At low densities a normal Fermi sea of nucleons (N) is approximated as a non-interacting gas. Under adiabatic compression, the system can become unstable with respect to the creation of real pions through the equilibrium process:
The process will occur for a sufficiently dense medium and a sufficiently attractive p-N interaction so that the total energy of the system is lowered through the incidence of a non-zero expectation value for the pion field in the ground state.
At normal nuclear density, ρ0 the combined effect of strong tensor forces and cancellation between the repulsive core and the attractive forces lead to a stable nuclear matter. At densities ρ>ρ0 certain forces whose influence were negligible at ρ0 become prominent and their effect indicate the possible existence of new phases of nuclear matter.
The condensation of a pion field in a sufficiently dense nuclear medium was
proposed in the 70s, first by A.B. Migdal and independently by Sawyer
and Scalapino (Akmal and Pandharipande, 1997; Barshey
and Brown, 1973; Baym and Flowers, 1974; Brown
and Weise, 1976). The existence of such a condensate of pseudoscalar particles
is of both nuclear and astrophysical interest since the Equation of State (EOS)
of dense matter and the properties of neutron stars are influenced by the presence
of such a condensed pion field. Migdal (1973a, b)
saw the incidence of a condensed pion field in a nuclear medium as an instability
of a Bose field. Since, his pioneering effort, different mechanisms have been
proposed to explain this instability or as being responsible for it.
The instability of the pion field can be described in terms of pion exchange
interactions. In symmetric nuclear matter, this instability results if the OPE
is sufficiently attractive at high momentum transfers, q~2-3 mπ.
This attraction is evidenced by the downward shift of unnatural parity states
from their unperturbed shell model position if repulsive correlations are not
strong enough to provide screening. Migdal (1978) investigated
the condensation of a pion field both in an external field and in a nuclear
medium. It is known that a vacuum Boson in a strong external field is restructured
with a lowering of the energy of the system. This restructuring can be regarded
as a phase transition, a pion condensed phase, since the energy gained is proportional
to the volume of the system.
The phenomenon of pion condensation has been approached from various perspectives
and directions. It has been approached from the perspective of the type of medium
in which the condensation occurs. In connection with this, various nuclear media
have been studied such as isosymmetric nuclear matter (Chanowitz
and Siemens, 1977; Migdal, 1973a, 1978)
neutron matter (Baym et al., 1975; Weise
and Brown, 1975; Wilde et al., 1978); neutron
stars (Au and Baym, 1974; Baym and
Flowers, 1974; Brown and Weise, 1976;
Khadkikar et al., 1995; Schaffner and Mishustin,
1996; Suh and Mathews, 2000).
It has also been studied from the perspective of the type of pion field, whether
charged, π± (Backman and Weise, 1975)
or uncharged, π0 (Sandler and Clark, 1981;
Tatsumi, 1980; Matsui et al.,
Condensation of a pion field has been investigated using the pion propagator
in the medium (Backman and Weise, 1975; Dawson
and Piekarewicz, 1991) or using the Equation of State (EOS) method (Au,
1976; Baym et al., 1975; Maxwell
and Weise, 1976). The pion propagator approach only helps us to obtain the
threshold conditions or critical parameters that signal the onset of an instability
in the system. It does not describe the system in the presence of the condensed
pion field. To be able to describe the system in the condensed phase we need
to derive its energy density.
A further perspective from which pion condensation has been studied is the
derivation of a finite temperature EOS for the condensed system by Toki
et al. (1978).
An approach describing a relativistic model of pion condensation has also been
carried out (Dawson and Piekarewicz, 1991; Kutschera,
1982; Nakano et al., 2001; Walecka,
The question of how best to model the π-π interactions of the condensed field and the interaction of the condensed pion field with the nucleons of the medium has resulted in the use of either the σ-model Lagrangian or the Weinberg Largrangian, both of which are chirally symmetric.
There were various objections that were made (Barshey and
Brown, 1973) against the existence of a pion condensed field in nuclei.
These objections have been shown to be baseless (Brown and
Weise, 1976; Migdal, 1973a, b,
1978; Oset et al., 1982;
Also, while the condensation of charged pions was accepted, Takatsuka
et al. (1978) reported that π0 condensation was not
seen as feasible because the energy required was thought to be prohibitive.
Working in the Alternating Layer Spin (ALS) scheme, they assumed that the existence
of a condensed π0 field is possible and results from the localization
of nucleons having a specific spin-isospin order.
CRITERIA FOR PION CONDENSATION
Green Function or Self-energy Approach
Condensation parameters are obtained using either the self-energy approach
or the Equation of State (EOS) formalism. The self-energy approach which is
also referred to as the Greens function approach enables us to describe
the medium at the threshold of condensation while the EOS method gives us a
description of the system in the presence of a condensed pion field.
There are variations of the Green function approach. In a series of papers
Migdal (1973a, b) studied the
conditions for the onset of a charged pion (p+, p-) condensate
in neutron matter. He specified the Green function of the uncondensed medium
and gave its inverse as:
the pion self-energy owing to interactions with the medium. If ωi
is a solution of the equation:
Then the condensation condition is:
Another criterion based on the Green function was derived by Bertsch
and Johnson (1975). The Green function and the equation of this theory is:
They contended that condensation will occur when two poles of D(w) on opposite sides of the real axis pinch together. This criterion is satisfied when two equal solutions of (5) can be found.
A final modification of the Green function approach was proposed by Wilde
et al. (1978) and it borrows from the works of Migdal
(1973a, b) and Bertsch and Johnson
(1975). In this scheme, the condition that a condensed pion field be formed
at a given density is satisfied when the following equations are satisfied:
Equation 6c is to be solved with the condition that:
Equations 6a and b are equivalent to
the double-root requirement of Bertsch and Johnson (1975).
Therefore, they give the condition for the realization of a condensed pion field.
The minimum density or critical density, rc for the onset of condensation
is obtained from Eq. 6c.
