Exploring photoproduction dynamics through polarization experiments

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Exploring photoproduction dynamics through

polarization experiments

F. Arash, G.R. Goldstein, M. J. Moravcsik

To cite this version:

Exploring photoproduction

dynamics

through

polarization

experiments (*)

F. Arash

(1,3),

G. R. Goldstein

₍₂₎

and M. J. Moravcsik

_{(3, ~)}

(1)

Department

of

_{Physics, Temple University, Philadelphia,}

PA 19122, U.S.A.

(2)

Department

of

Physics,

Tufts

University,

Medford, Mass 02155 U.S.A.

(3)

Department

of

_Physics

and Institute of Theoretical Science,

University

of

Oregon, Eugene,

Oregon

97403 U.S.A.

(Reçu

le Il décembre 1989,

accepté

le 21

_{février 1990)}

Abstract. 2014 The structure of

spin-0

meson

_{photoproduction}

from

_{spin-1 /2 particles}

is

given

in several useful

_optimal

frames Polarization

_experiments

for this reaction are then discussed from the

point

of view of

a) determining

the reaction

_{amplitudes, b) studying one-particle-exchange}

mechanisms,

c) detecting

patterns in the

_amplitude

structure. The

_existing

data are then

analyzed

in terms of the above

considerations,

and the need for future

experiments

is

specified.

Classification

Physics

Abstracts 13.60 -

13.75G - 13.88

1. Introduction.

Spin-0

meson

_{photoproduction}

from

_spin-1/2

hadrons has

_always

_played

an

_important

role in

our

_{longstanding quest}

to discover the laws

_{underlying strong}

interactions.

_{Experimentally,}

such reactions have been

_{relatively easily}

accessible

_through

the various electron accelerators.

Theoretically,

the reaction is

_{partly strong}

and

_{partly electromagnetic,}

and since the latter is

to a

_large

extend under our calculational

_control,

such reactions have formed an

_important

alternative to the

_entirely

_strong

_{interaction processes.}

_{Photoproduction}

reactions are also

favorable in the richness of their

_spin

structure,

being

neither

_overly

_simple

nor too

complicated. Unfortunately,

this

_advantage

has not been well

_exploited

in the

_past,

as we will

see, and in fact one of the aims of the

_present

paper is to describe the _ways in which this defect

can be remedied.

In any case,

however,

photoproduction played

a crucial role in the

_exploration

of the

pion-nucleon interaction at the lowest

_energies

in the very

early days

of

_{particle physics [1],}

and

interest in such reactions has continued into the

_present.

A number of

_present

_day

laboratories

_[2]

concern themselves with such reactions and

_{plan experimental}

_programs around them.

It

_is,

_therefore,

our aim to

_give

a

comprehensive exposition

of the

_polarization

structure of this reaction and of the uses of this structure for

_exploring

the

_{dynamics underlying}

such a

reaction. The outline will be as follows :

Section 2 : Formalisms

_{describing photoproduction.}

Section 3 : The

_{observable-amplitude}

structure for Lorentz-invariance

_only.

Section 4: The

_{observable-amplitude}

structure with Lorentz invariance and

_parity

conservation for several useful formalisms.

Section 5 :

_Determining

the

_amplitudes

from the observables.

Section 6 : Polarization

_signatures

of various

_dynamical

mechanisms. Section 7 : The

_analysis

of

_existing

data.

Section 8 :

_Looking

at the future.

2. Formalisms

_{describing photoproduction.}

There have been two

_{major types}

of

_{phenomenological}

formalisms for the

_description

of

photoproduction

reactions.

The first and

_{chronologically}

older one is the

_partial

wave cum

_multipole

formalism

_[3],

used from the very

beginnings

of the studies for

_{photoproduction.}

This formalism results in

parameters,

each of which

_gives

the reaction

_strength

in a

_{given angular}

momentum and

multipole

state of the

_{photon-hadron}

and meson-hadron

systems.

This

_description

is

_not

as

economical for this reaction as the

_{corresponding angular}

momentum

_{(phase shift) description}

is for elastic

_scattering,

since

_unitarity

does not reduce the number of

_parameters

_needed,

except

when Watson’s theorem can be used to link

_{photoproduction}

with the

_{corresponding}

elastic

_scattering.

The

_description,

_however,

still results in _economy at the lowest

_energies

where

_only

a _very few states need to be included. The

_description

also has some theoretical

advantages

in energy

regions

where certain

_angular

momentum and

_multipole

states dominate the

_dynamics,

such as at a few hundred MeV’s. Another theoretical

_advantage

is that certain

mechanisms in the interaction

_(such

as the meson-current

_term)

can be

_explicitely

included

[4],

thus

_making

the

_angular

momentum

_expansion

_converge better. An

_{important advantage}

of the

_description

is that it links data at different

_{angles (at}

a

_{given energy).}

The

_{disadvantages}

of this

_description

manifest themselves

_increasingly

as one _goes to

_higher

energies.

There the number of

_parameters

needed becomes very

large,

and since the method

requires

data in an

_{increasingly complete}

_range of

_angles,

the determination of the

_parameters

becomes

_{experimentally}

more and more difficult. At the

_{higher energies}

the distinctive role of

particular angular

momentum and

_multipole

states also subsides.

Furthermore,

in this method

there are

_always

uncertainties about where the

_angular

momentum and

_{multipole expansion}

can be cut off without

_mutilating

the

_analysis.

The other

_type

of formalism used for

_{phenomenological analyses}

of

_{photoproduction}

reactions is in terms of reaction

_{amplitudes [5].}

This method

_performs

the

_analysis

at every energy and

angle separately,

which

_represents

a loss of economy, at least at lower

_energies,

but offers

_{greater flexibility}

since data at

_{single angles}

can thus be treated. The number of

reaction

_amplitudes

for a

_given

reaction is

independent

of energy, and this

eventually

favors

this method over the first one in which the number of

_parameters

increases with energy

indefinitely.

