The Color-Flavor-Locked phase of the
Nambu–Jona-Lasinio model and applications in
quark stars
Kostas Adam
M.Sc. Thesis
Supervisor: Prof. dr. J. Smit
University of Utrecht, The Netherlands
December 20, 2006
Abstract
In this thesis we employ the Nambu–Jona-Lasinio (NJL) model, a low-energy effective
model for QCD, in finite temperature and chemical potential, in order to obtain an equation of state for cold dense matter. We focus in the high density-low temperature regime
where color-superconducting phases are expected to occur. Using the approximation of
three degenerate quark flavors and specializing in the color-flavor-locked (CFL) phase,
we introduce a pseudoscalar diquark condensate. Working in the mean field approximation, we obtain the grand-canonical potential and the resulting equation of state, in zero
temperature. A framework within which the theoretical predictions can be tested is the
interior of cold compact stars. As an application we look at the mass-radii relations a
star with a pure quark core described by our equation of state would have, by solving
numerically the Tolman-Oppenheimer-Volkoff equations.
Acknowledgements
This thesis would not be possible without the help and support of some people which I
would like to thank at this point. First of all I should mention my supervisor Prof. dr.
Jan Smit for his constant help and guidance throughout the time I was working on my
thesis. Not only he helped me in getting over various ‘sticky’ points and grasping concepts
involved in the thesis, but he introduced me in various other concepts and chapters of
quantum field theory that do not appear here in order to widen my understanding. It
would not be an exaggeration to claim that next to Jan I learned as much, if not more,
as I did throughout the rest of the M.Sc. I would also like to thank my parents for their
endless support in many ways through this tough master’s program. Last but not least I
would like to thank my brother and colleague for putting up with me and helping me out
whenever I needed help.
Contents
Acknowledgements
iii
Conventions
iv
Introduction
1
1 QCD basics
1.1 The QCD Lagrangian and symmetries . . . . . . . . . . . . . . . . .
1.2 Spontaneous Symmetry Breaking and the Nambu–Goldstone bosons
1.3 PCAC and the Goldberger - Treiman relation . . . . . . . . . . . . .
1.4 Asymptotic freedom and color confinement . . . . . . . . . . . . . .
1.5 Quark-Gluon Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 The Nambu-Jona–Lasinio model
2.1 Introduction - historical remarks
2.2 The NJL model . . . . . . . . . .
2.2.1 The Nf = 2 model . . . .
2.2.2 The Nf = 3 model . . . .
2.3 Fierz identities . . . . . . . . . .
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3 Effects of finite temperature and chemical potential
3.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . .
3.2 Thermal Field Theory . . . . . . . . . . . . . . . . . .
3.2.1 Bosons . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Fermions . . . . . . . . . . . . . . . . . . . . .
3.2.3 Interactions . . . . . . . . . . . . . . . . . . . .
3.3 The QCD phase diagram . . . . . . . . . . . . . . . .
3.4 Color superconductivity . . . . . . . . . . . . . . . . .
3.4.1 The Color-Flavor-Locked phase . . . . . . . . .
3.5 From QCD to the NJL model . . . . . . . . . . . . . .
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3
3
9
13
16
18
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20
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24
25
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29
29
31
32
33
37
38
41
42
44
CONTENTS
iii
4 The phase diagram of the NJL model for three degenerate
4.1 NJL model for Nf = 3. . . . . . . . . . . . . . . . . . . . . . .
4.2 The Grand Canonical Potential . . . . . . . . . . . . . . . . .
4.2.1 The Hubbard - Stratonovich transformation . . . . . .
4.2.2 The Mean Field Approximation . . . . . . . . . . . . .
4.2.3 Nambu - Gor’kov formalism . . . . . . . . . . . . . . .
4.2.4 The Grand potential in the MFA . . . . . . . . . . . .
4.2.5 Three degenerate flavors . . . . . . . . . . . . . . . . .
4.2.6 The 1/Nc expansion . . . . . . . . . . . . . . . . . . .
4.3 The Gap Equation for the diquark condensate . . . . . . . . .
4.3.1 Choice of Parameters . . . . . . . . . . . . . . . . . .
4.3.2 Numerical results . . . . . . . . . . . . . . . . . . . . .
4.4 Equation of State . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Zero Temperature Case . . . . . . . . . . . . . . . . .
5
Compact Stars
5.1 Introduction . . . . . . . . . . . . . . . . . .
5.2 Neutron and quark stars . . . . . . . . . . .
5.2.1 Electrical and color neutrality . . . .
5.3 The Tolman-Oppenheimer-Volkov equations
5.4 Numerical results . . . . . . . . . . . . . . .
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flavors.
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46
46
47
48
49
52
54
55
60
62
64
67
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70
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72
72
73
75
76
76
6 Concluding remarks
79
A Units
81
B Miscellanies
83
B.1 The Saddle Point method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
B.2 The Tolman-Oppenheimer-Volkoff equations . . . . . . . . . . . . . . . . . . 85
Conventions
• Greek indices take the values 0, 1, 2, 3 and will be used for denoting 4-vectors and
tensors. Latin indices take the values 0, 1, 2 or 0, 1, . . . , 8 and will be used as group
indices or to denote 3-vectors, unless otherwise stated. Repeated upper and lower
indices are summed over.
• The metric tensor is g µν = diag(+, −, −, −)
• The displacement 4-vector is defined as a contravariant 4-vector xµ ≡ (x0 , xi ) =
(t, x), with upper indices. The derivative with respect to xµ is defined as
∂
∂µ ≡
=
∂xµ
µ
∂
∂
, i
0
∂x ∂x
¶
= (∂0 , ∂i ) = (∂t , ∇)
In these conventions xµ = (x0 , −x) and ∂ µ = (∂t , −∇). Inner products of 4-vectors
are defined as
V · W = V µ Wµ = Vµ W µ = V 0 W0 + V i Wi = V 0 W 0 − V i W i = V 0 W 0 − V · W
where bold-face letters denote ordinary 3-vectors. The momentum 4-vector is defined
as pµ = (p0 , pi ) = (E, p) = i∂ µ = (i∂t , −i∇), such that
6 p = γ µ pµ = γ 0 p0 − γ · p = i(γ 0 ∂t + γ · ∇)
Also 6 ∂ = γ µ ∂µ = γ 0 ∂t + γ · ∇, where γ ≡ γ i .
• We use the chiral (Weyl) representation for Dirac gamma matrices:
Ã
γ0 =
0 1
1 0
!
Ã
, γk =
0
σk
−σ k 0
!
Ã
, γ 5 = γ5 =
−1 0
0 1
The σ i are the Pauli matrices:
Ã
1
σ =
0 1
1 0
!
Ã
2
, σ =
0 −i
i 0
!
Ã
3
, σ =
1 0
0 −1
!
.
!
.
CONTENTS
v
• In this representation γ 0 = γ0 , γ k = −γk , γ 0† = γ 0 , γ k† = −γ k
• The charge conjugation matrix is defined as C = iγ 2 γ 0 = −iσ 2 ⊗ σ 3 . The following
important relations hold:
C † = C −1 , C T = −C = C † , C 2 = −1, Cγ µ C † = −γ µT .
In this representation C † γ µ C = Cγ µ C † , γ µT = Cγ µ C, Cγ5 C −1 = γ5T .
• The Gell-Mann matrices λa , a = 1, . . . , 8 are 3 × 3 traceless hermitean matrices.
They are chosen to have the form:


λ1 = 


λ4 = 


λ7 = 
0
1
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
i


0
0 −i 0


0  , λ2 =  i 0 0
0
0 0 0


1
0 0 −i


0  , λ5 =  0 0 0
0
i 0 0


1 0
0


1
−i  , λ8 = √3  0 1
0 0
0




 , λ3 = 




 , λ6 = 

0

0 .
−2
1
0
0
0
0
0

0 0

−1 0  ,
0 0

0 0

0 1 ,
1 0
They satisfy q
the normalization tr (λa λb ) = 2δab . For our purposes we will use the
matrix λ9 = 23 11 as well.
In this thesis the Gell-Mann matrices referring to flavor SU (3) will be denoted as λ
and the ones referring to color SU (3) as t. For SU (2) we will use σ, τ respectively.
Introduction
This M.Sc. thesis deals mostly with physics at high densities. This regime of matter is
not yet fully explored and many exciting phenomena are predicted, such as the quarkgluon plasma and color-superconducting phases. Knowing more about the physics of high
densities, higher than nuclear densities, would answer to the childish question of what
happens to matter when it is squeezed harder and harder, as the authors of [44] write.
Since the densities involved are higher than the nuclear saturation density, which means
that quarks are going to be assumed to be closer to each other than in an ordinary
nucleus, one may naturally assume that the theory of the strong interactions should be
the theoretical framework within which research should be done.
This is indeed the case, however many problems arise when one considers the appropriate theory of the strong interactions, which is Quantun Chromodynamics (QCD) at finite
temperature and density. The problems arise due to the complexity of the calculations.
One then might think of Lattice gauge theory which has helped substantially in many other
cases by providing illuminating numerical simulations. However we’re unlucky in this case
as lattice calculations at high density (which means high chemical potential) are not yet
possible. What is left then to guide as in the regime of high density are various effective
models for QCD. Such models have their own advantages and their own shortcomings and
are applicable in some sub-regime of QCD matter.
As it is briefly mentioned in the abstract, a natural candidate for the occurrence of
the predicted phenomena of color-superconductivity is the interior of cold neutron stars or
compact stars more generally. If this is indeed the case, then we expect that the presence
of color-superconducting matter to leave it’s signature in some observable quantities of the
star, such as gamma ray bursts or mass-radii relations or the appearance of the so-called
‘glitches’ [44]. In this case we are interested in the physics of the strong interaction in the
low temperature - high density regime, in which a good effective model for QCD is the
Nambu-Jona–Lasinio model, see Chapter 2 for details. Using this model we can construct
an equation of state for the matter of the inner core of cold compact stars and deduce the
mass-radii relations via solving a general-relativistic set of coupled differential equations,
the TOV equations, see chapter 5.
CONTENTS
2
In the writing of this thesis effort has been done in order for the text to be understandable to someone with a basic knowledge of quantum field theory, statistical (many
body) field theory and general relativity. The text has been organized as follows:
In chapter 1 some bare essentials of QCD are given, starting by the definition of the
theory, it’s symmetries and two very important features: asymptotic freedom and color
confinement. The Partially Conserved Axial Current hypothesis and Quark Gluon Plasma
are covered briefly.
In chapter 2 the Nambu - Jona–Lasinio (NJL) model in zero temperature is introduced,
for two and three quark flavors. Later on we will focus in three degenerate quark flavors
(equal mass and chemical potential) in order to deduce an equation of state for the inner
core of a cold compact star.
In chapter 3 we deal with the NJL model in finite temperature and density. An
introduction to the necessary Thermodynamics is provided. We explain how to derive the
equation of state of the NJL model from which the relevant phase diagram is constructed.
The passage from QCD to the NJL model is also dealt with.
In chapter 4 the actual calculation is presented in details.
In chapter 5, after a brief introduction to compact objects, the TOV equations are
solved numerically and the final mass-radii relations of our simplified quark star is calculated.
Chapter 6 is a summary of our conclusions.
Finally, some useful information is given in two appendices. The material presented
there is necessary, however it would be tiring for the reader to present it in the main part
of the text.
Chapter 1
QCD basics
In this chapter we review some essential background knowledge from the quantum theory
of the strong interactions, Quantum Chromodynamics (QCD). QCD in itself is a fascinating never ending subject which poses questions that remain open even today. We will
cover, briefly, the most important aspects of this theory which will be necessary for the
understanding of the following chapters, without giving proofs (the reader may follow the
references to the literature for this purpose).
1.1
The QCD Lagrangian and symmetries
The theory of the strong interactions, Quantum Chromodynamics, is a non-Abelian gauge
field theory, with the gauge group SU (3) in the fundamental representation. It is described
by the Lagrangian density:
1
LQCD = ψ̄ (iγ µ Dµ − m̂) ψ − Tr Fµν Fµν
2
1 a µν a
= ψ̄i (iγ µ Dµ − mi ) ψi − Fµν
F
,
4
(1.1)
where i = 1 . . . Nf , with Nf being the number of flavors, a = 1 . . . 8 the group index and
Einstein’s summation convention is used. The fermionic quark field is denoted by ψ and
has six flavor (u, d, s, c, b, t), and three color (r, g, b) degrees of freedom. In equation (1.1)
summation over flavor, color, Dirac and space-time indices is implied. Each one of the
quark fields is a color triplet, for example the u quark is associated with the triplet:


ur


u =  ug  ,
ub
1.1 The QCD Lagrangian and symmetries
4
where ur , ug , ub are four component Dirac spinors. The mass matrix is diagonal in flavor
space, m̂ = diagf (mu , md , ms , mc , mb , mt ). The covariant derivative is defined as:
ta
,
(1.2)
2
and is responsible for the coupling of the fermionic quark fields with the gauge bosons of
the strong interactions, the gluons. The latter are described by Aµ , the gluon field, for
which the associated field strength tensor is:
Dµ = ∂µ − igAaµ
a
Fµν
= ∂µ Aaν − ∂ν Aaµ + gf abc Abµ Acν ,
(1.3)
where g is the QCD coupling constant and ta and f abc are the generators (Gell–Mann
matrices) and structure constants of SU (3) in color space, which we will denote as SU (3)c ,
satisfying the su(3) algebra:
·
¸
tc
ta tb
,
= if abc .
2 2
2
(1.4)
The Gell–Mann matrices are normalized by the relation Tr ta tb = 2δ ab and f abc are completely antisymmetric. The gluon field and the field strength tensor are Lie-valued vector
and tensor fields respectively:
Aµ = Aaµ
ta
;
2
a
Fµν = Fµν
ta
i
= [Dµ , Dν ] .
2
g
(1.5)
Writing out the Lagrangian (1.1) we see that it gives rise to a quark propagator, a gluon
propagator, a quark-gluon vertex as well as a 3-gluon and a 4-gluon vertex. The last two
vertices are due to the fact that gluons carry color and thus are self-interacting, which
is a typical feature of the non-Abelian gauge theories. We can also note that there is no
mass term for the gluon fields. The field equations for the quarks and gluons follow from
Hamilton’s principle of stationary action. They read:
∂µ F µν a
(iγ µ Dµ − mi ) ψi = 0 ,
Ã
!
a
X
t
µν
= J ν a = g f abc Abµ Fb +
ψ̄i γ ν ψi .
2
(1.6)
(1.7)
i
The bound states of QCD are quark-antiquark (q q̄) combinations, three quark combinations (qqq), glueballs and possibly loosely bound mixtures thereof. Free quarks are not
observed, something that is related to ‘color confinement’ (see the last section of this chapter). The particles occurring in the first case are called mesons and behave like bosons,
while in the second case they are called baryons and behave like fermions. This is justified
since quarks are fermions; their spin is 1/2. Collectively mesons and baryons are referred
1.1 The QCD Lagrangian and symmetries
5
to as hadrons. The various quantum numbers and other properties of the quarks can
be found in the Particle Data Group, [20]. Table 1 below lists some of the most basic
properties of quarks, taken from the 2006 version of the Particle Data Group tables.
quark name
charge/e
Mass [M eV /c2 ]
up, u
+ 23
1.5 − 3
down, d
− 13
− 13
+ 23
− 13
+ 23
3−7
strange, s
charm, c
bottom, b
top, t
95 ± 25
(1.250 ± 0.09) × 103
(4.2 ± 0.07) × 103
(174.2 ± 3.3) × 103
Table 1. Quark charges and masses.
Let us now turn to briefly mention the symmetries LQCD possesses. The Lagrangian
density is built by hand to have a local SU (3)c symmetry; the c index in SU (3) is used to
denote ‘color’, which suggests that the gauge transformation acts on color space only:
¾
½ a
t
ψ → ψ = exp ig ξa (x) ψ ,
2
0
(1.8)
which we also be written as ψ 0 = U ψ and
½
¾
ta
ψ̄ → ψ̄ = ψ̄ exp −ig ξa (x) ,
2
0
(1.9)
Suppose U (x) ∈ SU (3)c . Then the gauge field Aµ under a local SU (3) transformation,
transforms as:
Aµ → A0µ = U Aµ U † +
i
[∂µ U ] U † .
g
The covariant derivative, equation (1.2), is then defined in such a way that it is covariant
under a local SU (3)c transformation, i.e. Dµ0 ψ 0 = U Dµ ψ.
We already mentioned that the bound states in QCD are either q q̄ or qqq combinations
(mesons or baryons respectively) where q is one of the quarks. In both cases, the possible
states are color singlets, which means that under an SU (3) transformation the multiplication law for the state is a trivial multiplication by unity in color space. To clarify this
statement, suppose a baryon state is denoted by |bi and a meson state by |mi. Suppose
further that a three quark state where the first quark has color i, the second has color
j and the third k, is denoted by |i, j, ki and similarly for a quark - antiquark state. We
take the color indices to take the values 1, 2, 3 instead of red, green, blue and we suppress
all other quantum numbers and indices. Starting with the baryons and taking in account
1.1 The QCD Lagrangian and symmetries
6
Pauli’s principle, which requires baryon states to be antisymmetric in the interchange of
color indices, we write down the only antisymmetric combination for a baryon state:
1
|bi = √ ²ijk |i, j, ki ,
6
(1.10)
where ²ijk is the completely antisymmetric Levi-Civitta symbol. Assuming the quarks to
transform under SU (3) as ψ → ψ 0 = U ψ, the baryon state should transform as:
1
∗ ∗ ∗
|bi → |b0 i = √ |a, b, ci²ijk Uai
Ubj Uck
6
1
= √ |a, b, ci²abc det U ∗
6
1
= √ ²abc |a, b, ci
6
= |bi ,
(1.11)
since the determinant of an SU (3) matrix is equal to one. We can proceed in a similar
fashion for the mesons. Denoting a quark-antiquark pair as |i, j̄i, a meson would be the
linear combination:
1
|mi = √ (|1, 1̄i + |2, 2̄i + |3, 3̄i)
3
1
= √ δij |i, j̄i .
3
(1.12)
Under an SU (3) transformation, the meson state would transform as:
1
∗
|mi → |m0 i = √ δij |k, ¯liUki
Ulj
3
1
†
= √ δij |k, ¯liUli Uik
3
1
= √ δab |a, b̄i = |mi ,
3
(1.13)
thus mesons are color singlets as well. Hence the bound quark states are invariant under
color transformations. Thus it is said that matter is colorblind.
In addition to the local SU (3)c symmetry, there are six additional U (1)f global symmetries:
ψj → ψj0 = eiαj ψj ,
(1.14)
1.1 The QCD Lagrangian and symmetries
7
and similarly for the adjoint quark field (ψ̄j0 = ψ̄j e−iαj ). The conserved currents associated
with these symmetries are the flavor currents, Jµi = ψ̄i γµ ψi . This implies conservation of
the quark number, for each flavor. Thus, the strong interactions do not change flavor1 .
These symmetries are also referred to as U (1) “vector”, in order to distinguish them from
the “axial” ones (an axial vector is an object transforming as a vector under continuous
Lorentz transformations but has an additional minus sign under discrete parity transformations). The general form of a U (1)V transformation is ψ → e−iα ψ while an axial U (1)A
transformation is written generally as ψ → e−iαγ5 ψ. The U (1)A symmetry is spoiled by
the mass terms in the Lagrangian. The associated partially conserved axial current is
JµA = ψ̄i γµ γ5 ψi but does not correspond to any observed symmetry in the hadron spectra;
this is directly related to the so called “U (1) problem” which we will briefly mention later
on.
There remain two approximate symmetries of the QCD Lagrangian to be discussed;
isospin symmetry and, most important, chiral symmetry. Let us start with the first one.
As can be seen in Table 1, the masses of the u and d quarks are considerably smaller than
those of the rest of the quarks. Hence, at low energies, we can ignore all but the u and d
quark. The fermionic part of the Lagrangian (1.1) will take the form:
Lud = ū (iγ µ Dµ − mu ) u + d¯(iγ µ Dµ − md ) d ,
(1.15)
where the covariant derivative now is re-evaluated to contain the gluon fields evaluated
from equation (1.7) ignoring all but the up and down quark. We can combine those quark
fields in an isospin doublet, or isodoublet, as follows:
Ã
Q (x) =
u (x)
d (x)
!
.
(1.16)
mu + md
mu − md
Q̄Q −
Q̄σ 3 Q ,
2
2
(1.17)
The Lagrangian is then written as:
Lud = Q̄iγ µ Dµ Q −
where σ 3 is the Pauli matrix. Now the Lagrangian possesses two global U (1) symmetries:
0
Q → Q0 = e−iα Q ,
(1.18)
and
1
Actually this symmetry is exact only within the framework of the strong forces, as weak interactions
can change the flavor. In the Standard Model the weak forces play an important role; if we include the part
involving the Cabibbo-Kobayashi-Maskawa mixing matrix these six symmetries reduce to one, leading to
conservation of the overall quark number only.
1.1 The QCD Lagrangian and symmetries
8
3 σ3
Q → Q0 = e−iα
Q.
(1.19)
The so called isospin symmetry appears if we neglect the mu − md term which is negligible
since mu , md are very small. The isospin symmetry is a global SU (2) transformation:
k σk Q
Q → Q0 = e−iα
.
(1.20)
a
The current associated with this is symmetry via Noether’s theorem is Jaµ = Q̄γ µ σ2 Q.
Finally, we can turn to chiral symmetry. For clarity’s sake we will first exhibit chiral
symmetry in the two flavor case. First we need to define the left and right handed spinors
via the action of the left and right projection operators respectively:
1 − γ5
1 + γ5
, PR =
(1.21)
2
2
If we ignore the mass terms in the Lagrangian (1.15), and use the doublet Q introduced
above, the fermionic part of the Lagrangian can then be written as:
PL =
χ
Lud
= ūL iγ µ Dµ uL + ūR iγ µ Dµ uR + d¯L iγ µ Dµ dL + d¯R iγ µ Dµ dR
= Q̄L iγ µ Dµ QL + Q̄R iγ µ Dµ QR ,
(1.22)
2
where QL,R = PL,R Q. The projection operators further satisfy PL,R
= PL,R , PL + PR = 1,
PL PR = 0. The Lagrangian (1.22) is symmetric under U (2)L × U (2)R , which means
it is invariant under separate unitary 2 × 2 transformations that act on the left and
right components of the quark doublet. This symmetry is called chiral symmetry. The
generalization to more flavors is straightforward. Suppose Q is a multiplet of nf flavors,
and TL,R ∈ U (nf ) are U (nf ) transformations acting on QL,R such that QL → TL QL and
QR → TR QR . Then the Lagrangian (1.22) is invariant under U (nf )L × U (nf )R . The
original spinors Q, Q̄ would transform according to:
Q = QL + QR → Q0 = TL QL + TR QR = (TL PL + TR PR ) Q = T Q ,
(1.23)
³
´
Q̄ = Q̄L + Q̄R → Q̄0 = Q̄R TL† + Q̄L TR† = Q̄ TL† PR + TR† PL = Q̄T̄ .
(1.24)
and
The transformations for which TL = TR are called vector transformations. The special
case TL = TR† are the chiral transformations introduced above. Note that the general form
k k
of such transformations in the 2 × 2 case is q → q 0 = e−iσ θ γ5 /2 q.
1.2 Spontaneous Symmetry Breaking and the Nambu–Goldstone bosons
9
If one wishes to include the mass terms of the Lagrangian, the chiral and flavor symmetries are broken. A flavor symmetric mass term, m̂ = m1f , implying degenerate quarks
of equal mass, would leave the action invariant under vectorial flavor transformations
but the chiral symmetry would still be broken, unless m = 0, a case referred to as the
“chiral limit”. The U (nf )L × U (nf )R symmetry is broken down due to anomalies to
SU (nf )L × SU (nf )R × U (1)V .
1.2
Spontaneous Symmetry Breaking and the Nambu–Goldstone
bosons
Spontaneous Symmetry Breaking is one of the central aspects of the Standard Model and
Particle Physics. Detailed references can be found in any Quantum Field Theory text, like
[12, 24, 36]; here we’ll just summarize the important results and their applications to chiral
symmetry breaking. We begin by Noether’s theorem which is an essential ingredient. It
is generally known in physics that conservation laws can be attributed to symmetry principles. Familiar examples at the classical level are translational invariance in time versus
energy conservation or translational invariance in space versus momentum conservation.
At the quantum level the connection between conservation laws and symmetry is described
by Noether’s theorem: for a system described by the Lagrangian:
Z
L = d3 x L (φi (x), ∂µ φi (x)) ,
(1.25)
with the equation of motion given by the Euler-Lagrange equations:
∂µ
δL
δL
−
= 0,
δ (∂µ φi )
δφi
every one parameter continuous symmetry transformation that leaves the action S =
invariant implies the existence of a conserved current
∂ µ Jµ (x) = 0 .
(1.26)
R
Ldt
(1.27)
The corresponding charge is defined by:
Z
Q(t) =
d3 x J0 (x)
(1.28)
and is a constant of motion,
dQ
= 0.
dt
(1.29)
1.2 Spontaneous Symmetry Breaking and the Nambu–Goldstone bosons
10
Suppose the Lagrangian is invariant under some Lie group G, that is, under a transformation of the form
φi (x) → φ0i (x) = e−i²
a ta
φ(x) ,
which we take to be a global one, i.e. the ²a are space-time independent. The ta ’s are a
set of matrices satisfying the Lie algebra of the group G:
h
i
ta , tb = if abc tc ,
(1.30)
where f abc are the structure constants of the group. Assuming the action invariant under
such transformations, which implies δL = 0 one can derive the conserved currents:
Jµa = −i
δL
ta φj (x) .
δ (∂ µ φi (x)) ij
(1.31)
for which it holds that ∂ µ Jµa = 0. The conserved currents are then given by
Z
a
Q (t) =
d3 x J0a (x)
and are the generators of the symmetry group, i.e. they satisfy the same algebra:
h
i
Qa , Qb = if abc Qc .
(1.32)
These commutation relations are useful for classifying particle states. In fact similar
commutation relations hold even in the presence of symmetry breaking terms. Consider
a Lagrangian of the form
L = L0 + L1 ,
(1.33)
where the term L0 is invariant under the symmetry group while L1 is not. Even in this
case we can define the currents in the same way as before, only that now they’re not
conserved. The charges are also defined in the same way as before only that now they’re
time-dependent:
Z
a
Q (t) =
Z
d
3
x J0a (x)
= −i
d3 (x)
δL a
t φj
δ(∂ 0 φi ) ij
Due to the fact the conjugate canonical momentum is still defined by πi (x) = δL /δ(∂ 0 φi )
and satisfies the canonical commutation relations:
[πi (~x, t) , πj (~y , t)] = −iδ 3 (~x − ~y ) δij ,
(1.34)
1.2 Spontaneous Symmetry Breaking and the Nambu–Goldstone bosons
11
we can still write down charge algebras, even by not knowing the actual form of L1 . This
is done in detail in [12]. One derives:
h
i
Qa (t) , Qb (t) = if abc Qc (t) .
(1.35)
This means that even if the charge is time dependent, a any given time t the group algebra
is still satisfied; it is called charge algebra in this occasion. In the following we will examine
the free SU (3)f case for QCD, which is of special interest to us.
Suppose we are only concerned about the three lightest quarks. We can the define the
flavor triplet:

