m! m 2 m! 2 m 3 ) 2 2 @ ln [Zce(T,N,V )] 2instead )2ofg field(2 σ⇡ ) it is more convenientm! to use m* => + ! N =(2SN +1)(2+IN от+1)=4!T и µ определяется, экстремумоNZ =(2SNмтермодинамическогопотенциала+1)(2IN +1)=4, (максимум давления): 2 mechanical0 2 p = T 2 0 2 @pV Degeneracy factor N =(2SN +1)(2IN +1)=4,g!!0 m⇤ 2 3 Degeneracy=0 factor N =(2δ p(T,µSN) +1)(2ImNdP +1)=4(M ∗) ,gd!!k0 mM⇤∗ 1 (m m⇤) γN 1 (f+ + f )=0. (1.6) 3 nucleon spin SN =pand isospin IN =p 3 2 2 Apparently, in thermodynamic limit they should coincidedk + δ M ∗ , 2≡ g2 dM∗ − 2 (2π) } 2 2 − m m Degeneracy factor N =(2SN +1)(2IN +1)=4Degeneracy,g1 factor2!!0N =(2mp2⇤ SNT,1 +1)(2 µ =0 INp+1)=42,g! =0!!20k + mM⇤∗ ! 2 2 nucleonN spin SN = andk isospin+(m⇤I)N +==0g!!0 f+ +=0 k +(m⇤) g!!0 f ! = 3 2 2!0 " 0 nucleon spin S = 1 and isospin I = 1 Z (2⇡) ✓q1 !0 1 ◆ ✓q ◆ 2 2 SimilarlyN 2 to the ordinaryN 2nucleon gases, spin in theSN = hadronicПлотность2 and isospin барионного or nuclearIN = числа2 systems и плотность the энерги sourceимогутбытьнайденыиз of (1.4) - (1.6), ис- Other thermodynamic quantities can be found from identities: p p scalar density hard-core repulsion isThese connected conditions to the provide Pauli maximum blocking e ↵ofect3 pressure between3 and the =0 interacting=0 dk dk !0 2 2 allow one to recover the standardg!N g !thermodynamicsN [f+ f[f] + identities!m!f!0] =0m !0 =0 ) 3 3 ! fermionic constituentsI Law existingp + ✏ = interiorTs+ µn the composite) Z particles(2⇡) (2 (see,⇡) for instance, 3 [2]). 35 Z dk 2 g!N [f+ f ] m !0 =0 Similarly to the ordinary gases, in) the hadronicRecall(2 or⇡ nuclear)3 1-st systems L. van the! source Hove of axiom! This e↵ect appears due to the requirement of antisymmetrization2 of theZ wave function пользуя известныеSimilarly термодинамические to the ordinary тождестваm gases,2: in the hadronic or nuclearg systems the source of @p @hard-corep repulsion m is connected to the Pauli blocking e↵ect! betweeng the interacting = from(m 1-stm ⇤Eq.) 2 !0 = n ! Similarlyof to all the fermionic ordinaryII constituents gases, Law in thes =Similarly hadronic existing; n orto= in thenuclear the ordinary system systems2 gases, and the atin source the very hadronic of highm densities or nuclear2 it systems may theg! source of hard-core repulsion is connectedg to the(m Paulim blocking⇤)=) (m e↵ectm⇤m) between!0 = the! =n interactingn @T @fermionicµ constituents3 2 existing interior the composite2 particles! (see,) for02 instance,2 [2]). ∂Similarlyp to thed k above ordinaryg gases, ing the) hadronic orm! nuclearm! systems the source of hard-coreSimilarlylead repulsion to to the the is connected Mott ordinary e↵ect, toρ gases,( theT,µfermionic i.e.hard-core Pauli) to in a blocking dissociation constituents the repulsion hadronic= e↵γNect existing is of between connected composite or( interiorf nuclear+ the tof interacting) the the particles, systems Pauli composite blocking or even the particles e source the↵ect (see, clusters between of for(1.7) instance, the interacting [2]). 3 − ≡ !∂ µ"T This# e↵ect(2π appears) due− to the requirement of antisymmetrization of the wave function fermionic constituents existing interior thefermionic compositehard-core constituents particles repulsion existing (see, for is interior connected instance, the [2]). composite to the Pauli particles blocking (see, for e↵ instance,ect between [2]). the interacting hard-coreof repulsion particles is into connected their constituentsThis to e↵ theect appears Pauli [2].3 due blocking Therefore, to the requirement e↵ itect is between evidentdk of3 antisymmetrization that the atinteracting suciently of the wave function III Law s 0, if∂Tp dk0 of all fermionic∂ p constituents existing in the2 system2 and at very high densities2 it may2 ε(T,µ)! T ! + µ 2 = 2N k +(m⇤) +2 g!!0 f+ 2+ k +(m⇤) g!!0 f This e↵ect appears due to the requirementThis of= e antisymmetrization↵ectN3 appears duek to of the+(- thepm requirement wave⇤) + functiong!!30 of antisymmetrizationf+ + k +(m⇤) of theg! wave!0 functionf 2 2 of all fermionicfermionicdk ∂ T constituents constituents3 ∂ µ existing existing inZ the(2⇡) system interior✓ and the at composite very high◆ densities✓ particles it may (see, form◆ instance, m [2]). high densities one cannot ignore≡ ! the" hard-core(2µ ⇡)lead! to the repulsion" MottT e↵ect, i.e. or to the a dissociation finiteq of (e composite↵ective) particles size or of evenq the clusters ! 2 2 fermionic constituents existing interior theZ composite2 ✓q particles2 m2 (see,m2 ◆ for instance,✓q2dk3 [2]).2 ◆ m2 m2 of all fermionic constituents existing inN the system andk at very+(m high⇤) densities+ g!! !20 it mayfρ+ 2+ k +(m2 ⇤) 2 g!!0 f ! 2 !20 = ✏ is energy density (E/V ), s is entropylead density,of to all the fermionic2 Mottn3 is baryonic3 e↵ constituentsect,2 i.e. charge to a existing dissociation density3! in the of composite system= N and particles at veryk +( or highm even⇤) 2[f densities+ the+ f clusters]+ it2 may! (2Thism⇡) e↵dectk m appears due todk the0 requirement of3 antisymmetrizationm m of0 the2 wave function2 composite particles and theZ success! ✓2 ofq traditionalof particles2 2 into2 EoS their used constituents2 ◆ in22 the [2]. theory Therefore,✓q2Z (2⇡ of it) is real evident gases that! at2 suciently◆ 2 2 2 This e↵ect appears due to the requirement= γN ! of antisymmetrizationk +=MN (f+ f )+k of+(dn them U⇤( waven)) [f+P+ function(qMf∗)]+. (1.8)! lead to the Mott e↵ect, i.e. to a dissociationoflead particles of to composite the into Mott0 their3 e particles↵ect, constituents i.e. or∗ to even a dissociation [2]. the3 clusters Therefore, of composite it is evident particles that or ateven su0 theciently clusters of2 all(2 fermionicπ) high2 densities constituents one cannot(2−⇡− ignore) existing the hard-core in repulsion the− system or the finite and2 (e↵ective) at very size2 of high densities it may [9] based on the hard-core repulsion# approach$ tells usZ thatq this#0 is a fruitful framework ofof all particles fermionic into their constituents constituents existing [2].highof Therefore, densities particles in the one into it system cannotis their evident ignore constituents and that the at at hard-core very su [2].ciently high Therefore, repulsion densities it or is the evident it finite may (e that↵ective) at su sizeciently of lead to thecomposite Mott e particles↵ect, and i.e. the to success a dissociation of traditional EoS of used composite in the theory of particles real gases or even the clusters high densitiesalso one for cannot quantum ignoreИз systems.(1.5) the hard-coreследуетcompositehigh Hence, densitiesчто repulsion particles у we нуклона one start or and cannot( theантинуклон from the finite success ignore the (e)↵ective)распределение simplest the of traditional hard-core size case, of импульса repulsion EoS i.e. used the имеет or in the thequantum форму finite theory иде (e of↵-ective) real gases size of lead to the Mott e↵ect, i.e. to a dissociation of[9] composite based on the hard-core particles repulsion or approach even tells the us that clusters this is a fruitful framework of particles into their constituents [2]. Therefore, it is evident that at suciently compositeVdW particles EoS and [14, theального 15]. success The распределения of[9] traditional typicalcomposite based on form Ферми theEoS particles hard-core ofused во EoS‘внешнем in and the for repulsion the theory quantum пол successе’: of approachскалярное ofreal traditional quasi-particles gases tells поле us изменяет EoS that used this of массу massin is the a fruitful нуклона theorymp framework of real gases of particles into their constituents [2]. Therefore,also for quantum it is systems. evident Hence we that start from at the su simplestciently case, i.e. the quantum [9] based on the hard-core repulsion approach[9] basedhigh tells on densities us the that hard-core this one is a cannot repulsionSimilarly fruitful framework ignore to approach the ordinary the tells hard-core gases, us that in the this repulsion hadronic is a fruitful or or nuclear the framework finite systems (e the↵ective) source of size of and degeneracy(антинуклона factor dalso) isM for asна quantumэффективную follows systems. массу M Hence∗, авекторноеполедобавляетэнергию we start from the simplest case,U( i.e.