Pion-quark scattering model for lepton pair production

A model for lepton pair production from quark-antiquark annihilation is presented in which antiquarks are obtained from constituent pions and form factors are included to account for the transverse-momentum dependence. The single-lepton spectrum and, the lepton-pair spectra in mass squared, transverse momentum, and longitudinal momentum are calculated and compared with proton-and pion-beam experiments.


I. INTRODUCTION
Experiments have recently studied the spectrum of lepton pairs emitted in hadron collisions.
These' ' have varied all of the parameters of the pairs: mass squared, 0.04 (Q'( 120 GeV', trans- verse momentum, 0 (Q, -5 GeV/c; and Feynman- scaled longitudinal momentum, 0 (x~&.Some experiments"' have also included pion beams.These are complemented by experiments' which measure the single-lepton spectrum and go to larger q~& 6 GeV/c.In this paper we attempt to fit the continuum contribution to these processes with a model that is basically the Drell-Yan model'.quark-antiquark annihilation to a lepton pair via a virtual photon.To account for the Q, dependence we modify the Drell-Yan model by using constituent pions as the source of antiquarks or sea quarks' and incorporate the transversemomentum and related off-mass-shell dependence of the quarks in terms of a form factor describing the meson wave function. The Drell-Yan process as originally calculated uses on-mass-shell quark constituents of only the initial beams and yields a single-lepton spectrum that falls off with the canonical scaling power q, 4.This has been compared with data"' and found to be an order of magnitude smaller, although approaching the data at the largest q, -6 GeV/c.Since the p, /w spectrum ratio is approximately constant in q, , the single-lepton spectrum must fall faster in q~.This same situation arises with the single-pion spectrum falling as q~i nstead of the canonical q, scaling behavior.In order to provide a mechanism for this canonical scaling violation we introduce constituent pions as the predominant source of sea antiquarks' or quarks and include a form factor E, (k2) for the pion to dissociate to an off-shell quark or antiquark with virtual momentum squared k' (Fig. 1).This quark exchange also provides the scattering mechanism which creates virtual photons with momentum transverse to the beam or collinear constituent pion direction.In the modified Drell-Yan model, the high-Q, lepton pair requires that an annihilating quark or antiquark have a high k, =Q, , and this is the exchanged quark in Fig. 1.This high-k, quark is also off its mass shell by an order k'--k, '.
The form factor F, (k'}= A(-k'+ m') ' ' is used (with A having the dimensions of mass) for the off-shell quark or antiquark in a, pion.This describes effectively the distribution of quark trans- verse momentum and provides the extra damping in k, needed to agree with experiment.
The constituent pions are considered to have a central plateau or sea-type distribution P, &= (2.5/x)(1 -x)'.Pion-beam experiments are also studied with the same model (Fig. 2).The valence q and q in the pion are also supplemented by a sea resulting from pion constituents of the pion P, &, = 2/3(2.5/x)(1 -x P. The power in F, (k'} is chosen to be consistent with the constituent-interchange calculation of the calculation of the single-pion spectrum via q+ v-x+q (Fig. 3).There the form factor occurs twice in the amplitude giving the Edc/dp'o-A'p, ' behavior of the cross section at CERN ISR.Here the analogous process (Fig. 1) for lepton pair production has F, (k~) occurring only once in the amplitude.This gives for the single-lepton spectrum q'd&r/dq'~A'q, '.Since m' is large, however, the f/x ratio does not show significant deviation from a constant until q"&4GeV/c.The coupling strength A and mass m are ad- justed to fit the data, and good agreement is found between the A needed for lepton production experi- ments and the single-pion spectrum.
In Sec.II we present the detailed calculations of the model and in Sec.III we compare the numeri- cal results with the experiments.
ii Q tional momenta x"x, and probability distributions P,»(x,)dx"P, &~(x, )dx"respectively.The quark and antiquark annihilate to form a virtual photon of momentum Q" and mass squared Q'.Although the transverse momentum of the constituents is small and has been neglected, photons at nonzero Q, can be produced due to the interchanged anti- quark with k, =Q, .If we view this in the standard Drell-Yan way, the pion is actually providing an creates virtual photons at Qi= 0.
The off-shell quark exchange Fig. 1(a) is not by itself gauge invariant.Including the s-channel quark pole in Fig. 4(a) does not make it gauge in- variant due to the form factors at the pion vertices.The form factors mean that the pion ver- tex contains an internal structure, and the virtual photon must be coupled to the internal charges.However, we can isolate the contribution that this vertex adds to the quark sand t-channel poles to make a gauge-invariant amplitude.' The current from Fig. 1(a) is j, "= eX,u(q')y, E, (k')(-$m) 'y"u(x, P,) .'The current of the virtual photon coupling into the w"-vertex itself has several Lorentz-covariant terms, among which we need only j"' = eu(q')y, Q'u(x, P,)f(s, k', Q'), where s=(x, P, +x, P, )'.The amplitude f neces- sary to make j= j, + j"gauge invariant is obtained from setting Q' (j, + j")=0,giving f= X, E,(k')/Q and j"= e&,u (q')y, E, (k') && [(-ft' -m) 'y" +Q /Q']u(x, P,).
