some criticisms of phenomenological models

IL NUOVO CIMENTO VOL. 6A, N. 4 21 Dicembre 1971

Some Criticisms of Phenomenological Models.

~/[. BE]N'AYOUN a n d P. LERUST]~]

Laboratoire de Physique Atomique et Moldculaire, CoU4gc de .France - Paris

(ricevuto il 22 Matzo 1971)

S u m m a r y . - - We prove here that some phenomenological models often used to describe experimental data in the framework of many-channel scattering can be obtained as approximations of various kinds of the DID formalism we have previously developed. In this paper, we limit ourselves to the analysis of the rat ional approximation to which varying-width Breit-Wigner fol'mulae as well as the effective-range theory of Ross and Shaw are connected. Using the D/D formalism, we can easily prove rela- t ive irreducibil i ty propert ies of these models. We can, therefrom, deduce an explanation concerning the diverging descr ip t ions--obta ined through these models- -of the same experimental data. This is, in part icular , shown by means of an analysis of the S~l-wave of r:A" scattering; we are also led to emphasize some arguments to explain the deviation of the phenomenologieal Y*(1405) coupling constant from the unrefined SU~ predictions. We are thus led to ask a question about the very nature of a resonance: is i t a single pole of the T-matr ix or of the K-mat r ix?

1 . - I n t r o d u c t i o n .

The a i m of th i s w o r k is to g ive a un i f i ed d e s c r i p t i o n of a class of a p p r o x i -

m a t i o n s c u r r e n t l y used in t he p h e n o m e n o l o g i e ' d a n a l y s i s of e x p e r i m e n t a l d a t a .

U p to now, t h e m a n y - c h a n n e l processes h a v e been s t u d i e d us ing va r i ous m e t h -

ods, n a m e l y the one- or m a n y - l e v e l B r e i t - W i g n e r f o r m u l a e (wi th f ixed or

v a r y i n g wid ths ) , t he e f f ec t ive - range t h e o r y of R o s s a n d SgAW (1) (which can be

exp res sed in t h e fo rm of w a v e l e n g t h f o r m u l a e (zero range) , or of e f f e c t i v e -

(1) M. H. Ross and G. L. SHAW: Ann. o/ Phys., 9, 391 (1960); 13, 147 (1961).

30 - 1l Nuovo Cimento A. 441

4 4 2 M. Bt iNAYOUN a n d PII . LGRUSTE

range formulae), the Dalitz and Tuan models (2.~) and the weak Levinson

condition (4).

I t turns out tha t each one of these models can be expressed as a part icular

approximation of the D-function in the framework of the D/D formalism,

which we previously developed (5).

In the present paper, we shall limit ourselves to examining among the mod-

els just enumerated those which are rational (6~,~,~) (*).

In the second Section, we first recall some definitions, we afterwards

analyse the various rational approximations and then express the varying-

width Breit-Wigner formulae (BW), the Ross and Shaw models (RS) and the

weak Levinson condition (WLC) in the form of degree laws on the Ba(s)-funetions of the K-matr ix . We can then exhibit the reason why these models

cannot provide equivalent descriptions of the same phenomenology.

We also give the extension of the previous conclusions to some of the most

f requent ly used weight functions, and to any wave. In the third Section, we discuss the irreducibility properties of the models

we examined. In particular, we prove an impor tan t result concerning the

number of the D-function zeros in a ncighbourhood of the second threshold

when the problem only involves two channels. Finally, we discuss the results obtained and examine the implications of

the irreducibility properties of the rational models. We thus prove, by means of

the example of the ~3F Sly-waves, how failure of the parametr izat ion can lead

to erroneous physical conclusions. Using the review of DAL~TZ (3) on the K3~ scattering, we enforce these conclusions. I n this way, we are natural ly led

to ask a question about the physical meaning of a resonance: is it a pole of

the T-matr ix or of the K-matr ix?

