We obtain the production rates for c, b, and t quarks in deep-inelastic neutrino- (antineutrino-) nucleon interactions, in the standard six-quark model with left-handed couplings. The results are obtained with the most recent mixing parameters and we include a comparison between quark parametrizations. The excitations are calculated separately for each flavor, allowing the understanding of the role of threshold effects when considered through different rescaling variables.Facultad de Ciencias Exacta

Ducati, M.B.G.

García Canal, Carlos Alberto

Santángelo, Eve Mariel

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PHYSICAL REVIEW D VOLUME 31, NUMBER 11Excitation rates of heavy quarks1 JUNE 1985C. A. Garcia Canal and E. M. SantangeloLaboratorio de Fisica Teorica, Uniuersidad 1Vacional de La Plata, 1900La Plata, Republica ArgentinaM. B. Gay DucatiInstituto de Fisica, Universidade Federal do Rio Grande do Sul, 90000 Porto Alegre, Rio Grande do Sul, Brasil(Received 9 November 1984)We obtain the production rates for c, b, and t quarks in deep-inelastic neutrino- |,'antineutrino-)nucleon interactions, in the standard six-quark model with left-handed couplings. , The results areobtained with the most recent mixing parameters and we include a comparison between quarkparametrizations. The excitations are calculated separately for each flavor, allowing the under-standing of the role of threshold effects when considered through different rescaling variables.I. INTRODUCTIONThe question of production rates of heavy quarks inhigh-energy neutrino beams is still an object of study andits interest is made evident by the several experiments tak-ing place with such beams in different laboratories. 'From the theoretical point of view there exist open prob-lems, since, for example, no good explanation is availablefor the same-sign-dileptons rates.'In this paper, we present a study of the rates of c-, band t-quark production in deep-inelastic neutrino-nucleonand antineutrino-nucleon interactions. Our results wereobtained using the standard six-quark model with left-handed couplings and employing the most recent values ofthe Kobayashi-Maskawa mixing angles, which are quitestringent concerning ub and cb couplings.We present curves of excitation of heavy quarks distin-guishing all the possible channels of production, calculat-ed with two different QCD-dependent parametrizations"'of quark distributions and a simple quark-parton distribu-tion. The evolution until the free hadron states originat-ed at the quark level could not be completely explainedwithout the understanding of the fragmentation mecha-nism. In order to test fragmentation models, it is clearthat one needs a complete parametrization able to coverproperly the quark-excitation mechanism up to high ener-gies.To have a feeling about the regions of dominance of theproduction of the different flavors and to estimate futurerates, results are presented for energies as high as 1200GeV, even though the most recent experimental data at-tain 250 GeV.The aim of this work is to estimate, constrained to therecent values for the parameters involved, at which ener-gies new effects should be expected due to the excitationof heavy quarks. The analysis concerning the top quark isincluded because it seems that this flavor is showing up inIthe very recent UA1 experiment.In Sec. II we present the cross-section formulas in orderto fix our notation. In Sec. III we discuss the quark-distribution parametrizations used in our predictions. InSec. IV we present the very recent mixing parametersused. In Sec. V we discuss the fundamental role of rescal-ing, the variable which takes into account threshold ef-fects, in the calculation. In Sec. VI we present the results,together with some comments.II. CROSS SECTIONSThe differential cross section for neutrino-nucleon (iso-scalar target) interaction is given in terms of the Callan-Gross structure