Form factors and charge radii in a quantum chromodynamics-inspired potential model using variationally improved perturbation theory

Transcription

1 PRAMANA c Indian Academy of Sciences Vol. 8, No. journal of January 205 physics pp Form factors and charge radii in a quantum chromodynamics-inspired potential model using variationally improved perturbation theory BHASKAR JYOTI HAZARIKA, and D K CHOUDHURY,2 Centre for Theoretical Studies, Pandu College, Guwahati 78 02, India 2 Department of Physics, Gauhati University, Guwahati 78 0, India Corresponding author. bh53033@gmail.com MS received 2 August 203; revised 8 April 20; accepted 22 April 20 DOI: 0.007/s x; epublication: 2 November 20 Abstract. We use variationally improved perturbation theory VIPT) for calculating the elastic form factors and charge radii of D,D s,b,b s and B c mesons in a quantum chromodynamics QCD)-inspired potential model. For that, we use linear-cum-coulombic potential and opt the Coulombic part first as parent and then the linear part as parent. The results show that charge radii and form factors are quite small for the Coulombic parent compared to the linear parent. Also, the analysis leads to a lower as well as upper bounds on the four-momentum transfer Q 2, hinting at a workable range of Q 2 within this approach, which may be useful in future experimental analyses. Comparison of both the options shows that the linear parent is the better option. Keywords. Variationally improved perturbation theory; form factor; charge radii. PACS Nos 2.3. x; 2.3.Jh; 2.3.Pn. Introduction The potential model description in the nonperturbative regime of QCD is tremendously successful in providing both qualitative and quantitative descriptions of the hadron spectrum and the deacy modes. On this basis, we have pursued a nonrelativisic constituent quark model NRQM) [ ] for mesons containing a heavy light) quark and a light heavy) antiquark. Although nonrelativistic in nature, relativistic effect is to be introduced from outside [5,6] due to the light quarks involved. Basically, the model relies on the work of Rujula et al [7] who used a nonrelativistic treatment of potential model which was quite successful in describing different hadronic properties. We have solved the Schrödinger equation for the spin-independent Fermi Breit Hamiltonian used in Rujula et al [7]) consisting of linear-cum-coulombic potential for the ground state [2]. The solution, i.e. wavefunction has been obtained using different approximation methods like Dalgarno method [8] and variationally improved perturbation theory VIPT) [ ] Pramana J. Phys., Vol. 8, No., January 205 6

2 Bhaskar Jyoti Hazarika and D K Choudhury which is then used in predicting the Isgur Wise I W) function [2,2,3], mass, decay constant, charge radii [] etc. We note that with the linear-cum-coulomb potential of QCD, we have two options in choosing the parent or the child i.e. perturbation): i) we can consider the linear one as the perturbation i.e. the Coulombic one as the parent) and then ii) linear one as the parent i.e. Coulombic one as the perturbation). As we have already successfully used VIPT for both the options in the calculation of slope and curvature of I W function for D, D s,b,b s and B c mesons [2,3], extending it for predicting elastic form factors and charge radii which provide important insight on the distribution of different charge constituents of a hadron) definitely makes sense. It is well known that the form factor and charge radius are dependent on the momentum transform of the wavefunction. So, getting an appropriate wavefunction is very essential for a fruitful analysis. With the success of VIPT in the calculation of Isgur Wise function as pointed out in [2,3], one can expect a similar success here too. It is worthwhile to note that while investigating the form factor, one must take into account the proper range of four-momentum transfer Q 2. The Q 2 range usually determines the applicability of perturbative QCD pqcd) or nonperturbative QCD npqcd). So, an accurate selection of Q 2 range within the nonperturbative approach, which will also fall within the experimental regime, is necessary. This fecilitates a direct comparison between theory and experiment. This has been done both theoretically and experimentally since long [ ] for the light π,k etc. mesons. However, for the mesons which contain at least one heavy quark, very little has been investigated theoretically [20 22]. In the absence of any experimentaldata for them, our results may be helpful in future in the experimental set-up regarding the Q 2 range. As far as our model is concerned, the perturbative or nonperturbative regime of QCD can be interpreted through the relativistic factor ɛ = α s /3) []. The reality constraint on the form factor FQ 2 ) leads to the condition 0 <ɛ<, where ɛ 0 ɛ ) corresponds to the perturbative nonperturbative) limit of QCD. The ɛ limit demands large α s or low Q 2. So, discussing the nonperturbative effects of QCD with large confinement parameter b = 0.83 GeV 2 ), we must consider the low Q 2 limit of α s in this model. However, we have observed in ref. [] that large value of b prohibits the use of low Q 2 compelling one to involve with small α s which corresponds to the perturbative regime and thus cannot be accounted for in this nonperturbative approach. We reanalyse all these observations in this approach of VIPT for both the cases linear or confinement part as perturbation and Coulombic part as perturbation. We shall explore the possibility of incorporating significant value of α s even with large confinement. This work will also check the status of both confinement and Coulombic parts as perurbation and observe the consequences regarding the usable range of Q 2 to work in the absence of experimental data for the said mesons. The calculations are done with a fixed value of α s from V-scheme [23 25] with large confinement effect instead of variation in both. Even with this single value of α s and b, one can draw similar conclusion regarding the effective range of Q 2. The calculated form factors are plotted graphically to show their variation with Q 2 for both the cases. Basically, this work explores the possibility of improving the results for form factors and charge radii over those of ref. [,22] with the help of VIPT. In that sense, this work can be thought of as the improved version of that in ref. [], as it uses the reasonably new technique, VIPT, which has certain advantages [2,3] over the Dalgarno method used in 70 Pramana J. Phys., Vol. 8, No., January 205

