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Quasielastic electron scattering in the relativistic σω model
Nuclear Physics @ North-Holland
A445 (1985) 685-705 Publishing Company
QUASIELASTIC ELECTRON SCATTERING IN THE RELATIVISTIC vw MODEL HARUKI
KURASAWA*
Research Institute for Fundamental
and
TOSHIO
Physics, Kyoto
Received (Revised
SUZUKI
University,
Kyoto 606, Japan
8 March 1985 20 May 1985)
Nuclear response functions for quasielastic electron scattering are investigated in the random phase approximation for the relativistic lagrangian of the VW model. In contrast to the transverse response function, the longitudinal one is strongly affected by particle-hole correlations due to the (T- and o-meson exchanges. This result is different from the previous understanding in the oo model that the whole effect of the nuclear interactions on the response function would be incorporated into the nucleon effective mass. The Coulomb sum rule is also discussed in the relativistic model.
Abstract:
1. Introduction In recent years relativistic quantum field theories have been developed in order to study nuclear many-body problems ‘*‘). In those theories, mesonic degrees of freedom are treated explicitly as dynamical variables, and nucleons interact with each other through the exchange of virtual mesons, but not through static nucleonnucleon potentials. Various relativistic effects, such as relativistic propagations and retardation of particles and constraints of Lorentz covariance are taken into account explicitly. The simplest relativistic model is the (+o model where one takes only neutral scalar u- and vector o-mesons. As discussed by Walecka 3), this model can be solved exactly within the mean field approximation replacing the meson field operators
by their
classical
expectation
values.
In employing
phenomenological
coupling constants of the (+NN and oNN interactions, the mean field theory can reproduce well the binding energy and Fermi momentum of nuclear matter 334). Moreover, it has been shown that various well-known phenomena of finite nuclei can be explained as a relativistic effect within the same framework ‘*2). The (+w model with the mean field approximation has been applied also to the study of nuclear response to quasielastic electron scattering and has given a good agreement with experimental data 5-9). In this approximation, the whole effect of the oN interaction on the response function is incorporated into the nucleon effective mass, whereas the wN interaction has no effect on it. The coupling constant of the aN interaction to reproduce the saturation properties yields the effective mass as about 0.6 of the bare mass. Because of this small effective mass, the peak position * Fellow of the Yukawa
Foundation. 685
686
of the response
H. Kurasawa,
function
T. Suzuki / @&zsiehtic
appears
at a higher excitation
wider, and the absolute values of the function for the bare nucleon mass. Thus, the effective mean field approximation
electron scattering
by explaining
energy,
the width becomes
are reduced, compared mass plays an essential
the data for quasielastic
with those role in the
electron
scattering.
Although the mean field theory has reproduced well the experimental data, one should investigate effects not included in its approximation, in particular, when studying dynamical structure. In the discussion of nuclear response functions, it is important to explore the effects of particle-hole correlations. The aN interaction giving a small effective mass is expected to yield strong particle-hole correlations, whose effect on the response function would be opposite to that of the effective mass. Moreover, we should take account of another particle-hole correlation which is induced by the w-meson exchange. The purpose of the present paper is to explore how the response function for quasielastic electron scattering is affected by particle-hole correlations due to the (+- and W-meson exchanges. The particle-hole correlations will be taken into account according to the relativistic random phase approximation (RPA). In sect. 2, the relativistic RPA will be formulated using a Green function approach lo). We will derive the Dyson equation for baryon correlation functions by summing the set of ring diagrams. Each ring diagram is described using baryon propagators in the relativistic Hartree approximation. For the case of nuclear matter the Dyson equation is easily solved and the RPA correlation functions are expressed in terms of the Hartree ones corresponding to the lowest-order ring diagram. In this paper the Hartree correlation functions will be calculated neglecting the contributions from the vacuum polarization which are divergent. The analytic expressions for the Hartree correlation functions are derived in the appendix. The response functions for electron scattering are given by the imaginary part of the electromagnetic current-current correlation functions. Adopting the electromagnetic current operators for point-like free nucleons with the anomalous magnetic moments, we will compare, in sect. 3, the RPA response functions with the Hartree ones. It will be shown that the longitudinal response function is strongly affected by particle-hole
correlations
due to the V- and w-meson
exchanges.
On the other
hand, the transverse response function is hardly changed by the particle-hole correlation, although the transverse component of the w-meson exchange causes the particle-hole correlation in the transverse channel. In sect. 3, we will also discuss the relativistic Coulomb sum rule, comparing with the nonrelativistic one. Our results will be summarized in sect. 4. A brief report on the longitudinal response function has been presented elsewhere ‘l).
2. Relativistic We derive the RPA response from the relativistic lagrangian
RPA response
functions
functions for quasielastic electron scattering starting of the (+o model. The model contains nucleons I,!J
H. Kurasawa,
T. Suzuki / Quasielastic
687
electron scattering
C#Jand neutral vector w-mesons V,. The
interacting with neutral scalar a-mesons lagrangian density is given by 3)t
where M, m, and m, stand for the nucleon, u-meson and w-meson masses, respecthe vector meson field strength. The interactions tively, and VP, = t?,V, -a,V, are of Yukawa type with the scalar and vector coupling constants denoted by g, and g,. Those values will be determined phenomenologically so as to reproduce saturation properties of nuclear matter 3*4). For later convenience we rewrite the interaction lagrangian density as
where the index “a” ranges from -1 to 3 with g, =
2.1. RELATIVISTIC
gs
{ -&
r,=
1 l
V” =
YP ’
I
d,
V’”
fora=-l fora-y.
(2.3)
HARTREE APPROXIMATION
For the sake of completeness, we briefly summarize the relativistic Hartree approximation, which is discussed in detail in refs. 4*13). The relativistic Hartree equation is derived using a Green function approach, in which the contributions from tadpole diagrams to the baryon propagator are summed to all orders with the Dyson equation. Since the self-energies in this equation are divergent owing to the presence of antinucleons in the tadpole diagrams, the divergences should be renormalized by adding appropriate counterterms to the lagrangian, eq. (2.1) [ref. “>I. In the present work, however, we neglect the contributions from antinucleons to the self-energies. The resulting Hartree propagator Gn is identical to that of the mean field theory by Walecka “). The Hartree field t,!~nsatisfies the Dirac equation (i~-M*-yoE~)~H=O,
(2.4)
where the baryon effective mass M* and the vector self-energy 2: are given by d4k -
reJ%
(2?7)4 e
-
d4k
(2?T)4 e
’ We foIlow the conventions of Bjorken and Bell I*).
Tr G,(k) ,
z&k,Tr yOGn( k) .
(2.5)
(2.6)
H. Kurasawa, T. Suzuki / Quasielastic electron scattering
688
The factor
ei+ (a + +0) in eqs. (2.5) and (2.6) ensures that nucleons above the Fermi sea do not contribute to the above integrals. The Hartree propagator GH is
defined
in the usual manner G,(k)=
as d4(x-y)eik(x-y)GH(x-y)
I = -_I’
I
d4b-y) eik’“-y’(l~[~H(~)ICIH(y)I(), .
where 1 ) is the Hat-tree ground state of the symmetric wave number k,. We readily obtain GH( k) =
(R+ M*)
nuclear
matter
(2.7) with Fermi
(2.8)
~_;*2+ia+;S(I;,-Ek)ek k
where Ek = (M**+k*)“*
(2.9)
and k@ = (k,,, k) = (k,,- Xt, k). The ground-state distribution function is given by 0,‘ = O(kF- lkl). A s mentioned above, the integral in eq. (2.5) is divergent due to the antinucleon contribution coming from the first term in eq. (2.8). Omitting the antinucleon contribution we obtain the Hartree self-consistency condition: M*
(2.10)
(27r)3 (M*2+k2)1’2’ The vector 2k3,/(3r2).
self-energy
We use the free propagators involve the later discussion. way as eq. (2.7), are
density
ps =
meson propagators, since tadpole corrections to the meson only zero-momentum components *), which are irrelevant for The scalar and vector meson propagators, defined in the same given by
D,(k)= They are combined
is given by Ez = (gV/m,)2pB with the nucleon
’ k*-m$+is’
into a single
D:“(k) 5 x 5 matrix
=
-gp” + k*k”/mz k2-mz+ic
propagator
*
(2.11)
D”* given by (2.12)
2.2. RPA RESPONSE
FUNCTIONS
In order to evaluate we consider correlation
the RPA response functions for inelastic electron scattering, functions based on the Hartree approximation. For operators
689
H. Kurasuwa, T. Suzuki / Quasielffsric electron scattering
of the type A(x) = ~~(x)~~~~(x) and B(x) = ~~(~)~~~~(x), where r, and 17, are arbitrary 4 x4 matrices, we define the correlation function as G[A(x),
B(Y)] =
(IT[SA(x)B(y)ll)I(lSI),
(2.13)
with S=Texp
( rg, ’ j d4x :~&YY&)
K(x):)
(2.14)
,
where : : represents normal ordering with respect to the Hartree ground state, I ). In the perturbation expansion of eq. (2.13) we take only the ring diagrams and sum them to all orders. This is equivalent to the RPA ‘*). The summation of the ring diagrams leads to the integral equation for the correlation function, GM(x),
J xGo[A(x),
B(y)1= Go[W), B(y)1-
d4x’
%a86
d4y’
(2.15)
B(Y)],
F”(x’)lo,b(x’-y’)~[Fb(yl),
with G[A(x),
WY>I= GCA(x),%)I + Ur ~AGdx - x+))(Tr~&My-y’)) B(Y)I=T~[T,G~(~-Y)T,G,(Y
Go[@x),
-x)1,
F”(x) = ~~(~)~=~~(x) where (x+)@ =(x0+ E, x). The Hartree propagator and &btX-‘Y) by
J-
d4k
Dab(X-Y)
=
c2rl~
_
,
(2.16) (2.17) (2.18)
1
G,(x -y)
is given by eq. (2.7)
.
(2.19)
e -Ikcxy'&,(k)
The first term in eq. (2.15), G,[A, B], is the Hartree correlation function, which corresponds to the lowest-order ring diagram. The subtraction of the ground-state expectation value ([A( x)1) = -i Tr T,G,( x - x+) in eq. (2.16) ensures that disconnected diagrams are excluded. Fig. 1 shows eq. (2.15) diagrammatically.
Fig. 1. Diagrammatic representation of eq. (2.15). The solid lines with the arrows denote the baryon propagator G, in the Hartree approximation, while the dashed line corresponds to the free meson propagator Da*. The wavy lines represent the external fieids A and B.
