We calculate Ay in neutron-proton scattering for the interactions models
WJC-1 and WJC-2 in the Covariant Spectator Theory. We find that the recent 12
MeV measurements performed at TUNL are in better agreement with our results
than with the Nijmegen Phase Shift Analysis of 1993, and after reviewing the
low-energy data, conclude that there is no Ay problem in low-energy np
scattering.Comment: 5 pages, 2 figures, accepted by PL

Alfred Stadler

Braun

de Swart

Franz Gross

Gross

Machleidt

Salpeter

Stadler

Stoks

Weisel

Physics Letters B

English

arXiv

AbstractWe calculate Ay in neutron–proton scattering for the interactions models WJC-1 and WJC-2 in the Covariant Spectator Theory. We find that the recent 12 MeV measurements performed at TUNL are in better agreement with our results than with the Nijmegen Phase Shift Analysis of 1993, and after reviewing the low-energy data, conclude that there is no Ay problem in low-energy np scattering

Gross, Franz

Stadler, Alfred

Elsevier - Publisher Connector 

Physics Letters B 668 (2008) 163–166Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Is there an Ay problem in low-energy neutron–proton scattering?
Franz Gross a,b,∗, Alfred Stadler c,d
a Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
b College of William and Mary, Williamsburg, VA 23187, USA
c Departamento de Física da Universidade de Évora, 7000-671 Évora, Portugal
d Centro de Física Nuclear da Universidade de Lisboa, 1649-003 Lisboa, Portugal
a r t i c l e i n f o a b s t r a c t
Article history:
Received 29 July 2008
Accepted 19 August 2008
Available online 26 August 2008
Editor: W. Haxton
We calculate Ay in neutron–proton scattering for the interactions models WJC-1 and WJC-2 in the
Covariant Spectator Theory. We ﬁnd that the recent 12 MeV measurements performed at TUNL are
in better agreement with our results than with the Nijmegen Phase Shift Analysis of 1993, and after
reviewing the low-energy data, conclude that there is no Ay problem in low-energy np scattering.
© 2008 Elsevier B.V. Open access under CC BY license.In a recent paper, Braun et al. (designated BR08) report on a
new set of very precise measurements of Ay in elastic neutron–
proton scattering at 12.0 MeV neutron lab energy [1]. They com-
pare their experimental observables to the results produced by the
Nijmegen Phase-Shift Analysis [2] of 1993 (PWA93) and by the CD-
Bonn potential model [3], also ﬁtted to the data and producing
phase shifts consistent with PWA93. They observe that the pre-
dictions of the PWA93 as well as the CD-Bonn differ considerably
from the new data and interpret this as evidence for a signiﬁcant
shortcoming of the phase shifts and, consequently, of all poten-
tial models ﬁtted to them. In a model study, starting from the
CD-Bonn potential, they ﬁnd that increasing the charged-pion cou-
pling constant g2
π±/4π , while keeping the neutral-pion coupling
constant unchanged at g2
π0
/4π = 13.6, brings the theoretical pre-
dictions into better agreement with the new data.
In this Letter, we take up the issues raised in [1] and consider
them in light of a recently ﬁnished new analysis of np scattering
within the Covariant Spectator Theory (CST) [4,5]. As described in
detail in Refs. [5,6], the scattering amplitude M for two-nucleon
scattering in CST is obtained by solving a covariant integral equa-
tion of the form
M = V − V GM, (1)
where V is the irreducible kernel (playing the role of a potential)
and G is the intermediate state propagator. The Bethe–Salpeter
(BS) equation [7] has a similar structure, but in the BS theory the
four-momenta of the particles in intermediate states are subject
only to the conservation of total four-momentum P = p1 + p2, so
loop integrations over intermediate states are over four indepen-
* Corresponding author at: Thomas Jefferson National Accelerator Facility, New-
port News, VA 23606, USA.
E-mail address: gross@jlab.org (F. Gross).0370-2693 © 2008 Elsevier B.V.
doi:10.1016/j.physletb.2008.08.037
Open access under CC BY license.dent variables. The CST equation maintains four-momentum con-
servation, but constrains the energy components by placing one of
the two intermediate particles on its positive-energy mass shell.
Since P is ﬁxed, both intermediate energies become functions of
the three-momenta only, and all intermediate loop integrations re-
duce to three dimensions, as in the non-relativistic theory. In spite
of this considerable simpliﬁcation, the equation remains covariant
because the on-mass-shell constraint itself is a covariant condi-
tion. This framework has been generalized to other systems and
applied successfully to many problems, in particular also to the
three-nucleon system [8,9].
Our recent ﬁt to the np scattering data [5] uses the CST with a
kernel approximated by the sum of one-boson-exchange (OBE) in-
teractions. Our best model, designated WJC-1, uses 8 bosons with
27 parameters and provides a high precision ﬁt to the 2007 data
base (χ2/Ndata = 1.06). A new phase shift analysis emerges from
