PHYSICAL REVIEW D VOLUME 53, NUMBER 9 1 MAY 1996
Excited charmed baryon decays and their implications for fragmentation parameters
John K. Elwood California Institute of Technology, Pasadena, California 91125 ͑Received 9 November 1995͒
The production of the excited charmed baryon doublet ⌳*c via fragmentation is studied. An analysis of the subsequent hadronic decays of the doublet within the framework of heavy hadron chiral perturbation theory produces expressions for both the angular distribution of the decay products and the polarization of the final state heavy baryon in terms of various nonperturbative fragmentation parameters. Future experimental inves- tigation of this system will determine these parameters. In addition, recent experimental results are shown to fix one of the parameters in the heavy hadron chiral Lagrangian. ͓S0556-2821͑96͒02409-5͔
PACS number͑s͒: 13.87.Fh, 12.39.Fe, 12.39.Hg, 13.30.Eg
I. INTRODUCTION tude of the projection of j onto the axis of fragmentation, and not on its sign. That is, transverse may be preferred to lon- The production of a heavy quark at high energy via some gitudinal, but forward may not be preferred to back. Further, hard process is a relatively well understood phenomenon, as the light system may prefer to invest its angular momentum we may bring the full apparatus of perturbative QCD to bear in orbital channels as opposed to spin channels. These pref- on the problem. Less well understood is the subsequent frag- erences are catalogued by a set of fragmentation parameters: mentation of the heavy quark to form heavy mesons and A and 1 , defined in ͓1͔, and B and ˜ 1 , defined in the baryons. It is the dynamics of this process that we propose to following section. address in this paper. We imagine that a heavy quark with Let us consider a fragmentation process in which light mass mQӷ⌳QCD is produced on very short time scales in a degrees of freedom of spin j are produced. They then asso- 1 hard reaction. It then travels out along the axis of fragmen- ciate with the heavy quark spin sϭ 2 to form a doublet of 1 tation and hadronizes on a much longer time scale, at dis- total spin Jϭ jϮ 2. Two paths now lie open. The doublet ͑the tances of order 1/⌳QCD . The fractional change in the heavy two members of which have the same decay rate in the heavy quark’s velocity is therefore of order (⌳QCD /mQ), and van- quark limit͒ may decay rapidly enough that heavy quark spin ishes at leading order in the heavy quark limit. Likewise, the flip processes have no time to occur. Then the doublet states heavy quark spin couples to the light degrees of freedom via decay coherently, the heavy quark retains its initial polariza- the color magnetic moment operator tion in the final states, and the process begins anew with the decay products. On the other hand, heavy quark spin flip 1 ¯͑Q͒ a a ͑Q͒ processes may have time to occur, in which case the doublet hv G T hv , ͑1.1͒ mQ states decay incoherently, and the heavy quark polarization is altered. The two parameters responsible for determining which again vanishes in the heavy limit. We may therefore which regime we are in are the total decay rate out of the view the initial fragmentation process as leaving the heavy doublet, ⌫, and the mass splitting between the doublet states, quark velocity and spin unchanged. Notice that, in this limit, ⌬. The splitting ⌬ vanishes in the heavy quark limit, and is the