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ELSEVIER Nuclear Physics A 608 (1996) 243-304

N U C L E A R PHYSICS A

Correlated 7rTr and K K exchange in the baryon-baryon interaction

A. Reuber, K. Holinde, H.-C. Kim 1, j. Speth lnstitut fiir Kernphysik (Theorie), Forschungszentrum Jiilich GmbH, D-52425 Jfilich, Germany

Received 14 November 1995; revised 1 July 1996

Abstract

The exchange of two correlated pions or kaons provides the main part of the intermediate-range interaction between two baryons. Here, a dynamical model for correlated two-pion and two-kaon exchange in the baryon-baryon interaction is presented, both in the scalar-isoscalar (tr) and the vector-isovector (p) channel. The contribution of correlated zrTr and KK exchange is derived from the amplitudes for the transition of a baryon-antibaryon state (BB') to a ~sr or K K state in the pseudophysical region by applying dispersion theory and unitarity. For the B B - 7 -* 7rzr, K-K amplitudes a microscopic model is constructed, which is based on the hadron-exchange picture. The Born terms include contributions from baryon exchange as well as p-pole diagrams. The correlations between the two pseudoscalar mesons are taken into account by means of zrzr-KK amplitudes derived likewise from a meson-exchange model, which is in line with the empirical zrzr data. The parameters of the B B ' ~ zrTr, K K model, which are related to each other by the assumption of SU(3) symmetry, are determined by the adjustment to the quasiempirical N N --* zrzr amplitudes in the pseudophysical region. It is found that correlated K K exchange plays an important role in the tr-channel for baryon-baryon states with non-vanishing strangeness. The strength of correlated 7r,r plus KK exchange in the tr-channel decreases with the strangeness of the baryon-baryon system becoming more negative. Due to the admixture of baryon-exchange processes to the S U ( 3 ) - s y m m e t r i c p-pole contributions the results for correlated zrzr exchange in the vector-isovector channel deviate from what is expected in the naive S U ( 3 ) picture for genuine p-exchange. In present models of the hyperon-nucleon interaction contributions of correlated zrzr and KK exchange are parametrized for simplicity by single tr- and p-exchange. Shortcomings of this effective description, e.g. the missing long-range contributions, are pointed out by comparison with the dispersion-theoretic results. It is pointed out that further correlations, namely other K K - exchange processes like, e.g. in the fl ' = 1-, I = 0 channel (to, ~b) and zrK correlations (K*), remain to be investigated.

1 Present address: Institut ffir Theoretische Physik II, Ruhr-Universit~t Bochum, D-44780 Bochum, Germany.

0375-9474/96/$15.00 Copyright @ 1996 Elsevier Science B.V. All rights reserved PII S0375-9474(96)00256-4

244

1. Introduetion

A. Reuber et al . /Nuclear Physics A 608 (1996) 243-304

The study of the role of strangeness degrees of freedom in low energy nuclear physics is of high current interest since it should lead to a deeper understanding of the relevant strong interaction mechanisms in the non-perturbative regime of QCD. For example, the system of a strange baryon (hyperon Y) and a nucleon (N) is in principle an ideal

testing ground to investigate the importance of SU(3)flavor symmetry for the hadronic interactions. This symmetry is obviously broken already by the different masses of

hadrons sitting in the same multiplet. However, the important question arises whether (on the level of hadrons) it is broken not only kinematically but also dynamically, e.g. in the values of the coupling constants at the hadronic vertices. The answer cannot be given at the moment since the present empirical information about the YN interaction is too scarce and thus prevents any definite conclusions. Hopefully the situation will be improved by experiments of elastic ,~±p, Ap, and even ~p , currently performed at KEK [ 1,2].

Existing meson-exchange models of the YN interaction assume for the hadronic cou-

pling constants at least SU(3) symmetry, in case of models A and B of the Jiilich group [ 3] even SU(6) of the static quark model. This symmetry requirement provides relations between coupling constants of a meson multiplet to the baryon current, which strongly reduce the number of free model parameters. Specifically, coupling constants at the strange vertices are then connected to nucleon-nucleon-meson coupling constants, which in turn are fixed between close boundaries by the wealth of empirical NN scattering information. All YN interaction models can reproduce the existing empirical YN scattering data. Therefore at present the assumption of SU(3) symmetry for the coupling constants is not in conflict with experiment.

However, the treatment of the scalar-isoscalar meson sector, which provides the intermediate range baryon-baryon interaction, is conceptionally not very convincing

so far. The one-boson-exchange models of the Nijmegen group start from the existence of a broad scalar-isoscalar ~rTr resonance (e-meson, m~ = 760 MeV, F~ = 640 MeV), which is hidden in the experiment (e.g. 7rN --~ 7rTrN) under the strong p°-signal and can therefore not be identified reliably. For practical reasons the exchange of this broad e-meson is then approximated by the exchange of a sum of two mesons with sharp mass ml and m2; the smaller mass is around 500 MeV and thus corresponds to the phenomenological o--meson in conventional OBE models. The e-meson is then treated as SU(3) singlet (model D [4] ) or as member of a full nonet of scalar mesons (model F [5] and NSC [6]) . In the SU(3) framework the e-coupling strength is equal for all baryons in model D, while it depends on four open SU(3) parameters in models F and NSC, which are adjusted to NN and YN scattering data. In the latest Nijmegen model NSC [6] the scalar meson nonet includes apart from the e the isoscalar f0(975), the isovector a0(980) and the strange mesons x, which the authors identify [7] with scalar q2q2-states predicted by the MIT bag model [8]. This interpretation is however doubtful, at least for the f0(975) and a0(980). According to a recent theoretical analysis of the 7rTr, 7rr/, and KK system [9] in the meson-exchange framework the f0(975) is

A. Reuber et al . /Nuclear Physics A 608 (1996) 243-304 245

N N

N N a,)

N N

N,A N,A

N N h)

N N

I I, ~7"r /I-.. N,A| SaS. N,A

N N c)

Fig. 1. Two-pion exchange in the nucleon-nucleon interaction: (a) iterative boxes, (b) crossed boxes, (c) correlated two-pion exchange. The iterative box with an N N intermediate state is generated in the scattering equation by iterating the one-pion exchange. In OBE models all other contributions are parametrized by o'OBE and p-exchange. In the Bonn N N potential [ 10] the uncorrelated contributions (a) and (b) are evaluated explicitly whereas the correlated ~rTr exchange is parametrized by tr'- and p-exchange.

a K K molecule bound in the ~rTr continuum, while a0(980) is dynamically generated by the KK threshold. Thus both mesons do not appear to be genuine quark model

resonances, with the consequence that SU(3) relations should not be applied to these mesons. (The non-strange members of the scalar nonet are expected to be at higher

energies.) In the Bonn potential [ 10] the intermediate range attraction is provided by uncorre-

lated (Fig. la, b) and correlated (Fig. lc) 7r~- exchange processes with NN, NA and

dd intermediate states. It is known from the study of the rrTr interaction that the 7rTr correlations are important mainly in the scalar-isoscalar and vector-isovector channel.

The Bonn potential includes such correlations, however only in a rough way, namely in terms of sharp mass o-'- and p-exchange. One disadvantage of such a simplified treatment is that this parametrization cannot be transported into the hyperon sector in a well-defined way. Therefore in the YN interaction models of the Jiilich group [31,

which start from the Bonn N N potential, the coupling constants of the fictitious o-'- meson at the strange vertices (AAo J, X,~tr') are essentially free parameters. In view of

the little empirical information about the YN interaction this feature is not satisfactory.

This is especially true for an extension of the YN models to baryon-baryon channels with strangeness S = - 2 . So far there is no empirical information about these channels (apart from some data on -~- and AA-hypernuclei). Still there is a large interest in these channels initiated by the prediction of the H-dibaryon by Jaffe [ 11 ]. The H-dibaryon is a deeply bound 6-quark state with the same quark content as the AA system (uuddss)

and with 1S0 quantum numbers. For the experimental search it is important to know whether conventional deuteron-like AA states exist. An analysis of possible S = - 2 bound states in the meson-exchange framework could provide valuable information in this regard, but requires a coupled channels treatment of AA, ~ , and N ~ channels. An extension of the Jtilich YN models to those channels is only of minor predictive power since the strength of the important ~ o "~ vertex is completely undetermined and cannot

be fixed by empirical data. These problems can be overcome by an explicit evaluation of correlated 7rTr-exchange

processes in the various baryon-baryon channels. A corresponding calculation has been

246 A. Reuber et aL /Nuclear Physics A 608 (1996) 243-304

C D C D

A B A B

C D

A B

Fig. 2. Two-pion and two-kaon exchange in the baryon-baryon process A + B --~ C + D. The unshaded ellipse denotes the direct coupling of the two pseudoscalar mesons/z~ = ~r~-, KK, KK to the baryons without any correlation effects (cf. Fig. 3). The shaded circle in th__e lower diagram for the correlated exchange stands for the full off-shell amplitude of the process/z~ ~/~*/.t r.

done for the N N case (Fig. lc) [ 12]. Starting point was a field-theoretic model for

both the N N --~ ~rzr Born amplitudes and the 7rTr -KK interaction [ 13]. With the help

o f unitarity and dispersion relations the amplitude for the correlated ~r~ exchange in

the N N interaction has been determined, showing characteristic discrepancies to o -t- and

p-exchange in the (full) Bonn potential.

For the correct description of the 7rTr interaction in the scalar-isoscalar channel the

coupling to the K K channel is essential, which is obvious from the interpretation of

the f0 (975) as a K K bound state. Apart from the z r ~ r - K K interaction model the K K

channel is not considered in Ref. [ 12], i.e. the coupling of the kaon to the nucleon is not taken into account. In fact, this approximation is justified in the N N system [ 14] ;

it is however not expected to work in channels involving hyperons. The aim of the present paper is to give a microscopic derivation of correlated 7rTr-

exchange process being of relevance in the various baryon-baryon channels with S = 0, - 1, - 2 (Fig. 2). The K K channel is treated on an equal footing with the 7r~" channel

in order to determine reliably the influence of K K correlations in the considered t-

channels. It is pointed out that further correlations, namely other KK-exchange processes

like, e.g., in the JP = 1- , I = 0 channel (w, ~b) and 7rK correlations especially in the K*-channel remain to be investigated. Our results replace the phenomenological o -t- and p-exchange in the Bonn N N and Jtilich YN models by correlated processes and

in this way eliminate undetermined model parameters (e.g. tr ' coupling constants). Corresponding interaction models thus have more predictive power and should make a

A. Reuber et al./Nuclear Physics A 608 (1996) 243-304 247

. . . . . 4 S S

+

Fig. 3. Microscopic model for the B8 ~ ~ ¢rcr, KK Born amplitudes. The solid lines denote (anti-)baryons, the dashed lines the pseudoscalar mesons ¢rcr or KK. The sum over exchanged baryons X contains all members

l + j p + of the JP = i octet and the = 2 ~ decuplet which can be exchanged in accordance with the conservation of strangeness and isospin.

sensible treatment of S = - 2 baryon-baryon channels possible. The formal treatment is similar to that of Refs. [ 12,15,16] dealing with correlated ¢rcr

exchange in the N N interaction. Due to the inclusion of the KK channel and different baryon masses (e.g. in the N A channel) generalizations are however required at some

places. Starting point is a field-theoretic model for the baryon-antibaryon ( B B ' )

7rcr, K K Born amplitudes in the j v = 0 +, 1 - channels. Besides various baryon-exchange

terms the model includes also a p-pole term (cf. Fig. 3) in consistency to the ¢ r c r - K K

interaction model [ 13,17]. These Born amplitudes are analytically continued into the

pseudophysical region below the B B ~ threshold. The solution of a covariant scattering equation with full inclusion of ¢ r r r - KK correlations yields the B B --7 ~ ¢r¢r, K K

amplitudes in the pseudophysical region. In the N N ~ ¢rcr channel these amplitudes

are then adjusted to quasiempirical information [ 18,19], which has been obtained by analytic continuation of ¢rN and ¢rzr data. With the assumption of SU(6) symmetry for the coupling constants a parameter-free description of the other particle channels can then be achieved.

Via unitarity relations the products of B B ~ ~ ¢r~r, K K amplitudes fix the singularity

structure of the baryon-baryon amplitudes for ¢rrr- and KK-exchange. Assuming ana-

lyticity for the amplitudes dispersion relations can be formulated for the baryon-baryon amplitudes, which connect physical amplitudes in the s-channel with singularities and

discontinuities of these amplitudes in the pseudophysical region of the t-channel processes. With a suitable subtraction of uncorrelated contributions, which are calculatexl directly in the s-channel and therefore guaranteed to have the correct energy behavior, we

finally obtain the amplitudes for correlated ¢rrr- and KK-exchange in the baryon-baryon system.

In the next section we describe the underlying formalism which is used to derive correlated ¢rcr- and KK-exchange potentials for the baryon-baryon amplitudes in the or- and p-channel. Furthermore we present our microscopic model for the required B B ~ --~ 7r~r, K K amplitudes. Section 3 contains our results and also a comparison with those obtained from other models. The paper ends with some concluding remarks.

248 A. Reuber et al./Nuclear Physics A 608 (1996) 243-304

t

C,Pc D,PD

>

A'PA f B'PB S

Fig. 4. Two-particle scattering process.

2. Formalism

2.1. Kinematics and amplitudes

The kinematics of a two-body scattering process A + B ~ C + D (cf. Fig. 4)

is uniquely determined by the 4-momenta PA, PB, Pc, PD of the particles. Taking into

account the on-mass-shell relations (p2 = M2x, X = A . . . . . D) and the conservation of

the total 4-momentum (PA ÷ PB = PC + PO) only two independent Lorentz scalars can be built out of these momenta. For these Lorentz scalars one usually introduces the three

Mandelstam variables

s= (PA ÷ PB) 2 = (Pc ÷ PD) 2 ,

t = ( p C -- PA) 2 = (pB - pD ) 2,

u = ( p o - - P a ) 2 = ( P B - - P c ) 2 , (1)

which are related by

s + t + u = M~A + M~ + M~ + M ~ -- ~ . (2)

By crossing the scattering process A + B ~ C + D is closely related to two other processes, as indicated by Fig. 4:

A,pa + B,pB ---*C, pc + D, pD 's-channel',

A,pA + C , - P c - - + B , - - p B + D,pD ' t-channel', (3)

A,pa + D, --PD---+C, pc + B, --PB 'u-channel' .

Here, the channels are named according to the Mandelstam variable which denotes the squared total energy in the center-of-mass (c.m.) system.

For the s-channel process the particle 4-momenta in the c.m. system read

(EA) PA = Ps '

with Ex = x/M2x + (final) state can be

2 [ S - ( M A - ~ M B ) 2 ] IS Ps = 4s

~ = I s - ( M c + MD) 2] Is

4s

The scattering angle O, = g ( P s ,

s ( t - - u ) + (M2A--

A. Reuber et al./Nuclear Physics A 608 (1996) 243-304

EB P c = , PD = PB = --Ps ' qs --qs

p2 x. The modulus of the relative momentum Ps (qs) expressed in terms of s:

- (Ma - MB) 2]

- ( M c - - MD) e]

qs) is related to the Mandelstam variables by

249

( 4 )

of the initial

(5)

e / 1 -- - M A), : - M ~ ) Q . A = ~ ( M B

P . a = ( s - u ) / 4 .

(11)

In covariant field theory the scattering amplitude T for a general process with ni (n f) particles in the initial (final) state Ii) (I f ) ) is related to the (unitary) S-matrix by

(f[ S 1i) = ( f l i) - i ~ 6 ( 4 ) ( P f - e i ) ( f [ T 1i), (12) ~v/1V f lVi

Mandelstam variables

/ I 2 ---- t ,

p2= 1 2 ( M A + M2c ) -- t / 4 ,

Q2 1 ~ + M2o) - t / 4 = ~ ( M a

cos O,. = (6) 4S[Ps[[qs[

For the t-channel process A + C --~ B -4- D the c.m. 4-momenta of the particles are

EA Ec Po , --PB --qt PA = Pt --Pc --Pt qt

The analog of Eqs. (5) for the modulus of the relative momenta Pt, qt and the scattering

angle Ot = ~ (Pr, qt) now reads

2 [ ' - ( M A + M C ) 2] [ t - - (MA--MC) 2] Pt = 4t '

¢,__ 4t ' (8)

cosOt = t (u - s) + (M2A -- M2)(M2o - M 2) (9)

4tlP, llq, I Instead of the particle 4-momenta, usually the total 4-momentum and the following

three linear combinations are used to characterize the kinematics of a two-body scattering

process:

A = _ _ p C - - p t = P B - - P o , P - ~ l ( p a W p c ) , Q - - - - l ( p B + P o ) . (10)

The scalar products of these three momenta can again be expressed in terms of the


where Pi (Pf) denotes the total 4-momentum in the initial (final) state. The factors Nx (x = i, f ) are given by

nx 2Ej Nx = (2~r) 3n" 1-'[

j=l (2Mj)bj ' (13)

where the j th particle of channel x has mass M j, momentum p j, energy Ej = (M} + pj 2)1/2 and spin sj. (Spin and isospin quantum numbers are suppressed for the mo-

ment.) The exponent bj is given by

bj = ( 0 if 2Sj e v e n ,

I if 2sj odd. (14)

For a two-body scattering process A + B ~ C + D Eq. (12) reads

<Cpc, DPDI S IApA, BpB>

= (Cpc, DpDI A p t , BpB)

i /(2MA)bA(2MB)b"(2Mc)bc(2MD)bD6(4)(p c V r --AA E e O + P ° - P - P " )

x (Cpc, DpDI T Iapa, BpB>. (15)

Particles with spin (and helicity Ax) are described in the helicity basis according to the conventions of Jacob and Wick [20]. By separating off the helicity spinors ux(Px, ax) of the particles from the scattering amplitudes one obtains the transition matrix .A4. If, for instance, all four particles of the process A + B ~ C + D are spin-½ baryons, the transition matrix .A4 is a 16 x 16 matrix in spinor space and is defined by

<CPcAc,DPDADITIAPAaA,BpBAB>

=UC(Pc,AC)UD(PD, AD).A/IAB~CD(~ Q)UA(PA,AA)UB(pB,AB). (16)

Now the transition matrix A// can be constructed as a linear combination of the so-called kinematic covariants Oi, which are like .A// operators in spinor space.

