Nuclear Physics A 690 (2001) 473–493
www.elsevier.nl/locate/npe
Simultaneous measurements
of the p + d → (A = 3) + π reactions
GEM collaboration
M. Betigeri i , J. Bojowald a , A. Budzanowski d , A. Chatterjee i , J. Ernst g ,
L. Freindl d , D. Frekers h , W. Garske h , K. Grewer h , A. Hamacher a ,
J. Ilieva a,e , L. Jarczyk c , K. Kilian a , S. Kliczewski d , W. Klimala a,c ,
D. Kolev f , T. Kutsarova e , J. Lieb j , H. Machner a,∗ , A. Magiera c ,
H. Nann a,1 , L. Pentchev e , H.S. Plendl k , D. Protić a , B. Razen a,g ,
P. von Rossen a , B.J. Roy i , R. Siudak d , J. Smyrski c , R.V. Srikantiah i ,
A. Strzałkowski c , R. Tsenov f , K. Zwoll b
a Institut für Kernphysik, Forschungszentrum Jülich, Jülich, Germany
b Zentrallabor für Elektronik, Forschungszentrum Jülich, Jülich, Germany
c Institute of Physics, Jagiellonian University, Krakow, Poland
d Institute of Nuclear Physics, Krakow, Poland
e Institute of Nuclear Physics and Nuclear Energy, Sofia, Bulgaria
f Physics Faculty, University of Sofia, Sofia, Bulgaria
g Institut für Strahlen- und Kernphysik der Universität Bonn, Bonn, Germany
h Institut für Kernphysik, Universität Münster, Münster, Germany
i Nuclear Physics Division, BARC, Bombay, India
j Physics Department, George Mason University, Fairfax, VA, USA
k Physics Department, Florida State University, Tallahassee, FL, USA
Received 4 January 2000; revised 24 November 2000; accepted 5 December 2000
Abstract
A stack of annular detectors made of high-purity germanium was used to measure simultaneously
pd → 3 H π+ and pd → 3 He π0 differential cross sections at beam momenta of 750 MeV/c,
800 MeV/c, and 850 MeV/c over a large angular range. The extracted total cross sections for the
pd → 3 He π0 reactions bridge a gap between near threshold data and those in the resonance region.
cm /m yields
The ratio of the cross sections for the two reaction channels taken at the same η = pπ
π
2.11 ± 0.08 indicating that a deviation from isospin symmetry is very small. 2001 Elsevier Science
B.V. All rights reserved.
* Corresponding author.
E-mail address:
[email protected] (H. Machner).
1 On leave from IUCF, Bloomington, Indiana, USA.
0375-9474/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 5 - 9 4 7 4 ( 0 0 ) 0 0 6 9 9 - 0
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M. Betigeri et al. / Nuclear Physics A 690 (2001) 473–493
1. Introduction
The study of the two reactions
pd → 3 H π+
3
pd → He π
and
0
(1)
(2)
is of interest because of the underlying reaction mechanism. On the one hand, the deuteron
is the ideal case for the impulse approximation. In this approximation the incident proton
interacts with one target nucleon leaving the third nucleon as a spectator. Thus, both
reactions are dominated by the underlying two elementary reactions
pp → dπ+
0
np → dπ .
and
(3)
(4)
On the other hand, the struck nucleon is bound in the deuteron and its momentum
distribution will strongly influence the reaction since the presently chosen lowest
beam momentum of 750 MeV/c is still below the pion threshold in nucleon–nucleon
interactions.
The model of the underlying elementary reactions (3) and (4) goes back to Ruderman [1]. Although many groups have theoretically studied the two reactions along this
path, only moderate success has been achieved. The various approaches differ in the treatment of the bound-state wave functions, additional interactions, distortions, inclusion of
the deuteron D-state and so on. An excellent overview of the theoretical work is given by
Canton et al. [2].
Studies of pion production on the nucleon as well as on light nuclei induced by protons
have been shown to be dominated by excitation of the P -wave (1232) isobar. Only
very close to threshold one is sensitive to the interesting pion S-wave [3]. While for
the elementary reactions (3) and (4) , no interference between these two waves can
occur because of the symmetry in the entrance channel, reactions (1) and (2) show
very strong forward–backward asymmetry containing information on the phases. The
NN → dπ models are therefore especially successful in the resonance region where the
interference is small compared to the P -wave contribution. Germond and Wilkin [4]
showed that for collinear kinematics only two independent amplitudes exist. These were
fitted to forward and backward scattering angles, to differential cross-section data and to
deuteron tensor analyzing powers [6]. It was shown in Ref. [4] that close to threshold
the energy dependence of these two amplitudes is represented by a linear expansion
cm /m and θ the emission angle in the centre of
in terms of x = η cos(θ ) with η = pπ
π
mass system. However, when this model with the fitted parameters is extended to higher
energies, it can neither describe the total cross sections nor the differential cross sections
at forward and backward angles. The situation for the total cross section is shown in
Fig. 1. The data are from Refs. [7–11]. For the data from Ref. [10], charge symmetry
was assumed to hold. Pickar et al. [11] extracted total cross sections from the forward
and backward differential cross sections of Kerboul et al. [6] by making additional use
of some systematics. These data are omitted here since they are much larger than direct
M. Betigeri et al. / Nuclear Physics A 690 (2001) 473–493
475
Fig. 1. Excitation function for the indicated reaction. Data are shown as symbols with error bars and
are from Refs. [7–11]. The result of the Germond–Wilkin model is shown as dashed curve, and the
pp → dπ+ reaction divided by 160 as solid curve.
measurements. The near-threshold data with η 0.5 are reproduced by the Germond–
Wilkin model. The data in the resonance region follow the scaled pp → dπ+ cross section.
