spin observables at threshold for the reaction p-barp--\u003e lambda -bar lambda

$: Spin observables at threshold for the reaction p-barp--\u003e Lambda -bar Lambda$
PHYSICAL REVIEW C VOLUME 44, NUMBER 5 NOVEMBER 1991

ARTICLES

Spin observables at threshold for the reaction pp - AA

Frank TabakinDepartment of Physics 8 Astronomy, University of Pittsburgh, Pittsburgh, Pennsyluania 1$g60

Robert A. EisensteinNuclear Physics Laboratory, Department of Physics, Uniuersity of Ilhnois, Champaign, Illinois 61820

Yang LuDepartment of Physics 8 Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania XM60

(Received 3 June 1991)

The observables measured in two-body reactions, such as cross sections, polarizations, spin corre-lations, and the singlet fraction, take on simpler forms at threshold since only a few partial wavescontribute. A study of these observables at the pp ~ AA threshold is made in this paper. Our resultsare compared to the recent LEAR PS185 measurements, and several interesting angle-dependentfeatures of the data are seen to originate from normal threshold behavior. Using general thresholdproperties, some recently proposed physics models of this process are strictly tested in this energyregion. Several constraints on the transition amplitudes are deduced and excellent fits to the dataare achieved using a. simple scattering length approach. We find that S-wave amplitudes, S-P inter-ference, a, strong tensor interaction, and P-wave splitting all play appreciable roles in the reactionnear threshold.

I. INTRODUCTION

The pp ~ AA experiments [1] at the low-energy an-tiproton ring (LEAR) and elsewhere [2] have provideda wealth of information on the reaction cross sections,polarizations, and spin correlations for the AA produc-tion process. Results for the reaction pp —+ EA have alsoappeared [3], and other experiments studying processessuch as the pp ~ ZZ and pp ~ PP reactions [4] are inprogress. All of these cases involve strangeness produc-tion via the creation of one or more ss pairs. Recentprecise and detailed measurements of cross sections andspin observables, performed extremely close to thresh-old [1], represent an excellent experimental opportunityto shed light on the basic ss production mechanism atlow energies. The data have revealed some systematictrends which may provide helpful guidance to a betterunderstanding of this physics.

The most recent AA data [1] seem to indicate veryinteresting phenomena in the threshold region. These in-clude what is apparently a significant contribution due tohigher partial waves (E ) 0) all the way down to thresh-old; this behavior is seen in both the differential and totalcross sections. In addition, the polarization data show across over from positive to negat, ive values at an approx-imately constant value of the relative momentum trans-fer, t' 0.2 GeV/c, irrespective of the incident beamenergy. The polarization then becomes sizably negativeand remains so until returning to zero at 180 . Finally,

the experimentally observed [1] singlet fraction is essen-tially zero, indicating that the AA pair is always pro-duced with spins aligned. We know of no reason basedon fundamental symmetries that this should be so; itis apparently due to the reaction dynamics, notably thepresence of a strong tensor force in the pp I = 0 channel[5, 6] at these energies.

It is clear that to understand the new pp reaction datafully, we will need to ascertain the proper annihilationmechanism, determine the coupling between the various

pp reactions, and simultaneously extract the detailed spindependences of the basic interactions. In achieving thesedificult goals, systematics of the data such as those men-tioned above play a key role.

Recent theoretical calculations have ranged from t- ands-channel meson exchange analyses [7—17], to simplifiedquark-based mechanisms [18—25], to coupled-channelsapproaches [13—17]. The dominant physical effect in allof these calculations, as in the data itself, is the fun-damental annihilation process. In all cases annihilationhas been included phenomenologically using either opti-cal potentials or effective channels" methods. Incom-ing and outgoing waves are attenuated using either theeikonal approximation, distorted waves, or by flux trans-fer to coupled channels.

Provocative explanations for some of the observed be-havior have been offered [13], including the possible ex-istence of resonant behavior at or below the reactionthreshold. Also, total suppression of 9 waves via annihi-

1749 1991 The American Physical Society

FRANK TABAKIN, ROBERT A. EISENSTEIN, AND YANG LU

lation has been advocated as a mechanism for enhancingP-wave effects [26, 27]. However, before such explana-t;ions can be firmly endorsed, it is important to determinejust what is to be expected in reactions of this type un-der "normal" conditions; that is, simply by virtue of thefact that the reaction is taking place at or near thresh-old. That is the purpose of this paper. We present herea straightforward analysis of the general nature of spinobservables at threshold, focussing on the AA data.

After a brief overview of the pp interaction, stressingwhat is to be expected due to the strongly absorptivenature of the reaction in this energy range, we proceedto the threshold analysis. Expressions for the cross sec-tion, polarization, and spin correlation coefficients for thepp ~ AA reaction are reviewed. Our basic references arethe papers by Tabakin and Eisenstein [7] (TE) and byTabakin [8], and other work cited there, which developedthe relevant formulas in the helicity basis using the phaseconventions of 3acob and Wick [28]. Those studies useda density matrix approach to perform the ensemble av-eraging.

Since the physics of the threshold region is more read-ily apparent using LS states', we transform to that basis.Then, using the basic characteristics of threshold behav-ior, the possible angular dependencies in this region areextracted. Those forms lead to some general conclusionsconcerning the basic amplitudes at threshold which areshown to be consistent with what is observed in the re-cent data. A scattering length approximation is thenintroduced, and amplitude fits to the data are then pre-sented and analyzed in light of the general rules deducedearlier.

Many of the threshold and scattering length ideas weadopt were developed in classic references such as Wigner[29] or Blatt and Weisskopf [30]. One might reasonablyask why it is of current interest. There are several rea-sons: first, the relevant formulas for the behavior of spinobservables near threshold are not generally available inthe literature, although limited studies of the spin ob-servables in the pp ~ AA reaction [27, 31] and others [32]near threshold have appeared recently. Second, to ourknowledge no experiment has ever been done, on this orany other reaction, which has been able to determine spinparameters so close to threshold with such high precision.This is due entirely to the extraordinary capabilities ofthe LEAR facility and to the self-analyzing feature [33]of the pp ~ AA reaction.

tons from a nucleus, in a strongly absorptive situation,and the resulting proton polarization. Both the scatter-ing and the polarization are strongly dependent on thegeometry and the strength of the absorption.

An important feature of strong absorption is its abilityto attenuate wave functions. For elastic scattering, withthe strong absorption described by an optical potential,the product of the attenuated wave with the optical po-tential, Vy@', remains finite even if the optical potentialbecomes extremely large. From that viewpoint, or froma T—matrix N jD approach, one can deduce that even anextremely large optical potential leads to a finite elasticscat tering amplitude.

In contrast, for a reaction channel such as AA, whichis not the main cause of flux loss, the amplitude vanishesin the optical model for strong annihilation. Therefore,it is theoretically possible to have a total suppression ofthe S waves in AA production due to strong annihila-tion. Whether that occurs or not is, however, a dynam-ical question, which can be partly addressed by analysisof the data, as we proceed to do.

A conceptually correct theory of annihilation is pro-vided by coupled-channels methods. One characteristicof a coupled-channels approach is that 9 waves are notattenuated as much as for a local optical potential treat-ment, because the departure of flux takes place over anextended region due to the essentially nonlocal natureof a coupled-channels approach. In comparing the opti-cal model to the coupled-channels method, we see thatthere is a deep connection between S-wave suppressionand the method used to describe the basic annihilationmechanisms.

Whatever method is finally adopted for calculation,the results will depend in an essential way on the under-lying NN force. In particular, because the tensor forceis expected [5] to be strongly enhanced in the pp I = 0channel, we anticipate that the pp ~ AA reaction maybe a good place to study its eAects. Since it is well knownthat spin-orbit and tensor forces play an essential role indetermining spin observables, we expect such observablesto be a sensitive indicator of the presence of these com-ponents. Indeed, it has already been suggested [1,6] thatthe strong attenuation of the singlet fraction is due to thepresence of a strong tensor force. Another such indicatorwould be firm evidence for significant P-wave splitting,which also arises due to these forces, in the data. Weshall see below to what extent such splitting occurs.

