On tests of time reversal invariance in nucleon nucleon scattering

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On tests of time reversal invariance in nucleon nucleon scattering

J. Bystricky, F. Lehar, P. Winternitz

To cite this version:

J. Bystricky, F. Lehar, P. Winternitz. On tests of time reversal invariance in nucleon nucleon scattering.

Journal de Physique, 1984, 45 (2), pp.207-224. �10.1051/jphys:01984004502020700�. �jpa-00209747�

On

tests

of time reversal invariance in nucleon nucleon scattering

J.

Bystricky (1),

^{F. Lehar}

(1)

^{and P.} Winternitz

(2,3)

(1) D.Ph.P.E., CEN-Saclay, ⁹¹¹⁹¹ Gif sur Yvette, ^France

(2) ^Centre de recherche de mathématiques appliquées,Université ^de Montréal, Montréal,Québec,Canada ^H3C3J7 (Refu ^{le 22} juin 1983, accepté ^{le 14} septembre 1983)

Résumé. - On développe le formalisme de la diffusion nucléon-nucléon en assumant l’invariance de Lorentz et la conservation de la parité ^{mais hon} l’invariance dans le renversement du temps. ^La matrice de diffusion pp contient alors 6 amplitudes (une d’elles, g(E, 03B8) ^ne satisfait pas à I’IRT); celle de diffusion np, 8 amplitudes

(g(E,

0) ^{ne satisfait}

pas à I’IRT, f(E, 03B8) ^ne satisfait pas à l’invariance isotopique ^et h(E, 03B8) ^{ne satisfait aucune de ces deux} symétries).

Une analyse préliminaire ^{de l’etat} expérimental ^{actuel sur 1’IRT en} diffusion pp mène à des conclusions ^{non encore} définitives.

Abstract. ²⁰¹⁴ The nucleon-nucleon scattering formalism is developed assuming Lorentz invariance and parity

conservation but allowing ^{for a violation of} time reversal invariance. The pp scattering ^matrix involves six ampli-

tudes (one, g(E, 0) ^violates TRI) the np matrix eight

(g(E,

0) ^violates TRI, h(E, 0) ^{violates TRI and} isospin invariance, f(E, 03B8) ^violates isospin invariance). ^A preliminary analysis ^{of the} experimental ^status of TRI in pp scattering ^is performed and the results are surprisingly inconclusive.

Classification

Physics ^Abstracts

11.80 -11.30 -13.75C -13.85C

1. Introductioa

The _purpose of this article is to

provide

^{a detailed}

framework for

performing

^tests of time reversal

(TRI)

in elastic nucleon nucleon

scattering

and simulta-

neously

^to review the present ^status of TRI in the

elementary

nucleon nucleon interaction.

A recent experimental comparison

[1]

^{of the} proton

polarization

^{P in} the reactions ’He + 7Li -+ _p + 9Be and 3He + 9Be -+ _p + 11B ^{with the} ’He asymmetry A

in the inverse reactions

using

^a

polarized

proton ^beam reports ^a

large

^P-A difference (for ^{14 MeV 3He} ions).

This P-A difference, ^if confirmed,

directly implies

^a

serious breakdown of TRI.

In the strong interactions the consequences of this breakdown would be

extremely

dramatic since all current notions of the strong interactions assume

TRI, ^{at least}

implicitly

^via the CPT theorem. In

particular it is hard to

imagine

^that QCD ^could

survive in

anything

^{like its} present form if TRI does not hold.

The authors of reference 1

point

^out that theirs is the first

comparison

^{of P and A made in an inelastic} reaction, ^{and that}

previous

^{P-A tests} in elastic scat-

tering

^on 3He or 13C nuclei ^were

performed

^{in unfor-}

tunately

^{chosen kinematic}

regions (where

^{P A is}

small

indepently

^of

TRI).

^{To this one can add that}

tests of TRI via detailed balance ^are not sensitive to

possible

spin

dependent

violations

(e.g.

^{if different}

spin amplitudes

^{in an} inelastic reaction

acquire

different

phases

under time reversal).