(2) R. H. DALITZ and S. F. TUAN: Ann. o] Phys., 3, 307 (1960). (a) R. H. DALITZ: H.I~. 70003, Hyperon Resonances, 1970, Con]erenee at the Du]ce University, Durham, N. C. (4) B. L]~HEVRE and Pm LERUSTE: NUOVO Cimento, 66 A, 349 (1970). (5) M. BEN~.YOUN, J. LANCIEN and Pit. LERUSTE: dVuovo Cimento, 63A, 969 (1969); M. BENAYOVN and Pm LERUSTE: NUOVO Ci•ento, 65 A, 99 (1970). (6a))d. BENAYOUN and PH. LERUSTE: P.AM. 70.07 (Paris, 1970), or Nuovo Cime~to, 5A, 513 (1971). (60) M. BENhYOUN and PH. LERUSTE: P.AM. 70.08 (Paris, 1970), or Nuovo Cimento, 5A, 513 (1971). (~) M. BENAYOUN and Pm LERUST]~: P.AM. 70.09 (Paris, 1970), or Nuovo Cimento, 5 A, 513 (1971). (*) As we shall see hereafter, the rational inodcls can be defined as approximations of various kinds of thc Ba(s)- or D-hmction(s) on the whole complex plane. We exan|ine in another paper another kind of approximation in the framework of local models (2.a.7) to which the Breit-Wigner formulae with fixed widths nmst bc added. (~) M. BENAYOUN and Pn. LERUSTE: P.AM. 71.01 (Paris, 1971), or Nuovo Cimento, 5A, 513 (1971).

SOME CRITICISMS OF PIIENOMENOLOGICAL MODELS 443

The notat ions we shall use throughout this work have been defined in

ref. (5)~ however we recall some of them:

a) L, M, ..., N will denote subsets of thc whole channel set N.

b) l, m, ..., n will denote the respective cardinal numbers of the preceding subsets.

c) ~, #, ..., v will denote the ((sums))(5), which ac tual ly are a kind of

ordinal l~umber re la ted one-to-one to the corresponding subsets.

These ordinal numbers also define an unambigous indexing of the R iemann

surface ~ of the D-funetion. By Dr, we shall denote the branch of the D- funct ion on the sheet of ~ labeled by #.

By q,~ we mean the following product of channel momen ta :

q., - I I q~, V M e ~ ( N ) , iEM

with

q ~ = - l .

2. - T h e r a t i o n a l a p p r o x i m a t i o n .

2"1. Definitions and properties. - We first need to recall some classical results on many-va lued meromorph ic functions (s).

Definition 1. An algebraic function is called rational if its only singulari- ties on its Riemann surface ~ are poles (zeros and ~lgebraic branch points of ~ arc not singul~rities of ~) .

Definition 2. The order of a rat ional funct ion on ~ is the number of its poles on ~ , each one being counted with its mult ipl ic i ty .

Theorem 1. The number of zeros of a ra t ional function ](z) on ~ is equal to the order of [(z) on ~ .

We aiso have to recall tha t the D-funct ion of the D/D formal ism can be

expanded (5) us ing- - in the n-channel c a s e - - a set of 2 ~' one-valued meromorphic functions B~(s):

(2.1) Da -- ~ q.~B~,(s)(-- i) m (__)Card(L o M ) VL ~ J ( N ) ,

this formula being valid for any algebraic weight function q~; the questions

related to the l% 0 waves are not examined here (see ref. (6~)). Using this

(s) G. Sm~I~(~ER: Introduction to Rie'mann Sur/aces (New York, 1957).

4 4 4 M. B E N A Y O U N ~nd I: 'H. L E R U S T : E

relation together with Definitions 1 and 2 and Theorem 1, we can easily deduce the following Lemmata (6c):

Zemma A. Given any algebraic weight function, the assertion (~ D is ra-

t ional on ~ ~, ~ being its l~iemann surface, is equivalent to (~ the B~(s)-functions are one-valued rat ional functions on the complex s-plane )>.

.Semma B. The D-function being defined up to a one-valued meromorphic funct ion of s, the assertion (, the D-function is rat ional on ~ ~ is equivalent

to (( each one of the Ba(s)-functions can be chosen as a polynomial in s ~.

