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Nucleon-nucleon phase shift analysis
J. Bystricky, C. Lechanoine-Leluc, F. Lehar
To cite this version:
J. Bystricky, C. Lechanoine-Leluc, F. Lehar. Nucleon-nucleon phase shift analysis. Journal de Physique, 1987, 48 (2), pp.199-226. �10.1051/jphys:01987004802019900�. �jpa-00210432�
Nucleon-nucleon phase shift analysis
J. Bystricky (+), C. Lechanoine-Leluc (++) and F. Lehar (+) (+) DPhPE, CEN-Saclay, 91191 Gif sur Yvette, Cedex, France
(++) DPNC, Université de Genève, Geneva, Switzerland
(Requ le 26 mai 1986, accept6 le 2 octobre 1986)
Résumé. 2014 Une analyse en déphasages avec dépendance en énergie est présentée pour les réactions élastiques pp, np et pn entre 10 et 800 MeV. Pour l’analyse de la réaction pp, une solution unique a été trouvée présentant une dépendance en énergie lisse sauf pour le déphasage 1D2 qui montre un comportement résonnant vers la masse de 2,14 GeV. Pour l’analyse des réactions np et pn très peu de mesures de paramètres complexes de polarisation existent, si bien que la solution trouvée est moins précise surtout au-dessus de 500 MeV. Une analyse à énergie fixe à
1 GeV a aussi été effectuée, pour laquelle une unique solution a été trouvée. Le formalisme, la base de données utilisée et la compatibilité avec d’autres analyses sont discutées.
Abstract. 2014 An energy dependent phase-shift analysis is reported for pp, np and pn elastic scattering between 10
and 800 MeV. For the pp case a unique solution smoothly varying with energy is found. A resonant behaviour is observed only in the 1D2 phase shift near the mass 2.14 GeV. For np and pn scattering only a few complex polarization parameters are available ; therefore the solution is less precise, especially above 500 MeV. A fixed- energy analysis at 1 GeV is also reported providing a unique phase-shift set. The formalism, data base and
compatibility with other analyses are discussed.
Classification
Physics Abstracts
03.75C
1. Introduction.
Phase shift analyses (PSA) are still the most successfull
phenomenological approaches to the nucleon-nucleon
amplitude determination in the intermediate energy range. The PSA will provide excellent results for pp
scattering up to 1 000 MeV as soon as all SIN, LAMPF
and Satume II results from recent experiments will be
available. The np solution is not as satisfactory and
more data on several complex polarization parameters
(e.g. Aookk, Aoonn) are needed.
The present phase shift analysis is an updated and improved version of our previous one [1, 2]. Our approach to the analysis was guided by the following
considerations : (i) The amount of experimental data
up to 800 MeV is sufficient for an energy dependent
PSA. (ii) A fixed-energy PSA is fully justified if a
sufficient number of experiments are performed at practically the same energy. Otherwise, averaging of
values measured at different energies introduces sys- tematic uncertainties which cannot be correctly es-
timated. (iii) In an energy dependent analysis, one
should be careful to leave enough freedom to the phases in order to describe possible structures.
(iv) Model-dependent input to the analysis should be
minimized in order to detect any unpredicted
phenomena. In particular we did not introduce as data
the ratio of the real and imaginary parts of the spin- independent forward scattering amplitudes, non the
total elastic cross section, calculated by integrating the
differential cross sections (included in the analysis).
(v) Inelastic total cross sections were used as part of
the input. They were obtained by summing the cross
sections of all reaction dhannels open in the considered energy interval. Direct bubble chamber data on
Utot (inel) were also introduced. (vi) Certain noncon-
troversial kinematical and theoretical results were im-
plemented in the analysis. In particular we enforced the
correct threshold behaviour for each partial wave amplitude and used one pion exchange results for higher order partial waves.
Taking these considerations into account, we have divided the energy range from 10 to 800 MeV in four
overlapping intervals. In each of them our energy
expansion of the phase shifts allows two extrema. This leads to four independent analyses for each isospin
channel. The isospin I = 1 phase shifts were first
determined on the basis of pp data. The data are well distributed between different experimental quantities
and the resulting energy dependences of the phase
shifts are smooth functions of the energy. The pp phase
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004802019900
shifts were then used as fixed input in the np (pn) analysis, where only the isospin I= 0 phase shifts were
fitted from np and pn data. Here, the available data are mostly differential cross-sections. The I= 0 phase
shifts are therefore not so well determined, as illus-
trated by their energy dependences. In overlapping
energy regions, where 2 analyses give different sol- utions, either of them can be used to calculate predic- tions, as both of them describe equally well the existing
data. This is an analogy to a fixed-energy analysis which
results in two different solutions.
When results from Satume II will be available, the
pp analysis could be extended above 1 GeV. Meanwhile
we give fixed-energy PSA results at 1 GeV, where only
one solution remains.
2. Formalism
2.1 AMPLITUDES. - In the analysis we have used the
invariant scattering amplitudes a, b, c, d and e. The
expression of all observables in terms of these quantities
can be found in reference [3]. For each laboratory
kinetic energy T and centre-of-mass scattering angle 0, these amplitudes may be written in terms of singlet- triplet amplitudes [3-5] :
The expansion of the singlet-triplet amplitudes in Legendre polynomial series and partial wave amp- litudes a u, as well as their parametrization by nuclear
bar phase shifts BLJ and mixing parameters EJ were taken from reference [4] :
Up to the one pion production threshold s’s and B’s
are real, above the 6’s can be complex with Im 8 , 0.
These requirements are introduced to fulfill the unitari- ty condition. In principle a sixth parameter, cp J, should be introduced as discussed in reference [6]. We have
omitted it as our fit to the data is reasonably good with only « five parameter » representation. The six par- ameter approach has been tried and no sizable improve-
ment to the fit was observed at 1 GeV [7, Priv. Com.].
Other phase shift analyses in this energy region [7-13]
use only five parameters.
Denoting Legendre polynomials as P J (cos 0 ) and
their derivatives with respect to cos 0 as P J (cos 0
we write the resulting expansion of the invariant
scattering amplitudes as
where p is the centre-of-mass momentum. The a Lj for J 0 and aJ for J 0 are defined to be zero.
2.2 ONE PION EXCHANGE CONTRIBUTIONS. - The
expansion of the amplitudes was truncated at the total angular momentum Jmax. The higher angular momen-
tum states were replaced, as usual, by the Bom approximation of the one pion exchange contribution
(OPE) [5, 14]. The OPE invariant amplitudes, whose
J * Jmax contributions should be subtracted, are :
where
with
Ml, M2 mass of the beam and target nucleon respect-
ively,
0 mass of ir 0
u mass of charged pion,
E total energy,
f2 pion-nucleon coupling constant = 0.08.
The pion-nucleon coupling constant was introduced
as a fixed parameter.
2.3 ELECTROMAGNETIC CONTRIBUTIONS. - The Coulomb amplitudes and Coulomb-nuclear interfer-
ences were introduced as in references [4, 5]. Namely,
the Coulomb parts of the invariant pp scattering amplitudes are:
where
« - the hne structure constant = 1/137.036 M = the proton mass.
For angular momenta L higher than Lmax, we have
added to the amplitude e the magnetic moment correc-
tion [15] given by
where 4 p (= 2.7928456) is the anomalous magnetic
moment of the proton.
Another form of the electromagnetic contributions
given in reference [16] was also tested, but the resulting phase shifts were not significantly different from the
present analysis. The electromagnetic contributions calculated from the one photon exchange were studied
in detail in reference [17].
2.4 ENERGY DEPENDENCE OF PHASE SHIFTS. - The energy dependence of the phase shifts were fitted by polynomial expansion of the form
in each interval, where To is the central point of the
interval and aLJn are variable parameters. In all cases it turned out that n = 3 was sufficient, in some cases we
adopted n = 2, or n = 1. Proper threshold behaviour
was assured by multiplying equation (8) by OPE factors, obtained from the appropriately calculated
OPE elements of the K-matrix. The higher waves, as
mentioned above, were taken to be pure OPE ones.
It is well known that low-L OPE phase shifts do not correspond to the nuclear-bar phase shifts. We there-
fore have used for iSo and 3SJ phase shifts the scattering length and effective radius energy dependence. For
61 and 3D1 the energy dependence is properly taken
into account by multiplying the polynomials by arctg (T) 3/2 and arctg (T) 5/2, respectively.
Above the inelastic threshold, the phase shifts are
allowed to be complex. The imaginary parts of these phase shifts are then written:
where the threshold energy TLJ is proper to each phase shift, as well as auo, aul and aw which were treated as free parameters. Inelastic unitarity was imposed by constraining the Im 5 LJ to be non-negative.
The parametrization equation (9) of the threshold
behaviour of the imaginary parts of the phase shifts is in
agreement with general analyticity and unitarity re- quirement. The inelastic threshold energy T u was left
as a free parameter and was fitted in the lowest energy interval in which the corresponding phase shift SLJ
receives an inelastic contribution. The same form
equation (9) was used in all intervals, with T u already
found in the initial interval.
Continuity and the boundaries of the intervals was
not artificially imposed. For pp scattering the energy
dependences of the real and imaginary parts of the
phase shifts do however turn out to be close to
continuous as a result of the fit.
The isospin I = 1 phase shifts found in the pp
analysis were used as a fixed input to the np analysis.
The OPE parts of the pp phase shifts were replaced by
the np OPE and the multiplying polynomials were changed so that whole np phase shifts reproduce the pp
ones (see also Ref. [18]).
3. Data base.
3.1 ANALYSED DATA. - To denote the experimental quantities, we use a four subscript notation [3] : Xsrbt, where the subscripts s, r, b, t refer to the scattered, recoil, beam and target particle respectively.
In the present analysis we have used all the relevant data available in the compilation of reference [19] as
well as recently published data. These latter are listed in table I (pp) and table II (np).
In total 6966 independent pp data points and 6866 np data points were analysed in the energy dependent
PSA. For pp scattering 35 % of the data are spin independent measurements, 34 % are polarizations
(p = Aoono = Aooon) and 31 % are parameters with
two or three subscripts (correlations and Wolfenstein
parameters). For np scattering the corresponding per-
centages are 81 %, 14 % and 5 %, respectively. A more
detailed repartition of the data is shown in table III (pp) and in table IV (np). For each type of experiment
we give the total number of data points as well as their
occurrences. For the fixed energy solution 618 indepen-
dent data points in the energy range 970-1 040 MeV
were used as listed in table III.
3.2 INELASTIC CROSS-SECTIONS. - In the energy re-
gion from 290 to 1000 MeV there ’exist only few independent measurements of the total pp or np inelastic cross sections. On the other hand, about 260 pp and 100 np (pn) measurements of different inelastic channel cross sections are known in this energy range. In order to use this information, we have fitted
the energy dependence of the total cross-section of the
reaction j by an expression of the form [75] :
where Pi are polynomials of the energy T and the parameters Toj’ coj, ..., Cnj are fitted for each reaction j.
The total inelastic cross-section was then calculated as a sum of all the reaction cross-sections, fitted together
with the independent measurements of the atot (inelas- tic). The recent Dubna measurements of otot (np, inel) [58] were also taken into account. The atot (inelastic)
were introduced into the PSA in 5 MeV steps with the calculated errors.
Figure 1 compares our fitted pp and np reaction cross-sections to those obtained by Ver West and Arndt
[76]. For pp the agreement is excellent up to 750 MeV.
