Calculation of Quark-antiquark Potential Coefficient and Charge Radius of Light Mesons

Transcription

1 Applied Physics Reseach ISSN: Vol., No., May E-ISSN: Calculation of Quak-antiquak Potential Coefficient and Chage Radius of Light Mesons M.R. Shojaei (Coesponding autho ) Depatment of Physics Shahood univesity of Technology Shahood, , Islamic Republic of Ian Tel: M.R.Shojaei@shahoodut ac.i H.Tavakoli-Anbaan Depatment of Physics Shahood univesity of Technology Shahood, , Islamic Republic of Ian Tel: tavakoli.anbaan@gmail.com Abstact Mesons ae composed of one quak and one anti quak which ae consideed to be govened by quantum chomo dynamic. In the pesent pape, it is consideed that mesons ae made up entiely, at least in pinciple, of light quaks (u, d, s). Since mesons consisting of light quaks ae elativistic systems, we cannot use the Schödinge equation, and the theoy is athe limited. The light quak (u, d, s) mesons ae intinsically elativistic; because binding enegies (typically a few hunded MeV) ae not small compaed to the constituent masses. Theefoe they ae based on nonelativistic quantum mechanics. Fo the heavy quaks (c, b, t) though, the nonelativistic theoy should wok easonably well. Fo most pat of this pape we esticted ou attention to elativistic bound state of two paticles and consideed inteaction between quak and antiquak, the essential point is that at shot ange we expect a columbic potential and at lage distance we have to account fo quak confinement, the simplest is a linea potential and othe inteaction oscillato potential. Then Diac equation was solved with the cental potential analytically and the wave function of system was detemined. Finally, by using wavefunction we detemined the chage adius of p meson. Keywods: Quak model, Cental potential, Light meson, Diac equation. Intoduction The Constituent Quak Model (CQM) has been extensively applied to the desciption of meson popeties.in 966, Yoichio Nambu fist poposed the SU (3) gauge theoy, and quantum chomo dynamics (QCD), as a candidate fo the fundamental theoy of the stong inteaction, just afte the intoduction of the new quantum numbe colo. QCD has been established as the fundamental theoy of stong inteaction (Giannini, et al., 3) Howeve, in spite of its simple fom, QCD ceates thousands of hadons and leads to vaious inteesting phenomena such as colo confinement and oscillato potential. Mesons with heavy quak have been investigation by Schödinge equation seveal times, but mesons consisting of light quak ae elativistic systems (Giannini, et al., ). Since the light mesons ae elativistic, we should utilize Diac equation in ode to investigation mesons with light quak. Then we also used cental potentials (i.e. columbic, confiment and oscillato) to detemine static popeties of light mesons.( Bhadui. et al., 98) The pupose of the pesent pape is to povide a mechanism fo investigation of mesons and inteaction between quak and anti quak.( Tegen, et, al 98) Since the meson consists of one quak and one antiquak, the wavefunction ae dependent on two coodinates 48