u (x)


ψ =  d (x)  .
s (x)

(1.36)
Next we consider a global SU (3)f transformation, which in infinitesimal form is written
as:
ψi →
ψi0
µ
¶
λa
a ij
= δij + iα
ψj ,
2
(1.37)
Since this is still the free model in which gluons do not exist, the covariant derivative in
equation (1.1) has not yet been promoted to a covariant one. The Lagrangian density has
the form :
L = L0 + L1
¯ + ms s̄s .
= ψ̄iγ µ ∂µ ψ + mu ūu + md dd
(1.38)
The symmetry is broken by the mass terms, assuming they’re not equal. If they were
absent the symmetry invariance would be restored, and in fact L0 is invariant under a
larger group, containing the axial transformations, written in infinitesimal form:
ψi →
ψi0
µ
¶
λa
a ij
= δij + iβ
γ5 ψj .
2
(1.39)
Following the previous analysis, we can define a vector and an axial current associated
with those transformations:
λa
ψ (x) ,
2
(1.40)
λa
γ5 ψ (x) .
2
(1.41)
Vµa (x) = ψ̄ (x) γµ
Aaµ (x) = ψ̄ (x) γµ
1.2 Spontaneous Symmetry Breaking and the Nambu–Goldstone bosons
12
and the corresponding charges:
Z
a
Q (t) =
Z
3
d x
V0a (x)
;
5a
Q
(t) =
d3 x Aa0 (x) .
(1.42)
One can now calculate the equal time commutation relations as in equation (1.35), where
the structure constants refer to SU (3). However, we can define left and right-handed
charges and show that those charges reproduce the chiral SU (3)L × SU (3)R algebra.
Writing
¢
¢
1¡ a
1¡ a
Q − Q5a
; QaR =
Q + Q5a ,
(1.43)
2
2
we can see that the left handed charges generate the ’left’ part of the symmetry and
similarly for the right handed charges:
QaL =
h
i
QaL (t) , QbL (t) = if abc QcL (t) ;
h
i
QaR (t) , QbR (t) = if abc QcR (t) ;
h
i
QaL (t) , QbR (t) = 0 .
(1.44)
One can extend the charge algebra and define a current algebra. This is done by just
calculating the equal time commutation relations for the currents. It is important to note
that the symmetry currents (also referred to as Noether currents) are the ones entering
the Lagrangian (or linear combinations there of).
Finally we’re able to discuss SSB and the Goldstone theorem. The concept of a spontaneous symmetry breaking refers to the case that a Lagrangian or Hamiltonian is invariant
under some symmetry group G, but the ground state of the theory does not respect the
symmetry: if U ∈ G, and the Hamiltonian H is invariant under G, that is if
U HU † = H ,
then the symmetry is spontaneously broken if U |0i 6= |0i. An equivalent statement is that
the vacuum expectation values of some field operators are different than zero: h0|φi |0i 6= 0,
which is used more often, or even that the Noether charges do not annihilate the vacuum:
Qa |0i 6= |0i. Note that if the Lagrangian density is given, L = L0 − V , where V is the
potential of the system, then the ground state of the quantum theory will be the value
of the field (or fields) where the minimum of the potential is. The implication of the
spontaneously broken symmetry are then described by the Goldstone theorem:
• The Goldstone theorem
For every spontaneously broken continuous symmetry, the theory must contain a
1.3 PCAC and the Goldberger - Treiman relation
13
massless particle. The new massless particle is called Nambu–Goldstone boson. The
number of Goldstone bosons emerging is equal to the number of linearly independent
transformations that are broken.
As an example we can consider QCD with massless quarks.
The breaking of
SU (3)L × SU (3)R symmetry is such that the vacuum is only SU (3)V invariant; according
to the Goldstone theorem there should be eight massless Goldstone bosons, as the number
of generators (which equals the number of linearly independent transformations) is equal
to eight2 . Finally we note that the Nambu–Goldstone (NG) bosons emerge in case of a
global symmetry breaking, as stated earlier. In case the symmetry is a local gauge one,
the Meissner-Anderson-Higgs mechanism is triggered and the would-be NG bosons become
the longitudinal polarization of massive vector particles.
1.3
PCAC and the Goldberger - Treiman relation
As we already discussed, the partial symmetries of the Lagrangian or the Hamiltonian of a
theory are reflected in the current algebra, even in the case of the existence of a symmetry
breaking term in the Lagrangian. In case of a spontaneously broken symmetry the particle
spectrum that is realized is the part of the original symmetry that is also respected by the
ground state. In case of the SU (3)L × SU (3)R breaking down to SU (3)V , there should be
eight massless pseudoscalar mesons associated with the eight spontaneously broken axial
charges Q5a , equation (1.42). In reality there are no such massless particles observed;
instead eight light mesons, the pions, π, the kaons K and the η mesons are observed, out
of which the pions are the lightest; still this suggests that the SU (3)L × SU (3)R symmetry
must be spontaneously broken. Thus, we can write the Hamiltonian of the theory as:
H = H0 + λH 0 ,
(1.45)
where H0 is invariant under SU (3)L × SU (3)R while H 0 is not. Furthermore, since the
pions3 are lighter than the kaons, which contain the s quark, we can further decompose
the symmetry breaking term as:
λH 0 = λ1 H1 + λ2 H2 ,
(1.46)
where H1 is invariant under SU (2)L × SU (2)R and λ1 À λ2 . Due to the fact that
¯ and λ2 H2 = ms s̄s. In the rest
mu , md ¿ ms we can identify λ1 H1 = mu ūu + md dd
2
The number of generators for SU (N ) is N 2 − 1.
there exist three pions: π 0 which is a uū and a dd¯ combination, π − which has quark content of dū and
¯
which is ud.
3
π+
1.3 PCAC and the Goldberger - Treiman relation
14
we shall briefly mention the Goldberger-Treiman relation, which is a soft pion low energy
theorem, without giving any proof. The reader may consult [12, 24] and references therein
for more details.
Let us assume that λ1 = 0, which is the chiral symmetry limit. The pions have odd
parity and can be created by the axial isospin currents. Denoting the pion state by π b (p),
b = 1, 2, 3, one can show that the matrix element of the axial current in equation (1.41)
between the vacuum and a pion state of four-momentum p can be written as:
¯
¯
h0 ¯Aaµ (x)¯ π b (p)i = ipµ fπ δ ab e−ipx ,
or
¯
¯
h0 ¯Aaµ (0)¯ π b (p)i = ipµ fπ δ ab
(1.47)
where fπ is the pion decay constant in the π + → l+ + νl weak decay. It is measured
experimentally to be fπ ≈ 92.4 M eV . By taking the divergence we obtain:
¯
¯
h0 ¯∂ µ Aaµ (0)¯ π b (p)i = fπ m2π δ ab .
(1.48)
Then, with λ1 = 0, the chiral SU (2)L × SU (2)R symmetry of the Hamiltonian is exact
and the axial current is conserved, ∂ µ Aaµ = 0. According to equation (1.48) this implies
that mπ = 0, as required by Goldstone’s theorem.
If the symmetry however is explicitly broken, λ1 6= 0, the axial current is not conserved.
In this case
¯
¯
h0 ¯∂ µ Aaµ (0)¯ π b (p)i = fπ m2π h0 |φa (0)| π b (p)i .
where φa is the pion field operator and h0 |φa (0)| π b (p)i = δ ab . We are lead to the partially
conserved axial current hypothesis (PCAC):
∂ µ Aaµ = fπ m2π φa .
(1.49)
Furthermore, by using the explicit form of the axial vector current and the equation of
←
−
motion, in which case read iγ µ Dµ = m̂q, −iq̄ D µ γ µ = q̄ m̂, where m̂ = diag(mu , md ), one
can derive, see [36], chapter 19:
m2π = M 2
mu + md
,
fπ
(1.50)
where the mass parameter M is of the order of 400M eV . Thus, to give the pions a mass
of about 135M eV we need mu + md ≈ 10M eV . A detailed report regarding quark masses
can be found in [18].
1.3 PCAC and the Goldberger - Treiman relation
15
A useful formula relating quantities from the strong and weak interactions is the Goldberger - Treiman relation. It translates the PCAC hypothesis into relations connecting
physically measurable quantities, however additional assumptions have to be made. A
proof of it can be found in [24, 18]; for our purpose it suffices to say that it involves
calculating the matrix element of the axial vector current between nucleon states in low
energy pion-nucleon interactions. The latter are parametrized by a term in an effective
Lagrangian of the form:
LπN N = igπN N π a N̄ γ 5 σ a N ,
where gπN N is the effective pion-nucleon coupling constant. If gA is the nucleon - axial
vector current coupling, then it holds, see [12], chapter 5:
fπ gπN N (0) = mN gA (0) ,
where mN is the nucleon mass. In this formula gπN N (p) is the on-shell πN N vertex
function at zero momentum transfer. The physical pion-nucleon coupling constant is then
defined as gπN N = gπN N (m2π ). In order to have a formula applicable to the real world,
where pions are massive, we need to make a further assumption that the function gπN N (q 2 )
is smooth, such that we can extrapolate from the value q 2 = 0 to q 2 = m2π and arrive at
the Goldberger-Treiman relation:
fπ gπN N = mN gA (0) ,
(1.51)
The value of gA (0) is known experimentally to be gA ≈ 1.26. Equation (1.51) is satisfied
experimentally with 10% accuracy.
The same analysis leads to a generalized version of the Goldberger - Treiman relation. In
the SUL (3) × SUR (3) case the PCAC relation would read:
∂ µ Aaµ = fa m2a φa ; a = 1, 2, . . . , 8.
(1.52)
The fields φa represent the pseudoscalar mesons. More interesting low energy soft meson
theorems can be derived by considering other matrix elements of the currents.
Before concluding this section, we note some useful mass relations that are obtained
in the SU (3)L × SU (3)R case4 . Proofs can be found in [12] and [18].
4
It is referred to as the (3, 3∗ ) + (3∗ , 3) theory of chiral symmetry breaking, proposed by Gell-Mann,
Oakes, Renner and by Glashow and Weinberg.
1.4 Asymptotic freedom and color confinement
mu + md
¯ |0i
h0| ūu + dd
2
mu + ms
2
fK
m2K =
h0| ūu + s̄s |0i
2
mu + md
¯ |0i + 4ms h0| s̄s |0i .
fη2 m2η =
h0| ūu + dd
6
3
fπ2 m2π =
16
(1.53)
(1.54)
(1.55)
This formulas are known as the Gell-Mann–Oakes–Renner relations. We shall need them
later on, adapted to the NJL model. For an SU (3) symmetric vacuum h0|ūu|0i =
¯
h0|dd|0i
= h0|s̄s|0i ≡ µ3 and it can be shown that fπ = fK = fη = f which leads to
the Gell-Mann –Okubo mass relation:
4m2K = 3m2η + m2π ,
(1.56)
as well as to quark mass ratio formulae:
mu + md
m2π
1
=
≈
.
2
2
2ms
25
2mK − mπ
(1.57)
It should be emphasized that the quark masses which enter the above formulas are current algebra quark masses and are the masses appearing in the Lagrangian. They are
different from the constituent quark masses which appear in non-relativistic bound state
calculations for hadrons. See also [40].
1.4
Asymptotic freedom and color confinement
Last but not least we discuss two very important features of QCD, namely asymptotic
freedom and color confinement. For a deeper discussion on asymptotic freedom one would
have to talk about the renormalization group and the Callan - Symanzik equation, something that is outside the scope of this presentation. Here we merely summarize the most
important results, omitting the field - theoretic details and focusing on a more qualitative
description.
Asymptotic freedom refers to the phenomenon that the effective strong interaction
coupling constant becomes so small at short distances (or large momenta) that quarks
and gluons are considered asymptotically free. It was discovered in the early 70’s and and
in fact it was shown that asymptotic freedom is a general property of Yang-Mills theories.
The latter are described by a Lagrangian of the form:
1 a µν a
LY M = ψ̄ (iγ µ Dµ − m) ψ − Fµν
F
4
(1.58)
1.4 Asymptotic freedom and color confinement
17
of which QCD is a special case. In the beginning of this chapter we defined g as the QCD
coupling. Here, we define the strong coupling αs as:
g 2 (Q)
,
(1.59)
4π
in analogy with QED. It can be shown, see [12, 36, 40], that up to one loop order in
perturbation theory, αs has the following dependance on the momentum:
αs (Q2 ) =
¡ ¢
gs Q2
4π
αs =
=¡
¢ ³ 2´ ,
2
4π
11 − 3 nf ln Q
Λ2
(1.60)
where the parameter Λ which is called scale parameter has dimension of a mass and is the
fundamental momentum scale in QCD. It’s value is a measure for the energy scale where
the strong interactions coupling constant is large. As can be seen from the above formula,
αs → ∞ when Q2 → Λ2 . Asymptotic freedom is displayed as αs → 0 when Q2 becomes
very large. Note that equation (1.60) is a perturbative result; perturbation theory applies
in the regime where the coupling is small, that is in short distances or equivalently large
momentum transfers.
However, at distances equal to the hadron size, the coupling is strong and perturbation
theory cannot be applied anymore. In order to explore this regime one would need a nonperturbative approach to QCD. Lattice gauge theory, or Lattice QCD in this case, is such
an approach, see [40] and references therein. In lattice gauge theory the space-time continuum is discretized as the name suggests; it is replaced by a four-dimensional Euclidean
lattice. It is in the context of this theory that quark confinement is best understood, even
though the exact mechanisms of confinement are not perfectly understood even today. In
fact it is shown that all gauge theories with a compact gauge group (such as SU (3)) have
the property of confinement at sufficiently strong coupling: quarks can only exist confined
in bound states.
A useful qualitative picture for confinement emerges in the context of lattice gauge
theory, by considering the theory with gluons only. Static external sources with quark
numbers are then inserted, separated by a distance r. In QCD confinement is realized in
such a way that the quark-antiquark potential V (r) is proportional to the distance r:
V (r) ≈ σr , r → ∞ ,
(1.61)
where σ is called the string tension. The name is due to effective ‘string models’ for V (r).
Thus, the energy cost of separating color sources grows proportionally with distance. We
can also imagine ‘chromoelectric’ field lines between quark-antiquark pairs, in analogy with
the case of two opposite charges in electrodynamics. At large distances these lines will
form a tube with constant radius and energy density between the quark-antiquark pair.
1.5 Quark-Gluon Plasma
18
In this sense the q q̄ potential is said to be string-like at large distances5 . It is because
of this color confinement that quarks and gluons cannot exist as free particles and have
never been observed.
1.5
Quark-Gluon Plasma
In the 1970’s, after it was clear that hadronic matter consists of quarks and gluons and
the asymptotic behavior of the strong forces had been understood, it was suggested that
there might be a deconfined phase of quarks and gluons, at sufficiently high temperatures
and densities, the so-called ‘quark-gluon plasma’ (QGP). The arguments were roughly
that the infrared problems, which are related to long-range interactions, are absent in a
dense medium due to screening, and that the interaction is weak enough for a perturbative
approach. It was suggested that the quark model implies that super-dense states of matter,
e.g., in neutron stars and the early universe, should be composed of deconfined quarks,
rather than hadrons, see
An argument for the existence of deconfined quark matter follows directly from the
asymptotic behavior of the strong interactions. Since the strong force becomes arbitrarily weak as the quarks are squeezed closer to each other, matter should behave as an
ideal Fermi gas of quarks at asymptotically high densities and/or temperatures. A phase
transition from the confined hadronic phase to the deconfined phase of QGP is therefore
expected at sufficiently high temperatures, T , or quark number chemical potentials µ. The
possibility that Cooper pairs of quarks could form in dense QGP in analogy with Cooper
pairs in ordinary superconductivity was suggested already in the mid 1970s, however, the
idea did not get much attention until recently. Since quarks carry color instead of electrical
charge, this new alleged for of superconductivity was named ‘color superconductivity’.
Up to now there have been many attempts to experimentally create QGP, via heavy
ion collisions and there is still a number of ongoing experiments at the Relativistic Heavy
Ion Collider at the Brookhaven National Laboratory and CERN; in fact there were cases
when experimentalists thought they had created QGP, however the validity of their claim
is still ambiguous. The dedicated experiment ALICE which will run at the LHC (Large
Hadron Collider) at CERN is expected to have better chances in the formation of QGP. In
nature, QGP is thought to have existed in the early universe, a few microseconds after the
big bang, when temperature and density were high enough. Another possible scenario is
that QGP may exist in the interior of neutron stars, where the density is extremely high.
5
For the case of compact U (1) as a gauge group a Coulomb-like potential is predicted for short distances.
One speaks of a Coulomb and a confining phase, with a phase transition at the critical coupling, where the
string tension as a function of the coupling vanishes. There is no phase transition for SU (2) and SU (3)
theories.
1.5 Quark-Gluon Plasma
We will focus on the interior of neutron stars later on, in chapter 5.
19
Chapter 2
The Nambu-Jona–Lasinio model
As was outlined in the previous chapter, the behavior of the strong interactions changes
dramatically depending on the distances we’re trying to probe. QCD shows a weak coupling at short distances, allowing for asymptotic freedom, and a strong coupling at large
distances. The main means of exploring the physics of QCD are perturbation theory and
lattice simulations. However, there is another alternative, specially useful in regimes unaccessible to the previous methods, which is none other than simpler or ‘effective’ as they
are typically called, models. Such a model must display at least some of the essential
features of the theory it is supposed to simplify, yet it should be mathematically simpler
to work with. Then we should be able to investigate the consequences of the features we
have effectively isolated in the model. It basically consists in making ‘toy models’ of the
real theory we’re trying to study, in order to get an understanding of the physics involved.
The Nambu-Jona–Lasinio, or NJL, model is one such low-energy effective model for QCD.
2.1
Introduction - historical remarks
The NJL model, which was constructed in 1961, was, in it’s original form, see [34, 35], a
pre-QCD theory of nucleons that interact via an effective two-body interaction. At those
times quarks and QCD had not been conceived yet and thus they were using nucleons as
the building blocks of matter. However there were indications for the existence of partially
conserved axial currents and chiral symmetry. The NJL model was used to explain the
large nucleon mass, since approximate chiral symmetry, as we also saw, implies almost
massless fermions. The pioneering idea of Y. Nambu and G. Jona-Lasinio was that the
mass gap in the spectrum of the nucleon can be generated in a way similar to the energy gap
in the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity, which was developed
four years earlier, see [4, 16]. They used a Lagrangian of the form:
2.1 Introduction - historical remarks
L = ψ̄ (iγ µ ∂µ − m) ψ + G
21
h¡ ¢
¡
¢2 i
2
ψ̄ψ + ψ̄iγ5~σ ψ
,
where ψ is a nucleon field with point-like, chirally symmetric four-fermion interaction,
σ i are the Pauli matrices acting on flavor (isospin) space. Within this model, the self
energy of the interactions gives rise to a mass M , which is considerably larger than m and
remains so even in the chiral limit. The pions emerged as the Nambu-Goldstone bosons
of the spontaneously broken chiral symmetry for the case of two flavors, while the “pion
octet” for three flavors can be explicitly traced.
After the emergence of QCD, the NJL model was reinterpreted as an effective, schematic
in a way quark model. As an effective model, it carries the symmetries of QCD that are observed in nature and it should provide a tractable basis for the study of symmetry breaking
and the relevant fermion masses (obtained via chiral symmetry breaking). Furthermore,
the features of QCD which are consequences of symmetry properties only, should be also
be valid for NJL models. As such, the Goldberger-Treiman and other results of current
algebra should also and indeed do hold for NJL models.
In its third era, starting roughly at the end of the nineties, the Nambu-Jona–Lasinio
model has been employed to study ’color’ superconducting phases in deconfined quark
matter; it is this view of the NJL model that we will be mostly involved with in this
thesis. The actual mechanism of chiral symmetry breaking that occurs in the NJL model
resembles the BCS theory of superconductivity. In the NJL model one can argue that the
quark-antiquark interaction which arises from some complicated gluon exchange process,
can be attractive thus leading to a q q̄ pair condensation in vacuum.
Let us turn now to the shortcomings of the NJL model. As already mentioned, the
interactions between fermions are taken to be point-like, something that does not lead to
confinement. There are of course questions for which confinement may not be important,
such as color superconductivity. Another shortcoming that is also related to the local
character of the interactions is that the model is not renormalizable, hence, in order to
have the NJL as an effective model we need a regularization scheme. A regularization
scheme will define a length scale for the theory and may be expressed as a cutoff in the
quark momenta.
We should note that in the literature one can come across many different versions of
the NJL model; we will be referring to such models collectively as NJL models, provided
they describe quark interactions via four-point vertices, or n-point vertices in general1 . A
detailed list of the authors involved in studying the NJL model can be found in [28]. Here
we just note the most relevant articles related to the material in this thesis. We will be
working in the three flavors case, as our objective is to study the so-called ‘color flavor
1
For the Nf = 3 NJL models there are the so called ’t Hooft interactions which are 6-point vertices.
2.2 The NJL model
22
locked’, or CFL, phase which occurs in the three-flavor models. The U (3) flavor model
was first introduced by Kikkawa in 1976, see [27], while Hatsuda and Kunihiro, [30] as
well as Bernard, Jaffe and Meissner, [7] studied the SU (3) case. The last group of authors
has also initiated the discussion of finite temperature and density effects, [6].
2.2
The NJL model
In this section we introduce the NJL model. We will focus one the three flavor case,
however we present briefly the two flavor model as well. We’re interested in the constituent
quark masses, symmetry properties and a regularization scheme.
2.2.1
The Nf = 2 model
Starting with simplest case, we note the mostly used Lagrangian:
LNJL = ψ̄ (iγ µ ∂µ − m) ψ + G
h¡ ¢
¡
¢2 i
2
ψ̄ψ + ψ̄iγ5 σ i ψ
.
(2.1)
Here ψ stands for a quark field with two flavor and three color degrees of freedom. G is
the coupling constant, with dimension [G] = −2 in units of mass and σ i are the Pauli
matrices acting on flavor space. Writing out the Dyson equation for the selfenergy, i.e.
S(p) = S0 (p) + S0 (p)Σ(p)S(p), where S0 (p) = (γ µ pµ − m + i²)−1 is the bare propagator
and S(p) = (γ µ pµ − M + i²)−1 is the dressed propagator, Σ(p) is the self-energy in the
Hartree approximation, one gets:
Z
M = m + 2iG
d4 p
Tr S(p) = m + 8Nf Nc Gi
(2π)4
Z
d4 p
M2
,
(2π)4 p2 − M 2 + i²
(2.2)
where the trace appears due to the fermion loop and is taken over flavor, color and Dirac
indices, which are suppressed. Here Nf = 2, Nc = 3.
Figure 2.1 Diagrammatic representation of the Dyson equation. The bold lines denote the dressed
propagator and the thin ones the bare propagator.
2.2 The NJL model
23
Another important quantity is the ‘quark condensate’, which is given by the expectation
value:
Z
hψ̄ψi = −i
d4 p
Tr S(p) ,
(2π)4
(2.3)
which, combined with equation (2.2) gives:
M − m = −2Ghψ̄ψi .
(2.4)
In analogy with BCS superconductivity, equation (2.2) is called the ‘gap equation’ for the
mass. A few comments are in order. First we should comment that in the rest we will be
interested in the effects of finite temperature and chemical potential, in which case we will
derive the gap equation via the grand canonical potential in a mean field approximation.
However, equation (2.2) is useful in calculating the constituent quark mass M . Second, as
can be seen from the Lagrangian (2.1), we have two different types of four fermion vertices;
¡
¢2
the term G(ψ̄ψ)2 gives rise to a scalar vertex, while the term G ψ̄γ5 σ i ψ gives rise to
a pseudoscalar vertex. The self-energy is written in the Hartrre-Fock approximation as a
sum of self-energies of Hartree and Fock contributions, for each vertex separately. Thus
the self-energy, in coordinate space, is:
Σ = ΣH + ΣF
¡
¢ ¡
¢
= ΣSH + ΣPH + ΣSF + ΣPF
= 2iG [Tr S (x, x) − S (x, x)] + 2iG (iγ5 σ) [Tr (S (x, x) (iγ5 σ)) − S (x, x) (iγ5 σ)] , (2.5)
where H, F denote the Hartree and Fock terms respectively, S, P denote ‘scalar’ and
‘pseudo-scalar’ respectively. The factors of 2 are there because of the two different ways
one can connect external lines to a four fermion vertex. The propagator is defined as
iS(x, y) = hT ψ(x)ψ̄(x0 )i and obeys the equation
(iγ µ ∂µ − Σ) S(x, y) = δ 4 (x − y) ,
where differentiation is with respect to x. This equation has a solution in momentum
space,
S(p) =
γµp
1
.
µ − M + i²
Plugging the last equation in (2.5) and by using properties of traces of gamma matrices we
arrive at equation (2.2). We should further note that it has been shown, see [28], that in
the large Nc limit the Fock contribution can be ignored and in this sense the Hartree-Fock
2.2 The NJL model
24
approximation is just as good as the Hartree approximation. ’t Hooft [citation needed]
has shown that in this limit the NJL model follows from the QCD Lagrangian and can be
regarded as an effective theory of interacting quarks.
2.2.2
The Nf = 3 model
Let us turn to the three flavor model. The most commonly used Lagrangian is, [7, 10, 30]2 :
LNJL = ψ̄ (iγ µ ∂µ − m̂) ψ + Lsym + Ldet
£
¤
= ψ̄ (iγ µ ∂µ − m̂) ψ + G (ψ̄λa ψ)2 + (ψ̄iγ5 λa ψ)2
¾
½
¤
£
¤
£
,
− K det ψ̄(1 + γ5 )ψ + det ψ̄(1 − γ5 )ψ
f
f
(2.6)
where ψ = (u, d, s)T , is the quark field with three flavors, m̂ = diagf (mu , md , ms ) is the
corresponding mass matrix. G and K denote the couplings for the interactions. The
latter are of two types, given by Lsym and Ldet respectively. Lsym is a generalization of the
interactions of the Nf = 2 Lagrangian, and has a U (3)L × U (3)R symmetry.
q The matrices
λa , a = 1, . . . , 8 are the generators of SU (3)f . We also introduce λ0 = 23 1f . The other
interaction term, Ldet , corresponds to instanton-induced interactions and is a determinant
in flavor space, denoted by the subscript f . It was proposed by ’t Hooft in order to remove
the U (1)A symmetry that otherwise would be present, [28]. It is in general a 2Nf -point
interaction and for three flavors it leads to a 6-point vertex. To see that, consider that:
det(ψ̄Oψ) =
f
X
¯ j )(s̄Oψk ) ,
²ijk (ūOψi )(dOψ
(2.7)
i,j,k
where i, j, k are flavor indices. In the mean field approximation, in which we will be
working later on (see chapter 4), the 6-point vertex is turned into effective 4-point vertices
by closing two quark lines to form a quark loop contributing a factor of one of the chiral
condensates hψ̄i ψi i, i = u, d, s or combinations of them, see [29]. The presence of this
term is phenomenologically important in order to get the correct mass splitting for the
η, η 0 mesons, [10], while most probably it only affects the position of the phase boundaries
of the phase diagram, even though the full contribution of this term has not yet been
fully explored, [10]. Absence of this term implies no flavor mixing in the quark-antiquark
channel. We will omit this term in the calculations leading to the phase diagram in chapter
4.
2
A complete list of all the possible terms respecting the QCD symmetries for Nf = 3 can be found in
[29].
2.3 Fierz identities
25
The Lagrangian in equation (2.6) is the simplest most general case of a Lagrangian
respecting the QCD symmetries and possessing four and six point fermion interactions,
see [29] for details.
Dyson Equation and the Gap Equation for the Mass
Similar to the two flavor case, we can write down Dyson’s equation the self energy for
this case as well. The main difference is that the 6-point vertex will give two quark loops
instead of one. The relevant diagram can be seen in figure 2.2.
Figure 2.2 Diagrammatic representation of the Dyson equation, including the ’t Hooft term. The
bold lines denote the dressed propagator and the thin ones the bare propagator. Two quark loops are
present.
In a similar fashion to the previous paragraph, by writing out the Dyson equation we can
get the gap equation for the mass (or the chiral condensates). It reads, [10]:
Mi = mi − 4Gφi + 2Kφj φk ,
(2.8)
where i, j, k can be any one of the u, d, s quarks and φi = hψ̄i ψi i.
2.3
Fierz identities
In this section we examine the Fierz transformations and their application to the NJL
model, following [10, 28, 29]. The Fierz transformation is a technical device that enables
us to examine the effect of reordering fermionic field operators appearing in products of
Dirac bilinears. Consider the Lagrangian:
¡
¢2
Lint = gI ψ̄ΓI ψ = gI ΓIij ΓIkl ψ̄i ψj ψ̄k ψl ,
(2.9)
2.3 Fierz identities
26
where I is some channel (scalar, pseudoscalar, vector, axial vector, tensor) and Γ is some
local operator in Dirac, flavor and color space. Denoting by F the operator that performs
the Fierz transformation, we can rewrite this as:
F (Lint ) = −gI ΓIij ΓIkl ψ̄i ψl ψ̄k ψj = gI ΓIij ΓIkl ψ̄i ψ̄k ψl ψj ,
|
{z
} |
{z
}
Lex
(2.10)
Lqq
where we only used the fact that the ψ fields anticommute. If we restrict ourselves to
the Hartree approximation where the first field is contracted to the second and the third
to the fourth, then the first form of Lint in the equation above gives us the ‘exchange’
diagrams (Fock terms), while the second one corresponds to particle-particle or antiparticle
- antiparticle contributions. The original Lagrangian then is the ‘Hartree’ Lagrangian and
contains the ‘direct’ diagrams3 . By demanding that the Fierz-transformed Lagrangian is
equivalent to the initial, Lint we wish to rewrite the operators Γ as:
ΓIij ΓIkl =
X
I M M
CM
Γil Γkj ,
(2.11)
M
I . This way:
which implies we need to know the coefficients CM
Lex = −gI
X
I M M
CM
Γil Γkj ψ̄i ψl ψ̄k ψj = −gI
X
¡ M ¢2
I
CM
ψ̄Γ ψ .
(2.12)
M
M
The total quark-antiquark interaction would be the sum of the original plus the Fierztransformed Lagrangian:
Lq̄q = Lint + F (Lint ).
(2.13)
Later on we will be mostly interested in the so-called diquark condensates, for the introduction of which we need quark-quark interactions in our Lagrangian. To this end we can
write equations similar to the above but in the particle-particle channel:
ΓIij ΓIkl =
X
D
D
dID ΓD
ik Γlj =
X
¡
¢ ¡
¢
dID ΓD C ik CΓD lj .
(2.14)
D
Then Lqq introduced above can be written as:
Lqq = gI
X
D
i
X
¤h ¡
¢
£ ¡
¢
¡
¢ ¡
¢
dID ΓD C ik CΓD lj ψ̄i ψ̄kT ψlT ψj =
gI dID ψ̄i ΓD C ik ψ̄kT ψlT CΓD lj ψj .
| {z }
D
HD
(2.15)
The coefficients cIM and dID in the various channels are listed in [10]:
3
The terms ‘Hartree’, ‘Fock’ and ‘Hartree-Fock’ come from the context of Statistical Field Theory, see
for example [16].
2.3 Fierz identities
27
• quark-antiquark channel (corresponds to Lex ):
In Dirac space:








(11)ij (11)kl
(iγ5 )ij (iγ5 )kl
(γ µ )ij (γµ )kl
(γ µ γ5 )ij (γµ γ5 )kl
(σ µν )ij (σµν )kl


1
4
− 14
 
 
 
= 1
 
 
  −1
3
In U (N ) (flavor
or color); here λa ,
q
and λ0 = N2 11:
Ã
(11)ij (11)kl
(λa )ij (λa )kl
!


1
(11)il (11)kj
− 14
8


1
− 14 − 18   (iγ5 )il (iγ5 )kj 
4


µ


− 12
0 
1
  (γ )il (γµ )kj 



−1
− 12
0   (γ µ γ5 )il (γµ γ5 )kj 
−3
0
0 − 12
(σ µν )il (σµν )kj
(2.16)
2
a = 1, . . . , N − 1 are the generators of SU (N ),
Ã
=
− 14
1
4
1
4
1
−2
− 12
1
N
2
2 NN−1
2
!Ã
1
2
(11)il (11)kj
(λa )il (λa )kj
− N1
!
(2.17)
• quark-quark channel (corresponds to Lqq ):
In Dirac space:








(11)ij (11)kl
(iγ5 )ij (iγ5 )kl
(γ µ )ij (γµ )kl
(γ µ γ5 )ij (γµ γ5 )kl
(σ µν )ij (σµν )kl


 
 
 
=
 
 
 
