The scaling behavior observed in deep-inelastic electron scattering is related to the structure of the electric current commutation function in position space. We show that scaling is assured when that object has the following form, which is also consistent with Regge behavior: $\left\{i〈p\right|[j^{\mu },\mathrm{}(x), j^{\nu }\mathrm{}(0\left)\right]|p〉=[g^{\mu \nu }\square -{\partial }^{\mu }{\partial }^{\nu }\left]\left[\frac{1}{4{\pi }^2}\epsilon \left(x·p\right)\delta \left(x^2\right)\int 0^{\infty }d\omega \frac{cos\omega x·p}{{\omega }^2}{F}_L\left(\omega \right)+\epsilon \left(x·p\right)\theta \left(x^2\right)f_1(x^2, x·p)\right]\right\}\{+[p^{\mu }{p}^{\nu }\square -p·\partial ({\partial }^{\mu }{p}^{\nu }+{\partial }^{\nu }{p}^{\mu })+g^{\mu \nu }{\left(p·\partial \right)}^2\left]\left[\frac{1}{4{\pi }^2}\epsilon \left(x·p\right)\theta \left(x^2\right)\int 0^{\infty }d\omega \frac{sin\omega x·p}{\omega x·p}{F}_2\left(\omega \right)+\epsilon \left(x·p\right)\theta \left(x^2\right){\tilde{f}}_2(x^2, x·p)\right]\right\}$ In the above, $F_L=F_2-2\mathrm{}\omega F_1$, and the $F_i$ are the conventional scaling functions of Bjorken. The $f_i$ are arbitrary, except that $f_2(0, x·p)=0\mathrm{}$. It is also demonstrated that when the combination $T_1+\left(\frac{{\nu }^2}{q^2}\right)T_2$ of the conventional forward Compton amplitudes, as well as $T_2$, are unsubtracted, a new sum rule can be derived: $〈p\left|\right[j^0(0, \mathrm{x}), j^i\mathrm{}\left(0\right)\left]\right|p〉=\frac{i}{2\pi }{\partial }^i\delta \left(\mathrm{x}\right)\int o^{\infty }d\omega \frac{F_L\left(\omega \right)}{{\omega }^2}$ Finally, the consequences of the same unsubtractedness hypothesis for the electromagnetic self-mass of the target proton are discussed. The unsubtractedness hypothesis is consistent with present experimental results.

Close

PrevNext

View figure in article