Predictions for masses of Xi_b baryons
Marek Karliner, Boaz Keren-Zur, Harry J. Lipkin, Jonathan L. Rosner
aa r X i v : . [ h e p - ph ] J un EFI 07-16TAUP 2857/07WIS/09/07-JUNE-DPPANL-HEP-PR-07-41
Predictions for masses of Ξ b baryons Marek Karliner a , Boaz Keren-Zur a , Harry J. Lipkin a,b,c , and Jonathan L. Rosner da School of Physics and AstronomyRaymond and Beverly Sackler Faculty of Exact SciencesTel Aviv University, Tel Aviv 69978, Israel b Department of Particle PhysicsWeizmann Institute of Science, Rehovoth 76100, Israel c High Energy Physics Division, Argonne National LaboratoryArgonne, IL 60439-4815, USA d Enrico Fermi Institute and Department of PhysicsUniversity of Chicago, 5640 S. Ellis Avenue, Chicago, IL 60637, USA
ABSTRACT
The recent observation by CDF of Σ ± b ( uub and ddb ) baryons within 2 MeVof the predicted Σ b − Λ b splitting has provided strong confirmation for thetheoretical approach based on modeling the color hyperfine interaction.We now apply this approach to predict the masses of the Ξ b family ofbaryons with quark content usb and dsb – the ground state Ξ b at 5790to 5800 MeV, and the excited states Ξ ′ b and Ξ ∗ b . The main source ofuncertainty is the method used to estimate the mass difference m b − m c from known hadrons. We verify that corrections due to the details of theinterquark potential and to Ξ b –Ξ ′ b mixing are small.PACS codes: 14.20.Mr, 12.40.Yx, 12.39.Jh, 11.30.Hw Introduction
For many years the only confirmed baryon with a b quark was the isospin-zero Λ b .A recent measurement of its mass by the CDF Collaboration is M (Λ b ) = 5619 . ± . ± . b = bud , where the ud pair has spin andisospin S ( ud ) = I ( ud ) = 0. Now CDF has reported the observation of candidates forthe Σ ± b and Σ ∗± b [2] with masses consistent with quark model predictions [3, 4, 5, 6, 7], M (Σ − b ) − M (Λ b ) = 195 . +1 . − . (stat . ) ± . M (Σ + b ) − M (Λ b ) = 188 . +2 . − . (stat . ) ± . M (Σ b ) − M (Λ b ) = 192 MeV, to be comparedwith the prediction [5, 8] M Σ b − M Λ b = 194 MeV.The Σ ± b states consist of a light quark pair uu or dd with S = I = 1 coupledwith the b quark to J = 1 /
2, while in the Σ ∗± b states the light quark pair and the b quark are coupled to J = 3 /
2. The CDF sensitivity appears adequate to detectfurther heavy baryons, such as those with quark content bsu or bsd . The S-wavelevels of these quarks consist of the J = 1 / , − b and Ξ ′ (0 , − ) b and the J = 3 / ∗ (0 , − ) b . In this paper we predict the masses of these states and estimate thedependence of the predictions on the form of the interquark potential. This exercisehas been applied previously to hyperfine splittings of known heavy hadrons [9].We discuss the predictions for M (Ξ b ) in Section 2, starting with an extrapolationfrom M (Ξ c ) without correction for hyperfine (HF) interaction and then estimatingthis correction. In the Ξ b the light quarks are approximately in a state with S = 0,while another heavier state Ξ ′ b is expected in which the light quarks mainly have S = 1. There is also a state Ξ ∗ b expected with light-quark spin 1 and total J =3 /
2. Predictions for Ξ ′ b and Ξ ∗ b masses are discussed in Section 3. We estimate theeffect of mixing between light-quark spins S = 0 and 1 in Section 4, while Section 5summarizes. Ξ b mass prediction In our model the mass of a hadron is given by the sum of the constituent quark massesplus the color-hyperfine (HF) interactions: V HFij = v ~σ i · ~σ j m i m j h δ ( r ij ) i (2)where the m i is the mass of the i ’th constituent quark, σ i its spin, r ij the distancebetween the quarks and v is the interaction strength. We shall neglect the massdifferences between u and d constituent quarks, writing u to stand for either u or d .All the hadron masses (the ones used and the predictions) are for isospin-averagedbaryons and are given in MeV. 2he s and u quarks in Ξ q ( q standing for c or b ) are assumed to be in relative spin0 and the total mass is given by the expression:Ξ q = m q + m s + m u − v h δ ( r us ) i m u m s (3)The Ξ b mass can thus be predicted using the known Ξ c baryon mass as a startingpoint and adding the corrections due to mass differences and HF interactions:Ξ b = Ξ c + ( m b − m c ) − vm u m s h δ ( r us ) i Ξ b − h δ ( r us ) i Ξ c ! (4)The experimentally determined masses for the charmed-strange baryons Ξ c , Ξ ′ c ,and Ξ ∗ c are [10]:Ξ c = 2469 . ± . ′ c = 2577 ± ∗ c = 2646 . ± . . (5) The mass difference ( m b − m c ) can be obtained from experimental data using one ofthe following expressions: • We can simply take the difference of the masses of the Λ q baryons, ignoring thedifferences in the HF interaction: m b − m c = Λ b − Λ c = 3333 . ± . . (6) • We can use the spin averaged masses of the Λ q and Σ q baryons: m b − m c = ∗ b + Σ b + Λ b − ∗ c + Σ c + Λ c ! = 3330 . ± . . (7) • Since the Ξ q baryon has strangeness 1, it might be better to use masses ofmesons with S = 1: m b − m c = B ∗ s + B s − D ∗ s + D s ! = 3324 . ± . . (8) The HF interaction correction can also be based on Ξ c baryon experimental data: vm u m s h δ ( r us ) i Ξ b − h δ ( r us ) i Ξ c ! = v h δ ( r us ) i Ξ c m u m s h δ ( r us ) i Ξ b h δ ( r us ) i Ξ c − ! (9)= 2Ξ ∗ c + Ξ ′ c − c h δ ( r us ) i Ξ b h δ ( r us ) i Ξ c − ! = h δ ( r us ) i Ξ b h δ ( r us ) i Ξ c − ! (38 . ± .
5) MeV3owever, this expression requires the calculation of the δ function expectation values.These were calculated using 3-body wavefunctions obtained by a variational methodas described in [9]. The only input required for these calculations is the shape ofconfining potential, because the coupling constants cancel out when taking the ratioof the δ function expectation values. The potentials considered in this work are thelinear, Coulomb and Cornell (Coulomb + linear) potentials. We also wrote down theresults obtained without the HF corrections. Note that in the case of the Cornellpotential we have an additional parameter, which determines the ratio between thestrengths of the linear and Coulombic parts of the potential. In these calculationswe used the parameters extracted in [11] from analysis of quarkonium spectra (or K = 0 .
45 when using the parameterization described in [9]).As a test case we compared the values obtained from experimental data andvariational calculations for the ratio of contact probabilities in Ξ and Ξ c .2Ξ ∗ c + Ξ ′ c − c ∗ − Ξ) = 6 v h δ ( r us ) i Ξ c m u m s v h δ ( r us ) i Ξ m u m s = h δ ( r us ) i Ξ c h δ ( r us ) i Ξ (10)The results given in Table 1 show good agreement between data and theoreticalpredictions using the Cornell potential. h δ ( r us ) i Ξ c / h δ ( r us ) i Ξ Experimental data [10] 1 . ± . . ± . . ± . . ± . Comparison between experimental data and predictions of the ratio of u and s contact probabilities in Ξ and Ξ c (Eq. (10)). The final predictions for the Ξ b mass with the different assumptions regarding theconstituent quark mass differences and the confinement potentials are given in Table2. From previous experience we know that the predictions of the Coulomb potentialmodel show a very strong dependence on the quark masses which is not observed inthe data, hence one should probably give these predictions less weight. Ignoring theCoulomb potential, one gets a prediction for the Ξ b mass in the range of 5790 - 5800MeV. 4 b − m c = Λ b − Λ c Σ b − Σ c B s − D s Eq. (6) Eq. (7) eq. (8)No HF correction 5803 ± ± ± ±
11 5798 ±
11 5792 ± ± ± ± ± ± ± Predictions for the Ξ b mass with various confining potentials and methods ofobtaining the quark mass difference m b − m c Ξ ∗ b , Ξ ′ b mass prediction Ξ ∗ b + Ξ ′ b )/3 The s and u quarks of the Ξ ∗ q and Ξ ′ q baryons are assumed to be in a state of relativespin 1. We then findΞ ∗ q = m q + m s + m u + v h δ ( r qs ) i m q m s + h δ ( r qu ) i m q m u + h δ ( r us ) i m u m s ! (11)Ξ ′ q = m q + m s + m u + v − h δ ( r qs ) i m q m s + − h δ ( r qu ) i m q m u + h δ ( r us ) i m u m s ! The spin-averaged mass of these two states can be expressed as2Ξ ∗ q + Ξ ′ q m q + m s + m u + v h δ ( r us ) i m u m s , (12)and as for the Ξ b case, the following prediction can be given:2Ξ ∗ b + Ξ ′ b ∗ c + Ξ ′ c m b − m c ) + 2Ξ ∗ c + Ξ ′ c − c h δ ( r us ) i Ξ b h δ ( r us ) i Ξ c − ! . (13)The predictions obtained using the same methods described above are given in Table3. In this case it is clear that the effect of the HF correction is negligible. Thus thedifference between the spin averaged mass (2Ξ ∗ b + Ξ ′ b ) / b is roughly 150 − Ξ ∗ b − Ξ ′ b This mass difference is more difficult to predict, but it will be small due to the largemass of the b quark. Ξ ∗ q − Ξ ′ q = 3 v h δ ( r qs ) i m q m s + h δ ( r qu ) i m q m u ! (14)5 b − m c = Λ b − Λ c Σ b − Σ c B s − D s Eq. (6) Eq. (7) Eq. (8)No HF correction 5956 ± ± ± ± ± ± ± ± ± ± ± ± Predictions for the spin averaged Ξ ′ b and Ξ ∗ b masses with various confining poten-tials and methods of obtaining the quark mass difference m b − m c We can once again use the Ξ c hadron masses:Ξ ∗ b − Ξ ′ b Ξ ∗ c − Ξ ′ c = 3 v h δ ( r bs ) i m b m s + h δ ( r bu ) i m b m u ! v h δ ( r cs ) i m c m s + h δ ( r cu ) i m c m u ! = m c m b h δ ( r bs ) i Ξ b + m s m u h δ ( r bu ) i Ξ b ! h δ ( r cs ) i Ξ c + m s m u h δ ( r cu ) i Ξ c ! (15)This expression is strongly dependent on the confinement model. In the resultsgiven in Table 4 we have used m s m u = 1 . ± . m b m c = 2 . ± . ∗ b − Ξ ′ b No HF correction 24 ± ± ± ± Predictions for the mass difference between Ξ ∗ b and Ξ ′ b with various confiningpotentials. Ξ b and Ξ ′ b In estimates up to this point we have assumed that the light-quark spins in Ξ b andΞ ′ b are purely S = 0 and S = 1, respectively. The differing hyperfine interactions6etween the b quark and nonstrange or strange quarks leads to a small admixture ofthe opposite- S state in each mass eigenstate [12, 13, 14, 15]. The effective hyperfineHamiltonian may be written [14, 15] H eff = M + λ ( σ u · σ s + ασ u · σ b + βσ s · σ b ) , (16)where M is the sum of spin independent terms, λ ∼ / ( m u m s ), α = m s /m b , and β = m u /m b . The calculation of M / is straightforward, as the expectation value ofeach σ i · σ j in the J = 3 / J = 1 / × M / = " M − λ λ √ β − α ) λ √ β − α ) M + λ (1 − α − β ) . (17)The eigenvalues of H eff are thus M / = M + λ (1 + α + β ) , (18) M / , ± = M + λ [ − (1 + α + β ) ± λ (1 + α + β − α − β − αβ ) / . (19)In the absence of mixing ( α = β ) one would have M / = M + λ (1 + 2 α ), M / , + = M + λ (1 − α ), and M / , − = M − λ .To see the effect of mixing, we rewrite the expression for M / , ± , M / , ± = M − λ (1 + α + β ) ± λ − α + β ! + 34 ( α − β ) / (20)The effect of the mixing is seen in the term ( α − β ) . Expanding M / , ± to secondorder in small α − β , we obtain M / , ± ≈ (terms without mixing) ± λ ·
34 ( α − β ) − α + β m u = 363 MeV, m s = 538 MeV, and m b = 4900 MeV [16], one has α ≃ . β ≃ .
07, while the discussion in the previous section implies λ ≃
40 MeV [Eq. (10)].Hence the effect of mixing on our predictions is negligible, amounting to ± .
04 MeV.Since we use the Ξ c and Ξ ′ c masses as input for Ξ b , it is also important to checkthe mixing effects on the former. Since m b /m c ∼
3, this amounts to changing in theexpressions above α → α , β → β . The corresponding effect of mixing on Ξ c andΞ ′ c is ∼ . Summary
We have shown that predictions for M (Ξ b ) based on the masses of Ξ c , Ξ ′ c , and Ξ ∗ c liein the range of 5790 to 5800 MeV, depending on how the mass difference m b − m c isestimated. Wave function differences tend to affect these predictions by only a fewMeV. The spin-averaged mass of the states Ξ ′ b and Ξ ∗ b is predicted to lie around 150 to160 MeV above M (Ξ b ), while the hyperfine splitting between Ξ ′ b and Ξ ∗ b is predictedto lie in the rough range of 20 to 30 MeV. We look forward to the verification of thesepredictions in experiments at the Fermilab Tevatron and the CERN Large HadronCollider.Note added: After this work was completed we received notice of the Ξ − b obser-vation at the Fermilab Tevatron in the J/ψ Ξ − decay mode by the D0 Collaboration[17]. After the first version of this paper appeared [18], the CDF Collaboration re-leased their Ξ − b results in the same decay channel [19]. The reported masses, Gaussianwidths (due to instrumental resolution), and significances of the signal are summa-rized in Table 5 and in Fig. 1. The CDF Collaboration also observes a significantΞ − b → Ξ c π − signal with mass consistent with that found in the J/ψ Ξ − mode. TheD0 mass is consistent with all our predictions for the isospin-averaged mass, whilethat of CDF allows us to rule out the (previously disfavored [9]) prediction based onthe Coulomb potential. Both experiments also agree with a prediction in Ref. [4], M (Ξ b ) = M (Λ b ) + (182 . ± .
0) MeV = (5802 . ± .
3) MeV, where differences in wavefunction effects were not discussed and m b − m c was taken from baryons only, whereasin our work the optional value of m b − m c was obtained from B s and D s mesons whichcontain both heavy and strange quarks, as do Ξ b and Ξ c . See also Ref. [6] and TableIII therein for a compilation of earlier predictions for the Ξ b mass. That the value of m b − m c obtained from B and D mesons depends upon the flavor of the spectatorquark was noted in Ref. [5] where Table I shows that the value is the same for mesonsand baryons not containing strange quarks but different when obtained from B s and D s mesons. Some reasons for this difference were noted and the issue requires furtherinvestigation. Here we have updated the prediction of Ref. [4] using the recent CDF[1] value of M (Λ b ). Acknowledgements
J.L.R. wishes to acknowledge the hospitality of Tel Aviv University during the earlystages of this work. We thank Dmitry Litvintsev for providing his figure compar-ing theoretical predictions with measurements of the Ξ − b mass. This research wassupported in part by a grant from Israel Science Foundation administered by IsraelAcademy of Science and Humanities. The research of H.J.L. was supported in partby the U.S. Department of Energy, Division of High Energy Physics, Contract DE-AC02-06CH11357. The work of J.L.R. was supported by the U.S. Department ofEnergy, Division of High Energy Physics, Grant No. DE-FG02-90ER40560.8able 5: Observations of Ξ − b → J/ψ Ξ − at the Fermilab Tevatron. Errors on mass are(statistical, systematic). D0 [17] CDF [19]Mass (MeV) 5774 ± ±
15 5793 ± . ± . ± ∼ . σ . σ Fig. 1 (adapted from [19]). Comparison of theoretical predictions and experimentalresults for the Ξ − b mass from D0 [17] and CDF [19]. The theoretical predictions are denotedby the two horizontal bands, corresponding to Refs. [4] and [18], respectively. eferences [1] D. Acosta et al. [CDF Collaboration], Phys. Rev. Lett. , 202001 (2006)[arXiv:hep-ex/0508022].[2] J. Pursley, talk presented on behalf of the CDF Collaboration at the 11th In-ternational Conference on B -Physics at Hadron Machines (Beauty 2006), 25–29September 2006, University of Oxford, to be published in Nucl. Phys. B (Proc.Suppl.); I. Gorelov, presented on behalf of the CDF Collaboration at the SecondMeeting of the APS Topical Group on Hadron Physics (GHP 2006), 22–24 Oc-tober 2006, Nashville, TN; CDF Collaboration, public web page .[3] E. Bagan, M. Chabab, H. G. Dosch and S. Narison, Phys. Lett. B , 367(1992); B , 176 (1992); S. Capstick and N. Isgur, Phys. Rev. D , 2809(1986); R. Roncaglia, D. B. Lichtenberg and E. Predazzi, Phys. Rev. D , 1722(1995); N. Mathur, R. Lewis and R. M. Woloshyn, Phys. Rev. D , 014502(2002).[4] E. Jenkins, Phys. Rev. D , 4515 (1996); ibid. , 10 (1997).[5] M. Karliner and H. J. Lipkin, arXiv:hep-ph/0307243 (unpublished).[6] D. Ebert, R. N. Faustov and V. O. Galkin, Phys. Rev. D , 034026 (2005).[7] J. L. Rosner, Phys. Rev. D , 013009 (2007) [arXiv:hep-ph/0611207].[8] M. Karliner and H. J. Lipkin, arXiv:hep-ph/0611306 (unpublished).[9] B. Keren-Zur, arXiv:hep-ph/0703011, to be published in Annals of Physics.[10] W.-M. Yao et al., Journal of Physics G , 1 (2006).[11] E. Eichten et al. , Phys. Rev. D , 203-233 (1980).[12] K. Maltman and N. Isgur, Phys. Rev. D , 1701 (1980).[13] H. J. Lipkin, unpublished.[14] J. L. Rosner, Prog. Theor. Phys. , 1422 (1981).[15] J. L. Rosner and M. P. Worah, Phys. Rev. D , 1131 (1992).[16] S. Gasiorowicz and J. L. Rosner, Am. J. Phys. , 954 (1981).[17] V. Abazov et al. [D0 Collaboration], arXiv:0706.1690 [hep-ex], submitted forpublication to Phys. Rev. Lett.[18] M. Karliner, B. Keren-Zur, H.J. Lipkin and J.L. Rosner, hep-ph/0706.2163v1.[19] D. Litvintsev, on behalf of the CDF Collaboration, seminar at Fermilab, June15, 2007, http://theory.fnal.gov/jetp/talks/litvintsev.pdfhttp://theory.fnal.gov/jetp/talks/litvintsev.pdf