y66 ff3 fs7 fc0 sc0 ls4 ws2e">cast doubt on the claims of isospin breaking [7,8];
therefore, it must be investigated whether the newly-
obtained measurements invalidate those claims.
We must stress that the physical quantities appearing
here (i.e., the fit parameters of Sections 3.1-3.4, the scat-
tering lengths and volumes of Section 3.4.2, the phase
shifts of Section 3.4.3, etc.) are not purely-hadronic
quantities since they still contain residual em effects,
which the stage-I em corrections of Refs. [5] do not re-
move. These residual effects relate in particular to the
fact that, in principle, the (unknown) hadronic masses
should be used (in the hadronic part of the π interac-
tion) instead of the physical masses of the proton, of the
neutron, and of the charged and neutral pion. Unfortu-
nately, it is not possible at the present time to assess the
significance of these effects. As a result, we must retain
the cautious attitude of considering our physical quanti-
ties “em-modified hadronic” (as we did in Ref. [1]).
However, the repetitive use of this term is clumsy.
Therefore, we omit it, unless we consider its use neces-
sary as, for instance, in Section 4 and in the captions of
our tables and figures.
N
πp
p
th
j
2
2. Method
2.1. Formalism
The determination of the observables from the hadronic
phase shifts has been given in detail in Section 2 of Ref.
[1]. For scattering, one obtains the partial-wave
amplitudes from Equation (1) of that paper and deter-
mines the no-spin-flip and spin-flip amplitudes via Equa-
tions (2) and (3). Finally, the observables are evaluated
from these amplitudes via Equations (13) and (14). For
elastic scattering, the observables are determined
on the basis of Equations (15)-(20).
π
2.2. Minimisation Function and Scale Factors
Similarly to our previous PSA [1], we make use of the
minimisation function given by the Arndt-Roper formula
[9]. The contribution of the data set to the overall
is:
22
1
j
j
z
z





exp
ij
yth
ith
j
th
y
exp
y
exp
y
exp
2
exp
1
jth
N
jij ij
j
iij
zy y
y





(1)
where denotes the data point of the data
set, ij the corresponding fitted value (also referred to
as “theoretical”), ij the statistical uncertainty of the
ij data point,
j
z a scale factor for the relative nor-
malisation applying to the entire data set,
j
z
the cor-
responding uncertainty (reported or assigned), and
j
N
th
y
the number of data points in the data set. The fitted val-
ues ij are obtained by means of parameterised forms
of the s- and p-wave amplitudes. The values of the scale
factor
j
z
2
are determined (for each data set separately)
in such a way as to minimise
j
. For each data set,
there is a unique solution for
j
z

:



22
exp exp
1
22
exp
1
,
j
j
N
th
ij ijijj
i
jN
th
ij ijj
i
yy yz
z
yy z



(2)
which leads to




2
exp
2
2
min exp
1
2
2
exp exp
1
22
exp
1
j
j
j
th
Nij ij
j
iij
N
th th
ij ijijij
i
N
th
ij ijj
i
yy
y
yy yy
yy z


(3)
22
1min
N
j
j

2
(where N stands for The overall
the number of data sets used in the fit) is a function of the
parameters entering the modelling of the s- and p-wave
amplitudes; these parameters were varied until
at-
tained its minimal value 2
min
.
The part of
2
min
j
th
j
j
z
which represents the pure ran-
dom fluctuations in the measurements of the data
set (i.e., the “unexplained variation” in standard regres-
sion terminology) may be obtained from Equation (3) in
the limit
, which is equivalent to removing the
2
j
z
from the denominator of the second term term
on the right-hand side (rhs) of the expression; we will
2
j
denote this value by
s
t
. The variation which is con-
tained in
2
min
j

2
j
in excess of
s
t
must be associ-
ated with the contribution from the rescaling (floating) of
the data set as a whole; the expression for
22 2
min
jj j
s
cst
 




is


2
2
2exp exp
1
2
22
2
exp exp
11
j
jj
N
th th
jijijijij
i
jNN
sc
th th
ij ijij ijj
ii
zyyyy
yyyy z








.


(4)
Copyright © 2012 SciRes. JMP
E. MATSINOS, G. RASCHE 1371
The scale factors which minimise only the first term
on the rhs of Equation (1) are obtained from Equation (2)
in the limit j
z


:

2
exp exp
2
exp
j
j
N
th
ij ijij
i
jN
th
ij ij
i
yy y
z
yy
ˆ
1
1
ˆ. (5)
The scale factors of Equation (2) are appropriate when
investigating the goodness of the overall reproduction of
a data set in terms of a reference solution (yielding the
fitted values). On the contrary, those of Equation (5) give
maximal freedom to the baseline solution when deter-
mining the offset of a data set (with respect to that solu-
tion) and, as such, are more suitable when the emphasis
is placed on the absolute normalisation, rather than on
the overall reproduction, of a data set.
The statistical uncertainty in the evaluation of the scale
factors
j
z is given by

2
exp
1.
th
ij ij
yy
ˆ
1
ˆ
j
jN
i
z
(6)
When comparing the absolute normalisation of a data
set with a reference solution, the statistical uncertainty of
Equation (6) must be taken into account, along with the
normalisation uncertainty
j
z
of that data set. When
investigating the absolute normalisation of some specific
data sets, namely of those with suspicious absolute nor-
malisation (e.g., see Section 3.4.6), the total uncertainty
22
ˆ
ˆ
j
jj
zzz

1N
 will be used.
2.3. Statistical Tests
For a data set containing Nj data points, j meas-
urements have actually been made, the additional one
relating to the absolute normalisation of the data set.
Since the fit involves the fixing of each
j
z

