Kaon-induced deuteron disintegration with two-body final states

Nuclear Physics A423 (1984) 445-476 @ North-Holland Publishing Company

KAON-INDUCED DEUTERON DISINTEGRATION WITH TWO-BODY FINAL STATES O.V. MAXWELL TRIVMF, 4004 Wesbrook Mall, Vancouver, BC, Canada V6TZA3 Received 14 October 1983 (Revised 7 February 1984) Ahstraetr Integrated cross sections for the reactions K-d -) I-p and K-d + An have been obtained within a simple model that involves excitation of an S =-1, I =0 or 1 resonance (Y*) followed by a pseudoscalar meson exchange that deexcites the resonance and transfers momentum to the other nucleon. The model incorporates the eight well-established Y* below 1700 MeV with coupling strengths estimated from partial decay widths and SU(3) symmetry relations. Initial-state correlations are included through the deuteron wave function, obtained with the Reid NN interaction, while final-state correlations are ignored. When examined as functions of the laboratory kaon momentum, the cross sections computed with form factors omitted exhibit two prominent features: a narrow structure centred at 385 MeV/c, due almost entirely to A(1520) excitation, and a broader peak around 700 MeV/c, that is associated with several overlapping resonances above 1650 MeV. In principle, these two structures could provide coupling-strength information within a more quantitative model. Both peaks are quite sensitive to the inclusion of form factors, however, and may be difficult to detect if the phenomenologically required form factors are as long-ranged as the sparse data presently available indicate.

1. Introduction and overview With the acquisition of new data on (K-, rr-) strangeness exchange reactions at Brookhaven and CERN 14) and the possibility that high-quality kaon beams from “kaon factories” will become available in the foreseeable future ‘), theoretical interest in K-nucleon and K-nuclear physics has intensified in recent years [for a comprehensive review, see ref. 6)]. Among the many fields of interest in this area, the study of K- reactions with very light nuclei may prove especially illuminating as such light nuclei are simple enough to make possible detailed microscopic analyses of reaction mechanisms. On the one hand, this affords an opportunity to test particular dynamical models, such as the resonance excitation models employed extensively in r-nuclear studies, in a new regime. On the other hand, within the framework of a specific model, studies of K- reactions on light nuclei could provide new information concerning the Y* (A* or E*) resonance spectrum observed in K- nucleon reactions. Such information is difficult to extract from heavier systems due to the tendency of Fermi motion and nucleon recoil effects to smear out structures in the cross sections and angular distributions associated with particular resonances 6,7). 445

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As the simplest example of a K- nuclear system, the K- deuteron system is an especially attractive one from a theoretical point of view. Recently, most of the effort devoted to this system, both theoretical and experimental, has been oriented toward studies of the strangeness exchange reaction K-d + ApF

,

(1.1)

particularly in connection with the possible existence of an I = $ dibaryon resonance near the EN threshold ‘-lo). The two-nucleon absorption processes K-d + Z-p, K-d+ An

(1.2)

have received comparatively little attention. Early experimental 11712)and theoretical 13)work on these reactions, dealing mainly with K- absorption from K- deuteron atomic systems, indicates that they account for only 1% or so of the total absorption cross section. Reactions (1.2) are interesting, nevertheless, by virtue of their kinematic resemblance to the well-studied pion absorption process v+d-+pp.

(1.3)

In all three reactions, (1.2) and (1.3), momentum and energy conservation dictate a large momentum transfer between the two participating baryons, effectively suppressing the impulse contribution to the amplitude in favor of resonance rescattering processes. This similarity among reactions (1.2) and (1.3) suggests that simple dynamical models that have achieved some success in the description of pion absorption by the deuteron could be adapted to a study of the analogous kaon reactions. One model particularly well-suited for this purpose is the isobar rescattering model introduced originally by Brack, Riska and Weise (BRW) i4) and subsequently refined and extended in refs. 15,16).BRW proposed a reaction mechanism consisting of three contributions: (i) single-nucleon absorption (the impulse approximation); (ii) excitation of a p-wave VN A33 resonance followed by isovector meson exchange (7r +p), by means of which the requisite momentum transfer between the two nucleons is effected; and (iii) a phenomenological s-wave rescattering term, introduced to correctly reproduce the threshold behavior of the absorption amplitude. With the insertion of phenomenological form factors at the internal meson-baryon vertices and the inclusion of initial- and final-state correlations via the Schriidinger equation, this model provides reasonably good fits to both the integrated absorption cross section and spin-averaged angular distributions over a wide energy range 15*16). To handle K- absorption, one need only replace the isobar contribution to the amplitude in the BRW model by the analogous processes involving Y* excitations. In contrast to pion absorption, where just a single resonance is important at moderate energies above threshold, a whole spectrum of KN, &r, and AT resonances can

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contribute to the K-d amplitudes. Two of these, in fact, the p-wave AT resonance, X*(1385), and the 2% resonance, A*(1405), lie below the K-N threshold and can be expected to dominate the K- absorption amplitudes in the threshold and nearthreshold region. A further complication relative to the rd reaction lies in the greater variety of meson exchanges through which the intermediate Y*N state can deexcite: strange I = 4 and non-strange isoscalar exchanges, as well as non-strange isovector exchanges, are possible, depending upon the particular resonance excited and the particular YN final state considered. Clearly, the resonance rescattering contribution to the amplitude is considerably more complicated for K- absorption than for pion absorption. On the other hand, the occurrence of s-wave and p-wave resonances below the K-N threshold leads to some simplifications for the K-d reactions, as compared with the rd reaction. In particular, the possibility of A*(1409 excitation, as the dominant reaction mechanism at threshold, makes the introduction of a supplementary phenomenological s-wave term in the amplitude, as required in the Ird case, unnecessary here. The X*(1385), which dominates the K-d amplitude somewhat above threshold {centrifugal factors suppress its contribution at threshold), acts to reduce the significance of the single-nucleon absorption term. For Xp final states, this term is further suppressed by the smrllt magnitude of the XNK coupling strength. Thus, it is not unreasonable to expect the resonance rescattering terms alone to provide a fair description of the K-d absorption amplitudes, at least at the qualitative or semi-quantitative level. On the basis of these considerations, a resonance-excitation model analogous to the isobar rescattering terms in the BRW model has been developed to represent K-d absorption and employed to obtain the integrated cross sections for reactions (1.2) as functions of the laboratory K- momentum. All of the eight well-established ( *** or **** status) S = -1 resonances listed in the Particle Data Group tables I’) below 1700 MeV in the KN c.m. are incorporated in the reaction amplitudes. The less well-established resonances, aside from any questions regarding their existence, presumably couple weakly to the K-N channel (otherwise, they would be seen more clearly) and consequently should not figure prominently in the K-d reactions, For this reason they have been ignored in the present work. In line with the discussion above, we have also ignored single-nucleon absorption contributions to the amplitudes and have not included any s-wave background terms other than those associated with the excitation of s-wave resonances. The calculations reported here are intended to be mainly qualitative, at best semi-quantitative, in nature. They are meant to provide order-of-magnitude estimates of the cross sections for reactions (1.2) in different momentum regions and to uncover structures in the cross sections that could yield information concerning particular Y* resonances. With this in mind, two approximations have been made, which, while compromising the quantitative reliability of the results somewhat, greatly simplify the calculations while hopefully preserving the qualitative features

