- Email: [email protected]
Collective modes in a relativistic meson-nucleon system
Nuclear
Physics A501 (1989)
North-Holland,
729-750
Amsterdam
COLLECTIVE MODES IN A RELATIVISTIC MESON-NUCLEON SYSTEM* K.
LIM
and C.J.
HOROWITZ
Received 6 March Abstract:
Collective
modes in B relativistic
state is treated in a mean-field relativistic
random-phase
meson-branch
1989
meson-nucleon
approximation
approximation.
system are investigated.
while the meson propagators
Three types of collective
modes, and modes which indicate
an instability
also important
mixing
between
of the mean-tield of zero-sound
theory ground at wturation
models which often have zero-sound.
meson and nucleon-antinucleon
with a
modes are found: zero-sound,
state. The mixing of scalar and Lector mesons prevents the propagation density. This is in sharp contrast to nonrelati\istic
The baryon ground are described
excitations
ahich
There is
greatly
effect
meson branch modes.
1. Introduction The study of the relativistic system of coupled mesons and nucleons is a first step in the investigation of finite nuclear systems. In this context it is important to examine meson propagation through nuclear matter. In the medium, the meson propagators are quite different from the free propagators. This implies that in the long-wavelength limit, q<< 2k, , meson propagation can be considered as a collective mode rather than the propagation of a nearly free meson. In the meson-nucleon description of nuclear matter, a density fluctuation, corresponding to a collective mode, produces a fluctuation in the meson field. According to linear response theory this induced fluctuation of the meson fields, in turn, induces further fluctuations in their sources, which in the present case are the baryon current density for the vector meson and the baryon scalar density for the scalar meson. For the collective modes, the fluctuation in the baryon-current or the baryon-scalar density must be selfconsistently sustained without external perturbation. Therefore, the interpretation of the meson propagation in the long-wavelength limit as collective mode is reasonable, and the collective modes in nuclear matter are characterized by the poles of the meson propagators ’ 1. Previous works “1 show that there are plasma-like modes and zero-sound modes in nuclear matter. In the meson-nucleon relativistic many-body theory, particle-hole and antiparticle-particle excitations appear as collective modes of the system. Furthermore, relativistic effects allow the possibility for new exotic modes. l
Supported
in part by Department
0275.9474/S9/$03.50
CNorth-Holland
((3 Elsevier
of Energy contract
Science Puhli\hrr\
Physic\ Puhli\hing
Diti\ion)
I3.V
DE-FGOX7ER40365
K. Lim, C.J. Horowi~r
730
/ Collective
modes
Nonrelativistically, the zero-sound mode is possible only if the NN interaction potential has a repulsive core ‘), V(/q/ = 0) = f d-‘x V(x) > 0, which is also responsible for the saturation relativistic Therefore
of the system.
mean-field
theory
the zero-sound
However,
(MFT)
the mechanism
is different
modes in a relativistic
for the saturation
than in the nonrelativistic many-body
in the theory.
system might be qualita-
tively different. The study of the collective modes also provides an analysis of the stability of the uniform ground state in various approximations to meson-nucleon field theory ‘,‘). The existence of poles of the meson propagators at zero energy transfer indicates that the uniform ground state of nuclear matter is unstable against small perturbations of the density. In this paper we study the collective excitations ie3) of a coupled relativistic meson-nucleon system by examining the poles of the meson propagators. We compute the meson propagator in the one-loop approximation 2%4,7)using the meanfield approximation of the Walecka model to treat the baryon ground state. This problem was originally investigated by Chin ‘), who addressed it in the limit of small 4:. In this context, he was able to obtain qualitative predictions about general features of the collective excitations. However, for high-lying excitations his approach is an oversimplification because qt is never small in this case. Therefore, the extension of the results of ref. ‘) to this region is inappropriate. Here we extend the computation to the region where qi is not small and carry out the numerical computation for the poles of the meson propagators. Besides confirming the predictions of ref. “) in the small 4: limit, we obtain new features of the collective modes. In sect. 2, we summarize the theoretical background for the computation of the meson propagators in the one-loop approximation and the derivation of the dielectric functions. In sect. 3, we present our results on the collective modes. We have found three kinds of collective modes: zero-sound, meson-branch modes and modes which indicate instabilities of the MFT ground state. There is no zero-sound at the saturation density of nuclear matter; this is a relativistic effect which has no counterpart in a nonrelativistic
system. The absence
of zero-sound
at the saturation
density
is because
the attractive scalar interaction decreases the integrated strength of the effective vector potential through the form of scalar-vector mixing. Furthermore, the scalarvector mixing separates the two longitudinal modes of the meson branch. This effect is more impo~ant at higher /qi_ Both longitudinal and transverse instabilities are found at high density, while at low density only longitudinal instability exists. At small jg/, the scalar-vector mixing shifts the allowed values of density of longitudinal instability modes below the saturation density. Sect. 4 contains the conclusions.
2. Theoretical
background
In this section we describe the derivation represent the collective excitations in nuclear
of the dielectric function whose zeros matter. The formalism was originally
K. Lim, C.J. Horowirz / C‘ollecrive modes
developed
by Chin ‘). We describe
I) Y. This model is a renormalizable with each other by meson
nuclear
matter
quantum-field
exchange.
using the Walecka
731
model (QHD-
theory in which the nucleons
The lagrangian
interact
is
~=~[y,(irl~-gg,.V~“)-(M-g,~)]~+:(iJ,~iJ~~-mt~’) -bF,,,F~“i:mtV,V~‘t~.
(2.1)
Here 4 is the baryon field, $J is the scalar-meson (a) field, which couples to the baryon scalar density through the interaction term g,&/$, V’” is the vector-meson (w) field, which couples to the conserved baryon current density through g\.tjy,$V”, includes the term &Y arising from the renorand F,,, = it, V,, - dl,Vw. The lagrangian malization counterterms. We treat the baryon ground state in MFT approximation. In this approximation the meson fields are replaced by their expectation values, which are classical fields: 4 + (4) = &,
v,-XV,)-+,v,,.
(2.2)
This approximation should be increasingly valid as the density increases “). Note that the mean scalar-meson field & shifts the nucleon mass from M to the selfconsistent effective mass M” in both the Dirac and Fermi seas, M*=M-g,c,b,,.
(2.3)
The mass shift of the Dirac sea produces a vacuum fluctuation contribution to the energy density “). In the MFT this vacuum fluctuation effect is ignored (see eq. (2.14)) while in relativistic Hartree approximation (RHA) this effect is included ‘.“). We will use the MFT in the present work and neglect vacuum effects. We use the parameters g;=91.64,
m, = 550 MeV,
g; = 136.2,
m, = 783 MeV,
(2.4)
which reproduce (in MFT approximation) nuclear saturation at a density correbinding energy sponding to a Fermi momentum kF = 1.42 fm ’ and nuclear-matter of 15.75 MeV [ref. “)I. To compute the meson propagators, we sum over ring diagrams which consist of repeated insertions of the lowest order one-loop proper-polarization part. This procedure is equivalent to the relativistic random-phase approximation (RPA). The importance of summing the set of ring diagrams for the study of collective modes is well stated in conventional many-body physics I,‘)). Since we have both scalar and vector mesons in our model, besides being affected by the presence of the nuclear matter, the meson propagators must take into account the mutual interaction of the respective fields, namely, scalar-vector mixing. This arises when a particle-hole pair is excited by a vector meson and decays into a scalar meson. This scalar-vector
732
K. Lim,
<:J.
Hornwirz
/ Coktiw
m&s
mixing is a purely dells~ty-dependent effect. It complicates the summation because the Dyson’s equations for the scalar- and vector-meson propagators
process become
coupled. Therefore it is convenient to define a full scalar-vector meson propagator in the form of a 5 x 5 matrix with indices a, h ranging from -1 to 3, where -1 iJ‘,i, corresponds to the scalar meson and 0, 1, 2, 3 to the components of the four-vector. Dyson’s
equation
for 9 can be written
as a matrix
equation
(s = “ii”+ F/ TC’fl”T . . The lowest-order
scalar-vector
meson
propagator
(3.5) 9” is block-diagonal, (2.6)
1
the entries
being the noninte~cting
propagators
(2.71 where -I
D”(q)=q;_m;+in.
-
(2.8)
Note that the llPq,, term in Dam, does not contribute to the physical because of current conservation “I. The density dependent polarization insertion is also a 5 x 5 matrix
quantities
(2.9)
the indices p and I/ ranging from zero to three. Here the lowest order scalar, vector and scalar-vector-mixed polarization parts are defined as
(2.10) G(k)
is the self-consistent
= G,(k)+
RHA baryon
G,,(k),
propagator
given by
(2.1 I)
K. Litn,
C’.J. Horowit-_ / C’ollectiw
where k, is the Fermi momentum. the energy
and three momentum
The effective
mode
733
mass M* is given by eq. (2.3), and
are k:%P= ( /q” _ g, v”, k ) ,
(2.12)
E*(k)=\!k:+M”?.