ρ) нук thequantum p VdW EoShard-core [14, 15]. The repulsion typical form is connected of EoS for quantum to the Pauli quasi-particles blocking of e↵ massect betweenmp the interacting also for quantum systems. Hence we start from the simplest case, i.e. the quantum high densities one cannot ignoreVdW thealso EoS forhard-corecomposite [14, quantum 15]. Theparticles repulsion systems. typical Hence and form or the the we of EoSsuccess start finite for from (e quantum of↵ the traditionalective) simplest quasi-particles size case, EoS of i.e. used of the mass in quantum themp theory of real gases лону (-U(ρ) дляSimilarly антинуклона to). Важно theand degeneracy ordinary, однакоfermionic factor, что constituentsd gases,p в is(1.4) as followsи in(1.8) existing theпоявляются hadronicinterior the дополнитель composite or nuclear particles- systems (see, for instance, the source[2]). of VdW EoS [14, 15]. The typical formSimilarly ofVdW EoS for EoS to quantum [14, the 15]. ordinary quasi-particles The typical gases, form of in mass of the EoSm hadronicp for quantum or nuclear quasi-particles systems of the mass sourcem of composite particles and thep success(T,µ,nand degeneracyid of)= traditionalpid( factorT,⌫(µ,dp EoS nisid as)) used followsPint in(T,n theid theory) , of real gases(1) p [9] based on the hard-coreThis e↵ect appears repulsion due to the approach requirement tells of antisymmetrization us that this is of a the fruitful wave function framework ные члены, представляющие термодинамическ2 исогласованныевкладыполейвдавлениеиp(T,µ,n )=p (T,⌫(µ, n )) P (T,n ) , (1) and degeneracy factor dp is as followshard-corehard-coreand degeneracy repulsion repulsiondk factor is connectedk isdpofis connected all as fermionic follows to1 theid constituents to Pauliid the blocking Pauli existingid blockingint in e↵ theectid system between e↵ andect at the very between interacting high densities the itinteracting may p(T,µ,nid)=pid(T,⌫(µ, nid)) Pint(T,nid) , (1) [9] based on the hard-core repulsionpid(T,⌫)= approachalsodp for quantum tells us systems. thatE(k) this⌫ Hence is, a fruitful we start framework from the(2) simplest case, i.e. the quantum 3 плотностьfermionic энергии. Форма constituents этих(2⇡ дополнительн) 3 existingE(klead) ( toых interior theT вкладов Mott) thee↵ вect,2 давление composite i.e. to a(1.4) dissociation particlesуже согласована of composite (see, for particles instance, or even [2]). the clusters p(T,µ,nid)=pidfermionic(T,⌫(µ, nid)) constituentsZ Pint(T,npid(T,µ,n) , existinge id)=pdidk interior(+T,⇣⌫k(µ,(1) nid the)) 1 compositePint(T,nid) , particles (see,(1) for instance, [2]). 2 pid(T,⌫)=dp E(k) ⌫ , (2) also for quantum systems. Hencedk wek start from1 theof simplest particles into3 case, their constituents i.e.2 ( the ) [2]. quantum Therefore, it is evident that at suciently сраспределениямиФерми⌫This(µ, n eid↵)=ectµ appears(1.5)bp+через dueU( toT,n общее theid) те requirement, рмодинамическоеZ (2⇡ d)k3 E( ofkk) e antisymmetrization тождествоT 1+ ⇣(1.6)! Формулы(3) of the wave function pid(T,⌫)=dp This e↵ect appearsE(k) ⌫ pid, due(T,⌫ to)= thedp requirement(2) E of(k) ⌫ antisymmetrization, of(2) the wave function 3 ( ) high densities one3 cannot ignore the hard-core repulsion or the finite (e↵ective) size of (2⇡ ) 3 E(k) e T ⌫+(µ,⇣ nid)=µ bp+(2⇡U()T,n3 Eid(k)), ( T ) (3) VdW EoS [14, 15]. The typicalZ form of EoS for quantum quasi-particlesZ e of mass+ ⇣ mp (1.4) - (1.8)ofопределяют all fermionic поэтом constituentsуспециальныйкласстермодинаcomposite existing particles in the and system theмически success and самосогласованных of traditional at very highEoS used densities in the theory it may of real gases ⌫(µ, nid)=µ bpof+ allU(T,n fermionicid) , constituents⌫(µ, n )=µ existingbp+ U(3)(T,n in the) , system and at very high(3) densities it may where the constant b 4V =id16⇡ R3 is the excludedid volume of particles with the hard- and degeneracy factor dp is aslead follows to the Mott e↵ect,⌘ i.e.0 [9] to based3 a dissociationp on the hard-core of composite repulsion approach particles tells us or that even this the is a fruitful clusters framework УС ядерной16⇡ 3 материи. Это феноменологическое обобщение16⇡ 3 модели среднего поля. Модели это- where the constant b 4V0 = 3 Rp isleadcore thewhere radius excluded to the theR constant Mott volume(here V eb of↵isect, particles their4V0 i.e.= proper with toR a volume), theis dissociation the hard- excluded the relativistic volume of composite of energy particles of particleparticles with the with hard- or even the clusters ⌘ of particles intop their0 constituents⌘ 3 p [2]. Therefore, it is evident that at suciently core radius Rp (here V0 isго their класса proper определяют volume),core radiusся the~ двумяR relativistic(here функциямиV energyis~ 2 theirU(ρ of) proper2и particleP (M volume),∗). withОбщие the физические relativistic ограничения energy of particle with p(T,µ,n of)=momentum particlesp (T,k⌫(isµ, intopE n(k) their)) 0 kP constituents+ m(pT,nand) the, density [2]. Therefore, of point-like particles(1) it is evident is defined that at suciently highid densitiesid one cannotid ⌘ ignoreint the hard-coreid repulsion or the finite (e↵ective) size of ~ ~ 2 2 ~@pid(T,⌫) q ~ 2 2 momentum k is E(k) k + mp andas momentumn the(T, density⌫) k ofis 2 point-likeE(k.) The particlesk parameter+ mp isand defined⇣ switches the density between of point-like the Fermi particles (⇣ = 1), is defined the ⌘ на эти функции таковыid d:k k@⌫ 1 q high densities⌘ one cannot⌘ ignore the hard-core repulsion or the finite (e↵ective) size of @pid(T,⌫) composite particles@p andid(T,⌫) theq success of traditional EoS used in the theory of real gases as nid(T,⌫) .p Theid(T, parameter⌫)=dasp⇣ nswitches(T,⌫) between the.E( Thek Fermi) ⌫ parameter (⇣ =, 1),⇣ theswitches between the(2) Fermi (⇣ = 1), the @⌫ Bose (⇣id= 31) and the@⌫ Boltzmann (⇣ = 0) statistics. The interaction part of pressure ⌘ Z (2⇡ ) 3⌘E(k) e( T ) + ⇣ Bose (⇣ = 1) and the Boltzmann[9] (⇣composite= basedBose 0)U statistics.( on (ρ⇣)== the particles1) hard-core TheU and(ρ) interaction,U the and Boltzmann repulsion(ρ) the part success of (ρ approach⇣a pressure=for 0) of statistics. 0 traditional tellsa us The1 , that interaction EoS this is used a part fruitful in of the pressure framework theory of real gases Pint(T,nid) and the mean-fieldρU(T,nid) will be specified later. ⌫(µ, nid)=µ bp− +U−(T,nid) , →∞ ∼ ≤ ≤ (3) Pint(T,nid) and the mean-field U(T,n[9]id)P based willint(T,n be specifiedid on) and the the later.hard-core mean-field repulsionU(T,nb id) will approach be specified tells later. us that this is a fruitful framework U(ρ)ρ 0 ρ , for 0 b → ∼ ≤ also for quantum systems. Hencek we start from the simplest case, i.e. the quantum P (M ∗)= ak (M M ∗) with a2 < 0 . (1.9) k 2 − %≥ Например, верхнее неравенство a 1 для степени a, обеспечивает выполнение условия при- ≤ чинности при высоких барионных плотностях, то есть тот факт, что скорость звука не пре-