1
4
− 14
− 14
1
1
−3
1
1
3
1
4
1
4
1
4
− 12
1
2
− 14
− 14
− 12
0
− 18

(11)il (11)kj
(iγ5 )il (iγ5 )kj
(γ µ )il (γµ )kj









0 


 µ

1
(γ
γ
)
(γ
γ
)
0

5 il
µ 5 kj 
2
0 − 12
(σ µν )il (σµν )kj
(2.18)
1
8
and in U (N ) color or flavor space:
Ã
(11)ij (11)kl
(λa )ij (λa )kl
!
Ã
=
1
2
N −1
N
1
2
− NN+1
!Ã
(λS )ik (λS )lj
(λA )ik (λA )lj
!
,
(2.19)
where A are the antisymmetric generators, A ∈ {2, 5, 7}, and S the symmetric ones.
An interesting feature of the Fierz-transformed NJL Lagrangian is that a simple local color
current-current term leads to the four fermion terms in the Lagrangian (2.6), [28]. Thus,
since we are omitting six-point vertices in this thesis, we can consider as Lint :
¡
¢2
Lint = −g ψ̄γ µ ta ψ .
(2.20)
The color current-current interaction can be thought of as abstracted from the QCD
Lagrangian by converting the original SU (N )c gauge symmetry into a global symmetry of
the quark color currents. In the following chapter we will see that when we integrate out
2.3 Fierz identities
28
the gauge fields from the QCD partition function in order to get the NJL model as a low
energy effective theory, we will end up with an NJL type model with such an interacting
term. The Fierz transformed part of the initial Lagrangian ψ̄(iγ µ ∂µ − m̂)ψ + Lint in the
quark-antiquark and quark-quark channel then is:
Lex
Lqq =
¡ 2
¢ ·
¸
¡
¢2 ¡
¢2 1 ¡ µ
¢2 1 ¡ µ
¢2
Nc − 1
g ψ̄λa ψ + ψ̄iγ5 λa ψ −
=
ψ̄γ λa ψ −
ψ̄γ γ5 λa ψ
Nc2
2
2
·
¸
¡
¢2 ¡
¢2 1 ¡ µ
¢2 1 ¡ µ
¢2
1
−
g ψ̄ta λb ψ + ψ̄iγ5 ta λb ψ −
ψ̄γ ta λb ψ −
ψ̄γ γ5 ta λb ψ
,
2Nc
2
2
(2.21)
¢¡
¢
¢¡
¢ ¡
Nc + 1 h¡
g ψ̄iγ5 tA λA0 C ψ̄ T ψ T Ciγ5 tA λA0 ψ + ψ̄tA λA0 C ψ̄ T ψ T CtA λA0 ψ
2Nc
¢¡
¢ 1¡ µ
¢¡
¢i
1¡ µ
−
ψ̄γ γ5 tA λA0 C ψ̄ T ψ T Cγµ γ5 tA λA0 ψ −
ψ̄γ tA λA0 C ψ̄ T ψ T Cγµ tA λA0 ψ
2
2
¢¡ T
¢ ¡
¢¡
¢
Nc − 1 h¡
T
−
ψ Ciγ5 tS λS 0 ψ + ψ̄tS λS 0 C ψ̄ T ψ T CtS λS 0 ψ
g ψ̄iγ5 tS λS 0 C ψ̄
2Nc
¢¡
¢ 1¡ µ
¢¡
¢i
1¡ µ
−
ψ̄γ γ5 tS λS 0 C ψ̄ T ψ T Cγµ γ5 tS λS 0 ψ −
ψ̄γ tS λS 0 C ψ̄ T ψ T Cγµ tS λS 0 ψ ,
2
2
(2.22)
where the indices A, A0 and S, S 0 stand for the antisymmetric and symmetric generators
of SU (3) respectively. For later convenience we define as G the coefficient of (ψ̄λa ψ)2 and
H the coefficient of (ψ̄iγ5 tA λA0 C ψ̄ T )(ψ T Ciγ5 tA λA0 ψ). Then, for Nc = 3:
H
3
= .
G
4
(2.23)
Chapter 3
Effects of finite temperature and
chemical potential
3.1
Thermodynamics
In this section we review some essential Thermodynamic relations and the connection
with field theory in finite temperature and chemical potential, often referred to as ‘Thermal Field Theory’, see [5, 26]. Our main interest is to study color superconductivity inside
compact stars, thus we need an equation of state for matter inside such conditions. In
order to obtain an equation of state, i.e. a relation between energy density and pressure
as functions of temperature and chemical potential, we resort to the first law of Thermodynamics:
E = TS − PV +
N
X
µi Ni ,
(3.1)
i=1
where E stands for energy, P for pressure, T for temperature, S for entropy, µi for the
chemical potential of the species i and Ni the particle number for the species i. All of
these variables are thought of as functions of T and µ. By dividing with the volume V of
the system we get an equivalent formula where only densities are involved:
²(T, µ) = T s(T, µ) − p(T, µ) +
N
X
µi ni (T, µ) .
(3.2)
i=1
Next we need a framework within which we can calculate pressure and other thermodynamic variables as functions of T, µ. One naturally chooses the grand-canonical ensemble
of statistical mechanics since the independent variables needed to describe a system are
T, V, µ and it allows for particle and energy exchange with a ‘reservoir’. The relevant poten-
3.1 Thermodynamics
30
tial is the grand-canonical potential Ω(T, V, µ), which is derived from the grand-canonical
partition function Z(T, V, µ):
Ω(T, µ) = −
T
ln Z ,
V
(3.3)
and1
Z = Tr e−β (Ĥ−µi N̂i ) = e−βΩ ,
(3.4)
where H is the Hamiltonian and the trace here is over the Hilbert space. We use β =
1
1
kB T = T since in our system of units Boltzman’s constant is set to one, kB = 1. The
partition function may also be defined in terms of the statistical density matrix ρ̂:
h
³
´i
ρ̂ = exp −β H − µi N̂i ,
(3.5)
where summation over i is implied. The grand partition function is then defined as:
Z = Tr ρ̂ .
(3.6)
Another very important notion we will be using is the ensemble average of an operator Ô:
hÔi =
Tr (ρ̂Ô)
Tr (ρ̂Ô)
=
.
Tr ρ̂
Z
(3.7)
The partition function is the single most important quantity as all of the thermodynamic
properties can be derived from it by differentiating the grand potential, which also follows
from the partition function. This is done via the following formulae:
p(T, µ) = −Ω(T, µ) ,
n(T, µ) = −
s(T, µ) =
∂Ω(T, µ)
,
∂µ
∂Ω(T, µ)
.
∂T
(3.8)
(3.9)
(3.10)
1
In equation (3.3) we wrote Ω(T, µ) instead of Ω(T, V, µ); it will be clear from the arguments of Ω
if we’re referring to the potential or the relevant density. The pressure may also be used instead of the
density; see equation (3.8). We can also write equation (3.3) as:
1
ln Z(T, V, µ) .
β
We will be referring to Ω and Z as the ‘grand potential’ or ‘thermodynamic potential’ and ‘grand partition
function’ respectively.
Ω(T, V, µ) = −
3.2 Thermal Field Theory
31
By plugging these equations back to equation (3.2) we obtain the desired equation of state.
Thus we are left with the calculation of the grand partition function for the system under
study. This is where quantum field theory enters the game; in it’s finite temperature and
density form it allows us to compute the partition function given the Lagrangian of a
system.
3.2
Thermal Field Theory
Statistical field theory, or non-relativistic quantum field theory at finite temperature and
chemical potential, was devised in the late 50s and has been applied in problems where
there is a large number of particles, as in many condensed matter problems. Thermal field
theory is a relativistic version of statistical field theory. Although it was first conceived in
the 60s, it gained popularity and was studied more intensely later on in the 70s, the main
motivation being the electroweak phase transition and it’s implications in cosmology. In
the 80s, lattice gauge theory suggested the existence of a deconfined phase of quarks and
gluons, the ‘quark-gluon plasma’, something that inspired experiments in ultrarelativistic
heavy ion collisions, which in turn gave another boost in the study of thermal field theory.
As is the case with all the different forms of quantum field theory, there are two
different approaches to field theory at finite temperature and/or finite chemical potential:
the second quantization formalism and the path integral formalism. We will not bother
with the second quantization method; the reader may consult [1, 16]. The path integral or
functional approach is used mainly in modern times, mainly because of it’s advantages in
the treatment of non-perturbative phenomena like tunneling, instantons, in lattice gauge
theory and because gauge fields are treated in a much more effective manner. In this thesis
we do not have gauge fields; however we will also use the functional formalism. In this
section we summarize the formalism of thermal field theory. Standard textbooks for this
topic are [26, 5].
The path integral may be defined in real or imaginary time, however in finite temperature the imaginary time formalism is more convenient. This means we have to perform
an analytical continuation from real to imaginary time: t → −iτ, τ ∈ R (Wick rotation).
This also means that we change our metric from Minkowskian to Euclidean:
xµ xµ = t2 − x2 → −τ 2 − x2 = −xE µ xµE ,
(3.11)
The procedure of defining the partition function as a path integral goes along the same
lines with the derivation of the path integral formalism for the generating functional in
ordinary field theory, see [26]. As we will see below there is a correspondence between
the generating functional W for a quantum field theory described by the Lagrangian L
3.2 Thermal Field Theory
32
and the grand partition function Z for a many-particle system described by LE at finite
temperature and chemical potential2 . Below we recapitulate the main points for bosons
and fermions separately; the emphasis though is on fermions as we will encounter fermionic
path integrals later on.
3.2.1
Bosons
Suppose a theory for bosons is described by a Lagrangian L. In ordinary field theory the
associated action and generating functional are defined by:
Z
Z
S=
dtL =
d4 x L (φ, ∂µ φ) ,
(3.12)
[dφ] e iS ,
(3.13)
Z
W =N
where N is an irrelevant normalization constant. The functional measure is [dφ] =
Q
i dφi (x). The Hamiltonian associated to this theory is
Z
Z
3
H=
d x H (π, φ) =
h
i
d 3 x π φ̇ − L ,
(3.14)
∂L
where H is the Hamiltonian density and π(x) = ∂ φ̇(x)
is the canonical momentum.
In the context of finite temperature and chemical potential field theory and for a system
that admits some conserved charge, the Hamiltonian is modified according to:
H (π, φ) → H 0 (π, φ) = H (π, φ) − µN (π, φ) ,
(3.15)
where N (π, φ) is the conserved charge density and µ the chemical potential.
In order to calculate the partition function one starts by the definition, which can now
be written as:
Z
−β(Ĥ−µi N̂i )
Z = Tr e
≡
dφ hφ|e−β(Ĥ−µi N̂i ) |φi ,
(3.16)
Then the problem is shifted in evaluating the last integral and this is done the usual way;
one discretizes time and inserts a complete set of eigenstates of the φ and π fields in each
time point. The partition function is then written as a path integral over the phase space:
Z
Z=
[dπ]
p.b.c.
2
µZ
Z
[dφ] exp
Z
β
dτ
0
·
¸¶
∂
d x iπ φ − H (π, φ) + µN (π, φ)
.
∂τ
3
(3.17)
In fact many authors use the same symbol for the generating functional and the partition function, Z.
3.2 Thermal Field Theory
33
The exponent in the last equation is always quadratic in the momentum and thus the
momentum integrations can be done immediately since they are products of Gaussian
Z
N Z
Y
dπi
integrals. Here [dπ] ≡
. After the momentum integral has been performed,
2π
i=1
what remains is a path integral over the φ field which is cast in the form:
Z
[dφ] e SE ,
Z=N
(3.18)
p.b.c
in which
Z
SE =
Z
β
dτ
0
d 3 x LE ,
(3.19)
where LE is the Euclidean Lagrangian. The analogy between the generating functional
and the partition function becomes now obvious; compare equations (3.13) and (3.17)3 .
One thing that remains to be clarified is the boundary conditions which are of a periodic
nature, as the term ‘p.b.c.’ indicates; it is shorthand notation for ‘periodic boundary
conditions’. The periodicity of the boundary conditions is due to the trace operation: the
integral in equation (3.18) is such that φ(x, 0) = φ(x, τ ).
One can go on and define (thermal) Green’s functions for the free as well as for the
interacting theory, via perturbation theory, however this is outside the scope of this presentation as we will meet fermionic path integrals.
3.2.2
Fermions
In order to describe free fermions in zero chemical potential, we use the standard fermionic
Lagrangian:
L = ψ̄ (iγ µ ∂µ − m) ψ ,
(3.20)
where γ µ are the Dirac matrices, as defined in the ‘conventions’ part. It can also be written
as L = ψ̄ (i 6 ∂ − m) ψ, where Feynman’s notation for products of gamma matrices and
4-vectors is used. The Dirac conjugate spinor ψ̄ is defined as ψ̄ = ψ † γ 0 .
The momentum conjugate field is given by:
3
Note that the sign of the action in the exponential is conventional. Some Authors prefer to write
Z
Z[φ] = N
[dφ] e −SE .
p.b.c
When one passes from real to imaginary time there is an overall minus sign that appears naturally in front
of the Euclidean action. We prefer to leave it there, while the authors that use the above formula prefer
to extract it from the action.
3.2 Thermal Field Theory
34
Π=
∂L
= iψ † .
∂(∂ψ/∂t)
(3.21)
This suggests that the fields ψ and ψ † must be treated as independent fields. The Hamiltonian density follows as usually:
¡
¢
∂ψ
− L = ψ̄ −iγ i ∂i + m ψ .
∂t
In case of µ 6= 0 the Hamiltonian is modified according to:
H =Π
(3.22)
H → H 0 = H − µN
(3.23)
where now N = ψ † ψ = ψ̄γ 0 ψ is the particle number density. The partition function, can
be written in an way similar to equation (3.17) (we discard the normalization constant
µZ β Z
¶
Z
Z = [dψ̄][dψ] exp
dτ d 3 x {iΠ∂τ ψ − [H (Π, ψ) − µN (Π, ψ)]}
0
Z
=
 β

Z
Z
h ¡
i
¢
[dψ † ][dψ] exp  dτ d3 x ψ † −∂τ + iγ 0γ · ∇ − mγ 0 + µ 
0
a.b.c
µZ
Z
[dψ̄][dψ] exp
=
Z
β
dτ
0
a.b.c.
£ ¡
¢ ¤
d x ψ̄ −γ 0 ∂τ + iγ · ∇ − m + µγ 0 ψ
3
(3.24)
¶
,
(3.25)
R
R
R
where N = d3 x ψ † ψ = d3 x ψ̄γ 0 ψ = d3 x N . This can also be written in a form
reminiscent of real-time formalism if we define the Euclidean derivative operator ∂µE ≡
(i∂τ , ∇):
µZ
Z
[dψ̄][dψ] exp
a.b.c.
Z
β
dτ
0
¶
£ ¡ µ E
¢ ¤
0
d x ψ̄ iγ ∂µ − m + µγ ψ
.
3
Equations (3.24) and (3.25) are equivalent; the Jacobian coming from the change of variables in the path integral from ψ † to ψ̄ is equal to one, since det γ 0 = 11. Two important
points should be noted here. First we need to clarify the boundary conditions for ψ(x, τ ):
they are antiperiodic which means that ψ(x, 0) = −ψ(x, β), which we denoted by ‘a.b.c.’
below the integral in the last equation. Second, while the fields ψ̄, ψ in equation (3.23)
are operators, in the partition function they are Grassmann variables. This is a result of
the spin-statistics theorem and Pauli’s exclusion principle. It is also reflected in the fact
that fermions obey anticommutation relations, instead of commutation relations as in the
case of bosons:
n
o
ψ̂α (x, t), ψ̂β† (y, t) = δαβ δ(x − y)
(3.26)
3.2 Thermal Field Theory
35
while all other anticommutators vanish. The antiperiodicity required for fermion fields
can be verified by examining the thermal Greens functions. The latter are defined by:
G(x, y; τ, 0) = Z −1 Tr {ρ̂ Tτ [η̂(x, τ )η̂(y, 0)]}
(3.27)
where Tτ is the ‘imaginary time ordering operator’:
Tτ [η̂(τ1 )η̂(τ2 )] = η̂(τ1 )η̂(τ2 )θ(τ1 − τ2 ) ± η̂(τ2 )η̂(τ1 )θ(τ2 − τ1 )
(3.28)
where the plus corresponds to bosons and the minus sign to fermions. The field η̂ is some
generic field that can be bosonic or fermionic. In case of fermions it then holds that:
G(x, y; τ, 0) = −G(x, y; , τ, β) .
(3.29)
To prove that one uses the definition (3.27) and takes advantage of the cyclic property of
the trace. Heisenberg’s equation for the imaginary-time evolution is also used:
η̂(y, β) = eβ(H−µN ) η̂(y, 0)e−β(H−µN ) .
(3.30)
It is sometimes convenient to work in (iωn , p) space instead of (τ, x) space, in fact this
is the case for the calculations in the next chapter. The grand partition function can also
be calculated in (iωn , p) space. In order to pass to the frequency space we expand the
fermion fields as:
1 X X i(p x+ωn τ )
ψα (x, τ ) = √
e
ψα;n (p)
V n p
(3.31)
where the summation over n and p runs over positive and negative values. Note that
the normalization factor is the 4-dimensional volume in frequency-momentum space. The
frequencies ωn are the fermionic (odd) Matsubara frequencies4 , ωn = (2n + 1)πT . By
plugging this expansion in equation (3.25) and doing the algebra we get the partition
function in frequency-momentum space:
Z
[dψ̄][dψ]eS
XX
S=
ψ̄α;n (p) Dαα0 (p) ψα0 ;n (p)
Z=
n
(3.33)
p
¡
¢
D = β −iγ 0 ωn − γ · p − m + µγ 0 ,
4
(3.32)
(3.34)
Some note on conventions is needed here; some authors prefer to write ωn = (2n + 1)πiT and use ωn
instead of iωn .
3.2 Thermal Field Theory
36
where the measure is now
Y
[dψ̄] [dψ] =
dψ̄α;n (p) dψα;n (p) ,
n,p,α
The partition function is now a Gaussian integral over Grassmann fields, and by using the
standard formula for such integrals, viz.:
Z
[dη † ] [dη] e−η
† Dη
=
Z ÃY
N
!
dηi† dηi