(2.13)
and
The Feynman propagator G, involves the propagation of virtual particles antiparticles, and G,, describes the propagation of ‘holes’ inside the Fermi while correcting GF for the Pauli exclusion principle.
and sea,
In accordance with the MFT approximation we neglect the vacuum fluctuation effects; therefore, we take only the density-dependent part of these polarization insertions. In RHA one must include vacuum polarization insertions which are divergent and need to be renormalized. We will examine the effects of vacuum fluctuations in a future paper. The density-dependent polarization insertions are 17!‘( 4) = pigi
J~.“[G,~(“‘G,~‘“+““““(“‘G
+G,(k)G,,(k+q)l, II;,.(q)
= -ig;
5
~TrLu,G,,lk)y.,G,,(l+q~+y~G,,(k)y.,C;.,(k+q)
+ r,G(k)r,G,,(k+q)l, Lfp’C
q) = kg,
J
d”k -Tr[y,G(li)G(k+q)]. (27i-?
(2.14)
This procedure of taking only density-dependent part of the polarization insertions might cause problems calculating the damping of the collective modes because, when the imaginary part of the polarization insertions are nonzero, collective modes can decay into real particle-hole or particle-antiparticle pairs. For space like q,, the imaginary part of the vacuum polarization insertions are zero but for time like q, they are no longer zero. Therefore, we should include the imaginary part of the vacuum polarization insertions to study the damping. This will be discussed in sect. 3. The density-dependent polarization insertions in eq. (2.14) include the Pauli blocking of particle-antiparticle excitations besides the particle-hole excitations. This arises from the presence of antiparticle propagation in the Feynman part of the propagator, eq. (2.11). Therefore the terms G,,( li)G,( k + y) and G,( k)G,,( k + y) in eq. (2.14) include the Pauli blocking of NN excitations. These terms correct for particle-antiparticle excitations of the vacuum which are no longer possible in the medium. At zero density, in the limit q + 0 the vacuum contributions are canceled by renormalization counterterms. As the density and the momentum increase there will be vacuum corrections from the momentum and M* dependence of the vacuum
K. Lim, C.J. Horowitz / Collective
734
polarization insertions all of these corrections
and from the Pauli blocking. It is often difficult to include and it is common to take into account only the Pauli blocking
effects. We include only the Pauli blocking fluctuation effects and examine the limitations Pauli
blocking
of NN excitations
discussed in sect. 3. Baryon current conservation polarization insertions,
solution
implies
to the Dyson’s 9
Defining
the dielectric
the following
equation,
are
for the one-loop
flT,D,(q)q” = 0.
(2.15)
eq. (2.5), is (2.16)
det (1 - abOnD) ,
it follows that the full scalar-vectormeson vanishes,
equations
modes
E as F -
function
=
collective
g”p-‘g”.
= (I-
function
corrections neglecting the vacuum of this approach. The effects of the
for the density-dependent
q”Ii!,“,(q) The formal
modes
propagator
(2.17) has a pole when the dielectric
i.e. e=det(l-g”LrD)=O.
(2.18)
This is the eigencondition for determining the collective-excitation spectrum 2.‘). Therefore, to determine the dispersion relation of the collective excitation we need to know the polarization insertions. In a frame of reference where q = (q,0, 0), the current conservation conditions become q"17,M+qlIy=o.
q"II~~+q17~~=o,
From these equations polarization
and the symmetry
of the integrals,
(2.19)
we get the density-dependent
matrix
ny n,”
Ill,” IIC” II,“, II,“,
0
0
0
0
ng
0
00
IT: 0
I
(2.20)
.
0
,I’= By defining
the longitudinal
li.‘(l
0 i nr”
0 0 n:‘,
and transverse -A”II:)(l
0
II::0
dielectric
functions
as
-@n:‘)+$d”O”(n~)‘, (2.21)
K. Lim, C.J. Horowic
where n:‘=
I7& -n::
modes
735
and IT’,‘= n_g = II_::. We can write & =
Note that we have used the current When
/ Collrcriae
ELET.
z
conservation
191~ q = 0, Ily(q = 0) = 0 by the current
(2.22) conditions conservation
(2.19) to write eq. (2.21). (see eq. (2.19))
and
Z7y( q = 0) = 0 because of the symmetry of the integral. The density-dependent part of the lowest-order scalar, vector, and scalar-vector-mixed polarization insertions are evaluated analytically, and their analytic expressions are given in the appendix. They are substituted in eq. (2.21) to evaluate the dielectric functions. The collective modes obtained from the zeros of the dielectric functions are presented in the next section.
3. Results In this section we present numerical results for the collective modes. First, we consider undamped collective modes, which can be obtained by letting Im 17”= 0 and searching for the zeros of the real part of the dielectric functions. In the MFT approximation a non-zero Im nil indicates the decay of the collective modes into pairs for time-like real particle-hole pairs for space-like qp, or particle-antiparticle q, which includes the decays into Pauli blocked NN pairs. The decays of the collective modes into the Pauli blocked NN pairs will be cancelled exactly by the corresponding ones in the imaginary part of UUCUU~~polarization insertions when these are taken into account I’)). We will consider the damping region after having investigated the collective modes using the real part of the dielectric functions. We have found three kinds of collective modes: zero-sound modes, meson-branch modes and instabilities.
3.1.
ZERO-SOUND
MODE
This is a low-lying longitudinal collective mode with a dispersion relation typical of sound propagation q"= C,,q.However, zero-sound is physically different from ordinary first-sound. In the zero-temperature limit (T+O), because of the Pauli principle, the interparticle collision time increases as TV’ and ordinary sound propagation is no longer possible. On the other hand, following the Landau-Fermi liquid theory ‘,‘,‘).“), a collisionless zero-sound mode is possible in the limit of zero temperature. We do not find a zero-sound mode at low density. This result is in agreement with the prediction of ref. “). The reason for the absence of the zero-sound mode at low density is that the attractive scalar interaction decreases the integrated strength of the effective vector potential through the scalar-vector mixing. Removing the scalar-vector mixing (i.e. set flM =0) we find zero-sound even at low densities
K. Lim,
736
caused
by vector-meson
zero-sound
is only possible
poles
/ Cofkctioe modes
C.J. Harowilz
for q not
at high densities.
too large.
With
Typical
zero-sound
scalar-vector
in figs. 1, 2 and 3. From these figures we can see that even though is zero at q = 0, for small
nonzero
q and low density
mixing
modes are plotted the mixing
the scalar-vector
term
mixing
is
important for zero-sound. As 9 and density increase it is less important. According to ref. “), to have zero-sound at low density the condition gz/ rnt > gi/ rnt,
(3.1)
must be satisfied. This is not satisfied for the parameters in eq. (2.4). The zero-sound speed C,, calculated from Landau-Fermi liquid theory ‘), agrees with the microscopic calculation and is plotted in fig. 4. The zero-sound mode first appears at a density corresponding to a Fermi momentum kt. = 1.64 fm-‘. The absence of zerosound mode at the saturation density is a purely relativistic effect because, nonrelativistically the zero-sound mode is related to the repulsive core of the NN interaction. Therefore, any system which saturates from a repulsive core has zero-sound at saturation density. The zero-sound speed approaches the speed of light from below as the density increases; this is the causal limit expected on physical grounds. For k,< 2.73 fm ’ there are two zero-sound modes. Further zero-sound modes appear when k,> 2.73 fm-‘. (See fig. 1) One of these modes is not damped at low
oI/e/: 0
.
, 1
.‘.
kF
/ 2
i 3
.,
(fm-‘)
Fig. 1. Zero-sound mode and instability mode at y = 10 MeV yI, versus k, Solid curves are with scalar-vector mixing, dot-dashed and dashed curves are without mixing. Upper two solid curves represent the zero-sound, lower solid curve represents the instability mode. Dot-dashed curve represents poles from the vector meson, and the dashed curve shows the poles from scalar meson.
kF
(fm-‘)
Fig. 3. Longitudinal modes at y = 500 MeV q,, versus k, Solid curves include scalar-vector mixing, dot-dashed tunes and dashed curves without mixing. Note the difference in q,, scale compared to fig. I. Upper two solid curves are meson branch modes and the lower two are zero-sound. The lowest zero-sound branch is always damped. Dashed curves are from the vzalar-meson poles and the dot-dashed curves are from the vector-meson poles.
q and merges into the region of particle-hole damping as y increases. The other modes are always damped. (The lower curves of the zero-sound modes in figs. 1 and 2 are modes of this kind). At k, = 1.9 fm ’ two zero-sound modes are possible and the initially undamped mode is plotted in fig. Sa as a curve of q” versus q. The region where it is not damped can be seen more clearly from tig. 5b where the same mode is plotted as a curve of (q’)/q)/C,, versus q. At k, = 3 fm-’ four zero-sound modes are possible at small q. These modes are plotted as dashed curves (II) in fig. 7a. The upper two, which are not distinguishable from the third mode in this figure, are resealed and plotted in fig. 7b. Only the uppermost mode (see fig. 7b) is not damped at low q and the other modes are always damped. The zero-sound mode is damped if (3.2) where U, = lir/ EE is the Fermi velocity. In this region polarization insertions is no longer zero, and therefore decay into real particle-hole pairs. The critical values of modes are damped, are plotted versus k, in fig. 6. At a
the imaginary part of the the zero-sound mode can q for which the zero-sound given k, zero-sound is not
K. Lim, C.J. Horowitz / Collective
738
modes
1500
zz
1000
VECTOR . .
i?
