A new variational approach and its application toheavy quarkonia

Authors

  • R. Manzoor Centre For High Energy Physics, Punjab University,Lahore (54590), Pakistan.
  • J. Ahmed Centre For High Energy Physics, Punjab University,Lahore (54590), Pakistan.
  • A. Raya Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo

DOI:

https://doi.org/10.31349/RevMexFis.67.33

Keywords:

QCD, potential models, quarkonium systems

Abstract

By combining the variational principle with Heisenberg uncertaintyprinciple in an effective Hamiltonian for heavy flavored mesons, we in-troduce a framework to estimate masses and radii of these states froman analytical constraint. In a novel manner, a model for quark velocityand a model for quark momentum width are introduced. These kinemat-ical model parameters are obtained as analytical functions of inter quarkseparation in heavy quarkonia. The values of such quark parameters arethen used in the calculation of S-wave annihilation decay rates of \bar{c}c and\bar{b} b. To test the accuracy of our technique we first calculate the spin averaged masses, scalar radii and annihilation decay rates of charmoniumand bottomonium without and with relativistic corrections by solving theSchrödinger wave equation with the appropriate parametrization of the Song-Lin potential. The Schrödinger wave equation is solved numericallywith the matrix Numerov method and we observe a good agreement withthe experimental measurements and other theoretical calculations and extract strong running coupling constant for \bar{c}c and \bar{b}b systems. In non rel-ativistic settings, heavy meson spectra have been obtained and extended to rather higher excited states within our framework by using bare masses of c and b quarks which we have extracted from analysis of experimentaldata

References

i.This relation is motivated from Heisenberg uncertainty princi-ple and is of central importance in Sec.4.on VP.

A. Bazavovet al.for the HotQCD Collaboration,Phys.Rev. D90(2014) 094503.https://doi.org/10.1103/PhysRevD.90.094503

M. A. Escobedo,Effective Field Theory and Lattice QCDapproaches for hard probes in QCD matterProceedings ofthe International Conference on Hard and ElectromagneticProbes of High-Energy Nuclear Collisions, 2018. Preprint:arXiv:1812.06344.

S. Godfrey and K. Moats, Bottomonium mesons and strategiesfor their observation.Phys. Rev. D92(2015) 054034.https://doi.org/10.1103/PhysRevD.92.054034

B.-Q. Li and K.-T. Chao, Bottomonium spectrum with screenedpotential.Comm. Theor. Phys.52, 653 (2009).10.1088/0253-6102/52/4/20

W. J. Deng, H. Liu, L. C. Gui and X. H. Zhong, X. H. . Spec-trum and electromagnetic transitions of bottomonium.Phys.Rev. D95(2017) 074002.https://doi.org/10.1103/PhysRevD.95.074002

D. Ebert, R. N. Faustov and V. O Galkin,Phys. Rev.D,67(2003) 014027.https://doi.org/10.1103/PhysRevD.67.014027

J. Ferretti and E. Santopinto, Higher mass bottomonia.Phys.Rev. D,90(2014) 094022.https://doi.org/10.1103/PhysRevD.90.094022

Y. Li, P. Maris, X. Zhao and J. P. Vary, Heavy quarkonium ina holographic basis.Phys. Lett. B758(2016) 118.https://doi.org/10.1016/j.physletb.2016.04.065

J. Vijande, F. Fern ́andez and A. Valcarce, A. Constituent quarkmodel study of the meson spectra.J.Phys. G31(2005) 481.https://doi.org/10.1088/0954-3899/31/5/017

S. N. Gupta, S. F. Radford and W. W. Repko, W. W. . Semirel-ativistic potential model for heavy quarkonia.Phys. Rev. D,34(1986) 201.https://doi.org/10.1103/PhysRevD.34.201

J. L. Richardson, The heavy quark potential and the®;J=√systems.Phys. Lett. B,82(1979) 272.https://doi.org/10.1016/0370-2693(79)90753-6