this ﬁt, with phase shifts that differ signiﬁcantly in some cases
from the PWA93 analysis. Another model (WJC-2) was chosen to
be as simple as possible, and requires only 6 bosons with 15 pa-
rameters. It does almost as well, with χ2/Ndata = 1.12.
Before making a detailed assessment, we point out that the
errors quoted by BR08 combined systematic and statistical errors to-
gether in quadrature. A better procedure is to treat the system-
atic normalization error as an independent experimental degree of
freedom, so that the χ2 for the data set would be
χ2t =
N∑
i=1
(oi/Z − ti)2
(δoi/Z)2
+ (1− 1/Z)
2
(δsys/Z)2
, (2)
where oi and ti are the measured and the calculated value of
the observable at point i, δoi and δsys are the statistical errors at
point i and the systematic error, Z is a factor [chosen to minimize
Eq. (2)] by which the data and errors can be divided to correct for
the systematic error and improve the agreement with theory, and
164 F. Gross, A. Stadler / Physics Letters B 668 (2008) 163–166Fig. 1. Neutron–proton analyzing power at 12.0 MeV. Lines are WJC-1 (solid), WJC-2 (dashed) and Nijmegen PWA93 (dotted). Left panel shows BR08 data scaled by WJC-1
(solid circles) and WJC-2 (half ﬁlled squares). Right panel shows the WE92 [10] data (half ﬁlled squares) and the BR08 data (solid circles) both scaled by PWA93.Table 1
Statistical errors for the data set BR08 of Ref. [1]. The systematic error is 1.5%
Angle Ay Error(stat) Angle Ay Error(stat)
32.6 0.00854 0.00066 96.3 0.01198 0.00056
40.5 0.01231 0.00061 104.2 0.01110 0.00055
48.5 0.01451 0.00061 112.2 0.00662 0.00061
56.5 0.01443 0.00059 120.2 0.00558 0.00064
64.4 0.01560 0.00059 128.2 0.00483 0.00056
72.4 0.01659 0.00062 136.0 0.00372 0.00067
80.5 0.01470 0.00056 143.8 0.00287 0.00079
88.4 0.01386 0.00053
the last term is the additional contribution to the χ2 coming from
the renormalization of the data. We contacted the authors of BR08
who told us that the systematic error was about 1.5%, and pro-
vided us with the original statistical errors, reproduced in Table 1
[11].
Using the new errors given in Table 1 and the new models
(and phase shift analyses), we reach a different assessment of the
impact of the BR08 experiment. Figs. 1 and 2 present different
comparisons of the 12 MeV Ay BR08 measurements with the pre-
dictions of the Nijmegen PWA93, and models WJC-1 and WJC-2.
Table 2 gives the χ2/Ndata for each of these theories and Table 3
surveys the previous Ay measurements for all laboratory energies
less than 20 MeV. From these ﬁgures and tables we draw the fol-
lowing conclusions:
First, without even looking at the data, observe that the predic-
tions of both of our models WJC-1 and WJC-2 differ substantially
from the PWA93 prediction (the theoretical curves shown in both
panels of Fig. 1 are identical), especially at the smaller angles. This
is not unexpected; it is a consequence of the difference between
our phase shifts and those of PWA93. This may have signiﬁcant
impact on other observables as well.
Next, Table 2 shows that the scale factor Z differs for each
model, and hence there is no one “correct” data set. For example, to
accommodate PWA93, the data is increased by over 6% (four stan-
dard deviations), giving a large normalization contribution to the
χ2 but lowering the collective effect of the statistical errors on
each of the 15 measured points. This renormalization of the data
lowers the χ2/Ndata from about 6.6 to about 3.85. In the end, even
with this renormalization the PWA93 model is the least successful
in describing the data. For comparison, Fig. 1 (right panel) and Ta-
ble 2 also show previous Ay measurements performed at TUNL
at the same energy, designated WE92 [10]. Qualitatively, they ap-
pear to agree well with the new measurements, and also with the
theoretical models, with considerably smaller χ2/Ndata than BR08,
owing mainly to their larger statistical errors.Fig. 2. Ratios of the 12.0 MeV BR08 data for the analyzing power to the predictions
of WJC-1 (1), WJC-2 (2), and PWA93 (N).
Table 2
The average χ2 from Ref. [1] compared to the average χ2 obtained by scaling
the data and errors in Table 1 [with the scaling factor Z determined by mini-
mizing Eq. (2)]. The scaled WE92 [10] data, also measured at 12 MeV, are shown
for comparison. The Nijmegen rejection criteria say that a set with this many data
(15 points plus one normalization error) should be retained only if χ2/Ndata < 2.26.
Note that the BR08 data almost meets this criteria with model WJC-2
χ2/Ndata
Model BR-Ref. [1] BR-scaled Z(BR) WE92
PWA93(1)a 6.621 3.849 0.9367 1.71
PWA93(2)a 6.654 3.861 0.9365 1.71
WJC-1 3.631 3.014 0.9690 1.61
WJC-2 2.377 2.387 1.0029 1.16
a Our ﬁtting procedure uses the effective range expansion. The numbers shown
for PWA93(2) use WJC-1 parameters, which give the best ﬁt to the overall data set
at all energies. Nijmegen does not give 1 S0 parameters, but PWA93(1) uses 3 S1
parameters taken from Ref. [12], which does not ﬁt the overall data set as well.
