nucleon effective mass in cold dense matter
TRANSCRIPT
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International Journal of Modern Physics EVol. 21, No. 2 (2012) 1250009 (12 pages)c© World Scientific Publishing Company
DOI: 10.1142/S0218301312500097
NUCLEON EFFECTIVE MASS IN COLD DENSE MATTER
WANG QING-WU
Department of Physics, Sichuan University, Chengdu 610064, China
LI XIAO-YA
National Key Laboratory of Shock Wave and Detonation Physics,
Mianyang 621900, China
LU XIAO-FU
Institute of Theoretical Physics,
The Chinese Academy of Sciences, Beijing 100080, China
Department of Physics, Sichuan University, Chengdu 610064, China
Received 18 October 2011Revised 29 November 2011Accepted 5 January 2012
Published 24 February 2012
We study the effective mass of nucleon from chiral perturbation theory in the finitechemical potential. By including in the chiral Lagrangian a chemical potential conjugat-ing to the baryon number density, the calculation of integration over meson momentumhas to face complex plain which gives the chemical potential dependence of nucleon massin dense matter. The results indicate that the interaction between nucleons is attractiveat first and then repulsive as the chemical potential increases.
Keywords: Nucleon effective mass; chiral perturbation theory; dense matter.
1. Introduction
The effective mass of nucleon is of particular interest in exploring the phase diagram
of nuclear matter.1 It also plays an important role in studying compact objects,
such as neutron star.2 Since QCD theory becomes highly non-perturbative at low
energies, so many effective theory of QCD have been proposed. Among them, the
chiral perturbation theory works very well in studying the chiral extrapolation of
nucleon mass.3,4
Recently, an effective chiral Lagrangian had been proposed to explore the prop-
erty of cold dense nuclear matter.5 By including a chemical potential which relates
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to the baryon number density of QCD, the authors were able to study the baryon
number density and its susceptibility, pressure and quark condensate. Nevertheless,
the chemical potential dependence of the nucleon mass had been neglected in Ref. 5.
Starting from π −N Lagrangian6,7
Leff = Lπ + Lπ−N , (1)
the one-loop contribution to the self-energy of nucleon may be obtained and the
physical mass of the nucleon is determined then by the positions of the pole in the
two point function:∫
d4xeip.x〈0|Tψ(x)ψ(0)|0〉 = 1
m+Σ+ i/p. (2)
When cold dense matter with nonzero chemical potential is considered, an ad-
ditional term is introduced to the L , which leads to a replacement of the derivative
of the nucleon field5:
∂µΨ → ∂µΨ− δµ4µbΨ . (3)
With this replacement, the two point function is changed to
1
m+Σ(p′) + i/p′, (4)
where
p′ = p+ iµ . (5)
However, it has to be emphasized that because Σ(p) is divergent, we cannot get
Σ(p′) by a direct replacement. For the developed HBχPT, the Σ has to be separated
into two parts: infrared(I) and regular(R).6 The regular part R can be written in
the form:
R = O(p0) +O(p1) +O(p2) + · · · , (6)
where p is low energy parameter. The expansion coefficients are polynomials of ex-
ternal momenta. While the regular part can be included in the chiral Lagrangian
density, the infrared parts contain both the finite and infinite part. In the modified
minimal subtraction scheme, one obtains the renormalized finite part. In this pro-
cess, replacement of p→ p′ = p+iµ will produce other finite parts, see the following
illustration. Therefore, it is not correct to simply replace in the renormalized finite
part.
In the regularization process, to get a form of (k2 +R2) in the denominator of
the master integral
H =
∫
d4k1
((p− k)2 +m2)(M2 + k2), (7)
we need to use the Feyman parametrization
1
ab=
∫ 1
0
dz
[az + b(1− z)]2,
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Nucleon Effective Mass in Cold Dense Matter
and perform a shift k → k + zp.
When µ = 0, the shift k → k+ zp does not change the limits of the integral. In
the case µ 6= 0, the replacement p → p′ = p + iµ transforms the integration over
the fourth axis to complex plain:∫ ∞
−∞
d4k →∫ 1
0
dz
∫ ∞
−∞
d3k
∫ ∞−zp′
−∞−zp′
dk4 . (8)
The integration over k4 can be calculated as∫ ∞−zp′
−∞−zp′
dk4f(k, p′) =
∫ ∞−iµz
−∞−iµz
dk4f(k, p′)
=
∫ ∞−iµz
−∞−iµz
dk4f(k, p′)−
∫ ∞
−∞
dk4f(k, p′) +
∫ ∞
−∞
dk4f(k, p′)
=
∮
dk4f(k, p′) +
∫ ∞
−∞
dk4f(k, p′) · · · .
(9)
So, from Σ(p) to Σ(p′), a contour integral will appear. And we will see that
the integral is nonzero which makes the nucleon mass shifting from free mass and
varying as the chemical potential changes.