The

_{amplitude description}

also has the

advantage

of

_offering

_greater

_variety

and

_flexibility,

since the number of different

_amplitude

formalisms that can be used for the

_description

of a

given

reaction is very

large,

and so one can

_adjust

the choice of formalism to the

_specific

aim at hand.

In _{this paper}

_only

the

_amplitude

formalisms will be

used,

partly

because the other

above a GeV

_(or

even

_starting

from below

_{that) only}

the

_{amplitude description}

is

_practically

feasible.

Thus the next

_step

is the choice of

_amplitude

formalisms.

For a

given

reaction,

and for a

given

set of

_{symmetries holding}

for that

reaction,

the number of

_amplitudes

is the same for any formalism. This number

depends only

on the values of the

spins

of the

_{particles participating}

in the reaction and on which

_{symmetries (conservation}

laws) apply.

This has been discussed

_extensively

before,

and hence we will

_{just apply}

those

results to our

_present

reaction. In

_particular,

for the reaction of

the number of

_complex

reaction

_amplitudes

with the

_imposition

of Lorentz invariance alone is

8,

while if

_parity

conservation is also

_imposed,

the number reduces to 4.

Since the observables

_{depend linearly}

on bilinear

products

of the

complex amplitudes,

there are, for the case of Lorentz invariance

alone,

64 different bilinear

products

and hence 64

different

_{linearly independent}

observables. If non-linear

_dependences

are also

considered,

the number of

_independent

observables is

_only

2 x 8 - 1 = 15

_(because

one overall

_phase

factor is

_{always arbitrary).}

The

_{corresponding}

numbers for the case when both Lorentz invariance and

_parity

conservation are

imposed

are 16 and 7.

The

_relationship

between observables and the bilinear

_products

of

_amplitudes

is

_{given by}

a

(large)

matrix. In order to _{achieve economy and}

_simplicity

of structure, this matrix should be

as close to a

_diagonal

one as

_possible.

It can be shown that at

_{complete diagonalization}

of this

matrix is

_prohibited

by

_requirements

of

_{Hermiticity (which corresponds}

to the

_requirement

that observables be

_real).

The class of formalisms in which the matrix is as close to

_diagonal

as

possible [6]

is called «

_optimal

». In it this matrix consists of a

string

of small matrices

along

the main

_diagonal

and zeroes

_everywhere

else. The sizes of the small

matrices,

for any

four-particle

reaction,

are

_{1-by-1, 2-by-2, 4-by-4,}

or

_8-by-8.

If one of the four

_particles

has zero

_spin

(as

it is the case in our

_reaction),

there are no

_8-by-8

matrices.

In _{this paper} we will consider

_only

formalisms that are «

_{optimal »}

in the sense

_explained

above. Other

_amplitude

formalisms have no

_advantages

over

_optimal

ones and are less

economical and more

_complicated

and obscure in their structure.

_Although

some

_amplitude

formalisms used in the

_past

are

_non-optimal,

many others are

_optimal,

and will appear as

special

cases of our discussion.

The observables referred to above are of the

_{« primary »}

_type,

that

is,

in them the

polarization

states of each of the three

_particles

with

_spins

are

_specified.

In actual

_practice

one

deals often with

_experiments

in which the

_polarization

state of one or several

_particles

is

averaged

over, that

is,

in which

unpolarized

particles

are used. Such observables

_(which

can

be obtained from the

_primary

observables

_{by averaging)}

are called «

_secondary

» observables

[7].

The numbers of

_amplitudes

and observables in the various submatrices of the

_optimal

formalism can be ascertained

_easily.

The situation for the case when

_only

Lorentz invariance

is

_imposed

is

_given

in table I. This

_applies

to all

_optimal

formalisms.

When

_parity

conservation is also

_{imposed [8],}

the size of the submatrices and the related numbers

_depend

on which

_type

of

optimal

formalism we consider. Because of

_that,

and

because of the submatrix structure

_being

_{very much}

_simpler

_{anyway in that} case, tables

analogous

to table 1 for the

_{parity conserving}

case will not be

_given.

Optimal

formalisms differ from each other in the orientation of the

_quantization

axis for the

participating particles.

When

_only

Lorentz invariance

_applies,

each of these axes can

_point

in

Table I. - Submatrix structure in the

optimal formalism

with

_only

Lorentz invariance

imposed.

Legend for

the columns : A : Size of the submatrix

B : Number of submatrices with Re’s C : Number of submatrices with Im’s

D : Number of observables in such submatrices E : Which indices are the same ?

F : Number of observables

_including

_secondary

ones

is also

_imposed,

the orientation for a

_given

of these four axes must be either in the reaction

plane (« planar

formalism

_»)

or

_{perpendicular}

to the reaction

_{plane (« transversity}

forma-lism

»).

Unless time reversal invariance or identical

particle

constraints are

additionally

imposed,

however,

the four

_(or,

in our case,

three)

orientation axes can be chosen

independently

of each other. This is the case for the reaction under consideration. Thus in this

case we have the

_possibility

also of so-called

_hybrid

formalisms in which some of the

_particles

are in a

_planar

frame and others in a

_transversity

frame.

For the

_photon

(or,

for any zero-mass

_particle)

the

helicity

formalism

[9]

_appears to be the

most « natural » one since the zero-mass

_particle

can have

only

two

_polarization

states, which

are the

_positive

and

_{negative helicity}

states. For this reason all formalisms we will consider

have the

_photon

in the

_helicity

frame.

For the two

_{nucleon, however,}

we have a choice. There _appears to be no reason to treat the two nucleons

_differently

from each

_other,

_though

we are not

_{compelled by}

conservation laws

to do so. Thus we will consider

_only

formalisms in which the two nucleons are either both in

the

_transversity

frame or in the

_planar

frame.

Having

the two nucleons in the

_transversity

frame offers some

advantages.