Convention. Everywhere in the following, a polynomial of negative degree must be unders tood as being identical ly zero.

The physical meaning of the rat ional approximat ion is very simple. The resonances are described through poles of the T-matr ix, i.e. through the zeros of the D-function. The additional hypothesis is tha t the left-hand cuts can be described by means of a finite number of poles and zeros of D (Lemma A) or only by means of zero of D (Lemma B). In these cases, the left-hand cut

poles and (or) zeros can be distinguished from the resonance zeros only by their location on ~ (for the D-function) or in the complex s-plane (for the B~(s)- functions).

2"2. The degree laws o/ the phenomenological models. - Each of the phenomenological models we shall analyse hereaf ter proceeds from a rat ional approximat ion of the D-function everywhere on ~ , or equivalent ly of the B~(s)- functions on the whole complex s-plane.

The rat ional hypothesis for the D-function leads us to define degree laws (see ref. (~)) on the Ba(s)-functions.

2"2.1. T h e R o s s - S h a w m o d e l s (1). The effective-range theory of Ross and S~AW (1) results in a parametr iza t ion of the inverse of the K-mat r ix derived from its Taylor expansion (1) with respect to the total energy vari-

able E:

(2.2) g - ' = (Mo) . + ½R:~(E--Eo) + i i . . . .

To be consistent with the s-dependence of the K-matr ix , we shall rewrite

eq. {2.2) as follows:

(2.3) K -1 : (M0)i~ + ½Rij (s - -s o) + i i " ' "

with an obvious change in the parametr iza t ion {eb). l~ow, we need to recall formulae of the K-ma t r ix subdeterminants (5):

(2.4) AL(K) B~(s) -- VL e ~ ( ~ ) . = Bo(s) and A~(K_I ) B,_~(S)B,(s) '

SOME CRITICISMS OF PHENOMENOLOGICAL MODELS

In the zero-range (ZR) approximation (1), eq. (2.4) becomes

(2.5) g-',j = (Mo)~j

and the ZI~ degree law follows:

(2.6) d° B a = 0 , VL e ~ ( N ) .

In the effective-range theory (ER)

K 5 x : (Mo),¢ + 1 R , i ( s - - So) •

The ER degree law can be writ ten as

[ d°Bo = n , (2.7) / d ° B u = d ° B o - - m = n - - m , V M e ~ ( N ) .

The degree laws (2.6) and (2.7) are obviously independent of the wave number as well as the weight function.

2"2.2 . T h e d e g r e e l a w of t h e w e a k L e v i n s o n c o n d i t i o n (WLC). The weak Levinson condition (WLC) can be defined as follows (6o):

(2.8) S~j ~ , or T , ~ 0 .

I t can be proved tha t such a condition is equivalent to

(2.9) D .- , / ( s ) V2~(0, 1, 2 , . , 2 " - -1 )

That is, the asymptotic behaviour of the D-function is the same for each of its branches.

From now on, we shall consider the following weight functions:

(2.10a) l~:~ = ( s - si) ~ ,

( (8 - 8 , ) (8 - 8 . ) ) ~ , (2.1o~1

8

for q, of eq. (2.1). The degree law here depends on the weight function we choose. Taking

the asymptotic behavoiur of D~ as in eq. (2.1), we find the following degree laws,

445

44~ M. BENAYOUN a n d PII. LERUST:E

for any /~ (i, 2, ..., 2n--l):

(2.11) { a°B.(s)< a°Bo(S)--m/2

d°B , ( s ) < d°Bo(s) - - 1

for ]~i,

for ~ and ~ ,

val id only for the s-wave (*).

2"2.3. V a r y i n g - w i d t h B r e i t - W i g n e r f o r m u l a e (BW). I t can be

p roved (6~.,) t ha t the vary ing-wid th Brei t -Wigner formulae can be deduced f rom the WLC degree law by adding the following s t a t ement :

r ank (K) = I for any s a C .

