PION CORRELATIONS IN NUCLEAR MATTER

DOI: 10.1142/S0218301311017338

March 8, 2011 10:12 WSPC/143-IJMPE S0218301311017338

International Journal of Modern Physics EVol. 20, No. 1 (2011) 63–80c© World Scientific Publishing Company

PION CORRELATIONS IN NUCLEAR MATTER

P. K. PANDA∗ and J. DA PROVIDENCIA

Centro de Fısica Computacional, Departamento de Fısica,

Universidade de Coimbra,

P-3004-516 Coimbra, Portugal∗[email protected]

S. SARANGI

Faculty of Science & Technology, ICFAI University,

Jharkhand, Ranchi 834002, India

Accepted 17 May 2010

The saturation properties of the nuclear matter taking pion correlations into accountare studied. We construct a Bogoliubov transformation for the pion pair operators andcalculate the energy associated with the pion pairs. The pion dispersion relation isinvestigated. The correlation energy due to one-pion exchange in nuclear matter andneutron matter at random phase approximation using the generator coordinate methodis also studied. The techniques of the charged pion correlations are discussed in theneutron matter calculations. We observe that there is no sign of the pion condensation

in this model.

Keywords: Nuclear matter; generator coordinate method; random phase approximation.

PACS Number(s): 21.65.-f, 21.65.Mn, 21.65.Jk, 21.60.Jz

1. Introduction

Investigation of the possibility of microscopically occupied pion modes (pion con-

densates) in nuclear matter has been an interesting problem. Over the years, various

theoretical calculations1,2 have been carried out to study the occurrence and the

onset of the pion condensates. This has been a controversial point of discussion.

In recent years, relativistic mean field theory (RMF)3,4 has been quite successful

in describing nuclear matter and finite nuclei properties. The NN dynamics arises

in this model from the exchange of a Lorentz scalar isoscalar meson, σ, which

provides the mid-range attraction and a Lorentz vector isoscalar meson, ω, which

provides the repulsion. Also, the inclusion of the Lorentz vector isovector ρ-meson

takes care of the neutron–proton asymmetry in this model. With a small number of

parameters, the RMF model reproduces the nuclear matter saturation and describes

the bulk and the single-particle properties for the finite nuclei reasonably well.

63

http://dx.doi.org/10.1142/S0218301311017338


64 P. K. Panda, J. da Providencia & S. Sarangi

Despite the success of the RMF model, several open questions still remain un-

resolved. First, the meson fields are classical. Second, the attractive part of the

nuclear force is mediated through the hypothetical σ meson whose interpretation

as a physical particle or a resonance state is yet to be confirmed experimentally. Al-

ternatively, it has been suggested that the attractive part of the nuclear force could

be an effect of multi-pion exchanges.5–9 On aesthetic as well as phenomenological

grounds, this alternative approach is expected to add to our understanding.

The original Walecka model at the Hartree approximation does not contain a

dynamical description of the pion fields. However, the importance of the pions in NN

dynamics cannot be ignored. Realizing the essential role of pion in the description

of the nuclear medium, an alternative approach for the nuclear matter,6 deuteron,7

and for 4He 8 has been developed. In this method, the description of nuclear matter

and pure neutron matter using pion pairs through a squeezed coherent state type

of construction6–9 has been studied. The pion pair construction is based on a very

natural quantum mechanical formalism of classical fields. It simulates the effects

rendered by the σ-meson in RMF theory.

The generator coordinate method (GCM) is a technique of great physical ap-

peal which has been developed10 to describe collective oscillations in nuclei. Besides

being extensively used in nuclear structure physics, it often finds application in

various other branches of physics.11 In this work we investigate the pion condensa-

tion problem in nuclear matter and neutron matter using the generator coordinate

method. In a simplified model, the problem had been studied in Ref. 12 where a

coherent state description for the pions was used. Using the Gaussian overlap and

harmonic approximation, the Hamiltonian was diagonalized by a random phase

approximation (RPA) like canonical transformation.13 It is possible to extend this

methodology beyond the coherent state description. Besides, these earlier calcula-

tions included only up to one-pion exchange terms, in the present work, we have

included the two-pion exchange terms and have carried out an exact calculation

without any further approximation.