2
min
j
2
at the value
given in Equation (2), the proper number of degrees of
freedom (hereafter, the acronym DOF will stand for “de-
gree(s) of freedom”, whereas NDF for the “number of
DOF”), associated with the jth data set, is just Nj. This
implies that the quantity of Equation (3) is ex-
pected to follow the
distribution with
j
N

2
min
j
2
DOF.
The essential difference between the present study and
our previous PSA [1] is that only one statistical test for
each data set will be performed here, the one involving
its contribution to the overall
; in Ref. [1],
we instead performed tests for the shape and for the
normalisation of each data set.
The p-value which is evaluated on the basis of
and Nj will be compared with the confidence level min
p
for the acceptance of the null hypothesis (no statisti-
cally-significant effects); in case that the extracted p-value
is below min
p
, the DOF with the largest contribu-
2
mi
j
n value) will be eliminated in the sub- tion (to the
sequent fit. Only one point will be removed at each step,
and the optimisation will be repeated. Data sets which do
not give acceptable p-values (i.e., above min
p
) after the
elimination of two of their data points (the absolute nor-
malisation is also considered to be one data point) will be
removed from the database.
The second difference to Ref. [1], as far as the data
analysis is concerned, relates to the choice of the confi-
dence level which is assumed in the statistical tests; in
Ref. [1], min
p
was set to about (which is
equivalent to a 3
3
2.70 10
effect in the normal distribution).
Herein, we will instead adopt the min
p
value which is
associated with a 2.5
effect; this value is approxi-
mately equal to 2
1.24 10
2
1.00 10
, that is, slightly larger than
πp
πp
πp
πp
, which is “commonly” (among statisticians)
associated with the outset of statistical significance. In
any case, only a few data (five DOF of the elastic-
scattering databases) are affected by this more stringent
acceptance criterion.
3. Results
The repetitive use of the full description of the databases
is largely facilitated if we adhere to the following nota-
tion: DB+ for the database; DB for the
elastic-scattering database; DB+/ for the combined
elastic-scattering databases.
The initial DB+ comprises differential cross sections
(DCSs) [10-18], analysing powers (APs) [19,20], partial-
total cross sections (PTCSs) [21,22], and total (in fact
total-nuclear) cross sections (TCSs) [23,24]. The initial
DB+ consists of 364 data points, distributed among 54
data sets, 26 of which relate to the DCS, 3 to the AP, and
25 (all one- or two-point data sets) to the PTCS and TCS.
The initial DB consists of 336 data points distributed
among 36 data sets, i.e., 27 for the DCS ([12,13], [15-18],
and [25]) and 9 for the AP ([19] and [26-28]).
We now list the measurements which have not been
included in our analysis.
The self-consistency of the πp
elastic-scattering
DCS measurements of Ref. [3] will be addressed in
detail elsewhere [2]. This is an enormous piece of
experimental data, comparable in quantity to the da-
tabase we established in our UZH06 PSA. Prior to
their incorporation into a self-consistent set of data,
the self-consistency of these measurements (as well
as their compatibility with the established database)
must be verified; in Ref. [2], we will come to a nega-
tive result.
2
mi
j
n
Copyright © 2012 SciRes. JMP
E. MATSINOS, G. RASCHE
1372
The 70 πp
DCS measurements of Ref. [29] are
obvious outliers in all exclusive analyses of the low-
energy data; when using “traditional” statistics, the
inclusion of these values in a PSA is bound to intro-
duce spurious effects (e.g., drifting of the parameters
during the optimisation, entrapment of the minimi-
sation algorithms in local minima, failing fits, etc.).
We will further comment on these data in Section
3.4.6.
The 6 (3 for πp
at 94.50 MeV, 3 for πp
at
88.50 MeV) DCS measurements of Ref. [30] have not
appeared in a form which would enable their straight-
forward inclusion in our database. Furthermore, we
are not convinced that the original DCS data may be
retrieved by simply adding the contributions appear-
ing in columns 4 and 5 of Table 1 of Ref. [30].
Given that the 9 existing πp
PTCSs and TCSs
contain a component from CX scattering, they cannot
be used; the inclusion of these data in any part of the
analysis would perplex the discussion on the violation
of the isospin invariance in the hadronic part of the
πN interaction [4].
The inclusion of the 28 AP measurements of Ref. [31]
in the fits would necessitate substantial modifications
in the database structure and in the analysis software.
This is due to two reasons: 1) each of the three data
sets, to which the measurements of Ref. [31] must be
assigned, contains data taken at more than one beam
energy and 2) the last of the data sets contains meas-
urements for both elastic-scattering reactions. Given
the difficulty at present to include these measure-
ments in our fits, we can only use them in testing the
overall consistency of our approach across energies.
(If a significant amount of πp
measurements had
been acquired in that experiment, we could have in-
vestigated the consistency of our approach across re-
actions.) We will return to this subject in Section
3.4.5.
We will also not use in the fits the scattering length
obtained from the experimental result for the strong
shift of the 1s state in pionic hydrogen [32], after it
has been corrected in Ref. [33] also taking into ac-
count the proper contributions of the n
channel;
the difference to the cc
a
value of our UZH06 PSA
(which is almost identical to the value we will obtain
in this work) has been addressed in Ref. [33].
In order to give the data maximal freedom in the proc-
ess of identifying the outliers, the two elastic-scattering
reactions will be analysed separately using simple param-
eterisations of the s- and p-wave K-matrix elements. The
(small at low energies) d and f waves have been fixed
herein from the current SAID solution (WI08) [34]. In
the SAID analysis, the energy dependence of the d and f
phase shifts is determined from the region T > 100 MeV
(i.e., from energies where these contributions are sizable).
The largest of these phase shifts in the energy interval of
this analysis, D15, does not exceed 0.27˚.
For the purpose of fitting, the standard MINUIT
package [35] of the CERN library was used (FORTRAN
version). Each optimisation was achieved on the basis of
the (robust) SIMPLEX-MINIMIZE-MIGRAD-MINOS
chain. All fits of the present work terminated success-
fully.
3.1. Fits to the DB+ Using the K-Matrix
Parameterisations
The parameterisation, which we will now describe, was
introduced (and successfully applied to scattering)
in Ref. [36]. For elastic scattering, the s-wave
phase shift is parameterised as
πp
πp

