Mass-splittings of doubly heavy baryons in QCD
aa r X i v : . [ h e p - ph ] S e p Mass-splittings of doubly heavy baryons in QCD
R.M. Albuquerque a,b,1 , S. Narison b, ∗ a Instituto de F´ısica, Universidade de S˜ao Paulo, C.P. 66318, 05389-970 S˜ao Paulo, SP, Brazil. b Laboratoire de Physique Th´eorique et Astroparticules, CNRS-IN2P3 & Universit´e de Montpellier II,Case 070, Place Eug`ene Bataillon, 34095 - Montpellier Cedex 05, France.
Abstract
We consider (for the first time) the ratios of doubly heavy baryon masses (spin 3 / / α s corrections induced by the anomalous dimensions of the correlators are the main sources of the Ξ ∗ QQ − Ξ QQ mass-splittings,which seem to indicate a 1 / M Q behaviour and can only allow the electromagnetic decay Ξ ∗ QQ → Ξ QQ + γ but not to Ξ QQ + π . Ourresults also show that the SU(3) mass-splittings are (almost) independent of the spin of the baryons and behave approximately like1 / M Q , which could be understood from the QCD expressions of the corresponding two-point correlator. Our results can improvedby including radiative corrections to the SU(3) breaking terms and can be tested, in the near future, at Tevatron and LHCb. Keywords:
QCD spectral sum rules, baryon spectroscopy, heavy quarks.
1. Introduction
In a previous paper [1], we have considered, using double ra-tios [2–6] of QCD spectral sum rules (QSSR) [7, 8] (DRSR),the splittings due to SU(3) breakings of the baryons made withone heavy quark. In this paper, we pursue this project in thecase of doubly heavy baryons. The absolute values of the dou-bly heavy baryon masses of spin 1 / Ξ QQ ≡ QQu ) and spin3 / Ξ ∗ QQ ≡ QQu ) have been obtained using QCD spectral sumrules (QSSR) (for the first time) in [9] with the results in GeV: M Ξ ∗ cc (3 / = . , M Ξ ∗ bb (3 / = . . , M Ξ cc (1 / = . , M Ξ bb (3 / = . , (1)and in [10]: M Ξ bcu = . . (2)More recently [11, 12], some results have been obtained us-ing some particular choices of the interpolating currents. Thepredictions for M Ξ ∗ cc and M Ξ cc are in good agreement with theexperimental candidate M Ξ cc = . / / ∗ Corresponding author
Email addresses: [email protected] (R.M. Albuquerque), [email protected] (S. Narison) FAPESP thesis fellow within the France-Brazil bilateral exchange pro-gram.
2. The interpolating currents and the two-point correlatorFor the spin 1 / QQq baryons ¯ , and following Ref. [9], wework with the lowest dimension currents: J Ξ Q = ǫ αβλ h ( Q T α C γ q β ) + b ( Q T α Cq β ) γ i Q λ , (3)where q ≡ d , s are light quark fields, Q ≡ c , b are heavy quarkfields, b is a priori an arbitrary mixing parameter. Using the b -stability criterion of the QSSR results for the masses and cou-plings, the optimal values of these observables have been foundfor: b = − / , (4)in the case of light baryons [17] and in the range [1, 18–20]: − . ≤ b ≤ . , (5)for non-strange heavy baryons . The corresponding two-pointcorrelator reads: S ( q ) = i Z d x e iqx h |T J Ξ Q ( x ) J Ξ Q (0) | i≡ ˆ qF + F , (6)where : ˆ q ≡ / q . The invariants F j ( j = ,
2) obey the dispersionrelation: F j ( q ) = Z ∞ (2 M Q + mq ) dtt − q − i ǫ π Im F j ( t ) + . . . , (7)where . . . indicate subtraction constants and − q ≡ Q > For the spin 3 / QQq baryons ¯ , we also follow Ref. [9] andwork with the interpolating currents: J µ Ξ ∗ Q = r ǫ αβλ h Q T α C γ µ d β ) Q λ + ( Q T α C γ µ Q β ) q λ i (8) We use the notation in the Landau and Lifchitz’s book.
