Measurements of $ep\to e^{\prime }{\pi }^{+}{\pi }^{-}{p}^{\prime }$ cross sections with CLAS at $1.40\mathrm{GeV}<W<2.0$ GeV and $2.0{\mathrm{GeV}}^2<Q^2<5.0{\mathrm{GeV}}^2$

This paper reports new exclusive cross sections for $ep\to e^{\prime }{\pi }^{+}{\pi }^{-}{p}^{\prime }$ using the CLAS detector at Jefferson Laboratory. These results are presented for the first time at photon virtualities $2.0{\mathrm{GeV}}^2<Q^2<5.0{\mathrm{GeV}}^2$ in the center-of-mass energy range 1.4 GeV $<W<$ 2.0 GeV, which covers a large part of the nucleon resonance region. Using a model developed for the phenomenological analysis of electroproduction data, we see strong indications that the relative contributions from the resonant cross sections at $W<1.74$ GeV increase with $Q^2$. These data considerably extend the kinematic reach of previous measurements. Exclusive $ep\to e^{\prime }{\pi }^{+}{\pi }^{-}{p}^{\prime }$ cross section measurements are of particular importance for the extraction of resonance electrocouplings in the mass range above 1.6 GeV.

Figure 1

Vertex reconstruction projected onto the beam axis for Sector 2 of CLAS, before (dashed) and after (black) applying corrections to align the sectors of CLAS. The vertical lines show the region of the vertex event selection. The small peak at zero originates from an aluminum window 2 cm downstream of the target cell.

Figure 2

The position of electron events in the EC for the six sectors of CLAS for all events (light gray or red online) and selected events (black). The stripe seen in the lower left sector is from inefficient phototubes on a few scintillator strips of the EC. The same inefficiencies were introduced in the simulations of the detector acceptance.

Figure 3

The energy deposited in the inner ($E_{\mathrm{in}}$) and outer ($E_{\mathrm{out}}$) layers of the EC for all electron candidates. The line corresponds to 60 MeV, which separates the minimum ionizing pions (to the left) and electrons (to the right).

Figure 4

Ratio of the total energy in the EC $E_{\mathrm{tot}}$ and the track momentum versus momentum for $3.5{\mathrm{GeV}}^2<Q^2<4.2{\mathrm{GeV}}^2$ for data (left) and Monte Carlo (right).

Figure 5

Velocity difference $\mathrm{\Delta }\beta ={\beta }_{\mathrm{TOF}}-{\beta }_{\mathrm{DC}}$ for a sample of positively charged tracks versus momentum for assumed masses of a pion (top) or a proton (bottom).

Figure 6

Histogram of the correlation between the momentum $p$ and the polar angle $\theta$ for tracks of positively charged pions in Sector 3 of CLAS. The inefficient regions of the detector, shown between the bands of solid lines, were removed from the analysis.

Figure 7

Square of the missing mass $M_X^2$ for $ep\to e^{\prime }{\pi }^{+}{\pi }^{-}{p}^{\prime }$, showing a peak at the ${\pi }^{-}$ mass squared. The dashed histogram is from the Monte Carlo and the solid histogram is the data. The vertical lines show the applied cut.

Figure 8

The kinematic coverage of the ${\pi }^{+}{\pi }^{-}p$ events, shown as a scatter plot as a function of squared 4-momentum transfer $Q^2$ and center-of-mass energy $W$. Bins are shown within which the integrated and nine onefold differential ${\pi }^{+}{\pi }^{-}p$ cross sections were obtained.

Figure 9

Angular variables from the set defined by Eq. (13) for the description of the $ep\to e^{\prime }{\pi }^{+}{\pi }^{-}{p}^{\prime }$ reaction in the CM frame of the final state hadrons. (a) The ${\pi }^{-}$ spherical angles ${\theta }_{{\pi }^{-}}$ and ${\phi }_{{\pi }^{-}}$. Plane A is defined by the 3-momenta of the initial state proton and the final state ${\pi }^{-}$. Plane C represents the electron scattering plane. (b) The angle ${\alpha }_{\left[{\pi }^{-}p\right]\left[{\pi }^{+}{p}^{\prime }\right]}$ between the two defined hadronic planes A and B. Plane B is defined by the 3-momenta of the final state ${\pi }^{+}$ and $p^{\prime }$. The vectors $\vec{\gamma }$ and $\vec{\beta }$ are normal to the ${\pi }^{-}$ 3-momentum in the planes A and B, respectively.

Figure 10

A comparison between the measured event distributions (solid circles) and the simulated event distributions (open squares) within the framework of the JM06 model [30, 31, 32], which was further adjusted to reproduce the measured event distributions. These comparisons are shown for the representative bin of $W$ = 1.99 GeV and $Q^2=4.6{\mathrm{GeV}}^2$.

Figure 11

Impact of the interpolation of the fivefold ${\pi }^{+}{\pi }^{-}p$ differential cross sections into the blind areas of CLAS to the nine onefold differential cross sections at $W$ = 1.81 GeV and $Q^2=2.6{\mathrm{GeV}}^2$. The onefold differential cross sections obtained assuming zero fivefold differential cross sections and the interpolated values for these cross sections in the blind areas of CLAS are shown by the black squares and red circles, respectively. The error bars represent statistical uncertainties. To aid visualization, we have slightly shifted apart the two data sets.

Figure 12

The radiative correction factor $1/R$ for the representative bin $4.2{\mathrm{GeV}}^2<Q^2<5.0{\mathrm{GeV}}^2$ as a function of $W$. The solid magenta line represents a polynomial plus Gaussian fit.

Figure 13

Ratio of the elastic cross sections measured from the CLAS e1-6 data set to the Bosted parametrization [34] as a function of electron polar angle $\theta$ for each of the six sectors of CLAS. The regions where there are missing data are the result of $\theta$ vs $p$ cuts to remove problematic areas of the detector. The horizontal lines represent $\pm 10\%$ deviations of the ratio from unity.

Figure 14

Fully integrated cross sections for ${\pi }^{+}{\pi }^{-}p$ electroproduction off protons as a function of $W$ at photon virtualities $Q^2=2.2$, 2.6, 3.2, 3.8, and $4.6{\mathrm{GeV}}^2$. The error bars represent the statistical uncertainties.

Figure 15

The resonant contributions from the JM16 model [4, 16, 18] (red solid lines) in comparison with the CLAS results on the fully integrated ${\pi }^{+}{\pi }^{-}p$ electroproduction cross sections (points with statistical error bars) as a function of $Q^2$ in three $W$ bins near the central mass of the $N\left(1440\right)1/2^{+}$: $W$ = 1.41 GeV (left), $W$ = 1.44 GeV (center), and $W$ = 1.46 GeV (right). The systematic uncertainties of the measurements are shown by the bands at the bottom of each plot. The dashed black lines that form a band about the central solid red JM16 prediction represent the model uncertainties. The black solid lines represent the full cross sections from the JM16 model.

Figure 16

The resonant contributions from the JM16 model [4, 16, 18] (red solid lines) in comparison with the CLAS results on the fully integrated ${\pi }^{+}{\pi }^{-}p$ electroproduction cross sections (points with statistical error bars) as a function of $Q^2$ in three $W$ bins near the central mass of the $N\left(1520\right)3/2^{-}$: $W$ = 1.51 GeV (left), $W$ = 1.54 GeV (center), and $W$ = 1.56 GeV (right). The dashed black lines that form a band about the central solid red JM16 prediction represent the model uncertainties. The black solid lines represent the full cross sections from the JM16 model.

Figure 17

The resonant contributions from the JM16 model [4, 16, 18] (red solid lines) in comparison with the CLAS results on the fully integrated ${\pi }^{+}{\pi }^{-}p$ electroproduction cross sections (points with statistical error bars) as a function of $Q^2$ in the $W$ range from 1.70 GeV to 1.82 GeV: $W$ = 1.71 GeV (left) at the resonant maximum in Fig. 14, and $W$ = 1.79 GeV (center) and $W$ = 1.81 GeV (right) that are located between the four-star resonances with masses below 1.74 GeV and above 1.90 GeV. The dashed black lines that form a band about the central solid red JM16 prediction represent the model uncertainties. The black solid lines represent the full cross sections from the JM16 model.

Figure 18

The resonant contributions from the JM16 model [4, 16, 18] (red solid lines) to the nine onefold differential ${\pi }^{+}{\pi }^{-}p$ electroproduction cross sections in representative $W$ bins inside two $W$ intervals of distinctively different resonant content described in Sec. 3 at $Q^2=2.1{\mathrm{GeV}}^2$. The model uncertainties for the resonant contributions are shown by the thin red band at the bottom of each plot. The black solid lines represent the full cross sections from the JM16 model.

Figure 19

The resonant contributions from the JM16 model [4, 16, 18] (red solid lines) to the nine onefold differential ${\pi }^{+}{\pi }^{-}p$ electroproduction cross sections in representative $W$ bins inside two $W$ intervals of distinctively different resonant content described in Sec. 3 at $Q^2=3.2{\mathrm{GeV}}^2$. The model uncertainties for the resonant contributions are shown by the thin red band at the bottom of each plot. The black solid lines represent the full cross sections from the JM16 model.

Figure 20

The resonant contributions from the JM16 model [4, 16, 18] (red solid lines) to the nine onefold differential ${\pi }^{+}{\pi }^{-}p$ electroproduction cross sections in representative $W$ bins inside two $W$ intervals of distinctively different resonant content described in Sec. 3 at $Q^2=4.6{\mathrm{GeV}}^2$. The model uncertainties for the resonant contributions are shown by the thin red band at the bottom of each plot. The black solid lines represent the full cross sections from the JM16 model.