Our motivation for

reconsidering

^the

question

^of

TRI in NN

scattering

^{is that we find it}

extremely unlikely

that TRI could be violated in nuclear reac-

tions, ^without

being

^violated in the interaction of two nucleons.

Independently

^{of the outcome of the} present contraversy ^{over the P-A tests} in inelastic

scattering,

we find that the situation in NN

scattering

is far from conclusive and that further tests of TRI would be most valuable.

We

systematically develop

^{a 6}

amplitude

^formalism

for _pp

scattering

^{and an 8}

amplitude

formalism for the np ^case. Our notations and conventions are the same as in two

previous

^articles

[2,

3]. ^{The 8}

amplitude

formalism is also

appropriate

^{for the}

study

^{of inelastic}

reactions of the

type 2 + .1 -+ .1 + I ^{(e.g. Ap -+ I + n).}

In our

analysis

^{we make use of a recent}

compilation

of the world nucleon nucleon

scattering

^data

[4]

and also of ^a

recently performed phase

^shift

analysis [5].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01984004502020700

For a

general

^{review of the}

experimental

^{status of}

TRI in weak,

electromagnetic

^and strong interactions

we refer to Richter

[6].

^{An earlier}

analysis

^{of the} pp

scattering

^{data was}

performed by

^Thorndike [7].

A violation of TRI in NN

scattering

^is

expected

ⁱⁿ

certain models. Sudarshan [8]

proposed

^{a unified}

model of strong, ^{weak and} electromagnetic ^inter-

actions in which CP invariance, ^{and hence TRI is} violated in a

specific

^manner. The consequence of this model for TRI in NN

scattering [9, 10]

^{have been}

studied and

predictions

have been made

assuming

that TRI violation occurs in the lowest

possible angular

momentum state

only [11].

The formalism

presented

in this article should be of use in the

planning

^of

experiments directly testing

TRI in NN

scattering

^{on one} hand and in the recons-

truction of

scattering amplitudes

^from data, ^without the

assumption

^of TRI, ^on the other. Such a recons-

truction

would provide

^{the best}

possible

^{test of TRI.}

If

performed

^via

phase

^shift

analysis

^{it would}

provide

the

possible

^TRI violating

amplitudes

^as functions of the scattering

angle

^and

possibly

also of the energy.

A model

independent

^{test of TRI is best}

provided by

^a

reconstruction of the NN

scattering

matrix from a

complete

experiment

[12]. ^For pp

scattering

^{this was}

performed recently by

the Geneva _group

[13]

^at

579 MeV for six different

scattering angles (use

^was.

made of some formulas from a

preliminary

^version

of this _paper

[14]).

Their conclusion is that at the considered energy the TRI

violating amplitude

^does

not contribute more than 1

%

^{to the} differential

cross section.

2. The

scattering

^matrix.

Assuming only

Lorentz invariance

(or

Galilei inva-

riance)

^and

parity

conservation we can write the

scattering

matrix for the elastic

scattering

^{of two}

particles

^of

spin 1/2

^{in terms of 8}

complex

^functions

of the

energy E

^and

scattering angle

^0.

Using

^invariant

amplitudes a,

b, c, d,

e, f, g

^{and h we can write}

where

and

k;

^and

kf

^{are unit vectors in the} direction of the incident and scattered

particles

ⁱⁿ the centre-of-mass system

(c.m.s.).

For identical

particles

^{the Pauli}

principle requires

Thus

(2 . 3)

^must hold for pp and ⁿⁿ

scattering

^whereas for np

scattering

this reation would be a consequence of an additional

requirement, namely isotopic spin

invariance.

Electromagnetic

corrections, for instance due to one

photon exchange taking

the nucleon

electromagnetic

form-factors into account, will ^cause the

amplitude

^{f to}

differ from zero

[15-19J.

Time reversal invariance, ^on the other hand,

requires

that g and h vanish

Thus in proton proton

scattering precisely

^one

amplitude, namely g

⁼

g(E, 0),

^violates TRI, ^while

respecting parity

conservation. In neutron proton

scattering

^{there are two such}

amplitudes;

the second one h ⁼

h(E, 0)

also violates

isospin

invariance. For

f = g = h = 0 we

^{return to the five}

amplitudes

introduced

originally by

Wolfenstein

[20].