Since the added Q" vertex term is contracted with the current-conserving lepton current it contributes nothing to the cross section and will henceforth be dropped.The crossed diagram Fig. 1(b) and the s-channel pole terms Fig. 4 are made gauge invariant in the same way with the Q" vertex terms.
Compared with the data of where x, ' is obtained from x, by interchanging a and b.
With pion beams, the direct graph has contributions P, ~,(x,) and P;1,(x,) from on-shell q or q [Fig.2(a)] in place of P, »(x,) in the first term of Eq. (2.7).The cross graph [Fig.2(b)] has P, &, (x,) = 1.7(1 -x, )'/x"acentral plateau distribution which dominates low-x~calculations.In addition, we can have the crossed type graph with the initial pion as the source of the off-shell antiquark in- stead of a secondary pion [Fig.2(c}].This is given by replacing P, &, (x,) by 5(x, -1) in the sec- ond term of Eq. (2.7) and retaining the E, (k'2} off-shell form factor to allow it to contribute to QJ4 0 lepton pairs.We have considered the dia- gram with a direct-channel quark pole" (Fig. 4) that would make a pointlike theory gauge invariant.
The cross section with this s-channel quark prop- agator and pion form factor behaves like A'/t' with s=sx~~and at 90', s& vs (Q~'+Q')' '.The quark-exchange t-channel diagrams (Fig. 1) that we have used behave like A'/(k')', however, and are dominated by the smallest possible k'= -Q J'.
The s-channel quark diagrams are thus down by order Q, '/s from the f-channel exchanges.
III. RESULTS l000" l. 5& M& l, 9 GeV 1.9& M&2.$ 2.5& M& 2.7 We determine the parameters for these calculations as follows.The proton quark distribution functions P,f~(x) are those which fit electroproduc- tion and neutrino production." The distribution functions for quarks in pions are taken to be" P g, (x)=0.15/x.
A. Single-lepton spectrum We calculate q'd'o/dq' for large transverse mo- mentum muons at 90' and Ws = 23.7 GeV.%e com- pare with the cross section per nucleon of Boy- mond et a/.' for Cu and W targets and find that our calculation has the proper q, ' dependence at large q, (Fig. 5).The data are fitted with A='1. 8  GeV and m'=4 GeV'." xz& 0.3 is due to the pion beam directly producing an antiquark for the annihilation.
The data for the calculations in this subsection a.re fitted with m'= 1 GeV' and the normalization A =12.8 GeV.
C. Lepton-pair spectra xF 4 0 We calculate Q, 'do/dQ, per nucleus, for a Be target, integrated over x~f or both incident proton and v' beams at Ws = 16.8GeV (Fig. 8).We find good agreement with Anderson et a/.' in the mass bin not dominated by a resonance 1.13&M& 2.0 GeV.In the other nonresonant mass bin 0.45&M & 0.55 GeV the data exceeds our calculation by an order of magnitude and indicate additional sources of lepton pairs for very low Q'& 0.5 GeV'.
We also calculate Q'do/dx~and compare with the same experiment in the same mass bin (Fig. 9).
Including colored quarks, there is a factor of -' entering this cross section, similar to the -', arising in the quark annihilation process for lepton pairs, Eq. (2.1).The data" for v' produced at 90' with v s = 45.1 GeV are fitted in Fig. 13 with A = 12.8.'The value of A'determined from the lepton- pair spectrum is thus consistent with that needed to fit the pion spectrum in the constituent-interchange model.Our calculation shows the rise of (Q, ) with Q' but saturates at (Q, ) = 1.1 GeV jc (Fig. 12).We conclude that the constituent-pion-quark scattering model for lepton pair production includ- ing form factors can account for the dependence of the continuum produced single-lepton and lepton- pair spectra in transverse and longitudinal mo- mentum and in the lepton-pair mass.It also accounts for the relative normalization of pionversus proton-produced lepton-pair data, and is consistent with the constituent-interchange-model normalization for single-pion production.
*Work supported in part by the National Science Founda- tion.
FIG. 1.(a) Direct and (b) crossed diagrams for pp -L l X' via secondary-pion-quark scattering with quark interchange.
FIG.  3. Direct and crossed diagrams for secondary- pion-quark scattering in ppn X via constituent inter- change.
FIG.4.Quark bremsstrahlung or s-channel quark diagrams for pp l'l X via secondary-pion-quark scattering.
FIG. 7. Q d 0. /dq for muon pairs at 90' and up=27.4 GeV.Compared to the data of Hom et al. (Ref.4) for various mass bins.
FIG. 9. Q dg/dx~for muon pairs, per nucleus, for a Be target, for both incident proton and x' beams.Com- pared with the data ofAnderson et al. (Ref.2) at v g = 16.8GeV in a nonresonant mass bin.
. 11. dg/dx& for muon pairs, per nucleus, for a carbon target, for both incident proton and m' beams.Compared with the data of Anderson et al. (Hef.3) at v g =20.6 Gep in various mass bins. FIG