Nucleon and pion distribution functions in the valence region

An experimental and theoretical perspective is provided on the behavior of unpolarized distribution functions for the nucleon and pion on the valence-quark domain, namely, Bjorken $x\gtrsim 0.4$. This domain is a key to much of hadron physics; e.g., a hadron is defined by its flavor content and that is a valence-quark property. Furthermore, its accurate parametrization is crucial to the provision of reliable input for large collider experiments. The focus is on experimental extractions of distribution functions via electron and muon inelastic scattering, and from Drell-Yan interactions; and on theoretical treatments that emphasize an explanation of the distribution functions, providing an overview of major contemporary approaches and issues. Valence-quark physics is a compelling subject, which probes at the heart of our understanding of the standard model. There are numerous outstanding and unresolved challenges, which experiment and theory must confront. In connection with experiment, an explanation that an upgraded Jefferson Laboratory facility is well suited to provide new data on the nucleon is provided, while a future electron-ion collider could provide essential new data for the mesons. There is also great potential in using Drell-Yan interactions, at FNAL, CERN, J-PARC, and GSI, to push into the large-$x$ domain for both mesons and nucleons. Furthermore, it is argued that explanation, in contrast to modeling and parametrization, requires a widespread acceptance of the need to adapt theory: to the lessons learnt already from the methods of nonperturbative quantum-field theory and a fuller exploitation of those methods.

Figure 1

Feynman diagram representing deep inelastic lepton scattering from a hadron.

Figure 2

(Color online) Quark flavor dependence of the proton structure function according to the CTEQ6L (leading-order) evaluation at a scale $Q^2=10{\mathrm{GeV}}^2$ (318; 243). We plot the quantities $e_i^{2}x{q}_i\left(x\right)$ in order to emphasize the contribution to the structure function from the various quark flavors.

Figure 3

(Color online) Structure function of the proton at three values of $Q^2$, computed from the CTEQ6L leading-order evaluation.

Figure 4

Graphical representation of the leading-order DGLAP equations.

Figure 5

(Color online) Comparison of the proton’s $u\left(x\right)$ distributions in LO, NLO, and NNLO at $Q^2=10{\mathrm{GeV}}^2$.

Figure 6

(Color online) Empirical nonsinglet higher-twist coefficient, denoted $C_{HT}$, as a function of Bjorken $x$, for the proton (left panel) and deuteron (right panel), as obtained when the twist-2 contributions are treated at NLO, NNLO, and N3LO. Adapted from 67.

Figure 7

$Q^2$ vs $x$ for three values of $W$. Measurements at very high $x$ demand very high $Q^2$ to remain in the DIS region.

Figure 8

Proton structure function vs Bjorken $x$ with $W⩾2.0\mathrm{GeV}$. For ease of representation: the upper bands of data were multiplied by a factor of 10 and correspond to the domain $25⩽Q^2⩽40{\mathrm{GeV}}^2$; the next bands are scaled by a factor of 3 and cover the domain $15⩽Q^2⩽25{\mathrm{GeV}}^2$; the third-lowest bands are unscaled and cover $7⩽Q^2⩽15{\mathrm{GeV}}^2$; and the lowest bands are scaled by a factor of $1∕5$ and cover $2⩽Q^2⩽7{\mathrm{GeV}}^2$.

Figure 9

(Color online) The proton structure functions vs Bjorken $x$ with $W⩾2\mathrm{GeV}$. The dashed black curve represents the CTEQ6L evaluation with $Q^2=10{\mathrm{GeV}}^2$, while the solid curve indicates the result obtained when target-mass corrections are incorporated. From 230.

Figure 10

The ratio of the neutron-to-proton structure functions as a function of $x$. The results of three extraction methods are shown.

