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Electromagnetic excitation of the delta-baryon in quantum hadrodynamics
Nuclear Physics AS0zl(1989) 797-817 North-Holland, Amsterdam
ELECTROMAGNETIC EXCITATION OF THE DELTA-BARYON QUANTUM HADRODYNAMICS
IN
K. WEHRBERGER Nuclear Theory Center, Indiana University, Bloomington, IN47405, USA
C. BEDAU and F. BECK lnstitut fiir Kernphysik, Technische Hochschule Darmstadt D-6100 Darmstadt, Fed. Rep. Germany
Received 15 June 1989 Abstract: In photoabsorption and electron scattering at large energy transfer, the photon can excite a nucleon to a A-baryon. This gives rise to the A-peak in the spectrum. We investigate the cross section for this process in the framework of quantum hadrodynamics. The results on the A-peak depend on the coupling constants of the A to the scalar and vector mesons. We compare results in Hartree approximation with the data. I. Introduction Q u a s i e l a s t i c e l e c t r o n s c a t t e r i n g p r o v i d e s an i m p o r t a n t test for theories o f n u c l e a r structure. B e c a u s e o f the large e n e r g y a n d m o m e n t u m transfers i n v o l v e d relativistic t h e o r i e s are e s p e c i a l l y well s u i t e d for such investigations. T h e y p r o v i d e a c o v a r i a n t f r a m e w o r k o f n u c l e o n s i n t e r a c t i n g via d y n a m i c a l m e s o n e x c h a n g e a n d thus constitute a n effective field t h e o r y for the n u c l e a r m a n y - b o d y p r o b l e m . Q u a n t u m h a d r o d y n a m i c s 1) has r e c e n t l y b e e n u s e d b y a n u m b e r o f a u t h o r s 2-5) to c a l c u l a t e the s e p a r a t e d l o n g i t u d i n a l a n d t r a n s v e r s e r e s p o n s e functions. It was f o u n d t h a t R P A c o r r e l a t i o n s are i m p o r t a n t in o r d e r to i m p r o v e the a g r e e m e n t o f the l o n g i t u d i n a l r e s p o n s e with the d a t a , b u t leave the t r a n s v e r s e r e s p o n s e a l m o s t unaffected. W h i l e the overall a g r e e m e n t with the d a t a is fairly s a t i s f a c t o r y in the l o n g i t u d i n a l case, the s i t u a t i o n for the t r a n s v e r s e r e s p o n s e is u n c l e a r b e c a u s e o f the o v e r l a p p i n g A_peak; at large e n e r g y t r a n s f e r the virtual p h o t o n can excite a n u c l e o n to a za-baryon, a n d this has n o t b e e n i n c l u d e d in p r e v i o u s c a l c u l a t i o n s . The resulting A . p e a k is also a p r o m i n e n t f e a t u r e o f the d a t a o n p h o t o a b s o r p t i o n , i.e. in the limiting case o f real p h o t o n s . It is the a i m o f the p r e s e n t w o r k to use the e x p e r i m e n t a l i n f o r m a t i o n a v a i l a b l e on t h e s e p r o c e s s e s to o b t a i n i n f o r m a t i o n o n the d y n a m i c s o f the A in the nucleus. Specifically, we i n c o r p o r a t e the za as an effective p a r t i c l e in m e a n - f i e l d a p p r o x i m a t i o n to q u a n t u m h a d r o d y n a m i c s with effective c o u p l i n g s to the s c a l a r a n d v e c t o r m e a n fields, a n d we s h o w t h a t t h e d a t a on e l e c t r o m a g n e t i c e x c i t a t i o n o f the A p r o v i d e i n f o r m a t i o n on these c o u p l i n g s . This i n f o r m a t i o n is very i m p o r t a n t , since the influence o f the za-baryon on the e q u a t i o n o f state o f n u c l e a r m a t t e r d e p e n d s very m u c h o n t h e s e u n k n o w n c o u p l i n g c o n s t a n t s 6). 0375-9474/89/$03.50 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
K. Wehrberger et aL / Electromagnetic excitation
798
In sect. 2 we introduce the mean-field lagrangian with the A and we briefly review the role of the A for the equation of state of nuclear matter. In sect. 3 we evaluate the A-hole polarization propagator and discuss its relation to the response functions and cross sections for electron scattering and photoabsorption. The longitudinal and transverse cross sections for electroexcitation of free nucleons as evaluated in our relativistic approach are compared with the data in sect. 4. This serves as a check of our input. Results for the response functions of nuclear matter are given in sect. 5. We discuss the influence of the finite width of the A, of the Fermi motion of the nucleons, and of the choice of the A coupling constants. Results for finite nuclei, obtained in local-density approximation, are presented and compared with data and with results obtained in the nonrelativistic A-hole model in sect. 6. Sect. 7 contains our conclusions.
2. Role of the delta in mean-field theory When applying quantum hadrodynamics at large nuclear densities or medium to high energy transfers the possibility of the excitation of nucleon resonances has also be considered. The resonance lowest in mass is the spin-3-isospin -3 delta baryon at 1232 MeV. To incorporate the A in quantum hadrodynamics one has to extend the lagrangian density by a kinetic7'8) and an interaction term for the RaritaSchwinger spinor ~ a with mass Ma :
with
~a=~a"(iYA~A--M'a)~a+~a"%,(iYa~x - iq~a~'(3, ~
+ 3,~,) ~,
+ M a ) Y ' v~va
(2)
=O, + ig~v V~,,
(3)
M'a = Ma - ga da ,
(4)
~
where we have denoted the coupling constants of the A to the o-- and w-mesons with mean fields ~b and V~, by g a and go. a This lagrangian is meant to be used in mean-field approximation only, it is not renormalizable and the A is treated as a stable particle. Since the a is, however, a pion-nucleon resonance, a more fundamental approach would be to start from a renormalizable lagrangian for nucleons and pions and to generate the A dynamically. For our purposes the above mean-field lagrangian provides a convenient way of approximately taking into account this resonance as an effective particle. The couplings of the A to the ~r and o~ mean fields, which could in principle be derived in a more fundamental approach, are here to be considered as free parameters.
K. Wehrberger et al. / Electromagnetic excitation
799
Eq. (2) is a special case of a whole set of lagrangians which all yield the same Rarita-Schwinger field equations
(iya~ x - M ~ ) ~ , = O, ~
=0,
y~A =0.
(5)
Correspondingly, the noninteracting spin- 3 propagator contains a free parameter. For the special lagrangian density (1) the propagator is given by the commonly adopted form 8), qXyx+Maf ~ l v S"~-M~-~ ~g -3Y~'Y
2 q~q~ 3 M~
1
~_.~__~(q Y _q~y~)
}
(6)
Due to the simple structure of the chosen mean field interaction, the propagator S~ ~ in Hartree approximation is obtained by the replacements Ma -) M~ = Ma - g ~ b ,
k,~ ~ k'~ = k~, - 6,~og~ Vo,
(7) (8)
as long as no real A's are present in t h e ground state. Little is known about the couplings gsa and gva. From SU(6) symmetry the vector coupling is expected to be similar to the corresponding coupling gvN of the nucleon 9). A restriction on the couplings is obtained if one demands that no real A's are present in the ground state of nuclear matter at saturation density k N and zero temperature, which means that the equation ~2
N 2
N
x/MN + ( k F ) + g v
Vo=~/M'f+(kaF)2+gv A Vo
(9)
does not have a real solution for the Fermi m o m e n t u m kFa of the d. With the Hartree values for the nucleon couplings and meson fields 1o) one obtains rs ~<0.82rv + 0.71,
(10)
with A
gv
rv=--gvN,
g~
rs=--gN.
(11)
This is only a weak condition. I f for example rv = 1.0,
rs = 1.35,
(12)
a A-isomer state appears at about twice saturation density 6 ) , while nothing spectacular is predicted to h a p p e n for universal coupling (rs = rv = 1.0). Therefore information on these couplings from a different source would be very important.
800
K. Wehrberger et al. / Electromagnetic excitation
3. Delta-hole polarization propagator The ~/NA transition amplitude can generally be written in the form ~1) (/tlj, lN)-- q,'~(k)F~,,(q)qt(p).
(13)
Here k is the outgoing momentum of the A, p is the incoming momentum of the nucleon, q = k - p , and k~ is a nucleon spinor. This transition is predominantly a magnetic dipole, and the corresponding vertex function is, including an isospin factor, given by n) - ( M a + MN) x/~ GM(q:) C ~ -- MN((Ma + MN)2 _ q2)
x {(-q~g~ + qAg~,)Ma3,x~/5 + (q~k~ - q~k~gm,)Vs},
(14)
with the M1 form factor
q2 GM(q 2) = GM(0) (1 \
"~-2 //
0.71~eV2 )
kl
q2
~-1/2
3.5~-eV2]
GM(0) = 3.0.
,
(15) (16)
Here the second factor is the dipole fit to the electromagnetic form factor of the nucleon, and the third factor fits the small deviation of the yNA transition form factor from this dipole form. The central quantity determining the response of a system to a weak external perturbation is the corresponding polarization propagator. In Hartree approximation the total A-hole polarization propagator is given by the sum of the y(N, A)y and y(A, N ) y polarization propagators:
I - l ~ = - i f ~ d4 T rp( F ~ ( - q , - p ) S ~ ( p ) F ~ ( q , p ) G H ( p - q ) ) + ( q ~ - q ) .
(17)
Here GH is the density-dependent part of the nucleon propagator in Hartree approximation: ilr G . ( p ) = (3~'p * + M*) ~.. 8(p* - E*)O(kv-Ipl)
(18)
p~g e- p ~ - 8~,ogvN Vo,
(19)
with
M * = MN -- gN~b,
(20)
Ep~ = ~/p2 -I'-M~N2 ,
(21)
801
K . W e h r b e r g e r et al. / E l e c t r o m a g n e t i c e x c i t a t i o n
and S . is the A-propagator in Hartree approximation as given in eqs. (6), (7) and (8), assuming that no real A's are present in the ground state. The evaluation of the trace is straightforward, and we obtain -1 f d4p n~'~(q) = 6-~3 E*_q(p,2_M,a2+ie) ~( p*o - q o - E*,_q)O( k F - IP - ql)(M,~ +
~.,~) (22)
+(q~ -q), with
M., = ( M , a M . _ ( p , k . ) ) {p.p,. (q2 (qp,___))2,~M, ]a +(qp)Z(g~.-P:P2~4 \ Ma ] 2(qp)(qp') ~ PiP;-2(qp)q+.p. ) +--~'a (2M'aM* - (p'k*) ) t × {_q2p, p,. _ (qp)(qp,)g~.. + (qp,)q.p. + (qp)q,..~.}
+Ma
M'a
M2
{(qp')p,. - (qp)p'}{(qk*)p'~ - (qp')k*} - ~
(p'k*)
t 2 ¢ l x{(qp ¢) 2 g~.~-Z(qp t )qwt.+q p~.p.}
+ MaM'a{qZk*p. + (qp)(qk*)g~. - (qk*)q~p. - (qp)q.k*} + m a ( M a m N + (p'k*)){qZg~,. - q~q~} + MZ{(qp')q~k * + (qk*)p'~q~, - q zp~.~ , ,_. - (qp')(qk*)g~,~},
(23)
where k = p - q ,
etc.
