IC/93/25 ,~ \ IIMTERNATIOIMAL CENTRE FOR THEORETICAL PHYSICS
Ac EXCLUSIVE WEAK DECAYS IN A LIGHT CONE MODEL
INTERNATIONAL Faheem Hussain ATOMIC ENERGY Dongsheng Liu AGENCY and Salam Tawfiq UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION Ml RAM ARE-TRIESTE
T' ,*-1 'Dt*r ••%»• IC/93/25
1. INTRODUCTION International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization Recently much attention has been focussed on the study of heavy flavour baryons, such as Ac and A6- Experimentally, besides the existing data for charmed baryons, events INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS providing evidence for the semileptonic decays of the bottom baryon to the charm baryons have been observed in Z decays by the ALEPH and OPAL collaborations at LEP [lj. We hope that more measurements of the bottom baryon will be available in the near future. On the other hand, a lot of theoretical investigation for heavy baryon [2-6] decays have
At -* Ac EXCLUSIVE WEAK DECAYS IN A LIGHT CONE MODEL been carried out in the framework of the Heavy Quark Effective Theory (HQET) [7-10]. The HQET has a particularly strong predictive power when applied to the decays of heavy Faheem Hussain, Dongsheng Liu A-like baryons. This is because a heavy AQ is particularly simple in its spin structure International Centre for Theoretical Physics, Trieste, Italy due to the fact that the light degrees of freedom are in a scalar (spin zero) state. At the heavy quark limit, the heavy quark spin decouples from the gluon field and one can and identify the heavy quark spin with the spin of the baryon. A well-known consequence of Salam Tawfiq the HQET, to lowest order, for baryon weak decays is that the number of independent Department of Theoretical Physics, University of Trieste, Trieste, Italy. form factors governing the transition between the heavy A-like baryons decreases from six to one [2,5,6] and that this single form factor is normalized to unity at zero recoil. Even when the corrections arising from the finite quark mass are taken into account at ABSTRACT O(^), there are still five relations among the six form factors, apart from a dimensional parameter [11], Furthermore, in the case that the heavy quark mass limit is taken only for the parent baryon, while the daughter baryon is considered as light, just two independent An analysis of the weak transition form factors of heavy A In Section II of this paper, we first present the formalism needed to evaluate form factors in terms of the tight cone wavefunction and then show that the heavy quark symmetry is explicit in this formalism in the limit of infinite quark mass. In Section III we evaluate this single form factor within a valence dominance assumption. The study of the decay properties in terms of this form factor is reported in Section IV. Section V contains our conclusions. (4) with J = V or A, and correspondingly h, — f, or g,. The plus (minus) sign is associated LIGHT CONE FORMALISM AND HEAVY QUARK SYMMETRY 2. with V{A). When a particular reference frame is specified, further simpler relations can be set up. For instance, if we choose the initial baryon moving along the z-axis, we find that We begin with the general form for the matrix elements of the vector and axial the third form factors are only connected with the matrix elements of the left components vector currents, V = *Q.7(J*Q and A* = *g'7(i75*Q, inducing transitions between the of the currents between opposite helicity states, i+ heavy baryons BQ and Bey. Here, Q and Q' are the initial and final active quarks. Following the usual practice in heavy quark physics, we write the matrix elements as ,JL, (5) 1 (BQ.;X',V'\V' \BQ;X,V) = ux.(v')[- + v"f2 + v'"f3]Ux(v) Another example is that if we work in a Breit-like frame in which (B ;X\V'\A»\BQ;X,V) = uy(v'){~(' v"g + v'"g }y .