The effects of magnetic interaction between vector mesons and nucleons on the propagation (mass and width) of the \rho-meson in particular moving through very dense nuclear matter is studied and the modifications, qualitative and quantitative, due to the relevant collective modes (zero-sound and plasma frequencies) of the medium discussed. It is shown that the \rho-mesons produced in high-energy nuclear collisions will be longitudinally polarized in the region of sufficiently dense nuclear matter, in the presence of such an interaction

De, T

Dutta-Roy, B

Kundu, A

CERN Document Server

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SINP-TNP/95-08
June 1995
Tensor coupling and vector mesons
in dense nuclear matter
Abhee Kanti Dutt- Mazumder
1
, Binayak Dutta-Roy
2
,
Anirban Kundu
3
and Triptesh De
Theory Group,
Saha Institute of Nuclear Physics, 1/AF Bidhannagar,
Calcutta - 700 064, India
Abstract
The eects of magnetic interaction between vector mesons and
nucleons on the propagation (mass and width) of the -meson in par-
ticular moving through very dense nuclear matter is studied and the
modications, qualitative and quantitative, due to the relevant collec-
tive modes (zero-sound and plasma frequencies) of the medium dis-
cussed. It is shown that the -mesons produced in high-energy nuclear
collisions will be longitudinally polarized in the region of suciently
dense nuclear matter, in the presence of such an interaction.
1
E-mail: abhee@saha.ernet.in
2
E-mail: bnyk@saha.ernet.in
3
E-mail: akundu@saha.ernet.in
1. Introduction
The study of the properties of hadrons in hot and dense nuclear matter is
of cardinal importance in the understanding of various signals that probe the
dynamics of heavy ion collisions [1, 2], as also in the study of the equation of
state of superdense matter. Such investigations furthermore are of relevance
to various important issues of nuclear astrophysics dealing with the properties
of neutron stars, cooling of supernovae, the gravitational collapse of massive
stars etc [3, 4]. Despite numerous theoretical attempts to determine the
properties of vector mesons in dense nuclear matter, controversy still exists
regarding their in-medium eective masses and decay widths [5]; various
formalisms such as the NJL model, Walecka model or QCD sum rules [6, 7,
8, 9] give dierent results [10, 11, 12]. Calculation of mass shifts and decay
widths of mesons in nuclear matter has assumed particular importance in
view of proposals to make such measurements at CEBAF [13]. The essential
focus of this paper is to investigate the eect of the magnetic interaction on
vector mesons propagating in dense nuclear matter. In particular we study
the case of the  meson where this eect is expected to be more pronounced.
The paper is organised as follows: the formalism is rst set forth and this
is succeeded by a discussion of the results.
2
2. Formalism
The vector meson-nucleon interaction Lagrangian may be written as
L
int
= g

[

N



N  


2M

N



N@

]V


(1)
where V

= f!; g,  running from 0 to 3, indexes quantities relevant for !
when  = 0, while  = 1 to 3 refers to the  meson; 
0
= 1 and 
i
are the
isospin Pauli matrices. The coupling constants g

, g
!
and the \anomalous" or
tensor-coupling parameters 

and 
!
may be estimated [7] from the Vector
Meson Dominance (VMD) of nucleon form-factors or from the tting of the
nucleon-nucleon interaction data as done by the Bonn group [14]. In view of
the relatively small value of the iso-scalar magnetic moment of the nucleon
as compared to the iso-vector part, the tensor coupling is more important
for the  than it is for the !. In our calculation the handling of these two
mesons runs very similarly, the only essential dierence residing in the values
of the coupling parameters for the two cases. From the Lagrangian the V NN
vertex factor is given by
 


= g

[



 


