--- abstract: 'In order to facilitate comparison of results from different studies of structure function and correlation functions, we offer here some general expressions for a class of functions, known to be in very good agreement with experimental data, obtained within the framework of Regge theory. Although by no means all of these are available in the literature, yet it is important to provide these for possible future reference.' author: - 'M. Bishari$^{1,2}$, D.f. Haldane$^3$, D. S. Koltun$^4$, M. Levy$^5$ and A. Palev$^{5,6}$' --- The functions ============= For the first time we present here a unified framework for the description of several structure functions and correlation functions. The formulation is in terms of Regge theory and can be used in any Regge-based phenomenological model to study scattering processes. The motivation for this particular choice of Regge-based formulation is that it is in very good agreement with a wide range of data covering a wide range of physical systems; this is indeed the idea on which Regge theory is based. Our motivation for our current work was stimulated by several considerations, the following items to be dealt with in the framework of Regge theory. A detailed formulation of Regge-based expressions can be found in ref.[@hdlkp] and references therein. A.1 === Dynamical structure functions are defined by: $$\begin{aligned} \label{e1} F_1(q^2,x) &=& {Q^2\over \nu} \sum_f e_f^2 (q_f^2 - q_{min}^2)^2 \ C_{1/2}(x/x_0)\nonumber \\ F_2(q^2,x) &=& {Q^2\over \nu} \sum_f e_f^2 x (q_f^2 - q_{min}^2) \ C_{1/2}(x/x_0)\nonumber \\ F_3(q^2,x) &=& {Q^2\over \nu} \sum_f e_f^2 (q_f^2 - q_{min}^2) \ C_{3/2}(x/x_0)\end{aligned}$$ with $C_{1/2}(x/x_0) = \left[\frac{\nu}{Q^2} \int dx\ (F_1(x))^{{1\over 2}}\right]$, $C_{3/2}(x/x_0) = \left[\frac{\nu}{Q^2} \int dx\ (F_1(x))^{{1\over 3}}\right]$. Here the photon is virtual and $q_{min}$ corresponds to a cut off in the momentum transfer $q$, such that $q < q_{min}$. The structure function $F_1$ can be obtained from experiment in the range $0.1 < x < 0.6$. Therefore we need a cut off $x_0 = 0.1$. The average momentum transfer $\nu$ appearing in these formulae should be the virtual photon momentum at the corresponding value of $x$. We also have to take the photon momentum $Q$ large enough so that we can ignore $\gamma N$ corrections to the elementary nucleon-quark vertex. We have to mention the relations: $$\begin{aligned} & &Q^2F_1(Q^2) = \sum_f e_f^2\ C_{1/2}(Q^2)\left[\frac{\nu}{Q^2} \int dx\ (F_1(x))^{{1\over 2}}\right] \\ & &Q^2F_2(Q^2) = \sum_f e_f^2\ C_{1/2}(Q^2)\ x\ \left[\frac{\nu}{Q^2} \int dx\ (F_1(x))^{{1\over 3}}\right]\end{aligned}$$ where $C_{1/2}(Q^2) = \left[\frac{\nu}{Q^2} \int dx\ F_1(x)\right]$. A.2 === The photon nucleon $F_2$ structure function is given by: $$\begin{aligned} F_2(x) &=& 2 x \int^{\infty}_{x/x_0} F_1(\xi)d\xi \nonumber \\ &\approx& \frac{Q^2}{4\pi^2\alpha_{em}} \frac{2\ M_0^2(x) x^2}{x_0^2 \ln x_0^2} (1 + \sum_r e^{- \tau r(x)}),\end{aligned}$$ where $\tau = Q^2/4\pi^2\alpha_{em}M^2$, the sum is over all the resonances in the given channel and $e^{- \tau r(x)} $ is the exponential damping term. The expression for $F_2$ in the $W$ region can be obtained in a similar way. Note that $F_2$ in this region is dominated by the $\Delta$ resonance, which is the only resonance with a significant contribution to this region of the energy. $$\begin{aligned} F_2^{W}(\nu) &\approx& \frac{1}{3}\left(1 + 0.08\left(\frac{W}{M}\right)^2\right) \left(1 + \sum_r e^{- \tau r(\nu)} \right)\nonumber \\ & & \mbox{\ \ \ \ \ \ \ for} \ \ \frac{M^2}{4\pi\alpha_{em}}\ln \nu \ll \frac{W^2}{M^2} \ll 1, \end{aligned}$$ where $\tau = {W^2\over Q^2}$, the sum is over all the resonances in the given channel and $e^{- \tau r(\nu)} $ is the exponential damping term. A.3 === We next consider the spin-1 nucleon resonances $D$, $P$ and $F$. The corresponding invariant structure functions are given by: $$\begin{aligned} & & F_2^P(x) = 2 x\int^{\infty}_{x/x_0} F_1(\xi) d\xi \nonumber \\ & & \mbox{\ \ \ \ \ \ \ \ \ } = \frac{2M^2(x)x^2}{x_0^2 \ln x_0^2} \int^{\infty}_{0} dt\ \frac{e^{- t \frac{Q^2}{8M^2}}}{\left(1 + \frac{Q^2}{16M^2}\right)^2} \nonumber \\ & & \mbox{\ \ \ \ \ \ \ \ \ \ } = \frac{2M^2(x)x^2}{x_0^2 \ln x_0^2} (1 + \sum_r e^{- \tau r(x)}).\end{aligned}$$ The above equations also contain a sum over $r$ and hence, in this case, $\tau = Q^2/4\pi^2\alpha_{em}M^2$, the sum is over all the resonances in the given channel and $e^{- \tau r(x)} $ is the exponential damping term. A.4 === We next consider a spin-3/2 baryon. The invariant structure functions are given by: $$\begin{aligned} & & F_2^{D}(x) = 2 x\int^{\infty}_{x/x_0} F_1(\xi) d\xi \nonumber \\ & & \mbox{\ \ \ \ \ \ \ \ \ } = \frac{Q^2}{4\pi^2\alpha_{em}} \frac{2\ M^2(x)x^2}{x_0^2 \ln x_0^2} \int^{\infty}_{0} dt\ \frac{e^{- t \frac{Q^2}{8M^2}}}{1 + \frac{Q^2}{8M^2}} \nonumber \\ & & \mbox{\ \ \ \ \ \ \ \ \ \ } = \frac{Q^2}{4\pi^2\alpha_{em}} \frac{2\ M^2(x)x^2}{x_0^2 \ln x_0^2} (1 + \sum_r e^{- \tau r(x)}).\end{aligned}$$ We can now extend this result to the spin-5/2 baryons $H$, $G$ and $I$, as follows: $$\begin{aligned} F_2^{H}(x) = 2 x\int^{\infty}_{x/x_0} F_1(\xi) d\xi \approx {2\ M_0^2(x)\ x^2\over x_0^2 \ln x_0^2} \left(1 + \sum