Phenomenological extraction of two-photon exchange amplitudes from elastic electron-proton scattering cross section data

Background: The inconsistency in the results obtained from the Rosenbluth separation method and the high-$Q^2$ recoil polarization results on the ratio ${\mu }_p{G}_E^p/G_M^p$ implies a systematic difference between the two techniques. Several studies suggest that missing higher-order radiative corrections to elastic electron-proton scattering cross section ${\sigma }_R(\varepsilon ,Q^2)$ and in particular hard two-photon-exchange (TPE) contributions could account for the discrepancy.

Purpose: In this work, I improve on and extend to low and high $Q^2$ values the extractions of the $\varepsilon$ dependence of the real parts of the TPE amplitudes relative to the magnetic form factor, as well as the ratio $P_l/P_l^{\text{Born}}(\varepsilon ,Q^2)$ by using world data on ${\sigma }_R(\varepsilon ,Q^2)$ with an emphasis on precise new data covering the low-momentum region which is sensitive to the large-scale structure of the nucleon.

Method: I combine cross section and polarization measurements of elastic electron-proton scattering to extract the TPE amplitudes. Because the recoil polarization data were confirmed “experimentally” to be essentially independent of $\varepsilon$, I constrain the ratio $P_t/P_l(\varepsilon ,Q^2)$ to its $\varepsilon$-independent term (Born value) by setting the TPE contributions to zero. This allows for the amplitude $Y_M(\varepsilon ,Q^2)$ and ${\sigma }_R(\varepsilon ,Q^2)$ to be expressed in terms of the remaining two amplitudes $Y_E(\varepsilon ,Q^2)$ and $Y_3(\varepsilon ,Q^2)$ which in turn are parametrized as second-order polynomials in $\varepsilon$ and $Q^2$ to reserve as possible the linearity of ${\sigma }_R(\varepsilon ,Q^2)$ as well as to account for possible nonlinearities in the TPE amplitudes. Furthermore, I impose the Regge limit which ensures the vanishing of the TPE contributions to ${\sigma }_R(\varepsilon ,Q^2)$ and the TPE amplitudes in the limit $\varepsilon \to 1$.

Results: I provide simple parametrizations of the TPE amplitudes, along with an estimate of the fit uncertainties. The extracted TPE amplitudes are compared with previous phenomenological extractions and TPE calculations. The $P_l/P_l^{\text{Born}}$ ratio is extracted by using the new parametrizations of the TPE amplitudes and compared to previous extractions, TPE calculations, and direct measurements at $Q^2{=2.50(\mathrm{GeV}/c)}^2$.

Conclusions: The extracted TPE amplitudes are on the few-percentage-points level and behave roughly linearly with increasing $Q^2$ where they become nonlinear at high $Q^2$. Contrary to $Y_M$, which is influenced mainly by elastic contributions, I find $Y_E$ to be influenced by inelastic contributions at large $Q^2$ values. While $Y_E$ and $Y_3$ differ in magnitude, they have opposite sign and tend to partially cancel each other. This suggests that the TPE correction to ${\sigma }_R(\varepsilon ,Q^2)$ is driven mainly by $Y_M$ and to a lesser extent by $Y_3$ in agreement with previous phenomenological extractions and hadronic TPE calculations.

Figure 1

The reduced cross section ${\sigma }_R$ as a function of $\varepsilon$ for a sample of low- and high-$Q^2$ data points from the data of Refs. [60, 61, 62]. Also shown are my new fit based on Eq. (15) (solid black line) and the Rosenbluth fit (dashed red line) for comparison.

Figure 2

${[G_E^p/G_D\left(Q^2\right)]}^2$ (top) and ${[G_M^p/{\mu }_p{G}_D\left(Q^2\right)]}^2$ (bottom) as obtained by using Eq. (16) and the parametrization of the ratio $R$ from Eq. (10) (open dark-green squares). In addition, I compare the results to the extractions from several previous TPE calculations and phenomenological fits: AMT [65] (solid blue line), VAMZ [66] (solid magenta line), Bernauer [62] (long-dashed red line), ABGG [67] (dashed-dotted black line), Arrington $Y_{2\gamma }$ [45] (dashed-dotted magenta line), and Puckett (large-dotted blue line; the fit labeled “new” in Ref. [68]).

Figure 3

The TPE amplitude coefficients ${\alpha }_{(0,1,2)}\left(Q^2\right)$ and ${\beta }_{(0,1,2)}\left(Q^2\right)$ as a function of $Q^2$ as extracted from this work (open black circles) along with the fits (solid red lines) valid up to $Q^2=4{(\mathrm{GeV}/c)}^2$. Note that $Q^2$ points which yielded large TPE amplitudes were excluded for clarity. See text for details.

Figure 4

The extracted TPE amplitudes from this work: $Y_M$ (solid black line), $Y_E$ (solid red line), and $Y_3$ (solid dark-green line) as a function of $\varepsilon$ at the $Q^2$ value listed in the figure. In addition, I compare the results to previous hadronic calculations assuming different intermediate states: elastic labeled as “$Y_M$ elastic” (dashed-dotted black line), “$Y_E$ elastic” (dashed-dotted red line), “$Y_3$ elastic” (dashed-dotted dark-green line) from Ref. [28], and elastic $+\pi N$ with spin 1/2 and 3/2 channels labeled as “$Y_M$ spin” (long-dashed black line), “$Y_E$ spin” (long-dashed red line), “$Y_3$ spin” (long-dashed dark-green line) from Ref. [32].

Figure 5

The extracted TPE amplitudes from this work: $Y_M$ (top; solid black line), $Y_E$ (middle; solid red line), and $Y_3$ (bottom; solid dark-green line) as a function of $\varepsilon$ at $Q^2=2.50{(\mathrm{GeV}/c)}^2$. The error bands on the amplitudes are shown as very-long-dashed (black, red, dark-green) lines, respectively. In addition, I compare the results with phenomenological extractions from Ref. [46] “Guttmann Fit I” FIG. 5. (Continued) (dashed black line) and “Guttmann Fit II” (long-dashed black line), and to several previous hadronic TPE predictions: “BK: elastic” [28] (dashed red line), “BK: elastic $+\mathrm{\Delta }\left(1232\right)$” [30] (long-dashed red line), “BK: elastic $+P_{33}$” [31] (solid magenta line), and “BK: elastic $+\pi N$” [32] (dashed-dotted dark-green line). Also shown are calculations based on QCD factorization “KV” [18] (dotted blue line). In a leading twist QCD-type calculation the $Y_E$ amplitude cannot be calculated. See text for details.

Figure 6

The extracted TPE amplitudes from this work: $Y_M$ (top), $Y_E$ (middle), and $Y_3$ (bottom) as a function of $Q^2$ at fixed $\varepsilon =0.25$ (solid black lines). The error bands on the amplitudes are shown as very-long-dashed black lines. In addition, I compare the results with several previous hadronic TPE calculations assuming different intermediate states: elastic “BK: elastic” [28] (solid red line), elastic $+\mathrm{\Delta }$(1232) “BK: $\mathrm{\Delta }$ (1232)” [30] (solid magenta line), elastic $+\pi N\left(P_{33}\right)$ channel “BK: elastic $+\pi N\left(P_{33}\right)$” [31] (solid dark-green line), and elastic $+\pi N$ (spin 1/2 and 3/2 channels) “BK: elastic $+\pi N$ (spin 1/2 and 3/2)” [32] (dashed dark-green line).

Figure 7

The ratio $P_l/P_l^{\text{Born}}$ as a function of $\varepsilon$ as determined by using my parametrizations of the TPE amplitudes and Eq. (9c) at the $Q^2$ value listed in the figure (solid black line).

Figure 8

The extracted ratio $P_l/P_l^{\text{Born}}$ as a function of $\varepsilon$ at $Q^2=2.50{(\mathrm{GeV}/c)}^2$ from this work (solid black line). Also shown are the data points from the GEp-$2\gamma$ experiment Ref. [56] (solid red squares). The blue star indicates the value of $\varepsilon$ at which the data have been normalized to unity. In addition, I compare the results to previous fits from Ref. [46] “Guttmann Fit I” (dashed black line), and “Guttmann Fit II” (long-dashed black line), and to several previous theoretical predictions: “Hadronic” [13] (solid red line), “GPD” [23] (dashed red line), “KV-Soft Spectator” (solid blue line) and “KV-hard Spectator” (dashed blue line) [18], and “MV” [19] prediction band (solid dark-green lines).