How are we to interpret these large BR08 χ2? Does it mean
that all of the theories are wrong? In this context we must re-
member that a phase shift analysis gives the most general theoret-
ical description of scattering below the pion production threshold, and
while several theoretical models might generate the same set of
phase shifts, the best phase shifts provide the best model indepen-
dent description of the data. At low energies, only a few partial
waves contribute noticeably to the NN scattering amplitude, and
of those only the S-waves are not small. In principle, this leaves
the remaining partial waves with large uncertainties. However, the
low-energy phase shifts are linked to the much better determined
F. Gross, A. Stadler / Physics Letters B 668 (2008) 163–166 165phases at higher energies through the requirement of a smooth
continuous energy dependence, which limits their uncertainties
and relates observables at different energies. While a continuous
energy dependence arises automatically when a potential model
is used, it is imposed also in a multi-energy phase shift analysis,
although in a phenomenological way. Thus, any new data set effec-
tively competes with the entire phase shift analysis, which is based
in turn on all of the 3788 data currently included in the 2007 data
base. If a new data set disagrees with the predictions of the phase
shift analysis it also disagrees with all of the other data.
So the BR08 data disagrees with the latest phase shift analy-
ses (and therefore the entire 2007 np data base), but does this
represent a real incompatibility? Is the χ2/Ndata of this data set
too high to be tolerated in a quantiﬁable sense? The Nijmegen
group used a statistical criteria based on the following consider-
ation: from a given model (in our case a phase shift analysis or
a potential model) one calculates the observables contained in a
measured dataset. If the model described the observables exactly,
in other words, if we had the “correct” theory, one should obtain
χ2/Ndata ≈ 1 as a result of the unavoidable and random statis-
tical errors in the experimental data. This presupposes that each
individual data point in the set is normally distributed around the
theoretically predicted value with a standard deviation equal to the
statistical error.
One can then ask: based on the theoretical model, and allow-
ing for random ﬂuctuations of the size given by statistical errors,
what is the probability that one will obtain a χ2/Ndata as large
as or larger than the actually observed value for the data set?
If that probability becomes very small, for instance smaller than
some predeﬁned value, one may conclude that the model and the
data are not compatible with each other. By the same line of ar-
guments, getting very small values of χ2/Ndata can also be too
improbable, leading to a second criteria testing data sets for too
low χ2/Ndata.
The Nijmegen criteria adopts a critical probability of about
0.27%, in analogy to the usual “3σ -criteria” for testing individ-
ual normally distributed measurements. Accordingly, for Ndata = 16
(15 data points plus one common normalization error), the high-
est admissible value of χ2/Ndata is 2.26. From Table 2 we see that
the BR08 data set is not compatible, in this sense, with any of the
models, although it comes very close to WJC-2.
How signiﬁcant is this incompatibility, and could it be due, in
part, to a problem with the BR08 data themselves? Fig. 2 shows
the ratio of the BR08 Ay measurements to the theory for three
different theories. There is a rather pronounced disagreement be-
tween the data and all models at smaller angles, but the “break”
in the set at about 110◦ may be more signiﬁcant. The theories
are in good agreement with each other at these larger angles, yet
none of them can reproduce this behavior. Eliminating, for exam-
ple, the two data points at 112◦ and 120◦ would already bring the
χ2/Ndata of BR08 with respect to both WJC-1 and WJC-2 below
the critical value (although not for PWA93). A small increase in
the estimated statistical errors would produce a similar result. It is
not our intention here to second-guess the data analysis of [1], but
rather to point out that the disagreement of their data with the
WJC models is mild and may be partly due to the behavior of the
data set itself.
In this context it is interesting to examine the np database for
any other independent evidence of a problem with Ay at low en-
ergy. Table 3 shows all Ay data for E lab  20 MeV. All of these
sets (except the backward angle TO88 measurements which are ex-
cluded because the χ2 is too small) are accepted by the Nijmegen
criteria, and all agree well with both WJC models. Thus there is no
indication of a problem in the older Ay data.
One might argue that the previous Ay measurements were not
as precise as the BR08 data, for instance because they were notTable 3
Previous Ay (or P ) measurements for E lab  20 MeV. The tabulated (χ2/N)1 are
for model WJC-1; (χ2/N)2 are for model WJC-2
E lab Ref. N (χ2/N)1 (χ2/N)2 (χ2/N)PWA93
