- Email: [email protected]
Pion electroproduction in the 3He(e,é π+)3H reaction at intermediate energies
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A 626 (1997) 871-885
Pion electroproduction in the 3He(e,e Tr+)3H reaction at intermediate energies K.I. Blomqvist a, W.U. Boeglin a, R. Bthm a, M. Distler a, R. Edelhoff a, J. Friedrich a, R. Geiges a, M. Kahrau a, S. Kamalov b, M. Kirchbach a, M. Kohl c, M. Korn a, H. Kramer a, K.W. Krygier a, V. Kunde a, M. Kuss c'1, J.M. Laget d, A. Liesenfeld a, K. Merle a, R. Neuhausen a, E.A.J.M. Offermann a, Th. Pospischil a, M. Potokar e, C. Rangacharyulu f, A. Richter c'2, A.W. Richter a, A. Rokavec e, G. Rosner a, P. Sauer a, St. Schardt a, G. Schrieder c, T. Sudag, L. Tiator a, B. Vodenik e, A. Wagner a, Th. Walcher a, St. Wolf a a Institutfiir Kernphysik, Universitgit Mainz, D-55099 Mainz, Germany b Laboratory of Theoretical Physics, JINR Dubna, 141980 Dubna, Moscow region, Russia c lnstitutfiir Kernphysik, Technische Universitit't Darmstadt, D-64289 Darmstadt, Germany d CEA Saclay, DAPNIA/SPhN, F-91191 Gif-sur-Yvette Cedex, France e Institute "Jo~efStefan", University of Ljubljana, SI-61111 Ljubljana, Slovenia f Department of Physics, University of Saskatchewan, Saskatoon, SK, S7N 5E2, Canada g Department of Physics, Tohoku Universi~, Kawauchi, Sendai, 980, Japan
Received 12 June 1997; accepted 7 August 1997
Abstract
The differential coincident pion electroproduction cross section in the 3He(e,e'zr+ )3H reaction has been measured with high resolution with the three-spectrometer set-up at the Mainz Microtron (MAMI) electron accelerator. Measurements were performed at the four incident energies Eo = 855, 675, 600, and 555 MeV at fixed four-momentum transfer Q2 = 0.045 GeV2, with the pions detected in parallel kinematics. This enables a separation of the measured cross section into the transverse and longitudinal structure functions by means of the Rosenbluth method. The experimental data are compared to model calculations, in which the elementary pion production amplitude includes the standard Born-amplitudes and also delta and higher resonance terms. Threebody Fadeev wave functions are used and the final state interaction of the outgoing pion is taken into account. The calculation describes the data only after medium modifications of the delta and of the pion are included. (~) 1997 Elsevier Science B.V. PACS: 21.45.+v; 25.10.+s; 25.30.Rw; 27.10.+h Keywords: Coherent pion electroproduction;Longitudinal-transverseseparation; 3He; Mediumeffects; Delta
resonance region
0375-9474/97/$17.00 (~ 1997 Elsevier Science B.V. All rights reserved. PII S0375-9474(97) 00463-6
872
K.L Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
1. Introduction In intermediate-energy nuclear physics research, experiments involving pion production by electromagnetic probes are of particular interest. As direct measures of pion fields in nuclei and of the contribution of nucleonic excitations, these channels provide a good handle on nuclear dynamics. While measurements with real photons probe only the transverse structure of the hadronic system, the electroproduction process also probes longitudinal components. Separate access to these components is achieved by an appropriate choice of kinematical settings. This allows a detailed study of of the dynamics. Recently, with the advent of modern high duty-factor, high resolution intermediate energy electron accelerator facilities such as the Mainz Microtron (MAMI), such experiments under well defined exclusive kinematical conditions have become possible. As part of the scientific program of the Al-collaboration at MAMI, studies of the 3He(e,e'Tr±) reactions were initiated. The motivation for this work has been many-fold. An important impetus was to learn about possible delta isobar components in the 3He ground state from the comparison of the 7r+- to the ~--production in the break-up channels [ 1 ]. With the excellent missing mass resolution at the MAMI three-spectrometer set-up, the two-body final state (t~ +) could be clearly separated from the three- and four-body (dn~-+, pnn~r +) final states. This allowed to examine in a kinematically complete measurement the t~+-channel, which is of particular theoretical interest. It is the aim of the present paper to report on the results of the 3He(e,e'Tr+)3H reaction. With energy transfers in the A-resonance region, we determined the longitudinal and transverse production cross sections at fixed four-momentum transfer by the Rosenbluth method. The mass-3 system continues to be a subject of intensive physical interest. From the theoretical side [2], this system is amenable to detailed Faddeev type calculations, and it is of interest to test by experiments [ 3,4] corresponding calculations for 3He-3H, a pair of mirror nuclei very similar to the p-n system, in the spirit of the elementary particle approach [5]. This question has received renewed attention, as the forward-angle pion electroproduction measurements of Gilman et al. [6] with a deuterium target argued for a quenching of the production cross section on the nucleus compared to that on the proton. Section 2 of this paper describes the kinematics of the present experiment. Section 3 describes the spectrometer set-up and details of the cryogenic gas target used for this experiment. The data reduction procedures and the results of the measurements are presented in Section 4. In Section 5, a theoretical model based on the elementary particle approach [7] is described. The results of the model calculations are discussed and compared to the experimental data in Section 6.
I Present address: Jefferson Lab., 12000 Jefferson Ave., Newport News, VA 23606, USA. 2 Corresponding author. E-maih [email protected].
K.L
Blomqvistet al./Nuclear PhysicsA 626 (1997) 871-885
reaction plant ~
873
~
Fig. 1. Kinematicsof the (e,e'~r) reaction. The variables in the scattering and reaction planes are indicated. See text for details. 2. Kinematics In this section, as well as in Sections 3 and 4, all physical quantities are given in the laboratory system without referring to this system explicitly by some label. Pion electroproduction off a target nucleus can be visualized as in Fig. 1. The direction of the scattered electron and of the virtual photon define the scattering plane; their angles with respect to the incoming electron are denoted by 0e and 0~,, respectively. The pion direction is specified by its angle with respect to the virtual photon, 0~., and by the out-of-plane angle ~b~-. The threefold differential cross section can be written as (see Refs. [7-10] for details) d3o_ _ / "do-v d~2e, dEe, df2~r d~2~r '
la)
with do-v _ drrT dirE dO~r dO~ + e - - + [ 2 e ( l + e ) ] l / 2 c ° s 6 ~ r - ~
dO'LT
dOrTT d--~
lb)
Here, e and F denote the polarization and flux of the virtual photon of energy tot and three-momentum k~,
I
-Ik~'2
e= Ll+2--0T-tan
2 ~ ] -I ~ ,
O' Ee' kL 1 F = 2 7 r 2 Ee Q 2 1 _ e
(2)
Here, kL denotes the "photon equivalent energy" in Hand's convention [ I 1 ], related via kL = (SA -- M2A)/2MA to the Mandelstam variable SA = _Q2 + M~A + 2mTMa, where MA stands for the mass of the target nucleus. The positive quantity Q2 is introduced as Q2 = k~,2_ to~.2 The first two terms in Eq. ( l b ) are the transverse (T) and longitudinal (L) cross sections. These terms are independent of the azimuthal angle ~b~.. The third and fourth term, respectively, describe the longitudinal-transverse (LT) and transversetransverse (TT) interference structure (or response) functions. They vanish in parallel
874
K.L Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
Table 1 Kinematical settings of the 3He(e,e~rr+)3H experiment. The four-momentum transfer squared Q2 was 0.045 (GeV/c) 2 for all settings. The notation is: E0 = incident electron energy, a~r = energy transfer, 0c = electron scattering angle, • = virtual photon polarization, 0rr = pion detection angle relative to the electron beam &~ (MeV)
w:, (MeV)
0c
•
07r
855 675 600 555
368-430 368-425 395-432 363-398
19.5° 28.4° 36.4° 40.8°
0.79 0.63 0.48 0.47
19.6° 16.8° 13.8° 14.8°
kinematics (0~, = 0 °) since they appear proportional to sin 0~, and sin 2 0~, respectively. The emphasis of the measurement was to determine the longitudinal and transverse production cross sections in the delta resonance region. To this end, the pion was measured in parallel kinematics at fixed Q2 for different angle-energy-combinations for the electron in order to allow a Rosenbluth separation of the cross section. The kinematical settings are listed in Table 1.