(p q ) = p " q~
The longitudinal and transverse response functions of nuclear matter are proportional to the volume and to the imaginary parts of the corresponding A-hole polarization propagators: SL(Iql, to) = --Vim rto~o(Iql, o,), 7/"
sT(Iq[,
to) = 2 V i m 77"
n,~l(Iql, o,)
(24)
(for q in 3-direction). The cross section for quasielastic electron scattering is given in terms of the response functions by trM.,q.4SL+
--
+tan2½0
Sr ,
(25)
with O'M--
442E~ cos 2 ½0 4
(26)
802
K. Wehrberger et al. / Electromagnetic excitation
Here 0 is the scattering angle, Er is the energy of the outgoing electron and a is the fine-structure constant. Alternatively, eq. (25) can be written in terms of the longitudinal and transverse cross sections of the virtual photon: d2o "
dO dEf
o~ - K Er
1 -(Cr-r+ eo'e), 2,rr2 q2 Ei 1 - e
(27)
with K-=to+
e ---
(
q~ 2MN '
1-2
(28)
t a n 2 ½0
)1
(29)
,
q. 27./-2 o'v =
K
(30)
aST ,
O'L= ----~-- a
SL-
(31)
In the limit q~ ~ 0, crT is the total photoabsorption cross section for real photons. In the formalism presented so far the At-baryon is treated as a free particle. It does, however, decay into zrN with a width in free space of F = 115 MeV. A convenient and physically transparent way of introducing the width here is by averaging the response S(/x) obtained for fixed At-mass /z, with respect to tz with a suitable weight function, centered around tz = M ~ [ref. 12)]: S = - -1
f
d/z 2 S(/x)
M'~F . ( M'~2- g2)2 + M'~2r2
(32)
We conclude this section with a nonrelativistic estimate of the position and width of the quasielastic At-peak in nuclear matter. From energy and m o m e n t u m conservation for the excitation of a nucleon with m o m e n t u m k by a virtual photon with energy to and m o m e n t u m q to a At-baryon, we obtain to = ( M '~ - M * ) - ( gvN - g va) V ° +
kq
-~ M 'a
ke
2 M 'a M * (
M' - M*) a
•
(33)
The position of the A-peak depends on g# (via the effective A_mass) and gv~, and the width, being proportional to k F / M ' ~ , depends weakly on the scalar coupling. The width, however, is determined from both Fermi broadening and the A decay width, and the former dependence is therefore even weaker. It is nevertheless evident that the data on the A-peak contain information on the coupling constants of the A to the mean tr- and to-fields.
K. Wehrberger et al. / Electromagnetic excitation 08
I
. . . . qu
I
2
=
. . . .
-0.2
803
I
. . . .
I
.
-GeV z
0.6
"-S
&
0.4
0.2
0.0
i
, I ....
I ....
1.1
1.2 g (GeV)
.
.
.
1.3
1.4
Fig. 1. T r a n s v e r s e c r o s s s e c t i o n f o r e l e c t r o n scattering o n free p r o t o n s at fixed q ~ = - 0 . 2 G e V z as a f u n c t i o n o f c e n t e r - o f - m a s s e n e r g y E. T h e s o l i d line is o u r result, the d a t a are f r o m ref. t3).
4. Eleetroexcitation on free nucleons A basic requirement o f any model o f the A-excitation in nuclei is that it reproduces the measured transverse and longitudinal cross sections for electroexcitation of free nucleons. In the framework o f the formalism presented here, these cross sections can be obtained from eqs. (30) and (31) as the limiting case o f kF-->0 and for vanishing m e s o n fields. Our results are compared with the data from ref. 13) in figs. 1-5. In figs. 1-3 we have plotted the transverse cross section for three fixed values o f qZ as a function o f the c.m. energy E. The cross section is s o m e w h a t underestimated at the maximum. This is due to the fact that we have not included any background contributions. If we add the background contributions as evaluated in ref. 14), the 08
I
. . . . q~
2
I =
. . . .
-0.3
[
'
'-'
'
GeV a
0.6
0.4
0.2
0.0 1.1
1.2
1.3 (CeV)
Fig. 2. S a m e as fig. l , b u t at q ~ = - 0 . 3 G e V 2.
1.4
K. Wehrberger et al. / Electromagnetic excitation
804
0.8
'
'
~
qu
'
2
I
r
-0,4
E
,
,
GeV z
0.6
"S
&
0.4
0.2
0.0
1.1
Fig.
3.
1.2 Z (aeV)
Same
as
fig. 1,
but
at
1.3
1.4
q~ =-0.4GeV
2.
agreement with the data is quite good. In fig. 4 the longitudinal and transverse cross sections are plotted at fixed c.m. energy E = 1.22 GeV, i.e. a p p r o x i m a t e l y at the m a x i m u m o f the peak, as a function o f q~,. 2 Again the absence o f the background is evident. The g o o d a g r e e m e n t o f the q2 d e p e n d e n c e o f our result with the data indicates that the form factor was c h o s e n correctly. Since w e are here mainly interested in the effects o f the coupling constants o f the A to the or- and to-mesons on the d o m i n a n t resonant part o f the response, w e do not explicitly include the background in the present calculation, but w e always have to r e m e m b e r this w h e n w e c o m p a r e with data. 0.8
. . . .
I
. . . .
I
E =
. . . .
1.22
I
GeV
0.6
7
04
& b 0.2
0.0
.
.
i
.
.
i
i
i
I .... 0.2
I , , 0.4
_%2
i
, I , , 0.6
[GeVz]
Fig. 4. Cross section for electron scattering on free protons at fixed c e n t e r - o f - m a s s e n e r g y E = 1.22 G e V as a f u n c t i o n o f q .2. The solid line is our result and the squares are the data f r o m ref. ~3) for the transverse cross section. The l o n g i t u d i n a l cross section is s h o w n by the d a s h e d line and the d i a m o n d s .
805
K. Wehrberger et al. / Electromagnetic excitation 1.50
I
. . . .
I 2
q,~
1.25
=
. . . .
-0.3
/
\ \\
/
\
ii 0.75
\\
//
\\
/
¢
\
/ 0.50
....
-"'G-eV 2
// 1.00
I
\
/
m
\
0/25
0.00
1.1
1,2
1.3
1.4
z (g~v) Fig. 5. Transverse cross section for electron scattering on free protons with (solid line) and without (dashed line) the non-pole terms in the A-propagator. The data are from ref. t3).
The calculated longitudinal cross section s h o w n by the dashed line is very small but n o n z e r o and compatible with the data. In fig. 5 we show the i m p o r t a n c e o f the terms not proportional to g,~ in the A - p r o p a g a t o r (eq. 6). The transverse cross section without these terms is shown by the d a s h e d line. It is m u c h larger and in disagreement with the data. O u r results are o b t a i n e d with the p r o p a g a t o r as given in eq. (6). As already pointed out in sect. 2, the general s p i n ) p r o p a g a t o r contains a free parameter w. Different choices o f w differ by n o n - p o l e terms. In the present investigation we only need the imaginary part o f the polarization p r o p a g a t o r , i.e. the A is on mass shell. In this case the n o n - p o l e terms do not contribute and our results are therefore i n d e p e n d e n t o f the parameter w.
5.
Results
for
nuclear
matter
The response functions d e p e n d on the energy and m o m e n t u m transfer, on the d e c a y width o f the A, the m o m e n t u m distribution o f the target nucleons and the coupling constants o f the nucleons and A's to the m e s o n fields. This is especially transparent for scattering f r o m nuclear matter. Since the electromagnetic transition amplitude for A_excitation is equal for protons and neutrons, we here consider only symmetric nuclear matter. All response functions s h o w n are normalized to N = Z = 20, and they are s h o w n as a function o f energy transfer ~o at fixed m o m e n t u m transfer ]q[. The calculations are only extended up to ~o ~< Iql, since little is k n o w n a b o u t the NA transition form factor for time-like m o m e n t u m transfer. This is not really a restriction in the present context since we will not need this time-like region when discussing p h o t o a b s o r p t i o n and electron scattering.
806
K. Wehrberger et al. / Electromagnetic excitation
In fig. 6 we show the t r a n s v e r s e r e s p o n s e f u n c t i o n o f a free F e r m i gas at s a t u r a t i o n d e n s i t y for different values o f the d e c a y w i d t h o f the zl. T h e r e s p o n s e for zero width, i.e. stable zi, is s h o w n b y the d o t t e d line. In this case the w i d t h o f the r e s p o n s e f u n c t i o n is d e t e r m i n e d o n l y b y the F e r m i m o t i o n o f the n u c l e o n s a n d t h e r e f o r e p r o p o r t i o n a l to the F e r m i m o m e n t u m (see eq. (33)). The r e s p o n s e v a n i s h e s o u t s i d e a w e l l - d e f i n e d k i n e m a t i c a l region. I n c l u d i n g the d e c a y w i d t h o f the A m e a n s a v e r a g i n g with r e s p e c t to the /t-mass as in eq. (32) a n d thus l e a d s to a b r o a d e n i n g , a r e d u c t i o n o f the m a x i m u m , a n d a small shift o f the p e a k position. This is s h o w n for the free A - w i d t h , F = 115 MeV, b y the solid line. The d a s h e d line is o b t a i n e d with h a l f the free width, the d o t - d a s h e d line with d o u b l e the free width. A l t h o u g h the w i d t h o f the A is small c o m p a r e d to its mass, it has a large effect on the r e s p o n s e f u n c t i o n : the relevant scale is the w i d t h o f the r e s p o n s e f r o m F e r m i m o t i o n w h i c h is a p p r o x i m a t e l y the s a m e as the d e c a y width. The d e p e n d e n c e o f the r e s p o n s e on the F e r m i m o m e n t u m at fixed free d e c a y w i d t h is s h o w n in fig. 7. The d o t t e d line is o b t a i n e d in the limit o f zero density, w h e r e the w i d t h o f the r e s p o n s e is d e t e r m i n e d b y the d e c a y w i d t h only. The solid line shows the r e s p o n s e o f a free F e r m i gas at s a t u r a t i o n density. F e r m i m o t i o n gives rise to an a d d i t i o n a l b r o a d e n i n g . F o r h a l f a n d d o u b l e the s a t u r a t i o n d e n s i t y we o b t a i n the d a s h e d a n d d o t - d a s h e d lines. H e n c e f o r t h all results will be s h o w n for the free d e c a y w i d t h a n d at s a t u r a t i o n density. In fig. 8 we c o m p a r e the t r a n s v e r s e r e s p o n s e f r o m n u c l e o n k n o c k o u t a n d from A - e x c i t a t i o n . The d o t t e d lines show the r e s p o n s e o f a free F e r m i gas. D u e to the i n t e r a c t i o n with the s c a l a r a n d v e c t o r m e s o n fields the n u c l e o n s a n d za's a c q u i r e effective m a s s e s a n d energies. The solid lines s h o w the r e s p o n s e o f an interacting
f ....
I~
' ~ ' ' T
....
I. . . .
--
0.15
q = 500 MeV
" f
0.i0
.' / "'
', /
x ' \ , \i",
II
// 0.05
0.00
100
200
[MeV]
300
400
500
Fig. 6. Transverse response function from A-excitation for a free Fermi gas at momentum transfer Iql = 500 MeV as a function of energy transfer to, normalized to N = Z = 20. The dotted line shows the response for a stable A-baryon, and the solid line for the free width F = 115 MeV. The dashed and dot-dashed lines are obtained with half and double the free width.
K. Wehrberger et al. / Electromagnetic excitation ....
I ....
I ....
I ....
807
I ....
0.15
q = 500
MeV ',
0. i 0
,/-\ ,)
0.05
,. \,
.//
, ' / ,'
0.00
I00
200
300
400
500
[MeV]
Fig. 7. Transverse response function from zi-excitation for different values of the Fermi momentum. The dotted line shows the response for free nucleons in the limit of vanishing density, the solid line for a free Fermi gas at saturation density. The dashed and dot-dashed lines are obtained for half and double the saturation density. For all results the free width of the A was used. F e r m i g a s w i t h s t a n d a r d v a l u e s for t h e m e a n f i e l d s a n d n u c l e o n c o u p l i n g c o n s t a n t s lO) a n d t h e a s s u m p t i o n o f u n i v e r s a l c o u p l i n g f o r t h e A. T h e s t r e n g t h is s h i f t e d t o w a r d s l a r g e r e n e r g y t r a n s f e r a n d t h e v a l u e at t h e m a x i m u m
is r e d u c e d . T h i s is d u e to t h e
s m a l l e f f e c t i v e m a s s e s , M * ~ 0 . 5 4 M N a n d M ~ ~ 0 . 6 5 M a , g i v i n g rise to l a r g e r k i n e t i c e n e r g i e s at t h e s a m e t h r e e - m o m e n t u m
transfer. T h e d a s h e d l i n e is o b t a i n e d w i t h
i n t e r a c t i n g , effective, n u c l e o n s b u t free Zl's. T h e m a x i m u m o f t h e r e s p o n s e is s h i f t e d [ ....
I ....
I ....
0.15
q =
500
MeV
Nucleon
DelLa
0.10
/
2"
... ,
/
,
....-
0.05
-.