Clearly, in the intermediate region where the S–D-wave interference effects are expected
to be large, data are missing. We, therefore, performed measurements for reaction (2)
in that interval. In these experiments, also the production of pd → 3 H π+ (reaction (1))
was measured simultaneously. A comparison of reactions (1) and (2) permits the study of
isospin symmetry breaking. The beam energy was chosen to be below the resonance
in order to avoid problems due to different resonance masses. On the other hand, the
momentum seems to be sufficiently large to minimize Coulomb effects in the exit channel.
Isospin symmetry is known to be only an approximate symmetry. In addition to static
breaking due to Coulomb effects, the dynamic effects due to the mass differences between
the up and down quark are expected to contribute [12]. Isospin symmetry predicts a value
of 2.0 for the ratio of the cross sections. There have been several studies in the literature of
the ratio
R(θ ) =
dσ (θ, pd → 3 H π+ )/dΩ
dσ (θ, pd → 3 He π0 )/dΩ
(5)
for the same beam momentum. However, it is not clear whether this is the proper quantity
to study isospin symmetry conservation because of the different pion masses and mass
differences between 3 H and 3 He. The great advantage of the present work compared to
measurements with constant pion centre of mass momentum pπ is that a simultaneous
measurement avoids a lot of systematic errors. This principal advantage is lost when the
detector does not allow simultaneous measurements as in Refs. [14–17]. Furthermore,
these studies are restricted to only a few (up to 4) angles. All previous results for R are
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larger than 2. The two reactions were also measured almost over the full angular range by
the same group with the same apparatus at 500 MeV, however at different time [9,18]. We
found from these measurements an integral ratio R = 1.82 ± 0.18. It should be mentioned
that the authors never drew conclusions about charge independence from the cross-section
data but studied analyzing powers. Köhler [19] studied the ratio R(θ ) in the framework of
the Ruderman approach [1,13] and found the ratio of the form factors to be the dominant
source of deviation from 2.
2. Experimental procedure
We have measured reactions (1) and (2) simultaneously employing a detector with large
momentum and geometrical acceptance for both heavy recoiling nuclei at the same time.
Proton beams with momenta of 750 MeV/c, 800 MeV/c, and 850 MeV/c were extracted
from the COSY accelerator and focussed onto a target cell containing liquid deuterium. It
had a diameter of 6 mm and a thickness of 6.4 mm [20] with windows of 1.5 ţm mylar.
The excellent ratio of deuterium to heavier nuclei in the window material reduced empty
target events to a negligible level in contrast to all previous studies. The beam intensity
was measured by different monitor counter arrangements being individually calibrated
for each setting of the accelerator. The calibration was done by measuring the number
of scattered particles in the monitor counters as a function of the beam particles. The latter
were measured with the trigger hodoscope in the focal plane of the magnet spectrograph.
This number is of course much larger and leads to dead time in the hodoscope. The
beam intensity was then reduced by debunching the beam between the ion source and
the cyclotron injector. For sufficiently small beam intensity, the relation between monitors
and hodoscope is linear. Because of the chosen geometry, the counting rate in the monitor
counters was small, thus avoiding errors due to pile up. The detector system is the so
called GEM detector, consisting of the “GErmanium Wall” [21] and the Q3D2Q Magnetic
spectrograph [3] at COSY. Here we give only some additional details specific for this
experiment. The germanium wall consisted of three high purity germanium detectors with
radial symmetry with respect to the beam axis as shown in Fig. 2. The first detector (called
Quirl-detector) measures the position and the energy loss of the penetrating particles. The
active area of this diode is divided on both sides by 200 grooves. Each groove is shaped as
an Archimedes’ spiral covering an angular range of 2π with opposite directions on the front
and rear side, respectively. The energy detectors are mainly used for measuring the energy
loss of the penetrating particles or the total kinetic energy of stopped particles, respectively.
These detectors are divided into 32 wedges to reduce the counting rate per division leading
to a higher maximum total counting rate of the total detector.
Fig. 3 shows the response of the germanium wall for reaction particles from the
interaction of 850 MeV/c protons with deuterons. Clearly visible are bands attributed to
protons, deuterons, tritons, and 3 He. The latter two ions are produced in the two reactions
of present interest pd → 3 H π+ and pd → 3 He π0 . Protons and deuterons are from elastic
scattering and from pd → pdπ0 reactions. The faint areas for protons and deuterons are
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477
Fig. 2. The “germanium wall”.
Fig. 3. Energy loss in the Quirl as function of the total energy deposited in all detectors. The bands
visible are due to detected protons, deuterons, tritons and 3 He.
due to the situation where one particle with higher energy generates a trigger for a lowenergy particle falling below the discriminator thresholds. The information deduced from
the germanium wall are energy, emission vertex and particle type. They were converted
to a four-momentum vector. These measurements together with the knowledge of the four
momenta in the initial state yield the missing mass of the unobserved pion by applying
conservation of momentum, energy, charge and baryon number.
