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Regge of framework ol rvd nueu nomto nftr experimental future states. in these information of useful an provide could fhge xie ttso hs ayn.Teotie mass obtained The baryons. these of states excited higher of 5 ute efi h lp n necp fec eg iei ( in line Regge each of intercept and slope reasonably the and fix we available Further where observations experimental ss 0 / / c h asso rudadectdsae fΩ of states excited and ground of masses The c 2 r bevdwt hi unu numbers, quantum their with observed are .Nwcnie ttswt eaieor- relative with states consider Now 2. 3 2 (3090) iur ean nisclrtilti has it triplet color its in remains diquark .INTRODUCTION I. + + or .. n naua ( unnatural and ....) , rudadEctdsaemse of masses state Excited and Ground c 0 L epciey[7,where [17], respectively ttssc sΩ as such states b n obnn with combining and 1 = hc ae tese ounder- to easier it makes which ) 0 n Ω and c (3119) J uiOdcha ea Gandhi, Keval Oudichhya, Juhi 1 = c 0 (3119) te orsae were states four other / c ,3 2, J (3000) P ss J c 0 P / auso excited of values n n one and ) ur,w have we quark, and 2 eaiginto decaying 3 = J 0 Ω , stetotal the is / 2 S c J + Dtd a 5 2021) 25, May (Dated: s (3050) 5 , quarks 1 = 1 = / 2 I / . − / 2, 0 7 , 2 ,M J, c 0 , c 0 / Ω , 2 2 + 2]ms nlssi are u o h el excited newly the for called out fication carried is Ref. Ω analysis In mass (hCQM). of model [22] approach quark and non-relativistic constituent doubly using singly, hypercentral baryons [18–21] of heavy masses Refs. to state triply of method authors excited various The the mass are calculate spectrum. is there mass states far the hadronic So depict of analysis. properties spectrum the way prominent determine the of to one spectroscopy, hadron the study Ξ eain hydtriems agso oemesons some these predict of on and ranges using Based baryons mass hadrons and determine for ansatz. they inequali- equalities trejectory relations mass mass Regge quadratic quadratic quasilinear inequalities, and mass ties linear as quark constituent Hamil- etc. Hypercentral the 19] 29], [18, the [28, model [30], QCD model Lattice [27], tonian [26] model model Salpeter quark of , Relativistic masses like models, the nomenological predicted dou- have detected. the Ω authors of experimently none many been far have Although So baryons states. Ω these bly/triply of nature the mine iaetemse n eiuso hs el observed es- newly [25] these of Ref. residues baryons. Ω and heavy masses of the masses model timate quark the hamiltonian calculate Robert non-relativistic to model. used quark al.[24] et relativistic by QCD-motivated of hs tts h rudsaeadectdsaespectra state excited of and Ω spin-parity state of the ground The predict to states. favor these results The coupling. aia ur edi e.[]adrslspeitdthe predicted results and [5] Ref. J in field quark namical Ω a pcr fectdhaybroscnitn ftolight two of consisting baryons heavy ( excited of spectra n hs ayncsae sbudsaeof state bound as states assum- by baryonic parity these negative with ing states five of existence the n 5 and ln ihbt aua ( natural both with plane ) .. pnprte r nareetwt the with agreement in are spin-parities ....) , cc + ,d s d, u, P c c/b cc c 0 c P ∗ potedt hr r ieetrsac set to aspects research different are there date the Upto e ta.[1 eiesm sflms eain such relations mass useful some derive [31] al. et Wei 35) Ξ (3055), (3066) t fTcnlg,Srt uaa-907 India Gujarat-395007, Surat, Technology, of ute aynsae ihnQDsmrlsadtyt deter- to try and rules sum QCD within states n Ω and sinet ftesae Ω states the of assignments n Ω and n jyKmrRai Kumar Ajay and c 0 wave ayn ypooiganwshm fsaeclassi- state of scheme new a proposing by baryons / aynaeetmtdfo atc C ihdy- with QCD lattice from estimated are baryon n n ev ( heavy one and ) Ω 2 erhsadtesi-aiyassignment spin-parity the and searches eain n h asvlepredictions value mass the and relations − ls oohrtertclpredictions. theoretical other to close ,M n, c 0 0 ccc ++ epciey h uhr fRf 4 predicts [4] Ref. of authors The respectively. , , Ω , ccc ss Ω g osbesi-aiyt these to spin-parity possible a ign c ayn r acltdwti the within calculated are baryons dqak br ta.[3 eemn mass determine [23] al. et Ebert -diquark. 2 37)adΞ and (3077) c cc + ayn sn aiu hoeia n phe- and theoretical various using baryons ln n siaetemasses the estimate and plane ) Jls (3090) and iigculn hc osbeyond goes which coupling mixing 0 Ω n Ω and ccc ++ ,b c, J J c P 32)bros e.[32] Ref. baryons. (3123) P urs nteframework the in quarks, ) baryons c 1 = (3119) sinet fΞ of assignments c (3000) / 2 + 0 3 , ih1 with 0 / Ω , 2 − , c c (3050) qakand -quark / 2 − c (2980), 3 , 0 / and 2 jj − 2

