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Test of crossing symmetry on low-energy ππ S- and P-wave models using crossing relations on the physical region
Nuclcar Physics B52 ( 19731 506 -528 North-Itolland ['tthll~,hlng ( ompam
T E S T O F C R O S S I N G S Y M M E T R Y ON L O W - E N E R G Y rrTr S- A N D P-WAVE M O D E L S U S I N G C R O S S I N G R E L A T I O N S ON T H E P H Y S I C A L R E G I O N B. B O N N I I ' R and P GAURON In~tttut de Physique Vucl(atre. l)ll t~topt de Ph)~tque l h ( o r t q u c *
Recewed 13 Jul) 1972 (Revised 16 Ot_u~ber 19721
Ab',tra~-t We use the crossing relations on tht_"ph,,slcal re(non as a teq o1 the t.rosslng properties ol various nrr S- and P-x,,ave lllodcls In a ~,t l'len'le (.ontalflln~2 d~ d llrst Inpul the D- mcgon',, existence and elastic unlt,lrlt~, I hese reldllOrls ,ippear to be selettl',C but are unable to rcInovc %(lineambiguities like lhe up-dov, n one We also build a parametrl/atlon ol the partial x+a+cs espccl,llly suited for these ncx+ relations We find d 'Aide (+]as,, o| S-v+a',cs whlt.h lulfll the rcqulrenlents of elastlt, umtarlt+x, crossing symmetry and D-meson existence in the P-wave }lov+cver, their lox+-energ~, behd~'lotlr r C l l l a l n s i11 d g r e c l l l , 2 n t w i t h
[)rt2vltlt.lS prt.'dlt. 11o11'~
1. Introduction Various m o d e l s [1 3] using general principles [4] and the o - m e s o n ' s e x i s t e n c e have led to some p r e d l c t u ) n s a b o u t the rrTr l n t e r a c t m n , at least in the very low-energy rcglon [5]. The global c o n s i s t e n c y r c a c h e d m these d i f f e r e n t p a r a m e t r l z a t l o n s seems to indicate that the S-waves have zeroes and scattering lengths in rough a g r e e m e n t with c u r r e n t algebra p r e d i c t i o n s [6, 7]. The o or % m e s o n ' s e x i s t e n c e [8] seen) also to emerge f r o m this p a t t e r n , flowever, the posit(on and w i d t h o f the associated resonance are f o u n d to have a wide range o f possible values More generally, lugS-energy p r e d i c t i o n s b e c o m e increasingly m o d e l d e p e n d e n t , since the rigorous c o n s t r a i n t s c o m i n g from the general principles act on the non-physical strip only l t o w e v e r , f r o m this p o i n t o f w e w , some progress has been t h e o r e t i c a l l y m a d e with the w o r k s o f R o y [9] and Basdevant, l_c (,ulllou and Navelet [10], since thetr f o r m a h s m allows us to e x p r e s s the partial waves & r e c t l y on the physical regmn up to 1 G e V as f u n c t m n s o f the a b s o r p t w e parts * 1 aboratmre as,,O~.l¢'au ('NRS. Po,,tal address Laboratolre de Physique Th6orlquc et }lautcs 1 nergles Tour 16 - I atuhd de', Sciences - 9, Qual Saint-Bernard 75 - Paris 5e
B Bonnwr. P (,auron. (,'Josstng ~ mm~'rrv
507
In this paper, we first use these relations to test directly on the physical region the crossing properties of the 7rrr S and P partial waves given by some recent unitary models. Our purpose is to see If the preceding predictions about the low-energy domaul carl be confirmed. This seems to be n n p o r t a n t due to solne recent results of Piguet and Wanders [111, which conclude that tile rigorous constranats below threshold have a physical relevance restricted to a small interval above threshold. Our second aim Is to inveshgate by means of a snnple model fl" the crossing relations on the physical region can bring somethnlg new m the o-region A prtort tins Is not hopeless, since problems usually solved m a model-dependent way can be aw)lded, as for example the determination of the left-hand cut Moreover, tile highenergy behavlour of the absorptwe parts must be considered, and this allows one to inJect easily some Regge c o n t r i b u t i o n s to tile partial waves The plan of this work IS then the following, in sect. 2, we recall Roy's relations for elastic rcrr scattering, and we give an e s n m a t e of tile c o n t r i b u t i o n to the S- and P-waves of tile high-energy region, described by a Regge fom~ with p o m e r o n exchange. In sect. 3, we investigate the crossing properlles of the S- and P-waves given by some models, including our last work on this sublect [2]. In sect. 4, we extend this study to a p a r a m e t n z a t l o n of the S- and P-waves especially suited to these crossmng relanons. Our conclusion can be found m sect 5
2. Formalism, Crossing relations on the physical region 2 1 Notattons'attd d~Jtnittons The partial wave expansion of the total amphtude t'I(~ ', t, u) ot lsospm I m the schannel is
1,16, t, ,,) = ~
(2t + 1).t/(,0 pt(cos 0 ),
/=o where
(2 l)
1
J[<,) = ~ f
FI(~, t, u) P/
-1 cos 0 = I +
2t --
:~-4"
s', t, u, being the usual Mandelstam variables ( m = ~J = 1 ) Normahzatlons are chosen such that, [or s/> 4, d°l (s, t) = 167r i~ 12 d~~ - ~,)IF (s, t) ,
~ot ( s ) -
16n
x/s(~
I
4)
AI(s, o),
(2.2)
(2 3)
508
B. Bonnwr, t' (,auron, (.'tossing symmetry
where A l(~,t) lsthe absorptwe partmthe s-channel of Fl(s,t) Elastic umtarlty then reads l m / / ( s ) = p(s)lf/(s)l e ,
4~s~<16, ,ors) = [s/(s-4)]
(2.4) },
which nnphes the parametrlzatlon.
/l(s)=p
l(s) e ' a l s , n ~ [ ,
(2.5)
m terlns of the real phase shifts 6[. As we are mainly Interested m the S- and P-waves, we shall adopt the shorthand notation f/(s) for/if(s), (I = 0, 1, 2) and we define the scattering lengths al by.
aI .
hill { 1/(5) } . . . . . s-+4 (¼(s- 4)) /
(2.6)
2.2 Parttal wave representattons The Frmssart-Grlbov representation is valid for />/2, - 28 < s ~< 4. [towever, it is known that a representation of this type can be extended to the S- and P-waves, and tile subtraction constants which appear can be linked to the S-wave scattering lengths. As an example we shall recall the derivation of this subtracted Frmssart-Grlbov representation for tile S-wave ]N(S) of the neutral amphtude FN(S, t) = F°(s, t) + 2F2(s, t) winch is completely symmetric under s-, t- and u-exchange. In the partial waves expansion FN(*, o)=fN(s) + ~ (2/+ 1)//(s), (2.7) 1>2 we call replace for / ~> 2 the partial waves by their Frolssart-Gnbov representation f/(s) = n(~ 4-4)- a7f QI(1 +s
4 AN(S's) ds'
(2.8)
FN(S, o) = fN(S) + "/r1 f Ko(s, ' S) AN (s', s) ds',
(2.9)
winch leads to
4
where Ko(s, s ) = l , + ( ,. +~1 +_4 4-s2-l°g ( I +ss' 4) on the other hand, the forward dispersion relation gives
(2.10)
B. Bonnter, P Gauron, Crossing symmetr~
509
0o
FN(S'°)=ao +2a2 + lrtf
K l ( s ',S ) A N ( ~ ' °, ) d s ' ' 4
,
- 4) (J
(2.11) - 4)
KI(S,S) =(s,_~)(s,+s_4)(s,_
4) s'"
Comparison between (2.11) and (2.9) leads to the expected result
/N(s)=ao+ 2a9- + lIr f_ IKI (s',s) AN(S'' o) -- Ko(S',s)A N(S' . s)}. ds'. .
(v 12)
4 Computations along the sane hne can be made on the other amphtudes l"l(s, t) and
Fa(s , t) = 2F°(s, t) - SF2(s, t). (We work on the set FN, F 1, F A which has simpler crossing properties than the set F °, F 1, F 2, but fmal results apply to the partial waves )'/(s).) As the usual Frolssart-Gribov representation, expression (2.12) is valid for - 28 < s .K<4, thus forbidding direct evaluation of the S-wave on the physical region. Such an evaluation becomes possible by means of the representation gwen by Roy [9] which provides the analytic continuation o f ( 2 12) in the range 4 <~ s < 60 tlowever, the breakdown of (2.12) is due to the s dependence of the absorptwe part AN(S', s) and does not happen ifAN(S', s) is replaced by a fimte number of partial waves. In this approxmlation, the Frolssart-Gnbov or Roy's relations lead to stmdar results. More precisely, if the absorptive parts of FN(S', s), FA(S', s) and F l(s ', s) obey the relanons:
A N(S', s) = A N(S', o) = A N(s'),
AA(S', S) = AA(S', O) = AA(S') , A I ( S , S ) = A I ( S , O ) 1 +s - 4]
l(S')
I+
,
(2.13)
the two representations are found identical and lead to.
