Two-photon exchange corrections to $\gamma^*N\Delta$ form factors for $Q^2\leq 4 $ (Gev/$c$)$^2$

TL;DR

This paper calculates two-photon exchange corrections to gamma-star N Delta transition form factors at moderate to high Q^2, finding small effects on some form factors but significant corrections on others, impacting multipole analyses.

Contribution

It provides the first hadronic model estimate of TPE effects on gamma-star N Delta form factors, including comparisons with partonic calculations at high Q^2.

Findings

TPE effects on G*_M are negligible.

TPE effects on G*_E and G*_C are up to 15% at Q^2 ~ 4 GeV^2.

Significant TPE corrections suggest their inclusion in high Q^2 multipole analyses.

Abstract

We evaluate the corrections of the two-photon exchange (TPE) process on the $\gamma^* N\Delta$ transition form factors. The contributions of the TPE process to the $eN\rightarrow e\Delta(1232)\rightarrow eN\pi$ are calculated in a hadronic model with the inclusion of only the elastic nucleon intermediate states, to estimate its effects on the multipoles $M^{(3/2)}_{1^+}, E^{(3/2)}_{1^+}, S^{(3/2)}_{1^+}$ at the $\Delta$ peak. We find that TPE effects on $G^*_M$ is very small. $G^*_E,$ and $ G^*_C$ are also little affected at small $Q^2 $. For $G^*_E$, the TPE effects reach about $3 - 8\%$ near $Q^2\sim 4$ GeV$^2$, depending on the model, MAID or SAID, used to emulate the data. For $G^*_C$, the TPE effects decrease rapidly with increasing $\epsilon$ while growing with increasing $Q^2$ to reach $\sim 6 - 15\%$ with $Q^2 \sim 4$ GeV$^2$ at $\epsilon = 0.2$. Sizeable TPE corrections to $G^*_E$ and $G^*_C$ found here points to the need of including TPE effects in the multipole analysis in the region of high $Q^2$ and small $\epsilon$. The TPE corrections to $R_{EM}$ and $R_{SM}$ obtained in our hadronic calculation are compared with those obtained in a partonic calculation for moderate momentum transfer of $2<Q^2<4$ GeV$^2$.

Equations

$$G_M^{\ast }=N\mathrm{Im}{M}_{1^{+}}^{(3/2)}(Q^2,W=M_{\Delta }),$$

$$G_E^{\ast }=-N\mathrm{Im}{E}_{1^{+}}^{(3/2)}(Q^2,W=M_{\Delta }),$$

$$G_C^{\ast }=-(\frac{2M_{\Delta }}{q_{\Delta }})N\mathrm{Im}{S}_{1^{+}}^{(3/2)}(Q^2,W=M_{\Delta }),$$

$$R_{EM}\equiv (E_{1^{+}}^{(3/2)}/M_{1^{+}}^{(3/2)}){\mid }_{W=M_{\Delta }}=-G_E^{\ast }/G_M^{\ast }\to 1,$$

$$R_{SM}\equiv (S_{1^{+}}^{(3/2)}/M_{1^{+}}^{(3/2)}){\mid }_{W=M_{\Delta }}$$

$$=-(Q_{+}{Q}_{-}/4M_{\Delta }^2)(G_C^{\ast }/G_M^{\ast })\to const.$$

$${\mathcal{M}}^{2\gamma ,a}=-i\int \frac{d^{4}k}{(2\pi {)}^4}\overline{u}(p_3){\Gamma }_{\mu }^{\gamma ee}{S}^{(e)}(p_1-k){\Gamma }_{\nu }^{\gamma ee}u(p_1)$$

$$\times \frac{-i}{k^2+i\epsilon }\frac{-i}{(P_{\Delta }-P_N{)}^2+i\epsilon }\overline{u}(p_4){\Gamma }^{\pi N\Delta ,\alpha }(q)S_{\alpha \beta }^{(\Delta )}(P_{\Delta })$$

$$\times {\Gamma }_{\gamma N\to \Delta }^{\mu \beta }(P_{\Delta },P_{\Delta }-P_N)S^{(N)}(P_N){\Gamma }^{\gamma NN,\nu }(k)u(p_2),$$

$${\mathcal{M}}^{2\gamma ,ct}=-i\int \frac{d^{4}k}{(2\pi {)}^4}\overline{u}(p_3){\Gamma }_{\mu }^{\gamma ee}{S}^{(e)}(p_1-k){\Gamma }_{\nu }^{\gamma ee}u(p_1)$$

