Q\bar{Q} (Q\in\{b,c\}) spectroscopy using Cornell potential

() spectroscopy using Cornell potential

N. R. Soni nrsoni-apphy@msubaroda.ac.in    B. R. Joshi brijaljoshi99@gmail.com    R. P. Shah shahradhika61@gmail.com    H. R. Chauhan hemangichauhan29@gmail.com    J. N. Pandya jnpandya-apphy@msubaroda.ac.in Applied Physics Department, Faculty of Technology and Engineering,
The Maharaja Sayajirao University of Baroda, Vadodara 390001, Gujarat, India.
July 26, 2019
Abstract

The mass spectra and decay properties of heavy quarkonia are computed in nonrelativistic quark-antiquark Cornell potential model. We have employed the numerical solution of Schrödinger equation to obtain their mass spectra using only four parameters namely quark mass (, ) and confinement strength (, ). The spin hyperfine, spin-orbit and tensor components of the one gluon exchange interaction are computed perturbatively to determine the mass spectra of excited , , and states. Digamma, digluon and dilepton decays of these mesons are computed using the model parameters and numerical wave functions. The predicted spectroscopy and decay properties for quarkonia are found to be consistent with available experimental observations and results from other theoretical models. We also compute mass spectra and life time of the meson without additional parameters. The computed electromagnetic transition widths of heavy quarkonia and mesons are in tune with available experimental data and other theoretical approaches.

Cornell potential, decay properties, electromagnetic transition
pacs:
12.38.Bx; 12.39.Pn; 13.20.Gd; 13.40.Hq; 14.40.Pq

I Introduction

Mesonic bound states having both heavy quark and anti-quark (, and ) are among the best tools for understanding the quantum chromodynamics. Many experimental groups such as CLEO, LEP, CDF/D0 and NA50 have provided data and BABAR, Belle, CLEO-III, ATLAS, CMS and HERA-B are producing and expected to produce more precise data in upcoming experiments. Comprehensive reviews on the status of experimental heavy quarkonium physics are found in literature Eichten:2007; Godfrey:2008; Barnes:2009; Brambilla:2010; Brambilla:2014; Andronic:2016.

Within open flavor threshold, the heavy quarkonia have very rich spectroscopy with narrow and experimentally characterized states. The potential between the interacting quarks within the hadrons demands the understanding of underlying physics of strong interactions. In PDG pdg2016, large amount of experimental data is available for masses along with different decay modes. There are many theoretical groups viz. the lattice quantum chromodynamics (LQCD) Dudek:2007; Meinel:2009; Burch:2009; Liu:2012; McNeile:2012; Daldrop:2011; Kawanai:2013; Kawanai:2011; Burnier:2015; Kalinowski:2015; Burnier:2016, QCD Hilger:2014; Voloshin:2007, QCD sum rules Cho:2014; Gershtein:1995, perturbative QCD Kiyo:2013, lattice NRQCD Liu:2016; Dowdall:2013 and effective field theories Neubert:1993 that have attempted to explain the production and decays of these states. Others include phenomenological potential models such as the relativistic quark model based on quasi-potential approach Ebert:2011; Ebert:2009; Ebert:2005; Ebert:2002; Ebert:2003lepton; Ebert:2003gamma, where the relativistic quasi potential including one loop radiative corrections reproduce the mass spectrum of quarkonium states. The quasi potential has also been employed along with leading order radiative correction to heavy quark potential Gupta:1981; Gupta:1982kp; Gupta:1982; Pantaleone:1985, relativistic potential model Maung:1993; Radford:2007; Radford:2009 as well as semi relativistic potential model Gupta:1986. In nonrelativistic potential models, there exist several forms of quark anti-quark potentials in the literature. The most common among them is the coulomb repulsive plus quark confinement interaction potential. In our previous work Vinodkumar:1999; Pandya:2001; Rai:2008; Pandya:2014, we have employed the confinement scheme based on harmonic approximation along with Lorentz scalar plus vector potential. The authors of Devlani:2014; Parmar:2010; Rai:2002; Rai:2005; Rai:2006; Rai:2008prc; Patel:2008 have considered the confinement of power potential with varying from 0.1 to 2.0 and the confinement strength to vary with potential index . Confinement of the order have also been attempted FabreDeLaRipelle:1988. Linear confinement (Cornell potential) of quarks has been considered by many groups Eichten:1974; Eichten:1978; Eichten:1979; Quigg:1979; Eichten:1980; Barnes:2005; Sauli:2011; Leitao:2014; Godfrey:1985; Godfrey:2004; Godfrey:2015; Deng:2016cc; Deng:2016bb and they have provided good agreement with the experimental data for quarkonium spectroscopy along with decay properties. The Bethe-Salpeter approach was also employed for the mass spectroscopy of charmonia and bottomonia Sauli:2011; Leitao:2014; Fischer:2014. The quarkonium mass spectrum was also computed in the nonrelativistic quark model Lakhina:2006, screened potential model Deng:2016cc; Deng:2016bb and constituent quark model Segovia:2016. There are also other non-linear potential models that predict the mass spectra of the heavy quarkonia successfully Patel:2015; Bonati:2015; Gutsche:2014; Li:2009; Li:2009bb; Quigg:1977; Martin:1980; Buchmuller:1980.