Equation of State (EOS) Approach
In contrast to the previous approach-the Green function or self-energy model-which
gives parameters that will signal the onset of condensation, the equation of
state approach enables us to describe the nuclear medium in the presence of
a condensed pion field.
The procedure of this approach is to construct the energy density of the medium
and then minimize it with respect to various parameters of interest (Chad-Umoren
and Alagoa, 2006). This process then enables us to obtain a description
of the pion condensed system.
Since pions are bosons, the condensate is a single macroscopically and coherently
occupied mode of the pion field. The system is described by the expectation
value of the pion field operator
which destroys negative charge. The condensate is treated as a classical coherent
In the uncondensed phase charge conservation means that. <φ> = 0
In the presence of the condensed pion field we write the equation of state of isosymmetric nuclear matter in the form:
is the nucleon quasi-particle energies resulting from the π-N interaction. Eπ is the energy density of the condensed pion filed. ρ is the density of the medium; μπ is the pion chemical potential, u is the nucleon chemical potential, k is the condensed pion momentum, k the condensate angle in the σ model and φ is the condensate wave function.
Since pions are bosons, the condensate is a single macroscopically and coherently occupied mode of the pion field. The values of the pion field system is described by the expectation operator, φ (r, t).
The Hamiltonian density, H is defined by:
and the canonical momentum, π (x) corresponding to the field, φ (x) is:
The nucleon quasi-particle energies, E± Eq. 8 are obtained by diagonalizing the nucleon part of Eq. 9.To obtain the energy density, EN we use:
The ground state of the system is to be studied as a function of the baryon density, ρ. To do this the expression for the energy of the system is written in the form:
And v' = v-½2 μπ
where, E0 is the energy of nuclear matter in the absence of condensation. Then, condensation occurs if:
we are now in a position to obtain both the parameters of the system at the threshold of condensation and a description of the system beyond the threshold, that is in the presence of the pion condensate. This is done by minimizing Eq. 14 with respect to k and θ respectively, that is:
An EOS framework has been used to study the incidence of pion condensation
in isosymmetric (N = Z) nuclear matter using the chirally symmetric Weinberg
Lagrangian (Chad-Umoren, 2005). The realization of a pion
condensed system is influenced by the interactions undergone by the pions within
the medium. An appropriate EOS must incorporate these interactions. Chad-Umoren
(2005) used an EOS of the form:
where, E0 (ρ) is the energy of nuclear matter in the absence of condensation. Eπ (μπ, k, φ) is the energy of the condensed pion field, including contributions from π-π interactions. EπN (ρ0, μπ, v, k, φ) is the energy due to the interaction of the condensed pion field with the medium. ρ is the density of the medium; θ, the condensate angle; k, the condensed pion momentum; μπ the pion chemical potential; v the nucleon chemical potential and φ the pion field.
The Weinberg Lagrangian is given by:
The covariant derivatives in the Lagrangian are given explicitly by:
where, Ψ is the nucleon field and
the pion field. fπ is the p-N p-wave coupling constant; Fπ
the pion decay constant; mπ the pion mass.
The first term of the Lagrangian is the Dirac Lagrangian; the second term generates the p-N interaction in the pseudovector coupling mode; the p-p interactions are generated by the third term which is the kinematic Lagrangian for massless pions and by the nonlinear terms proportional to (1+Fπ-2 φ2)-1, Ep(mp, k, f) come from these. The last term is the symmetry breaking term.
FACTORS AFFECTING π-CONDENSATION
There are various effects that influence the condensation of a pion field in a nuclear medium. These are often accounted for in calculations in order to make the models realistic. These modifications can be divided into two groups, namely those that enhance condensation for instance, by lowering the density at which condensation occurs and those that inhibit condensation, for instance by raising the critical density for the onset of condensation.
Effects that favour an instability of the medium include nucleon resonances (isobars and N*), p-N p-wave interactions, higher order effects on the OPEP in the N-N interaction. These generate attractive forces which lower the condensate density.
On the other hand, effects that inhibit condensation includes p-N s-wave interactions, p-p interactions, short range N-N correlations, r-meson exchange. These generate repulsive forces which raise the condensate density, even to the point where condensation is not at all possible.
Chad-Umoren and Alagoa (2006) studied the phenomenon
in the presence of the N* resonance using a Hamiltonian of the form:
where, the vector current, Vμ(3) is:
the axial-vector current, Aμ(2) is:
and Δ is the mass difference operator.
Chad-Umoren et al. (2007a) investigated the
influence of nuclear correlations, while Chad-Umoren et
al. (2007b) considered the effect of including nucleon resonance and
nuclear correlations simultaneously in the description of pion condensed isosymmetric
Nuclear correlation effects have been studied in both pure neutron matter and
isosymmetric nuclear matter using a variational approach with hypernetted chain
summation principles (Akmal and Pandharipande, 1997).
The effects have also been investigated in isosymmetric nuclear matter using
a relativistic self-energy formulation (Nakano et al.,
An important quantity in pion condensation studies is the polarization operator, Π. It forms part of a transcedental equation whose solution gives the various branches of the spectrum of excitations having the pion quantum numbers. The spectrum of solutions contains both physically acceptable solutions representing particles and superfluous ones representing antiparticles. The criterion for selecting the correct solutions is:
An analysis of the spectrum shows that instability of the spin-acoustic branch sets in at some specific density of the medium. It has been found that the π-N interaction leads to an instability of the medium by forming nucleon spin-density waves. This is interpreted as the condensation of the spin-acoustic branch.
Sawyer and Scalapino in their pioneering work studied the nature of this instability
in the case of neutron star (Brown and Weise, 1976).
They assumed that at a specific density instability sets in with respect to
Using a Hamiltonian in which the nucleons interacted with a classical π¯-field,
they were able to demonstrate this instability. Their conclusion was later demonstrated
to be erroneous from the work of Migdal (1973a, 1978)
which showed that the Sawyer-Scalapino instability did not result from Eq.
