Fragmentation of Heavy Quarks Using Wave Function at the Origin

Transcription

1 Universal Journal of hysics and Application (): 59-65, DOI:.89/ujpa.. Fragmentation of eavy Quarks Using Wave Function at the Origin G. Rboroun,*,. Abdolmaleki hysics department, Razi University, Kermanshah 6749, Iran Department of hysics, Toyserkan branch, Islamic Azad University, Toyserkan, Iran *Corresponding Author: Copyright orizon Research ublishing All rights reserved. Abstract In this paper we investigate the fragmentation function for heavy quarkanium using a global potential model. We employ the invariant verte factor under parity, Lorentz transverse and invariant amplitude scattering for the fragmentation function. In the heavy quark limit we have obtained the S-wave analytical fragmentation functions for the heavy mesons at leading order perturbative QCD. The wave function of the bound state for mesons was set at origin of the framework. The fragmentation function is calculated with respect to the and t variables for m +m and eperimental scales. In order to eamine the accuracy of our calculations, the obtained results compared to the BFGW method. Keywords Fragmentation, J/Ψ, WFO, Global otential are done in perturbation QCD that quarks and gluons are fied with photons. Indeed, for eplain of the fragmentation function in our desired scale, we can used the Altarelli- arisi evolution equations and also the wave function at the origin R (). As the ground state of heavy q and q changes in non-relativity form, as systems including massive quarks would construct bounded states. So, for studying massive hadrons, evaluation of the Schrödinger equation is very important. In previous work, the potential provided by the artin, logarithmic, Cornell, Lichtenberg. Introduction Light quarks fragmentation has been studied in various models []. These models include statistical manner as if the fragmentation parameter increase we observe that the fragmentation function decreases. After b and c quarks were discovered, the fragmentation mechanism of these quarks was described by eterson in a quantum-mechanical model []. Another models used both on eperimental and theoretical approaches [, 4]. any models have been used to describe the eperimental fragmentation function that the most popular are the power and the eterson models []. In applied theoretical model the wave function at origin describe the heavy quark system in limited range of hadron. ere, in order to avoiding the limitation of nucleus size of hadrons, we have used invariant amplitude and wave function at origin using a global potential..the Wave Function at Origin (WFO) The wave function at origin that has been calculated by the Schrödinger equation is one of the proper tools for calculating of fragmentation function [5]. These calculations Figure. massive meson production The Feynman diagram for this massive meson production is shown in Fig. at leading order (LO). ere we would like to consider the process of a heavy quark fragmentation with respect to a wave function at the origin (WFO). New chances have been brought up to apply non-relativity methods that are successful for bounded states in high-energy physics, which have been considered in the discovery of massive flavor mesons. So, for studying these mesons, evaluation of the Schrödinger equation is very important [6]. For calculate of the WFO, we used a global potential [5] in the following form, as: α V ( r) = k( r + ) + c r k = 7585, α =.46, c = 8.875

2 6 Fragmentation of eavy Quarks Using Wave Function at the Origin The constants can be determined when we fit method with the eperimental data. The wave function at the origin with values calculated using the eact solution of the Schrödinger equation for the potential of the ground state is listed in Table. Table. The numerical values of the radial wave function at the origin, to the global potential in reference [5]. R ) for bb R () = Ψ() 4π, for S, S, state of b b cc, bc s (GeV ) s (GeV ) p (GeV ) ( R for cc R ) ( for bc. Fragmentation function In the arton model each particle carry out, X i = X i, system with respect fraction of the momentum-energy of meson with the constraint of i.in the infinite four- momentum frame, the output four-momentum of meson is which produces a jet. Then the four- momentum particles have these momentums as: = X, = X µ µ. () Therefore we can write for each particle four- momentums in the following form: = X, = X µ µ µ µ = ( ) µ µ Where we used the fragmentation parameter, z, as given by, adron = = Quark With the choice of the wave function at origin according to a global potential that is obtained from [4], the fragmentation function for massive quark has the following general form, as: D( z, µ ) = dn( z, µ ) on n (4) where dn ( z, µ ) gives the probability for the arton i to form a jet that includes a cc pair in the state labeled by n, and on is proportional to the probability for a point like cc pair in the n- state. According to the current- current interaction and hard amplitude scattering [9], the probability o = d d d T WFO n ( ) o n given by the following form: δ ( p + p + p p) ( p + p + p p ) f s T = TQ = W FO p p p p n f s C g mm o = C g m m W FO d d d i δ ( p + p + p p) ( + + ) p p p p p p p p () () (5)

3 Universal Journal of hysics and Application (): 59-65, 6 T ere is the hard scattering amplitude. Also the unpolarized fragmentation that average over initial quark spin and sum over final state particles has the following form for invariant amplitude: 4 = q L ( γ )( γ ) µν µ µ L = / U ( p ) U ( p ) U ( p ) U ( p ) Q µ µ ( γ )( γ ) Q L = / U ( p ) V ( p ) U ( p ) V ( p ) µν spin spin L µν Q Q µν 8 = + 4 q [( p. p )( p. p ) ( p. p )( p. p ) ] (6) Where, L µν Q is invariant under proper Lorentz transformations and parity operation.therefore the invariant amplitude in the infinite momentum limit can be written: The dot products in Eq.6 are given by: X = X= X. 8 = + 4 q - X (. ) = m + m + ( p. p )( p. p ) ( p. p )( p. p ) T X - (. ) = (m )+X m + T X X X (. ) = (m )+ (m ) X X (. ) = -( m + ) + ( m + )- - T T T Where Therefore, the fragmentation function for heavy meson with WFO of global potential [5] is obtained from equations (5, 6), as we have: D ( z, µ ) = ( + ) C g m m ( W FO ) qq qg f s d d d δ ( p + p + p p) ( + + ) p p p p p p p p Where qq, qg are the functions obtained from []. In order to solving integrals in phase space, we can integrate into two integrals elements. Thus we obtain the solution of phase space integrals according to the following equations, as: ( p p p) p = ( + ) ( + ) δ + d p p p p p p 4 = f(, z) d= f(, z) d d T T L T p f z T (, ) (7) (8) d = d d (9) ()

4 6 Fragmentation of eavy Quarks Using Wave Function at the Origin In the infinite momentum limit, the integration from longitudinal momentum convert to energy integration and also transverse integration convert to average transverse momentum, respectively. This means simply replace T with its average value. Finally, we have ( qq + ) qg D ( ) = ( ( π )( C g ( W FO ) X ) (( ) X ) ) ( ) ξ Λ = q = ( X ) ( m ) + X m + ( T ) 4 f s X X X Λ = X ( m + T) + m ( X ) X X X X ξ X This equation defines the fragmentation function for J ψ meson. () 4. Result and Discussion =m The eplicit form of Eq.() in our calculations, for J ψ meson at two case +m and by the following form, respectively, as: and 4( + ( z ) ) 4( + z ) K D( ) =.96( + ) z ( z ) F K = z + z z z + z z + z 4 (.7968( ) ( ) 4.9( ).54 ) F = ( z(.9485( z ) ( z ) z +.77 z ) (.9485( z ) +.77 z ) ( z ) ) z z ( ) 4( + ( z ) ) 4( + z ) K D( ) =.597( + ) z ( z ) F = ep., is given K = z + z z z + z z + z 4 (.7968( ) ( ) 4.68( ).57 ) F = ( z(.9485( z ) ( z ) z z ) ( z + z z.86(.9485( z ).8895 z ) + Firstly we eamined the behavior of Eq. with respect to t scales. Secondly, in figure, the result appears in various values ranging from - GeV for heavy quark fragmentation. Then we compared our fragmentation functions with the eperimental predications []. Therefore our comparison shows that the fragmentation functions in high t scale have same procedure with the eperimental predicate []. Therefore in Fig. we compared our fragmentation functions with BFGW predication []. )) () ()

5 Universal Journal of hysics and Application (): 59-65, 6 Figure a. Our Fragmentation function according to z for µ =.5 GeV scale compared with BFGW predication model (dash line) Figure b. The same as Fig.-a for µ =.969 GeV scale

6 64 Fragmentation of eavy Quarks Using Wave Function at the Origin z, our work in pt 5 z, our work in pt z, our work in pt z, our work in pt z, our work in pt z, our work in pt Figure. Fragmentation functions at different pt(gev) scales 5. Conclusion In the heavy quark limit we have obtained the eact analytical fragmentation functions for S-wave two heavy mesons at leading order perturbative QCD. The wave function at the origin (WFO) for bound state wave function of mesons has been used. In this model, the C quark mass is considered to be.959 Gev. Fragmentation functions have been calculated for LE energy. In this determination we used the verte invariant under parity, Laurence transfer, and hard scattering amplitude for heavy mesons. REFERENCES [] B. Andersson et. al., hys. Rept. 97 (98) B, R. Webber Nucl. hys. B 8 (984)49 [] C. eterson et. al., hys. Rev. D 7(98) 5 [] E.L.Berger and D.Jones, hys.rev.d(98) 5 S.D.Ellis,.Einhorn and C.Quigg, hys.rev.lett.6 (976) 6.Cacciari and.greco, hys.rev.lett.(994)586; K.Sridhar, A.D.artin and W.J.Stirling, phys.lett.b48(998); A.etrell, Nucl.phys.roc.Supp.86()5. FredCooper, et.al., arxiv:hep-ph/49v(4) [4] C.Albajar, et.al.,phys.lett.b56(99); F.Abe, et.al.,hys.rev.lett.69(99)74 F.Abe, et.al.,hys.rev.lett.7(99)57; RaphaelGranierde, Cassagnac, ENIXCollaboration,J.hys.G: Nucl.and art.hys.(4)s4; [5] G. R. Boroun and. Abdolmalki, hys. Scr. 8 (9) 65.

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