3. Experimental details The measurements were carried out at the high-resolution spectrometer set-up of the A1 collaboration at M A M I . The facility is already described elsewhere [12,13]. The experiment was performed with the cryogenic 3He gas target used earlier in electron scattering experiments at Saclay [ 14]. Fig. 2 shows a cross section of the target. The target cell is made o f a stainless steel foil of about 8 0 / z m thickness and it has a diameter of 80 ram. The helium gas is circulated through the closed loop with the help of a fan, below which the helium has thermal contact with liquid hydrogen in a heat exchanger produced by about 50 copper grids soldered into the passing tube. The electron beam deposits a power o f about 20 W in the target cell for incident beam currents of 3 0 / , A , typically used in the measurements. The gas temperature is monitored by two carbon resistors (Tc) located about 4 cm below and above the beam position. The temperature of the loop is measured by a silicon resistor (Tsi) attached to a copper block in the lower part o f the loop. The target was run at a pressure of 15 bar and a temperature o f 20 K, corresponding to an effective thickness of about 160 m g / c m 2. To avoid excessive heating of the target cell window by the narrow electron beam spot ( F W H M ~ 0.16 m m ) , the beam was wobbled over an area of 7 mm × 7 mm. This was achieved by a system of magnetic coils located about 8 m upstream o f the target chamber [ 15]. The current of the coils was digitized and recorded in the data stream, enabling event-by-event reconstruction of the beam position and thus a correction for the wobbling of the beam in the off-line analysis. The scattered electrons and the produced charged pions were detected in coincidence by two magnetic spectrometers, equipped with a set of four vertical drift chambers
K.L Blomqvist et aL /Nuclear Physics A 626 (1997) 871-885
875
"~He
J fan
target cell
heat exchanger
I
,
•
I
i0 cm
Fig. 2. Side view of the 3He target. The main componentsare the target cell (right) and the heat exchanger (left) which is filled with liquid hydrogen as the coolant for the helium gas. The system has two closed loops, one for the coolant and one for the helium. lot particle tracking and with two planes of segmented plastic scintillators and a gas Cherenkov detector for timing and particle identification. A logic combination of the signals from the scintillator and from the Cherenkov generated the trigger. The Cherenkov veto suppressed the positron background in the hadron spectrometer. Pions were distinguished from protons by their different energy losses in the scintillators. Problems associated with large singles rates ( ~ 1 5 0 kHz protons) were overcome by fast hardware vetos using signals from the scintillators and from the Cherenkov detectors. Detailed particle identification was done in the offline analysis. The singles rates from the electron spectrometer were scaled down and recorded along with the double coincidence events. This enabled us to monitor the luminosity to better than 0.5%. A separate measurement was performed to normalize the coincidence rates to the well-known elastic cross section [14,16]. Thus, the resulting cross sections do not depend on beam-current and target-density fluctuations, since identical cuts on the target vertex were applied to both singles and coincidence data. As shown in Table 1, the data were taken in four settings at three different polarizations
876
K.I. Blomqvist et al./Nuclear Physics A 626 (1997) 871-885 50 tt_ to
~" f~. :..,::..:
0 • ~:
<
-50 I
"~"" I
88
~ .... :""~':~"~" ::~.i":'!?:~'~ I
I
I
I
I
I
[~,~:. :.:,...:
[
k))QU?
!
I
.~1000
-
E 500 0
o 0
"l
-40
' -20 0' 2'o Vertex Yo (mm)
4'o
' ' -50 0' 50 Target length zo (mm)
Fig. 3. Result of the reconstruction of the scattering vertex from the backtracing of the electron in spectrometer B. The left (right) hand side is without (with) correction for the wobbling of the electron beam. The abscissa Y0 on the left-hand side is the coordinate at target perpendicular to the median plane of the spectrometer, while z4) in the figure on the right-hand side is the coordinate in the beam direction. The dots in the lower figure on the right-hand side show the z0-spectrum for a measurement with an empty target cell. The small peak at the right in the same figure is due to an additional shielding foil at the corresponding side of the target. covering the energy transfer w = 3 6 8 - 4 3 0 MeV, with electron and pion spectrometers located on either side of the incident electron beam. Events originating from the target cell walls were identified in the offiine analysis due to the good vertex resolution o f about 2 m m ( F W H M ) , which, in fact, was most important for this experiment• It enabled the identification o f events originating from the target cell walls (cf. Section 4) even at very forward angles. The excellent angle (/tO,/t~b ~ 2 mrad) and momentum ( / t p / p < 10 - 4 ) resolution o f the spectrometers yielded a missing mass resolution A M ,~ 700 keV which enabled a clean separation of the tT"r+-channel from the break-up events.
4. Data
analysis
The first step in the analysis was to select the events which originate in the target gas. This was done by backtracing of the particle trajectory in the electron spectrometer. Fig. 3 illustrates the vertex reconstruction. The upper figure on the left side demonstrates the correlation between the reconstructed transverse coordinate in non-dispersive direction, yo, and the beam position given by the current of the wobbler coils (in arbitrary channel numbers). The lower figure on the left side is a projection on the y0-axis without, the right side shows the corresponding graphs with correction for the wobbling o f the beam. On the right, in addition, Y0 has been converted to the beam direction zo. Here, one clearly sees the contribution from the target gas bounded by the wall of the target cell.
K.L Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
877
J 300 "-2.
.,.,.
,~ 2 0 0 100
~
(channels)
ouw
-v.~
~.v
Fig. 4. Energy-loss spectrum in the two scintillator planes of the hadron spectrometer. The sharp peak on the left-hand side is due to minimum ionizing particles, the broad peak at higher AEI and dE2 is due to protons. Thus, a clean separation of the protons was achieved.
Superposed on the lower right figure is the corresponding result of a measurement with an empty target cell where only the events from the target walls are seen: The peaks are separated by 80 ram, the diameter of the target cell. The small peak at the right side, slightly outside the target cell, stems from an additional thin temperature shielding foil. Fig. 3 confirms that the vertex reconstruction employed is reliable and that background events can be removed by a cut on z0. For particle identification in the hadron spectrometer, the information from the energy loss in the two scintillator planes was used [ 17]. Fig. 4 shows the clean distinction of protons from pions. It is also clear that the arrangement with two scintillator planes is far superior to just a single plane in this context. The valid events are selected by cuts on the signal heights in the two scintillator planes. There follow eight more steps to determine the cross section from the selected events. In the first step, the time resolution is optimized via the information about the particle trajectory. As a result, one gets a sharp peak riding on top of a uniform background of accidental coincidences. The typical width of this peak is 2 to 3 ns (FWHM). Then, setting a window on the peak corresponding to real coincidences, the missing mass spectrum is constructed. The subsequent steps involve background subtraction, normalization to the integrated luminosity and correction for pion-decay in flight and for detector efficiency, followed by the normalization for the acceptance of the spectrometers [ 18]. Finally, the two-fold differential cross section is obtained by normalization to the virtual photon flux. Absolute cross sections were determined by normalization to elastic scattering data [ 14,16]. For details of this analysis see Ref. [ 19]. For the second kinematical setting given in Table 1, Fig. 5 shows the double differential cross section as function of the missing mass (the mass of 3H in its ground state is subtracted). The narrow peak at the origin is due to the t~-+-channel. It is clearly separated from the continuum of the three- and four-body break-up channels, indicating
878
K.L Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
~>1.5
3He(e,e'~*)3H xO,2
E o =675 MeV 0 e = 28.4 ° O= =16.8" to =400 MeV
~o.5 13
~e '1o
o
I
I
I
I
25
50
75
1O0
Missing mass M (MeV/c2) Fig. 5.3He(e,e~cr ~ ) cross sections for an incident electron energy of Eo = 675 MeV. The narrow peak at the origin, scaled down by the factor 0.2, is due to the two-body t~r+ channel. It is clearly separated from the three- and four-body continuum. i
i
i
ZHe(e,e'=*)3H ................... •
30
~2o
~1o 0
012
014
016
018
Virtual photon polarization Fig. 6. The 3He(e,et~r + )3H cross section in the laboratory frame as function of the virtual photon polarization e. The circles represent the measured cross sections, and the shaded area indicates the error band associated with a line fit to the data (see Table 2). The lines represent the model calculations, described in the text. In particular, they illustrate the distortion and medium effects. The dotted line is the model calculation in PWIA including the Born and A-resonance terms with the free delta width Fa = 110 MeV. The short dashed line represents the corresponding DWIA calculation. The long dashed and solid lines are obtained by taking into account in addition the effects of the medium, where the parametrization of the in-medium propagator from Ref. [24] was used and where, in addition, an in-medium modification of the pion-pole term is taken into account. Here, mean values for the matter density of t5 = 0.06 fm -3 (dashed line) and/~ = 0.09 fm -3 (full line) are used. the excellent m i s s i n g mass resolution o f the set-up. Results on the break-up channels are published in Ref. [ 1 ]. In the present paper w e discuss the c o h e r e n t p i o n - p r o d u c t i o n to the t w o - h a d r o n final state, viz. tTr +. A c c o r d i n g to Eq. ( l b ) , the cross section can be separated into its longitudinal and transverse part f r o m m e a s u r e m e n t s p e r f o r m e d in parallel kinematics for at least two values o f the virtual p h o t o n ' s polarization • under o t h e r w i s e equal conditions ( R o s e n b l u t h separation). Fig. 6 shows the m e a s u r e d cross section plotted as a function o f e. The experimental data are c o m p a r e d with results o f theoretical calculations to be described below. T h e d e d u c e d longitudinal and transverse cross sections are listed in Table 2. Q u o t e d here are both the uncorrelated and the correlated ( s h o w n in brackets) errors f r o m the straight line fit.
K.I. Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
879
Table 2 Experimental and theoretical transverse (T) and longitudinal (L) cross sections of the 3He(e,etTr+)3H reaction in the laboratory frame (in/zb/sr). A-resonanceand pion propagators are taken into account without (a) and with (b, c) medium modifications [see Eqs. (13)-(15)1. (b) is calculated with/~ = 0.06 fm-3), (c) is calculated with p = 0.09 fm -3) (see text). To be in line with the normalization of the experimental cross sections chosen in the previous work [ 1] we calculated the theoretical values by using the photon three-momentum in the laboratory frame in place of the photon equivalent energy kL from Eq. (2) experiment
(dO'L/dI~-)lab
10.4 -F 0.3(4-1.7)
(dO'T/d12zr)lab
8.9 + 0.2(4-1.1)
Calculation
(a) (b) (c) (a) (b) (c)
PWIA
DWIA
8.16 10.54 11.86 23.24 15.62 11.87
5.47 7.13 8.07 18.86 11.94 9.03
5. Theoretical model If not stated otherwise explicitly, in Sections 5 and 6 all physical quantities are given in the photon-nucleus center o f mass (cm) system, which is equal to the pion-nucleus cm system. The longitudinal and transverse virtual-photon cross sections can be expressed in terms o f the nuclear structure function W,. with u = x, y, z
dO_'____T_T= ]k~l Wxx + Wyy d S2~ ~ 2 '
d, L
Ik l
dg2~r- ~:z'
(3a)
Qe
(('OY)2 Wzz'
(3b)
where k~- is the pion three-momentum a n d / ~ = ( M A / v / - ~ ) k L is the "photon equivalent energy". The cross sections in the laboratory (lab) frame are obtained by multiplying Eqs. (3a) and ( 3 b ) with the Jacobian
df2~ b = ik~rl(lk~bl(MA + o)lTab )
_ j~lab[klab
] ,
(4)
and using the following transformation properties o f the structure functions wlva:
"q- W~,ab, = Wxx + Wyy,
- -
.
