Model-independent analysis for determining mass splittings of heavy baryons
aa r X i v : . [ h e p - ph ] M a y Model-independent analysis for determining masssplittings of heavy baryons
Chien-Wen Hwang ∗ Department of Physics, National Kaohsiung Normal University,Kaohsiung, Taiwan 824, Republic of China
Abstract
We study the hyperfine mass differences of heavy hadrons in the heavy quark effect theory(HQET). The effects of one-gluon exchange interaction are considered for the heavy mesons andbaryons. Base on the known experimental data, we predict the masses of some heavy baryons ina model-independent way. ∗ Email: [email protected] . INTRODUCTION It is widely accepted that Quantum Chromodynamics (QCD) is the correct theory forstrong interactions. QCD is a renormalizable quantum field theory which is closely modeledafter quantum electrodynamics (QED), the most accurate physical theory we have to date.However, in the low-energy regime, QCD tells us that the interactions between quarks andgluons are strong, so that quark-gluon dynamics becomes non-perturbative in nature. Un-derstanding the structures of hadrons directly from QCD remains an outstanding problem,and there is no indication that it will be solved in the foreseeable future. In 1989, it wasrealized that, in low energy situations where the typical gluon momenta are small comparedwith the heavy quark mass ( m Q ), QCD dynamics becomes independent of the flavor andspin of the heavy quark [1, 2]. For the heavy flavors, this new symmetry called heavy quarksymmetry (HQS). Of course, even in this infinite heavy quark mass limit, low energy QCDdynamics remains non-perturbative, and what HQS can do for us is to relate otherwiseunrelated static and transition properties of heavy hadrons, and hence enormously reducesthe complexity of theoretical analysis. In other words, HQS allows us to factorize the com-plicated light quark and gluon dynamics from that of the heavy one, and thus provides aclearer physical picture in the study of heavy quark physics. Beyond the symmetry limit,a heavy quark effective theory (HQET) can be developed by systematically expanding theQCD Lagrangian in powers of 1 /m Q , with which HQS breaking effects can be studied orderby order [2, 3, 4].In the experimental area, all masses of s -wave charmed hadrons and bottomed mesonswhich containing one heavy quark are found at present. However, except the particle Λ b wasalready found in the early 1980’s, there has not been significant progress in searching s -wavebottomed baryons until these months. Recently some bottomed baryons were discovered atFermilab. They are the exotic relatives of the proton and neutron Σ ( ∗ )+ b and Σ ( ∗ ) − b by CDFcollaboration [5] and the triple-scoop baryon Ξ − b by D0 and CDF collaborations [6, 7]. It isreasonable that the remainder particles, which include Ξ ′ b , Ξ ∗ b , Ω b , and Ω ∗ b , will be observedin the foreseeable future. All these heavy hadrons provide a testing ground for HQET withthe phenomenological models to the low-energy regime of QCD. In this paper we focus onone static property, that is, the mass spectrum of heavy hadrons and combine HQET withthe known experimental data to predict the mass splitting of some heavy baryons. The2henomenological models are not needed here.The paper is organized as follows. In Sec. II brief introductory notes are given forHQET. In Sec. III we formulate the hyperfine mass splitting for heavy mesons and baryons.In Sec. IV we evaluate the numerical results and predict some mass differences betweenheavy baryons. Finally, the conclusion is given in Sec. V. II. HEAVY QUARK EFFECT THEORY
The full QCD Lagrangian for a heavy quark ( c , b , or t ) is given by L Q = ¯ Q ( iγ µ D µ − m Q ) Q, (1)where D µ ≡ ∂ µ − ig s T a A aµ with T a = λ a /
2. Inside a hadronic bound state containing aheavy quark, the heavy quark Q interacts with the light degrees of freedom by exchanginggluons with momenta of order Λ QCD , which is much smaller than its mass m Q . Consequently,the heavy quark is close to its mass shell, and its velocity does not deviate much from thehadron’s four-velocity v . In other words, the heavy quark’s momentum p Q is close to the“kinetic” momentum m Q v resulting from the hadron’s motion p µQ = m Q v µ + k µ , (2)where k µ is the so-called “residual” momentum and is of order Λ QCD . To describe theproperties of such a system which contains a very heavy quark, it is appropriate to considerthe limit m Q → ∞ with v and k being kept fixed. In this limit, it is evident that the quantity m Q v is “frozen out” from the QCD dynamics, so it is appropriate to introduce the “large”and “small” component fields h v and H v , which is related to the original field Q ( x ) by h v ( x ) = e im Q v · x P + Q ( x ) ,H v ( x ) = e im Q v · x P − Q ( x ) , (3)where P + and P − are the positive and negative energy projection operators respectively: P ± = (1 ± 6 v ) /
2. so that Q ( x ) = e − im Q v · x [ h v ( x ) + H v ( x )] . (4)It is clear that h v annihilates a heavy quark with velocity v , while H v creates a heavyantiquark with velocity v . In the heavy hadron’s rest frame v = (1 , ~ h v ( H v ) correspondto the upper (lower) two components of Q ( x ).3n terms of the new fields, the QCD Lagrangian for a heavy quark given by Eq. (1) takesthe following form L Q = ¯ h v iv · Dh v − ¯ H v ( iv · D + 2 m Q ) H v + ¯ h v i D ⊥ H v + ¯ H v i D ⊥ h v (5)where D µ ⊥ = D µ − v µ v · D is orthogonal to the heavy quark velocity, v · D ⊥ = 0. FromEq. (5), we see that h v describes massless degrees of freedom, whereas H v corresponds tofluctuations with twice the heavy quark mass. The heavy degrees of freedom representedby H v can be eliminated using the equations of motion of QCD. Substituting Eq. (4) into( i D − m Q ) Q ( x ) = 0 gives i Dh v + ( i D − m Q ) H v = 0 . (6)Multiplying this equation by P ± , one obtains − iv · Dh v = i D ⊥ H v , (7)( iv · D + 2 m Q ) H v = i D ⊥ h v . (8)The second equation can be solved schematically to give H v = 1( iv · D + 2 m Q − iǫ ) i D ⊥ h v , (9)which shows that the small component field H v is indeed of order 1 /m Q . One can insert thissolution back into Eq. (7) to obtain the equation of motion for h v . It is easy to check thatthe resulting equation follows from the effective Lagrangian L Q,eff = ¯ h v iv · Dh v + ¯ h v i D ⊥ iv · D + 2 m Q − iǫ ) i D ⊥ h v . (10) L Q,eff is the Lagrangian of the heavy quark effective theory (HQET), and the second termof Eq. (10) allows for a systematic expansion in terms of iD/m Q . Taking into account that P + h v = h v , and using the identity P + i D ⊥ i D ⊥ P + = P + h ( iD ⊥ ) + g s σ αβ G αβ i P + , (11)where G αβ = T a G αβa = ig s [ D α , D β ] (12)4s the gluon field strength tensor, one finds that L Q,eff = ¯ h v iv · Dh v + 12 m Q ¯ h v ( iD ⊥ ) h v + g m Q ¯ h v σ αβ G αβ h v + O ( 1 m Q ) . (13)The new operators at order 1 /m Q are O = 12 m Q ¯ h v ( iD ⊥ ) h v , (14) O = g m Q ¯ h v σ µν G µν h v , (15)where O is the gauge invariant extension of the kinetic energy arising from the off-shellresidual motion of the heavy quark, and O describes the color magnetic interaction of theheavy quark spin with the gluon field. It is clear that both O and O break the flavorsymmetry, while O breaks the spin symmetry as well. For instance, O would introduce acommon shift to the masses of pseudoscalar and vector heavy mesons, and O is responsiblefor the color hyperfine mass splittings δm HF .The full expansion of L Q,eff in iD/m Q can be organized as follow L Q,eff = X n =0 (cid:18) m Q (cid:19) n L n (16)where the first few terms are given by L = ¯ h v ( i v · D ) h v , L = ¯ h v ( iD ⊥ ) h v + g h v σ αβ G αβ h v , L = g ¯ h v σ αβ v γ iD α G βγ h v + g ¯ h v v α iD β G αβ h v . We must emphasize that this effective theory comes from first principle directly, and it isin terms of the power of 1 /m Q , which is small enough, to calculate the physical quantitieswhich concerning heavy quarks perturbatively, i.e., order by order. If one chooses the ap-propriate frame and phenomenological model, then one can handle many physical processessystematically. We also note that, as shown in Eq. (16), the information of heavy quarkflavor is only involved in the factor (1 / m Q ) n . In other words, all L n ’s are independent ofthe heavy quark flavor. We will use this property to evaluate the mass splittings of someheavy hadrons. 5 II. HYPERFINE MASS SPLITTING
First, we consider the hyperfine mass splitting between pseudoscalar and vector heavymesons. The operators that break HQS to order 1 /m Q are O and O given in Eqs. (14) and(15) respectively. O can be separated into a kinetic energy piece and a one-gluon exchangepiece: O = O k + O g (17)where O k ≡ − m Q ¯ h v [ ∂ µ ∂ µ + ( v · ∂ ) ] h v , (18) O g ≡ − g s m Q ¯ h v [( p + p ′ ) µ − v · ( p + p ′ ) v µ ] A µ h v , (19)Also, O can be reexpressed as O = − g s T a σ µν ∂ µ A aν . (20)With the 1 /m Q corrections included, the heavy meson masses can be expressed as M M = m Q + ¯Λ q − m Q ( λ q + d M λ q ) , (21)where λ q comes from O and λ q comes from O . λ q receive two different contributions, onefrom O k and the other from O g , thus λ q = λ k + λ q g . (22)The parameter ¯Λ q in Eq. (21) is the residual mass of heavy mesons in the heavy quarklimit. In other words, ¯Λ q is independent of heavy quark flavor. λ k comes from the heavyquark kinetic energy, λ q g and λ q are respectively the chromoelectric and chromomagneticcontributions. λ q parameterizes the common mass shift for the pseudoscalar and vectormesons, and λ q accounts for the hyperfine mass splitting. In both the non-relativistic andrelativistic quark models, the hyperfine mass splitting comes from a spin-spin interaction ofthe form H HF ∼ ~S Q · ~J l . (23)6 ABLE I: The s -wave heavy baryons and their quantum number, where the subscript l stands forthe quantum number of the two light quarks.state Λ Q Σ Q Σ ∗ Q Ξ Q Ξ ′ Q Ξ ∗ Q Ω Q Ω ∗ Q J P
12 + 12 + 32 + 12 + 12 + 32 + 12 + 32 + J l where ~S Q is the spin operator of the heavy quark and ~J l is the angular momentum operatorof the light degree of freedom. Thus d M = −h M ( v ) | ~S Q · ~J l | M ( v ) i = − S M ( S M + 1) − S Q ( S Q + 1) − J l ( J l + 1)] , (24)where S M is spin quantum number of the meson M . Consequently, d M = − d M = 3 for a pseudoscalar meson. Therefore, we obtain the hyperfine masssplitting, ∆ M V P ≡ M V − M P = 2 λ q m Q . (25)We next consider the hyperfine mass splitting among the baryons containing one heavyquark ( Q ) and two light quarks ( q , q ). Each light quark is in a triplet q = ( u, d, s ) ofthe flavor SU (3). Since 3 ⊗ ⊕ ¯3 and the lowest lying light quark state has n = 1and L = 0 ( S -wave), there are two different diquarks: a symmetric sextet ( ~J l = 1) andan antisymmetric antitriplet ( ~J l = 0). When the diquark combines with a heavy quark,the sextet contains both spin- ( B ) and spin- ( B ∗ ) baryons, and the antitriplet containsonly spin- ( B ¯3 ) baryons. The multiplets B ¯3 and B ( ∗ )6 are illustrated in Fig. 1 (a) and (b),respectively, and their quantum numbers are listed in TABLE I.By analogy with Eq. (21), the heavy baryon masses can be expressed as M B = m Q + ¯Λ q q J l − m Q ( λ q q + d B λ q q ) , (26)where ¯Λ q q J l is the residual mass of heavy baryons in the heavy quark limit. The proportionsof λ q , (for meson) to λ q q , (for baryon) are [8] λ q ∼ λ q q ,λ q ∼ N c λ q q , (27)7here N c is the color number. Thus d B = −hB ( v ) | ~S Q · ~J l ) |B ( v ) i = − S B ( S B + 1) − S Q ( S Q + 1) − J l ( J l + 1)] , (28)where S B is spin quantum number of the baryon B . Consequently, d B = 0 for a B ¯3 baryon, d B = 4 for a B baryon, and d B = − B ∗ baryon. Therefore, the hyperfine masssplittings of Λ Q , Σ Q , and Σ ∗ Q are∆ M Σ ∗ Q Σ Q = 3 λ ˜ q ˜ q m Q , (29)∆ M Σ Q Λ Q = − λ ˜ q ˜ q m Q + δ ¯Λ ˜ q ˜ q , (30)∆ M Σ ∗ Q Λ Q = λ ˜ q ˜ q m Q + δ ¯Λ ˜ q ˜ q , (31)where ˜ q is the u or d quark and δ ¯Λ q q = ¯Λ q q − ¯Λ q q . For the Ξ Q baryons, however, thecomplexity is increased because of the flavor SU (3) symmetry breaking. We consider theflavor SU (3) symmetry breaking and write down the hyperfine mass splittings of Ξ Q , Ξ ′ Q ,and Ξ ∗ Q as ∆ M Ξ ∗ Q Ξ ′ Q = 3 λ s ˜ q m Q , (32)∆ M Ξ ′ Q Ξ Q = − λ s ˜ q m Q + δ ¯Λ s ˜ q , (33)∆ M Ξ ∗ Q Ξ Q = λ s ˜ q m Q + δ ¯Λ s ˜ q . (34)It is worth to mention that, from Eqs. (30), (31), (33), and (34), δ ¯Λ q q are δ ¯Λ ˜ q ˜ q = M Σ Q + 2 M Σ ∗ Q − M Λ Q , (35) δ ¯Λ s ˜ q = M Ξ ′ Q + 2 M Ξ ∗ Q − M Ξ Q . (36)Finally, the hyperfine mass splitting of Ω Q and Ω ∗ Q is∆ M Ω ∗ Q Ω Q = 3 λ ss m Q . (37)We can use the hyperfine mass differences which are experimentally known for charmedbaryons to calculate the ones for bottomed baryons.8 V. NUMERICAL RESULTS
Now we consider the numerical results of the hyperfine mass splitting for heavy mesons.As mentioned above, λ q just relates to the light degrees of freedom and independent of theheavy quark mass m Q . Thus, from Eq. (25), we obtain∆ M B ∗ B ∆ M D ∗ D = m c m b = ∆ M B ∗ s B s ∆ M D ∗ s D s , (38)whatever the values of light quark masses ( m u , m d , m s ) and other parameters appearing inany phenomenological model are. Experimentally, the ratio of hyperfine mass splitting isgiven by [9] ∆ M B ∗ B ∆ M D ∗ D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) expt = 45 . ± . . ± .
12 = 0 . ± . , (39)∆ M B ∗ s B s ∆ M D ∗ s D s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) expt = 46 . ± . . ± . . ± . , (40)where we take the masses of D ∗ and D mesons to the average ones of their charged andneutral mesons. This agreement is not only a triumph of HQET, but also reveals that the1 /m Q corrections are enough here. In addition, from the experimental data shown in Eqs.(39) and (40), we also find that λ s − λ q λ s + λ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D s D = (0 . ± .
15) % , (41) λ s − λ q λ s + λ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B s B = (0 . ± .
67) % (42)This means the SU (3) breaking effect of the hyperfine mass splitting in heavy mesons isvery small.Next we consider the numerical results of mass difference between the heavy baryons B ∗ and B . From Eq. (29) and the ratio in Eq. (39), we predict∆ M Σ ∗ b Σ b = m c m b ∆ M exptΣ ∗ c Σ c = 20 . ± . , (43)where ∆ M exptΣ ∗ c Σ c = 64 . ± . M exptΣ ∗ b Σ b = 21 . ± . M Ξ ∗ b Ξ ′ b = m c m b ∆ M exptΞ ∗ c Ξ ′ c = 22 . ± . , (45)where ∆ M exptΞ ∗ c Ξ ′ c = 69 . ± . M Ω ∗ b Ω b = m c m b ∆ M exptΩ ∗ c Ω c = 22 . ± . , (47)where the value ∆ M exptΩ ∗ c Ω c = 70 . ± . λ s ˜ q − λ ˜ q ˜ q λ s ˜ q + λ ˜ q ˜ q = (3 . ± . , (49) λ ss − λ s ˜ q λ ss + λ s ˜ q = (0 . ± .
6) % . (50)These results reveal that the flavor SU (3) breaking effect of the hyperfine mass splitting isvery small in heavy baryons, as well as in heavy mesons.Finally we consider the numerical results of mass difference which is related to the heavybaryons B ¯3 . Combining the experimental values [5, 9, 10] and the theoretical evaluation of m Σ b [11], we have ∆ M exptΣ + c Λ + c = 166 . ± . , ∆ M Σ b Λ b = 191 . ± . . Then we get from Eq. (35) δ ¯Λ ˜ q ˜ q Σ c Λ c = 209 . ± . δ ¯Λ ˜ q ˜ q Σ b Λ b = 206 . ± . . δ ¯Λ ˜ q ˜ q Σ c Λ c − δ ¯Λ ˜ q ˜ q Σ b Λ b δ ¯Λ ˜ q ˜ q Σ c Λ c + δ ¯Λ ˜ q ˜ q Σ b Λ b = (0 . ± .
65) % (51)This result reveal that, as mention above, δ ¯Λ ˜ q ˜ q is just related to the light degrees of freedomand independent of the heavy quark flavors. Now we use the above argument to evaluate δ ¯Λ s ˜ q . From the data ∆ M exptΞ ∗ c Ξ c = 176 . ± . δ ¯Λ s ˜ q = 154 . +3 . − . MeV for Ξ ∗ c Ξ c system . (52)This result can be use to the bottomed sector due to it is also independent of the heavyflavor. Thus, we predict∆ M Ξ ′ b Ξ b = 139 . +3 . − . MeV , ∆ M Ξ ∗ b Ξ b = 161 . +3 . − . MeV . Combine the data M exptΞ b = 5792 . ± . M Ξ ′ b = 5932 . ± . , M Ξ ∗ b = 5954 . ± . . (53)Furthermore, we may use the Gell-Mann/Okubo formula to obtain the equal mass differenceequations M Ξ ′ Q − M Σ Q = M Ω Q − M Ξ ′ Q ,M Ξ ∗ Q − M Σ ∗ Q = M Ω ∗ Q − M Ξ ∗ Q . (54)The accuracy of Eq. (54) can be checked in charmed sector, the experimental data give M Ξ ′ c − M Σ c = 123 . ± . , M Ω c − M Ξ ′ c = 120 . ± . ,M Ξ ∗ c − M Σ ∗ c = 128 . ± . , M Ω ∗ c − M Ξ ∗ c = 121 . ± . , where the values of M Ξ ′ , ∗ c and M Σ ( ∗ ) c are taken from the average masses in charged andneutral cases. Therefore, we have the confidence to use Eq. (54) in bottomed sector. FromEqs. (47) and (53), we predict the masses of Ω b and Ω ∗ b M Ω b = 6053 . ± . , M Ω ∗ b = 6076 . ± . . (55)We summarize the predictions of this work and list the other theoretical calculations andthe experimental data in TABLE II. 11 ABLE II: Experimental data, the predictions of this work and the other theoretical calculation(in units of MeV).Experiment This work [8] [12] [13] [14]∆ M Σ ∗ c Σ c . ± . . ± . ± M Ξ ∗ c Ξ c . ± . M Ξ ∗ c Ξ ′ c . ± . . ± . ± M Ω ∗ c Ω c . ± . . ± . ± M Σ ∗ b Σ b . ± . . ± . . ± . +13 − ∆ M Ξ ∗ b Ξ b . +3 . − . ∆ M Ξ ∗ b Ξ ′ b . ± . . ± . +13 − ∆ M Ξ ′ b Ξ b . +3 . − . +35 − ∆ M Ω ∗ b Ω b . ± . . ± . ± M Ξ b . ± . . ± . M Ξ ′ b . ± . M Ξ ∗ b . ± . M Ω b . ± . . ± . M Ω ∗ b . ± . . ± . V. CONCLUSION
In this paper, based on HQET, we have presented a formalism to describe the hyperfinemass splittings of the heavy baryons. Furthermore, through the known experimental datain charmed sector, we predicted the hyperfine mass differences in bottomed sector. Theparameters appearing in this analysis are the ratio m c /m b and the residual mass of heavybaryons in the heavy quark limit ¯Λ q q . On the one hand the ratio m c /m b is fixed by theexperimental values of heavy mesons, and on the other hand the residual mass difference δ ¯Λ q q , due to it is independent of heavy flavor, is obtained by the known mass differences ofcharmed baryons. The prediction of ∆ M Σ ∗ b Σ b is in agreement with the experimental values,we expect the deviations of the other predictive mass differences are all small for the futureexperimental data. In addition, in both heavy meson and baryon systems, we find that theflavor SU (3) breaking effect of the hyperfine mass splitting is very small. Finally we also12stimated the masses of Ξ ′ b and Ξ ∗ b and used the Gell-Mann/Okubo formula to calculate themasses of Ω b and Ω ∗ b . The uncertainties of these four heavy baryon masses mainly comefrom the error of the measured value M exptΞ b . To get the more confidence in HQET, the moreprecise experimental data are needed. Acknowledgments
This work is supported in part by the National Science Council of R.O.C. under Grant No:NSC-96-2112-M-017-002-MY3. [1] N. Isgur and M.B. Wise, Phys. Lett. B , 113 (1989); B , 527 (1990)[2] H. Georgi, Nucl. Phys. B , 447 (1990).[3] M. Neubert, Phys. Rep. , 261 (1994).[4] T. Mannel, W. Roberts and Z. Ryzak, Nucl. Phys. B ,204 (1992).[5] T. Altonen et al. (CDF Collaboration), Phys. Rev. Lett. , 202001 (2007).[6] V. M. Abazov et al. (D0 Collaboration), Phys. Rev. Lett. , 052001 (2007).[7] T. Altonen et al. (CDF Collaboration), Phys. Rev. Lett. , 052002 (2007).[8] E. Jenkins, Phys. Rev. D , 4515 (1996).[9] W.-M. Yao et al., J. of Phys. G , 1 (2006) and 2007 partial update for edition 2008(URL:http://pdg.lbl.gov).[10] D. Acosta et al. (CDF Collaboration), Phys. Rev. Lett. , 202001 (2006).[11] C. W. Hwang, Eur. Phys. J. C , 793 (2007).[12] N. mathur, R. Lewis, and R. M. Woloshyn, Phys. Rev. D , 014502 (2002).[13] D. Ebert, R. N. Faustov, and V. O. Galkin, Phys. Lett. B , 612 (2008).[14] M. Karliner, B. Keren-Zur, H. J. Lipkin, and J. L. Rosner, arXiv: 0708.4027 [hep-ph]. ddQ )( (cid:43)(cid:82) (cid:289) Qdu },{ )( (cid:43)(cid:82) (cid:289) uuQ )( (cid:43)(cid:82) (cid:289) Qdu Q ],[ (cid:282)
Qsd Q ], [ (cid:285)
Qsu Q ], [ (cid:285)
Qsd }, { )( (cid:45)(cid:43)(cid:40)(cid:82) (cid:285)
Qsu }, { )( (cid:45)(cid:43)(cid:40)(cid:82) (cid:285) ssQ )( (cid:43)(cid:82) (cid:295) (a) (b) FIG. 1: The multiplets (a) B ¯3 and (b) B ( ∗ )6 ..