~ 500
0
0.5
1.5
1
kF
2.5
3
&d;
Fig. 3. Longitudinal and transverse modes at y = 1 GeV. Upper solid curves are the longitudinal meson branch modes with mixing, middle solid curves are zero-sound, and the lowest solid curve is the instability mode. Dotted curves labeled vector are from the vector-meson poles, the ones labeled scalar are from the scalar-meson poles without mixing. Dashed curve is the transverse meson branch mode. The instability from the transverse mode is not plotted in this picture
1.0
0.9
0.8
0
u
0.7
0.6
0.5
t 0
0.5
1
kF
1.5
2
2.5
3
(frn-“)
Fig. 4. Zero-sound speed which is obtained from Landau-Fermi liquid theory. See ref. ‘), No zero sound exist at low density (k,~ 1.64 fin-I). The sound speed approaches unity at high density.
K. Lim, C. J. Horowitz / Cbllective modes
ZERO SOUND
kF=1.9 fm-' M"=.25M
100 q
1.04
. ”
’
”
150
200
(MeV)
ZERO SOUND i
739
kF"1.9 fm-' M"=.25M
”
”
”
”
”
”
”
04 i ,’ /’ /’
DAMPED
NOT DAMPED 0.96
-
t 0
6
50
I
100 q
b
I,..
1.50
.
.
I
.
200
(MeV)
Fig. Sa, b. Solid curve represents the zero-sound mode at k, = 1.9 fm-‘. Dashed curve shows the upper limit of non-zero lm II”. Zero-sound is damped for y z 8X MeV. No zero-sound is found for y > 226 MeV at this density.
740
kF Fig. 6. The critical
values
(fm-‘)
of y for which the zero-sound modes are damped. damped in the region below the solid curve.
The zero-sound
is not
damped only for y less than a critical value of q. From this figure we can see that at very high densities the zero-sound mode is not damped only for very small q; for instance, at kfy= 3.3 fin- ’ and q = 1.2 MeV the zero-sound mode is already damped. Therefore the allowed regions of q and density for undamped zero-sound mode propagation are very restricted.
3.2. MESON-BRANCH
MODE
This is a high-lying excitation mode. The modes in this branch are like plasma excitations in QED and at zero density this branch reduces to the propagation of free mesons. We have both transverse and longitudinal modes. For a fixed ii,2 1.6 fin-‘, there are two longitudinal modes and one transverse mode (figs. 7a,8). In each branch there is a decrease in the excitation energy which seems to be related to the effective mass of the baryon; as the effective mass is decreased this effect is enhanced. In fig. 9, the longitudinal meson mode energies q” are displayed versus kF at a small fixed three-momentum (q = 1 MeV). The same energy curves without meson mixing are plotted in fig. 10. These two figures show that for small q the scalar-vector mixing is not strong enough to separate the two longitudinal meson modes. On the
LO~GrTU~INAL MODE: kF=3. fm-* M" =.0885M
600
400 4
ZERO SOUND i’.‘---T’
r
.‘I’
1.0002 0 Y -&oooo
t
2 tr‘ 0.9998
i0.9996
I
0
1200
kF=3.0 fm-' M*=.OBRM
(b)
1.0004
1000
800
(MeV)
15
10 q
(MeV)
‘.--r------
K. Lim, C.J. Horowitz
742
TRANSVERSE
/ Cdiectiue
MODE kFc2.6
modes
fm-’
M’=.l188M
500 -
0
200
400
600 q
a00
1000
1200
WV)
Fig. 8. Transverse mode at k, = 2.6 frn~~‘. Upper solid curve (I) is the meson branch mode and the lower curve (II) shows the instability. Shaded regions A-D show the damping regions and they are discussed in text. other
hand
at higher
q the mixing
becomes
more
effective
(see figs. 2, 3). If we
remove the mixing term, the scalar- and vector-meson modes are well separated at low density but they cross each other twice as the density increases (see figs. 2, 9). For instance at q = 1 MeV they cross each other at k, - 1.7 fm- ’ and at kF - 2.2 fm- ‘. As q increases, the first crossing point decreases a little but the second crossing point increases very much. The role of scalar-vector mixing is to separate the two meson modes at these points; otherwise it is not important in this meson branch. This conclusion is in disagreement with the result in ref. ‘) in which Chin said that the scalar-vector mixing is important at low densities and it is unimportant at high densities. We believe that this is due to the inappropriate extrapolation of the results of ref. “) to the region of large 4:. Also note that for wry snail three momentum (q = 1 MeV) the transverse mode, which is plotted in fig. 11, almost coincides with the vector-meson-like longitudinal mode of fig. 9; however, at high q the two are well separated (see fig. 3). An interesting peak is seen in fig. 9 for the vector-meson-like and the scalar-mesonlike mode. The peak of the vector-meson-like mode is at the energy qo- 2M* while the peak for the scalar-meson-like mode is at q,)-- M, where M is the free nucleon mass. For q = 0 and at given density the smallest amount of the energy that will excite a Pauli blocked NN pair is q. = 2M”. Therefore, we think that the peak of the vector-meson-like mode is related to the Pauli blocking of NI? excitations. Since
K. Lim, (1.1. Horowitz / Collective
modes
743
I”“,“‘-,“.’
,
I
*I
,
/
\\’ %
1500
’
/
.. .
c
;(?
A. y,.i, 1‘,A., \ ‘,\‘.
-
/
/
/
/I
, /
/
/
/
/
/II
I
/
I
>--
“/ ’ //,
01”““““““““““” -0.5
“////,,f,,
,
“////I;,/,
‘1
J
1.5
1
kF
2
2.5
3
(fm-*)
Fig. 9. Longitudinal meson branch modes at y = 1 MeV. Here dashed curve is vector-meson-like longitudinal mode and solid curve represents the scalar-meson-like longitudinal mode. Dot-dashed curves are scalar (I) and vector (II) meson-like longitudinal modes without Pauli blocking of Nfi excitations. The shaded region C is where the Im I7 ” ib non-zero. Note the peaks due to the Pauli blocking of NN excitation.
0
0.5
1
1.5
kF Fig. IO. Longitudinal
meson
branch
2
2.5
3
(fm-‘)
modes at q = 1 MeV without coincide with those of fig. 9.
scalar-vector
mixing.
These modes
144
K. Lim, C.J. Horowitz / Collective
modes
~~~‘~~~~‘~~~,‘~~~.‘,~~~
0
1
1.5
2
2.5
3
-1 kF
(fm
>
Fig. 11. Solid curve is the transverse meson branch mode at q = 1 MeV. This mode almost coinside with vector-meson-like longitudinal mode in fig. 9. Dot-dashed curve is the mode computed without Pauli blocking of Nrri excitations. The shaded region C is where Im I7” is non-zero.
we can write the Feynman propagator as the particle part plus the antiparticle part, we can remove the Pauli blocking of NN excitation process by removing Gr,( k)Gr,( k + q) and G,,( k)G,( k + q) terms from the density-dependent polarization insertions. Where GF( k) represents the antiparticle part of the Feynman propagator. The energy curves obtained from removing these contributions are plotted as dot-dashed curves in fig. 9. The upper curve (1) corresponds to the vector-meson-like, and the lower curve (II) represents the scalar-meson-like mode. Removing the antiparticle
states
causes
the peak
of both
scalar-
and
vector-meson
modes
to
disappear. The peak appears in meson modes due to the Pauli blocking of NN excitations. We believe these peaks will go away if the full virtual NN pairs are also included in a relativistic Hartree approximation. However, the full renormalized vacuum calculations may be difficult in many situations. Therefore these peaks suggest important limitations in the simple MFT calculations. Note that q. is proportional to k, at high densities, therefore, it is hard to excite a mode in this meson branch at very high density. The shaded regions B and C of figs. 7a, 8 are the region where Im II” is not zero and qr is time-like (see also the shaded region of figs. 9, 11). In these regions the meson modes can decay into particle-antiparticle pairs. B is enclosed by O(q,,(ET+E~,~,))B(E::+E~,,,,-q,)andCis enclosedby O(qS-4M*‘)O(E$+E:,_,,qJo(2k,
- q), where
Ef, ty = d(kF*q)7+M*7,
E; = ,,lk;+ M”’
(3.3)
K. Lim, C.J. Horowit;
However,
since
polarization
q, is time-like
insertions
745
/ C‘olleclioe modes
in these regions,
the imaginary
are not zero. Therefore,
the vacuum
part of the vacuum
polarization
insertions
should be included in the study of the damping of the collective modes in this branch. The imaginary part of the vacuum polarization insertions are non-zero if qt >4M*‘. This inequality holds in the regions A, B and C of figs. 7a, 8. The density-dependent meson modes in region C decay into Pauli blocked Nfi pairs which are cancelled by the counterpart of the vacuum polarization I’). Therefore, when the vacuum polarization is taken into account the meson modes are damped in the region A and B. In this case the threshold of particle-antiparticle pair production will be at q,,= M*+ ET and q = k,. Note that the q(, at the threshold can be less than 2M. (See for instance figs. 7a, 8) The shaded region in figs. 9, 11 corresponds to the region C of figs. 7a, 8. Thus in this small-q limit the modes decay into only Pauli blocked NN pairs. Therefore, including the vacuum polarization the meson-branch modes at small q might propagate without damping. One can see that the inclusion of vacuum polarization insertions will also change the real part of the collective modes. This will be discussed in a future publication.
