the structure of compact stars - istituto nazionale di

Dottorato di Ricerca in FisicaXXXII Ciclo

The Structure of Compact StarsSpring 2017

Omar Benhar

Contents

Introduction 1

1 The structure of white dwarfs 3

1.1 Energy-density and pressure of the degenerate Fermi gas . . . . . . . . . . . . . 5

1.2 Dynamical content of the equation of state . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Equation of state of the degenerate Fermi gas . . . . . . . . . . . . . . . . . . . . 10

1.4 Hydrostatic equilibrium and structure of white dwarfs . . . . . . . . . . . . . . . 11

1.5 Equilibrium equation in general relativity . . . . . . . . . . . . . . . . . . . . . . . 14

2 Introduction to neutron stars 17

2.1 Outer crust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 Inverse β-decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.2 Neutronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Inner crust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 The neutron star interior 29

3.1 Constraints on the nuclear matter EOS . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 The nucleon-nucleon interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 The two-nucleon system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 non relativistic many-body theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4.1 The few-nucleon systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

IV CONTENTS

3.4.2 Nuclear matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5 Relativistic mean field theory: the σ−ω model . . . . . . . . . . . . . . . . . . . 48

4 Exotic forms of matter 57

4.1 Stability of strange hadronic matter . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1.1 Hyperon interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Deconfinement and quark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.1 The bag model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.2 The equation of state of quark matter . . . . . . . . . . . . . . . . . . . . . 64

5 Cooling mechanisms 71

5.1 Direct Urca process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.1.1 Threshold of the direct Urca process . . . . . . . . . . . . . . . . . . . . . . 73

5.2 Modified Urca processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2.1 Neutron branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2.2 Proton branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Neutrino bremsstrahlung in nucleon-nucleon collisions . . . . . . . . . . . . . . 80

Appendixes 82

A Derivation of the one-pion-exchange potential 85

B Neutron β-decay 89

C Basic of QCD thermodinamics 93

D Phase-space decomposition 95

D.1 Calculation of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

D.2 Calculation of I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

CONTENTS V

Bibliography 100

VI CONTENTS

Introduction

Compact stars, i.e. white dwarfs, neutron stars and black holes, are believed to be the final stageof stellar evolution. They are different from normal stars, the stability of which against gravitationalcollapse is due to thermal pressure. In compact stars, the nuclear fuel necessary to ignite fusion reac-tions leading to heat production is no longer available, and the pressure needed for hydrostatic equi-librium is produced by different mechanisms, driven by quantum effects and interactions betweenthe constituents of matter in the star interior.

Compacts stars provide a unique laboratory to study the structure of matter under extreme condi-tions, in which all known forces - gravitation, electromagnetism, weak and strong interactions - playa role. Hence, understanding their properties is of paramount importance for both astrophysics andfundamental physics.

In 1844, the german astronomer Friedrich Bessel deduced that the star called Sirius had an unseencompanion, which was first observed two decades later by Alvan Graham Clark and named Sirius B.The mass of Sirius B was determined by applying Kepler’s third law to the orbit of the binary system,while its radius was obtained in the 1920s from the equation describing blackbody emission, usingthe measured spectrum and luminosity. The resulting values, M ∼ 0.75−0.95 M¯ (M¯ = 1.989×1033 gdenotes the mass of the sun) and R ∼ 18800 Km, comparable to the radius of a planet 1, revealed thatSirius B was a very compact object, of density reaching millions of g/cm3. In his book The InternalConstitution of the Stars, published in 1926, Sir Arthur Eddington wrote: “we have a star of mass aboutequal to the sun and radius much less than Uranus”. Sirius B is now known to belong to the class ofstellar objects called white dwarfs.

The observation of the first white dwarf triggered the early efforts aimed at understanding theproperties of matter at densities exceeding by many orders of magnitude the typical density of terres-trial macroscopic objects (ρ ∼ 20 g/cm3), the structure of which is mainly determined by electromag-netic interactions. In contrast, at very large density the role of electromagnetic interactions becomesnegligible and the structure of matter is determined by quantum statistics and nuclear interactions.

The discovery, made by Chandrasekhar in 1931, of an upper bound to the mass of white dwarfs,started the speculations on the fate of stars of mass exceeding the limiting value. Neutron stars, whoseexistence is said to have been first discussed by Bohr, Landau and Rosenfeld right after the discov-ery of the neutron, were predicted to occur in the aftermath of a supernova explosion by Baade andZwicky in 1934 [1] . Due to the large densities involved, up to ∼ 1015 g/cm3, the theoretical descriptionof neutron star matter must take into account the full complexity of the dynamcs of strong interac-tions, including the forces acting at nuclear and hadronic level, as well as the possible occurrence of

1Note that this is about four times bigger than the value resulting from more recent measurements

2 Introduction

a core of deconfined quark matter. In addition, the effects predicted by Einstein’s theory of generalrelativity, that play a negligible role in white dwarfs, in neutron stars become large.

These notes provide an introduction to the structure of matter at densities 104 <∼ ρ <∼ 1015 g/cm3,typical of white dwarfs and neutron stars. The focus will mainly be on the density region ρ > ρ0, whereρ0 = 2.67×1014 g/cm3 is the the central density of atomic nuclei, i.e. the largest density observed onearth under ordinary conditions. Describing this region necessarily requires a significant amountof extrapolation of the available empirical data on the properties of strongly interacting matter. Inaddition, as the density exceeds ∼ 2ρ0, new form of matter, whose occurrence is predicted by thefundamental theories of weak and strong interactions, may appear.

The classic references on the Physics of compact stars are the books of Shapiro and Teukolskii [2],Glendenning [3] and Weber [4]. A number of aspects of the structure of matter at high density havebeen also discussed in a somewhat more concise fashion by Leung [5].

These notes have been written using a system of units in which ħ= h/2π= c = KB = 1, where h, cand KB denote Plank’s constant, the speed of light and Boltzmann’s constant, respectively.

Chapter 1

The structure of white dwarfs

The formation of a star is believed to be triggered by the contraction of a self-gravitating hydrogencloud. As the density increases, the cloud becomes more and more opaque, and the energy releasedcannot be efficiently radiated away. As a consequence, the temperature also increases, and eventuallybecomes high enough (∼ 6×107 K)to ignite the nuclear reactions turning hydrogen into helium:

p +p → 2H+e++ν+0.4 MeV ,

e++e− → γ+1.0 MeV ,2H+p → 3He+γ+5.5 MeV ,

3He+3 He → 4He+2p +26.7 MeV .

Note that the above reactions are all exothermic, and energy is released in form of kinetic energy ofthe produced particles (1 MeV = 1.6021917× 10−6 erg). Equilibrium is reached as soon as gravitationalattraction is balanced by matter pressure.

When the nuclear fuel is exhausted the core stops producing heat, the internal pressure cannot besustained and the contraction produced by gravitational attraction resumes. If the mass of the heliumcore is large enough, its contraction, associated with a further increase of the temperature, can thenlead to the ignition of new fusion reactions, resulting in the appearance of heavier nuclei: carbon andoxygen. Depending on the mass of the star, this process can take place several times, the final resultbeing the formation of a core made of the most stable nuclear species, nickel and iron, at density ∼1014 g/cm3 (note that the central density of atomic nuclei, the highest density observable on earth,is ρ0 ≈ 2 × 1014 g/cm3). Even larger densities are believed to occur in the interior of neutron stars,astrophisical objects resulting from the contraction of the iron core in very massive stars (M > 4 M¯).

If the star is sufficiently small, so that the gravitational contraction of the core does not producea temperature high enough to ignite the burning of heavy nuclei, it will eventually turn into a whitedwarf, i.e. a star consisting mainly of helium, carbon and oxygen.

The over 2000 observed white dwarfs have luminosity L ∼ 10−2 L¯ (L¯ = 4×1033 erg s−1 is theluninosity of the sun) and surface temperature Ts ∼ 104 K. The radius of many white dwarfs has beendetermined from their measured flux, defined as

F (D) = L

4πD2 , (1.1)

4 The structure of white dwarfs

Nuclear fuel Main products Temperature Density Duration[K] [g/cm3] [yrs]

H He 6×107 5 7×106

He C, O 2×108 700 5×105

C O, Ne, Mg 9×108 2× 105 600Ne O, Mg, Si 1.7×109 4× 105 1O Si, S 2.3×109 107 0.5Si Fe 4×109 3× 107 0.0025

Table 1.1: Stages of nucleosynthesis for a star of mass ∼ 25 M¯

where D is the distance from the earth, obtained using the parallax method. Combining the aboveequation to the equation describing black body emission

L = 4πσR2T 4s , (1.2)

one obtains

R =√

F D2

σT 4s

. (1.3)

The measured values of mass and radius of three white dwarfs are given in Table 1.2. The cor-responding average densities are ∼ 107 g/cm3, to be compared to the typical density of terrestrialmacroscopic objects, not exceeding ∼ 20 g/cm3.

Mass Radius[M¯] [R¯]

Sirius B 1.000±0.016 0.0084±0.0002Procyon B 0.604±0.018 0.0096±0.000440 Eri B 0.501±0.011 0.0136±0.0002

Table 1.2: Measured mass and radius of three white dwarfs

The pressure required to ensure the stability of white dwarfs against gravitational collapse is pro-vided by a gas of noninteracting electrons at very low temperature. The properties of this systems will

1.1 Energy-density and pressure of the degenerate Fermi gas 5

be discussed in the following Sections,

1.1 Energy-density and pressure of the degenerate Fermi gas

Let us consider a system of noninteracting electrons uniformly distributed in a cubic box of vol-ume V = L3. If the temperature is sufficiently low, so that thermal energies can be neglected, thelowest quantum levels are occupied by two electrons, one for each spin state. Both electrons havethe same energy, i.e. they are degenerate. This configuration corresponds to the ground state of thesystem. A gas of noninteracting electrons in its ground state is said to be fully degenerate. At highertemperature, the thermal energy can excite electrons to higher energy states, leaving some of thelower lying levels not fully degenerate.

As the electrons are uniformly distributed, their wave functions must exhibit translational invari-ance. They can be written in the form

ψps(r) =φp(r)χs , (1.4)

where χs is a Pauli spinor specifying the spin projection and

φp(r) =√

1

Vei p·r , (1.5)

is an eigenfunctions of the the momentum, the generator of spacial translation, satisfying the peri-odic boundary conditions (x, y and z denote the components of the vector r, specifying the electronposition)

φp(x, y, z) =φp(x +nx L, y +ny L, z +nz L) , (1.6)

where nx ,ny ,nz = 0, ±1, ±2, . . .. The above equation obviously implies the relations (p ≡ (px , py , pz ) )

px = 2πnx

L, py =

2πny

L, pz = 2πnz

L, (1.7)

which in turn determine the momentum eigenvalues.Each quantum state is associated with an eigenvalue of the momentum p, i.e. with a specific

triplet of integers (nx ,ny ,nz ). The corresponding energy eigenvalue is (p2 = |p|2 = p2x +p2

y +p2z )

εp = p2

2me=

(2π

L

)2 1

2me(n2

x +n2y +n2

z ) , (1.8)

me being the electron mass (me = 9.11× 10−28 g, or 0.511 MeV). The highest energy reached, called theFermi Energy of the system, is denoted by εF , and the associated momentum, the Fermi momentum,is pF =p

2meεF .The number of quantum states with energy less or equal to εF can be easily calculated. Since each

triplet (nx ,ny ,nz ) corresponds to a point in a cubic lattice with unit lattice spacing, the number of mo-mentum eigenstates is equal to the number of lattice points within a sphere of radius R = pF L/(2π).The number of electrons in the system can then be obtained from (note: the factor 2 takes into ac-count spin degeneracy, i.e. the fact that there are two electrons with opposite spin projections sittingin each momentum eigenstate)

N = 24π

3R3 =V

p3F

3π2 , (1.9)


and the electron number density, i.e. the number of electrons per unit volume, is given by

ne = N

V= p3

F

3π2 . (1.10)

The total ground state energy can be easily evaluated from

E = 2∑

p<pF

p2

2me(1.11)

replacing (use (1.7) again and take the limit of large L, corresponding to vanishingly small level spac-ing) ∑

p<pF

→ V

(2π)3

∫p<pF

d 3p (1.12)

to obtain

E = 2V

(2π)3 4π∫ pF

0p2d p

p2

2me= N

3

5

p2F

2me. (1.13)

The resulting energy density is

ε= E

V= 1

(2π)3 4πp5

F

5me. (1.14)

From Eq.(1.10) it follows that the Fermi energy can be written in terms of the number densityaccording to

εF = p2F

2me= 1

2me

(3π2ne

)2/3. (1.15)

The above equation can be used to define a density n0 such that for ne À n0 the electron gas at giventemperature T is fully degenerate. Full degeneracy is realized when the thermal energy KB T (KB isthe Boltzman constant: KB = 1.38 × 10−16 erg/K) is much smaller than the Fermi energy εF , i.e. when

ne À n0 = 1

3π2 (2me KB T )3/2 . (1.16)

For an ordinary star at the stage of hydrogen burning, like the Sun, the interior temperature is∼ 107 K, and the corresponding value of n0 is ∼ 1026 cm−3. If we assume that the electrons come froma fully ionized hydrogen gas, the matter density of the proton-electron plasma is

ρ = (mp +me ) n0 ∼ 200 g/cm3 , (1.17)

mp being the proton mass (mp = 1.67 × 10−24 g). This density is high for most stars in the early stageof hydrogen burning, while for ageing stars that have developed a substantial helium core the density(mn denotes the neutron mass: mn ≈ mp ).

ρ = (mp +mn +me ) n0 ∼ 400 g/cm3 (1.18)

can be largely exceeded within the core. For example, white dwarfs have core densities of the order of107 g/cm3. As a consequence, in the study of their structure thermal energies can be safely neglected,the primary role being played by the degeneracy energy p2/2me .

1.1 Energy-density and pressure of the degenerate Fermi gas 7

The pressure P of the electron gas, i.e. the force per unit area on the walls of the box, is definedin kinetic theory as the rate of momentum transferred by the electrons colliding on a surface of unitarea. Consider first one dimensional motion along the x-axis. The momentum transfer associatedwith the reflection of electrons carrying momentum px off the box wall during a time d t is

d px

d t= 1

2× (2px )× (ne vx L2) . (1.19)

In the above equation, the second factor is the momentum transfer associated with a single electron,while the third factor is the electron flux, i.e. the number of electrons hitting the wall during the timed t . The factor 1/2 accounts for the fact that half of the electrons go the wrong way and do not collidewith the box wall. The resulting pressure is

P (px ) = 1

L2

d px

d t= ne px vx = ne p2

x

me. (1.20)

In the three dimensional case the electron carries momentum p =√

p2x +p2

y +p2z and we have to

repeat the calculation carried out for the x-projection of the momentum transfer. As the system isisotropic

p2x

me= px vx = 1

3(px vx +py vy +pz vz ) = 1

3(p ·v) = 1

3

p2

me, (1.21)

and Eq.(1.20) becomes

P (p) = 1

3ne (pv) == 1

3

N

V

p2

me. (1.22)

The pressure can therefore be obtained averaging over all momenta. We find (as usual, the factor2 accounts for the two possible spin projections)

P = 1

N2

∑p≤pF

P (p) = 2

3

1

(2π)3 4π∫ pF

0p2d p (pv) = p5

F

15π2me. (1.23)

Note that the above result can also be obtained from the standard thermodynamical definition ofpressure

P =−(∂E

∂V

)N

, (1.24)

using E given by Eq.(1.13) and (∂pF /∂V )N =−pF /(3V ).Eq.(1.23) shows that the pressure of a degenerate Fermi gas decreases linearly as the mass of the

constituent particle increases. For example, the pressure of an electron gas at number density ne is∼ 2000 times larger than the pressure of a gas of nucleons, neutrons and protons, at the same numberdensity.

So far, we have been assuming the electrons in the degenerate gas to be nonrelativistic. However,the properties of the system depend primarily on the distribution of quantum states, which is dictatedby translation invariance only, and is not affected by this assumption. Releasing the nonrelativivsticapproximation simply amounts to replace the nonrelativistic energy with its relativistic counterpart:

p2

2me→

√p2 +m2

e −me . (1.25)


The transition from the nonrelativistic regime to the relativistic one occurs when the electron energybecomes comparable to the energy assciated with the electron rest mass, me . It is therefore possibleto define a density nc such that at ne ¿ nc the system is nonrelativistic, while ne À nc correspondsto the relativistic regime. The value of nc can be found requiring that the Fermi energy at ne = nc beequal to me . The resulting expression is

nc = 23/2

3π2 m3e ∼ 1.6×1030cm−3 . (1.26)

The energy density of a fully degenerate gas of relativistic electrons can be obtained from (com-pare to Eqs.(1.13) and (1.14))

ε= 21

(2π3)4π

∫ pF

0p2d p

[√p2 +m2

e −me

]. (1.27)

while the equation for the pressure reads (compare to Eq.(1.23) and use again v = ∂εp /∂p with rela-tivistic εp )

P = 2

3

1

(2π)3 4π∫ pF

0p2d p

(p∂εp

∂p

). (1.28)

Carrying out the integrations involved in Eqs.(1.27) and (1.28) we find:

ε= πme

λ3e

[t(2t 2 +1

)√t 2 +1− ln

(t +

√t 2 +1

)− 8t 3

3

], (1.29)

and

P = πme

λ3e

[1

3t(2t 2 −3

)√t 2 +1+ ln

(t +

√t 2 +1

)], (1.30)

where λe = 2π/me c is the electron Compton wavelength and (see Eq.(1.15))

t = pF

me= 1

me

(3π2ne

)1/3. (1.31)

Eqs.(1.29) and (1.30) give the energy density and pressure of a fully degenerate electron gas as a func-tion of the variable t , which can in turn be written in terms of the number density ne according toEq.(1.31).

As a final remark, we briefly discuss the possible relevance of electrostatic interactions. In a fullyionized plasma their effect can be estimated noting that the corresponding energy is

Ec = Ze2

⟨r ⟩ ∝ Z e2 n1/3e , (1.32)

where Z e is the electric charge of the ions and ⟨r ⟩ ∝ n1/3e is the typical electron-ion separation dis-

tance. It follows that the ratio between Ec and the Fermi energy is given by

Ec

εF∝ 1

n1/3e

. (1.33)

The above equation shoes that, for sufficiently high density, the contribution of electrostatic interac-tions becomes negligibly small. As this condition is largely satisfied at the densities typical of whitedwarfs, electrons can be safely described as a fully degenerate Fermi gas.

1.2 Dynamical content of the equation of state 9

1.2 Dynamical content of the equation of state

The equation of state (EOS) is a nontrivial relation linking the thermodynamic variables specify-ing the state of a physical system. The best known example of EOS is Boyle’s ideal gas law, stating thatthe pressure of a collection of N noninteracting, pointlike classical particles, enclosed in a volume V ,grows linearly with the temperature T and the average particle density n = N /V .

The ideal gas law provides a good description of very dilute systems. In general, the EOS can bewritten as an expansion of the pressure, P , in powers of the density (from now on we use units suchthat Boltzmann’s constant is KB = 1):

P = nT[1+nB(T )+n2C (T )+ . . .

]. (1.34)

The coefficients appearing in the above series, that goes under the name of virial expansion, are func-tions of temperature only. They describe the deviations from the ideal gas law and can be calculatedin terms of the underlying elementary interaction. Therefore, the EOS carries a great deal of dynam-ical information, and its knowledge makes it possible to establish a link between measurable macro-scopic quantities, such as pressure or temperature, and the forces acting between the constituents ofthe system at microscopic level.

Figure 1.1: Behavior of the potential describing the interactions between constituents of a van derWaals fluid (the interparticle distance r and v(r ) are both given in arbitrary units).

This point is best illustrated by the van der Waals EOS, which describes a collection of particlesinteracting through a potential featuring a strong repulsive core followed by a weaker attactive tail(see Fig.1.1). At |U0|/T << 1, U0 being the strength of the attractive part of the potential, the van derWaals EOS takes the simple form

P = nT

1−nb−an2 , (1.35)

and the two quantities a and b, taking into account interaction effects, can be directly related to thepotential v(r ) through [6]

a =π∫ ∞

2r0

|v(r )| r 2dr , b = 16

3πr 3

0 , (1.36)


where 2r0 denotes the radius of the repulsive core (see Fig.1.1).In spite of its simplicity, the van der Waals EOS describes most of the features of both the gas and

liquid phases of the system, as well as the nature of the phase transition.

1.3 Equation of state of the degenerate Fermi gas

Equation (1.30) is the EOS of the degenerate electron gas, providing a link between its pressure Pand the matter density ρ, which is in turn related to the electron number density ne through

ρ = mp

Yene , (1.37)

where Ye is the number of electrons per nucleon in the system. For example, for a fully ionized heliumplasma Ye = 0.5, whereas for a plasma of iron nuclei Ye = Z /A = 26/56 = 0.464 (Z and A denote thenuclear charge and mass number, respectively).

The EOS of the fully degenerate electron gas takes a particularly simple form in the nonrelativisticlimit (corresponding to t ¿ 1), as well as in the extreme relativistic limit (corresponding to t À 1).From Eq.(1.30) we find

P = 8

15

πme

λ3e

(3π2Ye

mp

)5/3

ρ5/3 . (1.38)

for eF ¿ me and

P = 2

3

πme

λ3e

(3π2Ye

mp

)4/3

ρ4/3 . (1.39)

for eF À me .A EOS that can be written in the form

P ∝ ρΓ , (1.40)

is said to be polytropic. The exponent Γ is called adiabatic index, whereas the quantity n, definedthrough

Γ= 1+ 1

n, (1.41)

goes under the name of polytropic index.The adiabatic index, whose definition for a generic equation of state reads

Γ= d (l nP )

d (lnρ). (1.42)

is related to the compressibility χ, characterizing the change of pressure with volume according to

1

χ=−V

(∂P

∂V

)N= ρ

(∂P

∂ρ

)N

. (1.43)

through

Γ= 1

χP. (1.44)

1.4 Hydrostatic equilibrium and structure of white dwarfs 11

The compressibility is also simply related to speed of sound in matter, cs , defined as

cs =(∂P

∂ρ

)1/2

= 1

χρ. (1.45)

The magnitude of the adiabatic index reflects the so called stiffnes of the equation of state. Largerstiffness corresponds to more incompressible matter. As we will see in the following lectures, stiffnessturns out to be critical in determining a number of stellar properties.