690
H. Kurasuwa, T. Suzuki j’ Quasielastic electran scattering
From the definition eq. (2.13), the Fourier transform of G[A, B] becomes G[A(-k),
WV;
~1
J
J
B(y)] =2’ d(x,-y,) eiw(%-Ydd3Xd3ye-ik,x+ip’J’G[A(x), (2.20)
=(O~[A(-k)(H-w-i~)-‘B(k’)+B(k’)(N+o-i~)-’A(-k)](O),
where H is the total hamiltonian, HIO) = 0 and
IO)the correlated (RPA) ground state satisfying
A(k) =
Owing to the momentum conservation, G[A(-k),
d3X es‘“A(O, x) .
(2.21)
G is decomposed
B(k’); w] = S(k - k’)17,,,(A,
into B; k) ,
(2.22)
where k’” = (0, k). In terms of LT,, eq. (2.15) is written as %,(A,
B; k) =Ji’(A, B;
k)+xa(kP(A,
F”; kV&dL
B; kl
,
(2.23)
with
fl(A, R k) =
-&
Jd4p
Tr [f,G&+
1 XS=m*
k)fdMp)l
,
(2.24)
2 a=-1, (2.25)
xa =
1
z2
Here we have assumed k” f (0,O) and have used the conservation baryon current k,J7( F”, A) = k&17(A, P)
relation of the (2.26)
= 0.
Because of this conse~ation the gauge terms in the vector meson propagators not contribute to II,,. Eq. (2.23) can be solved for ITRpa, n,,,(A,
B) = fl(A, B)+xJI(A,
F”)( u-‘),JWb,
B) ,
do
(2.27)
where CJab= gab -xJI(F”,
Fb),
Uab( u-l)bc = a,, ,
(2.28)
with gab = l(a = b = -l), g”“(a = EL,b = v) and g-lfl = g’-’ = 0. As is verified by explicit calculation of the traces in ZI(F”, Fb), the matrix U can be divided into two parts: one is the longitudinal part, which is a 3 ~3 matrix, due to the scalar e-meson and the longitudinal and the time component of the vector
691
H. Kurasawa, T. Suzuki / Quasielastic electron scattering
w-mesons,
the other the transverse
part, which is a 2 x 2 matrix,
from the transverse
vector mesons. In the former matrix, the terms from the longitudinal are related to those from the time component through the current
vector meson conservation
relation, eq. (2.26). Therefore, the former matrix is also expressed by a 2 x 2 matrix. With the choice of the four-momentum as kp = (w, Ikl, 0, 0), we obtain IIRPA(A, B) = 17(A, B) + tX$i,(A, Here the longitudinal
B) + SIl:‘,(A,
B) .
(2.29)
part is given by
Mi$&( A, B) = s L
where the 2 X2 matrix
fl(A,
( UL)ob (a, b = -1,O) U = L
F”)( W.dT(Fb, is written
&(I -,CJL) i”(
1-
(2.30)
as
X./i?V&
XSiJL
(
B) ,
xm
(2.31)
>
with the abbreviations
(2.32) The vector coupling
strength
in eq. (2.30) for the longitudinal k2
i”=sXy=(21r)3
The transverse
1
part in eq. (2.29) is described
s: mz-k2
k2 k2’
part turns
out to be (2.33)
as (2.34)
where rr,
=
n(F2, F2)= n(F3, F3) *
(2.35)
Eqs. (2.30) and (2.34) provide us with the RPA correlation functions in terms of the Hartree ones and are the basic equations in the present study. The RPA response functions for electron scattering with the four-momentum transfer kp = (w, k) are proportional to the imaginary part of ITRPA(j, _!; k) for electromagnetic current operators 1 Although there is an ambiguity concerning the choice of _? [refs. 5’9)], we adopt here the current operators of point-like free nucleons with anomalous magnetic moments K, ( K~ = 1.793 and K, = -1.913): (2.36)
692
ff. Ku~ffsaw4
T. Suzuki / ~as~e~astie
electron scattering
where TVis 1 for protons and 0 for neutrons, and me,,=$[r,, ~~1. The longitudinal and transverse response functions, denoted by S,(k) and S,(k), respectively, are given by (2.37) (2.38) where Z denotes the proton number in symmetric nuclei, and p^= .& and .?r = .& = ?=’ for our choice of k = (l&l, 0,O). The propo~ional constants in front of the square brackets of the above equations are easily obtained by comparing eq. (2.20) with eq. (2.22). Eqs. (2.30) and (2.34) provide us with dispersion relations of undamped collective modes, which are expressed as det U,(k) = 0 for longitudinal modes and as (lx&r=)2 = 0 for transverse modes. We shall discuss those undamped modes later, but further discussions will be found in refs. 4,‘4). 2.3. CALCULATION
OF THE HARTREE
CORRELATION
FUNCTIONS
In this subsection we will give analytic expressions for the Hartree correlation functions which are needed to evaluate the RPA response functions, eqs. (2.37) and (2.38). Substituting eq. (2.8) into eq. (2.24), we obtain4) II(A, 3; k)=IIF(A,
l.3; k)+lTD(A,
B; k),
(2.39)
where iI,(A,
B; k) = -&
I&,(A,B;
k)=-
I
d4p
t(p, k)
d4&S(p,--E,) P
d4p~
__$iT
J
(2.40)
(p2-M*2+iE)((p+k)2-M*2+ie)’
t(p, -k)
t(p, k)
(p+k)2-M*2+i~+(p-k)2-M*2+ie 6(po-E,)6(p~+o-E,+,)t(p,
P
k),
(2.41)
P+k
with the trace factor t(p, k) = Tr [r,(p+K+
M*)TB(~+
M*)l .
(2.42)
The Feynman-propagator piece flF describes the most of the NR excitations, and the density-dependent piece flu the particle-hole excitations and a part of the NR ones. In the present paper, we ignore X7, in eq. (2.39) instead of a renormalization procedure, which will render nF finite. Note that the current conservation relation, eq. (2.26), holds even without flF for A = F, and 6. Hereafter the subscript D of fl, is omitted.
I$.
Kurusawa,
T
Suzuki /
Quasielastic
electron
scattering
693
It is verified by explicit calculations of the trace factors that the Hartree correlation functions, eq. (2.41), are expressed in terms of the following three integrals (n = 0, 1,2): Z,(k) = -2
1 ~(~~-E,)S(~~+O-E
d4Z+ S(po-- EJ P
(2pofo)”
(2P,- w)”
(p+k)2-M*2+ief(p-k)2-M*2+iE
(2.43)
The results are ZZv=c;s-k2Zo+Z,,
Z?s=-&+(4M”2-k2)I,, II,=-up.-?
&+k2
_
ZZ,,= 2M*Z,,
1 k2
(2.45)
,
~Zz+(4M**+k2)Zo
(2.44)
(2.46) fffF*,p^)=zz(~,
F,)=tn,-M*(c,+e,)k210,
ZZ(F2, &) = ZZ(& F’) =fZ&- M*(c,+
(2.47) (2.48)
c,)k2Zo,
(2.49)
n(dP^)=~~,-2M*cpk2Zo-(e~+c~)(w2p”,f~k2Z2-t2M*2k2Z0), ZZ(.f,, &) =~~~-2~*c~k21~-~k2(c~+c~)(~~+~~+(4~*’+
k2)Z,),
(2.50)
where & is related to the scalar density ps by 4 (J,,f*2_+)‘/2
(2.51)
and c, = u,/2M. The analytic expression for Z,(k) is derived in the appendix. The Hartree correlation functions given above includes a part of the Nij excitations and the particle-hole excitations with energy between max (0, E?‘) and Ey’, where E!_’ is defined in eq. (A.8). It should be noted that the particle states involved in the former excitations are within the Fermi sphere. This fact implies that the Pauli exclusion principle is violated in the Nfi excitations’. Since the fourmomentum transfer in electron scattering is space-like, k2 < 0, these Ns pairs, which lie in the time-like region, affect the RPA response functions through the real parts of the Hartree correlation functions. As we shall see later, however, these off-shell effects are small. Before closing this section, let us show the explicit forms of the Hartree response functions. Using eqs. (2.37), (2.49) and (A.27), the longitudinal response function ’ In fact, if we in&de the vacuum polarization part IIF the contribution from the NR pairs violating the Pauii principle to II, is cancelied by that from the same pairs to flF [refs. 4-‘5)].
H. Kurasawa,
694
T. Suzuki / Quasielastic
electron scattering
S’,“‘(k) is written as
S:H’W=
z
L[FL(k)B(max ,k,
16k;
(0,E’_‘)
< w
-F,(-k)fI(O
(2.52)
with (2.53)
where EF, E!_’ and f(w) are defined in the appendix as EF= (M*‘+
k$)l”,
E~:)=(M*2+(kFflk()2)1’2-EF,
f(“)=lk[( 1+$$)“‘. Similarly we obtain the transverse response function @‘(k) (2.52) FL with FT,
by replacing in eq.
3. Results and discussion
In numerical calculations of the response functions we use the following experimental values for the nucleon and meson masses: M = 939 MeV, m, = 783 MeV and m, = 554 MeV. For the coupling constants we will employ the values which are phenom~nologically determined by Walecka ‘) so as to reproduce observed saturation properties of nuclear matter, i.e. the binding energy = 15.75 MeV and kF = 1.42 fm-‘. His values are given by gz = 92.87 (g$‘47r = 7.39) and gz = 135.7 (gzf4w = 10.8). For these values the effective mass satisfying the Hartree self-consistency condition, eq. (2.10), turns out to be M* = 0.557M.
