Research ArticleBound State of Heavy Quarks Using a GeneralPolynomial Potential

HeshamMansour and Ahmed Gamal

Physics Department Faculty of Science Cairo University Giza Egypt

Correspondence should be addressed to Ahmed Gamal alnabolci2010gmailcom

Received 3 October 2018 Revised 15 November 2018 Accepted 27 November 2018 Published 17 December 2018

Academic Editor Sally Seidel

Copyright copy 2018 Hesham Mansour and Ahmed Gamal This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited The publication of this article was funded by SCOAP3

In the present work the mass spectra of the bound states of heavy quarks cc bb and 119861119888 meson are studied within the framework ofthe nonrelativistic Schrodingerrsquos equation First we solve Schrodingerrsquos equation with a general polynomial potential by Nikiforov-Uvarov (NU) method The energy eigenvalues for any L- value is presented for a special case of the potential The results obtainedare in good agreement with the experimental data and are better than previous theoretical studies

1 Introduction

The study of quarkonium systems provides a good under-stating of the quantitative description of quantum chro-modynamics (QCD) theory the standard model and par-ticle physics [1ndash7] The quarkonia with a heavy quarkand antiquark and their interaction are well described bySchrodingerrsquos equation The solution of this equation with aspherically symmetric potentials is one of the most impor-tant problems in quarkonia systems [8ndash11] These potentialsshould take into account the two important features of thestrong interaction namely asymptotic freedom and quarkconfinement [2ndash6]

In the present work an interaction potential in the quark-antiquark bound system is taken as a general polynomialto get the general eigenvalue solution In the next stepwe chose a specific potential according to the physicalproperties of the system Several methods are used to solveSchrodingerrsquos equation One of them is the Nikiforov-Uvarov(NU) method [12ndash14] which gives asymptotic expressionsfor the eigenfunctions and eigenvalues of the Schrodingerrsquosequation Hence one can calculate the energy eigenstates forthe spectrum of the quarkonia systems [12ndash15]

The paper is organized as follows In Section 2 theNikiforov-Uvarov (NU) method is briefly explained In Sec-tion 3 the Schrodinger equation with a general polynomial

potential is solved by the Nikiforov-Uvarov (NU) method InSection 4 results and discussion are presented In Section 5the conclusion is given

2 The Nikiforov-Uvarov (NU) Method [12ndash15]

The Nikiforov-Uvarov (NU) method is based on solving thehypergeometric-type second-order differential equation

Ψ (119904) + 120591 (119904)120590 (119904) Ψ (119904) + (119904)1205902 (119904)Ψ (119904) = 0 (1)

Here 120590(119904) and (119904) are second-degree polynomials 120591(119904) isa first-degree polynomial and 120595(s) is a function of thehypergeometric-type

By taking Ψ(119904) = 120593(119904)Υ(119904) and substituting in equation(1) we get the following equation

Υ (119904) + [2 (119904)120593 (119904) + 120591 (119904)120590 (119904)] Υ (119904)+ [ (119904)120593 (119904) + (119904)120593 (119904) 120591 (119904)120590 (119904) + (119904)1205902 (119904)] Υ (119904) = 0

(2)

HindawiAdvances in High Energy PhysicsVolume 2018 Article ID 7269657 7 pageshttpsdoiorg10115520187269657

2 Advances in High Energy Physics

By taking

2 (119904)120593 (119904) + 120591 (119904)120590 (119904) = 120591 (119904)120590 (119904) (119904)120593 (119904) = 120587 (119904)120590 (119904)

(3)

we get

120591 (119904) = 120591 (119904) + 2120587 (119904) (4)

where both 120587(119904) and 120591(s) are polynomials of degree at mostone

Also we one can take

(119904)120593 (119904) + (119904)120593 (119904) 120591 (119904)120590 (119904) + (119904)1205902 (119904) = 120590 (119904)1205902 (119904) (5)

where

(119904)120593 (119904) = [ (119904)120593 (119904)] + [ (119904)120593 (119904)]

2 = [120587 (119904)120590 (119904)] + [120587 (119904)120590 (119904)]

2

(6)

And

120590 (119904) = (119904) + 1205872 (119904) + 120587 (119904) [120591 (119904) minus (119904)]+ (119904) 120590 (119904) (7)

So equation (2) becomes

Υ (119904) + 120591 (119904)120590 (119904)Υ (119904) + 120590 (119904)1205902 (119904)Υ (119904) = 0 (8)

An algebraic transformation fromequation (1) to equation (8)is systematic Hence one can divide 120590(119904) by 120590(119904) to obtain aconstant ie

120590 (119904) = 120590 (119904) (9)

Equation (8) can be reduced to a hypergeometric equation inthe form

120590 (119904) Υ (119904) + 120591 (119904) Υ (119904) + Υ (119904) = 0 (10)

Substituting from equation (9) in equation (7) and solving thequadratic equation for 120587(119904) we obtain

1205872 (119904) + 120587 (119904) [120591 (119904) minus (119904)] + (119904) minus 119896120590 (119904) = 0 (11)

where

119896 = minus (119904) (12)

120587 (119904) = (119904) minus 120591 (119904)2plusmn radic( (119904) minus 120591 (119904)2 )2 minus (119904) + 119896120590 (119904)

(13)

The possible solutions for 120587(119904)depend on the parameter 119896according to the plus and minus signs of 120587(119904) [13] Since 120587(s)

is a polynomial of degree at most one the expression underthe square root has to be the square of a polynomial In thiscase an equation of the quadratic form is available for theconstant 119896 To determine the parameter 119896 one must set thediscriminant of this quadratic expression to be equal to zeroAfter determining the values of 119896 one can find the values of120587(119904) 119886119899119889 120591(119904)

Applying the same systematic way for equation (10) weget

119899 = minus119899 120591 (119904) minus 119899 (119899 minus 1)2 (119904) (14)

where 119899 is the principle quantum numberBy comparing equations (12) and (14) we get an equation

for the energy eigenvalues

3 The Schroumldinger Equation with a GeneralPolynomial Potential

The radial Schrodinger equation of a quark and antiquarksystem is

11988921198761198891199032 + [2120583ℎ2 (119864 minus 119881) minus 119897 (119897 + 1)1199032 ]119876 = 0 (15)

We will use a generalized polynomial potential

119881 (119903) = 119898sum119898=0

119860119898minus2119903119898minus2 119898 = 0 1 2 3 4 (16)

By substituting in equation (15) we get

11988921198761198891199032 + [2120583ℎ2 119864 minus 2120583ℎ2119898sum119898=0

119860119898minus2119903119898minus2 minus 119897 (119897 + 1)1199032 ]119876 = 0 (17)

Let

2120583ℎ2 119860119898minus2 = 119886119898minus2119887minus2 = 119897 (119897 + 1) + 119886minus21198870 = 1198860 minus 2120583ℎ2 119864 = 1198860 minus 1205980

(18)

and hence

11988921198761198891199032 + [119898sum119898=0

119860 (119886 119887)119898minus2119903119898minus2]119876 = 0 (19)

where

119898sum119898=0

119860 (119886 119887)119898minus2119903119898minus2= minus [119887minus2119903minus2 + 119886minus1119903minus1 + 1198870 + 1198861119903 + 11988621199032 + 11988631199033 + sdot sdot sdot]

(20)

Advances in High Energy Physics 3

Let r = 1x hence11988921198761198891199032 = 119889119889119903 (119889119876119889119903 ) = 119889119889119909 119889119909119889119903 (119889119876119889119903 ) = 119889119889119909 (1199094 119889119876119889119909 )

= 41199093 119889119876119889119909 + 1199094 11988921198761198891199092

(21)

And119898sum119898=0

119860 (119886 119887)119898minus2 ( 1119909)119898minus2

= minus[119887minus21199092 + 119886minus1119909 + 1198870 + 119898sum119898=1

119886119898 ( 1119909)119898]

(22)

By substituting in equation (19) we get

1199094 11988921198761198891199092 + 41199093 119889119876119889119909+ [minus119887minus21199092 minus 119886minus1119909 minus 1198870 minus 119898sum

119898=1

119886119898 ( 1119909)119898]119876 = 0

(23)

We propose the following approximation scheme on the term119886119898(1119909)119898 Let us assume that there is a characteristic radius(residual radius) 1199030 of the quark and antiquark system (whichis the smallest distance between the two quarks where theycannot collide with each other) This scheme is based onthe expansion of 119886119898(1119909)119898 in a power series around 1199030 iearound 120575 = 11199030 in the x-space up to the second orderso that the 119886119898 dependent term preserves the original formof equation (23) This is similar to Pekeris approximation[14 15] which helps to deform the centrifugal potential suchthat the modified potential can be solved by the Nikiforov-Uvarov (NU) method Setting 119910 = (119909 minus 120575) around 119910 = 0 (thesingularity) one can expand into a power series as follows

119898sum119898=1

119886119898 ( 1119909)119898 = 119898sum119898=1

119886119898(119910 + 120575)119898 =119898sum119898=1

119886119898120575119898 [1 + 119910120575]minus119898 (24)

119898sum119898=1

119886119898 ( 1119909)119898

asymp 119898sum119898=1

[119886119898120575119898 minus 119898119886119898120575119898+1 119909 + 119898 (119898 + 1) 1198861198982120575119898+2 1199092] (25)

By substituting from equation (25) in equation (23) dividingby 1199094 where 119909 = 0 and rearranging this equation we get

11988921198761198891199092 + 41199091199092 119889119876119889119909 + 11199094 [minus(1198870 +119898sum119898=1

119886119898120575119898)

+ ( 119898sum119898=1

119898119886119898120575119898+1 minus 119886minus1)119909

minus (119887minus2 + 119898sum119898=1

119898(119898 + 1) 1198861198982120575119898+2 )1199092]119876 = 0

(26)

We define

(1198870 + 119898sum119898=1

119886119898120575119898) = 119902

( 119898sum119898=1

119898119886119898120575119898+1 minus 119886minus1) = 119908

(119887minus2 + 119898sum119898=1

119898(119898 + 1) 1198861198982120575119898+2 ) = 119911

(27)

And hence equation (26) becomes

11988921198761198891199092 + (41199091199092 ) 119889119876119889119909 + 11199094 [minus119902 + 119908119909 minus 1199111199092]119876 = 0 (28)

Comparing with equation (1) we get

120591 = 4119909120590 = 1199092 = minus119902 + 119908119909 minus 1199111199092

(29)

And by substituting in equation (13) we get

120587 (119909) = minus119909 plusmn radic(1 + 119896 + 119911) 1199092 minus 119908119909 + 119902 (30)

Now one can obtain the value of the parameter 119896 by knowingthat120587(119909) is a polynomial of degree atmost one and by puttingthe discriminant of this expression under the square rootequal to zero

1199082 minus 4 (1 + 119896 + 119911) 119902 = 0 997888rarr 119896 = 11990824119902 minus 119911 minus 1 (31)

By substituting in equation (30) and taking the negative valueof 120587(119909) for bound state solutions one finds that the solutionis in agreement with the free hydrogen atom spectrumbecause of the Coulomb term

