As a natural extension of the formulation of asymptotic $\mathrm{SU}(3)$ symmetry previously presented, a new formulation of asymptotic $\mathrm{SU}(2)$ symmetry is proposed. Intuitively speaking, we assume that both the $\mathrm{SU}(3)$ and $\mathrm{SU}(2)$ symmetries are well realized among particles of extremely high momenta where masses are not important. This point of view is formulated by assuming that, only in the asymptotic limit, the matrix elements of the $\mathrm{SU}(2)$ generators ${V}_{{\ensuremath{\pi}}^{+}}$ and ${V}_{{\ensuremath{\pi}}^{\ensuremath{-}}}$ and also the $\mathrm{SU}(3)$ generator ${V}_{K}$ [which, in the symmetry limit, are the isotopic-spin raising and lowering and the $\mathrm{SU}(3)$ raising operators, respectively] behave, to a reasonably good approximation, as if the symmetries were not broken. Mass sum rules are obtained by using these asymptotic $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$ symmetries together with exotic charge commutators which involve the time derivative of ${V}_{{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}}$ and which do not depend explicitly on the specific parameters of symmetry breaking. Assuming that the basic (and not effective) $\mathrm{SU}(2)$-breaking interaction transforms like an $\mathrm{SU}(3)$ octet, the exotic commutators are $[{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}}_{{\ensuremath{\pi}}^{+}},{V}_{{\ensuremath{\pi}}^{+}}]=[{V}_{{K}^{0}},{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}}_{{\ensuremath{\pi}}^{\ensuremath{-}}}]=[{V}_{{K}^{+}},{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}}_{{\ensuremath{\pi}}^{+}}]=0$, etc. From them we reproduce almost all the $\mathrm{SU}(2)$ sum rules previously obtained on the assumption of effective octet dominance. However, contrary to previous results, the ${\ensuremath{\pi}}^{+}$ and ${\ensuremath{\pi}}^{0}$, for example, are no longer degenerate in mass. The mass difference is explained in terms of the ${\ensuremath{\eta}}^{0}\ensuremath{-}{\ensuremath{\pi}}^{0}$ and ${\ensuremath{\eta}}^{\ensuremath{'}0}\ensuremath{-}{\ensuremath{\pi}}^{0}$ mixings. A study is also made of the exotic commutators involving the axial-vector charges. The commutator $[{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}}_{{\ensuremath{\pi}}^{+}},{A}_{{\ensuremath{\pi}}^{+}}]=0$ is the least model-dependent one. From this we obtain an intermultiplet baryon mass formula involving the $a$ decuplet and the $b$ octet: ${({\ensuremath{\delta}}_{a})}^{2}\ensuremath{\simeq}{({p}_{b})}^{2}\ensuremath{-}{({n}_{b})}^{2}$ ($a$ and $b$ are arbitrary). (${p}_{b}$) and (${n}_{b}$) denote the masses of the proton and neutron of the $b$ octet, respectively. ${({\ensuremath{\delta}}_{a})}^{2}$ denotes the equal-squared-mass spacing of the $a$ decuplet, i.e., ${({\ensuremath{\delta}}_{a})}^{2}={({{\ensuremath{\Delta}}_{a}}^{++})}^{2}\ensuremath{-}{({{\ensuremath{\Delta}}_{a}}^{+})}^{2}={({{\ensuremath{\Delta}}_{a}}^{+})}^{2}\ensuremath{-}{({{\ensuremath{\Delta}}_{a}}^{0})}^{2}={({{\ensuremath{\Xi}}_{a}}^{*0})}^{2}\ensuremath{-}{({{\ensuremath{\Xi}}_{a}}^{*\ensuremath{-}})}^{2}$. The case of $a={\frac{3}{2}}^{+}$ and $b={\frac{1}{2}}^{+}$ coincides with the good sum rule of $\mathrm{SU}(6)$. For bosons, the exotic commutators such as $[{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}}_{{\ensuremath{\pi}}^{\ensuremath{-}}},{A}_{{K}^{0}}]=[{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}}_{{\ensuremath{\pi}}^{+}},{A}_{{K}^{+}}]=0$, which are more model dependent than the $[{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}}_{{\ensuremath{\pi}}^{+}},{A}_{{\ensuremath{\pi}}^{+}}]=0$, produce the following general intermultiplet mass sum rule: ${({{K}_{\ensuremath{\alpha}}}^{0})}^{2}\ensuremath{-}{({{K}_{\ensuremath{\alpha}}}^{+})}^{2}=\mathrm{const}\ensuremath{\simeq}0.004$ ${(\mathrm{GeV})}^{2}$ ($\ensuremath{\alpha}$ is arbitrary). Here $({{K}_{\ensuremath{\alpha}}}^{0})$ denotes the mass of the ${K}^{0}$ meson belonging to the $\ensuremath{\alpha}$ octet. For the ${0}^{\ensuremath{-}}$ and ${1}^{\ensuremath{-}}$ octets, it implies ${({K}^{0})}^{2}\ensuremath{-}{({K}^{+})}^{2}={({K}^{*0})}^{2}\ensuremath{-}{({K}^{*+})}^{2}$, which is not inconsistent with present experiment. We also show that both the commutators, $[{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}}_{{\ensuremath{\pi}}^{+}},{V}_{{\ensuremath{\pi}}^{+}}]=0$ and $[{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}}_{{\ensuremath{\pi}}^{+}},{A}_{{\ensuremath{\pi}}^{+}}]=0$, give, in general, the same mass sum rule when they are applied to the same $\mathrm{SU}(2)$ multiplet. Sum rules for the axial-vector semileptonic hyperon decay couplings in broken $\mathrm{SU}(3)$ and $\mathrm{SU}(2)$ symmetries are also derived.
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Oct 1, 2016
Niels G Gresnigt 
Open Access
