Higgs scalar potential in asymptotically safe quantum gravity

The effect of gravitational fluctuations on the quantum effective potential for scalar fields is a key ingredient for predictions of the mass of the Higgs boson, understanding the gauge hierarchy problem and a possible explanation of an"asymptotically"vanishing cosmological constant. We find that the quartic self interaction of the Higgs scalar field is an irrelevant coupling at the asymptotically safe ultraviolet fixed point of quantum gravity. This renders the ratio between the masses of the Higgs boson and top quark predictable. If the flow of couplings below the Planck scale is approximated by the Standard Model, this prediction is consistent with the observed value. The quadratic term in the Higgs potential is irrelevant if the strength of gravity at short distances exceeds a bound that is determined here as a function of the particle content. In this event, a tiny value of the ratio between the Fermi scale and the Planck scale is predicted.

The effect of gravitational fluctuations on the quantum effective potential for scalar fields is a key ingredient for predictions of the mass of the Higgs boson, understanding the gauge hierarchy problem and a possible explanation of an -asymptotically-vanishing cosmological constant. We find that the quartic self interaction of the Higgs scalar field is an irrelevant coupling at the asymptotically safe ultraviolet fixed point of quantum gravity. This renders the ratio between the masses of the Higgs boson and top quark predictable. If the flow of couplings below the Planck scale is approximated by the Standard Model, this prediction is consistent with the observed value. The quadratic term in the Higgs potential is irrelevant if the strength of gravity at short distances exceeds a bound that is determined here as a function of the particle content. In this event, a tiny value of the ratio between the Fermi scale and the Planck scale is predicted.

I. INTRODUCTION
The quantum effective potential for the Higgs field is the central quantity for understanding the electroweak symmetry breaking in the Standard Model of particle physics (SM). The vacuum expectation value of the Higgs field φ 0 is determined by the location of the minimum of the potential. It defines the Fermi scale. For given gauge couplings and Yukawa couplings it sets the mass of W and Z bosons as well as of quarks and charged leptons. In turn, the vacuum expectation value depends on two renormalizable couplings, the mass parameter m 2 H and the quartic scalar coupling λ H . The observable mass of Higgs boson obeys M H = √ 2λ H |φ 0 |. The renormalizable couplings of the SM can be extrapolated to momenta much larger than the Fermi scale. In renormalization group (RG) improved perturbation theory their running is computed with an expansion in loops. Let us now assume that the SM is part of an "effective low energy theory" model for scales below some transition scale k t where gravitational fluctuations decouple. Typically, k t is close to the Planck mass. In the absence of gravitational fluctuations for momenta smaller than k t all couplings are small and in the perturbative regime. For a given model the "initial values" of m 2 H (k t ) and λ H (k t ) can be extrapolated perturbatively to momenta of the order of the Fermi scale where they determine the observable quantities.
For possible predictions of the Fermi scale and the mass of the Higgs boson, the decisive question is the predictability of m 2 H (k t ) and λ H (k t ). For this issue gravitational fluctuations become important. For the flow of couplings at momenta larger than k t the gravitational fluctuations strongly influence the running of m 2 H and λ H . It has been argued that the gravitational fluctuations drive λ H to a fixed-point value close to zero, such that λ H (k t ) has a tiny value. The extrapolation to low momenta within the SM as effective low energy theory has predicted [1] the mass of the Higgs particle in accordance with later observation.
We aim here for a systematic investigation of the effects of gravitational fluctuations on the shape of the effective scalar potential. Beyond the Higgs sector of the SM this is relevant for other theories with scalars, such as grand unified theories. For cosmology, gravitational fluctuations play an important role for the shape of scalar potentials responsible for the inflationary epoch or dynamical dark energy. Since gravity is not perturbatively renormalizable, any investigation of the role of gravitational fluctuations at momentum scales larger than k t has to employ some suitable non-perturbative method.
In the present work we use the functional renormalization group (FRG) for the effective average action [2]. The FRG has proven to be a successful non-perturbative method for various systems in both condensed matter and elementary particle physics. A central object within its formulation is the scale-dependent quantum effective action or effective average action Γ k , which includes all effects of quantum fluctuations with momenta larger than an infrared (IR) cutoff k. The scale dependence of Γ k obeys an exact flow equation [2].
The FRG is capable of understanding quantitatively asymptotically safe renormalizable quantum field theories, which is crucial for studying gravitational fluctuations near and beyond the Planck scale. For asymptotically safe theories, the interactions do not vanish at the ultraviolet (UV) fixed point, such that perturbative renormalizability is often not given. Well studied examples for non-perturbative asymptotic safety are the Wilson-Fischer fixed point for three-dimensional scalar theories or four-dimensional theories with four-fermion interactions, for which FRG has proven quantitative reliability. Quantum gravity presumably belongs to this class of asymptotically safe theories. For the quantitative study of this work we assume asymptotic safety [3,4] as a working hypothesis, leaving the fixed-point value of the dimensionless Planck mass as a yet not fully quantitatively determined parameter. This is sufficient to obtain rather robust results for the effect of gravitational fluctuations on the effective scalar potential.
Quantum gravity coupled to elementary particles reveals a new predictive power for particle properties. This is connected to the number of relevant parameters at the fixed point, which may be smaller than the number of renormalizable couplings in the SM. This entails that certain relations among the SM couplings become, in principle, computable. "Initial values" of running couplings become fixed at the Planck scale if they correspond to irrelevant parameters at the fixed point. This allows a computation of observable quantities such as the Higgsboson mass and the top-quark mass in the low-energy regime [1,[60][61][62][63]. The fixed-point structure and the RG flow could also determine the potential of scalar fields whose time evolution characterizes the history of our Universe [64][65][66][67].
In this paper, we investigate quantum gravity effects on the effective scalar potential in asymptotically safe gravity. Taking proper account of gauge symmetries, in our case diffeomorphism symmetry, is crucial for quantitative reliability. For this purpose we concentrate on a "physical gauge fixing", which purely acts on the gauge modes among the metric fluctuations, leaving the physical fluctuations untouched [59]. We employ the physical metric decomposition [68], where the metric fluctuations are split into physical modes consisting of the tracelesstransverse tensor (graviton) and a scalar, and the gauge modes, which comprise a transverse vector and a scalar. Employing the physical gauge fixing the two-point function becomes block-diagonal in the physical and gauge modes. A simple relation between ghost and gauge modes allows to combine their contributions to a universal measure factor, which does not depend on the value of the scalar field [59].
Our paper can also be seen as a first application of the gauge invariant flow equation employing only one macroscopic metric field [59]. In fact, at the level of truncation employed here there is no difference between the background formalism with physical gauge fixing and the gauge invariant flow equation. The proposed universal measure contribution [59] comes out directly in our approximation for the background formalism. The flow equation for the effective potential is the same for the truncated background formalism and the gauge invariant flow equation. The contribution to the gauge invariant flow equation from physical fluctuations involves formally non-local projections. This projection is implicitly performed in the background field formalism by the inversion of the second functional derivative of the effective action in presence of the physical gauge fixing term. The relevant projected differential operators for the graviton and the physical scalar metric fluctuations are second order differential operators. No non-locality is encountered explicity.
The propagator and interations for the physical modes are derived here from a gauge invariant effective action. In the backgorund formalism this is an approximation, while for the gauge invariant flow equation this is a genuine property. We also compute the flow of the mass term and quartic coupling by taking derivatives of the flow equation for the effective potential. In the truncated background formalism this is an approximation, while for the gauge invariant flow equation this is an exact property. We do not enter here the discussion if the employed one-loop form of the gauge invariant flow equation is itself an approximation, or if it can be made exact by a suitable definition of the macroscopic field [59].
Our main results for the effects of gravitational fluctuations on the scalar effective potential are the following. (i) A UV fixed point for the cosmological constant (value of scalar potential at its minimum) exists provided the dimensionless squared Planck massM 2 p = M 2 p /k 2 is above a minimal valueM 2 p,c . This value depends on the number N of particle degrees of freedom, as shown in Fig. 1. (ii) The quartic scalar coupling λ H of the Higgs boson is an irrelevant coupling at the UV fixed point. For large k 2 the gravitational fluctuations drive it very close to zero, enforcing an "initial value for the low energy effective theory" λ H (k t ) ≈ 0. For a given low-energy theory at momentum scales below the Planck mass and a given observed mass of the top quark the mass of the Higgs boson M H becomes predictable. If the low energy theory is the SM, the predicted value [1] is M H = 126 GeV with a few GeV uncertainty. For a higher-loop computation and dependence on the top mass see refs. [69,70]. (iii) For small enoughM 2 p (red region in Fig. 1) the quadratic term in the scalar potential is also an irrelevant parameter. Then the model is predicted to be located on the critical phase transition surface of the vacuum electroweak phase transition, realizing self-organized criticality. The gauge hierarchy [71,72] of a tiny ratio between Fermi scale and Planck scale could then be explained by the resurgence mechanism [73].
First indications that a quartic scalar coupling could be an irrelevant parameter can be found in ref. [13,14,17,18,74,75]. At the time of the prediction [1] of the Higgs-boson mass, important uncertainties about the sign and magnitude of the anomalous dimension for the quartic coupling of the Higgs scalar persisted, however. Emphasis on the dominant role of the graviton fluctuations ("graviton approximation") has shown [76] the positive sign of the anomalous dimension A and estimated its magnitude to be of the order one. These are precisely the requirements for the prediction of the mass of the Higgs boson [1]. In the present paper we confirm the graviton domination by an explict computation of the contribution of all other fluctuations, including the universal measure term for a physical gauge fixing. This allows for a quantitative comparison with the dominant graviton contribution. We also show that extensions of the truncation do not alter the main conclusion that quartic scalar couplings are irrelevant parameters at the UV-fixed point for asymptotic safety.
The quantitative precision of the present approach allows for a first time an estimate under which circumstances the scalar mass term can be an irrelevant coupling [73]. Typically, this occurs for A > 2. The size of A depends strongly on the fixed point value of the dimensionless Planck mass. This value depends, in turn, on the precise particle content of the model and requires a computation of the flow equation with similar detail as the one for the effective potential investigated here. Only once this task is accomplished a definite statement on the predictive power of asymptotic safety for the gauge hierarchy will be possible.
This paper is organized as follows: In the next section, we present the flow equation for the scalar potential. The technical aspects are given in appendix A. In Section III, we analyze the fixed-point structure and the critical exponent for the cosmological constant. Section IV investigates the critical exponents for the scalar mass term and the quartic interaction of the scalar field. We address here the predictive power of quantum gravity for properties of the Higgs scalar. Section V discusses the rubustness of our results by extending the truncation and varying the cutoff function. Section VI is devoted to summarize our results and to discuss their robustness and remaining quantitative uncertainty.

