Λb POLARIZATION IN THE Z BOSON DECAYS. V.A. Saleev Samara State University, Samara, Russia Abstract 8 9 In theframework of the perturbativeQCDand thediquark modelof baryons we 9 have obtained the fragmentation functions for heavy quark that split into polarized 1 Λ baryons. We predict the longitudinal polarization asymmetry for the prompt Λ b b n produced in e+e− annihilation at the Z resonance and estimate the spin-1/2 and a J spin-3/2 beauty baryon production rate. 0 2 Introduction 3 v 0 The heavy baryon production in e+e− annihilation is an increasingly important subject 7 to study. The recent measurements near the threshold of the heavy quark production, as 3 2 well as at the Z resonance, give us the information about heavy baryon masses, life times, 0 decay modes [1, 2, 3] and polarizations [4, 5]. 7 9 The b quarks produced in e+e− annihilation at the Z peak are strongly polarized. h/ Accordingly tothe StandardModeltheright-left longitudinalasymmetry isAb = 0.94. p RL − This polarization is expected [6] to be only slightly (2%) reduced by the hard gluon - p emission before the hadronization into baryons. Therefore the measurement of the Λ b e polarization is very important for study the details of the hadronization processes. The h : firstmeasurement [4]gaveanintriguinglysmallvalueAΛb = 0.23+0.26,whichcontradicts v RL − −0.23 i the prediction that the large part of the initial polarization of the quark to transfer to the X Λ baryon. r b a The data on the Λ polarization may be described in part (AΛb 0.68) by means of b RL ≈ − a simple depolarization model, based on the cascade Λ partial production in the decays b of higher spin beauty baryon produced in the Z decays [7]. In this paper we propose the depolarizing mechanism occurring during the prompt Λ production via b quark fragmentation. Our approach base on the perturbative QCD, b the quark-diquark model of the baryons [8] and the nonrelativistic approximation, which is used successfully for the description of the large distance effects in the heavy quarko- nium production [9]. The same approach have been discussed in the our previous paper [10], where the predictions for the doubly heavy baryon production via the heavy quark fragmentation have been presented. Λ 1 The prompt production in Z decays b The fragmentation function Db→Λb(z,µ) at the initial scale µ = µo = mQ +2mD for the productionoftheΛ baryon, containing theQ = bquark andthescalar diquark D = (ud), b 1 is given by the following expression [10, 11]: 1 ∞ 2 Db→Λb(z,µo) = 16π2 Zsmindsqoli→m∞ ||MMo||2, (1) where is the matrix element for the production Λ baryon and antidiquark D¯ with the b totalfoMur-momentumq = p+q′ andinvariantmasss = q2 intheZ decay(Z Λ D¯¯b), b o → M is the matrix element for the production of a real b quark with the same three-momentum ~q. The lower limit in the integral (1) is M2 m2 s = + D , min z 1 z − where M = m +m is the baryon mass, m is the heavy quark mass, m is the diquark Q D Q D mass. The amplitude may be presented as follows: o M (Z b¯b) = U¯(q)ΓαV(q¯)ε (Z), (2) o α M → where ε (Z) is the polarization four-vector of the Z boson and α ig Γα = γα Cb Cb γ5 −2cosθ V − A W (cid:16) (cid:17) is the quark-boson vertex. In the axial gauge for the gluon propagator associated with the four-vector n = (1,0,0, 1): − k n +k n α β β α d (k) = g + , αβ αβ − (kn) the fragmentation contribution comes only from the Feynman diagram shown in Fig.1. After some obvious simplification we have