1.4 Hydrostatic equilibrium and structure of white dwarfs

Let us assume that white dwarfs consist of a plasma of fully ionized helium at zero temperature.The pressure of the system, P , is provided by the electrons, the contribution of the helium nuclei beingnegligible due to their large mass. For any given value of the matter density ρ, P can be computedfrom Eqs.(1.30) and (1.31) (in this case Ye = 0.5, implying ne = ρ/2mp ). The results of this calculationare shown by the diamonds in fig. 1.2. For comparison, the nonrelativistic (Eq.(1.38)) and extremerelativistic (Eq.(1.39)) limits are also shown by the solid and dashed line, respectively. Note that thevalue of matter density corresponding to nc defined in Eq.(1.26) is ρ ∼ 6.3 106 g/cm3.

Figure 1.2: Equation of state of a fully ionized helium plasma at zero temperature (diamonds). Thesolid and dashed line correspond to the nonrelativistic and extreme relativistic limits, respectively.

In order to show the sensitivity of the EOS to the value of Ye , in Fig. 1.3 the EOS of the fully ionizedhelium plasma (Ye = 0.5) is compared to that of a hydrogen plasma (Ye = 1).

The surface gravity of white dwarfs, GM/R (G is the gravitational constant), is small, of order∼ 10−4. Hence, their structure can be studied assuming that they consist of a spherically symmetricfluid in hydrostatic equilibrium and neglecting relativistic effects.

Consider a perfect fluid in thermodynamic equilibrium, subject to gravity only. The Euler equa-tion can be written in the form

∂v

∂t+ (v ·∇)v =− 1

ρ∇P −∇φ , (1.46)


Figure 1.3: Comparison between the EOS of a fully ionized plasma of helium (solid line) and hydro-gen (dashed line).

where ρ is the density and φ is the gravitational potential, satisfying Poisson’s equation

∇2φ= 4πGρ . (1.47)

Eq.(1.46) describes the motion of a fluid in which processes leading to energy dissipation, occurringdue to viscosity (i.e. internal friction) and heat exchange between different regions, can be neglected.For a fluid at rest (i.e. for v = 0), it reduces to

∇P =−ρ (∇φ) . (1.48)

For a spherically symmetric fluid, Eqs. (1.47) and (1.48) become

dP

dr=−ρdφ

dr, (1.49)

and1

r 2

d

dr

(r 2 dφ

dr

)= 4πGρ . (1.50)

Substitution of Eq. (1.49) in Eq. (1.50) yields

1

r 2

d

dr

(r 2

ρ

dP

dr

)=−4πGρ , (1.51)

implyingdP

dr=−ρ(r )

GM(r )

r 2 , (1.52)

with M(r ) given by

M(r ) = 4π∫ r

0ρ(r ′)r ′2dr ′ . (1.53)

1.4 Hydrostatic equilibrium and structure of white dwarfs 13

The above result simply state that, at equilibrium, the gravitational force acting on an volumeelement at distance r from the center of the star is balanced by the force produced by the spacialvariation of the pressure.

Given a EOS, Eq.(1.52) can be integrated numerically for any value of the central density ρc toobtain the radius of the star, i.e. the value of r corresponding to vanishingly small density. The masscan then be obtained from Eq.(1.53).

Figure 1.4: Dependence of the mass of a white dwarf upon its central density, obtained from theintegration of Eq.(1.52) using the equation of state of a fully ionized helium plasma.

For polytropic EOS Eqs.(1.52) and (1.53) reduce to the Lane-Emden equation, whose integrationwith the polytropic index n = 3/2, corresponding to the non relativistic regime, yields the relation

M = 2.79

Y 2e

(ρc

ρ

)1/2M¯ , (1.54)

where ρ denotes the matter density corresponding to the electron number density of Eq.(1.26). Theresulting values of M agree with the results of astronomical observations of white dwarfs. However, itis very important to realize that Lane-Emden equation predicts the existence of equilibrium configu-rations for any values of the star mass.

In 1931 Chandrasekhar pointed out that, due to the large Fermi enegies, the non relativistic treat-ment of the electron gas was not justified [7]. Replacing the EOS of Eq.(1.38) with its relativistic coun-terpart, Eq.(1.39), he predicted the existence of a maximum mass for white dwarfs. If the mass exceedsthis limiting value gravitational attraction prevails on the pressure gradient, and the star becomes un-stable against gravitational collapse.

The dependence of the mass of a white dwarf upon its central density, obtained from integrationof Eq.(1.52) using the equation of state of a fully ionized helium plasma, is illustrated in Fig.1.4. Thefigure shows that the mass increases as the central density increases, until a value M ∼ 1.44 M¯ isreached at ρ0 ∼ 1010 g/cm3. This value is close to the one found by Chandrasekhar. However, as wewill see in Chapter 2, at ρ ∼ 108 g/cm3 the neutronization process sets in, and the validity of thedescription in terms of a helium plasma breaks down. At ρ ≥ 108 g/cm3, matter does not support


pressure as effectively as predicted by the equation of state of the helium plasma. As a consequence,a more realistic estimate of the limiting mass, generally referred to as the Chandrasekhar mass, isgiven by the mass corresponding to a central density of 108 g/cm3, i.e. ∼ 1.2 M¯.

1.5 Equilibrium equation in general relativity

The Newtonian equilibrium equation employed in the case of white dwarfs represents a goodapproximation when the matter density does not produce an appreciable space-time curvature, inwhich case the metric is simply given by

d s2 = ηµνd xµd xν , (1.55)

with

η=

−1 0 0 00 1 0 00 0 1 00 0 0 1

. (1.56)

In Einstein’s theory of general relativity, Eq.(1.55) is repalced by

d s2 = gµνd xµd xν , (1.57)

where the metric tensor gµν is a function of space-time coordinates.The effects of space-time distortion are negligible when the surface gravitational potential fulfills

the requirement GM/R ¿ 1. This condition is satisfied by white dwarfs, having GM/R ∼ 10−4, butnot by neutron stars, to be discussed in the following Chapters, whose larger density leads to muchhigher values of GM/R, typically ∼ 10−1.

Relativistic corrections to the hydrostatic equilibrium equations (1.52) and (1.53) are obtainedsolving Einstein’s field equations

Gµν = 8πGTµν , (1.58)

where Tµν is the energy-momentum tensor and the Einstein’s tensor Gµν is defined in terms of themetric tensor gµν describing space-time geometry. Equations (1.58) state the relation between thedistribution of matter, described by Tµν, and space-time curvature, described by gµν.

Consider a star consisting of a static and spherically simmetric distribution of matter in chemical,hydrostatic and thermodynamic equilibrium. The metric of the corresponding gravitational field canbe written in the form (x0 = t , x1 = r, x2 = θ, x3 =ϕ)

d s2 = gµνd xµd xν = e2ν(r )d t 2 −e2λ(r )dr 2 − r 2(dθ2 + sin2θdϕ2) , (1.59)

implying

g =

e2ν(r ) 0 0 0

0 −e2λ(r ) 0 00 0 −r 2 00 0 0 −r 2 sin2θ

, (1.60)

ν(r ) e λ(r ) being functions to be determined solving Einstein’s equations

1.5 Equilibrium equation in general relativity 15

Under the standard assumption that matter in the star interior behave as an ideal fluid, the energy-momentum tensor can be written in the form

Tµν = (ε+P )uµuν−P gµν , (1.61)

where ε and P denote energy density and pressure, respectively, while

uµ =d xµdτ

≡ (e−ν(r ),0,0,0) , (1.62)

is the local four-velocity.The Einstein tensor Gµν, given by

Gµν = Rµν− 1

2gµνR , (1.63)

withR = gµνRµν , (1.64)

depends on gµν through the Christoffel symbols

Γλµν =1

2gαλ(gαµ,ν+ gαν,µ+ gµν,α) , (1.65)

appearing in the definition of the Ricci tensor Rµν

Rµν = Γαµα,ν−Γαµν,α−ΓαµνΓβαβ+ΓαµβΓβνα . (1.66)

From Eqs.(1.66) and (1.59) it follows that the nonvanishing elements of Rµν are

R00 =[−ν′′+λ′ν′− (ν′)2 − 2ν′

r

]e2(ν−λ) ,

R11 = ν′′−λ′ν′+ (ν′)2 − 2λ′

r,

R22 = (1+ rν′− rλ′)e−2λ−1 ,

R33 = R22 sin2θ , (1.67)

implying

R =[−2ν′′+2λ′ν′−2(ν′)2 − 2

r 2 + 4λ′

r− 4ν′

r

]e−2λ+ 2

r 2 . (1.68)

Substitution of Eqs.(1.61),(1.67) and (1.68) into Eqs.(1.58) leads to the system of differential equations(1

r 2 − 2λ′

r

)e−2λ− 1

r 2 = 8πGε(r ) ,(1

r 2 + 2ν′

r

)e−2λ− 1

r 2 = −8πGP (r ) ,[ν′′+ (ν′)2 −λ′ν′+ ν′−λ′

r

]e−2λ = −8πGP (r ) ,


where ε(r ) and P (r ) denote the space distribution of energy density and pressure, respectively.The above equation can be cast in the form originally derived by Tolman, Oppenheimer and Vol-

foff (TOV)dP (r )

dr=−ε(r )

GM(r )

r 2

[1+ P (r )

ε(r )

][1+ 4πr 3P (r )

M(r )

],

[1− 2GM(r )

r

]−1

(1.69)

withd M(r )

dr= 4πr 2ε(r ) . (1.70)

The first term in the right hand side of Eq.(1.69) is the newtonian gravitational force. It is the sameas the one appearing in Eq.(1.52), but with matter density replaced by energy density. The first twoadditional factors take into account relativistic corrections, that become vanishingly small in the limitpF /m → 0, m and pF being the mass and Fermi momentum of the star constituents, respectively.Finally, the third factor describes the effect of space-time curvature. Obviously, in the nonrelativisticlimit Eq.(1.69) reduces to the classical equilibrium equation (1.52).

Chapter 2

Introduction to neutron stars

The temperatures attained in stars with initial mass larger than ∼ M¯ are high enough to bringnucleosynthesis to its final stage (see Table 1.1), i.e. to the formation of a core of 56Fe. If the mass ofthe core exceeds the Chandrasekhar mass, electron pressure is no longer sufficient to balance gravi-tational contraction and the star evolves towards the formation of a neutron star or a black hole.

The formation of the core in massive stars is characterized by the appearance of neutrinos, pro-duced in the process

56Ni → 56Fe +2e++2νe . (2.1)

Neutrinos do not have appreciable interactions with the surrounding matter and leave the core re-gion carrying away energy. Thus, neutrino emission contributes to the collapse of the core. Otherprocesses leading to a decrease of the pressure are electron capture

e−+p → n + ve , (2.2)

the main effect of which is the disappearance of electrons carrying large kinetic energies, and ironphotodisintegration

γ+56 Fe → 134He +4n , (2.3)

which is an endothermic reaction.Due to the combined effect of the above mechanisms, when the mass exceeds the Chandrasekhar

limit the core collapses within fractions of a second to reach densities ∼ 1014 g/cm3, comparable withthe central density of atomic nuclei.

At this stage the core behaves as a giant nucleus, made mostly of neutrons, and reacts elestically tofurther compression producing a strong shock wave which trows away a significant fraction of matterin the outher layers of the star. Nucleosynthesis of elements heavier than Iron is believed to take placeduring this explosive phase.

The above sequence of events leads to the appearance of a supernova, a star the luminosity ofwhich first grows fast, until it reaches a value exceeding sun luminosity by a factor ∼ 109, and thendescreases by a factor ∼ 102 within few months. The final result of the explosion of the formation of anebula, the center of which is occupied by the remnant of the core, i.e. a neutron star.

The existence of compact astrophysical objects made of neutrons was predicted by Bohr, Landauand Rosenfeld shortly after the discovery of the neutron, back in 1932. In 1934, Baade and Zwicky

18 Introduction to neutron stars

first suggested that a neutron star may be formed in the aftermath of a supernova explosion. Finally,in 1968 the newly observed pulsars, radio sources blinking on and off at a constant frequency, wereidentified with rotating neutron stars.

The results of a pioneering study, carried out in 1939 by Oppenheimer and Volkoff within theframework of general relativity (i.e. using the equilibrium equations discussed in Section 1.5), showthat the mass of a star consisting of noninteracting neutrons cannot exceed ∼ 0.8 M¯. The fact thatthis maximum mass, the analogue of the Chandrasekhar mass of white dwarfs, turns out to be muchsmaller that the observed neutron star masses (typically ∼ 1.4 M¯) clearly shows that neutron starequilibrium requires a pressure other than the degeneracy pressure, the origin of which has to beascribed to hadronic interactions.

Unfortunately, the need of including dynamical effects in the EOS is confronted with the com-plexity of the fundamental theory of strong interactions. As a consequence, all available descriptionof the EOS of neutron star matter are obtained within models, based on the theoretical knowledge ofthe underlying dynamics and constrained, as much as possible, by empirical data.

The internal structure of a neutron star, schematically illustrated in Fig. 2.1, is believed to featurea sequence of layers of different composition and density.

Figure 2.1: Schematic illustration of a neutron star cross section. Note that the equilibrium densityof uniform nuclear matter corresponds to ∼ 2×1014g/cm3.

The properties of matter in the outer crust, corresponding to densities ranging from ∼ 107 g/cm3

to the neutron drip density ρd = 4×1011 g/cm3, can be obtained directly from nuclear data. On theother hand, models of the EOS at 4×1011 < ρ < 2×1014 g/cm3 are somewhat based on extrapolations

2.1 Outer crust 19

of the available empirical information, as the extremely neutron rich nuclei appearing in this densityregime are not observed on earth.

The density of the neutron star core ranges between ∼ ρ0 (= 2.67 ×1014 g/cm3) at the boundarywith the inner crust, and a central value that can be as large as 1÷ 4× 1015 g/cm3. All models ofEOS based on hadronic degrees of freedom predict that in the density range ρ0 <∼ ρ <∼ 2ρ0 neutron starmatter consists mainly of neutrons, with the admixture of a small number of protons, electrons andmuons. At any given density the fraction of protons and leptons is determined by the requirementsof equilibrium with respect to β-decay and charge neutrality.

This picture may change significantly at larger density with the appearance of heavy strangebaryons produced in weak interaction processes. For example, although the mass of the Σ− exceedsthe neutron mass by more than 250 MeV, the reaction n + e− → Σ− +νe is energetically allowed assoon as the sum of the neutron and electron chemical potentials becomes equal to the Σ− chemicalpotential.

Finally, as nucleons are known to be composite objects of size ∼ 0.5− 1.0 fm, corresponding toa density ∼ 1015 g/cm3, it is expected that if the density in the neutron star core reaches this valuematter undergoes a transition to a new phase, in which quarks are no longer clustered into nucleonsor hadrons.

The theoretical description of matter in the outer and inner neutron star crust will be outlined inthe following Sections, whereas the region corresponding to supranuclear density will be describedin Chapter 3.

2.1 Outer crust

A solid is expected to form when the ratio of Coulomb energy to thermal energy becomes large,i.e. when

Γ= Z2e2

TrLÀ 1 , (2.4)

with rL defined through

nI4πr 3

L

3= 1 , (2.5)

nI being the number density of ions. If the condition (2.4) is fulfilled Coulomb forces are weaklyscreened and become dominant, while the fluctuation of the ions is small compared to average ionspacing rL . From (2.4) it follows that a solid is expected to form at temperature

T < Tm = Z2e2

rL∝ Z2e2n1/3

I . (2.6)

For example, in the case of 56Fe at densities ∼ 107 g/cm3 solidification occurs at temperatures below108 K and Coulomb energy is minimized by a Body Centered Cubic (BCC) lattice. As the density fur-ther increases, rL decreases, so that the condition for solidification continues to be fulfilled. However,as matter density advances into the density domain, 107 - 1011 g/cm3, the large kinetic energy of therelativistic electrons shifts the energy balance, favouring inverse β-decay (i.e. electron capture) thatleads to the appearance of new nuclear species through sequences like

Fe → Ni → Se →Ge . (2.7)


This process is called neutronization, because the resulting nuclide is always richer in neutron con-tent than that initial one.

Before going on with the analysis of the neutronization process in the neutron star crust, we willduscuss a simple but instructive example, that will allows us to introduce some concepts and proce-dures to be used in the following Sections.

2.1.1 Inverse β-decay

Consider a gas of noninteracting protons and electrons at T = 0. The neutronization process isdue to the occurrence of weak interactions turning protons into neutrons through

p +e− → n +νe . (2.8)

Assuming neutrinos to be massless and non interacting, the above process is energetically favorableas soon as the electron energy becomes equal to the neutron-proton mass difference

∆m = mn −mp = 939.565−938.272 = 1.293 MeV . (2.9)

As a consequence, the value of ne at which inverse β-decay sets in can be estimated from√p2

Fe+m2

e =∆m , (2.10)

where (see Section 1.1)

pFe = (3π2ne )1/3 , (2.11)

leading to

ne = 1

3π2

(∆m2 −m2

e

)3/2 ≈ 7×1030cm−3 . (2.12)

The corresponding mass density for a system having Ye ∼ 0.5, is ρ ≈ 2.4 × 107 g/cm3.Now we want to address the problem of determining the ground state of the system consisting

of protons, electrons and neutrons, once equilibrium with respect to the inverse β-decay of Eq. (2.8)has been reached. All interactions, except the weak interaction, will be neglected. Note that process(2.8) conserves baryon number NB (i.e. the baryon number density nB ) and electric charge.

For any given value of nB , the ground state is found by minimization of the total energy densityof the systems, ε(np ,nn ,ne ), np and nn being the proton and neutron density, respectively, with theconstraints nB = np +nn (conservation of baryon number) and np = ne (charge neutrality).

Let us define the function

F (np ,nn ,ne ) = ε(np ,nn ,ne )+λB(nB −np −nn

)+λQ(np −ne

), (2.13)

where ε is the energy density, while λB and λQ are Lagrange multipliers.The minimum of F corrisponds to the values of np , nn and ne satisfying the conditions

∂F

∂np= ∂F

∂nn= 0 ,

∂F

∂ne= 0 (2.14)

2.1 Outer crust 21

as well as the aditional costraints∂F

∂λB= ∂F

∂λQ= 0 . (2.15)

From the definition of chemical potential of the particles of species i (i = p,n,e)

µi =(∂E

∂Ni

)V=

(∂ε

∂ni

)V

(2.16)

it follows that Eqs. (2.14) imply

µp −λB +λQ = 0 , µn −λB = 0 , µe −λQ = 0 , (2.17)

leading to the condition of chemical equilibrium

µn −µp =µe , (2.18)

where, in the case of noninteracting particles at T = 0,

µi = ∂ε

∂ni= 2

(2π)3

(∂pFi

∂ni

)∂

∂pFi

4π∫ pFi

0p2 d p

√p2 +m2

i

= 8π

(2π)3

(∂ni

∂pFi

)−1

p2Fi

√p2

Fi+m2

i =√

p2Fi+m2

i . (2.19)

Defining now the proton and neutron fraction of the system as

xp = np

nB= np

np +nn, xn = nn

nB= 1−xp (2.20)

we can rewritepFp = pFe =

(3π2xp nB

)1/3, pFn =

[3π2 (

1−xp)

nB]1/3

. (2.21)

For fixed baryon density, use of the above definitions in Eq. (2.19) and substitution of the resultingchemical potentials into Eq. (2.18) leads to an equation in the single variable xp . Hence, for any givennB the requirements of chemical equilibrium and charge neutrality uniquely determine the fractionof protons in the system.

Once the value of xp is known, the neutron, proton and electron number densities can be evalu-ated and the pressure

P = Pn +Pp +Pe (2.22)

can be obtained using Eq. (1.28).Figure 2.1.1 shows the proton and neutron number densities, np and nn (recall that ne = np ) as a

function of matter density ρ. It can be seen that in the range 105 ≤ ρ ≤ 107 g/cm3 there are protonsonly and lognp grows linearly with logρ. At ρ ∼ 107 g/cm3 neutronization sets in and the neutronnumber density begins to steeply increase. At ρ > 107 np stays nearly constant, while neutrons dom-inate.

The equation of state of the β-stable mixture is shown in the upper panel of Fig. 2.1.1. Its mainfeature is that pressure remains nearly constant as matter density increases by almost two orders ofmagnitude, in the range 107 ≤ ρ ≤ 109 g/cm3. The electron and neutron contributions to the pressureare shown in the lower panel of Fig. 2.1.1. Note that, since charge neutrality requires np =ne , theproton pressure is smaller than the electron pressure by a factor (mp /me ) ∼ 2000.


Figure 2.2: Number density of noninteracting protons and neutrons in β-equilibrium as a functionof matter density.

2.1.2 Neutronization

The description of β-stable matter in terms of a mixture of degenerate Fermi gases of neutrons,protons and electrons is strongly oversimplified. In reality, electron capture changes a nucleus withgiven charge Z and mass number A into a different nucleus with the same A and charge (Z−1). More-over, the new nucleus may be metastable, so that two-step processes of the type

5626Fe → 56

25Mn → 5624Cr (2.23)

can take place. In this case, chemical equilibrium is driven by the mass difference between neighbor-ing nuclei rather than the neutron-proton chemical potential difference.

The measured nuclear charge distributions and masses, ρch(r ) and M(Z , A), exhibit two very im-portant features

• The charge density is nearly constant within the nuclear volume, its value being roughly thesame for all stable nuclei, and drops from ∼ 90 % to ∼ 10 % of the maximum over a distance RT

∼ 2.5 fm (1 fm = 10×10−13 cm), independent of A, called surface thickness (see Fig. 2.4). It canbe parametrized in the form

ρch(r ) = ρ01

1+e(r−R)/D, (2.24)

where R = r0A1/3, with r0 = 1.07 fm, and D = 0.54 fm. Note that the nuclear charge radius isproportional to A1/3, implying that the nuclear volume increases linearly with the mass numberA.

• The (positive) binding energy per nucleon, defined as

B(Z,A)

A= 1

A

[Zmp + (A−Z)mn +Zme −M(Z,A)

], (2.25)

2.1 Outer crust 23

Figure 2.3: (A) Equation of state of a mixture of noninteracting neutrons, electrons and protons inβ-equilibrium. (B) Density dependence of the neutron (solid line) and electron (dashed line) contri-butions to the pressure of β-stable matter.

where M(Z,A) is the measured nuclear mass, is almost constant for A≥ 12 , its value being ∼ 8.5MeV (see Fig. 2.5).