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of a more complete analysis. First, we ignore the effect of correlations in the final two-baryon state. Such correlations exert a relatively modest influence in the rd system, at least in the evaluation of the integrated absorption cross section 14,i6); however, their effect in the K-d system could be important due to the strong conversion processes that couple the Xp and An final states. Second, we neglect the contributions from vector meson exchanges to the Y*N+ YN transition potentials in the reaction amplitudes. These vector exchanges are shorter-ranged than the pseudoscalar 7r and K exchanges, due to the larger meson masses involved, but may become important at the large momentum transfers associated with the two-nucleon absorption mechanism. In the rd system their incorporation tends to reduce the contribution from pseudoscalar exchange 14); however, for the spin-averaged absorption cross section, this effect is indistinguishable from that resulting from the inclusion of form factors at the internal mesonbaryon vertices 14,18).While the explicit calculation of vector exchange contributions to the amplitudes would certainly be desirable, this is extremely difficult, if not impossible, in practice within the present model due to the lack of information concerning the relevant coupling strengths. Even symmetry considerations are not of much use here because such considerations do not fix the relative phases between couplings involving resonances in different multiplets. Note that for pseudoscalar exchanges and pseudoscalar exchanges only, this phase is squared (the diagrams then involve two resonance-baryon-pseudoscalar vertices) and is thus of no consequence. Sect. 2 details the construction of the reaction amplitudes within the model outlined above. We employ relativistic expressions for both the resonance propagators and the Y*NK- resonance creation vertices and approximate the mesonexchange Y*N + YN transition amplitudes by non-relativistic transition potentials. Evaluation of the reaction-amplitude matrix elements is carried out in position space with initial-state correlations included in the form of non-relativistic deuteron wave functions. To obtain numerical results for the absorption cross sections, we must specify the various pseudoscalar coupling strengths that appear in the amplitude expressions. In general, these strengths are not very well known, especially those involving the higher-lying resonances. There have been a number of studies conducted over the past several years in which the existing K- nucleon data is utilized to extract resonance decay amplitudes 19-23).In principle, the results of any of these studies could be utilized to fix the coupling strengths required in the present work. Unfortunately, all of the analyses suffer somewhat from ambiguities in the data and even when based on the same data, are not always consistent with one another. Consequently, the coupling strengths so determined will depend to a certain extent on the particular set of decay amplitudes employed in the determination. An alternative approach, which avoids this ambiguity and is utilized here, is to extract the required couplings directly from the empirical widths and branching

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ratios, invoking SU(3) symmetry relations in cases where data is lacking or of very poor quality. This procedure does not yield values for the E*(1385)NK and A*(1405)NK coupling strengths however, since both the X*( 1385) and A*( 1405) lie below the KN threshold. These two couplings have been the subject of several studies which analyze (K, n) reactions in deuterium within a single-nucleon exchange model 24,25).We have adopted values for the .Z*(1385)NK and A”(1405)NK couplings obtained in the most recent of the analyses 25). Further details concerning our choice of coupling strengths appear in sect. 3. Sect. 3 also specifies the procedure employed here to extrapolate the resonance widths away from the peak positions of the resonances. In sect. 4, we present cross section results obtained within the simplest version of the model, in which mesons and baryons are assumed to interact through structureless pointlike vertices (no form factors) and distortions of the incoming K- are neglected (plane-wave approximation). These results exhibit two prominent features that may be accessible experimentally: a sharp peak centered at a K- lab momentum of 385 MeV/c and a broader structure around 700 MeV/c. Unfortunately, the two features are sensitive both to the inclusion of phenomenological form factors at the internal meson-baryon vertices and to the inclusion of Kdistortions, as evidenced by the results presented in sect. 5. In sect. 6 we conclude our discussion with a brief summary. 2. The reaction amplitudes As discussed above, we assume that the K-d absorption mechanism is dominated by rescattering processes, where a Y* resonance is excited at the initial absorption vertex and then deexcited through a virtual meson exchange. In such processes, the incident kaon can be absorbed by either component of the deuteron, and the virtual meson exchange may or may not involve a strangeness transfer. These four possibilities are represented diagrammatically in fig. 1, where the YN labels on the outgoing lines refer to either the Z-p or An final state. Note that the isospin degree I

2

I

2

I N

Y

2

I

2

Y

k EXCHANGE

-0

Fig. 1. Contributions to the reaction amplitude. Y* denotes a Z* or A* resonance, while YN represents either the J-p or An outgoing state.

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of freedom is treated explicitly here; once the final state is specified, the charge states of all particles in each diagram are fixed. In principle, several different mesons can participate in either the non-strange or strange exchange processes, but, as mentioned previously, we will include only those exchanges involving pions and K’s. We have drawn the diagrams in the figure so that the outgoing hyperon and nucleon are oriented in the same sense (Y left, N right) in all the diagrams. Fixing this orientation also fixes the location of the initial kaon absorption vertex for each meson exchange. To preserve the correct isospin symmetry of the initial deuteron state, it is necessary to include in the reaction amplitudes not only these diagrams, but also those diagrams with both the final YN orientations and locations of the absorption vertices reversed. Since each diagram of the second set differs from one of the first by just an interchange of labels 1 and 2, the contributions of the two sets of diagrams to the amplitudes are identical. Consequently we need calculate only the diagrams of fig. 1 and multiply by two. Each contribution to the reaction amplitude consists of a product of three elements - a K- absorption vertex, a resonance propagator, and a Y*N+ YN transition potential - summed over the spectrum of resonances included in the calculation. If the incident kaon wave function is incorporated in the amplitude as a plane wave and the integral over the nucleon c.m. performed, this sum of products can be expressed in the schematic form f(s, r) = F* [ Q;,(r)&(s) +

eiq’r’2U);Y*(W, q)

V&(-r)&(s)

e-i4’r’22)Z*(w,q)+] ,

(2.1)

where the labels 1 and 2 specify the location of the absorption vertex and the direction of the momentum transfer in accord with fig. 1; q, w, X& represent respectively the kaon momentum and energy and the total energy in the KN c.m.; r = r, - r,; and the sum over resonances includes a sum over charge states. Note here that the transition potentials V” and VK, as well as the propagators Dy+, depend explicitly upon the particular resonance excited. In terms of the reaction amplitude, the integrated cross section evaluated in the lab system (deuteron rest frame) can be approximated by the expression p E2+M~-M$ --- 1 aabs(q)- (4~)’ qlab 2E

da 5 C lZr)* 7 spins

(2.2)

valid to lowest order in qlab/E. Here hi:,,denotes the matrix element of T between the initial and final baryon states; p is the relative outgoing momentum, given approximately by p2=+(M$-M;+20),

(2.3)

~=~(E2-t$&-hf;-ii&

(2.4)

qlab and E are the lab kaon momentum

and total lab energy, related to the KN

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c.m. quantities by

+(Mi+&b)1’2; E = M,euteron

(2.6)

and the spin sums are to be carried out over all Si = 1 initial spin states and Sr = 0 and 1 final spin states. Corrections to eqs. (2.4) and (2.5) are of order (qlab/E)’ and hence, alter the lowest-order results by at most a few percent for qi&c 1 GeV/ c. In what follows, we will examine in some detail the various quantities that make up the amplitude in eq. (2.1), beginning with the absorption vertices. Explicit expressions for the initial and final states and details concerning the evaluation of the amplitude matrix elements have been relegated to an appendix.

