Inclusive photoproduction cross sections of charmed mesons and baryons

Z. Phys. C - Particles and Fields 11,211-222 (1981) for Physik C

and Finds �9 Springer-Verlag 1981

Inclusive Photoproduction Cross Sections of Charmed Mesons and Baryons*

M. Fontannaz 1, B. Pire 2, and D. Schiff 1

1 Laboratoire de Physique ThSorique et Hautes Energies**, Universit6 de Paris-Sud, Bgtiment 211, F-91405 Orsay, France 2 Centre de Physique Th6orique, Ecole Polytechnique***, F-91128 Palaiseau Cedex, France

Received 26 June 1981

Abstract. The various mechanisms which may be responsible for charm photoproduction are reviewed and compared to experimental data. The photon- gluon fusion process provides the dominant contribution to the total cross-section at present energies. Emphasis is given to the final state structure. Assuming that hadronization takes place through string fragmentation, we calculate rapidity distributions and correlations for charmed mesons and baryons.

I. Introduction

Photoproduction as well as hadronic production of charmed quarks may be described in the framework of perturbative QCD. It is indeed commonly assumed that the high masses involved establish the small distance nature of such processes thus allowing a perturbative approach ; the lowest order contributions to heavy Q(~ pair production are then given by the qEI-,Q0 and gg--'QO, subprocesses in the hadronic case and by gT--'QQ one in the photoproduction case. Such an approch leads to total cross-sections in agreement with data for hidden charm as well as for open charm production [1]. The theoretical calculations are, however, flexible and the prediction depend on the gluon distribution and on the assumed mass of the heavy quarks. A detailed discussion of these effects will be given in Sect. II.

The QCD subprocess does not specify, however, the final state hadronization, that is, the way c and quarks turn into charmed mesons and baryons. Further assumptions are needed and we shall borrow

* Dedicated to Vladimir Kislik (detained in a Soviet Camp after the authorities refused him the right to emigrate) ** Laboratoire associ6 au Centre de la Recherche Scientifique

*** Groupe de Recherche du C.N.R.S.

them frorn the dual model and the 1/N expansion approach to the strong interactions [2].

We are thus able to describe the D, /5, and A~ + production, the rapidity distributions and correlations of the charmed hadrons, the respective yields of mesons and baryons and the energy dependence of these quantities. These predictions are more complete than those previously obtained for the total cross-section only. They should lead, when compared with data, to a better understanding of the production mechanism and of the final state hadronization. The fusion mechanism implies, for instance, specific rapidity correlations, whereas the rapidity distribution of the 15 meson should allow to determine the ~ fragmentation function.

Our paper is organized as follows. Section II dis- cusses the various mechanisms involved in the photoproduction of charmed hadrons; the total cross- section is calculated in the ;~g fusion model and compared with data. In Sect. III we outline the model used to describe the c and ~ quark hadronization and we calculate the/5o rapidity distribution. In Sect. IV we extend the model to the production of charmed baryons and calculate rapidity distributions as well as relative yields of D o and A~ +. Rapidity correlations of charmed hadrons are also given. Section V is devoted to the possibility of producing charm particles from an intrinsic charm component inside hadrons, especially through diffractive dissociation. In the conclusion, we sum up the features of the model and discuss its application to virtual photoproduction.

II. Total Cross-Sections

A. Review of the Various Contributing Mechanisms

1) Photon-gluon Fusion Mechanism

The lowest order QCD process wich contributes to atot(?P-'cdX) is photon-gluon fusion : ~g-~cg (Fig. 1). It

O] 70-9739/81/001 !/02] 1/$02 40

212 M. Fontannaz et al.: Photoproduction of Charmed Mesons and Baryons

P

Fig. l. The photon-gluon fusion mechanism

(a) (b)

(c) Fig. 2a-e. Higher order contributions to charna photoproduction

V V

Fig. 3. Hard production of charm through lhe vector meson dominance model

p P p (a) (b)

Fig. 4a and b. Diffractive dissociation of the ~o dominated photon a in the Regge language through Pomeron exchange; b in the QCD language

has already been proposed by many authors [3] for explaining open charm photoproduction (in a similar way as its analogue in hadroproduction [1] : gg~cd) . We shall analyze its contribution in detail and show that it explains the data.

Another approach which has been suggested is the generalized vector dominance model [4] (GVDM). Both frameworks: GVDM and QCD yield similar

results [5] for the total charm photoproduction cross- section. We may, thus, assume that the infinite sum

?(V) + p ~ Vp is dual to the photoproduction of V = tp, ~p ' . . .

a free c~ pair and shall not add the two contributions.

2) Higher Order Corrections

The higher order term drawn in Fig. 2a is expected to be of order G/n compared to the lowest order. This is due to the fact that the only available scale is m~, which

/ QZ..~m2\ makes l adin corrections O/Co )

small. On the other hand, the higher order diagrams drawn in Fig. 2b, c involve light quarks : corrections of

order O Log may be of some importance. In order

to evaluate them, let us consider the corresponding photon-hadron-like corrections drawn in Fig. 3. Previous experience in similar problems, in large Pr hadron photoproduction [6], teaches us that in the present Q2 range: Q2_~m~ point-like and hadron-like photon contributions are comparable (for higher Q2, the point-like "'anomalous" contribution should be dominant). Let us then estimate the hadron-like contributions drawn in Fig. 3 by writing:

a(Tp--+ c-dX) = c~ ~ 4n v=~,o~,~,~, f?~ ~(Vp-~c-~x)

4n o - ~ ~176 (1) ~-c~Z~vZ~(n p-~c-dX)- 200

We shall assume, neglecting detailed effects due to D* decays and F production that

~(},p--, ~~ ~_ ~r(yp --, D - X) ~- �89 r ~ c-dX) . (2)

At energies E~ = 50GeV and E~= 100 GeV, the pion cross-section may be estimated from [7] to be of order 5 and 10pb respectively, leading to a contribution to r176 10 and 20nb which may be neglected.