II. FLOW OF THE SCALAR POTENTIAL
The flow equation for the effective scalar potential is extracted from the exact flow equation for the effective average action by taking space-and time-independent field values for the scalar field configuration. The flow is evaluated for a flat spacetime geometry that we take here to be Euclidean. The crucial quantity for the flow equation is the inverse propagator, which is given by the matrix of second functional derivatives Γ (2) k of the effective action, evaluated for the given scalar and metric fluctuation. Precision and robustness of results depend on the validity of the approximations used for Γ (2) k . Gravity is a local gauge theory, with gauge transformations associated to diffeomorphism or general coordinate transformations. It is a crucial issue to take the gauge symmetry properly into account. The fluctuations around any given metric configuration can be split into gauge fluctuations or gauge modes, and physical fluctuations or modes. The gauge modes correspond to the infinitesimal changes of the given metric induced by an infinitesimal gauge transformation. Existence of the UV fixed point and sign of the critical exponent of the scalar mass term, as a function of the fixedpoint valueMp for the running dimensionless Planck mass and the number of effective particle degrees of freedom N = NS + 2NV − 2NF . The grey line is the critical value of the Planck massMp,c. In the red-colored region the critical exponent of the scalar mass term becomes negative, i.e., this coupling is irrelevant, while it is relevant in the blue-colored region. The quartic scalar coupling is irrelevant whenever a fixed point exists.
We follow the standard treatment of functional integrals for gauge theories, implementing gauge fixing and the associated Faddeev-Popov determinant. We impose a particular "physical gauge fixing" [59]. A physical gauge fixing acts only on the gauge modes. Choosing a decomposition of the metric fluctuations into physical modes and gauge modes [68] the physical gauge fixing term renders Γ (2) k effectively block diagonal, with separate blocks for the physical modes and the gauge modes. Furthermore, imposing the physical gauge constraint on the fields in the effective action leaves a gauge invariant effective action [59]. We can therefore employ an ansatz where the effective action consists of a gauge invariant partΓ k plus a gauge fixing part. For the inverse propagator Γ (2) k the block for the physical modes is given by the second functional derivative ofΓ k . This is an important advantage, since gauge symmetry severely restricts the form ofΓ k .
We find that the contribution of the gauge modes, together with the contribution from the Faddeev-Popov determinant or the corresponding ghosts, results in a simple universal contribution to the flow equation. This "measure contribution" depends on the metric, but not on the values of scalar fields. For the flow of the effective potential it only concerns an overall constant, but not the field dependence.
What remains to be done is an effective approximation for the physical inverse propagatorΓ (2) k . This is done by making a ansatz for the gauge invariant effective action Γ k . We approximate the gravitational part ofΓ k by the Einstein-Hilbert action, with coefficient of the curvature scalar given by the running or scale dependent squared Planck mass M 2 p (k). The cosmological constant is included as part of the effective scalar potential, namely its value at the minimum. We discuss in the conclusions how this ansatz can also incorporate effects of higherderivative invariants inΓ k , as R 2 or R µν R µν . This is done by an adaptation of the definition of M 2 p (k). According to our assumption of asymptotic safety the running Planck mass has to scale at and near the UV fixed point proportional to k The fixed-point valueM 2 p * depends on the particular model. For the purpose of this paper we treat it as an unknown parameter. Some of the predictions depend on the precise value of this parameter, while others such as the quartic coupling λ H being an irrelevant parameter are independent of the precise value.
We first consider a single real scalar field φ coupled to gravity. The detailed steps of the computation along the lines sketched above are displayed in appendix A. We obtain for the flow of the effective potential U (ρ) at fixed ρ = φ 2 /2 a differential equation with a rather simple form, Hereπ 2 is the contribution of the graviton fluctuations corresponding to the traceless-transverse metric fluctuations, the termπ 0 combines the physical scalar fluctuations, both from φ and the physical scalar mode in the metric fluctuations. Finally,η is the measure contribution. Employing a Litim-type cutoff function [77], the terms are given bỹ Here we have defined the dimensionless quantities, with Z φ the coefficient of the scalar kinetic term, and primes denoting derivatives with respect toρ. The dimensionless ratio depends on ρ. The poles at v = 1 and v = 4 correspond to tachyonic instabilities in the graviton and the scalar mode of metric fluctuation propagator, respectively. They are not reached by the flow. We furthermore define η g = −∂ t lnM 2 p = 2 − ∂ t ln M 2 p and the anomalous dimension of the scalar field, η φ = −∂ t ln Z φ .
We will see that at the fixed point the minimum of U (ρ) occurs for ρ = 0. In the vicinity of this point we can neglect the term 3ρŨ 2 /M 2 p in the denominator of (3), such that also the effect of scalar fluctuations becomes block diagonal,π 0 π 0,g +π 0,φ , The first contributionπ 0,g arises from the scalar mode in the metric, while the second termπ 0,φ is the standard flow contribution from scalar fields in flat space [2]. The flow equation (2) holds at constant ρ. For a discussion of a fixed point and the behavior close to the fixed point we have to translate to the flow at constant ρ. Furthermore, we are interested in the flow of the dimensionless scalar potentialŨ (ρ). Generalizing to N real scalars with O(N ) symmetry, and employing the approximation (6) we obtain the beta function or flow generator forŨ as The dependence of the gravitational contributions onρ arises through the quantity v = v(ρ). Also the dimensionless Planck mass enters in (8) only through v. The first two terms on the right-hand side of (8) are the canonical scaling of the effective potential. We have extended the scalar sector to N scalars with SO(N )-symmetry. For scalar theories with SO(N ) symmetry the term ∼ (N − 1) arises from the fluctuations in the Goldstone directions. For the Higgs doublet one has N = 4. Furthermore, we have included in (8) the contribution of fluctuations beyond the gravitational degrees of freedom and the Higgs sector. For massless particles, as gauge bosons or chiral fermions, they contribute to ∂ t U a field independent term ∆N/(32π 2 ), where with ∆N S the number of additional scalars, N V the number of gauge bosons (with two physical degree of freedom each) and N F the number of Weyl fermions. For the SM-matter content, this number is ∆N SM = −66, for a grand unified gauge theory based on SO(10), one has ∆N GUT = N S + 10 with N S the total number of real scalars beyond a complex 10-representation. Around the origin atρ = 0 we expand Inserting into (8) yields the beta function for each coupling, Here we have defined the dimensionless renormalized parameters asṼ = U (ρ = 0) /k 4 ,m 2 In general, eqs. (11)-(13) receive contributions from the term 3ρŨ 2 /M 2 p in (3), neglected in (7) and (8). These contributions are proportional to higher orders of the coupling constants, e.g.,m 4 H orm 2 Hλ H . If the fixed point of matter interactions occurs form 2 H * =λ H * = 0, these terms do not contribute to the critical exponents defined below, see (16).
The effects of Yukawa couplings or gauge couplings to the Higgs sector correspond to the standard perturbative contributions to the beta functions. These effects are small and are not included in our discussion of the UV fixed point. In the present approximation the additional particles only influence the flow ofṼ , with no direct influence on (12) and (13).
Instead of the cosmological constantṼ , it is useful to introduce the beta function of the dimensionless quantity Eqs. (12)-(14) constitute a system of three coupled nonlinear differential equations. They are solved numerically. We employ η g = 0 as appropriate for the UV fixed point, and also neglect the presumably small scalar anomalous dimension η φ , which arises from the flow of the kinetic term for φ. The result of the numerical solution is shown in Fig. 1. For the purpose of this figure we define an effective N = ∆N + 4. ForM p * outside the grey region we find indeed a UV fixed point. For N ≥ −4 our assumption of asymptotic safety holds only if gravity is not too strong, such thatM p * remains above the lower bound indicated by the grey line in this Fig. 1.