In 90’s, the nonrelativistic potential models predicted not only the ground state mass of the tightly bound state of and in the range of 6.2–6.3 GeV Kwong:1990; Eichten:1994 but also predicted to have very rich spectroscopy. In 1998, CDF collaboration Abe:1998 reported mesons in collisions at = 1.8 TeV and was later confirmed by D0 Abazov:2008 and LHCb Aaij:2012 collaborations. The LHCb collaboration has also made the most precise measurement of the life time of mesons Aaij:2014. The first excited state is also reported by ATLAS Collaborations Aad:2014 in collisions with significance of .

It is important to show that any given potential model should be able to compute mass spectra and decay properties of meson using parameters fitted for heavy quarkonia. Attempts in this direction have been made in relativistic quark model based on quasi-potential along with one loop radiative correction Ebert:2011, quasistatic and confinement QCD potential with confinement parameters along with quark masses Gupta:1996 and rainbow ladder approximation of Dyson-Schwinger and Bethe-Salpeter equations Fischer:2014.

Moreover, the mesonic states are identified with masses along with certain decay channels, therefore the test for any successful theoretical model is to reproduce the mass spectrum along with decay properties. Relativistic as well as nonrelativistic potential models have successfully predicted the spectroscopy but they are found to differ in computation of the decay properties Quigg:1977; Eichten:1978; Martin:1980; Buchmuller:1980; Gershtein:1995; Rai:2002; Rai:2005; Rai:2006; Rai:2008prc; Parmar:2010. This discrepancy motivates us to employ nonrelativistic potential of the one gluon exchange (essentially Coulomb like) plus linear confinement (Cornell potential) as this form of the potential is also supported by LQCD Bali:2000; Bali:2001; Alexandrou:2002. We solve the Schrödinger equation numerically for the potential to get the spectroscopy of the quarkonia. We first compute the mass spectra of charmonia and bottomonia states to determine quark masses and confinement strengths after fitting the spin-averaged ground state masses with experimental data of respective mesons. Using the potential parameters and numerical wave function, we compute the decay properties such as leptonic decay constants, digamma, dilepton, digluon decay width using the Van-Royen Weiskopf formula. These parameters are then used to compute the mass spectra and life-time of meson. We also compute the electromagnetic ( and ) transition widths of heavy quarkonia and mesons.