An alternative to the work of Sawyer and Scalapino was proposed by Baym
and Flowers (1974) who derived the equilibrium thermodynamic conditions
that will be obeyed at finite temperature by a system containing a condensed
In this model, the condensate is regarded as a coherent macroscopic single mode of the pion field. It is described by a complex order parameter or condensate wavefunction.
The system is modelled with a Lagrangian density, L which is a sum of free pion Lagrangian density, Lπ(0), an interaction Lagrangian density, Lint which represents the interactions of the pions and terms independent of the pion field:
The expectation value, <H> of the Hamiltonian density, H satisfies:
where, μπ is the pion chemical potential.
The variation in ρπ is given by:
substituting Eq. 20 and 21 into 19
and comparing coefficients of δ<π*> and δ<φ*>
The condensate charge density is given by:
For the reaction n:p+π to proceed as an equilibrium process, we must have μn = μp + μπ, where, μn and μp are the neutron and proton chemical potentials respectively. The field equation from which the wavefunction of the condensate is determined is:
is the source of the condensed pion field and is given by:
The ground state energy is obtained by varying the Hamiltonian with respect to the condensed fields:
The case J(r) = 0 means the absence of a condensed pion field within the medium.
An important quantity in this formulation is the pion self-energy,
which is obtained from the expansion of
to first order in <φ>:
Dashen and Manassah (1974a, b)
working in the s- model showed that a phase transition occurs when the isospin
chemical potential, μ is equal to the mass of the pion, mπ.
They proceeded from there to obtain a general relation between the phase transition
and the symmetry breaking of the chiral Hamiltonian. This approach led to the
conclusion that the phase transition is dependent on both the symmetry breaking
term of the Lagrangian and the axial current renormalization, gA.
The dependence of pion condensation on the axial current renormalization has
since been further established (Chanowitz and Siemens, 1977;
Migdal, 1978; Riska and Sarafian,
1980; Toki et al., 1978).
As a first order approximation, Baym and Flowers (1974)
assumed that the coupling of the neutrons, protons and pions was through the
non-relativistic pseudo-vector pion-nucleon coupling. To obtain the threshold
parameters of the system at condensation, they constructed the pion field equation
and the electromagnetic current using the Lagrangian of the system. This led
to a source J of the pion field given by:
and a pion field equation of the form:
They obtained the following values for a pion condensed neutron matter:
The work of Baym and Flowers (1974) showed the need
to incorporate the effect of nuclear forces. They found that the parameters
of the pion condensed system were very sensitive to the correlations resulting
from nuclear forces. In neutron star matter, the p-n interaction is substantially
more attractive than the n-n interaction. This will encourage condensation,
as it will increase μn-μp in the uncondensed
state. On the other hand, the velocity dependence of nuclear forces reduces
the mean n-p attraction as the mean relative n-p momentum increases. This works
to inhibit condensation because the neutron and proton distributions become
separated in momentum space with increasing pion condensation, so that their
interaction energy becomes less negative.
At the threshold, this effect manifests in the reduction of the nucleon effective masses by the velocity dependent forces leading to a reduction of the nucleon densities of states, hence decreasing the effectiveness of the p-wave π-N attraction in lowering the pion self-energy.
Aside neglecting the effects of nuclear correlations, Baym
and Flowers (1974) also ignored the influence of Δ resonance on the
Migdal (1978) obtained estimates of the condensation
parameters such as the critical density and the energy using the gas approximation.
In this approximation, the influence of the pions on the medium is neglected
while account is taken of the effect of the medium on the pions. A second set
of values were obtained in an appropriate many-body scheme in which the principal
processes that affect the motion of the pions within the medium were considered.
The polarization operator for N = Z obtained, incorporated the virtual transition
of a pion into a nucleon and a nucleon-hole and contributions of transitions
into N* and a nucleon-hole. Also, this operator had earlier been obtained for
an N>>Z medium (neutron star) (Migdal, 1973a).
For such a medium s-wave π -N scattering are very important while they
are found to be inessential for the:
The essential focus of Wilde et al. (1978) was
to investigate if π-N scattering plays any significant role in the condensation
of a pion field in neutron matter. Their work used the off-shell model for π-N
scattering based upon current algebra (Gross and Surya,
1993; Scadron, 1981) and a dispersion theoretical
axial-vector nucleon amplitude dominated by the Δ isobar (Helgesson
and Randrup, 1995).
The effect of π-N scattering can be incorporated in the following manner:
Assuming that the π-meson is propagating through the medium by sequential
single scatterings from the nucleons (Kargalis et al.,
1995; Jain and Santra, 1992; Johnson
et al., 1991), then the lowest order approximation relates the pion
to the off-shell π-N amplitude, Tft:
and s, t, u are Mandelstam variables.
If there is a difference between the initial and final nucleon spins and momenta,
then the final state is not only a pion, but a pion plus a particle-hole excitation
(Kargalis et al., 1995; Oset
et al., 1982). In such a situation and for π- neutron
scattering the only contribution to (28) is from the s-channel, I = 3/2
non-spin-flip forward direction amplitude, F (Scadron, 1981;
Wilde et al., 1978). Thus, we have:
To obtain an expression for Eq. 29 suitable for computation
Wilde et al. (1978) split F3/2 into
the isospin even and isospin-odd t-channel amplitudes and applied dispersion
theory to the nucleon-pole contribution. The off-shell background amplitude,
evaluated with the help of Partial Conservation of Axial Current (PCAC) and
the algebra of currents.
Their theory, which neglected the effects of N-N correlations, showed that
π-N interactions enhance condensation. They also found that π-
condensation in neutron matter occurs at approximately nuclear matter density,
ρ0. This is in agreement with the pioneering works (Brown
and Wesie, 1976; Migdal, 1978; Migdal,
1973a, b). Sawyer and co-workers (Migdal,
1978) had predicted a π- condensate in neutron star matter
at a threshold density ρc ≈ ρ0. Migdal
predicted the same threshold density but for a π0 condensate
in nuclear and neutron star matter.
CURRENT-CURRENT CORRELATION FUNCTION, <J, J>, APPROACH
Considering a p-wave pion, the corresponding vertex operator is
This vertex creates particle-hole excitations of the type (Backman
and Weise, 1975).