3) Diffraction Dissociation

Photon-gluon fusion and higher orders discussed above will be shown to correspond to inelastic final states (Fig. 8). On the other hand, diffractive dissociation leads to a different type of final state structure with quasi-elastic production of an excited object which decays into charm-anticharm pairs (Fig. 4). We assume that the ratio of diffractive dissociation to inelastic production is the same here as in hadron-hadron collisions i.e. only of the order of a few percent leading to a negligible contribution to the total cross-section. Notice that the order of magnitude found: a few tens of nanobarns, is similar to the "elastic" ~p-photoproduction [8]. In this last case, the mechanism at work is also 7g~cK Below open charm

M. Fontannaz et al. : Photoproduction of Charmed Mesons and Baryons 213

threshold however, the final state structure is corn- p N2//.D ptetely different: the assumption of color evaporation ~ ~ II ~ b [-9] implies emission of soft gluons so that in fact the ]1 ~P photon-gluon mechanism in this case may be viewed as diffractive (Fig. 4b) and of the same type as the P diffractive dissociation graph drawn in Fig. 4a.

4) Intrinsic Charm

i) Diffractive Dissociation. Another source of diffractively produced DD or AcA~ pairs in either the photon or the proton direction is the possible presence of an "intrinsic charm" component [10-1 in the hadronlike photon (dominated by ordinary ~, co, ~b, ( . . . mesons) or in the proton. The former may be estimated from recent measurements of diffractive charm production in rc-p collisions at Fermilab [11]. The cross-section a(rc-p~DDX) is found equal to about 40~tb at

]/-S-= 20 GeV. Using (1) and (2), this implies a contri-

bution to o-(?p~b~ at ] / s=20GeV. It is easy to infer the cross-section variation with energy:

da 1 writing ~ oc ~-~ for the diffractive production of an

excited object of mass M 2, we get

(a-~)~ dM 2 _ x l ) s a = % ~ M2 = % L o g ( 1 (3) Mg M2 "

x 1 is the minimum value of x v for the recoiling hadron (Fig. 5) in diffractive events; we may take x~ -~0.8. The threshold energy M 2 is in the present case equal to" M~ =4m 2. This leads to a contribution to a(7p--,D~

25 nb at 1 ~ = 10 GeV. As for diffractive production in the proton direc-

tion, we may estimate its importance using formula (3) and rescaling the value of o- o given by Brodsky et al., in pp collisions by the ratio of yp to pp total cross- sections. The value of Mo 2 should now be taken to be M~=(mA++mb) z and x~ is now associated to the vector meson recoiling against the diffractively produced large mass object. This leads to a contribution

to a(Vp---,c-dX)~5Onb at ] /~=10GeV and 300nb at

= 20 GeV. When comparing models to data, it is thus nec-

essary to distinguish diffractive from non diffractive contributions; we shall discuss this question in the next subsection B

ii) Central Production. The hypothesis of an intrinsic charm component in the proton implies the possibility of producing charm in yp collisions via the Compton process ~,c-~9c (Fig. 6). Similarly, the intrinsic charm constituent of the photon may undergo a hard col- lision with a gluon or a quark in the proton. We shall show in Sect. IV that this type of contributions to the total cross-section is small and may be neglected.

Fig. 5. Diffractive dissociation of the light vector meson dominated photon into a Dr) pair

~ c

P g

Fig. 6. The Compton graph with the intrinsic charm component of the proton

B. Contribution of the Photon-Gluon Fusion Mechanism

The contribution to the total cross-section of the photon-gluon fusion mechanism is straightforwardly written under the form'

1 1. da ato,(Tp ~ b ~ ~ -2 0"tot(Y p ~ c'dX) = ~ j dy ~yy

with

do- _ rc p} ~ Q2) 93l 2 e y 0 ( 1 - x ) . (4) dy s ]/~ o ~ dp~G~ 4~ m•

y is the c or ~ quark rapidity, m I the charm quark

transverse mass: m I = ]/~l + m~ , G ~(x, Q2) is the gluon structure function with x given by

m• x = . ( 5 )

eY(]/S-mae ')

The definition of Q2 is somehow arbitrary as usually in leading log calculations; we shall perform the calculation for Q2 =4rn~ and Q2=m2.

The matrix element is given by

).

12rc with ~s(Q2)= 25 Log(Q2/A 2) (A=0.5 GeV) and g, t, fi

the Mandelstam invariants associated to the photon gluon fusion subprocess. Let us study the sensitivity of O'to t t o the value of me, the choice of G~(x, Q2) and the definition of QZ. For Gg(x, Q2) we shall successively try


-1000

"~ 500 (el

20 40 60 80 100 120 1/,0 160 El" (GeV)

Fig.?. The cross-section for 7p-*D~ Curvea: m~=l.2GeV, QZ:4m~, G(x,Q 2) taken from [-13]. Curveb: mc=l,6OeV, Q2=4m~, G(x,Q 2) idem. Curvec: m~=l.2GeV, Q2=m2, G(x,Q 2) idem. Curve d:mc= 1.2GeV, Qz =4m~, G(x, Q2) taken from [12]. The experimental points are from [-14] and [15]

the parametrization given by Owens and Reya E12] and by Baier et al. [13]. As can be seen from Fig. 7 the value of O-to t depends crucially on the three effects. No threshold effect has been included in the calculation except ~>4m~ z implicit in (4). In Sect. III, we shall discuss threshold behaviour in more details.