In the off-line analysis, soft gates were applied to the triton and 3 He loci in Fig. 3. This
leads to some background in the missing mass spectrum but avoids throwing away good
events. An example for 3 H and 3 He emission at a beam momentum of 850 MeV/c is shown
in Fig. 4. The data are shown as histograms. The good peak to background ratio is obvious
as well as the excellent statistics. The peak in the case of 3 He emission is broader than for
3 H emission. This is due to the larger energy straggling in the target of the doubly charged
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Fig. 4. Missing mass distributions for the two reactions at a beam momentum of 850 MeV/c.
particle compared to the one for single charged particle. In the final analysis this effect is
corrected. The data could be fitted by a second-order polynomial for the background and
a Gaussian with variable width for the peak. In order to remove this background, spectra
of observables not depending on the emission angle were analyzed. For 3 He the E–E
curve was linearized and projected to the E-axis. For 3 H the centre of mass momentum
was used. The resulting spectra show a peak on a smooth background. In a second step,
the background was fitted by smooth functions and subtracted for each bin in cos(θ ). The
efficiency of the analysis procedures were studied by Monte Carlo calculations [22].
For 3 He particles being stopped in the Quirl detector, the kinematical relation between
emission angle and energy was applied. Unfortunately, this was only possible for the runs
at 750 MeV/c and 800 MeV/c, while in the runs at 850 MeV/c a hardware coincidence
between the Quirl and the first energy detector was required. Tritons being emitted under
zero degree in the laboratory system with the smallest energy were detected in the magnetic
spectrometer applying hardware and software cuts as reported in Ref. [3]. Finally, the
data were corrected for reduced efficiencies due to nuclear absorption in the detector
material [23].
3. Experimental results
The measured angular distributions in the centre of mass system are shown in Figs. 5, 6,
and 7 for beam momenta of 750 MeV/c, 800 MeV/c, and 850 MeV/c, respectively.
All data are from the measurements with the germanium wall except the triton zero
degree measurement corresponding to cos(θ ) ≈ −1 in the centre of mass system. These
M. Betigeri et al. / Nuclear Physics A 690 (2001) 473–493
479
Fig. 5. Differential cross sections for the p + d → 3 H + π+ reaction at a beam momentum of
750 MeV/c are shown as full dots. The data are obtained by coincidence measurements with
the germanium wall or by the magnetic spectrograph (only cos(θ) = −1). In the case of the
p + d → 3 He + π0 reaction data obtained by coincidence measurements are shown as full dots,
those obtained by kinematic relation as full and open squares. The data with uncertain efficiency
because of the vicinity of the corresponding detector element to the central hole are shown by the
open symbol and are excluded from the further analysis. Calculations in the Locher–Weber model
are shown as solid curves.
points were obtained with the help of the magnetic spectrograph. Because of the small
acceptance left over for this device by the hole in the germanium wall the count rate was
rather small. An additional uncertainty is due to the size of the acceptance which may
result from a slightly inclined beam together with the rather thick target and a beam having
widths of σx = 0.66 mm and σy = 0.53 mm. We assume this additional uncertainty to
be 20% and add it to the count rate error in quadrature. For 3 He emission at smaller
energies for the 750 MeV/c and 800 MeV/c beam momenta the analysis was based on
the almost linear relation between kinetic energy and emission angle for the two-body
reaction, as mentioned above. This method leads to a larger background than in case
of coincidence measurements especially due to the triton events with not too different
kinematics. Therefore, the statistical error bars are much larger for these data which are
shown as squares in Figs. 5 and 6. These points show some deviations from the points
obtained from the coincidence measurements because of threshold effects. The Quirl had
a thickness of 1.3 mm. 3 He ions with approximately 70 MeV kinetic energy are stopped
in this detector. This corresponds to cos(θ ) ≈ −0.70 and cos(θ ) ≈ −0.86 for the beam
momenta of 750 MeV/c and 800 MeV/c, respectively. Ions with slightly larger energy can
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Fig. 6. Same as Fig. 5 for 800 MeV/c. In addition a calculation from Canton et al. [2] is shown as
long-dashed curve.
Fig. 7. Same as Fig. 6 for 850 MeV/c with an additional calculation by Ueda [24]. In the latter the
amplitudes have been normalized by a factor of 0.1.
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481
fall below thresholds in the trigger electronics or the ADC. This effect leads to a reduction
of counting rate for larger emission angles and to an increased counting rate in a similar
interval below these angles due to the smaller energy deposit of these ions in the Quirl
detector. (In principle a hole should appear but this is smeared out due to the longitudinal
straggling.) We have studied this effect with Monte Carlo calculations and found that
differential cross sections can be different up to 11%. The data are finally corrected for
this effect.
The shown error bars include the statistical uncertainty as well as the systematic one
from background subtraction. This is larger in the case of tritium emission than for 3 He
emission while the pure statistical uncertainty has the opposite behaviour.
The cross sections for both reactions show a backward peaking of the A = 3 nuclei
which corresponds to a forward peaking of the pion. The angular distribution for the
smallest beam momentum shows an exponential decrease with increasing cos(θ ). For
the larger beam momenta an additional component at forward angles shows up which
seems to be less angle dependent. Model calculations also shown in Figs. 5–7 will
be discussed further down. Only the statistical errors are shown. In addition, there are
systematic uncertainties due to target thickness (5%), luminosity calibration (7%) and
detector response and analysis method (3%). Only the last contribution is different for
the two reactions. The same is true for the energy-loss corrections in the target. This leads
to a total systematic error of up to 9.1%.
