Gravitational corrections to Yukawa systems

We compute the gravitational corrections to the running of couplings in a scalar-fermion system, using the Wilsonian approach. Our discussion is relevant for symmetric as well as for broken scalar phases. We find that the Yukawa and quartic scalar couplings become irrelevant at the Gaussian fixed point.


I. INTRODUCTION
The lack of renormalizability of Einstein's theory does not preclude the possibility of calculating quantum corrections to low energy processes due to graviton loops [1]. This effective field theory approach has been applied to calculate corrections to the gravitational potential [2] and the running of Newton's constant [3][4][5]. Graviton loops also contribute to the beta functions of matter couplings. This has been studied in the case of a scalar field in [6]. More recently, there has been considerable interest in (and controversy about) the corrections to the beta function of gauge couplings [7]. Aside from the intrinsic theoretical interest, such effects could have obvious applications to grand unified theories, whose characteristic energy scale is not too distant from the Planck scale, where gravity becomes strong. In fact, it has been argued recently [8] that in the determination of the GUT scale, quantum gravitational effects could be more important than two loop effects.
With these motivations in mind, and in the same spirit, we will calculate here the gravitational effects on the beta functions of a simple Yukawa theory, consisting of one scalar and N f fermion fields. We will do our calculations in flat Euclidean space, and therefore we will not calculate here the effect that the matter has on the running of the gravitational couplings (e.g. Newton's constant), but at least in the limit where the matter couplings are negligible, this effect is easily calculable [9].
In addition to the above, there is also another reason for studying this problem. If we look for a fundamental, as opposed to effective, theory of quantum gravity, there is now the concrete possibility that a purely field theoretic solution can be obtained, provided that the renormalization group has a fixed point with a finite number of UV attractive (relevant) directions. A theory with these properties is said to be asymptotically safe and has the same good properties (finiteness, predictivity) as, for example, QCD. The failure of perturbation theory means that the Gaussian fixed point of gravity does not have the desired properties. Work done in the last ten years has provided rather convincing evidence for the existence of a suitable nontrivial fixed point in pure gravity; see [10] for reviews. It is then important to make sure that this fixed point persists also when interacting matter is brought in.
In the case of scalar interactions, this was discussed in [11]. It was shown that there exists a "Gaussian matter fixed point", where the gravitational couplings are nonzero and slightly shifted relative to pure gravity, but all scalar selfinteractions are asymptotically free or zero. Our results imply that such a fixed point exists also in the presence of a Yukawa coupling.
Finally we mention that asymptotic safety may play a role also in the standard model. Some evidence for a nontrivial fixed point in Yukawa systems has appeared recently [12]. If this was the case, then the calculations presented here are necessary to complete the picture by including also the gravitational interactions.

II. YUKAWA SYSTEM
In this section we set up the calculation. The flow of the renormalized couplings will be computed on a flat Euclidean background using an exact flow equation. An infrared cutoff, denoted k, is introduced via a cutoff term ∆S k , in order to define a scale dependent generating functional of connected Green's functions: In flat space the cutoff term has the general form is constructed so as to suppress the contributions to the functional integral from the infrared modes of the field Φ. For a scalar φ, we choose R φ k (z) = k 2 r(z/k 2 ), with r(y) = (1−y)θ(1−y) [13], leading to the substitution −∂ 2 = z → P k (z) = z + k 2 r(z/k 2 ), a kind of cutoff-propagator. For a fermion ψ, defines the effective average action Γ k satisfying the renormalization group equation [14,15] where t = ln k and STr denotes a functional trace, including a factor −1 for fermions. We will restrict our considerations to functionals Γ k of the following form The theory contains a single scalar field with Lagrangian We choose the potential V to be even in φ. Then, where ω µcd is the spin connection and J cd = 1 4 [γ c , γ d ] are the O(4) generators. We will choose the O(4) gauge such that the vierbein is symmetric, so that all vierbein fluctuations can be written in terms of the metric fluctuations and there are no O(4) ghosts [16]. For the time being we keep the function H(φ) general . On the other hand we will set Z φ = Z ψ = 1 and neglect anomalous dimensions.
For gravity we have the Einstein-Hilbert Lagrangian Similarly to previous analyses, we shall work with the background-field method. We ex-pand around constant backgrounds, which we still denote g µν = δ µν , φ, ψ andψ with corresponding fluctuations h µν , ϕ, χ andχ. For diffeomorphisms we fix a covariant background gauge, with gauge fixing term and the ghost action term consequently given by We also employ the tensor decomposition , vector (v µ ) and scalar (σ,h) fluctuations of the metric.
In order to write the RG flow for our system using (2) we introduce a supermultiplet Υ T = (h ⊥ µν , v µ , c µ ,c µ , σ, h, ϕ, χ T ,χ) containing all the field fluctuations of the system and the matrix of operators Γ Fixing the form of the potentials and expanding around an appropriate basis of operators one may find the run-ning of any coupling of interest. We consider in the following local power-law potentials, expanding either around φ = 0 or φ = √ κ. Concerning h, from now on we restrict ourselves to a simple Yukawa interaction h = yφ.
a. Expansion around φ = 0. For a quartic potential inserting in (4) we find, in the gauge α = 0 and in the approximation λ 0 = 0, In general, the beta functions would depend nonpolynomially on λ 0 andG. In the approximation λ 0 = 0,G appears only polynomially: the highest power ofG occurs inλ 4 and is 2. In all other termsG appears at most linearly.
(11) We do not give here the O(α) corrections to these formulae. We notice that unlike in the expansion around φ = 0, here θ 0 appears only in its own beta function.
Up to orderG, there is no approximation involved in setting θ 0 = 0 in the beta functions of κ, θ 4 and y, as is natural in an expansion around flat space.