The available data come from two experiments: i) the WA4 collaboration data [14] with photon

energies 20 < E~ < 70 GeV, ii) the CIF collaboration data [15] with

E~ > 80 GeV. The comparison of the data with the fusion mecha-

nism is made somehow delicate, due to the presence of diffractive dissociation. Contrary to the situation in hadronic collisions, the photon-gluon fusion kinematics is such that the so-called central events overlap in part the forward region, populated by diffractive events. In particular, the acceptance of the CIF experiment favors the forward region so that some care is necessary to compare theory to data. We estimated above the forward diffractive production at Fermilab energies to be of order 100 nb. A conservative attitude is to consider the CIF data on Fig. 7 as an upper limit for the central b ~ production, to be compared with the photon-gluon fusion model. On the other hand, the WA4 collaboration does not see any very forward events xv>0.5 [5] whereas ~xr)---.5 in the CIF experiment. Another discrepancy between both experiments lies in the high value of the cross-section found by WA4 at large energy.

This discussion allows concluding from Fig. 7 that the general trend of the data agrees with the 79 fusion mechanism. A satisfactory fit is obtained with

me= 1.2 GeV. Notice that a similar low value of m c is also used in charm production in hadronic collisions [1]. From now on, in this paper, we shall adopt the following set of conventions" mc = 1.2 GeV, Q2 =4m2; Go(x ' Q2) taken from [13] - corresponding to Curve a in Fig. 7. The sensitivity of O-to t to the choice of conventions implies that, when investigating rapidity distributions, it is better to calculate normalized distributions :

dN 1 da dy ato t dy"

III. Inclusive Production of/~o

Having successfully described the total cross-section for charm photoproduction, we now turn to an at- tempt to treat the structure of the final state. This may be of course more model dependent. Topological arguments allow us however to handle the (long distance) structure of hadronization in the final state. This is what we are going to develop in order to get the rapidity distribution of the charmed mesons and baryons.

A. The Model

Hadronization of the produced partons is commonly seen, in the framework of QCD, as due to the sepa- ration of color fields. Describing completely this process is out of reach of the present understanding of the confinement problem. The ideas of the topological expansion allow us, however, to clarify the basic picture of final state interactions. The mechanism for jet production is then constrained by topological prescriptions, the leading topology corresponding to planar diagrams. We have used this picture, which has been emphasized by Veneziano [2], in a preceding paper [16] for the case of large PT reactions. It has also been applied to e+e - reactions [17]; recent results at PETRA [18] have indeed Shown that the final state structure of 3 jet events favors such a model In the particular case of interest here, the c~ pair hadronization proceeds through the fragmentation of two strings which stretch between the ~ and a quark q of the proton (string 1) and between the c quark and the remaining "diquark" of the proton (string 2). This configuration, which is the lowest order term of the 1IN expansion, is schematically represented on Fig. 8. Restricting ourselves to the observation of/5 ~ that is to string 1, we may write the inclusive cross-section as

1 9~ 2 dk 1 dk 2 da=~dxldx2G(xl,x2, QZ)~s ~ ~ co2

�9 5 ( k + K - k 1 - k2) D ( z +) dz +, (7)


~ k z

P K q - - v ~ ~ J------

~-~o

Fig. 8. A schematic representation of final state hadronization in the ~g fusion model

where G(xl, x2, Q2) is the joint probability of finding a gluon with momentum fraction x~ and a quark with momentum fraction Xz(1-x l ) [i.e. a diquark with momentum fraction ( 1 - x j) ( 1 -x2 ) ] in the proton, D(z +) describes the fragmentation of string 1, in its own center of mass system in to / ) ~ (Note that in such a model, the fragmentation of a patton is a meaningless concept, only a color neutral system being taken as a candidate for fragmentation.) z + is the light cone variable defined as

z+ = ~ (8a)

where p* is the /3 ~ momentum (p*ll its component parallel to k~) and P* the string momentum, the star denoting the string center of mass system (CM*) or any system deduced from it by a Lorentz transformation along the longitudinal axis k*. We neglect here the transverse momentum of fragments. In CM*, denoting the string invariant mass by Mz, one has

z + _ (p0 + pll), (8 b) M1

Since we take the diquark as massless, we also have

(p~ +pill* = . ( 8 c )

Let us now find the relations between t he / ) ~ momentum (described by its rapidity y and transverse part ps) and the ~ momentum (rapidity y~), in the 7P center of mass system, when z § is kept fixed. The easiest way to get these relations [16] is to go to the quark q rest frame. If the mass of this quark is negligible, the relations do not depend on the quark momentum�9 A straightforward calculation yields

p•177 = z + , (9)

m• y' = z + (10)

with

= + ( t 1 )

and the coordinate transformation reduces to

dk i • 1 = dp• dy/z + 2 . ( 12 )

We can now write the inclusive cross-section as

d~ _S dx~dx2G(xl,xz, Q2)~ ~ 2 dpidy

�9 6(~ + ? + fi - 2m 2) ~ dz + (13)

I

which may be written as

da _Sdx 2 G(xpx,z,Q 2) 1 ~ 2 D(z +) + dp• s - ]/~m• + ~ z+---~-dz

(14)

with

2 2 + x l=(m• 2rnZ~)e-V(z+ V-smi-e,m2). (15)