The nuclear structure function Wu~ in Eqs. (3a) and (3b) is defined as
(Sa)
(5b)
880
K.L Blomqvistet al./Nuclear PhysicsA 626 (1997) 871-885
Wu~ = (flJu[i)(flJ~[i)*,
(6)
where ti) and [f) are the initial photon-nucleus and final pion-nucleus states, respectively. Throughout this paper we use impulse approximation (IA), i.e. the nuclear many-body current J,, is expressed as sum of single-nucleon currents, i.e. Ju = (y~A p(i), y-~a j ( i ) ) . The space-like part of this current is conventionally evaluated in the photon-nucleon cm-system in terms of six scalar functions Fi (the so-called CGLN amplitudes [ 10,20] ) j = i F l ( O ' ) ± + F2(o" • k~r) (0" x ky) + i[F3(o"" lcr) + F4(o"" ~:~-)] (krr)_l_ + i [ f s ( o ' . k~,) + F6(o'-~:rr) ] (~:~) •
(7)
Here, ~::, and k,~ are the unit vectors in the direction of the photon and pion momenta (remember: in the photon- (or pion-)nucleon cm-frame), o'± = ~r - (~r. ~:r)~::, and (~:.)± = ~:~-- (~:~r "/q,)k~,. Gauge invariance implies that the time-like part of the electromagnetic current can be expressed in terms of the longitudinal current: p = ( kr. j ) / w y . When restricting the theoretical discussion to the parallel kinematics of the experiment, the terms with F3 and F4 do not contribute. The details of the complete formalism, with which the final calculation of the structure functions have been performed, are given in Ref. [7]. In particular, the pion final state interaction is included and realistic nuclear wave functions are used. For illustrative reasons, we discuss in this section the 3He(e,e'rr+)3H-reaction in the plane-wave (PW) approximation for the outgoing pion, i.e. we describe the process in PWIA. Furthermore, the contributions of the D-components in the tri-nucleon wave functions are neglected. With these simplifications, we obtain the following expressions for the transverse and longitudinal structure functions
Wx~ + Wyy 2
= IF' - F212 W2AF2(Q2)'
(8a)
Wzz : ( w~, ' z IF5 + F612wzFz(Q2), \ w~(p) ]
(8b)
where the factor WA is given by WA
~
=V
Sp
MA
S~A
(9/
This factor accounts for the replacement of the three-body by the one-body current and for the subsequent transformation from the (y,p) to the (y,3He) cm system. The virtual photon energy is given in the photon-proton and photon-nucleus cm-frames by
a)~,(p) =
st, _ Q2 _ 2 2 x / ~ Mr' ,
SA
wy =
--
Q2 _ M 2 2v/_~ ,
(10)
respectively. The nuclear formfactor F(Q 2) accounts in a simplified way for the overlap of the initial and final wave functions in the tri-nucleon system. It depends on the
K.1. Blomqvistet al./Nuclear Physics A 626 (1997) 871-885
881
momentum transferred to the recoiling nucleus: QA = k~,- k~.. For the small momentum transfers under discussion here, it is sufficiently accurate to take the simple Gauss form F(Q2A) = e x p ( - Q 2 b 2 / 6 ) , where b = 1.65 fm is the rms-radius of 3H. The scalar functions Fi in the expression for the single nucleon current j are normally expanded in terms of electric (Lt±,Et+) and magnetic (Mt~) multipoles, where 1 is the pion-nucleon orbital angular momentum and the sign determines the total angular 1 Accounting for the S- and P-wave contributions only, the momentum 1 as 1 = l =t= 3" following expressions for Fl - F2 and F s + F6 emerge FI - F2 =E0~ + Ml~ +3E1,
-
M 1
,
F5 + F6=L0* + L I - + 4 L I + .
(lla) (llb)
Using Eqs. ( l l a ) and ( l l b ) together with Eqs. (3a), (3b) and (8a), (8b), we find quite a simple expression for an estimate of the ratio of the transverse to the longitudinal cross section in parallel kinematics:
dO'T/dg2,r dO'L/df2~r
w~(p) IFJ - F2]2 w2(p) leo, + MI+ + 3E1+ - M1-[2 - - - Q2 IF5 4- F6[2 Q2 [Lo, 4- Ll- + 4Ll+ [ 2
(12)
Note that in our notation, cf. Eq. ( l b ) , the factor QZ/to~ is not included in a longitudinal polarization factor, instead, it is part of the longitudinal cross section. As a result, the cross section ratio in Eq. (12) does not depend on the reference frame. Within the approximations under discussion, this ratio can be given in terms of the elementary amplitudes alone. The transverse cross section is governed by the leading E0+ Born term and by the A(1232) excitation which dominates in the MI+ multipole, while the longitudinal cross section is mainly determined by the pion-pole term.
6. Results and discussion
The full numerical evaluation of the transverse and longitudinal cross sections for rr + electroproduction off 3He has been performed in analogy to Ref. [7]. The elementary pion production amplitude is calculated on the tree-level and includes the standard Born terms (the seagull term, the pion-pole term in the t-channel, the nucleon-pole terms in the s- and u-channels) and the A-resonance term (see Fig. 7). The electromagnetic form factors entering the calculation are the same as used previously in Ref. [21] in evaluating the p(e,¢rr +)n reaction. In addition, graphs containing higher resonance states have been included along the lines of Ref. [22]. However, they do not contribute significantly in our energy domain. Finally, also vector-meson exchange, containing wand p-meson poles, has been incorporated. An important aspect of the present calculation is the treatment of the three-body system. The initial and final bound-state wave functions for 3He and 3H, including D-wave components, have been obtained from solving the Faddeev equations with a realistic nucleon-nucleon interaction. As a refinement compared to a previous work [ 23 ],
K.L Blomqvistet al./NuclearPhysicsA 626 (1997)871-885
882
s-channel
t-channel
u-channel
co, p
contact term
/X
Fig. 7. The elementary pion productiondiagrams. Wavy,single, double (hatched) and dashed lines represent the virtual photon, the nucleon, the resonance and the pion, respectively. The upper diagrams show the Born-terms, the lower diagrams represent the w- and p-pole and the A-resonance. where only the PW was used, we now account for the final state interaction (FSI) by describing the outgoing pion in distorted wave (DW). The effect of single charge exchange meson currents has also been considered, but, for the present kinematics, its contribution turned out to be negligible. Results of the present calculation are listed in Table 2. At first, we notice that the PWIA calculation with a free delta width essentially reproduces the longitudinal cross section, but it overestimates the transverse component by a factor of 2.5. It is seen that the effect of FSI is to reduce the PWlA results for both the longitudinal and transverse cross sections by about 30%. At this stage, further refinements of modeling the process concern the influence of other resonances and medium modifications of the resonance and of the meson propagators. Of crucial importance for the absolute value of the transverse cross section is the effective width of the A(1232) resonance. For the free A-width of F~ ,.~ 110 MeV, the calculated transverse cross section still overestimates the observed one by more than a factor of two, and including a monopole form factor in the ¢rN vertex with a cutoff parameter A,~ = 800 MeV reduces dO'T/d12,~ only by about 20%. On the other hand, it is well known that the properties of the isobar in the medium differ appreciably from those at zero matter density due to many-nucleon correlation effects. In Ref. [24], the following parametrization of the (non-relativistic) A-propagator has been suggested for the energy region of interest here:
(x/-s-Ma
+ 1-~iFa) - '
--+(x/s-Ma+ l i F a X- , a ) -1
(13)
For the self-energy of the el we use the following parametrization taken from Ref. [24] ReXa = ( - 7 0 + 0.113wr) P--- [MeV] , t90 ImXa=-
Co
+CA2 ~
+CA3 -~0
(14a)
[MeV],
(14b)
K.L Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
883
with y = 1.7,
C a = 13.13,
CA2 = 14.4,
CA3 = 19.25,
while a and fl depend on p. For the density p we take the average density deduced from the measured charge distribution Pcharge for 3He, cf. Ref. [25,26]. Averaging over Pcharge and multiplying by A/Z, one gets for the average mass distribution ,63He,mass = 0.06 fm -3. Unfolding the finite size of the proton from pcharge yields the distribution of point-like nucleons Ppoint [25]. Averaging over Ppoint and multiplying by A/Z, one gets for the average mass distribution ,63He,point = 0.09 fin -3. For a and fl we get ce = 0.212(0.322), fl = 1.051(0.850) for ,6 = 0.09(0.06). In order to demonstrate the sensitivity to p, we have calculated the medium effect with both values for the density. Replacing the free d propagator by the in-medium one from Eqs. (14a) and (14b) results in a substantial suppression of the real part of the M1 + multipole contribution to Eq. (12), and, consequently, in a substantial reduction of the transverse cross section. All modifications considered above practically do not affect the value of the calculated longitudinal cross section dO-L/d~.. Indeed, the many-nucleon correlation effects associated with the self-energy of the/:'33 state enhance the longitudinal cross section by about 2% only. A somewhat stronger increase of the latter is achieved by replacing the free pion propagator in the pion-pole term by the in-medium quasi-pion propagator [27] according to
(q2 _ m ~ ) - ' ~ (q~ - ~ ( p ) q ~ - fl(p)m2) -' .