fN(s ) =ao + 2a2 + l f KNAN(S, ) ds',
(2.14)
4 co
I f ds'{KAAA(S) ÷ 9 K I A l(S')} ' fA(s) = (2ao--5a2) {~-(3s--4)} +-~n
(2.15)
4
fl(s) = (2a o - 5a2) { 6 (s-- 4)} + ~I f ds' 4
{KtAA(S,)+3K~A I(s')}
,
( ' '¢,)
5 I0
B Bonmcr, P. (;auron, Crossing svrnmetr~
where
1 K N - s~
2
1
s - s' - s '
4
[
4 t- ~ ( ' - ) o
1,1 + s
;~_'~ 41 ,
Ka-T2+¼(3s--4) 7 s--aT s_~0o l+s_4! A
-
~,,
$
, s4
KA=--4~7--
-
4 ( .1,+s~)7 - _4-
~,(S,~+S-~-
{l 1}
K'? = i-l.2(S - 4) - ~ - - ~
4+43 12(+---~4)
Kll = 4-~7-s ''
+~4QI
e
Q,,
1 +s~
'
'
1 +;:74- ,
4( 1 + s~
+ 5(]¢--- 6 + - --- s4
QI
(I + ;--4)2s''~
Moreover, in this approxnnation, the S-and P - a m p h t u d e s / l ( S ) can be seen to satisty identically the Balachandran-Nuyts-Roskies crossing symmetry relations [12 I These relations are our basic tool for the following sections of this work. 2..¢ l h g h - c n e r g ) ' ¢ o n t r t b u t t o n ~
The approximation (2.13) means that in the low-energy region the absorptwe parts contanl the S-and P-waves only. At higher energies, it is compatible with a Regge behav,our dominated by the pomeron exchange with a fiat tralectory. On the other hand, the kernels entering the dispersive integrals In relations (2 14) and (2.16) are converging hke ¢- ~. The approximation (2 13) is then consistent with these relations, and the integrals can he written as the sum of two terms as has been already polntedcmt m ref [10] 7t
4
l
77
4
77
S\l
In the first mtegral of the right-hand side, the absorptive parts are replaced by the S-and P-waves, and in the remaining one, by some high-energy contr%unon, in such a way that relations (2.14, 16) can be formally written, in an obvious notation
)/(~) =./s(0 +-"iP(~) + 1~(0
(2.17)
Our first task is then to give an estimate o f / / ( s ) , and for this we have chosen the Regge picture, with the following representation for the amphtude of isospm I t in the t channel =
(~t
/ t=O P(s, t) +/3,)( 1 + e t,ra(t)) I'(1 - c~(t)) - \511
cat'
,
(2 I ~)
B Bonnet, P. (;auron, Crossing symmetry
511
F It=2= 0 ,
Fit = l = 31 ( I
- e-Znn(tb
1-" ( 1
o¢(t)) \~;l ]
'
where (1) c~(t) ts the p- fo hnear trajectory given by o4t) = % + ~ ' t = 0.5 + 0 017 t " 01) s I is a sc',dmg factor of the order of 1 GeV2(s I = 50,u2), ( n 0 3o and/31 are the reduced residues supposed to be i n d e p e n d e n t of t They are evaluated by reqmrmg that expressions (2.18) and (2.19) give tile Brmt-Wlgner forms assomated w~th the 0 and fo mesons when t = m~ or mj,, 2 If we defme the reduced widths 7R by TR = mRl'R/m~r, we obtain ,
2
2
/3(i = 15~ 3,1o m~o Sl(mto - 4) /31 = 3 ~ ' 3 , p m o s t ( m ~ -
4) }
(2 20)
Numerically, with Sl = 50~2, m 0 = 765 MeV, I"o = 135 MeV, mlo = 1264 MeV, Pro = 150 MeV, we obtain /30 = 1 0 4 , 131 = 0 . 5 3 , and we shall then choose m the following/3 o = 2/31 = l 05; (w) P(s, t) is the c o n t r l b u t m n associated with the pomeron exchange We need only hn P(s, t) fi~r s > s~4 and we choose h n P(s, t) = s~J ,
where o can be hnked to the asymptotic total cross secuon. As we shall use the Sand P-waves only up to s ~ 60, we must take SM ~ 60, whmh hes between the O and fo resonances. We then shall adjust o m such a way that the absorptwe parts gwen by the S, P and D o waves lore approxmlatlvely thmr Regge form m the vlmmty of s ~ sxt. This gwes valucs of o m the range 0.05 ~< o ~< 0 11, aJld we have chosen the mean value o = 0.08, which gwes a good interpolation of A () and A I between the f(, and g resonances and an asymptotic total cross section ol 26 mb Using the preceding estmaates, It appears that for s' --+ + AAS
30S'
, ,42~s(~,o)--~
+31 ~/ Sl
, { co,' -/3, o,,~-7;31
,
,/.71 ~, ! ~, j-
512
B 13onnter, P. Gauron, Cros~tng symmet O'
tlL
'20
Ilg I. Ihgh-encrgy componentsf/~(s) of the S- and P-waves. ('ur',e', A, B and (" represent f(}~, f~ and f~l re,,pcctlvcly The corresponding asymptotic contributions t l ( S ) to the S-and P-waves are drawn on fig 1, for 4 < s ~< 40. It appears that they are neghglble for s ~< 20, but become quickly increasing. These qualitative considerations remain true when we vary the high-energy parameters a little around the given values. For subsequent use of relations (2.14),(2.16), we must then point out that the uncertainties associated with unportant asymptotic contributions restrict their range of applicability to s ~< 40 (850 MeV).
3. Test of the crossing properties of .,~me models In the previous section we have given an estimate of J / ( s ) Now, If we have a set of S-and P-waves on the physical region, we can use their imaginary parts in relation (27.17) to obtain as an output their real parts, given by the crossing properties of the 7rTr amphtude. But, on the other hand, we have these real parts as an input since we can compute them directly through the elastic umtarity relation. This is the crossing test already proposed in ref. [10]. Moreover, relations (2.14) and (2.16) enable us to calculate the analytic continuation oil 0 ~< s ~< 4 of any set of S-and P-waves and consequently to check the conrpatiblhty of these phase shifts with the Martin posltivity constraints [4] on tile non-physical strip. In fact, the coincidence between input and output in each isospln is restricted to tile elastic strip, but it carl in practice be extended to s ~ 36, and we reach again the limitation s ~ 40 found ill tile previous section. We have then tested, using the asymptotic contributions given sect. 2, some sets of S- and P-waves available in the hterature (a) The Bonnier-Gauron model [2]. (b) The Le Guxllou-Morel-Navelet inodel [3]. (c) The Kang-Lacoinbe - Vmh Mau model [13]. (d) The model-independent phase shifts of Morgan and Shaw [14] (e) The experimental phase shifts of Baton, Laurens and Reignier [15]. The first three models are built from the elastic unitarity, the correct s = 0 threshold behaviour and the p- meson's existence. They satisfy the five S and P crossing sum rules of Balachandran-Nuyts-Roskles and Martin's constraints on the non-physical strip llowever these conditions are fulfilled with different parametrxzatlons, es-
B Bonmer, P. Gauron, Cro~smg O,mmetry
513
-150"
o
12.8
,
,40 S ......-~
Fig. 2a. S-and P-wavesphase shifts selected m model I using eqs. (2.14) and (2.16) (ao = 0.18, a2 = 0.08). Bars give the hmlts of the imual set. peclally case (c), which is an unitanzation of the Venezlano amplitude. The physv cal pre&ctlons of these models are m rough agreement (0 a o - meson is found m all cases, but with different mass and width, (li) the models (a) and (b) agree for the low-energy I = 0 S-wave behav]our and the models (b) and (c) for the I = 2 one. These amplitudes being elastic unitary, eqs. (2.14) and (2.16) are used as a test of their crossing symmetry properties on the physical strip. The phase shifts and scattering lengths of case (d) are computed m a model rodependent way throughout ftxed-t dispersion relations and their derwatwes with respect to the m o m e n t u m transfer, both evaluated m the forward &rectlon. In case (e), we have only the phase-shifts and we have to search the scattering lengths a 0 and a 2 which provide the best fit to eqs. (2.14) and (2.16). The results can be seen on figs. 2- 6. The first conclusion which can be drawn is that eqs. (2.14) and (2.16) correlate tightly the scattering lengths and the phase shifts, more tightly than the usual disper-
514
B Bonnier P (,auron. Crossing4 s y m m e t r y
kJ _
~.8
_
,:~6
_
| l g 2b Resultsot thccrossmgtestonthephaseshfltsoi f,g. 2a ('ur~esUaretheumtarlty hmits± } . v @ ( s - 4 ) Curvesl~mputs) are the real parts of the partlal ~aves m each lsospm taken from tile model ( urves O (outputs) are the real paris of the partial ~,aves given by eq',. (2 14) and (2.16) slon relations or the Balachandran-Nuyts-Roskles relations For example, on fig. 6, beside the best fits given by a 0 = 0 22, a 2 = - 0 036 for d o w n - d o w n (511and c o = 0.20, a 2 = 0.04 tk)r down-up 6'c~, one can see the effect of changing a 0 and a 2 from the preceding values to a 0 = O. 16 and a 2 = 0 04. As a consequence, better results can be obtained, especially in case o f model (c), if the scattering lengths are shghtly deviated from their orlglnal determinations by the Balachandran-Nuyts-Roskles relations. A numerical o p t n m s a t l o n on the scattering lengths can be done, which leaves the phase shifts practically unchanged, but which is sufficient to bring an i m p o r t a n t m l p r o v e m e n t on the fulfilhng o f eqs. (2.14) and (2 16). For example, m m o d e l (c), it we pass from a 0 = 0.186 t o a 0 = 0.15, the soltltlOn 12) gives results comparable to those o f models (a) and (b) (fig. 4). It appears also that some solt|tlons roughly consistent with eqs (2 14) and (2 16) can be selected in each set o f phase shifts. Particularly, ~t seems that the up-down ambxgulty in the I = 0 phase shift cannot be removed. But we lack definite conclusions. It IS then useful to generate and study a more c o m p l e t e spectrum o f solutions. This is the purpose o f the following sections, where we build a model especially stated for the Frolssart-Grlbov representation o f the partial waves.
515
B. Bonntcr P. (,auron, Crossing ~vmtnetrr
'~
/
gl
~°
g
S.-~
g~
fig. 3a ( c n t r a l s o h m o n o I model 21a0 =1) 2116 a2 =
(I.0731 B a r ' , g l v c t h e h m l l ~ o t t h c m l t l a l
4. Model-dependent fit of the crossing relations on the physical region In expressions (2.14) and (2.16) we need a representatton of Im/l(S) on tile physical region only. However, we shall see that a reahsuc parametrl,,auon of hnj'/(s) must take mto account general propemes offl(s ) in the physical and unphyslcal sheets.
4 1 Analyttc pn)perttes ofp - 1 (s) lm fl(~) Let us define [161 the functions G/I~)by
Gl(S)
= {Jl(S)} 2 {1 +
2,p(s)fl(s) } 1
m such a way that, for s >~ 4, elastic umtarity Imposes
('4 1)
B Botmter, P (;auron, Cros~tttg ~)'mmetrv
516
Fig. 3b. Results ol the cros',lng test on the phase shifts o f llg. 3a.