3 Form factors and charge radii in a QCD-inspired potential model ref. []. Besides, the present work deals with the linear potential, both as perturbation and parent, unlike in ref. [] where only perturbation option was considered. As a result, the present work explores the effectiveness of VIPT in a more elaborate way for both the options at the same time. In the process, we shall try to find the possible lower and/or upper limits of Q 2 where the linear part as perturbation or parent is applicable. The rest of the paper is organized as follows: 2 contains the formalism, 3 the result and calculation, while includes the discussion and conclusion. 2. Formalism 2. VIPT with Coulombic potential as parent 2.. Wavefunction. We breifly reformulate the VIPT with the expression for the wavefunction corrected upto the first order of jth state given by [0,2] ψ 0) k H P j ψ0) j dv ψ j = ψ 0) j + k =j E 0) j E 0) k, ) where P is the variational parameter which is later optimized with respect to energy) considered instead of physical parameter P. For the Coulombic part as parent, the physical parameter is α = α s /3 and the optimized variational parameter is ᾱ [2]. As done in ref. [2], we consider the wavefunction for triple term consideration of the summation given by eq. ) above and rewrite the equation viz. eq. 5) of ref. [2]) with the subscript 0 n =,l = 0) being replaced by T : ψ T = ψ 0) 0 A +B μᾱ 0 r 2 2μᾱ 0 r 3 +D 3μᾱ 0 r where the different parameters are given by ) e μᾱ 0 r/2) + 2μ2 ᾱ 0 r μ2 ᾱ 0 r2 32 ) e μᾱ 0 r/3) ) μ3 ᾱ 0 3 r3 e μᾱ 0 r/), 2) 8 6 and c = μᾱ 0 3) π /3 [ μ μᾱ A = 3 0 α ᾱ 0 ) πᾱ 0 32b ] )/2 27 8μᾱ 0 ) [ μ 3μᾱ B = 0 α ᾱ 0 ) πᾱ 27b ] 0 ) / μᾱ 0 5) [ D = μᾱ 0 )3/2 36α ᾱ 0 ) π 5625ᾱ 0 38b 7825μ 2 ᾱ 3 0 ]. 6) Pramana J. Phys., Vol. 8, No., January 205 7

4 Bhaskar Jyoti Hazarika and D K Choudhury The wavefunctions for single and double term consideration can be obtained by putting B = D = 0andD = 0 respectively in the same eq. 2). However, we shall consider the relativistic version ɛ = 0) of the above wavefunction, viz. [2] ψ T,Rel r) = ψ T rμᾱ 0 ) ɛ. 7) The relativistic factor ɛ is given by [] ɛ = α s 3. 8) 2..2 The elastic charge form factor and charge radii. The form factor can be expressed as [26] where efq 2 ) = e i Q i Q i = + 0 r ψt,rel r) 2 sin Q i rdr, ) j =i Q mi. 0) Putting 2)and7)in) we get the formfactor as efq 2 ) = e i N 3 Ɣ3 2ɛ) [ q +q 2 +q 3 +q +q 5 +q 6 +q 7 +q 8 +q +q 0 ], ) where N 3 is the same normalization constant as appeared in eq. 5) of ref. [2] andthe different q i Q i ), si =, 2,...,0) are defined in Appendix. The charge radius is derived as [] r 2 = def Q2 )) dq 2 2) which in the present model is Q 2 = 0 r 2 =N 3 Ɣ3 2ɛ)[r + r 2 + r 3 + r + r 5 + r 6 + r 7 + r 8 +r + r 0 ], 3) the different ri 3 s i =, 2,..., 0) are defined in Appendix Status of linear potential as perturbation. The momentum transform of eq. 7)is [27,28] ψ T,Rel Q 2 ) = e i 2 + rψ T,Rel r) sin Q i r dr ) Q i π 0 = 2 e i N 3 π Ɣ3 2ɛ)[ ] p p 2 + p 3 + p. 5) The p i s which depend on Q 2 i,ɛ etc. are given in Appendix. 72 Pramana J. Phys., Vol. 8, No., January 205