II. GENERAL REMARKS ON ANNIHILATION III. OBSERVABLES

Before describing the amplitudes and the associatedobservables for the pp ~ AA reaction, let us note somegeneral aspects of such strongly absorptive productionprocesses, which have been studied extensively for nu-clear scattering and reactions. Indeed, the AA pro-duction is analogous to the double excitation processA + A ~ A + A', where A denotes a nucleus. Rulesrelating such reaction cross sections to the elastic scat-tering have been extracted [34] and similar ideas shouldapply here. Furthermore, those calculations were able toshow a direct link between the elastic scattering of pro-

It is well known that the helicity basis is a more naturalway to describe high-momentum initial states because itis both relativistically and rotationally invariant. It al-lows expression of transition amplitudes and their asso-ciated diagrams in a compact and convenient way. Theprocedures for carrying out a calculation in this basis arewell described in the literature, and have been applied tothe case of interest here in Refs. [7, 8]. However, sincethreshold behavior is mainly characterized by centrifugalbarrier effects, we wish to express the pp states in the L Sbasis and then examine the allowed angular dependence

SPIN OBSERVABLES AT THRESHOLD FOR THE REACTION pp —+AA 1751

of observables. Of course, the two bases are equivalentand are linked by a unitary transformation. We indicatebelow how to express those previous calculations in theL,S basis. Readers wishing to move directly to a compar-ison with the data are referred to Sec. III B.

A(0) = —,' ) (2J + 1)(A + B ) d„(0)

= &{+-;+-),

A. Amplitudes and observables8{0)= ~ ) (2J + l){A —B ) d, , (0)

J

We begin by expressing the pp ~ AA helicity ampli-tude in the center-of-momentum frame as

= -&(+-;-+),

C(0) = ).(2J + I)&' doo(0)

{0I&ln) = (A) A21&(0)l»A2) = &(AiA'»' A~»)

(3.1)

where the initial pp and the final AA momenta and he-licities are denoted by p, AqA2 and q, A&A2, respectively.The angle 0 is determined by cos0 = q p. The pro-jection of these helicity basis states onto eigenstates oftotal angular momentum can be accomplished using theexpressions

= &(++;++)+ 7 (++;——),

'D(0) = ) (2J + 1)D d, o(0)

= —»(+—;++),

S(0) = ) (2J + 1)E d, o(0)

(3.5)

2J+ 1iPAiA2) = ) iP JM A)A2) DM~(P') (3 2)

= &(++;++)—&(++; ——),

&(0) =) (2J+1)F' d'„(0)

all d

lq&|&'2) = ). Ie~~Ã&l)/ 4 &~~(i)J,M

(3.3)

(P i2 in) = ) (2J + l)(q A'A'i7 ipA Ag)

"&~~ (& '&) (3 4)

Here A = Aq —A2 and A': Ay A2 and D denotesthe Wigner functions [35]. The label J indicates the con-served total angular momentum for the final and initialstates. Using rotational invariance to remove the M de-pendence of the matrix element, and taking the initialbeam in the z direction (i.e. , p = z), the amplitude sim-plifies to

= 2 &(++;+—).

The right-hand sides of these equations are corrected ver-sions of Eq. (15) in TE. In these expressions the func-tion dMM, (0) is the Wigner "reduced d function" as de-fined by Brink and Satchler [35]. The amplitudes A(0)and C(0) + Z(0) describe no helicity flip, 17(0) and X(0)describe single helicity flip, and 8(0) and C(0) —f(0)describe double helicity flip. The quantity F(0) rep-resents the singlet-state amplitude (i.e. , So, 'Pq, Dp,etc. ), while the quantity A(0) + 8(0) is the JJ triplet-uncoupled amplitude, and the quantities A(0) —8(0),C(0), 17(0), and X(0) are the (J—1)g, (J+1)g triplet-coupled amplitudes. Only 0 and T must vanish at thecos 0 = +I end points. In addition, 8 is zero at 0' andA is zero at 180'.

Note that IS states are eigenstates of parity, whereasthe helicity states are not, However, eigenstates of paritycan be formed from the helicity states in various ways.One way is to define a basis in JM ) as

wIlere T denotes the partial-wave helicity amplitude.The helicity basis offers the advantage of including theangle dependence completely within the Wigner D func-tion for the case of particles with spin. This can be ex-pressed, using the conventions and symmetries [8] dis-cussed in TE, in terms of the following amplitudes:

2) I I I 0 03) ~g 0 0 I I4)) M (0 0 I —I)

(3.6)

1752 FRANK TABAKIN, ROBERT A. EISENSTEIN, AND YANG LU

whereI + —) denotes, for example, the helicity state

INNpJMAiAq) with {Ai, Aq) ~ (+1,—1). The I1) and

I 4) states have parity (—1) +; whereas for theI 2) and

I3) states the parity is (—1) . The G parity ofI 1) is given

by (—1)~+i+I, and for the I2), I3), I4) states by (—1)~+I;recall that in our case I = 0. Therefore, the

I 1) and I4)states, although of the same parity, dilfer in G parity andhence cannot couple, while the states

I 2) andI 3) can.

The relation between the LS basis and the helicitybasis NN states can now be expressed using the aboveIn JM) eigenstates of parity. We obtain

( JJ(J —1)J

3(J + 1)g)

o o 00 ag bg 0 I2)0 b, —a, O

&0 o o1) E, I4)),(3.7)

The usual notation of 2s+ 1 L& is used. The quanti-ties a~ and b~ are equal to J(J + 1)/(2J + 1) and

«J J/(2 J + 1), respectively. A direct relation between theLS and the helicity states results from multiplying thetwo matrices above:

((J —1)J

3(J + l)J&L

o o ia, a,

t, o o

+-)i—+)++)—-)),

(3.8)

The singlet I ~ state involves a combination of no-Hiphelicity states; the triplet-uncoupled Jp states involvethe helicity A = +1 states; and the triplet-coupled (J 61)g states involve the helicity A = 0 and +1 states.

Using the above equations we find the following usefulrel at ions:

~~ = (1I T' ll) =?»~,

&~ = (2IT'I2) =

J(J + 1)+ [?L I, +?L, L, ],2J+1 2 g 0 (3.10)

I

A~ is the triplet-uncoupled amplitude, BJ and C~ arethe triplet-coupled diagonal amplitudes, and DJ and FJare the off-diagonal or noncentral amplitudes. For two-nucleon elastic scattering, using time reversal, we haveD~ —F~, but for the pp ~ AA case they are distinctlynot equal; indeed, we expect the TpD amplitude to dom-inat, e over the TDg amplitude at threshold since the finalAA state is less likely to be in a D wave due to the effectof t, he centrifugal barrier.

In terms of the above amplitudes, we obtain the fol-lowing expressions for observables such as the differentialcross section:

C, = (3I T' I3) = J J+13L «l +

2J 3L «I

gJ(J+ 1)[Tl LII +I?I )iL i]2J+ 1

(3.9)

where «IIs = 2 p~ + m = 2/q + m~ is the totalcenter-of-momentum energy and

&(0) = [I&+~l'+ I& —~l'+ l~l'+ I&l'+ l~l'+ I&l')

The total cross section is given by

3+12J+1

Og ——(2I T I3) = gJ(J + 1)2J+1 [?3I.' J T3I "J]

+ ALII LI2J+ 1

I(mpmA)' ) (2J + 1)&',

ps i (3.1 1)

with

~' = [l~' I'+ l~' I'+ I&'I'+ Io'I'+ I&'I'+ IF'I']

(3.12)

Fg ——(3I T I2) = QJ(J + 1)2J+ 1

[?3L,I Z—?3I"J].

J+1 ~ J T2J+ 1 2J+1

The A polarization is

P„(e)Z(9) = 2Im[(A' + 8")X+C"D],

and the spin correlations are

(3.13)

Eg = (4I T I4) =?jJg.

Here the subscript on 2 denotes the SLJ quantum num-bers. The coupling terms for the triplet I' = J —1 andL" = J + 1 states are labeled simply by the subscriptL'L". Here we see that EJ is the singlet amplitude,

c-{0)&(b') = [I&l'+ l~l' —I&l' —l~l' —4«(&'~))cry(b) ~(0) = [I&l'+ l~l'+ I&l' —l~l'+ 4«(&'~)l

(0) z(0) = [z(0) —2lc

and

SPIN OBSERVABLES AT THRESHOLD FOR THE REACTION pp ~Ah 1753

C, (8) Z(8) = 2 Re[(A'+ 8")X—O''D]. (3.14)

The expression for C, corrects a sign error in Eq. (22)of TE. Because parity is conserved in strong interactions,the quantities C &, C„,C&, , and C,&

are all zero, andZX '

The spin correlations can be combined to form the av-erage singlet fraction, given by

SF=(-,'(1 —~i &z))= -'(1 —C —C„y—C„)= l~(8) l'/&(8) (3.15)

where (1 —o'i rrq)/4 is the singlet spin projection oper-ator and F is the singlet amplitude. The singlet fractionSF ranges from one for pure singlet to zero for purelytriplet cases; experiment [1] indicates that SF 0 forthe pp ~ AA reaction. This expression has been evalu-ated in the Jacob-Wick coordinate system, which diAersfrom the one used in Ref. [1].

