Electromagnetic Mass Splittings of theNandN*(1238 MeV)

Abstract

We examine the Dashen-Frautschi calculation of the neutron-proton mass difference ${\ensuremath{\delta}}_{n,p}$. Their $\mathrm{SU}(2)$ calculation considers the nucleon to be a $\ensuremath{\pi}N$ bound state with the dominant forces due to nucleon and ${N}^{*}(1238\ensuremath{-}\mathrm{MeV})$ exchange. ${\ensuremath{\delta}}_{n,p}$ then depends linearly on ${\ensuremath{\delta}}_{\ensuremath{-},++}$ (the mass difference between the ${N}^{*}'\mathrm{s}$ with charges - and + +) and the one-photon-exchange driving term $\ensuremath{\Gamma}$. [We note that this $\mathrm{SU}(2)$ model predicts ${\ensuremath{\delta}}_{0,+}=\frac{1}{3}{\ensuremath{\delta}}_{\ensuremath{-},++}$.] The ${N}^{*}$ is calculated as a $\ensuremath{\pi}N$ resonance with $N$ and ${N}^{*}$ exchange as the forces. This gives another relation among ${\ensuremath{\delta}}_{\ensuremath{-},++}$, ${\ensuremath{\delta}}_{n,p}$, and $\ensuremath{\Gamma}$. Now in the static Chew-Low theory with a linear $D$ function the $N\ensuremath{-}{N}^{*}$ reciprocal bootstrap conditions on the residues are exactly satisfied. In this case we show that ${\ensuremath{\delta}}_{n,p}$ (and ${\ensuremath{\delta}}_{\ensuremath{-},++}$) is infinite. (Following Gerstein and Whippman, this divergence is seen to be a general consequence of the static, linear-$D$, reciprocal bootstrap conditions.) Thus it is only the deviations from the static Chew-Low theory with linear $D$ which give a finite ${\ensuremath{\delta}}_{n,p}$. Dashen and Frautschi consider two such effects: (a) They show that the ${N}^{*}$ exchange force is suppressed (by a factor of 0.6) because of the detailed shape of the resonance. (b) The physical $D$ function must approach a constant at high energy, and they choose the simple rational form $D\ensuremath{\propto}\frac{(W\ensuremath{-}M)}{(W\ensuremath{-}\frac{7M}{3})}$ for the ${P}_{11}$ partial wave which simulates the $D$ function calculated by Bal\'azs. This choice for $D$ leads to an additional suppression of the ${N}^{*}$ exchange force. We concentrate our criticism on the nature of the $D$ function. We note that the Bal\'azs $D$ function corresponds to a ${P}_{11}$ partial wave with a negative definite phase shift, in contradiction to experiment. Using results of $\ensuremath{\pi}N$ phase-shift analyses, we calculate the $D$ functions and find that the ${N}^{*}$ exchange contribution to the binding of the nucleon is enhanced relative to the linear form for $D$. Depending on the high-energy behavior of these phase shifts, not only can the calculated ${\ensuremath{\delta}}_{n,p}$ have the wrong magnitude, but also the wrong sign. We conclude that the calculation of ${\ensuremath{\delta}}_{n,p}$ depends critically on the details of the strong interactions. On the other hand, the ratio $\frac{{\ensuremath{\delta}}_{\ensuremath{-},++}}{{\ensuremath{\delta}}_{n,p}}$ is insensitive to these details and is predicted to be \AA{}3. Thus a less ambitious point of view is to use the experimental value of ${\ensuremath{\delta}}_{\ensuremath{-},++}(=7.9\ifmmode\pm\else\textpm\fi{}6.8 \mathrm{MeV})$ to get a rough value of ${\ensuremath{\delta}}_{n,p}$ (or vice versa).