Asymptotic safety in $O(N)$ scalar models coupled to gravity

We extend recent results on scalar-tensor theories to the case of an O(N)-invariant multiplet. Some exact fixed point solutions of the RG flow equations are discussed. We find that also in the functional context, on employing a standard"type-I"cutoff, too many scalars destroy the gravitational fixed point. For d=3 we show the existence of the gravitationally dressed Wilson-Fisher fixed point also for N>1. We discuss also the results of the analysis for a different, scalar-free, coarse-graining scheme.


I. INTRODUCTION
In the last years many efforts have been devoted to studying the possibility of defining UV complete QFTs which may describe gravitational interactions, possibly in the presence of other matter fields. Such a program is defined within the paradigm of Asymptotic Safety [1] and can be seen as a bottom-up approach which seeks a consistent extension to UV of results of low energy effective field theories. It is related to the construction of the non perturbative RG flow of the effective action, which should show a UV interacting fixed point (in theory space) with a finite number of relevant directions.
In a recent paper [2] two of us discussed fixed-functional solutions in a scalar-tensor theory with action containing a potential V (φ) and a generic non-minimal interaction −F (φ)R. These functions were subjected to a renormalization group flow depending on a cutoff k. It was found that in generic dimension d the flow equations for the dimensionless functions v(ϕ) = k −d V (φ) and f (ϕ) = k 2−d F (φ) of the dimensionless field ϕ = k 1−d/2 φ admit some simple exact solutions where v = v 0 is constant and f = f 0 + f 2 ϕ 2 . These solutions can have interesting applications to cosmology in d = 4. [3]. Unfortunately it seems that one of these solutions, with f 0 and f 2 both nonzero, does not survive, at least in the same form, when one applies the "renormalization group improvement", while another, which remains unchanged by the "renormalization group improvement", has f 0 = 0 but f 2 < 0, making its physical relevance questionable.
In this paper we consider the case when there are N scalar fields, with an O(N )-invariant action of the general form where now V and F are functions of the radial degree of freedom and we use the notation φ = a φ a φ a and for the dimensionless rescaled field ϕ = a ϕ a ϕ a . (In section IV it will prove more convenient to think of v and f as functions of the variable ρ = ϕ 2 /2.) The remaining N − 1 angular degrees of freedom will be referred to as the Goldstone bosons. One of the main results of this note is to show that in the case when N > 1 there exists also another solution with f 0 = 0 and f 2 > 0.
The other main result will be the confirmation, on employing a standard cutoff of "type-I" (in the terminology of [4]), of the existence of an upper bound on the number of scalars in order to obtain a FP with positive f . Such bounds have been discussed earlier but only for minimally coupled fields. We will see that the existence of the potential and of the nonminimal interactions does not substantially change the picture.
In the same scheme for d = 3 we also discuss the existence of a non trivial gravitationally dressed fixed point and give a specific solution for N = 2. Finally we will employ an alternative coarse-graining scheme in order to test the scheme-dependence of the analytical scaling solutions.

II. FLOW EQUATIONS
For the construction of the flow equations we follow the same steps as in [2], which we briefly recall. The calculation is based on the background field method, but instead of the usual linear classical-background-plus-quantum-fluctuation split we use an exponential parametrization for the metric of the form g µν =ḡ µρ (e h ) ρ ν . This has some concrete practical advantages that will be mentioned later, but the original theoretical motivation is that it respects the non-linear nature of the metric. The fluctuation field is further decomposed into its irreducible spin-two, spin-one and spin-zero components h T T µν , ξ µ , σ and h. Then one calculates the second variation of the action with respect to the fields, which appears as a central ingredient in the flow equation. In order to have a diagonal Hessian we define new scalar fields that involve a mixture of σ and φ. We then choose a very simple "unimodular physical gauge" which amounts simply to suppressing the fields ξ and h. We describe briefly these steps in the Appendix. Finally we introduce a "type I" cutoff, which amounts to replacing all occurrences of −∇ 2 by −∇ 2 + R k (−∇ 2 ) [4]. In particular we will use the cutoff profile function R k (z) = k 2 − z θ k 2 − z [5]. Note that this cutoff is "spectrally adjusted", in the sense that it depends on some running couplings, and also depends explicitly on the background scalar field, in addition to the background metric. Later we shall consider also other types of cutoff.
In general dimension d and for any number of scalars, denoting by a dot the derivative with respect to RG time t = log k, the full flow equations arė The only difference between the case N = 1 discussed in [2] and the case of general N is the addition to the beta functionals of v and f of the contribution of the Goldstone modes, which can be easily identified by the factor N − 1. The "RG-unimproved" or "one-loop" flow equations can be obtained by replacing in the r.h.s.
which is equivalent to settingḞ andḞ to zero in the equation for the dimensionful functions V and F .

