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I=3/2 Kπ Scattering in the

arXiv:hep-ph/9207251v1 21 Jul 1992MIT-CTP-2108ORNL-CCIP-92-06UTK-92-4UMS/HEP 92-021I=3/2 Kπ Scattering in theNonrelativistic Quark Potential ModelT. 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 02139J. WeinsteinDepartment of Physics and AstronomyUniversity of MississippiUniversity, MS 38677We study I = 3/2 elastic Kπ scattering to Born order using nonrelativistic quarkwavefunctions in a constituent-exchange model.

This channel is ideal for the studyof nonresonant meson-meson scattering amplitudes since s-channel resonances donot contribute significantly.Standard quark model parameters yield good agree-ment with the measured S- and P-wave phase shifts and with PCAC calculations ofthe scattering length. The P-wave phase shift is especially interesting because it isnonzero solely due to SU(3)f symmetry breaking effects, and is found to be in goodagreement with experiment given conventional values for the strange and nonstrangeconstituent quark masses.Submitted to Phys.

Rev. DJune 1992

I. INTRODUCTIONThe derivation of hadronic interactions from QCD has been a goal of nuclear physics formany years. At present this appears to be a very difficult problem; even the more modestgoal of deriving hadronic interactions from the nonrelativistic quark model is difficult, duein part to the variety of mechanisms which contribute to scattering.

In a typical hadronicscattering process these mechanisms include s-channel resonance production, t-channel res-onance exchange, and nonresonant scattering. Despite the apparent complexity, there isconsiderable evidence that some scattering amplitudes are dominated by relatively simpleperturbative QCD processes.

One well known example is the short-range part of the NNinteraction; many groups have concluded that the NN repulsive core is due to the combinedeffects of the Pauli principle and the color magnetic spin-spin component of one gluon ex-change. Similarly, the intermediate-range attractive interaction may be due to a relativelysimple effect at the quark level, specifically a color-dipole interaction induced by the spatialdistortion of the three-quark clusters [1].

Of course one pion exchange dominates at suffi-ciently large separations, and in a more complete description one should adjoin this to theshort-range quark interaction.In this paper we discuss I = 3/2 Kπ elastic scattering in the nonrelativistic quark model.This process resembles NN scattering in that s-channel resonances are not expected to giveimportant contributions, assuming that multiquark resonances are not in evidence. TheBorn-order QCD scattering amplitude for this process involves one gluon exchange followedby quark exchange.

In a previous paper [2] it was shown that this simple description ofhadron-hadron scattering leads to good agreement with the nonperturbative variationalresults of Weinstein and Isgur [3] near threshold, and with the measured I = 2 ππ S-wavephase shift throughout the full range of Mππ for which data exists. A similar method was usedin Ref.

[4] to extract effective potentials for many low-lying meson-meson channels. Thesepotentials have recently been applied to several problems in low-energy meson physics.

Inparticular, Dooley et al. [5] used the results of Ref.

[4] to suggest that the IJP C = 00++θ(1720) may be a (K∗¯K∗)-(ωφ) vector-vector molecular bound state. Simple estimates ofbranching ratios in this model find good agreement with Particle Data Group values andpredict new decay modes.

In another application it has been argued that the f1(1420) “E”effect may be a threshold enhancement which is due to an attractive (K∗¯K)-(ωφ) interactionin the 01++ channel [4].The I = 3/2 Kπ channel is an ideal one for testing our model of Born scattering ampli-tudes, since we expect it to be largely unaffected by s-channel resonances. The success ofthe previous application of the model to I = 2 ππ scattering and the assumption of flavorsymmetry suggest that we may also find reasonable agreement with the experimental S-waveKπ phase shift.

The P-wave, however, is driven entirely by flavor symmetry breaking, andhence the interplay of these two waves provides an interesting and nontrivial test of themodel.II. METHODThe calculation is based on a standard quark model Hamiltonian of the form2

H =4Xi=1 p2i2mi+ mi!+Xi

We shall also ignore the contribution of Vconfto the scattering interaction. This may appear to be a questionable approximation; however,resonating group calculations have found that the exchange (scattering) kernel due to Vconfis much smaller than the corresponding kernel for the hyperfine term [7].

This result wasalso found in the variational calculation of Ref. [3] and the perturbative calculation of Ref.[4].

The latter reference noted that the small Vconf contribution to scattering is due to acolor-factor cancellation in the matrix element of Vconf. However, this result only appliesto certain channels; one should not neglect the effects of the confinement term in scatteringinvolving vector mesons.Although we calculate the scattering amplitude only to Born order, there is evidence thatthis is a useful and even accurate approximation in systems which are not dominated bys-channel resonances or t-channel meson exchange.