1
32 322
00 33
cot ,
c
qabc



q
(7)
where c and
are respectively the momentum and
the pion kinetic energy in the centre-of-mass (CM) sys-
tem. The 12-wave phase shift is parameterised accord-
ing to the form
p
32 2
13131
tan c
qd e.


(8)
Since the 32 wave contains the (1232) resonance,
a resonant piece in Breit-Wigner form is added t2
JORAM95 32.70 2 0.9527 5.7974 0.0551
JORAM95 45.10 4 0.9561 12.4590 0.0142
JORAM95 45.10 3 0.9462 9.2581 0.0260
JORAM95 68.60 7 1.0863 14.2673 0.0466
JORAM95 68.60 3 1.0314 2.2747 0.5174
JORAM95 32.20 20 1.0617 21.3392 0.3774
JORAM95 44.60 20 0.9462 29.7408 0.0742
JANOUSCH97 43.60 1 1.0420 0.1682 0.6817
JANOUSCH97 50.30 1 1.0364 0.1557 0.6931
JANOUSCH97 57.30 1 1.0830 4.5370 0.0332
JANOUSCH97 64.50 1 0.9962 0.0010 0.9753
JANOUSCH97 72.00 1 1.3045 4.8348 0.0279
ALDER83 98.00 6 1.0335 5.0919 0.5321
SEVIOR89 98.00 5 0.9882 1.5869 0.9028
HOFMAN98 86.80 11 1.0020 5.8362 0.8841
PATTERSON02 57.00 10 0.9365 11.0276 0.3554
PATTERSON02 66.90 9 0.9985 4.3343 0.8881
PATTERSON02 66.90 10 0.9479 18.0250 0.0545
PATTERSON02 87.20 11 0.9835 8.2469 0.6910
PATTERSON02 87.20 11 0.9945 5.0493 0.9288
PATTERSON02 98.00 12 0.9954 6.9731 0.8594
Copyright © 2012 SciRes. JMP
E. MATSINOS, G. RASCHE
1376
pled from the same distribution), the ratio
2
2
NDF
NDF


ND 321NDF

2
j
πp
F (11)
must be significantly different from 1. In this formula,
the subscripts “+” and “” denote the two elastic-scat-
tering reactions. The ratio F follows Fisher’s (F) distri-
bution. From the two final fits to the truncated databases
using the K-matrix parameterisations, we obtain the score
value of 1.110 for and DOF
in the two separate fits, which is translated into the p-
value of . Therefore, the claim about the dis-
similarity of the two databases cannot be sustained. As a
result, it makes sense to analyse the two reactions in
terms of a common optimisation scheme.
333F
1
1.7 10
In order to give the two elastic-scattering reactions
equal weight, we multiplied for each
data set by min
2
NN
N

