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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.

I. INTRODUCTION

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 experiments . 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 hadroproduction 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 calculations 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 containing a heavier new quark. Also the parton-fragmentation calculations a r e appropriate to mesons ra ther than baryons. However, the genera l qualitative conclusions will apply both to charmed mesons and to charmed baryons, of any masses .

The decay distributions that can be direct ly compared 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 momentum.

(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 mechanism,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 calculated. 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 semileptonic modes was originally discussedL5 in connection 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 independent 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 distributions. 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 leptoproduction 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 threshold, 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 distributions, 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 experimental 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 inclusive 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

POINTLIKE

- . .....-. DIPOLE

I

0 0.5 1.0

P,, (GeV)

POlNTLlKE MONOPOLE

-..... .DIPOLE

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 calculated 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 electromagnetic 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)

-POINTLIKE

4 -

d N - - d p ~

2 -

0 * 0 0.5 1 .O 1 5 2 0

P, (GeV)

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 except 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 distributions.

In multiparticle decay modes, the matrix element 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 multiparticle 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 momenta, 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 associated with the lepton tensor, though the dominant 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

4

- POINTLIKE ( a ) D-Kev 1

FORM- FACTOR EFFECTS 4 - 7

1 ---- MONOPOLE ( b ) D-Kneu 1

3

pK (GeV)

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)

D-Xlv. (lob)

We studied this possibility in connection with kaon multiplicity,15 where the kaon content of X' was relevant. But for present purposes X' is i r re le- vant: We a r e concerned only with the hadron multiplicity in X. In the present work we regard the first state (10a) a s part of the D-production mechansim, and the second stage a s being included in (i) o r (ii) above.

The simplest multiparticle effects occur in the kinematical boundaries that set the scales for the pt, PI, and m,, distributions. For uncorrelated K i n n final states, and p,, = 0, the bounds a r e

p,,(max) = [mD2 - (m, + n m d 2 ] / ( 2 m , ) , (11)

V . R A R G E R , T . G O T T S C H A L K , A N D R . J . N . P H I L L I P S

------ D-Kev

D - K a e v . . . . . . , . . . . . . . .

..

0 0.2 0.4 0.6 0.8 1 .O

pe (GeV) FIG. 4. Electron momentum distributions in the rest

frame of D for the various decay modes discussed in Sec. 111.

ptK (max) = h"2(un," wzK2, nz tn t ) / (2~nD) , (12)

where h(a, b, c ) = ( a - b - ~ ) ~ - 4bc, and Ip, / has the same bound a s p,. Fo r resonant hadron systems, the K* mass constraint res t r ic t s the boundaries even more: For example,

The bounds on p,, and m,, a r e more complicated, and may be found in Appendixes B and D. For nonzero ptD the kinematic limits in Eqs. ( l l ) , (12), and (14) must be boosted fo r p, to

where

(1 5b) For nonzero p,, the p , limits a r e boosted analog- ously.

Figure 4 compares electron nlomentum distr i-

TABLE I. Maximum and average momenta of the decay electron originating in semileptonic decay modes of a D meson at rest.

p,(max) (P,) Decay mode (GeV) (GeV) @,)/p,(max)

production by parton fragmentation in charged- current (CC) and neutral-current (NC) processes. For specific calculations, we take the GIM charm schemelg with the Weinberg-Salam n1odelz0 for neutral currents .

According to the quark-part on nm del (QPM), the c r o s s sections fo r leptoproducing a fast charmed quark c o r Z a r e

do (vN - vex)= - dxd y rirE) x i ' b ) [ s L 2 +,gR2(l - y12] ,

d o z ~ ~ \

dxdy - (vN - VEX) = (5) x~'(x)[,g,%gL2(1 - y)z],

d u dxdy

butions in the D res t f rame for the decay modes Here G and a, a r e the Fermi and fine-structure

described in Eqs. (7)-(9). As the sum of the final constants, x = Q2/(2Mv) and y = v / E a r e the usual

hadron masses increases, the kinematic endpoint scaling variables. u , 5 , and 5' a r e distributions

p,(max) decreases according to Eqs. (11) and (14). of valence and s e a quarks in an isospin-averaged Moreover, the high-p, tai ls of the K +nn distr i- nuclear target , in t e rms of which the usual proton but ions a r e suppressed by phase-space limitations. distributions a r e 2 +cP = 2(v+ 0, g = % = h = X = 5, For the uncorrelated multiparticle modes, this c = E = 6' [assuming SU(3) symmetry for the sea] . phase-space suppression i s more important than

g, and g, a r e given by g, = 1 - g sin28,, g, = - $ the kinematic endpoint in controlling the overall shape of the distributions, as indicated by the sin2tiw. 6, and t), a r e the Cabibbo and Weinberg

angles: sin2@, - 0.05, sin28, = 0.33. The QPM values of (p,)/p,(max) in Table I.

distributions of solution I11 in Ref. 21 a r e used in

IV. ENERGY SPECTRA IN LEPTOPRODUCTION our calculations.

The CC processes above a r e single-charm pro- In this section we show the effects of decay duction; we neglect CC production from the cc

multiplicity on energy spectra, and hence on ex- sea, because here the final charmed quark is perimental acceptance, assuming charmed particle slow. The electromagnetic and NC weak processes

16 - M U L T I H A D R O N S E M I 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 75 1

a r e associated production, in which the fast c o r i? quark is accompanied by a slow C o r c quark among the target fragments; we deal he re with the fast quark. The antineutrino neutral-current - v - c and 5 - C c ros s sections a r e equal to v - C and v - c c r o s s sections, respect.ively. The electromagnetic p ' - c , p i - E , e 1 - c , e* - E c ros s sections a r e a l l equal. Equations (18)-(20) a r e based on diagrams in which the current couples to Fc; associated Tc production in which the current couples to noncharmed quark.s i s more difficult t o estimate.

Since these reactions can only occur above charm threshold, a factor O(W - W,) should strictly be added in Eqs. (16)-(20), where W:'= BMEy(1 -x) + illZ i s the invariant hadron mass squared and W , i s the appropriate charm threshold. For single-charm production we assume W , - 2.8 GeV, while fo r associated production a value W , 2 4.7 GeV seems appropriate, motivated by the charmed- meson" and -baryon candidateszz observed. For production near threshold, it i s also better to use a slow-rescaling variable x' rather than x in the parton distributions: v(x) - v ( x l ) , etc.; with the prescription of Ref. 23

We take m, = 1.5 GeV. The factors governing y dependence in Eqs. (16)-(20) also change a s follows:

The c r o s s sections above a r e for producing a fast charmed quark c o r F, with lab energy v. We assume that the probability for a charmed particle Y to emerge with energy z v from the fragmentation of c i s D(z), and adopt the form

which f i ts uncharmed fragmentation for z , > 0.1. An equally good description of uncharmed fragments is obtained with

Use of this form in our calculations with the cutoff z > 0.1 yields essentially the same results a s Eq. (23). The energy spectrum for the fast charmed particle i s given by

where a, denotes the relevant QPM c ros s section and +(E) i s the incident lepton spectrum. Kine- matical l imits on the integration region a r e de-

scribed in Ref. 10; we also limit z 2 0.1 to exclude the target-fragmentation region. The normalization of D(Z) in Eq. (23) i s

in order that each charmed-quark event yields one charmed meson. The zma,, zmi, boundaries introduce an x, y dependence in the D(z) normalization, but this has a very small effect on the shape of the d o / d ~ , spectrum. The energy spectrum of the decay lepton is given by the convolution

Details a r e given in Appendix B. Following the parton-fragmentation and slow-

rescaling prescriptions above, we have calculated the decay lepton spectrum for various interesting situations, comparing the predictions with different multiparticle decay models. Fo r incident neutrino energies above about 20 GeV, the D- meson energy distribution is determined largely by D(z) and W,. Above the threshold region (ED > 4 GeV), the ED distributions a r e extremely simi- l a r for the CC and NC processes of Eqs. (16) and (18). Accordingly, the predicted decay distributions fo r p, (or p,) > 2 GeV a r e essentially the same with CC and NC production. The W distr i- butions a r e also similar for CC and NC processes a t W >> lV,. At low W, however, there a r e dramatic threshold differences. The energy lost to the decay neutrino is a relatively insignificant correction to the 1V distr ibnt~ons.