4. Discussion
4.1. Comparison with other data
We will first compare the present results with older data. This is done in Fig. 8. The
data from Refs. [25,27] are from pion absorption and are transformed assuming charge
symmetry as well as time reversal invariance and hence these data were transformed using
detailed balance. All previous data have larger error bars than the present ones. For the
charged exit channel, the older data seem to follow a steeper angular dependence than the
present data. The cross sections from Källne et al. [25] for the forward angles are smaller
than the present ones. For the neutral exit channel there is a striking agreement between the
present data and those of Cameron et al. [9], but the data from Carroll et al. [28] differ. The
data from Rössle et al. [10] are for the related nd → tπ0 reaction. They have extremely
large error bars. The trend in the latter data to rise at forward angles is not seen in the
present data and seems to be not statistically significant.
4.2. Forward–backward and total cross sections
In order to extract total cross sections as well as to extrapolate the present measurements,
analytic functions were fitted to the angular distributions. Legendre polynomials of
fourth and fifth order in the case of 850 MeV/c were found to yield the smallest
χ 2 /degree of freedom and the final results are from these fits. Also an exponential with
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Fig. 8. Comparison of the present data at 850 MeV/c (full dots) with data from Refs. [25–27] for the
3 H π+ reaction (left part) and from [9,10,28] for the 3 He π0 reaction (right part).
a constant could account for the data, however with much larger uncertainties. This may
have its origin in the different number of fitted parameters. The total cross section is
obtained by integrating the fitted angular distributions over the full solid angle. Due
to the ambiguity in the order of the Legendre polynomial and the corrections applied
to the data close to the detector acceptance limit a systematic uncertainty shows up.
In case of the 3 He emission the backward-angle range causes some errors. For the
two lower beam momenta the different method of measurement without coincidence
compared to the forward emitted particles introduced larger error bars and a specific
structure as is discussed above. The influence of this structure on the total cross section
is estimated by simulations. As a result, a systematic error in addition to the fit error
shows up. For 850 MeV/c no such data exist because of the coincidence requirement
in the hardware trigger. The influence of the missing data was studied by truncating
the corresponding triton angular distribution and the 3 He distribution from Ref. [9]
suggesting a systematic uncertainty of 10%. The total cross sections for both reactions
are compiled in Table 1 with the first error the statistical error and the second error
the systematical one as discussed. In addition, there is a systematic uncertainty of 9.1%
from target thickness, beam intensity measurements and cuts during the data evaluation
processes. The deduced cross-section values increase with beam momentum for both
reactions.
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Table 1
Total cross sections for the two reactions as functions of the beam momentum
Momentum (MeV/c)
σ 3 He π0 (ţb)
σ 3 H π+ (ţb)
750
800
850
12.48 ± 0.17 ± 0.38
14.43 ± 0.13 ± 0.38
15.54 ± 0.30 ± 1.55
25.42 ± 0.30 ± 1.3
28.05 ± 0.18 ± 0.20
34.90 ± 0.23 ± 0.05
The first error is the statistical uncertainty from the Legendre polynomial fit the second the
systematical uncertainty due to the ambiguities in the data at backward angles and degree of the
Legendre polynomial fitted. In addition there is an overall uncertainty of ±9.1% due to target
thickness, luminosity calibration and analysis method. The uncertainty from the fit contains the
systematical uncertainty resulting from background subtraction.
Fig. 9. Excitation function for the pd → 3 He π0
reaction at a 3 He emission angle of 0 degree. The
present results are derived by extrapolation of a
fit with an exponential plus a constant to the data.
Fig. 10. Same as Fig. 9 but for an emission angle
of the 3 He of 180 degrees. The present results
are derived by extrapolation of the fitted Legendre polynomials (full dots) including the uncertainties due to different extrapolation methods.
We proceed by extrapolating to cos(θ ) = −1 and cos(θ ) = +1 in the same way.
The results are shown in Figs. 9 and 10, respectively, together with the results from
Refs. [6,7,9,11]. Only the data from Kerboul et al. [6] are direct measurements; all other
data are obtained by us by extrapolation of fitted Legendre polynomials. For the distribution
from Ref. [9] at a momentum of 1090 MeV/c one point, being off the distribution by
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almost four standard deviations, was excluded from the analysis. The best fitting Legendre
polynomial had an order of 6 in this case. The extrapolation method leads to rather
large uncertainties especially for cos(θ ) = 1. Unfortunately, the measurement of the cross
sections for cos(θ ) = ±1, which was foreseen to be done with the magnet spectrometer,
failed, because of too thick material in the focal plane with respect to the large stopping
power of 3 He. The Legendre polynomials have the tendency to underestimate the cross
sections at cos(θ ) = 1. For these cases we show the more robust results of the exponential
fits. They are in nice agreement with the data from Kerboul et al. [6]. The present backward
angle results seem to be slightly smaller than the measurements by Kerboul et al. [6].
These large backward cross sections may be the origin for the too large total cross sections
extracted from them by Pickar et al. [11] as discussed in the introduction. The present data
agree with the result of Cameron et al. [9] as could be expected from Fig. 8.