W. Buchm ̈uller and S. H. Tye, Quarkonia and quantum chro-modynamics.Phys. Rev. D,24(1981) 132.https://doi.org/10.1103/PhysRevD.24.132

A. Martin, A fit of upsilon and charmonium spectra.Phys.Lett. B,93(1980) 338.https://doi.org/10.1016/0370-2693(80)90527-4

P. Gonz ́alez, A. Valcarce, H. Garcilazo and J. Vijande,Heavy meson description with a screened potential.Phys.Rev. D,68(2003) 034007.https://doi.org/10.1103/PhysRevD.68.034007

Y. B. Ding, K. T. Chao and D. H. Qin, Possible effects of colorscreening and large string tension in heavy quarkonium spec-tra.Phys. Rev. D51(1995) 5064.https://doi.org/10.1103/PhysRevD.51.5064

M. Beyer, U. Bohn, M. G. Huber, B. C. Metsch and J. Resag,Relativistic effects and the constituent quark model of heavyquarkonia.Z. Phys. C55(1992) 307.https://doi.org/10.1007/BF01482594

T. Wei-Zhao, C. Lu, Y. You-Chang and C. Hong, Bottomoniumstates versus recent experimental observations in the QCD-inspired potential model.Chin. Phys. C37(2013) 083101.https://doi.org/10.1088/1674-1137/37/8/083101

P. Gonz ́alez, Generalized screened potential model.J. Phys.G41(2014) 095001.https://doi.org/10.1088/0954-3899/41/9/095001

J. Segovia, P. G. Ortega, D. R. Entem and F. Fern ́andez,F. Bottomonium spectrum revisited.Phys. Rev. D93(2016)074027.https://doi.org/10.1103/PhysRevD.93.074027

T. Kawanai and S. Sasaki, Potential description of charmo-nium and charmed-strange mesons from lattice QCDPhys.Rev. D92(2015) 094503.https://doi.org/10.1103/PhysRevD.92.094503

Y. H. Zhuo, D. S. Jiang, W. J. Qian, and X. H. Gong, THEHEAVY QUARKONIUM SPECTRUM OF ANHARMONIC OSCILLATORS CONFINEMENT POTENTIAL.Chin. Phys.C11(1987) 202. A. O. Adelakun and A. D. Dele, Solution ofquantum anharmonic oscillator with quartic perturbation,Adv.Phys. Theor. Appl.27(2014) 38.

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flan-nery,Numerical Recipes: The art of Scientific Computing,3rd. ed. Cambridge University Press, New York, (2007). ISBN0521880688.

H. Mutuk, Spin averaged mass spectrum of heavy quarkoniumvia asymptotic iteration method.Can. J. Phys.97(2019) 1342.https://doi.org/10.1139/cjp-2018-0589

M. Pillai, J. Goglio, and T. G. Walker, Matrix Numerov methodfor solving Schr ̈odinger’s equation.Am. J. Phys.80(2012)1017.https://doi.org/10.1119/1.4748813

A. M. Yasser, G. S. Hassan, and T. A. Nahool, NumericalStudy of Heavy Meson’s Spectra Using the Matrix Numerov’sMethod.Int. J. New. Hor. Phys.2(2015) 33.http://dx.doi.org/10.12785/ijnhp/020106

J. Ahmed, R. Manzoor and A. Raya, Variational constraints ofmasses and radii of cπcMesons.Quant. Phys. Lett.6(2017) 1.http://dx.doi.org/10.18576/qpl/060204

X, Song and H. Lin, A new phenomenological potential forheavy quarkonium.Z.Phys. C34(1987) 223.https://doi.org/10.1007/BF01566763

J. Z. Wang, R. Q. Qian, X. Liu and T Matsuki, Are the Y statesaround 4.6 GeV from e+ e- annihilation higher charmonia?.Phys. Rev. D101(2020) 034001.https://doi.org/10.1103/PhysRevD.101.034001

A. Bazavov and J. H. Weber, Color screening in quantumchromodynamics.Prog. Part. Nucl. Phys.(2020) 103823.10.1016/j.ppnp.2020.103823

M. Chabab, and L. Sanhaji,Int. J. Mod. Phys. A,20(2005) 1863.https://doi.org/10.1142/S0217751X05023505

E. Leader and E. Predazzi, E.An Introduction to Gauge Theo-ries and Modern Particle Physics 2Cambridge University Press(1996). ISBN 9780511622601.