obtained the matrix element of the Z decay into Λ baryon [10]: b Ψ(0) 4δij F (k2) (Z Λ ¯bD¯) = g2 s M → b √2m s3√3(s m2)2 · D − Q np 2U¯(p) M(qˆ+m )+(s m2) ΓαV(q¯)ε (Z) (3) − Q − Q nk α (cid:20) (cid:18) (cid:19)(cid:21) where g2 = 4πα , 4δij/3√3 is the color factor of the diagram, F (k2) is the form factor s s s of the vertex g∗ DD¯, Ψ(0) is the Λ = (QD) wave function at the origin in the quark- b → diquark approximation. The gluon coupling to scalar diquarks was used in the following form [8]: Sa = ig Ta(q′ p ) F (k2), (4) ν − s − D ν s where Ta = λa/2 are Gell-Mann matrices, k = q′ +p is the gluon four-momentum, the D spinor U¯(p) describes Λ baryon, p = p +p , p = rp, p = (1 r)p, r = m /M. The b Q D D Q D − scalardiquarkformfactoratk2 > 0maybeparameterizedasinref.[12], wheretheauthors 2 ¯ fit successfully the angular distributions of the baryons in the processes γγ pp¯,ΛΛ and the widths of J/ψ pp¯,ΛΛ¯: → → Q2 F (k2) = s , (5) s Q2 k2 s − where Q2 = 3.22 GeV2 and form factor is restricted to value smaller than 1.3. We use s 2 also the parameterization (5) with the fixed full width at half maximum Γ 0.8 GeV . ≈ In the both cases there are no singularities in the physical region of the virtuality of the fragmenting quark or the square of the gluon four-momentum. The phenomenological diquark form factor parameterization (5) corresponds to the elastic vertex g∗ DD¯ [12]. However, in the fragmentation processes the contribution → ∗ ¯ from the inelastic vertex g Du¯d may be dominant. There are no any information → about inelastic diquark form factors. That is why we will use parameterization (5) to describe spin effects in the Λ production. As it will be shown, the choice of the form b factor parameterization is important only for the prediction the absolute values of the heavy baryon production rates, but the spin asymmetry doesn’t depend on a diquark form factor in the kinematic region, which is studied. The reduced mass of the heavy quark – diquark system is the same order as one for the system of the two charmed quarks and we can hope that the calculation of the parameter Ψ(0)using thepotentialapproach, willbewell-grounded andmaybedonewiththequark- diquark interaction potential, which has been fixed previously in calculating the heavy baryon mass spectrum [13]. The energy distribution of the right (R) or the left (L) longitudinally polarized Λ b baryons produced in the Z decays reduces at leading order in α to s dΓR,L (Z Λ X) = Γ(Zo b¯b)DR,L (z,µ), (6) dz → b → b→Λb where p +p 0 3 z = , q +q 0 3 p = (p ,0,0,p ) is 4-momentum of the Λ , q = (q ,0,0,q ) is 4-momentum of the frag- 0 3 b 0 3 menting b-quark and µ m /2. Let us define Z ≈ ∆Db→Λb(z,µ) = DbR→Λb(z,µ)−DbL→Λb(z,µ). (7) The unpolarized fragmentation function is Db→Λb(z,µ) = DbR→Λb(z,µ)+DbL→Λb(z,µ). (8) The longitudinal asymmetry AΛb may be presented as follows: RL AΛb (z,µ) = ∆Db→Λb(z,µ). (9) RL Db→Λb(z,µ) In the limit of m2 >> s,M2 we have got Z g2 Tr qˆΓ ˆq¯Γ gαβ +ZαZβ/m2 = m2(Cb2 +Cb2) , (10) α β − Z Z V A "4cos2θW# h i(cid:16) (cid:17) 3 g2 Tr qˆγ Γ ˆq¯Γ gαβ +ZαZβ/m2 = 2m2Cb Cb . (11) 5 α β − Z − Z V A"4cos2θW# h i(cid:16) (cid:17) Taking into account that the four-vector of the Λ polarization in the states with the b different longitudinal polarization is sR,L = (p /M,0,0,p /M) and using the following Λb ± 3 0 exact formulae 2 2 2 2pk = s m , k = r(s m ), − Q − Q 2 2 2 2 2pq = s m +2M (1 r), 2Zq = m +s m , − Q − Z − Q we can reduce the squared matrix elements to the form, which contains expressions (10) or (11). Omitting the next details, we write here the obtained results for the unpolarized fragmentation function at the scale µ = µ : o ∞ ds M2 4 2 2 Db→Λb(z,µo) = DoZsmin M2Fs(k ) s−m2Q! s m2 s m2 2 d +d − Q +d − Q , (12)  o 1 M2 ! 