.A4( P,Q) = ~"~ ci(s,t)Oi( P,Q) . (17) i

The Oi are built up from the Dirac y-matrices and the momenta P and Q (cf. Eq. (10) ) in such a way that their matrix dements are Lorentz-invariant quantities. The invariant amplitudes ci( s, t) are Lorentz scalars.

The number of independent kinematic covariants for a given scattering process corresponds to the number of independent helicity amplitudes and is determined by the dimension of the spinor space and invariance principles for the underlying interaction. For the scattering of four spin-½ baryons (A + B ~ C + D) there are in general eight independent kinematic covariants. For the elastic scattering (A + B ~ A + B) their


number is reduced to six due to time reversal invariance. For the 'superelastic' scattering of four identical particles (A + A ~ A + A) the number of independent kinematic covariants is further reduced to five due to the symmetry under particle exchange.

The set of kinematic covariants is not unique. However, for the forthcoming it is essential that the Oi are chosen in such a way that the invariant amplitudes ci(s, t) do not contain any kinematic singularities, but only 'physical' singularities demanded by unitarity. In the case of four spin-½ particles this condition is fulfilled by the set of eight covariants given in Ref. [21 ]. This set is based on the so-called Fermi covariants S,P,V,A,T:

S =- (9s = 14®14,

P - (gP = Y s ® Y s , V - (gv = yu ® yu ,

A =- OA =ySy~ ,®ySy u ,

T - 07" = l o" ® o "u" 2 Izv (Q6 = 14 ® y # P # - y~Qt, ® 14,

(97 = Y5 ® YsY~ Pu , (98 = y5yuQ # ® y5 •

(18)

These covariants are of the form Oi(P,Q) = o } l ) (Q ) ® (9}2)(p) and the matrix

elements have to be evaluated according to

uc(Pc , Ac )uo (po , Ao) (9i(P, Q)UA(PA, AA)UB(p~, AB)

m

2.2. Dispersion relations for baryon-baryon amplitudes of ¢r~ and KK exchange

For a general two-body scattering process the physical regions of the Mandelstam variables for the s-, t- and u-channel reaction are non-overlapping. Therefore the transition matrices in the three channels can be interpreted as independent branches of one operator A4 defined in the various kinematic regions. In the t-channel (A + C ~ B + D), for instance, the scattering amplitude is related to the invariant amplitudes by

(-B - PBAB, DPoAD I T IAPAAA,-C -- Pc Ac) 8

= E C i ( S , t ) ['Uc(--pC,~C)O~I)(Q)UA(PA,~A)] i=l

× ['UD(PD,/~D)O~2)(p)UB(--pB, AB)]. (20)

By introducing the concept of analyticity the invariant amplitudes in the three channels become closely related: the invariant amplitudes Ci(S,t) are supposed to be analytic functions (except for the physical singularities) in the whole complex st-plane. There- fore, if all these physical singularities are known the ci(s, t) can be deduced at any point in the complex Mandelstam plane by the formulation of dispersion integrals.

The singularity structure of the invariant amplitudes is completely determined by the unitarity of the S-matrix. In terms of the scattering amplitude unitarity of the S-matrix is expressed as

252 A. Reuber et a l . /Nuc lear Physics A 608 (1996) 243-304

(27"r) 4 i [ ( f l T l i ) - ( f l T f Ii)] = ~ N, 6 ( 4 ) ( P i - e n ) { f l T t In) (n lT] i ) ' (21)

n

where Pn and Pf = Pi denote the total 4-momenta. The summation in Eq. (21) is to be understood over all physical states n, i.e. states which are energetically accessible for the system with energy p/0.

The singularities of the baryon-baryon (A + B ~ C + D) amplitudes generated by (correlated) ¢rTr and KK exchange are most easily derived using the unitarity relation for the t-channel reaction A + C ~ D -4- B. For this, one has to restrict the summation in Eq. (21) to physical 7rrr and KK states; i.e. contributions of single-meson poles or of heavier two-meson and multi-meson (e.g. 3¢r, 4¢r) channels are disregarded. In the c.m. system (19 = p f = (v'7, 0)) the summation over the states n then reads

(2NTr~4t~(4)(pi - P n ) [ n ) ( n l

n

- (2¢r)------- ~ ~ _ Nu- ~ d3k t3(v / ' i -2 tou(k) )4 t%(k) 2 Ilt-~,k)(I.t-~,k[ Iz[x=rrTr,KK

327r 2 ~ _ t bt ft=~r zr ,K K

(22)

with a~j, (k) = J m 2 + k 2 and the on-shell momentum k**~ = ~ t / / ~ - m~. The symmetry I t I '

factor Nt, ~ is introduced in order to obtain the correct phase space in case of identical particles:

{ 1/2 i f / , t~ = rrcr, 1 i f / x ~ = KK. (23)

Now, the kinematic covariants Oi in Eq. (18) have been chosen in such a way [21 ] that the left-hand side of Eq. (21) yields the imaginary part of the invariant amplitudes, i.e.

i(D-B, q AD ABI T - T t I A-C, p aA ac)

= - 2 ~ Im [ci(s, t) ] -@(-p , AC)-ffD(q, AD) i

xOi(P, Q)ua(p, AA)vB(--q, AB) . (24)

Hence, the unitarity relation for the helicity amplitudes of the process A + C" --~ D + becomes in the c.m. system

Im [ ci( s, t) ] -Vc ( - P , Ac )-ffD ( q, /~D ) (.~i( P, Q )u A ( p, AA )VB(--q, AB ) i


1 ~t-- 6 4 r r 2 Z N ~ 4m2u 0( t - 4m 2)

_ t i.x fz=,n.,n" . K K

x f T ID-~,qaoaB)*(tz-~,k~,zl T [a -c ,p ,~ AaC) . (25)

From the beginning, the unitarity relations (21), (25) are just defined above the

kinematic threshold of the process A + C- ~ D + B, that is for t >>, to = m a x { M a +

M c , MB + M o } . However, they can be continued analytically into the pseudophysical region (4m 2 ~< t ~< to) as discussed in Ref. [22]. Below the kinematic threshold the

baryon-antibaryon momenta become imaginary. According to Ref. [22] the unitarity relation for the S-matrix, which can be written symbolically as [ S ( p ) ] * S ( p ) = 1, has

to be continued to complex momenta by [ S ( p * ) ] * S ( p ) = 1. As explained in Ref. [23], for imaginary momenta this is equivalent to evaluating the expression [S(p)] * S ( p ) = 1

with real dummy variables for the momenta and replacing these dummy variables at the very end (after having performed all complex conjugations) by the imaginary momenta.

The right-hand side of Eq. (25) obviously vanishes below the 7r~ threshold at t =

4m~. Since for the processes considered here the left-hand cuts, which are due to unitarity constraints for the u-channel process, do not extend up to t = 4m~ (for fixed s lying

inside the physical s-channel region), the invariant amplitudes c i ( s , t) are real-analytic

functions of t, i.e. c i ( s , t*) = c i ( s , t )*. According to Eq. (25) and

2 i lm [ C i ( S , f + iE) ] = c i ( s , t + iE) - Ci(S , t -- ie) , (26)

c i ( s , t) has a branch cut along the real t-axis extending from t = 4m~ to t = +oc. Corresponding statements hold for the K K branch cut.

For c.m. energies below 1 GeV the zrTr interaction is dominated by the J I = 00, 11

partial waves [ 13]. At low transferred momenta, relevant in low-energy baryon-baryon scattering, correlations between two exchanged pions are therefore only considerable

when the exchanged 7r¢r system is in a state with relative angular momentum J = 0 and isospin I = 0 ( 'o '-channel ') or J = l and I = 1 ( 'p-channel ' ) [12]. In our approach only the correlated part of two-pion and two-kaon exchange is evaluated by dispersion-

theoretic means. The uncorrelated part has to be calculated directly in the s-channel in order to include all t-channel partial waves and to guarantee the correct energy dependence of this contribution. In the A N channel, for instance, the iterative two-pion

exchange with an N £ intermediate state becomes complex above the N £ threshold. This behavior cannot be reproduced with a single-variable dispersion relation in the t-channel which will be applied to the correlated contribution (see below).

As stated before, we limit the following dispersion-theoretic considerations to the o- and p-channel of 7rqr ( K K ) exchange. For this, the B B ' ~ lx-~ amplitudes on the right-hand side of the unitarity relation (25) are decomposed into partial waves [24]. Choosing the coordinate system so that p points along the e-axis and q lies in the ~ - p l a n e the partial wave decomposition gives

254 A. Reuber et a l . /Nuc lear Physics A 608 (1996) 243-304

f d21,,,~,(,,-~, k~ l q <1*-~, k,~x,I p AAAC) T Io- , aDan)* T lAW, = Z 2 J + l

4rr d~*-ac'a°-as(c°sOt) (Ix-~l TJ(t) ID-B, aoa.>* J

x (Izgl U(t) lAW, aaac), (27)

where the on-shell momenta p = p( t ) and q = q(t) (see Eq. (8)) are suppressed as arguments of the partial wave decomposed T matrix elements. Note that the right- hand side depends on the Mandelstam variable s only via the angle Ot = Ot(s, t) (see Eq. (9)) between p and q. Now, by restricting the sum over J in Eq. (27) to J = 0 ( J = 1 ) the contribution c~ J) (s, t) = c.'[(s, t) (c~(s, t) ) of the rr~r and KK intermediate states to the discontinuity of the invariant amplitudes ci(s, t) in the o- (p) channel can be isolated. From the unitarity relation (25) we obtain

E Im[c~J)(s ' t)] "SC(-p'AC) uD(q'AD)('gi(P'Q)UA(p'AA)vB(--q'A") i

= E dJ (cosOt)H~-~(t) Aa -- AC ,AD -- AB

/ . t # = ~ , K R

X {Iz-~] TJ(t) ID-B, aoan)*(/x~l TJ(t) lAW, AAAC),

with the abbreviation

I

2J + 1 I t - 4m 2 ~ 3 Nt*-~ V t ~* O( t - 4m2u) "

(28)

(29)

Eq. (28) is a system of linear equations for the discontinuities Im[c~J)(s,t)]. Its solution provides the discontinuities as linear combinations of the following products of BB' --+ !~-~ helicity amplitudes:

F J = aoaB,aaac E H~-~ (t)(ix'~[TJ(t) ID-B'a°az)*(tz'~lTJ(t) [aW, aaac). laft=rrrr,K R

(30)

From the symmetry properties of the BB' --+ tz-~ helicity amplitudes, which are due to the parity invariance of the underlying strong interaction (see e.g. Ref. [20] ),

(tz'~l TJ(k ,q) IB-B 7, AA') = (/z~l TJ(k ,q) IBB -7, -A - A'), (31)

the following relations can be deduced:

F J = F J F J = F J avaB,aaac (-ao)(-aa),aaac = aoa~,(-aa)(-ac) (-ao)(-as),(-aa)(-ac), (32)

Therefore only four linearly independent F J exist for J > 0. For J = 0 there AoAB,AaAc is only one independent BB --7 --+ /.t-~ helicity amplitude, (/x~l VJ(k,q) IBB -7, ++) , and consequently only one independent F J Using the explicit representation of ADAB,)IAAC"


helicity spinors in Appendix A the evaluation of the matrix elements of the kinematic covariants (-9i in Eq. (28) is straightforward.

In case of une, qual baryon masses MB ~= MB, the analytic structure of the BB ~ -~ /z~ helicity amplitudes is much more involved compared to the case when the baryon masses are equal. This complicates an analytic continuation of the amplitudes to the pseudophysical region. The AT channel is the only channel where this problem arises. In order to facilitate an easy handling of the expressions we treat this channel approximately by setting the mass of the A and of the X equal to the average mass (MA + M z ) / 2 . This approximation is justified for two reasons: First the mass difference between A and X hyperon is small (77 MeV) on the baryonic scale. Second, since we evaluate the uncorrelated part directly in the s-channel the important energy dependence of this contribution is not affected by our approximation. The correlated contribution on the other hand is known to have a rather smooth energy dependence which is supposed not to change drastically by our approximation.

In the following we restrict ourselves therefore to the case where

MA = Mc ~-- M , MB = MD =-- M' . (33)

Solving the system of linear equations (28) yields that in the o--channel ( J = 0) all discontinuities Im [c~(s, t)] vanish except for the scalar component

M' M FY=O Im [c~(t)] = ++,++, qP

Im [c°~(s,t)] =Ira [c~(s,t)] =Im [c~a(s,t)] = Im [c~(s,t)] =Im [c~(s,t)] = 0 .

(34)

In contrast, in the p-channel only the axial-vector component vanishes:

( g FJ=I , FJ=I Im [CPs(S, t)] = -½g(t)v '~qp cosOt(s, t ) [ - 2 v ~ ( g ' + g ) , . ++,++ + +_+_)

+ v ~ ( t + 4M~M) J=l J=l (F++,+_ + F+_ ++) 1,

Im [c~ (s, t)] = g( t )qp cos Or(s, t) [ (M' + M) ( t - 4 M ' M ) F J=l + - - , + - -

--2 V~V~(p2 F+J?,+- + qZFl=l+-,++) l'" ,

M~M~ F g=l I m [ c ~ ( t ) ] = g ( t ) v ~ [ - v q ( M ' + M ) ( t / 4 - M ' Z - M 2 + , +-,+-

+2v~M'M(pZFS+?,+_ + qZVJ=l+_ ++) l'" ,

=0 ,

Im [c~(t)] = ¼g(t)t[ (M' + M) (t - 4 M ' M ) F j=l + - , + -

-2v~V~(p2 F+~+I,+- + qZFS=l+-,++)-I ,

- v ~ V + _ + _ I m [c~(t)] = g ( t ) v q [ ( M '2 M 2) J=l

_ 2 v ~ ( p 2 M' FY+=+l,+_ 2 M F J=' - q +_ ++)] , (35)

with


M ~ M g ( t ) ---- 2 q 2 p 2 ( M , + M ) t " (36)

The discontinuity of the pseudoscalar and the tensor component are obviously related

by

Im [ c ~ ( s , t ) ] - 4qp cos~9,im [cP(t) ] (9)__. _ _ u - Slm [c~.(t)] . (37) t t

Consequently, only four of the five non-vanishing discontinuities are linearly independent in alignment with the number of independent J--~ Faoan,AAaC "

Except for poles (corresponding to single-particle exchange) and cuts the invariant amplitudes c~ J) (s fixed, t) are real-analytic functions of t. Therefore, fixed-s dispersion

relations can be formulated for the c~J)(s , t ) . Since we want to restrict the dispersion- theoretic evaluation to the contribution of (correlated) ¢r~r and K K exchange to the baryon-baryon amplitudes we take into account only those singularities which are generated by ~Tr and K K intermediate states, namely the discontinuities of Eqs. (34) - (35) due to the ~-~r and K K unitarity cut ( 'right-hand cut' ). The left-hand cuts, which are due to unitarity constraints for the u-channel reaction, can be neglected in the baryon-baryon channels considered here, since they start at large, negative t-values (from which they extend to - c ~ ) and are therefore far away from the physical region relevant for low- energy s-channel processes. For identical baryons (e.g. N N ~ N N ) this is only true if the dispersion relations are applied only to the direct baryon-baryon amplitude and the antisymmetrization of the amplitudes is not taken into account from the very beginning

but just for the final s-channel amplitudes. Otherwise, crossing of the exchange diagram would result in a u-channel cut starting at u = 4m~ which could not be neglected in the dispersion integrals [ 16].

In this work the B B ~ --~ tz-~ amplitudes, which enter in Eq. (28), are derived from a microscopic model which is based on the hadron-exchange picture (see Section 2.4). Of course, this model has a limited range of validity: for energies far beyond tma x t ~ 100m~ it cannot provide reliable results. The dispersion integral for the invariant amplitudes extending in principle along the whole ~'Tr right-hand cut has therefore to be limited to an upper bound i (tmax). In addition left-hand cuts and unphysical cuts introduced for instance by the form factor prescription of the microscopic model for the B B ~

/z~ amplitudes are neglected. Because of these approximations of the exact expressions, which are necessary in order to obtain a solution of the physical problem, the formulation of either a dispersion relation or a subtracted dispersion relation might lead to different results for the amplitudes although both should be mathematically equivalent. However, the ambiguity which dispersion relation to choose can be avoided by demanding that the analytic structure of the resulting c~ J) (s, t) should agree as far as possible with the expressions for sharp o-- and p-exchange in the baryon-baryon interaction.