X †
ηi Dij ηj  = det D ,
exp −

i,j
i=1
we get for the partition function:
£ ¡
¢¤
Z = det D = det β −iωn − γ 0γ · p − mγ 0 + µ .
(3.35)
The determinant is to be carried out in all indices, Dirac, frequency and momentum.
The matrix D is actually the inverse propagator in frequency-momentum space which is
sometimes referred to as the Matsubara propagator βG−1
0 (ωn , p) = βD, the subscript ‘0’
denoting that we’re talking about the free theory.
As was outlined in the previous section, the thermodynamical properties are obtained
through the logarithm of the grand partition function or equivalently the grand canonical
potential, Ω = − β1 ln Z. The evaluation of the latter is done by following a standard
procedure which can be found in [26]; we will follow a similar procedure in the next
chapter, thus we only mention the final result for the free theory here:
Z
ln Z = 2V
h
i
h
io
d3 p n
−β(²−µ)
−β(²+µ)
β²
+
ln
1
+
e
+
ln
1
+
e
.
(2π)3
(3.36)
The factor of two in front of the right hand side is there due to the spin 1/2 nature of the
fermions. The first term in the integrand represents the zero point energy of the vacuum,
while the other two terms are the particle (+µ) and antiparticle (−µ) contributions. The
discrete sum has been replaced by an integral over the momenta in equation (3.36). The
corresponding rule is:
X
p
Z
→V
d3 p
.
(2π)3
(3.37)
When one passes from the momentum space to the frequency-momentum space, by making
the replacement p0 → iωn , the corresponding change of the momentum integral
Z∞
−∞
dp0
1X
→
2π
β n
(3.38)
3.2 Thermal Field Theory
37
is necessary. In this case the integral is being replaced by a discrete sum over the Matsubara
frequencies.
3.2.3
Interactions
Up to this point we only dealt with free theories. Interactions are introduced and treated
the usual way, via perturbation theory: the partition function is expanded in powers of
the interaction, provided the latter are weak enough for perturbation theory to apply. The
action is as usually written as”
S = S0 + SI ,
(3.39)
where S0 corresponds to the free field and SI contains the interactions. The partition
function can then be written as:
Z
Z=N
Ã
Z
[dψ̄][dψ] eS = N
[dφ]eS0
∞
X
1 n
S
n! I
!
,
(3.40)
n=0
and can also be written as a sum of the free and the interacting theory in perturbation
theory. As in zero-temperature field theory one can make use of diagrammatic techniques
and define Feynman rules for thermal field theory. However, we will do not need to do
that here, as it is not necessary for our purposes, the reader may consult [26]. We will
only need the self-energy of the fermion field. As in ordinary field theory, one can define
the full propagator G which corresponds to the full theory. Switching to position space,
G is written as:
R
[dψ̄][dψ]ψ̄ (x1 , τ1 ) ψ (x2 , τ2 ) eS
R
.
[dψ̄][dψ]eS
(3.41)
The thermal average is now taken with respect to the full action. Notice that we don’t need
the normalization constant anymore. The fermion self-energy Σ however is best defined
in (ωn , p) space as:
¤®
­ £
G (x1 − x2 ; τ1 − τ2 ) = Tτ ψ̄ (x1 , τ1 ) ψ (x2 , τ2 ) =
G−1 = G−1
0 + Σ =6p − m + Σ(ωn , p) .
(3.42)
where G0 is written in a form similar to the ordinary propagator for fermions by considering
the fields ψ̄, ψ instead of ψ † , ψ:
G−1
0 (ωn , p) =6p − m
where now p0 ≡ iωn + µ and Feynman’s notation is used, p6 = γ µ pµ .
3.3 The QCD phase diagram
3.3
38
The QCD phase diagram
In the next chapter we will try to reproduce a small part of the alleged QCD phase diagram
via the NJL model introduced in chapter 2, focusing at the low temperature and large
chemical potential regime. Before doing so however, some insight in the structure and
intricacies of the phase diagram is necessary. The structure of the QCD phase diagram
is one of the most exciting topics in the study of the strong forces and modern physics
in general. Most if our knowledge about it, as far as theory is concerned, comes from
ab-initio Monte Carlo simulations on the lattice. A full analytic phase diagram for QCD
is still impossible, as lattice calculations at finite, large chemical potential are not yet
possible [citation]. This means that we are restricted to studying simplified versions of
the full theory or effective models, in order to gain knowledge about the phase diagram.
Typical simplifications are assuming equal and sometimes unphysical masses and chemical
potential for all or some of the flavors, neglecting electromagnetism, or obtaining results
by extrapolating, often to unrealistic values of zero or infinite quark masses, see [10, 44]
for details.
Up to this point we have only mentioned the possibility for the existence of QGP;
quarks and gluons may be deconfined at high temperature or density, when the hadrons
strongly overlap with each other. In this picture, two distinct phases exist: the ‘hadronic
phase’ and the ‘QGP phase’. In the hadronic phase the quarks are confined, in contrast to
the QGP phase where the quarks are deconfined and all of the symmetries are unbroken.
On a schematic level, this ‘pre-color superconductivity’ phase diagram is depicted in figure
3.1. This picture remained as a standard result until the 1990’s, when the idea that color
superconductivity might play an important role in the structure of the phase diagram
was given more thought. Actually, even though the possibility of color superconductivity
had already been considered as early as the 70’s, see [3, 10] and references therein, the
relevance of this idea was ignored until the end of the 90’s. At that time, new approaches
revealed much larger gaps than expected earlier, of the order of 100M eV , something that
implies larger critical temperatures, which in turn suggests that the color superconducting
regime of the phase diagram is larger than it was initially thought off.
3.3 The QCD phase diagram
39
Figure 3.1 An early phase diagram for the strong interactions.
After the inclusion of color superconductivity in the phase diagram, many possibilities
emerged. It is expected that at large chemical potential, the up, down and strange quark
are paired in the so-called color-flavor locked (CFL) phase which breaks chiral symmetry; we will specialize later on in the CFL phase. In lower densities, a second colorsuperconducting phase (2SC) is expected; in this one only the up and down quarks are
paired. However, recently new phases have been suggested, such as the ‘LOFF’5 phase
(also referred to as crystalline color-superconductivity) or the ‘CFL-K’ phase (a CFL phase
with condensed kaons), see [10] and references therein. These new phases are expected to
replace, partially or even completely, the 2SC phase. The relevant, schematic also, phase
diagrams can be seen in figure 3.2.
Figure 3.2 Schematic QCD phase diagrams including color superconductivity. In the left diagram
the LOFF phase replaces the 2SC phase.
It should be emphasized that these diagrams are not exact as they are constrained by
a small number of safe theoretical and empirical results. In fact, there are only two
points in the phase diagram where direct empirical information is available; both belong in the hadronic phase where chiral symmetry is broken. The first corresponds to
(µ, T ) = (0, 0), the vacuum, and the second one to zero temperature and chemical potential that corresponds to nuclear saturation density, µ . 300M eV , see [10] for details.
This point marks the onset of what we should refer to as ‘dense matter’; the regime where
T = 0, µ . 300M eV connects to the vacuum. This point is the beginning of a first order
5
LOFF stands for Larkin-Ovchinnikov-Fulde-Ferrell, the authors that first studied this case.
3.3 The QCD phase diagram
40
phase boundary, separating a hadron gas phase at lower µ from a hadron liquid at larger
µ. In figure 3.2, left diagram, this phase boundary is denoted with a solid black line, while
in the diagram on the right it is the lower ‘E’ point. As can be seen on the diagrams this
phase boundary is expected to finish in a critical endpoint.
On the other hand, the chiral phase transition has been studied since 1984, see [37].
The results of those studies reveal a rather interesting behavior: the order of the phase
transition depends on the number of light quark flavors. Several cases are distinguished:
the transition is second order for massless up and down quarks and heavy strange quark
and becomes a crossover for light up and down quarks, and it is first order for small ms
or three degenerate flavors. The first two cases are depicted in figure 3.3 and the case of
three degenerate flavors is shown in figure 3.4. This last case is of special interest to us as
we will study the same case with the NJL model.
Figure 3.3 QCD phase diagrams for different cases: on the left, the up and down quarks are massless
and the strange quark is heavy. Chiral symmetry is broken in the hadronic phase and is restored
everywhere else. The transition changes from second to first order at the tricritical point. The transition
between the 2SC phase and the plasma is likely first order, as well as the one between the hadronic phase
and the 2SC phase. The diagram on the right corresponds to two light u, d and a heavy s flavor. The
second order transition is now a crossover. The CFL phase may be also present and the new ‘unlocking’
transition is first order. The diagrams are taken from [44].
3.4 Color superconductivity
41
Figure 3.4 QCD phase diagram for three degenerate flavors. Chiral symmetry is broken everywhere
between the solid line, which is a first order transition. The CFL and hadronic phase have the same
symmetries. The dashed line shows the critical temperature at which quark-quark pairing vanishes. The
question mark is there to show that new phase transition may be present at that regime. This diagram is
also taken from [44].
The methods that were used to produce these qualitative phase diagrams can be found in
[44] and references therein.
3.4
Color superconductivity
Ideas about color superconductivity emerged almost 20 years ago, [3], however only recently this phenomenon received the attention it deserves, [11, 44, 10]. Naively, one expects that at asymptotically high densities, due to the phenomenon of asymptotic freedom,
quarks would form a Fermi sea of free fermions. However, Bardeen, Cooper and Schrieffer
have shown that the Fermi sea of free fermions is unstable in presence of an arbitrary small
attractive interaction, [4]. In ordinary superconductors, the screened Coulomb interaction
between electrons is repulsive. Superconductivity arises due to the attractive effective
interactions of electrons via phonon exchange; this effect takes place at the Fermi sea for
opposite electron momenta. This attraction forces electron pairs to pair in the so-called
Cooper pairs.
Color superconductivity resembles in general the ordinary BCS theory. Here, the
attractive interactions arise already from the primary interactions, as in QCD one gluon
exchange (which can be thought of as the analogue of Coulomb scattering in QCD) is
attractive if the quarks are antisymmetric in color and the pair is said to be in the color 3̄
channel, [44]. The instanton interactions are also attractive in the 3̄ channel, which may
be of some importance at stronger coupling. The single gluon attractions however are
3.4 Color superconductivity
42
enough to lead to condensation in the color 3̄ channel. The condensates are quark Cooper
pairs, called diquarks. A diquark condensate is in general defined as the expectation value:
hψ T Oψi ,
(3.43)
where ψ is a quark field with spin, color and flavor degrees of freedom, and O is an operator
acting on Dirac, color, flavor space:
O = ODirac ⊗ Ocolor ⊗ Oflavor .
(3.44)
As we saw, a number of different superconducting phases arises in QCD. However, our
goal is to estimate the consequences of the presence of color-superconducting matter in
cold compact stars, namely in their mass-radii relations. Thus our focus is in the low
temperature-high density regime of the QCD phase diagram (or the NJL phase diagram
for our purposes), where the CFL phase is expected to occur. In the rest of this chapter
we will focus on condensation patterns and symmetries for the CFL phase.
3.4.1
The Color-Flavor-Locked phase
The general form of the condensate for the CFL phase has the form:
bβ
bβ
aα
aα
hψiL
(p)ψjL
(−p)²ab i = −hψiR
(p)ψjR
(−p)²ab i = ∆(p2 )²αβA ²ijA ,
(3.45)
where here color is denoted by Greek letters, flavor is denoted by the Latin i, j and spin
degrees of freedom by a, b. In the last equality the index A is summed, locking in this
way color and flavor. The form of the condensate can be anticipated by analyzing the
renormalization of couplings at the Fermi surface; however this is outside our scope. There
are many different treatments that show that this is indeed the dominant condensate in
three-flavor QCD, see [32, 44, 38] for details. For our purposes it suffices to say that the
essential physical reason is that the free energy is maximally decreased for the maximally
symmetric condensate, in which we have the maximally unbroken group which leads quarks
of all three flavors and colors to condense. The most symmetric condensate is of the form
of equation (3.45). It’s structure in color, flavor and Dirac space is fixed by the following
arguments, [11]:
• We want an attractive interaction, thus the condensate should be antisymmetric in
color indices.
• The condensate should be antisymmetric in spin indices in order to have total spin-0.
3.4 Color superconductivity
43
• Given the antisymmetry on color and spin indices, and because Pauli’s principle
requires overall antisymmetric condensate (since fermions are involved), we deduce
that the condensate should be antisymmetric in flavor indices as well.
We also know that the momenta in Cooper pair are opposite and the same holds for a
diquark condensate. Since the spin of the quarks is zero in total (they have opposite spin),
the pairing is done among quarks with the same helicity; this is denoted by the indices
R, L in equation (3.45).
The reasoning behind the name ‘color-flavor-locking’ is that the condensate is symmetric under simultaneous color and flavor transformations, but not in separate color or
flavor transformations.
Before proceeding we should note that the full form of the condensate in equation
(3.45) is actually, [32, 38]:
D
E
¡ ¢
¡ ¢
¡ ¢
bβ
aα
(−p) ²ab ∝ ∆ p2 ²αβA ²ijA + κ1 p2 δiα δjβ + κ2 p2 δjα δiβ .
ψiL(R)
(p) ψjL(R)
(3.46)
Note that since ²αβA ²ijA = δiα δjβ − δjα δiβ , the pure 3̄ condensate has κ1 = −κ2 . The extra
terms arise because the 3̄ condensate induces condensation in the color 6 channel, see
[44, 32]. It turns out however, according to [32], that the values of κ are rather small and
thus we can ignore them; condensation occurs mostly at the 3̄ channel and hence equation
(3.45) can be used safely.
As far as the symmetry breaking induced by the condensate, in three massless flavors
in QCD, the pattern is as follows:
SU (3)c × SU (3)L × SU (3)R × U (1)B −→ SU (3)c+L+R × Z2 .
(3.47)
The separate color and flavor symmetries are broken down to simultaneous color and flavor
transformations (denoted by SU (3)c+L+R ). The Z2 symmetries arise since the condensate
is left invariant by a change of sign of the left and/or right handed fields, [32]. Using Dirac
spinors the condensate in equation (3.45) can be written as
hψiα Cγ 5 ψjβ i ∝ ²αβA ²ijA ,
(3.48)
which is of the form hψ T Oψi, where we have omitted the color and flavor parts of the
operator ODIrac ⊗ Ocolor ⊗ Oflavor . The most important diquark condensates with which
we will be dealing in the next chapter are of the form:
∆AB = hψ T tA λB Ciγ 5 ψi ,
(3.49)
3.5 From QCD to the NJL model
44
where the λ’s and t’s are the 9 U (3) generators, which can be found in the ‘Conventions’
part; the λ’s refer to flavor space and the t’s to color space. The indices A, B take the values
2, 5, 7 since we want the condensate to be antisymmetric in flavor and color and the GellMann matrices λA , tA , A ∈ {2, 5, 7}, are antisymmetric. the non-vanishing condensates
that make up the CFL phase are ∆22 , ∆55 , ∆77 , see also [10]. Each of the condensate
is like a two-flavor color superconductor, being a color and flavor triplet in the scalar
channel. For example the condensate ∆22 corresponds to diquark pairs (ur , dg ), (ug , dr )
and similarly for the other condensates.
3.5
From QCD to the NJL model
Up to now we have only mentioned that the NJL model can be viewed as an effective low
energy model for QCD. In this section we will qualitatively justify the above statement.
The Lagrangian of QCD in real time and finite temperature and chemical potential may
be written as:
¡
¢
1 a µν a
LQCD = ψ̄i iγ µ Dµ − mi + µγ 0 ψi − Fµν
F
4
¡
¢
ta
1 a µν a
= ψ̄i iγ µ ∂µ − mi + µγ 0 ψi − gAaµ ψ̄γ µ ψ − Fµν
F
2
4
= Lf + gJaµ Aaµ + Lg ,
(3.50)
where Jaµ is a fermion color current. The QCD partition function can be written then as:
Z
R
4
d[ψ̄]d[ψ]d[Aaµ ] ei d x LQCD
Z
Z
R 4
µ a
iSf
= d[ψ̄]d[ψ] e
d[Aaµ ] ei d x (Lg +gJa Aµ )
Z
µ
= d[ψ̄]d[ψ] eiSf +iSint [Ja ] ,
ZQCD =
(3.51)
where
Z
Sint [Jaµ ]
= −i ln
d[Aaµ ]
½ Z
¾
¡
¢
4
µ a
exp i d x Lg + gJa Aµ
.
(3.52)
By adding a gauge fixing term and the ghost fields6 , the integral over the gauge fields can
be performed in a way such that Sint can be written as, see [8, 42]:
6
The Faddeev-Popov ghosts are auxiliary fields that need to be introduced in order for non-Abelian
gauge theories to be consistent, see [24, 36].
3.5 From QCD to the NJL model
Sint [Jaµ ] = Sint [0] +
ig 2
2
45
Z
b
d4 x d4 y Jµa (x)Gµν
ab (x − y)Jν (y) + · · ·
(3.53)
where Gµν
ab (x − y) is the gluon propagator corresponding to Lg :
µ
ν
Gµν
ab (x − y) = hAa (x)Ab (y)i .
(3.54)
In equation (3.53) 6-point interactions and higher have been neglected. In principle the
propagator can be evaluated in perturbation theory, at a specific gauge. Perturbation
theory is however valid only in high momentum transfers. At lower momentum nonperturbative effects, such as the formation of massive bound states of gluons in ‘glueballs’,
are expected to modify the propagator. One may then choose an ansatz for the propagator
that incorporates the effect of the glueballs. An accurate ansatz may be found ny fitting
to lattice simulations for the pure gluon propagator, however this has lead to non-local
interactions, see the references in [42]. A more crude approximation aiming in an effective
low energy theory is to consider:
(
Gµν
ab (p)
=
i
δ g µν
M 2 ab
0
, p2 < Λ2
,
, p2 > Λ2
(3.55)
where Λ is momentum cut-off. In finite temperature and chemical potential, the term
Sint [0] does not depend on the chemical potential, only on temperature and it represents
a constant contribution of the gluon part to the QCD pressure and can be neglected. The
QCD partition function may then be approximated by:
½ Z
¾ Z
½ Z
µ
¶¾
g2 µ a
4
4
ZQCD ≈ d[ψ̄]d[ψ] exp i d x LNJL = d[ψ̄]d[ψ] exp i d x Lg −
J J
.
2M a µ
(3.56)
The NJL type Lagrangian appearing here contains a local color current-current term, as
the one we Fierz-transformed in chapter 2.
Z
Chapter 4
The phase diagram of the NJL
model for three degenerate flavors.
We now have all the necessary theoretical ingredients to construct the phase diagram for
the NJL model. We will work on a simplified model, as the calculation for the full one is
rather long and complicated, thus exceeding the scope of this thesis. The simplification
is to assume degenerate quark flavors, which means equal masses and chemical potentials
for all quarks. This simplification makes the model unrealistic, however it captures all the
techniques used and most of the physics involved in the full model.