2.1. KAON ABSORPTION

VERTICES

The structure of the Y*NK- vertex for a particular resonance is dictated by its angular momentum and parity. Since four different angular momenta occur among the eight low-lying resonances considered here, four types of absorption vertices are required. Except for the 1’ resonances, we have adopted a minimal coupling scheme for these vertices; that is, we have assumed that mesons and baryons interact in the simplest fashion possible consistent with Lorentz invariance. For the 4’ resonances, however, the pseudovector vertex is employed in preference to the pseudoscalar one, since it seems to be more consistent with SU(3) symmetry for the couplings among the ground-state octet baryons and mesons 26). With this prescription, the relativistic interaction lagrangians assume the forms, up to phase factors of no significance, for Jp = iLz* y NK-

-

-(flm,)b+d%w4dk7 (f/m,,)a?*UNq,bK, (f/ma)~?*wNq&K,

for Jp = 1+ 2 for Jp=3+

(2.7)

2

for _I’=$-,

where f is the coupling strength, & the K- field, and u’ a spin-$ Rarita-Schwinger spinor related to the ordinary Dirac spinor u and the covariant spin-l field Ed by u,(M)

= C (lm~h~~M)~,(m)u(h) . rnh

(2.8)

To obtain the kaon absorption vertices in a form suitable for use in the reaction amplitudes, we expand the relativistic expressions above in the KN c.m., where the resonance is at rest. In this frame Ed reduces to the ordinary three-component unit vector, and the Rarita-Schwinger spinor can be recast in terms of a transition spin operator that transforms spin-$ objects into spin-$ objects. Introducing second quantized operators, at(Y*) and ai( in connection with the transition N+Y*

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on line i then yields the expression z$*(o, q)+ = a(Y*NK-)u:p(o,

q)at(Y*)ai(N)

(2.9)

with coupling strength and structure functions a(Y*NK-)

=- f*kNk($Ir; 77

4 --ql&* I;*)

(2.10)

and

&+(W, q) =-K(1

+w/2h&,K2)-‘(u

’

q) ,

U:,2+bJ, 4) = -K(S+ * 4) ,

&,2-b’, 4) = (~MNK)-'(S+.

q)(u ’ 4) ,

(2.11)

where 1&, Iy* and 1;. denote the nucleon isospin projection and Y* isospin and isospin projection respectively, and the quantity K is given by (2.12)

K2=[b’&+(M;+tf)1’2]/2hfN.

Note in eq. (2.10) that the coupling strength has been factored into an overall strength fpNK,hereafter referred to as the “reduced” coupling strength, which is common to all vertices that involve the same isospin multiplets, and a vector coupling coefficient, which specifies how the coupling varies within the iso-multiplets. Eqs. (2.11) represent the structure functions in a form that displays the partial-wave structure of the vertices. Thus, the negative-parity 4- and z- resonances clearly involve the s- and d-wave meson-baryon interactions respectively, while the two positive-parity resonances both originate from p-wave interactions. 2.2. RESONANCE

PROPAGATORS

After a Y* resonance is excited at the kaon absorption vertex, it propagates through the system before deexciting through virtual meson emission. In principle, this propagating resonance can interact any number of times with the other nucleon composing the deuteron before it deexcites; we will neglect such interactions here, however, and treat the intermediate Y” as a freely propagating resonance. Then the Y” propagator is just given by the relativistic Breit-Wigner expression Dy*( s) =

2M,* MC. - s - iMyJ(

s) ’

(2.13)

where My* is the peak position of the resonance and T(s) its energy-dependent width. In contrast to the As3 (1232), which decays almost exclusively into a single channel, several channels are generally accessible in Y* decays. The total width is thus

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comprised of a sum of partial widths,

ns)=r:r,fs)>

(2.14)

0

corresponding to the different channels into which the Y* can decay. At the peak position of a resonance, the partial widths are given empirically by the product of the observed branching ratios xn and the empirical value for the total width Temp: r, ( f = MC*) = x0&.,

(2.15)

.

Off the peak position, we require analytic expressions for the widths to take account of their energy dependence. For 2-body decays these are easily obtained within lowest-order perturbation theory, once the form of the meson-baryon-resonance interaction vertex is specified. In terms of the helicity-basis matrix elements (c@, 0#I’&$U>, where qu is the a-channel momentum, A the decay baryon helicity, J, P, M the spin, parity, and spin projection of the resonance, and the angles 8, Q, specify the orientation of qa with respect to the quantization axis, the lowest-order expression for the partial width in the resonance rest frame is just (2.16) The angular dependence of the integrand here is made explicit by reexpressing the helicity-basis matrix elements in terms of partial-wave amplitudes: (%h, e&&#)=+./Ci

&&e,

fff)rJ+).

(2.17)

Pe~orming the angular integrations in eq. (2.16) and summing over the helicities, noting that lsJp(~)12 IS . actually independent of h, then yields the simple expression

(2.19) for the channel momentum, where MB and I& denote the decay baryon and meson masses. For decays into pseudoscalar mesons and f’ baryons, the relevant interaction vertices are just given by eqs. (2.7) with the replacements N+ B and K+ M. Evaluating the corresponding partial-wave amplitudes and inserting into eq. (2.18)) we obtain the corresponding widths in terms of the reduced coupling strengths defined previously and a set of angular-momentum-dependent structure functions. Specifically, (2.20)

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with

(2.21) where (2.22)

k2=(J;+Mg)2-A&.

Note here that the isospin-projection dependence appearing in eq. (2.10) for the coupling strengths has been eliminated from eq. (2.20) by summing over the projections, utilizing the completeness of the vector coupling coefficients. Expressions analogous to eqs. (2.20) and (2.21) can be derived for other 2-body decays by repeating the procedure above with the appropriate interaction vertices. For decays of the type Y*+ B($+)+M(O-), for example, the relevant interaction lagrangians have the forms for Jp = $for

Jp

=

L+ 2

(2.23)

for Jp=s-, yielding for the widths, r[Y*+B@)+M(O-)]=fq,

t*BMqJ.+(s)

(2.24)

with

G/~-(S)

=

5,

(2.25)

where we have assumed an s-wave Y*BM coupling in the last of these expressions. We have omitted the expression for the Jp = 5’ case, since it is not required in what follows. To extract numerical values for the widths from eqs. (2.20)-(2.22) or (2.24)(2.25), it is necessary to specify values for the reduced coupling strengths. This problem will be addressed in some detail in sect. 3. In the remainder of this section,

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we will examine the last element comprising the reaction amplitude in eq. (2.1), the Y*N + YN transition potentials. 2.3. TRANSITION

POTENTIALS

In the last step of the two-nucleon absorption mechanism, the Y* resonance exchanges a meson with the surviving nucleon, thereby deexciting and providing the momentum transfer required to balance energy and momentum between the initial and final states. To represent this process, we employ transition potentials obtained by Fourier transformation from the corresponding momentum-space transition amplitudes. For a particular resonance, the complete amplitude has contributions from several meson exchanges; however, we will restrict ourselves to the pion and l? exchange contributions, as discussed earlier. Each of these consists of meson emission and absorption vertices and an intermediate meson propagator: V;,(k,,

k) = v;N”(-k)+(m2,-t)-‘~~*Y”(k),

Vfz(k,,

k) = v2’N”(-k)+(m$-

t)-‘v;*N”(k).