0
,. ,, ',.
//
,
/
0.00
/
/
. ,
,
f
.//
//
I00
200
300
400
500
Fig. 8. Transverse response function from nucleon knockout and A-excitation. The dotted lines show the response of a free Fermi gas, the solid lines show the response of an interacting Fermi gas, with standard nucleon coupling constants determined to fit ground-state properties and A-coupling constants from the assumption of universal coupling. The dashed line is obtained for an interacting Fermi gas of nucleons and for free deltas. All response functions are evaluated in Hartree approximation.
808
K, Wehrberger et al. / Electromagnetic excitation o.o6 ~-'
' '-'
I '-'
q = 500
~-
0,06
I ....
' ' I ' ' ' ' I ....
MeV
.
0.04
/
/ / /
0.02
/
/
,'
\
.
"~
\
/
"
'~
\
/
" \
/
// //
/
/
,-
\
/
\
/ i
O.O0
lOO
200
300
400
500
[MeV]
Fig. 9. Transverse response from Zl-excitation for universal vector coupling and different values for the scalar coupling of the A. The ratio rs of the scalar coupling of the Zl to the scalar coupling of the nucleon is 0.9 for the dotted line, 1.0 for the solid line, 1.1 for the dot-dashed line and 1.2 for the dashed line. t o w a r d s l a r g e r e n e r g y t r a n s f e r w i t h r e s p e c t t o t h e c a s e o f free n u c l e o n s A's by approximately The dependence
on the scalar and vector coupling
in m o r e d e t a i l in figs. 9 a n d 10, f o r f i x e d n u c l e o n the scalar coupling
and free
the difference of the mean vector and scalar potentials. c o n s t a n t s o f t h e A is s h o w n
coupling
constants. Increasing
ratio ( s e e eq. ( 1 1 ) ) f r o m r s = 1.0 (fig. 9, s o l i d l i n e ) to r s = 1.1
( d o t - d a s h e d l i n e ) d e c r e a s e s t h e e f f e c t i v e m a s s o f t h e A b y 10% o f t h e s c a l a r p o t e n t i a l , i.e. a b o u t 4 0 M e V . T h i s r e s u l t s in a c o r r e s p o n d i n g
0.08
0.06
....
T ....
I. . . .
q = 500
shift of the maximum
I ....
of the
I ....
MeV
J./ /
.
/
'> = ~x
ii
0.04
ii i 0.02 / /
O.O0
/
I /
/
/
.i
/
\
.
,
~.
.'"
/i \ \
."
\
.' . I k
100
200
300
400
i
b
L
i
500
[MeV]
Fig. 10. Transverse Response from A-excitation for universal scalar coupling and different values for the vector coupling of the A The ratio rv of the vector coupling of the za to the vector coupling of the nucleon is 1.1 for the dotted line, 1.0 for the solid line, 0.9 for the dot-dashed line and 0.8 for the dashed line.
809
K. Wehrberger et al. / Electromagnetic excitation 0.04
0.03
'
'1 ....
I ....
I ....
[ ....
q = 500 MeV
0.02
0.01
0.00
100
200
300
400
500
[~eV] Fig. 11. Longitudinal response from nucleon knockout and A_excitation for an interacting nucleon Fermi gas with standard nucleon couplings and with universal coupling for the z~. response function towards smaller energy transfer. The dashed and dotted lines show the results obtained for rs = 1.2 and 0.9. Similarly, an increase o f the vector coupling f r o m rv = 1.0 (fig. 10, solid line) to rv = 1.1 (fig. 10, dotted line) increases the effective energy by 10% o f the vector potential, i.e. about 30 MeV. This increased effective energy shifts the m a x i m u m o f the response towards larger energy transfer. The dashed and d o t - d a s h e d lines show the results obtained for rv = 0.8 and rv = 0.9. F o r completeness we s h o w in fig. 11 also the longitudinal response functions from n u c l e o n k n o c k o u t and A-excitation, for an interacting Fermi gas with standard n u c l e o n a n d universal A-couplings. As expected from the mainly transverse nature o f the interaction, the longitudinal A-response is almost negligible c o m p a r e d to the response from nucleon knockout.
6. Results for finite nuclei and comparison with data The nuclear matter results are not sufficient for a quantitative c o m p a r i s o n with the data on quasielastic electron scattering on nuclei. The simplest way to incorporate finite size effects is the local-density approximation. Here we use local densities and m e s o n fields as determined f r o m the solution o f the mean-field equations o f q u a n t u m h a d r o d y n a m i c s lO). The local response functions are a d d e d to obtain the response o f the nucleus. This is k n o w n to be a g o o d a p p r o x i m a t i o n for the quasielastic n u c l e o n p e a k 3). The main difference between the results obtained with the exact o n e - b o d y density and in local-density a p p r o x i m a t i o n is a shift towards larger energy transfer by the average separation energy o f about 10 MeV. Since there is no reason to expect this to be different for the A-peak, the local-density a p p r o x i m a t i o n is sufficient for the present investigation.
810
K. Wehrberger et aL / Electromagnetic excitation
In nuclei the w i d t h o f the A will be m o d i f i e d d u e to Pauli b l o c k i n g a n d the s p r e a d i n g w i d t h ,5). Pauli b l o c k i n g has a s t r o n g d e n s i t y a n d m o m e n t u m d e p e n d e n c e : it r e d u c e s the w i d t h to zero at s a t u r a t i o n d e n s i t y a n d very s m a l l m o m e n t u m o f the A, a n d has a l m o s t no effect on the w i d t h at very large m o m e n t u m . The s p r e a d i n g w i d t h , on the o t h e r h a n d , is n e a r l y m o m e n t u m i n d e p e n d e n t . A d y n a m i c a l i n c o r p o r a t i o n o f b o t h o f these effects w h i c h we e x p e c t to d e p e n d also on the c o u p l i n g s o f the A to the o-- a n d to-fields w o u l d t h e r e f o r e be d e s i r a b l e . F o r this first survey, however, we will a s s u m e t h a t the i n t e r p l a y o f b o t h effects leaves the free width on the average u n m o d i f i e d , a n d we will t h e r e f o r e use a m o m e n t u m a n d d e n s i t y i n d e p e n d e n t w i d t h o f 115 M e V everywhere. This a s s u m p t i o n is s u p p o r t e d by the fact t h a t o u r results r e p r o d u c e the w i d t h o f the d a t a well, b u t we s h o u l d k e e p in m i n d t h a t a m o d i f i c a t i o n o f the a v e r a g e w i d t h by 10% w o u l d c h a n g e the r e s p o n s e at the m a x i m u m b y a b o u t 10%. In fig. 12 we show the cross section for electrons with energy E = 620 M e V at scattering angle 0 = 60 ° as a f u n c t i o n o f e n e r g y t r a n s f e r to. The d a t a are f r o m ref. ,6). T h e d o t - d a s h e d line shows the result f r o m n u c l e o n k n o c k o u t only. Here c o r r e l a t i o n s d u e to o% to- a n d p - m e s o n s are i n c l u d e d in the r a n d o m - p h a s e a p p r o x i m a t i o n s). E x c e p t for the missing strength at large energy t r a n s f e r the d a t a are well r e p r o d u c e d . M o s t o f this missing strength can be p r o v i d e d b y the A - b a r y o n . The d a s h e d line shows the r e s p o n s e f r o m A - e x c i t a t i o n a l o n e in H a r t r e e a p p r o x i m a t i o n a s s u m i n g free n u c l e o n s a n d free A ' s , This result is very s i m i l a r to that o b t a i n e d in a nonrelativistic a p p r o a c h in ref. ,4), if the b a c k g r o u n d c o n t r i b u t i o n as e s t i m a t e d there is t a k e n into account. The cross section is, h o w e v e r , r e d u c e d s u b s t a n t i a l l y if we a s s u m e
le C 4
E = 620 MeV £, ~
I00
200
0
= 600
300
40{3
500
600
~0 (MeV)
Fig. 12. Cross section for electron scattering on '2C at incident electron energy E = 620 MeV and scattering angle 0 = 60° as a function of energy transfer w. The dot-dashed line is the contribution from nucleon knockout only as evaluated in random-phase approximation. The contributions from a-excitation only for universal coupling and for no coupling are shown by the solid and dotted lines. The dashed line shows the i'esult for noninteracting nucleons and al's. The data are from ref. ~6).
K. Wehrberger et al. / Electromagnetic excitation
811
effective n u c l e o n s a n d d ' s with universal c o u p l i n g . This is s h o w n b y the solid line. The s t r e n g t h is shifted t o w a r d s l a r g e r e n e r g y transfer, a n d o n l y a b o u t 60% o f the o b s e r v e d cross section at the m a x i m u m is a c c o u n t e d for. I f we a s s u m e interacting n u c l e o n s b u t free A's we o b t a i n the d o t t e d line. The m a x i m u m at the p e a k is a b o u t the s a m e as with free n u c l e o n s , b u t the p o s i t i o n o f the m a x i m u m is, as in fig. 8, shifted b y a p p r o x i m a t e l y the difference b e t w e e n the average v e c t o r a n d s c a l a r p o t e n t i a l s a n d is in d i s a g r e e m e n t with the data. In fig. 13 we s h o w the sum o f the c o n t r i b u t i o n from n u c l e o n k n o c k o u t a n d d - e x c i t a t i o n for s t a n d a r d n u c l e o n c o u p l i n g s a n d different d - c o u p l i n g s . The solid a n d d o t t e d lines are o b t a i n e d with u n i v e r s a l a n d v a n i s h i n g c o u p l i n g , respectively. T h e d a s h e d a n d d o t - d a s h e d lines s h o w the result for a ratio rs = 0.15 a n d 0.30 o f s c a l a r d - c o u p l i n g to s c a l a r n u c l e o n c o u p l i n g a n d v a n i s h i n g vector coupling. The small v a l u e rs = 0.15 is sufficient to shift the p e a k p o s i t i o n such that it agrees with the data, a n d it leaves the p e a k height a l m o s t u n m o d i f i e d c o m p a r e d with the result for free d. On the o t h e r h a n d , we see that the d a t a on the p o s i t i o n o f the p e a k r e q u i r e t h a t rs c a n n o t e x c e e d rv b y m o r e t h a n a b o u t 0.2. F o r 12C the s e p a r a t e d t r a n s v e r s e a n d l o n g i t u d i n a l cross sections are a v a i l a b l e not o n l y in the r e g i o n o f the n u c l e o n p e a k , b u t also for the A - p e a k . These d a t a from ref. 17) are s h o w n t o g e t h e r with o u r results in figs. 14-16. I n fig. 14 we s h o w the t r a n s v e r s e cross s e c t i o n at q~ = - 0 . 1 G e V 2 as a f u n c t i o n o f e n e r g y transfer. The lines s h o w the c o n t r i b u t i o n s f r o m n u c l e o n k n o c k o u t a n d A - e x c i t a t i o n s e p a r a t e l y a n d c o r r e s p o n d to t h o s e s h o w n in fig. 12. A g a i n the missing strength in the A - r e g i o n is a p p a r e n t . In fig. 15 we s h o w the sum o f the n u c l e o n a n d A - c o n t r i b u t i o n s , a n d the
12C
4
E = 620 MeV
~,~II .,~
"
0
100
o
=
'
200
6o ° ~r
300
(Mev)
I.L ±Z~TT
400
500
600
Fig. 13. Cross section for electron scattering on '2C at incident electron energy E =620 MeVand scattering angle 0 = 60° as a function of energy transfer to for standard nucleon and different zl-couplings. The lines are the results for the sum of the contribution from nucleon knockout and d-excitation. The dotted line shows the cross section for free A's, and the dashed and dot-dashed lines for no coupling to the vector field and a ratio r~= 0.15 and 0.30 of the scalar coupling of the A to the scalar coupling of the nucleon. The solid line is obtained for universal coupling. The data are from ref. t6).
812
K. Wehrberger et al. / Electromagnetic excitation . . . .
i
. . . .
I
. . . .
I
lZC,
j
. . . .
qu z
I
. . . .
I
CeV z
-0.1
=
. . . .
..
