Research ArticleDouble Charmonium Productions in Electron-PositronAnnihilation Using Bethe-Salpeter Approach

Hluf Negash 1 and Shashank Bhatnagar 2

1Department of Physics Samara University PO Box 132 Samara Ethiopia2Department of Physics University Institute of Sciences Chandigarh University Mohali 140413 India

Correspondence should be addressed to Hluf Negash h2002hlufyahoocomand Shashank Bhatnagar shashank bhatnagaryahoocom

Received 21 May 2018 Accepted 26 June 2018 Published 3 March 2019

Academic Editor Andrzej Okninski

Copyright copy 2019 Hluf Negash and Shashank Bhatnagar 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

We calculate the double charmonium production cross-section within the framework of 4 times 4 Bethe-Salpeter Equation in theelectron-positron annihilation at center of mass energy radic119904 = 106GeV that proceeds through the exchange of a single virtualphoton In this calculation we make use of the full Dirac structure of 4D BS wave functions of these charmonia with theincorporation of all the Dirac covariants (both leading and subleading) The calculated cross-sections for the double charmoniumproductions for final states (119869Ψ 120578119888) (Ψ1015840 120578119888) (119869Ψ 1205781015840119888) and (Ψ1015840 1205781015840119888) are close to experimental data and in broad agreement withresults of other theoretical models

1 Introduction

One of the challenging problems in heavy-quark physics isthe process of double charmonium production in electron-positron annihilation at B-factories [1ndash4] Many studies ofdouble quarkonium production process [5ndash9] have beenpreformed in order to understand this process and therebythe internal structures of quarkonia and interactions of thequark and antiquark inside the quarkonium In recent yearsthe quarkonium production has been studied in variousprocess at B-factories whose measurements were were madeby Babar and Belle collaborations [1ndash4] Among them thestudy of charmonium production in 119890+119890minus annihilation isparticularly interesting in testing the quarkoniumproductionmechanisms at center of mass energies radic119904 = 106119866119890119881The investigation of double charmonium production is veryimportant since these charmonium can be easily producedin experiments and hence their theoretical prediction canverify the discrepancy between different theoretical modelsand experimental data As regards the dynamical frameworkto investigate the double charmoniumproduction concernedmany approaches have been proposed to deal with the cross-section of double charmonium production [5ndash9] and the

pseudoscalar and vector charmoniumproduction process hasbeen recently studied in a Bethe-Salpeter formalism [10 11]However in these studies the complete Dirac structure of P(pseudoscalar) and V (vector) quarkonia was not taken intoaccount and calculations were performed by taking only theleading Dirac structures 1205745 and 119894120574120576 in the BSwave functionsof pseudoscalar and vector charmonia respectively

In these calculations we make use of the Bethe-SalpeterEquation (BSE) approach [12ndash16] which is a conventionalapproach in dealing with relativistic bound state problemsDue to its firm base in quantum field theory and being adynamical equation based approach it provides a realisticdescription for analyzing hadrons as composite objects andcan be applied to study not only the low energy hadronicprocesses but also the high energy production processesinvolving quarkonia as well

In our recent works [17ndash20] we employed BSE underCovariant InstantaneousAnsatz which is a Lorentz-invariantgeneralization of Instantaneous Approximation to inves-tigate the mass spectra and the transition amplitudes forvarious processes involving charmonium and bottomoniumThe BSE framework using phenomenological potentials cangive consistent theoretical predictions as more and more data

HindawiAdvances in High Energy PhysicsVolume 2019 Article ID 4029356 6 pageshttpsdoiorg10115520194029356

2 Advances in High Energy Physics

are being accumulated In our studies on 4 times 4 BSE in allprocesses except in [19] the quarkndashantiquark loop involveda single hadron-quark vertex which was simple to handleHowever for the transitions such as 119881 997888rarr 119875 + 120574 (where119881 vector and 119875 pseudoscalar quarkonium) which we havestudied in [19] and for the process of double charmoniumproduction 119890+119890minus 997888rarr 119881 + 119875 we will study in present workthe process requires calculation involving two hadron-quarkvertices due to which the calculation becomes more andmore difficult to handle However in the present work wedemonstrate an explicitmathematical procedure for handlingthis problem using the formulation of 4 times 4 Bethe-SalpeterEquation under Covariant Instantaneous Ansatz We will usethis framework for the calculation of cross-section for theproduction of double charmonia (119869120595 120578119888) (119869120595 1205781015840119888) (1205951015840 120578119888)and (1205951015840 1205781015840119888) in electron-positron annihilation that proceedsthrough a single virtual photon at center of mass energiesradic119904 = 106 GeV where such problems do not enter ourprevious papers on 4 times 4 BSE [17ndash22]

This paper is organized as follows In Section 2 weintroduce the detailed formulation of the transition ampli-tudes for the process of double charmonium production andthe numerical results of cross-sections Finally we give thediscussions and conclusions in Section 3

2 Formulation of Double CharmoniumProduction Amplitude

We start from the lowest-order Feynman diagrams for theprocess 119890+119890minus 997888rarr 119881 + 119875 as given in Figure 1 Thereare four Feynman diagrams for the production of doublecharmonium one of which is shown in Figure 1 while theother three can be obtained by reversing the arrows of thequark lines

The relativistic amplitude 1198721119891119894 for double charmoniumproduction corresponding to Figure 1 is given by the one-loop momentum integral as

1198721119891119894 = 27120587212057211989011989812057211990432119904 [] (1199012) 120574120583119906 (1199011)] int 1198894119902119886(2120587)4 int

1198894119902119887(2120587)4

sdot 119879119903 [Ψ (119875119886 119902119886) 120574120573119878119865 (1199021) 120574120583Ψ(119875119887 119902119887) 120574120573] 11198962

(1)

where 119904 is theMendelstam variable defined as 119904 = minus(1199011+1199012)2120572119890119898 = 11989024120587 is called the electromagnetic coupling constant120572119904 = 1198922119904 4120587 is the strong coupling strength and Ψ(119875119886 119902119886)and Ψ(119875119887 119902119887) are the conjugations of the BS wave function ofvector and pseudoscalar charmonium respectively From thefigure we can relate the momenta of the quark and antiquarkrespectively as

11990210158401 = 12119875119886 + 119902119886

1199023 = 12119875119886 minus 119902119886

1199024 = 12119875119887 + 119902119887

1199022 = 12119875119887 minus 119902119887

(2)

and the momenta for the propagators of gluon and quark aregiven respectively by

119896 = 12 (119875119886 + 119875119887) minus 119902119886 + 119902119887

1199021 = 119875119886 + 12119875119887 + 119902119887

(3)

As the quark and gluonpropagators depend upon the internalhadron momenta 119902119886 and 119902119887 the calculation of amplitude isgoing to involve integrations over these internal momentaand will be quite complex Hence following [17ndash19] wesimplify the calculation by reducing the 4-dimensionalexpression of BS amplitude 1198721119891119894 into 3-dimensional expres-sion of BS amplitude 1198721119891119894 and employing the heavy-quarkapproximation on the propagators themomenta 119896 and 1199021 canbe written as 119896 asymp (12)(119875119886 +119875119887) and 1199021 asymp 119875119886 + (12)119875119887 whichleads to 1198962 asymp 1199044 and 11990221 asymp 1199042 + 1198982119888 [10 11] Then withthe above approximation and applying the definition of 3DBS wave function in [17ndash20]120595(119902119886) = (1198942120587) int119872119886119889120590Ψ(119875119886 119902119886)and 120595(119902119887) = (1198942120587) int119872119887119889120590119887Ψ(119875119887 119902119887) the double charmo-nium production BS amplitude 1198721119891119894 can be written in theinstantaneous Bethe-Salpeter amplitude form as

1198721119891119894 = minus2101205872120572119890119898120572119904321199043 [] (1199012) 120574120583119906 (1199011)]

sdot int 1198893119902119886(2120587)3 int

1198893119902119887(2120587)3

sdot 119879119903 [120595 (119902119886) 120574120573 (minus119894119875119886 minus 1198942119875119887 + 119898119888) 120574120583120595 (119902119887) 120574120573]

(4)

where120595(119902119886) and120595(119902119887) are the relativistic BS wave function ofpseudoscalar and vector charmonium respectively We nowgive details of calculation of double charmonium productionfor the process 119890+119890minus 997888rarr 119881 + 119875 in the next section

(i) For the production process 119890+ + 119890minus 997888rarr 119881 + 119875The relativistic BS wave function of pseudoscalar andvector charmonium is taken from our recent papers[17ndash19] respectively as

120595119875 (119902119887) = 119873119875 [119872119887 +119875119887 + 119902119887119875119887119898119888 ] 1205745120601119875 (119902119887) (5)

for 119875119887 and 119872119887 are the momentum and mass of thepseudoscalar charmonium respectively and119873119875 is theBS normalization of the pseudoscalar charmoniumwhich is given in a simple form as

119873119875 = [16119872119887119898119888 int 1198893119902119887(2120587)3

1199022119887120596119887 1206012119875 (119902119887)]

minus12

(6)

Advances in High Energy Physics 3

Figure 1 Leading-order Feynman diagrams for the production of double charmonium in 119890+119890minus-annihilation

and

120595119881 (119902119886)= 119873119881 [119872119886120576 + 119902119886120576119872119886119898119888 + 120576119875119886 +

119875119886119902119886120576119898119888 minus 119875119886120576119902119886119898119888 ]sdot 120601119881 (119902119886)

(7)

where 120576 is the polarization vector of the vectorquarkonia 119875119886 and 119872119886 are the momentum and massof the vector quarkonia respectively and119873119881 is the BSnormalization of the vector quarkonia which is givenin a simple form as

119873119881 = [16119898119888119872119886 int 1198893119902119886(2120587)3

11990221198861205963119886 1206012119881 (119902119886)]

minus12

(8)

where120601119875(119902119887) and 120601119881(119902119886) are the radial wave functionsof 119875 and 119881 which are solutions of the 3D BSE forpseudoscalar and vector quarkonia respectively (see[18]) The adjoint BS wave functions for pseudoscalarand vector charmonia are obtained from 120595119875119881(119902119887119886) =1205740(120595119875119881(119902119887119886))+1205740 With the substitution of adjoint BSwave functions of 119875 and 119881 charmoni into the 3Damplitude (4)1198721119891119894 becomes