(15)
Using the parametrization from Ref. [28] one finds a ( p ) = 0.445 and fl(p) = 1.036. The resulting propagator-modification increases do'L/d12~ by about 20% over its vacuum value. Note that such an effect has earlier been suggested in Ref. [29] as a possible enhancement mechanism for nuclear longitudinal response functions. Finally let us consider the ratio of p(e,etTr+)n and 3He(e,e'Tr+)3H cross sections which is discussed in Ref. [30] in terms of a quenching of the latter with respect to the former. For an estimate of this ratio we use expressions (8a), (8b), (9) of our model. Then, the pion production cross section on 3He is given in the photon-nucleus cm-frame by ( d o ' / d ~ r ) 3 H e ~ W2A FZ(Q 2) rlcr( d o ' / d ~ . ) p = R (do'Idler)p,
(16)
where (do'/df2~r)p is calculated in the photon-proton cm system, and where we have introduced the quantity R as a short-hand notation for the ratio between the two cross sections. We have also introduced a distortion factor rl,~ which describes the effects of the pion final state interaction. From the PWIA and DWIA calculations mentioned above it follows that r/~. ~ 0.7. For the kinematics of the present experiment, the kinematical factor WA 2 and F2(Q 2) take the values 1.42 and 0.86, respectively. In the PWIA approach, i.e. for rl~r = 1, we obtain the ratio Rcm(PWIA) ~ 1.22, i.e. an enhancement rather than a quenching. However, accounting for the FSI of the pion leads to Rcm(DWIA) ~ 0.85, thus to a quenching. The renormalization of the zl propagator in the nuclear medium may result in a further quenching.
K.L Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
884
For a direct comparison between the two cross sections, one has to refer to a common reference system. To calculate the ratio in the laboratory frame, we have to introduce in Eq. (16) the ratio of the Jacobians (Eq. (4)) for the transformations from the pionnucleus and the pion-nucleon systems to the common laboratory system. This factor turns out to be rather important and it reduces the Rcm ratio by further 40%. This means, that, besides its strong model dependence, the quenching effect discussed in Ref. [30] is a non-covariant quantity. Our final expression for the ratio R in the laboratory frame is similar to that reported in Ref. [30]: Rlab =
( d~r/dl21,rab)3He ~ FZ(Q2A) r/~ (dtr/da~b)p
M AIk~(p ) I
Mplk~l
(17)
In the PWIA and in the DWIA approach, we obtain Rlab(PWIA) = 0.81 and Rlab(DWIA) = 0.56, respectively, which differ considerably from the corresponding ratios with the single cross sections calculated in the respective cm-frames.
Acknowledgements We are grateful to Philip Leconte from DAPNIA, CEA, for his help with the cryogenic target. This work has been supported by the Sonderforschungsbereich 201 of the Deutsche Forschungsgemeinschaft and by the BMBF under Contract Number 06DA665I. One of us (T.S.) acknowledges a grant from the Alexander-von-Humboldt Foundation.
References [ 1] K.I. Blomqvist, W.U. Boeglin, R. B6hm, M. Distler, R. Edelhoff, J. Friedrich, R. Geiges, M. Kahrau, M. Kom, H. Kramer, K.W. Krygier, V. Kunde, M. Kuss, J.M. Laget, A. Liesenfeld, K. Merle, R. Neuhausen, E.A.J.M. Offermann, Th. Pospischil, M. Potokar, C. Rangacharyulu, A. Rokavec, A. Richter, A.W. Richter, G. Rosner, P. Saner, St. Schardt, G. Schrieder, T. Suda, B. Vodenik, A. Wagner, Th. Walcher and St. Wolf, Phys. Rev. Lett. 77 (1996) 2396. 121 See for example E Cardarelli and J. Piekarewicz, Nucl. Phys. A 612 (1997) 429. 131 G. Audit, A. Braghieri, N. D'Hose, V. lsbert, S. Kerhoas, J.M. Laget, M. Mac Cormick, L.Y. Murphy, A. Panzeri, P Pedroni, T. Pinelli, G. Tamas, J. Ahrens, J.R.M. Annand, R. Crawford, P. Grabmayr, S.J. Hall and J.D. Kellie, Nucl. Phys. A 614 (1997) 461. [41 N. d'Hose, G. Audit, A. Bloch, N. de Botton, L. Ghedira, L. Jammes, J.M. Laget, J. Martin, E. Mazzucatto, C. Schuhl, G. Tamas, E. Vincent, M. Rodgers, P. Stoler, P. Argan, A. Braghieri and P. Pedroni, Nucl. Phys. A 554 (1993) 679. [ 51 C.W. Kim and H. Primakoff, in Mesons in Nuclei, Vol. 1, ed. M. Rho and D. Wilkinson (North Holland, Amsterdam, 1979) p. 67. 161 R. Gilman, M. Bernheim, M. Brussel, J. Cheminaud, J.-E Danel, J.-P. Didelez, M.-A. Duval, G. Fournier, R. Frascaria, R.J. Holt, H.E. Jackson, J.-C. Kim, E. Kinney, J.-M. LeGoff, R. Letourneau, A. Magnon, J. Morgenstern, C. Pasquier, J. Picard, D. Poizat, B. Saghai, J. Specht, P. Vernin and E. Warde, Phys. Rev. Lett. 64 (1990) 622. 171 S. Kamalov, L. Tiator and C. Bennhold, Few Body Syst. 10 (1991) 143. [81 M.M. Rosenbluth, Phys. Rev. 79 (1950) 615. [9] J. M. Laget, in New Vistas in Electro-Nuclear Physics, ed. E.L. Tomusiak, H.S. Caplan and E.T. Dressier (Plenum, New York, 1986) p. 361. 1101 D. Drechsel and L. Tiator, J. Phys. G 18 (1992) 449.
K.I. Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
885
[11] L. N. Hand, Phys. Rev. 129 (1963) 1834. 1121 J. Ahrens, H. Backe, D. von Harrach, K.H. Kaiser, E Klein, R. Neuhausen and Th. Walcher, Nucl. Phys. News 2 (1994) 5. [13] R. Neuhausen, Nucl. Phys. (Proc. Suppl.) B 44 (1995) 695; K.I. Blomqvist et al., to be published. 1141 A. Amroun, V. Breton, J.M. Cavedon, B. Frois, D. Goutte, EP. Juster, P. Leconte, J. Martino, Y. Mizuno, X.H. Phan, S.K. Platchkov, I. Sick and S. Williamson, Nucl. Phys. A 579 (1994) 596. 1151 W. Wilhelm, Diploma thesis, University of Mainz, 1993. 1161 C.R. Ottermann, G. KObschall, K. Maurer, K. R6hrich, C. Schmitt und V.H. Walther, Nucl. Phys. A 435 (1985) 688. [171 A.W. Richter, Dissertation, Mainz, Germany (1994). 118] M.O. Distler, Dissertation, Mainz, Germany (1996). 1191 M. Kuss, Dissertation, Darmstadt, Germany (1996). [201 G.F. Chew, M.L. Goldhaber, F.E. Low, and Y. Nambu, Phys. Rev. 106 (1957) 1345. 121 I K.I. Blomqvist, W.U. Boeglin, R. B6hm, M. Distler, D. Drechsel, R. Edelhoff, J. Friedrich, R. Geiges, O. Hanstein, M. Kahrau, M. Korn, H. Kramer, K.W, Krygier, V. Kunde, M. Kuss, A. Liesenfeld, K. Merle, R. Neuhausen, E.A.J.M. Offermann, Th. Pospischil, M. Potokar, A.W. Richter, A. Rokavec, G. Rosner, P. Saner, St. Schardt, H. Schmieden, S. Sirca, L. Tiator, B. Vodenik, A. Wagner, Th. Walcher and St. Wolf, Z. Phys. A 353 (1996) 415. [22] O. Hanstein, Diploma Thesis, University of Mainz, Germany (1993). [23] L. Tiator and D. Drechsel, Nucl. Phys. A 360 (1981) 208. I24] E. Oset and L.L. Salcedo, Nucl. Phys. A 468 (1987) 631; R.C. Carrasco and E. Oset, Nucl. Phys. A 536 (1992) 445. 1251 J.S. McCarthy, I. Sick and R.R. Whitney, Phys. Rev. C 15 (1977) 1396. [261 H. de Vries, C. W. de Jager and C. de Vries, At. Data and Nucl. Data Tables 36 (1987) 495. 127] See also T. Ericson and W. Weise, Pions and Nuclei (Clarendon, Oxford, 1988). [281 M. Kirchbach and A. Wirzba, Nucl. Phys. A 604 (1996) 695. 1291 M. Ericson and A.W. Thomas, Phys. Lett. B 128 (1983) 112. 1301 R.J. Loucks and V.R. Pandharipande, Phys. Rev. C 54 (1996) 32; R.J. Loucks, V.R. Pandharipande, and R. Schiavilla, Phys. Rev. C 49 (1994) 342.