120587 (119909) = minus119909 minus 1199082radic119902119909 + radic119902 (32)

Hence by substituting in equation (4) we get the following

120591 (119909) = [2 minus 119908radic119902]119909 + 2radic119902

where (2 minus 119908radic119902) lt 0

(33)

Substituting in equation (12) we obtain

= 119896 + (119909) = 11990824119902 minus 1199082radic119902 minus 119911 minus 2 (34)

Using equation (14) we obtain

119899 = minus119899 [2 minus 119908radic119902] minus 119899 (119899 minus 1)

= minus2119899 + 119908radic119902119899 minus 1198992 + 119899[5119901119905] (35)

4 Advances in High Energy Physics

Equalizing equations (34) and (35) we get

119902 = 11990824 [radic119911 + 94 + (119899 + 12)]2 (36)

By substituting the values of (119902 119908 119911 and 120575) in equation (36)we get

1205980 = 1198860 + 119898sum119898=1

1198861198981199030119898 minus (sum119898119898=11198981198861198981199030119898+1 minus 119886minus1)24 [radic(119897 (119897 + 1) + 119886minus2 + (12)sum119898119898=1119898(119898 + 1) 1198861198981199030119898+2) + 94 + (119899 + 12)]2

(37)

Equation (37) is the desired equation of the energy eigen-values in spherical symmetric coordinates with a generalpolynomial radial potential using theNikiforov-Uvarov (NU)method

A special case of the above potential was chosen todescribe the 119902119902 interaction namely

119881 (119903) = 119887119903 + 119886119903 + 1198891199032 + 1199011199034 (38)

where the first term is the Coulomb potential because the twoquarks are charged and the second term is the linear term in

119903 which means that 119881(119903) continues growing as 119903 997888rarr infin Itis this linear term that leads to quark confinement One ofthe striking properties ofQCDasymptotic freedom is that theinteraction strength between quarks becomes smaller as thedistance between them gets shorter

The third term is a harmonic term and the fourth is ananharmonic term and they are responsible also for quarkconfinement

For the above chosen potential we put 119886minus2 = 1198860 = 0 and119898 = 1 2 4 and hence

1205980 = 11988611199030 + 119886211990302 + 119886411990304 minus (119886111990302 + 2119886211990303 + 4119886411990305 minus 119886minus1)24 [radic119897 (119897 + 1) + 119886111990303 + 3119886211990304 + 10119886411990306 + 94 + (119899 + 12)]2

(39)

Now we can rewrite equation (39) in a different form whichdepends on the parameters of the potential as follows

119864 = 1198861199030 + 11988911990302 + 11990111990304 minus (2120583ℎ2) (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic(2120583119886ℎ2) 11990303 + (6120583119889ℎ2) 11990304 + (20120583119901ℎ2) 11990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2

(40)

4 Results and Discussion

In this section we will calculate the spectra for the boundstates of heavy quarks such as charmonium bottomoniumand BC meson To determine the mass spectra in three

dimensions we use the following relation

119872 = 119898119902 + 119898119902 + 119864 (41)

By substituting in equation (40) we get

119872 = 2119898119902 + 1198861199030 + 11988911990302 + 11990111990304 minus (2120583ℎ2) (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic(2120583119886ℎ2) 11990303 + (6120583119889ℎ2) 11990304 + (20120583119901ℎ2) 11990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2

(42)

Advances in High Energy Physics 5

It is clear that equation (42) depends on the potentialparameters (119886 119887 119889 119901 119886119899119889 1199030) which will be obtained fromthe experimental data

In the case of charmonium [Ψ = cc] the rest massequation is as follows

119872 = 2119898119888 + 1198861199030 + 11988911990302 + 11990111990304 minus 32743 lowast (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic3274311988611990303 + 9822911988911990304 + 3274311990111990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2 (43)

In the case of bottomonium [ = bb] the rest mass equationis as follows

119872 = 2119898119887 + 1198861199030 + 11988911990302 + 11990111990304 minus 10786 lowast (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic10786 lowast 11988611990303 + 32358 lowast 11988911990304 + 10786 lowast 11990111990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2 (44)

And in the case of themeson [B119888 = bc] the restmass equationis as follows

119872 = 119898119888 + 119898119887 + 1198861199030 + 11988911990302 + 11990111990304 minus 5024 lowast (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic5024 lowast 11988611990303 + 15072 lowast 11988911990304 + 5024 lowast 11990111990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2 (45)

Comparing our theoretical work with the experimental datawe found that the maximum errors are 0229 for thecharmonium 00742 for the bottomonium and 000123for the BC meson These may be due to errors in the mea-surements of the device The spin can also be taken intoaccount if one uses relativistic corrections and the appropriaterelativistic Schrodingerrsquos equation Our results are shown inTables 1 2 and 3 with a comparison between our resultsand those obtained in previous calculations in the literatureIn the charmonium system the maximum distance wherea quark and antiquark can approach each other is 1199030 =08043 119891119898 Similarly the maximum distances in the casesof the bottomonium system and 119861119862 meson system are 1199030 =047 119891119898 and 1199030 = 04256 119891119898 respectively The positive andnegative signs of the coefficient of the harmonic potentialrefer to the direction ofmotion of the oscillationThenegativesign of the coefficient of the Coulomb potential refers to thecharges of the two quarks but the positive sign refers to theexistence of another negative contribution The positive andnegative signs of the coefficient of the anharmonic potentialgive a correction to the linear potential

5 Conclusion

The mass spectra of the quarkonia (charmonium bottomo-nium and BC meson) using our potential were studiedwithinthe framework of the nonrelativistic Schrodingerrsquos equationby using the Nikiforov-Uvarov (NU) method In [16 17] theauthors used an iteration method and the same potentialwhen 119901 = 0 It is found that adding the anharmonic potentialgives a good accuracy and our work is comparable withthem In [18 19] the authors used also the same potentialwhen 119886 = 119901 = 0 We found that our work gives betterresults in comparison with experimental data In [20 2128 29] the authors used the same method and the samepotential when 119901 = 0 We noticed that our work is inbetter agreement with the experimental data In [22 23] theauthors used the Cornell potential only and used the samemethod It is found that their results are comparable withour work In [24 25] the authors used the same potentialwhen 119901 = 0 and the same method used in the presentwork We found that our work is comparable with them In[22 23 30ndash37] for the mass spectra of 119905ℎ119890 119861119862 meson there is

6 Advances in High Energy Physics

Table 1 Mass spectra of charmonium in comparison with other works [1199030 = 08043 119891119898 119886 = 303857 119866119890119881119891119898 119889 =minus07054 119866119890119881(119891119898)2 119887 = minus049842 119866119890119881119891119898 119875 = minus02379 119866119890119881(119891119898)4]Type present work [16 17] [18 19] [20 21] [22 23] [24 25] EXP [26 27]1s 30969 3078 3096 3096 3096 3078 309691p - 3415 3433 3433 3255 3415 -2s - 4187 3686 3686 3686 3581 -1d 36861 3752 3676 3770 3504 3749 368612p 37702 4143 3910 4023 3779 3917 37733s - 5297 3984 4040 4040 4085 -4s - 6407 4150 4358 4269 4589 -2d - - - 4096 - 3078 -4d 4160 - - - - - 41591g 4039 - - - - - 40396d 4263 - - - - - 4263

Table 2 Mass spectra of bottomonium in comparison with other works [1199030 = 047 119891119898 119886 = 107 119866119890119881119891119898 119889 = minus495 119866119890119881(119891119898)2 119887 =639286 119866119890119881119891119898 119875 = 71 119866119890119881(119891119898)4]Type present work [16 17] [18 19] [28 29] [22 23] [24 25] EXP [26 27]1s 94600 9510 9460 9460 9460 9510 946011p - 9612 9840 9811 9916 9862 -2s - 10627 10023 10023 10023 10038 -1d - 10214 10140 10161 9864 10214 -2p - 10944 10160 10374 10114 10390 -3s - 11726 10280 10355 10355 10655 -2d 1002 - - - - - 100234s 10571 12834 10420 10655 10567 11094 105793d 10358 - - - - - 103556S 110198 - - - - - 110195d 10873 - - - - - 10876

Table 3 Mass spectra of 119861119862 119898119890119904119900119899 in comparison with other works [1199030 = 04256 119891119898 119886 = 4196 119866119890119881119891119898 119889 = minus26064 119866119890119881(119891119898)2 119887 =95675 119866119890119881119891119898 119875 = 60631 119866119890119881(119891119898)4]Type present work [30ndash32] [33 34] [35ndash37] [22 23] EXP [26 27]1s 6227 6349 6264 6270 6278 62771p 6287 6715 6700 6699 6486 -2s 6714 6821 6856 6853 6866 -1d 6398 - - - 6772 -2p 6759 7102 7108 7091 6973 -3s 709 7175 7244 7193 7181 -4s 7386 - - - 7369 -2d 685 - - - 7128 -4d 7469 - - - - -1g 6728 - - - - -

no enough experimental data to comparewith In conclusioncomparing with the experimental data we found that ourresults are better than those given by previous theoreticalestimates

Data Availability

The information given in our tables is available for readers inthe original references listed in our work

Advances in High Energy Physics 7

Disclosure

Hesham Mansour is a Fellow of the Institute of Physics(FInstP)

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] F Halzem and A Martin Quarks and Leptons an IntroductoryCourse in Modern Particle Physics Wiley Publication 1984

[2] D Griffiths Introduction to Elementary Particles John Wiley ampSons New York NY USA 1987

[3] D H Perkins Introduction to High Energy Physics CambridgeUniversity Press Cambridge UK 2000

[4] B R Martin and G G Shaw Particle Physics The ManchesterPhysics Series 2nd edition 1997

[5] A Bettini Introduction to Elementary Particle Physics Cam-bridge University Press 2018

[6] J Rathsman Quark and Lepton Interactions Acta UniversitatisUpsalensis Uppsala 1996

[7] N Mistry Brief Introduction of Particle Physics Cornell Univer-sity 2011

[8] A T Paul Ralph LlewellynModern PhysicsWHFreeman 2012[9] BH Bransden andC J JoachainQuantumMechanics Prentice

Hall 2000[10] L D Landau and E M Lifshitz Quantum Mechanics Butter-

worth Heinemann 1981[11] D Griffiths Introduction to Elementary Particles John Wiley amp

Sons New York NY USA 2008[12] N ZettiliQuantumMechanics Concepts and Applications John

Wiley 2009[13] C Berkdemir Theoretical Concept of Quantum Mechanics In

Tech 2012[14] A F Nikiforov andV B Uvarov Special Functions ofMathemat-

ical Physics Birkhaauser Basel Switzerland 1988[15] H Karayer D Demirhan and F Buyukkılıc ldquoExtension of

Nikiforov-Uvarov method for the solution of Heun equationrdquoJournal of Mathematical Physics vol 56 no 6 Article ID063504 14 pages 2015