dynamics are also blind to the mass of the heavy quark, of the order of the rate for heavy quark spin flip processes which therefore acts as a static color source in its interactions within the doublet. We therefore expect that the situation with the light degrees of freedom. ⌫ӷ⌬ produces overlapping resonances which decay coher- This simple result may not apply to the ultimate products ently out of the multiplet, and that the opposite extreme of the strong fragmentation process, however, as was pointed ⌫Ӷ⌬ allows for incoherent decays and the influence of the out by Falk and Peskin ͓1͔. Specifically, the polarization of spin of the light degrees of freedom. the final state heavy baryons and mesons may not be deter- mined solely by the heavy quark spin, but may depend in addition on the spin of the light degrees of freedom involved II. THE CHARMED BARYON SYSTEM in the fragmentation process. This is the case when the initial In the charmed baryon system, the ground state is ob- fragmentation products decay to lower energy heavy baryons tained by putting the light diquark in an antisymmetric and mesons on a time scale long enough to allow interaction IϭSϭ0 state with spin-parity j Pϭ0ϩ. This yields the J P between the heavy quark spin and that of the light degrees of 1 ϭ ϩ baryon ⌳ϩ , with mass 2285 MeV. Alternatively, the freedom. We will find that this is indeed the case in the 2 c light quarks may form a symmetric IϭSϭ1 state with spin- ⌳* system. c parity j Pϭ1ϩ. The light spin then couples to that of the In this situation, one must know something about the spin P 3 ϩ 1 ϩ of the light degrees of freedom in order to proceed further. heavy quark to produce the symmetric J ϭ( 2 , 2 ) doublet (0,ϩ,ϩϩ) (0,ϩ,ϩϩ) The parity invariance of the strong interactions, coupled with (⌺c* ,⌺c ) with mass ͑2530 MeV, 2453 MeV͒. ( ) heavy quark spin symmetry, demands that formation of light Fragmentation through the ⌺c* system has already been degrees of freedom with spin j depends only on the magni- considered in ͓1͔; we concern ourselves here with the J P
0556-2821/96/53͑9͒/4866͑9͒/$10.0053 4866 © 1996 The American Physical Society 53 EXCITED CHARMED BARYON DECAYS AND THEIR . . . 4867
3 Ϫ 1 Ϫ also note that the rates above place us securely in the regime ϭ( 2 , 2 ) doublet (⌳c*1 ,⌳c1) that results when the light diquark is an IϭSϭ0 state with a single unit of orbital an- ⌫Ӷ⌬, so that we anticipate interaction of the heavy quark gular momentum. Allowing the light quarks to have both spin with the light degrees of freedom in decays to the ⌳c . spin and orbital angular momentum produces a tremendous This will allow us to shed some light on the parameter ˜ 1 . number of states, none of which have been observed to date. In the following section, we provide a brief review of heavy We ignore such states in the analysis that follows. hadron chiral perturbation theory before tackling the (⌳* ,⌳ ) decays. The fragmentation parameters A,B,1 , and ˜ 1 , may now c1 c1 be defined. A is taken to be the relative probability of pro- ducing any of the nine IϭSϭ1, j Pϭ1ϩ diquark states dur- III. HEAVY HADRON CHIRAL PERTURBATION ing fragmentation relative to that of producing the IϭSϭ0, THEORY j Pϭ0ϩ ground state. B is similarly the probability for pro- Heavy hadron chiral perturbation theory incorporates as- ducing any of the three IϭSϭ0, j Pϭ1Ϫ diquark states rela- pects of both ordinary chiral perturbation