The total cross sections are compared in Figs. 11 and 12 with data from Refs. [7–11]
for the neutral exit channel and from Refs. [18,25,27,29–31] for the charged exit channel. In addition to the cases mentioned above, the data from Aniol et al. [29] and Weber
et al. [27] were transformed using detailed balance. The present cross sections for the pd →
3 He π0 reaction seem to nicely interpolate between the near-threshold data and those in the
resonance region. The calculation employing the low-energy parameters of the Germond–
Wilkin model [4] shown as dashed curve and previously discussed in the introduction seem
to come close to the present data point at 750 MeV/c bombarding momentum. Also shown
is a calculation in the framework of the Locher–Weber prescription [5], which will be discussed below.
Fig. 11. Excitation function for the pd → 3 He π0 reaction. The present data (full dots) are compared
with other data indicated by different symbols. The inner error bars denote the statistical errors,
the outer bars the systematical errors. The long dashed curve is again the low-energy fit to the
Germond–Wilkin model, the solid curve is a calculation in terms of the Locher–Weber model. The
short-dashed and dotted curves are similar calculations but using a different 3 He form factor and
different normalisation (see Section 4.3).
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485
Fig. 12. Excitation function for the pd → 3 H π+ reaction. The present data are indicated by full
dots, others by different symbols. The statistical errors for the present data are in the size of the
symbols. Also shown is the systematical error. The data from Refs. [25,27,29] are from negative
pion absorption on 3 He. The curves are model calculations [2] using the Bonn B potential (solid
curve), Bonn A (dotted curve), and Paris interaction (dashed curve).
For the pd → 3 H π+ reaction, there are a few points in the range of the present data.
Almost all of them stem from studies of negative pion absorption transformed by detailed
balance assuming charge symmetry and time-reversal invariance. This disagreement was
already noted by the authors of the latter data [29].
4.3. Model comparison
As discussed in the introduction, there are numerous models for the mechanism of these
reactions. Here, we restrict ourselves to comparisons with published calculations for beam
momenta close to the present ones or perform such calculations in the very transparent
Locher–Weber model [5]. The differential cross section in this model is given by
2 dσ
dσ
= SK FD (q) − FE (q)
pp → dπ+
(6)
dΩ
dΩ
with S, a spin factor, K, a kinematical factor, and FD , the direct form factor and FE , the
exchange form factor, i.e. an elastic πd scattering after pion emission from the incident
proton. The form factors were evaluated with emphasis on the short-range components of
the deuteron and the triton. This is achieved by fitting the free parameters in a Hulthén
function to the deuteron-charge form factor and similarly for tritium, the parameters for
the Eckart function and the 3-pole function to the tritium-charge form factor. These form
factors are structure functions obtained from elastic electron scattering and should not be
mixed with its component the monopole-charge form factor, having a similar shape. The
latter is the Fourier transform of a monopole density containing both the s-state and d-state
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deuteron wave functions times the sum of the proton and neutron form factors. The figure
captions in Ref. [5] are somewhat confusing in this respect. It should be mentioned that
the contribution of the magnetic dipole form factor to this quantity is very small. Since the
experimental input into the fitting functions is rather old, we have compared these functions
with newer measurements and find nice agreement for the deuteron form factor with results
from Platchkov et al. [32] and for the 3 He form factor with those from Amroun et al. [33].
However, recent data from JLAB [34,35] show that the assumed Hulthén function is too
small for large momentum transfers.
We have fitted Legendre polynomials to the cross section of the elementary reaction
including new data in the threshold region [3]. The energy dependence of the Legendre
coefficients was fitted in three different but overlapping regions: threshold region,
resonance region and above, although that interval is presently not of interest. Since
the model neither treats the internal structure of the nuclei nor angular momenta, it
cannot predict spin observables. One model ambiguity is the choice of the momentum
of the struck nucleon. Since ignoring the Fermi motion in the deuteron leads to a wrong
resonance position, the second option of Ref. [5] was chosen, i.e. ignoring of the Fermi
motion in the triton. The emission angle was always assumed to be the same for both
reactions.
Calculations employing the 3-pole wave function were found to be always smaller in the
exponential part than those using the Eckart wave function. Here we present results only
for the Eckart wave function. Such calculations are shown in Figs. 5–7. A normalization
factor of 1.5 was always applied. For 3 He emission, just an isospin factor of 0.5 was used.
The small differences result from the different kinematics.
At 750 MeV/c, the data are larger than the calculations, whereas for the higher
beam momenta the part of the angular distribution showing an exponential dependence
is nicely reproduced. The strong increase in the calculations for forward angles is not
supported by the data. This increase is reduced if direct and exchange contributions
are added incoherently. The total cross-section prediction within this model is shown
in Fig. 11. It seems to work well in the resonance region, but fails in the threshold
region.
In order to study the momentum range being sensitive to the present reactions we have
introduced a step function in the integrals of the form factors. The results become almost
independent if the truncation omits the range larger than 4 fm−1 . However, in this range
the quadrupole form factor is already larger than the monopole form factor, thus making
the D-state important which is missing in the present model.