[16] R Kumar and F Chand ldquoAsymptotic study to the N-dimensional radial schrodinger equation for the quark-antiquark systemrdquo Communications in Theoretical Physics vol59 no 5 pp 528ndash532 2013

[17] M Abu-Shady ldquoHeavy Quarkonia and Mesons in the Cor-nell Potential with Harmonic Oscillator Potential in the N-dimensional Schrodinger Equationrdquo International Journal ofApplied Mathematics and Theoretical Physics vol 2 no 2 p 162016

[18] A Al-Oun A Al-Jamel and H Widyan ldquoVarious Propertiesof Heavy Quarkonia from Flavor-Independent Coulomb PlusQuadratic Potentialrdquo Jordan Journal of Physics vol 8 no 4 pp199ndash203 2015

[19] H Ke G Wang X Li and C Chang ldquoThe magnetic dipoletransitions in the (119888119887) binding systemrdquo Science China PhysicsMechanics and Astronomy vol 53 no 11 pp 2025ndash2030 2010

[20] N V Masksimenko and S M Kuchin ldquoDetermination of themass spectrum of quarkonia by theNikiforovndashUvarovmethodrdquoRussian Physics Journal vol 54 no 57 2011

[21] E Eichten K Gottfried T Kinoshita J Kogut K D Laneand T M Yan ldquoSpectrum of charmed quark-antiquark boundstatesrdquo Physical Review Letters vol 34 no 6 pp 369ndash372 1975

[22] S M Kuchin and N V Maksimenko ldquoTheoretical Estimationsof the Spin ndash Averaged Mass Spectra of Heavy Quarkonia andBc Mesonsrdquo Universal Journal of Physics and Application vol 7pp 295ndash298 2013

[23] R Faccini BABAR-CONF-07035 SLAC-PUB-13080[24] R Kumar and F Chand ldquoEnergy Spectra of the Coulomb

Perturbed Potential inN-Dimensional Hilbert Spacerdquo Phys Scrvol 85 Article ID 055008 2012

[25] Y Li P Maris and J P Vary ldquoQuarkonium as a relativisticbound state on the light frontrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 96 no 1 2017

[26] J Beringer J F Arguin and R M Barnett ldquoReview of particlephysicsrdquo Physical Review D Particles Fields Gravitation andCosmology vol 86 Article ID 010001 2012

[27] B Hong T Barillari S Bethke S Kluth and S MenkeldquoOverview of quarkonium production in heavy-ion collisionsat LHCrdquo EPJ Web of Conferences vol 120 p 06002 2016

[28] Z Ghalenovi A A Rajabi and A Tavakolinezhad ldquoMasses ofCharm and Beauty Baryons in the Constituent Quark ModelrdquoInternational Journal of Modern Physics E vol 21 no 6 ArticleID 1250057 2012

[29] J Niu L Guo H Ma and S Wang ldquoHeavy quarkoniumproduction through the top quark rare decays via the channelsinvolving flavor changing neutral currentsrdquo The EuropeanPhysical Journal C vol 78 no 8 2018

[30] A K Ray and P C Vinodkumar ldquoProperties of B119888 mesonrdquoPramana vol 66 no 5 pp 953ndash958 2006

[31] S Laachir and A Laaribi ldquoExact Solutions of the Helmholtzequation via the Nikiforov-UvarovMethodrdquo International Jour-nal of Mathematical Computational Physical Electrical andComputer Engineering vol 7 no 1 2013

[32] HCiftci R LHall andN Saad ldquoIterative solutions to theDiracequationrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 72 no 2 part A 022101 7 pages 2005

[33] E J Eichten and C Quigg ldquoMesons with beauty and charmSpectroscopyrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 49 no 11 pp 5845ndash5856 1994

[34] M Alberg and L Wilets ldquoExact solutions to the Schrodingerequation for potentials with Coulomb and harmonic oscillatortermsrdquo Physics Letters A vol 286 no 1 pp 7ndash14 2001

[35] D Ebert R N Faustov and V O Galkin ldquoProperties of heavyquarkonia andrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 67 no 1 2003

[36] B Ita P Tchoua E Siryabe and G E Ntamack ldquoSolutions ofthe Klein-Gordon Equation with the Hulthen Potential Usingthe Frobenius Methodrdquo International Journal of Theoretical andMathematical Physics vol 4 no 5 pp 173ndash177 2014

[37] H Goudarzi and V Vahidi ldquoSupersymmetric approach forEckart potential using the NU methodrdquo Advanced Studies inTheoretical Physics vol 5 no 9-12 pp 469ndash476 2011

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

Advances in High Energy Physics 3

Let r = 1x hence11988921198761198891199032 = 119889119889119903 (119889119876119889119903 ) = 119889119889119909 119889119909119889119903 (119889119876119889119903 ) = 119889119889119909 (1199094 119889119876119889119909 )

= 41199093 119889119876119889119909 + 1199094 11988921198761198891199092

(21)

And119898sum119898=0

119860 (119886 119887)119898minus2 ( 1119909)119898minus2

= minus[119887minus21199092 + 119886minus1119909 + 1198870 + 119898sum119898=1

119886119898 ( 1119909)119898]

(22)

By substituting in equation (19) we get

1199094 11988921198761198891199092 + 41199093 119889119876119889119909+ [minus119887minus21199092 minus 119886minus1119909 minus 1198870 minus 119898sum

119898=1

119886119898 ( 1119909)119898]119876 = 0

(23)

We propose the following approximation scheme on the term119886119898(1119909)119898 Let us assume that there is a characteristic radius(residual radius) 1199030 of the quark and antiquark system (whichis the smallest distance between the two quarks where theycannot collide with each other) This scheme is based onthe expansion of 119886119898(1119909)119898 in a power series around 1199030 iearound 120575 = 11199030 in the x-space up to the second orderso that the 119886119898 dependent term preserves the original formof equation (23) This is similar to Pekeris approximation[14 15] which helps to deform the centrifugal potential suchthat the modified potential can be solved by the Nikiforov-Uvarov (NU) method Setting 119910 = (119909 minus 120575) around 119910 = 0 (thesingularity) one can expand into a power series as follows

119898sum119898=1

119886119898 ( 1119909)119898 = 119898sum119898=1

119886119898(119910 + 120575)119898 =119898sum119898=1

119886119898120575119898 [1 + 119910120575]minus119898 (24)

119898sum119898=1

119886119898 ( 1119909)119898

asymp 119898sum119898=1

[119886119898120575119898 minus 119898119886119898120575119898+1 119909 + 119898 (119898 + 1) 1198861198982120575119898+2 1199092] (25)

By substituting from equation (25) in equation (23) dividingby 1199094 where 119909 = 0 and rearranging this equation we get

11988921198761198891199092 + 41199091199092 119889119876119889119909 + 11199094 [minus(1198870 +119898sum119898=1

119886119898120575119898)

+ ( 119898sum119898=1

119898119886119898120575119898+1 minus 119886minus1)119909

minus (119887minus2 + 119898sum119898=1

119898(119898 + 1) 1198861198982120575119898+2 )1199092]119876 = 0

(26)

We define

(1198870 + 119898sum119898=1

119886119898120575119898) = 119902

( 119898sum119898=1

119898119886119898120575119898+1 minus 119886minus1) = 119908

(119887minus2 + 119898sum119898=1

119898(119898 + 1) 1198861198982120575119898+2 ) = 119911

(27)

And hence equation (26) becomes

11988921198761198891199092 + (41199091199092 ) 119889119876119889119909 + 11199094 [minus119902 + 119908119909 minus 1199111199092]119876 = 0 (28)

Comparing with equation (1) we get

120591 = 4119909120590 = 1199092 = minus119902 + 119908119909 minus 1199111199092

(29)

And by substituting in equation (13) we get

120587 (119909) = minus119909 plusmn radic(1 + 119896 + 119911) 1199092 minus 119908119909 + 119902 (30)

Now one can obtain the value of the parameter 119896 by knowingthat120587(119909) is a polynomial of degree atmost one and by puttingthe discriminant of this expression under the square rootequal to zero

1199082 minus 4 (1 + 119896 + 119911) 119902 = 0 997888rarr 119896 = 11990824119902 minus 119911 minus 1 (31)

By substituting in equation (30) and taking the negative valueof 120587(119909) for bound state solutions one finds that the solutionis in agreement with the free hydrogen atom spectrumbecause of the Coulomb term

120587 (119909) = minus119909 minus 1199082radic119902119909 + radic119902 (32)

Hence by substituting in equation (4) we get the following

120591 (119909) = [2 minus 119908radic119902]119909 + 2radic119902

where (2 minus 119908radic119902) lt 0

(33)

Substituting in equation (12) we obtain

= 119896 + (119909) = 11990824119902 minus 1199082radic119902 minus 119911 minus 2 (34)

Using equation (14) we obtain

119899 = minus119899 [2 minus 119908radic119902] minus 119899 (119899 minus 1)

= minus2119899 + 119908radic119902119899 minus 1198992 + 119899[5119901119905] (35)

4 Advances in High Energy Physics

Equalizing equations (34) and (35) we get

119902 = 11990824 [radic119911 + 94 + (119899 + 12)]2 (36)

By substituting the values of (119902 119908 119911 and 120575) in equation (36)we get

1205980 = 1198860 + 119898sum119898=1

1198861198981199030119898 minus (sum119898119898=11198981198861198981199030119898+1 minus 119886minus1)24 [radic(119897 (119897 + 1) + 119886minus2 + (12)sum119898119898=1119898(119898 + 1) 1198861198981199030119898+2) + 94 + (119899 + 12)]2

(37)

Equation (37) is the desired equation of the energy eigen-values in spherical symmetric coordinates with a generalpolynomial radial potential using theNikiforov-Uvarov (NU)method

A special case of the above potential was chosen todescribe the 119902119902 interaction namely

119881 (119903) = 119887119903 + 119886119903 + 1198891199032 + 1199011199034 (38)

where the first term is the Coulomb potential because the twoquarks are charged and the second term is the linear term in

119903 which means that 119881(119903) continues growing as 119903 997888rarr infin Itis this linear term that leads to quark confinement One ofthe striking properties ofQCDasymptotic freedom is that theinteraction strength between quarks becomes smaller as thedistance between them gets shorter

The third term is a harmonic term and the fourth is ananharmonic term and they are responsible also for quarkconfinement

For the above chosen potential we put 119886minus2 = 1198860 = 0 and119898 = 1 2 4 and hence

1205980 = 11988611199030 + 119886211990302 + 119886411990304 minus (119886111990302 + 2119886211990303 + 4119886411990305 minus 119886minus1)24 [radic119897 (119897 + 1) + 119886111990303 + 3119886211990304 + 10119886411990306 + 94 + (119899 + 12)]2

(39)

Now we can rewrite equation (39) in a different form whichdepends on the parameters of the potential as follows

119864 = 1198861199030 + 11988911990302 + 11990111990304 minus (2120583ℎ2) (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic(2120583119886ℎ2) 11990303 + (6120583119889ℎ2) 11990304 + (20120583119901ℎ2) 11990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2