?:.: .-...
zoo
z50
300
350 (M~V)
400
450
500
Fig. 14. Transverse cross section for electron scattering on '2C at fixed q~ = -0.1 GeV 2 as a function of energy transfer to. As in fig. 12, the dot-dashed line is the contribution from nucleon knockout only, and the contributions from A-excitation alone for universal coupling and for no coupling are shown by the solid and dotted lines. The dashed line shows the result for noninteracting nucleons and A's. The data are from ref. IT).
12C,
200
I
L
250
300
t
i
,
ku z = - 0 . i GeV2
i
i 350
h
,
i
L 400
L
,
,
, 450
500
(~ev) 2 __ Fig. 15. Transverse cross section for electron scattering o n L 2 C at fixed q~, - -0.1 GeV 2 as a function of energy transfer w. As in fig. 13, the lines represent the sum of the contributions from nucleon knockout and A-excitation. The dotted line shows the cross section for free A's, and the dashed and dot-dashed lines for no coupling to the vector field and a ratio rs = 0.15 and 0.30 of the scalar coupling of the A to the scalar coupling of the nucleon. The solid line is obtained for universal coupling. The data are from ref. ,~).
813
K. Wehrberger et aL / Electromagnetic excitation
I .... 7-' ' '~- .... 12C,
6
[ ....
F ~
q2 = -0.1
~-~ GeV 2
4'
S
-2
200
250
300
350
400
450
500
Fig. 16. Longitudinal cross section for electron scattering on 12C at fixed q~ = -0.1 GeV2 as a function of energy transfer to. The dot-dashed line shows the result from nucleon knockout only, the solid line shows the result from A-excitation. The data are from ref. 17).
l i n e s c o r r e s p o n d to t h o s e in fig. 13. As s h o w n in fig. 16, t h e m e a s u r e d l o n g i t u d i n a l c r o s s s e c t i o n is c o m p a t i b l e w i t h z e r o , a n d o u r r e s u l t f o r t h e l o n g i t u d i n a l c r o s s s e c t i o n is i n d e e d a l m o s t z e r o e x c e p t f o r t h e tail f r o m t h e l o n g i t u d i n a l n u c l e o n p e a k w h i c h decreases rapidly with increasing energy transfer. T h e e s s e n t i a l l y t r a n s v e r s e c r o s s s e c t i o n f o r e l e c t r o m a g n e t i c e x c i t a t i o n o f t h e d is s u p r e s s e d at s m a l l e r s c a t t e r i n g angles. I n fig. 17 w e s h o w t h e cross s e c t i o n f o r 12C at i n c i d e n t e n e r g y E = 680 M e V a n d s c a t t e r i n g a n g l e 0 = 36 ° w i t h d a t a f r o m ref. ~6). H e r e t h e A - p e a k is i n d e e d m u c h s m a l l e r t h a n t h e n u c l e o n p e a k . As in fig. 13, t h e
12 C
3o
t
E = 680 MeV
/~ t t:~ % '~
,
0
= 360
\,
20
m
o
....
I ~ I tO0
.... ,800
I ....
300 co (MeV)
I .... 400
Fr, 500
, , 600
Fig. 17. Same as fig. 13, but for incident electron energy E =680 MeV and scattering angle 0 =36 °. The data are from ref. ,6).
K. Wehrberger et al. / Electromagnetic excitation
814
solid line shows the result obtained for standard nucleon coupling and universal A-coupling. The dotted, dashed and dot-dashed lines are for vanishing a vector coupling and rs = 0.0, 0.15 and 0.3. As in the previous results for a scattering angle of 60 ° the data are underestimated. Data for the electron scattering cross section in the region of the A-peak are available also for 4°Ca [ref. ,8)]. These are shown for E = 695 MeV and 0 = 60 ° in fig. 18. The lines have the same meaning as in figs. 13 and 17. Again the result for interacting nucleons and universal A-coupling (solid line) underestimates the data. The results for small scalar coupling and no vector coupling overestimate the data in the dip region although no background is included. For free za's we obtain the dotted line which at large energy transfer leaves no room for the background contribution which is expected to increase with increasing energy transfer. In figs. 19-21 we show the cross section per nucleon for photoabsorption on 2°8Pb with data from ref. ,9). In fig. 19 the dot-dashed line shows the cross section for free nucleons in the limit of vanishing density. As in fig. 12, the dashed line is the result for finite density as obtained from the solution of the mean field equations, but assuming noninteracting nucleons and A's in the evaluation of the response. The solid line shows the result for effective nucleons and effective A's with universal coupling, the dotted line for effective nucleons and free a ' s . The reduction of the response due to the small effective mass of the A in the case of universal coupling is similar as for the quasielastic electron scattering cross section. In fig. 20 the solid and dotted lines are the results for universal and vanishing coupling, respectively, and the dashed and dot-dashed lines show the results for rv = 0.15 and 0.3. Again comparatively good agreement with the data is obtained with vanishing vector and small scalar coupling rs = 0.15 of the A and rv = 0.0 and rs = 0.3 is incompatible with
4OCa 8
"
~.i~qu.
+,
7
I 2
E -
6 9 5 MeV
e =
60°
~\
/
~-t
5[ ,, ",
++,
\
,
~
\
+. ++. 0
100
20{3
300 co (MeV)
400
500
600
Fig. 18. S a m e as fig. 13, but for electron scattering on 40 C a at i n c i d e n t electron energy E = 695 MeV a n d s c a t t e r i n g angle O = 60 °. The d a t a are from ref. ts).
K. Wehrberger et al. / Electromagnetic excitation 0e
'
'
~-]
....
I
~
'
815
[ ....
0.5
0.4
0.3
0.2
0.1
0.0
ZOO
100
300 E (~v)
500
400
Fig. 19. Photoabsorption cross section per nucleon of 2°spb as a function of the photon energy E. The dot-dashed line shows the cross section for photoabsorption on free nucleons, the other lines are results for 2°spb obtained in local-density approximation. As in fig. 12, the results for free nucleons and free A's is shown by the dashed line, for effective nucleons and free /t's by the dotted line, and for effective nucleons and A's with universal coupling by the solid line. The data are from ref. 19).
t h e d a t a , I n fig. 21 w e s h o w t h e r e s u l t s f o r r v (dotted
line) and
peak position
r~= 1.30 (dot-dashed
require
1.0 a n d r~ = 1.0 ( s o l i d l i n e ) , r~ = 1.15
l i n e ) . It c a n b e s e e n t h a t t h e d a t a
the scalar and vector couplings
on the
o f t h e zl t o b e s i m i l a r .
0.6
0.5
0.4
0.3
0.2
0.1
0.0
i
loo
J
i
,
i
,
zoo
,
,
.
.
.
300
.
.
.
.
.
4oo
.
.
500
E (MeV)
Fig. 20. Photoabsorption cross section per nucleon of 2°8pb as a function of the photon energy E, with data from ref. ~9) and our results for standard nucleon couplings and different A-couplings obtained in local-density approximation. The dotted line shows the cross section for free A's, and the dashed and dot-dashed lines for no coupling to the vector field and a ratio r~ = 0.15 and 0.30 of the scalar coupling of the A to the scalar coupling of the nucleon. The solid line is obtained for universal coupling.
816
K. Wehrberger et al. / Electromagnetic excitation 06
-
~
-
~
~
-
~
-
~
-
~
'
I
'
'
'~---
0,5
04
0.3
02
0.1
O0
10(3
200
300
400
500
E (MeV)
Fig. 21. Sameas fig. 20, but for universal vectorcoupling rv= 1.0 and r~= 1.0 (solid line), rs = 1.15 (dotted line) and r~= 1.30 (dot-dashed line). 7.
Conclusions
We have extended previous calculations of the response functions for quasielastic electron scattering in quantum hadrodynamics to include the A-baryon. The response functions of nuclear matter have been evaluated as the imaginary parts of the corresponding particle-hole and A-hole polarization propagators. We use an effective relativistic A-propagator which includes the scalar and vector self-energies due to the coupling to the ~ and w mean fields. We employ a purely magnetic yNA vertex that reproduces the data on photoabsorption on free nucleons except for a background contribution which is not included in the present investigation. These results for nuclear matter have then been used for finite nuclei in local-density approximation. The strong dependence of the results on the coupling constants of the ,~-baryon to the scalar and vector fields has been demonstrated. The calculated cross sections generally underestimate the data. For universal coupling of the ~ the calculated cross section for quasielastic electron scattering on ~2C and photoabsorption on 2°sPb is only about 60% of the observed cross section at the maximum, while the peak position agrees quite well with the data. The discrepancy for the strength is somewhat smaller for 4°Ca. With no vector coupling and small scalar coupling the response is larger and the peak position agrees as well as with universal coupling. While a reliable model of the background contributions would be necessary to decide between these possibilities, the data on the peak position require very similar scalar and vector couplings: the difference between the ratios r~ and rv of the scalar and vector couplings of the a and the nucleon must be positive and can not be bigger than 0.2. For example, the values of rv = 1.0 and rs = 1.35 which would give rise to a A-condensate at twice the saturation density 6) are incompatible with these data.
