Excited state mass spectra of doubly heavy $\Xi$ baryons

TL;DR

This paper calculates the mass spectra, Regge trajectories, and magnetic moments of doubly heavy $\\Xi$ baryons using a hypercentral quark model, providing theoretical predictions for their excited states and properties.

Contribution

It introduces a comprehensive calculation of excited state spectra, Regge trajectories, and magnetic moments for doubly heavy $\\Xi$ baryons within a hypercentral quark model, comparing with other predictions.

Findings

Mass spectra for various excited states are predicted.

Regge trajectories are constructed with determined slopes and intercepts.

Ground state magnetic moments are calculated.

Abstract

In this paper, the mass spectra are obtained for doubly heavy $\Xi$ baryons, namely, $\Xi_{cc}^{+}$, $\Xi_{cc}^{++}$, $\Xi_{bb}^{-}$, $\Xi_{bb}^{0}$, $\Xi_{bc}^{0}$ and $\Xi_{bc}^{+}$. These baryons are consist of two heavy quarks($cc$, $bb$ and $bc$) with a light($d$ or $u$) quark. The ground, radial and orbital states are calculated in framework of Hypercentral constituent quark model with coul- omb plus linear potential. Our outcomes are also compared with other predictions, thus, the average possible range of excited states masses of these $\Xi$ baryons can be determined. The study of the Regge trajectories are performed in (n, $M^{2}$) and (J, $M^{2}$) planes and their slopes and intercepts are also determined. Lastly, the ground state magnetic moments of these doubly heavy baryons are also calculated.

Tables

Table 1. Table 1: Ground state masses of Ξ Ξ \Xi baryons are listed and compared. Our work shows the ground state masses of all baryons with both light quark combination as Ξ c ​ c + superscript subscript Ξ 𝑐 𝑐 \Xi_{cc}^{+} / Ξ c ​ c + + superscript subscript Ξ 𝑐 𝑐 absent \Xi_{cc}^{++} , Ξ b ​ b − superscript subscript Ξ 𝑏 𝑏 \Xi_{bb}^{-} / Ξ b ​ b 0 superscript subscript Ξ 𝑏 𝑏 0 \Xi_{bb}^{0} and Ξ b ​ c 0 superscript subscript Ξ 𝑏 𝑐 0 \Xi_{bc}^{0} / Ξ b ​ c + superscript subscript Ξ 𝑏 𝑐 \Xi_{bc}^{+} (in GeV).

Baryons	${\Xi }_{ccd}$/${\Xi }_{ccu}$	${\Xi }_{bbd}$/${\Xi }_{bbu}$			${\Xi }_{bcd}$/${\Xi }_{bcu}$
$J^P$	${\frac{1}{2}}^{+}$	${\frac{3}{2}}^{+}$	${\frac{1}{2}}^{+}$	${\frac{3}{2}}^{+}$	${\frac{1}{2}}^{+}$	${\frac{3}{2}}^{+}$
Our work	3.520/3.511	3.695/3.687	10.317/10.312	10.340/10.335	6.920/6.914	6.986/6.980
Exp.olive	3.519$\pm$0.009		-	–	-	-
Ref.yoshida	3.685	3.754	10.314	10.339	-	-
Ref.kwei	3.520	3.695	10.199	10.316	-	-
Ref.paula	3.610(09)(12)	3.694(07)(11)	-	-	-	-
Ref.brown	3.610	3.692	10.143	10.178	6.943	6.985
Ref.alex	3.561(22)	3.642(26)	-	-	-	-
Ref.aliev	3.720	-	9.960	-	6.720	-
Ref.roberts	3.678	3.752	10.322	10.352	7.014	7.064
Ref.Roberts2008	3.676	3.753	10.340	10.367	7.011	7.074
Ref.Giannuzzi2009	3.547	3.719	10.185	10.216	6.904	6.936
Ref.valcarce	3.579	3.656	10.189	10.218	-	-
Ref.ebert	3.620	3.727	10.202	10.237	6.933	6.980
Ref.Gershtein	3.478	3.610	10.093	10.133	6.820	6.900
Ref.kar	3.627	3.690	10.162	10.184	6.914
Ref.Ltang	3.519	3.620	9.800	9.890	6.650	6.690
Ref.albertus	3.612	3.706	10.197	10.136	6.919	6.986
Ref.1	3.510	3.548	10.130	10.144	6.792	6.827
Ref.wang	3.570	3.610	10.170	10.220	-	-

Table 2. Table 2: Radial excited states of doubly heavy Ξ Ξ \Xi baryons(in GeV). Column 4 and 5 show the masses with light quark d combination whereas Column 6 and 7 show the masses with light quark u.

Particle	State	$J^P$	A	B	A	B	yoshida	Roberts2008	Giannuzzi2009	valcarce	ebert	roberts
A$\to$without first order correction, B$\to$with first order correction
	2S		3.912	3.925	3.905	3.920	4.079	4.029	4.183	3.976	3.910	4.030
	3S	${\frac{1}{2}}^{+}$	4.212	4.233	4.230	4.159	4.206		4.640		4.154
${\Xi }_{ccd}$	4S		4.473	4.502	4.468	4.501
and	5S		4.711	4.748	4.708	4.748
${\Xi }_{ccu}$	2S		3.976	3.988	3.970	3.983	4.114	4.042	4.282	4.025	4.027	4.078
	3S	${\frac{3}{2}}^{+}$	4.244	4.264	4.238	4.261	4.131		4.719
	4S		4.492	4.520	4.488	4.519
	5S		4.724	4.759	4.720	4.759
	2S		10.605	10.612	10.603	10.609	10.571	10.576	10.751	10.482	10.441	10.551
	3S	${\frac{1}{2}}^{+}$	10.851	10.862	10.851	10.862	10.612		11.170		10.630
${\Xi }_{bbd}$	4S		11.073	11.088	11.075	11.090					10.812
and	5S		11.278	11.297	11.282	11.301
${\Xi }_{bbu}$	2S		10.613	10.619	10.611	10.617	10.592	10.578	10.770	10.501	10.482	10.574
	3S	${\frac{3}{2}}^{+}$	10.855	10.855	10.866	10.866	10.593		11.184		10.673
	4S		11.075	11.090	11.077	11.092					10.856
	5S		11.280	11.298	11.284	11.302
	2S		7.236	7.244	7.231	7.240			7.478			7.321
	3S	${\frac{1}{2}}^{+}$	7.495	7.509	7.492	7.507			7.904
${\Xi }_{bcd}$	4S		7.727	7.746	7.726	7.744
and	5S		7.940	7.963	7.940	7.964
${\Xi }_{bcu}$	2S		7.260	7.267	7.256	7.263			7.495			7.353
	3S	${\frac{3}{2}}^{+}$	7.507	7.521	7.505	7.518			7.917
	4S		7.734	7.752	7.733	7.752
	5S		7.944	7.968	7.945	7.969