In order to study the sensitivity of the model calculations to the form factors we have
chosen the options used by Fearing [13]: again the Hulthén form for the deuteron but
exponential, Gaussian, and Irving–Gunn forms for the 3 He using the range parameters
from Ref. [13] but the same normalization as Locher and Weber. In these calculations
the same pion centre-of-mass momentum in the elementary as well as the p + d reaction
was assumed. Exponential and Gaussian give almost the same results as the Eckart
function. The Irving–Gunn function leads to an almost symmetric angular distribution.
Such a behaviour is in disagreement with the data. It is only this function which yields
M. Betigeri et al. / Nuclear Physics A 690 (2001) 473–493
487
a different shape. The three others are very close to each other in the range up to
4 fm except for the Eckart function, which drops down for radii below the maximum
at 1 fm. The resulting excitation function for the choice of an exponential form is
also shown in Fig. 11. We have applied two normalizations: one to fit the present data
and the same curve multiplied by 1.5 to investigate the near threshold region. The
maximum is shifted towards smaller beam momenta when compared to the calculation
employing the Eckart form. This is due to the assumptions about the effective energy and
momentum in the pp → dπ+ subamplitude. The calculation with larger normalization
yields larger cross sections than the Eckart form. However, the resulting cross-section
values are below the experimental points. This is similar to the results obtained by
Falk [36]. His model is based on the same physical picture but more refined to also
predict spin observables. The calculation underestimates the cross-section data close to
threshold by a factor of two, but overestimates data in the resonance region by a factor of
approximately 2.6.
Canton and Schadow [2] performed much more rigorous calculations; again the
pp → dπ+ reaction is the underlying mechanism. The three-body wave function was
calculated from nucleon–nucleon potentials employing three-body calculations. In Figs. 6
and 7 the calculations are compared to the data. A normalization constant of 3 is
used. The exponential behaviour at backward angles is reproduced. Forward angles are
overestimated. Total cross sections as a function of the previously defined quantity η
are shown in Fig. 12 together with data. The calculations were performed for different
nucleon–nucleon potentials: Bonn A, Bonn B, and Paris potential. The quality of the
data in the resonance region does not allow discrimination between different potentials.
It is surprising that in the range below η = 1 all calculations with different potential
choices start to merge whereas it is just this range where there are large differences in
the calculations for the pp → dπ+ reaction [2]. It is interesting to note that the calculations
for this reaction agree only with the experimental data when a strong final state between
the deuteron and the pion is included in the calculation. The inclusion of this finalstate interaction increases the cross section by typically an order of magnitude which is
surprising with respect to other calculations. This model also fails to reproduce the large
differential cross sections shown in Fig. 10.
Ueda [24] attacked the problem by splitting the many-body process into coupled
multi-, three-, and two-body systems which were treated in a relativistic and unitary
approach. Numerical input is obtained by adjusting potential parameters to two-body
scattering amplitudes: NN−NN, NN−dπ, πd−πd, NN−N , πd−N , and N −N .
A calculation without further normalization is shown in Fig. 7. It should be mentioned
that a damping factor of 0.1 was applied to the otherwise too large amplitudes. The
large cross sections at backward angles are reproduced. The oscillations at forward
angles are not supported by the data. The success of this model may be that a lot of
different graphs like NN and N correlations (see Ref. [37]), multiple scattering effects
(Ref. [13]) and three-nucleon mechanism (Ref. [38]) are automatically included in this
approach.
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M. Betigeri et al. / Nuclear Physics A 690 (2001) 473–493
5. Isospin symmetry
It was already pointed out by Ruderman [1] that data of the present type, i.e. differential
cross sections for the two isospin related reactions, should allow for testing the validity
of isospin symmetry. However, the two isospin related reactions have different Q values
leading to differences even for the same beam momentum.The ratios for the total cross
sections yield 1.97 ± 0.05 for the three measurements (see Table 1). We compare the two
exit channels on the level of differential cross section. The mean ratios are 1.89 ± 0.04,
1.87 ± 0.03 and 1.64 ± 0.04 for the beam momenta of 750 MeV/c, 800 MeV/c and
850 MeV/c, respectively. However, the measured ratios are not constant, they show a
trend to decrease with increasing emission angle. This may point to different shapes of
the angular distributions for the two reactions. Already the different pion masses lead
to trivial deviations from 2 for the ratio Eq. (8). To take this trivial deviation in some
approximate way into account, we study the ratio for the differential cross sections instead
cm and the same η value. The angular distributions for the p + d → 3 H + π+
at the same pπ
cm and η value are obtained by linear interpolating the angular
reaction at the same pπ
cm and η values, respectively. The principal advantage
distributions for the neighbouring pπ
of cancellation of systematical errors is lost in part by doing so and the linear interpolation
may introduce another systematical error in the percent range. The results are 1.91 ± 0.04
cm = 108.8 MeV/c and 1.97 ± 0.03 for pcm = 136.6 MeV/c. The ratios for the
for pπ
π
two reactions obtained for constant η values are shown in Fig. 13. This procedure yields
for η = 0.807, 1.905 ± 0.034 and for η = 1.014, a ratio 2.145 ± 0.013. In all cases we
have corrected the 3 H + π+ data for the Coulomb effects by the usual Gamow factor
being 2 to 3% depending on the momentum. All three methods lead to a weighted mean
Fig. 13. The ratio of the differential cross sections for the two reactions as a function of the emission
angles in the centre of mass system for the indicated values of η. A constant of 2 was added to the
upper data set.