K. Wehrberger et al. / Electromagnetic excitation
817
T h e r e a r e several p o t e n t i a l l y i m p o r t a n t effects w h i c h have n o t yet b e e n i n c l u d e d . First, we h a v e u s e d the free w i d t h o f the A also in nuclei. This a p p r o x i m a t i o n m a y be j u s t i f i e d b e c a u s e the P a u l i b l o c k i n g a n d s p r e a d i n g t e n d to cancel each other, b u t this can b e verified o n l y b y d e t a i l e d c a l c u l a t i o n . W e have not i n c l u d e d a n y b a c k g r o u n d c o n t r i b u t i o n s . T h e r e are several sources: n o n - r e s o n a n t p i o n p r o d u c t i o n , p i o n - e x c h a n g e currents, t w o - n u c l e o n k n o c k o u t a n d a tail f r o m the e x c i t a t i o n o f h i g h e r r e s o n a n c e s . All these c o n t r i b u t i o n s i n c r e a s e with i n c r e a s i n g e n e r g y t r a n s f e r a n d can b e i m p o r t a n t in the r e g i o n o f the A - p e a k . F i n a l l y , we have not i n c l u d e d a n y c o r r e l a t i o n s . In the case o f the n u c l e o n p e a k c o r r e l a t i o n s d u e to tr-, to- a n d p - m e s o n s are k n o w n to b e i m p o r t a n t . I n the case o f the A - p e a k , h o w e v e r , there are n o tr a n d to c o r r e l a t i o n s , a n d also the f i r s t - o r d e r p i o n i c c o n t r i b u t i o n vanishes for e l e c t r o m a g n e t i c excitation. T h e r e r e m a i n h i g h e r - o r d e r p i o n i c c o r r e l a t i o n s a n d correl a t i o n s d u e to the p m e s o n . W h i l e a careful a n a l y s i s o f all t h o s e c o n t r i b u t i o n s is n e e d e d to extract the A - c o u p l i n g c o n s t a n t s with g o o d p r e c i s i o n , t h e y are s m o o t h f u n c t i o n s o f the e n e r g y t r a n s f e r a n d t h e r e f o r e not e x p e c t e d to c h a n g e the p o s i t i o n o f the A - p e a k a n d o u r c o n c l u s i o n a b o u t the difference o f the scalar a n d v e c t o r c o u p l i n g s o f the A - b a r y o n . This w o r k was s u p p o r t e d b y G S I D a r m s t a d t . O n e o f the a u t h o r s ( K . W . ) a c k n o w l edges also s u p p o r t f r o m the D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t a n d the I n d i a n a U n i v e r s i t y N u c l e a r T h e o r y Center. We t h a n k B. Serot a n d R. W i t t m a n for useful discussions.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1 H. Kurasawa and T. Suzuki, Nucl. Phys. A445 (1985) 685 K. Wehrberger and F. Beck, Phys. Rev. C37 (1988) 1148 K. Wehrberger and F. Beck, Nucl. Phys. A491 (1989) 587 C.J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. 62 (1989) 391 J. Boguta, Phys. Lett. B109 (1982) 251 D. Lurie, Particles and fields (Interscience, New York, 1968) R.D, Peccei, Phys. Rev. 176 (1968) 1812 M.A.B. Beg, B.W. Lee and A. Pais, Phys. Rev. Lett. 13 (1964) 514 C.J. Horowitz and B.D. Serot, Nucl. Phys. A368 (1981) 503 H.F. Jones and M.D. Scadron, Ann. of Phys. 81 (1973) 1 J.D. Jackson, Nuovo Cim. 34 (1964) 1644 K. B~itzner et al., Phys. Lett. B39 (1972) 575 J.H. Koch and N. Othsuka, Nucl. Phys. A435 (1985) 765 E.J. Moniz and A. Sevgen, Phys. Rev. C24 (1981) 224 P. Barreau et al., Nucl. Phys. A402 (1983) 515 D.T. Baran et al., Phys. Rev. Lett. 61 (1988) 400 Z.E. Meziani et al., Phys. Rev. Lett. 54 (1985) 1233 P. Carlos et al., Nucl. Phys. A431 (1984) 573
ELECTROMAGNETIC EXCITATION OF THE DELTA-BARYON QUANTUM HADRODYNAMICS
IN
K. WEHRBERGER Nuclear Theory Center, Indiana University, Bloomington, IN47405, USA
C. BEDAU and F. BECK lnstitut fiir Kernphysik, Technische Hochschule Darmstadt D-6100 Darmstadt, Fed. Rep. Germany
Received 15 June 1989 Abstract: In photoabsorption and electron scattering at large energy transfer, the photon can excite a nucleon to a A-baryon. This gives rise to the A-peak in the spectrum. We investigate the cross section for this process in the framework of quantum hadrodynamics. The results on the A-peak depend on the coupling constants of the A to the scalar and vector mesons. We compare results in Hartree approximation with the data. I. Introduction Q u a s i e l a s t i c e l e c t r o n s c a t t e r i n g p r o v i d e s an i m p o r t a n t test for theories o f n u c l e a r structure. B e c a u s e o f the large e n e r g y a n d m o m e n t u m transfers i n v o l v e d relativistic t h e o r i e s are e s p e c i a l l y well s u i t e d for such investigations. T h e y p r o v i d e a c o v a r i a n t f r a m e w o r k o f n u c l e o n s i n t e r a c t i n g via d y n a m i c a l m e s o n e x c h a n g e a n d thus constitute a n effective field t h e o r y for the n u c l e a r m a n y - b o d y p r o b l e m . Q u a n t u m h a d r o d y n a m i c s 1) has r e c e n t l y b e e n u s e d b y a n u m b e r o f a u t h o r s 2-5) to c a l c u l a t e the s e p a r a t e d l o n g i t u d i n a l a n d t r a n s v e r s e r e s p o n s e functions. It was f o u n d t h a t R P A c o r r e l a t i o n s are i m p o r t a n t in o r d e r to i m p r o v e the a g r e e m e n t o f the l o n g i t u d i n a l r e s p o n s e with the d a t a , b u t leave the t r a n s v e r s e r e s p o n s e a l m o s t unaffected. W h i l e the overall a g r e e m e n t with the d a t a is fairly s a t i s f a c t o r y in the l o n g i t u d i n a l case, the s i t u a t i o n for the t r a n s v e r s e r e s p o n s e is u n c l e a r b e c a u s e o f the o v e r l a p p i n g A_peak; at large e n e r g y t r a n s f e r the virtual p h o t o n can excite a n u c l e o n to a za-baryon, a n d this has n o t b e e n i n c l u d e d in p r e v i o u s c a l c u l a t i o n s . The resulting A . p e a k is also a p r o m i n e n t f e a t u r e o f the d a t a o n p h o t o a b s o r p t i o n , i.e. in the limiting case o f real p h o t o n s . It is the a i m o f the p r e s e n t w o r k to use the e x p e r i m e n t a l i n f o r m a t i o n a v a i l a b l e on t h e s e p r o c e s s e s to o b t a i n i n f o r m a t i o n o n the d y n a m i c s o f the A in the nucleus. Specifically, we i n c o r p o r a t e the za as an effective p a r t i c l e in m e a n - f i e l d a p p r o x i m a t i o n to q u a n t u m h a d r o d y n a m i c s with effective c o u p l i n g s to the s c a l a r a n d v e c t o r m e a n fields, a n d we s h o w t h a t t h e d a t a on e l e c t r o m a g n e t i c e x c i t a t i o n o f the A p r o v i d e i n f o r m a t i o n on these c o u p l i n g s . This i n f o r m a t i o n is very i m p o r t a n t , since the influence o f the za-baryon on the e q u a t i o n o f state o f n u c l e a r m a t t e r d e p e n d s very m u c h o n t h e s e u n k n o w n c o u p l i n g c o n s t a n t s 6). 0375-9474/89/$03.50 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
K. Wehrberger et aL / Electromagnetic excitation
798
In sect. 2 we introduce the mean-field lagrangian with the A and we briefly review the role of the A for the equation of state of nuclear matter. In sect. 3 we evaluate the A-hole polarization propagator and discuss its relation to the response functions and cross sections for electron scattering and photoabsorption. The longitudinal and transverse cross sections for electroexcitation of free nucleons as evaluated in our relativistic approach are compared with the data in sect. 4. This serves as a check of our input. Results for the response functions of nuclear matter are given in sect. 5. We discuss the influence of the finite width of the A, of the Fermi motion of the nucleons, and of the choice of the A coupling constants. Results for finite nuclei, obtained in local-density approximation, are presented and compared with data and with results obtained in the nonrelativistic A-hole model in sect. 6. Sect. 7 contains our conclusions.
2. Role of the delta in mean-field theory When applying quantum hadrodynamics at large nuclear densities or medium to high energy transfers the possibility of the excitation of nucleon resonances has also be considered. The resonance lowest in mass is the spin-3-isospin -3 delta baryon at 1232 MeV. To incorporate the A in quantum hadrodynamics one has to extend the lagrangian density by a kinetic7'8) and an interaction term for the RaritaSchwinger spinor ~ a with mass Ma :
with
~a=~a"(iYA~A--M'a)~a+~a"%,(iYa~x - iq~a~'(3, ~
+ 3,~,) ~,
+ M a ) Y ' v~va
(2)
=O, + ig~v V~,,
(3)
M'a = Ma - ga da ,
(4)
~
where we have denoted the coupling constants of the A to the o-- and w-mesons with mean fields ~b and V~, by g a and go. a This lagrangian is meant to be used in mean-field approximation only, it is not renormalizable and the A is treated as a stable particle. Since the a is, however, a pion-nucleon resonance, a more fundamental approach would be to start from a renormalizable lagrangian for nucleons and pions and to generate the A dynamically. For our purposes the above mean-field lagrangian provides a convenient way of approximately taking into account this resonance as an effective particle. The couplings of the A to the ~r and o~ mean fields, which could in principle be derived in a more fundamental approach, are here to be considered as free parameters.
K. Wehrberger et al. / Electromagnetic excitation
799
Eq. (2) is a special case of a whole set of lagrangians which all yield the same Rarita-Schwinger field equations
(iya~ x - M ~ ) ~ , = O, ~
=0,
y~A =0.
(5)
Correspondingly, the noninteracting spin- 3 propagator contains a free parameter. For the special lagrangian density (1) the propagator is given by the commonly adopted form 8), qXyx+Maf ~ l v S"~-M~-~ ~g -3Y~'Y
2 q~q~ 3 M~
1
~_.~__~(q Y _q~y~)
}
(6)
Due to the simple structure of the chosen mean field interaction, the propagator S~ ~ in Hartree approximation is obtained by the replacements Ma -) M~ = Ma - g ~ b ,
k,~ ~ k'~ = k~, - 6,~og~ Vo,
(7) (8)
as long as no real A's are present in t h e ground state. Little is known about the couplings gsa and gva. From SU(6) symmetry the vector coupling is expected to be similar to the corresponding coupling gvN of the nucleon 9). A restriction on the couplings is obtained if one demands that no real A's are present in the ground state of nuclear matter at saturation density k N and zero temperature, which means that the equation ~2
N 2
N
x/MN + ( k F ) + g v
Vo=~/M'f+(kaF)2+gv A Vo
(9)
does not have a real solution for the Fermi m o m e n t u m kFa of the d. With the Hartree values for the nucleon couplings and meson fields 1o) one obtains rs ~<0.82rv + 0.71,
(10)
with A
gv
rv=--gvN,
g~
rs=--gN.
(11)
This is only a weak condition. I f for example rv = 1.0,
rs = 1.35,
(12)
a A-isomer state appears at about twice saturation density 6 ) , while nothing spectacular is predicted to h a p p e n for universal coupling (rs = rv = 1.0). Therefore information on these couplings from a different source would be very important.
800
K. Wehrberger et al. / Electromagnetic excitation
3. Delta-hole polarization propagator The ~/NA transition amplitude can generally be written in the form ~1) (/tlj, lN)-- q,'~(k)F~,,(q)qt(p).
(13)
Here k is the outgoing momentum of the A, p is the incoming momentum of the nucleon, q = k - p , and k~ is a nucleon spinor. This transition is predominantly a magnetic dipole, and the corresponding vertex function is, including an isospin factor, given by n) - ( M a + MN) x/~ GM(q:) C ~ -- MN((Ma + MN)2 _ q2)
x {(-q~g~ + qAg~,)Ma3,x~/5 + (q~k~ - q~k~gm,)Vs},
(14)
with the M1 form factor
q2 GM(q 2) = GM(0) (1 \
"~-2 //
0.71~eV2 )
kl
q2
~-1/2
3.5~-eV2]
GM(0) = 3.0.
,
(15) (16)
Here the second factor is the dipole fit to the electromagnetic form factor of the nucleon, and the third factor fits the small deviation of the yNA transition form factor from this dipole form. The central quantity determining the response of a system to a weak external perturbation is the corresponding polarization propagator. In Hartree approximation the total A-hole polarization propagator is given by the sum of the y(N, A)y and y(A, N ) y polarization propagators:
I - l ~ = - i f ~ d4 T rp( F ~ ( - q , - p ) S ~ ( p ) F ~ ( q , p ) G H ( p - q ) ) + ( q ~ - q ) .
(17)
Here GH is the density-dependent part of the nucleon propagator in Hartree approximation: ilr G . ( p ) = (3~'p * + M*) ~.. 8(p* - E*)O(kv-Ipl)
(18)
p~g e- p ~ - 8~,ogvN Vo,
(19)
with
M * = MN -- gN~b,
(20)
Ep~ = ~/p2 -I'-M~N2 ,
(21)
801
K . W e h r b e r g e r et al. / E l e c t r o m a g n e t i c e x c i t a t i o n
and S . is the A-propagator in Hartree approximation as given in eqs. (6), (7) and (8), assuming that no real A's are present in the ground state. The evaluation of the trace is straightforward, and we obtain -1 f d4p n~'~(q) = 6-~3 E*_q(p,2_M,a2+ie) ~( p*o - q o - E*,_q)O( k F - IP - ql)(M,~ +
~.,~) (22)
+(q~ -q), with
M., = ( M , a M . _ ( p , k . ) ) {p.p,. (q2 (qp,___))2,~M, ]a +(qp)Z(g~.-P:P2~4 \ Ma ] 2(qp)(qp') ~ PiP;-2(qp)q+.p. ) +--~'a (2M'aM* - (p'k*) ) t × {_q2p, p,. _ (qp)(qp,)g~.. + (qp,)q.p. + (qp)q,..~.}
+Ma
M'a
M2
{(qp')p,. - (qp)p'}{(qk*)p'~ - (qp')k*} - ~
(p'k*)
t 2 ¢ l x{(qp ¢) 2 g~.~-Z(qp t )qwt.+q p~.p.}
+ MaM'a{qZk*p. + (qp)(qk*)g~. - (qk*)q~p. - (qp)q.k*} + m a ( M a m N + (p'k*)){qZg~,. - q~q~} + MZ{(qp')q~k * + (qk*)p'~q~, - q zp~.~ , ,_. - (qp')(qk*)g~,~},
(23)
where k = p - q ,
etc.
