Hadron structure at low $Q^2$

This review deals with the structure of hadrons, strongly interacting many-body systems consisting of quarks and gluons. These systems have a size of about $1\mathrm{fm}$, which shows up in scattering experiments at low momentum transfers $Q$ in the GeV region. At this scale the running coupling constant of quantum chromodynamics (QCD), the established theory of the strong interactions, becomes divergent. It is therefore highly intriguing to explore this theory in the realm of its strong interaction regime. However, the quarks and gluons cannot be resolved at the GeV scale but have to be studied through their manifestations in the bound many-body systems, for instance, pions, nucleons, and their resonances. The review starts with an overview of QCD at low momentum transfer and a summary of the theoretical apparatus describing the interaction of hadrons with electrons and photons. Experimental results are presented for the most significant observables studied with the electromagnetic probe: form factors, polarizabilities, excitation spectra, and sum rules. These experimental findings are compared and interpreted with various theoretical approaches to QCD, such as phenomenological models with quarks and pions, dispersion relations as a means to connect observables from different experiments, and, directly based on the QCD Lagrangian, chiral perturbation theory and lattice gauge theory.

Figure 1

Multipole notation for pion photoproduction.

Figure 2

(Color online) The isovector spectral functions in units of $m_{\pi }^{-4}$. Solid lines, $2\mathrm{Im}{G}_E^v\left(t\right)∕t^2$; dashed lines, $2\mathrm{Im}{G}_M^v\left(t\right)∕t^2$. Thin lines are from 213, thick lines include modern data for the pion form factor. From 74.

Figure 3

The world data for the four nucleon form factors according to 165. Solid line is the phenomenological fit given by Eqs. (38, 39). The other lines show the two dipole contributions and the bump-dip term of the fit separately. Note that the absolute value is plotted for negative quantities. See 165 for a listing of the data.

Figure 4

The nucleon form factors $G_E^p$, $G_M^p∕{\mu }_p$, and $G_M^n∕{\mu }_n$ as a function of $Q$, divided by the standard dipole form factor of Eq. (37). The solid line represents the fit of 165, the dashed line shows the smooth contribution of the two dipoles.

Figure 5

(Color online) The world data for the electric form factor of the neutron $G_E^n$ as a function of $Q^2$. Solid squares, $d(\vec{e},e^{\prime }\vec{n})p$ at MAMI, 183, and 296; solid triangles, $d(\vec{e},e^{\prime }\vec{n})p$ at Jefferson Laboratory, 278; solid diamond, $d(\vec{e},e^{\prime }\vec{n})p$ at BATES, 155; solid circle, $\vec{d}(\vec{e},e^{\prime }n)p$ at NIKHEF, 306; open triangles, $\vec{d}(\vec{e},e^{\prime }n)p$ at Jefferson Laboratory, 359; open diamonds, $\vec{d}(\vec{e},e^{\prime }n)p$ at BATES, 177; open squares, ${}^3\overrightarrow{\mathrm{He}}(\vec{e},e^{\prime }n)pp$ at MAMI, 318. Open circles are derived from the deuteron quadrupole form factor (327). Solid line, new fit of the phenomenological model of 165; dashed line, the result of dispersion relation (75; 279); and dotted line, the original Galster fit of Eq. (40). The new fit includes all shown data except for the data of 177 and values derived by 327. Data were updated by J. Friedrich and theory curves complemented by L. Tiator.

Figure 6

(Color online) The charge density distribution for the neutron $4\pi r^2{\rho }_E^n$ as a function of $r$. The notation for the lines corresponds to that of the form factors shown in Fig. 5. Courtesy of J. Friedrich and L. Tiator.

Figure 7

(Color online) The world data for the timelike form factor of the proton as function of the beam energy $E_{\text{beam}}$, extracted assuming $\mid G_E\mid =\mid G_M\mid$. Open symbols, data from $e^{+}{e}^{-}\to p\overline{p}$ (123; 135; 98, 99; 38, 39; 2; 53; 308); solid symbols, data from $\overline{p}p\to e^{-}{e}^{+}$ (62; 45; 60; 25; 31). From 322.

Figure 8

(Color online) The world data for the timelike form factor of the neutron as a function of the beam energy $E_{\text{beam}}$, extracted assuming $\mid G_E\mid =\mid G_M\mid$. Data are obtained by the FENICE Collaboration from the reaction $e^{+}{e}^{-}\to n\overline{n}$. Solid line is a perturbative QCD extrapolation using proton data (37, 39). From 322.