1198721119891119894 = minus2101205872120572119890119898120572119904321199043 [] (1199012) 120574120583119906 (1199011)]

sdot int 1198893119902119886(2120587)3119873119881120601119881 (119902119886)int

1198893119902119887(2120587)3119873119875120601119875 (119902119887) [119879119877]

(9)

where

[119879119877] = 119879119903[(minus119872119886120576 + 119902119886120576119872119886119898119888 minus 120576119875119886 minus119875119886119902119886120576119898119888

+ 119902119886120576119875119886119898119888 )120574120573 (minus119894119875119886 minus 1198942119875119887 + 119898119888) 120574120583 (119872119887 +119875119887

+ 119902119887119875119887119898119888 )1205745120574120573] (10)

Applying the trace theorem and evaluating trace overthe gamma matrices one can obtain the expression

[119879119877] = minus8119872119886120598]120575120583120582120576]119875119886120582119875119887120575+ 8119894119898119888 [(119902119886120576) 120598120582120575120590120583119875119886120582119875119887120573119902119887120590

+ 120598120588]120582120601 (119875119887120601119902119887120583 minus 119875119887120583119902119887120601)]sdot 8 (119902119886120576)1198721198861198982119888 [120598120583120590120573120575119875119886120573119875119887120575119902119887120590 + 120598120582120573120583120590119875119886120582119875119887120573119902119887120590]

+ 161198722119887119898119888 120598]120575120590120583120576]119875119887120575119902119887120590 minus 8 (119902119886120576)119898119888 120598120582120575120590120583119875119886120582119875119887120575119902119887120590+ 8119894119872119887120598120588]120582120583120576]119875119886120582119902119886120588

(11)

After somemathematical steps we can get the follow-ing expression

[119879119877] = minus16119898119888120598120583]120588120590120576120583119875120588119886119875120590119887 (12)

Thus we can express the amplitude1198721119891119894 for the vectorand pseudoscalar charmonium production as

1198721119891119894 = 2141205872120572119890119898120572119904119898119888321199043 [] (1199012) 120574120583119906 (1199011)]

sdot 120598120583]120588120590120576120583119875120588119886 119875120590119887 int 1198893119902119886(2120587)3119873119881120601119881 (119902119886)

4 Advances in High Energy Physics

sdot int 1198893119902119887(2120587)3119873119875120601119875 (119902119887)

(13)

The full amplitude for the process 119890++119890minus 997888rarr 119881+119875 canbe obtained by summing over the amplitudes of all thepossibilities in Figure 1 Then the unpolarized totalcross-section is obtained by summing over various119881 + 119875 spin-states and averaging over those of theinitial state 119890+119890minus which is given as

120590 = 132120587

radic119904 minus 16119898211988811990432 int 1

4sum119904119901119894119899100381610038161003816100381611987211990511990011990511988611989710038161003816100381610038162 119889 cos 120579 (14)

The total amplitude |119872119905119900119905119886119897|2 is given as

14sum119904119901119894119899

100381610038161003816100381611987211990511990011990511988611989710038161003816100381610038162

= 2301205874120572211989011989812057221199041198982119888341199045 [int 1198893119902119886(2120587)3119873119881120601119881 (119902119886)]

2

sdot [int 1198893119902119887(2120587)3119873119875120601119875 (119902119887)]

2

(15)

After integration over the phase space the total cross-section for vector and pseudoscalar charmoniumproduction is

120590119890+119890minus997888rarr119881119875 = 2271205873120572211989011989812057221199041198982119888341199044 (1 minus 161198982119888119904 )32

sdot [int 1198893119902119886(2120587)3119873119881120601119881 (119902119886)]

2

sdot [int 1198893119902119887(2120587)3119873119875120601119875 (119902119887)]

2

(16)

The algebraic expressions of the wave functions ofpseudoscalar and vector charmonium 120601119875(119881)(119902119887(119886))for ground (1S) and first excited (2S) states respec-tively that are obtained as analytic solutions ofthe corresponding mass spectral equations of thesequarkonia in an approximate harmonic oscillatorbasis are [18]

120601119875(119881) (1119878 119902) = 11205873412057332119875(119881)

119890minus119902221205732119875(119881)

120601119875(119881) (2119878 119902) = radic3231205873412057372

119875(119881)

(31205732119875(119881) minus 21199022) 119890minus119902221205732119875(119881) (17)

where the inverse range parameter 120573119875for pseudoscalar charmonium is 120573119875 =(4(1198981198881205962119902119902radic1 + 21198600(119873 + 32)))14 while

120573119881 for vector charmonium is 120573119881 =(2(1198981198881205962119902119902radic1 + 21198600(119873 + 32)))14 and thesetwo constants depend on the input parameters andcontain the dynamical information and they differfrom each other due to spin-spin interactions It canbe checked that our cross-sectional formula is in(16) scales as 1205722119890119898120572211990411989861199044 with 119898 being the mass of119888-quark where in (16) the wave functions 120601(119875119881)involve the inverse range parameters 120573119875119881 sim 11989812

Numerical Results We had calculated recently the massspectrum and various decays of ground and excited statesof pseudoscalar and vector quarkonia in [18ndash20] The sameinput parameters are employed in this calculation as in ourrecent works given in Table 1

Using these input parameters listed in Table 1 we calcu-late the cross-sections of pseudoscalar and vector charmo-nium production in our framework listed in Table 2

We wish to mention here that the 119887119887119887119887 as well as 119887119887119888119888production has not been observed so far but cross-sectionshave been predicted for 119890minus119890+ 997888rarr Υ + 120578119887 at center of massenergy radic119904 = 25 minus 30 GeV We thus wished to check ourcalculations for double bottomonium (Υ 120578119887) production inelectron-positron annihilation for sake of completeness Thesame can be studied with our framework used in this workwith little modifications Taking the mass of the 119887minusquark thatis fixed from our recent work on spectroscopy of 119887119887 states as119898119887 = 507GeV [18] and other input parameters the same thecross-section for production of ground states of 120578119887 and Υ forenergy range (radic119904 = 28 minus 30)GeV is given in Table 3

3 Discussions and Conclusion

Our main aim in this paper was to study the cross-sectionfor double charmonium production in electron-positroncollisions at center of mass energies radic119904 = 106119866119890119881 havingsuccessfully studied the mass spectrum and a range of lowenergy processes using an analytic treatment of 4 times 4 Bethe-Salpeter Equation (BSE) [18ndash22] This is due to the fact anyquark model model should successfully describe a rangeof processesmdashnot only the mass spectra and low energyhadronic decay constantsdecay widths but also the highenergy production processes involving these hadrons and allwithin a common dynamical framework and with a singleset of input parameters that are calibrated to the mass spec-trum

Further the exclusive production of double heavy char-monia in 119890+119890minus annihilation has been a challenge to under-stand in heavy-quark physics and has received considerableattention in recent years due to the fact that there is asignificant discrepancy in the data of Babar [2] and Belle[1] and the calculations performed in NRQCD [5 6] ofthis process at radic119904 = 106119866119890119881 where using leadingorder (LO) QCD diagrams alone leads to cross-sectionsthat are an order of magnitude smaller than data [1 24] These discrepancies were then resolved by taking intoaccount the Next-to-Leading-Order (NLO)QCD correctionscombined with relativistic corrections [23 24]) though the

Advances in High Energy Physics 5

Table 1 Input parameters for this study

1198620 1205960(119866119890119881) Λ(119866119890119881) 1198600 119898119888(119866119890119881)0210 0150 0200 0010 1490

Table 2 The total cross-sections of pseudoscalar and vector charmonium production for ground and first excited states with experimentaldata and other theoretical models (in units of fb)

Production Process 120590 (Our result) 120590 [1] 120590 [2] 120590 [8] 120590 [9] 120590 [10] 120590 [11]119890+119890minus 997888rarr 119869120595120578119888 20770 256plusmn28 176plusmn28 267 222plusmn42 223 2175119890+119890minus 997888rarr 1205951015840120578119888 11643 163plusmn46 163 153plusmn29119890+119890minus 997888rarr 1198691205951205781015840119888 11177 165plusmn30 164plusmn37 266 164plusmn31119890+119890minus 997888rarr 12059510158401205781015840119888 6266 160plusmn51 145 96plusmn18Table 3 The total cross-sections of pseudoscalar and vector bottomonium production for their ground states with predictions of othertheoretical models (in units of fb)

Production Process 120590(Our result) 120590[1] 120590[2] 120590[10] for (radic119904 = 25 minus 30)GeV120590[119890+119890minus 997888rarr Υ120578119887] 0155-0058 016-006

NLO contributions were larger than the LO contributions[10]

This process has also been studied in Relativistic QuarkModel [7 9] Light Cone formalism [8] and Bethe-SalpeterEquation [10 11] Relativistic corrections to cross-section inLight Cone formalism (in [8]) were considered to eliminatethe discrepancy between theory and experiment Furtherin an attempt to explain a part of this discrepancy [25]even suggested that processes proceeding through two virtualphotons may be important In Relativistic Quark Model [7]which incorporated relativistic treatment of internal motionof quarks and bound states improvements in the results wereobtained

Wewish tomention that in our framework of 4times4Bethe-Salpeter Equation (BSE) using the Covariant InstantaneousAnsatz where we treat the internal motion of quarks andthe bound states in a relativistically covariant manner weobtained results on cross-sections (in Table 2) for produc-tion of opposite parity 119888119888 states such as (119869Ψ 120578119888) (Ψ1015840 120578119888)(119869Ψ 1205781015840119888) (Ψ1015840 1205781015840119888) which are close to central values of data[1 2 4] using leading-order (LO) QCD diagrams alone Thisvalidates the fact that relativistic quark models such as BSEare strong candidates for treating not only the low energyprocesses but also the high energy production processesinvolving double heavy quarkonia due to their consistentrelativistic treatment of internal motion of quarks in thehadrons where our BS wave functions that take into accountall the Dirac structures in pseudoscalar and vector mesonsin a mathematically consistent manner play a vital role inthe dynamics of the process We also give our predictions oncross-section for double 119887119887 production at energies 28- 30GeVin Table 3 for future experiments at colliders