150e
_ 50"
t 16
128
140
7_
g~
Fig. 4a. Phase-shffls p r e d i c t e d by m o d e l 3. The s o l u t i o n s 1 and 2 are b o t h a s s o c i a t e d w i t h the same s c a t t e r i n g l e n g t h s ao = I). 186, a 2 = 0,045.
B Bonmer. 1' (;auron, Crossing symmetry
-
_
_
517
L2_
t
o
•
~
, .....
~
..
I'lg. 4b, c, d, e Results of the cro~,~mg test on the phase shflts ot fig. 4a, b ~olul,on I, c ,,olutmn 2 ,d . solutmn 1 w l t h a o = 0 15,a2 = - 0 0 4 5 . e solutmn 2 v, lth a0 = 0 16, a2 ~ 0036
Im fi(s)
= p(s) al(s) .
(4.2)
Gl(S )
has the following propemes. (1) It is meromorphlc m the s-plane cut along the neganve real axis and it Is real for s/> 0 (positive for s >~ 4); (u) its poles can be associated wxth resonances (ff not too far from the real ax~s) and virtual b o u n d states. A resonance (SR, 7R) reduces two complex conjugate poles at s+ = s R -+ Z3'R. Such a pair of poles occurs in the P-wave, the corresponding residue being c o m p u t e d from 7R in the Brmt-Wigner approximation. We postulate the same situation in the 1 = 0 S-wave, b u t in this case the parameters of the poles and the residue are left free. A virtual b o u n d state occurs at s = s v (0 < % < 4) when V
/l(Sv)
= 21"
Sv
4 - sv
We know from the work of Martin (1 7) the existence of these poles at least m the partial waves I = 0, 1 t> 2. In fact, from previous models 12, 3, 1 3], It appears that the S [=2 w a v e has such a singularity m the vicinity o f s = 0. In this case the singular part of Gi(s ) can be written Fv Gv(S) = s - Sv ' w~th
5 18
B Bonmer P. (,auron. O'ossmg ~rmmetrv
[ 'SU i50"
I00 ° 8,'D
J
i i
up
!; 12.8
ill,
14o
s -...~
A
I ]g. 5,]. Ph,]se-shHls s e l e c t e d b y M o r g a n ,rod Sha~,. I l h e best results u n d e r the c r o s s i n g test are airways o b t a i n e d w i t h the Lurve B for ,5 2)
I'v= ~
Sv
[4
sv--
t#
4(4
sv) d f(s)
(4 3)
,v
We shall then suppose such a singularity to occur in G2(s ), s v and I"v are free parameters to be computed by self-consistency, as will be explained later. (in) Gl(s) has known threshold behaviours at s = 0 and s = 4, derived from the behaviour of J/(s) at /he same points • Gl(S) = ai2 (4I (s -- 4) 2/+ O((s
G/(O
= - ~,
x/~Jl(O )
s i-(;
4)2/* I ) ,
+O(sx/-s)
(4.4)
(iv) GI(S ) has double zeroes when [l(S) has a zero on the non physical strip As such zeroes ~ e l n systematically Rmnd m the S-waves [2.3, 13], we shall suppose a priori their existence, but their location has to be computed by self consistency
B Bonnwr, P Gauron, O'ossmg symmetr;
4
-
LI~-
laO
5 19
,~8
,3G
f
I ~g 5b, c, d, e. Result', of the ~.ros,,mg tcst on the phase shflts e l ttg 5a b. ,,olunon UI). c solun o n BL 1, d. ,,olutlon BD.1. c ~,olutlon BI).2.
4.2 Practical use oJ"the eqs. (2 14) and (2.16) In the following section, we shall use tile analync properties of Gl(S) to propose a parametrlzatlon of these functions vahd in the whole cut s-plane. The freedom allowed in these parametn/ations (scattering lengths, pole a n d / e r o parameters) is to be restricted according to the following programme: For each set Gl(S), I =- 0, 1, 2 through (4.2), eqs. (2.14) and (2.16) give/l(S) for 0 -%<~ ~< 4 and ReJl(s) for 4 <~ .s <~ 40. But, on the other hand, (4.1) gives alsofl(s ) for 0 ~ ~ ~< 4, and elastic unltanty gives Re f/(~) for ~ ~> 4. These two representations o f / l ( S ) induced by the same GI( 0 must be ldenncal for ~ t> 0, then (l) We can compute by self-consistency all the parameters linked to tile non-physical region (zeroes and pole parameters), since a priori they appear m different manner m the two representations. (n) After this prehmmary ehmmation, the remaining parameters of GI(,O are fixed by mlmmi/mg the discrepancies between the two representanons of R e / i ( s ) on tile whole strip 0 ~< ~ ~ 40.
4. 3. Parametrtzatum of the/un~ ttons Gl(S) After having relnoved t i l e / e r o e s and poles o[ Gl(S ) we can expand the remaining part of Gl(S) in a power series in the variable ~"(~) which maps onto the u n i t circle the s-plane cut along the real negative ax]s
520
B ]]()littler, 1) GaltrOll, (ro351tl,t~ s v m m e l r v
,._.tSO °
~ up
g:
~g DOWN
50"
116
~Z~,
,40 S.-~
| lg 6,1 Central e\pcrtmental phase shifts of Baton, l aurens and Rmgnler.
r(s)
_V';- 2
(4.s1
v¢÷2"
According to the results of paragraph (4.1), we have chosen the representations
c,o(s ) = (s-s012 {(so --st 2 +(s--4/ ~0
Gl(s t = (~--412
{(so-s)2+ (Ss4) y
0"2(8t = (S-S2 t2
{S
5
' 2 11/' s1,
:}5
It : 0
-I ,,~=o g'l r"(s),
4
Sv}
2 g2 e'(s). tl = 0
In these expressions"
(4.6)
B Bonnier. P (,auron, O'osstng wmmetrv
s
,
I
521
[b
f
°
o
~
f Fig. 6b, c, d, e. Results o! the crossing test on the phase-shifts of llg 6a, b best fit tot downdown,h°(a0= 0.22, a;b= 0036),l,.bcst fit for down-upS°(ao = 0.20, a2 = 0 0 4 ) . d solution down-do~,.n for 6o with ao = 0 16, a2 = 0.04, e. solutl~ down-up for 6~ ~lth ao = 1).16, a 2 =
-
0 1)4
9
s o and 70 = mol'o/m~, sp and "to = real o/nl[r are the masses and widths o f the oand 0-mesons respectwely, s o, s 2 are the zeroes of the S-waves, s v is the wrtual b o u n d state pole. All the three are c o m p u t e d by self-consistency. The four first coefficlents,~/~tn each Isospm are fixed by the threshold conditions at s = 0 and s = 4. The remammggZ/are hnked to the residues at the poles. Then g42 is c o m p u t e d from I"v (eq. (4.3)), g l and g~ are determmed by lmposmg the mass (765 MeV) and width ( 125 -+ 15 MeV) of the o-meson, g~ and g~ either are free, or fixed as funct lon~ of one free parameter Sl) > 4, the value of s where we tmpose elastic u n l t a n t y of /0(s) together with the passage through 90 ° of the phase shift. Fwe (or stx) parameters remain free. ao, a2, so, 9'0, together with ~1) (or gO and
gO).
4.4. Computattons and results As has been explamed in subsect 4.2, we fix these parameters by the m i n m u / a tlon of the relative ×2 defined by
522
B. l J o t l t l l e r .
P
{;aur¢Jn.
¢. )¢Jsstn
t l'.fo"
_iN*
I f I
S.___~
I
| It: 7,1 ()LH Jli'~d ~,C| Ot ~,tl|l.lllon~
rncnt. 1 1 ",pcrmlcnt,d
[).l~,hL'd ,.LIr'~c'~ rcpre,,Cll! the.' ,,oiUtlon~, ~,L'],..'ct e d |:,~ [h0 c \ p c r l -
data f r o m rcl
,c': Z: Z: /=tl \ :It
[15}
t I
\pe[llncnldl tl,lta f r o m r c f [21l
--
-' .
where Jtt'(s x ) a n d / i f ( v v) ,ire the real pa, ts of the amphlt.aes J[(~) taken ai = ~x (0 < s \ < 40), and c o m p u t e d from (;l(S) through the elastic unitarlty relatum (or ec1 (4 5) for 0 < s < 4) and through eqs (2.14) and (2.16) respectively. Two sets of solutions are found, for whirl1 Ill each i,,,ospm the uintaHty ratio R [ = I(1) ~ /1t~,)/-1It I is always less than 2 × 10 with lwo exceptions (i) set I R 1 can reach 5 X 10 2 for 35/J 2 < s < 40~ 2. (n) set II R 0~.,tn reach 3 X 10 2 t o r 10~ 2 < s ' < 16~ 2. The first set (fig 7) p, obtained with lhe following values of tile parameters 13 5 ± 3 51a2 ,
a0=0
20:t:0.06,
<,<,=
a~ =
0 0 8 ± 001~,
1" = 4 9 5 ± 170MoV,
ao a
=
2~)±14.
523
B. Bonmer. P (,auron. Crossmg symmetry
U
-
_05
_
=D
i ~6___
u L
f
-
~
-
F,g. 7b. Results of the umtarlty and crossing test on the central solutmn of fig. 7a The 1 = 0 S-wave phase shift 60 xs of the up-down type and hes m a rather wide band, since It can vary from 65 ° up to 100 ° at the 0 position. The corresponding I = 2 Swave phase shift 602, small and negative, increases m magnitude when 6~ does, passmg from - 2 1 ° up to - 29 ° at the ,o-mass. The P-wave phase-shift 611 remains the same on the whole set and corresponds to the following values of the scattering length a 1 and the width Fo: a I =0.044+-0004, Fp =!35
+5MeV.
We have c o m p u t e d the D-waves from thmr real parts gwen by the Fro~ssart-Grxbov representation. They are very small, positwe for I = 0 and negative (except sometimes m the viclmty of the threshold) for I = 2. T h e y reach respectively + 3 ° and 1° at the ,o-mass and have scattering lengths d I -
d o = 0.0021 -+ 0 . 0 0 0 3 , d 2 -- 0.00025 + 0 . 0 0 0 2 5 . On the non-physical strip 0 ~< s < 4, the p o s m o n s o and s 2 of the zeroes o f the two S-waves are: So= 1.6-+0.7, wxth 4 ~ o + 5 ~ 2 = 12 +-O.Ol
32= 1.1 -+0.5,
B Bonnter. P GauroJt. Cro~stng symmetry
524
150"
J
tItLttlt /
O0 °
SO*
S-->.
