Phase shift analyses of pp elastic scattering at fixed energies between 0.83 and 1.8 GeV

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Phase shift analyses of pp elastic scattering at fixed

energies between 0.83 and 1.8 GeV

J. Bystricky, C. Lechanoine-Leluc, F. Lehar

To cite this version:

2747

Phase

shift

_analyses

of

_pp

elastic

_scattering

at

fixed

_energies

between 0.83

and

1.8 GeV

J.

_Bystricky

_(1),

C. Lechanoine-Leluc

₍₂₎

and F. Lehar

₍₁₎

(1)

DPhPE/SEPh, CEN-Saclay,

91191 Gif-sur-Yvette Cedex, France

(2)

DPNC, Université de Genève, Geneva, Switzerland

(Received

31

_July

1990,

accepted

17

_{September 1990)}

Résumé. 2014 Une

_analyse

en

_dephasage

à

énergie

_{fixe, pour} la diffusion

_élastique

_pp, a été réalisée à huit

_énergies

entre 0.84 et 1.8 GeV. A chacune de ces

_énergies,

un ensemble

complet

d’observables a été mesuré à SATURNE II. La

_dépendance

en

_énergie

obtenue pour les

différents

_déphasages

est

_comparée

à celle obtenue par d’autres

analyses

en

_déphasages

récentes.

Les résultats obtenus pour les

amplitudes

de diffusion a 0° sont

_comparés

à ceux basés sur les

relations de

dispersion.

La

_{dépendance angulaire}

des 5

_amplitudes

de diffusion est en bon accord avec celle obtenue par reconstruction directe.

Abstract. 2014 Fixed energy

phase

shift

analyses

have been carried out at

_{eight energies}

between

0.83 and 1.8 GeV for elastic pp

scattering.

At these

_{energies complete}

sets of observables have been measured at SATURNE II. The _energy

_dependences

of the

_phase

shifts are

_compared

with

other recent

_analyses.

The

_predictions

of the forward

_{scattering amplitudes}

are

_compared

with

dispersion

relation calculations. The

_{angular dependence}

of the 5 total

_{scattering amplitudes}

is in

good

agreement with results of the direct

amplitude

reconstruction.

J.

_Phys.

France 51

₍₁₉₉₀₎

2747-2776 15 DECEMBRE 1990,

Classification

Physics

Abstracts 13.75

1. Introduction.

Complete

sets of observables in _pp elastic

_scattering

were measured at

_{0.834, 0.874, 0.934,}

0.995,

1.095, 1.295,

1.596, 1.796, 2.096,

2.396 and 2.696 GeV

_using

the SATURNE II

polarized

proton

beam and the

_Saclay

frozen

_{spin polarized}

_target

_[1-16].

The choice of measured observables makes it

_possible

to determine

unambiguously

the

_scattering

amplitudes

analytically, provided

that these _parameters are determined with an infinite

precision.

In the _{present paper} the

_analysis

is limited to

_energies

below 1.8 GeV. At

_higher

energies

more

_experimental

data and further

study

are needed.

We have used the

_Saclay

data

_together

with other

_existing

data close to the

_{energies given}

above,

to _carry out fixed _{energy proton-proton}

_phase

shift

_analyses.

These include the available total and total elastic cross section

_data,

the total cross section differences

laT

and

_~~L

and the fitted total inelastic cross sections

[17].

The

existing

data sets of elastic differential cross sections at

_{higher energies}

were

_interpolated

to the central

_energies

of the

phase

shift

_{analysis (PSA).}

We use

_basically

the same _{PSA program} as in our

_previous

_papers

_{[18, 19].}

In adddition we

have introduced the « sixth »

_parameter

for each value of

_J,

i.e. the

_inelasticity

contributions

to the

_mixing

_parameters. The calculations show that this _parameter is small at all

_energies

and does not affect the PSA results

(see

Sect.

_5).

Below 0.6 GeV

_complete

sets of observables have been measured at PSI

_[20-22]

and have ensured a

_unique

PSA solution. At

_{higher energies complete}

sets of observables measured in

a

_large

_{angular region}

do not

_guarantee

a

_unique

_phase

shift solution. This is due to the

_large

number of

complex

phase

shifts to fit. To add further constraints we have made use of

additional

_information,

_namely

values of the ratio of the real to

_imaginary

_part of the

_spin

independent

forward

_scattering

_{amplitude [23]}

and the inelastic

_parts

of

_high

orbital

momentum

_partial

waves, as calculated

by

a

dispersion

relation method

by

the Paris _group

(PG)

[24-26].

