SU(3) mass-splittings of heavy-baryons in QCD

We extract directly (for the first time) the charmed (C=1) and bottom (B=-1) heavy-baryons (spin 1/2 and 3/2) mass-splittings due to SU(3) breaking using double ratios of QCD spectral sum rules (QSSR) in full QCD, which are less sensitive to the exact value and definition of the heavy quark mass, to the perturbative radiative corrections and to the QCD continuum contributions than the simple ratios commonly used for determining the heavy baryon masses. Noticing that most of the mass-splittings are mainly controlled by the ratio kappa=<\bar ss>/<\bar dd>of the condensate, we extract this ratio, by allowing 1 sigma deviation from the observed masses of the Xi_{c,b} and of the Omega_c. We obtain: kappa=0.74(3), which improves the existing estimates: \kappa=0.70(10) from light hadrons. Using this value, we deduce M_{Omega_b}=6078.5(27.4) MeV which agrees with the recent CDF data but disagrees by 2.4 sigma with the one from D0. Predictions of the Xi'_Q and of the spectra of spin 3/2 baryons containing one or two strange quark are given in Table 2. Predictions of the hyperfine splittings Omega*_Q- Omega_Q and Xi*_Q-Xi_Q are also given in Table 3. Starting for a general choice of the interpolating currents for the spin 1/2 baryons, our analysis favours the optimal value of the mixing angle b= (-1/5 -- 0) found from light and non-strange heavy baryons.


Introduction
QSSR [1,2] à la SVZ [3] has been used earlier in full QCD [4,5,6] and in HQET [7] for understanding heavy baryons [charmed (cqq), bottom (bqq), double charm (ccq), double bottom (bbq) and (bcq)] masses.Recent observations at Tevatron of families of b-baryons [8,9] and of the Ω * c baryon by Babar and Belle [10] have stimulated different recent theoretical activities for understanding their nature [11,12,13,14,15,16,17,18,19].QSSR results are in quite good agreement with recent experimental findings but with relatively large uncertainties.The inaccuracy of these results is mainly due to the value of the heavy quark mass and of its ambiguous definition when working to lowest order (LO) in the radiative α s corrections in full QCD and HQET 1 , where the heavy quark mass is the main driving term in the QCD expression of the baryon two-point correlator used in the QSSR analysis.Another source of uncertainty is the effect of the QCD continuum which parametrizes the higher baryon masses contributions to the spectral function and the ad hoc choices of interpolating baryon currents used in different literatures.In this paper, we shall concentrate on the analysis of the heavy baryons mass-splittings due to SU (3) breaking using double ratios (DR) of QCD spectral sum rules (QSSR), which are less sensitive to the exact value and definition of the heavy quark mass and to the QCD continuum contributions than the simple ratios used in the literature to determine the absolute value of heavy baryon masses.In this letter, we extend the previous analysis in [4,5] by including the new SU (3) breaking terms: m s and the ratio of the condensate κ ≡ ss / dd .For the spin 1/2 baryons ¯, and following Ref.[4], we work with the lowest dimension general currents: where we use standard notations; b is a priori an arbitrary mixing parameter.Its value has been found to be: in the case of light baryons [20] and in the range [4,5,6]: for non-strange heavy baryons, which do not favour the Ioffe choice b = −1 [21].The corresponding two-point correlator reads: where F 1 and F 2 are two invariant functions.
