P H Y S I C A L R E V I E W D V O L U M E 1 6 . N U M B E R 3 1 A U G U S T 1 9 7 7
Multihadron semileptonic decays of charmed mesons * V. Barger and T. Gottschalk
University of Wisconsin, Physics Department, Madison, Wisconsin 53706
R. J. N. Phillips Rutherford Laboratory, Chilton, Didcot, Oxon, England
(Received 17 March 1977)
We discuss and illustrate the effects of hadron multiplicity in the semileptonic decay of charmed mesons. Using specific models for the decay matrix elements, and for the production process where necessary, we evaluate the consequences of multiple-hadron decay products on distributions of practical interest: (i) energy distributions in e +e - colliding beam measurements, (ii) energy distributions in leptoproduction, (iii) transverse-momentum distributions in leptoproduction, photoproduction, and hadroproduction, and (iv) Kl invariant-mass distributions and other two-particle correlations. Theoretical estimates suggest that D + K l v and D -+ K*l v modes, in similar proportion, approximately saturate the inclusive semileptonic rate; this is consistent with existing data. Effects of hadronic form factors on decay rates and distributions are considered. Calculated acceptance corrections for undetected slow secondary muon in neutrino dimuon experiments increase the rates to 2-3% of charged-current events.
There i s g rea t interest in semileptonic decay modes
where 1 denotes p o r e , that can provide a d i s - tinctive s ignature f o r new "charmed" part ic les Y (we u s e "charm" generically t o denote a new hadronic quantum number). Such decays a r e be- lieved to manifest themselves in the neutrino- induced dimuon and be events,'-' in the i ~ . -induced dimuons,%nd in electron events f r o m e'e- col- l i s i o n ~ . ~ ' ~ They have been searched f o r in photo- productiong and a r e the b a s i s f o r charm t r i g g e r s in a wide range of high-energy hadron-hadron ex- periments . In a previous paperL0 we developed a fo rmal i sm f o r ev and p v semileptonic decays of a r b i t r a r y unpolarized part ic les , and discussed the single-particle decay distributions, dis t r ibu- tion broadening f r o m t r a n s v e r s e momentum of the charmed part ic le , and K1 invariant-mass d i s t r i - butions in Y-KLv decays. In the present paper we address fu r ther important phenomenological questions: the effects of hadronic fo rm fac tors , and the consequences of multiparticle decay modes f o r the observable distributions in leptoproduction, e'e- production, photoproduction, and hadropro- duction experiments .
Weak decay is likely only f o r the lightest par t i - c l e s of a new hadronic c lass . Prac t ica l calcula- tions in th i s paper generally assume a charmed meson of m a s s in .= 1.87 GeV, corresponding to the lightest of the new narrow s ta tes recently d i s - covered a t SPEAR." In one connection we a l so consider a heavier m a s s m .= 5.5 GeV, c o r r e s -
ponding to a hypothetical c l a s s of mesons contain- ing a heavier new quark. Also the parton-frag- mentation calculations a r e appropriate to mesons ra ther than baryons. However, the genera l quali- tative conclusions will apply both to charmed mesons and to charmed baryons, of any masses .
The decay distributions that can be direct ly com- pared with theory depend on the c h a r m production p r o c e s s considered:
(a) Invariant-mass distributions of p a r s of decay part ic les cLN/dmi, a r e always calculable, s ince they do not depend on the initial Y momen- tum.
(b) Energy distributions diV/dEi requ i re the initial momentum p, to be known. In special c a s e s such a s e'e- - Y ' Y - th i s may be t r ivial , but in genera l a specific dynamical model is needed. F o r instance, leptoproduction of a fast Y part ic le may be described by a parton-fragmentation mech- anism,10,12-14
(c) Specific t ransverse-momentum components may be accessible . In leptoproduction, the c h a r m momentum 5 , is expected to l i e mainly along the a x i s of momentum t rans fe r to the hadrons, s o that decay momentum distiributions d N / d p t i per - pendicular to th i s ax i s can be defined and calcula- ted. In pract ice with neutrino beams this ax i s is not accurately known, except that it l i e s i n the V-p plane, s o that distributions &/dpL i perpen- d icu la r t o this plane must be considered instead. In photoproduction and hadroproduction the beam direct ion gives the longitudinal axis s o that clN/ lipti c a n b e measured: However, both h e r e and in leptoproduction some t ransverse Y momentum is a l so expected, and must be integrated over. F o r e'e- production the ini t ia l Y momentum direct ion
16 - M U L T I H A D R O N S E M I L E P T O N I C
cannot generally be inferred, so that t ransverse decay distributions cannot be defined.