The transition amplitude .A4 '~ for the exchange of a scalar o--meson with mass m,~ between two JP = 1/2 + baryons A and B follows from the interaction Lagrangians ( X = A , B )

£ x x ~ ( x ) = g x x ~ b x ( X ) ~ b x ( x ) ~ b ~ ( x ) (38)


(see Appendix A for the hadronic field operators). The result is

r~ct) O- A'4~ ( t) = gAac, gBB~ t _---~ ~ s , (39)

where a form factor F~(t) has to be applied at each vertex since the exchanged o--meson is far away from its mass shell. This form factor is parametrized in the conventional monopole form

A~ 2 - - / T I , o -

F~r(t) - A2 ~ - t (40)

with a cutoff mass A~ assumed to be uniform for both vertices.

The construction of the amplitude for p-exchange in the transition A 4. B -~ C + D

( w i t h MA = Mc ~ M and MB = Mo -- M p) starts from the interaction Lagrangians

£xyp( x ) = gxypJ, x(X)~,~,y ( x) ~ ( x )

. f x r p - r " x ' t r -

(+h.c. if X v~ Y) (41)

with (XY) = ( A C ) , ( B D ) . According to the conventional Feynman rules, using Eq. ( 11 )

and the generalized Gordon decomposition [25] for the spinors of two Dirac particles

X and Y,

-ffx(P', A') [ ( m x + m v ) y ~ - (p ' + p )~ - io'U"(p ' - p)~] uv(p , A) = 0, (42)

the transition amplitude A/[ p turns out to be

--1 [fACpfBI)p (S A d P ( P ' Q ) = - L 4M~ " - u) Os + GacpGnDp Ov

- GACpfBDp -t- GBDpfACp 502] , (43) _~ GAcpft~op GBOofACp 06 + 2M,v 2MN J

with

M x + Mr ,~ Gxrp -~ gxrt, 4. 2~t-N Jx~'o (44)

and the 4-momenta P and Q defined in Eq. (10). 502 is one of the so-called perturbative

covariants introduced in Ref. [ 15] as

502 = --14 Q T/.LP/~ -- T/zO/z @ 14. (45)

502 can be expanded in terms of the Oi of Eq. (18) since they form a complete set of kinematic covariants. For the given baryon masses one finds

1 502 = 2 (M 4- M') [(u - s) ( Os 4, Op) - 4 M m ' O v 4, tOT 4" 2(M - Mt)06] .

(46)


After replacing 792 in Eq. (43) the final result for .M p reads

I F~ ~ t ~ I gacpfBDp ÷ fAcpgoDp (U -- S) OS A4P(P' Q) = t - m~ _ 4MNMtot

gACpgBDp Mr2 M2 ] × + gACpfBDp MNMtot + fACpgBDp MN'-Mtot ] Ov

MA ×(gACpfBDPM~Mtot fACPgBDPM-~tot) 06

GACpfBDp ÷ facpGnDp } × 4MIvMtot [ (u - s) Or, + t Or] (47)

with Mto t = M + M' and the form factor Fp(t) parametrized according to Eq. (40). By comparison of the discontinuities in the o'- and p-channel in Eqs. (34), (35) with

the transition amplitudes for sharp o'- and p-exchange it follows that c~, c~ and c~ obey an unsubtracted dispersion relation,

,= [ ] 1 f Im c}J)(s , t ') c}J)(s,t) . . . . . . ] dt p . (48)

¢r t t -- t 4=~

The tensor component of sharp p-exchange is proportional to t (cf. (47)) . In order to generate this factor t also for the tensor component of correlated ¢rcr and KK exchange a subtracted dispersion relation (subtraction point to and subtraction constant cr p (s, t0) = 0) is assumed for the invariant amplitude c~(s, t):

t tm~

c~(s, t) = t - - / I m [c~(s, t ' )] "/t 'dt,. (49) qr t t - t

4,,~

Similarly, the u - s dependence of the (pseudo-)scalar component of sharp p-exchange (cf. Eq. (47)) can be reproduced by assuming a subtracted dispersion relation for c~(s , t ) and cPe(s,t) with to(s) = Z - 2 s and ~,e(s , to(s) ) =0:

t

c~.l,(S,t) =u-s~. flm[c~'l"(~f')!/(u'-s)dt'- - - - , (50)

4m~

where, according to Eq. (2), s ÷ t + u = s + t ~ + u ~ = Z.

2.3. Baryon-baryon interaction arising from correlated zr~r and KK exchange

The invariant amplitudes constructed in the preceding section using dispersion theory still contain the uncorrelated contributions of ~r¢r and KK exchange. Investigating the problem of baryon-baryon scattering requires the knowledge of the on-shell scattering amplitude T, which is usually obtained as a solution of a scattering equation T = V+ ~,GT


that iterates the interaction kernel V. In general, V contains besides other terms one-pion and one-kaon exchange contributions as well as the contribution V2,~ from two-pion and two-kaon exchange. But iterating 7r- and K-exchange in the second order term I/UV also generates two-pion and two-kaon exchange contributions to the scattering amplitude. In order to avoid double counting these 'iterative' contributions therefore have to be left out from the dispersion-theoretically calculated ½,~. As stated above we even go beyond this and subtract all uncorrelated contributions from ½,~. By this the dispersion-theoretic calculations can be restricted to the o-- and p-channel (since only there significant correlations occur in the kinematic region considered), whereas the uncorrelated contributions are evaluated in the s-channel and therefore contain all t-channel partial waves.

In order to eliminate the uncorrelated contributions from V2,r we determine the discontinuities (~) Im[Ci,Born(S, t)] generated from the BB -7 ---+/z~ Born amplitudes V ~ (i.e. no 7rTr- KK correlations included) using as before the unitarity relation (28) (with T J replaced by V J) and subtract them finally from the full discontinuities Im[c~J)(s, t)]. (The contributions of the p-pole diagram to the BB' ~ tz-fi Born amplitudes must not be subtracted since the corresponding s-channel processes are not included explicitly in V.) Hence, for the invariant amplitudes of correlated ~rrr and KK exchange, P~J)(s, t), the (unsubtracted) dispersion relation (48) has to be modified to

t~ax . j -

6~j)(s,t ) = 1 f p~ ' ( s , t ' ) -~ J 7 ~ - t dt ' , ( Y ) = o - , p , (51) 4m2~

where the spectral function p}J) is given by

[c (J) " t')] . (52t p ~ ' ) ( s , " ) - Im [ c ~ ' ) ( s , t ' ) ] - I m L i,Bornl.S,

Corresponding expressions hold for the subtracted dispersion relations (49) and (50). Now the baryon-baryon helicity amplitudes arising from correlated rrcr and KK

exchange can be evaluated according to Eqs. (16), (17)

(CD, q Ac aol v( J) IAB, p AAAB) "2~r

='ffc(q, hc) -ffo(--q, Ao) ~)(J)t P "2,~ " ' Q ) UA(P, AA) uB(--p, AB)

with

]2~, r(y) (p,Q) = Z g } Y ) ( s , t ) O i ( P , Q ) ( (J) =o',p) . (53) i

The partial wave decomposition of these matrix elements then proceeds as usual (see e.g. Refl [ 3] ).

Of course, when iterating the baryon-baryon interaction kernel in a scattering equation ½,r,corr has to be known off-shell. However, dispersion theory applies only to on-shell amplitudes and does not provide any information on the off-shell behavior of the amplitudes. Therefore an arbitrary prescription for the off-shell extrapolation of V2~r,corr has


to be defined [ 12], which is certainly a drawback of the dispersion-theoretical derivation of this potential. Nevertheless the characteristic features of correlated 7rTr and KK exchange like the strength of ½~,eo~ in the various baryon-baryon channels can already be discussed by means of the unique on-shell amplitudes. Therefore we postpone the discussion of how to extrapolate ½~,corr off-shell to a subsequent work.

The dispersion-theoretic amplitudes for correlated 7r~r and KK exchange (Eqs. (34), (35)) have been constructed in such a way that their operator structure agrees as far as possible with sharp or- and p-exchange (39), (47). Therefore our results for the correlated exchange can be parametrized in terms of tr- and p-exchange, i.e. the products of coupling constants for o-- and p-exchange are replaced by effective coupling strengths G (J) (s, t), which contain the full s- and t-dependence of the dispersion-theoretic results. In the o--channel this gives for the elastic baryon-baryon process A + B ~ A + B

t lmax

( t - m 2)~~. . ( t - m ~ ) f Ps-~Zt'~ P dt' gAatrgBBo, , G A B . . . , A B ( t ) -- -F2- ~ Cs[ , l ) - ¢rF2(t ) J ~ . (54)

4m 2

Note that sharp o--exchange (Eq. (39)) would correspond to a spectral function

o" S Ps( , tt) --gaa~rgBB~rt~(t' -- m~) (55)

(except for form factors). This suggests to interpret the spectral function as a function that denotes the strength of an exchange process depending on the invariant mass of the exchanged system (here 7rTr, KK).

By comparing the coefficients of the kinematic covariants Oi in Eqs. (47) and (53) we obtain in the p-channel

( t - m ~ ) [4MM'C](S'~ c~( ' ) (M t ] , WGPAB--'co(t) -- -F~(-D _ + - M ) ~ ( t ) j gaBpgBDp

gABpfBDp___~WGPAB._~CD(t) = ( t - -m~) [ ~ : ] (s,_) + e (t) ,

( t - m ~ ) _ [ e f ( s , t ) ] fABpgBOp ~ 7VGPAB~CD(t) . . . . . MN FJ(t ) aM' ---u-s- e~(t) ,

f a ~ --+ ITGP tt~ = ( t - m z ) 4M~cPs(S't) ~ C~ (S~ t) BpJBDp AB---*CO~" " -FT('-D u - s (56)

Obviously the effective coupling strengths do not depend on s but only on t. This is only possible by choosing the subtracted dispersion relation (50) for ~,p(S, t), since the integrand of the dispersion integral becomes independent of s (Im[c~,e(s, t~)] cx cosOt ( s , t t) ~ u ~ - s). Therefore ~ P , e ( S , t ) depends on s only by the factor (u - s), which cancels out exactly when calculating the effective coupling strengths. It should be emphasized that the parametrization of ½~,~o~r discussed here does (so far) not contain any approximations.


2.3.1. lsospin crossing Up to now, in favor of a clear representation of the dispersion-theoretical calculation,

we have suppressed isospin degrees of freedom. The isospin structure of the BB -7 --~/z~ amplitudes will be discussed in the next sections when the microscopic model for these amplitudes is presented. However, when 'crossing' is applied to the baryon-baryon (s- channel) and baryon-antibaryon (t-channel) amplitudes it has to be kept in mind that the total isospin in the various channels is constructed from different combinations of particle isospins. The total isospin Is of the s-channel process A + B ~ C + D is composed out of [ IA ® IB ]/, or [ Ic ® lo ] I, and that of the t-channel process A + C- --+ D + B out of [ Ia ® I~ ] t, or [ 1 o ® I~ ] I,.

Consequently, besides the analytic continuation of the invariant amplitudes in s and t the recoupling of the particle isospins has to be taken into account when crossing amplitudes. Therefore, the isospin amplitudes AB--.CO T~ (ls) and TAC~DB(It) of the s- and t-channel processes being independent of the isospin projections m,. and mr are linearly related:

AB--.-*CD Z T~ (Is) = XAB.co(ls, lt)TAC~DS(lt) , (57) It

where XAB,CD(Is,It) is the so-called isospin-crossing matrix. Note that our isospin- crossing matrix differs from the XAB,CO (Is, It) introduced in Refs. [26,27] since their t-channel process (D + B ~ C + A) differs from the one in Section 2.1:

l"~lB+lo+lt--21~'~z :" It) (58) XAB,CO(Is, It) = ( - - l ) ABA,DCkIs, .

As shown in Ref. [26] the isospin-crossing matrix f(AB,cD(ls, It) can be expressed in terms of Clebsch-Gordan coefficients (or Wigner's 6j symbols)

f(aB'co(l~'It)=r/ar/D(2lt+l)(--l)Is+t'-IA+'°{ I A I B I ' } I D I c It

=r/ar/D E (--1)la+l°+ma+mt){lalBmamB[ Isms)(lclomcmol Ism~) m A ,m B ,

m C , m D ,hi t

× (Ic Iamc (--ma ) ] Itmt) ( Ioln (--rno ) ms[ ltmt) (59)

with m~ (Imsl <<. Is) arbitrary. For a particle A with isospin Ia and isospin projection ma the particle state IA) and the isospin state Ilm)a might differ in sign. The isospin state of the antiparticle A is generated by applying the G-parity operator G to Ilm)A [26]:

IIm),~ = r/a~ [Im)a. (60)

With the phase convention used here for the SU(3) field operators (e.g. when calculating the isospin factors of the BB ~ ~ i.t-~ amplitudes in the next section) the phase r/a, which is independent of ma, comes out to be [27]

r/A = ( --1 )la--Ya/2 , (61)

262 A. Reuber et al . /Nuclear Physics A 608 (1996) 243-304

Table 1 Isospin-crossing matrices X(ls , It) for the various baryon-baryon channels A + B ~ C + D considered in this paper. The t-channel reaction related by crossing reads A + C ---* D + B. In the matrix X(Is, It) Is increases from top to bottom and It from left to right. The last two columns contain the factors F,r and Fp (see text for further explanation)

A B C D Is It X (Is, It ) Fo- Fp

1/2 - 3 / 2 ' ~ N N N N 0,I 0,1 1/2 ! /2 J i /2 1/2

N A N A 1/2 0 -- 1/V"2 - l / v / 2

( 1 / v / g - 1 ) l /v/6 1/2 N 27 N 27 1/2, 3/2 0,I 1 /v~ 1/2

N 2" N A 1/2 I - - ~ - - -1 /v~

A A A A 0 0 1 1

1/3 --1 5/3 27 27 27 27 0,1,2 0,1,2 1/3 --1/2 - 5 / 6 ] 1/3 1/2

1/3 +1/2 1/6 ]

( - 1 / 2 3/2 ) - 1 / 2 - 1 / 2 N ..~ N ----- 0,1 0,1 - 1 / 2 --1/2

where Y denotes the hypercharge of particle A. Note that this phase convention differs from the one used in Ref. [26]. For the baryon-baryon processes considered in this work the isospin-crossing matrices are tabulated in Table 1.

By the partial wave decomposition the amplitude of correlated zrrr and K K exchange is separated into the contributions of the o-- ( I t = 0) and p-channel (It = 1 ). Suppressing the spin-momentum dependence the isospin amplitude can be written as

Ts( Is) = X( Is, O)T 7 + X( Is, 1)r~. (62)

The column X(ls ,O) (X(Is , 1)) of the isospin-crossing matrix agrees except for a constant factor Fo. (Fp) with the isospin factors for t-channel exchange of a o- (p) meson in the corresponding s-channel process. Conventionally these constant factors F,~ and Fp, which are also tabulated in Table 1, are extracted from the isospin-crossing matrix and put into the spectral functions (52), so that the isospin factors for the s-channel potential ½,~ agree with the isospin factors of tr- and p-exchange.

n

2.4. A microscopic model for the BB' --~ ~'Tr, K K transition amplitudes

In the preceding sections we have outlined the dispersion-theoretic calculation of correlated ~rTr and K K exchange in the baryon-baryon interaction starting from amplitudes for the transition of a baryon-antibaryon (BBO state to two pions (~'Tr) or a kaon and a antikaon (KK) . These amplitudes have to be known in the so-called pseudophysical region, i.e. for energies below the BB ~ threshold. However, in case of the process N N ~ zr~r, these amplitudes can be derived from empirical data by analytic continu-


ation of zrN and 7r,r scattering amplitudes, which are extracted from scattering data, into the pseudophysical region [ 18,28-30]. Corresponding analyses for the transitions YY~ ---, ,rTr, K K are out of sight since the required empirical information (e.g. 7rA scattering data) does not exist. An evaluation of correlated ~rzr and KK exchange in the YN or FY interaction therefore necessitates the construction of a microscopic model for the B B ' --~ 7mr, K K amplitudes. This model can be tested against the quasiempirical information for the N N --* 7mr amplitudes and then has to be extrapolated to the other channels of interest. In addition, only the use of a microscopic model for the

B B ~ --~ 7rTr, K K amplitudes allows a consistent treatment of medium modifications of the baryon-baryon interaction.

The microscopic model presented in the following is a generalization of the hadron- exchange model for the N N --~ 7rTr transition amplitudes of Ref. [ 17], where it was applied to the analysis of correlated two-pion exchange in the 7rN interaction. A main feature of the model presented here is the completely consistent treatment of its two components, namely the B B ' --~ , m r , K K Born amplitudes and the 7 r , r - K K correlations with respect to the o--, p-channels. Both components are derived in field theory from an ansatz for the hadronic Lagrangians.

The amplitudes for the processes B B ' --* 7r,r, K K are obtained from a scattering equation which can be written in operator form as

T = V + TGV, (63)

where the scattering amplitude T, the Born amplitude V and the Green operator G are operators in channel space of B B ' , 7rzr and K K .

