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MIT-CTP-2169 / ORNL-CCIP-92-15

arXiv:nucl-th/9212008v1 15 Dec 1992MIT-CTP-2169 / ORNL-CCIP-92-15Kaon-Nucleon Scattering Amplitudes and Z∗-Enhancementsfrom Quark Born DiagramsT.BarnesPhysics Division and Center for Computationally Intensive PhysicsOak Ridge National Laboratory, Oak Ridge, TN 37831-6373andDepartment of PhysicsUniversity of Tennessee, Knoxville, TN 37996-1200E.S.SwansonCenter for Theoretical PhysicsLaboratory of Nuclear Science and Department of PhysicsMassachusetts Institute of Technology, Cambridge, MA 02139We derive closed form kaon-nucleon scattering amplitudes using the “quark Borndiagram” formalism, which describes the scattering as a single interaction (here theOGE spin-spin term) followed by quark line rearrangement. The low energy I=0and I=1 S-wave KN phase shifts are in reasonably good agreement with experi-ment given conventional quark model parameters.

For klab > 0.7 Gev however theI=1 elastic phase shift is larger than predicted by Gaussian wavefunctions, and wesuggest possible reasons for this discrepancy. Equivalent low energy KN potentialsfor S-wave scattering are also derived.

Finally we consider OGE forces in the re-lated channels K∆, K∗N and K∗∆, and determine which have attractive interactionsand might therefore exhibit strong threshold enhancements or “Z∗-molecule” meson-baryon bound states. We find that the minimum-spin, minimum-isospin channelsand two additional K∗∆channels are most conducive to the formation of boundstates.

Related interesting topics for future experimental and theoretical studies ofKN interactions are also discussed.December 1992

I. INTRODUCTIONKaon-nucleon collisions allow one to address many interesting problems in nuclear andhadron physics [1]. (By “kaons” we refer to the K+ = u¯s and K0 = d¯s, generically K, asdistinct from the ¯K antikaons K−= s¯u and ¯K0 = s ¯d.) Three familiar examples which weshall discuss below are 1) the origins of nonresonant “nuclear” forces in a system distinct fromNN, 2) nuclear structure physics, using kaons as weakly scattered probes, and 3) searchesfor possible exotic Z∗baryon resonances which couple directly to KN.

More recently it hasbecome clear that an understanding of KN scattering in nuclear matter is important inother areas, such as the interpretation of strangeness production in nuclear collisions and intwo-kaon correlation measurements [2].Elastic KN scattering is a natural system for the study of nonresonant nuclear forces.Since the valence kaon wavefunction contains an ¯s antiquark which cannot annihilate againstthe nonstrange nucleon state, direct production of conventional baryon resonances is ex-cluded. KN scattering is further simplified by the absence of one pion exchange, so one canstudy the nonresonant, non-OPE part of hadron scattering in relative isolation.

Theoreticalstudies of KN nuclear forces are especially appropriate because there is already considerableexperimental information on the elastic amplitudes and two-body inelastic reactions such asKN→K∗N and KN→K∆[1,3–5]. These experimental amplitudes provide stringent testsfor models of hadronic interactions.

The dominant S-wave elastic phase shifts are moder-ately well established, and the higher partial waves up to L=4 have been determined orestimated [5]. The basic features of the elastic reaction are a strong repulsion in the I=1S-wave, a weaker repulsion in the I=0 S-wave, and an important spin-orbit interaction whichis evident in the P-waves.

The important low energy behavior of the I=0 S-wave, in par-ticular the scattering length, is unfortunately not yet very well known. The experimentalsituation should improve considerably with the development of new hadronic facilities suchas DAΦNE and KAON [6,7].KN scattering also has applications in nuclear physics; since the kaon-nucleon cross sec-tion is relatively small, kaon beams can be used as probes of nuclear structure.

It wouldobviously be useful to understand the mechanism and properties of the kaon-nucleon in-teraction for this application. In view of this application one topic in this paper will bethe derivation of effective low energy KN potentials from the nonrelativistic quark potentialmodel.Another reason for interest in KN collisions is the possibility of producing flavor-exotic Z∗baryon resonances.

If discovered, these might be resonances with the quark valence structureq4¯s [8], where q = u or d. Such multiquark hadrons were widely predicted in the early days ofthe quark model [9], but it now appears that multiquark basis states usually do not supportresonances, due to the “fall apart” effect [10,11]. The known exceptions are deuteronlike“molecule” states of hadron pairs, which should perhaps be classified as unusual nuclearspecies.

(Nuclei themselves are excellent examples of the tendency of multiquark systemsto separate into hadronic molecules.) In the meson-meson sector two K¯K molecule statesare reasonably well established [12], and there are several other meson-meson candidates[13].

In the antikaon-nucleon sector the Λ(1405) is an obvious candidate ¯KN molecule, andthere presumably are other molecule states in channels with attractive interactions. Boththe elastic reaction KN→KN and inelastic processes such as KN→K∗N and KN→K∆can2

be studied for evidence of exotic Z∗baryon resonances. With a realistic model of hadronicinteractions we might reasonably expect to predict the quantum numbers of exotic meson-baryon molecular bound states, should these exist.In this paper we apply the “quark Born diagram” formalism to KN scattering.

In thisapproach we assume conventional nonrelativistic quark model wavefunctions for the asymp-totic hadrons, and calculate the Hamiltonian matrix element for scattering due to a singleinteraction between constituents in different incident hadrons. To form color singlet finalstates at lowest order one must then exchange constituents.

The full Born amplitude isobtained by summing over all such processes coherently. (Similar constituent exchangemechanisms have been proposed for high energy hadron scattering [14], and there is strongexperimental evidence in favor of this mechanism from large-t exclusive reactions [15].) Thisnonrelativistic Hamiltonian matrix element is then combined with relativistic phase spaceand kinematics to give results for differential cross sections, partial wave amplitudes andother scattering observables.

In previous work we derived the elastic scattering amplitudesfor I=2 ππ [16], I=3/2 Kπ [17] and I=1 KK [16]. (These cases were chosen because they arefree of valence q¯q annihilation processes, which are known to be important if allowed.) Wefound good agreement with experimental ππ and Kπ S-wave phase shifts given conventionalquark model parameters.

We have also applied similar techniques to pseudoscalar-vectorand vector-vector meson channels [18], and the results may have important implications formeson spectroscopy [13]. In Appendix C of [16] we presented a diagrammatic representationof these techniques, with associated “Feynman rules” for the scattering diagrams.

KN elasticscattering is also annihilation free and affords a nontrivial test of the quark Born formalism.KN elastic scattering has previously been the subject of numerous theoretical investi-gations. Meson exchange models have been applied in several studies [19], but these aredifficult to justify fundamentally because the range of heavier meson exchange forces (≈0.2fm) is much smaller than the minimum possible interhadron distance for two distinct hadrons(≈1 fm) [11].

These models typically have many free parameters, which are not well estab-lished experimentally and are fitted to the data. Thus one is in effect simply parametrizingexperiment.

This type of model may be of theoretical interest as a parametrization of morefundamental scattering mechanisms which operate at the quark and gluon level, as it maybe possible to relate the predictions of these different approaches.A quark and gluon approach to scattering using the P-matrix and bag model wave-functions was proposed by Jaffe and Low [20]. They suggested interpreting the multiquarkclusters of the bag model not as resonances, but instead as the short distance parts ofhadron-hadron scattering states.

In principal this approach can be used to predict phaseshifts, but in practice it has mainly been used to interpret experimental phase shifts interms of P-matrix poles. This approach has been followed for KN by Roiesnal [21], whoconcluded that the KN data could indeed be interpreted in terms of poles approximatelyat the energies predicted by the bag model, but that the pole residues (coupling strengthsto asymptotic KN channels) did not agree well with predictions.

A more recent bag modelcalculation of KN scattering by Veit, Thomas and Jennings [22] used the cloudy bag model,which combines quark fields (in the baryon) with fundamental pseudoscalar meson fieldsin an effective lagrangian. This composite model leads to an I=1 S-wave phase shift anda scattering length which are very similar to our result, but their I=0 phase shift is muchsmaller than experiment.