Then the BW degree law can be expressed as

(2.12)

d°Bo =- c ~ l ,

O ~ d ° B , ~ c - - 1 ,

d°B# = - - 1 ,

if m = C a r d ( M ) = l ,

if m = Card ( M ) > 2 .

The degree law does not depend on the wave number . I f we take c = 1 in eqs. (2.12), we obtain the one-level formula, otherwise we have to deal with a c-level Brei t -Wigner formlfla (6).

F r o m the preceding analysis of the phenomenologic~l models which satisfy the ra t ional approx imat ion , we can conclude t h a t each one of these models results in a degree law on the B x ( s ) - f u n e t i o n s . Conversely, every degree law

can a pr io r i be t aken as a phenomenologieal model. We shall discuss la ter

h~rther propert ies of the models we have just studied.

2"3. Struc ture of the R i e m a n n surface. - The R iemann surface of any many-

channel process does not in general depend on the wave number , at least for

the previously listed weight functions. The R iemann surface ~ of the D-funct ion has the following propert ies (5.6):

a) ~ is a 2n-sheeted R i em ann surface.

b) The genus (s) of ~ uniquely depends on the weight function, as can

be seen h 'om Table I .

(*) I t can be proved (6c) that the degree laws for any wave are

d°B, < d°Bo - m(1-7 ~-)

d°B~ ~ d°Bo - - l

with a D-function which takes the threshold behaviours into account.

for k i ,

fo r q~ o r 0 ,

SOME CRITICISMS OF PItENOMENOLOGICAL MODELS

T A B L E I .

447

Weight function Genus of

ki gk =2~-2(n 3 )+ 1

o~ g~.-- 2- ~ ( n - 3 ) + 1

Oi go = 2 ~ - 1 ( n - - 2 ) + 1

c) We also recall tha t ~ can be mapped onto a plane(~a) by means

of a one-to-one conformal mapping (s), if and only if

g = 0 .

d) The ow;r-determination relations (5) always exist as soon as ~ > 3 .

Fur ther properties can be found in ref. (~).

2"4. Zerology and pology.

2"4.1. T h e m e t h o d . The problem is merely to find how many zeros

the D-function (2.1) possesses. I t follows from Theorem 1 that in the rational

approximation this number is eqmfl to the order of the D-function on ~ .

From eqs. (2.1) and (2.10), it follows (6) tha t these poles can only stand at

s - 0 or c~ in the complex s-plane. Therefore, the problem turns out to be

the counting of these poles, and then we only have to find the behaviour of

D at s - - O and s - - oo.

For a regular point Zo of ~ , if D behaves like

6¢ (2.13) D z~"~. ( z - z0) ~ ' p being in teger ,

then Zo is a pole of D of order p.

For an algebraic branch point Zl, if we have

6~ (2.14) D . . . . (z--z~) ~/~ ' q in teger ,

then z, is a pole of order q for D (*).

I t becomes easy to see (6) tha t the order of D on ~ is a function of the

number of channels involved by the process and also depends on the wave

number 1. We are interested in the zerology because, ~s wc previously assumed,

(*) At iniinity, which can be a branch point, eqs. (2.13) and (2.14) have to be writ- ten D z~, ~ z~ and 1 ) ~ ' o z q/2 (p, q > 0). Every 1)hysical branch point being a square root in z, wc reea.ll results for only such points.

M. B E N A Y O U N and e m L E R U S T E

the physical quanti t ies (resonances) are related to the zerology, and not to

the poles which have been removed. On the other hand, it is clear tha t when

fitting has to be performed, the number of zeros is related to the parametri .

zation.

2"4.2. S o m e g e n e r a l r e s u l t s . I t is easy to see tha t the asymptot ic

behaviour of the D-function is in t imate ly related to the relevant degree law

on the B~(s)-functions and to the weight function choosen; on the other hand,

it linearly depends on the wave number (e) involved. The main results are

gathered in the two following Tables. The number of zeros is denoted there

by Z; the zerologies are relative to the D-function (*).

TABL]~ II. - Zerology ]or 1 =/= 0 waves.