вышает скорость света.

Выбирая U(ρ) и P (M ∗), которые удовлетворяют (1.9), можно воспроизвести многие мо-

дели ядерного УС, известные из литературы. Модели [68, 69] соответствуют M ∗ = M (с

P (M ∗) 0) в (1.4) испециальнойформеU(ρ). Для моделей среднего поля [10, 11, 64, 70, 71] ≡ 2 2 U(ρ)=C ρ. Выбор a2 = 1/2 C , ak =0(k>3) соответствует линейной модели среднего V − s 7

7 m! 2 m 2 + ! N =(2SN +1)(2IN +1)=4, 2 0 2

m! 2 m 2 + !0 N =(2SN +1)(2IN +1)=4, 2 2Degeneracy factor N =(2SN +1)(2IN +1)=4,g!!0 m⇤

1 1 nucleon spin SN = 2 and isospin IN = 2 Degeneracy factor N =(2SN +1)(2IN +1)=4,g!!0 m⇤

1 1 nucleon spin SN = and isospin IN = 2 2 p p =0 =0 !0

7 p p 3 dk 2 =0 g=0!N [f+ f ] m !0 =0 7 3 ! !0 ) Z (2⇡) 2 ⇤ 2 2 2

f 3 m m dk m g! ◆ 2

0 g m! [2f =mf ] (2mm m! ⇤)=0 !0 = n 4MeV ! N + 2 ! 0 2 . ! 3 , + ! g N =(2SN)+1)(2IN +1)=4m , 2 0 ) 0

0 (2⇡) ! !

0 Z 2 2 g ! ! ± 2 ! ! 2 9 . m 2 2 )

⇤ m 3 g!

+1)=4 dk ]+ n ⇤ =0 = (m m ) ! = n m Degeneracy factor N2 =(20 SN2+1)(2IN +1)=4,g2 !!02 m⇤ =17

2 2

2 ! = k +(m ) + g ! f + k +(m ) g ! f ,g N N ⇤ ! 0 + ⇤ ! 0 0 ect between the interacting ! g ) m f 3 I ! g ↵ ! (2⇡) m +( cep Z ✓ ◆ ✓ ◆

2 ! q q + 1 1 2 T nucleon spin S2 = and2 isospin I = 3 2 2

= N N k m + m 2 m dk2 m m ] 2 ! 2 2 ! 2 2 f 0 2 2 q [

! = k +(m ) [f + f ]+ ! H N +

+1)(2 ⇤ ✓ ! 0 0 2 3 ] 3 +1)=4 ) =0 2 2 (2⇡) 2 2 N ⇤ + dk Z q f S

N 2 2 2 2 p + m = k +(m ) + g ! f + k +(m ) g ! f I

M/T N ⇤ ! 0 + ⇤ ! 0 ) f 3

6GeV (2⇡) . ◆ Z ✓q ◆ ✓q ◆ +( + p p =(2 0

f 2 2 3 dN 2 2 2 ! [ Walecka (σ-ω) model: normal nuclear=0 matter=0 377

N m m dk m m k ! ! 2 2 exp [+!M/T2 ] 2 +1)(2 3 exp [+ 2 2 H ) ! g 1 2 0 ) 3 q ! = N k +(m⇤) [f+ + f ]+ ! ⇤ ' 0 dM ⇠ 0 N

3 3 ⇡ =0 ⇠ + ) 3 2 2 (2⇡) 2 2 2 = dk S m 2 Z2 2

0 q (2 2 ⇡ g g! 2 g 2 m p ) dk

N 3 2 Cv = 2 285.9GeV , Cs = 2 377.6GeV ⇤ Having 2 parameters (2 Z

I m m

! dk

dN ! dM 2 = ' ' =(2 m m g [f f ] m ! =0 N

! N + 0 Z s !

( 3

N ) (2⇡) C 2 m N !

2 dN Z +( g g m , one can describe the properties of nuclear matter: 2 exp [+M/TH] 2 k = ⇠ dM 2 = q 2 2 0 m g 2 p=0 at T=0 and2 n=n_0 and W(n_0) = -16 MeV ! ) 9 MeV with experimental value ! g g ✓ 2 = (m 2m⇤) ! = n . ! 0 and isospin 2 0

! C = 285.9GeV , C = 377.6GeV2 2

3 v 2 s 2 2 g ) m

2 m m 1 2 ) 3 ! ! m

m ' ' 9GeV ⇡ . rovides a reasonable value for the critical endpoint temperature Pn/n = dk =18 (2 + 2 0 N = 285 cep Z ! S Similarly to the3 ordinary gases, in the hadronic or nuclear systems the source of T 2 ! ' N Compare T =18.9 MeV with experimentaldk value T =17 1MeV 2 cep 2 2 cep 2 2 m 2 !

2 = k +(m ) + g ! f + k +(m ) g ! f ! Degeneracy factor N ⇤ ! 0 + ± ⇤ ! 0 cep g 3 m hard-core repulsion is connected to the Pauli blocking e↵ect between the interacting

= Z (2⇡) ✓q ◆ ✓q ◆

= 2 2 3 2 2 T/T m m dk m m v fermionic! constituents!2 But2 = existingthere are interior threek2 problems:+( them composite)2 [f + f particles]+ ! ! (see,2 for instance,2 [2]). C Compare = Similarly to the ordinary gases, in the hadronic or nuclear systems the source of N ⇤ + 1.2 = T/Tcep = n/n0 0 2.5 3 0 ~ T 2 2 (2⇡) 2 2 ~ p Z nucleon spin supercrhard-corei repulsionti isc connecteda tofermionicl constituents the existingf Paulil interior blockingu the e i composited particles (see,This for instance, [2]). e↵ect(a) appears due to the requirementq of antisymmetrization(b) of the wave function Walecka 1.0 21..0 K_0 = 553 MeV is too huge! of all fermionic constituentsQvd existingdNW in the system and at very high densities it may gas 0.8 n liquid expn [+M/TH] mixed phase lead to the Mott e↵1.ect,5 i.e. todM a dissociation⇠ of composite particles or even the clusters + nU(n) d⇢ U2(⇢)+P (M2.⇤) cannotP reproduce(n)=2 d ⇢flowU(⇢ constraint!) nU(n)

0.6 g! 2 int g 2 ofCv particles= 2 285 into.9GeV their constituents, Cs = 2 [2].377.6GeV Therefore, it is evident that at suciently Z0 m! ' 1.0 m 'Z0 0.4 Walecka 3. m*/m =0.55 is too small!T = T QvdW 0.5 c 0.2 Compare T =18.9 MeV with experimental value T =17.9 0.4MeV cep should be m*/m =[0.6;cep 0.8] ± 0.0 0.0 0.Should0 0.5 obey1.0 a self-consistency1.5 2.0 2 condition.5 0.0 0.5 1.0 1.5 2.0 2.5 ~ ~ = T/Tcep n = n/n0 How can we improve this model?n

Figure 2: The scaled coexistence curves (a) and the scaled critical isotherms (b) calculated for the Walecka (dashed blue lines) and the QvdW (solid red lines) models. The solid circles correspond to the CP. Similarly to the ordinary gases, in the hadronic or nuclear systems the source of hard-coreWalecka repulsionHybrid is connectedQvdW Experiment to the Pauli [29] blocking e↵ect between the interacting T [MeV] fermionic18.9 constituents19.2 existing19.7 interior17.9 the0.4 composite particles (see, for instance, [2]). c ± 3 n [fm ] 0.070 0.071 0.072 0.06 0.01 c ± 3 p [MeV fm ] 0.48 0.50 0.52 0.31 0.07 c ± K0 [MeV] 553 674 763 250 - 315

Table I: The location of the CP and the value of the ground state incompressibility K0 within the Walecka, Hybrid and QvdW models, together with their experimental estimates.

intermediate between the Walecka and QvdW ones, they are not shown in the ﬁgures.

Let us introduce the following reduced variables: T T/T , n n/n ,andp p/p . The ⌘ c ⌘ c ⌘ c classical vdW equation of state (9) becomes independent of the interaction parameters a and b e e e when expressed in these reduced variables. This independence on the interaction parameters is known as the law of corresponding states. Although vdW equation do not describe real gases and liquids such a law is known to hold true for many real gases and liquids with rather good accuracy (see, e.g., Ref. [30]). The coexistence curves and critical isotherms in the reduced

9 Improving Walecka (σ-ω) model

1. One can add more meson fields, add nonlinear interaction for σ field => relativistic mean-field approach => difficulties with the flow constraint even having 10-15 parameters!