Although this cloudy bag approach gives promising numerical3

results, it does not provide us with an understanding of the scattering mechanism at thequark and gluon level.Studies of the dominant S-wave KN scattering amplitudes in terms of quark model wave-functions and quark-gluon interactions have been published by Bender and Dosch [23] (adi-abatic approach), Bender, Dosch, Pirner and Kruse [24] (variational generator coordinatemethod, GCM) and Campbell and Robson [25] (resonating group method, RGM). Thelarge spin-orbit forces evident in the KN P-wave data have also been studied using similarquark model techniques, first qualitatively by Pirner and Povh [26] and later in detail byMukhopadhyay and Pirner [27] (using GCM).

The assumptions regarding dynamics, thescattering mechanism, quark model wavefunctions and the parameters used in these calcu-lations are very similar to our assumptions in this paper. The most important differencesare that 1) our techniques are perturbative and allow analytic solution, and 2) we disagreeabout the size of the OGE contribution to KN scattering.

Specifically, we find that OGEalone suffices to explain the observed I=1 KN scattering length, whereas Bender et al. [24]conclude that OGE is too small, and that a Pauli blocking effect is dominant in I=1.

Camp-bell and Robson [25] similarly found that the experimental I=1 phase shift was larger thantheir predictions, which were based on generalizations of Gaussian wavefunctions and a fullOGE and confining interaction.II. CALCULATION OF KN AND RELATED SCATTERING AMPLITUDESa) Hamiltonian and hadron statesOur technique involves a Born order calculation of the matrix element of the Hamiltonianbetween asymptotic hadron states in the nonrelativistic quark model.

In the KN case thedominant interaction was previously found by Bender et al. [24] to be the spin-spin “colorhyperfine” term.

A similar conclusion has been reached for the NN interaction [11,28]. Herewe shall adopt this approximation and neglect the other OGE and confining terms.

Thus,our scattering amplitude is proportional to the matrix element ofHscat =Xa,i

Ourmomentum space Gaussian wavefunctions for the kaon and nucleon are conventional quarkmodel forms,φkaon(⃗prel) =1π3/4β3/2 exp−⃗p 2rel8β2(2)where⃗prel ≡(m¯q⃗pq −mq⃗p¯q)(mq + m¯q)/2,(3)4

andφnucleon(⃗p1, ⃗p2, ⃗p3) =33/4π3/2α3 exp−(⃗p 21 + ⃗p 22 + ⃗p 23 −⃗p1 · ⃗p2 −⃗p2 · ⃗p3 −⃗p3 · ⃗p1)3α2. (4)The parameters α and β are typically found to be ≈0.3−0.4 Gev in hadron phenomenology.These are relative momentum wavefunctions, and have an implicit constraint that the con-stituent momenta add to the hadron momentum.

In the full momentum space wavefunctionthere is an overall delta function that imposes this constraint;Φkaon(⃗pq, ⃗p¯q; ⃗Ptot) = φkaon(⃗prel) δ(⃗Ptot −⃗pq −⃗p¯q) ,(5)Φnucleon(⃗p1, ⃗p2, ⃗p3; ⃗Ptot) = φnucleon(⃗p1, ⃗p2, ⃗p3) δ(⃗Ptot −⃗p1 −⃗p2 −⃗p3) . (6)The normalizations are⟨Φkaon(⃗P ′tot)|Φkaon(⃗Ptot)⟩=Z Z Z Zd⃗p d⃗¯p d⃗p ′ d⃗¯p ′ Φ∗kaon(⃗p ′,⃗¯p ′; ⃗P ′tot)Φkaon(⃗p,⃗¯p; ⃗Ptot) = δ(⃗Ptot −⃗P ′tot)(7)and⟨Φnucleon(⃗P ′tot)|Φnucleon(⃗Ptot)⟩=Z Z Z Z Z Zd⃗p1 d⃗p2 d⃗p3 d⃗p1′ ⃗p2′ d⃗p3′ Φ∗nucleon(⃗p1′, ⃗p2′, ⃗p3′; ⃗P ′tot)Φnucleon(⃗p1, ⃗p2, ⃗p3; ⃗Ptot)= δ(⃗Ptot −⃗P ′tot) .

(8)Since these state normalizations are identical to those used in our previous study of Kπscattering [17] we can use the relations between amplitudes and scattering observables givenin that reference.The color wavefunctions for the asymptotic hadrons are the familiar color singlet states|meson⟩=Xı,¯ı=1,31√3 δı¯ı |ı¯ı⟩(9)and|baryon⟩=Xi,j,k=1,31√6 ǫijk |ijk⟩. (10)Our spin-flavor states for the meson and baryon are standard SU(6) states, but we havefound it convenient to write the baryon states in an unconventional manner, as the usualquark model conventions are unwieldy for our purposes.

First, to establish our notation, thespin-flavor K+ state is5

|K+⟩= 1√2|u+¯s−⟩−|u−¯s+⟩. (11)For quark model baryon states it is conventional to assign each quark a fixed location in thestate vector, as though identical quarks were distinguishable fermions.

One then explicitlysymmetrizes this state. Thus for example one writes the normalized ∆+(Sz = +3/2) stateas|∆+(+3/2)⟩= 1√3|u+u+d+⟩+ |u+d+u+⟩+ |d+u+u+⟩(12)and treats each basis state as orthogonal.

Note however that this is not the usual way torepresent multifermion states. In standard field theoretic usage each of these basis states isidentical to the others, to within an overall phase.

In this language the normalized ∆+(+3/2)state is simply|∆+(+3/2)⟩= 1√2 |u+u+d+⟩,(13)which we could equally well write as |u+d+u+⟩/√2 or |d+u+u+⟩/√2. The advantage of usingfield theory conventions becomes clear in calculating nucleon matrix elements.

For example,the usual quark model proton state is|P(+1/2)⟩=2|u+u+d−⟩−|u+u−d+⟩−|u−u+d+⟩+ 2|u+d−u+⟩−|u+d+u−⟩−|u−d+u+⟩+ 2|d−u+u+⟩−|d+u+u−⟩−|d+u−u+⟩√18 ,(14)and in comparison this state in field theory conventions is|P(+1/2)⟩=s23|u+u+d−⟩√2−s13 |u+u−d+⟩. (15)Use of the latter form, with all permutations of quark entries allowed in matrix elements,reduces the number of P→P terms from 81 (many of which are zero) to 4.

Of course theresults are identical, as these are just different conventions for the same state.b) Enumeration of quark line diagrams for KNNow we consider KN scattering. As explained in reference [16], we begin by determiningthe matrix element of the scattering Hamiltonian (1).

First we factor out the overall mo-mentum conserving delta function and then derive the remaining matrix element, which wecall hfi;f⟨KN|Hscat|KN⟩i ≡hfi δ(⃗Pf −⃗Pi) . (16)We will discuss one part of the calculation in detail to explain the techniques, and thensimply quote the full result.

Specializing to the spin up I=1 case K+P(+1/2)→K+P(+1/2),we require the matrix element of the scattering hamiltonian (1) between initial and final K+P6

states with color and spin-flavor wavefunctions given by (9,10) and (11,15) respectively. Sincethe kaon and proton states (11) and (15) are each the sum of two terms, the full amplitudefor K+P(+1/2)→K+P(+1/2) is a weighted sum of 16 subamplitudes.

We shall consider thesubamplitude for |u+¯s−⟩{|u+u+d−⟩/√2} →|u+¯s−⟩|u+u−d+⟩, which we call he.g.fi , in detailfor illustration.We begin by constructing all allowed quark line diagrams and their associated combi-natoric factors. First we arrange the initial and final states with their normalizations on ageneric scattering diagram,he.g.fi=(17)1√2u+¯s−u+u+d−u+¯s−u+u−d+.✲✲✛✛✲✲✲✲✲✲Now we connect the initial and final lines in all possible ways consistent with flavorconservation.For the d quark and the ¯s antiquark this choice is unique.For the finalmeson’s u quark however there are two choices for which initial baryon’s quark it originatesfrom.