Form factors in the hadronic weak current a r e often ignored in the literature, for simplicity. In Sec. I1 we study quantitatively the inclusion of a form factor in D -Klv and D -K*lv decays. The effect is minimal in p, , p,, and m , , distributions, since the form factor is smeared by an integration in these cases. The most significant form-factor effect is an increase of decay rates.
The possible importance of multiparticle semi- leptonic modes was originally discussedL5 in con- nection with the apparent large kaon multiplicity in neutrino pe events reported in Ref. 3; however, Ref. 5 reports a much smal ler kaon multiplicity, so this effect is in doubt. Later there appeared in- dependent evidence of nontrivial charm decay multiplicity in the electron spectra for e'e- - e + hadrons at DESY.7*8*Lfi
In the present paper we leave aside the question of kaon multiplicity, and consider wider questions raised by multiparticle decays in general. Whether the final hadrons a r e kaons o r pions o r whatever, increasing the multiplicity reduces the accessible phase space and affects the shapes of decay dis- tributions. In successive sections we investigate various important phenomenological consequences, using uncorrelated Y -K(nn) lv and resonance- dominated Y -K*lv examples a s prototypes; the lat ter a r e introduced in Sec. 111.
The laboratory energy distribution of the decay lepton depends both on the energy of the original Y and on the decay mechanism. In all the lepto- production and e+e- experiments, there i s a minimum-energy cut on the observed decay lepton, and some fraction of the events a r e excluded. Thus both the spectral shape and the event ra te itself depend partly on decay multiplicity. In Sec. IV we study leptoproduction, using the parton- fragmentation mechanism to calculate the energy spectrum of produced Y mesons, and relate the results to the observed dimuon and pe In Sec. V we consider e'e- production near thresh- old, where the Y energy i s rather small, and relate the results to recent e x p e r i n ~ e n t s ~ , ~ in the c.m. energy range E = 4.0 - 4.2 GeV. The expected kaon spect ra a r e also calculated in each case.
Transverse-momentum components p , o r p, of the decay particle ( transverse to a plane o r a line containing the longitudinal production axis) a r e affected only by the t ransverse momentum of the parent Y- l e s s controversial than i t s total energy. In Sec. VI we calculate p, and p , decay distribu- tions, for various three-particle and multiparticle decay mechanisms, assuming a reasonable p , distribution for Y; the results apply equally to production by lepton, photon, o r hadron beams.
D E C A Y S O F C H A R M E D M E S O N S 74 7
We compare the results with leptoproduced dimuon and pe
In Sec. VII we discuss correlations among final- s tate particles in charm semileptonic decay events, presenting specific results for distributions of (i) r r z , , , the KI invariant mass, (ii) e,,, the angle between the decay kaon and lepton, and (iii) @ , , p , the azimuthal angle between the prompt and decay leptons in neutrino dirnuon events. The experi- mental a,, t distributions from neutrino dimuon experimentsfk2 a r e satisfactorily reproduced by our charm-fragmentation-model calculations. In Sec. VIII we discuss theoretical semileptonic branching-ratio est imates f o r the D meson. We find that the modes D -Klv and D - K*Zv can easily account fo r 8Wo of the expected semileptonic in- clusive rate, and this is consistent with present experimental information. Section M summarizes our main results.
Calculations in this paper a r e based on a general procedure applicable to any spin-averaged decay problem. Technical details a r e described in Appendixes A-E.
11. HADRON FORM-FACTOR EFFECTS
Although pointlike hadronic couplings a r e often assumed fo r simplicity, in general a nontrivial form factor may be expected. For example, in the decay D - K l v , the effective hadronic matrix element i s
where lepton masses a r e ignored, f(q2) is a form factor, and ci=pD-p,.
The invariant single-particle decay distributions may be writtenL0
dh' E i dpi3 -= & ( ~ , k )
(3) where i, j , k a r e any permutation of K, 1, v and s jk = - ( p j + p k ) 2 i s the invariant j + k mass squared. In the present example the product of lepton and hadron tensors gives
within a normalization constant, and g , ( s ) =g ,(s) from the symmetry of Eq. (4). One can observe g, and g , via the p,, and p,, distributions, and g, via the m,, invariant-mass distribution, since''
V . B A R G E R . T . G O T T S C H A L K , A N D R . J . Y . P H I L L I P S
FORM-FACTOR EFFECTS D-*Kev
- . .....-. DIPOLE
0 0.5 1.0
FIG. 1. Effects of hadronic f o r m fac tors on the distributions of t ransversn momenta p,, ,p,, and the invariant m a s s r n ~ ~ for D -Kev decay. The distr ibutions a r e given in a r b i t r a r y units h e r e and in subsequent f igures.
(Note that the corresponding equation in Ref. 10 was incorrectly printed, but a l l curves were cal- culated correctly .)