Ii Methodology

Basically, the bound state of two body system within relativistic quantum field is described in Bethe-Salpeter formalism. But the Bethe-Salpeter equation is solved only in the ladder approximations. Also Bethe-Salpeter approach in harmonic confinement is successful in low flavor sectors Isgur:1978; VijayaKumar:1997. Therefore the alternative treatment for the heavy bound state is nonrelativistic. Also due to negligible momenta of quark and anti quark compared to mass of quark-antiquark system , which constitutes the basis of the nonrelativistic treatment for the heavy quarkonium spectroscopy. Here for the study of heavy bound state of mesons such as , and , the nonrelativistic Hamiltonian is given by

(1)

where

(2)

where and are the mass of quark and antiquark, is the relative momentum of the each quark and is the quark-antiquark potential of the type coulomb plus linear confinement (Cornell potential) given by

(3)

Here, term is analogous to the Coulomb type interaction corresponding to the potential induced between quark and antiquark through one gluon exchange that dominates at small distances. The second term is the confinement part of the potential where the confinement strength is the model parameter. The confinement term becomes dominant at the large distances. is a strong running coupling constant and can be computed as

(4)

where is the numbers of flavors, is renormalization scale related to the constituent quark masses as and is a QCD scale which is taken as 0.15 GeV by fixing = 0.1185 pdg2016 at the -boson mass.

The confinement strengths with respective quark masses are fine tuned to reproduce the experimental spin averaged ground state masses of both and mesons and are given in Table 1. We compute the masses of higher excited states without any additional parameters. Similar kind of work has been done by Patel:2008; Rai:2008prc; Parmar:2010 and they have considered different values of confinement strengths for different potential indices. The Cornell potential has been shown to be independently successful in computing the spectroscopy of and families. In this article, we compute the mass spectra of the and families along with meson with minimum number of parameters.

Using the parameters defined in Table 1, we compute the spin averaged masses of quarkonia. In order to compute masses of different states according to different values, we use the spin dependent part of one gluon exchange potential (OGEP) perturbatively. The OGEP includes spin-spin, spin-orbit and tensor terms given by Gershtein:1995; Barnes:2005; Lakhina:2006; Voloshin:2007 {dmath} V_SD (r) = V_SS (r) [S(S+1) - 32] + V_LS(r) (→L⋅→S) + V_T(r) [S(S+1)-3(S⋅^r) (S⋅^r)]

1.317 GeV 4.584 GeV 0.18 GeV 0.25 GeV
Table 1: Parameters for quarkonium spectroscopy

The spin-spin interaction term gives the hyper-fine splitting while spin-orbit and tensor terms gives the fine structure of the quarkonium states. The coefficients of spin dependent terms of the Eq. (II) can be written as Voloshin:2007

State Present PDG pdg2016 Ebert:2011 Deng:2016cc Patel:2015 Fischer:2014 LQCD Kalinowski:2015
2.989 2.984 2.981 2.984 2.979 2.925 2.884
3.094 3.097 3.096 3.097 3.096 3.113 3.056
3.602 3.639 3.635 3.637 3.600 3.684 3.535
3.681 3.686 3.685 3.679 3.680 3.676 3.662
4.058 4.039 3.989 4.004 4.011
4.129 4.039 4.030 4.077 3.803
4.448 4.421 4.401 4.264 4.397
4.514 4.427 4.281 4.454
4.799 4.811 4.459
4.863 4.837 4.472
5.124 5.155
5.185 5.167
3.428 3.415 3.413 3.415 3.488 3.323 3.412
3.468 3.511 3.511 3.521 3.514 3.489 3.480
3.470 3.525 3.525 3.526 3.539 3.433 3.494
3.480 3.556 3.555 3.553 3.565 3.550 3.536
3.897 3.918 3.870 3.848 3.947 3.833
3.938 3.906 3.914 3.972 3.672
3.943 3.926 3.916 3.996 3.747
3.955 3.927 3.949 3.937 4.021 4.066
4.296 4.301 4.146
4.338 4.319 4.192 3.912
4.344 4.337 4.193
4.358 4.354 4.211
4.653 4.698
4.696 4.728
4.704 4.744
4.718 4.763
4.983
5.026
5.034
5.049
Table 2: Mass spectrum of and wave charmonia (in GeV)
(5)
(6)
(7)