M. Betigeri et al. / Nuclear Physics A 690 (2001) 473–493
489
value close to two. However, it is not clear on which level one should compare. In
addition, trivial effects due to different pion masses or pion or beam momenta remain.
We may, therefore, compare matrix elements instead of differential cross sections. They
are calculated according to
2
dσ (θ ) (2sb + 1)(2sB + 1) pbcm Wa Wb WA WB
=
M(t)
(7)
cm
4
2
dΩ
p
s
(2π) h̄
a
for a reaction a + A → b + B. In Eq. (7), si denotes the spins, picm the momenta of the initial and final state in the c.m. system, Wi the total energies and s the total energy squared.
The matrix element |M| is a function of the four-momentum squared t = t[cos(θ )] and is
measured in fm3 MeV. Eq. (7) is the relativistic formulation of the well-known nonrelativistic result (see Eq. (4.8) in Ref. [39]).
Such elimination of the phase-space factor from the data should lead to a more rigourous
result as was pointed out by Silverman et al. [17]. The ratio of the matrix elements squared
are 2.06 ± 0.06, 2.02 ± 0.03 and 1.73 ± 0.04 for the same total energy s, 1.95 ± 0.04 and
cm , and 1.87 ± 0.03 and 2.08 ± 0.01 for the same η values.
1.98 ± 0.04 for the same pπ
cm .
It is interesting to note that the correction is smallest (less than 1%) for constant pπ
It is always the measurement at 850 MeV/c which has a ratio much smaller than 2.
cm or constant η is
The influence of this measurement on the results for constant pπ
small. If one neglects this one ratio, the mean values are 2.03 ± 0.02, 1.97 ± 0.03 and
cm and η, respectively. Which criterion is the best is not
2.02 ± 0.07 for constant s, pπ
clear. The latter two methods suffer from different measurements and the interpolation
procedure resulting in some systematic error being at least of the order of the statistical
error.
One may inspect the matrix elements directly. This is done in Fig. 14. The matrix
elements for the lowest beam momentum show an almost exponential dependence on the
four-momentum transfer squared. For the highest momentum, an additional component for
large momentum transfer shows up, the same as the differential cross sections. The data
definitely show a dependence on the beam momenta. This is in contrast to the conjecture
by Silverman et al. [17] that the invariant matrix element squared is independent of the
beam energy which, by the way, is not supported by any theory.
The y-axis for each beam momentum are adjusted to each other for the two reactions
by a factor of two, in order to get an estimate of the validity of isospin symmetry. There
is a common tendency in the data: the matrix element for the 3 He + π0 reaction is smaller
than the isospin corrected one for the 3 H + π+ reaction for small momentum transfer
and larger for large momentum transfer. This behaviour becomes more pronounced with
increasing beam momentum. This may have its origin in different slopes for different s
and t, the data decrease with increasing pion momentum. This leads to a t-dependence of
the ratio
R=
|M(p + d → 3 H + π+ , t)|2
|M(p + d → 3 He + π0 , t)|2
(8)
for data obtained at the same beam momentum. The ratios, not Coulomb corrected, show
an decrease with increasing momentum transfer which may have its origin in different
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M. Betigeri et al. / Nuclear Physics A 690 (2001) 473–493
Fig. 14. The deduced squared matrix elements for the two reactions and for the three bombarding
energies. The y-axes for the pd → 3 H π+ reactions are on the left side while those for the
pd → 3 He π0 reactions are on the right side. For each beam momentum, the two axes are adjusted
by a factor of two in order to simulate the naive isospin ratio.
kinematics. The ratio for 850 MeV/c is flat for high momentum transfers and in this region much smaller than two. Whether these effects result from interferences with Coulomb
effects or are due to increasing importance of resonance excitation with increasing bombarding energy needs further studies.
M. Betigeri et al. / Nuclear Physics A 690 (2001) 473–493
491
Fig. 15. The ratios of the matrix elements squared for the three beam momenta. A constant of 4 is
added to the results for 750 MeV/c and 2 for those at 800 MeV/c.
6. Conclusion
In summary, we have measured simultaneously the two reactions p + d → 3 H + π+ and
p + d → 3 He + π0 in the intermediate region between the near threshold and the resonance
range. A liquid deuterium target with very thin walls reduced empty-target corrections to
a negligible level. By the present method problems due to normalization to beam current,
target thicknesses, solid angle, dead time corrections, etc., are avoided. It is the first time
that the ratio of the two reactions is studied over a large angular range. The differential
cross sections as well as the total cross sections for the p + d → 3 He + π0 reaction bridge
the previous gap between data in the threshold region and the resonance region. However,
for the p + d → 3 H + π+ reaction, a discrepancy with data for the time-reversed reaction
shows up. This was already mentioned by Aniol et al. [29] and may have its origin in the
differences between initial and final states due to Coulomb effects in the two reactions.
It is found that models based on the input of p + p → d + π+ cross sections or amplitudes
yield total cross sections for the p + d reactions which are close to p + p → d + π+
cross sections scaled down by factors of 80 and 160 for the p + d → 3 H + π+ reaction
and p + d → 3 He + π0 reaction, respectively. Since these data follow a different energy
dependence than the present data, the calculations do the same. The enhancement of the
calculated differential cross sections for large momentum transfer is not supported by the
present data. It seems that none of the models discussed above is able to account for
the experimental data over a large energy range on an absolute base. It is worthwhile
to mention that the S-wave model of Locher and Weber [5] agrees best with the data in
a range where the D-wave in the deuteron contributes most strongly.
