The Decay constants of $B_c(nS)$ and $B^*_c(nS)$

The decay constants of the low lying S-wave $B_c$ mesons, i.e. $B_c(nS)$ and $B^*_c(nS)$ with $n\leq 3$, are calculated in the nonrelativistic quark model. The running coupling of the strong interaction is taken into account, and the uncertainties due to varying parameters and losing Lorentz covariance are considered carefully. As a byproduct, the decay constants of the low lying S-wave charmonium and bottomium states are given in the appendixes.


Introduction
The B c mesons are the only open flavor mesons containing two heavy valence quarks, i.e. one charm quark and one bottom anti-quark (or vice versa).The flavor forbids their annihilation into gluons or photons, so that the ground state pseudoscalar B c (1S) can only decay weakly, which makes it particularly interesting for the study of weak interaction.In the experimental aspect, the B c mesons are much less explored than the charmonium and bottomonium due to the small production rate, as the dominant production mechanism requires the production of both cc and bb pairs.The B c (1S) meson was first observed by CDF experiment in 1998 [1].In the later years, the mass and life time of B c (1S) were measured precisely, and its hadronic decay modes were also observed [2][3][4][5].The excited B c meson state was not observed until 2014, by the ATLAS experiment [6].The mass of B c (2S) was measured by LHCb experiment [7] and CMS experiment [8] independently in 2019.However, for the vector B c mesons, only the mass difference MeV is known [8].In the theoretical aspect, the mass spectrum and the decays of B c mesons are investigated by various methods.For example, the quark model [9][10][11][12][13][14], the light-front quark model [15][16][17], the QCD sum rule [18,19], the QCD factorization [17,[20][21][22][23], the instantaneous approximation Bethe-Salpeter equation [24,25], the continuum QCD approach [26][27][28], the lattice QCD [29] and other methods [30][31][32].The quark model, with the interaction motivated by quantum chromodynamics (QCD), is quite successful in describing the hadron spectrum and decay branching ratios, see Refs.[33,34] for an introduction.The nonrelativistic version of the quark model is suitable for heavy quark systems.It is not only phenomenologi-cal successful in describing mesons and baryons [35][36][37], but also powerful in predicting the properties of exotic hadrons, such as tetraquarks [38,39].
The decay constant carries the information of strong interaction in leptonic decay, and thus it is intrinsically nonperturbative.A presice determination of the decay constant is crucial for a precise calculation of the leptonic decay width.In this paper, we investigate the decay constants of low lying S-wave B c mesons, i.e.B c (nS) and B * c (nS) with n ≤ 3 in the nonrelativistic quark model.As the B c mesons are less explored, our result is significant for both theoretical and experimental exploring of the B c family.The work of Lakhina and Swanson [40] showed that two elements are important in calculating decay constants within nonrelativistic quark model, one is the running coupling of strong interaction, the other is the relativistic correction.Both of these two elements are taken into account in this paper.What's more, the uncertainty due to varying parameters and losing Lorentz covariance are considered carefully.
This paper is organized as following.In section 2, we introduce the framework of the quark model.The formulas for the decay constants in quark model are given in section 3.In section 4, the results of mass spectrum and decay constants are presented and discussed.Summary and conclusions are given in section 5. We also present the mass spectrum and decay constants of charmonium in Appendix A and those of bottomium in Appendix B for comparison.