7.6 WE92 5 2.10 2.23 2.51
10.0 HO88 13 0.76 1.25 0.82
11.0 MU71 1 0.04 0.13 0.03
12.0 WE92 9 1.61 1.16 1.71
13.5 TO77 1 0.04 0.53 0.02
14.1 BR81 11 0.36 0.42 0.37
WE92 6 0.63 0.66 0.67
14.5 FI77 9 0.87 0.88 0.99
14.8 TO77 1 0.35 1.67 0.41
16.0 TO77 1 0.04 0.28 0.03
WE92 6 1.20 1.39 1.10
16.2 GA72 3 0.17 0.18 0.18
16.4 BE62 4 0.69 0.71 0.69
JO74 4 0.90 1.08 0.89
16.8 MU71 1 0.03 0.07 0.02
16.9 MO74 5 0.63 0.52 0.59
TO88a 12 1.30 1.30 1.33
TO88b 5 0.03 0.05 0.06
17.0 WI84 7 0.52 0.55 0.57
18.5 WE92 5 0.56 0.47 0.62
19.0 WI84 7 0.62 0.62 0.62
All 111 0.85 0.90 0.90
a This set of forward angle measurements from 51.0 to 143.7 degrees is retained
in the ﬁt.
b This set of backward angle measurements from 136.5 to 166.5 degrees is ex-
cluded from the ﬁt because its error is too small.
Table 4
Values of the pion coupling constant used in the ﬁts described in this Letter
Coupling WJC-1 WJC-2 PWA93
g2
π0
/4π 14.608 14.038 13.567
g2
π±/4π 13.703 14.038 13.567
corrected for the polarization dependent eﬃciency of the neutron
detectors [1]. Moreover, they have larger errors and are therefore
not as restrictive as the new precise BR08 data. This may be true,
but it also means that the larger χ2 of BR08 is not a result of the
older Ay data pulling the ﬁts into a wrong direction, since they
could accomodate larger variations in the ﬁts without a prohibitive
increase in the overall χ2. The ﬁts must be dominated by other
data.
Our np data base contains 73 data sets below 20 MeV. Only
three of them are excluded after applying the Nijmegen criteria:
two because their χ2 is too small, and only a single set, consist-
ing of three total cross section measurements between 0.5 and
2.0 MeV, because of a very large χ2. So again, we ﬁnd no hints
for any problems in the database. On the contrary, the low-energy
data seem to be remarkably consistent with each other and with
the phase shift analyses.
This brings us back to the question if the BR08 data set should
be excluded from further ﬁts because of its incompatibility with
the rest of the data base. From the description given above, it
should be clear that, although the Nijmegen criteria leads to a
well deﬁned critical value for the maximum tolerated χ2/Ndata of
a data set, its choice is also somewhat arbitrary.
Instead of worrying too much about whether or not the Ni-
jmegen criteria should be relaxed a bit in order to accomodate the
BR08 data set, we decided simply to see if its inclusion in the data
base leads to any signiﬁcant changes in the resulting ﬁts. We found
that, because it adds only 16 points (15 plus the normalization er-
ror) to a data base of over 3700 points, it had essentially no effect
on the ﬁtting process. It is important to realize that the other 3700
points already ﬁx the phases to a very large extent. We must accept
166 F. Gross, A. Stadler / Physics Letters B 668 (2008) 163–166the fact that the BR08 data set has a large χ2 that cannot be re-
duced by further ﬁtting.
We are left with some ambiguities. We cannot conclude, as is
suggested in [1], that the BR08 data point to a need to increase the
pion coupling constants. As it turns out, both of the WJC models
have larger couplings than advocated by the Nijmegen group (see
Table 4), but the BR08 data were not used to obtain these results,
and these results are not altered by including the BR08 data in a
reﬁt. Still, this observation may explain why the BR08 data is closer
to models WJC.
One cannot escape the fact that it is inappropriate to draw
strong conclusions from a single data set. As new data sets are
added to the data base, they may slowly change the phase shifts,
but any new data set cannot be expected to exert much lever-
age, and if a new data set disagrees signiﬁcantly with the phase
shift analyses we are rather led to look for problems with the new
data set. Our conclusion at the moment is that the BR08 dataset is
marginally consistent with the WJC-2 phases, less consistent with
the WJC-1 phases (the best so far), and probably inconsistent with
the old PWA93 phases.Acknowledgements
We thank W. Tornow for helpful correspondence and for pro-
viding us with the errors in Table 1. This work is supported
by Jefferson Science Associates, LLC under US DOE Contract No.
DE-AC05-06OR23177. A.S. was supported by FCT under grant
No. POCTI/ISFL/2/275.
References
[1] R.T. Braun, et al., Phys. Lett. B 660 (2008) 161.
[2] V.G.J. Stoks, et al., Phys. Rev. C 48 (1993) 792.
[3] R. Machleidt, Phys. Rev. C 63 (2001) 024001.
[4] F. Gross, A. Stadler, Phys. Lett. B 657 (2007) 176.
[5] F. Gross, A. Stadler, Phys. Rev. C 78 (2008) 014005.
[6] F. Gross, J.W. Van Orden, K. Holinde, Phys. Rev. C 45 (1992) 2094.
[7] E.E. Salpeter, H.A. Bethe, Phys. Rev. 84 (1951) 1232.
[8] A. Stadler, F. Gross, Phys. Rev. Lett. 78 (1997) 26.
[9] A. Stadler, F. Gross, M. Frank, Phys. Rev. C 56 (1997) 2396.
[10] G.J. Weisel, et al., Phys. Rev. C 46 (1992) 1599.
[11] W. Tornow, private communication, 2008.
[12] J.J. de Swart, C.P.F. Terheggen, V.G.J. Stoks, nucl-th/9509032.


