A general Lorentz-covariant quark model of mesons, whose nonrelativistic limit corresponds to Isgur-Scora-Grinstein-Wise model, is constructed. It possesses the heavy quark symmetry and can be easily applied to calculation of form factors. Besides it can be engaged in novel tasks, such a the investigation of the two photon decay of scalar mesons. Its behaviour in the infinite momentum frame and the light cone is discussed

Horvatić, Davor

Tadić, Dubravko

Žganec, Srećko

Repository of Faculty of Science, University of Zagreb

ISSN1330–0016CODEN FIZBE7RELATIVISTIC QUARK MODELD. HORVATIC´, D. TADIC´ and S. ZˇGANECPhysics Department, University of Zagreb, Bijenicˇka c. 32, Zagreb, CroatiaReceived 16 February 1999; Accepted 12 July 1999A general Lorentz-covariant quark model of mesons, whose nonrelativistic limitcorrespond to Isgur-Scora-Grinstein-Wise model, is constructed. It possesses theheavy-quark symmetry and can be easily applied to calculation of form factors. Be-sides it can be engaged in novel tasks, such as the investigation of the two-photondecay of scalar mesons. Its behaviour in the infinite momentum frame and on thelight cone is discussed.PACS numbers: 12.39.Ba, 12.39.Ki UDC 539.126Keywords: quark, relativistic, decays, mesons1. IntroductionIt is useful to develop a fully covariant model that, in nonrelativistic limit(NRL), goes into the ISGW model. Such a covariant model can, to a great extent,retain the simplicity which was endearing and useful feature of the nonrelativisticquark model of Isgur, Scora, Grinstein and Wise (ISGW)[1]. That model has alsobeen employed [2-5] in the investigation of heavy-quark symmetry (HQS). Althoughthe ISGW model helped HQS investigations, this nonrealtivistic model was not ca-pable [4,5] of properly reproducing all of the heavy-quark effective-theory (HQET)relations among semileptonic meson decay form factors. It had to be ”relativized”to some extent [2-5]. In the covariant model HQS is produced by free particle Diracspinors used to describe valence quarks in the correlated quark models [2-7]. Thisapplies also to the models [8,9] in which valence quarks are bound to a center offorce.The SU(2) ”spin” symmetry [10,11] which holds in the effective field theory,holds in the covariant quark models also. One obtains HQS relations among formfactorsF1 = V = A2 =(MB +MD)22MBMD(1 + w)A1 ; w = vBvD (1.1)FIZIKA B 8 (1999) 2, 353–362 353horvatic´ et al.: relativistic quark modelHere MB,D are B,D meson masses and vi = Pi/Mi. The form factor F1 appearsin the B → D decay and A1 in the B → D∗ decay for example.The form factors calculated in a nonrelativistic model satisfy HQS only in thezero recoil limit, where w = 1.The covariant quark model leads to Isgur-Wise functions (IWF) in which rel-ativistic corrections [5] and the Wigner rotation [4] are automatically included.Defining the slope ρ of IWF byξ(w) = 1− ρ2(w − 1) +O((w − 1)2) (1.2)Reference 6 found ρ = 1.17. In the nonrelativistic model [1] one obtains [5] ρ = 0.56or ρ = 0.93 when some relativistic corrections are included. Further increase in ρis in agreement with Ref. 4 which concluded that Wigner rotation should increasethe