III. FIXED POINT AND CRITICAL EXPONENTS FOR THE SCALAR POTENTIAL
Let us now investigate the fixed-point structure and the critical exponents for the scalar potential U . To this end, we need the dimensionless Planck massM p , which enters directly in (12), (13) and indirectly through v 0 = 2Ṽ /M 2 p . In order to close the system the beta function forM p would be needed. The latter depends on the particle content of the theory. Its computation is also influenced by the truncations of the system, the choices of gauge parameters and the regulator. We assume here only that a fixed point of the Planck mass exists and treatM p as a free constant parameter. Since the Newton coupling is defined as G N = 1/(8πM 2 p ), a small value of the Planck mass corresponds to a strong interaction of gravity. A constantM p results in η g = 0.
We also assume that the system has a Gaussian-matter fixed point, namely, that a non-trivial fixed point is present in the gravity sector, while gauge and Yukawa couplings in the matter sector vanish at the fixed point. We will discuss in Sec. VI the possibility that the matter interactions have a non-trivial fixed point and their effects on the critical exponents. We neglect the small [33] anomalous dimension η φ . With these assumptions we obtain the fixed points and the critical exponents as functions ofM p .

A. Critical exponents
Before discussing the structure of the beta functions, we briefly recall the definition of the critical exponents. We denote the renormalized couplings that span the theory space by g = {g 1 , ..., g i , ...}. The RG equations are generally given by whereg i = g i k −di is a dimensionless coupling and d i is the mass dimension of g i . The first term on the righthand side reflects the canonical scaling, whereas the second one is the fluctuation contribution obtained from the flow equation. Suppose that there exists a non-trivial fixed pointg i * . The critical exponents characterize the RG flow in the vicinity of the fixed point. We therefore linearly expand the RG equation (15) The matrix T is the stability matrix and its eigenvalues, denoted by θ l , are the critical exponents. The solution to (16) isg where V l i is the matrix that diagonalizes the stability matrix and C l are constant coefficients given at a reference scale µ. Positive critical exponents correspond to relevant couplings, whereas the irrelevant couplings have negative critical exponent. As k is lowered, the irrelevant couplings flow towards their fixed-point values. Defining a theory at some UV fixed point, the irrelevant couplings take their fixed-point values, setting C l = 0 for all l with θ l < 0 in (17). The coefficients C l for the relevant parameters are the only free parameters of the theory.
Close to a fixed point withm 2 H * = 0,λ H * = 0 we linearize inm 2 H andλ H . Taking N = 4, the flow equations simplify to The stability matrix in the space of couplings v 0 ,m 2 H , λ H follows by taking derivatives at the fixed-point values The quantity depends only onM 2 p and v 0 , not on ∆N . The eigenvalues of T are simply the diagonal elements of the matrix (21). In a more complete setting with a beta function forM 2 p depending on v 0 andm 2 H , the extended stability matrix involves mixing effects with the sector describing the flow ofM 2 p , as well as possibly with other flowing parameters in the gravitational sector. These mixing effects are neglected in the present work. They concern only the critical exponent for v 0 . Since the beta functions form 2 H andλ H vanish form 2 H = 0,λ H = 0, the derivative of these functions with respect toM 2 p does not contribute at the fixed point. As a consequence, the critical exponents form 2 H andλ H are not affected by the mixing and remain to be given by 2 − A and −A.
These features allow for rather robust predictions of the critical exponents once the fixed-point values forM 2 p and v 0 are known. One only needs the computation of A in (22). The first term in (22) is the graviton contribution fromπ 2 . It typically exceeds the second term by more than a factor 20. This validates the "graviton approximation" for the computation of the critical exponents in ref. [59].

B. Cosmological constant
The value of the cosmological constant v 0 near the fixed point has a substantial influence on the size of the gravitational fluctuations. We determine here the fixedpoint value and the associated critical exponent.