This is so

because if we consider the task of

determining

phenomenologically

the

amplitudes

from

experimental

measurements, for all

_{parity conserving}

reactions this task can be most

_simply

accomplished

for the

_{transverity amplitudes.}

This is so because in that case the direction

normal to the reaction

_plane,

which is the

_{distinguished}

direction for the

_operation

_{of space}

reflection,

coincides with the orientation of the

_quantization

axes of the

_particles.

As a

_result,

a

_simpler

set of

_experiments

can be used to determine the

_amplitude

parameters.

Furthermore,

the determination of the

_magnitudes

of the

_amplitudes

from a subset of

experiments

can also be done more

_easily

in the

_transversity

frame.

One of the

formalisms, therefore,

that we will use to describe

_{photoproduction}

with the additional constraint of

_parity

conservation is the

_hybrid

frame in which the

_photon

is in the

helicity

frame and the two

_{spin-1/2 particles}

in the

_transversity

frame. We will call this formalism the

_hybrid

formalism.

The second formalism is the one in which all three

_particles

with

spin

are described in the

helicity

frame. This is a traditional formalism and hence deserves to be included.

The third formalism is one in which the

_photon

is described in the

_helicity

frame but the two

spin-1/2 particles

have their

_quantization

directions oriented in the

_{« planar-transverse »}

direction,

that

is,

in a direction in the reaction

_plane

which is

_{perpendicular}

to both the

helicity

direction and the

_transversity

direction. Such a

_{planar-transverse}

formalism has

attracted attention since in several other

_strong

interaction

reactions,

and at a broad range of

energies,

the relative

_phases

of the reaction

_amplitudes

in such a

_system

_appear to be

multiples

of 90

_{degrees [10],}

thus

_marking

a

_general

feature of

_strong

interactions hitherto not

predicted by

any model. In the other reactions this feature

appeared

in a _pure

planar-transverse

_frame,

that

is,

in a frame in which all

_particles

had

_quantization

axes in the

planar-transverse direction. This is

_slightly

different from our

_present

situation where the

photons

are described in the

_helicity

frame,

and so it is of interest to

_explore

whether the feature

observed in other reactions appears in such a

_{hybrid planar system}

or not.

Finally,

the fourth formalism we will use is the so-called

_{« magic »}

formalism

_[11],

which is

particularly

suited to

_explore

the

_degree

of dominance of t-channel

_{one-particle-exchange}

mechanisms. The «

_{magic »}

formalism is also a

_planar

one in which the

_quantization

axes of

particles

are in directions which can be obtained from the

_helicity

directions

_by

a rotation

_(in

the reaction

_{plane) by angles}

which are

_{given by expressions depending}

on the kinematic

parameters

of the reaction.

any other frame

by

linear

transformations,

which will be

given

in this paper.

Thus,

in

obtaining

the

_amplitudes

from the

_experimental

measurements, one has two

_options.

One is to determine the

_amplitudes

in

_{question directly}

from the

_experimental

observables.

In that case one needs to have measured the

_particular

set of observables which leads

_directly

to such a determination. For _{that purpose, tables} are

_{given linking}

the

_amplitudes

to observables in each of the four formalisms. The

_advantage

of this method is a

_possible

reduction of the uncertainties in the

_amplitude

parameters.

The drawback is that the

particular

set of

_experiments

needed to determine the

_amplitude

in a

_{given system}

_{may be}

experimentally

more cumbersome or difficult to _carry out.

The second

_option

is to determine the

_amplitudes

in a formalism in which this

determination is

_{experimentally}

the

_{simplest (which,}

almost

_invariably,

turns.out to be the

transversity system,

as mentioned

_above),

and then to use the transformation relations among

amplitudes

in different frames to obtain the

_amplitudes

in the desired formalism. The

advantage

of

_greater

_experimental

ease is

_coupled,

in this case,

by

the

disadvantage

of

additional uncertainties in the

_amplitudes

due to the transformation relations themselves.

More

_precisely,

in this second method the error matrix on the

amplitude

_parameters

will tend

to have

_{larger off-diagonal}

_elements,

thus

_resulting

in a less clear determination of the values

of the

_{amplitude parameters.}

Another

_disadvantage

of the second method is in the case when the set of

_experimental

data is not

_{complete enough}

to allow a

_complete

determination of all the

amplitude

_parameters.

This situation occurs

_{quite frequently}

in

_practice.

In such a situation the first of the above

methods _may

_{possibly yield}

some

_{partial knowledge}

of the

amplitude parameters,

while the

second method does not.

One can see,

therefore,

that there is a

good justification

to exhibit the

amplitude-observables relations in several formalisms and to

_give

the relations

_connecting

the

amplitudes

in various frames to the

_amplitudes

in other frames. We will do this in section 4. As said

_earlier,

all this is not needed in section

3,

since for Lorentz invariance

alone,

the

observable-amplitude

structure

_is,

_{formally speaking,}

identical for all

_optimal

formalisms.

3. The

_{observable-amplitude}

structure for Lorentz-invariance alone.

The

_generation

of the

_amplitude

relations in the

_optimal

formalism is

by

now a standard

procedure

since not

_only

has the

_general

formulation of it been available for a

_long

time but

also numerous

_{specific examples}

of it are available in the literature. It

is, therefore,

sufficient here to review the notation and then

_supply

the relations

_themselves.

The

_amplitudes

are denoted

_{by D ( a ;}

d,

b )

where d denotes the final

_{spin-1 /2 particle,}

a the

photon,

and b the initial

_{spin-1 /2 particle.}

As an abbreviation in the

_tables,

the

_amplitude

will

also be denoted

_by

the

_triplet

of indices in the

_following

fashion :

The _a, b and d denote the

_{spin projections along}

the

_quantization

axis. Since the

_photon

will

always

be taken in the

_helicity

frame,

a can assume the values + 1 and - 1

(denoted simply by

+ and

_-).

The indices b and

_d,

which refer to

_{spin/ 1 /2 particles,}

can assume values of

+ 1/2

and -

_1/2,

which will be denoted

_by

+ and -.