5 Form factors and charge radii in a QCD-inspired potential model Table. Values of lower limit of four-momentum transfer Q 2 0 with Coulombic parent taking single, double and triple terms in eq. ). We have to use Q 2 values above these. Meson D + D D + s B + B 0 B 0 s B + c Q 2 0,S Q 2 0,D Q 2 0,T Table 2. Values of charge radii fermi) for different mesons with Coulombic parent for single, double and triple terms in eq. ). The subscripts S, D, T correspond to single, double and triple terms respectively whereas F means finite mass consideration. The infinite mass limit subscript is used) is shown for the triple term alone. Meson D 0 D + D + s B + B 0 B 0 s B + c rs,f 2 / rd,f 2 / rt,f 2 / rt, 2 / If linear potential is treated as perturbation, then from eq. 5), the following inequality must be preserved: p >p 2 p 3 p. ) This inequality leads to a lower limit of Q 2, namely Q 2 0 [], above which one has to use the values of Q 2. Q 2 0 is determined from the condition p = p 2 p 3 p. 7) Due to the quark mass dependence, Q 2 0s have different values and they are shown table. In the Dalgarno method [], the lower limits Q 2 0 were large and the formalism failed to account for large confinement effect in the nonperturbative QCD regime where α s values were taken to be large. Only in the limit b 0, the Q 2 0 values were lowered and the formalism worked for low Q 2 range []. In this method of VIPT, the values of Q 2 0 are shown to be quite small even with large confinement effect enabling us to work in the nonperturbative QCD regime with large α s. We also note that for single-term consideration, only p 2 exists on the RHS of inequality and for double term consideration, both p 2 and p 3 exist. We have recorded the values of charge radii in table VIPT with linear potential as parent 2.2. Wavefunction. As pointed in refs [0,3], linear parent gives rise to Airy functions. The physical parameter is b and the optimized variational parameter is b.we Pramana J. Phys., Vol. 8, No., January

6 Bhaskar Jyoti Hazarika and D K Choudhury reproduce the analogous wavefunction in this case also for three-term consideration of eq. ) as was for the Coulombic parent: ψ T = N [ψ 0) 2μ) /3 ) + b b ρ 02 ρ 0 ) b ) r 2, α ψ 20 r) /3 r 2, 2μ) /3 ) + b b ρ 03 ρ 0 ) b ) r 3, α ψ 30 r) /3 r 3, 2μ) /3 ) ] + b b ρ 0 ρ 0 ) b ) r /3, α ψ 0 r), 8) r where N is the normalization constant as appeared in ref. [3]. We note that for single double) term consideration of eq. ), the third and fourth terms fourth term) are dropped from eq. 8) and normalization constants also change to different one eqs 7) and 2) of ref. [3]). For this case also, we take the relativistic version of the above wavefunction: ψ T,Rel = ψ T rμᾱ 0 ) ɛ. ) The zeros of the Airy function ρ 0n is given by eq. ) of ref. [3]as [ ] 3πn ) 2/3 ρ 0n = 20) 8 and + r k n,n = N n N n r k Ai 2μ b ) /3 ) r ρ 0n Ai 2μ b ) /3 r ρ 0n ) dr, 0 where N 2) n,n n are the normalization constants for n and n states respectively. Like the expressions, we have adopted the same values of b, b,ρ 0n from ref. [3]., Table 3. Values of charge radii fermi) for different mesons with linear parent for single, double and triple terms in eq. ). The subscripts S, D, T correspond to single, double and triple terms respectively whereas F means finite mass consideration. The infinite mass limit subscript is used) is shown for the single term alone. Meson D 0 D + D + s B + B 0 B 0 s B + c C S C S rs,f 2 / C D C D rd,f 2 / C T C T rt,f 2 / rs, 2 / Pramana J. Phys., Vol. 8, No., January 205

