Gauge-Yukawa theories: Beta functions at large $N_f$
Oleg Antipin Nicola Andrea Dondi Francesco Sannino Anders Eller Thomsen Zhi-Wei Wang




a]Oleg Antipin,
b]Nicola Andrea Dondi,
b]Francesco Sannino,
b]Anders Eller Thomsen,
b,c]Zhi-Wei Wang,


[a]Rudjer Boskovic Institute, Division of Theoretical Physics,
 Bijenička 54, HR-10000 Zagreb, Croatia
[b]CP^3-Origins, University of Southern Denmark,
Campusvej 55
5230 Odense M, Denmark
[c]Department of Physics, University of Waterloo,
Waterloo, ON, N2L 3G1, Canada

oantipin@irb.hr
dondi@cp3.sdu.dk
sannino@cp3.sdu.dk
aethomsen@cp3.sdu.dk
wang@cp3.sdu.dk

 
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 semi-simple groups. In this regime these theories develop an interacting ultraviolet fixed point that for the semi-simple 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 semi-simple 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 1/N_f for the Yukawa and quartic beta functions. Our work elucidates, consolidates and extends results obtained earlier in the literature.  We also acquire relevant knowledge about the dynamics of gauge-Yukawa theories beyond perturbation theory.  Our findings are applicable to any extension of the standard model featuring a large number of fermions such as asymptotic safety.  


	Preprint: CP^3-Origins-2018-011
 


Gauge-Yukawa theories: Beta functions at large N_f
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§ INTRODUCTION


The most general class 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 <cit.>, are well defined at arbitrarily short distances. Asymptotically free  <cit.>  and safe <cit.> QFTs are complementary examples of fundamental theories[The instanton analysis and contribution can be found in <cit.>.]. The recent discovery of four-dimensional, controllable, in the perturbative sense, asymptotically safe QFTs <cit.> has opened the way to novel dark and bright extensions of the SM <cit.>. More generally, it is interesting to investigate the short distance fate of the SM and its extensions including gravity <cit.>. 

To gain information beyond perturbation theory, one can use supersymmetry. A systematic investigation of non-perturbative constraints that  a supersymmetric, asymptotically safe QFT must abide, including a-maximization <cit.> and collider bounds <cit.>, appeared in <cit.> extending and correcting the results of <cit.>. Building upon results of <cit.>, the first evidence for non-perturbative, supersymmetric safety was  gathered in <cit.>  and further analyzed in <cit.>. 

Non-perturbative results can also be deduced for non-supersymmetric theories when considering specific  limits in theory space. For example, building upon the large N_f results of <cit.>, that gauge-fermion theories at any finite number of colors can be argued to develop a non-perturbative ultraviolet (UV) fixed point <cit.>. Consequently, one can extend the original conformal window, reviewed in <cit.>, to include an asymptotically safe phase <cit.>.

It is therefore timely to consider the dynamics of gauge-Yukawa theories at large N _f <cit.>. We investigate this here by elucidating, consolidating, and extending the results obtained earlier in the literature. The results are useful when searching for asymptotically safe extensions of the SM <cit.>.

The paper is organized as follows. In section <ref>, we will introduce our model and the renormalization conventions used throughout the paper. Section <ref> then proceeds to review the current status of large N_f computations for the gauge beta function. We then generalize the results to semi-simple gauge groups and in addition we present their phase diagrams. In section <ref>, we provide a detailed computation of the Yukawa and quartic coupling beta functions at the first nontrivial order in 1/N_f for generic, semi-simple gauge groups. Section <ref> concludes the paper. The explicit derivations of the various resummation formulas used in the large N_f computations can be found in the appendix <ref>.    




§ GAUGE-YUKAWA MODELS: NOTATION AND CONVENTIONS
  
We consider both Abelian and non-Abelian semi-simple gauge-Yukawa models featuring N_f vector-like fermions, Ψ_I  charged under the full gauge group.  Additionally,  the models contain two Weyl spinors, χ and ξ, and a complex scalar, ϕ, such that there is enough content to form a (chiral) Yukawa coupling  among these three fields, and for quartic scalar self-interactions to emerge[Gauge anomalies are  avoided by either adding new chiral fermions or by arranging  χ, ξ  in anomaly free representations of the gauge group. Our results are adaptable to a given gauge-anomaly free model. ]. The field content of the model is summarized in Table <ref> where we report the transformation of each matter field with respect to the gauge interactions. 
 
 

The Lagrangian of the theory reads 
	
    ℒ = -14 F^A_μν F^A,μν    + ∑_I=1^N_f i Ψ_Iγ^μ D_μΨ^I + i χ̅σ̅^μ D_μχ + i ξ̅σ̅^μ D_μξ + (D_μϕ )^† (D_μϕ )
        -  (  y_aijϕ^aχ^iξ^j + y^∗, aijϕ^∗_a χ̅_i ξ̅_j )  - 14λabcdϕ^∗_a ϕ^∗_b ϕ^c ϕ^d  ,

where the index I=1,⋯ N_f is the Ψ flavor index, i,j are gauge indices for χ and ξ,  and a, b, c, d are reserved for the gauged  scalar indices within a given representation that can be read off from the associated covariant derivative 

    D_μΨ_I^i = [∂_μ + i g   A^A_μ (T_Ψ^A)ij] Ψ_I^j  ,
    
    D_μχ^i = [∂_μ + i g   A^A_μ (T_χ^A)ij] χ^j   ,
    
    D_μξ^i = [∂_μ + i g  A^A_μ  (T_ξ^A)ij] ξ^j   ,
    
    D_μϕ^a = [∂_μ + i g A^A_μ (T_ϕ^A)ab] ϕ^b   .

In the most general version of the model, the gauge group is allowed to be semi-simple. The generalization of the covariant derivative in this case is straight forward. Gauge invariance imposes the following constraints

    0    = y_bij (T_ϕ^A)ba + y_akj (T_χ^A)ki + y_aik (T_ξ^A)kj , 
    
    0    = -λebcd (T_ϕ^∗ A)ea -λaecd (T_ϕ^∗ A)eb + λabed (T_ϕ^A)ec+ λabce (T_ϕ^A)ed ,

while the constraint on the Abelian charges reads 
	
    q_ϕ + q_χ + q_ξ =0.
 

To prepare for the large number of flavors limit, the gauge couplings for each gauge group G_α are rescaled as follows
	
    K_α = g_α^2 𝒩 S_2(R_Ψ^α)4π^2  d(R_Ψ^α),     where  𝒩 = N_f ∏_α d(R_Ψ^α).

Here the Dynkin index S_2(R_Ψ^α) is defined via the relation S_2(R_Ψ^α) δ^AB = [ T_R_Ψ^α^A T_R_Ψ^α^B]. In the fundamental representation of an (N) group we take it to assume the value 1/2.  The dimension of a given representation is indicated with d(R_Ψ^α). 



 §.§ Renormalization conventions

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

In the Lagrangian   (<ref>), the bare fields  renormalize according to:
	
    A_α,0^μ = Z_A_α^1/2μ^-ϵ/2 A_α^μ ,   Ψ_0 = Z_Ψ^1/2μ^-ϵ/2Ψ ,   ϕ_0 = Z_ϕ^1/2μ^-ϵ/2ϕ , 
    χ_0 = Z_χ^1/2μ^-ϵ/2χ ,   ξ_0 = Z_ξ^1/2μ^-ϵ/2ξ ,

while the bare couplings are given by 
	
    y_0,aij   = (Z_χ Z_ξ Z_ϕ)^-1/2μ^ϵ/2 (y_aij + δ y_aij)  , 
    λ_0abcd   = Z_ϕ^-2μ^ϵ (λabcd + δλabcd)  , 
     
    	g_α,0    = g̃_α,0μ^ϵ/2 = Z_K_α^-1/2μ^ϵ/2 g_α .
 
We use dimensional regularization with d=4-ϵ. The field renormalizations are expanded in terms of their ϵ poles, writing 
	
    Z_i = 1 + ∑_k=1^∞1ϵ^k Z_i^(k).
 
Similarly, the counter terms are expressed as
	
    δ y_aij = ∑_k=1^∞1ϵ^kδ y^(k)_aijδλabcd = ∑_k=1^∞1ϵ^kδλ^(k)abcd.

It is now possible to expresses the beta function for the couplings in terms of the field-strength renormalizations and the counter terms of the renormalized Lagrangian in the above notation. The beta functions, β_x =  x /lnμ, are given by
	
    β_y,aij   = (-12 + K_β∂∂ K_β + y_ekl2∂∂ y_ekl + λefgh∂∂λefgh) [ δ y^(1)_aij - Z_χ^(1) + Z_ξ^(1) + Z_ϕ^(1)2 y_aij], 
    β_λabcd   = (-1 + K_β∂∂ K_β + y_ekl2∂∂ y_ekl+  λefgh∂∂λefgh) [δλ^(1)abcd - 2 Z_ϕ^(1)λabcd], 
    β_K_α   = (-1 + K_β∂∂ K_β + y_ekl2∂∂ y_ekl + λefgh∂∂λefgh) [- Z_K_α^(1) K_α].


In order to practically evaluate the gauge field renormalization we make use of
	
    0 = div_ϵ[Z_A (1-Π_B({ x_0 })) ] ,

where { x_0 } represents the full set of bare couplings and Π_B is the bare, 1PI, 2-point function of the gauge bosons after having factorized out momenta and polarization structure; iΠ_B,μν(p)=i p^2 Δ_μν(p) Π_B(p^2) with  Δ_μν(p) = η_μν - p_μ p_ν/p^2. 
Similarly, to compute the fermion and scalar field renormalization, we rely on the following relations involving the bare, 1PI, 2-point fermion, -i Σ_B(p) and 2-point scalar, -i S_B(p^2), functions:
	
    0    = div_ϵ[Z_χ,ξ(1- σ̅· pΣ_χ(ξ),B( { x_0 }) ] ,
    
    	0    = div_ϵ[Z_ϕ(1-  p^2 S_B({ x_0 } )) ] .

Finally for the renormalization of the couplings we employ 
	
    0    = div_ϵ[ Z_ϕ^1/2 Z_χ^1/2 Z_ξ^1/2 Y_B ( { x_0 } ) ] 
    
    	0    = div_ϵ[ Z_ϕ^2Λ_B ( { x_0 } ) ]
 
where iY_B  , iΛ_B are the bare, 1PI, 3- and 4-point functions.  These are used to renormalize the Yukawa and quartic couplings. 

 


§ GAUGE-FERMION THEORY
 
We start with reviewing the large N  dynamics in gauge-fermion theory investigated some time ago in <cit.>. This means that we drop ϕ, χ, and ξ from the beginning. We will extend the analysis to include semi-simple gauge groups. The full dynamics including ϕ, χ, and ξ will be investigated in Section <ref>. 

Only a limited set of diagrams contribute when computing the RG functions in the large N limit. In general the order, (1/𝒩)^k, of a diagram in the large 𝒩 expansion can be determined as
	
    k =  powers of g_0^2  -  # of fermion loops.

It follows that dressing gauge lines with  Ψ fermion bubbles (a bubble chain) does not increase the order of a diagram. To obtain the contribution at a given order in 1/𝒩, it is sufficient to consider a small set of diagrams, but one has to sum over every number of bubbles on each gauge line. The resulting power series in K is so well behaved that it is often possible to obtain a closed form expression for the 1/ϵ pole. 

In the following computations we will need to have an expression for the bubble chain. Each elementary bubble stems from a bare, 1PI, Ψ-fermion loop that in MS  reads   
	
    i Π_μν(p) = i p^2 Δ_μν(p) Π_0(p^2),     where
       Π_0(p^2) = - 2 K_0 Γ_0(ϵ) (-4πμ^2p^2)^ϵ/2Γ_0(ϵ) =  Γ^2(2-ϵ2) Γ(ϵ2)Γ(4-ϵ) .


Note that  K_0 is related to g̃_0 like K_α is related to g_α in  (<ref>) . To avoid making the notation heavy we dropped a tilde on K_0.


The expression for a bubble chain with n > 0 bubbles and n + 1 free gauge propagators, D_μν(p), reduces to 
	
    D^(n)_μν(p)    = [ D_μμ_1(p) iΠ^μ_1 μ_2(p) ] [ D_μ_2μ_3(p) iΠ^μ_3 μ_4(p) ] ⋯ D_μ_2nν(p) 
       = -ip^2Δ_μν(p)  Π_0^n(p^2)  .
	
The chain is fully transverse in p  because the gauge-fixing parameter does not renormalize in MS. In our computations we  work in the Landau (Lorenz) gauge. This has the added benefit that D^(n=0)_μν(p) = D_μν(p). The discussion above applies to each individual gauge group α. 
 
 
 

 §.§ Large 𝒩 gauge beta function

To determine the gauge beta function one has to compute the divergent part of the 2-point function. 

The leading order (LO) contribution in 1/𝒩 is simply given by one Ψ bubble. The NLO contribution, on the other hand, is non-trivial and was computed in <cit.> by evaluating the diagrams shown in Fig. <ref>. The first two  diagrams of Fig. <ref> and Fig. <ref> yield 


    iδ^AB p^2 Δ^μν(p) Π^(n)_P(p)=    -( - i g̃_0)^4  [ T_Ψ^A T_Ψ^C T_Ψ^C T_Ψ^B ] μ^2ϵ∫^d k/(2π)^d^d ℓ/(2π)^d D^(n)_ρσ(ℓ) 
           ×[ γ^μi k/k^2γ^ρi (k - ℓ) /(k - ℓ)^2γ^σi k/k^2γ^νi(p - k)/(p-k)^2], 
    
    iδ^AB p^2 Δ^μν(p) Π^(n)_T(p)=    -( - i g̃_0)^4 Tr[ T_Ψ^A T_Ψ^C T_Ψ^B T_Ψ^C ] μ^2ϵ∫^d k/(2π)^d^d ℓ/(2π)^d D^(n)_ρσ(k-ℓ) 
           ×Tr[γ^μi (k - p) / (k- p)^2 γ^ρi (ℓ - p) /( ℓ - p)^2 γ^νi ℓ/ℓ^2γ^σi k/k^2] ,

where  n≥ 0 is the number of bubbles in the chain. These diagrams are present for all gauge groups[In the Abelian case one replaces the gauge generators with the fermion charges q_Ψ.]. For the purpose of summing the contributions from all n, it is useful to extract the coupling and group structure from the 2-point functions
	
    Π^(n)_P    =  d(G)𝒩K_0^n+2 A^(n)_P , 
    Π^(n)_T   = d(G)𝒩( 1 - 1/2C_2(G)/C_2(R_Ψ)) K_0^n+2  A^(n)_T  .

Here the functions A^(n)_P,T contain the loop structure of the respective diagrams and C_2(R_Ψ) is the quadratic Casimir of the representation R_Ψ. 

Going to the non-Abelian group we will have additional contributions from the gluon self-interactions, c.f. Fig. <ref> and Fig. <ref>. The coupling and group structure from their contribution to the 2-point function is parametrized by
	
    Π^(n)_G =  d(G)𝒩C_2(G)/C_2(R_Ψ)K_0^n+1  A^(n)_G  .


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





  §.§.§ Abelian beta function

We consider the case where the Ψ fermions are charged under a single (1) gauge group and determine the associated gauge beta function. In this case, we point out that the K coupling from Eq. (<ref>) reduces to K = g^2 q_Ψ^2 N_f /4 π^2 that agrees with earlier literature.  The resummation of the beta function was performed first by Palanques-Mestre and Pascual <cit.>. Including both the LO and the 1/𝒩 contributions to the 1PI 2-point function, they found[The function F̃ is related to the Mestre-Pascual result via F̃(n,ϵ) = F_MP(n,-ϵ/2).]
	
    Z_K Π_B    = Z_K Π_0(p^2) + Z_K K_0𝒩∑_n=1^∞K_0^n ( A^(n-1)_T  +2 A^(n-1)_P ) 
       = - 2K (-4πμ^2p^2)^ϵ/2Γ_0(ϵ) + 3K/4𝒩∑_n=1^∞( -2K_0/3)^n 1/(n+1)ϵ^nF̃(n+1,ϵ)  .

Notice that we have used the fact that for Abelian gauge theory Z_K = Z_A. The function F̃  encodes the diagram structure, and it turns out that the beta function depends only on F̃(0,ϵ). Using analyticity of F̃,  we can apply the resummation formula (<ref>) to obtain 
	
    Z_K^(1) = . Z_K Π_B |_1/ϵ = -2K3 - 1/2𝒩∫_0^K x  (K-x) F̃ (0,23x) ,

where
	
    F̃(0,x) = (1-x) (1-x3) (1+x2)Γ(4-x) 3 Γ^2(2-x2) Γ(3-x2) Γ(1+x2) .
	




Finally, applying   (<ref>)  the gauge beta function reads
	
    β_K    =  -K^2 ∂∂ K Z_K^(1) = 2K^2 /3[ 1 + 1/𝒩 F_1(K)  ]

to NLO in 1/𝒩. For later convenience we  introduced 
	
    F_1(K) = 3/4∫^K_0  x F̃( 0, 23 x )  .





  §.§.§ Non-Abelian beta function

Now we turn to the case where the Ψ fermions transform under a given representation R_Ψ of  a non-Abelian gauge group. In this instance, the gauge field 2-point function gets an additional contribution due to the gluon self-interaction. The NLO 2-point function is then given by 
	
    Π_B = Π_0 + K_0 d(G)/𝒩∑_n=1   { K_0^n[( 1 - C_2(G)/2 C_2(R_Ψ)) A^(n-1)_T + 2 A^(n-1)_P ] . 
           .  + K_0^n-1C_2(G)/ C_2(R_Ψ) A_G^(n-1)} .

In the non-Abelian case the gauge coupling renormalization is more involved. The computation can either be performed in the background field gauge or in ξ gauge provided that for the latter one includes the vertex renormalization. This computation was originally performed in <cit.> using an alternative method and later reviewed in <cit.> and the result reads
	
    β_K = 2K^23[1 + d(G)𝒩 H_1(K) ]  ,

where we have defined the functions

    H_1(K)    = -11 C_2(G)4 C_2(R_Ψ) +3/4∫_0^K  x F̃(0,23x) G̃(13x) ,
    G̃(x)    = 1+C_2(G)/C_2(R_Ψ)20-43x+32x^2-14x^3+4x^4/ 4( 2 x-1 )  (2 x-3 ) ( 1-x^2 )  .

While F̃ is the function obtained in the (1) case, the non-trivial part of G̃  stems from the gluon contribution[G̃(x)=d(R_Ψ)/d(G)I_2(x) when comparing with the result in <cit.>.]. Notice that H_1 reduces to F_1 in the case of an Abelian gauge group, so Eq. (<ref>) is valid for all simple gauge groups. 





 §.§ Extension to semi-simple gauge groups

 
Let us now generalize the result to the case where the vector-like fermions are charged under a semi-simple gauge group. To determine the mixed contribution to the gauge coupling renormalization Z_K_α it is sufficient to consider only the mixed diagrams appearing in the gauge 2-point function. In the ξ gauges the mixed contribution to the vertex and fermion field renormalization will cancel against each other.  To determine Z_A_α we employ equation (<ref>) to accommodate mixed gauge contributions. Starting from one gauge field, α, the 2-point function contains the usual terms present in Eq. (<ref>). These are unaffected by the presence of other gauge groups. Additionally, at NLO, it is possible to have a gauge bubble chain from a different gauge group β stretching across the fermion loop instead of the original α chain, as shown in Fig. <ref>. These are the only type of diagrams mixing the gauge groups, since a single fermion bubble cannot couple simultaneously to two different gauge groups. The new contribution to the 1PI 2-point function coming from the mixed diagrams with the group β is
	
    Z_A_αΔΠ_α,B   =  Z_A_α K_α,0d(G_β)𝒩∑_n=1^∞ K_β,0^n (A^(n-1)_T  +2 A^(n-1)_P ) 
       =3  d(G_β)/4𝒩 K_α∑_n=1^∞( -2K_β,0/3)^n F̃ (n+1,ϵ) /(n+1) ϵ^n .

To renormalize K_α,0, we have used the fact that Z_K_α = Z_A_α at LO in 1/𝒩. Once again, the 1/ϵ pole can be extracted using the resummation formula (<ref>). The new contribution to the gauge coupling renormalization is obtained as
	
    Δ Z_K_α^(1) = Δ Z_A_α^(1) = - d(G_β)/2 𝒩 K_α∫_0^K_β x (1-xK_β) F̃ (0, 23x)  .
 
The mixed contributions to the beta function read
	
    Δβ_K_α   = - K_α( K_α∂/∂ K_α + K_β∂/∂ K_β) Δ Z_K_α^(1)=   d(G_β)/ 2 𝒩 K_α^2 ∫_0^K_β x F̃(0, 23x) .

Taking into account the mixed contributions coming from all the different gauge groups to each beta function we find 
	
    β_K_α = 2K^2_α3[1 + d(G_α)𝒩 H_1^(α)(K_α) + 1𝒩∑_β≠α d(G_β) F_1(K_β) ].

Note that the H_1 functions are dependent on the specific gauge group and fermion representation as evident from Eq. (<ref>) (hence the superscript). In the case of an Abelian group H_1 reduces  to F_1.  
 
A test of our results consists in checking that when re-expanding the beta functions  given in Eq. (<ref>)  as functions of the couplings the coefficients agree with the state-of-the art  3-loop perturbative computation <cit.> which for G= G_α× G_β reads


    β_K_α^ 3-Loop   =2K^2_α3[1+1/𝒩(K_α(5C_2(G_α) +3C_2(R_Ψ^α) )/4S_2 ( R_Ψ^α) d(R_Ψ^α)^-1 - K_α^2 (79 C_2(G_α)+66C_2(R_Ψ^α))/288S_2(R_Ψ^α)d(R_Ψ^α)^-1)
       +1/𝒩(3K_β C_2(R_Ψ^β)/4S_2(R_Ψ^β)d(R_Ψ^β)^-1-11K_β^2 C_2(R_Ψ^β)/48S(R_Ψ^β)d(R_Ψ^β)^-1)-1/𝒩^2(17K_α C_2(G_α)^2/3S_2(R_Ψ^α)^2d(R_Ψ^α)^-2
       +K_α^21415C_2(G_α)^2+615C_2(G_α)C_2(R_Ψ^α)-288C_2(R_Ψ^α)^2/288S_2(R_Ψ^α)^2d(R_Ψ^α)^-2+⋯)
       -1/𝒩^3(K_α^22857C_2(R_Ψ^α)^3/288S_2(R_Ψ^α)^3d(R_Ψ^α)^-3+⋯)] .
















It is straightforward to check that the leading 1/𝒩 terms agree with the corresponding terms in Eq. (<ref>). 

In the derivation of Eq. (<ref>) we have assumed that the gauge group under which Ψ is charged contains at most one (1). If that were not the case, it would be possible for the fermion bubbles to couple to two different Abelian groups simultaneously. This would give a new class of diagrams, where the bubble chains would alternate between the two groups. In such a case one would also have to take into account kinetic mixing between the two gauge groups.  This has not be considered here.




 §.§ Safe phase diagrams

To conclude this section, we investigate the short distance fate of gauge-fermion theories  at 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 non-trivial zero of the  beta functions at NLO in 1/𝒩 <cit.>.




  §.§.§ Safe QCD
 
For single gauge groups, resembling QCD with many flavors, asymptotic safety is indeed a possibility <cit.>.   To elucidate this point while making this work self-contained, we briefly summarize here the salient points of how a UV fixed point emerges. To make our point clear, we consider an (N) gauge group with N_f flavors transforming according to the fundamental representation. From Eq. (<ref>), one shows that there is a fixed point a K^∗ = 3 up to exponentially vanishing corrections <cit.>.  This occurs because the G̃(x) function in (<ref>) has a pole in the integrand at x=1 (K=3), corresponding to a logarithmic singularity in the H_1(K) function.  The beta function will therefore  have a UV fixed point at K^*≈ 3  to leading order in 1/𝒩, which is obtained from the condition 1+d(G)H_1(K)/𝒩=0.  The new conformal window for these theories as function of number of flavors and colors extends the original infrared (IR) conformal window to also contain the asymptotically safe scenario <cit.>.   It is worth mentioning that to provide a rough estimate of the lower boundary of the asymptotically safe window, one can use the stability of the 1/𝒩 expansion <cit.> by estimating when the 1/𝒩^2 and higher corrections become relevant.  

 







  §.§.§ Safe Semi-simple Gauge Groups

 We now  investigate the semi-simple case   starting with the   G = (N_1) ×(N_2) example.  The associated phase diagram refers to the RG-flow  plotted in the plane of the two gauge couplings, K_1 and K_2, and it is presented in Fig. <ref>. The UV interacting fixed point, repulsive in all directions,  occurs for K_1^∗ = K_2^∗ =3 (the blue-dot) up to exponentially small corrections. Two more interacting fixed points occur for (K_1^∗ =3, K_2^∗ =0) and  (K_1^∗ =0, K_2^∗ =3) corresponding to the fixed points of each single gauge group. Finally we have the Gaussian IR fixed point at the origin of the coupling space. This analysis complements the perturbative analysis for semi-simple gauge groups investigated first in <cit.>. We therefore discover that there is a UV complete fixed point for semi-simple gauge theories with a two-dimensional critical surface. 

 
 The phase diagram for the semi-simple group G = (1) ×(N_2) is presented in Fig. <ref>. It is structurally identical to the (N_1) ×(N_2) case above with the difference that the UV fixed point for the (1) gauge couplings occurs at K_1^∗=15/2.



 
One can derive a rough estimate of the asymptotically safe conformal window for the semi-simple gauge group as well. We use again the stability of the 1/𝒩 expansion by estimating numerically the size of the known 1/𝒩^2 and 1/𝒩^3 corrections from  Eq. (<ref>). 
We expect it to be wider than safe QCD because the effective number of flavors 𝒩 = N_f d(R_Ψ^1)d(R_Ψ^2) is  larger. 
















§ YUKAWA AND SELF-COUPLING BETA FUNCTIONS

We now review and further elucidate the computation of the RG-functions of the Yukawa <cit.> and quartic couplings <cit.> of the model (<ref>) in  the presence of a large number of vector-like fermions.  Finally, the results of the running of the quartic coupling are extended to the case where the Ψ fermions transform under a semi-simple gauge group. We are interested in the case in which y and λ scale with 𝒩 as λ∼ y^2 ∼ 1/𝒩. This is the region for which a UV fixed point can appear due to the interplay between the large 𝒩 gauge contribution and leading  corrections stemming from the Yukawa and scalar self-coupling. With this scaling of the couplings it is  sufficient to consider  the 1-loop contributions from the Yukawa and quartic coupling to their beta functions. The counting will then ensure that higher loops will give corrections that are higher order in 1/𝒩. 

The leading 1/𝒩 contribution stemming from the Ψ fermions is obtained by dressing gauge lines with their bubbles. As in section <ref>, these diagrams can be resummed and the 1/ϵ pole extracted in a closed from. We will first discuss the new contribution to the fermion and scalar self energies before moving to compute the vertex corrections.  We shall see that it is straightforward to generalize the results to the semi-simple case except for the quartic self-coupling.  




 §.§ Fermion self-energy

To compute the Yukawa beta function we need first to compute the gauge correction to the fermion self-energy to LO in 1/𝒩 for the χ, ξ fermions.  




  §.§.§ Abelian case

We start with the Abelian case and then extend the result to the non-Abelian one.  At this order in 1/𝒩 the relevant diagram is shown in Fig. <ref>. 

For  the χ (identically for ξ) self-energy, the defining integral for the n-bubble diagram is
	
    -i Σ_χ^(n)(p) = ( i q_χg̃_0)^2 μ^ϵ∫^d k/(2π)^dσ̅^μiσ· (p-k)/(p-k)^2σ̅^ν D^(n)_νμ(k)  .
 
 The integral is known, and the diagram evaluates to 
	
    -i Σ^(n)_χ (p) = -3i/8 𝒩q^2_χ/q_Ψ^2σ̅· p ( -2 K_0/3)^n+1 3^n Γ_0^n(ϵ) Γ_ψ(n,ϵ) ( - 4 πμ^2/p^2)^(n+1)ϵ/2,

where we defined
	
    Γ_ψ(n,ϵ) = n (3 - ϵ)n+1Γ(2 - ϵ2 ) Γ(1 + n+12ϵ ) Γ(1 - n+1/2ϵ)Γ(2+ n2ϵ ) Γ(3 - n+2/2ϵ) .

Summing over all bubbles to obtain the total gauge contribution to the self-energy at 1/𝒩 and shifting the sum from n → n-1, we obtain 
	
    Σ_χ/σ̅· p = - 9/16 𝒩q^2_χ/q_Ψ^2∑_n=1^∞(- 2 K_0/3)^n 1n ϵ^n H_ψ(n, ϵ)  ,

where
	
    H_ψ(n,ϵ) = - 23(- 4 πμ^2/p^2)^nϵ/2 [3ϵ Γ_0(ϵ)]^n-1 (1-nϵ) n ϵ Γ_ψ(n-1,ϵ)  .

The contribution to the RG-function stems from the  1/ϵ pole which is extracted using the resummation formula (<ref>) and yields  
	
    Z_χ (ξ) ^(1) = Σ_χ(ξ)/σ̅· p  = 3/8 𝒩q_χ (ξ)^2/q_Ψ^2∫_0^K x   H_ψ(0,23x).

To arrive at the above relation between Z_χ and the 2-point function  we used the fact that Z_χ = 1 + 𝒪(1/𝒩). For the reader's convenience we also give the expression for H_ψ 
	
    H_ψ(0,x) = x(1-x3) Γ(4-x)3 Γ^2(2-x) Γ(3-x2 ) Γ(1+x2) .





  §.§.§ Non-Abelian case

The result for the non-Abelian gauge group case is obtained by replacing 	
	
    (q_χg̃_0)^2 ⟶g̃_0^2 (T_χ^A T_χ^A)ij = 4π^2 d(R_Ψ) K_0𝒩  S_2(R_Ψ) C_2(R_χ)  δij

in the n-bubble self-energy (<ref>). The rest of the computation follows the Abelian case  yielding the field-strength renormalization  
	
    Z_χ (ξ)^(1) =  3d(R_Ψ)/8 𝒩 S_2(R_Ψ) C_2(R_χ (ξ)) ∫_0^K x   H_ψ(0,23x)  .






 §.§ Scalar self-energy


We proceed to determine the  correction to the scalar self-energy at LO in 1/𝒩. This is a necessary step towards the full computation of the Yukawa and quartic self-coupling.  


  §.§.§ Abelian case

  Here the  diagrams that contribute contain a chain of n fermion bubbles as shown in Fig. <ref>. Analytically 
	
    -iS^(n)(p^2) = (ig̃_0 q_ϕ)^2 μ^ϵ∫^d k(2π)^di(p-k)^2 (2p-k)^μ D^(n)_μν(k) (2p-k)^ν .

The integral  yields the n-bubble contribution 
	
    -iS^(n)(p^2) = -i K_0 q_ϕ^2𝒩 q_Ψ^2 (-2K_0)^n p^2 (- 4πμ^2p^2)^(n+1) ϵ/2 (3 - ϵ) Γ_0^n(ϵ)  Γ_ϕ(n,ϵ) ,

where we have defined the function
	
    Γ_ϕ(n,ϵ) = Γ(1-n+12ϵ) Γ(2-ϵ2) Γ(n+12ϵ) 2  Γ(2+n2ϵ) Γ(3- n+22ϵ) .

Summing over S^(n)(p^2) and shifting the summation from n to n-1, one can rewrite the derivative with respect to p^2  in the form  
	
    p^2S(p^2)  = - 9 q_ϕ^24𝒩 q_Ψ^2∑_n=1^∞(-2 K_03)^n 1nϵ^n H_ϕ(n,ϵ)  .

Here we defined
	
    H_ϕ(n,ϵ) =  4 (- 4πμ^2p^2)^n ϵ/2 (1 - n2ϵ ) (1 - ϵ3 ) [3ϵ Γ_0(ϵ)]^n-1Γ_ϕ(n-1,ϵ)  .

The simple ϵ pole of interest for the RG-function,  is determined using   (<ref>) and it yields	

    Z_ϕ^(1) =  p^2 S(p^2)  = 3 q_ϕ^22 𝒩 q_Ψ^2∫_0^K x  H_ϕ(0, 23x)  ,

where	
    H_ϕ(0, x) = (1 - x3) Γ(4 - x) 3 Γ^2(2-x2) Γ(3- x2) Γ(1 + x2)  .




  §.§.§ Non-Abelian case

To determine the scalar self-energy for the non-Abelian case one replaces the (1) charges in (<ref>)  as follows 
	
    (q_ϕg̃_0)^2 ⟶g̃_0^2 (T_ϕ^A T_ϕ^A)ab = 4π^2 d(R_Ψ) K_0𝒩 S_2(R_Ψ) C_2(R_ϕ) δab .

The rest of the computation is identical to the Abelian case and yields 
	
    Z^(1)_ϕ = 3  d(R_Ψ)2 𝒩  S_2(R_Ψ) C_2(R_ϕ) ∫_0^K x  H_ϕ(0, 23x)  .






 §.§ Yukawa vertex

The only vertex diagram that contributes to the Yukawa beta function in the Landau gauge is shown in Fig. <ref>. The other diagrams vanish trivially in this gauge when the external momenta are set to zero. 



  §.§.§ Abelian case

With vanishing external momenta,  the analytic expression representing the diagrams  contributing to the Yukawa coupling with n bubbles on the gauge line is  


	
    iY^(n) =(-i y) (-iq_χg̃_0)  (i q_ξg̃_0) μ^ϵ∫^d k(2π)^dσ^μi σ̅· k k^2 -m^2iσ· kk^2 -m^2σ̅^ν D_μν^(n)(k)  .

A common mass has been added to the fermion propagators as an IR regulator, as it does not influence the divergent part of the diagram. One finds that
	
    iY^(n) = -i 3  y8 𝒩q_χ q_ξq_Ψ^2 (-2K_0)^n+1Γ_0(ϵ)^n (4πμ^2m^2)^(n+1)ϵ/2 (1-ϵ3) Γ(2 - n+12ϵ) Γ(n+12ϵ) Γ(2 - ϵ2) .

Next we sum over every number of fermion bubbles to find the full 1 / 𝒩 gauge contribution to the vertex, and cast the expression in a suitable form, yielding 
	
    iY = -i y  9 4𝒩q_χ q_ξq_Ψ^2∑_n=1^∞(- 2K_03)^n1nϵ^n H_y(n,ϵ) ,

where  
	
    H_y(n,ϵ) = (4πμ^2m^2)^nϵ /2[3ϵ Γ_0(ϵ)]^n-1 (1 - ϵ3) Γ(2 - n2ϵ) Γ(1+ n2ϵ) Γ(2 - ϵ2) .

Following the usual procedure the 1/ ϵ pole can be extracted in a closed form using  (<ref>), as H_y is sufficiently regular. The counter term for the Yukawa coupling is then extracted via 
	
    δ y^(1) = . Y |_1/ϵ = y 3 2 𝒩q_χ q_ξq_Ψ^2∫_0^K x  H_y(0,23 x)

with
	
    H_y(0,x) = (1-x/3) Γ(4-x)/6 Γ^3(2- x2) Γ(1 + x2) .





  §.§.§ Non-Abelian case

The previous result can be extended to the non-Abelian case provided that we use
	
    y q_χ q_ξg̃_0^2 ⟶ y_akl g̃_0^2 (T_χ^A)ki (T_ξ^A)lj  = y_akl   (T_χ^A)ki (T_ξ^A)lj4π^2 d(R_Ψ) K_0𝒩  S_2(R_Ψ)

in the Abelian expression  (<ref>). Employing the identity  
	
    y_akl (T^A_χ)ki (T^A_ξ)lj  = -y_aijC_2(R_χ) + C_2(R_ξ) - C_2(R_ϕ)2 ,

 we arrive at 
 	
    δ y^(1)_aij = -y_aij3 d(R_Ψ)2𝒩C_2(R_χ) + C_2(R_ξ) - C_2(R_ϕ)/2 S_2(R_Ψ) ∫_0^K x  H_y(0,23 x)  .





 §.§ Quartic vertex

We  evaluate the leading order gauge vertex contribution to the scalar self coupling. Such contributions first appear at 1/𝒩^ 2, and in the Landau gauge the only contribution stems from the diagram of Fig. <ref>. All other types of diagrams, see Fig. <ref>, contain at least one three-point gauge insertion on an external scalar leg. Since the gauge propagator is transverse in the Landau gauge, any such coupling will be proportional to the external momenta and vanish when this is taken to zero. Therefore these diagrams will not contribute to the vertex counter term. 

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






  §.§.§ Abelian computation

In order to evaluate the vertex contribution due to the diagrams in Fig. <ref>, we first denote by iΛ^(n,m) such a diagram with m bubbles on the one propagator and n-m bubbles on the other. Λ^(n,m) and Λ^(n,n-m) are indistinguishable, therefore we include a factor of 12 for each pair (n,m) to avoid double counting. This also agrees with the diagrams where n=2m, in which case the two bubble chains are indistinguishable and they receive a symmetry factor 12 from the Feynman rules. In the limit of vanishing external momenta, all permutations of the scalar legs count the same.  Moreover, in this limit, only the loop momenta passes through the bubble chains and the loop integrals will be indifferent to which propagator the bubbles are placed on. The divergent part of the 4-point function at vanishing external momenta is obtained from   
	
    i Λ = ∑_n=0^∞∑_m=0^n i Λ^(n,m)  +  permutation = 2∑_n=0^∞ (n+1) iΛ^(n,0).

It is thus clear that it is sufficient to evaluate only the diagrams with bubbles on one of the gauge lines.
  
To regulate the IR-divergence of the relevant diagrams, we consider non-vanishing external momenta, as given in Fig. <ref>, from which we obtain
	
    iΛ^(n,0) = 12(i 2 q_ϕ^2 g̃_0^2)^2 μ^2ϵ∫^d k(2π)^d D^μν(p-k) D^(n)_νμ(k)  ,
 
where the factor 12 is the aforementioned symmetry factor. Evaluating the integral one obtains the result
	
    iΛ^(n,0)   = iπ^24 𝒩^2q_ϕ^4q_Ψ^4 (-2 K_0)^n+2μ^ϵ( -4πμ^2p^2)^(n+1)ϵ/2Γ^n_0(ϵ) Γ_λ(n,ϵ)  ,

with 
	
    Γ_λ(n,ϵ) = (3-ϵ)(4 - ϵ + nϵ)(n+1)ϵΓ(1 - n+12ϵ) Γ(1 -ϵ2) Γ(1 + n+12ϵ) Γ(2 + n2ϵ) Γ(2 - n+22ϵ) .

At this point we may sum over all the different diagrams as indicated by Eq. (<ref>) to obtain the pole structure of the vertex.   By redefining the summation from n ⟶ n-2, one finds
	
    iΛ = i 54 π^2 μ^ϵ𝒩^2q_ϕ^4q_Ψ^4∑_n=2^∞(-2K_03)^n1ϵ^n-1 H_λ(n,ϵ) ,
 
where
	
    H_λ(n,ϵ) =  (-4πμ^2p^2)^(n-1)ϵ/2[3ϵ Γ_0(ϵ) ]^n-2(n-1)ϵ12 Γ_λ(n-2, ϵ)  .

It is now possible to resum the pole structure of the vertex contribution using (<ref>). The resulting leading order gauge contribution to the quartic counter term pole is
	
    δλ^(1) = Λ =  24 π^2 𝒩^2q_ϕ^4q_Ψ^4 μ^ϵ  K^2 H_λ(1,23 K)  ,
 
with
	
    H_λ(1,x) = (1 - x3) Γ(4-x) 6Γ^3(2 - x2) Γ(1 + x2) .
 




  §.§.§ Non-Abelian case
 
 In the non-Abelian case we have 

    iΛ^(n,0)abcd = 12(i g̃_0^2 )^2 T_ϕ^A, T_ϕ^BacT_ϕ^A, T_ϕ^Bbd μ^2ϵ∫^d k(2π)^d D^μν(p-k) D^(n)_νμ(k)  .
 
By  comparing this expression with the Abelian diagram of (<ref>) we can read off the non-Abelian result, paying attention to the fact that the color structure changes depending on the permutation of the external scalars. The contribution to the counter term in the non-Abelian theory is thus given by
	
    δλ^(1)abcd =  24 π^2 d^2(R_Ψ) 𝒩^2 S_2^2(R_Ψ) Aabcd  K^2  H_λ(1,23 K)  ,

	with 

    Aabcd = 18(T_ϕ^A, T_ϕ^BacT_ϕ^A, T_ϕ^Bbd + T_ϕ^A, T_ϕ^BadT_ϕ^A, T_ϕ^Bbc)  .





  §.§.§ Semi-simple gauge group


The quartic coupling contains mixed gauge-coupling contributions already at LO in 1/𝒩. If we consider the case where both the scalar and the vector-like fermions are charged under a semi-simple gauge group, then the quartic coupling receives mixed contributions of the type sketched in Fig. <ref>. For every pair of simple gauge groups (G_α, G_β), all the diagrams contain at least one power of  K_α,0 and K_β,0 respectively. Starting from the simple diagram where all fermion bubbles are put on  the  G_β gauge line, we have 
	
    iΛ^(n,0) =  (i 2 g̃_α,0g̃_β,0)^2 ( T^A_ϕ,α T^B_ϕ,β )ac (T^A_ϕ,α T^B_ϕ,β )bd μ^2ϵ∫^d k(2π)^d D_α^μν(p-k) D^(n)_β,νμ(k).

Recall here that the generators belong to different gauge groups and therefore they commute. Comparing this diagram to Eqs. (<ref>) and (<ref>)  leads to 
	
    iΛ^(n,0) = i ( T^A_ϕ,α T^B_ϕ,β )ac    (T^A_ϕ,α T^B_ϕ,β)bdπ^2 μ^ϵ2  𝒩^2d(R_Ψ^α)  d(R_Ψ^β)S_2(R_Ψ^α)   S_2(R_Ψ^β)
       (-2K_α,0) (-2K_β,0)^n+1(-4πμ^2p^2)^(n+1)ϵ/2Γ_0^n(ϵ) Γ_λ(n,ϵ).

We can distribute the fermion bubbles  in several ways on the two gauge lines. Each bubble gives a factor of K_α,0 or K_β,0 depending on the gauge line, but the kinematic part of the diagram remains unchanged in the limit of vanishing external momentum. Because the pole structure does not depend on the external momentum, the pole structure of a general diagram can be related to the diagram with fermion bubbles on only one of the gauge line. In particular 
	
    iΛ^(n,m) = i (K_α,0K_β,0)^mΛ^(n,0).

Thus summing over all possible bubbles and taking into account the different permutations of the external scalars, we arrive at 
	
    i Λabcd = i    B_α,βabcd108 π^2𝒩^2d(R_Ψ^α)  d(R_Ψ^β)S_2(R_Ψ^α)   S_2(R_Ψ^β) 
           ×∑_n=1^∞∑_m=0^n-1(-2K_α,03)^1+m(-2 K_β,03)^n-mH_λ(n+1,ϵ)   nϵ^n,

having defined the tensor 
	
    B_α,βabcd = 12[( T^A_ϕ,α T^B_ϕ,β )ac (T^A_ϕ,α T^B_ϕ,β )bd + ( T^A_ϕ,α T^B_ϕ,β )ad (T^A_ϕ,α T^B_ϕ,β )bc].
 
Employing Eq. (<ref>) and collecting the contributions from all the mixed terms, we find	
	
    δλ^(1)abcd =   ∑_α 24π^2 d^2(R_Ψ^α) 𝒩^2  S_2^2(R_Ψ^α) A_1abcd  K_α^2 H_λ(1,23 K_α)
        +∑_α < β B_α, βabcd48 π^2𝒩^2d(R_Ψ^α)  d(R_Ψ^β)S_2(R_Ψ^α)   S_2(R_Ψ^β)K_α K_βK_α -K_β∫_K_β^K_α x  H_λ(1, 23x)  ,

which naturally also contains the unmixed contributions.  




 §.§ Complete set of large 𝒩 beta functions
 
Having evaluated all relevant diagrams we now compute all the beta functions using Eq. (<ref>), starting with the Yukawa that reads   

    (β_y)_aij =   1/32 π^2[ (y_b y^†, b y_a)_ij + (y_a y^†, b y_b)_ij + 2[y_a y^†,b] y_bij] 
       - y_aij∑_α3  d(R^α_Ψ)/16 𝒩C_2(R_χ^α ) + C_2( R_ξ^α)/S_2(R_Ψ^α )  K_α  H_ψ (0,23 K_α) 
       - y_aij∑_α3  d(R_Ψ^α)4𝒩C_2(R_ϕ^α)/S_2(R_Ψ^α) K_α  H_ϕ(0, 23K_α) 
        - y_aij∑_α3  d(R_Ψ^α)2𝒩C_2(R_χ^α) + C_2(R_ξ^α) - C_2(R_ϕ^α)/2 S_2(R_Ψ^α) K_α  H_y(0,23 K_α) + 𝒪(1/𝒩^2)  .

This result includes the 1-loop terms from the matter sector. 

For the quartic scalar coupling, the beta function is  

    β_λabcd =   116π^2(2 λaecfλbfde +2 λaedfλbfce + λabefλefcd) +1/4π^2[y_d y^†,e] λabce
       - 1/4 π^2Tr[y^a y^†_c y^b y^†_d + y^a y^†_d y^b y^†_c ] - λabcd∑_α3  d(R_Ψ^α)𝒩C_2( R_ϕ^α)/S_2( R_Ψ^α)  K_α  H_ϕ(0, 23 K_α)
       + 48 π^2𝒩^2∑_α< β B_α,βabcdd(R_Ψ^α)  d(R_Ψ^β)S_2(R_Ψ^α)   S_2(R_Ψ^β)K_α K_βK_α - K_β[K_α  H_λ(1, 23 K_α) - K_β  H_λ(1, 23K_β) ] 
       + 24 π^2𝒩^2∑_α A_αabcd d^2(R_Ψ^α)  S_2^2(R_Ψ^α) [K_α^2  H_λ(1,23 K_α) + K_α^3 ∂∂ K_α H_λ(1,23 K_α)] + 𝒪(1/𝒩^3)  .


Finally the gauge beta function for the full model  of  Eq. (<ref>) is  

    β_K_α =  2K_α^23   [ 1 +  d(R_Ψ^α) /𝒩 S_2(R_Ψ^α )( 1/2 S_2( R_χ^α) + 1/2 S_2( R_ξ^α) + 1/4 S_2(R_ϕ^α) ) . 
            + d(G)/𝒩 H_1^(α) (K_α) + 1𝒩∑_β≠α d(G_β) F_1(K_β) ] + 𝒪(1/𝒩^2)  .

We note that the 1/𝒩 counting is consistent with the fact that λ∼ y^2 ∼ g^2 ∼ 1/𝒩.




 §.§ A mnemonic for Yukawa and quartic beta functions 
 
Since the beta functions of many phenomenological models are known to LO, it is convenient to rewrite the above Yukawa and quartic beta function Eq. (<ref>) and Eq. (<ref>) in a more compact form. With this prescription one can immediately obtain the bubble diagram contributions to known 1-loop beta functions by simply using the following recipe. 


The Yukawa beta function at large number of fermions can be written in the following compact form


    β_y=c_1y^3 + y ∑_αc_α K_α I_y(K_α),

with 

    I_y(K_α)=H_ϕ(0,23K_α)(1+K_αC_2(R_ϕ^α)/6(C_2(R_χ^α)+C_2(R_ξ^α))) ,

containing information about the resummed fermion bubbles and c_1, c_α are the standard 1-loop coefficients for the Yukawa beta function. Thus, when c_1, c_α are known, the total Yukawa beta function with bubble diagram contributions is straightforward.
Similarly, for the quartic coupling we write

    β_λ=c_1λ^2+λ∑_αc_α K_α I_λ g^2(K_α)+∑_α c'_α K_α^2I_g^4(K_α) +∑_α < βc_αβ K_α K_β I_g_1^2g_2^2(K_α, K_β) ,

with c_1, c_α, c'_α, c_αβ  the known 1-loop coefficients[Clearly these are not the same numerical coefficients appearing in (<ref>).] for the quartic beta function and the resummed fermion bubbles appear via 

    I_λ g^2(K_α)    =H_ϕ(0,23K_α)
    
    I_g^4(K_α)   =H_λ(1,23K_α)+K_αdH_λ(1,23K_α)/dK_α
    
    I_g_1^2g_2^2(K_α, K_β)   =1/K_α-K_β[K_α H_λ(1,23K_α)-K_β H_λ(1,23K_β)] .
 
It is thus also straightforward to obtain the total quartic beta function including the bubble diagram contributions when c_1, c_α, c_α', c_αβ are known.




 §.§ Pole structure of the beta functions

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 literature <cit.>, we move immediately to consider the semi-simple gauge-fermion theories. Here we observe that if the group structures contains an Abelian factor  the corresponding beta function is such that it still features a singularity for  K=15/2 regardless of the presence of other non-Abelian factors. This is so since the extra contribution assume the form of F_1 rather than H_1 (see Eq. (<ref>)). Thus, it is not possible to shift the resulting UV fixed point value of the Abelian gauge coupling away from the Abelian pole. This is clearly manifest in the phase diagram structure of Fig. <ref>.



For the Yukawa beta function we first observe, using Eqs. (<ref>),  (<ref>), and (<ref>),  that the following relations hold:

    H_ψ(0,x) = x H_0(x),    H_ϕ(0, x) = H_0(x),    H_y(0,x) = H_λ(1,x) = (1-x4) H_0(x),

	where 
	
    H_0(x) = (1 - x3) Γ(4-x)3 Γ^2(2 - x2) Γ(3 - x2) Γ(1 + x2) .

This means that they all inherit a pole at x=5 yielding 

    H_ψ(0,23K_α)∼1/15/2-K_α,   H_ϕ(0,23K_α)∼1/15/2-K_α,   H_y(0,23K_α)∼1/K_α-15/2 .

Thus the Yukawa coupling RG function Eq. (<ref>) near the pole will assume the following form:

    β_y=c_1y^3 + y K_α(1/K_α-15/2)(c_2+c_3K_α) ,

where c_1, c_2, c_3 are positive constants stemming from the group structure of the theory. 
It is clear that the three summation functions altogether provide large negative contributions when approaching the pole (i.e. K_α=15/2) from the left. The pole in the Yukawa beta function appears at the original Abelian gauge coupling location (see Eq. (<ref>)). This implies that if the gauge group features an Abelian factor, from  Eq. (<ref>),  we deduce that the Yukawa gauge coupling vanishes in the UV (free rather than safe). The situation changes dramatically when only non-Abelian gauge groups are involved. This is so since the non-Abelian gauge beta function reaches an UV fixed point  at K_α=3, which is clearly away from the Abelian pole, allowing for non-trivial UV zeros of the Yukawa beta function.  

Similarly to the Yukawa beta function the RG equation for the quartic coupling, due to the K_α H_ϕ(0,2/3K_α) term in Eq. (<ref>), receives a large negative contribution at the Abelian pole. 
This is made explicit by the relations  

    H_λ(1,23K_α)=H_y(0,23K_α)∼1/K_α-15/2,    ∂/∂ K_αH_λ(1,23K_α)∼-1/(K_α-15/2)^2 .

Thus, the  quartic coupling RG function Eq. (<ref>) near the first singularity assumes the form

    β_λ=c_1λ^2+c_2λ K_α(1/K_α-15/2)+c_3 K_α^2(1/K_α-15/2-1/(K_α-15/2)^2) ,

where c_1, c_2, c_3 denote positive constants (distinct from the Yukawa case). From Eq. (<ref>) we learn that the quartic beta function has an even more singular structure located at the Abelian pole. If the theory contains an Abelian gauge group one observes that the quartic coupling develops an explosive behavior (λ∝exp(N_f)) at the Abelian fixed point and the fixed point analysis cannot be trusted. 

The situation for the non-Abelian case resemble the Yukawa case. Here the non-Abelian UV fixed point is achieved at K_α=3 which is below and sufficiently away from the pole in the quartic coupling, allowing for (depending on the theory) the existence of UV fixed points in all couplings. 







§ CONCLUSION
 
We investigated gauge-Yukawa theories  at large number of  gauged fermion fields. We begun our analysis by reviewing the state-of-the-art of the  gauge-fermion theories. We considered also semi-simple 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 1/N_f Yukawa and quartic beta functions. We then discussed the pole structure of the system of RG equations. This has an immediate impact on the existence, location and stability of  related fixed points.  In particular one observes that when an Abelian gauge coupling is present in the theory, the Yukawa beta function is driven to be free while the quartic coupling becomes uncontrollable, de facto requiring a fully non-perturbative analysis near this point. The situation changes dramatically when only non-Abelian gauge couplings are present. Because the latter achieve a fixed point at a much lower value of the Abelian one (still appearing as the only pole in the Yukawa and quartic beta function) Yukawa and quartic couplings (depending on the theory) can still admit UV interacting fixed points. These results cannot be extended to the supersymmetric case <cit.> for a number of reasons.  The first reason is that the resummation procedure would have to respect supersymmetry and, in addition, it has already been proven in <cit.> that it is impossible to have an UV fixed point for any N_f in super QCD. 

Our work elucidates, corrects, consolidates, and extends results obtained earlier in the literature <cit.>. It also provides the stepping stone and the needed instruments for future theoretical and phenomenological extensions and analyses.  
 



 §.§ Acknowledgments

The work has been partially supported by the Danish National Research Foundation under grant DNRF:90, the Croatian Science Foundation under the project 4418 and the Natural Sciences and Engineering Research Council of Canada (NSERC). OA also acknowledges the partial support by the H2020 CSA Twinning project No.692194, RBI- T-WINNING.  Z.W. Wang thanks Robert Mann, Tom Steele, Cacciapaglia Giacomo and Emiliano Molinaro for helpful suggestions.





§ RESUMMATION FORMULAS
 
Here we present proofs for the four resummation formulas used for the large 𝒩 computations. Regardless of the quantity in question only the pole structure at ϵ→ 0 will be relevant. We will use the notation “ϵ→ 0∼” to mean equal divergent parts. Note that all of these resummations are only valid to LO in 1/𝒩.

All the resummation formulas rely on a function H(n,ϵ) being regular both for α = nϵ→ 0 with ϵ constant and for ϵ→ 0 with α constant. In general, the functions occurring in this paper can easily be checked to satisfy this condition, except, possibly, for the term [3ϵ Γ_0(ϵ)]^n with fixed α and ϵ→ 0. To see that this term is well behaved, it is sufficient to note that the base satisfy  
	
    (6 Γ^2(2-ϵ2) Γ(1 + ϵ2) Γ(4-ϵ))^α/ϵ - 1 = (1 + ϵ f(ϵ) )^α/ϵ,

for some regular function f that is regular in 0. To prove that this expression has no pole, observe that there must exist constants c, δ >0 such that 
	
    f(ϵ)≤ c,     ∀ϵ < δ.
 
It must then hold that 
	
    lim_ϵ→ 01 + ϵ  f(ϵ)^α/ϵ≤lim_ϵ→ 0_+(1 +c ϵ)^α/ϵ = lim_x →∞(1 + cx)^x α = e^c α

having substituted x= 1ϵ. It is thus found that [3ϵ Γ_0(ϵ)]^n is without a pole in ϵ→ 0 and fixed nϵ.




 §.§ First resummation

	
    ∑_n=1^∞( -2K_0/3)^n H(n+1,ϵ)/(n+1) ϵ^n1/ϵ∼ -23ϵ∫_0^K x (1 - xK) H(0, 23x)

The proof, originally due to <cit.>, is presented here to keep the work self-contained. We define 
	
    R(ϵ) = ∑_n=1^∞( -2K_0/3)^n H(n+1,ϵ)/(n+1) ϵ^n .

The resummation proceeds assuming that H(n,ϵ) can be written as a power series
	
    H(n,ϵ) = ∑_j=0^∞ H^(j)(ϵ) (nϵ)^j,
  
where every H^(j)(ϵ) are regular in ϵ. Such an expansion only exists, if H(αϵ, ϵ) is regular in both α and ϵ. Furthermore, the bare coupling is renormalized with K_0 = Z_K^-1 K. As this is a LO in 1/𝒩 computation, one may then expand 
	
    Z_K^-n = [1 - 2K3ϵ + O(1𝒩) ]^-n = ∑_k = 0^∞n + k -1k(2K3ϵ)^k + O(1𝒩).

Inserting all this back into Eq. (<ref>), we find
	
    R(ϵ) = ∑_n=1^∞∑_k=0^∞∑_j=0^∞( - 2K/3)^n+kn + k - 1k(-1)^k (n+1)^j-1/ϵ^n+k-j  H^(j)(ϵ),

where H^(j)(ϵ) are regular functions of ϵ. Defining m = n + k, the sums are redefined in terms of m and k. The only poles in ϵ occur for j < m, so the divergent part of R is given by
	
    R(ϵ) ϵ→ 0∼∑_m=1^∞( - 2K/3)^m∑_j=0^m-1H_j(ϵ)ϵ^m-j∑_k=0^m-1m - 1k  (-1)^k (m - k + 1)^j-1.

The sum greatly simplifies as the identity 
	
    ∑_k = 0^mmk (-1)^k (x -k)^j = 0 ,

valid for all integer 0≤ j < m and real numbers x, implies that the k sum vanish for all j≠ 0. Meanwhile the j=0 term evaluates to 
	
    ∑_k=0^m-1m - 1k(-1)^km-k+1 = (-1)^m-1m(m+1).

Performing the j and k sums then yield
	
    R(ϵ) ϵ→ 0∼ -∑_m = 1^∞( 2K/3)^mH^(0)(ϵ)ϵ^m1m(m+1).

Expanding now H^(0)(ϵ) = ∑_ℓ=0^∞ H_ℓ^(0)ϵ^ℓ and selecting the simple pole one finds 
	
    R(ϵ) 1/ϵ∼ - 1ϵ∑_ℓ = 0^∞( 2K/3)^ℓ + 1H^(0)_ℓ(ℓ+1)(ℓ+2).

Finally, before resumming the power series in H_ℓ^(0), the fraction is decomposed so that

    R(ϵ)    1/ϵ∼ -1ϵ[∑_ℓ = 0^∞( 2K/3)^ℓ + 1H^(0)_ℓ(ℓ+1) - 32K∑_ℓ = 0^∞( 2K/3)^ℓ + 2H^(0)_ℓ(ℓ+2)] 
       1/ϵ∼ -23ϵ[∫_0^K x   H(0, 23x) - 1K∫_0^K x  x H(0, 23x) ]  ,

taking a K derivative and resumming the power series before integrating again. This concludes the proof. 




 §.§ Second resummation
 
	
    ∑_n=1^∞(-2 K_03)^n 1nϵ^n H(n,ϵ) 1/ϵ∼ -23ϵ∫_0^K x  H(0,23x)
	
To prove this formula <cit.>, let 
	
    S(ϵ) = ∑_n=1^∞(-2 K_03)^n 1nϵ^n H(n,ϵ) .

Assuming again that H(n, ϵ) is sufficiently regular, it may be expanded according to Eq. (<ref>). At the same time, the bare coupling is renormalized by K_0= Z_K^-1 K. Then using the expansion Eq. (<ref>) one finds that
	
    S(ϵ)  = ∑_n=1^∞∑_k = 0^∞∑_j=0^∞(-2 K3)^n+kn + k -1k(-1)^k n^j-1ϵ^n + k - j H^(j)(ϵ).
	
Note that as H^j(ϵ) is regular in 0, only the terms in the sum with j≤ n + k - 1 will contribute to the pole structure. The sums are redefined with m = n + k and so
	
    S(ϵ) ϵ→ 0∼∑_m=1^∞(-2 K3)^m∑_j=0^m-1H^(j)(ϵ)ϵ^m-j∑_k = 0^m-1m -1k (-1)^k (m-k)^j-1.
	
According to the identity (<ref>), the k sum vanish for all j≠ 0. The sum is thus evaluated by keeping just the j=0 term[For j=0 one has to use the summation: 
    ∑_k=0^nnk(-1)^kn-k+1 = (-1)^n ∑_k=0^nnk(-1)^kk+1 = (-1)^nn+1∑_k=0^nn+1k+1 (-1)^k = (-1)^nn+1
 ]
	
    S(ϵ) ϵ→ 0∼  -∑_m=1^∞(2 K3)^m1mH^(0)(ϵ)ϵ^m .

Expanding H^(0)(ϵ) as power series, the sum gives 
	
    S(ϵ) ϵ→ 0∼  -∑_m=1^∞∑_ℓ=0^∞(2 K3)^m1mH^(0)_ℓϵ^m-ℓ,     where    H^(0)(ϵ)  = ∑_ℓ=0^∞ H^(0)_ℓϵ^ℓ .

As only the simple pole in epsilon is of interest to us, the resummation can now be concluded
	
    S(ϵ) 1/ϵ∼ -1ϵ∑_ℓ=0^∞(2 K3)^ℓ +11ℓ + 1 H^(0)_ℓ1/ϵ∼ -23ϵ∫_0^K x ∑_ℓ=0^∞(2 x3)^ℓ H^(0)_ℓ

The second to last step is done by taking a K derivative before reintegrating so that the power series of H^(0)(x) = H(0, x) can finally be resummed proving (<ref>).




 §.§ Third resummation
 

    ∑_n=2^∞(-2K_03)^n1ϵ^n-1 H(n,ϵ) 1/ϵ∼1ϵ(2K3)^2 H(1,23 K)

To prove that this is the case, let

    T(ϵ) = ∑_n=2^∞(-2K_03)^n1ϵ^n-1 H(n,ϵ).
 
Here too, the function H(n,ϵ) is expanded as a power series in nϵ as given in Eq. (<ref>) such that all H^(j)(ϵ) are regular in ϵ = 0. Simultaneously the bare couplings are renormalized K_0 = Z_K^-1 K_0 using Eq. (<ref>), and we write

    T(ϵ) = ∑_n=2^∞∑_j=0^∞∑_k=0^∞(-2K3)^n+kn+k-1k (-1)^k H^(j)(ϵ) n^jϵ^n+k-j-1.

The sums can be rewritten in terms of m =n + k so that 

    T(ϵ) = ∑_m=2^∞(-2K3)^m∑_j=0^∞H^(j)(ϵ)ϵ^m-j-1[∑_k=0^m-1m-1k (-1)^k (m-k)^j - (-1)^m-1] .

Here the last terms is a compensation for the fact that the k sum is taken to go all the  way to m-1 rather than just to m-2. Poles in ϵ will only appear for j≤ m-2 in which case the identity (<ref>) causes the first term in the sum to vanish. Thus, for the pole structure it holds that

    T(ϵ) ϵ→0∼∑_m=2^∞(2K3)^m∑_j=0^∞H^(j)(ϵ)ϵ^m-j-1 = ∑_m=2^∞(2K3)^mH(1, ϵ)ϵ^m-1,
 
where the last equality is due to a resummation of the H function. Re-expanding now H(1,ϵ) yields

    T(ϵ) ϵ→0∼(2K3)^2 ∑_m=0^∞∑_ℓ=0^∞(2K3)^mH_ℓϵ^m+1-ℓ,     where   H(1, ϵ) = ∑_ℓ=0^∞ H_ℓϵ^ℓ.

At this point one can immediately determine the simple ϵ pole, which is evaluated to

    T(ϵ) 1/ϵ∼1ϵ(2K3)^2 ∑_m=0^∞(2K3)^m H_m = 1ϵ(2K3)^2 H(1, 23 K).
 



 §.§ Fourth resummation


    ∑_n=1^∞∑_m=0^n-1(-2K_α,03)^1+m(-2K_β,03)^n-mH(n,ϵ) nϵ^n1/ϵ∼49ϵK_α K_βK_α - K_β∫_K_β^K_α x  H(0,23x)

To prove this, denote by
	
    U(ϵ) = ∑_n=1^∞∑_m=0^n-1(-2 K_α,03)^1+m(-2 K_β,03)^n-mH(n,ϵ) nϵ^n.

First the sum over m is performed by noting that 
	
    ∑_m=0^n-1   (-2K_α,03)^1+m( -2 K_β,03)^n-m
            = 23[1K_α,0 - 1K_β,0]^-1[( -2 K_α,0/3)^n - (- 2 K_β,0/3)^n ].

Now going from the bare to the renormalized couplings using K_α,0 = Z_K_α^-1 K_α = K_α (1 - 2 K_α3ϵ)^-1 at LO in 1/𝒩, it is beneficial to first consider the first term,
	
    [1K_α,0 - 1K_β,0]^-1  = - K_α K_βK_α - K_β.
    
This term is thus finite and does not contribute to the pole structure. The sum can be written as
	
    U(ϵ) = -23K_α K_βK_α -K_β∑_n=1^∞[ (-2K_α ,0/3)^n - (-2K_β, 0/3)^n ] H(n,ϵ) nϵ^n.

At this stage the resummation (<ref>) can be applied directly to obtain 
	
    U(ϵ) 1/ϵ∼49ϵK_α K_βK_α - K_β[∫_0^K_α x  H(0,23x) - ∫_0^K_β x  H(0, 23 x) ].
 




apsrev4-1