The present analysis also can be viewed as an extension of the mean field ap-

proach of Walecka with the classical fields replaced by quantum coherent states

for the pion pairs. This approach also has an advantage that the one-pion ex-

change correlation contributions are considered at RPA level using similar Bogoli-

ubov transformations. The criterion for pion condensation phase transition is that

the frequency of the RPA modes with the pion quantum numbers shows the ten-

dency to vanish and becomes imaginary. In the present model, we have observed no

sign of the pion condensations in the RPA modes. However, we are able to exactly

derive the nuclear potential up to two-pion exchange terms.

The outline of this paper is as follows. In Sec. 2, we derive a pion nucleon

Hamiltonian in nonrelativistic limit. We then construct the Glauber coherent states

of pions and derive the one-pion and two-pion exchanges potentials. Expressions

for two-body and three-body forces mediated by pion exchange are derived. In

Sec. 3, we then construct a Bogoliubov transformation for the pion pair operators


Pion Correlations in Nuclear Matter 65

and calculate the energy associated with the pion pairs. We next calculate the

correlation energies due to one-pion exchange in nuclear matter and neutron matter

at random phase approximation (RPA). Section 4 consists of the discussions of the

saturation properties of nuclear matter, the pion dispersion relation in the medium

and concluding remarks.

2. Formalism

2.1. Nonrelativistic Hamiltonian

The Lagrangian for the pion–nucleon system is taken as

L = ψ(iγµ∂µ −M +Gγ5φ)ψ − 1

2(∂µϕi∂

µϕi −m2ϕiϕi) , (1)

where

ψ =

(

ψI

ψII

)

is the doublet of the nucleon field with mass M , ϕi’s are pion fields and φ = τiϕi

represents the off-mass shell isospin triplet pion field with mass m. G is the pion–

nucleon coupling constant. Repeated indices indicate summation.

The representations of γ matrices are

γ =

(

0 σ

−σ 0

)

, γ0 =

(

1 0

0 −1

)

, γ5 =

(

0 −i−i 0

)

.

From the above Lagrangian, the equations of motions are

(E −M)ψI − (σ · p+ iGφ)ψII = 0 , (2)

(E +M)ψII − (σ · p− iGφ)ψI = 0 , (3)

where, E = i(∂/∂t) and p = i(∂/∂x). Eliminating the small component ψII from

Eqs. (2) and (3) we have

[(E2 −M2)− (E +M)(σ · p+ iGφ)(E +M)−1(σ · p− iGφ)]ψI = 0 . (4)

Equation (4) can be rewritten as

[E2 −M2 − p2 + iG[(σ · p), φ]−G2φ · φ]ψI = 0 . (5)

From Eq. (5), we can immediately identify the effective Hamiltonian for the nucleons

as

HN = ψ†I(x)

[

p2 +M2 − iG((σ · p)φ) +G2φ2]1/2

ψI(x)

' ψ†I(x)

[

εx −iG

2εx((σ · p)φ) + G2

2εxφ2]

ψI(x)

= H0N (x) +Hint(x) , (6)



where the single-particle nucleon energy operator εx is given by εx = (M2−∇2x)

1/2.

In the nonrelativistic assumption, we have to replace εx by M , when in a denomi-

nator and by M + p2

2M when not in a denominator. Now the effective Hamiltonian

becomes

H(x) = H0N (x) +Hint(x) +HM (x) , (7)

where the free nucleon part H0N (x) is given by

H0N (x) = ψ†(x)

(

M − ∇2x

2M

)

ψ(x) , (8)

the πN interaction Hamiltonian is provided by

Hint(x) = ψ†(x)

[

− iG

2M((σ · p) φ) + G2

2Mφ2]

ψ(x) , (9)

and the free meson part HM (x) is defined as

HM (x) =1

2

[

ϕ2i + (∇ϕi) · (∇ϕi) +m2ϕ2

i

]

. (10)

We expand the pion field operator ϕi(x) in terms of the creation and annihilation

operators of off-mass shell pions satisfying equal time commutation relation as

ϕi(x) =1√2ωx

(ai(x)† + ai(x)), ϕi(x) = i

√

ωx

2(ai(x)

† − ai(x)) , (11)

with energy ωx = (m2 −∇2x)

1/2.