B. Choice of D(z +)

We now have to specify the fragmentation function of the string. Assuming that only the half of the string (in CM*) which contains the ~ quark can fragment in a / ) ~ we have to impose that

z + > mo/M 1 . (16)

(Note that M 1 is, at fixed Pi and y, a function ofz +, so that this condition leads to a somewhat complicated expression for + Zmin=mo/M1.) Because of mass effects, the fragmentation function of the charmed string into a charmed meson is not much suppressed near z + = i. In the case of an infinitely heavy quark, a good approxi- mation would be a 6(z + - 1) form [19]. There is not much experimental information on this function Db~ +) (which we will assume to be scale invariant), but a rather fiat behaviour seems compatible with the data from e+e - annihilation [20]. In order to test the sensitivity of our computation to this model dependent input, we will choose two somewhat extreme shapes as follows

OO~(z+) = 1 - z + (1 + 2 o (z+-z+mi . ) , (17)

- - Z m i n )

+ + Z - - Z m i n { ~ ( Z + +

Db~(z+) = (1 + 2 - - Zmin) (18) - - Z m i n )

which are normalized by the relation 1

DN(z+)dz + = 1/2. (19) Z+min

C. Results

We present in Figs. 9 and 10 the rapidity distribution

dN _ 1 ~ , da(yp~ ~ X ) '

dy O'to t j api


-0.5 d N D~ 50GeV) dy

-0-4 .'7..S .....

-0.3

-0.2

-0.1 b) ~" , i t I r

-1.0 0. 1.0

Fig. 9. The rapidity distribution of the /30 for E~ = 50GeV. The Curvesa and b correspond to D ( z + ) o c ( 1 - z +) and to D(z+)ocz +

+ respectively. The dotted curves are obtained with a quark - - Zmin~

distribution given by formula (23), the full and dashed curves with a distribution given by formula (22), with ~=0.5 and c~=0.4, respectively

-0.5

-0.4

~ 6

dN (E.~=100 GeV) dy

{a) (b) (c} - 0.3

-0.2

-0.1

-1. 0. 1. 2. Fig.10

Fig. 10. The rapidity distribution of the / ) ~ for E~ = 100 GeV. Curve a corresponds to D(z +) oc 1 - z+, Curve b to D(z +) ocz + - Z+mi,, and Curve c t o D(z+)oc6(1-z+) . They are calculated with a quark distribution 5(x2- 0.5)

which is normalized to 1/2. Calculations have been pursued with m~= 1.2GeV. For simplicity, the two parton joint distribution function G(x 1, x 2, Qz) has been given a factorized form

G(xa ' x2 ' Q 2 ) = Go(Xl , Q Z ) G 2 ( x 2 ) ,

where Go(x 1, Q2) is the gluon structure function [13] and G2(x 2) describes the unknown distribution of a dressed light quark of momentum (1-x l )x2P. This latter will be taken as normalized by the two following sum rules

dx2G2(x2) = 1, (20)

(1 - x 1 ) x 2 G 2 ( x z ) d x 2 = (1 - x1)c( , (21)

where (1 - xa ) e is the proton momentum fraction carried by the quark. We will choose two different scale invariant forms for this function, namely

G2(x2) = a ( x 2 - 0c), (22)

G2(x2) = 12X2(1 - - X2) 2 . (23)

The latter choice corresponding to c~=0.4. Figures 9 and 10 show clearly that various forms of D(z § give distinct dN/dy distributions. The extreme case in which D(z+)oca(1-z +) may be read from the g quark rapidity distribution. One may also note that changing G2(x2) from (23) to a fi-function centered at the corresponding value of c~, does not change the general pattern of the rapidity distribution, except for a slight narrowing of the curve. The rapidity distribution is of course always peaked in the forward direction since the photon carries all its energy contrarily to the relatively low amount of energy of the gluon.

These conclusions indicate that, provided the photon-gluon fusion mechanism is indeed at work in the photoproduction of charmed mesons, the rapidity distribution of the /) ~ may well turn out to be a sensible test of the fragmentation mechanism of a charmed string.

IV. Associated Charm Production

A. Rapidity Correlations Using the model outlined in Sect. III, the formula

which gives the double inclusive cross-section for producing/) ~ with rapidity y and either D O or Af with rapidity Y, is easily obtained to be:

do- 2re 2 1

~(z • )dz; 7~(z~ ) dz2 (24) e y e g +

The joint probability distribution G(xl,x>Q 2) has + and + already been defined in Sect. III; z, z 2 are the

light cone fragmentation variables fo r / ) ~ and D~ +) respectively; M is either D O or A~ + mass. Energy momentum conservation implies equal transverse momenta for final charmed quarks which is denoted by kl• and the following (implicit) definition of k l i and x I :

]/~=eY/k2• + eY/k~• ~' (25)

- - '~1.1- ~ - m c [ - - - - _ _ . ( 2 6 )

xl 1/7 I1/ 2 / m o ' 2 + / - - i M 2

B. Choice of the Fragmentation Functions The fragmentation function fo r / ) ~ is taken, in agreement with experiment, to be constant with z[ , normal-


ized to

z+rnax 3~b~z H ~ + 1

+ z i n~in

with only half of the string fragmenting, as in Sect. III*. Let us now consider the fragmentation of string 2

with mass M 2 defined as M~=(kz+Q) z (see Fig.8). When m~ + m~ < M 2 < m o +mp the only charmed hadron produced is A]. We shall assume, for simplicity, that the whole string fragments symmetrically on the