For large t ~, contributions of the spectral functions of correlated zrzr and K K exchange to the dispersion integrals (48) - (50) values are suppressed by the denominator 1 / ( t ~-t) because in the physical s-channel t~<0. Since the unitarity cuts of the B B ~ states start far above the branch points of the ~rTr and K K cuts the contribution of the B B ' Green function GBT to the B B ~ --, 7mr, K K scattering amplitudes in Eq. (63) and to the spectral functions can be neglected. For the same reason the coupling to other mesonic channels like the p p channel can be renounced. Of course, by these approximations the range of validity of the microscopic model for the B B ---7 --~ I~-~ ( t x -~ = 1rTr, K K )

amplitudes is limited to low t' values. Therefore instead of integrating along the whole I 7r~r unitarity cut up to infinity the upper bound of the dispersion integrals, tmax, is set

to a value somewhere below the p p threshold at t t ~ 120m~ so that convergence of the integrals is achieved.

Taking into account only ,mr and K K intermediate states, i.e. neglecting Gs~ , we obtain for the two components of Eq. (63) which give the B B ' --~ Iz-~ amplitudes

In Fig. 5 this scattering equation for the B B ' --~ 1r~', K K amplitudes is represented symbolically.


I i I i I I ! I I i I i e I I

B g' B g'

~', _ _ , ' r t K , ~ , .K

-4-

Fig. 5. Model for the BB ~ --* tz I

~ K "K K R "~,K "~ ,K \ /

%% / \~ / \ \ / \ \ / /

11 \ \ \ I \ i I \ I I \ 1 \ 1 \ I \ \ \ i ~, I \

I \

"~ ~ "~ "~ K R "~,K "~,g

Fig. 6. Born amplitudes included in the model of Ref. [ 17] for the 7rTr-KK interaction.

The correlations T ~ / K - g _ ~ / r ~ are generated with a realistic model [ 17] of the

7 r ~ r - K K interaction. This meson-exchange model is a modified version of the so-called Jiilich ~r~r model [ 13]. The Born amplitudes of this model for the elastic 7rTr and K K channels as well as for the transition ~Tr --* K K are shown in Fig. 6. Besides

the t-channel (and in case of 7rTr ~ 7rzr also u-channel) exchanges of the vector mesons p, w, ~b, K* the s-channel exchanges (pole graphs) of the p, the scalar-isoscalar

• and the isoscalar tensor meson ]'2 are taken into account. The potentials derived from Fig. 6 are iterated in a coupled-channel calculation according to the prescription of Blankenbecler and Sugar [ 31 ]. The free parameters of the ~r~r- K K model of Ref. [ 17 ]

were adjusted to the empirical 7rTr phase shifts and inelasticities. A good agreement with the empirical data was achieved (cf. Fig. 7).

The scattering equation (64) is likewise solved using the ansatz [31] of Blanken- becler and Sugar (BbS) to reduce the 4-dimensional Bethe-Salpeter equation to a 3-dimensional equation which simplifies the calculation while retaining unitarity.

The total conserved 4-momentum P and the relative 4-momentum k' of the interme- d ia te /z ' /x ---7 = ¢r¢r, K K state are expressed in terms of the particle 4-momenta k ' ( 1 ) and k/(2) by

P = k ' ( 1 ) + k ' ( 2 ) , k' = ( k ' ( 1 ) - k ' ( 2 ) ) / 2 . (65)

Corresponding relations hold for the relative 4-momenta q of the initial B B ~ and k of the f i n a l / x ~ = ~rTr, K K state. In the center-of-mass system we have P = (V't 7, 0) with v/t 7 = EB (q) + EB, (q) , k ~ = k~( 1 ) = - k ~ ( 2 ) and, if the particles in the initial and final state are on their mass shell,

qo = ( E n ( q ) - E B , ( q ) ) / 2 , k0 = k~ = 0 (66)

with E B ( q ) := V ~ + M ~ .


I I • s I I • •

I I I I • •

I I I I Y I I I I

L = +

z ~ z E z E

265

K ~ ~ g K ~ ~ g I • s I ~ s

I i ~ 1 i

I I ¢¢ •• I

z [ E E z E z [

Fig. 7. ~mr phase shifts in the o'- and p-channel and the corresponding inelasticity in the o'-channel. (From Ref. 1171.

Now, according to the prescription of Blankenbecler and Sugar, the relativistic two- particle Green function, G#-~,, is replaced by a 3-dimensional Green function gu,g, .:~ 8(k~), which respects unitarity. Due to the 6 function 6(Uo), which sets the two intermediate particles (being of equal mass) equally off-mass-shell (k~(1) = k~(2)), the integration over k~ can be carried out immediately and the so-called Blankenbecler- Sugar equation for the helicity amplitudes is obtained:

D

(iz[t, kl T(t) IBB',q, zl,&) 1

= (tzfx, k I V(t)IB-BT, q, AIA2) + (27r)------- ~ Z Nl"~' tz' [~'=Ir~r,KR

[ d3ktT##,__,M,(k, kt; t) (tz'fz', k' I V(t) IBm, q, al •2) × (67) J OSk,(t--4os~, + ie)

where N#~, is defined as in Eq. (23) and Tu,x,,~uf, = Mu,f,,__,~,;~. Because of the rotational invariance of the underlying interactions the BbS equation

can be decomposed into partial waves [24]:

TJ ( k, q; t) I B~ 7, A~ &) 1

= (/z/21 VJ(k, q; t) IBB 7, AI A2) + (27r1------- ~ ~ Uu, ~, tz'/2'

J t . X f kt2dk ! T:'"'----~l't~ ( k, k , 1) (i,~'f_ztl g J ( k t, q, l) [o-n 7,/~l 1~2) (68)

tok,(t - 4to 2, + ie)

Here, q, k, k / denote the modulus of the corresponding 3-momenta. The extrapolation of the model for the NN --* or~r amplitudes to the other particle

channels is made under the assumption that the hadronic interactions are, except for the


Table 2 Particles considered in the ff~" - KK interaction model and the model for the BB ''7 ~ I~-~ Born amplitudes. Given are their masses and the relevant quantum numbers (B: baryon number, S: strangeness, J: spin, P: parity, 1: isospin)

Particle Mass (MeV) Bare mass (MeV) B S j e 1

139.57

K 495.82

e 1400.0

p 770.

.[2 1270.0

782.6

1020.0

K* 895.0

N 938.919

A 1115.68

2 1193.1

.,~ 1318.1

4 1232.

Y*----2(1385) 1385.

,,=* 1533.4

/2- 1672.5

1505.0

1151.26

1710.0

0 0 0 - 1 1 0 +1 0 -

0 0 0 ÷ 0

0 0 1- 1

0 0 2 + 0

0 0 1- 0

0 0 1- 0 1 0 +1 1-

0 l+ ! 2 2

- 1 l + 0 2

1+ 1 --1

-2 l+ l 2 2

0 3+ 3 2 2

- 1 3+ 1 2

-2 3+ l

1 - 3 -3+ 0 2

particle masses, SU(3)flavor symmetric. That means that the coupling constants at the various hadronic vertices are related to each other by SU(3) relations. In this way it comes out that all free parameters of the model for the B B ' --~ tz-~ amplitudes can be fixed by adjusting the N N ~ 7rTr amplitudes to the quasiempirical data [ 18,19] (see Section 3.1 ).

m

2.4.1. B B ' --~ ~rTr, K K Born ampl i tudes

In our model, the B B ~ ~ Iz-~ transition potentials are build up from a p-pole diagram I + j P 3 + and all diagrams in which a baryon out of the Je = ~ octet or the = ~ decuplet is

exchanged. Of course only those diagrams are considered which respect the conservation of isospin and strangeness. As an example Fig. 8 shows the Born amplitudes for the transition E ~ -+ ~r~r, K K . The particles occurring in this model are listed in Table 2 together with their masses and their basic quantum numbers.

In order to start from a maximum SU(3) symmetry our model differs slightly from the one presented in Ref. [ 17] for the N N - - , ~ amplitudes. Here, we include the exchange of the Y* ~ ,~(1385) in the N N --~ K K transition potential, which was neglected in Ref. [ 17]. In addition, the form factors at the hadronic vertices are chosen identical within a given SU(3) multiplet.

Starting point for the derivation of the various Born amplitudes are the following

360

tOO

i i f i i i i I I I

~'k~ SO0. (j pc =0++ ;I G=O+)

180: .................

120

60:

O: i

soo 900 12'oo Ee.=. (MeV)

, tz0

oo 60

a l l

0 300

1.20

I i I I ~ I i i I lit ~|| i

----o =l+

600 900 1200 Ec.=. (MeV)

1.00

,~O.BO-

~ o.0o-

• ~ 0.40-

0.20-

i

1200

i I

(J PC=O+÷;I°=O+)

0.00 300 600 900

E o.m. (MeV)


Fig. 8. Contributions to the Born amplitudes ~ --* ~-~', KK.


interaction Lagrangians, which are characterized by the j e quantum numbers of the hadrons involved. For simplicity, we suppress here the isospin or SU(3) dependence of the Lagrangians,

1 + 1 + - ' 7 @ ~ @0-- : ff..B'Bp(X) = fB'BpI/JB,(X)y5yI't~JB(X)O#(I/)p(X ) m~r+

(+h.c., if B ~ 4= B ) ,

3 + 1 + fDBp @ ~ @0-- : ff-.OBp(X) = - - - ~ B ( X ) ( g ~ u "~-XAylzyv)~,l/l?)(X)~V~p(X) + h . c . , mTr+

1 + 1 + ( 6 9 ) ® ~ ® 1- : £B,Bp(X) = g B ' B p ~ B , ( X ) y / z C B ( X ) q ¥ ¢ ( X )

fB'Bp-7- , , "or --

(+h.c., if B t 4: B ) ,

O- ® 0 - ® 1 - : ~.p,pp(X) =gp,ppq~p,(X)a/zq~pq~(x)(-q-h.c. , i f p ~ 4: p ) .

For the conventions used for the field operators and the Dirac y-matrices, see Ap- pendix A. The Lagrangian £onp includes an off-shell part, which is proportional to x~. As the parameter A, which occurs in the free Lagrangian [32] and then in the non-pole part of the propagator of a spin-~ particle (cf. Eq. (A.16)), the parameter x~ characterizing the strength of the off-shell part of the DBp coupling is not determined from first principles. However, it is known [33] that field-theoretic amplitudes derived with this most general ansatz depend only on a certain combination of A and x~, namely

1 + 4xa - - =: 1 + 4 Z . ( 7 0 ) 2 A + l

It follows that different pairs of (XA, A) values, which give the same value of Z, describe the same interaction theory. Therefore, without restricting the general validity of our results, we can set A = -1 (i.e. omitting the non-pole part of the spin -3 propagator) and select the interaction theory (characterized by Z) through x,~, which is finally adjusted to the quasiempirical data for N N ~ 7rcr.

In order to account for the extended structure of hadrons the vertex functions resulting from the Lagrangians (69) are modified by phenomenological form factors. For the baryon-exchange processes these form factors are parametrized in the usual multipole form

(nxa2x - M2x )nx Fx(p 2) = k, n~x--_-- ~ - , (71)

where p denotes the 4-momentum and Mx the mass of the exchanged baryon X. The two parameters, the so-called cutoff mass Ax and the power nx, are chosen uniquely for all BBtp (A8, n8) and for all BDp (Am, nl0) vertices in order to keep the number of parameters of our model as low as possible. The dependence of the form factor (71 ) on the power nx is quite weak [ 17]. We choose n8 = 1 and nl0 = 2. Finally, A8 and Am are adjusted to the quasiempirical data for the N N ~ ~'~" amplitudes in the pseudophysical region.


For the p-pole diagram we parametrize the form factor at the/z/zp vertex in the same way as is done in the model of the 1r~r-KK interaction in Ref. [ 17]:

[ 2 L (m~)(O) )z2 '~ nm, o n ~ p A ~ p

( k = I 2 2 2 ) ' \ nuupAm, p + 4wu(k ) (72)

where k is the relative 3-momentum of the two pseudoscalar mesons and k ~ ( t ) = 2 is the squared on-shell momentum of t h e / ~ state. t /4 - m~

For the dispersion-theoretic calculation of correlated 7mr and KK exchange the BB ~ --~ /z~ Born amplitudes are evaluated only for BB ~ states being on their mass shell. Therefore, there is no need for a form factor at the BB'p vertex as to assure convergence of the scattering equation (68). Therefore, we disregard this form factor, again in order to keep the parameters of the model as low as possible.

Now, taking into account that the BB ~ state is on mass shell and that due to the Blankenbecler-Sugar condition (Eq. (66)) the energy component of the relative momentum of t h e / ~ state always vanishes (k0 = 0) the 4-momenta at the external legs of the BB ~ ---* tz-fi Born diagrams read in the center-of-mass system:

(E/~ ) ' q / ~ ' = ( - q ) ' ( 7 ) ' ( - ) '

(73)

According to the usual Feynman rules [34] (for the various propagators, see also Appendix A) we obtain for the spin-momentum parts of the BB ~ ~ Iz-fi Born amplitudes VBT__+ug ( k, q; t):

(i) Exchange of a baryon X

VB~7_ - ~ ( k, q; t') =

(ii) Exchange of a baryon X

vBx~u~ (k, q; t') =

1 + w i t h j e = g and m o m e n t u m p = q - k

( _ f XB' ~ ysyaka (-~) ~ - ( ~ ~ M____fx ) m~r ~ / p2 _ M2 ~

× ( - - f xB~y5yvkV( IX) ) m~r+

3 + w i t h J e = ~ and m o m e n t u m p = q - k

[ _ fx, '~, ( g ~ + Xay,,ya)k~(-fi)] S~P(p,A mr,

[ f x s . ,_ ] x + ;

(iii) p-pole graph with bare mass m(v°)

f(0), \ ga~ V p - (k ,q ; t I) = ig(BO)B,p%~ + JBB'Po'AvPV]

BB'--*I~'~ 2Mu ]

x [-ig(u°)up(k(tz) - k(~)),r] •

= - I )

_ papa/(m~O))2

t t _ (m(°))2

(74)

(75)

(76)


These expressions can be further simplified by introducing the momenta given in Eq. (73) and contracting the ~-matrices. Finally, the corresponding B B ' --* ke-fi helicity amplitudes are obtained by applying ~)B~7__.~,f,(k,q; t) to the Dirac helicity spinors of the baryons (cf. Eqs. (A.2) and (A.9)):

( t z~ , k [ V ( t ) IBB',q,A, A2> =-6B,(--q, a2 ) l )nT__ ,u f , ( k , q ; t )uB(q , h l ) . (77)

The final results for the helicity amplitudes are summarized in Appendix B. In our model the coupling constants at the various hadronic vertices are related to

each other by SU(3) arguments. The SU(3) relations together with the isospin factors of the various Born amplitudes are given in Appendix C.

As discussed in the previous sections, for the dispersion-theoretic calculation of correlated ¢rvr and K K exchange the B B ' --+ 7rTr, K K have to be known in the pseudophysical region, i.e. for energies t' below the B B "-7 threshold (4m~ ~< t ~ < (MB + MB, )2). There- fore, after having derived the analytic expressions for these Born amplitudes in the physical region (v~ 7 > MB + MB,) they have to be continued analytically as functions of t ~ (and s) into the pseudophysical region. For this all energy-dependent quantities occurring in the expressions for the Born amplitudes (cf. Appendix B) have to be expressed as functions of t ~ and s.

If we adopt the approximation introduced in Section 2.2, namely that the masses of the baryon and of the antibaryon are equal (MB = MB,), the square of the relative 4- momentum and the one-particle energies of the B B ' state are given in the center-of-mass system for physical values of t by

q2( t ) = t / 4 - MZB,

EB = En, = v'7/2. (78)

The analytic continuation of these relations to the pseudophysical region is obvious. Note that if we would have allowed MB and MB, to be different the corresponding relations would look more involved:

q2( t ) = [t - (MB q- MB,) 2] It -- (MB -- MB,) 2] 4t

(EB + EB,) (E~ - E2,) EB ( t ) - +

2 2 (E , + EB,) t q- M 2 - - M 2,

2vq E B , ( t ) = t + M 2, - MZB (79)

2 : Baryon-exchange diagrams in which the mass M x of the exchanged baryon is suffi-

ciently smaller than the mass MB = Mn, of the external baryons (e.g. N-exchange in AA ---* KK or A-exchange in ~ ---* ¢rvr do not satisfy a Mandelstam representation as was already pointed out in Ref. [35] for the latter example. The non-validity of the Mandelstam representation becomes obvious when extrapolating the corresponding Born


amplitude in Eq. (74) to the pseudophysical region. For given t < 4(M~ - M2), the propagator of baryon X,

[ p 2 M 2 ] - ' = _ I t ~ 4 - M~ + M2x + k 2 - 2q. k ] - ' , (80)

then acquires a singularity at cosO = 0 (q . k imaginary) and the following off-shell momentum of t he / z~ state:

k 2 : M~ - M2x - t /4 > O. (81)

Since this problem would hinder an evaluation of correlated 7rTr and KK exchange and we do not see at present any proper solution, we eliminate the problem by the following approximation: In all baryon-exchange diagrams in which the mass of the exchanged baryon, Mx, is smaller than the mass of the external baryons, MB, Mx is increased by hand to MB. Again, since the uncorrelated contributions of two-pion and two-kaon exchange (e.g. iterative two-pion exchange in the £N channel with a AN intermediate state) are evaluated explicitly in the s-channel, these contributions are not affected by our approximation and thus have the correct energy dependence. Approximations are only made in the correlated part which have a much weaker energy dependence than the uncorrelated contributions.