First, as there is little evidence for flavormixing in meson spectroscopy outside the η −η′ system, one anticipates that neglectinghigher terms in the Born series (such as q¯q →gg →q¯q) is not a bad approximation. Inaddition, the Born-approximation I = 2 ππ effective potentials derived in Refs.

[2] and [4] arenumerically very similar to the nonperturbative potentials derived by Weinstein and Isgur.Finally, comparison of perturbative phase shifts to those found in a variational resonatinggroup calculation shows good numerical agreement [4].For simplicity we use single Gaussians for the asymptotic pion and kaon wavefunctions,ψπ(K)(r) = β2π(K)π!3/4e−β2π(K) r2/2 ,(3)where r = |⃗rq −⃗r¯q|. The corresponding momentum-space wavefunction φ(krel) is a functionof the magnitude of the relative momentum vector ⃗krel = (m¯q⃗kq −mq⃗k¯q)/(mq + m¯q).Flavor symmetry breaking is incorporated through unequal strange and nonstrange quarkmasses (we introduce a mass ratio ρ = mu/ms) and a meson width parameter ξ (defined byξ = β2π/β2K; ξ < 1 corresponds to a smaller kaon than pion).

Of course these parametersare related. For instance if we take Vconf(rij) = C + κr2ij/2 then ξ =q(1 + ρ)/2.

Standardquark model values for the constituent masses, mu = 0.33 GeV and ms = 0.55 GeV, giveρ = 0.6, and from the SHO relation above we might anticipate ξ ≈0.9. A fit to lightmeson spectroscopy in a Coulomb plus linear potential model with a contact hyperfine termfinds a similar ρ value of ρ = 0.58.

With this ρ a single-Gaussian variational calculation [4]actually finds a value for ξ slightly above unity, ξ = 1.05, because the stronger pion hyperfineattraction leads to a smaller pion than kaon despite the heavier strange quark mass. In anycase we expect ξ to be near unity.3

There are four Born-order quark exchange graphs for Kπ scattering, which we previouslyclassified as two “transfer” or “capture” processes in our discussion of ππ scattering [2].The transfer diagrams represent scattering due to a spin-spin hyperfine interaction betweena quark pair (T1) or an antiquark pair (T2). In the capture diagrams the interaction isbetween a quark-antiquark pair in different mesons, u¯s for C1 and u ¯d for C2.

We apply themethods of Ref. [2] (Appendix C) to obtain the Born-order Hamiltonian matrix element hfifor these diagrams, which ishfi =1(2π)34παs9m2uT1 + T2 + C1 + C2,(4)where the term contributed by each diagram isT1 = exp−(1 + ξ(1 + ζ)2)1 −µ2 k24β2π(5a)T2 = ρ 2√ξ1 + ξ!3exp−ξ1 + ξ1 + (1 −ζ)2 + 2(1 −ζ)µ k24β2π(5b)C1 = ρ 42 + ξ!3/2exp−12 + ξ1 + 3ξ −ξζ(1 −ζ) + (ξ −1 −3ξζ)µ k24β2π(5c)C2 = 4ξ1 + 2ξ!3/2exp−ξ1 + 2ξ3 −ζ + ζ2 + ξ(1 + ζ)2+(1 −3ζ −ξ(1 + ζ)2)µ k24β2π.

(5d)In these matrix elements µ = cos(θCM), where θCM is the center of mass scattering angle, thequark mass parameter ζ is (ms −mu)/(ms + mu) = (1 −ρ)/(1 + ρ), and k is the magnitudeof the asymptotic three-momentum of each meson in the CM frame. The matrix elementhfi is related to the ℓth partial-wave phase shift by [2]δ(ℓ) = −2π2kEπEK(Eπ + EK)Z 1−1 hfi(µ) Pℓ(µ) dµ ,(6)with the meson energies related to k by relativistic kinematics.

This result involves δ(ℓ)rather than sin δ(ℓ) because we choose to equate our Born amplitude to the leading term inthe elastic scattering amplitude (exp{2iδ(ℓ)} −1)/2i rather than to the full real part. Thephase shifts for all partial waves follow from this result through application of the integralR 1−1 eaµPℓ(µ)dµ = 2iℓ(a) where iℓis the modified spherical Bessel function of the first kind.III.

RESULTS AND DISCUSSIONOn evaluating the angular integrals (6) we find I = 3/2 Kπ phase shifts for all ℓin Bornapproximation given SHO wavefunctions; these are functions of the four free parameters βπ,αs/m2u, ρ = mu/ms, and ξ = β2π/β2K, and require the physical meson masses as input. In thefollowing discussion we shall fix the nonstrange quark mass to be mu = 0.33 GeV since thephase shifts actually involve the ratios given above rather than the absolute scale of mu.4

We proceed by fitting the predicted phase shifts to the S- and P-wave phase shift data ofEstabrooks et al. [8].