4.1
NJL model for Nf = 3.
Our starting point is the Fierz transformed Lagrangian density of the Nambu–Jona-Lasinio
(NJL) model for three flavors, Nf = 3, in finite temperature and chemical potential and
including terms in the flavor vector and axial vector channel and the diquark pseudoscalar
channel. This means that the lagrangian we will be using contains only the relevant terms
of the Lagrangian (2.21) and (2.22). By using (2.23):
LNJL = ψ̄ (iγ µ ∂µ − M + µγ0 ) ψ + G
= L0 + Lq̄q + Lqq ,
h¡
¢2 ¡
¢2 i 3 ¡
¢¡
¢
ψ̄λa ψ + ψ̄λa iγ5 ψ
+ G ψ̄tA λB iγ5 C ψ̄ T ψ T CtA λB iγ5 ψ
4
(4.1)
where M0 = diag(m0u , m0d , m0s ), µ = diag(µu , µd , µs ). The spinors ψ have Dirac, three
flavor and three color degrees of freedom. Their Dirac conjugate is ψ̄ = ψ † γ 0 . Theq
matrices
2
λa , ta are the 9 U (3) generators; λ1 , . . . , λ8 are the Gell-mann matrices, λ9 =
3 I and
similarly for the t matrices. The λ’s act on flavor space and the t’s on color space. They
are normalized as tr(λa λb ) = tr(ta tb ) = 2δab . The indices A, B take the values {2, 5, 7},
4.2 The Grand Canonical Potential
47
as we are interested in a flavor and color antisymmetric triplet in which the interaction is
attractive. The matrices λ2 , λ5 , λ7 are antisymmetric and couple (u, d), (u, s) and (d, s)
quarks respectively. C = iγ 2 γ 0 is the charge conjugation matrix; see the ‘Conventions’
part. The coupling constant G has dimensions in units of mass [G] = −2. In this model
instanton induced terms as well as terms with 6-point interactions are ignored. We also
do not treat spin-1 condensates. As it was outlined in chapter 3, the most important
condensates for the CFL phase are the diquark pseudoscalar condensates, see [23, 10, 9, 44]
and references therein, and we will focus only on them later on.
We now introduce the Nambu–Gor’kov spinors, for later convenience, which are defined
as:
1
Ψ= √
2
Ã
ψ
ψc
!
´
1 ³
, Ψ̄ = √
ψ̄ ψ̄c ,
2
(4.2)
where ψc is the charge conjugate spinor, ψc = C ψ̄ T . Also1 , ψ̄c = ψ T C (note than we can
also write ψc = −C † ψ̄ T and ψ̄c = −ψ T C † ). The products appearing in equation (4.1) are
tensor products; all terms are tensor products of matrices in color, flavor and Dirac space
respectively.
4.2
The Grand Canonical Potential
As has been outlined before, the single most important object we are interested in is the
grand-canonical potential Ω corresponding to the Lagrangian (4.1). Knowing Ω will enable
us to construct the phase diagram and derive gap equations for the condensates. In finite
temperature and density we have:
Ω=−
T
ln Z ,
V
(4.3)
where V is the volume, and:
µZ
Z
Z=N
d[ψ̄]d[ψ] exp
¶
Z
dτ
3
d x LNJL
,
(4.4)
where N is a normalization factor.
The first formal step we will take is to introduce condensates in some relevant channels,
such as the color vector and axial-vector channel for quark-antiquark condensates and the
scalar diquark channel. By doing this we expect to eventually obtain an expression for
1
Many different conventions for the charge
matrix exist in the literature; in our conventions
h¡ conjugation
¢ i∗ 0
¡
¡
¢∗
¢∗
† 0
T T
this is obtained as follows: ψ̄c = ψc γ = C ψ̄
γ = ψ̄C T γ 0 = ψ † γ 0 C † γ 0 = ψ T γ 0 C † γ 0 =
ψ T C, since γ 0 C † γ 0 = C.
4.2 The Grand Canonical Potential
48
Ω depending on the condensate values. This in turn will allow us, by using the methods
outlined in the previous chapter, to obtain gap equations for the condensates, as well as
an equation of state depending on them.
The condensates are introduced via a simple two-step method. First, we ‘integrate
in’ some properly chosen auxiliary fields in the partition function, a trick that goes under
the name of ‘Hubbard - Stratonovich’ transformation. For the second step we have two
equivalent alternatives: we may treat the introduced fields in the mean field approximation
(MFA), which accounts in setting the relative fields equal to their expectation value, or we
may perform the so-called ‘large 1/Nc expansion’ by applying the saddle point method, see
Appendix B, to the resulting partition function. In both cases we first need to ’bosonize’
the model. The term ‘bosonization’ refers to integrating out the fermionic degrees of
freedom and thus obtaining an effective action in which only the fields we introduced will
exist. The MFA is equivalent to the leading order terms in the large 1/Nc expansion, as
we will show later on.
4.2.1
The Hubbard - Stratonovich transformation
The Hubbard - Stratonovich transformation is performed by merely rewriting the exponent
of the interacting part of the Lagrangian in equation (4.4) as Gaussian integrals over some
properly chosen auxiliary fields. Thus we introduce the real fields αa , βa , a = 1, . . . , 9 and
the complex fields ∆AB , A, B ∈ {2, 5, 7}, and by using the standard formulas for Gaussian
integrals over real (x) and Grassmann (z) variables, namely
Z
1
1
−1
d[xi ] e− 2 xi Aij xj +xi Ji = (det A)−1/2 e 2 Ji Aij
Z
∗
∗
∗
∗
Jj
,
−1
ζj
d[zi∗ ]d[zi ] e−zi Hij zj +ζi zi +zi ζi = (det H) eζi Hij
,
we make the following substitutions:
½Z
¾
h¡
¢2 ¡
¢2 i
exp
dτ d x G ψ̄λa ψ + ψ̄λa iγ5 ψ
=
½Z
·
¸¾
Z
Z
¢
1 ¡ 2
αa + βa2 + αa (ψ̄λa ψ) + βa (ψ̄λa iγ5 ψ)
= d[αa ]d[βa ] exp
dτ d3 x −
4G
(4.5)
Z
3
and
4.2 The Grand Canonical Potential
49
¾
¢¡ T
¢
3 ¡
T
ψ tA λB Ciγ5 ψ
=
exp
dτ d x G ψ̄tA λB Ciγ5 ψ̄
4
(Z
"
¡
¢
¡ T
¢ #)
Z
Z
ψ̄tA λB Ciγ5 ψ̄ T
ψ tA λB Ciγ5 ψ
∆∗AB ∆AB
∗
3
∗
= d[∆AB ]d[∆AB ] exp
dτ d x −
+
∆AB + ∆AB
,
3G
2
2
½Z
Z
3
(4.6)
where the Dirac bilinear terms are treated as sources. Plugging in the last two equations
in equation (4.4) for the partition function, we get:
Z
ZHS =
d[ψ̄]d[ψ]d[αa ]d[βa ]d[∆∗AB ]d[∆AB ]×
( Zβ
exp
"
Z
¡
¢
¡
¢
d3 x ψ̄ (iγ µ ∂µ − M0 + µγ0 ) ψ − αa ψ̄λa ψ − βa ψ̄λa iγ5 ψ +
dτ
0
#)
¡ T
¢ ¡
¢
ψ tA λB Ciγ5 ψ
ψ̄tA λB Ciγ5 ψ̄ T
αa2 + βa2 |∆AB |2
+
∆AB −
−
2
2
4G
3G
µZ
¶
Z
Z
= d[ψ̄]d[ψ]d[αa ]d[βa ]d[∆∗AB ]d[∆AB ] exp
dτ d3 x LHS .
(4.7)
∆∗AB
This concludes the Hubbard - Stratonovich transformation. Notice that the 4-point interactions have been eliminated and the Lagrangian is quadratic in the quark fields.
4.2.2
The Mean Field Approximation
The next step we will take is to treat the fields introduced above in the mean field approximation. This consists in replacing the auxiliary fields by the values for which the action
is an extremum and neglecting fluctuations around those values, which implies dropping
the functional integral over those fields. In order to see that we start by the partition
function, written in the compact form:
Z
Z=
d[ψ̄, ψ, α, β, ∆∗ , ∆] eS[ψ̄,ψ,α,β,∆
∗ ,∆]
Taking advantage of the fact that the integral of a derivative of a function that vanishes
at the limits of integration is zero, we differentiate the partition function with respect to
α, β, ∆∗ , ∆, for example:
Z
δ S[ψ̄,ψ,α,β,∆∗ ,∆]
e
δαa
¶
µ
Z
δS
∗
∗
eS[ψ̄,ψ,α,β,∆ ,∆]
= d[ψ̄, ψ, α, β, ∆ , ∆]
δαa
0=
d[ψ̄, ψ, α, β, ∆∗ , ∆]
4.2 The Grand Canonical Potential
50
δS
which implies δα
= 0. Let us denote the value of the α field for which the previous
a
equation holds as α̃, the ‘classical’ value. Setting the functional derivatives of the action
with respect to the rest of the fields equal to zero we get:
¡
¢
α̃a = −2G ψ̄λa ψ
¡
¢
β̃a = −2G ψ̄λa iγ5 ψ
¡
¢
˜ AB = 3 G ψ T tA λB Ciγ5 ψ ,
∆
2
(4.8)
(4.9)
(4.10)
˜ = 0. By furthermore demanding that the derivative of the
where δS/δ β̃ = δS/δ ∆
logarithm of the partition function is zero, for example δδα̃ ln Z = 0 or equivalently
­ ®
R
1
S[ψ̄,ψ,α,...] = δS = 0 we get:
d[ψ̄, ψ, α, . . .] δS
Z
δ α̃ e
δ α̃
­
®
hα̃a i = −2G ψ̄λa ψ
­
®
hβ̃a i = −2G ψ̄λa iγ5 ψ
­
®
˜ AB i = 3 G ψ T tA λB Ciγ5 ψ ,
h∆
2
We then decompose the auxiliary fields in the following manner:
˜ AB + ∆0
αa = α̃a + αa0 , βa = β̃a + βa0 , ∆AB = ∆
AB
where the primed letters correspond to fluctuations around the classical values. This shift
leaves the measure of the path integral for the partition function unaltered:
Z
Z=
0
0
˜
0 ∗ ,∆+∆
0]
˜
d[ψ̄, ψ, α0 , β 0 , ∆0∗ , ∆0 ] eS[ψ̄,ψ,α̃+α ,β̃+β ,(∆+∆ )
The MFA then consists in neglecting the fluctuations of the fields and taking them equal to
their expectation values, i.e. taking αa = hα̃a i and so on. These vacuum expectation values
are then thought as the ‘condensates’, as a non vanishing vacuum expectation value of a
Dirac bilinear indicates condensation in the corresponding channel. The partition function
now is a path integral over the fermionic fields only:
Z
ZMFA =
˜ ∗ i,h∆i]
˜
d[ψ̄]d[ψ] eS[ψ̄,ψ,hα̃i,hβ̃i,h∆
Thus, we define 18 real condensates which we will denote as (αa , βa ) from now on and
9 complex ones, ∆AB , however only the ∆AB are relevant for the CFL phase2 , see [10, 23]:
2
The ∆’s are diquark condensates in the scalar channel. The pseudoscalar diquark condensates become
important in the more realistic case of explicitly broken chiral symmetry by the strange quark, see [9].
4.2 The Grand Canonical Potential
51
­
®
αa = −2G ψ̄λa ψ
®
­
βa = −2G ψ̄λa iγ5 ψ
®
3 ­ T
∆AB =
G ψ tA λB Ciγ5 ψ .
2
(4.11)
(4.12)
(4.13)
The Lagrangian in the mean field approximation now looks as LHS :
¡
¢
¡
¢
LMFA = ψ̄ (iγ µ ∂µ − M0 + µγ0 ) ψ − αa ψ̄λa ψ − βa ψ̄λa iγ5 ψ
+
¢ ¡
¢
¤ α2 + βa2 |∆AB |2
1£ ∗ ¡ T
∆AB ψ tA λB Ciγ5 ψ + ψ̄tA λB Ciγ5 ψ̄ T ∆AB − a
−
,
2
4G
3G
(4.14)
where now αa , βa , ∆AB are given by equations (4.11)-(4.13).
There is however, yet another, less formal but equivalent approach which leads to the
same Lagrangian as above, introduced first by the authors of [33]. The MFA does not
include variations of the fields around their vacuum expectation value. We may use another approximation, similar to the MFA, without performing the Hubbard - Stratonovich
transformation. The first step would be to expand the Dirac bilinears appearing in LNJL
in equation (4.1) as:
ψ̄ΓI ψ = CI + δ ψ̄ΓI ψ ,
(4.15)
where ΓI is some operator in Dirac, color and flavor space, i.e. ΓI = {λa , λa iγ5 } and CI
is a c-number corresponding to channel I . Similar equations will hold for the bilinears
of the form ψ T CΓJ ψ and ψ̄ΓJ C ψ̄ T where now ΓJ = tA λB iγ5 . This expansion includes
variations of the fields around the c-numbers CI , CJ . Then
¡
¢2
¡
¢2
ψ̄ΓI ψ = CI2 + δ ψ̄ΓI ψ + 2 CI δ ψ̄ΓI ψ .
(4.16)
Ignoring second order terms in the variation and replacing δ ψ̄ΓI ψ back from equation
(4.15), we get:
¡
¢2
ψ̄ΓI ψ ≈ −CI2 + 2CI ψ̄ΓI ψ .
(4.17)
Repeating the same steps for the rest bilinears we cast the partition function (4.4) in the
compact form:
Z
Z≈
R
d[ψ̄]d[ψ] e
·
µ
dτ d3 x ψ̄Dψ+2CI G ψ̄ΓI ψ+ 32 G CJ
ψ̄ΓJ C ψ̄ T
ψ T CΓJ ψ ∗
+
CJ
2
2
¶
¸
−GCI2 − 3G
|CJ |2
4
, (4.18)
4.2 The Grand Canonical Potential
52
where D = iγ µ ∂µ −M +µγ0 . We may now take the functional derivative of the logarithm of
the partition function with respect to CI,J equal to zero as we did before, i.e. δCδI,J ln Z = 0,
to obtain that:
hCI i = hψ̄ΓI ψi
and
hCJ∗ i = hψ̄ΓJ C ψ̄ T i , hCJ i = hψ T CΓJ ψi .
which shows that ‘C’ is the vacuum expectation value of the bilinear, which is the‘condensate’
(given it is different than zero). The partition function (4.18) is the same as ZHS if we
1
rescale CI as CI = − 2G
{α, β} for the two values of I and CJ as CJ = (2/3G)∆. The new
fields α, β have now dimension [α] = [β] = 1 in units of mass. This choice corresponds to
the substitutions:
¡
¢2
α2
αa
ψ̄λa ψ = − a2 −
ψ̄λa ψ
4G
G
¡
¢2
β2
βa
ψ̄λa iγ5 ψ = − a2 − ψ̄λa iγ5 ψ
4G
G
and
¡
¢¡
¢
¢ ¡
¢
¤
4
2 £ ∗ ¡ T
ψ̄τA λB Ciγ5 ψ̄ T ψ T τA λB Ciγ5 ψ = − 2 |∆AB |2 +
∆AB ψ τA λB Ciγ5 ψ + ψ̄τA λB Ciγ5 ψ̄ T ∆AB .
9G
3G
Plugging in the above formulas in the Lagrangian in equation (4.1) and setting α = hαi
and so on for β, ∆ we get the Lagrangian in the mean field approximation as in equation
(4.14).
The neglect of the fluctuations of the fields around their expectation value, which is
the central aspect of the MFA is also supported by the 1/Nc expansion, where Nc is the
number of colors, as we shall see in section 4.2.4.
4.2.3
Nambu - Gor’kov formalism
In order to introduce the Nambu–Gor’kov spinors in a generic Lagrangian of the type:
¡
¢2
L = ψ̄ (iγ µ ∂µ − M0 + µγ0 ) ψ + gi ψ̄Γi ψ ,
where Γi is some local operator in Dirac, flavor and color space, we first write:
4.2 The Grand Canonical Potential
53
·
¸
¤T
¢ 1¡
¢T 2
1
1£
1¡
L = ψ̄ (iγ µ ∂µ − M0 + µγ0 ) ψ+ ψ̄ (iγ µ ∂µ − M0 + µγ0 ) ψ +gi
ψ̄Γi ψ +
ψ̄Γi ψ
,
2
2
2
2
(4.19)
The second term in the right hand side is treated as follows:
³ ←
´T
¤T
£
ψ̄ (iγ µ ∂µ − M0 + µγ0 ) ψ = −ψ T iγ µ ∂ µ − M0 + µγ0 ψ̄ T
´
³
←
= ψ T −iγ µT ∂ µ + M0T − µT γ0T ψ̄ T .
The minus sign in the first line comes from the Grassmann nature of the fields. Eventually
we are interested for the action. Thus, the first term in the above equation would be written
as (after a partial integration with a vanishing boundary term):
Z
Z
β
dτ
0
d3 x T µT ← T
ψ iγ ∂ µ ψ̄ = −
(2π)3
Z
Z
β
dτ
0
d3 x T µT
ψ iγ ∂µ ψ̄ T ,
(2π)3
and hence the second term in the right hand side of equation (4.19) is:
¡
¢
ψ T iγ µT ∂µ + M0T − µT γ0T ψ̄ T = ψ T (iCγ µ C∂µ + M0 − µCγ0 C) ψ̄ T
= ψ̄c (iγ µ ∂µ − M0 − µγ0 ) ψc .
Equation (4.18) is now written by introducing the Nambu–Gor’kov spinors as:
³
´2
L = Ψ̄G̃−1
Ψ
+
g
Ψ̄
Γ̃
Ψ
,
i
i
0
(4.20)
Ã
!
Γi 0
where now Γ̃i =
, Γci = −CΓT
i C. This procedure is needed for the first line in
0 Γci
equation (4.14). The resulting Lagrangian is now written as:
1
1
ψ̄ (iγ µ ∂µ − M0 + µγ0 ) ψ + ψ̄c (iγ µ ∂µ − M0 − µγ0 ) ψc
2
2
¢
¡
¢
1
1¡
ψ̄αa λa ψ + ψ̄c αa λT
ψ̄βa λa iγ5 ψ + ψ̄c βa λT
−
a ψc −
a iγ5 ψc
2
2
¢ ¡
¢
¤ α2 + βa2 |∆AB |2
1£ ∗ ¡
+
∆AB ψ̄c tA λB iγ5 ψ + ψ̄tA λB iγ5 ψc ∆AB − a
−
2
4G
3G
= Ψ̄KΨ − V ,
(4.21)
LMFA =
where K (the full inverse propagator) is equal to:
4.2 The Grand Canonical Potential
Ã
K=
54
iγ µ ∂µ − M0 + µγ0 − αa λa + βa λa iγ5
∆AB tA λB iγ5
∗
µ
∆AB tA λB iγ5
iγ ∂µ − M0 − µγ0 − αa λTa + βa λTa iγ5
!
,
or, in explicit form (and in momentum space):
Ã
K(p) =
1c ⊗ K1
∆AB tA ⊗ λB ⊗ iγ5
∗
∆AB tA ⊗ λB ⊗ iγ5
1c ⊗ K2
!
,
(4.22)
where
¡
¢
K1 = 1f ⊗ γ0 p0 + γ i pi − µ ⊗ γ0 − (M0 + αa λa ) ⊗ 1d − βa λa ⊗ γ5 ,
(4.23)
¡
¢
¡
¢
T
K2 = 1f ⊗ γ0 p0 + γ i pi + µ ⊗ γ0 − M0 + αa λT
a ⊗ 1d − βa λa ⊗ γ5
(4.24)
and
αa2 + βa2 |∆AB |2
+
.
(4.25)
4G
3G
The matrix K is a 2×2 block matrix, where each block is a 36×36 matrix (3c ×3f ×4D = 36)
and hence K is a 72 × 72 matrix.
Note that the steps of performing the MFA and introducing the Nambu-Gor’kov spinors
are interchangeable.
V=
4.2.4
The Grand potential in the MFA
Our object is the thermodynamic potential and the gap equation, which will be obtained
by minimizing the thermodynamic potential with respect to the condensates. In finite
temperature and density we have:
Ω(T, V, µ) = Ω(T, µ)V = −
1
ln Z(T, V, µ) ,
β
(4.26)
where V is the volume. For the rest it is most convenient to work in (iωn , p) space, rather
than (τ, x) space. We can expand the Ψ spinors in momentum-frequency space, according
to:
1 X i(ωn τ +px)
Ψ= √
e
Ψn (p) ,
V n,p
(4.27)
thus obtaining the partition function in momentum space :
Z=
Z ÃY
n,p
!
dΨn (p) exp
(
X
n,p
)
Ψ̄n (p) βK (n, p) Ψn (p) − βV V
,
(4.28)
4.2 The Grand Canonical Potential
55
where ωn = (2n + 1)πT are the fermionic Matsubara frequencies. Carrying out the integration over the Grassmann fields we obtain:
Z = eTr ln(−βK)−βV V
(4.29)
where the trace now is extended in momentum-frequency space:
X Z d3 p 1
1X
Tr ≡
tr f tr c tr D ≡ V
Tr
2 n,p
(2π)3 2
n
and is sometimes referred to as functional trace. The factor of 1/2 in front of the trace
is due to the fact that it is now performed over Nambu-Gor’kov space. Using equation
(4.26) and the identity ln det A = tr ln A we obtain the thermodynamic potential:
Ω=V −T
XZ
n
d3 p 1
ln det βK ,
(2π)3 2
(4.30)
which is properly normalized by subtracting Ω0 = Ω(T = 0, µ = 0), so that the pressure
(which is P (T, µ) = −Ω(T, µ)) is zero at vacuum, P (0, 0) = −Ω(0, 0). What remains to be
done is the evaluation of the determinant, which is a rather lengthy and technical matter
if we consider many different condensates and different masses and chemical potential for
the quarks.
4.2.5
Three degenerate flavors
In order to simplify the case above, we make the following simplifications: assume equal
masses and chemical potential for the quarks and assume a simpler form for the condensates, restricting ourselves to the color-flavor locked phase (CFL). To be more specific,
now M0 = m1f , µ = µ1f . We set the condensates αa and βa equal to zero and take
∆22 = ∆55 = ∆77 and ∆AB = ∆∗AB . The matrix K in this case takes the simpler form:
Ã
K=
γ µ pµ − M0 + µγ0
∆AB tA λB iγ5
µ
∆AB tA λB iγ5
γ pµ − M0 − µγ0
!
,
where ∆AB tA λB = ∆22 t2 λ2 + ∆55 t5 λ5 + ∆77 t7 λ7 is a matrix in flavor-color space. The
locking of flavor and color takes place by assigning to the gap matrix the matrix structure:
l αβγ
(∆tλ)αβ
²jkl
jk = −∆γ ²
(4.31)
where Greek indices refer to color and Latin to flavor. Considering a transformation in
flavor-color space, V , Ψ → V Ψ, V = U ⊗ V , U ∈ SU 3c , W ∈ SU (3)f , under which
0
0
∆γl → Uγγ 0 W ll ∆γl0 , or ∆ → U ∆V T , we can diagonalize the gap matrix, i.e. ∆γl = δlγ 1.
Then:
4.2 The Grand Canonical Potential
56
αβγ
(∆tλ)αβ
²jkγ .
jk = −∆²
(4.32)
∆AB tA λB = ∆ (t2 λ2 + t5 λ5 + t7 λ7 ) .
(4.33)
We can also write:
Consistency with equation (4.32) is obtained by identifying ²αβ3 = i(λ2 )αβ , ²αβ1 = i(λ7 )αβ ,
²αβ2 = i(λ5 )αβ , ²jk3 = i(t2 )jk and similarly for the rest.
What we are aiming at is the calculation of the determinant of the matrix K. In order
to do so, we want to replace the operators in K by their eigenvalues. Then, if ξi are the
eigenvalues,
det K =
Y
ξi .
i
The matrix K is a matrix in color, flavor, Dirac space, ie. K = κf ⊗κc ⊗κD = κij καβ κP Q =
KiαP,jβQ . Suppose that the eigenvalue problem for each of the κ’s is solved separately,
f c D
D D
ie. κij efj = λf efj , καβ ecβ = λc ecβ , κP Q eD
Q = λ eP , then eiαP = ei eα eP and K eiαP =
λf λc λD eiαP . This means that the eigenvalues of K are the product of the eigenvalues in
flavor, color and Dirac space respectively.
The next step is to multiply K by Γ0 , det Γ0 = 1 and observe that det K = det Γ0 det K =
det Γ0 K, where:
Ã
Γ0 =
1f ⊗ 1c ⊗ γ0
0
0
1f ⊗ 1c ⊗ γ0
!
.
The resulting matrix is:
Ã
!
γ0 γ µ pµ − γ0 M0 + µ
γ0 ∆AB tA λB iγ5
K̃ = Γ0 K =
γ0 ∆AB tA λB iγ5
γ0 γ µ pµ − γ0 M0 − µ
Ã
! Ã
!
p0 0
γ0 γ i pi − γ0 M0 + µ
γ0 ∆AB tA λB iγ5
=
+
.
0 p0
γ0 ∆AB tA λB iγ5
γ0 γ i pi − γ0 M0 − µ
(4.34)
We can write the result in the form:
K̃ = p0 1 − HD .
(4.35)
HD stands for ’Dirac Hamiltonian’. In our formalism p0 = iωn . Now we are able to replace
the operators by their eigenvalues. Switching to Dirac’s notation we have αi = γ5 σ i ,
αi = γ 0 γ i , β = γ0 (not to be confused with β = 1/T ). By employing the helicity operator:
4.2 The Grand Canonical Potential
h=
57
1 σ i pi
,
2 |p|
(4.36)
we can write:
γ0 γ i pi − γ0 M0 + µ = αi pi − βM0 + µ
= γ5 σ i pi − βM0 + µ
= 2γ5 h |p| − βM0 + µ
Ã
!
2h |p| + µ
M0
=
.
M0
−2h |p| + µ
(4.37)
We would also like to know the eigenvalues of t2 λ2 + t5 λ5 + t7 λ7 . Suppose:
(t2 λ2 + t5 λ5 + t7 λ7 ) e = ηe .
We can determine the eigenvalues η using a computer program, like Mathematica. The
resulting 9 eigenvalues are:
η1 = −2, η2 = η3 = η4 = −1, η5 = ... = η9 = 1 ,
and the corresponding eigenvectors:
e1 = (1, 0, 0, 0, 1, 0, 0, 0, 1)T
e2 = (0, 0, 0, 0, 0, −1, 0, 1, 0)T
e3 = (0, 0, −1, 0, 0, 0, 1, 0, 0)T
e4 = (0, −1, 0, 1, 0, 0, 0, 0, 0)T
e5 = (−1, 0, 0, 0, 0, 0, 0, 0, 1)T
e6 = (0, 0, 0, 0, 0, 1, 0, 1, 0)T
e7 = (0, 0, 1, 0, 0, 0, 1, 0, 0)T
e8 = (−1, 0, 0, 0, 1, 0, 0, 0, 0)T
e9 = (0, 1, 0, 1, 0, 0, 0, 0, 0)T
The term γ0 ∆AB tA λB iγ5 can now be written as:
γ0 ∆AB tA λB iγ5 = ∆ (t2 λ2 + t5 λ5 + t7 λ7 ) γ0 iγ5 ,
Plugging in γ0 , γ5 we get:
4.2 The Grand Canonical Potential
Ã
i∆η
0 1
−1 0
58
!
Ã
=
ˆ
0
i∆
ˆ 0
−i∆
!
.
Then the matrix −HD is replaced by:






ˆ
−2h |p| + µ
−M0
0
i∆
ˆ
−M0
2h |p| + µ
−i∆
0
ˆ
0
i∆
−2h |p| − µ
−M0
ˆ
−i∆
0
−M0
2h |p| − µ



.


(4.38)
Now we can evaluate the determinant. This is done as follows
det K̃ = det (p0 1 − HD )
=
=
=
72
YY
(iωn − ²a )
n α=1
36
YY
(iωn − ²i ) (iωn + ²i )
n i=1
36
YY
¡ 2
¢
ωn + ²2i (−1) ,
(4.39)
n i=1
where we split the range of values for the index α by assigning the positive energy solutions
to the values α = 1, . . . , 36 and the negative ones to α = 37, . . . , 72. Because the energy
eigenvalues enter as ²2α we then take the product for the values i = 1, . . . , 36. Then, by
taking the logarithm (and by suppressing the sum over n as well as a constant coming
from ln(−1) = iπ) we get:
ln det K̃ =
36
XX
n
Hence, by writing (β K̃ =
1
ln det
2
1 −1
T G0 (iωn , p))
µ
¡
¢
ln ωn2 + ²2i .
i=1
we arrive at:
¶
µ 2
¶
36
1 −1
1 XX
ωn + ²2i
G (iωn , p) =
ln
,
T 0
2 n
T
(4.40)
i=1
where ²i are the eigenvalues of HD . Next one uses the standard result for Matsubara
summations of the form, see [26]:
µ 2
¶
³
´
²i
ωn + ²2i
T X
−²i /T
ln
=
+
T
ln
1
+
e
,
2 n
T2
2
(4.41)
4.2 The Grand Canonical Potential
59
to get a formula for the thermodynamic potential in the form:
Ω=
∆2
−
G
Z
³
´i
d3 p X h ²i
−²i /T
+
T
ln
1
+
e
,
(2π)3 i 2
We can determine the four eigenvalues of the HD matrix with the help of Mathematica.
After some algebra we can write them in the form:
sµ
¶2
q
2
2
±
p + M0 ± µ + ∆2 ηj2 .
Writing Ep =
∆2
Ω=
−
G
Z
p
p2 + M02 and Eµ± = Ep ± µ we obtain for the thermodynamic potential:
"
9
1q 2
1q 2
1q 2
d3 p X 1 q 2
2η2 +
2η2 −
2η2 −
E
+
∆
E
+
∆
E
+
∆
Eµ− + ∆2 ηj2
µ+
µ−
µ+
j
j
j
2
2
2
(2π)3 j=1 2
µ
¶
µ
¶
q
q
2 +∆2 η 2 /T
2 +∆2 η 2 /T
− Eµ+
− Eµ−
j
j
+ T ln 1 + e
+ T ln 1 + e
¶
µ
¶#
µ
q
q
+ T ln 1 + e
2 +∆2 η 2 /T
Eµ+
j
+ T ln 1 + e
2 +∆2 η 2 /T
Eµ−
j
.
(4.42)
Carrying out the sum and switching to polar coordinates we arrive at:
∆2
1
Ω=
− 2
G
2π
"
Z
dp p
2
¶
µ
¶
µ
q
q
2 +4∆2 /T
2 +∆2 /T
− Eµ+
− Eµ+
+ 8T ln 1 + e
T ln 1 + e
µ
¶
µ
¶
q
q
2 +4∆2 /T
2 +∆2 /T
− Eµ−
− Eµ−
+ T ln 1 + e
+ 8T ln 1 + e
¶
µ
¶
µ
q
q
2 +4∆2 /T
2 +∆2 /T
Eµ+
Eµ+
+ 8T ln 1 + e
+ T ln 1 + e
µ
¶
µ
¶#
q
q
+ T ln 1 + e
2 +4∆2 /T
Eµ−
+ 8T ln 1 + e
2 +∆2 /T
Eµ−
.
(4.43)
This is our final expression for the grand potential.It is normalized by subtracting Ω0 ≡
Ω(0, 0).
4.2 The Grand Canonical Potential
60
Figure 4.1: A plot of Ω(µ, ∆) for µ = 550M eV . The negative part of the diagram is shown in order
to exhibit the ‘Mexican hat’ shape the plot would have if it was defined for negative condensate and
chemical potential. The minimum value of ∆ is a bit less than
4.2.6
The 1/Nc expansion
In this section we want to briefly exhibit the equivalence between the MFA and the 1/Nc
expansion. Employing the Nambu-Gor’kov formalism leads to a convenient way of writing
the MFA Lagrangian, as equation (4.21) clearly exhibits. The main benefit of this formalism is that it allows for a straightforward integration of the Grassmann fields, in contrast
to equation (4.7) which contains terms that is not clear how to integrate over, as the
integration is over ψ̄, ψ and the integrand contains ψ̄, ψ, ψ T , ψ̄ T . The partition function
in the Nambu-Gor’kov formalism reads:
Z
d[Ψ]d[αa ]d[βa ]d[∆∗AB ]d[∆AB ] eS
Z=N
µZ
Z
=N
d[Ψ]d[αa ]d[βa ]d[∆∗AB ]d[∆AB ]
exp
dτ
0
¶
Z
β
3
d x LM F A
,
(4.44)
where N is a normalization factor. Note that we didn’t write the fermionic part of the
integration measure as
Z
£ ¤
d Ψ̄ d [Ψ] ,
(4.45)
as the original integration is over ψ, ψ̄ and the Nambu Gor’kov spinor Ψ includes both
ψ, ψ̄. Now it is indeed straightforward to do the fermionic path integral, thus arriving at
(we ignore the overall normalization factor):
4.2 The Grand Canonical Potential
(Z
Z
d[Ψ]d[αa ]d[βa ]d[∆∗AB ]d[∆AB ] exp
Z=
Z
=
61
Z
β
dτ
0
"
α2 + βa2 |∆|2
d3 x Ψ̄K (α, β, ∆∗ , ∆) Ψ − a
−
4G
3G
d[αa ]d[βa ]d[∆∗AB ]d[∆AB ] exp (Seff [α, β, ∆∗ , ∆]) ,
#)
(4.46)
where
Seff
!
Ã
Z β Z
Nc
αa2 + βa2 |∆|2
3
=
Tr ln(−K) −
−
dτ d x
2
4G
3G
0
"
Ã
!#
1
αa2 + βa2 |∆|2
= Nc
Tr ln(−K) − βV
−
,
2
4G̃
3G̃
(4.47)
and we used that det A = eTr ln A . The factor of 1/2 comes because the trace is over
Nambu-Gor’kov space and it is taken over flavor, Dirac and position space. We have
defined G̃ = GNc in order to extract an overall factor of Nc in the exponent. We do that
because now we can use the saddle point method and set up the 1/Nc expansion. By
restricting ourselves to the simplified case of three degenerate flavors, and thus setting
αa = βa = 0 and ∆∗AB = ∆AB , we get for the partition function:
Z
d[∆] e S̃eff [∆]
· µ
¶¸
Z
1
∆2
= d[∆] exp Nc
Tr ln(−K̃) − βV
2
3G̃
Z
≡ d[∆] eNc A(∆) .
Z=
(4.48)
Now we may employ the saddle point method, see Appendix B, to the above integral. We
start by Taylor expanding the function A(∆) around the saddle point ∆s :
¯
δA ¯¯
A(∆) = A(∆s ) + d x1E
(∆(x1E ) − ∆s )+
δ∆(x1E ) ¯∆s
¯
Z
¯
1
δ2A
4
4
¯ (∆(x1E ) − ∆s )(∆(x2E ) − ∆s ) + · · ·
+
d x1E d x2E
2
δ∆(x1E ) δ∆(x2E ) ¯∆s
¯
Z
¯
1
δnA
4
4
¯ (∆(x1E ) − ∆s ) · · · (∆(xnE ) − ∆s )+
+
d x1E . . . d xnE
n!
δ∆(x1E ) . . . δ∆(xnE ) ¯∆s
Z
4
+ O(∆n+1 ) .
(4.49)
Since ∆s is a saddle point the
¯ first order¯ derivative terms will vanish by virtue of the
δSeff ¯
δA ¯
saddle point equations: δ∆(x) ¯ = δ∆(x)
¯ = 0. By setting ∆ − ∆s = ∆0 , the partition
∆s
∆s
function takes the form:
4.3 The Gap Equation for the diquark condensate
(
Z
Z≈e
Seff [∆s ]
0
62
¯
¯
δ2A
¯
∆0 (x1E )∆0 (x2E ) + · · ·
δ∆0 (x1E ) δ∆0 (x2E ) ¯∆0 =0
)
¯
Z
nA
¯
δ
Nc
¯
d4 x1E . . . d4 xnE
∆0 (x1E ) · · · ∆0 (xnE )
+
n!
δ∆0 (x1E ) . . . δ∆0 (xnE ) ¯∆0 =0
d[∆ ] exp
Nc
2
Z
d4 x1E d4 x2E
(4.50)
˜ = √∆0 , the coefficient of the n-th order term
Finally, by rescaling the fields ∆0 as ∆0 → ∆
Nc
√
1− n
2
will be Nc , which for n > 2 scales as 1/ Nc and thus, in the large Nc limit terms of
order greater than 2 will have a vanishing contribution. The leading order term eSeff (∆s ) is
the same as in equation (4.30), which was derived via the mean field approximation. The
saddle point ∆s corresponds to the value of the order parameter (the gap) that minimizes
the grand potential. The saddle point equation emerges as the gap equation in the context
of the MFA.
4.3
The Gap Equation for the diquark condensate
The gap equation for the gap parameter is derived by minimizing the grand potential with
respect to the condensate:
δΩ
= 0.
δ∆
After some algebra we obtain the gap equation:
∆
1
= 2
G
π
Z
(4.51)
q

2 + 4∆2
E
µ−
∆
∆
+ q

dp p2 q
tanh 
tanh 
2T
2T
2
2
2
2
Eµ+ + 4∆
Eµ− + 4∆
q

q
#
2 + ∆2
2 + ∆2
E
E
µ+
µ−
2∆
 + q 2∆
 .
+q
tanh 
tanh 
2T
2T
2
2 + ∆2
Eµ+ + ∆2
Eµ−
"
q
2 + 4∆2
Eµ+

(4.52)
By using the formula
tanh
³x´
2
=1−
ex
2
,
+1
the Fermi-Dirac distribution function,
f (²) =
1
,
+1
e²/T
(4.53)
4.3 The Gap Equation for the diquark condensate
and the notation ²a± =
63
r³
p
´2
p2 + M02 ± µ + (a∆)2 , we can rewrite the gap equation in
the form:
·
¶¸
f (²2+ ) f (²2− ) 2f (²1+ ) 2f (²1− )
dp p
+
+
+
−2
+
+
+
.
²2+ ²2− ²1+ ²1−
²2+
²2−
²1+
²1−
(4.54)
The integral in (4.54) does not converge in the limit [0, ∞), which means that we cannot solve the gap equation analytically. We are then restricted to evaluate this integral
numerically. There are a few options one may choose from in order to regularize the
above integral, such as imposing a momentum ‘cut-off’, or the Pauli-Villars regularization
scheme, see [28]. We choose to introduce a momentum cut-off, Λ.
The zero temperature limit can be found in the same way as the T 6= 0 case, by starting
from the grand potential at zero temperature. The later can be found by making use of
the identities
∆
∆
= 2
G
π
Z
2
1
1
2
2
µ
³
´
−²/T
lim T ln 1 + e
= 0,
(4.55)
³
´
lim T ln 1 + e²/T = ² .
(4.56)
T →0
T →0
The result is:
Ω(µ) ≡ lim Ω(T, µ)
=
T →0
∆2 (µ)
G
1
− 2
2π
Z
dp p2
q
q
hq
³q
´i
2 + 4∆2 +
2 + 4∆2 + 8
2 + ∆2 +
2 + ∆2
Eµ+
Eµ−
Eµ+
Eµ−
.
(4.57)
The corresponding gap equation at zero temperature is obtained in the same way as before
and reads:
∆
1
= 2
G
π
Z


