# Galois correspondence for augmented monads

Johan Felipe García Vargas  Departamento de Matemáticas, Universidad de los
Andes, Bogotá, Colombia<EMAIL_ADDRESS>
###### Abstract.

We establish a Galois connection between sub-monads of an augmented monad and
sub-functors of the forgetful functor from its Eilenberg-Moore category. This
connection is given in terms of invariants and stabilizers defined through
universal properties. An explicit procedure for the computation of invariants
is given assuming the existence of suitable right adjoints. Additionally, in
the context of monoidal closed categories, a characterization of stabilizers
is made in terms of Tannakian reconstruction.

###### Key words and phrases:

Augmented monads; Grothendieck’s Galois Theory; Galois connection; Hopf
monads; Tannaka duality

###### 1991 Mathematics Subject Classification:

18C15, 18E50, 18M05, 18D15, 16T05

###### Contents

  1. 1 Introduction
  2. 2 Preliminars
  3. 3 Invariants and stabilizers for augmented monads
  4. 4 Computing invariants via right adjoints
  5. 5 Computing stabilizers in monoidal closed categories

##  1\. Introduction

The structural approach to Galois theory, as develop by Emil Artin and taught
in most algebra textbooks, place as fundamental theorem the correspondence,
given by invariants and stabilizers, between subgroups of symmetries and
intermediate structures.

In [4, Exposé V], Grothendieck reframed Galois theory as an equivalence
between a category of interest and the category of actions of a group. Since
the actions of a group $G$ form the Eilenberg-Moore category for the monad
$G\times\mathbf{\qm}$, Grothendieck’s formulation is a monadicity result. This
article proposes a path for recovering the Galois correspondence from a
monadicity result.

Our main hypothesis is that the monad possess an augmentation. Where, an
augmentation is a monad homomorphism to the identity of the base category.
This hypothesis is strong, because in the case of Hopf monads it implies that
the monad is $\otimes$-representable, as shown in [2, Section 5].

Morally, having an augmentation distinguish groups from groupoids. More
precisely, the monad induced by a groupoid has an augmentation only when every
morphism in the groupoid is an endomorphism. However, this makes sense,
because there is no clear meaning for invariants of a groupoid action when it
moves elements from a set to another.

Before presenting the details, let me exhibit the intuition behind them.
Imagine the monad $T$ as an algebraic gadget useful for “capturing the
symmetries” of some kind of structure defined over a category $\mathzc{C}$.
With this in mind, we regard the forgetful functor $U$ as the “universal
$T$-structure”. A Galois correspondence will emerge from comparing the action
of a part of the _symmetries_ , parametrized by a monad homomorphism
${h}:{S}\Rightarrow{T}$, over a part of the of the _structure_ , parametrized
by a natural transformation ${\alpha}:{V}\Rightarrow{U}$, against a trivial
action induced by an augmentation of the monad.

The next example illustrates the notions that we aim to generalize and might
be considered a trailer for the paper:

######  Example 1.1.

Let $G$ be a group. Consider the augmented monad $G\times\mathbf{\qm}$ in
$\operatorname{\mathzc{Sets}}$, where the augmentation is the projection to
the second component. The Eilenberg-Moore category
$\operatorname{\mathzc{Sets}}^{G}$ is the category of $G$-sets and equivariant
maps.

For any group homomorphism ${H}\xrightarrow{\phi}{G}$ the $H$-invariants
define a subfunctor of the forgetful
${U}:{\operatorname{\mathzc{Sets}}^{G}}\rightarrow{\mathzc{Sets}}$, mapping an
action $(X,{G\times X}\xrightarrow{r}{X})$ to the set inclusion

$\operatorname{Inv}H_{\phi}(X,r)=\\{x\in X:\;\phi(h)x=x\;\text{for all}\;h\in
H\\}\subseteq X.$

We will dissect this construction in the following way:

The action ${G\times X}\xrightarrow{r}{X}$ and the augmentation ${G\times
X}\xrightarrow{e\times X}{X}$ (the $e$ stands for erase, or for the constant
function to the group identity) parametrize for every $x\in X$ functions from
$G$ to $X$, mapping $g\mapsto(r(g,x)=gx)$ and $g\mapsto(e(g)x=x)$,
respectively.

Denoting by
${\left[G,X\right]}\xrightarrow{\left[\phi,X\right]}{\left[H,X\right]}$ the
function precomposing by ${H}\xrightarrow{\phi}{G}$, we can present
$\operatorname{Inv}H(X,r)$ as the equalizer of
${{X}}$${{\left[G,X\right]}}$${{\left[H,X\right]}.}$$\scriptstyle{\ulcorner
r\urcorner}$$\scriptstyle{\ulcorner e\otimes
X\urcorner}$$\scriptstyle{\left[\phi,X\right]}$ This presentation of
invariants is generalized in Lemma 4.1.