2. PSA formalism.

The formalism used is

_given

in reference

_[19].

In addition we have introduced the

_inelasticity

parameter

a J

(sixth parameter), defining

the

mixing

element of

coupled triplet spin

states in

the S matrix as

This

_parameter

_{only weakly}

affects the

_{phases 6 j, j _}

_{and ~y,y+1}

1 because

of the small values of the

_mixing

_parameters

_~y. The renormalized 7T-N

_coupling

constant

_{f 2}

= 0.08 _was used for

the OPE

phase

shifts.

_Higher

_{(J, ~ )}

_inelasticity

_{parameters calculated}

_by

the Paris

Group

were introduced as fixed values.

Normalization

factors,

multiplying

measured

values,

one for each data set, were also introduced as free

parameters.

Each set of data is fitted at its nominal _energy, i.e. the

_amplitudes

are calculated

_using

the

corresponding

wave number

_(local

_energy

dependence).

3. Data basis.

The

_present

notation of

_experiments

follows reference

_[27].

i.e. for an observable

Xsrbt

the

_subscripts

refer to the

_polarization

_components of the

_{scattered, recoil,}

beam and

target

particles, respectively.

A summary of all data used in these

analyses

is

given

in table I. If a

_given

set of data is

already

listed in the

_compilations

of references

_{[28, 29]}

the same notation is

kept.

Normalization factors are also

_given

in this table

_(see

also Sect.

_4).

One notices that it is

mainly

differential cross sections which are renormalized. Normalization factors reflect

incompatibility

between different data sets. The fact that the data are measured at

_energies

differing

from the central values of the PSA solutions _may

_slightly

affect the normalization. It

can be seen that more than 80 % of the

spin dependent

data were measured at SATURNE II.

The recent KEK

_analyzing

_{power data in the energy}

_region

from 0.4 to 2.2

GeV,

which show

some structure

_[30],

have not been used as we consider them to be

_preliminary

and

_requiring

confirmation. The Gatchina

_{A oo,,o}

and

_{A oonn}

data

[31] were

not

_introduced,

since the authors used for the _pp

_analyzing

_power a value different from that used in their

_previous

measurements. Therefore

_only

the

_previous

data from reference

_[32]

were used as in the

2749

Table I. - Data sets used in the PSA. Data listed in the

compilations [28, 29]

are

refered

to in

the same manner as in the

_{compilations [29].}

The comment «Int» stands

_for

the values

interpolated

between two

_neighbouring

_energies.

Numerical values

_{obtained for}

the

2751

2753

2755

2757

-2759

2761

4. Minimization

procedure

and

_spreads

of

_phase

shift values.

Initial

parameters

at 0.834 GeV were taken from our _energy

_dependent

PSA of reference

_[18].

At

_{higher energies}

the minimization

_procedure

was

performed

_{sequentially starting}

from the solution obtained at the

_previous

lower _energy. These solutions at each _energy will be referred to as our

_primary

solutions

_(PS)

in the rest of the _{paper. Two} solutions

_(A

and

_B)

were found at 1.295 GeV.

_They

differ

_mainly

in the real and

_imaginary

_parts

of

’So

and

_3Po.

At 1.596 GeV we obtain the same solution when the minimization

_procedure

starts either from solution A or B.

In order to estimate the

_stability

and

_precision

of the

phase

shifts,

their variation with the number of free

phases

was observed. We

_{progressively}

added

_{high-J phase}

shifts one after the

other from

_{Ism 116}

up to Im

_3L9.

At all

_energies

the

_{X 2 value}

decreases

_only

a little if _parameters

above

_{Re lI6}

are free. One

_exception

is the

_phase

shift

_3Jg

which decreases the

_{X 2}

value

considerably

and was varied

_starting

at 0.834 GeV. The

_imaginary

parts of

_’So

and

3Po

were also allowed to _vary, whereas in the PS

_they

are fixed to zero. For each

_{phase shift,}

we observed a

_spread

_which,

in

_general,

is not

_symmetric

around the value found in the PS. The range of values is

always

larger

than the error

_{given by}

error matrix. The _upper and lower

limit of the

spread

may be considered as an interval of

stability

for a

_{given phase}

shift. We

expect

that

_phase

shifts _may move inside these intervals when additional data

_points

will be

available. At 1.295 GeV we obtained different

_spreads

for the two solutions A and B.