2.2. Glauber coherent state

Let us consider the Hamiltonian involving pion terms from Eqs. (9) and (10). For

an N nucleon system, with the nucleons fixed at xβ , β = 1, . . . , N , we have

H =∑

q,j

ωqa†q,jaq,j −

∑

β

∑

q,j

eiq·xβG

2M√

2ωq

(iσ · qτj)β(a†q,j + a−q,j)

+∑

q,q′,j

ei(q+q′)·xβG2

2M 2ωq

(a†q,ja†−q′,j + a†q,ja−q′,j + a†−q,jaq′,j + a−q,jaq′,j),

(12)

with ωq = (m2+q2)1/2, q and j being the summation indices over momentum and

isospin space, respectively. We treat it in perturbation theory, the two-pion part

being the perturbation. Consider the Glauber coherent state

|Φ〉 = exp

∑

q,i

ζq,ia†q,i

|N〉 , (13)



where ζq,i is a one-body operator which acts on the N nucleon state |N〉 with i

being the isospin summation index. The expectation value of the unperturbed part

becomes

ε0 =∑

q,j

ωqζ∗q,jζq,j −

∑

β

∑

q,j

eiq·xβG

2M√

2ωq

(iσ · qτj)β(ζ∗q,j + ζ−q,j) . (14)

At the minimum of this expectation energy, we have

∂ε0∂ζ∗q,j

= ωqζq,j −∑

β

eiq·xβG

2M(iσ · qτj)β

1√

2ωq

= 0 , (15)

which gives

ζq,j =∑

β

eiq·xβG

2M(iσ · qτj)β

1

ωq

√

2ωq

. (16)

If we ignore the self-interactions, the one-pion exchange interaction energy reads

Vπ =∑

β 6=γ

∑

q,j

eiq·(xβ−xγ)G2

4M2(σ · qτj)β(σ · qτj)γ

1

2ω2q

=G2

4M2

∑

β 6=γ

(σβ · ∇β)(σγ · ∇γ)(τβ · τγ)∑

q,j

eiq·(xβ−xγ)1

2ω2q

=G2

4M2

∑

β 6=γ

Wπβγ , (17)

where

Wπβγ = (τβ · τγ)(σβ · xβγ)(σγ · xβγ)

(

v′′(|xβγ |)−1

|xβγ|v′(|xβγ |)

)

+σβ · σγ|xβγ |

v(|xβγ |) , (18)

and

xβγ ≡ xβ − xγ , x =x

|x| , v(|x|) =∑

q

eiq·x1

ω2q

.

The two-body interaction due to the two-pion terms becomes

Wππβγ =

1

2

∑

q,q′,j

ei(q+q′)·(xβ−xγ) G4

M3

1

ω2qω

2q′

((σ · qτj)β(σ · q′τj)β+(σ · qτj)γ(σ · q′τj)γ)

=3G4

M3

∑

q,q′

ei(q+q′)·(xβ−xγ)1

ω2qω

2q′

(q · q′)

=3G4

M3

∑

q

eiq·(xβ−xγ)1

ω2q

q ·∑

q′

eiq′·(xβ−xγ)

1

ω2q′

q′

=3G4

M3(v′(|xβγ |))2 . (19)



Similarly, three-body force becomes

Wππβγδ =

G4

2M3

∑

q,q′,j

eiq·(xβ−xγ)+iq′·(xβ−xδ)1

ω2qω

2q′

(σ · q)γ(σ · q′)δ(τγ · τδ)

=G4

2M3

∑

q

eiq·(xβ−xγ)1

ω2q

(σ · q)γ∑

q′

eiq′·(xβ−xδ)

1

ω2q′

(σ · q′)δ(τγ · τδ)

= − G4

2M3(σγ · xβγ)(σδ · xβδ)(τγ · τδ)(v′(|xβγ |) v′(|xβδ|)) , (20)

with the force being 13 (W

ππβγδ +Wππ

γδβ +Wππδβγ).

3. Correlation Energy Associated with Two Pions and Bogoliubov

Transformation

We now proceed to calculate the correlation energy associated with two pions for

the nuclear matter. The quadratic terms in the pion field in Eq. (9) provide the

isoscalar scalar interaction of nucleons and thus would simulate the effects of σ-

mesons of the Walecka model.

Let us consider a pion-pair creation operator in the momentum space given as

B† =1

2

∑

k

fka†kia

†−ki , (21)

with the ansatz function f(k). We then define the unitary transformation U as

U = e(B†−B) , (22)

and note that U , operating on vacuum, creates an arbitrarily large number of scalar

isospin singlet pairs of pions corresponding to squeezed coherent states. We will

show that this is the appropriate transformation to diagonalize the pion part of the

Hamiltonian. The “pion dressing” of nuclear matter is then introduced through the

state

|Ψ〉 = U |0〉 = e(B†−B)|0〉 . (23)

We obtain

aki = U †akiU = (cosh fk)aki + (sinh fk)a†−ki , (24)

which is a Bogoliubov transformation. Here U is a unitary and Hermitian operator.

