Grand Unified Theories in Renormalisable, Classically Scale Invariant Gravity

We analyze $SO(N)$ and $SU(N)$ gauge theories with scalars in adjoint and fundamental representations, coupled to renormalisable, classically scale invariant gravity. In the specific case of $SO(12),$ we show that the quantum field theory can be can be asymptotically free in all couplings (hence ultra-violet complete). For a region of parameter space, Dimensional Transmutation occurs, with the adjoint vacuum expectation value breaking $SO(12) \to SU(6)\otimes U(1)$ and producing a Low Energy Effective Theory having Einstein-Hilbert gravity. We verify that certain minima are locally stable and lie within the catchment basin of the ultraviolet fixed points.

perturbatively as in Yang-Mills theory or perturbatively, as in massless electrodynamics [15]. We are especially interested in models that have all couplings AF [16][17][18] in perturbation theory.
In a previous paper [18], we analysed an SO(10) gauge theory coupled to RNG, including a scalar multiplet in the adjoint representation. We showed that if we impose scale invariance on the classical theory, it can undergo Dimensional Transformation (DT), in a manner akin but not identical to the phenomenon observed by Coleman and Weinberg (CW) [15]. We showed that there is a region of parameter space such that the couplings are all asymptotically free 2 and remain perturbative at scales down to the point where DT occurs. At this scale the vacuum expectation value (vev) of the scalar multiplet breaks SO (10) to SU (5) ⊗ U (1), and the effective theory at lower scales has an Einstein R term arising from the dimensionless nonminimal coupling of the scalars to gravity.
Elsewhere [19], we revisited the classic flat space β-function calculations of CEL [7], which were motivated by the search for asymptotically free (and hence UV complete) gauge theories. We found a number of errors in the CEL results, including the cases of both a SO(N ) and a SU (N ) gauge theory coupled to two scalar multiplets: an adjoint and a fundamental. Here we extend those results to the classically scale invariant renormalisable gravity case to see if the UV completeness and DT phenomena we identified in [18] persist in these cases, which we regard as prototypes for a realistic Grand Unified Theory (GUT) due to the more complicated scalar sector. We consider in detail both the case SO(12) and the limit of large N for both SO(N ) and SU (N ). 2 The SO(N ) theory.
The action we shall consider is Here C is the Weyl tensor and G the Gauss-Bonnet term; a, b, c are dimensionless coupling constants. All the fields are real. χ i is in the defining (fundamental) representation with i : 1 · · · N and φ a is in the adjoint representation with a : 1 · · · N (N − 1)/2. It is convenient for some purposes to write φ a as Φ = T a φ a where T a are the N × N hermitian generators of SO(N ), normalised so that Tr T a T b = 1 2 δ ab . For an arbitrary irreducible representation of the generators R a , we write T R is determined for each representation by the normalisation Eq. (2.3).
The general form of the scale invariant scalar potential is 3 3 The SO(N ) β-functions

Classical gravity
Although quantum corrections due to gravity are not included in this approximation, the ξ 1,2 terms in the potential, Eq.(2.5), must still be present since, even in a background gravitational field having R = 0, ξ 1,2 undergo renormalisation. The flat space β-functions are (we suppress throughout a factor of (16π 2 ) −1 ) in each β-function).
In Eq. (3.1a), T S,F represent the values of T R for the scalars and fermions. For SO(N ), Although {ξ 1 , ξ 2 } are not relevant for flat space, in curved spacetime, matter self-interactions contribute to their one-loop β functions. Letting ξ i ≡ ξ i + 1/6, we have Here (and subsequently) we have set α ≡ g 2 . Eq. (3.3) obviously gives rise to fixed point solutions for β corresponding to conformal coupling, ξ 1 =ξ 2 =0. Assuming that det[β ij ] = 0, there are no other fixed point solutions. We shall see shortly that the one-loop gravity contribution changes that result.