2. One can add a phenomenological repulsion a la Van der Waals to weaken the vector meson repulsion

D. H. Rischke, M. I. Gorenstein, H. Stoecker and W. Greiner, Z. Phys. C 51, (1991) 485 => same difficulties remain up to huge value for nucleon hard-core radius R_N = 0.7 fm! 3. One can add another phenomenological attraction which depends on baryonic (vector, not a scalar!) density M. I. Gorenstein, D. H. Rischke, H. Stoecker, W. Greiner and K. A.Bugaev, J. Phys. G 19, (1993) 69

However, the problem is how to recover the 1-st L. Van Hove axiom? ! In Walecka model this occurred automatically, since the rule to calculate pressure from Lagrangian is known! 7

m! 2 m 2 + !0 N =(2SN +1)(2IN +1)=4, 2 2 7 7

Degeneracy factor =(2S +1)(2I +1)=4,g34 ! m⇤ m m N N N ! 0 ! 2 2 m! 2 m 2 + ! + N!0=(2SN +1)(2N =(2IN S+1)=4N +1)(2IN ,+1)=4, 2 0 1 2 2 21 зовалось импульсноеnucleon распределение spin SN = идеального2 and isospin газа нуклоновIN = 2, апривведении‘энергии сжа-

тия’ учитывалось взаимодействие между нуклоDegeneracyнами. Но factor взаимодействиеN =(2SN +1)(27 должноIN +1)=4 менять,g им!!0- m⇤ Degeneracy factor N =(2SN +1)(2IN +1)=4,g!!0 m⇤34 nucleon spin S = 1 and isospin I = 1 пульсное распределение идеального газа, и возникаетN 2 проблема совместностиN 2 релятивистско- 1 1 p p nucleonmзовалось spin! 2 SNm импульсное=2 and распределение isospin IN идеального= газа нуклонов, апривведении‘энергии сжа- + ! 2 N =(2SN +1)(2IN +1)=42 , =0 =0 34 го Ферми распределения2 0 нуклонов2 по импульсам ифункциональнойформы! ‘энергии сжатия’. тия’ учитывалось взаимодействие между нуклонами0 . Но взаимодействиеp p должно менять им- =0 =0 Для решениязовалось этойпульсное импульсное проблемы распределение распределение напомним идеальн, чтоого идеального газа в релятивистской, и возникает газа нуклонов проблема!0 модели, апривведении совместности среднего релятивистско поля‘энергии Ва- сжа- - Degeneracy factor N =(2SN +1)(2IN +1)=4,g!!0 3m⇤ dk 2 тия’ учитывалось взаимодействие междуp нуклонами.Ноp взаимодействиеdk3 должноµ менять им- лечки [10] (см. также1го Ферми[11]) распределениявзаимод1 ействие нуклонов описывается поg импульсам!N скалярнымифункциональнойформы[f+φ ивекторнымf ] m‘энергии!2 !U0 =0мезон сжатия-’. nucleon spin SN = 2 and isospin IN = 2 =0 g!N3=0 [f+ f ] m !0 =0 ) )Z (2⇡) (2⇡)3 ! пульсное распределениеДля решения этой идеальн проблемыого газа напомним!,0и возникает, что в релятивистской проблемаZ совместности модели среднего релятивистско поля Ва- - ными полями с вершинами мезон-барионного взаимодействия в функции Лагранжа: gs ψψφ¯ 2 µ лечки [10] (см. также [11]) взаимодействие2 описываетсяm скалярным φ ивекторнымg! U мезон- µ го Ферми распределения нуклонов по импульсамm 3 ифункциональнойформы= (m m⇤) !0 =‘энергииg!n сжатия’. ¯ p p dk 2 ) 2 и gv ψγ ψ Uµ. Для ядерной материи=0 в термодинамическом=0= (m mg равновесии⇤) 2 эти! мезонные= m! n поля, g!N 2 [f+ f ] m !0 =00 2 ¯ ными полями! с0 вершинами мезон-барионноg 3го взаимодействия в)! функции Лагранжаm : gs ψψφ Для решения этой проблемы) напомнимZ (2⇡, )что в релятивистской модели среднего! поля Ва- как обычно полагают,¯ являютсяµ постоянными классическими величинами. Скалярное поле и gv ψγ ψ Uµ. Дляdk3 ядерной материи вdk термодинамическом3 равновесии эти мезонныеµ поля, лечки [10] (см. также [11]) взаимодействие2 описывается2 скалярным2 φ ивекторным2 U2 мезон- g!N [f+ f2 ]= mN !0 =0 k +(m⇤) + g!!0 f+ + k +(m⇤) g!!0 f 3 ! (2⇡)3 описывает притяжениекак) обычно между полагаютZ (2 нукло⇡) , намиявляютсяm и изменяет постоянныZ нуклоннуюми✓q классическими массу величинамиMg◆! M✓q. Скалярное= M gs полеφ. ◆ = 3 (mm2 m⇤m) 2 dk!30 = →n ∗ − m2 ¯ m2 ными полями с вершинамиdk мезон2 -барионно! 2 го взаимодействия2 в функции2 2 2 Лагранжа: g!s ψψφ2 2 2 g !2 =2 )N k m+(m⇤) [f+2+ f ]+ 2! описываетm= притяжениеN между нуклоk намиg0+( иm изменяет⇤) + нуклоннуюg!!03 f+ массу+! M k M+(=mM⇤)gs φ0 . g!!0 f Нуклонное отталкивание описывается нулевой3 2 компонентой! 2 (Zпространственные(2⇡) компоненты2 2 ¯ µ = (m m⇤)(2⇡) !0 = n q → ∗ − и gv ψγ ψ Uµ.gДля2 ядернойZ материи) ✓q вm термодинамическом2 равновесии◆ ✓q эти мезонные поля, ◆ Нуклонное отталкивание описывается! нулевой компонентой (пространственные компоненты должны исчезнуть из-за трансляционнойm2 m2 инвариантностиdk)3векторного поля, которая добавm- 2 m2 как обычно полагают!3, являются2 постоянны2 ми классическимиdN2 величинами2 . Скалярное поле! 2 2 должны исчезнутьdk !0 из-за трансляционной = N инвариантностиk ) +(векторногоexpm [+⇤)M/T поля[fH+] ,+котораяf ]+ добав- !0 3 2 2 3 dM ⇠ 2 2 dk2 = N 2 k 2+(m⇤) + g!!(20 ⇡f)+ + k +(m⇤) g!!0 f 2 2 ляет U(ρ)=C ρ (Cv =2 const2)3 кэнергиинуклона2 2 Z(2 U(ρ) дляq антинуклона2 ). В [67] было = N v k +((2m⇡⇤))2 + g!!0 f+ + g! k +(m⇤) g2!!0 f g M 2 M = M gsφ описываетляет3 притяжениеU(Zρ)=C ρ между(✓Cv = Cconst нуклоv = ) 2кэнергиинуклонанами285 и.9GeV изменяет ◆, C нуклонную(s =U(✓ρ2) для377 массу антинуклона.6GeV ). В [67]◆было . Z (2⇡) ✓ v q ◆ m✓ ! − ◆ mq q2 2 q ' 3 − ' →2 ∗ −2 m2 m2 m dkm3 dk m2 m2 m m сформулировано! 2 обобщенное 2 ! 2 УС ядерной22 2 матери, которое2! 2 включает 22 и модель среднего! 2 по- 2 Нуклонное!0 сформулировано отталкивание=!N обобщенноеk описы+(= mвается⇤) УС[f+ ядерной+ нулевойf ]+ материk компонентой+(!0, котороеm ) [f включает(пространственные+ f ]+ и модель! среднего компоненты по- 2 2 0 (2⇡)3 N 3 dN2 2⇤ + 0 2 Z 2 q Compare(2Tcep⇡)=18.9 MeV with experimental value2Tcep =17.9 2 0.4MeV Z q exp [+M/TH] ± ля и чистодолжны феноменологическиеля исчезнуть и чисто феноменологические из-за модели трансляционной какчастныеслучаи модели ин кавариантностикчастныеслучаиdM ⇠ . Ограничиваясь) векторного. Ограничиваясь поля нуклонными, которая нуклонными добав- dN Thermodynamicallyстепенями2g свободы2 , expрассмотрим [+M/T ] следующее Self-consistent ядерноеg2 УС достаточно EoS общего for вида Nuclear в большом Matter ляет U(ρ)=Cv ρ! (Cv = const=)T/TкэнергиинуклонаHcep 2 = n/n0 ( U(ρ) для антинуклона2 ). В [67] было степенями свободыC, рассмотримv = 2dM ⇠285 следующее.9GeV ядерноеdN, Cs УС= достаточно2 377.6GeV общего вида в большом m! m − 2 '2 M/T' g! каноническом2 ансамблеg : 2 exp [+ H] Cv = 2 285.9GeVgeneralized , Cs = 2 377 pressure.6GeV ⇠ каноническомm! 'сформулировано ансамбле: обобщенноеm ' УС ядернойdM матери, которое включает и модель среднего по- ρ 2 3 2 2 n g! γN d k 2 k g 2 ляCv и= чистоp2(T,µ феноменологические)=285.9GeV , C моделиs = 2 ка(f кчастныеслучаи+377f )+.+6GeVnUρ U((ρnρ)) . Ограничиваясьd⇢dnU( U⇢)+(n)+P (MP⇤()M нуклонными∗) , mCompare! 3 T 2 =183 .9 MeVm with+ experimental value T =17(1.4).9 0.4MeV Compare Tcepγ =18.9d MeV'k with3 experimentalkcep(2π) value2 Tcep2 =17' .9 − 0.4MeV − cep N ! k + M ∗ ± Z0 !0 ± p(T,µ)= (f+ + f )+ρ U(ρ) dn U(n)+P (M ∗) , (1.4) степенями свободы3 , рассмотрим" следующее− ядерное УС достаточно общего вида в большом 3 ! (2π) k2 + M 2 − ! где f+ и f функцииideal распределенияgas∗ with M* ну massклонов и антинуклоновmean-field0 contributions = T/Tcep каноническомCompare= n/n0 Tcep ансамбле−"=18.:9 MeV with experimental value Tcep =17.9 0.4MeV = T/Tcep = n/n0 1 ± где f+ и f функции распределения нуклонов и2 антинуклонов2 − k + M ∗ µ U(ρ) ρ − generalized3 f distribution2exp function∓ ± +1 , (1.5) γN d k ± ≡ k⎡ ⎛" T ⎞ ⎤ p(T,µ)=n (fn+ + f )+ρ U(ρ) 1 dn U(n)+P (M ∗) , (1.4) = T/Tcep = n/n0 3 2 22 − − + nU(n) d⇢ U3(⇢)+(2P (πM) ⇤)k 2 +P⎣intM(n⎝)= dµ⇢ U(⇢)U(nUρ) (n) −⎠ ⎦ ! k + M∗∗ !0 µ Z0барионныйf exp химический потенциал∓Z0, и γN±число нуклонных+1 спин, -изоспиновыхµ baryonic состояний(1.5) chemical, ± ≡ ⎡ ⎛"" n T ⎞ ⎤ n где f+ и f функции распределения нуклонов и антинуклонов γN =4− для симметричной+ nU(n) ядернойd⇢ материиU(⇢)+. ЗависиP (Mмость⇤) эффективноP (йнуклонноймассыn)= potentiald⇢ U(M⇢)∗ nU(n) ⎣ ⎝ ⎠ ⎦ int µ барионный химический потенциалn , и γZ0N число нуклонных спин-изоспиновыхn 1 состоянийZ0 , от T и µ определяется экстремумо2мтермодинамическогопотенциала2 −(максимум давления): Should obey a self-consistency condition k + M ∗ µ U(ρ) 8 n-dependent+ nU(n)f interactiond⇢expU(⇢)+ pressureP (M ⇤)∓ ±Pint(n)=+1d⇢ U, (⇢) nU(n)(1.5) γ =4 ± ⎡ ⎛" T 3 ⎞ ⎤ M ∗ N для симметричной ядернойδ p(T,µ)Z0≡ материиdP. Зависи(M ∗) мость эффективноd k M ∗ йнуклонноймассыZ0 γN 3 (f+ + f )=0. (1.6) + δ M ∗ , ⎣ ≡ ⎝ dM∗ − (2π) 2 ⎠ 2 ⎦ − T, µ ! k + M ∗ от T и µ определяетсяµ барионный экстремумо химическиймтермодинамическогопотенциала потенциал, и γN число нуклонных спин(максимум-изоспиновых давления состояний): , " dPint(n) dU(n) ПлотностьShould барионного obey a self-consistency числа и плотность энерги conditionимогутбытьнайденыиз(1.4)= - (1.6),n ис- γ =4 3 M ∗ δ Np(T,µдля) симметричнойdP(M ядерной∗) материиd .kЗависимостьM ∗ эффективнойнуклонноймассыdn dn Should obey a self-consistencyγN condition3 (f+ + f )=0. (1.6) + δ TM ∗µ , ≡ dM∗ − (2π) 2 2 − от и MaximumопределяетсяT, µ pressure экстремумо withмтермодинамическогопотенциала! respect to keffective+ M ∗ mass (максимум давления): " 3 Плотность барионногоδ p числа(T,µ) и плотностьdP(M энерги∗) имогутбытьнайденыизd k M ∗ (1.4) - (1.6), ис- γN 3 (f+ + f )=0. (1.6) + δ M ∗ , ≡ dM∗ − (2π) 2 2 − T, µ ! k + M ∗ " Плотность барионного числа и плотность энергиимогутбытьнайденыиз(1.4) - (1.6), ис-

VdW EoS [14, 15]. The typical form of EoS for quantum quasi-particles of mass mp

and degeneracy factor dp is as follows

p(T,µ,n )=p (T,⌫(µ, n )) P (T,n ) , (1) id id id int id dk k2 1 pid(T,⌫)=dp E(k) ⌫ , (2) 3 Z (2⇡ ) 3 E(k) e( T ) + ⇣ ⌫(µ, n )=µ bp+ U(T,n ) , (3) id id

where the constant b 4V = 16⇡ R3 is the excluded volume of particles with the hard- ⌘ 0 3 p core radius Rp (here V0 is their proper volume), the relativistic energy of particle with momentum ~k is E(k) ~k2 + m 2 and the density of point-like particles is defined ⌘ p q as n (T,⌫) @pid(T,⌫) . The parameter ⇣ switches between the Fermi (⇣ = 1), the id ⌘ @⌫ Bose (⇣ = 1) and the Boltzmann (⇣ = 0) statistics. The interaction part of pressure Pint(T,nid) and the mean-field U(T,nid) will be specified later. Thermodynamically Self-consistent EoS for Nuclear Matter

Similarly to Walecka model, the self-consistency condition 35 35 provides fulfillment of thermodynamic identities35 пользуя известные термодинамические тождества: пользуя известные термодинамические тождества: пользуя известные термодинамические тождества: ∂ p d3k 3 nρ(T,µ) ∂ p = γN d k (f+ f ) , baryonic charge density (1.7) ∂ p d3k 3 − ρ(T,µ) ≡ !∂ µ"T = γN # (2π) (f+− f ) , (1.7) ρ(T,µ) = γN (f3+ f ) , − (1.7) ≡≡ !∂∂µµ"T #(2π(2)3π) − −− ! ∂"Tp # ∂ p 8 ε(T,µ) T ∂ p + µ ∂ p ≡ ∂∂pT ∂ p∂ µ εεε((T,µT,µ)) TT ! "µ++µ µ ! "T - p energy density ≡≡ !!∂∂TT"" !∂!µ∂"µ" ρ µ 3µ T T d k ρn 3 2 2 ρ = γN ddPk3 int(nk) + M dU((fn+) f )+ dn U(n) P (M ∗) . (1.8) d k3 2 2 ∗ 2 = γN (2π) k +2 =M ∗n(f+ f−)+− ddn⇢ U U((n⇢)) PP((−M∗⇤)). (1.8) = γN # 3 $k + M ∗ (f+− f )+#0 dn U(n) P (M ∗) . (1.8) # (2(2π)π)dn3$ dn− − − #Z − − # $ 00 #0 ИзИз(1.5)(1.5) следуетследует,, чточто у у нуклона нуклона(антинуклон(антинуклон) распределение) распределение импульса импульса имеет имеет форму форму иде- иде- Из (1.5) следует, что у нуклона (антинуклон) распределение импульса имеет форму иде- derive Eq. for ε from expression for pressure; use альногоального распределения распределенияHome work: Ферми Ферми во во‘внешнем‘внешнем пол поле’: скалярноее’: скалярное поле поле изменяет изменяет массу массу нуклона нуклона ального распределенияthermodynamic Ферми воidentities‘внешнем for пол idealе’: скалярное gas with chemical поле изменяет potential массу ν=µ-U(n) нуклона (антинуклона(антинуклона)) M нана эффективную эффективную массу массуM ∗M, авекторноеполедобавляетэнергию∗, авекторноеполедобавляетэнергиюU(ρ) нукU(-ρ) нук- (антинуклона) M на эффективную массу M ∗, авекторноеполедобавляетэнергиюU(ρ) нук- лонулону(-(-UU((ρρ)) длядля антинуклона антинуклона).).ВажноВажно, однако, однако, что, что в (1.4) в (1.4)и (1.8)и (1.8)появляютсяпоявляются дополнитель дополнитель- - лону (-U(ρ) для антинуклона). Важно, однако, что в (1.4) и (1.8) появляются дополнитель- ныеные члены члены,, представляющиепредставляющие термодинамическ термодинамическисогласованныевкладыполейвдавлениеиисогласованныевкладыполейвдавлениеи ные членыSimilarly, представляющие to the ordinary термодинамическ gases, inисогласованныевкладыполейвдавлениеи the hadronic or nuclear systems the source of плотностьплотность энергии энергии.. ФормаФорма этих этих дополнительн дополнительныхых вкладов вкладов в давление в давление(1.4) (1.4)уже согласованауже согласована плотностьсраспределениямиФермиhard-core энергии repulsion. Форма(1.5) этих isчерез connected дополнительн общее to термодинамическое theых Pauli вкладов blocking в давление тождество e↵ect(1.4)(1.6)! betweenужеФормулы согласована the interacting сраспределениямиФерми(1.5) через общее термодинамическое тождество (1.6)! Формулы сраспределениямиФерми(1.4) -fermionic (1.8) определяют constituents поэтом(1.5)успециальныйкласстермодина existingчерез общее interior термодинамическое the compositeмически particles тождество самосогласованных (see,(1.6)! for instance,Формулы [2]). (1.4) - (1.8)Thisопределяют e↵ect appears поэтом dueуспециальныйкласстермодина to the requirement of antisymmetrizationмически самосогласованных of the wave function (1.4)УС ядерной - (1.8) определяют материи. Это поэтом феноменологическоеуспециальныйкласстермодинаобобщение модели среднегомически поля. самосогласованныхМодели это- УС ядернойof all материи fermionic. Это constituents феноменологическое existingобобщение in the system модели and среднего at very поля high. Модели densities это it- may УСго ядерной класса определяют материи. сяЭто двумя феноменологическое функциями U(ρ) иобобщениеP (M ∗). Общие модели физические среднего ограничения поля. Модели это- го классаlead определяют to the Mottся двумя e↵ect, функциями i.e. to a dissociationU(ρ) и P ( ofM composite∗). Общие физические particles or ограничения even the clusters на эти функции таковы: го классаof определяют particles intoся двумя their constituentsфункциями U [2].(ρ) и Therefore,P (M ∗). Общие it is физические evident that ограничения at suciently на эти функции таковы: a на эти функцииhigh densities таковыU( ρ)= one: cannotU(ρ),U ignore(ρ) theρ hard-coreρ for repulsion 0 a or1 , the finite (e↵ective) size of − − →∞ ∼ ≤ ≤ a compositeU particles( ρ)= andU(ρ the),U success(ρ)ρ of traditionalb ρ for EoS 0 useda in1 the, theory of real gases U(ρ)ρ 0 →∞ ρ , for 0 b − − → ∼ a ≤ ≤ U( ρ)= U(ρ),U(ρ)ρ ∼ ρ for≤ 0 a 1 , [9] based on the hard-core repulsion→∞ approachb tells us that this is a fruitful framework − − U(ρ)ρ 0 ∼ ρk , for 0 ≤ b ≤ P (M ∗)= ak (M→ M ∗) with a2 < 0 . (1.9) also for quantum systems. Hence we− start∼ fromb the simplest≤ case, i.e. the quantum k%2U(ρ)ρ 0 ρ , for 0 b ≥ → k P (M ∗)= ak (M ∼ M ∗) with ≤a2 < 0 . (1.9) VdW EoS [14, 15]. The typical form of EoS for quantum quasi-particles of mass mp Например, верхнее неравенство a 1 для степениk 2 a, обеспечивает− k выполнение условия при- P (M≤ ∗)=%≥ ak (M M ∗) with a2 < 0 . (1.9) and degeneracy factor dp is as followsk 2 − Напримерчинности, приверхнее высоких неравенство барионныхa плотностях1 для%≥ степени, то естьa тот, обеспечивает факт, что скорость выполнение звука не условия пре- при- ≤ Напримервышает скорость, верхнее света неравенство. p(T,µ,na 1 для)= степениp (T,⌫(aµ,, обеспечивает n )) P ( выполнениеT,n ) , условия при- (1) чинности при высоких барионных≤ плотностяхid id, то есть тотid факт ,intчто скоростьid звука не пре- dk k2 1 чинностиВыбирая приU высоких(ρ) и P (M барионных∗), которые плотностях удовлетворяют, то(1.9),естьможно тот факт воспроизвести, что скорость многие звука мо- не пре- pid(T,⌫)=dp E(k) ⌫ , (2) вышает скорость света. 3 Z (2⇡ ) 3 E(k) e( T ) + ⇣ вышаетдели ядерного скорость УС света, известные. из литературы. Модели [68, 69] соответствуют M ∗ = M (с Выбирая U(ρ) и P (M ∗), которые⌫(µ, nid)= удовлетворяютµ bp+ U(1.9),(T,nidможно) , воспроизвести многие мо- (3) P (M ∗) 0) в (1.4) испециальнойформеU(ρ). Для моделей среднего поля [10, 11, 64, 70, 71] Выбирая≡ U(ρ) и P (M ∗), которые удовлетворяют (1.9), можно воспроизвести многие мо- M ∗ = M дели ядерного2 УС, известные из2 литературы16⇡ 3 . Модели [68, 69] соответствуют (с U(ρ)=whereC ρ the constanta = 1/b2 C 4Va0 ==0(k>R 3)is the excluded volume of particles with the hard- V . Выбор 2 ⌘s , k 3 p соответствует линейной модели среднего дели ядерного УС, известные− из литературы. Модели [68, 69] соответствуют M ∗ = M (с P (M ∗) 0) в (1.4) испециальнойформеU(ρ). Для моделей среднего поля [10, 11, 64, 70, 71] ≡core radius Rp (here V0 is their proper volume), the relativistic energy of particle with P (M ∗) 02 ) в (1.4) испециальнойформе2 2 U(ρ2). Для моделей среднего поля [10, 11, 64, 70, 71] U(ρ)=momentumC ρ. Выбор ~ka2is=E(1k/)2 C , ~kak+=0(mp k>and3) theсоответствует density of линейной point-like модели particles среднего is defined ≡ V − ⌘s 2 @pid(T,⌫) 2 q U(ρ)=Cas ρn.idВыбор(T,⌫) a2 = 1/2 C., Theak parameter=0(k>3)⇣соответствуетswitches between линейной the моделиFermi ( среднего⇣ = 1), the V ⌘ −@⌫ s Bose (⇣ = 1) and the Boltzmann (⇣ = 0) statistics. The interaction part of pressure Pint(T,nid) and the mean-field U(T,nid) will be specified later. 35

пользуя известные термодинамические тождества:

∂ p d3k ρ(T,µ) = γN (f+ f ) , (1.7) 3 − ≡ !∂ µ"T # (2π) − ∂ p ∂ p ε(T,µ) T + µ ≡ !∂ T "µ !∂ µ"T ρ 3 d k 2 2 = γN k + M ∗ (f+ f )+ dn U(n) P (M ∗) . (1.8) (2π)3 − − − # $ #0

Из (1.5) следует, что у нуклона (антинуклон) распределение импульса имеет форму иде- ального распределения Ферми во ‘внешнем поле’: скалярное поле изменяет массу нуклона 8

(антинуклона) M на эффективную массу M ∗, авекторноеполедобавляетэнергиюU(ρ) нук- 8 лону (-U(ρ) для антинуклона). Важно,Requirementsоднако, что в (1.4) и to(1.8) Mean-fieldпоявляютсяn дополнитель Potentials- dPint(n) dU(n) = n d⇢ U(⇢) P (M ⇤) 8 ные члены, представляющие термодинамическIndn contrastисогласованныевкладыполейвдавлениеи to Waleckadn model, our functions P_int(n) and P(M*) n are notZ0 restricted by some Lagrangian! плотностьdPint энергии(n) . ФормаdU этих(n дополнительн) ых вкладов в давление (1.4) уже согласована = n d⇢ U(⇢) P (M ⇤) ! dn dn сраспределениямиФерми(1.5) черезBut общееZ 0we have термодинамическое to payn for this freedom тождество and(1.6)! haveФормулы to formulate some dPint(n) dU(n) = n d⇢ Ugeneral(⇢) Pconditions(M ⇤) on these functions: (1.4) - (1.8) определяютdn поэтомуспециальныйкласстермодинаdn мически самосогласованных Z0 УС ядерной материи. Это феноменологическоеU( n)= U(обобщениеn)forodd модели functionn среднего of baryonic поляU(.n chargeМодели) n density этоa,a- 1 !1 ) !  го класса определяютU( n)=ся двумяU( функциямиn)forU(ρn) и P (M ∗). ОбщиеU( физическиеn) na,a ограничения1 !1 )a !  Uна( этиn функции)= U таковы(n)for: n U(n) n ,a 1 to obey causality condition, !1 ) !  i.e. speed of sound < speed of light

a b U( ρ)= forU(ρn),U0(ρ)ρ U(nρ ) forn 0 , 0a b1 , i.e. interaction must vanish at − − ! )→∞ ∼ ! ≤ ≤ b vanishing density U(ρ)ρ 0 ρ , for 0 b → ∼ ≤ k P (M ∗)= ak (M M ∗) with a2 < 0 . higher powers(1.9) than one can k 2 − %≥ get from Lagrangians Например, верхнее неравенство a 1 для степени a, обеспечивает выполнение условия при- Similarly to the≤ ordinary gases, in the hadronic or nuclear systems the source of чинностиhard-core при высоких repulsion барионных плотностях is connected, то есть to тот the факт Pauli, что blocking скорость звука e↵ect не between пре- the interacting SimilarlySimilarly toвышает the ordinary скоростьfermionic to the света gases, ordinary constituents. in the gases, hadronic existing in the interior or hadronic nuclear the or systems composite nuclear the particles systems source (see, of the for source instance, of [2]). hard-corehard-core repulsionВыбирая is repulsionThis connectedU(ρ e)↵иectP is(M appearsconnected to∗), theкоторые Pauli due удовлетворяют to to blocking the the Pauli requirement(1.9), e↵ blockingectможно between of воспроизвести antisymmetrization e↵ect the between interacting многие мо theof interacting the wave function fermionicfermionic constituentsдели ядерного constituentsof existing all УС fermionic, известные interior existing constituents из the литературы interior composite. existing theМодели particles composite[68, in the 69] (see,соответствуют system particles for and instance, (see,M at∗ = very forM [2]).( instance, highс densities [2]). it may