(40)

4 Results and Discussion

In this section we will calculate the spectra for the boundstates of heavy quarks such as charmonium bottomoniumand BC meson To determine the mass spectra in three

dimensions we use the following relation

119872 = 119898119902 + 119898119902 + 119864 (41)

By substituting in equation (40) we get

119872 = 2119898119902 + 1198861199030 + 11988911990302 + 11990111990304 minus (2120583ℎ2) (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic(2120583119886ℎ2) 11990303 + (6120583119889ℎ2) 11990304 + (20120583119901ℎ2) 11990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2

(42)

Advances in High Energy Physics 5

It is clear that equation (42) depends on the potentialparameters (119886 119887 119889 119901 119886119899119889 1199030) which will be obtained fromthe experimental data

In the case of charmonium [Ψ = cc] the rest massequation is as follows

119872 = 2119898119888 + 1198861199030 + 11988911990302 + 11990111990304 minus 32743 lowast (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic3274311988611990303 + 9822911988911990304 + 3274311990111990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2 (43)

In the case of bottomonium [ = bb] the rest mass equationis as follows

119872 = 2119898119887 + 1198861199030 + 11988911990302 + 11990111990304 minus 10786 lowast (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic10786 lowast 11988611990303 + 32358 lowast 11988911990304 + 10786 lowast 11990111990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2 (44)

And in the case of themeson [B119888 = bc] the restmass equationis as follows

119872 = 119898119888 + 119898119887 + 1198861199030 + 11988911990302 + 11990111990304 minus 5024 lowast (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic5024 lowast 11988611990303 + 15072 lowast 11988911990304 + 5024 lowast 11990111990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2 (45)

Comparing our theoretical work with the experimental datawe found that the maximum errors are 0229 for thecharmonium 00742 for the bottomonium and 000123for the BC meson These may be due to errors in the mea-surements of the device The spin can also be taken intoaccount if one uses relativistic corrections and the appropriaterelativistic Schrodingerrsquos equation Our results are shown inTables 1 2 and 3 with a comparison between our resultsand those obtained in previous calculations in the literatureIn the charmonium system the maximum distance wherea quark and antiquark can approach each other is 1199030 =08043 119891119898 Similarly the maximum distances in the casesof the bottomonium system and 119861119862 meson system are 1199030 =047 119891119898 and 1199030 = 04256 119891119898 respectively The positive andnegative signs of the coefficient of the harmonic potentialrefer to the direction ofmotion of the oscillationThenegativesign of the coefficient of the Coulomb potential refers to thecharges of the two quarks but the positive sign refers to theexistence of another negative contribution The positive andnegative signs of the coefficient of the anharmonic potentialgive a correction to the linear potential

5 Conclusion

The mass spectra of the quarkonia (charmonium bottomo-nium and BC meson) using our potential were studiedwithinthe framework of the nonrelativistic Schrodingerrsquos equationby using the Nikiforov-Uvarov (NU) method In [16 17] theauthors used an iteration method and the same potentialwhen 119901 = 0 It is found that adding the anharmonic potentialgives a good accuracy and our work is comparable withthem In [18 19] the authors used also the same potentialwhen 119886 = 119901 = 0 We found that our work gives betterresults in comparison with experimental data In [20 2128 29] the authors used the same method and the samepotential when 119901 = 0 We noticed that our work is inbetter agreement with the experimental data In [22 23] theauthors used the Cornell potential only and used the samemethod It is found that their results are comparable withour work In [24 25] the authors used the same potentialwhen 119901 = 0 and the same method used in the presentwork We found that our work is comparable with them In[22 23 30ndash37] for the mass spectra of 119905ℎ119890 119861119862 meson there is

6 Advances in High Energy Physics

Table 1 Mass spectra of charmonium in comparison with other works [1199030 = 08043 119891119898 119886 = 303857 119866119890119881119891119898 119889 =minus07054 119866119890119881(119891119898)2 119887 = minus049842 119866119890119881119891119898 119875 = minus02379 119866119890119881(119891119898)4]Type present work [16 17] [18 19] [20 21] [22 23] [24 25] EXP [26 27]1s 30969 3078 3096 3096 3096 3078 309691p - 3415 3433 3433 3255 3415 -2s - 4187 3686 3686 3686 3581 -1d 36861 3752 3676 3770 3504 3749 368612p 37702 4143 3910 4023 3779 3917 37733s - 5297 3984 4040 4040 4085 -4s - 6407 4150 4358 4269 4589 -2d - - - 4096 - 3078 -4d 4160 - - - - - 41591g 4039 - - - - - 40396d 4263 - - - - - 4263

Table 2 Mass spectra of bottomonium in comparison with other works [1199030 = 047 119891119898 119886 = 107 119866119890119881119891119898 119889 = minus495 119866119890119881(119891119898)2 119887 =639286 119866119890119881119891119898 119875 = 71 119866119890119881(119891119898)4]Type present work [16 17] [18 19] [28 29] [22 23] [24 25] EXP [26 27]1s 94600 9510 9460 9460 9460 9510 946011p - 9612 9840 9811 9916 9862 -2s - 10627 10023 10023 10023 10038 -1d - 10214 10140 10161 9864 10214 -2p - 10944 10160 10374 10114 10390 -3s - 11726 10280 10355 10355 10655 -2d 1002 - - - - - 100234s 10571 12834 10420 10655 10567 11094 105793d 10358 - - - - - 103556S 110198 - - - - - 110195d 10873 - - - - - 10876

Table 3 Mass spectra of 119861119862 119898119890119904119900119899 in comparison with other works [1199030 = 04256 119891119898 119886 = 4196 119866119890119881119891119898 119889 = minus26064 119866119890119881(119891119898)2 119887 =95675 119866119890119881119891119898 119875 = 60631 119866119890119881(119891119898)4]Type present work [30ndash32] [33 34] [35ndash37] [22 23] EXP [26 27]1s 6227 6349 6264 6270 6278 62771p 6287 6715 6700 6699 6486 -2s 6714 6821 6856 6853 6866 -1d 6398 - - - 6772 -2p 6759 7102 7108 7091 6973 -3s 709 7175 7244 7193 7181 -4s 7386 - - - 7369 -2d 685 - - - 7128 -4d 7469 - - - - -1g 6728 - - - - -

no enough experimental data to comparewith In conclusioncomparing with the experimental data we found that ourresults are better than those given by previous theoreticalestimates

Data Availability

The information given in our tables is available for readers inthe original references listed in our work

Advances in High Energy Physics 7

Disclosure

Hesham Mansour is a Fellow of the Institute of Physics(FInstP)

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] F Halzem and A Martin Quarks and Leptons an IntroductoryCourse in Modern Particle Physics Wiley Publication 1984

[2] D Griffiths Introduction to Elementary Particles John Wiley ampSons New York NY USA 1987

[3] D H Perkins Introduction to High Energy Physics CambridgeUniversity Press Cambridge UK 2000

[4] B R Martin and G G Shaw Particle Physics The ManchesterPhysics Series 2nd edition 1997

[5] A Bettini Introduction to Elementary Particle Physics Cam-bridge University Press 2018

[6] J Rathsman Quark and Lepton Interactions Acta UniversitatisUpsalensis Uppsala 1996

[7] N Mistry Brief Introduction of Particle Physics Cornell Univer-sity 2011

[8] A T Paul Ralph LlewellynModern PhysicsWHFreeman 2012[9] BH Bransden andC J JoachainQuantumMechanics Prentice

Hall 2000[10] L D Landau and E M Lifshitz Quantum Mechanics Butter-

worth Heinemann 1981[11] D Griffiths Introduction to Elementary Particles John Wiley amp

Sons New York NY USA 2008[12] N ZettiliQuantumMechanics Concepts and Applications John

Wiley 2009[13] C Berkdemir Theoretical Concept of Quantum Mechanics In

Tech 2012[14] A F Nikiforov andV B Uvarov Special Functions ofMathemat-

ical Physics Birkhaauser Basel Switzerland 1988[15] H Karayer D Demirhan and F Buyukkılıc ldquoExtension of

Nikiforov-Uvarov method for the solution of Heun equationrdquoJournal of Mathematical Physics vol 56 no 6 Article ID063504 14 pages 2015

[16] R Kumar and F Chand ldquoAsymptotic study to the N-dimensional radial schrodinger equation for the quark-antiquark systemrdquo Communications in Theoretical Physics vol59 no 5 pp 528ndash532 2013

[17] M Abu-Shady ldquoHeavy Quarkonia and Mesons in the Cor-nell Potential with Harmonic Oscillator Potential in the N-dimensional Schrodinger Equationrdquo International Journal ofApplied Mathematics and Theoretical Physics vol 2 no 2 p 162016

[18] A Al-Oun A Al-Jamel and H Widyan ldquoVarious Propertiesof Heavy Quarkonia from Flavor-Independent Coulomb PlusQuadratic Potentialrdquo Jordan Journal of Physics vol 8 no 4 pp199ndash203 2015

[19] H Ke G Wang X Li and C Chang ldquoThe magnetic dipoletransitions in the (119888119887) binding systemrdquo Science China PhysicsMechanics and Astronomy vol 53 no 11 pp 2025ndash2030 2010

[20] N V Masksimenko and S M Kuchin ldquoDetermination of themass spectrum of quarkonia by theNikiforovndashUvarovmethodrdquoRussian Physics Journal vol 54 no 57 2011

[21] E Eichten K Gottfried T Kinoshita J Kogut K D Laneand T M Yan ldquoSpectrum of charmed quark-antiquark boundstatesrdquo Physical Review Letters vol 34 no 6 pp 369ndash372 1975

[22] S M Kuchin and N V Maksimenko ldquoTheoretical Estimationsof the Spin ndash Averaged Mass Spectra of Heavy Quarkonia andBc Mesonsrdquo Universal Journal of Physics and Application vol 7pp 295ndash298 2013

[23] R Faccini BABAR-CONF-07035 SLAC-PUB-13080[24] R Kumar and F Chand ldquoEnergy Spectra of the Coulomb

Perturbed Potential inN-Dimensional Hilbert Spacerdquo Phys Scrvol 85 Article ID 055008 2012

[25] Y Li P Maris and J P Vary ldquoQuarkonium as a relativisticbound state on the light frontrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 96 no 1 2017

[26] J Beringer J F Arguin and R M Barnett ldquoReview of particlephysicsrdquo Physical Review D Particles Fields Gravitation andCosmology vol 86 Article ID 010001 2012

[27] B Hong T Barillari S Bethke S Kluth and S MenkeldquoOverview of quarkonium production in heavy-ion collisionsat LHCrdquo EPJ Web of Conferences vol 120 p 06002 2016

[28] Z Ghalenovi A A Rajabi and A Tavakolinezhad ldquoMasses ofCharm and Beauty Baryons in the Constituent Quark ModelrdquoInternational Journal of Modern Physics E vol 21 no 6 ArticleID 1250057 2012