theory and the tive to ground state production. The parameters and 1 heavy quark effective theory, and describes the low energy ˜ , on the other hand, encode the orientation of the light 1 interactions between hadrons containing a heavy quark and diquark angular momentum. The various helicity states of the light pseudo Goldstone bosons. It has been discussed the spin-parity 1ϩ and 1Ϫ diquarks are populated with the previously in a number of papers ͓3͔. probabilities For definiteness we consider the charmed baryon system. P 1 ϩ Members of the ground state J ϭ 2 antitriplet are de- 1 P͓1͔ϭP͓Ϫ1͔ϭ , P͓0͔ϭ1Ϫ for j Pϭ1ϩ, stroyed by the velocity dependent Dirac fields T (v), where 2 1 i ͑2.1͒ 0 ϩ ϩ T 1ϭ⌶c , T 2ϭϪ⌶c , T 3ϭ⌳c . ͑3.1͒ and P 1 ϩ The symmetric J ϭ 2 states are destroyed by the Dirac fields Sij(v) with components ˜ 1 P Ϫ P͓1͔ϭP͓Ϫ1͔ϭ , P͓0͔ϭ1Ϫ˜ 1 for j ϭ1 . 2 S11ϭ⌺ϩϩ , S12ϭͱ 1 ⌺ϩ , S22ϭ⌺0 , S13ϭͱ 1 ⌶ϩЈ , ͑2.2͒ c 2 c c 2 c
23 1 0Ј 33 0 The analysis of the excited D system in ͓1͔ has already in- S ϭͱ 2 ⌶c , S ϭ⍀c , ͑3.2͒ dicated that 3/2 , the analog of 1 for the light degrees of P 3 ϩ freedom in the meson sector, is likely close to zero. One and their symmetric J ϭ 2 counterparts by the correspond- *ij might also anticipate, therefore, that 1 would be close to ing Rarita-Schwinger fields S (v). Finally, we define zero. We will concentrate on ˜ 1 most heavily in what fol- Dirac and Rarita-Schwinger fields Ri(v) and R*i(v) to an- P 1 Ϫ P 3 Ϫ lows. nihilate the J ϭ 2 and J ϭ 2 excited antitriplet states re- The masses of the ⌳c*1 and ⌳c1 are naively expected to be spectively. In our analysis the components of interest will be 2 * * split by ϳ(⌳ QCD/mc)Ӎ30 MeV, in fortuitously close agree- R3ϭ⌳c1 and R3ϭ⌳c1, . 3 ment with the recently measured values M ⌳* ϭ2625 MeV As the heavy quark mass goes to infinity, the Jϭ 2 and c1 Jϭ 1 members of the sextet and excited antitriplet multiplets and M ϭ2593 MeV ͓2͔. Decay of the ⌳* to ⌳ via pion 2 ⌳c1 c1 c1 become degenerate. It is then useful to combine them to form emission is thus kinematically forbidden, and the corre- ij the superfields R and S , defined by sponding electromagnetic transition is very slow compared i with strong decays out of the doublet. Indeed, the dominant 1 5 Riϭͱ 3 ͑␥ϩv͒␥ RiϩR*i , ͑3.3͒ decay mode of both ⌳c*1 and ⌳c1 is to ⌳c via pion emission. As both (⌳* ,⌳c1) and ⌳c are Iϭ0 states, single pion emis- c1 ij 1 5 ij ij sion is forbidden by isospin conservation, and the dominant S ϭͱ3 ͑␥ϩv͒␥ S ϩS* . ͑3.4͒ modes are ⌳c*1 ⌳c and ⌳c1 ⌳c. The mass differ- → → If we are to discuss decay by emission, we must also ences (M ⌳* ϪM ⌳ )ϭ340 MeV and (M ⌳ ϪM ⌳ )ϭ308 c1 c c1 c incorporate the pseudo-Goldstone boson octet into our La- MeV are very close to threshold, and the pions produced will grangian. The Goldstone bosons are a product of the sponta- be soft. We therefore expect the decays to be accurately de- neous breakdown of the chiral flavor symmetry SU͑3͒ ϫ scribed by heavy hadron chiral perturbation theory. L SU͑3͒ R to SU͑3͒ V , its diagonal subgroup. They appear in The CLEO Collaboration recently measured the ⌳c1 the octet width to be ⌫ ϭ3.9ϩ1.4ϩ2.0 MeV, and placed a new upper ⌳c1 Ϫ1.2Ϫ1.0 bound on the ⌳c*1 width: ⌫⌳* Ͻ1.9 MeV ͓2͔. It is an inter- c1 Mϭ͚ aTa esting breakdown of the naive heavy quark approximation a that these rates are significantly different. The explanation is 0 ϩ ϩ that, at leading order in the heavy hadron chiral Lagrangian, /ͱ2ϩ/ͱ6 K ⌳* is connected to ⌳ only via an intermediate ⌺* , 1 Ϫ 0 0 c1 c c ϭͱ 2 Ϫ /ͱ2ϩ/ͱ6 K , whereas ⌳ is connected via an intermediate ⌺ . Kinemat- c1 c ͩ KϪ K¯0 Ϫ2/ͱ6 ͪ ics allows the ⌺c , but not the ⌺c* , to go on shell. The ⌳c1 thus enjoys a resonant amplification of its decay rate. We ͑3.5͒ 4868 JOHN K. ELWOOD 53