Themain objective of this study was to test the validationof our approach which provides a much deeper insight thanthe purely numerical calculations in 4 times 4 BSE that are verycommon in the literature We wish to mention that we havenot encountered any work in 4 times 4 representation of BSE

that treats this problem analytically On the contrary allthe other 4 times 4 BSE approaches adopt a purely numericalapproach of solving the BSE We are also not aware of anyother BSE framework involving 4times4 BS amplitude and withall the Dirac structures incorporated in the 4D hadronic BSwave functions (in fact many works used only the leadingDirac structures for instance see [10 11]) for calculations ofthis production process We treat this problem analyticallyby making use of the algebraic forms of wave functions120601(119875119881) derived analytically from mass spectral equations [18]for calculation of cross-section of this double charmoniumproduction process

This calculation involving production of double charmo-nia in electron-positron annihilation can be easily extendedto studies on other processes (involving the exchange of asingle virtual photon) observed at B-factories such as 119890minus 119890+minus gtΨ(2119878)120574 119890minus119890+minus gt 119869Ψ1205941198880 and 119890minus119890+minus gt Ψ(2119878)1205941198880 We furtherwish to extend this study to processes involving two virtualphotons such as the production of double charmonia in finalstates with 119862 = +1 such as 119890minus119890+ 997888rarr 119869Ψ119869Ψ) recentlyobserved at BABAR

These processes that we intend to study next are quiteinvolved and the dynamical equation based approachessuch as BSE are a promising approach And since theseprocesses involve quark-triangle diagrams we make use ofthe techniques we used for handling such diagrams recentlydone in [19 26 27] Such analytic approaches not only leadto better insight into the mass spectra and low energy decayprocesses involving charmonia but also their high energyproduction processes such as the one studied here

Note Added in Proofs We have just now come to know aboutthe recent experimental observation by CMS collaboration[28] of two excited B+c (2S) and B

lowast+c (2S) states in pp collisions

at radic119904 = 13 TeV at Large Hadron Collider (LHC) Thetheoretical study of this process will be a challenge tohadronic physics

6 Advances in High Energy Physics

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was carried out at Samara University Ethiopiaand at Chandigarh University IndiaThe authors thank theseinstitutions for the facilities provided during the course of thiswork

References

[1] K-F Chen W-S Hou I Adachi et al ldquoObservation ofan enhancement in e+eminus 997888rarr Υ(1S)120587+120587minus Υ(2S)120587+120587minus andΥ(3S)120587+120587minus production near radic119904 = 106 Gevrdquo Physical Reviewvol 82 Article ID 091106 2010 httpsarxivorgabs08103829

[2] B AubertM Bona Y Karyotakis et al ldquoEvidence for the 120578119887(1119878)meson in radiative Υ(2119878) decayrdquo Physical Review Letters vol103 Article ID 161801 2009

[3] K W Edwards R Janicek P M Patel et al ldquoStudy of B Decaysto Charmonium States 119861 997888rarr 120578119888119870 and 119861 997888rarr 120578119887119870rdquo PhysicalReview Letters vol 86 no 30 2001

[4] K Olive K Agashe andC Amsler ldquoReview of particle physicsrdquoChinese Physics C vol 38 no 9 Article ID 090001 2014

[5] G T Bodwinin E Braaten J Lee and C Yu ldquoExclusive two-vector-meson production from e+eminus annihilationrdquo PhysicalReview D vol 74 no 7 Article ID 074014 2006

[6] G T Bodwin D Kang T Kim J Lee and C Yu ldquoRelativisticCorrections to e+eminus 997888rarr J120595 + 120578c in a Potential Modelrdquo inProceedings of the AIP Conference vol 892 2007

[7] D Ebert and A P Martynenko ldquoRelativistic effects in the pro-duction of pseudoscalar and vector doubly heavy mesons frome+eminus annihilationrdquo Physical Review D Article ID 054008 2006

[8] V V Braguta A K Likhoded and A V Luchinsky ldquoExcitedcharmonium mesons production in e+eminus annihilation atradics=106 GeVrdquo Physical Review D vol 72 Article ID 0740192005

[9] D Ebert R N Faustov and V O Galkin ldquoRelativistic model ofhidden bottom tetraquarksrdquo Modern Physics Letters A vol 24no 08 pp 567ndash573 2009

[10] X-H Guo H-W Ke X-Q Li et al ldquoDouble heavy-quarko-nium production from electron-positron annihilation in theBethe-Salpeter formalismrdquo httpsarxivorgabs08040949

[11] E Mengesha and S Bhatnagar ldquoCross section for doublecharmonium production in electron-positron annihilation atenergyradic119904 = 106Gevrdquo International Journal of Modern Physicsvol E20 pp 2521ndash2533 2011

[12] A N Mitra and B M Sodermark ldquoA dynamical principle for3Dndash4D interlinkage in Salpeter-like equationsrdquo Nuclear PhysicsA vol 695 pp 328ndash352 2001

[13] A NMitra and S Bhatnagar ldquoA hadron-quark vertex functioninterconnection between 3D and 4D wave functionsrdquo Interna-tional Journal of Modern Physics A vol 7 p 121 1992

[14] A N Mitra ldquoQCD motivated BSE SDE framework for quarkdynamics underMarkov-Yukawa transversality AUnified viewof q anti-q and q q q systems 1rdquo in Proceedings of the IndianNational Science Academy vol 65 pp 527ndash584 1999

[15] S Bhatnagar D S Kulshreshtha and A N Mitra ldquoHow big arethe decay constants f 119875 of heavy-light mesonsrdquo Physics LettersB vol 263 no 3-4 pp 485ndash490 1991

[16] S Bhatnagar J Mahecha and Y Mengesha ldquoRelevance ofvarious Dirac covariants in hadronic Bethe-Salpeter wave func-tions in electromagnetic decays of ground state vector mesonsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 90 Article ID 014034 2014

[17] H Negash and S Bhatnagar ldquoMass spectrumand leptonic decayconstants of ground and radially excited states of 120578119888 and 120578119887 ina BethendashSalpeter equation frameworkrdquo International Journal ofModern Physics E vol 24 no 4 Article ID 1550030 2015

[18] H Negash and S Bhatnagar ldquoSpectroscopy of ground andexcited states of pseudoscalar and vector charmonium and bot-tomoniumrdquo International Journal of Modern Physics E vol 25no 8 Article ID 1650059 2016

[19] H Negash and S Bhatnagar ldquoRadiative decay widths of groundand excited states of vector charmonium and bottomoniumrdquohttpsarxivorgabs170306082

[20] H Negash and S Bhatnagar ldquoAnalytical calculations of groundand excited states of unequal mass heavy pseudoscalar andvector mesons mass spectra using Bethe-Salpeter formalismrdquohttpsarxivorgabs171107036

[21] E Gebrehana S Bhatnagar and H Negash ldquoAnalytic approachto calculations of mass spectra and decay constants of heavy-light quarkonia in the framework of Bethe-Salpeter equationrdquohttpsarxivorgabs190101888

[22] S Bhatnagar and L Alemu ldquoApproach to calculation of massspectra and two-photon decays of cc mesons in the frameworkof Bethe-Salpeter equationrdquo Physical Review D vol 97 ArticleID 034021 2018

[23] E Braaten and J Lee ldquoExclusive double-charmonium produc-tion from e+eminus annihilation into a virtual photonrdquo PhysicalReview D vol 67 Article ID 054007 2003

[24] Y J Zhang Y J Gao and K T Chao ldquoNext-to-leading-orderQCD correction to e+eminus 997888rarrJ120595+120578c at radics=106 GeVrdquo PhysicalReview Letters vol 96 Article ID 092001 2008

[25] G T Bodwin J Lee and E Braaten ldquoe+eminus annihilation intoJ120595+J120595rdquo Physical Review Letters vol 90 Article ID 1620012005

[26] E Mengesha and S Bhatnagar ldquoRadiative decays of equalmass vector mesons in a Bethe-Salpeter equation frameworkrdquoInternational Journal of Modern Physics E vol 21 Article ID1250084 2012

[27] E Mengesha and S Bhatnagar ldquoStrong decays of first radialexcitations of lightest vectormeson statesrdquo International JournalofModern Physics E vol 22 no 7 Article ID 1350046 p 15 2013

[28] A M Sirunyan et al ldquoObservation of two excited B+c states andmeasurement of the B+c (2S) mass in pp collisions at radic119904 = 13TeVrdquo httpsarxivorgabs190200571 2019

Advances in High Energy Physics 3

Figure 1 Leading-order Feynman diagrams for the production of double charmonium in 119890+119890minus-annihilation

and

120595119881 (119902119886)= 119873119881 [119872119886120576 + 119902119886120576119872119886119898119888 + 120576119875119886 +

119875119886119902119886120576119898119888 minus 119875119886120576119902119886119898119888 ]sdot 120601119881 (119902119886)

(7)

where 120576 is the polarization vector of the vectorquarkonia 119875119886 and 119872119886 are the momentum and massof the vector quarkonia respectively and119873119881 is the BSnormalization of the vector quarkonia which is givenin a simple form as

119873119881 = [16119898119888119872119886 int 1198893119902119886(2120587)3

11990221198861205963119886 1206012119881 (119902119886)]

minus12

(8)

where120601119875(119902119887) and 120601119881(119902119886) are the radial wave functionsof 119875 and 119881 which are solutions of the 3D BSE forpseudoscalar and vector quarkonia respectively (see[18]) The adjoint BS wave functions for pseudoscalarand vector charmonia are obtained from 120595119875119881(119902119887119886) =1205740(120595119875119881(119902119887119886))+1205740 With the substitution of adjoint BSwave functions of 119875 and 119881 charmoni into the 3Damplitude (4)1198721119891119894 becomes

1198721119891119894 = minus2101205872120572119890119898120572119904321199043 [] (1199012) 120574120583119906 (1199011)]

sdot int 1198893119902119886(2120587)3119873119881120601119881 (119902119886)int

1198893119902119887(2120587)3119873119875120601119875 (119902119887) [119879119877]

(9)

where

[119879119877] = 119879119903[(minus119872119886120576 + 119902119886120576119872119886119898119888 minus 120576119875119886 minus119875119886119902119886120576119898119888

+ 119902119886120576119875119886119898119888 )120574120573 (minus119894119875119886 minus 1198942119875119887 + 119898119888) 120574120583 (119872119887 +119875119887

+ 119902119887119875119887119898119888 )1205745120574120573] (10)

Applying the trace theorem and evaluating trace overthe gamma matrices one can obtain the expression

