A detailed discussion of our results on the phase structure of the ASSM and the nature of the UV fixed points have been presented in <Ref> and <Ref>, cumulating in an evaluation of the systematics in  <Ref>. In short, in the present approximation the ASSM lies in a regime of a novel UV fixed point with a non-trivial but flat Higgs potential with two relevant directions with the eigenvalues eq:RelevantEigenvaluesVH, see <Ref>. Consequently, the matter sector of the ASSM has as many relevant parameters as the SM, and the gravity sector shows another three relevant directions with overlap to the cosmological constant, the scalar curvature (Ricci scalar) term and the Ricci-scalar squared term. The distance of the solution to the standard Gaußian fixed point regime with one relevant parameter less (also called `predictive regime') is depicted in <Ref>. For sufficiently small top pole mass we first cross the standard Gaußian regime and the non-trivial stable regime, before entering the unstable regime. We remark, that while not excluded, we consider the distance of roughly 10 GeV as rather large. While the current approximation is not fully quantitative, significant changes would be required for entering the `predictive' regime. Note also, that this regime is but one step away from the unstable one, and we consider the relatively large distance from the latter as a positive outcome of the present analysis.

The current minimal ASSM does not provide an answer to fundamental open problems as for example the nature of dark matter or the fermion mass hierarchies. Nevertheless, the framework here presented shows great potential for BSM studies in the need of non-perturbative methods or mass dependence schemes. It provides an easily adaptable and RG-consistent playground in which the viability of models phenomenology can be tested. Moreover, this setup can be easily modified towards the study of effective field theories, different matter contents or other types of UV completions thanks to its versatility. Investigations of non-trivial Higgs potentials <cit.> which follow this line have already been performed.

In conclusion, the present comprehensive analysis of the ASSM constitutes a further important step towards the quantitative construction of the ASSM. Moreover, the present RG-consistent set-up can be systematically improved, important next steps in the SM sector of the ASSM are the inclusion of the CKM matrix, the inclusion of the full Higgs potential at all scales, the resolution of further pole masses, as well as general momentum dependences. On the gravity side, the momentum dependences of the fixed point solutions have to be used in the flow, gravity-induced matter vertices have to be added and a more complete tensor basis for the gravity vertices has to be used. All the above steps either have already been investigated in parts of the ASSM, or are work in progress. The present set-up also allows for a comprehensive study of scattering amplitudes giving access to the chiefly important question of unitarity in the ASSM. We hope to report on the respective progress soon.

§ ACKNOWLEDGMENTS

We thank T. Denz, F. Goertz, C. Wetterich, and M. Yamada for discussions and comments on the manuscript. Á. Pastor-Gutiérrez thanks T. Denz for technical support on VertEXpand. M. Reichert acknowledges support by the Science and Technology Research Council (STFC) under the Consolidated Grant ST/T00102X/1. This work is supported by EMMI, and is funded by Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster) and the DFG Collaborative Research Centre "SFB 1225 (ISOQUANT)".

§ EFFECTIVE ACTION OF THE ASSM

We provide the gauge-fixed classical action eq:SM+Gravity of the ASSM in <Ref> (gravity-matter sector) and <Ref> (pure gravity part). The classical gauge fixed-action of the ASSM can be split into a gravity-matter part and a pure gravity part

\begin{align}\label{eq:SM+Gravity}
S_\textrm{ASSM} = S_\textrm{SM} + S_\textrm{gravity} \,.
\end{align}

The gravity-matter part is the SM action in a given geometry defined by the metric $g_{\mu\nu}$. For the gauge fixing in the presence of a metric one usually introduces a linear split of the full metric into a background $\bar g_{\mu\nu}$ and a dynamical fluctuation $h_{\mu\nu}$ as shown in eq:linearsplit which is then also taken for the expansion of the pure gravity action in terms of quantum fluctuations in a given background. The prefactor of the fluctuation term is chosen such that the fluctuation field $h_{\mu\nu}$  has momentum dimension one and its kinetic term has the canonical form of a spin-two field, a more detailed discussion of general splits can be found in <cit.>.

§.§ Gauge-fixed Standard Model

The Euclidean SM action reads

\begin{align}
S_{\text{SM}}&= \frac{1}{4}\int_{x}  F^{a}_{\mu\nu}  F^{a}_{\mu \nu}+ \,\frac{1}{4}\int_{x} B_{\mu\nu}    B_{\mu \nu} +\, S^{\text{EW}}_{\textrm{gf}}+S^{\textrm{EW}}_{\textrm{ghosts}}\nonumber\\[1ex]
&\quad\,+\frac{1}{4}\int_{x} G^{b}_{\mu\nu}    G^{b}_{\mu \nu} +S^{SU(3)}_{\textrm{gf}}  +S^{SU(3)}_{\textrm{ghosts}}\nonumber\\[1ex]
&\quad\,+ \int_{x} (D_{\mu}  \Phi_i)^{\dagger }( D_{\mu }\Phi_i)+ V\!\left( \Phi ^{\dagger }_i\Phi_i \right)\nonumber\\[1ex]
&\quad\,+ \sum_{j=1,2,3 }  \int_{x}  \,  \bar \psi^{ L/ R}_{i\,, j} \slashed D \, \psi^{L/R}_{i\,, j} +S_{\textrm{Yukawa}} \,,
\label{eq:SMAction}
\end{align}

\begin{align}
\int_{x}&= \int \mathrm d^4 x \, \sqrt{\left| g \right| }  \,.
\end{align}

The covariant derivative is defined in a concise form as

\begin{align}\label{eq:D}
D_\mu =\partial_\mu -\imag \,{\bf g}\, {\cal A}_\mu\,, \qquad {\cal A}_\mu={\cal A}^i_\mu t^i\,,
\end{align}

with the Lie algebra generators $t^i$,

\begin{align}\label{eq:LieAlg}
[t^i\,,\, t^j ] &= \imag f^{ijk} t^k \,,
t^i& \in \textrm{u(1)}_\textrm{Y}\times\textrm{su(2)}_\textrm{L} \times\textrm{su(3)}_\textrm{C}\,,
\end{align}

and the structure constants $f^{ ijk}$ of the SM gauge group U(1)${}_\textrm{Y}\,\times\,$SU(2)${}_\textrm{L}\,\times\,$SU(3)${}_\textrm{C}$. The coupling matrix ${\bf g}$ in eq:D is diagonal in the Lie algebra with entries $g_\textrm{Y}, g_2, g_3$ in the respective subspaces. Here, $g_y$ is the hypercharge coupling, $g_2$ is the weak coupling, and $g_3$ is the strong coupling. For the weak hypercharge gauge coupling we employ the normalisation

\begin{align}\label{eq:g1-gY}
g_1= \sqrt{\frac{5}{3}}\, g_Y\,.
\end{align}

We compute $\vec g_1, \vec g_2, \vec g_3$ from the flow of gauge-matter vertices, where the vectors $\vec g_i$ comprise the couplings derived from the different gauge-matter vertices.

The field strength tensor ${\cal F}_{\mu\nu}$ reads

\begin{align}
{\cal F}_{\mu\nu} = \partial_\mu {\cal A}_\nu -  \partial_\nu {\cal A}_\mu - \imag [ {\cal A}_\mu\,,\,{\cal A}_\nu]\,.
\end{align}

The $\psi_{i, j}$ in eq:SMAction stands for a general fermion field representing quark $q_{i, j}$ and lepton $l_{i, j}$ doublets with their respective chiralities. The index $j$ is the generation index, and $i$ is the isospin eigenvalue. We have included all three generations of leptons and quarks. We only consider interactions within fermions of the same generations, hence, a diagonal CKM matrix. This approximation is also used in the effective action discussed below.

The classical Higgs potential $	V^{\ }_{\Phi}$ in  eq:SMAction is given by

\begin{align}\label{eq:HiggsPotClApp}
V^{\ }_{\Phi}(\rho)=  \mu_\Phi  \rho + \lambda_{\Phi} \rho^2 \,,
\end{align}

with the Higgs doublet $\Phi$ and its radial field $\rho$,

\begin{align}\label{eq:HiggsApp}
\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}

Here, $H$ is the Higgs field that acquires an expectation value and the ${\cal G}_i$ are the Goldstone modes.

The Yukawa terms of the SM read

\begin{align}\label{eq:yukawaAction}
S_{\textrm{Yukawa}}=& \int_x  y_{\textrm{u}}\, \bar{q }^{{L}}_{i\,,1} \, \tilde \Phi _i \, q^{{R}}_{1\,,1}  + y_{\textrm{d}}\, \bar{q }^{{L}}_{i\,,1}   \Phi_i q^{{R}}_{2\,,1} \notag\\
&+y_{\nu_e}\, \bar{l }^{{L}}_{i\,,1} \, \tilde \Phi _i \, l^{{R}}_{1\,,1} + y_{\textrm{e}}\, \bar{l }^{{L}}_{i\,,1}  \, \Phi _i \,  l^{{R}}_{2\,,1} \, +\textrm{h.c.} +\dots
\end{align}

where we have only shown explicitly one single generation of fermions and where

\begin{align}
\tilde \Phi_i \equiv i\,  \sigma_2  \Phi_i^*\,.
\end{align}

The interaction terms between Goldstones and fermions will be used to read off the background field dependence of the Yukawa couplings. This subject is discussed within our systematics overview in <Ref>. Moreover, we have checked that the Yukawa interaction and respective mass terms of the Neutrinos lead to negligible effects in the flow. Thus, we have dropped them for the results obtained here.

Finally, we have to specify the gauge fixings in the EW and QCD sectors. In the EW sector, we employ an $R_\xi$-gauge fixing which is adapted to the occurrence of massive modes in the broken phase

\begin{align}
S^{\text{EW}}_\text{gf}=&\int_{\bar x} \,\frac{1}{2 \, \xi}\left(\partial_\mu A_{\mu}^a -i\, g_2 \, \xi\, t^a_{ij} \,\left(\Phi_i^\dagger \,v_j -v_i^\dagger \,\Phi_j  \right) \right)^{\!2}\nonumber\\[1ex]
&\hspace{-.5cm}+\frac{1}{2\, \xi_Y}\left(\partial_\mu B_{\mu }-\frac{i}{2} \, g_{\text{Y}}\, \xi _Y \,\left(\Phi_i^\dagger \,v_i -v_i^\dagger \,\Phi_i  \right) \right)^{\!2}\, .
\label{eq:GFEW}
\end{align}

\begin{align}
\int_{\bar x}&= \int \mathrm d^4 x \, \sqrt{\left| \bar g \right| }  \,,
\end{align}

refers to the volume factor of the background metric, which is taken to be the flat metric here. The associated Faddeev-Popov ghost action reads

\begin{align}
S^{\textrm{EW}}_{\textrm{ghosts}}=&\int_{\bar x}\left[-\partial_\mu \bar{c}_{Y} \left(\partial_{\mu }c_{Y}\right)-\xi_Y g_{\text{Y}}^2  \, (\bar{c}_{\text{Y}} c_{\text{Y}}) \, \phi^{\dagger}_i \,v_i\right]\nonumber\\[1ex]
&+ \int_{\bar x}\Bigg[-\partial_{\mu}\bar{c}^a \partial_\mu  c^a -  \xi g_2^2 t^{a}_{ij} t^{b}_{jl} \,(\bar{c}^ac^{b} )\, \Phi^{\dagger }_i \, v_l \nonumber\\[1ex]
&\qquad \quad+ g_2\, A^c_{\mu } f^{abc}\partial _{\mu}\bar{c}^a c^b \Bigg] .
\label{eq:SM-EWghosts}
\end{align}

Here, $v_i$ are the components of the Higgs background field, see  eq:HiggsApp.

For the high-energy regime of QCD, we use the gauge-fixing and ghost action

\begin{align}
S^{SU(3)}_{\textrm{gf}}  +S^{SU(3)}_{\textrm{ghosts}}=\frac{1}{2\, \xi_3} \int_{\bar x} \left(\partial_\mu G^{c_a}_\mu \right)^2+\int_{\bar x}  \bar{c}^{c_a}\partial_\mu D_\mu ^{c_a c_b } c^{c_b} \,,
\end{align}

in the Landau limit, $\xi_3\to 0$.

