On the transverse momentum distribution of massive lepton pairs

Volume 77B, number 4,5

PHYSICS LETTERS

28 August 1978

ON THE TRANSVERSE MOMENTUM DISTRIBUTION OF MASSIVE LEPTON PAIRS

E.H. de GROOT, H. SATZ and D. SCHILDKNECHT

Department of Theoretical Physics, University of Bielefeld, Germany Received 25 April 1978

We show that the transverse momentum distribution of massive lepton pairs at low transverse momenta may be predicted from inclusive hadron production without adjustable parameters. The conventional distinction between low p± and high Pi physics should thus also be adopted for massive lepton pair production.

It is widely believed [1,2] that massive lepton pairs in hadron reactions are produced by a dynamical mechanism quite distinct from the one which governs the production of hadrons (e.g. pions) or low mass lepton pairs (e.g. m(p+U-) ~ m(00)). In this note we wish to point out that recently measured transverse momentum distributions [3] for massive lepton pairs show a striking similarity to the transverse momentum distributions observed in inclusive hadron and low mass lepton pair production. In fact, we will quantitatively and without any adjustable parameter predict the transverse moment dependence of massive lepton pairs in the low transverse momentum region from the transverse momentum behaviour observed in hadron production. Let us start with inclusive hadron production. In the low transverse momentum region (p± ~< 1.5 GeV), it is most simply described by an independent emission ansatz (uncorrelated jet model [4]) in which the production is essentially determined by phase space with a transverse energy cut-off. For the transverse momentum distribution of hadron h(e.g, h = 7r, p, p0) in pp -+ h + X at high center of mass energy ~ one obtains the result [ 5 - 7 ] do

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In this expression f(E±) is a universal function o f the transverse kinetic energy E± - ~ - m h : the 418

same f(E) describes all observed hadrons from lr to J/t~ ,[8]. An exponential ansatz f(E±) = exp(-XE±) with the 'hadronic scale" 2~-~ 6 GeV -1 (corresponding to an av average p± of 333 MeV for pions) gives satisfactory agreement with the measured hadron spectra. The second factor on the right hand side of eq. (1) is a kinetic correction term due to transverse momentum balance in the hadronic final state. It is obtained within the uncorrelated jet model in the low Pi approximation by applying the central limit theorem of statistical mechanics. The exponential factor in eq. (1) is equal to the probability that the hadronic system X, consisting of the average of (n)(s) pions, has the same absolute value of the transverse momentum as the observed hadron h. The gauss±an form in eq. (1) is valid as long as p2/~n)(p2)~ '~ 1. In the limits -+ o¢, we have (n) -+ ~ , the exponential in eq. (1) reduces to unity, and the p± distribution is simply given by f(E±). The average squared transverse momentum (p2)= of the produced pions appearing in the exponent of eq. (1) is calculated from the distribution f(x/p 2 + m 2 - m~r) and is thus identical to the average transverse momentum squared as measured in the limit s -+ oo. For inclusive hadron production, the gauss±an factor in eq. (1) is usually neglected, as it is Fairly unimportant at sufficiently high energies. For pions f(E±) in eq. (1) is a very steep function of p l so that the correction factor due to transverse momentum conservation only slightly modifies the slope of the distribution, while for large masses the two factors in eq. (1) are o f comparable magnitude. In fig. 1, the measured pion spec-

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28 August 1978

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trum [9] is compared with the uncorrelated jet model prediction (1) both at finite s and for s ~ oo. For later reference, we remark that fig. 1 shows the well-known departure from eq. (1) in the region of p j_ ~> 1.5 GeV. Let us now turn to lepton pair production, e.g. pp -+ p + p - + X. If we assume that lepton pair production is governed by the same kind of dynamical mechanism as hadron production, then we simply have to replace the hadron mass m h in eq. (1) by the lepton pair mass M(p+/J-). Such a procedure could be motivated by a production mechanism yielding hadronic clusters, which with a certain probability then decay into lepton pairs. However, such a specific picture is not necessary for the subsequent discussions. For sufficiently large mass M (M >~ 5 GeV), the exponential exp(-XE±) in eq. (1) may be expanded. Thus we predict a gaussian transverse m o m e n t u m distribution for massive lepton pair production,

il< > (2)

The energy available for the hadronic system X produced in conjunction with the lepton pair of mass M(g+g-) (M>~ 5 GeV) at rapidity y ~ 0 is approximately given by x / s - - M ( P + / J - ) . The average particle multiplicity (n) of the system X appearing in eq. (2) is thus to be taken as the multiplicity of a hadronic system of energy x / S - - M ( p + p - ) . As soon as the lepton pair of mass M(p+~ - ) takes a sizable fraction of the available energy x/~, the multiplicity (n) of the hadronic rest system X becomes fairly small. The decrease of the slope a(M, x/s) in eq. (2) due to the 1/M transverse kinetic energy term for large M at fixed energy V~, will thus be partially compensated by the increase of the 1/(n) term. Hence, the slope a(M, X/s) at fixed x/~ for sufficiently large M will be approximately constant, and the average transverse m o m e n t u m (qz),