Where and correspond to the vector and scalar part of the Cornell potential in Eq. (3) respectively. Using all the parameters defined above, the Schrödinger equation is numerically solved using Mathematica notebook utilizing the Runge-Kutta method Lucha:1998. The computed mass spectra of heavy quarkonia and mesons are listed in Table 27

State Present Ebert:2011 Deng:2016cc Patel:2015 Fischer:2014
3.755 3.813 3.808 3.798 3.869
3.765 3.807 3.805 3.796 3.739
3.772 3.795 3.807 3.794 3.550
3.775 3.783 3.792 3.792
4.176 4.220 4.112 4.425 3.806
4.182 4.196 4.108 4.224
4.188 4.190 4.109 4.223
4.188 4.105 4.095 4.222
4.549 4.574 4.340
4.553 3.549 4.336
4.557 4.544 4.337
4.555 4.507 4.324
4.890 4.920
4.892 4.898
4.896 4.896
4.891 4.857
3.990 4.041
4.012 4.068 3.999
4.017 4.071 4.037
4.036 4.093
4.378 4.361
4.396 4.400
4.400 4.406
4.415 4.434
4.730
4.746
4.749
4.761
Table 3: Mass spectrum of and wave charmonia (in GeV)
State Present PDG pdg2016 Godfrey:2015 Ebert:2011 Deng:2016bb Segovia:2016 Fischer:2014
9.428 9.398 9.402 9.398 9.390 9.455 9.414
9.463 9.460 9.465 9.460 9.460 9.502 9.490
9.955 9999 9.976 9.990 9.990 9.990 9.987
9.979 10.023 10.003 10.023 10.015 10.015 10.089
10.338 10.336 10.329 10.326 10.330
10.359 10.355 10.354 10.355 10.343 10.349 10.327
10.663 10.523 10.573 10.584
10.683 10.579 10.635 10.586 10.597 10.607
10.956 10.869 10.851 10.800
10.975 10.876 10.878 10.869 10.811 10.818
11.226 11.097 11.061 10.997
11.243 11.019 11.102 11.088 10.988 10.995
9.806 9.859 9.847 9.859 9.864 9.855 9.815
9.819 9.893 9.876 9.892 9.903 9.874 9.842
9.821 9.899 9.882 9.900 9.909 9.879 9.806
9.825 9.912 9.897 9.912 9.921 9.886 9.906
10.205 10.232 10.226 10.233 10.220 10.221 10.254
10.217 10.255 10.246 10.255 10.249 10.236 10.120
10.220 10.260 10.250 10.260 10.254 10.240 10.154
10.224 10.269 10.261 10.268 10.264 10.246
10.540 10.552 10.521 10.490 10.500
10.553 10.538 10.541 10.515 10.513 10.303
10.556 10.541 10.544 10.519 10.516
10.560 10.550 10.550 10.528 10.521 –
10.840 10.775 10.781
10.853 10.788 10.802
10.855 10.790 10.804
10.860 10.798 10.812
11.115 11.004
11.127 11.014
11.130 11.016
11.135 11.022
Table 4: Mass spectrum of and wave bottomonia (in GeV)