A particle-hole pair coupled to J = 0, T = 1 will couple to a pion according
to (Brown and Weise, 1976):
The threshold for π- condensation is signaled by a singularity
in the current correlation function, <J, J>, (Backman
and Weise, 1975; Brown and Weise, 1976) at a pion
frequency given by:
In the absence of isobars, Backman and Weise (1975)
have given an expression for <J, J>:
where, UN(k, ω) is the Lindhard function, gNN the reaction matrix.
When isobars are taken into account, the poles of <J, J> are determined by the secular equation:
Using this formalism, Backman and Weise (1975) have
obtained a critical density, ρc of 2ρ0 for π-
condensation in neutron matter. Their theory neglected the s-wave interaction
and the effect of Δ isobars.
The result of Backman and Wesie (1975) agrees with
that of Weise and Brown (1975) who studied the effects
of Δ isobars on the equation of state of pion condensed neutron matter.
They included relativistic corrections related to the Rarita-Schwinger description
of spin- 3/2 fields to take account of the fact that Δ
isobars become more influential with increase in baryon density.
Brown and Weise (1976) have used this approach to obtain
expressions for condensation threshold parameters and to show the relationship
between <J, J> and D(k, ω). They assumed an admixture of some π+
mesons with the π¯ mesons.
Solving the equation:
to satisfy the requirement of a double pole, Brown and
Weise (1976) have obtained the following critical parameters for the onset
Equation 35c gives the critical density at the threshold
of condensation that is approximately equal to nuclear matter density.
σ-MODEL OF PION CONDENSATION
The σ-model has been used to study the incidence of a condensed pion field
in neutron matter (Baym et al., 1975; Weise
and Brown, 1975), in neutron star matter (Au, 1976)
and in abnormal nuclear matter (Chanowitz and Siemens, 1977).
It has also been used to investigate the role of many-body effects on the EOS
of isosymmetric nuclear matter and neutron-rich matter (Prakash
and Ainsworth, 1987).
In this model, the nucleon mass within the medium is obtained through the coupling
of the nucleon with the scalar σ-field (Brown and Weise,
1976; Prakash and Ainsworth, 1987; Weise,
Due to its pseudoscalar nature, the pion field has zero expectaion value in
vacuum, but possesses a finite value, breaking the symmetry of the vacuum, when
the density of the medium is high enough so that there is a non-zero solution
for Eq. 2.13 (Brown
and Weise, 1976).
For infinite σ mass, using the renormalized tree approximation (Nyman
and Rho, 1976), the pion field with frequency (or chemical potential) ω
= μπ and momentum, k is given in terms of the condensate
angle, θ by:
and is accompanied by a σ field:
Therefore in the σ-model, π condensation means the realization of a finite value of the angle θ.
The σ model Lagrangian is given by Campbell et
where, γμ are 4x4 matrices and LSB is the symmetry breaking term.
Working within this model, Baym et al. (1975) have
derived an EOS for neutron matter in the presence of a condensed π-
The non-relativistic Hamiltonian of the system is a sum of two terms:
They ignored nuclear correlations and the contributions of N* resonance and gave the non-relativistic nucleon Hamiltonian, HnuČ.
The σ-model has been applied to a variety of pion condensation problems.
Before their collaborative work, Campbell et al.
(1975), Dashen and Manassah (1974a, b)
used a non-linear σ-model to prove that there is a phase transition to
a pion condensed mode when the isospin chemical potential, μπ
is equal to the pion mass, mπ. They used a Hamiltonian density
from which fermion terms had been removed and derived an effective Hamiltonian,
Heff given by:
They then went on to establish that Eq. 39 has a minimum
which is proof that there is a phase transition at:
They also established that this result was model-independent and holds generally in the SU(2) x SU(2) symmetry breaking formalism.
Chanowitz and Siemens (1977) evaluated Eq.
37 in neutron matter in order to establish the possibility of creating a
condensed pion field in abnormal nuclear matter. Abnormal nuclear matter as
proposed by Lee (1975) is a nuclear state in which at
high density the nucleon mass is zero or nearly zero.
Their computation followed standard procedure for deriving the EOS of a medium
in the presence of a condensed pion field. First, they obtained the effective
energy, Eeff of the system using the effective Hamiltonian Heff
derived from Eq. 37 which they minimized with respect to
the parameters of their theory.
With this they obtained the EOS of pion condensed abnormal nuclear matter in the form:
Equation 41 includes auxiliary equations which must be satisfied.
A is a classical field and λ2 = 50.
A pion condensed phase exists in such a medium if a minimum value of E(A,θ) can be found with θ≠0. They found out that pion condensation was very much dependent on the choice of parameters, especially the value of the renormalized axial charge of the nucleon in the medium, gA.
FINITE TEMPERATURE EQUATION OF STATE
Finite temperature has critical effects on high energy heavy-ion collisions
and in the formation of neutron stars at the center of a supernova (Akmal
et al., 1998; Khadkikar et al., 1995).
Such phenomena have substantial influence on pion condensation (Brown
et al., 1991; Helgesson and Randrup, 1995;
Tripathi and Faessler, 1983). For example, Krewald
and Negele (1980) sought to establish the existence of pion condensation
by studying spin-isospin instabilities in high energy heavy-ion collisions.
Such instabilities correspond to the onset of pion condensation (Akmal
et al., 1998; Helgesson and Randrup, 1995;
Kargalis et al., 1995; Migdal,
1978; Oset et al., 1982).
An EOS for a pion condensed medium that explicitly incorporates temperature
was derived by Toki et al. (1978) for neutron
matter and by Tripathi and Faessler (1983) for 16O.
Their work was based on the σ-model and had two basic assumptions, namely,
they neglected the thermal fluctuations of the σ field, but accounted for
the thermal fluctuations of the condensed pion field and assumed that the coupling
of the pions to the nucleon source function was responsible for the thermal
fluctuations of the condensed pion field.
