Abstract
Within the framework of non-relativistic QCD (NRQCD) factorization, we compute the matching coefficients between full Quantum Chromodynamics (QCD) and NRQCD for the heavy flavor-changing vector, axial-vector, scalar and pseudo-scalar currents up to next-to-next-to-next-to-leading order (N3LO). We accomplish the analytical expressions for the three-loop renormalization constants and the corresponding anomalous dimensions for all of the four NRQCD currents with two different heavy flavors. The three-loop QCD corrections to the matching coefficients turn out to be significantly larger than lower order corrections. By employing the scale relation, we approximately simulate the N3LO correction to the wave functions at the origin for the vector \( {B}_c^{\ast } \) meson and the pseudo-scalar Bc meson from the known result for the equal-mass heavy quarkonium in potential NRQCD (pNRQCD). We find large cancellations at the third order between the matching coefficients and the wave functions at the origin, and obtain the convergent decay constants of \( {B}_c^{\ast } \) and Bc up to N3LO. We present the complete perturbative QCD predictions for the decay constants, leptonic decay widths, and branching ratios of the beauty-charmed mesons.
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References
G.T. Bodwin, E. Braaten and G.P. Lepage, Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium, Phys. Rev. D 51 (1995) 1125 [Erratum ibid. 55 (1997) 5853] [hep-ph/9407339] [INSPIRE].
M. Beneke, A. Signer and V.A. Smirnov, Two loop correction to the leptonic decay of quarkonium, Phys. Rev. Lett. 80 (1998) 2535 [hep-ph/9712302] [INSPIRE].
A. Czarnecki and K. Melnikov, Two loop QCD corrections to the heavy quark pair production cross-section in e+e− annihilation near the threshold, Phys. Rev. Lett. 80 (1998) 2531 [hep-ph/9712222] [INSPIRE].
B.A. Kniehl, A. Onishchenko, J.H. Piclum and M. Steinhauser, Two-loop matching coefficients for heavy quark currents, Phys. Lett. B 638 (2006) 209 [hep-ph/0604072] [INSPIRE].
B.A. Kniehl, A.A. Penin, M. Steinhauser and V.A. Smirnov, Heavy quarkonium creation and annihilation with \( \mathcal{O}\left({\alpha}_s^3\ln \left({\alpha}_s\right)\right) \) accuracy, Phys. Rev. Lett. 90 (2003) 212001 [Erratum ibid. 91 (2003) 139903] [hep-ph/0210161] [INSPIRE].
P. Marquard, J.H. Piclum, D. Seidel and M. Steinhauser, Fermionic corrections to the three-loop matching coefficient of the vector current, Nucl. Phys. B 758 (2006) 144 [hep-ph/0607168] [INSPIRE].
P. Marquard, J.H. Piclum, D. Seidel and M. Steinhauser, Completely automated computation of the heavy-fermion corrections to the three-loop matching coefficient of the vector current, Phys. Lett. B 678 (2009) 269 [arXiv:0904.0920] [INSPIRE].
M. Beneke, Y. Kiyo and K. Schuller, Third-order correction to top-quark pair production near threshold I. Effective theory set-up and matching coefficients, arXiv:1312.4791 [INSPIRE].
P. Marquard, J.H. Piclum, D. Seidel and M. Steinhauser, Three-loop matching of the vector current, Phys. Rev. D 89 (2014) 034027 [arXiv:1401.3004] [INSPIRE].
F. Feng et al., Complete three-loop QCD corrections to leptonic width of vector quarkonium, arXiv:2207.14259 [INSPIRE].
J.H. Piclum, Heavy quark threshold dynamics in higher order, Ph.D. thesis, Hamburg U., Hamburg, Germany (2007) [INSPIRE].
M. Egner et al., Charm-quark mass effects in NRQCD matching coefficients and the leptonic decay of the Υ(1S) meson, Phys. Rev. D 104 (2021) 054033 [arXiv:2105.09332] [INSPIRE].
M. Egner et al., Three-loop nonsinglet matching coefficients for heavy quark currents, Phys. Rev. D 105 (2022) 114007 [arXiv:2203.11231] [INSPIRE].