2 ISSN: Applied Physics Reseach E-ISSN: Vol., No., May and, whee, ae the quak and antiquak position vecto (Lin, 98).Then the intenal quak and anti quak motion ae descibed by the elative coodinates x which is defined as follow: ( Shojaei, et al., 8) = In this model we conside the cental potentials between quak and antiq (Giannini, et al., 999) uak. Fist, fo small sepaations, the potential which has an attaction is columbic potential oiginating fom the colo chage (Rajabi, 5). k αs c V ( ) = =. () While at lage sepaations a linea tem gives ise to quak confinement (Rajabi, 6). V ( ) In this aticle we have added the oscillato potential as follow ( Shojaei, et al 9) V () = b. (3) = a (4) Hee the total potential is assumed as below: c v( ) = a + b. (5) Whee abcae,, potential coefficient.in section () we have calculated the elativistic wave function fo quaks and anti quak.the esults indicate that this potential is useful fo quaks having masses in the ange used in the phenomenological analysis of quak model and detemined chage adius of p meson.. Solution of Diac equation fo the meson The Diac equation may tansfom in vaious ways unde a Loentz tansfomation.the fom in common use fo scala cental potential ( u ) and vecto cental ( v ) is taken as (Shojaei,et al 7) ( σ. P) χ + ( m + u ( ) + v ( )) ϕ = εϕ ( σ. P) ϕ ( m + u ( ) v ( )) χ = εχ Since the meson consists of one quak and one antiquak then we can use the Diac equation fo the quak as follows: ( σ. p ) χ + ( m + u + v ) ϕ = εϕ (7-a) ( σ. p ) ϕ ( m + u v ) χ = εχ (7-b) We suppose the u( ) = v ( ).Now by combining two equations (7-a) and (7-b), we obtain (6) p ϕ ( ) + ( m ε ) ϕ ( ) + v ( m + ε) ϕ ( ) = (8) Fo the antiquak as befoe it would by p ϕ ( ) + ( m ε ) ϕ ( ) + v ( m + ε) ϕ ( ) = (9) By combination equation (8) and (9) we have: ( P + P ) ϕ( ) + ( m ε ) + ( m + ε) v ( ) ϕ( ) = () 49

3 Applied Physics Reseach ISSN: Vol., No., May E-ISSN: Whee j () is the uppe component wavefunction of light meson. Since we suppose c v ( ) = u ( ) = ( a + b ) depends only on the elative distance between quak and antiquak, it is moe appopiate to use the cente mass and elative coodinates R and instead of the coodinate and.the tansfomation fom, to, R is given by (Lin, 983). = m+ m R = m + m We can substitute the Laplacians based on the function of and with, R Eq. () in tems of elative coodinate becomes as follows. () which ae elated. Theefoe d d L ( Ω) = d d ( ) ϕ( ) ( m ε ) 4( ε m) v ( ) ϕ( ) () With v ( ) given by (5), and L ( Ω ) = l( l + ) is the gand obital opeato and l is the gand angula quantum numbe. Now fo the eigenfunction ϕ ( ), we make an ansatz, ( Shojaei, et al 8) With f ( ) and h ( ) as: ϕ ( ) = f ( )exp( h( )) (3) This implies f ( ) = h( ) = α + β + δ ln (4) h g + g = h h g x + + ( ) ( ) ( ) ( ) ( ) γ γ γ (5) Equations () and (5) yield α, β, d and the constaints between the potential paametes a, b and c. We have α = 4a ε + m (6-a) ε + mb (6-b) β = a βδ ( + ) (6-c) c = ( ε + m ) δ = l Taking δ = l esults a wavefunction which is well behaved at the oigin.theefoe the uppe component of Diac spino of the meson is as below: ε + l ( m) b ϕ( ) = exp( (4 a( ε + m)) ) (7) a 5

4 ISSN: Applied Physics Reseach E-ISSN: Vol., No., May The lowe component χ( ) of the Diac hype-cental spino can be found fom (7-a), (7-b), fo quak and antiquak we find χ ( σ.( p + p )) = ϕ ( m + ε) () () (8) We suppose p = p + p. We can (. ) ( ) σ p ϕ be witten as: σ. A σ. B = A. B + iσ.( A B) ( ) ( ) d Whee P = L andp = i d σ. ( σ. P ) ϕ ( ) = ( σ. )( σ. P) ϕ( ) Using the equationand substituting it in equation(8) σ. P ϕ( ) =. P + iσ.( P) ϕ( ) σ. ( ) d σ. P ϕ( ) i = + σ. L ϕ( ) d ( ) By eplace equation () in equation (8) and = ˆ the lowe component obtained as so (9) () () iσ. ˆ d σ. L χ( ) = ϕ( ) ( m + ε ) d () Theefoe the wave functions fo l = is as the following fom: ( ε + m) b exp( (4 a( ε + m)) ) a ψ ( ) = (3) ˆ iσ. d ( ε + m) b (exp( (4 a( ε + m)) )) ( m + ε ) d a By using this wavefunction we can detemine the static popeties of mesons. (Mass of meson, chage adius, magnetic moment ). In this pesent pape we calculated the chage adius of meson. 3. Chage Radius of p meson by Using Quak Model Let s takep meson chage-adius. The chage-adius < > is defined as. em p 3 q ψ ( ) ( ) γ ψ γ = d (4) Hee y() is the quak wave function given by (3).Using fom equation (6, ) we can detemined the potential coefficient and wavefunction. Then we calculated the chage adius of p meson in Table. Which these esults fom the quak- anti quak substuctue of the p meson. Although the shot lifetime of 5

5 Applied Physics Reseach ISSN: Vol., No., May E-ISSN: the p meson excludes the possibility of diectly measuing the chage adius of the p meson at pesent, one can define a diffactive adius fom the t dependence of diffactive p meson electo poduction coss section. 4. Conclusion An exact analytical solution fo potential in the fom of the confinent is pesented. In this aticle we have shown the complete inteaction including confinement, oscillato and coulomb in tems epoduces the position of the quak and antiquak. The cental potential is a good stating point fo constuction of an unpetubed states and leads to ealistic quak states, which shows the static popeties of meson which ae sensitive to the coected wave functions. The cental potential is a good stating point fo investigation of meson stuctue A consideable impovement in the desciption of the static popeties of nucleon is obtained with an isospin-dependent potential. By use this model we can investigation the othe mesons. As we suggested the method fo solving the wave function, essentially based on an ansatz fo the eigenfunctions, woks successfully fo a lage class of potentials fo many paticles inteacting in D- dimension and at least fo the study of the gound state of the system. Busied of these potentials we can conside the hypefine potentials and calculate the shift enegy. Finally one can use this model and detemine the static popeties of hadons. Acknowledgement This eseach was suppoted by the eseach gant of Shahood Univesity of technology Refeence Bhadui R K, Cohle L E and Nogami Y. (98). Quak Quak inteaction and the nonelativistic quak model. Phys. Rev Lett A, 44, Giannini M M, Feis M, Pizoo M and Santopinto E. (3). An oveview of the hypecental constituent quak model. Pog.Pat.Nucl.Phys, Giannini M M, Santopinto E and Vassallo A. (). The hypecental constituent quak model. Nucl. Phys.A., Giannini M M, Santopinto E. (999). The hypecental constituent quak model. Few Body Syst. Suppl., Lin C D. (98). Radial and angula coelations of doubly excited electons. Physical Review A, 5, Lin C D. (983). Radial and angula coelations of two excited electons. Physical Review A, Rajabi A A. (5). Exact analytical solution of the Schödinge equation fo an n-identical body foce system. Few-Body systems, Rajabi A A. (6). Bound state fo hypecental singula and exponential potentials. Commun. Theo. Phys, Shojaei M R, Rajabi A A and Hassanabdi H. (8). Hype-spheical hamonics and anhamonics in m-dimensional space. Int. J. Mod.Phys. E., Shojaei M R, Rajabi A A. (7). Hypecental constituent quak model and hypefine dependence potential. Ianian Jou of phys Reseach, Shojaei M R, Rajabi A A. (9). Thee body foce model of nonhypecental hamonics ans anhamonics potential in thee dimensional potentials. Mod. Phys. Lett., A Tegen R, Bockmann R. (98). Electomagnetic popeties of elativistic quaks in confining potentials. Z. physa, 37,

6 ISSN: Applied Physics Reseach E-ISSN: Vol., No., May Table. The chage adius of p meson fo diffeent quak mass ( # m 3( MeV ), b,c ae the of potential coefficient (a=) q m ( ) q fm - ( fm ) b( fm - ) c