The A and Z dependence of B(Z,A) can be parametrized according to the semiempirical-massformula

B(Z,A)

A= 1

A[ aVA−asA2/3 −ac

Z2

A1/3−aA

(A−2Z)2

4A+λ ap

1

A1/2] . (2.26)

The first term in square brackets, proportional to A, is called the volume term and describes the bulkenergy of nuclear matter. The second term, proportional to the nuclear radius squared, is associatedwith the surface energy, while the third one accounts for the Coulomb repulsion between Z protonsuniformly distributed within a sphere of radius R. The fourth term, that goes under the name of sym-metry energy is required to describe the experimental observation that stable nuclei tend to have thesame number of neutrons and protons. Moreover, even-even nuclei (i.e. nuclei having even Z andeven A − Z) tend to be more stable than even-odd or odd-odd nuclei. This property is accounted forby the last term in the above equation, where λ −1, 0 and +1 for even-even, even-odd and odd-oddnuclei, respectively. Fig. 2.5 shows the different contributions to B(Z,A)/A, evaluated using Eq. (2.26).

The semi-empirical nuclear mass fomula of Eq. (2.26) can be used to obtain a qualitative descrip-tion of the neutronization process. The total energy density of the system consisting of nuclei of mass


Figure 2.4: Nuclear charge distribution of 208Pb, normalized to Z/ρ(0) (Z = 82). The solid line hasbeen obtained using the parametrization of Eq. (2.24), while the diamonds represent the results of amodel independent analysis of electron scattering data.

number A and charge Z arranged in a lattice and surrounded by a degenerate electron gas is

εT (nB ,A,Z) = εe +(nB

A

)[M(Z,A)+εL] , (2.27)

where εe is the energy density of the electron gas, Eq. (1.27), nB and (nB /A) denote the number den-sities of nucleons and nuclei, respectively, and εL is the electrostatic lattice energy per site. As a firstapproximation, the contribution of εL will be neglected.

At any given nucleon density nB the equilibrium configuration corresponds to the values of A andZ that minimize εT (nB ,A,Z), i.e. to A and Z such that(

∂εT

∂Z

)nB

= 0 ,

(∂εT

∂A

)nB

= 0 . (2.28)

Combining the above relationships and using Eq. (2.26) one finds

Z ' 3.54 A1/2. (2.29)

Once Z is known as a function of A, any of the two relationships (2.28) can be used to obtain A asa function of nB . The mass number A turns out to be an increasing function of nB , implying that Zalso increases with nB , but at a slower rate. Hence, nuclei become more massive and more and moreneutron rich as the nucleon density increases.

The above discussion is obvioulsy still oversimplified. In reality, A and Z are not continuous vari-ables and the total energy has to be minimized using the measured nuclear masses, rather than theparametrization of Eq. (2.26), and including the lattice energy, that can be written as

εL =−K(Ze)2

rs(2.30)

2.1 Outer crust 25

Figure 2.5: Upper panel: A-dependence of the binding energy per nucleon of stable nuclei, eval-uated according to Eq. (2.26) with aV = 15.67 MeV, as = 17.23 MeV, ac = .714 MeV, aA = 93.15 MeVand ap = 11.2 MeV. Lower panel: the solid line shows the magnitude of the volume contribution tothe binding energy per nucleon, whereas the A-dependence of the surface, coulomb and symmetrycontributions are represented by diamonds, squares and crosses, respectively.

where rs is related to the number density of nuclei through (4π/3)r 3s = (nB /A)−1 and K = 0.89593 for a

BCC lattice, yielding the lowest energy. At fixed nucleon number density nB the total energy densitycan be written in the form

εT (nB ,A,Z) = εe +(nB

A

)[M(Z,A)−1.4442(Ze)2

(nB

A

)1/3]

, (2.31)

where, for matter density exceeding ∼ 106 g/cm3, the extreme relativistic limit of the energy densityof an electron gas at number density ne =ZnB /A (see Section 1.1)

εe = 3

4

(Z

nB

A

)4/3, (2.32)

has to be used.Collecting together the results of Eqs. (2.30)-(2.32) and expressing nB in units of nB0 = 10−9 fm−3

(the number density corresponding to a matter density ∼ 106 g/cm3), the total energy per nucleon,εT /nB , can be rewritten in units of MeV as

εT

nB= M(Z,A)

A+ 1

A4/3

[0.4578 Z4/3 − Z2

480.74

] (nB

nB0

)1/3

. (2.33)


Figure 2.6: Total energy per nucleon of a BCC lattice of 62Ni (dashed line) and 64Ni (solid line) nu-clei surrounded by an electron gas, evaluated using eq.(2.33) and plotted versus the inverse nucleonnumber density.

The average energy per nucleon in a nucleus is about 930 MeV. It can be conveniently written inunits of MeV in the form M(Z,A)/A = 930 + ∆. As long as we are dealing with nuclides that are notvery different from the stable nuclides, the values of ∆ are available in form of tables based on actualmeasurements or extrapolations of the experimental data.

In practice, εT /nB of Eq. (2.33) is computed for a given nucleus (i.e. for given A and Z) as a func-tion of nB , and plotted versus 1/nB (see Fig. 2.6). The curves corresponding to different nuclei arethen compared and the nucleus corresponding to the minimum energy at given nB can be easilyidentified. For example, the curves of Fig. 2.6 show the behavior of the energy per particle corre-sponding to 62Ni and 64Ni, having A−Z = 34 and 36, respectively. It is apparent that a first order phasetransition is taking place around the point where the two curves cross one another. The exact den-sities at which the phase transition occurs and terminates can be obtained using Maxwell’s doubletangent construction. This method essentially amounts to drawing a straight line tangent to the con-vex curves corresponding to the two nuclides. In a first order phase transition the pressure remainsconstant as the density increases. Hence, as all points belonging to the tangent of Maxwell’s construc-tion correspond to the same pressure, the onset and termination of the phase transition are simplygiven by the points of tangency. As expected, at higher density the nucleus with the largest number ofneutrons yields a lower energy.

It has to be pointed out that there are limitations to the approach described in this section. Someof the nuclides entering the minimization procedure have ratios Z/A so different from those corre-sponding to stable nuclei (whose typical value of Z/A is ∼ 0.5, as shown in Fig. 2.7) that the accuracyof the extrapolated masses may be questionable. Obviously, this problem becomes more and moreimportant as the density increases. The study of nuclei far from stability, carried out with radioac-tive nuclear beams, is regarded as one of the highest priorities in nuclear physics research, and new

2.2 Inner crust 27

Figure 2.7: Chart of the nuclides. The black squares represent the stable nuclei as a function of Z andN=A−Z. The solid lines correspond to the estimated proton and neutron drip lines.

dedicated experimental facilities are currently being planned both in the U.S. and in Europe.Table 2.1 reports the sequence of nuclides corresponding to the ground state of matter at subnu-

clear density, as a function of matter density.

2.2 Inner crust

Table 2.1 shows that as the density increases the nuclides corresponding to the ground state ofmatter become more and more neutron rich. At ρ ∼ 4.3 × 1011 g/cm3 the ground state correspondsto a Coulomb lattice of 118Kr nuclei, having proton to neutron ratio ∼ 0.31 and a slighltly negativeneutron chemical potential (i.e. neutron Fermi energy), surrounded by a degenerate electron gaswith chemical potential µe ∼ 26 MeV. At larger densities a new regime sets in, since the neutronscreated by electron capture occupy positive energy states and begin to drip out of the nuclei, fillingthe space between them.

At these densities the ground state corresponds to a mixture of two phases: matter consisting ofneutron rich nuclei (phase I), with density ρnuc, and a neutron gas of density ρNG (phase II).

The equilibrium conditions are

(µn)I = (µn)I I =µn (2.34)

and

µp =µn −µe , (2.35)

where (µn)I and (µn)I I denote the neutron chemical potential in the neutron gas and in the matter ofnuclei, respectively.


Nuclide Z N = A − Z Z/A ∆ ρmax

[MeV] [g/cm3]

56Fe 26 30 .4643 .1616 8.1 × 106

62Ni 28 34 .4516 .1738 2.7 × 108

64Ni 28 36 .4375 .2091 1.2 × 109

84Se 34 50 .4048 .3494 8.2 × 109

82Ge 32 50 .3902 .4515 2.1 × 1010

84Zn 30 54 .3750 .6232 4.8 × 1010

78Ni 28 50 .3590 .8011 1.6 × 1011

76Fe 26 50 .3421 1.1135 1.8 × 1011

124Mo 42 82 3387 1.2569 1.9 × 1011

122Zr 40 82 .3279 1.4581 2.7 × 1011

120Sr 38 82 .3166 1.6909 3.7 × 1011

118Kr 36 82 .3051 1.9579 4.3 × 1011

Table 2.1: Sequence of nuclei corresponding to the ground state of matter and maximum density atwhich they occur. Nuclear masses are given by M(Z,A)/A = (930 + ∆) MeV (from Ref. [10]).

The details of the ground state of matter in the neutron drip regime are specified by the densitiesρ, ρnuc and ρNG, the proton to neutron ratio of the matter in phase I and its surface, whose shape isdictated by the interplay between surface and Coulomb energies.

Recent studies suggest that at densities 4.3×1010 <∼ ρ <∼ .75×1014 g/cm3 the matter in phase I is ar-ranged in spheres immersed in electron and neutron gas, whereas at .75×1014 <∼ ρ <∼ 1.2×1014 g/cm3

the energetically favoured configurations exhibit more complicated structures, featuring rods of mat-ter in phase I or alternating layers of matter in phase I and phase II. At ρ >∼ 1.2×1014 g/cm3 there is noseparation between the phases, and the ground state of matter corresponds to a homogeneous fluidof neutrons, protons and electrons.

Chapter 3

The neutron star interior

Understandig the properties of matter at densities comparable to the central density of atomicnuclei (ρ0 ∼ = 2.7 × 1014 g/cm3) is made difficult by both the complexity of the interactions and theapproximations necessarily implied in the theoretical description of quantum mechanical many par-ticle systems. The situation becomes even more problematic as we enter the region of supranucleardensity, corresponding to ρ > ρ0, as the available empirical information is scarce, and one has tounavoidably resort to a mixture of extrapolation and speculation.

The approach based on non relativistic quantum mechanics and phenomenological nuclear hamil-tonians, while allowing for a rather satisfactory description of nuclear bound states and nucleon-nucleon scattering data, fails to fulfill the constraint of causality, as it leads to predict a speed ofsound in matter that exceeds the speed of light at large density. On the other hand, the approachbased on relativistic quantum field theory, while fulfilling the requirement of causality by construc-tion, assumes a somewhat oversimplified dynamics, not constrained by nucleon-nucleon data. Inaddition, it is plagued by the uncertainty associated with the use of the mean field approximation,which is long known to fail in strongly correlated systems, such as, for example a van der Waals fluid[11].

In this Chapter, after reviewing the phenomenological constraints on the EOS of cold nuclearmatter, we will outline the current understanding of the nucleon-nucleon interaction and the nonrelativistic and relativistic approaches employed to study the structure of neutron star matter at nu-clear and supernuclear density. As anticipated in Section 2.2, in this region a neutron star is believedto consist of a uniform fluid of neutrons, protons and electrons in β-equilibrium.

3.1 Constraints on the nuclear matter EOS

The body of data on nuclear masses can be used to constrain the density depencence predictedby theoretical models of uniform nuclear matter at zero temperature. Note that the zero temperaturelimit is fully justified, as the typical temperature of the neutron star interior is ∼ 109 K ∼ .01 MeV, tobe compared to nucleon Fermi energies of tens of MeV.

As we have seen in Section 2.1.2, the A-dependence of the nuclear binding energy is well de-scribed by the semiempirical formula (2.26). In the large A limit and neglecting the effect of Coulomb

30 The neutron star interior

repulsion between protons, the only term surviving in the case Z = A/2 is the term linear in A. Hence,the coefficient aV can be identified with the binding energy per particle of symmetric nuclear mat-ter, an ideal uniform system consisting of equal number of protons and neutrons coupled by stronginteractions only. The equilibrium density of such a system, n0, can be inferred exploiting satura-tion of nuclear densities, i.e. the fact that the central density of atomic nuclei, measured by elasticelectron-nucleus scattering, does not depend upon A for large A (see Fig. 3.1).

Figure 3.1: Saturation of central nuclear densities measured by elastic electron-nucleus scattering.

The empirical equilibrium properties of symmetric nuclear matter are(E

A

)n=n0

=−16 MeV , n0 ∼ .16 fm−3 . (3.1)

In the vicinity of the equilibrium density e = E/A can be expanded according to

e(n) ≈ e0 + 1

2

K

9

(n −n0)2

n20

, (3.2)

where

K = 9 n20

(∂2e

∂n2

)n=n0

= 9

(∂P

∂n

)n=n0

(3.3)

is the (in)compressibility module, that can be extracted from the measured excitation energies of nu-clear vibrational states. Due to the difficulties implied in the analysis of these experiments, however,empirical estimates of K have a rather large uncertainty, and range from ∼ 200 MeV (correspondingto more compressible nuclear matter, i.e. to a soft EOS) to ∼ 300 MeV (corresponding to a stiff EOS).

Unfortunately, the quadratic extrapolation of Eq. (3.2) cannot be expected to work far from equi-librium density. In fact, assuming a parabolic behavior of e(n) at large n (>> n0) leads to predict aspeed of sound in matter, cs , larger than the speed of light, i.e. (compare to Eq. (1.45))

c2s =

1

n

(∂P

∂e

)> 1 , (3.4)

3.1 Constraints on the nuclear matter EOS 31

regardless of the value of K .Equation (3.4) shows that causality requires(

∂P

∂ε

)< 1 , (3.5)

ε being the energy-density. For a relativistic Fermi gas ε∝ n4/3, and

P ≤ ε

3, cs ≤ 1

3. (3.6)

where the equal sign corresponds to massless fermions. In the presence of interactions the abovelimits can be easily exceeded. For example, modeling the repulsion between nucleons in terms of arigid core leads to predict infinite pressure at finite density.

The relation between microscopic dynamics and speed of sound in matter has been analyzed byZel’dovich in th early 60s within the framework of relativistic quantum field theory [12].

The model of Ref. [12] describes a system of charged scalar particles of mass M interacting with amassive vector field Aµ. The corresponding field equations are

2Aµ+m2v Aµ = jµ , (3.7)

where mv is the mass of the vector field. In the case of a point charge at rest at x = 0

j0(x) = qδ(3)(x) , (3.8)

q being the charge, and Eq.(3.7) has the solution

φ(x) = qe−mv |x|

|x| , A = 0 . (3.9)

Two charges at rest sitting at positions x1 and x2 repel each other, their interaction energy beinggiven by

qφ(x12) = q2 e−mv x12

x12, (3.10)

with x12 = |x1|−x2. Note that the effects of self interactions are included in the mass M .The total energy of a system of N charges can be found using the superposition principle, with

the result

E = N M + q2

2

N∑i 6= j=1

e−mv xi j

xi j. (3.11)

Denoting by n the density of charges and assuming that

n−1/3 < m−1v , (3.12)

we can resort to the mean field approximation, and write the energy of one charge in the form

e = M +q2 n

2

∫d 3x

e−mv x

x= M +2πq2 n

m2v

. (3.13)


Finally, from the above equation we can readily obtain the expresions of energy density and pressure

ε= ne = nM +2πq2 n2

m2v

, (3.14)

P = n2(∂e

∂n

)= 2πq2 n2

m2v

, (3.15)

implying that, in the limit of large n, P → ε.Assuming that the energy density be related to number density through the power law

ε= E

V= anν , (3.16)

energy per particle and pressure can be written

e = E

N= aνn−1 , (3.17)

P = n2(∂e

∂n

)= (n −1)anν = (n −1)ε . (3.18)

From the above relations we easily see that that ν= 4/3 corresponds to P = ε/3 and that the limitingcase cs = 1, i.e. (∂P )/(∂ε) = 1 is reached when ν = 2. Powers higher than ν = 2 are forbidden bycausality.

In the model proposed by Zel’dovich matter consists of baryons interacting through exchange ofa massive vector meson described by the lagrangian density

L V =−1

4FµνFµν− 1

2µ2 AµAµ , (3.19)

where Aµ ≡ (φ,A) and µ is the meson mass. The corresponding field equation is

(∂ν∂ν+µ2)Aµ = g Jµ , (3.20)

g being the coupling constant. In the simple case of a point source located at x = 0

Jµ ≡ (J0,J) ≡ (δ(x),0) , (3.21)

and the solution of (3.20) is

φ(x′) = ge−µ|x

′−x|

|x′−x| , A = 0 . (3.22)

Two charges at rest separated by a distance r repel each other with a force of magnitude

F =−g 2 d

dr

e−µr

r, (3.23)

and the corresponding interaction energy is

gφ= g 2 e−µr

r. (3.24)

3.2 The nucleon-nucleon interaction 33

In the case of N particles of mass M, as the equation of motion (3.20) is linear, we can use thesuperposition principle and write the total energy as (ri j = |ri − r j |)

E = N M + g 2N∑

j>i=1

e−µri j

ri j. (3.25)

Let us now make the further assumption that the average particle density be such that(1

n

)1/3

< 1

µ. (3.26)

The above equation implies that the meson field changes slowly over distances comparable to theaverage particle separation. If this is the case we can use the mean field approximation and rewriteEq. (3.25) in the form

e = E

N= M + g 2

2

∫d 3r

e−µr

r= m +2πg 2 n

µ. (3.27)

The corresponding expression of the energy density and pressure read

ε= ne = nM +2πg 2 n2

µ(3.28)

and

P = n2(∂e

∂n

)= 2πg 2 n2

µ. (3.29)

From the above equations it follows that, in the large n limit P → ε, inplying in turn cs → 1. In conclu-sion, Zel’dovich model shows that the causality limit, corresponding to ε∝ n2, is indeed attained in asimple semirealistic theory, in which nucleons are assumed to interact through exchange of a vectormeson.

3.2 The nucleon-nucleon interaction

The main features of the nucleon-nucleon (NN) interaction, inferred from the analysis of nuclearsystematics, may be summarized as follows.

• The saturation of nuclear density (see Fig. 3.1), i.e. the fact that density in the interior of atomicnuclei is nearly constant and independent of the mass number A, tells us that nucleons cannotbe packed together too tightly. Hence, at short distance the NN force must be repulsive. As-suming that the interaction can be described by a non relativistic potential v depending on theinterparticle distance, r, we can then write:

v(r) > 0 , |r| < rc , (3.30)

rc being the radius of the repulsive core.


• The fact that the nuclear binding energy per nucleon is roughly the same for all nuclei with A≥20, its value being

B(Z,A)

A∼ 8.5 MeV , (3.31)

suggests that the NN interaction has a finite range r0, i.e. that

v(r) = 0 , |r| > r0 . (3.32)

• The spectra of the so called mirror nuclei, i.e. pairs of nuclei having the same A and chargesdiffering by one unit (implying that the number of protons in a nucleus is the same as the num-ber of neutrons in its mirror companion), e.g. 15

7 N (A = 15, Z = 7) and 158 O (A = 15, Z = 8), exhibit

striking similarities. The energies of the levels with the same parity and angular momentumare the same up to small electromagnetic corrections, showing that protons and neutrons havesimilar nuclear interactions, i.e. that nuclear forces are charge symmetric.

Charge symmetry is a manifestation of a more general property of the NN interaction, called iso-topic invariance. Neglecting the small mass difference, proton and neutron can be viewed as twostates of the same particle, the nucleon (N), described by the Dirac equation obtained from the la-grangiam density

L =ψN

(iγµ∂µ−m

)ψN (3.33)

where

ψN =(

pn

), (3.34)

p and n being the four-spinors associated with the proton and the neutron, respectively. The la-grangian density (3.33) is invariant under the SU(2) global phase transformation

U = eiα jτ j , (3.35)

where α is a constant (i.e. independent of x) vector and the τ j ( j = 1,2,3) are Pauli matrices. Theabove equations show that the nucleon can be described as a doublet in isospin space. Proton andneutron correspond to isospin projections +1/2 and −1/2, respectively. Proton-proton and neutron-neutron pairs always have total isospin T=1 whereas a proton-neutron pair may have either T = 0 orT = 1. The two-nucleon isospin states |T,T3⟩ can be summarized as follows

|1,1⟩ = |pp⟩|1,0⟩ = 1p

2

(|pn⟩+ |np⟩)|1,−1⟩ = |nn⟩|0,0⟩ = 1p

2

(|pn⟩− |np⟩) .

Isospin invariance implies that the interaction between two nucleons separated by a distance r =|r1 − r2| and having total spin S depends on their total isospin T but not on T3. For example, thepotential v(r) acting between two protons with spins coupled to S = 0 is the same as the potentialacting between a proton and a neutron with spins and isospins coupled to S = 0 and T = 1.

3.3 The two-nucleon system 35

3.3 The two-nucleon system

The details of the NN interaction can be best studied in the two-nucleon system. There is onlyone NN bound state, the nucleus of deuterium, or deuteron (2H), consisting of a proton and a neutroncoupled to total spin and isospin S = 1 and T = 0, respectively. This is clear manifestation of the factthat nuclear forces are spin dependent.

Another important piece of information can be inferred from the observation that the deuteronexhibits a nonvanishing electric quadrupole moment, implying that its charge distribution is notspherically symmetryc. Hence, the NN interaction is noncentral.

Besides the properties of the two-nucleon bound state, the large data base of phase shifts mea-sured in NN scattering experiments (∼ 4000 data points corresponding to energies up to 350 MeV inthe lab frame) provides valuable additional information on the nature of NN forces.

The theoretical description of the NN interaction was first attempted by Yukawa in 1935. He madethe hypotesis that nucleons interact through the exchange of a particle, whose mass µ can be relatedto the interaction range r0 according to

r0 ∼ 1

µ. (3.36)

Using r0 ∼ 1 fm, the above relation yields µ∼ 200 MeV (1 fm = 197.3 MeV).

N(p2)

N(p2´) N(p1´)

N(p1)

π

Figure 3.2: Feynman diagram describing the one-pion-exchange process between two nucleons.The corresponding amplitude is given by Eq. (3.37).

Yukawa’s idea has been successfully implemented identifying the exchanged particle with the πmeson (or pion), discovered in 1947, whose mass is mπ ∼ 140 MeV. Experiments show that the pionis a spin zero pseudoscalar particle1 (i.e. it has spin-parity 0−) that comes in three charge states,denoted π+, π− and π0. Hence, it can be regarded as an isospin T=1 triplet, the charge states beingassociated with isospin projections T3=+ 1, 0 and −1, respectively.

The simplest π-nucleon coupling compatible with the observation that nuclear interactions con-serve parity has the pseudoscalar form i gγ5τ, where g is a coupling constant and τ describes the

1The pion spin has been deduced from the balance of the reaction π++2 H ↔ p +p, while its intrinsic paritywas determined observing the π− capture from the K shell of the deuterium atom, leading to the appearanceof two neutrons: π−+d → n +n.


isospin of the nucleon. With this choice for the interaction vertex, the amplitude of the process de-picted in Fig. 3.2 can readily be written, using standard Feynman’s diagram techniques, as

⟨ f |M |i ⟩ =−i g 2 u(p ′2, s′2)γ5u(p2, s2)u(p ′

1, s′1)γ5u(p1, s1)

k2 −m2π

⟨τ1 ·τ2⟩ , (3.37)

where k = p ′1−p1 = p2−p ′

2, k2 = kµkµ = k20 −|~k|2, u(p, s) is the Dirac spinor associated with a nucleon

of four momentum p ≡ (~p,E) (E=√

p2 +m2) and spin projection s and

⟨τ1 ·τ2⟩ = η†2′τη2 η

†1′τη1 , (3.38)

ηi being the two-component Pauli spinor describing the isospin state of particle i .In the nonrelativisti limit, Yukawa’s theory leads to define a NN interaction potential that can be

written in coordinate space as

vπ = g 2

4m2 (τ1 ·τ2)(σ1 ·∇)(σ2 ·∇)e−mπr

r

= g 2

(4π)2

m3π

4m2

1

3(τ1 ·τ2)

[(σ1 ·σ2)+S12

(1+ 3

x+ 3

x2

)]e−x

x

− 4π

m3π

(σ1 ·σ2)δ(3)(r)

, (3.39)

where x = mπ|r| and

S12 = 3

r 2 (σ1 · r)(σ2 · r)− (σ1 ·σ2) , (3.40)

is reminiscent of the operator describing the noncentral interaction between two magnetic dipoles.A detailed derivation of Eq.(3.39) can be found in Appendix A.

For g 2/(4π) = 14, the above potential provides an accurate description of the long range part(|r| > 1.5 fm) of the NN interaction, as shown by the very good fit of the NN scattering phase shifts instates of high angular momentum. In these states, due to the strong centrifugal barrier, the probabil-ity of finding the two nucleons at small relative distances becomes in fact negligibly small.

At medium- and short-range other more complicated processes, involving the exchange of two ormore pions (possibly interacting among themselves) or heavier particles (like the ρ and theωmesons,whose masses are mρ = 770 MeV and mω = 782 MeV, respectively), have to be taken into account.Moreover, when their relative distance becomes very small (|r| <∼ 0.5 fm) nucleons, being compositeand finite in size, are expected to overlap. In this regime, NN interactions should in principle be de-scribed in terms of interactions between nucleon constituents, i.e. quarks and gluons, as dictated byquantum chromodynamics (QCD), which is believed to be the fundamental theory of strong interac-tions.

Phenomenological potentials describing the full NN interaction are generally written as

v = vπ+ vR , (3.41)

where vπ is the one pion exchange potential, defined by Eqs. (3.39) and (3.40), stripped of the δ-function contribution, whereas vR describes the interaction at medium and short range. The spin-isospin dependence and the noncentral nature of the NN interactions can be properly described

3.4 non relativistic many-body theory 37

rewriting Eq. (3.41) in the form

vi j =∑ST

[vT S(ri j )+δS1vtT (ri j )S12

]PSΠT , (3.42)

S and T being the total spin and isospin of the interacting pair, respectively. In the above equation PS

(S = 0,1) are the spin projection operators

P0 = 1

4(1−σ1 ·σ2) , P1 = 1

4(3+σ1 ·σ2) , (3.43)

satisfyingP0 +P1 = 1 , PS |S′⟩ = δSS′ |S′⟩ , PSPS′ = PSδSS′ , (3.44)

andΠT are the isospin projection operators that can be written as in Eq. (3.43) replacing σ→τ . Thefunctions vT S(ri j ) and vT t (ri j ) describe the radial dependence of the interaction in the different spin-isospin channels and reduce to the corresponding components of the one pion exchange potentialat large ri j . Their shapes are chosen in such a way as to reproduce the available NN data (deuteronbinding energy, charge radius and quadrupole moment and the NN scattering phase shifts).

Substitution of Eq. (3.43) and the corresponding expressions for the isospin projection operatorsallows one to rewrite Eq. (3.42) in the form

vi j =6∑

n=1v (n)(ri j )O(n)

i j , (3.45)

whereO(n)

i j = 1, (τi ·τ j ), (σi ·σ j ), (σi ·σ j )(τi ·τ j ), Si j , Si j (τi ·τ j ) , (3.46)

and the v (n)(ri j ) are linear combination of the vT S(ri j ) and vtT (ri j ). Note that the operatos definedin Eq. (3.46) form an algebra, as they satisfy the relation

O(n)i j O(m)

i j =∑`

Knm`O(`)i j , (3.47)

where the coefficients Knm` can be easily obtained from the properties of Pauli matrices. Equations(3.47) can be exploited to greatly simplify the calculation of nuclear observables based on the repre-sentation (3.45)-(3.46) of the NN potential.

The typical shape of the NN potential in the state of relative angular momentum ` = 0 and totalspin and isospin S = 0 and T = 1 is shown in Fig. 3.3. The short range repulsive core, to be ascribed toheavy meson exchange or to more complicated mechanisms involving nucleon constituents, is fol-lowed by an intermediate range attractive region, largely due to two-pion exchange processes. Finally,at large interparticle distance the one-pion-exchange mechanism dominates. Note the similarity withthe van der Waals potential of Fig. 1.1.

3.4 non relativistic many-body theory

Within non relativistic many-body theory (NMBT), nuclear systems are described as a collectionof pointlike nucleons interacting through the hamiltonian

H =A∑

i=1

p2i

2m+

A∑j>i=1

vi j + . . . , (3.48)


Figure 3.3: Radial dependence of the NN potential describing the interaction between two nucleonsin the state of relative angular momentum `= 0, and total spin and isospin S = 0 and T = 1.

where pi denotes the momentum carried by the i-th nucleon, vi j is the two-body potential describingNN interactions the and the ellipsis refers to the possible existence of interactions involving morethan two nucleons.

The validity of the assumptions implied in the treatment of nucleons as pointlike particles, inspite of their finite size, can be gauged using the empirical information provided by elastic electron-proton scattering experiments. The measured cross section are in fact simply related to the protoncharge form factor, the Fourier transform of which yields the charge distribution in coordinate space.Figure 3.4 illustrates the overlap between the charge distributions of two protons separated by a dis-tance 1 < d < 2 fm. The results of theoretical calculations, yielding an average nucleon-nucleon sep-aration distance ∼ 2.5 fm in isospin symmetric nuclear matter at equilibrium desity, suggest that thefinite size of the nucleon may not play a critical role up to densities ρ <∼ 3ρ0.

Unfortunately, solving the Schrödinger equation

H |Ψ0⟩ = E0|Ψ0⟩ . (3.49)

for the ground state of a nucleus, using the hamiltonian (3.48) and the NN potential of Eqs.(3.45) and(3.46), is only possible for not too large A. The numerical solution is trivial for A=2 only. For A=3 Eq.(3.49) can still be solved using deterministic approaches, while for A>3 sthocastic methods, such asthe Green Function Monte Carlo method, have to be employed. The results of these calculations willbe briefly reviewed in the next Section.

3.4.1 The few-nucleon systems

The NN potential determined from the properties of the two-nucleon system can be used to solveEq. (3.49) for A > 2. In the case A = 3 the problem can be still solved exactly, but the resulting groundstate energy, E0, turns out to be slightly different from the experimental value. For example, for 3Heone typically finds E0 = 7.6 MeV, to be compared to Eexp = 8.48 MeV. In order to exactly reproduce


-2 -1 0 1 20.0

0.5

1.0

1.5

2.0

ρ ch

(fm-3

)

d = 1 fm

-2 -1 0 1 20.0

0.5

1.0

1.5

2.0ρ c

h (fm

-3)

1.5 fm

-2 -1 0 1 20.0

0.5

1.0

1.5

2.0

z (fm)

ρ ch

(fm-3

)

2 fm

Figure 3.4: Overlap between the charge distributions of two protons separated by a distance1 < d < 2 fm. The shaded area corresponds to the experimental error, while the dashed lines havebeen obtained using a simple dipole parametrization of the proton charge form factor.

Eexp one has to add to the nuclear hamiltonian a term containing three-nucleon interactions de-scribed by a potential Vi j k . The most important process leading to three nucleon interactions is thetwo-pion exchange associated with the excitation of a nucleon resonance in the intermediate state,depicted in Fig. 3.5.

The three-nucleon potential is usually written in the form

Vi j k =V 2πi j k +V N

i j k , (3.50)

where the first contribution takes into account the process of Fig. 3.5 while V Ni j k is purely phenomeno-

logical. The two parameters entering the definition of the three-body potential are adjusted in sucha way as to reproduce the properties of 3H and 3He. Note that the inclusion of Vi j k leads to a verysmall change of the total potential energy, the ratio ⟨Vi j k⟩/⟨vi j ⟩ being ∼ 2 %.

For A > 3 the Scrödinger equation is no longer exactly solvable. The ground state energy of nucleihaving A ≥ 4 can be estimated from Ritz principle, stating that the expectation value of the hamilto-nian in the trial state |ΨV ⟩ satisfies

EV = ⟨ΨV |H |ΨV ⟩⟨ΨV |ΨV ⟩ ≥ E0 , (3.51)


Figure 3.5: Diagrammatic representation of the process providing the main contribution to thethree-nucleon interaction. The thick solid line corresponds to an excited state of the nucleon.

E0 being the ground state energy. Obviously, the larger the overlap ⟨Ψ0|ΨV ⟩ the closer EV is to E0.In the variational approach based on Eq. (3.51) E0 is estimated carrying out a functional mini-

mization of EV . The trial ground state is written in such a way as to reflect the structure of the nuclearinteraction hamiltonian. For few nucleon systems it takes the form

|ΨV ⟩ = (1+U ) |ΨP ⟩ , (3.52)

where

|ΨP ⟩ = F |ΦA(J J3T T3)⟩ . (3.53)

In the above equations |ΦA(J J3T T3)⟩ is a shell model state, describing A independent particles cou-pled to total angular momentum J and total isospin T , with third components J3 and T3, while theoperators U and F take into acount the correlation structure induced by the two- and three-nucleonpotentials, respectively. The dominant correlation effects, associated with the NN potential vi j , aredescribed by the operator F , which is usually written

F =SA∏

j>i=1fi j , (3.54)

where S is the symmetrization operator and (compare to Eq. (3.45))

fi j =6∑

n=1f (n)(ri j )O(n)

i j . (3.55)

The shape of the radial correlation functions f (n) are determined by minimizing the expectation valueEV . In few nucleon systems this procedure is implemented choosing suitable analytical expressionsinvolving few adjustable parameters.

The main features of the f (n) are dictated by the behaviour of the corresponding component ofthe potential vi j . For example, due to the presence of the strong repulsive core f (n)(r ) ¿ 1 at r <∼ 1fm. A typical set of radial correlation functions is shown in Fig. 3.6.


0 1 2 3 4 5 6 7 8r (fm)

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

fc,utτ (2H)

fc,utτ (3H)

fc,utτ (4He)

fsp,utτ (6Li)

φp (6Li)

fc

10 utτ

φp

fsp

[fc(4He)]3

[fc(3H)]2

Figure 3.6: Correlation functions in few nucleon systems: the central and tensor-isospin compo-nents f (1) and f (4) are denoted by fc and utτ, respectively. The dashed lines show f 2

c (3H) and f 3c (4He)

to illustrate the large r behavior. The dot-dashed line marked φp shows the independent particlemodel wave function for 6Li.

The main difficulty of the variational approach is the calculation of the expectation value EV ,involving an integration over 3A space coordinate as well as a sum over the spin-isospin degrees offreedom, which makes the dimensionality of the problem very difficult to handle for A ≥ 8.

To understand this problem, let us write the variational state in the form

|ΨV ⟩ =∑nΨn(R)|n⟩ , (3.56)

where the sum includes all possible spin-isospin states, labelled by the index n, and R ≡ r1, . . . ,rAspecifies the space configuration of the system. For example, in the case of 3He (J = T = 1/2) onefinds

|1⟩ = | ↑ p ↑ n ↓ n ⟩|2⟩ = | ↓ p ↑ n ↓ n ⟩|3⟩ = | ↓ n ↑ p ↓ n ⟩. . . = . . . . . . , (3.57)

The possible spin states of A spin-1/2 particles are 2A and, since Z of the A nucleons can be protons,there are A!/Z !(A−Z )! isospin states. Hence, the sum over n in Eq. (3.56) involves

M = 2A A!

Z !(A−Z )!, (3.58)

contributions.


In the representation of Eq. (3.56) the nuclear hamiltonian H is a M ×M matrix whose elementsdepend upon R. To obtain EV one has to evaluate the M ×M integrals∫

dRΨ†n(R)HnmΨm(R) , (3.59)

whose calculation is carried out using the Monte Carlo (MC) method.The expectation value of any operator O in the stateΨV can be written in the form

⟨O⟩ = ∑m,n

∫dR

[Ψ†

m(R)Omn(R)Ψn(R)

Pmn(R)

]Pmn(R)

= ∑m,n

∫dR Omn(R)Pmn(R) , (3.60)

with

Omn = Ψ†m(R)Omn(R)Ψn(R)

Pmn(R), (3.61)

the probability distribution Pmn(R) being given by

Pmn(R) = |Re(Ψ†m(R)Ψn(R))| . (3.62)

Let Rp ≡ R1, . . . ,RNc be a set of Nc configurations drawn from the probability distribution of Eq.(3.60), i.e. such that the probability that a configuration R belongs to the set Rp is proportional toPmn(R). It then follows that∫

dR Omn(R)Pmn(R) = limNc→∞

1

Nc

Nc∑p=1

Omn(Rp ) . (3.63)

The above procedure, called Variational Monte Carlo (VMC) method, allows one to obtain esti-mates of the ground state energy E0 whose accuracy is limited by the statistical error associated withthe use of a finite configuration set and by the uncertainty in the choice of the trial wave function. Thesecond source of error can be removed using the Green Function Monte Carlo (GFMC) approach.

Let |Ψm⟩ be the complete set of eigenstates of the nuclear hamiltonian, satisfying

H |Ψm⟩ = Em |Ψm⟩ . (3.64)

The trial variational wave function can obviously be expanded according to

|ΨV ⟩ =∑nβn |Ψm⟩ , (3.65)

implying

limτ→∞e−Hτ|ΨV ⟩ = lim

τ→∞∑nβn e−Emτ|Ψm⟩ =β0 e−E0τ|Ψ0⟩ . (3.66)

Hence, evolution of the variational ground state to infinite imaginary time projects out the true groundstate of the nuclear hamiltonian and allows one to extract the corresponding eigenvalue.


The calculation is carried out dividing the imaginary time interval τ in N steps of length∆τ= τ/Nto rewrite

e−Hτ = (e−H∆τ)N

. (3.67)

The state at imaginary time (i + 1)∆τ is can be obtained from the one corresponding to τ = i∆τthrough the relation

|Ψi+1V ⟩ = e−H∆τ|Ψi

V ⟩ , (3.68)

that can be rewritten (|RST ⟩ specifies the configuration of the system in coordinate, spin and isospinspace)

⟨R ′S′T ′|Ψi+1V ⟩ =∑

ST

∫dR ⟨R ′S′T ′|e−H∆τ|RST ⟩⟨RST |Ψi

V ⟩ , (3.69)

or

Ψi+1V ,S′T ′(R) =∑

ST

∫dR GS′T ′,ST (R ′,R)Ψi

V ,ST (R) , (3.70)

The Green’s function appearing in the above equation, yielding the amplitude for the system to evolvefrom |RST ⟩ to |R ′S′T ′⟩ during the imaginary time interval ∆τ, is defined as

GS′T ′,ST (R ′,R) = ⟨R ′S′T ′|e−H∆τ|RST ⟩ . (3.71)

The GFMC approach has been succesfully employed to describe the ground state and the lowlying excited states of nuclei having A up to 8. The results of these calculations, summarized in Table3.1 and Fig. 3.7, show that the non relativistic approach, based on a dynamics modeled to reproducethe properties of two- and three-nucleon systems, has a remarkable predictive power.

A Z (Jπ;T ) VMC (ΨT ) VMC (ΨV ) GFMC Expt

2H(1+;0) –2.2248(5) –2.22463H((1/2)+;1/2) –8.15(1) –8.32(1) –8.47(1) –8.484He(0+;0) –26.97(3) –27.78(3) –28.34(4) –28.306He(0+;1) –23.64(7) –24.87(7) –28.11(9) –29.276Li(1+;0) –27.10(7) –27.83(5) –31.15(11) –31.997He((3/2)−;3/2) –18.05(11) –19.75(12) –25.79(16) –28.827Li((3/2)−;1/2) –31.92(11) –33.04(7) –37.78(14) –39.248He(0+;2) –17.98(8) –19.31(12) –27.16(16) –31.418Li(2+;1) –28.00(14) –29.76(13) –38.01(19) –41.288Be(0+;0) –45.47(16) –46.79(19) –54.44(19) –56.50

Table 3.1: Experimental and quantum Monte Carlo ground state energies of nuclei with A=2-8 inMeV. The columns marked VMC(ΨV ) and GFMC show the VMC and GFMC results, respectively.


-60

-55

-50

-45

-40

-35

-30

-25

-20E

nerg

y (M

eV)

VMCGFMC

Exp

0+

4He

0+

2+

6He

1+

3+

6Li

3/2−

7He3/2−1/2−

7/2−

7Li

0+

8He

2+1+

3+

4+

8Li0+

1+

8Be

α+d

α+t

α+α

α+2n 6He+n

6He+2n

7Li+n

Figure 3.7: VMC and GFMC energies of nuclei with A ≤ 8 compared to experiment.

3.4.2 Nuclear matter

In the case of neutron stars, correponding to A ∼ 1057, the computational techniques describedin Section 3.4.1 cannot be applied and approximations need to be made.

In the simplest scheme the complicated NN potential is replaced by a mean field. This amount tosubstituting

A∑j>i=1

vi j →A∑

i=1Ui , (3.72)

in Eq. (3.48), witht the potential U chosen in such a way that the single particle hamiltonian

h0 = p2

2m+U , (3.73)

be diagonalizable. Within this framework the nuclear ground state wave function reduces to a Slaterdeterminant, constructed using the A lowest energy eigenstates of h0:

|Ψ0⟩ = 1pA!

detφi , (3.74)

the φi′s (i = 1, 2,. . ., A) being solutions of the Schrödinger equation

h0|φi ⟩ = εi |φi ⟩ , (3.75)


and the corresponding ground state energy is given by

E0 =A∑

i=1εi . (3.76)

This procedure is the basis of the nuclear shell model, that has been succesfully applied to explainmany nuclear properties.

Matter in the neutron star interior, however, is a uniform, dense nuclear fluid, whose single par-ticle wave functions are known to be plane waves, as dictated by translational invariace. Shell effectsare not expected to play a major role in such a system. On the other hand, strong correlations betweennucleons induced by the NN potential, not taken into account within the mean field approximation,become more and more important as the density increases, and can not be disregarded.

Let us first consider symmetric nuclear matter, defined as a uniform extended system containingequal numbers of proton and neutrons which interact through strong interactions only. Neglecting,for the sake of simplicity, three-nucleon forces, the nuclear matter hamiltonian can be written as inEq.(3.48) with vi j denoting the NN potential. In absence of interactions, the wave function is a Slaterdeterminant of single particle states

ϕkστ(r) = 1pV

ek·r χσητ , (3.77)

whereχ andη are the Pauli spinors describing spin and isospin, respectively, and |k| < kF = (3π2n/2)1/3,n being the matter density.

The main problem associated with the application of many-body perturbation theory to nuclearmatter is the presence of the strongly repulsive core in the NN potential (see Section 3.2 and Fig. 3.3),that makes the matrix elements

⟨ϕk1′σ1′τ1′ϕk2′σ2′τ2′ |v12|ϕk1σ1τ1ϕk2σ2τ2⟩ , (3.78)

very large or even divergent. As we will see, this difficulty can be circumvented either through a properredefintion of the interaction potential or changing the basis of states describing the “unperturbed”system.

A. G-matrix perturbation theory

Within the first approach the hamiltonian is first split in two pieces according to

H = H0 +H1 , (3.79)

with

H0 =N∑

i=1(Ki +Ui ) , (3.80)

where K =−∇2/2m is the kinetic energy operator, and

H1 =N∑

j>i=1vi j −

N∑i=1

Ui , (3.81)


with the single particle potential generally chosen in such a way as to make the perturbative expan-sion rapidly convergent. The interaction hamiltonian H1 is then treated perturbatively, summing upinfinite set of diagrams to overcome the problems assciated with the calculation of the matrix ele-ments (3.78). This procedure leads to the integral equation defining the G-matrix

G(W ) = v − vQ

WG . (3.82)

The G-matrix, as diagrammatically illustrated in Fig. 3.8, is the operator describing NN scattering inthe nuclear medium. The quantity W appearing in Eq. (3.82) is the energy denominator associatedwith the propagator of the intermediate state, while the operator Q prevents scattering to states in theFermi sea, forbidden by Pauli exclusion principle.

G v= + + + ...

Figure 3.8: Diagrammatic representation of Eq. (3.82).

The state describing two interacting nucleons ψi j can be expressed in terms of G through theBethe-Golstone equation

ψi j =φi j − Q

WG ψi j , (3.83)

whereφi j =ϕiϕ j , withϕi =ϕkiσiτi given by Eq. (3.77), is the corresponding unperturbed state. FromEq. (3.83) it follows that the matrix elements of G between unperturbed states

⟨φi ′ j ′ |G|φi j ⟩ = ⟨φi ′ j ′ |v |ψi j ⟩ , (3.84)

are well behaved.Although the expansion in powers of G is still not convergent, the terms in the perturbative series

can be grouped in such a way as to obtain a convergent expansion in powers of the quantity

κ= n∑i j

∫d 3r |φi j (r )−ψi j (r )|2 , (3.85)

where the sum is extended to all states belonging to the Fermi sea. The definition shows that κ mea-sures the average distortion of the two-nucleon wave function produced by NN forces.