(2.26)

Here ko, k denote the energy and 3-momentum transferred from the meson emission vertex (labelled 1) to the absorption vertex (2) and t = ki- lk1’. The vertices in these expressions are obtained by a non-relativistic reduction of relativistic vertices having the same structure as the kaon absorption vertices given by eqs. (2.9)-(2.11) (with appropriate adjustments of coupling constants and baryon masses). To implement the reduction, we consider the limit in which k. and Ik( are both small compared with baryon masses. In this limit the factors K and (1+ ko/2MNK2)-’ in the relativistic expressions can be replaced by unity, thereby eliminating the k, dependence of those expressions. The meson emission vertices then assume the form v;*BM(k) = a(Y*BM)#(k) , (2.27) where the coupling strengths are given, in analogy with eq. (2.10), by a(Y*BM) =-f*r

(2.28)

(4 r;, f I$lly* I$*) 7r

and the non-relativistic

structure functions by

F’,%(k)= m,, j&+(k)

= S1 * k ,

d::-(k)

p$)2+(k)=u1. =

(2WJ1b1

k, . kM

. k),

(2.29)

with (B, M) =(N, P) or (Y, I?) and Jp denoting the spin, parity of the Y*. The corresponding expression for the absorption vertices is just vFNM(-k)+ = a(BNM)/&+(-k)+

= -a(BNM)(az

* k) .

(2.30)

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Although the reduction procedure eliminates the energy transfer dependence of the meson-baryon vertices, the transition amplitudes retain a k. dependence through the meson propagators. A proper treatment of this k, dependence would involve evaluation of an energy-loop integral and hence require detailed information concerning the energy dependence of the deuteron wave functions. An alternative approximate procedure, suggested by the kinematic requirement that the momentum transfer greatly exceed the energy transfer, is to simply fix k. at some value within the kinematically allowed range. Here we adopt the prescription, (2.31) i.e. we require the outgoing kinetic energy to be evenly divided between the two outgoing baryons (neglecting the deuteron binding energy). With k. fixed in terms of the incoming kaon energy, the transition amplitudes can be Fourier-transfo~ed to position space in a straightforward manner. The resulting expressions for the transition potentials are somewhat lengthy and consequently, are not listed here. They may be found in appendix B. 3. Coupling strengths and widths In the previous section, a general formalism was devefoped for the analysis of K-d absorption without identifying the particular resonances to be included in the reaction mechanism. To utilize this formalism in a numerical computation, these resonances must now be specified and the available information concerning them examined in some detail. In table 1, we list the eight resonance states included in the present calculations, together with their quantum number assignments, their symmetry classification under both SU(6) X O(3) and SU(3) XSU(2), and their empirical widths and decay branching ratios. Under SU(6) X O(3) they can all be

TABLE

1

S = -1 resonances included in the reaction amplitudes (masses and widths are in MeV) Branching ratios

Symmetry classbication Y* SU(6)xO(3) E(1385) A(l405) A(1520) A(1600) X(1660) A(1690) E(1670) A(1670)

;+tp,31 l-&It ?@A,,, fe(P01) f’(Pll) %-0%~) $-(JAJ t-(%,1

35 40 16 1.50 100 60 60 3.5

(56,0+) (70,1-I (70,1-I (56,0+) (56,0+) (70,1-I (7&l-? (70,1-j

SU(3)xSU(2) QO mainly ‘1 mainly ‘1 =8 ‘8 mainly 28 mainly ‘8 mixed

RN

0.45 0.23 0.2 0.25 0.1 0.2

&r

AT

0.12 0.88 1.0 0.42 0.4 no value no value 0.3 0.45 0.1 0.4

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assigned to one of two multiplets according to their parity: either the Cl+ground-state 56-plet (positive parity) or the l- 70-plet of I = 1 orbital excitations (negative parity). Under SU( 3) X SU( 2) the symmetry assignments are less certain. The classifications quoted in the table represent a compilation of assignments from several references 20,22,27),which, in general, agree on the classification of resonances that are pure or nearly pure symmetry states, but may differ considerably on the classification of mixed symmetry states. All three of the positive-parity resonances considered here appear to be almost pure symmetry states under SU( 3) X SU(2). In particular, the s(1385) is naturally assigned to the same decuplet as the A(1232), while the two 1’ resonances fn well into an SU(3) octet along with the N(1440) Roper resonance. Because these resonances fall clearly into multiplets of SU(3) X SU(2), one might expect the corresponding Y*BM couplings (B = a ground-state baryon; M = a ground-state meson) to satisfy SU(3) symmetry relations, at least approximately. The negative parity resonances, with the exception of the A (1670), can also be assigned to SU( 3) x SU(2) multiplets, but in contrast to the positive-parity resonances exhibit signi~~ant mixing into other symmet~ states. ConsequentIy, their coupling strengths may differ substantially from the predictions of SU(3) for the pure symmetry states. The width information in table 1 has been taken from the compilation of ref. I’). We have employed the nominal values quoted there for the total widths and for the branching ratios have chosen the central values within the ranges specified. There is a rather considerable spread in the uncertainties connected with these numbers. The uncertainties are reasonably small for the trio of low-lying resonances, moderate for the negative-parity states above 1650 MeV, and quite large for the pair of 1* resonances. A more quantitative discussion of these uncertainties, together with detailed reference lists, is contained in ref. I’). From their widths and branching ratios, the significance of the various resonances in K-d absorption can be predicted. The A(1520) provides a noteworthy example in this regard. Not only does it couple strongly to the KN channel, but it is quite narrow and hence should generate a sharp structure in the integrated absorption cross section. By contrast, the higher-lying resonances do not couple so strongly to the KN channel and due to their broader widths and close spacing, overlap one another to a significant extent. Consequently, their influence on the cross section may be difficult to discern on an individual basis. As mentioned previously, there is relatively little information concerning the KN system compared with the rrN system, and much of the available information is of rather poor quality. This is especially true of the Y*BM coupling strengths. Even for the lowest-lying resonances, the X( 1385) and the A (1405), the coupling strengths derived in various analyses of experimental data vary widely [see ref. 26) for a recent compilation]; for the higher-lying resonances, they are essentially unknown. Thus, in order to obtain numerical results, some prescription for estimating these coupling strengths is required.

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A particularly simple prescription from a computational point of view, and one which maintains close contact with the empirical data, combines the analytic expressions for the widths derived in sect. 2 with the experimental information summarized in table 1. We simply equate the widths given by eq. (2.20) with the empirical partial widths obtained from (2.15) and solve for f&, using eqs. (2.21) with s = MC*. Of course this determines only the magnitude of fY*aM. To fix the relative phases among the different couplings (different choices of B and M) of a particular resonance, we consult the detailed studies of EN amplitudes carried out in refs. 19-23). Although these studies may vary widely in their estimates of the amplitude magnitudes, as discussed in the introduction, they generally agree on the phases, which for those resonances largely comprised of a single symmetry state, are predicted to be just the phases of the SU(3) isoscalar factors. There is still one overall phase for each resonance (or, more correctly, SU(3) multiplet of resonances) that remains undetermined, but since this phase occurs at both resonance vertices, it is immaterial for the calculations discussed here. The prescription outlined above is inadequate in two cases. On the one hand, the two lowest resonances, the Z( 1385) and A (1409, lie below the l?N threshold and hence cannot decay into that channel. As estimates of the Y*NR coupling strengths for these resonances, we employ the values deduced from K-d scattering in ref. 25), using the appropriate SU(3) relations to fix the phases relative to the Y r couplings. The other case where the width based prescription is inadequate involves the 4’ pair of resonances, the A(1600) and X(1660). In this case, the difficulty lies in a lack of information. While the I?N branching ratios for these resonances are su~~iently well known to provide estimates for the Y*NK couplings, the YV branching ratios are either unknown or so uncertain as to preclude their usefulness for obtaining coupling-strength estimates. In the absence of this branching ratio information, we must resort to symmetry relations to estimate the Y*Y 7r couplings. Fortunately, as observed above, the 4” resonances are almost pure SU(3) octet states and lie close together in mass, so that SU(3) estimates of their coupling strengths should be fairly reliable. Since the interaction vertices for these resonances involve three octets, two independent SU( 3) couplings, one symmetric with strength 0, and one antisymmetric with strength F, must be considered. In terms of the ratio of these, the resonance couplings have the form fn*Nn=J$(3-2*)g,

fx*NR=-&(2a-

ljg,

f/i*&.=--2Lyg,

fz*x?r=2&(1-Lu)g,

f-*Ax=2J$cwg,

(3.1)

where g is an overall strength and (Y= D/( D + F). Inputting the A *NE and T*NE couplings, as deduced from the partial widths, fixes the two parameters, g and C-Y, enabling the other three couplings to be calculated. As a check on the procedure, we can also compute the N*NP coupling using the relation (3.2)

0. V, Maxwell

/ ~a~n-~n~~~e~

deuteron

459

~is~ntegr~ti~~

and compare the result with the coupling strength deduced from the empirical Roper width and its NV branching ratio. With a value (Y=0.75 we find that the SU(3) relations among the N*Nr, A*NIf, and X*Nl? couplings are well satisfied by the empirical values, suggesting that our symmetry estimates for the Y “Y 7r couplings are at least qualitatively reasonable. Of course, the actual values of all the 4 resonance couplings are subject to considerable uncertainty due to the uncertainties in the input widths. Our results for the Y*Nff, Y*&, and Y*A?r couplings of all eight resonances are summarized in table 2. In addition to these couplings, we also require values for the NNn; XNK, and ANR couplings for use in the transition potentials. The NNrr coupling strength employed here was taken from ref. *6) with a suitable factor included to conform with our conventions for the interaction vertices. To estimate the .ZNi( and ANIf coupling strengths, we used the NNv strength, together with the X% and Brr strengths given in ref. 26>,to fix the parameters fy and g for the ground-state octet. The I? couplings were then deduced from the symmetry relations, eqs. (3.1). This yields the values f&q,J4?r = 0.238,

f &/47T

= 0.011 ,

f&&4T

=0,159

(3.3)

with positive phases at the NNrr and XNil: vertices and a negative phase at the ANK vertex. Once again, it should be noted that all these coupling strengths are the reduced strengths with the isospin dependence removed. To obtain the couplings associated with particular charge states, we have to multiply by the appropriate isospin vector-addition coefficients. Thus, for example, in the usual baryon first convention,

(3.4)

TABLE

2

Squared coupling strengths as defined in sect. 2 (a minus in parentheses foliowing an entry indicates that f is negative)

f2/4v Y*

X(1385) A(1405) A(l520) A(1600) P(1660) Af1690) X( 16’70) A(1670)

Y*RN

Y*&r

Y*AT

0.086 (-) 0.20 1.454 0.0079 0.0028 (-) 0.192 0.095 (-) 0.014

0.076 0.158 1.075 0.012 (-) 0.0028 0.383 (-) 0.691(-) 0.025 (-)

0.125 (-)

0.0042 0.074 (-)

0. V. Maxwell / Kaon-induced

460

deuteron disintegration

In addition to coupling-strength estimates, some method is required for estimating the widths of resonances off their peak positions to permit the evaluation of resonance propagators over a wide energy range. For 2-body decays, the energy dependence of the partial widths is approximately described by the lowest-order perturbative expressions derived in sect. 2. However, only part of the total width of a resonance is due to 2-body decays; the remainder is due to decays into three or more particles. Many-body decays are considerably more complicated than 2-body decays, since they involve a larger number of vertices and additional integrations over decay product momentum distributions. Rather than attempt an explicit calculation of many-body decay amplitudes, we assume that the baryon-decay product in such a decay resonates strongly with all but one of the meson-decay products. Then the many-body decay can be treated as an effective 2-body decay into a lower-lying resonance state and a single meson. If we further suppose that 2-body channels energetically inaccessible at the resonance peak do not contribute to the width above the peak, the number of such resonance-meson channels is severely limited. In particular, for resonances below 1700 MeV, only the 1(1385)7r, A(1405)7r, and A (1520) 7~channels are energetically accessible. Symmetry considerations eliminate the last two channels [if both the A (1405) and A (1520) are regarded as pure SU(3) singlets], leaving just the Z( 1385) r channel, which is accessible in decays of any of the upper five resonances listed in table 1. In fig. 2 we have depicted in a diagrammatic fashion the decay width model suggested by these considerations. The indicated sum extends over all 2-body ground-state meson-baryon channels accessible at the energy considered with the proviso that channels inaccessible at the resonance peak are not included at any energy. For the A(1670), we have included AT decays in the sum, in addition to &r and NI? decays, since the empirical AT branching ratio for this resonance is substantial. The 2-body partial widths are calculated for arbitrary energies using eqs. (2.20)-(2.22) with coupling strengths determined as described previously. In accord with the discussion above, many-body decays are represented collectively in fig. 2 as a single %-body decay into an effective E(1385)~ channel. To estimate the corresponding partial width, we employ eqs. (2.24) and (2.25) using coupling strengths deduced from the total width at the resonance peak after subtracting the sum of 2-body partial widths. 4. Results without form factors Using the model described above, we have calculated the integrated cross sections for the absorption reactions (1.2) as functions of the laboratory kaon momentum. Y;

@-y

=

~~

\ /.

P_+

,y”

+

YW_

7-r

M Fig. 2. Model

for the resonance

widths

sizy*

with notation

as explained

in the text.

5. V. Maxweti/ Kaon-induced deuteron disintegration

461

27.5

22.0

2 -

16.5

E b

11.0

L-J.

5.5

0

I

0

12.5

I

250

I

I

I

375

500

q,,,

( MeV/c

I

625

750

875

1000

1

Fig. 3. @(K-d + Z-p) versus qrabwith ail resonances included in the reaction amplitude.

In this section, we present numerical results based on the simplest version of the model in which all meson-baryon couplings are treated as pointlike interactions and distortion of the incident K- is neglected. The Reid NN interaction was employed in these calculations to generate the initial deuteron wave function. In the final YN state, we included partial waves up to LyN = 4 and ignored correlations induced by the final-state interaction. The two final-state channels, Z-p and An, are then decoupled and could be treated independently. Figs. 3 and 4 display the integrated cross sections obtained when all eight Y* resonances are included in the amplitudes. Although quite small compared with the rrd absorption cross section over most of the energy range considered, the Kd cross sections exhibit two prominent features which may be accessible experimentally: a strong, narrow peak centered at 385 MeV/ c and a smaller, broader structure around 700 MeV/c. As comparison of figs. 3 and 4 reveals, neither the positions nor the strengths of these peaks depend much on the choice of final state. The partial-wave structure of the integrated cross sections is exhibited in figs. 5 and 6. Only contributions from the three most important final-state partial waves have been displayed here; the other partial waves included in the calculations are relatively unimportant. Note that in the vicinity of the first peak, the F-waves dominate the cross sections, in contrast with 7rd absorption where the ID2 wave is the most important one, Since the final-state F-waves are mainly associated with d-wave resonance excitation, this indicates that d-wave resonances, rather than p-wave resonances, are largely responsible for the first peaks in the cross sections.

462

0. V. Maxwell / Kuon-induced deuteron disintegration

-

22.5

I

I

I

I

18.0

2

13.5 E

-g

9.0

4.5

0

I

0

I

I

125

250

I

I

375

500

625

qlab

( MeVk)

I

I

750

875

I

I

1000

Fig. 4. Same as fig. 3 for K-d+ An.

22.5

1

I

I

I

I

I

18.0

wave

,F 2

13.5

: \I’.... - i

E

wave

i//p

-g

9.0

I

~

\

I

\

4.5

0

0

100

200

300

qlab Fig. 5. Contributions

400

500

600

--T.-

700

( MeVk 1

to a(K-d + X-p) from different X-p partial waves.

800

0. V. Maxwell f Gon-induced deuteron disintegration

0

100 200

300

400

500

q,,,

( MeVk 1

600

463

700

800

Fig. 6. Sameas fig. 5 for K-d+ AN. The threshold behavior of the cross sections, on the other hand, is due to s-wave resonance contributions, which are manifested primarily in the final-state P-waves. Both the P-wave and F-wave contributions are insensitive to the final-state channel. By contrast, the D-wave contributions obtained for the two final states are quite different, especially at low energies where the An result exceeds the X-p result by a factor of three to four. This difference can be understood by noting that at low energies the D-wave contributions to the cross sections result primarily from excitation of the Z(1385). Since the X(1385) couples more strongly to the AT channel than to the XIT channel and since fiNK greatly exceeds f&x, both the pion and if exchange interactions favor the deexcitation of intermediate X(1385)N states to An rather than 2-p. In figs. 7 and 8 the cross-section contributions from the lowest two resonances are shown explicitly. We see here that the X(1385) indeed couples with much greater strength to the An channel than to the X-p channel and that the difference in X( 1385) contributions to the cross sections is sufficient to account for the different D-wave behavior characterizing the two final states. Figs. 7 and 8 also confirm the connection between the threshold behavior of the cross sections and excitation of the s-wave A(1405). Fig. 9 illustrates the dramatic role played by the A( 1520) in K-d absorption, Since the effect of the A( 1520) on the absorption cross section is similar for the two fmal states, only the 2-p results are shown here. As expected, the narrow structure at 385 MeV/c is due almost exclusively to this one d-wave resonance.

464

0. V. Maxwell f Kacm-induced deuteron disintegratjon

0.8

0

75

0

150

225

300

375

qlab

( MeV/c

450

525

600

>

Fig. 7. u(K-d + 2-p) versus qieb with just the two lowest resonances incIuded in the reaction ampiitude.

-_.----A

-0

75

(1405)

150

ONLY

225

300

q lab

(

375

MeV/c

450 >

Fig. 8. Same as fig. 7 for K-d+An.

525

600

0. V. Maxwell j Kuon-induced deuteron disintegration

465

16.5

100

175

250

225

qlab

400

475

550

625

700

( MeVk)

Fig. 9. Effect of the A(1520) on o(K-d-, X-p). The label “lowest 2” indicates that just the X(1385) and A(l405) are included in the reaction amplitude, while the label “incl. A(l520)” indicates that .A(1520) is included as well.

The higher-lying resonances, which are broader than the A f 1520) and bunched together, do not individually affect the cross sections in such a dramatic way, as can be seen in figs. 10 and 11. In either final-state channel, the most important of these resonances is the X(1670), which provides the largest single cont~bution to the peak around 700 MeVf c. However, as a comparison of the X( 1670) contributions to the cross sections with the full cross sections reveals, other resonances are important in this energy region as well, in particular the A (1690) in the X-p channel and the X(1660) in the An channel. By contrast, neither the A(1600) nor the A (1670) seems to have much effect on K-d absorption. To a certain degree, these results, like those at lower energy, can be interpreted in terms of coupling strengths, but in this energy region the analysis is complicated by the number of closely spaced, overlapping resonances involved. It is of some interest to ascertain the extent to which our results depend upon the inclusion of l? exchange terms in the reaction mechanism. Because of the small magnitude of the XNK coupling, these terms have’little effect on the cross section in the 2-p channel. Numerically, their inclusion just reduces the strength of the A(lS20) peak somewhat, while leaving the cross section intact over the rest of the energy range. In the An channel, on the other hand, R exchange plays a central role in the determination of the cross-section structure. This is illustrated in fig. 12, where the cross sections with and without g exchange are compared. There one

466

0. V. Maxwell / Kaon-induced deuteron disintegration

5

4

r;3 E x2

I

0 450

525

600

675

750

825

q,,,

(MeWA

900

975

1050

Fig. 10. Effect of the higher-lying resonances on cr(K-d+ 8-p). The label “lowest 3” indicates that just the lowest three resonances are included; the label “all” that all eight resonances are included.

4.0

3.2

1.6

0.8 ---___ 450

525

600

675

750

q,,,

( MeVk)

825

Fig. 11. Same as fig. 10 for K-d+An.