The pseudo-pions aki are the results of the unitary transformation. It can also be

easily checked that the operator aki satisfies the standard bosonic commutation

relations:

[aki, a†k′j ] = δijδk,k′ , [a†ki, a

†k′j ] = [aki, ak′j ] = 0 , (25)

and also

aki|Ψ〉 = 0 . (26)



The inverse transformation is given by

aki = (cosh fk)aki − (sinh fk)a†−ki ≡ xkaki − yka

†−ki . (27)

In momentum space the effective Hamiltonian (7) may be rewritten as

H ≈∑

p,αη

εpc†p,αηcp,αη +

∑

q,j

ωqa†q,jaq,j

+∑

pq,jαη

G2

2MωqVc†p,αηcp,αη(a

†q,ja

†−q,j + aq,ja−q,j + 2a†q,jaq,j)

−∑

pq,jαα′ηη′

G

2M√

ωqVc†p+q,αηcp,α′η′(iσ.q)αα′τj(aq,j + a†−q,j) . (28)

Here p, α and η are, respectively, the momentum, spin, and isospin quantum num-

bers of the nucleon and q and j are, respectively, the momentum and isospin labels

of the pion. c†p,αη and cp,αη are the usual nucleon creation and annihilation oper-

ators with indices as specified above. The contribution of the terms quadratic in

pion field coming from the above Hamiltonian can be rewritten as

H2π =∑

q,j

ωqa†q,jaq,j

+∑

pq,jαη

G2

2Mωq

c†p,αηcp,αη

(

a†q,ja†−q,j + aq,ja−q,j + 2a†q,jaq,j

)

=∑

q,j

ωqa†q,jaq,j +

∑

q,j

G2ρ

2Mωq

(

a†q,ja†−q,j + aq,ja−q,j + 2a†q,jaq,j

)

=∑

q,j

(

ωq +G2ρ

Mωq

)

a†q,jaq,j +∑

q,j

G2ρ

2Mωq

(

a†q,ja†−q,j + aq,ja−q,j

)

=∑

q,j

ω′qa

†q,jaq,j +

∑

q,j

g′

2

(

a†q,ja†−q,j + aq,ja−q,j

)

, (29)

where ω′q = (ωq + G2ρ

Mωq

) = ωq + g′ with g′ = G2ρMωq

. Now the equation of motion for

the pions becomes

[H2π, a†q,j] = ωqa

†q,j . (30)

This gives

ω′qxqa

†q,j + g′xqa−q,j − ω′

qyqa−q,j − g′yqa†q,j = ωq(xqa

†q,j + yqa−q,j) . (31)

The characteristic equation is∣

∣

∣

∣

(ω′q − ωq) −g′

g′ −(ω′q + ωq)

∣

∣

∣

∣

= ω2q − ω′2

q + g′2 = 0 , (32)



which gives

ωq =√

ω′2q − g′2 with xq =

√

ω′q + ωq

2ωq

yq =

√

ω′q − ωq

2ωq

. (33)

Now

H2π =∑

q,j

ωqa†q,jaq,j +

3

2

∑

q

(ωq − ω′q) . (34)

We now have to include a term which corresponds to a phenomenological repulsion

energy between the pions of a “pair” in the above Hamiltonian H2π and is given by

HRm = A

∑

q,j

eR2

πq2

a†q,jaq,j , (35)

where the two parameters A and Rπ correspond to the strength and length scale,

respectively, of the repulsion and will be determined phenomenologically. This term

amounts to imposing a cutoff on the momentum q which accounts for the fact that

large momenta are not dynamically meaningful. The high-energy pion excitations

with higher momenta are not expected to be realistic. With this repulsion term, we

now have

ω′q =

(

ωq +AeR2

πq2

+G2ρ

Mωq

)

= ωq +AeR2

πq2

+ g′ , (36)

with

g′ =G2ρ

Mωq

, ωq =√

q2 +m2 and ωq =√

ω′2q − g′2 . (37)

After transformation the Hamiltonian in Eq. (28) becomes

H '∑

p,αη

εpc†p,αηcp,αη +

∑

q,j

ωqa†q,jaq,j +

3

2

∑

q

(ωq − ω′q)

−∑

pq,jαα′ηη′

gq√

ωqVc†p+q,αηcp,α′η′(iσ · q)αα′τj(aq,j + a†−q,j) , (38)

where

gq =G(xq − yq)

2M. (39)

In the next section, we consider RPA and calculate the correlation energy associated

with one-pion exchange for nuclear matter and neutron matter.