Nuclear matter calculations carried out within G-matrix perturbation theory include contribu-tions of order κ3 to the energy per nucleon. The results show that the convergence strongly dependson the choice of the single particle potential U .

B. CBF perturbation theory

In the alternative approach, called Correlated Basis Functions (CBF) perturbation theory, the non-perturbative effects arising from the short range repulsive core of the interaction potential are in-corporated in the basis functions. The unperturbed Fermi gas states |n0⟩ are replaced by the set ofcorrelated states

|n⟩ = F |n0⟩⟨n0|F †F |n0⟩1/2

, (3.86)

where F is the correlation operator, whose structure reflects the complexity of the NN potential. Inmost nuclaer matter applications F is written in the form

F =SA∏

j>i=1fi j , (3.87)

wherefi j =

∑n

f (n)(ri j )O(n)i j , (3.88)

S is the operator that symmetrizes the product on its right hand side and the operators O(n) aredefined in Eq. (3.47).

The correlated states (3.86) form a complete set but are not orthogonal to one another. However,they can be orthogonalized using standard techniques of many-body perturbation theory.

The radial shapes of the f (n)(r ) are determined minimizing the expectation value EV = ⟨0|H |0⟩.In nuclear matter, this procedure leads to a set of Euler-Lagrange equations, whose solutions satisfythe boundary conditions

limr→∞ f (n)(r ) =

1 n = 10 n > 1

. (3.89)

The short range behaviour of the two-nucleon correlation functions is such that the quantity

f †i j Hi j fi j = f †

i j

(p2

i

2m+

p2j

2m+ vi j

)fi j , (3.90)

which reduces to Hi j at large interparticle distances, is well behaved as r → 0.Once the correlated basis has been defined, the nuclear hamiltonian can be split in two pieces

according to Eq. (3.79), where H0 and H1 are now defined as the diagonal and off diagonal part of Hin the correlated basis, respectively. We can then write

⟨m|H0|n⟩ = δmn⟨m|H |n⟩ (3.91)

⟨m|H1|n⟩ = (1−δmn)⟨m|H |n⟩ . (3.92)

If the two-body correlation function has been properly chosen, i.e. if EV is close to the eigenvalue E0,the correlated states have large overlaps with the true eigenstates of the nuclear hamiltonian and the


matrix elements of H1 are small. Hence, the perturbative expansion in powers of H1 is expected to berapidly convergent.

The explicit calculation of matrix elements of H between correlated states involves prohibitivedifficulties, as it requires integrations over the coordinates of a huge number of particles. It is usuallyperformed expanding the matrix element in a series whose terms represent the contributions of sub-systems (clusters) containing an increasing number (2, 3, . . . , A) of nucleons. The terms in the seriescan be classified according to their topological structure and summed up to all orders solving a set ofcoupled integral equations, called Fermi Hyper-Netted Chain (FHNC) equations.

3.5 Relativistic mean field theory: the σ−ωmodel

The theoretical approach described in the previous section is based on the assumption that thedegrees of freedom associated with the carriers of the NN interaction can be eliminated in favor ofa static NN potential. While this procedure has proved exceedingly succesful at ρ ∼ ρ0, as matterdensity (and therefore the nucleon Fermi momentum) increases the relativistic propagation of thenucleons, as well as the retarded propagation of the virtual meson fields giving rise to nuclear forces,are expected to become more and more significant.

In principle, relativistic quantum field theory provides a well defined theoretical framework inwhich relativistic effects can be taken into account in a fully consistent fashion. Due to the com-plexity and non perturbative nature of the strong interaction, however, the ab initio approach to thenuclear many problem, based on the QCD lagrangian, involves prohibitive difficulties. In fact, eventhe structure of individual hadrons, like the proton or the π meson, is not yet understood at a fullyquantitative level in terms of the elementary QCD degrees of freedom. Let alone the structure ofhighly condensed hadronic matter at supernuclear densities.

It has to be pointed out, however, that when dealing with condensed matter it is often conve-nient to replace the lagrangian describing the interactions between elementary constituents, be itsolvable or not, with properly constructed effective interactions. For example, the properties of highlycondensed systems bound by electromagnetic interactions are most successfully explained using ef-fective interatomic potentials. In spite of the fact that the lagrangian of quantum electrodynamics isvery well known and can be treated in perturbation theory, nobody in his right mind would ever useit to carry out explicit calculations of the bulk properties of condensed matter.

The fact that most of the time nucleons in nuclear matter behave as individual particles interact-ing through boson exchange suggests that the fundamental degrees of freedom of QCD, quarks andgluons, may indeed be replaced by nucleons and mesons, to be regarded as the degrees of freedom ofan effective field theory.

In this section we will describe a simple model in which nuclear matter is viewed as a uniformsystem of nucleons, described by Dirac spinors, interacting through exchange of a scalar and a vectormeson, called σ and ω, respectively.

The basic element of the σ-ω model is the lagrangian density

L =LN +LB +Li nt , (3.93)

where LN , LB and Li nt describe free nucleons and mesons and their interactions, respectively. The

3.5 Relativistic mean field theory: the σ−ωmodel 49

dynamics of the free nucleon field is dictated by the Dirac lagrangian of Eq. (3.33)

LN (x) =ψ(x) (i∂/−m)ψ(x) , (3.94)

where the nucleon field, denoted by ψ(x), combines the two four-component Dirac spinors describ-ing proton and neutron, as in Eq. (3.34). The meson lagrangian reads

LB (x) = Lω(x)+Lσ(x)

= −1

4Fµν(x)Fµν(x)+ 1

2m2ωVµ(x)V µ(x)

+ 1

2∂µφ(x)∂µφ(x)− 1

2m2σφ(x)2 (3.95)

whereFµν(x) = ∂νVµ(x)−∂µVν(x) , (3.96)

Vµ(x) andσ(x) are the vector and scalar meson fields, respectively, and mω and mσ the correspondingmasses.

In specifying the form of the interaction lagrangian we will require that, besides being a Lorentzscalar, Li nt (x) gives rise to a Yukawa-like meson exchange potential in the static limit. Hence, wewrite

Li nt (x) = gσφ(x)ψ(x)ψ(x)− gωVµ(x)ψ(x)γµψ(x) , (3.97)

where gσ and gω are coupling constants and the choice of signs reflect the fact that the NN interactioncontains both attractive and repulsive contributions.

The equations of motion for the fields follow from the Euler-Lagrange equations associated withthe lagrangian density of Eq. (3.93). The meson fields satisfy

(2+m2σ)φ(x) = gσ ψ(x)ψ(x) (3.98)

and(2+m2

ω)Vµ(x)−∂µ(∂νVν) = gω ψ(x)γµψ(x) , (3.99)

while the evolution of the nucleon field is dictated by the equation[(/∂− gωγµV µ(x)

)− (m − gσφ(x)

)]ψ(x) = 0 . (3.100)

The above coupled equations are fully relativistic and Lorentz covariant. However, their solution in-volves prohibitive difficulties, that can not be circumvented using approximations based on pertur-bation theory. Here we will restrict ourselves to the discussion of a different scheme, known as meanfield approximation and widely used to solve Eqs. (3.98)-(3.100), that essentially amounts to treatφ(x)and Vµ(x) as classical fields.

We replace the meson field with their mean values in the ground state of uniform nuclear matter

φ(x) →⟨φ(x)⟩ , Vµ(x) →⟨Vµ(x)⟩ , (3.101)

where ⟨φ(x)⟩ and ⟨Vµ(x)⟩must be computed from the equations of motion. In uniform nuclear matterthe baryon and scalar densities, nB = ⟨ψ†ψ⟩ and ns = ⟨ψψ⟩, as well as the current jµ = ⟨ψγµψ⟩, are


constants, independent of x. In addition, rotation invariance implies ⟨ψγiψ⟩ = 0 (i = 1, 2, 3). As aconsequence, the mean values of the meson fields satisfy the relations

m2σ ⟨φ⟩ = gσ⟨ψψ⟩ (3.102)

m2ω ⟨V0⟩ = gω⟨ψ†ψ⟩ (3.103)

m2ω ⟨Vi ⟩ = gω⟨ψγiψ⟩ = 0 , i = 1, 2, 3 . (3.104)

The nucleon equation of motion, rewritten in terms of the mean values of the meson fields, reads[(/∂− gωγµ⟨V µ⟩)− (

m − gσ⟨φ⟩)]ψ(x) = 0 . (3.105)

In uniform matter, the nucleon states must be eigenstates of the four-momentum operator, that canbe written as

ψkei kx =ψkei kµxµ =ψkei (k0t−k·r) , (3.106)

the four-spinors ψk being solutions of[(/k − gωγµ⟨V µ⟩)− (

m − gσ⟨φ⟩)]ψk

= [γµ

(kµ− gω⟨V µ⟩)− (

m − gσ⟨φ⟩)]ψk = 0 . (3.107)

The above equation can be recast in a form reminiscent of the Dirac equation for a non interactingnuleon. Defining

Kµ = kµ− gω⟨V µ⟩ (3.108)

m∗ = m − gσ⟨φ⟩ , (3.109)

we obtain (/K −m∗)

ψk = 0. (3.110)

The corresponding energy eigenvalues can be easily using(/K +m∗)(

/K −m∗) = /K /K −m∗2 = KµKνγµγν−m∗2

= KµKνγµγν+γνγµ

2−m∗2

= KµK µ−m∗2 . (3.111)

Substitution in Eq.(3.110) yields (KµK µ−m∗2

)ψk = 0 , (3.112)

implying (KµK µ−m∗2

)= 0 , (3.113)

and

K 20 = (k0 − gω⟨V0⟩)2 = |k|2 +m∗2 = E 2

k . (3.114)

It follows that the energy eigenvalues associated with nucleons and antinucleons can be written

ek = gω⟨V0⟩+Ek , (3.115)


andek = gω⟨V0⟩−Ek , (3.116)

respectively. The above equations give the nucleon and antinucleon energies in terms of the meanvalues of the meson fields, which are in turn defined in terms of the ground state expectation valuesof the nucleon densities and current, according to Eqs. (3.102)-(3.104).

The ground state expectation value of an operator ψΓψ can be evaluated exploiting the fact thateach nucleon state is specified by its momentum, k, and spin-isospin projections. Denoting the av-erage ofψΓψ in a single particle state by ⟨ψΓψ⟩kα, where the index α labels the spin-isospin state, wecan write the ground state expectation value as

⟨ψΓψ⟩ =∑α

∫d 3k

(2π)3 ⟨ψΓψ⟩kα θ(eF −ek) , (3.117)

where the θ-function restricts the momentum integration to the region corresponding to energieslower than the Fermi energy eF .

To obtain the single particle average ⟨ψγµψ⟩kα, we use Eq. (3.110), implying

k0 = γ0(γ ·k+ gωγµ⟨V µ⟩+m∗)

. (3.118)

The quantity defined by the above equation can be regarded as the single nucleon hamiltonian, whoseeigenvalues are given by (compare to Eq. (3.115))

⟨k0⟩kα = ⟨ψ†k0ψ⟩kα = gω⟨V0⟩±Ek . (3.119)

The ground state expectation value of the baryon density can be readily evaluated from Eqs. (3.118)and (3.119) noting that

∂

∂⟨V0⟩⟨ψ†k0ψ⟩kα = ∂

∂⟨V0⟩(gω⟨V0⟩+Ek

)= gω

= ⟨ψ† ∂k0

∂⟨V0⟩ψ⟩kα = gω⟨ψ†ψ⟩kα , (3.120)

implying⟨ψ†ψ⟩kα = 1 . (3.121)

It follows that nB can be obtained using Eq. (3.117), leading to

nB = ⟨ψ†ψ⟩ = ν∫

d 3k

(2π)3 θ(eF −ek) , (3.122)

where ν is the degeneracy of the momentum eigenstate (ν = 2 and 4 for pure neutron matter andisospin symmetric nuclear matter, respectively).

The same procedure can be applied to calculate the ground state expectation value ⟨ψγiψ⟩ (i =1, 2, 3). Taking the derivative with respect to ki we find

∂

∂ki⟨ψ†k0ψ⟩kα = ∂

∂ki

(gω⟨V0⟩+Ek

)= ∂Ek

∂ki

= ⟨ψ† ∂k0

∂kiψ⟩kα = ⟨ψ†γ0γiψ⟩kα = ⟨ψγiψ⟩kα , (3.123)


leading to

⟨ψγiψ⟩ = ν

∫d 3k

(2π)3

(∂Ek

∂ki

)θ(eF −ek)

= ν

(2π)3

∫ ∏j 6=i

dk j

∫dEk θ(eF −ek) = 0 . (3.124)

The above result follows from the fact that, by definition, ek ≡ eF−gω⟨V0⟩ everywhere on the boundaryof the integration region. The vanishing of the baryon current had been anticipated noting that inuniform matter the mean values of the space components of the vector field vanish, i.e. that ⟨Vi ⟩ = 0.As a consequence, the energy eigenvalues depend upon the magnitude of the nucleon momentumonly, according to

ek = ek = gω⟨V0⟩+√

|k|2 + (m − gσ⟨φ⟩)2 , (3.125)

and the occupied region of momentum space is a sphere. Eq. (3.122) then shows that in symmetricnuclear matter, with Z=(A−Z)=A/2, the baryon density takes the familiar form nB = 2k3

F /(3π2), kF

being the Fermi momentum.Finally, the scalar density ns = ⟨ψψ⟩ can be evaluated from the derivative of ⟨ψ†k0ψ⟩kα with re-

spect to m:∂

∂m⟨ψ†k0ψ⟩kα = ∂Ek

∂m= ⟨ψ† ∂k0

∂mψ⟩kα = ⟨ψ†γ0ψ⟩kα = ⟨ψψ⟩kα , (3.126)

yielding

⟨ψψ⟩kα = (m − gσ⟨φ⟩)√|k|2 + (m − gσ⟨φ⟩)2

, (3.127)

and

⟨ψψ⟩ = ν

2π2

∫ kF

0k2dk

(m − gσ⟨φ⟩)√|k|2 + (m − gσ⟨φ⟩)2

. (3.128)

Collecting together the results of Eqs. (3.122), (3.124) and (3.128) we can rewrite the equations ofmotion (3.98)-(3.100) in the form

gσ⟨φ⟩ =(

gσmσ

)2 ν

2π2

∫ kF

0|k|2d |k| (m − gσ⟨φ⟩)√

|k|2 + (m − gσ⟨φ⟩)2, (3.129)

gω⟨V0⟩ =(

gωmω

)2

nB =(

gωmω

)2

νk3

F

6π2 , (3.130)

m2ω⟨Vi ⟩ = 0, i = 1,2,3 . (3.131)

Note that, while Eqs. (3.130) and (3.131) are trivial, Eq. (3.129) implies a self-consistency requirementon the mean value of the scalar field, whose value has to satisfy a transcendental equation.

To obtain the equation of state, i.e. the relation between pressure and density (or energy density)of matter, in quantum field theory we start from the energy-momentum tensor, that for a genericLagrangian L =L (φ,∂µφ) can be written

T µν = ∂L

∂(∂µφ)∂νφ− gµνL , (3.132)


gµν being the metric tensor.In a uniform system the expectation value of T µν, is directly related to the energy density, ε, and

pressure, P , through⟨Tµν⟩ = uµuν (ε+P )− gµνP , (3.133)

where u denotes the four velocity of the system, satisfying uµuµ = 1. It follows that in the referenceframe in which matter is at rest ⟨Tµν⟩ is diagonal and

ε= ⟨T00⟩ = ⟨ψγ0k0ψ⟩−⟨L ⟩ , (3.134)

P = 1

3⟨Ti i ⟩ = 1

3⟨ψγi kiψ⟩+⟨L ⟩ . (3.135)

Within the mean field approximation, the lagrangian density of the σ-ω model reduces to

LMF =ψ[i∂/− gωγ

0⟨V0⟩− (m − gσ⟨φ⟩)]ψ− 1

2m2σ⟨φ⟩2 + 1

2m2ω⟨V0⟩2 , (3.136)

implying

T µν

MF = iψγµ∂νψ− gµν[−1

2m2σ⟨φ⟩2 − 1

2m2ω⟨V0⟩2

]. (3.137)

As a consequence, Eqs. (3.134) and (3.135) become

ε=−⟨LMF ⟩+⟨ψγ0k0ψ⟩ (3.138)

P = ⟨LMF ⟩+ 1

3⟨ψγi kiψ⟩, (3.139)

where (use Eqs. (3.119), (3.125) and (3.130))

⟨ψγ0k0ψ⟩ = ν

2π2

∫ kF

0|k|2d |k|

[√|k|2 + (m − gσ⟨φ⟩)2 + gω⟨V0⟩

]= gω⟨V0⟩nB + ν

2π2

∫ kF

0|k|2d |k|

√|k|2 + (m − gσ⟨φ⟩)2

= g 2ω

m2ω

n2B + ν

2π2

∫ kF

0|k|2d |k|

√|k|2 + (m − gσ⟨φ⟩)2 , (3.140)

and (use eq.(3.124))

⟨ψγi kiψ⟩ = ⟨ψ(γ ·k)ψ⟩ = ν

2π2

∫ kF

0d |k| |k|4√

|k|2 + (m − gσ⟨φ⟩)2. (3.141)

Substitution of the above equations into Eqs. (3.138)-(3.139) finally yields (use Eq. (3.136) and theequation of motion for the nucleon field)

ε= 1

2

m2σ

g 2σ

(m −m∗)2 + 1

2

g 2ω

m2ω

n2B + ν

2π2

∫ kF

0|k|2d |k|

√|k|2 +m∗2 (3.142)

P =−1

2

m2σ

g 2σ

(m −m∗)2 + 1

2

g 2ω

m2ω

n2B + 1

3

ν

2π2

∫ kF

0d |k| |k|4√

|k|2 +m∗2(3.143)


The first two contributions to the right hand side of the above equations arise from the massterms associated with the vector and scalar fields, while the remaining term gives the energy densityand pressure of a relativistic Fermi gas of nucleons of mass m∗ given by (see Eq. (3.129))

m∗ = m − g 2σ

m2σ

ν

2π2

∫ kF

0|k|2d |k| m∗√

|k|2 +m∗2

= m − g 2σ

m2σ

m∗

π2

[kF e∗F −m∗2 ln

(kF +e∗F

m∗

)], (3.144)

with e∗F =√

k2F +m∗2. Equations (3.142)-(3.144) yield energy density and pressure of nuclear mat-

ter as a function of the baryon number density nB (recall: kF = (6π2nB /ν)(1/3)). The values of theunknown coefficients (m2

σ/g 2σ) and (m2

ω/g 2ω) can be determined by a fit to the empirical saturation

properties of nuclear matter, i.e. requiring

B

A= ε(n0)

n0−m =−16 MeV (3.145)

with n0 = .16 fm−3. This procedure leads to the result

g 2σ

m2σ

m2 = 267.1 ,g 2ω

m2ω

m2 = 195.9 . (3.146)

Figure 3.9: Fermi momentum dependence of the binding energy per nucleon of symmetric nuclearmatter (solid line) and pure neutron matter (dashed line) evaluated using the σ−ω model and themean field approximation.


Fig. 3.9 shows the binding energies of symmetric nuclear matter (solid line) and pure neutronmatter (dashed line) predicted by the σ−ω model, plotted against the Fermi momentum kF . Notethat pure neutron matter is always unbound.

Chapter 4

Exotic forms of matter

As the density increases up to values well beyond nuclear saturation density, different forms ofmatter, containing hadrons other than protons and neutrons, can become energetically favoured.Strange hadrons are produced through weak interaction processes, such as

p +e →Λ0 +νe , (4.1)

p +e →Σ0 +νe , (4.2)

n +e →Σ−+νe . (4.3)

For example, the reaction (4.1), leading to the appearance of a Σ−, sets in as soon as the sum of theproton and electron chemical potentials exceeds the rest mass of the produced baryon, MΣ−1 . Theresults of theoretical calculations suggest that this condition is typically fulfilled at densities n >∼ 2n0.

At even larger density (n > 4n0) a new transition, to a phase in which quarks are no longer clus-tered into hadrons is eventually expected to take place.

4.1 Stability of strange hadronic matter

The properties of baryons with non zero strangeness, also referred to as hyperons, are sum-marised in Table 4.1.

The mechanism responsible for the stability of strange hadronic matter matter—driven by theFermi-Dirac statistics obeyed by the constituent particles —is analog to the one leading to neutron-ization, discussed in Chapter 2.

Let us consider a system consisting of B baryonic species b1 . . .bB and L leptonic species `1 . . .`L ,in equilibrium with respect to the weak interaction processes

bi → b j +`k +ν` , (4.4)

b j +`k → bi +ν` , (4.5)

with i , j = 1. . .B and k = 1. . .L. The ground state of system, specified by the densities of the con-stituent particles, nbi and n`i , is determined through minimisation of the energy density, with the

1Recall that we always assume that neutrinos have vanishing chemical potentials.

58 Exotic forms of matter

Charge Mass [MeV] Valence quark structure Strangeness

Λ0 0 1115.7 ud s -1Σ− -1 1197.4 dd s -1Σ0 0 1192.6 ud s -1Σ+ +1 1189.4 uus -1Ξ0 0 1314.8 d ss -2Ξ− -1 1321. 3 d ss -2

Table 4.1: Properties of strange Baryions.

constraints dictated from conservation of the baryon density, nB , and charge neutrality, implying

B∑i=1

nbi = nB , (4.6)

B∑i=1

Qbi nbi +L∑

i=1Qì nì = 0 , (4.7)

where Qbi and Qì denote the electric charge of the i -th baryonic and leptonic species, respectively.Minimization of energy density ε with respect to the densities nbi and nì with the above con-

straints results in L+B equations involving the chemical potentials of the constituents

µbi =∂ε

∂nbi

, µì =∂ε

∂nì

,

and two Lagrange multipliers, denoted λB and λQ . There are as many independent chemical poten-tials as there are conserved quantities. All other chemical potentials can be expressed in terms of theindependent ones.