Gravitational Corrections
The gravitational corrections to the matter β-functions have a universal form. We infer from Ref. [13] that the full β-functions are The β-function for the gauge coupling is unchanged by gravity (at least at one loop) and remains as in Eq. (3.1a). For the gravitational self-interactions we have (again of universal form): Here N 0 denotes the number of (real) scalars; N 1/2 , Dirac fermions; N 0 1 , massless vectors; N 1 , massive vectors. (For chiral or Majorana fermions, the coefficients of N 1/2 would be half those given above). In the last equation, we set x ≡ b/a = 1/w. It is sometimes convenient to use We shall eventually be interested in determining whether this model is asymptotically free in all couplings, requiring reduced couplings to approach UVFP's. In this case, we have To analyse the β-functions and seek FPs it is convenient to introduce rescaled couplings a ≡ a/α, x i ≡ λ i /α, with consequent "reduced" β-functions, For b g > 0, as is required for asymptotic freedom of α, we see that a has a UVFP at Hence a → 0 at high energies, and, since UVFPs only exist when b g is small, a is small at its FP.
The β x i are given by In Eq.(3.11), the ∆β terms are identical to the ∆β terms in Eq.(3.6), except that a is replaced by a.
In terms of reduced couplings, the classical gravity β-functions for β ξ 1,2 become Adding gravitational corrections, we have where ∆β ξ (ξ ) is given by Eq. (3.6e) with a → a : While the matter contributions β (0) ξ j vanish for conformal coupling, the gravitational corrections vanish for minimal coupling, ξ = 0. When the two are combined, the FPs respect neither limit.

The SO(12) Fixed Points
Our first goal in this class of theories is the identification of cases such that the renormalization group evolution of the couplings approaches a fixed point at high energies (UVFP). In the flat space limit, we showed that a minimum value of N=12 is required. For N = 12 and with the minimum possible value of b g , which is b g = 1/6, we found a UVFP with results for the quartic couplings in the flat space limit as follows: As mentioned earlier, even with classical gravity, ξ 1,2 both nevertheless undergo renormalisation. However, it is easy to see from Eq. (3.3) that they have a FP for the conformal values For curved space we expect to find a FP with similar values of the quartic couplings and values of ξ 1,2 both close to the conformal value (zero). This is basically because, as explained in Ref. [18], the FP result for a/α is necessarily small. The result for N = 12 is Note that the large value of x at the FP does not invalidate perturbation theory since b = ax is only O(1) at the FP.
There is another FP with similar values of x i , ξ 1,2 and a small value of x, which is a saddle point. We shall not elaborate on that. 5 The SO(N ) large N limit As we have seen, requiring the existence of a UVFP leads to a minimum value of N . It is therefore natural to consider the large N limit, first discussed in non-abelian gauge theories by 't Hooft. To retain finite couplings in the large N limit, we need to rescale the couplings. Since we cannot solve the theory exactly in this limit, we shall have to require that these rescaled couplings remain perturbatively small at all relevant scales.
Assuming b g = b g N, we found that for the quartic scalar couplings, scaling behavior requires In Ref. [20], we argued that the ambiguity in the rescaling of λ 4 reflects a nonuniformity of the limiting behavior that is best resolved by setting p 4 = 2.

Gravitational coupling rescaling-A
For the remaining couplings associated with the gravitational interactions, the natural rescaling takes the form We have incorporated both the rescaling with powers of N and forming ratios of couplings to α in the above equations. Note that since the scalings of a and b are identical, we have x = x. Defining y n ≡ λ n /α, the reduced β-functions in the large N limit are as follows: For the quartic couplings, and for the gravitational couplings, After the rescaling chosen in Eqs. (5.1), (5.2), the β-functions for the quartic couplings are the same as in flat space; hence their values at any FP will be the same as in flat space.
We found UVFPs for a range of values of b g from Eq. (5.3) above in Ref. [19] with results as shown in Table 1. (There was an calculational error in Ref. [19] so that the results in Table 1 differ slightly from the corresponding Table there). It is easy to see that these remain FPs if the remaining couplings are ξ 1 = ξ 2 = x = 0. However, from Eq. (5.4b) we then see that these FPs are destabilised by the gravitational corrections since b 2 > 13/120.  Note, however, that the results in Table 1 correspond to various specific values of b g . There are a finite possible number of values of b g for any given N , but the number of values of b g tends to infinity in the large N limit. We may therefore allow arbitrary values of b g in the search for a FP.
We now seek a FP with nonzero ξ 1 . We may first solve Eq.(5.3a), Eq.(5.3b) and Eq.(5.4c) for y 1 , y 2 and b g . We can then solve Eqs. (5.3c)-(5.3e) for y 3 , y 4 , y 5 . This leads to two new FPs, shown in Table 2. Unfortunately, neither of these FPs is UV stable.
So it seems that with the most natural rescaling (in terms of powers of N ), gravitational corrections destabilise the UVFP. In the next subsection we explore an alternative rescaling, albeit with a similar conclusion.

