Characteristics of Charged Pions in the Quark Model with Potential Which Is

Kuchin and Maksimenko Journal of Theoretical and Applied Physics 2013, 7:47 http://www.jtaphys.com/content/7/1/47

RESEARCH Open Access Characteristics of charged pions in the quark model with potential which is the sum of the Coulomb and oscillator potential Sergey M Kuchin1* and Nikolay V Maksimenko2

Abstract

In this paper, we calculate the polarizability of the charged pions in the nonrelativistic quark potential model. Keywords: Schrödinger equation; Characteristics of charged pions; Polarizability

Introduction were also computed in the context of the nonrelativistic The theoretical description of the present experimental quark model [4-12] and other potential models [13-17]. data on the interaction of photons with hadrons at high In this article, we look at the mass spectrum, mean energies and large momentum transfer is carried out, square radius, leptonic decay constant, and electric po- mainly in the framework of perturbation theory of larizability of charged pions in the quark model with po- quantum chromodynamics. However, the structural de- tential, which is the sum of the Coulomb and oscillator grees of freedom, which appear in low energy, cannot be potential. This potential is used, for example, in [18,19] reduced to simple ideas about the interaction of electro- to describe the mass spectrum of quarkonia. In addition, magnetic fields with hadrons. The response of the quark in [20-27], they conducted research into this building, degrees of freedom for the action of the electromagnetic which contains the linear part. The good agreement field can be determined phenomenologically on the basis between the results of this work with the experimental data of self-consistent description of the polarizabilities, root is of interest for the use of this potential for the description mean square radius of the charge, and other electromag- of the component systems, consisting of not only heavy netic properties of hadrons. The unique sensitivity of but also of light quarks. these values to theoretical models puts them among the In this paper, we calculate the characteristics of the most important characteristics, with which you can set pions in the quark model with the potential, which is the features of the structure of hadrons, exhibited at low the sum of the Coulomb and oscillator potential. The energies. Therefore, the study of the electromagnetic results are in good agreement with the experimental characteristics of related systems in the framework of data. potential models is becoming one of the most effective methods for studying the characteristics of the quark- Solution of the Schrodinger equation quark interaction. To find the wave function of the relative motion of At present, there are quite a number of theoretical stud- aquarkandanantiquark,wesolvetheSchrodinger ies providing the electric polarizabilities of charged had- equation: rons, including mesons. Among them, we can mention calculations that make use of effective Lagrangians [1,2] ΔΨ þ 2μ½Ε−UðÞr Ψ ¼ 0; ð1Þ and current algebra [3]. Nucleon and meson polarizabilities where μ is the reduced mass, U(r)isthequark- antiquark interaction potential, and r is the relative * Correspondence: [email protected] 1 coordinate. I.G. Petrovsky Bryansk State University - Novozybkov Branch, Novozybkov U r 243020, Russia As the potential ( ) is spherically symmetric, the var- Full list of author information is available at the end of the article iables in the Schrodinger Equation 1 are separable [28],