[29] J Niu L Guo H Ma and S Wang ldquoHeavy quarkoniumproduction through the top quark rare decays via the channelsinvolving flavor changing neutral currentsrdquo The EuropeanPhysical Journal C vol 78 no 8 2018

[30] A K Ray and P C Vinodkumar ldquoProperties of B119888 mesonrdquoPramana vol 66 no 5 pp 953ndash958 2006

[31] S Laachir and A Laaribi ldquoExact Solutions of the Helmholtzequation via the Nikiforov-UvarovMethodrdquo International Jour-nal of Mathematical Computational Physical Electrical andComputer Engineering vol 7 no 1 2013

[32] HCiftci R LHall andN Saad ldquoIterative solutions to theDiracequationrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 72 no 2 part A 022101 7 pages 2005

[33] E J Eichten and C Quigg ldquoMesons with beauty and charmSpectroscopyrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 49 no 11 pp 5845ndash5856 1994

[34] M Alberg and L Wilets ldquoExact solutions to the Schrodingerequation for potentials with Coulomb and harmonic oscillatortermsrdquo Physics Letters A vol 286 no 1 pp 7ndash14 2001

[35] D Ebert R N Faustov and V O Galkin ldquoProperties of heavyquarkonia andrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 67 no 1 2003

[36] B Ita P Tchoua E Siryabe and G E Ntamack ldquoSolutions ofthe Klein-Gordon Equation with the Hulthen Potential Usingthe Frobenius Methodrdquo International Journal of Theoretical andMathematical Physics vol 4 no 5 pp 173ndash177 2014

[37] H Goudarzi and V Vahidi ldquoSupersymmetric approach forEckart potential using the NU methodrdquo Advanced Studies inTheoretical Physics vol 5 no 9-12 pp 469ndash476 2011

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

4 Advances in High Energy Physics

Equalizing equations (34) and (35) we get

119902 = 11990824 [radic119911 + 94 + (119899 + 12)]2 (36)

By substituting the values of (119902 119908 119911 and 120575) in equation (36)we get

1205980 = 1198860 + 119898sum119898=1

1198861198981199030119898 minus (sum119898119898=11198981198861198981199030119898+1 minus 119886minus1)24 [radic(119897 (119897 + 1) + 119886minus2 + (12)sum119898119898=1119898(119898 + 1) 1198861198981199030119898+2) + 94 + (119899 + 12)]2

(37)

Equation (37) is the desired equation of the energy eigen-values in spherical symmetric coordinates with a generalpolynomial radial potential using theNikiforov-Uvarov (NU)method

A special case of the above potential was chosen todescribe the 119902119902 interaction namely

119881 (119903) = 119887119903 + 119886119903 + 1198891199032 + 1199011199034 (38)

where the first term is the Coulomb potential because the twoquarks are charged and the second term is the linear term in

119903 which means that 119881(119903) continues growing as 119903 997888rarr infin Itis this linear term that leads to quark confinement One ofthe striking properties ofQCDasymptotic freedom is that theinteraction strength between quarks becomes smaller as thedistance between them gets shorter

The third term is a harmonic term and the fourth is ananharmonic term and they are responsible also for quarkconfinement

For the above chosen potential we put 119886minus2 = 1198860 = 0 and119898 = 1 2 4 and hence

1205980 = 11988611199030 + 119886211990302 + 119886411990304 minus (119886111990302 + 2119886211990303 + 4119886411990305 minus 119886minus1)24 [radic119897 (119897 + 1) + 119886111990303 + 3119886211990304 + 10119886411990306 + 94 + (119899 + 12)]2

(39)

Now we can rewrite equation (39) in a different form whichdepends on the parameters of the potential as follows

119864 = 1198861199030 + 11988911990302 + 11990111990304 minus (2120583ℎ2) (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic(2120583119886ℎ2) 11990303 + (6120583119889ℎ2) 11990304 + (20120583119901ℎ2) 11990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2

(40)

4 Results and Discussion

In this section we will calculate the spectra for the boundstates of heavy quarks such as charmonium bottomoniumand BC meson To determine the mass spectra in three

dimensions we use the following relation

119872 = 119898119902 + 119898119902 + 119864 (41)

By substituting in equation (40) we get

119872 = 2119898119902 + 1198861199030 + 11988911990302 + 11990111990304 minus (2120583ℎ2) (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic(2120583119886ℎ2) 11990303 + (6120583119889ℎ2) 11990304 + (20120583119901ℎ2) 11990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2

(42)

Advances in High Energy Physics 5

It is clear that equation (42) depends on the potentialparameters (119886 119887 119889 119901 119886119899119889 1199030) which will be obtained fromthe experimental data

In the case of charmonium [Ψ = cc] the rest massequation is as follows

119872 = 2119898119888 + 1198861199030 + 11988911990302 + 11990111990304 minus 32743 lowast (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic3274311988611990303 + 9822911988911990304 + 3274311990111990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2 (43)

In the case of bottomonium [ = bb] the rest mass equationis as follows

119872 = 2119898119887 + 1198861199030 + 11988911990302 + 11990111990304 minus 10786 lowast (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic10786 lowast 11988611990303 + 32358 lowast 11988911990304 + 10786 lowast 11990111990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2 (44)

And in the case of themeson [B119888 = bc] the restmass equationis as follows

119872 = 119898119888 + 119898119887 + 1198861199030 + 11988911990302 + 11990111990304 minus 5024 lowast (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic5024 lowast 11988611990303 + 15072 lowast 11988911990304 + 5024 lowast 11990111990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2 (45)

Comparing our theoretical work with the experimental datawe found that the maximum errors are 0229 for thecharmonium 00742 for the bottomonium and 000123for the BC meson These may be due to errors in the mea-surements of the device The spin can also be taken intoaccount if one uses relativistic corrections and the appropriaterelativistic Schrodingerrsquos equation Our results are shown inTables 1 2 and 3 with a comparison between our resultsand those obtained in previous calculations in the literatureIn the charmonium system the maximum distance wherea quark and antiquark can approach each other is 1199030 =08043 119891119898 Similarly the maximum distances in the casesof the bottomonium system and 119861119862 meson system are 1199030 =047 119891119898 and 1199030 = 04256 119891119898 respectively The positive andnegative signs of the coefficient of the harmonic potentialrefer to the direction ofmotion of the oscillationThenegativesign of the coefficient of the Coulomb potential refers to thecharges of the two quarks but the positive sign refers to theexistence of another negative contribution The positive andnegative signs of the coefficient of the anharmonic potentialgive a correction to the linear potential

5 Conclusion

The mass spectra of the quarkonia (charmonium bottomo-nium and BC meson) using our potential were studiedwithinthe framework of the nonrelativistic Schrodingerrsquos equationby using the Nikiforov-Uvarov (NU) method In [16 17] theauthors used an iteration method and the same potentialwhen 119901 = 0 It is found that adding the anharmonic potentialgives a good accuracy and our work is comparable withthem In [18 19] the authors used also the same potentialwhen 119886 = 119901 = 0 We found that our work gives betterresults in comparison with experimental data In [20 2128 29] the authors used the same method and the samepotential when 119901 = 0 We noticed that our work is inbetter agreement with the experimental data In [22 23] theauthors used the Cornell potential only and used the samemethod It is found that their results are comparable withour work In [24 25] the authors used the same potentialwhen 119901 = 0 and the same method used in the presentwork We found that our work is comparable with them In[22 23 30ndash37] for the mass spectra of 119905ℎ119890 119861119862 meson there is

6 Advances in High Energy Physics

Table 1 Mass spectra of charmonium in comparison with other works [1199030 = 08043 119891119898 119886 = 303857 119866119890119881119891119898 119889 =minus07054 119866119890119881(119891119898)2 119887 = minus049842 119866119890119881119891119898 119875 = minus02379 119866119890119881(119891119898)4]Type present work [16 17] [18 19] [20 21] [22 23] [24 25] EXP [26 27]1s 30969 3078 3096 3096 3096 3078 309691p - 3415 3433 3433 3255 3415 -2s - 4187 3686 3686 3686 3581 -1d 36861 3752 3676 3770 3504 3749 368612p 37702 4143 3910 4023 3779 3917 37733s - 5297 3984 4040 4040 4085 -4s - 6407 4150 4358 4269 4589 -2d - - - 4096 - 3078 -4d 4160 - - - - - 41591g 4039 - - - - - 40396d 4263 - - - - - 4263

Table 2 Mass spectra of bottomonium in comparison with other works [1199030 = 047 119891119898 119886 = 107 119866119890119881119891119898 119889 = minus495 119866119890119881(119891119898)2 119887 =639286 119866119890119881119891119898 119875 = 71 119866119890119881(119891119898)4]Type present work [16 17] [18 19] [28 29] [22 23] [24 25] EXP [26 27]1s 94600 9510 9460 9460 9460 9510 946011p - 9612 9840 9811 9916 9862 -2s - 10627 10023 10023 10023 10038 -1d - 10214 10140 10161 9864 10214 -2p - 10944 10160 10374 10114 10390 -3s - 11726 10280 10355 10355 10655 -2d 1002 - - - - - 100234s 10571 12834 10420 10655 10567 11094 105793d 10358 - - - - - 103556S 110198 - - - - - 110195d 10873 - - - - - 10876

Table 3 Mass spectra of 119861119862 119898119890119904119900119899 in comparison with other works [1199030 = 04256 119891119898 119886 = 4196 119866119890119881119891119898 119889 = minus26064 119866119890119881(119891119898)2 119887 =95675 119866119890119881119891119898 119875 = 60631 119866119890119881(119891119898)4]Type present work [30ndash32] [33 34] [35ndash37] [22 23] EXP [26 27]1s 6227 6349 6264 6270 6278 62771p 6287 6715 6700 6699 6486 -2s 6714 6821 6856 6853 6866 -1d 6398 - - - 6772 -2p 6759 7102 7108 7091 6973 -3s 709 7175 7244 7193 7181 -4s 7386 - - - 7369 -2d 685 - - - 7128 -4d 7469 - - - - -1g 6728 - - - - -

no enough experimental data to comparewith In conclusioncomparing with the experimental data we found that ourresults are better than those given by previous theoreticalestimates

Data Availability

The information given in our tables is available for readers inthe original references listed in our work

Advances in High Energy Physics 7

Disclosure

Hesham Mansour is a Fellow of the Institute of Physics(FInstP)

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] F Halzem and A Martin Quarks and Leptons an IntroductoryCourse in Modern Particle Physics Wiley Publication 1984

[2] D Griffiths Introduction to Elementary Particles John Wiley ampSons New York NY USA 1987

[3] D H Perkins Introduction to High Energy Physics CambridgeUniversity Press Cambridge UK 2000

[4] B R Martin and G G Shaw Particle Physics The ManchesterPhysics Series 2nd edition 1997

[5] A Bettini Introduction to Elementary Particle Physics Cam-bridge University Press 2018

[6] J Rathsman Quark and Lepton Interactions Acta UniversitatisUpsalensis Uppsala 1996