Similarly the initial meson’s u quark can attach to either of two final baryon u quarks.Thus we have four quark line diagrams. We may immediately simplify the diagrams; sincethe baryon wavefunctions are symmetric, we may permute any two initial or final baryonlines and obtain an equivalent diagram.

We use this symmetry to reduce all diagrams to a“standard form” in which only the meson’s quark and the upper baryon quark are exchanged.The two choices for the initial baryon’s spin up u+ quark are thus equivalent, and contributean overall combinatoric factor of two. The final baryon’s quarks however give inequivalentdiagrams, one being nonflip (u+(K) →u+(P)) and the other spin flip (u+(K) →u−(P)).

(No polarization selection rules are being imposed yet, only flavor conservation.) Thus ouramplitude leads to the line diagramshe.g.fi = 1√2 · 2·+(18)✲✲✛✛✲✲✲✲✲✲u+¯s−u+u+d−u+¯s−u+u−d+✁✁✁✁✁✁✁❆❆❆❆❆❆❆✲✲✛✛✲✲✲✲✲✲u+¯s−u+u+d−u+¯s−u−u+d+.✁✁✁✁✁✁✁❆❆❆❆❆❆❆7

We next “decorate” each of these line diagrams with all possible single interactions (1)between a quark (or antiquark) in the initial meson and a quark in the initial baryon. Thereare six of these per line diagram (two choices in the meson times three in the baryon), so wehave a total of twelve scattering diagrams to evaluate.

However in this case all but one aretrivially zero. Note that in the first line diagram we must flip the spins of u and d quarks inthe initial baryon to have a nonzero contribution.

This however is not part of our scatteringinteraction, which operates between pairs of constituents in different initial hadrons. The⃗Si · ⃗Sj interaction either flips a pair of spins in different incident hadrons (through S+S−or S−S+ terms) or leaves all spins unchanged (through SzSz).

Thus, the transition in thefirst line diagram cannot occur through a single ⃗Si · ⃗Sj interaction. For the second diagramhowever there is a single nonvanishing transition, in which the initial meson’s u+ quark andthe baryon’s d−quark interact through the spin flip operator;he.g.fi=√2·(19).✲✲✛✛✲✲✲✲✲✲u+¯s−u+u+d−u+¯s−u−u+d+✁✁✁✁✁✁✁❆❆❆❆❆❆❆rrc) Independent quark and gluon diagrams and their spin and color factorsFinally we require the spin, color, overall phase and spatial factors associated with thisand the other independent diagrams.

There are only four independent quark and gluondiagrams, since all others can be obtained from these by permutation of lines. These fourdiagrams areD1=(20),✲✲✛✛✲✲✲✲✲✲✁✁✁✁✁✁✁❆❆❆❆❆❆❆rr8

D2=(21),✲✲✛✛✲✲✲✲✲✲✁✁✁✁✁✁✁❆❆❆❆❆❆❆rrD3=(22),✲✲✛✛✲✲✲✲✲✲✁✁✁✁✁✁✁❆❆❆❆❆❆❆rrD4=(23).✲✲✛✛✲✲✲✲✲✲✁✁✁✁✁✁✁❆❆❆❆❆❆❆rrThe spin factor is simply the matrix element of ⃗Si · ⃗Sj for scattering constituents i and j,evaluated between the initial and final (q¯q)(qqq) spin states. This is 1/2 if both spins i andj are antialigned and both flip, +1/4 if the spins are aligned and neither flips, and −1/4 ifthey are antialigned and neither flips.

All other cases give zero. All spectator spins mustnot flip or the overall spin factor is trivially zero.The color factor can be evaluated using the states (9), (10) and standard trace techniques,as in (51) of reference [16].

The result for each diagram isIcolor(D1) = +4/9 ,(24)Icolor(D2) = −2/9 ,(25)Icolor(D3) = −4/9 ,(26)Icolor(D4) = +2/9 . (27)9

d) “Diagram weights” for KN scatteringWe conventionally write the meson-baryon hfi matrix elements as row vectors whichdisplay the numerical coefficient of each diagram’s spatial overlap integral. Thus,hfi =w1, w2, w3, w4(28)representshfi = w1 Ispace(D1) + w2 Ispace(D2) + w3 Ispace(D3) + w4 Ispace(D4) .

(29)This notation is useful because the diagram weights {wi} are group theoretic numbers thatobey certain symmetries, whereas the spatial overlap integrals are complicated functions thatdepend on the specific spatial wavefunctions rather than the symmetries of the problem.As an illustration, our practice subamplitude he.g.fi ishe.g.fi =√2 · 12·−29· Ispace(D2) . (30)(using the spin and color matrix elements given above), which we abbreviate ashe.g.fi =0, −√29 , 0, 0.

(31)This completes our detailed derivation of he.g.fifor the subprocess |u+¯s−⟩{|u+u+d−⟩/√2} →|u+¯s−⟩|u+u−d+⟩.Proceeding similarly, we have derived the weights for the full KN elastic scattering am-plitudes, given the states (11) and (15) and their isospin partners. These arehKNfi (I = 0) =0, 16, 0, 16(32)andhKNfi (I = 1) = 13,118, 13,118.

(33)For numerical estimates of these amplitudes we require the spatial overlap integrals, whichwe shall evaluate explicitly with Gaussian wavefunctions.e) Spatial overlap integralsThe spatial overlap integrals represented by the four diagrams D1 . .

. D4 may be deter-mined using the diagrammatic techniques discussed in Appendix C of reference [16].

Theseare formally 30-dimensional overlap integrals (three dimensions times ten external lines),but twelve integrations are eliminated by external momentum constraints and an additionalnine are eliminated by the unscattered spectator lines. This leaves a nontrivial 9-dimensionaloverlap integral for each diagram.

We give the initial meson a label A, with three-momentumalso called A and quark three-momentum a and antiquark momentum ¯a, and we similarlylabel the initial baryon B, the final meson C and the final baryon D. Since we choose toevaluate these integrals in the CM frame we use the momentum substitutions B = −A and10

D = −C. We also introduce a nonstrange to strange quark mass ratio ρ = mq/ms.

Withthese substitutions the four spatial overlap integrals areIspace(D1) = +8παs3m2q1(2π)3Z Z Zd⃗a d⃗b1 d⃗b2 φA(2a −2ρA1 + ρ) φ∗C(2a + 2C1 + ρ −2A)· φB(b1, b2, −A −b1 −b2) φ∗D(b1 + A −C, b2, −A −b1 −b2) ,(34)Ispace(D2) = +8παs3m2q1(2π)3Z Z Zd⃗b1 d⃗c d⃗d1 φA(2c −2A1 + ρ −2C) φ∗C(2c −2ρC1 + ρ)· φB(b1, c, −A −b1 −c) φ∗D(d1, A −C + b1 + c −d1, −A −b1 −c) ,(35)Ispace(D3) = +8παs3m2qρ1(2π)3Z Z Zd⃗a d⃗b2 d⃗c φA(2a −2ρA1 + ρ) φ∗C(2c −2ρC1 + ρ)· φB(a −A + C, b2, −a −b2 −C) φ∗D(a, b2, −a −b2 −C) ,(36)Ispace(D4) = +8παs3m2qρ1(2π)3Z Z Zd⃗a d⃗b1 d⃗c φA(2a −2ρA1 + ρ) φ∗C(2c −2ρC1 + ρ)· φB(b1, c, −A −b1 −c) φ∗D(A −C −a + b1 + c, a, −A −b1 −c) . (37)There are many equivalent ways to write these integrals which arise from different choicesof the variables eliminated by momentum constraints.Note that the overall coefficients of these integrals are positive, although the coefficientof Hscat (1) is negative.

This is because there is an overall phase factor of (−1) for eachdiagram D1 . .

. D4, due to anticommutation of quark creation and annihilation operators inthe matrix element.

Here we incorporate this phase, which we call the “signature” of thediagram [16], in the spatial overlap integrals. The signature is equal to (−1)Nx, where Nx isthe number of fermion line crossings.

For diagrams D1 . .

. D4 above Nx = 3, so the signatureisIsignature = (−1) .

