Electron-deuteron deep-inelastic scattering with spectator nucleon tagging and final-state interactions at intermediate $x$

We consider electron-deuteron deep-inelastic scattering (DIS) with detection of a proton in the nuclear fragmentation region (“spectator tagging”) as a method for extracting the free neutron structure functions and studying their nuclear modifications. Such measurements could be performed at a future electron-ion collider (EIC) with suitable forward detectors. The measured proton recoil momentum ($\lesssim 100$ MeV in the deuteron rest frame) specifies the deuteron configuration during the high-energy process and permits a controlled theoretical treatment of nuclear effects. Nuclear and nucleonic structure are separated using methods of light-front quantum mechanics. The impulse approximation to the tagged DIS cross section contains the free neutron pole, which can be reached by on-shell extrapolation in the recoil momentum. Final-state interactions (FSIs) distort the recoil momentum distribution away from the pole. In the intermediate-$x$ region $0.1<x<0.5$ FSIs arise predominantly from interactions of the spectator proton with slow hadrons produced in the DIS process on the neutron (rest frame momenta $\lesssim 1\mathrm{GeV}$, target fragmentation region). We construct a schematic model describing this effect, using final-state hadron distributions measured in nucleon DIS experiments and low-energy hadron scattering amplitudes. We investigate the magnitude of FSIs, their dependence on the recoil momentum (angular dependence, forward/backward regions), their analytic properties, and their effect on the on-shell extrapolation. We comment on the prospects for neutron structure extraction in tagged DIS with an EIC. We discuss possible extensions of the FSI model to other kinematic regions (large/small $x$). In tagged DIS at $x\ll 0.1$ FSIs resulting from diffractive scattering on the nucleons become important and require separate treatment.

Figure 1

Physical picture of FSI in electron-deuteron DIS with proton tagging, $e+d\to e^{\prime }+p\left(p_p\right)+X$, in the deuteron rest frame. A slow hadron in the final state produced by DIS on the active neutron scatters from the spectator proton, changing its momentum compared to the IA. The fast component of the DIS final state does not interact strongly with the spectator.

Figure 2

Inclusive electron scattering from the deuteron with an identified proton in the deuteron fragmentation region, $e+d\to e^{\prime }+p+X$ (“tagged DIS”).

Figure 3

Tagged DIS in the collinear frame, Eq. (2.33). The deuteron momentum ${\mathit{p}}_d$ and the vector $\mathit{q}$ are collinear and define the $z$ axis, with $\mathit{q}$ pointing in the negative $z$ direction. The initial and final electron momenta lie in the $xz$ plane and have positive $x$ component. ${\varphi }_p$ is the angle of the transverse ($xy$) component of the recoil momentum, measured relative to the positive $x$ axis.

Figure 4

Physical region of recoil proton momentum phase space in the variables ${\alpha }_p$ and $t^{\prime }$.

Figure 5

Deuteron $pn$ LF wave function.

Figure 6

Current matrix element of tagged DIS on the deuteron in the IA.

Figure 7

The deuteron spectral function in the IA, $S_d\left[\mathrm{IA}\right]$, with the pole factor $R/{\left(t^{\prime }\right)}^2$ extracted, as function of $t^{\prime }$, for ${\alpha }_p=1$. The spectral function is evaluated in the nonrelativistic approximation Eq. (4.43) using the AV18 deuteron wave function [56]. Dashed line: $S$-wave contribution only. Solid line: Sum of $S$ and $D$ waves.

Figure 8

Current matrix element in DIS on the nucleon with an identified hadron $h$ in the nucleon fragmentation region, $eN\to e^{\prime }+h+X^{\prime }$. The LF plus momenta of the final state are expressed as fractions of $p_N^{+}$.

Figure 9

Momentum distributions of nucleons ($M_h=M_N$) in the nucleon fragmentation region in DIS, $eN\to e^{\prime }+h+X$. The contours show the allowed values of $p_h^z$ and $p_h^x$ (with $p_h^y=0$, so that $|{\mathit{p}}_{hT}|=|p_h^x|$) for given constant values of ${\zeta }_h$. The contours thus describe the intersection of the allowed three-momentum cones with the transverse $x$ plane.

Figure 10

Minimal three-momentum $|{\mathit{p}}_h|\left(\min \right)$ of nucleons ($M_h=M_N$) in the nucleon fragmentation region in DIS, Eq. (5.6), as a function of the variable $\xi =x+O(M_N^2/Q^2)$, Eq. (5.2).

Figure 11

Current matrix elements in tagged DIS on the deuteron. (a) IA current. (b) FSI between a slow DIS hadron $h$ and the spectator.

Figure 12

Momentum distributions in the radial integrals in the rescattering integral, Eq. (6.59). Shown are the integrands as functions of the intermediate proton momentum $p\equiv |{\mathit{p}}_{p1}|$, evaluated with the AV18 wave functions. Red and blue lines: $S$ and $D$ waves. Solid and dashed lines: Momentum dependence without and with the $t_1$ dependence of the rescattering amplitude.

Figure 13

The ratio of the linear FSI and IA deuteron spectral functions, $S_d\left[\mathrm{FSI}\right]/S_d\left[\mathrm{IA}\right]$, Eqs. (6.53) and (6.54), as a function of the cosine of the recoil momentum angle in the deuteron rest frame, $cos{\theta }_{pq}$, for several values of the recoil momentum modulus $p_p\equiv \left|{\mathit{p}}_p\right|$, as indicated on the plot. The plot shows the $S_d\left[\mathrm{FSI}\right]$ obtained from the $S$ wave only and the total from $S+D$ waves; the $S_d\left[\mathrm{IA}\right]$ in the denominator is always the total from $S+D$ waves. Note that the ratio shown here includes only the term linear in the FSI amplitude, not the quadratic one.

Figure 14

The ratio of the distorted spectral function to the IA spectral function, $S_d\left[\mathrm{dist}\right]/S_d\left[\mathrm{IA}\right]$, Eqs. (6.52)–(6.55), as a function of the cosine of the recoil momentum angle in the deuteron rest frame, $cos{\theta }_{pq}$, for several values of the recoil momentum modulus $p_p\equiv \left|{\mathit{p}}_p\right|$. The distorted spectral function includes the IA, FSI, and ${\mathrm{FSI}}^2$ terms.

Figure 15

The ratio of the distorted spectral function to the IA spectral function, $S_d\left[\mathrm{dist}\right]/S_d\left[\mathrm{IA}\right]$, Eqs. (6.52)–(6.55), as a function of $-t^{\prime }$, for a fixed value ${\alpha }_p=1$. The plot shows separately the IA, IA + FSI, and IA + FSI + ${\mathrm{FSI}}^2$ results.

Figure 16

The distorted spectral function $S_d\left[\mathrm{dist}\right]$, Eqs. (6.52)–(6.55), with the pole factor $R/{\left(t^{\prime }\right)}^2$ extracted, as function of $t^{\prime }$, for ${\alpha }_p=1$ (cf. Fig. 7). The plot shows separately the IA, IA + FSI, and IA + FSI + ${\mathrm{FSI}}^2$ results.

Figure 17

Momentum distribution of the deuteron with the AV18 wave function [56]. The histogram shows the probabilities to find a nucleon with momentum $p_1<p<p_2$, Eq. (A12), for $p_{1,2}$ in steps of 50 MeV. The constant values shown at $p_1<p<p_2$ give the value of $P_d(p_1<p<p_2)$ for that range. The median momentum is indicated by a vertical line.