3.3. INSTABILITIES
In addition to the zero-sound and the meson-branch modes, we found collective modes at low energy transfer q,,. In fig. 12 these modes are plotted for q,,= 0, q versus k,. At low density only the longitudinal mode (solid curve) shows this 2000
I
I
I
1 // /
T
1500
-
1000
-
2
/
/
/
/
/
/
rT 500
/ \
-
\
”
0
1
KF Fig. 12. Poles of the meson propagator dielectric function and the dashed
.
”
2
I-
---_
------__
”
”
3
‘I”
4
(fm-‘)
at yI, = 0 MeV. Solid curves show the zeros from the longitudinal curves show the zeros from the transverse dielectric function.
746
K. Lim, C..l. Horon+tz / Collecrive
modes
instability. It appears at low k, and low q. For the longitudinal modes this instability comes from the scalar-meson poles and the scalar-vector mixing tends to shrink the allowed region of this mode (see figs. 1, 2). For instance at q = 500 MeV this mixing completely removes the mode generated q this mixing is the relevant factor that brings
by the scalar-meson poles. At small the density of the instability below
the saturation density. The existence of these modes at q. - 0 indicates that uniform low-density nuclear matter is unstable against perturbations of the density at long wavelengths. This represents the liquid-vapor phase transition of nuclear matter below the saturation density. At high density both transverse and longitudinal modes exist. This kind of longitudinal mode appears at densities corresponding to k,a 2.6 fm-’ and q 900 MeV and the momentum increases as the density increases (solid curve in fig. 12). For this high q the mixing seems less important than at low q. The transverse mode apperars at kFa 1.81 fin-‘; q starts at intermediate wavelength (q - SO0 MeV) and it spreads into a very wide region as the density increases (dashed curve in fig. 12). The existence of the transverse instability at high density is due to the attractive particle-hole interaction mediated by the exchange of transverse w-mesons. We will refer to this as w-meson condensation for transverse modes. In refs. 4*“) the instabilities of the meson-nucleon system in the one-loop approximation including vacuum fluctuation effects (RHA) are investigated. The RHA also has the instability at low density, low q but no high-density medium-q case exists. RHA ground state is also found to be unstable at large q because of vacuum polarization I’). (See fig. 1 of ref. I’)). Region D of figs. 7a, and 8 shows the nonzero imaginary part of the densitydependent polarization insertions for space-like q,*. This region is enclosed by - E$) -q,,)O(q,,-(E%_,E~))f?(qo). Where ET,,, and ET are given in NE>+, eq. (3.3). The instability modes and the zero-sound modes can decay into particlehole pairs in this region. Figs. 7a and 8 show that the instability modes are damped in most of the cases. 4. Conclusions We investigated collective modes in the one-loop approximation to the Walecka model and found three kinds of collective modes: zero-sound, meson-branch modes and modes with zero energy transfer, which indicate an instability of the MFT ground state. We found three important results about the zero-sound modes; first, zero-sound mode is a vector-meson-like longitudinal mode and it is absent at the saturation density. This is a relativistic effect that has no analog in nonrelativistic systems. It arises because the scalar-vector mixing of longitudinal mode decreases the strength of the vector meson and pushes the allowed density of zero-sound above the saturation density. Second, the zero-sound speed approaches unity from below at high density, and third, except at very low q.
at high density
there
is no undamped
zero-sound
mode
K. Lim, C.J. Horowirz / C’ollec~ive modes
As for the meson at a given density
branch
sible for the separation the region where when the scalar-
modes;
(k, 2 1.64 fin-‘).
747
first, there are typically
two longitudinal
We saw that the scalar-vector
of the scalar-meson
mixing
modes is respon-
mode from the vector-meson
mode in
they cross. Therefore the scalar-vector mixing is not important and vector-meson modes are well separated. This is expected
because the effects of the scalar-vector mixing must be stronger when scalar and vector modes are closer. Furthermore, we have shown that the Pauli blocking of NN excitations strongly effect the meson modes in this MFT approximation. Finally, the longitudinal instability at low density represents the liquid-vapor phase transition of nuclear matter below the saturation density. The scalar-vector mixing is the important factor shifting this instability below the saturation density. At high density transverse instability represents w-meson condensation. At low density the meson branch modes are the only undamped collective modes while at high density it seems that there is no undamped collective mode except for the zero-sound mode at extremely small q. However, the inclusion of the vacuum polarizations might change this picture. Vacuum polarization effects will be included in a future paper. The existence of poles in the meson propagator at zero energy transfer indicates the instability of uniform ground state of the meson-nucleon model of nuclear matter and this may give the upper limit for the domain of validity of the various approximation of meson-nucleon field theory 4,5). The collective modes are also important for the study of the linear response of the nuclear matter to various probes I’), for the calculation of RPA energy ‘,‘j) and to study meson propagation through nuclear matter. We are grateful
to Brian
D. Serot for his valuable
comments
on the manuscript.
Appendix
Here we present one-loop polarization (2.14) in sect. 2. - Real part:
the analytic expressions for the density-dependent part of insertions which are obtained from doing the integrals in eq.
7
ReIll’(y,,,y)=$$
57
k,Ec-(3M*‘-i9t)In
+$(4M*‘-9;)
Renf’(a,,y)=ti
-$
]n IPI+:(~M*‘-~;)‘~~
(4M”‘-9;)
In IcyI
1
ik,;E;_:y’]n(~)+~(E:‘+~)]n,o,
77l9’ _ E;(3q;+4ET2) 249
~
(2M”‘+
9;)(4M*”
- q;).] I
where h is the isospin
degeneracy,
A==
2,
for nuclear
matter
i 1,
for neutron
matter,
and N, ,R and .I are cY=
q; -4( qoJF; - qk,)’ q~-4(~l~~~~+qk~)~’
p = (q;+2q$,.)‘-4q,‘,E;’ (q; -2qk,.)‘-4q;Ef”
+tanrl;w.bTl
-k,q;dq;,(4M”‘-q;) q:E;-4q,,M*‘(q,,E;+qk,)
II
’
Where the first tan ’ is the branch with values in [O, rr] and the second tan-’ is the branch with values in [ --;TT, ST]. Note that .1 is zero at qi = 0. When the argument of any tan -’ is imaginary, this becomes a branch of the logarithm function. -Imaginary part:
The plus sign or the minus sign in the right-hand
sides of the expressions
corresponds
to space-like qr((qo(i q), or time-like qM(jq,,j> q), respectively. For space-like q,. the quantities in the right-hand side of these equations by
are given
___E,,-\;~;+&,f:%‘zzE~, E,,,:,,] ,
E, = min[ E;,
ES -)q,,),
IS,,,,, = max [M*,
I?,.],
E,.=;[q,~l-4M”‘/q~-/q~t/].
For time-like
qU, they are
if q: <4M*‘,
/_iVin [
E:,
E, =
j(/q,,/+qdq)],
otherwise.
0, min (IS,., 15:) , M*,J(lq,,l-qJ1
if qL 5 4M*’ 3 otherw-ise , -4M”‘/q:
References 1) A.L. Fetter 2nd J.D. Walecka,
Quantum
theory of many-particle
systems
(McGraw-Hill,
New York.
1971)
3) S.A. Chin, Ann. Phys. 108 (1977) 301 3) T. Matsui. Nucl. Phya. A370 (1981) 365 41 R.J. Furnstahl and C.J. Horowitz, Nucl. Phys. A485 (1948) 632 5) R.J. Perry, fhys. Lett. B199 (lY87) 489 6) H. Kurasuwa and 1. Suzuki, Nucl. Phys. A445 i 1985)6X5 7) C.J. Horowitz, Phys. Lett. B208 11985) 8 8) B.D. Serot and J.D. Walecka, Ad\. in Nucl. Phys. 16, ed. J. Negeie and E. Vogt 1986)
Y) 11. Pines, The many-body
problem
(Benjamin,
New York,
1961)
f Plenum.
New York,
750
I(, tim,
C.J. H0rnWili / C’ollectioe modes
IO) T. Matsui and B.D. Serot, Ann. Phys. 144 (1982) 107 11) G. Baym and S.A. Chin, Nucl. Phys. A262 (1976) 527 12) C.J. Horowitz, 3rd Conf. on the intersections between particle 1988 13) X. Ji, Phys. Lett. B208 (19%) 19
and nuclear
physics,
Rockport
Maine,
Physics A501 (1989)
North-Holland,
729-750
Amsterdam
COLLECTIVE MODES IN A RELATIVISTIC MESON-NUCLEON SYSTEM* K.
