Colloquium: Roper resonance: Toward a solution to the fifty year puzzle

Discovered in 1963, the Roper resonance appears to be an exact copy of the proton except that its mass is 50% greater and it is unstable. These features of the Roper have been very difficult to explain so that for half a century this lightest excited state of the proton has defied understanding. The last decade has presented a new challenge, viz. precise information on the proton-to-Roper electroproduction transition form factors. Reaching to momentum transfer $Q^2\approx 4.5{\mathrm{GeV}}^2$, the data probe a domain within which hard valence-quark degrees of freedom could be expected to determine form factor behavior. An explanation of the Roper should combine an understanding of all these things. This is a prodigious task, but a ten-year international collaborative effort, involving experimentalists and theorists, has presented a candidate solution to the puzzle. Namely, the observed Roper is at the heart of the proton’s first radial excitation, consisting of a dressed-quark core augmented by a meson cloud that reduces the core mass by approximately 20% and materially alters its electroproduction form factors on $Q^2<2m_N^2$, where $m_N$ is the proton’s mass. This Colloquium describes the experimental developments which enabled electroproduction data to be procured within a domain that is the purview of strong quantum chromodynamics, thereby providing challenges and opportunities for modern theory, and surveys the developments in reaction models and QCD theory that have enabled this picture of the Roper resonance to be drawn.

Figure 1

Solid blue lines: level ordering produced by the Hamiltonian in Eq. (2.1). The $(56^{\prime },0^{+})$ level represents a supermultiplet that is completed by the states in the following representations of $SU(3)\times O(3)$: $(56,2^{+})$, $(20,1^{+})$, $(70,2^{+})$, and $(70,0^{+})$. Dashed green lines and shaded bands: pole mass and width of the nucleon’s two lowest-lying $J=1/2$ excitations, determined in a wide ranging analysis of available data ([144]). For the purposes of this illustration, $\hslash \omega$ is chosen so that the proton-$N(1535)1/2^{-}$ splitting associates the $N(1535)1/2^{-}$ state with the $(70,1^{-})$ supermultiplet, as suggested in quantum mechanics by its spin and parity.

Figure 2

If an arbitrary anharmonic potential, restricted only insofar as it can be written as a sum of two-body terms, is added to $H_0$ in Eq. (2.1), then at first order in perturbation theory the $n=2$ harmonic oscillator supermultiplet is split as indicated here. [$E_0$ is roughly the original $(56^{\prime },0^{+})$ energy and $\mathrm{\Delta }$ is a measure of the shape of the potential.]

Figure 3

Data on the (left panel) transverse and (right panel) longitudinal Roper resonance photo- and electrocoupling helicity amplitudes, Eqs. (3.2), as they were available in the last millennium. Data: open (red) circle—1998 estimate of $A_{1/2}$ at the photoproduction point ([79]), and error bar (gray)—our assessment of the true uncertainty in this value at that time. Solid squares and short-dashed (cyan) curves—results from a fixed-$t$ dispersion relation fit ([121]), where the error bars on the squares are our estimate of the systematic uncertainty. Model results: long-dashed (red) curves—nonrelativistic quark model ([147]; [87]) (incompatible with then-existing data); dotted curve (purple, left panel)—relativized quark model ([210]); and solid curve (green)—model constructed assuming the Roper is a hybrid system, constituted from three constituent quarks plus a type of gluon excitation ([153]), wherewith the longitudinal amplitude vanishes. Ordinate expressed in units of ${10}^{-3}{\mathrm{GeV}}^{-1/2}$.

Figure 4

Kinematics of ${\pi }^{+}$ electroproduction from a proton.

Figure 5

Cross-section data. (Left panels) $Q^2=0.45{\mathrm{GeV}}^2$ and (right panels) $Q^2=0.60{\mathrm{GeV}}^2$. (Upper panels) ${\gamma }^{*}p\to {\pi }^{0}p$ and (lower panels) ${\gamma }^{*}p\to {\pi }^{+}n$. The curves are results of global fits to these data using the (solid) UIM and (dashed) DR approaches. [Details provided elsewhere ([40]). The ordinate unit is $\mu \mathrm{b}$.]

Figure 6

First results from CLAS on the Roper helicity amplitudes ([40])—solid squares. All curves are results from various types of CQM: solid bold and solid thin—results obtained using, respectively, relativistic and nonrelativistic versions ([74]); dotted ([210]); dashed ([72]); dot-dashed, thin—quark-gluon hybrid model ([153]); and dot-dashed ([208]). Open circle in left panel—Estimate at photoproduction point from [113].

Figure 7

Lowest moment of the polar-angle dependence in the Legendre expansion of the total cross section ${\sigma }_{T+L}$ for the $n{\pi }^{+}$ and $p{\pi }^0$ electroproduction final states, where the solid (red) and dashed (blue) curves represent, respectively, DR and UIM fits ([37]). Evidently, while the $\mathrm{\Delta }(1232)$ is the most conspicuous feature at low $Q^2$ (left panels), the Roper resonance becomes prominent in the $n{\pi }^{+}$ final state at large $Q^2$, generating the broad shoulder centered near $W=1.35\mathrm{GeV}$ (lower right panel). N.B. The strong peak at 1.5 GeV owes two other resonances: $N(1520)3/2^{-}$ and $N(1535)1/2^{-}$.

Figure 8

(Upper panel) Transverse and (lower panel) longitudinal Roper resonance electrocoupling helicity amplitudes. Circles (purple)—analysis of single-pion final states ([41], [37]); triangles (green)—analysis of $ep\to e^{\prime }{\pi }^{+}{\pi }^{-}{p}^{\prime }$ ([171], [172]); square (upper panel, black)—CLAS Collaboration result at the photoproduction point ([103]) and triangle (black)—review of particle properties global average of this value ([207]). Square (lower panel, black)—$Q^2\simeq 0.1{\mathrm{GeV}}^2$ $\overrightarrow{e}p\to e^{\prime }\overrightarrow{p}{\pi }^0$ measurement ([205]).

Figure 9

Open circle (black): mass of the bare Roper state determined in the EBAC DCC analysis of $\pi N$ scattering ([140]; [143]; [206]). This bare Roper state, with full spectral weight at mass 1.763 GeV, splits and evolves following the inclusion of meson-baryon final-state interactions, with the trajectories in this complex-energy plane depicting the motion of the three, distinct daughter poles as the magnitude of those interactions is increased from zero to their full strength. The horizontal dashed lines (black) mark the branch cuts associated with all thresholds relevant to the solution of the DCC scattering problem in this channel. Solid star (green): mass of the dressed-quark core of the proton’s first radial excitation predicted by a three valence-quark Faddeev equation ([201]).

Figure 10

Illustrative collection of lQCD results for the mass of the nucleon (lower band) and its lightest positive-parity excitation as a function of $m_{\pi }^2$, where $m_{\pi }$ is the pion mass used in the simulation. The results depicted were obtained with different lattice formulations and varying methods for identifying the excited state, as described in the source material ([105]; [23]; [159]; [162]; [114]; [158]).

Figure 11

Existing results for the (upper panel) Dirac and (lower panel) Pauli proton-Roper transition form factors computed using the methods of lQCD ([156]) on anisotropic lattices with pion masses (in GeV): 0.39 (red squares), 0.45 (orange triangles), and 0.875 (green circles); and associated spatial lengths of 3, 2.5, and 2.5 fm. Open circles are empirical results from the CLAS Collaboration ([37]; [103]; [171], [172]).

Figure 12

Poincaré covariant Faddeev equation: a homogeneous linear integral equation for the matrix-valued function $\mathrm{\Psi }$, being the Faddeev amplitude for a baryon of total momentum $P=p_q+p_d$, which expresses the relative momentum correlation between the dressed quarks and diquarks within the baryon. The shaded rectangle demarcates the kernel of the Faddeev equation: single line, dressed-quark propagator; $\mathrm{\Gamma }$, diquark correlation amplitude; and double line, diquark propagator. Further details are provided in Sec. 5b.

Figure 13

Solid (blue) curve: process-independent running coupling ${\widehat{\alpha }}_{\mathrm{PI}}(k^2)$ ([55]). The shaded (blue) band bracketing this curve combines a 95% confidence-level window based on existing lQCD results for the gluon two-point function with an error of 10% in the continuum analysis of relevant ghost-gluon dynamics. World data on the process-dependent effective coupling ${\alpha }_{g_1}$, defined via the Bjorken sum rule ([26], [27], [28], [29], [30], [31]; [4], [5], [6], [7], [8], [9], [10], [11]; [12], [13]; [17], [18], [19]; [146]; [21]; [94], [95], [96]; [20]; [15]). The shaded (yellow) band on $k>1\mathrm{GeV}$ represents ${\alpha }_{g_1}$ obtained from the Bjorken sum by using QCD evolution ([127]; [24]; [98]) to extrapolate high-$k^2$ data into the depicted region ([94], [95]).

Figure 14

Dressed-quark mass function $M(p)$ in Eq. (5.1). Solid curves—gap equation results ([50]; [51]), “data”—numerical simulations of lQCD ([61]). The current quark of perturbative QCD evolves into a constituent quark as its momentum decreases. The constituent mass arises from a cloud of low-momentum gluons attaching themselves to the current quark. This is DCSB, the nonperturbative effect that generates a quark mass from nothing; namely, it occurs even in the chiral limit. Notably, the size of $M(0)$ is a measure of the magnitude of the QCD scale anomaly in $n=1$ point Schwinger functions ([184]), and experiments on $Q^2\in [0,12]{\mathrm{GeV}}^2$ at the upgraded JLab facility are sensitive to the momentum dependence of $M(p)$ within a domain that is here indicated approximately by the shaded region.

Figure 15

(Upper panel) Zeroth Chebyshev moment of all $S$-wave components in the nucleon’s Faddeev wave function, which is obtained from $\mathrm{\Psi }$ in Fig. 12, by reattaching the dressed quark and diquark legs. (Lower panel) Kindred functions for the first $J^P=1/2^{+}$ excited state. $S_1$ is associated with the baryon’s scalar diquark; the other two curves are associated with the axial-vector diquark, and here the normalization is chosen such that $S_1(0)=1$.

Figure 16

(Upper panel) $F_1^{*}$ as a function of $x=Q^2/m_N^2$. Gray band within black curves—dressed-quark-core contribution with up to 20% Faddeev amplitude renormalization from MB FSIs, implemented according to Eq. (5.6). The transition form factor curve with smallest magnitude at $x=1$ has the maximum renormalization. Green band within dotted green curves—inferred MB FSI contribution. The band demarcates the range of uncertainty arising from 0% to 20% renormalization of the dressed-quark core. Dashed blue curve—least-squares fit to the data on $x\in (0,5)$. Dot-dashed red curve—contact-interaction result ([214]). Long-dashed brown curve—light-front (LF) CQM result reconstructed from the helicity amplitudes in [39] using Eqs. (3.2). (Lower panel) $F_2^{*}(x)$ with the same legend. Data: circles (blue) ([37]), triangle (gold) ([103]), squares (purple) ([171], [172]), and star (green) ([207]).