Anomalous dimensions of conformal baryons

We determine the anomalous dimensions of baryon operators for the three-color theory as functions of the number of massless flavors within the conformal window to the maximum known order in perturbation theory. We show that the anomalous dimension of the baryon is controllably small, within the $\delta$ expansion, for a wide range of number of flavors. We also find that this is always smaller than the anomalous dimension of the fermion mass operator. These findings challenge the partial compositeness paradigm.

Figure 1

Value of the fixed-point coupling $a^{*}$ at four-, three- and two-loop order as a function of the number of flavors $n_f$. The four-loop fixed-point coupling reaches a finite value at the lower end of the four-loop conformal window at $n_f\approx 7.3$, while both the three- and two-loop fixed-point couplings diverge there.

Figure 2

Eigen-anomalous dimensions ${\gamma }_{\pm }^{\mathcal{B}}$ for three-quark composite operators compared to the anomalous dimension of the mass ${\gamma }_m$ at three and four loops as a function of $n_f$. The anomalous dimension ${\gamma }_{-}$ corresponds to the proton in QCD.

Figure 3

To estimate the convergence of the perturbative series for ${\gamma }_{-}^{\mathcal{B}}$, we evaluate its two- and three-loop expression using the two-, three- and four-loop values of the fixed-point value $a^{*}$.

Figure 4

Anomalous dimensions of the quark mass and of the three-quark operators ${\gamma }_{\pm }^{\mathcal{B}}$ as a function of the number of fundamental Dirac fermions $n_f$ in the physical $\delta =n_f^{\mathrm{AF}}-n_f$ expansion (solid lines). We compare it with the perturbative results (dashed lines) at three loops.