P 0 posed by Nambu [33, 34]. He assumed that the uniform TABLE I. Masses, width and J value of Ωc baryons reported interaction of quark and antiquark pair form a strong flux by LHCb [1]. tube and at the end of the tube the light quarks rotating Resonance Mass(MeV) Width(MeV) J P with the speed of light at radius R. The mass originating 0 ± 1 + in this flux tube is estimated as [35], Ωc 2695.2 1.7 - 2 0 3 + c ± Ω (2770) 2765.9 2.0 - 2 R 0 ± ± ± ± ? σ Ωc(3000) 3000.4 0.22 0.1 4.5 0.6 0.3 ? M =2 dr = πσR, (1) 0 ? 2 Ωc(3050) 3050.2±0.1 ± 0.1 0.8 ± 0.2 ± 0.1 ? 0 1 − ν (r) 0 ? Z Ωc(3065) 3065.6±0.1 ± 0.3 3.5 ± 0.4 ± 0.2 ? where σ represent the string tension, i.e the mass density 0 ? Ωc(3090) 3090.2±0.3 ± 0.9 8.7±0.1 ± 0.8 ? per unit length. Likewise,p the angular momentum of this 0 ? Ωc(3120) 3119±0.3 ± 0.9 1.1±0.8 ± 0.4 ? flux tube is evaluated as,

R 2 σrν(r) πσR ′ J =2 dr = + c , (2) 1 − ν2(r) 2 the same group extended this methodology to compute Z0 the masses of ground state and excited state of other ∗ ∗ one can also write, p doubly and triply charmed baryons such as Ωcc, Ξcc, Ωcc and Ωccc. In the present work we have used the same 2 M ′′ approach of Regge phenomenology with the assumption J = + c , (3) of linear Regge trejectories. We obtained relations 2πσ between intercept, slope ratios and baryon masses in ′ ′′ where c and c are the constants of integration. Hence both (J, M 2) and (n ,M 2) planes. Using these relations r we can say that, J and M 2 are linearly related to each we determine the mass spectra of Ω baryon and try to c other. The plots of Regge trejectories of hadrons in assign a possible spin-parity to these recently observed the (J, M 2) plane are usually called Chew-frautschi plots five baryonic states. Further we apply the same scheme [36]. Also hadrons lying on the same Regge line pos- for doubly and triply charmed omega baryons. Since sess the same internal quantum numbers. From Eq. (3), these baryons are yet to be observed by experiments the most general form of linear regge trejectories can be and therefore they are not listed in PDG [17] yet. This 0 + expressed as [31], motivate us to study the mass spectra of Ωc, Ωcc and Ω++ baryons. ′ ccc J = α(M)= a(0) + α M 2, (4)

The remainder of this paper is organised as follows. ′ where a(0) and α represents the intercept and slope of After introduction, in sec II we briefly explain the linear the trejectory respectively. These parameters for differ- Regge trejectories and extract the quadratic mass rela- ent quark constituents of a baryon multiplet can be re- tion in (J, M 2) plane to calculate the ground state and lated by two relations [31, 37–40], excited state masses of Ωc baryon. Further we extend this work and again derive some mass equations to eval- aiiq (0) + ajjq (0) = 2aijq (0), (5) uate the mass spectra for Ωcc and Ωccc baryons. In addi- tion, we estimate the masses of these baryons in (n,M 2) plane by calculating Regge slopes and intercepts of par- ticular regge lines. The discussions of our results appear 1 1 2 ′ + ′ = ′ , (6) in sec.III. α iiq α jjq α ijq

where i, j, q represent quark flavors, mi < mj and q de- II. THEORETICAL FRAMEWORK notes an arbitrary light or heavy quark. Using Eqs. (4) and (5) we obtain, Regge theory is one of the simplest and effective ′ ′ ′ 2 2 2 phenomenological approach to study the hadron spec- αiiq Miiq + αjjq Mjjq =2αijqMijq . (7) troscopy. Various theories were developed to understand the regge trejectory. Among them, one of the straightfor- Combining the relations (6) and (7) and after solving ward explaination for linear regge trejectories were pro- the quadratic equation, we obtain a pair of solutions as,