+ is quark and diquark sides. On the quark side, z a defined as in (8b):

+ (p,O + fill)* Z 2 - - M2 (27)

(p' is the A + momentum)

+ being equal to z,,~, = mAt /M v with Zmi . <Z 2 < 1; Zmi n On the diquark side, keeping the same conventions for longitudinal momenta the fragmentation variable is n O W :

(p,O _p,ll), z ; - (28)

M2

and z+z~ z 2 = m S m 2 . For the fragmentation functions, we choose the

simple form :

~A+~(Z+-)=DA+~(Z+-) when m a + + m ~ < M 2 <mD+m p

with

DA:(z • = C(1 - z-+). (29)

The normalization constant C being constrained to satisfy:

1 1

DA+~(z+)dz+ + ~ DA+~(z-)dz-=l. (30) Zmixa z m j n

When M z lies above the threshold for D+p production, both A~ + and D are produced. For the purpose of imposing some kind of threshold condition, let us introduce the 2-body phase-space factors PS(A~ + re) and PS(Dp) associated with the decay of the string of mass M a > m D + rnp into A~ + ~z and Dp respectively.

In a first step, we shall define the fragmentation functions in the following fashion - f o r D O production

D O + _ D O +

(Zz) -FDD (z2) when M2?>I~D+mv, (31)

where F D is a relative phase-space factor:

PS(Dp) FD = PS(Dp) + PS(A~ + re) (32)

* Strictly speaking, the upper limit Zm. x+ is only equal to 1 if neglecting the proton mass. The lower limit Z~m~,=mv/M ~

and DD~ +) characterizes the fragmentation of the string. As for/5 ~ we assume that only hatf of the string

+ + fragments namely Z2min<Z 2 < l (Z~min~mo/M2) and

+ is that the fragmentation function, constant with z2, normalized to

+ z2 rn~'x 1

DD~ + = - (33) + 2 " Z 2 m l n

As in Sect.III, we shall, for simplicity, assume ~D + = ~DDo _ for A~ + production

+ + + + ~Da~ -) when M2>mD+mp, (34)

where

PS(A~+ ~) FA~ = PS(Dp) +PS(A+~) (35)

and ~,A~: +, v ~tZ-) defined by (29). We should stress that the choices (30) and (33) for

normalization of the D's imply that apart from the relative phase-space factors, the relative weights for the production of D~ + :A~ + are 1:1:2. This may be considered as a reasonable assumption, which is in agreement with simple quantum number counting at threshold when the string M 2 may be viewed as a massive cqq object, made of the c quark and of the backward diquark qq, decaying into either D +, D o or A~ + .

In a second step we shall modify the F's by introducing an additional factor f responsible for the relative suppression of A~ + with respect to D for large string masses in the following way"

PS(Dp) FD = PS(Dp) + fPS(A~ + x)

fPS(A~ +u) (36)

FA~ = PS(Dp) +fPS(A + u)

defining the fragmentation functions with the same ansatz as in (31) and (34). Such a suppression factor is to account for the fact that when the string mass gets large, it should be more and more difficult to recom- bine the charmed quark c with the baryonic number (associated with the diquark). We assume

f = ( M ~ / M ~ / , (37)

where fi may be related to the intercept o fa baryonium trajectory [213; M o and fl are 2 parameters. We shall choose, for definiteness M o = m D + m v and try fl = 0 and fl = - 0.5 successively.

The expressions (36) for the F factors have been chosen such that the :D's satisfy the following simple features :

218 M. Fontannaz et al. : Photoproduction of Charmed Mesons and Baryons

jG \

Fig. l I. The two string structure of Af production from the diquark- charmed quark system. The dashed line represents the baryonic junction line [21] which migrated from the proton to the A~ +

i) obvious normalization condition:

1 1

S + S a+min Zmii1

1

+ ~ dz - i ! )A+~(z - )= l , 2mila

ii) presence of a threshold effect

(3s)

when M 2 = m D + mp : the D O fragmentation function behaves

�9 D O hke : ~ ocPS(Dp), iii) suppression of A + with respect to D production

at large string mass : when M 2 >~ m D +mp the fragmentation functions are respectively proport ional to

~Dooc 1 ~ ) a + O Q f 1 + f 1 + f "

The treatment presented here is of course very crude. In fact, the dual model approach would imply that a double string is stretched between the charmed quark c and the diquark* (Fig. 11). We feel, however, that our picture has the merit of exposing in a simple way, the main effects present in the problem. In particular, we are able to test the sensitivity of our results to the presence of a suppression factor. We do not consider, however, possible large values of M o which would produce a variation of the relative rate of D and A~ + production, with the string mass, in the sense of an enhancement followed by a suppression of A~ + with respect to D. This is a possible flexibility of the model which should be kept in mind, to be reexamined when more is known experimentally about the charmed hadron decay modes.

* We neglect associated A~+N production in a single string. Inferring, from the p/n ratio in e+e - annihilation, a value for A~+/D~ . 2, we assume this mechanism to be negligible in front of the one discussed here which leads to A+~/D ~ .5 (at least for energies E~ < 150 GeV)

Table 1. The total cross sections for 7p~f)~ and yp--,D --6 D~ in ixb for fl=0 (i.e. no suppression factor) and for fl=-0.5, M o = rn v + m;

/~=0 ~ = -0.5

Ey (GeV) b~ + A~ + D~ ~ b~ + A~ + 7)~ + D ~

20 0.065 0.020 0.057 0.024 40 0.14 0.065 0.10 0.085 60 0.19 0.092 0.12 0.13 80 0.23 0A1 0.13 0.17

100 0.26 0.13 0.14 0.19 150 0.32 0.16 0.14 0.25

C. Resul ts and Discussion

i) Total Cross-Sections. The integrated cross-section o - ( T p - + D ~ 1 7 6 and o - ( T p ~ A ] + b o + X ) may be readily obtained from (24) by integrating on y and Y. They do not depend on the shape of the D(z +-)

functions. In Appendix A, we shall indicate a simpler way to obtain these quantities.

