Quasilinear models of low-energy pion-pion scattering and the rigorous constraints below threshold

8~.A~

Nuclear Physics B46 (1972) 2 9 5 - 3 1 8 North-Holland Pubhshmg Company

QUASILINEAR MODELS OF LOW-ENERGY PION-PION SCATTERING AND THE RIGOROUS CONSTRAINTS BELOW THRESHOLD * 0 PIGUET and G WANDERS Instttut de Phystque Theorlque, Unn,erslt~ de Lausanne, Lausanne, Switzerland Received 4 April 1972 (Revised 9 June 1972)

Abstract Models of S- and P-wave pmn-plon scattering are investigated Thetr amphtudes satisfy elastic umtarlty and are quaslhnear m the sense that they are nearly hnear functions of s m the unphyslcal interval 0 < s < 4m~r The models are submitted to rigorous constraints holding m this interval and resulting from crossing symmetry and posltlvlty of the absorptxve parts If one imposes only the hnear parts of the amphtudes as input, these constraints have nearly no restrictive effect However, ff the p-meson is required and if the shape of the I = 2 S-wave Is fixed, the I = 0 S-wave is strongly restricted m a small but fimte domain above threshold (280 MeV < mTrn ~ 450 MeV) In the o-region, the hmxtatlons resulting from the constraints are no more slgmflcant One concludes that the rigorous constraints below tlareshold have a physical relevance which is restricted to a small energy interval above threshold

1 INTRODUCTION Low-energy p l o n - p l o n s c a t t e r i n g is one o f the few fields m h a d r o n physics w h e r e the p h e n o m e n o l o g l c a l d e s c r i p t i o n o f a process can be c o n n e c t e d d i r e c t l y to specific p r e d i c t i o n s resulting rigorously f r o m first principles Besides the general analytlcxty a n d u m t a r l t y p r o p e r t i e s o f the partial wave a m p h t u d e s r e f e r r e d f r o m the a x i o m s o f q u a n t u m field t h e o r y [ 1 ], we have a set o f e x a c t c o n s t r a i n t s o n the behavxour o f these a m p h t u d e s m the u n p h y s l c a l interval (0 ~< s ~< 4 m 2) (We set m n -_7 i from n o w o n ) These c o n s t r a i n t s are direct c o n s e q u e n c e s o f a n a l y t l c l t y , crossing s y m m e t r y [2] a n d posatxvlty o f the a b ~ o r p t w e p a r t s [ 3 - 9 ] . we shall call t h e m crossing a n d posltlvlty c o n s t r a i n t s T h e possible physical relevance o f such c o n d i t i o n s is due to the c o n n e c t i o n , w a a n a l y t i c c o n t m u a t m n , b e t w e e n the low-energy p a r t o f a p a r t m l wave a n d its b e h a w o u r b e l o w the elastic t h r e s h o l d As a m a t t e r o f fact, a large n u m b e r o f low-energy plon-pxon s c a t t e r i n g m o d e l s has * Work supported by the Fonds National de la Recherche Sclentlfique

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0 Plguet, G Wanders, Quaslhnear models

been proposed m the recent past The crossing and posltlvlty constraints are frequently used, either as basic ingredients [ 1 0 - 1 4 ] or as subsidiary tests [ 1 5 - 1 7 ] Studying these inodels, one gets the impression that there should be amplitudes which lulflll the rigorous constraints and which are compatible with experiment (It is needless to say that this unpression originates partly in our still rather Incomplete knowledge ot the plon-plon phase shifts ) In spite of this impressive production of models, we beheve that the real physical relevance ot the crossing and posltlVlty constraints is still unclear This opinion IS based on the following remarks We know from the work ot AtkInson [18] that a large class of amplitudes is compatible with the general requirements of analytlclty, crossing symmetry and unitarlty restricted to the two-body system Therefore the constraints alone cannot determine a unique model some input must be added Explicit physical requirements form, In general, a hrst visible part of this Input Scattering lengths can be imposed, or the existence of the o-meson The nature of the model used and the way It is parametrlzed constitute a second, more occult part of the input For instance, uniqueness can be simulated by hmiting adroitly the number of free parameters Our experimental Information is inalnly limited to a 500 MeV Interval around the o-meson Theretore, we are interested in the physical predictions of a model in this energy region, and not just above threshold The procedure of extrapolating a model from the unphyslcal interval 0 ~< s ~< 1 up to the o-meson is partly arbitrary It is evidently impossible to establish a clear distinction between the role of the crossing and posmvlty constraints and the effects of a particular parametrlzatlon and a specific extrapolation Nevertheless, one ol the motivations of the present investigation is to get some clarification of this problem To this end we consider a class of models depending on a suitably chosen number of parameters On the one hand, this number is not too large, in order to get significant restrictions on the values of the parameters trom the crossing and posltlvlty constraints and the physical requirements On the other hand, the number of parameters is large enough so that uniqueness cannot be artificially enforced The models we consider are sets of I = 0 and I = 2 S-waves and I = 1 P-waves which fulfill elastic unltarlty exactly We denote the three amplitudes of a model by ./i(s)(fo(s) andf2(s ) are the I = 0 and I = 2 S-waves, fl(s ) IS the I = 1 P-wave) Our models belong to a class which is characterized by a nearly linear behaviour of the amplltudesJ'i(s) In the unphyslcal interval 0 < s < 1 There are several reasons which explain our interest in these quasihnear models First, as a matter of fact, a large fraction of the proposed models exhibits a quasllInear behavIour without reqmrlng It exphcItly Therefore a systematic study of the quasillnear models in the context of the crossing and posltlVIty constraints is not a purely academic exercise In fact, the relevance of the positlvity constraints IS particularly problematic In the quasflinear case This IS due to the fact that the linear terms which dominate the amplitudes in (0, 1) are submitted to three and only three crossing constraints The

