Safe Pati-Salam

We provide an asymptotically safe Pati-Salam embedding of the Standard Model. Safety is achieved by adding to the theory gauged vector-like fermions and by employing recently developed large number-of-flavor techniques and results. We show that the gauge, scalar quartic and Yukawa couplings achieve an interacting ultraviolet fixed point below the Planck scale. The minimal model is a relevant example of a Standard Model extension in which unification of all type of couplings occurs because of a dynamical principle, i.e. the presence of an ultraviolet fixed point. This extension differs from the usual Grand Unified Theories scenario in which only gauge couplings unify and become free with the remaining couplings left unsafe. We find renormalization group flow solutions that match the Standard Model couplings values at low energies allowing for realistic safe extensions of the Standard Model.


Introduction
The recent discovery of four dimensional asymptotically safe quantum field theories [1,2] has opened the way to safe extensions of the Standard Model, starting with the envision of a safe rather than free QCD [3], to scenarios in which the gauge, the Yukawa and scalar quartic couplings are unified by a dynamical rather than a symmetry principle [5][6][7][8]. On the supersymmetric front, exact non perturbative results and constraints were first discussed coherently in [9], extending and correcting the results of [10] while opening the way to (non)perturbative supersymmetric safety in [11][12][13], and to the first applications for super GUT model building [11,14]. Simultaneously there has been much advancement in our understanding of the nonsupersymmetric dynamics of large number of flavors gauge-Yukawa theories [15][16][17][18][19]. This has led, for example, to enrich the original conformal window [20,21], reviewed in [22,23], with a novel asymptotically safe region [24]. The discovery led to the upgraded conformal window 2.0 of [24]. The large N f dynamics of gauge-fermion theories has been extended to gauge-Yukawa theories starting with the Yukawa sector [25][26][27] and for the first time to all couplings in [28,29]. A gaugeless study appeared in [30]. The results widened the palette of tools and theories at our disposal for novel large N f safe extensions of the SM [7,28,[31][32][33].
We use the acquired knowledge to construct a novel safe Pati-Salam extension by adding vector-like fermions and showing that all couplings acquire an UV fixed point at energies that are far from the onset of quantum gravity. The separation of scales allow us to investigate a condense-matter-like unification of the SM couplings before having to consider the gravitational corrections. The interplay with gravity has been investigated in several recent works [34][35][36][37][38] and it will not be considered here. Differently from the usual Grand Unified scenarios [39] in which only the gauge couplings unify because of their embedding into a larger group structure and then they eventually become free, in the present scenario we have that Yukawa and scalar self couplings are intimately linked because of the safe dynamics with their high energy behavior tamed by the presence of an interacting fixed point.
The paper is organized as follows: In section 2 we review and introduce the Pati-Salam [40] extension of the SM and build the minimal vector-like structure able to support a safe scenario. We develop the renormalization group (RG) equations and determine the couplings' evolution in section 3 . Here we analyze and classify the UV fixed point structure of the model. We offer our conclusions in section 4. In appendix A we summarize the one-loop RG equations for the Pati-Salam model investigated here.

Pati-Salam extension of the Standard Model
Consider the time-honored Pati-Salam gauge symmetry group G PS [40] with gauge couplings g 4 , g L and g R , respectively. Here the gauge group SU (4) ⊃ SU (3) C ⊗ U (1) B−L , where SU (3) C denotes the SM color gauge group, and the corresponding gauge couplings are related according to The gauge fields of G PS can be written as follows: In this parametrization, W 0 Lµ and W ± Lµ correspond to the electroweak (EW) gauge bosons, G 3µ , G 8µ , G ± 12µ , G ± 13µ and G ± 23µ are the SU (3) C gluons, B µ is the B − L gauge field, and X ± 1µ , X ± 2µ and X ± 3µ are leptoquarks. The SM quark and lepton fields are unified into the G PS irreducible representations where i = 1, 2, 3 is a flavor index. In order to induce the breaking of G PS to the SM gauge group, we introduce a scalar field φ R which transforms as the fermion multiplet ψ R , that is φ R ∼ (4, 1, 2): where the neutral component φ 0 The hypercharge Y is a linear combination between the diagonal generator of SU (2) R and the generator of B − L, namely with Tr I 2 R = 1/2 for the fundamental representation. Then, the EW gauge couplings g 2 and g Y result: We also introduce an additional (complex) scalar field Φ ∼ (1, 2, 2), with which is responsible of the breaking of the EW symmetry.