LIM
and C.J.
HOROWITZ
Received 6 March Abstract:
Collective
modes in B relativistic
state is treated in a mean-field relativistic
random-phase
meson-branch
1989
meson-nucleon
approximation
approximation.
system are investigated.
while the meson propagators
Three types of collective
modes, and modes which indicate
an instability
also important
mixing
between
of the mean-tield of zero-sound
theory ground at wturation
models which often have zero-sound.
meson and nucleon-antinucleon
with a
modes are found: zero-sound,
state. The mixing of scalar and Lector mesons prevents the propagation density. This is in sharp contrast to nonrelati\istic
The baryon ground are described
excitations
ahich
There is
greatly
effect
meson branch modes.
1. Introduction The study of the relativistic system of coupled mesons and nucleons is a first step in the investigation of finite nuclear systems. In this context it is important to examine meson propagation through nuclear matter. In the medium, the meson propagators are quite different from the free propagators. This implies that in the long-wavelength limit, q<< 2k, , meson propagation can be considered as a collective mode rather than the propagation of a nearly free meson. In the meson-nucleon description of nuclear matter, a density fluctuation, corresponding to a collective mode, produces a fluctuation in the meson field. According to linear response theory this induced fluctuation of the meson fields, in turn, induces further fluctuations in their sources, which in the present case are the baryon current density for the vector meson and the baryon scalar density for the scalar meson. For the collective modes, the fluctuation in the baryon-current or the baryon-scalar density must be selfconsistently sustained without external perturbation. Therefore, the interpretation of the meson propagation in the long-wavelength limit as collective mode is reasonable, and the collective modes in nuclear matter are characterized by the poles of the meson propagators ’ 1. Previous works “1 show that there are plasma-like modes and zero-sound modes in nuclear matter. In the meson-nucleon relativistic many-body theory, particle-hole and antiparticle-particle excitations appear as collective modes of the system. Furthermore, relativistic effects allow the possibility for new exotic modes. l
Supported
in part by Department
0275.9474/S9/$03.50
CNorth-Holland
((3 Elsevier
of Energy contract
Science Puhli\hrr\
Physic\ Puhli\hing
Diti\ion)
I3.V
DE-FGOX7ER40365
K. Lim, C.J. Horowi~r
730
/ Collective
modes
Nonrelativistically, the zero-sound mode is possible only if the NN interaction potential has a repulsive core ‘), V(/q/ = 0) = f d-‘x V(x) > 0, which is also responsible for the saturation relativistic Therefore
of the system.
mean-field
theory
the zero-sound
However,
(MFT)
the mechanism
is different
modes in a relativistic
for the saturation
than in the nonrelativistic many-body
in the theory.
system might be qualita-
tively different. The study of the collective modes also provides an analysis of the stability of the uniform ground state in various approximations to meson-nucleon field theory ‘,‘). The existence of poles of the meson propagators at zero energy transfer indicates that the uniform ground state of nuclear matter is unstable against small perturbations of the density. In this paper we study the collective excitations ie3) of a coupled relativistic meson-nucleon system by examining the poles of the meson propagators. We compute the meson propagator in the one-loop approximation 2%4,7)using the meanfield approximation of the Walecka model to treat the baryon ground state. This problem was originally investigated by Chin ‘), who addressed it in the limit of small 4:. In this context, he was able to obtain qualitative predictions about general features of the collective excitations. However, for high-lying excitations his approach is an oversimplification because qt is never small in this case. Therefore, the extension of the results of ref. ‘) to this region is inappropriate. Here we extend the computation to the region where qi is not small and carry out the numerical computation for the poles of the meson propagators. Besides confirming the predictions of ref. “) in the small 4: limit, we obtain new features of the collective modes. In sect. 2, we summarize the theoretical background for the computation of the meson propagators in the one-loop approximation and the derivation of the dielectric functions. In sect. 3, we present our results on the collective modes. We have found three kinds of collective modes: zero-sound, meson-branch modes and modes which indicate instabilities of the MFT ground state. There is no zero-sound at the saturation density of nuclear matter; this is a relativistic effect which has no counterpart in a nonrelativistic
system. The absence
of zero-sound
at the saturation
density
is because
the attractive scalar interaction decreases the integrated strength of the effective vector potential through the form of scalar-vector mixing. Furthermore, the scalarvector mixing separates the two longitudinal modes of the meson branch. This effect is more impo~ant at higher /qi_ Both longitudinal and transverse instabilities are found at high density, while at low density only longitudinal instability exists. At small jg/, the scalar-vector mixing shifts the allowed values of density of longitudinal instability modes below the saturation density. Sect. 4 contains the conclusions.
2. Theoretical
background
In this section we describe the derivation represent the collective excitations in nuclear
of the dielectric function whose zeros matter. The formalism was originally
K. Lim, C.J. Horowirz / C‘ollecrive modes
developed
by Chin ‘). We describe
I) Y. This model is a renormalizable with each other by meson
nuclear
matter
quantum-field
exchange.
using the Walecka
731
model (QHD-
theory in which the nucleons
The lagrangian
interact
is
~=~[y,(irl~-gg,.V~“)-(M-g,~)]~+:(iJ,~iJ~~-mt~’) -bF,,,F~“i:mtV,V~‘t~.
(2.1)
Here 4 is the baryon field, $J is the scalar-meson (a) field, which couples to the baryon scalar density through the interaction term g,&/$, V’” is the vector-meson (w) field, which couples to the conserved baryon current density through g\.tjy,$V”, includes the term &Y arising from the renorand F,,, = it, V,, - dl,Vw. The lagrangian malization counterterms. We treat the baryon ground state in MFT approximation. In this approximation the meson fields are replaced by their expectation values, which are classical fields: 4 + (4) = &,
v,-XV,)-+,v,,.
(2.2)
This approximation should be increasingly valid as the density increases “). Note that the mean scalar-meson field & shifts the nucleon mass from M to the selfconsistent effective mass M” in both the Dirac and Fermi seas, M*=M-g,c,b,,.
(2.3)
The mass shift of the Dirac sea produces a vacuum fluctuation contribution to the energy density “). In the MFT this vacuum fluctuation effect is ignored (see eq. (2.14)) while in relativistic Hartree approximation (RHA) this effect is included ‘.“). We will use the MFT in the present work and neglect vacuum effects. We use the parameters g;=91.64,
m, = 550 MeV,
g; = 136.2,
m, = 783 MeV,
(2.4)
which reproduce (in MFT approximation) nuclear saturation at a density correbinding energy sponding to a Fermi momentum kF = 1.42 fm ’ and nuclear-matter of 15.75 MeV [ref. “)I. To compute the meson propagators, we sum over ring diagrams which consist of repeated insertions of the lowest order one-loop proper-polarization part. This procedure is equivalent to the relativistic random-phase approximation (RPA). The importance of summing the set of ring diagrams for the study of collective modes is well stated in conventional many-body physics I,‘)). Since we have both scalar and vector mesons in our model, besides being affected by the presence of the nuclear matter, the meson propagators must take into account the mutual interaction of the respective fields, namely, scalar-vector mixing. This arises when a particle-hole pair is excited by a vector meson and decays into a scalar meson. This scalar-vector
732
K. Lim,
<:J.
Hornwirz
/ Coktiw
m&s
mixing is a purely dells~ty-dependent effect. It complicates the summation because the Dyson’s equations for the scalar- and vector-meson propagators
process become
coupled. Therefore it is convenient to define a full scalar-vector meson propagator in the form of a 5 x 5 matrix with indices a, h ranging from -1 to 3, where -1 iJ‘,i, corresponds to the scalar meson and 0, 1, 2, 3 to the components of the four-vector. Dyson’s
equation
for 9 can be written
as a matrix
equation
(s = “ii”+ F/ TC’fl”T . . The lowest-order
scalar-vector
meson
propagator
(3.5) 9” is block-diagonal, (2.6)
1
the entries
being the noninte~cting
propagators
(2.71 where -I
D”(q)=q;_m;+in.
-
(2.8)
Note that the llPq,, term in Dam, does not contribute to the physical because of current conservation “I. The density dependent polarization insertion is also a 5 x 5 matrix
quantities
(2.9)
the indices p and I/ ranging from zero to three. Here the lowest order scalar, vector and scalar-vector-mixed polarization parts are defined as
(2.10) G(k)
is the self-consistent
= G,(k)+
RHA baryon
G,,(k),
propagator
given by
(2.1 I)
K. Litn,
C’.J. Horowit-_ / C’ollectiw
where k, is the Fermi momentum. the energy
and three momentum
The effective
mode
733
mass M* is given by eq. (2.3), and
are k:%P= ( /q” _ g, v”, k ) ,
(2.12)
E*(k)=\!k:+M”?.