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and the equation can be reduced to the equation for ra- that includes deriving the lower and upper boundaries for dial wave functions Rnl(r): this quantity. Consider the equation d dR llðÞþ 1 2 nlðÞ r 1 r þ 2μðÞE−UðÞr − RnlðÞ¼r 0: r2 dr dr r2 ^ H jjΦ〉 ¼ E Φ〉 ð8Þ ð2Þ with a Hamiltonian consisting of a sum of two operators: ‘ ’ χ r rR Introducing reduced radial wave functions nl( )= nl ^ ^ ^ r ∫∞ χ ðÞr j2dr ¼ ; H ¼ H þΔ H; ð9Þ ( ) normalized by the condition, 0 nl 1 we re- 0 write Equation 2 as ^ where H0 is a Hamiltonian of an ‘unperturbed’ system ″ llðÞþ 1 ^ χ ðÞþr 2μðÞE−UðÞr − χ ðÞ¼r 0; ð3Þ while Δ H is a small additional term (a perturbation nl r2 nl operator). We also assume that Equation (8) has the form χ″ ðÞ¼r d2 ¼ d2 χ ðÞr : where nl dr2 dr2 nl ^ H jjψ 〉 ¼ εn ψ 〉; n ¼ 0; 1; 2; … ð10Þ The interaction potential between a quark and anti- 0 n n quark can be written as provided there are no perturbations. b According to the stationary perturbation theory, the UðÞ¼r ar2− þ c; ð4Þ r energy value additional to that of the ground state ε0 will be looked for in the form of a series: where a and b are nonnegative constants and r is the ðÞ1 ðÞ2 interquark distance. This potential has two parts: the first E ¼ ε0 þ Δε þ Δε þ ⋯ ð11Þ is ar2 accounts for quark confinement at large distances, b Respectively, the wave function will also be repre- while the second part − r0 which corresponds to the poten- sented as a series in a small parameter, which is part of tial induced by one gluon exchange between the quark and ^ antiquark that dominated at short distances. Δ H: Substituting the quark-antiquark interaction potential jΦ〉 ¼ jΨ〉 þ jΔΨ〉 þ ⋯ ð Þ U(r) in Equation 3, we obtain 12 b llðÞþ ε ≤ ε … ≤ ε ″ 2 1 When 0 1 n, we find that the value of add- χnlðÞþr 2μ E−ar þ −c − χnlðÞ¼r 0: (2) r 2μr2 itional energy Δε belongs to the interval [14]: ð Þ ÀÁ 5 F G2−F 2 ðÞ2 ≤Δε ≤ 0 ð13Þ The approximate solution of this equation was obtained ε0−ε1 Fε0−D in [29] using the method Nikiforov-Uvarov [30]. Here are the expressions obtained in [29] for the eigenvalues and where the following notation is introduced: D ¼ Ψ 0 eigenfunctions: ^ ^ ^ ^ 2 Δ HH0Δ HjΨ 0; F ¼ 〈Ψ 0 ΔH jΨ 0〉; μ b þ 8a 6a 2 δ3 Enl ¼ − hiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ c: ^ δ2 2 G ¼ Ψ jΔ H jΨ : ð14Þ ðÞÆnþ þ llðÞþþ 24μa 0 0 2 1 4 1 δ4 ð6Þ Therefore, in order to find the interval boundaries (13), it is necessary to determine the wave function of Ψ the ground state 0 as well as the energies of the ground pffiffiffiffiffiffi npffiffiffiffiffiffi B d B −pffiffiffi−1 2Ar 2 −2nþp2ffiffiffi −2 2Ar and first radially excited states. Unlike the case when it RnlðÞ¼r N nlr 2A e −r r 2Ae ; dr is needed to find an exact value of Δε(2), it is not re- ð7Þ quired to completely solve an unperturbed problem in the case at hand. A ¼ μ −E þ c þ 6a ; B ¼ μ b þ 8a ; δ≡ 1 ; r − Correction Δε(2) to the ground-state energy of a bound where δ2 δ3 and r 0 0 system, when the role of perturbation is played by the the characteristic radius of the meson, n =0,1,2,… external stationary field with strength E, is related to the electric polarizability of system a as follows [4]: Method of estimating electric polarizabilities 0 a In this section, we shall give a general method of estimat- ðÞ 0 Δε 2 ¼ − E2: ð15Þ ing the static electric polarizability of a bound system [14] 2 Kuchin and Maksimenko Journal of Theoretical and Applied Physics 2013, 7:47 Page 3 of 5 http://www.jtaphys.com/content/7/1/47

