}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${M_{1}A^{j}_{0}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
16.16472pt\hfil&\hfil\hskip 33.36885pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{}{}}{}{}{{}{}}}}}{}{{{{{}}{
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}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
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}\hbox{$\scriptstyle{a^{1,0}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-21.45554pt}{4.2592pt}\pgfsys@lineto{-21.45554pt}{-12.88086pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{M_{1}p^{j}_{0}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{30.60413pt}{-6.02673pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{p^{j}_{1}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.09082pt}{-23.38052pt}\pgfsys@lineto{18.22249pt}{-23.38052pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{18.22249pt}{-23.38052pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{2.39021pt}{-18.93095pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{a^{1,0}_{j}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\text{for every
$j\in J$}.$

Indeed, the morphisms $a^{1,0}_{j}\cdot M_{1}p^{j}_{0}$ form a cone (in
$\mathscr{C}$) on the diagram yielding the component $A_{1}$ of the colimit
$A$: for every morphism $h=(h_{0},h_{1})\colon A^{j}\to A^{k}$ in the given
diagram scheme we have a commutative diagram

${M_{1}A^{j}_{0}}$${A^{j}_{1}}$${M_{1}A_{0}}$${M_{1}A^{k}_{0}}$${A^{k}_{1}}$$\scriptstyle{a^{1,0}_{j}}$$\scriptstyle{M_{1}h_{0}}$$\scriptstyle{h_{1}}$$\scriptstyle{M_{1}p^{j}_{0}}$$\scriptstyle{M_{1}p^{k}_{0}}$$\scriptstyle{a^{1,0}_{k}}$

Thus, $A$ carries a unique structure of an $\mathscr{I}$-object such that the
limit projections $p^{j}$ are morphisms of $\mathscr{I}$.

We still need to prove that $A$ is the limit of the $A^{j}$ in $\mathscr{I}$.
Given any cone $h^{j}\colon B\to A^{j}$ ($j\in J$) in $\mathscr{I}$, we know
that there is a unique morphism $h\colon B\to A$ of $\mathscr{K}$ such that
$p^{j}\cdot h=h^{j}$ for every $j\in J$. It suffices to show that $h$ is a
morphism in $\mathscr{I}$. For this consider the following diagram:

${M_{1}B_{0}}$${B_{1}}$${M_{1}B_{0}}$${B_{1}}$${M_{1}A_{0}}$${A_{1}}$${M_{1}A^{j}_{0}}$${A_{1}}$$\scriptstyle{b^{1,0}}$$\scriptstyle{M_{1}h^{j}_{0}}$$\scriptstyle{h^{j}_{1}}$$\scriptstyle{b^{1,0}}$$\scriptstyle{M_{1}h_{0}}$$\scriptstyle{\mathsf{id}}$$\scriptstyle{h_{1}}$$\scriptstyle{\mathsf{id}}$$\scriptstyle{a^{1,0}}$$\scriptstyle{M_{1}p^{j}_{0}}$$\scriptstyle{p^{j}_{1}}$$\scriptstyle{a^{1,0}_{j}}$

The outside commutes since $h^{j}$ is a morphism of $\mathscr{I}$. The left-
and right-hand parts commute by the definition of $h$, the lower part commutes
by the definition of $a^{1,0}$ and the upper part trivially does. Thus, the
middle square commutes when postcomposed by every limit projection
$p^{j}_{1}$, whence this square commutes, as desired.

For $\kappa$-filtered colimits, given a $\kappa$-filtered diagram of
$M_{1}$-algebras $A^{j}$ ($j\in J$), let $A$ be its colimit in $\mathscr{I}$
with colimit injections $\mathsf{in}^{j}\colon A^{j}\to A$. We use that
$M_{1}$ preserves the colimit in the first component; that is, we have a
$\kappa$-filtered colimit with the injections $M_{1}\mathsf{in}^{j}_{0}\colon
M_{1}A^{j}_{0}\to M_{1}A_{0}$ ($j\in J$) in
$\operatorname{\mathsf{Alg}}_{0}(\mathbb{M})$ (whence in $\mathscr{C}$). Then
we obtain a unique algebra structure $a^{1,0}\colon M_{1}A_{0}\to A_{1}$ such
that every $\mathsf{in}^{j}$ ($j\in J$) is a morphism in $\mathscr{I}$, i.e.
the following squares commute:

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}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{$\scriptstyle{a^{1,0}_{j}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{$\scriptstyle{M_{1}\mathsf{in}^{j}_{0}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
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{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{30.60413pt}{-8.74005pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\mathsf{in}^{j}_{1}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}\pgfsys@setdash{2.79985pt,1.59991pt}{0.0pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.00417pt}{-23.38052pt}\pgfsys@lineto{18.13582pt}{-23.38052pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{18.13582pt}{-23.38052pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{2.39021pt}{-21.02776pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{a^{1,0}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\text{for every
$j\in J$}.$

Indeed, the morphisms $\mathsf{in}^{j}_{1}\cdot a^{1,0}_{j}\colon
M_{1}A^{j}_{0}\to A_{1}$ form a cocone of the above diagram formed by the
$M_{0}$-algebras $M_{1}A^{j}_{0}$: for every morphism $h=(h_{0},h_{1})\colon
A^{j}\to A^{k}$ in the diagram scheme we have a commutative diagram

${M_{1}A^{j}_{0}}$${A^{j}_{1}}$${A_{1}}$${M_{1}A^{k}_{0}}$${A^{k}_{1}}$$\scriptstyle{a^{1,0}_{j}}$$\scriptstyle{M_{1}h_{0}}$$\scriptstyle{\mathsf{in}^{j}_{1}}$$\scriptstyle{h_{1}}$$\scriptstyle{a^{1,0}_{k}}$$\scriptstyle{\mathsf{in}^{k}_{1}}$

Thus, $A$ carries a unique structure of an $\mathscr{I}$-object such that the
colimit injections $\mathsf{in}^{j}$ are morphisms of $\mathscr{I}$.

We still need to prove that $A$ is the colimit of the $A^{j}$ in
$\mathscr{I}$. Given any cocone $h^{j}\colon A^{j}\to B$ ($j\in J$) in
$\mathscr{I}$ we know that there is a unique morphism $h\colon A\to B$ of
$\mathscr{K}$ such that $h\cdot\mathsf{in}^{j}=h^{j}$ for every $j\in J$. It
suffices to show that $h$ is a morphism in $\mathscr{I}$. For this consider
the following diagram:

${M_{1}A^{j}_{0}}$${A^{j}_{1}}$${M_{1}A_{0}}$${A_{1}}$${M_{1}B_{0}}$${B_{1}}$${M_{1}B_{0}}$${B_{1}}$$\scriptstyle{M_{1}\mathsf{in}^{j}_{0}}$$\scriptstyle{a^{1,0}_{j}}$$\scriptstyle{M_{1}h^{j}_{0}}$$\scriptstyle{\mathsf{in}^{j}_{1}}$$\scriptstyle{h^{j}_{1}}$$\scriptstyle{a^{1,0}}$$\scriptstyle{M_{1}h_{0}}$$\scriptstyle{h_{1}}$$\scriptstyle{b^{1,0}}$$\scriptstyle{\mathsf{id}}$$\scriptstyle{b^{1,0}}$$\scriptstyle{\mathsf{id}}$

The outside commutes since $h^{j}$ is a morphism in $\mathscr{I}$. The left-
and right-hand parts commmute by the definition of $h$, the upper part
commutes by the definition of $a^{1,0}$ and the lower part trivially does.
Thus, the inner square commutes when precomposed by every colimit injection
$M_{1}\mathsf{in}^{j}_{0}$, whence it commutes, as desired.

  6. (6)

It follows that $\mathscr{I}$ is complete, and therefore it is locally
presentable by A.5(1). Moreover, by the Adjoint Functor Theorem (Adámek and
Rosický, 1994, Thm. 1.66), $V$ has a left adjoint
$L\colon\mathscr{K}\to\mathscr{I}$.

From A.4(1) we obtain that for every $\kappa$-presentable object in
$\mathscr{K}$, i.e. a pair $A$ formed by $\kappa$-presentable objects
$(A_{i},a^{0,i})$, $i=0,1$, the object $LA$ is $\kappa$-presentable in
$\mathscr{I}$. We show that the these $\kappa$-presentable objects $LA$ form a
strong generator in $\mathscr{I}$. By A.5(2), it follows that $\mathscr{I}$ is
locally $\kappa$-presentable. Given an object $A$ of $\mathscr{I}$ we know
that $VA$ is an extremal quotient of a coproduct of $\kappa$-presentable
objects in $\mathscr{K}$, reprensented, say, by the extremal epimorphism

$\textstyle e\colon\coprod_{j\in J}A_{j}\twoheadrightarrow VA$

for some set $J$. Using A.4(1), that the left-adjoint $L$ preserves coproducts
and that $V$ reflects isomorphisms and is clearly faithful we thus see that
the adjoint transpose

$\textstyle\bar{e}\colon\coprod_{j\in J}LA_{j}\twoheadrightarrow A$

exhibits $A$ as an extremal quotient of the coproduct of the
$\kappa$-presentable objects $LA_{j}$.

  7. (7)

Now let $Q_{i}\colon\mathscr{I}\to\mathscr{C}$, $i=0,1$, be the obvious
forgetful projection functors. We have natural transformations
$\varphi^{i,j}\colon M_{i}Q_{j}\to Q_{i+j}$ for $i+j\leq 1$ with components
$\varphi^{i,j}_{A}=a^{i,j}$. The missing laws of an $M_{1}$-algebra
(homomorphy and coequalization) can be expressed by equifiers of the following
pairs of natural transformations

$\displaystyle\varphi^{1,0}\cdot\mu^{1,0}Q_{0},\varphi^{1,0}\cdot
M_{1}\varphi^{0,0}$ $\displaystyle\colon M_{1}M_{0}Q_{0}\to Q_{1},$
$\displaystyle\varphi^{1,0}\cdot\mu^{1,0}Q_{0},\varphi^{1,0}\cdot
M_{0}\varphi^{1,0}$ $\displaystyle\colon M_{0}M_{1}Q_{0}\to Q_{1}.$

Consequently, $\operatorname{\mathsf{Alg}}_{1}(\mathbb{M})$ is
$\kappa$-accessible and the inclusion functor
$I\colon\operatorname{\mathsf{Alg}}_{1}(\mathbb{M})\hookrightarrow\mathscr{I}$
is $\kappa$-accessible by A.5(6).

  8. (8)

To see that $\operatorname{\mathsf{Alg}}_{1}(\mathbb{M})$ is locall
$\kappa$-presentable, we prove that
$\operatorname{\mathsf{Alg}}_{1}(\mathbb{M})$ is closed under limits and
$\kappa$-filtered colimits in $\mathscr{I}$. This is a slight variation of the
standard argument that the Eilenberg-Moore algebras for a $\kappa$-accessible
monad $T$ is closed under limits and $\kappa$-filtered colimits.