{v) QI 2 3 iU> (1) + v — (6) where {A,u} and (A',u') are the helicities and the velocities of the initial and final baryons, respectively. The baryon spinor is normalized such that ti\'U\ = <5vA- the first axial form factor is expressed by the matrix element of the plus component of the axial current, In the light-cone formalism [14,19], a vector is written in terms of its plus, minus and two transverse components which are related to the usual components through Here v±_ = (vR,vL). From now on we use the Brodsky-Lepage approach of quantizing QCD on the (2) light-cone in the light-cone gauge A+ — 0 [14]. The advantages of this approach are well- For on-shell particles one has p~~ == known and will not be repeated here. The interested reader is referred to Ref. [14]. In the light-cone QCD, a hadronic state is described as a sum over all Fock states. The heavy In the following analysis, we are going to link the form factors to certain matrix BQ baryon with momentum (P+,P±) and helicity A is described by the state elements which can be evaluated in the Hght-cone QCD. To this end, we present a number of useful algebraic relations (here F is either / (8) or 75): where _ , , lv+±v lv+v'L-v'+vL uT(i" )Tu^v) = *- VtJ + t! + t uT(7;')7 rui(ir) = 0 (3) where the positive sign is associated with T — I and the negative sign with F = 75. More + useful relations are presented in the Appendix. Here the Fock space state vector |rt;i,P ,XiPx + k±n\i) represents a parton configuration with the quantum numbers of the BQ. The fraction of the longitudinal Substituting these relations in Eq.(l) leads to + momentum the ith parton carries is denoted by i; = pf /P , and fcj_; is the transverse relative momentum of the ith parton with respect to the direction of motion of the baryon. (UJ+W- $n,k{xi, £±j>^i) specifies the parton wavefunction representing the probability amplitude 3 for a certain parton configuration. One of the advantages of this description is that the dT is obtained from the measure in the Fock states, Eq. (8), after imposing energy momen- inner motion of the Fock state, represented by the wave function, is decoupled from the tum conservation, q — P — P', at the baryonic level and also detailed energy-momentum hadronic momentum P, except through the mass-shell condition P2 = M2. As we shall balance at the parton level. The actual expressions for dT will depend on the choice of see later, the evaluation of form factors in the light cone QCD formalism is simpler than frame. This will be illustrated later. In Eq. (13), uxQ and u\Q, are the light-cone spinors in other covariant descriptions of hadronic states. for the active quarks involved in the weak decay. As discussed in Ref. [14], unrenormaiized QCD perturbation theory indicates For the good current, F+ is just the gamma matrix -y+ and the spinor algebra that hadronic wavefunctions do not fall off sufficiently quickly as k\ —> oo. The usual is trivial. In this case the r.h.s. of Eq.(13) is just a sum of overlap integrals that is way [14] to get around this difficulty is to introduce an ultra violet cut-off A which simply quite analogous to the nonrelativistic result (see Fig.la). However, for the transverse 2 truncates the Fock space, excluding all states with k^_, > A . This is implied in Eq.(8) components the TM in general are more complicated expressions of the momenta. Also and in the rest of this section. However, in the next section, when we explicitly evaluate the vertex no longer conserves particle number since there will now be terms involving the form factors we will not use QCD perturbation theory to calculate the wavefunctions. transitions Q 4- W -+ Q' + g and Q + g + W^>Q'as shown in Fig. lb. Instead we will use model wavefunctions which parametrize the transverse distribution such that they fall off rapidly with increasing k2^. We now discuss the consequences of the heavy quark limit for the Fock state wavefunctions. The heavy quark limit can be used to constrain these wavefunctions. We The hadronic state, Eq.