State Present PDG pdg2016 Godfrey:2015 Ebert:2011 Deng:2016bb Segovia:2016 Fischer:2014
10.073 10.115 10.166 10.157 10.127 10.232
10.074 10.148 10.163 10.153 10.123 10.194
10.075 10.163 10.147 10.161 10.153 10.122 10.145
10.074 10.138 10.154 10.146 10.117
10.423 10.455 10.449 10.436 10.422
10.424 10.450 10.445 10.432 10.419
10.424 10.449 10.443 10.432 10.418
10.423 10.441 10.435 10.425 10.414
10.733 10.711 10.717
10.733 10.706 10.713
10.733 10.705 10.711
10.731 10.698 10.704
11.015 10.939 10.963
11.015 10.935 10.959
11.016 10.934 10.957
11.013 10.928 10.949
10.283 10.350 10.343 10.338 10.315
10.287 10.355 10.346 10.340 10.321 10.302
10.288 10.355 10.347 10.339 10.322 10.319
10.291 10.358 10.349 10.340
10.604 10.615 10.610
10.607 10.619 10.614
10.607 10.619 10.647
10.609 10.622 10.617
10.894 10.850
10.896 10.853
10.897 10.853
10.898 10.856
Table 5: Mass spectrum of and wave bottomonia (in GeV)
State Present Devlani:2014 Ebert:2011 Godfrey:2004 Monteiro:2016 PDG pdg2016
6.272 6.278 6.272 6.271 6.275 6.275
6.321 6.331 6.333 6.338 6.314
6.864 6.863 6.842 6.855 6.838 6.842
6.900 6.873 6.882 6.887 6.850
7.306 7.244 7.226 7.250
7.338 7.249 7.258 7.272
7.684 7.564 7.585
7.714 7.568 7.609
8.025 7.852 7.928
8.054 7.855 7.947
8.340 8.120
8.368 8.122
6.686 6.748 6.699 6.706 6.672
6.705 6.767 6.750 6.741 6.766
6.706 6.769 6.743 6.750 6.828
6.712 6.775 6.761 6.768 6.776
7.146 7.139 7.094 7.122 6.914
7.165 7.155 7.134 7.145 7.259
7.168 7.156 7.094 7.150 7.322
7.173 7.162 7.157 7.164 7.232
7.536 7.463 7.474
7.555 7.479 7.510
7.559 7.479 7.500
7.565 7.485 7.524
7.885 7.817
7.905 7.853
7.908 7.844
7.915 7.867
8.207
8.226
8.230
8.237
Table 6: Mass spectrum of and wave meson (in GeV)
State Present Devlani:2014 Ebert:2011 Godfrey:2004 Monteiro:2016
6.990 7.026 7.029 7.045 6.980
6.994 7.035 7.026 7.041 7.009
6.997 7.025 7.025 7.036 7.154
6.998 7.030 7.021 7.028 7.078
7.399 7.363 7.405
7.401 7.370 7.400
7.403 7.361 7,399
7.403 7.365 7.392
7.761 7.750
7.762 7.743
7.764 7.741
7.762 7.732
8.092
8.093
8.094
8.091
7.234 7.273 7.269
7.242 7.269 7.276
7.241 7.268 7.266
7.244 7.277 7.271
7.607 7.618
7.615 7.616
7.614 7.615
7.617 7.617
7.946
7.954
7.953
7.956
Table 7: Mass spectrum of and wave meson (in GeV)