(38)Note that a diagram in nonstandard form, such as the kaon’s quark line crossing to thesecond baryon quark, can have a (+1) signature; in the full hfi matrix element this iscompensated by a change in sign of the color factor.We explicitly evaluate these overlap integrals using the Gaussian wavefunctions (2) and(4). For Gaussians the integrals factor into products of three 3-dimensional integrals, andthe results are all of the formIspace(Di) = 8παs3m2q1(2π)3 ηi exp−(Ai −Biµ)P 2cm,(39)11

where P 2cm is the modulus of each hadron’s three-momentum in the CM frame, µ = cos(θCM)where θCM is the CM scattering angle, and the constants ηi, Ai and Bi are functions of α, βand ρ. Bi > 0 implies forward peaked scattering and Bi < 0 is backward peaking. The pureexponential dependence in P 2cm and P 2cmµ is a consequence of the Gaussian wavefunctionsand the contact interaction.

Introducing the ratio g = (α/β)2, these constants areη1 = 1(40)A1 = 2ρ2 + 4ρ + (3g + 2)6(1 + ρ)2α2(41)B1 = A1 ,(42)η2 = 12g7g + 63/2(43)A2 = (40g + 3)ρ2 + (32g + 6)ρ + (21g2 + 28g + 3)6(7g + 6)(1 + ρ)2α2(44)B2 = (−8g + 1)ρ2 + 2ρ + (7g2 + 8g + 1)2(7g + 6)(1 + ρ)2α2,(45)η3 = ρ6g + 33/2(46)A3 = (10g + 6)ρ2 + (8g + 12)ρ + (7g + 6)6(g + 3)(1 + ρ)2α2(47)B3 = (−g + 1)ρ2 + 2ρ + (g + 1)(g + 3)(1 + ρ)2α2,(48)η4 = ρ12g(2g + 3)(g + 2)3/2(49)A4 = (20g2 + 40g + 3)ρ2 + (4g2 + 14g + 6)ρ + (5g2 + 10g + 3)6(2g + 3)(g + 2)(1 + ρ)2α2,(50)B4 = (−4g2 −8g + 1)ρ2 + (−4g2 −6g + 2)ρ + (g2 + 2g + 1)2(2g + 3)(g + 2)(1 + ρ)2α2. (51)12

These results were derived at MIT and ORNL [29] independently using MAPLE and MAC-SYMA algebra programs respectively.Some important properties of these diagrams, specifically which are forward peaked orbackward peaked processes, and which diagrams dominate at high energies, can be inferredby inspection. The leading diagram in the high energy limit is D1, which is a forward peakedexponential in t. The other diagrams are exponentially suppressed in s and are also forwardpeaked, with the single exception of D4.

Note that for plausible values of g ≈1 and ρ ≈0.6this diagram leads to a backwards peak (B4 < 0). These properties have a simple commonorigin; since we are scattering through a hard delta function interaction, the only angulardependence comes from overlap suppression due to the spectator lines.

A spectator linewhich is required to cross into the other hadron gives an especially large suppression athigh energies and small angles. The amplitude for a crossing spectator line is maximum forbackscattered hadrons; in this case the crossing spectator is actually continuing to move ina new hadron with the same momentum vector as the hadron it originally resided in.The first diagram D1 has no crossing spectators, so it is not suppressed in s; only the hardscattered constituents are required to cross into different hadrons.

In diagrams D2 and D3one spectator line is required to cross to a different hadron, so there is some suppression withincreasing s. Since two spectators do not cross, they dominate the angular dependence, andthe scattering is forward peaked. Diagram D4, the backward peaking process, is qualitativelydifferent because two spectator lines are required to change hadrons, and only one spectatordoes not cross.

In this case “backwards” meson-baryon scattering actually corresponds toforward scattering for the two crossing spectator lines, which is obviously preferred. Thisdescription attributes backward peaks, which might otherwise appear counterintuitive, tothe obvious mechanism of “minimum spectator suppression” at the quark level.f) KN phase shifts and scattering lengthsGiven the diagram weights (32-33) and our results (40-51) for the Gaussian overlapintegrals, we have completed the derivation of the Hamiltonian matrix element hfi for KNelastic scattering.

Since we have used the same normalization for KN states as in our previousdiscussion of Kπ scattering [17] we can use the same relations derived there to relate hfi toscattering variables. First we consider the elastic phase shifts, which are given byδKNℓ= −2π2PcmEKEN(EK + EN)Z 1−1 hKNfiPℓ(µ)dµ .

(52)Using the integralR 1−1 ebµPℓ(µ)dµ = 2iℓ(b), we findδKNℓ= −4αs3m2qPcmEKEN(EK + EN)4Xi=1wi ηi exp(−AiP 2cm) iℓ(BiP 2cm) ,(53)where one specifies the isospin state I=0 or I=1 through the choice of the diagram weights{wi}.As we approach the KN threshold the S-wave phase shift is asymptotically linear in Pcm,and the coefficient is the scattering length aI. Since the exponential and the i0 Bessel functionare both unity in this limit, we recover a relatively simple result for the KN scattering length,aKNI= −4αs3m2qMKMN(MK + MN)4Xi=1wi ηi .

(54)13

Since the coefficients {ηi} are relatively simple functions, we can write these scatteringlengths as simple functions of αs/m2q, ρ = mq/ms, the meson-baryon relative scale parameterg = (α/β)2 and the physical masses MK and MN. The results areaKNI=1 = −4αs3m2qMKMN(MK + MN)· 13 +118 12g7g + 63/2+ 13 ρ6g + 33/2+ 118 ρ12g(2g + 3)(g + 2)3/2 (55)andaKNI=0 = −4αs3m2qMKMN(MK + MN)· 16 12g7g + 63/2+ 16 ρ12g(2g + 3)(g + 2)3/2 .

(56)The basic features of the low energy KN interaction, a repulsive I=1 S-wave and a repul-sive but less strong I=0 S-wave, are already evident in these formulas. (The parameterg = (α/β)2 is constrained by quark model phenomenology to be comparable to unity.) De-tailed numerical results for the scattering lengths and phase shifts and a comparison withexperiment are presented in the next section.III.

COMPARISON WITH EXPERIMENTa) Scattering lengthsBefore we discuss our numerical predictions we first review the status of the experimentalscattering lengths. Since there are unresolved disagreements between analyses in the I=0channel, we have compiled relatively recent (since 1980) single-energy S-wave phase shiftsfor our discussion.

These are in chronological order Martin and Oades [30] (Aarhus and UCLondon, 1980); Watts et al. [31] (QMC and RAL, 1980); Hashimoto [32] (Kyoto and VPI,1984); and Hyslop et al.

[5] (VPI, 1992). The I=1 data set analysed by Arndt and Roper [33](VPI, 1985) was incorporated in the 1992 VPI simultaneous analysis of I=0 and I=1 data,so we shall not consider it separately.

The energy dependent parametrizations of Corden etal. [34] and Nakajima et al.

[35] are not included in our discussion.In Fig.1 we show these experimental I=0 and I=1 S-wave phase shifts versus Pcm = |⃗Pcm|.The linear low energy behavior which determines the scattering length is evident in theI=1 data, and Hyslop et al. cite a fitted value of aKNI=1 = −0.33 fm.

Previous analyses(summarized in [1] and [5]) have given values between −0.28(6) fm [36] and −0.33 fm [5,37].A more useful way to present the S-wave phase shift data is to display δI0/Pcm versus P 2cm;the intercept is the scattering length, and the slope at intercept determines the effectiverange. In Fig.2 we show the S-wave phase shifts in this manner; an I=1 scattering lengthof about −0.31(1) fm is indeed evident, which we shall take as our estimated experimentalvalue.14

Unfortunately the I=0 scattering length is much less well determined, as is evident inFigs.1 and 2.Previous (favored) solutions up to 1982 are summarized in Table 2.3 of[1], and range between +0.02 fm and −0.11+0.06−0.04 fm. There appear to be two sets of lowenergy values in the data of Fig.1, a smaller phase shift from the Aarhus-UCL and QMC-RAL collaborations and a larger one from from the Kyoto-VPI and VPI analyses.