The Scalar sector
The general scalar potential of the model defined above is given by: The quartic couplings λ 2,3 and λ RΦ2 , and the dimensional term µ 12 , carry a non-trivial phase in case CP symmetry is explicitly broken. We have also introduced the conjugate field Φ c ≡ τ 2 Φ * τ 2 , τ 2 being the standard Pauli matrix.

The Yukawa sector
The most general Yukawa Lagrangian for the matter fields ψ L/R is 1 In terms of the SM fermion fields Eq. (12) reads: Electroweak symmetry breaking is induced by a nonzero vev of Φ, which takes the form: with generally u 1 = u 2 . From Eq. (13) we have the fermion mass spectrum: where v ≡ u 2 1 + u 2 2 = 174 GeV and tan β ≡ u 1 /u 2 . In the case of a self-conjugate bi-doublet field Φ ≡ Φ c , one has u 1 = u 2 in Eq. (14) and equality between fermion masses is enforced at tree-level, namely In order to separate the neutrino and top masses in Eq. (15) and Eq. (16) we implement the seesaw mechanism [41][42][43][44] by adding a new chiral fermion singlet N L ∼ (1, 1, 1), which has Yukawa interaction The latter generates a Dirac mass term M R N L ν R , with M R ≡ y ν v R . The resulting Majorana mass term for the neutral fermion fields reads: with ν c L/R ≡ Cν T R/L and N c R ≡ CN L T . The mass spectrum consists of one massless with tan θ = m t /M R , and one Dirac neutrino N D with mass m D = m 2 t + M 2 R and chiral components: By adding a Majorana mass term for the singlet fermion N L the total lepton number is explicitly broken and the spectrum consists of three massive neutrinos. Taking M N m t , M R , the mass eigenstates result in one light active Majorana neutrino ν τ with mass and two quasi-degenerate heavy Majorana neutrinos N 1,2 with opposite CP parities and masses Threshold corrections may induce a sizable mass splitting between m τ and m b in Eq. (15) and Eq. (16), which depends on the G PS breaking scale v R , see Ref. [45].

The minimal model
In the simplest scenario where the field Φ is self-conjugate, the fermion spectrum is degenerate, see Eq. (16), and the scalar potential in Eq. (11) consists of the quartic couplings λ 1 , λ R1,2 and λ RΦ1,3 . As discussed above, by adding a new chiral fermion N L , which is a singlet under G PS , it is possible to induce a hierarchy between the top quark and neutrino masses via the seesaw mechanism, such that the correct light neutrino mass scale can be accommodated. Here, we further extend the matter content of the theory with a new vector-like fermion F ∼ (10, 1, 1) with mass M F and Yukawa interactions: In terms of the SU (3) C representations, the field F can be decomposed as where S, B and E denote a color sextet, triplet and singlet, respectively. Then, from Eq. (24) the fields B and E mix with the right-handed components of ψ R , b R and τ R , respectively, giving the overall Dirac mass terms: As a result of this mixing, the top quark becomes naturally heavier than the other SM fermions. In fact, in the limit m B m t , M F , the b quark and τ charged lepton masses satisfy the tree-level relation: Analogously, we have a new vector-like quark,B, and a new vector-like lepton,Ê, with corresponding masses M B and M E , which satisfy the tree-level relation: Gauge Couplings Yukawa Couplings Scalar Couplings SU (4) :

Renormalization group analysis
In this section, we perform the RG analysis of the Pati-Salam extension of the SM introduced above and discuss the relevant phenomenological implications. The gauge, Yukawa and scalar couplings in the minimal and extended realizations are listed in Tab. 1. The corresponding RG equations at one loop order are reported in appendix A.