IV. DISCUSSION.
The standard MS result for the beta function of the Yukawa coupling isẏ = 5y 3 16π 2 + . . .. On the other hand, neglectingG and λ 2 in (6) or neglectingG and κ in (11), we remain withẏ = y 3 8π 2 +. . .. The difference is due to the fact that here we neglect the anomalous dimensions of φ and ψ. Since their contribution is not very small, our results are not quantitatively accurate, but they should still give a reasonable qualitative picture of the gravitational corrections. We also stress that even though here we analyze a toy model, our result for the leading one loop gravitational correction applies also to realistic theories. In particular when the Yukawa couplings form a matrix y ij , every beta functionẏ ij will receive the same correction (27/16π)Gy ij . The inclusion of anomalous dimensions is currently under study. Switching off the gravitational corrections, our results are in agreement with those of [12], when the anomalous dimensions are neglected. Furthermore, the results forλ i in (6) are also in agreement with those of [11]. We have given in Eqs. (6) and (11) also the beta functions of the vacuum energy λ 0 and θ 0 . One can see the leading contributions, proportional to (3 − 4N f ), the difference between the number of bosonic and fermionic degrees of freedom.
Having used an expansion around flat space, gravity is off shell. This is the cause of the dependence of the results on the gauge parameter α (and β, the dependence on which we have computed but not reported here for simplicity). We note that the sign of the leading corrections does not change as long as α > 0; we have also checked that it remains the same at least for 0 ≤ β ≤ 1.8, which comprises the most popular gauge choices. Furthermore, there are arguments showing that if α was allowed to run, α = 0 would corresponds to a nonperturbative fixed point [17]. This suggests that the results obtained for α = 0 are probably the most reliable.
The procedure also generically depends on the choice of cutoff scheme, and in particular on the cutoff function r(y). The leading terms in the beta functions of λ 4 and y turn out to be independent on this choice, but not the gravitational corrections, which are related to a dimensionful coupling. In the results presented above we only used the cutoff r(y) = (1 − y)θ(1 − y), so the scheme dependence is not manifest, but the numerical coefficients of the gravitational correction would change if we used another cutoff function. We have checked that the leading gravitational correction is proportional to a single integral involving r(y), so that the ratio of the leading correction terms in (6) and (7) is independent of r. Furthermore, the sign of the gravitational correction would be the same for any choice of r(y) that satisfies the boundary and monotonicity conditions to be a good cutoff.
The system (6) has a (Gaussian) fixed point when λ 2 = λ 4 = y = 0. Without gravity both λ 4 and y are marginal, but the gravitational correction makes them irrelevant. In fact the critical exponents are 2 − (3 + 2α)G/π, −(3 + 2α)G/π and −(27 + 29α)G/16π, corresponding to the eigenvectors λ 2 − 3λ 4 /16π 2 , λ 4 and y respectively. (Note that the gravitational corrections depend on α but are always negative.) This is a remarkable result, because in the standard model these couplings are free parameters, to be determined by experiment, whereas here they are predicted to be zero at high energy. Any value they have at low energy is due to the nonlinearity of the RG flow. This result may change in the presence of other matter fields: it was shown in [11] that minimally coupled matter fields can change the sign of the critical exponent, making λ 4 relevant. Then its value at low energy would be a free parameter, while at high energy we would have asymptotic freedom.
All this holds both for positive and negative λ 2 . However for negative λ 2 we may obtain an improved perturbation series [12] expanding both v and h around the VEV. Then, the beta functions are those given in (11).
Most of the comments made above holds also in this case. The main difference lies in the fact that, in the absence of gravitational corrections, the fixed point now has θ 4 = y = 0 and κ = 3/32π 2 . Remarkably, the beta function of κ does not receive any gravitational correction, as was already noted in [6] for the potential (9) with θ 0 = 0, even taking into account the scalar field anomalous dimension. This is a general property: for any scalar potential v, using (10) and (4), We stress again that the beta function of κ obtained from the relationκ = −λ 2 /2λ 4 + λ 2λ4 /2λ 2 4 together with (6) has a G dependence in it. Also note that the general beta functional of h in (4) can be used to calculate the running of any term of the form φ nψ ψ, in particular of an explicit fermionic mass.
The gravitational corrections are of orderG = k 2 /M 2 Planck and therefore can be treated perturbatively at low energies. They may not be negligible at the GUT scale, though. Beyond the Planck scale the gravitational corrections seem to be large and unbounded. The theory may still be meaningful provided all couplings (in par-ticularG) reach a fixed point. It is known that in the Einstein-Hilbert truncation gravity has a nontrivial fixed point, also in the presence of minimally coupled matter fields. Since the Yukawa system has a Gaussian fixed point, one can conclude that the theory of gravity coupled to scalars and fermions also has a fixed point, which we may call a "Gaussian matter" fixed point. However, it is clear that to study the properties of this fixed point, in particular the critical exponents, it is necessary to calculate also the beta function ofG. There is also the possibility that the matter sector exhibits a nontrivial fixed point [12]. Preliminary results indicate that, as long as G * < ∼ 1, this fixed point would also exist in the presence of gravity. We plan to discuss these matters in more detail elsewhere.