3. Results and discussion

3.1. Determination of free parameters

B

During the construction of the microscopic model for the BB ~ ~ ix-~ amplitudes it proved to be essential to restrict the number of free parameters as much as possible, which can then be fixed by adjusting the model predictions to the quasiempirical NN --* ~Tr amplitudes. Only in this way one can hope to obtain a reasonable description of the other baryon-antibaryon channels, for which no empirical data exist.

As shortly outlined in Section 2.4 the 7r~r-KK interaction model has been developed independently before, with all parameters adjusted to fit the existing 7r~- scattering phase shifts. Therefore coupling constants and form factors at the ¢rTrp (°) and KKp (°) vertices occurring in the p(O~ pole terms are already determined. Assuming that the bare p-meson couples universally to the isospin current we can also fix all vector couplings ~(0) to 6BBPp

the baryonic vertices. Corresponding tensor couplings f~o),p will be related by SU(3)

¢(0) and the symmetry, with two parameters remaining, namely the coupling constant JNNp F/( F + D) ratio a v .m

The coupling of the pseudoscalar mesons 7r and K to the octet baryons is in the framework of SU(3) symmetry likewise determined by two parameters, the F/ (F + D) ratio ap and the coupling constant gN~,r. (For the latter we will take throughout g2NlWr/47r = 14.3). There is an additional freedom since SU(3) symmetry can be either


assumed for the pseudoscalar coupling constants gBB'p or the pseudovector ones fBB'p, which are related by

MB + MB, gs'sp = fB'Bv (82)

m,n-+

These two possibilities are not equivalent since, because of SU(3) breaking of baryon masses, both sets of coupling constants cannot be SU(3) symmetric at the same time. In this work we will assume SU(3) symmetry for the pseudosealar couplings, but we will also check the influence of the alternative possibility.

Finally, under the assumption of SU(3), the couplings fBov of ~ and K to the transition current between baryon octet and decuplet are determined by only one parameter, fN,a~. We will use f2z~/47r = 0.36 in the following.

In addition we have form-factor parameters at the hadronic vertices. In order to keep their number small we assume that the cutoff masses ABxu are independent of the exchanged baryon X within one SU(3) multiplet. Consequently, we have two additional parameters: As if X is a member of the baryon octet and A10 if it is in a decuplet. The power n8 = 1 and nl0 = 2 in the form-factor ansatz are sufficient to ensure convergence of the scattering equation. Finally, x,a (Eq. (69)) characterizing the off-shell part of the (3) + ® ( ½ ) + ® 0 - coupling is treated as a free parameter in our BB -7 ~ / ~ model.

In order to reduce the number of parameters further we even assume SU(6) symmetry, :(o) m to be 0.4. Thus we are left with four free parameters ~ NN?, x~, which fixes c~p and a,

A8 and Alo, which have been fixed by adjusting our theoretical predictions for the N N --~ ~rTr amplitudes to the quasiempirical results of H6hler and Pietarinen [ 18,19] given in the form of Frazer-Fulco amplitudes f~: ( t) , which, up to kinematic factors, correspond to the partial wave decomposed helicity amplitudes of Section 2.2, i.e.

fJ+(t) = 1 q M o (Trot] TJ( t ) ]N-N, ++) × F J 16zr 2 (qk) J

l qMN ('B"/r[ TJ( t ) IN-N, + - ) × V J (83) f J ( t ) = - 8¢r~ ( qk ) J------~

where k and q are the on-shell momenta of pions and nucleons. The factors F J = l ( _ ½) for J = 0 (1 ) are due to the transition from isospin amplitudes used in this

work to the Frazer-Fulco amplitudes, which are defined in isospin space as coefficients of the independent isospin operators ~5,~ and ½ [a-a, ~'~]. Since (Trlr I TV--°(t) INN, + - ) vanishes identically we have only one amplitude, fo+, in the o--channel whereas in the p-channel we have both fl+ and fl_.

Fig. 9 shows the predictions of our microscopic model for f0+, fl+ and fl_, in comparison to the quasiempirical results of Ref. [ 18] in the pseudophysical region t/> 4m~; Table 4 contains the chosen parameter values. (Note that the present model differs somewhat from our former model [ 17], e.g. by the inclusion of Y* exchange; therefore the values differ slightly from those given in Ref. [ 17] ). With only four parameters we obtain a very satisfactory reproduction of the quasiempirical data, especially in the p-channel. Some discrepancies occur in the o'-channel, which however have only small


120 t . . . . . . . . .

100 i

8 0 1

6o.' _ =";-:~,~= :o,

` = o (1) II

" " 20; D U D

o

-20 1'0 2'0 3'0 4'0 .50

t(~. ~)

5.0 , . , - , - . -

4 .0

.3,0

E

.---.... 2 .0 7+._+

n . - 1.0

0.0

-1.0 lb 2'0 3'0 4'0 50

t(m~)

5.0 , . , . . . . •

4.0

3.0

'E v

.-.---.-- 2 .0 - - i

r Y 1,0

0 .0

- 1 . 0 lb 2'0 3'0 4'o 50

t (m. ' )

60

~ 40

~-: 20 v

E

0

-20

• , • , • , • ,

D t = =

= ¢~¢1

=

D

= =

~b 2'0 go 4'0 50 t(~:)

L 10 20 .50 4 0 50

t (m~')

2.00

1.50

1.oo / \

- - 0 0O . . . . . 10 20 30 40

t (m~')

50

Fig. 9. NN --* ~'Tr, KK helicity amplitudes in Frazer-Fulco normalization. The solid lines are the results of our microscopic model. The squares denote the quasiempirical data [ 18,19] obtained by analytic continuation of the ~rN and 7r~" scattering amplitudes.


Table 3 Isospin factors T(1) for the various B B ~ --~ iz-~ Born amplitudes. In the column labelled 'process' the exchanged baryon (Y* -- X(1385) ) or a 'p ' in case of a p-pole diagram is given

B B ' --~ I~-~ Process T(0) T( I ) T(2)

NN ---* ~ " N +x,/6 +2

A + V / ~ -2/3 p -2

NN -+ KK A +1 +1

X , Y * +3 - I

p - 2

AA ~ ~ X, Y* - v ~

a ~ ~ r ~ N -v5 -=,-=* - v ~

--* ~rrr A +1 +1 +1

~,Y* +2 +1 - 1

p - 2

• 2 ~ / ~ N + ~ - 2

a + V / ~ +2/3 ~, -=* +v"6 +2

p - 2

A~ -'~ "n"n" ~Y, Y* --v/2

p +v~

_=,_=* +v5

p +v'~ = ~ --+ rrrr ,..~, =* -v/'6 --2

p +2

_=~-+ K~ a -1 +i 2?,Y* - 3 - 1

~ - --1 -1

p +2

influence on final results for the correlated ~'~r exchange in the s-channel reactions (NN, 7rN), as also discussed below. Furthermore it should be kept in mind that in this channel the quasiempirical information is plagued with considerable uncertainties.

m

3.2. The B B ' --* tz-~ transition potentials

m

Having fixed all parameters in our microscopic model for the B B ' --~ tz-~ amplitudes we will now first look at the transition potentials (Born amplitudes) in the various baryon-antibaryon channels. Figs. 10 and 11 show the contributions of the various


NN > ~ 2.00~

. ._j.5ot I

~oot > ~ t S ~ - A . . . . . . . . . -

t(m~) 80 100

0"80 l

~0'20 t / o 2'0 4'0 6'0

t (rn~) 8'0 tO0

0.90

0.60 q.)

v

"~o.3o

0.00

/ /

. / /

2'0 4'0 6'0 8'0 ~oo t(m~)

0.2C

%÷O.lO. E I

0.00

0.09

~"0,06 > (1)

v

~ o . o 3

0.00

0.09

0.06

"~0.03 r Y

0.00

NN > K /

" A

. . . . . . . . Y~ . . . . . . . .

:E

6'0 7'0 8'0 9'0 t (rn~)

100

/ /

/

/

6'0 7'o 8'0 9'0 loo

j / " A

/

/

/ /

y * - - . . . . . .

6'o 7'0 8'o 9'o loo t(@

m

Fig. 10. N N ---* ~'~', K K Born amplitudes.


0.60

T> 0.40.

"~'0.20 I

0.~

0.20

~-"0.10

O.OC

0.2

o.i

" ~ o.o

-0.1

Z Z > × ~

I I I

!

2'0 4'0 6'0 do t(m~)

100

/ /

/

I

2'0 4'0 6'0 8'0 ,00 t(m~)

/ A ~ ~

20 4b 6b t(m~)

8'0 ~oo

3.00

~_. 2"501

T> 2.00.

~,.°° 1 0.504

0.00

ZZ > KK :\ '\.\

60 70 80 90 100 t(m~)

0.2 . . . . . . . . . . . . . . . . . . . . . ."7 . . . . . . . . . . . . . . . . . . . . . . . . . .

7> 0.0 v

~-21 . ' ~ . . . "

.4 ' ~ , 6O

. . " "

7'0 8'0 9'o ~00 t (r~)

0.40[

._.0.30 t / . . / ~ . . . . . . . . . . . . . . ! . , " "

~o.2ot .,' / / . .~ ... o,ot! ii i; ..........................

. . . - ' "

0 . 0 0 ~

3¢

60 7O 80 90 1 O0 t(mg)

Fig. 11. ,~'~ ~ ~'~', K'K Born amplitudes.


Table 4 - - - - ( 0 ) . ( 0 ) Parameters of the microscopic model for the BB' --, I~-fi Born amplitudes. The four parameters JhNp/gNNp,

xa, A8 and Alo are adjusted to the quasiempirical NN --* ¢rcr amplitudes of Refs. [ 18,19]

Parameter Value

fN °) /n(0) 4.21 Npl,SNNp x,~ -0.823 A8 1779.1 MeV AI0 1704.7 MeV ap 2/5 (SU(6) [41])

m 2/5 (SU(6) 1411) o ' v

m _ _ - -

baryon-exchange processes to the N N --* 7rqr, K K and ~X --~ 7rTr, K K Born amplitudes above the /x~ thresholds, i.e. for t ~> 4m2u. Contributions of the p(0) pole terms are

not shown since they possess a singularity at the bare p-mass m~ °). This pole will be regularized only after iteration and coupling to the full zrrr amplitude and then leads to

the resonance structure at the physical p-mass rnp. The notation of the (Born) amplitudes follows that of the Frazer-Fulco amplitudes,

i.e.

VJ+ ( t) = (P'~I VJ ( t) I B~7, ++),

~ ( t ) = (~1 VJ(t) IBB-7, + - ) • (84)

The partial wave decomposed Born amplitudes are either even or odd functions of

the baryonic relative momenta (cf. Appendix B). Since the latter are imaginary in the pseudophysical region the Born amplitudes being odd functions (e.g. V~+) become

likewise imaginary. Apart from coupling constants and isospin factors the strengths of the various contri-

butions are strongly determined by mass ratios. Exchange baryons with the same mass M x as the outer baryon-antibaryon pairs (according to our approximation introduced

in Section 2.4.1 this case includes also those exchange baryons whose physical mass is lower than that of the outer baryons) produce, in the 7rTr amplitudes V~+ and V+ I , a typical structure at the ~Tr threshold. The strong rise of the amplitudes, which acquire a finite value at t = 4m~, is a direct signal of the so-called left-hand cut, which is

generated by the singularity of the corresponding u-channel pole graph. This cut starts just below the 7rTr threshold and extends from to = 4m2(1 - mZ~/Mx) along the real

axis to - ~ . Obviously, in the o--channel, the various pieces interfere constructively, whereas in the

p-channel also destructive interferences occur. The contributions generated by a spin -3 exchange baryon have opposite sign in V+ 1 and V_ l whereas both amplitudes are almost

equal when a spin-½ baryon is exchanged. The NN ~ 7r~ Born amplitudes in Fig. 10 built up by nucleon and A-exchange are

noticeably larger than the NN ~ KK Born amplitudes, which are suppressed because


400

2O0

? 0

~-2oo

-61111

400

0

-120~

-160C

i I I / s " ~

i t I / / /

I i II

2b 4'o ~ ~ 16o t (rn~)

m B

K K -+ K K

Fig. 12. ~rlr and KK amplitudes in the o'-channel ( J = 1 = 0) as a function of the squared c.m. energy t. The dashed (solid) line represents the real (imaginary) part of the amplitude calculated with the field-theoretical model of Ref. [ 17].

of the high mass of the exchange baryons A, 2, Y*. Due to different mass ratios this is no longer true in the hyperon-antihyperon channels. For instance, in the 2,~ channel shown in Fig. 11 the coupling via ,4-exchange to K K even dominates, due to the large coupling constant (f2aK = f24=) and the small mass difference between ,~ and .4 (Ma - M z ~ 39 MeV). Therefore already from these results it is to be expected that correlated K K exchange processes play a minor role in the N N system but are important for the interactions involving hyperons.

m

3.3. The B B ~ ~ Iz-~ transition ampl i tudes

m

The B B I ~ I z ~ helicity amplitudes, T:~(t), which are obtained from the solution of the BbS scattering equation (68), consist of Born terms and pieces containing meson- meson correlations. The latter part fixes the phase of the B B ~ --~ IX-~ amplitudes and generates the discontinuities of T ~ ( t ) along the unitarity cut. It contains 7rTr and KK correlations in the form of 7r~-, K K amplitudes. Corresponding predictions from our field-theoretic model, for the scalar-isoscalar on-shell amplitude TJ=°,t=°(t) are shown in Fig. 12. By comparing the 7rTr amplitude with the ~-~r phase shifts (cf. Fig. 7) we can see immediately that the vanishing of the real part of the amplitude at t = 37m 2 and t = 71m~ corresponds to the phase shifts acquiring the value of 90 ° and 270 °, respectively. The phase value of 180 ° (and the corresponding vanishing of the amplitude) occurs just below the KK threshold (t = 50.48m2). Remarkably, there is a steep rise of the

A. Reuber et al. /Nuclear Physics A 608 (1996) 243-304 279

imaginary part of the ~r,r amplitude at low energies, which has to vanish at the 7rqr threshold for unitarity reasons. We mention that, in the N N system, this part provides the dominant contribution to correlated 1rTr exchange. It should be noted that both the real and imaginary part of the K K --* K K amplitude (which in Fig. 12 are shown only in the physical region above the K K threshold) acquire a non-vanishing value at the K K threshold due to the open zrTr channel.

Figs. 13 and 14 show the resulting B B ~ ~ 7r~r amplitudes for B B ~ = N N , ~ . Besides the predictions of the full model (solid line) the figures also show the effect which is obtained when neglecting the B B ~ --~ K K Born amplitudes in the o'-channel (dashed line) and restricting oneself to the p-pole amplitudes in the p-channel (dash-dotted line). The B B ~ ~ K K amplitudes (dotted lines) are only shown for J = 0 since for J = 1 as exemplified for the 2 2 channel they provide almost negligible contributions, mainly because of the weak K K interaction in that partial wave. Note that in the or_ channel, due to the imaginary Born amplitudes, the role of the real and imaginary part of TO+ are in a certain sense interchanged: Im[T~+] contains the Born amplitudes whereas the discontinuity along the unitarity cut is contained in Re[T°+ ].

Obviously, in all channels, the structure of the 7r¢r amplitude of Fig. 12 can be rec- ognized in those B B --7 ~ ~r~r amplitudes TO+, which have been evaluated without the B B --7 ~ K - K Born amplitudes (dashed curves). The steep rise of Im[T~+] at the 7r~r threshold originates from the analogous behavior of the Born amplitudes (cf. Figs. 10 and 11). As follows from the unitarity relation for the B B ~ ~ 7rTr amplitudes below the KK threshold (tthr ~ 50.48m~) the discontinuity of the BB -7 ~ ~-~- amplitudes (contained in Re[T°+] ) must vanish at t ~ 50.43m~ since there T J=°J=° = 0. Further- more, those amplitudes T~+, which have been evaluated without the KK Born amplitudes, vanish at an additional, though for each channel different position in the energy region below the K K threshold. Interestingly this vanishing of the amplitudes is lost in the hyperon-antihyperon channels when the KK Born amplitudes are included; in case of the N N --* 7rTr channel on the other hand the resulting full amplitude still vanishes, at a slightly different position (t ~ 50.2m~). Note that the quasiempirical NN ---, ~-Tr amplitude f0+ in Fig. 9 has a comparable structure, however the amplitude vanishes already at a somewhat smaller value of t ~ 43m 2. Fig. 15 also clearly shows the cusp structure of the amplitudes at the K K threshold, which is however much stronger in the

hyperon-antihyperon channels. The B B ~ --~ K K Born amplitudes do not play any role in the N N --~ 7rrc channel, as

noted already in Refs. [ 14,17]. In the hyperon-antihyperon channels on the other hand, we have already at small t-values important contributions to the B B ~ --~ Ir l r amplitudes TO+, which lead to an enhancement of the amplitudes below the K K threshold and to a noticeably different energy dependence.