4. Correlation Energy Associated with One-Pion Exchange

4.1. Nuclear matter

We consider a Slater determinant of plane waves

|Φ〉 =∏

αη,|p|≤pF

c†p,αη|0〉 , (40)



with |0〉 is the absolute vacuum, pF is the Fermi momentum and

cp,α,η|0〉 = 0 . (41)

Excitations with momentum transfer q are coupled to excitations with momentum

transfer −q. Thus, the wavefunction |Ψ〉 which describes such excitations should

read

|Ψ〉 = expS|Φ〉 , (42)

where

Sqj = Uq

√

Nq

∑

α,α′,η,η′p∈Ωq

〈α, η|(σ · q)τj |α′, η′〉c†p+q,α,ηcp,α′,η′

+U−q

√

Nq

∑

α,α′,η,η′p∈Ω−q

〈α, η| − (σ · q)τj |α′, η′〉c†p−q,α,ηcp,α′,η′

= UqB†q,j + U−qB

†−q,j . (43)

In the above, Nq is the normalization factor insuring

Nq

∑

αα′ηη′,p∈Ωq

|〈α, η|σ · qτj |α′, η′〉|2 = 4Nq

∑

p∈Ωq

q2 = 1 , (44)

and the domain Ωq is defined by |p+q| > pF , |p| ≤ pF . Only positive energy states

are occupied. Here, |α, η〉 denotes the spin iso-spin eigenstate, σ3|α, η〉 = α|α, η〉,and τ3|α, η〉 = η|α, η〉. The determination of the Nq is now very simple. All we need

is the volume of the intersection of two spheres of radius pF with their centers being

a distance q apart. With these assumptions, we have

N−1q =

V

(2π)34πq3

(

p2F − q2

12

)

. (45)

The transformed unperturbed Hamiltonian becomes

H0 =∑

p,αη

εpc†p,αηcp,αη+

∑

q,j

ωqa†q,jaq,j+∆, where ∆ =

3

2

∑

q

(ωq−ω′q) . (46)

The pion–nucleon coupling from Eq. (38) reads,

Hint = −∑

pq,jαα′ηη′

〈αη|i(σ · q)τj |α′η′〉 gq√

ωqVc†p+q,αηcp,α′η′(aq,j + a†−q,j) . (47)

In order to proceed, it is convenient to bosonize the Hamiltonian H , restricted to

the subspace S. This is done by the replacement Sq,j → Bq,j where Bq,j satisfy

boson commutation relations. The bosonized Hamiltonian reads as

HB = E0 +∑

q,j

(εqB†q,jBq,j −Qq(B

†−q,j +Bq,j)(aq,j + a†−q,j) + ωqa

†q,j aq,j) .

(48)



The parameters of HB, namely εq and Qq, are fixed by expectation values such as

〈Φ|BqjHB†qj |Φ〉 = EHF + εq, 〈Φ|aqjHB†

qj|Φ〉 = Qq ,

and

〈Φ|aqjB−qjH |Φ〉 = Qq . (49)

The above Hamiltonian is diagonalized by a Bogoliubov transformation of the type

Θ†(n)qj = x

(n)1 B†

qj + x(n)2 a†qj + y

(n)1 B−qj + y

(n)2 a−qj , n = 1, 2 , (50)

which leads to excitation energies and the correlation energy.

In the above

εq = 4Nqq2∑

p∈Ωq

1

2M((p+ q)2 − p2) =

4Nqq2

2M

V

(2π)34πq2p3F

3

=1

2M

4qp3F3(p2F − q2/12)

, (51)

and

Qq =

√

N−1q

2ωqVgq . (52)

The eigenfrequencies are

Ω(±)q =

1√2

√

ε2q + ω2q ±

√

(ε2q − ω2q)

2 + 16εqωqQ2q . (53)

The correlation energy becomes

Ecorr =3

2

∑

q

(Ω(+)q +Ω(−)

q − εq − ωq) . (54)

4.2. Neutron matter

The correlated Fermion wavefunction may be written as τ = −1 for neutron and

τ = 1 for proton

|Ψ〉 = expS|Φ〉, ||Φ〉 =∏

α,p|≤pF

c†|p,α,−1|0〉 , (55)

where the correlation operator reads

S = UqB†q,0 + U−qB

†−q,0 + VqB

†q,+ + V−qB

†−q,+ , (56)

with the quasi boson operators

B†q,0 =

√

Nq

∑

α,α′,p∈Ωq

〈α,−1|(σ · q)τ0|α′,−1〉c†p+q,α,−1cp,α′,−1 , (57)

