Properties of excited charmed-bottom mesons

We calculate the spectrum of $B_c$ mesons using a non-relativistic quark potential model. Using the calculated wave functions, we compute the radiative widths of $B_c$ excited states. The strong decay widths are calculated in a modified $^3P_0$ model, assuming harmonic oscillator wave functions. The hadronic transition rates of $B_c$ mesons are calculated using the Kuang-Yan approach. These results are used to determine branching ratios of possible decay channels of several $B_c$ excited states. Calculated branching ratios are then combined with production cross section of $B_c$ states at the LHC to suggest strategies to find missing excited states of $B_c$ mesons.


I. INTRODUCTION
B + c meson is the lowest-mass bound state of a charm quark and a bottom antiquark. This pseudoscalar mesonic ground state has no electromagnetic or strong decays as it cannot annihilate into gluons or photons. It was first observed by the CDF Collaboration at Fermilab through a semi-leptonic decay mode B ± c → J/ψ l ± ν in pp collisions. The measured mass and life time of B c by CDF were 6.40 ± 0.39 ± 0.13 GeV and 0.46 +0. 18 −0.16 ± 0.03 ps respectively [1]. It has also been observed by LHCb [2] and D0 [3] experiments through its different decay channels. In 2014 an excited state of B c meson was observed by ATLAS experiment at LHC through the decay channel B ± c → J/ψ π ± in pp collisions. The measured mass of the excited state was found to be 6842 ± 4 ± 5 MeV [4], which is considered as the second S-wave state of B c . However, this excited state has not been confirmed by other experiments yet.
Excited B c states below BD threshold (≈ 7144 MeV) can only decay through radiative and hadronic transitions to B c ground state, which decay through weak interaction. There are at least two S-wave, two P -wave, and one D-wave B c multiplets lying below the threshold.
Each of these states cascades into B c ground state through emission of photons and/or pions only. This results into unique experimental signatures through which we can identify them. This is particularly important when a large sample of B c states is expected to be produced at LHC. To predict event rates of various decay chains of excited B c states lying below the BD threshold at LHC, we require a knowledge of branching ratios of their electromagnetic and hadronic transitions along with their production cross sections.
There have been many calculations of B c spectrum using non-relativistic and relativistic quark models [5][6][7][8][9][10]. The electromagnetic transitions of B c are predicted in Refs. [6,[8][9][10][11] and hadronic transitions are calculated in [8,12]. In Ref. [5], the open-flavor strong decay widths of B c mesons are predicted in the 3 P 0 model. Ref. [5]  The organization of the paper is as follows. First, we describe the potential model used to calculate the mass spectrum of charm-bottom mesons. In Sec. III, we review the 3 P 0 decay model and evaluate the strong decay amplitudes. E1 and M1 radiative transitions are calculated in Sec. IV. This is followed by the estimates in Sec. V of hadronic transitions based on the Kuang-Yan approach. We discuss the best strategies for searching the excited B c states lying below the BD threshold in Sec. VI, while our concluding remarks are given in Sec. VII.

II. MASS SPECTRUM
In this section we give the mass predictions of the non-relativistic quark model for charmbottom mesons. We use the standard "Coulomb+linear" potential, and spin-dependent corrections generated from vector gluon exchange and an effective scalar confinement interaction. The potential used in this paper is given by [13] Here α s is the strong coupling constant, b is the string tension, and T is the tensor operator with diagonal matrix elements given by The strong coupling constant α s used in this potential for B c mesons is taken to be 0.4.
This value was obtained by our fit to the masses of two experimentally known states of B c mesons (these are listed in 4 th column of Table I). The parameters σ = 0.84 GeV, b = 0.0945 .325 GeV and m s = 0.422 GeV were obtained from fits to light mesons [30], and m c = 1.4794 GeV is from charmonium sector. Finally m b = 4.825 GeV is taken from Ref. [14].
The meson states with quark and antiquark of unequal mass are not charge conjugation eigenstates. Therefore, states with J = L and S = 0, 1, i.e., |n 1 L J and |n 3 L J can mix via spin-orbit interaction. For example 1 P 1 and 3 P 1 states can mix through the following linear combination where φ 1P is the mixing angle. In the heavy quark limit m Q → ∞ the mixing angles become [15] φ m Q →∞ = arctan( L L + 1 ).
This implies φ nP = 35.3 • , φ nD = 39.2 • and φ nF = 40.89 • . The mixing angles in heavy quark limit for 1D and 1F states are close to those produced by Ref. [8] whereas for 1P state it is slightly different. The spectrum of B c states obtained by solving the radial Schrödinger equation through the shooting method [16] is reported in Tables I and II.

