Fully coupled functional equations for the quark sector of QCD

We present a comprehensive study of the quark sector of $2+1$ flavor QCD, based on a self-consistent treatment of the coupled system of Schwinger-Dyson equations for the quark propagator and the full quark-gluon vertex in the one-loop dressed approximation. The individual form factors of the quark-gluon vertex are expressed in a special tensor basis obtained from a set of gauge-invariant operators. The sole external ingredient used as input to our equations is the Landau gauge gluon propagator with $2+1$ dynamical quark flavors, obtained from studies with Schwinger-Dyson equations, the functional renormalization group approach, and large volume lattice simulations. The appropriate renormalization procedure required in order to self-consistently accommodate external inputs stemming from other functional approaches or the lattice is discussed in detail, and the value of the gauge coupling is accurately determined at two vastly separated renormalization group scales. Our analysis establishes a clear hierarchy among the vertex form factors. We identify only three dominant ones, in agreement with previous results. The components of the quark propagator obtained from our approach are in excellent agreement with the results from Schwinger-Dyson equations, the functional renormalization group, and lattice QCD simulation, a simple benchmark observable being the chiral condensate in the chiral limit, which is computed as $(245\mathrm{MeV}{)}^3$. The present approach has a wide range of applications, including the self-consistent computation of bound-state properties and finite temperature and density physics, which are briefly discussed.

Figure 1

Diagrammatic representation of the quark gap equation. Gray (blue) circles denote full propagators (vertices), and black dots denote classical vertices.

Figure 2

Diagrammatic representation of the quark-gluon SDE. Gray (blue) circles denote full propagators (vertices), and black dots denote classical vertices. The ellipses denote higher-order contributions: diagrams without perturbative one-loop counterparts.

Figure 3

The $2+1$-flavor gluon propagator, $G_A(p^2)$, and dressing function $1/Z_A(p)=p^2{G}_A(p)$. Lattice simulations [31, 83, 84], fRG-DSE approach [60, 61], and fRG approach [58]. The computations in [58, 60, 61] are based on the 2-flavor input fRG data from [81].

Figure 4

Quark dressing, $1/Z_q(p^2)$, and mass function, $M_q(p^2)$, in the chiral limit and for physical quark masses.

Figure 5

Numerical results for the coupled system of SDEs for quark-gluon vertex (light quark, left panel), and the light and strange quark propagators (right panel). (a) Quark-gluon couplings ${\overline{\lambda }}_i(p)$, (7.4). Black: classical tensor structure, (2.6). Grey: chirally symmetric nonclassical tensor structures. Red: chiral-symmetry breaking tensor structures. (b) Quark mass functions $M_q(p)$ and propagator dressings $1/Z_q(p)$ (inset) for $q=l,s$, see (2.7). Lattice results from [106]: light quark mass function $M_l(p)$ and dressing $1/Z_l(p)$.

Figure 6

Full results for dominant quark-gluon couplings ${\overline{\lambda }}_{1,4,7}$ (this work, black) from the present work in comparison to full results from [61] (fRG-DSE, red), lattice results (Lattice, green), and the approximation STI scaling: ${\lambda }_{1,2,6}$ from STI, and ${\lambda }_{4,5,6,7}$ from scaling relations (see discussion at the end of Sec. 9 and [60, 81]). The ${\lambda }_i$ are dressings of tensor structures in Eq. (4.11a) and measured in ${\lambda }_1$, ${\lambda }_4[(\mathrm{GeV}{)}^{-1}]$ and ${\lambda }_7[(\mathrm{GeV}{)}^{-2}]$. (a) Dominant quark gluon couplings ${\lambda }_{1,4,7}$: full solution SDE (this work, black), full solution [60] (fRG-DSE, red), couplings with STI, ${\overline{\lambda }}_1$, and scaling relations, ${\overline{\lambda }}_{4,7}$ (STI-scaling, blue). (b) Light quark mass function $M_l(p)$ and dressing function $1/Z_l(p)$ (inset): full solution (this work, full black), full solution [60] (fRG-DSE, dashed red), lattice results [106] (Lattice, green).