M. Bender,Phys. Rep. C75(1981) 205.

J. Z. Wang, Z. F Sun, X. Liu, and T. Matsuki, Higher bottomo-nium zoo.Eur. Phys. J. C,78(2018) 915.https://doi.org/10.1140/epjc/s10052-018-6372-1

A. B. Henriques, B. Kellett and R. G. Moorhouse, Ra-diative transitions and the P wave levels in charmonium.Phys. Lett. B64(1976) 85.https://doi.org/10.1016/0370-2693(76)90364-6

Y. B. Ding, K. T. Chao and D. H. Qin, Possible effects of colorscreening and large string tension in heavy quarkonium spec-tra.Phys. Rev. D51(1995) 5064.https://doi.org/10.1103/PhysRevD.51.5064

M. M. Brisudova, L. Burakovsky and T. Goldman, Effectof color screening on heavy quarkonia Regge trajectories.Phys. Lett. B460(1999) 1.https://doi.org/10.1016/S0370-2693(99)00732-7

S. M. Ikhdair and R. Sever, Heavy-quark bound states in po-tentials with the Bethe-Salpeter equation.Z. Phys. C56(1992)155. DOIhttps://doi.org/10.1007/BF01589718Rev. Mex. Fis.67(1) 33–53

S. M. Ikhdair and R. Sever, Bc meson spectrum andhyperfine splittings in the shifted large-N-expansion tech-nique.Int. J. Mod. Phys. A18(2003) 4215.10.1142/S0217751X03012370

M. Tanabashi et al.Phys. Rev. D98(2018) 030001.https://doi.org/10.1103/PhysRevD.98.030001

A. K. Rai, B. Patel and P. C. Vinodkumar, PropertiesofQπQmesons in nonrelativistic QCD formalism.Phys.Rev. C,78(2008) 055202.https://doi.org/10.1103/PhysRevC.78.055202

D. Ebert, R. N. Faustov and V. O. Galkin, Spectroscopy andRegge trajectories of heavy quarkonia and B c mesons.Eur.Phys. J. C71(2011) 1825.https://doi.org/10.1140/epjc/s10052-011-1825-9

N. R. Soniet al., Spectroscopy using the Cornell potential.Eur.Phys. J. C78(2018) 592.https://doi.org/10.1140/epjc/s10052-018-6068-6

S. Godfrey and N. Isgur, Mesons in a relativized quark modelwith chromodynamics.Phys. Rev. D,32(1985) 189.https://doi.org/10.1103/PhysRevD.32.18

B. Q. Liand K. T. Chao, Higher charmonia and X, Y,Z states with screened potential.Phys. Rev. D79(2009)094004.https://doi.org/10.1103/PhysRevD.79.094004

N. Akbar, M. A. Sultan, B. Masudand F. Akram, Higherhybrid bottomonia in an extended potential model.Phys.Rev. D95(2017) 074018.https://doi.org/10.1103/PhysRevD.95.074018

L. D. Soloviev, Masses and internal structure of mesons in thestring quark model.Phys. Rev. D61(1999) 015009.https://doi.org/10.1103/PhysRevD.61.015009

G. L. Wang, T. F Feng and X. G and Wu, Average speed andits powers v n of a heavy quark in quarkonia.Phys. Rev. D101(2020) 116011.10.1103/PhysRevD.101.116011

J. J Aubertet al., The ratio of the nucleon structure func-tions F2n for iron and deuterium.Phys. Lett. B123(1983)275.https://doi.org/10.1016/0370-2693(83)90437-9