[119879119877] = minus8119872119886120598]120575120583120582120576]119875119886120582119875119887120575+ 8119894119898119888 [(119902119886120576) 120598120582120575120590120583119875119886120582119875119887120573119902119887120590

+ 120598120588]120582120601 (119875119887120601119902119887120583 minus 119875119887120583119902119887120601)]sdot 8 (119902119886120576)1198721198861198982119888 [120598120583120590120573120575119875119886120573119875119887120575119902119887120590 + 120598120582120573120583120590119875119886120582119875119887120573119902119887120590]

+ 161198722119887119898119888 120598]120575120590120583120576]119875119887120575119902119887120590 minus 8 (119902119886120576)119898119888 120598120582120575120590120583119875119886120582119875119887120575119902119887120590+ 8119894119872119887120598120588]120582120583120576]119875119886120582119902119886120588

(11)

After somemathematical steps we can get the follow-ing expression

[119879119877] = minus16119898119888120598120583]120588120590120576120583119875120588119886119875120590119887 (12)

Thus we can express the amplitude1198721119891119894 for the vectorand pseudoscalar charmonium production as

1198721119891119894 = 2141205872120572119890119898120572119904119898119888321199043 [] (1199012) 120574120583119906 (1199011)]

sdot 120598120583]120588120590120576120583119875120588119886 119875120590119887 int 1198893119902119886(2120587)3119873119881120601119881 (119902119886)

4 Advances in High Energy Physics

sdot int 1198893119902119887(2120587)3119873119875120601119875 (119902119887)

(13)

The full amplitude for the process 119890++119890minus 997888rarr 119881+119875 canbe obtained by summing over the amplitudes of all thepossibilities in Figure 1 Then the unpolarized totalcross-section is obtained by summing over various119881 + 119875 spin-states and averaging over those of theinitial state 119890+119890minus which is given as

120590 = 132120587

radic119904 minus 16119898211988811990432 int 1

4sum119904119901119894119899100381610038161003816100381611987211990511990011990511988611989710038161003816100381610038162 119889 cos 120579 (14)

The total amplitude |119872119905119900119905119886119897|2 is given as

14sum119904119901119894119899

100381610038161003816100381611987211990511990011990511988611989710038161003816100381610038162

= 2301205874120572211989011989812057221199041198982119888341199045 [int 1198893119902119886(2120587)3119873119881120601119881 (119902119886)]

2

sdot [int 1198893119902119887(2120587)3119873119875120601119875 (119902119887)]

2

(15)

After integration over the phase space the total cross-section for vector and pseudoscalar charmoniumproduction is

120590119890+119890minus997888rarr119881119875 = 2271205873120572211989011989812057221199041198982119888341199044 (1 minus 161198982119888119904 )32

sdot [int 1198893119902119886(2120587)3119873119881120601119881 (119902119886)]

2

sdot [int 1198893119902119887(2120587)3119873119875120601119875 (119902119887)]

2

(16)

The algebraic expressions of the wave functions ofpseudoscalar and vector charmonium 120601119875(119881)(119902119887(119886))for ground (1S) and first excited (2S) states respec-tively that are obtained as analytic solutions ofthe corresponding mass spectral equations of thesequarkonia in an approximate harmonic oscillatorbasis are [18]

120601119875(119881) (1119878 119902) = 11205873412057332119875(119881)

119890minus119902221205732119875(119881)

120601119875(119881) (2119878 119902) = radic3231205873412057372

119875(119881)

(31205732119875(119881) minus 21199022) 119890minus119902221205732119875(119881) (17)

where the inverse range parameter 120573119875for pseudoscalar charmonium is 120573119875 =(4(1198981198881205962119902119902radic1 + 21198600(119873 + 32)))14 while

120573119881 for vector charmonium is 120573119881 =(2(1198981198881205962119902119902radic1 + 21198600(119873 + 32)))14 and thesetwo constants depend on the input parameters andcontain the dynamical information and they differfrom each other due to spin-spin interactions It canbe checked that our cross-sectional formula is in(16) scales as 1205722119890119898120572211990411989861199044 with 119898 being the mass of119888-quark where in (16) the wave functions 120601(119875119881)involve the inverse range parameters 120573119875119881 sim 11989812

Numerical Results We had calculated recently the massspectrum and various decays of ground and excited statesof pseudoscalar and vector quarkonia in [18ndash20] The sameinput parameters are employed in this calculation as in ourrecent works given in Table 1

Using these input parameters listed in Table 1 we calcu-late the cross-sections of pseudoscalar and vector charmo-nium production in our framework listed in Table 2

We wish to mention here that the 119887119887119887119887 as well as 119887119887119888119888production has not been observed so far but cross-sectionshave been predicted for 119890minus119890+ 997888rarr Υ + 120578119887 at center of massenergy radic119904 = 25 minus 30 GeV We thus wished to check ourcalculations for double bottomonium (Υ 120578119887) production inelectron-positron annihilation for sake of completeness Thesame can be studied with our framework used in this workwith little modifications Taking the mass of the 119887minusquark thatis fixed from our recent work on spectroscopy of 119887119887 states as119898119887 = 507GeV [18] and other input parameters the same thecross-section for production of ground states of 120578119887 and Υ forenergy range (radic119904 = 28 minus 30)GeV is given in Table 3

3 Discussions and Conclusion

Our main aim in this paper was to study the cross-sectionfor double charmonium production in electron-positroncollisions at center of mass energies radic119904 = 106119866119890119881 havingsuccessfully studied the mass spectrum and a range of lowenergy processes using an analytic treatment of 4 times 4 Bethe-Salpeter Equation (BSE) [18ndash22] This is due to the fact anyquark model model should successfully describe a rangeof processesmdashnot only the mass spectra and low energyhadronic decay constantsdecay widths but also the highenergy production processes involving these hadrons and allwithin a common dynamical framework and with a singleset of input parameters that are calibrated to the mass spec-trum

Further the exclusive production of double heavy char-monia in 119890+119890minus annihilation has been a challenge to under-stand in heavy-quark physics and has received considerableattention in recent years due to the fact that there is asignificant discrepancy in the data of Babar [2] and Belle[1] and the calculations performed in NRQCD [5 6] ofthis process at radic119904 = 106119866119890119881 where using leadingorder (LO) QCD diagrams alone leads to cross-sectionsthat are an order of magnitude smaller than data [1 24] These discrepancies were then resolved by taking intoaccount the Next-to-Leading-Order (NLO)QCD correctionscombined with relativistic corrections [23 24]) though the

Advances in High Energy Physics 5

Table 1 Input parameters for this study

1198620 1205960(119866119890119881) Λ(119866119890119881) 1198600 119898119888(119866119890119881)0210 0150 0200 0010 1490

Table 2 The total cross-sections of pseudoscalar and vector charmonium production for ground and first excited states with experimentaldata and other theoretical models (in units of fb)

Production Process 120590 (Our result) 120590 [1] 120590 [2] 120590 [8] 120590 [9] 120590 [10] 120590 [11]119890+119890minus 997888rarr 119869120595120578119888 20770 256plusmn28 176plusmn28 267 222plusmn42 223 2175119890+119890minus 997888rarr 1205951015840120578119888 11643 163plusmn46 163 153plusmn29119890+119890minus 997888rarr 1198691205951205781015840119888 11177 165plusmn30 164plusmn37 266 164plusmn31119890+119890minus 997888rarr 12059510158401205781015840119888 6266 160plusmn51 145 96plusmn18Table 3 The total cross-sections of pseudoscalar and vector bottomonium production for their ground states with predictions of othertheoretical models (in units of fb)

Production Process 120590(Our result) 120590[1] 120590[2] 120590[10] for (radic119904 = 25 minus 30)GeV120590[119890+119890minus 997888rarr Υ120578119887] 0155-0058 016-006

NLO contributions were larger than the LO contributions[10]

This process has also been studied in Relativistic QuarkModel [7 9] Light Cone formalism [8] and Bethe-SalpeterEquation [10 11] Relativistic corrections to cross-section inLight Cone formalism (in [8]) were considered to eliminatethe discrepancy between theory and experiment Furtherin an attempt to explain a part of this discrepancy [25]even suggested that processes proceeding through two virtualphotons may be important In Relativistic Quark Model [7]which incorporated relativistic treatment of internal motionof quarks and bound states improvements in the results wereobtained

Wewish tomention that in our framework of 4times4Bethe-Salpeter Equation (BSE) using the Covariant InstantaneousAnsatz where we treat the internal motion of quarks andthe bound states in a relativistically covariant manner weobtained results on cross-sections (in Table 2) for produc-tion of opposite parity 119888119888 states such as (119869Ψ 120578119888) (Ψ1015840 120578119888)(119869Ψ 1205781015840119888) (Ψ1015840 1205781015840119888) which are close to central values of data[1 2 4] using leading-order (LO) QCD diagrams alone Thisvalidates the fact that relativistic quark models such as BSEare strong candidates for treating not only the low energyprocesses but also the high energy production processesinvolving double heavy quarkonia due to their consistentrelativistic treatment of internal motion of quarks in thehadrons where our BS wave functions that take into accountall the Dirac structures in pseudoscalar and vector mesonsin a mathematically consistent manner play a vital role inthe dynamics of the process We also give our predictions oncross-section for double 119887119887 production at energies 28- 30GeVin Table 3 for future experiments at colliders

Themain objective of this study was to test the validationof our approach which provides a much deeper insight thanthe purely numerical calculations in 4 times 4 BSE that are verycommon in the literature We wish to mention that we havenot encountered any work in 4 times 4 representation of BSE

that treats this problem analytically On the contrary allthe other 4 times 4 BSE approaches adopt a purely numericalapproach of solving the BSE We are also not aware of anyother BSE framework involving 4times4 BS amplitude and withall the Dirac structures incorporated in the 4D hadronic BSwave functions (in fact many works used only the leadingDirac structures for instance see [10 11]) for calculations ofthis production process We treat this problem analyticallyby making use of the algebraic forms of wave functions120601(119875119881) derived analytically from mass spectral equations [18]for calculation of cross-section of this double charmoniumproduction process

This calculation involving production of double charmo-nia in electron-positron annihilation can be easily extendedto studies on other processes (involving the exchange of asingle virtual photon) observed at B-factories such as 119890minus 119890+minus gtΨ(2119878)120574 119890minus119890+minus gt 119869Ψ1205941198880 and 119890minus119890+minus gt Ψ(2119878)1205941198880 We furtherwish to extend this study to processes involving two virtualphotons such as the production of double charmonia in finalstates with 119862 = +1 such as 119890minus119890+ 997888rarr 119869Ψ119869Ψ) recentlyobserved at BABAR

