The excited bottom-charmed mesons in a nonrelativistic quark model

TL;DR

This paper predicts the mass spectrum, radiative transitions, and strong decay channels of excited bottom-charmed mesons using a nonrelativistic quark model, aiding future experimental searches at the LHC.

Contribution

It provides a comprehensive calculation of $B_c$ meson states up to the 6S level, including decay modes, based on recent experimental and lattice QCD data, which is novel in scope.

Findings

Predicted $B_c$ mass spectrum up to 6S multiplet.

Identified promising decay channels for experimental detection.

Suggested potential observation of higher excited states at the LHC.

Abstract

Using the newly measured masses of $B_c(1S)$ and $B_c(2S)$ from the CMS Collaboration and the $1S$ hyperfine splitting determined from the lattice QCD as constrains, we calculate the $B_c$ mass spectrum up to the $6S$ multiplet with a nonrelativistic linear potential model. Furthermore, using the wave functions from this model we calculate the radiative transitions between the $B_c$ states within a constituent quark model. For the higher mass $B_c$ states lying above $DB$ threshold, we also evaluate the Okubo-Zweig-Iizuka (OZI) allowed two-body strong decays with the $^{3}P_{0}$ model. Our study indicates that besides there are large potentials for the observations of the low-lying $B_c$ states below the $DB$ threshold via their radiative transitions, some higher mass $B_c$ states, such as $B_c(2^3P_2)$, $B_c(2^3D_1)$, $B_c(3^3D_1)$, $B_c(4^3P_0)$, and the $1F$-wave $B_c$ states, might be first observed in their dominant strong decay channels $DB$, $DB^*$ or $D^*B$ at the LHC for their relatively narrow widths.

Tables

Table 1. Table 1: Predicted masses (MeV) of B c subscript 𝐵 𝑐 B_{c} states compared with other model predictions and data. The mixing angles between B c ​ ( n 3 ​ L J ) subscript 𝐵 𝑐 superscript 𝑛 3 subscript 𝐿 𝐽 B_{c}(n^{3}L_{J}) and B c ​ ( n 1 ​ L J ) subscript 𝐵 𝑐 superscript 𝑛 1 subscript 𝐿 𝐽 B_{c}(n^{1}L_{J}) obtained in this work are presented in Table  2 .

	State	$J^P$	Ours	ZVR Zeng:1994vj	SJSCP Soni:2017wvy	MBV Monteiro:2016ijw	EQ Eichten:1994gt	EFG Ebert:2002pp	GI Godfrey:2004ya	KLT Kiselev:1994rc	Lattice Mathur:2018epb	Exp Sirunyan:2019osb
	$B_c\left(1^3{S}_1\right)$	$1^{-}$	6326 (input)	6340	6321	6357	6337	6332	6338	6317	6331$\pm 10$	$\cdots$
	$B_c\left(1^1{S}_0\right)$	$0^{-}$	6271 (input)	6260	6272	6275	6264	6270	6271	6253	6276$\pm 9$	6271
	$B_c\left(2^3{S}_1\right)$	$1^{-}$	6890	6900	6900	6897	6899	6881	6887	6902	$\cdots$	$\cdots$
	$B_c\left(2^1{S}_0\right)$	$0^{-}$	6871 (input)	6850	6864	6862	6856	6835	6855	6867		6871
	$B_c\left(3^3{S}_1\right)$	$1^{-}$	7252	7280	7338	7333	7280	7235	7272	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(3^1{S}_0\right)$	$0^{-}$	7239	7240	7306	7308	7244	7193	7250	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(4^3{S}_1\right)$	$1^{-}$	7550	7580	7714	7734	7594	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(4^1{S}_0\right)$	$0^{-}$	7540	7550	7684	7713	7562	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(5^3{S}_1\right)$	$1^{-}$	7813	$\cdots$	8054	8115	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(5^1{S}_0\right)$	$0^{-}$	7805	$\cdots$	8025	8097	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(6^3{S}_1\right)$	$1^{-}$	8054	$\cdots$	8368	8484	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(6^1{S}_0\right)$	$0^{-}$	8046	$\cdots$	8340	8469	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(1^3{P}_2\right)$	$2^{+}$	6787	6760	6712	6737	6747	6762	6768	6743	$\cdots$	$\cdots$
	$B_c\left(1P_1^{\prime }\right)$	$1^{+}$	6776	6740	$\cdots$	6734	6736	6749	6750	6729	$\cdots$	$\cdots$
	$B_c\left(1P_1\right)$	$1^{+}$	6757	6730	$\cdots$	6686	6730	6734	6741	6717	$6736\pm 24$	$\cdots$
	$B_c\left(1^3{P}_0\right)$	$0^{+}$	6714	6680	6686	6638	6700	6699	6706	6683	$6712\pm 25$	$\cdots$
	$B_c\left(2^3{P}_2\right)$	$2^{+}$	7160	7160	7173	7175	7153	7156	7164	7134	$\cdots$	$\cdots$
	$B_c\left(2P_1^{\prime }\right)$	$1^{+}$	7150	7150	$\cdots$	7173	7142	7145	7150	7124	$\cdots$	$\cdots$
	$B_c\left(2P_1\right)$	$1^{+}$	7134	7140	$\cdots$	7137	7135	7126	7145	7113	$\cdots$	$\cdots$
	$B_c\left(2^3{P}_0\right)$	$0^{+}$	7107	7100	7146	7084	7108	7091	7122	7088	$\cdots$	$\cdots$
	$B_c\left(3^3{P}_2\right)$	$2^{+}$	7464	7480	7565	7575	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(3P_1^{\prime }\right)$	$1^{+}$	7458	7470	$\cdots$	7572	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(3P_1\right)$	$1^{+}$	7441	7460	$\cdots$	7546	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(3^3{P}_0\right)$	$0^{+}$	7420	7430	7536	7492	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(4^3{P}_2\right)$	$2^{+}$	7732	7760	7915	7970	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(4P_1^{\prime }\right)$	$1^{+}$	7727	7740	$\cdots$	7942	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(4P_1\right)$	$1^{+}$	7710	7740	$\cdots$	7943	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(4^3{P}_0\right)$	$0^{+}$	7693	7710	7885	7970	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(1^3{D}_3\right)$	$3^{-}$	7030	7040	6990	7004	7005	7081	7045	7134	$\cdots$	$\cdots$
	$B_c\left(1D_2^{\prime }\right)$	$2^{-}$	7032	7030	$\cdots$	7003	7012	7079	7036	7124	$\cdots$	$\cdots$
	$B_c\left(1D_2\right)$	$2^{-}$	7024	7020	$\cdots$	6974	7009	7077	7041	7113	$\cdots$	$\cdots$
	$B_c\left(1^3{D}_1\right)$	$1^{-}$	7020	7010	6998	6973	7012	7072	7025	7088	$\cdots$	$\cdots$
	$B_c\left(2^3{D}_3\right)$	$3^{-}$	7348	7370	7399	7410	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(2D_2^{\prime }\right)$	$2^{-}$	7347	7360	$\cdots$	7408	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(2D_2\right)$	$2^{-}$	7343	7360	$\cdots$	7385	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(2^3{D}_1\right)$	$1^{-}$	7336	7350	7403	7377	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(3^3{D}_3\right)$	$3^{-}$	7625	7660	7761	7796	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(3D_2^{\prime }\right)$	$2^{-}$	7623	7650	$\cdots$	7783	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(3D_2\right)$	$2^{-}$	7620	7650	$\cdots$	7781	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(3^3{D}_1\right)$	$1^{-}$	7611	7640	7762	7761	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(1^3{F}_4\right)$	$4^{+}$	7227	7250	7244	$\cdots$	$\cdots$	$\cdots$	7271	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(1F_3^{\prime }\right)$	$3^{+}$	7240	7250	$\cdots$	$\cdots$	$\cdots$	$\cdots$	7266	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(1F_3\right)$	$3^{+}$	7224	7240	$\cdots$	$\cdots$	$\cdots$	$\cdots$	7276	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(1^3{F}_2\right)$	$2^{+}$	7235	7240	7234	$\cdots$	$\cdots$	$\cdots$	7269	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(2^3{F}_4\right)$	$4^{+}$	7514	7550	7617	$\cdots$	$\cdots$	$\cdots$	7568	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(2F_3^{\prime }\right)$	$3^{+}$	7525	7550	$\cdots$	$\cdots$	$\cdots$	$\cdots$	7571	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(2F_3\right)$	$3^{+}$	7508	7540	$\cdots$	$\cdots$	$\cdots$	$\cdots$	7563	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(2^3{F}_2\right)$	$2^{+}$	7518	7540	7607	$\cdots$	$\cdots$	$\cdots$	7565	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(3^3{F}_4\right)$	$4^{+}$	7771	7810	7956	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(3F_3^{\prime }\right)$	$3^{+}$	7779	7810	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(3F_3\right)$	$3^{+}$	7768	7800	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
	$B_c\left(3^3{F}_2\right)$	$2^{+}$	7730	7800	7946	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$

Table 2. Table 2: Mixing angles.

Mixing angle	ours	Godfrey:2004ya	Devlani:2014nda	Ebert:2002pp
${\theta }_{1P}$	${35.5}^{\circ }$	${22.4}^{\circ }$	${20.57}^{\circ }$	${20.4}^{\circ }$
${\theta }_{2P}$	${38.0}^{\circ }$	${18.9}^{\circ }$	${19.94}^{\circ }$	${23.2}^{\circ }$
${\theta }_{3P}$	${39.7}^{\circ }$	$\cdots$	${17.68}^{\circ }$	$\cdots$
${\theta }_{4P}$	${39.7}^{\circ }$	$\cdots$	$\cdots$	$\cdots$
${\theta }_{1D}$	${45.0}^{\circ }$	${44.5}^{\circ }$	$-{2.49}^{\circ }$	$-{35.9}^{\circ }$
${\theta }_{2D}$	${45.0}^{\circ }$	$\cdots$	$-{2.8}^{\circ }$	$\cdots$
${\theta }_{3D}$	${45.0}^{\circ }$	$\cdots$	$\cdots$	$\cdots$
${\theta }_{1F}$	${41.4}^{\circ }$	${41.4}^{\circ }$	$\cdots$	$\cdots$
${\theta }_{2F}$	${43.4}^{\circ }$	$\cdots$	$\cdots$	$\cdots$
${\theta }_{3F}$	${42.4}^{\circ }$	$\cdots$	$\cdots$	$\cdots$

Table 3. Table 3: Partial widths of the M ​ 1 𝑀 1 M1 transitions for the low-lying 1 ​ S 1 𝑆 1S -, 2 ​ S 2 𝑆 2S -, and 3 ​ S 3 𝑆 3S -wave B c subscript 𝐵 𝑐 B_{c} states compared with the other model predictions.

Initial	Final	$E_{\gamma }$ (MeV)					${\Gamma }_{\mathrm{M1}}$ (eV)				${\Gamma }_{\mathrm{M1}}$ (eV)
state	state	Eichten:1994gt	Ebert:2002pp	Godfrey:2004ya	Kiselev:1994rc	ours	Eichten:1994gt	Ebert:2002pp	Godfrey:2004ya	Kiselev:1994rc	Ours
$1^3{S}_1$	$1^1{S}_0$	72	62	67	64	55	134.5	73	80	60	57
$2^3{S}_1$	$2^1{S}_0$	43	46	32	35	19	28.9	30	10	10	2.4
	$1^1{S}_0$	606	584	588	649	591	123.4	141	600	98	1205
$2^1{S}_0$	$1^3{S}_1$	499	484	498	550	523	93.3	160	300	96	99
$3^3{S}_1$	$3^1{S}_0$			22		13			3		0.8
	$2^1{S}_0$			405		371			200		356
	$1^1{S}_0$			932		915			600		1885
$3^1{S}_0$	$2^3{S}_1$			354		341			60		152
	$1^3{S}_1$			855		855			4200		510

Table 4. Table 4: Partial widths of the M ​ 1 𝑀 1 M1 transitions for the higher n ​ S 𝑛 𝑆 nS -wave ( n = 4 , 5 , 6 𝑛 4 5 6 n=4,5,6 ) B c subscript 𝐵 𝑐 B_{c} states.

Initial state	Final state	$E_{\gamma }$(MeV)	${\Gamma }_{\mathrm{EM}}$(eV)	Initial state	Final state	$E_{\gamma }$(MeV)	${\Gamma }_{\mathrm{EM}}$(eV)
$4^1{S}_0$	$3^3{S}_1$	283	186	$4^3{S}_1$	$4^1{S}_0$	10	0.35
	$2^3{S}_1$	622	579		$3^1{S}_0$	305	252
	$1^3{S}_1$	1116	1122		$2^1{S}_0$	648	806
					$1^1{S}_0$	1171	2501
$5^1{S}_0$	$4^3{S}_1$	251	209	$5^3{S}_1$	$5^1{S}_0$	8	0.18
	$3^3{S}_1$	533	720		$4^1{S}_0$	268	210
	$2^3{S}_1$	861	1260		$3^1{S}_0$	553	675
	$1^3{S}_1$	1339	1893		$2^1{S}_0$	885	1316
					$1^1{S}_0$	1390	3107
$6^1{S}_0$	$5^3{S}_1$	230	225	$6^3{S}_1$	$6^1{S}_0$	8	0.18
	$4^3{S}_1$	481	849		$5^1{S}_0$	245	191
	$3^3{S}_1$	755	1613		$4^1{S}_0$	498	643
	$2^3{S}_1$	1073	2203		$3^1{S}_0$	774	1239
	$1^3{S}_1$	1536	2822		$2^1{S}_0$	1096	1917
					$1^1{S}_0$	1586	3772

Table 5. Table 5: The masses (MeV) of the final hadrons appearing in the strong decay processes of the B c subscript 𝐵 𝑐 B_{c} states. The masses are taken from the Particle Data Group  Tanabashi:2018oca if there are experimental data, otherwise we take the quark model predictions in Refs.  Lu:2016bbk ; Li:2010vx .

State	$1^1{S}_0$	$1^3{S}_1$	$1^3{P}_0$	$1P_1$	$1P_1^{\prime }$	$1^3{P}_2$
$B$	5279	5325	5683	5729	5754	5768
$B_s$	5367	5415	5756	5801	5836	5851
$D$	1870	2010	2252	2402	2417	2466
$D_s$	1968	2112	2344	2488	2510	2559

Table 6. Table 6: Partial widths of the E ​ 1 𝐸 1 E1 dominant radiative transitions for the 1 ​ P 1 𝑃 1P -, 1 ​ D 1 𝐷 1D -, and 1 ​ F 1 𝐹 1F -wave B c subscript 𝐵 𝑐 B_{c} states. For comparison, the predictions from the relativistic quark model  Ebert:2002pp , relativized quark model  Godfrey:2004ya , nonrelativistic constituent quark models  Eichten:1994gt ; Kiselev:1994rc are listed in the table as well.

Initial	Final	$E_{\gamma }$ (MeV)					${\Gamma }_{\mathrm{E1}}$ (keV)				${\Gamma }_{\mathrm{EM}}$ (keV)
state	state	Eichten:1994gt	Ebert:2002pp	Godfrey:2004ya	Kiselev:1994rc	ours	Eichten:1994gt	Ebert:2002pp	Godfrey:2004ya	Kiselev:1994rc	Ours
$1^3{P}_2$	$1^3{S}_1$	397	416	416	426	445	112.6	122	83	102.9	87
$1P_1^{\prime }$		387	405	399	412	433	0.1	13.7	11	8.1	40
$1P_1$		382	389	391	400	416	99.5	87.1	60	77.8	70
$1^3{P}_0$		353	355	358	366	377	79.2	75.5	55	65.3	96
$1P_1^{\prime }$	$1^1{S}_0$	455	463	462	476	484	56.4	147	80	131.1	74
$1P_1$		450	447	454	464	468	0	18.4	13	11.6	35
$1^3{D}_3$	$1^3{P}_2$	258	312	272	264	239	98.7	149	78	76.9	67
$1D_2^{\prime }$	$1^3{P}_2$		310	263	273	241		12.6	8.8	6.8	8.3
	$1P_1^{\prime }$		321	280	287	253		143	63	46.0	41
	$1P_1$		338	289	301	271		7.1	7	25.0	0.39
$1D_2$	$1^3{P}_2$		308	268	258	233		23.6	9.6	12.2	8.7
	$1P_1^{\prime }$		319	285	272	246		14.9	15	18.4	1.09
	$1P_1$		335	294	284	263		139	64	44.6	44
$1^3{D}_1$	$1^3{P}_2$	258	303	255	265	229	2.7	3.82	1.8	2.2	0.7
	$1P_1^{\prime }$	268	315	273	279	242	0	7.81	4.4	3.3	12
	$1P_1$	331	315	281	291	259	49.3	65.3	28	39.2	29
	$1^3{P}_0$	302	365	315	325	299	88.6	133	55	79.9	65
$1^3{F}_4$	$1^3{D}_3$			222		194			81		69
$1F_3^{\prime }$	$1^3{D}_3$			227		207			5.4		4.76
	$1D_2^{\prime }$			231		205			82		32
	$1D_2$			236		212			0.04		0.04
$1F_3$	$1^3{D}_3$			218		191			3.7		4.91
	$1D_2^{\prime }$			222		189			0.5		0.22
	$1D_2$			226		197			78		29
$1^3{F}_2$	$1^3{D}_3$			221		202			0.4		0.12
	$1D_2^{\prime }$			224		200			6.3		5.72
	$1D_2$			229		208			6.5		6.36
	$1^3{D}_1$			237		212			75		78

Table 7. Table 7: Partial widths of the E ​ 1 𝐸 1 E1 dominant radiative transitions for the 2 ​ D 2 𝐷 2D -wave B c subscript 𝐵 𝑐 B_{c} states.

Initial state	Final state	$E_{\gamma }$ (MeV)	${\Gamma }_{\mathrm{EM}}$ (keV)	Initial state	Final state	$E_{\gamma }$ (MeV)	${\Gamma }_{\mathrm{EM}}$ (keV)
$2^3{D}_1$	$1^3{P}_2$	528	8.13	$2^3{D}_3$	$1^3{P}_2$	540	32
	$1P^{\prime }$	540	7.6		$1P^{\prime }$	552	0.54
	$1P$	557	12.5		$1P$	568	1.23
	$1^3{P}_0$	596	41.8		$1^3{P}_0$	607	2.04
	$2^3{P}_2$	174	0.58		$2^3{P}_2$	186	54
	$2P^{\prime }$	184	10.15		$2P^{\prime }$	195	0.09
	$2P$	199	20.88		$2P$	211	0.23
	$2^3{P}_0$	225	46		$2^3{P}_0$	237	0.05
$2D_2$	$1^3{P}_2$	535	7.04	$2D_2^{\prime }$	$1^3{P}_2$	539	7.28
	$1P^{\prime }$	547	0.12		$1P^{\prime }$	551	19
	$1P$	564	22.6		$1P$	567	1.48
	$1^3{P}_0$	602	0.29		$1^3{P}_0$	606	0.3
	$2^3{P}_2$	181	6.33		$2^3{P}_2$	185	6.71
	$2P^{\prime }$	190	0.74		$2P^{\prime }$	194	29
	$2P$	206	34		$2P$	210	0.24
	$2^3{P}_0$	232	0.04		$2^3{P}_0$	236	0.05

Table 8. Table 8: Partial widths of the E ​ 1 𝐸 1 E1 dominant radiative transitions for the 2 ​ S 2 𝑆 2S -, 2 ​ P 2 𝑃 2P -wave B c subscript 𝐵 𝑐 B_{c} states. For comparison, the predictions from the relativistic quark model  Ebert:2002pp , relativized quark model  Godfrey:2004ya , nonrelativistic constituent quark models  Eichten:1994gt ; Kiselev:1994rc are listed in the table as well.

Initial	Final	$E_{\gamma }$ (MeV)					${\Gamma }_{\mathrm{E1}}$ (keV)				${\Gamma }_{\mathrm{EM}}$ (keV)
state	state	Eichten:1994gt	Ebert:2002pp	Godfrey:2004ya	Kiselev:1994rc	ours	Eichten:1994gt	Ebert:2002pp	Godfrey:2004ya	Kiselev:1994rc	Ours
$2^3{S}_1$	$1^3{P}_2$	151	118	118	159	102	17.7	7.59	5.7	14.8	6.98
	$1P_1^{\prime }$	161	130	136	173	115	0	0.74	0.7	1.0	1.56
	$1P_1$	167	146	144	185	133	14.5	7.65	4.7	12.8	4.62
	$1^3{P}_0$	196	181	179	219	174	7.8	5.53	2.9	7.7	3.48
$2^1{S}_0$	$1P_1^{\prime }$	119	84	104	138	96	5.2	4.40	6.1	15.9	6.38
	$1P_1$	125	101	113	150	114	0	1.05	1.3	1.9	5.33
$2^3{P}_2$	$1^3{D}_3$	142	75	118	127	129	17.8	2.08	6.8	10.9	14
	$1D_2^{\prime }$		77	122	118	127		0.139	0.6	0.5	0.93
	$1D_2$		79	127	133	135		0.285	0.7	1.5	1.1
	$1^3{D}_1$	142	84	135	126	139	0.2	0.035	0.1	0.1	0.13
	$2^3{S}_1$	249	270	272	232	265	73.8	75.3	55	49.4	50
	$1^3{S}_1$	770		778	817	785	25.8		14	25.8	52
$2P_1^{\prime }$	$1D_2^{\prime }$		66	113	108	117		1.49	5.5	3.5	1.05
	$1D_2$		68	123	123	125		0.172	1.3	2.5	0.03
	$1^3{D}_1$	131	73	121	116	129	0.4	0.07	0.2	0.3	1.27
	$2^3{S}_1$	239	259	258	222	255	5.4	10.4	5.5	5.9	25
	$1^3{S}_1$	760		769	807	777	2.1		0.6	2.5	26
	$2^1{S}_0$		303	289	257	274		90.5	52	58.0	36
	$1^1{S}_0$			825	871	825			19	131.1	44
$2P_1$	$1D_2^{\prime }$		47	108	97	101		0.023	0.8	1.2	0.006
	$1D_2$		49	103	112	109		0.517	3.6	3.9	0.84
	$1^3{D}_1$	125	54	116	105	113	0.3	0.204	1.6	1.6	1.45
	$2^3{S}_1$	232	241	253	211	240	54.3	45.3	45	32.1	34
	$1^3{S}_1$	754		761	796	762	22.1		5.4	15.3	40
	$2^1{S}_0$		285	284	246	258		13.8	5.7	8.1	19
	$1^1{S}_0$			820	860	811			2.1	3.1	25
$2^3{P}_0$	$1^3{D}_1$	98	19	93	80	86	6.9	0.041	4.2	3.2	5.6
	$2^3{S}_1$	205	207	231	186	214	41.2	34	42	25.5	53
	$1^3{S}_1$	729		741	771	738	21.9		1	16.1	41

Table 9. Table 9: Partial widths of the E ​ 1 𝐸 1 E1 dominant radiative transitions for the 3 ​ S 3 𝑆 3S -, 4 ​ S 4 𝑆 4S -, 3 ​ P 3 𝑃 3P -wave B c subscript 𝐵 𝑐 B_{c} states.

Initial state	Final state	$E_{\gamma }$ (MeV)	${\Gamma }_{\mathrm{EM}}$ (keV)	Initial state	Final state	$E_{\gamma }$ (MeV)	${\Gamma }_{\mathrm{EM}}$ (keV)
$3^1{S}_0$	$2P^{\prime }$	88	11.13	$3^3{S}_1$	$2^3{P}_2$	91	11.89
	$2P$	104	10.93		$2P^{\prime }$	101	2.92
	$1P^{\prime }$	450	1.74		$2P$	117	7.2
	$1P$	467	1.25		$2^3{P}_0$	144	5
					$1^3{P}_2$	450	1.58
					$1P^{\prime }$	462	0.7
					$1P$	479	1.72
					$1^3{P}_0$	518	1.73
$4^1{S}_0$	$1P^{\prime }$	727	1.93	$4^3{S}_1$	$1^3{P}_2$	724	1.88
	$1P$	743	1.7		$1P^{\prime }$	736	0.82
	$2P^{\prime }$	380	6.31		$1P$	752	1.37
	$2P$	395	5.14		$1^3{P}_0$	790	1.3
	$3P^{\prime }$	82	13		$2^3{P}_2$	380	5.78
	$3P$	98	17		$2P^{\prime }$	389	1.96
					$2P$	405	4.04
					$2^3{P}_0$	430	3.28
					$3^3{P}_2$	86	16
					$3P^{\prime }$	91	4.06
					$3P$	108	8.71
					$3^3{P}_0$	129	6.12
$3^3{P}_0$	$2^3{D}_1$	84	10.93	$3^3{P}_2$	$2^3{D}_3$	115	22
	$1^3{D}_1$	389	1.84		$2D^{\prime }$	116	1.57
	$3^3{S}_1$	166	45		$2D$	120	1.72
	$2^3{S}_1$	511	36		$2^3{D}_1$	127	0.23
	$1^3{S}_1$	1013	30		$1^3{D}_3$	421	9.07
					$1D^{\prime }$	419	1.06
					$1D$	427	1.16
					$1^3{D}_1$	431	0.94
					$3^3{S}_1$	209	43
					$2^3{S}_1$	552	39
					$1^3{S}_1$	1051	42
$3P_1$	$2D^{\prime }$	93	0.003	$3P_1^{\prime }$	$2D^{\prime }$	110	1.9
	$2D$	97	1.3		$2D$	114	0.05
	$2^3{D}_1$	104	2.39		$2^3{D}_1$	121	2.47
	$1D^{\prime }$	398	0.74		$1D^{\prime }$	414	0.19
	$1D$	405	0.31		$1D$	421	0.93
	$1^3{D}_1$	409	0.67		$1^3{D}_1$	425	0.61
	$3^3{S}_1$	187	26		$3^3{S}_1$	203	25
	$2^3{S}_1$	531	27		$2^3{S}_1$	546	22
	$1^3{S}_1$	1031	30		$1^3{S}_1$	1046	24
	$3^1{S}_0$	199	18		$3^1{S}_0$	216	30
	$2^1{S}_0$	548	19		$2^1{S}_0$	564	28
	$1^1{S}_0$	1078	23		$1^1{S}_0$	1093	32

Table 10. Table 10: Strong decay properties for the 4 ​ S 4 𝑆 4S -, 5 ​ S 5 𝑆 5S -wave B c subscript 𝐵 𝑐 B_{c} states. Γ t ​ h subscript Γ 𝑡 ℎ \Gamma_{th} and B r subscript 𝐵 𝑟 B_{r} stand for the partial widths and branching ratios of the strong decay processes, respectively.

	State	Decay mode	${\Gamma }_{th}$ (MeV)	$B_r(\%)$	State	Decay mode	${\Gamma }_{th}$ (MeV)	$B_r(\%)$
	$3^1{S}_0$(7239)	$B^{\ast }D$	161	100	$3^3{S}_1$(7252)	$BD$	28	21
						$B^{\ast }D$	105	79
		Total	161	100		Total	133	100
	$4^1{S}_0$(7540)	$B^{\ast }D$	0.14	0.1	$4^3{S}_1$(7550)	$BD$	4.53	2.7
		$B{D}^{\ast }$	34.9	18.3		$B^{\ast }D$	0.41	0.2
		$B^{\ast }{D}^{\ast }$	104	54		$B{D}^{\ast }$	17.0	10
		$B_s^{\ast 0}{D}_s^{+}$	6.7	3.5		$B^{\ast }{D}^{\ast }$	112	66
		$B_s^0{D}_s^{\ast +}$	5.8	3.1		$B_s^0{D}_s^{+}$	2.81	1.6
		$B_s^{\ast 0}{D}_s^{\ast +}$	15.5	8.1		$B_s^{\ast 0}{D}_s^{+}$	5.29	3.1
		$BD\left(1^3{P}_0\right)$	24	12.6		$B_s^0{D}_s^{\ast +}$	1.83	1.1
						$B_s^{\ast 0}{D}_s^{\ast +}$	26.9	16
		Total	191	100		Total	171	100
	$5^1{S}_0$(7805)	$B^{\ast }D$	24.5	5.9	$5^3{S}_1$(7813)	$BD$	15.81	3.9
		$B{D}^{\ast }$	1.5	0.4		$B^{\ast }D$	20.18	5
		$B^{\ast }{D}^{\ast }$	2.28	0.6		$B{D}^{\ast }$	2.65	0.7
		$B_s^{\ast 0}{D}_s^{+}$	1.62	0.4		$B^{\ast }{D}^{\ast }$	0.19	0.05
		$B_s^0{D}_s^{\ast +}$	4.65	1.1		$B_s^0{D}_s^{+}$	0.02	0.005
		$B_s^{\ast 0}{D}_s^{\ast +}$	5.75	1.4		$B_s^{\ast 0}{D}_s^{+}$	0.62	0.2
		$B\left(1^3{P}_0\right)D$	18.6	4.5		$B_s^0{D}_s^{\ast +}$	3.02	0.8
		$B\left(1^3{P}_2\right)D$	27.6	6.7		$B_s^{\ast 0}{D}_s^{\ast +}$	8.09	2
		$B\left(1P^{\prime }\right)D^{\ast }$	82	19.9		$B\left(1P^{\prime }\right)D$	18.96	4.7
		$B\left(1P\right)D^{\ast }$	6.2	1.5		$B\left(1P\right)D$	13.34	3.3
		$B\left(1^3{P}_2\right)D^{\ast }$	56.5	13.7		$B\left(1^3{P}_2\right)D$	16.1	4
		$BD\left(1^3{P}_0\right)$	23.5	5.7		$B\left(1^3{P}_0\right)D^{\ast }$	0.04	0.01
		$BD\left(1^3{P}_2\right)$	48.2	11.7		$B\left(1P^{\prime }\right)D^{\ast }$	53.93	13.4
		$B^{\ast }D\left(1P^{\prime }\right)$	70.9	17.2		$B\left(1P\right)D^{\ast }$	5.19	1.3
		$B^{\ast }D\left(1P\right)$	12.3	3.0		$B\left(1^3{P}_2\right)D^{\ast }$	96	24
		$B^{\ast }D\left(1^3{P}_2\right)$	25.7	6.2		$BD\left(1P^{\prime }\right)$	0.89	0.2
		$B_S\left(1^3{P}_0\right)D_S$	0.17	0.04		$BD\left(1P\right)$	0.63	0.2
		$B_S{D}_S\left(1^3{P}_0\right)$	0.56	0.14		$BD\left(1^3{P}_2\right)$	17.34	4.3
						$B^{\ast }D\left(1^3{P}_0\right)$	18.32	4.6
						$B^{\ast }D\left(1P^{\prime }\right)$	32.9	8.2
						$B^{\ast }D\left(1P\right)$	6.8	1.7
						$B^{\ast }D\left(1^3{P}_2\right)$	69.87	17
						$B_S\left(1P^{\prime }\right)D_S$	0.36	0.09
						$B_S\left(1P\right)D_S$	0.01	0.002
						$B_S^{\ast }{D}_S\left(1^3{P}_0\right)$	0.3	0.07
		Total	413	100		Total	401	100

Table 11. Table 11: Strong decay properties for the 6 ​ S 6 𝑆 6S -wave B c subscript 𝐵 𝑐 B_{c} states.

	State	Decay mode	${\Gamma }_{th}$ (MeV)	$B_r(\%)$	State	Decay mode	${\Gamma }_{th}$ (MeV)	$B_r(\%)$
	$6^1{S}_0$(8046)	$B^{\ast }D$	44.4	12	$6^3{S}_1$(8054)	$BD$	17.6	4.7
		$B{D}^{\ast }$	24.3	6.7		$B^{\ast }D$	31	8.3
		$B^{\ast }{D}^{\ast }$	24.3	6.7		$B{D}^{\ast }$	19.1	5.1
		$B_s^{\ast 0}{D}_s^{+}$	1.11	0.3		$B^{\ast }{D}^{\ast }$	37.9	10.2
		$B_s^0{D}_s^{\ast +}$	0.38	0.11		$B_s^0{D}_s^{+}$	1.78	0.5
		$B_s^{\ast 0}{D}_s^{\ast +}$	3.33	0.9		$B_s^{\ast 0}{D}_s^{+}$	1.22	0.3
		$B\left(1^3{P}_0\right)D$	11.3	3.1		$B_s^0{D}_s^{\ast +}$	0.09	0.02
		$B\left(1^3{P}_2\right)D$	4.85	1.3		$B_s^{\ast 0}{D}_s^{\ast +}$	2.96	0.8
		$B\left(1P^{\prime }\right)D^{\ast }$	28.3	7.8		$B\left(1P^{\prime }\right)D$	0.25	0.07
		$B\left(1P\right)D^{\ast }$	24.7	6.8		$B\left(1P\right)D$	11.1	3
		$B\left(1^3{P}_2\right)D^{\ast }$	20.6	5.7		$B\left(1^3{P}_2\right)D$	1.09	0.3
		$BD\left(1^3{P}_0\right)$	13.2	3.6		$B\left(1^3{P}_0\right)D^{\ast }$	10.9	3
		$BD\left(1^3{P}_2\right)$	28.9	8		$B\left(1P^{\prime }\right)D^{\ast }$	21.2	5.7
		$B^{\ast }D\left(1P^{\prime }\right)$	46.8	13		$B\left(1P\right)D^{\ast }$	17	4.6
		$B^{\ast }D\left(1P\right)$	41.4	11.4		$B\left(1^3{P}_2\right)D^{\ast }$	34	9.1
		$B^{\ast }D\left(1^3{P}_2\right)$	23.5	6.5		$BD\left(1P^{\prime }\right)$	9.37	2.5
		$B_s\left(1^3{P}_0\right)D_s$	5.5	1.5		$BD\left(1P\right)$	16.6	4.4
		$B_s\left(1^3{P}_2\right)D_s$	0.17	0.05		$BD\left(1^3{P}_2\right)$	14.1	3.8
		$B_s\left(1P^{\prime }\right)D_s^{\ast }$	0.88	0.24		$B^{\ast }D\left(1^3{P}_0\right)$	12.9	3.5
		$B_s\left(1P\right)D_s^{\ast }$	0.03	0.01		$B^{\ast }D\left(1P^{\prime }\right)$	30.5	8.2
		$B_s\left(1^3{P}_2\right)D_s^{\ast }$	0.02	0.01		$B^{\ast }D\left(1P\right)$	27.9	7.5
		$B_s{D}_s\left(1^3{P}_0\right)$	6.62	1.8		$B^{\ast }D\left(1^3{P}_2\right)$	39.9	10.7
		$B_s{D}_s\left(1^3{P}_2\right)$	2.47	0.68		$B_s\left(1P^{\prime }\right)D_s$	0.61	0.2
		$B_s^{\ast }{D}_s\left(1P^{\prime }\right)$	4.14	1.1		$B_s\left(1P\right)D_s$	3.06	0.8
		$B_s^{\ast }{D}_s\left(1P\right)$	0.23	0.06		$B_s\left(1^3{P}_2\right)D_s$	0.24	0.1
		$B_s^{\ast }{D}_s\left(1^3{P}_2\right)$	0.18	0.05		$B_s\left(1^3{P}_0\right)D_s^{\ast }$	0.001	0.0003
						$B_s\left(1P^{\prime }\right)D_s^{\ast }$	1.13	0.3
						$B_s\left(1P\right)D_s^{\ast }$	0.005	0.001
						$B_s\left(1^3{P}_2\right)D_s^{\ast }$	0.48	0.1
						$B_s{D}_s\left(1P^{\prime }\right)$	0.56	0.2
						$B_s{D}_s\left(1P\right)$	0.04	0.01
						$B_s{D}_s\left(1^3{P}_2\right)$	1.35	0.4
						$B_s^{\ast }{D}_s\left(1^3{P}_0\right)$	3.3	0.9
						$B_s^{\ast }{D}_s\left(1P^{\prime }\right)$	3.06	0.8
						$B_s^{\ast }{D}_s\left(1P\right)$	0.21	0.1
						$B_s^{\ast }{D}_s\left(1^3{P}_2\right)$	0.03	0.008
		Total	361	100		Total	372	100

Table 12. Table 12: Strong decay properties for the 3 ​ P 3 𝑃 3P -, 4 ​ P 4 𝑃 4P -wave B c subscript 𝐵 𝑐 B_{c} states.

	State	Decay mode	${\Gamma }_{th}$ (MeV)	$B_r(\%)$	State	Decay mode	${\Gamma }_{th}$ (MeV)	$B_r(\%)$
	$3^3{P}_0$(7420)	$BD$	9.6	3.5	$3^3{P}_2$(7464)	$BD$	22	11.1
		$B^{\ast }{D}^{\ast }$	255	93		$B^{\ast }D$	16	8.1
		$B_s^0{D}_s^{+}$	9.7	3.5		$B{D}^{\ast }$	3.4	1.7
						$B^{\ast }{D}^{\ast }$	146	74
						$B_s^0{D}_s^{+}$	2.7	1.4
						$B_s^{\ast 0}{D}_s^{+}$	7.8	4
		Total	274	100		Total	198	100
	$3P_1^{\prime }$(7458)	$B^{\ast }D$	13.6	7.3	$3P_1$(7441)	$B^{\ast }D$	9.3	4.3
		$B{D}^{\ast }$	32	17.2		$B{D}^{\ast }$	62	28.1
		$B^{\ast }{D}^{\ast }$	129	69.4		$B^{\ast }{D}^{\ast }$	145	65.8
		$B_s^{\ast 0}{D}_s^{+}$	11.1	6		$B_s^{\ast 0}{D}_s^{+}$	4.0	1.8
		Total	185	100		Total	220	100
	$4^3{P}_0$(7693)	$BD$	13.6	25.6	$4^3{P}_2$(7732)	$BD$	21.76	11.4
		$B^{\ast }{D}^{\ast }$	14	26.4		$B^{\ast }D$	30.1	15.8
		$B_s^0{D}_s^{+}$	7.16	13.5		$B{D}^{\ast }$	13.9	7.3
		$B_s^{\ast 0}{D}_s^{\ast +}$	4.6	8.7		$B^{\ast }{D}^{\ast }$	7.82	4.1
		$B\left(1P^{\prime }\right)D$	7.66	14.4		$B_s^0{D}_s^{+}$	0.84	0.4
		$B\left(1P\right)D$	0.44	0.83		$B_s^{\ast 0}{D}_s^{+}$	0.01	0.005
		$BD\left(1P\right)$	0.07	0.13		$B_s^0{D}_s^{\ast +}$	2.34	1.2
		$B^{\ast }D\left(1^3{P}_0\right)$	5.5	10.4		$B_s^{\ast 0}{D}_s^{\ast +}$	11.1	5.8
						$B\left(1P^{\prime }\right)D$	27.7	14.5
						$B\left(1P\right)D$	6.95	3.6
						$B\left(1^3{P}_2\right)D$	20.2	10.6
						$B\left(1^3{P}_0\right)D^{\ast }$	8.8	4.6
						$BD\left(1P^{\prime }\right)$	13.1	6.9
						$BD\left(1P\right)$	6.61	3.5
						$B^{\ast }D\left(1^3{P}_0\right)$	10.1	5.3
						$B^{\ast }D\left(1P\right)$	9.22	4.8
		Total	53	100		Total	190	100
	$4P_1^{\prime }$(7727)	$B^{\ast }D$	41.6	29.1	$4P_1$(7710)	$B^{\ast }D$	24.5	19.4
		$B{D}^{\ast }$	11.9	8.4		$B{D}^{\ast }$	3.7	2.9
		$B^{\ast }{D}^{\ast }$	6.55	4.6		$B^{\ast }{D}^{\ast }$	0.86	0.7
		$B_s^{\ast 0}{D}_s^{+}$	1.42	1.0		$B_s^{\ast 0}{D}_s^{+}$	4.4	3.5
		$B_s^0{D}_s^{\ast +}$	6.2	4.3		$B_s^0{D}_s^{\ast +}$	6.78	5.4
		$B_s^{\ast 0}{D}_s^{\ast +}$	9.09	6.3		$B_s^{\ast 0}{D}_s^{\ast +}$	6.66	5.3
		$B\left(1^3{P}_0\right)D$	10.4	7.3		$B\left(1^3{P}_0\right)D$	0.002	0.002
		$B\left(1P^{\prime }\right)D$	0.003	0.002		$B\left(1P^{\prime }\right)D$	0.4	0.3
		$B\left(1P\right)D$	0.01	0.01		$B\left(1P\right)D$	3.32	2.6
		$B\left(1^3{P}_2\right)D$	36.6	25.6		$B\left(1^3{P}_2\right)D$	15	11.9
		$B\left(1^3{P}_0\right)D^{\ast }$	0.02	0.01		$B\left(1^3{P}_0\right)D^{\ast }$	11.8	9.4
		$BD\left(1^3{P}_0\right)$	13.6	9.5		$BD\left(1^3{P}_0\right)$	0.1	0.08
		$BD\left(1P^{\prime }\right)$	0.009	0.006		$BD\left(1P^{\prime }\right)$	15.32	12.2
		$BD\left(1P\right)$	0.05	0.03		$BD\left(1P\right)$	23.03	18.3
		$B^{\ast }D\left(1^3{P}_0\right)$	0.1	0.07		$B^{\ast }D\left(1^3{P}_0\right)$	10.02	8.0
		$B^{\ast }D\left(1P\right)$	0.31	0.22
		$B_s{D}_s\left(1^3{P}_0\right)$	4.75	3.3
		$B_s\left(1^3{P}_0\right)D_s$	0.41	0.3
		Total	143	100		Total	126	100

Table 13. Table 13: Strong decay properties for the 2 ​ D 2 𝐷 2D -, 3 ​ D 3 𝐷 3D -wave B c subscript 𝐵 𝑐 B_{c} states.

	State	Decay mode	${\Gamma }_{th}$ (MeV)	$B_r(\%)$	State	Decay mode	${\Gamma }_{th}$ (MeV)	$B_r(\%)$
	$2^3{D}_1$(7336)	$BD$	0.55	1.0	$2^3{D}_3$(7348)	$BD$	41.6	22.1
		$B^{\ast }D$	6.24	10.9		$B^{\ast }D$	50.8	26.9
		$B{D}^{\ast }$	50.1	87		$B{D}^{\ast }$	9.29	4.9
		$B^{\ast }{D}^{\ast }$	0.48	0.8		$B^{\ast }{D}^{\ast }$	87	46.1
		$B_s^0{D}_s^{+}$	0.18	0.3		$B_s^0{D}_s^{+}$	0.013	0.01
		Total	57	100		Total	189	100
	$2D_2^{\prime }$(7347)	$B^{\ast }D$	57.1	34.7	$2D_2$(7343)	$B^{\ast }D$	38.2	27
		$B{D}^{\ast }$	66.8	40.7		$B{D}^{\ast }$	89	64
		$B^{\ast }{D}^{\ast }$	40.4	24.6		$B^{\ast }{D}^{\ast }$	12.3	9
		Total	164	100		Total	139	100
	$3^3{D}_1$(7611)	$BD$	25.2	28.2	$3^3{D}_3$(7625)	$BD$	19.3	17
		$B^{\ast }D$	5.65	6.3		$B^{\ast }D$	29.7	26.5
		$B{D}^{\ast }$	0.48	0.5		$B{D}^{\ast }$	20.8	18.6
		$B^{\ast }{D}^{\ast }$	19.5	21.9		$B^{\ast }{D}^{\ast }$	18.4	16.4
		$B_s^0{D}_s^{+}$	2.27	2.5		$B_s^0{D}_s^{+}$	1.45	1.3
		$B_s^{\ast 0}{D}_s^{+}$	3.16	3.5		$B_s^{\ast 0}{D}_s^{+}$	0.12	0.1
		$B_s^0{D}_s^{\ast +}$	1.82	2.0		$B_s^0{D}_s^{\ast +}$	2.94	2.6
		$B_s^{\ast 0}{D}_s^{\ast +}$	16.5	18.5		$B_s^{\ast 0}{D}_s^{\ast +}$	6.6	5.9
		$B\left(1P\right)D$	0.76	0.9		$B\left(1P^{\prime }\right)D$	0.001	0.001
		$B^{\ast }D\left(1^3{P}_0\right)$	13.9	15.6		$B\left(1P\right)D$	4.62	4.1
						$B^{\ast }D\left(1^3{P}_0\right)$	8.14	7.3
		Total	89	100		Total	112	100
	$3D_2^{\prime }$(7623)	$B^{\ast }D$	45.8	34.6	$3D_2$(7620)	$B^{\ast }D$	38.9	34.2
		$B{D}^{\ast }$	20.6	15.6		$B{D}^{\ast }$	13.8	12
		$B^{\ast }{D}^{\ast }$	21.1	16		$B^{\ast }{D}^{\ast }$	22.1	19
		$B_s^{\ast 0}{D}_s^{+}$	2.25	1.7		$B_s^{\ast 0}{D}_s^{+}$	3.89	3.4
		$B_s^0{D}_s^{\ast +}$	6.33	4.8		$B_s^0{D}_s^{\ast +}$	6.46	5.7
		$B_s^{\ast 0}{D}_s^{\ast +}$	9.07	6.8		$B_s^{\ast 0}{D}_s^{\ast +}$	11.6	10
		$B\left(1^3{P}_0\right)D$	12.1	9.1		$B\left(1^3{P}_0\right)D$	0.03	0.03
		$B\left(1P\right)D$	0.02	0.02		$B\left(1P\right)D$	2.82	2.5
		$BD\left(1^3{P}_0\right)$	14.4	10.9		$BD\left(1^3{P}_0\right)$	0.65	0.6
		$B^{\ast }D\left(1^3{P}_0\right)$	0.65	0.5		$B^{\ast }D\left(1^3{P}_0\right)$	13.6	12
		Total	132	100		Total	114	100

Table 14. Table 14: Strong decay properties for the 1 ​ F 1 𝐹 1F -, 2 ​ F 2 𝐹 2F -wave B c subscript 𝐵 𝑐 B_{c} states.

	State	Decay mode	${\Gamma }_{th}$ (MeV)	$B_r(\%)$	State	Decay mode	${\Gamma }_{th}$ (MeV)	$B_r(\%)$
	$1^3{F}_2$(7235)	$BD$	61.9	85	$1^3{F}_4$(7227)	$BD$	0.85	97
		$B^{\ast }D$	11.1	15		$B^{\ast }D$	0.03	3
		Total	73	100		Total	0.88	100
	$1F_3^{\prime }$(7240)	$B^{\ast }D$	15.1	100	$1F_3$(7224)	$B^{\ast }D$	8.53	100
		Total	15.1	100		Total	8.53	100
	$2^3{F}_2$(7518)	$BD$	45.1	20.2	$2^3{F}_4$(7514)	$BD$	8	6
		$B^{\ast }D$	19.2	8.6		$B^{\ast }D$	20.9	16
		$B{D}^{\ast }$	0.39	0.2		$B{D}^{\ast }$	37.7	29
		$B^{\ast }{D}^{\ast }$	151	68		$B^{\ast }{D}^{\ast }$	57	43
		$B_s^0{D}_s^{+}$	0.68	0.3		$B_s^0{D}_s^{+}$	4.48	3.4
		$B_s^{\ast 0}{D}_s^{+}$	3.63	1.6		$B_s^{\ast 0}{D}_s^{+}$	3.26	2.5
		$B_s^0{D}_s^{\ast +}$	3.17	1.4		$B_s^0{D}_s^{\ast +}$	0.05	0.04
		Total	223	100		Total	131	100
	$2F_3^{\prime }$(7525)	$B^{\ast }D$	45.2	25	$2F_3$(7508)	$B^{\ast }D$	43.9	25
		$B{D}^{\ast }$	41.0	23		$B{D}^{\ast }$	30.2	17
		$B^{\ast }{D}^{\ast }$	80.3	45		$B^{\ast }{D}^{\ast }$	90.2	52
		$B_s^{\ast 0}{D}_s^{+}$	7.19	4		$B_s^{\ast 0}{D}_s^{+}$	7.78	4.5
		$B_s^0{D}_s^{\ast +}$	4.53	3		$B_s^0{D}_s^{\ast +}$	2.57	1.5
		Total	178	100		Total	175	100

Table 15. Table 15: Strong decay properties for the 3 ​ F 3 𝐹 3F -wave B c subscript 𝐵 𝑐 B_{c} states.

	State	Decay mode	${\Gamma }_{th}$ (MeV)	$B_r(\%)$	State	Decay mode	${\Gamma }_{th}$ (MeV)	$B_r(\%)$
	$3^3{F}_2$(7730)	$BD$	32.1	14	$3^3{F}_4$(7771)	$BD$	2.82	1.6
		$B^{\ast }D$	16.1	7		$B^{\ast }D$	8.9	5.0
		$B{D}^{\ast }$	3.89	1.7		$B{D}^{\ast }$	20.2	11.4
		$B^{\ast }{D}^{\ast }$	72	31.5		$B^{\ast }{D}^{\ast }$	50.9	28.7
		$B_s^0{D}_s^{+}$	0.38	0.2		$B_s^0{D}_s^{+}$	3.2	1.8
		$B_s^{\ast 0}{D}_s^{+}$	0.11	0.05		$B_s^{\ast 0}{D}_s^{+}$	3.2	1.8
		$B_s^0{D}_s^{\ast +}$	2.09	0.9		$B_s^0{D}_s^{\ast +}$	0.4	0.2
		$B_s^{\ast 0}{D}_s^{\ast +}$	5.25	2.3		$B_s^{\ast 0}{D}_s^{\ast +}$	9.19	5.2
		$B\left(1P^{\prime }\right)D$	19.5	8.5		$B\left(1P^{\prime }\right)D$	19.4	10.9
		$B\left(1P\right)D$	5.03	2.2		$B\left(1P\right)D$	1.85	1.04
		$B\left(1^3{P}_2\right)D$	12.1	5.3		$B\left(1^3{P}_2\right)D$	12.3	6.9
		$B\left(1^3{P}_0\right)D^{\ast }$	2.56	1.1		$B\left(1^3{P}_0\right)D^{\ast }$	6.82	3.8
		$BD\left(1P^{\prime }\right)$	45.8	20		$B\left(1P^{\prime }\right)D^{\ast }$	0.02	0.01
		$BD\left(1P\right)$	2.3	1		$B\left(1P\right)D^{\ast }$	3.03	1.7
		$B^{\ast }D\left(1^3{P}_0\right)$	9.14	4		$BD\left(1P^{\prime }\right)$	9.19	5.2
		$B^{\ast }D\left(1P\right)$	0.48	0.2		$BD\left(1P\right)$	11.2	6.3
						$BD\left(1^3{P}_2\right)$	1.61	0.9
						$B^{\ast }D\left(1^3{P}_0\right)$	1.91	1.1
						$B^{\ast }D\left(1P^{\prime }\right)$	2.83	1.6
						$B^{\ast }D\left(1P\right)$	8.42	4.7
						$B_s\left(1P\right)D_s$		$\simeq 0$
						$B_s^{\ast }{D}_s\left(1^3{P}_0\right)$	0.02	0.01
		Total	228	100		Total	177	100
	$3F_3^{\prime }$(7779)	$B^{\ast }D$	33.6	11	$3F_3$(7768)	$B^{\ast }D$	39.9	12
		$B{D}^{\ast }$	34.4	11.3		$B{D}^{\ast }$	31.3	9.5
		$B^{\ast }{D}^{\ast }$	59.9	19.6		$B^{\ast }{D}^{\ast }$	63.5	19.3
		$B_s^{\ast 0}{D}_s^{+}$	4.2	1.4		$B_s^{\ast 0}{D}_s^{+}$	2.64	0.8
		$B_s^0{D}_s^{\ast +}$	1.69	0.6		$B_s^0{D}_s^{\ast +}$	2.02	0.6
		$B_s^{\ast 0}{D}_s^{\ast +}$	4.85	1.6		$B_s^{\ast 0}{D}_s^{\ast +}$	3.63	1.1
		$B\left(1^3{P}_0\right)D$	0.008	0.003		$B\left(1^3{P}_0\right)D$	0.01	0.003
		$B\left(1P^{\prime }\right)D$	0.01	0.003		$B\left(1P^{\prime }\right)D$	6.25	1.9
		$B\left(1P\right)D$		$\simeq 0$		$B\left(1P\right)D$	2.26	0.7
		$B\left(1^3{P}_2\right)D$	36.7	12		$B\left(1^3{P}_2\right)D$	27.6	8.4
		$B\left(1^3{P}_0\right)D^{\ast }$	0.08	0.03		$B\left(1^3{P}_0\right)D^{\ast }$	8.06	2.5
		$B\left(1P^{\prime }\right)D^{\ast }$	30.4	10		$B\left(1P^{\prime }\right)D^{\ast }$	8.69	2.6
		$B\left(1P\right)D^{\ast }$	7.75	2.5		$B\left(1P\right)D^{\ast }$	2.3	0.7
		$B\left(1^3{P}_2\right)D^{\ast }$	0.68	0.2		$BD\left(1^3{P}_0\right)$	0.6	0.2
		$BD\left(1^3{P}_0\right)$	0.11	0.04		$BD\left(1P^{\prime }\right)$	11.6	3.5
		$BD\left(1P^{\prime }\right)$	0.07	0.02		$BD\left(1P\right)$	16.6	5.1
		$BD\left(1P\right)$	0.56	0.2		$BD\left(1^3{P}_2\right)$	34.2	10
		$BD\left(1^3{P}_2\right)$	27.1	8.9		$B^{\ast }D\left(1^3{P}_0\right)$	1.73	0.53
		$B^{\ast }D\left(1^3{P}_0\right)$	0.13	0.04		$B^{\ast }D\left(1P^{\prime }\right)$	57.4	17.5
		$B^{\ast }D\left(1P^{\prime }\right)$	38.9	12.8		$B^{\ast }D\left(1P\right)$	8.23	2.5
		$B^{\ast }D\left(1P\right)$	19.2	6.3		$B_s\left(1^3{P}_0\right)D_s$	0.003	$\simeq 0$
		$B_s\left(1^3{P}_0\right)D_s$	1.38	0.45		$B_s{D}_s\left(1^3{P}_0\right)$	0.14	0.04
		$B_s\left(1P\right)D_s$		$\simeq 0$		$B_s^{\ast }{D}_s\left(1^3{P}_0\right)$	0.01	0.003
		$B_s{D}_s\left(1^3{P}_0\right)$	3.03	1.0
		$B_s^{\ast }{D}_s\left(1^3{P}_0\right)$	0.01	0.003
		Total	305	100		Total	329	100

Equations

$$V(r)=H_0(r)+H_{sd}(r),$$

$$H_0(r)=-\frac{4}{3}\frac{{\alpha }_s}{r}+br$$

$$H_{sd}(r)=H_{SS}+H_T+H_{LS},$$

$$H_{SS}=\frac{32\pi {\alpha }_s}{9m_q{m}_{\bar{q}}}{\tilde{\delta }}_{\sigma }(r){\mathbf{S}}_q\cdot {\mathbf{S}}_{\bar{q}}$$

$$H_T=\frac{4}{3}\frac{{\alpha }_s}{m_q{m}_{\bar{q}}}\frac{1}{r^3}\left(\frac{3{\mathbf{S}}_q\cdot \mathbf{r}{\mathbf{S}}_{\bar{q}}\cdot \mathbf{r}}{r^2}-{\mathbf{S}}_q\cdot {\mathbf{S}}_{\bar{q}}\right).$$

$$H_{LS}=H_{sym}+H_{anti},$$

$$H_{sym}=\frac{{\mathbf{S}}_{+}\cdot \mathbf{L}}{2}\left[\left(\frac{1}{2m_{\bar{q}}^2}+\frac{1}{2m_q^2}\right)\left(\frac{4}{3}\frac{{\alpha }_s}{r^3}-\frac{b}{r}\right)+\frac{8{\alpha }_s}{3m_q{m}_{\bar{q}}{r}^3}\right],$$

$$\displaystyle H_{anti}=\frac{\mathbf{S_{-}\cdot L}}{2}\left(\frac{1}{2m_{q}^{2}}-\frac{1}{2m_{\bar{q}}^{2}}\right)\left(\frac{4}{3}\frac{\alpha_{s}}{r^{3}}-\frac{b}{r}\right).\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$$

$$\frac{d^{2}u(r)}{d{r}^2}+2{\mu }_R\left[E-V_{q\bar{q}}(r)-\frac{L(L+1)}{2{\mu }_R{r}^2}\right]u(r)=0,$$

$$V_{q\bar{q}}(r)=V(r)+H_{SS}+H_{SL}+H_T,$$

$$M_{q\bar{q}}=m_q+m_{\bar{q}}+E.$$

$$\left(\begin{array}[]{c}B_{c}(nL^{\prime}_{J})\\ B_{c}(nL_{J})\\ \end{array}\right)=\left(\begin{array}[]{cc}\cos\theta_{nL}&\sin\theta_{nL}\\ -\sin\theta_{nL}&\cos\theta_{nL}\\ \end{array}\right)\left(\begin{array}[]{c}B_{c}(n^{1}L_{J})\\ B_{c}(n^{3}L_{J})\\ \end{array}\right),$$

$$H_e=-\sum _j{e}_j{\bar{\psi }}_j{\gamma }_{\mu }^j{A}^{\mu }(\mathbf{k},\mathbf{r}){\psi }_j,$$

$$H_{e}^{nr}=\sum_{j}\left[e_{j}\mathbf{r}_{j}\cdot\mbox{\boldmath$\epsilon$\unboldmath}-\frac{e_{j}}{2m_{j}}\mbox{\boldmath$\sigma$\unboldmath}_{j}\cdot(\mbox{\boldmath$\epsilon$\unboldmath}\times\hat{\mathbf{k}})\right]e^{-i\textbf{k}\cdot\textbf{r}_{j}},$$

$$\mathcal{A}$$

$$\Gamma =\frac{|\mathbf{k}{|}^2}{\pi }\frac{2}{2J_i+1}\frac{M_f}{M_i}\sum _{J_{fz},J_{iz}}|{\mathcal{A}}_{J_{fz},J_{iz}}{|}^2,$$

$$\hat{T}$$

$$⟨BC|T|A⟩=\delta ({\mathbf{P}}_A-{\mathbf{P}}_B-{\mathbf{P}}_C){\mathcal{M}}^{M_{J_A}{M}_{J_B}{M}_{J_C}}(\mathbf{P}).$$

$${\mathcal{M}}^{JL}(A\to BC)=\frac{\sqrt{4\pi (2L+1)}}{2J_A+1}\sum _{M_{J_B},M_{J_C}}⟨L0J{M}_{J_A}|J_A{M}_{J_A}⟩$$

$$\times ⟨J_B{M}_{J_B}{J}_C{M}_{J_C}|J{M}_{J_A}⟩{\mathcal{M}}^{M_{J_A}{M}_{J_B}{M}_{J_C}}(\text{P}).$$

$$\displaystyle\Gamma=2\pi|\textbf{P}|\frac{E_{B}E_{C}}{M_{A}}\sum_{JL}\Big{|}\mathcal{M}^{JL}\Big{|}^{2},$$

$$\Delta m(2S)\simeq 20\mathrm{MeV},$$

$$\Gamma [B_c^{\ast }(2S)\to B_c\gamma ]\simeq 1.2\mathrm{keV},$$

$$Br[B_c^{\ast }(2S)\to B_c\gamma ]\sim 2\%.$$

$$Br[B_c(1P_1)\to B_c\gamma ]\sim 33\%,$$

$$Br[B_c(1P_1^{\prime })\to B_c\gamma ]\sim 65\%.$$

$$\Gamma [B_c(2^3{P}_2)\to DB]\simeq 760\mathrm{keV},$$

$$\Gamma [B_c(2^3{P}_2)\to B_c^{\ast }\gamma ]\simeq 52\mathrm{keV},$$

$$\Gamma [B_c(2^3{P}_2)\to B_c^{\ast }(2S)\gamma ]\simeq 50\mathrm{keV},$$

$$Br[B_c(2P_1)\to B_c\gamma ]\simeq 20\%,$$

$$Br[B_c(2P_1^{\prime })\to B_c\gamma ]\simeq 33\%$$

$$Br[B_c(2^3{P}_0)\to B_c^{\ast }\gamma ]\simeq 41\%.$$

$$Br[B_c(2^3{D}_1)\to B{D}^{\ast }]\simeq 87\%.$$

Full text

I Introduction

The $B_{c}$ states composed of a bottom-charmed quark-antiquark pair, as an important family of hadron spectra was predicted in theory about 40 years ago Eichten:1980mw , however, the experimental progress towards establishing the $B_{c}$ spectrum is not obvious. Except for the ground state $B_{c}$ meson observed in 1998 by the CDF Collaboration at Fermilab Abe:1998fb , until 2018, only the ATLAS Collaboration reported evidence of an excited $B_{c}$ state with a mass of $6842\pm 9$ MeV Aad:2014laa consistent with the values predicted for $B_{c}(2S)$, while it was not confirmed by the LHCb Collaboration by using their 8 TeV data sample Aaij:2017lpg . The poor situation of the observations and measurements of $B_{c}$ spectrum is due to that the production yields are significantly smaller than those of the charmonium and bottomonium ($c\bar{c}$ and $b\bar{b}$) states. Fortunately, the LHC provides good opportunities for our search for the excited $B_{c}$ states with its high collision energies and integrated luminosity. Very recently, two excited $B_{c}^{+}$ states were observed in the $B_{c}^{+}\pi^{+}\pi^{-}$ invariant mass spectrum by the CMS Collaboration Sirunyan:2019osb . Signals are consistent with the $B_{c}(2S)$ and $B^{*}_{c}(2S)$ states. These two states are well resolved from each other and are observed with a significance exceeding five standard deviations. The mass of $B_{c}(2S)$ meson, $6871\pm 2.8$ MeV, measured by the CMS Collaboration is inconsistent with the determination $6842\pm 9$ MeV by the ATLAS Collaboration. The reason is that the peak observed by ATLAS could be the superposition of the $B_{c}(2S)$ and $B^{*}_{c}(2S)$ states, too closely spaced with respect to the resolution of the measurement Sirunyan:2019osb .

The $B_{c}$ states as the only conventional heavy mesons with different flavors have aroused great interests in theory. Compared with the $c\bar{c}$ and $b\bar{b}$ spectra, the $B_{c}$ spectrum has several special features for the bottom-charmed quark-antiquark pair. (i) The $B_{c}$ states cannot annihilate into gluons, thus, the lowlying excited $B_{c}$ states below the $DB$ threshold are more stable with a narrow width less than a few hundred keV, they mainly decay via the electromagnetic or hadronic transitions between two different $B_{c}$ states. (ii) In the $B_{c}$ meson spectrum there are configuration mixings between the states with different total spins but with the same total angular momentum, such as ${}^{3}P_{1}-^{1}P_{1}$, ${}^{3}D_{2}-^{1}D_{2}$, and ${}^{3}F_{3}-^{1}F_{3}$ mixings via the antisymmetric part of the spin-orbit potential. (iii) Additionally, the $B_{c}$ states provide a unique window for studying the heavy-quark dynamics that is very different from those provided by the $c\bar{c}$ and $b\bar{b}$ states. In the past years, the $B_{c}$ mass spectrum has been predicted with various models Zeng:1994vj ; Eichten:1994gt ; Ebert:2002pp ; Godfrey:2004ya ; Kiselev:1994rc ; Monteiro:2016ijw ; Soni:2017wvy ; Godfrey:1985xj ; Tang:2018myz ; Baldicchi:2000cf ; Ikhdair:2003tt ; Ikhdair:2003ry ; Wang:2012kw ; Chen:2011qu ; Wei:2010zza ; Badalian:2009cx ; Rai:2008sc ; Guo:2008he ; Patel:2008na ; Bernotas:2008bu ; AbdElHady:2005bv ; AbdElHady:1998kc ; Li:2004gu ; Ikhdair:2004hg ; Fulcher:1998ka ; Motyka:1997di ; Devlani:2014nda ; Monteiro:2016rzi ; Eichten:2019gig . Furthermore, a few lattice calculations can be found in Refs. Davies:1996gi ; Mathur:2018epb ; Gregory:2009hq ; Dowdall:2012ab ; Allison:2004be . To estimate the production rates in experiments, the production of the excited $B_{c}$ states were often discussed in the literature Braaten:1996pv ; Cheung:1995ir ; Cheung:1995ye ; Cheung:1993qi ; Berezhnoy:1996ks ; Chang:2004bh ; Chang:2005bf ; Liao:2018nab ; Kai:2017cba ; Liao:2015vqa ; Liao:2014rca ; Liao:2011kd ; Yang:2010yg ; Chang:2007si . As the dominant decay modes, the electromagnetic transitions of the low-lying $B_{c}$ states were also widely estimated in the literature Eichten:1994gt ; Zeng:1994vj ; Kiselev:1994rc ; Fulcher:1998ka ; Ebert:2002pp ; Godfrey:2004ya ; AbdElHady:2005bv ; Devlani:2014nda ; Monteiro:2016ijw ; Soni:2017wvy ; Patnaik:2017cbl ; Simonis:2016pnh ; Wang:2015yea ; Wang:2013cha ; Ebert:2002xz . However, the studies of the OZI-allowed strong decays for the high-lying $B_{c}$ states are confined only to a few calculations Kiselev:1996un ; Ferretti:2015rsa ; Monteiro:2016rzi ; Eichten:2019gig .

The successes of the observations of the radially excited $B_{c}$ states $B_{c}(2S)$ and $B^{*}_{c}(2S)$ by the CMS Collaboration Sirunyan:2019osb have demonstrated that more excited $B_{c}$ states are to be discovered in future LHC experiments. Stimulated by the great discovery potentials of the missing $B_{c}$ states in future experiments, in present work we carry out a systematical study of the $B_{c}$ spectrum. First, using the newly measured masses of $B_{c}(1S)$ and $B_{c}(2S)$ from the CMS Collaboration Sirunyan:2019osb and the $1S$ hyperfine splitting determined from the lattice QCD Gregory:2009hq ; Dowdall:2012ab ; Mathur:2018epb as constrains, we calculate the $B_{c}$ mass spectrum up to the $6S$ multiplet with a nonrelativistic linear potential model. The slope parameter of the linear potential has been well determined in our previous study of the charmonium states Deng:2016stx . To involve the spin-dependent corrections of the spatial wave functions, following the method adopted in Refs. Deng:2016ktl ; Deng:2016stx we treat the spin-dependent interactions as nonperturbative terms in our calculations. With this nonperturbative treatment, we can reasonably include the effect of spin-dependent interactions on the spatial wave functions, which is essential for us to gain reliable predictions of the decay behaviors.

Then, with the available wavefunctions from the potential model, we evaluate the electromagnetic transitions between the $B_{c}$ states within a nonrelativistic constituent quark model developed in our previous works Deng:2016ktl ; Deng:2016stx . With this approach the possible higher EM multipole contributions to a EM transition process can be included naturally. Considering the fact that the higher $B_{c}$ states lying above the $DB$ threshold may have enough possibilities to be produced at LHC, and they are easy to be established in the $D^{(*)}B^{(*)}$ hadronic final states, thus to give useful references for the LHC observations, we further calculate the OZI-allowed strong decays of the higher $B_{c}$ states within the widely used ${}^{3}P_{0}$ model Micu:1968mk ; LeYaouanc:1972vsx ; LeYaouanc:1973ldf . It is found that $B_{c}(2^{3}P_{2})$, $B_{c}(2^{3}D_{1})$, $B_{c}(3^{3}D_{1})$ together with the $1F$-wave $B_{c}$ states might be first observed in their dominant strong decay channels $DB$, $DB^{*}$ or $D^{*}B$ at LHC for their relatively narrow width.

This paper is organized as follows. In Sec. II, the $B_{c}$ mass spectrum is calculated within a nonrelativistic linear potential model. Then, with the obtained $B_{c}$ spectrum the radiative transitions between the $B_{c}$ states are estimated in Sec. III within a nonrelativistic constituent quark model. In Sec. IV, the OZI-allowed two-body strong decays of the excited $B_{c}$ state are also studied within the ${}^{3}P_{0}$ model. In Sec. V, we focus on the calculation results and discuss some strategies for looking for the $B_{c}$ states in future experiments. Finally, a summary is given in Sec. VI.

II mass spectrum

To describe a bottom-charmed meson system, we adopt a nonrelativistic linear potential model. In this model, the effective quark-antiquark potential is written as the sum of the spin-independent term $H_{0}(r)$ and spin-dependent term $H_{sd}(r)$, i.e.,

[TABLE]

where

[TABLE]

includes the standard color Coulomb interaction and the linear confinement. The spin-dependent part $H_{sd}(r)$ can be expressed as Eichten:1980mw ; Kiselev:1994rc ; Godfrey:2004ya

[TABLE]

where

[TABLE]

is the spin-spin contact hyperfine potential. Here, we take $\tilde{\delta}_{\sigma}(r)=(\sigma/\sqrt{\pi})^{3}e^{-\sigma^{2}r^{2}}$ as suggested in Ref. Barnes:2005pb . The tensor potential $H_{T}$ is adopted as

[TABLE]

For convenience in the calculations, the potential of the spin-orbit interaction $H_{LS}$ is decomposed into symmetric part $H_{sym}$ and antisymmetric part $H_{anti}$,

[TABLE]

with

[TABLE]

In these equations, $\mathbf{L}$ is the relative orbital angular momentum of the $q\bar{q}$ system; $\mathbf{S}_{q}$ and $\mathbf{S}_{\bar{q}}$ are the spins of the quark $q$ and antiquark $\bar{q}$, respectively, and $\mathbf{S}_{\pm}\equiv\mathbf{S}_{q}\pm\mathbf{S}_{\bar{q}}$; $m_{q}$ and $m_{\bar{q}}$ are the masses of quark $q$ and antiquark $\bar{q}$, respectively; $\alpha_{s}$ is the running coupling constant of QCD; and $r$ is the distance between the quark $q$ and antiquark $\bar{q}$. The five parameters in the above equations ($\alpha_{s}$, $b$, $\sigma$, $m_{b}$, $m_{c}$) are determined by fitting the spectrum.

We can get the masses and wave functions by solving the radial Schrödinger equation

[TABLE]

with

[TABLE]

where $\mu_{R}=m_{q}m_{\bar{q}}/(m_{q}+m_{\bar{q}})$ is the reduced mass of the system, and $E$ is the binding energy of the system. Then, the mass of a bottom-charmed state is obtained by

[TABLE]

In this work, to reasonably include the corrections from these spin-dependent potentials to both the mass and wave function of a meson state, we deal with the spin-dependent interactions nonperturbatively. We solve the radial Schrödinger equation by using the three-point difference central method Haicai from central ($r=0$) towards outside ($r\to\infty$) point by point. This method was successfully to deal with the spectroscopies of $c\bar{c}$ and $b\bar{b}$ Deng:2016ktl ; Deng:2016stx . To overcome the singular behavior of $1/r^{3}$ in the spin-dependent potentials, following the method of our previous works Deng:2016ktl ; Deng:2016stx , we introduce a cutoff distance $r_{c}$ in the calculation. Within a small range $r\in(0,r_{c})$, we let $1/r^{3}=1/r_{c}^{3}$.

Finally, it should be mentioned that the antisymmetric part of the spin-orbit potential, $H_{anti}$, can let the states with different total spins but with the same total angular momentum, such as $B_{c}(n^{3}L_{J})$ and $B_{c}(n^{1}L_{J})$, mix with each other. Thus, as mixing states between $B_{c}(n^{3}L_{J})$ and $B_{c}(n^{1}L_{J})$, the physical $B_{c}$ states $B_{c}(nL)$ and $B_{c}(nL^{\prime})$ are expressed as

[TABLE]

where $J=L=1,2,3\cdots$, and the $\theta_{nL}$ is the mixing angle. In this work $B_{c}(nL^{\prime})$ corresponds to the higher mass mixed state as often adopted in the literature.

In this work the parameter set is taken as $\alpha_{s}=0.5021$, $b=0.1425$GeV2, $m_{b}=4.852$ GeV, $m_{c}=1.483$ GeV, $\sigma=1.3$ GeV and $r_{c}=0.16$ fm. To consistent with our previous study Deng:2016stx , the charmed quark mass $m_{c}$ and the slope for the linear confining potential are taken the determinations, i.e., $m_{c}=1.483$ GeV and $b=0.1425$ GeV2. The other three parameters ($m_{b}$, $\alpha_{s}$, $\sigma$) are determined by fitting the masses of the $B_{c}$, $B_{c}^{*}$ and $B_{c}(2S)$ mesons. The masses of $B_{c}$ and $B_{c}(2S)$ are taken from the recent measurements of the CMS Collaboration Sirunyan:2019osb . Although the $B_{c}^{*}$ meson is still not measured in experiments, the mass difference between the $B_{c}^{*}$ and $B_{c}$ is predicted to be around 55 MeV from lattice QCD Gregory:2009hq ; Dowdall:2012ab ; Mathur:2018epb . Thus, combining it with the measured mass 6271 MeV for $B_{c}$, in present work we estimate the mass of $B_{c}^{*}$ as $\sim 6326$ MeV. The cutoff distance $r_{c}$ is determined by the mass of $B_{c}(1^{3}P_{0})$. To determined the mass of $B_{c}(1^{3}P_{0})$, we adopt a method of perturbation, i.e., we let $H=H_{0}+H^{\prime}$, where $H^{\prime}$ is a part which contained the term of $1/r^{3}$. By solving the equation of $H_{0}|\psi_{n}^{(0)}\rangle=E_{0}|\psi_{n}^{(0)}\rangle$, we can get the energy $E_{0}$ and wave function $|\psi_{n}^{(0)}\rangle$, then, we obtain the mass of $B_{c}(1^{3}P_{0})$, $M=m_{b}+m_{c}+E_{0}+\langle\psi_{n}^{(0)}|H^{\prime}|\psi_{n}^{(0)}\rangle$.

By solving the radial Schrödinger equation and with the determined parameter set, we obtain the masses of the bottom-charmed states, which have been listed in Tab. 1 and shown in Fig. 1. For comparison, the other model predictions in Refs. Zeng:1994vj ; Eichten:1994gt ; Ebert:2002pp ; Godfrey:2004ya ; Kiselev:1994rc ; Monteiro:2016ijw ; Soni:2017wvy are listed in the same table as well.

It is found that the masses of the low-lying $1S$-, $2S$-, $3S$-, $1P$-, $2P$-, $1D$-wave $B_{c}$ states predicted in this work are compatible with the other potential model predictions. For the higher mass states, such as $4S$-, $5S$-, $6S$-, $3P$-, $4P$-, $2D$-, $2F$-, $3F$-wave states, the masses predicted by us are very close to those predicted with a relativistic model in Ref. Zeng:1994vj , while are about $100-200$ MeV smaller than those predicted in Refs. Monteiro:2016ijw ; Soni:2017wvy . Furthermore, the hyperfine splitting between $B_{c}^{*}(2S)$ and $B_{c}(2S)$ is predicted to be $19$ MeV, which is slightly smaller than $30-45$ MeV predicted in previous works Gregory:2009hq ; Dowdall:2012ab ; Mathur:2018epb ; Zeng:1994vj ; Soni:2017wvy ; Monteiro:2016ijw ; Eichten:1994gt ; Ebert:2002pp ; Godfrey:2004ya ; Kiselev:1994rc . Finally, it should be pointed out the mixing angles for ${}^{3}P_{1}-^{1}P_{1}$, ${}^{3}D_{2}-^{1}D_{2}$, and ${}^{3}F_{3}-^{1}F_{3}$ have obvious model dependencies (see Tab. 2).

III RADIATIVE TRANSITIONS

We use the nonrelativistic constituent quark model as adopted in Refs. Deng:2016ktl ; Deng:2016stx ; Wang:2017hej ; Xiao:2017udy ; Lu:2017meb ; Wang:2017kfr ; Yao:2018jmc to calculate the radiative transitions between the $B_{c}$ states. In this model, the quark-photon EM coupling at the tree level is taken as

[TABLE]

where $A^{\mu}$ represents the photon field with three momenta $\mathbf{k}$; while $e_{j}$ and $\mathbf{r}_{j}$ stand for the charge and coordinate of the constituent quark $\psi_{j}$, respectively. In order to match the nonrelativistic wave functions of the $B_{c}$ states, we adopt the nonrelativistic form of Eq. (13), which is given by  Brodsky:1968ea ; Li:1997gd ; Zhao:2002id ; Xiao:2015gra ; Zhong:2011ti ; Zhong:2011ht ,

[TABLE]

where $\boldsymbol{\epsilon}$ is the polarization vector of the final photon, $m_{j}$ and \mbox{\boldmath\sigma\unboldmath}_{j} stand for the constituent mass and Pauli spin vector for the $j$th quark. The helicity amplitude $\mathcal{A}$ can be expressed as

[TABLE]

Finally, we obtain the partial decay width of a radiative transition by

[TABLE]

where $J_{i}$ is the total angular momentum of an initial meson, and $J_{fz}$ and $J_{iz}$ are the components of the total angular momenta along the $z$ axis of initial and final mesons, respectively. $M_{i}$ and $M_{f}$ correspond to the masses of the initial and final $B_{c}$ states, respectively.

The radiative decays properties for the $B_{c}$ states have been listed in Tables  3- 7. For a comparison, some other predictions of the low-lying $B_{c}$ states from Refs. Eichten:1994gt ; Ebert:2002pp ; Godfrey:2004ya ; Kiselev:1994rc are also given in the tables.

IV strong decays

In this work, we use the ${}^{3}P_{0}$ model Micu:1968mk ; LeYaouanc:1972vsx ; LeYaouanc:1973ldf to calculate the OZI-allowed strong decays of the bottom-charmed mesons. In this model, it assumes that the vacuum produces a quark-antiquark pair with the quantum number $0^{++}$ and the heavy meson decay takes place via the rearrangement of the four quarks. The transition operator $\hat{T}$ in this model can be written as

[TABLE]

where $\gamma$ is a dimensionless constant that denotes the strength of the quark-antiquark pair creation with momentum $\mathbf{p}_{3}$ and $\mathbf{p}_{4}$ from vacuum; $b_{3i}^{\dagger}(\mathbf{p}_{3})$ and $d_{4j}^{\dagger}(\mathbf{p}_{4})$ are the creation operators for the quark and antiquark, respectively; the subscriptions, $i$ and $j$, are the SU(3)-color indices of the created quark and anti-quark; $\phi_{0}^{34}=(u\bar{u}+d\bar{d}+s\bar{s})/\sqrt{3}$ and $\omega_{0}^{34}=\frac{1}{\sqrt{3}}\delta_{ij}$ correspond to flavor and color singlets, respectively; $\chi_{{1,-m}}^{34}$ is a spin triplet state; and $\mathcal{Y}_{\ell m}(\mathbf{k})\equiv|\mathbf{k}|^{\ell}Y_{\ell m}(\theta_{\mathbf{k}},\phi_{\mathbf{k}})$ is the $\ell$-th solid harmonic polynomial. The factor $(-3)$ is introduced for convenience, which will cancel the color factor.

For an OZI allowed two-body strong decay process $A\to B+C$, the helicity amplitude $\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}(\mathbf{P})$ can be derived as follow

[TABLE]

Using the Jacob-Wick formula Jacob:1959at , one can convert the helicity amplitudes $\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}(\mathbf{P})$ to the partial wave amplitudes $\mathcal{M}^{JL}$ via

[TABLE]

In the above equations, ($J_{A}$, $J_{B}$ and $J_{C}$), ($L_{A}$, $L_{B}$ and $L_{C}$) and ($S_{A}$, $S_{B}$ and $S_{C}$) are the quantum numbers of the total angular momenta, orbital angular momenta and total spin for hadrons $A,B,C$, respectively; $M_{J_{A}}=M_{J_{B}}+M_{J_{C}}$ , $\mathbf{J}\equiv\mathbf{J}_{B}+\mathbf{J}_{C}$ and $\mathbf{J}_{A}\equiv\mathbf{J}_{B}+\mathbf{J}_{C}+\mathbf{L}$. In the c.m. frame of hadron $A$, the momenta $\mathbf{P}_{B}$ and $\mathbf{P}_{C}$ of mesons $B$ and $C$ satisfy $\mathbf{P}_{B}=-\mathbf{P}_{C}\equiv\mathbf{P}$.

Then the strong decay partial width for a given decay mode of $A\to B+C$ is given by

[TABLE]

where $M_{A}$ is the mass of the initial hadron $A$, while $E_{B}$ and $E_{C}$ stand for the energies of final hadrons $B$ and $C$, respectively. The details of the ${}^{3}P_{0}$ model can be found in our recent paper Gui:2018rvv .

In the calculations, the wavefunctions of the initial $B_{c}$ states are adopted our quark model predictions. Furthermore, we need the wavefunctions of the final hadrons, i.e., the $B^{(*)}$, $B_{s}^{(*)}$, $D^{(*)}$, $D_{s}^{(*)}$ mesons and some of their excitations, which are adopted from the quark model predictions of Refs. Lu:2016bbk ; Li:2010vx .

In this work, for the masses of the light constituent $u$, $d$ and $s$ quarks, we set $m_{u}=m_{d}=0.33$ GeV, $m_{s}=0.45$GeV; while for the heavy $b$ and $c$ quarks, their masses are taken to be $m_{b}=4.852$ GeV and $m_{c}=1.483$ GeV as the determinations in the calculations of the $B_{c}$ mass spectrum. The masses of the final hadron states in the decay processes are adopted from the Particle Data Group Tanabashi:2018oca if there are measured values, otherwise we take the quark model predictions of Refs. Lu:2016bbk ; Li:2010vx (see Table 5). There is no experimental data which can be used to determine the quark pair creation strength, thus, in this work we adopt a typical value $\gamma=0.4$ that gives a reasonably accurate description of the overall scale of decay widths of both light and heavy mesons Ackleh:1996yt ; Godfrey:2016nwn ; Barnes:2002mu ; Barnes:2005pb ; Godfrey:2015dva ; Close:2005se . The strong decays properties for the bottom-charmed states are presented in Tab. 10 to  15.

V discussion

V.0.1 $S$-wave states

Recently, signals of two excited $\bar{b}c$ states $B_{c}(2S)$ and $B^{*}_{c}(2S)$ were observed in the $B_{c}^{+}\pi^{+}\pi^{-}$ invariant mass spectrum by the CMS Collaboration at LHC. These two states are well resolved from each other and are observed with a significance exceeding five standard deviations. The mass of $B_{c}(2S)$ meson is measured to be $6871\pm 2.8$ MeV. Furthermore, a more precise mass of $B_{c}(2S)$, $M(B_{c}^{+})=6871.1\pm 0.5$ MeV, is measured by the CMS Collaboration as well. Combining these newest measurements, we predict that the mass of $B_{c}(2S)$ might be $\sim 6890$ MeV, and the mass hyperfine splitting between $B^{*}_{c}(2S)$ and $B_{c}(2S)$,

[TABLE]

is slightly smaller than $30-45$ MeV predicted in previous works (see Table 1). The predicted masses for the other higher $S$-wave states compared with other works are also given in Table 1. Obvious differences can be found in various theoretical predictions.

The $M1$ transitions of the low-lying $S$-wave states $B_{c}^{*}(2S)$ and $B_{c}^{(*)}(1S)$ were often discussed in the literature for these transitions which might be used to establish them in experiments. In this work we also calculate their $M1$ transitions. Our results compared with the some other predictions are listed Table 3. Obvious model dependence can be seen in various calculations. Our predicted partial width

[TABLE]

for the $M1$ transition $B_{c}^{*}(2S)\to B_{c}\gamma$ is about an order of magnitude larger than that predicted in Refs. Eichten:1994gt ; Ebert:2002pp ; Kiselev:1994rc , and about a factor 2 larger than the value predicted within the GI model Godfrey:2004ya . Combining our calculations of the EM transitions $B_{c}^{*}(2S)\to 1P\gamma$ and the strong transitions $B_{c}^{*}(2S)\to B_{c}^{*}\pi\pi$ predicted in Godfrey:2004ya , the total decay width of $B_{c}^{*}(2S)$ meson is estimated to be $\Gamma_{total}\sim 75$ keV, then the branching fraction for $M1$ transition $B_{c}^{*}(2S)\to B_{c}\gamma$ is predicted to be

[TABLE]

The fairly large branching fraction may give a good opportunity for us to observe the $B_{c}^{*}(2S)$ via the $M1$ transition $B_{c}^{*}(2S)\to B_{c}\gamma$. This process may be used to determined the mass of $B_{c}^{*}(2S)$ in future experiments.

The masses of $3S$-wave states $B_{c}(3^{1}S_{0})$ and $B_{c}(3^{3}S_{1})$ are predicted to be $\sim 7.24$ GeV and $\sim 7.25$ GeV, respectively, which are just above the $DB^{*}$ threshold. Their radiative and strong decay properties are estimated in this work. The results for the $M1$ transitions, $E1$ dominant transitions and strong decays of the $3S$-wave states are given in Tables 3, 7 and 10, respectively. There are only a few works about the radiative and strong decay properties of the $3S$-wave states Godfrey:2004ya ; Kiselev:1996un ; Ferretti:2015rsa ; Eichten:2019gig . The $M1$ transitions of the $3S$-wave states roughly agree with the predictions in Ref. Godfrey:2004ya , except that our predicted partial width $\Gamma[3^{3}S_{1}\to 1^{1}S_{0}+\gamma]\simeq 510$ eV for the $M1$ transition $3^{3}S_{1}\to 1^{1}S_{0}+\gamma$ is about an order of magnitude smaller than that in Ref. Godfrey:2004ya . The strong decay widths of $B_{c}(3^{1}S_{0})$ and $B_{c}(3^{3}S_{1})$ predicted by us are comparable to those predicted in recent works Ferretti:2015rsa ; Eichten:2019gig . Both $B_{c}(3^{1}S_{0})$ and $B_{c}(3^{3}S_{1})$ might be broad states with a width of $\sim 100$ MeV. The $B_{c}(3^{1}S_{0})$ dominantly decay into $DB^{*}$ channel, while $B_{c}(3^{3}S_{1})$ dominantly decay into both $DB$ and $DB^{*}$ channels. The production rates of the $3S$-wave $B_{c}$ states in $pp$ collisions at the LHC may be comparable with those of the $2S$-wave $B_{c}$ states Eichten:2019gig , thus, the $3S$-wave $B_{c}$ states may have large potentials to be established in the $DB^{*}$ final states.

The higher $S$-wave states $B_{c}(n^{1}S_{0})$ and $B_{c}(n^{3}S_{1})$ ($n\geq 4$) are far from the $DB$ threshold, thus many OZI-allowed two-body strong decay channels are open. There are few discussions of the decay properties of the higher mass $S$-wave states in the literature. To know some decay properties of these higher $S$-wave states, in this work we give our predictions of the $M1$ transitions and strong decays of $B_{c}(nS)$ ($n=4,5,6$), which are listed in Table 4 and  10, respectively. It is found these higher mass $S$-wave states are broad states with a width of $\sim 100-400$ MeV. Combining $M1$ transitions of higher $S$-wave states with their strong decays, we found that the branching fractions of the $M1$ transitions $B_{c}(nS)\to B_{c}(1S)+\gamma$ may reach up to a sizeable value $\mathcal{O}(10^{-5})$.

V.0.2 $P$-wave states

The masses of $1P$-wave states $B_{c}(1P)$ might lie in the range of $(6710,~{}6790)$ MeV, which are consistent with the other predictions with potential models Eichten:1994gt ; Ebert:2002pp ; Zeng:1994vj ; Godfrey:2004ya ; Kiselev:1994rc , and the recent lattice calculations Mathur:2018epb . The $1P$-wave $B_{c}(1P)$ states mainly decays via the $E1$ dominate transitions $1P\to 1S$. We have calculated the partial decay widths for the $EM$ transitions $1P\to 1S$, our results compared with some other predictions are listed in Table 6. Most of our results are compatible with the predictions in Eichten:1994gt ; Ebert:2002pp ; Godfrey:2004ya ; Kiselev:1994rc , except our predicted partial decay widths of $\Gamma[B_{c}(1P_{1})\to B_{c}\gamma]\simeq 35$ keV and $\Gamma[B_{c}(1P_{1}^{\prime})\to B_{c}^{*}\gamma]\simeq 40$ keV are about a factor of $3-5$ larger than the predictions in Refs. Ebert:2002pp ; Godfrey:2004ya ; Kiselev:1994rc . The $B_{c}(1P_{1})$ and $B_{c}(1P_{1}^{\prime})$ states might be first found in the $B_{c}\gamma$ final state via their radiative transitions. The branching fractions for $B_{c}(1P_{1})$ and $B_{c}(1P_{1}^{\prime})$ decay into $B_{c}\gamma$ are predicted to be

[TABLE]

While the $B_{c}(1^{3}P_{0})$ and $B_{c}(1^{3}P_{2})$ states dominantly decay into $B_{c}^{*}\gamma$ final state with a decay rate $\sim 100\%$, thus, they have good potentials to be found via the radiative decay chains $B_{c}(1^{3}P_{0})\to B_{c}(1^{3}S_{1})\gamma\to B_{c}(1^{1}S_{0})\gamma\gamma$ and $B_{c}(1^{3}P_{2})\to B_{c}(1^{3}S_{1})\gamma\to B_{c}(1^{1}S_{0})\gamma\gamma$, respectively.

For the $2P$-wave states $B_{c}(2P)$, their masses might lie in the range $(7100,~{}7160)$ MeV, which are consistent with the other model predictions in the literature Zeng:1994vj ; Eichten:1994gt ; Ebert:2002pp ; Godfrey:2004ya ; Kiselev:1994rc ; Soni:2017wvy ; Monteiro:2016ijw . The masses for $B_{c}(2^{3}P_{0})$ and $B_{c}(2P_{1})$ are slightly lower than the $DB$ mass threshold, while $B_{c}(2P^{\prime}_{1})$ and $B_{c}(2^{3}P_{2})$ slightly lie above the $DB$ mass threshold. The $B_{c}(2^{3}P_{2})$ state mainly decay into the $DB$ channel, while its radiative decay rates into the $B_{c}(n^{3}S_{1})\gamma$ ($n=1,2$) are also sizeable. Their partial widths are predicted to be

[TABLE]

Thus, the total width of $B_{c}(2^{3}P_{2})$ is $\Gamma_{\mathrm{total}}[B_{c}(2^{3}P_{2})]\simeq 880$ keV. The $B_{c}(2^{3}P_{2})$ state may have potentials to be observed in the $DB$ and $B_{c}\gamma$ final states. While for $B_{c}(2^{3}P_{0})$, $B_{c}(2P_{1})$ and $B_{c}(2P^{\prime}_{1})$ states, their decays are governed by the $EM$ transitions. The radiative decay properties of these states have been given in Table 8. With these predictions, the total widths for $B_{c}(2^{3}P_{0})$, $B_{c}(2P_{1})$ and $B_{c}(2P^{\prime}_{1})$ are estimated to be $\Gamma_{\mathrm{total}}[B_{c}(2^{3}P_{0})]\simeq 100$ keV, $\Gamma_{\mathrm{total}}[B_{c}(2P_{1})]\simeq 120$ keV, and $\Gamma_{\mathrm{total}}[B_{c}(2P_{1}^{\prime})]\simeq 133$ keV, respectively. The branching fractions for $B_{c}(2P_{1})\to B_{c}\gamma$, $B_{c}(2P^{\prime}_{1})\to B_{c}\gamma$ and $B_{c}(2^{3}P_{0})\to B_{c}^{*}\gamma$ are predicted to be

[TABLE]

The large branching fractions indicate that $B_{c}(2P_{1})$ and $B_{c}(2P^{\prime}_{1})$ may be established in the $B_{c}\gamma$ channel, while $B_{c}(2^{3}P_{0})$ may be observed via the radiative decay chain $B_{c}(2^{3}P_{0})\to B_{c}^{*}\gamma\to B_{c}\gamma\gamma$. It should be pointed out that the $B_{c}(2P_{1})$, $B_{c}(2P^{\prime}_{1})$ and $B_{c}(2^{3}P_{2})$ states may lie above the $B^{*}D$ threshold, so they may have fairly large strong decay widths $\mathcal{O}(10-100)$ MeV into $B^{*}D$ and/or $BD$ channels as predicted in Ref. Monteiro:2016rzi .

For the higher $P$-wave states $B_{c}(nP)$ ($n=3,4$), many OZI allowed strong decay channels are open (see Table 12), thus, these states usually are broad states with a width of $\mathcal{O}(100)$ MeV, except the $B_{c}(4^{3}P_{0})$ state has a relatively narrow width of $\mathcal{O}(10)$ MeV. The $B_{c}(4^{3}P_{0})$ state may be first observed in the $DB$ channel, the branching fraction for the process $B_{c}(4^{3}P_{0})\to DB$ can reach up to $\sim 20\%$.

V.0.3 $D$-wave states

The masses of the $1D$-wave states $B_{c}(1D)$ is predicted to be $\sim 7.02$ GeV in this work. The mass splitting between the $1D$-wave states is no more than $15$ MeV. The masses predicted by us are consistent with the results in Refs. Zeng:1994vj ; Eichten:1994gt ; Godfrey:2004ya . The $1D$-wave states mainly decay via the $EM$ transitions, which have been given in Table 6. It is seen that our main results are in reasonable agreement with the other predictions. Our study indicates that the $B_{c}(1^{3}D_{3})$ state may have a relatively large potential to be observed via the radiative decay chain $B_{c}(1^{3}D_{3})\to B_{c}(1^{3}P_{2})\gamma\to B_{c}(1^{3}S_{1})\gamma\gamma\to B_{c}(1^{1}S_{0})\gamma\gamma\gamma$, and the branching fraction for this chain is estimated to be $\sim 100\%$. The optimal decay chain for the observations of $B_{c}(1^{3}D_{1})$ is $B_{c}(1^{3}D_{1})\to B_{c}(1^{3}P_{0})\gamma\to B_{c}(1^{3}S_{1})\gamma\gamma\to B_{c}(1^{1}S_{0})\gamma\gamma\gamma$, and the branching fraction for this chain is estimated to be $\sim 60\%$. The optimal decay chains for the observations of $B_{c}(1D_{2})$ are $B_{c}(1D_{2})\to B_{c}(1P_{1})\gamma\to B_{c}(1^{3}S_{1})\gamma\gamma\to B_{c}(1^{1}S_{0})\gamma\gamma\gamma$ and $B_{c}(1D_{2})\to B_{c}(1P_{1})\gamma\to B_{c}(1^{1}S_{0})\gamma\gamma$, and the branching fraction for these chains are estimated to be $\sim 50\%$ and $\sim 30\%$, respectively. While for the observations of $B_{c}(1D_{2}^{\prime})$, the optimal decay chains are $B_{c}(1D_{2}^{\prime})\to B_{c}(1P_{1}^{\prime})\gamma\to B_{c}(1^{3}S_{1})\gamma\gamma\to B_{c}(1^{1}S_{0})\gamma\gamma\gamma$ and $B_{c}(1D_{2}^{\prime})\to B_{c}(1P_{1}^{\prime})\gamma\to B_{c}(1^{1}S_{0})\gamma\gamma$, and the branching fraction for these chains are estimated to be $\sim 35\%$ and $\sim 47\%$, respectively.

The masses of the $2D$ states are predicted to be $\sim 7.34$ GeV, which is very close to the $D_{s}B_{s}$ threshold. Their decays are governed by the strong decay modes, such as $DB$, $DB^{*}$, $BD^{*}$ or $B^{*}D^{*}$. Their strong decay properties predicted by us have been listed in Table 13. There are few discussions about the radiative decays of the $2D$-wave $B_{c}$ states in the literature. In this work, we also calculate their radiative decay properties, our results are given in Table 7. It is found that the $B_{c}(2^{3}D_{1})$ state has a relatively narrow width of $\Gamma\sim 58$ MeV. The decays of $B_{c}(2^{3}D_{1})$ are governed by the $BD^{*}$ mode with a branching fraction

[TABLE]

The other three $2D$ states $B_{c}(2^{3}D_{3})$, $B_{c}(2D_{2})$ and $B_{c}(2D_{2}^{\prime})$ are broad states with a width of $\sim 100-200$ MeV. The $B_{c}(2^{3}D_{3})$ state mainly decays into $DB$, $DB^{*}$, and $B^{*}D^{*}$ channels. While the $B_{c}(2D_{2})$ and $B_{c}(2D_{2}^{\prime})$ states dominantly decay into $DB^{*}$, $BD^{*}$ or $B^{*}D^{*}$ channels. Combing the strong and radiative decay properties with each other, it is found that the branching fractions of the dominant $EM$ decay processes $B_{c}(2D)\to B_{c}(nP)$ ($n=1,2$) are $\mathcal{O}(10^{-4})$. The observations of the $DB$, $DB^{*}$, $BD^{*}$ or $B^{*}D^{*}$ final states might be useful to search for these missing $2D$ states in future experiments.

The higher $3D$-wave states $B_{c}(3D)$ are also studied in present work. The masses predicted by us are about $7.62$ GeV, which are comparable with those predicted in Ref. Zeng:1994vj , while about 150 MeV smaller than those predicted in Refs. Soni:2017wvy ; Monteiro:2016ijw . The strong decay properties are shown in Table 13. It is found that these higher $3D$-wave states have a width of $\sim 100$ MeV. These higher states might be observed in their dominant strong decay channels.

V.0.4 $F$-wave states

The masses of the $1F$-wave states $B_{c}(1^{3}F_{4})$, $B_{c}(1F_{3})$, $B_{c}(1F_{3}^{\prime})$ and $B_{c}(1^{3}F_{2})$ are predicted to be $\sim 7.23$ GeV, which are comparable to those predicted in Refs. Zeng:1994vj ; Soni:2017wvy ; Godfrey:2004ya . These $1F$ wave states lie above the mass threshold of $DB$ and $B^{*}D$, while below the $D^{*}B$ threshold. From our predictions of the strong decay properties for these $1F$ wave states (see Table 14), it is found that the $B_{c}(1^{3}F_{4})$ state might be a very narrow state with a width of $\sim 1$ MeV, its decays are governed by the $DB$ mode. Both $B_{c}(1F_{3})$ and $B_{c}(1F_{3}^{\prime})$ are narrow states with a width of $\sim 10$ MeV, they dominantly decay into the $DB^{*}$ channel. The $B_{c}(1^{3}F_{2})$ should be a relatively broad state with a width of $\sim 70$ MeV, it mainly decays into the $DB$ channel with a branching fraction of $Br[B_{c}(1^{3}F_{2})\to DB]\simeq 85\%$. To look for the missing $1F$-wave $B_{c}$ states, the $DB$ and $B^{*}D$ final states are worth observing.

The predicted masses for the $2F$- and $3F$-wave $B_{c}$ states are $\sim 7.5$ GeV and $\sim 7.8$ GeV, respectively, which are comparable with the predictions in Refs. Zeng:1994vj ; Godfrey:2004ya . There are many strong decay channels for these higher mass $F$-wave states. Our predictions of their strong decay properties have been listed in Tables 14 and  15. It is found that the higher mass $F$-wave states might be broad states with a width of $\sim 100-300$ MeV.

VI summary

In this paper, we have calculated the $B_{c}$ meson spectrum up to the $6S$ states with a nonrelativistic linear potential model by further constraining the model parameters with the mass of $B_{c}(2S)$ newly measured by the CMS Collaboration. As important tasks of this work, the radiative transitions between the $B_{c}$ states and the OZI allowed two body strong decays for the higher mass excited $B_{c}$ states are evaluated with the wavefunctions obtained from the linear potential model. Our calculations may provide useful references to search for the excited $B_{c}$ states. The main results are emphasized as follows.

For the $S$-wave states, the $2S$ hyperfine splitting is predicted to be $m[B_{c}^{*}(2S)]-m[B_{c}(2S)]\simeq 19$ MeV. The mass of the newly observed $B_{c}^{*}(2S)$ state might be determined via the $M1$ transition $B_{c}^{*}(2S)\to B_{c}\gamma$ in future experiments. The $3S$-wave states $B_{c}(3^{1}S_{0})$ and $B_{c}(3^{3}S_{1})$ are about $50$ MeV above the $DB^{*}$ threshold, their widths are estimated to be $\sim 100$ MeV. Since production rates of the $3S$-wave $B_{c}$ states in $pp$ collisions at the LHC are comparable with those of the $2S$-wave $B_{c}$ states Eichten:2019gig , both $B_{c}(3^{1}S_{0})$ and $B_{c}(3^{3}S_{1})$ states may have large possibilities to be established in the $DB^{*}$ final state, while $B_{c}(3^{3}S_{1})$ might be observed in the $DB$ final state as well.

For the $P$-wave states, it is found that the decays of the $2P$-wave states, $B_{c}(2^{3}P_{0})$, $B_{c}(2P_{1})$ and $B_{c}(2P^{\prime}_{1})$ together with all of the $1P$-wave states are governed by the $E1$ transitions, their typical decay widths are $\sim 100$ keV. It should be possible to observe these $P$-wave states via their dominant radiative decay processes with the higher statistics of the LHC. The $B_{c}(2^{3}P_{2})$ state is just $\sim 20$ MeV above the $DB$ threshold. It mainly decays into the $DB$ channel with a very narrow width of $\Gamma\sim 1$ MeV, so it has a large potential to be first observed in the $DB$ final state. The predicted masses of $3P$-wave states are in the range of (7420,7470) MeV. They are broad states with widths of $\sim 200$ MeV, and strongly couple to the $B^{*}D^{*}$ final state. It is interesting found that the $4P$-wave states $B_{c}(4^{3}P_{0})$, $B_{c}(4P_{1})$ and $B_{c}(4P^{\prime}_{1})$ with a mass around $7.7$ GeV may have relatively narrow widths $\mathcal{O}(100)$ MeV, these higher $P$-wave states might be first observed in their dominant channel $DB$ or $DB^{*}$.

The $1D$-wave states mainly decay via the $EM$ transitions. Our study indicates that these $1D$-wave states may have a relatively large potential to be observed via the radiative decay chains. For example, to look for the $B_{c}(1^{3}D_{3})$ state, the $B_{c}(1^{3}D_{3})\to B_{c}(1^{3}P_{2})\gamma\to B_{c}(1^{3}S_{1})\gamma\gamma\to B_{c}(1^{1}S_{0})\gamma\gamma\gamma$ is worthy to be searched, for the branching fraction of this chain is estimated to be $\sim 100\%$. The masses of the $2D$ and $3D$ states are predicted to be $\sim 7.34$ and $7.62$ GeV, respectively. Their decays are governed by the strong decay modes, such as $DB$, $DB^{*}$, $BD^{*}$ or $B^{*}D^{*}$. These higher $D$-wave states usually have a width of $\mathcal{O}(100)$ MeV. The observations of the $DB$, $DB^{*}$, $BD^{*}$ or $B^{*}D^{*}$ final states might be useful to search for these missing $2D$ and $3D$ states in future experiments.

For the $F$-wave states, one should pay more attention to $1F$-wave $B_{c}$ states in future observations. They have a mass of $\sim 7.23$ GeV, lie between the $DB$ and $B^{*}D$ mass thresholds. They are narrow states with a width of several MeV to several ten MeV, and dominantly decay into $DB$ or $B^{*}D$ channels. For example the $B_{c}(1^{3}F_{4})$ state might be a very narrow state with a width of $\sim 1$ MeV, its decays are governed by the $DB$ mode. To look for the missing $1F$-wave $B_{c}$ states, the $DB$ and $B^{*}D$ final states are worth observing.

Finally, it should be pointed out the strong decay widths of the excited $B_{c}$ states predicted in this work may have large uncertainties, for the parameter $\gamma$ cannot be directly determined by the strong decay processes of $B_{c}$ states. Fortunately, the uncertainties of the total strong decay widths of the excited $B_{c}$ states do not affect the important information, such as the dominant decay modes and corresponding decay rates, for our searching for the excited $B_{c}$ states in future experiments. Furthermore, the mixing angles for ${}^{3}P_{1}-^{1}P_{1}$, ${}^{3}D_{2}-^{1}D_{2}$, and ${}^{3}F_{3}-^{1}F_{3}$ have obvious model dependencies. The uncertainties of the mixing angles also affect our predictions of the decay properties of the mixed states.

Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grants No. 11775078, No. U1832173, No. 11705056, and No. 11405053.