For limits, given a diagram of $M_{1}$-algebras $A^{j}$ ($j\in J$), let $A$ be
its limit in $\mathscr{I}$ with limit projections $p^{j}\colon A\to A^{j}$. We
show that $A$ is an $M_{1}$-algebra, that means that it satisfies homomorphy
and coequalization. For the latter consider the following diagram (structure
morphisms of $A^{j}$ have their index as a subscript, e.g. $a^{1,0}_{j}\colon
M_{1}A^{j}_{0}\to A^{j}_{1}$):

${M_{1}M_{0}A^{j}_{0}}$${M_{1}A^{j}_{0}}$${M_{1}M_{0}A_{0}}$${M_{1}A_{0}}$${M_{1}A_{0}}$${A_{1}}$${M_{1}A^{j}_{0}}$${A^{j}_{1}}$$\scriptstyle{\mu^{1,0}_{A^{j}_{0}}}$$\scriptstyle{M_{1}a^{0,0}_{j}}$$\scriptstyle{a^{1,0}_{j}}$$\scriptstyle{M_{1}M_{0}p_{0}^{j}}$$\scriptstyle{\mu^{1,0}_{A_{0}}}$$\scriptstyle{M_{1}a^{0,0}}$$\scriptstyle{M_{1}p_{0}^{j}}$$\scriptstyle{a^{1,0}}$$\scriptstyle{M_{1}p_{0}^{j}}$$\scriptstyle{a^{1,0}}$$\scriptstyle{p_{1}^{j}}$$\scriptstyle{a^{1,0}_{j}}$

The outside commutes due to coequalization of $A^{j}$ and the four inner
trapezoids by the naturality of $\mu^{1,0}$ and since
$p^{j}=(p^{j}_{0},p^{j}_{1})$ is a morphism in $\mathscr{I}$. Thus, the
desired middle square commutes when postcomposed by the projection
$p^{j}_{1}$. Since limits in $\mathscr{I}$ are formed componentwise, the
$p^{j}_{1}$ ($j\in J$) form a limit cone whence a jointly monic family. This,
implies that the desired inner square commutes. That $A$ satisfies homomorphy
is shown analogogously.

For $\kappa$-filtered colimits, given a $\kappa$-filtered diagram of
$M_{1}$-algebras $A^{j}$ ($j\in J$), let $A$ be its colimit in $\mathscr{I}$
with colimit injections $\mathsf{in}^{j}\colon A^{j}\to A$. We show that $A$
is an $M_{1}$-algebra, that means that it satisfies homomorphy and
coequalization. For the former, observe first that since $M_{0}M_{1}$ is
$\kappa$-accessible and colimits in $\mathscr{I}$ are formed componentwise we
have a colimit cocone $M_{0}M_{1}\mathsf{in}^{j}_{0}\colon A^{j}_{0}\to
A_{0}$. Now consider the following diagram (again we put the index of $A^{j}$
in the subscript of structure morphisms)

${M_{0}M_{1}A_{0}^{j}}$${M_{1}A^{j}_{0}}$${M_{0}M_{1}A_{0}}$${M_{1}A}$${M_{1}A_{0}}$${A_{1}}$${M_{1}A^{j}_{0}}$${A_{1}^{j}}$$\scriptstyle{\mu^{0,1}_{A_{0}^{j}}}$$\scriptstyle{M_{0}a^{1,0}_{j}}$$\scriptstyle{M_{0}M_{1}\mathsf{in}^{j}_{0}}$$\scriptstyle{a^{1,0}_{j}}$$\scriptstyle{M_{1}\mathsf{in}^{j}_{0}}$$\scriptstyle{\mu^{0,1}_{A}}$$\scriptstyle{M_{0}a^{1,0}}$$\scriptstyle{a^{1,0}}$$\scriptstyle{a^{1,0}}$$\scriptstyle{a^{1,0}_{j}}$$\scriptstyle{M_{1}\mathsf{in}^{j}_{0}}$$\scriptstyle{\mathsf{in}^{j}_{1}}$

The outside commutes since $A^{j}$ is an $M_{1}$-algebra, and the four inner
trapezoids commute due to the naturality of $\mu^{0,1}$ and the fact that
$\mathsf{in}^{j}$ is a morphism in $\mathscr{I}$. Thus, the desired inner
square commutes when precomposed by every colimit injection
$M_{0}M_{1}\mathsf{in}^{j}_{0}$, whence it commutes as desired. That $A$
satisfies coequalization is shown analogously

  9. (9)

The desired result now follows from the Reflection Theorem (Adámek and
Rosický, 1994, Thm. 2.48). ∎

###### Corollary A.1.

The functor
$(-)_{1}\colon\operatorname{\mathsf{Alg}}_{1}(\mathbb{M})\to\operatorname{\mathsf{Alg}}_{0}(\mathbb{M})$
taking $1$-parts is $\kappa$-accessible.

Indeed, this follows from the proof of A.6 since we have the commutative
diagram of functors

${\operatorname{\mathsf{Alg}}_{1}(\mathbb{M})}$${\operatorname{\mathsf{Alg}}_{0}(\mathbb{M})}$${\mathscr{I}}$${\mathscr{C}}$$\scriptstyle{(-)_{1}}$$\scriptstyle{I}$$\scriptstyle{U_{0}}$$\scriptstyle{Q_{1}}$

where the functor $Q_{1}I$ preserves and the forgetful functor $U_{0}$ creates
$\kappa$-filtered colimits.

###### Corollary A.2.

The functor $\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}\colon\operatorname{\mathsf{Alg}}_{0}(\mathbb{M})\to\operatorname{\mathsf{Alg}}_{0}(\mathbb{M})$
has a final coalgebra.

Indeed, since $\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}=(-)_{1}\cdot
E$, where $E$ is a left adjoint (see (5.1)) we see that $\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}$
is a $\kappa$-accessible endofunctor on the locally $\kappa$-presentable
category $\operatorname{\mathsf{Alg}}_{0}(\mathbb{M})$. It follows that the
category of all coalgebra for $\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}$
is locally presentable (Adámek and Rosický, 1994, Exercise 2j). (Though note
that the index of presentability may not be $\kappa$.) In particular, this
category is complete and therefore has a terminal object.

#### Proof of 7.5

###### Remark A.8.

Suppose that $\mathscr{C}$ is a category with a (regular epi, mono)
factorization system (e.g. a regular category (Borceux, 1994, Def. 2.1.1));
that means that every morphism $f$ of $\mathscr{C}$ has a factorization
$f=m\cdot e$ into a regular epimorphism $e$ followed by a monomorphism $m$.
Moreover, the following unique diagonal fill-in property holds: whenever we
have $m\cdot f=g\cdot e$ for a regular epimorphism $e$ and a monomorphism $m$,
then there exists a unique diagonal $d$ such that $m\cdot d=g$ and $d\cdot
e=f$:

${A}$${B}$${C}$${D}$$\scriptstyle{e}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{d}$$\scriptstyle{m}$

Now suppose further that $F\colon\mathscr{C}\to\mathscr{C}$ preserves
monomorphisms. Then every coalgebra morphism $h\colon(A,\gamma)\to(B,\delta)$
can be factorized into a coalgebra morphism carried by a regular epi and one
carried by a monomorphism in $\mathscr{C}$. Indeed, take the factorization $h$
into a regular epimorphism $e\colon A\twoheadrightarrow C$ followed by a
monomorphism $m\colon C\rightarrowtail B$ in $\mathscr{C}$, and observe that
the unique diagonal fill-in property yields a unique coalgebra structure
$\alpha\colon C\to FC$ such that $e$ and $m$ are coalgebra morphisms (here one
uses that $Fm$ is a monomorphism):

${A}$${FA}$${C}$${FC}$${B}$${FB}$$\scriptstyle{\gamma}$$\scriptstyle{e}$$\scriptstyle{h}$$\scriptstyle{Fe}$$\scriptstyle{Fh}$$\scriptstyle{\alpha}$$\scriptstyle{m}$$\scriptstyle{Fm}$$\scriptstyle{\delta}$

Towards the proof of 7.5, we generalize the game to be played on any
$\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}$-coalgebra
$(A,\gamma)$. Positions of D then are pairs in $A\times A$; positions of S are
relations $Z\subseteq A\times A$. A move from $(x,y)\in A\times A$ to such a
$Z$ is admissible for D if the relations in $Z$ entail equality of $\gamma(x)$
and $\gamma(y)$; in categorical formulation, this means that $\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}c_{Z}(\gamma(x))=\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}c_{Z}(\gamma(y))$
where $c_{Z}\colon A\to C_{Z}$ is the coequalizer of the pair
$\ell^{*}_{0},r^{*}_{0}\colon M_{0}Z\to A$ for the projections $\ell,r\colon
Z\to A$. Again, S just picks from $Z$; any player who cannot move, loses, and
infinite matches are won by D. We then claim, generalizing the claim of the
theorem, that $x,y\in A$ are behaviourally equivalent iff D wins $(x,y)$.

###### Proof (7.5).

($\Leftarrow$) We denote the winning region of $D$ by $\sim_{\mathcal{G}}$. It
is easy to see that $\sim_{\mathcal{G}}$ is a congruence on the
$M_{0}$-algebra $A$ (this is similar as in the proof of 7.12), so we have a
unique $M_{0}$-algebra structure on $A/{\sim_{\mathcal{G}}}$ such that the
quotient map $e\colon A\twoheadrightarrow A/{\sim_{\mathcal{G}}}$ is an
$M_{0}$-algebra homomorphism. We write elements of $A/{\sim_{\mathcal{G}}}$ in
the form $[x]_{D}$ for $x\in A$. We shall define a coalgebra structure
$\gamma_{D}\colon A/{\sim_{\mathcal{G}}}\to\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}(A/{\sim_{\mathcal{G}}})$
such that the square below commutes in
$\operatorname{\mathsf{Alg}}_{0}(\mathbb{M})$:

${A}$${\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}A}$${A/{\sim_{\mathcal{G}}}}$${\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}(A/{\sim_{\mathcal{G}}})}$$\scriptstyle{\gamma}$$\scriptstyle{e}$$\scriptstyle{\mkern
1.5mu\overline{\mkern-2.5muM\mkern-1.5mu}\mkern
1.5mu_{1}e}$$\scriptstyle{\gamma_{D}}$

That means that it suffices to show that

$\gamma_{D}([x]_{D})=\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}e(\gamma(x))\qquad\text{for
every $x\in A$}$

is well-defined; indeed, since $\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}e\cdot\gamma$
is an $M_{0}$-algebra homomorphism and $\sim_{\mathcal{G}}$ a congruence,
$\gamma$ is then clearly an $M_{0}$-algebra homomorphism, too, and the above
diagram commutes so that $e$ becomes a coalgebra morphism that identifies all
$x,y$ such that D wins $(x,y)$).

So let D win $(x,y)$; we have to show that

$\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}e(\gamma(x))=\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}e(\gamma(y)).$

Let $Z$ be the winning move of D at $(x,y)$ (that is, $Z$ is D’s first move of
the corresponding match). Since S can pick any element of $Z$, D wins on every
element of $Z$, so that $e$ factors through the above coequalizer $c_{Z}\colon
A\to C_{Z}$: there is a unique $M_{0}$-algebra homomorphism $h\colon C_{Z}\to
A/{\sim_{\mathcal{G}}}$ such that $h\cdot c_{Z}=e$. Hence, $\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}e$
factors through $\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}c_{Z}$
via $\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}h$.
Since $Z$ is an admissible move, $\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}c_{Z}$
identifies $\gamma(x)$ and $\gamma(y)$; hence, so does $\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}e$,
as required.