Is there an Ay problem in low-energy neutron–proton scattering? 

Crossref

Is there an <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msub><mml:mi>A</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:math> problem in low-energy neutron–proton scattering?

We calculate Ay in neutron–proton scattering for the interactions models WJC-1 and WJC-2 in the Covariant Spectator Theory. We find that the recent 12 MeV measurements performed at TUNL are in better agreement with our results than with the Nijmegen Phase Shift Analysis of 1993, and after reviewing the low-energy data, conclude that there is no Ay problem in low-energy np scattering

Repositório Científico da Universidade de Évora

Physics Letters B 668 (2008) 163–166Contents lists available at ScienceDirectPhysics Letters Bwww.elsevier.com/locate/physletbIs there an A y problem in low-energy neutron–proton scattering?Franz Gross a,b,∗, Alfred Stadler c,da Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USAb College of William and Mary, Williamsburg, VA 23187, USAc Departamento de Física da Universidade de Évora, 7000-671 Évora, Portugald Centro de Física Nuclear da Universidade de Lisboa, 1649-003 Lisboa, Portugala r t i c l e i n f o a b s t r a c tArticle history:Received 29 July 2008Accepted 19 August 2008Available online 26 August 2008Editor: W. HaxtonWe calculate A y in neutron–proton scattering for the interactions models WJC-1 and WJC-2 in theCovariant Spectator Theory. We find that the recent 12 MeV measurements performed at TUNL arein better agreement with our results than with the Nijmegen Phase Shift Analysis of 1993, and afterreviewing the low-energy data, conclude that there is no A y problem in low-energy np scattering.© 2008 Elsevier B.V. All rights reserved.In a recent paper, Braun et al. (designated BR08) report on anew set of very precise measurements of A y in elastic neutron–proton scattering at 12.0 MeV neutron lab energy [1]. They com-pare their experimental observables to the results produced by theNijmegen Phase-Shift Analysis [2] of 1993 (PWA93) and by the CD-Bonn potential model [3], also fitted to the data and producingphase shifts consistent with PWA93. They observe that the pre-dictions of the PWA93 as well as the CD-Bonn differ considerablyfrom the new data and interpret this as evidence for a significantshortcoming of the phase shifts and, consequently, of all poten-tial models fitted to them. In a model study, starting from theCD-Bonn potential, they find that increasing the charged-pion cou-pling constant g2π±/4π , while keeping the neutral-pion couplingconstant unchanged at g2π0/4π = 13.6, brings the theoretical pre-dictions into better agreement with the new data.In this Letter, we take up the issues raised in [1] and considerthem in light of a recently finished new analysis of np scatteringwithin the Covariant Spectator Theory (CST) [4,5]. As described indetail in Refs. [5,6], the scattering amplitude M for two-nucleonscattering in CST is obtained by solving a covariant integral equa-tion of the formM = V − V GM, (1)where V is the irreducible kernel (playing the role of a potential)and G is the intermediate state propagator. The Bethe–Salpeter(BS) equation [7] has a similar structure, but in the BS theory thefour-momenta of the particles in intermediate states are subjectonly to the conservation of total four-momentum P = p1 + p2, soloop integrations over intermediate states are over four indepen-* Corresponding author at: Thomas Jefferson National Accelerator Facility, New-port News, VA 23606, USA.E-mail address: gross@jlab.org (F. Gross).0370-2693/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2008.08.037dent variables. The CST equation maintains four-momentum con-servation, but constrains the energy components by placing one ofthe two intermediate particles on its positive-energy mass shell.Since P is fixed, both intermediate energies become functions ofthe three-momenta only, and all intermediate loop integrations re-duce to three dimensions, as in the non-relativistic theory. In spiteof this considerable simplification, the equation remains covariantbecause the on-mass-shell constraint itself is a covariant condi-tion. This framework has been generalized to other systems andapplied successfully to many problems, in particular also to thethree-nucleon system [8,9].Our recent fit to the np scattering data [5] uses the CST with akernel approximated by the sum of one-boson-exchange (OBE) in-teractions. Our best model, designated WJC-1, uses 8 bosons with27 parameters and provides a high precision fit to the 2007 database (χ2/Ndata = 1.06). A new phase shift analysis emerges fromthis fit, with phase shifts that differ significantly in some casesfrom the PWA93 analysis. Another model (WJC-2) was chosen tobe as simple as possible, and requires only 6 bosons with 15 pa-rameters. It does almost as well, with χ2/Ndata = 1.12.Before making a detailed