Figure 7

Strong coupling ${\alpha }_{l\overline{l}A}(p)$ [see (2.6)] in comparison to the one- and two-loop counterparts, adjusted at the RG scale $\mu =40\mathrm{GeV}$ (left panel) and $\mu =4.3\mathrm{GeV}$ (right panel).

Figure 8

Numerical results for the coupled system of SDEs for different UV cutoffs $\mathrm{\Lambda }=50$, 100, 1000, 5000 GeV, based on the gluon input data [60, 61] (fRG-DSE in Fig. 3). (a) Light quark mass function $M_l(p)$ and dressing $1/Z_l(p)$ (inset) for different UV cutoffs. (b) Quark-gluon coupling ${\overline{\lambda }}_1(p)$ of the classical tensor structure for different UV cutoffs.

Figure 9

Numerical results for the coupled system of SDEs with $\mu =40\mathrm{GeV}$ for different gluon input data, Fig. 3 in Sec. 5b: [60, 61] (fRG-DSE), [58] (fRG), [31, 83, 84] (lattice fit). (a) Light quark mass function $M_l(p)$ and dressing $1/Z_l(p)$ (inset) for different gluon input. (b) Dominant quark-gluon couplings ${\overline{\lambda }}_{1,4,7}(p)$ for different gluon input.

Figure 10

The $\mu$ independence of $M_l(p)$ and ${\overline{\lambda }}_{1,4,7}(p)$, and multiplicative renormalizability of $Z_l(p)$. The RG scales differ by an order of magnitude: $\mu =40\mathrm{GeV}$ and $\mu =4.3\mathrm{GeV}$. (a) $M_l(p)$ and $Z_l(p)$ using the fRG-DSE input, for $\mu =40\mathrm{GeV}$ (black-solid) and $\mu =4.3\mathrm{GeV}$ (red-dashed and blue-dashed (rescaled)). The blue dot indicates the rescaling factor. (b) Results for the dominant quark-gluon couplings ${\overline{\lambda }}_{1,4,7}$, obtained with the fRG-DSE input, and renormalised at $\mu =40\mathrm{GeV}$ and $\mu =4.3\mathrm{GeV}$.

Figure 11

Quark-gluon couplings ${\overline{\lambda }}_i(p)$, defined at the symmetric point; see (7.4). The ordering in the legends is reflecting their strength. (a) Quark-gluon couplings from chirally symmetric tensor structures ${\mathcal{T}}_{1,5,6,7}$. (b) Quark-gluon couplings from the chiral-symmetry–breaking tensor structures ${\mathcal{T}}_{2,3,4,8}$.

Figure 12

Lack of quantitative reliability without quark-gluon couplings ${\overline{\lambda }}_{4,7}$ on the right-hand sides of the quark-gluon SDEs and the quark gap equation. (a) Quark dressings $M_l(p)$ and $1/Z_l(p)$ without ${\overline{\lambda }}_4$ (dashed, red), and without ${\overline{\lambda }}_7$ (dotted, blue) in the vertex SDEs in comparison to the full solution (full, black). (b) Quark dressings $M_l(p)$ and $1/Z_l(p)$ without ${\overline{\lambda }}_4$ (dashed, red), and without ${\overline{\lambda }}_7$ (dotted, blue) in the quark gap equation in comparison to the full solution (full, black).

Figure 13

Quantitative reliability of approximations: (a) Quark-gluon couplings ${\lambda }_{1,2,6}$ from STI, and ${\lambda }_{4,5,6,7}$ from scaling relations; see text and [60, 81]. (b) ${\lambda }_{1,4,7}$.

Figure 14

${\mathrm{\Lambda }}_{\mathrm{QCD}}(\mu )=p_{\mathrm{sing}}$, defined by the singularity of ${\alpha }_s^{(2\mathrm{loop})}(p)$, for $\mu \in (1,40)\mathrm{GeV}$. The RG scale $\mu =4.3\mathrm{GeV}$ is indicated with a dashed vertical line. The inset shows the full $\beta$-function, ${\beta }_{l\overline{l}A}({\alpha }_s)$, in comparison to its two-loop counterpart. The red vertical band indicates the transition momentum regime, in which the full $\beta$-function starts to deviate from the perturbative two-loop $\beta$-function.