Fig ,~,1. Our s e c o n d ,,el of s o l u t i o n s
The second set (fig. 8) is less satisfactory than the first one from the p o m t o f view of the umtarity. It is obtained with the following values o f the parameters a 0 = 0.137 -+ 0 . 0 1 3 ,
s o = 31 -+ 8~ 2 ,
a,~=-0.055-+0.005,
1~o
-
IO00-+200MeV,
ao
--
2.5+_0.5
a2
The 60 phase shift is of the, up-up type and passes through 90 ° at 20/.t 2 < s < 30~ 2, 0 but the associated pole IS very far trom the real axis. It must be stressed that Go(s) b e c o m e s negative and violates badly the posltwlty o f hn ]o(S) for s ~ 45. A l t h o u g h this region is out of the range o f vahdlty of our model, such a thing throws some suspicion on these solutlons. The 60 phase shift is s o m e w h a t smaller m magnitude than in the preceding case. The P-wave is agam stable but corresponds now to a I = 0.0335 -+ 0.001 , F /9 = I I O M e V .
525
B Bonnter, P. (;auron, Crossing symmetry
/
,,
I Ig 8b Results ol the unlt,trlty and crossing lest on the central solution o f fi,, 8d (a~ = 0 138, a~ ~ 0 06).
The D-waves are unchanged and the zeroes of the S-waves on the unphyslcal region are s o = 1.4 + 0 . 3 ,
~2 = 1.25 -+ 0 . 2 .
Some comments can be made about these results. First, we remark that the updown ambiguity of 6 c)above the p-mass cannot be removed by cqs. (2.14) and (2.16) and elastic unttarity this ts m agreement with the conclusion of sect. 3. Below the p, 8fl~is of the up or between type. Secondly, we can try to discrmunate atnong all the solutions which are obtained. For this purpose, we have chosen for the tmle being the expertmental point of view and we have checked how these solutions fit (a) the cross secttons o ( n + n - ~ n + n - ) , o ( n - n 0 ~ n n O) (ref [15]) and o(n+n ~ n%r °) (ref. [18]), (b) the ratio [19]: R 1 = o ( n + n - ~ nOnO) l o ( n + n + -+ n + n + ) , R 2 = o(n+rr
~ n ° n O ) / o ( n + n - -~ n+n -),
(c) the determination of 60 - ~511at low energy froth the Ke4 decay [20] The three tests rule out the second set of phase shifts and reduce appreciably the first one. The remaining solutions can be seen on fig. 7. The corresponding values of the parameters are.
526
B Bonmer, 1: (;auron, ('rovsmg 9i'mmetr~:
a(~ = 0.23 -+ 0 . 0 3 ,
~ = 1 3 4 + 1 2~ 2 ,
s()= 1 0 4 ± 0 . 1 4 .
a 2 = - 0.067 + 0 . 0 0 3 ,
I" =495 ±75 MeV,
s 2= 1 55 - + 0 . 1 0 ,
d(1~00021
d~ ~ 0 00027
with a I = 0.0448 -+ 0 0 0 2 ,
,
0
The vahte h n ( m K) h ~ ( m k ) turn~ out to be 57 ° + 5 ° We can point out that tile final 8{)) phase shifts are in good agreement with the experimental resuhs of Malanmd and Schlem [21] and with the conclusions o f Scharengulvel, G u t a y and M i l e r [22] from the study o f the forward-backward asynmletry They also agree pertectly with the t h e o t e u c a l d e t e r m i n a t i o n of Iv ( h n l h m , blorel and Navelet 131 over the whole energy range and wtth our preceding one 121 over the low-energy region
5. Conclusion In tile first part of thl~, work, we have shown that the models fulfllhng the S- and P-crossing relations of Balachand, an-Nuyts-Roskles do not satisfy systematically the crossing relations on the physical regmn. In this respect, these last relations appear to be very selective But on the otl,er hand, and this is the second part o f this work, they lead to a new representation of the partial waves which alh)ws a wide class of solutmns In particular, the crossing relatums on the physical region together with elastic t l n l t a r l t y a n d p - m e s o n e x i s t e n c e does n o t p r o v i d e a n e w d e c i s i v e test to d i s c l l t l l l n a t e between dlfterent predictions hke the up-down ambiguity m the d e t e r m i n a t i o n o f 8 0o above the p-mass. C o m p a r e d to some prehmlnary hopes [10], this tact is disappointing and give a c o n f n m a t l o n of the role of ttle p a r a m e t r l / a t t o n m the tlnal result. Ill fact, in this scheme, as we do not COlltpttte exact so[utloils, the results seem to be very sensltlve to the criterion used for their selection If we require the consistency between nlput and o u t p u t (In the sense o f sect (4.4)) on the physical region only, oul paranletrl/atton allows a largm ,,pectruln ot solution ~, 12~;1 " We have chosen to take into account as tar as possible the constraints reduced by the elastic u m t a n t y and crossing. I his had led us to use the analytm c ( m t m u a t l o n of the functions G/(a) oil the unphyslcal strip. We are then able to require the consistency between input and o u t p u t not only on the physical regum, but also on the unphysical one. In particular, the v ~ 0 threshold behavumr can be put into the game, as m our previous model [2] * IIo'a ever. v,c CdlltlOl o b l d l n aS large a ,,pe~.Irum ,is the (}lie o|'lldlned b v ]~dsdcvdnt, [ nL~egat and Pcter,,en [23 l, o w i n g to o u r mlHal I1~ pc~thc'qs dimout the d l , , t n b u t u m of the / t r o t , , a n d virtual b(~ulld M,tte,, In the S w a v e s It c,lll hi) ,,.ho~,n that otlr LhoI(C, LonMxlelll v. lth the rcsult~ ot rcls I2, 3 13 J. restrict,, a przort the s c a t t e r i n g length,, m the range 0 - at) ~ (I 35 a n d - () 15 ~ a 2 "-. (I \t,e .arc g r a t c l u l to Dr la;asde',ant lot ,t clarfl,,In'a dlst.u,,,,um un Ibis s u b l c t t
527
B Bonnu'r, P. Gauron, (.~'ossmg ~vmmetrv
In o r d e r to t a k e i n t o a c c o u n t s o m e r e s u l t s o f t h e s e m o d e l s [2, 3], we h a v e also IiIlpO,,cd a prl()tl s o m e g e n e r a l l e a l m c s like a 2 n e g a t i v e F v e n m t h , s r e s t r i c t e d t r a m e w o r k , we h a v e f o u n d a w i d e class o f s o l u t u m s , b u l w i t h ,t COll)inon l o w - e n e r g y b e h a v l o u J a n d In g o o d a g r e e m e n t w l l h s o l n e p r e d i c t i o n s p r e v u > u s l y n),(de [2, 3], t h o u ~ } t h e e l a s t i c u m l a t ) t y a n d c r o s s i n g are u n p o s e d m a very different way m tins work and m ref
12, 3]. T i n s s e e m s to u s a v e r y e n c o u r a g -
111g (dOt I l o w c v e t , tt a p p e a r s also t]lal we need s o m e e x p e r m m n t a l
i n f o r l l l a t l o n s 111
a d d i t i o n to tlae p - m e s o n ill o r d e r to select a u n i q u e class o f p h a s e - s h l t Is t o t t h e 280 850 MeV range. We a,e h a p p y to t h a n k P r o f e s s o r R V m h M a u a n d Dr. L L u k a s / u k f o r h e l p f u l dtsc uqMons
References Ill 12t 131 141
(, Aubcrson, O. Ihgucl and (. ~3,andcr,,, I'h', ~ t otters 28B (1968) 41. B l,~onrncr ,rod P (,auron, Nuul. l'h',', B36 (1972) 11 I ( 1 c (,ull]ou, A, Morel ,rod II. Na'.clct, Ntltp, o ('lmcntt) 5A (1971) 6'r,9 ,Set' lor CXdlllplc (a) A ,Martin ,rod I ( 11cun~, '~ndI.VtltlI 3 properties ,,'Jd bt)ur~tls ot the ',tdtlCllll~' ,tmplitudc% I.)o~.umcnts on Modern Ph', s~t s ( 1971 ).