The results with f i=0 and f i = - 0 . 5 are listed in Table 1. They are not too sensitive to small variations of M o : the effect being simply a rescaling of a ~5~176 by a factor (M'o/Mo) -2~. Notice that when f l=0, i.e. f = 1, the difference between o-oAt and r is only due to phase space effects; this is illustrated in Table 1 where both quantities become equal when E~ increases (with our simplifying assumptions concerning the various D production modes, o- ba+~ = 2 a b~ and o-bD=4a ~5~176 When fl = - 0.5, the cross-section for A c production is suppressed with respect to D production already when E~ ~> 40 GeV ; when E~ = 20 GeV, the string is small enough to fragment easily into a baryon.

There are few experimental data to compare with. Do < 0.4 ~tb and The WA4 collaboration [14] gives O-to t

O-OOA: = 0.58 +_ 0.30 gb* when 40 < E~ < 70 GeV. The results of Table 1 are only indicative and may change when the two parameters M 0 and fl are varied. It seems, however, difficult to obtain a good agreement with the quoted experimental numbers: the predicted A~ + production is smaller than the data and o-Do is at least as big as ag-OA; contrarily to the data (although large error bars enable marginal compatibility of theory with experiment). These difficulties may be connected with the surprisingly high value of the total cross-section in the corresponding energy range.

On the other hand, the CIF collaboration [15] with E~ > 80 GeV ((E~) ~ 120 GeV) gives the following

* This value depends appreciably on the assumed branching ratio of charmed baryons to protons [5]. Then it cannot be considered as definitely established


results :

a DO = 0.390_+ 0,190 ~b

a A~+ ~0.2 gb.

At least, half of the Ac'S may be assumed to be of diffractive origin since the particle to antiparticle ratio for A~ + is quoted to be 1.0+_0.4. It is reasonable to infer from the numbers above, that o -A2 to be compared with Table 1 is of the order of: o -A} -~0.1 gb. This is smaller than the value predicted by the model even when

= -0.5" o-~ r b---0.3 lxb. It should, however, be kept in mind that the acceptance of the apparatus is such that it may miss a large fraction of the A~ + produced via the photon-gluon mechanism. This is illustrated in Fig. 12 where we show the A~ + rapidity distribution, peaked in the backward region.

Similarly, some of the observed D~ are of diffrac- DO tive origin; we estimated in Sect. II O'diff ~'~0.1 gb whlch

Do yields anona~ff_~0.30lab in agreement with Table l, when f l=-0 .5 . (Notice that the D o produced via photon-gluon fusion have a forwardly peaked rapidity distribution, so that there may be less acceptance biases in this case.)

ii) Rapidity Distribution. Let us calculate the rapidity distribution of either D O or Ac +, integrating formula (24) on y. The results are plotted in Figs. 12 and 13 for E~=50 and 100GeV. The distributions dN/dY are normalized in the following way:

dNA2 /dy= daa+~ /dy/(a a+~ + a D~

and similarly for D O so that

(dN A~ /d Y+ dN~176 Y) d Y= 1.

The D o and A~ distributions are peaked in opposite directions" D o in the forward and A{ in the backward direction as expected from the fragmentation mechanism kinematics. This effect would be less pronounced for an asymmetric choice (z + ~ z - ) of the string fragmentation function into A{, favouring the quark side. The suppression factor pulls the string which fragments into A~ towards smaller masses and has, thus, the effect of slightly displacing the A~ distribution towards negative rapidities; it also reduces its magnitude, the D o distribution being correspondingly en- hanced. Notice that the shape of the A~ distribution is particularly relevant in discussing experimental acceptance problems*.

* For the WA4 collaboration data, the A~ + distribution shown in Fig. 13 is in rough agreement with the assumptions [5] made on the experimental distributions which led to the quoted numbers for er #~ The CIF data were already discussed above, when we noticed that the acceptance may exclude a significant amount of A~ + events

- 0 . 5

dN dY f - ~

/ \ - o,~ / \

- 0 . 3 I II/ " " ~ I . . " . . ."

-0.2 //....

( (b) I i I I I I

-2. -1. 0. 1. Y

Fig. 12. The rapidity distribution at E~=100GeV of the A~ +, Curves a and of the D ~ Curves b. The full and dotted lines have been obtained with / ? = - 0 . 5 (see text) and for a quark distribution function given respectively by formula (23) and by 6(xz-0.4). The dashed curves correspond to /?=0 and the quark distribution of formula (23)

-0.5 dN du

-0.4

_2

-0.3

-0.2

-02

(a) (b) f I I

-1 [ I

0

\

1

Fig. 13. The rapidity distribution at E~ = 50 GeV of the Ar + and D O (same conventions as in Fig. 12)

As a last step, we should ask the question whether the obtained distributions are characteristic of the 7g fusion mechanism. The intrinsic charm component yields DD and AcYl c pairs in the forward and backward directions but in only a small amount as shown in Sect. V.

iii) Rapidity Correlations. The photon gluon fusion mechanism is responsible for very specific rapidity correlations; an example is shown on Fig. 14. Observing/~o at y= t.5 or -0.75 implies D O rapidity distributions peaked in the backward and forward region respectively. On the contrary, diffractive mechanisms would produce D and b in the same rapidity hemisphere.