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remaining crossmg constraints and all positivlty constraints affect only the small corrections to the linear terms, which are, In some sense second-order effects The question is to know how these second-order effects propagate Into the physical region Finally, the hypothesis of quaslhnearity allows an approximate treatment of the crossing and posltlVlty constraints With this approximation schelne a relatively large number of parameters can be Included Into the models, the size of the numerical calculanons remaining within reasonable hlnlts The main conclusion of our investigations is that the use of the constramts in the unphyslcal interval (0, 1) does not allow well dehned predictions concernmg the pregion This holds it the linear parts of our amphtudes are imposed It remains true If we impose the p-meson and fix the shape of the I = 2 S-wave We are lett with a large family of I = 0 S-waves containing resonating as well as non resonating amphtudes On the other hand we find out that all the members of this family have the same behavIour in a relatively small, but finite, interval above threshold This means that, combining our various mputs with the crossing and posItlVity constramts we get signihcant restrictions on the phase shifts in the interval 1 < s <~ 2 5 (280 MeV < r n ~ ~ 450 MeV) Our results are in contradiction with claims, according to which the crossing and posItlvlty constraints lead to unambiguous predictions up to the p-region, 1or instance, nearly unique S-waves once the I = 1 P-wave is given [12] [towever, our conclusions confirm those obtained by Brander [20] In the case ot neutral models In our opinion, the apparent uniqueness of some models IS due to the use ol a relatively small number of independent parameters Thus, we conclude that true predictions concerning the p-region cannot be obtained from the general requirements ot analytlcity and unItarity combined with the nowadays available constraints in (0, 1) The existence o1 more powerful, still unknown, rigorous constraints in (0, 1), which could change this situation IS quite dubious Therefore, complementary theoretical ingredients are needed Besides the well-known expressions for the left-hand cut discontinuities, other tools have been developed recently [21] Their possible use has been described in ret [22] In the present work, we restricted ourself purposely to the rigorous constraints below threshold in order to clarity the question of their relevance Our models are defined in sect 2 The Implications ol the threshold behavlour at s = 0 and of crossing symmetry are presented In sect 3 Sect 4 IS devoted to the quasihnearlty and posltlvlty constraints Our results are described In the tollowing sections In sect 5 we show that the imposition oI the linear parts ot a quasihnear model does not restrict significantly the physical content ot the model The ettects of imposing the p-meson are discussed in sect 6 and the shape ot the I = 2 S-wave IS fixed in sect 7 Whereas the I = 0 S-wave IS still largely undetermined in the region of the p-meson, it is nearly unique In the very low-energy region The effect ot modifying the asymptotic form of the model is studied In sect 8 Further tests, comparisons with other models and final comments are given In sect 9

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298

2 D E F I N I T I O N O F THE MODELS Our S- and P-waves have the form -

fl(S) = al

&IGI(S) 1 - &IDI(S) SI +

(2 1)

The q u a s l h n e a n t y o f f / ( s ) corresponds to the fact that the linear term al(S - Sl) is d o m i n a n t in the interval (0, 1) As we k n o w that fl(s ) vamshes at s = 1, we take s I = 1 and we write G l ( s ) = (1 - s)al(s )

(2 2)

To get a similar n o t a t i o n for all lsospin states, we set

Gi(s ) = $ l ( S ) n l ( s ) ,

(2 3)

where

~I (s)

/ 1

for I = 0 2 ,

1(1 -- s)

for I = 1

(2 4)

To characterize q u a s l h n m a n t y more precisely, we assume that the dominance o f

al(S - Sl) is due to the smallness ofalGl(S) and alDl(s) in (0, 1) Specifically, we shall reqmre

I~iGl(S)l < ~ 0(~) ,

lalOl(S)l <~ O(~6o) ,

0 ~< s ~< 1

(2 5)

Elastic u m t a n t y

1 _

Im fl(S

_

V~-

1

(2 6)

,

lS o b t a i n e d lfH1(s ) is regular at s = 1 and if

DI(S ) = - ~

2 [s sI + alGl(S)] F(s) + 2 Ql(s) ,

(2 7)

where F(s) is the Chew-Mandelstam function F ( s ) = - - ~ ( s - 1) f

ds'A~Cs, " 1

-1

1,,s,_s,

= V 1-

/

arctg

s

,

(28)

and Ql(S) is regular at s = 1 In order to get a p a r a m e t r l z a t l o n of the functions Hl(S ) and Ql(S) which is hkely to be valid on the whole unphyslcal interval (0, 1) as well as m the low-energy region, we use an Ansatz which has a correct analytmal structure at both branch points s = 0

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IL l d

c

b

I I

a

c

'

a 1=

d

=~ "

b

I s-plane

v-plane

Fig 1 The change of variable s = cos2(~ fro) maps the physical sheet of the cut s-plane onto the strip A (0 < Re v < 1) of the o-plane The upper hp a of the right-hand cut has its linage on the negatwe imaginary o-axis The images of the hps b, c and d are as shown m the figure The strip B IS the image of the second sheet of the elastic threshold and the strip C corresponds to the second sheet of the left-hand branch point s = 0 and s = 1 One k n o w n s that these branch points are of the square-root type Therefore, we p a r a m e t n z e HI and QI m terms of a variable o which umformazes the Rlem a n n surface generated by s = 0 and s = 1 (ref [23]) (fig 1) We take s = cos 2 (-~7to) The regularity functions of o

of Hj(s) and Qi(s)

(2 9)

at s = 1 lmphes that these functions are even

HI(--O) =Hi(o ) , QI(-o) = QI(O) (2 10) In our models, Hi(v ) and Ql(O) are polynomials of stxth degree In 02 These polynomials are such that the expansion ofD1(s ) m powers ofx/'s has the form Dl(S) = di, 0 + di,2s2 + di,3s3 + O(s -~) (2 l l ) Tlus means that the n u m e r a t o r f u n c t i o n Hi(v ) is responsible for the left-hand cut o f ~ ( s ) up to terms of order sk F u r t h e r m o r e , the expansion ofDl(s ) has n o hnear term, it can be seen that Gl(S ) and Dl(S ) in (2 I) can always be adjusted m order to achieve ttus It is c o n v e m e n t to construct Hl(v ) and Ql(O) as linear c o m b i n a t i o n s o f basle polynomials Rn(v ) m 02 which have a specific behavlour at s = 0 We take

RO(O)=

1,

R2(O ) = w 2 + 2 w 3 - 6 w 5 /r3

= s + r 2 , 4s 2 + r 2 , 6s 3 + O ( s -~)

R 3(0) = w 3 + -~~'w 4 - 3 _ w5 = s~ + r3,4 s2 + r3,6 s3 + O (s -~) 7r2 R4(o ) = w 4 + 4 w 5 = s 2 + r4,6s3 + O(s ~')

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RS(U) = - - ~ w 4 = s-~ + r5,4s2 + r5,6s3 + O(s~), 7

