^{1}

^{,*}

^{2}

^{,†}

^{2}

^{,‡}

^{2}

^{,§}

^{2,3}

^{,∥}

^{3}-Origins,

^{3}.

We consider the dynamics of gauge-Yukawa theories in the presence of a large number of matter constituents. We first review the current status for the renormalization group equations of gauge-fermion theories and extend the results to semisimple groups. In this regime these theories develop an interacting ultraviolet fixed point that for the semisimple case leads to a rich phase diagram. The latter contains a complete asymptotically safe fixed point repulsive in all couplings. We then add two gauged Weyl fermions belonging to arbitrary representations of the semisimple gauge group and a complex, gauged scalar to the original gauge-fermion theory allowing for new Yukawa interactions and quartic scalar self-coupling. Consequently, we determine the first nontrivial order in

The most general classes of four-dimensional, renormalizable quantum field theories (QFTs) are in the form of gauge-Yukawa theories. Their dynamics underlies the standard model (SM) interactions and those of any of its sensible extensions. It is therefore paramount to gain a deeper understanding of their dynamics, which is often limited to perturbation theory.

Fundamental theories are those gauge-Yukawa theories that, according to Wilson

The instanton analysis and contribution can be found in

To gain information beyond perturbation theory, one can use supersymmetry. A systematic investigation of nonperturbative constraints that a supersymmetric, asymptotically safe QFT must abide, including a-maximization

Nonperturbative results can also be deduced for nonsupersymmetric theories when considering specific limits in theory space: for example, building upon the large

It is therefore timely to consider the dynamics of gauge-Yukawa theories at large

The paper is organized as follows. In Sec.

We consider both Abelian and non-Abelian semisimple gauge-Yukawa models featuring

Gauge anomalies are avoided by either adding new chiral fermions or by arranging

Summary of the field content of the model. The first two columns detail the transformation of each field under Lorentz and flavor symmetry.

The Lagrangian of the theory reads

To prepare for the large-number-of-flavors limit, the gauge couplings for each gauge group

We now briefly summarize our renormalization conventions to prepare for the computations of the renormalization group (RG) functions in the model. We denote all bare fields and couplings with subscript 0.

In the Lagrangian

In order to practically evaluate the gauge field renormalization we make use of

We start with reviewing the large

Only a limited set of diagrams contribute when computing the RG functions in the large

In the following computations we need to have an expression for the bubble chain. Each elementary bubble stems from a bare, 1PI,

The expression for a bubble chain with

To determine the gauge beta function one has to compute the divergent part of the 2-point function. The leading order (LO) contribution in

In the Abelian case one replaces the gauge generators with the fermion charges

Feynman diagrams for gauge field renormalization at order

Going to the non-Abelian group we have additional contributions from the gluon self-interactions, cf., Figs.

We now review the final results for the gauge beta functions for the Abelian and non-Abelian gauge groups.

We consider the case where the

The function

Now we turn to the case where the

Let us now generalize the result to the case where the vectorlike fermions are charged under a semisimple gauge group. To determine the mixed contribution to the gauge-coupling renormalization

Feynman diagrams for the 2-point functions giving mixed terms to the beta functions.

A test of our results consists in checking that when reexpanding the beta functions given in Eq.

It is straightforward to check that the leading

In the derivation of Eq.

To conclude this section, we investigate the short distance fate of gauge-fermion theories at a large number of matter fields. Here asymptotic freedom is lost and unless an interacting UV fixed point emerges, the underlying theory can be viewed, at best, as an effective low-energy description of physical phenomena. In this regime asymptotic safety is dynamically achieved due to the collective effect of the many fermions present in the theory. This is reflected in the emergence of a nontrivial 0 of the beta functions at NLO in

For single gauge groups, resembling QCD with many flavors, asymptotic safety is indeed a possibility

We now investigate the semisimple case starting with the

Phase diagrams of semisimple gauge theories consisting of two non-Abelian groups (left) and an Abelian and a non-Abelian group (right).

The phase diagram for the semisimple group

One can derive a rough estimate of the asymptotically safe conformal window for the semisimple gauge group as well. We use again the stability of the

We now review and further elucidate the computation of the RG functions of the Yukawa

The leading

To compute the Yukawa beta function we need first to compute the gauge correction to the fermion self-energy to LO in

We start with the Abelian case and then extend the result to the non-Abelian one. At this order in

LO gauge contribution to the fermion self-energy.

The result for the non-Abelian gauge group case is obtained by replacing

We proceed to determine the correction to the scalar self-energy at LO in

Here the diagrams that contribute contain a chain of

LO gauge contribution to the scalar self-energy.

To determine the scalar self-energy for the non-Abelian case one replaces the U(1) charges in

The only vertex diagram that contributes to the Yukawa beta function in the Landau gauge is shown in Fig.

Contributions to the Yukawa vertex at order

With vanishing external momenta, the analytic expression representing the diagrams contributing to the Yukawa coupling with

The previous result can be extended to the non-Abelian case provided that we use

We evaluate the leading order gauge vertex contribution to the scalar self-coupling. Such contributions first appear at

We proceed by computing the diagrams in the Abelian theory before considering the non-Abelian one as well as the semisimple gauge groups.

In order to evaluate the vertex contribution due to the diagrams in Fig.

To regulate the IR divergence of the relevant diagrams, we consider nonvanishing external momenta, as given in Fig.

In the non-Abelian case we have

The quartic coupling contains mixed gauge-coupling contributions already at LO in

The mixed gauge term contributing to the quartic vertex in a semisimple gauge theory.

Having evaluated all relevant diagrams we now compute all the beta functions using Eq.

For the quartic scalar coupling, the beta function is

Finally the gauge beta function for the full model of Eq.

Since the beta functions of many phenomenological models are known to LO, it is convenient to rewrite the above Yukawa and quartic beta function Eqs.

Clearly these are not the same numerical coefficients appearing in

We now elucidate the pole structure of the resummed beta functions, which is a characteristic feature of the theories investigated here.

Since the pole structure of beta function in theories with a simple gauge group has been discussed already in the literature

For the Yukawa beta function we first observe, using Eqs.

Similarly to the Yukawa beta function the RG equation for the quartic coupling, due to the

The situation for the non-Abelian case resemble the Yukawa case. Here the non-Abelian UV fixed point is achieved at

We investigated gauge-Yukawa theories at a large number of gauged fermion fields. We began our analysis by reviewing the state of the art of the gauge-fermion theories. We considered also semisimple groups and by discussing their RG phase diagram we discovered a complete asymptotically safe fixed point which turns out to be repulsive in all gauge couplings.

Subsequently we enriched the original gauge-fermion theories by introducing two Weyl gauged fermions transforming according to arbitrary representations of the gauge group and further added a complex gauged scalar. The latter is responsible for the presence of Yukawa and quartic scalar self-coupling interactions. On par with the gauge sector, we determined the leading

Our work elucidates, corrects, consolidates, and extends results obtained earlier in the literature

The work of N. A. D., A. E. T., F. S., and Z.-W. W. is partially supported by the Danish National Research Foundation under the Grant No. DNRF:90, the Croatian Science Foundation under Grant No. 4418 and the Natural Sciences and Engineering Research Council of Canada (NSERC). O. A. also acknowledges the partial support by the H2020 CSA Twinning Grant No. 692194, RBI-T-WINNING. Z.-W. W. thanks Robert Mann, Tom Steele, Cacciapaglia Giacomo and Emiliano Molinaro for helpful suggestions.

Here we present proofs for the four resummation formulas used for the large

All the resummation formulas rely on a function

For