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Twist-two parton distribution amplitudes at a resolving scale $\zeta=2 \,$GeV$=:\zeta_2$. \textbf{A} solid (green) curve – pion $\Leftarrow$ emergent mass generation is dominant; \textbf{B} dot-dashed (blue) curve – $\eta_c$ meson $\Leftarrow$ Higgs mechanism is the primary source of mass generation; \textbf{C} solid (thin, purple) curve -- asymptotic profile, 6x(1 - x); and \textbf{D} dashed (black) curve – ``heavy-pion'', \emph{i.e}.\ a pion-like pseudo-scalar meson in which the valence-quark current masses take values corresponding to a strange quark $\Leftarrow$ the boundary, where emergent and Higgs-driven mass generation are equally important.

Lattice-QCD computations of the pion’s electromagnetic charge radius (green circles \cite{Wang:2018pii}, red down-triangle \cite{Chambers:2017tuf}, cyan cross \cite{Koponen:2017fvm}) as a function of $m_\pi^2$, compared with a continuum theory prediction \cite{Chen:2018rwz} (blue curve within bands, which indicate response to reasonable parameter variation). The continuum analysis establishes $f_\pi r_\pi \approx\,$constant, from which it follows that the size of a Nambu-Goldstone mode decreases in inverse proportion to the active strength of the dominant mass generating mechanism. The empirical value of $r_\pi$ is marked by the gold star.

Sullivan processes. In these examples, a nucleon's pion cloud is used to provide access to the pion's (a) elastic form factor and (b) parton distribution functions. $t = –(k-k^\prime)^2$ is a Mandelstam variable and the intermediate pion, $\pi^\ast(P=k-k^\prime)$, $P^2= –t$, is off-shell.

Virtuality-dependence of pion twist-two PDA. Solid (blue) curve: $v_\pi =0$ result; and dot-dashed (green) curve, PDA at $v_\pi=31$. Even this appreciable virtuality only introduces a modest rms relative-difference between the computed PDAs; namely, 13\%. Measured equivalently, the zero virtuality result differs by 34\% from that appropriate to QCD's asymptotic limit (dotted, red curve).

Geometric acceptances for detection of leading neutrons and the decay products of $\Lambda$ and $\Sigma$ particles in the integrated JLEIC detector concept, to tag the pion and kaon Sullivan processes.

Ratio of the component of the $F_2$ structure function related to the pion Sullivan process as compared to the proton $F_2$ structure function in the low-$t$ vicinity of the pion pole, as a function of $t$ for various values of Bjorken-$x$.

A sample EIC extraction of valence quark, sea quark and gluon PDFs in the pion, at a scale $Q^2 =10\,$GeV$^2$. The extraction is done with the following assumptions on PDFs: the $u$ PDF equals the $\bar d$ PDF in the pion and the $\bar u$ PDF is the same as the other sea quark PDFs ($d$, $s$ and $\bar s$). The extraction at $x_\pi < 10^{-2}$, at this $Q^2$ scale, is constrained by the existing HERA data.

\emph{Upper panel}. Two dressed-quark mass functions distinguished by the amount of DCSB: emergent mass generation is 20\% stronger in the system characterized by the solid green curve, which describes the more realistic case. \emph{Lower panel}. $F_\pi(Q^2)$ obtained with the mass function in the upper panel: $r_\pi = 0.66\,$fm with the solid green curve and $r_\pi = 0.73\,$fm with the dashed blue curve. The long-dashed green and dot-dashed blue curves are predictions from the QCD hard-scattering formula, obtained with the related, computed pion PDAs. The dotted purple curve is the result obtained from that formula if the asymptotic profile is used for the PDA: $\varphi(x)=6x(1-x)$.

\emph{Upper panel}. Two dressed-quark mass functions distinguished by the amount of DCSB: emergent mass generation is 20\% stronger in the system characterized by the solid green curve, which describes the more realistic case. \emph{Lower panel}. $F_\pi(Q^2)$ obtained with the mass function in the upper panel: $r_\pi = 0.66\,$fm with the solid green curve and $r_\pi = 0.73\,$fm with the dashed blue curve. The long-dashed green and dot-dashed blue curves are predictions from the QCD hard-scattering formula, obtained with the related, computed pion PDAs. The dotted purple curve is the result obtained from that formula if the asymptotic profile is used for the PDA: $\varphi(x)=6x(1-x)$.

Projected EIC pion form factor data as extracted from a combination of electron-proton and electron-deuteron scattering, each with an integrated luminosity of $20\,{\rm fb}^{-1}$ -- black stars with error bars. Also shown are projected JLab 12-GeV data from a Rosenbluth-separation technique -- orange diamonds and green triangle. The long-dashed green curve is a monopole form factor whose scale is determined by the pion radius. The black solid curve is the QCD-theory prediction bridging large and short distance scales, with estimated uncertainty \cite{Chen:2018rwz}. The dot-dashed blue and dotted purple curves represent the short-distance views \cite{Lepage:1979zb, Efremov:1979qk, Lepage:1980fj}, comparing the result obtained using a modern DCSB-hardened PDA and the asymptotic profile, respectively.

Ratio of valence $u$-quark PDFs in the pion and the kaon at $\zeta = 5.2\,$GeV$=:\zeta_5$. Data are from Drell-Yan measurements \cite{Badier:1980jq}. The computed results are taken from Ref.\,\cite{Chen:2016sno}, with the dashed, solid, and dot-dashed curves representing, respectively, $0$, $5$\%, $10$\% of the kaon's light-front momentum carried by glue at the scale, $\zeta_K = 0.51\,$GeV. For the projected EIC data (brown points drawn at $u_K(x)/u_\pi(x)=1.2$) we assumed $u$-quark dominance. (For reference, the horizontal dotted line marks $u_K(x)/u_\pi(x)=1$.)

Pion valence-quark momentum distribution function, $x {\mathpzc u}^\pi(x;\zeta_5)$: dot-dot-dashed (grey) curve within shaded band -- lQCD result \cite{Sufian:2019bol}; long-dashed (black) curve -- early continuum analysis \cite{Hecht:2000xa}; and solid (blue) curve embedded in shaded band -- modern, continuum calculation \cite{Ding:2019lwe}. Gluon momentum distribution in pion, $x g^\pi(x;\zeta_5)$ -- dashed (green) curve within shaded band; and sea-quark momentum distribution, $x S^\pi(x;\zeta_5)$ -- dot-dashed (red) curve within shaded band. (In all cases, the shaded bands indicate the size of calculation-specific uncertainties, as described elsewhere \cite{Ding:2019lwe}.) Data (purple) from Ref.\,\cite{Conway:1989fs}, rescaled according to the analysis in Ref.\,\cite{Aicher:2010cb}.

Projected uncertainties for measurements of the $u$-quark to pion (\emph{upper panel}) and kaon (\emph{lower panel}) fragmentation function at EIC for an integrated luminosity of $10\,{\rm fb}^{-1}$, for the large $z$ region, $z > 0.5$, and transverse momentum $k_\perp$ (as picked up in the fragmentation process) of $k_\perp = 0.1, 0.3, 0.5\,$GeV, respectively.

Projected uncertainties for measurements of the $u$-quark to pion (\emph{upper panel}) and kaon (\emph{lower panel}) fragmentation function at EIC for an integrated luminosity of $10\,{\rm fb}^{-1}$, for the large $z$ region, $z > 0.5$, and transverse momentum $k_\perp$ (as picked up in the fragmentation process) of $k_\perp = 0.1, 0.3, 0.5\,$GeV, respectively.