(S), is normalized as shall consider baryons, like A{, and Ac> which contain only one heavy quark. Since the + 3 i creation of a heavy quark-antiquark pair is suppressed in the heavy quark limit, we shall (BQ.; P', X'\BQ; P, A} = 2P (2v) S (E - £ assume that there is only one heavy quark parton in each Fock state. which leads to the normalization for the parton wave functions In the rest of this paper we will be concerned with discussing the form factors in (10) the transition AQ —* AQ', where Ag has the quantum numbers Q [ud]. For such a heavy 16TT3 a,X J baryon the helicity of the heavy quark can be identified with the helicity of the baryon at the heavy quark limit, where the heavy quark moves with the same velocity as the baryon As far as the current operator responsible for weak decays is concerned, we refer and carries its spin. This is because in the AQ the light degrees of freedom are in a total to the plus component of the current operator as a good current, since it only contains the spin zero state. Therefore positive projections of the quark fields, namely that rl>nMo ? A) = 0 (14) upto order k^qjM or AQCD/M. where *(i) = A±* with A = 7°7±/2. The advantage of this property arises from the ± Let us now apply these properties to the form factors. Firstly, we consider the fact that, in the light cone quantization, the positive projected quark field can be expanded case that only the parent AQ is heavy while the daughter baryon may be either heavy or in terms of a set of creation and annihilation operators. The matrix elements of a good light. That is, at this stage, we impose constraints only on the parent baryon. In fact, this current turn out to have a simple form. This simplification does not occur in the transverse turns out to be sufficient to ignore some of the form factors. currents __ J± = *Q'1±T*Q = *<-"(l + ?g.)7j.r(l + Pg)*S,+) (12) Because of Eq.( 14), the sum over the helicities of the initial heavy quark is reduced (where PQ = jgjr[iD±_ • 3j_ +/3m]) which contain both the positive and negative projected to one term and Eq.(5) becomes fields (^Q = PQ^Q )• In general, the analysis for their matrix elements is more involved. (15) Substituting the heavy quark current (at the moment we restrict ourselves to the vector current) between the Fock state expansions, Eq. (8), for the baryonic states gives the weak decay matrix element which reads formally A detailed analysis in a reference frame in which the initial heavy baryon moves (13) along the z-axis shows that the coupling of a left handed current with a helicity down quark, TLui, does not depend on the longitudinal momentum of the quark and only contains small T components compared to the heavy quark mass Af, such as PQ±/M, p/M and qjM where independent form factors incorporate all the effects of the low-energy QCD interactions on PQL is the transverse momentum of the heavy quark, p is the momentum of a light quark the weak decay of heavy AQ baryons. One remark which should be stressed here is that parton and q is the gluon parton momentum. All of them are of the order of A.QCD/M we have set up four equations for the six form factors in the light-cone QCD at the limit and vanish as long as the decaying quark is infinitely heavy. Hence we get the immediate where the decaying quark has a mass much larger than the QCD-scale, which should be result a reasonable approximation scheme for At, as well as for Ac, decays. These results are in /3orff3 2££ ^ o when M -> co . (16) agreement with the heavy quark symmetry analysis of Refs. [2,5,6], M The results above, Eqs. (16) and (18), are valid for heavy to light decays. It is Recalling that these form factors are connected with the transverse current, we realize that easy now to proceed to the heavy to heavy transitions. Here we take the heavy quark limit form factors associated with good currents are sufficient to describe heavy baryon decays. in the final AQ> baryon as well. In this limit, the velocity of the daughter heavy quark is Hence, the non-trivial form factors are obtained from the matrix elements the same as the daughter baryon, and also the wavefunction «/v,t(^Q' =1) is zero*. + (T \J \ T) = Thus we are left with just one form factor (17) in the heavy quark limit for the initial h = 9i = h = n = (21) Again, using the heavy quark symmetry, Eq. (14), in the velocity Breit frame, Eq. (6), we find that Eqs. (17) give As one can see, this result, Eq. (21), for heavy to heavy A transitions is the same as the heavy quark symmetry analysis of Refs. [2,5,6]. M f\ +h = 9\ = 3. RELATIVISTIC CONSTITUENT QUARK MODEL The symmetry results of the last section are as far as one can go without knowing the detailed Fock state wavefunctions. In this section, we compute