900

975

1050

467

0. V. Maxwell / Kaon-induced deuteron disintegration

22.5 II

18.0

r;

13.5

E 75

9.0

4.5

0 0

125

250

375 q,,,

500

625

750

875

1000

( MeVk)

Fig. 12. a(K-d + An) versus qlabwith and without l? exchange terms in the transition potentials. All eight resonances are included in the reaction amplitudes.

sees that, as a consequence of isospin conservation, which forbids A * + An transitions through pion exchange, the suppression of k exchange completely eliminates both the threshold peak and the peak at 385 MeV/c! 5. Inclusion of form factors and K- distortions The results reported in the previous section were obtained using pointlike mesonbaryon interactions and a plane-wave representation of the’ incident K- wave function. For the reactions under study, however, neither of these approximations is apt to be very good. On the one hand, since the momentum transferred in the Y*N+YN transitions greatly exceeds the energy transferred, the virtual mesons exchanged are far off-shell, and the finite spatial extent of the meson-baryon vertices cannot be safely ignored. On the other hand, the large size of the total kaon-nucleon cross section (compared with the K-d + YN cross section) in the momentum range of interest suggests that optical distortion of the incident kaon could also be important. In view of these considerations, we repeated the calculations described above with both off-shell and distortion effects included in an approximate matter. The off-shell effects were treated by means of phenomenological form factor insertions at the internal vertices. In momentum space these factors have the form (5.1)

468

0. V. Maxwell / Kaon-induced

deuteron disintegration

where A and MM are the form-factor mass and mass of the exchanged meson; k is the 4-momentum transfer; and L is the meson-baryon orbital angular momentum, equal to Lye, the resonance orbital angular momentum, at the emission vertex and 1 at the absorption vertex. Inserting one such form factor at each internal vertex yields transition amplitudes of the same form as eqs. (2.26), but with the meson propagators replaced by (see subsect. 2.3 for notations) (M*-

k2)-‘+

~~*~~(k*)(M~-

k’)-‘~~~~(k*).

(5.2)

The product of denominators

here can be expanded in a series of partial fractions and then the Fourier transform to position space performed. The resulting transition potentials are similar to those given in appendix B but contain additional terms with ranges of the order of (A*- k,)* 1’2. For the sake of simplicity, we have set the form-factor masses at all vertices equal so that inclusion of form factors in the calculations introduces just a single additional parameter. Kaon distortion effects were incorporated in the simplest manner possible, through use of the eikonal approximation. In a strict sense, this is a high-energy approximation; however, detailed comparisons of DWIA eikonal results with full coupledchannel DWBA results for K- capture in nuclei reveal it to be reliable even at the moderate energies of interest here 28-3’). Certainly it is adequate for a semiquantitative estimate of distortion effects. Implementation of the eikonal approximation amounts to the replacement of the plane-wave expression for the K- wave function by the distorted-wave expression e&.p3 eiq.r+i5(r),

(5.3)

where the function e(r) is just the integral of the K- deuteron optical potential along a straight-line trajectory with constant impact parameter b:

w=-;I’--m U,,(b,

z’) dz’ .

(5.4)

With use of the optical theorem, the optical potential I&, can be expressed in terms of the total K-N cross section a, the ratio (Yof the real to imaginary part of the forward K-N amplitude, and the nucleon number density p(r). In particular, after averaging over isospin 28), (5.5) By now there exists fairly extensive data for both K- proton scattering and Kdeuteron scattering, which permit extraction of the K-p and K-n amplitudes over the whole momentum range of interest. In table 3 we have listed the values of IY and @ employed in the present treatment. The cu-values were taken from the work of Baillon et al. 32) (averaging over isospin), while the cross sections represent a compilation of results reported in refs. 32-34). The density factor appearing in eq.

0. V. Maxwell / Kaon-induced deuteron disintegration

469

TABLE 3 Isospin-averaged values for u, the ratio of the real to imaginary part of the forward K-N amplitude, and D, the totai K-N cross section, for various laboratory kaon momenta [data from refs. 32-34)] a

qI~b(MeV/c)

200

0.09

250 300 350 380 400 450 500 600 700 800 900 1000

0.13 0.19 0.27 0.27 0.13 0.32 0.48 0.66 0.79 0.61 0.47 0.18

@WI

100 76.8 58.1 49.9 55.3 54.5 38.5 33.6 29.4 30.3 34.3 39.2 41.7

(5.5) is simply related to the square of the deuteron wave function averaged over spin projections, i.e. p(r)

=$ x

d’r’ I#%(r-r’)]2i@(r+r’)]

Md

=

(Pry

c r.&(Zr),

(5.6)

where r and r’ are the nucleon positions, Md is the deuteron angular momentum projection, and uh is the radial wave function defined in eq. (A.l) of appendix A. Note here that p is normalized so that

J

d3rp( r) = 2 .

(5.7)

Evaluation of the reaction amplitude matrix elements with eikonal distortions included proceeds exactly as in appendix A, but with the Bessel function in eq. (A.4) replaced by the complex radial function that results from expansion of the complete distorted-wave function in partial waves. The effects of form factors and K- distortion on the results obtained previously are displayed in figs. 13-15 for the 2-p final state. The An results are affected by form factors and distortion in a very similar manner and hence need not be discussed explicitly. The first figure, fig. 13, illustrates the influence of distortions alone on the cross section. Comparison of the two curves here reveals that distortions affect the results in an almost momentum-independent manner, reducing the cross section

0. V. Maxwell / Kaon-induced

470

27.5

deuteron disintegration

T

/NO

r;

DISTORTIONS

16.5

E -G

11.0

,DISTORTIONS

750

550

350 9 ,ab Fig. 13. u(K-d+Z-p)

IN C L

( MeVk

950

1

versus qlsb with and without K- distortions. All eight resonances are included in the reaction amplitudes.

25

20

Ll

15

E b

IO

5

0

! I

0.5

1.3

A

I

I

2.1

2.9

3.7

(GeV)

Fig. 14. o(K-d + E-p) with and without K- distortions at q,ab = 385 MeV/c versus A(GeV). All eight resonances are included in the reaction amplitudes.

0. V. Maxwell / Kaon-induced

0.5

1.3

471

deuteron disintegration

2.1

2.9

3.7

h(GeV) Fig. 15. Same as fig. 14 at q,ab = 700 MeV/ c.

by a factor two to three. The magnitude of this reduction indicates that K- distortion will have to be treated rather carefully in any analysis that aims at quantitative results. Form-factor effects are exhibited in figs. 14 and 15, where the cross sections with and without distortion are plotted at the positions of the two peaks as functions of the form-factor mass. As these figures make evident, incorporation of form factors in the model substantially alters the results obtained. The effect is particularly pronounced at the A(l520) peak, but even at the higher-energy peak inclusion of 1.2 GeV mass form factors reduces the unmodified cross section by a factor four. By contrast, BRW found that form factors of this mass reduce the absorption cross section in the rd system by only a factor two or so 14). The pronounced form factor sensitivity of the K-d absorption cross sections, as compared with the rrd cross sections, has two explanations. On the one hand, since mk - (Mu-MN) > m,, the momentum transfers involved in K-d absorption exceed those in 7rd absorption, making the denominator in eq. (5.1) larger for the kaon reactions. More significantly, the resonances responsible for the peaks in the K-d cross section are mainly d-wave resonances, rather than p-wave resonances, and thus, are more effectively suppressed by form factors, due to the power L appearing in eq. (5.1). Unfortunately, the background contribution to K-d absorption arises primarily from the p-wave X( 1385), so that form factors not only reduce the overall normalization of the cross sections, but also diminish the strengths of the peaks relative to the background. This could make the structures discussed in the previous section quite difficult to observe if the form factors required are very long-ranged.