Note however that there may be a discrepancy between this data andearlier I = 3/2 Kπ results [9,10] near threshold; two S-wave data sets are shown in Fig. 1.As an initial “benchmark” prediction we first neglect flavor-symmetry violation (exceptfor the use of physical meson masses in kinematics and phase space) and employ the sameparameters we previously used to describe I = 2 ππ scattering in Ref.

[2]; αs = 0.6, mu =0.33 GeV and βπ(fitted to ππ) = 0.337 GeV, and we set ms = mu and βK = βπ so that ρ = 1and ξ = 1. The resulting S-wave phase shift is shown as a dotted line in Fig.

1. Although theshape of the predicted phase shift is qualitatively correct, evidently the predicted magnitudeis somewhat larger than the data at invariant masses above 0.9 GeV.Of course this flavor-symmetric parameter set is unrealistic because it does not assume aheavier strange quark; allowing ms to vary yields the value ρ = 0.677 in a fit to the S-wavedata of Estabrooks et al.

; this is close to the ρ ≈0.33 GeV/0.55 GeV = 0.6 expected fromq¯q quark model spectroscopy. The resulting phase shift is shown as a dashed line in Fig.

1,and the agreement is impressive. The same parameter set gives a P-wave phase shift whichis shown as a dashed line in Fig.

2. Evidently the agreement with experiment is again quitegood.

Note that the predicted P-wave phase shift is zero for ρ = 1, so the S-wave data areconsistent with approximate flavor symmetry (which implies ππ S-wave ≈Kπ S-wave) andthe P-wave data are consistent with the expected amount of flavor symmetry breaking (seenin the nonzero Kπ P-wave).Although we have found a satisfactory description of the data simply by using ππ param-eters and physical meson masses and fitting ms, it is of interest to investigate the sensitivityof our results to changes in the other parameters and to determine their global optimumvalues. Fixing ξ = 1 and βπ = 0.337 GeV and fitting αs and ρ to the Estabrooks et al.S-wave data gives αs = 0.634 and ρ = 0.604, again consistent with standard quark modelvalues.

A global fit to the 33 S- and P-wave data points of Estabrooks et al. with all fourparameters free gives ρ = 0.789, αs = 0.577, βπ = 0.293 GeV and ξ = 0.568.

The ratherlarge mu/ms in this fit is partially compensated by a spatially small kaon wavefunction, butas the phase shifts are rather insensitive to ξ, and we expect a value closer to unity, this bestfit probably gives less realistic parameter values than the single-parameter fit which foundρ = 0.677. The phase shifts predicted by the the global four-parameter set are shown assolid lines in Figs.

1 and 2. Note that the four-parameter S-wave phase shift is essentiallyindistinguishable from the one-parameter (ms) fit (dashed line); the most important differ-ence in the predictions of the two parameter sets is in the P-wave, which is not yet very welldetermined experimentally.Estabrooks et al.

also reported measurements of the I = 3/2 Kπ D-wave phase shift. Wepredict a small negative D-wave phase shift in accord with the data, although the magnitudeof our D-wave is somewhat smaller than is observed.

A similar discrepancy in the I = 2 ππD-wave was previously noted [2,4]. It should be stressed that the D-wave is qualitativelydifferent from the P-wave; it is not driven by flavor symmetry breaking and is an intrinsicallysmall effect at these energies, so that other contributions which we have neglected may beimportant here.

Possible contributions to this higher partial wave include the confinement,spin-orbit and tensor interactions. The departure of the actual q¯q wavefunction from theassumed single Gaussian may also be important, although this effect was investigated inRef.

[4] for I = 2 ππ scattering and was found to be small.5

Weinberg [11] used PCAC to predict an I = 3/2 Kπ scattering length ofa(3)S = −mKmπmK + mπ18πf 2(7)in his original PCAC paper.Here, f is the pseudoscalar decay constant which may beidentified with fπ in the flavor symmetric limit. The quark Born approximation for thescattering length may be extracted from our expression for the S-wave phase shift, and isa(3)S = −mKmπmK + mπ2αs9m2u1 + 4ξ1 + 2ξ!3/2+ ρ 42 + ξ!3/2+ ρ 2√ξ1 + ξ!3.

(8)With our various parameter sets we find the following values for the scattering length a(3)S :−0.092/mπ (ρ = 1, ξ = 1); −0.077/mπ (ρ = 0.677, αs = 0.6); −0.078/mπ (ρ = 0.604, αs =0.634); −0.076/mπ (global fit). These are compared to the PCAC prediction, one-loop chiralperturbation theory and various model calculations in Table I.

Experimental values for thescattering length range from −0.07/mπ to −0.14/mπ, and are also summarized in Table I.Note that we may also interpret our scattering length as a quark Born formula for fπ if weassume the PCAC relation (7). With the flavor-symmetric parameter set we find fπ = 80MeV, in reasonable agreement with the experimental value of 93 MeV.Our theoretical values for the scattering length are consistent with most experimentalresults, but not with the most recent, which is that of Estabrooks et al.