the single form factor In deriving relations (18), we have used the algebraic relations, Eq. (3). We also have the in heavy AQ to Agi decays based on the dominance of valence quarks in the Fock-state momentum conservations relations in the Breit frame (Eq.6): expansion. This approach is similar in spirit to the relativistic constituent quark model IQ.) = M(I-XQ) used by Dziembowski and co-workers [16,20] to successfully calculate the electromagnetic form factors of nucleons. -« - 2(1-X )MV (19) Q ± In a valence quark dominant approximation, the AQ wavefunction used in our for the active quarks, Q and Q', and present work has the form 'x'i = Mi, *,[(!+ ^TsCluI",* i = 1,2,3 (22) ku + 2x,Mvx (20) where the factor [(1+ /))fsC]®u\ is the spin projector for the AQ baryon, which guarantees for a!l other partons. the antisymmetry of the helicity amplitude under the interchange of the two light quarks in a scalar state [2,5,21]. We take the 3rd quark as the heavy, active quark. Dziembowski and We have, thus, found two relations among these form factors and, therefore, there remain only two independent form factors. Also, in the light-cone QCD, they are just a The equal velocity assumption for the final heavy quark, X'Q = -j^-, requires that M' — sum of overlap integrals over Fock states, very similar to the non-relativistic form. The two = M — rriQ. This is the assumption usually made in the heavy quark effective theory. Weber [22] have also shown that the helicity structure in Eq.(22) follows from solving the Now, for the longitudinal momentum distribution fli(i,-) and 2 2 It is well-known that such a wavefunction overlap calculation for the electromag- • v') = [1 + Cl(« •«'-!) + c2(v • v> - l) ]exp[-p (V • v' - 1)] (27) 2 2 netic form factors of the nucleon is restricted to low values of q , i.e. for large recoil, with p = ••Hj-, where m is the light quark mass. The coefficients Cj and c2 are given by since it fails to reproduce the correct power-law fall off at high q2 as expected by pertur- 1 1 - batjve QCD. However, in the case of heavy hadron decays, as Isgur has noted, [17], the momentum transfer of the light degrees of freedom is actually quite small. Most of the momentum transfer is due to the mass difference between the active heavy quarks. Hence, since the form factor in the heavy to heavy case, depends essentially on the overlap of the light quark distributions, we believe our calculations to be valid over the whole q2 range, The structure underlying this form factor is transparent. The exponential factor comes i.e. from high to low recoil. from the transverse Gaussian distribution and the polynomial is from the light quark helicity part. The exponential part of the form factor is the same as in the meson decay The choice of the distribution , Eqs. (23) is also dictated by the fact that the case since we use similar Gaussian distributions as in Eq. (23) [17,24] with the same final form of the form factor is well-behaved in the limit, M, M' —+ oo. On the other hand parameter p2 = ^. The slope of the form factor, ~d^jd{v • f'), at the zero recoil point, if one uses another distribution, for example, as in Ref. 20, which has the form v • v' = 1, is approximately p2 if p2 ~ 1. This exponential form for the meson form factor + has been used by the ARGUS [25] collaboration to fit the decays B° —> D* i~vt. They obtain p - 1.37 ± 0.19 ± 0.08 with |Vj | = 0.050 ± 0.008 ± 0.002. These are the values we exp - (24) c will use in our numerical calculations in Sec. IV. We also investigate the sensitivity of the form factors, as well as exclusive decay rates, by using various values of the p-parameter. then it leads to a form factor which blows up in the limit. With this single form factor available, we can also take the leading -^-y corrections Then, substituting Eq. (22), (23) in eq. (21) we find that the single form factor into account, using the expansion of the heavy quark effective Lagrangian [11]. Thus, ; = fi = gi) has the form including -~^ corrections, we havve A 2 (29) irriQ. v • v' + 1 3 TT A 2 h = 92 = - (30) (25) 2roQ. v • v' + 1 3 7T with where in the Breit-like frame we have lQg(u • V> + y/(u • V1)2 - 1) (31) 2 2 q = (M - M') - 