2M


@



] (2)
We shall consider the propagation of vector mesons in dense nuclear mat-
ter at zero temperature. As such we shall be following the usual methods of
relativistic quantum eld theory, with the vacuum replaced by the ground
state of nuclear matter at zero temperature, specied by the Fermi momen-
tum (k
F
) corresponding to its density. Such a eld theoretic approach to
the study of many body problems was developed, in the context with which
we are concerned, by Matsubara, Galitskii and Migdal [15, 16, 17], and the
relativistic generalization provided by Fradkin [18], and most importantly in
the approach being rather closely followed in this paper, by Chin [19]. Fur-
thermore, the Hartree or Mean-Field energy density for dense nuclear matter
goes as k
6
F
, while the correlation energy varies as k
4
F
and this leads us to the
position that we may use the mean-eld description for nuclear matter at
very high densities (that being our present concern).
3
The second order polarization tensor 

, for the vector meson arising
from the nucleon loop (Fig. 1), is thus calculable from the Lagrangian and
yields



=
 i
(2)
4
Z
d
4
k Tr[i 


iG(k + q)i

 


iG(k)] (3)
where (; ) are the isospin indices and G(k) is the in-medium nucleon prop-
agator
G(k) = G
F
(k) +G
D
(k) (4)
where
G
F
(k) = (k



+M

)[
1
k
2
 M
2
+ i
] (5)
and
G
D
(k) = (k



+M

)[
i
E

(k)
(k
0
  E

(k))(k
F
  j
~
kj)] (6)
with M

denoting the eective mass of the nucleon in the medium and
E

(jkj) =
q
j
~
kj
2
+M
2
. The rst term in G(k) namely G
F
(k) is the same as
the free propagator of a spin
1
2
Fermion, except for the fact that the eective
mass of the nucleon is to be used, while the second part, G
D
(k), involving
(k
F
 j
~
kj), arises from Pauli blocking, describes the modications of the same
in the nuclear matter at zero temperature [20], as it deletes the on mass-shell
propagation of the nucleon in nuclear matter with momenta below the Fermi
momentum.
In a similar vein the polarization insertions can also be written as sum of
two parts:


(q) = 
F

(q) + 
D

(q); (7)

F

(q) =
 i
(2)
4
Z
d
4
k Tr[ 

G
F
(k + q)

 

G
F
(k)] (8)

D

(q) denotes the density dependent part of the polarization and 
F

(q)
denotes the free part i.e. it contains the eect of `Dirac sea'. The density
dependent part has a natural cut o because of the (k
F
  j
~
kj) function
whereas the free part is regularized by adding appropriate counter-terms to
the Lagrangian. As known from ordinary quantum eld theory the eect of
4
the nite part of 
F
(q) is to render the coupling constant momentum depen-
dent and this modies the interaction subtantially as short distances. Here
the procedure of regularization and renormalization is very similar except
that the nucleon mass M would have to be replaced by the eective nucleon
mass in nuclear matter (M

); the tensor coupling, however, belongs to the
genre of unrenormalizable theories and has to be tackled in the spirit of an
eective theory. In the case of ordinary nuclear matter densities this short
distance modication is important as discussed in Ref [7], however, in the
present context where we concentrate on the long wavelength limit, i.e. low
momentum transfer to study the collective modes in dense nuclear matter,
the eect of vacuum polarization would in general be small as pointed out
by Chin [19].
The real part of the density dependent piece of the polarization is given
by

D

=
g
2
v

(2)
4
Z
d
4
k
E

(k)
(k
0
  E

(k))(k
F
  j
~
k j)

h
T

(k   q; k)
(k   q)
2
 M
2
+
T

(k; k + q)
(k + q)
2
 M
2
i
(9)
Although the form of 
D

(q) is the same as that of Chin [19], here,
however, it has three part corresponding to vector-vector, vector-tensor and
tensor-tensor terms. Hence the self energy can be written as