The observables are denoted

_by

L

_{(uv, UV ; E fJ),}

here u and v refer to the

_photon,

U and V to the initial

_{spin/ 1 /2}

_particle,

and and n to the final

_{spin-1 /2}

_particle.

In the

_primary

observables,

for each of the

_arguments

in

L,

we can have the four

possibilities

of

The

_relationship

between these states and the more traditional Cartesian

_arguments

for the

spin-1 /2 particles

is

_{given by}

where z is the

_quantization

direction.

The

_{corresponding}

relations for the

_photon

are somewhat more

involved,

_partly

on account

of the zero-mass of the

photon

and

_partly

on account of a traditional

_{nomenclature,}

originating

in classical

_optics,

for the various states of

_{polarized light.}

This is therefore discussed in detail in

_appendix

A.

The

_secondary

observables

_contain,

for the

_{spin-1/2 particles,}

R and 1 as

_{before, but,}

instead of + +

_{and - - ,}

the

_unpolarized

states A =

_{( + + ) + ( - - )}

and the vector

polarization along

the

_quantization

axis, à

=

(+ + ) - ( - - ).

In the matrices

_connecting

the bilinear

_products

of

_amplitudes

and the

_experimental

observables,

for the case of Lorentz invariance alone all non-zero numbers are + 1 or

- 1,

which therefore are

_denoted,

for the sake of

_{simplicity, by}

+ and -,

respectively.

In the tabulated

_{relationships}

the

_headings

of the columns

_give

two

_amplitudes,

one below

the

_other,

which denote

_DD*,

with D

_being

the upper and D * the lower one. The

designations

of the rows

_give

the

observables,

with

L (

) omitted,

that

is,

showing only

the

arguments.

The

_{relationships}

are

_given

in table II.

Only

the

secondary

observables are indicated.

4.

_{Observables-amplitude}

structure for Lorentz-invariance with

_parity

conservation.

While the

_{observable-amplitude relationship}

was the same for all

_optimal

formalisms,

regardless

of the

_quantization

directions of the

_particles,

when

_only

Lorentz invariance was

imposed

on the

_reaction,

the situation is different when

_parity

conservation is also

_imposed.

We therefore have to consider the various formalisms of interest one

_by

one. As

_explained

in

section

_2,

four

_particular

formalisms will be discussed. In all of them the

_photon

is in the

helicity

system,

and the differences among the four formalisms are therefore in the

quantization

axes of the two fermions.

In

_particular,

in what we will call the

_{hybrid system,}

the fermions are in the

_transversity

frame. The second formalism is the

_{all-helicity system}

in which the two fermions are also in

the

_helicity

frame. The third formalism has the two fermions in the

_planar

transverse

direction,

that

is,

their

_quantization

axes are normal to both the transverse and the

_helicity

directions.

_Finally,

the

_fourth,

the so-called «

_{magic »}

frame has the

_quantization

axes of the

two fermions in the reaction

_plane

but rotated from the

_helicity

directions

_by

certain

_angles

which can be calculated

(as

functions of the kinematic

_parameters)

from a well defined and

known formula.

As far as the structure of the

_relationship

between observables and

_{amplitude products}

is

concerned,

the above four cases can be classed into two. In one the two fermions have

quantization

axes in the

_transversity

direction,

in the other in the reaction

_plane.

The latter class includes the

_{all-helicity,}

the

planar

transverse, and the

magic

formalisms. These three will

_only

differ from each other in the labels of the observables but not in the structure of the submatrices which connect observables with bilinear

_products

of

_amplitudes.

Thus it is useful

to discuss first these two

_classes,

and then later turn to the differentiation of the various formalisms within the

_planar

class.

As we discussed

_earlier,

under

parity

conservation the

_{eight independent amplitudes}

reduce

to four. Because all of the formalisms we are interested in have the

_photon

in the

_helicity

Table II. - The

These

_equalities

are different for the

_hybrid

and the

_planar

_cases, but are the same for all

formalisms in the

_planar

class.

Indeed,

the

_{relationships}

are as follows :

For the

_hybrid

case :

For the

_planar

case :

In these

relations,

the upper

signs

hold when the

_product

of the four intrinsic

_parities

of the

particles

in the reaction is

_{+ 1,}

while the lower

_signs

hold when this

_product

is

- 1. For

photoproduction

of

_pions

or kaons when both fermions are nucleons or

_hyperons,

this

_product

is + 1.

The notation of the

_amplitudes

and of the observable

_arguments,

for the latter

_using

the conventional

_symbols

often used

_{by experimentalists,}

is

given

in table III. Not

_given

are the

corresponding

observable

_arguments

for the

_magic

_formalism,

since those

_arguments,

expressed

in terms of this conventional

_notation,

are

_complicated

and

_dependent

on

kinematic

_{parameters. Fortunately}

this

_complication

in no _way

_implies

that the

_experiments

well suited for this

_particular

formalism are also

_complicated.

_{Since, however,}

the

_magic

frame is

_simply

a

_special

case of the

_planar

formalism which is included in our tables and

results,

for _any

_{particular application}

of the

_magic

frame

only

a small amount of

straightforward

calculation is needed to obtain

_specific

result for that frame from the results

given

in the tables of this paper.

The actual

_{relationships}

between observables and

_{amplitude products}

are

_given

in

tables IV-VII. We note from the remark at the bottom of table IV that results for the two

different values of the

_product

of intrinsic

_parities

are _very

_similar,

_{differing only}

in the

_signs

of some of the observables.

5. The

_{expérimental}

détermination of the

_amplitudes.

Since we want to determine four

_complex

_amplitudes,

and since one overall

_phase

is

_arbitrary

and

_{unmeasurable,}

we need at least seven measurements to eliminate all continuous

ambiguities

from the

_amplitude

determination. Since the

_relationship

between observables and

_amplitudes

is

_bilinear,

not all sets of seven

_{linearly independent}

observables will eliminate

the continuum of

_ambiguities,

and even if a

_given

set of seven

_does,

there may still be discrete

ambiguities

left which will then be eliminated

_by

additional measurements.