It is worth noting that for the purposes of data anal-ysis the above equations for the various observables canbe reduced to simpler expressions. We form the squaresof 0-dependent amplitudes and use the properties of thereduced d functions [35] to contract the resulting bilinearproducts. We find that the quantities do/dA, C ~X(8),C&& X(8), and C„Z(8)all behave like functions of theform

The j = 1 (vector) term is found to vanish due to therestrictions on the C,&

imposed by the conservation ofparity mentioned above. The J = 2 spherical tensoryields the following combination of the Cartesian spincorrelations:

[C + C„„—2C„],(3.20)

(~")&(8) = — -' [I(8) —4 l~l']

-2 l~l'+ I'Dl' —2 I&l'],

Again we take advantage of the restrictions due to parityin writing these equations. In comparing these expres-sions to the data, we take into account diR'erent defini-tions for the c.m. coordinate system. Note that (Cz+i)has the form of Eq. (3.17), whereas (C 0) and (C +

) fol-low Eq. (3.16).

These forms have simpler behavior at threshold thando the C,&. For later use in making the required ex-pansions, we write out the (C ) explicitly in terms ofamplitudes:

) aL, PL, (cos 8),L

(3.16)

while P&(8)Z(8) and C, Z(8) behave like functions ofthe form

(Cz+') X(8) = p 2 Re[(A' + 8')P —C"D],

(C+ )Z(8) = —[l'Dl + 4 Re(A'8)].

(3.21)

) ~1 PL(cos 8), (3.17)

involving the associated I egendre polynomials with M =1. The quantities aL, and bL, are combinations of thehelicity amplitudes A, . . . , I" appearing above, and arereal. The quantities P& and C~, equal zero at 0' and180' because of the PL functions. This behavior occursbecause at those angles the reaction plane (and thereforethe y axis) is not defined.

A simpler way to organize the spin correlations andthe associated singlet fraction is to form spherical tensoroperators C~M defined by

t JM[

(A) X (A)]JM

(Im, lm,ljI)~tA&~&;&,

t7l 1 fAg

(3.18)

[1 —4SF] =— [C..+C»+ C„].(3.19)

and to take the usual ensemble averages using densitymatrix procedures, just as is done in TE to obtain theC,~ listed above. Here we use the 0 operators in thespherical basis: oo —a, and oui ——p(0 + ioz)/~2.We find that the resulting expression for (C ) is simplyrelated to the singlet fraction:

Of course the (C"M), the C;&, and the amplitudesA, . . . , X are 8 dependent. Note that the singlet am-plitude Z contributes only to the (Coo) and hence to thesinglet fraction, and that the (C2M) values depend solelyon the triplet amplitudes. The (CiM) all vanish. Recallthat the amplitudes B and T are zero at cos0 = +1 andthat A(180') = 8(0') = 0. Thus (C ), (C ') and P„must also vanish at the cos 0 = +1 end points.

We prefer to study the above products, which wecall the "polarization profile, "

P&(8) 2'(8), and the "spin-correlation profiles, " C M(8) Z(8), because they are sim-ply related to the basic amplitudes.

B. Total and difFerential cross sections at threshold

Using the above results, we switch now to the LS basis.The conserved quantum numbers for the LS basis are L,S, j, I, parity, and G parity. The reaction pp ~ AA hasonly I = 0 transitions, but all singlet ( Lg) and triplet(sLg, s [j + 1]g) states can contribute.

Near threshold the final AA state contains at most afew partial waves of low L value. (The kinematic quan-tities relevant to the data to be examined are given inTable I.) Since j is conserved, only a few of the manypossible initial states will be present. Accordingly, werestrict ourselves to all LS states with j ( 1, plus the

P2 state (in order to include all P waves).


TABLE I. Kinematic quantities for the experiments reported in Ref. [1]. In order, the columns

are the incident laboratory momentum; the total c.m. energy; the incident particle c.m. momentum;the final-state particle c.m. momentum; the excess energy above threshold e = ~s —2m~, the

squared minimum four- momentum transfer; and the density-of-states factor p = 2irq(m„rnA) /psdefined in Eq. (3.24). The masses used to calculate these quantities are taken from the ParticleData Group [37]: m~ = 0.93827231 GeV and mA ——1.11563 GeV.

(GeV/c)

1.435 951.436 951.445 351.476 51.507 61.5461.695

~s(GeV)

2.23152.23182.23482.24582.25682.27032.3230

(GeV/c)

0.60380.60410.60680.61690.62680.63890.6846

(GeV/c)

0.01620.02560.06290.12750.16910.20970.3232

(MeV)

0.23640.58833.5460

14.52325.49539.05691.725

&min

(GeV/c)'

0.34520.33460.29580.23950.20950.18430.1307

100p(GeV)

3.7185.862

14.2928.2136.4843.8360.22

It has been suggested in one study [16] that the F2 ~Pp transition is dominant even at low energies. This

suggests the existence of a st, rong and long-range ten-sor force in their interaction. AVe do not include theF~ ~ P2 transition in our calculation, since our goal

is to test if a simpler basis is consistent with the presentdata. Hence we have only the following seven contribut-ing stat, es:

3p

Si + Di,3p

3pJ=0J= 1

J=2. (3.22)

The notation is as follows: both uncoupled and di-

agonal transitions, pp(2 +1L~) ~ AA(2s+1LJ), are de-

scribed by amplitudes with only one subscript, as in

T~g+qL, J. For oA'-diagonal, or noncentral, transitions ofthe type (L + l)~ ~ (L ~ l)g, the amplitude is de-noted in the general case by two subscripts. In our lim-

+pP2(z) + bP3(z)]. (3.23)

Higher-order terms in z = cos0 are missing because ofthe restricted basis. The coe%cients in this expressionare given by

ited basis t, he only coupled states are for J = 1: we write

T3D1 3s1 = TDs for the pp( Si ) ~ AA( Di) case, and

Tas1,3D1 = TSD ««he pp( Di) ~ AA( Si) case. Thisnot, ation incorporates angular momentum, parity, and G-parity conservation, which require that initial- and final-state J values be the same, that there be no mixing be-tween odd and even L values, and that t = (—I) + +

be conserved. In this discussion this means that only t,he

Si and Di waves can mix.The difkrential cross section for the pp ~ AA reaction,

using the above basis, has the form [see also Eq. (3.16)]:

do q~ (m„m ) [o:Po(z) + PP1(z)

dQ 2psp

cr =+ ITisol + I'T3POI + 3[173sil + I'T3Dil + ITDsl + ITsDI + I71Pil + I'T3P1] ]+51+3P21,

p =+«[v2(3&3Pl T3P2 2'T3PQ)(7sD+TDs)'+6+1P1Ti*so

+( 4 T3PO + s T3P2 + 3 +3P1) +3D1 + {2+3PO + 10 T3P2 + 6 +3P1) +3si ]&

and

7 = +2 (l&»il'+ l&»il')+ 6 l»Pil + 2 IT3P21'

+Re [(4+&Po + 9 +&Pi )T3P2 + 6 (+~D1 +351 + TDs +sD ) 3~2(+~D + +&s) +3D11

Two facts are of interest: first, we note that the isotropicterm o is n, ot limited to L = 0 amplitudes, as mightbe naively expected. The point is that with spin in-cluded the angular distribution by itself does not spec-ify the orbital angular momentum of the contributingstates. Second, since the cross section must be posi-tive definite everywhere, there are interesting relationsamong the coe%cients that can be found by examining

this expression at 0 = 0', 90', and 180'. We learn thatn+p+pkb & 0, that a & 2p, and, since the cross sectionis larger at forward angles than at backward ones, thatcr + p & P + b & 0. From the above expression, or fromthe positive nature of the tot, al reaction cross section, we

find that o. & 0. We also find that —n & p ( 2o, , which

limits the range of y.Fits to the data [1] using this equation are shown in

SPIN OBSERVABLES AT THRESHOLD FOR THE REACTION pp ~AA 1755

Fig. I and the results for the above coefficients (whichwere allowed to vary freely for each data set) are listedin Table II. A discussion is given in Sec. V.

The total cross section at threshold is obtained by in-tegrating Eq. (3.23) over all solid angle:

Combining this with the result for o above, we see thatthe total cross section has the expected form in termsof the amplitudes. We also define by this equation thephase space factor p.