Nuclear Physics A 626 (1997) 871-885
Pion electroproduction in the 3He(e,e Tr+)3H reaction at intermediate energies K.I. Blomqvist a, W.U. Boeglin a, R. Bthm a, M. Distler a, R. Edelhoff a, J. Friedrich a, R. Geiges a, M. Kahrau a, S. Kamalov b, M. Kirchbach a, M. Kohl c, M. Korn a, H. Kramer a, K.W. Krygier a, V. Kunde a, M. Kuss c'1, J.M. Laget d, A. Liesenfeld a, K. Merle a, R. Neuhausen a, E.A.J.M. Offermann a, Th. Pospischil a, M. Potokar e, C. Rangacharyulu f, A. Richter c'2, A.W. Richter a, A. Rokavec e, G. Rosner a, P. Sauer a, St. Schardt a, G. Schrieder c, T. Sudag, L. Tiator a, B. Vodenik e, A. Wagner a, Th. Walcher a, St. Wolf a a Institutfiir Kernphysik, Universitgit Mainz, D-55099 Mainz, Germany b Laboratory of Theoretical Physics, JINR Dubna, 141980 Dubna, Moscow region, Russia c lnstitutfiir Kernphysik, Technische Universitit't Darmstadt, D-64289 Darmstadt, Germany d CEA Saclay, DAPNIA/SPhN, F-91191 Gif-sur-Yvette Cedex, France e Institute "Jo~efStefan", University of Ljubljana, SI-61111 Ljubljana, Slovenia f Department of Physics, University of Saskatchewan, Saskatoon, SK, S7N 5E2, Canada g Department of Physics, Tohoku Universi~, Kawauchi, Sendai, 980, Japan
Received 12 June 1997; accepted 7 August 1997
Abstract
The differential coincident pion electroproduction cross section in the 3He(e,e'zr+ )3H reaction has been measured with high resolution with the three-spectrometer set-up at the Mainz Microtron (MAMI) electron accelerator. Measurements were performed at the four incident energies Eo = 855, 675, 600, and 555 MeV at fixed four-momentum transfer Q2 = 0.045 GeV2, with the pions detected in parallel kinematics. This enables a separation of the measured cross section into the transverse and longitudinal structure functions by means of the Rosenbluth method. The experimental data are compared to model calculations, in which the elementary pion production amplitude includes the standard Born-amplitudes and also delta and higher resonance terms. Threebody Fadeev wave functions are used and the final state interaction of the outgoing pion is taken into account. The calculation describes the data only after medium modifications of the delta and of the pion are included. (~) 1997 Elsevier Science B.V. PACS: 21.45.+v; 25.10.+s; 25.30.Rw; 27.10.+h Keywords: Coherent pion electroproduction;Longitudinal-transverseseparation; 3He; Mediumeffects; Delta
resonance region
0375-9474/97/$17.00 (~ 1997 Elsevier Science B.V. All rights reserved. PII S0375-9474(97) 00463-6
872
K.L Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
1. Introduction In intermediate-energy nuclear physics research, experiments involving pion production by electromagnetic probes are of particular interest. As direct measures of pion fields in nuclei and of the contribution of nucleonic excitations, these channels provide a good handle on nuclear dynamics. While measurements with real photons probe only the transverse structure of the hadronic system, the electroproduction process also probes longitudinal components. Separate access to these components is achieved by an appropriate choice of kinematical settings. This allows a detailed study of of the dynamics. Recently, with the advent of modern high duty-factor, high resolution intermediate energy electron accelerator facilities such as the Mainz Microtron (MAMI), such experiments under well defined exclusive kinematical conditions have become possible. As part of the scientific program of the Al-collaboration at MAMI, studies of the 3He(e,e'Tr±) reactions were initiated. The motivation for this work has been many-fold. An important impetus was to learn about possible delta isobar components in the 3He ground state from the comparison of the 7r+- to the ~--production in the break-up channels [ 1 ]. With the excellent missing mass resolution at the MAMI three-spectrometer set-up, the two-body final state (t~ +) could be clearly separated from the three- and four-body (dn~-+, pnn~r +) final states. This allowed to examine in a kinematically complete measurement the t~+-channel, which is of particular theoretical interest. It is the aim of the present paper to report on the results of the 3He(e,e'Tr+)3H reaction. With energy transfers in the A-resonance region, we determined the longitudinal and transverse production cross sections at fixed four-momentum transfer by the Rosenbluth method. The mass-3 system continues to be a subject of intensive physical interest. From the theoretical side [2], this system is amenable to detailed Faddeev type calculations, and it is of interest to test by experiments [ 3,4] corresponding calculations for 3He-3H, a pair of mirror nuclei very similar to the p-n system, in the spirit of the elementary particle approach [5]. This question has received renewed attention, as the forward-angle pion electroproduction measurements of Gilman et al. [6] with a deuterium target argued for a quenching of the production cross section on the nucleus compared to that on the proton. Section 2 of this paper describes the kinematics of the present experiment. Section 3 describes the spectrometer set-up and details of the cryogenic gas target used for this experiment. The data reduction procedures and the results of the measurements are presented in Section 4. In Section 5, a theoretical model based on the elementary particle approach [7] is described. The results of the model calculations are discussed and compared to the experimental data in Section 6.
I Present address: Jefferson Lab., 12000 Jefferson Ave., Newport News, VA 23606, USA. 2 Corresponding author. E-maih [email protected].
K.L
Blomqvistet al./Nuclear PhysicsA 626 (1997) 871-885
reaction plant ~
873
~
Fig. 1. Kinematicsof the (e,e'~r) reaction. The variables in the scattering and reaction planes are indicated. See text for details. 2. Kinematics In this section, as well as in Sections 3 and 4, all physical quantities are given in the laboratory system without referring to this system explicitly by some label. Pion electroproduction off a target nucleus can be visualized as in Fig. 1. The direction of the scattered electron and of the virtual photon define the scattering plane; their angles with respect to the incoming electron are denoted by 0e and 0~,, respectively. The pion direction is specified by its angle with respect to the virtual photon, 0~., and by the out-of-plane angle ~b~-. The threefold differential cross section can be written as (see Refs. [7-10] for details) d3o_ _ / "do-v d~2e, dEe, df2~r d~2~r '
la)
with do-v _ drrT dirE dO~r dO~ + e - - + [ 2 e ( l + e ) ] l / 2 c ° s 6 ~ r - ~
dO'LT
dOrTT d--~
lb)
Here, e and F denote the polarization and flux of the virtual photon of energy tot and three-momentum k~,
I
-Ik~'2
e= Ll+2--0T-tan
2 ~ ] -I ~ ,
O' Ee' kL 1 F = 2 7 r 2 Ee Q 2 1 _ e
(2)
Here, kL denotes the "photon equivalent energy" in Hand's convention [ I 1 ], related via kL = (SA -- M2A)/2MA to the Mandelstam variable SA = _Q2 + M~A + 2mTMa, where MA stands for the mass of the target nucleus. The positive quantity Q2 is introduced as Q2 = k~,2_ to~.2 The first two terms in Eq. ( l b ) are the transverse (T) and longitudinal (L) cross sections. These terms are independent of the azimuthal angle ~b~.. The third and fourth term, respectively, describe the longitudinal-transverse (LT) and transversetransverse (TT) interference structure (or response) functions. They vanish in parallel
874
K.L Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
Table 1 Kinematical settings of the 3He(e,e~rr+)3H experiment. The four-momentum transfer squared Q2 was 0.045 (GeV/c) 2 for all settings. The notation is: E0 = incident electron energy, a~r = energy transfer, 0c = electron scattering angle, • = virtual photon polarization, 0rr = pion detection angle relative to the electron beam &~ (MeV)
w:, (MeV)
0c
•
07r
855 675 600 555
368-430 368-425 395-432 363-398
19.5° 28.4° 36.4° 40.8°
0.79 0.63 0.48 0.47
19.6° 16.8° 13.8° 14.8°
kinematics (0~, = 0 °) since they appear proportional to sin 0~, and sin 2 0~, respectively. The emphasis of the measurement was to determine the longitudinal and transverse production cross sections in the delta resonance region. To this end, the pion was measured in parallel kinematics at fixed Q2 for different angle-energy-combinations for the electron in order to allow a Rosenbluth separation of the cross section. The kinematical settings are listed in Table 1.