Weight function k~+l. (~o)~+1. ~i~+1

WLC Z = c . 2 " Z = 2 ~-1. Z = 2 n-

d°Bo = c e > l + 1 . ( 2 c + n ( 2 1 + 1)) . ( c + n ( 2 1 + 1)) c ~ l c ~ l

BW b = d ° B i ~ e - I Z = 2'*. Z = 2". d O B o = c > l 1) If O ~ b ~ c - - l - - 1 . (1+ 2(/~c)) .(2/+ l + c ) i stands for the Z = c.2" sum of the set {i} 2) If b~0 , b ~ c - - l

Z = 2 " ( b - - l + 1)

RS Z = 2 ~-1" Z = 2 ~-1. Z = 2 "+1" (ER approximation) .(2/+ 1)n . n . (21~- 3) . n . ( 2 1 ~ - 1)

RS Z = 2n-1(2/+ 1)n (ZR approximation)

TABL]~ I I I . - Zerology ]or the s-wave.

Weight function ki o Qi

WLC or BW c ~ 1 Z = c .2" Z = 2"-1(2c÷ n) Z = 2" (c+ n)

RS (ER approximation) Z = n-2" Z = 3n'2 ~-1 Z = n '2 n+l

RS (ZR approximation) Z = n-2 n-1

2"4.3. Z e r o s of t h e D - f u n c t i o n i n t h e n e i g h b o u r h o o d of t h e

s e c o n d t h r e s h o l d in t h e t w o - c h a n n e l c a s e . In this case, eq. (2.1)

can be rewri t ten as

(2.15) Do ---- (Bo - - i k l B 1 ) - - i k2(B2 - - i k l B 3 ) •

(*) No te added i n proo]s. - We do not take here into account questions related with causality, which arise for l ~ 0 waves. They will be examined in another paper.

SOM]~ CRITICISMS OF PHENOMENOLOGICAL MODELS

The second threshold s2 is a branch point of ~ ; it will be a zero of Do if

(B o - - i k i B 1) . . . . = 0 .

Using the T- invar iance proper t ies (~), we deduce

(2.16) B l ( s ) : Bo(s) : 0 a t s = s2 ,

because of He rmi t i an ana ly t ic i ty of the B~(s)-functions.

I f B 2 - - i k l B 3 is not zero for s ~ - s 2 , s2 is only a s i m p l e pole of the ampli tude.

I f we want a pole of order 2, we have to require

(2.17) B2(s~) : B3(s2) : 0 .

This result mus t be unders tood as follows. Let the set of four functions

Bo, B1, B2, B3 be given; we should have more than one zero in a certain neighbourhood of s~ if definite slight var ia t ions of the ra t ional functions Ba(s )

c a n b r i n g more t han one zero of D a t s : s2, i .e . if the var ied B~(s)-funetions

sat isfy eqs. (2.16) ~nd (2.17). :Now, we only have to look a t the various degree

laws to see tha t such a possibil i ty is forbidden in the Ross-Shaw models which

necessarily imply in this case

B3 : cons tant ¢=- 0 .

Such a result has i m p o r t a n t consequences in phenomenology, as will be seen hereafter .

449

3. - D i s c u s s i o n a n d c o n c l u s i o n s .

3"1. I r r e d u c i b i l i t y p r o p e r t i e s . - F r o m the above analysis of the various

ra t ional models used to describe phenomenology, we have derived the properties ga thered hereaf ter .

3"1.1. T h e v a r y i n g - w i d t h B r e i t - W i g n e r f o r m u l a e . They a~re a

special form of the D I D formal ism where the D-funct ion (or the B~(s)-functions) satisfies the following restr ict ions:

a) the polynomia l approx ima t ion holds for the B~(s)-functions;

b) the weak Levinson condition is general ly satisfied (');

(*) This is always true apart from thc case when we chose k i as weight function and when the wave number is nonzero.