w
πp
and for each elastic-scattering data set by
,
2
NN
N

w
N N
2
where and represent the NDF in the two da-
tabases; we then added these quantities for all the data
sets to obtain the overall
value. The application of
these “global” weights for the two reactions was made as
a matter of principle; given that the values of N
and
are very close, the effect of this weighting on our
results is very small.
N
2
The common fit to the truncated DB+/ (detailed in
Tables 1 and 2) was made, using the 14 parameters of
the K-matrix parameterisations given in Sections 3.1 and
3.2. We did not find any additional data points (or data
sets) which had to be removed; we concluded that the
truncated DB+/ are self-consistent. The common fit to
the data yielded a
value of 792.4 for 654 DOF. The
set of the excluded DOF represents 4.57% of the initial
database. In the following, we will use this truncated
DB+/.
3.4. Common Fits to the Truncated DB+/ Using
the ETH Model
The modelling of the hadronic part of the interac-
tion on the basis of the K-matrix parameterisations of
Sections 3.1 and 3.2 is suitable as a first test of the
self-consistency of the two elastic-scattering databases
and as an efficient method for the identification of the
outliers, yet neither does it provide insight into the un-
derlying physical processes nor can it easily incorporate
the important theoretical constraint of crossing symmetry.
In order to accomplish these two tasks, we will next in-
volve in the analysis a model based on Feynman dia-
grams, namely the ETH model. This model was intro-
duced in Ref. [38] and was developed further throughout
the early 1990s. The ability of the ETH model to account
for the low-energy elastic-scattering data has been
convincingly demonstrated over the past two decades.
πN
πp
Δ
ππ
G
The main diagrams on which this isospin-invariant
model is based are graphs with scalar-isoscalar (I = J = 0)
and vector-isovector (I = J = 1) t-channel exchanges, as
well as the N and s- and u-channel graphs. The main
contributions to the partial-wave amplitudes from these
diagrams have been given in detail in Ref. [38]. The
small contributions from the six well-established four-
star s and p higher baryon resonances with masses up to
2 GeV have also been analytically included in the model;
in fact, the only significant contributions come from the
Roper resonance. The tensor component of the I = J = 0
t-channel exchange was added in Ref. [8]; after this (in-
significant) modification, no changes have been made to
the model.
The I = J = 0 t-channel contribution to the amplitudes
is approximated in the model by a broad resonance,
characterised by two parameters,
and
K
. Its exact
position has practically no effect on the description of the
scattering data or on the fitted values of
πN G
and
K
; for a long time, it has been fixed at 860 MeV. The
I = J = 1 t-channel contribution is described by the
-
meson, with
775.49m
MeV [6]; this contribution
introduces two additional parameters: G
and
K
.
The contributions of the s- and u-channel graphs with an
intermediate N involve the coupling constant
π
πNN
N
N
g
and one additional parameter x representing the
pseudoscalar admixture in the vertex; for pure
pseudovector coupling,
πNN
0x
. Finally, the contributions
of the graphs with an intermediate state introduce the
coupling constant π
N
g
and the parameter Z (which is
associated with the spin-1/2 admixture in the
field).
The higher baryon resonances do not introduce any pa-
rameters.
When a common fit of the ETH model to the data is
made using all eight aforementioned parameters, it turns
out that there is a strong correlation between G
, G
,
and x; due to this correlation, it is not possible to deter-
mine the values of all three quantities. We have chosen to
set x to 0; this choice is usually adopted in effective-field
theoretical models of low-energy scattering. The
common fits of the ETH model to the truncated DB+/
will be performed on the basis of seven parameters:
πN
G
,
G
K
K
,
,
, π
N
N
g
, πΔ
N
g
, and Z.
3.4.1. Model Parameters
The choice of the probability value below which data
points must be excluded is difficult. We have adopted
Copyright © 2012 SciRes. JMP
E. MATSINOS, G. RASCHE 1377
here the value of min
p
corresponding to a 2.5
effect
in the normal distribution. Recognising the subjective-
ness in this choice, we consider that, in order to have
confidence in the reliability of our analysis, it is neces-
sary to verify that the fitted values of the seven model
parameters remain stable over a reasonably broad range
of min
p
values; we followed the same strategy in Ref.
[1]. Thus, in addition to min
2
p
1.2410
, the analysis
was performed with a database reduced by using the
min
p
values of about (equivalent to a
3
10
2.703
effect in the normal distribution) and 2
4.55 10
(equivalent to a 2
effect in the normal distribution)1.
Table 3 shows the values of the seven model parame-
ters for the common fits to the truncated DB+/ for the
three selected values of min
p
. The uncertainties shown
correspond to 2
min
p
1.24 10
. In fact, when the uncer-
tainties are calculated with the Birge factor 2NDF
included, they do not vary much with the value of min
p
.
As min
p
is increased, the truncated database fitted shrinks
and so the raw uncertainties increase. However, the fac-
tor 2NDF
decreases as the fit quality improves
(despite the decrease in NDF) and the two effects largely
compensate. Table 3 shows that the results of the fit are
reasonably stable as the criterion for the rejection of data
points is varied2.
We see from Table 3 that the values of π
N
N
g
are
compatible; converted to the usual pseudovector cou-
pling constant3, our result for 2
min
p
1.2410 is

,
4π
NN c
p
fg
m





2
22
ππ
0.0726 14
24π
NN
where c
denotes the mass of the charged pion and
p
m
2
that of the proton. This result agrees well with the
value we had obtained in our previous PSA [1].
The correlation (Hessian) matrix for the seven pa-
rameters of the ETH model is given in Table 4; the num-
bers correspond to the fit for min
p
1.24 10
. This
matrix, together with the uncertainties given in Table 3,
enables the determination of predictions (and of their
associated uncertainties) for the threshold constants, for
the hadronic phase shifts and amplitudes, and for the
observables at any combination of the energy and of the
scattering angle. Table 3 shows that the value of
K
is
consistent with 0; the quality of the fit would deteriorate
very little if this parameter were set to 0. The value of
G
is very little correlated with the values of the other
five parameters. However, these parameters (G
,
K
,
π
N
N
g
, πΔ
N
g
, and Z) are all strongly correlated with each
other. (As expected, the correlations among the model
parameters are smaller when no floating of data sets is
allowed in the fit.)
Our results for the seven model parameters have
shown remarkable stability over the years, from the pe-
riod when the fits were performed to old, outdated phase
shifts (e.g., those of Refs. [39]) to the present times when
the low-energy measurements are directly fitted to.
The database itself has changed significantly over the
past twenty or so years, with the important contributions
from experiments performed at the meson factories by
different research groups, which made use of different
apparatuses and techniques in their experiments. Our
method of applying the em corrections has also changed.
Finally, various approaches have been implemented in
the optimisation (e.g., in the choice of the minimisation
function, from “standard”
πN
2
functions [36], to robust
statistics without any rejection of data [8], to the use of
the Arndt-Roper formula [9] along with pruning the
databases [1]). The observed stability is indicative of the
robustness one obtains in the results when involving the
ETH model in the analysis of the low-energy data.
πN
πN
2
3.4.2. Threshold Constants
From the parameters of the ETH model and their uncer-
tainties given in Table 3 for min
p
1.24 10, as well
as the correlation matrix given in Table 4, we deter-
mined the isoscalar and isovector s-wave scattering
lengths and the isoscalar(isovector)-scalar(vector) p-
wave scattering volumes. The results are:




12 321
00
12 321
00
12 3212 323
11 11
12 32 12323
11 11
1232 1232
1111
120.0033 12,
33
11 0.07698 60,
33
12240.2039 19,
33 33
11 220.1728 18,
3333
1212
3333
c
c
c
c
aa
aa
aaaa
aa aa
aaaa



 


 

 




 



3
1232 12 323
11 11
0.183019,
1111 0.06724 83.
3333
c
c
aaaa

 

min
p
1It must be mentioned that, in Ref. [1], the results for the different
levels had been obtained by applying cuts to the distribution of the
residuals, as this distribution came out in the 3
min
p2.7010

min
p
3
solu-
tion. On the contrary, the analysis here is performed separately for the
different levels.
(12)
min
p
2The effects, which are seen when increasing the value from the
equivalent of a 2.5 to a
effect in the normal distribution, are due
to the removal of the large JORAM95
p
π data set at 44.60 MeV in
the analysis of the data using the K-matrix parameterisations of Section
3.1. A third data point from this data set (the measurement at 14.26˚)
must
b
e excluded at the last step of the iteration, thus resulting in the
removal of the entire data set; were this data set not excluded, there
would have been almost no change in the parameter values shown in
Table 3.
3Some authors redefine 2
πNN
f
, absorbing in it the factor . 4π
Copyright © 2012 SciRes. JMP
E. MATSINOS, G. RASCHE
Copyright © 2012 Sci JMP
1378
min
p2.min
p1.
Table 3. The values of the seven parameters of the ETH model obtained from the common fits to the truncated combined π±p
elastic-scattering databases for three values of pmin (the confidence level in the statistical tests); these three pmin values
correspond to a 3, 2.5, and 2σ effect in the normal distribution, respectively. The uncertainties correspond to the fit for pmin
1.24 × 102.
3
7010
2
2410
 2
min
p4.5510
 uncertainty
(GeV2) 27.43 27.48 27.37 0.86
G
K
0.014 0.016 0.075 0.034
G
(GeV2) 54.71 54.67 55.98 0.61
K
0.66 0.66 1.35 0.41
πNN
g
12.84 12.84 13.07 0.12
g
πN 29.78 29.77 29.31 0.26
Z 0.550 0.552 0.439 0.056
Table 4. The correlation matrix for the seven parameters of the ETH model for the common fit to the truncated combined
π±p elastic-scattering databases for pmin 1.24 × 102.
G
K
G
K
πNN
g
πN
g
Z
G 1.0000 0.5095 0.0886 0.0378 0.1030 0.1602 0.1999
K
0.5095 1.0000 0.7314 0.7974 0.8847 0.9210 0.7176
G
0.0886 0.7314 1.0000 0.9044 0.9038 0.8487 0.8977
K
0.0378 0.7974 0.9044 1.0000 0.9522 0.9284 0.9530
πNN
g
0.1030 0.8847 0.9038 0.9522 1.0000 0.9497 0.9216
g
0.1602 0.9210 0.8487 0.9284 0.9497 1.0000 0.9001
πN
Z
0.1999 0.7176 0.8977 0.9530 0.9216 0.9001 1.0000
Converting these results to the familiar spin-isospin
quantities, we obtain
Res.
 
 
 
321 121
3
3
3 13,
110 66,
.0796 16.
c
c
c
00
32 3 12
11
32312
11
0.0737 16,0.157
0.2090 20,0.03
0.04124 78,0
c
c
c
aa
aa
aa





 
 



(13)
Our results for the s-wave scattering lengths 32
a0
and
12
a
πp
0 in Equations (13) are compatible with those ob-
tained in Refs. [1,8,36]; these values have been very sta-
ble over the last fifteen years. The large quantity of the
elastic-scattering data below 100 MeV obtained at pion
factories since 1980, when analysed without influences
from the data obtained at higher energies, leads to results
for the s-wave scattering lengths (and hadronic phase
shifts) which are significantly different from those ex-
tracted via dispersion relations after also including the
charge-exchange database in the analysis and using
the measurements up to the few-GeV region.
From the results in Equations (13), we obtain