($\Rightarrow$) Let $h\colon(A,\gamma)\to(B,\delta)$ be an $\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}$-coalgebra
morphism; it suffices to show that the kernel

$\operatorname{\mathsf{ker}}h=\\{(x,y)\in A\times A\mid h(x)=h(y)\\}$

of $h$ is contained in $\sim_{\mathcal{G}}$. To see this, it suffices to show
that D can maintain the invariant $\operatorname{\mathsf{ker}}h$, i.e. ensure
that her positions always remain in $\operatorname{\mathsf{ker}}h$ if the game
starts in a position in $\operatorname{\mathsf{ker}}h$. But when at a position
$(x,y)\in\operatorname{\mathsf{ker}}h$, she can clearly ensure that the next
position is still in $\operatorname{\mathsf{ker}}h$ by just playing
$Z:=\operatorname{\mathsf{ker}}h$.

We proceed to show that $Z$ is admissible. Let $\ell,r\colon Z\to A$ be the
obvious projection maps. Since the kernel is clearly a congruence, $Z$ is an
$M_{0}$-algebra, and $\ell,r\colon Z\to A$ are $M_{0}$-algebra homomorphims.
(In fact, $Z$ is the pullback of $h$ along itself, and the forgetful functor
$\operatorname{\mathsf{Alg}}_{0}(\mathbb{M})\to\mathbf{Set}$ creates limits.)
Using 6.2(1) (and noting that it holds for arbitrary algebras $A$ in lieu of
$M_{0}X$) we take the coequalizer $c_{Z}\colon A\twoheadrightarrow C_{Z}$ of
the pair $\ell,r$, and we now prove that $\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}c_{Z}\cdot\gamma$
merges the pair $\ell,r$. By A.3, the unique morphism $m\colon
C_{Z}\rightarrowtail B$ such that $h=m\cdot c_{Z}$ is monic. By A.8 we know
that $C_{Z}$ carries a unique coalgebra structure $\alpha\colon
C_{Z}\to\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}C_{Z}$
such that $c_{Z}$ and $m$ are coalgebra morphisms. Using the former fact and
that $c_{Z}$ merges $\ell$ and $r$ we obtain the following commutative
diagram:

${A}$${FA}$${Z}$${C}$${FC}$${A}$${FA}$$\scriptstyle{\gamma}$$\scriptstyle{c_{Z}}$$\scriptstyle{\mkern
1.5mu\overline{\mkern-2.5muM\mkern-1.5mu}\mkern
1.5mu_{1}c_{Z}}$$\scriptstyle{\ell}$$\scriptstyle{r}$$\scriptstyle{\alpha}$$\scriptstyle{\gamma}$$\scriptstyle{c_{Z}}$$\scriptstyle{\mkern
1.5mu\overline{\mkern-2.5muM\mkern-1.5mu}\mkern 1.5mu_{1}c_{Z}}$

The commutativity of its outside is the desired equality.∎

#### Proof of 7.12

It is straightforward to see that $\sim_{\mathcal{G}}$ is an equivalence
relation. For instance, transitivity is seen as follows. Assume that D wins on
$(s,t)$ and on $(t,u)$, with winning (first) moves $Z,Z^{\prime}$,
respectively. Then $Z\cup Z^{\prime}$ is a winning move for D on $(s,u)$,
where we exploit that by the assumption that $\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}$
preserves monos, we do not need to worry about $Z\cup Z^{\prime}$ possibly
using more variables than needed in the top-level decompositions of $s$ and
$u$ (cf. 6.8). Symmetry and reflexivity are easier. We write $[t]^{D}=q(t)$.
The $M_{0}$-algebra structure of $U$ is then given by

$f^{U}(([t_{i}]^{D})_{i\in I})=[f((t_{i})_{i\in I})]^{D}$

for a depth-0 operation $f$ of arity $I$. Well-definedness is seen similarly
as transitivity above.

For well-definedness of $\zeta$, suppose that $t_{1}\sigma,s_{1}\sigma$ are
top-level decompositions of infinite-depth terms such that D wins on
$(t_{1}\sigma,s_{1}\sigma)$, again w.l.o.g. with
$t_{1},s_{1}\in\operatorname{\mathscr{T}}_{\\!\Sigma,1}(X)$ for the same $X$,
and with the same $\sigma$. Let
$Z\subseteq\operatorname{\mathscr{T}}_{\\!\Sigma,1}(X)\times\operatorname{\mathscr{T}}_{\\!\Sigma,1}(X)$
be D’s winning move. Then $Z$, being an admissible move of D, entails
$t_{1}=s_{1}$. Moreover, since S can pick any element $(u,v)\in Z$ as a
response, D wins on each of the subsequent positions $(u\sigma,v\sigma)$, so
$u$ and $v$ are identified under $q\cdot\sigma$. It follows that
$M_{1}(q\cdot\sigma)([t_{1}])=M_{1}(q\cdot\sigma)([s_{1}])$. This shows well-
definedness; it is then clear by construction that $\zeta$ is an
$M_{0}$-homomorphism.

Finally, let $\gamma\colon A\to\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}A$
be an $\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}$-coalgebra
on an $M_{0}$-coalgebra $A$; we abuse $A$ to denote also the carrier of $A$.
For each $x\in A$, the successor structure $\gamma(x)\in\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}A$
is represented as a term
$g_{x}\in\operatorname{\mathscr{T}}_{\\!\Sigma,1}(A)$. Infinite unrolling of
$A$ thus produces an infinite-depth term $h_{x}$ for each $x\in A$. We claim
that the map $h\colon A\to U$ given by $h(x)=[h_{x}]^{D}$ is the unique
$\mathrlap{\hskip
2.0pt\overline{\scalebox{0.8}[1.0]{\hphantom{$M$}\vphantom{\rule{0.0pt}{5.5pt}}}}}M_{1}$-coalgebra
morphism from $(A,\gamma)$ to $(U,\zeta)$.

First note that the map $h$ is independent of the choice of representing terms
$g_{x}$. This is seen by letting D play, in
$\mathcal{G}_{\infty}^{\mathsf{syn}}$, a strategy based on maintaining the
invariant that the present state is a pair of unrollings, under differently
chosen representing terms (where the choice of representing terms may change
during the unrolling process), of the same state in $A$. Like in the ‘only if’
direction of the proof of 7.5, D can maintain the invariant by playing it in
every move. To see that $h$ is an $M_{0}$-homomorphism, let $f$ be a depth-0
operation of arity $I$; then $\gamma(f(x_{i})_{i\in I})=f(\gamma(x_{i})_{i\in
I})$ has $f((g_{x_{i}})_{i\in I})$ as a representing term, so the unrolling
$h(f(x_{i})_{i\in I})$ arises by applying $f$ to the unrollings $h(x_{i})$.

Finally, being a coalgebra morphism $A\to U$ amounts precisely to the
unrolling property, so $h$ is a coalgebra morphism, and unique as such. ∎

### Proof of Proposition 8.5

###### Proof.

We first show that $Z$ is admissible at $(s,t)$, i.e.
$Z\vdash(\alpha_{X}\cdot\gamma)^{*}_{0}s=(\alpha_{X}\cdot\gamma)^{*}_{0}t$,
where we write $Z\vdash$ to indicate equational derivability from $Z$. This
follows from the construction of $\bar{Z}$ since Condition 1 ensures that

$Z\vdash s_{a}+t_{a}=\sum_{x\in s_{a}}x+t_{a}=t_{a}.$

Likewise Condition 2 ensures that $Z\vdash s_{a}+t_{a}=s_{a}$. Thus, $Z\vdash
s_{a}=t_{a}$; whence,
$Z\vdash(\alpha_{X}\cdot\gamma)^{*}_{0}s=(\alpha_{X}\cdot\gamma)^{*}_{0}t$.

Moreover, Spoiler thanks to Condition 1 (resp. Condition 2) can play the
positions either of the form $(x\lor t^{\prime},t^{\prime})$ or $(y\lor
s^{\prime},s^{\prime})$. In either cases, $M_{0}!$ will map the left and right
term to $1$ since $M_{0}1=1$. So, in hindsight, if
$(s,t)Z(s^{\prime},t^{\prime})$ is a winning match for Duplicator, then
$s^{\prime}\mathrel{Z}t^{\prime}$. Thus, repeating the above argument
$n$-times give the desired result. Lastly, $Z$ is also a winning strategy in
the infinite game because $(s^{\prime},t^{\prime})\in Z$ whenever
$(s,t)Z(s^{\prime},t^{\prime})$ is a match. ∎


# Mass-Metallicity Relation during the Epoch of Reionization in the CROC
Simulations

Isaac Noel  New Trier High School; Winnetka, IL 60093, USA  Hanjue Zhu (朱涵珏)
Department of Astronomy & Astrophysics; The University of Chicago; Chicago, IL
60637, USA  Nickolay Y. Gnedin  Particle Astrophysics Center; Fermi National
Accelerator Laboratory; Batavia, IL 60510, USA  Kavli Institute for
Cosmological Physics; The University of Chicago; Chicago, IL 60637, USA
Department of Astronomy & Astrophysics; The University of Chicago; Chicago, IL
60637, USA

###### Abstract

The low-redshift mass-metallicity relation (MZR) is well studied, but the
high-redshift MZR remains difficult to observe. To study the early MZR
further, we analyze the Cosmic Reionization on Computers (CROC) simulations
with a focus on the MZR from redshifts 5 to 10. We find that, across all
redshifts, CROC galaxies exhibit similar stellar-phase and gas-phase MZRs that
flatten with higher stellar mass. We attribute this flattening to the
inaccurate star formation and feedback modeling in CROC (star formation is
overly suppressed in massive CROC galaxies). In addition, we show that the
ratio between stellar metallicity and gas metallicity ($Z_{*}/Z_{gas}$)
decreases as stellar age increases, meaning that in CROC galaxies, gas
accretion rate is lower than metal production rate. With JWST we will be able
to compare our predictions to observations of the Epoch of Reionization and
understand better early galaxy formation.

††journal: ApJ

##  1 Introduction

As star formation continues in galaxies, stellar death through supernovae
sends newly formed metals into the surrounding gas. The metals in the gas in
turn enhance the metallicity of the next generation of stars. The metallicity
of stars in a galaxy gives us information on when the galaxy acquired its
star-forming gas, where stars are formed, and the impact of inflows and
outflows on the mass and metals of the galaxy.

A well-studied and well-observed property of galaxies at low redshift is the
galaxy mass-metallicity relation (MZR). As discovered in both observations and
numerical simulations, there is a tight correlation between galaxy mass and
stellar metallicity as well as galaxy mass and gas metallicity. While we see
an abundance of evidence for the MZR at low redshifts (Maiolino & Mannucci
(2019) summarizes recent observational efforts, including Harikane et al.
(2020); Faisst et al. (2016); Cullen et al. (2019)), it is uncertain whether
this scaling relation exists at high redshifts, because strong emission lines
used to measure galaxy metallicity at low redshifts, such as [OIII], are
redshifted out of the current observational wavelength limit. However, with
the successful commisioning of JWST (Gardner et al., 2006; Rigby et al.,
2022), we will soon be able to study the high redshift MZR in detail.