functions as=K +(1—y) F2(g)dx dg+ —(y' —y ) + (1—y) F,(g) (2.1)and for antineutrino-nucleon interaction as0dx dy=K —(y —y)+(1 —y) F2(g)(2.2)where K=6 ME/sr, 6 being the Fermi constant, M thenucleon mass, and E the neutrino (antineutrino) incidentenergy; x and y are the scaling variables, and g the rescal-ing variable, introduced in order to take properly into ac-count threshold effects, as will be discussed later. F2 andI'2 are the structure functions, related to quarks and anti-quarks, respectively.In terms of quark distributions, expressions (2.1) and(2.2) give rise tod 2~vX —+I Xdx dg=Kg (1—y)+ [[u(g)+d(g)]Vd ++2s(g)V, +2b(g)VI, I+ (1—y)+ —(y' —y) [[u(g)+d(g)]V, '+2c(g)V, '+2t(g)V, '} (2.3)31 2748 1985 The American Physical Society31 EXCITATION RATES OF HEAVY QUARKS 2749andd2 vN~l+X=Kg' (1—y)+ —(y —y) {[u(g)+d(g)]V„+2c(g)V,+2t(g) V, jdxdy g+ (1—y)+ t[u(g)+d(g)]Vd +Zs(g) V, +2b(g) Vb j (2.4)where u(g), d(g), etc. , are the quark distribution functions, depending on the rescaling variable g, and V; is the sum of arow or column of the Kobayashi-Maskawa matrix, so2 2 2 2~u = ~ad + ~.. + ~.b1/d = ~ud + &ed + ~td2 2 2 2(2.5a)(2.5b)(2.5c)The partial productions are immediately obtained from the general cross-section expressions, Eqs. (2.3) and (2.4), sincethe couplings are obtained fromT&ud ~.s V.b d(uct) Vd V- V& s~~b b(2.6)and so we have, for instance,vN —+I" cXdx dpvN —+I +tXdx dp=Kg (1—y)+ t[u(g)+d(g)]Vd'+2s(g) V„'+2b(g)V,a'j (2.7a)(2.7b)vN —+I +bXdxdy=Kg (1—y)+ —(y —y) I[u(g)+d(g)]V„b+2c(g)Vt, +2t(g)Vtd j (2.7c)The quark distribution functions depend on the rescal-ing variable, and when considering QCD-dependentparametrizations, on the QCD parameters A and Q .III. QUARK DISTRIBUTION FUNCTIONSOur results were obtained from three differentparametrizations for quark distribution functions. Thefirst QCD-based parametrization is the standard Buras-Gaemers one and the second one considered is that re-cently proposed in Ref. (5), whose parameters include asecond-order dependence on the QCD evolution parameters =in[in(Q /A )/ln(go /A )],having the general form for valence quarks,x[uv(x, Q )+dv(x, Q )]=C,x '(1 —x) '(1+ytx), (3.1a)and for SU(3)-symmetric sea,»(x~1 )=Asx (1—x) (1+asx+/3sx+ysx )(3.1b)These QCD parametrizations are obtained by fixing pa-rameters both from the momentum sum rules and fittingfrom available data. We expect that the results obtainedusing Ref. 5 could be more predictable, since thatVN~C-I / ~ ~~ ~ ~ 0~ ~ ~ ~10III:vd~c vs~c———DUKE-OWENS'--"-- BURAS-GAEMERS---- AGUIAR ET AL(mqf+mq;)'2M&~ ~ ~ 01~ ~ ~~ ~ ~I ~ ~ +~ ~ ~0~ +~ ~~It~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ \ ~ 0~ 0 ~ ~ ~ ~ ~ ~ y~ ~I102, I I I I I I I I I I I liI I I I I I200 600 1000 200 600 1000 200 600 1000E (GeV)FIG. 1. Total and partial production rates of c quarks fromneutrino-nucleon interactions for E„upto 1200 CieV, obtainedwith three different quark parametrizations. The QCD parame-ters are A =0.25 CxeV and Qo —1.8 GeV' for Ref. 4 andA =0.16GeV and Qo =4CreV~ for Ref. 5, and m, =1.9oeV.parametrization was constructed using more recent exper-imental data, which covers energy values up to 200 GeV.As a matter of comparison we also present results froma naive quark-parton pararnetrization; since we are in-terested in the high-energy domains, this permits us toseparate scaling violation from QCD effects and threshold2750 GARCIA CANAL, SANTANGELO, AND GAY DUCATI 31effects coming with the rescaling. This can be seen inFig; 1, for charm excitation.IV. MIXING PARAMETERSDue to the large amount of data coming from differentexperiments, providing more and more constraints, themixing parameters are a subject of continuous analysisand discussion, reaching more precise and, in some cases,more stringent values. Very low values seem to be con-firmed for the off-diagonal terms of the mixing matrix re-lated to the b-quark couplings V„b and V,&. The experi-mentally determined large B lifetime, ' with valuesaround 1 psec, has a direct relation in those parameterssince the b-quark lifetime is given by'%N ~cX/// r10IIIlorio- L1III I I I I200 600 1000wS~ C///II:I:I:l:,fo r~rI~'. II:lgt.