Final-state interactions in inclusive deep-inelastic scattering from the deuteron

We explore the role of final-state interactions (FSIs) in inclusive deep-inelastic scattering from the deuteron. Relating the inclusive cross section to the deuteron forward virtual Compton scattering amplitude, a general formula for the FSI contribution is derived in the generalized eikonal approximation, utilizing the diffractive nature of the effective hadron-nucleon interaction. The calculation uses a factorized model with a basis of three resonances with mass $W<2$ GeV and a continuum contribution for larger $W$ as the relevant set of effective hadron states entering the final-state interaction amplitude. The results show sizeable on-shell FSI contributions for Bjorken $x\gtrsim 0.6$ and $Q^2\lesssim 10$ GeV${}^2$, increasing in magnitude for lower $Q^2$, but vanishing in the high-$Q^2$ limit because of phase-space constraints. The off-shell rescattering contributes at $x\gtrsim 0.8$ and is taken as an uncertainty on the on-shell result.

Figure 1

Forward virtual Compton scattering amplitude from a nucleus $A$, with $q$ and $p_A$ the photon and target four-momenta and $p_X$ the four-momentum of the produced state $X$.

Figure 2

Forward virtual Compton scattering amplitude for the deuteron, composed of (a) the Born diagram and (b) the rescattering contribution. The gray blob in the intermediate state represents the effective rescattering interaction of the hadronic debris ($X_1$) and the spectator nucleon ($S_1$) to the final hadronic state ($X_2$) and nucleon ($S_2$). The deuteron momentum in the Born diagram is given by $p_D=p_i+p_s$ and in the FSI diagram by $p_D=p_{i_1}+p_{s_1}=p_{i_2}+p_{s_2}$.

Figure 3

Maximum value of Bjorken $x$ allowed for the application of the eikonal approximation as a function of the invariant mass $W$ of the state $X^{\prime }$.

Figure 4

$F_2$ proton (solid lines) and neutron (dashed lines) structure functions at $Q^2=2$ GeV${}^2$ for the Alekhin leading twist [54] (black), SLAC [55] (blue), and Christy and Bosted (CB) [56] (red) parametrizations of $F_2$.

Figure 5

Ratio of the deuteron $F_2^D$ structure function with FSIs to that computed in the plane-wave (PW) approximation at $Q^2=5{\mathrm{GeV}}^2$, using only the on-shell amplitude in Eq. (32) (a) and including also the off-shell contribution of Eq. (37) (b). The results with the resonance region contributions alone (black solid lines) are compared with those including a continuum component with a Gaussian distribution (blue dashed lines) and a uniform distribution (red dotted lines). The arrow along the $x$ axis indicates the boundary at $W=2$ GeV between the resonance and DIS regions for free nucleon kinematics.

Figure 6

Ratio of the deuteron $F_2^D$ structure function including FSIs to that computed in the PW approximation, at several values of $Q^2$ from 2 to 50 GeV${}^2$. The on shell only results from Eq. (32) (blue shaded bands) and those including off-shell contributions from Eq. (37) (orange shaded bands) span the range of models for the distribution of intermediate-state masses shown in Fig. 5. The arrows along the $x$ axis indicate the $W=2$ GeV point at each $Q^2$ value for free nucleon kinematics.

Figure 7

Ratio of the deuteron $F_2^D$ to isoscalar nucleon $F_2^N$ structure functions at $Q^2=2$ GeV${}^2$ (a) and $Q^2=20$ GeV${}^2$ (b). The PW results (black solid curves) are compared with those including FSIs with the on-shell amplitude [Eq. (32)] (blue dashed curves) and with the addition of the off-shell contribution of [Eq. (37)] (red dotted curves). The arrows along the $x$ axis indicate the $W=2$ GeV boundary for free nucleon kinematics. The nucleon resonance structures are clearly visible in the ratio at $Q^2=2$ GeV${}^2$.

Figure 8

Ratio of the deuteron $F_2^D$ structure function with and without FSIs at $Q^2=2$ GeV${}^2$ (a) and $Q^2=20$ GeV${}^2$ (b). The results using the Paris deuteron wave function [57] (black solid curves) are compared with those employing the CD-Bonn [58] (blue dashed curves) and WCJ-1 [59] (red dotted curves) wave functions.