For the spin 3/2 baryons ¯, we follow Ref. [5] and work with the interpolating currents: where an anti-symmetrization over colour indices is understood.The normalization in Eq. ( 5) is chosen in such a way that in all cases one gets the same perturbative contribution.The corresponding two-point correlator reads: In the following, the contribution of the heavy quark condensate m Q QQ will not appear [5] as it is cancelled by a part of the gluon condensate contribution due to the heavy quark relation . Again due to this relation, the one due to m s QQ is numerically negligible because of the extra (1/12π) loop and 1/M Q factors compared to the one due to m s qq and due to the four-quark condensate contributions.The same loop factor also numerically suppresses the contributions of α s G 2 and m s α s G 2 compared to the other ones.These negligible SU (3) breaking contributions will not be considered in the following.
2. The spin 1/2 two-point correlator in QCD T ¯he Λ Q (Qqq) and Ξ Q (Qsq) baryons The expression for Λ Q has been (first) obtained in the chiral limit m q = 0 in [5], and the one of Ξ Q including SU (3) breaking in [14].One can notice that due to the expression of the current the m s corrections vanish to leading order in α s for the perturbative term, while the D = 6 condensates for the SU (2) case of [5] needs the following replacement in the SU (3) case: where ρ = 2 ∼ 3 indicates the violation of the four-quark vacuum saturation [22,1,30].The additional SU (3) breaking corrections for the Ξ Q are [14]: -F 1 : -F 2 : where x ≡ m 2 Q /s and sGs ≡ g sσ µν λ a /2G µν a s .T ¯he Σ Q (Qqq) and Ω Q (Qss) baryons The expression for Σ Q has been (first) obtained in [4].The additionnal SU (3) breaking terms for the Ω Q are: -F 1 : -F 2 : T ¯he Σ Q (Qqq) and Ξ ′ Q (Qsq) baryons The expression for the Ξ ′ Q tends to the one of the Σ Q in the chiral limit m q,s → 0 and is very similar with one of the Ω Q .The SU(3) breaking corrections read: F 2 : We have checked the existing results in [4] obtained in the chiral limit and all our previous results agree with these ones.
3. The spin 3/2 two-point correlator in QCD The QCD expression of the two-point correlator for the Σ * Q (Qqq) has been (first) obtained in the chiral limit m u,d = 0, to LO in α s and up to the contributions of the D = 6 condensates in [4].In this letter, we extend the previous analysis by including the new SU (3) breaking m s correction terms and consider the SU (3) breaking of the ratio of quark condensates ss = qq like we did for the spin 1/2 case.
T ¯he Ω * Q (Qss) baryons Compared with the expression of the Σ * Q (Qqq) in [5], the additionnal SU (3) breaking terms for the Ω * Q are: -F 1 : -F 2 : We have checked the existing results in [5] obtained in the chiral and SU (2) limits and agree with these ones.

Form of the sum rules and QCD inputs
We parametrize the spectral function using the standard duality ansatz: "one resonance"+ "QCD continuum".The QCD continuum starts from a threshold t c and comes from the discontinuity of the QCD diagrams.
Transferring its contribution to the QCD side of the sum rule, one obtains the finite energy Laplace/Borel sum rules: where τ ≡ 1/M 2 is the sum rule variable and λ B ( * ) q and M B ( * ) q are the heavy baryon residue and mass.Notice that, within our choice of the interpolating currents, we may have a contamination due to the negative parity states contribution, which we can quantify from the ratio of sum rules associated to ( [7].We shall check (a posteriori) that this ratio is negligible at the stability regions.On the other, the observed negative parity states are systematically heavier than the positive ones such that they can be legitimately included into the QCD continuum contribution.Consistently, we also take into account the SU (3) breaking at the quark and continuum threshold: where q 1,2 ≡ q or/and s depending on the channel.mqi are the running light quark masses.m Q is the heavy quark mass, which we shall take in the range covered by the running and on-shell mass (see Table 1) because of its ambiguous definition when working to lowest order of perturbative QCD.One can estimate the baryon masses from the following ratios: where at the τ -stability point : These quantities have been used in the literature for getting the baryon masses and lead to a typical uncertainty of 15-20% [4,5,6] 2 .In order to circumvent these problems, we work with the double ratio of sum rules (DR) [23]: which take directly into account the SU (3) breaking effects.These quantities are obviously less sensitive to the choice of the heavy quark masses, to the perturbative radiative corrections and to the value of the continuum threshold than the simple ratios R i and R 21 2 3 .Analogous DR quantities have been used successfully (for the first time) in [23] for studying the mass ratio of the 0 ++ /0 −+ and 1 ++ /1 −− B-mesons, in [24] for extracting f Bs /f B , in [25] for estimating the D → K/D → π semi-leptonic form factors and in [26] for extracting the strange quark mass from the e + e − → I = 1, 0 data.For the numerical analysis whe shall introduce the RGI quantities μ and mq [27]: 2 More accurate results quoted in the recent QSSR literature [14,15] do not take into account the uncertainties due to the heavy quark mass definitions and to the arbitrary choice of the baryonic interpolating currents. 3One may also work with the double ratio of moments Mn based on different derivatives at q 2 = 0 [23].However, in this case the OPE is expressed as an expansion in 1/m Q , which for a LO expression of the QCD correlator is more affected by the definition of the heavy quark mass to be used. where ) is the first coefficient of the β function for n flavours.We have used the quark mass and condensate anomalous dimensions reviewed in [1].We shall use the QCD parameters in Table 1.At the scale where we shall work, and using the paramaters in the table, we deduce: which controls the deviation from the factorization of the four-quark condensates.We shall not include the 1/q 2 term discussed in [28,29],which is consistent with the LO approximation used here as the latter has been motivated for a phenomenological parametrization of the larger order terms of the QCD series.5. κ ≡ ss / dd from the spin 1/2 baryons As a preliminary step of the analysis, we check the different results obtained in full QCD and in the chiral limit [4,5]: which we confirm.However, we have not tried to improve these results for the absolute values of the masses due to the ambiguity in the definition of the heavy quark mass input mentioned earlier at LO, which induces large errors.
¯Ξc (csq)/Λ c (cqq) -Optimal choice of the currents: we start from the general choice of interpolating currents given in Eq. ( 1).We study the b-behaviour of the predictions in Fig. 1a), by fixing τ =0.35 GeV −2 and t c = 15 GeV 2 .The result presents b-stability around b = 0, which is in the range given in Eq. ( 3) obtained from light baryons and heavy non-strange baryons.We consider this value as the optimal choice of the interpolating currents.However, this generous range does not favour the ad hoc choice around 1 used in the existing literature [14,15].Therefore, in this channel, we shall work with: τ stabilities: we show in Fig. 1b) the τ behaviour of the different DR at fixed tc = 15 GeV −2 and b = 0.
t c stabilities and choice of the sum rules: fixing b = 0 from the previous analysis, we study in Fig. 1c) the t cbehaviour of predictions.Among the three sum rules, we retain r sd 21 which is the most stable in t c and then less affected by the higher state contributions.
-Results: From this r sd 21 sum rule, we can deduce the DR: where the value κ = 0.7 has been taken.We have considered the mean value of r sd 21 from t c = 10 to 20 GeV 2 .The errors are due respectively to the values of t c , τ = (0.35 ± 0.05) GeV −2 , m c , m s and the factorization of the four-quark condensate ρ.The errors due to b and some other SU(3) symmetric QCD parameters are negligible.The large error due to κ compiled in Table 1 is not included in Eqs.(27).Using as input the data [9]: and adding the different errors quadratically, one can deduce: which agrees nicely with the data [9]: M exp Ξc = (2467.9± 0.4) MeV .
For improving the existing value of κ , we allow a 1σ deviation of the DR prediction from the experimental value.In this way, we deduce: ¯Ξb (bsq)/Λ b (bqq) We repeat the previous analysis in the case of the bquark.The analysis of the ratio of sum rules shows similar curves than for the charm case except the obvious change of scale.Using r sd 21 , we illustrate the analysis using κ = 0.74.In this way, we obtain: where the errors come from t c from 45 to 80 GeV 2 , τ = (0.18±0.05)GeV −2 , m b , m s and the factorization of the four-quark condensate ρ.From this result, and using: we deduce: which agrees quite well with the data [9]: Allowing a 1σ deviation from the data, we deduce: ¯Ωc (css)/Σ c (cqq) We do an analysis similar to the one in the previous section.The result for the c-quark is shown in Fig 2 .One can notice that the optimal choice of the current is the same as in Eq. ( 2) which we fix to the value b=0.

One can notice from Fig 2a) and Fig 2b)
, that only r sd 2 presents simultaneously τ and b stabilities from which we extract the optimal result.Using κ = 0.78, the final result from r sd 2 is: The errors are due respectively to the values of τ = (1.0 ∼ 1.2) GeV −2 , m c , m s and the factorization of the four-quark condensate ρ.The one due to t c from 10 to 20 GeV 2 is negligible.Using this previous result together with the experimental averaged value [9]: one can deduce: in good agreement with the data: Now, we study the influence of κ on the mass prediction.Allowing a 1σ deviation from the experimental mass, we deduce the estimate:  ¯Final value of κ Taking the (arithmetic) mean value of κ ≡ ss / dd from the different channels Ξ c,b [Eqs.(31) and (36)] and Ω c [Eq. ( 41)], we deduce: which we can consider as an improved estimate of this quantity compared with the existing one from the light mesons [31,1] and baryons [20]: κ = 0.700(100) . (43)

The mass of the Ω b (bss)
We repeat the previous analysis of the Ω c (css) in the case of the b-quark.The curves present the same qualitative behaviour as in the case of the charm, where, only r ds 2 survives the different tests of stabilities.The optimal value is taken at the extremum τ = (0.25 ± 0.05) GeV −2 and in the t c stability region.Then, we obtain: The errors are due respectively to the values of τ , m b , m s and the factorization of the four-quark condensate ρ.The last error is due to κ.The one due to t c in the t c stability region is negligible.Using this value together with the experimental averaged value [9]: one can deduce the result:  which we compile in Table 2.This result agrees within the errors with the one from the CDF collaboration [37]: 6054.4(6.9)MeV but disagrees by about 2.4 σ with the D0 value [38]: M D0 Ω b = 6165.0(13.0)MeV given in the same table.

The mass of the Ξ ′
c,b (Qsq) We do a similar analysis for the Ξ ′ c , which we show in Fig. 4, where we have only retained the r ds 2 which satisfies all stability tests.We obtain: The errors are due respectively to the values of τ = (0.9 ± 0.1) GeV −2 , m c , m s , ρ, b = −(0.4± 0.2), and κ.Using the experimental value of M Σc , we obtain: which we compile in Table 2. Our prediction is in good agreement (1σ) with the data [9]: A similar analysis is done for Ξ ′ b , which is sumarized in Fig. 5.One can notice that here the b-stability occurs at 0 in this channel and there is no sharp selection between r ds 1 and r ds 2 .We obtain the mean from r ds 1 and r ds 2 : r sd where the sources of the errors are τ = (0.5 ± 0.1)  error comes from the choice of the sum rules.Using the experimental value of M Σ b , we predict: which we compile in Table 2.

The masses of the spin 3/2 baryons
As a preliminary step of the analysis, we check the different results obtained in [5]: and confirm them.However, like in the spin 1/2 case, we have not tried to improve these (old) results.
¯Ξ * c (csq)/Σ * c (cqq) We repeat the previous DR analysis for the case of the Ξ * c .We show in Fig. 6a) the τ -behaviour of the mass predictions for t c = 14 GeV 2 .From this analysis, we do not retain r sd 21 which differs completely from r ds 1 and r ds 2 , while we do not consider r ds 1 which is τ -instable.We show in Fig. 6b) the t c -behaviour of r ds 2 given τ =0.9 GeV −2 .We deduce the optimal value: The errors are due respectively to the values of τ = (0.9 ± 0.1) GeV −2 , m c , m s , ρ (factorization of the fourquark condensate) and κ ≡ ss / dd = 0.74±0.03.The ones due to some other parameters are negligible.Using the data [9]: and adding the different errors quadratically, we deduce the results in Table 2.
¯Ξ * b (bsq)/Σ * b (bqq) We extend the analysis to the case of the bottom quark.The curves are qualitatively similar to the charm case.We deduce: The sources of the errors are the same as for the Ξ * c , where here τ = (0.25 ± 0.05) GeV −2 and m c replaced by m b .The ones due to some other parameters are negligible.Using the averaged data [9]: and adding the different errors quadratically, we deduce: which we report in Table 2.
¯Ω * c (css)/Σ * c (cqq) We pursue the analysis to the case of the Ω * c (css).We show the τ -behaviour of the different DR in Fig. 7a).From this figure, we shall not retain r ds 21 which differs  2 for a given optimal τ = 0.9 GeV −2 .
completely from r ds 1 and r ds 2 .Given the optimal value of τ = 1 GeV −2 , we show the t c -behaviour of the DR r ds 1 and r ds 2 in Fig. 7b).The final result is the mean from r ds 1 and r ds 2 : The 1st error is due to the choice of r sd i .The other ones are due to τ = (1.0 ± 0.2) GeV −2 , m c , m s , ρ and κ.The other QCD parameters give negligible errors.Using the averaged data in Eq. ( 56), and adding the different errors quadratically, one can deduce the result in Table 2.

¯Ω *
b (bss)/Σ * b (bqq) We repeat the previous analysis in the b-channel.The curves are qualitatively analogue to the ones of the charm.We shall not consider r sd 21 because of its incompatibility with the other ones.From the mean of r ds 1 and r ds 2 , we deduce: where the sources of the errors are the same as for Ω * c , where τ = (0.30 ± 0.05) GeV −2 here and m c replaced by m b .Using the averaged data in Eq. (56), and adding the different errors quadratically, one can deduce: which we report in Table 2.

Hyperfine mass-spilltings
Combining the results for spin 1/2 and spin 3/2 given in Table 2, we deduce in  Table 2 QSSR predictions of the strange heavy baryon masses in units of MeV from the double ratio (DR) of sum rules with the QCD input parameters in Table 1 and using as input the observed masses of the associated non-strange heavy baryons.We have used κ ≡ ss / dd = 0.74±0.03fixed from the experimental Ξ c,b and Ωc masses.
agree within the errors with the ones from quark models [11] and 1/N c expansion [19] and seems to behave like 1/m b despite the large errors.On the other hand, we predict with a better precision and Ω * b will shed light on the quark mass behaviour of these massdifferences, which we also plan to study in a future work.

Summary and Conclusions
We have directly extracted (for the first time) the heavy baryons (charmed C = 1 and bottom B = −1) masssplittings due to SU (3) breaking using double ratios (DR) of QCD spectral sum rules(QSSR), which are less sensitive to the exact value and the definition of the Table 3 QSSR predictions of the strange heavy baryon hyperfine splittings in units of MeV from the double ratio (DR) of sum rules with the QCD input parameters in Table 1 and using as input the predicted values in Table 2.We have added the errors quadratically.heavy quark mass, to the perturbative radiative corrections and to the QCD continuum contributions than the simple ratios commonly used in the current literature for determining the heavy baryon masses: R ¯emarking that the leading term controlling the masssplittings is, in most of the cases, the ratio κ ≡ ss / dd of the condensate rather than the running mass ms , we use as input the observed masses of the Ξ c,b and Ω c , for extracting κ.We obtain the mean value from Eqs. ( 31), ( 36) and (41): κ = 0.738 (29) [Eq.( 42)] ,

Hyperfine
which we can consider as an improved estimate of this quantity compared with the existing one κ = 0.7 ± 0.1 compiled in Table 1 from the light mesons [31,1] and baryons [20].U ¯sing this value of κ, we give predictions of the Ξ ′ c,b , Ω b and spin 3/2 baryons masses which are summarized in Table 2.These predictions are in good agreement with the experimental masses and can be considered as improvements of existing QSSR results based on the simple ratio of moments [14,15].O ¯ur result for the Ω b favours the one observed by CDF [37] but disagrees within 2.4σ with the one from D0 [38].M ¯ost of predictions agree with the ones from different approaches (quark models [11], lattice calculations [18] and large N c [19]).Our predictions for the not yet observed states Ξ ′ b , Ξ * b and Ω * b given in Table 2 can serve in a near future as a test of the QSSR approach.W ¯e show in Table 3 our predictions for the hyperfine splittings.Our results agree with the observed values, while the ones for not yet observed states can serve as a test of the QSSR approach.From our results, we expect that the Ω * Q can only decay electromagnetically to Ω Q + γ due to the available phase space, while the Ξ * Q can, in addition, decay hadronically to Ξ Q + π.O ¯ne can also notice that, if we have used a SU (3) symmetric quark condensates ss ≃ dd , the predictions would be systematically lower by about (70 ∼ 100) MeV than the predictions given in Table 2.In this case, the agreement with the observed masses in different channels cannot be achieved.It would be interesting to understand analogous effects of κ using some other approaches.

Table 1
[1] input parameters.For the heavy quark masses, we use the range spanned by the running MS mass mQ(MQ) and the on-shell mass from QSSR compiled in page 602,603 of the book in[1].

Table 3
the values of the hyperfine mass-spilittings.From our analysis, one expects that the Ω * Q can only decay electromagnetically to Ω Q + γ due to phase space, while the Ξ * Q can, in addition, decay hadronically to Ξ Q + π.On one hand, our result for the