R6(v) = w 6 = s 3 + O ( ~ ) ,

(2 12) where w = 17r(1 - v 2) As the expansion offl(s ) has no s~ term, no s~- terms are needed in Hl(S ) and Dl(S ) and anRl(V ) is unnecessary [23, 11,9] We do not restrict the generahty of the Ansatz (2 1) by reqmring not only the expansion (2 11) Of Dl(S), but also the absence o f R 4 a n d R 6 terms lnHl(V ) Therefore we write

HI(V) = hi, 0 + hl,2R2(v) + hi,3R3(o) + hl,5R5(v)

(2 13)

We take h 1,0 = 0, this choice definles c~1 as the slope offl(S ) at s = 0 Using (2 13), (2 7) and (2 3), the condition (2 1 1) lmphes the following form of

QI(V) Qi(o) =

sI + (1 + ~si)R2(o ) + o~I[hl, 0 + (hi, 2 - ~hz, o)R2(o )

+ hi,3R3(v ) + (ttl,5 _ 1 hi,3 _ 61, lhl,3)RS(V)]

(2 14)

+ 3 rr [qI, O + qI,4R4 (°) + qI,6R6 (°)]

3 THRESHOLD BEHAVIOUR AT s = 0 AND CROSSING SYMMETRY The parameters al and sI (s 1 = 1) defining the linear part al(S - si) offl(s ) will be considered as fixed input parameters Then our models contain the twenty free parameters {hi, o , h l , 2 , h i 3 , h l , 5 l I = 0,2 }, { h l , 2 , h l , 3 , h l , 5 }, {qI, O, qI,4, qI,6 [I = 0, i, 2} Among the rigorous constraints our models have to satisfy, there are exactly eleven equalities, the remaining constraints are given by inequalities Six equalities determine the behavlour of J'l(S) at s = 0 and five equalities result from crossing symmetry The expansions of Imfl(S) at s = 1 and s = 0 are related by crossing symmetry [23, 11, 9] The expansions of Imfi(s)(I-- 0, 1, 2) around s = 0 have no s~ terms, the coefficients of the s~ and s~ terms are determined by the coefficients of (s-1)-~ and (s 1)9 appearing in the expansions of I m f i ( s ) (I = 0,2) at s = 1 In the evaluation of the coefficients appearing in these relations our hypothesis of quasIhnearlty will be used explicitly If (2 5) IS satisfied, we may approximate fl(s) by

fi(s)=ai(s - si)+a2[Gl(s)+(s

- Sl)Dl(S)],

for 0~
(3 1)

These functions depend linearly on the parameters of the model We take advantage of this circumstance by adopting the following scheme In our forthcoming calculations In a first step, the constraints are applied to fi(s) In this step, the constralnts are approximated by sets of equalities or inequalities which are linear in the

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301

parameters They define an approximate domain of allowed values In a second step, we check how well the resulting models satisfy the exact constraints and reject those models which are too unsatisfactory As we shall always use small values of o~I ~t can also be v~ewed as a second-order approximation m o~I In th~s approximation, we obtain a correct behavxour at the threshold s = 0 ff h0, 3 = a [ 4 A 0 + 5 A 2 ] ~ , h0, 5 = h0, 3 - ~o [490 + 5A 2]

hl, 3 = 2 [ - 8 A 0 + 5 A 2 ] , ,

h 1,5 --

~h

1,3

h2, 3=-~[8A0+A2] ,(3 2)

~3 [ - 890 + 592] '

(3 3)

h2, 5 = h2, 3 - ~ [ 8 9 0 + 9 2 ] , w~th AI=(1 -Sl)2,

9i=(1 -sI)(3+si)

Therefore, the input parameters s! the set ofstx parameters { h L n l I = 0, The plon-plon S- and P-waves have the coefficients fLn of the expansions

(I = 0, 2) determine a first approximation of 1, 2,n = 3, 5} to satisfy five crossing constraints They relate o f f i ( s ) m Jacob1 polynomials [2]

f i ( s ) = 2(1 - s) l ~ (l + n + 1)fi, nP(Zl+l'O)(Zs - 1) n=0

(3 4)

Before we write the crossing constraants, we complete the defimaon of our models by gwmg an intrinsic definmon of the hnear term al(S - si) I f I = O, 2 we reqmre that these terms coincide with the sum of the first two terms m the expansions (3 4) I f / = 1, the first term o f ( 3 4) has to be equal to al(S - 1) These reqmrements are expressed by the following orthogonahty conditions 1

f

ds(1 - _ ~]l+1 , Pn(2l+1 t ' 0)(2 _ s - 1) [fl(s) - oLI(S - Sl)] = 0 ,

(3 5)

0 for n=0,1, n=0

•=0,2, ,

I=1

/=0, ,

l=1

Now the five crossing conditions involving S- and P-waves trnply three relations between the a 1 and s 1 a0 = 6a 1 ,

4a2 = - 3 a 1

s 2 = 4 ( 3 - 4s0)

(3 6)

These relations reduce the number of independent input parameters to two for mstance a t and s o The remalnmg two crossmg condmons give

0 Ptguet, G Wanders,Quasdmearmodels

302

2f0,2 -- 53'2,2 - 6fl,l = 0 ,

(3 7)

23'0,3 - 5f2, 3 + 15fl, 2 = 0

Once (3 2) and (3 3) have been taken Into account, we have tourteen free parameters {hi, o,hi,211 =0,2 },hl,2, {ql, o,ql,4,ql,6ll=O, 1,2} The conditions (3 6) and (3 7) produce a further reduction of this number to seven We choose the following seven independent free parameters

hl,2,ql,4,ql, 6, {ql,4,ql,61I =0,2}

4 QUASILINEARITY AND POSIT1VITY CONSTRAINTS It has already been stated that the parameters ~1 and s o are our input parameters In tact, we shall consider a unique value of r~1 c~1 = 0 0 3 5

(41)

This choice ensures a realistic value of the P-wave scattering length a 1 (ref [24]) (a I ~- c~1 In a quaslhnear model) Eqs (4 1) and (3 6) Imply c~0 = 0 2 1 0 ,

o~2 = --0 105

(4 2)

The parameters hl, 3 and hi, 5 ot Hi(v ) depend, In the approximation (3 2) and (3 3), on s o only It turns out that quasihnearlty cannot hold if these parameters are too large Therefore the condition (2 5) restricts the possible values of s o One gets - 1 ~
(4 3)