Iii Decay properties

The mass spectra of the hadronic states are experimentally determined through detection of energy and momenta of daughter particles in various decay channels. Generally, most phenomenological approaches obtain their model parameters like quark masses and confinement/Coulomb strength by fitting with the experimental ground states. So it becomes necessary for any phenomenological model to validate their fitted parameters through proper evaluation of various decay rates in general and annihilation rates in particular. In the nonrelativistic limit, the decay properties are dependent on the wave function. In this section, we test our parameters and wave functions to determine various annihilation widths and electromagnetic transitions.

iii.1 Leptonic decay constants

The leptonic decay constants of heavy quarkonia play very important role in understanding the weak decays. The matrix elements for leptonic decay constants of pseudoscalar and vector mesons are given by

(8)
(9)

where is the momentum of pseudoscalar meson, is the polarization vector of meson. In the nonrelativistic limit, the decay constants of pseudoscalar and vector mesons are given by Van Royen-Weiskopf formula VanRoyen:1967

(10)

Here the QCD correction factor Braaten:1995; Berezhnoy:1996

(11)

With = 2 and = 8/3. Using the above relations, we compute the leptonic decay constants and for charmonia, bottomonia and mesons and are listed in Table 813.

State Patel:2008 Krassnigg:2016 Lakhina:2006 LQCDBecirevic:2013 QCDSR Becirevic:2013
440.406 350.314 363 378 402 387(7)(2) 309 39
350.056 278.447 275 82 240
313.354 249.253 239 206 193
290.673 231.211 217 82
274.367 218.241 202
261.698 208.163 197
Table 8: Pseudoscalar decay constant of charmonia (in MeV)
State Patel:2008 Krassnigg:2016 Lakhina:2006 LQCDBecirevic:2013 QCDSR Becirevic:2013
448.096 325.876 338 411 393 418(8)(5) 401 46
353.855 257.34 254 155 293
316.064 229.857 220 188 258
292.829 212.959 200 262
276.177 200.848 186
263.266 191.459 175
Table 9: Vector decay constant of charmonia (in MeV)
State Patel:2008 Krassnigg:2016 Pandya:2001 Lakhina:2006
646.025 744 756 711 599
518.803 577 285 411
474.954 511 333 354
449.654 471 40
432.072 443
418.645 422
Table 10: Pseudoscalar decay constant of bottomonia (in MeV)
State Patel:2008 Krassnigg:2016 Lakhina:2006 Wang:2005 LQCDColquhoun:2014
647.250 706 707 665 498 649(31)
519.436 547 393 475 366 481(39)
475.440 484 9 418 304
450.066 446 20 388 259
432.437 419 367 228
418.977 399 351
Table 11: Vector decay constant of bottomonia (in MeV)
State Patel:2008 Ebert:2002 Gershtein:1995 Eichten:1994 Monteiro:2016
432.955 465 503 460 500 554.125
355.504 361
325.659 319
307.492 293
294.434 275
284.237 261
Table 12: Pseudoscalar decay constant of meson (in MeV)
State Patel:2008 Ebert:2002 Gershtein:1995 Eichten:1994
434.642 435 433 460 500
356.435 337
326.374 297
308.094 273
294.962 256
284.709 243
Table 13: Vector decay constant of meson (in MeV)

iii.2 Annihilation of heavy quarkonia

Digamma, digluon and dilepton annihilation decay rates of heavy quarkonia are very important in understanding the dynamics of heavy quarks within the mesons. The measurement of digamma decay widths provides the information regarding the internal structure of meson. The decay , was reported by CLEO-c Ecklund:2008, BABAR Lees:2010 and then BESIII Ablikim:2012 collaboration have reported high accuracy data. LQCD is found to underestimate the decay widths of and when compared to experimental data Dudek:2006; ChenT:2016. Other approaches to attempt computation of annihilation rates of heavy quarkonia include NRQCD Bodwin:1994; Khan:1995; Schuler:1997; Bodwin:2006; Bodwin:2007, relativistic quark model Ebert:2003gamma; Ebert:2003lepton, effective Lagrangian Lansberg:2009; Lansberg:2006 and next-to-next-to leading order QCD correction to in the framework of nonrelativistic QCD factorization Sang:2015.

The meson decaying into digamma suggests that the spin can never be one Landau:1948; Yang:1950. Corresponding digamma decay width of a pseudoscalar meson in nonrelativistic limit is given by Van Royen-Weiskopf formula VanRoyen:1967

(12)
(13)
(14)

where the bracketed quantities are QCD next-to-leading order radiative corrections Kwong:1988; Barbieri:1981.