and are conveniently incorporated into the Lagrangian via sion, we demand that our Lagrangian be invariant under such the dimensionless fields ⌺ϵe2iM/f and ϵeiM/f, where a transformation. The corresponding shifts induced in the f ϭ f ϭ93 MeV, the pion decay constant, at lowest order in fields are ͓4͔ chiral perturbation theory. The goal is to combine these fields to produce a Lorentz ⑀” ⑀ R invariant, parity even, heavy quark spin symmetric, and light ␦R ϭ R Ϫ , ͑3.18͒ 2M M chiral invariant Lagrangian. To this end, we now assemble various transformation properties of the fields. Under parity, P, the superfields transform as ⑀” ⑀ S ␦S ϭ S Ϫ , ͑3.19͒ 2M M Ϫ1 PR͑rជ,t͒P ϭ␥0R ͑Ϫrជ,t͒, ͑3.6͒
ជ Ϫ1 ជ ⑀” PS ͑r,t͒P ϭϪ␥0S ͑Ϫr,t͒, ͑3.7͒ ␦T ϭ T . ͑3.20͒ 2M Ϫ1 PT ͑rជ,t͒P ϭ␥0T ͑Ϫrជ,t͒. ͑3.8͒ Invariance of the Lagrangian under these shifts further They also obey the constraints restricts the terms that may appear, and leaves us with the following form for the most general Lorentz invariant, parity v Rϭv S ϭ0; v”RϭR ; even, heavy quark spin symmetric, and light chiral invariant Lagrangian: v”S ϭS ; v”T ϭT . ͑3.9͒ ͑0͒ i The Rarita-Schwinger components obey the additional con- Lv ϭ͕R͑Ϫiv•Dϩ⌬M R͒Ri straints ij i ϩS ij͑Ϫi•Dϩ⌬MS ͒S ϩT i•DT i ␥R* ϭ␥S *ijϭ0. ͑3.10͒ i i jk ϩig1⑀S ik ͑A ͒j͑S ͒ We are also interested in how the various fields transform i i under chiral SU͑3͒. The ⌺ and fields obey ϩig2⑀R ͑A ͒j͑R ͒j i j kl ijk l ⌺ L⌺R†, ͑3.11͒ ϩh1͓⑀ijkT ͑A ͒iS ϩ⑀ S kl͑A͒jT i͔ → ϩh ͓⑀ Ri AjS klϩ⑀ijkS AlR ͔͖, ͑3.21͒ LU†͑x͒ϭU͑x͒R†, ͑3.12͒ 2 ijk • l kl • j i →
where L and R are global SU͑3͒ matrices, and U(x)isa where ⌬M RϭM RϪM T is the mass splitting between the local member of SU͑3͒ V . If we further define the vector and excited and ground state antitriplets, and ⌬M S ϭM S ϪM T axial vector fields is the corresponding splitting between the sextet and the ground state antitriplet. 1 † † - ͔, ͑3.13͒ In defining the velocity dependent heavy fields which ap ץϩ ץ V ϭ 2 ͓ pear above, a common mass must be scaled out of all heavy i † † ͔, ͑3.14͒ fields ץϪ ץ A ϭ 2 ͓ we find that, under chiral SU͑3͒, ϪiM•x Hϭe H , ͑3.22͒ U†͒, ͑3.15͒ץV UVU†ϩU͑ → despite the different masses of the various heavy baryons. In A UAU†. ͑3.16͒ the above analysis we have chosen MϭM . → ⌳c It is also instructive at this point to examine the term The only constraint imposed on the heavy fields is that proportional to h , which allows single transitions be- they transform according to the appropriate sextet or antitrip- 2 tween the excited antitriplet and sextet states. This term in- let representation under transformations of the SU͑3͒ sub- V duces only S-wave transitions, although naive angular mo- group. mentum and parity arguments would allow D-wave There remains one final symmetry to aid us in construct- transitions as well. The D-wave transitions are induced by a ing our Lagrangian, and that is symmetry under reparametri- higher dimension operator which is therefore suppressed by zation of the heavy field velocity. The momentum of a heavy further powers of M and does not appear