Electromagnetic Mass Splittings of theNandN*(1238 MeV)

We examine the Dashen-Frautschi calculation of the neutron-proton mass difference ${\delta }_{n,p}$. Their $\mathrm{SU}\left(2\right)\mathrm{}$ calculation considers the nucleon to be a $\pi N$ bound state with the dominant forces due to nucleon and $N^{*}(1238\mathrm{}-\mathrm{MeV})$ exchange. ${\delta }_{n,p}$ then depends linearly on ${\delta }_{-,++}$ (the mass difference between the $N^{*}\text{'}\mathrm{s}$ with charges - and + +) and the one-photon-exchange driving term $\Gamma$. [We note that this $\mathrm{SU}\left(2\right)\mathrm{}$ model predicts ${\delta }_{0,+}=\frac{1}{3}{\delta }_{-,++}$.] The $N^{*}$ is calculated as a $\pi N$ resonance with $N$ and $N^{*}$ exchange as the forces. This gives another relation among ${\delta }_{-,++}$, ${\delta }_{n,p}$, and $\Gamma$. Now in the static Chew-Low theory with a linear $D$ function the $N-N^{*}$ reciprocal bootstrap conditions on the residues are exactly satisfied. In this case we show that ${\delta }_{n,p}$ (and ${\delta }_{-,++}$) 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 ${\delta }_{n,p}$. 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 $D\propto \frac{\mathrm{}(W-M)\mathrm{}}{(W-\frac{7M}{3})}$ for the $P_{11}$ partial wave which simulates the $D$ function calculated by Balázs. 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 Balázs $D$ function corresponds to a $P_{11}$ partial wave with a negative definite phase shift, in contradiction to experiment. Using results of $\pi N$ 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 ${\delta }_{n,p}$ have the wrong magnitude, but also the wrong sign. We conclude that the calculation of ${\delta }_{n,p}$ depends critically on the details of the strong interactions. On the other hand, the ratio $\frac{{\delta }_{-,++}}{{\delta }_{n,p}}$ is insensitive to these details and is predicted to be Å3. Thus a less ambitious point of view is to use the experimental value of ${\delta }_{-,++}(=7.9\mathrm{}\pm 6.8 \mathrm{MeV})$ to get a rough value of ${\delta }_{n,p}$ (or vice versa).