Colloquium: The Shape of Hadrons

This Colloquium addresses the issue of the shape of hadrons and, in particular, that of the proton. The concept of shape in the microcosm is critically examined. Special attention is devoted to properly define the meaning of shape for bound-state systems of near massless quarks. The ideas that lead to the expectation of nonsphericity in the shape of hadrons, the calculations that predict it, and the experimental information obtained from recent high-precision measurements are examined. Particular emphasis is given to the study of the electromagnetic transition between the nucleon and its first excited state, the $\Delta (1232)$ resonance. The experimental evidence is critically examined and compared with lattice calculations, as well as with effective-field theories and phenomenological models.

Figure 1

Tomographic view of the deformed nucleus ${}^{154}\mathrm{Gd}$ derived from the study of the rotational bands of this nucleus using electron scattering. The plot shows the contours of equal charge density revealing its shape. This spin zero nucleus has a vanishing quadrupole moment and therefore what is revealed through this reconstruction is its intrinsic shape (From 74).

Figure 2

The transition densities characterizing the moving charges involved in excitations in the deformed nucleus ${}^{154}\mathrm{Gd}$, derived from the study of this nucleus using electron scattering, demonstrate that vibrations along the two axes of the ellipsoidal shape have different spatial extent From 74.

Figure 3

Coupling of a spacelike virtual photon to a relativistic many-body system, as a proton. Upper panel: Diagonal transition where the photon couples to a quark, in the leading $3q$ Fock component (left), or in a higher $5q$ Fock component (right). Lower panel: Process where the photon creates a $q\overline{q}$ pair leading to a nondiagonal transition between an initial $3q$ state and a final $5q$ state in the proton.

Figure 4

Schematic view of the projection of the charge density along the line of sight (perpendicular to the figure), for a hadron polarized along the direction of ${\overrightarrow{S}}_{\perp }$. The position of the (quark) charge inside the hadron is denoted by $\overrightarrow{b}$.

Figure 5

Two-dimensional charge density ${\rho }_{T1}$, according to Eq. (5), for a deuteron polarized along the positive $x$ axis. The light (dark) regions correspond with largest (smallest) densities. The density is calculated from empirical information for the deuteron EM FFs (2). From 42.

Figure 6

Three-dimensional contour plot of the $\rho$ meson, of spin projection $s_z=0$, density-density correlator (red surface), showing all positions where the correlator is reduced to half its value at the origin. As can be seen by comparing to a sphere (green transparent surface) of radius of approximately 0.5 fm, the $\rho$ meson surface extends outside the sphere at the poles whereas at the equator is inside the sphere, showing the cigarlike $\rho$-meson shape. Adapted from 9.

Figure 7

The detection of the deexcitation radiation pattern from a system allows one to isolate the contributing multipoles. The isolation of the multipole FFs was achieved for the first time in the ${}^{12}\mathrm{C}(e,e^{\prime }\gamma )$ and ${}^{15}\mathrm{N}(e,e^{\prime }\gamma )$ reactions where the $E2$ and $M1$ FFs were isolated through the tagging of the decay radiation. The radiation pattern and the isolated FFs for this transition in ${}^{15}\mathrm{N}$. From 120.

Figure 8

The precise range of uncertainty that is allowed by the Bates and MAMI data (103; 134, 132; 136, 137) for the ${\sigma }_{\mathrm{LT}}$ response of the $p(e,e^{\prime }p){\pi }^0$ reaction, at $Q^2=0.127{\mathrm{GeV}}^2$ and at the position of the ${\Delta }^{+}(1232)$ resonance, is shown as a function of the c.m. angle between the proton and virtual photon ${\theta }_{pq}^{*}$. Current phenomenological models [MAID (47), Sato-Lee (SL) (128, 129), and DMT (93; 94), continuous curves] predict this satisfactorily within a $2\sigma$ confidence level. The corresponding calculations for “spherical” nucleon and $\Delta$ (dashed curves using same color coding for the various models) cannot describe the data; they are excluded to $2\sigma$ confidence level.

Figure 9

Upper panel: disconnected. Lower panel: connected diagrams. With $h$ and $h^{\prime }$ we denote hadronic states and with $\mathcal{O}$ the operator of interest.

Figure 10

Lattice QCD results for the quark transverse charge density ${\rho }_{T(3/2)}^{\Delta }$ in a ${\Delta }^{+}(1232)$ which is polarized along the positive $x$ axis. The light (dark) regions correspond to the largest (smallest) values of the density. In order to see the deformation more clearly, a circle of radius 0.5 fm is drawn for comparison. The density is obtained from quenched lattice QCD results at $m_{\pi }=410\mathrm{MeV}$ for the $\Delta$ EM FFs (6; 7).

Figure 11

Quark-model picture of $M1$, $E2$, and $C2$ amplitudes in the $N\to \Delta$ transition induced by the interaction of a photon (real or virtual) with a single quark in the nucleon. Presence of quadrupole amplitudes in the transition requires $N$ and/or $\Delta$ wave functions to have a $D$-wave component (indicated by a nonspherical shape).

Figure 12

Results from MAMI (left) and LEGS (right) have yielded precise measurements of the resonant quadrupole amplitude at the photon point. A most sensitive probe is the polarization asymmetry $\Sigma$, which has been measured precisely at both MAMI and LEGS. The derived multipoles from the MAMI cross sections yield an accurate measurement of EMR.

Figure 13

The experimentally derived values for $M_{1+}^{3/2}$ ($M1$) (top panel), CMR (middle panel), and EMR (lower panel) compared to phenomenological model results for $M1$, and to lattice for EMR and CMR. The derived multipole ratios are shown without the model error that is of the order or larger than the depicted experimental error From 11.

Figure 14

Lattice results for the ${\Delta }^{+}(1232)$ form factor $G_{M1}$ (upper panel) and $G_{E2}$ (lower panel) at the smallest pion mass in three simulations. The lines show the fits to an exponential form of the quenched lattice results and to the results obtained using the hybrid action. The error band is calculated using a jackknife analysis on the fitted parameters From 6.

Figure 15

Quadrupole contribution to the transverse charge density for the $p\to {\Delta }^{+}$ transition, when $N$ and $\Delta$ are polarized along the $x$ axis with spin projection $+1/2$. For the $N\to \Delta$ EM FFs, the phenomenological MAID2007 (47) parameterization is used. From 41.

Figure 16

$Q^2$ dependence of the $N\to \Delta$ FF $G_{M1}^{\star }$, at the lightest pion mass for each type of simulation (4). Quenched results are shown with the asterisks, results with $N_F=2$ Wilson with the open squares, results using the hybrid action with the dotted squares and using DWF with the filled circles. Experimental data from 18; 3; 17; 138; 19; 113; 90; 136; 132 are shown with the filled triangles. Lattice data are from 10, 11.

Figure 17

The $m_{\pi }$ dependence of EMR (upper panel) and CMR (lower panel) at $Q^2=0.1{\mathrm{GeV}}^2$. The data points are from MAMI (127) (circle), and BATES (113; 134) (squares). The filled diamonds are quenched lattice results (5). The open diamond near $m_{\pi }\simeq 0$ represents their extrapolation assuming linear dependence in $m_{\pi }^2$. The solid squares at $m_{\pi }=0.36\mathrm{GeV}$ are hybrid lattice results (10) at $Q^2\simeq 0.04{\mathrm{GeV}}^2$ (upper points) and at $Q^2\simeq 0.2{\mathrm{GeV}}^2$ (lower points). The solid curves are a $\chi \mathrm{EFT}$ result (123). The error bands represent the estimate of theoretical uncertainty for the $\chi \mathrm{EFT}$ calculation.