M. Beneke et al., Next-to-next-to-next-to-leading order QCD prediction for the top antitop S-wave pair production cross section near threshold in e+e− annihilation, Phys. Rev. Lett. 115 (2015) 192001 [arXiv:1506.06864] [INSPIRE].
M. Beneke et al., Leptonic decay of the Υ(1S) meson at third order in QCD, Phys. Rev. Lett. 112 (2014) 151801 [arXiv:1401.3005] [INSPIRE].
E. Braaten and S. Fleming, QCD radiative corrections to the leptonic decay rate of the Bc meson, Phys. Rev. D 52 (1995) 181 [hep-ph/9501296] [INSPIRE].
D.S. Hwang and S. Kim, QCD radiative correction to the decay of Bc and B∗c, Phys. Rev. D 60 (1999) 034022 [INSPIRE].
J. Lee, W.L. Sang and S. Kim, Relativistic corrections to the axial vector and vector currents in the \( \overline{b}c \) meson system at order alphas, JHEP 01 (2011) 113 [arXiv:1011.2274] [INSPIRE].
A.I. Onishchenko and O.L. Veretin, Two loop QCD corrections to Bc meson leptonic constant, Eur. Phys. J. C 50 (2007) 801 [hep-ph/0302132] [INSPIRE].
L.-B. Chen and C.-F. Qiao, Two-loop QCD corrections to Bc meson leptonic decays, Phys. Lett. B 748 (2015) 443 [arXiv:1503.05122] [INSPIRE].
W. Tao, R. Zhu and Z.-J. Xiao, Next-to-next-to-leading order matching of beauty-charmed meson Bc and B∗c decay constants, Phys. Rev. D 106 (2022) 114037 [arXiv:2209.15521] [INSPIRE].
F. Feng et al., Three-loop QCD corrections to the decay constant of Bc, arXiv:2208.04302 [INSPIRE].
W.-L. Sang, H.-F. Zhang and M.-Z. Zhou, Decay constant of B∗c accurate up to \( O\left({\alpha}_s^3\right) \), Phys. Lett. B 839 (2023) 137812 [arXiv:2210.02979] [INSPIRE].
W. Tao, R. Zhu and Z.-J. Xiao, Three-loop QCD matching of the flavor-changing scalar current involving the heavy charm and bottom quark, Eur. Phys. J. C 83 (2023) 294 [arXiv:2301.00220] [INSPIRE].
D.J. Broadhurst and A.G. Grozin, Matching QCD and HQET heavy-light currents at two loops and beyond, Phys. Rev. D 52 (1995) 4082 [hep-ph/9410240] [INSPIRE].
V.V. Kiselev and A.I. Onishchenko, Two loop anomalous dimensions for currents of baryons with two heavy quarks in NRQCD, hep-ph/9810283 [INSPIRE].
G. Bell, M. Beneke, T. Huber and X.-Q. Li, Heavy-to-light currents at NNLO in SCET and semi-inclusive \( \overline{B}\to {X}_s{l}^{+}{l}^{-} \) decay, Nucl. Phys. B 843 (2011) 143 [arXiv:1007.3758] [INSPIRE].
W. Tao, Z.-J. Xiao and R. Zhu, Next-to-leading order QCD calculation of Bc to charmonium tensor form factors, Phys. Rev. D 105 (2022) 114026 [arXiv:2204.06385] [INSPIRE].
A. Pineda and J. Soto, Effective field theory for ultrasoft momenta in NRQCD and NRQED, Nucl. Phys. B Proc. Suppl. 64 (1998) 428 [hep-ph/9707481] [INSPIRE].
M. Beneke, Perturbative heavy quark-anti-quark systems, PoS hf8 (1999) 009 [hep-ph/9911490] [INSPIRE].
N. Brambilla, A. Pineda, J. Soto and A. Vairo, Potential NRQCD: an effective theory for heavy quarkonium, Nucl. Phys. B 566 (2000) 275 [hep-ph/9907240] [INSPIRE].