′ α jjq 1 2 2 2 2 2 2 2 2 2 ′ = 2 × [(4Mijq − Miiq − Mjjq ) ± (4Mijq − Miiq − Mjjq) − 4MiiqMjjq ]. (8) α 2Mjjq iiq q

This is the important relation we obtained between slope ratios and baryon masses. Eq. (8) can also be 3 expressed in terms of baryon masses by introducing k, where k can be any quark flavor,

′ ′ ′ αjjq αkkq αjjq ′ = ′ × ′ , (9) αiiq αiiq αkkq

we have,

2 2 2 2 2 2 2 2 2 [(4Mijq − Miiq − Mjjq )+ (4Mijq − Miiq − Mjjq ) − 4MiiqMjjq ] 2 q 2Mjjq (10) 2 2 2 2 2 2 2 2 2 2 [(4Mikq − Miiq − Mkkq)+ (4Mikq − Miiq − Mkkq) − 4MiiqMkkq]/2Mkkq = . 2 2 2 q 2 2 2 2 2 2 2 [(4Mjkq − Mjjq − Mkkq)+ (4Mjkq − Mjjq − Mkkq ) − 4MjjqMkkq ]/2Mkkq q

This is the general relation we have derived in terms of two and one charmed quark (ssc). So of baryon masses which can be used to predict the mass keeping this configuration in mind we put i = u, j = s, of any baryon state by knowing all other masses. q = c, k = u in Eq. (10) and we get,

0 A. Masses of Ωc baryon

In the present work we used Eq. (10) to evaluate the 0 0 ground state masses of Ωc baryon. Since Ωc is composed

2 2 2 2 2 2 2 2 2M + (MΩ0 − 2M + ) − M 0 − M ++ = (4M + − M ++ − M 0 ) − 4M 0 M + , (11) Ξc c Ξc Ωc Σc Ξc Σc Ωc Ωc Ξc h i q

+ P + + substituting the masses of Ξc (J = 1/2 , 3/2 ) and From Eq. (4) one can have, ++ P + + Σc (J = 1/2 , 3/2 ) baryon from PDG [17] into 0 relation (11), we obtain the ground state masses of Ω 2 1 c MJ+1 = M + ′ . (14) P 1 + J α baryon such as 2.702 GeV for J = 2 and 2.772 GeV r + for J P = 3 . From Eq. (8) we can write , Using this equation we calculate the masses of excited 2 0 ′ states of Ωc baryon, which are presented in Table II and αssc 1 2 2 2 Table III for natural and unnatural parity states, respec- ′ = × [(4M + − M ++ − M 0 ) 2 Ξc Σc Ωc α 2M 0 tively. We compared our results with other theoritical uuc Ωc (12) predictions and experimental observations where avail- 2 2 2 2 2 2 + (4M + − M ++ − M 0 ) − 4M ++ M 0 ]. able. Ξc Σc Ωc Σc Ωc q ′ Putting′ the values of masses in above equation, we get αssc/αuuc. We can determine the slope by using Eq. (4), B. Masses of Ωcc and Ωccc

′ (J + 1) − J ′ ′ α = , The values of α /α from Eq. (8) should be a real M 2 − M 2 jjq iiq J+1 J number. So we can write Eq. (8) as, ++ for Σc (uuc) it will be, 2 2 2 |4Mijq − Miiq − Mjjq | > 2MiiqMjjq , (15) ′ 1 this inequality relation can be simplified to, αuuc = 2 2 , (13) M ++ 3 − − M ++ 1 + Σc ( 2 ) Σc ( 2 ) 2Mijq > (Miiq + Mjjq ). (16) ′ −2 P 1 + we get αssc = 0.54835 GeV for J = 2 . Similarly ′ Many theoretical studies shows that the slopes of −2 P 3 + we get αssc = 0.66524 GeV for J = 2 . Regge trejectories decreases with increasing quark masses 4

0 This relation can be used to calculate the upper and TABLE II. Masses of excited states of Ωc baryon (in GeV) in lower limit for baryon masses M . Further to estimate 2 J− 1 ijq (J, M ) plane with natural parities P =(−1) 2 . the deviation of relation (19), we introduce a parameter 2 2 2 2 2 2 called δ which is used to replace the signs of inequalities States 1 S 1 1 P 3 1 D 5 1 F 7 1 G 9 1 H 11 2 2 2 2 2 2 to equal signs. For baryons it is denoted by δb and Present 2.702 3.049 3.360 3.645 3.871 4.156 ij,q LHCb [1] 3.050 given by, Belle [2] 3.050 b 2 2 2 PDG [17] 2.695 3.050 δij,q = Miiq + Mjjq − 2Mijq, (20) Ref. [20] 2.696 2.968 3.251 3.519 3.783 Ref. [21] 2.695 3.024 3.299 3.565 here also i, j and q represents the arbitrary light or heavy Ref. [23] 2.698 3.054 3.297 3.514 3.705 quarks. Ref. [24] 2.718 2.977 3.196 Now from Eqs. (5) and (6) we can write, Ref. [30] 2.731 3.033 Ref. [41] 2.698 3.026 3.218 Ref. [42] 2.718 3.056 3.273 aiiq(0) − aijq (0) = aijq (0) − ajjq (0), (21)

0 TABLE III. Masses of excited states of Ωc baryon (in GeV) 1 1 1 1 2 J+ 1 in (J, M ) plane with unnatural parities P =(−1) 2 . ′ − ′ = ′ − ′ , (22) αiiq αijq αijq αjjq 4 4 4 4 4 4 States 1 S 3 1 P 5 1 D 7 1 F 9 1 G 11 1 H 13 2 2 2 2 2 2 Present 2.772 3.055 3.314 3.555 3.779 3.991 based on these equations we introduce two parameters, PDG[17] 2.765 Ref. [20] 2.766 2.962 3.241 3.503 3.756 1 1 λx = annn(0) − annx(0),γx = ′ − ′ ; (23) Ref. [21] 2.765 3.010 3.276 3.532 αnnx αnnn Ref. [23] 2.776 3.051 3.283 3.485 3.665 Ref. [24] 2.776 3.014 3.206 where n represents light nonstrange quark (u or d) and Ref. [30] 2.779 3.057 x denotes i, j or q. From Eqs. (21-23)we have, Ref. [41] 2.765 3.022 3.237 Ref. [42] 2.766 3.014

aijq (0) = annn(0) − λi − λj − λq,

′ ′ 1 1 (24) [37, 38, 43–46]. For m >m , we can write α /α < 1. ′ = ′ + γi + γj + γq. j i jjq iiq α α Therefore Eq. (8) is, ijq nnn ′ Since in baryon multiplets for nnn and ijq states we αjjq 1 2 2 2 ′ = 2 × [(4Mijq − Miiq − Mjjq) can write from Eq. ??eq:4), αiiq 2Mjjq (17) ′ 2 2 2 2 2 2 2 + (4Mijq − Miiq − Mjjq ) − 4MiiqMjjq] < 1, J = annn(0) + αnnnMnnn, ′ (25) 2 q J = aijq (0) + α M , the above relation gives another inequality equation as, ijq ijq

2 2 2 solving Eqs. (24) and (25) we have, 2Mijq

′ so from Eqs. (16) and (18) we have, 2 2 1 M = (α M +λ +λ +λ ) ′ + γ + γ + γ . ijq nnn nnn i j q α i j q  nnn  (26) M + M M 2 + M 2 iiq jjq

b 2 2 2 δij,q = Miiq + Mjjq − 2Mijq

′ ′ 2 1 2 1 = (α M +2λ + λ ) ′ +2γ + γ + (α M +2λ + λ ) ′ +2γ + γ nnn nnn i q α i q nnn nnn j q α j q  nnn   nnn  ′ 2 1 − 2(α M + λ + λ + λ ) ′ + γ + γ + γ nnn nnn i j q α i j q  nnn  = 2(λi − λj )(γi − γj ). (27) 5

b So, we can say that δij,q is independent of quark flavor solving Eqs. (28) and (29) we have, q. Hence by keeping i and j fixed and changing the q quark, using Eq. (20) we have some following relations 3 + for 2 states : 2 2 2 2 2 2 ∗ ∗ ∗ ∗ ∗ ∗ (I) i = u, j = s, q = u, c (MΩcc − MΞcc ) + (MΞ − MΣ ) = (MΩc − MΣc ), (31)

(3/2)+ 2 2 2 δ = M + M ∗ − 2M ∗ us ∆ Ξ Σ (28) 2 2 2 ∗ ∗ ∗ = MΣc + MΩc − 2MΞc ; 1 + similarly, its corresponding relation for 2 baryonic (II) i = u, j = c, q = u ,s states is,

(3/2)+ 2 2 2 δ = M + M ∗ − 2M ∗ uc ∆ Ξcc Σc (29) 2 2 2 = M ∗ + M ∗ − 2M ∗ ; Σ Ωcc Ξc 2 2 2 2 2 2 (MΩcc − MΞcc ) + (MΞ − MΣ) = (MΩc − MΣc ), (32) (III) i = s, j = c, q = u ,c

(3/2)+ 2 2 2 δ = M ∗ + M ∗ − 2M ∗ sc Ξ Ξcc Ξc (30) 2 2 2 using Eqs. (11) and (32) we have the expression, ∗ ∗ ∗ = MΩc + MΩccc − 2MΩcc ;

2 2 2 2 2 2 2 2 2 2 2 4M + − M + + M ++ − 2M ++ − M 0 + M + − 2M ++ M + − M ++ + M 0 − M + + M ++ Ξc Ωcc Ξcc Σc Ξ Σ Σc Ωcc Ξcc Ξ Σ Σc .  2 q 2 2 2 2 2 2 2 2 2 2 2 2 = − 4M + − M + + M ++ − 2M ++ − M 0 + M + − 4M ++ M + − M ++ + M 0 − M + + M ++ . Ξc Ωcc Ξcc Σc Ξ Σ Σc Ωcc Ξcc Ξ Σ Σc r     (33)

Using the above expression we have calculated the hence using relation (8) we have, + ground state masses of Ωcc baryon. Inserting the masses ′ + ++ MΞc , MΣc , MΞcc and MΞ, MΣ from PDG[17] and αscc 1 2 2 2 ′ = × [(4M + − M + − M + ) 0 + 2 Ξc Σ Ωcc MΩc (calculated above) in Eq. (33), we get MΩcc =3.752 αuus 2M + Ωcc . (34) P 1 + + GeV for J = 2 . Similarly we get MΩcc =3.816 GeV for 2 2 2 2 2 2 + + (4M + − M + − M + ) − 4M + M + ]. J P = 3 . Since the quark configuration of Ω++ is scc, Ξc Σ Ωcc Σ Ωcc 2 cc q Further using Eqs. (34) and (14) we calculate the ex- + cited state masses of Ωcc baryon in the same way we 0 have done for Ωc. Our predicted masses are labeled with ”Present” for both natural and unnatural parity states presented in Table IV and Table V respectively, where our predictions are reasonably close to other theoretical predictions. Similarly for triply charmed omega baryon (Ωccc), using relations (30) and (33) we have,

2 2 2 2 2 2 2 2 2 2 2 2 2 2 − − − Σ − ++ − − 3MΞcc MΩccc + 6MΞc 4MΣc MΩc + MΞ + 2M 2MΣc MΩccc 3MΞcc + MΩc + MΞ + 2 MΞc + MΣc MΣ  q  − 2 − 2 2 − 2 − 2 2 2 2 − 2 2 − 2 2 2 2 2 − 2 = 3MΞ MΩ + 6MΞ 4MΣ MΩ + MΞ + 2MΣ 8M ++ MΩ 3MΞ + MΩ + MΞ + 2 MΞ + MΣ MΣ . r cc ccc c c c Σc ccc cc c c c    (35)

P 3 + ++ Since the ground state (J = 2 ) mass of Ξcc is not [32] and inserting all other masses from latest PDG [17] ∗ + confirmed by PDG yet, so we take M = 3.695 GeV ∗ P 3 Ξcc into Eq. (35), we obtain MΩccc =4.841 GeV for J = 2 . 6

0 + ++ 2 C. Masses of Ωc , Ωcc and Ωccc baryons in (n, M ) TABLE IV. Masses of excited states of Ωcc baryon (in GeV) Plane. 2 J− 1 in (J, M ) with natural parities P =(−1) 2 .

2 2 2 2 2 2 States 1 S 1 1 P 3 1 D 5 1 F 7 1 G 9 1 H 11 The general equation for linear Regge trejectories in 2 2 2 2 2 2 2 Present 3.752 3.975 4.186 4.387 4.579 4.763 (n,M ) plane can be expressed as, Ref. [18] 3.650 3.972 4.141 4.296 2 Ref. [24] 3.815 4.052 4.202 n = β0 + βM , (36) Ref. [26] 3.719 Ref. [30] 3.832 4.086 4.264 where n = 1,2,3....is the radial principal quantum num- Ref. [31] 3.650 4.174 ber, β and β0 are the slope and intercept of the trejec- Ref. [32] 3.650 3.910 4.153 4.383 tories. The Regge slope (β) and Regge intercept (β0) Ref. [47] 3.778 4.102 are assumed to be same for all baryon multiplets lying Ref. [48] 3.710 on the single Regge line. So first of all these param- Ref. [49] 3.637 eters (β and β0) are calculated, and with the help of Ref. [50] 3.732 3.986 2S+1 them the excited states masses (N LJ , where N, L, Ref. [51] 3.804 S denote the radial excited quantum number, the orbital quantum number and the intrinsic spin, respectively) of 0 + ++ Ωc, Ωcc and Ωccc baryons are estimated lying on these 0 TABLE V. Masses of excited states of Ωcc baryon (in GeV) Regge trejectories. For Ωc baryon, using the slope equa- 2 J+ 1 2 2 in (J, M ) with unnatural parities P =(−1) 2 . tion β(S) =1/(MΩc(2S)−MΩc(1S)), where MΩc(1S)=2.702

4 4 4 4 4 4 GeV (calculated above) and taking MΩc(2S)=3.164 GeV states 1 S 3 1 P 5 1 D 7 1 F 9 1 G 11 1 H 13 P + 2 2 2 2 2 2 from [21] for J = 1/2 state, we get β(S) = 0.3689 Present 3.816 4.094 4.354 4.599 4.832 5.055 −2 Ref. [18] 3.810 3.958 4.122 4.274 GeV . Using the following equations, Ref. [24] 3.876 4.152 4.230 2 1= β0(S) + β(S)MΩc(1S), Ref. [29] 3.822 (37) Ref. [31] 3.808 4.313 2 2= β0(S) + β(S)MΩc(2S), Ref. [32] 3.809 4.058 4.294 4.516 Ref. [48] 3.760 we have β0(S) = -1.6939. In the same way we can write, Ref. [49] 3.762 Ref. [50] 3.765 1= β + β M 2 , Ref. [51] 3.850 0(P ) (P ) Ωc(1P ) 2 Ref. [52] 3.730 4.134 4.204 2= β0(P ) + β(P )MΩc(2P ), Ref. [53] 3.800 2 (38) 1= β0(D) + β(D)MΩc(1D), 2 2= β0(D) + β(D)MΩc(2D). Excited state masses for Ωccc baryon are calculated in the With the help β and β , we calculate the masses similar manner as we have calculated for Ω0 and Ω+ , only (S) 0(S) c cc of excited Ω0 baryon for n=3,4,5... Tables VII and VIII by changing the quark constituents i.e. i ,j and q. Our c shows our calculated radial and orbital excited state calculated masses of excited Ω are presented in Table ccc masses of Ω baryon in (n,M 2) plane for natural and VI. Our results are in accordance with other theoritical c unnatural parity states respectively. In the similar man- predictions. ner we estimate the mass spectra for Ωcc (see Tables IX 2 and X) and Ωccc(see Table XI) baryons also in (n,M ) plane. TABLE VI. Masses of excited states of Ωccc baryon (in GeV) 2 J+ 1 in (J, M ) plane with unnatural parities P =(−1) 2 . III. RESULT AND DISCUSSION 4 4 4 4 4 4 states 1 S 3 1 P 5 1 D 7 1 F 9 1 G 11 1 H 13 2 2 2 2 2 2 Present 4.841 5.081 5.310 5.530 5.741 5.945 In the framework of regge phenomenology , we have Ref. [24] 4.965 5.331 obtained the radial and orbital excited states of singly, Ref. [26] 4.803 doubly and triply charmed omega baryons. In the present Ref. [31] 4.818 5.302 0 Ref. [32] 4.834 5.301 work we have focused on studying the masses of Ωc Ref. [50] 4.773 baryon by obtaining the mass relations derived from Ref. [51] 4.930 regge phenomenology. Ground state as well as excited 2 Ref. [54] 4.827 state masses are obtained successfully in both (J, M ) Ref. [55] 4.758 5.300 and (n,M 2) planes for the following baryons. Ref. [56] 4.761 5.396 • 2 Ref. [57] 4.880 Ωc baryon :Our calculated masses in (J, M ) 0 plane for Ωc baryon are shown in Table II and 7

0 + TABLE VII. Masses of excited states of Ωc baryon (in GeV) in TABLE IX. Masses of excited states of Ωcc baryon (in GeV) in 2 J− 1 2 J− 1 (n, M ) plane with natural parity P =(−1) 2 . The masses (n, M ) plane with natural parity P =(−1) 2 . The masses from [21] are taken as input. from [18] are taken as input.

2S+1 2S+1 N LJ Present [23] [20] [24] [55] [30] N LJ Present [47] [30] [24] [32] 2 2 1 S 1 2.702 2.698 2.696 2.718 2.699 2.731 1 S 1 3.752 3.778 3.832 3.815 3.650 2 2 2 2 2 S 1 3.164 [21] 3.088 3.165 3.152 3.159 3.227 2 S 1 4.028 [18] 4.075 4.227 4.180 2 2 2 2 3 S 1 3.566 3.489 3.540 3.292 3 S 1 4.310 4.321 4.295 2 2 2 2 4 S 1 3.928 3.814 3.895 4 S 1 4.564 2 2 2 2 5 S 1 4.259 4.102 4.238 5 S 1 4.804 2 2 2 2 6 S 1 4.566 4.569 6 S 1 5.033 2 2 2 2 1 P 3 3.049 3.054 2.968 2.986 3.033 1 P 3 3.975 4.102 4.086 4.052 3.910 2 2 2 2 2 P 3 3.408 [21] 3.433 3.334 3.056 2 P 3 4.259 [18] 4.345 4.201 4.140 2 2 2 2 3 P 3 3.732 3.752 3.687 3 P 3 4.525 2 2 2 2 4 P 3 4.031 4.036 4.029 4 P 3 4.776 2 2 2 2 5 P 3 4.309 4.362 5 P 3 5.015 2 2 2 2 1 D 5 3.360 3.297 3.251 1 D 5 4.186 4.264 4.202 4.153 2 2 2 2 2 D 5 3.680 [21] 3.626 3.604 2 D 5 4.407 [18] 2 2 2 2 3 D 5 3.974 3.752 3.947 3 D 5 4.617 2 2 2 2 4 D 5 4.248 4.282 4 D 5 4.818 2 2 2 2 5 D 5 4.505 5 D 5 5.011 2 2

0 + TABLE VIII. Masses of excited states of Ωc baryon (in GeV) TABLE X. Masses of excited states of Ωcc baryon (in GeV) 1 2 J+ 1 2 J+ in (n, M ) plane with unnatural parity P = (−1) 2 . The in (n, M ) plane with unnatural parity P = (−1) 2 . The masses from [21] are taken as input. masses from [18] are taken as input.

2S+1 2S+1 N LJ Present [23] [20] [24] [55] [30] N LJ Present [47] [30] [24] [58] 4 4 1 S 3 2.772 2.768 2.766 2.776 2.768 2.779 1 S 3 3.816 3.872 3.883 3.876 3.824 2 2 4 4 2 S 3 3.197 [21] 3.123 3.208 3.190 3.202 3.257 2 S 3 4.096 [18] 4.174 4.263 4.188 4.163 2 2 4 4 3 S 3 3.571 3.510 3.564 3.285 3 S 3 4.358 4.265 2 2 4 4 4 S 3 3.910 3.830 3.911 4 S 3 4.605 2 2 4 4 5 S 3 4.222 4.114 4.248 5 S 3 4.839 2 2 4 4 6 S 3 4.513 6 S 3 5.063 2 2 4 4 1 P 5 3.055 3.051 2.962 3.041 3.057 1 P 5 4.094 4.220 4.152 2 2 4 4 2 P 5 3.393 [21] 3.427 3.328 3.477 2 P 5 4.247 [18] 2 2 4 4 3 P 5 3.700 3.744 3.681 3.620 3 P 5 4.394 2 2 4 4 4 P 5 3.983 4.028 4.023 4 P 5 4.537 2 2 4 4 5 P 5 4.248 5 P 5 4.676 2 2 4 4 1 D 7 3.314 3.283 3.241 1 D 7 4.354 2 2 4 4 2 D 7 3.656 [21] 3.611 3.594 2 D 7 4.391 [18] 2 2 4 4 3 D 7 3.968 3.938 3 D 7 4.427 2 2 4 4 4 D 7 4.258 4.273 4 D 7 4.464 2 2 4 4 5 D 7 4.529 4.594 5 D 7 4.500 2 2

Table III for natural and unnatural parity states MeV and also they are in good agreement with respectively. Firstly we compared our predicted Refs.[20, 21, 23, 41]. Further the excited state ground state masses with experimently available masses are compared with other theoretical out- 2 2 comes. For 1 P 3 and 1 P 5 states our results are data [1, 2, 17] and other theoretical predictions 2 2 [20, 21, 23, 24, 30, 41, 42]. Our results are very in accordance with Refs.[21, 23, 30, 41, 42] with 2 mass difference of range 5-50 MeV and for 1 D 5 - close to [17] with a slight mass difference of 7-8 2 8

++ estimated results for natural and unnatural parity TABLE XI. Masses of excited states of Ωccc baryon (in GeV) states respectively in (J, M 2) plane. Our calculated 2 J+ 1 − 2 2 in (n, M ) plane with unnatural parity P = ( 1) . The ground state (1 S 1 ) mass is in good agreement masses from [19] are taken as input. 2 with the predictions of Refs.[24, 26, 47, 48, 50, 51] 2S+1 2 N LJ Present [24] [32] [31] [55] with mass difference of 25-65 MeV, and for 1 P 3 - 4 2 1 S 3 4.841 4.965 4.834 4.818 4.758 2 7 2 1 F states our results are in accordance with 4 2 2 S 3 5.300 [19] 5.313 2 Refs.[24, 31, 32, 50] with mass difference of 20- 4 4 3 S 3 5.722 3 2 70 MeV (see Table IV). Similarly for 1 S state 4 2 4 S 3 6.115 2 our predicted mass is very close to the results of 4 5 S 3 6.484 Refs.[29, 31, 32, 53] having slight mass difference of 2 4 4 4 6 S 3 6.834 7-16 MeV, and for 1 P 5 -1 F 9 states our results are 2 2 2 4 in accordance with other theoritical outcomes (see 1 P 5 5.081 2 Table V). In the same manner we compared our 4 2 P 5 5.553 [19] 2 calculated radial and orbital excited state masses 4 3 P 5 5.987 2 2 evaluated in (n,M ) plane for both natural and 4 4 P 5 6.393 unnatural parity states and our results are consis- 2 4 5 P 5 6.774 tent with other theoretical and phenomenological 2 studies (see Tables IX and X). 4 1 D 7 5.310 5.331 5.301 5.302 5.300 2 4 2 D 7 5.961 [19] • Ωccc baryon : For triply charmed omega baryon, 2 4 3 D 7 6.547 we calculate the ground state and excited state 2 4 2 2 4 D 7 7.085 masses in both (J, M ) and (n,M ) planes shown 2 4 in Table VI and Table XI respectively. The masses 5 D 7 7.585 2 ++ the ground state Ωccc vary in the range 4.750-4.950 GeV in other theoretical references (Table VI). Our 4 predicted mass for (1 S 3 ) state shows few MeV dif- 2 4 4 2 1 F 7 and 1 D 7 -1 F 9 states our predictions are 2 2 2 ference with Refs. [26, 31, 32, 54]. For excited state consistent with Refs. [20, 23] with mass differ- very few results are availabe from previous theoret- ence of 30-65 MeV. Similarly in (n,M 2) plane, ical predictions [24, 31, 32, 55], and our calculate our calculated results are shown in Tables VII and masses are in accordance with them. VIII for natural and unnatural parity states re- spectively. Our predicted masses are in agreement We have successfully employed Regge phenomenological with Refs.[20, 23, 24]. The experimentally observed approach to calculate the masses of singly, doubly and 0 state Ωc(3050) having mass 3.050 GeV is close to triply charmed Ω baryons. This study will definitely help 0 our prediction 3.049 GeV, so we assigned Ωc(3050) future experimental studies at LHCb, Belle II and the up- as 1P state with J P =3/2− for S = 1/2. Other coming facility PANDA, to identify these baryonic states 0 0 0 from resonances. four states Ωc(3000) , Ωc(3066) , Ωc(3090) and 0 Ωc(3119) are not identified in this work (it may 2 4 belong to remaining 1P states such as 1 P 1 , 1 P 1 2 2 ACKNOWLEDGMENTS 4 2 2 and 1 P 3 [4]). Since we get masses for 1 S 1 ,1 P 3 , 2 2 2 2 4 4 4 1 D 3 ,... and 1 S 3 , 1 P 5 , 1 D 7 ,... states for nat- 2 2 2 2 One of the authors Juhi Oudichhya inspired by the ural and unnatural parity respectively in (J, M 2) work of Ke-Wei Wei, Bing Chen, Xin-Heng Guo, De-Min plane lying on Regge line. Li, Bing Ma, Yu-Xiao Li, Qian-Kai Yao, Hong Yu on Regge phenomenology and would like to thank them for • Ωcc baryon : Table IV and Table V shows our their valuabe contributions to this field.

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