Once al and s o have been chosen, the allowed domain of the remaining seven andependent parameters h 1,2, q 1,4, q 1,6 and {ql,4, qI,6 I1 = 0, 2 } as determined by the quaslhnearlty requirement and the positivlty constraints One finds empirically that quasillneanty is ensured on the whole interval (0, 1) xt conditions of the type (2 5) are imposed at s = 0 and s = 1 tODl(S) alone It has already been recognized by other authors [12] that, among the restrictions resulting from the many available posItiVlty constraints, the most important ones are due to a relatively small number of these constraints Experience shows that xt is convenient to select the following four 7r°rr° constraints P1

J00(0) ~>f00(0 797),

(ret [31),

P2

100(0 801) >f00(O 0534),

(ref [ 5 0 , (4 4)

t

P3

lo0(0 424) > 0,

P4

f00(0) <

[2f00(1)

(rets [3, 4 ] ) , f00(0 5)

f00(O)],

(ret [7]),

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303

(foo(S) = ~ fo(s) + ~ f2(s)) and one constraint revolving other isospm combinations P5

(fo-f2-1

7 1 5 f l ) ( S = O 8 4 7 2 ) > ( ~ J O + 32f 2 __ 5 554fl)(S = O1 201) ,

(ref [6])

(4 5)

l f w e apply the conditions (2 5), (4 4) and (4 5) to the functionsfl(s ) (defined in (3 1)), we obtain a system of inequalities which are hnear in the seven free parameters They define a convex polyhedron ,5(c~1, So) In the seven-dimensional space of our seven parameters

5 FIRST RESULTS The summits ot the polyhedron ,5(0 035, 0 5) have been calculated explicitly Fig 2 shows the projections of the summits of this polyhedron onto the planes (q0,4, q0,6) and (q2,4, q2,6) The projection of the polyhedron itself xs the polygonal convex envelope of this set of points (extending possibly to infinity) We see that the quasdmearity and posltlvity constraints Impose quite strong restrictions to the possible values ot our parameters However, these restrwttons are

not strong enough to impose stgnlfteant hmltattons on the scattering amphtudes m the physical regton This can be recognized directly from the diagrams of fig 2 In the region of the p-meson (s ~ 7 5), the polynomials R4(v ) and R6(v ) dominate RZ(V), R3(v ) and R5(v ) (fig 3), and ctgS/(s) has the following approximate form

2

ctg 61(s ) ~ ~ log (V/s + X / ~ )

+-'I' ] s ~I - ql'4RI'4(u) - ql'6Rl'6(v)

(5 1)

Therefore 6I(S ) IS essentially determined by qL4 and qL6 in the p-region The hnes ctg 6i(s ) = const are nearly straight lines in the plane (qL4, qL6) Then, the inspection of fig 2a shows that 8o(S ) may take practically any value between 0 ° and 180 ° at s = 8 According to fig 2b, 62(8) can have any value in the interval (0 °, - 180 °) A (ql,4, ql,6) diagram would lead to similar conclusions for 81(8) We have seen that large and negative values of 82(8) are allowed This corresponds to a rapid fall-off between s = 0 and s = 8 which is hkely to violate a causahty condltion of the type of Wigner's inequality [25] In fact, the amphtudesf2(s ) which have a large and negative phase shift at s = 8 have complex poles in the physical sheet due to zeros m the denominator of (2 1) These amplitudes have to be excluded in any case because they are not holomorphlc in the physical sheet Thus we eliminate the models which have 82(8 ) < 90 ° Fig 2a shows the effect of this ehmlnation on the I = 0 wave, ~t does not induce a new significant restriction of the possible values of 60(8 ) The different asospln states are not strongly correlated by the crossing and posltiv~ty constraints in the region of the p-meson

304

0 Pzguet, G Wanders, Quaslhnear models

Cl2,4

Fig 2 The projections ol the summits oI the polyhedron 4(0 035, 0 5) onto the plane (qo,4q0,6) (fig 2a) and (q~,4q~,6) (fig 2b) These plots are direct outputs of a low-accuracy computer calculation, they give a schematic p~cture of za, A has summits whose projections would be below the bottom of the tlgure Some points represent several superposed summits The hnes on which ~o(8) ~ 1°, ~ 90 °, ~ 179 ° are drawn In llg 2a Sxmflarly, fig 2b shows the hnes 62(8) ~ - 1 °, 20 °, ~. - 9 0 °, ~ - 1 7 9 ° The asterisks are projections of summits with 62(8) < - 9 0 ° and which are rejected The black dots give all the accepted summits with 65(8) > - 9 0 ° The approxImate shapes of the projections of A ,tself are also indicated

Fig 4 gwes a few e x a m p l e s o f phase shifts whach m a y be o b t a i n e d m the case s O = 0,3 T h e y c o r r e s p o n d to m o d e l s in w h i c h the p a r a m e t e r s qL6 (I = 0, 1, 2) have b e e n set e q u a l to zero By c o n s t r u c t i o n , these m o d e l s are s u c h t h a t the h n e a n z e d a p p r o x i m a t i o n f / ( s ) fulfill the crossing and posatlvlty c o n s t r a a n t s e x a c t l y T h e r e f o r e , the true amphtudesfl(s ) satisfy these c o n & t l o n s o n l y to some a p p r o x i m a t i o n T h e q u a l i t y ot tins a p p r o x l m a t x o n has b e e n t e s t e d for all m o d e l s o f fig 4 as well as for all m o d e l s w i n c h will be displayed expllcxtly T h e s e c h e c k s c o n c e r n (x) the s = 0 t h r e s h o l d c o n & t l o n s [23, 1 1, 9] (11) the o r t h o g o n a h t y a n d crossing condxtlons (3 5) a n d (3 7) ( m ) a set o f 71 k n o w n posatwtty c o n s t r a i n t s [ 5 - 7 , 9], w h i c h i n c l u d e s the five c o n s t r a i n t s (4 4) a n d (4 5)

0 Plguet, G Wanders, Quaszhnear models

305

R3

s 5o.