In this section we will comment on how the continuum of

_ambiguities

can be eliminated

_by

the various sets of seven measurements.

As for other

reactions,

the determination of the seven

_{amplitude parameters (apart}

from

discrete

_ambiguities)

can be

decomposed

into several

steps,

and it is useful to do so if definite

Table III. -

Notation

_{for amplitudes}

and observable arguments

for

some

of

the

important

formalisms.

In the «

_{hybrid » formalism}

the

photon

is in the

_{helicity frame}

and the two

_fermions

in the

_transversity

_frame.

In the «

_{all-planar » formalism}

the

_photon

is in the

_helicity

_frame

and the two

_fermions

in a

_general

_{planar frame.}

The other two

_formalisms

are

special

cases

of

the

« all

_planar

»: the

_{« all-helicity »}

_formalism

hds the two

_fermions

also in the

helicity

frame,

while the

_{« semi-planar-transverse »}

has the two

_fermions

in the

planar-transverse

optimal

frame.

The above observable _arguments

_pertain

to the fermions. For the situation in the case of the

photon,

see the

appendix.

done

_{quite accurately (much}

more so than one can determine in later

_steps

the relative

phases

of the

_amplitudes),

and in the determination of the

_magnitudes

there are never any discrete

ambiguities

which arise

_exclusively

in the determination of the relative

_phases.

What set of

_experiments

are needed to determine the four

_magnitudes

will

_depend

on the

formalism used. It was shown on other reactions that the best

_system

for the

phenomenologi-cal determination of

_amplitudes

for a

parity-conserving

reaction is the

transversity

frame. We

do not have a _pure

_transversity

frame for our

_present

reaction since the

photon

is

always

taken

in the

_helicity

_frame,

but even

_{taking just}

the two fermions in the

_transversity

frame

will,

as we will see,

represent

some

_advantages

from this

_point

of view.

It is also known

_[12]

that the

_magnitudes

can

always

be determined

through polarization

measurements in which the various

_particles

are

polarized

_along

their

_quantization

directions. It is evident that once one carries out a

_complete

set of

_experiments,

one that is sufficient to determine the

_amplitudes

in some

_frame,

that set can also determine the

amplitudes

in _any

Table IV. - The

relationship

between

secondary

observables and the bilinear

_{products of}

reaction

_{amplitudes for}

_spin-zero

boson

_{photoproduction from spin-1/2 fermions,}

with Lorentz invariance and

_parity

conservation, in the

_{hybrid optimal formalism}

in which the

_photon

is in a

helicity

frame

and the

_fermions

in a

_transversity

_frame.

For notation, see the text and the text

and the

_caption

_of

table II.

This table is for the

product

of the intrinsic

parities being

+ 1. If the

_product

is - _1, the

_signs

of the observables in which the

photon

argument is R or 1 should be reversed.

footing

from the

_point

of view of the determination of the entire set of

_{amplitudes, they}

differ in

_merit,

as

_explained

_above,

when it comes to

_obtaining

information from a

_{yet incomplete}

set of

_experiments.

Furthermore,

frames can also differ from each other in the

_{precision by}

which

_{they yield}

amplitude parameters,

since the error matrix connected with

_amplitude

_parameters

also

becomes transformed as we _{go from} one frame to another.

These

_general

remarks will now be demonstrated on the

_particular

situation we have at

hand for our

photoproduction

reaction. For this purpose we look at tables IV-VII. In the

_hybrid

formalism,

the four

_magnitudes

can be determined

_by

a set of

_experiments

using only unpolarized photons,

and the fermion

_{polarizations}

are

_only

in the N direction.

Table V. - The

relationship

between

_secondary

observables and the bilinear

products

of

reaction

_{amplitudes for spin-zero}

boson

_{photoproduction from spin-1/2 fermions,}

with Lorentz invariance and

_parity

conservation, in the

all-planar optimal formalism.

For notation, see the

text and the

_{caption of}

table II. For a note on intrinsic

_parities,

see table IV.

Finally,

to determine also the

third

relative

_{phase (e.g.}

between 1 and

_3,

or between 1 and

4,

or between 2 and

_3,

or between 2 and

4),

we need also

_experiments

with

_{polarized photons.}

We have a choice of whether we want to use

_{plane polarized}

or

_{circularly polarized photons.}

Thus,

in the

_hybrid

formalism,

the

_amplitude

determination is

_neatly

divided into three

steps,

each

_{distinguished}

from the others

_by

the

_type

of

_experiment

_needed,

and each

_step

yielding

definite information on a subset of the seven

_{amplitude parameters.}

In any

all-planar

formalism

_(that

is,

also in the

all-helicity

and

_{semi-planar-transverse}

formalisms)

the situation is less favorable. The subset of

experiments determining only

the

magnitudes

is of course also

_present,

and it can involve

_only

one

_type

of fermion

_polarization

(namely à).

The

_{subset, however,}

_requires

some measurements with

_{polarized photons,}

in this case

_{circularly polarized.}

One can also choose a set with

_{linearly polarized photons,}

but then the fermion

_{polarizations}

are not in the 4 direction.

Once the

_magnitudes

are determined in the

_{all-planar frame, however,}

we can determine the relative

_phases

without

_{polarized photons.}

This process can

_again

be done

stepwise,

each

step consisting

of two measurements, and

yielding

two relative

_phases.

Table VI. - The

relationship

between

secondary

observables and the bilinear

_{products of}

reaction

_amplitudes

_for

_spin-zero

boson

_{photoproduction}

_{from spin-1/2 fermions,}

with Lorentz invariance and

_parity

conservation, with

_{all particles}

in the

_{helicity frame.}

For notation, see the

text and the

_caption

_of

table II. For a note on intrinsic

_parities,

see table IV.

measurements with

_{polarized photons}

were

_available,

but various measurements with

unpolarized photons

but

_polarized

nucleons were.