BelowPcm = 0.4 Gev the Kyoto-VPI and VPI results are larger than Aarhus-UCL and QMC-RALby about a factor of two. The VPI group actually cite a scattering length of aKNI=0 = 0.0fm, although this requires rapid low energy variation below the first experimental point(compare their Fig.1(a) with the I=1 phase shift in their Fig.2(a), which is constrained byexperiment at lower energy and shows the expected √Tlab ∝Pcm S-wave dependence).

Sincethe I=1 phase shift is close to linear for Pcm < 0.4 Gev (klab < 0.7 Gev), we will assumethat the zero I=0 scattering length quoted in [5] is an artifact of their fit, and that theactual I=0 phase shift is approximately linear in Pcm for Pcm < 0.4 Gev. We can then readthe I=0 scattering length from the intercept in Fig.2.

From the figure we see that a naiveextrapolation to threshold leads to scattering lengths of about −0.09(1) fm and −0.17(2)fm respectively from the two sets of references. In summary, the experimental phase shiftsshown in Fig.2 suggest to us the scattering lengthsaKNI=1(expt.

)= −0.31(1) fm ;aKNI=0(expt. )Aarhus−QMC−RAL−UCL= −0.09(1) fm ,aKNI=0(expt.

)Kyoto−V P I= −0.17(2) fm . (57)We emphasize that the I=0 values are our interpretation of the data from Fig.2, and thereferences cited quote smaller scattering lengths that we believe the data does not support.As the values of the I=0 scattering length and low energy phase shifts are controversial, anaccurate determination should be a first priority at a kaon facility.To compare our predictions with experiment we first use a “reference parameter set”with conventional quark model parameters.

The hyperfine strength is taken to be αs/m2q =0.6/(0.33)2 Gev−2, and the nonstrange to strange quark mass ratio is ρ = mq/ms =0.33 Gev /0.55 Gev = 0.6. The remaining parameter in the scattering length formulasis g = (α/β)2, the ratio of baryon to meson width parameters squared.

These parame-ters are rather less well determined phenomenologically. For baryons, values in the rangeα = 0.25 −0.41 Gev have been used in nonrelativistic quark model studies [38–40].

Isgurand Karl [41] originally used α = 0.32 Gev for spectroscopy, but Copley, Karl and Obryk[42] had earlier found that the photocouplings of baryon resonances required a somewhatlarger value of α = 0.41 Gev, which may be a more realistic estimate [38,39] because it isless sensitive to short distance hyperfine matrix elements. This larger value was also foundby Koniuk and Isgur [43] for baryon electromagnetic transition amplitudes.

Here we takeα = 0.4 Gev as our reference value. For mesons, studies of various matrix elements have ledto values of β = 0.2 −0.4 Gev [39].

In our previous study of I=2 ππ scattering we found abest fit to the S-wave phase shift data with β = 0.337 Gev. Here we use a similar β = 0.35Gev as our reference value; if the quark Born formalism is realistic we should use essentiallythe same meson parameters in all reactions.15

With our reference parameter set and physical masses MK = 0.495 Gev and MN = 0.940Gev, our formulas (55) and (56) giveaKNI=1(ref. set) = −0.35 fm(58)andaKNI=0(ref.

set) = −0.12 fm . (59)In view of our approximations, the parameter uncertainties, and the uncertainties inthe I=0 data, these scattering lengths compare rather well with experiment.

Note that ourconclusions differ from those of Bender et al. [24], who reported that the OGE contributionto I=1 scattering was too small to explain the observed S-wave phase shift.

We discuss thisdisagreement further in the appendix.Now suppose we attempt to fit our estimated experimental values of the scattering lengths(57) by varying our parameter set. It is useful to fit the ratio aKNI=0/aKNI=1, since this involvesonly ρ and the width parameter g. We have fixed ρ = 0.6, and in any case we find thataKNI=0/aKNI=1 is insensitive to ρ, so only g remains as an important parameter.

In Fig.3 we showthe predicted ratio of KN scattering lengths as a function of α/β. The two experimentalratios assuming the values in (57) are also indicated.

The larger ratio aKNI=0/aKNI=1 = 0.17/0.31requires α/β = 1.91, rather far from typical quark model values. Fitting the smaller ratioaKNI=0/aKNI=1 = 0.09/0.31 requires α/β = 1.02, which is more representative of quark modelparameters.

An accurate determination of the I=0 KN scattering length through direct lowenergy measurements, rather than by extrapolation, would be a very useful experimentalcontribution; this would allow a more confident test of our results and those of other models(as shown for example in Table 6-4 of Hyslop [37]).b) S-wave phase shiftsThe S-wave KN phase shifts predicted by (53) with ℓ= 0 given the “reference parameterset” αs/m2q = 0.6/(0.33)2 Gev−2, ρ = mq/ms = 0.6, α = 0.4 Gev and β = 0.35 Gev areshown as dashed lines in Fig.4. As we noted in the previous section, this parameter setgives reasonable scattering lengths, although the I=0 scattering length is not yet very wellestablished experimentally.At higher energies the reference parameter set predicts an I=1 phase shift that retreatsmore quickly with energy than is observed experimentally; in Fig.4 we see a rapid departureof theory and experiment above Pcm = 0.4 Gev (klab = 0.7 Gev).

This is near the openingof the inelastic channels K∆and K∗N, as indicated in Fig.4.Two possible reasons forthis discrepancy are 1) inelastic effects of the channels K∆, K∗N and K∗∆, which shouldbecome important just where theory and experiment part, and 2) short distance componentsin the meson and baryon wavefunctions that are underestimated by the smooth Gaussianwavefunctions (2) and (4).Although inelastic effects are certainly important experimentally [3,4], the most impor-tant low energy inelastic process is P-wave K∆production [4]. Hyslop et al.

[5] similarlyfind relatively small inelasticities in the I=1 KN S-wave, with η ≥0.9 for Pcm ≤0.68 Gev.At the end of this range our predicted phase shift given the reference parameter set is only16

about half the observed value, so it appears unlikely that the discrepancy is mainly due toinelastic channels.A second possible reason for the discrepancy is a departure of the hadron wavefunctionsfrom the assumed single Gaussian forms at short distances; both the meson q¯q states and thebaryon qq substates experience attractive short distance interactions from the color Coulomband hyperfine terms (for spin singlets), which will lead to enhancements of the short distancecomponents of their wavefunctions and increased high energy scattering amplitudes. If thisis the principal reason for the discrepancy, we would expect a global fit to the S-wave phaseshifts to prefer a smaller hadron length scale.In Fig.4 we show the result of a three-parameter fit to the full 1992 VPI I=0,1 energy independent S-wave data set [5], lettingαs/m2q, α and β vary and holding ρ = 0.6 fixed.The fit is shown as solid lines, and isevidently quite reasonable both near threshold and at higher energies.

The fitted parametersare αs/m2q = 0.59/(0.33)2 Gev−2, α = 0.68 Gev and β = 0.43 Gev; the hyperfine strengthis a typical quark model result but the width parameters α and β are about 1.5 timesthe usual quark model values. Thus, a fit to the S-wave VPI data using single Gaussianwavefunctions requires a hadron length scale about 0.7 times the usual scale.

This result islargely independent of the data set chosen, since it is driven by the large I=1 phase shift,which shows little variation between analyses. Evidently the predicted S-wave phase shiftsat higher energies are indeed very sensitive to the short distance parts of the wavefunction;this supports our conjecture that the discrepancy at higher energies is an artifact of oursingle Gaussian wavefunctions.

A calculation of these S-wave phase shifts using realisticCoulomb plus linear plus hyperfine wavefunctions is planned [38] and should provide a veryinteresting test of the quark Born formalism.c) Higher-L partial waves, spin-orbit and inelastic effectsIn addition to the S-wave phase shifts, higher-L KN elastic phase shifts and propertiesof the inelastic reactions KN→K∗N, KN→K∆and KN→K∗∆have been the subjects ofexperimental investigations. These studies have found important effects in the L>0 partialwaves which are beyond the scope of the present paper.One especially interesting effect is a remarkably large spin-orbit interaction in the I=0 KNsystem; the L=1 states have a large, positive phase shift for J=1/2 and a weaker, negativephase shift for J=3/2 (see [5] and references cited therein).This spin-orbit interactioncannot arise in our quark Born amplitudes given the approximations we have made in thispaper; since we have incorporated only the spin-spin hyperfine interaction in single hadronicchannels, our phase shifts (53) are functions of the total hadronic L and S but not J. Somebut not all of this spin-orbit interaction may simply require incorporation of the OGE spin-orbit term; Mukhopadhyay and Pirner [27] found that the quark spin-orbit interaction wassufficient to explain the sign and magnitude of some of the weaker KN spin-orbit forces,but that the I=0, J=1/2 phase shift was much too large to be explained as an OGE force.The strong KN spin-orbit forces might conceivably be due to couplings to inelastic channels;since the available mixing states and their couplings to KN are J-dependent, they mightlead to effective spin-orbit forces at the hadronic level, even if we do not include spin-orbitforces at the quark level.

We hope to treat this interesting possibility in a future study ofcoupled channel effects using the quark Born formalism.Since we do not have a model of the large spin-orbit effect it is not appropriate to include17

a detailed discussion of our predicted amplitudes and cross sections at higher energies, wherehigher partial waves are important. In the interest of completeness however we will brieflydiscuss our predicted differential cross section at high energy, since we previously noted thatwe found an exponential in t in I=2 ππ scattering, reminiscent of diffraction in magnitudebut not in phase [16].

The differential cross section in this unequal mass case is related tothe hfi matrix element (29) bydσdt = 4π5hs2 −(M2N −M2K)2 i2s2h(s −(MN + MK)2)(s −(MN −MK)2)i |hfi|2 . (60)For KN scattering in the high energy limit only the contribution from diagram D1 (20)survives, and we findlims→∞dσdt = 4πα2s9m4qw21 exp{A1t} .

(61)Thus we again find an exponential in t at high energy, with a slope parameter (41) that isnumerically equal tob = A1 = 3.7 Gev−2(62)given our reference parameter set. This is similar to the observed diffractive I=1 KN slopeparameter [44] ofb(expt.) ≈5.5 −5.9 Gev−2 .

(63)The normalizations of the theoretical and experimental I=1 high energy differential crosssections however differ by about an order of magnitude, and are 1.8 mb Gev−2 (referenceparameter set) versus ≈15 mb Gev−2 (experiment [44]). We noted a similar tendency for thereference parameter set to underestimate high energy amplitudes in our discussion of the S-wave phase shifts, which we attributed to the single Gaussian wavefunction approximation.One interesting prediction is that I=0 KN scattering should have no diffractive peak inthe high energy limit, since it has has w1 = 0; unfortunately there is no I=0 high energydata to compare this prediction with.

A serious comparison with high energy scattering willpresumably require the use of wavefunctions with more realistic high momentum componentsas well as the incorporation of inelastic channels, which may strongly affect the elasticamplitudes.IV. KN EQUIVALENT POTENTIALSSufficiently close to threshold our quark Born scattering amplitudes can be approxi-mated by local potentials.

These potentials are useful in applications such as multichannelscattering and investigations of possible bound states, which are easiest to model using aSchr¨odinger equation formalism with local potentials. There are many ways to define anequivalent low energy potential from a scattering amplitude such as hfi; several such pro-cedures are discussed in [18,45] and in Appendix E of [16].

Of course effective potentials18

extracted using different definitions can appear to be very different functions of r althoughthey lead to similar low energy scattering amplitudes.One approach to defining an equivalent potential is to derive a potential operator Vop(r)which give the scattering amplitude hfi in Born approximation. This “Born-equivalent po-tential” technique is discussed in reference [45] and in Appendix E of [16]; it has been testedon the OGE interaction, from which one recovers the correct Breit-Fermi Hamiltonian atO(v2/c2) [45].

To derive the Born-equivalent potential we reexpress our scattering amplitudein the CM frame as a function of the transferred three-momentum ⃗q = ⃗C −⃗A and an orthog-onal variable ⃗P = ( ⃗A + ⃗C)/2. We then expand the scattering amplitude in a power series in⃗P and equate the expansion to the Born expression for nonrelativistic potential scatteringthrough a general potential operator Vop(r), which may contain gradient operators.

Theleading term, of order P0, gives the Born-equivalent local potential V (r).In this meson-baryon scattering problem our Hamiltonian matrix elements are of theformhfi = 8παs3m2q1(2π)34Xi=1wiηi exp−(Ai −Biµ)P 2cm. (64)Making the required substitutions P 2cm = ⃗P2 + ⃗q 2/4 and P 2cmµ = ⃗P2 −⃗q 2/4 and Fouriertransforming with respect to ⃗q as in [16] gives the equivalent low energy KN potentialVKN(r) =8αs3√πm2q4Xi=1wiηi(Ai + Bi)3/2 exp−r2/(Ai + Bi).

(65)Thus our Born-equivalent meson-baryon potentials are sums of four Gaussians, one fromeach inequivalent quark Born diagram, weighted by the diagram weights of that channel.The potentials for I=0 and I=1 with our reference parameter set αs = 0.6, mq = 0.33Gev, ρ = mq/ms = 0.6, α = 0.40 Gev and β = 0.35 Gev are shown in Fig.5. They arerepulsive and have a range of about 0.3 fm, as one would expect for a short range “nuclear”core.

The potentials at contact are rather similar in this formalism, and the relative weaknessof I=0 scattering is a result of its shorter range. This is an effect of the backward peakingdiagram D4, which leads to a very short range potential with a large value at contact, andcarries higher weight in I=0 scattering.Although these Born-equivalent potentials are convenient for use in a meson-baryonSchr¨odinger equation, the actual KN potentials are so strong that they reproduce somefeatures of the interaction only qualitatively.For example, the Born diagrams give anI=1 scattering length of −0.35 fm, but the Born-equivalent potential (65) for I=1 in theSchr¨odinger equation for KN leads to a scattering length of only about −0.22 fm.

Thediscrepancy is due to higher order effects of VKN(r) in the Schr¨odinger equation; we haveconfirmed that the ratio of hfi and VKN(r) scattering lengths approaches unity in the small-αs limit.In a multichannel study one might modify VKN(r) (65) to give the input hfiscattering lengths, perhaps through a change in the overall normalization, as a way ofproviding a more realistic potential model of the quark Born amplitudes.19

V. RESULTS FOR K∆, K∗N AND K∗∆; PROSPECTS FOR Z∗-MOLECULESThe channels K∆, K∗N and K∗∆are interesting in part because they may supportmolecular bound states if the effective interaction is sufficiently attractive. In contrast thelow energy KN interaction is repulsive in both isospin states.

These “Z∗-molecules” wouldappear experimentally as resonances with masses somewhat below the thresholds of ≈1.7Gev, ≈1.85 Gev and ≈2.1 Gev. Even if there are no bound states, attractive interactionswill lead to threshold enhancements which might be misidentified as Z∗resonances just abovethreshold.Plausible binding energies of hadronic molecules can be estimated from the uncertaintyprinciple and the minimum separation allowed for distinct hadrons as EB ∼1/(Mhad ·1 fm2) ∼50 Mev.

In comparison, the best established molecules or molecule candidateshave binding energies ranging from 2.2 Mev (the deuteron, which has a repulsive core)through 10-30 Mev (the f0(975), a0(980) and Λ(1405)). (The f0(1710), with a bindingenergy relative to K∗¯K∗of about 75 Mev, appears plausible but is a more controversialcandidate [13].) Finally, the state f2(1520) seen by the Asterix [46], Crystal Barrel [47]and Obelix [48] collaborations in P¯P annihilation is an obvious candidate for a nonstrangevector-vector molecule, with a (poorly determined) binding energy relative to ρρ thresholdof perhaps 20 Mev.Several candidate Z∗resonances which might be meson-baryon molecule states have beenreported in KN partial wave analyses.

The 1986 Particle Data Group compilation [49] (themost recent to review the subject of Z∗resonances) cited two I=0 candidates, [Z0(1780), 12+]and [Z0(1865), 32−] and four I=1 possibilities, [Z1(1725), 12+]; [Z1(1900), 32+]; Z1(2150) andZ1(2500). However the evidence for these states is not strong, and the PDG argue that thestandards of proof must be strict in this exotic channel.

For this reason these states wereonly given a one star “Evidence weak; could disappear.” status. The 1986 PDG also notedthat “The general prejudice against baryons not made of three quarks and the lack of anyexperimental activity in this area make it likely that it will be another 15 years before theissue is decided.”.

The 1992 PDG compilation [50] makes a similar statement, with “15years” revised to “20 years”.In their recent analysis of the data Hyslop et al. [5] summarize some previous claims andreport evidence for “resonancelike structures” [Z0(1831), 12+]; [Z0(1788), 32−]; [Z1(1811), 32+]and [Z1(2074), 52−].

The negative parity candidates Z0(1788) and Z1(2074) have quantumnumbers and masses consistent with S-wave K∗N and K∗∆molecules respectively. We wouldnot normally expect P-wave molecules; odd-L is required to couple to positive parity KNchannels, and the centrifical barrier suppresses binding due to these short range forces.However, threshold effects which resemble resonances might arise in the full multichannelproblem, and the very strong spin-orbit force evident in the P01 and P03 KN partial wavesmay be sufficient to induce binding in some channels.

A clarification of the status of Z∗candidates through the determination of experimental amplitudes for the processes KN →K∗N, KN →K∆and KN →K∗∆in addition to the elastic KN reaction will be an importantgoal of future studies at kaon factories.All the S-wave (I,JP) quantum numbers, in which molecule bound states are a priorimost likely, are as follows;20

K∆(≈1.6 −1.7 Gev) :(2, 32−) ; (1, 32−) ;K∗N(≈1.75 −1.85 Gev) :(1, 32−) ; (1, 12−) ;(0, 32−) ; (0, 12−) ;K∗∆(≈2.0 −2.1 Gev) :(2, 52−) ; (2, 32−) ; (2, 12−) ;(1, 52−) ; (1, 32−) ; (1, 12−) .We can use our detailed model of meson-baryon scattering in the (q¯s)(qqq) system (q =u, d) to identify channels which experience attractive interactions as a result of the colorhyperfine term. These we again show as weight factors which multiply each of the fourdiagrams D1 .

. .

D4. Since the overlap integrals these weights multiply are all positive andof comparable magnitude, the summed weight can be used as an estimate of the sign andrelative strength of the interaction in each channel.

Positive weights correspond to a repulsiveinteraction. Our results for the hfi “diagram weights” for all K∆, K∗N and K∗∆channels in(I,Stot) notation are given below.

We also give the numerical values we find for the scatteringlength in each channel given our reference parameter set and masses MK∗= 0.895 Gev andM∆= 1.210 Gev.K∆(2, 32) = 16+ 3, −1, +3, −1;(66)a = −0.38 fm . (67)K∆(1, 32) = −13 K∆(2, 32) = 118−3, +1, −3, +1;(68)a = +0.13 fm .

(69)K∗N(1, 32) = 127+ 7, +1, −5, 0;(70)a = −0.08 fm . (71)K∗N(1, 12) = 154+ 26, +5, +2, −3;(72)a = −0.39 fm .

(73)K∗N(0, 32) = 19+ 1, +1, +1, 0;(74)21

a = −0.22 fm . (75)K∗N(0, 12) = 118−4, +5, −4, −3;(76)a = +0.15 fm .

(77)K∗∆(2, 52) = 13+ 1, −1, −1, +1;(78)a = +0.14 fm . (79)K∗∆(2, 32) = 118+ 11, −1, −1, −9;(80)a = −0.20 fm .

(81)K∗∆(2, 12) = 19+ 7, +1, +1, +3;(82)a = −0.86 fm . (83)K∗∆(1, Stot) = −13 K∗∆(2, Stot)∀Stot ;(84)K∗∆(1, 52) = 19−1, +1, +1, −1;(85)a = −0.05 fm .

(86)K∗∆(1, 32) = 154−11, +1, +1, +9;(87)a = +0.07 fm . (88)K∗∆(1, 12) = 127−7, −1, −1, −3;(89)a = +0.29 fm .

(90)22

Evidently attractive forces arise from the OGE spin-spin interaction in the minimum-spin,minimum-isospin channels,K∆:(1, 32) ;K∗N :(0, 12) ;K∗∆:(1, 12) .The two exceptions to this rule are the K∗∆channelsK∗∆:(2, 52)andK∗∆:(1, 32) ;although their weights sum to zero, variations in the detailed overlap integrals lead to at-tractive OGE-hyperfine forces in these two channels as well.For our reference parameter set we find no molecular bound states; the attractive forcesare too weak to induce binding. The experimental situation at present is rather confused;some references claim evidence for resonances in several channels (see for example [5,32,37]),whereas other references such as [30] and [31] conclude that the same phase shifts are non-resonant.

Our results do not support the most recent claims of resonances [5], since theS-wave quantum numbers of our attractive channels do not correspond to those of the neg-ative parity candidates [Z0(1788), 32−] and [Z1(2074), 52−]. However our negative result maybe an artifact of our approximations, including the neglect of spin-orbit effects and couplingsbetween channels.

The spin-orbit effects are known from experiment to be very important,and might be sufficient to lead to Z∗-molecule bound states or strong threshold enhance-ments in the attractive channels. Our negative result is based on strong assumptions on theform of the interaction; this should be relaxed in future theoretical work, and should not beused to argue against experimental searches for possible Z∗meson-baryon molecules.VI.

SUMMARY AND CONCLUSIONSIn this paper we have applied the quark Born diagram formalism to KN scattering. In thisapproach one calculates hadron-hadron scattering amplitudes in the nonrelativistic quarkpotential model assuming that the amplitude is the coherent sum of all OGE interactionsfollowed by all allowed quark line exchanges; this is expected to be a useful description ofreactions which are free of q¯q annihilation.

The model has few parameters, here αs/m2q,ρ = mq/ms and the hadron wavefunction parameters, and with Gaussian wavefunctions thescattering amplitudes can be derived analytically. The model was previously applied to I=2ππ and I=3/2 Kπ scattering with good results.23

KN scattering is an important test of this approach because it is also annihilation-free(at the valence quark level) and the meson and baryon wavefunction parameters and theinteraction strength are already reasonably well established. Thus there is little freedomto adjust parameters.

We find good agreement with the experimental low energy I=0 andI=1 phase shifts given standard quark model parameters. (The experimental I=0 scatteringlength is usually claimed to be very small; we disagree with this interpretation of the dataand argue in support of a larger value.) A resolution of the disagreements between differentI=0 KN phase shift analyses, especially at very low energies, is an important task for futureexperimental work.Hyslop [37] also suggests additional experimental work on the I=0KN system.

At higher energies we find that the single Gaussian S-wave phase shifts fallwith energy more quickly than experiment given standard quark model parameters; weattribute most of this effect to departures of the hadron wavefunction from single Gaussiansat short distances, perhaps in response to the attractive color hyperfine interaction. We haveconfirmed that a smaller hadronic length scale (about 0.7 times the usual nonrelativisticquark potential model scale) gives S-wave phase shifts which are in good agreement withexperiment at all energies.We have investigated the possibility of Z∗-molecule meson-baryon bound states by ex-tending our calculations to all channels allowed for K∆, K∗N and K∗∆.

Although we dofind attractive interactions in certain channels, in no case is the corresponding interhadronpotential sufficiently strong to form a bound state. Of course this result may be an artifact ofour approximations, in particular the assumption of keeping only the spin-spin color hyper-fine term and the single channel approximation.

The effect of relaxing these approximationswould be a very interesting topic for future study.There are additional effects in the L>0 KN system which are known to be important ex-perimentally, which are not incorporated in our calculations of single channel color hyperfinematrix elements. The most important of these is a very large spin-orbit force, which it hasnot been possible to explain as an OGE interaction [22,27].

Both this spin-orbit interactionand the Z∗candidates may be strongly affected by coupled channel effects, which we plan toinvestigate in future work. Since much is already known experimentally about the reactionsKN→K∗N and KN→K∆, it should be possible to test predictions of the quark Born dia-grams for these channel couplings using existing data sets.