These processes that we intend to study next are quiteinvolved and the dynamical equation based approachessuch as BSE are a promising approach And since theseprocesses involve quark-triangle diagrams we make use ofthe techniques we used for handling such diagrams recentlydone in [19 26 27] Such analytic approaches not only leadto better insight into the mass spectra and low energy decayprocesses involving charmonia but also their high energyproduction processes such as the one studied here

Note Added in Proofs We have just now come to know aboutthe recent experimental observation by CMS collaboration[28] of two excited B+c (2S) and B

lowast+c (2S) states in pp collisions

at radic119904 = 13 TeV at Large Hadron Collider (LHC) Thetheoretical study of this process will be a challenge tohadronic physics

6 Advances in High Energy Physics

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was carried out at Samara University Ethiopiaand at Chandigarh University IndiaThe authors thank theseinstitutions for the facilities provided during the course of thiswork

References

[1] K-F Chen W-S Hou I Adachi et al ldquoObservation ofan enhancement in e+eminus 997888rarr Υ(1S)120587+120587minus Υ(2S)120587+120587minus andΥ(3S)120587+120587minus production near radic119904 = 106 Gevrdquo Physical Reviewvol 82 Article ID 091106 2010 httpsarxivorgabs08103829

[2] B AubertM Bona Y Karyotakis et al ldquoEvidence for the 120578119887(1119878)meson in radiative Υ(2119878) decayrdquo Physical Review Letters vol103 Article ID 161801 2009

[3] K W Edwards R Janicek P M Patel et al ldquoStudy of B Decaysto Charmonium States 119861 997888rarr 120578119888119870 and 119861 997888rarr 120578119887119870rdquo PhysicalReview Letters vol 86 no 30 2001

[4] K Olive K Agashe andC Amsler ldquoReview of particle physicsrdquoChinese Physics C vol 38 no 9 Article ID 090001 2014

[5] G T Bodwinin E Braaten J Lee and C Yu ldquoExclusive two-vector-meson production from e+eminus annihilationrdquo PhysicalReview D vol 74 no 7 Article ID 074014 2006

[6] G T Bodwin D Kang T Kim J Lee and C Yu ldquoRelativisticCorrections to e+eminus 997888rarr J120595 + 120578c in a Potential Modelrdquo inProceedings of the AIP Conference vol 892 2007

[7] D Ebert and A P Martynenko ldquoRelativistic effects in the pro-duction of pseudoscalar and vector doubly heavy mesons frome+eminus annihilationrdquo Physical Review D Article ID 054008 2006

[8] V V Braguta A K Likhoded and A V Luchinsky ldquoExcitedcharmonium mesons production in e+eminus annihilation atradics=106 GeVrdquo Physical Review D vol 72 Article ID 0740192005

[9] D Ebert R N Faustov and V O Galkin ldquoRelativistic model ofhidden bottom tetraquarksrdquo Modern Physics Letters A vol 24no 08 pp 567ndash573 2009

[10] X-H Guo H-W Ke X-Q Li et al ldquoDouble heavy-quarko-nium production from electron-positron annihilation in theBethe-Salpeter formalismrdquo httpsarxivorgabs08040949

[11] E Mengesha and S Bhatnagar ldquoCross section for doublecharmonium production in electron-positron annihilation atenergyradic119904 = 106Gevrdquo International Journal of Modern Physicsvol E20 pp 2521ndash2533 2011

[12] A N Mitra and B M Sodermark ldquoA dynamical principle for3Dndash4D interlinkage in Salpeter-like equationsrdquo Nuclear PhysicsA vol 695 pp 328ndash352 2001

[13] A NMitra and S Bhatnagar ldquoA hadron-quark vertex functioninterconnection between 3D and 4D wave functionsrdquo Interna-tional Journal of Modern Physics A vol 7 p 121 1992

[14] A N Mitra ldquoQCD motivated BSE SDE framework for quarkdynamics underMarkov-Yukawa transversality AUnified viewof q anti-q and q q q systems 1rdquo in Proceedings of the IndianNational Science Academy vol 65 pp 527ndash584 1999

[15] S Bhatnagar D S Kulshreshtha and A N Mitra ldquoHow big arethe decay constants f 119875 of heavy-light mesonsrdquo Physics LettersB vol 263 no 3-4 pp 485ndash490 1991

[16] S Bhatnagar J Mahecha and Y Mengesha ldquoRelevance ofvarious Dirac covariants in hadronic Bethe-Salpeter wave func-tions in electromagnetic decays of ground state vector mesonsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 90 Article ID 014034 2014

[17] H Negash and S Bhatnagar ldquoMass spectrumand leptonic decayconstants of ground and radially excited states of 120578119888 and 120578119887 ina BethendashSalpeter equation frameworkrdquo International Journal ofModern Physics E vol 24 no 4 Article ID 1550030 2015

[18] H Negash and S Bhatnagar ldquoSpectroscopy of ground andexcited states of pseudoscalar and vector charmonium and bot-tomoniumrdquo International Journal of Modern Physics E vol 25no 8 Article ID 1650059 2016

[19] H Negash and S Bhatnagar ldquoRadiative decay widths of groundand excited states of vector charmonium and bottomoniumrdquohttpsarxivorgabs170306082

[20] H Negash and S Bhatnagar ldquoAnalytical calculations of groundand excited states of unequal mass heavy pseudoscalar andvector mesons mass spectra using Bethe-Salpeter formalismrdquohttpsarxivorgabs171107036

[21] E Gebrehana S Bhatnagar and H Negash ldquoAnalytic approachto calculations of mass spectra and decay constants of heavy-light quarkonia in the framework of Bethe-Salpeter equationrdquohttpsarxivorgabs190101888

[22] S Bhatnagar and L Alemu ldquoApproach to calculation of massspectra and two-photon decays of cc mesons in the frameworkof Bethe-Salpeter equationrdquo Physical Review D vol 97 ArticleID 034021 2018

[23] E Braaten and J Lee ldquoExclusive double-charmonium produc-tion from e+eminus annihilation into a virtual photonrdquo PhysicalReview D vol 67 Article ID 054007 2003

[24] Y J Zhang Y J Gao and K T Chao ldquoNext-to-leading-orderQCD correction to e+eminus 997888rarrJ120595+120578c at radics=106 GeVrdquo PhysicalReview Letters vol 96 Article ID 092001 2008

[25] G T Bodwin J Lee and E Braaten ldquoe+eminus annihilation intoJ120595+J120595rdquo Physical Review Letters vol 90 Article ID 1620012005

[26] E Mengesha and S Bhatnagar ldquoRadiative decays of equalmass vector mesons in a Bethe-Salpeter equation frameworkrdquoInternational Journal of Modern Physics E vol 21 Article ID1250084 2012

[27] E Mengesha and S Bhatnagar ldquoStrong decays of first radialexcitations of lightest vectormeson statesrdquo International JournalofModern Physics E vol 22 no 7 Article ID 1350046 p 15 2013

[28] A M Sirunyan et al ldquoObservation of two excited B+c states andmeasurement of the B+c (2S) mass in pp collisions at radic119904 = 13TeVrdquo httpsarxivorgabs190200571 2019

4 Advances in High Energy Physics

sdot int 1198893119902119887(2120587)3119873119875120601119875 (119902119887)

(13)

The full amplitude for the process 119890++119890minus 997888rarr 119881+119875 canbe obtained by summing over the amplitudes of all thepossibilities in Figure 1 Then the unpolarized totalcross-section is obtained by summing over various119881 + 119875 spin-states and averaging over those of theinitial state 119890+119890minus which is given as

120590 = 132120587

radic119904 minus 16119898211988811990432 int 1

4sum119904119901119894119899100381610038161003816100381611987211990511990011990511988611989710038161003816100381610038162 119889 cos 120579 (14)

The total amplitude |119872119905119900119905119886119897|2 is given as

14sum119904119901119894119899

100381610038161003816100381611987211990511990011990511988611989710038161003816100381610038162

= 2301205874120572211989011989812057221199041198982119888341199045 [int 1198893119902119886(2120587)3119873119881120601119881 (119902119886)]

2

sdot [int 1198893119902119887(2120587)3119873119875120601119875 (119902119887)]

2

(15)

After integration over the phase space the total cross-section for vector and pseudoscalar charmoniumproduction is

120590119890+119890minus997888rarr119881119875 = 2271205873120572211989011989812057221199041198982119888341199044 (1 minus 161198982119888119904 )32

sdot [int 1198893119902119886(2120587)3119873119881120601119881 (119902119886)]

2

sdot [int 1198893119902119887(2120587)3119873119875120601119875 (119902119887)]

2

(16)

The algebraic expressions of the wave functions ofpseudoscalar and vector charmonium 120601119875(119881)(119902119887(119886))for ground (1S) and first excited (2S) states respec-tively that are obtained as analytic solutions ofthe corresponding mass spectral equations of thesequarkonia in an approximate harmonic oscillatorbasis are [18]

120601119875(119881) (1119878 119902) = 11205873412057332119875(119881)

119890minus119902221205732119875(119881)

120601119875(119881) (2119878 119902) = radic3231205873412057372

119875(119881)

(31205732119875(119881) minus 21199022) 119890minus119902221205732119875(119881) (17)

where the inverse range parameter 120573119875for pseudoscalar charmonium is 120573119875 =(4(1198981198881205962119902119902radic1 + 21198600(119873 + 32)))14 while

120573119881 for vector charmonium is 120573119881 =(2(1198981198881205962119902119902radic1 + 21198600(119873 + 32)))14 and thesetwo constants depend on the input parameters andcontain the dynamical information and they differfrom each other due to spin-spin interactions It canbe checked that our cross-sectional formula is in(16) scales as 1205722119890119898120572211990411989861199044 with 119898 being the mass of119888-quark where in (16) the wave functions 120601(119875119881)involve the inverse range parameters 120573119875119881 sim 11989812

Numerical Results We had calculated recently the massspectrum and various decays of ground and excited statesof pseudoscalar and vector quarkonia in [18ndash20] The sameinput parameters are employed in this calculation as in ourrecent works given in Table 1

Using these input parameters listed in Table 1 we calcu-late the cross-sections of pseudoscalar and vector charmo-nium production in our framework listed in Table 2