Figure 11

(Color online) Dressed-quark mass function $M\left(p\right)$: curves—Dyson-Schwinger equation (DSE) results, obtained as explained by 54 and 55; data—numerical simulations of unquenched lattice-QCD (75). In the momentum dependence of $M\left(p\right)$, one observes the current quark of perturbative QCD evolving into a constituent quark as its momentum becomes smaller. The constituent-quark mass arises from a cloud of low-momentum gluons attaching themselves to the current quark. This is dynamical chiral-symmetry breaking (DCSB): an essentially nonperturbative effect that generates a quark mass from nothing, namely, it occurs even in the chiral limit. Adapted from 52.

Figure 12

Feynman diagram representing the Drell-Yan interaction.

Figure 13

Feynman graphs of NLO corrections to the Drell-Yan process. Top row, Compton-like scattering in QCD, viz., $gq\to {\gamma }^{\ast }$ (and also $g\overline{q}\to {\gamma }^{\ast }$, not shown explicitly). Middle row, QCD radiative correction, viz., $q\overline{q}\to {\gamma }^{\ast }$. Bottom diagram, QCD vertex correction. Adapted from 290.

Figure 14

(Color online) Ratios of the measured Drell-Yan $pp$ (squares) and $pd$ (triangles) cross sections to NLO calculations based on the MRST 2001 PDF fit (240), plotted as a function of beam momentum fraction, here labeled $x_1$, with averaging over the target momentum fraction. The shaded area between the solid lines represents the experimental uncertainty ranges (242) on $4u\left(x\right)+d\left(x\right)$ in the MRST 2001 PDF fit. The data are from the FNAL E866 experiment. Adapted from 335 and 336.

Figure 15

(Color online) $x$ dependence of $\overline{d}∕\overline{u}$. The solid squares represent the measurements from FNAL E866 (188; 326), while the triangle indicates the CERN NA51 measurement (34). The shaded region indicates the uncertainty from the CTEQ6 evaluation of the world data.

Figure 16

The $\overline{u}$ quark distribution function in the ${\pi }^{-}$. The data are from a LO analysis of the FNAL E615 experiment (122). The curve reproduces a Dyson-Schwinger equation calculation (190), described in Sec. 6B3, which is consistent with QCD expectations. The discrepancy is disturbing and hitherto unresolved.

Figure 17

(Color online) Feynman diagram of the parton picture of DIS on the pion, where one hard gluon exchanges prior to absorption of the highly energetic photon, so that the struck parton carries most of the momentum of the pion. Evaluation of this diagram leads to a ${(1-x)}^2$ dependence of the pion structure function at very high $x$ (146; 148).

Figure 18

(Color online) The pion distribution function. Dashed curve, fit from the LO analysis of the FNAL E615 data; solid curve, NLO fit to the E615 data (342); and dotted curve, calculation of the distribution function using a Dyson-Schwinger equation approach (190), which manifestly incorporates the momentum dependence of the dressed-quark mass function that is shown in Fig. 11.

Figure 19

(Color online) The decay angular distribution of the Drell-Yan process in the Collins-Soper frame.

Figure 20

The $\nu$ or $\cos 2\varphi$ component of the Drell-Yan angular distributions obtained with a pion (NA10 and E615) or proton (E866) beam plotted as a function of the transverse momentum of the lepton pair. The dotted curve depicts a pQCD prediction (117) [Eq. (3.17) and associated discussion]. The solid curve, which represents the Boer-Mulders effect (70), was adjusted to describe the NA10 data [Eq. (3.22) and associated discussion].

Figure 21

(Color online) Ratio of kaonic to pionic Drell-Yan cross sections, which is usually interpreted as the ratio of the $\overline{u}$ quark distribution function in the $K^{-}$ to that in the ${\pi }^{-}$. Data from 32, which are obtained from a sample of dimuon events with invariant mass $4.1<M<8.5\mathrm{GeV}$, and solid curve from 306, an NJL-model result evolved from $Q_0=0.5$ to $4.5\mathrm{GeV}$ (see also Fig. 42 and associated discussion).