(2.13)
and
The Feynman propagator G, involves the propagation of virtual particles antiparticles, and G,, describes the propagation of ‘holes’ inside the Fermi while correcting GF for the Pauli exclusion principle.
and sea,
In accordance with the MFT approximation we neglect the vacuum fluctuation effects; therefore, we take only the density-dependent part of these polarization insertions. In RHA one must include vacuum polarization insertions which are divergent and need to be renormalized. We will examine the effects of vacuum fluctuations in a future paper. The density-dependent polarization insertions are 17!‘( 4) = pigi
J~.“[G,~(“‘G,~‘“+““““(“‘G
+G,(k)G,,(k+q)l, II;,.(q)
= -ig;
5
~TrLu,G,,lk)y.,G,,(l+q~+y~G,,(k)y.,C;.,(k+q)
+ r,G(k)r,G,,(k+q)l, Lfp’C
q) = kg,
J
d”k -Tr[y,G(li)G(k+q)]. (27i-?
(2.14)
This procedure of taking only density-dependent part of the polarization insertions might cause problems calculating the damping of the collective modes because, when the imaginary part of the polarization insertions are nonzero, collective modes can decay into real particle-hole or particle-antiparticle pairs. For space like q,, the imaginary part of the vacuum polarization insertions are zero but for time like q, they are no longer zero. Therefore, we should include the imaginary part of the vacuum polarization insertions to study the damping. This will be discussed in sect. 3. The density-dependent polarization insertions in eq. (2.14) include the Pauli blocking of particle-antiparticle excitations besides the particle-hole excitations. This arises from the presence of antiparticle propagation in the Feynman part of the propagator, eq. (2.11). Therefore the terms G,,( li)G,( k + y) and G,( k)G,,( k + y) in eq. (2.14) include the Pauli blocking of NN excitations. These terms correct for particle-antiparticle excitations of the vacuum which are no longer possible in the medium. At zero density, in the limit q + 0 the vacuum contributions are canceled by renormalization counterterms. As the density and the momentum increase there will be vacuum corrections from the momentum and M* dependence of the vacuum
K. Lim, C.J. Horowitz / Collective
734
polarization insertions all of these corrections
and from the Pauli blocking. It is often difficult to include and it is common to take into account only the Pauli blocking
effects. We include only the Pauli blocking fluctuation effects and examine the limitations Pauli
blocking
of NN excitations
discussed in sect. 3. Baryon current conservation polarization insertions,
solution
implies
to the Dyson’s 9
Defining
the dielectric
the following
equation,
are
for the one-loop
flT,D,(q)q” = 0.
(2.15)
eq. (2.5), is (2.16)
det (1 - abOnD) ,
it follows that the full scalar-vectormeson vanishes,
equations
modes
E as F -
function
=
collective
g”p-‘g”.
= (I-
function
corrections neglecting the vacuum of this approach. The effects of the
for the density-dependent
q”Ii!,“,(q) The formal
modes
propagator
(2.17) has a pole when the dielectric
i.e. e=det(l-g”LrD)=O.
(2.18)
This is the eigencondition for determining the collective-excitation spectrum 2.‘). Therefore, to determine the dispersion relation of the collective excitation we need to know the polarization insertions. In a frame of reference where q = (q,0, 0), the current conservation conditions become q"17,M+qlIy=o.
q"II~~+q17~~=o,
From these equations polarization
and the symmetry
of the integrals,
(2.19)
we get the density-dependent
matrix
ny n,”
Ill,” IIC” II,“, II,“,
0
0
0
0
ng
0
00
IT: 0
I
(2.20)
.
0
,I’= By defining
the longitudinal
li.‘(l
0 i nr”
0 0 n:‘,
and transverse -A”II:)(l
0
II::0
dielectric
functions
as
-@n:‘)+$d”O”(n~)‘, (2.21)
K. Lim, C.J. Horowic
where n:‘=
I7& -n::
modes
735
and IT’,‘= n_g = II_::. We can write & =
Note that we have used the current When
/ Collrcriae
ELET.
z
conservation
191~ q = 0, Ily(q = 0) = 0 by the current
(2.22) conditions conservation
(2.19) to write eq. (2.21). (see eq. (2.19))
and
Z7y( q = 0) = 0 because of the symmetry of the integral. The density-dependent part of the lowest-order scalar, vector, and scalar-vector-mixed polarization insertions are evaluated analytically, and their analytic expressions are given in the appendix. They are substituted in eq. (2.21) to evaluate the dielectric functions. The collective modes obtained from the zeros of the dielectric functions are presented in the next section.
3. Results In this section we present numerical results for the collective modes. First, we consider undamped collective modes, which can be obtained by letting Im 17”= 0 and searching for the zeros of the real part of the dielectric functions. In the MFT approximation a non-zero Im nil indicates the decay of the collective modes into pairs for time-like real particle-hole pairs for space-like qp, or particle-antiparticle q, which includes the decays into Pauli blocked NN pairs. The decays of the collective modes into the Pauli blocked NN pairs will be cancelled exactly by the corresponding ones in the imaginary part of UUCUU~~polarization insertions when these are taken into account I’)). We will consider the damping region after having investigated the collective modes using the real part of the dielectric functions. We have found three kinds of collective modes: zero-sound modes, meson-branch modes and instabilities.
3.1.
ZERO-SOUND
MODE
This is a low-lying longitudinal collective mode with a dispersion relation typical of sound propagation q"= C,,q.However, zero-sound is physically different from ordinary first-sound. In the zero-temperature limit (T+O), because of the Pauli principle, the interparticle collision time increases as TV’ and ordinary sound propagation is no longer possible. On the other hand, following the Landau-Fermi liquid theory ‘,‘,‘).“), a collisionless zero-sound mode is possible in the limit of zero temperature. We do not find a zero-sound mode at low density. This result is in agreement with the prediction of ref. “). The reason for the absence of the zero-sound mode at low density is that the attractive scalar interaction decreases the integrated strength of the effective vector potential through the scalar-vector mixing. Removing the scalar-vector mixing (i.e. set flM =0) we find zero-sound even at low densities
K. Lim,
736
caused
by vector-meson
zero-sound
is only possible
poles
/ Cofkctioe modes
C.J. Harowilz
for q not
at high densities.
too large.
With
Typical
zero-sound
scalar-vector
in figs. 1, 2 and 3. From these figures we can see that even though is zero at q = 0, for small
nonzero
q and low density
mixing
modes are plotted the mixing
the scalar-vector
term
mixing
is
important for zero-sound. As 9 and density increase it is less important. According to ref. “), to have zero-sound at low density the condition gz/ rnt > gi/ rnt,
(3.1)
must be satisfied. This is not satisfied for the parameters in eq. (2.4). The zero-sound speed C,, calculated from Landau-Fermi liquid theory ‘), agrees with the microscopic calculation and is plotted in fig. 4. The zero-sound mode first appears at a density corresponding to a Fermi momentum kt. = 1.64 fm-‘. The absence of zerosound mode at the saturation density is a purely relativistic effect because, nonrelativistically the zero-sound mode is related to the repulsive core of the NN interaction. Therefore, any system which saturates from a repulsive core has zero-sound at saturation density. The zero-sound speed approaches the speed of light from below as the density increases; this is the causal limit expected on physical grounds. For k,< 2.73 fm ’ there are two zero-sound modes. Further zero-sound modes appear when k,> 2.73 fm-‘. (See fig. 1) One of these modes is not damped at low
oI/e/: 0
.
, 1
.‘.
kF
/ 2
i 3
.,
(fm-‘)
Fig. 1. Zero-sound mode and instability mode at y = 10 MeV yI, versus k, Solid curves are with scalar-vector mixing, dot-dashed and dashed curves are without mixing. Upper two solid curves represent the zero-sound, lower solid curve represents the instability mode. Dot-dashed curve represents poles from the vector meson, and the dashed curve shows the poles from scalar meson.
kF
(fm-‘)
Fig. 3. Longitudinal modes at y = 500 MeV q,, versus k, Solid curves include scalar-vector mixing, dot-dashed tunes and dashed curves without mixing. Note the difference in q,, scale compared to fig. I. Upper two solid curves are meson branch modes and the lower two are zero-sound. The lowest zero-sound branch is always damped. Dashed curves are from the vzalar-meson poles and the dot-dashed curves are from the vector-meson poles.
q and merges into the region of particle-hole damping as y increases. The other modes are always damped. (The lower curves of the zero-sound modes in figs. 1 and 2 are modes of this kind). At k, = 1.9 fm ’ two zero-sound modes are possible and the initially undamped mode is plotted in fig. Sa as a curve of q” versus q. The region where it is not damped can be seen more clearly from tig. 5b where the same mode is plotted as a curve of (q’)/q)/C,, versus q. At k, = 3 fm-’ four zero-sound modes are possible at small q. These modes are plotted as dashed curves (II) in fig. 7a. The upper two, which are not distinguishable from the third mode in this figure, are resealed and plotted in fig. 7b. Only the uppermost mode (see fig. 7b) is not damped at low q and the other modes are always damped. The zero-sound mode is damped if (3.2) where U, = lir/ EE is the Fermi velocity. In this region polarization insertions is no longer zero, and therefore decay into real particle-hole pairs. The critical values of modes are damped, are plotted versus k, in fig. 6. At a
the imaginary part of the the zero-sound mode can q for which the zero-sound given k, zero-sound is not
K. Lim, C.J. Horowitz / Collective
738
modes
1500
zz
1000
VECTOR . .
i?