M.E. Luke, A.V. Manohar and I.Z. Rothstein, Renormalization group scaling in nonrelativistic QCD, Phys. Rev. D 61 (2000) 074025 [hep-ph/9910209] [INSPIRE].
N. Brambilla, A. Pineda, J. Soto and A. Vairo, Effective field theories for heavy quarkonium, Rev. Mod. Phys. 77 (2005) 1423 [hep-ph/0410047] [INSPIRE].
A.A. Penin and A.A. Pivovarov, Top quark threshold production in γγ collision in the next-to-leading order, Nucl. Phys. B 550 (1999) 375 [hep-ph/9810496] [INSPIRE].
A.A. Penin and A.A. Pivovarov, Analytical results for e+e−→ \( t\overline{t} \) and γγ → \( t\overline{t} \) observables near the threshold up to the next-to-next-to leading order of NRQCD, Phys. Atom. Nucl. 64 (2001) 275 [hep-ph/9904278] [INSPIRE].
M. Beneke, Y. Kiyo, A. Penin and K. Schuller, NNNLO correction to the toponium and bottomonium wave-functions at the origin, eConf C0705302 (2007) TOP01 [arXiv:0710.4236] [INSPIRE].
C. Peset, A. Pineda and M. Stahlhofen, Potential NRQCD for unequal masses and the Bc spectrum at N3LO, JHEP 05 (2016) 017 [arXiv:1511.08210] [INSPIRE].
J.-M. Shen et al., Renormalization group improved pQCD prediction for Υ(1S) leptonic decay, JHEP 06 (2015) 169 [arXiv:1501.04688] [INSPIRE].
Y. Schroder, The static potential in QCD to two loops, Phys. Lett. B 447 (1999) 321 [hep-ph/9812205] [INSPIRE].
K. Schuller, Third-order QCD corrections to heavy quark pair production near threshold, Ph.D. thesis, RWTH Aachen U., Aachen, Germany (2008) [INSPIRE].
M. Beneke, Y. Kiyo and K. Schuller, Third-order Coulomb corrections to the S-wave Green function, energy levels and wave functions at the origin, Nucl. Phys. B 714 (2005) 67 [hep-ph/0501289] [INSPIRE].
M. Beneke, Y. Kiyo and K. Schuller, Third-order non-Coulomb correction to the S-wave quarkonium wave functions at the origin, Phys. Lett. B 658 (2008) 222 [arXiv:0705.4518] [INSPIRE].
M. Beneke, Y. Kiyo and A.A. Penin, Ultrasoft contribution to quarkonium production and annihilation, Phys. Lett. B 653 (2007) 53 [arXiv:0706.2733] [INSPIRE].
M. Beneke, Y. Kiyo, A. Maier and J. Piclum, Near-threshold production of heavy quarks with QQbar_threshold, Comput. Phys. Commun. 209 (2016) 96 [arXiv:1605.03010] [INSPIRE].
M. Beneke, A. Maier, J. Piclum and T. Rauh, The bottom-quark mass from non-relativistic sum rules at NNNLO, Nucl. Phys. B 891 (2015) 42 [arXiv:1411.3132] [INSPIRE].
A.A. Penin and N. Zerf, Bottom quark mass from Υ sum rules to \( \mathcal{O}\left({\alpha}_s^3\right) \), JHEP 04 (2014) 120 [arXiv:1401.7035] [INSPIRE].
H.S. Chung and D. Nogradi, The f𝜚/m𝜚 and fπ/m𝜚 ratios and the conformal window, Phys. Rev. D 107 (2023) 074039 [arXiv:2302.06411] [INSPIRE].
A.A. Penin, A. Pineda, V.A. Smirnov and M. Steinhauser, \( M\left({B}_c^{\ast}\right) \) − M (Bc) splitting from nonrelativistic renormalization group, Phys. Lett. B 593 (2004) 124 [Erratum ibid. 677 (2009) 343] [hep-ph/0403080] [INSPIRE].
C. Anzai, D. Moreno and A. Pineda, S-wave heavy quarkonium spectrum with next-to-next-to-next-to-leading logarithmic accuracy, Phys. Rev. D 98 (2018) 114034 [arXiv:1810.11031] [INSPIRE].