Table 3. Table 3: P and D state masses of Ξ c ​ c subscript Ξ 𝑐 𝑐 \Xi_{cc} baryon(in GeV).

State	A	B	A	B	yoshida	Roberts2008	valcarce	ebert	kwei	Gershtein	roberts	paula
	${\Xi }_{cc}^{+}$		${\Xi }_{cc}^{++}$
A$\to$ without correction and B$\to$ with correction
$\left(1^2{P}_{1/2}\right)$	3.853	3.865	3.846	3.861	3.947	3.910	3.880	3.838			4.073	3.892(47)(48)
$\left(1^2{P}_{3/2}\right)$	3.834	3.847	3.828	3.842	3.949	3.921		3.959	3.786	3.834	4.079	3.989(58)(58)
$\left(1^4{P}_{1/2}\right)$	3.862	3.875	3.856	3.871
$\left(1^4{P}_{3/2}\right)$	3.843	3.856	3.837	3.851
$\left(1^4{P}_{5/2}\right)$	3.818	3.890	3.817	3.888	4.163	4.092		4.155	3.949	4.047	4.089
$\left(2^2{P}_{1/2}\right)$	4.138	4.161	4.134	4.140	4.135	4.074	4.018	4.085
$\left(2^2{P}_{3/2}\right)$	4.121	4.144	4.118	4.140	4.137	4.078	4.197
$\left(2^4{P}_{1/2}\right)$	4.146	4.169	4.143	4.167
$\left(2^4{P}_{3/2}\right)$	4.130	4.152	4.126	4.149
$\left(2^4{P}_{5/2}\right)$	4.108	4.183	4.104	4.181	4.488
$\left(3^2{P}_{1/2}\right)$	4.395	4.426	4.393	4.409	4.149
$\left(3^2{P}_{3/2}\right)$	4.381	4.411	4.379	4.409	4.159
$\left(3^4{P}_{1/2}\right)$	4.402	4.433	4.400	4.432
$\left(3^4{P}_{3/2}\right)$	4.388	4.419	4.386	4.417
$\left(3^4{P}_{5/2}\right)$	4.369	4.399	4.412	4.396	4.534
$\left(4^2{P}_{1/2}\right)$	4.633	4.671	4.633	4.671
$\left(4^2{P}_{3/2}\right)$	4.620	4.658	4.620	4.657
$\left(4^4{P}_{1/2}\right)$	4.640	4.678	4.639	4.678
$\left(4^4{P}_{3/2}\right)$	4.627	4.664	4.626	4.664
$\left(4^4{P}_{5/2}\right)$	4.610	4.646	4.609	4.646
$\left(5^2{P}_{1/2}\right)$	4.857	4.901	4.858	4.902
$\left(5^2{P}_{3/2}\right)$	4.845	4.889	4.846	4.889
$\left(5^4{P}_{1/2}\right)$	4.863	4.908	4.864	4.909
$\left(5^4{P}_{3/2}\right)$	4.851	4.895	4.852	4.896
$\left(5^4{P}_{5/2}\right)$	4.835	4.878	4.835	4.879
$\left(1^4{D}_{1/2}\right)$	4.053	4.077	4.043	4.071
$\left(1^2{D}_{3/2}\right)$	4.026	4.049	4.019	4.044
$\left(1^4{D}_{3/2}\right)$	4.035	4.058	4.027	4.053
$\left(1^2{D}_{5/2}\right)$	4.002	4.024	3.998	4.019	4.043	4.115	4.047		4.391	4.034	4.050	4.388
$\left(1^4{D}_{5/2}\right)$	4.011	4.033	4.006	4.029	4.027	4.052
$\left(1^4{D}_{7/2}\right)$	3.982	4.002	3.979	3.998	4.097			4.187	4.089	4.393
$\left(2^4{D}_{1/2}\right)$	4.311	4.345	4.311	4.342
$\left(2^2{D}_{3/2}\right)$	4.289	4.321	4.287	4.318
$\left(2^4{D}_{3/2}\right)$	4.296	4.329	4.295	4.326
$\left(2^2{D}_{5/2}\right)$	4.270	4.299	4.267	4.297	4.164	4.091
$\left(2^4{D}_{5/2}\right)$	4.277	4.307	4.275	4.305
$\left(2^4{D}_{7/2}\right)$	4.253	4.280	4.249	4.278	4.394
$\left(3^4{D}_{1/2}\right)$	4.554	4.592	4.553	4.592
$\left(3^2{D}_{3/2}\right)$	4.534	4.571	4.532	4.570
$\left(3^4{D}_{3/2}\right)$	4.541	4.578	4.539	4.578
$\left(3^2{D}_{5/2}\right)$	4.516	4.552	4.514	4.551	4.348
$\left(3^4{D}_{5/2}\right)$	4.523	4.559	4.521	4.558
$\left(3^4{D}_{7/2}\right)$	4.500	4.535	4.498	4.534
$\left(4^4{D}_{1/2}\right)$	4.780	4.825	4.781	4.826
$\left(4^2{D}_{3/2}\right)$	4.762	4.806	4.762	4.806
$\left(4^4{D}_{3/2}\right)$	4.768	4.812	4.768	4.813
$\left(4^2{D}_{5/2}\right)$	4.745	4.788	4.745	4.788
$\left(4^4{D}_{5/2}\right)$	4.751	4.795	4.751	4.795
$\left(4^4{D}_{7/2}\right)$	4.731	4.772	4.730	4.772

Table 4. Table 4: P and D state masses of Ξ b ​ b subscript Ξ 𝑏 𝑏 \Xi_{bb} (in GeV).