(p q ) = p " q~
The longitudinal and transverse response functions of nuclear matter are proportional to the volume and to the imaginary parts of the corresponding A-hole polarization propagators: SL(Iql, to) = --Vim rto~o(Iql, o,), 7/"
sT(Iq[,
to) = 2 V i m 77"
n,~l(Iql, o,)
(24)
(for q in 3-direction). The cross section for quasielastic electron scattering is given in terms of the response functions by trM.,q.4SL+
--
+tan2½0
Sr ,
(25)
with O'M--
442E~ cos 2 ½0 4
(26)
802
K. Wehrberger et al. / Electromagnetic excitation
Here 0 is the scattering angle, Er is the energy of the outgoing electron and a is the fine-structure constant. Alternatively, eq. (25) can be written in terms of the longitudinal and transverse cross sections of the virtual photon: d2o "
dO dEf
o~ - K Er
1 -(Cr-r+ eo'e), 2,rr2 q2 Ei 1 - e
(27)
with K-=to+
e ---
(
q~ 2MN '
1-2
(28)
t a n 2 ½0
)1
(29)
,
q. 27./-2 o'v =
K
(30)
aST ,
O'L= ----~-- a
SL-
(31)
In the limit q~ ~ 0, crT is the total photoabsorption cross section for real photons. In the formalism presented so far the At-baryon is treated as a free particle. It does, however, decay into zrN with a width in free space of F = 115 MeV. A convenient and physically transparent way of introducing the width here is by averaging the response S(/x) obtained for fixed At-mass /z, with respect to tz with a suitable weight function, centered around tz = M ~ [ref. 12)]: S = - -1
f
d/z 2 S(/x)
M'~F . ( M'~2- g2)2 + M'~2r2
(32)
We conclude this section with a nonrelativistic estimate of the position and width of the quasielastic At-peak in nuclear matter. From energy and m o m e n t u m conservation for the excitation of a nucleon with m o m e n t u m k by a virtual photon with energy to and m o m e n t u m q to a At-baryon, we obtain to = ( M '~ - M * ) - ( gvN - g va) V ° +
kq
-~ M 'a
ke
2 M 'a M * (
M' - M*) a
•
(33)
The position of the A-peak depends on g# (via the effective A_mass) and gv~, and the width, being proportional to k F / M ' ~ , depends weakly on the scalar coupling. The width, however, is determined from both Fermi broadening and the A decay width, and the former dependence is therefore even weaker. It is nevertheless evident that the data on the A-peak contain information on the coupling constants of the A to the mean tr- and to-fields.
K. Wehrberger et al. / Electromagnetic excitation 08
I
. . . . qu
I
2
=
. . . .
-0.2
803
I
. . . .
I
.
-GeV z
0.6
"-S
&
0.4
0.2
0.0
i
, I ....
I ....
1.1
1.2 g (GeV)
.
.
.
1.3
1.4
Fig. 1. T r a n s v e r s e c r o s s s e c t i o n f o r e l e c t r o n scattering o n free p r o t o n s at fixed q ~ = - 0 . 2 G e V z as a f u n c t i o n o f c e n t e r - o f - m a s s e n e r g y E. T h e s o l i d line is o u r result, the d a t a are f r o m ref. t3).
4. Eleetroexcitation on free nucleons A basic requirement o f any model o f the A-excitation in nuclei is that it reproduces the measured transverse and longitudinal cross sections for electroexcitation of free nucleons. In the framework o f the formalism presented here, these cross sections can be obtained from eqs. (30) and (31) as the limiting case o f kF-->0 and for vanishing m e s o n fields. Our results are compared with the data from ref. 13) in figs. 1-5. In figs. 1-3 we have plotted the transverse cross section for three fixed values o f qZ as a function o f the c.m. energy E. The cross section is s o m e w h a t underestimated at the maximum. This is due to the fact that we have not included any background contributions. If we add the background contributions as evaluated in ref. 14), the 08
I
. . . . q~
2
I =
. . . .
-0.3
[
'
'-'
'
GeV a
0.6
0.4
0.2
0.0 1.1
1.2
1.3 (CeV)
Fig. 2. S a m e as fig. l , b u t at q ~ = - 0 . 3 G e V 2.
1.4
K. Wehrberger et al. / Electromagnetic excitation
804
0.8
'
'
~
qu
'
2
I
r
-0,4
E
,
,
GeV z
0.6
"S
&
0.4
0.2
0.0
1.1
Fig.
3.
1.2 Z (aeV)
Same
as
fig. 1,
but
at
1.3
1.4
q~ =-0.4GeV
2.
agreement with the data is quite good. In fig. 4 the longitudinal and transverse cross sections are plotted at fixed c.m. energy E = 1.22 GeV, i.e. a p p r o x i m a t e l y at the m a x i m u m o f the peak, as a function o f q~,. 2 Again the absence o f the background is evident. The g o o d a g r e e m e n t o f the q2 d e p e n d e n c e o f our result with the data indicates that the form factor was c h o s e n correctly. Since w e are here mainly interested in the effects o f the coupling constants o f the A to the or- and to-mesons on the d o m i n a n t resonant part o f the response, w e do not explicitly include the background in the present calculation, but w e always have to r e m e m b e r this w h e n w e c o m p a r e with data. 0.8
. . . .
I
. . . .
I
E =
. . . .
1.22
I
GeV
0.6
7
04
& b 0.2
0.0
.
.
i
.
.
i
i
i
I .... 0.2
I , , 0.4
_%2
i
, I , , 0.6
[GeVz]
Fig. 4. Cross section for electron scattering on free protons at fixed c e n t e r - o f - m a s s e n e r g y E = 1.22 G e V as a f u n c t i o n o f q .2. The solid line is our result and the squares are the data f r o m ref. ~3) for the transverse cross section. The l o n g i t u d i n a l cross section is s h o w n by the d a s h e d line and the d i a m o n d s .
805
K. Wehrberger et al. / Electromagnetic excitation 1.50
I
. . . .
I 2
q,~
1.25
=
. . . .
-0.3
/
\ \\
/
\
ii 0.75
\\
//
\\
/
¢
\
/ 0.50
....
-"'G-eV 2
// 1.00
I
\
/
m
\
0/25
0.00
1.1
1,2
1.3
1.4
z (g~v) Fig. 5. Transverse cross section for electron scattering on free protons with (solid line) and without (dashed line) the non-pole terms in the A-propagator. The data are from ref. t3).
The calculated longitudinal cross section s h o w n by the dashed line is very small but n o n z e r o and compatible with the data. In fig. 5 we show the i m p o r t a n c e o f the terms not proportional to g,~ in the A - p r o p a g a t o r (eq. 6). The transverse cross section without these terms is shown by the d a s h e d line. It is m u c h larger and in disagreement with the data. O u r results are o b t a i n e d with the p r o p a g a t o r as given in eq. (6). As already pointed out in sect. 2, the general s p i n ) p r o p a g a t o r contains a free parameter w. Different choices o f w differ by n o n - p o l e terms. In the present investigation we only need the imaginary part o f the polarization p r o p a g a t o r , i.e. the A is on mass shell. In this case the n o n - p o l e terms do not contribute and our results are therefore i n d e p e n d e n t o f the parameter w.
5.
Results
for
nuclear
matter
The response functions d e p e n d on the energy and m o m e n t u m transfer, on the d e c a y width o f the A, the m o m e n t u m distribution o f the target nucleons and the coupling constants o f the nucleons and A's to the m e s o n fields. This is especially transparent for scattering f r o m nuclear matter. Since the electromagnetic transition amplitude for A_excitation is equal for protons and neutrons, we here consider only symmetric nuclear matter. All response functions s h o w n are normalized to N = Z = 20, and they are s h o w n as a function o f energy transfer ~o at fixed m o m e n t u m transfer ]q[. The calculations are only extended up to ~o ~< Iql, since little is k n o w n a b o u t the NA transition form factor for time-like m o m e n t u m transfer. This is not really a restriction in the present context since we will not need this time-like region when discussing p h o t o a b s o r p t i o n and electron scattering.
806
K. Wehrberger et al. / Electromagnetic excitation
In fig. 6 we show the t r a n s v e r s e r e s p o n s e f u n c t i o n o f a free F e r m i gas at s a t u r a t i o n d e n s i t y for different values o f the d e c a y w i d t h o f the zl. T h e r e s p o n s e for zero width, i.e. stable zi, is s h o w n b y the d o t t e d line. In this case the w i d t h o f the r e s p o n s e f u n c t i o n is d e t e r m i n e d o n l y b y the F e r m i m o t i o n o f the n u c l e o n s a n d t h e r e f o r e p r o p o r t i o n a l to the F e r m i m o m e n t u m (see eq. (33)). The r e s p o n s e v a n i s h e s o u t s i d e a w e l l - d e f i n e d k i n e m a t i c a l region. I n c l u d i n g the d e c a y w i d t h o f the A m e a n s a v e r a g i n g with r e s p e c t to the /t-mass as in eq. (32) a n d thus l e a d s to a b r o a d e n i n g , a r e d u c t i o n o f the m a x i m u m , a n d a small shift o f the p e a k position. This is s h o w n for the free A - w i d t h , F = 115 MeV, b y the solid line. The d a s h e d line is o b t a i n e d with h a l f the free width, the d o t - d a s h e d line with d o u b l e the free width. A l t h o u g h the w i d t h o f the A is small c o m p a r e d to its mass, it has a large effect on the r e s p o n s e f u n c t i o n : the relevant scale is the w i d t h o f the r e s p o n s e f r o m F e r m i m o t i o n w h i c h is a p p r o x i m a t e l y the s a m e as the d e c a y width. The d e p e n d e n c e o f the r e s p o n s e on the F e r m i m o m e n t u m at fixed free d e c a y w i d t h is s h o w n in fig. 7. The d o t t e d line is o b t a i n e d in the limit o f zero density, w h e r e the w i d t h o f the r e s p o n s e is d e t e r m i n e d b y the d e c a y w i d t h only. The solid line shows the r e s p o n s e o f a free F e r m i gas at s a t u r a t i o n density. F e r m i m o t i o n gives rise to an a d d i t i o n a l b r o a d e n i n g . F o r h a l f a n d d o u b l e the s a t u r a t i o n d e n s i t y we o b t a i n the d a s h e d a n d d o t - d a s h e d lines. H e n c e f o r t h all results will be s h o w n for the free d e c a y w i d t h a n d at s a t u r a t i o n density. In fig. 8 we c o m p a r e the t r a n s v e r s e r e s p o n s e f r o m n u c l e o n k n o c k o u t a n d from A - e x c i t a t i o n . The d o t t e d lines show the r e s p o n s e o f a free F e r m i gas. D u e to the i n t e r a c t i o n with the s c a l a r a n d v e c t o r m e s o n fields the n u c l e o n s a n d za's a c q u i r e effective m a s s e s a n d energies. The solid lines s h o w the r e s p o n s e o f an interacting
f ....
I~
' ~ ' ' T
....
I. . . .
--
0.15
q = 500 MeV
" f
0.i0
.' / "'
', /
x ' \ , \i",
II
// 0.05
0.00
100
200
[MeV]
300
400
500
Fig. 6. Transverse response function from A-excitation for a free Fermi gas at momentum transfer Iql = 500 MeV as a function of energy transfer to, normalized to N = Z = 20. The dotted line shows the response for a stable A-baryon, and the solid line for the free width F = 115 MeV. The dashed and dot-dashed lines are obtained with half and double the free width.
K. Wehrberger et al. / Electromagnetic excitation ....
I ....
I ....
I ....
807
I ....
0.15
q = 500
MeV ',
0. i 0
,/-\ ,)
0.05
,. \,
.//
, ' / ,'
0.00
I00
200
300
400
500
[MeV]
Fig. 7. Transverse response function from zi-excitation for different values of the Fermi momentum. The dotted line shows the response for free nucleons in the limit of vanishing density, the solid line for a free Fermi gas at saturation density. The dashed and dot-dashed lines are obtained for half and double the saturation density. For all results the free width of the A was used. F e r m i g a s w i t h s t a n d a r d v a l u e s for t h e m e a n f i e l d s a n d n u c l e o n c o u p l i n g c o n s t a n t s lO) a n d t h e a s s u m p t i o n o f u n i v e r s a l c o u p l i n g f o r t h e A. T h e s t r e n g t h is s h i f t e d t o w a r d s l a r g e r e n e r g y t r a n s f e r a n d t h e v a l u e at t h e m a x i m u m
is r e d u c e d . T h i s is d u e to t h e
s m a l l e f f e c t i v e m a s s e s , M * ~ 0 . 5 4 M N a n d M ~ ~ 0 . 6 5 M a , g i v i n g rise to l a r g e r k i n e t i c e n e r g i e s at t h e s a m e t h r e e - m o m e n t u m
transfer. T h e d a s h e d l i n e is o b t a i n e d w i t h
i n t e r a c t i n g , effective, n u c l e o n s b u t free Zl's. T h e m a x i m u m o f t h e r e s p o n s e is s h i f t e d [ ....
