text:
\section{Introduction}
\label{sec:intro}

Classical gravity described by General Relativity (GR) has passed many experimental tests, from the scale of the Solar System~\cite{Will:2014kxa} and binary pulsars~\cite{Wex:2014nva, Kramer:2016kwa} to the coalescence of binary black holes (BHs) \cite{Abbott:2016blz,TheLIGOScientific:2016src,TheLIGOScientific:2016pea,Abbott:2017vtc,LIGOScientific:2019fpa} and neutron stars (NSs) \cite{Abbott:2018lct}.
Despite its observational success, certain theoretical aspects of GR (e.g., its nonrenormalizability and its prediction of singularities \cite{Hawking:1969sw}) impede progress toward a complete theory of quantum gravity; yet, strong-field modifications of the theory may alleviate these issues \cite{Stelle:1976gc}.

Gravitational wave (GW) observations probe the nonlinear, strong-field behavior of gravity and thus can be used to search for (or constrain) deviations from GR in this regime.
Because detectors are typically dominated by experimental noise, sophisticated methods are required to extract GW signals.
The most sensitive of these techniques rely on modeled predictions of signals (gravitational waveforms), which are matched against the data.
This same approach can be adopted to test gravity with GWs; to do so requires accurate signal models that faithfully incorporate the effects from the strong-field deviations one hopes to constrain~\cite{Will:2014kxa}.
Ideally, these models would be agnostic about details of the strong-field modifications to GR, so that a single test could constrain a variety of alternative theories of gravity.

This work establishes a framework for such tests given the hypothetical scenario in which the gravitational sector manifests phase transitions, with only one phase corresponding to classical GR.
This proposal comprises an attractive target for binary pulsar and GW tests of gravity; if the transition between phases arises only in the strong-gravity regime (e.g., in the presence of large curvature, relativistic matter, etc.), then such a theory could generate deviations from GR in compact binary systems while simultaneously evading stringent constraints set by weak-gravity tests.
We consider the case wherein the ``new'' phases arise via spontaneous symmetry breaking in the gravitational sector.
Similar phase transitions occur in many areas of contemporary physics---perhaps the most famous example is the electroweak symmetry breaking through the Higgs field~\cite{Englert:1964et,Higgs:1964pj,Guralnik:1964eu}---so it is sensible to consider their appearance in gravity as well.
As a first step, we focus on a simple set of such gravitational theories, in which the transition from GR to a new phase most closely resembles the spontaneous magnetization of a ferromagnet; however, these theories can also be extended to instead replicate the standard Higgs mechanism in the gravitational sector \cite{Coates:2016ktu,Ramazanoglu:2018tig}.

Specifically, we investigate the nonlinear \textit{scalarization} of nonrotating compact objects (BHs and NSs), which arises from spontaneous symmetry breaking of an additional scalar component of gravity~\cite{Damour:1996ke,Sennett:2017lcx}.
\textit{Spontaneous scalarization}---the scalarization of a single, isolated object---has been found in several scalar extensions of GR, including massless \cite{Damour:1993hw,Stefanov:2007eq,Doneva:2010ke,Palenzuela:2015ima,Mendes:2016fby,Herdeiro:2018wub,Fernandes:2019rez} and massive \cite{Chen:2015zmx,Ramazanoglu:2016kul,Sagunski:2017nzb} scalar-tensor (ST) theories and extended scalar-tensor-Gauss-Bonnet (ESTGB) theories \cite{Doneva:2017bvd,Silva:2017uqg,Antoniou:2017acq,Antoniou:2017hxj,Doneva:2017duq,Brihaye:2019kvj}.
Similar phenomena can also occur for vector \cite{Ramazanoglu:2017xbl, Ramazanoglu:2019gbz}, gauge \cite{Ramazanoglu:2018tig}, and spinor \cite{Ramazanoglu:2018hwk} fields.
In contrast, \textit{dynamical scalarization}---scalarization that occurs during the coalescence of a binary system---has been demonstrated and modeled only for NS binaries in ST theories \cite{Barausse:2012da,Palenzuela:2013hsa,Shibata:2013pra,Sampson:2014qqa,Taniguchi:2014fqa,Zimmerman:2015hua,Sennett:2016rwa,Sennett:2017lcx}.

A scalarized compact object emits scalar radiation when accelerated, analogous to an accelerated electric charge.
In a binary system, the emission of scalar waves augments the energy dissipation through the (tensor) GWs found in GR, hastening the orbital decay.
Radio observations of binary pulsars \cite{Wex:2014nva, Kramer:2016kwa,Shao:2017gwu,Anderson:2019eay} and GW observations of coalescing BHs and NSs \cite{Abbott:2018lct,LIGOScientific:2019fpa} are sensitive to anomalous energy fluxes, and thus can be used to constrain the presence of scalarization in such binaries.
Binaries containing spontaneously scalarized components emit scalar radiation throughout their entire evolution.
In contrast, dynamically scalarizing binaries transition from an unscalarized (GR) state to a scalarized (non-GR) state at some critical orbital separation, only emitting scalar waves after this point.
Because this is a second-order phase transition~\cite{Sennett:2017lcx}, the emitted GWs contain a sharp feature corresponding to the onset of dynamical scalarization.
This feature cannot be replicated within typical theory-agnostic frameworks used to test gravity~\cite{Arun:2006hn,Yunes:2009ke, Li:2011cg, Agathos:2013upa}, as these only consider smooth deviations from GR predictions, e.g., modifications to the coefficients of a power-series expansion of the phase evolution.

While one could attempt to model dynamical scalarization phenomenologically by adding nonanalytic functions to such frameworks \cite{Sampson:2013jpa,Sampson:2014qqa}, in this paper, we propose a complementary theory-agnostic approach.
We focus on a specific non-GR effect, here scalarization, but remain agnostic toward the particular alternative theory of gravity in which it occurs.
The basis for our framework is effective field theory.
Scalarization arises from strong-field, nonlinear scalar interactions in the vicinity of compact objects; the details of this short-distance physics depends on the specific alternative to GR that one considers.
By integrating out these short-distance scales, we construct an effective point-particle action for scalarizing bodies in which the relevant details of the modification to GR are encapsulated in a small set of form factors.
The coefficients of these couplings offer a concise parametrization ideal for searches for scalarization with GWs.
The essential step in constructing this effective theory is identifying the fields and symmetries relevant to this phenomenon.
Starting from the perspective that scalarization coincides with the appearance of a tachyonic scalar mode of the compact object, we derive the unique leading-order effective action valid near the critical point of the phase transition.
Though this effective action matches that of Ref.~\cite{Sennett:2017lcx}---which describes the scalarization of NSs in ST theories\footnote{Dynamical scalarization was also modeled at the level of equations of motion in Refs.~\cite{Palenzuela:2013hsa, Sennett:2016rwa}.}---the approach described here is valid for a broader range of non-GR theories.

Our proposed parametrization of scalarization is directly analogous to the standard treatment of tidal interactions in compact binary systems.
Tidal effects enter GW observables through a set of parameters that characterize the response of each compact object to external tidal fields~\cite{Flanagan:2007ix}.
These parameters are determined by the structure of the compact bodies---for example, the short-distance nuclear interactions occurring in the interior of a NS.
This description of tidal effects is applicable to a broad range of nuclear models (i.e., NS equations of state) and offers a more convenient parametrization of unknown nuclear physics for GW measurements~\cite{TheLIGOScientific:2017qsa,Abbott:2018wiz} than directly incorporating nuclear physics into GW models.
From the perspective of modeling compact binaries, the primary difference between tidal effects and scalarization is that the latter is an inherently nonlinear phenomenon, necessitating higher-order interactions in an effective action.

Beyond offering a convenient parametrization for GW tests of gravity, our effective action also elucidates certain generic properties of scalarization phenomena.
Using a simple analysis of energetics based on the effective theory, we argue that \emph{any theory that admits spontaneous scalarization must also admit dynamical scalarization}.
Additionally, this type of analysis can provide further insights regarding the (nonperturbative) stability of scalarized configurations and the critical phenomena close to the scalarization phase transition.
We illustrate these points by applying our energetics analysis to a simple Einstein-Maxwell-scalar (EMS) theory in which electrically charged BHs can spontaneously scalarize, complementing previous results for NSs in ST theories \cite{Sennett:2017lcx}.

The paper is organized as follows.
In Sec.~\ref{sec:effaction}, we first review the mechanism of scalarization as the spontaneous breaking of the $\mathbb{Z}_2$ symmetry of a scalar field driven by a linear scalar-mode instability.
Then, we construct an effective worldline action for a compact object interacting with a scalar field valid near the onset of scalarization.
In Sec.~\ref{sec:matching}, we discuss how the relevant coefficients in the action can be matched to the energetics of an isolated static compact object in an external scalar field, demonstrating the procedure explicitly with BHs in the EMS theory of Ref.~\cite{Herdeiro:2018wub}.
In Sec.~\ref{sec:strongeffects}, we employ the effective action to further investigate scalarization in this EMS theory: we examine the stability of scalarized configurations, compute the critical exponents of the scalarization phase transition, and predict the frequency at which dynamical scalarization occurs for binary BHs.
We also argue that dynamical scalarization is as ubiquitous as spontaneous scalarization in modified theories of gravity.
Finally, in Sec.~\ref{sec:conclusions}, we summarize the main implications of our findings, and discuss future applications of our framework.
The appendixes provide a derivation of a more general effective action and details on the construction of numerical solutions for isolated BHs in EMS theory.\footnote{Throughout this work, we use the conventions of
  Misner, Thorne, and Wheeler~\cite{Misner:1974qy} for the metric
  signature and Riemann tensor and work in units in which the speed of
  light and bare gravitational constant are unity.}

\section{Linear mode instability and effective action close to critical point}
\label{sec:effaction}

In this section, we review the connection between the appearance of an unstable scalar mode in an unscalarized compact object and the existence of a scalarized state for the same body. We then derive an effective action close to this critical point at which this mode becomes unstable.

As an illustrative toy model for this discussion, we consider the modified theory of electrodynamics introduced in Ref.~\cite{Herdeiro:2018wub} (hereafter referred to as EMS theory for brevity), whose action is given by
\begin{equation} \label{EMSaction}
  S_\text{field} = \int d^4x \, \frac{\sqrt{-g}}{16\pi} \left[ R - 2 \partial_\mu \scalar \partial^\mu \scalar - f(\scalar) F^{\mu\nu} F_{\mu\nu} \right],
\end{equation}
where $R$ is the Ricci scalar, $g$ is the determinant of the metric $g_{\mu\nu}$, and $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ is the electromagnetic field tensor.
In this paper, we consider two choices of scalar couplings:
\begin{align}
f_1(\phi) =& e^{-\alpha\scalar^2}, \label{eq:f1coupling} \\
f_2(\phi) = &\left(1+\alpha \scalar^2 -\frac{1}{2} \alpha^2 \phi^4\right)^{-1}, \label{eq:f2coupling}
\end{align}
where $\alpha$ is a dimensionless coupling constant.
While the two couplings have the same behavior near $\phi =0$, their behavior for large field value differs drastically.
The former choice was used in Ref.~\cite{Herdeiro:2018wub} to construct stable scalarized BH solutions, whereas we introduce the latter in this work to demonstrate a theory in which no stable scalarized BH configurations exist (see Sec.~\ref{sec:strongeffects}).

The absence of any linear coupling of $\phi$ to the Maxwell term implies that any solution in Einstein-Maxwell (EM) theory, i.e., with $\scalar = 0$, also solves the field equations of Eq.~\eqref{EMSaction}; however, stable solutions in EM theory may be unstable in EMS theory.
To see this, we write the scalar-field equation schematically as
\begin{equation}
  \Box \scalar = m_\text{eff}^2 \, \scalar , \qquad \label{eq:EffMass}
  m_\text{eff}^2 = \frac{f'(\phi)}{4 \phi} F^{\mu\nu} F_{\mu\nu} .
\end{equation}
We consider an electrically charged BH, for which ${F^{\mu\nu} F_{\mu\nu} < 0}$ and thus the effective-mass squared $m_\text{eff}^2$ is negative for $\alpha <0$.
One can decompose $\scalar$ into Fourier modes with frequency $\omega$ and wave vector $\vct{k}$, which satisfy the dispersion relation $\omega^2 \approx \vct{k}^2 + m_\text{eff}^2(\vct{k})$, where curvature corrections have been dropped for simplicity.
We see that if $m_\text{eff}^2(\vct{k})$ is sufficiently negative, then $\omega^2$ is also negative, leading to a tachyonic instability.
The critical point at which this tachyonic instability first appears can be determined by identifying linearly unstable quasinormal scalar modes of the EM solution~\cite{Silva:2017uqg,Myung:2018vug} or by constructing sequences of fully nonlinear, static scalarized solutions (as we do here)~\cite{Herdeiro:2018wub,Myung:2018vug}.