at leading order in hadron is written pϭMϩk, where k is termed the residual the heavy hadron Lagrangian. This absence of D-wave tran- momentum of the hadron. If we make the following shifts in sitions simplifies the way in which the distributions de- and k: ( ) pend on ˜ 1 in the ⌳c*1 decay process. Finally, we comment ϩ⑀/M; k kϪ⑀, ͑3.17͒ quickly on the errors induced by keeping only leading order → → terms. The relevant expansion parameter in our analyses is 2 2 2 with •⑀ϭ0, then p p and ϩO(1/M ). Therefore, (p /M), so that we expect our results to be valid to if we are working only→ to leading→ order in the ͑1/M͒ expan- ϳ(200/2285)Ӎ10%. 53 EXCITED CHARMED BARYON DECAYS AND THEIR . . . 4869
IV. THE PARAMETER h2
The term proportional to h2 in the leading order Lagrang- ian is responsible for the tree-level decay ⌳c1 ⌺c, the rate for which is easily calculated to be →
2 M ͉h2͉ ⌺c ⌫͑⌳ ⌺ ͒ϭ ͑M ϪM ͒2 c1 c 4 f 2 M ⌳c1 ⌺c → ⌳c1
ϫ ͑M ϪM ͒2Ϫm2 , ͑4.1͒ ͱ ⌳c1 ⌺c as was done previously in ͓4͔. The ⌺c may then decay to ⌳c through the term proportional to h1 , producing a decay rate ⌫(⌳c1 ⌳c) that scales like the combination 2 2 → 2 ͉h1͉ ͉h2͉ . A quick calculation allows us to express ͉h1͉ in terms of the partial width ⌫(⌺ ⌳ ), c→ c 2 M ͉h1͉ ⌳c ⌫͑⌺ ⌳ ͒ϭ ͓͑M ϪM ͒2Ϫm2 ͔3/2, ͑4.2͒ c c 12 f 2 M ⌺c ⌳c → ⌺c
( ) which is by far the dominant contribution to ⌫ . We may FIG. 1. Feynman diagrams contributing to ⌳c*1 ⌳c1 at ⌺c → therefore view ⌫(⌳ ⌳ ) as a function of h and leading order in the heavy hadron chiral Lagrangian. c1→ c 2 ⌫⌺ . This decay is dominated by the pole region where ⌺c is c We now wish to calculate the double-pion distributions in close to being on shell, and its rate coincides with that for the decays of these states to the ground state ⌳ . The differ- ⌳ ⌺ as ⌫ 0. In this narrow width approximation, c c1→ c ⌺c→ ential decay rate may be written we obtain 2 ϩ Ϫ 2 d⌫ ͉M fi͉ ⌫͑⌳c1 ⌳c ͒ϭ4.6͉h2͉ MeV. ͑4.3͒ 2 2 2 2 → ϭ 5 ͱ͑E1Ϫm͒͑E2Ϫm͒ d⍀1d⍀2 8M ͑ ͒M ͑2͒ ⌳c*1 ⌳c The result is modified slightly if we allow the ⌺c to have a finite width. The ⌺c is not expected to have a width greater ϫ␦͑M ͑ ͒ϪE1ϪE2ϪM ͒dE1dE2 , ͑5.3͒ ⌳c*1 ⌳c than a few MeV. Setting ⌫ ϭ2 MeV, we find ⌺c
ϩ Ϫ 2 where ⍀1 and ⍀2 contain the angular variables for the two ⌫͑⌳c1 ⌳c ͒ϭ4.2͉h2͉ MeV. ͑4.4͒ → pions and E1 and E2 are their energies. A glance at the expression above indicates that we are conserving three mo- Comparison with the CLEO measurement ͓2͔ mentum, but not energy. The explanation is simply that, in ⌫͑⌳ ⌳ ϩϪ͒ϭ3.9ϩ1.4ϩ2.0 MeV ͑4.5͒ the infinite mass limit, the charm baryon recoils to conserve c1→ c Ϫ1.2Ϫ1.0 momentum, but carries off a negligible amount of energy in then yields a central value of ͉h ͉Ӎ0.9 in the narrow width the process. 2 * approximation, or ͉h ͉Ӎ1.0 with ⌫ ϭ2 MeV. Let us first address the case of ⌳c1 and ⌳c1 decay to 2 ⌺c 0 0 ⌳c . The relevant Feynman diagrams which arise from the Lagrangian ͑3.21͒ are shown in Fig. 1. In calculating the V. PRODUCTION AND DECAY OF ⌳c1 AND ⌳c*1 decays between ⌳c*1 and ⌳c1 states of definite helicity, we find two distinct angular patterns, depending only on the The probabilities for fragmentation to the ⌳c1 and ⌳c*1 states of various helicities may be expressed in terms of the change in the component of spin along the fragmentation axis, ⌬S , between the initial and final state heavy hadrons: parameters ˜ 1 and B once the initial polarization of the z heavy quark is given. For simplicity, we assume that the initial charm quark is completely left-hand polarized in the 3 F ͑⍀ ,⍀ ͒ϭ ͓cos2 ϩcos2 ϩ␣ cos cos ͔, analysis that follows. With this assumption, the relative 1 1 2 322 1 2 1 2 populations of the ⌳c*1 and ⌳c1 states are ͑5.4͒