(I)) 1.1 . Ba,,dc',,mt ,rod J I~,mgnmr, lhon-pton lnlcrdt, tlon. I,c~.turcs Note,,, I lcr~.c,.z, Nm.q St h~(~l (1971.)) [5] ] or ,1 dl',.t tl,,'..ion of till'., p,)lnt (a) I.l)lllc3,Nut.I Phy,,. B25 (1971) 227 (h) (). Br,mdcr. ( 1 RN prcprml I II - I q36 ( 19711 [61 I or ,, rcvlc~.~, ol thc',c toplt.~, ',co (a) I I Pctcrscn, Ph3,', Rcp 2 ( ( 1 9 7 1 ) 155. (h) I) M()t,~,in. ]~,uthcrtt)rd I al),*r,,~)r', plcprlnl RPI);( ,311 ( 1971 ) [71 \l.R. Pcnnm,_'t~mandP Ihmd, N u o ' , o ( m l c n l o 3A (1971) 548
[8 19 IS0 [11 112
Rc,,lev, ol PartRIc,, I)ropcrtlc,.. Part]~lc I)ata ~'roup, Ph3.', Letter', (1972) S \| I~,o~. Ph', ,,. l.ctter'~ 36B ( 1971 ) 353
I I B,p.de'..mt, J.( [,t., (,udlou and I1 N,p.'elct, Nlu~vo ( illlcrlto 7,\ (1972) 36q O lqgucI and (, ~Aandcr,,, I, np,'cr,,tt,.5 de I au,,,mnc prcpr.n! 119721 A P t'l,d,mthandr.m.and I Nu.~t,,,Ph~,,, Rcv 1 7 2 ( 1 9 6 8 ) 1821, R Ro',klc',. Ph.', ', Letter', ~;01:t (1969) 42. \ u o ' , o Cm',cnIo 6 5 , \ (197(I) 467
113 k K,mg, ~.1 La(_ombe .and R. Vmh \l,m, Ph'.', Re', 1)4 (1971) 31")1)5 114 1) \h)rt:an and (, ShJ:~,, Phys. Rt'~, 1)2 (1971)) 52() I l S I P. Baton, (. Lmrcn,, and J RcL,.2mcr. Nm.I Ph3.', B a, (1967) 349, t'h3 s. 1 citers 3:H'I (1970) 525, 33B (1970) S28 [16 \~. Zlmlncrm,mn Ntto~,o ( iiIlento 21 (249) 1961 117 A Martin N u . v o ( m l c n t o 4 7 A ( 1 9 6 7 ) 2 6 5 118 J R Bcn,,mgcr ct al. Ph,,,,,. letter,, 36B (1971) 134 119 1). ( hne, K.J Braun ,rod I R qthcrcr l)rot.cedlm.x ()l the Argtmnc ( unlerem.c on rr,rr ,rod Krr interactions. Argonne. 1969 120 A Z.',lberstcln el al , I'h~s I citer,, 38B (1972) 457
528
1:1 B o n n t e r P (,aurott Crosstng s v m m e t r ~
[21] I. \ 1 a l d m u d a n d P Schlem. Proteedmg~ of the ,~rgonncConl o n r r ~ a n d Krrmtcracnon~, Argonne, 1969 [22~ J II Scharenguwel, LJ Guta~ and 1) \1 Mdler, Nucl. Phys B22 (1970) 16. [23 ] J L Ba',devant. C D I roggatt and J / Petersen. CI RN preprmt 1 I| 1518, 1519 (1972)
T E S T O F C R O S S I N G S Y M M E T R Y ON L O W - E N E R G Y rrTr S- A N D P-WAVE M O D E L S U S I N G C R O S S I N G R E L A T I O N S ON T H E P H Y S I C A L R E G I O N B. B O N N I I ' R and P GAURON In~tttut de Physique Vucl(atre. l)ll t~topt de Ph)~tque l h ( o r t q u c *
Recewed 13 Jul) 1972 (Revised 16 Ot_u~ber 19721
Ab',tra~-t We use the crossing relations on tht_"ph,,slcal re(non as a teq o1 the t.rosslng properties ol various nrr S- and P-x,,ave lllodcls In a ~,t l'len'le (.ontalflln~2 d~ d llrst Inpul the D- mcgon',, existence and elastic unlt,lrlt~, I hese reldllOrls ,ippear to be selettl',C but are unable to rcInovc %(lineambiguities like lhe up-dov, n one We also build a parametrl/atlon ol the partial x+a+cs espccl,llly suited for these ncx+ relations We find d 'Aide (+]as,, o| S-v+a',cs whlt.h lulfll the rcqulrenlents of elastlt, umtarlt+x, crossing symmetry and D-meson existence in the P-wave }lov+cver, their lox+-energ~, behd~'lotlr r C l l l a l n s i11 d g r e c l l l , 2 n t w i t h
[)rt2vltlt.lS prt.'dlt. 11o11'~
1. Introduction Various m o d e l s [1 3] using general principles [4] and the o - m e s o n ' s e x i s t e n c e have led to some p r e d l c t u ) n s a b o u t the rrTr l n t e r a c t m n , at least in the very low-energy rcglon [5]. The global c o n s i s t e n c y r c a c h e d m these d i f f e r e n t p a r a m e t r l z a t l o n s seems to indicate that the S-waves have zeroes and scattering lengths in rough a g r e e m e n t with c u r r e n t algebra p r e d i c t i o n s [6, 7]. The o or % m e s o n ' s e x i s t e n c e [8] seen) also to emerge f r o m this p a t t e r n , flowever, the posit(on and w i d t h o f the associated resonance are f o u n d to have a wide range o f possible values More generally, lugS-energy p r e d i c t i o n s b e c o m e increasingly m o d e l d e p e n d e n t , since the rigorous c o n s t r a i n t s c o m i n g from the general principles act on the non-physical strip only l t o w e v e r , f r o m this p o i n t o f w e w , some progress has been t h e o r e t i c a l l y m a d e with the w o r k s o f R o y [9] and Basdevant, l_c (,ulllou and Navelet [10], since thetr f o r m a h s m allows us to e x p r e s s the partial waves & r e c t l y on the physical regmn up to 1 G e V as f u n c t m n s o f the a b s o r p t w e parts * 1 aboratmre as,,O~.l¢'au ('NRS. Po,,tal address Laboratolre de Physique Th6orlquc et }lautcs 1 nergles Tour 16 - I atuhd de', Sciences - 9, Qual Saint-Bernard 75 - Paris 5e
B Bonnwr. P (,auron. (,'Josstng ~ mm~'rrv
507
In this paper, we first use these relations to test directly on the physical region the crossing properties of the 7rrr S and P partial waves given by some recent unitary models. Our purpose is to see If the preceding predictions about the low-energy domaul carl be confirmed. This seems to be n n p o r t a n t due to solne recent results of Piguet and Wanders [111, which conclude that tile rigorous constranats below threshold have a physical relevance restricted to a small interval above threshold. Our second aim Is to inveshgate by means of a snnple model fl" the crossing relations on the physical region can bring somethnlg new m the o-region A prtort tins Is not hopeless, since problems usually solved m a model-dependent way can be aw)lded, as for example the determination of the left-hand cut Moreover, tile highenergy behavlour of the absorptwe parts must be considered, and this allows one to inJect easily some Regge c o n t r i b u t i o n s to tile partial waves The plan of this work IS then the following, in sect. 2, we recall Roy's relations for elastic rcrr scattering, and we give an e s n m a t e of tile c o n t r i b u t i o n to the S- and P-waves of tile high-energy region, described by a Regge fom~ with p o m e r o n exchange. In sect. 3, we investigate the crossing properlles of the S- and P-waves given by some models, including our last work on this sublect [2]. In sect. 4, we extend this study to a p a r a m e t n z a t l o n of the S- and P-waves especially suited to these crossmng relanons. Our conclusion can be found m sect 5
2. Formalism, Crossing relations on the physical region 2 1 Notattons'attd d~Jtnittons The partial wave expansion of the total amphtude t'I(~ ', t, u) ot lsospm I m the schannel is
1,16, t, ,,) = ~
(2t + 1).t/(,0 pt(cos 0 ),
/=o where
(2 l)
1
J[<,) = ~ f
FI(~, t, u) P/
-1 cos 0 = I +
2t --
:~-4"
s', t, u, being the usual Mandelstam variables ( m = ~J = 1 ) Normahzatlons are chosen such that, [or s/> 4, d°l (s, t) = 167r i~ 12 d~~ - ~,)IF (s, t) ,
~ot ( s ) -
16n
x/s(~
I
4)
AI(s, o),
(2.2)
(2 3)
508
B. Bonnwr, t' (,auron, (.'tossing symmetry
where A l(~,t) lsthe absorptwe partmthe s-channel of Fl(s,t) Elastic umtarlty then reads l m / / ( s ) = p(s)lf/(s)l e ,
4~s~<16, ,ors) = [s/(s-4)]
(2.4) },
which nnphes the parametrlzatlon.
/l(s)=p
l(s) e ' a l s , n ~ [ ,
(2.5)
m terlns of the real phase shifts 6[. As we are mainly Interested m the S- and P-waves, we shall adopt the shorthand notation f/(s) for/if(s), (I = 0, 1, 2) and we define the scattering lengths al by.
aI .
hill { 1/(5) } . . . . . s-+4 (¼(s- 4)) /
(2.6)
2.2 Parttal wave representattons The Frmssart-Grlbov representation is valid for />/2, - 28 < s ~< 4. [towever, it is known that a representation of this type can be extended to the S- and P-waves, and tile subtraction constants which appear can be linked to the S-wave scattering lengths. As an example we shall recall the derivation of this subtracted Frmssart-Grlbov representation for tile S-wave ]N(S) of the neutral amphtude FN(S, t) = F°(s, t) + 2F2(s, t) winch is completely symmetric under s-, t- and u-exchange. In the partial waves expansion FN(*, o)=fN(s) + ~ (2/+ 1)//(s), (2.7) 1>2 we call replace for / ~> 2 the partial waves by their Frolssart-Gnbov representation f/(s) = n(~ 4-4)- a7f QI(1 +s
4 AN(S's) ds'
(2.8)
FN(S, o) = fN(S) + "/r1 f Ko(s, ' S) AN (s', s) ds',
(2.9)
winch leads to
4
where Ko(s, s ) = l , + ( ,. +~1 +_4 4-s2-l°g ( I +ss' 4) on the other hand, the forward dispersion relation gives
(2.10)
B. Bonnter, P Gauron, Crossing symmetr~
509
0o
FN(S'°)=ao +2a2 + lrtf
K l ( s ',S ) A N ( ~ ' °, ) d s ' ' 4
,
- 4) (J
(2.11) - 4)
KI(S,S) =(s,_~)(s,+s_4)(s,_
4) s'"
Comparison between (2.11) and (2.9) leads to the expected result
/N(s)=ao+ 2a9- + lIr f_ IKI (s',s) AN(S'' o) -- Ko(S',s)A N(S' . s)}. ds'. .