~ 500
0
0.5
1.5
1
kF
2.5
3
&d;
Fig. 3. Longitudinal and transverse modes at y = 1 GeV. Upper solid curves are the longitudinal meson branch modes with mixing, middle solid curves are zero-sound, and the lowest solid curve is the instability mode. Dotted curves labeled vector are from the vector-meson poles, the ones labeled scalar are from the scalar-meson poles without mixing. Dashed curve is the transverse meson branch mode. The instability from the transverse mode is not plotted in this picture
1.0
0.9
0.8
0
u
0.7
0.6
0.5
t 0
0.5
1
kF
1.5
2
2.5
3
(frn-“)
Fig. 4. Zero-sound speed which is obtained from Landau-Fermi liquid theory. See ref. ‘), No zero sound exist at low density (k,~ 1.64 fin-I). The sound speed approaches unity at high density.
K. Lim, C. J. Horowitz / Cbllective modes
ZERO SOUND
kF=1.9 fm-' M"=.25M
100 q
1.04
. ”
’
”
150
200
(MeV)
ZERO SOUND i
739
kF"1.9 fm-' M"=.25M
”
”
”
”
”
”
”
04 i ,’ /’ /’
DAMPED
NOT DAMPED 0.96
-
t 0
6
50
I
100 q
b
I,..
1.50
.
.
I
.
200
(MeV)
Fig. Sa, b. Solid curve represents the zero-sound mode at k, = 1.9 fm-‘. Dashed curve shows the upper limit of non-zero lm II”. Zero-sound is damped for y z 8X MeV. No zero-sound is found for y > 226 MeV at this density.
740
kF Fig. 6. The critical
values
(fm-‘)
of y for which the zero-sound modes are damped. damped in the region below the solid curve.
The zero-sound
is not
damped only for y less than a critical value of q. From this figure we can see that at very high densities the zero-sound mode is not damped only for very small q; for instance, at kfy= 3.3 fin- ’ and q = 1.2 MeV the zero-sound mode is already damped. Therefore the allowed regions of q and density for undamped zero-sound mode propagation are very restricted.
3.2. MESON-BRANCH
MODE
This is a high-lying excitation mode. The modes in this branch are like plasma excitations in QED and at zero density this branch reduces to the propagation of free mesons. We have both transverse and longitudinal modes. For a fixed ii,2 1.6 fin-‘, there are two longitudinal modes and one transverse mode (figs. 7a,8). In each branch there is a decrease in the excitation energy which seems to be related to the effective mass of the baryon; as the effective mass is decreased this effect is enhanced. In fig. 9, the longitudinal meson mode energies q” are displayed versus kF at a small fixed three-momentum (q = 1 MeV). The same energy curves without meson mixing are plotted in fig. 10. These two figures show that for small q the scalar-vector mixing is not strong enough to separate the two longitudinal meson modes. On the
LO~GrTU~INAL MODE: kF=3. fm-* M" =.0885M
600
400 4
ZERO SOUND i’.‘---T’
r
.‘I’
1.0002 0 Y -&oooo
t
2 tr‘ 0.9998
i0.9996
I
0
1200
kF=3.0 fm-' M*=.OBRM
(b)
1.0004
1000
800
(MeV)
15
10 q
(MeV)
‘.--r------
K. Lim, C.J. Horowitz
742
TRANSVERSE
/ Cdiectiue
MODE kFc2.6
modes
fm-’
M’=.l188M
500 -
0
200
400
600 q
a00
1000
1200
WV)
Fig. 8. Transverse mode at k, = 2.6 frn~~‘. Upper solid curve (I) is the meson branch mode and the lower curve (II) shows the instability. Shaded regions A-D show the damping regions and they are discussed in text. other
hand
at higher
q the mixing
becomes
more
effective
(see figs. 2, 3). If we
remove the mixing term, the scalar- and vector-meson modes are well separated at low density but they cross each other twice as the density increases (see figs. 2, 9). For instance at q = 1 MeV they cross each other at k, - 1.7 fm- ’ and at kF - 2.2 fm- ‘. As q increases, the first crossing point decreases a little but the second crossing point increases very much. The role of scalar-vector mixing is to separate the two meson modes at these points; otherwise it is not important in this meson branch. This conclusion is in disagreement with the result in ref. ‘) in which Chin said that the scalar-vector mixing is important at low densities and it is unimportant at high densities. We believe that this is due to the inappropriate extrapolation of the results of ref. “) to the region of large 4:. Also note that for wry snail three momentum (q = 1 MeV) the transverse mode, which is plotted in fig. 11, almost coincides with the vector-meson-like longitudinal mode of fig. 9; however, at high q the two are well separated (see fig. 3). An interesting peak is seen in fig. 9 for the vector-meson-like and the scalar-mesonlike mode. The peak of the vector-meson-like mode is at the energy qo- 2M* while the peak for the scalar-meson-like mode is at q,)-- M, where M is the free nucleon mass. For q = 0 and at given density the smallest amount of the energy that will excite a Pauli blocked NN pair is q. = 2M”. Therefore, we think that the peak of the vector-meson-like mode is related to the Pauli blocking of NI? excitations. Since
K. Lim, (1.1. Horowitz / Collective
modes
743
I”“,“‘-,“.’
,
I
*I
,
/
\\’ %
1500
’
/
.. .
c
;(?
A. y,.i, 1‘,A., \ ‘,\‘.
-
/
/
/
/I
, /
/
/
/
/
/II
I
/
I
>--
“/ ’ //,
01”““““““““““” -0.5
“////,,f,,
,
“////I;,/,
‘1
J
1.5
1
kF
2
2.5
3
(fm-*)
Fig. 9. Longitudinal meson branch modes at y = 1 MeV. Here dashed curve is vector-meson-like longitudinal mode and solid curve represents the scalar-meson-like longitudinal mode. Dot-dashed curves are scalar (I) and vector (II) meson-like longitudinal modes without Pauli blocking of Nfi excitations. The shaded region C is where the Im I7 ” ib non-zero. Note the peaks due to the Pauli blocking of NN excitation.
0
0.5
1
1.5
kF Fig. IO. Longitudinal
meson
branch
2
2.5
3
(fm-‘)
modes at q = 1 MeV without coincide with those of fig. 9.
scalar-vector
mixing.
These modes
144
K. Lim, C.J. Horowitz / Collective
modes
~~~‘~~~~‘~~~,‘~~~.‘,~~~
0
1
1.5
2
2.5
3
-1 kF
(fm
>
Fig. 11. Solid curve is the transverse meson branch mode at q = 1 MeV. This mode almost coinside with vector-meson-like longitudinal mode in fig. 9. Dot-dashed curve is the mode computed without Pauli blocking of Nrri excitations. The shaded region C is where Im I7” is non-zero.
we can write the Feynman propagator as the particle part plus the antiparticle part, we can remove the Pauli blocking of NN excitation process by removing Gr,( k)Gr,( k + q) and G,,( k)G,( k + q) terms from the density-dependent polarization insertions. Where GF( k) represents the antiparticle part of the Feynman propagator. The energy curves obtained from removing these contributions are plotted as dot-dashed curves in fig. 9. The upper curve (1) corresponds to the vector-meson-like, and the lower curve (II) represents the scalar-meson-like mode. Removing the antiparticle
states
causes
the peak
of both
scalar-
and
vector-meson
modes
to
disappear. The peak appears in meson modes due to the Pauli blocking of NN excitations. We believe these peaks will go away if the full virtual NN pairs are also included in a relativistic Hartree approximation. However, the full renormalized vacuum calculations may be difficult in many situations. Therefore these peaks suggest important limitations in the simple MFT calculations. Note that q. is proportional to k, at high densities, therefore, it is hard to excite a mode in this meson branch at very high density. The shaded regions B and C of figs. 7a, 8 are the region where Im II” is not zero and qr is time-like (see also the shaded region of figs. 9, 11). In these regions the meson modes can decay into particle-antiparticle pairs. B is enclosed by O(q,,(ET+E~,~,))B(E::+E~,,,,-q,)andCis enclosedby O(qS-4M*‘)O(E$+E:,_,,qJo(2k,
- q), where
Ef, ty = d(kF*q)7+M*7,
E; = ,,lk;+ M”’
(3.3)
K. Lim, C.J. Horowit;
However,
since
polarization
q, is time-like
insertions
745
/ C‘olleclioe modes
in these regions,
the imaginary
are not zero. Therefore,
the vacuum
part of the vacuum
polarization
insertions
should be included in the study of the damping of the collective modes in this branch. The imaginary part of the vacuum polarization insertions are non-zero if qt >4M*‘. This inequality holds in the regions A, B and C of figs. 7a, 8. The density-dependent meson modes in region C decay into Pauli blocked Nfi pairs which are cancelled by the counterpart of the vacuum polarization I’). Therefore, when the vacuum polarization is taken into account the meson modes are damped in the region A and B. In this case the threshold of particle-antiparticle pair production will be at q,,= M*+ ET and q = k,. Note that the q(, at the threshold can be less than 2M. (See for instance figs. 7a, 8) The shaded region in figs. 9, 11 corresponds to the region C of figs. 7a, 8. Thus in this small-q limit the modes decay into only Pauli blocked NN pairs. Therefore, including the vacuum polarization the meson-branch modes at small q might propagate without damping. One can see that the inclusion of vacuum polarization insertions will also change the real part of the collective modes. This will be discussed in a future publication.