0

25

-5

5

10

-10"

Fig 3 The polynomials Rn(v), n = 2, 3,

,6 defined in (2 12), as functions of s

One finds that the condmons (1) and (11) are vxolated by an amount whxch is systematically consistent with the order of magmtude m cz1 of the difference between f l ( s ) and f/(s) The posmvlty constraints (111) are satisfactorily verified m general, a large number of mequalmes ~s frequently saturated In some exceptional cases, a set of amphtudes has to be ehmlnated because one of the posltwlty constraints is badly violated The c~rcumstance that the violations of our constraints are correctly predicted by considering powers of a 1 Is an mdlcaUon that we would reduce these violations by including consxstently terms of higher order m a I m our approxlmatlon scheme Therefore we beheve that every model which passes our tests with the expected accuracy Is m the nelghbourhood of a model which verifies all constraints exactly Fig 4 shows that we may have small solutions, hke (c) whose phase shifts remain relatively small To our knowledge, such small solutions have not been mentioned until now We may have large soluUons as well, hke (a), with resonating I = 0 and I = 1 waves It Is worthwhile to notice that the transmon between small and large solutions is continuous. We conclude from these first results that the mere ~mposmon o f a 1 and s o as mput parameters does not restrict sxgmficantly the physical content of our models Further input constraints have to be added, this will be done in sect 6

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306

180°~ (~] f !I !

I t/ !

I

"l So=O 3

QI= 0 035

I

I

I

bl

I I ! I I

•

90

I

a

l;

/ c

/ Ill ~'44t ..

~-'C--.

~=l="

,

C

,

S

,

v

m~ I

400

I

I

600

I

i

800

MeV

Fig 4 T h e p h a s e shifts ol t h r e e m o d e l s c o r r e s p o n d i n g t o s o = 0 3 a n d a~ = 0 0 3 5 T h e s o h d h n e s m t h e u p p e r h a l f p l a n e r e p r e s e n t t h e 1 = 0 p h a s e shifts a n d t h e s o h d h n e s in t h e l o w e r h a l f p l a n e give t h e I = 2 p h a s e shifts T h e d a s h e d h n e s are t h e I = I p h a s e shifts T h e c u r v e s c o r r e s p o n d i n g t o a s a m e m o d e l a r e i d e n t i f i e d b y t h e s a m e l e t t e r a, b o r c

6 IMPOSING THE p-MESON As a first physxcal input, we impose an I = 1 resonance by reqmrmg

ctg61(s°)=O"

d

ctg 8 l(s°) =

1 ~X/~p

(6 1)

We take m o -- 765 MeV (s o = 7,5) and Fp = 120 MeV (= 0 43 in our umts) Condv tlons (6 1) are hnear equataons m the parameters ot our models These condxtxons alone do n o t ensure an acceptable p-meson This is so because

0 Ptguet, G Wanders, Quasdmear models

307

our parametrlzatlon allows a zero o f f l ( s ) m the physical region whereas the experimental results exclude such a zero, at least b e t w e e n mrrTr = 500 MeV and mTr" = 1100 MeV Therefore, in order to get I = 1 waves which are qualitatively correct, we choose our parameters in such a way that f l ( s ) lS non-vanishing in (1,10) A closer analysis shows that the position o f the physical-region zero o f f l ( s ) is directly c o n n e c t e d to the value o f s 0 If the zero has to be above s ~ 10, we m u s t have 0 3 ~ sO ~ 2

(6 2)

We notice that current algebra gwes s o ~ 0 125, which is outside this interval This does n o t m e a n that the current algebra predictions, quaslllnearity and the Pmeson are incompatible Simply, our parametrlzatlon of the I = 1 wave IS such that we c a n n o t reproduce b o t h the current algebra results and a decent o-meson As we are primarily interested in a rather exhaustive discussion of a well defined class o f models, the fact that this class doesn't include the current algebra models should be considered as a m i n o r defect Once (6 1) has been taken into account, the n u m b e r of i n d e p e n d e n t parameters lS reduced to 5 hl,2, {qL4, qI,6 II = 0,2} The quaslhnearlty and posltIVlty constraints define an admissible p o l y h e d r o n Ap(cq, SO) in the fwe-dlmensmnal space of these parameters The summits of AO(0 035, SO) have been c o m p u t e d for s o = 0 5 and s o = 0 8 It appears that Ap(0 035, 0 5) has projections o n t o the (qI,4, qI,6) 18¢ 8t

490

,

890

,

890 M~V

mlxn

So=05 Q'1=0035a

180° 51 400

~

600

8?o M~V

mww

So=08 Q1=0035

~

/

//C i

90

iI

90°

d

I

Ii i

// / //

z¢ / / '

e /

/

/

// // a

O~

S

~

b

b Fig 5 Models which have an Imposed p-meson and a slowly varying 6:(s) eq = 0 035, so = 0 5 (fig 5a) and so = 0 8 (fig 5b) The conventions used are the same as m fig 4

308

0 Ptguet, G Wanders, Quasthnear models

planes, (I = 0,2), winch are much smaller than the projections of &(O 035, 0 5) Some characteristic phase shifts of Ao(O 035, 0 5) and &o(O 035, 0 8), corresponding to models which passed the tests of sect 5, are shown in fig 5 We see that once a slowly varying 62(s ) is required, tins phase shift is constrained to be negative and relatively small Slmllarfly, 60(s) is positive and relatively small, of the order of 10 ° i f s 0 = 0 8 The situation is strikingly different for s o = 0 5, where the spectrum ot the possible I = 0 phase shifts becomes very broad We may have large resonatmg I = 0 phase shifts, as well as relatively small phase sinfts Again, the transition between these possibilities is continuous Furthermore, one sees In fig 5 that conditions (6 1) define a nearly unique (5l(s) The main conclusion of this discussion is that our quaslllnear models cannot reproduce the large observed I = 0 phase shifts if So lS too large On the other hand, if s o is small enough we may have large as well as small phase shifts It s o = 0 5, the imposition of the p-meson does not reduce drastically the interval ot allowed values of 60(8 ) Whereas we had 0 ~ 60(8 ) <~ 180 ° without the p-meson, we now have 10 ° ~ 60 (8) ~ 170 ° Therefore, it appears that the crossing and positivIty constraints do not have very restrictive implications in the p-region This point will be investigated further in sect 7

7 A F U R T H E R INPUT, THE I = 2 S-WAVE We have seen that, once the p-meson is imposed and 62(s ) lS slowly varying, little freedom Is left m the I = 2 state Therefore, we may practically fix 62(s ) without sensible loss of generality We do this by reqmring 62(3 ) = - 9 59 ° ,