3. Experimental details The measurements were carried out at the high-resolution spectrometer set-up of the A1 collaboration at M A M I . The facility is already described elsewhere [12,13]. The experiment was performed with the cryogenic 3He gas target used earlier in electron scattering experiments at Saclay [ 14]. Fig. 2 shows a cross section of the target. The target cell is made o f a stainless steel foil of about 8 0 / z m thickness and it has a diameter of 80 ram. The helium gas is circulated through the closed loop with the help of a fan, below which the helium has thermal contact with liquid hydrogen in a heat exchanger produced by about 50 copper grids soldered into the passing tube. The electron beam deposits a power o f about 20 W in the target cell for incident beam currents of 3 0 / , A , typically used in the measurements. The gas temperature is monitored by two carbon resistors (Tc) located about 4 cm below and above the beam position. The temperature of the loop is measured by a silicon resistor (Tsi) attached to a copper block in the lower part o f the loop. The target was run at a pressure of 15 bar and a temperature o f 20 K, corresponding to an effective thickness of about 160 m g / c m 2. To avoid excessive heating of the target cell window by the narrow electron beam spot ( F W H M ~ 0.16 m m ) , the beam was wobbled over an area of 7 mm × 7 mm. This was achieved by a system of magnetic coils located about 8 m upstream o f the target chamber [ 15]. The current of the coils was digitized and recorded in the data stream, enabling event-by-event reconstruction of the beam position and thus a correction for the wobbling of the beam in the off-line analysis. The scattered electrons and the produced charged pions were detected in coincidence by two magnetic spectrometers, equipped with a set of four vertical drift chambers
K.L Blomqvist et aL /Nuclear Physics A 626 (1997) 871-885
875
"~He
J fan
target cell
heat exchanger
I
,
•
I
i0 cm
Fig. 2. Side view of the 3He target. The main componentsare the target cell (right) and the heat exchanger (left) which is filled with liquid hydrogen as the coolant for the helium gas. The system has two closed loops, one for the coolant and one for the helium. lot particle tracking and with two planes of segmented plastic scintillators and a gas Cherenkov detector for timing and particle identification. A logic combination of the signals from the scintillator and from the Cherenkov generated the trigger. The Cherenkov veto suppressed the positron background in the hadron spectrometer. Pions were distinguished from protons by their different energy losses in the scintillators. Problems associated with large singles rates ( ~ 1 5 0 kHz protons) were overcome by fast hardware vetos using signals from the scintillators and from the Cherenkov detectors. Detailed particle identification was done in the offline analysis. The singles rates from the electron spectrometer were scaled down and recorded along with the double coincidence events. This enabled us to monitor the luminosity to better than 0.5%. A separate measurement was performed to normalize the coincidence rates to the well-known elastic cross section [14,16]. Thus, the resulting cross sections do not depend on beam-current and target-density fluctuations, since identical cuts on the target vertex were applied to both singles and coincidence data. As shown in Table 1, the data were taken in four settings at three different polarizations
876
K.I. Blomqvist et al./Nuclear Physics A 626 (1997) 871-885 50 tt_ to
~" f~. :..,::..:
0 • ~:
<
-50 I
"~"" I
88
~ .... :""~':~"~" ::~.i":'!?:~'~ I
I
I
I
I
I
[~,~:. :.:,...:
[
k))QU?
!
I
.~1000
-
E 500 0
o 0
"l
-40
' -20 0' 2'o Vertex Yo (mm)
4'o
' ' -50 0' 50 Target length zo (mm)
Fig. 3. Result of the reconstruction of the scattering vertex from the backtracing of the electron in spectrometer B. The left (right) hand side is without (with) correction for the wobbling of the electron beam. The abscissa Y0 on the left-hand side is the coordinate at target perpendicular to the median plane of the spectrometer, while z4) in the figure on the right-hand side is the coordinate in the beam direction. The dots in the lower figure on the right-hand side show the z0-spectrum for a measurement with an empty target cell. The small peak at the right in the same figure is due to an additional shielding foil at the corresponding side of the target. covering the energy transfer w = 3 6 8 - 4 3 0 MeV, with electron and pion spectrometers located on either side of the incident electron beam. Events originating from the target cell walls were identified in the offiine analysis due to the good vertex resolution o f about 2 m m ( F W H M ) , which, in fact, was most important for this experiment• It enabled the identification o f events originating from the target cell walls (cf. Section 4) even at very forward angles. The excellent angle (/tO,/t~b ~ 2 mrad) and momentum ( / t p / p < 10 - 4 ) resolution o f the spectrometers yielded a missing mass resolution A M ,~ 700 keV which enabled a clean separation of the tT"r+-channel from the break-up events.
4. Data
analysis
The first step in the analysis was to select the events which originate in the target gas. This was done by backtracing of the particle trajectory in the electron spectrometer. Fig. 3 illustrates the vertex reconstruction. The upper figure on the left side demonstrates the correlation between the reconstructed transverse coordinate in non-dispersive direction, yo, and the beam position given by the current of the wobbler coils (in arbitrary channel numbers). The lower figure on the left side is a projection on the y0-axis without, the right side shows the corresponding graphs with correction for the wobbling o f the beam. On the right, in addition, Y0 has been converted to the beam direction zo. Here, one clearly sees the contribution from the target gas bounded by the wall of the target cell.
K.L Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
877
J 300 "-2.
.,.,.
,~ 2 0 0 100
~
(channels)
ouw
-v.~
~.v
Fig. 4. Energy-loss spectrum in the two scintillator planes of the hadron spectrometer. The sharp peak on the left-hand side is due to minimum ionizing particles, the broad peak at higher AEI and dE2 is due to protons. Thus, a clean separation of the protons was achieved.
Superposed on the lower right figure is the corresponding result of a measurement with an empty target cell where only the events from the target walls are seen: The peaks are separated by 80 ram, the diameter of the target cell. The small peak at the right side, slightly outside the target cell, stems from an additional thin temperature shielding foil. Fig. 3 confirms that the vertex reconstruction employed is reliable and that background events can be removed by a cut on z0. For particle identification in the hadron spectrometer, the information from the energy loss in the two scintillator planes was used [ 17]. Fig. 4 shows the clean distinction of protons from pions. It is also clear that the arrangement with two scintillator planes is far superior to just a single plane in this context. The valid events are selected by cuts on the signal heights in the two scintillator planes. There follow eight more steps to determine the cross section from the selected events. In the first step, the time resolution is optimized via the information about the particle trajectory. As a result, one gets a sharp peak riding on top of a uniform background of accidental coincidences. The typical width of this peak is 2 to 3 ns (FWHM). Then, setting a window on the peak corresponding to real coincidences, the missing mass spectrum is constructed. The subsequent steps involve background subtraction, normalization to the integrated luminosity and correction for pion-decay in flight and for detector efficiency, followed by the normalization for the acceptance of the spectrometers [ 18]. Finally, the two-fold differential cross section is obtained by normalization to the virtual photon flux. Absolute cross sections were determined by normalization to elastic scattering data [ 14,16]. For details of this analysis see Ref. [ 19]. For the second kinematical setting given in Table 1, Fig. 5 shows the double differential cross section as function of the missing mass (the mass of 3H in its ground state is subtracted). The narrow peak at the origin is due to the t~-+-channel. It is clearly separated from the continuum of the three- and four-body break-up channels, indicating
878
K.L Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
~>1.5
3He(e,e'~*)3H xO,2
E o =675 MeV 0 e = 28.4 ° O= =16.8" to =400 MeV
~o.5 13
~e '1o
o
I
I
I
I
25
50
75
1O0
Missing mass M (MeV/c2) Fig. 5.3He(e,e~cr ~ ) cross sections for an incident electron energy of Eo = 675 MeV. The narrow peak at the origin, scaled down by the factor 0.2, is due to the two-body t~r+ channel. It is clearly separated from the three- and four-body continuum. i
i
i
ZHe(e,e'=*)3H ................... •
30
~2o
~1o 0
012
014
016
018
Virtual photon polarization Fig. 6. The 3He(e,et~r + )3H cross section in the laboratory frame as function of the virtual photon polarization e. The circles represent the measured cross sections, and the shaded area indicates the error band associated with a line fit to the data (see Table 2). The lines represent the model calculations, described in the text. In particular, they illustrate the distortion and medium effects. The dotted line is the model calculation in PWIA including the Born and A-resonance terms with the free delta width Fa = 110 MeV. The short dashed line represents the corresponding DWIA calculation. The long dashed and solid lines are obtained by taking into account in addition the effects of the medium, where the parametrization of the in-medium propagator from Ref. [24] was used and where, in addition, an in-medium modification of the pion-pole term is taken into account. Here, mean values for the matter density of t5 = 0.06 fm -3 (dashed line) and/~ = 0.09 fm -3 (full line) are used. the excellent m i s s i n g mass resolution o f the set-up. Results on the break-up channels are published in Ref. [ 1 ]. In the present paper w e discuss the c o h e r e n t p i o n - p r o d u c t i o n to the t w o - h a d r o n final state, viz. tTr +. A c c o r d i n g to Eq. ( l b ) , the cross section can be separated into its longitudinal and transverse part f r o m m e a s u r e m e n t s p e r f o r m e d in parallel kinematics for at least two values o f the virtual p h o t o n ' s polarization • under o t h e r w i s e equal conditions ( R o s e n b l u t h separation). Fig. 6 shows the m e a s u r e d cross section plotted as a function o f e. The experimental data are c o m p a r e d with results o f theoretical calculations to be described below. T h e d e d u c e d longitudinal and transverse cross sections are listed in Table 2. Q u o t e d here are both the uncorrelated and the correlated ( s h o w n in brackets) errors f r o m the straight line fit.
K.I. Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
879
Table 2 Experimental and theoretical transverse (T) and longitudinal (L) cross sections of the 3He(e,etTr+)3H reaction in the laboratory frame (in/zb/sr). A-resonanceand pion propagators are taken into account without (a) and with (b, c) medium modifications [see Eqs. (13)-(15)1. (b) is calculated with/~ = 0.06 fm-3), (c) is calculated with p = 0.09 fm -3) (see text). To be in line with the normalization of the experimental cross sections chosen in the previous work [ 1] we calculated the theoretical values by using the photon three-momentum in the laboratory frame in place of the photon equivalent energy kL from Eq. (2) experiment
(dO'L/dI~-)lab
10.4 -F 0.3(4-1.7)
(dO'T/d12zr)lab
8.9 + 0.2(4-1.1)
Calculation
(a) (b) (c) (a) (b) (c)
PWIA
DWIA
8.16 10.54 11.86 23.24 15.62 11.87
5.47 7.13 8.07 18.86 11.94 9.03
5. Theoretical model If not stated otherwise explicitly, in Sections 5 and 6 all physical quantities are given in the photon-nucleus center o f mass (cm) system, which is equal to the pion-nucleus cm system. The longitudinal and transverse virtual-photon cross sections can be expressed in terms o f the nuclear structure function W,. with u = x, y, z
dO_'____T_T= ]k~l Wxx + Wyy d S2~ ~ 2 '
d, L
Ik l
dg2~r- ~:z'
(3a)
Qe
(('OY)2 Wzz'
(3b)
where k~- is the pion three-momentum a n d / ~ = ( M A / v / - ~ ) k L is the "photon equivalent energy". The cross sections in the laboratory (lab) frame are obtained by multiplying Eqs. (3a) and ( 3 b ) with the Jacobian
df2~ b = ik~rl(lk~bl(MA + o)lTab )
_ j~lab[klab
] ,
(4)
and using the following transformation properties o f the structure functions wlva:
"q- W~,ab, = Wxx + Wyy,
- -
.