3.1. LONGITUDINAL
RESPONSE
FUNCTION
As a typical example, we show in fig. 2 the numerical results of S,(k) given by eq. (2.37) as a function of w at Ikl = 2.5kF = 700 MeV. The dotted curve represents the response of the relativistic free Fermi gas, which is given by eq. (2.52) with M” = M; The Hartree response function in the CTWmodel obtained by eq. (2.52) with M” = 0.557M is shown by the dashed curve. In the Hartree approximation, the nuclear interaction affects the response function only through the effective
H. Kurasawa, T. Suzuki / Quasielastic electron scattering
r
I
&(k)/Z
3
[MeV-‘I
695 /
;...,,
X 1om3
2
1
0 Fig. 2. Longitudinal response functions as a function of the energy transfer w at the momentum transfer Ikl = 2.5k, (k, = 1.42 fm-I). The dotted curve shows the response of the relativistic free Fermi gas (g, = g, = 0), while the dashed one is obtained by the Hartree approximation with gz = 92.87 (M* = 0.557M). The dot-dashed curve is the RPA result with gf =92.87 and gt=O, the two-dot-dashed one with g: = 0 and gt = 135.7, and the solid one with gt = 92.87 and gt = 135.7. The arrow indicates the position of the collective mode in the case of g: = 0 and gt = 135.7. The thin solid curve shows the results in ref. ‘I).
mass ‘), which does not depend on the wN interaction. In comparing the dashed curve with the dotted one, it is easily seen that the small effective mass shifts the peak position to higher excitation energy region and reduces the absolute values of the response function. These effects of the effective mass were essential for the Hartree approximation in order to explain the experimental data 5-9). Let us study the effects of the particle-hole correlations on the response function. First we explore those due to the a-meson exchanges only, since the effective mass comes from the aN interaction. Setting gf = 92.87 and g: = 0 in MIRPA of eq. (2.37) we obtain the RPA response function as shown by the dot-dashed curve. One can see that the shift of the peak position coming from the effective mass is perfectly cancelled by the strong attractive particle-hole correlations. In fact, if we neglect the w-meson exchange, the system with k, = 1.42 fm-’ becomes unstable against the density fluctuation, when gz 2 77.8 for Ikl = 2.5 k,. Since the present value, gf = 92.87, is slightly larger than the critical value, the dispersion equation det l_J,= 0 has a complex root near the real axis in the upper-half complex w-plane 16). The sharp peak of the dot-dashed curve reflects the existence of this root. Next let us take into account particle-hole correlations due to the w-meson exchanges, too. The RPA response function for gf=92.87 and gz= 135.7 is drawn by the heavy solid curve in fig. 2. It shows that the w-meson exchanges produce a strong repulsive effect on the response function. For reference, the RPA result is also shown in fig. 2 by the two-dot-dashed curve setting gt= 135.7 but gz=O in
696 811RPA
H. Kurasawa,
of eq. (2.37) and replacing
two-dot-dashed
curve is reduced,
tion is due to the existence indicated
T. Suzuki / Quasielastic
electron scattering
M* in II and ?XIRPA by M. The strength compared
of the undamped
zero-sound
mode whose
at w/ kF= 1.372. The energy-weighted
by the arrow
of the
with those of other curves. This reducstrength
position
is
carried
by
this mode is 62.72 (MeV), while that of the continuum states 74.22 (MeV). It is interesting to note that most of the strength is thus distributed over the same energy region as that for the case of gt = 92.87 and gt = 135.7. We show in fig. 3 the results at (kl= 1.5kF=420 MeV in the same way as in fig. 2. The arrow denotes the position of the zero-sound in the case of gf = 0 and gt = 135.7. Its energy-weighted strength, 32.52 (MeV), is approximately equal to that exhausted by the continuum states. The RPA result with gf = 92.87 and g: = 0, which is shown by the dot-dashed curve, exhibits a considerable reduction of the strength. This results from the fact that the value gz = 92.87 is far from the stable region, gf< 12.5 for gz = 0 and Ikl = 1.5kF, and the dispersion equation det U,= 0 has a complex root. The solid curve calculated with gf = 92.87 and g!j = 135.7, for which the system is stable, shows again an important role of the w-meson exchange in the RPA response function. I
6
0
SJk)/Z
I
IMeV’l
0.5 Fig. 3. Same as fig. 2, except
w/k,
1.0
for Ikl= 1.5k,.
From the above two examples it is easily seen that the longitudinal response function is greatly affected by the particle-hole correlation not only due to the a-meson exchange, but also due to the w-meson exchange. This is quite different from the Hat-tree result 5-9) that the wN interaction has no contribution to the response functions, while the whole effect of the UN interaction on them can be when the solid curves are incorporated into the effective mass M *. Furthermore, compared with the dashed ones, the strength of the RPA results is more reduced than the Hartree ones. This reduction comes from the ground-state correlations
H. liurasawa,
which are included is inadequate excitations
dynamical
that in the present
is partially
included.
that the Hartree
properties
RPA calculations
Comparing
697
electron scattering
in the RPA. Thus, it is concluded
for understanding
We have noted
7: Suzuki / Quasielastic
the present
approximation
of the longitudinal the off-shell
response.
effect of NN
results with those in ref. I’),
where Nfi excitations are completely neglected, we can see that the off-shell effect does not alter qualitative features of the response function. The previous RPA results for gt=92.87 and gg = 135.7 in ref. ‘l) are shown by the thin solid curves in figs. 2 and 3.
3.2. TRANSVERSE
RESPONSE
The transverse response and 5, respectively. Curve
FUNCTIONS
functions at (kl= 2Sk, and 1.5kF are shown in figs. 4 designations are the same as those of fig. 2. Since there
Fig. 4. Transverse response functions as a function of o at Ikl= 2.5k,. The curves are drawn in the same way as in fig. 2.
is no particle-hole correlation due to the u-meson exchange, the RPA with g,=O is nothing but the Hartree approximation. The particle-hole correlations are induced through the transverse part of the w-meson exchanges. We see from figs. 4 and 5, however that, in contrast to the longitudinal response function, the particle-hole correlation hardly produces an effect on the response function. This comes from the fact that while the w-meson is isoscalar, the transverse response for highmomentum transfer is mainly caused through the isovector magnetization current. The transverse response function in the 5~0 model is thus dominated by the effective mass even in the RPA. This does not imply, however, that one can extract simply the value of the effective mass from the experimental data. When we discuss
698
0
0.5 Fig. 5. Same as fig. 4, except
experimental exchanges
data, it is necessary
to further
for Ikl= lSk,.
take into account
which yield spin- and isospin-dependent
3.3. COULOMB
SUM
We consider
1.0
W/k,
at least the p-meson
particle-hole
correlations.
RULE
the Coulomb
Here GU is the contribution
sum for electron
scattering,
from the undamped
which is given by
collective
mode, if it exists in the
space-like momentum transfer region, and is given by the residue of S,( k) at w = w,,,, satisfying det U,_(w,,,,, k) = 0. The range of the integral in eq. (3.1) is restricted to the space-like momentum transfer region, because the time-like region cannot be reached
experimentally
by electron
scattering.
The off-shell effect of NR excitations
on eq. (3.1) is small, as was discussed in subsect. 3.1. Fig. 6 shows the results with the Hat-tree approximation using gs= 92.87. The solid and dashed curves are calculated with and without the anomalous magnetic moments K, respectively. The non-relativistic sum rule value, CNR, for the Fermi gas is given by 17) C
NR
=
%l/k&#&43, I 1,
O< lki
(3.2)
This is also illustrated by the dot-dashed curve in fig. 6. The values of the relativistic model approaches to C,, in the long-wavelength limit (lkl-, 0). This fact is verified directly using eq. (2.52)?. The difference between the dot-dashed curve and the ’ It is worthwhile to be k2/2EF, rather
noting that the value of the energy-weighted than the non-relativistic value k*/2M*.
sum of eq. (2.52) for lkl-+ 0 turns out
H. Kurasawa,
T. Suzuki / Quasielastic
-I dwS,(k)/Z
‘O
I
/
/.
electron scattering
699
,_________------.--
0 Fig. 6. Coulomb sum defined as the space-like energy integral of the longitudinal response function in the Hartree approximation with g: = 92.87. The solid and dashed curves represent the results with and without the anomalous magnetic moments, respectively. The dot-dashed curve shows the non-relativistic sum rule value given by eq. (3.2).
dashed one, whose value converges to i when Ikl + co, is due to the relativistic effect on the part of the positive-energy spinors. It is seen from the solid curve that the anomalous magnetic moments in the time component of the four-current fairly reduces the sum value at Ikl b 1.5kFt. The relativistic effects on the Coulomb sum rule, thus, produce a large reduction of the sum rule value. Using the VOJ model in the mean field theory which is equivalent to the present Hartree theory, Walecka 19) also calculated the Coulomb sum rule. He found that the sum rule value is dominated by the contribution from the anomalous magnetic moments
at lklz2k,
as (3.3)
This is considerably different from our result shown by the solid curve in fig. 6. In deriving eq. (3.3), however, the anticommutation relations {$a(x), I&(Y)} = yt,S(x - y) are used. This implies that the on-shell NN contributions are also included in eq. (3.3). Since K, in the time component of the nuclear four-current appears with the “odd” operator y [ref. “)I, the on-shell NN excitations through the anomalous magnetic moments yield a large contribution to eq. (3.3). Thus, a great part of the value of eq. (3.3) comes from NN excitations in the time-like region which cannot be seen by electron scattering. A similar result has already been pointed out in ref. ‘*), but see the previous footnote. ’ A similar calculation in ref. ‘s) found the enhancement of the sum value due to the anomalous magnetic moment. It seems, however, that there is a mistake in ref. I*), since we can reproduce its value by replacing {l+ (M*/M)K,} by {l - (A~*/M)K~} in eq. (2.54).
700
H. Kurasawa,
T. Suzuki / Quasielastic
electron scattering
Next we discuss the effects of the particle-hole
correlations
on the Coulomb
given by eq. (3.1). In fig. 7 is shown the RPA and Hat-tree results sum with use of the same curve designations obtained
from
the RPA
calculations
without
as in fig. 2. The dot-dashed the w-meson
sum
of the Coulomb
exchange.
curve is We have
mentioned in the discussion of figs. 2 and 3 that det U, = 0 without the W-meson exchange has an imaginary solution. The imaginary solution, however, disappears beyond a certain value of lkl, since the strength of the attractive force given in eq. (2.25) decreases with increasing Ikl. For the case of 92.87, this transition occurs at Ikl = 2.58kr. Th e s h arp peak of the dot-dashed curve is as a result of the transition, and hence its value at Ikl< 2.58kF has no meaning.
1.0
/do~(h)/Z
0
2
I a/l+
4
Fig. 7. Coulomb sum defined as the space-like energy integral of the longitudinal response functions. Curve designations are the same as in fig. 2. The two-dot-dashed curve denoted by “zero sound” shows the contribution from the undamped zero-sound mode to the upper two-dot-dashed curve.
The upper
two-dot-dashed
curve in fig. 7 shows the RPA results
with gf = 135.7
but without the a-meson exchange. In the region of Ik( < 2.84kF for this case, the repulsive correlations due to the W-meson exchange yield the undamped zero-sound mode. The value of the RPA sum, therefore, is given by adding the strengths of the continuum particle-hole states and the zero-sound state. The contribution from the latter state to the sum value is shown by the lower two-dot-dashed curve in fig. 7. In taking account of the correlations due to both u- and W-meson exchanges with gz = 92.87 and gz = 135.7, w e obtain the RPA results as shown by the solid curve. It is seen that the solid curve is between the two-dot-dashed curve and the dot-dashed one for Ik( > 2.58kF where the latter has a meaning. This is due to the fact that in the RPA attractive forces induce the ground-state correlations to enhance the sum value, while repulsive ones reduce the value. In the present case, the net effect of the ground-state correlations on the Coulomb sum produces about 10% reduction
701
H. Kurasawa, T. Suzuki / Quasielastic electron scattering
of the Hat-tree
result
which
is shown
by the dashed
solid curve with the dotted one for the free Fermi reduced by about 18% around /k/ = 1.56~~
curve.