State	A	B	A	B	yoshida	Roberts2008	valcarce	ebert	kwei	roberts	Others
	${\Xi }_{bb}^{-}$		${\Xi }_{bb}^0$
A$\to$ without correction and B$\to$ with correction
$\left(1^2{P}_{1/2}\right)$	10.507	10.514	10.504	10.511	10.476	10.493	10.406	10.368		10.691
$\left(1^2{P}_{3/2}\right)$	10.502	10.509	10.499	10.506	10.476	10.495		10.408	10.474	10.692	10.390wang
$\left(1^4{P}_{1/2}\right)$	10.510	10.517	10.506	10.514
$\left(1^4{P}_{3/2}\right)$	10.505	10.512	10.501	10.509							10.430aliev
$\left(1^4{P}_{5/2}\right)$	10.514	10.521	10.512	10.518	10.759				10.588	10.695
$\left(2^2{P}_{1/2}\right)$	10.758	10.770	10.757	10.770	10.703	10.710	10.612	10.563
$\left(2^2{P}_{3/2}\right)$	10.754	10.766	10.753	10.765	10.704	10.713		10.607
$\left(2^4{P}_{1/2}\right)$	10.760	10.772	10.760	10.772
$\left(2^4{P}_{3/2}\right)$	10.756	10.768	10.755	10.767
$\left(2^4{P}_{5/2}\right)$	10.751	10.763	10.763	10.776	10.973	10.713
$\left(3^2{P}_{1/2}\right)$	10.985	11.001	10.986	11.002	10.740			10.744
$\left(3^2{P}_{3/2}\right)$	10.981	10.997	10.982	10.998	10.742			10.788
$\left(3^4{P}_{1/2}\right)$	10.987	11.003	10.988	11.004
$\left(3^4{P}_{3/2}\right)$	10.983	10.999	10.984	11.000
$\left(3^4{P}_{5/2}\right)$	10.978	10.994	10.991	11.007	11.004
$\left(4^2{P}_{1/2}\right)$	11.194	11.214	11.197	11.217				10.900
$\left(4^2{P}_{1/2}\right)$	11.191	11.210	11.194	11.213
$\left(4^2{P}_{3/2}\right)$	11.196	11.216	11.199	11.219
$\left(4^4{P}_{3/2}\right)$	11.193	11.212	11.796	11.215
$\left(4^4{P}_{5/2}\right)$	11.188	11.208	11.202	11.222
$\left(5^2{P}_{1/2}\right)$	11.390	11.413	11.395	11.418
$\left(5^2{P}_{3/2}\right)$	11.387	11.410	11.392	11.415
$\left(5^4{P}_{1/2}\right)$	11.392	11.415	11.397	11.420
$\left(5^4{P}_{3/2}\right)$	11.389	11.412	11.394	11.417
$\left(5^4{P}_{5/2}\right)$	11.385	11.407	11.399	11.423
$\left(1^4{D}_{1/2}\right)$	10.665	10.677	10.663	10.675
$\left(1^2{D}_{3/2}\right)$	10.658	10.670	10.656	10.668
$\left(1^4{D}_{3/2}\right)$	10.660	10.672	10.659	10.670						11.011
$\left(1^2{D}_{5/2}\right)$	10.652	10.663	10.650	10.661	10.592	10.676			10.742	11.002
$\left(1^4{D}_{5/2}\right)$	10.654	10.666	10.652	10.664
$\left(1^4{D}_{7/2}\right)$	10.647	10.658	10.644	10.656		10.608			10.853	11.011
$\left(2^4{D}_{1/2}\right)$	10.897	10.913	10.897	10.913
$\left(2^2{D}_{3/2}\right)$	10.891	10.907	10.891	10.907
$\left(2^4{D}_{3/2}\right)$	10.893	10.909	10.893	10.909
$\left(2^4{D}_{3/2}\right)$	10.886	10.901	10.886	10.901		10.712
$\left(2^2{D}_{5/2}\right)$	10.888	10.903	10.888	10.903	10.613
$\left(2^2{D}_{5/2}\right)$	10.881	10.896	10.881	10.896		11.057
$\left(3^4{D}_{1/2}\right)$	11.109	11.130	11.113	11.133
$\left(3^2{D}_{3/2}\right)$	11.105	11.125	11.108	11.127
$\left(3^4{D}_{3/2}\right)$	11.107	11.126	11.110	11.129
$\left(3^4{D}_{3/2}\right)$	11.101	11.120	11.102	11.122
$\left(3^2{D}_{5/2}\right)$	11.103	11.122	11.105	11.124	10.809
$\left(3^2{D}_{5/2}\right)$	11.097	11.116	11.099	11.118
$\left(4^4{D}_{1/2}\right)$	11.310	11.333	11.314	11.337
$\left(4^2{D}_{3/2}\right)$	11.306	11.328	11.310	11.332
$\left(4^4{D}_{3/2}\right)$	11.307	11.330	11.311	11.334
$\left(4^4{D}_{3/2}\right)$	11.302	11.324	11.305	11.328
$\left(4^2{D}_{5/2}\right)$	11.303	11.325	11.307	11.330
$\left(4^2{D}_{5/2}\right)$	11.298	11.320	11.302	11.324

Table 5. Table 5: P and D state masses of Ξ b ​ c subscript Ξ 𝑏 𝑐 \Xi_{bc} (in GeV).