7 Form factors and charge radii in a QCD-inspired potential model The elastic charge form factor and charge radii. Putting 8) and) inthe definition of form factor ) given above, the form factor is foundto be efq 2 ) = [ ] e i N C C Q2 i. 22) 6 The coefficients C, C saregivenintable3. They are of course different for single, double or more than two-term consideration. Numerical integrations are done in getting the above result. The corresponding charge radius is obtained by using eqs 8) and) in2) which are recorded in table Status of Coulombic potential as perturbation. The momentum transform of ) is ψ T,Rel Q 2 ) = e i 2 + rψ T,Rel r) sin Q i r dr 23) Q i π 0 = [ 2 e i N 3 Ɣ3 2ɛ) p π + p 2 + p 3 + p ] Q2 i 6 p 5 + p 6 + p 7 + p 8 ). 2) The p i si =, 2,...,8) are given in Appendix. If Coulombic potential is treated as perturbation then from eq. 2) the following inequality must be preserved: p + p 2 + p 3 + p > Q2 i 6 p 5 + p 6 + p 7 + p 8 ). 25) This inequality leads to an upper limit of Q 2, namely Q 2 0, below which one have to use the values of Q 2. Q 2 0 is determined from the condition: p + p 2 + p 3 + p = Q2 i 6 p 5 + p 6 + p 7 + p 8 ). 26) Table. Values of upper limit of four-momentum transfer Q 2 0 with linear parent taking single, double and triple terms in eq. ). We have to use Q 2 values lower than these. Meson D + D D + s B + B 0 B 0 s B + c Q 2 0,S Q 2 0,D Q 2 0,T Pramana J. Phys., Vol. 8, No., January

8 Bhaskar Jyoti Hazarika and D K Choudhury Table 5. Values of b for relativistic case only. The values are the same as recorded in table 2 of ref. [3]. Mesons Reduced mass μ α = α s /3 b with relativistic effect D D s B B s B c Table 6. Prediction of r 2 /2 fermi) for finite and infinite mass consideration in other models. The subscript F ) means finite infinite) mass limit. Meson D 0 D + D + s B + /B B 0 B 0 s / B 0 s B + c /B c rf 2 /2 [] B ) B s 0) 0.236B c ) r 2 /2 [0] F B + ) Bs 0) 0.207B+ c ) r 2 /2 [0] B + ) Bs 0) B+ c ) rf 2 /2 [] B + ).2 r 2 /2 [] B + ).2.7Bs 0).2Bs 0).3B+ c ) c ) Different values of the upper limit Q 2 0s for different terms, e.g., single, double etc. are shown in table. 3. Calculations and results We have listed b in table 5 while the lower and upper limits of Q 2 0 for single-, doubleand triple-term considerations are given in tables and. In table 3, we record the charge radii for single-, double- and triple-term consideration for Coulombic potential as parent; whereas the same is recorded for linear potential as parent in table 2. The infinite mass consideration shown by the subscript is also included for triple single) term consideration for Coulombic linear) parent. Table 6 shows charge radii of different mesons obtained from other models and data. The α s values are taken from the V -scheme [23 25] and the integrations are done numerically for all these calculations. Figures and 2 show the variation efq 2 ) vs. Q 2 for D, D s and B c mesons for both the options.. Discussion and conclusion We have analysed elastic form factors and charge radii in a QCD-inspired potential model with Cornell potential using the VIPT under two scenarios linear potential i) 76 Pramana J. Phys., Vol. 8, No., January 205