To better interpret the forthcoming JWST data, we need theoretical
understanding of the high-z galaxy properties and their relations. This is why
we turn to numerical simulations to study the high redshift MZR. MZR at both
low-z and high-z have been analyzed by multiple groups using simulations
employing different numerical methods (e.g. Ma et al., 2016; Torrey et al.,
2019; Langan et al., 2020; Henriques et al., 2013, 2015; Lu et al., 2015).
Many of these works rely on “zoom-in” simulations, which treat galaxy feedback
and baryon cycle models with great detail and produce realistic galaxies.
However, they are limited to a small sample size of galaxies which may fail to
demonstrate large-scale emerging patterns.

In this paper, we use the Cosmic Reionization on Computers (CROC) simulations
(Gnedin, 2014), which we will explore further in the methodology section. The
CROC simulations give a large sample of galaxies at $z>5$, and model well the
relevant reionization physics. Therefore, CROC is suitable for studying the
MZR during the Epoch of Reionization.

This paper has defined goals. We focus on the high redshift gas-phase and
stellar-phase mass-metallicity relation and then make comparisons with current
observations of nearby dwarf galaxies. We organize the paper as follows.
Section 2 describes the CROC simulations, and how we measure the gas-phase and
stellar-phase metallicities. In Section 3, we show both the gas-phase and
stellar-phase MZR of the CROC galaxies. In addition, we study if the history
of cosmic reionization affects the MZR.

##  2 Methodology

###  2.1 CROC Simulations

We utilize data from the CROC simulations which use Adaptive Refinement Tree
(ART) code (Kravtsov et al., 1997). The CROC simulations are large-scale
cosmological simulations that model the intergalactic medium (IGM) and resolve
galaxies at high redshifts with a focus on cosmic reionization. They account
for a plethora of physical processes relevant to cosmic reionization,
including gas dynamics and star formation. More details are described in the
first CROC methodology paper (Gnedin, 2014).

From CROC, we use six simulation boxes with volumes of $40\,h^{-1}{\rm Mpc}$
comoving (cMpc). Each box includes simulation snapshots from redshifts 5 to
10. In the results we show, we include galaxies in all of the six boxes. We
also analyze a seventh box that is strongly overdense so that we can analyze
the impacts of reionization histories on the MZR. At $z=5$, each box includes
$1.4*10^{5}$ halos above the mass of $10^{5}{\rm\,M}_{\odot}$. We use the
Rockstar (Behroozi et al., 2013) halo finder to identify dark matter halos in
CROC, and use the “yt” package to analyze these dark matter halos (Turk et
al., 2011). To avoid errors that arise when the size of our halos approach the
simulation resolution, we ignore halos with $M_{*}<10^{5}{\rm\,M}_{\odot}$.

###  2.2 Measuring Stellar and Gas-Phase Metallicities

We analyze the gas cells within a certain distance away from each dark matter
halo center to determine the gas-phase metallicities ($Z_{\textrm{gas}}$).
Gas-phase metallicity is mass-weighted. Since gas metallicity is related to
star formation activity, we use only star particles within $R_{90}$, the
radius within which contains 90% of the stars in each galaxy. We also test our
choice of radius by using only star particles within $R_{50}$ and see a
negligible difference in the results.

Stellar metallicities ($Z_{*}$) are much simpler to obtain - we use all
stellar particles within $R_{90}$ and measure $Z_{*}$ of a galaxy as a mass-
weighted average of the stellar particles’ metallicity.

Note that the metallicities we present in this paper are all in absolute
units, not solar units.

We define the stellar mass of a halo, $M_{*}$, as the sum of stellar mass of
all the star particles in the halo within the $R_{90}$ of the halo.

##  3 Results

In this section, we explore the gas-phase and stellar-phase metallicities of
halos in the CROC simulation at high redshifts (5 $<z<$ 10) and examine how
they are impacted by their intergalactic environment. We then show how
$Z_{\textrm{gas}}$ and $Z_{*}$ evolve with redshift and DM halo age before
comparing our results to observations.

###  3.1 The Gas-Phase MZR

Figure 1: The evolution of the gas-phase mass-metallicity relation at
different redshifts. We distinguish between redshifts by displaying lines of
different color. The shaded regions represent the 25th to 75th percentile of
the distribution. We find a clear MZR at redshifts 5 to 10, and that higher
redshift galaxies have lower metallicities across our galaxy mass range.

In Figure 1, we see that the halos display a clear MZR which flattens with
higher mass. This change in slope contrasts findings from the FirstLight
Simulations (Langan et al., 2020) that predict a flat MZR. In addition, we see
that the MZR has a redshift dependence, with higher redshifts having lower
metallicities. This trend agrees with predictions from IllustrisTNG and FIRE
simulations (Torrey et al., 2019; Ma et al., 2016), and contrasts findings in
Langan et al. (2020).

Galactic chemical evolution is tightly connected to the interplay between star
formation, gas outflows, and gas recycling and accretion. For example, the
stellar mass-halo mass relation indicates that star formation efficiency
increases with galaxy mass, suggesting that more massive galaxies should more
metal-enriched. Meanwhile, galactic winds tend to blow away more gas and
metals from lower mass galaxies because of their shallow potential wells. Our
finding that gas metallicity increases with halo stellar mass is consistent
with these models. The flattening of the MZR at high mass could be attributed
to halo mergers; however, we note that in CROC, star formation is suppressed
in massive galaxies, likely because the CROC model for star formation and
feedback mis-models star formation in high-mass halos (Zhu et al., 2020).

The MZR serves as a test for star formation and feedback models implemented in
the simulations. Comparison between current simulation predictions and future
high-z observational data will inform us on the usefulness of our current
theoretical models.

###  3.2 The Stellar-Phase MZR

Figure 2: The evolution of the stellar-phase mass-metallicity relation at
different redshifts. We distinguish between redshifts by displaying lines of
different color. The shaded regions represent the 25th to 75th percentile of
the distribution. We observe a similar trend and redshift dependence as the
gas-phase MZR.

We see a relation that is similar to the gas-phase MZR in Figure 2. The main
difference is that the gas-phase MZR is higher than the stellar MZR at all
redshifts. This difference is expected, as stellar metallicities reflect the
time-averaged galactic metallicity across the galaxy star formation history,
and gas metallicities give information about the current state of metal
enrichment in galaxies. Combining this with the fact that star formation
enriches the gas with metals as time progresses, we expect the stellar
metallicity to be lower than the gas metallicity.

###  3.3 Comparing $Z_{*}$ and $Z_{\textrm{gas}}$ as the DM halos age

Figure 3: $Z_{*}/Z_{gas}$ as a function of normalized stellar age (age of halo
/ age of universe at redshift). The shaded region represent the 25th to 75th
percentile of the distribution. The spread is similar across redshifts so we
only show it at $z=5$. In bins with small sample sizes, individual galaxies
are shown in points with colors corresponding to redshifts. Other than the
sharp upward curve at high stellar ages (which can be attributed to outliers),
the plot displays a downward slope.

We now analyze the ratio between stellar metallicity and gas metallicity as
halos age. This ratio gives information about the relative rate at which
galaxies produce metals and replenish gas. We normalize the age of halos by
the age of the universe at a given redshift. By normalizing the halo age, we
can remove the redshift dependence that is inherent in the age of the
universe. The age of the universe is calculated assuming a flat universe with
$H_{0}$ = 68.14 km/s/Mpc and $\Omega_{m}=0.3036$. As shown in Figure 3, the
ratio between $Z_{*}$ and $Z_{\textrm{gas}}$ decreases with the normalized
halo age.

We see in the plot that $Z_{*}/Z_{gas}$ is independent of redshift. This means
that the relative metal production and gas accretion rate is similar at
$5<z<10$. The negative slope in the plot shows us that $Z_{gas}$ rises faster
than $Z_{*}$. If metals entered the gas at a constant rate, the
$Z_{*}/Z_{gas}$ ratio would be a constant because the gas metallicity and
stellar metallicity would rise at the same rate. The negative slope tells us
that as halos age, the metal production rate is higher than the rate at which
the stars accrete pristine matter.

###  3.4 Environmental effects on the MZR

Figure 4: Stellar metallicity as a function of stellar mass. The dotted lines
show the MZR in realistically dense simulations while the dotted lines show
the MZR in overdense simulations.

CROC simulations have high-fidelity radiative transfer modules implemented,
making them suitable for exploring the effect of the global radiative feedback
on the mass-metallicity relation.

To study how the intergalactic environment affects the MZR, we study DM halos
from an overdense simulation. As shown in Figure 4 in dashed lines, the MZR in
the overdense boxes is not very much different from the MZR observed in the
realistic boxes. We see the same redshift dependence as well as the same
flattening. From our results, we can conclude that the MZR is not impacted by
intergalactic interactions and that it is likely driven by physical processes
on a galactic scale, which agrees with the conclusions we have made so far.

###  3.5 Comparison to Observation

Figure 5: Left: The gas-phase MZR from our simulations compared to the MZR
observed in nearby dwarf galaxies. Right: The stellar-phase MZR from both our
simulations and observations. The observational data points are from Lee et
al. (2006); Kirby et al. (2013); McQuinn et al. (2015); Berg et al. (2016);
Simon (2019); McQuinn et al. (2020).

In this section, we compare our predictions with the MZR measured in nearby
dwarf galaxies. Dwarf galaxies are considered to be the driving force of
cosmic reionization and they are also most sensitive to the effects of
reionization as the reionization process kills star formation in them. In
Figure 5, we plot our predictions along with observed gas and stellar
metallicites from Lee et al. (2006); Kirby et al. (2013); McQuinn et al.
(2015); Berg et al. (2016); Simon (2019); McQuinn et al. (2020).

The left panel of Figure 5 shows the gas metallicity as function of stellar
mass. Our predictions are reasonable when considering the observed MZR at low
redshifts. The redshift dependence exhibited by the high-redshift galaxies
continues into the present as the low-redshift MZR lies above the high-
redshift MZR with reasonable distance. We also see the low-redshift gas
metallicities take a similar shape to our predictions, exhibiting flattening
at higher masses while maintaining a steeper rise around $M\approx
10^{6}M_{\odot}$. However, when comparing to observation it’s important that
we acknowledge the systematic uncertainties present in both observations and
simulations.

In the right panel of Figure 5, we plot the stellar metallicity with stellar
mass. Similarly to the gas-phase MZR, here is a fair agreement between
observations and the simulation. However, we note that observationally, there
is a much larger scatter in the dwarf galaxy stellar metallicities. A recent
paper from Naidu et al. (2022) has further made the distinction and showed
that the tidally shredded “disrupted dwarfs” have lower metallicities by
roughly 0.3 to 0.4 dex than the largely intact “surviving dwarfs”.

##  4 Summary and Discussion

We have analyzed data from the CROC simulations to predict the mass-
metallicity relation at high redshifts ($5<z<10$).

Our main results are:

  1. 1.

The MZR exists at high redshifts (5-10).

  2. 2.

The MZR is not affected by the cosmic reionization histories.

  3. 3.

As galaxies age, they accrete metal-poor gas slower than they replenish the
metals through star formation.