We wish to mention here that the 119887119887119887119887 as well as 119887119887119888119888production has not been observed so far but cross-sectionshave been predicted for 119890minus119890+ 997888rarr Υ + 120578119887 at center of massenergy radic119904 = 25 minus 30 GeV We thus wished to check ourcalculations for double bottomonium (Υ 120578119887) production inelectron-positron annihilation for sake of completeness Thesame can be studied with our framework used in this workwith little modifications Taking the mass of the 119887minusquark thatis fixed from our recent work on spectroscopy of 119887119887 states as119898119887 = 507GeV [18] and other input parameters the same thecross-section for production of ground states of 120578119887 and Υ forenergy range (radic119904 = 28 minus 30)GeV is given in Table 3

3 Discussions and Conclusion

Our main aim in this paper was to study the cross-sectionfor double charmonium production in electron-positroncollisions at center of mass energies radic119904 = 106119866119890119881 havingsuccessfully studied the mass spectrum and a range of lowenergy processes using an analytic treatment of 4 times 4 Bethe-Salpeter Equation (BSE) [18ndash22] This is due to the fact anyquark model model should successfully describe a rangeof processesmdashnot only the mass spectra and low energyhadronic decay constantsdecay widths but also the highenergy production processes involving these hadrons and allwithin a common dynamical framework and with a singleset of input parameters that are calibrated to the mass spec-trum

Further the exclusive production of double heavy char-monia in 119890+119890minus annihilation has been a challenge to under-stand in heavy-quark physics and has received considerableattention in recent years due to the fact that there is asignificant discrepancy in the data of Babar [2] and Belle[1] and the calculations performed in NRQCD [5 6] ofthis process at radic119904 = 106119866119890119881 where using leadingorder (LO) QCD diagrams alone leads to cross-sectionsthat are an order of magnitude smaller than data [1 24] These discrepancies were then resolved by taking intoaccount the Next-to-Leading-Order (NLO)QCD correctionscombined with relativistic corrections [23 24]) though the

Advances in High Energy Physics 5

Table 1 Input parameters for this study

1198620 1205960(119866119890119881) Λ(119866119890119881) 1198600 119898119888(119866119890119881)0210 0150 0200 0010 1490

Table 2 The total cross-sections of pseudoscalar and vector charmonium production for ground and first excited states with experimentaldata and other theoretical models (in units of fb)

Production Process 120590 (Our result) 120590 [1] 120590 [2] 120590 [8] 120590 [9] 120590 [10] 120590 [11]119890+119890minus 997888rarr 119869120595120578119888 20770 256plusmn28 176plusmn28 267 222plusmn42 223 2175119890+119890minus 997888rarr 1205951015840120578119888 11643 163plusmn46 163 153plusmn29119890+119890minus 997888rarr 1198691205951205781015840119888 11177 165plusmn30 164plusmn37 266 164plusmn31119890+119890minus 997888rarr 12059510158401205781015840119888 6266 160plusmn51 145 96plusmn18Table 3 The total cross-sections of pseudoscalar and vector bottomonium production for their ground states with predictions of othertheoretical models (in units of fb)

Production Process 120590(Our result) 120590[1] 120590[2] 120590[10] for (radic119904 = 25 minus 30)GeV120590[119890+119890minus 997888rarr Υ120578119887] 0155-0058 016-006

NLO contributions were larger than the LO contributions[10]

This process has also been studied in Relativistic QuarkModel [7 9] Light Cone formalism [8] and Bethe-SalpeterEquation [10 11] Relativistic corrections to cross-section inLight Cone formalism (in [8]) were considered to eliminatethe discrepancy between theory and experiment Furtherin an attempt to explain a part of this discrepancy [25]even suggested that processes proceeding through two virtualphotons may be important In Relativistic Quark Model [7]which incorporated relativistic treatment of internal motionof quarks and bound states improvements in the results wereobtained

Wewish tomention that in our framework of 4times4Bethe-Salpeter Equation (BSE) using the Covariant InstantaneousAnsatz where we treat the internal motion of quarks andthe bound states in a relativistically covariant manner weobtained results on cross-sections (in Table 2) for produc-tion of opposite parity 119888119888 states such as (119869Ψ 120578119888) (Ψ1015840 120578119888)(119869Ψ 1205781015840119888) (Ψ1015840 1205781015840119888) which are close to central values of data[1 2 4] using leading-order (LO) QCD diagrams alone Thisvalidates the fact that relativistic quark models such as BSEare strong candidates for treating not only the low energyprocesses but also the high energy production processesinvolving double heavy quarkonia due to their consistentrelativistic treatment of internal motion of quarks in thehadrons where our BS wave functions that take into accountall the Dirac structures in pseudoscalar and vector mesonsin a mathematically consistent manner play a vital role inthe dynamics of the process We also give our predictions oncross-section for double 119887119887 production at energies 28- 30GeVin Table 3 for future experiments at colliders

Themain objective of this study was to test the validationof our approach which provides a much deeper insight thanthe purely numerical calculations in 4 times 4 BSE that are verycommon in the literature We wish to mention that we havenot encountered any work in 4 times 4 representation of BSE

that treats this problem analytically On the contrary allthe other 4 times 4 BSE approaches adopt a purely numericalapproach of solving the BSE We are also not aware of anyother BSE framework involving 4times4 BS amplitude and withall the Dirac structures incorporated in the 4D hadronic BSwave functions (in fact many works used only the leadingDirac structures for instance see [10 11]) for calculations ofthis production process We treat this problem analyticallyby making use of the algebraic forms of wave functions120601(119875119881) derived analytically from mass spectral equations [18]for calculation of cross-section of this double charmoniumproduction process

This calculation involving production of double charmo-nia in electron-positron annihilation can be easily extendedto studies on other processes (involving the exchange of asingle virtual photon) observed at B-factories such as 119890minus 119890+minus gtΨ(2119878)120574 119890minus119890+minus gt 119869Ψ1205941198880 and 119890minus119890+minus gt Ψ(2119878)1205941198880 We furtherwish to extend this study to processes involving two virtualphotons such as the production of double charmonia in finalstates with 119862 = +1 such as 119890minus119890+ 997888rarr 119869Ψ119869Ψ) recentlyobserved at BABAR

These processes that we intend to study next are quiteinvolved and the dynamical equation based approachessuch as BSE are a promising approach And since theseprocesses involve quark-triangle diagrams we make use ofthe techniques we used for handling such diagrams recentlydone in [19 26 27] Such analytic approaches not only leadto better insight into the mass spectra and low energy decayprocesses involving charmonia but also their high energyproduction processes such as the one studied here

Note Added in Proofs We have just now come to know aboutthe recent experimental observation by CMS collaboration[28] of two excited B+c (2S) and B

lowast+c (2S) states in pp collisions

at radic119904 = 13 TeV at Large Hadron Collider (LHC) Thetheoretical study of this process will be a challenge tohadronic physics

6 Advances in High Energy Physics

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was carried out at Samara University Ethiopiaand at Chandigarh University IndiaThe authors thank theseinstitutions for the facilities provided during the course of thiswork

References

[1] K-F Chen W-S Hou I Adachi et al ldquoObservation ofan enhancement in e+eminus 997888rarr Υ(1S)120587+120587minus Υ(2S)120587+120587minus andΥ(3S)120587+120587minus production near radic119904 = 106 Gevrdquo Physical Reviewvol 82 Article ID 091106 2010 httpsarxivorgabs08103829

[2] B AubertM Bona Y Karyotakis et al ldquoEvidence for the 120578119887(1119878)meson in radiative Υ(2119878) decayrdquo Physical Review Letters vol103 Article ID 161801 2009

[3] K W Edwards R Janicek P M Patel et al ldquoStudy of B Decaysto Charmonium States 119861 997888rarr 120578119888119870 and 119861 997888rarr 120578119887119870rdquo PhysicalReview Letters vol 86 no 30 2001

[4] K Olive K Agashe andC Amsler ldquoReview of particle physicsrdquoChinese Physics C vol 38 no 9 Article ID 090001 2014

[5] G T Bodwinin E Braaten J Lee and C Yu ldquoExclusive two-vector-meson production from e+eminus annihilationrdquo PhysicalReview D vol 74 no 7 Article ID 074014 2006

[6] G T Bodwin D Kang T Kim J Lee and C Yu ldquoRelativisticCorrections to e+eminus 997888rarr J120595 + 120578c in a Potential Modelrdquo inProceedings of the AIP Conference vol 892 2007

[7] D Ebert and A P Martynenko ldquoRelativistic effects in the pro-duction of pseudoscalar and vector doubly heavy mesons frome+eminus annihilationrdquo Physical Review D Article ID 054008 2006

[8] V V Braguta A K Likhoded and A V Luchinsky ldquoExcitedcharmonium mesons production in e+eminus annihilation atradics=106 GeVrdquo Physical Review D vol 72 Article ID 0740192005

[9] D Ebert R N Faustov and V O Galkin ldquoRelativistic model ofhidden bottom tetraquarksrdquo Modern Physics Letters A vol 24no 08 pp 567ndash573 2009

[10] X-H Guo H-W Ke X-Q Li et al ldquoDouble heavy-quarko-nium production from electron-positron annihilation in theBethe-Salpeter formalismrdquo httpsarxivorgabs08040949

[11] E Mengesha and S Bhatnagar ldquoCross section for doublecharmonium production in electron-positron annihilation atenergyradic119904 = 106Gevrdquo International Journal of Modern Physicsvol E20 pp 2521ndash2533 2011

[12] A N Mitra and B M Sodermark ldquoA dynamical principle for3Dndash4D interlinkage in Salpeter-like equationsrdquo Nuclear PhysicsA vol 695 pp 328ndash352 2001

[13] A NMitra and S Bhatnagar ldquoA hadron-quark vertex functioninterconnection between 3D and 4D wave functionsrdquo Interna-tional Journal of Modern Physics A vol 7 p 121 1992

[14] A N Mitra ldquoQCD motivated BSE SDE framework for quarkdynamics underMarkov-Yukawa transversality AUnified viewof q anti-q and q q q systems 1rdquo in Proceedings of the IndianNational Science Academy vol 65 pp 527ndash584 1999

[15] S Bhatnagar D S Kulshreshtha and A N Mitra ldquoHow big arethe decay constants f 119875 of heavy-light mesonsrdquo Physics LettersB vol 263 no 3-4 pp 485ndash490 1991