assessment, we point out that theerrors quoted by BR08 combined systematic and statistical errors to-gether in quadrature. A better procedure is to treat the system-atic normalization error as an independent experimental degree offreedom, so that the χ2 for the data set would beχ2t =N∑i=1(oi/Z − ti)2(δoi/Z)2+ (1 − 1/Z)2(δsys/Z)2, (2)where oi and ti are the measured and the calculated value ofthe observable at point i, δoi and δsys are the statistical errors atpoint i and the systematic error, Z is a factor [chosen to minimizeEq. (2)] by which the data and errors can be divided to correct forthe systematic error and improve the agreement with theory, and164 F. Gross, A. Stadler / Physics Letters B 668 (2008) 163–166Fig. 1. Neutron–proton analyzing power at 12.0 MeV. Lines are WJC-1 (solid), WJC-2 (dashed) and Nijmegen PWA93 (dotted). Left panel shows BR08 data scaled by WJC-1(solid circles) and WJC-2 (half filled squares). Right panel shows the WE92 [10] data (half filled squares) and the BR08 data (solid circles) both scaled by PWA93.Table 1Statistical errors for the data set BR08 of Ref. [1]. The systematic error is 1.5%Angle A y Error(stat) Angle A y Error(stat)32.6 0.00854 0.00066 96.3 0.01198 0.0005640.5 0.01231 0.00061 104.2 0.01110 0.0005548.5 0.01451 0.00061 112.2 0.00662 0.0006156.5 0.01443 0.00059 120.2 0.00558 0.0006464.4 0.01560 0.00059 128.2 0.00483 0.0005672.4 0.01659 0.00062 136.0 0.00372 0.0006780.5 0.01470 0.00056 143.8 0.00287 0.0007988.4 0.01386 0.00053the last term is the additional contribution to the χ2 coming fromthe renormalization of the data. We contacted the authors of BR08who told us that the systematic error was about 1.5%, and pro-vided us with the original statistical errors, reproduced in Table 1[11].Using the new errors given in Table 1 and the new models(and phase shift analyses), we reach a different assessment of theimpact of the BR08 experiment. Figs. 1 and 2 present differentcomparisons of the 12 MeV A y BR08 measurements with the pre-dictions of the Nijmegen PWA93, and models WJC-1 and WJC-2.Table 2 gives the χ2/Ndata for each of these theories and Table 3surveys the previous A y measurements for all laboratory energiesless than 20 MeV. From these figures and tables we draw the fol-lowing conclusions:First, without even looking at the data, observe that the predic-tions of both of our models WJC-1 and WJC-2 differ substantiallyfrom the PWA93 prediction (the theoretical curves shown in bothpanels of Fig. 1 are identical), especially at the smaller angles. Thisis not unexpected; it is a consequence of the difference betweenour phase shifts and those of PWA93. This may have significantimpact on other observables as well.Next, Table 2 shows that the scale factor Z differs for eachmodel, and hence there is no one “correct” data set. For example, toaccommodate PWA93, the data is increased by over 6% (four stan-dard deviations), giving a large normalization contribution to theχ2 but lowering the collective effect of the statistical errors oneach of the 15 measured points. This renormalization of the datalowers the χ2/Ndata from about 6.6 to about 3.85. In the end, evenwith this renormalization the PWA93 model is the least successfulin describing the data. For comparison, Fig. 1 (right panel) and Ta-ble 2 also show previous A y measurements performed at TUNLat the same energy, designated WE92 [10]. Qualitatively, they ap-pear to agree well with the new measurements, and also with thetheoretical models, with considerably smaller χ2/Ndata than BR08,owing mainly to their larger statistical errors.Fig. 2. Ratios of the 12.0 MeV BR08 data for the analyzing power to the predictionsof WJC-1 (1), WJC-2 (2), and PWA93 (N).Table 2The average χ2 from Ref. [1] compared to the average χ2 obtained by scalingthe data and errors in Table 1 [with the scaling factor Z determined by mini-mizing Eq. (2)]. The scaled WE92 [10] data, also measured at 12 MeV, are shownfor comparison. The Nijmegen rejection criteria say that a set with this many data(15 points plus one normalization error) should be retained only if χ2/Ndata < 2.26.Note that the BR08 data almost meets this criteria with model WJC-2χ2/NdataModel BR-Ref. [1] BR-scaled Z(BR) WE92PWA93(1)a 6.621 3.849 0.9367 1.71PWA93(2)a 6.654 3.861 0.9365 1.71WJC-1 3.631 3.014 0.9690 1.61WJC-2 2.377 2.387 1.0029 1.16a Our fitting procedure uses the effective range expansion. The numbers shownfor PWA93(2) use WJC-1 parameters, which give the best fit to the overall data setat all energies. Nijmegen does not give 1 S0 parameters, but PWA93(1) uses 3 S1parameters taken from Ref. [12], which does not fit the overall data set as well.How are we to interpret these large BR08 χ2? Does it meanthat all of the theories are wrong? In this context we must re-member that a phase shift analysis gives the most general theoret-ical description of scattering below the pion production threshold, andwhile several theoretical