State	A	B	A	B	roberts	State	A	B	A	B	roberts
	${\Xi }_{bc}^0$		${\Xi }_{bc}^{+}$				${\Xi }_{bc}^0$		${\Xi }_{bc}^{+}$
A$\to$ without correction and B$\to$ with correction
$\left(1^2{P}_{1/2}\right)$	7.151	7.160	7.146	7.156	7.390	$\left(1^4{D}_{1/2}\right)$	7.322	7.336	7.318	7.334
$\left(1^2{P}_{3/2}\right)$	7.140	7.149	7.135	7.144	7.394	$\left(1^2{D}_{3/2}\right)$	7.307	7.321	7.303	7.318
$\left(1^4{P}_{1/2}\right)$	7.157	7.166	7.152	7.161	7.399	$\left(1^4{D}_{3/2}\right)$	7.312	7.326	7.706	7.308	7.324
$\left(1^4{P}_{3/2}\right)$	7.146	7.155	7.141	7.150		$\left(1^2{D}_{5/2}\right)$	7.294	7.308	7.290	7.304
$\left(1^4{P}_{5/2}\right)$	7.131	7.175	7.126	7.171		$\left(1^4{D}_{5/2}\right)$	7.299	7.313	7.702	7.295	7.309
						$\left(1^4{D}_{7/2}\right)$	7.293	7.296	7.708	7.278	7.292
$\left(2^2{P}_{1/2}\right)$	7.410	7.425	7.407	7.422		$\left(2^4{D}_{1/2}\right)$	7.559	7.425	7.558	7.579	7.579
$\left(2^2{P}_{3/2}\right)$	7.401	7.415	7.397	7.412		$\left(2^2{D}_{3/2}\right)$	7.547	7.567	7.545	7.565
$\left(2^4{P}_{1/2}\right)$	7.415	7.430	7.411	7.426		$\left(2^4{D}_{3/2}\right)$	7.551	7.571	7.549	7.570
$\left(2^4{P}_{3/2}\right)$	7.405	7.420	7.402	7.417		$\left(2^2{D}_{5/2}\right)$	7.536	7.555	7.534	7.553	7.538
$\left(2^4{P}_{5/2}\right)$	7.393	7.408	7.419	7.434		$\left(2^4{D}_{5/2}\right)$	7.540	7.559	7.538	7.558
						$\left(2^4{D}_{7/2}\right)$	7.526	7.545	7.523	7.544
$\left(3^2{P}_{1/2}\right)$	7.643	7.664	7.642	7.662		$\left(3^4{D}_{1/2}\right)$	7.779	7.804	7.777	7.804
$\left(3^2{P}_{3/2}\right)$	7.635	7.655	7.634	7.654		$\left(3^2{D}_{3/2}\right)$	7.768	7.792	7.7668	7.792
$\left(3^4{P}_{1/2}\right)$	7.647	7.668	7.646	7.666		$\left(3^4{D}_{3/2}\right)$	7.772	7.782	7.770	7.796
$\left(3^4{P}_{3/2}\right)$	7.639	7.659	7.638	7.658		$\left(3^2{D}_{5/2}\right)$	7.758	7.786	7.757	7.781
$\left(3^4{P}_{5/2}\right)$	7.629	7.648	7.653	7.673		$\left(3^4{D}_{5/2}\right)$	7.762	7.772	7.761	7.785
						$\left(3^4{D}_{7/2}\right)$	7.774	7.772	7.749	7.772
$\left(4^2{P}_{1/2}\right)$	7.859	7.884	7.859	8.015		$\left(4^4{D}_{1/2}\right)$	7.985	7.801	7.859	7.884	7.797
$\left(4^2{P}_{1/2}\right)$	7.852	7.876	7.852	7.877		$\left(4^2{D}_{3/2}\right)$	7.975	8.002	7.976	8.004
$\left(4^2{P}_{3/2}\right)$	7.863	7.888	7.863	7.888		$\left(4^4{D}_{3/2}\right)$	7.987	8.006	7.979	8.008
$\left(4^4{P}_{3/2}\right)$	7.856	7.880	7.856	7.880		$\left(4^2{D}_{5/2}\right)$	7.969	7.993	7.996	7.994
$\left(4^4{P}_{5/2}\right)$	7.846	7.870	7.869	7.895		$\left(4^4{D}_{5/2}\right)$	7.958	7.996	7.970	7.998
						$\left(4^4{D}_{7/2}\right)$		7.985	7.958	7.986
$\left(5^2{P}_{1/2}\right)$	8.062	8.091	8.064	8.092
$\left(5^2{P}_{3/2}\right)$	8.05528	8.084	8.057	8.085
$\left(5^4{P}_{1/2}\right)$	8.0652	8.094	8.067	8.096
$\left(5^4{P}_{3/2}\right)$	8.059	8.087	8.060	8.088
$\left(5^4{P}_{5/2}\right)$	8.050	8.078	8.073	8.079

Table 6. Table 6: F state masses of Ξ c ​ c subscript Ξ 𝑐 𝑐 \Xi_{cc} , Ξ b ​ b subscript Ξ 𝑏 𝑏 \Xi_{bb} and Ξ b ​ c subscript Ξ 𝑏 𝑐 \Xi_{bc} baryons(in GeV).