In the following, we will first derive the formula of the chemical dependence of
nucleon mass in the developed HBχPT in Sec. 2. Then, we will show our numerical
results and give our discussion in Sec. 3. Finally, we will give a short summary in
Sec. 4.
2. Self Energy: The Formula
2.1. µ = 0
Up to order of O(p4), the minimal relativistic effective π −N Lagrangian is given
in Refs. 3, 5 and 8:
L(1)πN = −ψ(/D +m)ψ +
1
2gAψ/uγ5ψ ,
L(2)πN = c1〈χ+〉ψψ − c2
4m2〈uµuν〉(ψDµDνψ + h.c.) +
c32uµu
µψψ ,
L(4)πN = − e1
16〈χ+〉2ψψ + · · · ,
(10)
with
Dµψ = ∂µψ + Γµψ ,
Γµ =1
2[u+, ∂µu] ,
χ+ = u+χu+ + uχ+u ,
u2 = U, uµ = iu+∂µUu+ .
(11)
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(c) (b) (c)
Fig. 1. One-loop graphs contributing to the self-energy of nucleon.
And the self-energy of nucleon up to order of O(p4) can be written directly in
Euclidian space as5,6,9,10:
Σ = −4c1M2 +Σa +Σb +Σc + e1M
4 +O(p5) , (12)
where
Σa =3g2A4F 2
π
(m− i/p)(M2I − i(m+ i/p)/pI(1) −∆N ) , (13)
Σb =3M2∆π
F 2π
(
2c1 +p2
4m2c2 − c3
)
, (14)
Σc = −4c1M2∂Σa
∂m(15)
and
∆π =M2
8π2lnM
m.
For some higher order expressions see Refs. 3 and 11. The Σa term is from one-loop
diagram shown in Fig. 1. The computation is as follows:
Σa =3g2A4F 2
π
1
(2π)4
∫
d4ki/kγ5m− i(/p − /k)
m2 + (p− k)21
M2 + k2i/kγ5 . (16)
The numerator of integrand in Eq. (16) is reduced to:
Σa → i/kγ5[m− i(/p − /k)]i/kγ5− = /k[m+ i(/p/k)]/k
= −(m− i/k)k2 − i/k/p/k
= −(m− i/k)k2 − i(2p · k − /p/k)/k
= −(m− i/p)k2 + i(k2 − 2p · k)/k . (17)
Further, we can express it as:
Σa → −(m− i/p)(k2 +M2) +M2(m− i/p) + i(−m2 − p2)/k
+ i[(p− k)2 +m2]/k . (18)
The first term on the r.h.s of Eq. (18) denotes the scalar nucleon propagator
(−∆N ) at the origin, which does not contain any infrared singularities and is equal
to zero in the infrared regularization scheme. The third term does not contribute
in view of the factor m + i/p. The integral over k of the last term is also equal
to zero as [(p − k)2 +m2] being canceled. So, in the case µ = 0, only the second
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Nucleon Effective Mass in Cold Dense Matter
term contribute to Σa. Details of the calculations of the integration over k and its
renormalization are given below.6
I = − 1
8π2
α√1− Ω2
1 + 2αΩ+ α2ArcCos
[
− α+Ω√1 + 2αΩ + α2
]
− 1
16π2
α(α +Ω)
1 + 2αΩ + α2(2 lnα+ 1) , (19)
I(1) = − 1
2p2[(−p2 −m2 +M2)I +∆π −∆N ] . (20)
Here
α =M
m,
Ω =p2 −m2 −M2
2mM
(21)
and m is the nucleon mass in the chiral limit.
In the following, we will show that the ∆N and I both contribute additional
terms to Σa if chemical potential is included.
2.2. µ 6= 0 and the contour integration
In the case µ 6= 0, the derivative in Eq. (11) changes to
Dµψ = ∂µψ − δµ4µψ + Γµψ . (22)
The scalar nucleon propagator is
∆N (µ) =
∫
d4k
(2π)41
m2 + (p′ − k)2
=
∫
d3k
(2π)4
∫ ∞−iµ
−∞−iµ
d4k1
m2 + k2
=
∫
d3k
(2π)4
(∫ ∞−iµ
−∞−iµ
−∫ ∞
−∞
+
∫ ∞
−∞
)
dk41
m2 + k2
=
∫
d3k
(2π)4
∮
c
dk41
m2 + k2+∆′
N (p′) . (23)
Here, we use the prime in ∆′N to indicate that the term is integrated from −∞
to ∞. ∆′N is zero in the infrared regularization because it does not contain any
infrared singularities.6 So we have
∆N (µ) =
∫
d3k
(2π)4
∮
c
dk41
m2 + k2
= − 1
4π2
∫
√µ2−m2
0
d|k| |k|2√
m2 + |k|2. (24)
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The other term which contributes to Σa and is proportional to M2 can be
written as:
∆M2 =3g2A4F 2
π
1
(2π)4M2(m− i/p′)
∫
d4k1
[m2 + (p′ − k)2](M2 + k2). (25)
Making use of the Feynman parametrization, one gets:
∆M2 = C
∫
d4k1
[m2 + (p′ − k)2](M2 + k2)
= C
∫ 1
0
dz
∫
d4k1
[m2 + (p′ − k)2]z + (M2 + k2)(1 − z)2
= C
∫ 1
0
dz
∫
d4k1
[(k − zp′)2 +R2]2, (26)
with
C =3g2A4F 2
π
1
(2π)4M2(m− i/p) ,
R2 = −p′2z2 + p′2z +m2z −M2z +M2 .