models might generate the same set ofphase shifts, the best phase shifts provide the best model indepen-dent description of the data. At low energies, only a few partialwaves contribute noticeably to the NN scattering amplitude, andof those only the S-waves are not small. In principle, this leavesthe remaining partial waves with large uncertainties. However, thelow-energy phase shifts are linked to the much better determinedF. Gross, A. Stadler / Physics Letters B 668 (2008) 163–166 165phases at higher energies through the requirement of a smoothcontinuous energy dependence, which limits their uncertaintiesand relates observables at different energies. While a continuousenergy dependence arises automatically when a potential modelis used, it is imposed also in a multi-energy phase shift analysis,although in a phenomenological way. Thus, any new data set effec-tively competes with the entire phase shift analysis, which is basedin turn on all of the 3788 data currently included in the 2007 database. If a new data set disagrees with the predictions of the phaseshift analysis it also disagrees with all of the other data.So the BR08 data disagrees with the latest phase shift analy-ses (and therefore the entire 2007 np data base), but does thisrepresent a real incompatibility? Is the χ2/Ndata of this data settoo high to be tolerated in a quantifiable sense? The Nijmegengroup used a statistical criteria based on the following consider-ation: from a given model (in our case a phase shift analysis ora potential model) one calculates the observables contained in ameasured dataset. If the model described the observables exactly,in other words, if we had the “correct” theory, one should obtainχ2/Ndata ≈ 1 as a result of the unavoidable and random statis-tical errors in the experimental data. This presupposes that eachindividual data point in the set is normally distributed around thetheoretically predicted value with a standard deviation equal to thestatistical error.One can then ask: based on the theoretical model, and allow-ing for random fluctuations of the size given by statistical errors,what is the probability that one will obtain a χ2/Ndata as largeas or larger than the actually observed value for the data set?If that probability becomes very small, for instance smaller thansome predefined value, one may conclude that the model and thedata are not compatible with each other. By the same line of ar-guments, getting very small values of χ2/Ndata can also be tooimprobable, leading to a second criteria testing data sets for toolow χ2/Ndata.The Nijmegen criteria adopts a critical probability of about0.27%, in analogy to the usual “3σ -criteria” for testing individ-ual normally distributed measurements. Accordingly, for Ndata = 16(15 data points plus one common normalization error), the high-est admissible value of χ2/Ndata is 2.26. From Table 2 we see thatthe BR08 data set is not compatible, in this sense, with any of themodels, although it comes very close to WJC-2.How significant is this incompatibility, and could it be due, inpart, to a problem with the BR08 data themselves? Fig. 2 showsthe ratio of the BR08 A y measurements to the theory for threedifferent theories. There is a rather pronounced disagreement be-tween the data and all models at smaller angles, but the “break”in the set at about 110◦ may be more significant. The theoriesare in good agreement with each other at these larger angles, yetnone of them can reproduce this behavior. Eliminating, for exam-ple, the two data points at 112◦ and 120◦ would already bring theχ2/Ndata of BR08 with respect to both WJC-1 and WJC-2 belowthe critical value (although not for PWA93). A small increase inthe estimated statistical errors would produce a similar result. It isnot our intention here to second-guess the data analysis of [1], butrather to point out that the disagreement of their data with theWJC models is mild and may be partly due to the behavior of thedata set itself.In this context it is interesting to examine the np database forany other independent evidence of a problem with A y at low en-ergy. Table 3 shows all A y data for E lab  20 MeV. All of thesesets (except the backward angle TO88 measurements which are ex-cluded because the χ2 is too small) are accepted by the Nijmegencriteria, and all agree well with both WJC models. Thus there is noindication of a problem in the older A y data.One might argue that the previous A y measurements were notas precise as the BR08 data, for instance because they were notTable 3Previous A y (or P ) measurements for E lab  20 MeV. The tabulated (χ2/N)1 arefor model WJC-1; (χ2/N)2 are for model WJC-2E lab Ref. N (χ2/N)1 (χ2/N)2 (χ2/N)PWA937.6 WE92 5 2.10 2.23 2.5110.0 HO88 13 0.76 1.25 0.8211.0 MU71 1 0.04 0.13 0.0312.0 WE92 9 1.61 1.16 1.7113.5 TO77 1 0.04 0.53 0.0214.1 BR81 11 0.36 0.42 0.37WE92 6 0.63 0.66 0.6714.5 FI77 9 0.87 0.88 0.9914.8 TO77 1 0.35 1.67 0.4116.0 