Large-N beta function
In order to ensure asymptotic safety in the UV for all the system in Tab. 1, we employ the 1/N F expansion approach developed in [15][16][17]46], first applied to the whole SM in [31]. More specifically, we introduce N F 1 vector-like fermions, which transform non-trivially under G PS . In this framework, the RG equations receive a contribution at leading order in the 1/N F expansion of the relevant Feynman diagrams, which are resumed as shown in Fig 1 (only gauge coupling cases are shown). This non-perturbative effect induces an interacting fixed point for both the Abelian and non-Abelian gauge interactions of the SM [31]. The fixed point is guaranteed by the pole structure occurred in the expressions of the summation [16,17].
In the present scenario, we consider three sets of vector-like fermions charged under G PS , with the following charge assignment: where the N F 2L vector-like fermions are chosen in the adjoint representation of SU (2) L to avoid fractional electrical charges. We have also chosen each set of vector-like fermions to have non-trivial charges only under one simple gauge group to avoid the extra contributions in the summation of semi-simple group.

Large-N gauge beta function and gauge coupling unification
To the leading 1/N F order for each set, the higher order (ho) contributions (i.e. the bubble diagrams in Fig. 1) to the RG functions of the gauge couplings are calculated in [17], while for the abelian case they were first computed in [15]. Here we list a short summary of the results. The higher order contributions are give by: with the functions H 1i and the t'Hooft couplings A i given by The Dynkin indices are T R = 1/2 (N ci ) for the fundamental (adjoint) representation. The RG functions of the gauge couplings (see Appendix A) including the contributions of bubble diagrams resummation are listed below: where the β 1loop are denoted as the original one loop RG beta functions of the three gauge couplings without bubble diagram contributions while β tot α 2L , β tot α 2R , β tot are the total RG beta functions including the higher order bubble diagram contributions up to 1/N F order. The reason that only one loop RG beta functions of the gauge couplings are used will be clear later on.
Thus, the UV fixed point for the gauge coupling sub-system (g 4 , g L , g R ) is guaranteed by the pole structure in the bubble diagram summation. For all the non-abelian gauge groups, the pole in the function H 1 i , and thus the UV fixed point of the non-abelian gauge couplings, always occurs at A i = 3. In particular, if one chooses the vector-like fermion representation with A 2L = A 2R = A 4 , gauge coupling unification is guaranteed. This is shown in Fig. 2, where we set N F 2L = 35 and N F 2R = N F 4 = 140. The IR initial conditions of g L , g R and g 4 are obtained by using the matching conditions of Eq. (2) and Eq. (9) and the SM couplings are running from the EW scale to the Pati-Salam symmetry breaking scale. For simplicity, we have assumed all the vector-like fermions were introduced at the Pati-Salam symmetry breaking scale v R . The latter is most strongly constrained by the kaon decay K L → µ ± e ∓ (see e.g. [45,47]). Using the current upper limit Br (K L → µ ± e ∓ ) < 4.7 × 10 −12 provided in [48], we obtain the lower limit v R 2000 TeV (see also e.g. [49]). In order to make closer connection to low energy phenomenology, in this work we choose the Pati-Salam symmetry breaking scale exactly at 2000 TeV.

Large-N Yukawa and quartic beta function
In the previous section, we have only considered the bubble diagram contributions in the gauge couplings subsystem. However, the bubble diagrams can directly contribute also to the quartic and Yukawa beta functions (see e.g. [25,29]). In the following, we provide a brief review of the procedure following [29].
The bubble diagram contributions to known 1-loop beta functions of quartic and Yukawa couplings can be obtained by employing the following recipe. The Yukawa beta function at large number of fermions can be written in the compact form containing information about the resumed fermion bubbles and c 1 , c α are the standard 1-loop coefficients for the Yukawa beta function while C 2 (R α φ ), C 2 (R α χ ), C 2 (R α ξ ) are the Casimir operators of the corresponding scalar and fermion fields. Thus, when c 1 , c α are known, the full Yukawa beta function including the bubble diagram contributions can be obtained. Similarly, for the quartic coupling we write with c 1 , c α , c α , c αβ the known 1-loop coefficients for the quartic beta function and the resumed fermion bubbles appear via .
Thus we have now the full quartic beta function including the bubble diagram contributions when c 1 , c α , c α , c αβ are known. Following the above recipe, the bubble diagram improved Yukawa beta function β y , for example, can be written as The bubble diagram improved quartic beta function β λ R1 reads

UV fixed point solutions in the gauge-Yukawa-quartic system
To prove the existence of a fixed point of the whole system in Tab. 1, we are entitled to assume the gauge couplings at the UV fixed point as background values (i.e. constants  in the RG functions of other couplings). This is so because at the UV fixed point they only depend on the choice of N F . By using the one loop RG functions in appendix A augmented with the large-N corrections (i.e. Eq. (34) and Eq. (35)), we can now set {β i = 0} where i denotes all the Yukawa and scalar couplings presented in Tab. 1. Our investigation and beta functions are consistent with the large-N limit, computations and results established in [29,31]. We impose CP invariance, that implies: Im (λ 2 ) = Re (λ 3 ) = Re (λ RΦ2 ) = 0. This symmetry requires y = ±y c , leading to top and bottom mass degeneracy, which is lifted when including the new vector-like fermion F ∼ (10, 1, 1) (see Eq. (24)). We have also checked that, when breaking the CP symmetry safety is lost, because the overall RG system is over-constrained.
The analysis unveils several UV candidate fixed points for different choices of N F . For example, for N F 2L = 40, N F 2R = 150, N F 4 = 200, we discover 30 sets of UV candidate fixed point. However the scalar potential is unbounded for several candidates UV fixed points. We therefore require the following vacuum stability conditions (see e.g. [50]) to be satisfied: These conditions are quite constraining, reducing to 5 the original set of 30 UV fixed point candidates. We find instructive to compare the effects of the resumed diagrams on the Yukawa and self-couplings corrections with respect to the non-resumed case. Consider the same value of the number of vector-like fermions discussed above (i.e. N F 2L = 40, N F 2R = 150, N F 4 = 200). We now select two sample UV fixed point solutions summarized in Tab. 2. We indicate with BF (the first row) and AF (the second row) represent respectively before and after involving the bubble diagram contributions in the Yukawa and quartic RG beta functions. We discover that most of the couplings are not shifted much except for λ R2 and λ 4 . The solutions listed in Tab. 2 satisfy the vacuum stability condition Eq. (39). For a different sample value of the number of vector-like fermions (N F 2L = 40, N F 2R = 80, N F 4 = 100), we also find a set of UV fixed point solutions (satisfying the vacuum stability conditions) with the fixed point values of λ R2 and λ 4 that are corrected the most among all couplings (see Tab. 3).
So far y F was asymptotically free ( see Tab. 2 and Tab. 3,) and we now exhibit the case in which y F = 0 in the UV. This case is shown in Tab. 4 in which we have a UV safe solution for y F for (N F 2L = 40, N F 2R = 130, N F 4 = 130). Interestingly this solution owes its existence to the bubble diagram contributions for the Yukawa and quartic RG beta functions. Thus, the large-N contributions for the Yukawa and quartic couplings add novel safe possibilities in which all Yukawa couplings are safe.
We now determine which fixed point is relevant/irrelevant (UV repulsive/attractive) following the convention according to which the RG flows towards the IR. The results are summarized in Tab. 5. We consider the cases that abide the vacuum stability conditions.      irrelevant (Irev) characteristics are listed. The symbol "×" denotes the corresponding coupling is turned off for simplification. From scenario 1 to 5, the complication of the scenario is gradually increased. y F = 0 is due to the asymptotically free solution we choose while λ 3 , λ RΦ2 and λ RΦ3 are chosen to be zero at all scale for simplification.
We use "×" to represent that the couplings are turned off to simplify the system. We gradually increase the complexity of the system from scenario 1 to 5 where more and more couplings are involved. Scenario 4 (two sample cases, for example, will be the BF row and AF row in Tab. 2) and scenario 5 (one sample case will be the AF row in Tab. 4) possess all the couplings involved in our Pati-Salam model. The value 0 denotes a zero value solution at the fixed point. There are two distinct cases in which a specific coupling can be zero at the UV fixed point: the coupling can be asymptotically free or can vanish at all scales. For example, y F is asymptotically free and therefore it leads to interesting physics in the IR while λ 3 , λ RΦ 2 and λ RΦ 3 can be set to zero at all energies, with the current approximations. This is what is assumed in the last row of Tab. 5 to simplify the analysis. We employed two approaches to determine the RG flow of the system: the IR to UV approach and the UV to IR approach. In the IR to UV approach, the RG flow of the irrelevant couplings is constrained on certain trajectories, the separatrices. 2 Thus, we can solve the set of equations β i = 0 (i corresponding to all the irrelevant couplings) and solve for all the irrelevant couplings as function of the relevant couplings. The IR initial conditions of the relevant couplings are compatible with the phenomenological constraints while preserving UV safety. For the UV to IR approach one simply starts from the UV fixed point and attempts to run towards the IR. Here we use the fact that the gauge couplings have RG functions that are sufficiently decoupled from the  other couplings. Thus, we can run the remaining couplings along the determined gauge coupling RG trajectories.
We report our results in Figs. 3 and 4 where we show the running of the gauge, Yukawa and scalar couplings by using the UV to IR approach for (N F 2L = 40, N F 2R = 130, N F 4 = 130). The corresponding UV fixed point solution is the one shown in the AF row of Tab. 4. As mentioned above the RG flows of the gauge couplings are determined once the IR conditions are given. The IR initial conditions for g L , g R and g 4 are obtained by using the matching conditions of Eq. (2) and Eq. (9) and the SM couplings are running from the EW scale to the Pati-Salam symmetry breaking scale. For simplicity, the vector-like fermions masses are taken to be the Pati-Salam symmetry breaking scale v R = 2000 TeV. From Figs. 3 and 4, it is clear that all couplings (i.e. gauge, Yukawa and scalar quartic) achieve a safe UV fixed point. The transition scale, above which, the UV fixed point is reached is about 0.5 × 10 9 GeV for all the couplings. Note that we could shift this transition scale significantly by increasing (the scale will decrease) or decreasing (the scale will increase) the number of vector-like fermions.

Conclusions
Models in which scalar degrees of freedom are fundamental a lá Wilson [51,52] require the presence of scale invariance at short distances [6,53,54]. A complete safe or free theory can therefore support elementary scalars as fundamental fields. Fundamentality and naturality are complementary concepts. Short distance scale invariance implies fundamentality while (near) long distance conformality and/or controllably broken sym-  metries help with naturality [6,53]. A coherent search of safe extensions of the SM has only recently begun. In this work we have constructed a realistic safe extension of the SM in which we add vector-like fermions to the time-honored Pati-Salam framework. Recent progress in the large-N safe dynamics of gauge-Yukawa theories has proven instrumental for the success of the project. In particular we have shown that the gauge, scalar quartic and Yukawa couplings achieve an interacting ultraviolet fixed point below the Planck scale. The minimal model is a relevant example of a Standard Model extension in which unification of all type of couplings occurs because of a dynamical principle, i.e. the presence of an ultraviolet fixed point. There are several aspects that deserve further investigation from a more in depth phenomenological study of the quark and lepton flavour sector to baryogenesis.