—~ -Ct ———DUKE -OWENS) -"""--EIURAS-GAEMERS....".j AGUIAR ET AL.. ~ "" (mq&+mqI)2MI &I 1' I I I I I I I I I200 600 1000 200 600 1000E~ (GeV j6 5(2 95I +cb I +6 33 I Vub I ')192mand so values lower than the previous ones are obtainedfor V„band V,b. Such low values for the mixing pararne-ters strongly suppressed the theoretical predictions forsame-sign-dilepton production rates via b production. "In our calculation we considered the most recent set ofmixing parameters given by0.945 0.231 0.005FIG. 2. Total and partial production rates of c quarks fromantineutrino-nucleon interactions for E„upto 1200 GeV. Theparameters are the same as for Fig. 1.In any process of quark excitation the invariant massW recoiling against the scattered lepton must satisfyW) 8,h, whereW =M +2MEy(1 —x) .0. 1 0.972 0.0440.024 0.069 0.99(4.1) This can be written, in terms of x and y, considering theabove limit, asin order to have an upper-bound estimation for the pro-duction rates for each flavor.8;h —M2 2y(1 —x) )V. RESCALING%e face at this point the question of rescaling, whichhas different formulations' ' as the excitation processoccurs from a light-to-light, light-to-heavy, or heavy-to-heavy quark. From those formulations we should con-clude that the most general one is the slow rescaling of theform(mq+mq )g'=x + (5.1)where mq denotes the final-quark mass and m~ theinitial-quark mass, since this expression takes into accountthe production of heavy-to-heavy quarks, when it becomeskinematically allowed. It also reduces to the usual' re-scaling variable when m& is negligiblel20'usual =x +2M@(5.2)A squared threshold invariant massW,„'+(m,+m, )'+M' (5.3)corresponds to the rescaling (5.1).The introduction of a rescaling variable, of any type, totake into account the effect of mass thresholds on thestructure functions implies a detailed analysis of the xand y variation ranges.and so8'th —M2M8',h —M2ME8',h —M2MEywhere we can easily see the limitations imposed by 8;hon the upper x bound and lower y bound.vN ~bX vLI ~bX vc ~bX-51010// r'/ rj /j /I I1 IIl IIIIIl IIIII Iii11ll. lIil I I I I I200 600 1000///IIlIl!I200 600 1000/// r// II /I /II/CI Il Imqf -mq;l i ~ 2M'(mql +mq;)II =X+ 2M'II t I I I I200 600 1000E (GeV)FIG. 3. Total and partial b production rates from neutrinosobtained with the parametrization (Ref. 5) showing the effect ofdifferent rescalings with mb —4.9 GeV.EXCITATION RATES OF HEAVY QUARKS 2751vN~bX vc~bXITlql —mq ~g-x+2M&~N~tX---—m) = 30 GeV—"—mt =40 GeVIo'-!IIIIIIIIII5IIo'200/tIIIIIIIIIII I I I I I I I I600 1000 200 600 1000(mqf + mqI)( =x+ 2M@/'/ // // /IiI I I I I I I200 600 1000E (GeV)FIG. 4. Total and partial bottom-quark production ratesfrom antineutrinos, calculated with Ref. 5 showing the effect ofdifferent rescalings.10IO10//////IIIIIIIIltII500I1000//IIIIII1500F~ (GeV)FIG. 5. Total top-quark production rates from neutrinoswith E up to 1500 GeV, obtained using parametrization (Ref.5) for m, =30 and 40 GeV.The threshold considerations have a direct consequence in the integral domains'" we must calculate in order to obtainthe cross sections. It is now clear that the rescaling variable is a function of the masses of all the quarks involved.The general form of the integration is11 —(m +m ) /2MEyq q.Z(qf)= gf, dy f dxw(x, g(q;, qf), Q )Vqq (5.4)where we normalize the result in terms of the single muoncross section. The function M includes the quark distri-bution functions and multiplicative factors [see Eqs. (2.3)and (2.4)].V. RESULTSThe main purpose of this work is to provide the pro-duction rates of heavy quarks in neutrino- andantineutrino-nucleon high-energy mteractions, with QCDparametrizations including charm sea distribution but noother heavy sea (b, t) contribution. The reason is thatsince the charmed sea contribution is probably' as muchas 10% of the whole light sea, the distributions of bottomand top quarks should be strongly suppressed due to theirhigher masses.We include in Fig. 1 the partial production rates for cquark, for neutrinos, coming from d and s quarks, andthe total rate, comparing the results for the three usedparametrizations. In Fig. 2 we present the curves ob-tained for c production by antineutrinos.Figure 3 shows the bottom rates for neutrinos, and Fig.4 for antineutrinos, considering the parametrization ofRef. 