450 M. B~NA~OUN and PH. LERUSTE

c) we necessarily have factorization of the T- and K-matr ix for every complex energy s, because

rank (T) = rank (K) = 1 , Vs e C ,

i.e. we h~ve: Card ( M ) > 2 implies B , ( s ) ~ 0 ;

d) in the two-channel case, the D-function can have two zeros in a neigh-

bourhood of the second threshold.

3"1.2. T h e R o s s a n d S h a w m o d e l s . These models obey another set of restrictions with respect to the D-function of the D/D formulism:

a) The polynomial approximat ion is assumed for the Ba(s)-functions.

b) The weak Levinson condition always is violated by the zero-range

approximation and generally holds for the effective-range approximation (*).

c) B~ always being a nonzero constant, this forbids factorization of the T and K matrices outside poles. Indeed, we have

rank (T) = rank (K) = n , a.e.~

for an n-channel process.

d) The D-function cannot have more than one zero in a neighbourhood

of the second threshold in the two-chunnel ease (**).

I f we now compare the properties of these two kinds of models, we can easily

find that the two last points are opposite. The first two points are either

identical or noncontradictory.

3"2. Physical consequences. - To discuss the consequences of the irreduci-

bility properties, we first list some of them:

a) The zero-range as well as the effective-range formulae(~) have long

been used to describe processes involving two or more channels (9"11). In some

cases the description thus derived gives a correct view of the experimental

data (9.~,); in others (1o), the description obtained fails. To explain this disa-

greement, one is often led to introduce new physical hypotheses.

(*) This is always the case, except when k~ is taken as weight function and for waves other than l = 0. (**) In another paper (7), we show that this fact also holds for the CSL Dalitz and Tuan (2) model, though it proceeds from another kind of approximation. (o) A. VV. tIENDItY and R. G. MOORtIOUSE: Phys. Left., 18, 171 (1965). (lo) p. 1N. DOBSON: Proceedings of the Oxford Iuternational Co~]erence (Didcot, Bcrks., 1965), p. 167. (xx) p. vow HIPPI.m and JAP. KWAN KIM: Phys. Rev..Lett., 20, 1303 (1968).

SOME CRITICISMS OF PI tENOMENOLOGICAL MODELS 451

Using the zero-range approximat ion , DOBSON (~0) fails when he analyses the ~q product ion in the f ramework of a two-channel process. ()n the other hand, H~:NDRY and MOORHOUSE ('~) succeed in their f i t t ing only by introducing explicit- ly ((r~rc) Ae-cttannel) or implici t ly (by t~king (M~,);, as complex matrices) a

th i rd channel in the process.

b) On the other hand, UCmYA~X-CA~IPm~LL and Lo(~AN (~2) (*), as well

as LEFIEVRE and LEmrST1~" (4), succeeded in their two-channel descriptions of

the SH-wave of 7~JY' scat ter ing and t he ~ product ion, by means of a varying-

width Brei t -Wigner formula; note tha t the fits in ref. (a) are be t te r than those

in ref. (12) because the A~*(1700) is tt~ken into account as an elastic background.

Then, we are led to opposite physical conclusions according" to the method used. W h a t we are now going to describe is in fact independent of the very nature of the Swwave , for which more complicated models t han the two-ch~mnel

one have been proposed (~3). Our purpose s imply is to show how m~them,~tical constraints lead artificially to se t t ing up physicid hypotheses.

I t can easily be proved tha t the two-channel models of ref. (4,10.12) have

always to deal with four zeros of tlw D-function: the zerology of DOBSO:¢

(see Fig. 1) is, however, complete ly different f rom those in ref. (4,~-~) (see Fig. 2)

with respec t to the phase shifts of Fig. 3 and 4 (**).

I t can also be p roved tha t the zerologies of Fig. 1 arc the best tha t can be obtained via any of the Ross and Shaw models (x). There two zeros can only provide a description of the background inste,~d of Fig. 2, where two zeros par t ic ipa te to provide the dominant physical effects. We can thus un- derstand the consequence of the hypothesis of HENDRY and MOORKOUSE~ who open a new (qlannel and can by this w~y avoid the above-ment ioned disagree-

men t (the general result of Subsect. 2"4.3 does not hold any longer in the three- channel c~se).