Although one might expect OPEforces to be important in coupling KN to inelastic channels, such as in I=1 KN→K∗N, theOPE contribution to this process has been found experimentally to be small near threshold[4]. Thus experiment suggests that interquark forces such as OGE and the confining inter-action may be more important than meson exchange in coupling KN to inelastic channels.We plan to evaluate these offdiagonal couplings in detail in a future study.ACKNOWLEDGMENTSWe acknowledge useful contributions from R.Arndt, W.Bugg, S.Capstick, F.E.Close,G.Condo,H.Feshbach,N.Isgur,R.Koniuk,M.D.Kovarik,K.Maltman,B.R.Martin,D.Morgan,G.C.Oades,R.J.N.Phillips,B.Ratcliff,R.G.Roberts,L.D.Roper,D.Ross,S.Sorensen, J.Weinstein and R.Workman.

This work was sponsored in part by the UnitedStates Department of Energy under contracts DE-AC02-76ER03069 with the Center forTheoretical Physics at the Massachusetts Institute of Technology and DE-AC05-840R2140024

with Martin Marietta Energy Systems Inc.25

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VII. APPENDIXThe procedure we advocate for describing hadronic interactions involves simply calcu-lating the Born order scattering amplitude for a given process using the constituent quarkmodel.

In cases where many channels contribute or one wishes to obtain nonperturbativeinformation (such as the possible existence of hadronic bound states) then one must extractan effective potential (or effective potential matrix in the case of multichannel problems)from the Born amplitude and integrate the appropriate Schr¨odinger equation. The validityof this procedure and its relationship to the resonating group method are the subjects ofthis appendix.Several theoretical complications arise when considering scattering of composite particles.For example, the Hamiltonian may be partitioned in many different ways correspondingto different rearrangement channels.

Thus if i represents the initial channel consisting ofhadrons a and b, and f represents the final channel with hadrons c and d, then we may writeH = Hi + Vi = Hf + Vf,(91)whereHi = −12µab∇2R + Ha + Hb(92)andHf = −12µcd∇2R + Hc + Hd. (93)Here ⃗R is the appropriate interhadron coordinate and µab is the reduced mass of the con-stituent masses of hadrons a and b.

The Hamiltonians Ha and Hb describe the hadronicwavefunctions in the initial channel. Thus we haveHaφa,n(ξa) = ǫa,nφa,n(ξa)(94)and similarly for the other hadrons.In general the wavefunction must be antisymmetrized appropriately and this meanssumming over the various rearrangement channels with the correct weights.

Thus we takethe incoming wavefunction to be φa(ξa)φb(ξb)ψ0(R) where ψ0 is a plane wave and define theantisymmetrization operator asA =1√ηXP(−)PP(95)where P is a quark exchange operator. The Born series for the process ab →cd may thenbe written as⟨ˆf|t|ˆi⟩= 1ηXP P ′(−)P(−)P ′⟨Pφcφdψ0|VP ′ + VPGcVP ′ + VPGcVcGcVP ′ + .

. .|P ′φaφbψ0⟩.

(96)Here Gc is the Green function in a general channel c.The Born order expression is28

⟨ˆf|T|ˆi⟩=XP(−)P⟨φcφdψ0|Vi|Pφaφbψ0⟩. (97)This is the “prior” form of the T-matrix.

The “post” form uses the potential in the finalchannel rather than the initial channel.The expressions are equivalent if the hadronicwavefunctions are exact and the T-matrix is evaluated on energy shell. Note that theseconditions must also hold if the effective potential matrix is to be Hermitian.Little is known about the convergence properties of the Born series.

With no exchange theconditions for convergence are probably similar to those for simple potential scattering [51].Of course exchange scattering is necessarily present when describing hadronic interactionsfrom the quark level. At high energy there is evidence that the lowest few Born terms can beaccurate [51].

However, since the potential is strong enough to cause binding in the initialand final states, the series will diverge at low energies. Nevertheless the strategy of extractingan effective potential can be useful when the Born approximation is not accurate or evenwhen it diverges.

We shall return to this point below. Despite the theoretical problems, thesmall nuclear binding energy, the small phase shifts seen in K+N scattering, and the lack ofquark model state mixing evidenced in most of the meson spectrum all suggest the utilityof the Born approximation.It should be stressed that the Born approximation can be useful even when the effectivepotential is very strong.This will be true if the Born term carries information on thedominant physics.

Then the Born order scattering amplitude may be Fourier transformed toyield an effective potential which contains all of the dominant physics and may be integratedexactly. Since hadronic interactions must involve constituents, this may not be carried outin general, however it will be accurate if the new physics induced at higher order in the Bornseries (such as polarization effects) does not dominate at low energy.

As will be discussedbelow, this appears to be true in many cases.We now turn to the relationship of this approach to the resonating group method. In thefollowing we shall restrict our attention to the single channel case.

The resonating groupAnsatz is thenΨ = φa(ξa)φb(ξb)ψ(R)(98)where ψ is an unknown function of the interhadron distance. This Ansatz must be antisym-metrized.

For later convenience, we choose to separate the identity permutation. Thus thewavefunction isˆΨ = AΨ =1√η1 +XP ̸=I(−)PPφaφbψ.

(99)Varying the Schr¨odinger equation with respect to ψ and rewriting the resulting expressionfor ψ in Lippmann-Schwinger form yields the following equation.ψ(R) = ψ0(R) + 2µabZG0(R, R′)VD(R′)ψ(R′)dR′−XP ̸=I(−)PZG0(R, R′)φ∗a(ξa)φ∗b(ξb)h∇2R + k2reliP[φa(ξa)φb(ξb)ψ(R′)]dξadξbdR′29

+ 2µabXP ̸=I(−)PZG0(R, R′)φ∗a(ξa)φ∗b(ξb) Vi(ξa, ξb, R′)P[φa(ξa)φb(ξb)ψR′)]dξadξbdR′ (100)wherek2rel = 2µab(E −ǫa −ǫb)(101)andVD(R) =Zφ∗a(ξa)φ∗b(ξb)Vi(ξa, ξb, R)φa(ξa)φb(ξb)dξadξb,(102)and G0 is the Green function for ∇2R + k2rel.The permutation operator in the third term implies that ∇2R+k2rel is a Hermitian operatorand may be safely applied to the left. Thus the third term (the kinetic and energy exchangekernels) simplifies to−XP ̸=I(−)PZφaφbP[φaφbψ(R)]dξadξb ≡ZN(R, ˜R)ψ(˜R)d˜R(103)where N(R, ˜R) is the normalization kernel.

Because of nontrivial permutation operatorsthis expression is damped as R →∞and hence it does not contribute to scattering.We may now iterate Eq. (??) to see that it corresponds to the full Born series (96) withthe sum over intermediate states restricted to the appropriate single channel.

In particularthe R →∞limit of the first term in the series corresponds to the Born order T-matrix ofEq. (97) (for the case of elastic scattering).Eq.

(??) indicates that setting Vi = 0 in a resonating group calculation should yielda null phase shift.

However if one uses approximate hadronic wavefunctions (as is almostalways the case in resonating group calculations) then a residual spurious phase shift willremain. We note that Bender et al.

[24] employ single Gaussian hadronic wavefunctions sothat one expects small phase shifts upon setting Vi = 0. This is indeed what they foundfor I=0 K+N scattering.

However they obtained rather large phase shifts for the I=1 case.Since the Hamiltonian is independent of the isospin the Gaussian wavefunctions should havebeen equally effective in both cases and one must conclude that there is likely to be anerror in the I=1 calculation. Maltman [52] has concluded that there are indeed errors in thehyperfine matrix elements in this reference.Solving the single channel resonating group equation is similar to the process of extractingan effective potential from the Born scattering amplitude and integrating it exactly.

Bothmethods treat the single channel subspace nonperturbatively and hence are successful whenthe single channel approximation is a good one. Both methods fail if (off-channel) virtualparticles, polarization, wavefunction distortion, and similar effects dominate the low energybehavior of the system.

As stated above, this does not seem to happen in practice. Thesimilarity of hadron scattering amplitudes obtained from the single channel resonating groupmethod and from integrating effective potentials has been demonstrated for several cases inRef.

[18].30

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