Naturalness of the Coleman-Glashow Mass Relation in the 1/Nc Expansion: an Update

A new measurement of the Ξ mass verifies the accuracy of the ColemanGlashow relation at the level predicted by the 1/Nc expansion. Values for other baryon isospin mass splittings are updated, and continue to agree with the 1/Nc hierarchy. 11.15.Pg, 14.20.-c,13.40.Dk Typeset using REVTEX [email protected] [email protected] The recent measurement of the Ξ mass 1314.82± 0.06± 0.2 MeV by the NA48 collaboration [1] represents a significant improvement over the 30-year-old value 1314.9± 0.6 MeV [2]. The Ξ mass now is known to an uncertainty comparable to that of the other baryons of the lowest-lying spin-1/2 octet. This improvement makes it possible to test the precision of the famous Coleman-Glashow (CG) mass relation [3] ∆CG = (p− n)− (Σ − Σ) + (Ξ − Ξ) = 0. (1) Using the old and new experimental values for Ξ yields ∆CG = 0.39± 0.61 and 0.29± 0.26 MeV, respectively: For the first time, ∆CG has been measured to have a nonzero value, though only at the one-sigma level. It is of theoretical interest to understand the size of this breaking. In this note, we observe that the experimental value agrees with the theoretical accuracy of the CG relation as predicted in the 1/Nc expansion of QCD [4]. The mass spectrum of the baryon spin-1/2 octet and spin-3/2 decuplet was analyzed in Ref. [4] in a combined expansion in 1/Nc and flavor-symmetry breaking. It was found that all of the baryon mass splittings have a natural explanation in terms of powers of 1/Nc, SU(3) breaking ǫ, and isospin breaking ǫ (from md−mu) or ǫ (from electromagnetic effects). Our analysis differs from the standard flavor-symmetry breaking analysis in that it incorporates the enhanced symmetry of baryons present in the large-Nc limit. Large-Nc baryons respect an exact SU(6) spin-flavor symmetry [5–7]. For arbitrary Nc, the ground state baryons fill the Nc-quark completely symmetric representation of the spin-flavor algebra, which for Nc = 3 reduces to the usual 56-plet of SU(6). The spin-flavor symmetry is broken by corrections of subleading order in 1/Nc, while flavor symmetry is broken in the usual manner. Our analysis in Ref. [4] showed that the CG mass combination is O(ǫǫ/N c ) relative to the average mass of the baryon 56 spin-flavor multiplet, which is of order NcΛQCD. For Nc = 3, this result implies that the CG mass combination is predicted to be an order of magnitude smaller than expected from an SU(3) flavor symmetry-breaking analysis alone. In this work, we update the experimental values of mass combinations affected by the new mass measurement of the Ξ. First, we briefly review notation introduced in Ref. [4]: The isospin I combinations of baryon masses are denoted by a subscript I. Thus, the I = 0 and I = 1 mass combinations of the Ξ and Ξ masses are denoted by Ξ0 = 1 2 (Ξ + Ξ), Ξ1 = (Ξ 0 − Ξ), (2) respectively. Using the new value of the Ξ mass changes the experimental values of these mass combinations to Ξ0 = 1318.07± 0.12 (was 1318.11± 0.31) MeV, Ξ1 = −6.50± 0.25 (was − 6.4 ± 0.6) MeV. (3) The improvement in the experimental value of the I = 0 mass combination Ξ0 is small, and does not appreciably affect the numerical evaluation performed in Ref. [4] of I = 0 For a recent review, see Ref. [8] and references therein.