III. SCALING SOLUTIONS
We list here some simple analytic solutions that exist in any dimension. Making the ansatz that v and f are both constant leads to a fixed point FP1. Using the simple, unimproved equations, it has the following coordinates: We note that the first fraction in f * is positive for d < 6.17. Thus for N = 1 this is an upper bound on the dimension dictated by positivity of Newton's constant. Having additional scalars lowers this bound. The bound becomes lower than four between N = 11 and N = 12. Thus, the fixed point has negative Newton's constant when there are more than 11 scalars, a result that is in rough agreement with previous calculations [6] that also used a similar cutoff.
If we include the RG improvement the coordinates of FP1 change to: Now, for N = 1 positivity of Newton's constant requires d < 5.73, and the bound becomes lower than four between N = 14 and N = 15. Thus, the fixed point has negative Newton's constant when there are more than 14 scalars.
If we make the ansatz that v is constant and f is of the .
We observe that while f 2 is always positive, f 0 becomes negative when either d or N become too large. For example for fixed N = 2 this happens at d ≈ 5.8 and for fixed d = 4 this happens for N ≈ 5.6. The solution is probably unphysical in these cases.
The same ansatz does not yield a solution for the improved equations. We suspect that there may be a solution with very similar properties but different functional form. The search of such generalization is beyond the scope of this paper.
If we make the ansatz that v is constant and f is proportional to ϕ 2 (i.e. that f 0 = 0), the where the upper sign corresponds to FP3 and the lower sign corresponds to FP4. We note that FP3 has a finite N → 1 limit, while in the case of FP4 there is a divergence. In fact for N = 1 the fixed point FP3 had already been seen in [2], but FP4 does not exist in that case. Imposing that f > 0 we find that FP3 is always unphysical while FP4 is acceptable for 0 < N < 15.33 in d = 3 and 1 < N < 11.25 in d = 4.

IV. LARGE-N LIMIT
All the solutions described in the previous section have the property that the function f * is not positive if the number of scalars exceeds a certain limit. In order to confirm that this behavior persists we consider here the large-N limit of the flow. It is very convenient to change variable and use ρ = ϕ 2 /2 as the argument of the functions v and f . It is easy to see that in the large-N limit both the potentials v and f as well as ρ scale linearly in N . In particular in such a limit the flow equations, in terms of all the quantities already rescaled by N , read simplẏ The fixed point equation for the scalar potential v is the same as in flat space. There is a solution with a constant potential v and one with a nontrivial one which is known analytically [7]. In the first case the solution to Eqs.
where a is a constant of integration. For such a line of fixed points it is clear that, for d > 2 and for any value of a, the function f is unphysical since either the constant part or the coefficient of ρ are negative.
In order to deal with the second case with the non trivial solution for v, its equation is best written, for d not an even integer, in terms of w(ρ) = v (ρ) in the implicit form Therefore the scalar sector presents the Wilson Fisher type global scaling solution for 2 < d < 4.
Also the fixed point equation for f can be further simplified in terms of w and defining a new which can be solved analytically and gives the implicit solution: which is always negative for d > 2 and therefore such a fixed point for 2 < d < 4 appears in the large N limit to lead to negative gravitational interactions in the current formulation. We also note that in the asymptotic region ρ → ∞ one has f (ρ) = − 2 3d(d−2) ρ.

V. STABILITY ANALYSIS
We discuss here the linearization of the flow in d = 4 around the fixed point FP1. We begin with the simpler "RG-unimproved" case, which yields the following linearized equations: In this case the critical exponents are equal to their classical values: The solution FP1 for the full fixed point equations in d = 4 has coordinates v * = N +4 128π 2 , f * = 169−12N 2304π 2 . Linearizing the full flow equations around this solution we get the eigenvalue equations for the eigenperturbations: We can study the spectrum of the leading eigenvalues analytically. We find four relevant and one marginal direction for N < 45 4 , while for 45 4 < N < 169 12 there are only two relevant and one marginal directions, since θ 2 and θ 4 become negative. In particular We note that the eigenvalues come in groups within which they are shifted by two. This behavior had already been observed and explained in [8].