Centrifugal barrier effects

2z'g 2OT (mpmA) cr = pn.ps

The contribution of each partial wave depends on thecentrifugal barrier [30], and whether or not there is spe-

0.2 0.25

0.16

L

0.12

0.08

0.04

1.4360 Ge V/ca = 0.236 MeV

0.2

CA

0.15

Q0.1

0.05

1.4

0-0.5 0

cos 80.5

0-0.5 0 0.5

O.S

tI)

0.6

0.4O

0.2

1.4454 Ge V/ca = 3.546 MeV

2.5

2

1.5

1

0.5

1.4765 Ge V/ce, = 14.523 MeV

6

-0.5 0cos 8

0.50

14

-0.5 0cos 8

0.5

5

4V)

3

O

O

1.508 = 2

12

10V)

8—6

04

1.5460 Ga = 39.056

0-0.5 0

cos 8

40

35

30

v) 25

20

1510

0.5

1 .6950 Ge V/ca = 91.725 Me Y

-0.5 0cos 8

0.5

00 0.5

cos 8

PIG. 1. Differential cross section data from Ref. [Ij fitted to Eq. (3.23), ~here i~ „=3. An independent set of parameterswas used for each data set; see Table II. All of the data except the set at 1.695 GeV/c are well represented by the truncatedbasis given in Eq. (3.22); as seen in this figure the 1.695 GeV/c data require a larger basis (I „=6).


TABLE II. Fits to the diR'erential cross section data [1] using Eq. (3.23). See also Fig. 1. The errors are absolute. Thereduced y indicates that L,„=3 gives a good account of all the data except for the case at 1.6950 GeV/c. The best fit for

that case requires L,„=6; the reduced X is then 0.547 and the total cross section is 69.9 + 1.0 pb (see Ref. [1]).

+lab(GeV/c)

CTz' = PO!

Value Error Value Error Value Error Value Error

2Xred

1.435 951.436 951.445 351.476 51.507 61.5461.695

0.8261.424.6712.924.944.463.7

0.0590.0670.1970.3670.6160.9422.20

0.4090.2280.7200.8210.9931.091.26

0.1320.0860.0880.0610.0560.0450.033

0.5100.4820.4800.6630.7191.101.14

0.1720.1160.1100.0710.0630.0530.037

0.2190.4460.2720.3660.3310.5620.646

0.2020.1360.1120.0780.0670.0550.034

0.790.631.130.690.871.205.93

cial physics (such as resonant behavior) going on in thatchannel. Because this reaction is endothermic, the be-havior of the cross section near threshold is expected tohave an explicit dependence on the small final AA mo-mentum q. To see this dependence, consider the formof the radial matrix elements in Born approximation.Schematically we have, for a local interaction potential

~,, (qr) V(r) j,, (pr) "d.,where the jg are spherical Bessel functions of order 8,and p, 8, and q, SJ are the linear and orbital angular mo-

menta for the incoming pp and outgoing AA particles,respectively. For small outgoing AA momentum q, we re-place jt (qr) with its limiting form (qr) &/(2lf + I)!!,andnote that the rest of the integral cannot be approximatedfurther since the incoming pp momentum is quite largeeven at the AA threshold. In this way we see that nearthreshold this matrix element behaves as q ~ rather thanas p '

q f, as would occur if both momenta were small,This helps to account for a natural "early" appearance ofEy ) 0 contributions in the threshold region Therefo. rethe appearance of P loaves quite near threshold is not unexpected, although their specific magnitude and splittingdepends on detailed dynamics. Similar remarks can bemade about the other transitions. We use these thresh-old dependences when we introduce the scattering lengthapproximation in Eq. (4.1).

2. Special limiting cases for the cross section

To examine the cross section formulas more closely, weomit the singlet amplitudes (based on the experimentalresults for the singlet fract, ion mentioned earlier), and we

temporarily neglect the D& final state because it is notas likely to contribute near threshold as the remainingSi, Pp, Py, and P2 amplitudes due to the centrifugal

barrier. We find the instructive limits

er = IT3POI + 3( l&n»l + l&asil )+5 I&»2i'+ I&»l',

p = «[2'Ta'si (T3PQ+ 3T3P1+ 5T3P2)

+v 2?sD (3%&P1—'T3P2 —2T3Po)]&.

v = -', ITaP1 I'+ —2, IT3P21'

+Re [ (4 +3PQ + 9 +3P1 )T3P2]

N ow the experimental differential cross section data showa strong forward peaking with scattering angle, even veryclose to threshold (see Fig. 1). In order to obtain thatbehavior with the above coefficients (recall that p+ b )0), we apparently must have p ) 0; this allows the p termto add constructively at forward angles and destructivelyat, backward angles. Let us assume that the first term in

P is dominant; then to have a positive P the phase, Pp,of the averaged P-wave amplitude T3~p + 3T3~~ + 5T3~2must be within 90' of the phase, Pasi, of the amplitude73S $ . Thus peaking of the cross section is obtained ifthe TsD ( 3 T3Pi —T3P2 —2 TaPO) term is small and thesephases satisfy the condition

143si —Apl &—2

We refer to this as the cross section peaking rule. Later,we shall see that the SD amplitude is not small, but theaveraged P-wave amplitude T3I o+3T3I x+5T3I 2 is muchlarger than the term (3'T3Pi T3P2 2?3PO). There-fore, peaking of the cross section constrains the natureof the P-wave splitting to some extent. The constraint—o, ( p & 2n also limits the the phase angle between theamplitudes 4T3&p + 9T3&~ and T3+2.

If one further assumes that the S wave is totally sup-pressed, p would be zero since both ZgD and 'Tasi wouldbe zero. Then the forward peaking is lost, unless the pa-rarneter b of Eq. (3.23) and its associated T3P2 and TaDiamplitudes are exceptionally large. With the last caseperhaps unlikely, we conclude that there is a need forsome S-wave contribution to provide the forward crosssection peak due to S- and P-wave interference. Thislast remark is directly related to the dynamics of annihi-lation and to absorption in initial and final states.

C. Polarization at threshold

VVe now proceed to discuss spin observables, which areparticularly sensitive to L & 0 effects even near thresh-


old. The polarization profile data shown in Fig. 2 wereobtained by convoluting the fits to the differential crosssections (see Fig. 1 and Table II) with the polarizationdata [1]. The errors were propagated using Gaussianstatistics assuming that the measured quantities were un-

cor related.We return to our truncated seven-state basis,

Eq. (3.22), and once more use explicit expressions forthe d functions. We find that the polarization profile hasthe general form [see Eq. (3.17)]

P&(0) X(8) = sin 0 [np + Pp cos 0 + yp cos 8]= (~P + —,'~P) Pi'(~) + 3&P P2 (~)

+ 1's» Ps (&) (3.25)

where the coeKcients are given by

1+SD (4 T3PO 3 T3P1 T3P2)

2

l1+DS (6 +3P1 '4 T3P2 2 +3PO)

( . 1~P —+9 lm

I'Tssi +3D1+ T3D1 TsD + +sD TDs

2

+—T3P2 (2%3PD + 3 %3Pl)),

and

pP = +27 1m[T3P2 73D1].

30

20

10A

0oV

-10

-20

l| II

~i&1.4454 Ge V/ca = 3.546 MeV

30

20

10Ao

CL 0V

-10

-20

1.4765 Ge V/ca = 14 523 MeV

-30-0.5 0

cos 80.5 0

cos 80.5

Ao

CLV

100

50

0

1.5076 Ge V/ca = 25.495 MeV

120

90

60A

Co 30V

1.546 Ge V/cs = 39.056 MeV

-30

-50-0.5 0

cos 80.5

-60-0.5 0

cos 00.5

100

Ao

CLV

50

0

1.695 Ge Y/ca = 91.725 Me V

-50-0.5 0

cos 8

FIG. 2. The polarization data from Ref. [1] fitted to Eq. (3.25). The ordinates give the "polarization profile. " Thecoeflicients are given in Table III. The zeros are at ~t'~ = 0.16, 0.16, 0.17, 0.17, and 0.19 (GeV/c) in order of increasingincident laboratory momentum.


There are several noteworthy features of Eq. (3.25).First, it has zeros at 0' and 180' (due to the overall sinefunction) because at those angles the reaction plane isundefined T. his is a general property of the polarizationSecond, the fact that we are near threshold enters via thepolynomial in coso. This polynomial has at most twoadditional physically meaningful zeros, which depend onthe coef5cients and hence on the threshold amplitudes.Third, at 90' the polarization is a direct measure of o,~.Other angles provide additional information; for exam-ple, P„X(45')—P„Z(135')= P~. Fourth, it is seenthat o.~ vanishes without P-wave splitting. A few spe-cial cases are now considered.

Case 1. If we now restrict the pp ~ AA transitions tothe sS, — and sPo+ terms, then P~ and p~ will be zero,and o.p ——2Im?3»?3+o ~ In this case, there are no zerosin the polarization except at 0' and 180'. This is clearlynot a feature of the data, and on this basis we concludethat including only the pp ~ AA sSi- and Po+ partialwaves is not valid at these lou energies. We stress thatour amplitudes TQL J refer to the pp ~ AA system, andnot to the underlying qq system.