B ˜ 1 2 ˜ 1 P͓⌳* ͔ϭ , ͑1Ϫ˜ ͒, ,0 , ͑5.1͒ 3 c1 1ϩAϩB 2 3 1 6 F ͑⍀ ,⍀ ͒ϭ ͓sin2 ϩsin2 ͫ ͬ 2 1 2 642 1 2
B 1 1 ϩ␣ sin1sin2cos͑2Ϫ1͔͒, ͑5.5͒ P͓⌳ ͔ϭ ͑1Ϫ˜ ͒, ˜ , ͑5.2͒ c1 1ϩAϩB ͫ3 1 3 1ͬ where 1 and 2 are the angles between the two pion mo- 3 1 1 3 where the helicity states for ⌳c*1 read Ϫ 2,Ϫ 2, 2, 2 from left to menta and the fragmentation axis, and 1 and 2 are the 1 1 right, and those for ⌳c1 read Ϫ 2, 2. azimuthal angles of the pion momenta about this axis. These 4870 JOHN K. ELWOOD 53
FIG. 2. The variation of the coefficient ␣ as a
function of the width of ⌺*c .
( ) angles are defined in the rest frame of the decaying ⌳c*1 . The number ␣ arises from interference between the two graphs depicted in Fig. 1, and is defined in ͑5.6͒ below. Its dependence on the width ⌫ is plotted in Fig. 2. To the order we are ⌺c* working, ␣ϭ1.3 for any reasonable value of ⌫ : ⌺c*
␣ϵ␣1 /␣2 ;
M ϪM ⌳c*1 ⌳c ␣1ϭ dE1 dE2␦͑M⌳* ϪM⌳ ϪE1ϪE2͒ ͵ ͵ c1 c m 2 2 2 2 2 2E1E2͑E Ϫm ͒͑E Ϫm ͓͒͑M ϪM ϪE1͒͑M ϪM ϪE2͒ϩ͑⌫ /2͒ ͔ 1 2 ⌺c* ⌳c ⌺c* ⌳c ⌺c* ϫ ; ͓͑M ϪM ϪE1͒͑M ϪM ϪE ͒ϩ͑⌫ /2͒2͔2ϩ͑⌫ /2͒2͑E ϪE ͒2 ͩ ⌺* ⌳c ⌺* ⌳c 2 ⌺* ͚* 1 2 ͪ c c c c
M ϪM 2 2Ϫ 2 3/2 2Ϫ 2 1/2 ⌳c*1 ⌳c E1͑E2 m͒ ͑E1 m͒ ␣2ϭ dE1 dE2␦͑M⌳* ϪM⌳ ϪE1ϪE2͒ 2 2 . ͑5.6͒ ͵ ͵ c1 c Ϫ Ϫ ϩ m ͑M M E2͒ ͑⌫ /2͒ ͩ ⌺c* ⌳c ⌺c* ͪ
The normalized differential rates (1/⌫)(d⌫/d⍀1d⍀2) for the various decays are then given in terms of F1 and F2 by 1 d⌫ 1 1 1 1 ⌳* ϩ ⌳ ϩ , ⌳* Ϫ ⌳ Ϫ ϭF ͑⍀ ,⍀ ͒, ͑5.7͒ ⌫ d⍀ d⍀ ͫ c1ͩ 2ͪ → cͩ 2ͪͬ ͫ c1ͩ 2ͪ → cͩ 2ͪͬ 1 1 2 1 2 ͭ ͮ
1 d⌫ 3 1 1 1 1 ⌳* ϩ ⌳ ϩ , ⌳* ϩ ⌳ Ϫ , ⌳* Ϫ ⌫ d⍀ d⍀ ͫ c1ͩ 2ͪ → cͩ 2ͪͬ ͫ c1ͩ 2ͪ → cͩ 2ͪͬ ͫ c1ͩ 2ͪ 1 2 ͭ 1 3 1 ⌳ ϩ , ⌳* Ϫ ⌳ Ϫ ϭF ͑⍀ ,⍀ ͒. ͑5.8͒ → cͩ 2ͪͬ ͫ c1ͩ 2ͪ → cͩ 2ͪͬͮ 2 1 2
The decays ⌳* (Ϯ 3) ⌳ (ϯ 1 ) are forbidden. A similar calculation for ⌳ decays yields c1 2 → c 2 c1 1 d⌫ 1 1 1 1 ⌳ ϩ ⌳ ϩ , ⌳ Ϫ ⌳ Ϫ ϭG ͑⍀ ,⍀ ͒, ͑5.9͒ ⌫ d⍀ d⍀ ͫ c1ͩ 2ͪ → cͩ 2ͪͬ ͫ c1ͩ 2ͪ → cͩ 2ͪͬ 1 1 2 1 2 ͭ ͮ
1 d⌫ 1 1 1 1 ⌳ ϩ ⌳ Ϫ , ⌳ Ϫ ⌳ ϩ ϭG ͑⍀ ,⍀ ͒, ͑5.10͒ ⌫ d⍀ d⍀ ͫ c1ͩ 2ͪ → cͩ 2ͪͬ ͫ c1ͩ 2ͪ → cͩ 2ͪͬ 2 1 2 1 2 ͭ ͮ where
3 G ϭ ͓cos2 ϩcos2 ϩ cos cos ͔, ͑5.11͒ 1 322 1 2 1 2
3 G ϭ ͓sin2 ϩsin2 ϩ sin sin cos͑ Ϫ ͔͒. ͑5.12͒ 2 642 1 2 1 2 2 1
The ratio  is defined analogously to ␣ in ͑5.6͒, but with the substitutions M ⌳* M ⌳ , M ⌺* M ⌺ , and ⌫⌺* ⌫⌺ , that is, c1→ c1 c → c c → c by removing all stars in ͑5.6͒. Its dependence on ⌫ is shown in Fig. 3. That  is much smaller than ␣ is easily understood. ⌺c Both ␣ and  arise from the interference between Feynman graphs, but in the case of ⌳c1 decay, the intermediate ⌺c may go 53 EXCITED CHARMED BARYON DECAYS AND THEIR . . . 4871
FIG. 3. The variation of the coefficient  as a function of the width of ⌺c .
on shell, and in fact, the rate is dominated by this region of phase space. The ⌳c1 decay is thus essentially a two-step process, and interference effects are therefore relatively unimportant. The steep dependence of  on the intermediate state width does not significantly limit our predictions since it is numerically small. We now take into account the initial populations of the various helicity states, as displayed in ͑5.1͒ and ͑5.2͒, and allow them to decay incoherently in light of the relation ⌫ ( )Ӷ(M ϪM ). This produces, after summing final state helicities, ⌳c*1 ⌳c*1 ⌳c1 the following double pion distributions for decay through ⌳c*1 and ⌳c1 states separately:
1 d⌫͑⌳c*1 only͒ 3 1 1 2␣ ␣ ϭ ϩ ͑cos2 ϩcos2 ͒ϩ cos cos ϩ ͱ͑1Ϫcos2 ͒͑1Ϫcos2 ͒cos͑ Ϫ ͒ ⌫ d⍀ d⍀ 322 ͫ3 2 1 2 3 1 2 6 1 2 2 1 ͬ 1 2 ͭ 1 3 ␣ ␣ ϩ˜ Ϫ ͑cos2 ϩcos2 ͒Ϫ cos cos ϩ ͱ͑1Ϫcos2 ͒͑1Ϫcos2 ͒cos͑ Ϫ ͒ , ͑5.13͒ 1ͫ2 4 1 2 2 1 2 4 1 2 2 1 ͬͮ
1 d only 1 ⌫͑⌳c1 ͒ 2 2 ϭ 2 ͕2ϩ͓ͱ͑1Ϫcos 1͒͑1Ϫcos 2͒cos͑2Ϫ1͒ϩcos1cos2͔͖. ͑5.14͒ ⌫ d⍀1d⍀2 32
Combining both ⌳c*1 and ⌳c1 decays incoherently yields 1 d⌫͑combined͒ 1 4 4␣  ␣  ϭ ϩcos2 ϩcos2 ϩ ϩ cos cos ϩ ϩ ͱ͑1Ϫcos2 ͒͑1Ϫcos2 ͒cos͑ Ϫ ͒ ⌫ d⍀ d⍀ 322 ͫ3 1 2 ͩ 3 3 ͪ 1 2 ͩ 3 3 ͪ 1 2 2 1 ͬ 1 2 ͭ 3 ␣ ϩ˜ 1Ϫ ͑cos2 ϩcos2 ͒Ϫ␣ cos cos ϩ ͱ͑1Ϫcos2 ͒͑1Ϫcos2 ͒cos͑ Ϫ ͒ . ͑5.15͒ 1ͫ 2 1 2 1 2 2 1 2 2 1 ͬͮ Note from Fig. 3 that  approaches zero as the width ⌫ vanishes. This means that the double pion distribution ͑5.14͒ ⌺c resulting from ⌳ decay becomes isotropic in this limit. This is easily understood as follows. As ⌫ approaches zero, c1 ⌺c ⌳c1 decay is entirely dominated by production of a real intermediate ⌺c as discussed above, a process which may occur only via S-wave pion emission. The subsequent single pion decay of the ⌺c is also isotropic if ⌳c helicities are summed over, as previously observed in ͓1͔. Integration of the combined distribution over azimuthal angles produces
1 d⌫͑combined͒ 1 4 4␣  3 ϭ ϩcos2 ϩcos2 ϩ ϩ cos cos ϩ˜ 1Ϫ ͑cos2 ϩcos2 ͒Ϫ␣ cos cos , ⌫ d cos d cos 8 ͫ3 1 2 ͩ 3 3 ͪ 1 2ͬ 1ͫ 2 1 2 1 2ͬ 1 2 ͭ ͮ ͑5.16͒
which is plotted for a variety of ˜ 1 values in Figs. 4–6. for fragmentation through ⌳c*1 alone, Alternatively, we may prefer to integrate over pion angles and observe instead the polarization of the final ⌳c . We then find the population ratios 1 ⌳c͑ϩ 2 ͒ 2Ϫ˜ 1 ϭ , ͑5.18͒ 1 ⌳c͑Ϫ 2 ͒ 1ϩ˜ 1 1 ⌳c͑ϩ 2 ͒ 2Ϫ˜ 1 ϭ , ͑5.17͒ 1 ⌳c͑Ϫ 2 ͒ 4ϩ˜ 1 for fragmentation through ⌳c1 alone, and 4872 JOHN K. ELWOOD 53
FIG. 4. Normalized differential decay rate for the case ␣ϭ1.3, ϭ0.08, and ˜ 1ϭ0.
1 For the case of a completely left-handed initial heavy quark, ⌳c͑ϩ 2 ͒ 4Ϫ˜ 1 ϭ , ͑5.19͒ we find 1 5ϩ2˜ ⌳c͑Ϫ 2 ͒ 1
A͑1ϩ41͒ϩB͑1ϩ4˜ 1͒ϩ9 for the incoherent combination of the two. To be consistent, P ϭ . ͑5.22͒ 9͑AϩBϩ1͒ however, we must include also the effects of initial fragmen- tation to (⌺* ,⌺c) and ⌳c . This analysis was already carried c 1 out in ͓1͔, and including such effects leaves us with This function may never fall below 9, so that the initial po- larization information may never be entirely obliterated by
1 the fragmentation process. As a first guess as to what polar- ⌳c͑ϩ 2 ͒ 2A͑2Ϫ1͒ϩ2B͑2Ϫ˜ 1͒ ϭ . ͑5.20͒ ization we may actually expect to measure, we may use the 1 A͑5ϩ2 ͒ϩB͑5ϩ2˜ ͒ϩ9 ⌳c͑Ϫ 2 ͒ 1 1 value 1ϭ0, suggested by experimental study of the charmed meson system ͓1͔, and Aϭ0.45, the default Lund value ͓5,9͔. If we further assume that the light degrees of We may define the polarization of the final state ⌳c in terms P ϩ P Ϫ 1 freedom fragment to j ϭ1 and j ϭ1 states indiscrimi- of the relative production probabilities for ⌳c(ϩ 2 ) and nately so that AϭB, we find that P ranges from 0.58 to 0.79 1 ⌳c(Ϫ 2 )as as ˜ 1 ranges from 0 to 1. For a heavy quark with initial polarization P, the above results for P are simply multiplied 1 1 by P. It is not unreasonable, therefore, to expect a significant Prob͓⌳c͑Ϫ 2 ͔͒ϪProb͓⌳c͑ϩ 2 ͔͒ Pϭ . ͑5.21͒ fraction of the initial heavy quark’s polarization to be ob- 1 1 Prob͓⌳c͑Ϫ 2 ͔͒ϩProb͓⌳c͑ϩ 2 ͔͒ servable in the final state ⌳c .
