More asymptotic safety guaranteed

We study interacting fixed points and phase diagrams of simple and semisimple quantum field theories in four dimensions involving non-Abelian gauge fields, fermions and scalars in the Veneziano limit. Particular emphasis is put on new phenomena which arise due to the semisimple nature of the theory. Using matter field multiplicities as free parameters, we find a large variety of interacting conformal fixed points with stable vacua and crossovers inbetween. Highlights include semisimple gauge theories with exact asymptotic safety, theories with one or several interacting fixed points in the IR, theories where one of the gauge sectors is both UV free and IR free, and theories with weakly interacting fixed points in the UV and the IR limits. The phase diagrams for various simple and semisimple settings are also given. Further aspects such as perturbativity beyond the Veneziano limit, conformal windows, and implications for model building are discussed.

Figure 1

The phase space of parameters (61) for the partially interacting fixed points ${\mathrm{FP}}_2$ (upper panel) and ${\mathrm{FP}}_3$ (lower panel) where one of the two gauge sectors remains interacting and all Yukawa couplings vanish. The inset indicates the different parameter regions and conditions for existence, including whether the non-interacting gauge sector is asymptotically free $(B^{\prime }>0)$ or infrared free $(B^{\prime }<0)$, see Tables 6, 7.

Figure 2

The phase space of parameters for the partially interacting fixed points ${\mathrm{FP}}_4$ and ${\mathrm{FP}}_5$, where the gauge and Yukawa coupling in one gauge sector take interacting fixed points while those of the other sector remain trivial. The insets indicate the different parameter regions and conditions for existence, and whether the non-interacting gauge sector is asymptotically free $(B^{\prime }>0)$ or infrared free $(B^{\prime }<0)$, see Tables 6, 7.

Figure 3

The phase space of parameters for the interacting fixed point ${\mathrm{FP}}_6$ (red) where both gauge sectors take interacting and physical fixed points while all Yukawa couplings vanish. The eigenvalue spectrum at the fixed point always displays exactly two relevant eigenvalues of $O(\epsilon )$ and two irrelevant eigenvalues of order $O({\epsilon }^2)$, see Table 8. Note that this fixed point invariably requires asymptotic freedom for both gauge sectors (see main text).

Figure 4

The phase space of parameters for the fixed points ${\mathrm{FP}}_7$ and ${\mathrm{FP}}_8$ where two gauge and one of the Yukawa couplings take interacting and physical fixed points, while the other Yukawa coupling remains trivial. The inset indicates the signs for $\epsilon$ and $P\epsilon$, together with the sign for the eigenvalue ${\vartheta }_2$, Table 8 (see main text).

Figure 5

The phase space of parameters for the fully interacting fixed point ${\mathrm{FP}}_9$ where all gauge and all Yukawa couplings are nontrivial. The coloured regions relate to the portions of parameter space where the fully interacting fixed point is physical. The inset provides additional information including the sign for the eigenvalue ${\vartheta }_4$ (see main text).

Figure 6

The “phase space” of quantum field theories with fundamental action (25) expressed as a function of field multiplicities and written in terms of $(P,R)$, see (61). The 22 different parameter regions are indicated by roman letters. Theories with parameters in region X are dual to those in region Xb under the exchange of gauge groups following the map (46). Further details on fixed points and their eigenvalue spectra per parameter region are summarized in Figs. 7, 8, and 9.

Figure 7

Shown are the fixed points and eigenvalue spectra of quantum field theories with Lagrangean (25) for the 17 parameter regions with $\epsilon <0$ and $P>0$ in Fig. 6. Scalar selfinteractions are irrelevant at fixed points. All cases display complete asymptotic freedom in the UV. Red shaded slots indicate eigenvalue spectra which arise due to the semisimple character of the theory. In the deep IR, various types of interacting conformal fixed points are achieved depending on whether both, one, or none of the Yukawa couplings $Y$ and $y$ vanish (from left to right). Regimes with “strong coupling only” in the IR are indicated by a hyphen.

Figure 8

Same as Fig. 7, covering the 10 parameter regions with $P<0$ of Fig. 6. Notice that ${\mathrm{FP}}_6$ is absent throughout. Exact asymptotic safety (AS) is realized in the cases 22 and 23. Red shaded slots indicate eigenvalue spectra which arise due to the semisimple character of the theory. For the cases 18–21 and 24–27, partial asymptotic freedom (pAF) or partial asymptotic safety (pAS) is observed whereby one gauge sector decouples entirely at all scales. The latter theories are only UV complete in one of the two gauge sectors and must be viewed as effective rather than fundamental.

Figure 9

Same as Figs. 7 and 8, covering the 17 parameter regions where $\epsilon >0$ and $P>0$ in Fig. 6. Asymptotic freedom is absent in both gauge sectors implying that ${\mathrm{FP}}_2$, ${\mathrm{FP}}_3$, ${\mathrm{FP}}_6$, ${\mathrm{FP}}_7$, and ${\mathrm{FP}}_8$ cannot arise. Partial asymptotic safety (in one gauge sector) is observed in case 28,31,32,35,37,40,41, and 44, whereby the other gauge sector remains free at all scales (pAS). All models must be viewed as effective rather than fundamental theories. All theories become trivial in the IR.

Figure 10

Phase diagrams of asymptotically free semisimple gauge theories (two gauge groups) coupled to matter without Yukawas, covering (a) asymptotic freedom and the Gaussian (G) without interacting fixed points and trajectories running towards strong coupling and confinement, (b) the same, with an additional Banks-Zaks (BZ) fixed point, (c) two BZ fixed points, one of which turned into an IR sink for all trajectories, or (d) three BZ fixed points, the fully interacting one now becoming the IR sink. Axes show the running gauge couplings, fixed points (black) are connected by separatrices (red), and red-shaded areas cover all UV free trajectories with arrows pointing from the UV to the IR.

Figure 11

Phase diagrams of UV complete and weakly interacting simple gauge theories coupled to matter with a single Yukawa coupling, covering (a) asymptotic freedom with the Gaussian UV fixed point and no other weakly interacting fixed point, (b) asymptotic freedom with a Banks-Zaks (BZ) fixed point, (c) asymptotic freedom with a Banks-Zaks and an IR gauge-Yukawa (GY) fixed point, and (d) asymptotic safety with an UV gauge-Yukawa fixed point. Axes display the running gauge and Yukawa couplings, fixed points (black) are connected by separatrices (red), and red-shaded areas cover all UV free trajectories with arrows pointing from the UV to the IR [34, 35]. Examples are given by (63), (67) (see main text).

Figure 12

“Primitives” for phase diagrams of simple gauge-Yukawa theories with asymptotic freedom (AF) or asymptotic safety (AS), corresponding to the different setting shown in Fig. 11. Arrows point from the UV to the IR and connect the different fixed points. Open arrows point towards strong coupling in the IR. The number of outgoing red arrows gives the dimensionality of the UV critical surface. The separate UV safe trajectory towards strong coupling in case (d) is not indicated. Yukawa-induced IR unstable directions in (a,b) or gauge Yukawa fixed points in (c,d) are absent as soon as Yukawa interactions are switched off from the outset.

Figure 13

Schematic phase diagram for asymptotically free semisimple gauge theories (25) with Banks-Zaks type fixed points without Yukawas and exact IR conformality. Field multiplicities correspond to the cases (a) 1–4, (b) 5–13, and (c) 14–17 of Fig. 7, respectively, with scalars decoupled. RG flows point from the UV to the IR (top to bottom). At each fixed point, the dimensionality of the UV critical surface is given by the number of outgoing red arrows. All UV free trajectories terminate at ${\mathrm{FP}}_2$, ${\mathrm{FP}}_6$, and ${\mathrm{FP}}_3$, respectively, which act as fully attractive IR sinks. The topology of the phase diagram (b) is the “square” of Fig. 12, representing Fig. 10. The phase diagrams (a) and (c), representing Fig. 10, cannot be constructed from the primitives in Fig. 12.

Figure 14

Asymptotic freedom and schematic phase diagram for semisimple gauge-Yukawa theories with field multiplicities as in case 9 of Fig. 7. RG flows point from the UV to the IR (top to bottom). Besides the Gaussian UV fixed point (${\mathrm{FP}}_1$), the theory displays all eight weakly interacting fixed points, see Table 3. At each fixed point, the dimensionality of the UV critical surface is given by the number of outgoing red arrows. ${\mathrm{FP}}_9$ is fully attractive and acts as an IR sink. The topology of the phase diagram is the square of Figs. 11, 12; see main text.

Figure 15

Asymptotic freedom and schematic phase diagrams for semisimple gauge-Yukawa theories with field multiplicities as in case 8 of Fig. 7. Flows point from the UV to the IR (top to bottom). The theories display five weakly interacting fixed points besides the Gaussian UV fixed point (${\mathrm{FP}}_1$). The unavailability of ${\mathrm{FP}}_5$, ${\mathrm{FP}}_8$, and ${\mathrm{FP}}_9$ implies that some trajectories escape towards strong coupling (short arrows), and none of the fixed points acts as a complete IR attractor. The topology of the phase diagram is the “direct product” of Figs. 11, 12 with Figs. 11, 12; see main text. The IR unstable direction is removed provided that the Yukawa coupling $y\equiv 0$, in which case the singlet mesons $h$ decouple.

Figure 16

Asymptotic safety and schematic phase diagram of semisimple gauge-Yukawa theories with field multiplicities as in case 22 of Fig. 8. Besides the partially interacting UV fixed point (${\mathrm{FP}}_5$), the theory displays five weakly interacting fixed points. The Gaussian (${\mathrm{FP}}_1$) takes the role of a crossover fixed point and ${\mathrm{FP}}_4$ takes the role of an IR sink. The topology of the phase diagram is the direct product of Figs. 11, 12 with Figs. 11, 12; see main text.