7.7 Discussion of the Result and
Calculation of the Scattering Length
a1-a3
Since the main topic of the measurements with the LH2 target was the
calibration of the calorimeter with photons [Slo98] an optimal setup for a more
precise determination of the Panofsky ratio was not achieved. In order to
reduce the error due to acceptance correction and software cuts, the use of a
properly designed collimator in front of the calorimeter seems to be
preferable. Furthermore the response function of photons must be examined in
more detail, despite the fact that the energy resolution and shower parameters
could be matched well by the simulation. The shape of the distribution of
SCX-photons is very sensitive to an assumed energy resolution function. Thus
even a small deviation between the simulated response and the `real' response
function (which is not significant for monoenergetic photons) resulted in a
relatively high systematical error for this measurement. Although major effort
was put into the understanding of the shower distribution of photons at
different energies a large contribution to the measurement error could not be
avoided.
Author
|
Year,
Reference
|
P
|
error
|
Panofsky et al.
|
[Pan50]
|
(0.94)
|
(0.30)
|
Sargent et al.
|
1955
[36]
|
1.10
|
0.50
|
Cassels
et al.
|
19571
|
1.50
|
0.15
|
Fischer
et al.
|
19581
|
(1.87)
|
(0.10)
|
Kuehner
et al.
|
19591
|
1.60
|
0.17
|
Koller
|
19591
|
1.46
|
0.10
|
Dunaitsev
et al.
|
19601
|
(1.40)
|
(0.08)
|
Derrick
et al.
|
19601
|
1.47
|
0.1
|
Samios
|
19601
|
(1.62)
|
(0.06)
|
Jones
et al.
|
19611
|
1.56
|
0.05
|
Cocconi
et al.
|
[Coc61]
|
1.533
|
0.021
|
Ryan
|
[Rya63]
|
1.51
|
0.04
|
Spuller
et al.
|
[Spu77]
|
1.546
|
0.009
|
This
work
|
|
1.546
|
0.010
|
Table 7-3 Compilation of all published determinations of the Panofsky ratio.
The values in brackets have not been used to calculate the weighted average.
A weighted average of all the measurements of P (listed in Table 7-3)
which are in good agreement gives 1.543±0.006. For the calculation of the
s-wave pion-nuclear cross-section b1, the transition
amplitude at threshold contributes the largest uncertainty. This uncertainty
is limited by the extrapolation to the threshold; but is expected to be reduced
by further measurements with energies closer to the p N threshold and
electron-deuteron scattering [Han97]. However a more accurate P value
can reduce ambiguities in the determination of either or b1.
Since the theoretical calculation of is model dependent, we have taken the
most actual determination of by the E643 collaboration [Kov97]. Their value[37] of
-31.5±0.8*10-3/m p coincides with the
theoretical prediction using either low energy theorem [Bae70] or chiral
perturbation theory [Ber96]. The s-wave scattering length also is available
through spectroscopy of pionic atoms [Sig96]. Nevertheless, the determined
value for a1-a3 differs for both approaches. A recent
value from pionic-atom spectroscopy is at -0.2760±0.0125
[Sch97] as opposed to -0.253[38],
both in units of m p -1. A refined calculation of the
p N s-wave scattering length following Weinberg's current algebra
[Ber96] puts the limits between -0.288 and -0.264. On the other hand
a calculation of the SCX scattering amplitude f based on a compilation
of all available p N scattering data at low energies gives
-0.0248±0.0045 [Mat97] for the extrapolation to the threshold.
However, the latest analysis of the width of the 3p-1s line of pionic Hydrogen
and Deuterium including Doppler-shift corrections determined
-0.2604±0.0043 (preliminary) [Bad98]. Taking P from this
work one obtains -0.252±0.006 for a1-a3
that translates into -0.085±0.002 for b1.
Thus there is evidence that the systematical discrepancy in the obtained
scattering length disappears; although a further reduction of the measurement
errors is needed. In fact, this calls for further studies of the s-wave
scattering amplitude at threshold.
[36] Quoted accordingly to [Rya63]
[37] So far, only the statistical error is
given; a more detailed analysis is to be expected in 1999
[38] No error given [Han97], but typically of
about 3%