Now that JWST has successfully established itself at L2, we can expect high-
quality spectroscopic data that allows high-z metallicity measurements. These
observations will open the door to theories surrounding galaxy formation and
evolution at cosmic dawn.

From new empirical data, we can learn about how we predict the MZR. In the
literature, there are many predictions for the high-redshift MZR with all of
them being from modern simulations. Each simulation uses different methods and
prioritizes different physical processes. Confirming or denying any
simulation’s prediction will not only give us insight into the state of the
universe at cosmic dawn, but it would also give us information that would lead
to changes in our simulation methods. Perhaps we will see another MZR
entirely, indicating that there is an aspect of high-redshift galaxies that we
have yet to consider. There is much to discover about the universe, especially
with the trove of JWST’s new data.

Fermilab is operated by Fermi Research Alliance, LLC, under Contract No. DE-
AC02-07CH11359 with the United States Department of Energy. This work used
resources of the Argonne Leadership Computing Facility, which is a DOE Office
of Science User Facility supported under Contract DE-AC02-06CH11357. An award
of computer time was provided by the Innovative and Novel Computational Impact
on Theory and Experiment (INCITE) program. This research is also part of the
Blue Waters sustained-petascale computing project, which is supported by the
National Science Foundation (awards OCI-0725070 and ACI-1238993) and the state
of Illinois. Blue Waters is a joint effort of the University of Illinois at
Urbana-Champaign and its National Center for Supercomputing Applications.

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# Open heavy flavor production via semi-leptonic decayed muons in Pb+Pb
collisions at $\mbox{$\sqrt{s_{NN}}$}=2.76$ TeV with the ATLAS detector at the
LHC

ATLAS Collaboration

###### Abstract

Measurements of heavy quark production and suppression in ultra-relativistic
nuclear collisions probe the interactions of heavy quarks with the hot, dense
medium created in the collisions. ATLAS has measured heavy quark production in
$\mbox{$\sqrt{s}$}=2.76$ TeV Pb+Pb collisions via semi-leptonic decays of open
heavy flavour hadrons to muons. Results obtained from an integrated luminosity
of approximately $7~{}\mu\mathrm{b}^{-1}$ collected in 2010 are presented for
the per-event muon yield as a function of muon transverse momentum,
$p_{\mathrm{T}}$, over the range of $4<p_{\mathrm{T}}<14$ GeV. Over that
momentum range single muon production results largely from heavy quark decays.
The centrality dependence of the muon yields is characterized by the “central
to peripheral” ratio, $R_{\rm CP}$. Using this measure, muon production from
heavy quark decays is found to be suppressed by a centrality-dependent factor
that increases smoothly from peripheral to central collisions. Muon production
is suppressed by approximately a factor of two in central collisions relative
to peripheral collisions. Within the experimental errors, the observed
suppression is independent of muon $p_{\mathrm{T}}$ for all centralities.

###### keywords:

Heavy Ion , Heavy flavor suppression , $R_{\rm CP}$

##  1 Introduction

Collisions between lead ions at the LHC are thought to produce strongly
interacting matter at temperatures well above the QCD critical temperature. At
such temperatures, strongly interacting matter is expected to take the form of
“quark-gluon plasma.” High-$p_{\mathrm{T}}$ quarks and gluons generated in
hard-scattering processes during the initial stages of the nuclear collisions
are thought to lose energy in the quark-gluon plasma resulting in “jet
quenching” [Majumder:2010qh] . Since the energy loss results from the
interaction of a quark or gluon with the medium, jet quenching is thought to
provide a valuable tool for probing the properties of the quark-gluon plasma
[Wang:1994fx, Baier:1998yf, Gyulassy:2000fs]. Measurements of heavy quarks are
an important complement to studies of light quark and gluon quenching. The
contributions from radiative [Djordjevic:2003zk] and collisional
[Wicks:2005gt, Djordjevic:2006tw] energy loss in weakly coupled calculations
are expected to depend on the heavy quark mass. In particular, the mass of
heavy quarks is expected to reduce radiative energy loss for quark transverse
momenta less than or comparable to the quark mass ($m$),
$p_{\mathrm{T}}\lesssim m$, through the dead-cone effect [Dokshitzer:2001zm].
However, measurements of heavy quark production at RHIC via semi-leptonic
decays to electrons showed a combined charm and bottom suppression in Au+Au
collisions comparable to that observed for inclusive hadron production
[Adare:2006nq, Abelev:2006db, Adare:2010de]. There is disagreement in the
theoretical literature regarding the interpretation of the RHIC heavy quark
suppression measurements [Wicks:2005gt, Djordjevic:2011tm, Gossiaux:2008jv,
Uphoff:2011ad] particularly regarding the role of non-perturbative effects
[Adil:2006ra, vanHees:2007me, Horowitz:2008ig]. It is clear that measurements
of heavy quark quenching at the LHC are an essential complement to inclusive
jet [Aad:2010bu, Chatrchyan:1327643] or single hadron measurements
[CMS:2012aa, Aamodt:2010jd].

This paper presents ATLAS measurements of inclusive muon production in Pb+Pb
collisions at $\mbox{$\sqrt{s}$}=2.76$ TeV from an integrated luminosity of
approximately $7~{}{\rm\mu b^{-1}}$ collected in 2010\. The measurements were
performed for several intervals of collision centrality over the muon
transverse momentum range $4<p_{\mathrm{T}}<14$ GeV and the yields compared to
those in a peripheral bin using $R_{\rm CP}$:

$\mbox{$R_{\rm
CP}$}\equiv\frac{\mbox{$N_{\mathrm{coll}}$}^{\mathrm{periph}}}{\mbox{$N_{\mathrm{coll}}$}^{\mathrm{cent}}}\frac{\mathrm{d}n^{\mathrm{cent}}}{\mathrm{d}n^{\mathrm{periph}}},$
(1)

where $\mathrm{d}n^{\mathrm{cent}}$ and $\mathrm{d}n^{\mathrm{periph}}$
represent the per-event differential rate for the same hard-scattering
observable in central and peripheral collisions, respectively.
$\mbox{$N_{\mathrm{coll}}$}^{\mathrm{cent}}$ and
$\mbox{$N_{\mathrm{coll}}$}^{\mathrm{cent}}$ represent $N_{\mathrm{coll}}$
values calculated for the corresponding centralities (see Sec. 3). Over this
$p_{\mathrm{T}}$ range, muon production in _p_ +_p_ collisions results
predominantly from a combination of charm and bottom quark semi-leptonic
decays; all other sources including $J/\psi$ decays contribute less than 1% of
the prompt muon yield [Aad:2011rr]. The results presented here complement
previous measurements by the CMS Collaboration [Chatrchyan:2012np] of bottom
quark suppression performed using measurements of non-prompt $J/\psi$
production.

##  2 Experimental setup

Muons were detected by combining independent measurements of the muon
trajectories from the inner detector (ID) and the muon spectormeter (MS),
which covers $|\eta|<2.5$ and $|\eta|<2.7$ in pseudorapidity respectively.

Two forward calorimeters [Aad:2008zzm] placed symmetrically with respect to
$z=0$ and covering $3.2<|\eta|<4.9$ are used in this analysis to characterize
Pb+Pb collision centrality. They are composed of tungsten and copper absorbers
with liquid argon as the active medium; each calorimeter has a total thickness
of about 10 interaction lengths.

Minimum bias Pb+Pb collisions were identified using measurements from the zero
degree calorimeters (ZDCs) and the minimum-bias trigger scintillator (MBTS)
counters [Aad:2008zzm]. The ZDCs are located symmetrically at $z=\pm 140$ m
and cover $|\eta|>8.3$. In Pb+Pb collisions the ZDCs measure primarily
“spectator” neutrons, which originate from the incident nuclei and do not
interact hadronically. The MBTS detects charged particles over
$2.1<|\eta|<3.9$ using two counters placed at $z=\pm 3.6$ m. Each counter is
azimuthally divided into 16 sections, and the MBTS provides measurements of
both the pulse heights and arrival times of ionization energy deposits in each
section.

Events used in this analysis were selected for recording by the data
acquisition system using a logical OR of ZDC and MBTS coincidence triggers.
The MBTS coincidence required at least one hit in both sides of the detector,
and the ZDC coincidence trigger required the summed pulse height from each
calorimeter to be above a threshold set below the single neutron peak.

##  3 Event selection and centrality

In the offline analysis, Pb+Pb collisions were required to have a primary
vertex reconstructed from charged particle tracks with $p_{\mathrm{T}}>500$
MeV. The tracks were reconstructed from hits in the inner detector using the
standard track reconstruction algorithm [Cornelissen:2008zzc] with settings
optimized for the high hit density in heavy ion collisions [ATLAS:2011ah].
Additional requirements of at least one hit in each MBTS counter, a time
difference between the two MBTS detectors of less than 3 ns, and a ZDC
coincidence were imposed, yielding a total of 53 million minimum-bias Pb+Pb
events. The combination of the MBTS, ZDC, and the primary vertex requirements
eliminates efficiently both beam-gas interactions and photo-nuclear collisions
[Djuvsland:2010qs]. Previous studies [ATLAS:2011ah] indicate that this
combination of ZDC trigger and offline requirements select minimum-bias
hadronic Pb+Pb collisions with an efficiency of $98\pm 2\%$. The inefficiency
is concentrated in the very most peripheral collisions not used in this
analysis.

The centrality of Pb+Pb collisions was characterized by $\Sigma
E_{\mathrm{T}}^{\mathrm{FCal}}$, the total transverse energy measured in the
forward calorimeters (FCal). For the results presented in this paper, the
minimum-bias $\Sigma E_{\mathrm{T}}^{\mathrm{FCal}}$ distribution was divided
into centrality intervals according to the following percentiles of the
$\Sigma E_{\mathrm{T}}^{\mathrm{FCal}}$ distribution ordered from the most
central to the most peripheral collisions: 0-10%, 10-20%, 20-40%, 40-60%, and
60-80%. The centrality intervals were determined after accounting for very
peripheral events lost due to the minimum-bias trigger efficiency. A standard
Glauber Monte-Carlo analysis [Alver:2008aq, Miller:2007ri] was used to
estimate the average number of participating nucleons,
$\langle\mbox{$N_{\mathrm{part}}$}\rangle$, and the average number of nucleon-
nucleon collisions, $\langle\mbox{$N_{\mathrm{coll}}$}\rangle$, for Pb+Pb
collisions in each of the centrality bins. The results are shown in Table 1.
The $R_{\rm CP}$ measurements presented in the paper use the 60-80% centrality
bin as a common peripheral reference.

The $R_{\rm CP}$ calculation uses the ratio
$\mbox{$R_{\mathrm{coll}}$}\equiv\langle N_{\mathrm{coll}}\rangle/\langle
N_{\mathrm{coll}}^{\mathrm{60-80}}\rangle$, where $\langle
N_{\mathrm{coll}}^{\mathrm{60-80}}\rangle$ is the average number of collisions
in the 60-80% centrality bin. The $R_{\mathrm{coll}}$ uncertainties have been
directly calculated by evaluating the changes in $R_{\mathrm{coll}}$ under
variations of the minimum-bias trigger efficiency, variations of the
parameters of the Glauber calculation such as the nucleon-nucleon cross-
section, the nuclear radius, and nuclear thickness, and variations in the
modeling of the $\Sigma E_{\mathrm{T}}^{\mathrm{FCal}}$ distribution
[ATLAS:2011ah]. The $R_{\mathrm{coll}}$ values and uncertainties are also
reported in Table 1.