Digluon annihilation of quarkonia is not directly observed in detectors as digluonic state decays into various hadronic states making it a bit complex to compute digluon annihilation widths from nonrelativistic approximations derived from first principles. The digluon decay width of pseudoscalar meson along with the QCD leading order radiative correction is given by Lansberg:2009; Kwong:1988; Barbieri:1981; Mangano:1995

(15)
(16)
(17)

The vector mesons have quantum numbers and can annihilate into dilepton. The dileptonic decay of vector meson along with one loop QCD radiative correction is given by VanRoyen:1967; Kwong:1988

(18)

Here, is the electromagnetic coupling constant, is the strong running coupling constant in Eq. (4) and is the charge of heavy quark in terms of electron charge. In above relations, corresponds to the wave function of wave at origin for pseudoscalar and vector mesons while is the derivative of wave wave function at origin. The annihilation rates of heavy quarkonia are listed in Table 14 - 19.

State Li:2009 Ebert:2003gamma Lakhina:2006 Kim:2004 PDG pdg2016
8.582 5.618 8.5 5.5 7.18 7.140.95 5.10.4
4.498 2.944 2.4 1.8 1.71 4.440.48 2.151.58
3.200 2.095 0.88 1.21
2.512 1.644
2.074 1.358
1.768 1.158
11.477 11.687 2.5 2.9 3.28 2.340.19
3.107 1.412 0.31 0.50 0.530.4
13.235 13.477 1.7 1.9
3.582 1.628 0.23 0.52
14.580 14.847 1.2
3.944 1.792 0.17
15.746 16.034
4.257 1.935
16.807 17.114
4.541 2.064
Table 14: Digamma decay width of and wave charmonia (in keV)
State Li:2009bb Godfrey:1985 Ebert:2003gamma Lakhina:2006 Kim:2004
0.366 0.527 0.214 0.35 0.23 0.384 0.047
0.223 0.263 0.121 0.15 0.07 0.191 0.025
0.180 0.172 0.906 0.10 0.04
0.157 0.105 0.755
0.141 0.121
0.129 0.050
0.021 0.050 0.0208 0.038
0.0060 0.0066 0.0051 0.008
0.022 0.037 0.0227 0.029
0.0058 0.0067 0.0062 0.006
0.022 0.037
0.0059 0.0064
0.023
0.0061
0.023
0.0062
Table 15: Digamma decay width of and wave bottomonia (in keV)
State Patel:2015 Kim:2004 PDG pdg2016
27.325 40.741 22.37 19.60 26.73.0
25.081 37.394 16.74 12.1 14.70.7
25.502 38.022 14.30
26.365 39.310
27.345 40.711
28.353 42.273
25.022 47.468 9.45 10.40.7
6.775 3.430 2.81 2.030.12
28.855 54.738 10.09
7.810 3.954 7.34
31.788 60.302
8.599 4.353
34.330 65.125
9.281 4.699
36.642 69.511
9.900 5.012
Table 16: Digluon decay width of and wave charmonia (in MeV)
State Parmar:2010 Gupta:1996gg
5.602 17.945 12.46
4.029
3.641
3.473
3.385
3.336
0.295 5.250 2.15
0.079 0.822 0.22
0.306
0.082
0.314
0.084
0.320
0.085
0.326
0.087
Table 17: Digluon decay width of and wave bottomonia (in MeV)
State Shah:2012 Patel:2008 Radford:2007 Ebert:2003lepton PDG pdg2016
6.437 2.925 4.95 6.99 1.89 5.4 5.547 0.14
3.374 1.533 1.69 3.38 1.04 2.4 2.359 0.04
2.400 1.091 0.96 2.31 0.77 0.86 0.07
1.884 0.856 0.65 1.78 0.65 0.58 0.07
1.556 0.707 0.49 1.46
1.326 0.602 0.39 1.24
Table 18: Dilepton decay width of charmonia (in kev)
State Shah:2012 Radford:2009 Patel:2008 Ebert:2003lepton Gonzalez:2003 PDGpdg2016
1.098 1.20 1.33 1.61 1.3 0.98 1.340 0.018
0.670 0.52 0.62 0.87 0.5 0.41 0.612 0.011
0.541 0.33 0.48 0.66 0.27 0.443 0.008
0.470 0.24 0.40 0.53 0.20 0.272 0.029
0.422 0.19 0.44 0.16
0.387 0.16 0.39 0.12
Table 19: Dilepton decay width of bottomonia (in keV)