Alternatively,

^the

scattering

^{matrix can be described in terms}

of eight helicity amplitudes [21]. Using

^standard

notations and conventions we express the

helicity amplitudes

^{in terms} of the invariant ^{ones as}

(0

^{is the c.m.s.}

scattering angle).

For identical

particles

^{we have}

Time reversal invariance

implies

Finally, nucleon nucleon

phase

^shift analysis ^{is best}

performed using

^the

singlet-triplet representation.

Generalizing

the notations of

Stapp et

^al. [22] ^we put

where we set ms

and ms equal

^{to S for the}

singlet

spin ^{state and to} 1, 0, - ^{1 for the}

corresponding

^three pro-

jections

^{in the}

triplet

^state.

We then have

For identical

particles

^{we have}

Time reversal invariance

implies

A

phase

^shift

expansion

^{of the}

singlet triplet amplitudes (2 . 8)

^can be written as

The

phases

^and normalization are so chosen that the formulas reduce to those

of Stapp et

^al.

[22]

for pp ^scatter-

ing

^{with TRI} conserved; ^the Clebsch-Gordan coefficients and

spherical

^{harmonics are as} in reference 23.

For

fixed j we parametrize

^the

partial

^wave

amplitudes R¿,I’S’

^as

and

All other elements of the matrix

Rjis,rs,

^vanish because of

angular

^{momentum or}

parity

conservation.

If the

particles

^are identical, the

singlet-triplet mixing

^is forbidden, i.e. the new

mixing

angle _{y j} ^{vanishes :}

For np scattering _yj ^{is a measure of} isospin ^violation.

If TRI is assumed, then the matrices (2.13) ^and

(2.14)

^are

symmetric,

^i.e.

If both

(2.15)

^and

(2.16)

hold, ^{then formulas}

(2.13)

^and (2.14) ^{reduce to those of} Stapp et ^al. [22]. The notation

2 yj

^{for the}

singlet-triplet mixing

angle ^{was chosen to} coincide with Gersten

[17].

Note that the

parametrization (2.13)

^and

(2.14)

^{of a 2 x 2}

unitary

matrix is somewhat

arbitrary.

E.g. ⁱⁿ

(2.13)

^{the TRI}

breaking phase

^shift _T. ^was introduced

by

^a rotation of the usual [22]

phase

^shift parameters :

and

similarly

^for (2.14)

Binstock et al.

[ 11 introduced

^{a different TRI}

breaking phase

^shift parameter. Instead of a TRI

breaking

rotation

t j they

introduced a TRI

mixing angle Àj’ ^putting

Similarly

^{we could}

parametrize

^{the 1} = j ^matrix

S 2 = R4

^{+ I as}

In the

purely

^elastic

region,

^{below the}

pion production

threshold in NN

scattering,

^all

phases

^and

mixing angles

are real. Above this threshold inelasticities must be introduced, e.g.

by allowing

^the parameters ^{to become} complex.

Finally,

^{let us}

give

^the

partial

^wave

expansions

^of the TRI and

isospin violating amplitudes explicity :

For _np

scattering,

^the

expansions

^{stand as} written. The summation is over all indicated values

of l or j.

^The

electromagnetic amplitude f EM

^{can be taken from the one}

photon approximation

^and

b,il ⁶¹

^and gj parameters

are to be

interpreted

^{as the « nucleon bar »}

phase

^{shifts of}

Stapp et

^al.

[22].

For pp

scattering

^{we have}

f

^{= h = 0. In terms of the}

phase

^shift parameters ^{this means} 7j ^{= 0 in equa-} tion 2.14

and yj

⁼ pj ^{= 0 in}

equation

^{2.20. The}

amplitude g

^{must be} odd under the

exchange

^{0 - n - 0,}

hence the summation is over even values

of j only.