Table 1: Results of Glauber model evaluation of $\langle\mbox{$N_{\mathrm{part}}$}\rangle$, $\langle\mbox{$N_{\mathrm{coll}}$}\rangle$, and the $N_{\mathrm{coll}}$ ratio, $R_{\mathrm{coll}}$for each centrality. The quoted uncertainties represent the systematic uncertainties in the Glauber calculation.  Centrality | $\langle\mbox{$N_{\mathrm{part}}$}\rangle$ | $\langle\mbox{$N_{\mathrm{coll}}$}\rangle$ | $R_{\mathrm{coll}}$
---|---|---|---
0–10% | $356\pm 2$ | $1498\pm 133$ | $56.7\pm 6.2$
10–20% | $261\pm 4$ | $923\pm 83$ | $34.9\pm 3.8$
20–40% | $158\pm 4$ | $441\pm 37$ | $16.7\pm 1.5$
40–60% | $69.3\pm 3.5$ | $130\pm 10$ | $4.9\pm 0.2$
60–80% | $22.6\pm 2.3$ | $26.5\pm 2$ | $1$

##  4 Muon reconstruction

Muons used in this analysis were obtained by combining separate measurements
of the muon trajectory in the ID and the MS using the same algorithms that
were developed for _p_ +_p_ collisions [Aad:2011rr]. The ID tracks were
required to have a hit in the first pixel layer, at least 7 hits in the
microstrip detector, no missing hits in the microstrip detector in active
sensors, and a maximum of one missing hit in the pixel detector. The tracks
were also required to point to within 5 mm of the primary vertex in both the
transverse and longitudinal directions.

The reconstructed muon momenta were obtained from a combined fit to the muon
trajectories in the ID and MS. The results presented here use muons having
$4<p_{\mathrm{T}}<14$ GeV and $|\eta|<1.05$. The lower limit of the
$p_{\mathrm{T}}$ range was chosen to be near the plateau of the muon
efficiency curve. The upper $p_{\mathrm{T}}$ limit results both from the
limited statistics of the data and the desire to limit this measurement to a
muon $p_{\mathrm{T}}$ range where the contribution of $W$ decays to the muon
spectrum is less than 1% [Aad:2011rr, ATLAS-CONF-2011-078]. The muon $\eta$
interval was chosen to avoid transitional regions in calorimeter coverage
where the muon performance is degraded. The momentum resolution is 5% over the
measured $p_{\mathrm{T}}$ range. The number of reconstructed muons passing the
described cuts in each $p_{\mathrm{T}}$ and centrality bin are listed in Table
2.

Table 2: Approximate number of reconstructed muon candidates in the
pseudorapidity interval $|\eta|<1.05$ in each $p_{\mathrm{T}}$ and centrality
bin.

Centrality [%] $p_{\mathrm{T}}$ [GeV] 0–10 10–20 20–40 40–60 60–80 4–5 49k 36k
43k 17k 4.7k 5–6 19k 15k 18k 7.4k 2.0k 6–7 8.6k 6.5k 8.2k 3.4k 0.93k 7–8 4.1k
3.1k 3.9k 1.6k 0.46k 8–9 2.2k 1.7k 2.1k 0.78k 0.21k 9–10 1.2k 0.93k 1.1k 0.46k
0.13k 10–14 1.7k 1.31k 1.5k 0.62k 0.15k

The performance of the ATLAS detector and offline analysis in measuring muons
in Pb+Pb collisions was evaluated using a Monte Carlo (MC) data set [atlassim]
originally prepared for studies of jet performance. The sample also provides
enough events containing high-$p_{\mathrm{T}}$ muons from heavy flavour decays
and background sources for the purposes of this analysis. The MC data set was
obtained by overlaying GEANT4-simulated [Agostinelli:2002hh]
$\mbox{$\sqrt{s}$}=2.76$ TeV  _p_ +_p_ dijet events on to 1 million
GEANT4-simulated minimum-bias Pb+Pb events obtained from version 1.38b of the
HIJING event generator [Wang:1991hta]. HIJING was run with default parameters,
except for the disabling of jet quenching. To simulate the effects of elliptic
flow [ATLAS:2011ah] in Pb+Pb collisions, a parameterized $\cos{2\phi}$
modulation that varies with centrality, $\eta$ and $p_{\mathrm{T}}$ as
determined by previous ATLAS measurements [ATLAS:2011ah] was imposed on the
particles after generation [Masera:2009zz]. The _p_ +_p_ events were obtained
from the ATLAS MC09 tune [:2010ir] of the PYTHIA event generator
[Sjostrand:2006za]. One million PYTHIA hard scattering events were generated
for each of five intervals of $\hat{p}_{\mathrm{T}}$, the transverse momentum
of outgoing partons in the $2\rightarrow 2$ hard scattering, with boundaries
$17,35,70,140,280$ and 560 GeV. The _p_ +_p_ events for each
$\hat{p}_{\mathrm{T}}$ interval were overlaid on the same sample of HIJING
events.

Figure 1:  Efficiency ($\varepsilon$) for reconstructing heavy flavour semi-
leptonic decay muons as a function of muon $p_{\mathrm{T}}$ for three
collision centrality bins. The efficiency is shown over a wider
$p_{\mathrm{T}}$ range than that used in the analysis. The dotted line
indicates the plateau efficiency value, $\mbox{$\varepsilon$}=0.8$.

The efficiency for reconstructing muons associated with heavy flavour decays
was evaluated using the MC sample described above for different bins in
collision centrality. Figure 1 shows the fraction of muons from semi-leptonic
charm and bottom decays reconstructed as a function of muon $p_{\mathrm{T}}$,
$\mbox{$\varepsilon$}(p_{\mathrm{T}})$, for a subset of the centrality bins:
0-10%, 20-40%, and 60-80%. For $p_{\mathrm{T}}\gtrsim 5$ GeV, the
reconstruction efficiency is found to be constant within the statistical
uncertainties. The $\mbox{$\varepsilon$}(p_{\mathrm{T}})$ values for each
centrality bin were fit to a constant over the range $5<p_{\mathrm{T}}<14$ GeV
to estimate the “plateau” efficiency. The results for all centrality bins were
consistent with $\mbox{$\varepsilon$}_{\mathrm{plat}}=0.80\pm 0.02$. For
$4<p_{\mathrm{T}}<5$ GeV, a statistically significant variation of
$\varepsilon$ with centrality is observed and the efficiencies were taken
directly from the results shown in Fig. 1. Those vary between $0.67\pm 0.02$
in peripheral collisions (60-80%) to $0.64\pm 0.02$ in central collisions
(0-10%).

Figure 2:  Simulated $C$ distributions, $\mathrm{d}P/\mathrm{d}\mbox{$C$}$,
for signal and background muons for the full (0-80%) centrality range and for
the 0-10% and 60-80% centrality bins for muons in the momentum range of 5 – 6
GeV.

##  5 Muon signal extraction

The measured muons consist of both “signal” muons and background muons. The
signal muons are those that originate directly from the Pb+Pb collision, from
vector meson decays, and from heavy quark decays. The background muons arise
from pion and kaon decays, muons produced in hadronic showers in the
calorimeters, and mis-associations of MS and ID tracks. Previous studies have
shown that the signal and background contributions to the reconstructed muon
sample can be discriminated statistically [Aad:2011rr, ATLAS-CONF-2011-003].
For this analysis a weighted combination of two discriminant quantities has
been used [ATLAS-CONF-2011-003]. The fractional momentum imbalance,
$\mbox{$\Delta p_{\mathrm{loss}}$}/\mbox{$p_{\mathrm{ID}}$}$, quantifies the
difference between the ID and MS measurements of the muon momentum after
accounting for the energy loss of the muon in the calorimeters [ATLAS-
CONF-2010-075]. It is defined as:

$\dfrac{\mbox{$\Delta
p_{\mathrm{loss}}$}}{\mbox{$p_{\mathrm{ID}}$}}=\dfrac{\mbox{$p_{\mathrm{ID}}$}-\mbox{$p_{\mathrm{MS}}$}-\mbox{$\Delta
p_{\mathrm{calo}}$}(p,\eta,\phi)}{\mbox{$p_{\mathrm{ID}}$}},$ (2)

where $p_{\mathrm{ID}}$ and $p_{\mathrm{MS}}$ represent the reconstructed muon
momenta from the ID and MS, respectively and $\Delta p_{\mathrm{calo}}$
represents a momentum and angle-dependent average energy loss of muons in the
calorimeter obtained from MC simulations. The “scattering significance”, $S$,
characterizes deflections in the trajectory resulting from (e.g.) decays in
flight. For each measurement along the muon trajectory, $i$, the deviation in
azimuthal angle of the measured hit position from the fitted trajectory,
$\Delta\phi_{i}$, was calculated and compared to the expected deviation from
multiple scattering, $\phi^{\mathrm{msc}}$, to obtain the signed quantity
$\mbox{$s_{i}$}\equiv q\Delta\phi_{i}/\mbox{$\phi^{\mathrm{msc}}$}$, where $q$
represents the track charge. The TRT measurements were not used when
calculating $s_{i}$ due to the high occupancy of the TRT in Pb+Pb collisions.
Then, the scattering significance at the $k{\rm th}$ layer, $S(k)$, was
calculated according to:

$\displaystyle\mbox{$S(k)$}=\frac{1}{\sqrt{n}}\left(\sum_{i=1}^{k}s_{i}-\sum_{j=k+1}^{n}s_{j}\right).$
(3)

Here $n$ represents the total number of measurements of the muon trajectory;
typically $n=16$. $S(k)$ yields a statistically significant value when the
muon is deflected between measurements $k$ and $k+1$. An overall scattering
significance, $S$, was obtained from the maximum value of $S(k)$:

$\displaystyle S=\max{\left\\{|\mbox{$S(k)$}|,k=1,2,...\right\\}}.$ (4)

The effectiveness of $\mbox{$\Delta
p_{\mathrm{loss}}$}/\mbox{$p_{\mathrm{ID}}$}$ and $S$ in discriminating
between signal and background muons varies with momentum in a complementary
manner. $S$ is more effective at lower muon momenta where the deflections
resulting from (e.g.) in-flight decays are greater. $\mbox{$\Delta
p_{\mathrm{loss}}$}/\mbox{$p_{\mathrm{ID}}$}$ is more effective at larger muon
momenta where in-flight decays are less important and a larger fraction of the
muons are generated in hadronic showers. To take advantage of the
complementarity of the two discriminants, a “composite” discriminant was
formed [ATLAS-CONF-2011-003] according to:

$\mbox{$C$}=\left|\dfrac{\mbox{$\Delta
p_{\mathrm{loss}}$}}{\mbox{$p_{\mathrm{ID}}$}}\right|+rS$ (5)

where $r$ is a parameter that controls the relative contribution of the two
discriminants to $C$. MC studies indicate that $r=0.07$ provides optimal
separation between signal and background muons over the muon momentum range
studied here [ATLAS-CONF-2011-003].