§.§ Gauge-fixed gravity

The classical Einstein-Hilbert action is given by

\begin{align}
S_{\text{EH}}=& \, \frac{1}{16 \pi G_\text{N}}\int_{x}  \left(2\Lambda -R \right)+ \frac{1}{2\alpha}\int_{\bar x} \bar g_{\mu \nu} F_\mu F_\nu \notag\\[1ex]
& + \int_{\bar x} {\bar c}_h ^{\,\mu}\, {\cal M}_{\mu \nu}\,  c_h ^\nu \,,
\end{align}

with the linear de-Donder gauge fixing

\begin{align}\label{eq:deDonder}
F_\mu =& \frac{1}{16 \pi} \left(\bar D^\nu h_{\mu \nu} - \frac{1}{2} \bar D_\mu h^\nu _\nu \right) \,.
\end{align}

Here, $\bar D $ is the covariant derivative only dependent on the background metric. The Faddeev-Popov operator for the gauge fixing eq:deDonder reads,

\begin{align}\label{eq:GravFPghosts}
{\cal M}_{\mu \nu} = \bar D ^\sigma \left(g_{\mu \nu }D_\sigma + g_{\rho \nu }D_\mu \right)- \bar D _\mu D_\nu \,.
\end{align}

§ FLOW EQUATIONS IN THE ASSM

In this Appendix, we describe the projection procedure used to derive the flows for the couplings and anomalous dimensions from the functional flow eq:Flow for the effective action. The anomalous dimensions are discussed in <Ref>, the flow of matter couplings except the Higgs self couplings is discussed in <Ref>. The flow of the Higgs potential is discussed in <Ref>, and that of the pure gravity parameters is discussed in <Ref>. Finally, in <Ref> we discuss the flows in the strongly correlated IR sector of QCD.

In short, we perform field derivatives of the functional flow  eq:Flow to obtain flows for correlation functions $\partial_t \Gamma^{(n)}(p_1,\ldots,p_n)$, and all flowing parameters are obtained by evaluating the flow of the respective correlation functions (or its momentum derivative in the case of anomalous dimensions) at vanishing momentum, $p_1=\ldots=p_n=0$, in the spirit of the derivative expansion, with the exception of the gravity couplings. The symbolic expressions of the flows on $n$-point functions were derived using the Mathematica package DoFun <cit.>.

In the present work, we use RG-adapted <cit.> regulators

\begin{align}\label{eq:Regulator}
R_{\phi}(p) &=  Z_{\phi} R^{(0)}_{\phi}(p)\,,
R^{(0)}_{\phi}(p)&= P_{\phi}(p) r(p^2/k^2)\,,
\end{align}

with the classical dispersions $P_{\phi}(p)$, and the shape function of the Litim-type regulator <cit.>,

\begin{align}
r(x)&= (1-x)\,\Theta(1-x)\,,
\end{align}

with the Heaviside step function $\Theta$. The Litim-type regulator leads to analytic flows. It is also optimised for the 0th order derivative expansion <cit.>, however, it is not the optimal regulator beyond this order. Optimal regulators can be constructed by functional optimisation <cit.>. Moreover, in the light of highly interesting further investigations of the resonance structure of the ASSM, scattering processes and the quantitative determination of (pole) masses there are further constraints on the analytic structure of the regulator also for timelike momenta, for related discussions in scalar theories, QCD and pure gravity see <cit.>.

§.§ Anomalous dimensions

The anomalous dimensions

\begin{align}
\eta_{\phi_i} = -\frac{\partial_t Z_{\phi_i}}{Z_{\phi_i}}\,,
\end{align}

are obtained from the flow of the respective two-point functions $\partial_t\Gamma^{(2)}$. For leptons and quarks, we define

\begin{align}\label{eq:etaferms}
\eta_\psi = 	 \left.  \frac{\imag     \tr\Bigl[ \slashed p\, \partial_t \Gamma^{(2)}_{\psi\bar\psi}(p)\Bigr] }{Z_\psi\, p^2 \tr \id }\right|_{p=0},
\end{align}

where the trace sums over all internal and group indices. Note also that the evaluation at $p=0$ is tantamount to a $p^2$-derivative at $p=0$ of the numerator.

For the Higgs field we use

\begin{align}\label{eq:etaHiggs}
\eta_\Phi = 	 \left. \frac{    \tr\Bigl[ \partial_t \Gamma^{(2)}_{\Phi\Phi}(p)-\partial_t\Gamma^{(2)}_{\Phi\Phi}(0)\Bigr]}{Z_\Phi\, p^2 \tr  \id } \right|_{p=0},
\end{align}

where the trace sums over all internal and group indices.

For the gauge fields, we use

\begin{align}\label{eq:etaA}
\eta_{\cal A} = 	 \left.  \frac{ \tr\, \Pi^\bot(p) \Bigl[  \partial_t \Gamma^{(2)}_{{\cal A}{\cal A}}(p)-\partial_t \Gamma^{(2)}_{{\cal A}{\cal A}}(0)\Bigr]}{Z_{\cal A}\, p^2 \tr \, \Pi^\bot(p) }  \right|_{p=0},
\end{align}

where the trace sums over all internal and group indices. In eq:etaA we have used the transversal projection operator

\begin{align}\label{eq:Pbot}
\Pi^\bot_{\mu\nu}(p) =\delta_{\mu\nu} -\frac{p_\mu p_\nu}{p^2}\,.
\end{align}

The anomalous dimensions of the auxiliary ghost fields are extracted with the flows

\begin{align}\label{eq:etaGhosts}
\eta_{\cal C} = 	 \left. \frac{\tr  \Bigl[  \partial_t \Gamma^{(2)}_{{\cal C}{\cal C}}(p)\Bigr]}{Z_{\cal C}\, p^2 } \right|_{p=0}\,,
\end{align}

Finally, we remark that the wave functions completely drop out of the diagrams as they are multiplicative factors in all vertices and (inverse) propagators. The global wave function factor $Z_{\phi_i}$ due to the external legs each carrying a $Z^{1/2} _{\phi_i}$ is cancelled by the $1/Z_{\phi_i}$ in the definition of $\eta_{\phi_i}$. The only remnant dependence on $Z_{\phi}$ arises from the $t$-derivatives  of the cutoff line $G_\phi\, \partial_t R_\phi\, G_\phi$ with propagators $G_\phi$ and regulators $R_\phi$. We are led to

\begin{align}
\frac{1}{Z_\phi} \partial_t R_\phi(p)= \bigl[ \partial_t -\eta_\phi\bigr] R_\phi^{(0)}\,,
\end{align}

more details on the structure of the equations for the anomalous dimension and possible systematic extensions have been deferred to <Ref>.

§.§ Flows of matter couplings

All couplings considered here are computed from three-point functions. The only exceptions are the Higgs mass $\mu_{\Phi}$ and self-couplings $\lambda_{\Phi,\,2n}$ in eq:ClassHiggsCouplings. The flow of the four-point function avatars of the gauge couplings are identified with that computed from the three-point function of the gauge fields coupled to fermions. In the present approximation, this identification is a consequence of gauge consistency and can be derived from the respective Slavnov-Taylor identities which collapse into Ward identities in the present approximation. However, the present projections also overlap with non-classical tensor structures hence the gauge couplings from different correlation functions differ, and the Slavnov-Taylor identities are more complicated in improved approximations.

The flows of the couplings of three-point functions can be obtained from eq:ApproxShort1 with

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

The left-hand side is simply given by

\begin{align}\nonumber
\partial_t \frac{\Gamma_k^{(n)}(p_1,...,p_n;\vec\lambda_k)} { \prod_{i=1}^n \sqrt{Z_{\phi_i,k}} } = &\, \frac12 \sum_{i=1}^n \eta_{\phi_i,k}   S^{(n)}(p_1,...,p_n;\vec \lambda_k)\\[1ex]
&	\,	+	\textrm{Flow}^{(n)}(p_1,...,p_n;\vec \lambda_k) \,,
\label{eq:FlowCouplings}\end{align}

where the second line is simply the flow diagrams of the given $n$-point function divided by the wave functions $\sqrt{Z_{\phi_i}}$ for each of its external legs,

\begin{align}
\textrm{Flow}^{(n)}(p_1,...,p_n;\vec \lambda_k) = \frac{\partial_t \Gamma_k^{(n)}(p_1,...,p_n;\vec \lambda_k)} { \prod_{i=1}^n \sqrt{Z_{\phi_i,k} }} \,.
\end{align}

As in the case of the anomalous dimensions this division, together with the use of RG-adapted regulators, eliminates all $Z_\phi$-dependence from the diagrams at the expense of an $\eta_\phi$-dependence.

Given the flow of a three-point function of the fields $\phi_{i_1}, \phi_{i_2}, \phi_{i_3}$ with the coupling $\lambda_{123} = \lambda_{\phi_{i_1} \phi_{i_2}\phi_{i_3}}$ we arrive at

\begin{align}\nonumber
& \Bigl(\partial_t -  \frac12 \sum_{j=1}^3 \eta_{\phi_{i_j},k}\Bigr)   \lambda_{123} \\[1ex]
&\, \hspace{1.5cm}=  \tr \,\Bigl[ P_{123 }( {\bm p} ) \textrm{Flow}^{(n)}_{\phi_{i_1} \phi_{i_2}\phi_{i_3}}\Bigr]( {\bm p=0} )\,,
\end{align}

where $P_{123}$ is a projection operator or rather procedure such as used for the anomalous dimensions. We performed consistency checks to confirm the veracity of our setup. We found the expected universal results up to one loop in agreement with the literature <cit.>.

In the solutions to the flows presented in <Ref>, we made different gauge fixing choices for each of the phases. In the symmetric phase, we chose the Landau gauge $\xi_Y=\xi=0$. Note that in this regime the gauge dependency appears beyond one loop. In the broken phase, to facilitate a correct cancellation between the unphysical modes of the Goldstone bosons and EW ghosts and longitudinal polarisations we chose the Feynman gauge $\xi_Y=\xi=1$.

§.§ Flow of the Higgs potential

In this section we  derive the flow of the scalar field parameters: couplings, curvature mass and vacuum expectation value. Starting from the renormalised potential eq:HiggsPot and its dimensionless version eq:FPVHiggsPot, the minimum $\bar \rho_0 = \frac{ Z_H v^2}{2\, k^2}$ is defined by

\begin{align}\label{eq:rho0}
\left.\partial_{\bar\rho}\, u\left(\bar\rho\right)\right|_{\bar \rho_{0}} &= 0\,.
\end{align}

Taking a $t$-derivative, we obtain

\begin{align}
\partial_{t}\bar \rho_{0}=&-\left.\frac{ \partial_{\bar\rho} \left(\left.\partial_{t}\right|_{\rho}\,u\left(\bar\rho\right)-\left.\partial_t\right|_{\rho} \bar \rho \,\partial_{\bar\rho} u\left(\bar\rho\right) \right)}{\partial_{\bar\rho} u\left(\bar\rho\right)} \right|_{\rho_0}\,.
\end{align}

Introducing the definition of the diagrammatic flows

\begin{align} \label{eq:flow-Higgs-pot}
\left. \partial_t\right|_{\rho}	V_{\Phi,\textrm{eff}}(\rho)&= \text{Flow}\left[	V_{\Phi,\textrm{eff}}\right]\notag\\
&= k^4 \left(\left.\partial_{t}\right|_{\rho}\,u\left(\bar\rho\right)-\left.\partial_t\right|_{\rho} \bar \rho \,\partial_{\bar\rho} u\left(\bar\rho\right) \right),
\end{align}

\begin{align}
\left. \partial_{t}\right|_\rho\bar \rho= -\left(2+\eta_\Phi\right)\bar \rho .
\end{align}

we reach the final expression for the flow of the vacuum expectation value of the Higgs field in the broken ($\mu_\Phi<0$) regime.

Dimensionless Newton coupling $g_h$ and graviton mass parameter $\mu_h$ as a function of the RG scale $k$. Above the Planck scale, the couplings approach an interacting fixed point while in the IR both coupling decrease quadratically towards the Gaußian fixed point.