To see how the determination of matter composition works, consider, as an example, a system ofprotons (p), neutrons (n), electrons (e), muons (µ), and hyperons Σ− and Λ0. In addition to Eqs.(4.6)and (4.7), which in this case take the form

nn +np +nΛ+nΣ = nB ,

andnp −ne −nmu −nΣ = 0 ,

we obtain B +L−2 = 4 equations involving the chemical potentials

µp =µn −µe ,

µΣ− =µn +µe ,

µΛ0 =µn ,

µµ =µe ,

4.1 Stability of strange hadronic matter 59

showing that there are in fact only two independent chemical potentials, e.g. µn and µp .In principle, for any given baryon density, nB , the densities of the constituent particles can be

determined from the above equations. Treating all constituents as non interacting particles, one findsthat Σ− and Λ0 appear at densities ∼ 4n0 and ∼ 8n0, respectively, n0 being the equilibrium density ofisospin-symmetric nuclear matter.

Note that the density at which the production of a strange hadron is expected to occur do notdepend on its mass only. Let us consider the Λ0 and Σ− hyperons, whose appearance becomes ener-getically favoured as soon as the threshold conditions

µn +µe = MΣ− ,

µn = MΛ0 ,

are fulfilled. From the above relations it follows that, in spite of the larger mass, the threshold densityof Σ− production is in fact lower than that ofΛ0 production if the electron chemical potential is suchthat

µe > MΣ− −MΛ0 ≈ 80 MeV .

Treating the electron as a non interacting particle, the corresponding electron density can be easilyobtained, and the threshold condition takes the form

µe =√

pF2e +m2

e ≈ pF e = (3π2ne )1/3 > 80 MeV ,

implyingne >∼ 2×10−3 fm−3 .

Under the reasonable assumption that the electron density be of the order of one percent of thebaryon density, the above estimate corresponds to nB > 0.2 fm−3 >∼ n0, a density that is certainlyreached in compact stars.

4.1.1 Hyperon interactions

While lepton interactions can be safely neglected, the interactions of baryons, including hyper-ons must be properly taken into account in the calculation of the corresponding chemical poten-tials. The maximum mass of a star consisting of non interacting neutrons, protons and leptons, ob-tained by solving the Tolman-Oppenheimer-Volkoff equations discussed in Section 1.5, turns out tobe Mmax ∼ 0.8 M¯—to be compared with the canonical value of the neutron star mass, M ∼ 1.4 M¯,resulting from observations—and the appearance of strange baryons, leading to a softening of theEoS, makes things worse, bringing the value of Mmax down to ∼ 0.7 M¯.

In principle, the EoS of strange hadronic matter can be studied using either non relativistic many-body theory or relativistic mean field theory, as discussed in Sections 3.4 and 3.5, respectively. How-ever, the former approach requires the determination of the potentials describing hyperon-nuleon(YN) and hyperon-hyperon (YY) interactions, while the latter involves the YN and YY coupling con-stants.

Experimental data providing information on YN and YY information are scarce and not very ac-curate. Studies of hypernuclei and Λp scattering suggest that the Λp interaction is weaker than thepn and pp interactions.


Some insight on the relation between strong interactions in the nucleon and hyperon sectors canbe obtained from an admittedly crude argument based on the quark model of hadrons. According tothis model, the π-meson is a quark-antiquark system, and the composition of the three charge statesis given by

π+ = ud , π− = du , π0 = 1p2

(uu −dd) ,

where u and d denote the up and down quark, respectively. As discussed in Chapter 3, the intermediate-range NN force can be described in terms of exchange of a quark-antiquark system, modelled by ascalar meson whose structure can be written in the form

σ= 1p2

(uu +dd) ,

which is essentially the number operator of non-strange quarks. Since nucleons contain three non-strange quarks, while the Λ and Σ− only contain two, it follows that the coupling constants of the Λpor Σp interaction mediated by the σ-meson is 2/3 of the corresponding pp coupling constant. In thelanguage of nuclear many body theory, this argument leads to predict the following simple relationsbetween the interaction potentials

vΛN = vΣN = 2

3vN N ,

vΛΛ = vΛΣ = vΣΣ = 4

9vN N ,

where N denotes either a proton on or a neutron.More advanced model of the YN potentials, referred to as Jülich and Nijmegen models, have been

developed exploiting all the available information on YN scattering and hypernuclear properties. Un-fortunately, the predictions of the two models are appreciably different. Using the Jülich potentialleads to predict Σ− and Λ appearance at density ∼ 2n0 and ∼ 3n0, respectively, whereas the corre-sponding quantities obtained from the Nijmegen potential are ∼ 1.5n0 and ∼ 4n0. Figure 4.1 illus-trates the typical composition of matter resulting from a calculation carried out using the formalismof nuclear many-body theory.

Recently, it has been suggested that the inclusion of a purely phenomenological three body force,involving two nucleons and aΛ hyperon, is needed to achieve an accurate description of theΛ bind-ing energy in a variety of nucleons. The results of a quantum Monte Carlo calculation performedtaking into accountΛN N interactions are displayed in Fig. 4.2

4.2 Deconfinement and quark matter

The ab initio description of the QCD phase diagram would require numerical simulations on alattice [?, ?]. However, application of this approach in the region corresponding to non vanishingchemical potential presents severe difficulties. Therefore, most studies of the properties of quarkmatter are carried out within simplified models, inspired to QCD and involving a set of of parameters,the values of which are determined in such a way as to reproduce the observed properties of hadrons.

4.2 Deconfinement and quark matter 61

Figure 4.1: Composition of strange hadronic matter obtained from non relativistic nuclear many-body theory. The two sets of lines have been obtained with and without inclusion of YY interactions.The possible occurrence of three body interactions involving hyperons is neglected.

Hypernuclei

B R [M

eV]

A-2/3

208

89

40

28

1612

9

76

54

3

s

p

df

g

emulsion

(/+,K+)

(e,e’K+)

(K-,/-)

AFDMC

0.0

5.0

10.0

15.0

20.0

25.0

30.0

0.0 0.1 0.2 0.3 0.4 0.5

Calculations can ber performed also for excited states: more information available on the structure of the interaction (e.g. spin orbit? charge symmetry breaking?)

• AFDMC calculations by D. Lonardoni & al., unpublished. • Experimental data re-analyzed by J. Millener.

Figure 4.2: Λbinding energy as a function of A−2/3. The results of quantum Monte Carlo calculations,labelled AFDMC, are compared to data collected using different experimental techniques.


4.2.1 The bag model

In this section, we will analyse a model developed in the 1970s at the Massachusetts Instituteof Technology (MIT)—referred to as bag model—in which the main features of QCD are taken intoaccount in a rather crude fashion. It is remarkable that, in spite of its simplicity, the MIT bag modelprovides a reasonable description of the hadronic spectrum.

Within the MIT bag model, the fundamental properties of QCD, that is, asymptotic freedom andconfinement, translate into two simple hypoteses:

• the QCD vacuum exerts a pressure on any quark aggregate of vanishing color charge, thus con-fining it to a finite region in space: the bag;

• within the bag, gluon-excgange interactions between quarks are negligible, or tractable at low-est order of perturbation theory.

In the simplest implementation of the above scenario, in which the quarks within the bag aredescribed of non-interacting particles obeying Fermi statistics, the thermodynamic potential can becomputed explicitly.

Consider a system consisting of N f types of fermions with masses mi , described by the DiracLagrangian density

L =N f∑i=1

q i (i∂/−mi )q i . (4.8)

The formalism suitable to describe fermion fields, satisfying anticommutation rules, is based onthe use of Grassman variables ηi (i = 1, . . . , N ), whose gaussian integral is given by [?, ?]∫

dη†1dη1 ·dη†

N dηN eη†i Ai jη j = det(A) . (4.9)

Exploiting the techniques descrided in Appendix C we can therefore write the partition functionof the system in the form [?]

Z =∏i

∫iDq i †

∫Dq i exp

[∫ β

0dτ

∫d 3x q i

(−γ0∂τ+ i~γ ·~∇−mi +µiγ

0)q i

], (4.10)

where the integration is extended to all paths satisfying the antiperiodicity conditions q i (~x,0) =−q i (~x,β).We can now transform to momentum space using the expansion

q iα(~x,τ) = 1p

V

∑n,~p

e i (~p·~x+Ei nτ)q iαn(~p) , (4.11)

where α is the index associated with the components of Dirac’s spinors, and the antiperiodicity con-straint requires

Ei n = (2n +1)πT . (4.12)

Using Eq.(4.9) we obtain the partition function

Z = ∏i ,α,~p,n

∫iD(q i

αn)†∫

D q iαn exp i

∑i n~p

(q iαn)†Dαρ q i

ρn = det(D) (4.13)


withD =−iβ

[(−i Ei n +µi )−γ0~γ ·~p −miγ

0] . (4.14)

In addition, exploiting the operatorial relation

ln[det(D)] = Tr [ln(D)] (4.15)

we can now derive from Eq.(4.14) the thermodynamic potential, defined as

Ω=−T ln Z =−PV . (4.16)

The results is

Ω = −T ln[det(D)] =−2T∑

i ,n,~p

lnβ2[(Ei n + iµi )2 +E 2i p ]

= −T∑

i ,n,~p

lnβ2[E 2i n + (Ei p −µi )2]+ lnβ2[E 2

i n + (Ei p +µi )2] , (4.17)

where Ei p =√

|~p|2 +m2i . Using Eq.(4.12). we can rewrite the logarithms appearing on the right hand

side of Eq.(4.17) in the form

ln[(2n +1)2π2 +β2(Ei p ±µi )2]

=∫ β2(Ei p±µi )2

1

dθ2

θ2 + (2n +1)2π2 + ln[1+ (2n +1)2π2] , (4.18)

which allows us to perform the sum over n, because

∞∑n=−∞

1

θ2 + (2n +1)2π2 = 1

θ

(1

2− 1

eθ+1

). (4.19)

By integrating over the variable θ we obtain an expression for the logarithm of the partition functionwhich involves terms independent of β and the chemical potentials µi . These contributions do notaffect the thermodynamics, because, after being exponentiated, can be included in the normalizationof the partition function.Neglecting these terms we find

Ω=−6T VN f∑i=1

∫d 3p

(2π)3 [βEi p + ln(1+e−β(Ei p−µi ))+ ln(1+e−β(Ei p+µi ))] , (4.20)

where the factor 6 = 2× 3 accounts for the degeneracy associated with colour and spin degrees offreedom, the second and third terms on the right hand side represent the contributions arising fromparticles and antiparticles, respectively, and the first term contributes to the vacuum energy.

From Eq.(C.6), establishing the relation between pressure, thermodynamic potential and parti-tion function, we finally obtain

P0 = 6 T∑

i

∫d 3p

(2π)3

ln[1+expβ(Ei p −µi )]+ ln[1+expβ(Ei p +µi )]

. (4.21)

Within the bag model, we have to add to the pressure P0, corresponding to a gas of non inter-acting quarks and antiquarks, three additional contributions: δP glue, describing the pressure of a


gluon gas (which can be computed using a technique similar to that employed to obtain Eq. (4.21)—),δP pert taking into account perturbative corrections arising from strong interaction processes associ-ated with gluon exchange, and the constant term −B , which parametrises the pressure exerted by theQCD vacuum on the perturbative vacuum within the bag. Therefore, we can write

P = P0 +δP glue +δP pert −B , (4.22)

with [?]

δP gluoni =−16∫

d 3p

(2π)3 ln[1−e−βp ] . (4.23)

Consider now the simplest implementation of the model, in which the number of flavours islimited to two—u and d , with mu = md = 0 and µu = µd = µ—and perturbative gluon exchange isneglected. Under these simplifying assumptions, the calculation of the pressure can be carried outanalytically, with the result [?]

P =−B +37π2

90T 4 +µ2T 2 + µ4

2π2 . (4.24)

The factor 37 in front of the term proportional to T 4 results from the sum 16+21, where 16 = 8×2 isthe number of gluonic degrees of freedom and 21 is obtained by multiplying the number of degreesof freedom associated with quarks and antiquarks (24 = 2×3×2×2) by the factor 7/8 typical of Fermistatistics.

Equation (4.24) can be exploited to obtain a qualitative description of the QCD phase diagram.Assuming that a pion gas with vanishing potential provides a schematic representation of the hadronicphase, we can write the corresponding pressure in the form [?]

Phad = 3π2

90T 4 , (4.25)

where 3 is the number of pionic degrees of freedom, that is, the number of isospin states. Gibb’sequilibrium condition is represented by the curve of Fig. 4.3, computed using the value B = 57.5MeV/fm3, resulting from a fit to the masses and magnetic moments of light hadrons [?].

4.2.2 The equation of state of quark matter

The thermodynamic functions describing a many-particle system at temperature T = 1/β can beobtained from the grand partition function, defined as

Z = Tr

[exp

(−β(H −∑

iµi Ni )

)], (4.26)

where H is the hamiltonian operator and µi and Ni are the chemical potential and the number oper-ator associated with particles of species i , respectively.

For example, pressure (P ) and energy density (ε) can be defined in terms of the Gibbs free energy,Ω, which is in turn related to Z through

Ω=− 1

βln Z , (4.27)


Figure 4.3: Diagramma di fase del bag model con N f = 2, ottenuto ponendo mu = md = 0, µu =µd =µ e B = 57.5 MeV/fm3 (corrispondente a B 1/4 = 145 MeV). La fase adronica è descritta come un gas di

pioni a potenziale chimico nullo.

according to

P =−ΩV

, (4.28)

and

ε=Ω−∑i

(∂Ω

∂µi

)T,V

, (4.29)

where V is the volume occupied by the system.In the case of quark matter, due to the properties of QCD described in Section XXXX, Ω consists

of two contributions. One of them, that will be denoted Ωper t , is tractable in perturbation theory,while the second one takes into acount nonperturbative effects induced by the properties of the QCDvacuum.

The relation (4.28) between pressure and Gibbs free energy and the interpretation of the bag con-stant discussed in Section 4.2.1 suggest that within the MIT bag model the difference betweenΩ andΩper t can be identified with the bag constant, i.e. that

Ω=Ωper t +V B . (4.30)

The perturbative contribution can be expanded according to

Ωper t =V∑

f

∑nΩ(n)

f , (4.31)


where the index f specifies the quark flavor, while Ω(n)f is the n-th order term of the perturbative

expansion in powers of the strong coupling constant, αs .The EOS of quark matter can be obtained from the relations linking pressure and energy density

toΩ:

P =−ΩV

=−B −∑f

∑nΩ(n)

f . (4.32)

ε= ΩV

−∑f

(∂Ω

∂µ f

)=−P +∑

fµ f n f , (4.33)

n f and µ f being the density and chemical potential of the quarks of flavor f .The lowest order contribution at T = 0, reads (compare to Eqs. (1.29) and (1.30))

Ω(0)f = − 1

π2

[1

4µ f

√µ2

f −m2f

(µ2

f −5

2m2

f

)

+3

8m4

f logµ f +

√µ2

f −m2f

m f

], (4.34)

where m f is the quark mass.Substituting Eq. (4.34) into Eqs. (4.32) and (4.33) and taking the limit of massless quarks one finds

the EOS

P = ε−4B

3. (4.35)

The contribution of first order in αs , arising from the one-gluon exchange processes discussed inAppendix ??, reads

Ω(1)f = 2αs

π3

[3

4

(µ f

√µ2

f −m2f −m2

f logµ f +

√µ2

f −m2f

m f

)2

−1

2(µ2

f −m2f )2

]. (4.36)

The chemical potentials appearing in Eqs.(4.34) and (4.36) can be written

µ f = eF f +δµ f =√

m2f +p2

F f+δµ f , (4.37)

where the first term is the Fermi energy of a gas of noninteracting quarks of mass m f at density n f =p3

F f/π2, whereas the second term is a perturbative correction of orderαs , whose explicit expression is

δµ f =2αs

3π2

[pF f −3

m2F f

eF f

log

(eF f +pF f

m f

)]. (4.38)

Including bothΩ(0)f andΩ(1)

f in Eqs. (4.32) and (4.33) and taking again the limit of vanishing quarkmasses one finds

P = 1

4π2

(1− 2αs

π

)∑fµ4

f −B (4.39)


ε= 3

4π2

(1− 2αs

π

)∑fµ4

f +B . (4.40)

Comparison to Eq. (4.35) shows that the EOS of a system of massless quarks in unaffected by one-gluon exchange interactions.

For any baryon density, quark densities are dictated by the requirements of baryon number con-servation, charge neutrality and weak equilibrium. In the case of two flavors, in which only the lightup and down quarks are present, we have

nB = 1

3(nu +nd ) , (4.41)

2

3nu − 1

3nd −ne = 0 (4.42)

µd =µu +µe , (4.43)

where ne and µe denote the density and chemical potential of the electrons produced through

d → u +e−+νe . (4.44)

Note that we have not taken into account the possible appearance of muons, as in the density regionrelevant to neutron stars µe never exceeds the muon mass.

As the baryon density increases, the d-quark chemical potential reaches the value µd = ms , ms

being the mass of the strange quark. The energy of quark matter can then be lowered turning d-quarksinto s-quarks through

d +u → u + s . (4.45)

In presence of three flavors, Eqs. (4.41)-(4.43) become

nB = 1

3(nu +nd +ns) , (4.46)

2

3nu − 1

3nd − 1

3ns −ne = 0 (4.47)

µd =µs =µu +µe . (4.48)

Unfortunately, the parameters entering the bag model EOS are only loosely constrained by phe-nomenology and their choice is somewhat arbitrary.

As quarks are confined and not observable as individual particles, their masses are not directlymeasurable and must be inferred from hadron properties. The Particle Data Group reports masses ofa few MeV for up and down quarks and 60 to 170 MeV for the strange quark. At typical neutron starsdensities heavier quarks do not play a role.

The strong coupling constant αs can be obtained from the renormalization group equation

αs = 12π

(33−2N f ) ln(µ2/Λ2), (4.49)

where N f = 3 is the number of active flavors, Λ is the QCD scale parameter and µ is an energy scaletypical of the relevant density region (e.g. the average quark chemical potential). UsingΛ∼ 100÷200MeV and setting µ=µd ∼µu at a typical baryon density nB ∼ 4n0 one gets αs ∼ 0.4÷0.6.


The values of the bag constant resulting from fits of the hadron spectrum range between ∼ 57MeV/fm3, with Λ= 220 MeV, and ∼ 350 MeV/fm3 , with Λ= 172 MeV. However, the requirement thatthe deconfinement transition does not occur at density ∼ n0 constrains B to be larger than ∼ 120−150MeV/fm3, and lattice results suggest a value of ∼ 210 MeV/fm3.

Figure 4.4 shows the energy density of neutral quark matter in weak equilibrium as a functionof baryon density, for different values of B and αs . Comparison between the dotdash line and thosecorresponding to αs 6= 0 shows that perturbative gluon exchange, whose inclusion produces a sizablechange of slope, cannot be simulated by adjusting the value of the bag constant and must be explicitlytaken into account.

Figure 4.4: Energy density of neutral quark matter in weak equilibrium as a function of baryon num-ber density. The solid and dashed lines have been obtained setting αs = 0.5 and B = 200 and 120MeV/fm3, respectively, while the dashdot line corresponds toαs = 0 and B = 200 MeV/fm3. The quarkmasses are mu = md = 0, ms = 150 MeV.

The composition of charge neutral quark matter in weak equilibrium obtained from the MIT bagmodel is shown in Fig. 4.5. Note that at large densities quarks of the three different flavors are presentin equal number, and leptons are no longer needed to guarantee charge neutrality.


Figure 4.5: Composition of charge neutral matter of u, d and s quarks and electrons in weak equilib-rium, obtained from the MIT bag model setting mu = md = 0, ms = 150 MeV, B = 200 MeV/fm3 andαs = 0.5.

Chapter 5

Cooling mechanisms

The physical processes leading to neutron star cooling are driven by the emission of neutrinos(and antineutrinos) produced through a variety of mechanisms, depending on the different forms ofmatter present in the crust and the core of the star.

In this Chapter, we will briefly describe the emission processes taking place in the core, wherematter is believed to consist mainly of neutrons, with a small fraction of protons and leptons, theabundance of which is determined by the requirements of weak equilibrium and charge neutral-ity. These processes turn out to be dominant, and can be described within a realistic and consistentmodel of the microscopic dynamics.

5.1 Direct Urca process

The most efficient neutrino and antineutrino production mechanisms, dubbed direct Urca pro-cesses, are the simplest weak processes taking place in matter consisting of neutrons, protons andleptons: neutron β-decay and electron capture by protons

n → p +e + νe , p +e → n +νe . (5.1)

At β-equilibrium, the above reactions take place at the same rate, and the chemical potentials ofthe participating particles satisfy the condition1

µn =µp +µe . (5.2)

Because at equilibrium the rate of production of neutrinos and antineutrinos is the same, the totalneutrino emissivity, QD , is simply twice the emissivity associated with neutron β-decay, discussed inAppendix B

QD = 2∫

dpn

(2π)3 dWi→ f Eν fn (1− fp ) (1− fe ) . (5.3)

1We will always assume that neutrinos and antineutrinos be non degenerate, i.e. that they have vanishingchemical potential.

72 Cooling mechanisms

In the above equation dWi→ f is the differential probability of β-decay, f j is the equilibrium Fermi-Dirac distribution of the particles of species j and Eν denotes the neutrino energy. Note that thecalculation of QD requires the evaluation of a twelve-dimensional integration, which can be reducedto a eight-dimensional one exploiting energy and momentum conservation.

The calculation of Eq.(5.3) can be greatly simplified, and treated analytically, using the approxi-mation known as phase space decomposition, which is expected to be applicable to strongly degen-erate fermions. This scheme is based on the tenet that, as at the temperatures typical of neutron starsboth nucleons and leptons are strongly degenerate, the main contribution to the emissivity is pro-vided by particles occupying quantum states in the vicinity of the Fermi surface. As a consequence,one can set |p j | ≈ pF j , implying in turn that the energy scale relevant to the process is driven by thetemperature, i.e. that Eν ∼ T . As the neutrino momentum is also of order T , it can safely be ne-glected, with respect to the momenta of the degenerate fermions, in the argument of the momentumconserving δ-function.

The differential transition probability reads

dWi→ f = 2πδ(En −Ep −Ee −Eν)δ(pn −pp −pe ) |M f i |2 4πE 2νdEν

dpp

(2π)3

dpe

(2π)3 , (5.4)

with |M f i |2 = 2G2(1+3g 2A), where G = GF cosθc , GF and θc being the Fermi constant and Cabibbo’s

angle, respectively, while g A is the axial vector coupling constant.Equation (5.4) can be cast in the form

dWi→ fdpn

(2π)3 = 1

(2π)8 δ(En −Ep −Ee −Eν)δ(pn −pp −pe ) (5.5)

×|M f i |2 4πE 2νdEν

3∏j=1

pF j m∗j dE j dΩ j , (5.6)

wheredΩ j is the differential solid angle specifying the direction of the momentum p j , m∗j = pF j /vF j

is the effective mass of the particles of species j and vF j = (∂E j /∂p j )p=pF j denotes the correspondingFermi velocity.

