The Asymptotically Safe Standard Model:
From quantum gravity to dynamical chiral symmetry breaking

Álvaro Pastor-Gutiérrez
Max-Planck-Institut für Kernphysik P.O. Box 103980, D 69029, Heidelberg, Germany
Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

Jan M. Pawlowski
Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung mbH, Planckstr. 1, 64291 Darmstadt, Germany

Manuel Reichert
Department  of  Physics  and  Astronomy,  University  of  Sussex,  Brighton,  BN1  9QH,  U.K.

We present a comprehensive non-perturbative study of the phase structure of the asymptotically safe Standard Model. The physics scales included range from the asymptotically safe trans-Planckian regime in the ultraviolet, the intermediate high-energy regime with electroweak symmetry breaking to strongly correlated QCD in the infrared. All flows are computed with a self-consistent functional renormalisation group approach, using a vertex expansion in the fluctuation fields. In particular, this approach takes care of all physical threshold effects and the respective decoupling of ultraviolet degrees of freedom. Standard Model and gravity couplings and masses are fixed by their experimental low energy values. Importantly, we accommodate for the difference between the top pole mass and its Euclidean analogue. Both, the correct mass determination and the threshold effects have a significant impact on the qualitative properties, and in particular on the stability properties of the specific ultraviolet-infrared trajectory with experimental Standard Model physics in the infrared. We show that in the present rather advanced approximation the matter part of the asymptotically safe Standard Model has the same number of relevant parameters as the Standard Model, and is asymptotically free. This result is based on the novel UV fixed point found in the present work: the fixed point Higgs potential is flat but has two relevant directions. These results and their analysis are accompanied by a thorough discussion of the systematic error of the present truncation, also important for systematic improvements.

§ INTRODUCTION

One of the most challenging open tasks in contemporary high energy physics is its ultraviolet (UV) closure including quantum gravity. In the past three decades, asymptotically safe gravity <cit.> has been established as a viable option for this endeavour, for reviews see <cit.>.

Asymptotically safe matter-gravity systems potentially include the Asymptotically Safe Standard Model (ASSM), the UV-complete unification of the Standard Model (SM) with gravity. This minimal set-up for the fundamental theory of matter and gravity with an absence of new physics implies that our current understanding of particle physics may hold up to the Planck scale. Hence, all masses and couplings converge at an interacting asymptotically safe UV fixed point. The solidified existence of such a minimal set-up would also allow for a systematic and well-controlled extension toward UV-complete Beyond Standard Model scenarios including the potential exclusion of such UV-safe embeddings in specific models.

This highly interesting programme has been set-up and pursued in a number of works concerning the matter content compatible with asymptotic safety <cit.>, as well as the relation to the IR physics <cit.>, and beyond the SM physics <cit.>. The running of SM couplings and full Higgs potentials has also been studied without the inclusion of gravity in similar frameworks <cit.>. While the physical mechanisms in the UV and potential scenarios have been well understood, the control over the mechanisms is still lacking. For example, it is necessary to consider higher-order curvature invariants such as $R^2$ in order to reliably determine a bound on the field content compatible with asymptotic safety <cit.>. Also the gravity contributions to the matter couplings are only partially under control: while it has been well understood that gravity supports asymptotic freedom of the gauge coupling or is vanishing at leading order <cit.>, the gravity contribution to the Yukawa coupling carries more intricacies  <cit.>. Moreover, so far, the full ASSM flows were investigated in perturbative threshold-free RG flows for the matter fields, in particular for sub-Planckian physics, as well as a (classical) $\phi^4$-approximation of the Higgs effective potential. For reliable predictions on the nature of the ASSM, a consistent RG flow  including physical threshold effects is needed.

The ASSM covers physics at vastly different momentum scales ranging from trans-Planckian momenta with asymptotic safety in the UV to sub-Fermi momenta with strongly correlated quantum chromodynamics (QCD) in the infrared (IR). Its parameters or rather the specific UV-IR trajectory are fixed by their experimental values measured in the IR, which also fixes the respective UV fixed point. Importantly, the UV landscape includes several physically distinct fixed-point classes. They range from fixed points with stable asymptotically free matter parts (or shifted Gaußian fixed points), fixed points with fully interacting matter parts over unstable FPs to regimes without fixed points. These differences not only concern the existence, stability and interaction nature of the UV regime, but also the number of relevant directions. This can lead to predictive trajectories with fewer parameters in the matter sector of the ASSM than in the SM. Trivial examples are boundary trajectories between the stable and unstable FP classes as well as the Gaußian FP class with a flat Higgs potential with only one relevant parameter, the Higgs mass.

While many of the SM parameters have little impact on the UV fixed point class, if varied largely about the experimental value, there are few whose precise value and correct IR-UV trajectory has a huge and qualitative impact on both, high energy physics below the Planck scale and the trans-Planckian UV physics. In particular, the values and correct scale dependence of the top quark and Higgs masses have a qualitative impact on the UV fixed point class of the ASSM, as well as the high energy stability of the Higgs potential.

In short, reliable access to even the qualitative physics properties of the ASSM require quantitative control of its physics at all scales. Accordingly, this task requires a systematic, self-consistent approach able to treat non-perturbative physics both in the UV and IR. In the present work, we add to this task by computing the UV-IR flow of the ASSM self-consistently within the functional renormalisation group (fRG) approach. Here, self-consistency refers to two important aspects: Firstly, all coupling parameters, whose flows are computed are fed back to the flow, which leads to full resummations. This property is required for the rapid convergence of physics results and is mandatory for a precision determination of the IR parameters of the ASSM. Secondly, all flows are computed within the renormalisation scheme inherent to the fRG approach. In particular, this scheme incorporates physical threshold effects naturally and self-consistently due to its Wilsonian nature. In terms of standard RG schemes this can be phrased as follows: the fRG scheme leads to independence of the physics results from the RG-point already in relatively simple approximations.

In the present work, we add substantially to this endeavour by considering for the first time a fully consistent fRG system of all couplings of the ASSM. In particular, the physical thresholds of the respective matter and gravity dynamics are included. This is required for reliable and quantitative access to the physics of electroweak spontaneous symmetry breaking and strong chiral dynamical symmetry breaking. Furthermore, we take into account that the Higgs mass parameter is related but not equivalent to the experimentally measured pole mass. We compute the pole mass of the top quark in the present non-perturbative setting and adjust the Higgs mass parameter such that the pole mass takes its experimental value. This is essential for a correct estimate of the metastability scale of the Higgs potential.

Finally, we compute the full effective Higgs potential in the trans-Planckian regime within a high order of the Taylor expansion in the Higgs field. This allows us to reveal the presence of a novel UV fixed point for the Higgs potential that is present for a large parameter range: while the full effective potential is flat, is features two relevant directions. One of them is aligned with the Higgs mass operator, but the operator of the most relevant direction is non-polynomial in the Higgs field. In the current approximation, it is this novel non-trivial fixed point that is connected to the physical SM in the IR; where the parameters are fixed by experimental observables.

In contradistinction, the standard Gaußian fixed point potential features one relevant direction, the Higgs mass. Naturally, for a vanishing coupling of the most relevant non-polynomial operator this FP is embedded in the novel FP as a one-dimensional sub-manifold. Parameter values in this sub-manifold are not connected to the physical SM in the IR: the resulting Higgs and top mass are roughly $2.9$ GeV from the central experimental values. Whether this distance is close or feasible is subject to the evaluation and interpretation of the systematic error of the present approximation. This is but one of the reasons for the rather detailed description of the systematic error estimates in the present work.

The physical thresholds and the dynamics of spontaneous symmetry breaking are specifically important for two interrelated reasons: Firstly, in the absence of any thresholds the infrared sector of the SM is not accessible, and the SM parameters have to be chosen for cutoff values at least above the electroweak scale. Such a procedure requires the identification of momentum and cutoff scales, which typically work qualitatively but not quantitatively.  Secondly, the fRG renormalisation scheme differs significantly from the standard $\overline{\textrm{MS}}$ and MOM schemes used in particle physics, and the use of the respective perturbative $\beta$-functions may only work qualitatively. In particular, the naive identifications of momentum and cutoff scales for physical thresholds such as the electroweak symmetry breaking scale may not be suitable. Indeed, we shall observe in the present computation, that it is nearly off by one order of magnitude. This has a significant impact on the parameter choices and hence on the selected UV-IR trajectory as well as potentially on the fixed point class.

To wrap up, the present work includes for the first time the sub-Planckian physics of spontaneous symmetry breaking, both in the electroweak sector and in QCD. Moreover, all physical threshold effects are included, and both properties are necessary (but not sufficient) for a precision determination of the matter parameters of the ASSM. Finally, we determine the pole mass of the top quark instead of estimating it by Euclidean running masses. This precision determination including that of the pole mass is of qualitative importance for the high energy stability range of the Higgs sector, as well as the UV fixed point physics. For the latter, we find in the present approximation novel fixed point properties: the matter fixed point is asymptotically free (Gaußian or shifted Gaußian), but with a flat Higgs potential with two relevant directions. Our analysis includes, also for the first time, a thorough error analysis of the present truncation, which suggests that the present findings have to be corroborated within systematic extensions of the truncation.

This work is organised as follows. In <Ref> we discuss the fRG approach to the ASSM and detail our approximation. In <Ref>, we discuss the existence, stability, and physics properties of the UV fixed point and the ASSM. This also includes a  part of the systematic error analysis, while most of it is distributed in the work.  In <Ref> we present our results on the full phase structure of the ASSM ranging from the asymptotically safe regime with the Reuter fixed point to the deep IR with QCD and chiral symmetry breaking, summarised in <Ref>. The novel property of a flat effective Higgs potential at the UV fixed point with two relevant directions is discussed in <Ref>. There we also discuss the fixed point landscape for general values of the gravity fixed point values. In <Ref> we evaluate the predictivity of the UV scenario developed in the present work, including an error analysis of the main sources of the systematic error. In <Ref> we conclude with a summary of our results, including a short discussion of some consequences. Most of the technical details have been deferred to Appendices.

§ ASYMPTOTICALLY SAFE STANDARD MODEL

In the Asymptotically Safe Standard Model (ASSM), the coupled SM–quantum-gravity system is fully described as a quantum field theory. The underlying classical action consists of the classical gauge-fixed action of the SM and the gauge-fixed Einstein-Hilbert action of General Relativity,

\begin{align}\label{eq:SM+GravityIntro}
S_\textrm{ASSM}[\phi] = S_\textrm{SM}[\phi]  + S_\textrm{gravity}[\phi_\textrm{grav}]  \,,
\end{align}

for the explicit form see <Ref> (SM)  and <Ref> (gravity).

The field content $\phi$ is given by the gauge fields ${\cal A}_\mu$ of the SM gauge groups, $SU(3)_C \times SU(2)_L \times U(1)_Y$, as well as the matter fields, leptons and quarks, $l, \bar l, q, \bar q$ and the scalar doublet, $\Phi$. The metric field $g_{\mu\nu}$ is split into a flat Euclidean background metric $\bar g_{\mu\nu}=\delta_{\mu\nu}$ and a fluctuation field $h_{\mu\nu}$, which carries the dynamics of the metric,

\begin{align}\label{eq:linearsplit}
g_{\mu\nu}= \delta_{\mu\nu} + \sqrt{16\pi G_\text{N}}\, h_{\mu\nu} \,,
\end{align}

We define the dynamical fluctuation superfield, which also includes the auxiliary ghost fields stemming from the gauge-fixing sector,

\begin{align}\label{eq:SuperfieldSplit}
\phi=(\phi_\textrm{grav},\, \phi_\textrm{SM})\,,
\end{align}

\begin{align}\nonumber
\phi_\textrm{grav}=&\,(h_{\mu\nu}, c_\mu,\bar c_\mu)\,,\\[1ex]
\phi_\textrm{SM}=&\,({\cal A}_\mu,\, {\cal C},\, \bar {\cal C}, \,l, \,\bar l,\, q,\,\bar q,\, \Phi)\,.
\label{eq:SuperfieldGravMat}\end{align}

The matter superfield $\phi_\textrm{SM}$ comprises the gauge and ghost fields of the SM gauge group  U(1)${}_\textrm{Y}\,\times\,$SU(2)${}_\textrm{L}\,\times\,$SU(3)${}_\textrm{C}$,

\begin{align} \label{eq:SuperfieldGauge}
{\cal A}_\mu&=( B_\mu,\, A_\mu^a,\, G_\mu^b)\,,
{\cal C}&= (c^a,\, c^b)\,,
\end{align}

with the hypercharge gauge field $B_\mu$, the weak gauge fields $A_\mu^a$ with $a=1,2,3$, and the gluons $G_\mu^b$ with $b=1,...,8$. The field $\cal C$ contains the respective ghost fields of the weak and strong gauge groups. $\phi_\textrm{SM}$ also contains the three families of quarks $q$ and leptons $l$,

\begin{align} \label{eq:SuperfieldFerms}
l&= (e,\,\nu_e, \,\mu,\, \nu_\mu,\, \tau,\,\nu_\tau)\,.
\end{align}

The full quantum effective action of the ASSM, $\Gamma[\phi]$ contains all interaction terms that are compatible with the symmetries of the ASSM.

§.§ Functional RG for the ASSM

We are interested in the full UV-IR phase structure of the ASSM, ranging from the trans-Planckian asymptotically safe UV regime ruled by gravity to the electroweak (EW) scale with EW symmetry breaking, and finally to the deep IR to strongly correlated QCD with strong chiral symmetry breaking and confinement. For covering this vast range of scales and different physics we use the functional renormalisation group (fRG) approach, which already has proven its applicability in all these regimes. In this approach, an IR regulator is introduced in the path integral, which suppresses quantum fluctuations below a given IR cutoff scale $k$, leading to the respective scale-dependent effective action $\Gamma_k[\phi]$. Then, the full effective action $\Gamma[\phi]=\Gamma_{k=0}[\phi]$ is obtained by successively integrating out momentum fluctuations at the scale $k$. The respective flow equation for $\Gamma_k$, the Wetterich equation <cit.>, reads

\begin{align}\label{eq:FlowEq}
\partial_t \Gamma_k \left[\phi \right] &= \frac{1}{2} \text{Tr}\!\left[\frac{1}{\Gamma_k^{(2)} +R_k } \partial_t R_k \right] ,
\end{align}

\begin{align}\label{eq:DefGn}
\Gamma^{(n)}_{\phi_{i_1}\cdots \phi_{i_n}}[\phi](p_1,...,p_n) =\frac{\delta\,\Gamma[\phi]}{\delta \phi_{i_1}(p_1)\cdots \delta\phi_{i_n}(p_n)}\,.
\end{align}

<Ref> provides us with a full non-perturbative setup, that enables us to access the full UV-IR phase structure of the ASSM.

§.§ Approximations of the quantum effective action

In general, eq:Flow cannot be solved for the full effective action of an interacting theory, leaving aside the ASSM. Hence, we have to approximate the full effective action and its flows. In the present section, we summarise the approximations and the reliability considerations behind them, more details can be found in <Ref>.

To begin with, we aim at an accurate description of the full dynamics of the gauge and scalar sectors, and specifically that of the EW sector. Hence, the scale dependence (average momentum dependence) of all primitively divergent vertices in this sector is taken into account. Within this setup, we can, for the first time, reproduce the EW transition U(1)${}_\textrm{Y}\,\times\,$SU(2)${}_\textrm{L} \to \textrm{U(1)}_\textrm{EM}$ including all threshold effects, starting from the asymptotically safe initial conditions.

§.§.§ Momentum symmetric point approximation

Below the Planck scale, quantum gravity effects quickly decouple and we are left with the quasi-perturbative quantum dynamics of the SM. Its effects are captured well by only considering the average momentum running (with $k$) of the primitively diverging vertices of the SM. To that end we consider general vertices $\Gamma_k^{(n)}(p_1,\dots, p_n)$ defined at symmetric points with

\begin{align}\label{eq:SymPoints}
\left|\frac{p_i^\mu p_j^\mu}{p^2}\right| &=\cos\theta\,.
\end{align}

It has been shown both for strongly correlated QCD <cit.>, and in gravity <cit.>, that the system of flow equations of symmetric point vertices is well-approximated by only feeding back these vertices in the diagrams: effectively these vertices represent a close system of flow equations, and define a good approximation $\Gamma_k^{(\textrm{SP})}[\phi]$ of the full effective action if only being interested in symmetric point physics. Note however, that it can be shown in QCD, that this approximation is limited to vertices and regimes without resonant momentum channels in vertices such as the scalar and pseudo-scalar meson channels in QCD for momentum and cutoff scales $k,p\lesssim 1$ GeV.

We also emphasise that $\Gamma_k^{(\textrm{SP})}[\phi]$ does not suffice to give access to important physics information such as S-matrix elements for generic scattering momenta related to the full vertices $\Gamma^{(n)}(p_1,...,p_n)$. However, the latter can be obtained in a systematic way from $\Gamma_k^{(\textrm{SP})}[\phi]$ by their flows. Importantly, feeding back the full scattering vertices to the flow of $\Gamma_k^{(\textrm{SP})}[\phi]$ leads to subleading effects.

In a final step, we introduce a further approximation to the symmetric-point vertices: they only depend on the symmetric-point momentum, the mass gaps $\vec m^2$ of the ASSM, and the cutoff scale $k$. For cutoff scales above the mass gaps, the (dimensionless) dressings can only depend on the ratio $p^2/k^2$ and we can trade one for the other. This property is also at the root of mass-independent RG-schemes, where (in asymptotic regimes) one can read off the momentum dependence of couplings from the RG-scale dependence. In turn, for cutoff scales below the mass threshold of specific fields, their contribution to the flows and the physics is suppressed, while the remaining flow still only depends on $p^2/k^2$. In consequence, for symmetric-point vertices, the cutoff dependence (at $p=0$) reflects well the momentum dependence with

\begin{align}
k=\alpha\, p\,,
\end{align}

and hence supports a low-order derivative expansion with vertices $\Gamma_k^{(n)}(p=0)$ with an approximate effective action $\Gamma_k^{(\textrm{av})}[\phi]$ of these average vertices. As for the symmetric-point vertices, this expansion holds in the absence of resonant vertex structures or more generally strong momentum and angular dependences of vertices. It is well-tested by now both in the asymptotically safe UV regime of gravity-matter systems <cit.> and also holds true in the SM including QCD <cit.>. We remark that within such an expansion the mass parameter extracted at $p=0$ corresponds to the Euclidean curvature mass and not to the pole mass. However, it has been shown that in the absence of strong momentum dependences of the wave-function renormalisation or anomalous dimensions the two agree well, see <cit.>. This intricate topic is discussed further in <cit.>.

§.§.§ Couplings of the ASSM

In our approximation, all primitively divergent vertices, as well as all wave functions of the SM, are taken into account with scale-dependent dressings. Moreover, we consider an effective Higgs potential $V_{\Phi,\textrm{eff}}$ in a high-order Taylor expansion about the flowing minimum. In short, we parametrise

\begin{align}\label{eq:ApproxShort1}
\Gamma^{(n)}_k(p_1,...,p_n)= \prod_{i=1}^n \sqrt{Z_{\phi_i,k}} \, S^{(n)}(p_1,...,p_n;\vec \lambda_k)\,,
\end{align}

where $S^{(n)}$ are the $n$th derivatives of the classical ASSM action in eq:SM+GravityIntro, augmented with higher-order Higgs-self interactions. In eq:ApproxShort1 the classical couplings are substituted by their running quantum analogues,

\begin{align}\label{eq:ApproxShort2}
\vec \lambda_k=(\vec g_{1,k} \,,\,
\vec g_{2,k}\,,\,
\vec g_{3,k}\,,\,
\vec y_{q,k}\,,\,
\vec y_{l,k}\,,\,
\vec \lambda_{\Phi,k}\,,\,
\vec G_{k}\,,\,
\vec \Lambda_{k})\,.
\end{align}

Here the vector on the coupling indicates that they represent a set of avatars of this coupling originating from different vertex functions, which will be explained below. The $g_i$ with $i=1,2,3$ are related to the hypercharge gauge coupling with $g_1 \equiv \sqrt{5/3}\, g_Y $, the weak gauge coupling $g_2$, and the strong gauge coupling $g_3$. The classical dispersions are augmented with wave functions,

\begin{align}
Z_{\phi,k}=(Z_{{\cal A},k}\,,\, Z_{{\cal C},k}\,,\, Z_{l,k}\,,\, Z_{q,k}\,,\, Z_{\Phi,k}\,,\, Z_{h,k}\,,\, Z_{c,k})\,.
\end{align}

The wave-function factors in eq:ApproxShort1 encode the running of the attached fields, and the coupling parameters  eq:ApproxShort2 are indeed running couplings (and masses), and not only vertex factors. From here on we drop the subscript $k$ on all couplings and wave functions and their scale dependence is implied.

In the Higgs sector, we go significantly beyond the approximation in eq:ApproxShort1, and consider the full Higgs potential within a high-order of a Taylor expansion about the flowing minimum. We parametrise the Higgs doublet $\Phi$ as

\begin{align}\label{eq:Higgs}
\Phi&= \frac{1}{\sqrt{2}}\begin{pmatrix} \mathcal{G}_1 +i \mathcal{G}_2 \\  v +H  +i \mathcal{G}_3  \end{pmatrix} \,,
\rho &= \tr \Phi^\dagger\Phi\,,
\end{align}

where $v$ is the flowing minimum, and $H$ is the (radial) fluctuation Higgs field with a vanishing expectation value $\langle H\rangle =0$, as the latter is explicitly carried by $v$.  The ${\cal G}_i$ with $i=1,2,3$ are the Goldstone modes. Then, the Higgs potential is a function of $\rho-v^2/2$. In the symmetric regime with a vanishing minimum of the potential, $v=0$, we use the parametrisation

\begin{align}\label{eq:HiggsPot}
V_{\Phi,\textrm{eff}}(\rho)= \sum_{n=1}^{N_\textrm{max}} \lambda_{\Phi, 2n} \, Z_\Phi^n \rho^n \,,
\end{align}

where the running mass parameter $\mu_\Phi>0$ and the quartic Higgs self-coupling, already  present in the classical Higgs potential, are

\begin{align}\label{eq:ClassHiggsCouplings}
\mu_\Phi &= \lambda_{\Phi,2}\,,
\lambda_\Phi &= \lambda_{\Phi,4}\,,
\end{align}

and the expansion starts at $n=1$ with the mass term.

In turn, in the broken regime with $v>0$, we parametrise

\begin{align}\label{eq:HiggsPotBroken}
V_{\Phi,\textrm{eff}}(\rho)= \sum_{n=2}^{N_\textrm{max}} \lambda_{\Phi, 2n}\, Z_\Phi^n \left( \rho-\frac{v^2}{2}\right)^{\!n} \,.
\end{align}

The lowest term with $n=2$ encodes the classical Higgs potential.  At the spontaneous symmetry breaking cutoff scale $k_\textrm{SSB}$, below which EW symmetry breaking kicks in, the flow is switched from eq:HiggsPot for $k> k_\textrm{SSB}$ with a flowing $\mu_\Phi$ to eq:HiggsPotBroken for $k<k_\textrm{SSB}$ with a flowing $v_k$.

In the following we will refer to both $\mu_\Phi$, $v^2$ as $\lambda_{\Phi,2}$, and hence the set of coupling parameters of the Higgs potential is always given by $\vec \lambda_\Phi = \{\lambda_{\Phi,2n}\}$ with $n=1,...,N_\textrm{max}$. It is understood that $\lambda_{\Phi,2}=v^2$ is taken in the broken regime while $\lambda_{\Phi,2}=\mu_\Phi$ is taken in the symmetric regime. More details on the EW symmetry breaking and the respective scales can be found in <Ref>.

In the UV-regime we will consider a high-order Taylor expansion with the maximal monomial power

\begin{align}\label{eq:Nmax}
N_{\textrm{max}} = 17\,.
\end{align}

This rather high-order of the Taylor expansion allows us to discuss the convergence of the potential at the UV fixed point. For the sake of simplicity, we do not consider such a general potential but the classical one for scales below $ k\le 10^{17}$. Such non-trivial potentials at the EW and metastability scales will be discussed in detail elsewhere.

The flows of the couplings and wave functions in the approximation are extracted at vanishing momentum, which captures the average momentum dependence as discussed in detail at the beginning of this section. For the details of the projection onto the flow equations for all couplings, see <Ref>.

We close this section with a discussion of the gauge and gravity couplings. The set of coupling parameters eq:ApproxShort2 comprises different avatars of gauge and gravity couplings as required for the flow of different gauge and gravity vertices. For gravity, we consider vertex couplings of the $n$-graviton scattering vertices proportional to the classical tensor structure,

\begin{align}
\Gamma^{(n)}_{h\cdots h} \simeq Z_h^{n/2}\, S^{(n)}_{h\cdots h}(G_{n}\,,\,\Lambda_n)\,.
\end{align}

Note that $G_{n}$ and $\Lambda_n$ are the vertex couplings of the classical tensor structure. While they should not be confused with the Newton coupling $G_\text{N}$ and cosmological constant $\Lambda$, they have the scale and momentum running of couplings. The different $G_n$ and $\Lambda_n$ are related by modified Slavnov-Taylor identities, see <cit.>. In the classical limit or rather classical regimes, the effective action reduces to the Einstein-Hilbert action, and all these couplings agree,

\begin{align}\label{eq:classicalGR}
\Lambda_n&=\Lambda\,.
\end{align}

Within our approximation, we use eq:classicalGR throughout.

We also emphasise that the vertex gauge couplings defined in eq:ApproxShort2 are subject to two-loop universality (in mass-independent RG schemes). However, in the presence of threshold effects and/or non-perturbative physics they differ genuinely. Specifically, we then have to consider avatars of the gauge couplings for different vertices. In the present approximation to the ASSM this concerns the quark-gluon couplings as well as the matter-gauge couplings in the EW sector. For example, the quark-gluon couplings are

\begin{align}\label{eq:quark-gluon}
\vec g_3 = (g_{3, u}\,,\, g_{3, d}\,,\, g_{3, s}\,,\, g_{3, c} \,,\, g_{3, b}\,,\, g_{3, t})\,,
\end{align}

where the field subscripts $q_i=u,d,s,c,b,t$ indicate the quarks in the respective quark-gluon vertex. We consider the strong isospin symmetric approximation with

\begin{align}
\end{align}

for all scales. Moreover, for momentum scales beyond the top threshold, all avatars converge towards a unique strong vertex coupling, $g_{3,q_i}=g_3$. In turn, in the presence of non-perturbative physics and threshold effects the two-loop universality of gauge couplings (in mass-independent RG schemes) does not hold anymore, and the matter-gauge couplings do not necessarily agree anymore, and also differ from the pure gauge couplings.

§.§ Strongly coupled UV & IR regimes

In the IR regime for $k, p\lesssim 10$ GeV, the approximation eq:ApproxShort has to be improved both in the gluonic sector for including confinement, as well as in the matter sector for including dynamical spontaneous chiral symmetry breaking, see <cit.>. Here, the epithet dynamical refers to the fact, that the quasi-goldstone modes of strong spontaneous chiral symmetry breaking in QCD are effective IR degrees of freedom, the pions, and not a fundamental field such as the Higgs field $\Phi$ in the ASSM. In turn, for more quantitative access to the asymptotically safe regime, momentum dependences for the pure gravity sector are required, see <cit.>. The two asymptotic regimes are discussed below in <Ref>.

§.§.§ Confinement and strong chiral symmetry breaking

At momentum scales $p, k\lesssim 10$ GeV, strongly correlated IR-QCD starts getting relevant and even two-loop perturbative approximations successively lack reliability, for a detailed discussion see <cit.>. However, in this regime, we can resort to results in functional approaches for 2 and 2+1 flavour computations that meet lattice 2 and 2+1 flavour benchmarks in the IR regime of physical QCD, see in particular <cit.>. Hence, below an interface scale $k_\textrm{inter}$ with

\begin{align}\label{eq:kinter}
5\,\textrm{GeV} \lesssim 	k_\textrm{inter} \lesssim 15\,\textrm{GeV}\,,
\end{align}

we use IR-QCD flows as an external input in our system of ASSM flow equations: it has been shown in <cit.> that below $k\approx 10$ GeV two-loop resummed approximations successively lose their reliability, a -weak- limit being given by $k\approx 5$ GeV. In turn, for $k\gtrsim k_\textrm{inter}$, the approximation to the ASSM flows discussed here suffices. Consequently, we use the stability of the results under a variation of $k_\textrm{inter}$ as a consistency and reliability check of the interface flows, see <Ref>.

Specifically, we utilise results from <cit.> as the approximation of the respective computation resembles most that used here, and the results can be used directly. This concerns the running of the quark masses including dynamical chiral symmetry breaking, as well as the running of the dressing of the gluon propagator, whose mass gap is related to the physical mass gap of QCD.

In terms of the correlation functions considered here, this amounts to using the gluon dressing $Z_{A}$ or rather the anomalous dimension

\begin{align}\label{eq:etaA}
\eta_{A}=-\frac{\partial_t Z_{A}}{Z_{A}}\,,
\end{align}

from <cit.>. This implements the physical gapping of glue interactions for small momentum scales and suffices to describe the respective decoupling of the glue sector from the matter sector semi-quantitatively. It has been shown in <cit.> that the anomalous dimension is well-described as a function of $\alpha_s$ and the mass gaps of the theory. This allows us to write the full anomalous dimension as

\begin{align}\label{eq:ASSSM-2+1}
\eta_{A}^{(\textrm{ASSM})}\approx  \eta_{A}^{(2+1)}\left( \vec \alpha^{(\textrm{ASSM})}_{s}, \vec {\bar m}^2\right) + \eta^{(c,b,t)}_A\,,
\end{align}

where $\vec \alpha^{(\textrm{ASSM})}_s$ is the vector of all strong fine structure constants derived from different vertices, see eq:Avatarsalphas in <Ref>. The vector $\vec {\bar m}^2$ is the vector of all mass gaps in the QCD sector, see eq:etaapprox+barm. The second term on the right-hand side, $\eta^{(c,b,t)}_A$, comprises the contributions from the heavier quarks $c,b,t$. The relation eq:ASSSM-2+1 works quantitatively for mapping the anomalous dimension from pure glue to two-flavour QCD, and from two flavour QCD to 2+1 flavour QCD, see  <cit.>. The approximation becomes better for larger $N_f$ and heavier quarks.

While the use of the relation in eq:ASSSM-2+1 is required for a quantitative description of the full IR dynamics of QCD, for the applications in the present work it suffices to use a semi-quantitative approximation derived in <Ref>, see eq:ASSSM-2+1LinearApp,

\begin{align}
\partial_t Z_{A}^{(\textrm{ASSM})}&\approx \partial_t Z_{A}^{(2+1)} - \eta^{(c,b,t)}_A Z_{A}^{(\textrm{ASSM})} \,. \label{eq:ASSSM-2+1Linear}
\end{align}

where $Z_{A}^{(\textrm{ASSM})}$ is determined by a UV-IR consistency condition at the interface scale $k_\textrm{inter}$ in the range eq:kinter, see eq:ZAinter. In <Ref> it is also discussed, that the results show a small dependence on $ k_\textrm{inter}$ within this range, see <Ref>.

Spontaneous chiral symmetry breaking requires the inclusion of the resonant scalar–pseudo-scalar four-quark interaction channel as in <cit.>. Effectively this leads to an additional mass contribution to the quark masses,

\begin{align}\label{eq:ConsitutentMasses}
m_q &= \frac{y_{q} \,v}{\sqrt{2}}
&& \rightarrow &
\frac{y_{q} \,v}{\sqrt{2}} &+ \Delta M^\textrm{const}_{q}\,.
\end{align}

The term $ \Delta M^\textrm{const}_{q}$ is taken from the results for the $u,d$ quarks in <cit.>, computed in a strong isospin symmetric approximation: $y_u=y_d$. Then, the QCD contribution is given by the full physical constituent quark mass $M^\textrm{const}_{q}$ without the current quark contribution, $\Delta M^\textrm{const}_{q}=M^\textrm{const}_{u}-y_u v$, where $y_u v$ is evaluated at $k=0$. We also use that the resonant interaction channel does not directly enter the flows of the other parameters considered in our approximation.

At large cutoff scales $k\gtrsim 10$ GeV, the approximation used in <cit.> matches that used here, so the QCD part of the present ASSM investigation flows into strongly correlated QCD. In combination this allows us to flow the ASSM down to $k=0$, including confinement and strong chiral symmetry breaking, leading to a semi-quantitative IR closure of the ASSM.

§.§.§ Asymptotically safe UV regime

In the asymptotically safe regime for $k\gtrsim M_\textrm{Pl}$, we utilise the results of <cit.> where pure gravity flows with momentum dependences of propagators and vertices at a symmetric point have been considered. We emphasise that the momentum dependences of the vertices imply the inclusion of higher-order curvature invariants and also the difference between background and fluctuation correlation functions is resolved.

Here, we consider the momentum-dependent couplings

\begin{align}\label{eq:GravCouplings}
g_{h,n}(p)&=G_{h,n} k^2\,,
\lambda_{h,\, n} &=\Lambda_{n}/k^2\,,
\end{align}

of the curvature tensor structures $(\sqrt{g}\, {\cal R})^{(n)}$ and the volume-form tensor structures  $(\sqrt{g} )^{(n)}$ respectively. They are part of the $n$-point vertices $\Gamma^{(n)}_{h\cdots h}$ of the fluctuation graviton $h$, and the present approximation is built upon the assumption of their dominance over other tensor structures in the vertices. While reminiscent of Newton coupling and the cosmological constant, these coupling parameters are vertex couplings and should not be confused with the former physical observables, see <cit.>.

Moreover, for the fixed-point analysis, we take into account the momentum-dependent wave functions of the fluctuation graviton, $Z_{h}(p)$, and the gravity ghost, $Z_{c}(p)$.

We identify the avatars of the vertex couplings, see eq:classicalGR, and compute the flow of

\begin{align}  \label{eq:GravCouplings1}
Z_{h}&\,,Z_c \,,
g_h &= g_{h,3}\,,
\mu_h&=-2 \lambda_{h,\, 2}\,,
\end{align}

where all the avatars of the Newton coupling are identified with that of the three-point vertex. In <cit.>, it has been shown that they are of similar size. The flow of $g_h$ and the anomalous dimensions are evaluated bilocally between $p=0$ and $p=k$. The couplings of the volume form tensor structures are of sub-leading importance and we use $\lambda_{h,\, n\geq 3}\approx 0$. This approximation has been chosen due to their small fixed-point values for $n=3,4,5$.

In <cit.>, it has been shown that matter-gravity systems admit an effective universality and a close perturbative behaviour. This entails that we can (approximately) identify the different gravity-matter vertex couplings of the $n$ matter fields $\Phi_{i_1},..., \Phi_{i_n}$ with $m$ gravitons $h$ with the uniform one in pure gravity, $g_h$,

\begin{align} \label{eq:eff-uni}
g^{\ }_{\Phi_1\cdots\Phi_i \,h\cdots h} =g_h\,,
\end{align}

also used for the back coupling of matter to gravity. We only consider gravity-matter couplings that arise from the dispersion of the matter fields (kinetic and mass term), that is two identical or conjugate matter fields and one to three fluctuating gravitons.

In summary, this approximation is informed by the results on the momentum dependence of vertices and propagators in <cit.> and the effective universality and close perturbative behaviour of matter-gravity couplings. It hence reflects the state-of-the art approximations for matter-gravity systems in the literature, except for gravity-induced higher order couplings, see, e.g., <cit.>. The respective extension will be considered elsewhere.

A final important novel ingredient is the non-trivial dimensionless Higgs potential $u$ considered in the fixed-point analysis. It derives from eq:HiggsPot by rescaling all dimensionful quantities by respective powers of the cutoff scale $k$. Restricting ourselves to the  parametrisation of the Higgs potential in the symmetric regime, eq:HiggsPot, we are led to

\begin{align}\label{eq:FPVHiggsPot}
u(\bar\rho ) = \frac{V_{\Phi,\textrm{eff}}}{k^4}=  \sum_{n=1}^{N_\textrm{max}} \,\bar \lambda_{\Phi,2n} \, \bar \rho^n\,,
\end{align}

with $N_\textrm{max}=17$, see eq:Nmax. The dimensionless radial Higgs field $\bar \rho$ and  couplings $\bar\lambda_{2n}$ are given by

\begin{align}\label{eq:barrho}
\bar\rho &= Z_\Phi \frac{ \tr \Phi^\dagger \Phi}{k^2}\,,
\bar \lambda_{\Phi,2n} &= \frac{\lambda_{\Phi,2n}}{ k^{4-2n}}\,.
\end{align}

In eq:barrho, we have also absorbed the wave-function factors $Z_\Phi^n$ into the definition of $\bar\rho$, for more details see  <Ref>. This allows us to map out the non-trivial UV phase structure of the ASSM with unstable potentials, trivial Gaußian potentials with one relevant direction, non-trivial flat potentials with two relevant directions, and stable as well as semi-stable potentials in <Ref>.

§.§ Intermediate high energy regime

For momentum scales above $k_\textrm{inter}\approx 10$ GeV, the fermionic matter sector hosts all three families of quarks and leptons. We use the fact that for cutoff scales $k\gtrsim 10$ GeV only the $t$ quark experiences a threshold effect due to its mass, while the other quarks, $q_l=d,u,s,c,b$, and all leptons $l$ can be assumed to be quasi-massless. Hence, for the sake of simplicity, we do not consider the running of the gauge couplings separately as they agree for $k\gtrsim 10$ GeV, but identify all couplings with

\begin{align} \label{eq:GaugeHighEnergy}
\end{align}

The approximation eq:GaugeHighEnergy ensures that all fields have the required universal running above the respective thresholds, and settle at their correct IR value at $k=0$. Below the mass threshold $m_{\phi_i}$ of a given field $\phi_i$, the running of its couplings is approximated with that of the lightest fields of the same type. This guarantees that the light fields have the correct running above the threshold, and is of sub-leading consequence for the dynamics triggered by the heavier field $\phi_i$: all diagrams with internal lines of the field $\phi_i$ are suppressed with powers of the running scale $k$ over its mass, $k^2/m^2_{\phi_i}\to 0$. Consequently, these contributions are suppressed.

A similar approximation could be applied to the set of Yukawa couplings. We have refrained from using it here, as quantitative access to the top Yukawa coupling is chiefly important for an accurate determination of the top pole mass to the per cent level. In conclusion, we include the running of all Yukawa couplings separately.

§.§ Wrap-up of the approximation

In summary, the approximation of the full effective action of the ASSM, is given by the classical one with running couplings and wave functions and a full effective Higgs potential within a high-order Taylor expansion in the trans-Planckian regime. In the asymptotic IR and UV regimes, we also take into account momentum-dependences for the gravity part in the asymptotic safety regime and for QCD in the strongly correlated IR regime. Then, the flow of the dynamical fluctuation parameters is solved self-consistently, the running parameters are fed back into the flow and importantly, we consider all physical threshold effects. The flow solved here is RG-consistent <cit.>, and can be systematically improved.

§ THE ASSM: EXISTENCE & STABILITY

A highly non-trivial aspect of general gravity-matter systems and the ASSM, in particular, is the reliability of predictions in the asymptotically safe regime. A necessary but not sufficient reliability criterion is the requirement that results are stable under changes of the regularisation and improvements of the approximation and also pass benchmark tests in simpler subsystems. It has been argued in <cit.> that the Reuter fixed point is always present for minimally coupled matter systems. Therefore, any approximation to the full system of flow equations that do not accommodate this property lacks full predictive power. In <cit.>, this was illustrated in the example of Yang-Mills gravity system. It was shown that the presence of the required Reuter fixed point including asymptotically free Yang-Mills theory seemingly depends on the regulator choice. Moreover, the sign of the graviton contributions of the gluon anomalous dimension $\eta_{a}(p)$ depends on the value of the graviton mass parameter as well as the momentum $p$.

In the current work, we extend this analysis to the full ASSM and provide a reliability analysis of the UV regime. We are interested in the scenario where the marginal matter couplings run into the asymptotically free UV fixed points and hence pure matter contributions are subleading in the scaling regime around the fixed point. Then, the system shows competing effects between graviton tadpole diagrams and diagrams with more vertices. These competing effects originate in the inherent momentum dependences of the matter-graviton vertices and they complicate the determination of the sign of the $\beta$-functions due to the non-trivial momentum dependence. The latter also implies that a derivative expansion about vanishing momentum has to be done with care. For works including momentum dependences see <cit.>, and the recent review <cit.>. An expansion in powers of curvature invariants as is done in most applications of heat-kernel methods is tantamount to a derivative expansion at vanishing momenta, see <cit.>. Consequently, the same reliability arguments apply and to access the full momentum dependence requires the inclusion of invariants with full covariant momentum dependence, see <cit.>. Most heat-kernel computations additionally use the background-field approximation which is lifted in the present fluctuation approach.

Most expansion schemes at work in asymptotically safe gravity explicitly or implicitly use expansions about specific backgrounds, in most cases the flat background. An optimal choice for such a background would be the (minimal) solution of the equations of motion, leading to an on-shell expansion. Notably, the flat background, or equivalently an expansion in powers of curvature invariants in the background field approximation, is an off-shell expansion.  Off-shell expansion schemes may show an unphysical behaviour of the system at least in low orders of the expansion scheme if the off-shell expansion point is too far away from the on-shell background. A further intricacy is the fact, that commonly used expansions about a given background have a finite radius of convergence and one may see convergence towards an unphysical behaviour. Finally, the theory may not exist for all possible metric backgrounds. A respective analysis requires the evaluation of the fixed point and the flow for all backgrounds. In the fluctuation approach the computation of background-dependent vertices was initiated in <cit.>, for respective works in the background field approximation see e.g. <cit.>.

In the present case, we do not expect the approximation to provide stable results for general background fields but only in the vicinity of the equations of motion (on-shell). Even though the present approximation together with the accompanying stability checks is qualitatively more elaborate than those used in previous works we expect instabilities for far off-shell configurations.

§.§ Beta functions of marginal matter couplings

In the following, we restrict our analysis to the flow of the three-point couplings $\lambda_i=(g_1, g_2, g_3, y)_i$ in the vicinity of their Gaußian fixed point. The UV  properties of the other avatars of the gauge couplings as well as the other Yukawa couplings follow similarly. We drop the subleading pure matter contributions, and we arrive at

\begin{align}\label{eq:flowASmatter-flow}
\partial_t \lambda_i({\bf p})	=\beta_{\lambda_i}^{\textrm{grav} }({\bf p})\,,
\end{align}

where ${\bf p}=(p_1,p_2)$ are two of the three independent four-momenta of a three-point function. The gravity contributions to these $\beta$-functions have been previously computed. For example, the gravity contribution to the gauge coupling was computed within the fRG in <cit.> and within perturbation theory in  <cit.>. Here we go beyond previous fRG works since we derive this contribution for the first time from the gauge-quark vertex. The gravity contribution to the Yukawa coupling was computed in <cit.> and we are in agreement with the results from, e.g., <cit.>. The gravity contribution to the quartic scalar coupling was studied in <cit.>. The right-hand side of eq:flowASmatter-flow reads

\begin{align}	\label{eq:flowASmatter-defbeta}
\beta_{\lambda_i}^{\textrm{grav} }({\bf p}) =\eta^{\textrm{grav} }_{\lambda_i}({\bf p}) \, \lambda_i({\bf p}) + F_{\lambda_i}^{\textrm{grav} }({\bf p})\,,
\end{align}

\begin{align}\label{eq:flowASmatter-defeta}
\eta^{\ }_{\lambda_i}({\bf p})=\frac12 \sum_{j_i} \eta_{\phi_{j_i}}(p_{j_i})\,.
\end{align}

<Ref> comprises the sum of (1/2 of) the momentum-dependent anomalous dimensions of the fields $\phi_{j_i}(p_{j_i})$ whose scattering is described in the vertex at hand, and $\eta^{\textrm{grav}}_{\lambda_i}({\bf p})$ is the matter-gravity part.

The second term in eq:flowASmatter-defbeta, $F^\textrm{grav}_{\lambda_i}$, stands for the diagrams of the vertex flow, see also <Ref>. We can distinguish two classes of diagrams, the class of matter graviton tadpoles / polarisation diagrams (tp), and genuine three-point function triangle diagrams (tri). The vertices in these diagrams depend on more momenta than the original couplings, for example, the two-graviton–two-quark–gluon vertex depends on four independent four-momenta. In the present approximation, these momentum dependences cannot be fed back properly. In the tadpole diagrams, the two graviton legs have the momenta $q$ and $-q$ and therefore we identify $\lambda_i({\bf p},q,-q) = \lambda_i({\bf p})$, and similarly for the polarisation diagrams where always two matter legs have the momenta $p_1$ and $p_2$. This leads us to

\begin{align}\label{eq:Flowtops}
F_{\lambda_i}^{\textrm{grav} }({\bf p}) =  F_{\lambda_i=1}^{\textrm{tp}}({\bf p})\,\lambda_i({\bf p})+ F_{\lambda_i}^{\textrm{tri} }({\bf p})\,,
\end{align}

where the tadpole and polarisation diagrams only include the dressings $\lambda_i({\bf p})$ as a prefactor in the involved graviton-matter vertices. In turn, the triangle diagrams include the genuine matter three-point vertices, and the loop momentum runs through the coupling, and hence this coupling cannot be pulled out in front of the diagram. In these diagrams, we can for example choose to identify the momenta of the coupling with the internal momenta $q_1=q, q_2=-p_1-q$, leading to

\begin{align}\label{eq:flowASmatter-flow-couptriangle}
\left.	\lambda_i(q_1,q_2)\right|_{\rm tri} \stackrel{{\bf p}\to 0}{\longrightarrow} \lambda_i(q,-q)\,.
\end{align}

This also entails that the coupling at an exceptional momentum with $p_1=0$ is relevant for this investigation, and not that at the symmetric point.

The coupling $\lambda_i({\bf p})$ in eq:flowASmatter is the renormalisation group invariant dressing of the vertex at hand. In the flow equation approach, respective couplings are typically evaluated at a symmetric point: feeding back only this dressing in an average momentum approximation can be shown to lead to quantitative results in the absence of resonant interaction channels, for more details see <cit.>. Further interesting couplings are defined at exceptional momenta with soft momenta for one of the fields.

In order to investigate the scaling regime in the vicinity of the UV fixed point for $k\to\infty$, we evaluate eq:flowASmatter at a fixed momentum ${\bf p}$. The dimensionless quantities in eq:flowASmatter are functions of ${\bf p}/k\to 0$ and we can safely use this limit for all fixed momenta, and hence the right-hand side of eq:flowASmatter-flow reduces to $\beta_{\lambda_i}^{\textrm{grav} }=\beta_{\lambda_i}^{\textrm{grav} }({\bf p}=0)$, with

\begin{align}\label{eq:flowASmatter-flow-Approx1}
\beta_{\lambda_i}^{\textrm{grav} }=\Bigl(\eta^{\textrm{grav} }_{\lambda_i}  +F_{\lambda_i=1}^{\textrm{tp} }+ \gamma^{\textrm{tri}} F_{\lambda_i=1}^{\textrm{tri} }\Bigr)\, \lambda_i\,,
\end{align}

\begin{align}\label{eq:trianglezero}
F_{\lambda_i=1}^{\textrm{tp/tri}} &= \left. F_{\lambda_i}^{\textrm{tp/tri} }\right|_{\lambda_i({\bf p})=1}(0)\,,
\gamma^{\textrm{tri}} &= \frac{ F_{\lambda_i}^{\textrm{tri} }(0)}{ \lambda_i \,F_{\lambda_i=1}^{\textrm{tri} }}\,.
\end{align}

The first two terms in the parenthesis on the right-hand side of eq:flowASmatter-flow-Approx1 only involve the anomalous dimension at vanishing momentum $\eta^{\textrm{grav} }_{\lambda_i}=\eta^{\textrm{grav} }_{\lambda_i}(0)$, the tadpole and polarisation diagrams $F_{\lambda_i=1}^{\textrm{tp} }$ and the coupling $\lambda_i=\lambda_i(0)$ at vanishing momentum. In contradistinction, the coupling $\lambda_i(q_1, q_2)\to \lambda_i(q, -q)$ in the flow term $F_{\lambda_i}^{\textrm{tri} }(0)$   is integrated over loop momenta $q^2$. Hence, it is sensitive to all momenta smaller than the cutoff scale as $q^2\lesssim k^2$. This is taken into account with the prefactor $\gamma^{\textrm{tri}}$ which is cutoff-independent in the scaling regime.  This relative prefactor changes the balance between the triangle diagrams and the tadpole and polarisation diagrams. It may increase or suppress the contribution of the triangle diagram, subject to the momentum dependence of the coupling $\lambda_i(q,-q)$ in the vicinity of the fixed point. The integrand of the triangle diagram involves a factor $q^3$ from the radial momentum integration and a factor $q^4$ from the vertices, leading to the factor $(q/k)^7$ in the integrand. This suppresses very efficiently the small loop momentum regime of the vertex dressing $\lambda_i(q,-q)$ in comparison to the regime with $q^2\approx k^2$ in the triangle diagram.

<Ref> is readily solved which leads us to

\begin{align}
\lambda_{i,k_1} =\lambda_{i,k_2} \,\exp( \int_{k_2}^{k_1} \frac{\mathrm dk}{k}\bigl[\eta^{\textrm{grav} }_{\lambda_i}  +F_{\lambda_i}^{\textrm{tp} }+\gamma^{\textrm{tri}} F_{\lambda_i}^{\textrm{tri} }\bigr])\,,
\end{align}

where both cutoff scales $k_1$ and $k_2$ are in the scaling regime. The present Gaußian fixed point scenario for the matter couplings $\lambda_i$ is sustained for

\begin{align}
\eta^{\textrm{grav} }_{\lambda_i}  +F_{\lambda_i}^{\textrm{tp} }+\gamma^{\textrm{tri}} F_{\lambda_i}^{\textrm{tri} } < 0\,.
\end{align}

This concludes the derivation of the UV flows of marginal matter couplings in the ASSM.

§.§ Variations of the regulator and cutoff scales

We are now in the position to access the stability of the Gaußian fixed point as well as the reliability of the results in the present approximation. A necessary condition is the independence of UV fixed-point nature from the chosen regulator. A given approximation typically only works for a given class of regulators. In turn, extreme choices of regulators potentially invalidate any approximations. In this context, we have to take into account that in the present approximation minimally coupled systems do not show the Reuter fixed point for all cutoff choices  <cit.>. Therefore we only expect to see a stable Gaußian fixed point for specific choices of regulators.

We analyse the fixed-point scenario under specific variations of the regulators. For the sake of simplicity, we only consider relative changes of the cutoff scales $k_\textrm{grav}$ for pure gravity and $k_\textrm{mat}$ for matter fields. To that end, we introduce

\begin{align}\label{eq:kgrav-kmat}
k &= \min (k_\textrm{grav}, k_\textrm{mat}) \,,
k_\textrm{mat} &= \gamma_\textrm{mg} k_\textrm{grav}\,.
\end{align}

The standard choice is $\gamma_\textrm{mg}=1$, but the qualitative features of the system should be independent of it. Its choice can be constrained by optimisation arguments in a given approximation: in the present approximation, we have dropped momentum dependences of the effective action. Then, optimal regulators are those that minimise the momentum transfer in diagrams as such a transfer (strong momentum dependences) cannot be captured by the present approximation that relies on $p^2 \approx k^2$ for all momenta. While this implies $\gamma_\textrm{mg}=1$ for different fields of the same kind, this is by far not obvious within a system that either contains both fermions and bosons or shows vastly different anomalous dimensions. By considering a variation of the relative cutoff scale between the matter and gravity sector, we provide a first exploratory study of the stability of the system.

Technically, we proceed as follows: the fixed point analysis is done in dimensionless variables. These variables are defined by multiplying the dimensionful ones with appropriate powers of the gravity cutoff scale $k$. This implies that for $\gamma_\textrm{mg}<1$ the only change in our system of flow equation originates in the matter shape functions and vice versa for $\gamma_\textrm{mg}>1$. For $\gamma_\textrm{mg}<1$, the matter shape functions are given by

\begin{align}\label{eq:RescaledShape}
r_\textrm{mat}(x) =  r_\textrm{flat}\left( \frac{1}{\gamma_\textrm{mg}^2} x \right)\,,\qquad x=\frac{p^2}{k^2}\,,
\end{align}

\begin{align}
r_\textrm{flat}(x)= \left( \frac{1}{x}-1\right)\,\Theta(1-x)\,.
\end{align}

In a rough first estimate, this change introduces relative factors $\gamma_\textrm{mg}^4$ or even higher powers in the diagrams with a scale-derivative $\partial_t R_\textrm{mat}$, if evaluated at vanishing external momentum. The power four is obtained for momentum-independent couplings, the higher power is obtained for  momentum-dependent couplings, triggered by

\begin{align}
\int_0^{\gamma_\textrm{mg} } \!\mathrm d  q \, q^3  ( q^2)^n = \frac{1}{4+2n}  \gamma_\textrm{mg}^{4+2n}\,,
\end{align}

with the highest power $n=2$ for the triangle diagrams in eq:flowASmatter-flow-Approx1. For this estimate we have neglected the subleading effect that the graviton and matter propagators have a non-trivial momentum dependence for $ q^2> 1$ and $ q^2 > \gamma^2_\textrm{mg}$. In summary, the main effect of the choice eq:RescaledShape is the suppression of diagrams with $\partial_t R_\textrm{mat}$ ($\gamma_\textrm{mg}<1$) or $\partial_t R_\textrm{grav}$ ($\gamma_\textrm{mg}>1$). If diagrams are contributing with different sings, this may introduce a change in the sign of given $\beta$-functions, entailing a qualitative change of the nature of the fixed point.

We note that such an enhancement or suppression can also be obtained by varying the graviton mass parameter $\mu_h$: large positive masses  $\mu_h\to \infty$ suppress diagrams with $\partial_t R_\textrm{grav}$ as they contain one further graviton propagator. In turn, for $\mu_h\to -1$, these diagrams are enhanced. As discussed in <cit.>, an on-shell analysis requires a non-trivial metric solution which compensates effects of the mass parameter but also changes the vertices. This implies that the physical consequences of a suppression or enhancement that originates in the value of the mass parameter, should be taken with a grain of salt: they are based on an off-shell background that can easily destroy the fixed-point properties. On-shell backgrounds with constant curvature were calculated in <cit.> within the vertex expansion and the on-shell background curvature turned out to be $\mathcal{O}(1)$ in units of the cutoff scale. This suggests that a natural estimate for the on-shell analysis is given by small values of the graviton mass parameter.

§.§ Stability of the Gaußian matter fixed point

We proceed with the stability analysis of the Gaußian matter fixed point. The pivotal part of the full system of flow equations is the matter-gravity part of the $\beta$-functions: for graviton couplings this part may destabilise the Reuter fixed point, leaving us with no UV closure. In turn, for matter couplings, the matter-gravity part is required for the existence of respective UV fixed points. Therefore the following discussion is restricted to the matter-gravity part as this is already conclusive.

Now we use that in the UV regime the matter-gravity parts of all matter-gauge couplings are identical as are that of all Yukawa couplings

\begin{align}
\beta^\textrm{grav}_{g_i}&\to \beta^\textrm{grav}_g\,,
\beta^\textrm{grav}_{y_{q,l}}&\to \beta^\textrm{grav}_y\,,
\end{align}

and we can restrict our analysis to the pair $(\beta^\textrm{grav}_g, \beta^\textrm{grav}_y)$. For the non-Abelian gauge couplings, the pure matter part of the $\beta$-function is negative, $\beta_{g_{2,3}}^\textrm{mat}\propto g_{2,3}^3$, which entails asymptotic freedom with $g_{2,3}\to 0$. Thus, the gravity part $\beta_{g_{2,3}}^\textrm{grav}\propto g_{2,3}$ dominates the asymptotically safe UV regime and is required to be negative for supporting the Gaußian fixed point with asymptotic freedom.

The pure matter parts of the U(1) coupling is positive $\beta^\textrm{mat}_{g_1} >0$ and hence the matter-gravity parts has to be negative, $\beta^\textrm{grav}_{g_1}<0$. In the Yukawa beta function, the Yukawa interaction contributes positively $\beta^\textrm{mat}_y\propto y^3>0$ while the Yukawa-gauge interaction contributes negatively $\beta^\textrm{mat}_y\propto y \, g^2<0$. For the Gaußian matter fixed point, the matter-gravity parts have to be negative, $\beta^\textrm{grav}_y<0$. In summary, in all cases negative matter-gravity parts are required, $\beta^\textrm{grav}_{g}, \beta^\textrm{grav}_y<0$.

The analysis is facilitated further as we focus on the leading order contribution and neglect the anomalous dimensions stemming from regulator insertions in the diagrams. Then the dimensionless Newton coupling, $g_h$, is simply an amplitude factor in $\beta_g, \beta_y$.  In the limits $\gamma_\textrm{mg} \to 0$ and $\gamma_\textrm{mg} \to \infty$, also the graviton mass parameter only changes the amplitude of the flow as all diagrams left have the same number of graviton propagators. For all other values of $\gamma_\textrm{mg}$, we have a competition of diagrams with one and two graviton propagators and the value of the graviton mass parameter can change the sign of the contribution. For the analysis, we use $\mu_h = 0$. The results for other values $\mu_h$ can be inferred by a simple rescaling of $\gamma_\textrm{mg}$ via $\gamma_{\textrm{mg},\mu_h=0} = (1+\mu_h) \gamma_{\textrm{mg}}$. Accordingly, the fixed point values of the pure gravity sector are irrelevant for the analysis.

Matter-gravity contributions to the gauge beta functions $\beta_{g}^{\text{grav}}$ derived from the gauge-fermion vertex for $g=1$, $g_h=1$, $\mu_h=0$, and different values of $\gamma_\textrm{mg}$. For $\gamma_\textrm{mg}= 0$, $\beta_{g}^{\text{grav}}$ vanishes due to an exact cancellation between the fermion anomalous dimension and the three-point diagrams.

We consider relative rescalings of matter and gravity cutoffs with $\gamma_\textrm{mg}$ defined in eq:kgrav-kmat in the regime

\begin{align}
\gamma_\textrm{mg}\in [0,\infty)\,,
\end{align}

which indirectly allows us to also access variations of the shape functions. As the present approximation does not allow for momentum transfers over large momentum scales, the limits $\gamma_\textrm{mg}\to 0,\infty$ can only support the UV fixed point if matter fluctuations dominate the matter-gravity part ($\gamma_\textrm{mg}\to 0$), or gravity fluctuations dominate the matter-gravity part  ($\gamma_\textrm{mg}\to \infty$). These are two physically different UV scenarios whose existence is accessed in these limits:

(i) For $\gamma_\textrm{mg}\to 0$ discussed in <Ref>, the matter sector is not IR regularised and quantum fluctuations of the matter fields are included at all scales. The matter propagators lack the IR mass introduced by the regulator and hence are enhanced relative to the graviton propagator. Therefore, this limit is a natural one for systems that are dominated by matter fluctuations (matter matters <cit.>).

(ii) For $\gamma_\textrm{mg}\to \infty$ discussed in <Ref>, the gravity sector is not IR regularised and quantum fluctuations of the graviton are included at all scales. The graviton propagator lacks the IR mass introduced by the regulator and hence is enhanced relative to the matter propagators. Therefore, this limit is a natural one for systems that are dominated by gravity fluctuations (gravity rules <cit.>).

The results of <cit.> entail that minimally coupled gravity matter systems should show a gravity-dominated Reuter fixed point apart from other fixed points. Evidently, an ASSM fixed point with a Gaußian or shifted Gaußian matter sector falls into the gravity-dominated class of fixed points, and we shall see in <Ref> that the ASSM fixed point with a Gaußian matter sector is supported in this limit, while the Gaußian matter fixed point is not present in the matter-dominated limit of the matter-gravity part as shown in  <Ref>. Note also, that this limit cannot work for the full system as it accesses the UV sector of the ASSM without UV fluctuations of gravity. This is simply the UV sector of the SM which is UV unstable.

This leaves us with the question in which regime for $\gamma_\textrm{mg}$ we lose the ASSM fixed point. In <Ref> we show that this happens for $\gamma_\textrm{mg}\approx 1$, which in our opinion is non-trivial support for the Gaußian fixed point scenario dominated by gravity fluctuations: stability only in one of the extreme limits $\gamma_\textrm{mg}\to 0,\infty$, while it may show the correct physical behaviour also cast serious doubts on the approximation used. This is discussed in <Ref>.

Matter-gravity contribution to the Yukawa beta function $\beta_y^{\text{grav}}$ for $y = 1$, $g_h=1$, $\mu_h=0$, and different values of $\gamma_\textrm{mg}$. For $\gamma_\textrm{mg}\approx 1.6$, $\beta_y^{\text{grav}}$ turns negative at $p/k\sim 0.5$, and for $\gamma_\textrm{mg}\approx2.55$ we observe a sign change at $p=0$. Larger/smaller values of $g_h$ increase/decrease the amplitude of $\beta_y^{\text{grav}}$.

§.§.§ Matter matters: no matter cutoff with $\gamma_\textrm{mg}\to 0$

All diagrams with $\partial_t R_\textrm{mat}$ drop out. This restates diffeomorphism and gauge consistency in the matter sector of the ASSM in terms of standard matter parts of the Slavnov-Taylor identities. Interestingly in this limit, the $\beta$-function of the minimal background gauge couplings vanishes as has been shown in <cit.>: the non-trivial cancellation of the graviton tadpole diagram and three-point function diagram in the graviton contribution to the background propagator is based on a kinematic identity rooted in diffeomorphism invariance of the gauge sector in this limit as well as the transversality of the kinetic term of the gauge field.

It is a highly non-trivial result of the present work, that this non-trivial kinematic identity at work in the flow for the gluon two-point function in <cit.> extends to the same result for the gauge-fermion coupling,

\begin{align}\label{eq:KinID3poiint}
\beta_g^\textrm{grav}(p=0) = 0\,, \qquad \textrm{for} \qquad \gamma_\textrm{mg}=0\,,
\end{align}

see also the red dot-dashed curve in <Ref>. As for the two-point functions of the gauge fields the identity eq:KinID3poiint is rooted in diffeomorphism invariance of the gauge sector for $\gamma_\textrm{mg}=0$ and transversality of the gauge field.

We emphasise that it is the above combination of diffeomorphism and gauge symmetry leading to this non-trivial result. For the Yukawa couplings, this cancellation between tadpole, polarisation and triangle diagrams is not present and the graviton tadpole dominates. Its contribution to the $\beta$-function is positive and we are led to

\begin{align}\label{eq:betakmat=0}
\beta_y^{\textrm{grav}} > 0\,, \qquad \textrm{for} \qquad \gamma_\textrm{mg}=0\,,
\end{align}

see also the red dot-dashed curve in <Ref>.

In conclusion, the gauge-gravity part $\beta_g^{\textrm{grav}}$, eq:KinID3poiint, supports the asymptotically free Gaußian fixed point of the non-Abelian gauge couplings $g_{2,3}$ and allows for a stable Gaußian as well as unstable interacting fixed point for the U(1) coupling $g_1$. Conversely, the matter-gravity part $\beta_y^\textrm{grav}$, eq:betakmat=0, is positive and no combined fixed point is present. One may speculate that the absence of the kinematic identity in the Yukawa coupling is a truncation artefact that originates in the lack of (modified) diffeomorphism invariance of the present approximation, also missing in other approximations used in the literature.

§.§.§ Gravity rules: no gravity cutoff with $\gamma_\textrm{mg}\to \infty$

All diagrams with $\partial_t R_\textrm{grav}$ drop out. This reinstates diffeomorphism consistency in the pure gravity sector of the ASSM in terms of standard gravity parts of the Slavnov-Taylor identities. In this limit, the graviton tadpole is not present and the other diagrams lead to negative contributions to the $\beta$-function. In summary, we are led to

\begin{align}\label{eq:betakgrav=0}
\beta^\textrm{grav}_{g,y}(0) < 0\,, \qquad \textrm{for} \qquad \gamma_\textrm{mg}=\infty\,,
\end{align}

see also the green dashed curves in <Ref>. This entails, that in the gravity-dominated UV scenario we encounter a Gaußian fixed point for all gauge and Yukawa couplings.

It is important to note that for $\gamma_\textrm{mg}\neq 0$ the absence of the graviton tadpole can also be simulated by a sufficiently large positive graviton mass parameter $\mu_h\gg 0$ or a sufficiently negative cosmological constant $\Lambda\ll 0$ in the background field approximation. This limit suppresses the graviton propagator and leads to an artificial suppression of gravity: on-shell a large graviton mass parameter or large negative cosmological constant does not lead to a suppression of gravity fluctuations, and already classically gravitons are massless in the presence of large curvatures. It is suggestive that this option is a mere truncation artefact, but this has to be investigated in better approximations.

As in the matter-dominated limit discussed in <Ref>, a change of the graviton mass parameter only leads to a change of the amplitude of $\beta_{g,y}$, but does not change the sign. In short, the stability of the Gaußian fixed point in the gravity-dominated scenario persists off-shell.

§.§.§ Crossover regime at $\gamma_\textrm{mg}\approx 1$

So far we have analysed two limits $\gamma_\textrm{mg}=0,\infty$ and differences are expected between these extreme choices. In the gravity-dominated limit with $\gamma_\textrm{mg}\to \infty$, the Reuter fixed point for the gravity couplings exists and all matter couplings have a stable Gaußian fixed point, while the Reuter fixed point is absent in the matter-dominated limit for $\gamma_\textrm{mg} =  0$ and only some matter couplings have a stable Gaußian fixed point.

It is now decisive to investigate the crossover from $\beta_y>0$ to $\beta_y<0$. Remarkably, it takes place in the regime with  $\gamma_\textrm{mg}=\mathcal{O}(1)$: This hints at the fact, that the $\gamma_\textrm{mg}$-dependence is caused by the low order of the approximation rather than being an artefact of an extreme choice of regulators.

For $\gamma_\textrm{mg}$ larger than $\gamma_\textrm{mg}\approx 1.6$ the $\beta$-function first dips below zero at momenta $p\approx k/2$, before it turns negative for $p=0$ for $\gamma_\textrm{mg}> \gamma_\textrm{mg}^\textrm{stab}$ with

\begin{align}\label{eq:gammambThresholds}
\gamma_\textrm{mg}^\textrm{stab}&\approx 2.55\,,
\end{align}

see <Ref>. The value $\gamma_\textrm{mg}^\textrm{stab}$ is very close to unity. This also entails that sign change may also be obtained by an appropriate change of the shape function of gravity and matter fields while keeping $\gamma_\textrm{mg}=1$. In our opinion, this gives a hint for the viability of the stable Gaußian fixed point scenario by the fact that the crossover between the two regimes takes place for $\gamma_\textrm{mg}=\mathcal{O}(1)$.

§.§ Existence of the ASSM

In short, the highly relevant question of the UV stability of the ASSM and its fixed-point properties cannot be answered conclusively in the present approximation. Future analyses should include a discussion of the on-shell or off-shell properties or instabilities. This entails, that we are in high demand for quantitative and qualitative upgrades, and a comprehensive fixed-point analysis should include all possible scenarios. In our opinion, despite the deficiencies of the present analysis, it still provides non-trivial hints for the existence of the ASSM with a stable Gaußian fixed point for all gauge and Yukawa couplings.

In the remainder of this work, we discuss the UV-IR and the fixed-point properties of the ASSM in the present existence scenario. This entails a sub-leading behaviour of the graviton tadpole contributions to the flows of the matter couplings in the vicinity of the UV fixed point. Below the Planck scale, these contributions are also suppressed, and we can safely drop the respective diagrams. Then, the ASSM fixed point persists for all $\gamma_\textrm{mg}$ and we choose $\gamma_\textrm{mg}=1$ for the sake of simplicity. This approximation can be summarised as

\begin{align}\label{eq:ApproxWork}
\gamma_\textrm{mg}=1\,,\qquad\textrm{and} \qquad F_y^\textrm{grav,tad}=F_g^\textrm{grav,tad} \approx 0\,,
\end{align}

i.e., we set the gravity-tadpole contributions to the running of the gauge and Yukawa couplings to zero.

§ SUB-FERMI TO TRANS-PLANCKIAN ASSM

As a first application of the fRG set-up of the ASSM we compute its UV-IR phase structure. This includes the determination of the fundamental parameters by the respective experimental observables that give access to a comprehensive error analysis including both,  experimental and systematic errors.

§.§ Fixing the coupling parameters of the ASSM

The physics trajectory of the ASSM is determined by matching its fundamental parameters to their experimental IR values. These observables have to be determined from $S$-matrix elements at $k=0$ for the respective scattering momenta.  We need to fix the following SM and gravity couplings

\begin{align}\label{eq:ASSM-Parameters}
&(g_1\,,\, g_2\,,\, g_3)\,,
&(\vec y_q \,,\,\vec y_l)\,,
&\vec \lambda_{\Phi}\,,
& (g_h \,, \mu_h)\,,
\end{align}

where we have paired the parameters of the different sectors of the ASSM, the gauge couplings, Yukawa couplings, parameters of the Higgs potential, and the Newton coupling. All SM parameters except the strong coupling $g_3$ and the top Yukawa coupling $y_t$ are determined at $p=0$ for the physical cutoff scale $k=0$.

For the EW gauge couplings, we use the experimental values for $p\to 0$ to fix the electron avatars $g_1=g_{1,e}$ and $g_2=g_{2,e}$ at vanishing momentum

\begin{align}\label{eq:ValuesEW-GaugeCouplings}
g_1&= 0.446\,,
g_2&=  0.618\,.
\end{align}

For the masses (except the top mass) we ignore the difference between the Euclidean curvature masses and the experimental pole masses, $M_{\phi_i}^{(\textrm{pole})}$. This leads us to

\begin{align} 	\label{eq:ValuesHiggscouplings}
\lambda_{\Phi,4}&=0.129\,,
v&= 246\,\textrm{GeV} \,,
\end{align}

for the Higgs self-coupling and Higgs vacuum expectation value, and

\begin{align} \nonumber
y_{b}&= 0.0240\, , &   y_{c}&= 0.00734\,,    \\[1ex]\nonumber
y_{s}&= 5.46\cdot 10^{-4}\, , &   y_{u}&= 1.32\cdot 10^{-5}\,,    \\[1ex]\nonumber
y_{d}&= 2.76\cdot 10^{-5}\, , &   y_{\tau}&= 0.0102\,,   \\[1ex]
y_{\mu}&= 6.06 \cdot 10^{-4}\, , &   y_{e}&= 2.93 \cdot 10^{-6}\,,
\label{eq:ValuesMassesYuk}
\end{align}

for the Yukawa couplings at $p=0$ except for the top Yukawa coupling $y_t$. With the vacuum expectation value $v$ in eq:ValuesHiggscouplings and the Yukawa couplings in eq:ValuesMassesYuk, all curvature masses except the top mass match the experimental pole massed from <cit.>.

The Newton coupling and cosmological constant are given by

\begin{align} \label{eq:IR-grav-param}
G_\text{N} &= (1.22\cdot10^{19}\,\textrm{GeV})^{-2}\,,
\Lambda&\approx 0\,.
\end{align}

It is left to determine the remaining two parameters, $y_t, g_3$. We first discuss the top Yukawa coupling $y_t$. While the identification of Euclidean curvature masses with the respective pole mass is a qualitatively reliable approximation for low-lying masses, see <cit.> for a case study, we cannot use it for the top quark mass or rather its Yukawa coupling $y_t$: the phase structure of the ASSM including its UV asymptotics is very sensitive to its precise value. The experimental pole mass from cross-section measurements is given by <cit.>,

\begin{align}\label{eq:topMassPole}
M_{t,\textrm{pole}}^{(\textrm{exp})}= 172.5 \pm 0.7\,\text{GeV} \,,
\end{align}

with a decay width of

\begin{align}\label{eq:topMassWidthPDG}
\Gamma_{t,\textrm{pole}}^{(\textrm{exp})}= 1.42^{+0.19}_{-0.15}\,\text{GeV} \,.
\end{align}

In <Ref>, we analytically compute the time-like momentum dependence of the top mass function $M_t(p)$ at vanishing cutoff scale, $k=0$. The access to the analytic momentum dependence is rooted in the simple one-loop exact structure of the flow of the top quark propagator. The present determination includes full resummations of diagrams due to the iterative structure of the flow, and hence the result is very stable under further improvements of the approximation.

This allows us to determine the pole mass of the top as a function of the Euclidean mass parameter $m_t$, and for the experimental value of the pole mass, eq:topMassPole, we arrive at the Euclidean mass parameter at vanishing cutoff scale, $k=0$,

\begin{align}\label{eq:topMassCurvature}
&&\leftrightarrow &
y_{t}&=0.950^{+0.005}_{-0.001}\, ,
\end{align}

see <Ref>. Having fixed the Euclidean mass parameter, the decay width is a prediction and we find

\begin{align} \label{eq:topWidthTheo}
\Gamma^{(\textrm{theo})}_{t, \textrm{pole}}=  1.72^{+0.09}_{-0.41} \,  \text{GeV} \,,
\end{align}

in quantitative agreement with the experimental value eq:topMassWidthPDG from the PDG. Note, that the error in eq:topMassCurvature constitutes a systematic error estimate while the error in eq:topWidthTheo only describes the relative weighting of QCD and non-QCD contributions, see <Ref> for details.

The error in $m_t$ in eq:topMassCurvature  is dominated by the uncertainty of the value of the strong coupling $g_3$. To begin with,  it cannot be determined at $k,p=0$: In the deep IR, QCD is strongly correlated and the strong coupling cannot be defined by scattering experiments. For this reason experimental values are only present at and above the $\tau$-scale $p=1.77$ GeV, see <cit.>. Moreover, the definition of the strong coupling in the IR is subject to threshold effects and non-universality beyond two loops, see the discussion in <Ref>. For this reason, we fix its value at the perturbative $Z$-scale. Here, its experimental value ($pp/p\bar p$) is given by <cit.>

\begin{align}\label{eq:alphaMSbarMZ}
g_{3,k=M_Z}^{\overline{\textrm{MS}}}& \approx 1.22
&&  \leftrightarrow &  \bar\alpha_s&:=\frac{\left(g_{3,k=M_Z}^{\overline{\textrm{MS}}}\right)^{\!2}}{4\pi}\approx 0.118 \,,
\end{align}

matching the strong coupling in an $\overline{\textrm{MS}}$ or MOM-scheme. In the 2+1 flavour QCD computation in <cit.>, it has been shown that the momentum-dependent (up-)quark-gluon coupling $g_3(p)$ at $k=0$ agrees quantitatively with that at $g_{3,k}(p=0)$ for $k=p$ for perturbative and semi-perturbative momenta, $k\gtrsim 4$ GeV. This has been checked with lattice results as well as the momentum-dependent fRG and DSE results from <cit.> and relates to the logarithmic running of the (up-)quark-gluon coupling in this regime far from the up-quark threshold.

It has been shown in <cit.>, that the MOM${}^2$ RG-scheme, used in the fRG, leads to an enhancement of the avatars of the running gauge coupling $\alpha_s(p)$. In pure Yang-Mills, the respective enhancement factor for $\alpha_s$ is approximately $4/3$, <cit.>, dropping to an enhancement factor of approximately 1.2 in the 2+1 flavour case (measured in the regime between 10-40 GeV). A linear extrapolation from these two values can be done for the ASSM quark content leading to a scaling factor of 1.07. This analysis suggests identifying

\begin{align}\label{eq:g3}
g_{3,k=M_Z} = \sqrt{1.07\,( 4 \pi  \bar\alpha_s)}= 1.26 \,.
\end{align}

However, being short of a full analysis, we show results within an estimate of the systematic error of the choice in eq:g3 with

\begin{align}\label{eq:alphasIntervallMain}
\alpha_{s,k=M_Z}\in \bigl[\bar\alpha_s  ,\, 1.10 \,\bar\alpha_s \bigr] \,,
\end{align}

see also eq:alphasIntervall  and the detailed discussion in  <Ref>. There and in <Ref> further estimates and consequences are discussed.

In particular, the above scale setting of the strong coupling, and its uncertainty, has a sizeable impact on the top Yukawa coupling and therefore in the high energy development of the ASSM flows. For example, the seemingly small uncertainty of the strong coupling has a huge effect on the metastability scale $k_\textrm{meta}$ at which the quartic Higgs-self coupling turns negative. We discuss this point in detail and perform a systematic error analysis in <Ref>.

§.§ The ASSM trajectory

Scale dependence of the gauge couplings, the top Yukawa coupling and the Higgs self-coupling in the ASSM. We have omitted here the flow of the remaining Yukawas and the flow of the curvature mass and vacuum expectation value. In the UV regime beyond the Planck scale (indicated by a violet vertical line), all matter couplings approach the Gaußian fixed point, including a flat Higgs potential with two relevant directions. Below the $k_{\text{SSB}}$ scale (indicated by a red vertical line), the top threshold is clearly visible in the  $y_t,\,\lambda_{\Phi, 4}$ and $g_2$ flows. Moreover, for the latter two the Higgs, $W^\pm$ and $Z^0$ bosons decoupling also play a quantitative role. For scales above $k=10^{17}$ GeV the flow of $\lambda_{\Phi, 4}$ is considered within the full Taylor expanded potential.

The scale-dependence of the ASSM parameters is depicted in <Ref>. We span momentum scales from $k\to 0$ and $p=0$ in the deep IR with chiral symmetry breaking and confinement to $k\to\infty$ far beyond the Planck scale in the asymptotically safe UV fixed-point regime. While the cutoff-scale dependence is not directly a physical momentum dependence, it relates to the physical momentum dependence at the symmetric point. For a detailed discussion see <Ref> and <cit.>.

For cutoff scales below roughly $1$ TeV, the ASSM enters the symmetry broken phase, as the effective potential of the Higgs field, $V_{\Phi,\textrm{eff}}$, develops a non-trivial minimum,

\begin{align}\label{eq:SSB}
\mu_\Phi(k< k_\textrm{SSB})&<0\,,
\end{align}

The non-trivial background of the Higgs renders all SM fields except photons and gluons massive. Consequently, this leads to a decoupling of the fluctuations of the respective fields below their mass thresholds, clearly visible in <Ref>.

The Euclidean masses of the SM fields read,

\begin{align}\label{eq:euclideanmasses}
m_{W^\pm} &=  \frac{g_2 \, v}{2}\,,
m_Z &=  \frac{g_2 \, v}{2\, \cos(\theta_W)}\,,
m_H &= \sqrt{2 \, \lambda_{\Phi, 4}}\, v\,,
\end{align}

for the EW and Higgs bosons. The quark/lepton fields with/without the constituent mass contribution are given in eq:ConsitutentMasses. The scale dependence of all Euclidean particle masses is shown in <Ref>. From this Figure, the intricacy of the particle masses can be understood. These evolutions are important for the correct determination of observables at colliders, for example, the weak mixing angle is especially sensitive to those.

Finally, dynamical strong chiral symmetry breaking has a sizeable impact on the (constituent) quark mass function eq:ConsitutentMasses, in particular for the light flavours $u,d,s$. The respective  $u$-quark-gluon exchange coupling, defined in eq:exchangecoupling, is depicted in <Ref>. It decays for cutoff scales below $\sim 1$ GeV as a consequence of the QCD mass gap. Note that the avatars of the quark-gluon exchange couplings for the different quarks vary slightly due to the different quark wave functions, see eq:Avatarsalphas. Moreover, the strong exchange couplings defined in eq:Avatarsalphas carry the physical threshold effect of the respective gluon exchange. This threshold is not present in the definition of the electroweak gauge couplings: the wave functions of $W^\pm,Z^0$  are the prefactors of the kinetic terms, while the masses originate from the interaction of the electroweak gauge bosons with the Higgs. These couplings are also called effective charges, e.g. <cit.>. A related definition in QCD, that does not show the threshold of the mass gap in QCD, is given by the process-independent coupling or effective charge, see e.g. <cit.>. In turn, the respective electroweak exchange couplings with the physical mass thresholds enter directly the ASSM flows, as do the quark-gluon exchange couplings, and directly show the decoupling of fluctuations below the mass thresholds. Their definition is provided in eq:exchangecoupling in <Ref>. For an illustration of the threshold effect and their similarity with the quark-gluon exchange couplings, they are also depicted there in <Ref>.

The exchange gauge-fermion couplings show the decoupling of fluctuations in the flows of the fermion mass terms due to the mass gaps of the gauge bosons. A second source for the decoupling of fluctuations is given by the mass thresholds of the fermions themselves. Finally, the electroweak fluctuations are suppressed due to the small electroweak couplings. All these threshold effects and the dominant nature of the strong fluctuations below the electroweak scale are visible in <Ref>, where we show the flows of the ASSM fermion masses below $k\leq k_\textrm{SSB}=940$ GeV, see eq:SSB. This regime covers all matter thresholds in the ASSM.

For $k\to k_{\textrm{SSB}}$ from below, the Higgs expectation value $v$ drops sharply as a function of the cutoff scale, and all fields become massless. We note in passing, that while this is reminiscent of a second order phase transition, it should not be confused with it. It is a standard reflection of spontaneous symmetry breaking in the RG flow in the absence of an explicit one. In particular, it does not carry directly the order of the thermal electroweak phase transition.

Flow of all SM Euclidean masses. Plain lines correspond to the EW and Higgs bosons, dashed to quarks and dotted to the leptons. The different particle decoupling scales are indicated by vertical lines. For quarks, the constituent masses as defined in eq:ConsitutentMasses and obtained from <cit.> are also accounted. At scales around the $k_{\text{SSB}}$ the vacuum expectation value of the Higgs drops and all masses vanish.

Above the EW cutoff scale, $k>k_\textrm{SSB}$, all fields are effectively massless and the $\beta$-functions and anomalous dimensions of all marginal (logarithmically running) parameters reduce to their universal form, and agree with that computed with perturbative methods, as do the respective trajectories, see e.g. <cit.>. In turn, the flow of the dimensionful Higgs mass parameter shows the expected quadratic running and hence is subject to quadratic fine-tuning in the UV. We emphasise that this is technically challenging but carries no conceptual intricacy.

At sufficiently large cutoff scales the quartic Higgs self-coupling changes sign and turns negative, see <Ref>. This indicates a metastable regime or the potential importance of higher-order couplings. In the present ASSM setup, this happens at

\begin{align}\label{eq:kMeta}
k_\textrm{meta}=1.2 \cdot 10^{10}\,\textrm{GeV} \,.
\end{align}

<Ref> compares well with the values obtained in perturbative computations within the $\overline{\textrm{MS}}$-scheme: As discussed before, a direct translation of the RG scales is not straightforward, as is their relation to momentum scales. However, a conservative estimate of these uncertainties is well below an order of magnitude, and hence we identify RG scales. This entails that eq:kMeta is well compatible with

\begin{align}
10^{9}\,\textrm{GeV}\lesssim k_\textrm{meta}\lesssim 10^{11} \,\textrm{GeV}\,,
\end{align}

in the $\overline{\textrm{MS}}$-scheme, see e.g. <cit.>. A conclusive analysis requires the inclusion of higher-order couplings in this regime <cit.>. Importantly, the metastability scale $k_\textrm{meta}$ is very sensitive to the size of the Yukawa coupling and hence the determination of the pole mass of the top quark, discussed in detail in <Ref>. There it is shown that the largest systematic error in the determination of the pole mass can be attributed to the uncertainty in the determination of the running QCD coupling $g_{3,k=M_Z}$, that stems from the requirement of a self-consistent RG scheme for all scales.

In the present work, we use the estimate eq:g3  with a relative factor 1.07 in comparison to the $\overline{\textrm{MS}}$-value. This is a linear extrapolation of the Yang-Mills and 2+1 flavour to the ASSM quark content with the error estimate eq:alphasIntervallMain, see also the discussion in <Ref>. Interestingly,for $\alpha_{s\,,k=M_Z}$ being roughly 15% larger than the $\overline{\textrm{MS}}$-value, the metastability scale approaches the Planck scale $M^2_\textrm{pl}=G_\text{N}^{-1}$. For the strong coupling $g_{3\,,k=M_Z}$ this entails a $7\%$ enhancement,

\begin{align} \label{eq:MetaAtPlanck}
g_{3,k=M_Z} &= 1.31\,,
k_\textrm{meta} &= M_\textrm{pl}\,.
\end{align}

An analysis of the respective systematic error is deferred to <Ref>, where we also show $k_\textrm{meta}$ as a function of the strong gauge coupling $\alpha_{s,k=M_Z}$, see <Ref>.

Above the Planck scale, the ASSM quickly gets dominated by gravity fluctuations, as is the ASSM UV fixed point. This has been discussed in particular in <cit.>. Moreover, matter contributions of the fixed point equations of the pure gravity coupling and mass parameters $g_h$ and $\mu_h$ are simply closed matter loops, and the external gravitons couple via avatars of the Newton coupling. Consequently, the fixed point equations for these couplings are closed and only depend on the number of matter fields of a given kind. In the ASSM, this leads us to the gravitational fixed point

\begin{align}\label{eq:PureGravityFP}
g^*_h &=0.147\,,
\mu^*_h &=-0.656\,,
\end{align}

with the cosmological constant avatars $\vec \lambda_{h,n}^*\approx 0$ for $n\geq 3$. With the fixed point values of the pure gravity parameters at hand, we can discuss the qualitative properties of the trans-Planckian asymptotic safety regime. For the physical trajectory, quantum gravity effects grow large at about the Planck scale, and the flow is dominated by gravity fluctuations, see <Ref> and <cit.>.

In the regime dominated by asymptotically safe quantum gravity, the gauge and Yukawa couplings are attracted towards the Gaußian fixed point. This is straightforward for the non-Abelian gauge couplings since they are asymptotically free without gravity and gravity is always supporting asymptotic freedom <cit.>. For the Abelian hypercharge coupling, the gravity fluctuation must exceed the matter fluctuation that drives the hypercharge coupling into a Landau pole, which indeed happens here.

The UV Higgs sector requires special care, and it is discussed in detail in <Ref>. Depending on the pure gravity fixed point values eq:PureGravityFP it has different stability and relevance properties. For the values computed in the present approximation, eq:PureGravityFP, we are left with a peculiar situation, which to our knowledge has not been discussed before: for increasing $N_\textrm{max}$ we see a convergence of the full potential to a flat one for fields within the validity regime of the Taylor expansion, see <Ref>. For each $N_\textrm{max}$ we have two relevant parameters whose Eigenvectors have maximal overlap with $\mu_\Phi$ and $\lambda_{\Phi,\textrm{max}}$. This is reminiscent of a standard Gaußian fixed point, but at the latter only $\mu_\Phi$ is relevant while all $\lambda_{\Phi,2 n>2}$ are irrelevant. In <Ref> we show $\lambda_{\Phi, 4}$, which is obtained from the UV-IR flow of the potential eq:HiggsPotBroken within the high (converged) order of the Taylor expansion with $N_\textrm{max}=17$, see eq:Nmax. This flow is initiated in the vicinity of the UV fixed point and the full system is flown down to cutoff scales below the Planck scale: $k\ge 10^{17}$ GeV. Below the Planck scale, gravity fluctuations and the higher-order ($n\ge 3$) Higgs self-couplings decouple rapidly. We have checked this also numerically, and $k=10^{17}$ GeV is more than one order of magnitude below the decoupling regime. Hence the higher-order couplings can safely be dropped for $k<10^{17}$ GeV, which is done here for reducing the numerical effort. Matter effects on the expanded potential at cutoff scales close to the EW scale have already been studied <cit.>, and their investigation in the current framework will be done elsewhere. In this work, for cutoff scales $k\le 10^{17}$ GeV, we continue within the $\Phi^4$-approximation.

In summary, at the ASSM UV fixed point, we have a flat Higgs potential

\begin{align}\label{eq:scalarFP}
\vec \lambda^*_{\Phi}=0 \,,
\end{align}

and in particular the coupling parameters $\mu^*_\Phi$ and $\lambda^*_{\Phi}$ of the classical Higgs potential, eq:HiggsPotClApp. Moreover, the ASSM has as many relevant parameters in the matter sector as we have in the SM, and an additional three relevant couplings in the gravity sector, that overlap with $G_\text{N},\, \mu_h,\, \Lambda_3$.  This concludes our discussion of the physical trajectory of the ASSM. For the first time, we connect the asymptotically safe UV regime of the full SM coupled to asymptotically safe gravity to the deep IR regime of the SM with IR-QCD with confinement and dynamical strong chiral symmetry breaking.

ASSM fixed point Higgs potential (left) and its critical exponents (right). The Higgs potential converges towards a flat potential within the reliability regime, which is indicated by the vertical lines as defined in eq:Reliu. In contradistinction to a standard Gaußian Higgs potential with one relevant direction, we find two relevant directions related to the two negative eigenvalues (right) for all approximations, and the eigendirections have strong overlap with $\lambda_{\Phi,2}$ (blue triangle) and $\lambda_{\Phi,2N_\textrm{max}}$ (blue circle), see <Ref>. The values rapidly converge with increasing $N_\textrm{max}$, see eq:RelevantEigenvaluesVH. This rapid convergence is also seen for the lower lying positive eigenvalues, while the convergence of the large ones would require larger $N_\textrm{max}$. The parameter space where such a non-trivial flat potential exist is indicated as the green regime in <Ref>, which includes the ASSM fixed point.

§ FIXED-POINT HIGGS POTENTIAL AND THE PROPERTIES OF THE ASSM FIXED POINT

In this section, we discuss the emergence of the full fixed-point potential of the Higgs field within a high-order Taylor expansion, see eq:FPVHiggs. This enables us to discuss the fixed-point properties of the ASSM, and in particular the number of relevant parameters and their stability. Moreover, we discuss the location of the ASSM fixed point in the fixed-point landscape, including a discussion of the reliability and a systematic error of the respective result.

In particular, the analysis in this section reveals a novel non-trivial fixed point of the ASSM that has not been seen before: the Taylor expansion of the Higgs fixed point potential $u^*(\bar\rho)$ converges to a flat potential, $u^*(\rho)\equiv 0$ with two relevant directions in contradistinction to a standard Gaußian fixed point with one relevant direction, see <Ref>. Such a potential cannot be identified within a low order of the Taylor expansion, and certainly, it cannot be seen within the standard $\Phi^4$ approximation. Moreover, the two relevant directions cannot be revealed in a global analysis of such a fixed point without a full stability analysis including the computation of the relevant eigen perturbations, see <Ref>. In short, it is easily overlooked. Interestingly, a rather similar fixed-point pattern has been seen very recently for an effective potential $u((\partial_\mu\varphi)^2)$ in a shift-symmetric setting, see <cit.>. While there the novel fixed point has been attributed to shift symmetry, in the present case the flat non-trivial fixed point is revealed for the ASSM with its explicit breaking of shift symmetry.

Besides being a novel fixed point, its relevance comes from the fact that in the current approximation this fixed point is indeed connected with the physical SM (with the parameters fixed by experimental observables) in the IR. In turn, the standard Gaußian fixed point is obtained with a vanishing coupling for the most relevant (non-polynomial) operator. However, it is connected to the physical SM within the present approximation: the resulting Higgs and top mass are roughly $2.9$ GeV from the central experimental values. Whether this distance is close or feasible is subject to the evaluation and interpretation of the systematic error of the present approximation. This is but one of the reasons for the rather detailed description of the systematic error estimates in the present work.

We emphasise in this context that the stability aspects of such an analysis are only fully conclusive if the following three aspects are taken into account:

First, within an Einstein-Hilbert truncation, the gravity-matter fixed point with minimally coupled matter can be mapped to the pure gravity fixed point, see <cit.>. Any approximation that shows instabilities in this approximation, lacks credibility also in improved ones. It is the ${\cal R}^2$, ${\cal R}_{\mu\nu}^2$ and ${\cal C}_{\mu\nu\rho\sigma}^2$ tensor structures, that can trigger instabilities or qualitative changes similar to the Bank-Zaks in QCD as also argued in <cit.>.

Secondly, a full Higgs potential is required. In fact, the $\Phi^4$ truncation of the Higgs potential shows an unstable fixed point potential with $\mu^{ *}_{\Phi}= 0.286$ and $\lambda^*_{\Phi,4}= -10.8$, which is qualitatively different from the solution it converges to, as we will show in this section.

Finally, one has to include gravity-induced matter interactions as discussed in <cit.> that are relevant for the discussion of a potential weak gravity bound, see <cit.>.

The current work takes the two first aspects partially into account: we consider a high-order Taylor expansion of the Higgs potential, and the gravity vertices used contain higher order momentum dependences that are generated by the higher order curvature invariants, for a respective discussion, see <cit.>, where it is argued that the ${\cal R}_{\mu\nu}^2$ is approximately absent at the fixed point.

§.§ Higgs fixed-point potential

We use a high-order Taylor expansion of the dimensionless Higgs potential $u(\bar\rho)$ about vanishing field, see eq:FPVHiggs. At the UV fixed point the other SM couplings are vanishing, see <Ref>, and the gravity couplings take their fixed-point values, see eq:PureGravityFP. The fixed-point equation for the Higgs potential $u(\bar \rho)$ reads

\begin{align} \label{eq:FPpotential}
\textrm{FP}_u(u^*)&=0\,,
&& \textrm{with}&
\textrm{FP}_u(u)&={\cal D}\,u -\textrm{ Flow}_u\,,
\end{align}

where $\textrm{Flow}_u [u(\bar \rho),\bar \rho]$ comprises the flow diagrams for  $V_{\Phi,\textrm{eff}}$ divided by $k^4$, and

\begin{align}\label{eq:Du}
{\cal D}\,u= \Bigl[4 -\left( 2+\eta_\phi \right)\bar{\rho} \,\partial_{\bar{\rho}} \Bigr] u(\bar\rho) \,.
\end{align}

The operator ${\cal D}$ generates full scalings on the potential $u$ including the anomalous part, and hence ${\cal D}\,u$ comprises the scaling parts of the fixed-point equation. Note that the part $4$ entails the canonical dimension of the potential, and in general we have $4\to d_\varphi$ for ${\cal D} \varphi$, where $d_\varphi$ is the full scaling dimension of the function $\varphi(\bar\rho)$.

As the Yukawa and gauge couplings vanish at the fixed point, $\textrm{Flow}_u$ only receives contributions from Higgs and graviton loops. In the present work, we use a high-order Taylor expansion of the potential. While this converges rather quickly for small $\bar\rho$, we have to carefully evaluate the reliability regime and the radius of convergence of the expansion. Accordingly we define the reliability regime by $\bar\rho \in [0,\bar\rho_\textrm{max}]$ with

\begin{align}\label{eq:Reliu}
\bar\rho_\textrm{max} = \max_{\bar \rho} \left\{\bar\rho\;\bigg|\; \frac{ \left|\textrm{FP}_u(u^*)\right|}{\sqrt{({\cal D} \,u^*)^2+\textrm{Flow}_u^2(u^*)}  } \leq 1\% \right\}\!,
\end{align}

i.e., we aim at a relative accuracy of 1% in our fixed-point search. Furthermore, we only take into account potential fixed-point solutions for which the lower critical exponents display convergence, in particular the relevant (negative) ones. Naturally, the higher ones are less stable as they are sensitive to the dropped even higher-order couplings $\Lambda_{\Phi,n}$ with $n>N_\textrm{max}$. Fixed-point potentials, that do not satisfy these criteria are not considered.

At the ASSM fixed point, we find one non-trivial fixed-point solution, displayed in <Ref> for $N_\textrm{max}=2,...,17$. We observe that the solution converges towards a flat potential within the reliability regime of the Taylor expansion and only outside of the regime we encounter an instability. Moreover, the reliability regime is growing with each order of the expansion. This is observed in the $N_\textrm{max}$-dependence of the ratio of the highest-order coupling and the second-highest one, $r_u(N_\textrm{max})=|\lambda_{\Phi,2 (N_\textrm{max}-1)}/\lambda_{\Phi,2 N_\textrm{max}}|$. For $\bar\rho = r_u$, the highest-order term is equal to the second-highest one, and we expect convergence for $\bar\rho\ll r_u(N_\textrm{max})$. We find that $r_u$ grows and reaches $r_u\approx 0.1$ at $N_\textrm{max}=17$. A simple extrapolation to $N_\textrm{max}\to\infty$ leads us to $r_u\approx 0.23$. We expect the same tendency for the convergence radius of the Taylor expansion.

The critical exponents that are displayed in the right panel of <Ref> converge well, in particular, the most relevant ones. Importantly, the fixed point potential has two relevant parameters with eigenvalues given by

\begin{align}\label{eq:RelevantEigenvaluesVH}
\theta_1 &= -1.93\pm 2 \cdot 10^{-3}   \,,
\theta_2 &= -0.811\pm 7 \cdot 10^{-5} \,.
\end{align}

Flat potentials with two relevant directions are qualitatively different from the standard Gaußian potential. The latter potential only has one relevant direction at the ASSM fixed point with the eigenvalue $\theta_{\Phi^2,\text{Gau\ss}} = - 2 + \frac{3 g_h^*}{(1+\mu_h^*)^2\pi}  \approx-0.811$ equal to $\theta_2$ in eq:RelevantEigenvaluesVH. Naturally, this FP is embedded in the two-dimensional critical surface of the non-trivial flat fixed point if using a vanishing coupling for the operator related to $\theta_1$.

Weighted and normalised eigenvectors at order $N_\text{max} = 17$. The most relevant eigenvalue $\theta_1=-1.93$ has the largest overlap with the highest order coupling $\lambda_{\Phi,34}$. Otherwise we observe a clear correspondence between order of the coupling and relevance, i.e., $\theta_2=-0.811$ has the largest overlap with $\mu_{\Phi}$, $\theta_3=1.20$ with $\lambda_{\Phi,4}$, etc.

This begs the question of how the solution in <Ref> can converge towards the flat potential but retain a stable second relevant direction. For that purpose, we investigate eigenvectors $v^{(i)}$ and eigenfunction $\phi_i(\bar\rho)$, which are related with

\begin{align}\label{eq:Eigenfunctions}
\varphi_i(\bar \rho) = c_i(N_\textrm{max}) \sum_{l=1}^{N_\textrm{max}}  v^{(i)}_l\bar\rho^l \,,
\end{align}

with an $N_\textrm{max}$-dependent normalisation $c_i$. The latter is requires as the FP potential vanishes in the limit $N_\textrm{max}\to\infty$, and so does the norm of all eigenvectors $v^{(i)}$. Accordingly the normalisation $c_i$ has to be chosen such that the eigenvectors $\bar v^{(i)}$ converge with a finite norm, leading also to converging eigenfunctions $\varphi_i$. Accordingly, a convenient choice for $c_i$ is given by the inverse of a specific $v^{(i)}_{l_i}(N_\textrm{max})$ with a fixed $l_i$.  It is suggestive  to choose $l_i=1$ for all $i$. However, since potentially $v^{(i)}_{1}=0$ for some eigenfunctions $\varphi_i$, this choice may not work globally. In the present case, we are only interested in $i=1,2$, where we indeed can choose

\begin{align}
c_i(N_\textrm{max})&=\frac{1}{ v^{(i)}_1}\,,
&& \textrm{for} & i&=1,2\,.
\end{align}

We remark that the above construction only works if the respective eigenfunctions are well-defined in the limit $N_\textrm{max}\to\infty$. We shall see that this is indeed the case, which provides further non-trivial evidence for the existence of the novel ASSM fixed point with a flat Higgs fixed point potential with two relevant directions.

Before we come to our results, we also briefly discuss the use and computation of the eigenfunctions from their respective fixed-point equations for the sake of completeness: Firstly, any potential can be expanded about the fixed-point solution in terms of the eigenfunctions,

\begin{align}
u= u^* + \sum_i b_i \varphi_i\,,
\end{align}

where the eigenfunctions carry the scaling of the respective eigenperturbation with the eigenvalue $\theta_i$. Hence, the flow of the $\varphi_i$ is given by

\begin{align}\label{eq:Eigenequation}
\partial_t  	\varphi_i(\bar \rho)  = \theta_i \,	\varphi_i(\bar \rho) \,.
\end{align}

This entails that the relevant perturbations die out with $\exp\{ \theta_i t\}$ when approaching the UV fixed point as $\theta_i>0$. In turn, the irrelevant ones grow with $\exp\{ \theta_i t\}$ as $\theta_i<0$.

We have shown above how the eigenfunctions $\varphi_i$ can be constructed from the $v^{(i)}$. They can also be derived from the respective fixed-point equation. For this derivation, we start with a potential that simply is the sum of the fixed-point potential $u^*$ and a perturbation in a given eigendirection $i$ with

\begin{align}
u=u^* +\varepsilon\,  \varphi_i\,,
\end{align}

with an infinitesimal parameter $\varepsilon$. This is inserted in the flow and expanded in $\varepsilon$. The zeroth order in $\varepsilon$ is simply the fixed-point equation for the Higgs potential, leading to $u^*\equiv 0$. The linear order in $\varepsilon$ gives us the differential equation for the eigenperturbations $\varphi_i$ with

\begin{align}
{\cal D}\,{\varphi_i} - \textrm{Flow}_u(\varepsilon\, \varphi_i)\big|_{\mathcal{O}(\varepsilon)}=0\,,
\label{eq:FPvrphi} \end{align}

where the flow term leads to further terms linear in $\varphi_i$, $\varphi_i'$, and $\varphi_i''$. The full scaling ${\cal D}\,{\varphi_i}$ of the eigenfunction $\varphi_i$ reads

\begin{align}\label{eq:Dvarphi}
{\cal D}\,{\varphi_i}(\bar\rho) = \Bigl[(4-\theta_i) -\left( 2+\eta_\phi \right)\bar{\rho} \,\partial_{\bar{\rho}} \Bigr] {\varphi_i}(\bar\rho) \,.
\end{align}

This allows us to define a reliability regime analogously to eq:Reliu by simply substituting $u^*\to \varphi_i$ and ${\cal D}u^*$ with $ \labelcref{eq:Dvarphi}$. A more conservative estimate arises from using the reliability regime of the Taylor expansion of $u^*$. Evidently, the latter leads to a smaller reliability regime since $u^*\to 0$ and the denominator in eq:Reliu is vanishing in the large $N_\textrm{max}$ limit, apart from the absolute convergence of the FP equation. Hence, we will resort to the reliability estimates from the potential throughout as a conservative estimate.

We further introduce the weighted and normalised eigenvectors <cit.> of the critical exponents. This rescaling and normalisation are determinants to quantify the real overlap of each operator within each direction. The `ad hoc' determined eigenvectors are known to be spuriously dominated by the highest orders of the expansion and therefore an equal weight of each operator must be imposed. The normalised eigenvector associated with the $i$-th critical exponent is defined as

\begin{align}
\bar{w}^{(i)}_l =& \frac{\zeta_l\,  v^{(i)}_l}{\sqrt{\sum_{m=0}^{N-1}\left|\zeta_m  v^{(i)}_m\right|^2 }}\,,
\label{eq:baromega}\end{align}

where $\zeta_l$ are the rescaling vectors to the `ad hoc' determined eigenvector $ v^{(i)}_l$ with directions $l\subset (\mu_{\Phi}, \lambda_{\Phi, 4}, \lambda_{\Phi, 6}, \dots)$, see <cit.> for a detailed derivation.

The coefficients of the weighted eigenvectors are depicted in  <Ref>. We observe that the eigenvector $\bar\omega^{(2)}$ of the eigenvalue $\theta_2= -0.811$ has the largest overlap with $\mu_\Phi^2$. Indeed, solving the fixed point equation for the respective eigenfunction $\varphi_2(\bar\rho)$, we find the exact solution

\begin{align}
\varphi_2(\bar\rho) =c_2 \bar \rho\,,
\end{align}

so the eigenvector $v^{(2)}$ of the full solution points exactly in the $\mu_\Phi^2$ direction, as does $\bar\omega^{(2)}=(1,0,....)$. In turn, the numerical solution of the fixed-point equation for $\varphi_2$ and the eigenvector $\bar\omega^{(2)}$ derived from the fixed point potential $u^*$ and depicted in <Ref>  also includes higher order terms. The latter originate in the subleading corrections in $\theta_2$ in comparison to the anomalous dimensions in the FP equation and give an estimate for the numerical accuracy of the $\theta_i$. These terms grow large for large $\bar\rho$ and are an additional source for the finite validity range of the current Taylor expansion observed in the fixed point potential, see the left plot in <Ref>.

The eigenvectors $\bar\omega^{(i)}$ of the higher eigenvalues $\theta_{i>2}$ are not exactly aligned with one coupling as is the case for $\bar\omega^{(2)}$. Hence, the respective eigenfunctions $\varphi_i$ are not monomials. However, the normalised eigenvector $\bar\omega^{(3)}$ of the eigenvalue $\theta_3=1.20$ has the (by far) largest overlap with $\lambda_4$, the eigenvector $\bar\omega^{(4)}$ of the eigenvector $\theta_4=3.37$ overlaps dominantly with  $\lambda_6$, and in general the normalised eigenvectors $\bar\omega^{(i)}$ of $\theta_i$ have the largest overlap with $\lambda_{2 i-2}$ for $i\leq 2$. This matches the expectation from the mass dimension of the operator.

Eigenfunction $\varphi_1(\bar \rho)$ as function of $\bar\rho$ for the most relevant critical exponent $\theta_1$  for different expansion orders $N_\textrm{max}=2,...,17$. We also depict the reliability regime of the Higgs fixed-point potential $u_*$ with the vertical lines at $\bar \rho\lesssim \bar\rho_\textrm{max}$ as defined in eq:Reliu. The rapid convergence of the Taylor expansion for $\bar \rho\lesssim \bar\rho_\textrm{max}$ and even beyond is clearly seen.

The eigenvector $\bar\omega^{(1)}$ of the most relevant direction with the eigenvalue $\theta_1=-1.93$ has the largest overlap with the highest-order coupling, that is with $\lambda_{N_\textrm{Max}}$. In the present case with a flat effective potential this is expected and almost required: in a Taylor expansion this is arranged (for small enough fields) by a cancellation between all monomials, leading to relatively large Taylor coefficients which trigger the observed relevance ordering. Moreover, this cancellation is solely due to the most relevant direction, which accordingly cannot be close to a polynomial, and hence features growing Taylor coefficients. Accordingly, a necessary (though not sufficient) condition for the global existence of the present solution is the convergence of the Taylor coefficients. This is well observed in the present high-order Taylor expansion and is concisely depicted in the convergence of the respective eigenfunction $\varphi_1(\bar\rho)$ in <Ref>. There we also depict the reliability regime of the Higgs fixed-point potential with the vertical lines as defined in eq:Reliu. Moreover, we also use the ratio of the highest-order coefficients, $r_{\varphi_1}(N_\textrm{max}) = |v^{(1)}_{N_\textrm{max}-1}/v^{(1)}_{N_\textrm{max}}|$ for an estimate of the $N_\textrm{max}$-dependence of the convergence radius for $\varphi_1$. Similarly to $r_u$, it grows with $N_\textrm{max}$ and settles quickly at $r_{\varphi_1}\approx 30$, which is also our estimate for $N_\textrm{max}\to \infty$. This quick saturation is different to the steadily rising $r_u$. However, in combination, this suggests a freezing in of the convergence radius at a large $\rho_\textrm{max}$. The highly interesting and relevant question of the global solution and that of a good analytic fit goes beyond the scope of the present work and will be discussed elsewhere.

To conclude the study of the ASSM fixed point potential, we analysed the potential at a fixed field value $\bar\rho\subset\left[0,\bar\rho_\textrm{max}\right]$ as a function of the expansion order $N_\textrm{max}$. The functional behaviour is roughly given by $1/N_\textrm{max}$ for any $\bar\rho$ chosen and thus the potential value in the $N_{\textrm{max}}\to \infty$ limit is compatible with zero. This provides further evidence that the fixed-point potential is indeed converging towards a flat potential with two relevant directions.

§.§ Higgs potential landscape

We now proceed with an analysis of the landscape of possible Higgs fixed-point potentials in dependence of the gravity parameters, $g_h^*$ and $\mu_h^*$. This analysis explores the stability of the Higgs fixed-point solution under the variation of gravity parameters given that there is a big uncertainty on the gravity fixed-point values. We investigate the regime $g_h^* \in [0,1]$ and $\mu_h^* \in [-1,0]$. We will argue later that $g_h^*$ and $\mu_h^*$ can be combined into an effective gravity coupling and therefore the range of parameters gives us a complete view on the Higgs fixed-point solutions. The resulting phase structure of the UV fixed point is depicted in <Ref>.

Apart from the standard Gaussian fixed point potential, that exists for all values of $g_h^*$ and $\mu_h^*$, we found three further, non-trivial, types of fixed-point potentials. The first is the non-trivial flat potential discussed in the previous section, see <Ref>, whose two-dimensional critical manifold naturally includes the standard Gaußian FP with one relevant direction mentioned above. This non-trivial FP with a flat fixed-point potential exists in a rather large parameter space, depicted in green in <Ref>. At the upper and lower boundary of the green region, one of the critical exponents approaches zero and it is suggestive that the fixed point vanishes due to a fixed-point collision. At the upper boundary, the second smallest eigenvalue $\theta_2$ approaches zero, while at the lower boundary, the third smallest eigenvalue $\theta_3$ approaches zero.

We also find a stable fixed-point potential with no relevant directions. This fixed-point potential exists only in the small lime-green band where it exists simultaneously with the non-trivial flat potential discussed before. We display the potential and the critical exponents of point (B) in <Ref> in <Ref>. The upper boundary of this region coincides with the upper boundary of the green region: it is precisely the collision of these two fixed-point solutions that determine the end of these regions. At the lower boundary, the stable potential solution is lost, resembling a fixed point going to infinity.

The stable fixed-point potential has no relevant directions, and hence all parameters of the IR Higgs potential are fixed by the other couplings: It is a predictive fixed point, and specifically the ratio between Higgs and top mass as well as the ratio between EW and Planck scale are predictions. However, we find that in the current approximation the potential does not enter a symmetry-breaking regime in the IR and can therefore not be connected to SM physics.

The third fixed-point potential is unstable with one relevant direction, present in the red region of <Ref>. We display the potential and the critical exponents of point (A) in <Ref> in <Ref>. Due to the instability of the potential, we consider this solution to be unphysical.

In summary, in the present approximation only the non-trivial solution in the green region is physical and compatible with the SM in the IR. We remark that the stability analysis here only takes into account the convergence regime of the Taylor expansion, no conclusion about the global stability properties of the potentials can be drawn.

Fixed-point phase diagram of the scalar potential. In the green region, we observe the existence of a flat potential with two relevant directions, while in the lime region additionally a stable potential with zero relevant directions exists. In the red region, we find an unstable potential with one relevant direction. The boundary curves are described with  eq:Boundary1,eq:BoundaryLinesFrak. The effective gravity coupling is approximately given by $g_h^*/(1+\mu_h^*)^2$, see eq:geff, and it grows from the green to the red region. The blue circle segment avoids this redundancy as it is close orthogonal to the boundary lines. The Higgs potential corresponding to (SM) is shown in <Ref>, while the potentials for (A) and (B) are shown in <Ref> in <Ref>.

The boundary between the red and lime-green region is determined by a fixed point collision and the eigenvalue corresponding to the $\Phi^2$ direction approaching zero. This allows us to analytically determine the boundary. For a fixed order in the Taylor expansion, both, the non-trivial flat potential and the stable potential, become flatter and we can evaluate the eigenvalue of the $\Phi^2$ direction for vanishing Higgs couplings. Then the eigenvalue is given by

\begin{align}\label{eq:AnalyticBoundLandscape}
\theta_{\Phi^2} = -2+\frac{3\, g_h^*}{(1+\mu_h^*)^2\,\pi} \,,
\end{align}

where $-2$ is the canonical value and the second term is the shift due to gravity. Solving for $\theta_{\Phi^2} = 0$, we obtain the boundary

\begin{align}\label{eq:Boundary1}
\frac3{2\pi}\,g_h^*	=	(1+\mu^*_h)^{2}\,,
\end{align}

which is precisely the boundary line between the red and lime-green region in <Ref>.

The lower boundary of the green region is given by $\theta_{\Phi^4} =0$, which we can again evaluate for vanishing Higgs couplings in the vicinity of the boundary

\begin{align}\label{eq:AnalyticBoundLandscape2}
\theta_{\Phi^4} = \frac{3\, g_h^*}{(1+\mu_h^*)^2\,\pi} \,.
\end{align}

Therefore the boundary extends up to vanishing Newton coupling. Note also, that the solution does not exist on the boundary, $g_h^*=0$.

We could not obtain the lower boundary of the lime-green region analytically. In fact, we found that the boundary is well described by a fractional power

\begin{align}\label{eq:BoundaryLinesFrak}
0.6\,g_h^*	=	(1+\mu^*_h)^{1.85}\,,
\end{align}

which indicates a non-trivial competition between diagrams with $g_h^*/(1+\mu^*_h)^2$ and those with $g_h^*/(1+\mu^*_h)$.

The above analysis emphasises, that <Ref> is effectively a one-dimensional plot since typical diagrams contain

\begin{align}
(g_h^*)^n & \,[G_h(\mu_h^*)]^m\,,
n,m\in & \mathbb{N}\,.
\end{align}

Typical combinations are $(n,m) = (1,1)$ and  $(n,m) = (1,2)$, a detailed analysis can be found in <cit.>. Accordingly, a smaller Newton coupling $g_h^*$ is similar to a larger mass parameter $\mu_h^*$, as the latter decreases the fixed-point propagator $G_h(\mu_h^*)$. We define the effective fixed point (Newton) coupling

\begin{align}\label{eq:geff}
\hat g^* \approx \frac{g_h^*}{(1+\mu^*_h)^{2}}\,,
\end{align}

which grows from the green to the red region

\begin{align} \label{eq:geffOrder}
\hat g^*_\textrm{green} \lesssim 		\hat g^*_\textrm{lime-green} < 		\hat g^*_\textrm{red}\,,
\end{align}

potentially implying a weak gravity bound $	\hat g^* < 	\hat g^*_\textrm{red}$. Note that in an on-shell approximation the two-dimensional plot <Ref> collapses to a line as this approximation lacks any $\mu_h$-dependence, and eq:geffOrder follows.

§ IR-PREDICTIVITY & SYSTEMATICS

In this section, we discuss the physics of the UV fixed point that has been presented in <Ref> and also evaluate the systematic error estimate based on the current truncation. In the present approximation, the ASSM lies in a regime with a non-trivial but flat Higgs potential with two relevant directions. Consequently, the matter sector of the ASSM has as many relevant parameters as the SM. Additionally, the gravity sector shows in the present approximation another two relevant directions: the cosmological constant and the scalar curvature (Ricci scalar). The number of relevant directions in the gravity sector increases to three, including the Ricci-scalar squared term, upon improving the truncation <cit.>.

The non-trivial flat Higgs potential has two relevant parameters but nonetheless, we need to study which IR values can be reached from this fixed point. We observed that the fixed point is approached from below, see <Ref>. This translates into the requirement of a negative quartic Higgs-self coupling around the Planck scale. Lowering the top mass leads to a larger metastability scale and eventually to a positive quartic coupling at the Planck scale. In summary, the values of $M_H$ and $M_t$ that can be reached are displayed by the green regime in <Ref>. The boundary of this green region is given by the plain white line which contains all points that connect to the standard Gaußian Higgs potential in the UV. The white dashed lines in <Ref> display our estimated error on this Gaußian line. The standard Gaußian Higgs potential has a higher predictivity since it has one relevant parameter less and therefore it relates the values of the Higgs and top masses, as displayed in <Ref>. The values of $M_H$ and $M_t$ in the red region connect neither to the non-trivial flat Higgs potential nor to the standard Gaußian Higgs potential. In fact, we did not find any viable UV completion for these points.

The blue ellipses in <Ref> display the experimental uncertainties of $M_H$ and $M_t$ at $1\sigma$, $3\sigma$, and $5\sigma$. The ellipses are in the green region and the central values have an approximate distance of 2.9 GeV to the standard Gaußian fixed point marked by a white line. Due to this vicinity, it is highly important to discuss the uncertainties going into the present computation. The location of the white line depends predominantly on the top mass, while the gravity fixed-point values and the threshold effect around the Planck scale only play a strongly subleading role. In consequence, we focus now on the uncertainties going into the top mass determination. For illustrative purposes, we show green and red shaded regimes for $\pm$1,2,3,4,5% variations of the top mass determination in <Ref>.

Values of $M_H$ and $M_t$ for which the standard Gaußian fixed-point Higgs potential is reached (white line). The values in the green region can be approached from the non-trivial flat Higgs potential presented in <Ref> while the red region cannot.
The white shadings show 1%, …,5% error in the overall computation, including the pole mass estimation. The blue ellipsis displays the SM experimental measurement at $1\sigma$, $3\sigma$, and $5\sigma$. The upper and lower dashed lines correspond to the Gaußian fixed-point trajectory for $\alpha_{s, k=M_Z}= 1.1\, \bar \alpha_s$ and $\alpha_{s, k=M_Z}= \bar \alpha_s$ respectively.

We have identified four systematic error sources within our computation, and below we discuss their influence on the top-mass determination:

(i) The first source is the general truncation error on the running of the couplings. Here we neglected, for example, higher-order contributions due to the anomalous dimension, see the discussion in <Ref>. We observed that the inclusion of the next order in the anomalous dimensions only had a subleading effect. Furthermore, our truncation includes all IR threshold effects, and therefore we believe that this source of error is subleading, significantly smaller than 1 GeV.

(ii) The second systematic error source concerns the location of the UV gravity fixed point. We expect the errors on the values of the gravity fixed point to be rather large, and this influences the running of the SM couplings in particular through threshold effects around the Planck scale. We tested this error source by varying the UV fixed-point values and found a very subleading effect on the location of the standard Gaußian line in <Ref>. The reason is the threshold effect at the Planck scale is washed out through the long RG running to the IR. We estimate that this effect is smaller than $0.1$ GeV. Importantly, the gravity fixed-point values determine the existence of the non-trivial but flat Higgs fixed-point potential, see <Ref> and thus the existence of a green regime in <Ref>.

(iii)  The third error source relates to the conversion from Euclidean curvature mass to pole mass discussed in detail in <Ref>: the wave function $Z_t(p)$ of the top quark has a negligible cutoff dependence for $k\leq k_\textrm{SSB}=930$ GeV, see <Ref>. This also entails a negligible momentum dependence for $p\lesssim 10^3$ GeV, see  <Ref> and <Ref>. Hence, we have approximated the wave function with a constant one in this regime also for timelike momenta of the same size, $Z_t(p)\approx 1$ for at least $|p^2|\leq (2 10^2\,\textrm{GeV})^2$. From the minimal variation of the cutoff dependence in <Ref> in this regime, we estimate the respective systematic error with $0.5\%$, which translates into an error of roughly $0.8$ GeV for the top pole mass determination, see also  <Ref>. An additional uncertainty stems from the strong coupling, as discussed in the next point. For the translation of Euclidean curvature mass to pole mass, this leads to an error of ${}^{+0.9}_{-0.2}$ GeV, see eq:topMassCurvature.

(iv) The largest systematic error stems from the uncertainty on the strong coupling, due to the translation of the $\overline{\textrm{MS}}$-value at $p=M_Z$, $\bar\alpha_s$, to the MOM${}^2$ value used here, $\alpha_s$, see <Ref>. From results in pure Yang-Mills and $2+1$ flavour QCD <cit.>, we extrapolated that the conversion factor is given by $\alpha_{s,k=M_Z} =  1.07 \,\bar \alpha_{s}$ in the ASSM, see eq:g3. This choice corresponds to the central white line in <Ref>.  We use the interval $\alpha_{s,k=M_Z}\in [\bar\alpha_s  ,\, 1.10 \,\bar\alpha_s ]$, see eq:alphasIntervallMain, to estimate our error in this identification. Using $\alpha_{s,k=M_Z} = \bar \alpha_{s}$ shifts the standard Gaußian line by $-2.6$ GeV in the top mass, see the lower white dashed line in <Ref>, while using $\alpha_{s,k=M_Z} = 1.1\,\bar \alpha_{s}$ shifts the standard Gaußian line by $0.9$ GeV in the top mass, see the upper white dashed line in <Ref>, which touches the 5$\sigma$ band of the experimental error. For $\alpha_{s,k=M_Z} \approx 1.15\,\bar \alpha_{s}$, the standard Gaußian line goes through the central experimental values since the metastability scale reaches the Planck scale, see eq:MetaAtPlanck, and the non-trivial flat potential needs to be approached from below, see <Ref>. A better control over this error would require a fully self-consistent computation of the strongly coupled IR-QCD sector with all SM flavours.

In summary, the experimental values of the Higgs and top mass can be reached from the non-trivial flat Higgs potential with two relevant directions eq:RelevantEigenvaluesVH. The standard Gaußian fixed point has a distance of roughly 2.9 GeV from the central experimental values, which is similar to the sum of the errors listed above. This gives us confidence that the ASSM is not in the red regime, which we consider as non-trivial evidence for the existence of the ASSM. With the systematic errors listed above, we also consider it viable, while not most likely, to reach the SM values of $M_H$ and $M_t$ from the standard Gaußian Higgs potential.

In the present work, we have solely discussed the minimal ASSM. Hence, physics properties such as the stability considerations and the location and properties of the UV fixed points may change even qualitatively with the introduction of beyond Standard Model (BSM) degrees of freedom. These are relevant for a comprehensive discussion of open cosmological questions such as dark matter. A survey of asymptotically safe BSM theories, and in particular the impact on the stability of the Higgs potential and the physics properties of the UV fixed point is a natural and highly relevant extension of the present work. We hope to report on this in the near future.

§ CONCLUSIONS

In the present work, we have presented a comprehensive analysis of the phase structure of the Asymptotically Safe Standard Model (ASSM). The present analysis includes several qualitative  improvements:

Firstly, RG-consistent flows have been derived and all flows carry the physical threshold effects required for a quantitative determination of the physical point in the parameter space of the ASSM. Secondly, in the deep IR the present approximation to the ASSM contains QCD with confinement and chiral symmetry allowing for a determination of most parameters at the physical cutoff scale $k=0$ at vanishing momentum, $p=0$. Thirdly, we have directly computed the pole mass of the top quark, which eliminates all ambiguities that go with mapping Euclidean curvature masses to the physical pole masses. Finally, our approximation includes a non-trivial Higgs potential for the transplanckian regime which is required for a predictive stability analysis of the Higgs fixed point potential.