The amplitudes T~: in the p-channel are dominated by the resonance structure in the region of the physical p-mass at t = 30.4m 2. We stress that by multiplication with the zr~r correlations in the scattering equation also the baryon-exchange Born terms lead to resonant contributions. However, in all channels, the discontinuity of the amplitudes mainly arises (to at least 60%) from both p pole terms and those contributions generated

3.50

3.00

~-"2.50 >

2.00

6-.-~1.50

"~1.00 E~

0.50

0.00

1.60

2..50

.. . . . . . . . . . . . . . . . . . . . . . . . - ' "

20 40 60 80 100 t(m~)

1.20

.J'~.

-0.40

- 0 . % ~ o t(m~)

0.80

0.40

T-~ 0.00

2.00

-1.00

1.50

1.00

v 0.50

T-2 0.00 r Y

-0.50

-1.50 10 20 30 40 50 60

t ( ~ )

280 A. Reuber et al. /Nuclear Physics A 608 (1996) 243-304

C" :~ 2

o + I---

E 0 i

-21

2.00

,_..,1.50 ~>

~'~-'1.00

E - - 0.50

o.oc 0

4.00,

3.50

,.,,3.00 i

> 2.50

=,~-2.00

~...~_~ 1.50. E

- - 1.00.

o.5oi o.Oeo

. . . . . . . ' ' " • . , . . ' "

20 4 ' 0 ' 6 ' 0 8'0 160

t ( ~ )

~o 20 30 4o 50 60 t (m~)

1 4 0

t(m~)

Fig. 13. Helicity amplitudes T J, for NN ~ ~'~ (solid) and NN ~ KK (dotted). For J = 1 the small

NN ~ KK amplitudes are suppressed• The dashed lines in the ~r-channel denote the NN ---* ~rTr amplitudes, that are calculated without the NN ~ KK Born amplitudes. In the p-channel the dash-dotted lines are obtained if only the p-pole graphs is included in the N N ~ IX~ Born amplitudes.

o

E,

~"

~-.

=--~,

g ~

N

~

• 3

~ g N

Re(T_

1) (M

eV-')

o I

o

"'..

.

Im(T_

1) (M

eV-')

Re(T;

) (Me

V-')

i i

o "\

Im(T~

) (ME

W')

p .o

.-

-,

~ ,,

o

o o 8

o ~o

N

o

i i

i

Re(T °

) (Me

V-')

-Im(T

+ °)

MeV-

') .

o ,

i ,

, ,

i ,

, ,

i .

..

.

o o

f i

I

t~

r~

~D

4~


0 . 6 0 .

0.40.

i o.~

O.OC

~"+-0~

-0.~

\

x

f

-0.6~ t .- " -

49.5 50.o 5o.5 5 .o t(m~)

Fig. 15. NN ~ l r~ amplitude T~+ calculated with the full model in the energy range around the KK threshold (dotted). The solid (dashed) line denotes the real (imaginary) part of T~+.

by them in the scattering equation (dash-dotted curves in Figs. 13 and 14). As shown in Fig. 14 for the 2 2 -* KK transition, the BB ~ --~ KK on-shell amplitudes

(dotted curves) are completely unimportant in the p-channel. On the other hand, in the o--channel, the BB ~ --* K K amplitudes can acquire large values especially near the KK

threshold, due to the corresponding behavior of the KK amplitude T J=°J=° (cf. Fig. 12), which lead to non-negligible contributions to the spectral functions.

3.4. The spectral functions

m

Based on the BB' --+ IZ-~ amplitudes in the pseudophysical region determined in the last section we can now in principle evaluate the spectral functions p~'P (defined in Eq. (51 ) ) of correlated zrTr and KK exchange for all baryon-baryon channels containing the octet baryons N, A, 2: and ~ . In this work however we restrict ourselves to those channels in which experimental information is available at present or in the near future. Besides the baryon-baryon channels with strangeness S = 0 and S = -1 , for which numerous ( N N ) and scarce (NA, N £ ) scattering data exist, we will also consider the S = - 2 channels AA, ~ and N ~ , which are relevant both for the description of AA-

and ~-hypernuclei, respectively, and for the study of the H-dibaryon [ 11 ] predicted in the (S = - 2 , J = I = 0) channel.

Fig. 16a (Fig. 16b) shows the spectral function '~ ' Ps(BB ) in the tr-channel predicted by the full microscopic model for the S = 0 , - 1 BB ~ channels NN, NA, N 2 (S =

- 2 channels AA, ~.,~, N ~ ) . Due to isospin conservation the 7rTr and KK exchanges respectively contribute to the NA and AA interaction only in the it-channel and to the N A - N ~ transition only in the p-channel. Up to near the KK threshold at t = 50.48m~ p~ is negative in all channels. Therefore, as expected, correlated 7r,r and K K exchange provides attractive contributions in all channels. Especially at small t-values, which determine the long-range part of the correlated exchanges and therefore yield for low- energy processes in the s-channel the main contributions to the dispersion integral, the spectral function p ~ ( N N ) is by about a factor of 2 larger than the results for p°~(NA)


4

2.

0

i~ -2

-41 - - 6 "

- - 8 '

-10

, - , - , - , -

k. . . . . . . ,}.4..1....."

2'0 4'0 60 80 ~ 100 t(m )

2

0

-2

~-6 -8

-10

0

, . ~ , - . . , . , .

\ 1 .

"\ I

II i

2'0 4'0 6b 8'0 16o t(m )

Fig. 16. Spectral function p~(t) for the scalar component of correlated 1rcr and KK exchange in the scallar-isosealar channel of various baryon-baryon processes: (a) NN (solid), NA (short dashed), NE (dotted); (b) AA (long dashed), ,~Z (dash-dotted), NE (dash-double-dotted).

4

2 / /t .........

O k , , , ~ "

t,:.- , ' / J . , , . . :

° t ,,7

t(m~)

Fig. 17. Spectral function p~(t) in the NN (solid) and N.,~ (dotted) channel derived with the full model (cf. Fig. 16). If the contributions of the BB -7 ---* K-K Born amplitudes are neglected the dashed (NN) and the dash-dotted lines (NZ) are obtained.

and p ~ ( N 2 ) , which have about the same size. Due to the sizable contributions of the K K

Born ampli tudes to the hyperon-ant ihyperon amplitudes found already in the last section,

p ~ ( N A ) as well as p ~ ( N ~ ) show a noticeably different t-dependence than p ~ ( N N ) . Below the K K threshold the spectral functions are broadened to larger t-values, therefore

the overall range o f this correlated exchange is reduced. This is even more true for the

S = - 2 channels A A and ~ in Fig. 16b since there the hyperon-ant ihyperon ampli tudes

enter quadratically. The effect of the BB' ---} K K Born amplitudes is once more shown

in Fig. 17 for the N N and N 2 channel. Whereas they have small influence on p ~ ( N N ) they provide important contributions to p~(N,~) already near the zmr threshold.


--4' I I= E

-12

• • . . ' f

",.."

10 2'0 3'0 40 50 t

121

~'= 8 E

%> 4

0

0

L'- 1'0 2'0 go 4'0 50

t (rng)

[E 4.

-4 I

- i . , • , • , -

', /

?0 2'0 3'0 4'0 50 t(m~)

0 .80

0 .40

~ O . O O

- 0 . 40

- 0 . 80

1'0 20 30 40 50 t(mg)

Fig. 18. Spectral functions p~( t ) ( i = S, V,T, 6 ) for the contribution of correlated ~-~ and KK exchange in the p-channel to the NN (solid), N2~ (dotted), ~2~ (dash-dotted) and N ~ interaction (dash-double-dotted) as well as to the NA - N.~ transition amplitude (short dashed)•

Fig. 18 shows the corresponding spectral functions in the p-channel (p~, p~,, p~. and p~) for the various particle channels, which are non-vanishing. (Note that pP depends linearly on the four functions shown, cf. Eq. (37) , and is therefore not shown), p~ contributes only in BB p channels with different masses and therefore vanishes in the

P S N N and 2~27 channel. According to Eq. (35) Ps( , t) depends on s through the factor cosOt(s , t ) . In order to enable a comparison of the results for p~(s, t ) in the various particle channels we have throughout set s on the threshold of the s-channel process in question, i.e. s = (MB + MB,) 2. Since the p-channel is dominated by the resonant pieces in the 7rTr channel and the KK channel does not play any role the pP(BB ~) possess almost the same t-dependence in the various particle channels.

For the NN channel, the spectral functions for correlated 7r~" and KK exchange can be derived either from the NN ~ ~r~- amplitudes of our microscopic model or, alternatively, from the quasiempirical results obtained in Refs. [ 18,19]. As before we have then to subtract the uncorrelated pieces evaluated from the microscopic model for


E E - 4

--8

-12"

0

- i w - i • i I .

' I I l l ;

I\~ i l l /

\ I \ ]

~'0 2b go 4'o 50 t (m9

-8

-12 lb 2'0 go 4'0 5o t(mD

12

8 I g : E v

~ 4

1'0 ~ go t(m )

12

50 0

/,

Z/'

fo io 3'o 4'o 5o t(m )

Fig. 19. NN spectral functions p~( t ) and p~(t) (i = S, V,T). The solid lines are derived with the microscopic model of Section 2.4 for the BB --7 --* ix-~ amplitudes. The results obtained with the quasiempirical NN --* 7rTr amplitudes of Refs. [ 18,19] are denoted by the dashed lines.

the N N ~ ~rTr Born amplitudes. Therefore the results derived from the quasiempirical amplitudes depend on parameters (e.g. gNN~) of the microscopic model.

In Fig. 19 we show the N N spectral functions in the or- as well as in the p-channel

obtained from our microscopic model (solid line) and the quasiempirical amplitudes (dashed line). As expected already from the comparison of the amplitudes in Section 3.1

the results agree quite well in the p-channel. On the other hand, some discrepancies occur in the o--channel. At smaller t-values, which determine the long-range part of the correlated exchanges, the theoretical model yields somewhat more attraction than the quasiempirical results. Larger discrepancies occur at higher t-values, which are however of minor relevance for the correlated exchange in the N N interaction. Namely, above t 30m 2 the correlated exchanges are of shorter range than to-exchange, which generates the strong repulsive inner part of the N N potential (as well as of the other baryon- baryon potentials). Therefore the short-ranged parts of the correlated ~-~r exchanges are


completely masked by the repulsive oJ-exchange and have a small influence only on N N observables. Furthermore one has to realize that the quasiempirical results obtained by extrapolation of data from the physical region of the s- and u-channel into the pseudophysical region of the t-channel have considerable uncertainties.

3.5. The potential o f correlated ~rTr and K K exchange

Using the spectral functions of the last section we can now evaluate the dispersion integrals, Eqs. (48 ) - (50 ) , in order to obtain the invariant amplitudes in the s-channel. Eq. (53) then provides the (on-shell) baryon-baryon interaction due to correlated 1r~" and K K exchange.

3.5.1. The effective coupling constants o',p The results will be presented in terms of the effective coupling strengths GAn__.c o ( t ) ,

which have been introduced in Eqs. (54), (56) to parametrize the correlated processes by (sharp mass) or- and p-exchange. We stress once more that this parametrization does not involve any approximations as long as the full t-dependence of the effective coupling strengths are taken into account. The parameters of o- and p-exchange (mass of the exchanged particle: m~, mp; cutoff mass: A,~, A a) are chosen to have the same values in all particle channels, m,~ and m a have been set to the values used in the Bonn-Jtilich models of the N N [I0] and YN [3] interactions, i.e. m,~ = 550 MeV, mp = 770 MeV. The cutoff masses have been chosen such that the coupling strengths in the S = 0, - 1 baryon-baryon channels vary only weakly with t. The resulting values (A,~ = 2.8 GeV, A t, = 2.5 GeV) are quite large compared to the values of the phenomenological parametrizations used in Refs. [ 10,3] and thus represent very hard

' in the dispersion form factors. Note that, unless stated otherwise, the upper limit tma x integrals is put to 120m~. Fig. 20 shows the effective coupling strengths G~AB(t) in the baryon-baryon channels considered here, as function of - t . With the exception of G~z( t ) (dash-dotted curve) the effective o- coupling strengths vary only weakly with - t , which proves that 550 MeV is a realistic choice for the o'-mass. In the .,~X channel the suitable mass lies somewhat higher due to the strong coupling to the K K channel generated by A-exchange, cf. Fig. 11. If we compare the relative strengths of effective o'- exchange in the various baryon-baryon channels we observe the same features observed already for the spectral functions: The scalar-isoscalar part of correlated ¢r~r and KK exchange is in the N N channel about twice as large as in both YN channels and by a factor 3-4 larger than in the S = - 2 channels.

For sharp mass o'-exchange the following equation holds:

G~lv~lvN(t)G~x__.$~(t) = [G~v.~Nz(t) 12 .

In other words, the three processes are determined by two coupling constants g~nu :

and g~.~ -- ~ , such that G~__,Nz(t ) = g~N~g~z~. For correlated exchanges this is not necessarily true anymore; here we have in general


W " 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ i ~ : ~ : ~ ~ ' -

~'o 2'o 3'o - t

40

Fig. 20. Effective tr coupling strengths G~B__,AB/(4Zr ) for correlated ~r~" and KK exchange as a function of the squared 4-momentum transfer t < 0 in the baryon-baryon channels: NN (solid), NA (short dashed), N2' (dotted), AA (long dashed), 2~? (dash-dotted) and N_ ~ (dash-double-dotted).

t t O" O" 2 GNN~NN(t)G~z__.~z(t) >/ [GN~__+N~(t) ] ,

which is just a consequence of the Schwarz inequality relation

The equality holds if both functions have the same t-dependence. This is roughly fulfilled for N N and AA, but not for ~ 2 ; therefore the equality approximately holds in the first

but not in the second case. Consequently, in general, effective vertex couplings for correlated exchanges are not well defined since they might take different values in

different baryon-baryon channels. Table 5 contains the coupling strengths at t = 0. Besides the results for our full model

for correlated ¢rcr and K K exchange the table includes results obtained when neglecting the BB ~ KK Born amplitudes. Obviously the inclusion of these amplitudes provides

only 15% of the coupling strengths in the N N channel; it is much more important in the channels with strangeness, in fact providing the dominant part in the S = - 2 channels. Furthermore the table contains results obtained when uncorrelated contributions involv-

ing spin-½ baryons only are subtracted from the discontinuities of the invariant baryon- baryon amplitudes in order to avoid double counting in case a simple OBE model is used

in the s-channel. For the full Bonn N N model contributions involving spin-~ baryons have to be subtracted too (as done in general in this work) since corresponding contributions are already treated explicitly in the s-channel. Obviously processes involving spin-~ baryons increase the 'true' correlated contribution by about 30% in all channels.

In the p-channel the spectral functions are dominated by the resonant contributions in the region of the p resonance. Therefore the effective coupling strengths 'JGPAB__,CO (t)

( i j = W, VF, T~, T/') vary even more slowly with t than those in the o--channel. Because of this weak t-dependence it is for the moment sufficient to consider only the values of coupling strengths at t = 0. They are shown for the N N channel in Table 6, for YN in


Table 5 Effective tr coupling strengths G~B~AB(t = 0) for correlated Ir~r and KK exchange in the various baryon- baryon channels (the meaning of the various rows is given in the text)

G~B~AB/41r

NN NA N,Y AA ,Y,Y N ~

Full model 5.87 2.82 2.58 1.52 1.72 1.19 Without KK Born terms 5.07 1.80 1.06 0.64 0.22 0.35 Subtractions for OBE model 7.77 3.81 3.15 2.00 2.31 1.52

Table 6 Effective p coupling strengths ijf~.p VNN~N N ( i j = W, V/;, TV, 7T) at t = 0 for correlated zr~" and KK exchange in the NN channel (the meaning of the various rows is given in the text)

iJGPN.__.NN/47r ( i j = W I T , TI(,1T)

NN

W W,,TV 77"

Full model 1.00 5.35 28.91 Without baryon exchange 0.52 2.17 9.13 Subtraction for OBE model 0.33 5.58 35.23

Table 7 The same as Table 6, for the hyperon-nucleon channels N.Y and NA-N,Y

ij p GNy...Nr,/4Zr ( i j = W, VI~,TI(,TT)

N,Y NA --, N,Y

W V/" TV T/" W V/" TV 7T

Full model 1.64 1.92 8.95 1 0 . 1 5 -0.15 3.97 -0.81 21.42 Without baryon exchange 1.03 1.12 4.34 4.70 0.00 1.86 0.00 7.83 Subtraction for OBE model 1.53 1.52 9.87 9.85 -0.52 4.54 -1.61 25.60

Table 7 and in Table 8 for the S = - 2 baryon-baryon channels. Note that for the equal (unequal) mass case the results are given in terms of 3

(4) coupling strengths. For equal masses the present description in terms of correlated 7rTr exchange is more involved compared to sharp mass p-exchange since the latter can be characterized by two parameters only, the vector and tensor coupling constant. Also, as in the scalar channel, there is no definite relation between coupling strengths in the various channels so that vertex coupling constants cannot be uniquely extracted but depend on the channel chosen. (Thus it is not surprising that our pNN coupling strengths, which are determined in the NN system, are not fully consistent with the vector and tensor coupling constants derived by H6hler and Pietarinen [18,19] from ~ N scattering, though both calculations agree of course qualitatively.)


