From quarks and gluons to hadrons: Chiral symmetry breaking in dynamical QCD

We present an analysis of the dynamics of two-flavor QCD in the vacuum. Special attention is paid to the transition from the high-energy quark-gluon regime to the low-energy regime governed by hadron dynamics. This is done within a functional renormalization group approach to QCD amended by dynamical hadronization techniques. These techniques allow us to describe conveniently the transition from the perturbative high-energy regime to the nonperturbative low-energy limit without suffering from a fine-tuning of model parameters. In the present work, we apply these techniques to two-flavor QCD with physical quark masses and show how the dynamics of the dominant low-energy degrees of freedom emerge from the underlying quark-gluon dynamics.

Figure 1

Regeneration of four-quark interactions from the RG flow.

Figure 2

The regulated gluon propagator from pure Yang-Mills theory as a function of the momentum for various $k$. We use this as an external input for our QCD computations.

Figure 3

The regulated ghost dressing function from pure Yang-Mills theory as a function of the momentum for various $k$. We use this as an external input for our QCD computations. The labeling is the same as in Fig. 2.

Figure 4

The UV and IR branches of ${\eta }_A^{\mathrm{YM}}$, ${\eta }^{+}$ and ${\eta }^{-}$, as a function of the strong coupling.

Figure 5

The UV and IR branches of ${\eta }_{A,k}^{\mathrm{YM}}(k)$, which is defined in (59).

Figure 6

Comparison of the momentum-dependent gluon dressing function $Z_{A,0}(p)$ and $Z_{A,k=p}$.

Figure 7

The running of the different strong couplings in comparison to the 1-loop running. Since perturbation theory breaks down at the scale where the strong couplings start to deviate from each other, we show the 1-loop running only down to 1 GeV.

Figure 8

Yukawa coupling as a function of the RG scale for various initial scales $\mathrm{\Lambda }$ and initial conditions $h_{\mathrm{\Lambda }}$.

Figure 9

Comparison between the quenched and the unquenched running gluon propagators $1/Z_{A,k}^{\mathrm{YM}}(k^2)$ and $1/Z_{A,k}(k^2)$ as defined in Eq. (61). We also show the curve for QCD (reduced) where the gluon propagator is a direct sum of Yang-Mills propagator and vacuum polarization; see Eq. (50).

Figure 10

Dressing function (red) and mass (blue) of the quark as function of the RG scale at vanishing momentum. We compare our present model (solid) to the quenched model (dashed) with the parameters fixed to match those of [3].

Figure 11

The renormalized quark, pion and sigma masses as a function of the RG scale. The inset figure shows the masses for a larger range of scales. The shaded gray area indicates which fields contribute dynamically: masses within the gray area exceed the cutoff scale and the corresponding fields are therefore decoupled from the dynamics. On the other hand, fields with masses within the white area are dynamical.

Figure 12

Dimensionless RG-invariant propagators as functions of the RG scale.

Figure 13

Wave-function renormalization of the mesons.

Figure 14

The masses $m_{\pi /\sigma ,k}=\sqrt{{\mathrm{\Gamma }}_{\sigma /\pi }^{(2)}(0)}=k{Z}_{\varphi ,k}^{1/2}{\overline{m}}_{\pi /\sigma ,k}$ of the mesons.