Gravitational coupling rescaling-B
There is an alternative rescaling giving rise to a non-trivial large N limit as follows: The quartic coupling rescaling is done in the same way as in the previous subsection. Note that now x = N 2 x. The resulting reduced β-functions (in the large N limit) take the form (where we have y n ≡ λ n /α as before), Unlike the previous case, there is no FP corresponding to Table 1 with ξ 1 = ξ 2 = x = 0. However, with ξ 1 = ξ 2 = 0 we do reproduce Table 1 with a FP at x = 10/(3 b 2 ). However, once again this is not a UVFP.

The SU (N ) theory
In this case we have the scalar potential is now a complex multiplet in the defining (fundamental) representation, and T a are no longer (all) antisymmetric; they are again normalised so that The SU (N ) β-functions

Classical gravity
As was the case with SO(N ), the potential Eq. (6.1) still must include the ξ 1,2 -terms. The flat space β-functions are [19] We also have

Gravitational Corrections
As we indicated in the SO(N ) discussion above, the gravitational corrections to the matter β-functions have a universal form; Eq. (3.5) remains valid in the SU (N ) case. There are minor changes to the gravitational corrections so that we have: The SU (N ) discussion proceeds in the same manner as the SO(N ) case. In Ref. [19] we showed that the minimum value of N consistent with the existence of a UVFP is N = 9, when b min g is 4/3. For the reduced couplings x i = λ i /α we found a flat space UVFP with 9 SU (N ) Large N limit We can define a large N limit in a similar way to the SO(N ) case. Once again we set As before there are distinct options for the rescaling of the gravitational corrections.

Gravitational coupling rescaling-A
With identical rescalings to those described in the corresponding SO(N ) case, we find for the reduced quartic couplings y i ≡ λ i /g 2 , :

2c)
β y 4 = y 5 y 1 + 2 y 2 + 2 y 3 + y 2 5 + 3 + y 4 y 1 + 4 y 2 + 2 y 3 − (9 − b g ) , Just as in the SO(N ) case, any flat space UVFP would necessarily imply a corresponding FP with ξ 1 = ξ 2 = x = 0. We found such FPs for specific values of b g in Ref. [19], as shown in Table 3.   At first sight, this result is surprising but can be understood (or at least made plausible), by inspection of the respective Dynkin diagrams. This means, of course that the FPs are essentially identical in the two cases. This is immediately apparent in the comparison of the first two rows of Table 1 with the corresponding rows in Table 3. (Recall that b g = 1/6 in the SO(N ) case corresponds to b g = 1/3 in the SU (N ) case).
In the same limit the gravitational couplings are: As we remarked above, any flat space UVFP would necessarily imply a corresponding FP with ξ 1 = ξ 2 = x = 0, but one destabilised by Eq. (9.3b). However, once again we see from Eq. (9.3b) that such a FP is unstable with respect to fluctuations in x. Now let us seek a FP for nonzero ξ 1 . Just as in the SO(N ) case, we find two real FPs, as shown in Table 4. These results are essentially identical to the corresponding ones for  SO(N ); because, of course, the large N β-functions in the two cases are essentially identical, differing only by redefinitions of the coupling y 5 and overall rescaling. Once again we do not find any real FPs in this case.