Our conclusion is tha t we have to fit exper imenta l da ta through different kinds of models, to test if any possible d isagreement proceeds f rom physics or if i t is ins tead a direct consequence of the model used.

c) We are then led to improve the :~ssertion of DALITZ (a) when he discusses the relat ive independence of the phenomenology with respect to the very values of the K - m a t r i x paramete rs .

(12) F. UCIIIYAMA-CAMPBELL ~1~11(| R. K. LO~AN: Phys. Rev., 149, 1220 (1966). {*) Their (( effective-range )) fits cannot a priori be accepted because they par~metrize (qKq) -1 a,nd not K -1. Then m) comp~trison can bc performed with the fits in ref. (9.10). (13) M. NIiVEU: cotnmunication at the Con/e;rence de Physique des Hautes E~Tergies, Aix-en-Provenee, /970. (**) These two Figures lmve been taken from ref. (4) for convenience. Fit~urc 3 is also the result of the N D -1 model of BALL el al. (14); we do not a~nalyse this model, which is not rational with respect to our detiniti(m in Sect. 2. (14) g. S. BALL: Phys. Rev., 149, 1191 (1966); J. S. BALL, I{. G. GARG a,td G. 1~. SHAW: Phys. Rec., 177, 2258 (1969).

4 ~ 2 ]~I. BENAYOUN and P m LERUSTE

she•physIm u l Im u she~et 3

/cal .,,.~,,,,~physical

Fig. 1. Fig. 2.

Fig. 1. - Structure of the four-zero D-function. Case of the zero-range two-channel model. In this Figure by • we indicate the location of the zeros of the D.function.

Fig. 2. - Structure of the four-zero D-function. Case of the varying-width Breit- Wigner two-channel model. In this Figure by • we indicate the location of the zeros of the D-function.

~90,0

0.0

O0 0.8 04 0 I

200

' \ I I

400 600 800 1000 Trt(MeV)

90.0

~= 50.0

0 0.8

-% O.4 o [

200

, -/

I I i

400 600 800 1000 T:(M eV)

Fig. 3. Fig. 4.

Fig. 3 . - Bali 's results for the Sll-wave (through an N D - : two-channel model).

Fig. 4 . - Lefievre-Leruste results for the S::-wave (varying-width Breit-Wigner model). The dashed curve is the fit without the ~*(1700) elastic background.

We can also give a reason to use models other than those involving the

effective range to fit the Yo*(1405) coupling constant . The argument is inti- mate ly related to those we discussed above for the S11-wave of ~A ~ scattering.

N o w we k n o w t h a t Ross a n d S h a w m o d e l s c a n n o t p r o v i d e t w o zeros of t h e

D-functions near the second threshold (this is the K S one; if I = 0, a good a p p r o x i m a t i o n is to t a k e K A ~ a n d ~:Z in to a c c o u n t a n d neg lec t t h e AT:~: channe l ) .

On t h e o t h e r h a n d , t he o t h e r m e t h o d used to fi t t he se d a t a is a t w o - c h a n n e l

p o t e n t i a l t h e o r y (15) w h i c h g ives a s ingle zero of D n e a r t h e s econd t h r e sh -

(15) H . W . WYLD: Phys. Rev., 155, 1049 (1967); H. W. WYLD and R. K. LOGAN: Phys. Rev., 158, 1467 (1967).

SOME CRITICS ON PHENOMENOLOGICAL MODELS 4 5 ~

old. Then we only have results proceeding from models involving one pole of

T- -ze ro of D - - i n the neighbourhood of the relevant threshold. Consequently,

we cannot conclude that the difference between the S U3 predictions and the

results of this kind of model derives from theory ~nd not from the models.

We are thus strengthened in our opinion by the discussion of the Swwave.

In the case when two poles of T are necessary to describe the Y*(1405)

resonance, we are led to ask again a question concerning the very nature of a

resonance: is it one pole or a set of poles the T-matrix? Or is it merely a single

pole of the K-matr ix (~)?