slope by about 20%.A relativistic model can provide a qualitatively correct description of the twophoton decays of scalar mesons.Model can be used to illustrate either the infinite momentum frame (IMF), orlight cone (LC) behaviour also. In this short review this last point will be onlybriefly sketched.2. Covariant quark model with correlated quarksThe state of a pseudoscalar meson H is given by|H(E, ~P ,M)〉 = N∑c,s1,s2∫4mQmdd4pδ4(p2 −m2Q)Θ(e)d4qδ(q2 −m2d)Θ(ǫ)d4K·F (K)δ(4)(p+ q +K − P )Θ(E)φ(l⊥) uQγ5vd d+d (~q, c, s2)b+Q(~p, c, s1)|0〉 (2.1)Here lµ = (pµ − qµ)/2, lµ⊥(P ) = lµ − Pµ(P · l)/M2, pµ = (e, ~p ), qµ = (ǫ, ~q ). Thevalence quarks are represented by the on-mass-shell Dirac spinors u and v. The seafunction F (K) is for exampleF (K) = δ(4)[Kµ − PµM(P νM(P − (p+ q))ν](2.2)orF (K) = δ(4)[Kµ − PµM(P νM(P − (p+ q))ν]e−αK2(2.3)Some discusion of the second possibility can be found in Appendix B. The mesonwave function can have the Gaussian formφ(lµ⊥) =1π3/4β3/2Se+(lµ⊥)2/2β2S (2.4)354 FIZIKA B 8 (1999) 2, 353–362horvatic´ et al.: relativistic quark modelThis construction insures that in the NRL model goes in the well known ISGWmodel [1]. HQS does not depend on a particular form of F (K).The meson decay constant fH , for example, is with (2.2) given by1(2π)32PµfH = 〈0| : ψ¯2(0)γµγ5ψ1(0) : |H(E, ~P ,M)〉= 3N(~P )∫d3pm1MeEMm2Eǫ− ~P · ~qφ(l⊥)[m2pµ +m1qµm1m2]|~q=−~p+~PM(p‖)T(2.5)Here p‖ = (Ee− ~P~p)/M and T = 1 +√m22 −m21 + (p‖)2/p‖.One obtains the same numerical value for fH in any frame, including IMF ,where:Pµ = (E, 0, 0, P ) , P →∞,x =pzP, ~pT = (px, py),fH = 3N(2π)52∞∫0pTdpT∞∫0dx2m1MxφIMF (l⊥), (2.6)φIMF (l⊥) =1π3/4β3/2exp{ 12β2[−~p 2T2(1 +m21x2M2+~p 2T2x2M2)− M24(x− m21xM2)2]}.Due to the presence of the quark-gluon sea, which also carries part K of the totalmomentum P , the momentum carried by the antiquark, i.e. qz/P , is not (1− x).3. Two photon decays of scalar mesonsHere the usage of small quark masses means the avodiance of the weak bindinglimit approximation in its strictest sense [1]. That might better mimick the realquark fields which should appear in the photon emitting quark loop (Fig. 1) inthe first order of QED/QCD expansion. A necessary modification of the model isdiscussed in Appendix A.Such reparametrized model is applied to study two photon decays of a0(980),f0(980), f0(1370) and χc0 mesons. The experimental data [12], on strong decaysof a0(980), f0(980) and f0(1370) do not lead to the final understanding of the realstructure of these mesons [13-26]. Ideas exists that these states are KK¯ molecules[13-18]. It seems that qq¯ structure is prefered by their decays into light pseudoscalars[18-25].FIZIKA B 8 (1999) 2, 353–362 355horvatic´ et al.: relativistic quark modelThe analisys was caried out with the quark wave functionφf (lµ⊥) =1(1− l2⊥4β2f,H)2. (3.1)The dipol form (3.1) was found to be a better choice than the exponential formused earlier [6]. See also some remarks in Appendix A concerning the parameter(s)βf,H (f = u, d, s, c). For simplicity we have used (2.2). The spinor form in (2.1) isreplaced for scalar mesons byu¯Qvd (3.2)The scalar meson state is normalized by〈H(E, ~P ,M)|H(E, ~P ,M)〉 = 2E = 3N(0)2∑fC2f∫d3pǫe(φf (l⊥)q‖)2(pq−m1m2)(3.3)Such model states are consistent with the very general requirement, i.e.