62(4)= - 1 1 98 °

(7 1)

Taking these restrictions into account, we are left with models forming a threeparameter family The complete construction of the allowed domain of the free parameters becomes feasible In order to appreciate the relevance ot the posItlVIty constraints, it is instructive to distinguish the restrictions due to the quasllineanty requirement alone from the combined effects of posltlVity and quaslhnearIty To tins end we have constructed the polyhedron Ao, 2 (0 035, 05) defined by the quaslllnearlty conditions (2 5) alone This polyhedron characterizes the set of quaslhnear models which are crossing symmetric, have a correct behavlour at s = 0 (In the hnearlzmg approximation fi(s) ~Ji(s)), produce a p-meson and have a 62(s ) fulfilling (7 1), Ap, 2 is convemently represented in the space of the_ independent parameters h2,2, q0,4, q0,6 Fig 6 gives a qualitative picture of Ap, 2, its projection onto the (q0,4, q0,6) plane is drawn accurately In fig 7 If we require the positivity conditions (4 4) and (4 5) m a d d m o n to the quasilinearity reqmrements, we get the polyhedron Ap, 2 Figs 6 and 7 show that dip, 2 ts substancIally smaller than Ap, 2, the positivIty constraints are quite efficient In

0 P1guet, G Wanders, Quasthnear models

2,5,7 I

....

~. ~1

4 5,TX'k---.~

~[ h2,2 /

935 [

,

,,, , 5 S 10 ~ . . _

'

1j235 \ ' ~. -+" y ' 7 7 ~

5 8 ' ~ ~ ¢ m ' - ' - ' - LJ4 510 ~

i '

309

I I

' I

/ /

~"%~,

/

"-.__41o11~

14,7 . .1.4. .8. /

I

I

I ~

1411 \

q04

'

I

o I

"q 1,3 4 \

Fig 6 A quahtatlve picture ot ~0,2 and Ap,2 (polyhedron with shaded faces) and ot their proJections onto the plane (q0,4, qo,6) At each summit of these polyhedrons, three mequahtles become equalmes The summits are identified by three numbers which indicate which mequahtles become equahtles The numbers 1 to 7 correspond to seven quaslhneanty conditions of the type (2 5) and the numbers 8, 9, 10, 11 correspond to the posltWlty constraints P5 (4 5), P1, P3, P4 (4 4)

reducing the allowed domain of the parameters We see that only three posltlvlty constraints (4 5), P3, P4 and P5, contribute effectively to determine the shape of Ap,2 The phase shifts of seven characteristic models belonging to Ap, 2 are gwen m fig 8 We see that even with the I = 1 and I = 2 phase shifts practically fixed, the freedom m the choice of the I = 0 phase shift remains very large Fig 9 shows the effect of adding the posltlvlty constraints to the quaslllnearlty conditions The extreme I = 0 phase_shifts provided by Ap, 2 have been drawn together with the extreme cases allowed by Ao, 2 We see that above 600 MeV, the specific restrictions due to the posltlvity conditions become very weak We may already conclude that, in the frame o f our quasthnear models, the postttvlty constramts have no relevant tmphcatlons m the regton o f the p-meson This conclusion will be confirmed in sect 8 However, fig 9 suggests that the situation might be different at lower energies To see more clearly how our models behave just above threshold, k ctg 80(s ) of various models is drawn m figs 10 and 11 Fig 11 proves that the restrlcUons due to positiwty are sigmficatave below 450 MeV Whereas crossing symmetry and quasflmearlty alone restrict the scattering length a 0 to 0 092 m~ 1 < a 0 < 0 130 m~1, the inclusion of posltlwty constraints imposes 0 103 m~-1 < a 0 < 0 108 m~-1 This is the range o f / = 0 scattering lengths compatible with a l = 0 035 and s o = 0 5, once the

0 Ptguet, G Wanders, Quasthnear models

310

2,5 7

457\

q 0,,6

\\

+6

\

\ \

%

147

-4

-3

-2

-1

'~4,1~

1

2 '

-1

So=05

Q I = O 035

-2

-3

¸

- 4

Fig 7 The projectzons of ~p,2 and '~p,2 (shaded area) onto the plane (qo,4, qo,6) The summits of these projections are identified according to the same notation as m fig 7, 60(8) > 90 ° approximately above the hne a and (d/ds)6o(8) > 0 approximately above the hne b

p-meson and the I = 2 phase shafts have been imposed Furthermore, fig 11 gwes some insight into the nature of the restrictions lmphed by posltwaty Rather than excluding some values of the phase shift 60(s ) at gwen energles, they exclude some shapes of the curve representing 60(s ) For instance, the

0 Ptguet, G Wanders, Quasthnear models

180°I61 . 4oo,

,

690

~?o

,

311

MeV

rn~rr

~ (11=0035/ ~ //~B 5810 458

4510

So=O 5

9o °

----2

-30°.

;

'

'

;

S

--I--0 ..... I=]

---I=2 Fig 8 Models with the o-meson, 6~(3) and 62(4) imposed, 62(s) and til(s) are nearly umquely determined The 1 = 0 phase shifts corresponding to fwe summzts of ,xp,2 (same notations as m figs 6 and 7) and to two interior points of/'0,2 (A, B) are given

m o d e l (1 2 3) gwes a (50(4) which is in the band allowed by posltlVlty Nevertheless, this m o d e l is forbidden because its (50(2) is too small

8 A MODIFIED PARAMETRIZATION The parameters o f our models are adjusted m such a way that a set of constramts involving the region below threshold are verified A priori, we do n o t k n o w the size of the d o m a i n above threshold where these constramts restrict s]gmficantly the form o f our a m p h t u d e s The results of sect 7 indicate that this d o m a i n does n o t e x t e n d up to the o-regzon

0 Ptguet G Wanders, Quasthnear models

312

18

400

I

600

,

,

800

MeV

~ °.

rn.n

35

So=O 5 0.1=0035

z

o

90 / f

l/

~

~

5

8

S

]=0 ......