Moreover these ranges for the same

_phase

shifts do not

_{always overlap}

and even

_{by increasing}

the number of

_parameters

one never obtains a

_unique

solution. One can conclude that new

data are needed.

The

_stability

of any free parameter,

including

the normalization

factors,

may be studied.

We observed that the

_spreads

of all norms

_depended

_very little on the number of free

_phase

shifts and we

_give

in table I the normalization factors obtained for the PS.

5. Results.

Numerical values of the

phase

shifts for the

_PS,

and the

_{corresponding}

spreads

at 8

_energies

are

_given

in table II. In table IIa the values

_quoted

in

_parentheses

are

_phases

fixed to the OPE

values,

in table IIb these

_phases

are fixed to the PG

_predictions.

_Spreads

for

_phase

shifts

3L8

and

_3L9

are not

_given

since

they

cannot be well determined. The energy

dependence

of

different

_phase

shifts and their

_spreads

are

_plotted

in

_figures

1 to 16. Solid lines are the results

of our

previous

_energy

dependent

PSA

[18]

in the _energy

region

from 0.7 to 1.3 GeV. For this

analysis only

a few data from SATURNE II were

_available,

_mainly

the

_spin

_{correlations,}

and no

_complete

data set had been measured. The PG

_predictions

are

_given

_by

dashed

_lines,

the

OPE

_phase

shifts are

_presented

_by

dotted lines and the two solutions of Arndt et al.

_(SP89

and

SM89,

Ref.

[33])

are shown as dot-dashed lines.

Let us discuss the behaviour of

phase

shifts.

- The

spread

intervals for

_’So

are in

_{general larger}

than for other

_phase

shifts. No

imaginary

part of

’So

is

required

for the PS at _{any energy. This} is in agreement with the reference

_[33].

-

Re

_3Po

has small

_spreads.

Im

_3PO

reaches a maximum at 1.295 GeV

_{(solution A)}

and is

small at

_{high energies.}

Its

_spread

is

_{large only}

at 0.934 and 1.295 GeV

_{(sol. A).}

- Re

3Pl

shows a minimum at 1.3 GeV. Im

_3p,

has a maximum close to 1.6 GeV and then reaches zero at 1.8 GeV.

_Spread

intervals confirm this structure.

-

Re

_3P2

decreases from 0.84 GeV on, in contrast to our

_previous

results from

Table II. - Values

_of

the

_{phase shifts}

in

_degrees.

Values

_quoted

in brackets are the

_fixed

OPE values

(Tab.

IIa)

and theoretical results

_of

Paris

_{Group [24-26] (Tab. lIb).}

The second row at each energy

gives

limits

_of

the

_phase

_{shift spread. If}

this value

_is

zero and does not

_change

_only

0.0 is

_given.

2763

2765

- Re

lD2

was

_badly

known above 1.3 GeV. This

_phase

shift is now well

_determined,

even

if its

_spread

intervals are

_{relatively large.}

It crosses zero at about 0.95 GeV and decreases with

energy to

_{large negative}

values. Im

_{1 D2}

is smooth and

_fairly

constant.

- 1m

3F2

has a minimum around 1.1 GeV in _agreement with PG

predictions.

- Re

3F3

has a broad minimum at about 1.3

GeV,

then increases

_slowly.

Its

_imaginary

_part

is

_practically

constant _up to 1.8 GeV. This is in contrast to the fast increase observed in reference

[ 18].

The increase was

_probably

due to the fact that the lack of data did not allow us

to determine

enough

inelastic

_{phases (e.g.}

neither Im

_3F4,

nor Im

_3H4)

and the inelastic

_part

of interaction affected

mainly

Im

_3F3.

The energy

dependence

of Im

3F3

also differs from the two

Arndt

_predictions.

- Re

3F4

is

_{slightly larger}

than found in

_{[19, 33]}

and

_slowly

decreases with energy. The Im

_3F4

follows the solutions of reference

_[33]

and is

_larger

than the PG calculations

_[23-25].

It becomes

important

above 1.3 GeV.