A temperature range of
was considered, leading to the assumption that only the negative quasiparticle
energy levels, E-were filled. Their choice of this range was influenced by an
earlier work showing that the thermal expectation value <σ> is inversely
proportional to the temperature and disappears beyond a certain temperature,
a modified Hartree approximation in three-dimensions they found
Temperature is incorporated into the model using the grand partition function.
are the Hamiltonian, charge and baryon number operators respectively.
The thermodynamic potential:
Unlike zero temperature σ-model EOS calculations in which equations such
as Eq. 38 is minimized to obtain the energy density of the
pion condensed system, equilibrium conditions in the case of finite temperature
demands that Eq. 43 be minimized with respect to k and θ.
Additional minimization with respect to fπ is carried out in
the work of Toki et al. (1978). The rationale
is that their work treats the chiral radius fπ as a variational
parameter. Minimization with respect to fπ gives the effective
nucleon mass, m* as a function of density. They found that both fπ
and m* are inversely proportional to the density. This nontrivial dependence
of the effective nucleon mass, m* on the nuclear density was also observed in
the work of Dawson and Piekarewicz, (1991) using a relativistic
approach to pion condensation. The effect of a decreasing fπ
is that s-wave π-N interaction is increased while p-wave interaction is
unaffected. This will work to inhibit condensation. Their treatment of gA
follows that of Au and Baym (1974) where gA
is taken as a dependent parameter (gA = 2fπ (f/m)).
Following the minimization procedure, the energy density of a pion condensed neutron matter at finite temperature is:
X is determined by:
where, n(z) is the Fermi distribution function and is given by:
It was shown in the work of Toki et al. (1978)
that for neutron matter at low temperatures, the transition to a condensed pion
phase resembles a transition of van der Waals type. That is, there is a region
of negative compressibility exhibited by the equation of state. Using various
models, they showed that such a region exists only up to a critical temperature,
Tc of about 50 MeV.
The work of Tripathi and Faessler (1983) examined the
role played by Δ isobars on the condensation of pions at finite temperature
in finite nuclei such as 16O. Their work demonstrates that the influence
of the Δ isobar and renormalizations resulting from thermal excitations
increase proportionately with rise in temperature.
At asymptotically high temperatures or at asymptotically high densities, perturbative
QCD can be used to determine the properties of strongly interacting matter.
Using chiral perturbation theory (Loewe and Villavicencio,
2005) and Lattice QCD (Kogut and Sinclair, 2002),
it has been demonstrated that the condensation of charged pions occurs if the
isospin chemical potential is greater than the pion mass. Models such as the
Nambu-Jona-Lasinio (NJL) approach that incorporate quarks as microscopic degrees
of freedom make it possible to simultaneously study the effects of finite baryon
chemical potential and isospin chemical potential. Such a model has been used
to investigate the effect of charge neutrality on pion condensation at finite
temperature and density (Andersen and Kyllingstad, 2007).
The Lagrangian of the NJL model is given by:
This Lagragian has both a global SU (NC) symmetry and a U(1)B symmetry. The SUI(2)-symmetry of the Lagrangian is broken to UB(1) x UI3(1) by the inclusion of the isospin chemical potential.
To allow for both chiral and charged pion condensates non-zero expectation values are introduced for the fields σ and π1:
are quantum fluctuating fields.
The thermodynamic potential, Ω is defined by:
where, V is the volume and Seff the effective action. Explicitly we have:
In the limit T→0, Eq. 49 becomes:
Minimizing the thermodynamic potential, Ω we obtain the values of M and ρ. That is, the following gap equations are solved:
Andersen and Kyllingstad (2007) showed that chiral symmetry
is restored by finite values of the chemical potential and that there exists
a temperature dependent charge pion condensate for small chemical potentials.
EFFECT OF CHIRAL SYMMETRY ON PION CONDENSATION
Chiral symmetry (spin-isospin SU(2) x SU(2) symmetry) is considered to be intrinsically
present in nature because of the smallness of the pion mass, mπ
(Weise, 1993; Mishustin et al.,
1993). The mass of the pion is a measure of the degree of chiral symmetry
breaking, because exact chiral symmetry means mπ = 0.
Now, the axial current, Aiμ (x) is:
Conservation of the axial current is an indication of exact chiral symmetry.
But PCAC theorem states that the axial current is almost conserved and its divergence is proportional to the pion field:
Campbell et al. (1975) in their pioneering work
using the σ-model have shown that the particular form of the chiral symmetry
breaking LSB in Eq. 37 was critical for pion condensation.
Two types of symmetry-breaking in the baryon sector were used by them. These
were derived using PCAC and the divergence of the axial vector current and given
Known respectively as cosθ symmetry breaking and sin2θ symmetry breaking.
An expression for the energy of the condensed system was sought in the form:
with the condensed phase treated as a state of chiral rotation on the normal ground state.
The effective Hamiltonian of the system is obtained from Eq.
37 and is:
which in momentum space leads to an eigenvalue equation for single particle energy levels, E(p):
where, α and β are 4 x 4 matrices.
Equation 58 has the form of a Dirac equation in an external
field. Using the Foldy-Wouthuysen transformation (Amore
et al., 1996) to decouple the spin states, the Dirac Hamiltonian
density, HD is obtained.
To demonstrate the effect of symmetry breaking, Eq. 55
are substituted in turn into Eq. 56. The resultant expressions
are then minimized with respect to k and θ.
The two types of symmetry breaking were studied at three condensation angles of:
Under Sin2q symmetry breaking, the first case, θ = 0 gave the energy of the ground state of the uncondensed system. The case θ = θ0 ≠ 0 or ≠/2 led to two phase transitions-the first and second order phase transitions. The second order phase transition is possible only when the baryon density satisfies the following condition:
In the case of the cosq symmetry breaking, Campbell et
al. (1975) did not report the second order phase transition as in the
Sinθ symmetry breaking. Another important
consequence that was observed is in the nature of the variation of the condensation angle, θ. For the Sinθ symmetry breaking we have:
while for the cosq symmetry breaking we have:
We deduce from the work of Campbell et al. (1975)
that the quantitative description of a pion condensed system is dependent on
the type of symmetry breaking term.