Quark models of this process are sometimes based [36]on an underlying uu ~ ss Si- and/or Po+ quark an-nihilation and creation mechanism. Then the 1 is as-sociated with "gluon" and the 0+ with "vacuum" quan-tum numbers. When this basic process is accompaniedby the diquark pairs necessary to make the composite:&ystem, and the initial- and final-state interactions, themodel generates pp ~ AA transitions beyond just basicthe 5& — and Pp+ par tial waves, and thereby can gener-ate zeros in the polarization. Thus a major challenge ofthe quark model, and a measure of its success, is its abil-ity to generate a set of pp ~ AA transitions beyond justthe S&- and Po+ terms inherent in the uu ~ ss transi-tion. Comparison of those calculations with the restric-tions implied by the data will hopefully allow extractionof information about the basic ss production dynamics.Polarization zeros, and hence the required transitions,have been obtained in studies [18] based on underlyingSi- plus Po+ transition quark dynamics.

Case 8. If the P wave splitting -(which explicitly man-ifests itself in the structure of n~) were zero, indicating

that all of the P-wave amplitudes were equal, then a~would be zero and the polarization would have a zero atexactly 90'. In addition, the last term in P~ would bepurely real and would thus not contribute to the polar-ization.

However, P wav-e splitting occurs when L . S and/ortensor forces are present. In a simple meson exchangemodel the most prominent contributor to the L. S forcearises from the exchange of vector mesons. While thedata (see Fig. 2) at the various incident energies showzeros at a t' value of about 0.2 (GeV/c), these do notgenerally correspond to zeros at 90'. Thus me conclude,on the basis of the observed polarization and the naturaloccurrence of P waves near threshold, that there is a sub-stantial P-wave splitting present in the da/a. Althoughit is tempting to attribute this to K* or I~2 exchange,the role of L . S and tensor force splitting due to initial-and final-state distortions clouds this interpretation.

Case 9. If the 9 waves were greatly suppressed, wewould expect o.~ to be quite small unless the P-wavesplitting is very large. Here we assume that the?3D~ andTD~ states are not large due to the centrifugal barrier. Inthat case, the polarization again would have a zero near90'. In this "small n~" limit, in order to get positivepolarization at small angles we must have Pt ) 0. Fromthis we deduce a rule that the phase of?3~~ lags behindthe phase of (22gy o + 3Ts+$) in this case.

However, the value of n~ cannot be zero since the po-larization nodes in the data are not generally at 90'. Ifthe S waves were totally suppressed, then cr~ would de-pend only on the P-wave splitting and on the?3D~ am-plitude. If this amplitude is small, a small value of o.~would result. In the case of total 5-wave suppressionthe polarization would stay at or near 90' even as theincident momentum varied, in disagreement with experi-ment. Therefore, full S-wave suppression is ruled out, asit was due to the forward peaking of the differential crosssection.

Furthermore, if the ?3~2 amplitude were suppressedalong with the S wave, then not only would the polariza-tion node be at exactly 90', but the polarization profilewould be antisymmetric about 90'. Neither of these prop-erties occurs in the data. We conclude ]hat there must

TABLE III. Coefficients for the fit of Eq. (3.25) to the polarization data [Ij. The top rows give the results for fits withI, = 3 and the second rows contain the fits for I,„=4. An extra term (6~ cos 8) is added to Eq. (3.25) for the latter fits.The errors are absolute.

Pleb(GeV/c)

1.4765

Value

—0.249

Error

7.46

Value

24.2

Error

14.7

Value

28.5

Error

28.6

Value Error

2Xred

3.30

1.5076 —14.4—17.6

12.515.3

55.529.1

21.546.2

91.3111

42.557.5 68.8 101

4.102.80

1.546 —24.1—27.2

5.904.80

63.324.8

13.618.0

83.5111

26.723.4 114 42.7

2.791.73

1.695 —33.2—35.4

3.993.55

23.71.47

9.1212.8

50.070.3

18.017.9 70.2 32.1

1.911.39


be both P2 and S w-ave contributions to the processCase g. With P wa-ve splitting apparently present, as-

pects of the tensor force mixing of 9' and 0 waves be-comes important in these expressions. It is difFicult atthis stage to make quantitative estimates of their size, buttheir presence suggests a role for pseudoscalar [I1 (494)]and tensor [I~&(1430)] mesons.

Case S. Consider the case where all of the seven statesin our restricted basis are contributing. Note that atbackward angles (0 ) vr/2), where the polarization profileis often large and negative (see Fig. 2), we must haveO'P —pP + pP ( 0. At forward angles, we must haveAP + pP + TP ) 0. [These results can be seen from aninspection of the derivative of Z(0) P„(0)near forwardand backward angles. ] From these conditions, we deducethat to get positive forward polarization and negativebackward polarization, we must have —pP ( nP + pP (Pp. Only a positive P is consistent with this condition.

The best fits using Eq. (3.25) are shown in Fig. 2, andthe values for the coeKcients np, Pp, and pp (whichwere allowed to vary freely) are provided in Table III.The results are discussed in Sec. V.

D. Spin correlations at threshold

We again confine ourselves to the restricted basis de-fined in Eq. (3.22). We find that simpler expressions re-sult if we use the spherical tensor basis described aboveinstead of the Cartesian tensor to describe the spin cor-relations. We have converted the data (at 1.546 and1.695 GeV/c) to these forms using Eqs. (3.20), takinginto account the coordinate system used here versus thatof Ref. [1]. Using the same methods as before, we findfor C22.

C = sin 0 [a22+ P22cos0],

where

(3.26)

~22 = -- [I'T3»l' —2I&snl'+3 I'T3»l +31'TSP21]

+3 Re (Tsni + ~2Tsn)

and

1T391 + +DS 7SD

2

3 ~+~ 3~] 3P2

p22 = +9 Re [( T3'D i + ~2Tsn ) r»2 ]

C:slil 0 [ct'21 + P21 cos 0 + T21 cos 0],

where

(3.27)

This simple expression arises since rather few ampli-tudes from our basis contribute to this "stretched" I =2, M = 2 tensor. This expression has nodes at 0' and180' due to the sine function, and possibly one more atcos 0 = —n22/P22. If o.22 vanishes, that node would occurexactly at 90'. Since this is inconsistent with the data,we conclude that o.g2 must be nonzero. At 90' we seethat C" = n22.

We see too that if the 'Tsp2 amplitude vanishes, P22would be zero. Then C has no nodes other than atits end points. The existence of such a node in C22 istherefore direct evidence of a T3+2 contribution in ourrestricted basis. Note also that P-wave splitting per sehas little effect on this coefFicient. Apparently C is moresensitive to noncentral Tsa, TDs effects.

Turning now to an evaluation of C~ we find

1 1+21 —Re T3sl T3D1 Tns (3+3P1 73P2 2 73Pp) + (Tsn Tns) (4+3Pp + 3 +3P1 7T3P2)

and

P21 = 3pani( —3 ~TsnI + 3ReI 'T3'P2(3 6P1 —273Pp —'6P2)+ &gn(Tsni +46si)2

'Us( +&so + 4'6+i ) —'Zsi&sai)2

721 —9 «[+3P2( 2 ~~ Tsn T3D 1 ) ] ~

This expression has nodes at 0' and 180' due to the sine function, and possibly two others due to the polynomial incos0. At 90' we have C = o.2i, other combinations of measurements can determine the other coefFicients.

Here the spin-orbit splitting does play a role since equal P-wave amplitudes will cause o, p~ to vanish. In that case,C has one node at exactly 90', and possibly one more, other than those at the end points. That second nodedisappears if there is no T3+g contribution. Here we see that this spin correlation depends sensitively on the spin-orbitsplitting, the P2 amplitude, and also on the noncentral force aspects.