On the other hand, for every natural transformation
${\alpha}:{V}\Rightarrow{U}$ the stabilizer of $V_{\alpha}$ is the subgroup of
$G$ defined by

$\operatorname{Stab}V_{\alpha}=\\{g\in
G:\;g\alpha_{M}(v)=\alpha_{M}(v)\;\text{for
every}\;M\in\operatorname{Sets}^{G}\;\text{and every}\;v\in VM\\}.$

Inspired by Tannakian reconstruction, we can also dissect the construction of
stabilizers. Notice that each $g\in G$ gives a natural endomorphism of $U$
obtained by acting, explicitly ${\rho(g)}:{U}\Rightarrow{U}$ is defined by
$\rho(g)_{M}(x)=r_{M}(g,x)=gx$ for every $\operatorname{G}$-set $M=(X,r_{M})$
and every $x\in X$. The augmentation similarly defines a natural endomorphism
of $U$ for each $g\in G$, ${\epsilon(g)}:{U}\Rightarrow{U}$ defined by
$\epsilon(g)_{M}(x)=e(g)x=x$ for every $\operatorname{G}$-set $M=(X,r_{M})$
and every $x\in X$, which is clearly the identity of $U$.

Denoting by
${\operatorname{End}(U)}\xrightarrow{\operatorname{Nat}({\alpha},{U})}{\operatorname{Nat}(V,U)}$
the function precomposing by ${\alpha}:{V}\Rightarrow{U}$, we can present
$\operatorname{Stab}V_{\alpha}$ as the equalizer of
${{G}}$${{\operatorname{End}(U)}}$${{\operatorname{Nat}(V,U)}.}$$\scriptstyle{\rho}$$\scriptstyle{\epsilon}$$\scriptstyle{\operatorname{Nat}(\alpha,V)}$
This presentation of stabilizers is generalized in Lemma 5.10.

The well known Galois connection between invariants and stabilizers is
generalized in Corollary 3.7.

The structure of the document is straightforward. Section 2 contains three
preliminars one about lifting functors to Eilenberg-Moore categories, one
about monoidal closed categories and Hopf monads and one about Galois
connections. Section 3 introduces invariants and stabilizers as universal
representants for the fix relation. Section 4 is about computing invariants
and Section 5 about computing stabilizers in monoidal closed categories.

##  2\. Preliminars

######  Notation 2.1.

We write compositions from right to left. Given natural transformations
${\alpha}:{F}\Rightarrow{F^{\prime}}$ and
${\gamma}:{F^{\prime}}\Rightarrow{F^{\prime\prime}}$, where
${F,F^{\prime},F^{\prime\prime}}:{\mathzc{C}}\rightarrow{\mathzc{D}}$, we
denote their vertical composition by
${\gamma\cdot\alpha}:{F}\Rightarrow{F^{\prime\prime}}$. If additionally given
${\beta}:{G}\Rightarrow{G^{\prime}}$, with
${G,G^{\prime}}:{\mathzc{D}}\rightarrow{\mathzc{E}}$, we denote the horizontal
composition by ${\beta\star\alpha}:{GF}\Rightarrow{G^{\prime}F^{\prime}}$. We
will abbreviate identities of functors (and of objects) as
$\mathrm{id}_{G}=G$. We will also omit the $\star$ when whiskering. For
instance, $\beta\star\alpha=(G^{\prime}\alpha)\cdot(\beta F)=(\beta
F^{\prime})\cdot(G\alpha).$

###  2.1. Monadic lifting to Eilenberg-Moore categories

It is well known, see [5, Chapter VI] for instance, that every adjunction
yields a monad and that the Eilenberg-Moore construction allow to decompose a
monad into an adjunction in a “universal” way. Where by universal, we mean
that every other adjunction, yielding the same monad, is included in it. It is
less well known, that the lift of functors to Eilenberg-Moore categories can
be understood in terms of natural transformations of the base categories. We
will make a quick review of this and refer the reader to [1, Chapter 2] for
the details.

We denote monads by $(T,\mu,\eta)$ and adjunctions by $(\eta,L\dashv
R,\varepsilon)$ as displayed in the next figure:

${\mathzc{C}}$${\mathzc{C}}$$\scriptstyle{T}$$\scriptstyle{\mathrm{Id}_{\mathzc{C}}}$$\scriptstyle{TT}$$\scriptstyle{\eta}$$\scriptstyle{\mu}$
(a) Monad.

${\mathzc{C}}$${\mathzc{D}}$$\scriptstyle{L}$$\scriptstyle{R}$$\scriptstyle{RL}$$\scriptstyle{LR}$$\scriptstyle{\dashv}$$\scriptstyle{\eta}$$\scriptstyle{\varepsilon}$
(b) Adjunction.

Figure 1. Structure of monads and adjunctions.

######  Definition 2.2.

Let $(T,\mu,\eta)$ be a monad over a category $\mathzc{C}$. The Eilenberg-
Moore category of $T$, denoted $\mathzc{C}^{T}$, has $T$-actions as objects
and $T$-morphisms as morphisms.

  1. (a)

A $T$-action is a pair $M=(X,{TX}\xrightarrow{r_{M}}{X})$, where $X$ is in
$\mathzc{C}$ and $r$ satisfies:

associativity $\displaystyle r_{M}\mu_{X}$ $\displaystyle=rTr_{M},$ and unity,
$\displaystyle r_{M}\eta_{X}$ $\displaystyle=X.$

  2. (b)

A $T$-morphism ${M}\xrightarrow{f}{N}$, between $T$-actions $M=(X,r)$ and
$N=(Y,r^{\prime})$, is a morphism ${X}\xrightarrow{f}{Y}$ satisfying
$fr=r^{\prime}Tf$.