$$\times \frac{-i}{k^2+i\epsilon }\frac{-i}{(P_{\Delta }-P_N{)}^2+i\epsilon }\overline{u}(p_4){\Gamma }^{\pi N\Delta ,\alpha }(q)S_{\alpha \beta }^{(\Delta )}(P_{\Delta })$$

$$\times {\Gamma }_{\gamma \gamma N\Delta }^{\mu \nu \beta }(P_{\Delta },p_2,k,p_4-p_2-k)u(p_2),$$

$${\Gamma }_{\gamma \gamma N\Delta }^{\mu \nu \beta }(P_{\Delta },p_2,k,\overline{k})$$

$$=e\{(2p_2+k{)}^{\nu }\frac{F_1(k)}{(p_2+k{)}^2-M_N^2}{\Gamma }_{\gamma N\to \Delta }^{\mu \beta }(P_{\Delta },\overline{k})$$

$$+(2p_2+\overline{k}{)}^{\mu }\frac{F_1(\overline{k})}{(p_{\Delta }-k{)}^2-M_N^2}{\Gamma }_{\gamma N\to \Delta }^{\nu \beta }(P_{\Delta },k)\},$$

$$\frac{d{\sigma }^{1\gamma }}{d{\Omega }_{\pi }}=\{{\sigma }_0^{1\gamma }+\sqrt{2ϵ(1+ϵ)}{\sigma }_{LT}^{1\gamma }\cos \varphi +ϵ{\sigma }_{TT}^{1\gamma }\cos 2\varphi \},$$

$$\frac{d{\sigma }^{ex}}{d{\Omega }_{\pi }}\simeq \frac{d{\sigma }^{1\gamma }}{d{\Omega }_{\pi }}=C|{\mathcal{M}}^{1\gamma }(X_{1^{+}}^{1\gamma },Z_{l^{\pm }}^{1\gamma }){|}^2$$

$$\frac{d{\sigma }^{ex}}{d{\Omega }_{\pi }}\simeq \frac{d{\sigma }^{1\gamma +2\gamma }}{d{\Omega }_{\pi }}$$

$$=C\{|{\mathcal{M}}^{1\gamma }(X_{1^{+}}^{1\gamma +2\gamma },Z_{l^{\pm }}^{1\gamma +2\gamma }){|}^2$$

$$+2Re[{\mathcal{M}}^{1\gamma \ast }(X_{1^{+}}^{1\gamma +2\gamma },Z_{l^{\pm }}^{1\gamma +2\gamma }){\mathcal{M}}^{2\gamma }],$$

$$\frac{d{\bar{\sigma }}^{ex}}{d{\Omega }_{\pi }}$$

$$\frac{d{\bar{\sigma }}^{ex,i+1}}{d{\Omega }_{\pi }}$$

$$\displaystyle\chi_{i}^{2}(Q^{2},\epsilon)\equiv\sum_{\theta_{\pi},\phi}\Big{(}\frac{\mathcal{N}_{i+1}}{\delta d\sigma^{ex}(\theta_{\pi},\phi)}\Big{)}^{2},$$

$${\mathcal{N}}_{i+1}$$

Full text

The Jones-Scadron form factors, magnetic dipole $G^{*}_{M}$, electric quardrupole $G^{*}_{E}$, and Coulomb quardrupole $G^{*}_{C}$, which describe electromagnetic transition between the first two lowest baryon states, nucleon and the $\Delta(1232)$ resonance, are of fundamental interest. They are proportional to the three multipoles $M_{1^{+}}^{(3/2)},E_{1^{+}}^{(3/2)},S_{1^{+}}^{(3/2)}$ at the resonance peak Pascal07 , which are all purely imaginary. Namely, on the resonance peak $W=M_{\Delta}$, one has

[TABLE]

where $N=\frac{8}{e}(\frac{\pi k_{\Delta}M_{\Delta}\Gamma_{\Delta}}{3q_{\Delta}}\frac{Q_{+}}{Q_{-}})^{1/2}(\frac{M_{N}}{M_{N}+M_{\Delta}})$, with $e^{2}/4\pi\simeq 1/137$, $Q_{\pm}\equiv[(M_{\Delta}\pm M_{N})^{2}+Q^{2}]^{1/2}$, $\Gamma_{\Delta}$ the $\Delta$ width, and $M_{N}$ and $M_{\Delta}$ are the nucleon and $\Delta$ masses, respectively. $q_{\Delta}$ and $k_{\Delta}$ denote the magnitude of the virtual photon and pion three momentum in the $\Delta$ rest frame at the resonance position, respectively.