Finally, we have for C2 a significantly more complicated expression:

C = [cl'2p + p2p cos 0 + p2p cos 0 + 62p cos 0 ]

where

(3.28)


1 2cr2O = ( (9 I'T3»l 8 IT3POI l&3P2I ) + 3(2ITSDI 4 I+DSI I'T3Dil ) )

2 63 1+«V'6'T3D1 Tssi + (TsD —&Ds ) + &q+2(8 T3Po ~ T3P1 + T3P2)

'2 6

+2 +3'T3"si (2 2as —TsD ) + v 6?~s&SD

P2O = 2 (3 T3P1 2 T3PO +3P2)Tssi + 2 (3 +3Pl + 5T3P2 8+3PO)+3D1

1 9 1+ / —(4 +3PO + 3 +3P1 7+3P2)(7DS + +SD) + 2+Ds 7SD + +3D1g'2 2 2

9Q20— {IT3P11 2 I+SD I' —31&»1I' —l&3P21

+2 «[2 (U 2 T3sl —TDS)7sD + 3 (9 T3P1 —8'GPQ —T3P2 )T3'P2

+ )/2 +3Di( 3 +Ds + +SD ~~Tssl) ]))

b~o = 27 «[Z P, ( V2TSD —73Di )]

While this expression is not simple, we again see the man-ifestation of P-wave splitting in these coeKcients. Thiscorrelation can have three nodes, while at the 0' and 180'end points C is not necessarily zero. The coefFicient 62ovanishes if the P2 amplitude equals zero; in that casethere is one fewer node. If o, 2o vanishes, then C has anode at exactly 90'. These expressions clearly depend onboth spin-orbit splitting and tensor, or noncentral force,effects. At 90', Q2 = a~o and the difference between the0' and the 180' values is 2(P2O + b2o), which might be auseful, albeit complicated, constraint on the amplitudes.

For the special limit having only Si and Po ampli-tudes, the spin-correlation tensors take on a particularcharacter. The value of C is then zero; hence, a nonzeroresult is additional evidence for the failure of a purely S~and Po annihilation model. The tensor C in this spe-cial case has only an o.2~ coe%cient and hence has nonodes other than at the 0' and 180' end points. For C

only the n21 and P21 coefficients would be nonzero andhence there is at most one node. Failure to satisfy any ofthese conditions indicates that the dynamics contribute

to more than just the Si and Po amplitudes.The best fits using Eqs. (3.26)—(3.28) are shown in

Fig. 8, and the values for the coefficients n2M, P2M, y2M,and 62M (which were allowed to vary freely) provided inTable IV. The results are discussed in Sec. V.

IV. THE SCATTERING LENGTH LIMIT

Further insight into the allowed structure of spin ob-servables at threshold can be gleaned from the previ-ous expressions by introducing the scattering length limit

[30], the origin of which is discussed in Sec. III Il 1. Inthis approximation, each amplitude is written with anexplicit momentum dependence:

&sl.z = &si.z qL (4 1)

Here ASI.J is a complex number which is assumed to beenergy independent; in general, it could depend on en-

ergy and include resonance and/or subthreshold effects.Indeed, the full determination of the amplitude requires a

TABLE IV. Coefficients for the fits of Eqs. (3.26)—(3.28) to the spin-correlation profiLe data [1] at 1.546 and 1.695 GeV/c.The errors are absolute.

0bser vable

+22+21~20

(GeV/c)

1.5461.5461.546

&2MValue

—60.80.453

21.3

Error

11.54.52

10.6

P2 MValue

—19.6—96.6

16.6

Error

35.710.740.7

"(2MValue

—114—78.9

Error

20.135.5

~2MValue

—121

Error

77.8

2Xred

1.600.200.55

+22C21~20

1.6951.6951.695

—52.815.428.9

9.4015.015.7

—173—47.5

84.1

27.033.549.4

—121—86.5

66.156.2 —167 95.7

1.703.21.89


dynamical theory and a coupled-channels study for thatpurpose is underway [15]. However, we consider here thesimple case of energy-independent "scattering lengths"AsL, g to gain insight into the normal behavior of thevarious observables at threshold. If a reasonable fit canbe achieved with energy-independent scattering lengths,then these data do not require special resonance and /orsubthreshold effects to explain them.

Substituting amplitudes of this form into Eq. (3.24)yields a total cross section of the form

(2xq & 2 T T 2 T 4oT =I (mpmA) (ao + rz2 q + tr4 q )s )

where the coe%cients are

n~ = 3(l&sDil'+ I&Dsl')

Here q is the momentum of a A in the fina1 AA state;it is related to the "excess energy" e by the formulaq =

& [a(e+ 4mA)] ~, where c = ~s —2rnA. Thus nearthreshold q e ~ . Clearly az, a2, and Q4 represent the8-, P , a-nd D wa-ve contributions to the cross section.

It is straightforward to make a similar expansion forthe diAerential cross section, Eq. (3.23). We find the fol-lowing explicit momentum dependence for the coe%cientsrr, P, y, andb:

A =Qo +Q2q +QgqT

P = biq+ 4q',

and

no ——l&iso I' + 3 l&ss, I' + 3 l&sD I',n2 = l~s~ol'+ 3 l&»il'+ 3 l&3P1 I' + ~ l&3P2I', p = C2q +Cqq2 4

b=d3q .

(4 3)

100 200

0

100

es 0

ItIII

-50

-100

I*C1.5460 GeV/c

s = 39.056 MeV

-100

-200

l*C1.6950 Ge Y/e

e = 91.725 MeV

-1500

cos 80.5

-3000

cos 80.5

50 50

0(3

-50

-100

*C2 1

1.5460 Ge V/ca = 39.056 Me

-50

-100l*C

1.6950 GeV/ca = 91.725 MeV

-0.5 0cos 8

50-0.5 0

cos 80.5

50 50

0U

ol 0

-50

-100

-1 50

I C1.5460 Ge Y/c

g = 39.056 MeV

-50

-100

-150

l + C2 2

1.6950 Ge V/ca = 91.725 MeV

-0.5 0cos e

0.5 -0.5 0cos 8

0.5

FIG. 3. The spin-correlation data from Ref. [1] plotted in the spherical basis of Eq. (3.20). The solid curves are fits usingEqs. (3.26)—(3.28); the fitted coefficients appear in Table IV.


The coefficients aTo, a&, and a4T are given in Eq. (4.2).The remaining expressions are

bi ——+Re [ +2 (3 A3P1 —A3P2 —2 A3PQ)+sD+6 A1P1A1SO

+( 2 A3PO + 10 A3P2 + 6 A3P1) A3S, ],

F13 = +Re [ 'i'(3 +3P1 +3P2 2 +3Po)~Ds+(4 A3PQ + s+3P2 + 3 +3P1) +3D11~

and

54d3 ——+—R [A3PgA'D, ].

5

The results of fitting this scattering length formula tothe data are given via the coefficients in Table V and areshown in Fig. 4. The discussion appears in Sec. V.

Turning now to the polarization profile, Eq. (3.25), thescattering length limit gives the following dependence onthe AA momentum:

e2 = +- l&3P11 + 6 l&»11 + - l&3P213 2 2 7 2

2 2+Re [ (4 +3po + ~ +spl )+3pg+6 ( A3D1 A3S1 + ADS ASD—3~2 ASD &3D, ],

e4 = +- l&3D1 I' —Re [3%»DS &3D1]2

Py(0) &(0) = qsino [(ao + 2 q )+ ~1 q os~

+ez (q cos9) ], (4.c)

where

~Q +™3$'$ 2 3PO + 3 +3P]. 5 +3P2

A1'D [4 A3P A0 33P1 A3P2))2

0.2 0.25

0.16I

th0.12

Cl0.08

0.04

IlIi

1.4360 GeV/ca = 0.236 MeV

0.2

0.15

0. 1

1.4370 Ge V/ca = 0.588 MeV

-0.5 0cos 8

0.5 -0.5 0cos e

0.8

CO

0.6

0.4o

O.. $$~

1.4454 Ge V/ca = 3.546 MeV

IlI) Il Il I&

2.5

2

1.5o0

0.5

1.4765 G eV/c14.523 M e V

7

-0.5 0cos 8

0.5 0cos 0

0.5

6

5

4

1.5076 Ge Y/cc = 25.495 Me Y

3

2

y. —e ++ —+~-++

0

10

8

6

4

2

1.5460 Ge Y/cg = 39.056 M e V

+~e~%-~ + I O. 0—.~ 0'

-0.5 0cos 8

0.5 -0.5 0cos 8

0.5

FIG. 4. Differential cross section data from Ref. [1] fitted to the scattering length approximation, Eq. (4.3). All of the dataexcept that from 1.695 GeV/c are included, and are well described by the eight real parameters listed in Table iv".

44 SPIN OBSERVABLES AT THRESHOLD FOR THE REACTION pp —+AA 1763

TABLE V. Coefficients for the fit of the scattering lengthapproximation formulas for the differential cross section[Eqs. (3.23) and (4.3)] and the polarization [Eq. (4.4)] to thedata [1]. The two fits are independent of each other; the re-duced y values for the two fits are 1.37 and 1.94, respectively.The errors are absolute.

aoa4

b3

C4

Value

23.329 8306 809

24 260

Error

0.953540857

7250

a2b1

C2

l3

Value

1 317215.7

1 3505 909

Error

14626.1

264500

P

bP127.6

1 53927.8

156a2 —5757c2 10 540

7831560

P 1a2 + Im

l &Ds (2 A3Pp 6 +3P1 + 4 +3P2)

+27 ™[+3P2 +3D 1] .

