Analytic phase-shift analysis

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Analytic phase-shift analysis

D. Atkinson, G. Mahoux, F.J. Yndurain

To cite this version:

D. Atkinson, G. Mahoux, F.J. Yndurain. Analytic phase-shift analysis. Journal de Physique, 1973,

34 (7), pp.495-497. �10.1051/jphys:01973003407049500�. �jpa-00207411�

LE JOURNAL ^DE PHYSIQUE

ANALYTIC PHASE-SHIFT ANALYSIS

D. ATKINSON

Institute for Theoretical

Physics,

^{PO Box}

800, Groningen,

Netherlands G. MAHOUX

Service de

Physique Théorique, CEA-Saclay,

^{BP n°}

2,

⁹¹¹⁹⁰

Gif-sur-Yvette,

France F. J.

YNDURAIN

Departamento

^de

Fisica,

Universidad Autónoma de

Madrid,

Canto

Blanco, Madrid-34, Spain (Reçu

^{le 24}

janvier 1973)

Résumé. ²⁰¹⁴ En dessous du seuil

inélastique,

^on peut déduire de la seule connaissance des sec-

tions efficaces différentielles et

polarisations

des réactions

élastiques 03C0

⁺ p ^{et 03C0 -} p, la section effi-

cace différentielle et la

polarisation d’échange

^de

charge,

^ainsi que ^tous les

paramètres A

^{et R.}

Par

ailleurs,

au-dessus du seuil

inélastique,

^{la mesure de toutes} les observables n’est _pas suffisante pour réduire une

ambiguité

continue dans la détermination des

déphasages.

Abstract. ²⁰¹⁴ From a

knowledge

^{of the 03C0 +} p and 03C0 - _p elastic differential cross-sections and

polarizations

below the inelastic

threshold,

^{we can}

predict

^the

charge-exchange

cross-section and

polarization,

and all the A and R parameters. However, in the inelastic

region,

^a measurement of all observables is not sufficient to remove a continuum

ambiguity

^{in the} determination of the

phase-

shifts.

Tome 34

NO 7 JUILLET 1973

Classification

Physics ^Abstracts

10.10 ^- 10.42

Surprising though

it may seem, ^a measurement of the differential cross-sections and

polarizations fort +

_p and - _p elastic

scattering (at a given

energy below the inelastic

threshold)

may be sufficient ^to determine

completely

^{all the}

phase-shifts,

and therefore in

particular

^the

charge-exchange

cross-section and

polarization,

^{as well as} the A and R

parameters

^for all

charge

^{states. On the other}

hand,

^{above the inelastic}

threshold,

^a

complete

^{set of} measurements

(differential cross-sections, polarizations, A

^{and R}

parameters,

and total

cross-sections,

^{for all}

charge states),

^leaves

a continuum

ambiguity

^{in the} determination of the

phase-shifts.

Since the

scattering

^{is described}

by

^four

complex functions,

^{which are}

analytic

^{in the}

complex

^{cos 8-}

plane,

^{and which are} constrained

by

^{four real}

unitarity equations

below the inelastic

threshold,

^one

might

indeed

expect

^{that the} measurement of

only

^four

real

quantities

should suffice for the

complete

^deter-

mination of the

system.

^We shall show

presently

^that

this is true, ^{and in}

particular

that the four real _quan- tities _may be taken to be the above two cross-sec-

tions and two

polarizations.

^An

interesting

alternative

is to take the

experimentally

accessible

cross-sections,

7C+ p --+ n+ P, n - p --> 71- p, 7T’

^{p --+}

^nO

^{n, and}

the

7ruz

_p

polarization [1].

Above the inelastic

threshold,

^the

unitarity

conditions are

only inequality constraints,

^{and we can show that a}

complete

^set

of measurements leaves undetermined one real function of x ⁼ cos 9. The

corresponding phenomenon

^for

scattering

^without

spin

^or

isospin

has been discussed

recently [2], [3].

We consider first

n +

_p

scattering,

since this is not

encumbered

by isospin complications. (Exactly

^the

same

equations,

^{and so the same}

conclusions,

^obtain

for

K+

_p

scattering.)

^{We write the}

amplitude

^as

the differential cross-section

(multiplied by q2)

^as

and the

polarization

^as

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01973003407049500

496

We consider the elastic

unitarity

constraint as a

mapping

^{of the}

imaginary

parts, a ^and

b, of g

and

h,

^{into a’ and b’}

[4] :

where

We find conditions on

Q(x)

^and

P(x)

^{for the}

existence,

and

eventually

^local

uniqueness,

^{of a fixed}

point a = a’, b = b’.