The model
The framework has been introduced elsewhere, see for example Refs.[10,35,36].We recapitulate the framework here for completeness and to specify the details.
The masses and wave functions are obtained by solving the radial Schrödinger equation, where r is the distance between the two constituent quarks, R(r) is the radial wave function, µ m = m m m+ m is the reduced mass with m and m being the constituent quark masses, L is the orbital angular moment quantum number.V is the potential between the quarks and E is the energy of this system.Then the meson mass is where n is the main quantum number, M L is the magnetic quantum number of orbital angular momentum, and Y LM L (θ, φ) is the spheric harmoics.In this paper a bold character stands for a 3-dimension vector, for example, r = r.
The potential could be decomposed into H SI is the spin independent part, which is composed of a coulombic potential and a linear potential, where b is a constant and α s (Q 2 ) is the running coupling of the strong interaction.The other three terms are spin dependent.
is the spin-spin contact hyperfine potential, where s and s are the spin of the quark and antiquark respectively, and δσ (r) = ( σ √ π ) 3 e −σ 2 r 2 with σ being a parameter.
is the tensor potential.H SO is the spin-orbital interaction potential and could be decomposed into a symmetric part H SO+ and an anti-symmetric part H SO-, i.e.
where S ± = s±s, and L is the orbital angular momentum of the quark and antiquark system.
In equations (3)∼( 8), the running coupling takes the following form, where Λ QCD is the energy scale below which nonperturbative effects take over, β = 11− 2 3 N f with N f being the flavor number, Q is the typical momentum of the system, and α 0 is a constant.Eq. ( 9) approaches the one loop running form of QCD at large Q 2 and saturates at low Q 2 .In practice α s (Q 2 ) is parametrized by the form of a sum of Gaussian functions and transformed into α s (r) as in Ref. [34].
It should be mentioned that the potential containing 1 r 3 is divergent.Following Refs.[35,36], a cutoff r c is introduced, so that 1 for r ≤ r c .Herein r c is a parameter to be fixed by observables.Most of the interaction operators in Eq. ( 2) are diagonal in the space with basis |JM J ; LS except H SO-and H T , where J, L and S are the total, orbital and spin angular momenta quantum number, M J is the magnetic quantum number.The anti-symmetric part of the spin-orbital interaction, H SO-, arising only when the quark masses are unequal, causes 3 L J ↔ 1 L J mixing.The tensor interaction, H T , causes 3 L J ↔ 3 L J mixing.The former mixing is considered in our calculation while the later one is ignored, as the mixing due to the tensor interaction is very weak [34].
There are 8 parameters in all: m, m, N f , Λ QCD , α 0 , b, σ and r c .m and m are fixed by the mass spectrum of charmonium and bottomium, see Appendix A and Appendix B. N f and Λ QCD are chosen according to QCD estimation.N f = 4 for charmonium and B c mesons, and N f = 5 for bottomium mesons.In this work we vary Λ QCD in the range 0.2 GeV < Λ QCD < 0.4 GeV, then α 0 , b, σ and r c are fixed by the masses of B c (1 ) and B c (1 3 P 0 ).For the B c meson masses, the experiment values [41] or the lattice QCD results [29] are referred, see Table 2.

Decay constant
The decay constant of a pseudoscalar meson, f P , is defined by where |P (p) is the pseudoscalar meson state, p µ is the meson 4-momentum, and j µ5 (x) = ψγ µ γ 5 ψ(x) is the axial vector current with ψ(x) being the quark field.In quark model the pseudoscalar meson state is described by where k, k and p are the momentum of the quark, antiquark and meson respectively, E p = √ M 2 + p 2 is the meson energy, N c is the color number, S(= S + ) is the total spin and M S is its z-projection (in the case of pseudoscalar meson, S = M S = 0), b † ks and d † ks are the creation operator of the quark and antiquark respectively.χ SM S ss is the spin wave function, and Φ mk−m k m+ m = k r is wave function in momentum space, k r is the relative momentum between the quark and antiquark.While Φ(k r ) = d 3 rΦ(r)e −ikr •r , we use the same symbol for wave functions in coordinate space and momentum space.
The decay constant is Lorentz invariant by definition, Eq. ( 10).However, |P (p) defined by Eq. ( 11) is not Lorentz covariant, and thus leads to ambiguity about the decay constant.Let the 4-momentum to be p µ = (E p , p) and p = (0, 0, p), we could obtain the decay constant by comparing the temporal (µ = 0) component or the spacial (µ = 3) component of Eq. (10).The decay constant obtained with temporal component is where l + = l + mp m+ m , l − = l − mp m+ m , E l + = (l + ) 2 + m 2 , and Ēl − = (l − ) 2 + m2 .The decay constant obtained with spacial component is The Lorentz covariance is violated in two aspects.Firstly, Eqs. ( 12) and ( 13) lead to different results.Secondly, f P varies as the momentum p = |p| varies.Losing Lorentz covariance is a deficiency of nonrelativistic quark model and covariance is only recovered in the nonrelativistic and weak coupling limits [40].Herein we treat the center value as the prediction, and the deviation is treated as the uncertainty due to losing Lorentz covariance.
The decay constant of a vector meson, f V , is defined by where M V is the vector meson mass, µ is its polarization vector, j µ (x) = ψγ µ ψ(x) is the vector current, the vector meson state is the same as Eq. ( 11) except S = 1 and M S = 0, ±1 (we use the quantum number to present the value of the angular momentum).With p µ = (E p , 0, 0, p), the polarization vector is We will get three different expressions for f V in nonrelativistic quark model.Let µ = µ 0 and µ = 0 (temporal), Let µ = µ 0 and µ = 3 (spacial longitudinal), Let µ = µ + or µ − and µ = 1 or 2 (spacial transverse), Again the center value is treated as the prediction of f V , and the deviation is treated as the uncertainty due to losing Lorentz covariance.