2 M2 !    where 32α 2 Ψ(0) 2 s D = | | , (13) o 27rM3 1 (4 r)z (1 r)z2 z3 d = 4(1 r), d = − − − − , d = , (14) o − 1 1 z(1 r) 2 (1 z(1 r))2 − − − − The result for the ∆Db→Λb(z,µo) may be written as for the Db→Λb(z,µo) but we don’t present it here because of it’s unwieldy. The recalculating of the fragmentation functions DRL(z,µ ) from the start point µ = Λb o µ to the µ > µ may be done using Gribov-Lipatov-Altareli-Parisi (GLAP) evolution o o equations ∂D 1 dy z µ (z,µ) = PQ→Q ,µ D(y,µ), (15) ∂µ Zz y y ! where PQ→Q is the splitting function at leading order in αs: 4α (µ) 1+x2 s PQ→Q(x,µ) = , (16) 3π 1 x ! − + 1 ′ ′ f(x) = f(x) δ(1 x) f(x)dx. + − − 0 Z Note that at leading order in α one has s PQ→Q(z,µ)dz = 0, Z and the evolution equation implies that the fragmentation probability Pb→Λb = Db→Λb(z,µ)dz Z 4 aswellasthetotalasymmetryAΛb don’tevolvewiththescaleµ. However, thez dependence of the AΛb (z,µ) changes drasticRaLlly when the scale µ growth from µ = µ to µ− m /2. RL o ≈ Z First ofall, wediscuss heretheresults concerning spineffects intheprompt Λ produc- b tion, which don’t depend on the value of Ψ(0). Figs. 2-4 demonstrate the dependence of the polarization asymmetry AΛb on the diquark mass, the form factor parameterization, RL z and QCD evolution scale µ. We can see in Fig.2 that AΛb = Ab at the realistic values of the diquark mass (m = 0.6 0.9 GeV correspoRnLds6 to RALΛb = 0.90 0.87) and only at the limit of m D 0 on−e has AΛb Ab = 0.94|. ROLu|r result c−orrects the assumption, which is D → RL ≈ RL − used usually, that the Λ baryon retain a large part of the initial polarization of the b b quark [14]. The predictions for the asymmetry as a function of z strongly depends on the diquark mass at µ = µ , but at the large µ this dependence vanishes (Fig.3). o AΛb (z,µ) may be sensitive to the choice of the form factor parameterization at the RL small µ (see Fig.4), however at the scale µ = m /2 the results corresponding the different Z form factor parameterizations are equal. The measurement of the AΛb versus z at the RL different µ is a exact test of the hadronization model. However, it may be done exper- imentally if we can to separate the prompt Λ production in the b quark fragmentation b ∗ ∗ from the Λ produced in the cascades: b Σ ,... plus the hadronical decays Σ ,... Λ . b → b b → b This opportunity was shown recently in the study of the heavy quarkonium production at FNAL [15], where the vertex detector technique was used. The Figs.5 show our results for the AΛc versus z at the different µ = µ ,m /2 and RL o Z m = 0.6 GeV. We have obtained AΛc = 0.58 opposite to Ac = 0.67. The probabil- D RL − RL − ity of the heavy quark spin turning during the hadronization is about 13% for the c-quark and only 4% for the b-quark. The both values are independent from the diquark form factor and were obtained at m = 0.6 GeV. We see that the simple rule ( m /m ) is D D Q ∼ satisfied. Λ 2 The cascade production in the Z decays b UsingthevalueoftheradialpartoftheΛ baryonwavefunction R(0) 2 = 0.8GeV3 (α = b s | | 0.4, m = 0.6 GeV), which was calculated in the quark-diquark approximation [13], we D 2h.a5ve· 1f0o−u3ndintthheetcoatsael opfrotbhaebfiloirtmy