At sufficiently large four-momentum transfer squared $Q^{2}$, perturbative QCD (pQCD) predicts that only helicity-conserving amplitudes contribute Brodsky81 , leading to $G^{*}_{M},G^{*}_{E},G^{*}_{C}$ scaling as $Q^{-4},Q^{-4}$, and $Q^{-6}$, respectively. It follows that

[TABLE]

In the nonpertubative regime with low $Q^{2}$, a symmetric SU(6) quark model would allow the electromagnetic excitation of the $\Delta$ to proceed only via $M1$ transition. However, the tensor component of the one-gluon exchange interaction between quarks would induce a $D-$state in the $\Delta$, which leads to a deformed $\Delta$ and the photon can excite a nucleon through electric $E2$ and Coulomb $C2$ quardrupole transitions, resulting in nonvanishing $E_{1^{+}}^{(3/2)}$ and $S_{1^{+}}^{(3/2)}$ multipoles. Experiments give, near $Q^{2}=0$, $R_{EM}=-(2.5\pm 0.5)\%$ Beck97 , a clear indication of $\Delta$ deformation. Below $Q^{2}\leq 6$ Gev2, $R_{EM}$ remains small and negative, while $R_{SM}$ continues to become more negative with increasing $Q^{2}$, indicating that pQCD limit is nowhere in sight. The intriguing difference in the behaviors of the $R_{EM}$ in the perturbative and nonperturbative domains remains to be understood.

The multipoles are extracted from pion electroproduction experiments based on one-photon exchange (OPE) approximation. OPE approximation has been widely used to analyze most of the electromagnetic nuclear reactions. The validity of OPE approximation has recently been under heavy scrutiny Carlson07 ; Arrington11 ; Yang13 . It was prompted by the sustantial difference in the ratio of proton electric and magnetic form factors extracted from $ep$ elastic scattering via Rosenbluth technique Arrington03 ; Qattan05 and polarization transfer measurements Jones00 ; Puckett10 ; Zhan11 , for $Q^{2}<6$ GeV2. The two-photon exchange (TPE) corrections as estimated by hadronic and partonic calculations show that TPE effects can account for more than half of that discrepancy.

It is hence important to determine how much TPE effects would affect the extraction of multipoles from pion electroproduction. Specifically we will be concerned with only the multipoles related to $N\Delta$ transition in this study, namely, how the extraction of $E_{1^{+}}^{(3/2)},M_{1^{+}}^{(3/2)}$, and $S_{1^{+}}^{(3/2)}$, or equivalently the transition form factors, would be affected in the presence of TPE. This question was addressed in Pascal06 , where a partonic approach, with the use of $N\Delta$ generalized parton distributions, was employed to estimate the TPE effects. For $2<Q^{2}<4$ GeV2 at $\epsilon=0.2$, they found that the TPE corrections on $R_{EM}$ and $R_{SM}$, are small, lying between $-(0.2-0.6)\%$ level. However, it is known that the partonic approach is applicable only for $Q^{2}$ large comparable to a typical hadronic scale and becomes questionable for $Q^{2}$, which in the current case, less than $\sim 2-3$ GeV2. In these lower $Q^{2}$ region, hadronic approach as developed in Blunden03 would be more reliable, which motivates this investigation.

In this work, we present results of a hadronic calculation of the TPE corrections, as depicted in Fig. 1, where only the elastic $N$ intermediate states are considered, to the process $eN\rightarrow eN\pi$ on the $\Delta$ peak. The intermediate nucleons are assumed to be on-mass-shell, which is justified in the study of TPE effects in $ep$ elastic scatterings within hadronic approach in Blunden03 .

As in Zhou07 ; Zhou10 ; Zhou15 , we choose Feynman gauge and neglect electron mass $m_{e}$ in the numerators to obtain the amplitude of box diagram in Fig. 1(a) as