D. Moreno Torres, Non-relativistic bound states in QCD: towards the N3LL heavy quarkonium spectrum, Ph.D. thesis, Autonoma U., Barcelona, Spain (2019) [INSPIRE].
C. Peset Martín, Effective field theories for heavy quarkonia and hydrogen-like bound states, Ph.D. thesis, Autonoma U., Barcelona, Spain (2016) [INSPIRE].
A. Pineda, Next-to-leading ultrasoft running of the heavy quarkonium potentials and spectrum: spin-independent case, Phys. Rev. D 84 (2011) 014012 [arXiv:1101.3269] [INSPIRE].
S. Bekavac, A. Grozin, D. Seidel and M. Steinhauser, Light quark mass effects in the on-shell renormalization constants, JHEP 10 (2007) 006 [arXiv:0708.1729] [INSPIRE].
P. Marquard et al., \( \overline{M}S \)-on-shell quark mass relation up to four loops in QCD and a general SU(N) gauge group, Phys. Rev. D 94 (2016) 074025 [arXiv:1606.06754] [INSPIRE].
M. Fael, K. Schönwald and M. Steinhauser, Exact results for \( {Z}_m^{\textrm{OS}} \) and \( {Z}_2^{\textrm{OS}} \) with two mass m 2 scales and up to three loops, JHEP 10 (2020) 087 [arXiv:2008.01102] [INSPIRE].
C. Duhr and F. Dulat, PolyLogTools — polylogs for the masses, JHEP 08 (2019) 135 [arXiv:1904.07279] [INSPIRE].
A. Mitov and S. Moch, The singular behavior of massive QCD amplitudes, JHEP 05 (2007) 001 [hep-ph/0612149] [INSPIRE].
K.G. Chetyrkin, B.A. Kniehl and M. Steinhauser, Decoupling relations to \( O\left({\alpha}_s^3\right) \) and their connection to low-energy theorems, Nucl. Phys. B 510 (1998) 61 [hep-ph/9708255] [INSPIRE].
T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, The four loop beta function in quantum chromodynamics, Phys. Lett. B 400 (1997) 379 [hep-ph/9701390] [INSPIRE].
V. Shtabovenko, R. Mertig and F. Orellana, FeynCalc 9.3: new features and improvements, Comput. Phys. Commun. 256 (2020) 107478 [arXiv:2001.04407] [INSPIRE].
C. Brønnum-Hansen and C.-Y. Wang, Two-loop helicity amplitudes for W/Z boson pair production in gluon fusion with exact top mass dependence, JHEP 05 (2021) 244 [arXiv:2101.12095] [INSPIRE].
X. Chen et al., Heavy-quark-pair production at lepton colliders at NNNLO in QCD, arXiv:2209.14259 [INSPIRE].
X. Chen et al., Complete two-loop electroweak corrections to e+e− → HZ, arXiv:2209.14953 [INSPIRE].
R. Zhu, Y. Ma, X.-L. Han and Z.-J. Xiao, Relativistic corrections to the form factors of Bc into S-wave charmonium, Phys. Rev. D 95 (2017) 094012 [arXiv:1703.03875] [INSPIRE].
F. Feng, Apart: a generalized Mathematica apart function, Comput. Phys. Commun. 183 (2012) 2158 [arXiv:1204.2314] [INSPIRE].
A.V. Smirnov and F.S. Chuharev, FIRE6: Feynman Integral REduction with modular arithmetic, Comput. Phys. Commun. 247 (2020) 106877 [arXiv:1901.07808] [INSPIRE].
J. Klappert, F. Lange, P. Maierhöfer and J. Usovitsch, Integral reduction with Kira 2.0 and finite field methods, Comput. Phys. Commun. 266 (2021) 108024 [arXiv:2008.06494] [INSPIRE].