[7] N Mistry Brief Introduction of Particle Physics Cornell Univer-sity 2011

[8] A T Paul Ralph LlewellynModern PhysicsWHFreeman 2012[9] BH Bransden andC J JoachainQuantumMechanics Prentice

Hall 2000[10] L D Landau and E M Lifshitz Quantum Mechanics Butter-

worth Heinemann 1981[11] D Griffiths Introduction to Elementary Particles John Wiley amp

Sons New York NY USA 2008[12] N ZettiliQuantumMechanics Concepts and Applications John

Wiley 2009[13] C Berkdemir Theoretical Concept of Quantum Mechanics In

Tech 2012[14] A F Nikiforov andV B Uvarov Special Functions ofMathemat-

ical Physics Birkhaauser Basel Switzerland 1988[15] H Karayer D Demirhan and F Buyukkılıc ldquoExtension of

Nikiforov-Uvarov method for the solution of Heun equationrdquoJournal of Mathematical Physics vol 56 no 6 Article ID063504 14 pages 2015

[16] R Kumar and F Chand ldquoAsymptotic study to the N-dimensional radial schrodinger equation for the quark-antiquark systemrdquo Communications in Theoretical Physics vol59 no 5 pp 528ndash532 2013

[17] M Abu-Shady ldquoHeavy Quarkonia and Mesons in the Cor-nell Potential with Harmonic Oscillator Potential in the N-dimensional Schrodinger Equationrdquo International Journal ofApplied Mathematics and Theoretical Physics vol 2 no 2 p 162016

[18] A Al-Oun A Al-Jamel and H Widyan ldquoVarious Propertiesof Heavy Quarkonia from Flavor-Independent Coulomb PlusQuadratic Potentialrdquo Jordan Journal of Physics vol 8 no 4 pp199ndash203 2015

[19] H Ke G Wang X Li and C Chang ldquoThe magnetic dipoletransitions in the (119888119887) binding systemrdquo Science China PhysicsMechanics and Astronomy vol 53 no 11 pp 2025ndash2030 2010

[20] N V Masksimenko and S M Kuchin ldquoDetermination of themass spectrum of quarkonia by theNikiforovndashUvarovmethodrdquoRussian Physics Journal vol 54 no 57 2011

[21] E Eichten K Gottfried T Kinoshita J Kogut K D Laneand T M Yan ldquoSpectrum of charmed quark-antiquark boundstatesrdquo Physical Review Letters vol 34 no 6 pp 369ndash372 1975

[22] S M Kuchin and N V Maksimenko ldquoTheoretical Estimationsof the Spin ndash Averaged Mass Spectra of Heavy Quarkonia andBc Mesonsrdquo Universal Journal of Physics and Application vol 7pp 295ndash298 2013

[23] R Faccini BABAR-CONF-07035 SLAC-PUB-13080[24] R Kumar and F Chand ldquoEnergy Spectra of the Coulomb

Perturbed Potential inN-Dimensional Hilbert Spacerdquo Phys Scrvol 85 Article ID 055008 2012

[25] Y Li P Maris and J P Vary ldquoQuarkonium as a relativisticbound state on the light frontrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 96 no 1 2017

[26] J Beringer J F Arguin and R M Barnett ldquoReview of particlephysicsrdquo Physical Review D Particles Fields Gravitation andCosmology vol 86 Article ID 010001 2012

[27] B Hong T Barillari S Bethke S Kluth and S MenkeldquoOverview of quarkonium production in heavy-ion collisionsat LHCrdquo EPJ Web of Conferences vol 120 p 06002 2016

[28] Z Ghalenovi A A Rajabi and A Tavakolinezhad ldquoMasses ofCharm and Beauty Baryons in the Constituent Quark ModelrdquoInternational Journal of Modern Physics E vol 21 no 6 ArticleID 1250057 2012

[29] J Niu L Guo H Ma and S Wang ldquoHeavy quarkoniumproduction through the top quark rare decays via the channelsinvolving flavor changing neutral currentsrdquo The EuropeanPhysical Journal C vol 78 no 8 2018

[30] A K Ray and P C Vinodkumar ldquoProperties of B119888 mesonrdquoPramana vol 66 no 5 pp 953ndash958 2006

[31] S Laachir and A Laaribi ldquoExact Solutions of the Helmholtzequation via the Nikiforov-UvarovMethodrdquo International Jour-nal of Mathematical Computational Physical Electrical andComputer Engineering vol 7 no 1 2013

[32] HCiftci R LHall andN Saad ldquoIterative solutions to theDiracequationrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 72 no 2 part A 022101 7 pages 2005

[33] E J Eichten and C Quigg ldquoMesons with beauty and charmSpectroscopyrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 49 no 11 pp 5845ndash5856 1994

[34] M Alberg and L Wilets ldquoExact solutions to the Schrodingerequation for potentials with Coulomb and harmonic oscillatortermsrdquo Physics Letters A vol 286 no 1 pp 7ndash14 2001

[35] D Ebert R N Faustov and V O Galkin ldquoProperties of heavyquarkonia andrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 67 no 1 2003

[36] B Ita P Tchoua E Siryabe and G E Ntamack ldquoSolutions ofthe Klein-Gordon Equation with the Hulthen Potential Usingthe Frobenius Methodrdquo International Journal of Theoretical andMathematical Physics vol 4 no 5 pp 173ndash177 2014

[37] H Goudarzi and V Vahidi ldquoSupersymmetric approach forEckart potential using the NU methodrdquo Advanced Studies inTheoretical Physics vol 5 no 9-12 pp 469ndash476 2011

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

Advances in High Energy Physics 5

It is clear that equation (42) depends on the potentialparameters (119886 119887 119889 119901 119886119899119889 1199030) which will be obtained fromthe experimental data

In the case of charmonium [Ψ = cc] the rest massequation is as follows

119872 = 2119898119888 + 1198861199030 + 11988911990302 + 11990111990304 minus 32743 lowast (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic3274311988611990303 + 9822911988911990304 + 3274311990111990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2 (43)

In the case of bottomonium [ = bb] the rest mass equationis as follows

119872 = 2119898119887 + 1198861199030 + 11988911990302 + 11990111990304 minus 10786 lowast (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic10786 lowast 11988611990303 + 32358 lowast 11988911990304 + 10786 lowast 11990111990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2 (44)

And in the case of themeson [B119888 = bc] the restmass equationis as follows

119872 = 119898119888 + 119898119887 + 1198861199030 + 11988911990302 + 11990111990304 minus 5024 lowast (11988611990302 + 211988911990303 + 411990111990305 minus 119887)24 [radic5024 lowast 11988611990303 + 15072 lowast 11988911990304 + 5024 lowast 11990111990306 + 119897 (119897 + 1) + 94 + (119899 + 12)]2 (45)

Comparing our theoretical work with the experimental datawe found that the maximum errors are 0229 for thecharmonium 00742 for the bottomonium and 000123for the BC meson These may be due to errors in the mea-surements of the device The spin can also be taken intoaccount if one uses relativistic corrections and the appropriaterelativistic Schrodingerrsquos equation Our results are shown inTables 1 2 and 3 with a comparison between our resultsand those obtained in previous calculations in the literatureIn the charmonium system the maximum distance wherea quark and antiquark can approach each other is 1199030 =08043 119891119898 Similarly the maximum distances in the casesof the bottomonium system and 119861119862 meson system are 1199030 =047 119891119898 and 1199030 = 04256 119891119898 respectively The positive andnegative signs of the coefficient of the harmonic potentialrefer to the direction ofmotion of the oscillationThenegativesign of the coefficient of the Coulomb potential refers to thecharges of the two quarks but the positive sign refers to theexistence of another negative contribution The positive andnegative signs of the coefficient of the anharmonic potentialgive a correction to the linear potential

5 Conclusion

The mass spectra of the quarkonia (charmonium bottomo-nium and BC meson) using our potential were studiedwithinthe framework of the nonrelativistic Schrodingerrsquos equationby using the Nikiforov-Uvarov (NU) method In [16 17] theauthors used an iteration method and the same potentialwhen 119901 = 0 It is found that adding the anharmonic potentialgives a good accuracy and our work is comparable withthem In [18 19] the authors used also the same potentialwhen 119886 = 119901 = 0 We found that our work gives betterresults in comparison with experimental data In [20 2128 29] the authors used the same method and the samepotential when 119901 = 0 We noticed that our work is inbetter agreement with the experimental data In [22 23] theauthors used the Cornell potential only and used the samemethod It is found that their results are comparable withour work In [24 25] the authors used the same potentialwhen 119901 = 0 and the same method used in the presentwork We found that our work is comparable with them In[22 23 30ndash37] for the mass spectra of 119905ℎ119890 119861119862 meson there is

6 Advances in High Energy Physics

Table 1 Mass spectra of charmonium in comparison with other works [1199030 = 08043 119891119898 119886 = 303857 119866119890119881119891119898 119889 =minus07054 119866119890119881(119891119898)2 119887 = minus049842 119866119890119881119891119898 119875 = minus02379 119866119890119881(119891119898)4]Type present work [16 17] [18 19] [20 21] [22 23] [24 25] EXP [26 27]1s 30969 3078 3096 3096 3096 3078 309691p - 3415 3433 3433 3255 3415 -2s - 4187 3686 3686 3686 3581 -1d 36861 3752 3676 3770 3504 3749 368612p 37702 4143 3910 4023 3779 3917 37733s - 5297 3984 4040 4040 4085 -4s - 6407 4150 4358 4269 4589 -2d - - - 4096 - 3078 -4d 4160 - - - - - 41591g 4039 - - - - - 40396d 4263 - - - - - 4263

Table 2 Mass spectra of bottomonium in comparison with other works [1199030 = 047 119891119898 119886 = 107 119866119890119881119891119898 119889 = minus495 119866119890119881(119891119898)2 119887 =639286 119866119890119881119891119898 119875 = 71 119866119890119881(119891119898)4]Type present work [16 17] [18 19] [28 29] [22 23] [24 25] EXP [26 27]1s 94600 9510 9460 9460 9460 9510 946011p - 9612 9840 9811 9916 9862 -2s - 10627 10023 10023 10023 10038 -1d - 10214 10140 10161 9864 10214 -2p - 10944 10160 10374 10114 10390 -3s - 11726 10280 10355 10355 10655 -2d 1002 - - - - - 100234s 10571 12834 10420 10655 10567 11094 105793d 10358 - - - - - 103556S 110198 - - - - - 110195d 10873 - - - - - 10876

Table 3 Mass spectra of 119861119862 119898119890119904119900119899 in comparison with other works [1199030 = 04256 119891119898 119886 = 4196 119866119890119881119891119898 119889 = minus26064 119866119890119881(119891119898)2 119887 =95675 119866119890119881119891119898 119875 = 60631 119866119890119881(119891119898)4]Type present work [30ndash32] [33 34] [35ndash37] [22 23] EXP [26 27]1s 6227 6349 6264 6270 6278 62771p 6287 6715 6700 6699 6486 -2s 6714 6821 6856 6853 6866 -1d 6398 - - - 6772 -2p 6759 7102 7108 7091 6973 -3s 709 7175 7244 7193 7181 -4s 7386 - - - 7369 -2d 685 - - - 7128 -4d 7469 - - - - -1g 6728 - - - - -

no enough experimental data to comparewith In conclusioncomparing with the experimental data we found that ourresults are better than those given by previous theoreticalestimates