I=1

- - -

I=2

Dg 9 Models with the o-meson, 62(3) and 62(4) Imposed Comparison ot the effects of posltlVlty and quasflmanty with the effects ot quaslhneanty alone The extreme I = 0 phase shnfts provided by the summits of Ap, ~ (quaslhneanty + posmvxty) (hmlts of hatched area) are shown together with the extreme phase shifts produced by the summits of ~o,2 (quasfllneanty alone) Same notations as m figs 6 and 7

We submit this conclusion to a further test To thls end, we m o d i f y the parametnzatlon o f one o f our models in such a way that it remains practacally unchanged m the interval (0,1) Thus, we get a new m o d e l which verifies the constraints as well as the lmtlal m o d e l Nevertheless, b o t h models have different extrapolations into the physical region The size of the interval in which these extrapolations are nearly coincident measures the efficiency o f our constraints Specifically, our change of p a r a m e t n z a t l o n consists xn the replacement o f the last two terms o f Q 0 ( v ) in f0(s ) (cf (2 1) (2 7) and (2 14))

qo,4R4(o) + q 0 , 6 R 6 ( o ) ,

(8 1)

0 P1guet, G Wanders, Quasdmear models

k ctg 300

313

50 400

500

MeV

mww

So=0 5

Gl=0 035

14 B

5

181o

q"~ 0

I

"

3

is

Fig ] 0 The / = 0 near threshold content o i the models wxth the p-meson, 62(3) and 62(4) imposed Plot o f k ctg 6o(s) for stx summits o f Ap, 2 (same notations as m rigs 6 and 7)

by the rational function as 2 1 + bs 1 + cs 2

(8 2)

The parameters a, b and c are fixed by reqmrxng the new model,f0(s ), to fit the pnmltwe one, fo(S), m (0,1) We notice that an effect of this substitution IS to modify the asymptotac form of fo(s) We have cgt 60(s)= 2 logs+ O ( ( 1 7 s)12),

(8 3)

314

0 Ptguet, G Wanders, Quasthnear models

k ctg 3O0

5 o

500 i

400

So=O 5

MeV

inn

Q1=0035

s

Fig 1 1 The I = 0 near threshold content of the models with the p-meson, 62(3) and 65(4) tmposed Comparison of the effects of posltlvlty and quasdmeanty with the effects ot quaslhnearlty alone The extreme curves of fig 10 (posltwlty + quasflmeanty) (hmlts ot hatched area) have been drawn together with k ctg 60(s) of all summits of Kp,2 (quasxhneanty alone)

ctg60(s ) =

logs---+O C

(8 4)

According to (8 3) 60(s ) tends to 0 or n as s ~ oo As the logarithmic term of (8 4) starts to d o m i n a t e at very high s, (s ~ 103), g0(s) staballzes first near the value - arctg (c/ab) over a large interval Fig 12 shows the effects of the modified parametrlzatlon on the central model A of Ap, 2 (fig 8) We see that ~0(s) differs substantially from 60(s ) above 450 MeV

0 Plguet, G Wanders, Quasthnear models

315

61

,8I°

4?0

9o]

,

6~o

,

8?o

M~v

m~w

So=05

O'

Gl=0035

}

S__...

~ BATONet al

~ BAiLLONet al Fig 12 Effects of a modified parametrlzatlon and comparison with experimental data The modlflcanon of the parametnzatIon, described In sect 8, transform the I = 0 phase shift A of fig 8 (dashed curve of this figure) Into the curve represented by a sohd line The experimental points are taken from ref [30], Baton et al [31], Balllon et al [321

T h e r e f o r e , we c o n c l u d e d e f l n m v e l y t h a t our crossing and poslttvtty constraints, combmed with the tmposttlon o f the o-meson and the I = 2 S-wave do not restrwt the I = 0 S-wave m the p-region In the f r a m e o f o u r q u a s l l l n e a r m o d e l s , 60(s ) ts strongly eonstramed up to 450 MeV only

9 F U R T H E R T E S T S , C O M P A R I S O N WITH O T H E R M O D E L S A N D F I N A L COMMENTS As was m e n t i o n e d in sect 1, t h e r e are c o n s t r a i n t s revolving t h e p I o n - p l o n amplit u d e s in the physical region w h i c h m a y lead t o a selection a m o n g the m o d e l s satisfying the u n p h y s l c a l region c o n d i t i o n s We did n o t e x a m i n e this q u e s t i o n in detail In o r d e r t o see w h a t m a y h a p p e n , we j u s t e v a l u a t e d the following e x a c t s u m rule [26]

0 Plguet, G Wanders, Quasdmear models

316

Table 1 The evaluatmn of the sum rule/or the eight summits of Ap,2 (cf hgs 6 and 7)

Summit

Lelt-hand side

Right-hand side

Discrepancy (%)

14 8 1 4 11 4 10 11 1 10 11 1 8 10 4 5 10 5 8 10 458

-0 -0 -0 -0 -0 -0 -0 -0

0 11247 0 10972 0 10691 0 10874 - 0 I0570 0 08819 0 08823 0 09491

15 2 3 2 10 20 30 30

2a 0

5a 2 -

12828 10820 10364 10702 11632 10697 11930 12961

18al_

1 ~ ds - s2(s - 1) 327r2

3s- 1 X [2A0(s, 0) - 3 ~ A l ( s ,

0) - 5A2(s, 0)] ,

(9 1)

for the summits of the polyhedron Ao,2(0 5) the right-hand side of (9 1) being approximated by the S- and P-wave contributions Table 1 gives the results of this test Clearly, small I -- 0 phase shifts, like (1 4 11) are favoured against the large resonating ones, of the type (5 8 10) (fig 8) As the linear part o f f l ( S ) does not enter into (9 1) and as the uncertainty on ( l l ( S ) - oQ(s - sI) ) is of the order ofoq3, a 10% discrepancy between the left- and right-hand side of (9 1) cannot be excluded Therefore, definite conclusions cannot be drawn from table 1 Nevertheless, we have an indication that it might be difficult to find a quaslhnear model with s = 0 5 reproducing the large observed I = 0 phase shift and fulfilling the sum rule (9 1) Several of our quaslhnear models exhlbat a qualitative resemblance with existing models, especially at low energies If we select our I = 0 S-wave phase shifts which are near 90 ° at the o-mass, we notice that these phase shifts are Increasing functions In the p-region, lndlcatlng a rather broad resonance (F ~ 250 MeV) Similar resonantlng 60(s ) are produced by the models of Bonmer and Gauron [11] On the other hand, various proposed models (for instance [ 1 2 - 1 4 ] ) have a large 60(s ) which is rather flat, staying near or below 90 ° over a large interval in the p-region Fig 12 indicates that we may get such a behavlour by modifying our original parametrlzatlon We conclude from these remarks that the up down ambiguity cannot be resolved with the help of the unphyslcal region constraints alone The existing models correspond m general to values of the input parameters a 1 and sO which differ from ours In particular, sO IS often below 0 3, which is the minimal value compatible with our parametrizatlon (cf (6 2)) Therefore, our models

