The New Σ b multiplet and magnetic moments of Λ c and Λ b

The recent measurement by CDF M(Σb)−M(Λb) = 192MeV is in striking agreement with our theoretical prediction M(Σb)−M(Λb) = 194MeV. In addition, the measured splitting M(Σb)−M(Σb) = 21 MeV agrees well with the predicted splitting of 22 MeV. We discuss the theoretical framework behind these predictions and use it to provide additional predictions for the masses of spin2 and spin2 baryons containing heavy quarks, as well as magnetic moments of Λb and Λc. e-mail: marek@proton.tau.ac.il e-mail: ftlipkin@weizmann.ac.il 1 I. STRIKING AGREEMENT WITH CONSTITUENT QUARK MODEL PREDICTIONS A new challenge demanding explanation from QCD is posed by the remarkable agreement shown in Fig. 1 between the experimental masses 5808 MeV and 5816 MeV of the newly discovered Σ+b and Σ − b and the 5814 MeV quark model prediction [1] from meson masses MΣb − MΛb MΣ − MΛ = (Mρ − Mπ) − (MB∗ − MB) (Mρ − Mπ) − (MK∗ − MK) = 2.51 (1) This then predicts that the isospin-averaged mass splitting is MΣb − MΛb = 194 MeV and M(Σb) = 5814 MeV, using the most recent CDF Λb mass measurement [3] M(Λb) = 5619.7± 1.2 (stat.)± 1.2 (syst.) MeV. An analogous prediction for MΣc − MΛc , eq. (17) is discussed at the end of Sec. II A below. The challenge is to understand how and under what assumptions one can derive from QCD the very simple model of hadronic structure at low energies which leads to such accurate predictions – constituent quarks plus spin-dependent color-magnetic hyperfine interactions with the same flavor dependence for quark-antiquark interactions in mesons and quark-quark interactions in baryons. CDF obtained the masses of the Σ−b and Σ + b from the decay Σb → Λb +π by measuring the corresponding mass differences [2] M(Σ−b ) − M(Λb) = 195.5 +1.0 −1.0 (stat.) ± 0.1 (syst.) MeV (2) M(Σ+b ) − M(Λb) = 188.0 +2.0 −2.3 (stat.) ± 0.1 (syst.) MeV with isospin-averaged mass difference M(Σb) − M(Λb) = 192 MeV. The final values for Σ−b and Σ + b are [2] M(Σ−b ) = 5816 +1.0 −1.0 (stat.) ± 1.7 (syst.) MeV (3) M(Σ+b ) = 5808 +2.0 −2.3 (stat.) ± 1.7 (syst.) MeV with isospin-averaged mass M(Σb) = 5812 MeV. There is also the prediction for the spin splittings, good to 5% M(Σ∗b) − M(Σb) = M(B) − M(B) M(K∗) − M(K) · [M(Σ) − M(Σ)] = 22 MeV (4) Ref. [1] used an older value M(Λb) = 5624 MeV [4], yielding M(Σb) = 5818 MeV. 2 to be compared with 21 MeV from the isospin-average of CDF measurements [2] M(Σ b ) = 5837 +2.1 −1.9 (stat.) ± 1.7 (syst.) MeV (5) M(Σ b ) = 5829 +1.6 −1.8 (stat.) ± 1.7 (syst.) MeV Fig. 1. Experimental results from CDF for M(Σ+b ) − M(Λb) and M(Σ − b ) − M(Λb) compared with the theoretical prediction in Ref. [1]. These results relating meson and baryon masses have been obtained without any explicit model for the hyperfine interaction beyond their flavor dependence. We also note that they relate experimental masses of mesons and baryons containing quarks of four different flavors, u, d, s, b with no free parameters. It is difficult to believe that these relations and others given below with five different flavors u, d, s, c, b are accidental when they relate so many experimentally observed masses of mesons and baryons. This suggests that any model for hadron spectroscopy which treats mesons and baryons differently or does not yield agreement 3 with data for all five flavors is missing essential physics. So far only the constituent quark model shows this achievement. That meson and baryon masses must be related because they are made of the same quarks was first pointed out by Sakharov and Zeldovich [5] in a paper that was completely ignored until the same work was independently rediscovered [6]. There followed a number of successes of quark model relations [1,7] which still have no explanation from QCD. These works tend to express relations in terms of effective quark masses which have not found any simple explanation from QCD. It is interesting to note that the new successful relations, eqs. (1) and (4), relate meson and baryon masses without explicitly mentioning constituent quark masses, even though the latter are used to derive these relations, as described in [1]. This approach follows the famous Gell-Mann’s recipe of 1964: “... We may compare this process to a method sometimes employed in French cuisine: a piece of pheasant meat is cooked between two slices of veal, which are then discarded.” We shall extend this line of thought here