From there on, the 2 7T-channel contributions, not taken into account in reference [76] become significant.
For np the disagreement is apparent from 400 MeV on,
and becomes more and more pronounced with increas- ing energy. This discrepancy comes mainly from a
different treatment of at (np => npir") , which is the most dominant and the least measured reaction. The bubble-chamber measurements of Qtot (inel) (e.g.
[58]), were not taken into account in [76]. Our treat-
ment of inelastic pp and np cross-sections will be discussed in a separate article.
3.3 FITTING PROCEDURE. - All experimental data
were fitted according to the standard x2-method, including the error matrix calculation (see e.g. ref. 77)
with statistical errors taken from publications. The systematic errors and the discrepancies in the normali- zations between the PSA and the data were taken into account by introducing variable normalization factor
multiplying each data set. They were kept free only if
their values were different from one by more than their
errors. We found that most of the remaining normaliz-
ation factors apply to differential cross-sections. Our
X 2 sum thus reflects correctly the systematic experimen-
tal uncertainties. Experimental points for which the
x2-contribution was larger than 10 were omitted from the analysis. These represent about 3 % of pp and 2 % of np data.
Several incompatible sets exist for 4u L (pp )
measurements. This is clearly illustrated e.g; in
figure 12 of reference [25]. Since the SIN 4u L data [25]
were obtained simultaneously with .4uL ( pp => 7T+d)
measurements which are in good agreement with the values extracted in a separate experiment for Aookk ( pp =:> dw + ) [78], confidence in the normaliz- ation of the AoL (pp elastic ) results appears justified.
We have therefore renormalized all other measure- ments on the SIN data in the 500 MeV interval and used this normalization in the other intervals as fixed parameters. In the 500 MeV interval the Saclay data
Table I. - Recent pp measurements from 1981 to 1986.
Table I (continues)
Table II. - Recent np/pn measurements from 1981 to 1986.
Table III. - Summary of pp data points between 10 and 800 MeV.
[27] turn out to need no normalization, being in
excellent agreement. The TRIUMF data [24] were
found too large by 23.2 % and the LAMPF data too
small by 1.7 %. In the 260 MeV interval the TRIUMF data at 202.7 and 325.5 MeV have been omitted due to
large X 2 values, showing that a common normalization cannot be applied to the overall set.
The apparent disagreement between the RICE- LAMPF, RICE ZGS data [22] and other Aory measure-
ments was found to be due to different treatment of Coulomb-Nuclear interference corrections. No normal- ization of any åUr data set was needed âÎ all, when the
two higher energy TRIUMF [24] data points are
omitted (see discussion in Ref. [23]).
The Argonne Aoonn measurement at 697.6 MeV [79]
is well fitted with a normalization of 24.9 %. More recent LAMPF [46] and Saclay [43] measurements confirm this normalization factor.
Table IV. - Summary of np/pn data points between 10 and 800 MeV used in PSA.
The np total cross-sections measured at LAMPF [55]
and at SIN [56] are nicely fitted without normalization.
This shows that above the pion production threshold a good compatibility exists between available forward-
scattering data, i.e. differential cross-sections at small
angles [63, 80], new total cross-sections [55, 56] and our
calculated inelastic cross-sections (see Eq. (10)). Other
differential cross-section data are also well fitted with normalization factors close to one, namely about 2 300 data points measured at LAMPF [81]. A disagreement
is found in the shape of the angular distributions of
Aoono measured at TRIUMF [82] and at LAMPF [83].
In this case, the normalization could not remove the
discrepancies, therefore we have omitted the Aoono data
from reference [82]. Several Aoonn ( np ) data points at
665 MeV from LAMPF [84] were also omitted.
A detailed discussion of the np data itself would
greatly increase the volume of this PSA article. Such a
discussion is being submitted separately to the J.
Physique.
4. Analysis.
4.1 COMPARISON WITH PREVIOUS ANALYSIS. - The present phase shift analysis differs from our previous
one in the following aspects.