Fixed point as a function of the Planck mass
We first look for a possible fixed point of the cosmological constant as a function of the Planck mass by setting the right-hand side of (18) to zero. We concentrate on N = 4, as appropriate for the Higgs doublet. The resulting quadratic equation for v 0 * admits two solutions: One is a UV fixed point, the other is an IR one. For instance, in the graviton approximation, namely, taking only the contribution of the traceless transverse mode π 2 into account, one finds for These values obtain small corrections from the scalar mode in the metric. In Fig. 2, we plot the dependence of the fixed-point value of v 0 on the Planck mass for N = 4 and ∆N = 0. We show both, the full result (blue) and the graviton approximation (only π 2 , red). We conclude that the graviton approximation is valid up to corrections ∼ 10%.
In the limitM p * → ∞, the UV fixed point v (UV) 0 * merges to the Gaussian fixed point, while the IR one converges to one. In contrast, for decreasingM p * these two fixed points approach each other. They merge at a critical valueM p,c = 0.217. ForM p <M p,c , no fixed point exists and quantum gravity is not asymptotically safe. The critical valueM p,c depends on the particle content in the UV. We plotM p,c as a function of N in Fig. 1.
Here N stands for 4 + ∆N . We include the range of negative N , as relevant for a sufficient number of fermions, see (9). Indeed, gauge bosons and scalars give a positive contribution to N , while for fermions the contribution is negative. The valueM 2 p,c is obtained by imposing at v 0 the simultaneous requirements ∂ t v 0 = β v = 0 and ∂β v /∂v 0 = 0. For a given N asymptotic safety is only realized forM p >M p,c (N ), as visible in Fig. 1. According to (21) and (22) the valueM p,c corresponds to A = 4. In the vicinity of this critical value both the mass term and the quartic scalar coupling are irrelevant couplings, with critical exponents −2 and −4, respectively.
The IR fixed point for v 0 always exits for a givenM p * > M p,c , since the flow of v 0 is always stopped before the pole of the beta function at v 0 = 1 is reached. On the other hand,M p depends on v 0 . The functionM p * (v 0 ) corresponds to a curve in Fig. 2 that is not computed in the present work. The UV fixed point corresponds to the intersection of this curve with the curve shown in Fig. 2. If there exist two intersection points, both with the solid and the dashed line, both an UV and an IR fixed point exist. If present, one could alternatively define the theory at the IR fixed point. Since ∂β v /∂v 0 > 0 at the IR fixed point one infers from (21) and (22) that A > 4. Thus all three parameters v 0 ,m 2 H andλ H are irrelevant couplings. In the present paper we do not pursue this possible alternative and rather concentrate on the UV fixed point.
OnceM 2 p increases as this coupling moves away from the UV fixed point, the IR fixed point for v 0 approaches the pole in the beta function. Indeed, the value v 0 = 1 corresponds to a pole of the propagator of the graviton. The existence of the IR fixed point close to v 0 = 1 induces strong fluctuation effects of the graviton in the IR regime and could be a key point to resolve the cosmological constant problem [76,78].
For general ∆N we parametrize the ratio between the scalar and tensor gravitational contributions by The fixed points for v 0 occur for with For large negative ∆N the UV fixed point occurs for a As long as z∆N remains small as compared to one, one has w s ≈ 1/5 and The IR fixed point approaches one, with The approximations (28) and therefore . (32) For large ∆N this results in a value of z c ∆N close to one such that the second term on the right-hand side of (32) can be neglected. Therefore the critical boundary forM 2 p increases ∼ ∆N , The corresponding critical v 0c approaches one, such that the graviton contribution is enhanced. The graviton approximation becomes rather accurate for values ofM 2 p and v 0 in the vicinity of (33) and (34).

Critical exponent
With the value of the UV fixed point, we obtain the critical exponent of the cosmological constant or v 0  We display in Fig. 3 various approximations. Besides the total contribution (blue) and the graviton approximation (red) we show the total contributions of the gravitational degrees of freedom without contributions of other particles, i.e., N = 0 and ∆N = 0. This corresponds in (2) toπ 2 +π 0,g +η (dashed green line). The total gravitational contribution is again well approximated by the graviton approximation. Finally, the green dashed line omits the measure contributionη.

IV. PREDICTIONS OF ASYMPTOTIC SAFETY FOR THE PROPERTIES OF THE HIGGS SCALAR
As mentioned in the introduction, a UV fixed point for quantum gravity can predict those renormalizable couplings in the SM that correspond to irrelevant couplings at the fixed point. We find that the quartic coupling of the Higgs scalar is always irrelevant, and the scalar mass term is irrelevant for a certain range of the fixed-point value of the Planck mass.

A. Effective low energy theory
So far we have concentrated on the flow equations in the vicinity of the UV fixed point of asymptotically safe quantum gravity. This corresponds to a constant value ofM 2 p in the flow equation (8). Let us now assume that the Planck mass corresponds to a relevant parameter in quantum gravity. At short distances, it scales according to the UV-fixed-point behavior, M 2 p (k 2 ) =M 2 p * k 2 . For small k, it deviates from this scaling behavior and takes a fixed value M 2 . This results in the qualitative behavior The transition scale is found as A more complete treatment smoothens the transition. Details of the threshold behavior are not important for our purpose. One may use the simple form For k below the transition scale k t the dimensionless cou-plingM 2 p increases rapidlỹ As a result, the gravitational contributions in (12) and (13) -not in (11) -become rapidly tiny and can be neglected. This leads to a simple picture. For k < k t the flow enters the regime of an "effective low energy theory" for which the effect of gravitational fluctuations can be neglected, except for the cosmological constantṼ . This effective low energy theory may be the SM or a possible extension of it.
To rather good accuracy the flow of dimensionless couplings asm 2 H andλ H can be divided into two regimes. For the UV regime k > k t it follows the flow in the vicinity of the UV fixed point. In contrast, for the IR regime k < k t , the flow is given by the low energy effective theory. In this picture the "initial values" of couplings for the IR flow, e.g., their values at k t , are determined by their final values of the UV flow. In case of irrelevant couplings the initial values for the IR flow are simply the UV-fixed-point values. They are therefore predicted. Following the IR flow from k t in the vicinity of M down to observable energy scales leads then to predictions for observable quantities.

B. Scalar mass term
Next, we turn to the behavior of the scalar mass term in the vicinity of the UV fixed point. Depending on the value ofM 2 p * this can be a relevant or an irrelevant parameter. This issue has important consequences for the gauge hierarchy problem [71,72]. Therefore, we start from the discussion of this problem in the context of flow equations.

scalar mass flow and gauge hierarchy problem
Let us first discuss RG improved perturbation theory in the SM. The RG equation of the scalar mass term at one-loop level is given by where the first term on the right-hand side reflects the canonical scaling and the second one is the anomalous dimension, which reads More precisely, the couplingm 2 H measures the distance from the critical surface of the (almost) second-order vacuum electroweak phase transition [79]. In (41), λ, y t , g, and g are the quartic coupling of the Higgs field, top-Yukawa coupling, SU(2) gauge coupling and U(1) gauge coupling, respectively. Using the values of the couplings at the Fermi scale, the anomalous dimension γ m ≈ 0.027 is much smaller than two.
For the marginal couplings in (41) the scale dependence is logarithmic. Neglecting their runnings the solution of (40) reads A frequent discussion of the gauge hierarchy problem relies on the fact that the one-loop correction to the scalar mass involves a quadratic divergence as the UV cutoff is sent to infinity. In perturbation theory, the observed Higgs mass is given by the cancelation between the squared bare mass and the quadratic divergence. The quadratic divergence strongly depends on the cutoff scheme. It is not present for dimensional regularization, while the momentum cutoff regularization and the Pauli-Villars type cutoff yield different values, depending on the precise implementation.
In terms of the RG, the quadratic divergence indicates the position of the phase boundary in the space of bare couplings. This boundary or "critical surface" separates the symmetric and broken phases and corresponds to the massless (critical) situation [73,[79][80][81][82][83][84]. The position of the phase boundary depends on the precise definition and choice of the bare couplings and on the precise regularization. It changes under a coordinate change in "theory space" if the latter is parametrized by the bare couplings. Different choices of the cutoff scheme also correspond to a coordinate transformation in theory space. In quantum field theory the precise choice of bare couplings is usually not of much interest.
On the other hand, the deviation from the phase boundary corresponds to a renormalized coupling. Its behavior is independent of the precise choice of microphysics as regularizations and the precise definition of bare couplings. This explains why the flow equation (19) only involves renormalized couplings, while no trace of the quadratic divergence appears. The vanishing of the right-hand side reflects the basic property of a secondorder phase transition. No trajectory can cross the phase boundary. Couplings on the critical surface stay on the critical surface.