〈0|V µ|H(E, ~P ,M)〉 ≡ 0 (3.4)(a) (b)f fq qp pk2 p-k1 k1 k1 p-k2 k20 0Fig. 1. Two-photon decay. Full lines are valence quarks, wavy lines are emittedphotons and blobs symbolize scalar meson states.The amplitude M for the transition f0 → 2γ is determined from the leadingdiagrams shown in Fig. 1. A loop corresponds to each quark flavor. For example,for the flavor d the amplitude corresponding to the diagram in Fig. 1a is determinedbyM = − 3N(0)mdω(2π)12∫p2dp sinθdθm2de21(1 + p2/(4β2d,f0))2(~ǫ1~ǫ2)2p2(ωcos2θ + esin2θ)e2 − p2cos2θ≡ (~ǫ1~ǫ2) · Id,d(f0) (3.5)356 FIZIKA B 8 (1999) 2, 353–362horvatic´ et al.: relativistic quark modelThe calculation of the decay widthΓ =132πMH|MH |2 (3.6)requires the summation over photon polarization states, as well as the summationover quark flavors wich are connected with the meson quark structure;|f0(980)〉 = cos θ√2(|uu〉+ |dd¯〉) + sin θ|ss¯〉 (3.7)|a0(980)〉 = 1√2(|uu¯〉 − |dd¯〉) (3.8)|f0(1370)〉 = − sin θ√2(|uu¯〉+ |dd¯〉) + cos θ|ss¯〉 (3.9)|χc0(3415)〉 = |cc¯〉 (3.10)The gauge invariance of all results can be explicitly tested by showing that theamplitude M does not change under gauge transformationǫiµ → ǫiµ + Λkiµ (3.11)Using the model parameters detarmined and described in Appendix A, and thestates (3.7)-(3.10) corresponding to mixings given [25] given bycos θ = 1 ; sin θ = 0 (3.12)orcos θ =13; sin θ =2√23(3.13)one ends with Table 1.The results indicate the importance of the quarkonium structure qq¯ [1] in themeson state.The experimental error in f0(980) → 2γ rate is rather large. Although,the larger theoretical prediction in Table 1, seems to be in better agreement withexperiments, the smaller one, which corresponds to the ideal mixing, cannot beruled out. However, the f0(1370) decay into pions indicates the presence of thelight qq¯ combinations [18,21,22]. Our result also agrees with the nonideal mixing asconsidered by Lanik [24]. Our approach has some analogy with Deakin et al. [24]who used constituent quark masses and concluded that theoretical results dependstrongly on the numerical values of those masses.FIZIKA B 8 (1999) 2, 353–362 357horvatic´ et al.: relativistic quark modelTABLE 1. Decay widths.Meson Mixing Γtheory(keV ) Γexp(keV )a0(980) 0.137 0.26± 0.08f0(980) (3.12) 0.380 0.56± 0.11f0(980) (3.13) 0.534 0.56± 0.11f0(1370) (3.12) 0.348f0(1370) (3.13) 0.145χc0(3415) 4.608 4.0± 2.84. Covariant quark model and light cone variablesThe LC field theory which may allow the derivation of a constituent quarkpicture [27] has prompted an extensive study of LC quark models [28,29,31].Any covariant quark model can be written using LC variables. The state (2.1)for example becomes (mQ = md = m):|H(P+, ~P⊥,M)〉 = N∑c,s1,s2∫dp+d2~p⊥mp+∫dq+d2~q⊥mq+δ(q+ + p+ − 2P+M(p‖))·δ(~q⊥+~p⊥−2~P⊥M(p‖))MmP+q‖φ(l⊥)uQγ5vdd+d (q+, ~q⊥, c, s2)b+Q(p+, ~p⊥, c, s1)|0〉. (4.1)Here p+ = e + p3, ~p⊥ = (p1, p2), p‖ = ((p+P− + p−P+)/2 − ~p⊥ ~P⊥)/M, p− =(m2 + ~p2⊥)/p+ etc.Although based on the same physics the models [6] and [28–32] are not identical.Some differences appear in parametrization of