Hadronic structure on the light front. VIII. Light scalar and vector mesons

We use the QCD instanton vacuum model to discuss the emergence of the light scalar and vector mesons on the light front. We take into account both the instanton and anti-instanton single and molecular interactions on the light quarks, in the form of nonlocal effective interactions. Although the molecular induced interactions are suppressed by a power of the packing fraction, they are still sufficient to bind the vector mesons, while keeping most of the scalar spectrum relatively unchanged. We explicitly derive the light front distribution amplitudes and partonic functions for the scalar and vector mesons, and compare them after pertinent QCD evolution, to the available empirical and lattice measured counterparts. The Dirac electric form factors for both the pion and $\rho$ meson are derived, and shown to compare well with current data.

Figure 1

Constituent mass as a function of the current mass with different scalar couplings $g_S$ for a fixed instanton size $\rho =0.31\mathrm{fm}$.

Figure 2

Constituent quark mass versus the scalar coupling $g_S$.

Figure 3

Quark condensate as a function of the scalar coupling $g_S$.

Figure 4

Quark constituent mass in chiral limit as a function of instanton density in the presence of different $\xi$.

Figure 5

The effective coupling $g_S$ in $\sigma$ channel as a function of instanton density.

Figure 6

Sigma mass versus the current quark mass. solid line. The dashed line is the 2M threshold.

Figure 7

Pion mass versus the current quark mass, solid line. The dashed line is the 2M threshold.

Figure 8

Vector masses $m_{\omega ,\rho }$ versus the current quark mass, in solid lines, for different vector couplings $g_V$. The dashed line is the $2M$ threshold in units of the instanton size $\rho$.

Figure 9

Vector masses $m_{\omega ,\rho }$ versus the vector coupling as a solid line, in units of the instanton size $\rho$. The dashed line is the $2M$ threshold.

Figure 10

Vector mesons PDFs versus parton-$x$: transverse polarization (solid green), longitudinal polarization (solid red). and unpolarized (dashed blue).

Figure 11

Parton density functions in the chiral limit.

Figure 12

Helicity distribution function for the quark (solid blue) and the antiquark (solid brown) in a $+\text{polarized vector meson}$, versus parton-$x$.

Figure 13

The unevolved pion DA versus parton-$x$ at low resolution, in the QCD instanton vacuum.

Figure 14

The evolved pion DA using the NLO ERBL equation to $\mu =2\mathrm{GeV}$ (solid green), compared with the lattice calculation (RQCD) (shaded blue) [65], Dyson-Schwinger result [66] (solid purple), and the asymptotic QCD result (dashed black). The experimental data points (red squared points) are extracted from ${\pi }^{-}$ into dijets via diffractive dissociation with invariant dijet mass 6 GeV [67] and normalized in [68].

Figure 15

Unevolved longitudinal vector DA in the QCD instanton vacuum at low resolution with $\mu \sim 1/2\rho$.

Figure 16

Evolved longitudinal $\rho$ DA to $\mu =2\mathrm{GeV}$ in solid green, compared with the lattice in RBC/UKQCD Collaboration [72] in filled red and the lattice [73] in filled blue. The QCD asymptotic result of $6x\overline{x}$ is in dashed black and the QCD sum result [71] is in filled purple.

Figure 17

Unevolved transverse DA for a vector meson in the QCD instanton vacuum at a resolutio $\mu \sim 1/2\rho$.

Figure 18

Evolved transverse DA for the rho meson at $\mu =2\mathrm{GeV}$ in solid green, and compared with the lattice result at $\mu =2\mathrm{GeV}$ [73] in filled blue and the QCD sum rule result also at $\mu =2\mathrm{GeV}$ [75] in filled purple.

Figure 19

The pion twist-2 distribution amplitude at $\mu =0.313\mathrm{GeV}$, $\mu =2\mathrm{GeV}$, and $\mu =\infty$ (asymptotic form).

Figure 20

The longitudinally polarized vector twist-2 distribution amplitude at $\mu =0.313\mathrm{GeV}$, $\mu =2\mathrm{GeV}$, and $\mu =\infty$ (asymptotic form).

Figure 21

The transversely polarized vector twist-2 distribution amplitude at $\mu =0.313\mathrm{GeV}$, $\mu =2\mathrm{GeV}$, and $\mu =\infty$ (asymptotic form).

Figure 22

Our results in solid blue (undressed) and solid orange (dressed) for the pion EM form factor are compared with the JLab measurements in red squares [81] and the Cornell measurements in purple dots [82, 83]. The recent lattice calculations are shown in blue triangles [84], and the Dyson-Schwinger results are shown in dashed green [85].

Figure 23

Our calculations are compared with the recent lattice calculation [86] and the model analysis by Bethe-Salpeter equation using the random-phase approximation in the NJL model [78].

Figure 24

Pion EM form factor in dashed blue (undressed) and in solid orange (dressed) in comparison to the vector EM form factor in solid green.