When
gas model,
comparing
the RPA result
the is
4. Summary We have derived the RPA response functions for quasielastic electron scattering from the relativistic lagrangian of the CTWmodel, using a Green function approach. Although the model lagrangian is renormalizable, we have neglected divergent terms in the Hartree baryon propagator and in the Hartree correlation functions. Adopting the electromagnetic current operators of point-like free nucleons with the anomalous magnetic moments, the RPA response functions have been calculated analytically. It is shown that the longitudinal response function is strongly affected by the particle-hole correlation not only due to the g-meson exchange but also due to the w-meson exchange. This result is quite different from the previous understanding 5-9) that the whole effect of the aN interaction on the response function would be incorporated into the effective mass, while the wN interaction would have no effect. In using the coupling constants to reproduce saturation properties of nuclear matter 3), the effect of the c-meson exchange on the response function is largely cancelled by the one from the repulsive correlation due to the w-meson exchange. The net effect dominates the peak position of the RPA response function, and yields about 10% reduction of the absolute values of the Hartree response function. The particle-hole correlation, which affects the transverse response function, is induced through the transverse component of the w-meson exchange. In contrast to the longitudinal response function, however, the transverse one is little affected by the correlation. This is due to the fact that the transverse response in electron scattering is induced mainly through the isovector magnetization current. Consequently, the transverse response function in the ITW model is dominated by the nucleon effective mass. We have also discussed the Coulomb sum rule which is defined as the space-like energy integral of the longitudinal response function. Comparing with the nonrelativistic sum rule value of the Fermi gas, the relativistic result is quenched by about 40-50% in the region of 1.5kFX< Ikl=~ 4.5 kF. This quenching is mainly attributed to the three sources. Those are the relativistic factor of the positive-energy spinors, the spin-dependent time component of the nuclear four-current and the ground-state correlations due to the w-meson exchanges. Finally it should be noted that when one compares the RPA response functions with experimental data for quasielastic electron scattering, at least the p-meson exchanges also must be taken into account. It is straightforward, however, to include those exchanges in the present scheme. The application of the present RPA to finite nuclei and to other phenomena including the particle-hole correlations would be done by slight modifications at a formal level. Examination of problems is in progress.
702
H. Xurasowa,
T. Suzuki j Quasielastic electron scattering
Grateful acknowledgements are due to Dr. S. Nishizaki for useful discussions. One of the authors (H.K.) thanks the Yukawa Foundation for financial support. Appendix We explicitly derive the analytic expression for I,,(k) defined in eq. (2.43), i.e. I,(k)=J,(k)+J,(-k)+iK,(k),
where
J
(A.11
k~ d3
(2E,+w)” p o Ep E:,, -(Ep+w)2-ic’
J,(k)=2
J
(A.21
k,
K,,(k) = -2r
d3p+
0
6(o+E,-E,+,)(2E,+w)“. P
L4.3)
r+k
First we calculate Re I,,. Changing the integral variable cos f3=p - k/lpi )k( into x=E pfk f Ep, J,, is written as
with E, = E,,*lkl= [M*2+ (p f /kj)2]1’2 and p = IpI. By integration by parts we obtain
J
kF
(2E,+
w)n+lL(k)
-
dp(2E,+
@)“+I
-k,
1
+d(E++E,)
E+-Ep-w-i&
(A.51
dp
where we have defined
J
EL-'
L(k)=
J
E$+'
dx
E”’
*
x-w-is
+
dxli,
(A.6)
E!?’
with EF= EkF= (My2+ k;)1’2f E!-’ = &&cl-
% Jk, dpdp J-k=
G=,
E$+’ = Ek&l+
(A.71 EF .
(A.81
E~-)dx[-X+O+f( J Q+) dx [X+W-f(x)]“+l J
The change of the variable p into x = E, rt Ep yields
dp
-k,
d(E+-E,) dp
d(E++E,)
(2EP+o)“+1
E+-E,--w-is= (2E,+o)“+’
E++Ep+o-iE=
+ 8(2k,_ lk,j
Ei-)
x-o-i&
EL+’
x+w-ia
J E(i’dxEX+ w-“,“‘-- 5,; w+fr’+l, =%/2
(A-9)
(A.10)
H. Kurasawa,
where f(x)
T. Suzuki / Quasielastic
electron scattering
703
is given by (A.ll)
Using
eqs. (AS),
Re L(k)=i
(A.9) and (A.lO),
we obtain
-[(2EF+w)“+‘L,(k)+(2E,-w)“+‘L,(-k)+Al,(k)], .:l
(A.12)
with (E!_‘-w)(EJ+‘+w) (E.-j_w)(E’+j+-w)
L(k)=ReL(k)=ln
(A.13)
,
A1~(k)=((-l)“j~;~dx-~~;dx)P(X-;~;)n+’ -(j(2k,_Ikl)
j-@+‘dxP(X-Onn+xr~~-W+f)““,(o--w), 2&/2 (A.14)
where P stands for the Cauchy principal For n = 0, eq. (A.14) becomes
values.
Ar,(k)=(-[E~;dx+jE~~dx+20(2k,-~k~) Denotingf(x)
by y, and noticing
0, Al, turns
thatf(E5))
{2zI)2xf(x)&. = E!+‘,f(E!+‘)
(A.15)
= IE>‘I andf(2E,,,)
=
out to be AS(k)=2([~~~dy-[E~~dg)P(&-fi)
>
(A.16)
with LY= k2(1 -4M*‘/k2). Hence,
the final expression Re I,(k)
=?
Ikl
(A.17)
for Re 1, is given by (2E,+o)L,(k)+(2E,-w)L,(-k)
(A.18) In a similar
way we obtain Re I,(k)=z
214[
(2EF+w)‘LR(k)+(2EF-a)‘&-k)
-ct(L,(k)+L,(-k))+4(E$+‘E~,-‘-E’t’Et’) -(jE~~dx+JE~~dx)(2x+~&)],
(A.19)
704
H. Kurasawa,
Re Iz(k)=x
3lkl
T. Suzuki / Quasielastic
electron scattering
(2E,+o~)~L,(k)+(2E,-w)~L,(-k)
Although the integrals appearing in the above equations can easily be written in terms of elementary functions, these expressions are not shown here since they are long. Next we evaluate the imaginary part of I,, which is given by ImI,(k)=Im(J,,(k)+J,(-k))+&(k). Using eqs. (AS),
(A.21)
(A.6), (A.9) and (A.lO) we immediately
obtain
+0(2k,-~k~)[(2E,-w)“-‘-(f(w))“+‘]0(0
+[(2EF-W)“+l + 0(2kF-
Ikl)Kf(4)“+1
-(-f(w))““]8(2E,,,
with
e(a
(A.23)
Here we have omitted terms disappearing for w > 0, noting that E’_’ < 0 for Ikl< 2k,. In the same way as in deriving eq. (A.4) we have E+-E
..,,,=-$
I
:’
dp-$2E,+o)“B(E,-w-E,) p
with x = E,,+k - I&.. Employing
‘dxS(x-W) E_-
(A.24)
E,
the identity
~(2E,fo)“B(E,-o-E,)=&$([(2E,+w)“+1-(2E,-o)”t’] P
(A.25)
x@(-b-w-E,)l,
integration
by parts in eq. (A.24) yields,
for w > 0,
K.(k)=-~--&[(2E~-~)‘“+1-(j-(4)“illB(2E,-w-f(4)
=-~-&[(2E,-o)n+1-
(f(w))““]f3(O
I
;z’dx8(x-o)
<-EL-‘)8(2k,-Ik(), (A.26)
H. Ku~sawu,
T. Suzuki / Quasielastic electron scattering
705
where we have used the equivalence between 2 EF - w -f (w ) > 0 and -E !_’ < o < -E’_‘. Substituting eqs. (A.22) and (A.26) into eq. (A.21) we obtain finally, for o > 0, Im &(k)=$
&{[(2&+w)“+’
-(f(w))““]@(max
(0, EC’) < w < EC’)
-8(2k,-lk1)[2E,-w)“+‘-(f(w))“+‘]e(O
-(-f(w))“+‘]B(Ei?
~(2k,-Ikl)[(ftw))“+‘-
< Er’)
(-f(0))“*‘]8(2E~,2
-=cw <
E?‘)} .
(A.27)
References 1) M.R. Anastasio, L.S. Celenza, W.S. Pong and CM. Shakin, Phys. Reports 100 (1983) 327, and references therein 2) B.D. Serot and J.D. Walecka, The relativistic nuclear many-body problem, Adv. Nucl. Phys., in press 3) J.D. Walecka, Ann. of Phys. 83 (1974) 491; Lecture Notes in Physics, vol. 92, Nuclear interactions (Springer, Berlin, 1978) p. 294 4) S.A. Chin, Ann. of Phys. 108 (1977) 301 5) R. Rosenfelder, Ann. of Phys. 128 (1980) 188 6) J.V. Noble, Phys. Rev. Lett. 46 (1981) 412 7) A. Hotta, P.J. Ryan, H. Ogino, B. Parker, G.A. Peterson and R.P. Singhal, Phys. Rev. C30 (1984) 87 8) G. Do Dang and Nguyen Van Giai, Phys. Rev. C30 (1984) 731 9) T. de Forest, Jr., Phys. Rev. Lett. 53 (1984) 895 10) A.L. Fetter and J.D. Walecka, Quantum theory of many-particle systems (McGraw-Hill, New York, 1971) 1 i) H. Kurasawa and T. Suzuki, Phys. Lett. 154B (1985) 16 12) J.D. Bjorken and SD. Drell, Relativistic quantum mechanics (McGraw-Hill, New York, 1964) 13) C.J. Horowitz and B.D. Serot, Nucl. Phys. A368 (1981) 503 14) T. Matsui, Nucl. Phys. A370 (1981) 365 15) T. Matsui and B.D. Serot, Ann. of Phys. 144 (1982) 107 16) T. Yukawa and H. Kurasawa, Phys. Lett. 129B (1983) 162 17) T. de Forest, Jr. and J.D. Walecka, Adv. in Phys. 15 (1966) 1 18) T. Matsui, Phys. Lett. 1328 (1983) 260 19) J.D. Walecka, Nucl. Phys. A399 (1983) 387
A445 (1985) 685-705 Publishing Company
QUASIELASTIC ELECTRON SCATTERING IN THE RELATIVISTIC vw MODEL HARUKI
KURASAWA*
Research Institute for Fundamental
and
TOSHIO
Physics, Kyoto
Received (Revised
SUZUKI
University,
Kyoto 606, Japan
8 March 1985 20 May 1985)
Nuclear response functions for quasielastic electron scattering are investigated in the random phase approximation for the relativistic lagrangian of the VW model. In contrast to the transverse response function, the longitudinal one is strongly affected by particle-hole correlations due to the (T- and o-meson exchanges. This result is different from the previous understanding in the oo model that the whole effect of the nuclear interactions on the response function would be incorporated into the nucleon effective mass. The Coulomb sum rule is also discussed in the relativistic model.