Other works have been done to further show the effect of symmetry breaking
on pion condensation (Tatsumi, 1980). His work used
the σ-model within the alternating-layer-spin (ALS) structure (Takatsuka
et al., 1978).
The ALS structure refers to the observation that has been made that when a π field is condensed, the nucleons of the medium become localized one-dimensionally in the same direction as the condensate momentum. In this state, the spin direction changes alternately layer by layer. This formalism therefore relates pion condensation to the structure of the nucleon system.
In the study of Tatsumi (1980), the usual s-model Lagrangian,
Eq. 37, is modified by introducing polar coordinates to obtain:
In this approach, the symmetry breaking terms, Eq. 55 become:
and the Hamiltonian density is:
where, in this case ρ is the chiral radius.
It was found that the Hamiltonian density, Eq. 61 played
a more crucial role in determining the critical density, ρc
than the symmetry breaking terms, Eq. 60.
Functionally minimizing the resultant energy density with respect to each field, two sets of coupled field equations for the two sets of symmetry breaking are obtained:
By solving these coupled equations, the ground state energy density for a pion condensed medium in the ALS structure are:
where, Jn(x) are Bessel functions;
are constants and d is the layer distance.
The results gave a critical density, ρc~9ρ0
for the condensation of a π0 field in neutron matter for both
cases of symmetry breaking. This seems to show that unlike the work of Campbell
et al. (1975), the critical density is independent of the choice
of symmetry breaking term. Tatsumi (1980) attributed
this surprising result to the anharmonicity of the Hamiltonian, Eq.
In an earlier work on π0 condensation within the ALS structure
coupling rather than the σ-model, Takatsuka et al.
(1978) reported a critical density, ρc of about 0.85ρ0
for neutron matter and 0.4ρ0 for symmetric nuclear matter. Modification
of the OPEP was found to substantially raise these values.
RELATIVISTIC MODEL OF PION CONDENSATION
A relativistic field theory approach to pion condensation was developed by
authors (Chin, 1976; Dawson and
Piekarewicz, 1991; Kutschera, 1982; Walecka,
1975). The Lagrangian density, L of the model consists of a free Lagrangian
density, L0 and an interaction Lagrangian density, Lint:
The relativistic approach is different from the s-model approach enventhough both begin with a Lagrangian.
In this approach, the nucleon field, Ψ is coupled to four meson fields, viz s, ω, π and r fields. The interaction Lagrangian, Lint is:
where, gσ, gω, gπ and gρ are the coupling constants for the respective meson fields.
Their work showed the formation of a condensed pion field for ρ≥2ρ0
This model was extended by Kutschera (1982) who included
the π-ρ interaction and made the assumption that the coupling constants
for the π-ρ and ρ-N interactions were the same.
In the limit of zero baryon density, the energy density is:
The inclusion of the π-ρ interaction led to the conclusion that the model did not predict a condensed pion field in symmetrical nuclear matter.
Kutscheras conclusion was investigated by Glendenning
and Hecking (1982). They declared it erroneous and contended that the error
resulted from a wrong coupling of the ρ- meson to the isospin conserved
current, Jm. They stressed that when ρ and π mesons are
involved, a satisfactory theory can only be obtained when the ρ meson is
coupled to the entire conserved isospin current, instead of only to the first
two terms as done in Kutscheras work.
Dawson and Piekarewicz (1991) studied the stability
of uniform nuclear matter against pion condensation in a relativistic random
phase approximation (RPA) to the Walecka model. The essential feature of the
Walecka model is that nucleons interact via the exchange of σ and ω
An important quantity which enables us gain an understanding of the pion propagation
in the nuclear medium is the pion self-energy, Π(q). All the physical information
about the modification of the pion propagator as it moves within the many-body
environment is contained in the pion self-energy (Oset et
al., 1982; Dawson and Piekarewicz, 1991). Dawson
and Piekarewicz (1991) began their evaluation of the pion self-energy in
the pseudovector representation using the axial polarization tensor defined
as a time-ordered product of axial-vector currents:
where, Ψ0 is the exact nuclear ground state and Jaμ5 is the isovector axial-vector current given by:
In symmetric nuclear matter, the mean field approximation of Eq.
Due to the nature of its structure, the nucleon propagator influences the pion
self-energy. But in the mean-field approximation to the Walecka model used by
Dawson and Piekarewicz (1991), the only contribution
to the nucleon self-energy comes from its interaction with the valence positive-energy
nucleons in the medium. Neglecting the vacuum polarization part of the pion
self-energy, a consistent linear response to the mean-field is obtained.
Within the nuclear medium, the relevant pion propagator is obtained as a solution
of Dysons equation which can be derived by iterating the pion self-energy,
Eq. 70 to all orders:
A condensed pion field is said to exist in a medium when there are poles in
the in-medium pion propagator. In the space-like region of the propagator, the
poles correspond to zeros of the dimesic function, ∈ (w, q), which, in
the static limit used by Dawson and Piekarewicz (1991)
Three different sets of mean-field models were used, namely Waleckas original model, a stiff model and a soft model. No evidence for pion condensation was found in the mean-field approximation to the Walecka model. With the soft model, a qualitative agreement with conventional nonrelativistic calculations is observed. It is found in this model that for the Landau parameter g' in the range 0≤g'≤0.9, there is always a critical nuclear density, ρc for which pion condensation occurs.
For the stiff model, the third model considered, it is also found that for
certain values of g' (g'〈0.29), there is a critical density of the medium
for the onset of condensation. An interesting aspect of this work is the discovery
that unlike conventional nonrelativistic cases of pion condensation, the stiff
model of Dawson and Piekarewicz (1991) showed that there
is an upper critical density at which the condensate disappears and the normal
state is restored.
