Quark-hadron duality and parity violating asymmetry of electroweak reactions in the Δ region

A dynamical model [T. Sato and T.-S. H. Lee, Phys. Rev. C 54, 2660 (1996); 63, 055201 (2001); T. Sato, D. Uno, and T.-S. H. Lee, ibid. 67, 065201 (2003)] of electroweak pion production reactions in the Δ(1232) region has been extended to include the neutral current contributions for examining the local quark-hadron duality in neutrino-induced reactions and for investigating how the axial $N\text{−}\Delta$ form factor can be determined by the parity violating asymmetry of $N(\overrightarrow{e},e^{\text{'}})$ reactions. We first show that the recent data of $(e,e^{\text{'}})$ structure functions $F_1$ and $F_2$, which exhibit the quark-hadron duality, are in good agreement with our predictions. For possible future experimental tests, we then predict that the structure functions $F_1,F_2$, and $F_3$ for $(\nu ,e)$ and $(\nu ,{\nu }^{\text{'}})$ processes also show the similar quark-hadron duality. The spin-dependent structure functions $g_1$ and $g_2$ of $(e,e^{\text{'}})$ have also been calculated from our model. It is found that the local quark-hadron duality is not seen in the calculated $g_1$ and $g_2$, while our results for $g_1$ and some polarization observables associated with the exclusive $p(\overrightarrow{e},e^{\text{'}}\pi )$ and $\overrightarrow{p}(\overrightarrow{e},e^{\text{'}}\pi )$ reactions are in reasonably good agreement with the recent data. In the study of parity violating asymmetry A of $N(\overrightarrow{e},e^{\text{'}})$ reactions, the relative importance between the nonresonant mechanisms and the Δ excitation is investigated by taking into account the unitarity condition. Predictions are made for using the data of A to test the axial $N\text{−}\Delta$ form factors determined previously in the studies of $N({\nu }_{\mu },{\mu }^{-}\pi )$ reactions. The predicted asymmetry A are also compared with the parton model predictions for future experimental investigations of quark-hadron duality.

Figure 1

Nonresonant pion production mechanisms of the SL model. See text for how these mechanisms are related to the electromagnetic current, charged current, and neutral current.

Figure 2

Δ excitation mechanism.

Figure 3

Structure functions $F_1$ (left) and $F_2$ (right) of $(e,e^{\text{'}})$ for the proton target at $Q^2=1.5$ (GeV/c)${}^2$. Dashed curves are calculated by using the CTEQ6 parton distribution functions at $Q^2=10$ (GeV/c)${}^2$. Solid curves in the region near $\xi ~0.6$ are calculated from the SL model. Data are from Liang et al. [10].

Figure 4

Structure functions $F_1$ (left) and $F_2$ (right) of $(e,e^{\text{'}})$ for the proton target. Dashed curves are calculated from using the CTEQ6 parton distribution functions at $Q^2=10$ (GeV/$c){}^2$. Solid curves are the results at $Q^2=0.7,1.5,2.5,3.5$ (GeV/$c){}^2$ (from left to right) calculated from the SL model. Data are from Liang et al. [10].

Figure 5

Structure functions ${\mathit{xF}}_1$ (left) and $F_2$ (right) of $(e,e^{\text{'}})$ for the proton (top) and an $I=0$ deuteron like target (bottom). Dashed curves are calculated by using the CTEQ6 parton distribution functions at $Q^2=10$ (GeV/c)${}^2$. Solid curves are the results at $Q^2=0.4,1,2,4$ (GeV/c)${}^2$ (from left to right) calculated from the SL model.

Figure 6

Structure functions ${\mathit{xF}}_1$ (left), $F_2$ (center), and ${\mathit{xF}}_3$ (right) of $(\nu ,e)$ for the proton (top) and an $I=0$ deuteron like target (bottom). The dashed curves are calculated from using the CTEQ6 parton distribution functions at $Q^2=10$ (GeV/c)${}^2$. The solid curves are the results at $Q^2=0.4,1,2,4$ (GeV/c)${}^2$ (from left to right) calculated from the SL model.

Figure 7

Structure functions ${\mathit{xF}}_1$ (left), $F_2$ (center), and ${\mathit{xF}}_3$ (right) of $(\nu ,{\nu }^{\text{'}})$ for the proton (top) and an $I=0$ deuteron like target (bottom). The dashed curves are calculated from using the CTEQ6 parton distribution functions at $Q^2=10$ (GeV/c)${}^2$. The solid curves are the results at $Q^2=0.4,1,2,4$ (GeV/c)${}^2$ (from left to right) calculated from the extended SL model described in this work.

Figure 8

Solid curves are the structure functions $F_2$ of $p(e,e^{\text{'}})$ as a function of the scaling variable ξ calculated at $Q^2=10,15,20$ (GeV/c)${}^2$ (from left to right) from the SL model. The dashed curve is calculated by using the CTEQ6 parton distributions at $Q^2=20$ (GeV/c)${}^2$.

Figure 9

Beam asymmetry $A_{{\mathrm{LT}}^{\text{'}}}$ (defined in Ref. 22) vs ${\varphi }_{\pi }^{*}$ of $p(e,e^{\text{'}}{\pi }^{+})n$ reaction at $Q^2=0.4$ (GeV/c)${}^2$ and $W=1.22$ GeV predicted by the SL model compared with the data of Joo et al. [22]. $({\theta }_{\pi }^{*},{\varphi }_{\pi }^{*}$) are the pion angles in the center of mass frame of the final $\pi N$ system.