Distributions for $C$ were obtained from the PYTHIA$+$HIJING Monte Carlo
sample for the different centrality and muon $p_{\mathrm{T}}$ bins used in
this analysis. The $C$ distributions were obtained separately for signal muons
from charm and bottom decays and for background muons generated by pions and
kaons. Figure 2 shows the resulting $C$ distributions for muons with
$5<p_{\mathrm{T}}<6$ GeV in the 0-10% and 60-80% centrality bins. As
demonstrated by the figure, the signal and background $C$ distributions were
found to vary only weakly with collision centrality. As a result, for the
template fitting procedure described below, the analysis was carried out using
centrality-integrated (0-80%) signal and background $C$ distributions which
are also shown in Fig. 2.

Figure 3: Examples of the template fits to measured $C$ distributions (see
text) in 0-10% (left) and 60-80% (right) centrality bins for two
$p_{\mathrm{T}}$ intervals: $5<p_{\mathrm{T}}<6$ GeV (top) and
$10<p_{\mathrm{T}}<14$ GeV (bottom). The curves show the contributions of
signal and background sources based on the corresponding $C$ distributions
obtained from MC including the shift, re-scaling, and smearing modifications
(see text).

The MC signal and background $C$ distributions were used to perform a template
fit to the $C$ distributions in the data. Specifically, a hypothetical
probability density distribution for $C$, $\mathrm{d}P/\mathrm{d}\mbox{$C$}$
is formed assuming that a fraction, ${f_{\mathrm{S}}}$, of the measured muons
is signal,

$\dfrac{\mathrm{d}P}{\mathrm{d}\mbox{$C$}}=\left.\mbox{${f_{\mathrm{S}}}$}\dfrac{\mathrm{d}P}{\mathrm{d}\mbox{$C$}}\right|_{\mathrm{S}}+(1-\mbox{${f_{\mathrm{S}}}$})\left.\dfrac{\mathrm{d}P}{\mathrm{d}\mbox{$C$}}\right|_{\mathrm{B}},$
(6)

where $\mathrm{d}P/\mathrm{d}\mbox{$C$}|_{\mathrm{S}}$ and
$\mathrm{d}P/\mathrm{d}\mbox{$C$}|_{\mathrm{B}}$ represent the Monte Carlo $C$
distributions from signal and background sources, respectively.

To account for possible differences in the $C$ distribution between data and
MC due to (e.g.) inaccuracies in the MC description of multiple scattering,
the fits allow for limited adaptation of the MC templates to the data. In
particular, the fit allows a shift and re-scaling of the $C$ distribution,
$\mbox{$C$}^{\prime}=a+\mbox{$\langle
C\rangle$}+b\left(\mbox{$C$}-\mbox{$\langle C\rangle$}\right)$, where $\langle
C\rangle$ is the usual mean of the $C$ distribution. In addition, a Gaussian
smearing of the MC $C$ distribution was included such that the data $C$
distribution was fit to

$\dfrac{\mathrm{d}P^{\prime}}{\mathrm{d}\mbox{$C$}^{\prime}}\equiv\left(\mbox{${f_{\mathrm{S}}}$}\left.\dfrac{\mathrm{d}P}{\mathrm{d}\mbox{$C$}^{\prime}}\right|_{\mathrm{S}}+(1-\mbox{${f_{\mathrm{S}}}$})\left.\dfrac{\mathrm{d}P}{\mathrm{d}\mbox{$C$}^{\prime}}\right|_{\mathrm{B}}\right)\otimes\dfrac{e^{-{\mathcal{C}}^{\prime
2}/2\sigma^{2}}}{\sqrt{2\pi}\sigma}$ (7)

A kernel estimation method [Cranmer:2000du] was used to produce the unbinned
probability density distribution, $dP^{\prime}/d\mbox{$C$}^{\prime}$ from the
MC signal and background samples. The signal fraction, ${f_{\mathrm{S}}}$, was
then evaluated for each centrality and muon $p_{\mathrm{T}}$ bin using
unbinned maximum likelihood fits with four free parameters,
${f_{\mathrm{S}}}$, $a$, $b$, and $\sigma$. The fits were performed using
MINUIT [James:1975dr] as implemented in RooFit[Verkerke:2003ir]. Examples of
the resulting template fits are shown in Fig. 3 for the 0-10% and 60-80%
centrality bins and for two muon $p_{\mathrm{T}}$ intervals. Typical fitted
values for the $a,b,$ and $\sigma$ parameters in Eq. 7 are, $a\sim 0.02$,
$b\sim 0.95-1.05$, and $\sigma\sim 0.002$. The combination of the MC signal
and background $C$ distributions are found to describe well the measured $C$
distributions. The description remains good even if the adaptation of the
template described in Eq. 7 is removed, though the fit results may change (see
below).

The signal muon fractions extracted from the template fits for all
$p_{\mathrm{T}}$ and centrality bins are shown in Fig. 4. The statistical
uncertainties on ${f_{\mathrm{S}}}$ from the fits represent $1\sigma$
confidence intervals that account for the limited statistics in the data $C$
distributions and correlations between the fit parameters. The uncertainties
in the fit results due to the finite MC statistics were evaluated using a
pseudo-experiment technique in which new pseudo-MC $C$ distributions with the
same number of counts as the original MC $C$ distributions were obtained by
statistically sampling the MC distributions. The resulting pseudo-MC
distributions were then used to perform template fits. The procedure was
repeated eight times for each $p_{\mathrm{T}}$ and centrality bin and the
standard deviation of the resulting ${f_{\mathrm{S}}}$ values in each bin was
combined in quadrature with the statistical uncertainty from the original fit
to produce a combined statistical uncertainty on ${f_{\mathrm{S}}}$. The
fractional uncertainties on ${f_{\mathrm{S}}}$ due to MC statistics are found
to be below $2\%$ and are typically much smaller than the uncertainties from
the template fits.

Figure 4:  Fractions of signal muons, ${f_{\mathrm{S}}}$, in the measured muon
yields as a function of muon $p_{\mathrm{T}}$ in different bins of collision
centrality. The points are shown at the mean transverse momentum of the muons
in the given $p_{\mathrm{T}}$ bin. The shaded boxes indicate systematic errors
(see text) due to possible data-MC template matches, hadron composition, the
template fitting (i.e. combination of $dP/dC$, $K/\pi$ and “Fit” entries in
Table 3), which can vary from point-to-point. The error bars show combined
statistical and systematic uncertainties.

The sensitivity of the measured ${f_{\mathrm{S}}}$ values to the adaptation of
the MC $C$ distributions to the data was evaluated by performing the fits
fixing the $a$ and $\sigma$ parameters to zero and $b$ to one. Those fits
yielded systematically lower ${f_{\mathrm{S}}}$ values than the values
obtained from the default fits. The reductions varied from as much as 18% for
the 60-80%, 4–5 GeV bin to about 3% for the highest $p_{\mathrm{T}}$ bins. The
differences between the two values for each centrality and $p_{\mathrm{T}}$
bin were used to estimate systematic uncertainties on the fit due to data-MC
matching of the $C$ distributions. Typically, those errors are largest for the
lowest $p_{\mathrm{T}}$ bin and decrease with increasing $p_{\mathrm{T}}$, but
the 6–7 GeV and 7–8 GeV bins also show enhanced sensitivity to the template
adaptation. That is particularly true for the 60-80% 7–8 GeV $p_{\mathrm{T}}$
bin which suffers from low statistics.

The use of the centrality-integrated MC $C$ distributions in the template
fitting provides another potential source of data-MC mismatch. To evaluate the
impact of this choice, the analysis was performed using centrality-selected MC
$C$ distributions. The reduction in template statistics worsened the
statistical accuracy of the fits, particularly for the most peripheral bin.
However, in all cases the differences from the default results were within the
uncertainty estimates described in the preceding paragraph. Based on that
observation and the judgment that the adaptation of the template distribution
can likely account for the small centrality differences observed in the $C$
distributions, no additional systematic error is assigned to account for the
use of the centrality-integrated templates.

Figure 5:  Invariant differential muon per-event yields calculated according
to Eq. 9 as a function of muon $p_{\mathrm{T}}$ for different bins in
collision centrality. The points are shown at the mean transverse momentum of
the muons in the given $p_{\mathrm{T}}$ bin. The statistical errors are
smaller than the sizes of the points in all bins and are not shown. The error
bars indicate combined statistical and systematic errors (see text).

The MC background $C$ distributions depend on the primary hadron composition
particularly the relative proportions of kaons and pions ($K/\pi$) which may
differ from that in the data. To test the sensitivity of the results to such a
difference, new MC background $C$ distributions were obtained by separately
doubling the $\pi$ and $K$ contributions. The new $\mbox{$C$}_{\mathrm{B}}$
distributions were used in the template fits and the resulting differences in
extracted ${f_{\mathrm{S}}}$ used to estimate the $K/\pi$ systematic
uncertainties on the ${f_{\mathrm{S}}}$ values. The resulting (fractional)
uncertainties are typically of order 1%, but they can be as large as 5% in the
lowest $p_{\mathrm{T}}$ bin.

Potential systematic uncertainties resulting from the template fitting
procedure were evaluated using a simple cut procedure applied to the $C$
distributions. For a given centrality and $p_{\mathrm{T}}$ bin, a cut was
applied at a chosen value of $C$, ${C_{\mathrm{cut}}}$, and the fraction of
muons in the data below the cut, $f^{<}$ was calculated. For an appropriate
choice of cut, that fraction represents most of the signal muons with a modest
contamination of background muons. The MC signal and background $C$
distributions were used to evaluate the fraction of the signal (background)
muons below the cut, $f^{<}_{\mathrm{S}}$ ($f^{<}_{\mathrm{B}}$). Then, the
fraction of signal muons evaluated using this cut method,
$f_{\mathrm{S}}(\mbox{${C_{\mathrm{cut}}}$})$, was estimated:

$f_{\mathrm{S}}(\mbox{${C_{\mathrm{cut}}}$})=\dfrac{f^{<}-f^{<}_{\mathrm{B}}}{f^{<}_{\mathrm{S}}-f^{<}_{\mathrm{B}}}.$
(8)

$f_{\mathrm{S}}^{\mathrm{cut}}$ values were obtained using
${C_{\mathrm{cut}}}$ values of 0.15, 0.2, and 0.25. The results from the three
${C_{\mathrm{cut}}}$ values agree typically to within a few %. The averages of
the three $f_{\mathrm{S}}^{\mathrm{cut}}$ values for each $p_{\mathrm{T}}$ and
centrality bin were compared to the results of the template fit where the
adaptation was disabled. These two different estimates of the signal fraction
agreed typically to better than 1%, but differences as large as 5% were
observed for the 60-80% centrality bin. These differences were used to
establish systematic uncertainties associated with the template fitting
procedure.

Table 3 summarizes the contributions to the systematic error in the
determination of the signal fraction, ${f_{\mathrm{S}}}$, for three sample
$p_{\mathrm{T}}$ bins in 0-10% and 60-80% centrality bins.