Preprint submitted to Physics Letters B October 24, 2018 he corresponding two-point correlator reads: S µν ( q ) = i Z d x e iqx h |T J µ Ξ ∗ Q ( x ) J ν Ξ ∗ Q (0) | i≡ g µν ( ˆ qF + F ) + . . . , (9)
3. The two-point correlator in QCD
The expressions of the two-point correlator using the previ-ous interpolating currents have been obtained in the chiral limit m q = h ¯ ss i condensate contributions. We shall use the same normalizationsas in [9]. For the spin 1 / ¯ , these corrections read:Im F m s | pert = m s m Q π (1 − b ) (cid:20) L v (2 x − + v x + + x ! (cid:21) , Im F m s | pert = m s m Q π L v (cid:20) x (5 b + b + − b + b + (cid:21) + v (cid:20) b + b + + x (5 b + b + + x (1 − b ) (cid:21) , Im F m s | ¯ ss = m s h ¯ ss i π v (cid:20) b + b + + (cid:16) b + b + (cid:17) x (cid:21) + v (1 + b ) , Im F m s | ¯ ss = m s m Q h ¯ ss i π (cid:16) − b (cid:17) v + v ! . (10) For the spin 3 / ¯ , these corrections read:Im F m s | pert = m s m Q π x (cid:20) L v (cid:16) x − (cid:17) x + v (6 x + x + (cid:21) , Im F m s | pert = m s m Q π x (cid:20) L v ( x + x − x + v (12 x + x + x + (cid:21) , Im F m s | ¯ ss = − m s h ¯ ss i π " v (4 x − − v , Im F m s | ¯ ss = − m s m Q h ¯ ss i π v (2 x − , (11)with: x ≡ m Q t , v ≡ √ − x , L v ≡ log + v − v ! , (12)
4. Form of the sum rules and QCD inputs
We shall work with the exponential sum rules [7, 24, 25]: F i ( τ ) = Z ∞ t q dt e − t τ π Im F i ( t ) , ( i = , , (13)( τ ≡ / M is the sum rule variable) from which, one can derivethe following ratios: R qi ( τ ) ≡ − dd τ ln F i = R ∞ t q dt t e − t τ Im F i ( t ) R ∞ t q dt e − t τ Im F i ( t ) , ( i = , , R q ( τ ) ≡ F F = R ∞ t q dt e − t τ Im F ( t ) R ∞ t q dt e − t τ Im F ( t ) , (14)used in the sum rule literature for extracting the baryon masses.We parametrize the spectral function using the standard dualityansatz: “one resonance” + “QCD continuum”. The QCD con-tinuum starts from a threshold t c and comes from the disconti-nuity of the QCD diagrams, which is consistent with a matchingof the QCD and the experimental sides of the sum rules for large t . The value of t c is not arbitrary as its value obtained inside theregion of t c -stablity of the sum rule is correlated to the groundstate mass and coupling [21]. This simple duality model hasbeen successfully tested in the literature when a complete datafor the spectral functions are available, like e.g., e + e − to I = . Transferring the QCDcontinuum contribution to the QCD side of the sum rules, oneobtains the finite energy inverse Laplace sum rules: | λ B ( ∗ ) q | M B ( ∗ ) q e − M B ( ∗ ) q τ = Z t c t q dt e − t τ π Im F ( t ) , | λ B ∗ q | e − M B ( ∗ ) q τ = Z t c t q dt e − t τ π Im F ( t ) , (15)where λ B ( ∗ ) q and M B ( ∗ ) q are the heavy baryon residue and massfrom which one can derive the FESR analogue of the ratios ofsum rules. Consistently, we also take into account the SU(3)breaking at the continuum threshold : √ t c | S U (3) ≃ (cid:16) √ t c | S U (2) ≡ √ t c (cid:17) + ¯ m s . (16)¯ m s is the running strange quark mass. As we do an expansionin m s , we take the threshold t q = m Q for consistency. m Q is theheavy quark mass, which we shall take in the range covered by More involved parametrizations of the continuum can also be proposed(see e.g. [22] for non-resonant final states within ChPT or [23] for a t-dependent t c model.). However, it is easy to check in the harmonic oscillator model dis-cussed in [23] that the 5% uncertainties induced e.g. by a t -dependent contin-uum model on the ratio of moments will be negligible in the double ratio ofsum rule (DRSR) defined in Eq. (20) as the leading corrections coming