Figure 10

Transverse-longitudinal interference term ${\sigma }_{{\mathrm{LT}}^{\text{'}}}$ (defined in Ref. 22) vs $\cos {\theta }^{*}$ for $p(e,e^{\text{'}}{\pi }^0)p$ (top) and $p(e,e^{\text{'}}{\pi }^{+})n$ (bottom) at $W=1.22$ GeV predicted by the SL model compared with the data of Joo et al. [22]. ${\theta }_{\pi }^{*}$ is the pion scattering angle in the center of mass frame of the final $\pi N$ system.

Figure 11

Beam-target asymmetries $A_t$(upper half) and $A_{\mathrm{et}}$(lower half) (as defined in Ref. 21) of $p(e,e^{\text{'}}{\pi }^0)p$ reaction as a function of $\cos {\theta }^{*}$ integrated over ${\varphi }^{*}$ at $W=1.22$ GeV and $0.5<Q^2$ $<0.9$ (GeV/c)${}^2$ (left), $0.9$<$Q^2$ $<1.5$ (GeV/c)${}^2$ (right). $({\theta }^{*},\varphi *$) are the pion angles in the center of mass frame of the final $\pi N$ system. Solid curves are the predictions of the SL model. Data are from Biselli et al. [21].

Figure 12

Beam-target asymmetries $A_t$(upper half) and $A_{\mathrm{et}}$(lower half) (as defined in Ref. 21) of $p(e,e^{\text{'}}{\pi }^0)p$ reaction as a function of ${\varphi }^{*}$ integrated over $\cos {\theta }^{*}$ at $W=1.22$ GeV and $0.5<Q^2$ $<0.9\hspace{1em}(\mathrm{GeV}/c){}^2$ (left), $0.9<Q^2$ $<1.5$ (GeV/c)${}^2$ (right). $({\theta }^{*},{\varphi }^{*}$) are the pion angles in the center of mass frame of the final $\pi N$ system. Solid curves are the predictions of the SL model. Data are from Biselli et al. [21].

Figure 13

Spin-dependent structure functions $g_1$ (left) and $g_2$ (right) for the proton target. Dashed curves are from the fits to the deeply inelastic scattering data as explained in the text. Solid curves are the results at $Q^2=0.21,0.35,0.62,0.92,1.37$ (GeV/c)${}^2$ (from left to right) calculated from the SL model. Data for $g_1$ are from Fatemi et al. [23].

Figure 14

Parity violating aymmetry terms ${\Delta }_A$(solid curve) and $-{\Delta }_V$(dotted curve) [defined in Eqs. (53, 54)] of $p(\overrightarrow{e},e^{\text{'}})$ as a function of invariant mass W calculated by the extended SL model described in this work. Results are for incident electron energy $E=1$ GeV and scattering angle $\theta ={110}^{ˆ}$.

Figure 15

Parity violating asymmetry terms ${\Delta }_A$(solid curves) and ${\Delta }_V$(dotted curves) [defined in Eqs. (53, 54)] of $p(\overrightarrow{e},e^{\text{'}})$ as a function of the incident electron energy $E_e$ calculated by the extended SL model described in this work. Results are for invariant mass $W=1.232$ GeV and electron scattering angles $\theta ={60}^{ˆ}$ (left) and $\theta ={110}^{ˆ}$ (right).

Figure 16

Parity violating asymmetry term ${\Delta }_A$ [defined in Eq. (54)] of $p(\overrightarrow{e},e^{\text{'}})$ as a function of $Q^2$ evaluated with an invariant mass $W=1.232$ GeV and electron scattering angle $\theta ={110}^{ˆ}$. Solid curve is from the extended SL model described in this work. Dashed curve is from turning off the pion cloud effect on the axial $N\text{−}\Delta$ transition. Dotted curve near the bottom is from keeping only the nonresonant amplitude in the calculation.

Figure 17

$Q^2$ dependence of the scaled parity violating asymmetry $A/Q^2$ of $p(\overrightarrow{e},e^{\text{'}})$ calculated from the extended SL model described in this work. Results are for $W=1.232$ GeV and incident electron energy $E=0.8$ (solid curve), 1.5 (dotted), 2.0 (dashed), and 4.0 (dash-dotted).

Figure 18

Scaled parity violating asymmetry $A/Q^2$ of $p(\overrightarrow{e},e^{\text{'}})$ calculated from the extended SL model described in this work. Solid (dashed) curve is from using the axial $N\text{−}\Delta$ form factor $G_{N,\Delta }^A$ of Ref. 3 (Refs. 27). Results are for invariant W = 1.232 GeV and electron angle $\theta ={110}^{ˆ}$.

Figure 19

Scaled parity violating asymmetry $A/Q^2$ of $p(\overrightarrow{e},e^{\text{'}})$ as a function of the scaling variable ξ predicted by the extended SL model described in this work (solid curves) compared with that calculated from Eqs. (55, 56, 57, 58, 59) using the CTEQ6 parton distribution functions (dashed curves). Results are for $E=4$ GeV, $Q^2=0.5$ (GeV/c)${}^2$ (top left), $E=6$ GeV, $Q^2=1$ (GeV/c)${}^2$ (top right), $E=8$ GeV, $Q^2=5$ (GeV/c)${}^2$ (bottom left), and $E=12$ GeV, $Q^2=10$ (GeV/c)${}^2$ (bottom right).