TO77 1 0.04 0.28 0.03WE92 6 1.20 1.39 1.1016.2 GA72 3 0.17 0.18 0.1816.4 BE62 4 0.69 0.71 0.69JO74 4 0.90 1.08 0.8916.8 MU71 1 0.03 0.07 0.0216.9 MO74 5 0.63 0.52 0.59TO88a 12 1.30 1.30 1.33TO88b 5 0.03 0.05 0.0617.0 WI84 7 0.52 0.55 0.5718.5 WE92 5 0.56 0.47 0.6219.0 WI84 7 0.62 0.62 0.62All 111 0.85 0.90 0.90a This set of forward angle measurements from 51.0 to 143.7 degrees is retainedin the fit.b This set of backward angle measurements from 136.5 to 166.5 degrees is ex-cluded from the fit because its error is too small.Table 4Values of the pion coupling constant used in the fits described in this LetterCoupling WJC-1 WJC-2 PWA93g2π0/4π 14.608 14.038 13.567g2π± /4π 13.703 14.038 13.567corrected for the polarization dependent efficiency of the neutrondetectors [1]. Moreover, they have larger errors and are thereforenot as restrictive as the new precise BR08 data. This may be true,but it also means that the larger χ2 of BR08 is not a result of theolder A y data pulling the fits into a wrong direction, since theycould accomodate larger variations in the fits without a prohibitiveincrease in the overall χ2. The fits must be dominated by otherdata.Our np data base contains 73 data sets below 20 MeV. Onlythree of them are excluded after applying the Nijmegen criteria:two because their χ2 is too small, and only a single set, consist-ing of three total cross section measurements between 0.5 and2.0 MeV, because of a very large χ2. So again, we find no hintsfor any problems in the database. On the contrary, the low-energydata seem to be remarkably consistent with each other and withthe phase shift analyses.This brings us back to the question if the BR08 data set shouldbe excluded from further fits because of its incompatibility withthe rest of the data base. From the description given above, itshould be clear that, although the Nijmegen criteria leads to awell defined critical value for the maximum tolerated χ2/Ndata ofa data set, its choice is also somewhat arbitrary.Instead of worrying too much about whether or not the Ni-jmegen criteria should be relaxed a bit in order to accomodate theBR08 data set, we decided simply to see if its inclusion in the database leads to any significant changes in the resulting fits. We foundthat, because it adds only 16 points (15 plus the normalization er-ror) to a data base of over 3700 points, it had essentially no effecton the fitting process. It is important to realize that the other 3700points already fix the phases to a very large extent. We must accept166 F. Gross, A. Stadler / Physics Letters B 668 (2008) 163–166the fact that the BR08 data set has a large χ2 that cannot be re-duced by further fitting.We are left with some ambiguities. We cannot conclude, as issuggested in [1], that the BR08 data point to a need to increase thepion coupling constants. As it turns out, both of the WJC modelshave larger couplings than advocated by the Nijmegen group (seeTable 4), but the BR08 data were not used to obtain these results,and these results are not altered by including the BR08 data in arefit. Still, this observation may explain why the BR08 data is closerto models WJC.One cannot escape the fact that it is inappropriate to drawstrong conclusions from a single data set. As new data sets areadded to the data base, they may slowly change the phase shifts,but any new data set cannot be expected to exert much lever-age, and if a new data set disagrees significantly with the phaseshift analyses we are rather led to look for problems with the newdata set. Our conclusion at the moment is that the BR08 dataset ismarginally consistent with the WJC-2 phases, less consistent withthe WJC-1 phases (the best so far), and probably inconsistent withthe old PWA93 phases.AcknowledgementsWe thank W. Tornow for helpful correspondence and for pro-viding us with the errors in Table 1. This work is supportedby Jefferson Science Associates, LLC under US DOE Contract No.DE-AC05-06OR23177. A.S. was supported by FCT under grantNo. POCTI/ISFL/2/275.References[1] R.T. Braun, et al., Phys. Lett. B 660 (2008) 161.[2] V.G.J. Stoks, et al., Phys. Rev. C 48 (1993) 792.[3] R. Machleidt, Phys. Rev. C 63 (2001) 024001.[4] F. Gross, A. Stadler, Phys. Lett. B 657 (2007) 176.[5] F. Gross, A. Stadler, Phys. Rev. C 78 (2008) 014005.[6] F. Gross, J.W. Van Orden, K. Holinde, Phys. Rev. C 45 (1992) 2094.[7] E.E. Salpeter, H.A. Bethe, Phys. Rev. 84 (1951) 1232.[8] A. Stadler, F. Gross, Phys. Rev. Lett. 78 (1997) 26.[9] A. Stadler, F. Gross, M. Frank, Phys. Rev. C 56 (1997) 2396.[10] G.J. Weisel, et al., Phys. Rev. C 46 (1992) 1599.[11] W. Tornow, private communication, 2008.[12] J.J. de Swart, C.P.F. Terheggen, V.G.J. Stoks, nucl-th/9509032.

Is there an Ay problem in low-energy neutron–proton scattering?

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Is there an Ay problem in low-energy neutron-proton scattering?

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