The deduced isospin ratio of 1.64 up to 1.89 for the differential cross sections is smaller
than the one found by Silverman et al. [17] of 2.36 ± 0.11. The measured ratio of the matrix
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M. Betigeri et al. / Nuclear Physics A 690 (2001) 473–493
elements squared for the two lower beam momenta of 2.03 ± 0.02 is in agreement with the
result R = 2.13 ± 0.13 obtained by Harting et al. [15] at a beam energy of 591 MeV.
At this energy the differences in the kinematics are expected to be small. Köhler [19]
estimated that, if the effect of the Coulomb force in the 3 He wave function is reduced,
the ratio changes from 2 to 2.14 ± 0.02 for 600 MeV. Introducing such an additional
correction leaves almost no room for isospin symmetry breaking. Such calculations for
different beam momenta and with modern wave functions will be helpful in answering
the question on the sensitivity of isospin symmetry breaking in the present two reactions.
Further measurements will improve the present error bars.
Acknowledgements
We are grateful to the COSY operation crew for their efforts making a good beam.
Support by BMBF Germany (06 MS 568 I TP4), Internationales Büro des BMBF (X081.24
and 211.6), SCSR Poland (2P302 025 and 2P03B 88 08), and COSY Jülich is gratefully
acknowledged.
References
[1] M. Ruderman, Phys. Rev. 87 (1952) 383.
[2] L. Canton, G. Cattapan, G. Pisent, W. Schadow, J.P. Svenne, Phys. Rev. C 57 (1998) 1588;
L. Canton, W. Schadow, Phys. Rev. C 56 (1997) 1231.
[3] M. Drochner et al., Nucl. Phys. A 643 (1998) 55.
[4] J.-F. Germond, C. Wilkin, J. Phys. G 16 (1990) 381.
[5] M.P. Locher, H.J. Weber, Nucl. Phys. B 76 (1974) 400.
[6] C. Kerboul et al., Phys. Lett. B 181 (1986) 28.
[7] V.N. Nikulin et al., Phys. Rev. C 54 (1996) 1732.
[8] A. Boudard et al., Phys. Lett. B 214 (1988) 6.
[9] J.M. Cameron et al., Nucl. Phys. A 472 (1987) 718.
[10] M. Dutty, Diploma thesis, Freiburg, 1981;
E. Rössle et al., in: R.D. Bent (Ed.), Proc. Conf. on Pion Production and Absorption in Nuclei,
AIP Conf. Proc., Vol. 79, 1982, p. 171.
[11] M.A. Pickar et al., Phys. Rev. C 46 (1992) 397.
[12] A.M. Bernstein, Phys. Lett. B 442 (1998) 20, and references therein.
[13] H.W. Fearing, Phys. Rev. C 16 (1977) 313, and references therein.
[14] A.V. Crewe et al., Phys. Rev. 118 (1960) 1091.
[15] D. Harting et al., Phys. Rev. 119 (1960) 1716.
[16] J.W. Low et al., Phys. Rev. C 23 (1981) 1656.
[17] B.H. Silverman et al., Nucl. Phys. A 444 (1985) 621.
[18] J.M. Cameron et al., Phys. Lett. 103B (1981) 317.
[19] H.S. Köhler, Phys. Rev. 118 (1969) 1345.
[20] V. Jaeckle, K. Kilian, H. Machner, Ch. Nake, W. Oelert, P. Turek, Nucl. Instrum. Methods A 349
(1994) 15.
[21] M. Betigeri et al., Nuclear Instrum. Methods 421 (1999) 447.
[22] W. Garske, PhD thesis, Universität Münster, 2000.
[23] H. Machner et al., Nucl. Instrum. Methods A 437 (1999) 419.
M. Betigeri et al. / Nuclear Physics A 690 (2001) 473–493
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
T. Ueda, Nucl. Phys. A 505 (1989) 610.
J. Källne, J.E. Bolger, M.J. Devereaux, S.L. Verbeck, Phys. Rev. 24 (1981) 1102.
G.J. Lolos et al., Nucl. Phys. 386 (1982) 477.
P. Weber et al., Nucl. Phys. A 534 (1991) 541.
J. Caroll et al., Nucl. Phys. A 305 (1978) 502.
K.A. Aniol et al., Phys. Rev. C 33 (1986) 1714.
E. Aslanides et al., Phys. Rev. 39 (1977) 1654.
W. Dollhopf et al., Nucl. Phys. A 217 (1977) 381.
S. Platchkov et al., Nucl. Phys. A 510 (1990) 740.
A. Amroun et al., Nucl. Phys. A 579 (1994) 596.
L.C. Alexa et al., Phys. Rev. Lett. 82 (1999) 1374.
D. Abbott et al., Phys. Rev. Lett. 82 (1999) 1379.
W.R. Falk, Phys. Rev. C 50 (1994) 1574, Phys. Rev. C 61 (1994) 034005.
A.M. Green, M.E. Sainio, Nucl. Phys. A 329 (1979) 477.
J.M. Laget, J.F. Lecolley, Phys. Lett. B 194 (1987) 177.
D.H. Perkins, Introduction to High Energy Physics, Addison-Wesley, 1987.
493