D

(q) = 
vv

(q) + 
vt+tv

(q) + 
tt

(q) (10)
The 
D

(q) functions in this case are as follows

vv

=
g
2
v

3
Z
k
F
0
d
3
k
E

(k)
K

 Q

(k  q)
2
q
4
  4(k  q)
2
(11)

vt+tv

=
g
2
v

3
(
kM

4M
)2q
4
Q

Z
k
F
0
d
3
k
E

(k)
1
q
4
  4(k  q)
2
(12)

tt

=  
g
2
v

3
(
k
4M
)
2
(4q
4
)
Z
k
F
0
d
3
k
E

(k)
K

+Q

M
2
q
4
  4(k  q)
2
(13)
5
where K

= (k

 
k:q
q
2
q

)(k

 
k:q
q
2
q

) and Q

= ( g

+
q

q

q
2
). It is clear that
the form for the polarization tensor conforms to the requirement of current
conservation, i.e.
q


D

= 
D

q

= 0 (14)
In order to evaluate 
D

conveniently, we choose
~
k to be along the x axis i.e.
q = (q
0
; j ~q j; 0; 0), and k:q =j k jj q j  E

(k)q
0
, where  is the cosine of the
angle between
~
k and ~q. After  integration the non-vanishing components

D

are as shown below
0
B
B
B
B
@

00

01
0 0

10

11
0 0
0 0 
22
0
0 0 0 
33
1
C
C
C
C
A
(15)
Also for isotropic nuclear matter we have 
D
22
= 
D
33
and 
D
01
= 
D
10
, and
hence taking all this into account we have only two non-vanishing indepen-
dent component of 
D

, linear combinations of which gives us the longitudi-
nal and transverse components of 
D

, namely, 
D
L
(q) =  
D
00
+
D
11
(q) and

D
T
(q) = 
D
22
= 
D
33
.
For collective excitations, the wavelength of the oscillations must be
greater than the interparticle spacing. Thus for super-dense matter q << k
F
and we can simplify the denominators of the above integrations by neglecting
q
4
compared to 4(k:q)
4
[19]. In this approximation the integrations can easily
be performed to give results in a closed form, viz.,

D
L
(q
0
; q) =  
g
2
v

2
k
F

F
(1  c
2
0
)(c
0
=v
F
)

D
T
(q
0
; q) =
1
2
g
2
v

2
k
3
F

F
[1 + (1 
c
2
0
v
2
F
)(c
0
=v
F
)]
 = 1  (

2M
)
2
q
2
(16)
where
(x) =  1 +
1
2
x ln j
(x+ 1)
(x  1)
j (17)
6
and c
0
=
q
0
j~qj
, v
F
=
k
F

F
and 
F
is the Fermi energy 
F
=
q
M
2
+ k
2
F
. In the
limit ! 0, of course, the results of Ref[19] ensue.
The eective mass of the  mesons in hadronic matter is described by
the poles of the in-medium  meson propagator. For short wavelengths the
oscillation of meson elds can be regarded as the usual meson propagation,
as in that regime the eect of Pauli blocking is not appreciable, but in the
long wavelength limit medium eects can be substantial and these can be
beautifully interpreted as arising from the meson `picking up' the collective
modes of the nucleonic Fermi uid [21]. The vector meson propagator is
calculated by summing over ring diagrams, a diagrammatic equivalent of the
random phase approximation (RPA), which consist of repeated insertions
of the lowest order polarization, as illustrated in Fig. 2. We make use of
Dyson's equation to carry out the summation
D

(q) = D
0

(q) +D
0

(q)

(q)D

(q) (18)
The poles are determinable from the equation
det[


 D
0



] = 0 (19)
The bracketed term is nothing but the dielectric tensor of the system



= 


 D
0



(20)
the determinant of which is the dielectric function of the system . The eigen-
condition can now be expressed as
(q) = 0: (21)
As 

(q) is already known, (q) can be calculated immediately . The rel-
evance of the set of ring diagrams and the origin of such an eigen-condition
can be understood from linear response theory [22], where the uctuation of
the current density, the source term for the meson eld in nuclear matter,
given in terms of the polarisation tensor, is `picked up' by the vector eld.
We have already shown that 
D

(q) manifestly satises the current con-
servation conditions, and with the choice of q

already discussed, assumes
7
a particularly simple structure eq. (15). The longitudinal and tranverse
dielectric functions are dened as

T
(q) = 1  D
0

D
22
= 1 D
0

33
= 1  D
0

D
T
(22)

L
(q) = (1  D
0

D
00
)(1 D
0

11
) D
0

D
01
D
0

D
10
(23)
The explicit form of the longitudinal and transverse dielectric functions can
be obtained with the help of eqs. (16) and (17). to yield,

T
= 1 +
1
q
2
 m
2
1
2
g
2
v

2
k
3
F

F
[1 + (1  (
c
0
v
F
)
2
)(c
0
=v
F
)] (24)

L
= 1 +
1
q
2
 m
2
g
2
v

2
k
3
F

F
(1  (
c
0
v
F
)
2
)(c
0
=v
F
) (25)
The eigenmodes of the vector meson are given by
(q) = 
2
T
(q)
L
(q) (26)
corresponding to the three degrees of fredom of a massive vector particle.
The two identical (or degenerate) transverse collective modes are each given
by

T
(q) = 0 (27)
and the single longitudinal mode by

L
(q) = 0 (28)
which yield the relevant dispersion curves. Solutions to be sought may be
classied as space-like (j~qj > q
0
) or time-like (j~qj < q
0
), the latter being the
`particle' modes, while the former corresponds, as we shall see, to zero sound.
3. Results and discussions
As already stated earlier, for collective excitations the wavelengths of the
collective oscillations must be greater than the interparticle spacings, to wit,
j~qj << k
F
. Furthermore, the stability of the collective modes requires on the
8
one hand that the dispersion curves be such that q
2
0
be non-negative or else q
0
would be imaginary and accordingly correspond to physically unacceptable
exponentially growing uctuations; on the other hand these modes must fur-
thermore be such that they are not dissipated through nucleon-antinucleon
decay (anticipating that the mass of the rho-meson, for instance, in dense
nuclear matter would make this energetically possible) for the time-like solu-
tions, or the creation of a nucleon-nucleon-hole pair (that is the absorption
of the rho to lift a nucleon from below the Fermi sea to one above above
it, creating thereby a hole-particle pair) in the case of a space-like mode.
Accordingly, as has already been discussed in Ref [19, 20], for the collective
modes to be undamped the following condition must be fullled
q
0
j~qj
> v
F
thus
we can expand  for c
0
=v
F
>> 1 . In this case we can retain only rst few
terms of the expansion which gives (c
0
=v
F
) =
1
3c
2
0
(as v
F
! 0). In this limit
for the longitudinal modes one arrives at the dispersion relation
q
4
0
(1 +


2

2
4M
2
)  q
2
0
(

2
+ j~qj
2
+m
2
+


2

2
2M
2
j~qj
2
) + 

2
j~qj
2
(1 +

2
j~qj
2
4M
2
) = 0 (29)
from the equation 
L
= 0, while for the transverse modes one obtains
q
4
0
(1 


2

2
8M
2
) q
2
0
(

2
+ j~qj
2
+m
2
 


2

2
4M
2
j~qj
2
) 
1
2


2
j~qj
2
(1+

2
j~qj
2
4M
2
) = 0: (30)
Here 
, given by 

2
=
1
3
g
2
V

2
k
3
F

F
, can be interpreted as the relativistic general-
isation of the familiar plasma frequency [19] with e
2
replaced by g
2
V
=4.
One of the solutions of the dispersion relations for the longitudinal mode
is such that as j~qj ! 0, q
0
! 0. For such low lying excitations, we have
q
2
0
=


2
j~qj
2


2
+m
2
+ (1 +


2

2
2M
2
)j~qj
2
(31)
when j~qj ! 0, c
0
=
q
0
j~qj
approaches a constant. This form of the dispersion
relation is characteristic of acoustic (sound) propagation. At zero tempera-
ture ordinary sound propagation is not possible in a Fermi uid. Following
Landau, one therefore identies this branch of collective modes as zero sound
branch which dier in nature from ordinary sound in that what is involved is
9
not a breathing of the Fermi sphere, but its shape change without alteration
of the volume, the limiting Fermi surface having the form of a surface of
revolution elongated in the forward direction of the propagation of the wave
[21].
On the other hand branches corresponding to high lying excitations are
obtained for both the longitudinal and the transverse modes. In the static
limit, these yield
q
2
0T
=
m
2
+ 

2
1 


2

2
8M
2
(32)
q
2
0L
=
m
2
+ 

2
1 +


2

2
4M
2
: (33)
These modes correspond to collective oscillations `picked up' by the meson
and the branches (transverse and longitudinal) are known as the particle or
mesonic branches. Here again 
 is the plasma frequency, but unlike the case
discussed by Chin [19] where in the static limit q
2
0T
= q
2
0L
= m
2
+ 

2
, the
two modes are split. Though similar in form, our results are dierent from
those of Chin [19] because of the inclusion of tensor coupling.
The main dierence resides in the fact that in the presence of the `mag-
netic' coupling the transverse mode of the mesonic branches not only lies
higher (as is also the case in Chin's discussion) but is split from the longitu-
dinal branch even in the static limit (j~qj = 0). This is because the interaction
between two `magnetic dipoles' does not possess azimuthal symmetry. Fur-
thermore the longitudinal branch is substantially lowered and the transverse
mode elevated in high density nuclear matter compared to the pure vector-
coupling case. The non-monotonic nature of the dispersion curve for the
transverse mode is an interesting feature which appears in the presence of
the magnetic coupling. Nevertheless some of these amusing aspects may not
be of much physical relevance for two reasons, namely, for large values of q
0
and j~qj the collective mode ceases to have any physical meaning (the wave-
length is less than the interparticle spacing), and for values of q
0
and j~qj not so
large (but larger than 2k
F
) the mode is unstable, and the meson decays into
a nucleon-antinucleon pair. For high nuclear matter density, the transverse
mode, even in the static limit, is unstable, as is evident from Fig. 3. This
10
shows that in the presence of magnetic coupling, in the regime of high den-
sity nuclear matter, the produced  will be entirely longitudinally polarized.
On the other hand, the observed attening of the zero-sound branch indi-
cates a reduction of the repulsive force between nucleons due to the magnetic
interaction, as is evident from eq. (31). Of course, the whole analysis has
been done for symmetric nuclear matter; the case where the nuclear matter
is asymmetric is under study.
In summary, one can say that in the presence of the magnetic interaction,
leads, apart from other eects, to a suppression and even the absence of the
transverse mode of  in nuclear matter which is dense enough and this feature
should be observable in high-energy nuclear collisions.
Acknowledgements
The authors thank Professors Ashoke Chatterjee, Raj Kumar Moitra and
Radhey Shyam for useful discussions.
11
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[21] D. Pines, The Many-body Problems (Benjamin, New York, 1961), p.
266.
[22] A.L. Fetter and J.D. Walecka, Quantum Theory of Many Particle Sys-
tems (McGraw-Hill, New York, 1971).
13
Figure Captions
1. Self-energy diagram of the -meson.
2. Diagrammatical representation of the Dyson equation.
3. Dierent collective modes with and without the magnetic interaction.
The solid lines represent the dispersion curves with magnetic coupling
 taken to be 6, while the dotted curves correspond to the case without
magnetic coupling. The density of nuclear matter used for illustration
is that corresponding to k
F
= 3:2 fm
 1
. The value of g
V
taken is 2.6
and the units used are MeV.
14


Tensor coupling and vector mesons in dense nuclear matter

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