I ....
I ....
0.15
q =
500
MeV
Nucleon
DelLa
0.10
/
2"
... ,
/
,
....-
0.05
-.
0
,. ,, ',.
//
,
/
0.00
/
/
. ,
,
f
.//
//
I00
200
300
400
500
Fig. 8. Transverse response function from nucleon knockout and A-excitation. The dotted lines show the response of a free Fermi gas, the solid lines show the response of an interacting Fermi gas, with standard nucleon coupling constants determined to fit ground-state properties and A-coupling constants from the assumption of universal coupling. The dashed line is obtained for an interacting Fermi gas of nucleons and for free deltas. All response functions are evaluated in Hartree approximation.
808
K, Wehrberger et al. / Electromagnetic excitation o.o6 ~-'
' '-'
I '-'
q = 500
~-
0,06
I ....
' ' I ' ' ' ' I ....
MeV
.
0.04
/
/ / /
0.02
/
/
,'
\
.
"~
\
/
"
'~
\
/
" \
/
// //
/
/
,-
\
/
\
/ i
O.O0
lOO
200
300
400
500
[MeV]
Fig. 9. Transverse response from Zl-excitation for universal vector coupling and different values for the scalar coupling of the A. The ratio rs of the scalar coupling of the Zl to the scalar coupling of the nucleon is 0.9 for the dotted line, 1.0 for the solid line, 1.1 for the dot-dashed line and 1.2 for the dashed line. t o w a r d s l a r g e r e n e r g y t r a n s f e r w i t h r e s p e c t t o t h e c a s e o f free n u c l e o n s A's by approximately The dependence
on the scalar and vector coupling
in m o r e d e t a i l in figs. 9 a n d 10, f o r f i x e d n u c l e o n the scalar coupling
and free
the difference of the mean vector and scalar potentials. c o n s t a n t s o f t h e A is s h o w n
coupling
constants. Increasing
ratio ( s e e eq. ( 1 1 ) ) f r o m r s = 1.0 (fig. 9, s o l i d l i n e ) to r s = 1.1
( d o t - d a s h e d l i n e ) d e c r e a s e s t h e e f f e c t i v e m a s s o f t h e A b y 10% o f t h e s c a l a r p o t e n t i a l , i.e. a b o u t 4 0 M e V . T h i s r e s u l t s in a c o r r e s p o n d i n g
0.08
0.06
....
T ....
I. . . .
q = 500
shift of the maximum
I ....
of the
I ....
MeV
J./ /
.
/
'> = ~x
ii
0.04
ii i 0.02 / /
O.O0
/
I /
/
/
.i
/
\
.
,
~.
.'"
/i \ \
."
\
.' . I k
100
200
300
400
i
b
L
i
500
[MeV]
Fig. 10. Transverse Response from A-excitation for universal scalar coupling and different values for the vector coupling of the A The ratio rv of the vector coupling of the za to the vector coupling of the nucleon is 1.1 for the dotted line, 1.0 for the solid line, 0.9 for the dot-dashed line and 0.8 for the dashed line.
809
K. Wehrberger et al. / Electromagnetic excitation 0.04
0.03
'
'1 ....
I ....
I ....
[ ....
q = 500 MeV
0.02
0.01
0.00
100
200
300
400
500
[~eV] Fig. 11. Longitudinal response from nucleon knockout and A_excitation for an interacting nucleon Fermi gas with standard nucleon couplings and with universal coupling for the z~. response function towards smaller energy transfer. The dashed and dotted lines show the results obtained for rs = 1.2 and 0.9. Similarly, an increase o f the vector coupling f r o m rv = 1.0 (fig. 10, solid line) to rv = 1.1 (fig. 10, dotted line) increases the effective energy by 10% o f the vector potential, i.e. about 30 MeV. This increased effective energy shifts the m a x i m u m o f the response towards larger energy transfer. The dashed and d o t - d a s h e d lines show the results obtained for rv = 0.8 and rv = 0.9. F o r completeness we s h o w in fig. 11 also the longitudinal response functions from n u c l e o n k n o c k o u t and A-excitation, for an interacting Fermi gas with standard n u c l e o n a n d universal A-couplings. As expected from the mainly transverse nature o f the interaction, the longitudinal A-response is almost negligible c o m p a r e d to the response from nucleon knockout.
6. Results for finite nuclei and comparison with data The nuclear matter results are not sufficient for a quantitative c o m p a r i s o n with the data on quasielastic electron scattering on nuclei. The simplest way to incorporate finite size effects is the local-density approximation. Here we use local densities and m e s o n fields as determined f r o m the solution o f the mean-field equations o f q u a n t u m h a d r o d y n a m i c s lO). The local response functions are a d d e d to obtain the response o f the nucleus. This is k n o w n to be a g o o d a p p r o x i m a t i o n for the quasielastic n u c l e o n p e a k 3). The main difference between the results obtained with the exact o n e - b o d y density and in local-density a p p r o x i m a t i o n is a shift towards larger energy transfer by the average separation energy o f about 10 MeV. Since there is no reason to expect this to be different for the A-peak, the local-density a p p r o x i m a t i o n is sufficient for the present investigation.
810
K. Wehrberger et aL / Electromagnetic excitation
In nuclei the w i d t h o f the A will be m o d i f i e d d u e to Pauli b l o c k i n g a n d the s p r e a d i n g w i d t h ,5). Pauli b l o c k i n g has a s t r o n g d e n s i t y a n d m o m e n t u m d e p e n d e n c e : it r e d u c e s the w i d t h to zero at s a t u r a t i o n d e n s i t y a n d very s m a l l m o m e n t u m o f the A, a n d has a l m o s t no effect on the w i d t h at very large m o m e n t u m . The s p r e a d i n g w i d t h , on the o t h e r h a n d , is n e a r l y m o m e n t u m i n d e p e n d e n t . A d y n a m i c a l i n c o r p o r a t i o n o f b o t h o f these effects w h i c h we e x p e c t to d e p e n d also on the c o u p l i n g s o f the A to the o-- a n d to-fields w o u l d t h e r e f o r e be d e s i r a b l e . F o r this first survey, however, we will a s s u m e t h a t the i n t e r p l a y o f b o t h effects leaves the free width on the average u n m o d i f i e d , a n d we will t h e r e f o r e use a m o m e n t u m a n d d e n s i t y i n d e p e n d e n t w i d t h o f 115 M e V everywhere. This a s s u m p t i o n is s u p p o r t e d by the fact t h a t o u r results r e p r o d u c e the w i d t h o f the d a t a well, b u t we s h o u l d k e e p in m i n d t h a t a m o d i f i c a t i o n o f the a v e r a g e w i d t h by 10% w o u l d c h a n g e the r e s p o n s e at the m a x i m u m b y a b o u t 10%. In fig. 12 we show the cross section for electrons with energy E = 620 M e V at scattering angle 0 = 60 ° as a f u n c t i o n o f e n e r g y t r a n s f e r to. The d a t a are f r o m ref. ,6). T h e d o t - d a s h e d line shows the result f r o m n u c l e o n k n o c k o u t only. Here c o r r e l a t i o n s d u e to o% to- a n d p - m e s o n s are i n c l u d e d in the r a n d o m - p h a s e a p p r o x i m a t i o n s). E x c e p t for the missing strength at large energy t r a n s f e r the d a t a are well r e p r o d u c e d . M o s t o f this missing strength can be p r o v i d e d b y the A - b a r y o n . The d a s h e d line shows the r e s p o n s e f r o m A - e x c i t a t i o n a l o n e in H a r t r e e a p p r o x i m a t i o n a s s u m i n g free n u c l e o n s a n d free A ' s , This result is very s i m i l a r to that o b t a i n e d in a nonrelativistic a p p r o a c h in ref. ,4), if the b a c k g r o u n d c o n t r i b u t i o n as e s t i m a t e d there is t a k e n into account. The cross section is, h o w e v e r , r e d u c e d s u b s t a n t i a l l y if we a s s u m e
le C 4
E = 620 MeV £, ~
I00
200
0
= 600
300
40{3
500
600
~0 (MeV)
Fig. 12. Cross section for electron scattering on '2C at incident electron energy E = 620 MeV and scattering angle 0 = 60° as a function of energy transfer w. The dot-dashed line is the contribution from nucleon knockout only as evaluated in random-phase approximation. The contributions from a-excitation only for universal coupling and for no coupling are shown by the solid and dotted lines. The dashed line shows the i'esult for noninteracting nucleons and al's. The data are from ref. ~6).
K. Wehrberger et al. / Electromagnetic excitation
811
effective n u c l e o n s a n d d ' s with universal c o u p l i n g . This is s h o w n b y the solid line. The s t r e n g t h is shifted t o w a r d s l a r g e r e n e r g y transfer, a n d o n l y a b o u t 60% o f the o b s e r v e d cross section at the m a x i m u m is a c c o u n t e d for. I f we a s s u m e interacting n u c l e o n s b u t free A's we o b t a i n the d o t t e d line. The m a x i m u m at the p e a k is a b o u t the s a m e as with free n u c l e o n s , b u t the p o s i t i o n o f the m a x i m u m is, as in fig. 8, shifted b y a p p r o x i m a t e l y the difference b e t w e e n the average v e c t o r a n d s c a l a r p o t e n t i a l s a n d is in d i s a g r e e m e n t with the data. In fig. 13 we s h o w the sum o f the c o n t r i b u t i o n from n u c l e o n k n o c k o u t a n d d - e x c i t a t i o n for s t a n d a r d n u c l e o n c o u p l i n g s a n d different d - c o u p l i n g s . The solid a n d d o t t e d lines are o b t a i n e d with u n i v e r s a l a n d v a n i s h i n g c o u p l i n g , respectively. T h e d a s h e d a n d d o t - d a s h e d lines s h o w the result for a ratio rs = 0.15 a n d 0.30 o f s c a l a r d - c o u p l i n g to s c a l a r n u c l e o n c o u p l i n g a n d v a n i s h i n g vector coupling. The small v a l u e rs = 0.15 is sufficient to shift the p e a k p o s i t i o n such that it agrees with the data, a n d it leaves the p e a k height a l m o s t u n m o d i f i e d c o m p a r e d with the result for free d. On the o t h e r h a n d , we see that the d a t a on the p o s i t i o n o f the p e a k r e q u i r e t h a t rs c a n n o t e x c e e d rv b y m o r e t h a n a b o u t 0.2. F o r 12C the s e p a r a t e d t r a n s v e r s e a n d l o n g i t u d i n a l cross sections are a v a i l a b l e not o n l y in the r e g i o n o f the n u c l e o n p e a k , b u t also for the A - p e a k . These d a t a from ref. 17) are s h o w n t o g e t h e r with o u r results in figs. 14-16. I n fig. 14 we s h o w the t r a n s v e r s e cross s e c t i o n at q~ = - 0 . 1 G e V 2 as a f u n c t i o n o f e n e r g y transfer. The lines s h o w the c o n t r i b u t i o n s f r o m n u c l e o n k n o c k o u t a n d A - e x c i t a t i o n s e p a r a t e l y a n d c o r r e s p o n d to t h o s e s h o w n in fig. 12. A g a i n the missing strength in the A - r e g i o n is a p p a r e n t . In fig. 15 we s h o w the sum o f the n u c l e o n a n d A - c o n t r i b u t i o n s , a n d the
12C
4
E = 620 MeV
~,~II .,~
"
0
100
o
=
'
200
6o ° ~r
300
(Mev)
I.L ±Z~TT
400
500
600
Fig. 13. Cross section for electron scattering on '2C at incident electron energy E =620 MeVand scattering angle 0 = 60° as a function of energy transfer to for standard nucleon and different zl-couplings. The lines are the results for the sum of the contribution from nucleon knockout and d-excitation. The dotted line shows the cross section for free A's, and the dashed and dot-dashed lines for no coupling to the vector field and a ratio r~= 0.15 and 0.30 of the scalar coupling of the A to the scalar coupling of the nucleon. The solid line is obtained for universal coupling. The data are from ref. t6).