〈Ψ Note that, when the ground state | 0 is spherically π− 1 Δε(1) ψ ðÞ¼ξ pffiffiffi ½jju↑d↓〉− u↓d↑ ; ð22Þ symmetric, the value of is zero, i.e., 2 ΔεðÞ1 ¼ G ¼ : ð Þ 0 16 where u and d are antiquarks. Using Equations 13 and 16, we find that the value of The relation static electric polarizability a lies within the interval 0 e2 Æ 2 Æ π ðÞe −e π ¼ ð23Þ F2=E2 F=E2 1 2 9 2 ≤a ≤ 2 : ð Þ D−Fε 0 ε −ε 17 0 1 0 was also used in the calculations. Pion static polarizabilities π In order to fix the model parameters of interquark po- Compton electric polarizability of meson tential and quark masses, we shall use experimental data As was, for example, shown in [4], generalized electric a on lepton decay constants, masses, and electromagnetic polarizability can be represented as a sum of two parts: radius of charged pions. For experimental values [31,32], a ¼ a þ Δa: ð24Þ we have 0

S The quantity a is called static polarizability and is re- M1 ¼ 139:57018 Æ 0:00035 MeV; 0 exp lated with the induced electric dipole moment in the ap- S M2 ¼ 1300 Æ 100 MeV; proximation of its pointness; i.e., a deformed composite exp system is described as a pointlike dipole. Δa f πÆ ¼ 130:70 Æ 0:10 Æ 0:36 MeV; The term takes into account the structure of a composite system and is expressed in the leading ap- 2 2 rπÆexp ¼ ðÞ0:420 Æ 0:014 fm : proximation through the rms radius of that same system. The quantity Δa is relativistic in nature and can be These data lead us to the following parameter values: explained as describing a transition from Thomson scat- 3 mu = md = 0.176 GeV; a = 0.00766 GeV ; b = 0.3; δ = tering on pointlike particles to that on composite ones 0.142 GeV; c = −0.982 GeV. with an electromagnetic radius [33]. For a spinless sys- The masses of mesons, lepton decay constant, and mean tem, this term is written as follows: square radius are determined by the following formulas: arhi2 M ¼ m þ E ; ð Þ Δa ¼ ; ð25Þ nl 2 nl 18 3M jRðÞj2 b where r is a pion electromagnetic radius, a is a fine 2 3 0 3 f p ¼ 1− ; ð19Þ structure constant, and M for the nonrelativistic case is πMp 2π the sum of the masses of the constituent particles. DEX Calculating Δa within the framework of the given 2 2 r ¼ eiðÞri−Rc:m : ð20Þ model with pointlike quarks, we find that, in the ap- proach proposed, the term related with a pion electro- Calculations by formula (17)-obtained values of parame- magnetic radius has the following value: ters give the following interval for the static polarizability: πÆ − Δa ¼ 5:81 Â 10 4 fm3: aπÆ ¼ ðÞÂ: Æ : −4fm3: 0 0 21 0 18 10 Thus, we obtain the next value Compton polarizabil- To estimate polarizability, a nonrelativistic operator of ities of charged pions in this model: the electric dipole interaction was used: −4 3 aπÆ ¼ ðÞÂ6:02 Æ 0:18 10 fm : 1 DE ¼ ðÞe1−e2 ðÞrE ; 2 We compare the results of our calculations with where ei are quark charge operators acting upon that the experimental values obtained in [34,35]. In [34], part of the wave function which depends on the unitary the following value of Compton polarizability is −4 3 spin and, for π± − mesons, have the following form [4]: aπÆ ¼ ðÞÂ6:8 Æ 1:4 Æ 1:2 10 fm ; and in Equation 3 a Æ ¼ ðÞÂ: Æ : : ÂÃ 35, it is π 6 5 1 1 10 fm Thus, the results πþ 1 ψ ðÞ¼ξ pffiffiffi d↑u↓〉− d↓u↑ ; ð21Þ of our calculations are in good agreement with the 2 experimental values. Kuchin and Maksimenko Journal of Theoretical and Applied Physics 2013, 7:47 Page 4 of 5 http://www.jtaphys.com/content/7/1/47

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doi:10.1186/2251-7235-7-47 Cite this article as: Kuchin and Maksimenko: Characteristics of charged pions in the quark model with potential which is the sum of the Coulomb and oscillator potential. Journal of Theoretical and Applied Physics 2013 7:47.

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