Quantum gravity effects
We next add the effects of the gravitational quantum fluctuations in the range k M . The beta function of the scalar mass becomes Comparing with (40), we see that A corresponds to the gravitational contribution to the anomalous dimension. Neglecting for simplicity the small value of γ m as compared to A, the critical exponent of the scalar mass parameter reads An important observation is that the sign of A is positive. Gravitational fluctuations lower the value of the critical exponent θ m . As long as A stays smaller than two the scalar mass termm 2 H remains a relevant parameter, θ m > 0. In this case the distance from the vacuum electroweak phase transition, as measured by the value of the Fermi constant, cannot be predicted. It is simply a free parameter specifying the theory. A dramatic change occurs for A > 2. In this event the scalar mass termm 2 H turns out to be an irrelevant coupling. The flow trajectory is always towards the phase-transition surface -an example of "self-organized criticality". If asymptotically safe gravity is realized in a model leading to A > 2, it predicts thatm 2 (k) vanishes for k = M . It not only explains why the ratiom 2 (k) /k 2 =m 2 H , evaluated at k = M , is a tiny quantity as required by the observed The vacuum electroweak transition is not an exact second-order phase transition. This is due to the running gauge and Yukawa couplings that prevent the realisation of exact scale symmetry in the effective low energy theory below the Planck mass. The dominant effect is believed to be due to the running strong gauge coupling. Chiral symmetry breaking induces a quark-antiquark condensate qq . For the light quarks this condensate sets a scale of the order 100 MeV, implying a lower bound on the Fermi scale of the same order of magnitude. The detailed effects of the scale violation in the top-quark-Higgs-scalar sector are not known quantitatively. If they are not substantially larger than the effects of the light quark condensate, asymptotically safe quantum gravity coupled to the SM predicts a W -boson mass around 100 MeV in case of A > 2. This is not compatible with observation.
For A > 2 extensions would be needed, as discussed in the resurgence mechanism [73]. This may either involve new particles with masses near or below the Fermi scale, or a more complicated UV-fixed-point structure for quantum gravity. We recall in this context that a definition of the theory at the IR fixed point for v 0 (which becomes then the effective UV fixed point) implies A > 4. The scalar mass term is always irrelevant for this setting.
In Fig. 4, we show the dependence of the critical exponent of the scalar mass term on the the fixed-point value of the Planck massM p for N = 4 and ∆N = 0. It falls below zero (A > 2) if the strength of gravity exceeds a certain bound, orM p becomes small enough. This behavior extends to other values of ∆N , as shown in Fig. 1 where N stands for 4 + ∆N . The red region in this figure occurs for θ m < 0, A > 2.
For large positive ∆N the physical scalar metric fluctuation can be neglected, yielding for the condition A = 2 the relation In this approximation the fixed-point relation between z and 1 − v 0 inferred from (18) for ∂ t v 0 = 0 reads With the line A = 2 occurs for For large ∆N this comes very close to the critical value (33) for which the UV fixed point disappears. Only a small region with negative critical exponent θ m remains.

C. Quartic scalar coupling
We finally investigate the quantum gravity effects on the quartic scalar coupling. The critical exponent of the quartic scalar coupling is given by the beta function in linear order inλ H , β λ = −θ λλH , The dependence of the critical exponent onM p is shown in Fig. 5.
The irrelevance of the quartic coupling at the fixed pointλ H * = 0 means that the coupling constant vanishes for k above the Planck scale. In other words, the boundary condition of the RG equation of the the quartic coupling is given by the fixed-point valueλ H * = 0 at k t close to the Planck scale. Withλ H (k t ) = 0 the flow of λ H (k) can be followed in the effective low energy theory for k < k t . Assuming that this IR flow is well approximated by the SM, the mass of the Higgs boson can be predicted as a function of the top-quark Yukawa coupling. The result of this prediction [1] was m H = 126 GeV with only a few GeV uncertainty. It agrees well with the observed value of the Higgs boson mass m H = 125 GeV.
In the flow equation forλ H we have neglected effects of Yukawa couplings and gauge couplings. They contribute to the wave function renormalization, leading to a small anomalous dimension η φ . Yukawa and gauge couplings also shift the fixed pointλ H * to a tiny non-zero value. Forλ H * = 0 the terms ∼λ 2 H in β λ contribute to the anomalous dimension, as seen from the last term in (13). This effectively enhances the anomalous dimension and makes the critical exponent θ λ more negative. A nonzero fixed-point valueλ H * could also be induced by nonminimal scalar interactions involving higher derivatives that may be generated by gravitational fluctuations. In any case, even for nonzeroλ H at the UV fixed point this value will be small. It therefore has only little impact on our results.
In summary, the negative sign of the critical exponent for the quartic scalar coupling seems to be a rather robust finding. For a given low energy model the mass of the Higgs scalar is predictable in asymptotically safe quantum gravity.

V. ROBUSTNESS OF RESULTS
In view of the far reaching consequences of our findings for the predictivity of quantum gravity for SM parameters, some tests of the robustness of these results seem appropriate. Possible errors are connected with a possibly insufficient truncation of the exact flow equation. Typical tests are the extension of the truncation and the sensitivity to the choice of the infrared cutoff function. Gauge dependence is a minor issue in our approach since we are bound to employ a physical gauge fixing that only acts on the gauge degrees of freedom in the metric. Within this class of physical gauge fixings the dependence on the precise gauge fixing is small [59,85]. At the present stage the main uncertainty concerns the fixed-point value of the dimensionless Planck mass. We will see that truncation errors can typically be compensated by a change of this value. The error analysis will become more meaningful at a later stage when the fixedpoint value for the Planck mass is also calculated. Nevertheless, our two main statements seem rather robust: (i) The quartic scalar coupling is an irrelevant parameter at the UV fixed point. (ii) There exists a range of fixedpoint values for the Planck mass for which the scalar mass term is also an irrelevant parameter.