The flow of the scalar coupling $\lambda_{\Phi, 4}$ and the curvature mass $\mu_\Phi$ have been obtained from the flow of scalar two- and four-point functions. For the higher $n$-point functions, this becomes tedious due to the large number of diagrams and external legs. Instead, we derive the flow of the scalar couplings $\lambda_{\Phi, 2n}$ directly from eq:flow-Higgs-pot. Considering a vanishing background for simplicity, the flow of any arbitrary $\lambda_{\Phi, 2n}$ can be derived by performing field derivatives of the flow of the effective potential,

\begin{align}
\left.\left(\left.\partial_t\right|_{\bar \rho}  \lambda_{2n}\right)\right|_{\bar \rho=0}&= \frac{1}{n!}\partial^n_{\bar \rho}\left[\frac{1}{k^4}\text{Flow}\left[	V_{\Phi,\textrm{eff}}\right]\right]\notag\\
&\qquad+ \lambda_{2n} \left(n\, \eta_\Phi  +\left(2\,n-4\right)\right)\,.
\end{align}

This procedure gives identical results to the derivation from $n$-point functions. On the technical side, it is important to take properly into account the mixing between the Higgs field and the scalar modes of the graviton.

§.§ Flow of gravity parameters

We extract the flow of the Newton coupling $g_h$ from the transverse-traceless graviton three-point function and the flow of the graviton mass parameter $\mu_h$ from the transverse-traceless graviton two-point function. All gravity flows and anomalous dimensions take into account momentum dependences and are evaluated bilocally between $p=0$ and $p=k$. The flows are identical to <cit.> with the approximation $\lambda_{h,n\geq3}=0$, and the identification of all avatars of the Newton coupling with the avatar from the three-graviton vertex, see eq:eff-uni. This entails that the matter fields only contribute in their minimally coupled versions to gravity flows and in the end only enter via their field multiplicities. In the SM, these multiplicities are given by $N_s = 4$, $N_f = 45/2$, and $N_v = 12$. In the IR, we tune the Newton coupling and the graviton mass parameter such that we reach the physical value of the Newton coupling and a vanishing cosmological constant, see eq:IR-grav-param. In <Ref>, we display the trajectories of $g_h$ and $\mu_h$.

§.§ Infrared regime of QCD

The IR regime of QCD with confinement and dynamical chiral symmetry breaking has been evaluated both qualitatively and quantitatively with functional approaches, for recent reviews see <cit.>. We are solely interested in the running of the quark-gluon coupling, which is dominated by the gluon anomalous dimension $\eta_{A,k}$ and its IR flow below an interface scale $k_\textrm{inter}$ defined in eq:kinter.

It has been argued and checked in <cit.>, that the gluon anomalous dimension $\eta_{A,k}$ is well described as a function of the running coupling $\alpha_s$ and the mass gaps of quarks and gluons,

\begin{align}\label{eq:etaapprox+barm}
\eta_{A,k} &\approx  \eta_{A}(\vec \alpha_{s,k},  \vec{\bar m}_{k}^2)\,,
{\bar m}^2&=\frac{m^2}{k^2}\,,
\end{align}

\begin{align}\label{eq:QCDgaps}
\vec{m}_k^2=(m^2_{A,\textrm{gap}}, \vec{m}_{q,k}^2)
\end{align}

The gluon mass gap in eq:QCDgaps should not be confused with a gluon mass. The vector $\vec \alpha_{s,k}$ again comprises all avatars of the strong coupling; related to the primitively divergent part of the ghost-gluon, three-gluon, four-gluon and quark-gluon vertices. In the matter sector, we have avatars for each quark flavour, that differ below the mass thresholds of the quarks,

\begin{align}
\alpha_{A q_i\bar q_i} &= \frac{1}{4\pi} \frac{\lambda_{A q_i \bar q_i}}{Z_A^{1/2} Z_{q_i}}\,,
&&\textrm{with} &
\end{align}

and in the pure glue sector we have three- and four-gluon couplings as well as the ghost-gluon coupling,

\begin{align}
\alpha_{A^n} &=\frac{1}{4\pi} \frac{\lambda_{A^n}}{Z_A^{n/2}}\,,
\alpha_{A\bar c c} &=\frac{1}{4\pi} \frac{\lambda_{A \bar c c}}{Z_A^{1/2} Z_c}\,,
\end{align}

where the $\lambda$'s are the dressings of the classical tensor structures of the respective vertices <cit.>. All the avatars of the strong fine structure constants agree at two loops (in mass-independent RG schemes), $(\alpha_s)_i=\alpha_s$, but run differently with the cutoff scale and momenta in the IR due to physical threshold effects introduced by the gluon and quark mass gaps $\vec m^2$, eq:QCDgaps.

The quark-gluon exchange coupling $\alpha^\text{(ex)}_{A u\bar u}$ defined in eq:Avatarsalphas as a function of $k$ for interface scales  $k_\textrm{inter}=5,\,15$ GeV. The chosen $k_\textrm{inter}$ does not affect severely the IR dynamics.

The above couplings are exchange couplings that describe the scattering of quarks ($\alpha_{s,A q\bar q}$), gluons ($\alpha_{s,A^3}$), and ghosts ($\alpha_{s,A c\bar c}$). These couplings occur directly in diagrams with the respective gluon lines, and carry directly the decoupling of gluons in the diagrams below the mass gap of QCD. In contradistinction, the electroweak $g_i$'s, depicted in <Ref>, do not include the exchange propagators and are also called effective charges, see e.g. <cit.>. For the definition and computation of a respective charge in QCD see <cit.>.

The exchange couplings of the quark-gluon coupling $\alpha^\text{(ex)}_{A u\bar u}$ (green, solid),  the electron-$W^\pm$ coupling $\alpha^\text{(ex)}_{W^\pm e^-\bar \nu_e}$ (red, dash-dotted), and the photon-electron coupling $\alpha^\text{(ex)}_{A^\gamma e^- e^+}$ (blue, dashed) as defined in eq:Avatarsalphas,eq:exchangecoupling. The decrease of the strong and weak coupling towards small momenta is triggered by the QCD mass gap and the mass of the $W$ boson, respectively. The electromagnetic coupling does not decrease in the IR due to the absence of a photon mass.

For the explicit computation, we utilise the 2+1 flavour IR-QCD results of <cit.>. For that purpose, we separate the 2+1 flavour part from the contributions $\eta^{(c,b,t)}_A$ of the heavier quarks

\begin{align}\label{eq:2+1-rest}
\eta_{A,k}^{(\textrm{ASSM})} =  \eta_{A,k}^{(\textrm{2+1})} + \eta^{(c,b,t)}_{A,k}\,,
\end{align}

where the 2+1 flavour contribution in eq:2+1-rest depends on the full coupling and mass gaps of the ASSM,

\begin{align}
\eta_{A,k}^{(\textrm{2+1})}  = \eta_{A,k}^{(\textrm{2+1})} \left(\vec \alpha^{(\textrm{ASSM})}_{s,k},  \left(\vec{\bar m}^{(\textrm{ASSM})}_{k}\right)^2\right).
\end{align}

The strong coupling is a multiplicative factor in all (QCD) diagrams and hence we have

\begin{align}
\eta_{A,k}(\alpha_{s,k},  \vec{\bar m}_{k}^2) \approx \eta_{A,k}( \vec{\bar m}_{k}^2)\alpha_{s,k} \,,
\end{align}

where $\alpha_{s,k}$ is the strong coupling avatar of the lightest field, the ghost,

\begin{align}\label{eq:Onealphas}
\alpha_{s,k} =\alpha_{A\bar c c,k}\,.
\end{align}

The identification of all avatars in eq:Avatarsalphas with the ghost-gluon coupling fails in the IR below the respective mass thresholds, see <cit.>.  There, however, the contributions to $\eta_{A,k}$ drop out. Furthermore, the quark masses in the 2+1 flavour computation are tuned to the physical ones. In summary, this supports the approximation

\begin{align}\label{eq:ASSSM-2+1App}
\eta_{A,k}^{(\textrm{ASSM})}\approx   \eta_{A,k}^{(2+1)}\frac{\alpha^{(\textrm{ASSM})}_{s,k}} {\alpha^{(2+1)}_{s,k}} + \eta^{(c,b,t)}_{A,k}\,,
\end{align}

also displayed in eq:ASSSM-2+1. Here, $\eta_{A,k}^{(2+1)}$ is now the genuine 2+1 flavour result and the difference between the ASSM strong fine structure constant and that in 2+1 flavour QCD is carried by the ratio of the couplings eq:Onealphas. The contribution from the heavier quarks, $\eta^{(c,b,t)}_{A,k}$, is simply that of the ASSM.

Finally, we use that in leading order the ghost propagator dressing and ghost-gluon vertex dressing are insensitive to the number of quarks at all scales, and hence they drop out of the ratio. This leaves us with our final equation for the anomalous dimension,

\begin{align}
\eta_{A,k}^{(\textrm{ASSM})}\approx  -\frac{\partial_t Z_{A,k}^{(2+1)}}{Z^{(\textrm{ASSM})}_{A,k}} + \eta^{(c,b,t)}_{A,k}\,,
\label{eq:ASSSM-2+1LinearApp}
\end{align}

where $\partial_t Z_{A,k}^{(2+1)}$ is the flow of the wave-function renormalisation in 2+1 flavour QCD, not in the ASSM. This can be re-arranged as a flow equation for the gluon wave-function renormalisation,

\begin{align}
\partial_t Z_{A,k}^{(\textrm{ASSM})}\approx &\, \partial_t Z_{A,k}^{(2+1)} - \eta^{(c,b,t)}_{A,k} Z_{A,k}^{(\textrm{ASSM})}	\,.
\end{align}

The right-hand side depends on the external input $\partial_t Z_{A,k}^{(2+1)}$ from 2+1 flavour QCD taken from <cit.>, while all the other terms are that in the ASSM.

The flow eq:ASSSM-2+1LinearApp is now solved for IR-QCD scales below the interface scale $k_\textrm{inter}$ in the range eq:kinter. The flow  eq:ASSSM-2+1LinearApp requires as an input the gluon wave-function renormalisation $Z_{A,k}^{(\textrm{ASSM})}$ at the interface scale. This input is fixed by the consistency condition that the anomalous dimension computed above and below $k_\textrm{inter}$ have to agree at the interface scale,

\begin{align}\label{eq:etaConsistency}
\left( \eta_{A,k_\textrm{inter}^+}^{(\textrm{ASSM})} - \eta_{A,k_\textrm{inter}^-}^{(\textrm{ASSM})}\right) &= 0\,,
k^\pm&= \lim_{\epsilon\to 0} k\pm \epsilon\,.
\end{align}

Inserting eq:ASSSM-2+1LinearApp into eq:etaConsistency leads us to

\begin{align}\label{eq:ZAinter}
Z^{(\textrm{ASSM})}_{A,k^-_\textrm{inter} } = - \frac{\partial_t Z_{A,k^-_\textrm{inter}}^{(2+1)} } {\eta_{A,k_\textrm{inter}^+}^{(\textrm{ASSM})}-\eta^{(c,b,t)}_{A,k^+_\textrm{inter}}} \,,
\end{align}

where the denominator is nothing but the 2+1 flavour part of the gluon anomalous dimension in the ASSM. This provides us with the initial condition for solving eq:ASSSM-2+1LinearApp, also discussed in <Ref>. For interface scales  $5\,\textrm{GeV} \lesssim 	k_\textrm{inter} \lesssim 15\,\textrm{GeV}$, see eq:kinter, we find a small dependence of the IR coupling on the interface scale.  This is depicted in <Ref>.

It is convenient to define exchange couplings not only for gluonic couplings but also for the other gauge fields,

\begin{align}\label{eq:exchangecoupling}
\alpha^{(\textrm{ex})}_i= \frac{g_i^2}{4 \pi} \left(\frac{1}{1 +m_i^2/k^2} \right),
\end{align}

which account for the masses of the mediating field in the scattering process with the intermediate gauge field coupled via $g_i$: the prefactor $g_i^2/(4 \pi)$ are the fine structure constants derived from the $g_i$'s shown in <Ref>. The factor $1/(1+m_i^2/k^2)$ constitutes the dimensionless exchange propagator of the gauge field.

In <Ref>, we depict the strong, weak, and hypercharge exchange couplings from fermion-gauge vertices. For scales below the threshold of the propagating field, the exchange coupling decreases reflecting the suppression of the interaction in the diagrams.

UV potential (left) and critical exponents (right) at different Taylor expanded orders for the gravitational fixed point (A) in <Ref> with $g_h= 0.219$ and $\mu_h = -0.730$. The vertical dashed lines in the left panel show the reliability regimes at each order of the Taylor expansion with a relative error of $1\%$ as defined in eq:Reliu. The fixed point has one relevant direction with lowest critical exponent min$\left[\theta_i^{(\text{A})}\right]=-1.39$.