Figure 22

Feynman diagram for a possible DIS interaction with a virtual pion in the proton.

Figure 23

The expected statistical error for a future experiment to measure the ${\pi }^{+}$ structure function at an electron-ion collider. Adapted from 195.

Figure 24

(Color online) (Color online) The absolute Drell-Yan cross section is proportional to $4u\left(x\right)+d\left(x\right)$ in the beam proton. The data are from the FNAL E866 experiment. The solid circles are projected data from experiment E906 at FNAL. Adapted from 335, 336, and 289.

Figure 25

Five parton distribution evaluations at NLO in the $\overline{\text{MS}}$ factorization scheme at $Q^2=10{\mathrm{GeV}}^2$ in the valence region. There are three families of curves: the $d\left(x\right)∕u\left(x\right)$ ratios, the $xu\left(x\right)$ distributions, and the $xd\left(x\right)$ distributions.

Figure 26

“Handbag diagram” contribution to the virtual-photon-nucleon forward Compton scattering amplitude: ${\gamma }^{\ast }\left(q\right)N\left(p\right)\to {\gamma }^{\ast }\left(q\right)N\left(p\right)$. The intermediate state (or nucleon remnant in the DIS process) here is assumed to be a scalar or axial-vector diquark, with masses $M_s$ and $M_v$, respectively.

Figure 27

A bag-model result for the $x$-weighted sum of valence-quark distributions in the proton. The model parameters are the following: bag radius $R=0.8\mathrm{fm}$; scalar and axial-vector diquark masses, $M_s=0.55\mathrm{GeV}$ and $M_v=0.75\mathrm{GeV}$; and infrared resolving scale $Q_0=\mu =0.26\mathrm{GeV}$. Adapted from 314.

Figure 28

$F_2^n\left(x\right)∕F_2^p\left(x\right)$. Left panel, computed in the model of 199: dot-dashed curve, at hadronic scale $Q_0\sim 0.4\mathrm{GeV}$; dotted curve, leading-order evolution to $Q_0^2=10{\mathrm{GeV}}^2$; and solid curve, next-to-leading-order evolution. Right panel—computed in the model developed by 169. In both panels—triangles, CTEQ5 fit to experimental data (218); straight line, Eq. (6.1). Using Eq. (2.45) and assuming $s_v\left(x\right)=0$ on the valence-quark domain, the large-$x$ behavior in these panels corresponds to $d_v\left(x\right)∕u_v\left(x\right)\gtrsim 1.0$. Adapted from 276.

Figure 29

Depiction of three invariant-mass sectors conjectured by 212 to contribute to deep inelastic scattering. Left image $\mathcal{A}$: low $w$—diquark spectators. Central image, $\mathcal{B}$: intermediate $w$—compound spectator system involving a pion (dashed line), constituent quarks and diquarks. Right image, $\mathcal{C}$: large $w$—Regge-trajectory model for $q\overline{q}$ scattering from a constituent quark. Adapted from 212.

Figure 30

Results illustrative of the pion-cloud model described in connection with Fig. 29. Left panel: Total valence-quark distribution function. Complete calculation, combining all sectors (solid curve); $\mathcal{A}+\mathcal{B}$ (dashed curve); and $\mathcal{A}$ alone (dash-dotted curve). Right panel: Contribution from sector $\mathcal{A}$ to the ratio of valence-quark distribution functions. $m_{0^{+}}=1.0\mathrm{GeV}$, $m_{1^{+}}=1.2\mathrm{GeV}$ (solid curve) and $m_{0^{+}}=0.6\mathrm{GeV}$, $m_{1^{+}}=0.8\mathrm{GeV}$ (dashed curve). Both panels: Computed results are reported at $Q_0^2=10{\mathrm{GeV}}^2$ after LO evolution from $Q_0=1\mathrm{GeV}$, whereat $\alpha ∕\left[2\pi \right]\simeq 0.03$. Parametrizations of data (246; 217) are provided for comparison (shaded band). Adapted from 212.

Figure 31

Results obtained in a topological soliton model whose infrared resolving scale is argued to be $Q_0\sim 0.6\mathrm{GeV}$. Left panel: $(x∕2)\{u_v(x;Q_0)+d_v(x;Q_0)\}$, proportional to the valence-quark distribution: contribution from the single discrete (valence) energy-level bound in the soliton background (solid curve) and from the Dirac continuum (dotted curve). Right panel: $(x∕2)\{\overline{u}(x;Q_0)+\overline{d}(x;Q_0)\}$, proportional to the antiquark distribution. Both panels: squares, NLO data parametrization drawn from 175, associated with $Q_0=0.56\mathrm{GeV}$. Adapted from 130: ${\Lambda }_{\mathrm{UV}}=0.56\mathrm{GeV}$, the calculated nucleon mass is $1170\mathrm{MeV}$, and $N_c=3$ with $1∕N_c$ corrections neglected.

Figure 32

Depiction of the general handbag diagram contribution to deep inelastic scattering from the pion. As we saw in Fig. 26, with an explicit expression for the single-flavor removal amplitude $\mathcal{H}(k,\mathcal{l})$ one defines a model framework within which to complete the calculation.

Figure 33

(Color online) Pion’s valence-quark distribution function calculated in a Nambu–Jona-Lasinio model that is regularized by using a hard cutoff (306). Under one-loop evolution, a resolving scale of $Q_0=0.5\mathrm{GeV}$ optimizes agreement with data parametrizations.

Figure 34

Valence $u$-quark (left-panel) and $d$-quark (right-panel) distributions computed in a scalar-diquark picture of the nucleon based on an NJL model (260). The results exhibit a marked sensitivity to the manner in which the model is regularized. In this study the TR scheme is judged best. Both schemes lead to unphysical behavior on $x\gtrsim 0.8$. NLO evolution of the model results is performed from $Q_0=0.4\mathrm{GeV}$. The parametrization is that of 246. Adapted from 260.

Figure 35

$F_2^n\left(x\right)∕F_2^p\left(x\right)$ computed by 259. $r_a$ measures the strength of axial-vector diquark correlations in the proton. It is a free parameter. Used in conjunction with $m_Q=0.4\mathrm{GeV}$ and $m_{0^{+}}=0.68\mathrm{GeV}$, the model-preferred value is $r_a=0.25$, at which the axial-vector correlation contributes just 7% to the proton’s normalization. $r_a=0$ reproduces the scalar-diquark only model (Fig. 34). With $r_a=2∕3$, the axial-vector diquark contributes 39% to the proton’s normalization. NLO evolution of the model results is performed from $Q_0=0.4\mathrm{GeV}$. An explanation of the experimental points is provided by 257. Adapted from 259.

Figure 36

(Color online) Valence-quark distribution functions computed by 111, which differs from 260 primarily in the choice of regularization prescription. The marked sensitivity to regularization scheme in NJL-type models is evident through comparison with Fig. 34. NLO evolution of the model results is performed from $Q_0=0.4\mathrm{GeV}$. “Empirical” denotes the parametrization of 241. Adapted from 111.

Figure 37

Handbag diagram contribution to the virtual Compton scattering process: ${\gamma }^{*}\left(q\right)T\left(p\right)\to \gamma \left(q^{\prime }\right)T^{\prime }\left(p^{\prime }\right)$. Left panel—kinematic variables in a symmetric reference frame. Right panel—light-front representation of $H^q$ in Eq. (6.48). Adapted from 74.

Figure 38

(Color online) Current-quark-mass dependence of ${⟨x⟩}_{Q_0}^{u-d}$, $Q_0=2\mathrm{GeV}$, computed directly through numerical simulations of lattice-regularized QCD: circles (272); diamonds (273); boxes (187); right triangles (88); and left triangles (177). The empirical value is marked by an asterisk. The clover and domain wall fermion results are in excellent agreement, except at the lightest pion mass, a discrepancy which can likely be attributed to finite-size effects. The mixed-action results appear to differ from the others by a normalization factor, which is perhaps understandable given that 187 does not use the fully nonperturbative renormalization procedure employed in the other studies. Adapted from 347.

Figure 39

“Handbag” contributions to the virtual-photon-pion forward Compton scattering amplitude. $\pi$, dashed line; $\gamma$, wavy line; $\mathcal{S}$, internal solid line, dressed-quark propagator [Eq. (6.73)]. The filled circles represent the pion’s Bethe-Salpeter amplitude ${\Gamma }_{\pi }$, in Eq. (6.74) and the dressed-quark-photon vertex ${\Gamma }_{\mu }$, discussed in Appendix , depending on which external line they begin or end. Adapted from 190.

Figure 40

(Color online) Solid curve, pion’s valence-quark momentum distribution computed by 190. Dashed curve, $A{(1-x)}^{{\eta }_2}$, with the parameters drawn from Eq. (6.97).

Figure 41

(Color online) Compilation of results for the valence-$u$-quark distribution in the pion: solid curve, DSE result (190); dotted-dashed curve, NJL model (see Sec. 6A6); short-dashed curve, instanton model (136); dash-dot-dotted curve, light-front constituent-quark model (156); squares, Drell-Yan data presented by 122; and long-dashed curve, reanalysis of that Drell-Yan data described by 342, which is also shown in Fig. 18. All calculations evolved at leading order to $Q_0=4.0\mathrm{GeV}$ using a four-flavor value of ${\Lambda }_{\mathrm{QCD}}=0.204\mathrm{GeV}$, except that from 342, which is reported at $Q_0=5.2\mathrm{GeV}$.

Figure 42

(Color online) Solid curve, preliminary Dyson-Schwinger equation prediction for the ratio of kaon-to-pion valence-quark distribution functions at $Q_0=5\mathrm{GeV}$, evolved at leading order from the dashed curve, which is the DSE prediction at $Q_0=0.57\mathrm{GeV}$ (267; 268). Dotted and dash-dotted curves, our reevaluation of $u_v^K\left(x\right)∕u_v^{\pi }\left(x\right)$ using the model of 306; filled circles, the ratio of kaon-to-pion Drell-Yan cross sections obtained from a sample of dimuon events with invariant mass $4.1<M<8.5\mathrm{GeV}$ (32); thin dashed line, the curve $u_v^K\left(x\right)=u_v^{\pi }\left(x\right)$; and thin dashed-dotted curve, fit to the data given in Eq. (3.24).

Figure 43

Poincaré covariant Faddeev equation. $\Psi$ is the Faddeev amplitude for a nucleon of total momentum $P=p_q+p_d$. It expresses the relative-momentum correlation between the dressed quark and diquarks within the nucleon. The shaded region demarcates the kernel of the Faddeev equation, in which the single line denotes the dressed-quark propagator; $\Gamma$ is the diquark Bethe-Salpeter-like amplitude; and the double line is the diquark propagator. [Full explanation provided in Appendix A of 112, from which this figure is adapted.]

Figure 44

Vertex which ensures a conserved current for on-shell nucleons described by the Faddeev amplitudes ${\Psi }_{i,f}$ obtained from the Faddeev equation shown in Fig. 43. The single line represents the dressed-quark propagator; the double line, the diquark propagator; and $\Gamma$ is the diquark Bethe-Salpeter amplitude. The remaining vertices are described in the text: the top-left image is diagram 1, the top right, diagram 2 etc., with the bottom-right image, diagram 6. In connection with DIS, the photon line is equated with a zero-momentum insertion. [A full explanation of the diagrams depicted here is provided in Appendix C of 112, from which this figure is adapted.]