[TABLE]

where $\Gamma^{\gamma ee}_{\mu}=-ie\gamma_{\mu}$, and $\Gamma^{\gamma NN}_{\mu}$=$ie\langle P(p^{\prime})|J^{em}_{\mu}|P(p)\rangle$, the proton e.m. current matrix element. $\Gamma^{\pi N\Delta,\alpha}(q)=(f_{\pi N\Delta}/m_{\pi})q^{\alpha}T^{\dagger}$ denotes the $\pi N\Delta$ transition vertex function with $f^{2}_{\pi N\Delta}/4\pi=0.36$ and $T^{\dagger}$ the isospin $1/2\rightarrow 3/2$ transition operator. $S^{(e,N,\Delta)}$ denote the propagators of electron, proton, and $\Delta$, respectively, as specified by the superscript. The forms of the $S^{(\Delta)}$ and the $\gamma N\rightarrow\Delta$ transition vertex function $\Gamma^{\mu\beta}_{\gamma N\rightarrow\Delta}$ can be found in Zhou15 . The realistic form factors are used for $\Gamma^{\gamma NN}_{\mu}$ and $\Gamma^{\mu\beta}_{\gamma N\rightarrow\Delta}$ as in Zhou15 ; Zhou10 . Amplitude for the** cross-box diagram can be written down similarly. A contact term $M^{2\gamma,ct}$, as depicted in Fig. 1b, is needed because of the requirement of current conservation. Following the prescription suggested in Kondra06 , we obtain

[TABLE]

with

[TABLE]

where $\overline{k}=p_{4}-p_{2}-k$, and $F_{1}$ is the Dirac form factor of the nucleon. The inclusion of contact term of Eq. (4) makes the full amplitude gauge invariant as discussed in Kondra06 . We have also checked numerically that the full amplitude does not dependent on the gauge parameter. It is also essential to ensure the sum to be free of IR divergence. The packages FEYNCALC Mertig91 and LoopTools Hahn99 are used to carry out the analytical and numerical calculations, respectively.

Within the OPE approximation, the fivefold $eN\rightarrow eN\pi$ differential cross section, with both unpolarized initial and final states, can be expressed as $d^{5}\sigma^{1\gamma}/d\Omega_{f}dE_{f}d\Omega_{\pi}\equiv\Gamma d\sigma^{1\gamma}/d\Omega_{\pi}$, with $\Gamma$ the virtual photon flux factor and

[TABLE]

where $\sigma_{0}^{1\gamma}=\sigma_{T}^{1\gamma}+\epsilon\sigma_{L}^{1\gamma}$, and $\epsilon$ the transverse polarization of the virtual photon. The superscript $1\gamma$ is used to emphasize that the quantities are defined within the OPE approximation scheme, a convention to be followed hereafter. $E_{f},\Omega_{f}$ denote the energy and solid-angle of the scattered electron in the lab frame, respectively, and $\phi$ is the tilt angle between the electron scattering plane and the reaction plane, $d\Omega_{\pi}$ is the pion solid-angle differential measured in the c.m. frame of the final pion and nucleon.

The OPE differential cross sections \sigma^{1\gamma}_{T,L,LT,TT}$${}^{\prime}s are all functions of multipoles, which depend on $W$, $Q^{2}$ and pion polar angle $\theta_{\pi}$ in $\pi N$ c.m. frame, but $\epsilon$-independent. The multipoles are determined in multipole analysis, e.g., MAID MAID07 or SAID SAID07 , by fitting the experimental data as,

[TABLE]

where $d\sigma^{ex}/d\Omega_{\pi}$ is measured experimentally. Here $X^{1\gamma}_{1^{+}}=(E_{1^{+}}^{(3/2),1\gamma},M_{1^{+}}^{(3/2),1\gamma},S_{1^{+}}^{(3/2),1\gamma})$ denote the multipoles pertaining to the $\Delta$ excitation channel of $(3/2,3/2)$, $Z^{1\gamma}_{l^{\pm}}$ represents all other multipoles, and $C$ is a kinematical factor.

With the TPE effects included, the analysis of the experimental data should be performed by using,

[TABLE]

where the term $|\mathcal{M}^{2\gamma}|^{2}$ has been neglected. $(X^{1\gamma+2\gamma}_{1^{+}},Z^{1\gamma+2\gamma}_{l^{\pm}})$ are the multipoles determined from the OPE plus TPE approximation of Eq. (8), as referred to by the superscript $1\gamma+2\gamma$, a notation to be followed hereafter. Obviously, they must deviate from $(X^{1\gamma}_{1^{+}},Z^{1\gamma}_{l^{\pm}})$ of Eq. (7) based on OPE. Eq. (6) still holds for $d\sigma^{1\gamma+2\gamma}/d\Omega_{\pi}$ but the cross sections $\sigma^{1\gamma+2\gamma}_{T,L,LT,TT}$’$s$ would become $\epsilon$-dependent Pascal07 ; Pascal06 .

In principle, one should try to determine the multipoles $X^{1\gamma+2\gamma}_{1^{+}}$ and $Z^{1\gamma+2\gamma}_{l^{\pm}}$ in the presence of TPE by fitting the data with Eq. (8). The obtained values of the multipoles would represent the genuine multipoles as would be defined within the OPE approximation scheme, with TPE effects removed, from the data.

Extraction of $X^{1\gamma+2\gamma}_{1^{+}},Z^{1\gamma+2\gamma}_{l^{\pm}}$’s from data via Eq. (8) is beyond the scope of the present study. To proceed, two approximations will be made. First, we assume that only the multipoles X^{1\gamma+2\gamma}_{1^{+}}$${}^{\prime}s will be much affected in the presence of TPE depicted in Fig. 1. This can be justified because the final $\pi N$ pair there arised only from the decay of $\Delta$ and would be in the state with $(J=3/2,I=3/2)$ only. The multipoles $Z^{1\gamma+2\gamma}_{l^{\pm}}$ will then be taken to be unchanged and fixed, i.e., $Z^{1\gamma+2\gamma}_{l^{\pm}}=Z^{1\gamma}_{l^{\pm}}$ and Eq. (8) is reduced to depend only on the three multipoles of X^{1\gamma+2\gamma}_{1^{+}}$${}^{\prime}s. The Fermi-Watson theorem requires that these three multipoles should all have the phase given by the $\pi N\,\,P_{33}$ phase shift, which is $\pi/2$ on the $\Delta$ peak. So the three multipoles $X^{1\gamma+2\gamma}_{1^{+}}$’s will all become purely imaginary in Eq. (8). Hereafter, $X_{1^{+}}$ will be taken to denote the imaginary part of $X^{1\gamma+2\gamma}_{1^{+}}$ for brevity.

Eq. (8) is then simplified to

[TABLE]

where a TPE-corrected cross section $d\bar{\sigma}^{ex}/d\Omega_{\pi}$ is introduced. Dependence on Z_{l^{\pm}}$${}^{\prime}s in $\mathcal{M}^{1\gamma}$ in Eq. (9) is not shown for simplicity since they remain fixed. We like to emphasize here that the \sigma^{ex}_{T,L,LT,TT}$${}^{\prime}s, are in principle $\epsilon$-dependent. Only with precisely determined d\sigma^{ex}/d\Omega_{\pi}$${}^{\prime}s and a complete theory for $\mathcal{M}^{2\gamma}$ would lead to $\epsilon$-independent \bar{\sigma}^{ex}_{T,L,LT,TT}$${}^{\prime}s. $d\bar{\sigma}^{ex}/d\Omega_{\pi}$ is only then expressible in the form of $|\mathcal{M}^{1\gamma}|^{2}$.

To proceed, we approximate the data $d\sigma^{ex}/d\Omega_{\pi}$ with the use of one of the existing $eN\rightarrow eN\pi$ models, MAID MAID07 or SAID SAID07 , to be denoted as $\sigma^{MAID/SAID}_{T,L,LT,TT}$. There is a caveat here with such an approximation. All existing models, like MAID and SAID, are all based on OPE approximation and the resulting cross sections \sigma^{1\gamma}_{T,L,LT,TT}$${}^{\prime}s and multipoles would hence be $\epsilon$-independent. Approximating $\epsilon$-dependent \sigma^{ex}_{T,L,LT,TT}$${}^{\prime}s by $\epsilon$-independent \sigma^{MAID/SAID}_{T,L,LT,TT}$${}^{\prime}s would give rise to X_{1^{+}}$${}^{\prime}s determined from Eq. (9) to be $\epsilon$-dependent.

Once $d\sigma^{ex}/d\Omega_{\pi}$ is given, Eq. (9) then can be solved for X^{1\gamma}_{1^{+}}$${}^{\prime}s by iteration via,

[TABLE]

We start with values of multipoles given by MAID or SAID, i.e., $X^{0}_{1^{+}}=X_{1^{+}}$(MAID/SAID) in the first iteration $i=0$, depending on which model is employed to approximate $d\sigma^{ex}/d\Omega_{\pi}$ in Eq. (9). It should be noted that both the l.h.s. and r.h.s. depend on $\theta_{\pi}$ and $\phi$.

Next, we have to determine the three multipoles X^{i+1}_{1^{+}}$${}^{\prime}s from Eq. (10) for fixed $Q^{2}$ and $\epsilon$ at the $i-$iteration. Upon first glance, one could in principle write down three equations for each of the \sigma_{0,LT,TT}$${}^{\prime}s and solve for the three variables X_{1^{+}}$${}^{\prime}s. These three equations are all quadratic equations in X_{1^{+}}$${}^{\prime}s. It turns out that there are a few angles where no real solutions exist for this coupled algebraic equations. The solutions show rapid variations w.r.t. $\theta_{\pi}$ in the neighbourhood of these angles. The reason can be traced to the approximation we make to replace $d\sigma^{ex}/d\Omega_{\pi}$ by ($d\sigma^{ex}/d\Omega_{\pi}$)(MAID/SAID) in (10).

We hence turn to least-square method. As reported in Zhou17 , results obtained with such minimization procedure show strong sensitivity to the angle-independent weights attached to each of the three cross sections \sigma_{0,LT,TT}$${}^{\prime}s. We now understand that this sensitivity arises from the problem described in the last paragraph. Accordingly, we decide to follow the fitting method adapted in MAID MAID07 . At the $i$-th iteraction, we minimize $\chi^{2}(Q^{2},\epsilon)$ defined as

[TABLE]

with

[TABLE]

where $d\sigma^{ex}/d\Omega_{\pi}=$($d\sigma^{ex}/d\Omega_{\pi}$)(MAID/SAID). $X^{i}_{1^{+}}$’s are kept fixed while $X^{i+1}_{1^{+}}$’s are varied in the minimization of $\chi^{2}_{i}$. In Eq.(12) $\delta d\sigma^{ex}(\theta_{\pi},\phi)$ is the total error of $d\sigma^{ex}(\theta_{\pi},\phi)$ which also depends on $Q^{2}$ and $\epsilon$. In our analysis, the experimental errors at $Q^{2}=2.8$ GeV2, $\epsilon=0.56$ and $Q^{2}=4$ GeV2, $\epsilon=0.5$ provided in Frolov99 , are used. Either set of errors give rise to nearly identical results. We choose to use the ones at $Q^{2}=2.8$ GeV2, $\epsilon=0.56$ for all other values of $Q^{2}$ and $\epsilon$ considered.

We will show only the ratios $X^{1\gamma+2\gamma}_{1^{+}}/X^{1\gamma}_{1^{+}}\equiv G^{*,1\gamma+2\gamma}/G^{*,1\gamma}$ between the TPE-corrected, or the genuine OPE values $X^{1\gamma+2\gamma}_{1^{+}}(\propto G^{*,1\gamma+2\gamma})$ obtained here, and the input OPE values $X^{1\gamma}_{1^{+}}(\propto G^{*,1\gamma})$ given by the models (MAID, SAID) used to emulate the experimental data. They will be labelled as MAID and SAID, respectively. Results for $M^{3/2}_{1^{+}}$ will not be shown as the TPE effects on it are found to be very small with both models. We do not show results above $Q^{2}>4$ GeV2 as the validity of hadronic approach adopted here might be questionable in those high $Q^{2}$-region. The results, obtained with MAID and SAID, are presented for $0<\epsilon<0.9$ at $Q^{2}=0.127$ and 2.8 GeV2, in Fig. 2, and for $0<Q^{2}<4$ GeV2 with $\epsilon=$ 0.2 and 0.5, in Fig. 3, respectively. The results with MAID are denoted by the solid and dotted (red) curves, while the results with SAID are denoted by the dashed and dashed-dot (blue) curves, respectively.

In Fig. 2, one sees that at small $Q^{2}=0.127$ GeV2, the TPE corrections to both $E^{3/2}_{1^{+}}(G^{*}_{E})$ and $S^{3/2}_{1^{+}}(G^{*}_{C})$ are less than $1\%$ and stay flat for all values of $\epsilon$, irrespective of the model used. As $Q^{2}$ grows, TPE effects begin to increase and dependence on the model used develops. For $E^{3/2}_{1^{+}}(G^{*}_{E})$, the TPE corrections eventually reach about 3% and 8% at $4$ GeV2 in the case of MAID and SAID, respectively, as seen in Fig. 3(a), with mild sensitivity w.r.t. $\epsilon$. The TPE corrections to $S^{3/2}_{1^{+}}(G^{*}_{C})$ at $Q^{2}=2.8$ GeV2, as depicted in Figs. 2(b) show considerable sensitivity not only to model but also $\epsilon$, decreasing from around 7.5% and 15% near $\epsilon=0$, for SAID and MAID, respectively, to only 2% as $\epsilon$ approaches 0.9. Fig. 3(b) shows how TPE corrections for $S^{3/2}_{1^{+}}(G^{*}_{C})$ grow with increasing $Q^{2}$ to reach about 15% and 6%, respectively, at $\epsilon=0.2$ and $Q^{2}=4$ GeV2, for MAID and SAID. Sizeable TPE corrections to $E^{3/2}_{1^{+}}(G^{*}_{E})$ and $S^{3/2}_{1^{+}}(G^{*}_{C})$ found here point to the need of including TPE effects in the multipole analysis of data in the region of high $Q^{2}$ and small $\epsilon$.

It is straightforward to obtain the values for the TPE-corrected ratios $R^{1\gamma+2\gamma}_{EM,SM}$ from the results presented in Fig. 3. The difference $\delta R_{EM,SM}$ between $R^{1\gamma+2\gamma}_{EM,SM}$ and the model ratios $R^{1\gamma}_{EM,SM}$, i.e., $\delta R_{EM}\equiv R^{1\gamma+2\gamma}_{EM}-R^{1\gamma}_{EM}$ and $\delta R_{SM}=R^{1\gamma+2\gamma}_{SM}-R^{1\gamma}_{SM}$, for $0<Q^{2}<4$ GeV2 are shown in Fig. 4, where the solid (red) and dashed (blue) curves refer to the results obtained with MAID and SAID, respectively. We first note that the TPE corrections $\delta R_{EM,SM}$ are almost equal with the two models except for $\delta R_{EM}$ when $Q^{2}>2$ GeV2. This is in contrast to Figs. 2 and 3 where model dependence grows rapidly with increasing $Q^{2}$ after $Q^{2}\sim 1$ GeV2. For both $\epsilon=$ 0.2 and 0.5, $\delta R_{EM}$ is negligible for small $Q^{2}$ and becomes more negative toward $-0.1\%$ and $-0.2\%$ when $Q^{2}$ approaches $Q^{2}=4$ GeV2, in the case of MAID and SAID, respectively. The TPE effects for $\delta R_{SM}$ is considerably larger than for $\delta R_{EM}$. It also starts near zero for $Q^{2}\sim 0$ but decreases rapidly to reach $\sim-1.4\%$ and $\sim-0.7\%$, for $\epsilon=$ 0.2 and 0.5, respectively. Magnitude-wise, they are comparable to the current experimental errors Ungaro06 .

The results of the partonic calculation of Pascal06 for $\delta R_{EM/SM}$’s, denoted by black triangles, are included in Fig. 4 for comparison. The regions of validity of the hadronic and partonic approaches are known to be different except possible overlap in the range of $2<Q^{2}<4$ GeV2. It is easily seen that, in this region, our results for $\delta R_{EM}$ at $\epsilon=$ 0.2 obtained with both models are considerably smaller. However, for $\epsilon=$ 0.5, our results obtained with MAID almost coincide with those of Pascal06 , while results obtained with SAID are distinctly smaller than partonic results. In the case of $\delta R_{SM}$, our values are substantially more negative than the partonic results, for both $\epsilon=$ 0.2 and 0.5.

The differences between our results and those of Pascal06 for the R_{EM,SM}$${}^{\prime}s, as shown in Fig. 4, can be dissected as follows. We first point out that there are two more differences between the two calculations besides partonic vs. hadronic approach. First, only the $\Delta$ pole diagram is considered for $\mathcal{M}^{1\gamma}$ in Pascal06 , to evaluate the interference effects between OPE and TPE. In other words, the background contribution to $\mathcal{M}^{1\gamma}$, which consists of Born terms in PV coupling and t-channel $(\rho,\omega)$ vector-meson exchanges Yang85 , are not included in the evaluation of $Re[\mathcal{M}^{1\gamma*}\mathcal{M}^{2\gamma}]$ in Eq. (9). In fact, it was found in KY99 that both the background terms and the pion cloud effects contribute significantly to $M^{(3/2)}_{1^{+}}$ and $E^{(3/2)}_{1^{+}}$ at $Q^{2}=0$. In addition, truncated multipole expansion (TME) is employed in Pascal06 to estimate the values of $R^{1\gamma+2\gamma}_{EM,SM}$. It is known that the use of TME and model fitting used here give rise to considerable difference in the extraction of $R^{1\gamma}_{EM,SM}$, a feature seen in Mertz01 ; Sparveris07 .

To summarize, we investigate the effects of two-photon exchange processes in $eN\rightarrow e\Delta(1232)\rightarrow eN\pi$ in low $Q^{2}$ region, in a hadronic approach. Only the elastic nucleon intermediate states are included in the present study. We focus on the $\Delta$ peak to estimate their effects on the $\gamma^{*}N\Delta$ transition form factors. We emulate the experimental pion electrproduction data with two existing phenomenological models, MAID and SAID. After subtracting out the interference of one-photon and two-photon exchanges from the data, the reminder is used to extract the ”genuine” one-photon exchange multipoles $M^{(3/2)}_{1^{+}},E^{(3/2)}_{1^{+}},S^{(3/2)}_{1^{+}}$ at $W=M_{\Delta}$. This gives us the three $\gamma^{*}N\Delta$ form factors, $G^{*}_{M}$, $G^{*}_{E}$, and $G^{*}_{C}$, for $0<Q^{2}<4$ GeV2.

We find that TPE effects on $G^{*}_{M}$ are very small. Both $G^{*}_{E}$ and $G^{*}_{C}$ are also little affected at small $Q^{2}<0.5$ GeV2. However, the TPE effects on $G^{*}_{E}$ and $G^{*}_{C}$ grow with $Q^{2}$ and the sensitivity w.r.t. $\epsilon$ and the data model used appears. For $G^{*}_{E}$, the TPE effects reach about 3% and 8% at $Q^{2}\sim 4$ GeV2, depending on whether MAID and SAID is used to emulate the data, respectively, with mild dependence on $\epsilon$. For $G^{*}_{C}$, the TPE effects obtained with both MAID and SAID decrease rapidly with increasing $\epsilon$ while grow with increasing $Q^{2}$ and reach $\sim 15\%$ and $\sim 6\%$ as $Q^{2}\rightarrow 4$ GeV2 at $\epsilon=0.2$, respectively, for MAID and SAID. Sizeable TPE corrections to $G^{*}_{E}$ and $G^{*}_{C}$ found here points to the need of including TPE effects in the multipole analysis of data in the region of high $Q^{2}$ and small $\epsilon$.

Our extracted TPE corrections for $\delta R_{EM}\equiv R^{1\gamma+2\gamma}_{EM}-R^{1\gamma}_{EM}$ are very small at $\epsilon=0.2$ and $0.5$, for both MAID and SAID models, up to $Q^{2}\leq 4.0$ GeV2. This feature is similar with results of the partonic calculation of Pascal06 , except our results are only about one third in magnitude given in Pascal06 for $\epsilon=0.2$. However, our TPE corrections for $R_{SM}$, independent of the models used, are considerably larger in magnitude than the results of Pascal06 , reaching $\sim-1.4\%$ and $\sim-0.7\%$ for $\epsilon=$ 0.2 and 0.5, respectively.

Besides hadronic $\it vs.$ partonic approach, the differences between our results and those of Pascal06 for $\delta R_{EM/SM}$’s could be attributed to two other simplifications used in Pascal06 . First, in Pascal06 only the $\Delta$ pole contribution is included in the OPE amplitude in the evaluation of the interference between OPE and TPE amplitudes. In addition, TME is invoked in the extraction of the rations $R_{EM/SM}$’s.

As the TPE effects on $G^{*}_{E}(\sim E^{(3/2)}_{1^{+}})$ and $G^{*}_{C}(\sim S^{(3/2)}_{1^{+}})$ found in this study are not small, more precision measurements on $ep\rightarrow e\Delta(1232)\rightarrow ep\pi^{0}$ in the region of $2<Q^{2}<4$ GeV2 will be very desirable. It is important to have data taken for the same $Q^{2}$ but at different values of $\epsilon$. The $\epsilon$-dependence in the resulting multipoles will be clear signature for the TPE effects.

We have considered only the elastic nucleon intermediate states in the present study. Similar TPE effects arising from the inclusion of higher resonances like $\Delta$ in the intermediate states should be further pursued. TPE effects on the transition form factors of other higher resonances will also be an interesting question to explore.

We thank Dr. Lothar Tiator for helpful communications, regarding MAID. We also thank Dr. T.-S.H. Lee for careful reading of the manuscript and suggestions. This work is supported in part by the National Natural Science Foundation of China under Grant No. 11375044, the Fundamental Research Fund for the Central Universities under Grant No. 2242014R30012 for H.Q.Z. and the National Science Council of the Republic of China (Taiwan) for S.N.Y. under grant No. NSC101-2112-M002-025. H.Q.Z. would like to gratefully acknowledge the support of the National Center for Theoretical Science of the National Science Council of the Republic of China (Taiwan) for his visits in January, 2016 and February, 2017. He also greatly appreciates the warm hospitality extended to him by the Physics Department of the National Taiwan University during the visits.