	${\Xi }_{ccd}$		${\Xi }_{ccu}$			${\Xi }_{bbd}$		${\Xi }_{bbu}$			${\Xi }_{bcd}$		${\Xi }_{bcu}$
State	A	B	A	B	kwei	A	B	A	B	kwei	A	B	A	B
$\left(1^4{F}_{3/2}\right)$	4.216	4.247	4.213	4.242		10.805	10.820	10.804	10.819		7.468	7.487	7.466	7.485
$\left(1^2{F}_{5/2}\right)$	4.185	4.215	4.182	4.210		10.797	10.812	10.796	10.811		7.451	7.469	7.448	7.467
$\left(1^4{F}_{5/2}\right)$	4.158	4.186	4.190	4.219		10.790	10.804	10.798	10.813		7.456	7.474	7.453	7.472
$\left(1^2{F}_{7/2}\right)$	4.166	4.194	4.163	4.191		10.799	10.814	10.791	10.806		7.440	7.458	7.437	7.455
$\left(1^4{F}_{7/2}\right)$	4.194	4.225	4.154	4.182	4.267	10.792	10.806	10.789	10.803	11.004	7.435	7.453	7.432	7.450
$\left(1^4{F}_{9/2}\right)$	4.133	4.159	4.129	4.156	4.413	10.784	10.797	10.783	10.797	11.112	7.421	7.439	7.418	7.436
$\left(2^4{F}_{3/2}\right)$	4.462	4.494	4.460	4.497		11.023	11.022	11.024	11.043		7.692	7.715	7.691	7.715
$\left(2^2{F}_{5/2}\right)$	4.435	4.468	4.433	4.468		11.016	11.035	11.018	11.036		7.677	7.700	7.676	7.699
$\left(2^4{F}_{5/2}\right)$	4.442	4.475	4.440	4.476		11.018	11.036	11.019	11.038		7.681	7.704	7.680	7.703
$\left(2^2{F}_{7/2}\right)$	4.410	4.445	4.415	4.450		11.010	11.028	11.013	11.031		7.667	7.690	7.666	7.689
$\left(2^4{F}_{7/2}\right)$	4.417	4.452	4.408	4.443		11.012	11.030	11.011	11.029		7.663	7.686	7.662	7.685
$\left(2^4{F}_{9/2}\right)$	4.388	4.424	4.386	4.420		11.005	11.022	11.006	11.023		7.651	7.674	7.650	7.672

Table 7. Table 7: Fitted slopes and intercepts of the regge trajectories in (n, M 2 superscript 𝑀 2 M^{2} ) plane.

Baryon	$J^P$	State	$\beta$	${\beta }_0$
${\Xi }_{cc}$	${\frac{1}{2}}^{+}$	S	0.409$\pm$0.0138	-5.181$\pm$0.247
	${\frac{1}{2}}^{-}$	P	0.457$\pm$0.004	-6.819$\pm$ 0.073
	${\frac{3}{2}}^{-}$	P	0.456$\pm$0.003	-6.729$\pm$0.075
	${\frac{5}{2}}^{+}$	D	0.462$\pm$0.003	-7.408$\pm$0.058
${\Xi }_{bb}$	${\frac{1}{2}}^{+}$	S	0.193 $\pm$0.006	-20.627$\pm$0.721
	${\frac{1}{2}}^{-}$	P	0.206$\pm$0.004	-22.914$\pm$0.540
	${\frac{3}{2}}^{-}$	P	0.207 $\pm$0.045	-22.850$\pm$0.054
	${\frac{5}{2}}^{+}$	D	0.210$\pm$0.004	-23.900$\pm$0.461
${\Xi }_{bc}$	${\frac{1}{2}}^{+}$	S	0.265 $\pm$0.009	-12.777$\pm$0.501
	${\frac{1}{2}}^{-}$	P	0.289$\pm$0.005	-14.832$\pm$0.298
	${\frac{3}{2}}^{-}$	P	0.288 $\pm$0.005	-14.738$\pm$0.305
	${\frac{5}{2}}^{+}$	D	0.291$\pm$0.004	-15.522$\pm$0.215

Table 8. Table 8: Fitted parameters α 𝛼 \alpha and α 0 subscript 𝛼 0 \alpha_{0} are slope and Intercept of parent and daughter Regge trajectories.

Baryon	Trajectory	$\alpha$	${\alpha }_0$	$\alpha$	${\alpha }_0$
${\Xi }_{ccd}$	parent	0.609$\pm$0.0607	-6.694$\pm$1.022	0.859$\pm$0.042	-10.650$\pm$0.648
	1 daughter	0.726$\pm$0.042	-10.197$\pm$0.737	0.865$\pm$0.016	-12.649$\pm$0.288
	2 daughter	0.751$\pm$0.041	-12.351$\pm$0.799	0.893$\pm$0.019	-15.069$\pm$0.368
	3 daughter	0.795$\pm$0.0319	-14.935$\pm$0.680	0.907$\pm$0.014	-17.292$\pm$0.298
${\Xi }_{bbd}$	parent	0.304$\pm$0.0156	-31.433$\pm$1.733	0.322$\pm$0.013	-33.518$\pm$1.481
	1 daughter	0.347$\pm$0.0112	-38.094$\pm$1.312	0.354$\pm$0.005	-38.947$\pm$0.561
	2 daughter	0.365$\pm$0.007	-41.989$\pm$0.945	0.377$\pm$0.003	-43.419$\pm$0.344
	3 daughter	0.391$\pm$0.006	-46.983$\pm$0.772	0.402$\pm$0.003	-48.306$\pm$0.326

Table 9. Table 9: Magnetic Moment(in nuclear magnetons) of J P superscript 𝐽 𝑃 J^{P}   1 2 + superscript 1 2 \frac{1}{2}^{+} and 3 2 + superscript 3 2 \frac{3}{2}^{+} doubly heavy baryons.

Baryons	Magnetic moment	Our	patel	103	104	101
${\Xi }_{cc}^{+}$	$\frac{4}{3}{\mu }_c$- $\frac{1}{3}{\mu }_d$	0.784	0.859	0.722	0.80	0.785
${\Xi }_{cc}^{++}$	$\frac{4}{3}{\mu }_c$- $\frac{1}{3}{\mu }_u$	0.031	-0.133	0.114	-0.12	-0.208
${\Xi }_{bb}^{-}$	$\frac{4}{3}{\mu }_b$- $\frac{1}{3}{\mu }_d$	0.196	0.190	0.086	0.215	0.251
${\Xi }_{bb}^0$	$\frac{4}{3}{\mu }_b$- $\frac{1}{3}{\mu }_u$	-0.663	-0.656	-0.432	-0.630	-0.742
${\Xi }_{bc}^0$	$\frac{2}{3}{\mu }_b$+$\frac{2}{3}{\mu }_c$-$\frac{1}{3}{\mu }_d$	0.354	0.476	0.068	0.480	0.518
${\Xi }_{bc}^{+}$	$\frac{2}{3}{\mu }_b$+$\frac{2}{3}{\mu }_c$-$\frac{1}{3}{\mu }_u$	-0.204	-0.400	-0.236	-0.369	-0.475
${\Xi }_{cc}^{+\ast }$	$2{\mu }_c$+${\mu }_d$	0.068	-0.168	0.163	0.035	-0.311
${\Xi }_{cc}^{++\ast }$	$2{\mu }_c$+${\mu }_u$	2.218	2.749	2.001	-	2.670
${\Xi }_{bb}^{-\ast }$	2${\mu }_b$+${\mu }_d$	-1.737	-0.951	-0.652	-1.029	-0.522
${\Xi }_{bb}^{0\ast }$	2${\mu }_b$+${\mu }_u$	-1.607	1.576	0.916	1.507	1.870
${\Xi }_{bc}^{0\ast }$	${\mu }_b$+${\mu }_c$+${\mu }_d$	-0.372	-0.567	-0.257	-0.508	-0.712
${\Xi }_{bc}^{+\ast }$	${\mu }_b$+${\mu }_c$+${\mu }_u$	1.562	2.052	1.414	2.022	2.270

Equations

$$m_{\rho }=\frac{2m_1{m}_2}{m_1+m_2},$$

$$m_{\lambda }=\frac{2m_3(m_1^2+m_2^2+m_1{m}_2)}{(m_1+m_2)(m_1+m_2+m_3)}$$

$$H=\frac{P_x^2}{2m}+V(x)$$

$$\sum _{i<j}V(r_{ij})=V(x)+....$$

$$\left[\frac{-1}{2m}\frac{d^2}{d{x}^2}+\frac{\frac{15}{4}+\gamma (\gamma +4)}{2m{x}^2}+V(x)\right]{\varphi }_{\gamma }(x)=E{\varphi }_{\gamma }(x)$$

$$V(x)=V^0(x)+\left(\frac{1}{m_{\rho }}+\frac{1}{m_{\lambda }}\right)V^{(1)}(x)+V_{SD}(x)$$

$$V^{(0)}(x)=\frac{\tau }{x}+\beta x\&V^{(1)}(x)=-C_F{C}_A\frac{{\alpha }_s^2}{4x^2}$$

$$V_{SD}(x)=V_{SS}(x)(\overrightarrow{S_{\rho }}.\overrightarrow{S_{\lambda }})+V_{\gamma S}(x)(\vec{\gamma }\cdot \vec{S})$$

$$+V_T(x)\left[S^2-\frac{3(\vec{S}\cdot \vec{x})(\vec{S}\cdot \vec{x})}{x^2}\right]$$

$$n=\beta M^2+{\beta }_0$$

$$J=\alpha M^2+{\alpha }_0$$

$$m_i^{eff}=m_i\left(1+\frac{⟨H⟩}{{\sum }_i{m}_i}\right)$$

$${\mu }_B=\sum _i⟨{\varphi }_{sf}|{\mu }_{iz}|{\varphi }_{sf}⟩)$$

$${\mu }_i=\frac{e_i{\sigma }_i}{2m_i^{eff}}$$

Full text

††journal: Eur. Phys. J. C

1 Introduction

Doubly heavy baryons have two families: $\Xi$ and $\Omega$. $\Omega$ has a light strange quark while $\Xi$ has up or down quark(s) with two heavy quarks(c and b). Our previous work zalak1 exhibited the mass spectra, magnetic moments and Regge trajectories of doubly heavy $\Omega$ baryons while in present paper, we established the $\Xi$ baryon family with six members. The only experimental evidence comes for $\Xi_{cc}^{+}$ by the SELEX experiment. They reported a ground state at 3520 MeV containing two charm quarks and a down quark olive ; selex ; selex1 . It’s yet to be confirmed from the other experiments babar ; belle ; lhcb ; focus . Recently, Hamiltonian model yoshida , Regge phenomenology kwei , Lattice QCD paula ; brown ; alex ; mathur , QCD sum rule aliev , variational approach roberts and many more Roberts2008 ; Giannuzzi2009 ; valcarce ; ebert ; Gershtein ; kar ; Ltang ; albertus ; 1 ; wang ; patel are providing new results in field of doubly heavy baryons. Many of them have only calculated the ground state masses while some of them have also shown the excited states.

We have used the QCD inspired Hypercentral constituent quark model(hCQM) with coloumb plus linear potential. The first order correction is also taken into account to the potential and calculation has been performed by solving six dimensional hyper radial Schr$\ddot{o}$-dinger equation numerically zalak ; zalak1 ; zalak3 . We have calculated the mass spectra of radial excited states upto 5S and orbital excited states for 1P-5P, 1D-4D and 1F-2F states. As per our knowledge, all the theoretical approaches have considered the $m_{u}$=$m_{d}$ so far but the light quark masses are different in our model. Thus, we have obtained the mass spectra with u & d quarks combinations for these baryons. Obtained masses were used in formation of Regge trajectories in (n, $M^{2}$) and (J, $M^{2}$) planes. The determination of slope and intercept of Regge trajectories of these baryons are very important as it provides better understanding of the dynamics of strong interactions in production of charmed and bottom baryons at high energies.

The paper is organized as follows. We give brief introduction in sect.1 and explained our Hypercentral Constituent Quark Model in sect. 2. We present our mass spectra results of all doubly heavy $\Xi$ baryons in sect. 3. Regge trajectories and magnetic moments are discussed in sect. 4. At last, our conclusion is in sect. 5.

2 The Model

The methodology for the determination of excited masses follow the same pattern as in our previous work zalak1 and Refs. their-in. Therefore, we discuss model very briefly in present paper. Starting with the Jacobi co-ordinates of three quark baryons that are given in terms of mass($m_{i}$) and co ordinates($\vec{r_{i}}$) below Bijker . The quark masses are taken in calculations as $m_{u}$ = 0.338, $m_{d}$ = 0.350, $m_{c}$ = 1.275 and $m_{b}$ = 4.67 (in GeV). The co-ordinates $\vec{\rho}$ and $\vec{\lambda}$ are with the respective reduced masses are given by

[TABLE]

[TABLE]

The Hamiltonian of three body baryonic system in the hCQM is then expressed as

[TABLE]

The hyper radius $x=\sqrt{\rho^{2}+\lambda^{2}}$ is a collective co ordinate and therefore the hypercentral potential contains also the three-body effects. where, $m=\frac{2m_{\rho}m_{\lambda}}{m_{\rho}+m_{\lambda}}$, is the reduced mass and $x$ is the six dimensional radial hyper central coordinate of the three body system. In present paper, the confining three-body potential is chosen within a string-like picture, where the quarks are connected by gluonic strings and the potential strings increases linearly with a collective radius $r_{3q}$ as mentioned in ginnani2015 . Accordingly the effective two body interactions can be written as

[TABLE]

In the hypercentral approximation, the potential is only depends on hyper radius(x). More details can be seen in references ginnani2015 ; M. Ferraris . We consider a reduced hypercentral radial function, $\phi_{\gamma}(x)=x^{\frac{5}{2}}\Psi_{\gamma}(x)$ where $\Psi_{\gamma}$(x) is the hypercentral wave function and $\gamma$ is the grand angular quantum number. Thus, six dimensional hyperradial Schrodinger equation reduces to,

[TABLE]

For the present study we consider the hypercentral potential $V(x)$ as the color coulomb plus linear potential with first order correction koma ; 11 is given below.

[TABLE]

[TABLE]

[TABLE]

Here, $\tau$ is the hyper-Coulomb strength corresponds to the strong running coupling constant $\alpha_{s}$. $\beta$ is the string tension of the confinement part of potential. $C_{F}$ and $C_{A}$ are the Casimir charges of the fundamental and adjoint representation with values $\frac{2}{3}$ and 3. The spin-dependent part, $V_{SD}(x)$ contains three types of the interaction terms 12 : the spin-spin term $V_{SS}(x)$, the spin-orbit term $V_{\gamma S}(x)$ and tensor term $V_{T}(x)$. The detail of the terms are given in zalak .

3 Mass Spectroscopy: $\Xi_{cc}$, $\Xi_{bb}$ & $\Xi_{bc}$

We begin with calculating the ground state masses of doubly heavy $\Xi_{cc}$, $\Xi_{bb}$ & $\Xi_{bc}$ baryons111We compare our ccd, bbd and cbd baryon combination masses with others for whole mass spectra discussion. The masses are computed for both parities $\frac{1}{2}^{+}$ and $\frac{3}{2}^{+}$ mentioned in Table 1. As we know, the ground state of $\Xi_{cc}^{+}$ is found experimentally as $\Xi_{cc}(3520)^{+}$; but its $J^{P}$ value is still undefined. Our prediction suggests that it would be $J^{P}=\frac{1}{2}^{+}$; similar suggestion given by refs. kwei ; Ltang ; 1 . The other ground state, with $J^{P}=\frac{3}{2}^{+}$ is found as 3.695 GeV by us. The value is nearer to other predictions kwei ; kar ; albertus and lattice paula ; brown calculations. In case of $\Xi_{bb}$ baryon, our ground state outcomes(both parities) are matched(with yoshida ; roberts ) very well. Our predicted ground states of $\Xi_{bc}$ are very close to refs. ebert ; albertus ; brown . We have also calculated the ground state spectra of ccu, bbu and bcu baryons. They are close to (8MeV, 5MeV and 6MeV difference, respectively) the results of d quark combinations.

Moving towards the excited states, the radial excited states are calculated from 2S-5S for $J^{P}$=$\frac{1}{2}^{+}$ and $\frac{3}{2}^{+}$. These radial excited states of $\Xi_{cc}$, $\Xi_{bb}$ & $\Xi_{bc}$ are mentioned in Table 2. P, D and F states with their different isospin splittings are computed and the excited state masses from 1P-5P, 1D-4D and 1F-2F are shown in Tables  3, 4, 5 and  6. One can observe that B masses are few MeV higher than A masses in each case of every baryonic systems. The results of different theoretical approaches for all the systems are also compared in the respective tables. Notice that, the heavy quarks combination(cc,bb and bc) with light quarks(u and d) for all three baryon have represented individualy in Table  1- 6 .

The excited states of doubly heavy $\Xi$ family are unknown experimentally. Starting from radial excited states mentioned in Table  2, we have compared our results with refs. yoshida ; Roberts2008 ; roberts ; valcarce ; ebert ; Giannuzzi2009 . For 2S state of the system $\Xi_{cc}$ with $J^{P}$ values $\frac{1}{2}^{+}(\frac{3}{2}^{+})$ have lowest prediction is 3.910 (4.027) (by ebert ) and highest is 4.183(4.282) (byGiannuzzi2009 ). Specifically, our 2S predictions are close to ebert .In similar way, while analyzing 2S state of $\Xi_{bb}(\frac{1}{2}^{+})$ and $\Xi_{bb}(\frac{3}{2}^{+})$ baryon, the lowest to highest range of masses are found to be (10441-10751)MeV and (10482-10770) MeV, respectively whereas our model sugeested masses are near to ref. Roberts2008 among all. The next baryon is $\Xi_{bc}$ and the 2S state with both isospins show more than 100 MeV difference with others(roberts ; Giannuzzi2009 . Though Refs. yoshida ; Giannuzzi2009 ; ebert have computed 3S state(with $\frac{1}{2}^{+}$ and $\frac{3}{2}^{+}$) for $\Xi_{cc}$ and $\Xi_{bb}$ baryons and only Giannuzzi2009 has computed for $\Xi_{bc}$ baryon, but their obtained mass difference with us is large(except yoshida for $\Xi_{cc}$).

In case of $\Xi_{cc}$, our 1P state $J^{P}=\frac{1}{2}^{-}$ show difference of 57 MeV(with Roberts2008 ), 27 MeV (with valcarce ), 12 MeV(with ebert ) and 39 MeV(with paula ), while $J^{P}=\frac{3}{2}^{-}$ shows 87 MeV(with Roberts2008 ) and 0 MeV(with Gershtein ).

For $\Xi_{bb}$, our 1P state $J^{P}=\frac{1}{2}^{-}$ and $J^{P}=\frac{3}{2}^{-}$ are only 14 and 7 MeV higher than ref. Roberts2008 whereas Ref. yoshida masses for $J^{P}=\frac{1}{2}^{-}$ and $J^{P}=\frac{3}{2}^{-}$ are 31, 26 MeV lower than our prediction. Our 2P state is few MeV higher than yoshida ; Roberts2008 . Our 1D-2D states have $\approx$35 MeV and $\approx$178 MeV difference with Roberts2008 . The P and D states of $\Xi_{bc}^{0}$ baryons are given in table 5 and it follow the same description mentioned in zalak1 and refs. therein. We have compare our results with recent paper kwei for 1P and 1D. Their values are higher than us. The rest spectra (2P-5P and 2D-4D) is performed by us for completeness.

F state masses for all three doubly heavy baryons are given in table 6. Apart from us, Ref. kwei has also calculated the 1F state of $\Xi_{cc}$ and $\Xi_{bb}$ for $J^{P}$= $\frac{7}{2}^{-}$ and $\frac{9}{2}^{-}$. For $\Xi_{cc}$, their masses are 73 and 280 MeV higher while for other system 212 and 328 MeV higher than our predictions. We did not find any other F state calculations for $\Xi_{bc}$ systems.

4 Regge Trajectories and Magnetic Moments

As discussed in section 3, we have calculated the 1S-5S, 1P-5P, 1D-4D and 1F-2F state masses for all doubly heavy $\Xi$ baryons. The obtained masses were very much useful in constructing Regge trajectories in (n, $M^{2}$) and (J, $M^{2}$) planes. Where, n is principal quantum number and J is a total spin. The Regge trajectories are presented in Figs. 1- 5 for $\Xi_{cc}(ccd)$, $\Xi_{bb}(bbd)$ and $\Xi_{bc}(bcd)$ baryons. Similar trajectories can also be plotted for the rest of the baryons. Straight lines were obtained by the linear fitting in all figures. The ground and radial excited S states ($J^{P}=\frac{1}{2}^{+}$) and the orbital excited P ($J^{P}=\frac{1}{2}^{-}$), D ($J^{P}=\frac{5}{2}^{+}$) and F ($J^{P}=\frac{7}{2}^{-}$) states are plotted in Fig 1- 3 from bottom to top. We use,

[TABLE]

Where, $\beta$ and $\beta_{0}$ are slope and intercept, respectively. The fitted slopes and intercepts are given in Table 7. We use natural parity $J^{P}=\frac{1}{2}^{+}$, $J^{P}=\frac{3}{2}^{-}$, $J^{P}=\frac{5}{2}^{+}$, $J^{P}=\frac{7}{2}^{-}$ and unnatural ($J^{P}=\frac{3}{2}^{+}$, $J^{P}=\frac{5}{2}^{-}$, $J^{P}=\frac{7}{2}^{+}$, $J^{P}=\frac{9}{2}^{-}$) parity masses and plotted graphs for $\Xi_{cc}$ and $\Xi_{bb}$ baryon states[See Fig.  4 and  5]. For that we use,

[TABLE]

Where, $\alpha$ and $\alpha_{0}$ are slope and intercept, respectively. The fitted slopes and intercepts are given in Table 8. We observe that the square of the calculated masses fit very well to the linear trajectory and almost parallel and equidistant in S, P, D and F states. We can determine the possible quantum numbers and prescribe them to particular Regge trajectory with the help of our obtained results.

To obtain magnetic moments of the $\Xi$ family, we need to calculate their effective masses first. As the combination of quarks in baryons changes, its binding interaction affects and $m_{i}^{eff}$ differs. The effective mass for each of the constituting quark $m_{i}^{eff}$ can be defined as

[TABLE]

where, $\langle H\rangle$ = E + $\langle V_{spin}\rangle$. Thus, the magnetic moment of baryons with bound quarks are given as [zalak1 and Ref. their in]

[TABLE]

where

[TABLE]

$e_{i}$ is a charge and $\sigma_{i}$ is the spin of the respective constituent quark corresponds to the spin flavor wave function of the baryonic state. Using these equations and our obtained ground state masses(mentioned in Table  1), we calculated magnetic moments of all six $\Xi$ baryons. The spin flavor wave function and magnetic moments are given in Table  9. Our obtained ground state magnetic moments are also compared with others shown in Table  9.

5 Conclusion

The Hypercentral constituent quark model is used to construct the mass spectra of doubly heavy $\Xi$ baryons. Ground states as well as excited state masses are obtained successfully. Mass difference between the light quarks (u and d) is 12 MeV in our model. So, it is obvious that when we move towards the calculations of the excited states the baryon masses would also have a very small mass difference. For the sake of completeness we calculated whole mass spectrum for all six doubly heavy baryon and noticed that it hardly differs less than $\approx$10 MeV; which can be observed in Tables  1- 6. The ground state of $\Xi_{cc}$ is experimental known and while comparing our ground states of $\Xi_{cc}^{++}$,$\Xi_{cc}^{+}$ baryons we define the state with parity $J^{P}$= $\frac{1}{2}^{+}$.

We successfully plotted Regge trajectories of present work in both (n, $M^{2}$) and (J,$M^{2}$) planes and fortunately assigned the quantum number for each cases of six $\Xi$ baryons. The magnetic moments of the ground states are also calculated using obtained masses. We can observe that our obtained results are close to other predictions(except $\Xi_{cc}^{++}$,$\Xi_{cc}^{+*}$, $\Xi_{bb}^{0*}$ baryons).

This study will definitely help future experiments and other theoretical models to identify the baryonic states from resonances. We would like to extend this model to calculate the mass spectra and other properties of triply heavy baryons.

Acknowledgements.

A. K. Rai acknowledges the financial support extended by DST, India under SERB fast track scheme SR/FTP/PS-152/2012. We are very much thankful to Prof. P. C. Vinodkumar and Dr. Kaushal Thakkar for their valuable suggestions throughout the work.