1
21 803 11,
c
12 32
00
0.0
33
cc
aaa
4.6



in good agreement with the value extracted in Ref. [1].
We have already discussed [1,33] the general disagree-
ment (which, at present, is equivalent to a
effect
in the normal distribution) of our value, extracted
from elastic-scattering measurements, with the
result obtained in the pionic-hydrogen experiments at the
Paul Scherrer Institut (PSI) [32], after the corrections of
Ref. [33] are applied. We comment further on this issue
in Section 6.1 of Ref. [4].
cc
a
πp
3.4.3. Hadronic Phase Shifts
The s- and p-wave hadronic phase shifts, obtained from
the common fit to the truncated DB+/ using the ETH
model, are given in Table 5. These phase shifts are also
shown in Figures 1-6, together with the current SAID
solution (WI08) [34] and their five single-energy values
(whenever available).
It is evident from Figures 1 and 2 that our values of
the s-wave hadronic phase shifts 32
0
and
12
0
differ
significantly from the SAID results. Our values of 32
0
are less negative, but converge towards the SAID values
as the energy approaches 100 MeV; for 12
0
, our values
are consistently smaller.
For the p-wave hadronic phase shifts 32
1
,
32
1
, and
12
1
, inspection of Figures 3-5 shows that there is gen-
eral agreement between the two solutions (the differences
do not exceed about 0.1˚). The significant difference in
E. MATSINOS, G. RASCHE 1379
Table 5. The values of the six s- and p-wave em-modified hadronic phase shifts (in degrees), obtained on the basis of the re-
sults of Tables 3 (for pmin 1.24 × 102) and 4.
T (MeV) 3/
0
2
1/ 2
0
(S31)
3/
1
(S11) 2
3/
1
(P33) 2
1/ 2
1
(P31)
1/ 2
1
(P13)
(P11)
20 2.375 (34) 4.189 (27) 1.2787 (94) 0.2239 (45) 0.1588 (37) 0.3687 (80)
25 2.772 (36) 4.673 (29) 1.817 (12) 0.3083 (63) 0.2153 (52) 0.486 (11)
30 3.164 (37) 5.105 (30) 2.431 (15) 0.3996 (83) 0.2747 (68) 0.602 (14)
35 3.555 (37) 5.496 (31) 3.122 (18) 0.497 (11) 0.3361 (86) 0.714 (17)
40 3.949 (37) 5.852 (33) 3.892 (21) 0.599 (13) 0.399 (11) 0.820 (20)
45 4.345 (37) 6.180 (34) 4.744 (23) 0.706 (16) 0.463 (13) 0.918 (23)
50 4.746 (37) 6.482 (37) 5.683 (24) 0.816 (18) 0.527 (15) 1.006 (27)
55 5.151 (38) 6.760 (39) 6.715 (26) 0.931 (21) 0.591 (17) 1.084 (30)
60 5.561 (39) 7.018 (42) 7.845 (27) 1.049 (25) 0.655 (20) 1.150 (34)
65 5.977 (41) 7.256 (46) 9.081 (29) 1.169 (28) 0.719 (23) 1.204 (38)
70 6.397 (44) 7.476 (51) 10.433 (31) 1.293 (32) 0.782 (26) 1.244 (42)
75 6.823 (48) 7.679 (56) 11.909 (35) 1.419 (36) 0.844 (29) 1.271 (46)
80 7.254 (53) 7.865 (61) 13.519 (40) 1.547 (41) 0.906 (32) 1.283 (50)
85 7.690 (59) 8.036 (67) 15.277 (48) 1.678 (45) 0.966 (36) 1.281 (55)
90 8.131 (67) 8.192 (74) 17.193 (59) 1.811 (50) 1.026 (39) 1.263 (60)
95 8.577 (75) 8.334 (81) 19.282 (73) 1.946 (55) 1.084 (43) 1.229 (66)
100 9.028 (85) 8.462 (89) 21.556 (90) 2.083 (61) 1.141 (48) 1.178 (71)
3/2
0+
Figure 1. The em-modified hadronic phase shift
(S31) from the present work (solid curve); the band around our solu-
tion indicates 1σ uncertainties. Shown also is the current SAID solution (WI08) [34] (dashed curve), along with their five sin-
gle-energy values (at T = 20, 30, 47, 66, and 90 MeV).
Copyright © 2012 SciRes. JMP
E. MATSINOS, G. RASCHE
1380
1/ 2
0+
Figure 2. The em-modified hadronic phase shift
(S11) from the present work (solid curve); the band around our solu-
tion indicates 1σ uncertainties. Shown also is the current SAID solution (WI08) [34] (dashed curve), along with their five sin-
gle-energy values (at T = 20, 30, 47, 66, and 90 MeV).
3/2
1+
Figure 3. The em-modified hadronic phase shift
(P33) from the present work (solid curve); the band around our solu-
tion indicates 1σ uncertainties. Shown also is the current SAID solution (WI08) [34] (dashed curve), along with their five sin-
gle-energy values (at T = 20, 30, 47, 66, and 90 MeV). To enable the meaningful comparison of the values contained in this
figure, an energy-dependent baseline δR (= (0.20 × T + 1.54) T × 102, with T in MeV and δR (T) in degrees) was subtracted
from all data.
Copyright © 2012 SciRes. JMP
E. MATSINOS, G. RASCHE 1381
3/2
1
Figure 4. The em-modified hadronic phase shift
(P31) from the present work (solid curve); the band around our solu-
tion indicates 1σ uncertainties. Shown also is the current SAID solution (WI08) [34] (dashed curve).
1/ 2
1+
Figure 5. The em-modified hadronic phase shift
(P13) from the present work (solid curve); the band around our solu-
tion indicates 1σ uncertainties. Shown also is the current SAID solution (WI08) [34] (dashed curve).
Copyright © 2012 SciRes. JMP
E. MATSINOS, G. RASCHE
Copyright © 2012 SciRes. JMP
1382
1/ 2
1
Figure 6. The em-modified hadronic phase shift
(P11) from the present work (solid curve); the band around our solu-
tion indicates 1σ uncertainties. Shown also is the current SAID solution (WI08) [34] (dashed curve), along with their five sin-
gle-energy values (at T = 20, 30, 47, 66, and 90 MeV).
the p-wave part of the interaction occurs for 12
1
; our
values are more negative. The values of 12
1
, obtained in
UZH06 and ZUAS12, are slightly different. This change
is almost entirely due to the current use of a lower
N
value; at the time when the UZH06 PSA was carried out,
the Particle-Data Group recommended the
N
πN
πN
value of
227.5 MeV.
Our phase-shift values are expected to be of interest in
analyses involving the low-energy interaction, as
well as in the determination of the
term (e.g.,
see Ref. [40]).
We will now express our criticism concerning the
SAID results at low energies.
One has the impression that new πN measurements
enter the SAID database without regard of whether
they comprise a self-consistent set and/or of whether
they are at least marginally-compatible with the data
which are already part of their database. Such a strat-
egy could be less problematic, if they had imple-
mented robust statistics in their data analysis; instead,
their results are obtained with a “standard” 2
func-
tion (i.e., the Arndt-Roper formula), and are thus ex-
pected to be sensitive to the presence of outliers in the
database (in particular, to the presence of one-sided
outliers).
The SAID results at low energies are literally swamp-
ed by the measurements at higher energies. In case
that the floating of the data sets is allowed (as is when
using the Arndt-Roper formula in the optimisation), it
is unavoidable that the low-energy behaviour of the
πN amplitudes will be influenced from higher ener-
gies. As a result, the low-energy experiments will be
scaled systematically in such a way as to match the
trends of the amplitudes suitable for the higher-en-
ergy data.
We have not found a published plot from the SAID
group, showing the energy dependence of their scale
factors below 100 MeV.
The distribution of the normalised residuals and the
function which is used in the optimisation are inti-
mately connected. In case that a 2
minimisation
function is used, the distribution of the normalised re-
siduals must, in a self-consistent analysis, be a Gaus-
sian centred at 0. Any effects observed in the distri-
bution of the normalised residuals (e.g., significant
offset, asymmetry of the distribution) are indicative of
problems during the fitting procedure. We have not
found this information in the SAID reports or in their
web site.
By forcing the data from all three reactions into an
isospin-invariant analysis scheme, SAID cannot ex-
plore a possible violation of the isospin invariance in
E. MATSINOS, G. RASCHE 1383
the hadronic part of the πN interaction.
3.4.4. Scale Factors and Normalised Residuals
We will now comment on the distribution of the scale
factors
j
z
πp
πp
obtained from the common fits using the
ETH model. In a “healthy” fit made on the basis of the
Arndt-Roper formula, the data sets which must be scaled
“upwards” should (more or less) be balanced by those
which must be scaled “downwards”. Additionally, the
energy dependence of the scale factors over the energy
range of the analysis should not be significant. If these
prerequisites are not fulfilled, the parametric forms used
in the fits cannot adequately reproduce the data over the
entire energy range. For both the (
Figure 7) and
(
Figure 8) elastic-scattering data sets, the values of
j
z which lie above and below 1 roughly balance each
other and there is no discernible energy dependence4.
Evidently, there is no subrange of the entire 30 to 100
MeV energy interval in which the data is better or worse
fitted than for the rest of the range.
A second issue which must be investigated in a fit in-
volving the minimisation of any χ2 function is whether
the distribution of the normalised residuals rij, defined as
exp
exp
th
j
ij ij
ij
zy y
y