812
K. Wehrberger et al. / Electromagnetic excitation . . . .
i
. . . .
I
. . . .
I
lZC,
j
. . . .
qu z
I
. . . .
I
CeV z
-0.1
=
. . . .
..
?:.: .-...
zoo
z50
300
350 (M~V)
400
450
500
Fig. 14. Transverse cross section for electron scattering on '2C at fixed q~ = -0.1 GeV 2 as a function of energy transfer to. As in fig. 12, the dot-dashed line is the contribution from nucleon knockout only, and the contributions from A-excitation alone for universal coupling and for no coupling are shown by the solid and dotted lines. The dashed line shows the result for noninteracting nucleons and A's. The data are from ref. IT).
12C,
200
I
L
250
300
t
i
,
ku z = - 0 . i GeV2
i
i 350
h
,
i
L 400
L
,
,
, 450
500
(~ev) 2 __ Fig. 15. Transverse cross section for electron scattering o n L 2 C at fixed q~, - -0.1 GeV 2 as a function of energy transfer w. As in fig. 13, the lines represent the sum of the contributions from nucleon knockout and A-excitation. The dotted line shows the cross section for free A's, and the dashed and dot-dashed lines for no coupling to the vector field and a ratio rs = 0.15 and 0.30 of the scalar coupling of the A to the scalar coupling of the nucleon. The solid line is obtained for universal coupling. The data are from ref. ,~).
813
K. Wehrberger et aL / Electromagnetic excitation
I .... 7-' ' '~- .... 12C,
6
[ ....
F ~
q2 = -0.1
~-~ GeV 2
4'
S
-2
200
250
300
350
400
450
500
Fig. 16. Longitudinal cross section for electron scattering on 12C at fixed q~ = -0.1 GeV2 as a function of energy transfer to. The dot-dashed line shows the result from nucleon knockout only, the solid line shows the result from A-excitation. The data are from ref. 17).
l i n e s c o r r e s p o n d to t h o s e in fig. 13. As s h o w n in fig. 16, t h e m e a s u r e d l o n g i t u d i n a l c r o s s s e c t i o n is c o m p a t i b l e w i t h z e r o , a n d o u r r e s u l t f o r t h e l o n g i t u d i n a l c r o s s s e c t i o n is i n d e e d a l m o s t z e r o e x c e p t f o r t h e tail f r o m t h e l o n g i t u d i n a l n u c l e o n p e a k w h i c h decreases rapidly with increasing energy transfer. T h e e s s e n t i a l l y t r a n s v e r s e c r o s s s e c t i o n f o r e l e c t r o m a g n e t i c e x c i t a t i o n o f t h e d is s u p r e s s e d at s m a l l e r s c a t t e r i n g angles. I n fig. 17 w e s h o w t h e cross s e c t i o n f o r 12C at i n c i d e n t e n e r g y E = 680 M e V a n d s c a t t e r i n g a n g l e 0 = 36 ° w i t h d a t a f r o m ref. ~6). H e r e t h e A - p e a k is i n d e e d m u c h s m a l l e r t h a n t h e n u c l e o n p e a k . As in fig. 13, t h e
12 C
3o
t
E = 680 MeV
/~ t t:~ % '~
,
0
= 360
\,
20
m
o
....
I ~ I tO0
.... ,800
I ....
300 co (MeV)
I .... 400
Fr, 500
, , 600
Fig. 17. Same as fig. 13, but for incident electron energy E =680 MeV and scattering angle 0 =36 °. The data are from ref. ,6).
K. Wehrberger et al. / Electromagnetic excitation
814
solid line shows the result obtained for standard nucleon coupling and universal A-coupling. The dotted, dashed and dot-dashed lines are for vanishing a vector coupling and rs = 0.0, 0.15 and 0.3. As in the previous results for a scattering angle of 60 ° the data are underestimated. Data for the electron scattering cross section in the region of the A-peak are available also for 4°Ca [ref. ,8)]. These are shown for E = 695 MeV and 0 = 60 ° in fig. 18. The lines have the same meaning as in figs. 13 and 17. Again the result for interacting nucleons and universal A-coupling (solid line) underestimates the data. The results for small scalar coupling and no vector coupling overestimate the data in the dip region although no background is included. For free za's we obtain the dotted line which at large energy transfer leaves no room for the background contribution which is expected to increase with increasing energy transfer. In figs. 19-21 we show the cross section per nucleon for photoabsorption on 2°8Pb with data from ref. ,9). In fig. 19 the dot-dashed line shows the cross section for free nucleons in the limit of vanishing density. As in fig. 12, the dashed line is the result for finite density as obtained from the solution of the mean field equations, but assuming noninteracting nucleons and A's in the evaluation of the response. The solid line shows the result for effective nucleons and effective A's with universal coupling, the dotted line for effective nucleons and free a ' s . The reduction of the response due to the small effective mass of the A in the case of universal coupling is similar as for the quasielastic electron scattering cross section. In fig. 20 the solid and dotted lines are the results for universal and vanishing coupling, respectively, and the dashed and dot-dashed lines show the results for rv = 0.15 and 0.3. Again comparatively good agreement with the data is obtained with vanishing vector and small scalar coupling rs = 0.15 of the A and rv = 0.0 and rs = 0.3 is incompatible with
4OCa 8
"
~.i~qu.
+,
7
I 2
E -
6 9 5 MeV
e =
60°
~\
/
~-t
5[ ,, ",
++,
\
,
~
\
+. ++. 0
100
20{3
300 co (MeV)
400
500
600
Fig. 18. S a m e as fig. 13, but for electron scattering on 40 C a at i n c i d e n t electron energy E = 695 MeV a n d s c a t t e r i n g angle O = 60 °. The d a t a are from ref. ts).
K. Wehrberger et al. / Electromagnetic excitation 0e
'
'
~-]
....
I
~
'
815
[ ....
0.5
0.4
0.3
0.2
0.1
0.0
ZOO
100
300 E (~v)
500
400
Fig. 19. Photoabsorption cross section per nucleon of 2°spb as a function of the photon energy E. The dot-dashed line shows the cross section for photoabsorption on free nucleons, the other lines are results for 2°spb obtained in local-density approximation. As in fig. 12, the results for free nucleons and free A's is shown by the dashed line, for effective nucleons and free /t's by the dotted line, and for effective nucleons and A's with universal coupling by the solid line. The data are from ref. 19).
t h e d a t a , I n fig. 21 w e s h o w t h e r e s u l t s f o r r v (dotted
line) and
peak position
r~= 1.30 (dot-dashed
require
1.0 a n d r~ = 1.0 ( s o l i d l i n e ) , r~ = 1.15
l i n e ) . It c a n b e s e e n t h a t t h e d a t a
the scalar and vector couplings
on the
o f t h e zl t o b e s i m i l a r .
0.6
0.5
0.4
0.3
0.2
0.1
0.0
i
loo
J
i
,
i
,
zoo
,
,
.
.
.
300
.
.
.
.
.
4oo
.
.
500
E (MeV)
Fig. 20. Photoabsorption cross section per nucleon of 2°8pb as a function of the photon energy E, with data from ref. ~9) and our results for standard nucleon couplings and different A-couplings obtained in local-density approximation. The dotted line shows the cross section for free A's, and the dashed and dot-dashed lines for no coupling to the vector field and a ratio r~ = 0.15 and 0.30 of the scalar coupling of the A to the scalar coupling of the nucleon. The solid line is obtained for universal coupling.
816
K. Wehrberger et al. / Electromagnetic excitation 06
-
~
-
~
~
-
~
-
~
-
~
'
I
'
'
'~---
0,5
04
0.3
02
0.1
O0
10(3
200
300
400
500
E (MeV)
Fig. 21. Sameas fig. 20, but for universal vectorcoupling rv= 1.0 and r~= 1.0 (solid line), rs = 1.15 (dotted line) and r~= 1.30 (dot-dashed line). 7.
Conclusions
We have extended previous calculations of the response functions for quasielastic electron scattering in quantum hadrodynamics to include the A-baryon. The response functions of nuclear matter have been evaluated as the imaginary parts of the corresponding particle-hole and A-hole polarization propagators. We use an effective relativistic A-propagator which includes the scalar and vector self-energies due to the coupling to the ~ and w mean fields. We employ a purely magnetic yNA vertex that reproduces the data on photoabsorption on free nucleons except for a background contribution which is not included in the present investigation. These results for nuclear matter have then been used for finite nuclei in local-density approximation. The strong dependence of the results on the coupling constants of the ,~-baryon to the scalar and vector fields has been demonstrated. The calculated cross sections generally underestimate the data. For universal coupling of the ~ the calculated cross section for quasielastic electron scattering on ~2C and photoabsorption on 2°sPb is only about 60% of the observed cross section at the maximum, while the peak position agrees quite well with the data. The discrepancy for the strength is somewhat smaller for 4°Ca. With no vector coupling and small scalar coupling the response is larger and the peak position agrees as well as with universal coupling. While a reliable model of the background contributions would be necessary to decide between these possibilities, the data on the peak position require very similar scalar and vector couplings: the difference between the ratios r~ and rv of the scalar and vector couplings of the a and the nucleon must be positive and can not be bigger than 0.2. For example, the values of rv = 1.0 and rs = 1.35 which would give rise to a A-condensate at twice the saturation density 6) are incompatible with these data.
K. Wehrberger et al. / Electromagnetic excitation
817
T h e r e a r e several p o t e n t i a l l y i m p o r t a n t effects w h i c h have n o t yet b e e n i n c l u d e d . First, we h a v e u s e d the free w i d t h o f the A also in nuclei. This a p p r o x i m a t i o n m a y be j u s t i f i e d b e c a u s e the P a u l i b l o c k i n g a n d s p r e a d i n g t e n d to cancel each other, b u t this can b e verified o n l y b y d e t a i l e d c a l c u l a t i o n . W e have not i n c l u d e d a n y b a c k g r o u n d c o n t r i b u t i o n s . T h e r e are several sources: n o n - r e s o n a n t p i o n p r o d u c t i o n , p i o n - e x c h a n g e currents, t w o - n u c l e o n k n o c k o u t a n d a tail f r o m the e x c i t a t i o n o f h i g h e r r e s o n a n c e s . All these c o n t r i b u t i o n s i n c r e a s e with i n c r e a s i n g e n e r g y t r a n s f e r a n d can b e i m p o r t a n t in the r e g i o n o f the A - p e a k . F i n a l l y , we have not i n c l u d e d a n y c o r r e l a t i o n s . In the case o f the n u c l e o n p e a k c o r r e l a t i o n s d u e to tr-, to- a n d p - m e s o n s are k n o w n to b e i m p o r t a n t . I n the case o f the A - p e a k , h o w e v e r , there are n o tr a n d to c o r r e l a t i o n s , a n d also the f i r s t - o r d e r p i o n i c c o n t r i b u t i o n vanishes for e l e c t r o m a g n e t i c excitation. T h e r e r e m a i n h i g h e r - o r d e r p i o n i c c o r r e l a t i o n s a n d correl a t i o n s d u e to the p m e s o n . W h i l e a careful a n a l y s i s o f all t h o s e c o n t r i b u t i o n s is n e e d e d to extract the A - c o u p l i n g c o n s t a n t s with g o o d p r e c i s i o n , t h e y are s m o o t h f u n c t i o n s o f the e n e r g y t r a n s f e r a n d t h e r e f o r e not e x p e c t e d to c h a n g e the p o s i t i o n o f the A - p e a k a n d o u r c o n c l u s i o n a b o u t the difference o f the scalar a n d v e c t o r c o u p l i n g s o f the A - b a r y o n . This w o r k was s u p p o r t e d b y G S I D a r m s t a d t . O n e o f the a u t h o r s ( K . W . ) a c k n o w l edges also s u p p o r t f r o m the D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t a n d the I n d i a n a U n i v e r s i t y N u c l e a r T h e o r y Center. We t h a n k B. Serot a n d R. W i t t m a n for useful discussions.
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