Figure 9

(Color online) The ratio of the electric and magnetic form factors of the proton ${\mu }_p{G}_E^p∕G_M^p$ compared to relativistic calculations in the framework of the CQM. Dotted line, light-cone front CQM (126); thick solid line, light-cone front CQM (164); dash-dotted line. light-cone front CQM with pointlike constituent quarks (119; 120); dashed line, Goldstone boson exchange between pointlike constituent quarks (104); and thin solid line, covariant spectator CQM (192). Data have been taken at Jefferson Laboratory by 315 (circles) and 176 (squares). From 309.

Figure 10

(Color online) The transverse and longitudinal cross sections for charged pion production as a function of $Q^2$. Data are from 63. Dashed lines, cross sections predicted by MAID2007 (146); solid lines, fit to the data with MAID2007 (see text for details). Courtesy of L. Tiator.

Figure 11

(Color online) The strangeness form factors $G_E^s$ and $G_M^s$ at $Q^2=0.1{\mathrm{GeV}}^2$ as obtained by the SAMPLE, A4, HAPPEX, and G0 experiments. The bands represent the $1\sigma$ (statistical plus systematical) error of the individual experiments, the ellipse is the combined $2\sigma$ area for all measurements. From 251.

Figure 12

(Color online) The $1\sigma$ band of HAPPEX-II together with the $1\sigma$ and $2\sigma$ ellipses derived by 3. The theoretical predictions with their estimate errors are identified by the following numbers: [18] 297, [19] 198, [20] 199, [21] 336, [22] 261, and [23] 257, 258.

Figure 13

(Color online) The form factor combination $G_E^s\left(Q^2\right)+\eta G_M^s\left(Q^2\right)$ as obtained by kinematical extrapolation from A4, HAPPEX II, and G0 experiments. The point HAPPEX III indicates the error bar for a planned measurement. From 251.

Figure 14

(Color online) The pion form factor $F_{\pi }$ multiplied with $Q^2$. Crosses, 26, 28 ($\pi \text{−}e$ scattering at CERN); asterisks, 4 (DESY); diamonds, 109, 110 (DESY); circles, 345 (Jefferson Laboratory); and squares, 220 (Jefferson Laboratory). Experimental results are compared to four model calculations as indicated by the lines. From 220.

Figure 15

Typical intermediate states contributing to Compton scattering off the nucleon. Top row: (a) Direct and (b) crossed Born diagrams with intermediate nucleons, a typical resonance excitation in the (c) $s$-channel and (d) its crossed version. Bottom row: mesonic contributions with photon scattering off (e) an intermediate pion, (f) the pion pole diagram, and a correlated two-pion exchange such as (g) the “$\sigma$ meson.”

Figure 16

Differential cross section for Compton scattering off the proton as a function of the photon laboratory energy $E_{\gamma }$ and at scattering angle ${\theta }_{\mathrm{lab}}\approx 135°$. Lines show the results of fixed-$t$ subtracted dispersion relations (solid), Klein-Nishina (small dots), Powell (dashed), Powell plus ${\pi }^0$ pole (large dots), and LEX including the leading order contributions of ${\alpha }_{E1}^p$ and ${\beta }_{M1}^p$ (dash-dotted). Experimental data are from 162; 277; and 295. From 148.

Figure 17

Differential cross section for Compton scattering off the proton as a function of the laboratory photon energy $E_{\gamma }$ and at different scattering angles ${\theta }_{\mathrm{lab}}$. Solid lines, fixed-$t$ subtracted DRs; dashed lines, fixed-$t$ unsubtracted DRs; dotted lines, hyperbolic subtracted DRs; and dash-dotted lines, hyperbolic unsubtracted DRs. All results are shown for fixed values of ${\alpha }_{E1}^p+{\beta }_{M1}^p=13.8$, ${\alpha }_{E1}^p-{\beta }_{M1}^p=10$, and ${\gamma }_{\pi }^p=-37$. Data are from 295 (solid circles), 162 (squares), 277 (triangles), and 375 (open circles). From 148.

Figure 18

Contour plot of ${\chi }^2+1$ for different measurements of ${\alpha }_{E1}^p$ and ${\beta }_{M1}^p$. The three dashed ellipses are obtained from 295; 162; and 277 as indicated. Dashed area corresponds to the measurement of 375 at $\theta =180°$, the area between the other two straight lines to the Baldin sum rule. The small ellipse drawn with the solid line is a common fit to all data. From 295.

Figure 19

Contributions to the reaction $e+p\to e^{\prime }+p^{\prime }+\gamma$, (a) virtual Compton scattering on the proton, (b) Bethe-Heitler process. The blob in (a) represents both nucleon intermediate states (Born terms) and excited states of the nucleon (non-Born terms).

Figure 20

Comparison between the unpolarized VCS structure functions calculated by dispersion relations (304) (left column) and HBCPT (206, 204) at $O\left(p^3\right)$ (right column). Top row: Result for $P_{LL}-P_{TT}∕\epsilon$ with $\epsilon =0.62$ (solid lines) and $\epsilon =0.9$ (dash-dotted lines) compared to the result for ${\alpha }_{E1}$ only and $\epsilon =0.62$ (dashed lines). The dispersive results for $\epsilon =0.62$ and 0.9 are obtained with ${\Lambda }_{\alpha }=1.79$ and $0.7\mathrm{GeV}$, respectively. Bottom row: Results for $P_{LT}$ (solid line) compared to the result for ${\beta }_{M1}$ only (dashed line). Data are from 295 (open triangles), 107 (solid triangles), and 317 (squares). From 148, updated by B. Pasquini.

Figure 21

Beam single spin asymmetry for VCS as a function of the photon scattering angle. The dispersive predictions are shown for different pairs of $({\Lambda }_{\alpha },{\Lambda }_{\beta })$ given in GeV. Solid line, (1.4, 0.7); dash-dotted line, (1.4, 0.4); long-dashed line, (0.8, 0.7); and short-dashed line, (0.8, 0.4). From 76.

Figure 22

Results for $P_{LL}-P_{TT}∕\epsilon$ (left panel) and $P_{LT}$ (right panel). Dashed lines: dispersive $\pi N$ contributions. Dotted lines: asymptotic contributions with ${\Lambda }_{\alpha }=0.71\mathrm{GeV}$ and ${\Lambda }_{\beta }=0.51\mathrm{GeV}$. Solid lines: full results. Data are from 295 (open triangles), 107 (solid triangles), 317 (squares), and 256 as obtained by the LEX (open circles) and DR (closed circles) analysis. The inner error bars describe the statistical error, the outer error bars include systematical errors. From 148, updated by B. Pasquini.

Figure 23

Comparison of experiment and theory for the generalized polarizabilities (GPSs). Left panel: The electric GP ${\alpha }_{E1}\left(Q^2\right)$ obtained from the DR formalism with ${\Lambda }_{\alpha }=0.7\mathrm{GeV}$ (solid line) and ${\Lambda }_{\alpha }=1.79\mathrm{GeV}$ (dash-dotted line). The solid line is the sum of the asymptotic (dashed line) and dispersive (dotted) contributions. Right panel: The magnetic GP ${\beta }_{M1}\left(Q^2\right)$ obtained from the DR formalism with ${\Lambda }_{\beta }=0.51\mathrm{GeV}$ (solid line) and ${\Lambda }_{\beta }=0.63\mathrm{GeV}$ (dash-dotted line). The dash-dotted line is the sum of the asymptotic (dashed line) and dispersive (dotted line) contributions. Data are from 295 (open triangles), 107 (solid triangles), 317 (squares), and 256 as obtained by the LEX (open circles) and DR (solid circles) analysis. The inner error bars describe the statistical error, the outer error bars also include systematical errors. From 256.

Figure 24

(Color online) Predicted double-polarization asymmetries as a function of the photon scattering angle ${\theta }_{\gamma \gamma }$ and for the following fixed kinematic values: $q=600\mathrm{MeV}$, $q^{\prime }=111.5\mathrm{MeV}$, $\epsilon =0.62$, and $\Phi =0°$. In order to highlight the model dependence, the (known) Bethe-Heitler and Born contributions to the asymmetry have been subtracted. Solid lines, results of dispersion relations for ${\Lambda }_{\alpha }=1\mathrm{GeV}$ and ${\Lambda }_{\beta }=0.6\mathrm{GeV}$ (148); dotted lines, predictions of HBCPT at $O\left(p^3\right)$ (204); and dashed lines, HBCPT at $O\left(p^4\right)$ (237).

Figure 25

The spin-flip GPs (without the ${\pi }^0$-pole contribution) from several calculations. Solid lines, dispersive $\pi N$ contribution (304); short-dashed lines, $O\left(p^3\right)$ HBCPT (204); long-dashed lines, $O\left(p^4\right)$ HBCPT (237); dash-dotted lines, linear $\sigma$ model (285); and dotted lines, nonrelativistic CQM (305). For visibility, the tiny CQM results for $P^{(L1,L1)1}$ and $P^{(M1,L2)1}$ are multiplied by a factor of 100. From 148, updated by B. Pasquini.

Figure 26

The real and imaginary parts of the $s$-wave amplitude $E_{0+}$ for ${\pi }^0$ photoproduction at threshold energies. The MAMI data of 70 and 168 are represented by circles, the SAL data of 77 by triangles. Dashed lines, predictions of CPT at $O\left(p^3\right)$ (89, 90); and solid lines, results from dispersion relations (200). The solid horizontal line at about $-2.2$ shows the prediction of 130 and 351. Courtesy of R. Beck.

Figure 27

The angular distribution of the photon asymmetry $\Sigma$ for the reaction $p(\vec{\gamma },{\pi }^0)p$. Dashed line, results of dispersion relations (200); dotted line, prediction of CPT (90); and full line, empirical fit to the data with Eqs. (25, 26). From 330.

Figure 28

The total cross section for the $p(e,e^{\prime }p){\pi }^0$ reaction as a function of $Q^2$ for several values of the excitation energy $\Delta W=W-M-m_{{\pi }^0}$. Data at $Q^2=0.05$ and $0.10{\mathrm{GeV}}^2$ are from 281 and 141, respectively. The solid line represents the prediction of CPT (91) and the dashed line the result of the phenomenological model MAID (144). From 281.

Figure 29

The separated cross sections ${\sigma }_0$, ${\sigma }_{\mathit{LT}}$, and ${\sigma }_{\mathit{TT}}$ as well as the beam helicity asymmetry $A_{\mathit{LT}}^{\prime }$ at ${\theta }_{\pi }^{*}=90°$. Solid line, HBCPT (91); dashed line, MAID (144); and dash-dotted line, DMT model (235). From 365.

Figure 30

The total photoabsorption cross section ${\sigma }_T$ on the proton as a function of the photon laboratory energy $\nu$. The various lines represent the MAID results (144) for the total cross section (solid line), one-pion channels (dashed line), more-pion channels (dashed-dotted line), and $\eta$ channel (dotted line). Solid circles, total cross section from MAMI (275); open circles, data from Daresbury (44); and open triangles, two-pion production (108). Courtesy of J. Ahrens.

Figure 31

The partial and total cross sections for the absorption of photons on the proton as a function of the photon laboratory energy $\nu$ obtained at MAMI. Open circles, total cross section as in Fig. 30 (275); solid and open triangles, $n{\pi }^{+}$ decay channel (275); open diamonds, $p{\pi }^0$ decay channel (275); asterisks, two-pion production (108); and open squares, $p\eta$ channel (253). Courtesy of J. Ahrens.

Figure 32

Resonance contributions to the total photoabsorption cross section of the proton. Calculations are from the unitary isobar model MAID2003 (144). Courtesy of R. Beck and L. Tiator.

Figure 33

The ratios $\mathrm{EMR}=R_{\mathit{EM}}$ and $\mathrm{CMR}=R_{\mathit{EM}}$ together with the amplitude $M_{1+}^{3∕2}$ of the $N\Delta \left(1232\right)$ transition as a function of $Q^2$. Data are from 71 (asterisks), 342 (diamonds), 255 (solid triangles), 312 (open triangles), 340 (solid circles), 157 (open circles), and 232 (squares). Theoretical predictions are represented by solid lines, MAID2007 (146); dotted lines, SAID (46); long-dashed lines, DMT (236); short-dashed lines, Sato-Lee model (325); dot-dashed lines, CEFT (298, 299); and crosses, predictions with error bars from lattice QCD (18). Adapted from 340, by L. Tiator.

Figure 34

The helicity amplitude $A_{1∕2}$ for the $S_{11}(1535⟨a⟩)$ in the $p(e,e^{\prime }p)\eta$ reaction. Data are represented by solid circles, 136; crosses, 42; and open circles, earlier publications as specified by 136. Predictions are based on quark models and given by the solid line (118) and dotted line (14). From 136.

Figure 35

(Color online) The helicity amplitudes $A_{1∕2}$ and $S_{1∕2}$ of the $P_{11}\left(1440\right)$ as a function of $Q^2$. The triangles are from the analysis of 54 based on data of the CLAS Collaboration at Jefferson Laboratory and obtained from dispersion relations (open triangles) and the unitary isobar model (solid triangles). The solid line is the global solution of MAID2007 (146), a fit to the full data basis, and the solid circles are local fits to the data in the particular energy region. The data point at $Q^2=0$ is from the PDG (374). From 146.

Figure 36

Spins and helicities in the reaction $\vec{\gamma }+\vec{N}\to N^{*}$. Open arrows denote the projections of the spin on the photon momentum, $S_z$, the helicities $h$ are the projections on the respective particle momentum, and the photon has right-handed helicity, $\lambda =+1$. Left, $N(S_z=1∕2,h=-1∕2)\to N^{*}(S_z=h=3∕2)$; right, $N(S_z=-1∕2,h=1∕2)\to N^{*}(S_z=h=1∕2)$.

Figure 37

The helicity difference $\Delta \sigma ={\sigma }_{3∕2}-{\sigma }_{1∕2}$ for the proton as a function of the photon energy $\nu$. The experimental data are from MAMI (5, 6) (solid circles) and ELSA (153, 154) (open circles). The various lines represent MAID results (144) for the total helicity difference (solid line), one-pion channels (dashed line), more-pion channels (dash-dotted line), and $\eta$ channel (dotted line). From 150.

Figure 38

The total photoabsorption cross section ${\sigma }_T$ (upper panel) and the helicity difference $\Delta \sigma ={\sigma }_{3∕2}-{\sigma }_{1∕2}$ (lower panel) for the deuteron. Experimental data are from 12 (solid circles), 276 (open circles), and 44 (asterisks). Theoretical predictions are from MAID03 (dashed lines) and 41 (solid lines). From 12.

Figure 39

The total cross section ${\sigma }_T$ and the helicity difference $\Delta \sigma ={\sigma }_{3∕2}-{\sigma }_{1∕2}$ for the reaction $\vec{\gamma }\vec{p}\to n{\pi }^{+}{\pi }^0$. Theoretical predictions are shown by solid lines (208), dashed lines (291), and dotted lines (216). Data are from MAMI (8, 9). Figure completed by L. Tiator.

Figure 40

The two-photon exchange diagram. The blob represents the possible intermediate states of the hadronic system.

Figure 41

The helicity difference $\Delta \sigma ={\sigma }_{3∕2}-{\sigma }_{1∕2}$ for the proton as a function of the cm energy $W$ for different values of $Q^2$ (144). The total helicity difference (solid line) shown as well as contributions from one-pion (dashed line), more-pion (dash-dotted line), and $\eta$ (dotted line) production. From 150.

Figure 42

The $Q^2$ dependence of the integral $I_1^p$ defined by Eq. (135). Open circles show the resonance contribution $(W<2\mathrm{GeV})$, the solid symbols also include the DIS contribution. Data are from 121, CLAS Collaboration (circles), 1, SLAC (diamonds), and 15, HERMES (triangles). Solid line, MAID including all channels up to $W=2\mathrm{GeV}$; dashed line, one-pion channel only; dotted, $O\left(p^4\right)$ prediction of HBCPT (229); dash-dotted, relativistic baryon CPT (83, 84); and dash-dot-dotted, interpolating formula of 35; asterisk, sum rule value at $Q^2=0$. From 150.

Figure 43

The $Q^2$ dependence of the integral $I_1^n$ defined by Eq. (135). Data are from 22, Jefferson Laboratory E94-010 Collaboration (circles). For further notation see Fig. 42. From 150.

Figure 44

The $Q^2$ dependence of the integral $I_2^n$ defined by Eq. (135). Neutron data were obtained by the Jefferson Laboratory E94-010 Collaboration (24). Dash-dot-dotted line, Burkhardt-Cottingham sum rule of Eq. (135). For further notation see Fig. 42. From 150.

Figure 45

The $Q^2$ dependence of the neutron integral $I_{\mathit{LT}}^n$ defined by Eq. (132). Open circles, the resonance contribution $(W<2\mathrm{GeV})$ measured by 22, 24 at Jefferson Laboratory; and closed solid circles, resonance contribution plus estimate for DIS region. See Fig. 42 for further notation. From 150.

Figure 46

The $Q^2$ dependence of the generalized neutron polarizability ${\gamma }_{\mathit{TT}}^n$ defined by Eq. (131). Open circles are data of the Jefferson Laboratory E94-010 Collaboration (24). See Fig. 42 for further notation. From 150.

Figure 47

The $Q^2$ dependence of the generalized neutron polarizability ${\delta }_{\mathit{LT}}^n$ defined by Eq. (133). Note that ${\delta }_{\mathit{LT}}^n$ has been multiplied by a factor of ${10}^4{Q}^6∕16{\alpha }_{\mathrm{em}}{M}_p^2$ in order to compensate for its rapid decrease with increasing $Q^2$. Open circles are data of the Jefferson Laboratory E94-010 Collaboration (23). See Fig. 42 for further notation. From 150.