5 and distinguishing the effects due to different re-scaling variables.In Fig. 5 we include the results obtained for top-quarkproduction rates, considering the top-quark mass at 30and 40 GeV. Notice that for masses greater than 40 GeVthe possibility of top-quark-excitation determination inneutrino- (antineutrino-) nucleon interactions could be ex-cluded due to the very low rates at present energies.All the curves show a smooth behavior with the energyof the incoming neutrino (antineutrino). The importanceof threshold effects is clear and for that reason differentrescaling variables could give rise to different predictionsfor bottom-quark excitation rates. It is interesting to notethat the effect of the rescaling is much less important inantineutrino bottom-quark production (see Fig. 4) than inthe neutrino process (see Fig. 3), because in the former thedominant contribution is from u distribution which isfrom valence. In the latter case both contributions arefrom the sea. Predictions for top-quark miss of 30 and40 GeV are provided (Fig. 5) because these seem to be inthe range of values experimentally indicated.The curve corresponding to charm production from dquarks coming from the parametrization of Ref. 5 showsa slow decreasing for high energies, probably due toparameter-fixing precision, and we think the effect shouldnot be taken seriously. It just shows the need of aparametrization able to cover a larger range of energies,and having simultaneously a good fit at already experi-mentally covered energy ranges.2752 GARCIA CANAL, SANTANGELO, AND GAY DUCATI 31ACKNOWLEDGMENTSWe would like to acknowledge H. Fanchiotti, H. Vucet-ich, and C. Camarata for illuminating discussions. One ofthe authors (M.B.Cx.D.) is very grateful for the kind hospi-tality of the Theoretical Physics Department of La PlataUniversity. This work was partially supported by Con, -selho Nacional de Desenvolvimento CientificoTecnologico (CNPq), Brasil, and by Consejo Nacional deInvestigaciones Cientificas y Technicas (CONICET), Ar-gentina.~C. Baltay et al. , Phys. Rev. Lett. 52, 1948 (1984); M. Jonkeret a/. , Phys. Lett. 998, 265 (1981); J. E. Kim et al. , Rev.Mod. Phys. 53, 211 (1981).2V. Barger, in Proceedings of the 21st International Conferenceon High Energy Physics, Paris, 1982, edited by P. Petiau andM. Porneuf [J.Phys. (Paris) Colloq. 43, C3 (1984)].P. Langacker, work presented at XXIInd International Confer-ence on High Energy Physics, Leipzig, 1984 (unpublished).4A. J. Buras and K. J. F. Gaemers, Nucl. Phys. 8132, 249(1978).5D. W. Duke and J. F. Owens, Phys. Rev. D 30, 49 (1984).C. E. M. Aguiar, C. A. Garcia Canal, and J. A. MartinsSimoes, Lett. Nuovo Cimento 36, 69 (1983).7R. Bock, work presented at XXIInd International Conferenceon High Energy Physics, Leipzig, 1984 (unpublished).8E. Fernandez et al. , Phys. Rev. Lett. 51, 1022 (1983).N. S. Lockyer et al. , Phys. Rev. Lett. 51, 1316 (1983).~ Ling-Lie Chau and Way-Yee Keung, Phys. Rev. D 29, 592(1984).J. Smith and C, H. Albright, Phys. Lett. 858, 119 (1979).Choy-Heng Lai, Phys. Rev. D 18, 1422 (1978).R. J. N. Phillips, Nucl. Phys. 8212, 109 (1983).D. M. Scott and K. Tanaka, Phys. Rev. D 21, 1771 (1980).

Excitation rates of heavy quarks

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Excitation rates of heavy quarks

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