9 Form factors and charge radii in a QCD-inspired potential model Figure. Variation of efq 2 ) vs. Q 2 for D, D s and B c mesons with Coulombic parent. as perturbation i.e. Coulombic part as parent) and ii) as parent i.e. Coulombic part as perturbation). We summarize our findings below: ) The form factor efq 2 ) decreases with the increase of Q 2 as it should) for both the scenarios. 2) The form factor is either very small for D-sector mesons) or small for B-sector mesons) with linear perturbation as compared to those with linear parent. The charge radius is also observed to be smaller with linear perturbation than with linear parent. 3) We used a fixed set of values for α s under V -scheme [23 25] in the calculation. For example, it is 0.63 for the D, D s mesons which is larger than the value 0.26 for the B,B s,b c mesons. This consideration directly results in the unexpectedly smaller values of charge radii for D, D s mesons as compared to the B,B s,b c mesons. Larger α s values are responsible for smaller charge radius. ) While checking the status of confinement as perturbation or Coulombic part as perturbation, i.e. linear parent) we obtain a lower or upper) limit on Q 2.Itshows that the former option is valid for high Q 2 and the latter one for low Q 2 as it should. 5) In the present analysis, even with large b, the lower limit of Q 2 for linear perturbation) is really small as shown in table for fixed α s. We have seen that for α s = 0.63, the lower limit of Q 2 for D, D s mesons are respectively , , whereas with α s = 0.26, the lower limit of Q 2 for B,B s,b c mesons are Pramana J. Phys., Vol. 8, No., January

10 Bhaskar Jyoti Hazarika and D K Choudhury Figure 2. Variation of efq 2 ) vs. Q 2 for D, D s and B c mesons with linear parent. respectively 0.05, 0.07, These values for B,B s,b c mesons will be lowered if we put α s > This is clearly advantageous over the Dalgarno method with linear perturbation as done in ref. [] where the formalism broke down for large b. Thus, this approach allows a large value of α s Q 2 ) in the limit Q 2 0evenwith large confinement, an important feature absent in ref. []. 6) Further, if we look at eq. ), consideration of different terms leads to different charge radii and the limiting values of Q 2 for both the cases. The charge radii and the lower limit of Q 2 decrease with more terms for the linear part as perturbation whereas the charge radii increase and upper limit of Q 2 decreases for the linear parent. 7) The infinite mass consideration in this work shows that the charge radii are larger than those for finite-mass consideration to agree well with other models table 6). The above list as a whole suggests the success of VIPT over the Dalgarno method [,22] as far as large confinement and limiting values of Q 2 are concerned. The difference in the values of form factors and charge radii for both the cases may be attributed to the use of same α s i.e. Q 2 ) under V -scheme for both the scenarios as the Coulombic potential is dominant for large Q 2 i.e. low r) and the linear potential in the low Q 2 i.e. large r) regime. It may be noted that we have used the low Q 2 assumption in the calculation of form factors and this clearly effects the upper limit of Q 2 corresponding to the validity of linear parent. The larger value of α s for D-sector as compared to B-sector is also another point to be taken into account. Although, the linear parent has shown more flexibility and hence is the better option than the linear perturbation in VIPT, it has used terms up to 78 Pramana J. Phys., Vol. 8, No., January 205

11 Form factors and charge radii in a QCD-inspired potential model a particular order in r in the integration involved with Airy function which is an infinite series). This may lead to loss of certain information as far as physics is concerned. In the absence of any experimental results for these mesons, it is quite difficult to make a direct conclusion but there is clear indication that one must be careful in choosing the parameter α s Q 2 ) as well as the confinement parameter in the calculation of form factor and charge radius within the QCD framework. The above discussion led to the conclusion that there is scope to use this approach in the study of meson decays. The lower and upper limits on Q 2 i.e. range of Q 2 )inthis analysis may be useful in the experimental set-up to investigate cross-section and form factor in future for these mesons. Further, from the model-specific values of form factors and charge radii, the method allows to investigate the behaviour of α s with respect to Q 2 in the nonperturbative regime of QCD. Appendix Expressions for q i s c q = μ 2 ᾱ + Q 2, A) i ) ɛ) [ q 2 = A 2 μ 2 ᾱ +Q 2 3 2ɛ)μᾱ i ) ɛ) μ 2 ᾱ +Q 2 + 2ɛ)3 ] 2ɛ)μ2 ᾱ i ).5 ɛ) μ 2 ᾱ + Q 2, i )2 ɛ) A2) q 3 = B 2 3 2ɛ)μᾱ μ2 ᾱ ) ɛ) 3 μ2 ᾱ ).5 ɛ) + 2ɛ)3 2ɛ)μ2 ᾱ ) 27 μ2ᾱ 2 ɛ) 85 2ɛ) 2ɛ)3 2ɛ)μ3 ᾱ 3 ) + Q 2 i 8 μ2ᾱ 2.5 ɛ) + Q 2 i + 6 2ɛ)5 2ɛ) 2ɛ)3 2ɛ)μ ᾱ ) 72 μ2ᾱ 3 ɛ) A3) + Q 2 i q = D ) μ2ᾱ ɛ) + Q 2 i 33 2ɛ)μᾱ 32 μ2ᾱ + Q 2 i ).5 ɛ) + 7 2ɛ)3 2ɛ)μ2 ᾱ ) 256 μ2ᾱ 2 ɛ) 5 2ɛ) 2ɛ)3 2ɛ)μ3 ᾱ 3 ) + Q 2 i 536 μ2ᾱ 2.5 ɛ) + Q 2 i Pramana J. Phys., Vol. 8, No., January 205 7