Table 3: Systematic uncertainty contributions from different sources for sample $p_{\mathrm{T}}$ bins in 0-10% and 60-80% centrality bins. “$dP/dC$” represents the systematic uncertainties due to potential data-MC mismatches in the template, “Fit” represents the estimated systematic on the fitting procedure evaluated using the cut method. “$K/\pi$” represents the uncertainty due to the hadron composition and “$\varepsilon$” represents the uncertainty on the efficiency estimation.  Centrality | $p_{\mathrm{T}}$ | Uncertainty [$\%$]
---|---|---
[%] | [GeV] | $dP/dC$ | Fit | $K/\pi$ | $\varepsilon$ | Total
0-10 | 4–5 | 4 | 0 | 5 | 3 | 7
7–8 | 5 | 0.5 | 0.5 | 2 | 5.5
10–14 | 4 | 1 | 1 | 2 | 5
60-80 | 4–5 | 18 | 1 | 5 | 3 | 19
7–8 | 14 | 5 | 0.5 | 2 | 15
10–14 | 4 | 4 | 2 | 2 | 6

##  6 Results

The invariant differential per-event muon yields for a given centrality bin
were obtained according to

$\left.\dfrac{1}{\mbox{$N_{\mathrm{evt}}$}}\dfrac{d^{2}N}{p_{\mathrm{T}}dp_{\mathrm{T}}d\eta}\right|_{\mathrm{cent}}=\dfrac{1}{N_{\mathrm{evt}}^{\mathrm{cent}}}\dfrac{N_{\mathrm{S}}^{\mathrm{cent}}(p_{\mathrm{T}})}{\varepsilon^{\mathrm{cent}}(p_{\mathrm{T}})\;p_{\mathrm{T}}\Delta
p_{\mathrm{T}}\Delta\eta}$ (9)

where $N_{\mathrm{evt}}$ is the number of sampled Pb+Pb collisions in the
centrality interval, $N_{\mathrm{S}}^{\mathrm{cent}}$ is the number of signal
muons in the centrality bin and $\varepsilon^{\mathrm{cent}}$ is the muon
reconstruction efficiency discussed in the previous section. Figure 5 shows
the resulting distributions for the five centrality bins included in this
analysis. The yields increase by more than an order of magnitude between
peripheral and central collisions as expected from the geometric enhancement
of hard scattering rates.

The muon $R_{\rm CP}$ is calculated from the ratio of the invariant
differential yields between a given centrality bin and the 60-80% bin
multiplied by $1/\mbox{$R_{\mathrm{coll}}$}$. Since the kinematic and phase
space factors cancel between the numerator and denominator, the $R_{\rm CP}$
calculation reduces to

$\left.\mbox{$R_{\rm
CP}$}(p_{\mathrm{T}})\right|_{\mathrm{cent}}=\dfrac{1}{\displaystyle
R_{\mathrm{coll}}^{\mathrm{cent}}}\left(\dfrac{\dfrac{1}{N_{\mathrm{evt}}^{\mathrm{cent}}}\dfrac{N_{\mathrm{S}}^{\mathrm{cent}}}{\varepsilon^{\mathrm{cent}}}}{\dfrac{1}{N_{\mathrm{evt}}^{60-80}}\dfrac{N_{\mathrm{S}}^{60-80}}{\varepsilon^{60-80}}}\right).$
(10)

The resulting $R_{\rm CP}$ values are shown in Fig. 6. The $R_{\rm CP}$ values
vary weakly with $p_{\mathrm{T}}$ and the points for each centrality interval
are consistent with a $p_{\mathrm{T}}$-independent $R_{\rm CP}$ within the
uncertainties on the points. The $R_{\rm CP}$ varies strongly with centrality,
increasing from about 0.4 in the 0-10% centrality bin to about 0.85 in the
40-60% bin.

Figure 6:  Muon $R_{\rm CP}$ as a function of $p_{\mathrm{T}}$ calculated
according to Eq. 10 for different centrality bins. The points are shown at the
mean transverse momentum of the muons in the given $p_{\mathrm{T}}$ bin. The
error bars include both statistical and systematic uncertainties. The
contribution of the systematic uncertainties from $R_{\mathrm{coll}}$ and
$\varepsilon$, which are fully correlated between $p_{\mathrm{T}}$ bins, are
indicated by the shaded boxes.

One disadvantage of $R_{\rm CP}$ is the poor statistics in peripheral bins,
which have small geometric enhancement of hard scattering processes compared
to central collisions. Statistical errors in the reference peripheral
$p_{\mathrm{T}}$ bin then influence the $R_{\rm CP}$ results for that bin for
all centralities. Such an effect can be observed in Fig. 6 for the 7–8 GeV
$p_{\mathrm{T}}$ bin. Alternatively, the centrality variation can be
characterized by comparing different centrality bins to the most central bin
which has the highest statistics producing a peripheral-to-central ratio,
$R_{\rm PC}$. The $R_{\rm PC}$ is defined analogously to Eq. 10 but with the
60-80% reference replaced by the 0-10% centrality bin. Figure 7 shows the
resulting $R_{\rm PC}$ values as a function of $p_{\mathrm{T}}$ for different
collision centralities. The figure provides similar insights as Fig. 6, but
the $R_{\rm PC}$ allows a stronger statement about the $p_{\mathrm{T}}$
dependence of the suppression. Namely, the shape of the muon spectra in the
0-10%, 10-20%, 20-40% and 40-60% centrality bins remains the same, within
uncertainties, while over those centrality bins the suppression is changing by
a factor of nearly 2.

The variation of $R_{\rm CP}$ with centrality as characterized by the average
number of participants, $\langle N_{\mathrm{part}}\rangle$, is shown in Fig. 8
for the different $p_{\mathrm{T}}$ bins included in the analysis. The $R_{\rm
CP}$ decreases smoothly from peripheral to central collisions. The centrality
dependence is observed to be the same for all $p_{\mathrm{T}}$ bins within the
experimental uncertainties.

Figure 7:  Muon peripheral to central ratio, $R_{\rm PC}$, as a function of
$p_{\mathrm{T}}$ calculated according to Eq. 10 for different centrality bins.
The points are shown at the mean transverse momentum of the muons in the given
$p_{\mathrm{T}}$ bin. The error bars include both statistical and systematic
uncertainties. The contribution of the systematic uncertainties from
$R_{\mathrm{coll}}$ and $\varepsilon$, which are fully correlated between
$p_{\mathrm{T}}$ bins, are indicated by the shaded boxes.

##  7 Discussion

CMS has previously reported measurements of $R_{\rm AA}$ for non-prompt
$J/\psi$ produced in $b$ decays in $\mbox{$\sqrt{s}$}=2.76$ TeV Pb+Pb
collisions [Chatrchyan:2012np]. The $R_{\rm AA}$ in the 0-20% centrality bin
($\mbox{$\langle N_{\mathrm{part}}\rangle$}=308$) is consistent with the 0-10%
($\mbox{$\langle N_{\mathrm{part}}\rangle$}=356$) muon $R_{\rm CP}$ presented
here in the $10<p_{\mathrm{T}}<14$ GeV bin, for which the muon yield is
expected to be dominated by $b$ decay contributions. However, the $R_{\rm AA}$
reported by CMS in the 20-100% bin ($\mbox{$\langle
N_{\mathrm{part}}\rangle$}=64$) falls well below the $R_{\rm CP}$ measurements
presented here. Since the CMS results are in terms of $R_{\rm AA}$ and this
analysis uses centrality 60-80% as the peripheral reference, we have repeated
the analysis using a 70-90% bin as reference and find no evidence that our
choice of peripheral reference can account for this difference.

Figure 8:  Muon $R_{\rm CP}$ as a function of $\langle
N_{\mathrm{part}}\rangle$ calculated according to Eq. 10 for different bins in
muon $p_{\mathrm{T}}$. The error bars show combined statistical and systematic
uncertainties. The sets of points for the different $p_{\mathrm{T}}$ bins are
successively displaced by $\Delta\mbox{$N_{\mathrm{part}}$}=6$ for clarity of
presentation.

The measurements presented in this paper of the suppression of muons from
heavy flavour decay differ appreciably from measurements of single hadron
suppression at the LHC at comparable transverse momenta [Aamodt:2010jd,
CMS:2012aa, ATLAS-CONF-2011-079]. The single hadron $R_{\rm CP}$ reported by
CMS [CMS:2012aa] in the 0-5% centrality bin relative to the 50-90% centrality
interval is a factor of approximately two smaller than the 0-10% muon $R_{\rm
CP}$ value reported here, while that in the 5-10% centrality bin is lower by
about 50%. The single hadron $R_{\rm CP}$ is also observed to vary much more
rapidly with $p_{\mathrm{T}}$ than the muon $R_{\rm CP}$ shown in Fig. 6.
Interpretation of this difference may be complicated by the indirect
relationship between the transverse momentum of the observed particle and the
parent parton for both the hadrons and the muons. Nonetheless, the clear
difference between semi-leptonic muon and charged hadron suppression at the
LHC should be contrasted with the situation at RHIC where the experimental
data do not indicate such a difference [Abelev:2006db].

PHENIX has reported a semi-leptonic electron $R_{\rm AA}$ for
$p_{\mathrm{T}}>4$ GeV in the 0-10% centrality bin of $0.30\pm
0.02(\mathrm{stat})\pm 0.04(\mathrm{syst})$ [Adare:2010de]. STAR has reported
similar results for $\mbox{$R_{\rm AA}$}(p_{\mathrm{T}})$ [Abelev:2006db]. The
RHIC semi-leptonic electron results show more suppression than the
measurements reported here, while the natural expectation would be that the
suppression would be greater in Pb+Pb collisions at the LHC due to the higher
initial energy densities [Horowitz:2011gd].

##  8 Conclusions

This paper has presented ATLAS measurements of muon production and suppression
in $\mbox{$\sqrt{s}$}=2.76$ TeV Pb+Pb collisions in a transverse momentum
range dominated by heavy flavor decays, $4<p_{\mathrm{T}}<14$ GeV, and over
the pseudo-rapidity range $|\eta|<1.05$. The fraction of prompt muons was
estimated using template fits to the distribution of a quantity capable of
distinguishing statistically between signal and background. The
$p_{\mathrm{T}}$ spectra of signal muons were evaluated in five centrality
bins: 0-10%, 10-20%, 20-40%, 40-60%, and 60-80%. The centrality dependence of
muon production was characterized using the central-to-peripheral ratio,
$R_{\rm CP}$, calculated using the 60-80% centrality bin as a peripheral
reference. The results for $R_{\rm CP}$ indicate a factor of about 2.5
suppression in the yield of muons in the most central (0-10%) collisions
compared to the most peripheral collisions included in the analysis (60-80%).
No significant variation of $R_{\rm CP}$ with muon $p_{\mathrm{T}}$ is
observed. The $R_{\rm CP}$ decreases smoothly from peripheral to central
collisions. The central muon $R_{\rm CP}$ for $10<p_{\mathrm{T}}<14$ GeV
agrees with the central (0-10%) non-prompt $J/\psi$ $R_{\rm AA}$ measured by
CMS. The central $R_{\rm CP}$ is 1.5-2 times larger than that measured for
charged hadrons in a comparable $p_{\mathrm{T}}$ range and using comparable,
though not identical, centrality ranges. The central $R_{\rm CP}$ indicates
weaker suppression than observed in the semi-leptonic electron $R_{\rm AA}$
measured over the same $p_{\mathrm{T}}$ range in $\mbox{$\sqrt{s}$}=200$ GeV
Au+Au collisions at RHIC.