[16] S Bhatnagar J Mahecha and Y Mengesha ldquoRelevance ofvarious Dirac covariants in hadronic Bethe-Salpeter wave func-tions in electromagnetic decays of ground state vector mesonsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 90 Article ID 014034 2014

[17] H Negash and S Bhatnagar ldquoMass spectrumand leptonic decayconstants of ground and radially excited states of 120578119888 and 120578119887 ina BethendashSalpeter equation frameworkrdquo International Journal ofModern Physics E vol 24 no 4 Article ID 1550030 2015

[18] H Negash and S Bhatnagar ldquoSpectroscopy of ground andexcited states of pseudoscalar and vector charmonium and bot-tomoniumrdquo International Journal of Modern Physics E vol 25no 8 Article ID 1650059 2016

[19] H Negash and S Bhatnagar ldquoRadiative decay widths of groundand excited states of vector charmonium and bottomoniumrdquohttpsarxivorgabs170306082

[20] H Negash and S Bhatnagar ldquoAnalytical calculations of groundand excited states of unequal mass heavy pseudoscalar andvector mesons mass spectra using Bethe-Salpeter formalismrdquohttpsarxivorgabs171107036

[21] E Gebrehana S Bhatnagar and H Negash ldquoAnalytic approachto calculations of mass spectra and decay constants of heavy-light quarkonia in the framework of Bethe-Salpeter equationrdquohttpsarxivorgabs190101888

[22] S Bhatnagar and L Alemu ldquoApproach to calculation of massspectra and two-photon decays of cc mesons in the frameworkof Bethe-Salpeter equationrdquo Physical Review D vol 97 ArticleID 034021 2018

[23] E Braaten and J Lee ldquoExclusive double-charmonium produc-tion from e+eminus annihilation into a virtual photonrdquo PhysicalReview D vol 67 Article ID 054007 2003

[24] Y J Zhang Y J Gao and K T Chao ldquoNext-to-leading-orderQCD correction to e+eminus 997888rarrJ120595+120578c at radics=106 GeVrdquo PhysicalReview Letters vol 96 Article ID 092001 2008

[25] G T Bodwin J Lee and E Braaten ldquoe+eminus annihilation intoJ120595+J120595rdquo Physical Review Letters vol 90 Article ID 1620012005

[26] E Mengesha and S Bhatnagar ldquoRadiative decays of equalmass vector mesons in a Bethe-Salpeter equation frameworkrdquoInternational Journal of Modern Physics E vol 21 Article ID1250084 2012

[27] E Mengesha and S Bhatnagar ldquoStrong decays of first radialexcitations of lightest vectormeson statesrdquo International JournalofModern Physics E vol 22 no 7 Article ID 1350046 p 15 2013

[28] A M Sirunyan et al ldquoObservation of two excited B+c states andmeasurement of the B+c (2S) mass in pp collisions at radic119904 = 13TeVrdquo httpsarxivorgabs190200571 2019

Advances in High Energy Physics 5

Table 1 Input parameters for this study

1198620 1205960(119866119890119881) Λ(119866119890119881) 1198600 119898119888(119866119890119881)0210 0150 0200 0010 1490

Table 2 The total cross-sections of pseudoscalar and vector charmonium production for ground and first excited states with experimentaldata and other theoretical models (in units of fb)

Production Process 120590 (Our result) 120590 [1] 120590 [2] 120590 [8] 120590 [9] 120590 [10] 120590 [11]119890+119890minus 997888rarr 119869120595120578119888 20770 256plusmn28 176plusmn28 267 222plusmn42 223 2175119890+119890minus 997888rarr 1205951015840120578119888 11643 163plusmn46 163 153plusmn29119890+119890minus 997888rarr 1198691205951205781015840119888 11177 165plusmn30 164plusmn37 266 164plusmn31119890+119890minus 997888rarr 12059510158401205781015840119888 6266 160plusmn51 145 96plusmn18Table 3 The total cross-sections of pseudoscalar and vector bottomonium production for their ground states with predictions of othertheoretical models (in units of fb)

Production Process 120590(Our result) 120590[1] 120590[2] 120590[10] for (radic119904 = 25 minus 30)GeV120590[119890+119890minus 997888rarr Υ120578119887] 0155-0058 016-006

NLO contributions were larger than the LO contributions[10]

This process has also been studied in Relativistic QuarkModel [7 9] Light Cone formalism [8] and Bethe-SalpeterEquation [10 11] Relativistic corrections to cross-section inLight Cone formalism (in [8]) were considered to eliminatethe discrepancy between theory and experiment Furtherin an attempt to explain a part of this discrepancy [25]even suggested that processes proceeding through two virtualphotons may be important In Relativistic Quark Model [7]which incorporated relativistic treatment of internal motionof quarks and bound states improvements in the results wereobtained

Wewish tomention that in our framework of 4times4Bethe-Salpeter Equation (BSE) using the Covariant InstantaneousAnsatz where we treat the internal motion of quarks andthe bound states in a relativistically covariant manner weobtained results on cross-sections (in Table 2) for produc-tion of opposite parity 119888119888 states such as (119869Ψ 120578119888) (Ψ1015840 120578119888)(119869Ψ 1205781015840119888) (Ψ1015840 1205781015840119888) which are close to central values of data[1 2 4] using leading-order (LO) QCD diagrams alone Thisvalidates the fact that relativistic quark models such as BSEare strong candidates for treating not only the low energyprocesses but also the high energy production processesinvolving double heavy quarkonia due to their consistentrelativistic treatment of internal motion of quarks in thehadrons where our BS wave functions that take into accountall the Dirac structures in pseudoscalar and vector mesonsin a mathematically consistent manner play a vital role inthe dynamics of the process We also give our predictions oncross-section for double 119887119887 production at energies 28- 30GeVin Table 3 for future experiments at colliders

Themain objective of this study was to test the validationof our approach which provides a much deeper insight thanthe purely numerical calculations in 4 times 4 BSE that are verycommon in the literature We wish to mention that we havenot encountered any work in 4 times 4 representation of BSE

that treats this problem analytically On the contrary allthe other 4 times 4 BSE approaches adopt a purely numericalapproach of solving the BSE We are also not aware of anyother BSE framework involving 4times4 BS amplitude and withall the Dirac structures incorporated in the 4D hadronic BSwave functions (in fact many works used only the leadingDirac structures for instance see [10 11]) for calculations ofthis production process We treat this problem analyticallyby making use of the algebraic forms of wave functions120601(119875119881) derived analytically from mass spectral equations [18]for calculation of cross-section of this double charmoniumproduction process

This calculation involving production of double charmo-nia in electron-positron annihilation can be easily extendedto studies on other processes (involving the exchange of asingle virtual photon) observed at B-factories such as 119890minus 119890+minus gtΨ(2119878)120574 119890minus119890+minus gt 119869Ψ1205941198880 and 119890minus119890+minus gt Ψ(2119878)1205941198880 We furtherwish to extend this study to processes involving two virtualphotons such as the production of double charmonia in finalstates with 119862 = +1 such as 119890minus119890+ 997888rarr 119869Ψ119869Ψ) recentlyobserved at BABAR

These processes that we intend to study next are quiteinvolved and the dynamical equation based approachessuch as BSE are a promising approach And since theseprocesses involve quark-triangle diagrams we make use ofthe techniques we used for handling such diagrams recentlydone in [19 26 27] Such analytic approaches not only leadto better insight into the mass spectra and low energy decayprocesses involving charmonia but also their high energyproduction processes such as the one studied here

Note Added in Proofs We have just now come to know aboutthe recent experimental observation by CMS collaboration[28] of two excited B+c (2S) and B

lowast+c (2S) states in pp collisions

at radic119904 = 13 TeV at Large Hadron Collider (LHC) Thetheoretical study of this process will be a challenge tohadronic physics

6 Advances in High Energy Physics

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was carried out at Samara University Ethiopiaand at Chandigarh University IndiaThe authors thank theseinstitutions for the facilities provided during the course of thiswork

References

[1] K-F Chen W-S Hou I Adachi et al ldquoObservation ofan enhancement in e+eminus 997888rarr Υ(1S)120587+120587minus Υ(2S)120587+120587minus andΥ(3S)120587+120587minus production near radic119904 = 106 Gevrdquo Physical Reviewvol 82 Article ID 091106 2010 httpsarxivorgabs08103829

[2] B AubertM Bona Y Karyotakis et al ldquoEvidence for the 120578119887(1119878)meson in radiative Υ(2119878) decayrdquo Physical Review Letters vol103 Article ID 161801 2009

[3] K W Edwards R Janicek P M Patel et al ldquoStudy of B Decaysto Charmonium States 119861 997888rarr 120578119888119870 and 119861 997888rarr 120578119887119870rdquo PhysicalReview Letters vol 86 no 30 2001

[4] K Olive K Agashe andC Amsler ldquoReview of particle physicsrdquoChinese Physics C vol 38 no 9 Article ID 090001 2014

[5] G T Bodwinin E Braaten J Lee and C Yu ldquoExclusive two-vector-meson production from e+eminus annihilationrdquo PhysicalReview D vol 74 no 7 Article ID 074014 2006

[6] G T Bodwin D Kang T Kim J Lee and C Yu ldquoRelativisticCorrections to e+eminus 997888rarr J120595 + 120578c in a Potential Modelrdquo inProceedings of the AIP Conference vol 892 2007

[7] D Ebert and A P Martynenko ldquoRelativistic effects in the pro-duction of pseudoscalar and vector doubly heavy mesons frome+eminus annihilationrdquo Physical Review D Article ID 054008 2006

[8] V V Braguta A K Likhoded and A V Luchinsky ldquoExcitedcharmonium mesons production in e+eminus annihilation atradics=106 GeVrdquo Physical Review D vol 72 Article ID 0740192005

[9] D Ebert R N Faustov and V O Galkin ldquoRelativistic model ofhidden bottom tetraquarksrdquo Modern Physics Letters A vol 24no 08 pp 567ndash573 2009

[10] X-H Guo H-W Ke X-Q Li et al ldquoDouble heavy-quarko-nium production from electron-positron annihilation in theBethe-Salpeter formalismrdquo httpsarxivorgabs08040949

[11] E Mengesha and S Bhatnagar ldquoCross section for doublecharmonium production in electron-positron annihilation atenergyradic119904 = 106Gevrdquo International Journal of Modern Physicsvol E20 pp 2521ndash2533 2011