(v 12)
4 Computations along the sane hne can be made on the other amphtudes l"l(s, t) and
Fa(s , t) = 2F°(s, t) - SF2(s, t). (We work on the set FN, F 1, F A which has simpler crossing properties than the set F °, F 1, F 2, but fmal results apply to the partial waves )'/(s).) As the usual Frolssart-Gribov representation, expression (2.12) is valid for - 28 < s .K<4, thus forbidding direct evaluation of the S-wave on the physical region. Such an evaluation becomes possible by means of the representation gwen by Roy [9] which provides the analytic continuation o f ( 2 12) in the range 4 <~ s < 60 tlowever, the breakdown of (2.12) is due to the s dependence of the absorptwe part AN(S', s) and does not happen ifAN(S', s) is replaced by a fimte number of partial waves. In this approxmlation, the Frolssart-Gnbov or Roy's relations lead to stmdar results. More precisely, if the absorptive parts of FN(S', s), FA(S', s) and F l(s ', s) obey the relanons:
A N(S', s) = A N(S', o) = A N(s'),
AA(S', S) = AA(S', O) = AA(S') , A I ( S , S ) = A I ( S , O ) 1 +s - 4]
l(S')
I+
,
(2.13)
the two representations are found identical and lead to.
fN(s ) =ao + 2a2 + l f KNAN(S, ) ds',
(2.14)
4 co
I f ds'{KAAA(S) ÷ 9 K I A l(S')} ' fA(s) = (2ao--5a2) {~-(3s--4)} +-~n
(2.15)
4
fl(s) = (2a o - 5a2) { 6 (s-- 4)} + ~I f ds' 4
{KtAA(S,)+3K~A I(s')}
,
( ' '¢,)
5 I0
B Bonmcr, P. (;auron, Crossing svrnmetr~
where
1 K N - s~
2
1
s - s' - s '
4
[
4 t- ~ ( ' - ) o
1,1 + s
;~_'~ 41 ,
Ka-T2+¼(3s--4) 7 s--aT s_~0o l+s_4! A
-
~,,
$
, s4
KA=--4~7--
-
4 ( .1,+s~)7 - _4-
~,(S,~+S-~-
{l 1}
K'? = i-l.2(S - 4) - ~ - - ~
4+43 12(+---~4)
Kll = 4-~7-s ''
+~4QI
e
Q,,
1 +s~
'
'
1 +;:74- ,
4( 1 + s~
+ 5(]¢--- 6 + - --- s4
QI
(I + ;--4)2s''~
Moreover, in this approxnnation, the S-and P - a m p h t u d e s / l ( S ) can be seen to satisty identically the Balachandran-Nuyts-Roskies crossing symmetry relations [12 I These relations are our basic tool for the following sections of this work. 2..¢ l h g h - c n e r g ) ' ¢ o n t r t b u t t o n ~
The approximation (2.13) means that in the low-energy region the absorptwe parts contanl the S-and P-waves only. At higher energies, it is compatible with a Regge behav,our dominated by the pomeron exchange with a fiat tralectory. On the other hand, the kernels entering the dispersive integrals In relations (2 14) and (2.16) are converging hke ¢- ~. The approximation (2 13) is then consistent with these relations, and the integrals can he written as the sum of two terms as has been already polntedcmt m ref [10] 7t
4
l
77
4
77
S\l
In the first mtegral of the right-hand side, the absorptive parts are replaced by the S-and P-waves, and in the remaining one, by some high-energy contr%unon, in such a way that relations (2.14, 16) can be formally written, in an obvious notation
)/(~) =./s(0 +-"iP(~) + 1~(0
(2.17)
Our first task is then to give an estimate o f / / ( s ) , and for this we have chosen the Regge picture, with the following representation for the amphtude of isospm I t in the t channel =
(~t
/ t=O P(s, t) +/3,)( 1 + e t,ra(t)) I'(1 - c~(t)) - \511
cat'
,
(2 I ~)
B Bonnet, P. (;auron, Crossing symmetry
511
F It=2= 0 ,
Fit = l = 31 ( I
- e-Znn(tb
1-" ( 1
o¢(t)) \~;l ]
'
where (1) c~(t) ts the p- fo hnear trajectory given by o4t) = % + ~ ' t = 0.5 + 0 017 t " 01) s I is a sc',dmg factor of the order of 1 GeV2(s I = 50,u2), ( n 0 3o and/31 are the reduced residues supposed to be i n d e p e n d e n t of t They are evaluated by reqmrmg that expressions (2.18) and (2.19) give tile Brmt-Wlgner forms assomated w~th the 0 and fo mesons when t = m~ or mj,, 2 If we defme the reduced widths 7R by TR = mRl'R/m~r, we obtain ,
2
2
/3(i = 15~ 3,1o m~o Sl(mto - 4) /31 = 3 ~ ' 3 , p m o s t ( m ~ -
4) }
(2 20)
Numerically, with Sl = 50~2, m 0 = 765 MeV, I"o = 135 MeV, mlo = 1264 MeV, Pro = 150 MeV, we obtain /30 = 1 0 4 , 131 = 0 . 5 3 , and we shall then choose m the following/3 o = 2/31 = l 05; (w) P(s, t) is the c o n t r l b u t m n associated with the pomeron exchange We need only hn P(s, t) fi~r s > s~4 and we choose h n P(s, t) = s~J ,
where o can be hnked to the asymptotic total cross secuon. As we shall use the Sand P-waves only up to s ~ 60, we must take SM ~ 60, whmh hes between the O and fo resonances. We then shall adjust o m such a way that the absorptwe parts gwen by the S, P and D o waves lore approxmlatlvely thmr Regge form m the vlmmty of s ~ sxt. This gwes valucs of o m the range 0.05 ~< o ~< 0 11, aJld we have chosen the mean value o = 0.08, which gwes a good interpolation of A () and A I between the f(, and g resonances and an asymptotic total cross section ol 26 mb Using the preceding estmaates, It appears that for s' --+ + AAS
30S'
, ,42~s(~,o)--~
+31 ~/ Sl
, { co,' -/3, o,,~-7;31
,
,/.71 ~, ! ~, j-
512
B 13onnter, P. Gauron, Cros~tng symmet O'
tlL
'20
Ilg I. Ihgh-encrgy componentsf/~(s) of the S- and P-waves. ('ur',e', A, B and (" represent f(}~, f~ and f~l re,,pcctlvcly The corresponding asymptotic contributions t l ( S ) to the S-and P-waves are drawn on fig 1, for 4 < s ~< 40. It appears that they are neghglble for s ~< 20, but become quickly increasing. These qualitative considerations remain true when we vary the high-energy parameters a little around the given values. For subsequent use of relations (2.14),(2.16), we must then point out that the uncertainties associated with unportant asymptotic contributions restrict their range of applicability to s ~< 40 (850 MeV).
3. Test of the crossing properties of .,~me models In the previous section we have given an estimate of J / ( s ) Now, If we have a set of S-and P-waves on the physical region, we can use their imaginary parts in relation (27.17) to obtain as an output their real parts, given by the crossing properties of the 7rTr amphtude. But, on the other hand, we have these real parts as an input since we can compute them directly through the elastic umtarity relation. This is the crossing test already proposed in ref. [10]. Moreover, relations (2.14) and (2.16) enable us to calculate the analytic continuation oil 0 ~< s ~< 4 of any set of S-and P-waves and consequently to check the conrpatiblhty of these phase shifts with the Martin posltivity constraints [4] on tile non-physical strip. In fact, the coincidence between input and output in each isospln is restricted to tile elastic strip, but it carl in practice be extended to s ~ 36, and we reach again the limitation s ~ 40 found ill tile previous section. We have then tested, using the asymptotic contributions given sect. 2, some sets of S- and P-waves available in the hterature (a) The Bonnier-Gauron model [2]. (b) The Le Guxllou-Morel-Navelet inodel [3]. (c) The Kang-Lacoinbe - Vmh Mau model [13]. (d) The model-independent phase shifts of Morgan and Shaw [14] (e) The experimental phase shifts of Baton, Laurens and Reignier [15]. The first three models are built from the elastic unitarity, the correct s = 0 threshold behaviour and the p- meson's existence. They satisfy the five S and P crossing sum rules of Balachandran-Nuyts-Roskles and Martin's constraints on the non-physical strip llowever these conditions are fulfilled with different parametrxzatlons, es-
B Bonmer, P. Gauron, Cro~smg O,mmetry
513
-150"
o
12.8
,
,40 S ......-~
Fig. 2a. S-and P-wavesphase shifts selected m model I using eqs. (2.14) and (2.16) (ao = 0.18, a2 = 0.08). Bars give the hmlts of the imual set. peclally case (c), which is an unitanzation of the Venezlano amplitude. The physv cal pre&ctlons of these models are m rough agreement (0 a o - meson is found m all cases, but with different mass and width, (li) the models (a) and (b) agree for the low-energy I = 0 S-wave behav]our and the models (b) and (c) for the I = 2 one. These amplitudes being elastic unitary, eqs. (2.14) and (2.16) are used as a test of their crossing symmetry properties on the physical strip. The phase shifts and scattering lengths of case (d) are computed m a model rodependent way throughout ftxed-t dispersion relations and their derwatwes with respect to the m o m e n t u m transfer, both evaluated m the forward &rectlon. In case (e), we have only the phase-shifts and we have to search the scattering lengths a 0 and a 2 which provide the best fit to eqs. (2.14) and (2.16). The results can be seen on figs. 2- 6. The first conclusion which can be drawn is that eqs. (2.14) and (2.16) correlate tightly the scattering lengths and the phase shifts, more tightly than the usual disper-
514
B Bonnier P (,auron. Crossing4 s y m m e t r y
kJ _
~.8
_
,:~6
_
| l g 2b Resultsot thccrossmgtestonthephaseshfltsoi f,g. 2a ('ur~esUaretheumtarlty hmits± } . v @ ( s - 4 ) Curvesl~mputs) are the real parts of the partlal ~aves m each lsospm taken from tile model ( urves O (outputs) are the real paris of the partial ~,aves given by eq',. (2 14) and (2.16) slon relations or the Balachandran-Nuyts-Roskles relations For example, on fig. 6, beside the best fits given by a 0 = 0 22, a 2 = - 0 036 for d o w n - d o w n (511and c o = 0.20, a 2 = 0.04 tk)r down-up 6'c~, one can see the effect of changing a 0 and a 2 from the preceding values to a 0 = O. 16 and a 2 = 0 04. As a consequence, better results can be obtained, especially in case o f model (c), if the scattering lengths are shghtly deviated from their orlglnal determinations by the Balachandran-Nuyts-Roskles relations. A numerical o p t n m s a t l o n on the scattering lengths can be done, which leaves the phase shifts practically unchanged, but which is sufficient to bring an i m p o r t a n t m l p r o v e m e n t on the fulfilhng o f eqs. (2.14) and (2 16). For example, m m o d e l (c), it we pass from a 0 = 0.186 t o a 0 = 0.15, the soltltlOn 12) gives results comparable to those o f models (a) and (b) (fig. 4). It appears also that some solt|tlons roughly consistent with eqs (2 14) and (2 16) can be selected in each set o f phase shifts. Particularly, ~t seems that the up-down ambxgulty in the I = 0 phase shift cannot be removed. But we lack definite conclusions. It IS then useful to generate and study a more c o m p l e t e spectrum o f solutions. This is the purpose o f the following sections, where we build a model especially stated for the Frolssart-Grlbov representation o f the partial waves.