To illustrate form-factor effects we compare three cases:
(i) f = 1, pointlike case, (ii) f = ~ 2 ~ / ( 4 ~ + m2), monopole shape, (iii) f = m4/(yZ + tnZ)', dipole shape,
with rn = 2 GeV, a likely charmed vector- o r axial- vector-meson mass. The n~onopole shape (ii) i s strongly favored by analogy to the pion electro- magnetic form factor in the timelike region and
4 ........ DIPOLE
dN - dPe
pD= 0.5 GeV
0 0.5 1.0 1.5 2.0 P, (GeV)
d N - - d p ~
0 * 0 0.5 1 .O 1 5 2 0
FIG. 2 Effects of hadronlc form fac tors on electron and kaon momentum d ~ s t r ~ b u t ~ o n s for D -Kev decay w ~ t h p,= 0 5 GeV
the K, - nev form factor, both of which have been measured and a r e well fitted by monopole forms with the corresponding p o r K* vector-meson mas^.'^.'^ The dipole shape i s shown simply a s an extreme example, and because charmed-baryon decays a r e expected to have dipole form factors.
Figure 1 shows form-factor effects in D - Ke v decay fo r p,, , piK, and t?z,, distributions where p , is the momentum normal to a plane containing pD; expressions for p, distributions a r e given in Appendix B. The form-factor effect in g, and g , (which give p,, and m,,) i s minimized, since it i s smeared out by the integral in Eq. (3). In g, (which gives p,,), there is no integration over q2, and the form factor makes i t s fullest effect.
Figure 2 i l lustrates p,,p, distributions from D - Klv at p, = 0.5 GeV. This pD value represents the average D momentum of DESY e'e- charm events7" at E,,=4.0 -4 .2 GeV, assuming the mechanisms eie- -DD, D*D, D5*, D*B*. The cusp in the p, distribution is associated with the nonvanishing of the p, spectrum at p,(max) in the D res t frame; averages over the D-production spectrum would wash out this cusp.
The mean values of the D -Klv distributions above a r e
Pointlike Monopole Dipole
From these values and Figs. 1 and 2 we conclude that neglect of the hadron form factor i s a quite acceptable approximation in a l l distributions ex- cept for the extreme and unlikely possibility of a
16 - M U L T I H A D R O N S E M l L E P T O N I C D E C A Y S O F C H A R M E D M E S O N S
dipole form factor in the case of kaon distribu- tions.
In multiparticle decay modes, the matrix ele- ment can contain a form-factor dependence on q2 = ( p -pX) ' , among other hadronic variables. However, when the final hadron state X contains a kaon, the physical range of q2 is always less than in the X = K example above: 0 6 - q2 s (m , - m,)'. Hence the form factor has even less effect here.
Figure 3 illustrates form-factor effects in p, and pK distributions for the multiparticle decay modes D -Knev and D - Kt(O.89)ev-Knev. These curves a r e computed from matrix elements given in Appendix A.
Inclusion of a form factor increases the partial widths, since the form factor i s greater than unity in the decay region, assuming the coupling is prescribed at q2 = 0. Defining
we obtain the values
This increase in the decay rates is the most significant form-factor effect.
111. MULTIPARTICLE DECAY EXAMPLES
To illustrate the general implications of multi- particle modes, we consider two kinds of D -Xlv decay mechanism, each with one final kaon a s suggested by the Glashow-Iliopoulus-Maiani (GIM) m e ~ h a n i s m . ' ~ In each case we assume the lepton current has conventional V-A form, and neglect lepton masses.
(i) Uncorrelated hadyon system: X= K +nn. Here we take the hadron current matrix element to be proportional to the sum of final hadron mo- menta, which is equivalent to the form
For X =K this form is essentially unique (up to a form factor). For multihadronic decays Eq. (7) has sufficient structure to represent effects as- sociated with the lepton tensor, though the do- minant multibody effects come from phase space.
(zi) Resonant hadron system: Xz=K *(0.89) o r K*(1.42), with subsequent K*-KT decay calculated in the narrow-width appyoximation. The K* decay mode affects r a n d n distributions but not lepton distributions. We take the matrix elements to have the minimal forms
- POINTLIKE ( a ) D-Kev 1
FORM- FACTOR EFFECTS 4 - 7
1 ---- MONOPOLE ( b ) D-Kneu 1
FIG. 3. Effects of a monopole hadronic form factor on electron and kaon momentum distributions in the r e s t frame of D for decay modes D -Kev, D-Knev, and D -tK*(0,89)eu.
- POINTLIKE ---- MONOPOLE
( a ) D-Keu -
where and $I,, a re J = 1 and J = 2 spin functions. It i s sometimes useful to distinguish a third
class of decay mechanism. (iii) Sequential strong-weak decays. Here an
initial charmed resonance D* decays first strongly to D:
D* - x ' D , (1 Oa)