Figure 5 shows uN/dp, and cUV/dp, distributions for vN - p-Xdecay modes D - K i v and D - K*(O.8Y)lv at incident neutrino energies of 50 and 150 GeV. Figure 6 compares the BNL-Columbia (BNL-COL) uN/dp, data5 for vN - ,a-e'X with the predictions of our fragmentation model. Figure 7 shows the p+ spectrum for VN - p-p'X' charged-current production, fo r the quadrupole triplet neutrino beam used in the Harvard-Pennsylvania-Wisconsin- Fermilab (HPWF) dimuon measurement.' In this experiment the p + energy must exceed 4 GeV to be counted. Table I1 summarizes numerically the fraction of events remaining after a cut E , > E,(min), in these calculations, for different decay modes. From this table, we see that this cut alone reduces the dimuon ra te by a factor of 3 to 6 depending on the decay mode.

Figure 7 shows the p - spectrum for 5N - gtp-X, averaged over the antineutrino spectrum of the HPWF experiment.' These results , based,on c - quark excitation, fai l to account for the high-p,- tail of the distribution, which may well be a s - sociated with b -quark production. l3 ,23324

V . B A R G E R , T . G O T T S C H A L K , A N D R . J . N . P H I L L I P S

FIG. 5. Momentum distributions of the decay lepton and kaon from charged-current D production and subsequent semileptonic decay. D meson production is computed via parton fragmentation using Eq. (16) and Eq. (23) . The curves a r e for incident neutrino energies E = 5 0 and 150 GeV.

Pursuing this speculation, we d igress to calcu- butions a r e shown in Fig. 8(b). Here the b quark late the p - spectrum based on b-quark excitation. leads to a relatively flat distribution, but there The c ros s section for producing the b quark in is l e s s evidence for this in the data. The xVi, dis- 7N i s tribution of the p + in i7N - p'p-X must exhibit

valence characterist ics if b production i s dominant K ( ~ ~ + p t b ~ ) = 7 dxdy

(GZME) 2r[u(x) + ( ( r ) ] . (27) here.24 Higher-statistics ZN - p *p-X data will be required to differentiate between c-quark o r b-

We again use Eqs. (23) and (25) with slow rescaling to generate the corresponding B-meson (i.e., b$ o r b s state) energy spectrum. In these caleu- lations, we take m,= 5 GeV, m , = 5.5 GeV, and W , = 6.4 GeV. The decay lepton spectrum 1s calculated from Eq. (26), assuming a B -plv decay mode. Figure 8(a) contrasts the diV/dp,- spectrum f rom b excitation with that f rom c excitation. The b-quark model leads to a broader distribution, in better agreement with the data. The p,,- dis t r i -

30

20 d N d P,+ BNL-COL DATA

10

0 0 4 8 12 16

FIG. 6 . Momentum distribution of the decay positron f rom vN-p 'etX dilepton events. The incident neutrino spectrum and data a r e from Ref. 5.

quark production mechanisms. Figure 9 shows the decay muon (pz) spectrum

for pN - p l p & at 150 GeV, calculated from c excitation with approximate energy and angular acceptance cuts appropriate to the experiment of Ref. 6. In this experiment, EPz had to exceed 12 GeV.

It i s also interesting to see how this E l acceptance changes with incident energy. Figure 10 shows the fraction of events with E , > E , a s a function of incident beam energy E, for VN - p-p+X dimuon production, based on the D - Klv and D - K*(0.89)lv decay modes. The illustrated acceptance cuts a r e relevant to the following experiments: BNL-COL p-e' (El > 0.3 GeV), Berke- ley-CERN-Hawaii-Wisconsin (BCHW) p-e+ (E, > 0.8 GeV), HPWF p + g - (Ez > 2.5 GeV, in progress); Caltech-Fermilab (CITF) p'p- (El > 3 GeV), HPWF p'p- ( E l > 4 GeV). These detection-efficiency predictions depend on the assumed form of D(z) and the cutoff z,, which largely determine the E D spectrum. The corresponding uncertainty in the detection-efficiency est imates i s greatest fo r E , s 50 GeV. Efficiency corrections for 7N - p+p-X a r e essentially identi- ca l to these vN results.

Given the decay lepton detection efficiencies in

M U L T I H A D R O N S E M I 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

FIG. 7. Momentum distributions for the decay muon in neutrino and antineutrino dimuon reactions. The curves a r e averaged over the appropriate incident neutrino o r antineutrino spectrum and compared with the data from Ref. 1.

Fig. 10 the true ra tes for dileptonprctduction inneu- trino experiments differ appreciably from the observed rates. Figure 11 shows the energy dependence of data fo r a ( v N - ~ - L + X ) / O ( V N - g-X) and o(5N - p+lX) /u (ZN - p ' X ) , corrected according to D -K*(0.89)lv decay male. The data a r e from a compilation given in Ref. 25. The corrections to the raw data a r e summarized in Table 111. The E > 50 GeV points in Fig. 11 a r e factors of 2 to 4 higher than the measured values, owing to the limited acceptance for the secondaly muon. Al- though these acceptance corrections a r e substan- tial, they a r e likely realistic. Figure 12 shows the lepton detection efficiency a s a function of the parent D energy for D - K*(O.89)1v; for E , > 3 GeV, the acceptance correction becomes

TABLE 11. Estimated detection efficiencies of neutrino and muon dilepton experiments.

v H P W F ti H P W F p Ref. 6 Decay mode E, > 4 GeV E , > 4 GeV E , > 12 GeV

plus angle cuts

FIG. 8. Predictions for decay lepton p and p, distributions in F-induced dimuon events based on production of mesons containing c quarks (solid curves) and and b quarks (dashed curves). The data and incident antineutrino spectrum a r e from Ref. 1.

V . B A R G E R , T . G O T T S C R . - I L K , A N D R . J . N. P H I L L I P S

0 20 p ( G ~ v ) ~ '

60

P2 FIG. 9. Energy distribution of the decay rnuor~ i n ~ . v - @ ~ p ~ X electromagnetlc interactions at 150 GeV. The data a r e

from Ref. 6; ap1~rosimate energy and angular acceptance cuts of the experiment a r e included in the calculated curves.

FIG. 10. Fraction of uX-p-p+X charged-current dimuon events with the decay lepton 1 * energy exceeding rhc Indicated cutoff a s a function of the incident neutrino encrgj . The curves a r e based on the Q P M , current- fragmentation prescription of Eqs. (16) and (23) and assume D - -Klu , D -K*(0.89)1 u , and B --pl l l decay modes.

smal l only fo r a very-high-energy parent D, which i s relatively unlikely in fragmentation production models.

The curves in Fig. 11 represent GIM charm predictions for B,uc/o fo r several values of the semileptonic branching ratio B,.The data require a mean semileptonic branching rat io of charm particles

where I denotes e i t i ~ e ~ e o r p , not the sum over both. This estimate is based on Q P M calculations of uc; asymptotic-freedom correctionsZF to a, wil l modify this B, estimate somewhat. The result f o r B, in Eq. (28) i s consistent with theoretical expectations based on asymptotic-freedom calcula- t i o n ~ . ~ ~

Alternatively we consider the possibility that the - v . -pi JIX events a r e largely due to b -quark production. Figure 13 compares b-quark predictions with the 'ij data, corrected f o r detection efficiency losses using the B -pLv results in Fig. 10. With the Q P M production c ros s section, a branching ratio

would be indicated. At the moment there i s insuf- ficient experimental information on dilepton production in neutrino experiments t o disentangle the possible new-quark o r neutral-heavy-lepton2* components.

M U L T I H A D R O N S i E M I 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

o p-e+ BCHW 0 p-e+ BNL-COL_ a p-p+ CITF

E ,,, GGeV 1 Evis (GeV) FIG. 11. Ratio of dilepton to single-lepton rates versus incident beam energy for neutrino- and antineutrino-induced

dileptons. The experimental ra tes have been correc ted according to the appropriate slow-lepton detection efficiencies of Fig. 1 0 for D-K*(0.89)1u. The curves a r e based on G I M charm production with the indicated values B, for the semileptonic inclusive branching ratio of D --lux for the relevant lepton in each case.

In Figs. 11 and 13, the d a t a points were plotted F o r E ,, 50 GeV, the missing energy is only a t the experimental mean visible energy; this dif- 3-4% of the visible energy, and is thus a very f e r s f r o m the incident energy E,+M by the energy modest correct ion. c a r r i e d off by the unobserved decay neutrino. F igure 14 shows the rat io of average missing V. ENERGY SPECTRA IN ete- PRODUCTION

energy to visible energy calculated Eor D -Klv Recent measurements of e'e- - e + hadrons and and D -K*(0.89)lv decays. The cunres in Fig. 14 e'e- - e K & in the DASP and P L U M spec t rometers a r e based on o u r charm-fragmentation model; a t DESY i n the range E,,,= 4.0-4.2 GeV, s t rong- Em,,,,, is the average energy of the decay neutrino ly suggest charm production with senlileptonic a t fixed incident energy E, and E ,, = E ,- EminiW decay. The main c h a r m production modes a r e

TABLE III. Experimental values for U ( I J - p-E')/u(v- p-) and n(ii-- 1'1 -)/u(B- p') and estimated acceptance correction factors associated with detection losses of slow decay leptons.

E E , (min) Acceptance correction (GeV) (Gel') l o 2 x Data D- K * I ~ B-plu Experiment

Keutrino

Antineutrino

GG 11 BCH W BNL C I T F C I T F HPWF C I T F C I T F C I T F

58 3 .O 0.310.2 3.8 1.9 C I T F 70 3 .O 1 . 0 f 0 . 5 3.2 1 .8 C I T F

150 3 .O 1 . 0 1 0 . 7 1.9 1.4 C I T F 180 3 .O 2 . 2 f 0 . 9 1.8 1.3 C I T F 190 3.0 1 . 8 f 1.3 1.7 1.2 C I T F

V . B A R G E R , T . G O T T S C H A L K , A N D R . J . N . P H I L L I P S 16 -

ED (GeV) FIG. 12 . Fraction of decay leptons from D-K*lv with

energy above the indicated cutoff versus energy of the parent D.

argued to be e i e - -DE, DE*, D*Z, D*E* in a s - cending order of i m p o i - t a n ~ e , ~ ~ with tn,, i.. 2.01 GeV and strong D* -D;r transitions. The weakly decaying D(1.87) mesons therefore have low momentum, with effective mean value near 0.5 G e ~ / c The observed decay electron spec t ra cut off above 0.7 GeV/c, unlike the predictions fo r D - e v or D - K e u decay, suggesting that higher multiplicities a r e involved.lfi A quantitative comparison of different decay -mode predictions with the DESY data would clearly be helpful. We have already pre- sented a brief account elsewhere'" the present section contains fuller details.

We calculate D-X1u decay with the X =Ki.ni i (uncorrelated) a.nd X = K* (resonant) mechanisms described in Sec. 111. Figure 15 shows the elec- t ron spectra, calculated at the appropriate mean

E,,, (GeV) FIG. 13. Ratio of antineutrino-induced dilepton to

single-lepton ra tes , based on b-quark production and the semileptonic decay mode B-pEu.

E,is (GeV)

FIG. 14. (Average missing energy)/(visible energy) as a function of visible energy, for D-Klw and D- K*(0.89) x 1v decays.

momentum p, = 0.5 ~ e v / c , compared with DASP and PLUTO For the PLUTO e'e- -eK,X results , we have subtracted the background measured from entrr-x events that did not fit the kaon mass; this explains the small negative fluctua- tions. The curves show the expected qualitative effects discussed in Sec. 111: a decrease in the momentum of the spectruin peak with increasing m X 2 , and a suppression of the high-p, tai l in the K t nn uncorrelated multiparticle modes. The data do not agree with the D - K e v prediction; i t seems

4

3

dN - ~ - ~ * ( 0 . 8 9 ) e v

- 2

dPe ......-. D- ~"*(1.42)ev

1

0

0 0.4 0.8 1.2 1.6

pe (GeV)

FIG. 15. Electron momentum distributions from D-Xev decays at p,= 0.5 GeV, compared with DASP and PLUTO data. DASP and PLUTO data have lower cutoffs a t p, = 0.2 Gelr (shaded area) and p, =0.3 GeV, respectively.

16 - M U L T I H A D R O N S E M I 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

that this cannot be the dominant mode. Other authors have recently reached s imi lar conclu- s i o n ~ . ~ ~ The K* (0.89)e v mode gives reasonable agreement with the data, a s do the Knev and Knnev uncorrelated modes. However, the present measurements a r e ra ther insensitive to very high multiplicities o r high-mass K* modes, that peak below the lower cutoff momentum.

Could the experimental measurements contain some bias against low-multiplicity decays?

(i) The DASP E , spectrum in Fig. 1 5 i s based on events with detected charged prong number n,, 3 4. Fo r heavy leptons Li the dominant decay modes a r e expected to give only one charged pal ilcle each, so efe-- L+L- events a r e largely excluded by this cut. However, the measured e+ hadron prong distribution is rather broad; i t peaks at n , = 5; there a r e 22 events with n,,a 4 compared to only 6 events with n, = 2 , 3 ; the lat ter 6 events a r e expected to contain LfL- and other background. From this we conclude that rather few charm events have been lost by the n,, 3 4 cut, and that if low-multiplicity semileptonic modes have been strongly suppressed, they cannot have been dominant in the f i r s t place.

(ii) The PLUTO data a r e fo r eP&Yfinal states, requiring a visible KO,, which biases against some low-multiplicity modes. For example, Do - Klv can have no KO, whereas Do -K*lv contains KO with branching fraction a. On the other hand, Df -Klv contains he always, whereas D'-K*Zv has with branching fraction f only. If semileptonic Do, D+ decays contributed equally in the complete reaction, there would be little net bias (remember also that KO can come from the "other" charmed particle decay too). However, their mass difference" favors Do over D+ a s the end product of D* -Dn decay .2g We conclude that DO

probably contributes much more strongly than Dt to the semileptonic decays, and hence that D -Klv modes have considerably reduced acceptance. However, this argument does not apply to the DASP data, where the d N / d p , spectrum i s also in- consistent with D - Ke v.

We have ignored possible contributions from heavy-lepton L o r charmed F-meson decays, in comparing with the data above. The former i s known to give a much harder E , ~ p e c t r u r n , ~ ' and therefore cannot be contributing significantly; however, we expect relatively little LZ production a t 4.0-4.2 GeV if m, - 2 GeV, and the prong num- be r cuts tend to exclude these events With estimated mass m,- 2 GeV we expect F production to be similarly kinematically suppressed: there a r e also arguments for a dvnamical - suppression of FF versus DD states near thresh-

pK (GeV) FIG. 16. Kaon momentum distributions for po=0.5

GeV.

Predictions for a!N/dp, at p, = 0.5 GeV fo r the various semileptonic modes a r e shown in Fig. 16.

VI. TRANSVERSE-MOMENTUM DISTRIBUTIONS

A universal feature of high-energy hadron production is that the momenta p , of secondary hadrons transverse to the incident beam axis remain predominantly small. The same is t rue fo r leptoproduction, where the longitudinal axis i s defined by the momentum transfer from the leptons to the hadron system. The shape of the p , distribution seems to depend mainly on the produced hadron mass , and is approximately represented by the universal behavior

with b = 6 GeV-' f o r z = n, p, K, p , # with any beam o r target, at any fixed longitudinal momentum p,, (see e.g. Ref. 33). This formula ignores a weak dependence of slope on p, , . Since the p , dependence of charmed meson production has not yet been measured, we shall assume it to be adequately described by Eq. (30) with rn, =m, = 1.87 GeV. More sophisticated universal formulas have been proposed,34 but they give very s imi lar predictions for p,, 2 GeV, where most of the events a r e expected to lie.

Given the p , distribution of Y , we can calculate the p , distributions of Y-decay products t rans- ve r se to the same longitudinal axis. We can also

758 V . R ; \ R G E R , 1'. G O T T S C H A L K , A N D R . J . N . P H I L L I P S - 16

calculate p , distributions, t ransverse to a plane have compared p,, distributions fo r the decay containing this axis; this i s useful in some neu- modes D - Klu and D -K*lu using several values trino processes where one cannot accurately inea- of b in Eq. (30). Results of these calculations a r e su re the momentum transfer axis itself, but only stiown in Fig. 20. For p,, l e s s than i t s maximum a plane in which i t l ies (defined by incident neu- value in the D res t f rame the smearing effects of trino and final direct lepton directions). Appendix tlhl /rlktD2 a r e not strongly b -dependent. B describes the details of these calculations. Insofar a s the p, , distribution i s universal, the VII TWO-PARTICLE CORRELATIONS

results apply equally to all hadroproduction, photoproduction, and leptoproduction experiments, at all high energies, using the appropriate longitudinal axis in each case.

Figure 17 shows the p , distributions of the decay lepton and kaon, for the decay modes described in Sec. 111, averaged over the p,, distribution of Eq. (30). Figures 18 and 19 show the corresponding p,, and p,, distributions. The y,, calculations a r e compared with data from the HPWF dimuon experiment.

We note that the smearing i s dominated by the smal l p , , behavior of ilN/(fp,,'. To investigate the sensitivity of the decay product t ransverse- momentum distributions to that of the parent, we

FIG. 17. Distributions of lepton and kaon momenta transverse to the longitudinal axis of hadroproduction, photoproduction, o r leptoproduction for various decay modes. The curves a r e smeared according to the P ,, distributions of Eq. (30) for b = 6 .

We consider here experimentally interesting correlations aniong final-state particles in semileptonic charm decay events. In particular, we present detailed calculations for (i) mKe the kaon- elec+~.on invariant mass, (ii) cos@,,, the angle hetureen the decay kaon and electron, and (iii) q, ,~, the azimuthal mgle (in a plane perpendicular to the neutrino beam axis) between the prompt and slow leptons in neutrino dimuon experiments.

The invariant mass rn,, of the decay kaon plus electron system has the advantage of being a Lorentz scalar , independent of the parent Y-meson mon~entum. so that no model of the Y production mechanisnl i s needed. It has the difficulty, however, that one must make sure that the measured kaon comes from Y decay and not from the production mechanism. For example, in neutrino charm production from the AX sea, at least one strange particle i s expected among the target fragments; also, in e'e- production the "other" charmed particle will often yield a decay kaon; such background kaons have to be excluded.

The method of calculating ru,, for a given multihadron decay mode i s described in Appendix D. Our results for the decay modes described in Sec. I11 a r e shown in Fig. 21. Note that these distr i- butions display qualitative multiplicity-dependent properties comparable to those for the p, distr i- butions discussed in Sec. 111: a decrease in mK,(max) with increasing final-hadron mass, and additional phase-space suppression of the high- m,, tai ls of the K + n x distributions.

The distribution in the angle OK, between the decay kaon and electron fo r the three-body decay D - K e u has the following simple form in the D res t frame:

where F i s the squared matrix element for the decay,

and

M U L T I H A D R O N S E M I 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 759

FIG. 18. Distributions of decay lepton momentum pLJ transverse to the production plane of neutrino experiments, smeared according to the p t ~ distribution of Eq. (30) for b - 6 . The data a r e f rom Ref. 1 . A single event at pip- = 1.9GeV has been excluded from the figure ( s ee Fig. 8) .

E,= s/2(mD -EK +p,cosb,,), (34)

t=mD2-2mDE, . (35)

For p, 2 0, complications from adclitional angle, energy integrations and nontrivial integration l imits make analytic generalizations of Eq. (31) quite un- wieldy. To compute OK, distributions, a s well a s other correlations with o r without various energy cuts, we use a Monte Carlo method described in

0 0 0.4 0.8 1.2 1.6

pLK (GeV)

FIG. 19. Distributions of kaon momentum pLK perpendicular to the production plane in neutrino experiments. The curves a r e smeared according to the p,, distribution of Eq. (30) with b = 6.

Appendix E; results for OK, distributions for D -Kev a r e also given in Ref. 35.

Figure 22 shows OK, distributions calculated a t p, = 0.5 and 1.25 GeV, for the decay modes D -Kev, D -Knev, and D -K*(0.89)ev -Knev. At low p, the distributions fo r D -Kev a r e ra ther strongly peaked at negative cos8,,. Asp , in- c reases , relativistic effects shift the cosOKe distributions toward cosOK, = 1. This trend i s already evident in calculations for p, = 1.25 GeV.

Using the techniques of Appendix E, minimum energy cuts can straightforwardly be incorporated into calculations of expected distributions. We have recalculated the curves of Figs. 21 and 22 incorporating typical cutoffs p,, p, > 0.24 GeV, and have found that such cuts cause little change in the calculated distributions.

Using the charm production mechanism of Eqs. (16) and (23) and the p , distribution of Eq. (30) fo r the produced D meson, we can calculate cor re lations between the prompt and decay leptons in neutrino-induced dilepton events. Of particular interest i s the azimuthal angle q5 between the leptons, a s determined in a plane perpendicular to the incident neutrino direction. In Fig. 23 we compare calculated q5 distributions with data from the HPWF and CITF experiment^"^; smearing effects from p,, make essentially no change in these results. The calculations include the decay lepton energy acceptance cuts, which a r e quite important in the C#I distribution. (The acceptance cuts remove contributions from low-energy D

V . B A R G E R , T . G O T T S C H A L K , A N D R . J . N . P H I L L l P S

FIG. 20. Distribution of lepton momentum ptl transverse to the longitudinal axis for the decay rnodes D-Klv and D-K*Lv. The distributions a r e smeared according to thept, distribution Eq. (29) for the indicated b values.

mesons, which tend to give a more isotropic 6 a s discussed in Sec. V. Here we attempt to under- distribution.) The charm-fragmentation model stand the basis of this relative suppression. provides a satisfactory description of the experi- The total D semileptonic decay width can be mental results. estimated from the pointlike free-quark decay

matrix elements for c - Xlv and c - X l v : VIII. THEORETICAL BRANCHING-RATIO ESTIMATES

The DORIS data indicate that D - K l u i s not the dominant semileptonic decay mode of the D meson, 1 i, p,=O.S , GeV

\ ---- 0-Kev -

- 0- K*ev

- 1 c0seKe 0

0.4 0.8 1.2 1.6 2.0

Ke (GeV)

4 . , 1 , 1 l ! 0

0-0 Kev 3

D- K*ev

3

- I 0 1

cos eKe FIG. 22. Distributions of c 0 s 0 ~ , for the decay modes

D-Kev, D-Krev, and D -K*(0.89)ev-Knevat p,= 0.5 GeV and p D = 1 .25 GeV.

- D - K * ( o . B ~ ) ~ u - - L KT

D - K * * ( ~ . ~ z ) ~ v - - d N 2 -

d m ~ e - 1 -

0 0.4 0 .8 1.2 1.6 2 . 0

Ke (GeV) FIG. 21. Invariant-mass dlstribtuins m,.

16 - M U L T I H A D R O N S E M I 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 761

4 'O - HPWF DATA ( Ep+ >4 G

D - K*$U

DATA (E,+ >3 GeV) ii 0 , I u

1 0 - I

cos FIG. 23. Distributions of A@, the azimuthal angle

between the prompt and slow leptons projected onto a pIane normal to the incident neutrino direction. The curves are based on the Q P M current-fragmentation calculations and the decay modes D-Klv and D --K*(0.89)lv averaged over the neutrino spectra of the HPWF and CITF experiments.

Here

and I = e o r p. We assign physical-hadron masses to the quarks 112, =in,, m , =m,, and m,=m, to ensure correct phase-space bounds. The resulting inclusive semileptonic width is r(D - Xlv) = 4.8 x 10" sec-'. We note in passing that this rate i s rather sensitive to our mass assignments.

The D - Klw partial width is given by

From Eqs. (36) and (39) we obtain the branching fraction

Including form-factor effects based on Eq. (6), this fraction is increased to 0.33. According to this estimate, the dominance of mul1;ihadron semi-

leptonic modes is not at all surprising. For D -K*(0.89)2v the decay ra te is36

I?(D - K*lv) = ~ , ( 1 1 ~ , / 4 r n ~ * ) ~ r n ~ ~ l 1(~nX*/mD)~os2eC ,

(41)

where nzo is the constant parameter in Eq. (8) and

With the natural choice m, = m, + m,,, the K* branching fraction is

Form-factor effects based on Eq. (6) increase this number to 0.43. Thus the KLv and K*lw modes can easily account fo r 60% (assuming no form factor) to 80% (with form factors) of the total semileptonic rate.

Semileptonic decays into nonresonant multiparticle final states a r e more difficult to reliably e ~ t i r n a t e . ~ ' For equal constant matrix elements, phase-space ratios yield

For matrix elements given by Eq. (7) withj(q2) = 1, the corresponding results a r e

On this basis i t seems unlikely that uncorrelated multihadron modes will account for a very significant fraction of the semileptonic decay.

Assuming that Klv and K*(0.89)lv modes saturate the semileptonic decay and that r ( D -KEY)/ T(D-K*lv)r 0.75, we make a final model comparison with the DORIS data in Fig. 24. The agreement i s quite satisfactory.

IX. CONCLUSIONS

We have made a detailed investigation of hadron- multiplicity effects in semileptonic decays of charmed particles. Our main results a r e as follows:

(i) Hadronic form factors do not appreciably al ter decay distributions; their most significant effect i s to increase decay rates.

(ii) The simplest multiparticle effects a r e associated with kinematic boundaries: Increases in the mass of the hadron decay products lead to more restricted scales of the distributions. For uncor-

V . B A R G E R , T . G O T T S C H A L K , A N D R . J . N . P H I L L I P S 16 -

APPENDIX A: SEMILEPTONIC DECAY FORMALISM

We consider the semileptonic decay process

Y - Z+vX (Al)

where X may be a single- o r a multihadron system. As discussed in Ref. 10 the process (Al) i s described by the squared matrix element

F=Lu,Wua, (-42)

where L a b and Wua a r e the spin-averaged tensors for the L V and YX vertices, respectively,

FIG. 24. Electron momentum distribution for semi- Lab=PtaPu6 + P d r ~ - ~ c r a P r ' Pv- E~6y6P~yf i~6 . leptonic decay of D at pD= 0.5 GeV assuming that the modes D - Kl v and D - K*l v dominate the semileptonic (A3)

width of D with branching ratios 43 and 57% respec- For a single-particle state X, the hadron tensor tively. The data are from Refs. 7 and 8. can be parametrized with structure functions W,:

related multiparticle modes, phase-space effects suppress the distributions near the maximum kinematic endpoints.

(iii) Decay lepton distributions f rom charmed particles produced in deep-inelastic neutrino and muon scattering a r e largely controlled by the fragmentation function. Assuming the fragmentation distribution for charmed fragments i s similar to uncharmed fragments, we obtain a reasonable description of the slow-lepton distributions with D - Kl v and D - K*I v charmed-meson decay modes.

(iv) c-quark fragmentation may not satisfactorily account for antineutrino events with high-energy secondary leptons; 6-quark production gives broader distributions and could account for these events .

(v) Estimated acceptance corrections for undetected slow secondary muons in neutrino dimuon experiments substantially increase the reported r a t e s (to 2-3% of charged-current rates). A charm semileptonic branching ratio B,(charm) -0.25 +0.10 i s then required in the GIM scheme. On the other hand, a 6-quark branching ratio B,(b) < 0.1 limit would be imposed by antineutrino data, assuming the QPM production c ros s section.

(vi) Theoretical estimates suggest that D - K l v and D - K*lv modes in similar proportion approximately saturate the inclusive semileptonic rate; this i s consistent with electron energy spectra f rom DESY experiments.

(vii) Smearing of transverse-momentum distributions of charm decay leptons and kaons due to the t ransverse momentum of the parent i s an ap- preciable, but not dominant, effect.

(viii) The azimuthal correlation between prompt and decay leptons in neutrino-induced dilepton events i s adequately described by the charm-fragmentation model.

where q = -(P, +Pu), P=Pr , and the metric i s a . 6 * = Z . b +a,b,.

In the limit of zero lepton mass, the W,, W, contributions to F vanish, leaving

For higher final-hadron multiplicities, a com- pletely general structure-function representation of Wcla i s unwieldly. However, simple generalizations of Eq. (A5) suffice for the multiparticle modes considered in this paper. Given the form of F, the various experimental distributions a r e calculated by the techniques described in Appendix B, C, D, and E.

For the decay models described in Sec. 111, the squared spin-averaged matrix element F takes simple forms:

(i) Uncorrelated Izadron system: X = K + na. The matrix element Eq. (7) corresponds to

(ii) Resonant hadron system: Y -K*l V, K* -KT. In the narrow-width approximation for K t we obtain

where F describes Y-K*eu . The form of F' depends on the spin of K*. The minimal spin couplings assumed in Eqs. (8) and (9) give, for spin 1,

and for spin 2,


= -3myz +3(py -Px**)Z/mx**, where

( i i i ) Strong-weak sequential decay: Y* - YA, Y - Blu, where B may be a multiparticle state. The decay rate i s proportional to

F=6((pr* +rnyz)F, (F10)

where F describes Y-Blu.

APPENDIX B: SINGLE-PARTICLE DISTRIBUTIONS

For the general decay process

with unpolarized Y , the invariant distribution function for X i s defined as

where s = - (py - pX)' and F i s the squared matrix element for the decay ( B l ) .

Appendix C discusses a general technique for in- tegrating Eq. (B2). In addition, Appendix C contains the functions F for the specific models discussed in Appendix A.

In the following, we abbreviate E q . ( B l ) as Y - X A , with

so that

The formulas given in the remainder of this appendix assume the absolute range o f s given in Eq. (B4) . However, for the resonant hadron systems discussed in Appendix A, the mass-shell constraint reduces the range of s for kaon and pion distributions. These addition restrictions must be added to the s l imits given below.

Let us consider different kinds of single-particle distributions in turn.

Momentum distributions perpendicular to a plane

Let PI denote momentum perpendicular to a given plane (e.g., the plane of the incident u and prompt I in inelastic neutrino scattering). At fixed p,, PIX, one can write

Smax myZ +mXz - 2EL yELX + 2PL yPIX,

S,, ' mAz , (B6)

ELX 1 (mXz +pLX2)1'z, ELY =(myZ +pLyz) l ' z .

If 6, lies entirely in the given plane, Eqs. (B5) and (B6) can be used with PLY = O to compute the dN/dpIx distribution, which then lies in the range

However, i f the initial Y production i s described by

Eq. (B5) must be folded with the production distribution to give

(B9) smb, smax are still as given in Eq. (B6). The l imits on pL are obtained by setting smh =s,,, which gives the following:

For rnx + 0,

P??= P P I X - YELx,

PY$= PPlm +YELX

with

For mx = 0,

with

For the sequential decay process

Z - YB - Y X A (B14)

[where A and B may be multiparticle states in the sense o f Eq. ( ~ 4 ) ] , we must f irst calculate dN/dh,, using Eqs. (B5)-(B9) with A, X , Y replaced by B, Y, 2. Then dN/dpLx can be computed f rom Eqs. (B8) and (B9) . Moreover, i f & l ies in the reference plane, the maximum value of PLY i s

p F = hl'z(mzz, my2, mBz)/(2rnz) (I3151

f rom Eq. (B7) . Substituting this value in Eq. (B10) gives the limit

764 V . B A R G E R , T . G O T T S C H A L K , . 4 N D R . J . N . P H I L L I P S - 16

( m Z 2 + my2 - m B 2 ) ~ " 2 ( m y 2 , mX2, mA2) Equation ( B 1 6 ) is more res t r i c t ive than the l imit p y y = 4 nzZ m y 2

fix" -' ~ " ' [ m , ~ , mx2, ( m ~ +mB)21/(2mz) ( m y 2 +tnx2 -mA2)h1'Z(mZZ, n z g , m B 2 )

. ( B 1 6 ) 4 m Z my2 f o r general Z - X A B decays.

Momentum distributions perpendicular to a line

Let pt denote momentum perpendicular t o a given line. Calculation of dN/dzptx(dN/dptx)/(2.iipt,) is more complicated than diV/dp,, and r e q u i r e s some additional r e m a r k s and notation. Choose the z ax is and y ax i s t o l i e along the re fe rence line and t h e ctx direction, respectively. F o r brevity, denote the x and y components of cty by px, py. Then for given p,, and cty we have

where

smax = m y Z +mx2 + 2PtxPy - 2EtxEt Y ,

smm = m ~ ~ , ( B I B )

E t X = ( m X 2 + p t X z ) 1 ~ 2 , E t y = ( m y 2 + p ~ + p y 2 ) 1 ' 2 .

If F y l i e s along the re fe rence line, setting cty = 0 in ( B 1 7 ) gives the complete distribution dN/d2p tx . More generally, if Y production i s descr ibed by

the complete diV/d2ptX distribution i s given by

T h e l imi t s on P, a r e determined by f i r s t setting Energy distributions g, =srn,, a t fixed p,, and P, l imi t s come f r o m set-

At fixed E,, Eq. ( B 2 ) gives t ing smk = at pr = 0. T h i s gives the following:

F o r m x + 0, diV

p ~ ' " ' ~ ' ' = pptx * y E t x , ( B 2 1 ) where

m, p y x /"" = * - [(prmax- pr)(pY - p ~ i " ) J l / z smaX = m y 2 +mx2 - 2 E y E x +2pyPX

Etr

smin = max ( B 2 4 ) with 13, y a s in Eq. (B11) .

F o r mx = 0, nzyZ +mx2 - 2 E y E x - 2pypx .

p p a x , , F o r p y =0, Eq. ( B 2 3 ) reduces to

with a a s in Eq. (B13) . A s with p, distributions, If E y has t h e distribution the above formal i sm can be used to compute Pt distributions fo r sequential decays. Moreover, Eq. dN/dEy =f ( E d , (B26)

( B 1 6 ) a l so gives fly fo r sequential modes. the complete dLV/dEX distribution i s


dN of Ref. 38; equations f r o m this reference will be prefixed with a K . F o r the decay process Eq . ( B l ) define the variables

By""" dE n f ( E ,) 1 gx(s)ds . ( 8 2 7 )

S . mm s =-(PY -PxI2 ,

Setting s,, =sn, gives the following limits: F o r mx + 0 ,

Emyax= PEx + YPX , (B28)

my, f o r E X < E i ,

@Ex - ypx, f o r E x > E i . F o r mx = 0 ,

ETA'= m ,

( m , f o r Ex< E:, (B29)

a, p, y a r e a s given previously, and

E: =(m$ +mx2 - m A 2 ) / ( 2 m y ) . 0330)

APPENDIX C: SINGLE-PARTICLE INVARIANT FUNCTIONS

General f o r m a l i s m s . The technique used t o evaluate the integral Eq . ( B 2 ) i s essentially that

7

~ , = - ( p ~ - p ~ - C p ~ ) ~ , l ~ r ~ n - 2 I =1

( C 1 ) u , = - ( p y - p x - p , + 1 ) 2 , 1 5 ~ ~ n - 2

t r = - ( p Y - p r + 1 ) 2 . O C r s n - 2

(here p , = P A ) . F o r th ree- and four-particle decays, a l l s c a l a r products p i . p , can be written as l inear combinations of the above variables with constant coefficients. F o r five o r more part ic les in the final s ta te , some of the products requ i re nonlinear expansions.

With the squared mat r ix element F written a s

F = F ( s r , u , , t , ) , ( C 2 )

g x ( s ) i s given by a (312 - 5 ) dimensional integral, E q . (K5) . The integrand of E q . ( K 5 ) i s mildly singular a t the endpoints of the u , , t , integrations. F o r purposes of numerical computations, i t is convenient t o make the change of variables

t ,=A ,+B,6 , , (C3)

where

With the above change of variables , g x ( s ) is given by

where the integration l imi t s a r e given by t O A = n ~ y 2 + m , 2 - ( s + m y 2 - m , 2 ) ( s + m l ~ s l ) / 2 s ,

U : =u,~*U,.q

with

If F has s imple t , dependence, some of the integrations in Eq. (C8) can be performed analytically.

F o r a given decay model, the p rec i se f o r m of

766 V . B A R G E R , T . G O T T S C H A L K , A N D R. J . N . P H I L L I P S - 16

F depends on the ordering of the final-state par - t i c les . We next s u m m a r i z e the integrands used in the various models discussed in this paper . Un- l e s s otherwise noted, the integration l imits a r e those given in E q s . ( C 9 ) - ( C l l ) .

Electyon invariant f i~nc i ions . We have the fol-

lowing: Y - K e v .

F = (We .py)(2P,, . P y ) + m y 2 ( 2 ~ e . P u ) . ( ' 3 2 )

Take p, = p , in E q . (C12) :

F = s t , - m y 2 m K Z . ((33)

Y - K * e v ( @ e v ) . In the narrow-width approximation, the subsequent decay of K* does not affect the electron distributions:

F = -(2Pe .P,)

+ [ ( 2 p Y . p e ) t 2 p y . p , ) + m y L ( 2 P , .PI, ) l / ( 2 m K * 2 ) .

((214)

Take p, =p , , p, = p , in Eq . ( C 1 4 ) :

F = -s + m y 2 i 2 +rnK*' + to ( s - 2 m K * 2 ) / ( 2 m K * 2 ) .

((215)

Y -K**eu.

F = -Wl(2P;P1,

+ w 2 [ ( 2 p , .PY)(2PU . P Y ) +my2(2pe .p , )1 / (2mY2) 9

(C16)

W , = - 3 t n y 2 + 3 ( p K * * . p y ) Z / m K * * 2 ,

LV, = m y 2 ~ , / m , * i ' + ) n y 2 ( 1 +pK** .py /mK**"2 .

Take p, = p K t * , p 2 = P , in E q . ( C 1 6 ) :

F = tow, + w 2 [ ( s - n ~ ~ ) ( r n , + + ~ - S - to ) - t o m Y Z ] / ( 2 f i z y 2 ) ,

( C 1 7 ) where W , , W 2 a r e given above with

2 p K * * . p y = t O - m y 2 -m,**'. (C18)

Y - K n e v , Y-Knaev . Take P , = p , in E q . (C12) :

F = s t , - s , n ~ , ~ . ((39)

Y - B A - B K e v , Y - B A - B K n e v . Assume a narrow width for the intermediate s t a t e A :

F = 6 ( ( p y - p B y + m A 2 ) F , ( C 2 0 )

F = (2PA.P,)(2PA .P,) +m,Z(2pe a @ , ) .

Take p, = p B , p, = p u in E q . (C20) :

F = 6 ( t , - m A 2 ) F ,

where we substitute s , = m K 2 in the A - K e u case . The mass-shel l constraint resu l t s in a restr ic t ion of the s , s , physical regions. Let 3. be the solu- tions of t i = m A 2 . Then, the new s , l imits a r e

s ; = max icz m a '

The range of s i s res t r i c ted by the requirement

Kaon inz~ariant fitnctions. We have the following: 1'- K c v . Take p, = p e . P Z = p , in Eq. (C12) :

+ [ ( 2 p . - p e ) ( 2 P y - p , ) +mY"2Pe . ~ , ) ] / i 2 m ~ * ~ ) .

Take p, = p T , p 2 =p, ,p, = p , in E q . (C25) :

F = 6( t , - t 0 ) F ,

F = s , - n z y 2 / 2 + t 1 / 2 + t , ( m y 2 -s l - t 1 ) / ( 2 m K * ' ) ,

( C 2 6 ) - t O = s l + m y Z + n ~ ~ ~ + m ~ ' - s - m ~ * ~ .

((227)

In correspondence to Eq. ( C 2 2 ) , the s , s, physical .-

region i s determined by solution of t = t o . Y - K * * e v - K n e v .

with W , , W , a s given in Eq. ( C 1 6 ) . Take p, =p , , p2 = p e , p, = p , in Eq. (C28) :

F = d ( t , - t , ) F ,

F = W , S ,

+ ~ , [ ( t , - m y 2 ) ( m K * * " s s , - t , ) - m y 2 s , 1 / 2 m y 2 . ( ~ 2 9 )

The s t ruc ture functions W,, W 2 a r e evaluated using

2 p y - p K * * = s l - m K * f 2 - m y 2 . ( C 3 0 )

The s , s , physical region is again reduced by the -

constraint t o =to, where - - F = ( s , - m A 2 ) ( t , + s + s 2 - m y 2 -u l - s ,) ( ~ 2 1 ) t o = ~ 1 + m y 2 + m K 2 + m n ~ s - m K t * 2 . (C31)

+ m A 2 ( s + 1 , - m y 2 - u , ) , Y - K n e v , Y - K n a e v . Take p , = p e , p, = p , in Eq.


((32): Noting that s,-, = m n ,,-,2, we consider the func-

~ = ( t ~ - m ~ ~ ) ( t ~ - m , ~ ) + m , ~ ( s ~ + u , - s - s , ) ,

(C32)

where we substitute s , = m n 2 in the Y -Knev case. Y - B A - B K e v . Take p , =p,, p , =p,, P, = p , in

Eq. (C20):

F = 6 ( t , - m A 2 ) F ,

F = ( t l + s - s 1 -u1 -myZ) (my2+m,2-m,2+ul - s - t , )

- s1mA2 . (c33) The mass-shell constraint again results in a re- striction of the s-s , physical region, as in Eqs. (C22) and (C23).

Decays to more than four final p(zrtic1es. For such decays, not all the scalar products pi .PI o f final momenta can be expressed as simple linear combinations of the invariants s , , jc, , t r ; some nonlinearity is inevitable. This i s because no five vectors p, can be linearly independent of four-dimensional space-time, leading to the vanishing of the Gram determinants

det[pi .p ,]=O, i , j = l , . . . , 5 (C34)

which give nonlinear constraints (see Ref . 39).

APPENDIX D: INVARIANT-MASS DISTRIBUTIONS

General formalisms. We consider here the distributions dN/dm,, where

ma , = -(P.+P*)' . ( D l )

Evaluation of these distributions is accomplished by the invariant integration techniques of Re f . 38; we again prefix equations from this reference with a K.

For the general decay process

Y - A , A , . . . A , , (D2

we define the variables

tions

to obtain

dh' -= 2mn,n-lg(mn,n-12) . dm,,,-1 (D5)

With the squared matrix element F expressed in terms of the variables s , , t ,, y , ,g(s,-,) is given b y a (3n - 8)-dimensional integral derived from Eq. (K42), with an interchange in s j integration order according to Eq. (K17). As was the case with evaluation of single-particle invariant functions, numerical integration is more efficient after the change of variables,

where

With these variables we obtain, up to an overall constant, the result

768 V . B A R G E R , T . G O T T S C H A L K , A N D R . J . N . P H I L L I P S

The integration limits for Eq. (D11) a re F = 6(t2 -mK*2)F,

n -2 (D20)

) I ! . ] ' , F=my2 + t2 - u, - u2 +(u,u, -my2 t2 ) / (~mK*2) ,="+I 0312) The mass-shell constraint res t r ic ts the s j - u,

s: = [ ( s r - , )1 /2 - m r ] 2 ,

u + = u ? A * u ~ B 9

where

physical region through the solution of t', =m,*'.

( D 1 3 ) Replacing K* by @, n by R, Eq. (D20) can be used to describe F- @ev-Kxev .

Y-K**ev-Knev. Take fi2 =p,, p, = f i e , p, =p , in Eq. (C28):

zlTA = m r 2 +m,+12 F =6(t2 -ma**Z)F, - ( ~ , + r n , + ~ ~ - s,+,)(mY2 + S T - s: ) / ( 2 s r ) , F = Wl(my% t2 - uu, - u2)

Z C r B = A ~ / ' ( s , , m,.+12, . S , + ~ ) ~ ' / Y ~ y Z , s , , s: ),'(2sT) . -4 &(t2 - ulu2/mr".

(D14)

For the decay

Y - A K L :

\ - ,

Evaluation of the structure functions of Eq. (C16) i s completed with

(Dl51 2pY . P,@+ = -24, - U, . (D22)

the m,, range i s (taking m , = O ) Y -Knev . Take P, =P,, P, =Pe, P4=PK in Eq. m K 2 4 m K E 2 s ( m y -mAYa. ( ~ 1 6 ) (C12):

Here A may be a multiparticle state in the sense F=(u2 -my2) ( s , -my2 ) +my2( t2 +s, - u , -mK2) .

of Eq. (B4). For the sequential decay mode (D23)

Y - Btv- KALu,

the upper bound on m,, i s

(Dl71 Y-Knnev. Take P, =PN, P4 =Pep Ps =PK in Eq. (C12):

(my2 -m2) F = (u, - mY2)(sl - my2) +my2( t3 + S , - t2 - mK2) . m,: c m K 2 + [rnB2 + n k 2 -mA2

2mB (D24)

+ ~ ~ ' ~ ( m ~ , ~ m ~ t ~ m ~ ~ ) ] . Y - K A -KBev.

(D18) F = 6 ( ( p , - pK)2 +nzA2)F,

Since a zero-energy lepton i s kinematically allowed - (D25) F = ( ~ P A ' Pe)(2PA ' t)u) +mAZ(2Pe ' PI,) ,

in the decay scheme (D17), the lower bound on m,, i s still m,. Take PI =A, P, =PK, P4 =pe in Eq. (D25):

Details for speczfic models. We have the follow- F=6(u2 -mA2)F ,

ing: (D26)

Y - Kev. Take pl =p,, P2 =Pe in Eq. (C12):

Y-K*eu-Knev . Take P,=P,, P,=P,, P,=PK in Y-KA-KB,B2ev . Take P, =P,, P4=PK, P 5 = P e in E q . (C25): Eq. (D25):

APPENDIX E: TWOPARTICLE CORRELATIONS where A may be a multiparticle state. Define the variables

General formalism s = -(P, - P*)2,

We consider correlations among decay products a t fixed momentum of the parent particle. In order to = -(PY - to use the formalism of Appendix C, we adopt the S1 = - (PY - P X - notation

Y - X A I A , The invariant two-particle function h ( s , to, sl) i s de-

(E 1) fined by


dN=* a h ( s , I,,, s,) . E x El

Conservation of four-momentum in ( E l ) yields the following physical s , s,, to region:

(m, +rnA)' S S s ( m y - m X ) 2 ,

mA2 S S , ~ ( & - r n , ) ~ , ( E 4 )

tA - tB S to -i tA + tB , where

Unlike the single-particle distributions discussed in Appendix B, i t i s not feasible t o present analytic expressions f o r general two-particle distributions. We give a Monte Car lo procedure f o r correlat ion calculations.

We wr i te the net momentum t rans fe r to part ic les A , and A,

T h e decay ( E l ) can be described by the var iab les E Y , Ex , El , Bt77, 4 where

and + is the azimuthal angle of Gl about E, measured f r o m the (kt, Gy) plane. It i s useful to introduce the additional var iables

and henceforth t o wr i te pj f o r 161. Then Eq. ( E 3 ) can be rewri t ten a s

with

s =my2 +mx2 - 2EyEx +2PYpxP,

s, = s +m12 - 2QE, +2kpl?7, @ l o )

to=my2 +m:-2EyEl +2Pyp1d .

At fixed P y , a n event i s a choice of Ex, P, E l , 17, and cp consistent with the physical region res t r i c - t ions of Eq. ( E 4 ) . Any des i red two-particle quantity can be computed f r o m the chosen variables and binned with the weight

W = ( E i - E,)(Pt - P-)(E; - E;)(q+ - 7 7 - )

X (@' - @-)P,Pxh(s, 4, s,) , ( E l l )

where + and - supersc r ip t s denote maximum and minimum values, respectively.

We presen t next a summary of the l imi t s on the

variables Ex, B, El , q , and 4. T h e prescr ipt ion below a s s u m e s A i s a multiparticle s tate . Modifi- cations f o r single-particle A a r e given la te r .

E x l i m i t s . Define

EJ = ( m y Z +mx2 - ( m , +mA)z) / (2my) , ( E 12)

p;t = ~ " ~ ( m ~ ~ , m ~ ~ , (nz, + m A I 2 ) / ( 2 m y ) ,

EC, =myE$/mx. (E l31

Then a t fixed py,

E ' ;=(EYE:+PyPZ) /m~, ( E 1 4 )

E , = i m,, E Y ~ E C , (El51 ( E Y E : - P Y P : ) / ~ Y , EY 2 EC.

p l imi t s .

p' = m i n ( E 16) +BEYEX - m y 2 -mX2) /2PYPX,

P-=max

il ' i -l

(E 17) ( ~ m i n + ~ E Y E x - m y 2 -mX2) /2pYPX.

where s,,, s,, a r e given in Eq. ( E 4 ) .

El l imi t s . At given E,, Ex , 8 , compute s according to Eq. (E lO) , and

Q = E y - E 1 , (El81

k = ( & 2 - ~ ) " ~ .

Define

E:= ( s +m,Z - mA2)/2&, (El91

P: = hU2(s , m12, mA2)/2&,

Q, = &Ej+/m,. (E20)

T h e allowed range of El is then

E: =(QE; + k p ~ ) / & , (E21)

E;=

m,, QSQC (E22)

(QE: - k p : ) 6 , Q 2 Q C .

17 l irnits.

17' = min (E23) ( s , I ' ,,, +2QE1 - m12 - s ) / 2 k P 1 ,

q - = m a x f ( E 2 4 ) (s,,, +2QE1 -m12 - s ) / 2 k p 1 ,

where s , ,, , s , ,,, a r e given in Eq. ( E 4 ) . @ l i m i t s . The azimuthal angle + has the l imi t s

ot = 2 r , @ - = O . (E 2 5)

to evaluation. The expression f o r to in t e r m s of the event var iab les of Eq. ( E l l ) is

to =A + B c o s q , 0326)

7 70 V . B A R G E R , T . G O T T S C H A L K , A N D R . J . N . P H I L L I P S - 16

where Two-particle invariant functions

Let F be the squared matrix element for the de- A =my2 +m12 - 2ErEl +2prP15q,

( E 2 7 ) cay ( E l ) . Construction of h(s, s,, to) depends on the multiplicity of the state A.

B=2prp1[(1 - n/lultiparticle state A. Write ( E l ) as

with

E =(Pr - PPx)/k

Y - X A I A z * * * A n - (E29)

Use the variables o f Eq. ( C l ) and the variable (E28) transformation of Eq. (C3):

with integration limits for s,, u, as given in Eqs. Since this i s a spin-averaged quantity, the decay (C9) and (C11). K* - Kn i s isotropic in the K* rest frame with

Single-particle state A. In this case, h i s given

by EK = (me2 +mKZ - mn2)/(2mK*) . 0335)

h(s , sl, to) = F(s , to)d(sl - m ~ ' ) . (E31) Then generating an event for Y - K*ev yields values for Ce, GK*. Choosing a random direction for cK in

The modifications in the general Monte Carlo pro- the K* rest frame, and subsequently boosting along cedure described previously are as follows. (i) q jK* b y y = E,*/m,* gives cK in the decay frame to i s no longer a free variable, but i s specified by

complete description of the event. 77 =(mA2 +2E,Q- s -?~z ,~ ) / (Zp ,k ) . (E32) Y - Kne v. The choice for pl , Px here depends on

the quantity being calculated. For total hadron en- The factor (17' - q-) in Eq. ( E l l ) i s correspondingly ergy take px =p,. pl =p,, so that E , = Er - Ex - El. replaced by 1/(2p1k). With F as in Eq. (C12),

( i i ) In Eq. (E22), E; i s given by (QE: - kp:)/& for all values of Q. h = (st, - slmr2)~"2(sl , mK2, mT2)/s1 . (E36)

For Ke correlations, take px =P,, PI = P K , SO that Details for specific examples

F = t1s - mr2u1 . W e summarize here the forms of h used in the 033'7)

two-particle examples discussed in this paper. The integrations of Eq. (E30) are easily performed Y - Kev. Take px =Be, P1 =PK in Eq. (C12): to give

Y-K*ev-Knev. Because of the K* mass shell x [s(QO +Q2tO) + u l ~ ( s Q 1 +sQ3f0 - ~ Z ~ ' ) ] / S ~ , (E38) constraint, it i s better to describe this decay by an extension of the Y -Kev method than by the general where u l ~ i s as given in Eq. ( C l l ) , and the coeff i- multiparticle formalism. Consider f irst the decay cients Q, are determined by decomposing A, o f Eq. Y-K*ev. Take & = f i e , pl=pK* in Eq. (C14): (C3) as

F = to(2mK*2 - s ) + (mr2 - s ) ( s - mK*2) . (E34) Al = QO + Qlul + Q2t0 + Q3u1t0. W39)

*Work supported in part by the University of Wisconsin Research Committee with funds granted by the Wiscon- sin Alumni Research Foundation, and in part by the Energy Research and Development Administration under Contract No. E(l1-1)-881, COO-564.

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