- The real and

imaginary

parts

of

_lG4

are close to our

_previous

solution of reference

_[18]

but the

_imaginary

_part is smaller than the PG calculations.

Spread

intervals are small. - A

positive

value of Re

_3H4

at 1.8 GeV seems to be confirmed

_by

its small

_spread.

Values

of PS for Im

_3H4

are

_larger

than

_{predicted by}

the

_PG,

whereas Im

_3H5

are smaller. The

_spreads

of Im

_3H4

are

_large.

--

3H6

agrees with the SP89

predictions

of Arndt and its

_imaginary

_part

is

_larger

than the PG

_prediction.

- Re

lI~

is close to the OPE values below 1.3 GeV. Then it increases and follows the SP89 solution of Arndt.

- The free

mixing

parameters

~2, E4

and E6

are rather small. The

_rapid

increase of

E2, observed in reference

[19]

and in the SP89 solution of Arndt is not found in the

_present

PSA.

-

Inelasticity

parameters

a j for

J =

2, 4,

6 do not affect the

X 2

values of the PSA solutions

at _{any energy and} were set to zero. As an

_example,

at 1.8 GeV we obtain _{« 2} = 6° ± 18.9° and

the

_{total X 2}

value decreases

_by

0.1 units.

To obtain the PS solutions real

phase

shifts with

_high-J

were fixed to OPE values and the

imaginary

parts

of

_II6,

_3J6,7,

_iKg

and

_{3Lg, 9}

were fixed to the values

_{given by}

the PG. All these values are

_given

in Table IIb in

_parentheses.

Their

_spread

intervals almost

_always

contain the OPE or PG

_{predictions. Exceptions}

are the real and

_imaginary

_parts

of

3Jg.

Their

spread

intervals are

_{systematically}

out of

_predictions

and therefore

_they

were varied at all

_energies.

6.

_Angular

_dependence

of

_{scattering amplitudes.}

The

_{scattering amplitudes}

a,

b,

c and d at 0 = 0° for the PS _are listed in table III.

They

are

compared

with the

_dispersion

relation

predictions

of Kroll

_[23]

_given

in the same table in

parentheses.

One can see that there exists an excellent _agreement with the

imaginary

parts

of

all

_amplitudes,

since

_they

are determined in both cases

_by

the measured values of total cross

sections, åCTT

and

_AUL.

For the real

_parts,

the

_agreement

is not

_excellent;

one observes a

similar energy

dependence,

but different absolute values. When

trying

to

_incorporate

these

predictions

as a theoretical

input,

the X 2

value increases

considerably.

Recall that we used the

ratio of the real to

_imaginary

_part

of the

spin

independent

forward

_amplitude

_{( a}

+

_{b )}

as

_input

data

_(see

Tab.

_IV).

The

angular dependence

of the five total

amplitudes (including

electromagnetic

contri-butions)

are shown in another _paper

(Ref. [48] Figs.

1 to

₈₎

which describes the direct

2767

Table III. - Real and

Imaginary

parts

of

the invariant

_scattering

_amplitudes

a,

b,

c and e at

0 = D °. Values

quoted

in brackets were calculated

by

Kroll

[23]

_using

dispersion

relations and are not used in the _present

_analyses.

Table IV. - Values

of

X 2,

number

of

data

_points

used in the

analyses (data

points

with

7.

_Argand

_diagrams.

The

_{Argand diagrams}

for most

_important

PS

_phase

shifts are shown in

_figure

17. The

_points

at

different

_energies

are connected

_by

_straight

lines. The

diagrams

_present no evident structures.

Only

diagrams

for

_{lD2, 3F3}

and

_lG4

undertake a

_slight

anticlockwise

_motion,

but no conclusion

about a resonance-like behaviour for these

phase

shifts can be made.

8.

_Quality

of the fit.

In

_general,

the data are well fitted

by

the _present

analyses.

This can be seen from table

IV,

where

_{the ~ 2 values}

per

degree

of freedom are

_given

for the

eight

energies.

In the same table we

_quote

the fitted values of the total cross

_{sections 0"0 tot,}

_{U1 1 tot}

= - åUT/2, - åUL

as well as

the ratio of the real and

_imaginary

_parts

of the

spin

independent scattering amplitudes

- , - , .- , - , -.

Fig. 1.

Fig. 2.

Fig.

1. - Re

’So

and Im

_1So

_phase

shifts as functions of kinetic energy. The errors _represent square roots

of the

_diagonal

elements of the error matrix. The

_rectangles

are

phase

shift

_{spreads (see text). Meaning}

of _{symbols : (Black dots)} present PSA ;

(Open circle)

present PSA

_(solution

B at 1.295

_{GeV) ; (Full}

line) PSA _{[18] ;}

_{(Dot-dashed lines)}

solution SP89 _(below 1.3 _GeV) and solution SM89

_(below

1.6

_GeV)

of Arndt et al.

_[33].

_Symbols are the same in

_{following figures.}

z

Fig.

2. - Real and

2769

Fig.

3. -

Real and

imaginary

parts of

_3pl.

_Fig.

4. - Real and

imaginary

parts of

_3P2.

Fig. 5. - Real and

imaginary

parts of

ID2.

The dashed line refers to the Paris _group calculations

[24-26]

(as in

_{following figures).}

Fig.

6. - Real and

Fig.

7. - Real and

_imaginary

_parts of

_3F3.

The dotted line is OPE contribution

_(as

in

_following

figures).

Fig.

8. - Real and

_imaginary

_parts of

_3F4.

2771

Fig.

11. - Real and

imaginary partis

of

_3Hs.

_Fig.

12. - Real and

imaginary

parts of

_3H6’

Fig.

13. - Real and

_imaginary

parts of

_lI6

and

’j6-Fig.

14. - Real and

Fig.

15.

_{- Mixing}

_{parameters 82} and E4.

Fig.

16.

_Mixing

_{parameters £6} and _E8.

2773

The

_{present phase}

shift

_analyses

describe well the

_negative

values of the pp

analyzing

power

Aoono

=

Aooon

close to

8c~

= 90° at

energies

1.3,

1.6 and 1.8 GeV.

They

describe

fairly

well a maximum in

_Aoosk

at - 75° CM below

1.3,

but not so well at 1.6 GeV.

_Note,

that the

Aoosk

data were determined with

_{relatively large}

errors at all

_energies.

At

1.8 GeV,

Aaosk

has a maximum around 40° CM which is

_slightly

shifted

_by

the PSA to

_{larger angles.}

The

Aoonn

energy

dependence

at

_{90° CM,}

which shows a minimum at - i .3 GeV and a maximum at

1.8 GeV is well described

by

our

_analyses.

9. Conclusions.

The _present PSA is based on a

_large

number of data measured

_mainly

at SATURNE II. It

increases the domain of

knowledge

of

_phase

shifts _up to 1.8 GeV. A similar extension for np

elastic

_scattering

to

_energies

_up to 1.1 GeV can be

anticipated

when all the SATURNE II

data are available. Phase shifts between 0.83 and 1.8 GeV are smooth and _may be

interpolated

if

_predictions

at an intermediate _energy is needed. The PSA uses as

_input

the

high-J imaginary

parts

of

phase

shifts from the Paris group

calculations,

the OPE contributions to the real

_parts

of

_phase

shifts for J > 6 and the ratio of real to

_imaginary

_parts

of the

_{spin independent}

forward

_amplitudes

determined from

dispersion

relations. It can be

suggested

that if new data become

available,

the

phase

shifts will

change

within limits

_{given by}

spread

intervals. No conclusion about a resonance-like behaviour can be made. Predictions of

observables and

_{scattering amplitudes}

are available on

_request.

As for

previous

Saclay-Geneva

analyses,

the

_present

results will be also available as an

option

in the SAID _program

of Arndt et al.

References

[1]

BYSTRICKY J., DERÉGEL _J., LEHAR _F., DE _{LESQUEN A.,} VAN ROSSUM _L., FONTAINE J. M.,

PERROT F., HASEGAWA T., NEWSOM C., ONEL Y., PENZO A., Lettere al Nuovo Cimento 40

(1984)

466.

[2]

BYSTRICKY J., CHAUMETTE _P., DEREGEL _J., FABRE _J., LEHAR _F., DE _LESQUEN A., VAN ROSSUM

L., FONTAINE J. M., PERROT F., BALL J., HASEGAWA T., NEWSOM C. R., PENZO A., ONEL

Y., AZAIEZ H., MICHALOWICZ A., Nucl.

Phys.

B 258

₍₁₉₈₅₎

483.

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