fromlight-quark flavour-independent terms will largely cancel out. This cancella-tion of the QCD continuum contribution will be signaled by the large range of t c -stability region obtained in our analysis. As we have done an expansion in terms of m s , the quark threshold has tobe taken at 4 m Q but not at (2 m Q + m s ) . τ -stability pointof the FESR analogue of the ratios of sum rules, one obtains: M B ( ∗ ) q ≃ q R qi ≃ R q , ( i = , . (17)These predictions lead to a typical uncertainty of 10-15% [9,19, 20], which are not competitive compared with predictionsfrom some other approaches, especially from potential models[14]. In order to improve the QSSR predictions, we work withthe double ratios of finite energy sum rules (DRSR) : r sdi ≡ s R si R di ( i = ,
2) ; r sd ≡ R s R d . (18)which take directly into account the SU(3) breaking e ff ects.These quantities are obviously less sensitive to the choice of theheavy quark masses and to the value of the continuum thresholdthan the simple ratios R i and R
21 6 . For the numerical analysiswe shall introduce the RGI quantities ˆ µ and ˆ m q [26]:¯ m q ( τ ) = ˆ m q (cid:16) − log √ τ Λ (cid:17) / − β h ¯ qq i ( τ ) = − ˆ µ q (cid:16) − log √ τ Λ (cid:17) / − β h ¯ qGq i ( τ ) = − ˆ µ q (cid:16) − log √ τ Λ (cid:17) / − β M , (19)where β = − (1 / − n /
3) is the first coe ffi cient of the β function for n flavours. We have used the quark mass andcondensate anomalous dimensions reviewed in [8]. We shalluse the QCD parameters in Table 1:– We shall not include the 1 / q term discussed in [27, 28],whichis consistent with the LO approximation used here as the latterhas been motivated for a phenomenological parametrization ofthe larger order terms of the QCD series.– We have used the value of κ ≡ h ¯ ss i / h ¯ dd i from [1] whichwe consider as improvements of the ones from light mesonsystems [8, 29–31], where the one from the scalar channelsu ff ers from the unknown nature of the κ meson, while theone from the pseudoscalar channel depends on the theoreticalappreciation of the π ′ meson contribution into the spectralfunction [8, 22]. However, the di ff erent estimates agree eachothers within the errors. To be conservative, we have multipliedthe original error in [1] by 2.– For the gluon condensate, we have used the estimate fromheavy quarkonia and e + e − data [21, 24, 32–37]. We donot expect that the estimate from τ -decays is reliable as itscontribution acquires an extra- α s term in the τ width compared Analogous DRSR quantities have been used successfully (for the first time)in [3] for studying the mass ratio of the 0 ++ / − + and 1 ++ / −− B-mesons, in [4]for extracting f B s / f B , in [5] for estimating the D → K / D → π semi-leptonicform factors and in [6] for extracting the strange quark mass from the e + e − → I = , One may also work with the double ratio of moments M n based on di ff er-ent derivatives at q = / m Q , which for a LO expression of the QCD correlator is morea ff ected by the definition of the heavy quark mass to be used. to the one in the two-point correlator [38], while its value,in this process, can also be a ff ected by the treatments of thelarge order PT series [27, 32]. However, the e ff ect of thegluon condensate is not important in our analysis of the SU(3)breaking as it disappears like some other flavour-independentcontributions in the DRSR.– For the heavy quark masses, we use the range spanned by therunning MS mass m Q ( M Q ) and the on-shell mass from QSSRcompiled in page 602, 603 of the book in [8]. Table 1:
QCD input parameters. The values of Λ , ˆ m s and µ d have been obtained from α s ( M τ ) = . m s (2) = . .