∆
∆
2∆
2∆
.
+q
+q
+q
dp p2  q
2 + 4∆2
2 + 4∆2
2 + ∆2
2 + ∆2
Eµ+
Eµ−
Eµ+
Eµ−
(4.58)
We could have also arrived at the previous equation directly from (4.52) by taking the
T → 0 limit and using the Heaviside step function via:
θ (x) =
h
³ x ´i
1
lim 1 + tanh
,
2 t→0
t
(4.59)
4.3 The Gap Equation for the diquark condensate
for x =
q
2 +(α∆)2
Eµ±
2
Ãq
2 +(α∆)2
Eµ±
. Then lim tanh
2T
T →0
!
64
Ãq
= 2θ
2 +(α∆)2
Eµ±
!
2
− 1 = 1, since
the square root is always positive and thus we reproduce equation (4.58).
From equation (4.57) we can also calculate Ω0 :
9
Ω0 = − 2
π
Z
dp p
2
q
p2 + M02
= −P0 ,
(4.60)
where we used that ∆(0) = 0. This integral diverges and we will circumvent this by
integrating up to a momentum cut-off Λ. Then it can be calculated analytically and Ω0
yields:
·
µ
¶¸
q
9
Λ
2
2
4
−1
2
2
Ω0 = − 2 Λ(2Λ + M0 ) Λ + M0 − M0 sinh
.
8π
M0
(4.61)
It is clear that Ω0 is a constant depending on our choice for the cut-off parameter Λ.
4.3.1
Choice of Parameters
The parameters introduced so far are the current quark masses appearing in equation
(4.1) and which are taken equal for all three flavors m0u = m0d = m0s = m0 , the coupling
constant G and the three-momentum cut-off Λ, introduced to regulate divergent integrals.
In order to numerically solve the gap equation and calculate an equation of state we
need a set of values for those parameters. The physical quantities in our arsenal that
may help us to fix the free parameters are the pion mass and decay constant, mπ , fpi
¯ hs̄si. The former are known accurately
respectively, and the chiral condensates hūui, hddi,
from measurements of the decay π − → µ− + νµ , and are taken as fπ = 92.4M eV and
mπ = 135M eV . The value of the chiral condensates however is not accurately known; it is
deduced from sum rules at a renormalization scale of 1GeV , which give values in the range
1
1
of 190 . hūui 3 . 260M eV , and from lattice calculations, suggesting hūui 3 ≈ −230M eV ,
see [10] and references therein.
Even though we did not include the chiral condensates in the version of the NJL model
we’re studying as we are restricting ourselves in the CFL phase, we will use the constituent
quark masses in order to have better results. This means that we have to consider the
chiral condensates in order to fix our parameters. To this end, the pion decay constant
and the chiral condensate can be calculated within the framework of the NJL model. The
pion decay constant is calculated from the vacuum to one-pion axial vector current matrix
element, see figure 4.2. below:
4.3 The Gap Equation for the diquark condensate
65
Figure 4.2: The vacuum to one pion axial vector matrix element.
One would have to start from an interacting Lagrangian of the form (we momentarily turn
back to the two flavors notation, as only the up and down quarks are relevant):
Lπqq = igπqq ψ̄γ5 σ · π ψ .
(4.62)
Then the Goldberger-Treiman relation is derived in a way similar to the way we outlined
in Chapter 1 and reads, see [28]:
2
fπ2 gπqq
= M2 ,
(4.63)
where M is the constituent quark mass, given by equation (2.8) for K = 0:
M = m0 − 4Ghūui .
(4.64)
The index i has been dropped as the mass and condensates are equal for all three flavors in
our case. The coupling constant gπqq may be derived in the framework of the NJL model,
from scattering diagrams like the one below, figure 4.2:
Figure 4.3: ud → u0 d0 scattering.
The last term of equation (2.1), (ψ̄γ5 σψ)2 is the one that is responsible for the excitation
of the isovector-pseudoscalar channel to be identified a the pion. The effective interaction
stemming from the exchange of the pions expressed to leading order in Nc as an infinite
4.3 The Gap Equation for the diquark condensate
66
sum of terms in the Random Phase Approximation, see [28, 41] for details; we will only
mention the results here. The coupling constant may be expressed as:
2
gπqq
= −4iGNc I(0) ,
(4.65)
where I is the integral:
Z
I(k 2 ) =
1
d4 p
£
¤£
¤.
(2π)4 (p + 12 k)2 − M 2 (p − 21 k)2 − M 2
(4.66)
By regularizing with a 3-momentum cut-off, the pion decay constant is written, [28]:
Z
d3 p
1
p
(2π)3 p2 + M 2
· p
µ ¶¸
3M 2
Λ
2
−1
2
2
,
=
Λ Λ + M − M sinh
2
4π
M
fπ2 = 3M 2
(4.67)
Another interesting result is that the pion mass can be expressed as:
mπ = −
1
m0
,
M 4iGNf Nc I(m2π )
(4.68)
which clearly exhibits that the pion mass vanishes for zero quark current mass; thus we
need to have a non vanishing m0 .
Furthermore, other current-algebra results such as the Gell-Mann–Oakes–Renner relations,
equations (1.52)-(1.55), also hold in the context of the NJL model, see [28, 41] for details:
1
¯
m2π fπ2 = − (m0u + m0d )hūu + ddi
2
= −2m0 hūui .
(4.69)
Now we’re ready to make our choice for our parameters, by fitting fπ and mπ to
their empirical values in equation (4.67) for arbitrary Λ. We set m0 = 5.5M eV which
reproduces the pion mass. Then by using equations (4.63), (4.64) and (4.69) we get our
set of parameters:
m0 = 5.5M eV , M = 368M eV ,
1
G
= 2.344 , −hūui 3 = 241M eV.
Λ2
(4.70)
4.3 The Gap Equation for the diquark condensate
4.3.2
67
Numerical results
Using a computer program like Mathematica we can numerically solve the gap equations,
equations (4.52) and (4.58), and thus investigate on the behavior of the gap as the temperature and chemical potential changes. We will not bother with the details of the numerical
calculations here; they are given in Appendix B.2.
A three dimensional plot of ∆ = ∆(T, µ) can be seen in figure 4.4.
Figure 4.4: Plot of ∆(T, µ)
However, in order to be able to conclude for the order of the phase transitions we will also
plot solutions to the gap equation at zero temperature and finite chemical potential, and
at fixed chemical potential and finite temperature. The relevant figures are 4.5 and 4.6
respectively. By obtaining finer graphs around the critical points we can see that the phase
transition of the gap as a function of the chemical potential at zero temperature seems
to be 2nd order, while the phase transition of the gap as a function of the temperature
at fixed chemical potential is of 1st order. In the latter case we used two values for the
chemical potential, namely µ = 400M eV and µ = 600M eV , corresponding to two slices
in the ∆ − T plane in figure 4.4.
4.3 The Gap Equation for the diquark condensate
a
68
b
Figure 4.5: Plot of ∆(µ). In the left panel we see the full range for the solution of the gap equation.
In the right panel we see a detail of the graph on the left, around the critical point. We can see that the
gap reaches zero smoothly. In figure a the step between two points is 2M eV , while in b it is 1eV . The
maximum value the gap acquires is 118.5M eV . The phase transition is most likely second order, however
it could also be a cross-over ending in a second order transition.
The gap equation for fixed µ and finite temperature is shown in figure 4.6. The step
between two successive points is 0.2M eV in that graph. Reducing the step down to 5eV
did not show any difference; there is a clear jump in the condensate values, indicating a
first order phase transition.
Figure 4.6: Plot of ∆(T, µ) for fixed µ: µ = 400M eV corresponds to the black (lower) line and
µ = 600M eV , close to the cut-off value Λ = 605M eV , corresponds to the red (upper) line. The step is set
to 0.2M eV . The maximum value of the gap is 118.5M eV . The jump in the values of ∆ indicates a first
order phase transition.
Finally we can make a plot of the phase diagram in the T − µ plane with the order
parameter, the gap, appearing as a phase boundary; the CFL phase corresponds to where
∆ 6= 0. The relevant plot and a detail of it around the critical point is seen in figure 4.7.
4.4 Equation of State
69
a
b
Figure 4.7: The phase diagram in the T − µ plane. In the upper panel we see the full range for the
diagram with the gap as an order parameter. In the lower panel we see a detail of the graph above,
around the critical point. The step between successive points in the µ axis in figure b is 1eV .
Again, it is not safe to conclude about the order of the phase transition, as it could be a
cross-over or a cross-over ending in a first order phase transition. It is interesting however
to compare figure 4.6a with figure 3.4, which is an approximate diagram for QCD for three
degenerate flavors, including the chiral condensates.
4.4
Equation of State
After obtaining the thermodynamic potential for our simplified case, we now turn to the
equation of state (EoS from now on) this model leads to. Our final object is to solve the
Tolman-Oppenheimer-Volkov equations (see following section); thus we need the energy
density as a function of pressure. This will be obtained via the following thermodynamic
relations:
² (T, µ) = T s (T, µ) − P (T, µ) +
and
X
i
µi ni (T, µ)
(4.71)
4.4 Equation of State
70
P (T, µ) = −Ω (T, µ) ,
n (T, µ) = −∂Ω (T, µ) /∂µ,
s (T, µ) = ∂Ω (T, µ) /∂T .
(4.72)
Before turning to the zero temperature limit, we need to evaluate the number and entropy
density; then we shall worry with the T = 0 case. To this end, equation (4.58) will take
the form:
² (µ) = −P (µ) +
X
i
µi ni (µ) .
(4.73)
The number density is:
n (T, µ) = −
∂Ω (T, µ)
∂µ
∆∆0
1
+ 2
= −2
G
2π
(
Z
dp p2
³ ² ´ E + 4∆∆0
³² ´
Eµ+ + 4∆∆0
µ−
2+
2−
tanh
tanh
−
+
²2+
2T
²2−
2T
)
·
³ ² ´ E + ∆∆0
³ ² ´¸
Eµ+ + ∆∆0
µ−
1+
1−
tanh
tanh
+8
−
,
²1+
2T
²1−
2T
(4.74)
r³
p
q
´2
2 + (a∆)2 and ∆0 = ∂∆(µ) .
p2 + M02 ± µ + (a∆)2 = Eµ±
∂µ
In a similar fashion we can obtain the entropy density, however we will be working in the
zero temperature case and as equation (4.60) shows, we do not need the entropy density.
where again ²a± =
4.4.1
Zero Temperature Case
In order to find the T = 0 limit of equation (4.61), we make use of the identity (4.55)
again:
n (µ) ≡ lim n (T, µ)
T →0
∆∆0
1
= −2
+ 2
G
2π
(
Z
2
dp p
·
¸)
Eµ+ + ∆∆0 Eµ− + ∆∆0
Eµ+ + 4∆∆0 Eµ− + 4∆∆0
−
+8
−
²2+
²2−
²1+
²1−
(4.75)
Lets turn to the pressure now; it is equal to minus the normalized grand potential at zero
temperature:
P (µ) = − [Ω(µ) − Ω0 ] ,
(4.76)
4.4 Equation of State
71
where Ω(µ) is given by equation (4.57) and Ω0 by equation (4.61). Now we have all the
ingredients to calculate the equation of state, equation (4.64), which for equal chemical
potentials for the three quark flavors yields:
²(µ) = −P (µ) + 3µ n(µ) .
(4.77)
The relevant plot can be seen in figure 4.8.
a
b
c
d
Figure 4.8: Plots of the pressure (a), energy (b), number density (c) in units of M eV 4 and the
equation of state, in units of M eV f m−3 , (d) corresponding to equations (4.75) through (4.77)
respectively.
Chapter 5
Compact Stars
Up to this point we have calculated an equation of state for color-superconducting quark
matter within the framework of the NJL model, focusing on the high density-low temperature regime. In nature however, such extreme conditions do not occur often. Then the
question of how would one test color-superconductivity arises naturally. One possibility
for matter to be in such conditions is thought to be the interior of compact stars, such as
neutron stars. In this chapter we will review some basic properties of compact stars. After
a short introduction, we will focus on the Tolman-Oppenheimer-Volkoff (TOV) equations
for the hydrostatic equilibrium of compact stars, the solution of which will give us insight
on the mass-radius relationship. The TOV equations are a system of differential equations
and can be solved once the equation of state for the matter which the star consists of is
known, from an underlying microscopic theory. In the previous chapter we calculated the
equation of state of the NJL model for matter consisting of three degenerate quark flavors
including diquark condensates; now we may apply the TOV equations in order to explore
the effects of color superconductivity in observable quantities, such as the mass-radius
relations.
5.1
Introduction
Compact stars, a term which is used to refer collectively to white dwarfs, neutron stars
and black holes, are some of the densest objects in the universe. They are the end point of
the stellar evolution of massive stars; as the thermonuclear fusion reactions diminish in the
core of stars, the thermal pressure within the core decreases. Eventually, a critical point is
reached, at which the thermal pressure cannot overcome the gravitational forces and the
star starts to collapse under it’s own gravity. Depending on the progenitor star’s initial
mass, a white dwarf, neutron star or black hole will emerge. Theoretically, the upper limit
of a white dwarfs’ mass, the ‘Chandresekhar limit’, is 1.4M¯ , where M¯ is the solar mass.
5.2 Neutron and quark stars
73
When that limit is exceeded the star will collapse in a neutron star or even a black hole.
The corresponding mass threshold for the collapse of star into a neutron star or black hole,
sometimes referred to as the ‘Oppenheimer-Volkoff limit’ is not accurately known, as the
collapse is not yet fully understood; it is believed to be about 1 − 3M¯ . If those limits are
not exceeded, then stable cores will for,, at least in white dwarfs and neutron stars; in the
former the degenerate pressure from electrons halts the collapse, while in the latter the
density due to the gravitational collapse is increasing until the nucleons (neutrons mostly)
start to overlap and the core is stabilized; what is left is what we call a neutron star. In
ordinary stars the gravitational collapse is equilibrated by thermal gas pressure. A stable
star is said to be in ‘hydrostatic equilibrium’ when the gradient of the pressure is balanced
by gravity. In table 5.1 the mass, radius and mean density of compact stars can be seen.
Object
Sun
White dwarf
Neutron star
Black hole
Mass (M¯ )
Radius (Km)
Mean density (g/cm3 )
1
1.4
1∼3
arbitrary
∼ 7 × 106
∼ 104
∼ 10
2GM c2
1
107
. 1015
∼ M/R3
Table 5.1: Properties of the three types of compact stars traditionally treated in astrophysics.
M¯ ' 2 × 1030 Kg is the mass of the Sun.
All three types of compact objects are basically static over the lifetime of the universe
(exceptions are accreting neutron stars and white dwarfs and black holes with size permitting evaporation via the emission of ‘Hawking radiation’). In the rest we will focus
on neutron stars and quark stars; the reader may consult [39] for more information on
compact objects.
5.2
Neutron and quark stars
Neutron stars were first theoretically proposed by Baade and Zwicky in 1934, in their
work on supernovae. The first calculations on neutron stars though were done in 1939 by
Oppenheimer and Volkov focusing on theoretical predictions of their properties, assuming
matter to be composed by an ideal gas of neutrons at high density. Their discovery however
took place after almost 20 years, in 1967 when Bell and Hewish discovered radio pulsars
which were identified by Gold as rotating neutron stars. Since that time and mostly due
to their extreme density neutron stars have become laboratories for testing fundamental
physics, including relativistic theories of gravity and the properties of matter at extreme
densities among others, see [17]. Typical mean values for the density of a neutron star can
5.2 Neutron and quark stars
74
be seen in Table 5.1; the reader should compare with nuclear saturation density, which is
defined as the density of a nucleon of range 1.2f m, and is equal to ρ0 = 1.5 × 1014 g/cm3 .
Let us turn to the temperature of a neutron star. At birth, the temperature in the
interior of the star reaches up to 1011 K and within a few days it cools down to less than
1010 K, mainly by neutrino emission, which is the single most important mechanism in
the cooling process of neutron stars with an age < 105 years, see [11]. Throughout it’s
early life, the star has a temperature in the range of 108 − 109 K, while the temperature
on the surface of the star is about 10% of the inner temperature. Many more properties
of neutron stars as well as information on their formation can be found in the reference
work [39], while references [21, 17] offer more up to date information. In figure 5.1, taken
from [15], a cross section of a neutron star is seen, as well as the many scenarios about
the structure of the inner core that can be found in recent literature. In the upper right
corner the ‘traditional’, or pre-color-superconductivity era neutron star structure can be
seen: it consists of a Helium-Hydrogen atmosphere, a 56 F e crust, a layer of neutrons,
protons and electrons with different sub-layers of increasing density, after which the inner
core begins. The exact composition of the later is of course still unknown and many
different scenarios are being examined. In the context of this quest many different kinds
of theoretical compact objects have emerged, aside of the traditional neutron stars; we