515
B. Bonntcr P. (,auron, Crossing ~vmtnetrr
'~
/
gl
~°
g
S.-~
g~
fig. 3a ( c n t r a l s o h m o n o I model 21a0 =1) 2116 a2 =
(I.0731 B a r ' , g l v c t h e h m l l ~ o t t h c m l t l a l
4. Model-dependent fit of the crossing relations on the physical region In expressions (2.14) and (2.16) we need a representatton of Im/l(S) on tile physical region only. However, we shall see that a reahsuc parametrl,,auon of hnj'/(s) must take mto account general propemes offl(s ) in the physical and unphyslcal sheets.
4 1 Analyttc pn)perttes ofp - 1 (s) lm fl(~) Let us define [161 the functions G/I~)by
Gl(S)
= {Jl(S)} 2 {1 +
2,p(s)fl(s) } 1
m such a way that, for s >~ 4, elastic umtarity Imposes
('4 1)
B Botmter, P (;auron, Cros~tttg ~)'mmetrv
516
Fig. 3b. Results ol the cros',lng test on the phase shifts o f llg. 3a.
150e
_ 50"
t 16
128
140
7_
g~
Fig. 4a. Phase-shffls p r e d i c t e d by m o d e l 3. The s o l u t i o n s 1 and 2 are b o t h a s s o c i a t e d w i t h the same s c a t t e r i n g l e n g t h s ao = I). 186, a 2 = 0,045.
B Bonmer. 1' (;auron, Crossing symmetry
-
_
_
517
L2_
t
o
•
~
, .....
~
..
I'lg. 4b, c, d, e Results of the cro~,~mg test on the phase shflts ot fig. 4a, b ~olul,on I, c ,,olutmn 2 ,d . solutmn 1 w l t h a o = 0 15,a2 = - 0 0 4 5 . e solutmn 2 v, lth a0 = 0 16, a2 ~ 0036
Im fi(s)
= p(s) al(s) .
(4.2)
Gl(S )
has the following propemes. (1) It is meromorphlc m the s-plane cut along the neganve real axis and it Is real for s/> 0 (positive for s >~ 4); (u) its poles can be associated wxth resonances (ff not too far from the real ax~s) and virtual b o u n d states. A resonance (SR, 7R) reduces two complex conjugate poles at s+ = s R -+ Z3'R. Such a pair of poles occurs in the P-wave, the corresponding residue being c o m p u t e d from 7R in the Brmt-Wigner approximation. We postulate the same situation in the 1 = 0 S-wave, b u t in this case the parameters of the poles and the residue are left free. A virtual b o u n d state occurs at s = s v (0 < % < 4) when V
/l(Sv)
= 21"
Sv
4 - sv
We know from the work of Martin (1 7) the existence of these poles at least m the partial waves I = 0, 1 t> 2. In fact, from previous models 12, 3, 1 3], It appears that the S [=2 w a v e has such a singularity m the vicinity o f s = 0. In this case the singular part of Gi(s ) can be written Fv Gv(S) = s - Sv ' w~th
5 18
B Bonmer P. (,auron. O'ossmg ~rmmetrv
[ 'SU i50"
I00 ° 8,'D
J
i i
up
!; 12.8
ill,
14o
s -...~
A
I ]g. 5,]. Ph,]se-shHls s e l e c t e d b y M o r g a n ,rod Sha~,. I l h e best results u n d e r the c r o s s i n g test are airways o b t a i n e d w i t h the Lurve B for ,5 2)
I'v= ~
Sv
[4
sv--
t#
4(4
sv) d f(s)
(4 3)
,v
We shall then suppose such a singularity to occur in G2(s ), s v and I"v are free parameters to be computed by self-consistency, as will be explained later. (in) Gl(s) has known threshold behaviours at s = 0 and s = 4, derived from the behaviour of J/(s) at /he same points • Gl(S) = ai2 (4I (s -- 4) 2/+ O((s
G/(O
= - ~,
x/~Jl(O )
s i-(;
4)2/* I ) ,
+O(sx/-s)
(4.4)
(iv) GI(S ) has double zeroes when [l(S) has a zero on the non physical strip As such zeroes ~ e l n systematically Rmnd m the S-waves [2.3, 13], we shall suppose a priori their existence, but their location has to be computed by self consistency
B Bonnwr, P Gauron, O'ossmg symmetr;
4
-
LI~-
laO
5 19
,~8
,3G
f
I ~g 5b, c, d, e. Result', of the ~.ros,,mg tcst on the phase shflts e l ttg 5a b. ,,olunon UI). c solun o n BL 1, d. ,,olutlon BD.1. c ~,olutlon BI).2.
4.2 Practical use oJ"the eqs. (2 14) and (2.16) In the following section, we shall use tile analync properties of Gl(S) to propose a parametrlzatlon of these functions vahd in the whole cut s-plane. The freedom allowed in these parametn/ations (scattering lengths, pole a n d / e r o parameters) is to be restricted according to the following programme: For each set Gl(S), I =- 0, 1, 2 through (4.2), eqs. (2.14) and (2.16) give/l(S) for 0 -%<~ ~< 4 and ReJl(s) for 4 <~ .s <~ 40. But, on the other hand, (4.1) gives alsofl(s ) for 0 ~ ~ ~< 4, and elastic unltanty gives Re f/(~) for ~ ~> 4. These two representations o f / l ( S ) induced by the same GI( 0 must be ldenncal for ~ t> 0, then (l) We can compute by self-consistency all the parameters linked to tile non-physical region (zeroes and pole parameters), since a priori they appear m different manner m the two representations. (n) After this prehmmary ehmmation, the remaining parameters of GI(,O are fixed by mlmmi/mg the discrepancies between the two representanons of R e / i ( s ) on tile whole strip 0 ~< ~ ~ 40.
4. 3. Parametrtzatum of the/un~ ttons Gl(S) After having relnoved t i l e / e r o e s and poles o[ Gl(S ) we can expand the remaining part of Gl(S) in a power series in the variable ~"(~) which maps onto the u n i t circle the s-plane cut along the real negative ax]s
520
B ]]()littler, 1) GaltrOll, (ro351tl,t~ s v m m e l r v
,._.tSO °
~ up
g:
~g DOWN
50"
116
~Z~,
,40 S.-~
| lg 6,1 Central e\pcrtmental phase shifts of Baton, l aurens and Rmgnler.
r(s)
_V';- 2
(4.s1
v¢÷2"
According to the results of paragraph (4.1), we have chosen the representations
c,o(s ) = (s-s012 {(so --st 2 +(s--4/ ~0
Gl(s t = (~--412
{(so-s)2+ (Ss4) y
0"2(8t = (S-S2 t2
{S
5
' 2 11/' s1,
:}5
It : 0
-I ,,~=o g'l r"(s),
4
Sv}
2 g2 e'(s). tl = 0
In these expressions"
(4.6)
B Bonnier. P (,auron, O'osstng wmmetrv
s
,
I
521
[b
f
°
o
~
f Fig. 6b, c, d, e. Results o! the crossing test on the phase-shifts of llg 6a, b best fit tot downdown,h°(a0= 0.22, a;b= 0036),l,.bcst fit for down-upS°(ao = 0.20, a2 = 0 0 4 ) . d solution down-do~,.n for 6o with ao = 0 16, a2 = 0.04, e. solutl~ down-up for 6~ ~lth ao = 1).16, a 2 =
-
0 1)4
9
s o and 70 = mol'o/m~, sp and "to = real o/nl[r are the masses and widths o f the oand 0-mesons respectwely, s o, s 2 are the zeroes of the S-waves, s v is the wrtual b o u n d state pole. All the three are c o m p u t e d by self-consistency. The four first coefficlents,~/~tn each Isospm are fixed by the threshold conditions at s = 0 and s = 4. The remammggZ/are hnked to the residues at the poles. Then g42 is c o m p u t e d from I"v (eq. (4.3)), g l and g~ are determmed by lmposmg the mass (765 MeV) and width ( 125 -+ 15 MeV) of the o-meson, g~ and g~ either are free, or fixed as funct lon~ of one free parameter Sl) > 4, the value of s where we tmpose elastic u n l t a n t y of /0(s) together with the passage through 90 ° of the phase shift. Fwe (or stx) parameters remain free. ao, a2, so, 9'0, together with ~1) (or gO and
gO).
4.4. Computattons and results As has been explamed in subsect 4.2, we fix these parameters by the m i n m u / a tlon of the relative ×2 defined by
522
B. l J o t l t l l e r .
P
{;aur¢Jn.
¢. )¢Jsstn
t l'.fo"
_iN*
I f I
S.___~
I
| It: 7,1 ()LH Jli'~d ~,C| Ot ~,tl|l.lllon~
rncnt. 1 1 ",pcrmlcnt,d
[).l~,hL'd ,.LIr'~c'~ rcpre,,Cll! the.' ,,oiUtlon~, ~,L'],..'ct e d |:,~ [h0 c \ p c r l -
data f r o m rcl
,c': Z: Z: /=tl \ :It
[15}
t I
\pe[llncnldl tl,lta f r o m r c f [21l
--
-' .
where Jtt'(s x ) a n d / i f ( v v) ,ire the real pa, ts of the amphlt.aes J[(~) taken ai = ~x (0 < s \ < 40), and c o m p u t e d from (;l(S) through the elastic unitarlty relatum (or ec1 (4 5) for 0 < s < 4) and through eqs (2.14) and (2.16) respectively. Two sets of solutions are found, for whirl1 Ill each i,,,ospm the uintaHty ratio R [ = I(1) ~ /1t~,)/-1It I is always less than 2 × 10 with lwo exceptions (i) set I R 1 can reach 5 X 10 2 for 35/J 2 < s < 40~ 2. (n) set II R 0~.,tn reach 3 X 10 2 t o r 10~ 2 < s ' < 16~ 2. The first set (fig 7) p, obtained with lhe following values of tile parameters 13 5 ± 3 51a2 ,
a0=0
20:t:0.06,
<,<,=
a~ =
0 0 8 ± 001~,
1" = 4 9 5 ± 170MoV,
ao a
=
2~)±14.
523
B. Bonmer. P (,auron. Crossmg symmetry
U
-
_05
_
=D
i ~6___
u L
f
-
~
-
F,g. 7b. Results of the umtarlty and crossing test on the central solutmn of fig. 7a The 1 = 0 S-wave phase shift 60 xs of the up-down type and hes m a rather wide band, since It can vary from 65 ° up to 100 ° at the 0 position. The corresponding I = 2 Swave phase shift 602, small and negative, increases m magnitude when 6~ does, passmg from - 2 1 ° up to - 29 ° at the ,o-mass. The P-wave phase-shift 611 remains the same on the whole set and corresponds to the following values of the scattering length a 1 and the width Fo: a I =0.044+-0004, Fp =!35
+5MeV.
We have c o m p u t e d the D-waves from thmr real parts gwen by the Fro~ssart-Grxbov representation. They are very small, positwe for I = 0 and negative (except sometimes m the viclmty of the threshold) for I = 2. T h e y reach respectively + 3 ° and 1° at the ,o-mass and have scattering lengths d I -
d o = 0.0021 -+ 0 . 0 0 0 3 , d 2 -- 0.00025 + 0 . 0 0 0 2 5 . On the non-physical strip 0 ~< s < 4, the p o s m o n s o and s 2 of the zeroes o f the two S-waves are: So= 1.6-+0.7, wxth 4 ~ o + 5 ~ 2 = 12 +-O.Ol
32= 1.1 -+0.5,
B Bonnter. P GauroJt. Cro~stng symmetry
524
150"
J
tItLttlt /
O0 °
SO*
S-->.
Fig ,~,1. Our s e c o n d ,,el of s o l u t i o n s
The second set (fig. 8) is less satisfactory than the first one from the p o m t o f view of the umtarity. It is obtained with the following values o f the parameters a 0 = 0.137 -+ 0 . 0 1 3 ,
s o = 31 -+ 8~ 2 ,
a,~=-0.055-+0.005,
1~o
-
IO00-+200MeV,
ao
--
2.5+_0.5
a2
The 60 phase shift is of the, up-up type and passes through 90 ° at 20/.t 2 < s < 30~ 2, 0 but the associated pole IS very far trom the real axis. It must be stressed that Go(s) b e c o m e s negative and violates badly the posltwlty o f hn ]o(S) for s ~ 45. A l t h o u g h this region is out of the range o f vahdlty of our model, such a thing throws some suspicion on these solutlons. The 60 phase shift is s o m e w h a t smaller m magnitude than in the preceding case. The P-wave is agam stable but corresponds now to a I = 0.0335 -+ 0.001 , F /9 = I I O M e V .
525
B Bonnter, P. (;auron, Crossing symmetry
/
,,
I Ig 8b Results ol the unlt,trlty and crossing lest on the central solution o f fi,, 8d (a~ = 0 138, a~ ~ 0 06).
The D-waves are unchanged and the zeroes of the S-waves on the unphyslcal region are s o = 1.4 + 0 . 3 ,
~2 = 1.25 -+ 0 . 2 .
Some comments can be made about these results. First, we remark that the updown ambiguity of 6 c)above the p-mass cannot be removed by cqs. (2.14) and (2.16) and elastic unttarity this ts m agreement with the conclusion of sect. 3. Below the p, 8fl~is of the up or between type. Secondly, we can try to discrmunate atnong all the solutions which are obtained. For this purpose, we have chosen for the tmle being the expertmental point of view and we have checked how these solutions fit (a) the cross secttons o ( n + n - ~ n + n - ) , o ( n - n 0 ~ n n O) (ref [15]) and o(n+n ~ n%r °) (ref. [18]), (b) the ratio [19]: R 1 = o ( n + n - ~ nOnO) l o ( n + n + -+ n + n + ) , R 2 = o(n+rr
~ n ° n O ) / o ( n + n - -~ n+n -),
(c) the determination of 60 - ~511at low energy froth the Ke4 decay [20] The three tests rule out the second set of phase shifts and reduce appreciably the first one. The remaining solutions can be seen on fig. 7. The corresponding values of the parameters are.
526
B Bonmer, 1: (;auron, ('rovsmg 9i'mmetr~:
a(~ = 0.23 -+ 0 . 0 3 ,
~ = 1 3 4 + 1 2~ 2 ,
s()= 1 0 4 ± 0 . 1 4 .
a 2 = - 0.067 + 0 . 0 0 3 ,
I" =495 ±75 MeV,
s 2= 1 55 - + 0 . 1 0 ,
d(1~00021
d~ ~ 0 00027
with a I = 0.0448 -+ 0 0 0 2 ,
,
0
The vahte h n ( m K) h ~ ( m k ) turn~ out to be 57 ° + 5 ° We can point out that tile final 8{)) phase shifts are in good agreement with the experimental resuhs of Malanmd and Schlem [21] and with the conclusions o f Scharengulvel, G u t a y and M i l e r [22] from the study o f the forward-backward asynmletry They also agree pertectly with the t h e o t e u c a l d e t e r m i n a t i o n of Iv ( h n l h m , blorel and Navelet 131 over the whole energy range and wtth our preceding one 121 over the low-energy region
5. Conclusion In tile first part of thl~, work, we have shown that the models fulfllhng the S- and P-crossing relations of Balachand, an-Nuyts-Roskles do not satisfy systematically the crossing relations on the physical regmn. In this respect, these last relations appear to be very selective But on the otl,er hand, and this is the second part o f this work, they lead to a new representation of the partial waves which alh)ws a wide class of solutmns In particular, the crossing relatums on the physical region together with elastic t l n l t a r l t y a n d p - m e s o n e x i s t e n c e does n o t p r o v i d e a n e w d e c i s i v e test to d i s c l l t l l l n a t e between dlfterent predictions hke the up-down ambiguity m the d e t e r m i n a t i o n o f 8 0o above the p-mass. C o m p a r e d to some prehmlnary hopes [10], this tact is disappointing and give a c o n f n m a t l o n of the role of ttle p a r a m e t r l / a t t o n m the tlnal result. Ill fact, in this scheme, as we do not COlltpttte exact so[utloils, the results seem to be very sensltlve to the criterion used for their selection If we require the consistency between nlput and o u t p u t (In the sense o f sect (4.4)) on the physical region only, oul paranletrl/atton allows a largm ,,pectruln ot solution ~, 12~;1 " We have chosen to take into account as tar as possible the constraints reduced by the elastic u m t a n t y and crossing. I his had led us to use the analytm c ( m t m u a t l o n of the functions G/(a) oil the unphyslcal strip. We are then able to require the consistency between input and o u t p u t not only on the physical regum, but also on the unphysical one. In particular, the v ~ 0 threshold behavumr can be put into the game, as m our previous model [2] * IIo'a ever. v,c CdlltlOl o b l d l n aS large a ,,pe~.Irum ,is the (}lie o|'lldlned b v ]~dsdcvdnt, [ nL~egat and Pcter,,en [23 l, o w i n g to o u r mlHal I1~ pc~thc'qs dimout the d l , , t n b u t u m of the / t r o t , , a n d virtual b(~ulld M,tte,, In the S w a v e s It c,lll hi) ,,.ho~,n that otlr LhoI(C, LonMxlelll v. lth the rcsult~ ot rcls I2, 3 13 J. restrict,, a przort the s c a t t e r i n g length,, m the range 0 - at) ~ (I 35 a n d - () 15 ~ a 2 "-. (I \t,e .arc g r a t c l u l to Dr la;asde',ant lot ,t clarfl,,In'a dlst.u,,,,um un Ibis s u b l c t t
527
B Bonnu'r, P. Gauron, (.~'ossmg ~vmmetrv
In o r d e r to t a k e i n t o a c c o u n t s o m e r e s u l t s o f t h e s e m o d e l s [2, 3], we h a v e also IiIlpO,,cd a prl()tl s o m e g e n e r a l l e a l m c s like a 2 n e g a t i v e F v e n m t h , s r e s t r i c t e d t r a m e w o r k , we h a v e f o u n d a w i d e class o f s o l u t u m s , b u l w i t h ,t COll)inon l o w - e n e r g y b e h a v l o u J a n d In g o o d a g r e e m e n t w l l h s o l n e p r e d i c t i o n s p r e v u > u s l y n),(de [2, 3], t h o u ~ } t h e e l a s t i c u m l a t ) t y a n d c r o s s i n g are u n p o s e d m a very different way m tins work and m ref
12, 3]. T i n s s e e m s to u s a v e r y e n c o u r a g -
111g (dOt I l o w c v e t , tt a p p e a r s also t]lal we need s o m e e x p e r m m n t a l
i n f o r l l l a t l o n s 111
a d d i t i o n to tlae p - m e s o n ill o r d e r to select a u n i q u e class o f p h a s e - s h l t Is t o t t h e 280 850 MeV range. We a,e h a p p y to t h a n k P r o f e s s o r R V m h M a u a n d Dr. L L u k a s / u k f o r h e l p f u l dtsc uqMons
References Ill 12t 131 141
(, Aubcrson, O. Ihgucl and (. ~3,andcr,,, I'h', ~ t otters 28B (1968) 41. B l,~onrncr ,rod P (,auron, Nuul. l'h',', B36 (1972) 11 I ( 1 c (,ull]ou, A, Morel ,rod II. Na'.clct, Ntltp, o ('lmcntt) 5A (1971) 6'r,9 ,Set' lor CXdlllplc (a) A ,Martin ,rod I ( 11cun~, '~ndI.VtltlI 3 properties ,,'Jd bt)ur~tls ot the ',tdtlCllll~' ,tmplitudc% I.)o~.umcnts on Modern Ph', s~t s ( 1971 ).
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[21] I. \ 1 a l d m u d a n d P Schlem. Proteedmg~ of the ,~rgonncConl o n r r ~ a n d Krrmtcracnon~, Argonne, 1969 [22~ J II Scharenguwel, LJ Guta~ and 1) \1 Mdler, Nucl. Phys B22 (1970) 16. [23 ] J L Ba',devant. C D I roggatt and J / Petersen. CI RN preprmt 1 I| 1518, 1519 (1972)