8) MeV and m d (2) = . κ and h α s G i have been multiplied by 2. Parameters Values Ref. Λ ( n f =
4) (324 ±
15) MeV [13, 32] Λ ( n f =
5) (194 ±
10) MeV [13, 32]ˆ m s (114 . ± .
8) MeV [6, 8, 13, 29]ˆ µ d (263 ±
7) MeV [8, 29] κ ≡ h ¯ ss i / h ¯ dd i (0 . ± .
06) [1, 8, 29, 31] M (0 . ± .
1) GeV [3, 17, 39] h α s G i (6 ± × − GeV [21, 24, 32–37] m c (1 . ∼ .
47) GeV [8, 13, 29, 37, 40] m b (4 . ∼ .
72) GeV [8, 13, 29, 37, 40]
5. The Ξ ∗ QQ / Ξ QQ mass ratio -0.4 -0.2 0 0.2 0.40.90.9511.051.11.151.21.25 Figure 1:
Charm quark: b -behaviour of the di ff erent DRSR given τ = . − and t c =
25 GeV . r / dot-dashed line (red); r / continuous line(green); r / dashed line (blue). We have used m c = .
26 GeV and the otherQCD parameters in Table 1.
We extract the mass ratio using the DRSR analogue of the onein Eq. (18) which we denote by: r / i ≡ s R i R i : i = , r / ≡ R R , (20)where the upper indices 3 and 1 correspond respectively to thespin 3 / / / v (where v is the heavy quark ve-locity), which signals a coulombic correction and would requirea complete treatment of the non-relativistic coulombic correc-tions which is beyond the aim of this paper. Therefore, in ouranalysis, we truncate the QCD series at the dimension-4 con-densates until which we have calculated the m s corrections. We3hall only include the e ff ect of the mixed condensate (if nec-essary) for controlling the accuracy of the approach or for im-proving the τ or / and t c stability of the analysis. T¯ he charm quark channel to lowest order in α s Fixing τ = − and t c =
25 GeV , which are insidethe τ - and t c -stability regions (see Figs. 2 and 3), we show inFig. 1 the b -behaviour of r / which shows that r / and r / arevery stable but not r / . We then disfavour r / . Some commonsolutions are obtained for: b ≃ − . , and b ≃ + . , (21)which are inside the range given in Eq. (5). For definiteness,we fix b = − .
35 (the other value b = . τ -dependence of the result in Fig. 2. We havechecked in Fig. 2b that the inclusion of the mixed condensatecontribution does not a ff ect the result from r / i ( i = ,
2) ob-tained by retaining only the dimension-4 condensates (Fig. 2a)but a ff ects the one from r / . Therefore, we shall only retain theresults from r / i ( i = ,
2) and show their t c -dependence in Fig.3. The large stability in t c confirms our expectation of the weak Figure 2:
Charm quark: a) τ -behaviour of r / : dot-dashed line (red), r / :continuous line (green) and r / dashed line (blue) with b = − .
35 and t c = . b) the same as a) but when the mixed condensate is included.
15 20 25 30 35 40 45 500.99920.99940.99960.9998
Figure 3:
Charm quark: t c -behaviour of r / : dot-dashed line (red) and r / :continuous line (green) with b = − .
35 and τ = . − . t c -dependence of the DRSR and then on the non-sensitivity ofthe results on the exact form of the QCD continuum includingan eventual slight t -dependence of t c advocated in [23]. In thesefigures, we have used m c = .
26 GeV. We have also checkedthat the results are insensitve to the change of the charm mass to m c = .
47 GeV. From these previous analysis, we deduce tolowest order from r / i ( i = , M Ξ ∗ cc M Ξ cc = . . (22)The tiny error is the quadratic sum due to h α s G i , m c and α s . T¯ he bottom quark channel to lowest order in α s We extend the analysis to the case of the bottom quark.
Figure 4:
Bottom quark: a) τ -behaviour of r / : dot-dashed line (red), r / :continuous line (green)and r / : dashed line (blue) with b = − .
35 and t c = ; b) the same as a) but when the mixed condensate is included into theOPE.
80 100 120 140 160 180 2000.9999920.9999950.999997111.000011.00001
Figure 5:
Bottom quark: t c -behaviour of r / : dot-dashed line (red) and r / :continuous line (green) with b = − .
35 and τ = . − . The corresponding curves are qualitatively similar to the charmquark one. We take b = − .
35 like in the case of the charmquark. The τ -stability is reached for τ ≥ . − as shown inFig. 4, where we also see that r / is more a ff ected by the mixedcondensate contributions than r / i . Therefore, we shall elimi-nate it from our choice. Another argument raised later aboutthe radiative corrections does not also favour r / . In Fig. 4,we study the t c -stability of r / i which is reached for t c ≥ . Within these optimal conditions, one deduces from r / i to lowest order: M Ξ ∗ bb M Ξ bb = . . (23)4 ¯ stimate of the O ( α s ) corrections Radiative corrections due to α s are known to be large in thebaryon two-point correlators [17, 41]. However, one can easilyinspect that in the simple ratios R i and R i these huge correc-tions cancel out, while its only remain the one induced by theanomalous dimension of the baryon operators. Including theanomalous dimension γ = /
3) for the spin 1 / / t / m Q , which is a crude approximation but very informative: F i ( τ ) | pert ≈ ( α s ( τ )) − γβ A i τ − (cid:18) + K i α s π (cid:19) , (24)where β is the first coe ffi cient of the β -function; A i is a knownLO expression; K i is the radiative correction which is knownin some cases of light and heavy baryons [17, 41]. From theprevious expression in Eq. (24), one can derive the ratio of sumrules defined in Eq. (14) and then the DRSR in Eq. (18): r / i | NLOpert ≃ r / i | LOpert × + α s π + O (cid:16) α s , M Q τ (cid:17) . (25) Figure 6:
Charm quark: a) τ -behaviour of r / : dot-dashed line (red) and r / : continuous line (green) with b = − . t c =
25 GeV where radiativecorrections have been included. Bottom quark: b) the same as in a) but for thebottom quark. We use b = − .
35 and t c =
100 GeV It is important to notice for r / i that the radiative correction hasbeen only induced by the ones due to the anomalous dimen-sions, while the one due to K i cancels out to this order. This isnot the case of r / where the radiative correction is only due to K − K and needs to be evaluated which is beyond the aim ofthis letter. Therefore, in the following, we shall only considerthe results from r / i . In our numerical analysis, we shall in-clude the α s correction into the complete LO expressions of thecorrelators. We show the τ -dependence of the DRSR in Fig. 6.We shall take the range of τ -values where the LO expressionshave τ -stability which is (0.7-1) GeV − for charm and (0.5-0.8)GeV − for bottom (see Figs. 2 and 4). One can also notice thatthe NLO DRSR for charm presents a τ -extremum in the aboverange (0.7-1) GeV − of τ rendering its prediction more reliablethan for the bottom channel case. We can deduce : M Ξ ∗ cc M Ξ cc = . α s (16) m c , M Ξ ∗ bb M Ξ bb = . α s (2) m b . (26) This would correspond to the mass-splittings (in units of MeV): M Ξ ∗ cc − M Ξ cc = , M Ξ ∗ bb − M Ξ bb = , (27)if one uses the experimental value 3.52 GeV of the Ξ cc masswhich agrees with the QSSR prediction in Eq. (1). For the Ξ bb mass, we have used the central value 9.94 GeV in Eq. (1) . The ccq mass-splitting is comparable with the one of about 70 MeVfrom potential models [10, 14] but larger than the one of about24 MeV obtained in [16] . The bbq mass-splitting also agreeswith potential models and seems to indicate a 1 / M b behaviourwhich is also seen on the lattice [42]. Our result excludes thepossibility that M Ξ ∗ QQ ≥ M Ξ QQ + m π , indicating that it can onlydecay electromagnetically: M Ξ ∗ QQ → M Ξ QQ γ , M Ξ ∗ QQ M Ξ QQ π . (28)A future discovery of the Ξ ∗ cc and Ξ ∗ bb can infirm or support ourpredictions given to that order of QCD perturbative series. Weconsider the previous results as an improvement of the formerones deduced from the mass values in Eq. (1) obtained by [9]: M Ξ ∗ cc M Ξ cc ≃ . ± . , M Ξ ∗ bb M Ξ bb = . ± . . (29)
6. The Ω QQ / Ξ QQ mass ratio We use the DRSR in Eq. (18) where their QCD expressions
Figure 7: Ω cc / Ξ cc : a) τ -behaviour of r sd ( cc ): continuous line (green) and r sd ( cc ): dot-dashed line (red) in the charm quark channel for b = − . t c = and m c = .
26 GeV. b) t c -behaviour of r sd ( cc ) for τ = − : dot-dashed line (red) can be obtained from the one of the two-point correlator in [9]and the new quark mass corrections in Eq. (10). One can alsodeduce from Eq. (24) that the light-flavour independent radia-tive corrections including the one due to the anomalous dimen-sions disappear in the SU(3) breaking DRSR, while the mostrelevant radiative corrections are the one corresponding to the m s and h ¯ ss i terms which are beyond the scope of the LO anal-ysis in this paper. We show in Fig. 7a the τ -behaviour of theDRSR for m c = .
26 GeV and b = − .
35 for a given t c = . We have not shown r sd ( cc ) which is the lesser stable5 Figure 8: Ω bb / Ξ bb : a) τ -behaviour of r sd ( bb ): dot-dashed line (red) in thebottom quark channel for b = − . t c =
100 GeV and m b = .
22 GeV. b) t c -behaviour of r sd ( bb ) for τ = . − : dot-dashed line (green) among the three. We see that the most stable result is given by r sd ( cc ). We show in Fig. 7b the t c -behaviour of r sd ( cc ) for agiven τ = − . We deduce from the previous analysis: r sd ( cc ) ≡ M Ω cc M Ξ cc = . m c (2) ¯ ss (4) m s , (30)where the sub-indices indicate the di ff erent sources of errors(the parameters not mentioned induce negligible errors). Thisratio corresponds to : M Ω cc − M Ξ cc = , (31)where we have taken the experimental value M Ξ cc ≃ .
52 GeVfrom [13]. The errors induced by the other parameters in Table1 are negligible. We perform an analogous analysis in the b -channel, which we show in Fig. 8. In this case, we obtain: r sd ( bb ) ≃ . m b (3) ¯ ss (10) m s , (32)which corresponds to: M Ω bb − M Ξ bb = , (33)when we take the value M Ξ bb ≃ / m Q of the mass split-tings from the c to the b quark channels. This behaviour canbe qualitatively understood from the QCD expressions of thecorresponding correlator, where the m s corrections enter like m s / m Q , and which can be checked using some alternative meth-ods.
7. The Ω ∗ QQ / Ξ ∗ QQ mass ratio We pursue our analysis for the spin 3 / r sd and r sd are quite stable versus τ from τ ≥ . Figure 9: Ω ∗ cc / Ξ ∗ cc : a) τ -behaviour of r sd ( cc ) ∗ : dot-dashed line (red) and r sd ( cc ) ∗ : continuous line (green) in the charm quark channel for t c =
20 GeV and m c = .
26 GeV. b) t c -behaviour of r sd ( cc ) ∗ and r sd ( cc ) ∗ for τ = . − Figure 10: Ω ∗ bb / Ξ ∗ bb : a) τ -behaviour of r sd ( bb ) ∗ : dot-dashed line (red) and of r sd ( bc ): continuous line (green) in the bottom quark channel for t c =
100 GeV and m b = .
22 GeV. b) t c -behaviour of r sd ( bb ) ∗ for τ = . − GeV − . In Fig. 9b, we show the t c -behaviour of r sd and r sd given τ . We deduce at the stability regions: r sd ( cc ) ∗ ≡ M Ω ∗ cc M Ξ ∗ cc = . ¯ ss (4) m s (6) m c (1) t c , (34)where the errors coming from other parameters than ¯ ss are neg-ligible. This implies: M Ω ∗ cc − M Ξ ∗ cc = , (35)where we have used M Ξ ∗ cc ≃ .
58 GeV from Eq. (31) and theexperimental value of M Ξ cc . We show in Fig. 10 the analogousanalysis for the bottom channel. We deduce: r sd ( bb ) ∗ ≡ M Ω ∗ bb M Ξ ∗ bb = . ¯ ss (10) m s (4) τ (10) m b , (36)where the error is again mainly due to h ¯ ss i , the others beingnegligible. This implies: M Ω ∗ bb − M Ξ ∗ bb = , (37)6here we have used M Ξ ∗ bb ≃ .
96 GeV using our predictionin the previous section. This result agrees with the potentialmodel one of about 60 MeV given in [10]. Again like in thecase of spin 1 / ff erences appearsto behave like the inverse of the heavy quark masses, whichcan be inspected from the QCD expressions of the two-pointcorrelator. One can also observe that the mass-splittings arealmost the same for the spin 1 / /
8. The Ω bc / Ξ bc mass ratio -0.1 -0.05 0 0.05 0.11.00621.00641.00661.00681.007 Figure 11: Ω bc / Ξ bc : b behaviour of r sd ( bc ): dot-dashed line (red); r sd ( bc ):continuous line (green), and r sd ( bc ): dashed line (blue) for t c =
50 GeV , τ = . − , m c = .
26 GeV and m b = .
22 GeV.
Figure 12: Ω bc / Ξ bc a): τ -behaviour of r sd ( bc ): dot-dashed line (red) and r sd ( bc ): continuous line (green) for k = − . t c =
50 GeV and m c = . t c -behaviour of r sd ( bc ) and r sd ( bc ) for τ = . − and k = − . The Ξ ( bc ) and the Ω ( bc ) spin 1 / J Ξ bc = ǫ αβλ h ( c T α C γ d β ) + k ( c T α Cd β ) γ i b λ , J Ω bc = J Λ bc ( d → s ) , (38)where d , s are light quark fields, c , b are heavy quark fields and k is a priori an arbitrary mixing parameter. The expression ofthe corresponding two-point correlator has been obtained in thechiral limit m d = m s = m s -corrections for the PT and quark condensate contributions. The expressions of these corrections are:Im F m q | pert = − m s m c (1 − k )128 π t L h ( m c t − m b m c ) i + L m c t − λ bc (cid:20) t + (cid:16) m c − m b (cid:17) t − m c + m b m c + m b (cid:21) Im F m s | ¯ ss = m s ß(1 + k )32 π t λ bc [ t + ( m b + m c ) t − m b − m c ) ] + λ bc [( m b + m c ) t − m b + m c )( m b − m c ) t + ( m b − m c ) ] , (39)Im F m s | pert = − m s m c m b (1 + k )64 π t (cid:20) L [( m b + m c ) t − m b m c ] − L (cid:16) m b − m c (cid:17) t − λ bc h s + m b + m c i (cid:21) ImF m s | ¯ ss = m s m b ß(1 − k )16 π t (cid:20) λ bc (cid:16) t − m b + m c (cid:17) + λ bc [ m b t + (cid:16) m c + m b m c − m b (cid:17) t + (cid:16) m b − m c (cid:17) ] (cid:21) , (40)where: v = s − m b m c ( t − m b − m c ) , λ / bc = ( t − m b − m c ) v L =
12 log 1 + v − v L = log ( m b + m c ) t + ( m b − m c )( λ / bc − m b + m c )2 m b m c t . (41)Like in previous sections, we study the di ff erent ratios of mo-ments in Figs. 11 and 12. As one can see in Fig. 11a, r sd ( bc )and r sd ( bc ) are quite stable in k and present common solutionsfor : k = ± . , (42)inside the range given in Eq. (5), while r sd ( bc ) does not inter-sect with the other DRSR. The τ and t c behaviours given in Fig.12a,b are also very stable from which we deduce the DRSR: r sd ( bc ) ≡ M Ω bc M Ξ bc = . . ¯ ss (1 . m s (1) m Q , (43)where the errors coming from other parameters are negligible.This implies: M Ω bc − M Ξ bc = , (44)7here we have used the QSSR central value M Ξ bc ≃ .
86 GeVin Eq. (2). The size of the mass-splitting can be comparedwith the potential model prediction about (70-89) MeV givenin [10, 15].
Table 2:
QSSR predictions for the doubly heavy baryons mass ratios and splittings, whichwe compare with the Potential Model (PM) range of results in [10, 15]. The PM predictionfor the spin 3 / /
2. The mass inputs are in GeV andthe mass-splittings are in MeV.
Mass ratios Mass inputs Mass plittings PM Ξ ∗ cc / Ξ cc = . Ξ cc = . Ξ ∗ cc − Ξ cc = Ξ ∗ bb / Ξ bb = . Ξ bb = . Ξ ∗ bb − Ξ bb = Ω cc / Ξ cc = . Ξ cc = . Ω cc − Ξ cc = Ω bb / Ξ bb = . Ξ bb = . Ω bb − Ξ bb = Ω ∗ cc / Ξ ∗ cc = . Ξ ∗ cc = . ∗ ) Ω ∗ cc − Ξ ∗ cc = Ω ∗ bb / Ξ ∗ bb = . Ξ ∗ bb = . ∗ ) Ω ∗ bb − Ξ ∗ bb = Ω bc / Ξ bc = . Ξ bc = . Ω bc − Ξ bc = ∗ ) We have combined your results for the mass-splittings with the experi-mental value of M Ξ cc and with the central value of M Ξ bb in Eq. (1).
9. Conclusions
Our di ff erent results are summarized in Table 2 and agreein most cases with the potential model predictions given in[10, 14]:– The mass-splittings between the spin 3 / / / M Q .– For the SU(3) mass-splittings, our results, derived in Eqs. (31)and (33) for the spin 1 / /
2, indicate that the splittings due to the SU(3) breaking arealmost independent on the spin of the heavy baryons but ap-proximately behave like 1 / M Q . These mass-behaviours can bequalitatively understood from the QCD expressions of the cor-responding correlators where the leading mass corrections be-have like m s / m Q .– Finally, we obtain, in Eq. (44), the SU(3) mass-splittings be-tween the Ω ( bcs ) and Ξ ( bcd ), which is about 1 / Acknowledgements
We thank Marina Nielsen, Jean-Marc Richard and Valya Za-kharov for some discussions. R.M.A acknowledges the LPTA-Montpellier for the hospitality where this work has been done.This work has been partly supported by the CNRS-IN2P3within the Non-perturbative QCD program in Hadron Physics.
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