12 Bhaskar Jyoti Hazarika and D K Choudhury ɛ)5 2ɛ) 2ɛ)3 2ɛ)μ ᾱ ) 6 μ2ᾱ 3 ɛ) 7 2ɛ)6 2ɛ)5 2ɛ) 2ɛ)3 2ɛ)μ5 ᾱ 5 ) 2288 μ2ᾱ 3.5 ɛ) + Q 2 i q 5 = 2c A 3 2ɛ)μᾱ ) μ2ᾱ ɛ) ) + Q 2 i 2 μ2ᾱ.5 ɛ) + Q 2 i q 6 = 2c B 23 2ɛ)μᾱ ) ɛ) μ2ᾱ 3 μ2ᾱ + 2 2ɛ)3 2ɛ)μ2 ᾱ ) 27 μ2ᾱ 2 ɛ) q 7 = 2c 33 2ɛ)μᾱ D ) 25μ2ᾱ ɛ) 25μ2ᾱ ).5 ɛ) ).5 ɛ) A) A5) A6) + 2ɛ)3 2ɛ)μ2 ᾱ ) 2 ɛ) 5 2ɛ) 2ɛ)3 2ɛ)μ3 ᾱ 3 ) 2.5 ɛ) A7) 32 25μ2ᾱ +Q 2 i μ2ᾱ +Q 2 i 53 2ɛ)μᾱ q 8 = 2AB ) 25μ2ᾱ ɛ) ) + Q 2 36 i 6 25μ2ᾱ.5 ɛ) + Q 2 36 i ɛ)3 2ɛ)μ2 ᾱ ) 2 ɛ) ɛ) 2ɛ)3 2ɛ)μ3 ᾱ 3 ) 25μ2 ᾱ 36 +Q 2 i 27 25μ2ᾱ 2.5 ɛ), 36 +Q 2 i A8) q = 2AD 53 2ɛ)μᾱ ) μ2ᾱ ɛ) ) + Q 2 i μ2ᾱ.5 ɛ) + Q 2 i + 2ɛ)3 2ɛ)μ2 ᾱ ) 8 μ2ᾱ 2 ɛ) 35 2ɛ) 2ɛ)3 2ɛ)μ3 ᾱ 3 ) +Q 2 i 768 μ2ᾱ 2.5 ɛ) +Q 2 i + 6 2ɛ)5 2ɛ) 2ɛ)3 2ɛ)μ ᾱ ) 536 μ2ᾱ 3 ɛ) A) + Q 2 i 80 Pramana J. Phys., Vol. 8, No., January 205

13 Form factors and charge radii in a QCD-inspired potential model q 0 = 2BD 253 2ɛ)μᾱ ) μ2ᾱ ɛ) ) 8 μ2ᾱ.5 ɛ) + 5 2ɛ)3 2ɛ)μ2 ᾱ ) 86 25μ2ᾱ 2 ɛ) ɛ) 2ɛ)3 2ɛ)μ3 ᾱ 3 ) + Q 2 i μ2ᾱ 2.5 ɛ) + Q 2 i ɛ)5 2ɛ) 2ɛ)3 2ɛ)μ ᾱ ) 3 ɛ) μ2 ᾱ + Q2 i 7 2ɛ)6 2ɛ)5 2ɛ) 2ɛ)3 2ɛ)μ ᾱ ) 3.5 ɛ) μ2 ᾱ + Q2 i A0) Expressions for r i s r = 3c + m ) 2 i μ 2 ᾱ ) ɛ 2 2 2ɛ), A) r 2 = 3A 2 + m ) 2 i [2 2ɛ) μ 2 ᾱ ) ɛ 2 3μᾱ 3 2ɛ) 2 μ 2 ᾱ ) ɛ μ 2 ᾱ 2ɛ) 2 3 2ɛ)μ 2 ᾱ ) ɛ 3 ], r 3 = 3B 2 + m i ) ) 2 ɛ 2 μ 2 ᾱ 2 2ɛ) μᾱ 3 2ɛ) 2 μ 2 ᾱ A2) ) ɛ 2.5 ) ɛ 3 + μ2 ᾱ μ 2ɛ) 2 2 ᾱ 3 2ɛ) 8μ3 ᾱ 3 μ2ᾱ 8 5 2ɛ)2 2ɛ)3 2ɛ) ) ɛ μ ᾱ μ2ᾱ ɛ)2 5 2ɛ) 2ɛ)3 2ɛ) ) ɛ A3) r = 3D + m i ) 2 μ 2 ᾱ + 57μ2 ᾱ μ2ᾱ 2ɛ) 2 3 2ɛ) 32 ) ɛ 2 2 2ɛ) μᾱ ɛ)2 ) ɛ 3 μ 2 ᾱ ) ɛ 2.5 Pramana J. Phys., Vol. 8, No., January 205 8