offactthoer b(5)quaanrdkPfrba→gΛmb e≈nta2t.5io·n1i0n−t2oitfhteheΛsb:ingPubl→aΛribtie≈s are restricted using the fixed width Γ. In spite of this deference the shapes of the frag- mentation functions Db→Λb(z,µ) are the same for the both form factors (Fig. 6). The obtained value Pb→Λb gives the number of Λb per the Z decay which is smaller than it was measured at LEP [3]. We conclude that the main part of the Λ baryons are produced in b the cascade processes via the decays of the higher spin b baryons. The DELPHI collabo- ∗ ration has presented evidence for Σ baryon production [16] and measured the rate to be b Pb→Σ∗ = 4.8 1.6%, the total b baryon production rate to be Pb→baryon = 11.5 4.0% [2]. b ± ± ∗ We can estimate the fragmentation probability for the b quark that split into Σ b baryon in the our approach. However, as it was demonstrated in the case of the Λ baryon b production, the predicted absolute production rate is strongly depends on diquark form factor. It will be more reliable to calculate the ratio of the fragmentation probabilities for 5 the b quark that split into vector diquark states to scalar diquark states, using deferent form factors. As it was shown in ref. [7] this parameter (called A) controls the Λ b polarization in the cascade production. The gluon coupling to axial vector diquarks is presented by the following expression: Vµb = igsTb ε∗DεD¯(q′ −pD)µF1(k2)−[(q′εD¯)ε∗Dµ −(pDε∗D)εD¯µ]F2(k2)− (cid:26) ∗ ′ ′ 2 (εDq )(εD¯pD)(q −pD)µF3(k ) , (17) (cid:27) ∗ where εD,εD¯ are the diquark polarization four-vectors. F1,F2 and F3 are form factors depending on the momentum transfer squared k2 = (q′ +p )2. The same as in ref. [12] D we put for k2 > 0: Q2 2 F (k2) = V , (18) 1 Q2 k2! V − 2 2 2 F (k ) = (1+κ)F (k ), F (k ) = 0 2 1 3 where Q = 1.50 GeV2 and κ being the anomalous chromomagnetic moment of the vector V diquark. The anomalous magnetic moment of the (ud) vector diquark was determined in ref.[12], but it is not obviously that the chromomagnetic and magnetic moments are equal. We will choose κ = 0 for the simplicity. The amplitude for heavy quark fragmentation into spin-3/2 baryons, corresponding to fusion of the heavy quark and the axial vector diquark, can be written as a sum of two parts, which are proportional to F and F form factors: 1 2 1 2 = + , (19) M3/2 M3/2 M3/2 Ψ(0) 4δij F (k2) 1 2 1 = g (20) M3/2 √2m s3√3(s m2)2 D − Q (np) 2Ψ¯µ(pQ)εD¯µ"−M(qˆ+mQ)+(s−m2Q)(nk)#ΓαV(q¯)εα(Z), Ψ(0) 4δij F (k2) 2 2 2 = g (21) M3/2 √2m s3√3(s m2)2 D − Q 1 s m2 rΨ¯σ(pQ)εD¯λ"qσγλ(qˆ+mQ)+ (−kn)Q(kλnσ −kσnλ)#ΓαV(q¯)εα(Z). Here, the Ψ¯ (p) is the Rarita-Schwinger spinor for the spin-3/2 baryons. The procedure µ of the calculation for the fragmentation function Db→Σ∗(z,µ) is the same as for the Λb b baryons. Using the set of parameters: R(0) 2 = 0.8GeV3, α = 0.4, m = 0.6 GeV, κ = 0, s D | | we have found that the ratio A of the fragmentation probabilities for the b quark that split into vector diquark states to scalar diquark states is approximately equal to 5 1 ± 6 independently on the choice of the diquark form factor. Accordingly the formula (5.10) ref.