have pure quark stars, as is the case we will study later on, hybrid stars, which consist of a
crust as the traditional neutron stars and a core consisting of hadronic and quark matter
and so on.
In a theoretical treatment of a compact object as the ones described above, aiming at a
high degree of accuracy and contact with reality, one would have to include all the different
layers of hadronic and quark matter and the corresponding particles, like leptons, such as
the electron and muon, other than nucleons (in an analogy n : p = 1/9, see [17], 1 ) and
the possibility for condensation in other modes such as the Kaon and so on (see figure 5.1,
[10] and references therein), as far as the inner core is concerned. Including those layers
of matter means that one would have to have an equation of state for each one of them.
Many different models exist for the description of hadronic matter under such conditions,
such as the ‘Walecka model’, [31], or the non-relativistic many body ‘Brueckner-HartreeFock’ model, the ’Urbana’ model, see [10] and references therein. Moreover, the equation
of state for quark matter should have been obtained under the most general assumptions,
like different mass and chemical potential for each quark flavor, including the ‘t Hooft term
in the Lagrangian etc. Then the transitions from one state of matter to another could be
studied and realistic results could have been be obtained; however, such a detailed work
lies outside the scope of this thesis. In the literature there can be found many different
1
Actually this justifies the traditional name ‘neutron star’.
5.2 Neutron and quark stars
75
approaches for the existence of compact objects that could be neutron stars with a colorsuperconducting core (with condensate(s) in some channel(s)), or pure ‘quark’ stars etc,
see [10, 31] and references therein.
Figure 5.1: A cross-section of a compact object which could be a ‘traditional’ neutron star, or a
‘hyperon’ star, or a ‘quark’ star and so one as the arrows indicate. The picture is taken from [15].
5.2.1
Electrical and color neutrality
Up to now we have derived an equation of state for quark matter at high densities and low
temperatures, conditions that might be met in the interior of compact stars. Typically, as
was mentioned in the previous section, there are leptons and nucleons (mostly n) inside
neutron stars (or other candidates as the ones in figure 5.1). This normally imposes extra
conditions, such as electrical and color neutrality as well as β equilibrium (equal rates in
the reaction n → p + e− + ν̄e both ways). Electrical neutrality implies nu = nd = ns and
color neutrality is realized as nr = nb = ng . In fact the correct restriction would be that
5.3 The Tolman-Oppenheimer-Volkov equations
76
matter is a color singlet; however it was shown in [2] that this implies color neutrality.
Electric neutrality is imposed as:
1
1
2
(5.1)
nQ = nu − nd − ns − nl = 0 ,
3
3
3
where nl is the number density of leptons that might be present. In our case however,
we are going to assume a quark star, corresponding to the lower left arrow in figure 5.1,
omitting the hadronic matter layers; thus only u, d, s quarks with equal mass and chemical
potential are present. In this case color and electric neutrality are automatically satisfied
and no leptons are present.
5.3
The Tolman-Oppenheimer-Volkov equations
The ‘TOV equations of hydrostatic equilibrium’ for a spherical star are a set of a differential
equations for the pressure and the mass of a star, given the equation of state for the matter
the star consists of, namely:
£
¤
[p (r) + ² (r)] M (r) + 4πr3 p (r)
dp(r)
=−
,
dr
r [r − 2 M (r)]
Z
M (r) = 4π
r
r02 ² (r) dr0 ,
(5.2)
(5.3)
0
where p and ² are the pressure and energy density respectively and we are using the
gravitational units c = G = 1, see Appendix A. M (r) has the interpretation of ‘mass
inside radius r’. Then the radius R and the gravitational mass MG of the star are defined
as the as the value of r where the pressure vanishes and the corresponding value of M (r):
p(R) = 0 , MG = M (R) .
(5.4)
for a proof see Appendix B.3 and [43, 15]. For a given equation of state ² = ²(p) coming
from some microscopic theory (the NJL model in our case), the TOV equations can be
integrated numerically starting from the center of the star and moving towards the surface.
In this manner, by varying the central pressure pc = p(0) one can obtain a curve MG (R)
relating mass and radii for the specific equation of state and different initial conditions.
5.4
Numerical results
In this section we numerically integrate the TOV equations for the quark matter EoS
obtained in section 4.4.1, equation (4.77), which now must be expressed in gravitational
units. Then we set ² = ²(p(r)) in equations (5.2) and (5.3) (which should not be confused
5.4 Numerical results
77
with p(µ)). In order to avoid working with very large or small numbers we rescale M (r)
as:
M → M̃ =
M
,
M¯
(5.5)
where M¯ = 1.4766km, see Appendix A for the units. Then we may integrate equations
(5.2) and (5.3) numerically, using Mathematica. The solution depends of course on the
initial conditions for the central pressure, which corresponds to a central baryon density,
as can be seen in figure 5.2. This plot is constructed by using the equation of state.
Figure 5.2: Pressure as a function of baryon density rescaled in units of ρ0 = 0.17f m−3 .
For each value of the central pressure, a solution of the TOV equations is obtained. Then,
for each solution, we solve numerically the equation R = p−1 (0), where R is the radius of
the star, and then calculate MG ≡ M (R) from equation (5.3). Thus, for each value of the
central pressure, we get a set of values (MG , R); then by continuously varying the central
pressure pc = p(0) we can get a graph of MG (R). Our result is shown in figure 5.3.
5.4 Numerical results
78
Figure 5.3: The gravitational mass (in units of M¯ ) for different values of R (in km), for the equation
of state shown in figure 5.2.
As can bee seen above, our solutions range from 0.1 − 0.26 M¯ for the total mass and
3.91 to 5.025km for the radius. Both mass and radius are smaller than that of an average
neutron star.
Chapter 6
Concluding remarks
In this thesis we have calculated an equation of state for dense, cold, quark matter, within
the framework of the NJL model for three degenerate flavors. Following the modern trends
and trusting the plausibility of the arguments of the majority of the authors involved in
this topic, we introduced a pseudoscalar diquark condensate which reproduces the CFL
phase; the latter is thought to be the most favored candidate at the high density-low
temperature regime we are interested in. Working in the mean field approximation, we
derived the grand-canonical potential and the desired equation of state. The latter was
employed in order to calculate the mass-radii relation of a quark star configuration, by
integrating numerically the TOV equations.
Our results can be separated in two parts; the first part consists of the gap equation,
the EoS and the phase diagram of the NJL model and the second part involves the TOV
equation and the mass-radii relation for our quark star configuration. As far as the first
part is concerned, our results are quite satisfying: the diquark gap predicted is about
120MeV, in accordance with the literature, the EoS has a form similar to the ones found
in the literature and leads to phase diagram which has the form predicted by others, using
the NJL model or other approximations to QCD. As far as the mass-radii relations of a
quark star however, our approximation leads to masses and radii smaller than what is
predicted (on average) for a quark star in the CFL phase. Furthermore, we did not study
the stability of such star configurations; there are indications that pure quark stars might
be unstable; nevertheless this matter has not settled yet, as there are contradicting results
in the literature. This divergence in our results, compared to those of others, is however
plausible, as the approximation we used is somewhat crude for compact star calculations.
The next step towards realistic results would be to use the NJL model in at least 2+1
flavors (two degenerate up and down quarks plus the strange quarks), allowing for a large
current strange quark mass and to include the chiral condensates, to start with. This
would not lead to different behavior of matter on a qualitative basis, however it would be
80
better suited for calculations on compact stars. To that end, including equations of state
for hadronic matter and a leptonic contribution as well is of equal importance. Using the
CFL phase in 2+1 flavors would also imply imposing electrical and color neutrality as
well as β−equilibrium constraints, which are automatically satisfied in our case. The way
to deal with such a model however is not much different of what is presented here; the
calculations will be similar with more terms entering the game.
Appendix A
Units
In this thesis we have been using two different systems of units in order to simplify the
calculations; the so-called ‘natural’ or ‘Planck’ system and the ‘gravitational’ system of
units, see [20].
• The ‘natural’ system is defined by setting ~ = c = 1. In finite temperature field
theory however it is convenient to also set kB = 1. This means that:
~ = 1 = 6.5821 × 10−22 MeV s = 1.05457 × 10−34 kg
(A.1)
c = 1 = 2.99792458 × 108 m s−1
(A.2)
kB = 1 = 8.617343 × 10−5 ev K .
(A.3)
Connection with the SI system of units is done via 1eV = 1.60217653 × 1019 J. The
following conversion factors proved to be useful in this thesis:
1MeV3 = 1.3015 × 10−7 fm−3 .
(A.4)
1MeV4 = 2.3201 × 105 g cm−3 .
(A.5)
1MeV fm−3 = 1.7827 × 1012 g cm−3 ,
(A.6)
1MeV = 1.1605 × 1010 K .
(A.7)
In this system mass, energy and temperature can be expressed in units of energy,
MeV.
82
• The gravitational system of units is defined by setting c = G = 1:
c = 1 = 2.99792458 × 108 m s−1 ,
(A.8)
G = 1 = 6.6742 × 10−11 m3 kg−1 s−2 .
(A.9)
In the gravitational system it is most convenient to express quantities in km. The
following conversion factors are useful in calculations:
1kg = 7.4260 × 10−31 km ,
(A.10)
1g cm−3 = 7.4237 × 10−19 km−2 ,
(A.11)
1MeV fm−3 = 1.3234 × 10−6 km−2 .
(A.12)
A useful constant is the solar mass in km:
1M¯ = 1.4766km .
(A.13)
Appendix B
Miscellanies
B.1
The Saddle Point method
The saddle point method or the stationary point approximation, also referred to as the
steepest descent or Laplace’s method1 , is a way to approximate integrals of the form:
Zb
dx eM f (x) , M → ∞ ,
(B.1)
a
where f (x) is some twice differentiable function and the limits of integration a, b could
also be infinite. The idea behind this method is that if f (x) has a global maximum
at x0 , then the main contribution to the integral will stem from the values of f at the
neighborhood of that point. The reasoning is that for large M , the function eM f (x) is a
very fast ascending function; thus the main contribution to the integral will come from
f (x0 ) and the contribution of the values of f around the neighborhood of x0 will appear as
correction terms. Obviously this method works better for functions for which f (x) cannot
be very close to f (x0 ) unless x is close to x0 . In principle the same method can be applied
in case of integrals like:
Zb
dx e−M f (x) ,
lim
M →∞
a
only that now f must have a global minimum.
Considering the first case and assuming x0 6= a, b, we start by Taylor expanding f (x)
around x0 :
³
´
1
f (x) = f (x0 ) + f 0 (x0 ) + f 00 (x0 ) (x − x0 )2 + O (x − x0 )3 ,
2
1
this is actually a special case of a real function f .
(B.2)
B.1 The Saddle Point method
84
Since x0 is a global extremum, it is a stationary point, and thus f 0 (x0 ) = 0, and since it
is a maximum f 00 (x0 ) < 0. We can then approximate f (x) by truncating the above series
at second order terms, viz:
¯
1 ¯¯ 00
f (x0 )¯ (x − x0 )2
2
Plugging this back to the integral in (B.1) we get:
f (x) ≈ f (x0 ) −
Zb
Zb
dx e
M f (x)
≈ e
M f (x0 )
a
M
dx e− 2 |f
00 (x )|(x−x )2
0
0
(B.3)
,
(B.4)
a
where, if the limits of integration are taken to infinity (which can be assumed so because
the exponential decays very fast away from x0 ), the last integral is Gaussian, thus:
s
Zb
2π
eM f (x0 ) .
M |f 00 (x0 )|
dx eM f (x) ≈
a
(B.5)
This result reproduces our initial claim; the main contribution comes from eM f (x0 ) , scaled
by the square root term which involves f 00 (x0 ).
A typical example where this method can be applied, taken from calculus, is Stirling’s
law of large numbers. The starting point is:
Z∞
dx e−x xN .
N ! = Γ (N + 1) =
0
Next we rewrite this by making the change of variables x = N z, dx = N dz:
Z∞
Z∞
−N z
dz N e
N
(N z)
=N
N +1
0
dz eN (ln z−z) .
0
f 0 (1)
If we define f (z) = ln z − z, we see that
= 0 and f 00 (1) = −1 < 0. Approximating
f (z) as f (z) ≈ −1 − (1/2)(z − 1)2 and plugging it back to the above integral we get:
Z
N! ≈ N
N +1 −N
e
r
∞
dz e
0
−N
(z−1)2
2
=
2π −N N +1 √
e N
= 2πN N N e−N ,
N
which can be recast in the commonly used form:
ln N ! ≈ N ln N − N ,
by ignoring the term
1
2
ln 2πN .
B.2 The Tolman-Oppenheimer-Volkoff equations
B.2
85
The Tolman-Oppenheimer-Volkoff equations
We basically want to compute the pressure, density and gravitational fields within a static,
spherically symmetric star. This means that p = p(r); ² = ²(r). Since we will focus on
neutron stars, which are extremely dense objects and thus the gravitational fields extreme,
our framework will be general relativity.
The metric tensor will be taken to be:
£
¤
gµν = diag A(r), r2 , r2 sin2 θ, −B(r) ,
(B.6)
with inverse:
h
i
¡
¢−1
g µν = diag A−1 (r) , r−2 , r2 sin2 θ
, −B −1 (r) ,
where A(r), B(r) functions to be determined. The energy-momentum tensor is taken to
be of the form of a perfect fluid:
Tµν = p gµν + (² + p)Uµ Uν ,
(B.7)
with ², p the energy density and pressure respectively. The 4-velocity is defined so as
gµν U µ U ν = −1, and since the fluid is at rest we take Uµ to be:
³
´
´ ³
p
p
Uµ = 0, 0, 0, −1/ −g tt = 0, 0, 0, − B (r) .
The Einstein equations,
1
1
Tµν − gµν T = −
Rµν ,
2
8πG
where Rµν , T are the Ricci tensor and the trace of Tµν respectively, read:
Rrr
Rθθ
Rtt
B 00
2B
B0
4B
³
A0
A
B0
B
´
0
A
− rA
³ 0
´
0
r
= −1 + 2A
− AA + BB + A1
³
´
00
B 0 A0
B0
B0
= −B
+
+
2A
4A
A
B − rA
=
−
+
= −4πG (² − p)
= −4πG (² − p) r2
= −4πG (² + 3p) B ,
while from the zeroth component of the hydrostatic equilibrium relation:
−
we get:
√
∂p
∂
= (p + ²) λ ln −g00 ,
λ
∂x
∂x
B.2 The Tolman-Oppenheimer-Volkoff equations
86
B0
2p0
=−
,
B
p+²
(B.8)
where the prime denotes ∂/∂r. Combining these equations we can form an equation for
A(r) alone:
∂ ³r´
= 1 − 8πG².
∂r A
(B.9)
For bounded A(0) the solution is:
·
2G M (r)
A (r) = 1 −
r
¸−1
,
(B.10)
where
Z
r
M(r) ≡
4πr02 ²(r0 ) dr0 .
(B.11)
0
Finally, using the Einstein equation, as well as eq.’s (B.8) and (B.10), we can derive our
fundamental differential equation:
−r
2 ∂p (r)
∂r
·
p (r)
= GM (r) ² (r) 1 +
² (r)
¸·
4πr3 p (r)
1+
M (r)
¸·
2GM (r)
1−
r
¸−1
.
(B.12)
This equation is essentially the Tolman-Oppenheimer-Volkov equation with relativistic
corrections.
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Index
asymptotic freedom, 16
baryons, 4
Chandrasekhar limit, 72
charge algebra, 11
chiral symmetry, 7
chiral transformations, 8
color confinement, 16
color superconductivity, 18, 41
color-flavor locking, 22
compact stars, 72
Cooper pairs, 41
current algebra, 12
effective potential, 48
equation of state, 69
flavor transformations, 8
gap equation, 23, 62
for NJL model, Nf = 2, 23
Gell-Mann–Oakes–Renner relations, 16
Gell-Mann–Okubo mass relation, 16
Gell-Mann-Oakes–Renner relations, 66
glueballs, 4
gluons, 4
Goldberger-Treiman relation, 14, 15, 65
generalized, 16
grand canonical ensemble
partition function for, 30
potential for, 30
Green’s functions
thermal, for fermions, 35
hadrons, 5
Hubbard - Stratonovich transformation, 48
Hydrostatic equilibrium, 73
isospin symmetry, 7
Lagrangian, 3
for QCD, 3
Yang-Mills, 17
Matsubara
frequencies for fermions, 35
propagator, 36
Mean Field Approximation, 48
Mean Field Approximation (MFA), 49
mesons, 4
Nambu–Goldstone bosons, 13
Nambu-Gor’kov spinors, 47
neutron stars, 73
PCAC, 14
pions, 13
decay constant, 13
QCD, 3
Lagrangian for, 3
QCD phase diagram
QGP phase, 38
2SC, 39
Color-Flavor Locked phase, 39
hadronic phase, 38
INDEX
quark gluon plasma, 18
quark star, 73
quarks, 3
constituent masses for, 16, 23
current masses for, 16
saddle point method, 83
self-energy
for fermions, 37
Spontaneous Symmetry Breaking, 9
stationary point approximation, 83
Tolman-Oppenheimer-Volkoff equations, 69,
72, 76, 85
91