Data Availability

The information given in our tables is available for readers inthe original references listed in our work

Advances in High Energy Physics 7

Disclosure

Hesham Mansour is a Fellow of the Institute of Physics(FInstP)

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] F Halzem and A Martin Quarks and Leptons an IntroductoryCourse in Modern Particle Physics Wiley Publication 1984

[2] D Griffiths Introduction to Elementary Particles John Wiley ampSons New York NY USA 1987

[3] D H Perkins Introduction to High Energy Physics CambridgeUniversity Press Cambridge UK 2000

[4] B R Martin and G G Shaw Particle Physics The ManchesterPhysics Series 2nd edition 1997

[5] A Bettini Introduction to Elementary Particle Physics Cam-bridge University Press 2018

[6] J Rathsman Quark and Lepton Interactions Acta UniversitatisUpsalensis Uppsala 1996

[7] N Mistry Brief Introduction of Particle Physics Cornell Univer-sity 2011

[8] A T Paul Ralph LlewellynModern PhysicsWHFreeman 2012[9] BH Bransden andC J JoachainQuantumMechanics Prentice

Hall 2000[10] L D Landau and E M Lifshitz Quantum Mechanics Butter-

worth Heinemann 1981[11] D Griffiths Introduction to Elementary Particles John Wiley amp

Sons New York NY USA 2008[12] N ZettiliQuantumMechanics Concepts and Applications John

Wiley 2009[13] C Berkdemir Theoretical Concept of Quantum Mechanics In

Tech 2012[14] A F Nikiforov andV B Uvarov Special Functions ofMathemat-

ical Physics Birkhaauser Basel Switzerland 1988[15] H Karayer D Demirhan and F Buyukkılıc ldquoExtension of

Nikiforov-Uvarov method for the solution of Heun equationrdquoJournal of Mathematical Physics vol 56 no 6 Article ID063504 14 pages 2015

[16] R Kumar and F Chand ldquoAsymptotic study to the N-dimensional radial schrodinger equation for the quark-antiquark systemrdquo Communications in Theoretical Physics vol59 no 5 pp 528ndash532 2013

[17] M Abu-Shady ldquoHeavy Quarkonia and Mesons in the Cor-nell Potential with Harmonic Oscillator Potential in the N-dimensional Schrodinger Equationrdquo International Journal ofApplied Mathematics and Theoretical Physics vol 2 no 2 p 162016

[18] A Al-Oun A Al-Jamel and H Widyan ldquoVarious Propertiesof Heavy Quarkonia from Flavor-Independent Coulomb PlusQuadratic Potentialrdquo Jordan Journal of Physics vol 8 no 4 pp199ndash203 2015

[19] H Ke G Wang X Li and C Chang ldquoThe magnetic dipoletransitions in the (119888119887) binding systemrdquo Science China PhysicsMechanics and Astronomy vol 53 no 11 pp 2025ndash2030 2010

[20] N V Masksimenko and S M Kuchin ldquoDetermination of themass spectrum of quarkonia by theNikiforovndashUvarovmethodrdquoRussian Physics Journal vol 54 no 57 2011

[21] E Eichten K Gottfried T Kinoshita J Kogut K D Laneand T M Yan ldquoSpectrum of charmed quark-antiquark boundstatesrdquo Physical Review Letters vol 34 no 6 pp 369ndash372 1975

[22] S M Kuchin and N V Maksimenko ldquoTheoretical Estimationsof the Spin ndash Averaged Mass Spectra of Heavy Quarkonia andBc Mesonsrdquo Universal Journal of Physics and Application vol 7pp 295ndash298 2013

[23] R Faccini BABAR-CONF-07035 SLAC-PUB-13080[24] R Kumar and F Chand ldquoEnergy Spectra of the Coulomb

Perturbed Potential inN-Dimensional Hilbert Spacerdquo Phys Scrvol 85 Article ID 055008 2012

[25] Y Li P Maris and J P Vary ldquoQuarkonium as a relativisticbound state on the light frontrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 96 no 1 2017

[26] J Beringer J F Arguin and R M Barnett ldquoReview of particlephysicsrdquo Physical Review D Particles Fields Gravitation andCosmology vol 86 Article ID 010001 2012

[27] B Hong T Barillari S Bethke S Kluth and S MenkeldquoOverview of quarkonium production in heavy-ion collisionsat LHCrdquo EPJ Web of Conferences vol 120 p 06002 2016

[28] Z Ghalenovi A A Rajabi and A Tavakolinezhad ldquoMasses ofCharm and Beauty Baryons in the Constituent Quark ModelrdquoInternational Journal of Modern Physics E vol 21 no 6 ArticleID 1250057 2012

[29] J Niu L Guo H Ma and S Wang ldquoHeavy quarkoniumproduction through the top quark rare decays via the channelsinvolving flavor changing neutral currentsrdquo The EuropeanPhysical Journal C vol 78 no 8 2018

[30] A K Ray and P C Vinodkumar ldquoProperties of B119888 mesonrdquoPramana vol 66 no 5 pp 953ndash958 2006

[31] S Laachir and A Laaribi ldquoExact Solutions of the Helmholtzequation via the Nikiforov-UvarovMethodrdquo International Jour-nal of Mathematical Computational Physical Electrical andComputer Engineering vol 7 no 1 2013

[32] HCiftci R LHall andN Saad ldquoIterative solutions to theDiracequationrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 72 no 2 part A 022101 7 pages 2005

[33] E J Eichten and C Quigg ldquoMesons with beauty and charmSpectroscopyrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 49 no 11 pp 5845ndash5856 1994

[34] M Alberg and L Wilets ldquoExact solutions to the Schrodingerequation for potentials with Coulomb and harmonic oscillatortermsrdquo Physics Letters A vol 286 no 1 pp 7ndash14 2001

[35] D Ebert R N Faustov and V O Galkin ldquoProperties of heavyquarkonia andrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 67 no 1 2003

[36] B Ita P Tchoua E Siryabe and G E Ntamack ldquoSolutions ofthe Klein-Gordon Equation with the Hulthen Potential Usingthe Frobenius Methodrdquo International Journal of Theoretical andMathematical Physics vol 4 no 5 pp 173ndash177 2014

[37] H Goudarzi and V Vahidi ldquoSupersymmetric approach forEckart potential using the NU methodrdquo Advanced Studies inTheoretical Physics vol 5 no 9-12 pp 469ndash476 2011

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

6 Advances in High Energy Physics

Table 1 Mass spectra of charmonium in comparison with other works [1199030 = 08043 119891119898 119886 = 303857 119866119890119881119891119898 119889 =minus07054 119866119890119881(119891119898)2 119887 = minus049842 119866119890119881119891119898 119875 = minus02379 119866119890119881(119891119898)4]Type present work [16 17] [18 19] [20 21] [22 23] [24 25] EXP [26 27]1s 30969 3078 3096 3096 3096 3078 309691p - 3415 3433 3433 3255 3415 -2s - 4187 3686 3686 3686 3581 -1d 36861 3752 3676 3770 3504 3749 368612p 37702 4143 3910 4023 3779 3917 37733s - 5297 3984 4040 4040 4085 -4s - 6407 4150 4358 4269 4589 -2d - - - 4096 - 3078 -4d 4160 - - - - - 41591g 4039 - - - - - 40396d 4263 - - - - - 4263

Table 2 Mass spectra of bottomonium in comparison with other works [1199030 = 047 119891119898 119886 = 107 119866119890119881119891119898 119889 = minus495 119866119890119881(119891119898)2 119887 =639286 119866119890119881119891119898 119875 = 71 119866119890119881(119891119898)4]Type present work [16 17] [18 19] [28 29] [22 23] [24 25] EXP [26 27]1s 94600 9510 9460 9460 9460 9510 946011p - 9612 9840 9811 9916 9862 -2s - 10627 10023 10023 10023 10038 -1d - 10214 10140 10161 9864 10214 -2p - 10944 10160 10374 10114 10390 -3s - 11726 10280 10355 10355 10655 -2d 1002 - - - - - 100234s 10571 12834 10420 10655 10567 11094 105793d 10358 - - - - - 103556S 110198 - - - - - 110195d 10873 - - - - - 10876

Table 3 Mass spectra of 119861119862 119898119890119904119900119899 in comparison with other works [1199030 = 04256 119891119898 119886 = 4196 119866119890119881119891119898 119889 = minus26064 119866119890119881(119891119898)2 119887 =95675 119866119890119881119891119898 119875 = 60631 119866119890119881(119891119898)4]Type present work [30ndash32] [33 34] [35ndash37] [22 23] EXP [26 27]1s 6227 6349 6264 6270 6278 62771p 6287 6715 6700 6699 6486 -2s 6714 6821 6856 6853 6866 -1d 6398 - - - 6772 -2p 6759 7102 7108 7091 6973 -3s 709 7175 7244 7193 7181 -4s 7386 - - - 7369 -2d 685 - - - 7128 -4d 7469 - - - - -1g 6728 - - - - -

no enough experimental data to comparewith In conclusioncomparing with the experimental data we found that ourresults are better than those given by previous theoreticalestimates

Data Availability

The information given in our tables is available for readers inthe original references listed in our work

Advances in High Energy Physics 7

Disclosure

Hesham Mansour is a Fellow of the Institute of Physics(FInstP)

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] F Halzem and A Martin Quarks and Leptons an IntroductoryCourse in Modern Particle Physics Wiley Publication 1984

[2] D Griffiths Introduction to Elementary Particles John Wiley ampSons New York NY USA 1987

[3] D H Perkins Introduction to High Energy Physics CambridgeUniversity Press Cambridge UK 2000

[4] B R Martin and G G Shaw Particle Physics The ManchesterPhysics Series 2nd edition 1997

[5] A Bettini Introduction to Elementary Particle Physics Cam-bridge University Press 2018

[6] J Rathsman Quark and Lepton Interactions Acta UniversitatisUpsalensis Uppsala 1996

[7] N Mistry Brief Introduction of Particle Physics Cornell Univer-sity 2011

[8] A T Paul Ralph LlewellynModern PhysicsWHFreeman 2012[9] BH Bransden andC J JoachainQuantumMechanics Prentice

Hall 2000[10] L D Landau and E M Lifshitz Quantum Mechanics Butter-

worth Heinemann 1981[11] D Griffiths Introduction to Elementary Particles John Wiley amp

Sons New York NY USA 2008[12] N ZettiliQuantumMechanics Concepts and Applications John

Wiley 2009[13] C Berkdemir Theoretical Concept of Quantum Mechanics In

Tech 2012[14] A F Nikiforov andV B Uvarov Special Functions ofMathemat-

ical Physics Birkhaauser Basel Switzerland 1988[15] H Karayer D Demirhan and F Buyukkılıc ldquoExtension of