§ UV HIGGS POTENTIALS AT DIFFERENT GRAVITY PARAMETERS

In <Ref>, we have discussed the rich landscape of Higgs fixed-point potentials. Here, we study the potentials at the gravity parameters (A) with

\begin{align}
g_h^* &= 0.219  \,,
\mu_h^* &= -0.730\,,
\end{align}

and (B) with

\begin{align}
g_h^*&=0.190   \,,
\mu_h^*&=-0.695\,,
\end{align}

marked in <Ref> which characterise each of the regimes found.

The potential at point (A) belongs to the red regime and is shown in <Ref>. The scalar mass parameter and all scalar couplings $\lambda_{\Phi,\,2n}$ are negative. In particular, for the lowest-order expansion, we find

\begin{align}
\mu^{ *}_{\Phi} &=-0.230  \,,
\lambda^*_{\Phi,4} &= -2.22\,.
\end{align}

The critical exponents show a reasonable convergence pattern and in particular, the six lowest values are well converged. Remarkably, the higher-order critical exponents always form complex conjugate pairs. All critical exponents are strongly shifted towards irrelevance compared to their canonical value, for example, $\theta_{17} = \mathcal{O}(100)$ while the canonical value is 30. Moreover, a single relevant direction is found. Recall from the discussion in <Ref> that the transition from green to red regimes is given by the disappearance of a relevant direction. To further investigate this regime and in particular the boundary to the green region, it would be interesting to use a non-trivial background for the scalar field.

Point (B) falls in a regime where two fixed-point potentials exist. Additionally to the non-trivial flat potential discussed in <Ref>, a stable potential is found, shown in <Ref>. The fixed-point values  for the lowest order expansion read

\begin{align}
\mu^{ *}_{\Phi}&=0.0487\,,
\lambda^*_{\Phi,4}&= -0.0758 \,.
\end{align}

Note their similar absolute magnitude. This property is preserved for higher orders and shows the peculiarity of this solution. This potential is stable within the reliability regime of the Taylor expansion but no statements can be made beyond this regime.

The critical exponents display a very quick convergence even at low orders of the Taylor expansion, see the right panel of <Ref>. Compared to the critical exponents of point (A), the critical exponents remain close to their canonical values. Interestingly, this potential has no relevant directions as the minimal critical exponent reads

\begin{align}
\textrm{min}\!\left[\theta_i^{(\text{B})}\right]=0.043\,.
\end{align}

Therefore this fixed point shows maximal predictivity as all SM parameters related to the Higgs potential are predicted by the fixed point, namely, the ratio between the top and Higgs mass as well as the ratio between the Planck and EW scale. However, the trajectory emanating from this fixed-point potential does not lead to a physical IR limit in the present truncation. Particularly, this trajectory shows a negative quartic coupling for all scales below the Planck scale and does not generate a $k_{\textrm{SSB}}$ scale.

UV potential (left) and critical exponents (right) at different Taylor expanded orders for the gravitational fixed-point (B) in <Ref> with $g_h= 0.190$ and $\mu_h = -0.695$. The vertical dashed lines in the left panel show the reliability regimes at each order of the Taylor expansion with a relative error of $1\%$ as defined in eq:Reliu. The minimum critical exponent is min$\left[\theta_i^{(\text{B})}\right]=0.043$ and therefore, the fixed point (B) has no relevant direction and shows a maximal predictivity. As discussed in <Ref>, the IR trajectory of this FP does not lead to a physical IR.

§ SYSTEMATICS

In this section, we discuss the systematics of the present approximations as well as consistency checks and partial systematic error estimates. Specifically, we comment on the general structure of the flows in the combined vertex and derivative expansion, and the systematic extension of the current approximation in <Ref>. In <Ref>, we discuss specifics of the gravity-matter intersection couplings and the embedding of this part in other works as well as the algorithmic effort, mainly due to this sector and pure gravity. In <Ref> we discuss the systematic error introduced by adjusting the present MOM${}^2$ strong coupling to $\overline{\textrm{MS}}$  values. In <Ref>, we discuss the effects of a non-trivial background on the Higgs- and Goldstone-type of Yukawa couplings.

§.§ Anomalous dimensions

In this expansion scheme the flows projected on the dimensionless couplings $\vec{\lambda} =(\mu, g_3, \lambda_3,..., )$ of the gravity-SM system (including the mass parameters) read

\begin{align}\label{eq:flowgs-split}
\left( \partial_t +d_{\lambda_i} +\frac12 \sum_{n=1}^{n_i} \eta_{\phi_{i_n}}(0) \right) \lambda_i = &\, F^{(0)}_{\lambda_i}+ \sum_{n}\eta_{\phi_{n}}(k) F^{(1)}_{\lambda_i, n} \,,
\end{align}

where  $d_{\lambda_i}$ are the canonical dimensions of the couplings $\lambda_i$ and we have split the right-hand side into the contributions at vanishing anomalous dimensions $ F^{(0)}_{\lambda_i}$ and proportional to the anomalous dimensions  $ F^{(1)}_{\lambda_i}$. The anomalous dimension on the left-hand side is evaluated at $p=0$  and arises from the scale derivative of the wave-function renormalisation. On the right-hand side, the anomalous dimensions are evaluated at the cutoff momenta $k$ and come from the regulator insertion in the diagrams. The latter is an approximation as the anomalous dimensions are integrated over the loop momentum $q$. Since the integrands of the flow  peak at about $q^2 \approx k^2$, it is a well-working approximation to approximate $\eta_{\phi_i}(q^2)\to \eta_{\phi_i}(k^2)$.

The equation for the momentum-dependent anomalous dimensions has a similar structure,

\begin{align}\label{eq:floweta-split}
\eta_{\phi_{i}}(p) &= F^{(0)}_{\eta_{\phi_{i}}}(p) + \sum_{n}\eta_{\phi_{n}}(k) F^{(1)}_{\eta_{\phi_{i}},n}(p) \,.
\end{align}

Due to the loop momentum approximation $\eta_{\phi_i}(q^2)\approx \eta_{\phi_i}(k^2)$ we only obtain a closed set of equations for $p=k$. Its solution can be obtained by inverting the resulting matrix from coupled system of anomalous dimensions. Given the large field content of the ASSM, the complete system of coupled equations cannot be solved exactly and we resort to an iterative process. Specifically, we replace $\eta_{\phi_{n}}(k)\to \eta_{\phi_{n}}(0)$ on the right-hand of  eq:flowgs-split and, at lowest order, use eq:floweta-split evaluated at $p=0$ with $\eta_{\phi_{n}}(k)=0$. At the next order, we do not set $\eta_{\phi_{n}}(k)=0$ in eq:floweta-split but instead replace $\eta_{\phi_{n}}(k)\to \eta_{\phi_{n}}(0)$ and use again eq:floweta-split evaluated at $p=0$ with $\eta_{\phi_{n}}(k)=0$ for it. This defines the iterative process that we use. We used the first-order approximation of this iteration and found a good agreement with the zeroth-order approximation. This highlights the subleading role of the higher-order anomalous dimensions in the sub-Planckian regime of the ASSM, and in consequence we use the zeroth-order approximation for the results presented here.

§.§ Gravity-matter couplings

In our implementation of the interplay of gravity and matter, we distinguish the effects of matter on gravity from the gravity on matter. For the former we rely on particular studies of scalar-fermion- <cit.> and gauge-gravity <cit.> systems to compute the effect of the SM matter content on the Newton coupling and the cosmological constant. The results there presented are easily extendable to arbitrary matter contents and provide a consistent RG-scheme with inclusion of mass thresholds.

For the effects of gravity on matter, we employed the same truncation as the cited literature and proceeded to compute the full entangled SM-gravity system. The number of $n$-point functions relevant to the theory significantly increases and needs for automatization. For example, the SM alone presents $\sim 130$ $n$-point functions while coupled to gravity $\sim 270$. For the derivation of the $n$-point functions we relied on the Mathematica package VertEXpand <cit.>.

There are some subtle points in the inclusion of gravity along the EW sector and non-trivial backgrounds of the Higgs field that require further explanation. The gauge-fixing and ghost actions do not produce graviton interactions due to the background metric used in these terms. It is important to note that as this does not affect standard gauge-fixings (as the used in the strong sector), it does in the special $R_\xi$-gauge. Since the full metric is implicit in the contraction of the covariant derivatives applied on the scalar kinetic term, mixings between Goldstone, gauge and graviton fields do not cancel with the corresponding term generated from an $R_\xi$-gauge-fixing action containing the full metric. This naturally creates additional diagrams contributing to the self-energy of gauge and Goldstone bosons.

In general, these diagrams do not pose a severe problem as they are proportional to the background of the Higgs field. Moreover, in the EW broken phase, the gravitational coupling is negligible, and at Planckian scales, the flow of the curvature mass indicates that $\mu_{\Phi}>0$, e.g. a vanishing trivial minimum.

§.§ IR-QCD

In <Ref>, we have discussed the determination of the strong coupling in the present approach: its value in the $\overline{\textrm{MS}}$-scheme at $k=M_Z$ is given by

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

It has been shown in <cit.>, that the MOM${}^2$ RG-scheme used in the fRG, leads to an enhancement of the running gauge couplings $\alpha_s(p)$ compared to the $\overline{\textrm{MS}}$ value. 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). Moreover, in these works it is also shown that $\alpha_{s,k}(0)\lesssim \alpha_{s,k=0}(p=k)$, leading to a further (subleading) enhancement.

Metastability scale as a function of $\alpha_{s,k=M_Z}/\bar \alpha_s$. Here, $\bar \alpha_s\approx 0.118$ is the $\overline{\textrm{MS}}$-value with at the EW scale $p=M_Z$ (and $k=0$), see eq:alphaMSbarMZApp. For each value of $\alpha_{s,k=M_Z}$ the remaining parameters of the ASSM are set to reproduce the experimental values. Throughout this work, we use  $\alpha_{s,k=M_Z}  = 1.07 \,\bar \alpha_s$ marked by a plain line, see eq:alphasCentral.  The shaded interval, defined in eq:alphasIntervall, is used as a systematic error estimate.

Being short of a full analysis, which requires the full implementation of IR-QCD, we use a linear extrapolation in flavour numbers, which leads to an enhancement factor of 1.07 in the ASSM compared to the $\overline{\textrm{MS}}$ value at the $M_Z$ scale

\begin{align}\label{eq:alphasCentral}
\alpha_{s,k=M_Z}  = 1.07 \,\bar \alpha_s \,.
\end{align}

Furthermore, we use a variation of the strong MOM${}^2$ coupling $\alpha_s$ at $k=M_Z$ as an error estimate for our computation. We vary the strong coupling in the range

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

The main impact of this variation is the shift of the zero crossing of the quartic Higgs coupling, commonly referred to as metastability scale $k_\text{meta}$. In <Ref>, we show $k_\textrm{meta}$ as a function of the strong coupling eq:alphasIntervall. For our central value eq:alphasCentral, we find a metastability scale slightly above $10^{10}$ GeV in good agreement with perturbative computation in the $\overline{\textrm{MS}}$-scheme <cit.>. Interestingly, for $1.15\,\alpha_s$ this scale hits the Planck scale,

\begin{align}\label{eq:MetaPlanck}
k_\textrm{meta}(1.15 \, \bar\alpha_s ) = \frac1{\sqrt{G_\text{N}}}\,,
\end{align}

where $G_\text{N}$ is the physical value of the Newton constant at $p=0$. The metastability scale is computed in a $\phi^4$ approximation of the Higgs potential, which is a good approximation below the Planck scale. Around the Planck scale, this approximation cannot be trusted but we nonetheless might expect the disappearance of a metastability scale for a large enough strong coupling. This concludes our systematic error analysis of the strong coupling.

Flow of the top-quark Godstone (plain) and Higgs (dashed) Yukawa couplings defined in eq:GoldstoneHiggsYukawas and their difference (dot-dashed) eq:diffGoldstoneHiggs. The difference is given by the scalar-field dependence of the Yukawa coupling. In the symmetric phase both couplings agree due to equal vanishing background.