M. Beneke and G. Buchalla, B c meson lifetime.Phys.Rev. D,53(1996) 4991.https://doi.org/10.1103/PhysRevD.53.4991

B. Schmookleret al., CLAS Collaboration,Nature,566(2019) 354.https://doi.org/10.1038/s41586-019-0925-9

W. Kwong, P. B. Mackenzie, R. Rosenfeld and J. L. Rosner,Quarkonium annihilation rates.Phys. Rev. D,37(1988) 3210.https://doi.org/10.1103/PhysRevD.37.3210

W. Kwong and J. L. Rosner, D-wave quarkonium levels of the?family.Phys. Rev. D38(1988) 279.https://doi.org/10.1103/PhysRevD.38.279

V. A. Novikov, L. B. Okun, M. A. Shifman, A. I. Vain-shtein, M. B. Voloshinand V. I. Zakharov, Charmonium andgluons.Phys. Rep.41(1978) 1-133 .https://doi.org/10.1016/0370-1573(78)90120-5

W. Kwong, P. B. Mackenzie, R. Rosenfeldand J. L. Rosner,Quarkonium annihilation rates.Phys. Rev. D37(1988) 3210.https://doi.org/10.1103/PhysRevD.37.3210

E. S. Ackleh and T. Barnes, Two-photon width of singletpositronium and quarkonium with arbitrary total angular mo-mentum.Phys. Rev. D45(1992) 232.https://doi.org/10.1103/PhysRevD.45.232

A. Deur, S. J. Brodsky and G. F. de T ́eramond, The QCD run-ning coupling,Prog. Part. Nucl. Phys.90(2016) 1.10.1016/j.ppnp.2016.04.003

C. Patrignaniet al.,for the Particle Data Group, Review ofparticle physics.Chin. Phys. C40(2016) 100001.https://doi.org/10.1088/1674-1137/40/10/100001

T. Bhavsar, M. Shah and P. C. Vinodkumar, Status ofquarkonia-like negative and positive parity states ina relativistic confinement scheme.Eur. Phys. J. C78(2018) 227.https://doi.org/10.1140/epjc/s10052-018-5694-3

P. P. D’ Souza, M. Bhat, A. P. Monteiroand K. B. Vijaya Kumar,Realistic Results of Low-Lying Charmonium Using An Instan-ton Potential. Preprint: arXiv:1703.10413.

V. Kher and A. K. Rai, Spectroscopy and decay properties ofcharmonium.Chin. Phys. C42(2018) 083101.10.1088/1674-1137/42/8/083101

E. J. Eichten, K. Lane and C. Quigg, B-meson gateways tomissing charmonium levels. Physical review letters,Phys. Rev.Lett.89(2002) 162002.https://doi.org/10.1103/PhysRevLett.89.162002

H. Negash and S. Bhatnagar, Spectroscopy of ground and ex-cited states of pseudoscalar and vector charmonium and bot-tomonium. International Journal of Modern Physics E,Int. J.Mod. Phys. E25(2016) 1650059.https://doi.org/10.1142/S0218301316500592

J. T. Laverty, S. F. Radford and W. W. Repko,∞∞andggdecay rates for equal mass heavy quarkonia.Preprint:arXiv:0901.3917.

Bhagyesh, K. B. V. Kumar and A. P. Monterio,J. Phys.G38(2011) 085001.https://doi.org/10.1088/0954-3899/38/8/085001

L. Bai-Qing and C. Kuang-Ta, Bottomonium spectrum withscreened potential.Commun. Thero. Phys.52(2009) 653.10.1088/0253-6102/52/4/20

S. Godfrey and K. Moats, Bottomonium mesons and strategiesfor their observation.Phys. Rev. D92(2015) 054034.https://doi.org/10.1103/PhysRevD.92.054034