Dimensional Transmutation in the SO(N) model
In either the SO(N ) or the SU (N ) case, a comprehensive analysis of the behaviour of the effective action with two distinct scalar multiplets and five independent quartic scalar couplings would be a formidable undertaking. We choose to make the crucial assumption that DT occurs via the development of a vev for the adjoint representation only, just as we analysed in Ref. [18]. The precise details differ, however, because the analysis involves the behaviour of the couplings under renormalisation, and the renormalisation of the adjoint self couplings are, of course, affected by the presence of the other multiplet. In Ref. [18] we focused our attention on the SO(10) theory. We showed it was asymptotically free both without and with gravitational interactions, and that in the latter case, the adjoint developed a vev via DT, with symmetry breaking uniquely determined to be SO(10) → SU (5) ⊗ U (1). With the addition of a multiplet of scalars in the fundamental representation, we showed in Ref. [19] that asymptotic freedom is not sustained in the flat space case; the minimum value of N necessary becomes N = 12. We have seen above that this remains true when gravitational interactions are included. Therefore we need to generalise our previous discussion to accommodate N > 10.
We begin by assuming that the background metric is well-approximated by the de Sitter metric, with constant R > 0. Then if Φ is constant and non-zero, we showed in Ref. [18] that, in the SO(10) case, symmetry breaking occurs in the SU (5) ⊗ U (1) direction with where 1 is the 5 ⊗ 5 identity matrix, and r ≡ 2T 2 /(N R) = (φ a φ a )/(N R).
It is easy to see that if we assume that χ does not get a vev, then for any even N , Φ takes the same block form, that is proportional to with N/2 × N/2 blocks. The residual symmetry after such spontaneous symmetry breaking (SSB) is SU (N/2) ⊗ U (1) for arbitrary r. Thus the classical action for (even N ) takes the off-shell value S cl (λ i , r) where V 4 is an angular volume, and {λ i } is the complete set of couplings. Here, we introduce the symbol ζ 1 for the sum of the first two couplings, In order for the action per unit volume to be bounded from below, we must have ζ 1 > 0. We shall also need the derivatives Here (and subsequently) We can determine the extremal values of r at tree level by solving S cl = 0. Therefore, r = 0 or r = r 0 with which, since ζ 1 > 0, yields r 0 real only if ξ 1 > 0. Assuming this requirement is satisfied, the classical curvature S cl is negative at r = 0, so the unbroken solution is a maximum. At r 0 , the curvature takes the value S (os) cl At the minimum, the value of the classical action is Recall that, for N = 12, the UVFP, Eq. (4.3), has ξ 1 ≈ 0, or ξ 1 ≈ −1/6. This is a generic result, so it is important to establish that it is possible to fulfill this condition and still have ξ 1 > 0 at the DT scale. (This did in fact occur the simpler model [18] that did not include the scalar χ in the fundamental.) Generally, we want to determine whether DT can occur. Including radiative corrections, the effective action takes the generic form where ρ ≡ √ R. All coupling constants are denoted by the set {λ i }. In writing the effective action in this form, we have assumed that Φ is spacetime independent; it is not necessary to assume that the breaking pattern is to SU (N/2) ⊗ U (1).
We seek an extremum of the effective action such that (r, ρ) = ( r , ρ ) = (r 0 , v), that is solutions to where we choose µ = ρ ≡ v. These results are exact to all orders in the loop expansion. In order to determine stability, we shall also need the matrix of second derivatives on-shell: As before, we implicitly set µ = v after performing the derivatives. The second variation on-shell can be written (10.14) Given our conventions, these equations Eq. (10.13) are also exact to all orders in the loop expansion, but their leading nonzero contributions vary from tree level for those involving S cl , to one-loop for B, to two-loop 5 for C. This is the characteristic "see-saw" pattern, so this matrix has two eigenvalues i that may be approximated as 1 is determined by the classical curvature and given by Eq.(10.9), and 2 , although of order 2 , is determined by one-loop results.
In Ref. [14], we used the renormalisation group to show that (to leading order) so that the conditions for an extremum corresponding to DT in this model (and others of this general form) are r = r 0 and are the "on-shell" contributios to B 1 , S cl with "on-shell" corresponding to r = r 0 . Notice that Such an extremum corresponds to a minimum if ζ 1 > 0 and 2 > 0, and we showed, again using the renormalisation group, that [14,17,18] 19) or using Eq. (10.16), Remarkably, 2 can also be written an observation which is not particularly obvious and that we shall explain in detail elsewhere in a more general discussion of the RG and effective actions in this kind of theory.