Abstract:
1. Introduction In recent years relativistic quantum field theories have been developed in order to study nuclear many-body problems ‘*‘). In those theories, mesonic degrees of freedom are treated explicitly as dynamical variables, and nucleons interact with each other through the exchange of virtual mesons, but not through static nucleonnucleon potentials. Various relativistic effects, such as relativistic propagations and retardation of particles and constraints of Lorentz covariance are taken into account explicitly. The simplest relativistic model is the (+o model where one takes only neutral scalar u- and vector o-mesons. As discussed by Walecka 3), this model can be solved exactly within the mean field approximation replacing the meson field operators
by their
classical
expectation
values.
In employing
phenomenological
coupling constants of the (+NN and oNN interactions, the mean field theory can reproduce well the binding energy and Fermi momentum of nuclear matter 334). Moreover, it has been shown that various well-known phenomena of finite nuclei can be explained as a relativistic effect within the same framework ‘*2). The (+w model with the mean field approximation has been applied also to the study of nuclear response to quasielastic electron scattering and has given a good agreement with experimental data 5-9). In this approximation, the whole effect of the oN interaction on the response function is incorporated into the nucleon effective mass, whereas the wN interaction has no effect on it. The coupling constant of the aN interaction to reproduce the saturation properties yields the effective mass as about 0.6 of the bare mass. Because of this small effective mass, the peak position * Fellow of the Yukawa
Foundation. 685
686
of the response
H. Kurasawa,
function
T. Suzuki / @&zsiehtic
appears
at a higher excitation
wider, and the absolute values of the function for the bare nucleon mass. Thus, the effective mean field approximation
electron scattering
by explaining
energy,
the width becomes
are reduced, compared mass plays an essential
the data for quasielastic
with those role in the
electron
scattering.
Although the mean field theory has reproduced well the experimental data, one should investigate effects not included in its approximation, in particular, when studying dynamical structure. In the discussion of nuclear response functions, it is important to explore the effects of particle-hole correlations. The aN interaction giving a small effective mass is expected to yield strong particle-hole correlations, whose effect on the response function would be opposite to that of the effective mass. Moreover, we should take account of another particle-hole correlation which is induced by the w-meson exchange. The purpose of the present paper is to explore how the response function for quasielastic electron scattering is affected by particle-hole correlations due to the (+- and W-meson exchanges. The particle-hole correlations will be taken into account according to the relativistic random phase approximation (RPA). In sect. 2, the relativistic RPA will be formulated using a Green function approach lo). We will derive the Dyson equation for baryon correlation functions by summing the set of ring diagrams. Each ring diagram is described using baryon propagators in the relativistic Hartree approximation. For the case of nuclear matter the Dyson equation is easily solved and the RPA correlation functions are expressed in terms of the Hartree ones corresponding to the lowest-order ring diagram. In this paper the Hartree correlation functions will be calculated neglecting the contributions from the vacuum polarization which are divergent. The analytic expressions for the Hartree correlation functions are derived in the appendix. The response functions for electron scattering are given by the imaginary part of the electromagnetic current-current correlation functions. Adopting the electromagnetic current operators for point-like free nucleons with the anomalous magnetic moments, we will compare, in sect. 3, the RPA response functions with the Hartree ones. It will be shown that the longitudinal response function is strongly affected by particle-hole
correlations
due to the V- and w-meson
exchanges.
On the other
hand, the transverse response function is hardly changed by the particle-hole correlation, although the transverse component of the w-meson exchange causes the particle-hole correlation in the transverse channel. In sect. 3, we will also discuss the relativistic Coulomb sum rule, comparing with the nonrelativistic one. Our results will be summarized in sect. 4. A brief report on the longitudinal response function has been presented elsewhere ‘l).
2. Relativistic We derive the RPA response from the relativistic lagrangian
RPA response
functions
functions for quasielastic electron scattering starting of the (+o model. The model contains nucleons I,!J
H. Kurasawa,
T. Suzuki / Quasielastic
687
electron scattering
C#Jand neutral vector w-mesons V,. The
interacting with neutral scalar a-mesons lagrangian density is given by 3)t
where M, m, and m, stand for the nucleon, u-meson and w-meson masses, respecthe vector meson field strength. The interactions tively, and VP, = t?,V, -a,V, are of Yukawa type with the scalar and vector coupling constants denoted by g, and g,. Those values will be determined phenomenologically so as to reproduce saturation properties of nuclear matter 3*4). For later convenience we rewrite the interaction lagrangian density as
where the index “a” ranges from -1 to 3 with g, =
2.1. RELATIVISTIC
gs
{ -&
r,=
1 l
V” =
YP ’
I
d,
V’”
fora=-l fora-y.
(2.3)
HARTREE APPROXIMATION
For the sake of completeness, we briefly summarize the relativistic Hartree approximation, which is discussed in detail in refs. 4*13). The relativistic Hartree equation is derived using a Green function approach, in which the contributions from tadpole diagrams to the baryon propagator are summed to all orders with the Dyson equation. Since the self-energies in this equation are divergent owing to the presence of antinucleons in the tadpole diagrams, the divergences should be renormalized by adding appropriate counterterms to the lagrangian, eq. (2.1) [ref. “>I. In the present work, however, we neglect the contributions from antinucleons to the self-energies. The resulting Hartree propagator Gn is identical to that of the mean field theory by Walecka “). The Hartree field t,!~nsatisfies the Dirac equation (i~-M*-yoE~)~H=O,
(2.4)
where the baryon effective mass M* and the vector self-energy 2: are given by d4k -
reJ%
(2?7)4 e
-
d4k
(2?T)4 e
’ We foIlow the conventions of Bjorken and Bell I*).
Tr G,(k) ,
z&k,Tr yOGn( k) .
(2.5)
(2.6)
H. Kurasawa, T. Suzuki / Quasielastic electron scattering
688
The factor
ei+ (a + +0) in eqs. (2.5) and (2.6) ensures that nucleons above the Fermi sea do not contribute to the above integrals. The Hartree propagator GH is
defined
in the usual manner G,(k)=
as d4(x-y)eik(x-y)GH(x-y)
I = -_I’
I
d4b-y) eik’“-y’(l~[~H(~)ICIH(y)I(), .
where 1 ) is the Hat-tree ground state of the symmetric wave number k,. We readily obtain GH( k) =
(R+ M*)
nuclear
matter
(2.7) with Fermi
(2.8)
~_;*2+ia+;S(I;,-Ek)ek k
where Ek = (M**+k*)“*
(2.9)
and k@ = (k,,, k) = (k,,- Xt, k). The ground-state distribution function is given by 0,‘ = O(kF- lkl). A s mentioned above, the integral in eq. (2.5) is divergent due to the antinucleon contribution coming from the first term in eq. (2.8). Omitting the antinucleon contribution we obtain the Hartree self-consistency condition: M*
(2.10)
(27r)3 (M*2+k2)1’2’ The vector 2k3,/(3r2).
self-energy
We use the free propagators involve the later discussion. way as eq. (2.7), are
density
ps =
meson propagators, since tadpole corrections to the meson only zero-momentum components *), which are irrelevant for The scalar and vector meson propagators, defined in the same given by
D,(k)= They are combined
is given by Ez = (gV/m,)2pB with the nucleon
’ k*-m$+is’
into a single
D:“(k) 5 x 5 matrix
=
-gp” + k*k”/mz k2-mz+ic
propagator
*
(2.11)
D”* given by (2.12)
2.2. RPA RESPONSE
FUNCTIONS
In order to evaluate we consider correlation
the RPA response functions for inelastic electron scattering, functions based on the Hartree approximation. For operators
689
H. Kurasuwa, T. Suzuki / Quasielffsric electron scattering
of the type A(x) = ~~(x)~~~~(x) and B(x) = ~~(~)~~~~(x), where r, and 17, are arbitrary 4 x4 matrices, we define the correlation function as G[A(x),
B(Y)] =
(IT[SA(x)B(y)ll)I(lSI),
(2.13)
with S=Texp
( rg, ’ j d4x :~&YY&)
K(x):)
(2.14)
,
where : : represents normal ordering with respect to the Hartree ground state, I ). In the perturbation expansion of eq. (2.13) we take only the ring diagrams and sum them to all orders. This is equivalent to the RPA ‘*). The summation of the ring diagrams leads to the integral equation for the correlation function, GM(x),
J xGo[A(x),
B(y)1= Go[W), B(y)1-
d4x’
%a86
d4y’
(2.15)
B(Y)],
F”(x’)lo,b(x’-y’)~[Fb(yl),
with G[A(x),
WY>I= GCA(x),%)I + Ur ~AGdx - x+))(Tr~&My-y’)) B(Y)I=T~[T,G~(~-Y)T,G,(Y
Go[@x),
-x)1,
F”(x) = ~~(~)~=~~(x) where (x+)@ =(x0+ E, x). The Hartree propagator and &btX-‘Y) by
J-
d4k
Dab(X-Y)
=
c2rl~
_
,
(2.16) (2.17) (2.18)
1
G,(x -y)
is given by eq. (2.7)
.
(2.19)
e -Ikcxy'&,(k)
The first term in eq. (2.15), G,[A, B], is the Hartree correlation function, which corresponds to the lowest-order ring diagram. The subtraction of the ground-state expectation value ([A( x)1) = -i Tr T,G,( x - x+) in eq. (2.16) ensures that disconnected diagrams are excluded. Fig. 1 shows eq. (2.15) diagrammatically.
Fig. 1. Diagrammatic representation of eq. (2.15). The solid lines with the arrows denote the baryon propagator G, in the Hartree approximation, while the dashed line corresponds to the free meson propagator Da*. The wavy lines represent the external fieids A and B.