4MM'vl = ^ax - (26) and A = M\Ql — i . Also g1 = /j + f2 = 9 10 4. At -» Ac DECAY PROPERTIES As one can see in Fig. 2, the leading corrections to the form factors due to finite c-quark mass are quite large near q^ax or zero recoil. However, the total decay rate is not With the form factors presented in Sec. Ill, we are able to calculate the decay sensitive to form factors in this region due to the dramatic phase space suppression. The 2 properties for the process At —> A.ct~i>t- The zeroth order form factor, £(q ), and the main contribution to the total decay rate comes from the large recoil area where the l/mc •£- corrected form factors, Eqs. (29) and (30), are plotted in Fig. 2 with p = 1.37. As corrections are relatively small. a comparison, the dipole form factor is also shown. We find that our zeroth order form 2 We hope that future experiments will provide more detailed measurements of life- factor £{q ) is close to the dipole form. In Fig. 3, we show the form factors /] and f 2 2 times and q and lepton energy distributions to enable one to distinguish between different varying with the parameter p. models for Aj decays. This will enable us to evaluate V(,c to a higher accuracy. The differential (^-distribution of the decay rate is plotted in Fig.4, for the mass- !ess lepton limit, using the •£- corrected form factors, f\, J2, Abdus Salam, the Inter- state A . The remaining 26% would then be filled by semiieptonic transitions to excited c national Atomic Energy Agency and UNESCO for hospitality at the International Centre charm baryon states. for Theoretical Physics, Trieste. As pointed out in Ref. [4], it is known that the total non-leptonic bottom baryon decay rate cannot be reliably estimated because of the presence of W exchange and Pauli interference contributions in addition to the non-leptonic free quark decay rate. One expects the total semi-leptonic branching ratio of bottom baryons to be roughly 5%. These estimates would then yield an exclusive Aj, —> \c (ground state) branching ratio of (3—4)%. There is a considerable theoretical uncertainty in the determination of \Vf^\ from fitting the £-»£»* decay on the basis of HQET [25]. The reason is that \Vbc\ using HQET is essentially determined by the data points with the largest q1 values, where the statistical precision is poor. Consequently there is a considerable theoretical uncertainty 2 in our predictions for the decays rates arising from the uncertainties in |Vic| and p . 5. CONCLUSION In conclusion, we would like to stress a number of points. We have demonstrated that, as expected, the light-cone approach reproduces the Heavy Quark Symmetry results in the analysis of the form factors in heavy A-like baryon decays . Further we have shown that the good current is sufficient for the evaluation of the independent form factors. This makes the calculation of these form factors significantly easier, at least formally. 11 12 Appendix A REFERENCES 1. ALEPH Collab., D. Decamp et al., Phys. Lett. B278 (1992) 209; In this appendix we present matrix elements of some useful -/-matrices between OPAL Collab., P.D. Acton et al., Phys. Lett. B281 (1992) 394; light-cone spinors. ALEPH Collab., D. Buskulic et al., Phys. Lett. B294 (1992) 145; We use the standard representation of the 7-matrices. ALEPH Collab., D. Buskulic et al., Phys. Lett. B297 (1992) 449. n_fl 0 1 ,_[ 0 a'] TO 1] 2. F. Hussain, J.G. Korner, M. Kramer and G. Thompson, Z. Phys C51 (1991) 321; 7 7 l (.41) - [0 -ij' ~ [-<7 0 J' [1 oj N. Isgur and M.B. Wise, Nucl. Phys. B348 (1991) 276; with The light-cone spinors it\(v) and the antispinors vx(v), with velocity uand helicity 3. F. Hussain and J.G. Korner, Z. Phys. C51 (1991) 607. A, (T,l), are defined by [14] 4. F. Hussain, J.G. Korner and R. Migneron, Phys. Lett. B248 (1990) 406 (Erra- 1 . tum Phys. Lett. B252 (1990) 723. and 5. F. Hussain, D.S. Liu, M. Kramer, J.G. Korner and S. Tawfiq, Nucl. Phys. B370 (1992) 259. v\(v) - +a_L-v1 - (A3) 6. T. Mannel, W. Roberts and Z. Ryzak, Phys. Lett. B248 (1990) 392; Nucl. Phys. where 0 — 70 and a; = flf' and B355 (1991) 38; Phys. Lett. B255 (1991) 593. m r 0 i 1 7. N. Isgur and M.B. Wise, Phys. Lett. B237 (1990) 527; 0 U4) 0 1 