3.3. INSTABILITIES
In addition to the zero-sound and the meson-branch modes, we found collective modes at low energy transfer q,,. In fig. 12 these modes are plotted for q,,= 0, q versus k,. At low density only the longitudinal mode (solid curve) shows this 2000
I
I
I
1 // /
T
1500
-
1000
-
2
/
/
/
/
/
/
rT 500
/ \
-
\
”
0
1
KF Fig. 12. Poles of the meson propagator dielectric function and the dashed
.
”
2
I-
---_
------__
”
”
3
‘I”
4
(fm-‘)
at yI, = 0 MeV. Solid curves show the zeros from the longitudinal curves show the zeros from the transverse dielectric function.
746
K. Lim, C..l. Horon+tz / Collecrive
modes
instability. It appears at low k, and low q. For the longitudinal modes this instability comes from the scalar-meson poles and the scalar-vector mixing tends to shrink the allowed region of this mode (see figs. 1, 2). For instance at q = 500 MeV this mixing completely removes the mode generated q this mixing is the relevant factor that brings
by the scalar-meson poles. At small the density of the instability below
the saturation density. The existence of these modes at q. - 0 indicates that uniform low-density nuclear matter is unstable against perturbations of the density at long wavelengths. This represents the liquid-vapor phase transition of nuclear matter below the saturation density. At high density both transverse and longitudinal modes exist. This kind of longitudinal mode appears at densities corresponding to k,a 2.6 fm-’ and q 900 MeV and the momentum increases as the density increases (solid curve in fig. 12). For this high q the mixing seems less important than at low q. The transverse mode apperars at kFa 1.81 fin-‘; q starts at intermediate wavelength (q - SO0 MeV) and it spreads into a very wide region as the density increases (dashed curve in fig. 12). The existence of the transverse instability at high density is due to the attractive particle-hole interaction mediated by the exchange of transverse w-mesons. We will refer to this as w-meson condensation for transverse modes. In refs. 4*“) the instabilities of the meson-nucleon system in the one-loop approximation including vacuum fluctuation effects (RHA) are investigated. The RHA also has the instability at low density, low q but no high-density medium-q case exists. RHA ground state is also found to be unstable at large q because of vacuum polarization I’). (See fig. 1 of ref. I’)). Region D of figs. 7a, and 8 shows the nonzero imaginary part of the densitydependent polarization insertions for space-like q,*. This region is enclosed by - E$) -q,,)O(q,,-(E%_,E~))f?(qo). Where ET,,, and ET are given in NE>+, eq. (3.3). The instability modes and the zero-sound modes can decay into particlehole pairs in this region. Figs. 7a and 8 show that the instability modes are damped in most of the cases. 4. Conclusions We investigated collective modes in the one-loop approximation to the Walecka model and found three kinds of collective modes: zero-sound, meson-branch modes and modes with zero energy transfer, which indicate an instability of the MFT ground state. We found three important results about the zero-sound modes; first, zero-sound mode is a vector-meson-like longitudinal mode and it is absent at the saturation density. This is a relativistic effect that has no analog in nonrelativistic systems. It arises because the scalar-vector mixing of longitudinal mode decreases the strength of the vector meson and pushes the allowed density of zero-sound above the saturation density. Second, the zero-sound speed approaches unity from below at high density, and third, except at very low q.
at high density
there
is no undamped
zero-sound
mode
K. Lim, C.J. Horowirz / C’ollec~ive modes
As for the meson at a given density
branch
sible for the separation the region where when the scalar-
modes;
(k, 2 1.64 fin-‘).
747
first, there are typically
two longitudinal
We saw that the scalar-vector
of the scalar-meson
mixing
modes is respon-
mode from the vector-meson
mode in
they cross. Therefore the scalar-vector mixing is not important and vector-meson modes are well separated. This is expected
because the effects of the scalar-vector mixing must be stronger when scalar and vector modes are closer. Furthermore, we have shown that the Pauli blocking of NN excitations strongly effect the meson modes in this MFT approximation. Finally, the longitudinal instability at low density represents the liquid-vapor phase transition of nuclear matter below the saturation density. The scalar-vector mixing is the important factor shifting this instability below the saturation density. At high density transverse instability represents w-meson condensation. At low density the meson branch modes are the only undamped collective modes while at high density it seems that there is no undamped collective mode except for the zero-sound mode at extremely small q. However, the inclusion of the vacuum polarizations might change this picture. Vacuum polarization effects will be included in a future paper. The existence of poles in the meson propagator at zero energy transfer indicates the instability of uniform ground state of the meson-nucleon model of nuclear matter and this may give the upper limit for the domain of validity of the various approximation of meson-nucleon field theory 4,5). The collective modes are also important for the study of the linear response of the nuclear matter to various probes I’), for the calculation of RPA energy ‘,‘j) and to study meson propagation through nuclear matter. We are grateful
to Brian
D. Serot for his valuable
comments
on the manuscript.
Appendix
Here we present one-loop polarization (2.14) in sect. 2. - Real part:
the analytic expressions for the density-dependent part of insertions which are obtained from doing the integrals in eq.
7
ReIll’(y,,,y)=$$
57
k,Ec-(3M*‘-i9t)In
+$(4M*‘-9;)
Renf’(a,,y)=ti
-$
]n IPI+:(~M*‘-~;)‘~~
(4M”‘-9;)
In IcyI
1
ik,;E;_:y’]n(~)+~(E:‘+~)]n,o,
77l9’ _ E;(3q;+4ET2) 249
~
(2M”‘+
9;)(4M*”
- q;).] I
where h is the isospin
degeneracy,
A==
2,
for nuclear
matter
i 1,
for neutron
matter,
and N, ,R and .I are cY=
q; -4( qoJF; - qk,)’ q~-4(~l~~~~+qk~)~’
p = (q;+2q$,.)‘-4q,‘,E;’ (q; -2qk,.)‘-4q;Ef”
+tanrl;w.bTl
-k,q;dq;,(4M”‘-q;) q:E;-4q,,M*‘(q,,E;+qk,)
II
’
Where the first tan ’ is the branch with values in [O, rr] and the second tan-’ is the branch with values in [ --;TT, ST]. Note that .1 is zero at qi = 0. When the argument of any tan -’ is imaginary, this becomes a branch of the logarithm function. -Imaginary part:
The plus sign or the minus sign in the right-hand
sides of the expressions
corresponds
to space-like qr((qo(i q), or time-like qM(jq,,j> q), respectively. For space-like q,. the quantities in the right-hand side of these equations by
are given
___E,,-\;~;+&,f:%‘zzE~, E,,,:,,] ,
E, = min[ E;,
ES -)q,,),
IS,,,,, = max [M*,
I?,.],
E,.=;[q,~l-4M”‘/q~-/q~t/].
For time-like
qU, they are
if q: <4M*‘,
/_iVin [
E:,
E, =
j(/q,,/+qdq)],
otherwise.
0, min (IS,., 15:) , M*,J(lq,,l-qJ1
if qL 5 4M*’ 3 otherw-ise , -4M”‘/q:
References 1) A.L. Fetter 2nd J.D. Walecka,
Quantum
theory of many-particle
systems
(McGraw-Hill,
New York.
1971)
3) S.A. Chin, Ann. Phys. 108 (1977) 301 3) T. Matsui. Nucl. Phya. A370 (1981) 365 41 R.J. Furnstahl and C.J. Horowitz, Nucl. Phys. A485 (1948) 632 5) R.J. Perry, fhys. Lett. B199 (lY87) 489 6) H. Kurasuwa and 1. Suzuki, Nucl. Phys. A445 i 1985)6X5 7) C.J. Horowitz, Phys. Lett. B208 11985) 8 8) B.D. Serot and J.D. Walecka, Ad\. in Nucl. Phys. 16, ed. J. Negeie and E. Vogt 1986)
Y) 11. Pines, The many-body
problem
(Benjamin,
New York,
1961)
f Plenum.
New York,
750
I(, tim,
C.J. H0rnWili / C’ollectioe modes
IO) T. Matsui and B.D. Serot, Ann. Phys. 144 (1982) 107 11) G. Baym and S.A. Chin, Nucl. Phys. A262 (1976) 527 12) C.J. Horowitz, 3rd Conf. on the intersections between particle 1988 13) X. Ji, Phys. Lett. B208 (19%) 19
and nuclear
physics,
Rockport
Maine,