^The

electromagnetic

corrections can at this stage

only

^be

treated in a somewhat

hybrid

way. Thus, ^{we have}

The

phase

^{shifts are} approximated ^as

where

0l,

is the usual Coulomb phase ^shift [22] ^and

6fl 87

^are

^again

^« nucleon bar »

phase

parameters

[22].

3. Experimental

quantities.

Using

^{the same notations as in two}

previous

^articles [2, 3] ^{we write a}

general experimental quantity

^as

where Q is the differential cross section for

unpolarized particles.

The order of labels from left to

right

is scattered

particle,

^recoil

particle,

^{beam and} target. ^{For «} pure »

experiments

^{in the c.m.s.} all labels take the values 0, n, I,

or m. In the I.s. we introduce the usual three sets of basis vectors

where k, ^{k’ and k" are unit vectors} in the directions of the initial scattered and recoil

particle

^l.s. momenta,

respectively.

The Bohr rule

for 2 + -1 2 -+ 2 I + 2

^scattering

is a consequence of

parity

conservation alone and hence holds for the

scattering

^matrix

(2 .1 )

under considera- tion. Relation

(3.3)

^{allows us to reduce} the number of observables to be calculated

by

^{a factor of two.} Indeed,

in the c.m.s. we have

where

[li], [mi], [1f]

^and

[m f]

denote the numbers of I and m labels in the initial and final states,

respectively

^{and a}

label a’ is

equal

^{to n, 0, m, or 1, when a is}

equal

^{to o, n, I or m. In the I.s. we have}

where

[kj, [s;], [k f]

^and

[s f]

denote the number of k and s type ^labels in the initial and final states

respectively,

and a label x’ is

equal

to n, o, k(k’, k"), ^or

s(s’,

s") ^{when x is}

equal

to o, n, s(s’, s") ^or k(k’, k"),

respectively.

In tables I and II we

give expressions

^{for all «} pure » ^c.m.s. and l.s.

experiments

^{in terms of the}

amplitudes

a, ..., h. The observables 0,

Cnnoo, Dnono, Knoon, Pnooo, Ponoo’ Aoono

^and

Aooon

^{are the same in} both systems and ^are omitted from table II.

Altogether

⁶⁴

linearly independent experiments

^exist

The tables

(and

all results of this

paper)

^are valid for the elastic

scattering

^{of two} nonidentical

spin 1 /2 particles

for which TRI is not assumed

(e.g.

^elastic np --> np scattering ^{in which both} isospin ^{and TRI violation is}

allowed).

They

^{can also be}

interpreted

^{to describe an inelastic}

two-body

reaction of the

type i + L --, -L

⁺

-L (e.g.

^{A +} p - E + + n)

independently

of TRI or its violation.

2 2 2 21

Formulas for elastic _pp or nn

scattering,

^{in which TRI} violation is allowed but the Pauli

principle

^{is assumed,}

are obtained

by

putting

Formulas of reference 3 are obtained

by

putting g ^{= h =} 0; those of reference 2

by putting f

^{= g = h = 0.}

Table I. ^- Centre

of

^mass experimental quantities ^{in terms}

of

scattering

amplitudes.

Table I (continued)

Table II. ^-

Laboratory

system experiments ^{in terms}

of scattering amplitudes.

Table II (continued)

Table II

(continued)

All vectors and

angles

involved in transformations between the c.m.s. and l.s. are given ^on

figure

^{1 of refe-}

rence 2. We have

where

R1

^and

R2

^{denote a} relativistic spin rotation. The

angles

^{a and}

fl

for elastic scattering ^{are related to the}

relativistic spin rotation for the scattered and recoil particles

where 0 is the c.m.s.

scattering angle, 0,

^and

02

^{are the l.s.}

scattering

and recoil

angles.

^{In the} nonrelativistic limit we have a ⁼

0, p = n/2.

^{For an inelastic two}

body

reaction the

angles

^a

and P depend

^{in a}

quite compli-

cated _way ^on the _energy and

scattering angle.

^{We do not}

give

^the

expression

here but refer _e.g. to [24] ^{for a}

general

treatment of the Wigner

spin

^rotation.

