Determination of the Interaction Term in Deuteron Nucleus

Abstract

In this paper, the contribution of the interaction in the structure function of the deuteron is investigated. For this study we use two different methods, constituted nucleons model and the DGLAP evolution equations. We obtain the interaction term using calculated proton, neutron and deuteron structure functions. The extracted results are in agreement with experimental data. The results achieved from the DGLAP equations show better consequence with experimental data. Also the ratio ${{F_{2}^{d}} \mathord{\left/ {\vphantom {{F_{2}^{d}} {(F_{2}^{n} + F_{2}^{p})}}} \right. \kern-0em} {(F_{2}^{n} + F_{2}^{p})}}$ is studied and compared with the experimental data and the deuteron correction factor.

APPENDIX A

With respect to the optical theorem, the deuteron cross section is related to the imaginary part of the amplitude for elastic scattering in the forward direction as

$${{\sigma }_{d}} = \frac{{4\pi }}{k}{\text{Im}}F(0),$$

(3.1)

where $k$ is the wave number of the incident particle. The free-nucleon cross sections can be written as

$${{\sigma }_{{p,n}}} = \frac{{4\pi }}{k}{\text{Im}}{{f}_{{p,n}}}(0).$$

(3.2)

The deuteron cross section may be written in the following form

$${{\sigma }_{d}} = {{\sigma }_{p}} + {{\sigma }_{n}} + \delta \sigma ,$$

(3.3)

where the correction term $\delta \sigma$ dependent upon the nucleon elastic scattering amplitudes and the deuteron form factors [6]. Equation (3.3) is based on the Glauber’s multiple scattering theory, where $\sigma _{i}^{,}s(i = d,p,n)$ are total cross sections for deuteron and free nucleons in fm² (which ${{(1\;{\text{fm}})}^{2}} = 10\;{\text{mb}}$). When the high-energy interactions are purely absorptive, then the interaction term is defined into the shadowing term as

$$\delta \sigma = - \frac{1}{{4\pi }}{{\sigma }_{p}}{{\sigma }_{n}}{{\left\langle {{{r}^{{ - 2}}}} \right\rangle }_{d}}.$$

(3.4)

Here the quantity $\left\langle {{{r}^{{ - 2}}}} \right\rangle$ is the average separation of the nucleons (in fm^–2). The average inverse square of the neutron-proton distance in the deuteron ground state is defined by

$${{\left\langle {{{r}^{{ - 2}}}} \right\rangle }_{d}} = \frac{1}{{2\pi }}\int {S({\mathbf{q}}){{d}^{2}}{\mathbf{q}}} ,$$

(3.5)

which the form factor $S({\mathbf{q}})$ is defined due to the deuteron ground-state wave function $\varphi ({\mathbf{r}})$ by

$$S({\mathbf{q}}) = \int {{{e}^{{i{\mathbf{q}}.{\mathbf{r}}}}}} {{\left| {\phi ({\mathbf{r}})} \right|}^{2}}d{\mathbf{r}},$$

where $\hbar {\mathbf{q}}$ is the momentum transferred to the target nucleon. The relative coordinates are introduced by ${\mathbf{r}} = {{{\mathbf{r}}}_{1}} - {{{\mathbf{r}}}_{2}}$, where the indices 1 and 2 label the two nucleons of the deuteron. Indeed Eq. (3.5) holds only if one supposes that the hadron-nucleon amplitude is purely imaginary and the momentum transfer is essentially transversal (small scattering angles).

In the inelastic electron-deuteron scattering, dominant process is $e + d \to ek$, which the measured cross section is defined by $\tfrac{{{{\partial }^{2}}\sigma }}{{\partial {{\Omega }_{e}}\partial E_{e}^{'}}}$. Within the quark-parton model, the nuclear reduced cross section is related to the structure functions ${{F}_{2}}$ and $R$ by

$$\begin{gathered} \frac{{{{\partial }^{2}}\sigma }}{{\partial {{\Omega }_{e}}\partial E_{e}^{'}}}\frac{{4{{\alpha }^{2}}E{\kern 1pt} {{'}^{2}}{{\cos}^{2}}({\theta \mathord{\left/ {\vphantom {\theta 2}} \right. \kern-0em} 2})}}{{{{Q}^{4}}}} \\ \times \,\,\frac{1}{\nu }{{F}_{2}}(x,{{Q}^{2}})\left( {1 + \frac{{1 - \varepsilon }}{\varepsilon }\frac{1}{{1 + R(x,{{Q}^{2}})}}} \right), \\ \end{gathered} $$