the model wave function. The relativecoordinate k2 of Refs. 28, 29, 31 and 32 is in the notation of Ref. 6k2 = −14(p− q)2 (4.2)In Refs. 27, 28, 29, 31 and 32, a different spinor basis uLC(p) is employed, which isconnected to textbook spinors u(~p,E) [18] through:uLC(p)s ∼ us(~p,E) + v−s(−~p,−E) (4.3)LC spinors depend only on p+ and pR,L = (p1 ± ip2)/2 combinations.358 FIZIKA B 8 (1999) 2, 353–362horvatic´ et al.: relativistic quark modelWhile an u spinor can be constructed [33] by boosting the Pauli spinor χ, theuLC spinor is produced by a more complex Melosh transformationuLC =12√mp+(UM −ZMZM −UM)(χ0)(4.4)UM =(p+ +m −pLpR p+ +m)ZM =(p+ −m pLpR −p+ +m)This is only a formal difference as model [6] can be easily constructed by using uLC .The results (formfactors, widths) do not change.Another important difference lies in the model description of sea effects, whichin Ref. 6 are contained in F (K) (2.2, 2.3). The δ4 functions appearing in (2.1) and(2.2) are replaced in Refs. 27, 28, 29, 31 and 32 by a different construct.Appendix AReference 1 has determined the meson wave function parameter β from varia-tional principle assuming the quark-antiquark interactionV (r) = −4α3 r+ b r + c (A1)α = 0.5, b = 0.18GeV 2, c = −0.84GeVThey had used the nonrelativistic quark kinetic energies e.g. EQ = ~p2/2mQ withlargish constituent masses m¯i If procedure is carried out with the same potentialbut with the relativistic energies eQ =√~p 2 +m2Q [34] simmilar β’s can be foundfor mQ < mQ. This is ilustrated in Fig 2.One finds βrel > βISGW for mQ < 0.25GeV . At about mQ = 0.3GeV β’s arecomparable. Thus in a relativistic model one might find a wave fnction wich isconsistent with ISGW function, while model quark masses are small.For a scalar meson in the rest frame the model ( mock meson ) mass M0 isM0 =∑f C2f∞∫04πp2 dp2p2φ2fp2+m2f(2ǫf + V (p))∑f C2f∞∫04πp2 dp2p2φ2fp2+m2f(A2)FIZIKA B 8 (1999) 2, 353–362 359horvatic´ et al.: relativistic quark modelHere summation goes over quark flavors, and the model wave functions has dipoleform (3.1). Masses are:mu = md = 0.015GeV ; ms = 0.120GeV ; and mc = 1.4GeVIf one omits the potential and requires that M = M0, where M is the real mesonmass ( Table 1 ), one finds for β parametersβ = 0.288; 0.288; 0.262; 0.380; 0.399; 0.346 (A3)[These values correspond to mesons listed in the first column of the Table 1.]β (GeV)m  (GeV)pFig. 2. ISGW [1] and Lucha [34] model β’s for |qq¯〉Alternatively one can use the potential (A1) and find β’s which minimase theexpresion (A2) [1,34]. The results, obtained for simplicity for mf = 0, are in res-onable agreement with (A3). For the first two two cases one finds β = 0.256 whichagrees within 12%. The choice describes mock meson states for which P 2 = M2.Appendix BSome more realistic forms of the sea function, as for example (2.3) would alsoinfluence model predictions. This possibilities have not yet been fully exploited.Some interesting preliminary results, concerning the Isgur-Wise function (IWF)and the meson decay constant fM are listed below.Using parameters from [1] and (2.2) one obtains for the slope ρ (1.2)ρ ∼= 1.07With (2.3) and α = 1/2β2 one findsρ ∼= 1.17360 