H. Georgi, Phys. Lett. B240 (1990) 447.; OJ L-U J.D. Bjorken, invited talk at Les Rencontres de la Valle d'Aosta, La Thuile, Aosta Valley, Italy March 1990, SLAC-PUB-5278. It is straightforward to show that — v S. E. Eichten and F. Feinberg, Phys. Rev. D23 (1981) 2724; 1 VR G.P. Lepage and B.A. Thacker, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 199; U\{v) - (AS) v+-l E. Eichten, ibid. 170; E. Eichten and F. Feinberg, Phys. Rev. Lett. 43 (1979) 1205; W.E. Caswell and G.E. Lepage, Phys. Lett.B167 (1986) 437; Similarly, the antispinors v\(v) are given by M.B. Voloshin and M.A. Shifman, Sov. J. Nucl. Phys. 47 (1988) 511. (AS) 9. F. Hussain, J.G. Korner, K. Schilcher, G. Thompson and Y.L. Wu, Phys. Lett. 249B (1990) 295. 10. J.G. Korner and G. Thompson, Phys. Lett. B264 (1991) 185. where vR'L = u1 ± iv2. 11. H. Georgi, B. Grinstein and M.B. Wise, Phys. Lett. B252 (1990) 456. The spinors are eigenstates"of /fc, i.e. /JU^ = "A and they are normalized such P. Cho and B. Grinstein, SSCL-Preprint-111 and HUTP-92/A012. that (Al) 12. T. Mannel, W. Roberts and Z. Ryzak, Phys. Lett. B255 (1991) 593. The matrix elements fiV^>r^i|^ for various choices of 7 are listed in Table 2. 13. J.G. Korner and M. Kramer, preprint DESY 91-123 and MZ-TH/91-05. Table 3 represents some other important bilinears which have been used in Sec. III. 14. G.P. Lepage and S.J. Brodsky, Phys. Rev. D22 (1980) 2157; Phys. Lett B87 13 14 (1979) 359; Phys. Rev. Lett. 43 (1979) 545; 1625(E); Review article in "Per- Table Captions turbative Quantum Chromodynamics", Ed. A.H. Mueller, World Scientific, Sin- gapore (1989), pp. 93-240; G.P, Lepage, S-J. Brodsky, Tao Huang and P.B. Mackenzie, published in the 10 1 Table 1 Partial helicity and total rates (in units of lO ^" ) for At —> Ac£ve. Mass values "Proceedings of the Banff Summer Institute" 1981. used are: MAi = 5.60 GeV, MXc = 2.28 GeV, mb = 4.73 GeV, mc = 1.55 GeV. 15. M. Wirbel, B. Stech and M. Bauer, Z. Phys. C29 (1985), 637. We take Vic = 0.050 and Sti^sl - o.U. 16. Z. Dziembowski, Phys. Rev. D37 (1988) 778; Table 2 matrix elements Z. Dziembowski, H.J. Weber, L. Mankiewicz and A. Szczepaniak, Phys. Rev. Table 3 Light cone bilinears used in Sec. III. Here a< = x + -^ with z' = 1,2 and 3 and D39 (19S9) 3257. t mi = TTI2 = m is the light quark mass, fcj^ represents the ith quark momenta 17. N. Isgur, Phys. Rev. D43 (1991) 810. transverse to the direction of the hadron. 18. M. Neubert and V. Rieckert, Nucl. Phys. B382 (1992) 97. 19. P.A.M. Dirac, Rev. Mod. Phys. 21 (1949) 392. 20. W. Konen and H.J. Weber, Phys. Rev. D41 (1990) 2201. 21. A. Salam, R. Delbourgo and J. Strathdee, Proc. Roy. Soc. A284 (1965) 146; S. Weinberg, Phys. Rev. 133B (1964) 1318; 139B (1965) 597; G. Feldman and P.T. Matthews, Ann. Phys. N.Y. 40 (1966) 19; F. Hussain, Ph.D. Thesis, Imperial College, London (1966); T. Gudehus, Phys. Rev. 184 (1969) 1788; F. Hussain, J.G. Korner and G. Thompson, Ann. Phys. (NY) 206 (1991) 334, 22. Z. Dziembowski and H.J. Weber, Phys. Rev. D37 (1988) 1289. 23. S. Weinberg, Phys. Rev. 150 (1966) 1313. 24. N. Isgur, D. Scora, B. Grinstein and M.B. Wise, Phys. Rev. D39 (1989) 799. 25. H. Albrecht, et al, The ARGUS collaboration, preprint DESY 92-146. 15 16 Table 1 Table 3 A2 v A V^S^«* 1.10 5.52 0.62 2.72 8.86 r T T + I 1.37 4.16 0.53 2.22 6.91 T i I 1.64 2.82 0.43 1.68 4.93 i T •i- dipole 4.2 0.70 2.0 6.90 i i i i i FQD 6.2 0.59 2.49 9.28 Table 2 T-T l-l 1-1 2 v+v "2 i!+v'+ 2 t^+f'- ± 1 17 18 Figure Captions Fig. 1 Diagrams for matrix elements a) only for good currents J+; b) additional terms contributing to y,^ ^ +. Fig. 2 The single zeroth order form factor £ and the form factors /t and f2 with the leading corrections for Af, —* Ac weak s.l. decays. The value of p is taken as 1.37. For comparison the dipole form factor is also plotted. Fig. 3 Form factors fi and f2 with different values of p. 2 Fig. 4 Differential ( -rates for the process At —» Ac££y with p = 1.37. Also shown are partial helicity rates. Fig. 5 Differential E^-spectra for Aj> -* \c(.vc. Aiso shown are partial helicity rates. Fig-l 19 20 1.5 1.5 h p=1.10 Ab~- Ao p=1.6p=1.374 *r 0.5 -O.i 4 8 12 q2 [GeV] Fig.2 Fig.3 21 22 T 1 O tS I O V <0 in X) T3 •a Fig. Fig. 5 23