Conformal window of gauge theories with four-fermion interactions and ideal walking technicolor

We investigate the effects of four-fermion interactions on the phase diagram of strongly interacting theories for any representation as function of the number of colors and flavors. We show that the conformal window, for any representation, shrinks with respect to the case in which the four-fermion interactions are neglected. The anomalous dimension of the mass increases beyond the unity value at the lower boundary of the new conformal window. We plot the new phase diagram which can be used, together with the information about the anomalous dimension, to propose ideal models of walking technicolor. We discover that when the extended technicolor sector, responsible for giving masses to the standard model fermions, is sufficiently strongly coupled the technicolor theory, in isolation, must have an infrared fixed point for the full model to be phenomenologically viable. Using the new phase diagram we show that the simplest one family and minimal walking technicolor models are the archetypes of models of dynamical electroweak symmetry breaking. Our predictions can be verified via first principle lattice simulations.

Figure 1

NJL model critical line in the $(\alpha ,g)$ plane. It is assumed to separate the chiral spontaneously broken ($\mathrm{S}\chi \mathrm{SB}$) phase, which is the region above the line, from the unbroken one (Sym.) [96].

Figure 2

Effect of the four-fermion interactions on the lower bound of the conformal window of $SU(N)$ gauge theories with Dirac fermions transforming according to the fundamental representation: From left to right these figures correspond to $N=2$, 3, 4.

Figure 3

Effect of the four-fermion interactions on the lower bound of the conformal window of $SU(N)$ gauge theories with Dirac fermions transforming according to the two index symmetric representation: From left to right these figures correspond to $N=3$, 4.

Figure 4

Effect of the four-fermion interactions on the lower bound of the conformal window of $SU(N)$ gauge theories with Dirac fermions transforming according to the adjoint and two index antisymmetric representation: From left to right these figures correspond to the adjoint for $N=2,\cdots$ and two index antisymetric for $N=4$.

Figure 5

Effect of the four-fermion interactions on the anomalous dimension on the critical line for an $SU(N)$ gauge theories with Dirac fermions transforming according to the two fundamental representation: From left to right these figures correspond to $N=2$, 3, 4.

Figure 6

Effect of the four-fermion interactions on the anomalous dimension on the critical line for an $SU(N)$ gauge theories with Dirac fermions transforming according to the two index symmetric representation: From left to right these figures correspond to $N=3$, 4.

Figure 7

Effect of the four-fermion interactions on the anomalous dimension on the critical line for an $SU(N)$ gauge theories with Dirac fermions transforming according to the adjoint and two index antisymmetric representation: From left to right these figures correspond to the adjoint representation for $N=2$ and the two index antisymmetric one for $N=4$.

Figure 8

Phase diagram for $SU(N)$ gauge theories with (left panel) and without (right panel) four-fermion interactions with Dirac fermions transforming according to different representations of the gauge group. From top to bottom, we plot the conformal window for theories with fermions transforming according to the: (i) fundamental representation (grey), (ii) two-index antisymmetric (blue), (iii) two-index symmetric (red), (iv) adjoint representation (green) as a function of the number of flavors and the number of colors. The shaded areas depict the corresponding conformal windows. The upper solid curve represents loss of asymptotic freedom, the lower solid curve indicates when the zero in the two-loops beta function disappear, and finally above the dashed line, corresponding to $N_f^{\mathrm{crit}}[r,g]$ chiral symmetry restores. We have chosen $g=0.75$ for all the dashed lines in the left panel. The right panel is obtained imposing $g=0$ and it is identical to the one obtained in 3, 70.