_Using

the

_hybrid

frame one could then

extract from the six available

_experiments

six of the seven

_{amplitude parameters.}

A similar

analysis

in,

say, the

all-helicity

frame would have

_yielded

no

_amplitude

_parameter

at

_all,

_only

six

_complicated

constraints among the seven

amplitudé

_parameters.

We should remember that these considerations

_apply

to

_{photoproduction}

_processes

regardless

of whether the

_product

of intrinsic

_parities

is + 1 or -

_1,

since that switch

only

changes

the

_signs

of some of the observables but not the way the observables and the

amplitude products decompose

into submatrices.

Some of these

_{possibilities}

for the determination of the reaction

_amplitudes

are shown in table VIII.

-6. The

_{polarization signatures}

for various

_dynamical

mechanisms.

The

_dynamical

mechanism that has been most successful in

_describing

an enormous range of data in

_particle

reactions of all varieties and

_energies

is the one

_{particle exchange (OPE)}

Table VII. - The

relationship

between

_secondary

observables and the bilinear

_{products of}

reaction

_{amplitudes for}

_spin-zero

boson

_{photoproduction from spin-1/2 fermions,}

with Lorentz invariance and

_parity

conservation. The

_photon

is in the

_{helicity frame,}

the

_fermions

in the

planar-transverse

optimal

frame.

For notation, see the text and the

_{caption of}

table II. For a

note on intrinsic

_parities,

see table IV.

amplitudes,

the existence of

_specific

OPE leads to _{constraints among the}

_amplitudes

that reflect the

_spin

and

_parity

of the

_{exchange [14].}

In the

_following

we will consider those

constraints for

_{general amplitudes.}

We will also construct the «

_magic

frame »

_amplitudes

for which the constraints take

_{especially simple}

forms. In the course of this discussion the

special

properties

of the massless

_photon

will be encountered.

The reaction in

_question,

y + N --* 7r

+ N,

viewed in the t-channel becomes

y + _{p -}

N

+ N with the

_spin

structure

_{1 + 0 --> 1/2 + 1/2.}

Because

_helicity

0 is not allowed for the massless

_photon,

there are 8

_{spin amplitudes,}

which reduce to 4 due to

_parity

conservation

(as

we assume for the remainder of this

_section).

For definiteness let the t-channel reaction

_amplitudes

be written as

_{D (a;}

d,

b

_),

where a,

d,

b are the

_photon,

_nucleon,

anti-nucleon

_{spin components (along}

some

specific directions)

respectively,

and those

_components

take the

_{values ± 1,}

_{± 1 /2, ± 1 /2, respectively.}

The four

independent amplitudes

will be chosen as

_{D( +}

_{1 ;}

+

_1/2,

+

_{1/2 ),}

_{D(+ 1 ;}

+

_{1/2, - 1/2 ),}

D (+

1 ;

-

1/2, + 1/2 ), D( +

1 ;

-

1/2, -

1/2 ) .

Note that

parity

allows us to choose the

complete

set of four

_amplitudes

so that the

_{photon argument}

is

always

positive.

Consider the

_coupling

of a

_spin

zero

_{exchanged particle.}

At the

N

+ N vertex there can

_only

be 1

_unique

_coupling

of the scalar or

_pseudoscalar

to the

_1/2

+

_1/2

_system.

This is also true at

Table VIII. - Some

of

the many

experimental procedures for

the determination

of

the reaction

amplitudes for

spin-0

boson

_{photoproduction}

with

_{spin-1/2 fermions,}

in the three

_formalisms

considered in detail in this _paper.

Step

# 1 :

_{Magnitudes only :}

0 or ~ or A.

_Step

#2 : Two

_{phases : 2022}

or M or A.

_Step

*3 : The third

_{phase :}

x or + or

T.

hybrid

amplitude

for

_spin

0

_exchange.

For

_{higher spin exchange,}

vector, axial vector and

_beyond,

the number of

_{independent amplitudes}

becomes

_{unrestricted,}

i.e.

four,

although parity

conser-vation still

_imposes

some additional constraints.

To

_study

_{these various restrictions and express OPE in the} most

_transparent

formulation we

the

_(planar)

frame for the s-channel _process in which the

_spins

of the external

particles

are

quantized along

the directions

_{given by}

their

_{corresponding}

s- to t-channel

_{crossing angles}

[15].

More

_precisely,

the

_quantization

axis for

_particle

A in A + B - C + D is

_{given by}

a

counter-clockwise rotation in the

scattering plane

from the momentum direction of A

_through

an

_angle

_{X A} which is the

_{crossing angle}

for

_particle

_A,

and

_similarly

for

particles B,

C,

D. The

magic

frame has the salubrious

_{interpretation}

that it

_corresponds

to the t-channel

_helicity

frame. In that frame the

_{exchanged particle helicity}

is conserved in its

_coupling

to the external

particles

helicities

_(without

any orbital

angular

momentum

_component).

The

_{crossing angles}

for the

_{photoproduction}

reaction can be obtained from the

_general

formulae of reference

_{[15] ;}

the results are

where m is the nucleon mass, » is the

pion

mass, and s and t are the usual Mandelstam

kinematic variables. Note that because of the zero mass of the

_photon,

the

_{crossing angle}

is 0 for the

_photon

for all

_energies

and

_angles

in the

_scattering.

This leads to the

_important

conclusion that the matic frame for

_{photoproduction always}

leaves the

_{photon spin qùantized}

along

its momentum, i.e.

_helicity

is the

_{appropriate quantization}

for the

_photon.

The

magic

frame

_amplitudes

with a

single spin J exchange satisfy

the

_simple

factorization

equation [14]

from which it follows that

or

where the

_{signs depend}

on the

_parity

of the

_{exchanged particle}

as well as the overall intrinsic

parity.