B†q,+ =

√

N ′q

∑

α,α′,|p|≤pF

〈α, 1|(σ · q)τ+|α′,−1〉c†p+q,α,1cp,α′,−1 , (58)



where τ0 = τ3 and τ+ = (τ1 + iτ2)/2. Here, Nq and N ′q are normalization factors

insuring

Nq

∑

α,α′,p∈Ωq

|〈α,−1|σ · qτj |α′,−1〉|2 = 2Nq

∑

p∈Ωq

q2 = 1 , (59)

N ′q

∑

α,α′,|p|≤pF

|〈α, 1|σ · qτj |α′,−1〉|2 = 2N ′q

∑

p≤pF

q2 = 1 . (60)

As earlier, the domain Ωq is defined by |p+ q| > pF , |p| ≤ pF . The determination

of the normalization, Nq and N ′q, is now very simple. To compute Nq, all we need

is the volume of the intersection of two spheres of radius pF , their centers being a

distance q apart. We found

N−1q =

V

(2π)32πq3

(

p2F − q2

12

)

, N ′−1q =

V

(2π)38π

3q2p3F . (61)

The kinetic energy for the particle–hole pairs with momentum q becomes

εq = 2Nqq2∑

p∈Ωq

1

2M((p+ q)2 − p2) =

2Nqq2

2M

V

(2π)34πq2p3F

3

=1

2M

2qp3F3(p2F − q2/12)

, (62)

ε′q = 2Nqq2∑

p≤pF

1

2M((p+ q)2 − p2) =

q2

2M. (63)

The pion–nucleon interaction becomes

Hint = −∑

pq,αα′ηη′

〈αη|i(σ · q)τ0|α′η′〉 gq√

ωqVc†p+q,αηcp,α′η′(aq,0 + a†−q,0)

−∑

pq,αα′ηη′

〈αη|i(σ · q)τ+|α′η′〉 gq√

ωqVc†p+q,αηcp,α′η′(aq,+ + a†−q,−)

−∑

pq,αα′ηη′

〈αη|i(σ · q)τ−|α′η′〉 gq√

ωqVc†p+q,αηcp,α′η′(aq,+ + a†−q,−) . (64)

The effective bosonized Hamiltonian containing two-pion exchange becomes

HB = E0 +∑

q

εqB†q,0Bq,0 + ε′qB

†q,+Bq,+ + ωq

∑

j

a†q,jaq,j

−∑

q

Qq(B†q,0 +Bq,0)(aq,0 + a†−q,0)

−∑

q

Q′q(B

†q,+(aq,+ + a†−q,−) +Bq,+(a−q,− + a†q,+)) , (65)



where

Qq =

√

N−1q

2ωqVgq, Q′

q =

√

N ′−1q

2ωqVgq . (66)

The RPA equations are easily obtained. For the modes with charge,

[HB, (XqB†q,+ + ζqa

†q,+ + ηqa−q,−)]

= B†q,+(Xqε

′q + ζqQ

′q − ηqQ

′q + a†q,+(XqQ

′q + ζqωq) + a−q,−(XqQ

′q − ηqωq)

≡ Ωq(XqB†q,+ + ζqa

†q,+ + ηqa−q,−) . (67)

The characteristic is a cubic and becomes∣

∣

∣

∣

∣

∣

∣

∣

(ε′q − Ωq) Q′q −Q′

q

Q′q ωq − Ωq 0

Q′q 0 −ωq − Ωq

∣

∣

∣

∣

∣

∣

∣

∣

=−Ω3q+ε

′qΩ

2q+ω

2qΩq+2Q′2

qωq−ε′qω2q = 0, (68)

where the solutions are

Ω(1)q =

ε′

3+

21/3(−ε′2 − 3ω2)

3(−2ε′3 − 54Q′2ω + 18ε′ω2 +√

4(−ε′2 − 3ω2)3 + (−2ε′3 − 54Q′2ω + 18ε′ω2)2)1/3

− 1

3× 21/3

× (−2ε′3 − 54Q′2ω + 18ε′ω2 +

√

4(−ε′2 − 3ω2)3 + (−2ε′3 − 54Q′2ω + 18ε′ω2)2)1/3 ,

(69)

Ω(2)q =

ε′

3

− (1 + i√3)(−ε′2 − 3ω2)

3× 22/3(−2ε′3 − 54Q′2ω + 18ε′ω2 +√

4(−ε′2 − 3ω2)3 + (−2ε′3 − 54Q′2ω + 18ε′ω2)2)1/3

+(1 − i

√3)

6× 21/3(−2ε′

3 − 54Q′2ω + 18ε′ω2 +

√

4(−ε′2 − 3ω2)3 + (−2ε′3 − 54Q′2ω + 18ε′ω2)2)1/3,

(70)

Ω(3)q =

ε′

3

− (1− i√3)(−ε′2 − 3ω2)

3× 22/3(−2ε′3 − 54Q′2ω + 18ε′ω2 +√

4(−ε′2 − 3ω2)3 + (−2ε′3 − 54Q′2ω + 18ε′ω2)2)1/3

+(1 + i

√3)

6× 21/3(−2ε′

3 − 54Q′2ω + 18ε′ω2 +

√

4(−ε′2 − 3ω2)3 + (−2ε′3 − 54Q′2ω + 18ε′ω2)2)1/3.

(71)



In Eqs. (69) through (71), the suffixes q has been kept silent on the right handside

for the sake of brevity. Similarly

[HB, (YqB−q,+ + ζqa†−q,+ + ηqaq,−)] = Ωq(YqB−q,+ + ζqa

†−q,+ + ηqaq,−) . (72)

This leads to a similar equation with Ωq replaced by −Ωq, so that the eigenfrequen-

cies occur in pairs, ±Ωq. The correlation energy for the charged modes becomes

E′corr =

1

2

∑

q

(|Ω(1)q |+ |Ω(2)

q |+ |Ω(3)q | − ε′q − 2ωq) . (73)

The eigenfrequencies of uncharged modes are

Ω(±)q =

1√2

√

ε2q + ω2q ±

√

(ε2q − ω2q)

2 + 16εqωqQ2q . (74)

The correlation energy becomes

Ecorr =1

2

∑

q

(Ω(+)q +Ω(−)

q − εq − ωq) , (75)

which are given in earlier section.

5. Results and Discussion

We apply the formalism developed above to the cases of nuclear matter and neu-

tron matter and study the saturation properties with pion correlations taken into

account. We also explore the possibility of pion condensation in these cases. First,

we calculate the binding energy for nuclear matter. The free nucleon kinetic energy

density is obtained using Eq. (7):

hf = 〈Φ|Tr[ρNH′N (x)]|Φ〉 =

γk3f6π2

(

M +3

10

k2fM

)

. (76)

In the above equation, spin degeneracy factor γ = 4 (2) for nuclear matter (neutron

matter) and kf represents the Fermi momenta of the nucleons. The Fermi momenta

kf and the nucleon density ρ are related by kf = (6π2ρ/γ)1

3 . It is well known that

the short range interaction mediated by the isoscalar vector ω mesons plays a crucial

role in determining the saturation density. Here we introduce the energy of repulsion

by the simple form5,6

Table 1. Parameters ofthe model obtained self-consistently at saturationdensity.

a Rπ λω

(MeV) (fm) (fm2)

14.58 1.45 3.07



hω = λωρ2 , (77)

where the parameter λω is to be fixed using the saturation properties of nuclear

matter as described in Ref. 9. Thus we finally write down the binding energy per

nucleon EB of the nuclear matter:

EB =E0

ρ−M , (78)

where

E0 = hf +3

2

∑

q

ωq + hω . (79)

In the above, E0 is the energy density of nuclear matter or neutron matter without

one-pion correlation. The expression for E0 contains the three model parameters a,

Rπ, and λω as introduced in Eqs. (35) and (77). These parameters are determined

self-consistently through the saturation properties of nuclear matter at saturation

density ρ0 = 0.15 fm−3 with and without the correlations and are specified in

Table 1. While pressure P vanishes at saturation density for nuclear matter, the

values of binding energy per nucleon are chosen to be −16 MeV. In the numerical

calculations, we have used the nucleon mass M = 940 MeV, the pion masses m =

140 MeV and the omega meson mass, mω = 783 MeV, and the π − N coupling

constant G2/4π = 14.6.

We now discuss the results obtained in our calculation. We first construct a

Bogoliubov transformation for the pion-pair operator and calculate the expectation

value of the Hamiltonian with quadratic terms in pions (H2π of Eq. (29)). Thus

Fig. 1. Binding energy of nuclear matter and neutron matter as a function of nuclear density.The correlation is related to one-pion exchange.