Tables 6-8 include results obtained when only the p-pole terms are considered in

the B B ' ~ lx-~ Born amplitudes. In this case the effective coupling strengths are still SU(3) symmetric, i.e. they roughly fulfill the relations

g.YAp = O, f2~Ap = 2VI3 (gNNp q- fNNp) ,

gzzp = 2gNNp, fZZp = _ 2 (3gNNp -- 2fNNp) , (85)

g~--=p = gNIVp , f ~_--p = --~ (6gNNp + f NUp) •

with a,,e = 1 and a v= = 0.4. (We remind the reader that we have chosen the bare couplings to exactly obey SU(3) symmetry.) Obviously the influence of (SU(3) broken) baryon masses on the p-pole contributions is small, probably because our calculations are performed in the pseudophysical region, far below the baryon-baryon thresholds. As expected however the effective coupling strengths of the complete calculation do not respect SU(3) symmetry anymore, due to the sizable influence of the (non-pole) baryon- exchange processes.

Doing again a restricted subtraction of spin-½ baryon contributions only (suitable for an OBE model) we now do not have a unique trend for the change of coupling strengths in all channels, as found before in the scalar case. In case of the vector coupling strength ( W ) in the N N system the restricted subtraction leads even to a smaller value. This is not too surprising if one realizes that the p-vector coupling strengths are obtained from differences of approximately equal spectral functions so that well-controlled changes in the spectral functions can lead sometimes to large modifications of coupling strengths, in arbitrary direction.

At this point we would like to make a remark about the sensitivity of our results to ! the upper limit in the dispersion integral, tma x. It is in fact quite small: Lowering trnax

from the generally used value of 120m 2 to 80m 2 the effective o--coupling strengths are

increased by less than 10%; in the p-channel the variations are even smaller. Moreover, ratios of coupling strengths in the various channels are practically unchanged.

Two points remain to be addressed: (i) For the couplings of baryons to pseudoscalar mesons we have assumed in our cal-

culations SU(3) symmetry to be realized for pseudoscalar-type coupling, cf. Sec- tion 2.4.1. Results are different when the same symmetry is assumed for couplings of pseudovector type, since (apart from the NNrr coupling) BB'Iz couplings are then increased by a factor (MB + M ' ~ ) / 2 M u leading to much stronger o--coupling strengths in the strange baryon-baryon channels, see Table 9.

_ m = 0.4 predicted by the static (ii) We have assumed for the F / ( F + D) ratios ap - aL,

quark model. If we change Otp by about 10% (to 0.45) the resulting changes in the effective ~r-coupling strengths are much less than 10%. The reason is that part of the Born amplitudes are increased while others are decreased, so that the total effect is quite small. The situation is completely different for av', which determines the bare tensor couplings #0) in the p pole graph. The same increase of oe~' (to JBB'p 0.45) leads to strong modifications in the p coupling strengths, especially for the tensor-tensor part.


Table 8 The same as Table 6, for the baryon-baryon channels with strangeness S = -2

O'GPAB~AB/4cr ( i j = W, V~, TI(,TI')

~ N~

W Er, TV TT W V/" TV T/"

Full model 2.78 3.14 4.02 0.87 -1.92 4.71 -10.38 Without baryon exchange 2.06 2.23 2.42 0.52 - 1.05 2.17 - 4.43 Subtraction for OBE model 3.09 2.64 4.38 0.99 -2.30 5.40 -12.09

3.5.2. Comparison with other models

In the N N channel we can also determine the effective tr- and p-coupling strengths from the quasiempirical N N ~ 7rTr amplitudes [ 18,19]. Corresponding spectral func-

tions have already been discussed before. We now show in Fig. 21 the results for the

product of effective coupling strengths and form factors obtained from our microscopic

model and the quasiempirical amplitudes [ 18,19], in comparison to values used in the (full) Bonn potential [ 10]. Since the quasiempirical amplitudes are available only up

' = 50m 2, a corresponding cutoff is used in the dispersion integral. If we use the to tma x same lower cutoff also for our model corresponding results essentially agree with those

obtained for the quasiempirical amplitudes. Obviously the slightly stronger increase of

the spectral function p~ of the microscopic model at small f roughly compensates for the larger maximum of the quasiempirical spectral function at t ~ ,~ 20m~. Furthermore

we obtain a considerable reduction of strength in the o--channel from inclusion of the repulsive contributions above t ~ = 50m 2 whereas in the p-channel such pieces have only a very small influence. Especially in the p-channel our results are considerably

larger than the values used in the Bonn potential. The reason is the form factor with ANNp = 1.4 GeV used in the Bonn potential, which reduces the strength at t = 0 by as much as 50%. One final point remains to be addressed: Our present results differ considerably (by up to 30%) from our former calculations based on a different microscopic model for the N N ~ ~rTr amplitudes. The reason for these discrepancies (in the spectral functions and effective coupling strengths) is that the subtraction of the uncorrelated terms from the discontinuities is model dependent. Both models differ in the parametrization of A-exchange (75); furthermore the model of Ref. [ 12] does not include the p-pole term in the transition amplitude. Still both models provide a similar description of the quasiempirical data.

The average size of the effective coupling strengths is only a rough measure of the strength of correlated 7rTr and K K exchange in the various particle channels. The precise energy dependence of the correlated exchange as well as its relative strength in the different partial waves of the s-channel reaction is determined by the spectrum of the exchanged invariant masses, i.e. the spectral functions, leading to a different t-dependence of the effective coupling strengths.

Fig. 22 shows the on-shell N N potentials in spin-singlet states with angular momen-

A. Reuber et aL /Nuclear Physics A 608 (1996) 243-304 291

~b E~

z

25

20

15

10

0.80

0.60.

~,..~o,4o.

t'.~ Q,.

°~0.2ff

1'0 2'0 3b 4o 0.0~ 1'0 2b - t ( m ~ ) - t ( m ~ )

-4.... t ,.,,_ta.

15 _ _ . . . . . . . . . . . . . .

lb 2'0 3b 40 , lb 2b go - t ( m ~ ) - t (ra~)

40

~-=

40

Fig. 21. Effective strength of the N N interaction due to correlated 1r,r and KK exchange in the tr and p-channel as a function of the 4-momentum transfer t < O. Shown are g2crNN -~ G ~r g2p NN--~NN" N N - - W G P NN--~NN"

IT p f1TGP I W ~ P 11/2 including the form factors. T h e f 2 N N =-- GNN_~NN and fpNN/gpNN =-- t NN-~NN/ ~NN--NN*

solid (dotted) line is derived from the microscopic model for correlated ~'~r and K K exchange using tmaxt = 120m 2 (tma x ' = 50m 2 only for g2~,NN ). The dashed line follows from the quasiempiricai NN ~ ,r~r amplitudes [ 18,19]. The effective strength of tr'- and p-exchange in the Bonn potential [ 10] is denoted by the dash-dotted line.

tum L = 0, 2, and 4, which are generated by the scalar-isoscalar part of correlated zr~" and K K exchange. As expected it is attractive throughout. Slight differences occur between the potentials derived from the microscopic model for the BB' ---+ i~-~ amplitudes and those determined from the quasiempirical N N ---, 1r,r amplitudes, which can be traced to differences in the spectral function p ~ ( t ) (cf. Fig. 19): For small t' the microscopic input is larger; therefore the corresponding potential in high partial waves (which is dominated by the small-t' behavior) is by about 20% larger than the quasiempirical result. In the 1S0 partial wave, on the other hand, medium and short ranged exchange processes characterized by larger t'-values contribute. In this region the microscopic amplitudes are considerably weaker; furthermore they contain the repulsive contributions above the K K threshold (cf. Fig. 16). Consequently, the resulting potential is somewhat


>

? o

-12 0

% °

5'0 100 1½0' 200' 250' 300 Tiob(MeV)

-1

>

b -2

3

\ , , ~'.~....

ZI2

5'0 100 150' 200 250' 300 T,o (MeV)

O

-.&.......~... . . . . . .

" ~ ' , ,, ~ ' " 'N \ - , "%

, , " % ,

5'o ~6o 1~o 280 2~o 300 T,~(MeV)

Fig. 22. The (r-like part of the NN on-shell potential in various partial waves as a function of the kinetic energy in the laboratory system. The solid line is derived from our microscopic model for correlated Irlr and KK exchange (with tma x l = 120m2). The dotted lines are obtained if this dispersion-theoretic result is parametrized by o'-exchange and the coupling strength G~N~NN(t ) is subsequently set to the constant value at t = 0 The dispersion-theoretic calculation using the quasiempirical NN ~ 7rTr amplitudes of Refs. [ 18,19] gives the dashed lines. Finally, the dash-dotted lines correspond to (r~-exchange used in the Bonn potential [ 10].

less attractive in the 1S0 partial wave. In agreement with Ref. [ 12] our present results (evaluated either from the microscopic model or the quasiempirical amplitudes) are stronger than o'S-exchange of the full Bonn potential. The difference is especially large in high partial waves since o J-exchange, which corresponds to a spectral function proportional to 8(t I - m~), does not contain the long-range part of the correlated processes. Indeed if we parametrize our results derived from the microscopic model by o--exchange as before (Section 3.5.1) but use for the effective coupling strength G~VN__.NN(t) the constant value at t = 0 we obtain rough agreement with our unapproximated result in the Is0 partial wave but underestimate it considerably in high partial waves. Obviously the replacement of correlated 7rTr and KK exchanges by an exchange of a sharp mass o-- meson with t-independent coupling strength cannot provide a simultaneous description of low and high partial waves.

A. Reuber et al.INuclear Physics A 608 (1996) 243-304 293

Table 9

Effective ~r coupling strengths G~o_._,ao(t = 0) depending on the F / ( F + D) ratio otp. ps (pv) indicates that SU (3)-symmetry is assumed for the pseudoscalar (pseudovector) coupling constants gsB'u ( fnB,u ). The column 'ps, Otp = 0.40' agrees with the results of the full model in Table 5

G~e__,AB/4~r (ap)

Otp N N NA N2" A A 2~2 " N ~

ps 0.35 5.86 2.83 2.52 1.49 1.69 1.21 0.40 5.87 2.82 2.58 1.52 1.72 1.19 0.45 5.90 2.84 2.70 1.59 1.80 1.19

pv 0.35 6.00 3.36 3.16 2.04 2.37 1.64 0.40 6.00 3.31 3.28 2.04 2.46 1.57 0.45 6.00 3.31 3.45 2.10 2.62 1.54

Table 10 Strength of the or-like contributions (at t = 0) to the various baryon-baryon interactions relative to the N N channel. In case of the dispersion-theoretic result for 7rlr and KK exchange only those uncorrelated contributions are subtracted which are generated already in the s-channel by the iteration of an OBE potential (cf. Table 5). In case of the OBE models the numbers are extracted from the coupling constants of the o-- (OBEPT & Jiilich A with inclusion of form factors and averaging over the several isospin channels) or the e-meson (Nijmegen models and the ,-,=,=~e coupling from SU(3))

N N NA N2," AA .$2 ~ N - ~

zr~ + KK 1 0.49 0.41 0.26 0.30 0.19

OBEPT [10] and Jiilich A I3] 1 0.45 0.63 0.34 0.66 Nijmegen D [4] 1 1 1 1 1 I Nijmegen F [5] 1 0.74 0.61 0.55 0.37 0.41 Nijmegen SC [6] 1 0.58 0.45 0.34 0.20 0.10

It is interesting to compare our results for the effective o-- and p- coupling strengths in baryon-baryon channels with non-vanishing strangeness with those used in the hyperon- nucleon interaction models of the Nijmegen [4-6] and Jiilich [3] groups. These are (with the exception of the Jiilich model B which will not be considered in the following) OBE models, i.e. o- (and p) exchange effectively include uncorrelated processes involving the A-isobar. Therefore we have to use for comparison dispersion-theoretic results in which only uncorrelated processes involving spin-½ particle intermediate states have been subtracted. Table 10 shows the relative coupling strengths in the different baryon- baryon channels for various models, in the o--channel. Apart from the Nijmegen model D [4] , in which the scalar e-meson is treated as an SU(3) singlet and therefore couples with the same strength to all channels, the interaction is by far strongest in the N N

channel for all remaining models, and it becomes weaker with increasing strangeness. Obviously, the Nijmegen soft core model [6] is nearest to the dispersion-theoretic predictions. Table 11 shows the analogous results in the p-channel for the vector-vector ( W ) and tensor-tensor (TT) components. There are sizable differences between the effective coupling constants from correlated exchange and the coupling constants of the


Table 11 Strength of the p-like contributions (at t = 0) to the interaction in the various baryon-baryon channels with S = 0, - 1 . The values for ~r~r and KK exchange correspond to the effective p coupling strengths in the third row of Tables 6-8 times the form factors for effective p-exchange. For the OBE models the values are determined from the coupling constants of the p-meson (in case of OBEPT and Jiilich A under consideration of form factors)

NN N,~ NA ---, N2~

W TF W TT W 7T

Ir~r + KK 0.28 28.87 1.25 8.07 -0.43 20.97 OBEPT[ 10] & Jiilich A[3] 0.50 18.57 0.79 8.87 0 9.84 Nijmegen DI4] 0.35 23.20 0.71 15.51 0 17.82 Nijmegen F[5] 0.63 27.34 1.25 28.76 0 14.96 Nijmegen SC[6] 0.79 14.16 1.59 7.79 0 11.85

OBE models as well as among the OBE models themselves. The latter models assume the vector coupling to the isospin current to be universal, which fixes its relative strength

in the different particle channels (apart from form factors included in the Jtilich model): In the N27 channel it is twice as large as in the N N channel and vanishes for the transition NA --, N 2 . The correlated exchange result deviates strongly, which is another

manifestation that SU(3) symmetry does not hold for correlated exchanges, even if it is assumed for the bare pBB I couplings present in our microscopic model.

4. Summary and outlook

An essential part of baryon-baryon interactions is the strong attraction of medium

range, which in one-boson-exchange models is parametrized by an exchange of a fictitious scalar-isoscalar meson with a mass of about 500 MeV. In extended meson- exchange models this part is naturally generated by two-pion-exchange processes. Be- sides uncorrelated processes correlated terms have to be considered in which both pions

interact during their exchange; in fact these terms provide the main contribution to the intermediate-range interaction.

In the scalar-isoscalar channel of the 7r~r interaction the coupling to the K K channel plays a strong role, which has to be explicitly included in any model meant to be realistic for energies near and above the K K threshold. As kaon exchange is an essential part of hyperon-nucleon interactions a simultaneous investigation of correlated 7r~" and K K

exchanges is clearly suggested. In this work we have therefore derived the correlated 7r~- as well as K K exchange contributions in various baryon-baryon channels. Starting point of our calculations was a microscopic model for the transition amplitudes of the baryon-antibaryon system ( B B ' ) into two pseudoscalar mesons (TrTr, KK) for energies below the B B I threshold. The correlations between the two mesons have been taken into account by means of zrTr-KK amplitudes (determined likewise field-theoretically [ 13,17 ] ), which provide an excellent reproduction of empirical zrzr data up to 1.3 GeV.


With the help of unitarity and dispersion-theoretic methods we have then determined the baryon-baryon amplitudes for correlated 7rTr and KK exchange in the JP = 0 + (or) and JP = 1- (p) t-channel.

In the o--channel the strength of correlated ~rzr and KK exchange decreases with the strangeness of the baryon-baryon channels becoming more negative. In the N N channel the scalar-isoscalar part of correlated exchanges is by about a factor of 2 stronger than

in both hyperon-nucleon channels (AN, ,~N) and by a factor 3 to 4 stronger than in the S = - 2 channels (AA, ,~,~, N Z ) . The influence of KK exchange is strong in baryon-baryon channels with non-vanishing strangeness while it is small in the NN

channel. This feature can be traced to different coupling constants and isospin factors and especially to the different masses involved in the various baryon-antibaryon channels.

The role of correlated KK exchange is small in the p-channel. Here the correlations are dominated by the (genuine) p-resonance in the 7rTr interaction. Among the various BBt-zrTr, K K Born amplitudes the direct coupling of the p-resonance to the baryons in the form of a p-pole graph provides the dominant contribution to correlated exchange.

It turns out that our results depend only slightly on the upper limit (cutoff) introduced in the dispersion integral. Some uncertainty results from applying SU(3) and SU(6) relations to either pseudoscalar or pseudovector 7r and K coupling constants, respectively. Note that the same problem occurs already in OBE models of the hyperon-nucleon interaction. Ultimately it has to be decided by comparison with experiment which pro- cedure is to be preferred. Moreover, if instead of SU(6) symmetry SU(3) symmetry is assumed only, the results for correlated exchanges depends on the F / ( F + D) ratios a v and a~. While the dependence on ap is only weak variation of a m leads to noticeable changes in the model predictions for the correlated exchange in the p-channel. Also here a final decision about the correct choice of ap and aL m can be made only by comparison with experiment. Again these parameters occur already in OBE hyperon-nucleon models. Therefore no new parameters are introduced when including correlated 7rTr and KK exchange in baryon-baryon interaction models. On the contrary, the elimination of single o-- and p-exchange reduces the number of free parameters and thus enhances the predictive power of corresponding interaction models.

Our results can be represented in terms of suitably defined effective coupling strengths. It turns out that the resulting values in the various baryon-baryon channels are not connected by SU(3) relations. For example, although we have even assumed SU(6) symmetry tbr the coupling strength of the bare p to the baryon current sizable baryon exchange processes destroy this symmetry in the final effective couplings. Consequently, the assumption of SU(3) symmetry for single o-- and p-exchange is not supported by our findings.