T. Peraro, FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs, JHEP 07 (2019) 031 [arXiv:1905.08019] [INSPIRE].
K.G. Chetyrkin and F.V. Tkachov, Integration by parts: the algorithm to calculate beta functions in 4 loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
V. Shtabovenko, FeynCalc goes multiloop, J. Phys. Conf. Ser. 2438 (2023) 012140 [arXiv:2112.14132] [INSPIRE].
M. Gerlach, F. Herren and M. Lang, tapir: a tool for topologies, amplitudes, partial fraction decomposition and input for reductions, Comput. Phys. Commun. 282 (2023) 108544 [arXiv:2201.05618] [INSPIRE].
M. Fael, K. Schönwald and M. Steinhauser, Relation between the \( \overline{M}S \) and the kinetic mass of heavy quarks, Phys. Rev. D 103 (2021) 014005 [arXiv:2011.11655] [INSPIRE].
X. Liu and Y.-Q. Ma, AMFlow: a Mathematica package for Feynman integrals computation via auxiliary mass flow, Comput. Phys. Commun. 283 (2023) 108565 [arXiv:2201.11669] [INSPIRE].
X. Liu, Y.-Q. Ma and C.-Y. Wang, A systematic and efficient method to compute multi-loop master integrals, Phys. Lett. B 779 (2018) 353 [arXiv:1711.09572] [INSPIRE].
X. Liu and Y.-Q. Ma, Multiloop corrections for collider processes using auxiliary mass flow, Phys. Rev. D 105 (2022) L051503 [arXiv:2107.01864] [INSPIRE].
X. Liu, Y.-Q. Ma, W. Tao and P. Zhang, Calculation of Feynman loop integration and phase-space integration via auxiliary mass flow, Chin. Phys. C 45 (2021) 013115 [arXiv:2009.07987] [INSPIRE].
Z.-F. Liu and Y.-Q. Ma, Automatic computation of Feynman integrals containing linear propagators via auxiliary mass flow, Phys. Rev. D 105 (2022) 074003 [arXiv:2201.11636] [INSPIRE].
Z.-F. Liu and Y.-Q. Ma, Determining Feynman integrals with only input from linear algebra, Phys. Rev. Lett. 129 (2022) 222001 [arXiv:2201.11637] [INSPIRE].
R. Bonciani and A. Ferroglia, Two-loop QCD corrections to the heavy-to-light quark decay, JHEP 11 (2008) 065 [arXiv:0809.4687] [INSPIRE].
A.I. Davydychev, P. Osland and O.V. Tarasov, Two loop three gluon vertex in zero momentum limit, Phys. Rev. D 58 (1998) 036007 [hep-ph/9801380] [INSPIRE].
T. de Oliveira, D. Harnett, A. Palameta and T.G. Steele, Applications of diagrammatic renormalization methods in QCD sum-rules, Phys. Rev. D 106 (2022) 114023 [arXiv:2208.12363] [INSPIRE].
S. Groote, J.G. Korner and O.I. Yakovlev, Two loop anomalous dimensions of heavy baryon currents in heavy quark effective theory, Phys. Rev. D 54 (1996) 3447 [hep-ph/9604349] [INSPIRE].
J. Henn, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Massive three-loop form factor in the planar limit, JHEP 01 (2017) 074 [arXiv:1611.07535] [INSPIRE].
M. Fael, F. Lange, K. Schönwald and M. Steinhauser, Singlet and nonsinglet three-loop massive form factors, Phys. Rev. D 106 (2022) 034029 [arXiv:2207.00027] [INSPIRE].
A. Grozin, J.M. Henn, G.P. Korchemsky and P. Marquard, The three-loop cusp anomalous dimension in QCD and its supersymmetric extensions, JHEP 01 (2016) 140 [arXiv:1510.07803] [INSPIRE].
M.A. Özcelik, Pseudoscalar quarkonium hadroproduction and decay up to two loops, Ph.D. thesis, IJCLab, U. Paris-Saclay, Orsay, France (2021) [INSPIRE].
K.G. Chetyrkin, J.H. Kuhn and C. Sturm, QCD decoupling at four loops, Nucl. Phys. B 744 (2006) 121 [hep-ph/0512060] [INSPIRE].
W. Bernreuther and W. Wetzel, Decoupling of heavy quarks in the minimal subtraction scheme, Nucl. Phys. B 197 (1982) 228 [Erratum ibid. 513 (1998) 758] [INSPIRE].