Substutution of Eq.(5.5) in the definition of the neutrino emissivity, Eq.(5.3), leads to the result

QD = 2

(2π)8 T 8 A I |M f i |23∏

j=1pF j m∗

j , (5.7)

with

A = 4π∫

dΩ1 dΩ2 dΩ3δ(pn −pp −pe) , (5.8)

and

I =∫ ∞

0d xν x3

ν

[3∏

j=1

∫ ∞

−∞d x j f j

]δ(x1 +x2 +x3 −xν) . (5.9)

5.1 Direct Urca process 73

The definition of A, Eq.(5.8), involves the integration on angular variables, while the calculationof I , defined by Eq.(5.9), requires integration over the adimensional variables xν = Eν/T and x j =(E j −µ j )/T . The details of the calculations of both A and I are described in Appendix ??. The finalresults are

A = 32π3

pF n pF p pFeΘnpe , I = 457π6

5040, (5.10)

where the function

Θnpe =

1 se pF n ≤ pF p +pFe

0 otherwise, (5.11)

enforces the threshold condition that must be satisfied to fulfill he requirement of momentum con-servation.

The resulting expression of the emissivity is

QD = 457

10080G2

F cos2θc (1+3g 2A)

m∗n m∗

p m∗e

ħ10 c3 (KB T )6 Θnpe

' 4.00×1027(

ne

n0

)1/3 m∗n m∗

p

m2n

T 69 Θnpe ergcm−3 s−1 , (5.12)

where, as usual, n0 = 0.16 f m−3 is the equilibrium density of isospin symmetric nuclear matter, ob-tained from extrapolation of the central density of atomic nuclei and the semi empiric mass formula,while T9 denotes the temperature measured in units of 109K. Note that the emissivity depends ontemperature according to the power law QD ∝ T 6

9 . The power six can be easily explained on the basisof simple phase space considerations.

Integration over the momenta of the three strongly degenerate fermions participating in the pro-cess contributes a term ∝ T 3, as the integration region is restricted to a thin shell of size ∼ T aroundthe Fermi surface. On the other hand, the integration over the momentum of the non degenerate neu-trino is not restricted, so that its contribution is ∝ T 3. Taking into account that the energy conservingδ-function brings about a factor T −1, that cancels the factor T arising from the neutrino energy, onegets the expected ∝ T 6 dependence. This example illustrates how neutrino emissivity in neutron starmatter is strongly affected by degeneracy.

5.1.1 Threshold of the direct Urca process

The main feature of the direct Urca process is the occurrence of a threshold, reflected by the pres-ence of the step function Θnpe in Eq. (5.12). Because the neutrino emissivity associated with thisprocess, when active, dominates by many order of magnitude, it is very important to understand theimplications of the threshold condition.

As we have already seen, the occurrence of the direct Urca process requires that the Fermi mo-menta of the degenerate fremions, pF n , pF p e pFe , fulfill the triangular condition pF n ≤ pF p +pFe . Atdensities around the equilibrium density of isospin symmetric nuclear matter, n0, this requirementis not met. However, several models of the nuclear equation of state predict that the Fermi momentapF p and pFe can grow faster than pF n with increasing density, thus allowing the onset of the process.


The Urca process is active when the proton fraction reaches a critical value. In the case of matterconsisting of neutrons, protons and electrons (npe matter), electric neutrality implies pF p = pFe andthe triangular condition simplifies to pF p ≥ pF n/2. It follows that the proton and neutron densitiesmust be such that np ≥ nn/8. In terms of proton fraction this inequality reads

xp = np

nn +np≥ 1

9. (5.13)

If the electron chemical potential exceeds the muon rest mass, mµ = 105.7 Mev, a different kindof direct Urca process is energetically allowed, in addition to the one described above

n → p +µ+ νµ , p +µ→ n +νµ . (5.14)

The emissivity associated with the above processes can be written in the same form as Eq. (5.12), sincethe condition of β-equilibrium requires µµ = µe , implying in turn m∗

µ = m∗e . The threshold function

alone is modified toΘnpµ, as the appearance of muons affects the critical value of the proton fraction,which grows until it reaches the upper limit xp = 1/[1+ (1+2−1/3)3] ≈ 0.148 when µµÀ mµ.

As mentioned in the previous Chapters, early models of neutron star matter were largely basedon the non interacting Fermi gas description. Within this oversimplified picture, the proton fractionnever exceeds the threshold of direct Urca processes. However, in the 1980s [14] and early 1990s [15]it was shown that inclusion of the effects of strong interactions – through models of nuclear dynamicspredicting large symmetry energies – leads to a sizeable increase of the proton fraction. The resultingvalues of xp turn out to be above the direct Urca threshold in a density region which is believed to beattained in the neutron star core.

Figure 5.1 shows the proton, electron and muon fractions in cold, i.e. T = 0, nuclear matter ob-tained from the equation of state of Ref. [16]. It clearly appears that the horizontal lines correspond-ing to the thresholds of direct electron and muon Urca processes are crossed at densities in the range0.6. n . 0.9 fm−3.

The density dependence of the of the neutron Fermi momentum is displayed in Fig. 5.2. Forcomparison, the quantities pF p +pFe and pF p +pFµ are also shown. It can be seen that the triangularcondition on the Fermi momenta is fulfilled at density ∼ 0.7 fm−3 and ∼ 0.9 fm−3 for the electron andmuon Urca process, respectively.

5.2 Modified Urca processes

If the direct Urca process is not allowed, the most efficient mechanism of neutrino emission isthe modified Urca process, in which momentum conservation is made possible by the presence of anadditional, spectator, nucleon.

The modified Urca reactions involve a nucleon-nucleon collision associated with β-decay or cap-ture

n +n → p +n +e + νe , p +n +e → n +n +νe , (5.15)

n +p → p +p +e + νe , p +p +e → n +p +νe . (5.16)

where the two lines correspond to the neutron and proton branches, respectively. As we will see, theaddition of the spectator nucleon dramatically slows down the reaction rate.

5.2 Modified Urca processes 75

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x

n [fm-3

]

proton fractionelectron fraction

muon fraction

Figure 5.1: Proton, electron and muon fractions predicted by the model of Ref. [16]. The twohorizontal lines show the thresholds of the direct Urca processes involving electrons andmuons.

In the following, modified Urca processes will be denoted by the label M N , where M stands formodified, whereas N = n and p refers to the neutron or proton branch. Both branches include thedirect and inverse reactions, the rates of which at equilibrium are the same. Hence, only one rate is tobe computed, and multiplied by a factor two to get the total emission rate.

Neutrino emissivity can be written in the usual form

QM N = 2∫ [

4∏j=1

dp j

(2π)3

]dpe

(2π)3

dpν(2π)3 Eν (2π)4δ(E f −Ei )

× δ(p f −pi ) f1 f2 (1− f3) (1− f4) (1− fe )1

2|M f i |2 , (5.17)

where the indices i and f refer to the initial and final states and |M f i |2 is the squared modulus of theampletude associated with the process. The additional factor 2 is a symmetry factor, the inclusion ofwhich is needed to avoid double counting in collisions involving identical particles.

One can proceed following the same procedure employed in the discussion of the direct Urcaprocess (see Appendix D), and rewrite the emissivity associated with the direct reaction in the form

QM N = 1

(2π)14 T 8 A I ⟨|M f i |2⟩5∏

j=1pF j m∗

j , (5.18)


0.5

1

1.5

2

2.5

3

3.5

0.2 0.4 0.6 0.8 1

p [fm

-1]

n [fm-3

]

pFnpFp + pFepFp + pFµ

Figure 5.2: Density dependence of pF n , pF p +pFe and pF p +pFµ, obtained from the modelof Ref. [16]. The triangular conditions on the Fermi momenta are fulfilled at densities largerthan the values at which the dashed and dotted lines cross the solid line.

with

A = 4π

[5∏

j=1

∫dΩ j

]δ(P f −Pi ) , (5.19)

⟨|M f i |2⟩ =4π

A

[5∏

j=1

∫dΩ j

]δ(P f −Pi ) |M f i |2 (5.20)

and

I =∫ ∞

0d xν x3

ν

[5∏

j=1

∫ ∞

−∞d x j f j

]δ

(5∑

j=1x j −xν

). (5.21)

Note that, as the magnitudes of the momenta of the degenerate fermions, p j are set to be equal tothe corresponding Fermi momenta, the quantities A and ⟨|M f i |2⟩ only involve integrations over theangles specifying the directions of the particle momenta.

In the evaluation of A, the integration over the neutrino momentum can be readily carried out,as pν is neglected in the δ-function. Because, in general, |M f i |2 depends on the momenta, we haveintroduced the the squared modulus of the angle-averaged matrix element, ⟨|M f i |2⟩.


1e+12

1e+14

1e+16

1e+18

1e+20

1e+22

1e+24

1e+26

1e+28

2 4 6 8 10 12 14 16

Q [

erg

s−

1 c

m−

3]

ρ14 [g cm−3

]

T = 108 K

T = 3108 K

T = 109 K

Figure 5.3: Neutrino emissivity obtained from the dynamical model of Ref [16]. The directUrca process set in at density ∼ 1.3 g cm−3. Below threshold the dominant emission mecha-nism is the modified Urca process.

Calculation of the integral of Eq. (5.21) yields (see Appendix D)

I = 11513π8

120960. (5.22)

At this stage the neutron and proton branches require separate discussions.

5.2.1 Neutron branch

Let us consider the first reaction of Eq. (5.15). We label with 1 and 2 the neutrons in the initialstate, while 3 and 4 refer to the neutron and proton in the final state, respectively. The calculation of

Figure 5.4: Feynman diagram illustrating the first reaction of Eq. (5.15)


A can be performed, with the result

An = 2π(4π)4

p3F n

. (5.23)

Recall that the above expression should be modified at high density, if direct the Urca process sets in.In this case, modified Urca processes would give a negligible contribution to the emissivity.

The evaluation of the matrix elements M f i involves non trivial difficulties, as it includes stronginteractions. The nucleon-nucleon interaction, discussed in Chapter 2, features a long range com-ponent arising from one-pion-exchange (OPE) processes, taken into account within Yukawa’s theory,and a short range, strongly repulsive, component, that is generally described within phenomenolog-ical approaches.

The long range contribution has been estimated in the classic work of Friman and Maxwell [17],treating nucleons as non relativistic particles and assuming that all lepton momenta can be neglected.The result, obtained averaging over the direction of the neutrino momentum, can be written in theform

|M M Nf i |2 = 16G2

E 2e

(fπ

mπ

)4

g 2A FA , (5.24)

where mπ is the pion mass, fπ ≈ 1 and

FA = 4Q41

(Q21 +m2

π)2+ 4Q4

2

(Q22 +m2

π)2+ (Q1 ·Q2)2 −3Q2

1Q22

(Q21 +m2

π)(Q22 +m2

π), (5.25)

with Q1 = p1 −p3 e Q2 = p1 −p4.The first contribution arises from the amplitude of the reaction with 1 → 3 and 2 → 4, whereas

the second one is associated with the exchange process, in which 1 → 4 and 2 → 3. Finally, the thirdcontribution takes into account interference between the two amplitudes. In addition, Friman andMaxwell neglect the proton momentum, setting |Q1| = |Q2| ≈ pF n e Q1 ·Q2 ≈ p2

F n/2. Their final resultreads

|M Mnf i |2 = 16G2

(fπ

mπ

)4 g 2A

E 2e

21

4

p4F n

(p2F n +m2

π)2. (5.26)

Being independent of the directions of the momenta, the above expression can be moved out of theintegral appearing in Eq, (5.20), with the result ⟨|M Mn

f i |2⟩ = |M Mnf i |2. Comparison between the above

result and the exact solution, obtained through numerical integration, shows that the approximationemployed in Ref. [17] are remarkably accurate. The discrepancy turns out to be few percent at n ∼ n0,and ∼ 10% at n ∼ 3n0.

In conclusion, the emissivity associated with the neutron branch of the modified Urca processobtained by Frimam and Maxwell is

QMn = 11513

30240

G2F cos2θC g 2

A m∗3n m∗

p

2π

(fπ

mπ

)4 pF p (kB T )8

ħ10 c8 αn βn

≈ 8.1×1021(

m∗n

mn

)3(

m∗p

mp

) (np

n0

)1/3

T 89 αn βn er g cm−3 s−1 , (5.27)

with αn = 1.13 e βn = 0.68.


5.2.2 Proton branch

Let us now consider the second reaction of (5.16). We will label 1 e 2 the initial state protons 3 and4 the final state proton and neutron, respectively.

The calculation of A turns out to be more compilex, with respect to the case of the neutron branch.We find

Ap = 2(2π)5

pF n p3F p pFe

(pFe +3pF p −pF n

)2ΘM p , (5.28)

where ΘM p = 1 if the proton branch is allowed by momentum conservation and ΘM p = 1 otherwise.The triangular condition translates into the inequality pF n < 3pF p +pFe . Note that the expression ofAp should also be modified at densities larger than that corresponding to the threshold of the directUrca process. However, in this density region the contribution of the modified Urca processes turnsout to be negligible.

The calculation of the squared matrix element yields a result identical to (5.24). Replacing |Q1| =|Q2| ≈ pF n −pF p (maximum momentum trasfer) and Q1 ·Q2 =−(pF n −pF p )2, one obtains an expres-sion similar to Eq. (5.26), apart from the replacements of 21/4 with 6 e pF n with pF n−pF p . Neglectingthe angular dependence of the matrix element has been shown to be a good approximation in boththe neutron and neutron branches.

The proton branch emissivity can be obtained from the corresponding neutron branch resultexploiting the rescaling rule

QM p

QMn=

⟨|M M pf i |2⟩

⟨|M Mnf i |2⟩

(m∗

p

m∗n

)2(pFe +3pF p −pF n)2

8 pFe pF pΘM p

≈(

m∗p

m∗n

)2(pFe +3pF p −pF n)2

8 pFe pF pΘM p . (5.29)

In applications, it is common practice setting ⟨|M M pf i |2⟩ = ⟨|M Mn

f i |2⟩, as illustrated by that the last lineof Eq. (5.29).

The main difference between the two branches is the presence of a threshold for the protonbranch. In npe matter, this condition reduces to pF n < 4pF p , corresponding to a critical protonfraction x = 1/65 = 0.0154, which is believed to be attained almost everywhere in the neutron starcore. Once the modified Urca process is active in the proton branch, the associated emissivity slowlygrows, starting from zero and eventually becoming comparable to the the emissivity associated withthe neutron branch, in the vicinity of the threshold of the direct Urca process.

In conclusion, it has to be pointed out that the temperature dependence of the emissivity of themodified Urca process is a power law T 8. The additional factor T 2, with respect to the direct Urcaprocess, is to be ascribed to the presence of two additional degenerate fermions.


5.3 Neutrino bremsstrahlung in nucleon-nucleon collisions

To complete the discussion of neutrino emission processes in the neutron star core, we need toconsider neutron bremsstrahlung associated with nucleon-nucleon collisions

n +n → n +n +ν+ ν , n +p → n +p +ν+ ν , p +p → p +p +ν+ ν . (5.30)

Figure 5.5: Feynman diagram illustrating the first reaction of 5.30.

In these reactions, the scattering process driven by strong interactions leads to the productionof a neutrino-antineutrino pair, of any flavour. Neutrino bremsstrahlung has no threshold, and istherefore active at any densities. In addition, unlike the processes discussed above, does not changethe composition of matter.

The emissivity associated with bremsstrahlung can be written in the form

QN N =∫ [

4∏j=1

dp j

(2π)3

]dpν

(2π)3

dp′ν

(2π)3 ων (2π)4δ(E f −Ei )

× δ(P f −Pi ) f1 f2 (1− f3) (1− f4)1

s|M f i |2 , (5.31)

where the index j labels nucleons, pν and p′ν are the neutrino and antineutrino momenta, ων =

Eν+E ′ν is the energy carried by the neutron pair and the symmetry factor s (s = 1 for the np channel

and s = 4 for the nn and pp channels) is meant to avoid double counting. In the non relativistic limit,the squared matrix element, after spin summation, reads

|M f i |2 = |M f i |2/ω2ν . (5.32)

The factor ω2ν in the denominator comes from the propagator of the virtual nucleon appearing in the

Feynman diagram.Neglecting the momenta of the neutrinos with respect to those of the nucleons, |M f i |2 become

independent of pν e p′ν. As a consequence, the integration over the neutrinos phase-space yields∫ ∞

0E 2νdEν

∫ ∞

0E ′νdE ′

ν · · · =1

30

∫ ∞

0ω5νdων · · · , (5.33)

5.3 Neutrino bremsstrahlung in nucleon-nucleon collisions 81

and the emissivity takes the form (we follow the decomposition approximation again)

QN N = (2π)4

(2π)18

1

30A I

1

s⟨|M f i |2⟩T 8

4∏j=1

m∗j pF j , (5.34)

con

A = (4π)2∫

dΩ1 dΩ2 dΩ3 dΩ4δ(p1 +p2 −p3 −p4) , (5.35)

⟨|M f i |2⟩ =(4π)2

A

∫dΩ1 dΩ2 dΩ3 dΩ4δ(p1 +p2 −p3 −p4) |M f i |2 , (5.36)

I =∫ ∞

0d xν x4

ν

[4∏

j=1

∫ +∞

−∞d x j f j

]δ

(4∑

j=1x j −xν

)= 164π8

945. (5.37)

Recall that x j = (E j−µ j )/T is the dimensionless nucleon energy, while xν =ων/T is the correspondingvariable for neutrinos. The result of the angular integrations is

Ann = (4π)5

2 p3F n

, Anp = (4π)5

2 p2F n pF p

, App = (4π)5

2 p3F p

. (5.38)

Phase-space decomposition provides the rules to rescale the different processes

Qnp

Qnn = 4⟨|M np

f i |2⟩⟨|M nn

f i |2⟩

(m∗

p

m∗n

)2pF p

pF n,

Qpp

Qnn = 4⟨|M pp

f i |2⟩⟨|M nn

f i |2⟩

(m∗

p

m∗n

)4pF p

pF n. (5.39)

Within the OPE dynamical model, the squared matrix element turns out to be

|M N Nf i |2 = 16G2

F g 2A

(fπ

mπ

)4

FN N , (5.40)

with

FN N = Q41

(Q21 +m2

π)2+ Q4

2

(Q22 +m2

π)2+ Q2

1Q22 −3(Q1 ·Q2)2

(Q21 +m2

π)(Q22 +m2

π)(5.41)

for nn e pp processes, and

FN N = Q41

(Q21 +m2

π)2+ 2Q4

2

(Q22 +m2

π)2−2

Q21Q2

2 − (Q1 ·Q2)2

(Q21 +m2

π)(Q22 +m2

π), (5.42)

for the np process. Recall that Q1 = p1−p3, Q2 = p1−p4 and , in the strong degeneracy limit, Q1·Q2 = 0.After averaging the squared matrix element over the directions of the nucleon momenta, one

obtains

⟨|M N Nf i |2⟩ = 161,G2

F g 2A

(fπ

mπ

)4

⟨FN N ⟩ , (5.43)


where

⟨FN N ⟩ = 3− 5

qarctan q + 1

1+q2 + 1

q√

2+q2arctan

(q√

2+q2

), (5.44)

for nn or pp processes, and

⟨FN N ⟩ = 1− 3 arctan q

2q+ 1

2(1+q2)+ 2 p4

F n

(p2F n +m2

π)2−

(1− arctan q

q

)2 p2

F n

p2F n +m2

π

, (5.45)

with qN = 2pF n/mπ, for the np process.Note that Eq. (5.44) has been derived within the model under consideration, while Eq. (5.45) has

been obtained assuming pF p ¿ pF n . However, this approximation is applicable at densities n . 3n0.To summarise the calculation of neutrino emissivity, we can refer again to the treatment of Friman

and Maxwell. In the nn channel they neglect the exchange contribution to the squared matrix ele-ment (5.44), average over the directions of the nucleon momenta and set n = n0. Finally, they plug⟨|M f i |2⟩ obtained within this scheme into the expression of Qnn , with an arbitrary correction fac-tor βnn , meant to take into account all neglected effects (correlations, repulsive component of thenucleon-nuhcleon interaction . . . ). The same procedure is applied to the np contribution, in whichthe interference term of Eq. (5.45) is omitted.

The pp process, not taken into account in Ref. [17], has been studied by Yakovlev and Levenfish[18]. Their results read

Qnn = 41

14175

G2F g 2

A m∗4n

2πħ10 c8

(fπ

mπ

)4

pF nαnn βnn (kB T )8 Nν

≈ 7.5×1019(

m∗n

m∗n

)4 (nn

n0

)1/3

αnn βnn NνT 89 er g cm−3 s−1 , (5.46)

Qnn = 82

14175

G2F g 2

A m∗2n m∗2

p

2πħ10 c8

(fπ

mπ

)4

pF p αnp βnp (kB T )8 Nν

≈ 1.5×1020

(m∗

n m∗p

mn mp

)2 (np

n0

)1/3

αnp βnp NνT 89 er g cm−3 s−1 , (5.47)

Qpp = 41

14175

G2F g 2

A m∗4p

2πħ10 c8

(fπ

mπ

)4

pF p αpp βpp (kB T )8 Nν

≈ 7.5×1019

(m∗

p

m∗p

)4 (np

n0

)1/3

αpp βpp NνT 89 er g cm−3 s−1 , (5.48)

where mπ is the π0 mass and Nν is the number of neutrino flavours. The dimensionless flavoursαN N

are obtained from estimates of the matrix elements at n = n0. Their values are αnn = 0.59, αnp = 1.06,αpp = 0.11.

The emissivities of all process turn out to be comparable, with Qpp <Qnp <Qnn .Note that the emissivity is always ∝ T 8. This result can be explained with the usual phase-space

consideration. The four degenerate nucleons contributecontribuiscono a power T 4 and the two neu-trinos a power T 6. The squared matrix element is proportional to ω−2

ν , i.e. to T −2, which remove thepower T 2 in excess.