The tachyonic instability drives the body away from the unscalarized solution, thereby breaking the symmetry $\scalar \rightarrow - \scalar$ in Eq.~\eqref{EMSaction}.
For a stable scalarized equilibrium configuration to exist, this instability must saturate in the nonlinear regime \cite{Ramazanoglu:2017yun}.
These two conditions---the existence of a tachyonic instability and its eventual saturation---are satisfied in all of the theories discussed previously~\cite{Damour:1993hw,Stefanov:2007eq,Doneva:2010ke,Palenzuela:2015ima,Mendes:2016fby,Chen:2015zmx,Ramazanoglu:2016kul,Sagunski:2017nzb,Doneva:2017bvd,Silva:2017uqg,Antoniou:2017acq,Antoniou:2017hxj,Doneva:2017duq,Brihaye:2019kvj,Herdeiro:2018wub,Fernandes:2019rez,Ramazanoglu:2017xbl,Ramazanoglu:2019gbz,Ramazanoglu:2018tig,Ramazanoglu:2018hwk}.
The only difference between these theories is the form of $m_\text{eff}^2$; for example, in ST theories $m_\text{eff}^2$ depends on the stress-energy tensor, while in ESTGB it depends on the Gauss-Bonnet invariant.
Indeed, theories which meet these two criteria can be straightforwardly constructed, which is the reason why scalarization is such a ubiquitous phenomenon.

\begin{figure*}
\includegraphics{matchingfig}
\caption{\label{fig:matching} An illustration of the approach used in this paper (read from left to right).
  (\emph{Left}) The fields describing an isolated compact object in equilibrium are decomposed into short- (UV) and long- (IR) wavelength modes, depicted in blue and red, respectively.
  (\emph{Center}) We integrate out the UV modes via an IR projection.
  (\emph{Right}) By matching asymptotics, we identify the coarse-grained compact object with an effective point-particle model that describes the IR sector of the full theory.
  Section~\ref{sec:matching} contains a detailed description of this procedure and the definitions of all quantities shown above.
}
\end{figure*}

An even simpler perspective on scalarization arises from a coarse-grained, or effective, theory. Let us derive it explicitly.
We start by splitting the fields into the short- (or ultraviolet, UV) and long- (or infrared, IR) wavelength regimes separated by the object's size $\sim R$, i.e., $\scalar = \scals + \scalf$, and spatially average over (integrate out) the UV parts.
This effectively shrinks the compact object to a point and its effective action is given by an integral over a worldline $\y^\mu(\sig)$, where $\sig$ is the proper time (see Fig.~\ref{fig:matching} for a schematic illustration).
Dynamical short-length-scale processes like oscillations of the object are represented by dynamical variables on the worldline.
For simplicity, we assume that we can also average over fast oscillation modes and only retain the monopolar mode associated to a linear tachyonic instability, denoted by $\q(\sig)$.
This mode $\q(\sig)$ can indeed be excited by IR fields, since its frequency (or effective mass) vanishes at the critical point.

Effective actions are usually constructed by making an ansatz respecting certain symmetries and including only terms up to a given power in the cutoff between IR and UV scales.
The relevant symmetries here are diffeomorphism, U(1)-gauge, worldline-reparametrization, time-reversal\footnote{Time reversal is an approximate symmetry of compact objects in an adiabatic setup, like the inspiral of a binary system. In this case, a compact object's entropy remains approximately constant.}, and scalar-inversion invariance.
The last reads $\scalar \rightarrow -\scalar$ in the full theory, so in the effective action it decomposes into simultaneous IR $\scals \rightarrow - \scals$ and UV $\q \rightarrow -\q$ transformations.
The IR fields are of order $\scals \sim \Order(R/r)$ on the worldline, where $r$ is the typical IR scale (e.g., the separation of a binary).
Now, the oscillator equation for the mode $q(\sig)$ driven by the IR field $\scals$ can be schematically written as
\begin{equation}
  c_{\dot{\q}^2} \ddot{\q} + V'(q) = \scals(y) , \quad
  V(q) = \frac{c_{(2)}}{2} q^2 + \frac{c_{(4)}}{4!} q^4 + \dots ,\label{eq:effEOM}
\end{equation}
where $\dot{~} = d / d \sig$ and the $c_{\dots}$ are constant coefficients determined by the UV physics.
(The singular self-field contribution to $\scals(y)$ must be removed using some regularization prescription.)
The normalization of $\q$ is chosen to fix the coefficient of $\scals(y)$; for all that follows, we simply assume that $c_{\dot{\q}^2} >0$.
Close to the critical point, the quadratic term in $V$ is negligible, and thus from Eq.~\eqref{eq:effEOM}, one finds that for equilibrium configurations ($\dot{\q} =0$), $\q$ scales as $\q^3 \sim \scals$.
More generally, the mode $\q$ oscillates around this equilibrium point provided that the IR field evolves slowly relative to the frequency of the mode, i.e., $\dot{\phi}^\text{IR} = \dot{y}^\mu \partial_\mu \scals \ll \dot{\q} / \q$; this condition is satisfied for binary systems on quasicircular orbits (which we restrict our attention to in this work), but could be violated for highly eccentric orbits.
For small perturbations around equilibrium, one finds that $\dot{\q} \sim \delta \sqrt{\scals}$ and $\ddot{\q} \sim \delta \scals$ where $\delta \equiv (\q-\q_0)/\q_0 \ll 1$ is the fractional deviation from the equilibrium point $\q_0$.

Using the scaling relations derived above, we construct the most generic effective action for a nonrotating compact object with a dynamical mode $\q(\sig)$ described by Eq.~\eqref{eq:effEOM} close to the critical point, up to order $ \Order(R^2/r^2)$
\begin{align}
  S_\text{CO}^\text{crit} &= \int d\sig \bigg[
  \frac{c_{\dot{\q}^2}}{2} \dot{\q}^2 + \scals(y) \q - c_{(0)}
  - \frac{c_{(2)}}{2} \q^2 - \frac{c_{(4)}}{4!} \q^4 \nnlq
  + c_A A_\mu^\text{IR}(\y) \dot{\y}^\mu
  + \Order\bigg(\frac{R^2}{r^2}\bigg) \bigg]\,, \\
  &= \int d\sig \bigg[
  \frac{c_{\dot{\q}^2}}{2} \dot{\q}^2 + \scals(y) \q - m(q) \nnlq
  + c_A A_\mu^\text{IR}(\y) \dot{\y}^\mu
  + \Order\bigg(\frac{R^2}{r^2}\bigg) \bigg]\,, \label{SCO}
\end{align}
where CO stands for compact object and for later convenience we define
\begin{equation}
  m(q) \equiv c_{(0)} + V(\q) .
\end{equation}
Terms containing time derivatives of $\scals$ all enter at higher order in $R/r$ than we work, e.g., $\ddot{\phi}^\text{IR} \q \sim \dot{\phi}^\text{IR} \dot{\q} \sim \scals \ddot{\q} \sim \Order(R^2 / r^2)$, and thus are absent in Eq.~\eqref{SCO}.
A reparametrization-invariant action is obtained by inserting $d\sig = d \sigma \sqrt{ - g_{\mu\nu}^\text{IR}(\y) d \y^\mu/ d \sigma d\y^\nu / d \sigma}$, where $\sigma$ is an arbitrary affine parameter, and replacing derivatives $d/d\tau$ accordingly.
The complete effective action reads
\begin{align}
  S_\text{eff} = S_\text{field}^\text{IR} + S_\text{CO}^\text{crit}, \label{EFTaction}
\end{align}
where more copies of $S_\text{CO}^\text{crit}$ can be added depending on the number of objects in the system and $S_\text{field}^\text{IR}$ is given by Eq.~\eqref{EMSaction} with IR labels on the fields.
The equations of motion and field equations are obtained by independent variations of $y^\mu(\sigma)$, $q(\sigma)$, and $\scals(x)$, $g_{\mu\nu}^\text{IR}(x)$, $A_\mu^\text{IR}(x)$.

The simplicity of $S_\text{CO}^\text{crit}$ is striking, but we recall that it is only valid close to the critical point of a monopolar, tachyonic, linear instability of a scalar mode.
(A more generic effective action valid away from the critical point is discussed in Appendix \ref{sec:appendixA}.)
Despite its simplicity, the effective action (\ref{EFTaction}) is theory agnostic, in the sense that it is constructed assuming only the scalar-inversion symmetry and that the nonrotating compact object hosts such a mode; in particular, it should hold for the cases studied in Refs.~\cite{Damour:1993hw,Stefanov:2007eq,Doneva:2010ke,Palenzuela:2015ima,Mendes:2016fby,Chen:2015zmx,Ramazanoglu:2016kul,Sagunski:2017nzb,Doneva:2017bvd,Silva:2017uqg,Antoniou:2017acq,Antoniou:2017hxj,Doneva:2017duq,Brihaye:2019kvj,Herdeiro:2018wub,Fernandes:2019rez,Ramazanoglu:2017xbl,Ramazanoglu:2019gbz,Ramazanoglu:2018tig,Ramazanoglu:2018hwk} and similar work to come.
We emphasize that strong-field UV physics at the body scale is parametrized through the numerical coefficients $c_{\dots}$, which can be matched to a specific theory and compact object, or be constrained directly from observations (analogous to tidal parameters \cite{Abbott:2018lct,Abbott:2018wiz}).

\section{Matching strong-field physics into black-hole solutions}
\label{sec:matching}
As an illustrative example of the effective-action framework derived above, we now compute the sought-after coefficients $c_{\dots}$ for BHs in EMS theory.

For this purpose, we match a BH solution in the full theory~\eqref{EMSaction} to a generic solution of the coarse-grained effective theory~\eqref{EFTaction} for an isolated body.
Schematically, the former represents the full solution at all scales, while the latter only represents its projection onto IR scales. We focus first on BH solutions of the full theory~\eqref{EMSaction}, restricting our attention to equilibrium/static, electrically charged, spherically symmetric solutions.

In EMS theory, this family of solutions is characterized by three independent parameters, which we take to be the electric charge $\e$, the BH entropy $\s$, and the asymptotic scalar field $\scal0$, assuming a vanishing asymptotic electromagnetic field and an asymptotically flat metric.
The electric charge is globally conserved by the U(1) symmetry of the theory and the entropy remains constant under reversible processes, which we have implicitly restricted ourselves to by assuming time-reversal symmetry in the effective action.
Thus, we use a sequence of solutions with fixed $\e$ and $\s$ to represent the response of a BH to a varying scalar background $\scal0$.
Since EMS theory modifies electrodynamics, but not gravity or the coupling to gravity, the entropy of a charged BH is the same as in GR, i.e., it is proportional to the horizon area.
See also Ref.~\cite{Herdeiro:2018wub} for the first law of BH thermodynamics in EMS theory.

The asymptotic behavior of the fields take the form
\begin{equation}
  \left(\begin{array}{c}
     \scalar \\ A_0 \\ g_{00}
  \end{array}\right)
  =
  \left(\begin{array}{c}
  \scal0 \\ 0 \\ -1
  \end{array}\right)
  + \left(\begin{array}{c}
  \Q(\phi_0) \\
  - \e e^{\alpha \scal0^2} \\
  2 \mn(\phi_0)
  \end{array}\right) \frac{1}{|\vct{X}|}
  + \Order(|\vct{X}|^{-2}) , \label{eq:UVsols}
\end{equation}
where $\Q(\scal0)$ is the scalar charge of the BH and $\mn(\scal0)$ is its gravitational mass.\footnote{The quantities $\Q(\scal0)$ and $\mn(\scal0)$ describing the asymptotic behavior of the solution also depend on the parameters $\e$ and $\s$, but we suppress the dependence in our notation for brevity. Derivatives of $\Q$ and $\mn$ are taken holding $\e$ and $\s$ constant.}
We construct these solutions numerically (see Appendix~\ref{sec:appendixB} for details), and then compute $\scal0, \mn,$ and $ \Q$ directly from their asymptotic behavior.

\begin{figure*}
  \includegraphics[width=\linewidth]{spontscal}
  \caption{\label{fig:sombrero}The potential $V(\q)$ for $\alpha=-8$ and different electric charges $\e$. The left and right panels correspond to BHs in EMS theory with coupling $f_1(\phi)$~[Eq.~\eqref{eq:f1coupling}] and $f_2(\phi)$~[Eq.~\eqref{eq:f2coupling}], respectively. The numerical solutions and polynomial fits are indicated with points and lines, respectively.}
\end{figure*}

Next, we turn our attention to the description of these BH solutions in the effective theory~\eqref{EFTaction}.
We set up the solution under the same boundary conditions as the numerical sequence described above; in particular, we look at isolated equilibrium configurations, $y^\alpha \partial_\alpha g^\text{IR}_{\mu \nu} \approx y^\alpha \partial_\alpha A_0^\text{IR} \approx y^\alpha \partial_\alpha \scals \approx \dot{\q} \approx 0$.
We construct a coordinate system $\vct{x}$ in which the worldline has spatial components $\vct{y}=\vct{0}$, so that all fields are independent of time.
Furthermore, relying on the fact that both solutions are asymptotically flat, we choose the coordinates $\vct{x}$ such that they match the numerical coordinates $\vct{X}$ in the asymptotic region, i.e., $\vct{x} = \vct{X} + \Order(|\vct{X|}^{-1})$.
Then working to linear order in the fields, we find
\begin{equation}
  \left(\begin{array}{c}
     \scals \\ A_0^\text{IR} \\ g_{00}^\text{IR}
  \end{array}\right)
  =
  \left(\begin{array}{c}
  \scal0 \\ 0 \\ -1
  \end{array}\right)
  + \left(\begin{array}{c}
  q \\
  - c_A e^{\alpha [\scals(y)]^2} \\
  2 [ m(\q) - \scals(y) \q ]
  \end{array}\right) \frac{1}{|\vct{x}|}
  + \dots . \label{eq:IRsols}
\end{equation}
These fields are singular when evaluated on the worldline, $\vct{x} = \vct{y} = \vct{0}$.
This can be cured by appropriately regularizing the solution; here, we simply keep the finite part and drop the singular self-field part, e.g., $\scals(y) = \phi_0$.
The situation is analogous to the singular fields that arise in electrostatics when an extended source is approximated by a point charge.