Indeed, the present S U3 classification of ht~drons defines a resonance by

means of a set of quantum numbers (Y, J , P, I), the mass being fixed phenome-

nologic~lly as ~ pole of T. When m a n y decay channels ~re open, this proper ty

is taken into account by means of a p~rtial width for each channel.

I f our idea can be improved, i.e. if the Y*(1405) can be well fitted by two

poles of T near the second threshold, then the question would be what to

put into SU3: every pole of T, or the corresponding pole of the K-matr ix?

Though DALITZ (2) chooses the first possibility, we prefer the second.

Our reason can be expressed us follows. Let u many-channel process be given;

the occurrence of a set of poles of T is due to the same reason as the existence of a set of partial widths for each resonance (~).

If this ~ssertion is generally academic because most of these poles cannot

be seen, there exist some cases--especially the Y*(1405)--for which, because

of the threshold effects, more than one pole can be observed. These cases con- sti tute a test of our hypothesis.

~Te gratefully thank g . BARLOUTAUD Of C.E.lq.-Saclay for fruitful discus-

sions, and for having drawn our at tent ion to questions related to the models herein called rational.

~Yote added in proo/s.

We have just received a paper by G. RAJASEKARAN (An empirical test o/ composite hadrons, Tara Institute of Bombay preprint) which also deals with the problem of defining a resonance.

Using field-theoretical methods (Lee model), he defines a pole of the T-matrix as a resonance when, in its neighbourhood, it can be found a pole of the K-matrix. Hence a resonance can bc related with the ficld-thcoretical <( elementary )> particle. Otherwise, this author proves that we have to deal with <( composite )~ hadrons.

Roughly speaking, the elementary-particle case corresponds to the Breit-Wigner degree law. The composite-hadron case is then related with the zero-range model degree law, where the K-matrix has no pole.

(,8) See the note added at the end of the paper.

454 M. BENAYOUN ~nd PII. LERUSTE

• R I A S S U N T O (*)

Si d i m o s t r a qui che ,~lcuni model l i fenomenologic i u s a t i spesso pe r descr ivere i da t i s p e r i m e n t a l i hello s c h e m a dcllo s c a t t e r i n g di lnol t i cana l i possono essere o t t e n u t i come app ros s imaz ion i di va r i t ip i dcl fo rma l i smo D/D d a noi p r c c e d e n t e m e n t e svf luppa to . I n ques to ar t ieo lo ci l imi t i amo a l l ' ana l i s i delle app ros s imaz ion i r az iona l i a cui sono con- hesse sia le fo rmule di B r e i t - W i g n e r di ampiezza va r i ab i l c sia la t eo r i a del raggio effet- t i vo di Ross e Shaw. F a c e n d o uso del fo rmal i smo D/D, si d i m o s t r a f ac i lmen tc le pro- p r i e t h di i r r edue ib i l i th r e l a t i v a di ques t i modell i . Si pub ind i d e d u r r e u n a spiegazione delle d i s c o r d a n t i deser iz ioni degli s tessi d~t i spe r imen ta l i , o t t e n u t e t r a m i t c ques t i modell i . Si d i m o s t r a cib, in pa r t i eo la re , pe r mezzo di u u ' a n a l i s i dello s c a t t e r i n g n~N" in onde SH; si ~ p o r t a t i a m e t t c r e in r i l ievo a lcuni a r g o m e n t i p e r spiegare la der ivaz ione della cos t an t c di a e e o p p i a m e n t o dello Y*(1405) f enomenolog ica d,~llc p red iz ion i di S U a non raff ini te . Si ~ cosi c o n d o t t i a pors i u n ' i n t e r r o g a t i v o sulla n a t u r a p rop r i a di u n a r i s o n a n z a : essa

u n polo singolo del la m a t r i c c T o del la m a t r i c e K ?

(*) Traduzione a cura della Redazione.

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(*) IlepeeeOeno peOanquefi.

some criticisms of phenomenological models

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