and show that some kind of meson-baryon or light antiquark-diquark symmetry exists which describes many relations between meson and baryon states without explicitly mentioning quark masses. These are not simply described by QCD treatments, which tend to treat meson and baryon structures very differently. We first consider a meson M(q̄Qi) containing a light color-antitriplet antiquark q̄ (u or d) and a quark of some fixed flavor Qi, i = (u, s, c, b). We also consider a baryon containing the same quark Qi and two light quarks coupled to a diquark of spin S. We denote such a baryon by B([qq]SQi) and focus on a transformation between the two, M(q̄Qi) ⇐⇒ B([qq]SQi) (6) We find that the mass difference between the two is independent of the quark flavor i for all four flavors (u, s, c, b) when the contribution of the hyperfine interaction energies is removed, as in eq. (8). This mass relation has already been noted [7] in the case of spin-zero diquarks, S = 0, M(N) − M̃(ρ) = M(Λ) − M̃(K) = M(Λc) − M̃(D ) = M(Λb) − M̃(B ) 323 MeV ≈ 321 MeV ≈ 312 MeV ≈ 310 MeV (7) where M̃(Vi) ≡ (3MVi + MPi)/4 (8) is the meson mass without the hyperfine contribution. 4 The first equality in eq. (7) is seen to be an algebraically equivalent rewriting of the original SZL relation [5,6] between baryon and meson mass differences that cancel out the hyperfine interaction, MΛ − MN = 177 MeV = 3(MK∗ − Mρ) + MK − Mπ 4 = 179 MeV (9) the other two equalities in eq. (7) are an extension of the same idea to heavy flavors. However, these relations were all interpreted in terms of effective quark masses. We bypass the mass question here by noting that they relate states which transform into one another by the transformation (6). Why masses of boson and fermion states related by this transformation should satisfy a simple relation like (7) remains a challenge for QCD and perhaps indicate some kind of boson-fermion or antiquark-diquark supersymmetry. Following this interpretation we note that the equation (7) involves meson and baryon states related by the transformation (6) for the case S = 0 in which the antiquark is transformed into a diquark with spin zero. We now show that new successful relations between meson and baryon masses are obtained by applying the same procedure to the case S = 1 with spin-one diquarks. We first define the baryon analogue of the meson M̃(Vi) to obtain the linear combination M̃(Σi) of baryon masses without the hyperfine contribution between the diquark and the additional quark. M̃(Σi) ≡ 2MΣ∗ i + MΣi 3 ; M̃(∆) ≡ 2M∆ + MN 3 (10) Then M̃(∆) − M̃(ρ) = M(Σ) − M̃(K) = M̃(Σc) − M̃(D ) = M̃(Σb) − M̃(B ) 517.56 MeV ≈ 512.45MeV ≈ 523.95 MeV ≈ 512.45 MeV (11) II. SPIN SPLITTINGS AND MAGNETIC MOMENTS A. Ratios of Hyperfine splittings We now relate the mass ratios in mesons and baryons from the hyperfine splittings in the same way given in the original SZL papers [5,6,8], noting that the flavor dependence of the splittings can be expressed without including quark masses. Let Vox(qiqj), and Vhyp(qiq̄j) denote the color magnetic energies respectively of the quarkquark and quark-antiquark systems with flavors i and j. It is sufficient for our purpose to require that these color magnetic energies satisfy a simple flavor dependence relation [5,6,8], 5 Vhyp(qiq̄j) Vhyp(qiq̄k) = Vhyp(qiqj) Vhyp(qiqk) (12) This then leads to the relations; M∆ − MN MΣ∗ − MΣ = 1.53 = Vhyp(ud) Vhyp(us) ≈ Vhyp(ud̄) Vhyp(us̄) = Mρ − Mπ MK∗ − MK = 1.61 (13) MΣ∗ − MΣ MΣ∗c − MΣc = 2.84 = Vhyp(us) Vhyp(uc) ≈ Vhyp(us̄) Vhyp(uc̄) = MK∗ − MK MD∗ − MD = 2.81 (14) M∆ − Mp MΣ∗c − MΣc = 4.36 = Vhyp(ud) Vhyp(uc) ≈ Vhyp(ud̄) Vhyp(uc̄) = Mρ − Mπ MD∗ − MD = 4.46 (15) MΣ∗ − MΣ MΣ∗ b − MΣb = 9.11 = Vhyp(us) Vhyp(ub) ≈ Vhyp(us̄) Vhyp(ub̄) = MK∗ − MK MB∗ − MB = 8.69 (16) The presence of a fourth flavor gives us the possibility of obtaining a new type of mass relation between mesons and baryons. The Σ − Λ mass difference is believed to be due to the difference between the u − d and u − s hyperfine interactions. Similarly, the Σc − Λc mass difference is believed to be due to the difference between the u− d and u− c hyperfine interactions. We therefore obtain the relation MΣc−MΛc MΣ−MΛ =2.16= Vhyp(ud)−Vhyp(uc) Vhyp(ud)−Vhyp(us) ≈ Vhyp(ud̄)−Vhyp(uc̄) Vhyp(ud̄)−Vhyp(us̄) = (Mρ−Mπ)−(MD∗−MD) (Mρ−Mπ)−(MK∗−MK) = 2.10 (17) The original derivation [1] assumed that hyperfine interactions were inversely proportional to the products of quark masses,