Nakano et al. (2001) have used the Green function
approach to investigate the condensation of neutral pions in an isosymmetric
nuclear matter medium. Their work accounts for the effect of the particle-hole
or D-hole excitations within the medium by giving the relativistic pion self-energy,
P(k, k0) as a sum of the nucleon particle-hole and D-hole excitations:
In the random phase approximation these self-energies are:
where, t and T are isospin operators and the G and Gmu are nucleon and delta propagators respectively which are defined in particle-hole-antiparticle (PHA) representation. Expressions for the pion self-energy are obtained when use is made of the explicit forms of these propagators.
The interaction Lagrangians are:
These Lagrangians ignore the effects of N-N short-range correlations. Nucleon
correlations modify the pion self-energies and lead to the use of the Landau-Migdal
parameters g'NN, g'NΔ and g4'ΔΔ.
Values assigned to these parameters are often influenced by the so-called universality
assumption (Nakano et al., 2001) which holds
means that spin-isospin correlations in the NN and ND channels are not much
different from each other (Oset et al., 1982).
An increasing number of experimental data on isovector spin dependent transitions
in nuclei at low momentum transfer or Gamow-Teller transitions are consistent
with this view (Toki, 2002; Nakano
et al., 2001).
The data on Gamow-Teller transitions contain elements of substantial quenching. The range of the Landau-Migdal parameters can be determined if it is assumed that the quenching results from a mixing of the D-hole excitations with the particle-hole excitations and the Gamow-Teller strength disperses to the high energy region. The quenching factor is then given by:
Using expressions of the form of Eq. 6, Nakano
et al. (2001) showed that the critical density, ρc for
the neutral pion condensation in the N = Z medium ranges from 1.5ρ0
to 4ρ0 for the range of 0.0<g'ΔΔ<1.0.
They reported that their work also shows that results obtained for Gamow-Teller
quenching based on the universality assumption are inconsistent with earlier
results from one-boson exchange models for D-hole interaction and microscopic
EFFECT OF PION CONDENSATION ON THE GRAVITATIONAL STABILITY OF NEUTRON STARS
Neutron stars result from supernova implosions (Brown and
Weise, 1976) and they are very dense. The gravitational stability of such
high density mater is determined by the balance between the pressure and the
gravitational force (Weise, 1977).
Neutron stars have masses ranging between about 0.1M0 and about
0.75 M0 with radii of the order of 10 km. Neutron star structure
is determined by the form of the Equation of State (EOS) (Lattimer
and Prakash, 2006). The maximum allowable mass of a neutron star, Mmax,
follows the form of the EOS. This critical mass is a key quantity that affects
gravitational phenomena at very high densities.
When the mass of the star is greater than Mmax, it will possess
insufficient pressure to withstand gravitational collapse, possibly into a black
hole (Baym and Pethick, 1975; Ruffini,
2000). That is, gravitational instability sets in when the mass exceeds
a certain critical value.
Neutron star models are constructed using the Tolman-Oppenheimer-Volkoff (TOV) equation:
Where P(r) is the local pressures, ρm(r) is the mass density.
The mass inside a sphere of radius r is given by:
G is the gravitational constant.
The EOS of a neutron star is given by:
When the EOS, ρ(ρ) is specified, the TOV equation is then solved to determine the star structure and properties such as the mass and radius as functions of central density.
From Eq. 3 the effect of pion condensation is incorporated through the energy density of the system, E (ρ).
Brown and Weise (1976) have obtained an EOS of the
Where, E0(ρ) is the energy density of neutron matter in the absence of condensation. S(ρ) measures the strength of the condensed pion field and is determined by:
The work neglected the dependence of density on gA*. They used an
EOS for normal neutron matter and obtained a maximum star mass, Mmax
of 1.66M0. The region of stable neutron stars extends up to Mmax.
Beyond Mmax, the star becomes unstable against gravitational collapse.
The presence of a pion condensate softens the equation of state at high density
(Suh and Mathews, 2000) and this leads to a reduction
of Mmax, the magnitude of which depends on the effective axial vector
coupling strength, gA*.
The softening of the EOS due to the incidence of pion condensation also results
in the following effects (Suh and Mathews, 2001): (1)
enhanced rate of neutron star cooling via neutrinos, (2) a possible phase transition
of neutron stars to a superdense state, (3) sudden glitches in pulsar periods
and (4) Furthermore, if the condensation of the pion fields occurs in a strong
magnetic field, it may significantly affect starquakes.
The presence of the pion condensate affects other properties of the neutron
star such as the radius and moment of inertia. Now, in addition to neutrons
and protons, the EOS of a pion condensed neutron star also involves the presence
of Δ-isobars. That is, the total Hamiltonian of the system are quasi-particle
states having Δ-isobar component. The isobars increase with increasing
density. In the presence of a pion condensed field, a massive neutron star tends
to be smaller than one without pion condensate at a given mass MG.
At critical mass, the radius decreases from about 8.5 km to between 6.5 and
7.5 km (in the presence of condensation). This is somewhat less than twice the
Schwarzschild radius, RS which has a value of 2GMG/c2.
The presence of the pion condensate also reduces the moment of inertia, especially
for densities close to the critical density for condensation. The cooling of
neutron stars is sped up by pion condensation (Campbell
et al., 1975). In this process, a baryon picks up energy equal to
the pion chemical potential, μπ from a condensed pion and
decays into another baryon and a lepton pair.
Brown and Weise (1976) included the short range correlations
between nucleons in their EOS with this the pion condensate has modest effects
on the critical star mass and moment of inertia. The work did not include the
incidence of strange baryons such as Δ and Σ hyperons and Y* resonances.
These particles will further soften the EOS thereby also reducing Mmax.
Engvik et al. (1994), have used the relativeistic
Dirac-Brueckner-Hartree-Fock (RDBHF) approach to derive an EOS for neutron stars
with effect of magnetic field on poin conddenstation. This procedure involves
the nucleon-nucleon (NN) interaction taken from meson exchange models and the
renormalized NN potential accounted for by the reaction matrix G and given by
the Bethe-Goldstone integral equation:
Where ω is the energy of the interacting nucleons, V the free NN potential, H0 the unpertubed energy of the intermediate scattering states and Q the Pauli operator which prevents scattering into occupied states. The single particle (sp) properties are described using the Dirac equation.