When these definite J t-channel

_partial

wave

_amplitudes

are combined with the

relevent rotation functions it is

_only

relation

_(4.2)

that survives since both

_amplitudes

are

multiplied by

the same

dj, 0 (0,).

The final relation then becomes

This relation between the

_magic

frame

_{amplitudes provides}

a test of the

_single

OPE

_analogous

to the relation that has been used and tested for nucleon-nucleon

_{scattering [14].}

The

special

case of

spin

0

exchange requires

some further

_development.

To

begin

with,

constructed. For

_{pseudoscalar exchange}

the boson vertex must have the form

_{E (k ) .}

_k’,

where k and k’ are the

_photon

and

_pion

momenta,

respectively.

Then the

_coupling

vanishes for

photon

linear

_{polarization perpendicular}

to the

_scattering

_{plane, e(1- ) (k )}

oc

_y

oc

e +

+

e (- ).

That translates into the statement that

_{D (+ 1 ; d, b ) = - D (- 1 ; d, b ).}

The latter is related

to

_{D( +}

_{1 ;}

-

d, -

b ) by parity,

so for

_{pseudoscalar exchange}

Furthermore the

_coupling

to the fermion vertex in the

_magic

frame

_imposes

the restriction of d =

b,

so that

_{D( + 1 ;}

_{+ ,}

_{+ ) = D (+ 1 ; - , - )}

are non-zero, whereas the other 2

amplitudes

vanish. These results are correct whether the

_pseudoscalar

is a

_pion

or a heavier meson. Of course, when it is a

_pion,

_gauge invariance

_requires

additional contributions

(nucleon poles

in the s- and

u-channels),

but this does not alter the

_preceding

conclusions.

Lastly,

for

_{completeness,}

we write the relations between the

_magic

frame

_amplitudes

{D(... )}

and the

_helicity

_{amplitudes {Dh(... )},}

,

where the

_angles

are

_given

from

equations (4.1).

7.

_Analysis

of the

_existing

data and conclusions.

One could summarize the situation of the

_{photoproduction}

data in one sentence : there is no

energy and

angle

at which a sufficient number of different

_{polarization experiments}

have been

carried out to allow a direct determination of the

_{photoproduction amplitudes,}

without

dynamical assumptions.

The

_existing

data can

be,

somewhat

arbitrarily,

divided into two

_parts.

The old data before

1970,

starting

at the

_beginning

of the

1950’s,

consisted almost

_exclusively

of

_unpolarized

differential cross section measurements. These data were then

analyzed

with the

help

of the

phase

shift

_analogue

in

_{photoproduction, namely}

the

_{multipole amplitude}

series.

In this method of

_analysis

it is assumed that

_only

a finite number of

_multipole

states will contribute to the process at the lower

energies

at which the data are

_available,

and to

_simplify

the

_analysis

further,

the Watson theorem which links

_{photoproduction amplitudes}

with

scattering amplitudes using unitarity

is used. This work is

_by

now

old,

is limited in _energy, is

constrained

_by

the above

_{assùmptions,}

and hence will not be further referred to here. The second

_part

of the data are those taken after 1970. The various

_{experiments (other}

than

unpolarized

differential cross

_sections)

with

_{specifications}

and references are listed in

table IX. In these

_experiments

measurements other than

_unpolarized

differential cross section

appear much more

_frequently,

but

_still,

the

_types

of observables measured are limited.

Combining

the information of table IX with our

_analysis

of the

_{experimental requirements}

for the determination of the

_amplitudes,

_and,

_{particularly,}

with the content of table

_VIII,

it is easy for

experimenters

to deduce what

_experiments,

and at what

_energies

and what

_angles,

would be the most useful to

_complete

the

_forming

of

_complète

sets of

_experiments.

In view of the marked

_patterns

found in the

_{photoproduction amplitudes (even}

if not

completely determined) by

reference

_[13],

as well as the

_striking

_pattern

of reaction

amplitudes

found in other reactions

[16],

it would be most desirable to be able to determine

Table IX.

_{- Photoproduction}

data since 1970. The notation

_for

the observables in terms

_of

the

Appendix

A. Photon

_polarization

states.

The

_{complications}

in the discussion of

_{photon polarization}

states

_originate

with the historical

fact that this

_subject

was first covered in the context of classical

_{electromagnetic}

waves, and much later in the context of the

_{quantized photon}

as a

_particle.

The two

_{terminologies}

are

quite

different,

though

the same term

_{(« polarization »)}

is used in both.

The

_primary,

« natural » state in classical

_optics

is the linear

polarization,

which denotes the E vector

_pointing

in a

_given

direction in the

_{plane perpendicular}

to the direction in which the

light

wave _goes. In this context the circular

_polarization

state is a

_secondary,

derived

_quantity.

Thus here the E vector is

_pictured

as

_{« polarized}

».

In _contrast, the « natural » states in the

_description

of the

_photon

as a massless

_particle

of

spin

1 are the two

_helicity

_states, that

_is,

the states in which the

_spin

of this massless

_particle

points

in the direction of

_propagation

or

_opposite

to it. As it turns _out, these

_primary

states

correspond,

in the classical

_picture,

to the

_secondary

states of circular

_{polarization,}

whereas

the

_primary

states of the classical

_{picture (the}

linear

_{polarization states) correspond}

to

composite,

quasi

fictitous states in the

_particle

framework. We see that in this

_{particle picture}

it is the

_spin

of the

_particle

that is

_pictured

as

being

«

_polarized

».

One can,

nevertheless,

construct the translation from one

_language

to the

other, since,

after

all,

there can be no essential difference between the two. We will now review this

correspondence

between the two

_{terminologies}

and also connect them with the states used in the

_language

of the

_optimal

formalism.

Let us denote the direction of

propagation

of the

photon

as z, and the two directions

perpendicular

to it as x and y, so _{x, y} and z form the three axes of a usual Cartesian coordinate

system.

In classical

_optics,

then,

we denote

_by

x

_{and ÿ}

the linear

_polarization

states of the

photon

in the x

_{and y}

directions.