P (M ∗) 0) в (1.4) испециальнойформеU(ρ). Для моделей среднего поля [10, 11, 64, 70, 71] This e↵ectThis appears e↵ect due≡ appearslead to the to the due requirement Mott to the e↵ requirementect, of i.e. antisymmetrization to a dissociation of antisymmetrization of of composite the wave ofparticles function the wave or even function the clusters 2 2 U(ρ)=C ρ. Выбор a2 = 1/2 C , ak =0(k>3) соответствует линейной модели среднего of all fermionicof all fermionic constituentsVof particles constituents existing into− in their existings the system constituents in the and system at [2]. very Therefore, and high at densities very it is high evident it densities may that itat may suciently lead to thelead Mott to e the↵ect, Motthigh i.e. e densities to↵ect, a dissociation i.e. one to cannot a dissociation of composite ignore the of hard-coreparticles composite or repulsion particles even the or or clusters the even finite the (e clusters↵ective) size of composite particles and the success of traditional EoS used in the theory of real gases of particlesof into particles their into constituents their constituents [2]. Therefore, [2]. Therefore, it is evident it that is evident at su thatciently at suciently [9] based on the hard-core repulsion approach tells us that this is a fruitful framework high densitieshigh one densities cannot one ignore cannot the ignore hard-core the repulsion hard-core or repulsion the finite or (e the↵ective) finite size (e↵ective) of size of also for quantum systems. Hence we start from the simplest case, i.e. the quantum compositecomposite particles and particles the success and the of traditional success of traditional EoS used in EoS the used theory in of the real theory gases of real gases VdW EoS [14, 15]. The typical form of EoS for quantum quasi-particles of mass mp [9] based on[9] the based hard-core on the repulsionhard-core approach repulsion tells approach us that tells this us is that a fruitful this is framework a fruitful framework and degeneracy factor dp is as follows also for quantumalso for systems. quantum Hence systems. we Hence start from we start the simplest from the case, simplest i.e. the case, quantum i.e. the quantum

VdW EoS [14, 15]. The typical form of EoSp(T,µ,n for quantumid)=pid(T, quasi-particles⌫(µ, nid)) Pint of( massT,nid)m, (1) VdW EoS [14, 15]. The typical form of EoS for quantum quasi-particles p of mass mp dk k2 1 and degeneracy factor dp is as follows and degeneracy factor dp is as followspid(T,⌫)=dp E(k) ⌫ , (2) 3 Z (2⇡ ) 3 E(k) e( T ) + ⇣ ⌫(µ, nid)=µ bp+ U(T,nid) , (3) p(T,µ,nid)=p(T,µ,npid(T,⌫)=(µ,p nid())T,⌫(Pµ,int n(T,n)) id)P, (T,n ) , (1) (1) id id id int id dk k2 1 2 dk 16⇡ k3 1 pidwhere(T,⌫)= thepd constantp(T,⌫)=bd 4V = E(k)R⌫ is the, excluded volume, of particles(2) with(2) the hard- id 3 p 0 3 p E(k) ⌫ (2⇡ ) 3⌘E(k) (3 T ) Z Z (2e⇡ ) 3 E(k+) ⇣e( T ) + ⇣ core radius Rp (here V0 is their proper volume), the relativistic energy of particle with ⌫(µ, nid)=⌫µ(µ, nbp)=+ Uµ(T,nbpid)+, U(T,n ) , (3) (3) id id

16⇡ 3 16⇡ 3 where thewhere constant theb constant4V0 =b R4pVis= the excludedR is the volume excluded of particles volume of with particles the hard- with the hard- ⌘ 3⌘ 0 3 p core radiuscoreRp radius(here VR0 pis(here theirV proper0 is their volume), proper the volume), relativistic the relativistic energy of particle energy with of particle with momentum ~k is E(k) ~k2 + m 2 and the density of point-like particles is deﬁned ⌘ p q as n (T,⌫) @pid(T,⌫) . The parameter ⇣ switches between the Fermi (⇣ = 1), the id ⌘ @⌫ 36

2 поля (далее ‘модель Валечки’) [10]. Для a2 = 1/2 C , a3 =0, a4 =0, ak =0(k 5) данное − s ̸ ≥ обобщение воспроизводит нелинейную модель среднего поля [64, 70, 71].

При T =0находим общее соотношение для моделей (1.4) - (1.8):

2 3 3 2 2 dW U(ρ)+! π ρ + M ∗ = M + W (ρ)+ρ , (1.10) " 2 dn "$ % # которое соответствует теореме Гугенгольца–ван Хова [68] для взаимодействующего ферми

газа при нулевой температуре. Так как dW =0при плотности насыщения, то мы dn ρ=ρo & ' получаем аналог соотношения Вайскопфа [69] между энергией Ферми и энергией связи на частицу 36

2 2 3 поля (далее ‘модель Валечки’) [10]. Для a2 = 1/2 Cs , a3 =0, a4 =0, a3k =0(2 k 5) данное2 − U(ρo)+̸ ! π ρo ≥+ Mo∗ = M + Wo 922 MeV. (1.11) " 2 ≈ обобщение воспроизводит нелинейную модель среднего поля [64, 70," 71].$ % # 37 Уравнение (1.11) дает нам отношение между U(ρo) и M ∗ ипоэтомутолькооднаизэтих При T =0находим общее соотношение для моделей (1.4) - (1.8): o феноменологического УС. Для этого достаточно показать, что данное УС имеет разумные 2величин (например, Mo∗) независима. 3 физические свойства3. Это2 можно показать2 так. dW U(ρ)+! π ρ + M ∗ = M + W (ρ)+ρ , (1.10) " 2 Рассмотрим теперь новоеdn УС ядерной материи [47] из обобщенного класса моделей сред- Хотя фактор" несжимаемости$ % Ko не может быть выбран независимо от M ∗, находим, что # o которое соответствуетразумные теореме значения M Гугенгольцаo∗ приводятнего к поля значениям–ван(1.4) ХоваK -o,[68] (1.8).которыедляМожно лежат взаимодействующего в экспериментально выбрать опре ферми- деленном диапазоне (см. таблицу ).dWПервая строка в таблице соответствует модели Валечки, газа при нулевой температуре. Так как =0при плотности насыщения, то мы 1 dn ρ=ρo 1 2 2 2 2 P (M ∗)= C (M M ∗) ,U(ρ)=C ρ C ρ 3 . для которой получаем слишком маленькое& ' значение M, ислишкомбольшоезначениеs Ko. v d (1.12) получаем аналог соотношения Вайскопфа [69] между энергией Ферми− 2 и энергией− связи на −