Performing a shift k → k + zp′, one obtains:
∆M2(µ) = C
∫ 1
0
dz
∫
d3k
∫ ∞−iµz
−∞−iµz
dk41
[k2 +R2]2. (27)
With the same technique in calculating ∆N (µ), we express ∆M2(µ) as
∆M2(µ) = C
∫ 1
0
dz
∫
d3k
∮
c
dk41
[k2 +R2]2+∆′
M2(p′) . (28)
The residue is in k4 = −i√R2 + k2 and the integration over k4 gives |k| an up
limit:
|k|2max = µ2z2 −R2 .
After the contour integration, one obtains in the end
∆M2(µ) = C · (2πi) · 4π ·∫ 1
0
dz
∫ |k|max
0
|k2| · i
4(√
|k|2 +R2)3+∆′
M2(p′)
= 2π2C
∫ 1
0
dz
(
|k|max√
|k|2max +R2− ln
|k|max +√
|k|2max +R2
R
)
+∆′M2(p′) .
(29)
When µ = 0,
|k|2max = −R2 = p2z2 − (m2 + p2)z −M2(1 − z) . (30)
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Nucleon Effective Mass in Cold Dense Matter
With p2 = −m2N and m about 0.88 GeV, |k|2max is not larger than zero on the
interval 0 < z < 1. This makes the integral of d3k equal to zero and only ∆′M2(p′)
remains. ∆N (µ = 0) is also zero, so we can back to the theory of Becher and
Leutwyler.6 The integration over z is complicated and we can only assort to nu-
merical computation.
The Σb term contains only integral of the meson loop. So, the replacement
p → p′ does not contribute additional term in the integration. The Σc is again
derived from the Σa(µ). Finally, the chemical potential dependence of the physical
mass of the nucleon is then determined by the pole of 1/(m+Σ(p′) + i/p′) with
Σ(µ, p′) = −4c1M2 +Σa +Σb +Σc + e1M
4 +O(p′5) ,
Σa(µ, p′) = ∆M2(µ)− 3g2A
4F 2π
(m− i/p′)∆N (µ) ,
Σb(µ, p′) =
3g2A∆π
4F 2π
(
2c1 −p′2
4m2c2 − c3
)
,
Σc(µ, p′) = −4c1M
2 ∂Σa(µ, p′)
∂m.
(31)
3. Numerical Results and Discussions
The three terms all contribute to the effective mass of nucleon, but the term ΣB
does not include the infrared regularization and the term ΣC with a coefficient
M2 which suppresses its effect on the effective mass. So, it is the term ΣA which
mainly affects the chemical potential dependence of the nucleon effective mass.
The physical mass of the free nucleon is determined by the pole of the two point
function at µ = 0 and which can be used to find the chiral limit mass m reversely.
The other parameters are taken from other published papers, see Table 1.3,5 When
µ and m are given, the pole of 1/(m+Σ(p′) + i/p′) can be calculated and then the
µ dependence of nucleon mass is obtained.
The numerical results near the real axis are illustrated in Fig. 2. The figure
shows about four regions:
• When µ < m, the effective mass of nucleon ms is almost equal to the free mass;
• When µ > m and µ < 1.02 GeV, ms decreases first and then increases as µ
increases;
• When µ > 1.02 GeV, the complex solution appears;
• When µ > 1.06 GeV, the solution disappears and end with a lump on the axis.
Table 1. Parameters used in fitting the nucleon mass (GeV).
mN M m Fπ gA c1 c2 c3 e1
0.938 0.137 0.89 0.0924 1.27 −0.978 3.3 −4.7 −1
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Fig. 2. Chemical potential dependence of nucleon mass. The error bar indicates the imaginarypart of the solution. The points of µ > 1.06 GeV show the positions of limited maximum of thetwo point function.
Fig. 3. The complex solutions with large imaginary part and its dependence on chemical poten-
tial. The solutions with positive imaginary part are nonphysical.