Results
We take Eq. ( 1) as an eigenvalue problem, and solve it using the gaussian expansion method [42].Three parameter sets are used in our calculation, which are listed in Table 1.The B c mass spectra corresponding to these three parameter sets are listed in Table 2 from column three to five.The parameters are fixed by the masses of ) and B c (1 3 P 0 ), where the experiment values [41] (column seven) or the lattice QCD results [29] (column eight) are referred.The others are all outputs of the quark model explained from Eqs. (2) to (9).We also list the result of a previous nonrelativistic quark model [10] using a constant α s in column six.Comparing the results using different parameters, we see that the deviation is larger as n is larger.The deviation from the center value is about 30 MeV for 3S states and 50 MeV for 3P states.[41] or the lattice QCD results [29] are referred).1, where the underlined values are used to fix α0, b, σ and rc.The sixth column is the result of a previous nonrelativistic quark model using a constant αs.M expt. is the recent lattice QCD result [29].3.
Mixing angles of the nP1 and nP 1 (n = 1, 2, 3) states.The second to fourth columns are our results corresponding to the three parameter sets in Table 1.The fifth column is the result of a previous quark model using a constant strong coupling [10].

state
where θ nP is the mixing angle.We always choose B c (nP 1 ) to be the state nearer to B c (n 1 P 1 ), i.e. the mixing angle is in the range 0 ≤ θ nP ≤ 45 • .Our results of the mixing angles are listed from the second to fourth column in Table 3, and the previous quark model results using a constant strong coupling [10] are listed in the fifth column.We see that the mixing angle depends on the parameters.While a running coupling affects θ 1P very little, the mixing angles of the radial excited mesons from a running coupling are much smaller than that from a constant α s , i.e. the mixing of the radial excited mesons is much weaker.
As explained in section 3, we get two different expressions for f P and three for f V , and they depend on the momentum of the meson, due to losing Lorentz covari-ance.This is illustrated in Fig. 1, where the left panel is f Bc(1 1 S 0 ) and the right panel is f Bc(1 3 S 1 ) .The dependence on the meson momentum is weak upto 2 GeV, thus the main uncertainty comes from the different expressions (Eqs.( 12) and ( 13) for f P , Eqs. ( 18), ( 19) and ( 20) for f V ).We treat the central value as the predicted decay constant, and the deviation from central value as the uncertainty due to losing Lorentz covariance.Our results of decay constants of B c (nS) and B * c (nS) corresponding to the three parameter sets and its uncertainties are listed in Table 4.We see that the uncertainty due to losing Lorentz covariance is smaller for higher n states.Comparing the results from different parameters, the uncertainty due to varying the parameter is smaller than the former one in most cases.
Our final prediction for the decay constant together with both uncertainties are listed in Table 5.We also compare our result with others.f DSE c b is the result from Dyson-Schwinger equation approach [26,28].f lQCD c b is one of the lattice QCD result [43], other lattice QCD results are almost consistent with this one.The sixth and seventh columns are results from other potential models [44,45].The eighth column is the result from a lightfront quark model [46].These results are almost consistent except that our predictions for the radial excited mesons are smaller than that of Ref. [45].The main difference is that Ref. [45] uses the nonrelativistic limit van Royen and Weisskopf formula to calculate the decay constants, and this results in a larger decay constant [40].The reliability of our results could also be supported by the mass spectra and decay constants of the charmonium and bottomium, which are presented in the appendixes.We can see from Table A1, Table A2, Table B1 and Table B2 that our results are overall consistent with other results.is the lattice QCD results [43].The sixth and seventh columns are results from other potential models [44,45].The eighth column is the result from a light-front quark model [46].