iii.3 Electromagnetic transition widths

The electromagnetic transitions can be determined broadly in terms of electric and magnetic multipole expansions and their study can help in understanding the non-perturbative regime of QCD. We consider the leading order terms i.e. electric () and magnetic () dipoles with selection rules and for the transitions while and for transitions. We now employ the numerical wave function for computing the electromagnetic transition widths among quarkonia and meson states in order to test parameters used in present work. For transition, we restrict our calculations for transitions among waves only. In the nonrelativistic limit, the radiative and widths are given by Eichten:1974; Eichten:1978; Radford:2009; Brambilla:2010; Li:2010 {dmath} Γ(n^2S+1L_iJ_i →n^,2S+1L_fJ_f + γ) = 4 αe⟨eQ2ω33 (2 J_f + 1) S_if^E1 —M_if^E1—^2 {dmath} Γ(n^3S_1 →n^,1S_0+ γ) = αeμ2ω33 (2 J_f + 1) S_if^M1 —M_if^M1—^2 where, mean charge content of the system, magnetic dipole moment and photon energy are given by

(19)
(20)

and

(21)

respectively. Also the symmetric statistical factors are given by

(22)

and

(23)

The matrix element for and transition can be written as

(24)

and

(25)

The electromagnetic transition widths are listed in Table 20 - 25 and also compared with experimental results as well as theoretical predictions.

Transition Present Radford:2007 Ebert:2002 Li:2009 Deng:2016cc PDG pdg2016
21.863 45.0 51.7 74 22 26.3 2.6
43.292 40.9 44.9 62 42 25.5 2.8
62.312 26.5 30.9 43 38 25.2 2.9
36.197 8.3 8.6 146 49
31.839 87.3
64.234 65.7
86.472 31.6
51.917
46.872 1.2
107.088 2.5
163.485 3.3
178.312
112.030 142.2 161 167 284 371 34
146.317 287.0 333 354 306 285 14
157.225 390.6 448 473 172 133 8
247.971 610.0 723 764 361 257 2.80
70.400 53.6 61
102.672 208.3 103
116.325 358.6 225
163.646 309
173.324 20.8 74
210.958 28.4 83
227.915 33.2 101
329.384 134
161.504 423 486 272 202 42
93.775 142 150 138 81 27
5.722 5.8 5.8 7.1 24.8
165.176 317.3 297 342 285
50.317 65.7 62 70 91
175.212 62.7 252 284 350
205.93 335 575 362
Table 20: transition width of charmonia (in keV)
Transition Present Radford:2007 Ebert:2002 Li:2009bb Deng:2016bb PDG pdg2016
2.377 1.15 1.65 1.67 1.09 1.22 0.11
5.689 1.87 2.57 2.54 2.17 2.21 0.19
8.486 1.88 2.53 2.62 2.62 2.29 0.20
10.181 4.17 3.25 6.10 3.41
3.330 1.67 1.65 1.83 1.21 1.20 0.12
7.936 2.74 2.65 2.96 2.61 2.56 0.26
11.447 2.80 2.89 3.23 3.16 2.66 0.27
0.594 0.03 0.124 0.07 0.097 0.055 0.010
1.518 0.09 0.307 0.17 0.0005 0.018 0.010
2.354 0.13 0.445 0.15 0.14 0.20 0.03
3.385 0.03 0.770 1.24 0.67
13.981 3.07 11.0 4.25
57.530 31.2 29.5 38.2 31.8
54.927 27.3 37.1 33.6 31.9
49.530 22.1 42.7 26.6 27.5
72.094 37.9 54.4 55.8 35.8
28.848 16.8 18.8 18.8 15.5 15.1 5.6
26.672 13.7 15.9 15.9