A. Extension of truncation
So far, we have analyzed the Einstein-Hilbert truncation for the gravity sector. Extensions of the truncation are possible in various directions. They are, however, restricted by diffeomorphism symmetry of the effective action. One possibility is the inclusion of higher-order curvature invariants. Only the quadratic invariants influence the propagator of the metric fluctuations in flat space, which is the only quantity involved for the computation of the flow of the effective scalar potential. From the quadratic curvature invariants it is only the squared Weyl tensor that enters in the transverse traceless mode of the metric propagator, which constitutes the dominant graviton contribution. Denoting by D the coefficient of the squared Weyl tensor, the inverse graviton propagator becomes at the fixed point The flow is dominated by momenta q 2 ≈ k 2 , such that D becomes important only for A positive D * lowers the anomalous dimension A, while a negative D * enhances it. The sign of A is not changed as long as gravity is stable for positive G −1 g in the Euclidean domain. The inclusion of D modifies the factor (1−v 0 ) −2 in the first term of (22) . The crucial property A > 0 also holds within the perturbatively renormalizable quartic gravity [86,87].
As a second point, we investigate a possible dependence of the effective Planck mass on the scalar field, extending the truncation by a non-minimal interaction between the scalar field and the curvature, We consider here a constant ξ, i.e., assumed to be at its fixed-point value. In the following we derive the shift in the fixed-point values of m H and λ H , the changes in the stability matrix as well as new contributions due to the mixing in the scalar sector. As a main result we find that the dependence of the critical exponents on the non-minimal coupling is small as displayed in Fig. 6. The flow equation (3) for the effective potential remains valid if we replaceM 2 p →M 2 p + ξρ, v = 2Ũ /(M 2 p + ξρ). The additional dependence of v onρ, modifies the flow equation for the mass termm 2 H = ∂Ũ /∂ρ|ρ =0 and the quartic coupling λ H = ∂ 2Ũ /∂ρ 2 |ρ =0 [88]. On the other hand, the flow equation for v 0 = 2Ṽ /M 2 p depends on ξ only indirectly through the dependence of the scalar contribution π 0 onm 2 H . We first neglect the mixing between the scalar modes, which will be added below. The flow equation (19) for m 2 H gets extended to Similarly, the flow equation for λ H (20) becomes where all quantities are evaluated atρ = 0 and we take N = 4 for the number of scalars in (13). The fixed-point value ofm 2 H is now non-vanishing (assuming A = 2), with For the graviton approximation one may use the simple relation The stability matrix T receives additional off-diagonal entries form 2 H * = 0,λ H * = 0, where and Here the second lines in (64) and (65) use the approximation (61). For E one obtains which is of the order ξ 2 . The corrections from ξ are small as long as For an order of magnitude estimate this holds for Since for ξ = 0 the scaling solutions form 2 H * andλ H * occur for non-zero values, there is also a contribution to the flow equation from the mixing in the scalar sector. This only concerns a subleading term. For small ρŨ 2 /(M 2 p + ξρ) the mixing contributes an additional term toπ 0 , where we take η g = η φ = 0. This does not contribute to the flow of V or v 0 , but the contribution to the flow ofm 2 H andλ H vanishes only form 2 H = 0,λ H = 0. One finds, with H 0 = H(ρ = 0), and For small ξ this shifts the fixed-point valuem 2 H * by a small amount ∼ ξ 2 , while the shift inλ H * is ∼ ξ 3 . In leading order in ξ these shifts can be neglected.
The dominant contribution of the mixing effect to the stability matrix T is a shift in the diagonal terms for δm 2 H and δλ H (not for δv) As discussed before, it vanishes for ξ → 0,m 2 H * → 0. With we may neglect in leading orderm 2 H * in H 0 and employ eq. (61) form 2 H * in (72). We observe that H 0 is positive such that A is enhanced for negativem 2 H * . For A < 2 one hasm 2 H * < 0 such that the critical exponent for the scalar mass term moves closer to zero by the mixing effect.
The inclusion of the non-minimal coupling ξ further modifies the off-diagonal parts in the inverse propagator of the spin-0 metric fluctuation in (A24), such that This effect induces a shift in the anomalous dimension, adding to (72) a further piece with Even if the fixed-point values ofm 2 H * andλ H * are zero, this modification differs from zero as long as ξ = 0. A characteristic quantity for the field dependence of the effective Planck mass is ξ/M 2 p . For ξ/M 2 p 1 the field dependence is weak, while for ξ/M 2 p > ∼ 1 the Planck mass varies very rapidly with the scalar field. Fig. 6 displays the dependence of the critical exponents on ξ/M 2 p * by evaluating the eigenvalues of the stability matrix (63) including the shift (75) in the diagonal elements. We use the value of the dimensionless Planck massM p * = 0.22 at which we have the anomalous dimension A ≈ 2.78 for vanishing ξ. The dominant effect arrises from the nonzero values ofm 2 H * andλ H * . In a realistic setting also gauge and Yukawa couplings influence these fixed-point values. If the difference ofm H * andλ H * from zero can be neglected, the effect of ξ is much smaller, given by (76). As a main result we find that the dependence of the critical exponents on ξ is small, supporting our conclusions from the previous sections.

B. Regulator dependence
The functional flow equation is exact for an arbitrary choice of the regulator function R k . Provided that R k obeys the requirements for an efficient IR cutoff, with ∂ t R k decaying fast for high momenta, the results for observable quantities should not depend on the choice of R k . In any practical calculation, they do however and this is due to the choice of a truncation. This observation can be used for a test of validity of a given truncation. The dependence on R k should disappear for a "perfect truncation", and any remaining dependence can be taken as some form of measure for the error induced by the truncation.
For a general cutoff function the graviton contribution to the flow of the potential is given by the threshold func- with 4 0 (w) defined in terms of the propagator G q 2 as with x = q 2 /k 2 and For the Litim cutoff one takes with and This results in the threshold function 4 0 (w) = For η M = 2, corresponding to η g = 0, we recover (3). For a general cutoff function one replaces in the flow equation forŨ and similar for the scalar contributions. For the anomalous dimension A one has to replace Since A is dominated by the graviton contribution a good estimate of the effect of a general cutoff function is the multiplication of eq. (22) by a factor f , For the Litim cutoff, f = 1. Correspondingly, the value ofM 2 p * needed to realize a given A has to be multiplied by f . This feature is not surprising. A dominant effect of a change in the IR-cutoff function can be viewed as an effective rescaling of k, which may be absorbed by a multiplicative redefinition of k. The dimensionless ratiõ M 2 p = M 2 p /k 2 is directly affected by such a rescaling. This property demonstrates that our general results hold independently of the precise choice of the cutoff function.
Let us investigate the regulator dependence numerically. To this end, we use an exponential interpolating cutoff function for the explicit regulator comparison This cutoff function has the limits r int (x, b = 1, n → ∞) = r Litim (x) = (x −1 − 1)θ(1 − x) as well as r int (x, b = 0, n = 1) = r exp (x) = (exp(x) − 1) −1 . We further use the regulator at the values r int (x, b = 1, n = 2) as well as r int (x, b = 1 2 , n = 1). The results are displayed in Fig. 7. One can see that the fact that the critical exponent of the scalar mass θ m = 2 − A becomes negative for a certain value of the dimensionless Planck mass is not changed. The dominant effect is indeed a simple rescaling ofM 2 p * . For a given model and a given truncationM 2 p * will depend sensitively on the chosen cutoff function. For a valid truncation this cutoff dependence should drop out in the final value or θ m . A computaion of the flow equation for M 2 p will be needed for this check.