ij
r
(see Equation (1)), evaluated on the basis of the optimal
parameter values, is Gaussian5. In practice, one fits a
Gaussian function to the distribution of the normalised
residuals, i.e.,

2
e,
Br r
fr A

2
and investigates the quality of the fit (expressed through
the corresponding
value and the NDF in the fit), as
well as the asymmetry of the fitted distribution (e.g., ex-
pressed through the deviation of the extracted value of
r from 0). The distribution of the normalised residuals
is shown in Figure 9, along with the optimal Gaussian
function. The 2
value of this fit was 33.2 for 22 DOF
in the fit, whereas

2
1.9 4.710r

0.038B
2
, i.e., compatible
with 0. For the sake of completeness, we also give the
optimal value and the uncertainty of the parameter B:
; the expectation value for B is 0.5. 0.544
3.4.5. Reproduction of the MEIER04 Measurements
We will now discuss the MEIER04 measurements (which
have not been included in our fits). We have created
Monte-Carlo predictions for the AP corresponding to
each of their 28 data points. For the three experimental
data sets, the resulting values of min
were 12.5, 7.0,
and 16.1, for 12, 6, and 10 DOF, respectively. The val-
ues of the scale factor for the three data sets (in the
same order) are 1.011, 0.973, and 1.039; the reported
normalisation uncertainty of the data is 3.5%. It is evi-
dent that our hadronic phase shifts reproduce the MEI-
ER04 measurements very well; even the smallest
p-value does not fall below . This is a good
test of the consistency of our approach in describing the
energy and angular dependence of the elas-
tic-scattering data.
2
9.6 10
πp
πp
πN
2
3.4.6. Reproduction of the BERTIN76 Measurements
We will now comment on the reproduction of the old
measurements of Ref. [29], which are “traditionally”
considered outliers in almost all modern PSAs. Reported
in Ref. [29] were DCS measurements obtained in
a broad angular interval, at seven beam energies (20.80,
30.50, 39.50, 51.50, 67.40, 81.70, and 95.90 MeV). The
experimental group did not report any normalisation un-
certainties for their measurements; it cannot be excluded
that, at those times, the absolute normalisation was not
seriously investigated in the experiments.
We assigned rough normalisation uncertainties to the
BERTIN76 data, on the basis of the values obtained from
the modern low-energy experiments which properly re-
ported this quantity, and analysed these measurements as
if they comprised the entire DB+ at low energies. The
analysis in terms of our general K-matrix parameterisa-
tions of Section 3.1 revealed that the data set at 67.40
MeV had to be eliminated due to its (very bad) shape.
We were able to fit the remaining data successfully, and
obtained the final
value of 55.0 for 53 DOF in the
fit.
We subsequently investigated how well the phase-shift
solution of the present study reproduces the BERTIN76
data; it failed. The basic problem with the BERTIN76
measurements lies with their absolute normalisation, not
with their shape (though the shapes of the data sets at
39.50 and at 95.90 MeV do not pass our mi n
p
criterion).
For instance, for the data set at 20.80 MeV, j
ˆ1.349z
when the overall uncertainty ˆ
Δ
j
z
ˆ
(defined at the end of
Section 2.2) is equal to 0.081; this is the most striking
discrepancy in the data. The extracted
j
z
ˆ1.027z
factors de-
crease almost linearly with T, reaching j
at
95.90 MeV. In view of these large energy-dependent
effects in their absolute normalisation, we will continue
excluding the BERTIN76 data in our PSAs.
4. Discussion
In the present work, comprising the first of three papers
4The linear fit to the scale factors for the π+p reaction yields the inter-
cept of 1.012 ± 0.018 and the slope of (0.9 ± 2.4) × 104 MeV1; the
linear fit to the scale factors for the πp elastic-scattering reaction yields
the intercept of 1.031 ± 0.018 and the slope of (4.5 ± 2.4) × 104
MeV1.
5Of course, the values of