A.G. Grozin et al., Simultaneous decoupling of bottom and charm quarks, JHEP 09 (2011) 066 [arXiv:1107.5970] [INSPIRE].
P. Bärnreuther, M. Czakon and P. Fiedler, Virtual amplitudes and threshold behaviour of hadronic top-quark pair-production cross sections, JHEP 02 (2014) 078 [arXiv:1312.6279] [INSPIRE].
A.G. Grozin, P. Marquard, J.H. Piclum and M. Steinhauser, Three-loop chromomagnetic interaction in HQET, Nucl. Phys. B 789 (2008) 277 [arXiv:0707.1388] [INSPIRE].
M. Gerlach, G. Mishima and M. Steinhauser, Matching coefficients in nonrelativistic QCD to two-loop accuracy, Phys. Rev. D 100 (2019) 054016 [arXiv:1907.08227] [INSPIRE].
S. Abreu, M. Becchetti, C. Duhr and M.A. Ozcelik, Two-loop form factors for pseudo-scalar quarkonium production and decay, JHEP 02 (2023) 250 [arXiv:2211.08838] [INSPIRE].
K.G. Chetyrkin, J.H. Kuhn and M. Steinhauser, RunDec: a Mathematica package for running and decoupling of the strong coupling and quark masses, Comput. Phys. Commun. 133 (2000) 43 [hep-ph/0004189] [INSPIRE].
B. Schmidt and M. Steinhauser, CRunDec: a C++ package for running and decoupling of the strong coupling and quark masses, Comput. Phys. Commun. 183 (2012) 1845 [arXiv:1201.6149] [INSPIRE].
A. Deur, S.J. Brodsky and G.F. de Teramond, The QCD running coupling, Nucl. Phys. 90 (2016) 1 [arXiv:1604.08082] [INSPIRE].
F. Herren and M. Steinhauser, Version 3 of RunDec and CRunDec, Comput. Phys. Commun. 224 (2018) 333 [arXiv:1703.03751] [INSPIRE].
Particle Data Group collaboration, Review of particle physics, PTEP 2022 (2022) 083C01 [INSPIRE].
N. Mathur, M. Padmanath and S. Mondal, Precise predictions of charmed-bottom hadrons from lattice QCD, Phys. Rev. Lett. 121 (2018) 202002 [arXiv:1806.04151] [INSPIRE].
M. Gómez-Rocha, T. Hilger and A. Krassnigg, Effects of a dressed quark-gluon vertex in vector heavy-light mesons and theory average of the \( {B}_c^{\ast } \) meson mass, Phys. Rev. D 93 (2016) 074010 [arXiv:1602.05002] [INSPIRE].
B.-B. Zhou, J.-J. Sun and Y.-J. Zhang, Leptonic decays of heavy vector and pseudoscalar mesons, Commun. Theor. Phys. 67 (2017) 655 [INSPIRE].
Y. Yang et al., Purely leptonic decays of the ground charged vector mesons, Eur. Phys. J. C 81 (2021) 1110 [arXiv:2109.05650] [INSPIRE].
E.B. Gregory et al., A prediction of the B∗c mass in full lattice QCD, Phys. Rev. Lett. 104 (2010) 022001 [arXiv:0909.4462] [INSPIRE].
L.P. Fulcher, Phenomenological predictions of the properties of the Bc system, Phys. Rev. D 60 (1999) 074006 [hep-ph/9806444] [INSPIRE].
S.M. Ikhdair and R. Sever, Spectroscopy of Bc meson in a semirelativistic quark model using the shifted large N expansion method, Int. J. Mod. Phys. A 19 (2004) 1771 [hep-ph/0310295] [INSPIRE].
B. Martín-González et al., Toward the discovery of novel Bc states: radiative and hadronic transitions, Phys. Rev. D 106 (2022) 054009 [arXiv:2205.05950] [INSPIRE].
HPQCD collaboration, B-meson decay constants: a more complete picture from full lattice QCD, Phys. Rev. D 91 (2015) 114509 [arXiv:1503.05762] [INSPIRE].