4. Direct tests of time reversal invariance.

All

possible

^{direct tests of TRI can be read} off from tables I and II. If we restrict ourselves to observables ^involv-

ing

at most two

spin

components, ^{we find 12}

experiments measuring

the interference between g ^{or h and one of}

the

amplitudes

^a,

..., f

^{A further}

experiment

^{is sensitive to} the interference between _g and h. In the ^{c.m.s. we} have for _np

scattering :

Further tests of TRI would involve 3 and 4 component tensors, e:g. the _square moduli of g ^{and hare :}

For pp

scattering

^{we have}

f

^{= h = 0 so}

only

5 of the relations 4.1 survive and ^are

independent.

Translating

the relations into the I.s. we obtain

(for

^np

scattering) :

For identical

particles

the situation is much

simpler

^and

only

⁵ independent

experimental

^{tests of TRI}

involving

at most two component ^{tensors exist.} For pp scattering ^we

simplify

relations 4. 3 to obtain;

No tests of TRI have ^so far been

performed

ⁱⁿ np

scattering.

In pp

scattering

^the

polarization

asymmetry relation has been tested at

quite

^{a few} energies

(see below).

^The

only

^{other test}

performed

[24, 25] ^{was based on}

the

depolarization

^tensor relation 4.4b.

Although

^four components of the Wolfenstein tensor

Dpoio ^(or Doqok)

are involved in relation 4.4b. Handler et al. used an

ingenious experimental

setup which made it

possible

^to

verify

^the relation while

performing only

^two

scattering experiments.

Since the same is true for some of the other TRI

testing relations,

^{let us} discuss these setups ^{in some} detail,

performing

all considerations

directly

^{in the}

laboratory systeni.

4.1 DEPOLARIZATION TENSOR FOR POLARIZED BEAM IN pp ^OR np SCATTERING (see

Fig.

la). ^{- Consider an}

initial beam of nucleons with

polarization

^vector

PB

^{in the}

scattering plane

^{and an}

unpolarized

target :

Let the

polarization

of the scattered nucleon be

analysed by rescattering

^{on a zero}

spin

target ^with

analysing

power 1’1 (the polarization-asymmetry equality

^{for this} target follows from

parity

^and

angular

^momentum

conservation).

Let the normal n, ^to the

analysing plane

lie in the studied

scattering plane,

^{i.e. we are}

detecting

^an

up-down

asymmetry. ^{Place a}

magnetic

^field

parallel

^to the normal n between the target nucleon and the

spinless analyser.

^The

magnetic

field will rotate the scattered

particles

^{linear momentum k’}

through

^{some .fixed}

angle

and its

polarization through

^some additional

angle 4/ (about

the normal n). After this rotation we have

where +

( - )

^{refers to down}

(up)

scattering. ^The

up-down

asymmetry ^{on the}

analyser

^{will be}

given

^as

[2,

3]

Now consider two such asymmetry measurements, I and II

performed

^{for the same value} of the l.s. scatter-

ing

angle

81,

^{but for different} values of the initial

polarization angles x,

^{and xI, and the}

magnetic

field rotation

Fig. ^{1. -} Experimental set-ups ^{for tests of TRI} involving ^the depolarization ^{D or} polarization transfer K tensor. In all four

cases T denotes the target ^{M a} magnetic ^field parallel ^to the normal _n, A1 ^and A2 ^denote spinless analysers ^with analysing

powers

Pi

^and

P2,

respectively. ^{The unit vectors} kD, SD k; ^and sD point ^to the directions to which the vectors k’, s’, k", ^s"

were rotated by ^the magnetic field M. The unit vectors

kp, sp, kp,

sp ^{point to the} directions to which the corresponding polari-

zation components ^were rotated, 4/ ^{is the} angle ^{of the} additional spin ^rotation. P, and PT ^are the beam and target polarizations lying ^{in the} scattering plane, x is the angle between the initial particle polarization and the beam direction k. Figures ^{1 a-I d}

show the set-ups ^for

Dpoio(np

^{and pp),}

Dpqok(np

^{and pp),}

KOqiO(PP),

^and

Kpook(pp),

respectively.

angles 4/,

^and

1/111’

^Choose

We then obtain

If TRI holds, ^then

AI + All

^{= 0. In} different notations this

corresponds

^{to the}

experiment performed by

^Handler

4.2 DEPOLARIZATION TENSOR FOR POLARIZED TARGET IN pp ^OR np SCATTERING

(Fig. lb).