1
j
zz
j
, appearing in the second term on
the rhs of Equation (1), must also be included in the distribution of the
no
r
malised residuals.
Copyright © 2012 SciRes. JMP
E. MATSINOS, G. RASCHE
Copyright © 2012 SciRes. JMP
1384
Figure 7. The scale factors zj for the π+p data, obtained from the common fit to the truncated combined π±p elastic-scattering
databases using the ETH model (see Section 3.4). The values, corresponding to the two data sets which were freely floated
(see Table 1), have not been included; in the case of free floating,
20
jsc
χ.
Figure 8. The scale factors zj for the πp elastic-scattering data, obtained from the common fit to the truncated combined π±p
elastic-scattering databases using the ETH model (see Section 3.4). The value, corresponding to the data set which was freely
floated (see Table 2), has not been included; in the case of free floating,
20
jsc
χ.
E. MATSINOS, G. RASCHE 1385
Figure 9. The distribution of the normalised residuals, obtained from the common fit to the truncated combined π±p elastic-
scattering databases using the ETH model (see Section 3.4). Also shown (solid curve) is the optimal Gaussian fit to the data.
dealing with issues of the pion-nucleon () interaction
below pion laboratory kinetic energy of 100 MeV, we
report the results of a new phase-shift analysis (PSA) of
the elastic-scattering data, using the electromag-
netic (em) corrections of Refs. [5].
πN
πp

2
min
j
2
There are two main differences to the approach we set
forth in our previous PSA [1], both pertaining to the me-
thod for the exclusion of outliers, single data points and
entire data sets, in the optimisation phase. We now per-
form only one test for each data set on the only rele-
vant quantity, namely on its contribution to the
overall
. In Ref. [1], we instead performed several
tests (on shape, normalisation, etc.) for each data set; the
use of only one test led to the exclusion of fewer data
compared to Ref. [1]. The second difference concerns the
imposition of a more stringent acceptance criterion of the
null hypothesis in the statistical tests. Herein, we raised
the minimal p-value (min
p
) for the acceptance of the null
hypothesis from the equivalent of a 3 to a 2.5
effect
in the normal distribution; the latter value is closer to the
“common” choice (of most statisticians) as the outset of
statistical significance.
Similarly to Ref. [1], we first investigated the self-
consistency of the low-energy elastic-scattering
databases, via two separate analyses carried out (first) on
the and (subsequently) on the elastic-scat-
tering data using simple K-matrix parameterisations. We
found that it is possible to obtain self-consistent data-
bases by removing the measurements of two and
one data sets, as well as a few single data points;
the removal of these outliers resulted in enormous reduc-
tions in the minimal
πp
πp
πp
πp
πp
2
values for the separate fits to
the two elastic-scattering databases using the K-matrix
parameterisations. We give all the details concerning the
accepted data sets in Tables 1 and 2; these details may be
useful in other analyses. The aforementioned results were
obtained without imposing any theoretical constraints,
save for the expected low-energy behaviour of the s- and
p-wave K-matrix elements and the Breit-Wigner form
(see Equations (9) and (10)) for the contributions of the
resonant terms.
The ETH model of Ref. [38], based on s- and u-chan-
nel diagrams with N and
in the intermediate states,
and
and
t-channel exchanges, was subsequently
fitted to the truncated combined elastic-scattering
databases. The model-parameter values showed reason-
able stability when subjected to different criteria for the
removal of data (see Table 3). Our result for the pseu-
dovector coupling constant is
πp
πNN 0.07260.0014
.
On the basis of the results of the model fits, we obtained
the em-modified hadronic scattering lengths and volumes
(see Section 3.4.2), as well as the s- and p-wave em-
modified hadronic phase shifts up to T = 100 MeV (see
Table 5 and Figures 1-6). Large differences in the
s-wave part of the interaction were found when compar-
Copyright © 2012 SciRes. JMP
E. MATSINOS, G. RASCHE
1386
ing our hadronic phase shifts with the current SAID solu-
tion (WI08) [34] (see Figures 1 and 2); there is general
agreement in the p waves, save for the em-modified ha-
dronic phase shift 12
1
Apart from analysing our results in terms of the as-
sumed confidence level in the statistical tests, we also
investigated the possibility of a bias in the analysis. To
this end, we examined the energy dependence of the
scale factors
.
j
zπp
πp
ˆ
, shown in Figures 7 (for the data)
and 8 (for the elastic-scattering data), as well as
the characteristics of the distribution of the normalised
residuals (Figure 9). We did not find any significant de-
viation for these quantities from the expectations in a
successful optimisation.
We are grateful to G. J. Wagner for drawing our atten-
tion to the statistical uncertainties of the scale factors
j
z
for free floating (see Equation (6)). We would like to
thank W. S. Woolcock for his comments and sugges-
tions.
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