472

0. V. Maxwell / ~aon-induced 6.

S~rna~

deuteron disintegration

and conclusions

Within the framework of a simple resonance rescattering model that yields a semiquantitative description of rd absorption, we have investigated the integrated cross sections for the Kd+YN reactions for a range of kaon momenta from threshold up to 1 GeV/c. With form factors omitted, the results exhibit two distinct features superimposed on a l-4 mb background, depending on whether or not distortion of the incident kaon wave function is considered. The lower-lying of these is quite prominent and is associated with a single resonance, the d-wave A(1520), suggesting that empirical studies of K-d absorption in the appropriate momentum range could improve our understanding of this resonance. The other structure, higher up in energy, is broader and less prominent. In contrast with the lower peak, it reflects the influence of several overlapping resonances and consequently may not be useful for extracting information regarding particular resonances. The inclusion of form factors substantially reduces both peaks, not only in absolute magnitude, but also in magnitude relative to background. In fact if phenomenological constraints dictate form factors too long-ranged, the peaks may not be detectable. Unfortunately, the empirical data on the K-d + YN reactions are very meager at present. There does not yet exist a systematic experimental study of these reactions over an extended momentum range, to the author’s knowledge, but just a few measurements at specific momenta 11*12+25). One of these was obtained in a bubblechamber experiment invol~ng stopped kaons ‘I) and has little relevance here, not only because stopped kaons were employed, but also because the ratio of absorption rates with two- and three-body final states was measured, rather than the integrated cross sections. In another experiment, the absorption cross sections in both the Xp and An channels were determined at two different momenta using in-flight kaons “). At 340 MeV/c values of 300+ 100 pb and 360~1~150 ub were obtained in the Xp and An channels, respectively; while at 400 MeV/c, the measured value was 200 f 30 kb in both channels. More recently, values between 70 and 95 pbf 10% have been obtained for the K-d-,X-p cross section at several momenta between 686 and 844 MeV/c [ref. *‘>I. From a theoretical point of view, these values are quite small, especially since the momenta involved lie near the two peaks seen in the theoretical results. Moreover, the ratio of observed cross sections at the two lower momenta appears to contradict the model, which predicts a larger cross section at 400 MeV/c than at 340 MeV/c. This last disagreement is probably not a serious one, however, because the narrowness of the A( 1520) peak makes the theoretical results quite energy dependent in this momentum range. Shifting both momenta upward by just a few MeV/c, for example, would bring the observed cross-section ratios in agreement with the theoretical ones. To accommodate the observed cross sections within our model, we must employ rather long-range form factors, even when the effects of K- distortion are included.

0. V. Maxwellj ~a#n-sauced deuteron disin~egras~n

473

Such long-range form factors are roughly consistent with those required to fit the rrd absorption cross section when vector meson contributions to the NA + NN transition potential are omitted 14,18). This appears to indicate that an explicit treatment of vector exchanges in the K-d system would have an effect similar to that in gd absorption, namely, to reduce the pseudoscalar contributions in a manner consistent with the use of shorter-ranged form factors. Unfortunately, as discussed in the introduction, such an explicit treatment of vector exchanges within the present model requires considerably more information concerning the KN resonances than is currently available. It is a pleasure to acknowledge stimulating and useful conversations Dillig, B.K. Jennings, and E. Veit.

with M.

Appendix A MATRIX

ELEMENTS

To derive expressions for the matrix elements of the reaction amplitude, eq. (2. l), it is convenient to couple the initial deuteron state,

(A.11 where Inn, 1 =O)

=~[lp(l)n(2))-ln(l)p(2))17

lwciMi=Ml)= c (1 ml 9 m2

6

41

MJXl,,

~f&n2G~

64.2) ,

(A.3)

with the incident kaon wave functions appearing in (2.1). The result, after expanding the latter in partial waves and employing the addition theorem of spherical harmonics, is

(A-4) with the quantization

axis parallel to q and

The final YN state consists of a plane wave multiplied by channel-dependent spin and isospin functions. Removing the cm. part, expanding in partial waves, and

474

0. V. Maxwell / Kavn-induced deutervn disintegration

coupling the spin and orbital angular momenta projection, J&f,, yields

to total angular momentum

and

in the isospin notation of sect. 2. Note that the first vector coupling coefficient here ensures that isospin is conserved. Having constructed the initial and final states, we can now write down the matrix elements of (2.1) in an explicit form. We just combine eqs. (A.4)-(A.6) with eq. (2.9) for the resonance creation vertices and the expressions of appendix B for the transition potentials. The result is

(A.7) with the radial and isospin functions given by ahl,l(qp, r) = I,J&r)Gpr)

(A.@

and ~~~,~*~(~(Y*~pK-)a!(Y*~Y~)cu(Nn~)-a(Y”-nK-)a(Y*-Yn)cu(Nprr), I? &yN,y*= ~(Y*-nK-)~(Y*-N~)ff(Yp~)-ff(Y*‘pK-)~(Y*’N~)ff(Yn~),

(A.9)

and all other quantities defined as in sect. 2 or appendix B. The angular momentum matrix elements here can be handled in a straightforward manner using standard techniques, as discussed, for example, in appendix A of ref. 16), while evaluation of the radial integrals must be accomplished numerically. Note, finally, that eq. (A.7) includes an extra factor of 2 to take account of the diagrams not shown in fig. 1 (see discussion in sect. 2). Appendix B TRANSITION POTENTIALS The transition potentials employed in eq. (2.1) are defined as the Fourier transforms of the Y*N+ YN transition amplitudes:

475

0. V. Maxwell / Kaon-induced dertteron disintegration

Assembling expressions

Vg from eqs. (2.26)-(2.31)

and performing

the integrals yields the

Q:,(r)=& a(Y*Yw)a(NNrr)Y;;(r), 03.2) where &W%?(o*. em

=

r

4

Y~bbld

J&g-

,

/&cw,>

~zlwCLM~)+f(~t *~*)YdCLd)~,

Jp

/&LwI~

~2lY2hd+&~* ’ 4 Yoh4~)~,

Jp =;+

$+

[or, SIP ~zlY&d

-+-$WI, fL4Y1bd1,

=I+

(B.3)

Jp =$- ,

with s1 denoting the transition spin operator, PM = (I&- !c$“~, and (MM, MB) = (m,, My) or (mk, I\lfN). The spin-operator functions are defined by ~[A,~]=(A*~)(~.~)-~(A,~), &A, B, C] = (A * i)(B* ;)(C* :) -@[A,

B, C] ,

l2[A,B,C]=(A.B)(C.t)+(A.C)(B.F)+(A.i)(B~C),

03.4)

and the radial Yukawa functions by YO(pr) = e- cLr, Y,(pr) =

(

l+:

)

Y&M) = 1+ 3+ i W

es*‘, 3 2 e-” (V) ) ’

15 15 Y&M) = 1+ L+/.&r (/.&#+m i ) e-“‘*

(B-5)

Note in eq. (2.31) of sect. 2 that k. exceeds m, when q&>

mk-t(My--&+2m.,,)*.

03.6)

p,, is then imaginary and the radial functions above complex, corresponding standing waves, rather than decaying exponentials.

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