(3.6)

which describes the inelastic scattering off a nuclei. Here $\theta$ the scattering angle, $\nu = E - E{\kern 1pt} '$ is the energy transfer between the incident and final electron, $x = {{{{Q}^{2}}} \mathord{\left/ {\vphantom {{{{Q}^{2}}} {2M\nu }}} \right. \kern-0em} {2M\nu }}$ is the Bjorken scaling variable, $\varepsilon = [1 + 2(1 + {{{{\nu }^{2}}} \mathord{\left/ {\vphantom {{{{\nu }^{2}}} {{{Q}^{2}}}}} \right. \kern-0em} {{{Q}^{2}}}}){{\tan}^{2}}{{(\theta } \mathord{\left/ {\vphantom {{(\theta } 2}} \right. \kern-0em} 2}){{]}^{{ - 1}}}$ is the polarization of the exchange virtual photon [30]. However, as described in literatures, the ratio of the deuteron and the proton cross sections are dependent on the ratio $\tfrac{{F_{2}^{d}}}{{F_{2}^{p}}}$ [10]. Indeed total cross section is defined as ${{\sigma }_{{{\text{total}}}}} = \iint {\tfrac{{{{\partial }^{2}}\sigma }}{{\partial {{\Omega }_{e}}\partial E_{e}^{'}}}}\left( { \equiv \tfrac{{{{\partial }^{2}}\sigma }}{{\partial x\partial {{Q}^{2}}}}} \right)$. The neutral current double differential cross section is give by the expression

$$\frac{{{{\partial }^{2}}\sigma }}{{\partial x\partial {{Q}^{2}}}} = \frac{{2\pi {{\alpha }^{2}}{{Y}_{ + }}}}{{{{Q}^{4}}x}}{{\sigma }_{r}},$$

(3.7)

where the reduced cross section is defined as

$${{\sigma }_{r}} \equiv {{F}_{2}}(x,{{Q}^{2}}) - \frac{{{{y}^{2}}}}{{{{Y}_{ + }}}}{{F}_{L}}(x,{{Q}^{2}})$$

(3.8)

and ${{Y}_{ + }} = 1 + {{(1 - y)}^{2}}$. To transition from elastic to inelastic scattering we assume that a similar relation to Eq. (1.1) exists for the total cross sections in the inelastic process. Then the interaction function expressed in the general form of $f(r)$. According to the double differential cross sections, we can express the double deuteron differential cross section in terms of double nucleon differential cross sections as the interaction term is summarized in the last sentence.

$$\begin{gathered} \frac{{{{\partial }^{2}}{{\sigma }_{d}}}}{{\partial x\partial {{Q}^{2}}}} = \frac{{{{\partial }^{2}}{{\sigma }_{p}}}}{{\partial x\partial {{Q}^{2}}}} + \frac{{{{\partial }^{2}}{{\sigma }_{n}}}}{{\partial x\partial {{Q}^{2}}}} - \frac{1}{{4\pi }}{{\sigma }_{n}}{{\sigma }_{p}} \\ \times \,\,\left( {\frac{1}{{{{\sigma }_{p}}}}\frac{{{{\partial }^{2}}{{\sigma }_{p}}}}{{\partial x\partial {{Q}^{2}}}} + \frac{1}{{{{\sigma }_{n}}}}\frac{{{{\partial }^{2}}{{\sigma }_{n}}}}{{\partial x\partial {{Q}^{2}}}}} \right)f(r) \\ = \frac{{{{\partial }^{2}}{{\sigma }_{p}}}}{{\partial x\partial {{Q}^{2}}}} + \frac{{{{\partial }^{2}}{{\sigma }_{n}}}}{{\partial x\partial {{Q}^{2}}}} - \frac{1}{{4\pi }}{{\sigma }_{n}}{{\sigma }_{p}}\eta (x,{{Q}^{2}}), \\ \end{gathered} $$

(3.9)

where $f(r)$ is a unknown function. With respect to Eq. (3.7), the double differential cross sections change to the reduced cross sections and all the coefficients are summarized in the unknown $F$ function.

$$\frac{{\sigma _{r}^{d}}}{{\sigma _{r}^{p}}} = 1 + \frac{{\sigma _{r}^{n}}}{{\sigma _{r}^{p}}} + \mathcal{F}(x,{{Q}^{2}}).$$

(3.10)

Here $\mathcal{F}(x,{{Q}^{2}})$ must be dimensionless. Now, with respect to Eq. (3.8) at low inelasticity, the above equation (i.e., Eq. (3.10)) is simplified as follows

$$\frac{{F_{2}^{d}}}{{F_{2}^{p}}} = 1 + \frac{{F_{2}^{n}}}{{F_{2}^{p}}} + {\text{Interaction}}\,\,{\text{Term}}\,\,({\text{IT}}).$$

(3.11)