[12] A N Mitra and B M Sodermark ldquoA dynamical principle for3Dndash4D interlinkage in Salpeter-like equationsrdquo Nuclear PhysicsA vol 695 pp 328ndash352 2001

[13] A NMitra and S Bhatnagar ldquoA hadron-quark vertex functioninterconnection between 3D and 4D wave functionsrdquo Interna-tional Journal of Modern Physics A vol 7 p 121 1992

[14] A N Mitra ldquoQCD motivated BSE SDE framework for quarkdynamics underMarkov-Yukawa transversality AUnified viewof q anti-q and q q q systems 1rdquo in Proceedings of the IndianNational Science Academy vol 65 pp 527ndash584 1999

[15] S Bhatnagar D S Kulshreshtha and A N Mitra ldquoHow big arethe decay constants f 119875 of heavy-light mesonsrdquo Physics LettersB vol 263 no 3-4 pp 485ndash490 1991

[16] S Bhatnagar J Mahecha and Y Mengesha ldquoRelevance ofvarious Dirac covariants in hadronic Bethe-Salpeter wave func-tions in electromagnetic decays of ground state vector mesonsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 90 Article ID 014034 2014

[17] H Negash and S Bhatnagar ldquoMass spectrumand leptonic decayconstants of ground and radially excited states of 120578119888 and 120578119887 ina BethendashSalpeter equation frameworkrdquo International Journal ofModern Physics E vol 24 no 4 Article ID 1550030 2015

[18] H Negash and S Bhatnagar ldquoSpectroscopy of ground andexcited states of pseudoscalar and vector charmonium and bot-tomoniumrdquo International Journal of Modern Physics E vol 25no 8 Article ID 1650059 2016

[19] H Negash and S Bhatnagar ldquoRadiative decay widths of groundand excited states of vector charmonium and bottomoniumrdquohttpsarxivorgabs170306082

[20] H Negash and S Bhatnagar ldquoAnalytical calculations of groundand excited states of unequal mass heavy pseudoscalar andvector mesons mass spectra using Bethe-Salpeter formalismrdquohttpsarxivorgabs171107036

[21] E Gebrehana S Bhatnagar and H Negash ldquoAnalytic approachto calculations of mass spectra and decay constants of heavy-light quarkonia in the framework of Bethe-Salpeter equationrdquohttpsarxivorgabs190101888

[22] S Bhatnagar and L Alemu ldquoApproach to calculation of massspectra and two-photon decays of cc mesons in the frameworkof Bethe-Salpeter equationrdquo Physical Review D vol 97 ArticleID 034021 2018

[23] E Braaten and J Lee ldquoExclusive double-charmonium produc-tion from e+eminus annihilation into a virtual photonrdquo PhysicalReview D vol 67 Article ID 054007 2003

[24] Y J Zhang Y J Gao and K T Chao ldquoNext-to-leading-orderQCD correction to e+eminus 997888rarrJ120595+120578c at radics=106 GeVrdquo PhysicalReview Letters vol 96 Article ID 092001 2008

[25] G T Bodwin J Lee and E Braaten ldquoe+eminus annihilation intoJ120595+J120595rdquo Physical Review Letters vol 90 Article ID 1620012005

[26] E Mengesha and S Bhatnagar ldquoRadiative decays of equalmass vector mesons in a Bethe-Salpeter equation frameworkrdquoInternational Journal of Modern Physics E vol 21 Article ID1250084 2012

[27] E Mengesha and S Bhatnagar ldquoStrong decays of first radialexcitations of lightest vectormeson statesrdquo International JournalofModern Physics E vol 22 no 7 Article ID 1350046 p 15 2013

[28] A M Sirunyan et al ldquoObservation of two excited B+c states andmeasurement of the B+c (2S) mass in pp collisions at radic119904 = 13TeVrdquo httpsarxivorgabs190200571 2019

6 Advances in High Energy Physics

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was carried out at Samara University Ethiopiaand at Chandigarh University IndiaThe authors thank theseinstitutions for the facilities provided during the course of thiswork

References

[1] K-F Chen W-S Hou I Adachi et al ldquoObservation ofan enhancement in e+eminus 997888rarr Υ(1S)120587+120587minus Υ(2S)120587+120587minus andΥ(3S)120587+120587minus production near radic119904 = 106 Gevrdquo Physical Reviewvol 82 Article ID 091106 2010 httpsarxivorgabs08103829

[2] B AubertM Bona Y Karyotakis et al ldquoEvidence for the 120578119887(1119878)meson in radiative Υ(2119878) decayrdquo Physical Review Letters vol103 Article ID 161801 2009

[3] K W Edwards R Janicek P M Patel et al ldquoStudy of B Decaysto Charmonium States 119861 997888rarr 120578119888119870 and 119861 997888rarr 120578119887119870rdquo PhysicalReview Letters vol 86 no 30 2001

[4] K Olive K Agashe andC Amsler ldquoReview of particle physicsrdquoChinese Physics C vol 38 no 9 Article ID 090001 2014

[5] G T Bodwinin E Braaten J Lee and C Yu ldquoExclusive two-vector-meson production from e+eminus annihilationrdquo PhysicalReview D vol 74 no 7 Article ID 074014 2006

[6] G T Bodwin D Kang T Kim J Lee and C Yu ldquoRelativisticCorrections to e+eminus 997888rarr J120595 + 120578c in a Potential Modelrdquo inProceedings of the AIP Conference vol 892 2007

[7] D Ebert and A P Martynenko ldquoRelativistic effects in the pro-duction of pseudoscalar and vector doubly heavy mesons frome+eminus annihilationrdquo Physical Review D Article ID 054008 2006

[8] V V Braguta A K Likhoded and A V Luchinsky ldquoExcitedcharmonium mesons production in e+eminus annihilation atradics=106 GeVrdquo Physical Review D vol 72 Article ID 0740192005

[9] D Ebert R N Faustov and V O Galkin ldquoRelativistic model ofhidden bottom tetraquarksrdquo Modern Physics Letters A vol 24no 08 pp 567ndash573 2009

[10] X-H Guo H-W Ke X-Q Li et al ldquoDouble heavy-quarko-nium production from electron-positron annihilation in theBethe-Salpeter formalismrdquo httpsarxivorgabs08040949

[11] E Mengesha and S Bhatnagar ldquoCross section for doublecharmonium production in electron-positron annihilation atenergyradic119904 = 106Gevrdquo International Journal of Modern Physicsvol E20 pp 2521ndash2533 2011

[12] A N Mitra and B M Sodermark ldquoA dynamical principle for3Dndash4D interlinkage in Salpeter-like equationsrdquo Nuclear PhysicsA vol 695 pp 328ndash352 2001

[13] A NMitra and S Bhatnagar ldquoA hadron-quark vertex functioninterconnection between 3D and 4D wave functionsrdquo Interna-tional Journal of Modern Physics A vol 7 p 121 1992

[14] A N Mitra ldquoQCD motivated BSE SDE framework for quarkdynamics underMarkov-Yukawa transversality AUnified viewof q anti-q and q q q systems 1rdquo in Proceedings of the IndianNational Science Academy vol 65 pp 527ndash584 1999

[15] S Bhatnagar D S Kulshreshtha and A N Mitra ldquoHow big arethe decay constants f 119875 of heavy-light mesonsrdquo Physics LettersB vol 263 no 3-4 pp 485ndash490 1991

[16] S Bhatnagar J Mahecha and Y Mengesha ldquoRelevance ofvarious Dirac covariants in hadronic Bethe-Salpeter wave func-tions in electromagnetic decays of ground state vector mesonsrdquoPhysical Review D Particles Fields Gravitation and Cosmologyvol 90 Article ID 014034 2014

[17] H Negash and S Bhatnagar ldquoMass spectrumand leptonic decayconstants of ground and radially excited states of 120578119888 and 120578119887 ina BethendashSalpeter equation frameworkrdquo International Journal ofModern Physics E vol 24 no 4 Article ID 1550030 2015

[18] H Negash and S Bhatnagar ldquoSpectroscopy of ground andexcited states of pseudoscalar and vector charmonium and bot-tomoniumrdquo International Journal of Modern Physics E vol 25no 8 Article ID 1650059 2016

[19] H Negash and S Bhatnagar ldquoRadiative decay widths of groundand excited states of vector charmonium and bottomoniumrdquohttpsarxivorgabs170306082

[20] H Negash and S Bhatnagar ldquoAnalytical calculations of groundand excited states of unequal mass heavy pseudoscalar andvector mesons mass spectra using Bethe-Salpeter formalismrdquohttpsarxivorgabs171107036

[21] E Gebrehana S Bhatnagar and H Negash ldquoAnalytic approachto calculations of mass spectra and decay constants of heavy-light quarkonia in the framework of Bethe-Salpeter equationrdquohttpsarxivorgabs190101888

[22] S Bhatnagar and L Alemu ldquoApproach to calculation of massspectra and two-photon decays of cc mesons in the frameworkof Bethe-Salpeter equationrdquo Physical Review D vol 97 ArticleID 034021 2018

[23] E Braaten and J Lee ldquoExclusive double-charmonium produc-tion from e+eminus annihilation into a virtual photonrdquo PhysicalReview D vol 67 Article ID 054007 2003

[24] Y J Zhang Y J Gao and K T Chao ldquoNext-to-leading-orderQCD correction to e+eminus 997888rarrJ120595+120578c at radics=106 GeVrdquo PhysicalReview Letters vol 96 Article ID 092001 2008

[25] G T Bodwin J Lee and E Braaten ldquoe+eminus annihilation intoJ120595+J120595rdquo Physical Review Letters vol 90 Article ID 1620012005

[26] E Mengesha and S Bhatnagar ldquoRadiative decays of equalmass vector mesons in a Bethe-Salpeter equation frameworkrdquoInternational Journal of Modern Physics E vol 21 Article ID1250084 2012

[27] E Mengesha and S Bhatnagar ldquoStrong decays of first radialexcitations of lightest vectormeson statesrdquo International JournalofModern Physics E vol 22 no 7 Article ID 1350046 p 15 2013

[28] A M Sirunyan et al ldquoObservation of two excited B+c states andmeasurement of the B+c (2S) mass in pp collisions at radic119904 = 13TeVrdquo httpsarxivorgabs190200571 2019

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