We can understand that as so: When µ < m, the baryon number density is zero and
the nucleons can be regarded as independent of the chemical potential. When more
nucleons are involved, they show an attractive interaction to each other and the
effective mass decreases as µ increases. This is mainly contributed from the scalar
nucleon propagator. When more particles are involved, they begin to squeeze each
other which makes their effective mass increasing.
The dependence of the effective mass on the parameters we used is shown in
Fig. 4. These parameters are used to determine the value of the chiral limit mass
m. For parameter c1 we have used the up and down limit listed in Ref. 3. With the
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Nucleon Effective Mass in Cold Dense Matter
Fig. 4. Chemical potential dependence of nucleon mass with different parameters.
0.0 0.2 0.4 0.6 0.8 1.0 1.2Ms
200
400
600
800
Pole
Μ0.8
0.0 0.2 0.4 0.6 0.8 1.0 1.2Ms
200
400
600
800
Pole
Μ0.9
(a) (b)
0.0 0.2 0.4 0.6 0.8 1.0 1.2Ms
200
400
600
800
Pole
Μ1.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2Ms
200
400
600
800
Pole
Μ1.02
(c) (d)
Fig. 5. Poles locating on the real axis. When µ > 1.02 GeV the real solution disappears.
changing of the chiral limit mass, the point where the nucleon effective mass begin
to decrease moves. But the shapes of those figures are similar.
The two point function has also complex poles. When µ > m+M , πN resonance
may come into being. So, ms has imaginal part which corresponds to the width of
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Μ0.90
0.60.8
1.01.2R
0.5
0.0
0.5I
0
10
02
Μ1.02
0.60.8
1.01.2R
0.5
0.0
0.5I
0
10
2020
(a) (b)
Μ1.05
0.60.8
1.01.2R
0.5
0.0
0.5I
0
10
02
Μ1.09
0.60.8
1.01.2R
0.5
0.0
0.5I
0
10
02
(c) (d)
Fig. 6. Two-dimensional illustrations of the poles. R and I mean the real and imaginary axes,respectively.
the particle. We show all the poles on the interval 0.6 < µ < 1.2 of several chemical
potantial in Fig. 6. The integral of Eq. (16) obeys the dispersion relation, which
can be expressed as6
H(s) = H(s0) +s− s0π
∫ ∞
s+
ds′
(s− s0)(s− s′)ImH(s′) . (32)
Here,
s = p′2, s+ = (m+M)2 ,
ImH(s) =2Mm
√Ω2 − 1
16πsθ(s− s+) .
(33)
So, when the effective mass is larger than the value m+M , the complex solu-
tions must appear. This means that the particle state can decay. Usually this state
is called virtual state. However, this state appears in the second Riemann sheet.
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Nucleon Effective Mass in Cold Dense Matter
The second Riemann sheet continues at the cut line which ranges from S+ to in-
finity. The solution is physical virtual state only at low half plane. As the chemical
potential keeps raising, then for high chemical potential the baryon number den-
sity of nucleons becomes high. Thus, the π meson radiation becomes strong. The
π −N virtual states become easily to form. For this case, all nucleons are situated
in virtual states and no stable states exist.
This picture is similar to the result from potential model, but is different from
the one obtained in relativistic mean field theory(RMFT). In RMFT the nucleon
effective mass is defined as the free mass minus a quantity in direct proportion to
the strength of σ meson. Since the σ meson provides only attractive interaction and
its strength increases while the chemical potential increases, the effective mass is
monotonously decreasing.2
Actually, the solutions have other solutions. One is below 0.4 GeV on the real
axis, see Fig. 5. As it is far from the chiral scale m, it can be neglected. Another
pair of solutions has large imaginary part which means the states are easy to decay.
Its dependence on chemical potential is illustrated in Fig. 3.
For higher chemical potential, the chiral symmetry is restored. It is out of the
capabilities of this chiral model and some other methods are required, such as the
DSE method in Refs. 12–14. Further, the effective mass begins to vary when µ is
smaller than 938 MeV. Will it lead the baryon number density n(µ) to have the
same behavior as ms? If the answer is positive, the result is different from the
calculations in Refs. 15 and 16, but is consistent with the discussion in Ref. 5.
4. Summary
In this paper, we use an effective chiral Lagrangian to study the chemical depen-
dence of nucleon mass in cold dense nuclear matter. After calculating the one-loop
contribution to the self-energy, we have deduced the chemical dependence of the
nucleon mass. The results are reasonable in the low energy regions. We will study
the equation of state in future work and see whether it implies phase transition in
this region.
Acknowledgments
This work was supported in part by the Fundamental Research Funds for the
Central Universities (No. 2020204134010), the Key Research Plan of Theoreti-
cal Physics and Cross Science of China (Grant No. 90503011) and the National
Natural Science Foundation of China (Grant No. 10575050).
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