Model analysis of the $\mathit{ep}\to e^{\text{'}}p{\pi }^{+}{\pi }^{-}$ electroproduction reaction on the proton

Recent CLAS data on the $p{\pi }^{+}{\pi }^{-}$ electroproduction off protons at $1.3<W<1.57$ GeV and $0.25<Q^2<0.6\hspace{0.5em}{\mathrm{GeV}}^2$ have been analyzed using a meson-baryon phenomenological model. By fitting nine onefold differential cross-section data for each $W$ and $Q^2$ bin, the charged double-pion electroproduction mechanisms are identified from their manifestations in the observables. We have extracted the cross sections from amplitudes of each of the considered isobar channels as well as from their coherent sum. We also obtained nonresonant partial wave amplitudes of all contributing isobar channels that could be useful for advancing a complete coupled-channels analysis of all meson electroproduction data.

Figure 1

The mechanisms of JM05 model [51, 52] contributing to $p{\pi }^{+}{\pi }^{-}$ electroproduction in kinematic area covered by recent CLAS measurements [10] $W<1.6$ GeV and $0.25<Q^2<0.6\hspace{0.5em}{\mathrm{GeV}}^2$: three-body mechanisms (a), the processes contributing to $\pi \Delta$ isobar channels (b), and $2\pi$ direct production mechanisms (c).

Figure 2

The kinematics variables for description of $ep\to e^{\text{'}}{p}^{\text{'}}{\pi }^{+}{\pi }^{-}$ reaction in the center-of-mass frame of the final hadrons (first assignment presented in the Sec. 2). The top plot (a) shows ${\pi }^{-}$ spherical angles ${\theta }_{{\pi }^{-}}$ and ${\phi }_{{\pi }^{-}}$ while the bottom plot (b) shows angle ${\alpha }_{[p{\pi }^{-}][p^{\text{'}}{\pi }^{+}]}$ between two planes: one of them (plane A) is defined by the three-momenta of the initial proton and the final ${\pi }^{-}$, and a second (plane B) is defined by the three-momenta of the two others final hadrons ${\pi }^{+}$ and $p$.

Figure 3

Manifestation of additional contributions to the $\pi \Delta$ isobar channels with respect to the resonant and Born terms, parametrized by extra contact terms, described in Sec. 3. The data [10] at $W=1.33$ GeV [six panels at the left side (a)], $W=1.36$ GeV [six panels at the right side (b)] and $Q^2=0.425\hspace{0.5em}{\mathrm{GeV}}^2$ are shown by full symbols. Full JM06 calculations are shown by solid lines, while the same cross sections, calculated taking off the contributions from additional contact terms in $\pi \Delta$ channels, are shown by dotted lines.

Figure 4

A signature of direct $2\pi$ production mechanisms, seen in the previous CLAS $p{\pi }^{+}{\pi }^{-}$ data [9]. The data on ${\pi }^{-}$ angular distributions at $W=1.49\hspace{0.5em}\mathrm{GeV},Q^2=0.65\hspace{0.5em}{\mathrm{GeV}}^2$ (a) and at $W=1.49\hspace{0.5em}\mathrm{GeV},Q^2=0.95\hspace{0.5em}{\mathrm{GeV}}^2$ (b) are shown by points with error bars. The full calculation is shown by solid lines. The dashed lines correspond to the contributions from JM05 direct $2\pi$ production [51].

Figure 5

A signature of direct $2\pi$ production in the recent CLAS $p{\pi }^{+}{\pi }^{-}$ data at low $W$ and $Q^2$ [10, 11]. ${\pi }^{-}$ center-of-mass angular distributions at $W=1.49\hspace{0.5em}\mathrm{GeV},Q^2=0.325\hspace{0.5em}{\mathrm{GeV}}^2$ (a), $W=1.49\hspace{0.5em}\mathrm{GeV},Q^2=0.375\hspace{0.5em}{\mathrm{GeV}}^2$ (b), $W=1.49\hspace{0.5em}\mathrm{GeV},Q^2=0.425\hspace{0.5em}{\mathrm{GeV}}^2$ (c), $W=1.46\hspace{0.5em}\mathrm{GeV},Q^2=0.525\hspace{0.5em}{\mathrm{GeV}}^2$ (d) and their fit within the framework of the JM05 model [52]. Full calculations are shown by solid lines, while the contributions from direct $2\pi$ processes of JM05 model are shown by dash lines.

Figure 6

Direct $2\pi$ production mechanisms of JM06 model. Top (a) [bottom (b)]: the diagrams on the left side, describing direct $2\pi$ production mechanisms of JM05 model, were replaced by diagrams, shown in the middle and right parts of the plot. Contact interaction in down (upper) vertex was replaced by additional particle-exchange amplitudes. Sets of diagrams in the middle and right parts of the plot correspond to various assignments for the four-momenta squared running over propagators in exchange amplitudes [see definitions Eqs. (7) and (9) of $t_i\hspace{0.5em}(i=1,2,3)$ for diagrams in top/bottom rows, respectively].

Figure 7

The recent CLAS $p{\pi }^{+}{\pi }^{-}$ data at $W=1.51\hspace{0.5em}\mathrm{GeV},Q^2=0.43\hspace{0.5em}{\mathrm{GeV}}^2$ and their fit within the framework of JM05 model shown by solid lines. The contributions from direct $2\pi$ production, resulting in discrepancies in the description of ${\pi }^{+}$ and $p$ angular distributions, are shown by dashed lines.

Figure 8

Fit of recent CLAS charged double-pion data [10] at $W=1.51$ GeV and $Q^2=0.43\hspace{0.5em}{\mathrm{GeV}}^2$ after discussed in Sec. 3 improvements in parametrization of direct $2\pi$ production mechanisms. Full JM06 calculations are shown by solid lines, while the contributions from ${\pi }^{-}{\Delta }^{++}$, ${\pi }^{+}{\Delta }^0$ isobar channels, and direct $2\pi$ production are shown by dot-dash, dot, and dash lines, respectively.

Figure 9

Fit of CLAS data on $p{\pi }^{+}{\pi }^{-}$ electroproduction [10] at $W=1.46$ GeV and $Q^2=0.33\hspace{0.5em}{\mathrm{GeV}}^2$ within the framework of JM06 model. Selected in the fitting procedure calculated differential cross sections are shown by bunches of dashed lines.

Figure 10

The contributions from various isobar channels to the fully integrated $p{\pi }^{+}{\pi }^{-}$ electroproduction cross sections. The recent CLAS data are shown by full symbols. Shadowed areas represent the systematical uncertainties. Full calculations within the framework of JM06 are shown by solid lines. The contributions from ${\pi }^{-}{\Delta }^{++},{\pi }^{+}{\Delta }^0$ channels are shown by dashed and dot-dashed lines, respectively. The contributions from direct $2\pi$ production are shown by dotted lines.

Figure 11

Differential cross sections for the coherent sum of $\pi \Delta$ channel amplitudes at $W=1.46$ GeV and $Q^2=0.325\hspace{0.5em}{\mathrm{GeV}}^2$ are shown by bars connected by curves. The bar sizes represent uncertainties for these cross sections. The cross section values corresponding to the best data fit are shown by solid lines. Full symbols are the measured cross sections.

Figure 12

Contributions from various isobar channels to charged double-pion electroproduction obtained from the CLAS data fit at $W=1.46$ GeV and $Q^2=0.325\hspace{0.5em}{\mathrm{GeV}}^2$. The cross sections for ${\pi }^{-}{\Delta }^{++}$, ${\pi }^{+}{\Delta }^0$ isobar channels are shown by bars connected by solid and dash lines, respectively. Full symbols are the measured cross sections. The uncertainties of bars represent the spread of the cross sections calculated with JM06 parameters determined from the fit of measured differential cross sections, as described in Sec. 4. The values connected by solid lines represent the ${\pi }^{-}{\Delta }^{++}$ and ${\pi }^{+}{\Delta }^0$ channel cross sections, estimated with JM06 parameters, that correspond to the best data fit of minimal ${\chi }^2$/d.p..

Figure 13

Resonant and nonresonant contributions, determined from the recent CLAS data fit within the framework of the JM06 model. The data are shown by full symbols. Shadowed areas represent the systematic uncertainties. Full JM06 calculations are shown by full solid lines. The contributions from $N^{*}$’s and nonresonant mechanisms are shown by dot-dashed and dashed lines, respectively.

Figure 14

Resonant (dot-dash lines) and nonresonant (dot lines) mechanism contributions to the onefold differential cross sections at $W=1.51$ GeV and $Q^2=0.425\hspace{0.5em}{\mathrm{GeV}}^2$. The solid lines represent the full JM06 calculation.

Figure 15

Born amplitudes in JM06 model.