In addition to a solution for the fields, a variation of $q$ in the effective action leads to
\begin{equation}
  \scal0 = \frac{d m}{d \q} = \frac{d V}{d \q} = c_{(2)} \q + \frac{c_{(4)} \q^3}{3!} + \Order\bigg(\frac{R^2}{r^2}\bigg) . \label{eq:qEOM}
\end{equation}

The matching now consists of identifying the IR-scale fields in the solution of the full theory~\eqref{eq:UVsols} with the fields predicted in the IR effective theory~\eqref{eq:IRsols}.
We extract the IR-scale fields from the former solution using an appropriate IR projector $P^\text{IR}[\cdot]$, such that the matching conditions are given explicitly as $P^\text{IR}[\scalar] = \scals$ (and likewise for the other fields).
Such a projector is most easily formulated in the Fourier domain, so we first compute the (spatial) Fourier transform of the fields~\eqref{eq:UVsols}, denoted by a tilde
\begin{equation}
  \tilde{\scalar}(\vct{K}) = \scal0 \delta(\vct{K}) + \frac{4\pi Q(\scal0)}{\vct{K}^2} + \Order(|\vct{K}|^{-1}) . \label{eq:phiTilde}
\end{equation}
We employ the simple projector $P^\text{IR}[\tilde{\scalar}] \equiv \tilde{\scalar}(\vct{K}) \Theta(K^\text{IR} - |\vct{K}|)$, where $\Theta$ is the Heaviside function and $K^\text{IR}$ is the cutoff scale.
Applying this projection to Eq.~\eqref{eq:phiTilde} and taking the inverse Fourier transform, one finds that $P^\text{IR}[\scalar]$ takes the same form as Eq.~\eqref{eq:UVsols} on scales longer than the cutoff, i.e., for $|\vct{X}|   \gg 1/ K^\text{IR}$.
Then, our matching conditions $P^\text{IR}[\scalar] = \scals,P^\text{IR}[A_0] = A_0^\text{IR}, P^\text{IR}[g_{00}] = g_{00}^\text{IR} $ reduce to
\begin{equation}
  \Q(\scal0) = \q, \quad
  \e = c_A, \quad
  \mn(\scal0) = m(\q) - \scal0 \q, \label{eq:match}
\end{equation}
where we have used $\scals(y) = \scal0$ as discussed above.
Note that the last equation and $\m'(q) = \scal0$~\eqref{eq:qEOM} reveal that the two measures of energy $\mn$ and $\m$ are related by a Legendre transformation of the conjugate variables $(\q,\scal0)$.
Hence, we find that $\Q = \q = -\mn'(\scal0)$, in agreement with the first law of BH thermodynamics \cite{Gibbons:1996af, Cardenas:2017chu}.

While $\mn(\scal0)$ is the gravitational mass of the system, $\m(\q)$ represents the ``gravitational free energy'' of the body (see also Sec.~III.A of Ref.~\cite{Sennett:2017lcx}).
That is, $\m$ is the mass/energy with the potential energy $- \scal0 \q$ (due to the external scalar field) subtracted from $\mn$.
We find below that $\m$---not $\mathcal{M}$---serves as better representation of ``point-particle mass'' found in the Lagrangian or Hamiltonian description of a binary system; of course, both quantities reduce to the standard ADM mass in GR in the absence of scalarization.
Furthermore, away from the critical point, it is not necessary to treat the mode $q$ as a dynamical variable.
This means that we can set $\dot{\q} = 0$ and remove $\q$ from the action \eqref{SCO}.
The latter is achieved by virtue of the Legendre transformation between $m(\q)$ in Eq.~\eqref{SCO} and $\mn(\scals(y))$,
\begin{multline}\label{eardley}
  S_\text{CO} = \int d\sig \bigg[-\mn(\scals(y))
  + \e A_\mu^\text{IR}(\y) \dot{\y}^\mu + \dots \bigg].
\end{multline}
We see that $\mn$ plays the role of the Eardley mass \cite{eardley1975observable} in the action now.
We note that since the Eardley mass and $\m(q)$ are related by a Legendre transformation, they contain the same information.

To compute the values of the various $c_{\dots}$, we numerically construct a sequence of BH solutions as described above and extract the functions $\mn(\scal0)$ and $\Q(\scal0)$.
From there we obtain $\m(\q)$ and $V(\q)$ numerically from Eq.~\eqref{eq:match}, as illustrated in Fig.~\ref{fig:sombrero}.
Each curve indicates a BH sequence with a different electric charge-to-mass ratio $\e / m^{(0)}$, where $m^{(0)} \equiv c_{(0)}$ is the mass of the isolated BH with no scalar charge.
The points indicate the numerically computed solutions, which are calculated by solving the field equations with different boundary conditions for the scalar field  (see Appendix~\ref{sec:appendixB}).
The solid lines are polynomial fits of the form \eqref{eq:effEOM}, from which we extract the values of the coefficients $c_{(2)}$ and $c_{(4)}$.

It is remarkable that from equilibrium solutions one can fix the potential of a dynamical (nonequilibrium) mode to order $\q^4$.
This connection is nontrivial, and it breaks down when one relaxes the assumption of being close to the critical point.
For instance, if terms like $(\scals)^2$ are included in Eq.~(\ref{SCO}), then $\Q \neq q$; or consider the case of a minimally coupled scalar field ($\alpha = 0$),  wherein no-hair theorems \cite{Hawking:1972qk,Sotiriou:2011dz} guarantee that $\mn(\scal0) = \text{const}$---the energy of an equilibrium BH obviously does not encode any information about dynamical modes.

Having described how to compute the coefficients $c_{\ldots}$ in the effective action~\eqref{SCO}, the following section illustrates how this action can be used to study spontaneous and dynamical scalarization.

\section{Modeling strong-gravity effects within the theory-agnostic framework}
\label{sec:strongeffects}

In this section, we show how spontaneous and dynamical scalarization can be understood from the effective theory~\eqref{SCO}, based on a simple analysis of energetics.
We use this effective action to investigate the properties of these critical phenomena, namely their critical exponents.
Though we use EMS theory to make quantitative predictions throughout this section, we emphasize again that the qualitative behavior we find should hold generically for all theories in which spontaneous scalarization occurs.
More specifically, theories in which scalarized configurations are stable are analogous to EMS theory with scalar coupling $f_1(\phi)$, whereas those where such configurations are unstable correspond to the coupling $f_2(\phi)$ (see the following subsection for details).
Extending the predictions made below to other theories only requires computation of the effective mode potential $V(q)$, as described in the previous section, and the inclusion of any new long-range fields not present in EMS theory that impact the motion of binary systems.

\subsection{Spontaneous scalarization}

Recall that a spontaneously scalarized object is one that hosts a nonzero scalar charge even in the absence of an external scalar field $\phi_0=0$.
From Eq.~(\ref{eq:qEOM}) we see that $\scal0 = 0$ corresponds to extrema of $V(\q)$ for equilibrium configurations.
Furthermore, since $V(\q)$ is the oscillation-mode potential, $\q$ dynamically evolves into a minimum of $V(\q)$ [see Eq.~\eqref{eq:effEOM}].
Thus, the existence of spontaneously scalarized configurations is indicated by nontrivial extrema of $V(q)$, and the stability of these configurations depends on whether such points are local minima (stable) or maxima (unstable).
For example, the left panel Fig.~\ref{fig:sombrero} depicts the appearance of spontaneously scalarized BH solutions as one increases the charge-to-mass ratio in EMS theory with coupling $f_1(\phi)$.
Without enough electric charge (e.g., the red and orange curves, with $c_{(2)}>0$), the EM (unscalarized) solutions are the only stable BH solutions, but by increasing the charge beyond a critical value (e.g., the green and blue curves, with $c_{(2)}<0$), the EM solution becomes unstable and the stable solutions instead occur at nonvanishing values of $\q$.

Our approach allows one to determine the values of the coupling $\alpha$ and the electric charge $\e$ at which spontaneous scalarization first occurs ($c_{(2)} = 0$) using only sequences of equilibrium BH solutions.
A more direct approach employed in the past was to search for instabilities of linear, dynamical scalar perturbations on a (GR) Reissner-Nordstr\"{o}m background~\cite{Myung:2018vug}.
We find that the two methods provide the same predictions.
For the choice of coupling $f_1(\phi)$, we compute the critical coupling as a function of the electric charge $\alpha_\text{crit}(\e)$ where $c_{(2)} = 0$ and find that our results agree with the predictions of Ref.~\cite{Myung:2018vug} at the onset of the linear instability of the $\ell=0$ scalar mode to within $1\%$.
For theories in which scalarized solutions are easy to construct, like the EMS theory considered here, our approach can more efficiently compute this critical point than a perturbative stability analysis.
We see that the effective potential $V(\q)$ provides strong indications for a linear scalar-mode instability and its nonlinear saturation.

The same energetics argument reveals drastically different behavior in the EMS theory with coupling $f_2(\scalar)$, depicted in the right panel of Fig.~\ref{fig:sombrero}.
Recall that $f_1(\scalar)\approx f_2(\scalar)$ for small field values, but the two choices differ in the nonlinear regime, which will dramatically impact the stability of scalarized solutions.
This distinction is reflected in our effective action by the sign of $c_{(4)}$; this coefficient is positive for the choice of coupling $f_1(\scalar)$ and negative for $f_2(\scalar)$.
Our simple energetics arguments reveal that above some critical electric charge (e.g., the green and blue curves with $c_{(2)}<0$), no spontaneously scalarized solutions exist, whereas below this value (e.g., the red and orange curves with $c_{(2)}>0$) spontaneously scalarized solutions may exist, but are unstable to scalar perturbations.
In the former case (the green and blue curves), there is no sign of a nonlinear saturation of the tachyonic instability of the EM solution; no stable equilibrium solutions seem to exist.
However, it is impossible to infer how the unstable EM solutions would evolve using our effective theory, since the assumption of time-reversal symmetry (or constant BH entropy) will likely break down.
Numerical-relativity simulations are needed to answer this question (or the construction of a more generic effective theory).

The importance of nonlinear interactions in stabilizing spontaneously scalarized solutions has been studied extensively in the context of ESTGB theories \cite{Blazquez-Salcedo:2018jnn, Minamitsuji:2018xde, Silva:2018qhn}.
For those theories, exponential couplings (equivalent to our $f_1$) or quartic couplings ($f \sim  - \phi^2 +\phi^4$) provide stable scalarized solutions, whereas with quadratic couplings ($f \sim  - \phi^2$), all scalarized solutions are unstable.
Interestingly, quadratic couplings predict stable scalarized solutions in EMS theories \cite{Myung:2019oua}, but our analysis suggests that stability is not guaranteed for generic couplings.
The stability analyses in these references involve studying linearized perturbations on scalarized backgrounds.
Though technically only valid near the critical point of the spontaneous scalarization phase transition and for small $q$, our approach offers a much easier alternative for estimating stability.
We find that our approach correctly reproduces the findings of these stability analyses for scalarized BHs in EMS theories~\cite{Myung:2018jvi,Myung:2019oua}.

\subsection{Critical exponents of phase transition in gravity}

The point-particle action~\eqref{SCO} also offers some insight into the critical behavior that arises near the onset of spontaneous scalarization.
For this discussion, we restrict our attention to the scalar coupling $f_1(\phi)$, for which spontaneously scalarized configurations are stable.
Considering the various coefficients $c_{\ldots}$ as functions of the electric charge $\e$ and entropy $\mathcal{S}$ of a BH and the overall coupling constant $\alpha$, the effective potential $V(\q)$ corresponds precisely to the standard Landau model of second-order phase transitions \cite{Landau:1937,*LandauTranslation}.
Compared to the archetypal example of ferromagnetism, the role of temperature $T$ is played by either $\e$ or $\alpha$.
This connection reveals that (i) spontaneous scalarization is a second-order phase transition and (ii) the critical exponents characterizing this phase transition match the universal values predicted by the Landau model.
Point (i) was already demonstrated for NSs in ST theories in Ref.~\cite{Sennett:2017lcx}, but point (ii) is new to this work; we elaborate on (ii) below.

Critical exponents dictate how a system behaves close to a critical point (e.g., the location of a second-order phase transition).
Such phenomena have first been discovered in GR in the context of critical collapse \cite{Choptuik:1992jv,Abrahams:1993wa,Evans:1994pj,Choptuik:1996yg}, but also appear in perturbations of extremal BHs \cite{Gralla:2017lto,Gralla:2018xzo}.
Applied to the current example of spontaneous scalarization, we study how the structure of the BH solutions varies as we approach the critical point at which spontaneous scalarization first occurs, parametrized by $\xi \rightarrow 0$ where $\xi$ could be either $\xi = (\alpha - \alpha_c) / \alpha_c$ at fixed $\e$ (identifying temperature as $T \sim - 1 / \alpha$) or $\xi = (\e - \e_c) / \e_c$ at fixed $\alpha$ (identifying $T \sim 1/\e$).
For example, the critical exponent $\beta$ of a Landau model is given by the scaling of the order parameter $\q \propto \xi^\beta$ as $\xi \to 0^+$.

The effective potential $V(\q)$ in Eq.~\eqref{eq:effEOM} depends on the properties of the BH solution; this dependence is suppressed in the notation used in the previous section, but here we explicitly restore it.
In particular, close to the critical point, the potential takes the form
\begin{align} \label{eq:EffPotentialTau}
V(\q;\xi) = \frac{c_{(2)}(\xi) }{2} \q^2 + \frac{c_{(4)} (\xi)}{4!} \q^4.
\end{align}
If $c_{(2)}(\xi) $ and $c_{(4)}(\xi) $ are analytic functions, they must take the following form near $\xi = 0$,
\begin{align}
c_{(2)}(\xi) = a \,\xi + \Order\left(\xi^2\right), \qquad
c_{(4)} (\xi)= b + \Order\left(\xi\right),
\end{align}
where $a$ and $b$ are positive constants.
Then, the minima of $V(\q;\xi)$ occur at
\begin{align}
\q=\pm \sqrt{\frac{-6c_{(2)}}{c_{(4)}}} = \pm \sqrt{\frac{-6a}{b}} \, \xi^{1/2},
\end{align}
thus, $q \propto \xi^{1/2}$ as $\xi \rightarrow 0^+$.
For this system, $q$ represents an order parameter, and thus the critical exponent $\beta$ is 1/2.

\begin{figure}
\includegraphics[width=\columnwidth]{expbeta}
\caption{\label{fig:expbeta}Scalar charge $\q$ as a function of $\xi\equiv (\alpha-\alpha_c)/\alpha_c$ for different electric charges with coupling $f_1(\phi)$~[Eq.~\eqref{eq:f1coupling}].
The numerical solutions are indicated with points, while the solid lines are best fits with slope 1/2.
We see that the solutions agree well with the expected scaling $\q \propto \xi^{1/2}$.
}
\end{figure}

We numerically confirm this claim by computing the scalar charge $\q$ of electrically charged BHs as a function of $\xi$ near the critical point.
Fixing the electric charge $\e$, we first determine the critical coupling $\alpha_\text{crit}$ at which $c_{(2)}$ vanishes.
We then compute $\q$ for couplings just below this value, i.e., for $\xi = (\alpha - \alpha_\text{crit})/\alpha_\text{crit} \gtrsim 0$.
The dependence of $\q$ on $\xi$ is depicted in Fig.~\ref{fig:expbeta}; we find that $\q \propto \xi^{1/2}$ agrees well with our numerical results.

Similarly, we compute the other standard critical exponents describing the phase transition.
In particular, both the analytic model~\eqref{eq:EffPotentialTau} and our numerical solutions indicate that $\gamma = 1$ where $\chi=d\q/d\scal0  \propto |\xi|^{-\gamma}$ for $\xi \rightarrow 0^{\pm}$, and $\delta = 3$ where $\q\propto \scal0^{1/\delta}$ at $\xi = 0$.
These findings are consistent with the Landau model for phase transitions.
It would be interesting to find a correspondence to a correlation length in the future, so that all standard critical exponents can be studied.
The introduction of a correlation length (becoming infinite as $c_{(2)} \rightarrow 0$) as another scale next to the size of the compact object could also allow for a more formalized power counting for the construction of the effective action close to the critical point.

\subsection{Dynamical scalarization}

We now employ our effective action to study the dynamical scalarization of binary systems in EMS theory, only considering the scalar coupling $f_1(\phi)$ except where noted.
For this purpose, we integrate out the remaining IR fields from the complete action (i.e., the field part, suitable gauge-fixing parts, and a copy of $S_\text{CO}^\text{crit}$ for each body).\footnote{Strictly speaking, the IR fields are split again into body-scale and radiation-scale parts, and we integrate out the body-scale fields~\cite{Goldberger:2004jt}. This would be necessary for a treatment of radiation from the binary using effective-field-theory methods~\cite{Goldberger:2009qd}.}
We employ a weak-field and slow-motion (i.e., post-Newtonian, PN) approximation.
These approximations are not independent here, since a wide separation of the binary (weak field) implies slow motion due to the third Kepler law for bound binaries.
The leading order in this approximation is just the Newtonian limit of the relativistic theory we are considering.
Therefore, the Lagrangian of the binary to leading order (LO) reads\footnote{We have also gauge-fixed the worldline parameters to the coordinate time $\sigma=t$ as usual in the PN approximation.}
\begin{equation}
  \begin{split}
  L^\text{LO} &= - \m_A \left[ 1 - \frac{\dot{\bm{y}}_A^2}{2} \right] - \m_B \left[ 1 - \frac{\dot{\bm{y}}_B^2}{2} \right]
  + \frac{c_{\dot{\q}^2,A}}{2} \dot{\q}_A^2 \nl
  + \frac{c_{\dot{\q}^2,B}}{2} \dot{\q}_B^2
  + \frac{\m_A \m_B}{r} + \frac{\e_A\e_B}{r} + \frac{\q_A \q_B}{r},
  \end{split}
\end{equation}
where $A$ and $B$ label the bodies, $r = |\bm{y}_A - \bm{y}_B|$ is their separation, and in this section the dot $\dot{~}$ indicates a derivative with respect to coordinate time.
We have suppressed the dependence of $m_A$ on $q_A$ for brevity, but recall that the ``free energy'' of each body takes the form
\begin{align}
m_A(q_A) &= m_{(0),A} + V(q_A) \nonumber\\
&= c_{(0),A} + \frac{c_{(2),A}}{2} q_A^2 + \frac{c_{(4),A}}{4!} q_A^4 .
\end{align}
Notice that $m_A(q_A)$ plays the role of the body's mass in the binary Lagrangian because (i) it couples to gravity like a mass, see Eq.~\eqref{SCO}, and (ii) it is independent of the fields, so that for the purpose of integrating out the fields it can be treated as a constant.
The Hamiltonian for the binary can be obtained via a Legendre transformation
\begin{equation}
\begin{split}
H^\text{LO} &= \m_A + \m_B
+ \frac{\bm{p}_A^2}{2 \m_A} + \frac{\bm{p}_B^2}{2\m_B}
+ \frac{p_{q,A}^2}{2 c_{\dot{\q}^2,A}} + \frac{p_{q,B}^2}{2 c_{\dot{\q}^2,B}} \nl
- \frac{\m_A \m_B}{r} + \frac{\e_A\e_B}{r} - \frac{\q_A \q_B}{r},
\end{split}
\end{equation}
with the pairs of canonical variables $(\bm{y}_{A/B}, \bm{p}_{A/B})$ and $(q_{A/B}, p_{q,A/B})$.

Following Ref.~\cite{Sennett:2017lcx}, let us now consider a special case that allows simple analytic solutions for the scalar charges of the bodies.
We henceforth assume that the scalar charges evolve adiabatically $p_{q,A/B} \approx 0$ and that the two bodies are identical, i.e., $\q\equiv \q_A = \q_B$, $c_{(2)}\equiv c_{(2),A} = c_{(2),B}$, etc.
The Hamiltonian in the center-of-mass system $\bm{p} \equiv \bm{p}_A = - \bm{p}_B$ now reads
\begin{equation}
H^\text{LO,adiab.} = 2 \m
+ \frac{\bm{p}^2}{\m}
- \frac{\m^2}{r} + \frac{\e^2}{r} - \frac{\q^2}{r}.
\end{equation}
Under these assumptions and recalling again that $m=m(q)$, the equation of motion for the scalar charge $\q$ is given by
\begin{equation}\label{qEOM}
0 \approx \dot{p}_{\q} = \frac{\partial H^\text{LO,adiab.}}{\partial \q} = 2 z \left(c_{(2)} \q + \frac{c_{(4)}}{6} \q^3\right) - \frac{2\q}{r},
\end{equation}
with the redshift
\begin{equation}
z \equiv 1-\frac{\mathbf{p}^2}{2 m^2}-\frac{m}{r} .
\end{equation}
For simplicity, we neglect relativistic corrections to the redshift from here onward, i.e., $z \approx 1$; restoring these corrections does not affect the qualitative behavior that we describe.
Equation~\eqref{qEOM} has three solutions: an unscalarized solution with $\q=0$ and a two scalarized solutions with nonzero $q$ of opposite signs.
The condition for stability of these solutions is that they are located at a minimum of the energy of the binary,
\begin{equation}
0 \leq \frac{\partial^2H^\text{LO,adiab.}}{\partial \q^2} \approx 2c_{(2)} - \frac{2}{r} + c_{(4)}\q^2,
\end{equation}
which is violated for $\q=0$ when $1/r > c_{(2)}$.
Hence, the stable solutions are given by
\begin{equation}\label{Qsoln}
\q = \left\{ \begin{tabular}{lll}
$0$ & \quad\text{for} & $1 / r \leq c_{(2)}$ \\
$\pm \sqrt{\dfrac{6}{c_{(4)}}} \sqrt{\dfrac{1}{r} - c_{(2)}}$ &
\quad\text{for} & $1 / r \geq c_{(2)}$
\end{tabular} \right. ,
\end{equation}
which contain a phase transition at $r = 1 / c_{(2)}$ corresponding to the spontaneous breaking of the $q\rightarrow -q$ symmetry of the effective action.
Recall that $c_{(2)} < 0$ corresponds to the case where each object is spontaneously scalarized, for which the bottom condition in Eq.~\eqref{Qsoln} always holds, and thus $q \neq 0$ over all separations.

\begin{figure}
	\includegraphics[width=\columnwidth]{Q_Omega_alpha8}
	\caption{\label{fig:QOmega} Scalar charge $Q_\text{tot}$ of an equal mass binary as a function of orbital frequency $\Omega$ or GW frequency $f_\text{GW}=\Omega / \pi$ for coupling $f_1(\phi)$~[Eq.~\eqref{eq:f1coupling}]  with $\alpha=-8$.}
\end{figure}

Restricting our attention to circular orbits, we plot in  Fig.~\ref{fig:QOmega} the total scalar charge of the binary $\Q_\text{tot}\equiv \q_A+\q_B$ as a function of orbital frequency, given by Kepler's law as
\begin{equation}\label{eq:Kepler}
\Omega^2=\frac{1}{r^3}\left(\m_A+\m_B\right)\left(1+\frac{\q_A\q_B}{\m_A\m_B}-\frac{\e_A\e_B}{\m_A\m_B}\right).
\end{equation}
For simplicity, we only show the positive scalar charge branch of solutions.
The frequency is shown both as the dimensionless combination $M\Omega$ with $M\equiv \m_A^{(0)}+\m_B^{(0)}$ and as the equivalent GW frequency $f_\text{GW} = \Omega / \pi$ for a $(30 M_\odot + 30 M_\odot)$ binary system.
The plotted curves correspond to solutions with coupling constant $\alpha=-8$, and the colors correspond to different values of the electric charge.
The scalar charge vanishes below the onset of dynamical scalarization; the scalar charge grows abruptly at some critical frequency $\Omega_\text{scal}$ (as evidenced by kinks in the plotted curves) because dynamical scalarization is a second-order phase transition---see Ref.~\cite{Sennett:2017lcx} for a more detailed argument that dynamical scalarization is a phase transition.

In Fig.~\ref{fig:paramspace}, we depict the scalarization of binary systems for various charge-to-mass ratios and couplings $\alpha$.
The solid lines indicate the critical frequency $\Omega_\text{scal}$ at which dynamical scalarization begins; the heavily shaded regions above these lines correspond to dynamically scalarized binaries after the onset of this transition.
The critical point $c_{(2)}=0$, corresponding to $\Omega_\text{scal} \rightarrow 0$, represents the division between binaries that dynamically scalarize and those whose component BHs (individually) spontaneously scalarize; we depict all spontaneously scalarized configurations with a lighter shading.
Thus, we see that our effective action, which was matched to isolated objects and models spontaneous scalarization, predicts dynamical scalarization as well.
Refined predictions can be obtained by perturbatively calculating the binary Lagrangian to higher PN orders and also the emitted radiation; both are possible using effective-field-theory techniques \cite{Goldberger:2004jt,Goldberger:2009qd,Kuntz:2019zef} or more traditional methods where extended bodies are represented by point particles, e.g., Refs.~\cite{Blanchet:2013haa,Bernard:2018hta}.

\begin{figure}
	\includegraphics[width=\columnwidth]{dynscalfreq_stacked}
	\caption{\label{fig:paramspace} Scalarization of binary systems with various electric charges for scalar coupling $f_1(\phi)$~[Eq.~\eqref{eq:f1coupling}] with different coupling strengths $\alpha$. Lightly and heavily shaded regions indicate spontaneously and dynamically scalarized configurations, respectively. The solid lines depict the onset of dynamical scalarization $\Omega_\text{scal}$ as a function of electric charge.}
\end{figure}

The same calculation can be repeated for binary systems with the choice $f(\scalar) = f_2(\scalar)$.
As before, the unscalarized $q=0$ solution is stable for above $r=1/c_{(2)}$.
However, because $c_{(4)} < 0$, no stable dynamically scalarized branch exists below that separation; instead, the system becomes ``dynamically'' unstable after this critical point.
The phase diagram for this choice of coupling takes the same form as Fig.~\ref{fig:paramspace}, but here the shaded regions correspond to scenarios in which no stable configuration exists.
At the onset of instability, the scalar radiation will likely grow rapidly and the GW frequency will decrease more rapidly compared to the EM case.
However, long-term predictions are not possible with our effective theory since the assumption of time-reversal symmetry likely breaks down, as for the unstable isolated BHs.

\section{Conclusions}
\label{sec:conclusions}

In this paper, we developed a simple energetic analysis of spontaneous and dynamical scalarization, based on a strong-field-agnostic effective-field-theory approach (extendable beyond the scalar-field case \cite{Ramazanoglu:2017xbl,Ramazanoglu:2018tig,Ramazanoglu:2018hwk} in the future).
We demonstrated our analysis for BHs with modified electrodynamics here, complementing the study of NS in ST gravity from Ref.~\cite{Sennett:2017lcx}.
The theory-agnostic nature of our approach allowed us to draw general conclusions about scalarization
As an example, we found that dynamical scalarization generically occurs in theories that admit spontaneous scalarization.
Specific examples of theories for which our findings apply include those discussed in Refs.~\cite{Stefanov:2007eq,Doneva:2010ke,Ramazanoglu:2016kul,Doneva:2017bvd,Silva:2017uqg,Antoniou:2017acq,Antoniou:2017hxj,Doneva:2017duq,Ramazanoglu:2017xbl,Ramazanoglu:2019gbz,Ramazanoglu:2018tig,Ramazanoglu:2018hwk}.

The recent discovery of spontaneous scalarization in ESTGB theories~\cite{Doneva:2017bvd,Silva:2017uqg,Antoniou:2017acq,Antoniou:2017hxj,Doneva:2017duq} has sparked significant interest in this topic.
Our work predicts that dynamical scalarization can occur in binary systems in these theories and allows one to straightforwardly estimate the orbital frequency at which it occurs using only information derivable from isolated BH solutions.
Such information is valuable for guiding numerical-relativity simulations in these theories, for which there has been recent progress~\cite{Witek:2018dmd}.
Eventually higher PN orders in specific theories could be added to our model to derive more accurate predictions, and the framework could be extended to massive scalars or other types of new fields.

We demonstrated how scalarization, as an exemplary strong-gravity modification of GR, is parametrized by just a few constants in the effective action (which vanish in GR).
Hence, our effective action provides an ideal foundation for a strong-field-agnostic framework for testing dynamical scalarization.
Ultimately, one needs to incorporate self-consistently such effects into gravitational waveform models.
This undertaking will require the computation of dissipative effects (analogous to the standard PN treatment in GR) and the mode dynamics during scalarization, characterized (in part) by $c_{\dot{\q}^2}$.
The effective action can also, in principle, be extended to include other strong-field effects influencing the inspiral of a binary, e.g., phenomena like floating orbits \cite{Cardoso:2011xi,Endlich:2016jgc} or induced hair growth~\cite{Horbatsch:2011ye,Wong:2019yoc}.

Independent of GW tests of GR, further study of dynamical scalarization could offer some insight into the nonlinear behavior of merging binary NSs in GR.
As discussed in the Introduction, scalarization and tidal interactions enter models of the inspiral dynamics in a similar manner; in fact, the effective action treatment of dynamical tides (in GR) \cite{Steinhoff:2016rfi} is completely analogous to the approach adopted here for scalarization.
Unlike the case with tides, our model of scalarization includes nonlinear interactions via the $q^4$ term.
Nonlinear tidal effects could be relevant for GW observations of binary NSs~\cite{Reyes:2018bee,Weinberg:2018icl}, but are difficult to handle in GR.
Furthermore, mode instabilities also occur for NSs in GR \cite{Gaertig:2011bm,Andersson:2000mf}.
Dynamical scalarization can be used as a toy model for these types of effects, and further exploration of this non-GR phenomenon could improve gravitational waveform modeling in GR.

Our effective action approach allowed us to study the critical phenomena at the onset of scalarization; further study could also provide insight into critical phenomena in GR.
The critical exponents we obtained numerically agree with the analytic predictions from Landau's mean-field treatment of ferromagnetism \cite{Landau:1937,*LandauTranslation}.
Only missing here is a proper definition of correlation length, which we leave for future work.
In GR, the BH limit of compact objects has been suggested to play the role of a critical point and lead to the quasiuniversal relations for NS properties \cite{Yagi:2016bkt,Yagi:2013bca}.
These quasiuniversal relations are invaluable for GW science because they reduce the number of independent parameters needed to describe binary NSs, improving the statistical uncertainty of measurements.
In the BH limit (for nonrotating configurations), the leading tidal parameter vanishes \cite{Damour:2009vw,Binnington:2009bb,Kol:2011vg}, like $c_{(2)}$ at the critical point here.
Better theoretical understanding of the origin of these universal relations could help improve their accuracy; utilizing information from the critical phenomena at the BH limit is a compelling idea, and scalarization could serve as a toy model in that regard.

\acknowledgments
We thank Emanuele Berti, Caio Macedo, N\'{e}stor Ortiz, and Fethi Ramazanoğlu for useful discussions.


text:
\section{Introduction}
\label{sec:intro}
\input{intro}

\section{Fixed-Phase Diffusion Monte Carlo}
\label{sec:fpdmc}
\input{fpdmc}

\section{Spin-Orbit Interactions and Dynamic Spins}
\label{sec:so_dyn_spin}
\input{so_dyn_spin}

\section{Results}
\label{sec:results}
\input{results}

\section{Conclusions}
\label{sec:conclusions}
\input{conclusions.tex}

\section*{References}
\bibliographystyle{model1-num-names}

\subsection{Tungsten Oxide}
The tungsten atom and many tungsten containing compounds have a different electronic structure than their isovalent counterparts containing Cr and Mo.
For example, the ground state configuration of W is [Xe$4f^{14}$]$6s^25d^4$, whereas both Cr and Mo have $ns^1(n-1)d^5$ ground state configurations \cite{melton1}.
The favored configuration and state results from subtle balance between electron correlation as well as the splitting due to spin-orbit. Interestingly,
this trend manifests itself also in the
WO molecule.
CrO and MoO have $^5\Pi$ molecular ground states, and some of the original theoretical work on WO predicted a $^5\Pi$ ground state \cite{nelin_bauschlicher}.
However, experimental determination as well as contracted multi-reference configuration interaction (CMRCI) calculations show a $^3\Sigma^-$ ground state for WO \cite{ram2001}.
Our results described below confirm this conclusion using both scalar relativistic and spin-orbit Hamiltonians, indicating that this effect is predominantly driven by correlations with smaller impact from the spin-orbit effects.

We first discuss the determination of the ground state without the spin-orbit interaction. For experiments the estimation
of the equilibrium bond length
 was found to be 1.65807(6)~\AA\,  \cite{ram2001}.
The CMRCI calculations predict a minimum at 1.67~\AA\,  for the $^3\Sigma$ state, which is only in error by $\sim 0.01$~\AA.
In order to determine the ground state, we calculate the $^3\Sigma$ and $^5\Pi$ states at $r_e = 1.67$~\AA.
We also tested various nodal surfaces, namely PBE and hybrid PBE0 which includes (HF) exchange, which in many cases can have a significant impact on the nodal surface, especially for transition metals.
The results are shown in Table \ref{tab:WO_arep_gs}.
\begin{table}
    \centering
    \caption{Total energies for WO for the $^3\Sigma$ and $^5 \Pi$ states at $r_e = 1.67$~\AA. These results only utilize a scalar relativistic Hamiltonian and neglects spin-orbit.}
    \begin{tabular}{c|ccc}
	\hline\hline
	$\Psi_T$        & SCF & VMC & FN-DMC \\
	\hline
	$^3\Sigma$ PBE  & -83.501631 & -83.2377(2) & -83.3149(3) \\
	$^3\Sigma$ PBE0 & -83.429649 & -83.2397(1) & -83.3130(3) \\
	$^5\Pi$ PBE     & -83.484707 & -83.2227(2) & -83.2870(4) \\
	$^5\Pi$ PBE0    & -83.398745 & -83.2244(1) & -83.2862(4) \\
	\hline\hline
    \end{tabular}
    \label{tab:WO_arep_gs}
\end{table}
For the $^3\Sigma$ state, the PBE nodal surface results in the lowest energy overall by 2~mHa.
Clearly, the $^3\Sigma$ state is correctly predicted to be the ground state, even in the absence of spin-orbit.
Now that the ground state has been determined, we can determine the dissociation energy $D_e$ at equilibrium.
We calculate the W atom and O atom within FNDMC, and the dissociation energy can be determined by $D_e = E(\textrm{WO}) - E(\textrm{W}) - E(\textrm{O})$.
Note that our dissociation energy does not include the zero-point motion.
We find the dissociation energy to be $D_e^{FNDMC} = 6.80(2)$~eV.
To see how this compares to the experimental results, we first note that the experimental vibrational frequency is $\omega_e = 1065.6231(52)$~cm$^{-1}$ \cite{ram2001} and the
experimental binding energy (including zero point motion) is $D_0^{expt} = 6.85(44)$~eV \cite{nist}.
Correcting for the zero point motion, the experimental dissociation energy becomes $D_e^{expt} = 6.91(44)$~eV.
Without the spin-orbit correction, the FNDMC dissociation energy falls well within the experimental error bars.
Regarding the excited state $^5\Pi$, previous CASSCF (complete active space
self-consistent field) calculations predicted a minimum a $r_e = 2.11$~\AA \,and $D_e$ = 2.42~eV whereas CISD found $r_e = 1.99$~\AA \, and $D_e = 3.25$~eV \cite{nelin_bauschlicher}.
The more recent CMRCI finds an equilibrium bond length which is shifted toward the $^3\Sigma$ minimum at $r= 1.72$~\AA.
We calculate the $^5\Pi$ state at this geometry within FNDMC using PBE0 nodes and find a total energy of -83.2848(3)~Ha.
Comparing with the results in Table \ref{tab:WO_arep_gs}, it is clear that FNDMC does not predict $r_e = 1.72$~\AA\, to be the minimum for the $^5\Pi$ state, and in our
calculations it has a lower energy at $r_e = 1.67$~\AA.

\begin{figure}[t]
    \centering
    \caption{FPSODMC total energies of
    WO molecule as a function of spin time step $\tau_s$/effective spin mass $\mu_s$ for the lowest

    states with $^3\Sigma$ and $^5\Pi$ symmetries. Note that the within the error bars the difference between the states remains very similar regardless of the time step
}
\includegraphics[width=0.75\textwidth]{figures/WO}
    \label{fig:WO}
\end{figure}

We now turn to the FPSODMC results with explicit inclusion of spin-orbit and two-component spinors.
With the spin-orbit interaction included, the $^3\Sigma$ and $^5\Pi$ states are split.
We take our trial wave functions from the lowest energy spin-orbit state coming from each of the term symbols calculated from COSCI, which we indicate by $^3\Sigma$ and $^5\Pi$ respectively.
As above, we first investigate the difference between the ground states of the two manifolds at $r_e = 1.67$~\AA.
The results are shown in Figure \ref{fig:WO}.
As mentione above we extrapolate in the spin time step to find the desired expectation values. Note that the time step for the spatial degrees of freedom was conservatively small $\tau=10^{-3}$ a.u..
It is clear that
 FPSODMC predicts the $^3\Sigma$ symmetry to be the ground state, in agreement with
 CMRCI results quoted above.
In order to predict the dissociation energy, we calculate the ground states of individual atoms and find a dissociation energy of $D_e^{FPSODMC} = 6.52(2)$~eV.
We note that the predicted dissociation energy for the case without spin-orbit and with spin-orbit differ by roughly $0.3$~eV.
While the quality of the nodal surface or phase can shift the predicted value slightly (we use DFT nodal surfaces whereas we use COSCI phases due to difficulties in generating an open-shell DFT trial wave function within the relativistic framework), another factor contributes more strongly to this difference.
While the spin-orbit interaction tends to lower the ground state energy when it splits a term symbol into various states, this effect is much less pronounced in the WO molecular system.
The spin-orbit splitting leads to lower
energy for an individual atom, ultimately resulting in a reduced estimation of the dissociation energy than what the scalar relativistic FNDMC will predict.
Additionally, we note that the prediction from FPSODMC agrees with experiment to the experimental uncertainty.
We also investigate the location of the $^5\Pi$ minimum.
At $r_e = 1.72$~\AA\, as predicted by CMRCI, FPSODMC finds a total energy of -83.3705(5)~Ha.
If we compare to the total energy shown in Figure \ref{fig:WO} which saturates to -83.3724(7)~Ha, we find that $r_e = 1.72$~\AA\,  is not the equilibrium geometry for this excited state, in agreement with the scalar relativistic FNDMC calculations.

\subsection{Tungsten Dimer}
Unlike WO which differs from its isovalent counterparts, W$_2$ is expected to follow the trend of Mo$_2$ and Cr$_2$ and form a $^1\Sigma$ ground state configuration.
To the best of our knowledge, experimental data does not exist for the binding energy of the W$_2$.
However a number of model estimates are in the literature predicting a wide range of binding energies in eV, namely 4.69(89) \cite{dimer1}, 5.62(1.23) \cite{dimer2}, 4.68 \cite{dimer3} and 5.00(69) \cite{dimer4}.
From the scatter in the various estimates, an adopted prediction of the binding energy is given as $D_0^{est} = 5(1)$~eV \cite{morse}.
More recent B3LYP calculations predict a $^1\Sigma$ ground state over a $^3\Sigma$ calculation with an equilibrium bond length of 2.048~\AA\, \cite{kraus}.
Multi-reference perturbation theory estimates an equilibrium bond length of $r_e = 2.0561$~\AA\, and $D_e$ = 4.5110~eV \cite{angeli} in the $^1\Sigma$ state.
Furthermore, a CASPT2 study found a dissociation energy of 5.37~eV and $r_e = 2.010$~\AA\, \cite{borin}.
We aim to add DMC among the predictions of the dissociation energy for W$_2$.
A summary of the predictions are listed in Table \ref{tab:w2}.
Although there is consistent discrepancy among various theories for the dissociation energies, the vibrational frequencies are in reasonable agreement and are quite small.
Experimental detection of W$_2$ in an argon matrix found $\omega_e$ = 336.8~cm$^{-1}$ \cite{hu}, and theoretical estimates include $\omega_e = 401.2$~cm$^{-1}$ \cite{kraus}, $\omega_e = 326.69$~cm$^{-1}$ \cite{angeli}, and $\omega_e = 354$~cm$^{-1}$ \cite{borin}.
Because of the very small vibrational frequencies, the binding well is relatively shallow and energetic changes for geometric differences on the order of $0.01$~\AA\, as is seen between the various other predictions, will be difficult to discern within statistical errors within DMC, so we adopt a geometry of the intermediate bond length of $r = 2.048$~\AA\, to perform all our calculations.

\begin{figure}[t]
    \centering
    \caption{The difference in the charge density between DHF and PBE0 trial wave functions for the outermost spinors $\pi^4 \sigma_{z^2}^2 \sigma_s^2 \delta^4$. In red, we show where PBE0 has a greater charge density and in blue where DHF shows a greater charge density. The black spheres indicate the location of the Tungsten atoms.  }
    \input{figures/W2}
    \label{fig:density}
\end{figure}

\begin{table}[!h]
   \centering
   \caption{FNDMC (AREP) and FPSODMC (REP)  dissociation energies of W$_2$ compared with other methods. In the fixed-node DMC calculations the trial nodes are from Slater determinants built from PBE0 single-particle orbitals. }
    \begin{tabular}{lccr}
       \hline\hline
       Method & $D_0/$eV   & $r_e$/\AA & Ref.    \\
       \hline
       FNDMC/PBE0$^\dagger$          &  5.34(2)   & 2.048$^*$  &  This Work \\
       PBE$^\dagger$            &  4.57      & 2.048$^*$  &  This Work \\
       PBE0$^\dagger$           &  2.54      & 2.048$^*$  &  This Work \\
       B3LYP$^\dagger$          &            & 2.048      & \cite{kraus} \\
       SC-NEVPT3$^\dagger$      &  4.5110    & 2.056      & \cite{angeli} \\
       CASPT2 $^\dagger$        &  5.37      & 2.010      & \cite{borin} \\
       FPSODMC/DF$^\ddagger$    &  2.51(2)   & 2.048$^*$  & This Work \\
       FPSODMC/COSCI$^\ddagger$ &  3.03(2)   & 2.048$^*$  & This Work \\
       FPSODMC/PBE0$^\ddagger$  &  4.17(2)   & 2.048$^*$  & This Work \\
       Estimated                &  5(1)      &            & \cite{morse} \\
       \hline
    \end{tabular}\\
$^*$ Indicates the calculation was performed at this geometry. \\
$^\dagger$ Indicates a spin-averaged calculation\\
$^\ddagger$ Indicates an explicit spinor calculation
    \label{tab:w2}
\end{table}

We now turn to our DMC results.
Utilizing a scalar relativistic Hamiltonian within FNDMC, we test both a PBE and PBE0 nodal surface in order to minimize the DMC energy.
PBE0 results in a lower energy by 3~mHa, and produces a dissociation energy of 5.34(2)~eV.
Since W$_2$ is a closed shell molecule, when we include spin-orbit within FPSODMC we first try a single slater determinant built from DHF spinors.
However, this only results in underbinding with dissociation energy of 2.51(2)~eV that
results from rather poor approximation of the phase and consequently a larger fixed-phase error.
We attempt to improve the fixed-phase approximation by using the COSCI expansion where the anti-bonding spinors are included in the active space.
While this lowers the total energy within FPSODMC, the dissociation energy increases only to 3.03(2)~eV. As has been seen in other transition metal systems \cite{wagner07}, HF orbitals tend to localize charge too much
close to the ions
\cite{wagner2007a,kolorenc2010}.
This is due to the fact that the only way for HF to lower the energy is through exchange, which becomes larger when the charge density
and particular states become more localized.
The same bias is then built-in also in the excited virtual one-particle space.
This results in a well-known
 "too ionic" bias of the HF or HF-based wave functions
 and it is expected that the same persists
 with spin-orbit included.
We therefore consider further possibilities for improving the trial phases here.
Since W$_2$ is a closed-shell molecule, we can utilize a single Slater determinant built from DFT relativistic spinors.
In Figure \ref{fig:density}, we illustrate the difference in the relativistic charge density between the PBE0 and DHF trial wave functions.
As in the case of other FNDMC treatments of transition metals, HF localizes more on the atoms whereas the PBE0 pushes more charge density into the bonding region.
For the PBE0 trial phase, we find a significantly lower energy resulting in a dissociation energy of 4.17(2)~eV.
To ensure the quality of the trial phase, we also try LDA, PBE, and B3LYP trial wave functions.
All of these result in the same total energy as PBE0 within statistical error bars.
Note that compared to the FNDMC without spin-orbit, there is a significant contribution to the dissociation energy coming from spin-orbit.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
Let us recall the well-known topological Tverberg theorem  ~\cite{bar-shl-sz}:
{\Thm  Let $d\geq 1, \;r\geq 2$ a prime and  $N=(d+1)(r-1)$. Consider the $N$-simplex $\triangle_N$. Suppose $f: \triangle_N \to \mathbb R^d$ is a continuous map. Then there exists a family $\sigma_1,\sigma_2,\cdots\sigma_r$ of $r$ disjoint faces of $\triangle_N$ such that} $$\bigcap^{r}_{1}f(\sigma_i)\neq\phi.$$

\bigskip\noindent The above Theorem is a generalization of the classical Tverberg Theorem~\cite{tve} which says that if we have $N=(d+1)(r-1)+1$ points in $\mathbb R^d$ in general position, then we can partition them into $r$ disjoint subsets $F_1, F_2,\cdots , F_r$ such that the convex hulls of all these $F_i$'s have a non-empty intersection. The classical Tverberg Theorem has been proved for all positive integers $r\geq 1$ whereas the topological Tverberg Theorem is proved only when $r$ is a prime power. When $r$ is not a prime power, a counterexample to the topological Tverberg theorem has recently been published (see \cite {fri}).

\bigskip\noindent What would be the colored version of the topological Tverberg Theorem ?
This question was first studied by B$\acute{a}$r$\acute{a}$ny, F$\ddot{u}$redi and Lov\'{a}sz~\cite{bar-fur-lov}, who proved that if we take 21 points in the plane in general position such that 7 are red, 7 are blue and 7 are  green, then we can decompose them into 7 triangles, each triangle having vertices of different colors, such that their intersection is non empty. Later, B$\acute{a}$r$\acute{a}$ny and Larman~\cite{bar} studied the above question further and proved the following general result : If we take $3r$ points in the plane in general position where $r$ are of red color, $r$ are of blue color and  $r$ are of green color, then we can find $r$ disjoint  triangles, each triangle having vertices of different colors, such that their intersection will be non empty. For higher dimensional Euclidean spaces $\mathbb R^d$, B$\acute{a}$r$\acute{a}$ny and Larman asked the question:  Given $r$ and $N\geq (d+1)r$ points in $\mathbb R^d$, determine the smallest $t$ such that  if we take $d+1$ color classes of size $\leq t$, we can find  a family of $r$ disjoint $d$-simplices having vertices of different colors, such that their intersection is non empty.  $\check{Z}$ivaljevi$\acute{c}$ and Vre$\acute{c}$ica~\cite{vre-ziv} introduced the concept of chessboard complexex and showed that when $r$ is a prime, the colored Tverberg Theorem holds for $t\geq 2r-1$, and also for any $r\geq 2$ due to Bertrand's postulate that there is a prime between $r$ and $2r$.  This colored Tverberg Theorem of ZV attracted a lot of attention, but could not be considered very satisfactory for two reasons. The first reason was that the classical Tverberg Theorem does not appear as a special case of this Theorem, and the second reason was that several colored points were left out unaccounted. In 2009, Blagojevi$\acute{c}$, Matschke and Ziegler(see ~\cite{bmz}~\cite{bmz2}~\cite{bmz3}) formulated and proved a remarkable new colored topological Tverberg Theorem which is stated as follows:

{\Thm (BMZ)  Let $d\geq 1,\; r\geq 2$ prime and $N=(d+1)(r-1)$. Consider the $N$-simplex $\triangle_N$ whose vertices are colored into disjoint color classes
$$C_0,C_1,\cdots ,C_m\;\;\;m\geq d+2$$ such that for each $i$, the size $|C_i|$ of these colored classes is at most $r-1$. Then given any continuous map $f: \triangle_N\to\mathbb R^d$ there exists a family of $r$ disjoint rainbow faces $\sigma_1,\sigma_2, \cdots ,\sigma_r$ of $\triangle_N$ such that
$$f(\sigma_1)\cap\cdots\cap f(\sigma_r)\neq \phi.$$
}

Here  a face $\sigma_i$ is said to be a {\bf rainbow} face if each vertex of $\sigma_i$ is of different color, i.e., $|\sigma_i\cap C_j|\leq 1\;\;\forall\; i,\; 1\leq i\leq r$ and $\forall\; j,\;\;1\leq j\leq m$.

\bigskip\noindent The above Theorem, to be called the BMZ Theorem in the sequel, turns out to be very interesting topological colored Tverberg Theorem since the classical Tverberg Theorem becomes a special case  of this Theorem when the size of each color class is one, and also since all points of $\triangle_N$ are accounted for.
In this  paper we present  a simple minded proof of the BMZ Reduction Lemma (see~\cite{bmz} page 3, Reduction of Theorem 2.1 to Theorem 2.2) with a motivational perspective. We also state the BMZ reduction Lemma for maps to manifolds using the classical definition of cohomological dimension over $\mathbb Z$ rather than the one employed in (\cite{bmz3}, Lemma 2.1). We remark here that BMZ Theorem for manifolds has been proved when $r$ is prime.  The corresponding Theorem when $r$ is a prime power is not resolved yet, though the topological Tverberg Theorem has already been proved for any prime power $r$. However, the Reduction Lemma is true for any $r$, prime or otherwise.

\section{Reduction Lemma}
Let $d\geq 1,\;r\geq 2$ and $N=(d+1)(r-1)$. Color the vertices of the simplex $\triangle_N$ by $m$ colors, $m\geq d+2$. Suppose the color classes are denoted by $C_0,C_1\cdots ,C_m$.  If $|C_i|\leq r-1,\;\forall \; i=1,2,\cdots ,m$ then we say that
$C_0,C_1\cdots ,C_m$ is a {\bf general coloring} of $\triangle_N$.  Since $|C_i|\leq r-1$ for each $i$, there has to be at least $d+2$ colors.
 Among all the general colorings of $\triangle_N$, let us consider the colorings
$$C_0,C_1\cdots ,C_{d+1}$$ where $|C_i|= r-1$ for every $i$ except one color class whose size is 1. This accounts for all the points of $\triangle_N$. Without loss of generality we may occasionally consider the last color class viz.,
$C_{d+1}$, to be of size 1. Such colorings of $\triangle_N$ will be called the {\bf special colorings} of $\triangle_N.$

\bigskip\noindent {\bf Note :} It may be noted that any general coloring $G_0,G_1\cdots ,G_m,\;\;m\geq d+1$, of the vertices of $\triangle_N$ can be obtained from a given special coloring $$C_0,C_1\cdots ,C_{d+1}$$ by removing certain number of vertices from each color class and creating new color classes (one or more) in finitely many steps.
It is assumed that when we remove the single vertex of $C_{d+1}$, this singleton class disappears creating  another class of the same color with possibly bigger size.

\bigskip\noindent In this process  while  the number of vertices of $\triangle_N,\;N=(d+1)(r-1)$ remains the same, the class sizes become smaller and smaller and the number of color classes becomes larger and larger. In the limiting case when the size of each  color class becomes 1,  the number of color classes will become $N+1=(d+1)(r-1)+1$.

\bigskip\noindent In terms of the above definitions we can state the Blogojevi$\acute{c}$-Matschke-Ziegler colored Theorem as follows (see \cite{bmz} Theorem 2.1):

{\Thm {\bf ( BMZ Theorem)}  Let $d\geq 1,\; r\geq 2$ prime and $N=(d+1)(r-1)$  and $C_0,C_1\cdots ,C_m$ be a general coloring  of $\triangle_N$. Then given any continuous  map $f:\triangle_N\to\mathbb R^d$ there exists a family  of $r$ disjoint rainbow  faces $\sigma_1, \sigma_2,\cdots , \sigma_r$ of $\triangle_N$ such that
$$ f(\sigma_i)\cap\cdots\cap f(\sigma_r)\neq\phi.$$}

{\Lemma {\bf (Reduction Lemma)}  Let $d\geq 1,\; r\geq 2$  and $N=(d+1)(r-1).$  If the BMZ Theorem is true for special colorings for the parameters $(d+k, r, \mathbb R^{d+k}),\;k\geq 0,$ then the BMZ theorem is true for all general colorings of $\triangle_N$ for the parameters $(d,r,\mathbb R^d)$. }

\bigskip\noindent  We now give a detailed proof of the above Lemma following a simple minded approach with a motivational perspective.

\bigskip\noindent Let us start with a special coloring  $C_0,\cdots ,C_d, C_{d+1}$ of $\triangle_N,\;\;N=(d+1)(r-1),$ and a continuous map  $f:\triangle_N\to\mathbb R^d$ and assume BMZ for special colorings. First we prove the following :

{\Prop  Let us remove one vertex from some color class, say $C_0$, and give it a new color creating an additional color class, say $C_{d+2}$ such that $|C_{d+2}|=1$. Then BMZ theorem is true for the parameters $(d,r,\mathbb R^d)$ for the coloring  $C_0, C_1\cdots , C_{d+1},C_{d+2};\;\; |C_0|=r-2, \;|C_i|=r-1, \; |C_{d+1}|=1=|C_{d+2}|$.}

\begin{proof}
To prove this, we add one new vertex to $C_0$ of the same color as that of  $C_0$, making it $C_0'$, add additional $r-2$ vertices of the color of $C_{d+2}$ to make it $C_{d+2}'$ so that $|C_{d+2}'|=r-1$ (see Fig 1). Thus we have the coloring
$$ C_0', C_1',\cdots , C_{d+1}', C_{d+2}',\;\;C_i\subset C_i'\;\;\;\forall \; i$$
and $|C_i'|=r-1\;\;\forall\;i\neq d+1,\;|C_{d+1}'|=1$. Now we consider a new simplex $\triangle_{N'},\;N'=(d+2)(r-1)$ and regard $\triangle_N$ as the front face of $\triangle_{N'}$. Thus, there are $r-1$ additional vertices in $\Delta_{N'}-\Delta_N$.

$$\bordermatrix{ & & &  &  & \cr
& & &  & &  \cr
 C_{d+1} & \bullet &   &\cr
  C_{d} & \bullet &  \bullet &  \bullet  &  \bullet &  \bullet  \cr
                & \vdots & \vdots & \ddots & \vdots & \vdots\cr
                C_{1} & \bullet &  \bullet &  \bullet  &  \bullet &  \bullet  \cr
                C_{0} & \bullet &  \bullet &  \bullet  &  \bullet &  \bullet  } \hspace{1cm}\longrightarrow \hspace{1cm} \bordermatrix{ & & &  &  & \cr
                              C_{d+2}' & \bullet  &  *  &  *  &  *  &  *  \cr
                 C_{d+1}' & \bullet &   &\cr
                  C_{d}' & \bullet &  \bullet &  \bullet  &  \bullet &  \bullet \cr
                                & \vdots & \vdots & \ddots & \vdots & \vdots\cr
                                C_{1}' & \bullet &  \bullet &  \bullet  &  \bullet &  \bullet \cr
                                C_{0}' & \bullet &  \bullet &  \bullet  &  \bullet &  *  } $$

\begin{center} Fig 1  \end{center}

\noindent Now we extend the map $f:\triangle_N \to \mathbb R^d$ to a map $f':\triangle_{N'}\to\mathbb R^{d+1}$ by embedding $\mathbb R^d\subset \mathbb R^{d+1}$ where the last coordinate of $\mathbb R^{d+1}$ is zero:  Let $e_1,\cdots , e_{d+1}$ be the standard basis of $\mathbb R^{d+1}$ and $v_0,\cdots , v_N$ be the vertices of $\triangle_N$ and $v_0,\cdots , v_N, v_{N+1},\cdots , v_{N'}$ be the vertices of $\triangle_{N'}$. Let $x\in\triangle_{N'}$ and write $x=\sum\limits^{N'}_{0}\lambda_iv_i$ in terms of barycentric coordinates. Define $f' : \triangle_{N'}\to\mathbb R^{d+1}$ by
$$f'(x)=f(\lambda_0v_0+\cdots +\lambda_{N-1} v_{N-1} +(\lambda_N+\lambda_{N+1}+\cdots + \lambda_{N'})v_N) +(\lambda_N+\cdots +\lambda_{N'})e_{d+1}   $$
Then note that $f'$ naps additional vertices to $e_{d+1}$, is continuous and $f'|\triangle_N=f$. Since BMZ is true for the parameters $(d+1,\;r,\;\mathbb R^{d+1})$ and $C_0',C_1',\cdots , C_{d+2}'  $ is a special coloring of $\triangle_{N'}$, by assumption of reduction lemma there exists a family $\bar{\sigma_1}, \cdots , \bar{\sigma_r}$ of $r$ disjoint rainbow faces of $\triangle_{N'}$ such that
$$f'(\bar{\sigma_1})\cap\cdots\cap f'(\bar{\sigma_r})\neq \phi$$
Now let us consider the faces $\sigma_i=\bar{\sigma_i}\cap\triangle_N$ for $1\leq i\leq r$ of $\triangle_{N}$.  Note that at least one of the faces $\bar{\sigma_i}$ must be completely inside $\triangle_N$, otherwise we will have $r$ vertices of $\triangle_{N'}$ which are outside $\triangle_N$, but there are only  $r-1$ such vertices by definition, a contradiction.
Hence $\bar{\sigma_1}\cap \triangle_N=\bar{\sigma_1}=\sigma_1$ is a rainbow face of $\triangle_N$. Therefore $$f'(\bar{\sigma_1})\cap\cdots\cap f'(\bar{\sigma_r})\subset\mathbb R^d\subset\mathbb R^{d+1}$$
Now we also claim that each of $\sigma_1,..., \sigma_r$ is nonempty, because if some $\bar{\sigma_i},\;i\geq 2$, does not intersect $\triangle_N$, then $f'(\bar{\sigma_i})$ will lie in the positive part of  the last axis of $\mathbb R^{d+1}$, which means $f'(\bar{\sigma_1})\cap f'(\bar{\sigma_i})=\phi$, a contradiction. This implies that
$$ f(\sigma_i)\cap\cdots\cap f(\sigma_r)=f'(\bar{\sigma_1})\cap\cdots\cap f'(\bar{\sigma_r}) \neq\phi.$$
\end{proof}

\bigskip\noindent The following is a simple generalization of Proposition 2.1

{\Prop   Let us remove $q$ vertices $2\leq q\leq r-2$ from some color class, say $C_0$, and introduce one new color class $C_{d+2}$ of size $q$ so that we have the new coloring of $\triangle_N, $
\begin{equation}
 C_0,C_1, \cdots , C_{d+!}, C_{d+2},\;|C_0|=r-q+1,\;  \\|C_i|=i-1,\;|C_{d+1}|=1,\;|C_{d+2}|=q ~~\tag{2.1}\label{eq:2.1}
\end{equation}
Then the BMZ Theorem is true for the coloring  (2.1) of $\triangle_N$ also.}
\begin{proof}
We add $q$ vertices to the color class $C_0$ of the same color and add $r-q-1$ new vertices to the class $C_{d+2}$ of the same color and get the coloring
\begin{equation}
C_0',\; C_1', \cdots , C_{d+1}' ,~ C_{d+2}' ~~\tag{2.2}
\end{equation} such that $|C_i'|=r-1,~~\forall~i\neq d+1$ and $|C_{d+1}'|=1$
Now consider the simplex $\triangle_{N'},~N=(d+2)(r-1)$ containing $\triangle_N$ as the front face and define a map $f:\triangle_{N'}\to\mathbb R^{d+1}$ by mapping all additional vertices of $\triangle_{N'}$ onto $e_{d+1}$, and extending linearly. Then note that $\triangle_{N'}-\triangle_N$ has again only $r-1$ vertices.
Then using the fact that BMZ is true for  $(d+1, r,\mathbb R^{d+1})$ for the special coloring (2.1), we find by the same argument as in Prop 2.1 that BMZ is true for $f:\triangle_N\to\mathbb R^d$ for the coloring (2.1) also.
\end{proof}

\bigskip\noindent The following covers yet another possibility.

{\Prop  If we remove two vertices from two different color classes, say $C-0, C-1$, and  add a new color class $C_{d+2}$ having these two additional vertices, we will have the coloring
\begin{equation} C_0,~C_1,~C_2,\cdots , C_{d+1},~C_{d+2}~~\tag{3.1\label{eq:3.1}}\end{equation} such that $|C_0|=|C_1|=r-2,~|C_i|=r-1,~|C_{d+1}|=1,~|C_{d+2}|=2$ Then the BMZ is true for the coloring  (3.1)
}

\begin{proof}
Once again we embed the colorings (3.1) into the coloring \begin{equation} C_0',~C_1',~C_2',\cdots , C_{d+1}',~C_{d+2}'~~\tag{3.2\label{eq:3.2}}\end{equation} where $|C_i'|= r-1,~|C'_{d+1}|=1,~|C'_{d+2}|=r-1$ Then (3.2) is again a special coloring for  $(d+1, r,\mathbb R^{d+1})$. We define a map $f': \triangle_{N'}\to\mathbb R^{d+1},~N'=(d+2)(r-1)$ extending $f$ linearly as before. Then, again as in Proposition 2.1, this yields BMZ for $f:\triangle_N\to\mathbb R^d,~N=(d+1)(r-1)$.
\end{proof}

\bigskip\noindent {\bf Note :} It is clear from the above propositions that so long as  we remove any number of vertices from anywhere and  create only one new color class, the arguments of Proposition 2.1 will work and we will get BMZ for such general colorings. The argument will have to be changed when the number of new color classes so created exceeds one, as in the following.

{\Prop  Let us remove two vertices from some color class say $C_0$, and add two new color classes $C_{d+2},~C_{d+3}$ of size one each  such that we have  the coloring
\begin{equation} C_0,~C_1,~C_2,\cdots , C_{d+1},~C_{d+2},~C_{d+3}~~\tag{4.1\label{eq:4.1}}\end{equation}  with $|C_0|=r-3~|C_{d+1}|=|C_{d+2}|=|C_{d+3}|=1$  and $|C_i|=r-1$ for all other $i$'s. Then BMZ is true for the coloring (4.1) also.
}

\begin{proof}
Add two new vertices to $C_0$ of the same color as that of $C_0$, add $r-2$ new vertices to each of $C_{d+2},~C_{d+3}$ to get a new coloring of $\triangle_{N'},~N'=(d+3)(r-1)$
\begin{equation} C_0',~C_1',~C_2',\cdots , C_{d+1}',~C_{d+2}',~C_{d+3}'~~\tag{4.2\label{eq:4.2}}\end{equation} where $|C'_i|=r-1$ for each $i\neq d+1$
and $|C_{d+1}'|=1$. Then  (4.2) is a special coloring of $\triangle_{N'}$. Let $N=N_1=(d+1)(r-1),~N_2=(d+2)(r-1),~N'=N_3=(d+3)(r-1)$ and treat $\triangle_N$ as a front face of $\triangle_{N_2},~\triangle_{N_2} $ as a front face of $\triangle_{N_3}=\triangle_{N'}$. Let $e_i,~1\leq i\leq d+2$ be the standard basis of $\mathbb R^{d+2}$. Define a map $f':\triangle_{N'}\to\mathbb R^{d+2}$ as follows. Map one removed vertex of $C_0$ and last $r-2$ vertices of $C_{d+1}'$ to $e_{d+1}$, map other removed vertex of $C_0$ and $r-2$ vertices of $C_{d+2}'$ to $e_{d+2}$. Then as in Proposition  2.1, extend $f$ to $f': \triangle_{N'}\to\mathbb R^{d+2}$ linearly. We have $\triangle_{N_1}\subset\triangle_{N_2}\subset\triangle_{N_3},~\mathbb R^d\subset\mathbb R^{d+1}\subset\mathbb R^{d+2}$ and $\triangle_{N_3}-\triangle_{N_2},~\triangle_{N_2}-\triangle_{N_1}$ each have only $r-1$ vertices. Now using the BMZ for the special coloring (4.2) and the continuous map $f':\triangle_{N_3}\to\mathbb R^{d+2}$, we find $r$ disjoint rainbow faces $\sigma_1, \sigma_2,\cdots , \sigma_r$ of $\triangle_{N_3}$ such that
$$ f(\sigma_i)\cap\cdots\cap f(\sigma_r)\neq\phi.$$
Now arguing as in Proposition 2.1 again, we find that $\sigma_1\cap \Delta_{N_2},~\cdots , \sigma_r\cap \Delta_{N_2}$ are rainbow faces of $\triangle_{N_2}$ such that their $f'$-images have non empty intersection. Intersecting above rainbow faces with $\Delta_{N_1}$ and arguing once more, we find that $\sigma_1\cap \Delta_{N_1},~\cdots , \sigma_r\cap \Delta_{N_1}$ are  rainbow faces of $\triangle_{N_1}$ whose $f'$-images and therefore $f$-images have non empty intersection.
\end{proof}

\bigskip\noindent Now let us start with a general coloring \begin{equation} C_0,~C_1,~C_2,\cdots , C_{d+1},\cdots , C_{d+k}, \; k\geq 1~~\tag{5.1\label{eq:5.1}}\end{equation} of $\triangle_N,~N=(d+1)(r-1)$ where $|C_i|\leq r-1$. We introduce  a new color class $C_{d+k+1}=\{\phi\}$.

\bigskip\noindent We extend these color classes to the classes  \begin{equation} C_0',~C_1',~C_2',\cdots , C_{d+1}',~C_{d+2}',\cdots ,C_{d+k+1}'~~\tag{5.2\label{eq:5.2}}\end{equation}  such that $|C'_{d+k+1}|=1$
and $|C_i|=r-1,~~\forall~i\neq d+k+1$. Clearly $C_0',~C_1',~C_2',\cdots , C_{d+1}',~C_{d+2}',\cdots ,C_{d+k+1}'$ is a special coloring of $\triangle_{N'}, ~N'=(d+k+1)(r-1)$. We can now think that $C_{d+1}'$ in (5.2) has been obtained  by removing some vertices from $C_0\cup\cdots\cup C_d$ and giving them a new color. In this way we obtain a new  coloring $C_0',~C_1',~C_2',\cdots , C_{d+1}'$ such that $|C_0'\cup\cdots C_{d+2}'|=(d+2)(r-1)=N_2$. We can now assume that $N=N_1\subset N_2$. Similarly we get numbers $N_3,~N_4,\cdots ,N_{k+1}$ such that $\Delta_{N_1}\subset \Delta_{N_2}\subset\cdots \subset \Delta_{N_{k+1}}$ and $|\triangle_{N_{i+1}}-\triangle_{N_i}|=r-1$ for each $i$. Now we can also assume that $$\mathbb R^d\subset\mathbb R^{d+1}\subset\cdots\subset\mathbb R^{d+k+1}$$ where the last coordinate is taken as zero everywhere.  Let $e_1,\cdots , e_d,~e_{d+1},\cdots , e_{d+k+1}$ be the standard basis for $\mathbb R^{d+k+1}$. Then as in Proposition 2.4, we can define a map $f':\triangle_{N_{k+1}}\to\mathbb R^{d+k+1}$ such that $f'$ maps first $r-1$ additional vertices to $e_{d+1}$, second $r-1$ additional vertices to $e_{d+2}$ etc.
Then we extend $f$ to $f':\triangle_{N'}\to\mathbb R^{d+k+1}$ linearly. By the BMZ for parameters $(d+k+1,r,\mathbb R^{d+k+1})$ for the special coloring (5.2), we find that we have a family $\sigma_1, \sigma_2,\cdots , \sigma_r$ of disjoint rainbow faces of  $\triangle_{N'}$ such that
$$ f'(\sigma_i)\cap\cdots\cap f'(\sigma_r)\neq\phi.$$  Then arguing  as in Proposition 2.4 repeatedly, we find that $\sigma_1\cap \triangle_N,~\cdots , \sigma_r\cap \triangle_N$ is a family of $r$ disjoint rainbow faces of $\triangle_N$ whose $f$-images will have non empty intersection.

\bigskip\noindent The above analysis proves the following

{\Prop Let $d\geq 1,~r\geq 2$ prime and $N=(d+1)(r-1)$. Let $C_0,\cdots , C_{d+k},\;\; k\geq 1$ be a general coloring of $\triangle_N$ where $|C_i|\leq r-1$. Then assuming  BMZ for special colorings of $\triangle_{N'},~~N'=(d+k)(r-1)$,  we find that  BMZ is true for any general coloring  of $\triangle_{N}$.}

\section{Reduction Lemma for maps to  Manifolds}
The BMZ Theorem for a $d$-manifold $M$ is stated as follows \cite{bmz3}:
{\Thm {\bf (BMZ for Manifolds)}  Let $d\geq 1, r\geq 2$ a prime $N=(d+1)(r-1)$. Suppose $$C_0,~C_1,\cdots, C_m,~~m\geq d+1$$ is a general  coloring of the simplex $\triangle_N$ such that $|C_i|\leq r-1$. Then for any given continuous map $f: \triangle_N\to M$, where $M$ is a $d$-manifold, we can find a family of $r$ disjoint rainbow faces  $\sigma_1, \sigma_2,\cdots , \sigma_r$ of $\triangle_N$ such that $$\bigcap f(\sigma_i)\neq \phi .$$ }

\bigskip\noindent Let $\mathbb I=[0,1]$ denote the unit interval and $\bar{M}=M\times \mathbb I^k$

\bigskip\noindent {\bf Reduction Lemma :(\cite{bmz3} Lemma 2.1) } {\it Suppose the BMZ Theorem for manifolds is true for all special colorings for the parameters $(d+k, r, M\times \mathbb I^k),~k\geq 0$ . Then the BMZ Theorem for maps to manifolds is true for all general colorings  for the parameters $(d,r,M)$.}

\begin{proof}
The proof of above Lemma is exactly parallel to the proof of Proposition 2.5 proved earlier except that we replace $\mathbb R^{d+k}$ everywhere by $M\times \mathbb I^k$ and $\mathbb R^d \subset \mathbb R^{k+1}\subset ...\subset \mathbb R^{d+k}$ by $M\subset M\times \mathbb I .... \subset  M\times \mathbb I^k$.
\end{proof}

\bigskip\noindent The reduction lemma as stated by BMZ in ~\cite{bmz3} says the following

\bigskip\noindent {\bf Reduction Lemma : } {\it It suffices to consider a manifold $M$ which satisfies the property that  \begin{equation}(r-1)\dim M>r . \hbox{cohdim} ~M~~\tag{*} \end{equation}
where cohdim $M$ means the cohomological dimension of $M$.}

\bigskip\noindent Here the cohomological dimension means the largest integer $d$ such that the $d$-dimensional cohomology of $M$ does not vanish. If we use the classical definition of cohomological dimension which conforms to the geometric intuition, the inequality (*) may appear a bit surprising. In order to state it in conformity with classical definition of cohomological dimension, let us recall (see~\cite{nag} Appendix, Cohomological Dimension Theory,  and also ~\cite{deo}, ~\cite{deo1} for details).

{\Def Let $X$ be any topological space and $H^*(X,\mathbb Z)$ denote the $\check{C}$ech cohomology of $X$ with coefficients in $\mathbb Z$. Then the largest integer $n$ (if it exists) such that $H^q(X,A;\mathbb Z)=0~\forall~q>n$ and for all closed sets $A$ of $X$, is called the cohomological dimension of $X$. Otherwise we say that cohomological dimension of $X$ is infinite. We write it as cohdim$(X)$.}

\bigskip\noindent The role of the pair $(X,A)$ where $A$ is a closed set is very important. Since $\mathbb R$ is contractible, all the positive dimensional cohomologies of $\mathbb R^n$ vanish. This does not mean that that the cohdim $\mathbb R^n$ is zero. In fact, let us see that cohdim $(\mathbb R^n)=n.$  Let $A$ be any closed subset of $\mathbb R^n$. Then it is well known that $H^q(\mathbb R^n, A; \mathbb Z)= 0~\forall~q>n$ and for all closed sets $A$ of $\mathbb R^n$, which implies that cohdim$\mathbb R^n\leq n.$ On the other hand, it is easy to see $H^n(\mathbb R^n,\mathbb S^{n-1};\mathbb Z)\neq 0$ and so cohdim$\mathbb R^n\geq n.$ Thus we find that $\dim \mathbb R^n=\hbox{cohdim}~\mathbb R^n=n$. Also, for a paracompact Hausdorff space $X$, cohdim is a local property ~\cite{deo}, which means if $M$ is a paracompact manifold, then
$$\dim M=\hbox{cohdim} M. $$
Now going back to the BMZ Reduction Lemma for manifolds in \cite{bmz3}, observe that $M\times \mathbb I^k$ is a manifold of dimension $d+k ~(\dim M=d)$ and since $M\times \mathbb I^k$ is homotopically equivalent to $M,$ we find that $H^q(M\times\mathbb I^k)=0~\forall~q>d$. Thus the manifold $M''$ needed at the end of the proof of Lemma 2.1 in~\cite{bmz3} satisfies the condition that \begin{equation}(r-1)\dim M'' >r.d  ~~\tag{**} \end{equation} where $\dim M''=d+k$ and $H^q(M'';\mathbb Z)=0~\forall~q>d$. The manifold $M''= M\times\mathbb I^k$ clearly satisfies this property for $k$ suitably large.

The author is thankful to G$\ddot{u}$nter M. Ziegler for some valuable comments on this expository article.