The relativistic EOS of the work was found to be too stiff predicting a maximum star mass, Mmax = 2.4M0 with a corresponding radius of R = 12 km. However, inclusion of pion condensation softened the EOS with corresponding reduction in the maximum mass, Mmax to 2.0 M0 with a corresponding radius of R = 10 km.
Also investigated is the effect of different proton fractions on the mass and
radius of neutron stars. Pion condensation increases proton abundance even up
to more than 40% protons, which is close to isosymmetric nuclear matter, producing
a softer EOS and smaller maximum mass though the masses are slightly larger
than the experimental values (Engvik et al., 1994),
Neutron star structure with effect of pion condensation has also been constructed
using variational chain summation techniques and the Argonne V18
two-nucleon interaction (Akmal et al., 1998).
Effect of Magnetic Field on Pion Condensation
The effect of magnetic field on pion condensation has also been investigated.
Following the usual approach neutron star matter is modelled as an ideal Fermi
gas composed of electrons, protons and neutrons. Such a system is described
completely by the number density in phase space for each species of particle
n given by:
The energy density of the system is given by:
The pressure, ρ is:
The incidence of neutral pion (π0) condensation in the presence
of a magnetic field has been investigated (Takahashi, 2006).
It is assumed that the system is a proton-neutron-electron-muon system so that
the Fermi gas model is applied. Also, the mean field approximation of the chiral
model at zero temperature was used. Assuming exact isospin symmetry, the equation
of motion is given by:
Where τ3 is the Pauli matrix for the isospin. The configuration is chosen such that the ground state expectation value of the nucleon spin is along the z-axis. Consequently, the magnetic field resulting from the aligned magnetic moments is also along the z-axis. The gauge used is:
This gives the magnetic flux density
Qτ = e (or 0) for the proton (or the neutron)
The dispersion relation is obtained from Eq. 87 and for
the neutron is given by:
The number density and the energy density are given respectively by:
The properties of the proton is affected by the presence of the magnetic field.
Under this condition, the proton is now found on the Landau level. And the procedure
to obtain the proton spectrum in the presence of the magnetic field is to substitute
with 2veB in the field free case. So that the dispersion relation for the proton
v is an integer specifying the proton Landau levels, vH gives the
harmonic oscillator mode of each component of the spinor and s = ± 1
gives the possible spin direction of the proton. For a pion condensed medium,
the protons in the Fermi sea have s = ± 1. The state with v = 0 is known
as the lowest Landau orbit and is of the s = -1 state. In the model of Takahashi
(2006) it is shown that when the pion condensate is so developed that the
Fermi energy is less than the proton mass m in vacuum, the v = 0 state is higher
in energy than the highest Landau level.
The number density and the energy density for the proton in the magnetic field are given respectively by:
Where m the maximum of pz of the proton the vth Landau level is given by:
The expressions for the properties of the electron in the magnetic field are similar to those of the protons except that the electron has a spin degree of freedom that is twice that of the proton and unlike the proton case, the lowest energy is for the lowest Landau orbit. So that for the electrons we have:
Where, the Fermi momentum in vth Landau level is given by:
Effect of Chemical Equilibrium, Charge Neutrallity and Magnetic Consistency
For the npeμ gas, the total nucleon number density and the energy density
are given respectively by:
Where, επ is the energy of the pion. Chad-Umoren
(2005) has used the Weinberg Langrangian to obtain επ
in the form:
In order to discuss the effect of the presence or absence of the magnetic field
Takahashi (2006) have adopted a simple kinetic term
for επ given by:
Following the usual EOS procedure (Chad-Umoren et al.,
2007a), the condensation parameters are obtained by minimizing the energy
per nucleon, ε'AN. In this case three conditions are imposed
on the procedure, namely:
||Chemical equilibrium: It must be attained i.e.,:
||Charge neutrality: This condition demands that the
total electric charge must be zero. That is:
||Magnetic consistency: Pion condensation and the presence
of internal magnetic fields each results in the softening of the EOS of
matter, however pion condensation has a more significant effect (Takahashi,
2006). Also, the EOS for the simultaneous presence of both condensation
and magnetism is relatively softer than when only one or the other is present.
Furthermore, π0 condensation (Takahashi,
2002) or the magnetic field (Suh and Mathews, 2001)
acting separately, increases the critical density for charged meson condensation.
Consequently, their simultaneous presence is expected to enhance this behaviour
An alternative approach is to study the pion condensed system under the influence
of a strong external magnetic field. Takahashi (2007)
considered such a system with a modification in the particle composition, made
up in this case of nucleons (p, n), the negative sigma (Σ¯) and the
leptons (e¯, μ¯). It is expected that among the hyperons, Σ¯
will have the more significant role in π0 condensation due to
its diagonal symmetric interaction.
The total Lagrangian density of the system is given by:
Electronic and muonic terms are then added to Eq. 101 with
the imposition of the charge neutrality and beta equilibrium conditions.
The dispersion relation of the charged baryon that is on the nth Landau orbit is given by:
CONCLUSION AND OUTLOOK
The phenomenon of pion condensation in various nuclear media, including isosymmetric nuclear matter and neutron stars, has been reviewed in this study. The review has shown the underlying microscopic processes that result in this phase transition. Various elements that enhance or inhibit the phenomenon has been investigated, including a detailed analysis of the mathematical principles of Green function formalism and Equation of State (EOS) approach that are usually used to study the phenomenon. Also, the review has discussed the various effects of pion condensation including such astrophysical consequences as its influence on the gravitational stability of neutron stars and the cooling of such stars.
For further research it will be interesting to investigate the influence of
pion condensation on the shell model structure of finite nuclei, the saturation
properties of nuclear matter, superfluidity and superconductivity of neutron
stars. It is now accepted that neutrons and protons in an npe gas are
superfluid and the charged pion condensate is also superfluid and superconductive
(Suh and Mathews, 2001; Sedrakian,