Fig. 2. The correlation energy from one-pion exchange in nuclear matter and in neutron matter

as a function of the nuclear density.

we calculate the binding energy per nucleon EB (Eq. (78)) without correlation due

to one-pion exchange. We next calculate the correlation energy due to the one-

pion exchange in nuclear matter (Eq. (54)) and neutron matter (Eq. (73)) at RPA

using the generator coordinate method. The binding energy per nucleon EB as

a function of the density of the system is often referred as the nuclear equation

of state (EOS). In Fig. 1, we present the EOS with and without correlation for

the nuclear matter and for neutron matter. The correlation is related to one-pion

exchange. As expected, the binding energy for nuclear matter with and without

correlation initially decreases with density and reaches a minimum at ρ/ρ0 = 1

and then increases. The binding energy curve for neutron matter also exhibits the

standard result.

In Fig. 2, we show the variation of the correlation energy due to one-pion ex-

change, Ecorr as a function of density for nuclear matter and for the neutron matter.

The correlation energy initially decreases with density, then increases and eventu-

ally saturates above the nuclear saturation density.

Dispersion relation of modes with the quantum numbers of the pions in nuclear

medium is an interesting aspect. In Fig. 3, we plot the dispersion relations arising

from two-pion coherent states at different densities. The different curves signify

increase in the effective mass of the pions with increasing nuclear density. The

increase of the pion-pair energy for k/kF > 0.6 is probably an artifact of the

repulsion term of Eq. (38).

For nuclear matter, we observe two RPA modes Ω± with quantum number of the

pions. In Fig. 4, we present the dispersion relation for the RPA mode Ω− at different

densities. At saturation density, Ω− increases very slowly with momentum. However

at ρ = 4ρ0, it increases fast. In the lower panel of Fig. 4, we have plotted Ω− for

smaller densities. At small densities, Ω− increases monotonically with momentum

and corresponds to zero sound modes.



ω

Fig. 3. The dispersion relation with the quantum number of the pions (ω versus k/kF ) for nuclearmatter. The different curves signify increase in the effective mass of the pions with increasingnuclear density.

Fig. 4. The dispersion relation of the lower RPA mode Ω− with the quantum number of thepions for high densities (upper panel) and for low densities (lower panel) for nuclear matter. Thiscorresponds to zero-sound mode.

In Fig. 5, we plot the dispersion relation for the higher RPAmode Ω+ at different

densities for nuclear matter. In lower panel of Fig. 5, we plot Ω+ versus momentum

for smaller nuclear densities.

We next present the RPA modes for the neutron matter. There are three modes

for exchange of the charged pions; two of them, Ω1 and Ω2 being equal. In Fig. 6,

we plot the distinct RPA frequencies versus momentum k at different densities.

All the three RPA frequencies increase with density. In the lower panel, we plot

|Ω3| which corresponds to zero-sound mode. It is found that there is no sign of



Fig. 5. The dispersion relation of the higher RPA mode Ω+ with the quantum number of thepions at high densities (upper panel) and at low densities (lower panel) for nuclear matter.

|Ω1|

|Ω3|

Fig. 6. The dispersion relation of the two distinct RPA modes |Ω1| and |Ω3| with the quantumnumber of the pions for neutron matter.

pion condensation since the RPA frequency is an increasing function of the density

which shows no tendency to vanish and to become imaginary.

To summarize, a pion nucleon Hamiltonian in nonrelativistic limit involving

up to two-pion exchange terms has been derived. One-pion and two-pion exchange

interactions using Glauber coherent states have been discussed. A Bogoliubov trans-

formation for the pion pair operators has been constructed and the energy associ-

ated with the pion pairs for the nuclear matter and neutron matter has been exactly

calculated. This is an extension of the mean field approach of Walecka where the

classical fields are replaced by the quantum coherent states for the pion pairs. We



have also calculated the correlation energies due to one-pion exchange in nuclear

matter and neutron matter at RPA using generator coordinate method. It is found

that there is no sign of pion condensation since the RPA frequency is an increasing

function of the density which shows no tendency to vanish and to become imaginary.

Acknowledgments

One of the author (PKP) thanks the hospitality and the friendly atmosphere pro-

vided to him during the stay at Departmento de Fisica, Universidade Coimbra. This

work was partially supported by FEDER and FCT (Portugal) under the projects

PDCT/FP/64707/2006 and CERN/FP/83505/2008.

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PION CORRELATIONS IN NUCLEAR MATTER

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