Baryon Isospin Mass Splittings Lai-Him Chan Louisiana State University

Isospin Symmetry SU(2) SU(2) U(1) Isotopic spin symmetry is the best approximate flavor symmetries. The symmetry is easily identified in the mass spectrum. Mass splittings within the isospin multiplets are at most a few MeV. n>p, there must be other isospin violation other than the electromagnetic interaction. The only explanation has been attributed to the intrinsic up and down quark mass difference. There is no logical reason why that mass difference should be of orders of a few MeV. Such small mass splittings have posted a great challenge both to the experimentalists to measure them accurately and to the theorists to calculate them.

Isospin mass splittings Calculation There has not yet been any direct first principle calculation of the isospin mass splittings from QCD. Lattice gauge calculation is not anticipated in the near future. Phenomenological models of various assumptions have been used to calculate the isospin splittings. In the light baryon sector where the SU(3) symmetry breaking is small and accurate isospin mass splittings data are available, the predictions from various models are not sufficiently different to be differentiable. In the heavy baryon sector where the predictions spread in a wide range, isospin mass splittings have not been available until very recent years.

Charmed-Hadron Isospin Mass Splittings (1977)

the future 30 or more years …………... Have to wait for 30 years

Charm-Beauty-Baryon Isospin Mass Splittings (1985)

isospin mass splittings In the heavy baryon sector where the predictions spread in a wide range,all models agree on For many years, it clearly disagrees with the experimental value, deduced from the measurement of by CLEOII and later by CLEO Meaningful comparison of various theoretical models cannot be taken seriously until this contradiction has finally been removed by recent experiment (CLEO1). The new value listed by Particle Data Group

Large Cancellation in Charm Baryon Isosplittings Lichtenberg pointed out that in the case of the isospin splitting c++- c0, there exists a large cancellation between the mu-md and the Coulomb contribution to the isospin splitting and a measurement of this splitting would be the best hope for distinguishing among different models. Four recently measured charm isospin mass splittings and the predictions of various theoretical models are shown in the following Table. The small magnitude of has not only validated Lichtenberg's large cancellation but also has eliminated,by many standard deviations, all but one model by Chan.

Isospin mass splittings of the charm baryon isotriplets in MeV.

Light Baryon Isospin Splittings (In MeV)

Heavy Baryon Isospin Splittings (In MeV)

Isospin Symmetry Breaking Hamiltonian

Strong induced contribution from There has not yet been any direct first principle calculation of the isospin mass splittings from QCD. Lattice gauge calculation is not anticipated in the near future. First Principle Phenomenological Approach Quark Mass Interpolation QCD is flavor independent except broken by quark masses. The discrete label of flavor quantum numbers should be replaced by the corresponding quark masses which can be connected by a continuous quark mass parameters. As the number of heavy flavors increases and the quark mass gap becomes ever larger, the description of broken discrete flavor symmetry has lost its usefulness. On the other hand, as the number of heavy quark mass increases, the dependence on the continuous quark mass parameter becomes better defined. It is then possible to set up a mass interpolation scheme to deduce the effect of from the information obtained at higher quark masses.

S-wave baryon ground states For the S-wave baryon ground states, there are three quarks with spin. The wave function should be totally symmetric with respect to the simultaneous interchange of any two pairs of quark mass and spin. More importantly, given the unusual pattern of the physical quark masses, three quarks in a baryon can always be arranged such that In that configuration, an eigenstate of , to a good approximation, is also a eigenstate of the mass matrix. The totally symmetric wave-function implies the absence of a state with and a state with If we can construct a baryon mass function satisfying the prescribed properties and predicting all flavor splittings (excluding isosplittings) to within 10%, then we can expect to use the same function to interpolate the isosplittings to 10% accuracy (~ 0.3MeV.)

Contributions to the baryon masses: The De Rujula-Glashow-Georgi model can be used to guide the interpolation Contributions to the baryon masses:

Fit all isospin splittings to ~10% Parameters Strong Mass Splittings Isospin Splittings Fit all isospin splittings to ~10% 0.3MeV Fit all mass splittings to ~10%

Baryon magnetic Moments In nuclear magneton

Quark mass interpolation as first principle phenomenology It is more useful to label quark by its mass rather than its flavor, trading a discrete quantum number with a continuous parameter which can be continued analytically from one flavor value to another flavor value. The model is only used as a very important and convenient tool to perform the quark mass interpolation from large flavor mass splittings down to a small isospin mass splitting. If two models, dynamical or phenomenological, agree on their predictions on the baryon strong mass splittings, they must also predict baryon isospin mass splittings consistent with each other. Therefore the predictions on the isosplittings is insensitive to if not completely independent of the particular model used for the quark mass interpolation. This method has been applied to extract the isospin baryon mass splittings explaining why a simple and perhaps naïve quark model can survive the competition of many years theoretical development and pass numerous experimental tests with great success with no exception.