R. Chaturvedi and A.K. Rai, Bc meson spectroscopy motivated by general features of pNRQCD, Eur. Phys. J. A 58 (2022) 228 [arXiv:2211.04099] [INSPIRE].
S. Godfrey, Spectroscopy of Bc mesons in the relativized quark model, Phys. Rev. D 70 (2004) 054017 [hep-ph/0406228] [INSPIRE].
A.A. Penin, V.A. Smirnov and M. Steinhauser, Heavy quarkonium spectrum and production/annihilation rates to order \( {\beta}_0^3\ {\alpha}_s^3 \), Nucl. Phys. B 716 (2005) 303 [hep-ph/0501042] [INSPIRE].
D.S. Hwang and G.-H. Kim, Decay constant ratios fη /fJ/ψ and fη /fυ, Z. Phys. C 76 (1997) 107 [hep-ph/9703364] [INSPIRE].
A.K. Rai, B. Patel and P.C. Vinodkumar, Properties of \( Q\overline{Q} \) mesons in non-relativistic QCD formalism, Phys. Rev. C 78 (2008) 055202 [arXiv:0810.1832] [INSPIRE].
H.S. Chung, \( \overline{M}S \) renormalization of S-wave quarkonium wavefunctions at the origin, JHEP 12 (2020) 065 [arXiv:2007.01737] [INSPIRE].
N. Brambilla and A. Vairo, The Bc mass up to order \( {\alpha}_s^4 \), Phys. Rev. D 62 (2000) 094019 [hep-ph/0002075] [INSPIRE].
A. Pineda, Review of heavy quarkonium at weak coupling, Prog. Part. Nucl. Phys. 67 (2012) 735 [arXiv:1111.0165] [INSPIRE].
S.J. Collins, T.D. Imbo, B.A. King and E.C. Martell, Relating quarkonium wave functions at the origin, Phys. Lett. B 393 (1997) 155 [hep-ph/9610547] [INSPIRE].
B. Patel and P.C. Vinodkumar, Properties of \( Q\overline{Q}\left( Q\epsilon b,c\right) \) mesons in Coulomb plus power potential, J. Phys. G 36 (2009) 035003 [arXiv:0808.2888] [INSPIRE].
C. Quigg and J.L. Rosner, Quantum mechanics with applications to quarkonium, Phys. Rept. 56 (1979) 167 [INSPIRE].
A. Grozin, M. Hoschele, J. Hoff and M. Steinhauser, Simultaneous decoupling of bottom and charm quarks, PoS LL2012 (2012) 032 [arXiv:1205.6001] [INSPIRE].
A.A. Penin, A. Pineda, V.A. Smirnov and M. Steinhauser, Spin dependence of heavy quarkonium production and annihilation rates: complete next-to-next-to-leading logarithmic result, Nucl. Phys. B 699 (2004) 183 [Erratum ibid. 829 (2010) 398] [hep-ph/0406175] [INSPIRE].
A. Pineda and A. Signer, Heavy quark pair production near threshold with potential non-relativistic QCD, Nucl. Phys. B 762 (2007) 67 [hep-ph/0607239] [INSPIRE].
T. Rauh, Higher-order condensate corrections to Υ masses, leptonic decay rates and sum rules, JHEP 05 (2018) 201 [arXiv:1803.05477] [INSPIRE].
J. Soto, Model independent results for heavy quarkonium, Mod. Phys. Lett. A 19 (2004) 1563 [hep-ph/0406104] [INSPIRE].
Acknowledgments
We thank J. H. Piclum and A. Onishchenko for many helpful discussions. This work is supported by NSFC under grant No. 11775117 and No. 12075124, and by Natural Science Foundation of Jiangsu under Grant No. BK20211267.
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Tao, W., Xiao, ZJ. & Zhu, R. Three-loop matching coefficients for heavy flavor-changing currents and the phenomenological applications. J. High Energ. Phys. 2023, 189 (2023). https://doi.org/10.1007/JHEP05(2023)189
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DOI: https://doi.org/10.1007/JHEP05(2023)189