3 частицу Таблица 1.1. Различные наборыТак констант в модель связи для Валечки фиксированных добавлено значений притяжениеρo =0.159 фм− в потенциал U(ρ). Такая модификация U(ρ) 8 W [65] и o =-16 МэВ. 2 до3 некоторой степени подобна моделям [72, 73]. Однако, вотличиеотэтихмоделей, данная 3 2 2 U(ρo)+! π ρo 2 + M2 o∗ =2 M +2 Wo 2 922 MeV. (1.11) "M2∗/M C (GeV− ) C (GeV− ) C≈ Ko (MeV) "$ %модельv теперьs объясняетd тот факт, что M ∗ = M вядернойсреде. Введение третьего парамет- # n dPint(n) dU0.543(n) 285.90 377.56 0 553 ̸ Уравнение (1.11) дает нам= отношениеn междуdU⇢(Uρo)(⇢и) Mo∗Pипоэтомутолькооднаизэтих(M ⇤) dn dn ра Cd позволяет выбирать Mo∗ свободно в дополнение к значениям ρo и W (ρo). Необходимо 0.600 257.40Z0 326.40 0.124 380 величин (например, Mo∗) независима. 0.635 подчеркнуть238.08 296.05, что, если0.183 потребовать300 , чтобы новая константа связи8 была целой степенью фун- Рассмотрим теперь новое УС ядерной материи [47] из обобщенногоa класса моделей сред- U( n)= U(n)for0.688 n 206.79 251.14U(n) 0.254n ,a2101 1 Simplestдаментальных!1 Realization ) единиц! , тоof толькоthe Model степень 3 от37 ρ для нового притяжения в U(ρ) способна него поля (1.4) - (1.8). Можно выбрать0.720 186.94 244.52 0.288 170 n dPint(n) dU(n) феноменологического УС. Дляудовлетворить этого достаточно условию показать= n , что(1.9). данноеДополнительный УСd⇢ имеетU(⇢) разумныеP (M ⇤) вклад в потенциал U(ρ) может быть полу- dn dn 1 1 1 2 b 2 2 2 2 2Z0 for n 0P (M ∗)=U(n) C n(M, 0 M ∗bU) ,U(n()=ρ)=CCnρ CCnρ3 3 . (1.12) !физические) свойства− 2!. Этоs можночен−  в показать приближении так. среднегоv v − поляd d из функции Лагранжа, содержащей дополнительный член Удивительно хорошая корреляция между значениями Ko винтервале210 300 МэВa и Хотя факторCompared несжимаемости to WaleckaKoUне( можетn)= model бытьU(n выбран)for there независимо isn additional от Mo∗,Uнаходим− (attractionn) , nчто,a and1 one Так в модель Валечки добавлено притяжение в потенциал U(ρ). Такая!1 модификация ) ! U(ρ) значениями Mo∗ вблизи 0, 65 нуклонM [64, 65],-нуклонногоозначает, что предст самодействияавленная модель видаadditional описывает пять parameter разумные значения Mo∗ приводят к значениям Ko, которые лежат в экспериментально опре- величин основного состояния ядерной материи только с тремя независимыми параметрами. 2 до некоторой степени подобна моделям [72, 73]. Однако, вотличиеотэтихмоделейb 3 , данная2 2 1 деленном диапазоне (см. таблицу ).forПерваяn строка0 вU таблице(n) n соответствует, 0 bU модели(2n)= ВалечкиCµ n, C n 3 3 This attraction is generated by a peculiar Lagrangian C ψγ¯ v ψ ψγ¯ µd ψ . (1.13) Плотность энергии в данной модели [47]!имеет) вид !  4 d модель теперь объясняетдля которой тот получаем факт слишком, что M маленькое∗ = M вядернойсреде значение M, ислишкомбольшоезначение. Введение третьего& K параметo. - ' 3 ̸ d k 11 33 4 4 11 2 2 2 2 2 2 2 2 3 22 2 2 ра Cd позволяетε(T,µ выбирать)=γN M ∗ свободноk + MКонечно∗ в(f дополнение+ f, )+нет никакойCCv кρn значениямC непосредственнойCd ρn 3+ρo иCCsW((MM(ρo)M. M∗Необходимо) физ⇤.) (1.14)ической3 мотивации для введения такого вкла- Таблица 1.1. Различные(2πo)3 наборы констант− связи2 дляv фиксированных4 d значений2 s ρo =0.159 фм− Similarly to the ordinary gases,! in" the hadronic− or2 nuclear− 4 systems 2 the− source of подчеркнуть, что[65], еслии Wo =-16 потребоватьМэВ. , чтобыда на новая теоретико константа-полевых связи былаоснованиях целой степенью. Однако фунтакое- основание, конечно, не требуется для hard-core repulsion is connectedДля дальнейшегоThese to анализа the parameters Pauli мы фиксируем blocking areK onormalized= e300↵ectМэВ between(сравни on properties с [59]). theОтметим interacting of, чтоnuclear при matter 2 1 2 2 2 2 даментальныхэтом единиц значении, то толькоKo эффективнаяM /M степеньC массаотMCo∗ρнаходитсядля нового в оченьC притяжения хорошемK согласии в U с( результатаρ) способна- fermionic constituents existing interior∗ thev (GeV composite3− ) s (GeV particles− ) d (see,o (MeV) for instance, [2]). ми нерелятивистских0.543 вычислений285.90 в рамках377.56 задачимногих0-тел FriedmanU553(ρ)и Pandharipande [65]. This e↵ectудовлетворить appears due условию to the(1.9). requirementДополнительный of antisymmetrization вклад в потенциал of theможет wave function быть полу- На рис.1.1япривожусвободнуюэнергиюнабарионизвычислений0.600Similarly257.40 to the ordinary326.40 gases,0.124 in the380 hadronic[65] ирезультатыданной orallowed nuclear systems the source of of all fermionicчен в приближении constituents allowed среднего existing поля in из theфункции system Лагранжа and at, содержащей very high дополнительный densities it may член rangeмодели of. Из рис.1.1hard-core0.635видно довольно repulsion238.08 хорошее is296.05 connected согласие0.183 при to the маленьких Pauli300 blockingρ исистематическиеrange e↵ect of between the interacting нуклон-нуклонного самодействия вида lead to the Mott e↵ect,отклоненияvalues i.e. to от a нерелятивистскихdissociationfermionic0.688 constituents206.79 of результатов composite251.14 existing при больших particles interior0.254 ρ. Это the or210 происходитcomposite even} the при particlesvalues clustersρ 3 ρ (see,o, for instance, [2]). ≥ } This e↵3ect appears due to the2 requirement of antisymmetrization of the wave function of particles into their constituents0.720 [2]. Therefore,2186.94 µ 244.52 it3 is evident0.288 that170 at suciently C ψγ¯ ψ ψγ¯ µ ψ . (1.13) of all fermionic4 d constituents existing in the system and at very high densities it may high densities one cannot ignore the hard-core& repulsion' or the ﬁnite (e↵ective) size of leadModel to the with Mott K_0 e↵ ect,=220-300 i.e. to a MeV dissociation obeys ofthe composite proton particlesflow constraint or even the clusters Конечно, нет никакой непосредственной физической мотивации для введения такого вкла- Удивительно хорошая корреляция между значениями Ko винтервале210 300 МэВ и composite particles and the successof ofparticles traditional into their EoS constituents used in [2].the Therefore,theory of− it real is evident gases that at suciently M , M [9] basedда on на the теоретико hard-coreзначениями-полевых repulsion основанияхo∗ вблизиhigh approach densities0 65. Однако[64, one 65], tells cannotозначаеттакое us, ignorethat основаниечто предст this theавленная hard-core, isконечно a fruitful модель repulsion, не описывает framework требуется or the пять ﬁnite для (e↵ective) size of величин основного состояния ядерной материи только с тремя независимыми параметрами. also for quantum systems. Hencecomposite we start particles from and the the simplest success of case, traditional i.e. the EoS used quantum in the theory of real gases Плотность энергии[9] в based данной on модели the[47] hard-coreимеет вид repulsion approach tells us that this is a fruitful framework VdW EoS [14, 15]. The typical form of EoS for quantum quasi-particles of mass m 3 p d alsok for quantum systems.1 Hence3 we start4 1 from the simplest case, i.e. the quantum 2 2 2 2 2 3 2 2 ε(T,µ)=γN k + M ∗ (f+ f )+ Cv ρ Cd ρ + Cs (M M ∗) . (1.14) (2π)3 − 2 4 2 and degeneracy factor dp is as! followsVdW" EoS [14, 15].− The typical− form of EoS for quantum− quasi-particles of mass mp

Для дальнейшегоand анализа degeneracy мы фиксируем factor dKpo is= as300 followsМэВ (сравни с [59]). Отметим, что при p(T,µ,nid)=K pid(T,⌫(µ, nidM)) Pint(T,nid) , (1) этом значении o эффективная масса o∗pнаходится(T,µ,n в)= оченьp ( хорошемT,⌫(µ, согласии n )) P с результата(T,n )-, (1) dk k2 1 id id id int id ми нерелятивистских вычислений в рамках задачимногих-тел dFriedmank k2и Pandharipande1 [65]. pid(T,⌫)=dp E(k) ⌫ , (2) 3 pid(T,⌫)=dp E(k) ⌫ , (2) (2⇡ ) 3 E(k) ( T ) 3 ( ) На рис.1.1япривожусвободнуюэнергиюнабарионизвычисленийZ e + ⇣Z (2⇡ ) 3[65]E(ирезультатыданнойk) e T + ⇣ ⌫(µ, n )=µ bp+ U(T,n ⌫)(,µ, nid)=µ bp+ U(T,nid) , (3) (3) моделиid. Из рис.1.1 видно довольно хорошееid согласие при маленьких ρ исистематические 16⇡ 3 отклонения от нерелятивистскихwhere the constant результатовb 4 приV0 = большихR ρis. Это the происходитexcluded volume при ρ of3 ρ particleso, with the hard- 16⇡ 3 ⌘ 3 p ≥ where the constant b 4V0 = 3 Rp is the excluded volume of particles with the hard- ⌘ core radius Rp (here V0 is their proper volume), the relativistic energy of particle with core radius Rp (here V0 is their proper volume), the relativistic energy of particle with momentum ~k is E(k) ~k2 + m 2 and the density of point-like particles is deﬁned ⌘ p q as n (T,⌫) @pid(T,⌫) . The parameter ⇣ switches between the Fermi (⇣ = 1), the id ⌘ @⌫ Summary

1. We discussed the necessary apparatus to describe the nuclear matter EoS

2. The properties of normal nuclear matter are used to normalize the phenomenological EoS

3. The Walecka model is presented and its mean-field approximation is applied to normal nuclear matter

4. A phenomenological generalization of Walecka model is discussed and the self-consistency condition is obtained