VI. THE GRAVITATIONALLY DRESSED WILSON-FISHER FIXED POINT
In [2] we looked for scaling solutions in d = 3 with a potential resembling that of the Wilson-Fisher fixed point. We concentrated on the unimproved equations, and found a solution for sufficiently small ϕ whose potential is almost indistinguishable from the Wilson-Fisher potential, and with a function f that starts out positive at φ = 0 but has negative second derivative, such that it crosses zero at some critical value ϕ ≈ 0.92. The analysis of the solution beyond this critical value proved hard and we were unable to establish its global existence. Perhaps more important, the Hessian becomes ill-defined at the critical point, so that the equations themselves become unreliable. A little later, using more powerful numerical techniques, a global solution was found for the improved RG equation [9]. The fixed point potential for this solution is again very similar to the Wilson-Fisher potential, but the function f now has positive second derivative and is positive everywhere, avoiding the issues mentioned above. With hindsight this solution can be seen also with the simpler techniques used in [2], such as Taylor expansion and the shooting method. Near the origin it has the expansion v = 0.00935540 − 0.0292660ϕ 2 + 0.00359119ϕ 4 + 1.14530ϕ 6 + · · · f = 0.0686040 + 0.172245ϕ 2 − 0.132631ϕ 4 + 0.390317ϕ 6 + · · · We have looked for generalizations of this solution for N > 1. Treating N as a continuous variable, candidate scaling solutions can be found with the shooting method. We show, as an example, in Fig. 1 the cases N = 1 and N = 2. A spiraling structure appears close to the fixed point which is characterized by a very sharp relative peak.
The asymptotic behavior of the solution for large ϕ is u as (ϕ) = Aϕ 6 + 23 For the case N = 2 there is a very good match between the numerical solution found with the shooting method around the origin and the asymptotic behavior given above for A 5.149 and However, this advantage comes at a price: the cutoff depends on running couplings and there is an explicit breaking of the scalar split symmetryφ a →φ a + a , δφ a → δφ a − a . As discussed in [10], already in pure scalar theory the presence of the scalar background field in the cutoff action can lead to unphysical results. This argument casts doubts on the use of these cutoffs, and it is important to explore also cutoffs that do not have a prefactor F . In the Einstein-Hilbert truncation, where F = 1/(16πG), such cutoffs were called "pure" [11]. Here we shall call them "scalar-free". It is important to keep in mind that any cutoff still depends on the background metric, and that this dependence will have to be dealt with by other means, see e.g. [12,13].
Here we only get rid of theφ-dependence and have the scalar split Ward identity automatically fulfilled. In particular we shall add in the diagonal entries for h T T µν and σ of the Hessian given in Eq. (A.8) a cutoff γ k d−2 (k 2 − z)θ(k 2 − z), with the same sign of the Laplacian term, to implement the coarse-graining procedure. For all the other terms we shall use the more common We do not report here the form of the flow equations in any dimension, which contains hypergeometric functions. For d = 4 the equations read: Unlike (II.1,II.2), these are transcendental equations and cannot be solved analytically. For a comparison we discuss only the scaling solution FP1 which is defined by v(ϕ) = v 0 and f (ϕ) = f 0 .
With this ansatz the fixed point conditions reduce to the following algebraic equations These can be solved numerically. The linearization of the flow equation around FP1 yields the following equations: This term is absent with the type-I cutoff and it is its presence with the scalar-free cutoff that gives rise to the logarithmic terms in the flow equation. In a proper bi-metric calculation the coefficient of −∇ 2 in the denominator of the ERGE would be Z h − γk d−2 , where Z h is the graviton wave function renormalization constant. The argument of the logarithmic term would then be Z h /γ and the beta function of f would probably be regular for f = 0.

VIII. DISCUSSION
This paper extends earlier investigations of scalar-tensor gravity, where we used the exponential parametrization of the metric. The a priori motivation of this parametrization is that it respects the nonlinear structure of the space of riemannian metrics. It appears a posteriori that it leads to better behaved flows: there are no infrared singularities [2,14] and the equations can be made gauge-independent even off-shell [14]. Quite generally, the use of the exponential parametrization in conjunction with the physical gauge ξ µ = ∇ µ h = 0 leads to simpler equations. In the theory considered here, they are sufficiently simple that analytic scaling solutions can be found. We have given here the form of such solutions in any dimensions and for any number of scalar fields. Relative to our earlier work, the advantage of having more than one scalar field is that there exists a global solution (FP4) with f ≥ 0 everywhere.
One interesting by-product of this analysis is the confirmation of upper bounds on the number of scalars, for compatibility with a fixed point with positive f . This result is not confirmed when one uses a scalar-free cutoff but we have argued that this is probably not physically correct.
As is often the case when taking the large N limit we have been able to construct the exact analytical solutions and show that they are unphysical in the type-I cutoff scheme.
We have also investigated in d = 3 the existence of the gravitationally dressed Wilson-Fisher scaling solution for various finite N. In particular we have shown that physically acceptable solution do exist for small N and given details for the case N = 2.
We hope that these tools will prove useful also in the search of global solutions for f (R) truncations, where several results have been obtained recently [15], but a clear picture is still missing.
Another important direction of investigation is the inclusion of other matter fields, in particular fermions [16][17][18]. If the picture we have found up to now will be confirmed also in more sophisticated truncations, the possibility to define a consistent QFT of matter-gravity interactions will be a guidance in the understanding of several aspects of fundamental/effective physics. We find already interesting to start to investigate possible implications at phenomenological level for example in cosmology.
The Hessian can then be diagonalised by the transformation and the diagonal Hessian reads The ghost terms induced by the gauge fixing are given by: