Hyperfine Splittings of Baryons Containing a Heavy Quark in the Skyrme Model

The Σc − Σc and Σ∗b − Σb hyperfine mass splittings are computed in the Skyrme model. The hyperfine splittings are suppressed by both 1/Nc and by 1/mQ, where Nc is the number of colors and mQ is the mass of the heavy quark. The Σc, Σ ∗ c , Σb, Σ ∗ b , and Λb masses are predicted in terms of the known values of the Λc, D, D , B and B masses. UCSD/PTH 92-26 hep-ph/9208238 August 1992 1 Baryons containing a heavy quark can be interpreted as bound states of mesons containing a heavy quark and chiral solitons of the SU(2)L ×SU(2)R nonlinear sigma model. This approach was originally proposed by Callan and Klebanov to treat baryons containing an s quark as soliton–K-meson bound states [1]. Early attempts were made to extend this method to baryons containing a heavy quark Q, such as the c or b quark [1][2]. A correct treatment of heavy baryons as soliton–heavy-meson bound states must incorporate the consequences of heavy quark symmetry [3]. In the heavy quark limit, the D and D (B and B) mesons are degenerate, so the effective chiral theory must include both the pseudoscalar (PQ) and vector meson (P ∗ Q) fields in order to respect the heavy quark symmetry. Properties of the heavy baryon bound states have been calculated to leading order in 1/mQ [3][4]. The Skyrme model successfully predicts the existence of ΛQ, ΣQ and Σ ∗ Q bound states [3] and leads to mass relations which are well-satisfied by the measured Λc,b and Σc,b masses [4]. In the heavy quark limit, the ΣQ and Σ ∗ Q are a degenerate multiplet with isospin one, spin of the light degrees of freedom one, and total spin 1/2 and 3/2, respectively. The energy splitting between these states is a 1/mQ effect. In this letter, the ΣQ −ΣQ mass difference is computed. It is instructive to first study the hyperfine splittings in the constituent quark model. The ΣQ−ΣQ and P ∗ Q−PQ mass differences are due to the hyperfine interaction generated by one-gluon exchange between constituent quarks [5], H = − κ Nc ∑ T i T A j Si · Sj mimj , (1) where T i , Si and mi are the color generator, spin and mass of the ı th quark, Nc is the number of colors, and the sum is over all pairs. The constant κ depends on the wave function of the hadron, and may be different for mesons and baryons. H has an overall factor of 1/Nc, since the gluon coupling constant is of order 1/ √ Nc. The pseudoscalar and vector mesons PQ and P ∗ Q are bound states of a heavy quark Q and a light antiquark q in a color singlet state, with spin zero and one, respectively. The hyperfine interaction Eq. (1) in the meson sector can be written in the form H = − κ 4mQmqNc [ (TQ + Tq) 2 − T 2 Q − T 2 q ] [ (SQ + Sq) 2 − S Q − S q ] . (2) For the vector meson P ∗ Q, (SQ+Sq) 2 = 2, and for the pseudoscalar meson PQ, (SQ+Sq) 2 = 0. The color factors can be written in terms of quadratic Casimirs defined by (T R T A R ) i j = c(R) δ i j , (3) 2 for the SU(Nc) representation R. This gives the hyperfine splitting P ∗ Q − PQ = κ 2mQmqNc [ c( ) + c( )− c(1) ] , (4) where the irreducible representations of SU(Nc) are denoted by Young tableaux. The hyperfine interaction Eq. (1) in the baryon sector can be written in the form H = κ ′ 2Nc [ c( ) + c( )− c( ) ] 