VI. DISCUSSION
We have computed quantum gravity predictions for the mass and couplings of the Higgs scalar within the asymptotic safety scenario. We consider the SM of particle physics coupled to gravity, with possible extensions of the particle content. The value of the dimensionless flowing Planck mass at the fixed point,M 2 p = M 2 p (k) /k 2 , is influenced by the particle content of the model. We treat it here as an unknown parameter, to be determined for any given model.
Our main findings are the following: (i) The quartic self-couplingλ H of the Higgs scalar is an irrelevant coupling. Its value at the Planck scale is predicted to be very close to zero. For a given low energy model below the Planck scale, where gravitational contributions decouple, this initial value at the Planck scale is mapped by the renormalization flow to the value at the Fermi scale. The ratio between Higgs-boson mass and top-quark mass is therefore predicted. This prediction works well if the low energy theory is the SM. The consistency of other low energy models has to be tested. (ii) The dimensionless mass termm 2 H for the Higgs scalar can be a relevant or an irrelevant coupling, depending on the fixed-point valueM 2 p and on the degrees of freedom, as shown in Fig. 1. Ifm 2 H is relevant, the value of the Fermi scale, or more precisely the ratio between Fermi scale and Planck scale M W /M p , cannot be predicted. For a relevantm 2 H the gauge hierarchy is a free parameter. In contrast, if m 2 H is irrelevant, the model predicts that nature is located on the critical surface of the vacuum electroweak phase transition, with only a small deviation induced by running gauge and Yukawa couplings. Depending on the model this may overpredict the gauge hierarchy to be 10 −40 instead of 10 −34 . (iii) For a given particle content, as expressed by N , we find a lower bound on the fixed point value ofM 2 p if N > −4. These results extend to other quantum field theories with scalar fields coupled to gravity, such as extensions of the SM or grand unified theories. The gravitational contribution to the critical exponents is universal. Quartic scalar couplings are irrelevant parameters at the UV fixed point.
Our findings are of high relevance for the interplay between quantum gravity and particle physics. The validity of approximations and the robustness of results should therefore be critically questioned. First of all, quantum gravity contributions to the effective scalar potential can be performed in flat space, allowing for the full use of Euclidean SO(4)-symmetry or Lorentz symmetry for Minkowski space. Second, our split of gravitational fluctuations into physical modes and gauge modes, together with a physical gauge fixing acting only on the gauge modes, makes the contributions of different modes very transparent. We find that the dominant contributions in the gravitational sector come from the fluctuations of the graviton or traceless transverse tensor fluctuations.
The quantity needed for a reliable computation of the dominant graviton contribution is the exact propagator of the graviton. This is not directly available, and at this point an approximation is made. In the Einstein-Hilbert truncation employed in the present paper the inverse graviton propagator is given by and the question arises if this is a reasonable approximation. The inverse graviton propagator at zero momentum is given by the effective potential For a diffeomorphism invariant formulation this is an exact relation enforced by diffeomorphism symmetry [78]. The right-hand side of the flow equation involves a momentum integral that is dominated by momenta with q 2 ≈ k 2 . We may therefore define the parameter M 2 p (k) by the graviton propagator at q 2 = k 2 , more precisely by This definition goes beyond the Einstein-Hilbert truncation since the contribution of higher-derivative invariants as the squared Weyl tensor can be included in G −1 q 2 = k 2 . A similar definition was already used in [41, 43-45, 48, 50, 52, 53] and the contributions of higherderivative invariants were in particular investigated in [48,53]. Since we treat the fixed-point value of M 2 p (k) /k 2 as a free parameter we may reinterpret our results as reflecting the definition (91). We note that pure graviton vertices do not appear in the computation of the flow of the effective potential. They would differentiate between different terms in the gravitational sector. All this suggests that our estimate of the dominant graviton contribution, which is not affected by any gauge fixing issues, is quantitatively rather robust.
Another robust result is the contribution from the measure sector (gauge fluctuations and ghosts) if physical gauge fixing is employed. The measure contribution results in a simple field-independent term for the flow of U . This term is necessary in order to account for the correct counting of physical degrees of freedom. The contributions from the matter sector are well understood as well. These are the well tested standard contributions to the flow in flat space. One can extend their contribution to include Yukawa and gauge interactions with the Higgs field. As long as gauge couplings and Yukawa couplings remain small in the UV-fixed-point region, these interaction effects on the flow of U correspond to the standard perturbative beta function. The effect of the matter fluctuations on the field dependence of U is much smaller than the gravitational contribution. As far as the critical exponents forλ H andm 2 H are concerned the interactions in the matter sector give only tiny corrections.
In the limit of a constantM 2 p * and constant scalar wave function renormalization the only term that is perhaps subject to somewhat larger truncation errors is the contribution from the physical scalar fluctuations of the metricπ 0,g . Unless strong cancellations occur this contribution is only a rather small fraction of the graviton, matter and measure contributions. The uncertainty in the computation of this term will only result in a rather modest quantitative uncertainty for our overall results.
Replacing for the scaling form of the coefficient of the curvature scalar the constantM 2 p * by a function F (ρ) of the scalar field induces additional terms in the flow equation for the mass term and quartic coupling, without affecting much the anomalous dimension A. A small shift in the fixed point value ofλ H and in the location of the critical surface has no sizeable impact on the critical exponents. Finally, the omitted flow of the scalar wave function renormalization shifts θ m from 2 − A to 2 − A − η φ , and θ λ from −A to −A − 2η φ . This would be a sizeable effect only of η φ is of a similar magnitude as A.
We conclude that at least our Euclidean computation of the flow of the effective potential for the Higgs scalar seems rather reliable. A computation directly in Minkowski space along the lines discussed in ref. [76] would be welcome, but we do not expect important modifications as compared to the Euclidean results. The scalar potential is particularly robust with respect to analytic continuation since no momenta are involved. The remaining big issue is the determination of the fixed-point value of the dimensionless Planck massM 2 p . This depends on v 0 , rendering the flow ofM 2 p and v 0 a coupled system. Only once the fixed point of the combined system is found it can be decided where a model is situated in the plane of Fig. 1 or on the curves of Figs. 2-5. Various computations give values ofM 2 p . In the context of the present paper the use of a physical gauge fixing is appropriate, and results from a gauge invariant formulation of the flow equation would be most welcome.
In this appendix we derive the flow equation for the effective scalar potential (2) as an approximation to the exact flow equation for the effective average action. The exact flow [2,[89][90][91][92] takes a simple one loop form.
Here, R k is an infrared regulator function and ∂ t = k∂ k . The trace in (A1) sums over momenta and internal space indices of a multi field Φ, and the matrix of second functional derivatives Γ (2) k is the full inverse propagator of Φ. For reviews see [93][94][95][96][97][98][99][100][101]. We apply this equation to a model of a singlet real scalar field and gravity and derive the flow equation for the effective scalar field. Generalizations for additional fields are found in the main text.

Setup
We investigate an effective action of the type Γ k = Γ gravity k + Γ Higgs k . (A2) In the gravity sector we employ the Einstein-Hilbert truncation where M p is the reduced Planck mass related to Newton's constant by M 2 p = 1/8πG N . The cosmological constant is included in the scalar effective potential. The metric is linearly expanded around a fixed background whereḡ µν is a constant background metric and h µν is a fluctuation field. We will later use a flat Euclidean background,ḡ µν = δ µν . The gauge fixing and the ghost action for diffeomorphism symmetry are given by where C andC are ghost and anti-ghost fields. A class of general gauge fixings is given by where h =ḡ µν h µν is the trace mode. Bars denote covariant derivatives etc. formed with the background metric. Note that there are two gauge-fixing parameters for diffeomorphism symmetry, α and β. The parameter β is dimensionless, whereas α has mass dimension minus two. For the physical gauge fixing they are given by β = −1 and α → 0. We first keep general α and β in order to see the particular role of the physical gauge fixing explicitly. Next, we turn to the effective action for the Higgs sector. In the SM the Higgs field is a component of the doublet field, coupled to the SU(2) L and U(1) Y gauge fields as well as to quarks and leptons. Near the UV fixed point the contributions from these couplings to the beta function are smaller than the ones of the graviton. All essential points can be understood by restricting the discussion to a single real scalar field with Z 2 symmetry as representing the physical mode of the Higgs boson. The effective action takes the standard form Γ Higgs We subsequently extend our findings to the SM or possible extensions. The effective potential U (ρ) depends only on the invariant ρ = φ 2 /2. The value of U at the minimum can be identified with the cosmological constant. We are interested in momenta much larger than the Fermi scale. The expectation value of φ can be neglected in this range, and we expand where m 2 H is the mass term of the Higgs boson, and λ H is the quartic coupling. The field-renormalization factor of φ is denoted by Z φ .