§.§ Yukawa couplings

The Yukawa couplings in the fermion-scalar interactions are subject to a scalar-field dependence $\vec{y}(\rho)$. Such is key to reproducing multi-scalar production from fermion scattering, see for example <cit.>. Although this dependency is not explicitly considered in the Yukawa couplings defined in eq:yukawaAction the flow equation is known to generate such non-trivial contributions. As the flow is evaluated on the equations of motion, the scalar field dependency is only generated in the modes with a non-trivial background. To quantify such difference, we take advantage of the previous point and define two Yukawa couplings, the coupling to Higgs and Goldstones

\begin{align}\label{eq:GoldstoneHiggsYukawas}
\partial_t y_t^{(H)} (\bar \rho)&= \text{Tr}\!\left[P_{H t\bar{t}}(p) \textrm{Flow}^{(3)}_{H t\bar{t}}\right] \notag\\
&\hspace{2cm}+ \frac{1}{2}(\eta_H +\eta_{t_L}+\eta_{t_R} )y_t^{(H)}(\bar \rho)\,,\notag\\
\partial_t y_t^{(\mathcal{G})} (\bar \rho)&= \text{Tr}\!\left[P_{\mathcal{G}^0 t\bar{t}}(p) \textrm{Flow}^{(3)}_{\mathcal{G} t\bar{t}}\right] \notag\\
&\hspace{2cm}+ \frac{1}{2}(\eta_{\mathcal{G}^0} +\eta_{t_L}+\eta_{t_R} )y_t^{(\mathcal{G})}(\bar \rho).
\end{align}

While the former contains the complete dependency in the broken phase, by subtracting the latter we find

\begin{align}\label{eq:diffGoldstoneHiggs}
\partial_t y_t^{(H)}(\bar \rho)-\partial_t y_t^{(\mathcal{G})}(\bar \rho)&=\bar{\rho}\, \left.\partial_t\right|_{\bar{\rho}}\left(\partial_{\bar{\rho}} y_t^{(\mathcal{G})} (\bar \rho) \right)\,.
\end{align}

In fact, one single Yukawa coupling enters the theory but distinguishing as in eq:GoldstoneHiggsYukawas and computing two different flows allows us to obtain the differences caused by a non-trivial background. All flows in eq:GoldstoneHiggsYukawas,eq:diffGoldstoneHiggs are shown in <Ref>. The flow of the Higgs Yukawa coupling shows a maximum at $k\sim 100$ GeV, which is not present in the Goldstone coupling. In the symmetric phase $k>k_\text{SSB}$, both couplings agree due to equal vanishing background. This shows that there is a small qualitative difference between the two couplings. In the SM flows in <Ref>, we did not include both couplings separately but assumed them to be equal. The Goldstone Yukawa couplings were chosen as their appearance in fermion-scalar vertices is predominant.

§ POLE MASSES

For $k\to 0$, the relevant parameters of the ASSM can be determined by respective particle physics measurements and the classical (low energy) Newton coupling and the cosmological constant. The matter and gauge couplings can be determined by selected cross sections with a scattering momentum of the order $\lesssim 10^2$ GeV, typically chosen close to the EW scale. Most of these couplings and masses are not a very sensitive input, with the exception of the pole mass of the top quark: it has a relatively large experimental error, and its value (as the largest mass parameter) has a considerable influence on stability estimates of the Higgs potential.

We argue now, that its determination can be obtained with a very small systematic error from the current approximation: to begin with, the missing momentum dependence of the matter couplings and propagators discussed in <Ref> has a subleading impact on the flow of couplings, wave-function renormalisation and masses at vanishing momentum. This has been checked thoroughly in IR-QCD ($p\lesssim 10$ GeV), where the genuine momentum dependence is far stronger than here due to the significantly larger coupling and the resonant hadronic correlations, see in particular <cit.>, utilising also the momentum dependent QCD correlation functions from  <cit.>.

However, while the feedback of the momentum dependences is subleading, they are required for a determination of the coupling strengths via scattering processes and the determination of the pole masses. Specifically, for the present task of computing the top-pole mass $M_{t,\textrm{pole}}$, the momentum-dependent mass function of the top quark is required. We parametrise the two-point function as

\begin{align}\label{eq:GaTop}
\Gamma^{(2)}_{t\bar t}(p) =Z_{t,k}(p) \left[  \imag \slashed{p} + M_{t,k}(p) \right].
\end{align}

with the cutoff- and momentum-dependent wave function $Z_{t,k}(p)$ and mass function $M_{t,k}(p)$. For momentum-independent dressing functions this reduces to the approximation used in the present work,

\begin{align}
Z_{t,k}= Z_{t,k}(p=0)\,,\qquad m_{t,k}= M_{t,k}(p=0)\,.
\end{align}

At $k=0$, eq:GaTop leads us to the propagator

\begin{align}
G_t(p) =\frac{1}{Z_{t}(p)} \frac{1}{p^2 +M_{t}(p)^2}\left[ - \imag \slashed{p}  +M_{t,k}(p)\right],
\end{align}

with $Z_{t}(p)= Z_{t,k=0}(p)$ and $M_{t}(p)= M_{t,k=0}(p)$. The pole mass $M_t^{(\textrm{pole})}$ is extracted at $k=0$ from

\begin{align} \label{eq:MtopPole}
\left[	p^2 + \,M_{t}(p)^2\right]_{ p^2 = -	\left[M_t^{(\textrm{pole})}\right]^2}=0\,,
\end{align}

or equivalently from

\begin{align} \label{eq:MtopPoleZ}
\left[Z_t(p) p^2 + \, Z_t(p) M_{t}(p)^2\right]_{ p^2 = -	\left[M_t^{(\textrm{pole})}\right]^2}=0\,,
\end{align}

which follows from  eq:MtopPole by multiplying with $Z_t(p)$.

In the following, we determine the pole mass by using  eq:MtopPoleZ:  The combination  $Z_t(p) M_{t}(p)^2$ is obtained analytically from the integrated flow of the top two-point function

\begin{align}\label{eq:dtZM}
\partial_t\! \left[ Z_t(p)M_t(p)\right]=\frac1{4}\tr_D\,\partial_t \Gamma^{(2)}_{t\bar t}(p) \,,
\end{align}

with the Dirac trace $\tr_D$. The analytic momentum dependence allows us to resolve the mass function for timelike momenta $p^2 < 0$.

The second ingredient in eq:MtopPoleZ is the wave function renormalisation $Z_t(p)$. Here we use that the momentum dependence of correlation functions ${\cal O}_{k=0}(p)$ with a mild cutoff dependence ${\cal O}_{k}(p=0)$ at vanishing momentum is typically well approximated by

\begin{align}\label{eq:Opk}
{\cal O}_{k=0}(p) \approx {\cal O}_{k=\alpha p}(p=0)\,.
\end{align}

This has also been discussed in <Ref>, and is well-tested in QCD, where quantitative correlation with the full momentum dependence as well as the cutoff dependence have been computed and compared, see in particular <cit.>. In the present case of the wave function renormalisation of the top quark, its cutoff dependence is negligible in the symmetry broken phase for $k\leq k_\textrm{SSB}= 930\,$GeV, see  eq:SSB. Indeed, the $k$-dependence of $Z_{t,k}$ is very small, as shown in <Ref>, where  $Z_{t,k}$ is shown for $k\lesssim k_\textrm{SSB}$.

In summary, the above analysis suggests that a quantitative estimate is already obtained with $Z_t(p)=1$ also for timelike momenta within the regime $|p^2|\leq (2 10^2\,\textrm{GeV})^2$. 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. A direct computation will be presented elsewhere, together with further pole mass determinations.

Wave-function renormalisation $Z_{t,k}$ of the top quark as a function of the cutoff scale $k$. The $k$-dependence in the broken phase is negligible, and is well approximated by $Z_{t,k}=1$, leading to the estimate $Z_t(p)=1$ in eq:MtopPoleZApprox.

Within this approximation we are led to

\begin{align} \label{eq:MtopPoleZApprox}
\left[p^2 + \, Z_t(p) M_{t}(p)^2\right]_{ p^2 = -	\left[M_t^{(\textrm{pole})}\right]^2}\approx 0\,,
\end{align}

and it is left to compute the combination $Z_t(p) M_{t}(p)^2$ for timelike momenta $p^2\leq 0$, in particular including the pole position. We utilise that the sub-Planckian structure of the flow facilitates the Wick rotation.  First, the mass function does not receive contributions for cutoff scales $k\geq k_\textrm{SSB}\approx 10^3$ GeV, see eq:SSB. For larger scales, the theory is chiral and hence $M_{t,k\geq k_\textrm{SSB}}(p)= 0$. Accordingly,

\begin{align}\label{eq:MassfunctionTotal}
Z_t (p) M_t(p) = \int_{k_\textrm{SSB}}^0 \frac{\mathrm dk}{k} \,\partial_t\!\left[ Z_t(p)  M_{t,k}(p)\right].
\end{align}

For scales $k\leq  k_\textrm{SSB}$, the SM couplings quickly settle at their IR value and can be taken to be $k$-independent with the exception of the strong coupling, see <Ref>.

Top mass parameter $m_{t,k}$ measured in units of $m_t=m_{t,k=0}$ as a function of the cutoff scale. The mass parameter saturates at cutoff scales $k\approx 200$ GeV, and the vertical lines indicate the borders of the cutoff range eq:ksat for $k_\textrm{sat}$. In the inlay, we depict the variation of $m_{t}$ required for the experimental pole mass eq:topMassPole as a function of the saturation scale $k_\textrm{sat}$ used for $g_{3,k=k_\textrm{sat}}$ in eq:DeltaMtDiagrams. The upper dashed line is obtained for $\alpha_s=\bar\alpha_s$, the lower dot-dashed line is obtained for $\alpha_s= 1.1\,\bar \alpha_s$ (see eq:alphasIntervall) and the central plain line for the ASSM flavour number extrapolation $\alpha_s= 1.07\,\bar \alpha_s$. This covers the uncertainty in mapping the $\overline{\textrm{MS}}$-value of the coupling to the MOM${}^2$ value used in the fRG computation.

The strong coupling changes significantly between $k_\textrm{SSB}$ and $k=0$, see <Ref>. We now use, that the flow of the top mass quickly saturates, see <Ref>, and does receive negligible contributions below the top scale. More specifically, the QCD contributions in the flow are proportional to

\begin{align}
\frac{\alpha_{s,k}}{\left(1+\frac{M_{t,k}^2}{k^2}\right)^{\!3}}\,,
\end{align}

and higher powers in the denominator. These terms drop rapidly for $k< M_{t,k}$ towards smaller cutoff scales despite the rising coupling, which is responsible for the saturation of the QCD part of the flow of $m_{t,k}$. Accordingly, we can estimate its contribution by using an average coupling at the saturation scale $k_\textrm{sat}$. We estimate this saturation scale $k_\textrm{sat}$ very conservatively as

\begin{align}\label{eq:ksat}
k_\textrm{sat} \in [170\,,\, 300]\,\textrm{GeV}\,,
\end{align}

and use a respective constant strong coupling $g_3(k_\textrm{sat} )$ in the following computation of the pole mass. Varying this coupling for cutoffs in the range eq:ksat provides us with a very conservative systematic error estimate. Moreover, the above argument implies that the minimal value

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

is a good choice for the computation of the mass parameter $m_t$ from the pole mass condition eq:MtopPoleZApprox. Finally, we have to also take into account the uncertainty in the size of the coupling due to mapping the $\overline{\textrm{MS}}$-coupling to that in the MOM${}^2$ scheme in the fRG. Hence, we additionally employ the conservative estimate eq:alphasIntervall and vary the coupling between its $\overline{\textrm{MS}}$-value $\bar\alpha_s$ and 1.1 $\bar\alpha_s$.

In summary this leaves us with the explicit cutoff dependence of the regulator in the loops contributing to $\partial_t\left[ Z_t(p)  M_{t,k}(p)\right]$. Then, the flow is (explicitly) a total derivative w.r.t. to the cutoff scale,

\begin{align}
\partial_t\left[ Z_t(p)  M_{t,k}(p)\right]= \frac{\mathrm d}{\mathrm d t}\left[ Z_t  M_{t}\right]_k^\textrm{1loop}(p) \,,
\end{align}

where $\left[ Z_t  M_{t}\right]_k^\textrm{1loop}(p)$ are the standard one-loop diagrams with all couplings and parameters taken at $k=0$ and the only cutoff dependence in the loops comes from the regulators, either $R_{k=0}= 0$ or $R_{k=k_\textrm{SSB}}$ in the second term. Consequently, in this regime the flow can be integrated analytically, and leads to a difference of standard one-loop diagrams $ \left[Z_t M_t\right]_k^\textrm{1loop}(p)$,

\begin{align}
\Delta\left[ Z_t  M_{t}\right]^\textrm{1loop}(p) = \left[ Z_t  M_{t}\right]_{k=0}^\textrm{1loop}(p)- \left[ Z_t  M_{t}\right]^\textrm{1loop}_{k=k_\textrm{SSB}}(p)\,.
\end{align}

Now we use that for $p\ll k_\textrm{SSB}$ the second term is well approximated by its value at $p=0$. This leads us to a simple one-loop expression with the full (fRG-resummed) couplings and vertices at $k\to 0$,

\begin{align}\label{eq:resummedMt}
\left[ Z_t  M_{t}\right]^\textrm{1loop}(p\lesssim  k_\textrm{SSB} ) =Z_t m_t + \Delta \left[ Z_t  M_{t}\right](p)\,,
\end{align}

with the wave function $Z_t=Z_{t,k=0}(p=0)$ and top mass parameter $m_t=M_{t,k=0}(p=0)$  in our approximation, and

\begin{align}
\Delta \left[ Z_t  M_{t}\right](p)
= \left[ Z_t  M_{t}\right]^\textrm{1loop}(p)-\left[ Z_t M_{t}\right]^\textrm{1loop}_t(0)\,.
\end{align}

The first term $Z_t m_t$ in eq:resummedMt is a direct result from the ASSM flow. The second term, $ \Delta \left[ Z_t  M_{t}\right](p) $ has the form of the one-loop gap equation for the $t$-quark, subtracted at $p=0$; however with the full coupling and mass parameters of the SM at $k=0$. It is a finite difference of loops and can be evaluated within dimensional renormalisation for computational convenience. Its momentum dependence can be read off from the scalar parts of the one-loop self-energy diagrams in eq:dtZM. Generically, the momentum structure of the diagrams reads

\begin{align}\nonumber
{\cal I}(p,m_i) =& \, \mu^{2\varepsilon}\int\! \frac{\mathrm d^d q }{(2 \pi)^d } \int_0^1 \mathrm  d x\\[1ex]\nonumber
& \hspace{.6cm}\times 	\left[\frac{1}{\left[q^2+\Delta(p)\right]^2} -\frac{1}{\left[q^2+\Delta(0)\right]^2} \right]\\[1ex]
=&\,-\frac{1}{(4 \pi)^2}\int_0^1 \!\mathrm d x \log\frac{\Delta(p)}{\Delta(0)}  \,,
\label{eq:DefofCalI}
\end{align}

where $d=4-2\varepsilon$, $\mu$ is the RG scale in dimensional regularisation, and $\Delta$ is

\begin{align}
\Delta(p)= &\, m_{i}^2 (1-x)+ x \, m_{\textrm{t}}^2 + x(1-x) p^2  \,.
\label{eq:Deltas}\end{align}

The right-hand side of eq:DefofCalI does not depend on $\mu$ as the integral does not require renormalisation in the first place and the use of dimensional regularisation was only for computational means.

In eq:DefofCalI, $\Delta(p)$ with $m_i=0$ occurs in the top self-energy diagrams with massless modes such as gluons and photons (at the relevant scales), and the general case with $m_i\neq 0$ is used in top self-energy diagrams with massive modes such as the intermediate vector bosons $Z^0,W^\pm$.

For general masses $m_i$, the self-energy diagrams are computed as

\begin{align}\nonumber
\mathcal{I}(p^2,&m_i)\\[1ex]\nonumber
=  & \,1+\frac12\left(  \frac{m_{\textrm{t}}^2+m_i^2}{m_{\textrm{t}}^2-m_i^2} + \frac{m_{\textrm{t}}^2-m_i^2}{p^2}\right) \log \frac{m_t^2}{m_i^2}\\[1ex]\nonumber
&\, -\frac{\sqrt{m_{i}^4+\left(m_{\textrm{t}}^2+p^2\right)^2+2 m_{i}^2 \left(p^2-m_{\textrm{t}}^2\right)}}{p^2}\\[1ex]
&\times  \arcoth\! \left[\frac{m_{\textrm{t}}^2+m_{i}^2+p^2}{\sqrt{m_{i}^4+\left(m_{\textrm{t}}^2+p^2\right)^2+2 m_{i}^2 \left(p^2-m_{\textrm{t}}^2\right)}}\right].
\label{eq:I}
\end{align}

For massless modes, $m_i=0$, eq:I reduces to

\begin{align}\label{eq:I0}
\mathcal{I}(p^2) = 1-\left( 1+ \frac{m_{\textrm{t}}^2}{p^2} \right) \log(1+\frac{p^2}{m_{\textrm{t}}^2})\,,
\end{align}

where we have introduced $\mathcal{I}(p^2) =	\mathcal{I}(p^2,0)$. With eq:I,eq:I0, we arrive at the final expression for the momentum-dependent mass function,

\begin{align}\nonumber
\Delta [ Z_t &(p) M_t (p ) ]\\[1ex]\nonumber
=& \,\frac{m_{\textrm{t}}}{16\, \pi^2}\,\Bigg[4 \,g^2_3\,\mathcal{I}(p^2)+ 3\, \left(\frac{2\, g_2 \sin \theta_W }{3}\right)^{\!2}\,\mathcal{I}(p^2)\\[1ex]\nonumber
&\hspace{.3cm}-\frac{h^2_t}{2}\, \mathcal{I}(p^2,m_H)+\frac{h^2_t}{2}\, \mathcal{I}(p^2,m_Z) +h^2_b\, \mathcal{I}(p^2,m_W)\\[1ex]
&\hspace{.3cm}	-\frac{g_Y^2 \left(1+2 \cos 2\theta_W\right)}{9}\, \mathcal{I}(p^2,m_Z) \Bigg]\,,
\label{eq:DeltaMtDiagrams}\end{align}

computed in the gauge eq:GFEW. The first two terms on the right-hand side stem from gluon and photon mediated diagrams, and the terms in the second line stem from Higgs, $\mathcal{G}^0$, $\mathcal{G}^{\pm}$, and $Z^0$. Vertices, wave functions and the mass function are given by the full couplings and wave functions at $k=0$ and $p=0$ except for the strong coupling $g_3(k_\textrm{sat})$ which is evaluated at $k_\textrm{sat}=170$ GeV, see eq:ksatChoice.

<Ref> with eq:DeltaMtDiagrams allows us to resolve eq:MtopPoleZApprox for the pole position. This fixes the Euclidean mass parameter to

\begin{align}\label{eq:TopMassParameterApp}
m_{t}= 165.4^{+0.9}_{-0.2}\, \textrm{GeV}\,,
\end{align}

for the PDG pole mass (cross section measurements) $M_t=172.5$ GeV, see eq:topMassPole. The error in eq:TopMassParameterApp is solely due to the error estimate of the value of the strong coupling used in eq:DeltaMtDiagrams. It is discussed below eq:MassfunctionTotal, and is summarised in the inlay of <Ref>: The upper variation comes from the variation of the saturation scale up to $k_\textrm{sat}=300$ GeV with $\alpha_s=\bar\alpha_s(300\,\textrm{GeV})$, while the lower error is related to the estimate of the maximal MOM${}^2$-value of the strong fine structure constant in eq:alphasIntervall with $\alpha_s=1.1\,\bar\alpha_s(170\,\textrm{GeV})$.

We consider the (systematic) error estimate in eq:TopMassParameterApp a very conservative one. Note also that the theoretical systematic error is of the size of the experimental one.

A first prediction concerns the decay width $\Gamma_t$ of the Higgs. We obtain

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

which agrees quantitatively with that in the PDG one, $\Gamma_{t,\textrm{pole}}^{(\textrm{exp})}= 1.42^{+0.19}_{-0.15}\,\text{GeV}$ in eq:topMassWidthPDG.  Note, that while the error in eq:TopMassParameterApp constitutes a systematic error estimate, the error in eq:MpoleWidth only describes the relative weighting of QCD and non-QCD contributions. In a pure QCD system, our analysis would lead to a vanishing error on the decay width in eq:MpoleWidth.


# Joint Matrix Completion and Compressed Sensing for State Estimation in Low-
observable Distribution System

Shweta Dahale, Balasubramaniam Natarajan S. Dahale and B. Natarajan are with
Electrical and Computer Engineering, Kansas State University, Manhattan,
KS-66506, USA, (e-mail<EMAIL_ADDRESS>bala@ksu.edu). This material is
based upon work supported by the Department of Energy, Office of Energy
Efficiency and Renewable Energy (EERE), Solar Energy Technologies Office,
under Award Number DE-EE0008767

###### Abstract

Limited measurement availability at the distribution grid presents challenges
for state estimation and situational awareness. This paper combines the
advantages of two sparsity-based state estimation approaches (matrix
completion and compressive sensing) that have been proposed recently to
address the challenge of unobservability. The proposed approach exploits both
the low rank structure and a suitable transform domain representation to
leverage the correlation structure of the spatio-temporal data matrix while
incorporating the power-flow constraints of the distribution grid. Simulations
are carried out on three phase unbalanced IEEE 37 test system to verify the
effectiveness of the proposed approach. The performance results reveal - (1)
the superiority over traditional matrix completion and (2) very low state
estimation errors for high compression ratios representing very low
observability.

###### Index Terms:

Distribution system state estimation, matrix completion, compressive sensing,
unobservability

##  I Introduction

Distribution grid operation is becoming more challenging due to an increase in
the penetration of distributed energy resources. Instances of reverse power
flow and undesired voltage rise has increased and will occur more frequently
in the future. Therefore, state estimation (SE) is critical for the monitoring
and control of a distribution grid. However, extending the conventional SE
approaches for the distribution grid is difficult mainly because the system is
highly unobservable at the grid edge. The estimation of the network states
with only a limited number of measurements is a major challenge. Furthermore,
the various characteristics of distribution grids such as low $X/R$ ratio [1],
limited bandwidth capacity, unbalanced operation [2], and cyber-security
issues hinders the successful adoption of conventional SE approaches in
distribution grids.

Weighted least squares (WLS) estimation represents the conventional approach
for distribution system state estimation (DSSE). To address the low-
observability issue of distribution system, historical data based pseudo-
measurements are used along-with the WLS approach. However, the inaccuracies
in pseudo-measurements impacts the state estimation performance [3]. Data
driven approaches proposed in [4] employ model-free structure to estimate the
voltage states. However, these approaches still require large number of PMUs
to be installed. Recently, sparsity-based approaches have been used for DSSE
to address the challenge of unobservability. These approaches exploit the
network structure to estimate the states at the current levels of measurement
availability. These approaches neither require pseudo-measurements nor any
extra metering devices. Compressive sensing (CS) based DSSE was one of the
first sparsity-based solution proposed in [5]. This approach exploits the
spatial and temporal sparsity of measurements in a linear transformation basis
[6], [7]. Matrix completion (MC) [8] based DSSE is an another alternative to
deal with limited system observability. This method leverages the standard
matrix completion along-with the power-flow constraints to acknowledge the
physical network constraints. A comparative analysis of these sparsity based
approaches for DSSE along-with their robust formulation is presented in [9].
Authors in [10] proposes a Gaussian process based approach along-with matrix
completion to deal with multi time-scale measurements in a smart distribution
system.

In order to accurately estimate the states with high probability using the
sparsity based approaches, the requirement of minimum number of measurements
must be satisfied. In compressive sensing, the reconstruction of length $N$
states using $M$ measurements where $M\ll N$ is possible by exploiting the
sparsity of states in a transformed basis. $K$-sparse states are recovered
accurately using $M\geq cK\mbox{log}(N/K)$ i.i.d gaussian measurements [11].
In a matrix completion approach, the minimum number of measurements ($m$)
required to recover the matrix of size $n_{1}\times n_{2}$ with high
probability is $m\geq Cn^{1.2}r\mbox{log}n$ where $n=max(n_{1},n_{2})$ and $r$
is the rank of matrix [12]. These requirements restrict the application of
sparsity based approaches for highly unobservable distribution system.
Furthermore, in the matrix completion approach the minimization of nuclear
norm requires solving a semi-definite program which becomes computationally
inefficient for large matrices. An efficient alternating minimization
algorithm is proposed in [13] that reformulates the matrix completion problem
with time-series data.

This paper proposes two unique approaches to estimate the system states when
availability of spatio-temporal measurements at the local control center is
very limited. We consider a commonly occurring practical scenario where the
sensors at specific spatial locations send data to the local center at a
particular sampling rate. The first proposed approach estimates the states by
performing matrix completion across space and compressive sensing across time
using an alternating minimization approach. This formulation estimates the
states in a single shot by exploiting the low rank property as well as
temporal sparsity in the measurements while incorporating the power-flow
constraints.

In the second approach, the compressive sensing and matrix completion are
performed in two stages. In the first stage, the compressed measurements from
a single sensor are recovered by exploiting the sparsity of the states in a
linear transformation basis. In the second stage, matrix completion across the
network is performed at individual time instants. The contributions of this
paper are as follows:

  * •

An efficient and unique state estimation formulation that combines matrix
completion and compressive sensing based approaches in a single powerflow
constrained optimization framework is proposed for the first time.

  * •

The proposed approach exploits the low rank property and compactness of
temporal data in the wavelet transform domain. We validate that both these
property (low-rank and DCT compactness) hold true for practical data.

  * •

The proposed algorithm effectively estimates the states with high fidelity in
very low observability region. We demonstrate the performance of the algorithm
for IEEE 37 unbalanced test system. Relative to the classical matrix
completion approaches, the error performance of the proposed approach offer
nearly 91% improvement at 10% of the fraction of available measurements.

##  II Background

Consider a power distribution grid with $|\mathcal{P}|$ three phase non-slack
buses. Sensors are deployed throughout the grid but due to communication and
other constraints, only a fraction of data is aggregated from these sensors
and used for DSSE. This section first reviews the classic matrix completion
and compressive sensing approaches for DSSE.

###  II-A Classic Matrix completion

In a classic matrix completion based approach for DSSE, a structured matrix
$\mathbf{M}$ is formed such that each column represents a phase and each row
represents a measurement associated with the phase of each bus. The matrix is
given as,

$[\Re(v_{i}),\Im(v_{i}),|v_{i}|,\Re(s_{i}),\Im(s_{i})]^{\intercal}$ (1)

where, $s_{i}$ and $v_{i}$ represents the apparent power injection and
voltages of the $i^{th}$ bus respectively. The term $\Re(\cdot)$ and
$\Im(\cdot)$ represents the real and imaginary part of a complex variable
respectively.

In a distribution system, the matrix $\mathbf{M}$ is partially observed. At
the local control center, only a subset of the measurement matrix i.e.
$P_{\Omega}(\mathbf{M})$ is available. The goal is to fill the missing entries
in the matrix by exploiting the relationship among the raw measurements.
Specifically, the missing elements of the matrix are filled by suitable low-
rank approximation augmented with the power-flow constraints [9]. The
corresponding optimization formulation is given as,

$\displaystyle\mathbf{\hat{X}}=\underset{\mathbf{X}\in\mathbb{R}^{5\times|\mathcal{P}|}}{\text{\enspace
argmin}}\enspace\|\mathbf{X}\|_{*}$ (2) subject to
$\displaystyle\|P_{\Omega}({\mathbf{M}})-P_{\Omega}(\mathbf{X})\|_{F}^{2}<\epsilon$
$\mathbf{v\approx B}\begin{bmatrix}\Re(\mathbf{s})\\\
\Im(\mathbf{s})\end{bmatrix}+\mathbf{w},$ (3) $\mathbf{|v|\approx
C\mathbf{missing}}\begin{bmatrix}\Re(\mathbf{s})\\\
\Im(\mathbf{s})\end{bmatrix}+\mathbf{|w|},$ (4)

where, (3) and (4) captures the linearized power-flow constraints given in
[14]. Here, the nuclear norm
$\|\mathbf{X}\|_{*}=\sum_{i=1}^{r=min(5,|\mathcal{P}|)}\sigma_{i}(\mathbf{X})$
is the sum of the singular values of the matrix $\mathbf{X}$. While matrix
completion exploits the spatial correlation by low rank approximation, it
fails to capture the temporal correlation of the states. Another sparsity
based approach that effectively captures spatial or temporal correlation is
compressive sensing discussed in II.B.

###  II-B Classic compressive sensing

Compressive sensing based DSSE exploits the temporal or spatial sparsity of
measurements or states in a linear transformation basis. Let the states
$\mathbf{z}\in\mathbb{R}^{N}$ be compressible in a linear transformation basis
such that,

$\mathbf{z}=\mathbf{Da}$ (5)

where a has at most $K$ $\ll$ $N$ significant coefficients i.e., z is
$K$-sparse in sparsifying basis $\mathbf{D}$. Compressed measurements are
achieved by taking $M\ll N$ random projections of $\mathbf{z}$,

$\mathbf{h}=\mathbf{\Phi}\mathbf{z};\mathbf{h}\in\mathbb{R}^{M},\mathbf{\Phi}\in\mathbb{R}^{M\times
N},$ (6)

where, $\mathbf{\Phi}$ is a random measurement/projection matrix (e.g., matrix
elements distributed as i.i.d. Gaussian random variable with mean $0$ and
variance $1/M$ or Bernoulli random variables).

In spatial CS, the states
$\mathbf{z}=[\Re(\mathbf{s}),\Im(\mathbf{s}),\tilde{\mathbf{v}}]^{\intercal}$
can be estimated by solving the following $l_{1}$ minimization problem

$\displaystyle\mathbf{\hat{a}}=$
$\displaystyle\underset{\mathbf{s}}{\text{\enspace
min}}\enspace\|\mathbf{s}\|_{1}$ (7) $\displaystyle\text{subject
to}\enspace\mathbf{\|h-\Phi Ds\|_{2}^{2}}<\epsilon$ $\mathbf{v\approx
B}\begin{bmatrix}\Re(\mathbf{s})\\\ \Im(\mathbf{s})\end{bmatrix}+\mathbf{w},$
(8)

The recovered states $\mathbf{\hat{z}}$ are given as
$\mathbf{\hat{z}}=\mathbf{D}\mathbf{\hat{a}}$. Here, $\|\mathbf{s}\|_{1}$
represents the $l_{1}$ norm and $\mathbf{B}$, $\mathbf{w}$ are as defined
earlier. The compression of the states $\mathbf{z}$ is indicated by the
compressed measurement ratio (CMR) given as $CMR=\frac{M}{N}$. In spatial CS,
in order to construct the projection matrix $\mathbf{\Phi}$, the elements of
$\mathbf{z}$ should be known apriori. However, it may not be practical to
construct this matrix. Another form of CS captures the temporal sparsity of
each states $\mathbf{z}$ whose optimization formulation is similar to (7)
except the powerflow constraints (8). Thus, it is makes sense to incorporate
matrix completion in space and compressive sensing in time to exploit the
sparsity of the states. Jointly incorporating both of these approaches would
aid in accurately recovering the states in low-observable conditions.

##  III Proposed Approach

In this section, we have proposed two approaches to jointly estimate the
states using matrix completion and compressive sensing based DSSE techniques.

###  III-A Joint MC-CS approach

Figure 1: Framework of the Joint MC-CS approach

The framework of the joint MC-CS approach is shown in Fig.1. Assume the
sensors located at a subset of buses send data to the local control center at
time $t=1,2,...,T$ at a predefined sampling rate. Let $\mathbf{M}^{t}$ denote
the measurement matrix at time $t$ whose structure is given in (1). A block
matrix $\mathbf{M}$ is constructed as,

$\mathbf{M}=[\mathbf{M}^{1};\mathbf{M}^{2};...,\mathbf{M}^{T}]\in\mathbb{R}^{m\times
n}$ (9)

Here, $m=5T$ and $n=|\mathcal{P}|$. The linearized powerflow constraints from
(3) and (4) at time $t=1,..,T$ can be written as,

$\mathbf{y\approx Ap+b}$
$\text{where}:\mathbf{y}=[\Re(\mathbf{v}^{1}),\Im(\mathbf{v}^{1}),|\mathbf{v}^{1}|,...,\Re(\mathbf{v}^{T}),\Im(\mathbf{v}^{T}),|\mathbf{v}^{T}|]^{\intercal},$
$\mathbf{A}=\begin{bmatrix}\mathbf{B_{1}}&\mathbf{B_{2}}&\mathbf{0}&\cdots&\mathbf{0}&\mathbf{0}\\\
\mathbf{B_{3}}&\mathbf{B_{4}}&\mathbf{0}&\cdots&\mathbf{0}&\mathbf{0}\\\
\mathbf{C_{1}}&\mathbf{C_{2}}&\mathbf{0}&\cdots&\mathbf{0}&\mathbf{0}\\\
\vdots&\vdots&\vdots&\ddots&\mathbf{0}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&\mathbf{0}&\cdots&\mathbf{B_{1}}&\mathbf{B_{2}}\\\
\mathbf{0}&\mathbf{0}&\mathbf{0}&\cdots&\mathbf{B_{3}}&\mathbf{B_{4}}\\\
\mathbf{0}&\mathbf{0}&\mathbf{0}&\cdots&\mathbf{C_{1}}&\mathbf{C_{2}}\\\
\end{bmatrix},$
$\mathbf{p}=[\Re(\mathbf{s}^{1})^{\intercal}\enspace\Im(\mathbf{s}^{1})^{\intercal}...\enspace\Re(\mathbf{s}^{T})^{\intercal}\enspace\Im(\mathbf{s}^{T})^{\intercal}]^{\intercal}$
$\text{and}\enspace\mathbf{b}=[\Re(\mathbf{w})^{\intercal},\Im(\mathbf{w})^{\intercal},|\mathbf{w}|^{\intercal}...\Re(\mathbf{w})^{\intercal},\Im(\mathbf{w})^{\intercal},|\mathbf{w}|^{\intercal}]^{\intercal}$

where, the term $[\mathbf{B_{1}\enspace B_{2}}]=\Re(\mathbf{B})$,
$[\mathbf{B_{3}\enspace B_{4}}]=\Im(\mathbf{B})$ and $[\mathbf{C_{1}\enspace
C_{2}}]=\mathbf{C}$.

It is important to note that the rows/columns of the matrix $\mathbf{M}$ is
observed to exhibit low rank feature and discrete cosine transform (DCT)
compactness properties as discussed next.

####  III-A1  Low rank property

The columns of the matrix $\mathbf{M}$ are dependent on each other as there
exists spatial correlation between different locations in a power grid.
Furthermore, the physics of power-flow relates the different measurements.
Hence, $\mathbf{M}$ possess low-rank property which can be evaluated by
calculating the SVD of the matrix as,

$\mathbf{M=U\Sigma V^{T}}$ (10)

where the matrix $\mathbf{U}$ $\in$ $\mathbb{R}^{m\times m}$,
$\mathbf{V}\in\mathbb{R}^{n\times n}$ and
$\mathbf{\Sigma}\in\mathbb{R}^{m\times n}$ containing $p=min(m,n)$ singular
values arranged in descending order ($\sigma_{1}>\sigma_{2}>...>\sigma_{p}$).
For our experiments involving practical data from IEEE 37 bus test system, it
can be inferred that the largest 5 singular values occupy about 99.9% of the
energy confirming the low-rank property of the measurement matrix. This
property has also been confirmed by other prior efforts on matrix completion
[8].

####  III-A2 DCT compactness analysis

In a distribution network, the loads are observed to be slowly changing over
time. The temporal data in the matrix $\mathbf{M}$ represented by
$\mathbf{x}_{i}$ is observed to exhibit sparsity in a linear transformation
basis [5]. Discrete cosine transform (DCT) enables to represent the data in a
fewer coefficients. The DCT matrix $\mathbf{D}$ = ’$\\{d(k,n)\\}$’ of
dimension $T\times T$ as defined in [15] can be split as,

$\mathbf{D}=\begin{bmatrix}\mathbf{D}_{1}\\\ \mathbf{D}_{2}\end{bmatrix}$

where $\mathbf{D}_{1}$ consists of first ${}^{\prime}j^{\prime}$ rows of
$\mathbf{D}$ and $\mathbf{D}_{2}$ consists of last ${}^{\prime}T-j^{\prime}$
rows. To exhibit temporal sparsity for the timeseries data $\mathbf{x}_{i}$,
only few DCT coefficients will capture most of the energy i.e.,

$\frac{\|\mathbf{D}_{1}\mathbf{x}_{i}\|_{2}}{\|\mathbf{x}_{i}\|_{2}}\approx
1,\enspace\frac{\|\mathbf{D}_{2}\mathbf{x}_{i}\|_{2}}{\|\mathbf{x}_{i}\|_{2}}\approx
0$

This property can be observed from the practical time-series data from IEEE 37
bus test system where 1-2 DCT coefficients occupy 99% of the energy, thus
proving the temporal data in matrix $\mathbf{M}$ is compact. However, due to
limited system observability, only limited entries of the matrix $\mathbf{M}$
are observed. In order to recover the complete matrix, the low-rank feature
property, DCT compactness and the linearized power-flow constraints are
exploited in an integrated optimization formulation corresponding to,

$\displaystyle\underset{\mathbf{X}}{\text{\enspace
min}}\enspace\|\mathbf{X}\|_{*}+\lambda_{1}\|P_{\Omega}(\mathbf{X})-P_{\Omega}(\mathbf{M})\|_{F}^{2}+\nu\|\mathbf{y-(Ap+b)}\|_{2}^{2}+$
(11) $\displaystyle\lambda_{2}\|\mathbf{s}\|_{2}$
$\displaystyle\textrm{s.t.}\quad\mathbf{y}=[a_{1}^{\intercal}\mathbf{X}\enspace
a_{2}^{\intercal}\mathbf{X}\enspace...\enspace
a_{3T-2}^{\intercal}\mathbf{X}\enspace a_{3T-1}^{\intercal}\mathbf{X}\enspace
a_{3T}^{\intercal}\mathbf{X}]^{\intercal},$
$\displaystyle\mathbf{p}=[c_{1}^{\intercal}\mathbf{X}\enspace
c_{2}^{\intercal}\mathbf{X}\enspace...\enspace
c_{2T-1}^{\intercal}\mathbf{X}\enspace
c_{2T}^{\intercal}\mathbf{X}]^{\intercal},$
$\displaystyle\mathbf{s}=\begin{bmatrix}\mathbf{D}_{2}\enspace(e_{4}^{\intercal}\enspace
reshape(\mathbf{X}(:,1),[5,T]))^{\intercal}\\\
\mathbf{D}_{2}\enspace(e_{5}^{\intercal}\enspace
reshape(\mathbf{X}(:,1),[5,T]))^{\intercal}\\\
\mathbf{D}_{2}\enspace(e_{1}^{\intercal}\enspace
reshape(\mathbf{X}(:,1),[5,T]))^{\intercal}\\\ \vdots\\\
\mathbf{D}_{2}\enspace(e_{1}^{\intercal}\enspace
reshape(\mathbf{X}(:,n),[5,T]))^{\intercal}\\\ \end{bmatrix}$

where, $m=5T$, $n=|\mathcal{P}|$, $\mathbf{X}\in\mathbb{R}^{m\times n}$,
$\mathbf{y}\in\mathbb{R}^{\frac{3}{5}mn}$,
$\mathbf{p}\in\mathbb{R}^{\frac{2}{5}mn}$
Here, $a_{3(t-1)+i}=e_{5(t-1)+i}$ and $c_{2(t-1)+i}=e_{5(t-1)+3+i}$ are the
standard basis vectors in $\mathbb{R}^{m},$ $e_{1}$, $e_{4}$ and $e_{5}$ are
the standard basis vectors in $\mathbb{R}^{5}$. The parameters
$\lambda_{1}\geq 0$, $\nu\geq 0$, $\lambda_{2}\geq 0$ are the tuning
parameters.
The matrix $\mathbf{X}$ can be factorized into two matrices $\mathbf{U}$ and
$\mathbf{V}$. The nuclear norm of $\mathbf{X}$ can be expressed by the
Frobenius norm of matrix $\mathbf{U}$ and $\mathbf{V}$ given as,

$\displaystyle{\|\mathbf{X}\|_{*}}=$
$\displaystyle\underset{\mathbf{U},\mathbf{V}}{\text{\enspace
min}}\enspace\|\mathbf{U}\|^{2}_{F}+\|\mathbf{V}\|^{2}_{F}$ (12)
$\displaystyle\text{subject to}\enspace\mathbf{X=UV}$

Substituting (12) in (11), we obtain the following optimization problem,

$\displaystyle\underset{\mathbf{U,V}}{\text{min}}\enspace\|\mathbf{U}\|^{2}_{F}+\|\mathbf{V}\|^{2}_{F}+\lambda_{1}\|P_{\Omega}(\mathbf{UV})-P_{\Omega}(\mathbf{M)}\|_{F}^{2}+$
(13)
$\displaystyle\nu\|f_{1}(\mathbf{UV})-(Af_{2}(\mathbf{UV})+\mathbf{b})\|_{2}^{2}+\lambda_{2}\|f_{3}(\mathbf{UV})\|_{2}$
$\displaystyle\textrm{s.t.}\quad
f_{1}(\mathbf{UV})=[a_{1}^{\intercal}\mathbf{X}\enspace
a_{2}^{\intercal}\mathbf{X}\enspace...\enspace
a_{3T-2}^{\intercal}\mathbf{X}\enspace a_{3T-1}^{\intercal}\mathbf{X}\enspace
a_{3T}^{\intercal}\mathbf{X}]^{\intercal},$ $\displaystyle
f_{2}(\mathbf{UV})=[c_{1}^{\intercal}\mathbf{X}\enspace
c_{2}^{\intercal}\mathbf{X}\enspace...\enspace
c_{2T-1}^{\intercal}\mathbf{X}\enspace
c_{2T}^{\intercal}\mathbf{X}]^{\intercal},$ $\displaystyle
f_{3}(\mathbf{UV})=\begin{bmatrix}\mathbf{D}_{2}\enspace(e_{4}^{\intercal}\enspace
reshape(\mathbf{X}(:,1),[5,T]))^{\intercal}\\\
\mathbf{D}_{2}\enspace(e_{5}^{\intercal}\enspace
reshape(\mathbf{X}(:,1),[5,T]))^{\intercal}\\\
\mathbf{D}_{2}\enspace(e_{1}^{\intercal}\enspace
reshape(\mathbf{X}(:,1),[5,T]))^{\intercal}\\\ \vdots\\\
\mathbf{D}_{2}\enspace(e_{1}^{\intercal}\enspace
reshape(\mathbf{X}(:,n),[5,T]))^{\intercal}\\\ \end{bmatrix}$
$\displaystyle\mathbf{X=UV}$

The matrix completion formulation in (13) is a non-convex problem. But using
alternating minimization algorithm [13], the problem becomes convex when
either $\mathbf{U}$ or $\mathbf{V}$ is fixed. This algorithm updates the
variables $\mathbf{U}$ and $\mathbf{V}$ at each iteration $k$ in an
alternating fashion while fixing the other factor. The update rules are given
by the following update equations as, [3] U^(k) ∥ U∥^2_F + λ_1 ∥ P_Ω(UV^(k-1))
- P_Ω(M)∥_F ^2 + ν ∥ f_1(UV^(k-1)) - (A f_2(UV^(k-1)) + b) ∥_2^2 \+ λ_2 ∥
f_3(UV^(k-1))∥_2 [2] V^(k) ∥ V∥^2_F + λ_1 ∥ P_Ω(U^kV)- P_Ω(M)∥_F ^2 + ν ∥
f_1(U^kV) - (A f_2(U^kV) + b) ∥_2^2 \+ λ_2 ∥ f_3(U^kV)∥_2 The alternating
minimization approach for the proposed approach is given in Algorithm 1.

###  III-B CS-MC Approach

The joint MC-CS approach proposed in section III.A directly incorporates the
raw information from each of the sensors at a particular bus for state
estimation. However, due to the network bandwidth limitation, it may not be
practical to collect all the temporal measurements. Furthermore, as stated
earlier, to solve (9) involving large matrices can be computationally
inefficient. Therefore, we propose a state estimation approach that alleviates
these drawbacks.

The framework of the CS-MC approach is shown in Fig.2. Assume the measurements
collected from each buses is $\mathbf{\Phi}\mathbf{x}$ where
$\mathbf{\Phi}\mathbf{x}\ll$ $\mathbf{x}$. The recovery of the states is
performed in two stages. Firstly, the incomplete temporal measurements are
recovered by compressive sensing using (7). It should be noted that no power-
flow constraints are used at this stage. Once the estimates of all the
temporal measurements are obtained, the second stage consists of recovering
the spatial states by classic matrix completion based state estimation using
(2). The state estimation is performed separately at each time steps along-
with the power-flow constraints in (3) and (4).

Figure 2: Framework of the CS-MC approach

Input: Measurement matrix $\mathbf{M}$, Set of known indices $\Omega$, DCT
matrix $\mathbf{D}$, system model $\mathbf{A}$, $\mathbf{b}$, number of
iterations $N$.

initialization: Compute the SVD of the matrix $\mathbf{M=U\Sigma V^{T}}$, set
$\mathbf{V}^{(0)}=\mathbf{\Sigma}^{0.5}\mathbf{V}$

1: for $k=1,..,N$ do

2: Solve (13)

3: Solve (13)

4: end for

5: return $\mathbf{X}=\mathbf{U}^{(N)}\mathbf{V}^{(N)}$

Algorithm 1 Alternating minimization Algorithm for matrix completion across
space and compressive sensing across time

##  IV Computational Complexity

In this section, the computational complexity of the proposed approaches for a
matrix $\mathbf{X}\in\mathcal{R}^{5T\times|\mathcal{P}|}$ is discussed. In the
joint MC-CS approach, in order to solve the power-flow constraints,
$\mathcal{O}(|\mathcal{P}|^{3}T^{3})$ computations are required at each
iteration. In a classic MC approach, the matrix
$\mathbf{X}\in\mathcal{R}^{5\times|\mathcal{P}|}$ is estimated separately for
each time $T$. Denoting $m$ and $n$ as the rows and columns of $\mathbf{X}$,
the main computation is involved in calculation of the SVD of the matrix which
is $\mathcal{O}(mn\cdot\mbox{min}({m,n}))$ along-with the power-flow
constraints $\mathcal{O}(n^{3})$ at each time. Hence, the overall worst case
computational complexity is $\mathcal{O}(|\mathcal{P}|^{3}T)$. The CS-MC
approach performs CS for each temporal state and classic matrix completion in
the next step. CS requires $\mathcal{O}(T\cdot k\cdot\mbox{min}(T,k))$
computations where $T$ denotes the time and $k$ as constraints. The overall
worst case complexity in both the steps is $\mathcal{O}(|\mathcal{P}|^{3}T)$.
This illustrates that the proposed joint MC-CS approach is computationally
expensive as compared to the conventional matrix completion as well as the CS-
MC approach.

##  V Simulation Results and Discussion

In this section, the efficacy of the proposed formulation is demonstrated on
the IEEE 37 unbalanced three phase test system. We characterize the
performance of power and voltage magnitude recovery using the mean absolute
percentage error (MAPE) metric and the voltage angle recovery using mean
integrated absolute error (MIAE) metric [9]. For simulation, we consider a
matrix that includes 8 time steps and whose entries are randomly available
representing different fractions of available data (FAD). The compression of
the temporal measurements in the CS-MC approach is indicated by CMR as defined
earlier. Fig. 3 shows the recovery performance of power (active and reactive)
using the two proposed approaches. Fig.4 -5 shows the comparative performance
of the proposed formulation with the classic MC in the recovery of voltage
magnitude and voltage angle states respectively. It can be inferred that the
proposed joint MC-CS approach as well as the CS-MC approach outperforms the
classic matrix completion at all FADs. The performance is superior especially
in the low observability region. This is due to the fact that this approach
exploits both the spatial correlation as well as temporal correlation of the
states. The joint MC-CS approach, although computationally expensive, is
superior than CS-MC approach. This is due to the fact that raw measurements
are used for estimating the states rather than the compressed measurements.
The recovery of states in CS-MC approach also depends on the CMR of the
temporal measurements. The error in the recovery performance increases as the
CMR is decreased. Fig. 3-5 shows the performance of CS-MC at 40% and 80% CMR.

Figure 3: Power recovery at different FADs Figure 4: Voltage magnitude
recovery at different FADs Figure 5: Voltage angle recovery at different FADs

##  VI Conclusion

This paper presents an efficient joint matrix completion and compressive
sensing approach to perform DSSE. The proposed approach employs an alternating
minimization approach to estimate an incomplete spatio-temporal matrix. The
efficacy of the proposed approach was demonstrated using the IEEE 37 bus test
system. It can be inferred from the simulation results that the proposed
approach has a superior performance relative to classic matrix completion and
provides lower state estimation errors at high compression ratios.

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