III. OPEN FLAVOR STRONG DECAYS
In the 3 P 0 model, the open-flavor strong decay of a meson takes place through production of a light qq pair (q = u, d, s) with vacuum quantum numbers (J P C = 0 ++ ). The interaction Hamiltonian for the 3 P 0 model is obtained from the nonrelativistic limit of where γ is a dimensionless pair production strength. The pair-production strength parameter γ is fitted to strong decay data. In the original 3 P 0 model introduced by Micu [18], new qq pair is produced by a constant pair production amplitude γ. Some variants of the 3 P 0 model include an effective pair production strength (γ ef f ) that suppresses heavy qq pair production [19,20].
The 3 P 0 model has been successfully applied to strong decays of light mesons [21], strange mesons [22,23], charmonium states [13], bottomonium states [24,25], open-charm [26,27] and open-bottom sectors [5,[28][29][30]. In this study, we have computed strong decay widths of kinematically allowed open-flavor decay modes of all the B c states mentioned in Table I and II using the 3 P 0 model. The interaction Hamiltonian for the 3 P 0 model can be written in terms of the creation operators as where b † and d † are the creation operators for quark and antiquark respectively. The pair production strength factor γ = 0.35 is obtained from a fit of known strong decay widths of the cc states above open-charm threshold [13]. In this work, we use a modified version of pair production strength that replaces γ with where m is the mass of the produced quark [19,20]. This mechanism suppresses those diagrams in which a heavy qq pair is created.
With the 3 P 0 model, we use simple harmonic oscillator (SHO) wavefunctions and SHO scale β is taken as parameter of the model. In refs. [26][27][28][29], the β parameter was obtained by equating the root mean squared (RMS) radius of a harmonic oscillator wavefunction to the RMS radius of the quark model wavefunction. In this work, we fit β of SHO wavefunctions to the wavefunctions obtained by numercially solving Schrödinger equation for the potential given in Eq. (1). The resulting β values, that are more accurate, are listed in Tables I-III for the initial B c mesons, and the final D, D s , B and B s mesons appearing in strong decays of B c excited states. These two methods of finding β are compared in our earlier work [30].
In Fig. 1 we show the dependence of strong decay widths of few decay channels on the value of SHO parameter β. Disk and rectangular marks on each curve corresponds to the β values To calculate the decay rate of a process A → B + C, we evaluate the matrix element BC|H I |A by using interaction Hamiltonian of Eq. (7). In general two different diagrams, shown in Fig. 2, contribute to the matrix element BC|H I |A . Using the relativistic phase space factor from Ref. [31] and performing the angular integration gives where P = |P B | = |P C |, M A is the mass of the initial meson, and E B = M 2 B + P 2 and E C = M 2 C + P 2 are the energies of the final mesons B and C respectively. Where available, we use experimental masses [17] of B c mesons; otherwise we use the theoretical masses given in Tables I and II Table III. The detailed formulism to calculate the matrix element BC|H I |A and decay amplitude M LS is described in our earlier work [30].