Nikiforov-Uvarov method for the solution of Heun equationrdquoJournal of Mathematical Physics vol 56 no 6 Article ID063504 14 pages 2015

[16] R Kumar and F Chand ldquoAsymptotic study to the N-dimensional radial schrodinger equation for the quark-antiquark systemrdquo Communications in Theoretical Physics vol59 no 5 pp 528ndash532 2013

[17] M Abu-Shady ldquoHeavy Quarkonia and Mesons in the Cor-nell Potential with Harmonic Oscillator Potential in the N-dimensional Schrodinger Equationrdquo International Journal ofApplied Mathematics and Theoretical Physics vol 2 no 2 p 162016

[18] A Al-Oun A Al-Jamel and H Widyan ldquoVarious Propertiesof Heavy Quarkonia from Flavor-Independent Coulomb PlusQuadratic Potentialrdquo Jordan Journal of Physics vol 8 no 4 pp199ndash203 2015

[19] H Ke G Wang X Li and C Chang ldquoThe magnetic dipoletransitions in the (119888119887) binding systemrdquo Science China PhysicsMechanics and Astronomy vol 53 no 11 pp 2025ndash2030 2010

[20] N V Masksimenko and S M Kuchin ldquoDetermination of themass spectrum of quarkonia by theNikiforovndashUvarovmethodrdquoRussian Physics Journal vol 54 no 57 2011

[21] E Eichten K Gottfried T Kinoshita J Kogut K D Laneand T M Yan ldquoSpectrum of charmed quark-antiquark boundstatesrdquo Physical Review Letters vol 34 no 6 pp 369ndash372 1975

[22] S M Kuchin and N V Maksimenko ldquoTheoretical Estimationsof the Spin ndash Averaged Mass Spectra of Heavy Quarkonia andBc Mesonsrdquo Universal Journal of Physics and Application vol 7pp 295ndash298 2013

[23] R Faccini BABAR-CONF-07035 SLAC-PUB-13080[24] R Kumar and F Chand ldquoEnergy Spectra of the Coulomb

Perturbed Potential inN-Dimensional Hilbert Spacerdquo Phys Scrvol 85 Article ID 055008 2012

[25] Y Li P Maris and J P Vary ldquoQuarkonium as a relativisticbound state on the light frontrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 96 no 1 2017

[26] J Beringer J F Arguin and R M Barnett ldquoReview of particlephysicsrdquo Physical Review D Particles Fields Gravitation andCosmology vol 86 Article ID 010001 2012

[27] B Hong T Barillari S Bethke S Kluth and S MenkeldquoOverview of quarkonium production in heavy-ion collisionsat LHCrdquo EPJ Web of Conferences vol 120 p 06002 2016

[28] Z Ghalenovi A A Rajabi and A Tavakolinezhad ldquoMasses ofCharm and Beauty Baryons in the Constituent Quark ModelrdquoInternational Journal of Modern Physics E vol 21 no 6 ArticleID 1250057 2012

[29] J Niu L Guo H Ma and S Wang ldquoHeavy quarkoniumproduction through the top quark rare decays via the channelsinvolving flavor changing neutral currentsrdquo The EuropeanPhysical Journal C vol 78 no 8 2018

[30] A K Ray and P C Vinodkumar ldquoProperties of B119888 mesonrdquoPramana vol 66 no 5 pp 953ndash958 2006

[31] S Laachir and A Laaribi ldquoExact Solutions of the Helmholtzequation via the Nikiforov-UvarovMethodrdquo International Jour-nal of Mathematical Computational Physical Electrical andComputer Engineering vol 7 no 1 2013

[32] HCiftci R LHall andN Saad ldquoIterative solutions to theDiracequationrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 72 no 2 part A 022101 7 pages 2005

[33] E J Eichten and C Quigg ldquoMesons with beauty and charmSpectroscopyrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 49 no 11 pp 5845ndash5856 1994

[34] M Alberg and L Wilets ldquoExact solutions to the Schrodingerequation for potentials with Coulomb and harmonic oscillatortermsrdquo Physics Letters A vol 286 no 1 pp 7ndash14 2001

[35] D Ebert R N Faustov and V O Galkin ldquoProperties of heavyquarkonia andrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 67 no 1 2003

[36] B Ita P Tchoua E Siryabe and G E Ntamack ldquoSolutions ofthe Klein-Gordon Equation with the Hulthen Potential Usingthe Frobenius Methodrdquo International Journal of Theoretical andMathematical Physics vol 4 no 5 pp 173ndash177 2014

[37] H Goudarzi and V Vahidi ldquoSupersymmetric approach forEckart potential using the NU methodrdquo Advanced Studies inTheoretical Physics vol 5 no 9-12 pp 469ndash476 2011

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

Advances in High Energy Physics 7

Disclosure

Hesham Mansour is a Fellow of the Institute of Physics(FInstP)

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] F Halzem and A Martin Quarks and Leptons an IntroductoryCourse in Modern Particle Physics Wiley Publication 1984

[2] D Griffiths Introduction to Elementary Particles John Wiley ampSons New York NY USA 1987

[3] D H Perkins Introduction to High Energy Physics CambridgeUniversity Press Cambridge UK 2000

[4] B R Martin and G G Shaw Particle Physics The ManchesterPhysics Series 2nd edition 1997

[5] A Bettini Introduction to Elementary Particle Physics Cam-bridge University Press 2018

[6] J Rathsman Quark and Lepton Interactions Acta UniversitatisUpsalensis Uppsala 1996

[7] N Mistry Brief Introduction of Particle Physics Cornell Univer-sity 2011

[8] A T Paul Ralph LlewellynModern PhysicsWHFreeman 2012[9] BH Bransden andC J JoachainQuantumMechanics Prentice

Hall 2000[10] L D Landau and E M Lifshitz Quantum Mechanics Butter-

worth Heinemann 1981[11] D Griffiths Introduction to Elementary Particles John Wiley amp

Sons New York NY USA 2008[12] N ZettiliQuantumMechanics Concepts and Applications John

Wiley 2009[13] C Berkdemir Theoretical Concept of Quantum Mechanics In

Tech 2012[14] A F Nikiforov andV B Uvarov Special Functions ofMathemat-

ical Physics Birkhaauser Basel Switzerland 1988[15] H Karayer D Demirhan and F Buyukkılıc ldquoExtension of

Nikiforov-Uvarov method for the solution of Heun equationrdquoJournal of Mathematical Physics vol 56 no 6 Article ID063504 14 pages 2015

[16] R Kumar and F Chand ldquoAsymptotic study to the N-dimensional radial schrodinger equation for the quark-antiquark systemrdquo Communications in Theoretical Physics vol59 no 5 pp 528ndash532 2013

[17] M Abu-Shady ldquoHeavy Quarkonia and Mesons in the Cor-nell Potential with Harmonic Oscillator Potential in the N-dimensional Schrodinger Equationrdquo International Journal ofApplied Mathematics and Theoretical Physics vol 2 no 2 p 162016

[18] A Al-Oun A Al-Jamel and H Widyan ldquoVarious Propertiesof Heavy Quarkonia from Flavor-Independent Coulomb PlusQuadratic Potentialrdquo Jordan Journal of Physics vol 8 no 4 pp199ndash203 2015

[19] H Ke G Wang X Li and C Chang ldquoThe magnetic dipoletransitions in the (119888119887) binding systemrdquo Science China PhysicsMechanics and Astronomy vol 53 no 11 pp 2025ndash2030 2010

[20] N V Masksimenko and S M Kuchin ldquoDetermination of themass spectrum of quarkonia by theNikiforovndashUvarovmethodrdquoRussian Physics Journal vol 54 no 57 2011

[21] E Eichten K Gottfried T Kinoshita J Kogut K D Laneand T M Yan ldquoSpectrum of charmed quark-antiquark boundstatesrdquo Physical Review Letters vol 34 no 6 pp 369ndash372 1975

[22] S M Kuchin and N V Maksimenko ldquoTheoretical Estimationsof the Spin ndash Averaged Mass Spectra of Heavy Quarkonia andBc Mesonsrdquo Universal Journal of Physics and Application vol 7pp 295ndash298 2013

[23] R Faccini BABAR-CONF-07035 SLAC-PUB-13080[24] R Kumar and F Chand ldquoEnergy Spectra of the Coulomb

Perturbed Potential inN-Dimensional Hilbert Spacerdquo Phys Scrvol 85 Article ID 055008 2012

[25] Y Li P Maris and J P Vary ldquoQuarkonium as a relativisticbound state on the light frontrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 96 no 1 2017

[26] J Beringer J F Arguin and R M Barnett ldquoReview of particlephysicsrdquo Physical Review D Particles Fields Gravitation andCosmology vol 86 Article ID 010001 2012

[27] B Hong T Barillari S Bethke S Kluth and S MenkeldquoOverview of quarkonium production in heavy-ion collisionsat LHCrdquo EPJ Web of Conferences vol 120 p 06002 2016

[28] Z Ghalenovi A A Rajabi and A Tavakolinezhad ldquoMasses ofCharm and Beauty Baryons in the Constituent Quark ModelrdquoInternational Journal of Modern Physics E vol 21 no 6 ArticleID 1250057 2012

[29] J Niu L Guo H Ma and S Wang ldquoHeavy quarkoniumproduction through the top quark rare decays via the channelsinvolving flavor changing neutral currentsrdquo The EuropeanPhysical Journal C vol 78 no 8 2018

[30] A K Ray and P C Vinodkumar ldquoProperties of B119888 mesonrdquoPramana vol 66 no 5 pp 953ndash958 2006

[31] S Laachir and A Laaribi ldquoExact Solutions of the Helmholtzequation via the Nikiforov-UvarovMethodrdquo International Jour-nal of Mathematical Computational Physical Electrical andComputer Engineering vol 7 no 1 2013

[32] HCiftci R LHall andN Saad ldquoIterative solutions to theDiracequationrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 72 no 2 part A 022101 7 pages 2005

[33] E J Eichten and C Quigg ldquoMesons with beauty and charmSpectroscopyrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 49 no 11 pp 5845ndash5856 1994

[34] M Alberg and L Wilets ldquoExact solutions to the Schrodingerequation for potentials with Coulomb and harmonic oscillatortermsrdquo Physics Letters A vol 286 no 1 pp 7ndash14 2001

[35] D Ebert R N Faustov and V O Galkin ldquoProperties of heavyquarkonia andrdquo Physical Review D Particles Fields Gravitationand Cosmology vol 67 no 1 2003

[36] B Ita P Tchoua E Siryabe and G E Ntamack ldquoSolutions ofthe Klein-Gordon Equation with the Hulthen Potential Usingthe Frobenius Methodrdquo International Journal of Theoretical andMathematical Physics vol 4 no 5 pp 173ndash177 2014

[37] H Goudarzi and V Vahidi ldquoSupersymmetric approach forEckart potential using the NU methodrdquo Advanced Studies inTheoretical Physics vol 5 no 9-12 pp 469ndash476 2011

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom