Cross-link relations between $\pi$ and $\rho$-meson channels and the QCD vacuum

We discuss cross-link relations between the $\pi$ and $\rho$-meson channels emerging from two different descriptions of the QCD vacuum: instanton physics and QCD sum rules with nonlocal condensates (NLCs). We derive in both schemes an intriguing linear relation between the $\pi$ and the ${\rho }^{\parallel }$-meson distribution amplitudes in terms of their conformal coefficients and work out the specific impact of the scalar NLC in these two channels. Using a simple model with Gaussian decay of the scalar NLC, we are able to relate it to the moments of the pion nonsinglet parton distribution function measurable in experiment—a highly nontrivial result. The implications for the pion and the ${\rho }^{\parallel }$-meson distribution amplitudes entailed by the obtained cross-link relations are outlined in terms of two generic scenarios.

Figure 1

Comparison of domains of pairs $(a_2^{(\pi )},a_2^{(\rho )})$ obtained in different approaches at ${\mu }^2=4{\mathrm{GeV}}^2$ obeying the linear relation given by Eq. (3). The blue diagonal line and the dashed lines around it originate from this relation and from the uncertainties of $M_3^{(\pi )}$ in (4). Lattice constraints for $a_2^{(\pi )}$ at NNLO provide the range [0.077, 0.125]—shown as a red segment with a central big red point along the horizontal axis. The ${\mathrm{N}}^3\mathrm{LO}$ regime corresponds to the interval [0.096, 0.135] [3] limited by the red end points. The range $a_2^{(\rho )}=0.132\pm 0.027$ from Ref. [22] is shown as a short red segment at the upper end of the vertical axis. The NLC-SR fiducial regions are shown for the pion by the green segment [6] and for the ${\rho }^{\parallel }$-meson by the blue segment along the vertical axis [20]. The projections of the end points of the green segment and its center (green dot) are denoted by blue points on the $a_2^{(\rho )}$ axis. The yellow points mark the locations of the platykurtic pion and $\rho$ DAs [17, 19], respectively. The black points $a_2^{(\pi )}=0.149$ and $a_2^{(\rho )}=0.092$ denote the coefficients of the DSE DAs for $\pi$ [23] and ${\rho }^{\parallel }$ [24].

Figure 2

Scalar condensate ${\mathrm{\Phi }}_S(x,M^2)$ as a function of $x\in [0,1]$ using the value ${\lambda }_q^2=0.4{\mathrm{GeV}}^2$; see Ref. [6]. The lighter shaded areas correspond to $M^2=0.6{\mathrm{GeV}}^2$, and the darker shaded areas correspond to $M^2=0.9{\mathrm{GeV}}^2$. An analogous notation is used for the thinner (green) and thicker (red) lines.

Figure 3

The central solid (red) line represents the mean value of $M_3$ ($y$ axis) $\frac{7}{6}{M}_3=0.171(22)$ at ${\mu }_0^2$ (using the more precise lattice estimate from Ref. [5]), while the dashed (red) lines mark its uncertainties from Ref. [4]. The thick (blue) line depending on $M^2$ ($x$ axis) is the rhs of Eq. (22) before averaging, i.e., ${\mathrm{\Phi }}_2(M^2)$, whereas the upper thinner (blue) line denotes its upper limit [27] (see Appendix pp2).

Figure 4

Snapshot of various $\pi$ and ${\rho }^{\parallel }$ DAs from different approaches at the scale $\mu =4{\mathrm{GeV}}^2$ in the $(a_2,a_4)$ plane. The large “rectangle” intersecting with the $a_2=0$ line denotes the region of ${\rho }^{\parallel }$ DAs determined with NLC-SR. It encloses the shaded platykurtic regime, where circle white star is the DA determined in Ref. [19]. The symbol filled star refers to the bimodal DAs from NLC-SR in Ref. [21] and filled diamond [20], respectively. The symbol filled black down-pointing triangle represents the ${\rho }^{\parallel }$ DA from the lightfront model in Ref. [30], and ♦ marks the asymptotic DA. Outside this rectangle, one has the following DAs: Box [31] and diamond [24]. The other large rectangle farther to the right contains the domain of the BMS pion DAs [6] obtained with NLC-SR, with bold times denoting the bimodal BMS pion DA. The pion DA represented by asterisk-circle was determined in Ref. [26] as the crossing point of the long middle line of this rectangle with the lattice result $a_2^{(\pi )}=0.116_{-20}^{+19}$ at ${\mathrm{N}}^3\mathrm{LO}$ with three-loop matching to the $\overline{\mathrm{MS}}$ scheme [3]. The dashed rectangle, crossing the $a_4^{(\pi )}=0$ line, shows the platykurtic range of pion DAs with bold plus marking the DA determined in Ref. [17]. The symbols triangle and filled triangle reproduce, respectively, the DAs obtained within holographic AdS/QCD [32] and a DSE-based approach [23], while ○ shows the pion DA from the instanton model in Ref. [33] after NLO evolution to the scale $4{\mathrm{GeV}}^2$. Similarly, filled circle represents the instanton-based ${\rho }^{\parallel }$ DA from Ref. [34].