690
H. Kurasuwa, T. Suzuki j’ Quasielastic electran scattering
From the definition eq. (2.13), the Fourier transform of G[A, B] becomes G[A(-k),
WV;
~1
J
J
B(y)] =2’ d(x,-y,) eiw(%-Ydd3Xd3ye-ik,x+ip’J’G[A(x), (2.20)
=(O~[A(-k)(H-w-i~)-‘B(k’)+B(k’)(N+o-i~)-’A(-k)](O),
where H is the total hamiltonian, HIO) = 0 and
IO)the correlated (RPA) ground state satisfying
A(k) =
Owing to the momentum conservation, G[A(-k),
d3X es‘“A(O, x) .
(2.21)
G is decomposed
B(k’); w] = S(k - k’)17,,,(A,
into B; k) ,
(2.22)
where k’” = (0, k). In terms of LT,, eq. (2.15) is written as %,(A,
B; k) =Ji’(A, B;
k)+xa(kP(A,
F”; kV&dL
B; kl
,
(2.23)
with
fl(A, R k) =
-&
Jd4p
Tr [f,G&+
1 XS=m*
k)fdMp)l
,
(2.24)
2 a=-1, (2.25)
xa =
1
z2
Here we have assumed k” f (0,O) and have used the conservation baryon current k,J7( F”, A) = k&17(A, P)
relation of the (2.26)
= 0.
Because of this conse~ation the gauge terms in the vector meson propagators not contribute to II,,. Eq. (2.23) can be solved for ITRpa, n,,,(A,
B) = fl(A, B)+xJI(A,
F”)( u-‘),JWb,
B) ,
do
(2.27)
where CJab= gab -xJI(F”,
Fb),
Uab( u-l)bc = a,, ,
(2.28)
with gab = l(a = b = -l), g”“(a = EL,b = v) and g-lfl = g’-’ = 0. As is verified by explicit calculation of the traces in ZI(F”, Fb), the matrix U can be divided into two parts: one is the longitudinal part, which is a 3 ~3 matrix, due to the scalar e-meson and the longitudinal and the time component of the vector
691
H. Kurasawa, T. Suzuki / Quasielastic electron scattering
w-mesons,
the other the transverse
part, which is a 2 x 2 matrix,
from the transverse
vector mesons. In the former matrix, the terms from the longitudinal are related to those from the time component through the current
vector meson conservation
relation, eq. (2.26). Therefore, the former matrix is also expressed by a 2 x 2 matrix. With the choice of the four-momentum as kp = (w, Ikl, 0, 0), we obtain IIRPA(A, B) = 17(A, B) + tX$i,(A, Here the longitudinal
B) + SIl:‘,(A,
B) .
(2.29)
part is given by
Mi$&( A, B) = s L
where the 2 X2 matrix
fl(A,
( UL)ob (a, b = -1,O) U = L
F”)( W.dT(Fb, is written
&(I -,CJL) i”(
1-
(2.30)
as
X./i?V&
XSiJL
(
B) ,
xm
(2.31)
>
with the abbreviations
(2.32) The vector coupling
strength
in eq. (2.30) for the longitudinal k2
i”=sXy=(21r)3
The transverse
1
part in eq. (2.29) is described
s: mz-k2
k2 k2’
part turns
out to be (2.33)
as (2.34)
where rr,
=
n(F2, F2)= n(F3, F3) *
(2.35)
Eqs. (2.30) and (2.34) provide us with the RPA correlation functions in terms of the Hartree ones and are the basic equations in the present study. The RPA response functions for electron scattering with the four-momentum transfer kp = (w, k) are proportional to the imaginary part of ITRPA(j, _!; k) for electromagnetic current operators 1 Although there is an ambiguity concerning the choice of _? [refs. 5’9)], we adopt here the current operators of point-like free nucleons with anomalous magnetic moments K, ( K~ = 1.793 and K, = -1.913): (2.36)
692
ff. Ku~ffsaw4
T. Suzuki / ~as~e~astie
electron scattering
where TVis 1 for protons and 0 for neutrons, and me,,=$[r,, ~~1. The longitudinal and transverse response functions, denoted by S,(k) and S,(k), respectively, are given by (2.37) (2.38) where Z denotes the proton number in symmetric nuclei, and p^= .& and .?r = .& = ?=’ for our choice of k = (l&l, 0,O). The propo~ional constants in front of the square brackets of the above equations are easily obtained by comparing eq. (2.20) with eq. (2.22). Eqs. (2.30) and (2.34) provide us with dispersion relations of undamped collective modes, which are expressed as det U,(k) = 0 for longitudinal modes and as (lx&r=)2 = 0 for transverse modes. We shall discuss those undamped modes later, but further discussions will be found in refs. 4,‘4). 2.3. CALCULATION
OF THE HARTREE
CORRELATION
FUNCTIONS
In this subsection we will give analytic expressions for the Hartree correlation functions which are needed to evaluate the RPA response functions, eqs. (2.37) and (2.38). Substituting eq. (2.8) into eq. (2.24), we obtain4) II(A, 3; k)=IIF(A,
l.3; k)+lTD(A,
B; k),
(2.39)
where iI,(A,
B; k) = -&
I&,(A,B;
k)=-
I
d4p
t(p, k)
d4&S(p,--E,) P
d4p~
__$iT
J
(2.40)
(p2-M*2+iE)((p+k)2-M*2+ie)’
t(p, -k)
t(p, k)
(p+k)2-M*2+i~+(p-k)2-M*2+ie 6(po-E,)6(p~+o-E,+,)t(p,
P
k),
(2.41)
P+k
with the trace factor t(p, k) = Tr [r,(p+K+
M*)TB(~+
M*)l .
(2.42)
The Feynman-propagator piece flF describes the most of the NR excitations, and the density-dependent piece flu the particle-hole excitations and a part of the NR ones. In the present paper, we ignore X7, in eq. (2.39) instead of a renormalization procedure, which will render nF finite. Note that the current conservation relation, eq. (2.26), holds even without flF for A = F, and 6. Hereafter the subscript D of fl, is omitted.
I$.
Kurusawa,
T
Suzuki /
Quasielastic
electron
scattering
693
It is verified by explicit calculations of the trace factors that the Hartree correlation functions, eq. (2.41), are expressed in terms of the following three integrals (n = 0, 1,2): Z,(k) = -2
1 ~(~~-E,)S(~~+O-E
d4Z+ S(po-- EJ P
(2pofo)”
(2P,- w)”
(p+k)2-M*2+ief(p-k)2-M*2+iE
(2.43)
The results are ZZv=c;s-k2Zo+Z,,
Z?s=-&+(4M”2-k2)I,, II,=-up.-?
&+k2
_
ZZ,,= 2M*Z,,
1 k2
(2.45)
,
~Zz+(4M**+k2)Zo
(2.44)
(2.46) fffF*,p^)=zz(~,
F,)=tn,-M*(c,+e,)k210,
ZZ(F2, &) = ZZ(& F’) =fZ&- M*(c,+
(2.47) (2.48)
c,)k2Zo,
(2.49)
n(dP^)=~~,-2M*cpk2Zo-(e~+c~)(w2p”,f~k2Z2-t2M*2k2Z0), ZZ(.f,, &) =~~~-2~*c~k21~-~k2(c~+c~)(~~+~~+(4~*’+
k2)Z,),
(2.50)
where & is related to the scalar density ps by 4 (J,,f*2_+)‘/2
(2.51)
and c, = u,/2M. The analytic expression for Z,(k) is derived in the appendix. The Hartree correlation functions given above includes a part of the Nij excitations and the particle-hole excitations with energy between max (0, E?‘) and Ey’, where E!_’ is defined in eq. (A.8). It should be noted that the particle states involved in the former excitations are within the Fermi sphere. This fact implies that the Pauli exclusion principle is violated in the Nfi excitations’. Since the fourmomentum transfer in electron scattering is space-like, k2 < 0, these Ns pairs, which lie in the time-like region, affect the RPA response functions through the real parts of the Hartree correlation functions. As we shall see later, however, these off-shell effects are small. Before closing this section, let us show the explicit forms of the Hartree response functions. Using eqs. (2.37), (2.49) and (A.27), the longitudinal response function ’ In fact, if we in&de the vacuum polarization part IIF the contribution from the NR pairs violating the Pauii principle to II, is cancelied by that from the same pairs to flF [refs. 4-‘5)].
H. Kurasawa,
694
T. Suzuki / Quasielastic
electron scattering
S’,“‘(k) is written as
S:H’W=
z
L[FL(k)B(max ,k,
16k;
(0,E’_‘)
< w
-F,(-k)fI(O
(2.52)
with (2.53)
where EF, E!_’ and f(w) are defined in the appendix as EF= (M*‘+
k$)l”,
E~:)=(M*2+(kFflk()2)1’2-EF,
f(“)=lk[( 1+$$)“‘. Similarly we obtain the transverse response function @‘(k) (2.52) FL with FT,
by replacing in eq.
3. Results and discussion
In numerical calculations of the response functions we use the following experimental values for the nucleon and meson masses: M = 939 MeV, m, = 783 MeV and m, = 554 MeV. For the coupling constants we will employ the values which are phenom~nologically determined by Walecka ‘) so as to reproduce observed saturation properties of nuclear matter, i.e. the binding energy = 15.75 MeV and kF = 1.42 fm-‘. His values are given by gz = 92.87 (g$‘47r = 7.39) and gz = 135.7 (gzf4w = 10.8). For these values the effective mass satisfying the Hartree self-consistency condition, eq. (2.10), turns out to be M* = 0.557M.