The overall factor q implies that P&(0) X will be zero at

1bi ——+9 im A3si +3D1 + +3D1 +sD

2

+Ay~ Aas + 3

Appal

(2Am 0 + 3Am s)),

zero center-of-mass momentum, again because the reac-tion plane disappears in this limit. The bracketed poly-nomial in (q cos 8) determines other important charac-teristics of the polarization. From it, we see that at verylow AA momenta, only a& will be nonzero. Hence atvery loco q the polarization does not have a node otherthan those at 0' and 180'; and from the fact that the po-larization is positive at forurard angles, toe conclude thata~) 0.

As the AA momentum increases, the higher-orderterms in q become more important, and an intermedi-ate node can appear, first at a value of

lcos0l = 1 when

q becomes large enough. Indeed, the appearance of thisnode first at larger angles would mean that ao and b&

must be of the same sign. As the momentum contin-ues to increase, the intermediate node would then moveto smaller angles. This behavior is clearly exhibited inthe data, from which we conclude that b& is positive.Thus, normal behavior of the polarization near thresholdis that a critical momentum, q„exists for the onset of anintermediate node.

The results of fitting this scattering length formula tothe polarization profile data are given in Table V andshown in Fig. 5. A full discussion appears in Sec. V.

The scattering length approximation can also be ap-plied to the spin correlation coefficients, although weomit the details. As in the case of the polarization, wefind that ZC contains only the term o.22 at very low mo-menta, and therefore initially has no intermediate nodes.At some critical momentum, a node can appear as theq term in P22 becomes larger. That node should alsooccur at the larger values of

lcos el; hence, its intermedi-

ate node should appear near 180' and then move in with

30

20

10A0

cL 0V

-10

-20

1.4454 GeV/ca = 3.546 MeV

A0

CLV

30

10

-10

-20

1.4765 Ge V/cs = 14.523 MeV

-30-0.5 0

cos 60.5 0

cos 0

100 150

A0

CLV

50

0

1.5076 GeV/ce = 25.495 Me Y

100

50A

OCLV

-50

1.5460 Ge V/ca = 39.056 MeV

-100

-0.5 0cos 6

0.5 -0.5 0cos e

0.5

FIG. 5. "Polarization profile" data. calculated from Ref. [1] fitted to the scattering length approximation, Eq. (4.4). All ofthe data except that from 1.695 GeV/c are included, and are well described by the four real parameters listed in Table V.


increasing q.In contrast, for X C

' one part of the Pqi term is presentat threshold. Thus, an intermediate node should appearat 90' even at threshold; there is no need for a criticalmomentum to be reached for that node. However, as theo, 2i and p~~ terms turn on, a second intermediate nodeshould appear beyond some critical AA momentum.

For XC, both the a~0 and y2o terms contain piecesthat are present at threshold; this allows a node to bepresent even at threshold; other nodes may appear aftercritical momenta are reached. The overall picture thenis that we will pass through some (low) critical momentabefore the onset of additional intermediate zeros, but thatZC and ZC can have zeros even at threshold. Thedata in Fig. 3 exhibit these behaviors. Further discussionappea& s below.

V. FITS TO AA DATA

As an illustration of the limits placed on the thresholdamplitudes, we now discuss the fits to the data shownin Figs. 1—5. The fitting process is done on three levels:first, we simply fit the I,egendre expansions, Eqs. (3.24)—(3.28), to each data set independently, obtaining the co-efficients given in Tables II—IV. Second, we take a mareambitious approach by fitting the diff'erential cross sec-tion and polarization data (separately) to the energy-independent scattering length approximation formulasgiven in Eqs. (4.3) and (4.4). The twelve real numbers saobtained are given in Table V. Third, we use the scat-tering lengths themselves [Eq. (4.1)) ta fit the data, thuslinking the de'erential cross section and polarization datato a common set of complex amplitudes. We now discussthe results obtained using these three fitting procedures.

A. Fits using independent Legendre expansions

It is readily seen in Figs. 1—3 and Tables II—IV thatexcellent fits to all of the experimental observables can beobtained using this approach. It is clear in the case of thediff'erential cross sections (Fig. 1) that Eq. (3.23) (whereL „=3) is an excellent representation af the dataexcept for the set at 1.695 GeV/c. This fit validates theuse of a truncated number of states as given in Eq. (3.22),and it also shows that at c = 91.7 MeV more states areneeded.

Several aspects of these fits are noteworthy. First, theparameters n, P, p, and 6 satisfy all of the peaking con-straints deduced earlier. Second, they also indicate anappreciable role far P and hence demanstrate the need forappreciable S-P interference, especially as pI~b increases.Third, the role of p is similar to that of P, indicating thepresence of sizable P-wave amplitudes. Finally, while 6 isat first small, it rapidly becomes competitive with P ando;, indicating significant Pg and some Di contributions.

The overall trend of these parameters with pi b is rea-sonable. The fact that our limited basis does not providethe proper forward peaking at ~ = 91.7 MeV is not sur-prising. This result clearly delineates the region wherethe basis needs to be extended with the associated higherpowers of cos 0; as higher powers are introduced that peak

is fit (L „=6 gives an excellent result).In the fits to the polarization data the parameters o.~,

P~, and yy were independently adjusted to fit the po-larization profile; see Fig. 2 and Table III. At the lowmomenta those three parameters suKced, correspondingto our truncated basis result. However, an additionalterm (b~ cos 0) corresponding to L „=4 improvedthe fits significantly. This shows that the polarization ismore sensitive to smaller amplitude admixtures than isthe cross section.

The behavior of the polarization profile is consistentwith the general equations we obtained, and with theanticipated onset and motion of the nodes deduced fromscattering length ideas. We see in the data that thereis no node at Iow incident momentum, but that onesets in (at large angles) when the incident momentum islarger. As the momentum is increased further, it movesto smaller angles.

The motion of this node with momentum is crucial toour conclusions about the role of the S-wave amplitude.As indicated earlier, if the node were fixed at about 90'irrespective of momentum, one might conclude that S-wave suppression is taking place. Indeed, the first data[1] published (at 1.477 and 1.508 GeV/c) has nodes near90'. For those cases, a zero value of a~ would be appro-priate, which could be achieved by a total suppression ofS waves. However, below and above that momentum re-gion the node is not at 90', indicating that total 5-wavesuppression is not a correct description.

The spin-correlation profile data is displayed in Fig. 3for the tensor spin correlations discussed earlier. At 1.546GeV/c the fits using the parameters aqM, P2M, p2~, andb2M show the behavior anticipated above in the scatter-ing length discussion: that C is simple and nodeless,whereas C2 has a single node. Since that node is near90', ap~ is apparently small at that momentum; how-ever, with P-wave splitting and nonzero S waves, thatnode should move to smaller angles with increasing mo-mentum. Only C can be, and is, nonzero at 0 and180'

Although the data at 1.695 GeV/c is already outsidethe scope of our truncated basis, especially for the po-larization, the spin-correlation data there shows that (1)C has developed a single node at a large angle and itis moving to smaller angles [see Eq. (3.26)]; (2) C~i hasdeveloped an additional node at a large angle and its firstnode is moving in [see Eq. (3.27)]; (3) C2O has three nodesas allowed by Eq. (3.28).

B. Energy-independent fits

Encouraged by the above results at each momentum,a global fit to the difI'erential cross section data at sixmomenta (excluding the 1.695 GeV/c case) was madeusing the scattering length formula, Eq. (4.3). The eightreal parameters ao, . . . , d3 so obtained, which are notthe scattering lengths themselves, are given in Table V.They generate the global fit to the 100 differential crosssection data points shown in Fig. 4. Quite a good resultis abtained (g„,d ——1.37), indicating that the scattering


TABLE VI. The scattering lengths defined in Eq. (4.1) for the global fit to the differential crosssection and polarization profile data. The defining equations are given in Eqs. (3.23) and (4.3) andEq. (4.4), respectively. The reduced y value for the fit is 1.75. The errors are absolute.

AmplitudeValue

Real partError

Imaginary partValue Error

&3SI+3D1&SD&DS

1.238.321 ~ 62

10.7

0.0650.000 260.0550.001 3

0.82655.31.77

16.7

0.0760.000 190.0450.000 23

+3PO

&3P~

—2.950.207.22

1 ~ 100.560.21

8.9513.78.19

1.050.420.000 037

length approach generally sufFices for all but the highestmomentum data. One of the notable features of the fit isthat the S-wave parameter a+o is an essential contributorto the cross section, indicating an important role for theS waves.