0 Ptguet, G Wanders, Quasthnear models

317

cannot be compared quantitatively with most of the existing models One exception is a model due to Krlnsky [13], which has s O = 0 44 and is based on an approxnnatlon scheme similar to ours In fact we were able to determine a quasihnear model which coincides with Krlnsky's model for 1 < s ~ 2 5 The results of the preceding sections tell us that a coincidence in a larger interval should not be expected The scattering lengths of the models selected in sect 7 are

Ol03m~l
a2=-0

077 m ~ l

(92)

In the linear approximation fl(s) = ~l(S - sl) we would have a 0 = 0 1 0 5 m ~ 1,

a2 =

0085m~ 1

(9 3)

We see that the non-linear terms reduce the absolute value o f a 2 by 10% whereas their effect on a 0 is at most a 3% correction Due to our choice ofc~ 1 and So, our scattering lengths do not fulfill the requirement ao/a 2 ~ - 3 5 which results from current algebra [19] and is compatible with experiment [27] On the other hand, the scattering lengths (9 2) are nearly on the empirical universal curve obtained by Morgan and Shaw [28] We may notice that the scattering lengths (9 2) are closer to this curve than the scattering lengths (9 3) of the hnear approximation According to the results of our analysis, a comparison of our models with experimental data in the p-region IS not very significant It simply shows whether our parametrlzatlon allows a fit of these data It is quite clear that our quaslhnear models cannot produce a 60(x ) which reproduces the experimental results below the p and is of the up-type above the 0 On the other hand the model obtained in sect 8 from a modified parametrlzatlon happens to reproduce surprisingly well the down-type solution (fig 13) However, this model is Incompatible with the recent Ke4 results [29] According to these results 60 - 61 > 8 ° at 320 MeV, whereas our model gives 60 - 61 ~ 5 5 ° As we have seen that our models have to be trusted at such low energies, we conclude that the I = 0 scattering length has to be larger than our a 0 ~- 0 1 The question of the physical relevance of the crossing and posxtivlty constraints below threshold was at the origin of the present Investigation We tried to get an answer to this question which is valid in the frame ot the quasihnear models Our results indicate that this answer is positive in the sense that the constraints restrict significantly the behavlour of the I = 0 S-wave in the low-energy region (1 < s ~< 2 5 or 280 MeV < mTrTr<.~ 450 MeV) once the I = 1 P-wave and the I = 2 S-wave have been fixed The size of this low-energy domain is comparable to the size of the unphysical interval (0 < s < 1) No miracle occurs at higher energies, there the shape of the phase shifts depends mainly on the parametrizatlon and is no longer efficiently restricted by the constraints One definite result seems to be that the zero of fo(S) (~ So) has to be below 0 5 in order to allow a large 60(s ) (cf sect 6) and a relatively large a 0

318

0 Ptguet, G Wanders, Quasthnear models

REFERENCES

[ 1 ] A general exposition of these results is given in A Martin, Lecture notes m physics, vol 3 (Sprmger-Verlag 1969) [2] A P Balachandran and J Nuyts, Phys Rev 172 (1968) 1821, R Roskles, Nuovo Clmento 65A (1970) 467, J L Basdevant, G Cohen-Tannoudjl and A Morel, Nuovo Clmento 65A (1970) 743 [31 A Martm, NuovoCtmento47A(1967) 265, Acta Phys Austrlaca, Suppl VII (1970) 71 [4] A K Common, Nuovo Clmento 53A (1968) 946 [5] A Martin, Nuovo Clmento 63A (1969) 167 [6] G Auberson, G Mahoux, O Brander and A Martin, Nuovo Clmento 65A (1970) 743 [7] G Auberson, Nuovo Clmento 68A (1970) 281 [8] O Plguet and G Wanders, Phys Letters 30B (1969) 418 [9] G Wanders, Springer tracts In modern physics 57 (1971) 22 [10] G Auberson, O Plguet and G Wanders, Phys Letters 28B (1968) 41 [11] B B o n m e r a n d P Gauron, Nucl Phys B21 (1970) 465, B36(1972) 11 [121 J C Le Gudlou, A Morel and tt Navelet, Nuovo Clmento 5A (1971) 659 [13] S Krmsky, Phys Rev D2(1970)1168 [14] J S K a n g a n d B W Lee, Phys Rev D3(1971)2814 [15] C L Jen and K Kang, Nuovo Clmento 3A (1971) 425 [16] H Kuhnelt and F Wldder, Nuovo Clmento 7A (1972) 47 [17] J L Basdevant and B W Lee, Phys Rev D2 (1970) 1680 [181 D Atkmson, Nucl Phys B7 (1968)355 [19] S Wemberg, Phys Rev Letters 17 (1966)616, J lhopoulos, Nuovo Clmento 53A (1968) 552 [20] O Brander, CERN preprlnt TH-1366 (1971) [21] SM Roy, Phys Letters 36B (1971) 353 [221 J L Basdevant, J C Le Gudlou and tt Navelet, Nuovo Clmento 7A (1972) 363 [23] G Wanders and O Plguet, Nuovo Clmento 56A (1968) 417 [24] M G Olsson, Phys Rev 162 (1967) 1338 [25] E Wlgner, Phys Rev 98 (1955) 145, B Bonmer and R Vlnh Mau, Phys Rev 165 (1968)1923 [26] G Wanders, Holy Phys Acta 39 (1966) 228, S Krlnsky, Phys Rev D4 (1971)1046 [27] L J Gutay, F T Melere and J H Scharengmvel, Phys Rev Letters 23 (1969) 431 [281 D Morgan and G Shaw, Phys Rev D2 (1970) 520 [29] A Zylbersztejn et al, Geneva-Saclay Collaboration, preprmt (January, 1972) [30] C Schmld, Rapporteur Talk at the Amsterdam Conf on elementary particles, 1971, CERN preprmt TH 1403 [31] J P B a t o n e t a l , P h y s Letters 33B (1970) 528 [32] P Badlon et al, Contribution to the Amsterdam Conf on elementary particles, 1971