3.1. LONGITUDINAL
RESPONSE
FUNCTION
As a typical example, we show in fig. 2 the numerical results of S,(k) given by eq. (2.37) as a function of w at Ikl = 2.5kF = 700 MeV. The dotted curve represents the response of the relativistic free Fermi gas, which is given by eq. (2.52) with M” = M; The Hartree response function in the CTWmodel obtained by eq. (2.52) with M” = 0.557M is shown by the dashed curve. In the Hartree approximation, the nuclear interaction affects the response function only through the effective
H. Kurasawa, T. Suzuki / Quasielastic electron scattering
r
I
&(k)/Z
3
[MeV-‘I
695 /
;...,,
X 1om3
2
1
0 Fig. 2. Longitudinal response functions as a function of the energy transfer w at the momentum transfer Ikl = 2.5k, (k, = 1.42 fm-I). The dotted curve shows the response of the relativistic free Fermi gas (g, = g, = 0), while the dashed one is obtained by the Hartree approximation with gz = 92.87 (M* = 0.557M). The dot-dashed curve is the RPA result with gf =92.87 and gt=O, the two-dot-dashed one with g: = 0 and gt = 135.7, and the solid one with gt = 92.87 and gt = 135.7. The arrow indicates the position of the collective mode in the case of g: = 0 and gt = 135.7. The thin solid curve shows the results in ref. ‘I).
mass ‘), which does not depend on the wN interaction. In comparing the dashed curve with the dotted one, it is easily seen that the small effective mass shifts the peak position to higher excitation energy region and reduces the absolute values of the response function. These effects of the effective mass were essential for the Hartree approximation in order to explain the experimental data 5-9). Let us study the effects of the particle-hole correlations on the response function. First we explore those due to the a-meson exchanges only, since the effective mass comes from the aN interaction. Setting gf = 92.87 and g: = 0 in MIRPA of eq. (2.37) we obtain the RPA response function as shown by the dot-dashed curve. One can see that the shift of the peak position coming from the effective mass is perfectly cancelled by the strong attractive particle-hole correlations. In fact, if we neglect the w-meson exchange, the system with k, = 1.42 fm-’ becomes unstable against the density fluctuation, when gz 2 77.8 for Ikl = 2.5 k,. Since the present value, gf = 92.87, is slightly larger than the critical value, the dispersion equation det l_J,= 0 has a complex root near the real axis in the upper-half complex w-plane 16). The sharp peak of the dot-dashed curve reflects the existence of this root. Next let us take into account particle-hole correlations due to the w-meson exchanges, too. The RPA response function for gf=92.87 and gz= 135.7 is drawn by the heavy solid curve in fig. 2. It shows that the w-meson exchanges produce a strong repulsive effect on the response function. For reference, the RPA result is also shown in fig. 2 by the two-dot-dashed curve setting gt= 135.7 but gz=O in
696 811RPA
H. Kurasawa,
of eq. (2.37) and replacing
two-dot-dashed
curve is reduced,
tion is due to the existence indicated
T. Suzuki / Quasielastic
electron scattering
M* in II and ?XIRPA by M. The strength compared
of the undamped
zero-sound
mode whose
at w/ kF= 1.372. The energy-weighted
by the arrow
of the
with those of other curves. This reducstrength
position
is
carried
by
this mode is 62.72 (MeV), while that of the continuum states 74.22 (MeV). It is interesting to note that most of the strength is thus distributed over the same energy region as that for the case of gt = 92.87 and gt = 135.7. We show in fig. 3 the results at (kl= 1.5kF=420 MeV in the same way as in fig. 2. The arrow denotes the position of the zero-sound in the case of gf = 0 and gt = 135.7. Its energy-weighted strength, 32.52 (MeV), is approximately equal to that exhausted by the continuum states. The RPA result with gf = 92.87 and g: = 0, which is shown by the dot-dashed curve, exhibits a considerable reduction of the strength. This results from the fact that the value gz = 92.87 is far from the stable region, gf< 12.5 for gz = 0 and Ikl = 1.5kF, and the dispersion equation det U,= 0 has a complex root. The solid curve calculated with gf = 92.87 and g!j = 135.7, for which the system is stable, shows again an important role of the w-meson exchange in the RPA response function. I
6
0
SJk)/Z
I
IMeV’l
0.5 Fig. 3. Same as fig. 2, except
w/k,
1.0
for Ikl= 1.5k,.
From the above two examples it is easily seen that the longitudinal response function is greatly affected by the particle-hole correlation not only due to the a-meson exchange, but also due to the w-meson exchange. This is quite different from the Hat-tree result 5-9) that the wN interaction has no contribution to the response functions, while the whole effect of the UN interaction on them can be when the solid curves are incorporated into the effective mass M *. Furthermore, compared with the dashed ones, the strength of the RPA results is more reduced than the Hartree ones. This reduction comes from the ground-state correlations
H. liurasawa,
which are included is inadequate excitations
dynamical
that in the present
is partially
included.
that the Hartree
properties
RPA calculations
Comparing
697
electron scattering
in the RPA. Thus, it is concluded
for understanding
We have noted
7: Suzuki / Quasielastic
the present
approximation
of the longitudinal the off-shell
response.
effect of NN
results with those in ref. I’),
where Nfi excitations are completely neglected, we can see that the off-shell effect does not alter qualitative features of the response function. The previous RPA results for gt=92.87 and gg = 135.7 in ref. ‘l) are shown by the thin solid curves in figs. 2 and 3.
3.2. TRANSVERSE
RESPONSE
The transverse response and 5, respectively. Curve
FUNCTIONS
functions at (kl= 2Sk, and 1.5kF are shown in figs. 4 designations are the same as those of fig. 2. Since there
Fig. 4. Transverse response functions as a function of o at Ikl= 2.5k,. The curves are drawn in the same way as in fig. 2.
is no particle-hole correlation due to the u-meson exchange, the RPA with g,=O is nothing but the Hartree approximation. The particle-hole correlations are induced through the transverse part of the w-meson exchanges. We see from figs. 4 and 5, however that, in contrast to the longitudinal response function, the particle-hole correlation hardly produces an effect on the response function. This comes from the fact that while the w-meson is isoscalar, the transverse response for highmomentum transfer is mainly caused through the isovector magnetization current. The transverse response function in the 5~0 model is thus dominated by the effective mass even in the RPA. This does not imply, however, that one can extract simply the value of the effective mass from the experimental data. When we discuss
698
0
0.5 Fig. 5. Same as fig. 4, except
experimental exchanges
data, it is necessary
to further
for Ikl= lSk,.
take into account
which yield spin- and isospin-dependent
3.3. COULOMB
SUM
We consider
1.0
W/k,
at least the p-meson
particle-hole
correlations.
RULE
the Coulomb
Here GU is the contribution
sum for electron
scattering,
from the undamped
which is given by
collective
mode, if it exists in the
space-like momentum transfer region, and is given by the residue of S,( k) at w = w,,,, satisfying det U,_(w,,,,, k) = 0. The range of the integral in eq. (3.1) is restricted to the space-like momentum transfer region, because the time-like region cannot be reached
experimentally
by electron
scattering.
The off-shell effect of NR excitations
on eq. (3.1) is small, as was discussed in subsect. 3.1. Fig. 6 shows the results with the Hat-tree approximation using gs= 92.87. The solid and dashed curves are calculated with and without the anomalous magnetic moments K, respectively. The non-relativistic sum rule value, CNR, for the Fermi gas is given by 17) C
NR
=
%l/k&#&43, I 1,
O< lki
(3.2)
This is also illustrated by the dot-dashed curve in fig. 6. The values of the relativistic model approaches to C,, in the long-wavelength limit (lkl-, 0). This fact is verified directly using eq. (2.52)?. The difference between the dot-dashed curve and the ’ It is worthwhile to be k2/2EF, rather
noting that the value of the energy-weighted than the non-relativistic value k*/2M*.
sum of eq. (2.52) for lkl-+ 0 turns out
H. Kurasawa,
T. Suzuki / Quasielastic
-I dwS,(k)/Z
‘O
I
/
/.
electron scattering
699
,_________------.--
0 Fig. 6. Coulomb sum defined as the space-like energy integral of the longitudinal response function in the Hartree approximation with g: = 92.87. The solid and dashed curves represent the results with and without the anomalous magnetic moments, respectively. The dot-dashed curve shows the non-relativistic sum rule value given by eq. (3.2).
dashed one, whose value converges to i when Ikl + co, is due to the relativistic effect on the part of the positive-energy spinors. It is seen from the solid curve that the anomalous magnetic moments in the time component of the four-current fairly reduces the sum value at Ikl b 1.5kFt. The relativistic effects on the Coulomb sum rule, thus, produce a large reduction of the sum rule value. Using the VOJ model in the mean field theory which is equivalent to the present Hartree theory, Walecka 19) also calculated the Coulomb sum rule. He found that the sum rule value is dominated by the contribution from the anomalous magnetic moments
at lklz2k,
as (3.3)
This is considerably different from our result shown by the solid curve in fig. 6. In deriving eq. (3.3), however, the anticommutation relations {$a(x), I&(Y)} = yt,S(x - y) are used. This implies that the on-shell NN contributions are also included in eq. (3.3). Since K, in the time component of the nuclear four-current appears with the “odd” operator y [ref. “)I, the on-shell NN excitations through the anomalous magnetic moments yield a large contribution to eq. (3.3). Thus, a great part of the value of eq. (3.3) comes from NN excitations in the time-like region which cannot be seen by electron scattering. A similar result has already been pointed out in ref. ‘*), but see the previous footnote. ’ A similar calculation in ref. ‘s) found the enhancement of the sum value due to the anomalous magnetic moment. It seems, however, that there is a mistake in ref. I*), since we can reproduce its value by replacing {l+ (M*/M)K,} by {l - (A~*/M)K~} in eq. (2.54).
700
H. Kurasawa,
T. Suzuki / Quasielastic
electron scattering
Next we discuss the effects of the particle-hole
correlations
on the Coulomb
given by eq. (3.1). In fig. 7 is shown the RPA and Hat-tree results sum with use of the same curve designations obtained
from
the RPA
calculations
without
as in fig. 2. The dot-dashed the w-meson
sum
of the Coulomb
exchange.
curve is We have
mentioned in the discussion of figs. 2 and 3 that det U, = 0 without the W-meson exchange has an imaginary solution. The imaginary solution, however, disappears beyond a certain value of lkl, since the strength of the attractive force given in eq. (2.25) decreases with increasing Ikl. For the case of 92.87, this transition occurs at Ikl = 2.58kr. Th e s h arp peak of the dot-dashed curve is as a result of the transition, and hence its value at Ikl< 2.58kF has no meaning.
1.0
/do~(h)/Z
0
2
I a/l+
4
Fig. 7. Coulomb sum defined as the space-like energy integral of the longitudinal response functions. Curve designations are the same as in fig. 2. The two-dot-dashed curve denoted by “zero sound” shows the contribution from the undamped zero-sound mode to the upper two-dot-dashed curve.
The upper
two-dot-dashed
curve in fig. 7 shows the RPA results
with gf = 135.7
but without the a-meson exchange. In the region of Ik( < 2.84kF for this case, the repulsive correlations due to the W-meson exchange yield the undamped zero-sound mode. The value of the RPA sum, therefore, is given by adding the strengths of the continuum particle-hole states and the zero-sound state. The contribution from the latter state to the sum value is shown by the lower two-dot-dashed curve in fig. 7. In taking account of the correlations due to both u- and W-meson exchanges with gz = 92.87 and gz = 135.7, w e obtain the RPA results as shown by the solid curve. It is seen that the solid curve is between the two-dot-dashed curve and the dot-dashed one for Ik( > 2.58kF where the latter has a meaning. This is due to the fact that in the RPA attractive forces induce the ground-state correlations to enhance the sum value, while repulsive ones reduce the value. In the present case, the net effect of the ground-state correlations on the Coulomb sum produces about 10% reduction
701
H. Kurasawa, T. Suzuki / Quasielastic electron scattering
of the Hat-tree
result
which
is shown
by the dashed
solid curve with the dotted one for the free Fermi reduced by about 18% around /k/ = 1.56~~
curve.
When
gas model,
comparing
the RPA result
the is
4. Summary We have derived the RPA response functions for quasielastic electron scattering from the relativistic lagrangian of the CTWmodel, using a Green function approach. Although the model lagrangian is renormalizable, we have neglected divergent terms in the Hartree baryon propagator and in the Hartree correlation functions. Adopting the electromagnetic current operators of point-like free nucleons with the anomalous magnetic moments, the RPA response functions have been calculated analytically. It is shown that the longitudinal response function is strongly affected by the particle-hole correlation not only due to the g-meson exchange but also due to the w-meson exchange. This result is quite different from the previous understanding 5-9) that the whole effect of the aN interaction on the response function would be incorporated into the effective mass, while the wN interaction would have no effect. In using the coupling constants to reproduce saturation properties of nuclear matter 3), the effect of the c-meson exchange on the response function is largely cancelled by the one from the repulsive correlation due to the w-meson exchange. The net effect dominates the peak position of the RPA response function, and yields about 10% reduction of the absolute values of the Hartree response function. The particle-hole correlation, which affects the transverse response function, is induced through the transverse component of the w-meson exchange. In contrast to the longitudinal response function, however, the transverse one is little affected by the correlation. This is due to the fact that the transverse response in electron scattering is induced mainly through the isovector magnetization current. Consequently, the transverse response function in the ITW model is dominated by the nucleon effective mass. We have also discussed the Coulomb sum rule which is defined as the space-like energy integral of the longitudinal response function. Comparing with the nonrelativistic sum rule value of the Fermi gas, the relativistic result is quenched by about 40-50% in the region of 1.5kFX< Ikl=~ 4.5 kF. This quenching is mainly attributed to the three sources. Those are the relativistic factor of the positive-energy spinors, the spin-dependent time component of the nuclear four-current and the ground-state correlations due to the w-meson exchanges. Finally it should be noted that when one compares the RPA response functions with experimental data for quasielastic electron scattering, at least the p-meson exchanges also must be taken into account. It is straightforward, however, to include those exchanges in the present scheme. The application of the present RPA to finite nuclei and to other phenomena including the particle-hole correlations would be done by slight modifications at a formal level. Examination of problems is in progress.
702
H. Xurasowa,
T. Suzuki j Quasielastic electron scattering
Grateful acknowledgements are due to Dr. S. Nishizaki for useful discussions. One of the authors (H.K.) thanks the Yukawa Foundation for financial support. Appendix We explicitly derive the analytic expression for I,,(k) defined in eq. (2.43), i.e. I,(k)=J,(k)+J,(-k)+iK,(k),
where
J
(A.11
k~ d3
(2E,+w)” p o Ep E:,, -(Ep+w)2-ic’
J,(k)=2
J
(A.21
k,
K,,(k) = -2r
d3p+
0
6(o+E,-E,+,)(2E,+w)“. P
L4.3)
r+k
First we calculate Re I,,. Changing the integral variable cos f3=p - k/lpi )k( into x=E pfk f Ep, J,, is written as
with E, = E,,*lkl= [M*2+ (p f /kj)2]1’2 and p = IpI. By integration by parts we obtain
J
kF
(2E,+
w)n+lL(k)
-
dp(2E,+
@)“+I
-k,
1
+d(E++E,)
E+-Ep-w-i&
(A.51
dp
where we have defined
J
EL-'
L(k)=
J
E$+'
dx
E”’
*
x-w-is
+
dxli,
(A.6)
E!?’
with EF= EkF= (My2+ k;)1’2f E!-’ = &&cl-
% Jk, dpdp J-k=
G=,
E$+’ = Ek&l+
(A.71 EF .
(A.81
E~-)dx[-X+O+f( J Q+) dx [X+W-f(x)]“+l J
The change of the variable p into x = E, rt Ep yields
dp
-k,
d(E+-E,) dp
d(E++E,)
(2EP+o)“+1
E+-E,--w-is= (2E,+o)“+’
E++Ep+o-iE=
+ 8(2k,_ lk,j
Ei-)
x-o-i&
EL+’
x+w-ia
J E(i’dxEX+ w-“,“‘-- 5,; w+fr’+l, =%/2
(A-9)
(A.10)
H. Kurasawa,
where f(x)
T. Suzuki / Quasielastic
electron scattering
703
is given by (A.ll)
Using
eqs. (AS),
Re L(k)=i
(A.9) and (A.lO),
we obtain
-[(2EF+w)“+‘L,(k)+(2E,-w)“+‘L,(-k)+Al,(k)], .:l
(A.12)
with (E!_‘-w)(EJ+‘+w) (E.-j_w)(E’+j+-w)
L(k)=ReL(k)=ln
(A.13)
,
A1~(k)=((-l)“j~;~dx-~~;dx)P(X-;~;)n+’ -(j(2k,_Ikl)
j-@+‘dxP(X-Onn+xr~~-W+f)““,(o--w), 2&/2 (A.14)
where P stands for the Cauchy principal For n = 0, eq. (A.14) becomes
values.
Ar,(k)=(-[E~;dx+jE~~dx+20(2k,-~k~) Denotingf(x)
by y, and noticing
0, Al, turns
thatf(E5))
{2zI)2xf(x)&. = E!+‘,f(E!+‘)
(A.15)
= IE>‘I andf(2E,,,)
=
out to be AS(k)=2([~~~dy-[E~~dg)P(&-fi)
>
(A.16)
with LY= k2(1 -4M*‘/k2). Hence,
the final expression Re I,(k)
=?
Ikl
(A.17)
for Re 1, is given by (2E,+o)L,(k)+(2E,-w)L,(-k)
(A.18) In a similar
way we obtain Re I,(k)=z
214[
(2EF+w)‘LR(k)+(2EF-a)‘&-k)
-ct(L,(k)+L,(-k))+4(E$+‘E~,-‘-E’t’Et’) -(jE~~dx+JE~~dx)(2x+~&)],
(A.19)
704
H. Kurasawa,
Re Iz(k)=x
3lkl
T. Suzuki / Quasielastic
electron scattering
(2E,+o~)~L,(k)+(2E,-w)~L,(-k)
Although the integrals appearing in the above equations can easily be written in terms of elementary functions, these expressions are not shown here since they are long. Next we evaluate the imaginary part of I,, which is given by ImI,(k)=Im(J,,(k)+J,(-k))+&(k). Using eqs. (AS),
(A.21)
(A.6), (A.9) and (A.lO) we immediately
obtain
+0(2k,-~k~)[(2E,-w)“-‘-(f(w))“+‘]0(0
+[(2EF-W)“+l + 0(2kF-
Ikl)Kf(4)“+1
-(-f(w))““]8(2E,,,
with
e(a
(A.23)
Here we have omitted terms disappearing for w > 0, noting that E’_’ < 0 for Ikl< 2k,. In the same way as in deriving eq. (A.4) we have E+-E
..,,,=-$
I
:’
dp-$2E,+o)“B(E,-w-E,) p
with x = E,,+k - I&.. Employing
‘dxS(x-W) E_-
(A.24)
E,
the identity
~(2E,fo)“B(E,-o-E,)=&$([(2E,+w)“+1-(2E,-o)”t’] P
(A.25)
x@(-b-w-E,)l,
integration
by parts in eq. (A.24) yields,
for w > 0,
K.(k)=-~--&[(2E~-~)‘“+1-(j-(4)“illB(2E,-w-f(4)
=-~-&[(2E,-o)n+1-
(f(w))““]f3(O
I
;z’dx8(x-o)
<-EL-‘)8(2k,-Ik(), (A.26)
H. Ku~sawu,
T. Suzuki / Quasielastic electron scattering
705
where we have used the equivalence between 2 EF - w -f (w ) > 0 and -E !_’ < o < -E’_‘. Substituting eqs. (A.22) and (A.26) into eq. (A.21) we obtain finally, for o > 0, Im &(k)=$
&{[(2&+w)“+’
-(f(w))““]@(max
(0, EC’) < w < EC’)
-8(2k,-lk1)[2E,-w)“+‘-(f(w))“+‘]e(O
-(-f(w))“+‘]B(Ei?
~(2k,-Ikl)[(ftw))“+‘-
< Er’)
(-f(0))“*‘]8(2E~,2
-=cw <
E?‘)} .
(A.27)
References 1) M.R. Anastasio, L.S. Celenza, W.S. Pong and CM. Shakin, Phys. Reports 100 (1983) 327, and references therein 2) B.D. Serot and J.D. Walecka, The relativistic nuclear many-body problem, Adv. Nucl. Phys., in press 3) J.D. Walecka, Ann. of Phys. 83 (1974) 491; Lecture Notes in Physics, vol. 92, Nuclear interactions (Springer, Berlin, 1978) p. 294 4) S.A. Chin, Ann. of Phys. 108 (1977) 301 5) R. Rosenfelder, Ann. of Phys. 128 (1980) 188 6) J.V. Noble, Phys. Rev. Lett. 46 (1981) 412 7) A. Hotta, P.J. Ryan, H. Ogino, B. Parker, G.A. Peterson and R.P. Singhal, Phys. Rev. C30 (1984) 87 8) G. Do Dang and Nguyen Van Giai, Phys. Rev. C30 (1984) 731 9) T. de Forest, Jr., Phys. Rev. Lett. 53 (1984) 895 10) A.L. Fetter and J.D. Walecka, Quantum theory of many-particle systems (McGraw-Hill, New York, 1971) 1 i) H. Kurasawa and T. Suzuki, Phys. Lett. 154B (1985) 16 12) J.D. Bjorken and SD. Drell, Relativistic quantum mechanics (McGraw-Hill, New York, 1964) 13) C.J. Horowitz and B.D. Serot, Nucl. Phys. A368 (1981) 503 14) T. Matsui, Nucl. Phys. A370 (1981) 365 15) T. Matsui and B.D. Serot, Ann. of Phys. 144 (1982) 107 16) T. Yukawa and H. Kurasawa, Phys. Lett. 129B (1983) 162 17) T. de Forest, Jr. and J.D. Walecka, Adv. in Phys. 15 (1966) 1 18) T. Matsui, Phys. Lett. 1328 (1983) 260 19) J.D. Walecka, Nucl. Phys. A399 (1983) 387