The nuclear structure function Wu~ in Eqs. (3a) and (3b) is defined as
(Sa)
(5b)
880
K.L Blomqvistet al./Nuclear PhysicsA 626 (1997) 871-885
Wu~ = (flJu[i)(flJ~[i)*,
(6)
where ti) and [f) are the initial photon-nucleus and final pion-nucleus states, respectively. Throughout this paper we use impulse approximation (IA), i.e. the nuclear many-body current J,, is expressed as sum of single-nucleon currents, i.e. Ju = (y~A p(i), y-~a j ( i ) ) . The space-like part of this current is conventionally evaluated in the photon-nucleon cm-system in terms of six scalar functions Fi (the so-called CGLN amplitudes [ 10,20] ) j = i F l ( O ' ) ± + F2(o" • k~r) (0" x ky) + i[F3(o"" lcr) + F4(o"" ~:~-)] (krr)_l_ + i [ f s ( o ' . k~,) + F6(o'-~:rr) ] (~:~) •
(7)
Here, ~::, and k,~ are the unit vectors in the direction of the photon and pion momenta (remember: in the photon- (or pion-)nucleon cm-frame), o'± = ~r - (~r. ~:r)~::, and (~:.)± = ~:~-- (~:~r "/q,)k~,. Gauge invariance implies that the time-like part of the electromagnetic current can be expressed in terms of the longitudinal current: p = ( kr. j ) / w y . When restricting the theoretical discussion to the parallel kinematics of the experiment, the terms with F3 and F4 do not contribute. The details of the complete formalism, with which the final calculation of the structure functions have been performed, are given in Ref. [7]. In particular, the pion final state interaction is included and realistic nuclear wave functions are used. For illustrative reasons, we discuss in this section the 3He(e,e'rr+)3H-reaction in the plane-wave (PW) approximation for the outgoing pion, i.e. we describe the process in PWIA. Furthermore, the contributions of the D-components in the tri-nucleon wave functions are neglected. With these simplifications, we obtain the following expressions for the transverse and longitudinal structure functions
Wx~ + Wyy 2
= IF' - F212 W2AF2(Q2)'
(8a)
Wzz : ( w~, ' z IF5 + F612wzFz(Q2), \ w~(p) ]
(8b)
where the factor WA is given by WA
~
=V
Sp
MA
S~A
(9/
This factor accounts for the replacement of the three-body by the one-body current and for the subsequent transformation from the (y,p) to the (y,3He) cm system. The virtual photon energy is given in the photon-proton and photon-nucleus cm-frames by
a)~,(p) =
st, _ Q2 _ 2 2 x / ~ Mr' ,
SA
wy =
--
Q2 _ M 2 2v/_~ ,
(10)
respectively. The nuclear formfactor F(Q 2) accounts in a simplified way for the overlap of the initial and final wave functions in the tri-nucleon system. It depends on the
K.1. Blomqvistet al./Nuclear Physics A 626 (1997) 871-885
881
momentum transferred to the recoiling nucleus: QA = k~,- k~.. For the small momentum transfers under discussion here, it is sufficiently accurate to take the simple Gauss form F(Q2A) = e x p ( - Q 2 b 2 / 6 ) , where b = 1.65 fm is the rms-radius of 3H. The scalar functions Fi in the expression for the single nucleon current j are normally expanded in terms of electric (Lt±,Et+) and magnetic (Mt~) multipoles, where 1 is the pion-nucleon orbital angular momentum and the sign determines the total angular 1 Accounting for the S- and P-wave contributions only, the momentum 1 as 1 = l =t= 3" following expressions for Fl - F2 and F s + F6 emerge FI - F2 =E0~ + Ml~ +3E1,
-
M 1
,
F5 + F6=L0* + L I - + 4 L I + .
(lla) (llb)
Using Eqs. ( l l a ) and ( l l b ) together with Eqs. (3a), (3b) and (8a), (8b), we find quite a simple expression for an estimate of the ratio of the transverse to the longitudinal cross section in parallel kinematics:
dO'T/dg2,r dO'L/df2~r
w~(p) IFJ - F2]2 w2(p) leo, + MI+ + 3E1+ - M1-[2 - - - Q2 IF5 4- F6[2 Q2 [Lo, 4- Ll- + 4Ll+ [ 2
(12)
Note that in our notation, cf. Eq. ( l b ) , the factor QZ/to~ is not included in a longitudinal polarization factor, instead, it is part of the longitudinal cross section. As a result, the cross section ratio in Eq. (12) does not depend on the reference frame. Within the approximations under discussion, this ratio can be given in terms of the elementary amplitudes alone. The transverse cross section is governed by the leading E0+ Born term and by the A(1232) excitation which dominates in the MI+ multipole, while the longitudinal cross section is mainly determined by the pion-pole term.
6. Results and discussion
The full numerical evaluation of the transverse and longitudinal cross sections for rr + electroproduction off 3He has been performed in analogy to Ref. [7]. The elementary pion production amplitude is calculated on the tree-level and includes the standard Born terms (the seagull term, the pion-pole term in the t-channel, the nucleon-pole terms in the s- and u-channels) and the A-resonance term (see Fig. 7). The electromagnetic form factors entering the calculation are the same as used previously in Ref. [21] in evaluating the p(e,¢rr +)n reaction. In addition, graphs containing higher resonance states have been included along the lines of Ref. [22]. However, they do not contribute significantly in our energy domain. Finally, also vector-meson exchange, containing wand p-meson poles, has been incorporated. An important aspect of the present calculation is the treatment of the three-body system. The initial and final bound-state wave functions for 3He and 3H, including D-wave components, have been obtained from solving the Faddeev equations with a realistic nucleon-nucleon interaction. As a refinement compared to a previous work [ 23 ],
K.L Blomqvistet al./NuclearPhysicsA 626 (1997)871-885
882
s-channel
t-channel
u-channel
co, p
contact term
/X
Fig. 7. The elementary pion productiondiagrams. Wavy,single, double (hatched) and dashed lines represent the virtual photon, the nucleon, the resonance and the pion, respectively. The upper diagrams show the Born-terms, the lower diagrams represent the w- and p-pole and the A-resonance. where only the PW was used, we now account for the final state interaction (FSI) by describing the outgoing pion in distorted wave (DW). The effect of single charge exchange meson currents has also been considered, but, for the present kinematics, its contribution turned out to be negligible. Results of the present calculation are listed in Table 2. At first, we notice that the PWIA calculation with a free delta width essentially reproduces the longitudinal cross section, but it overestimates the transverse component by a factor of 2.5. It is seen that the effect of FSI is to reduce the PWlA results for both the longitudinal and transverse cross sections by about 30%. At this stage, further refinements of modeling the process concern the influence of other resonances and medium modifications of the resonance and of the meson propagators. Of crucial importance for the absolute value of the transverse cross section is the effective width of the A(1232) resonance. For the free A-width of F~ ,.~ 110 MeV, the calculated transverse cross section still overestimates the observed one by more than a factor of two, and including a monopole form factor in the ¢rN vertex with a cutoff parameter A,~ = 800 MeV reduces dO'T/d12,~ only by about 20%. On the other hand, it is well known that the properties of the isobar in the medium differ appreciably from those at zero matter density due to many-nucleon correlation effects. In Ref. [24], the following parametrization of the (non-relativistic) A-propagator has been suggested for the energy region of interest here:
(x/-s-Ma
+ 1-~iFa) - '
--+(x/s-Ma+ l i F a X- , a ) -1
(13)
For the self-energy of the el we use the following parametrization taken from Ref. [24] ReXa = ( - 7 0 + 0.113wr) P--- [MeV] , t90 ImXa=-
Co
+CA2 ~
+CA3 -~0
(14a)
[MeV],
(14b)
K.L Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
883
with y = 1.7,
C a = 13.13,
CA2 = 14.4,
CA3 = 19.25,
while a and fl depend on p. For the density p we take the average density deduced from the measured charge distribution Pcharge for 3He, cf. Ref. [25,26]. Averaging over Pcharge and multiplying by A/Z, one gets for the average mass distribution ,63He,mass = 0.06 fm -3. Unfolding the finite size of the proton from pcharge yields the distribution of point-like nucleons Ppoint [25]. Averaging over Ppoint and multiplying by A/Z, one gets for the average mass distribution ,63He,point = 0.09 fin -3. For a and fl we get ce = 0.212(0.322), fl = 1.051(0.850) for ,6 = 0.09(0.06). In order to demonstrate the sensitivity to p, we have calculated the medium effect with both values for the density. Replacing the free d propagator by the in-medium one from Eqs. (14a) and (14b) results in a substantial suppression of the real part of the M1 + multipole contribution to Eq. (12), and, consequently, in a substantial reduction of the transverse cross section. All modifications considered above practically do not affect the value of the calculated longitudinal cross section dO-L/d~.. Indeed, the many-nucleon correlation effects associated with the self-energy of the/:'33 state enhance the longitudinal cross section by about 2% only. A somewhat stronger increase of the latter is achieved by replacing the free pion propagator in the pion-pole term by the in-medium quasi-pion propagator [27] according to
(q2 _ m ~ ) - ' ~ (q~ - ~ ( p ) q ~ - fl(p)m2) -' .