Physical metric fluctuations
A crucial quantity for the flow equation is the inverse propagator, i.e. the matrix of second functional derivatives of Γ k . The physical understanding, as well as calculational simplicity, is greatly enhanced if we split the metric fluctuations into physical and gauge fluctuations [68]. In flat space,ḡ µν = δ µν , one can use a momentum space representation.
Let us start with splitting the metric fluctuations into where f µν are the physical metric fluctuations, which satisfy the transverse constraint q µ f µν = 0. The physical metric fluctuations can be decomposed into two independent fields as where the graviton t µν is the transverse and traceless (TT) tensor, i.e., q µ t µν = δ µν t µν = 0. The tensor s µν is given as a linear function of a scalar field σ such that where we define the projection operator Similarly, the gauge modes or unphysical metric fluctuations a µν is decomposed into a transverse vector mode κ µ satisfying q µ κ µ = 0 and a scalar mode u. In summary, the metric fluctuations (A9) are parametrized by Using the linear combinations we obtain the York decomposition [102] of the fluctuation field where h = δ µν h µν . We see that the scalar modes s and h in the York decomposition are given as a mixture of the physical scalar mode σ and the gauge mode u. The connection of the physical metric fluctuations f µν to the gauge invariant Bardeen potentials generally used in cosmology can be found in ref. [68]. The decomposition yields Jacobians that read where a prime denotes a subtraction of the zero eigenmode. These contributions are taken into account by introducing auxiliary fields where χ µ is a real bosonic vector field, (φ, ϕ) are complex bosonic scalar fields and (ζ µ , ζ µ ) are vector anticommuting ghosts.
In flat space the matrix of second functional derivatives Γ (2) k becomes block diagonal in the different representations of the Lorentz group, e.g., t µν , κ µ and the scalar fields (φ, σ, u). We will see that for the physical gauge, β = −1 and α → 0, it also becomes block diagonal in the physical fluctuations and gauge fluctuations. Thus the scalar sectors decouple into separate sectors of (φ, σ) and the gauge mode u. Finally, for a vanishing expectation value of φ the physical scalar sector becomes also block diagonal since σ and φ belong to different representations of the Z 2 symmetry. This reflects the different representations of the Higgs doublet and the singlet contained in the metric.
For the t µν -mode, we get where the TT-projection operator reads The Hessian for κ µ is given by with P (v) the projection operator on the vector mode, In the (σ, u, φ)-basis, the Hessian for the scalar modes becomes where the spin-0 gravitational part is given by the 2 × 2 matrix The choice of the gauge parameter β = −1 eliminates the off-diagonal terms in the matrix (A23). Furthermore, the σ-mode becomes independent of α. Thus the whole sector of physical metric fluctuations becomes independent of α. For the choice β = −1 the gauge fixing function (A6) becomes Σ µ =∇ ν h νµ . Therefore, the choice β = −1 is a gauge fixing for which the gaugē ∇ ν h νµ = 0 satisfies the transverse condition within the Faddeev-Popov method.
Taking furthermore the limit α → 0, and therefore realizing the physical gauge, the u-mode, i.e., the lower right element of the matrix (A23), is dominated by q 2 /α. In this limit the finite part of this element (which involves U ) no longer contributes after the inversion of (Γ (2) k +R k ). The same holds for the mixing ∼ U φ/2 with the physical modes in the 3 × 3 matrix (A22). For the physical gauge one therefore deals with a decoupled gauge mode with inverse propagator q 2 /α, and two physical scalars with inverse propagator matrix Finally, for α → 0 the inverse propagator for the gaugevector mode also becomes independent of U We conclude that the total contribution of the gauge modes (vector + scalar) gives a contribution that is independent of φ and independent of all parameters in the effective action. A similar property holds for the ghost contribution and the contribution from the Jacobians, (A17) and (A18). For an explicit computation, we decompose the ghost field as where C ⊥ µ (C ⊥ µ ) is the transverse (anti-)ghost field and C (C) is the scalar (anti-)ghost field. The Hessians for the ghost fields is We have demonstrated the decoupling of the gauge fluctuations for the physical gauge choice for the particular case of a flat background geometry. This property holds actually for a general background geometry, as advocated in ref. [59].

Flow generator from physical fluctuations
Let us next investigate the structure of the flow equation in case of the decomposition presented in the previous subsection. Since the sector of physical fluctuations decouples from the ones for the gauge modes we can treat their contributions separately, provided we choose a cutoff function that respects this decomposition. This is achieved by a block-diagonal cutoff scheme with a physical cutoff R (ph) k that only acts on the fluctuations f µν and φ, and a gauge cutoff ∼ α −1 R (g) k that only involves the gauge modes of the metric. The contributions of the gauge fluctuations can be combined with the contributions of ghosts and Jacobians to a total measure contribution. We write the general structure as with π k the physical mode contribution and η k the measure contribution. For flat spacetime we employ cutoff functions that replace for each mode q 2 by P k (q 2 ) = q 2 + R k (q 2 ). This can be generalized by replacing q 2 with an appropriate covariant differential operator. From the TT-mode t µν one finds a contribution Here q = (2π) −4 d 4 q, and the factor 5 comes from the trace of P (t) , corresponding to the five independent degrees of freedom in t µν . The dimensionless quantity v(ρ) is defined as v(ρ) = 2U (ρ) M 2 p k 2 . (A30) The contribution of the two physical scalar fluctuations takes the form π 0 = 1 2 q∂ t ln 3ρU 2 (A31) where∂ t is the logarithmic derivative k∂ k acting only on M 2 p P k and Z φ P k , not on U or ρ. The mixing effects are small for the range of ρ where In this range π 0 = π 0,g +π 0,φ decouples into two separate parts. The gravitational scalar contributes This contribution is similar to the tensor contribution π 2 . It is suppressed by a factor 1/5, reflecting the single degree of freedom, and a smaller enhancement of the denominator for positive v. The contribution of φ is the standard expression for a real scalar theory. For the full SM it will be supplemented by contributions from the Goldstone directions, the gauge bosons and fermions.

Measure contribution
We next turn to the measure contribution η k . We have already seen that for a physical gauge fixing it is independent of ρ and U . This contribution depends therefore only on the background metric. We will establish that the total measure contribution takes for physical gauge fixing the simple overall form with D 1 and D 0 appropriate differential operators formed with the background metric. For a general metric they take the following forms, Such a simple form has been proposed in ref. [59], based on a direct regularisation of the Faddeev-Popov determinant.
Let us explicitly see that the contribution of the gauge modes and ghosts are given by (A35) in the present setup.
We first look at the contributions from the spin-1 gauge and ghost modes and the auxiliary field: The last two terms on the right-hand side are the contributions from the Jacobian (A17) associated with the gauge mode κ µ . For α → 0, the transverse vector metric fluctuation and the contributions from the Jacobian become δ (1) The contribution from the vector ghost mode takes the form One finds a simple relation between the contributions from the gauge mode and the ghost field [59], (1) k = 2δ (1) k . (A40) The total contribution from spin-1 gauge modes is given by A different normalization of the vector field does not change this result. If we redefine the transverse vector metric fluctuaion asκ µ = q 2 κ µ , the contributions from the Jacobian in (A37) are eliminated. Instead, the contributions from the transverse vector metric fluctuation (A38) should be multiplied by a factor 1/2. Consequently, the contributions from the spin-1 modes yield the result (A41) independently of the field normalization. Next, we discuss the contributions from the spin-0 modes involved in η k . We have The last term on the right-hand side corresponds to the contribution from the auxiliary fields for the Jacobian (A18) associated with the spin-0 ghost mode. Here, we denote the first term on the right-hand side corresponding to the gauge scalar mode of the metric fluctuation as The spin-0 ghost and the auxiliary fields give Within the spin-0 modes, a relation similar to (A40) holds, i.e., The total spin-0 contribution from the gauge and the ghost modes is Again, we may redefine the gauge scalar mode asũ = u/ q 2 , the contribution from the gauge scalar mode (A43) has to be multiplied by the factor 2. On the other hand, the contributions from the Jacobian corresponding to the gauge scalar mode are modified correspondingly. Therefore, the total contributions from the gauge scalar mode and the contributions from the Jacobian do not change from (A43) and accordingly the relation (A40) holds.
In a flat background, the contributions from the gauge modes are given by with since the differential operators D 1 and D 0 are simply given by q 2 multiplied with appropriate projectors.
(A49) With the approximation (A32), we have π 0 = π 0,g + π 0,φ . In the high-momentum range and for constant M 2 p and Z φ and small v, all fluctuations behave as for massless particles, with contributions ∼ q (∂ t P k )/P k . One can easily count the degrees of freedom: The physical modes (π 2 + π 0,g + π 0,φ ) have 7 = 5 + 1 + 1 degrees of freedom. From these degrees are subtracted the 4 = 3 + 1 degrees of freedom from (η 1 + η 0 ). The remaining 7 − 4 = 3 degrees of freedom correspond to the three propagating modes of the system, namely the two helicities from the graviton and one degree of freedom from the real scalar.
We finally evaluate the explicit form of the flow generator (A28) and (A49) for a flat background and consider a constant scalar field. Dividing out a total volume factor, ζ k generates directly the flow of the effective potential U , ∂ t U =π 2 +π 0 +η 1 +η 0 . (A50) We employ the Litim-type cutoff function [77] for the regulator With this regulator, one can perform the momentum integrations analytically and obtain the explicit form of the beta functions. Verifying that the δ-function in ∂ t R k does not contribute, we can use ∂ t P k = ∂ t R k = 2k 2 θ k 2 − p 2 . The momentum integrations of eqs. (A29), (A31), (A48) yields the flow equation (3) for the effective potential, withη k =η 0 +η 1 . The measure contributions are simply given bỹ (A52)