^{- Consider an}

unpolarized

^{beam and a} target ^with

polarization

^vector

PT

^{in the}

scattering plane

Let the

polarization

of the recoil nucleon be

analysed

^{on a}

spin

^zero

analyser

^with

analysing power P2

^{and let}

the normal n2 ^{to the}

analysing plane

lie in the

scattering plane.

Rotate the

polarization

of the recoil

particle through

^an

angle 4/

^about the normal n

by

^{means of a}

magnetic

field. After this rotation we have

where +

(-) again

^{refers to down}

(up) rescattering.

^The

up-down

asymmetry ^{on the}

analyser

^{will be}

where

P2

^{is the}

analysing

power ^on the

analyser.

Let two such

experiments

^be

performed

^{for the same recoil}

angle 92

and for Xi

and qli satisfying

We then have

Again,

^the

right

hand side vanishes if TRI holds

(see

(4.

3d)

^and

(4 . 4b)).

4.3 POLARIZATION TRANSFER FOR POLARIZED BEAM IN pp SCATTERING

(Fig.

1 c). ^{- Consider a beam of} pro- tons with

polarization

^vector

PB

^{as in}

(4.5)

^{and an}

unpolarized

proton target

PT

^{= 0. Let the}

polarization

^of

the recoil proton ^be

analysed

^{on a}

spin

^zero

analyser

and let the normal n2 ^to the

analysing

plane ^{lie in the}

scattering plane.

Rotate the

polarization

of the recoil

particle through

^an

angle t/J

about the normal n

by

^means

of a

magnetic

field After this rotation we

again

have relations 4.11. The

up-down

asymmetry ^{on the}

analyser

is

[2,

3]

Let two such

experiments

^be

performed

^{for the same recoil}

angle 02

and Xi

and Oi satisfying (4.13).

We then have

which

provides

^{a test} of the first of relations 4. 4c.

4.4 POLARIZATION TRANSFER FOR POLARIZED TARGET IN pp SCATTERING

(Fig.

1 d). ^{- Consider an} unpo- larized proton ^beam

PB

^{= 0 and a} proton target

polarized

^{as in}

(4 .1 U).

^{Let the}

polarization

of the scattered proton ^be

analysed

^{on a}

spin

^zero

analyser

and let the normal n, ^to the

analysing plane

lie in the scattering plane. Rotate the

polarization

of the scattered

particle through

^the

angle 0

^about the normal n

by

^{means of a}

magnetic

^field. After this rotation ^we have relations 4.6. The

up-down

asymmetry ^{on the}

analyser

^is

[2,

3]

Let two such

experiments

^be

performed

^{for the same}

scattering angle 81 and xi and 4/, satisfying (4.8).

^We

then have

providing

^{a test of the second} of relations 4.4c.

The

polarization

^{transfer tests}

(4. 3e)-(4. 3h)

^for np

scattering

^{are more}

complicated They

involve 8 quan- tities each; these ^can be obtained

by performing

⁴

experiments

^{in each case. The same} holds for the np

depola-

rization test

(4. 3m).

5. Discussion of time reversal invariance tests.

Some conclusions ^can be drawn

directly

^{from the}

formalism

presented

^above.

1) ^{TRI tests should be}

performed

ⁱⁿ

kinematically

favorable regions. ^For pp scattering ^such

regions

^are

characterized

by

^{the fact that the} quantities

are all

large.

^In

particular,

^{P-A tests should be}

perform-

ed where the

amplitude d

^{is known to be}

large (from

other

experiments

^{and from}

phase

^shift

analysis).

Similarly,

^{the D and K} tensor tests should be

perform-

ed where c and b,

respectively,

^{are known to be}

large.