FIZIKA B 8 (1999) 2, 353–362horvatic´ et al.: relativistic quark modelObviously the sea is noticable.The sea contribution could lead to better description of meson states. That canbe illustrated in the calculation of the meson decay constants. With (2.2) one findsfπ = 461MeV (fπexp = 131.7MeV ) and fK = 412MeV (fKexp = 160.6MeV ).However by using (2.3) with α = αs/2Λ2, Λ = 0.25GeV , and αs = 0.6(0.55) forπ(K) meson, one finds fπ = 132MeV and fK = 160MeV .References1) N. Isgur, D. Scora, B. Grinstein, and M.B. Wise, Phys. Rev. D 39 (1989) 799; D. Scora,N. Isgur, Phys. Rev. D 52 (1995) 2783;2) N. Isgur, Phys. Rev. D 43 (1991) 810;3) C.O. Dib and F. Vera, Phys. Rev. D 47 (1993) 3938;4) F. E. Close and A. Wambach, Nucl. Phys. B 412 (1994) 169;5) J. F. Admundson, Phys. Rev. D 49 (1994) 373;6) D. Tadic´ and S. Zˇganec, Phys. Rev. D 52 (1995) 6466;7) L. Micu, Czech J. Phys. 41 (1991) 129;8) D. Tadic´ and S. Zˇganec, Phys. Rev. D 50 (1994) 5853, and references therein;9) T. Biswas and F. Rohrlich, Nuovo Cimento A 88 (1985) 57;10) B. Grinstein, Ann. Rev. Nucl. Part. Sci 42 (1992) 101;11) M. Neubert, Phys. Rep. 245 (1994) 259;12) R. M. Barnett et al., Phys. Rev. D 54 (1996) 1;13) D. Morgan and M. R. Pennington, Phys. Rev. D 48 (1993) 5422;14) B. S. Zou and D. V. Bug, Phys. Rev. D 48 (1993) 3948;15) G. Jansen et al. Phys. Rev. D 52 (1995) 2690;16) J. Weinstein and N. Isgur, Phys. Rev. D 41 (1990) 2236;17) F. E. Close et al., Phys. Lett. B 319 (1993) 291;18) N. A. To¨rnquist, Z. Phys. C 68 (1995) 647;19) A. Bramon and S. Narison, Mod. Phys. Lett. A 4 (1989) 1113;20) M. Genovese, Nuovo Cim. A 107 (1994) 1249;21) A. S. Deakin et al., Mod. Phys. Lett. A 9 (1994) 2381;22) J. Babckock and J. L. Rosner, Phys. Rev. D 14 (1976) 1286;23) A. Bramon, Phys. Lett. B 333 (1994) 153;24) J. Lanik, Phys. Lett. B 306 (1993) 139;25) S. Fajfer, Z. Phys. C 68 (1995) 81;26) N. N. Achasov et al., Phys. Lett. B 108 (1982) 134;27) S. J. Brodsky and D. G. Robertson, in Proc. “ELFE Summer School and Workshopon Confinment Physics”, Cambridge (1995);28) P. L. Chung, F. Coester and W. N. Polyzou, Phys. Lett. B 205 (1988) 545;29) W. Jasus, Phys. Rev. D 41 (1990) 3394;FIZIKA B 8 (1999) 2, 353–362 361horvatic´ et al.: relativistic quark model30) P. A. M. Dirac, Rev. Mod. Phys. 21 (1949) 392;31) C. R. Ji et al., Phys. Rev. D 45 (1992) 4214;32) Z. Dziembowski, Phys. Rev. D 37 (1988) 786;33) J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New-York (1964);34) W. Lucha, H. Rupprecht and F. F. Scho¨berl, Phys. Rev. D 44 (1991) 242.RELATIVISTICˇKI MODEL KVARKOVAIzveli smo poopc´eni Lorentz-kovarijantni model mezona cˇiji nerelativisticˇki limesodgovara Isgur-Scora-Grinstein-Wiseovom modelu. Model ima tesˇko-kvarkovskusimetriju i mozˇe se primijeniti za racˇunanje faktora oblika. Osim toga u ovom semodelu mogu rjesˇavati nove zadac´e kao sˇto je dvofotonski raspad skalarnih mezona.Raspravljamo svojstva modela u sustavu beskonacˇnog impulsa i na svjetlosnomstosˇcu.362 FIZIKA B 8 (1999) 2, 353–362

Relativistic quark model

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Relativistic quark model

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