14 Bhaskar Jyoti Hazarika and D K Choudhury μ3 ᾱ 3 μ2ᾱ ɛ)2 2ɛ)3 2ɛ) ) ɛ μ ᾱ μ2ᾱ 6 6 2ɛ)2 5 2ɛ) 2ɛ)3 2ɛ) ) ɛ 3μ5 ᾱ 5 μ2ᾱ ɛ)2 6 2ɛ)5 2ɛ) 2ɛ)3 2ɛ) r 5 = 2c A + m ) 2 i μ 3 2 ᾱ r 6 = 2c B + m ) 2 i μ2ᾱ 3 r 7 = 2c D + m i μᾱ 3 2ɛ) 2 ) ɛ 2 2 2ɛ) ) ɛ 2.5.5μᾱ 3 2ɛ) 2 μ 2 ᾱ ) ɛ 2 2 2ɛ) μ2ᾱ 2μᾱ 3 2ɛ) 2 ) ɛ μ2 ᾱ μ2ᾱ 2ɛ) 2 3 2ɛ) ) ) 2 ɛ 2 25μ2ᾱ 3 2 2ɛ) 25μ2ᾱ ) ɛ μ2 ᾱ 25μ2ᾱ 32 2ɛ)2 3 2ɛ) ) ɛ 3 3μ3 ᾱ 3 25μ2ᾱ ɛ)2 2ɛ)3 2ɛ) ) ɛ.5 A) A5) ) ɛ 3 A) ) ɛ 3.5 A7) 82 Pramana J. Phys., Vol. 8, No., January 205

15 Form factors and charge radii in a QCD-inspired potential model r 8 = 2AB + m i ) ) 2 ɛ 2 25μ2ᾱ 3 2 2ɛ) 36 ) ɛ μᾱ 3 2ɛ) 2 25μ 2 ᾱ 36 ) ɛ μ2 ᾱ 25μ2ᾱ 2ɛ) 2 3 2ɛ) 36 ) ɛ 3.5 μ3 ᾱ 3 25μ2ᾱ 5 2ɛ)2 2ɛ)3 2ɛ) A8) 36 r = 2AD + m ) 2 i [0.75 μ2 ᾱ )ɛ 2 2 2ɛ) ) ɛ 2.5 5μᾱ μ2ᾱ 3 2ɛ)2 + 3μ2 ᾱ 2ɛ) 2 3 2ɛ) 8 μ2ᾱ 3μ3 ᾱ ɛ)2 2ɛ)3 2ɛ) ) ɛ 3 μ2ᾱ + μ ᾱ ɛ)2 5 2ɛ) 2ɛ)3 2ɛ) r 0 = 2BD + m ) 2 i μ2ᾱ 0.75 ) ɛ 2.5 μ 7μᾱ 2 ᾱ 3 2ɛ)2 + 5μ2 ᾱ 2ɛ) 2 3 2ɛ) 288 μ2ᾱ 83μ3 ᾱ ɛ)2 2ɛ)3 2ɛ) ) ɛ 3.5 ) ɛ μ2ᾱ ) ɛ 2 2 2ɛ) ) ɛ 3 μ2ᾱ + 33μ ᾱ ɛ)2 5 2ɛ) 2ɛ)3 2ɛ) ) ɛ 3.5 μ 2 ᾱ μ5 ᾱ ɛ)2 6 2ɛ)5 2ɛ) 2ɛ)3 2ɛ) ) ɛ A) ) ɛ.5 μ2ᾱ. A20) Pramana J. Phys., Vol. 8, No., January