The success of the energy-independent approach sug-gests that the underlying dynamics might be describedby "normal" ideas. However, the discrepancies seen atthe lower momenta appear to be quite interesting. Un-fortunately, it is not possible to draw a clear conclusionsince those data are sparse, with larger errors, so that theleast-squares search stressed the higher-momentum mea-surements. This may account for the rather worse fits atthe lower momenta. However, it may also be true thatthe discrepancies are indicators of other physics takingplace there.

Adding in the 40 data points from 1.695 GeV/c raises

y„d from 1.37 to 3.38. This is a clear indication thatthe simple scattering length formula is not valid at thatexcess energy unless one extends the basis, or possiblyadds effective range terms to Eq. (4.1).

A global fit to the polarization data (excluding the1.695 GeV/c set), using the four-parameter Eq. (4.4) tofit 33 data points was also made, obtaining the resultsshown in Fig, 5 and Table V. These four parameters areindependent of those obtained for the difI'erential crosssection results. A good fit is obtained, again indicatingthat the scattering length approach suKces for the lowerexcess energy region. All of the data except those at 1.695GeV/c are included. Indeed, it is seen that ag ) 0, andthat a& and b&+ are of the same sign. Of course, a fullamplitude analysis provides the link between them, andwe turn our attention to that approach below.

0.025

0.015Scattering Amplitudes

at 3 Momenta

0 ~ 005

3p

0 3p

1

C'g) -0

~ 005

E

-0.015

3D

multiplying TsLJ by (2pgp /4z/100hci ). In the case ofelastic scattering, this leads to amplitudes of the formexp(ib) sin b. There is an overall phase ambiguity (+i)which can be resolved only by a dynamical theory; herewe make a convenient, but arbitrary, choice. [The ampli-tudes PAL, J have the units grab/GeV and the scatteringlengths AsL, J have the units gpb/(GeV) + .j

The results, shown in Fig. 6 in the form of Arganddiagrams, display some important features. Note thatthe Si and SD amplitudes are nonzero even at the low-est momenta, where all other amplitudes are very small.Indeed, the (off'-diagonal) SD amplitude is larger thanthe sSt amplitude, which indicates the presence of astrong tensor force. This fact suggests that the impli-cations of including the I'"2 ~ P2 transition should beinvestigated. Of course, this would greatly complicateour analysis, since, for consistency, we would have to in-clude all terms that can interfere with the I"~ ~ P2transition. Therefore, we leave this question for a future

C. The complex scattering length fit

In this last fitting procedure, we adjusted the sevencomplex scattering lengths in Eqs. (4.3) and (4.4) to fitthe difI'erential cross section and polarization data simul-taneously for the first six momenta of Table I. The resultsare given in Table VI, where we see that a good fit is ob-tained (g2,d

——1.75).In order to discuss the physics content, we convert

the amplitudes to the usual (unitless) S-matrix form by

-0.025

0.03 0 ~ 02

Real Part

0.03 0.04

FIG. 6. Argand diagram of the unitless amplitudes foundusing the complex scattering lengths from Eq. (4.1) inEqs. (4.3) and (4.4) that fit all of the differential cross sec-tion and polarization data except those from 1.695 GeV/c.See Table VI. The line markers indicate the amplitude valuesat 1.436, 1.477, and 1.546 GeV/c.


study. At this stage, we find that good fits to the datacan be obtained with our simple amplitude analysis with-out including I"2 ~ P2 transition. However, since largeI'" ~ P transitions have been found in recent dynamicaltheories, we are studying this problem further [16].

In addition, the P and D waves increase rapidly withmomentum, as expected, with the Pi amplitude lead-ing the way. The P waves display large splitting, whichroughly follows an L . S order (st, s Pi, s Po) for theimaginary part. Their real parts reveal a ( P2, Po, Pi)splitting, which is neither a spin orbit, nor a quadraticspin orbit, nor a tensor force ordering. Another sig-nificant aspect is that the averaged P-wave amplitude(7spo + 373/&i + 573/7/) and the combination (47s~p +97s+i) (see Sec. III B 2) are quite large at the higher mo-

menta, whereas (3'TgI i —7s~2 —27sy»p) remains small.This confirms the cross section peaking rule discussedearlier. This illustrates that the P-wave splitting is con-strained not only by the polarization data, but also bythe peaking of the diAerential cross section. Understand-ing this splitting clearly requires a dynamical theory.

Finally, we remark that since these are direct ampli-tude fits, all distortion eH'ects are implicitly included;however, since the amplitudes are not the result of dy-namical equations, they are not assured of being analyti-cally correct with respect to causality and fIux conserva-tion restrictions. Nevertheless, fitting the data directlydoes give us some insight as to the limitations providedby the threshold restrictions discussed earlier.

VI. CONCLUSIONS

Our study of the AA reaction near threshold has led toseveral conclusions based on the general angular struc-ture of the observables. We see from the excellent fitto the cross section data that the truncated basis ofEq. (3.22) sufIices for all but the highest energy data re-ported in Ref. [1]. At the higher energy (e = 91.7 MeV),additional partial waves are required, as expected.

Similar ideas apply for the polarization, except that itis more sensitive to small partial waves. Hence the trun-cated basis needs to be expanded at a lower energy thanfor the difI'erential cross section. Nevertheless, our trun-cated basis sufIices at the lower energies as evidenced bythe reasonably good fits to P& X. For the spin correlationswe expect that the sensitivity to small partial waves iseven greater.

Several other conclusions concerning o(0) and P& 7were presented here. The forward peaking of o(0) wasattributed to $' —P interference and it followed that(nonvanishing) S waves must appear, which implicitlyconstrains the net annihilation in pp. The nonvanishingnature of the S-wave amplitudes is further demonstratedby our polarization study; we find that both 3S~ and 3P2waves are needed to obtain the correct node structure forPy X.

The zeros in P& X also show that use of only the sS&

and Po partial waves to describe the pp ~ AA reaction

does not yield polarization zeros in agreement with thedata. It was concluded that quark-based models mustgenerate more than just the 5~ and Po pp ~ AA partialwaves inherent in the uu ~ 88 mechanism; this is possiblewithin these models [18] when the associated finite sizeand composite nature of hadrons is taken into account, .

The nodal structure of P& 2 demonstrates not only thatP waves are required, but also that there must be sub-stantial P-wave splitting to avoid a fixed zero at 90'.Constraints on the parameters specifying the angle de-

pendence of the polarization were deduced and verifiedby direct fitting.

While the origin of the fixed zero in the polarizationat ~t'~ 0.2 (GeV/c) does not emerge directly from ouranalysis, it seems clear that it is an artifact of the strongabsorption taking place in this reaction. It is quite likelythat an explanation based on geometrical ideas such asthose expressed in Ref. [34] will emerge.

The allowed forms of the tensor spin-correlation pro-files were deduced and their nodal structure wit, hin thetruncated basis was analyzed. Structures of this kindare observed in the data, and the proper spin-correlationprofiles were fit to them.

The scattering length expansion treatment led not onlyto excellent fits to data, but also to a description of thenatural onset and motion of nodes in the polarization andspin correlations. The very fact that a scattering lengthfit could be achieved indicates that the data behave "nor-mally" for a near-threshold endothermic production re-action. From the complex scattering length fit we learnthat there is a need for a strong tensor interaction, andthat P-wave splitting is present but not of simple charac-ter. We also see that not only the polarization, but alsothe difI'erential cross section constrains the nature of theP-wave splitting.

Evidence for resonances and jor quark dynamics wouldhave to arise from deviations from this normal behavioror from detailed dynamical studies, which are underway.In particular, it is too early to say whether the deviationsobserved here between the scattering length parametriza-tion and the lowest-energy difI'erential cross section dataare indicative of new phenomena.

Applications of these amplitude methods t, o spin ob-servables in KA and EK production, and more generallyto other reactions such as qq ~ ct." and meson electro-production, are also in progress. We hope such studieswill also reveal simple and general characteristics for spinobservables at threshold and thus lead to further insightsabout these processes.

ACKNOWI EDG MENTS

We recognize with thanks the continued interest in thisproject shown by our colleagues Niko Hamann, DavidHertzog, Tord 3ohansson, Kurt Kilian, and Walter Oel-ert. Their many comments and suggestions have addedin important ways to this work. We also would liketo thank Mary Alberg, Norman Austern, Carl Dover,Ernest Henley, Dale Kunz, and Joseph Speth for helpful


remarks. R.A.E. thanks the University of Illinois Cen-ter for Advanced Study for an Associateship during the1990—91 academic year, which made it possible for himto participate in this work. This research was also sup-

ported in part by the US National Science Foundationunder Grants Nos. NSF-PHY-89-20423 to the Universityof Pittsburgh, and NSF-PHY-89-21146 to the Universityof Illinois.

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spin observables at threshold for the reaction p-barp--\u003e lambda -bar lambda

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