2)

Only

^two of the five possible ^{TRI tests in} pp

scattering

^{have even been}

performed, namely (4.4a)

and

(4.4b),

the second one only ^{at one}

specific

energy

and

angle (E

^{= 430} MeV, ^{0 =} 300). ^These

provide

limits on

rather than

directly

on

I g 1.

3) ^It would be desirable to perform ^{a new}

phase

shift

analysis

^{of all} pp scattering ^{data without assum-}

ing

TRI. This would in

principle provide

^{the am-}

plitude

g(E,

0)

^{as a} function of

scattering angle

^and

energy. This is

unfortunately

very hard to do,

mainly

because TRI is

usually

built into the manner that data

are presented

(e.g. Aoono

⁼

Pnooo

^is

explicitly imposed).

A less ambitious

approach

^{is to assume} that the TRI

conserving phases

^are known and then to obtain values and limits for the additional TRI

violating phases.

This has been

attempted

^before

[7, 11]

^{and we}

do this below in section 6, using ^a

considerably

^more complete ^{set of data.}

4) ^{The most} convincing ^{test of TRI} would involve

a direct reconstruction of all scattering

amplitudes

from a

complete experiment

^{at different} energies ^and

angles.

As mentioned in the introduction, ^{this has}

recently

^been

performed

^{for one} energy [13] in pp

scattering.

5)

^{It is} quite concievable that TRI could be violated in _np scattering and hence in nuclear reactions, ^without

being

violated in _pp scattering. ^{Thus the}

general

nucleon nucleon

scattering

matrix could contain a term

(where tiz

is the third component ^{of the}

isospin

^matrix

for the ith nucleon), ^while

g(E, 0)

^{could be}

identically

zero. Very little is known

experimentally

^{about TRI}

in _np

scattering.

^{Tests should be}

performed

ⁱⁿ

regions

where the

following quantities

^are

large :

For each of

the

tests in

(4. 3)

^the

region

^{must be so}

chosen that the

amplitude

^with which g ^or h is inter-

fering

^is

large.

6.

Preliminary analysis

^of

experimental

^data.

In order to illustrate the status of TRI in _pp scatter-

ing

^we have taken data ^on the A-P difference from

experiments specifically designed

^{to test TRI}

[26-34]

at 142, 213 and 635 MeV. These are

reproduced

ⁱⁿ

table III. We have fitted the asymmetry

Aaono

^and

polarization Pnooo by

^a

Legendre polynomial

expan- sion

The

resulting

^{curves are}

presented

^on

figures

2a, ^3a and 4a. The corridor of errors is

provided by

^{an error}

matrix calculated from the

experimental

^{errors. We}

see that close to 900 the asymmetry approaches ^zero

somewhat faster than the

polarization (this

^{is consist-}

enly

^true for many

energies

up ^to 1

GeV). Using

^the-

formula

we have determined the

angular dependence

^of

I g I sin ogd.

The results ^are shown on

figures

2b, ^3b

and 4b, for the three

energies

concerned. The diffe- rential cross section

a(0)

^{and the}

amplitude d(O)

^were

calculated from the

Saclay-Geneva phase

^shifts [5].

(I.e.

modifications of

a(0)

^and

d(O)

^{due to the} presence of TRI

violating phase

^{shifts were}

ignored).

^The quantities

(5 .1 )

^{are all}

reasonably

large for the consi- dered

energies.

^Indeed the minimal values

of (1- Dnono

are

approximately equal

^to 0.8, 0.6 and 0.3 at 142,

213 and 635 MeV

respectively,

^for

8cM

> 30°. The

quantity (1 - Knoon) is

consistently

larger

^than

(1

^-

Dnono)

^{in the} considered energy

region

^and (1 ⁺

Cnnoo)

^{is even}

^larger,

^since

Cnnoo

^is

positive

^{in this}

region

^of energies ^and

angles.

^{At low}

energies

^the

condition on (5 .1 c) becomes the

limiting

^{on since}

there we have

Cnnoo = Aoonn ’" -

^1.

Table III. ^-

Experimental results from

^{TRI tests} measuring ^{the P-A}

difference.