We work in a Banach _space of

analytic

^functions

characterized

by

^{the norm}

where

’1

^{is not}

greater

^{than the}

semi-major

axis of the

large

^Martin

ellipse,

^{as in our similar treatment}

[5]

of the

spinless

^{case. This space is}

sufficiently large

^to

encompass any

amplitude

^of

physical

interest. Since

Ai

^and

B,

^{become infinite if}

D(x) vanishes,

^{it is neces-}

sary ^to look for a

fixed-point

^{in a} ball that does not contain the

origin

^{of the Banach} space, but is centred about ^a

spin-independent

^solution

(à, 0) corresponding

to the above

equations

^{with the same}

u(x),

^{but with}

P(x)

⁼ 0. In order to

complete

the existence

proof,

we

exploit

^the

analyticity of a(z)

^and

b(z) (1 - Z2)-1J2

in the Martin

ellipse by writing

eq.

(6)

in the form

and

similarly

^for eq.

(7).

^We

thereby guarantee

^the

exponential

decrease of

Ai

^and

BI,

^for

large 1,

^and

consequently analyticity

^{in the}

z-plane.

^{For further}

details,

^{we refer to a}

forthcoming

paper

[6].

The introduction of other

charge-states

^is

straight forward,

^{and for} definiteness we consider as

given

the 7i" p - 7T’ p

cross-section, a -(x),

^and

polariza- tion, P - (x).

^{We define a new}

mapping

^{for the}

isospin 2 amplitudes,

which has the same form as eq.

(4)-(11), except

^{that in} eq.

(4), (5), (8)

^and

(9),

^a,

^{b, A}

^{and B}

refer to the I

= 2

state, whereas in _eq.

(6), (7), (10)

and

(11) they

^{refer to the} n - p - n - p

isospin

^combit

nation, namely [(I

⁼

2)

⁺

2(1

⁼

-!)]/3. (In

^{the case}

Ko

_{p -}

K°

_p, one

simply

^{has to} take the I = 0

ampli-

tudes in the

first,

^{and the} combination

in the second set of

equations).

^{When the cross-}

sections ₆₊ and a - are

sufficiently

^{close to the} pure S-wave _case, and the

polarizations

^{are small}

enough,

there is a

locally unique

^fixed

point

^{of the}

equations,

as may be shown

by applying

the Contraction

Mapping Principle

^{in our Banach} space

(12).

Above the inelastic

threshold,

^{the terms}

and

must be added

respectively

^{to the}

right-hand

^{sides of}

eq.

(8)

^and

(9) (for

each of the two

isospin states),

where _{11 l:f:} are the usual

inelasticity parameters.

^It

can now be shown

that,

^{even when a}

complete

^{set of}

independent quantities

^{is measured}

(which

^{amounts to}

seven real

functions),

^{still one} of the four functions

(14)

and

(15)

is left undetermined. This

indeterminacy corresponds precisely

^{to a}

global, x-dependent phase.

The constraints of the

optical

^{theorem are}

only

^finite-

dimensional,

^{and can}

easily

^{be met, and} this still leaves an infinite number of

the 11

¹ undetermined.

We wish to

acknowledge helpful

discussions with R. F.

Alvarez-Estrada,

^G.

Bart,

^{P. W.}

Johnson,

C.

Michael,

and R. L. Warnock.

Note added in

proof :

^{The fact that a} measurement of four real

quantities

may suffice for the

complete

determination of the non

system

^below the inelastic threshold does not

require

^{the use of cos}

0-analyticity.

A treatment

employing merely continuity

^{in cos 0}

can be found in a recent

preprint

of R. F. Alvarez- Estrada and B. Carreras

[7].

497

References and Footnotes

[1]

The fact that a

complete

^{set of} measurements is not necessary for the determination of all the scatter-

ing amplitudes

below the inelastic threshold will

always

^be encountered wherever a symmetry group like

spin

^or

isospin

^acts.

[2]

BowcocK, ^{J. E. and} HODGSON, ^{D. C. On} the Existence of Different

Amplitudes Giving

^{Rise to the}

Same Differential

Cross-Section, Birmingham preprint.

[3] ATKINSON,

D., JOHNSON, ^{P. W. and} WARNOCK, ^R. L., Determination of the

Scattering Amplitude

from the Differential Cross-Section and

Unitarity,

to be

published

in Comm. Math.

Phys.

[4]

The idea of

considering unitarity

^{as a}

mapping

^{on a}

suitable _space was introduced

by

Martin and Newton.

NEWTON, ^R. G., ^{J. Math.}

Phys.

⁹

(1968)

^2050.

MARTIN, A., Nuovo Ciments 59A

(1969)

^131.

[5]

ATKINSON, D., MAHOUX, G., YNDURÁIN, ^F J., Construction of a

Unitary Analytic Scattering Amplitude

from Measurable

Quantities.

^{I. Scalar}

Particles, Saclay preprint (1972).

[6] ATKINSON,

D., MAHOUX, G., YNDURÁKIN, ^F. J., ^Cons- truction of a

Unitary Analytic Scattering Ampli-

tude from Measurable

Quantities.

II. Introduction of

Spin

^and

Isospin (UA

of Madrid

preprint,

in

preparation).

[7] ALVAREZ-ESTRADA,

^R.

F., CARRERAS,

B., ^JEN pre-

print,

^Madrid.