IV. RADIATIVE TRANSITIONS OF B c MESONS
A. E1 Radiative Transitions E1 radiative partial widths are computed with the following expression [32] where

5.415
Here e c and e b are the electric charges of the charm quark and bottom anti-quark in units of |e|, m b and m c are the constituent masses of the charm and bottom quarks, α is the fine structure constant, ω is the final photon energy, M i is mass of the initial meson, and E f is the energy of the final state. Finally, the angular matrix element C f i is given by Eq. 10 is the result of Ref. [32] except for our inclusion of the relativistic phase space factor E f /M i from Refs. [13,26]. The matrix elements n ′2S ′ +1 L ′ J ′ |r|n 2S+1 L J are obtained using the quark model wavefunctions obtained in Sec. II. Wavefunction corrections due to perturbative spin-dependent interactions were neglected in this computation, as in Refs. [13,26]. Results for E1 radiative transitions for B c mesons are given in Tables VII -XIV along with the matrix elements so that an interested reader can reproduce our results.

B. M1 Radiative Transitions
The M1 radiative partial widths are evaluated using the following expression [33] where j 0 (x) is a spherical Bessel function. The definitions of the other parameters are the same as in the E1 radiative transitions. The results for M1 radiative transitions for B c mesons are given in Tables VII -XIV.

V. E1-E1 HADRONIC TRANSITIONS
Hadronic transitions are needed to estimate branching ratios and the event rates of decay chains of B c states lying below BD threshold. The differential rate for E1-E1 hadronic transitions from an initial meson state Φ ′ to the final meson state Φ and a system of light hadrons h is given by [34,35] dΓ where J ′ , J are the total angular momentum and l ′ , l are the orbital angular momentum of initial and final meson states respectively, { ... ... } is a 6-j symbol, M 2 is the invariant mass squared of the light hadron system and A k (l ′ , l) are the reduced matrix elements [34]. Here we use scaling argument to predict the hadronic rates for cb mesons using measured rates of cc and/or bb as input. When measured rates are not available, we use predicted rates of hadronic transition of bb states. The scaling law for E1-E1 hadronic transitions is given by [25,34] Γ(cb) where r 2 (QQ) is the expectation value of the square of the interquark separation and p is the phase space factor depending on the masses of initial and final states. The phase space respectively, where G and H are defined in Ref. [35].
For the B c (2S) → B c (1S) + ππ reduced rates, we rescale the measured reduced rates of [17] and take their average value.
In Tables VII -XIV, we combine the widths of radiative decays, strong decays, and hadronic transitions to calculate the total widths and the branching ratios. These BR's are used in the next section to give estimates for the number of events expected at the LHC for different decay chains of B c states below the threshold. In these tables c P = cos φ nP , s P = sin φ nP , c D = cos φ nD , s D = sin φ nD , c F = cos φ nF , and s F = sin φ nF , with n being the principal quantum number.

VI. EXPERIMENTAL SIGNATURES AND SEARCH STRATEGIES
Our calculated masses of B c states show that there are three S-wave, two P -wave, two  Table III of Ref. [38], we obtain the production cross sections 5.5 and 9.3 nb for the 1 1 S 0 and 1 3 S 1 states respectively.
The production cross sections of the 2 1 S 0 and 2 3 S 1 states are obtained by multiplying the corresponding values of 1S states with the factor |R 2S (0)/R 1S (0)| 2 ≃ 0.6 [36]. In Ref. [38] production cross sections of 1P and 2P states are calculated using the fragmentation approach. However, for LHC they report only total cross section that include the contribution from 1S, 2S, 1P and 2P states. The reported value 33.8 nb for kinematic cuts p T (B c ) > 10 GeV and |y(B c )| < 2.5 implies that total contribution of 1P and 2P states is 10.2 nb. We divide this value over eight 1P and 2P states using the distribution reported for Tevatron in Fig. 3 and 4 of Ref. [38]. It is pointed in Ref. [37] that the distribution is not much changed at the LHC energy. In Ref. [39], it is shown that total fragmentation probability for ab-quark to produce the D-wave B c mesons is about 2 × 10 −5 , equivalent to 2% of the total inclusive cross section of all of B c states lying below BD threshold. These estimates of the cross sections are used to predict the number of events of various decay chains of excited B c states.
As the excited states below the BD threshold eventually decay into the B c ground state, therefore it is important to observe this state in order to reconstruct the events of originally produced states. Prominent weak decay modes of B c ground states are summarized in Table   10 of Ref. [8]. We assume that ground state B c is observed through two golden channels: i) B ± c → J/ψ + π ± → l − l + π ± having combined BR of 0.013% and detection efficiency of ∼ 2% and ii) B ± c → J/ψl ± ν l → l ′− l ′+ l ± ν l having combined BR of 0.21% and detection efficiency of ∼ 4% (See Table XV for the branching ratios). We calculate the number of events of various decay chains of excited B c states below BD threshold at LHC for integrated luminosity L = 100 fb −1 . The values reported in Tables XVI -XXII include the events observed through both of the golden channels. The decay chains having yield less than 100 are not included in these tables. We include a factor of 2 to incorporate both charge conjugate states of B c . It is noted that our mass calculations show that 3S, 2P , 2D, and 1F states are below but close to BD threshold (energy gap < 0.15 GeV). These results significantly differ from the mass predictions given in Refs. [5,6,8]. Ref. [5] shows that 3S, 2 3 P 2 , 2D, and 1F states are above BD threshold, whereas Ref. [8] shows that three multiplets of 2P states are above BD threshold along with 3S, 2D, and 1F states. Ref. [6] also shows that 3S and three multiplets of 2P states are above BD threshold. The result is that these states, that are expected to be observed through radiative and hadronic transition according to our mass predictions, are unlikely to be observed in these references. This also gives significantly different branching ratios and predicted strong decay widths. Thus the experimenters should treat our predictions of branching ratios and strong decay widths of the states close to BD threshold cautiously. Besides this caveat, there are no results available for the production cross sections of 3S, 2D, and 1F states in pp collision, therefore, we abstain to make any predictions of event rates for these states.
These results show that in LHC it is possible to produce sufficient number of events corresponding to different decay chains of the excited B c states below BD threshold. The task of event reconstruction become much easier when an excited B c state directly decays to the ground state through E1/M1 or hadronic transitions. Tables VII and VIII show that 1 3 S 1 , 2 3 S 1 , 1P 1 , 1P ′ 1 , 2P 1 , 2P ′ 1 states can directly decay to B c ground states through E1/M1 transitions. All these direct transitions also appear in the tables XVI, XVII, XIX and XX of decay chains as their yield is much higher than 100. The case of 2 3 S 1 → 1 1 S 0 + γ is particulary interesting. Only 2650 events are expected in this case owing to small value of its BR (≈ 2.75%) despite having relatively large production cross section of 2 3 S 1 state. Therefore, the best way to search this state is via 2 3  given in the tables XVI-XXII of decay chains can help experimentalists in adopting the best strategies to discover and study properties of the excited B c states below BD threshold.

VII. CONCLUDING REMARKS
In this paper we studied the properties of charmed-bottom mesons including masses, radiative transitions, hadronic transitions and the OZI allowed strong decays. We have computed the spectrum of B c mesons upto 2F states with a non-relativistic quark model that incorporates scalar confinement and one gluon exchange spin-dependent interactions.
These eigenfunctions were then used to obtain E1 and M1 radiative transitions. Strong decay amplitudes of excited B c states above the BD threshold have been obtained using the modified 3 P 0 pair creation model and fitted SHO wave functions. The hadronic transition rates for B c mesons have been predicted using the Kuang-Yan approach. The total decay widths of excited B c states have been predicted by summing the radiative, hadronic and strong widths. The branching ratios of different final states are estimated by using the total widths. These branching ratios are then combined with production rates at the LHC to estimate the number of events of various decay chains of excited B c states. We expect that the predictions presented in this work will be help experimentalists find the excited B c states at LHC and measure their properties.   DB * 3 S 1 = −0.0083c P + 0.0117s P 1.03 14.21           J/ψ → l + l − 11.9 ± 0.06 [17]