Dimensional Transmutation in the SO(12) model
Even after restriction to N = 12, b g = 1/6, this relatively simple model is still extremely complicated, both analytically and numerically, for several reasons. In our classically scaleinvariant version, it involves many fields, and only the possible "directions" of SSB in this field space is determined, depending on relations among the 10 dimensionless coupling constants of the bosonic sector. The scale of SSB is determined at one-loop order, and the determination of its character (and correspondingly, the dilaton mass) is determined at two-loops. The numerical results for the UV behavior were given in Sec. 4, showing that such a model can be AF for a certain range of coupling constants.
We now wish to show that the model can undergo DT and that it is locally stable for at least a portion of the DT surface. This will require fleshing out Eq. (10.23) in greater detail. From Eq. (3.7b), we have for N = 12, As mentioned in Sec. 4, even specifying b g = 1/6 does not yield a unique model, since several different arrangements of the fermion content are still possible. In Sec. 4, we arbitrarily chose to focus on the case of 52 two-component fermions in the fundamental representation N so that N f = 312 and b 1 = 917/36, b 2 = 59. As noted earlier, as the running scale decreases, we must hope that ξ 1 will run from near zero to the region where ξ 1 = ξ 1 − 1/6 > 0. For this reason, exploring the DT surface may be simpler in terms of ξ 1 rather than ξ 1 . Further, B 1 depends on the couplings (x 4 , x 5 ) only via the linear combination 6 x 4 + x 5 /24 ≡ z 4 . After making these notational changes, we find that B We note that z 1 enters only as the ratio ξ 1 /z 1 , and a appears only as the combination aξ 2 1 /z 1 , which may well turn out to be very small.
To determine 2 , we need to evaluate either Eqs. (10.19), (10.20) or Eq. (10.21) for N =12. From Eq. (11.2), we see that, when expressed in terms of the 7 rescaled variables {a, x, ξ 1 , ξ 2 , z 1 , x 2 , z 4 }, B (os) 1 is independent of {x 3 , x 5 }. Consequently, we require 7 reduced β-functions. The actual determination of 2 is unavoidably complicated and not very illuminating. For completeness, In Appendix A, we give the relevant β-functions and the resulting formula for 2 .
For our purpose here, we shall be satisfied to demonstrate a region of parameter space such that 2 > 0 and our other requirements (such as ξ 1 > 0) are satisfied and that the couplings there proceed to run to the UVFP we found at high energies. (A complete description of solutions of B (os) 1 = 0 remains a formidable undertaking). The simplest way to identify such a region is to start with as many couplings as possible (consistent with requirements at the DT scale) already near the UVFP. The most obvious problem with this is that, since we plan to generate the E-H term using the adjoint vev, we require ξ 1 > 0, whereas at the UVFP ξ 1 ≈ −1/6. We can however choose to try ξ 2 at or near zero.
In Table 5, we give two examples of points satisfying B 1 = 0, 2 > 0, and flowing to the UVFP.  This paper continues a series in which we have developed the theory of renormalisable quantum gravity coupled to matter fields, including in particular a clear demonstration that generation of scalar vacuum expectation values can occur via Dimensional Transmutation (DT). We have also claimed to identify a large class of such theories that exhibit asymptotic freedom, and consequently represent a possible UV completion of Einstein gravity (in combination with DT). This claim remains controversial because of disagreement [13] concerning the correct sign of the coefficient b of the R 2 term in the Lagrangian, Eq. (2.2). We believe that the sign we adopt (which results in AF for the b-coupling) is required for convergence of the Path Integral.
In another previous paper [19], we addressed the issue of AF in SU (N ) and SO(N ) gauge theories in flat space in the presence of adjoint and fundamental scalar representations. We identified some errors in the original treatment, and also added a discussion of a large N limit, where for a limited range of (small) values ofb g we showed that a UVFP existed.
In this paper we generalise this work by coupling these theories to renormalisable gravity. In both the SO(N ) and SU (N ) cases, the inclusion of these interactions has essentially no effect on the minimum value of N required for a UVFP, nor in the basic features of this FP. We described the SO(10) case (with only an adjoint scalar) in detail in Ref. [18]. With both scalar representations, however, the minimum value of N for AF becomes N = 12. 7 We argued, that in the SO(12) case, the theory with both scalar representations would exhibit a region of parameter space consistent with Dimensional Transmutation, in a similar way to the case with an adjoint only, and, moreover, that from parts of this region, the couplings flowed to a UVFP (which we identified) at high energies. In Appendix A, we present the expression for 2 , defined in Eqs. 10.19-10.21, which is required to be positive within a subregion of the DT surface.
The large N limit is interesting. We identified two distinct ways to implement the limit for the gravitational couplings. In both cases, however, the UVFP we identified in flat space is destabilized by inclusion of these couplings.
To sum up: we have demonstrated that it is feasible to construct a realistic Grand Unified Theory with a non-trivial set of scalar field representations (adjoint+scalar) for both SU (N ) and SO(N ) cases, with complete asymptotic freedom and (we argue) Dimensional Transmutation to a low energy theory with gravitational self-interactions described by the Einstein term. However the cases SO(10) and SU (5) are excluded. Moreover, the minimum value of N required is higher for both SU (N ) and SO(N ) than in the case with only an adjoint scalar, making it not unlikely that it will be higher still with a more complicated set of scalar representations. Problems with the scenario remain, most obviously the generation of the electroweak scale, and the issue of unitarity, which we hope to address elsewhere [21].