[7] one has 1+(1+4w )A/9 AΛb = 1 Ab , (22) RL 1+A RL where the parameter w gives the probability that spin-1 diquark has maximum an- 1 gular momentum j3 = 1 along the fragmentation axis. The authors ref. [7] used ± A = 0.45, w = 0, as it is motivated by the Lund fragmentation model [17], and ob- 1 tained that AΛb = 0.68. If we put A = 5 in (22), as it follows from our results, we find that AΛb = R0L.24 w−hich is in a good agreement with the value of the measurement. In cRoLnclu−sion we want note that the prediction of the cascade model [7] for the AΛb RL in the Z decays was obtained using the assumption that the turning of the b quark spin may be ignored, as is appropriate to the heavy quark approximation. Our results show that the probability of the b quark overturning during the hadronization may equal to 4-7 % and the more careful analysis of the depolarization in the cascade Λ production is b needed. When our work was completed the result of the computing the heavy baryon pro- duction rates at CLEO and LEP energies, which was obtained a similar way, has been presented [18]. Acknowledgments We are grateful to S. Gerasimov and A. Likhoded for discussions the problems of the heavy baryon physics and G. Goldstein for the valuable information on obtained results. References [1] Y. Kubota et al., Phys.Rev.Lett. 73, 3225 (1993); P. Avery et al., Phys.Rev. D43, 3599 (1991); H. Albrecht et al., Phys.Lett. 207, 109 (1988). [2] P. Abreu et al., Z.Phys. C68, 541 (1995). [3] D. Buskulic et al., Phys.Lett. B359, 236 (1995). [4] D. Buskulic et al., Phys.Lett. B365, 367 (1996); Phys.Lett. B365, 437 (1996). [5] D. Buskulic et al. Preprint CERN-PPE/96-04, 1996. [6] J.G. Ko¨rner J.G. et al., Z.Phys. C63, 575 (1994); M. Tung, Phys.Rev. D52, 1353 (1995). [7] A.F. Falk, M. Peskin, Phys.Rev. D49, 3320 (1994). [8] M. Anselmino M. et al., Rev.Mod.Phys. 65, 1199 (1993). [9] G.T. Bodwin, E. Braaten and G.P. Lepage, Phys.Rev. D51, 1125 (1995). S.S. Ger- shtein et al., Usp.Fiz.Nauk 165, 3 (1995); G.A. Schuler, Preprint CERN-TH.7170/94, 1994; 7 [10] A.P. Martynenko, V.A. Saleev, Phys.Lett. B385, 297 (1996). [11] E. Braaten, K. Cheung and T.C. Yuan, Phys.Rev. D48, 4230 (1993). [12] P. Kroll et al., Phys.Lett. B316, 546 (1993); R. Jacob, P. Kroll, Phys.Lett. B315, 463 (1993). [13] D. Ebert, R.N. Faustov, V.O. Galkin, A.P. Martynenko and V.A. Saleev, Z.Phys. C76, 111 (1997). [14] F. Close, J.G. Ko¨rner, R.J.N. Phillips and D.J. Summers, J.Phys, G18, 1716 (1992) [15] F. Abe et al., Nucl.Instr.Meth. A271, 387 (1988); D. Amidi et al., Nucl. Instr. Meth. Phys. Res., Sect. A350, 73 (1994). [16] P. Abreu et al., High Energy Physics Conference ”EPS-HEP”, Brusselles, 1995, Ref. eps0565. [17] T. Sjo¨strand, Comput. Phys. Commun. 39,347 (1986). [18] A.D. Adamov and G.R. Goldstein, hep-ph 9612443, 9706491. Figure captions 1. Diagram used for description of the process Zo Λ ¯bD¯. b → 2. The asymmetry AΛb as a function of the ratio m /M. RL D 3. The asymmetry AΛb as a function of z at µ = µ (curves 1 and 2) and µ = m /2 RL o Z (curves 3 and 4). The solid lines correspond to m = 0.9 GeV, the dotted lines D correspond to m = 0.6 GeV. D 4. The asymmetry AΛb as a function of z at m = 0.6 GeV and µ = µ (curves 1 RL D o and 2), m /2 (curves 3 and 4). The solid lines correspond to diquark form factor Z parameterization(5), thedottedlinescorrespondtoformula(5)withthefixedwidth Γ = 0.8 GeV2. 5. The asymmetry AΛc as a function of z at m = 0.6 and µ = µ (curve 1), m /2 RL D o Z (curve 2). 6. The fragmentation function Db→Λb(z,µ) normalized to unity at µ = µ0 (curves 1 and 2) and µ = m /2 (curves 3 and 4). The solid and dotted lines correspond to Z the different form factor parameterizations. 8 Figure 1: Diagram used for description of the process Zo Λ ¯bD¯. b → 9 Figure 2: The asymmetry AΛb as a function of the ratio m /M. RL D 10