Lang and Porod [10]note that the Estabrooks et al. S-wave phase shift agrees with previous data for mKπ >∼1GeV, but is somewhat larger in magnitude than previous measurements for mKπ <∼1 GeV.Presumably this leads to their rather large scattering length.

It would clearly be useful toresolve this experimental discrepancy, since only in this mass region is there any indicationof a possible disagreement between the S-wave phase shift and our predictions. It would alsobe very useful to improve the accuracy of the P-wave measurement, which is a sensitive testof flavor symmetry breaking, and to extend the S-wave phase shift measurements to higherinvariant masses as a test of the extremum predicted and perhaps observed near 1.4 GeV.IV.

CONCLUSIONSWe have calculated I = 3/2 Kπ elastic scattering phase shifts using a Born-orderconstituent-exchange description in the framework of the nonrelativistic quark potentialmodel.Extensive previous work leads us to believe that one gluon exchange combined withquark exchange may accurately describe nonresonant hadron scattering in certain channelsincluding I = 3/2 Kπ, and that the Born approximation to the scattering amplitude is anacceptable one. This reaction is appropriate for testing this model of scattering becauset-channel pion exchange is forbidden by G-parity and the experimental phase shift shows noevidence for s-channel resonance formation.Since this model was previously found to describe the related I = 2 ππ S-wave phaseshift accurately [2], approximate flavor symmetry leads us to expect that the predictedI = 3/2 Kπ S-wave phase shift should at least be in qualitative agreement with the data.6

This is indeed found to be the case. The agreement is considerably improved when flavorsymmetry is broken by assigning the strange quark a mass consistent with standard quarkmodel values.

The P-wave Kπ phase shift however is generated entirely by flavor symmetrybreaking effects (primarily by the strange-nonstrange quark mass difference in our model),and is not present in I = 2 ππ scattering. The very reasonable result we find for the P-wavephase shift using fitted S-wave parameters is therefore a nontrivial and successful test of themodel.

Finally, the model predicts an S-wave scattering length of about −0.077/mπ, whichis in the range of reported experimental values and is commensurate with the predictions ofchiral perturbation theory.Although we find a small negative D-wave phase shift as has been reported experimen-tally, the magnitude and energy dependence are not well reproduced. This, however, is asmall contribution to the scattering amplitude, and the various other scattering mechanismswhich have been neglected in this calculation may be significant in this case, and should beinvestigated in future.Weinstein and Isgur have also studied S-wave I = 3/2 Kπ scattering in the nonrelativisticquark model, using a nonperturbative variational technique [21].

(Their method does notallow extraction of higher partial waves at present.) They find good agreement with theS-wave data, although they must first scale the range and strength of their effective Kπpotentials.

We perform no such scaling but do employ relativistic phase space; the factthat both methods agree well with experiment suggests that their scaling may actually becompensating for kinematic effects above threshold. This conclusion is supported by a recentreanalysis of the Weinstein-Isgur variational calculations [22].The obvious extension of this work is to meson-baryon and baryon-baryon scattering,with the caveat that one should specialize to channels such as K+-nucleon and baryon-baryonin which q¯q pair creation and annihilation is unimportant.

These topics are currently underinvestigation.ACKNOWLEDGMENTSThis research was sponsored by the Natural Sciences and Engineering Research Council ofCanada; the United States Department of Energy under contracts DE-AC05-840R21400 withMartin Marietta Energy Systems Inc., DE-FG05-91ER40627 with the Physics Departmentof the University of Tennessee, DE-AC02-76ER03069 with the Center for Theoretical Physicsat the Massachusetts Institute of Technology; and by the State of Tennessee Science AllianceCenter under contract R01-1062-32.7

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FIGURESFIG. 1.

S-wave Kπ Phase Shift. The filled squares are data from [8]; the open squares are from[9] (second solution).

The dotted line corresponds to ρ = 1, the dashed line to ρ = 0.677, and thesolid line corresponds to the global fit (see text).FIG. 2.

P-wave Kπ Phase Shift.The data are from [8].The dashed line corresponds toρ = 0.677 and the solid line to the global fit (see text).9

TABLESTABLE I. Experimental and Theoretical Values for the S-wave Scattering Length.a(3)S · mπRef.comments−0.071(10)[9]experimental−0.076(10)[12]experimental−0.084(11)[13]experimental−0.086(24)[14]experimental−0.091(9)[15]experimental−0.110(16)[16]experimental−0.138(7)[8]experimental−0.05[17]1-loop chiral pert.

theory−0.05[18]pointlike meson model−0.06[19]crossing-symmetric Regge model−0.07[11]PCAC−0.074[20]coupled channel model−0.077—this work (central value)10

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