Dispersive approach to two-photon exchange in elastic electron-proton scattering

We examine the two-photon exchange corrections to elastic electron-proton scattering within a dispersive approach, including contributions from both nucleon and $\mathrm{\Delta }$ intermediate states. The dispersive analysis avoids off-shell uncertainties inherent in traditional approaches based on direct evaluation of loop diagrams and guarantees the correct unitary behavior in the high-energy limit. Using empirical information on the electromagnetic nucleon elastic and $N\mathrm{\Delta }$ transition form factors, we compute the two-photon exchange corrections both algebraically and numerically. Results are compared with recent measurements of $e^{+}p$ to $e^{-}p$ cross section ratios from the CLAS, VEPP-3, and OLYMPUS experiments, as well as with polarization transfer observables.

Figure 1

Contributions to elastic electron-nucleon scattering from (a) one-photon exchange and (b) two-photon exchange amplitudes, with particle momenta as indicated. For TPE we have $q_1+q_2=q$. Only the $s$-channel “box” diagram is drawn. The “crossed-box” contribution, which can be obtained by applying crossing symmetry $s\to u$, is implied.

Figure 2

Visualization of the Passarino-Veltman functions from the TPE amplitudes that have an imaginary part, for (a) $b_0\left(s\right)$; (b) $c_0(s;\mathrm{\Lambda })$; (c) $d_0(s;{\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2)$, where $s$ is the total invariant mass of the system. Here external legs have been amputated. The double line indicates a hadronic state of invariant mass $W$, and the dotted line indicates that the electron and hadronic states are on-shell. For elastic scattering, $c_0(s;\mathrm{\Lambda })$ is the same whichever photon propagator is pinched, and $d_0(s;{\mathrm{\Lambda }}_2,{\mathrm{\Lambda }}_1)=d_0(s;{\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2)$.

Figure 3

(a) The contour around the unit circle in the physical region, $-1\le cos\theta \le 1$. For every pole $z_{\mathrm{i}}$ inside the contour (denoted by a dot), there is a corresponding pole at $z_{\mathrm{o}}=1/z_{\mathrm{i}}$ lying outside the contour (denoted by a cross). For monopole form factors, interior poles in $Q_1^2$ appear along the real $z$ axis between 0 and 1, and corresponding poles in $Q_2^2$ along the line at angle $\theta$. (b) One possible choice of contour in the unphysical region, $cos\theta <1$, which we denote as ${\mathrm{\Gamma }}_{\mathrm{CD}}$. For monopole form factors, interior poles in $Q_1^2$ lie along the real $z$ axis between 0 and 1, and corresponding poles in $Q_2^2$ lie between $\beta$ and 0.

Figure 4

(a) The circular contour $\mathrm{\Gamma }$ (shown in red) in the unphysical region is valid for form factors having poles $Q_p^2$ satisfying the condition $R>cos\eta$, where $Q_p^2/Q^2=R{e}^{i\eta }$. For comparison, the light gray line is the contour ${\mathrm{\Gamma }}_{\mathrm{CD}}$ of Fig. 3. (b) Visualization of the condition on the poles of the form factors in the complex $Q_i^2$ plane, using $Q^2=4{\mathrm{GeV}}^2$ as an example. Any form factors with poles in the shaded region of a circle centered at $\{\frac{1}{2}{Q}^2,0\}$ of radius $\frac{1}{2}{Q}^2$ are not allowed. This will always occur for a large enough value of $Q^2$, since the limit $Q^2\to \infty$ is the whole spacelike region.

Figure 5

$Q^2$ dependence of the proton (a) electric and (b) magnetic form factors, scaled by the dipole form factor, $G_D\left(Q^2\right)$ [see Eq. (59)], for the Kelly [53] (dashed blue curves), Venkat et al. [50] (solid red curves), AMT [49] (dot-dashed green curves), and Bernauer et al. [52] (dotted black curves) parametrizations.

Figure 6

Imaginary parts of the TPE contributions from nucleon intermediate states to the generalized $F_1^{\prime }$ (solid red curves), $F_2^{\prime }$ (dashed blue curves), and $G_a^{\prime }$ (dotted green curves) form factors as a function of energy $E$, for fixed $Q^2=3{\mathrm{GeV}}^2$: (a) illustrating the low energy behavior, $E\to 0$, and (b) showing the asymptotic behavior at large $E$.

Figure 7

Dispersive results for the real parts of the TPE contributions from nucleon intermediate states to the generalized $F_1^{\prime }$ (solid red curve), $F_2^{\prime }$ (dashed blue curve), and $G_a^{\prime }$ (dotted green curve) form factors as a function of energy $E$, for fixed $Q^2=3{\mathrm{GeV}}^2$. The scattering threshold is at $E=1.97$ GeV. For $G_a^{\prime }$ the dispersive results are compared with the direct (off-shell) loop calculation (dot-dashed green curve).

Figure 8

Dispersive TPE correction to the cross section, ${\delta }_N$, from nucleon intermediate states as a function of $\varepsilon$ for fixed values of $Q^2$ (= 0.5, 1, 2 and 3 ${\mathrm{GeV}}^2$). The corrections are relative to the Mo-Tsai infrared result [7].

Figure 9

(a) ${\gamma }^{*}N\mathrm{\Delta }$ transition form factors $g_1, g_2$, and $g_3$ versus $Q^2$, as used in this analysis [60, 61]; (b) magnetic $G_M^{*}$ form factor, scaled by the dipole form factor, $3\times G_D$; (c) electric to magnetic dipole ratio $R_{\mathrm{EM}}$ (in percent); and (d) Coulomb to magnetic dipole ratio $R_{\mathrm{SM}}$ (in percent). In (b), (c), and (d) the transition form factors from Aznauryan [60, 61] (solid red curves) used in this analysis are compared with data from Ungaro et al. [62] (circles), Frolov et al. [63] (diamonds), and Joo et al. [64] (triangles), and with the recent parametrization by Zhou andYang [14].

Figure 10

TPE contributions from $\mathrm{\Delta }$ intermediate states to the (a) imaginary part and (b) real part of the generalized $F_1^{\prime }$ (solid red curves), $F_2^{\prime }$ (dashed blue curves), and $G_a^{\prime }$ (dotted black curves) form factors as a function of energy $E$, for fixed $Q^2=3{\mathrm{GeV}}^2$.

Figure 11

Dispersive TPE correction to the cross section, ${\delta }_{\mathrm{\Delta }}$, from $\mathrm{\Delta }$ intermediate states as a function of $\varepsilon$ for fixed values of $Q^2$ (= 0.5, 1, 2, and 3 ${\mathrm{GeV}}^2$).

Figure 12

Comparison of the $\mathrm{\Delta }$ intermediate state contribution to the TPE cross section, ${\delta }_{\mathrm{\Delta }}$, for the dispersive (solid red curves) and loop (dashed blue curves) calculations at fixed $Q^2=3{\mathrm{GeV}}^2$: (a) versus $\varepsilon$, and (b) versus electron energy $E$. Note that $\varepsilon =0.9$ corresponds to $E=5.9$ GeV for this $Q^2$. In (b) the loop corrections using a dipole approximation to the ${\gamma }^{*}N\mathrm{\Delta }$ form factor with mass 0.75 GeV is also shown (dotted green curve). For simplicity only the dominant magnetic contribution has been considered.

Figure 13

Ratio $R_{2\gamma }$ of $e^{+}p$ to $e^{-}p$ cross sections as a function of $\varepsilon$ for (a) fixed $\langle Q^2\rangle =0.85{\mathrm{GeV}}^2$ and (b) fixed $\langle Q^2\rangle =1.45{\mathrm{GeV}}^2$, and as a function of $Q^2$ for (c) fixed $\langle \varepsilon \rangle =0.45$ and (d) fixed $\langle \varepsilon \rangle =0.88$. The contributions with nucleon only (dashed blue curves) and the sum of nucleon and $\mathrm{\Delta }$ (solid red curves) intermediate states are compared with data from CLAS at Jefferson Lab (circles) [25], with the statistical and systematic uncertainties indicated by the (black) inner and (gray) outer error bars, respectively.

Figure 14

Ratio $R_{2\gamma }$ of $e^{+}p$ to $e^{-}p$ cross sections as a function of $\varepsilon$ for fixed energy (a) $E=1.594$ GeV and (b) $E=0.998$ GeV. The contributions with nucleon only (dashed blue curves) and the sum of nucleon and $\mathrm{\Delta }$ (solid red curves) intermediate states are compared with data from the VEPP-3 experiment (triangles) [26], with the statistical and systematic uncertainties indicated by the (black) inner and (gray) outer error bars, respectively.

Figure 15

Ratio $R_{2\gamma }$ of $e^{+}p$ to $e^{-}p$ cross sections as a function of $\varepsilon$ for fixed energy $E=2.01$ GeV. The contributions with nucleon only (dashed blue curves) and the sum of nucleon and $\mathrm{\Delta }$ (solid red curves) intermediate states are compared with data from the OLYMPUS experiment [28] (squares). The statistical and systematic uncertainties (correlated and uncorrelated) are indicated by the (black) inner and (gray) outer error bars, respectively.

Figure 16

(a) Ratio of the total longitudinal recoil proton polarization $P_L$ to the Born contribution $P_L^{\left(0\right)}$ and (b) ratio $R_{TL}$ of the transverse to longitudinal polarizations relative to the Born contribution, as a function of $\varepsilon$ for fixed $Q^2=2.49{\mathrm{GeV}}^2$. The dispersive calculation including nucleon only (dashed blue curves) and the sum of nucleon and $\mathrm{\Delta }$ (solid red curves) intermediate states is compared with the data (circles) from the GEp2$\gamma$ experiment at Jefferson Lab [31], with the statistical and systematic (correlated and uncorrelated) uncertainties indicated by the (black) inner and (gray) outer error bars, respectively.