-70

-60

_50

-/~0

_30

_2(3

-lO

_2-

Do~-o do (nb) dYdy

-1 t' O 1 I'

Fig. 14. Rapidity correlations between the D O and ~0. Curves a and b correspond to y , , = 1.5 and yb = -0.75 respectively

V. Intrinsic Charm Contributions

Experimental data on charm hadroproduction at FNAL and at the ISR seem to indicate a relatively hard behaviour of the charmed content of the proton. The concept of an "intrinsic" charmed sea, in contrast to the perturbative QCD evolving sea, has thus been proposed and developed by Brodsky et al. [10]. Following this idea, we shall explore the consequence of the assumption of a charm content of hadrons parametrized (neglecting scaling violations effects) by

G(x)=6x2E(1-x)(1 + lOx+x2)+6x(1 +x) Logx] (39)

for the proton (a somewhat different expression being proposed for mesons). This distribution is normalized to a 1% admixture of luudc~) component in a proton. As already discussed in Sect. I, the total cross-section due to this intrinsic component is small compared to the ?g fusion contribution. We want, in this section, to briefly present the structure of these events. Let us investigate separately diffractive events and central production due to hard charmed quark-photon collisions calculated in the QCD framework.

model of Das and Hwa [22], D and/5 are produced by recombining c and ~ (or ~ and q) and their x F distributions are obtained with a mean value given by

(x F) = (x~) + (x~) (40)

which is of the order of 0.5. This corresponds to a rapidity distribution peaked near y-- 1, which is super- posed, but in a small amount, to the "central" contribution. In the case of diffractively produced A~3~ pairs, however, the situation is different: the contribution due to the 7g fusion mechanism is peaked in the backward direction and is thus clearly distinguishable from the diffractive one.

In the proton direction, the total charm production cross-section due to the intrinsic charm in the proton is of the order of 200nb (see Sect.II) which would approximately yield, assuming simple charge counting, 35nb for /5~ production (idem for /5~176 to be compared with 140nb (190nb for /)~ ~ for the ?g fusion mechanism (Table 1). The corresponding distributions are predicted [10] to be peaked near x~=4/7 and 3/7 in the proton direction for A + and /5 respectively (slightly lower value for D) presumably out of reach of the experimental apparatus.

B. Central Production

The Compton process: ?c~gc represented in Fig. 6 contributes to the total cross- section an amount a~ given by

aC=~dxGc/p(x) 1 !lJi~26(~+'[+fi-2m~)dy~dk• (41) 4~

where (yt,k~l) describes the final charmed quark momentum and

~[J~c 2 2 6411/~-m 2 mZ-fi] 4~ =2aG(Q )~[2~m~fi-fi + ~-m2/

A. Diffractive Events The intrinsic charm component yields D/5 and AcA c pairs in the forward direction (DD and Ac/5 pairs in the backward direction). Let us estimate their relative amount compared to the 7g fusion mechanism predictions. For E~ = 100 GeV, for instance, in, the forward region, one expects DD ~ production of the order of a few tens ofnanobarns (see Sect. II) to be compared to about 400nb for the 7g fusion mechanism (Table 1). The shape of such events has been extensively studied in [10] and we just recall the results with some comments. In the forward direction, the photon com- ponents considered are of the type [q77c-d>, q being any light quark. In a model inspired by the recombination

\ m c - - u . .~c ~

and

me2 + xs/2 + l/m + (x 2

~" = m~ - 2rnil [ ] ~ + (x I/s/2) 2 chy~ + x ]/sshy 1/2]

with

m212 -- 2 2 - - m c q - k • .

(42)


It should be noted that the matrix element of (42) does not present any pole, even at k• = 0 since the minimum value of ~ is

Smi n - - D2 c +

It is therefore legitimate to pursue such a calculation in the framework of perturbative QCD. The definition of Q2 is, as usual, ambiguous and we will take

Q2 =4m~1.

The resulting cross-section, which decreases with energy, turns out to be negligibly small, of the order of 5 nb at E~ -~ 50 GeV.

C. Conclusion

The overall conclusion of this section is that photoproduction Of charmed mesons and baryons at medium energies ( l / s -~ l~20GeV) is not the right place to look for the intrinsic charm of the proton or of the vector meson dominated photon, its contribution being hidden by the photon gluon fusion mechanism.

VI. Conclusion

In conclusion, the 79 fusion model explains an impor- tant part of charm photoproduction. Measurements of rapidity distributions and correlations will yield further informations about the relevance of this mechanism. Although we explicitly treated the real photoproduction case, the results of our work may also be applied to virtual photoproduction. Charm production has been observed in muon experiments [23] and the obtained total cross-sections are not in disagree- ment with the 79 fusion model [1] including extrapo- lation t o Q2 = 0. These experiments give however fewer insight on the final state structure of this reaction since they trigger on multimuon events, thus losing a great deal of information through the semileptonic decays of charmed hadrons. Providing the decay processes are known, one should be able to predict rapidity distributions as in the case of real photoproduction.

Acknowledgements. We acknowledge the participation of B. Andersson to numerous discussions which initiated this study. We also thank X. Artru, A.Capella, F. Richard, and P. Roudeau for discussions and comments.