5.3 Neutrino bremsstrahlung in nucleon-nucleon collisions 83

1e+09

1e+10

1e+11

1e+12

1e+13

1e+14

1e+15

1e+16

1e+17

1e+18

2 3 4 5 6 7 8 9 10

Q [

erg

s−

1 c

m−

3]

ρ14 [g cm−3

]

T = 3108 K

total

mod. Urcan

p

nn

npbrems.

pp

ep

brems.ee

Figure 5.6: Density dependence of the neutrino emissivity in the neutron star core, associ-ated with different mechanisms. The density range is restricted to the region in which thedirect Urca process is not allowed. The most effective mechanism is the modified Urca pro-cesses, while the contribution arising from bremsstrahlung of neutrino-antineutrino pairsin nucleon-nucleon collisions is more than one order of magnitude smaller. The emissivityfrom bremsstrahlung in scattering processes driven by electromagnetic interactions appearsto be negligible.

10

12

14

16

18

20

22

24

8 8.2 8.4 8.6 8.8 9 9.2 9.4

lg Q

[e

rg s

−1 c

m−

3]

lg T [K]

ρ = 2ρ0

mod. Urca

np

nn np

pp

brems

ep

ee

Figure 5.7: Log-log plot illustrating the temperature dependence of the neutrino emissivityassociated with the different processes discussed in the text.

Appendix A

Derivation of the one-pion-exchangepotential

Let us consider the process depicted in Fig. 3.2. The corresponding S-matrix element reads

S f i = (−i g )2 m2

(E1E2E1′E2′)1/2(2π)4δ(4)(p1 +p2 −p1′ −p2′)

× η†1′τη1 u1′ iγ5u1

i

k2 −m2π

η†2′τη2 u2′ iγ5u2 (A.1)

where mπ is the pion mass, k = p2 −p2′ and ηi denotes the two-component Pauli spinor describingthe isospin state of particle i . Equation (A.1) can be rewritten in the form

S f i = i g 2 m2

(E1E2E1′E2′)1/2(2π)4δ(4)(p1 +p2 −p1′ −p2′)

× ⟨τ1 ·τ2⟩ u2′γ5u2u1′γ5u11

k2 −m2π

, (A.2)

with ⟨τ1 ·τ2⟩ = η†2′τη2 η

†1′τη1.

Substituting the nonrelativistic limit

u2′γ5u2 = (E2′ +m)1/2 (E2 +m)1/2

2m

[χ†

s′σ ·p2

E2 +mχs −χ†

s′σ ·p2′

E2′ +mχs

]≈ χ†

s′σ(p2 −p2′)

2mχs =χ†

s′(σ ·k)

2mχs (A.3)

and the similar expression for u1′γ5u1 in the definition of S f i and replacing (use Ei ≈ Ei ′ ≈ mπ andk2 = (Ei −Ei ′)2 −|k|2 ≈−|k|2)

1

k2 −m2π

≈− 1

|k|2 +m2π

(A.4)

we obtain

S f i ≈ −ig 2

4m2 (2π)4δ(4)(p1 +p2 −p1′ −p2′) ⟨τ1 ·τ2⟩

× χ†1′χ

†2′

(σ1 ·k)(σ2 ·k)

|k|2 +m2π

χ2χ1 (A.5)

86 Derivation of the one-pion-exchange potential

The corresponding potential in momentum space is

vπ(k) = − g 2

4m2

(σ1 ·k)(σ2 ·k)

k2 +m2π

τ1 ·τ2

=(

fπmπ

)2 (σ1 ·k)(σ2 ·k)

k2 +m2π

τ1 ·τ2 (A.6)

with g 2/4π= 14 and

f 2π = g 2 m2

π

4m2 ≈ 4π×14(140)2

4× (939)2 ≈ 4π×0.08 ≈ 1 (A.7)

The configuration space vπ(r) is obtained from the Fourier transform:

vπ(r) = − f 2π

m2π

∫d 3k

(2π)3 τ1 ·τ2 σ1 ·k σ2 ·k1

(|k|2 +m2π)

e−i k·r

= f 2π

m2π

τ1 ·τ2 σ1 ·∇ σ2 ·∇∫

d 3k

(2π)3

1

(|k|2 +m2π)

e−i k·r

= 1

4π

f 2π

m2π

τ1 ·τ2 σ1 ·∇ σ2 ·∇ e−mπr

r. (A.8)

The Laplacian of the Yukawa function,

yπ(r ) = e−mπr

r= 4π

∫d 3k

(2π)3

1

(|k|2 +m2π)

e i k·r, (A.9)

involves a δ-function singularity at the origin, as can be easily seen from

(−∇2 +m2π) yπ(r ) = 4πδ(r) . (A.10)

Gradients in Eq. (A.8) have to be evaluated taking this singularity into account. For this purpose it isconvenient to rewrite

σ1 ·∇ σ2 ·∇yπ(r ) =(σ1 ·∇ σ2 ·∇− 1

3σ1 ·σ2 ∇2

)yπ(r )

+ 1

3σ1 ·σ2 ∇2 yπ(r ) , (A.11)

as the δ-function contribution to ∇2 yπ(r ) does not appear in the first term, yielding(σ1 ·∇σ2 ·∇− 1

3σ1 ·σ2∇2

)yπ(r ) (A.12)

=(σ1 · r σ2 · r− 1

3σ1 ·σ2

)(m2π+

3mπ

r+ 3

r 2

)yπ(r ),

where r = r/|r|. The laplacian ∇2 yπ(r ) in the second term of Eq. (A.11) can be replaced with m2πyπ(r ) −

4π δ(r ) using Eq. (A.10).

87

Carrying out the calculation of the derivatives in Eq. (A.8) in this way we find

vπ(r) = 1

3

1

4πf 2π mπ τ1 ·τ2

[Tπ(r )S12

+(Yπ(r )− 4π

m3π

δ(r)

)σ1 ·σ2

], (A.13)

with

Yπ(r ) = e−mπr

mπr, (A.14)

and

Tπ(r ) =(1+ 3

mπr+ 3

m2πr 2

)Yπ(r ) , (A.15)

i.e. Eqs. (3.39) and (3.40).

88 Derivation of the one-pion-exchange potential

Appendix B

Neutron β-decay

Consider the reactionn −→ p +e + ν , (B.1)

illustrated by the Feynman diagram of Fig. B.1, where pn ≡ (En ,pn), pp ≡ (Ep ,pp ), pe ≡ (Ee ,pe ) andpν ≡ (Eν,pν) denote the neutron, proton, electron and antineutrino four momenta, respectively.

n(pn)

p(pp) e(p

e) ν

−

(pν)

Figure B.1: Feynman diagram illustrating the neutron β-decay process (B.1)

The Fermi interaction hamiltonian, driving the decay process, is given by

H = Gp2

jµ`µ , (B.2)

with G =GF cosθC , GF = 1.463×10−49 erg cm3 and θC ∼ 13 deg being the Fermi constant and Cabibbo’sangle, respectively. The weak leptonic current `µ has the form

`µ = ur ′(pe )γµ(1−γ5)vr (pν) , (B.3)

where ur (p) e vr (p) denote the four-spinors associated with the positive and negative energy solu-tions of the Dirac equation, describing a non interacting fermion of mass m with four-momentum p

90 Neutron β-decay

and spin projection r . They are normalised is such a way as to satisfy the relations

ur ′(p)ur (p) = vr ′(p)vr (p) = 2m δr r ′ . (B.4)

In the non relativistic limit, fully justified in the present context, the hadronic weak current reducesto

jµ =χ†s′

(δµ0gv +δµi g Aσi

)χs , (B.5)

where σi (i=1,2,3) are Pauli matrices, χs and χs′ are Pauli spinors satisfying the relations χ†s′χs = δss′

and the values of the vector and axial coupling constants are gv = 1 and g A = 1.26.The transition rate from the initial state |i ⟩ to the set of accessible final states | f ⟩ can be obtained

using Fermi’s golden rule

dWi→ f = 2π δ(En −Ep −Ee −Eν)∑

spi ns

∣∣H f i∣∣2 dpp

(2π)3

dpe

(2π)3

dpν(2π)3 , (B.6)

where ∑spi ns

∣∣H f i∣∣2 = (2π)3 G2

2δ(pn −pp −pe −pν)

1

2Ee

1

2EνJλµLλµ (B.7)

withJλµ =

∑spi ns

jλ† jµ (B.8)

andLλµ = ∑

spi ns`λ

†`µ . (B.9)

Substituting the leptonic weak current in Eq. (B.9) and exploiting the completeness relations ful-filled by Dirac’s spinors, we find

Lλµ = ∑r r ′

ur ′(pe )γλ(1−γ5)vr (pν)vr (pν)γµ(1−γ5)ur ′(pe )

= 8[

pλe pµ

ν +pµe pλ

µ− gλµ(pe pν)+ iελσµρpeσpνρ

], (B.10)

where gλµ = diag(1,−1,−1,−1) and ελσµρ are the metric tensor and the fully antisymmetric unit ten-sor.

The explicit expression of Jλµ is obtained using Eq. (B.5), implying

j 0 =χ†s′χs = δss′ , j i = g Aχ

†s′σ

iχs , (i = 1,2,3) . (B.11)

Carrying out the spin sum we readily find that the the tensor Jλµ is diagonal, and its non vanishingcomponents are given by

J00 = 2, J11 = J22 = J33 = 2g 2A . (B.12)

Finally, substitution of Eqs. (B.10) and (B.12) in the expression of the differential decay rate leads to

dWi→ f = (2π)4δ(En −Ep −Ee −Eν)δ(pn −pp −pe −pν)

× 2G2 [1+cosθ+ g A

2(3−cosθ)] dpp

(2π)3

dpe

(2π)3

dpν(2π)3 , (B.13)

91

where θ is the angle between the lepton momenta pe and pν.Neglecting the contribution of pν in the argument of the δ-function expressing momentum con-

servation1, we can perform the cosθ integration, yielding the final result

dWi→ f = 2π δ(En −Ep −Ee −Eν)δ(pn −pp −pe )

× 2G2 (1+3g A

2)4πE 2ν dEν

dpp

(2π)3

dpe

(2π)3 . (B.14)

1This approximation is fully justified for emission of antineutrinos of energy Eν ∼ T , T being the tempera-ture, in dense degenerate matter.

92 Neutron β-decay

Appendix C

Basic of QCD thermodinamics

Lo studio delle diverse fasi della QCD a temperatura e potenziale chimico diversi da zero, richiedela conoscenza delle grandezze termodinamiche, definite a partire dalla funzione di partizione grancanon-ica Z . Per un sistema a temperatura T =β−1 la cui dinamica è descritta dall’hamiltoniana H

Z = Tr exp[−β(H −∑iµi Ni )] , (C.1)

dove gli Ni sono gli operatori associati alle grandezze conservate (per esempio, nel caso della QED,la differenza tra il numero di elettroni e quello di positroni) e µi i potenziali chimici corrispondenti.Dalla (C.1) seguono immediatamente le relazioni

P =−T∂ lnZ

∂V, (C.2)

Ni = T∂ lnZ

∂µi, (C.3)

S = ∂ T lnZ

∂T, (C.4)

E =−PV +T S +∑iµi Ni = T

∂ lnZ

∂T+T

∑iµi∂ lnZ

∂µi, (C.5)

dove P , S ed E sono, rispettivamente, pressione, entropia ed energia del sistema.Le equazioni (C.2)-(C.5) mostrano che è conveniente introdurre il potenziale termodinamico

Ω=−T lnZ =−PV , (C.6)

dal quale si ottengono in modo semplice le grandezze termodinamiche tramite derivate parziali.In teoria quantistica dei campi, la traccia nella definizione di Z , ovvero la somma sugli elementi

diagonali della matrice densità, si effettua usando il formalismo degli integrali funzionali. Per esem-pio, nel caso semplice di un di campo scalare, la cui densità lagrangiana è

L = 1

2∂µ∂µφ− 1

2m2φ2 , (C.7)

94 Basic of QCD thermodinamics

non ci sono cariche conservate e si ottiene [?, ?]

Z =N

∫Dφ eS (C.8)

ovvero l’integrale esteso a tutti i cammini periodici, tali cioè che φ(~x,0) = φ(~x,τ), dell’esponenzialedell’integrale d’azione per tempi immaginari τ= i t

S =∫ β

0dτ

∫d 3x L (φ,∂φ/∂τ) =−1

2

∫ β

0dτ

∫d 3x

[(∂φ

∂τ

)2

+ (∇)φ)2 +m2φ2

]. (C.9)

Si noti che il fattore di normalizzazione N che compare nel secondo membro dalla (C.8) è irrilevante,poichè la presenza di un fattore moltiplicativo non cambia la termodinamica del sistema.

Il calcolo della funzione di partizione nel caso del campo di Dirac ψ(x), che verrà discusso indettaglio nel Capitolo ??, da il risultato [?]

Z =∫

iDψ†∫

Dψexp∫ β

0dτ

∫d 3x ψ

(−γ0 ∂

∂τ+ i~γ ·~∇−mi +µiγ

0)ψ , (C.10)

dove iψ† e ψ vanno considerati come variabili dinamiche indipendenti e l’integrale è esteso a tuttii cammini che soddisfano alla condizione di antiperiodicità ψ(~x,0) = −ψ(~x,β), come richiesto dalleregole di anticommutazione dei campi fermionici.

Appendix D

Phase-space decomposition

In the is Appendix, we describe the elements entering the calculation of neutrino emissivity asso-ciated with nucleonic Urca processes, carried out using the phase-s[ace decomposition method.

On account of the strong degeneracy of nucleons and electrons, the main contribution to theintegral yielding the emissivity

QD = 2∫

dpn

(2π)3 dWi→ f fn(1− fp )(1− fe ) , (D.1)

arises from quantum states with momenta within a thin shell around the Fermi surface. As a conse-quence, we can set |pi | = pFi in the integrand. Moreover, the typical energy exchanged in the reactionsis ∼ T .

Because the energy carried by the neutrino is Eν ∼ T , the corresponding momentum, being of thesame order, is much smaller than the momenta of the degenerate fermions involved in the process.Hence, it can be safely neglected in the argument of the momentum-conserving δ-function. Theresulting expression is

dWi→ fdPn

(2π)3 = (2π)4

(2π)12 δ(En −Ep −Ee −Eν)δ(pn −pp −pe )

× |M f i |2 4πE 2νdEν

3∏j=1

pF j m∗j dE j dΩ j . (D.2)

We now transform to the new variables

xν = EνT

, x j =E j −µ j

T( j = n, p, e) ,

and replace x j →−x j for j = p, e, to exploit the property of the Fermi distribution 1− f (x j ) = f (−x j ).In addition, we introduce the quantity

A = 4π∫

dΩ1dΩ2dΩ3δ(pn −pp −pe ) . (D.3)

96 Phase-space decomposition

Using the above definitions, Eq.(D.1) can be rewritten in the form

QD = 2

(2π)8 |M f i |2 A

[T 7

∫ ∞

0d xνx3

ν

(∫ ∞

− µ1T

∫ µ2T

−∞

∫ µ3T

−∞d x j f j

)

× 1

Tδ(x1 +x2 +x3 −xν)

] 3∏j=1

pF j m∗j . (D.4)

As the system is strongly degenerate, we can replace µ j /T →∞, the associated error being expo-nentially small. Thus, we finally obtain

QD = 2

(2π)8 |M f i |2 T 6 AI3∏

j=1pF j m∗

j , (D.5)

where

I =∫ ∞

0d xνx3

ν

[3∏

j=1

∫ ∞

−∞d x j f j

]δ(x1 +x2 +x3 −xν) . (D.6)

D.1 Calculation of A

Let us start rewriting the momentum conserving δ-function in spherical coordinates. To do this,we exploit the fundamental property of Dirac’s δ∫

δ(a−b)da =∫δ(ax −bx )d ax

∫δ(ay −by )d ay

∫δ(az −bz )d az = 1.

The requirement that the above relation still hold true in spherical coordinates, i.e. that∫δs(a−b)da =

∫δs(a−b) a2d a dΩa = 1,

implies

δs(a−b) = δ(a −b)δ(Ωa −Ωb)

a2 .

In the case under consideration the above expression becomes

A = 4π∫

dΩ1dΩ2dΩ3δ(pn −|pp +pe |)δ(Ω1 −Ω2+3)

p2n

= 4π∫

dΩ2dΩ3

δ(pn −|pp +pe |)p2

n. (D.7)

The radial δ-function can be transformed into a δ-function for one of the angular variables. We firstrewrite the δ-function in the form

δ( f (cosθ2)) = δ[pn − (p2p +p2

e +2pp pe cosθ2)12 ] ,

with the z-axis chosen along the direction electron momentum pe . The root of the function f (cosθ2)is

cosθ2 =p2

n −p2p −p2

e

2pp pe= a ,

D.2 Calculation of I 97

and| f ′(a)| = pp pe

pn.

It follows that

δ(pn −|pp +pe |)p2

n= 1

p2n

1

| f ′(a)| δ(cosθ2 −a) = 1

pn pp peδ(cosθ2 −a) ,

and replacing in (D.7) we obtain

A = 32π3

pF n pF p pFeΘnpe , (D.8)

where the function Θnpe takes into account the triangular condition on the Fermi momenta of thedegenerate fermions.

D.2 Calculation of I

Le us rewrite I in the form

I =∫ ∞

0d xνx3

ν J (xν) , (D.9)

with

J (xν) =∫ ∞

−∞

3∏j=1

d x j (1+ex j )−1δ

(3∑

j=1x j −xν

), (D.10)

and start with the calculation of J .Using the definition of the δ-function in terms of its Fourier transform

δ(x) = 1

2π

∫ ∞

−∞e i zx ,

we find

J (xν) = 1

2π

∫ ∞

−∞d z

∫ ∞

−∞

3∏j=1

d x j (1+ex j )−1e i z(x j−xν)

= 1

2π

∫ ∞

−∞d z e−i zxν

(∫ ∞

−∞d x (1+ex )−1e i zx

)3

= 1

2π

∫ ∞

−∞d z e−i zxν

[f (z)

]3 , (D.11)

where

f (z) =∫ ∞

−∞d x (1+ex )−1e i zx . (D.12)

To obtain the expression of f (z), consider the integral

K =∮

d x (1+ex )−1e i zx , (D.13)


-ℜx

6Im x

- -

−R R

−R +2iπ −R +2iπ

× iπ

Á

¬

Figure D.1: Integration contour employed for the evaluation of K (see Eq.(D.13)).

to be evaluated using Cahchy’s theorem, with the contour shown in Fig. D.1.As R →∞ the contributions of the vertical sides become negligible. Along the real axis, ¬, K = f (z),

whereas along the path marked with Á

K =∫ ∞

−∞d x (1+eℜx+2iπ)−1e i z(ℜx+2iπ) = e−2πz f (z) .

It follows thatK = f (z)−e−2πz f (z) = (1−e−2πz ) f (z) .

The value of K can be obtained using the mot hod of residues. From

ResK (z) = limx→iπ

(x − iπ)e i zx

1+ex = limx→iπ

(x − iπ)e i zx

1−∑∞n=0

1n! (x − iπ)n

= limx→iπ

− e i zx∑∞n=1

1n! (x − iπ)(n−1)

=−e−πz ,

it follows thatK (z) = (1−e−2πz ) f (z) =−2iπe−πz ,

. and using the above result we find

f (z) = π

i sinhπz. (D.14)

Using the above results, we can write J in the form

J (xν) =− 1

2iπ

∫ ∞−iε

−∞−iεd z e−i zxν

( π

sinhπz

)3, (D.15)

where the quantity −iε, with ε = 0+ takes into account the presence of a pole of third order in theitegrand. To determine a suitable integration contour, we carry out the transformation z = z ′ − i ,leading to

J (xν) = e−xν

2iπ

∫ ∞−iε+i

−∞−iε+id z e−i zxν

( π

sinhπz

)3. (D.16)

Summing Eqs. (D.15) and (D.16) we obtain an integration along the contour shown in Fig. D.2

(1+exν) J (xν) =− 1

2iπ

[∫ ∞−iε

−∞−iε+

∫ ∞−iε+i

−∞−iε+i

]d z e−i zxν

( π

sinhπz

)3

=− 1

2iπ

∮d z e−i zxν

( π

sinhπz

)3. (D.17)

D.2 Calculation of I 99

-ℜz

6Im z

- -

× i

×

Figure D.2: Integration contour employed for the evaluation the integral of Eq.(D.17).

From Eq. (D.17) it follows that

(1+exν) J =−Res

[e−i zxν

( π

sinhπz

)3]

z=0

=− limz→0

1

2

d 2

d z2

[z3e−i zxν

( π

sinhπz

)3]

=− limz→0

1

2

−x2

νe−i zxν( π

sinhπz

)3+6i xνe−i zxν

( π

sinhπz

)3(πz3 coshπz

sinhπz− z2

)+

[6z −18πz2 coshπz

sinhπz+3π2z3 1

(sinhπz)2 +9π2z3(

coshπz

sinhπz

)2] π3e−i zxν

(sinhπz)3

.

Let us consider the first line of the above result. In the z → 0 limit, the second contribution van-ishes, as can be easily seen substituting the first order expansion

coshπz

sinhπz≈ 1

πz,

while the first contribuito reduces to −x2ν. The second line must be expanded up to the zero-th order

term. The resulting contributions are

1.6π3z

(πz)3[

1+ (πz)2

2

] ≈ 6

z2

(1− 1

2π2z2

)= 6

z2 −3π2;

2.3π5z3

(πz)5[

1+ 5(πz)2

6

] ≈ 3

z2 − 5

2π2 ;

3.9π5z3

(πz)5[

1+ 5(πz)2

6

] (1+π2z2) ≈ 9

z2 + 3

2π2 ;

4. − 18π4z2

(πz)4[

1+ 2(πz)2

3

] (1+ 1

2π2z2

)≈−18

z2 +3π2 .

and their sum is −π2. Hence

(1+exν) J (xν) = x2ν+π2

2,


implying

J (xν) = π2 +x2ν

2(1+exν). (D.18)

Finally, the integral

I =∫ ∞

0d xνx3

ν J (xν) ,

that can be found in Ref. [13], Eq. (3.411), is given by∫ ∞

0

x2n−1

epx +1d x = (1−21−2n)

(2π

p

)2n |B2n |4n

,

where Bn is a Bernoulli number. In the case under consideration, correponding to p = 1 e n = 2, 3, weuse B4 =− 1

30 e B6 = 142 , to find

I = 457π6

5040. (D.19)

Substitution of the above results in Eq. (5.7), yields the expression of the neutrino emissivity of Eq. (5.12).

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