C. Y. Wong, Molecular states of heavy quark mesons.Phys.Rev. C69(2004) 055202.https://doi.org/10.1103/PhysRevC.69.055202

D. Ebert, R. N. Faustov and V. Galkin, Spectroscopy and Reggetrajectories of heavy quarkonia and B c mesons.Eur. Phys. J.C71(2011) 1825.https://doi.org/10.1140/epjc/s10052-011-1825-9

M. A. Sultan, N. Akbar, B. Masud and F. Akram, Higherhybrid charmonia in an extended potential model.Phys.Rev. D90(2014) 054001.https://doi.org/10.1103/PhysRevD.90.054001Rev. Mex. Fis.67(1) 33–53

T. Wang, G. L. Wang, W. L. Ju and Y. Jiang, Annihilation rateof 2-+ charmonium and bottomonium.JHEP2013(2013) 110.10.1007/JHEP03(2013)110

B. Andersson, G. Gustafson, G. Ingelmanand T Sj ̈ostrand, Par-ton fragmentation and string dynamics.Phys. Rep.97(1983)31.https://doi.org/10.1016/0370-1573(83)90080-7

B. Andersson, S. Mohantyand F. Soderberg,Recent develop-ments in the Lund model, Proceedings of the XXXVI AnnualWinter School on Nuclear and Particle Physics 2002. Preprinthep-ph/0212122.

J. N. Pandya, N. R. Soni, N. Devlani and A. K. Rai, Decayrates and electromagnetic transitions of heavy quarkonia.Chin.Phys. C39(2015) 123101.https://doi.org/10.1088/1674-1137/39/12/123101

V. R. Debastiani and F. S. Navarra,Chin. Phys. C43(2018)013105.https://doi.org/10.1088/1674-1137/43/1/013105

H. Abramowiczet al.,Eur. Phys. J. C78(2018) 473.10.1140/epjc/s10052-018-5848-3

H. Abramowiczet al.,Eur. Phys. J. C73(2013) 2311.https://doi.org/10.1140/epjc/s10052-013-2311-3

B. Aubertet al.,Phys. Rev. D81(2010) 032003.https://doi.org/10.1103/PhysRevD.81.032003

O. Samoylovet al.,Nucl. Phys. B876(2013) 339.https://doi.org/10.1016/j.nuclphysb.2013.08.021

C. Schwandaet al., Phys. Rev. D78(2008) 032016.https://doi.org/10.1103/PhysRevD.78.032016

H. Abramowiczet al.,JHEP2014(2014) 127.10.1007/JHEP09(2014)127

J. Abdallahet al.,Eur. Phys. J. C55(2008) 525.https://doi.org/10.1140/epjc/s10052-008-0631-5

, J. Abdallahet al.,Eur. Phys. J. C46(2006) 569.https://doi.org/10.1140/epjc/s2006-02497-6

A. H. Mahmoodet al.,Phys. Rev. D67(2003) 072001.https://doi.org/10.1103/PhysRevD.67.072001

J. Abdallahet al.,Eur. Phys. J. C45(2006) 35.https://doi.org/10.1140/epjc/s2005-02406-7

B. Aubertet al., Phys. Rev. Lett.93(2004) 011803.https://doi.org/10.1103/PhysRevLett.93.011803

T. Barnes, S. Godfrey and E. S. Swanson, Higher charmonia.Phys. Rev. D72(2005) 054026.https://doi.org/10.1103/PhysRevD.72.054026Rev. Mex. Fis.67(1) 33–53

Downloads

Published

2021-01-07

How to Cite

[1]
R. Manzoor, J. Ahmed, and A. Raya, “A new variational approach and its application toheavy quarkonia”, Rev. Mex. Fís., vol. 67, no. 1 Jan-Feb, pp. 33–53, Jan. 2021.

Issue

Vol. 67 No. 1 Jan-Feb (2021): Revista Mexicana de Física

Section

08 High Energy Physics

License

Copyright (c) 2021 Rahila Manzoor, Jamil Ahmed, Alfredo Raya

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Authors retain copyright and grant the Revista Mexicana de Física right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.