~±) - ½X/~ ( 1 / ~ ) ) ,

(3)

will become fairly independent of the lepton pair mass. Looking at the data [3] from the CFS collaboration shown in figs. 2, 3 and 4, we first of all observe that the mentioned qualitative predictions of eq. (2) are in fact fulfilled by the data. Figs. 2 and 3 show the expected gaussian transverse m o m e n t u m dependence for low q± (q± ~< 1.5 GeV), where eq. (2) is expected to be valid. At large q±, one observes deviations of the same qualitative nature as in the case of pion production shown in fig. 1. The predicted flattening of the average transverse momentum (q±) as a function of the lepton pair mass is seen in the experimental data of fig. 4. For the quantitative comparison of eq. (2) with experiment, we use 7, = 6 GeV -1 in accord with inclusive 2 = 0.19 GeV 2 hadron spectra. This value of X implies (P±)o. 2 2 for pions and (pz)oo = 0.43 GeV for nucleons. The difference of (p2)oo for pions and nucleons enters the numerical evaluation of eq. (2), as the expression (n) X (p2) has to be understood as Z i (t/i) (p_L2i),where the sum runs over all different particle types. To obtain the average number of pions in the system X, we use an em419

Volume 77B, number 4,5

PHYSICS LETTERS

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pirical fit [10] to the number of charged particles produced at energy ( ~ - ' - M) in ordinary hadron reactions: (n) c = 0.88 + 0.88 l n ( v ~ ' - M) + 0.472 l n 2 ( ~ -- M).

(4) The result obtained from eq. (2) is compared with the experimental data in figs. 2, 3 and 4. We see that, for q j_ not too large the q± distribution is reasonably well reproduced by the hadronic ansatz (2) using the universal hadronic scale X = 6 GeV -1 . As the overall normalization is undetermined within this scheme, for figs. 2 and 3, at q± = 0, we have used the simple exponential

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Fig. 4. The average transverse momentum of lepton pairs [3, 11 ] as a function of the lepton pair mass M ( / ~ - ) compared with the prediction from eqs. (2) and (3). Predictions at different energies and the prediction for s ~ ,~ are also indicated.

Volume 77B, number 4,5

PHYSICS LETTERS

In fig. 3 we also show the q± distributions to be expected for s ~ ~ . It should be noted that the approach to this limit is rather slow, however. On the other hand, the existence o f a limiting (s ~ ~ ) distribution for (q±) as a function o f M constitutes a crucial difference between our hadronic picture and QCD arguments [2], which generally provide for any large M a divergent (q±> as s ~ oo. The average value of q± as a function of the lepton pair mass M for different energies x/s is shown in fig. 4. The experimental values of (q±> lie above the prediction from eq. (3) in accord with the deviations at large q± as seen in fig. 3. The flat (q±) of the data as a function o f the lepton pair mass is nicely reproduced, however. The energy dependence seen in fig. 4 moreover shows the strong influence of the kinematic correction factor for large lepton pair mass. Let us conclude b y summarizing the main points: (1) The q± distribution of massive lepton pairs at low q i is described by an independent emission ansatz with X = 6 GeV - 1 just as ordinary inclusive hadron production: the q± distribution o f lepton pairs is hadronlike. With increasing energy x/~at fixed lepton pair mass, the qx distribution will become flatter, the limit s -> oo being reached very slowly, however. (2) Just as in inclusive hadron production, the cross section for lepton pair production at large qz is larger than the one predicted from a universal exponential transverse energy law. Thus, if one discriminates "low Pz" and "large p±" physics in inclusive hadron production, then the same discrimination should also be adopted for massive lepton pair production.

28 August 1978

(3) The flattening off of (q±) as a function of the lepton pair mass for M(IJ+/J - ) ~ 5 GeV is a consequence of the transversely limited phase space of the other produced particles. At any finite energy x/~, for sufficiently large M(tJ+p-), the limited number o f secondaries (n) (which number even decreases substantially as M increases) produced in conjunction with the lepton pair can provide the necessary m o m e n t u m balance only at the cost of a considerable damping of the cross section.

References [1] S.D. Drell and T.M. Yan, Phys. Rev. Lett. 25 (1970) 316. [2] K. Kinoshita et al., Phys. Lett. 68B (1977) 355; G. Altarelli et al., Phys. Lett. 76B (1978) 351; H. Fritzsch and P. Minkowski, Phys. Lett. 73B (1978) 30; J. Hinchcliffe and C.H. LleweUyn-Smith, Phys. Lett. 66B (1977) 281; K. Kajantie and R. Raitio, Helsinki-Preprint HU-TFT-7721, October 1977. [3] D.M. Kaplan et al., Phys. Rev. Lett. 40 (1978) 435. [4] A. Krzywicki, Nuovo Cimento 31 (1964) 1067; L.v. Hove, Rev. Mod. Phys. 36 (1964) 655. [5] R. Baier et al., Nuovo Cimento 28A (1975) 455; E.H. de Groot, Nucl. Phys. B40 (1972) 295. [6] C. Michael and L. Vanryckeghem, Liverpool preprint, . LTH 31. [7] H. Satz, Phys. Rev. D17 (1978) 914. [8] E.G: K. Kinoshita et al., Phys. Rev., to be published (BI-TP 77[14). [9] B. Alper et al., Nucl. Phys. B100 (1975) 237. [10] W. Thomd et al., Nucl. Phys. B129 (1977) 365. [11] J.G. Branson et al., Phys. Rev. Lett. 38 (1977) 1334.

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