(15)
Using the parametrization from Ref. [28] one finds a ( p ) = 0.445 and fl(p) = 1.036. The resulting propagator-modification increases do'L/d12~ by about 20% over its vacuum value. Note that such an effect has earlier been suggested in Ref. [29] as a possible enhancement mechanism for nuclear longitudinal response functions. Finally let us consider the ratio of p(e,etTr+)n and 3He(e,e'Tr+)3H cross sections which is discussed in Ref. [30] in terms of a quenching of the latter with respect to the former. For an estimate of this ratio we use expressions (8a), (8b), (9) of our model. Then, the pion production cross section on 3He is given in the photon-nucleus cm-frame by ( d o ' / d ~ r ) 3 H e ~ W2A FZ(Q 2) rlcr( d o ' / d ~ . ) p = R (do'Idler)p,
(16)
where (do'/df2~r)p is calculated in the photon-proton cm system, and where we have introduced the quantity R as a short-hand notation for the ratio between the two cross sections. We have also introduced a distortion factor rl,~ which describes the effects of the pion final state interaction. From the PWIA and DWIA calculations mentioned above it follows that r/~. ~ 0.7. For the kinematics of the present experiment, the kinematical factor WA 2 and F2(Q 2) take the values 1.42 and 0.86, respectively. In the PWIA approach, i.e. for rl~r = 1, we obtain the ratio Rcm(PWIA) ~ 1.22, i.e. an enhancement rather than a quenching. However, accounting for the FSI of the pion leads to Rcm(DWIA) ~ 0.85, thus to a quenching. The renormalization of the zl propagator in the nuclear medium may result in a further quenching.
K.L Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
884
For a direct comparison between the two cross sections, one has to refer to a common reference system. To calculate the ratio in the laboratory frame, we have to introduce in Eq. (16) the ratio of the Jacobians (Eq. (4)) for the transformations from the pionnucleus and the pion-nucleon systems to the common laboratory system. This factor turns out to be rather important and it reduces the Rcm ratio by further 40%. This means, that, besides its strong model dependence, the quenching effect discussed in Ref. [30] is a non-covariant quantity. Our final expression for the ratio R in the laboratory frame is similar to that reported in Ref. [30]: Rlab =
( d~r/dl21,rab)3He ~ FZ(Q2A) r/~ (dtr/da~b)p
M AIk~(p ) I
Mplk~l
(17)
In the PWIA and in the DWIA approach, we obtain Rlab(PWIA) = 0.81 and Rlab(DWIA) = 0.56, respectively, which differ considerably from the corresponding ratios with the single cross sections calculated in the respective cm-frames.
Acknowledgements We are grateful to Philip Leconte from DAPNIA, CEA, for his help with the cryogenic target. This work has been supported by the Sonderforschungsbereich 201 of the Deutsche Forschungsgemeinschaft and by the BMBF under Contract Number 06DA665I. One of us (T.S.) acknowledges a grant from the Alexander-von-Humboldt Foundation.
References [ 1] K.I. Blomqvist, W.U. Boeglin, R. B6hm, M. Distler, R. Edelhoff, J. Friedrich, R. Geiges, M. Kahrau, M. Kom, H. Kramer, K.W. Krygier, V. Kunde, M. Kuss, J.M. Laget, A. Liesenfeld, K. Merle, R. Neuhausen, E.A.J.M. Offermann, Th. Pospischil, M. Potokar, C. Rangacharyulu, A. Rokavec, A. Richter, A.W. Richter, G. Rosner, P. Saner, St. Schardt, G. Schrieder, T. Suda, B. Vodenik, A. Wagner, Th. Walcher and St. Wolf, Phys. Rev. Lett. 77 (1996) 2396. 121 See for example E Cardarelli and J. Piekarewicz, Nucl. Phys. A 612 (1997) 429. 131 G. Audit, A. Braghieri, N. D'Hose, V. lsbert, S. Kerhoas, J.M. Laget, M. Mac Cormick, L.Y. Murphy, A. Panzeri, P Pedroni, T. Pinelli, G. Tamas, J. Ahrens, J.R.M. Annand, R. Crawford, P. Grabmayr, S.J. Hall and J.D. Kellie, Nucl. Phys. A 614 (1997) 461. [41 N. d'Hose, G. Audit, A. Bloch, N. de Botton, L. Ghedira, L. Jammes, J.M. Laget, J. Martin, E. Mazzucatto, C. Schuhl, G. Tamas, E. Vincent, M. Rodgers, P. Stoler, P. Argan, A. Braghieri and P. Pedroni, Nucl. Phys. A 554 (1993) 679. [ 51 C.W. Kim and H. Primakoff, in Mesons in Nuclei, Vol. 1, ed. M. Rho and D. Wilkinson (North Holland, Amsterdam, 1979) p. 67. 161 R. Gilman, M. Bernheim, M. Brussel, J. Cheminaud, J.-E Danel, J.-P. Didelez, M.-A. Duval, G. Fournier, R. Frascaria, R.J. Holt, H.E. Jackson, J.-C. Kim, E. Kinney, J.-M. LeGoff, R. Letourneau, A. Magnon, J. Morgenstern, C. Pasquier, J. Picard, D. Poizat, B. Saghai, J. Specht, P. Vernin and E. Warde, Phys. Rev. Lett. 64 (1990) 622. 171 S. Kamalov, L. Tiator and C. Bennhold, Few Body Syst. 10 (1991) 143. [81 M.M. Rosenbluth, Phys. Rev. 79 (1950) 615. [9] J. M. Laget, in New Vistas in Electro-Nuclear Physics, ed. E.L. Tomusiak, H.S. Caplan and E.T. Dressier (Plenum, New York, 1986) p. 361. 1101 D. Drechsel and L. Tiator, J. Phys. G 18 (1992) 449.
K.I. Blomqvist et al./Nuclear Physics A 626 (1997) 871-885
885
[11] L. N. Hand, Phys. Rev. 129 (1963) 1834. 1121 J. Ahrens, H. Backe, D. von Harrach, K.H. Kaiser, E Klein, R. Neuhausen and Th. Walcher, Nucl. Phys. News 2 (1994) 5. [13] R. Neuhausen, Nucl. Phys. (Proc. Suppl.) B 44 (1995) 695; K.I. Blomqvist et al., to be published. 1141 A. Amroun, V. Breton, J.M. Cavedon, B. Frois, D. Goutte, EP. Juster, P. Leconte, J. Martino, Y. Mizuno, X.H. Phan, S.K. Platchkov, I. Sick and S. Williamson, Nucl. Phys. A 579 (1994) 596. 1151 W. Wilhelm, Diploma thesis, University of Mainz, 1993. 1161 C.R. Ottermann, G. KObschall, K. Maurer, K. R6hrich, C. Schmitt und V.H. Walther, Nucl. Phys. A 435 (1985) 688. [171 A.W. Richter, Dissertation, Mainz, Germany (1994). 118] M.O. Distler, Dissertation, Mainz, Germany (1996). 1191 M. Kuss, Dissertation, Darmstadt, Germany (1996). [201 G.F. Chew, M.L. Goldhaber, F.E. Low, and Y. Nambu, Phys. Rev. 106 (1957) 1345. 121 I K.I. Blomqvist, W.U. Boeglin, R. B6hm, M. Distler, D. Drechsel, R. Edelhoff, J. Friedrich, R. Geiges, O. Hanstein, M. Kahrau, M. Korn, H. Kramer, K.W, Krygier, V. Kunde, M. Kuss, A. Liesenfeld, K. Merle, R. Neuhausen, E.A.J.M. Offermann, Th. Pospischil, M. Potokar, A.W. Richter, A. Rokavec, G. Rosner, P. Saner, St. Schardt, H. Schmieden, S. Sirca, L. Tiator, B. Vodenik, A. Wagner, Th. Walcher and St. Wolf, Z. Phys. A 353 (1996) 415. [22] O. Hanstein, Diploma Thesis, University of Mainz, Germany (1993). [23] L. Tiator and D. Drechsel, Nucl. Phys. A 360 (1981) 208. I24] E. Oset and L.L. Salcedo, Nucl. Phys. A 468 (1987) 631; R.C. Carrasco and E. Oset, Nucl. Phys. A 536 (1992) 445. 1251 J.S. McCarthy, I. Sick and R.R. Whitney, Phys. Rev. C 15 (1977) 1396. [261 H. de Vries, C. W. de Jager and C. de Vries, At. Data and Nucl. Data Tables 36 (1987) 495. 127] See also T. Ericson and W. Weise, Pions and Nuclei (Clarendon, Oxford, 1988). [281 M. Kirchbach and A. Wirzba, Nucl. Phys. A 604 (1996) 695. 1291 M. Ericson and A.W. Thomas, Phys. Lett. B 128 (1983) 112. 1301 R.J. Loucks and V.R. Pandharipande, Phys. Rev. C 54 (1996) 32; R.J. Loucks, V.R. Pandharipande, and R. Schiavilla, Phys. Rev. C 49 (1994) 342.