Summary and conclusion
In summary, we calculated the decay constants of B c (nS) and B * c (nS) mesons (n = 1, 2, 3) in the nonrelativistic quark model.Our approach can be distinguished from other quark model studies by three points: (1) The effect of a running strong coupling is taken into account.We use the form, Eq. ( 9), which approaches the one loop running form of QCD at large Q 2 and saturates at low Q 2 .A running coupling affects the wave function of Eq. ( 1), so it has a considerable effect on the mixing angles and the decay constants.
(2) The ambiguity due to losing Lorentz covariance is discussed in detail.We get two different expressions for f P and three different expressions for f V in nonrelativistic quark model as a result of losing Lorentz covariance.The central value is treated as the prediction, and the deviation is treated as the uncertainty.We also find that the uncertain-ties due to losing Lorentz covariance decrease as n increases.
(3) We use three parameter sets, and the uncertainties due to varying the parameters are given.In most cases, this uncertainty is smaller than the former one.
Comparing our results with those from other approaches, we see that they are in good agreement.In the appendixes, we compare the decay constants of charmonium and bottomium from our calculation and those from other approaches.The overall agreement also raises the credibility of our approach.All in all, the decay constants of B c (nS) and B * c (nS) mesons (n = 1, 2, 3) are predicted, with the uncertainties well determined.And thus establish a good basis to study the decays of B c mesons. is our nonrelativistic quark model result, with the parameters listed in the caption of Table B1 is the results from Dyson-Schwinger equation (DSE) approach, where f η b (1 1 S 0 ) and f Υ (1 3 S 1 ) are from Ref. [28], f η b (2 1 S 0 ) and f Υ (2 3 S 1 ) are from Ref. [26], and the underlined values are inputs.f lQCD c b is the lattice QCD results, where f η b (1 1 S 0 ) is from Ref. [47], f Υ (1 3 S 1 ) and f Υ (2 3 S 1 ) are from Ref. [50].The seventh and eighth columns are other potential model results [44,45].The ninth column is a light front quark model result [46].f expt.b b is the experiment value and the vector meson decay constant is estimated by Eq. (A1).
. The uncertainties due to losing Lorentz covariance are listed in the parenthesis.f DSE b b

Table 1 .
Three paremater sets used in our calculation.mcandmbare fixed by the mass spectrum of charmonium and bottomium respectively, see TableA1and TableB1in the appendix.N f and ΛQCD are chosen according to QCD estimation.α0, b, σ and rc are fixed by the masses of Bc(1 1 S0), Bc(2 1 S0), B * c (1 3 S1) and Bc(1 3 P0) (the experiment values

Table 2 .
Mass spectrum of Bc mesons (in GeV).The third to fifth columns are our results corresponding to the three parameter sets in Table

Table 5 .
[26]y constants of Bc(nS) and B * c (nS) (in GeV).fQM c b is our prediction, where the first uncertainty is due to losing Lorentz covariance and the second uncertainty is due to varying the parematers.fDSEcbis the results from Dyson-Schwinger equation approach, f Bc(1 1 S 0 ) and f B * c (1 3 S 1 ) are from Ref.[28], f Bc(2 1 S 0 ) and f B * c (2 3 S 1 ) are from Ref.[26].f lQCD