Conformal windows of $Sp(2N)$ and $SO(N)$ gauge theories

We study the nonperturbative dynamics of nonsupersymmetric asymptotically free gauge theories with fermionic matter in distinct representations of the $SO(N)$ and $Sp(2N)$ gauge groups. We use different analytic methods to unveil the associated conformal windows for the relevant matter representations. We propose a direct test for confronting and establishing the validity of the analytic methods used to constrain the conformal windows. By comparing the resulting windows for $SU$, $Sp$, and $SO$ a pleasing universal picture emerges.

Figure 1

Rainbow approximation for the fermion self-energy function. The boson is a gluon.

Figure 2

Phase diagram of $Sp(2N)$ gauge theories with $2N_f$ fundamental Weyl fermions. Left panel: The upper solid (blue) line corresponds to the loss of asymptotic freedom, and it is labeled by AF; the dashed (black) curve corresponds to the SD prediction for the breaking/restoring of chiral symmetry. The solid grey (magenta in color) line corresponds to the ACS bound stating that the conformal region should start above this line. According to the all-orders BF, the conformal window cannot extend below the solid (blue) line, as indicated by the arrows. This line corresponds to the anomalous dimension of the mass reaching the maximum value of 2. Right panel: The BF line is plotted assuming the value of the anomalous dimension to be one.

Figure 3

Phase diagram of $Sp(2N)$ gauge theories with $N_f$ adjoint Weyl fermions. Left panel: The upper solid (red) line corresponds to the loss of asymptotic freedom, and it is labeled by AF; the dashed (black) curve corresponds to the SD prediction for the breaking/restoring of chiral symmetry. According to the all-orders BF, the conformal window cannot extend below the solid (red) line, as indicated by the arrows. This line corresponds to the anomalous dimension of the mass reaching the maximum value of 2. Right panel: The BF line is plotted assuming the value of the anomalous dimension to be one.

Figure 4

Phase diagram of $Sp(2N)$ gauge theories with $N_f$ two-index antisymmetric Weyl fermions. Left panel: The upper solid (blue) curve corresponds to the loss of asymptotic freedom, and it is labeled by AF; the dashed (black) curve corresponds to the SD prediction for the breaking/restoring of chiral symmetry. According to the all-orders BF, the conformal window cannot extend below the solid (blue) curve, as indicated by the arrows. This curve corresponds to the anomalous dimension of the mass reaching the maximum value of 2. Right panel: The BF curve is plotted assuming the value of the anomalous dimension to be one.

Figure 5

Phase diagram, from top to bottom, for $Sp(2N)$ gauge theories with $N_{\mathrm{Wf}}=2N_f$ Weyl fermions in the vector representation (light blue), $N_{\mathrm{Wf}}=N_f$ in the two-index antisymmetric representation (light red), and finally in the two-index symmetric (adjoint) [light green]. The arrows indicate that the conformal windows can be smaller, and the associated solid curves correspond to the all-orders beta function prediction for the maximum extension of the conformal windows.

Figure 6

Phase diagram of $SO(N)$ gauge theories with $N_f$ fundamental Weyl fermions. Left panel: The upper solid (blue) line corresponds to the loss of asymptotic freedom, and it is labeled by AF; the dashed (black) curve corresponds to the SD prediction for the breaking/restoring of chiral symmetry. The solid grey (magenta in color) line corresponds to the ACS bound stating that the conformal region should start above this line. According to the all-orders BF, the conformal window cannot extend below the solid (blue) line, as indicated by the arrows. This line corresponds to the anomalous dimension of the mass reaching the maximum value of 2. Right panel: The BF line is plotted assuming the value of the anomalous dimension to be one.

Figure 7

Phase diagram of $SO(N)$ gauge theories with $N_f$ Weyl fermions in the two-index symmetric representation. Left panel: The upper solid (blue) curve corresponds to the loss of asymptotic freedom, and it is labeled by AF; the dashed (black) curve corresponds to the SD prediction for the breaking/restoring of chiral symmetry. According to the all-orders BF, the conformal window cannot extend below the solid (blue) curve, as indicated by the arrows. This curve corresponds to the anomalous dimension of the mass reaching the maximum value of 2. Right panel: The BF curve is plotted assuming the value of the anomalous dimension to be one.

Figure 8

Phase diagram of $SO(N)$ gauge theories with $N_f$ Weyl fermions in the vector representation, in the two-index antisymmetric (adjoint) and finally in the two-index symmetric representation. The arrows indicate that the conformal windows can be smaller, and the associated solid curves correspond to the all-orders beta function prediction for the maximum extension of the conformal windows.