Appendix

In this appendix, we derive a useful formula for the integrated two particle cross-section which explicitly exhibits independence with respect to the fragmentation function shape. The fundamental idea is to consider the charm photoproduction process as the

production of two charmed strings: after the c~ pair has been produced by 7 - g fusion, the color neutral objects formed, on the one hand, by the ~ antiquark and a "dressed" quark q of the remaining baryon, and on the other hand by the c quark and the remaining "diquark" Q, are supposed to evolve independently. Denoting by M~ and M 2 the invariant mass of these two strings, the differential cross-section da/dMZldM~ can be straightforwardly written as

da

dMZidM~ 1 1 1 1

- 2 (2rt) z ,~,= G(x)dx ~ I~12 a(~ + ~ + ~ - 2m 2)

k * .a(M~- q 2 + \ 2 \ d 3 k

(2+k~))g)(M2-(q 2 2 ) ) ~ (a l ) \

with the notations explained in Fig. 8, and assuming that the available momentum has been equally parti- tionned between the "quark" and "diquark" parts (this last hypothesis can easily be relaxed with a few changes in the following formulae). One gets

S=XS

fi_

1 1 - x [2xM2 - m2(1 + x)]

1 [2xM~-mf(1 +x)] . 1 - - x

(A2)

Equation (A l) can be easily transformed, by use of the variable transformations

d3k j2E1 --+rcdy ldm~l

@1 dm~ l ~ d[dfi/~ (A 3)

d{dfi~dM~dM~ 4x 1 - - x

which yield

da

dM~dM~ = s ~ i G(x) 119~12s

with [gJ~[ 2 given by formula (6). The total two charmed particle cross-sections

a(yp~A~+DX) and a(Tp~DDX) are then straightforwardly obtained by integration, including the phase space factors F given in (36):

cr= ~_dM~ ~ 2 2 2 2 dM2da/dMldMzF(Mz) (A 5) m&in

2 with Mm~ n taken equal either to (ma~+m~) z or to (rod + r%)2.

222 M. Fontannaz et al. : Photoproduction of Charmed Mesons and Baryons

References 1. R.J.N.Phillips: XX Int. Conf. on High Energy Physics (Madison

1980), ed. L.Duraud, LPondrom and references therein 2. G. Veneziano : Proc. of the XII Rencontre de Moriond (1977), ed.

J.Tran Thanh Van 3. U Jones, H.Wyld: Phys. Rev. D17, 759, 1782, 2332 (1978)

M.Gltick, E.Reya: Phys. Lett. 79B, 453 (1978); 83B, 98 (1979) M.Gliick, J.Owens, E.Reya: Phys. Rev. Di7, 2324 (1978) H.Fritzsch, K.Streng: Phys. Lett. 72B, 404 (t978) M.Shifman, A.Vainshtein, V.Zacharov: Nucl. Phys. B136, 157 (1978) J. Babcock, D. Sivers, S.Wolfram : Phys. Rev. D 18, 162 (1978)

4. J.Sakurai: Proc. of the 1972 McGill Univ. Summer School (unpublished) D.Schildknecht : Proc. of the 8 th Rencontre de Moriond (1973), ed. J.Tran Thanh Van F.Close et al. :Phys. Lett. 62B, 212 (1976) ; Nucl. Phys. Bl17, 134 (1976) B.Margolis: Phys. Rev. D17, 1310 (1978) Y. Afek et al. : Phys. Rev. D22, 93 (1980)

5. P.Roudeau: Thesis, University of Orsay (1980) (unpublished) 6. M.Fontannaz, A.Mantrach, B.Pire, D.Schiff: Z. Phys. C -

Particles and Fields 6, 241, 357 (1980) 7. J.Cooper et al. : Proc. of the 16th Rencontre de Moriond (1981),

ed_ J.Tran Thanh Van

8. U.Camerini et al. : Phys. Rev. Lett. 35, 483 (1975) B.Gittelman et al. : Phys. Rev. Lett. 35, 1616 (1976)

9. C.Carlson, R.Suaya: Phys. Rev. DIS, 760 (1978) 10. S.Brodsky, H.Hoyer, C. Peterson, N.Sakai: Phys. Lett. 93B,

451 (1980) 11. L.Koester et al.: XX Int. Conf. on High Energy Physics

(Madison 1980), ed. L.Durand and L.Pondrom 12. J.Owens, E.Reya: Phys. Rev. DI7, 3003 (1978) 13. R.Baier, J. Engels, B.Petersson : Z. Phys. C - Particles and Fields

2, 265 (1979) 14. D. Aston et al. :Phys. Lett. 94B, 113 (1980)

15. S.Wojcicki: Proc of the XX Int. Conf. on High Energy Physics (Madison 1980), ed. L.Durand and L.Pondrom

16. M.Fontannaz, B.Pire, D.Schiff: Nucl. Phys. B162, 317 (1980)

17. B.Andersson, G.Gustafson, T.Sj/Jstrand: Phys. Lett. 94B, 211 (1980)

18. W.Bartel et al.: Phys. Lett. 101B, 129 (1981) 19. J.Bjorken: Phys. Rev. D17, 171 (1978) 20. P.Rapidis: Phys. Lett. 84B, 507 (1979)

M.Piceolo et al. : Phys. Let1. 86B, 220 (1979) 21. X.Artru, M.Mekhfi: Phys. Rev. D22, 751 (1980) 22. K.Das, R.Hwa: Phys. Lett. 68B, 459 (1977) 23. A.Clark et al. : Phys. Rev. Lett. 43, 187 (1979)

J.J.Aubert et al. :Phys. Lett. 89B, 267; 94B, 96, 101 (1980)

Documents

Inclusive photoproduction cross sections of charmed mesons and baryons