Fig. 1. - Inelastic pp and np total cross-sections. The full lines are our calculations, the dash-dotted line is that of Ver West and Arndt [76].
1. The data basis has been considerably enlarged in
both pp and np scattering. Above 400 MeV the new
data are mainly due to Saclay experiments.
2. The fixed energy analysis of pp scattering at
1 GeV has now become unique. The original 9 different possible solutions first collapsed to 4, mainly as a result
of the Argonne and Gatchina experiments. The new Saclay data permitted the present unique deter-
mination.
3. Correct threshold behaviour were implemented at
the elastic and inelastic thresholds (see Sect. 2).
4.2 ENERGY INTERVALS. - The analysis was per- formed in the energy range from 10 to 800 MeV. The entire range was divided into four overlapping inter-
vals : 10-220 MeV (« 80 Mev »), 130-450 Mev (« 260 MeV »), 380-610 MeV (« 500 MeV ») and 520-800
MeV (« 670 MeV »). In each of them our energy
expansion allows two extrema for every phase shift in
order to permit a good fit of possible structures. If the last parameter in equation (8) for a phase shift was
found to be less than its error then the parameter was
set to zero.
An independent analysis of pp data was performed in
each energy interval to find the isospin I =1 phase
shifts. The I =1 phase shifts were used as fixed input
to the np analysis. The analyses were performed once
more independently in each energy interval defined above.
4.3 RESULTS AND DISCUSSIONS. - Phase shift val-
ues found in the energy dependent PSA are given in
tables V-IX. The solution at 1 GeV is given in table X.
Both real and imaginary phase shifts are calculated in units of degrees. Phase shifts for which the real parts
are not mentioned in the tables, were set equal to the
OPE (see Eq. (4)) and magnetic moment (see Eq. (6))
contributions. All other phase shifts were fitted. The
number of experimental data points used and the
X 2-values for each analysis are summarized in table XI.
The value in the first column is the energy To in equation (8).
The starting values for the pp scattering analysis were
taken from reference [2]. A unique solution was found
in each energy interval. The thresholds of the imaginary
parts of all S, P, D, F and G waves were studied. The thresholds of the Im ’So and 3F4 are found to be higher
than 800 MeV and the 1m 3P1 is compatible with zero
up to 800 MeV. The Im 3Po, also compatible with zero
below 800 MeV, is determined with large error and give no improvement of the PSA fit.
At 1 GeV only one solution (Tabl. X) remains after introducing the recent Saclay data [44, 48]. The number
of different experimental quantities at 1 GeV is smaller
than that necessary for a direct reconstruction of the
scattering matrix. Consequently, values of some phase
shifts may change after introducing data recently
measured at Satume II.
For the I = 0 analysis, slightly modified pp phase
shifts were used as fixed input and only the isospin
I = 0 phase shifts were left free. The inelasticities in the S, P, D and F waves were studied. Only the imaginary parts of 3SJ, lpl, 3D1 and 3D2 should be
significantly different from zero, but in the analysis only the two last ones were considered.
The energy dependence of the S, P, D, F, G, H and I phase shifts and mixing parameters together with their
« error bars » are shown in figures 2-10 for pp and in
figures 11-17 for np and pn scattering. The « corridor of
errors » shown in some of figures was calculated as the square root of the corresponding diagonal element of
the error matrix. It is much narrower than the error
corresponding to the « confidence level 1 u». The
meaning of the indicated corridor is that it indicates the
regions in which further experiments would be particu- larly fruitful.
In the pp scattering, the continuity between solutions
in different energy intervals is very good. The point at
1000 MeV is our fixed-energy solution. An interesting
behaviour is observed in the real part of 3P2 above
400 MeV. Small discontinuities in pp phase shifts have
a strong influence in the np analysis.
Table V. - Real parts of pp phase shifts in degrees.
Table V (continued).