FIG. 5. Normalized differential decay rate for the case ␣ϭ1.3, ϭ0.08, and ˜ 1ϭ0.7. 53 EXCITED CHARMED BARYON DECAYS AND THEIR . . . 4873
FIG. 6. Normalized differential decay rate for the case ␣ϭ1.3, ϭ0.08, and ˜ 1ϭ1.
The parameters A and B are also of phenomenological produces distributions in ⌳c1 decay that are not symmetric ϩ Ϫ interest. Accurate association of ⌳c with final state pions with respect to the and momenta. Indeed, if we should measure the number of zero, one, and two pion events boldly accepted the central values of the sigma masses in the ratio: above, we would proceed to calculate an enhancement in the 2 coefficient of cos Ϫ by approximately 10% with respect 2 : : ϭ1:A:B. 5.23 to that of cos ϩ in ͑5.4͒ above, and a similar enhancement ⌳c ⌳c ⌳c ͑ ͒ 2 2 for the coefficient of sin Ϫ relative to that of sin ϩ in ͑5.5͒. In light of the errors listed in ͑5.25͒ and the order to Information on A and B may also be obtained by measur- which we are working, however, such a conclusion would be ing the relative number of fragmentation events containing inappropriate. The ϩϪ distributions are, within the accu- * ⌺c as opposed to those containing ⌺c . Direct fragmentation racy of this calculation, indistinguishable from those of the * * to (⌺c ,⌺c) produces them in the ratio ⌺c :⌺cϭ2:1. This neutral pions. ratio will be diminished, however, by ⌳c1 that decay to real ⌺c on their way to ⌳c . The decays of ⌳c*1 are kinematically forbidden from producing such an enhancement in the ⌺c* population. In the narrow width approximation for ⌺c ,we VI. CONCLUDING REMARKS find In this paper, we have studied fragmentation through the
(⌳c*1 ,⌳c1) system, and have calculated the resultant double events with ⌺c* 2 pion decay distributions in the well satisfied limit ϭ . ͑5.24͒ ( ) events with ⌺c B ⌫(⌳c*1 )Ӷ(M ⌳* ϪM ⌳ ). In so doing, we have introduced 1ϩ c1 c1 ͫ Aͬ the fragmentation parameters ˜ and B, and have shown 1 how ˜ 1 may be extracted from pion angular data. We have An accurate measurement of such departure from naive spin also found that the final state ⌳c particles produced in the counting could provide information on this interesting ratio, fragmentation process should retain a significant fraction of (B/A), and would be especially useful for checking the pre- the initial heavy quark’s polarization, allowing a test of the dictions of various fragmentation models. standard model’s predictions for heavy quark polarization in A few remarks are in order concerning the decays to such hard processes. ϩ Ϫ ⌳c . This case is slightly more complicated than the Experimental determinations of the parameters are ex- 0 0 tremely important in testing various ideas about fragmenta- case because the propagator connecting ⌳c*1 to ⌳c ( )0 ( )ϩϩ tion. Chen and Wise ͓6͔ have estimated 3/2 using the may be either ⌺c* or ⌺c* . This fact, coupled with the mc /mb 0 limit of a perturbative QCD calculation of different ⌺c masses, b B**→ done by Chen ͓7͔, and have found that → c ϩϩ 3/2ϭ29/114. That this admittedly oversimplified approach M͓⌺c ͔ϭ2453.1Ϯ0.6 MeV, gives reasonable agreement with the experimentally sug-
ϩ gested 3/2Ͻ0.24 ͓1͔ is of significant interest. Yuan ͓8͔ has M͓⌺c ͔ϭ2453.8Ϯ0.9 MeV, ͑5.25͒ augmented this analysis with a calculation of the dependence of on the longitudinal and transverse momentum frac- 0 3/2 M͓⌺c ͔ϭ2452.4Ϯ0.7 MeV, tions of the meson. Furthermore, fragmentation models such 4874 JOHN K. ELWOOD 53 as the Lund model make predictions for parameters related to ACKNOWLEDGMENTS A ͓5,9͔. Similar predictions will be possible for the remain- ing fragmentation parameters discussed in this paper, in ei- Insightful discussions with Peter Cho, Jon Urheim, Alan ther a limiting case of QCD, or in a model such as Lund, and Weinstein, and Mark Wise are acknowledged and appreci- the experimental extraction of these parameters will there- ated. This work was supported in part by the Department of fore provide nontrivial constraints on such methods. Deter- Energy under Contract No. DE-FG03-92-ER40701 mination of ˜ 1 may in fact soon be possible at CLEO ͓10͔. ͑Caltech͒.
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