With this model constructed in the present work it is now possible to take correlated ~-Tr and K K exchange (in the o-- and p-channel) reliably into account in the various baryon-baryon channels. Especially in channels in which only little empirical information exists the elimination of phenomenological o-- and p-exchange considerably enhances the predictive power of baryon-baryon interaction models.

At this point we should of course realize that further correlations exist, which should


be investigated along the same lines as done in this paper: First, there are additional KK correlations in the (JP = 1- , I = 0) channel (w, ~b). Second, correlations between

and K should be taken into account, which are presumably most important in the K* channel but should also be present for example in the scalar (x) channel. Such correlations are expected to play a strong role in the S = - 2 sector.

Fortunately we can do the corresponding investigations in a straightforward way. The formalism contains these additional channels, and we have from our model [ 13] the required additional K K and also 7rK correlations (only a small extension of the K K model is required, namely it should include s-channel to and ~b pole terms). Corresponding investigations will be the subject of a subsequent publication. The only new parameters would be the bare (to, ~b, K*) couplings, which we would of course relate by SU(3) to the p. Remaining SU(3) parameters have to be fixed by empirical data in the N N and YN channel. The interaction can then be extended to other baryon-baryon channels with S = -2 . In this way in the frame of the Bonn-Jiilich models the possibility arises for the first time to make sensible statements about the existence of bound baryon-baryon states with strangeness S = -2 , which should be of some importance regarding the analysis of H-dibaryon experiments.

Appendix A. Conventions

As far as possible the conventions in this work are chosen in accordance with Ref. [34]. The helicity spinors u ( p , A ) ( v ( p , A ) ) of a Dirac particle of spin-½ and mass M are solutions of the free Dirac equation in momentum space for positive (negative) energy and helicity A

(~ - M)u(p, a) =0, (lJ + M ) v ( p , A) = 0 ,

with the 4-momentum pU (p0 -- Ep = +v/-M--2 + p 2)

convention of Ref. [36] the helicity spinors read (A = -t-1/2)

2Alp ]/ep IA),

v(p,a)-- f ~ ( - l P l / ' P ) l -a) 2a

where ep = M + Ep They are normalized according to

"if(p, A )u (p , A') = ~aa,,

~(p , h ) v ( p , A ~) = -Saa , ,

with ~(p, A) = u t (p, A)y0.

(A.1)

and 1~ - p l ' y u. With the phase

(A.2)

(A.3)


If p lies in the xz-plane and encloses the polar angle 0 with the ez-axis the eigenstates la) of the helicity operator,

o" p 7 It, I la> = a la), (A.4)

are related to the Pauli spinors Xm by

where

~r. Lz X=1:1/2 = ~X+1/2. (A.6)

The same phase convention as in Ref. [36] is adopted for the helicity spinors (A.2); namely, the particle and antiparticle spinors are related by the charge conjugation C [34]:

v(p, A) = C-fir(p, A) = iyzu*(p, A). (A.7)

Helicity eigenstates of two-particle systems in the center-of-mass frame are built as a product of two helicity spinors according to the phase convention of Jacob and Wick [20]. In case of two spin -1 particles 1 and 2, the helicity spinors (A.2) are used for particle 1 (momentum p ) and for particle 2 (momentum - p ) the following spinors are used:

E / - - ~ ( 1 ) I12) (1.8) u ( - p , A2) = V 2M k, 2&lpl/ep

e 2 ~ (Ip_2&l/ev) I - a2) ' (A.9) v( -p , a2) =

where

IA2} = exp ( - 2 o ' 2 0 ) X_ & . (A.10)

The hadronic field operators can be expanded in momentum space solutions of the corresponding equation of motion. For Dirac particles ( j e = ½+) and (pseudo)scalar particles (je = 0+), see Ref. [34]. For spin-1 and spin-~- particles the field operators ~b ~ and ~M', respectively, then read

~ U ( x ) - (2rr)3/21 ~a J[d3kl----~e~z(k''~)~/2wk [a(k"~)e-ik'x +at(k'A)eik'x] , (A.I 1)

I ~A f ~ p [b(p,a)u~Z(P,A)e-ip'X+dt(p,a)vU(P,A) eipx] ¢ ~ ( x ) - (2,rr)3/2 d3p

(A.12)


with the polarization vector e~(k, A), cf. Ref. [37]. Note that the hermitian conjugated component of the spherical operator a is related to the component of the hermitian operator a t by [38]

a(k , A) t = ( - 1 ) a a t ( k , - ,~) . (A.13)

The Rarita-Schwinger spinors [37] u~(p, A) and v~'(p, A) are solutions of the Rarita- Schwinger equation

(1~ - M)u~(P, A) = O, (IJ + M)v~(p, A) = O, (A.14)

),/,u/~(p,A) = O, y/~v/~(p,A) = O.

The creation and destruction operators of bosons (fermions) follow the usual (anti-) commutator relations.

For spin-1 and spin -3 particles [32] the Feynman propagators Sv(k) and S~'(p; A), which are derived from the time-ordered product of the corresponding field operators [34], read in momentum space

_g/Z~ + k/,k~ /m 2 Sv(k)= k 2 - m 2q-ie ' (A.15)

S ~ (p; A) - p2 -- M 2 + ie - ~ P p

q-Sn~o~n_pole (p; A) (A.16)

with the non-pole part

Sn~on_pole(P;a)_ 1 a + l [ / ~ ½ ( A + I ) i J + A M ] 3M 2 2A + 1 ['Y 2-,4~ 1 V~ - (P/Z3/~ +P~3/~) "

(A.17)

Without restricting the general validity of the results the parameter A is set in our work to A = -1 so that the non-pole part vanishes (see Section 2.4.1 ).

m

A p p e n d i x B . M a t r i x e l e m e n t s o f B B ' ---+ tz-fi B o r n a m p l i t u d e s

In order to evaluate the helicity amplitudes (77) of the BB --7 --,/z-~ (/z-~ = 1rTr, KK) Born terms (74)-(76) the coordinate system is conveniently chosen such that the relative momentum q of the BB ~ state points along the z-axis and the relative momentum k of the two pseudoscalar mesons /z~ lies in the xz-plane, i.e. the components of the two momenta read ( sino)

q = , k = 0 , ( B . 1 )

k cos O

with the scattering angle v ~ = ~ (p, k).


As explained in Section 2.4 the B B ' ~ Iz-~ amplitudes need to be evaluated only for BB' states being on their mass shell:

x/~ = EB + Ea, = Mv/-M-~a + q 2 + x /MB, + q2. (B,2)

Therefore, in the following, t is suppressed as an argument of the helicity amplitudes. In case of the B B ' ~ 7r~r amplitudes one has in principle to take into account also the

exchange graph (with the external pion lines exchanged) arising from the symmetrized 7rTr states. However, this can be simply done by considering the selection rule for the 7rTr states: ( - 1 ) J+t = +1 (J: total angular momentum, I: total isospin), i.e. multiplying the direct, partial wave decomposed B B --7 --~ 1rTr amplitude by a factor (1 + ( - 1 )s+/).

In the final results for the BB' --+ tx-~ Born amplitudes given below we introduced the following abbreviations:

Ol+ ~ ~B + 6 B ' ,

fl+ : 6BeB ' + q2,

with ea -= MB + Ea.

O r _ --~ E B - - 6 B , ,

(B.3) f l - = 6 B e B ' - - q2,

l + B.1. Exchange o f a jr, = ~ baryon

(lz-~, kl Vx ]BB', q + +)

= C { - q [ M x ( t -4-4k2)cr+ - qo(t - 4k2) or- + (t + 4k2)fl_ ]

+ k cos O[ 4Mxx/~fl+ + ( t - 4k2)fl_] + 8k2qcos 2 Off_} , (B.4)

(tx-~, k] Vx ]BB', q + - )

= -Cks inO{ -4q2v / t~ r+ + (t - 4k2)fl+ + 4Mxx/Tf l_ + 8kqcosOf l+} , (B.5)

where

C : = - f B x u f B ' x g F~(P 2) (B.6) 8m~ x / eneB ,MBMs, q2 o - E2x '

w i t h E x = ~ / p 2 + M ~ , p = q - k .

3 + B.2. Exchange o f a JP = ~ baryon

( ~ , kl vx IBB ' , q + -4-)

= C { 1 { q [ M x A ( t , k ) a + + q o A ( t , k ) a _ + 2Mxk2x/Tfl+ + A ( t , k ) f l _ ] q02 - E2x

+k cos 0 [ 8Mxq2k2a+ + 8qoq2k2a-

+2Mxv/~(q2 _ M 2 _ q2 _ k2)fl+ + (8k2q2 _ A( t, k ) ) f l _ ]

+2k2qcos z 0[-2Mxq2a+ - 2q0q2cr- + Mxx/ t f l+ - 2(q 2 + 2k2)fl-]

300

with


+4k3q 2 cos 3 0fl_ }

+x~{ -q[ Mx( t + 4k2)a+ - 2qota_ ] + 4kcos O[ g x x / t f + - 2k2 f _ ]

+8kZq COS 2 Off_ }

+ x ~ { - q [ 2 M x ( t + 4kZ)a+ - qo(t - 4k2)a- + (t + 4k2)f_]

+kcos O[+8Mxx/~f+ + (t - 4kZ)fl_ ] + 8kZqcos 2 Off_}} (B.7)

(le'~, k[ Vx [BB', q + - )

= Ck sin 0 q~ - ~x { - [2M~v~qZa+ + 2Mxq°q2v~°t-

- A ( t , k)fl+ + 2Mxv~(q 2 - M2x - k2)fl_l

+2kq cos 0 [ 4k2 fl+ - Mx v/tf l- ]

-4k2q 2 cos 2 Off+ } + 4x~ {2k2fl+ - Mxx/~ f - - 2kq cos Off+ }

+xZ {-[-4q2v 'Ta+ + ( t -4k2)f l+ + 8Mxv/Tfl_ ] - 8kqcosOfl+ } ~ (B.8) J

c := fBx.fB'x~ VZx (p2), 2 2 12muMx~/eBeB,M B MB,

A(t , k) :=q2t - 4k2M 2 - M2t - 4k 4 .

(B.9)

(B.10)

B.3. p-pole graph

For the p-pole graph the p-propagator and the form factor Ftz~p(k 2) (cf. Eq. (72)) do not depend on the scattering angle 0. Therefore, the partial wave decomposition can easily be performed analytically. The results are

J BB'p./'~[:t (le-~, k[ V~ [B-B 7, q + +) = 8ylCpk [gBB'p~-- + v ,t-'+ , (B.1 I) d

.(0) ta f (0 ) J BB', Vqf (leg, k] VJp [B-B 7, q + - ) = 6jl v~Cpk [SBB';~'+ + ~ - , (B.12)

where

4~r o (°) o , ~ p Ft,~, ( k 2) Cp := 3 X/eBeB,MBMB, t - - (m~°)) 2 " (B.13)

Appendix C. SU(3) relations for coupling constants

The microscopic model for the BB ---7 -~ 1r~, KK amplitudes presented in Section 2.4 imposes SU(3) symmetry to the coupling constants at the hadronic vertices in order

A. Reuber et al, /Nuclear Physics A 608 (1996) 243-304 301

to keep the number of free parameters as low as possible. The SU(3) relations for

the coupling constants are derived by constructing an SU(3)-invariant interaction La-

grangian from particle field operators, which possess a well-defined behavior under SU(3) transformations. This method is well established [39] and makes use of the SU(3) Clebsch-Gordan coefficients and the so-called isoscalar factors which are tabu-

lated in Ref. [39]. Therefore, we give in the following only the final results relevant for this work.

The coupling of the pseudoscalar meson octet (¢r, r/8, K, K) to the current of the 1 + j e = ~ baryon octet (N, A, 2:, ~ ) is described by the Lagrangian [39]

~B 'Bp = gNNlr ( N ?'rN) • ~r + gNN78 ( N t N ) r18

+gANI¢ [(N t K ) A + At(K t N)] + gZNX [ (NtzK) " ~ + ~ t . (Ktz.N)]

[X t . v ra + A t ~ " ~] -igz~z~(Xt X ~ ) " zr +gzA~

"f-gaatl8 A t Avis + g~,78 ~'* " ~,r18

+g-=~,r ( ~* ~" ~ ) • ~" + g~-=n8 ( .~t ~ ) rla, (C. 1 )

where the coupling constants being of the pion and the kaon are given by

1 gNN~" = g , gANg = - - ' ~ g ( 1 + 2c~) ,

2 g~ 'ux = g( 1 - 2a) , gzaTr = - ~ g ( 1 -- a ) ,

1 gxz~ = 2 g a , g.~AX = - " ~ g ( 4 a -- 1) ,

g ~ x x = - g , g~E,r = - g ( 1 - 2 a ) . (C.2)

Hence, the coupling of 7r and K to the baryon octet is determined by two parameters: the coupling strength g8 and the so-called F / ( F + D ) ratio ap.

The transition from pseudoscalar to vector mesons can be simply made by the re- placements

7r --+ p, K ~ K*, r/--~ ¢, r//---* w. (C.3)

However, as can be seen from the spin-momentum part of the interaction Lagrangian equation (41) the vector mesons couple to the baryon octet in two different ways: via the vector and the tensor coupling with coupling constants g and f , respectively. Now, it is not clear which coupling constants underlie SU(3) symmetry relations. Besides the canonic assumption [4,6] that the SU(3) relations apply to g and f , sometimes [5,3] the electric (g) and the magnetic (G = g + f ) coupling constants are subject


tO SU(3). In fact, both assumptions are in a certain sense equivalent [40]. For given SU(3) parameters (gl,gs,a~o) and ( f l , f s , a~ f ) the parameters ( G l , G 8 , a m) of the magnetic coupling can be chosen such that both coupling schemes lead to the same tensor couplings f = G - g.

Extending the SU(3) symmetry to SU(6), predictions for the F / ( F + D) ratios of the different multiplet couplings can be derived [41]:

e 1, m = 0.4. (C.4) t:~p = 0.4, a v = a v

a , e = 1 corresponds to the usual assumption of universal electric coupling of the p-meson to the isospin current [42], demanding for instance the equality gNNp and grrp.

The coupling of the pseudoscalar meson octet to the current for the transition between 3 + _ 1 + the JP = 7 baryon decuplet (A, y*, ~*, /'2) and the Je ~ baryon octet is given by

the Lagrangian [43]

•DBp = fNa~r(N tTA) " ~r + f N r . K ( N t T K ) • Y* + f,V4K.,~ t • ( K t T A )

+ f Ar.lrAty* . ~ -ifzr*~ (.~t x Y*) • "a" + f~.,Ts Zt. Y'r/8

+fA=.KA t (~ t ~ . ) + f Z Z . K , ~ t . (~ t~ .~ . ) + f zT.r ( ~ tTR ) . y .

+ f~= .~r (~ t~ '~*) • ~- + f_=_=*,18 (~t--~*)r/8

+ f ~ a K ( ~ t K ) / 2 + h.c. (C.5)

with the ~r and K coupling constants

fN~. = f ,

fNY* K = - - f ~ X/6, f~ar = -- f ,

fAr*zr = f /v / '~ , f,vr*Ir = - - f /v / '~ , fAZ*K = f / v : 2 ,

fz_=*r = f / x ~ 6 , f_=r*r = - f l Y / - 6 , f_~_=., = - f lY~-6 ,

fZaK = f . (C.6)

Obviously, the coupling constants now depend only on one parameter, the coupling strength f .

The SU(3) part of the transition amplitudes is obtained by evaluating the matrix elements of the appropriate products of the SU(3) Lagrangians (C.1) and (C.5). If the transition amplitudes are evaluated between states of definite total isospin the results for the SU(3) part can be separated into the product of the corresponding coupling constants and the so-called isospin factor. For instance, the isospin factor T~-#2__,uu, ( I ) for the

transition of a baryon-antibaryon state BIB2 (with B1, B2 E {8B}) to a pseudoscalar meson /.t and its antiparticle ~ follows from

T~Xt-~2~g~, ( I ) gBlXl~gB2Xl~' = (tZ-~, In] ( T.U ) x IB1B2, lm)

= ~ (1B, I~2m~,m~21 lm)(l~l-ffm~m-ffl I n ) mB I ,rob 2


× (lxl~m~,, -~l-ffm-ffl ( £ L ' ) x I Blln, roB,, B21B2 m~2), ( C.7 )

with m deno t ing the z - c o m p o n e n t o f the total isospin I and a cor responding notat ion

for the par t ic le isospins In~, I ~ , 1~, I~,. The index X in ( L U ) x indicates that the two

Lagrangians L and L I are coupled by contract ing the field operators o f the exchanged

baryon i somul t ip le t X (o r o f the p -meson in case o f the p -po l e d iagram) .

The isospin factors obta ined in this way for the processes inc luded in the Born terms

o f the BB ~ --~ Ix-~ mode l (cf. Fig. 8) are l is ted in Table 3. Note that due to the

phase conven t ion (61 ) in t roduced in connect ion with the i sospin-cross ing matr ix in

Sect ion 2.3.1 some isospin factors (e.g. those for N N --~ 1r~r) differ in sign f rom the

ones usual ly used in the l i terature [ 12,14].

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