We will also introduce the

_{angle .0}

to describe the direction of a

_{plane polarized photon (in}

the x-y

plane)

in such a _way that in the x

direction 0 =

0,

and in

the y

direction

.0 =

90

_degrees.

A

_{plane polarized photon}

in

the 0

direction can then be described as

Still in this classical

_{optical terminology,}

the circular

_polarization

states are

complex

linear

combinations of the

_{plane polarized}

states.

_Denoting

the

_{right circularly polarized photon}

as

e(+ ),

and the left

_{circularly polarized photon}

as

_{e (-}

_),

we have

These relations can of course be inverted to

_give

Now we turn to the

_{particle description}

of the

_photon,

in which

_{by density}

matrix

_{of a spin-1}

particle

is,

in

_{general, given by}

a

_3-by-3

matrix. For a massless

_{spin-1 particle (and,}

for that matter, for any massless

particle

of

arbitrary spin),

the

density

matrix reduces to a

_2-by-2

matrix,

which is best

_given

in the

_helicity

frame. In

it,

the

_density

matrix of a

_{photon polarized}

Table A.1.

_{- Summary}

_{of photon}

_polarization

states in the two

_{terminologies}

_of

classical

_optics

and

_quantized

_particles.

_{P p}

and

_Pc

denote the

degree

of planar

and circular

_{polarization,}

respectively.

The notation

_of

In _contrast, the

_density

matrix for a

_{circularly polarized photon}

is

where

_{P, is}

+ 1 for

_right

and - 1 for left circular

_polarization

of maximum

_degree,

that

is,

for

We can see from these two

_density

matrices that the + + state of the

_optimal

formalism

correspond

to a

_{right circularly polarized photon,}

while the - - state to a left

_circularly

polarized photon.

We can also

_identify

the Re

_{(+ - )}

and Im

_{(+ - )}

states now with

_linearly

polarized

states. In

_particular,

for a

_{photon plane polarized}

in the directions

_{of c/J = 0,}

_180,

360,

etc.

_degrees

we

_{get -}

Re

_{(+ - ),}

for

plane polarization

in the directions of

c/J = 90,

270,

etc.

_degrees

we

_get

+ Re

_{(+ -}

_),

for

_{plane polarization}

in the directions of

c/J =

45,

225,

etc.

_degrees

we

_get

+ Im

_{(+ -}

_{), and}

for

_{plane polarization}

in the directions of

c/J = 135, 315,

etc.

_degrees

we have - Im

_{(+ - ).}

Possibly

at the risk of further

_confusing

the

_issue,

we

_might

mention that for a massive

spin-1

_particle,

the state in which the

_particle

is

_polarized

in the x direction is described

_(in

terms of the three states with the

_quantization

axis in the direction of

_propagation)

as

This reduces to our

_{previous expression}

for a

_photon

because the

_photon

does not have a

e ( 0 )

state

_(and

hence the normalization factor

2 1/2

also

_changes

to

_2).

The results of this discussion can be summarized in table

_A.l,

in which the three

customarily

used direction

_designators

_(L

for

_{longitudinal,}

i.e. z in our

notation,

S for

sideways,

i.e. x in our

notation,

and N for

normal,

_{i.e. y}

in our

notation)

also appear.

In this last context it is

_important

to _note,

_however,

that the L-S-N notation is

_meaningful

only

when the

_photon

is considered in a reaction in which the reaction

_plane

is

_defined,

whereas the rest of the above discussion

_applies

also to a

_{single photon}

which is not

_part

of the reaction. The results of the

_{table, therefore,}

as far as

_{they pertain}

to the connections between L-S-N and _x-y-z rest on the convention of

_identifying

S

with 0 =

0.

An observation of considerable

_practical

interest can be derived from the above results. In

terms of the

_optimal

_formalism,

+ + and - - are

_primary

observable

_arguments

for the

photon,

but it turns out that these are at the same time also

_secondary

observable

_arguments

in that

_they

denote

_{circularly polarized photons}

which can be created in actual

experiments.

The consequence of this is the

general

statement that in reactions

_involving

_photons,

experiments involving

circular

_{polarized photons}

are related to the

_{amplitude products}

in a

way which is

simpler

than it is the case for

_linearly

polarized photons,

and thus

efforts

should

always

be made to have

circularly polarized photons

available,

since

_{they yield simpler}

information.

References

[1]

For _this,

_by

now classical, area of

particle physics,

see for

example

BETHE H. A. and DE

HOFFMANN F., Mesons and _Fields, Vol. II, Row Peterson

₍₁₉₅₅₎

or G. Kallen,

Elementary

Particle

_{Physics (Addison Wesley, 1964).}

[2]

For

_example

SLAC, Cornell, Bonn,

Orsay,

Frascati, Kharkov and Bates.

[3]

See for

_example

CHAU Y. C., DOMBEY N. and MOORHOUSE R. G.,

Phys.

Rev. 163

₍₁₉₆₇₎

1632.

[4]

MORAVCSIK M. J.,

Phys.

Rev. 104

₍₁₉₅₆₎

1451 ;

KNAPP E. _A., KENNEY R. W. and PEREZ-MENDEZ V.,

Phys.

Rev. 114

₍₁₉₅₉₎

605.

[5]

See for

_example

WALKER R. L.,

Phys.

Rev. 182

₍₁₉₆₉₎

1729.

[6]

GOLDSTEIN G. R. and MORAVCSIK M. J., Ann.

_Phys.

N. Y. 98

₍₁₉₇₆₎

128.

[7]

MORAVCSIK M. J. and GOLDSTEIN G. R.,

Phys.

Rev. D 31

₍₁₉₈₅₎

2986,

appendix.

[8]

GOLDSTEIN G. R. and MORAVCSIK M. J., Ann.

_Phys.

N. Y. 142

₍₁₉₈₂₎

219.

[9]

JACOB M. and WICK G. C., Ann.

_Phys.

7

₍₁₉₅₉₎

404.