16 Expressions for p i s Bhaskar Jyoti Hazarika and D K Choudhury c p =, A2) μ2 ᾱ + Q 2 i ) 3 ɛ) 2 p 2 = A 3 2ɛ)μᾱ ) 3 ɛ) ) ɛ) A22) μ2ᾱ + Q 2 2 i 2 μ2ᾱ + Q 2 2 i p 3 = B μ2ᾱ ) 3 ɛ) ɛ)μᾱ μ2ᾱ 2 2ɛ)3 2ɛ)μ2 ᾱ ) 5 ɛ) 27 μ2ᾱ + Q 2 2 i ) ɛ) 2 A23) p = D 0.25 μ2ᾱ ) 3 ɛ) ɛ)μᾱ μ2ᾱ ) ɛ) 2 2ɛ)3 2ɛ)μ2 ᾱ ) 5 ɛ) 32 μ2ᾱ + Q 2 2 i 5 2ɛ) 2ɛ)3 2ɛ)μ3 ᾱ 3 ) 6 ɛ). 768 μ2ᾱ + Q 2 2 i A2) Expressions for p i s p = n b μ) 2/3, p 2 = n 2 b μ) 2/3, p 3 = n 3 b μ) 2/3, p = n b μ) 2/3, p 5 = n 5 b μ) 2/3, p 6 = n 6 b μ) 2/3, A25) A26) A27) A28) A2) A30) 8 Pramana J. Phys., Vol. 8, No., January 205

17 Form factors and charge radii in a QCD-inspired potential model p 7 = n 7 b μ ) 2/3, A3) p 8 = n 8 b μ ) 2/3. A32) Each of the constants n,n 2,...,n 8 are different for different mesons and they have been obtained by numerical integration. References [] D K Choudhury, P Das, D D Goswami and J N Sharma, Pramana J. Phys., 5 5) [2] D K Choudhury and N S Bordoloi, Int. J. Mod. Phys. A 5, ) [3] D K Choudhury and N S Bordoloi, Mod. Phys. Lett. A 26, 3 200) [] D K Choudhury and N S Bordoloi, Mod. Phys. Lett. A 72), ) [5] J J Sakurai, Advanced quantum mechanics Addison-Willey Publishing Company, Massachusetts, 86) p. 28 [6] C Itzykson and J Zuber, Quantum field theory, International Student Edition McGraw Hill, Singapore, 86) p. 7 [7] A D Rujula, H Georgi and S L Glashow, Phys. Rev. D 2, 7 75) [8] A K Ghatak and S Lokanathan, Quantum mechanics McGraw Hill, 7) p. 2 [] S K You, K J Jeon, C K Kim and K Nahm, Eur. J. Phys., 7 8) [0] I J R Aitchison and J J Dudek, Eur. J. Phys. 23, ) [] F M Fernandez, Eur. J. Phys. 2, ) [2] B J Hazarika and D K Choudhury, Pramana J. Phys. 75, ) [3] B J Hazarika and D K Choudhury, Pramana J. Phys. 78, ) [] S J Brodsky, SLAC-PUB-503 8) [5] C J Bebek et al, Phys. Rev. D7, 3 78) [] C R Ji and F Amiri, Phys. Rev. D2, 376 0) [7] T Applequist and E Poggio, Phys. Rev. D0, ) [8] H Pagels and S Stoker, Phys. Rev. D20, 27 7) [] N Isgur and C H Llewellyn Smith, Phys. Lett. B 27, 535 8); CERN-TH 503/88 [20] J N Pandya and P C Vinodkumar, Pramana J. Phys. 57, ) [2] C W Hwang, Eur. Phys. J. C 23, ) [22] N S Bordoloi and D K Choudhury, Ind. J. Phys. 826), ) [23] M Peter, Phys. Rev. Lett. 78, 603 7); Nucl. Phys. B 50, 7 7) [2] Y Schroeder, Phys. Lett. B 7, 32 ) [25] Y Schroeder, Nucl. Phys. Proc. Suppl. 86, ) [26] D P Stanley and D Robson, Phys. Rev.D2, ); Phys. Rev. D26, ) [27] S Flugge, Practical quantum mechanics Springer-Verlag, New York, Heidelberg, Berlin, 7) [28] H B Dwight, Tables of integrals and other mathematical data McMillan Company, ) Pramana J. Phys., Vol. 8, No., January