  3. (c)

The forgetful functor ${U^{T}}:{\mathzc{C}^{T}}\rightarrow{\mathzc{C}}$,
defined by

$U^{T}({(X,r)}\xrightarrow{f}{(Y,r^{\prime})})=({X}\xrightarrow{f}{Y}),$

has a left adjoint ${F^{T}}:{\mathzc{C}}\rightarrow{\mathzc{C}^{T}}$. This
functor is known as the free functor and it is defined by

$F^{T}X=(TX,{TTX}\xrightarrow{\mu_{X}}{TX})\quad\text{ and }\quad
F^{T}({X}\xrightarrow{f}{Y})={(TX,\mu_{X})}\xrightarrow{Tf}{(TY,\mu_{Y})}.$

Notice that $T=U^{T}F^{T}$, hence the unit of the adjunction
${\eta}:{\mathrm{Id}_{\mathzc{C}}}\Rightarrow{U^{T}F^{T}}$ is equal to the
unit of the monad ${\eta}:{\mathrm{Id}_{\mathzc{C}}}\Rightarrow{T}$. Moreover,
the evaluation
${\varepsilon}:{F^{T}U^{T}}\Rightarrow{\mathrm{Id}_{\mathzc{C}^{T}}}$ is given
by $\varepsilon_{M}={(TX,\mu_{X})}\xrightarrow{r}{(X,r)}$, for any $T$-action
$M=(X,{TX}\xrightarrow{r}{X})$.

  4. (d)

Suppose that there is another adjunction $(\eta,L\dashv R,\upsilon)$ with
${R}:{\mathzc{D}}\rightarrow{\mathzc{C}}$ such that $T=RL$ and $\mu=R\upsilon
L$. The comparison functor ${K}:{\mathzc{D}}\rightarrow{\mathzc{C}^{T}}$ is
defined by

$KA=(RA,{RLRA}\xrightarrow{R\upsilon_{A}}{RA}),$

for every object $A$ in $\mathzc{D}$. It is easy to verify that $KA$ is a
$T$-action and obviously $U^{T}K=R$.

  5. (e)

A functor ${R}:{\mathzc{D}}\rightarrow{\mathzc{C}}$ is called monadic if it
has a left adjoint , $L\dashv R$, and the comparison functor is an equivalence
from $\mathzc{D}$ to the Eilenberg-Moore category $\mathzc{C}^{T}$ of the
induced monad $T=RL$.

######  Definition 2.3.

Let $(T,\mu,\eta)$ and $(T^{\prime},\mu^{\prime},\eta^{\prime})$ be monads
over $\mathzc{C}$ and $\mathzc{C}^{\prime}$, respectively. Given a functor
${G}:{\mathzc{C}}\rightarrow{\mathzc{C}^{\prime}}$, we say that:

  * •

A lift of $G$ is a functor
${\overline{G}}:{\mathzc{C}^{T}}\rightarrow{\mathzc{C}^{\prime T^{\prime}}}$
such that $U^{T^{\prime}}\overline{G}=GU^{T}$.

  * •

A lifting data for $G$ is a natural transformation
${\gamma}:{T^{\prime}G}\Rightarrow{GT}$ such that

$\displaystyle\gamma\cdot(\mu^{\prime}G)$ $\displaystyle=(G\mu)\cdot(\gamma
T)\cdot(T^{\prime}\gamma),$ and $\displaystyle\gamma\cdot(\eta^{\prime}G)$
$\displaystyle=G\eta.$

  * •

When $\mathzc{C}=\mathzc{C}^{\prime}$, an homomorphism of monads
${h}:{T^{\prime}}\Rightarrow{T}$ is a lifting data for
$\mathrm{Id}_{\mathzc{C}}$. In that case, the above axioms simplify to
$h\cdot\mu^{\prime}=\mu\cdot(h\star h)$ and $h\cdot\eta^{\prime}=\eta$.

######  Theorem 2.4.

[1, Theorem and definition 2.27] Let $T$, $T^{\prime}$ and $G$ as in the
previous definition. There is a biyection between lifts of $G$ and lifting
data for $G$.

As a special case of the above theorem, taking the identity as base functor
(i.e $\mathzc{C}=\mathzc{C}^{\prime}$ and $G=\mathrm{Id}_{\mathzc{C}}$), we
obtain the following corollary:

######  Corollary 2.5.

Let $T$ and $T^{\prime}$ be monads over $\mathzc{C}$. There is a biyection
between monad homomorphisms ${h}:{T^{\prime}}\Rightarrow{T}$ and functors
${H}:{\mathzc{C}^{T}}\rightarrow{\mathzc{C}^{T^{\prime}}}$ such that
$U^{T^{\prime}}H=U^{T}$.

The next proposition gives a criterion for lifting natural transformations:

######  Proposition 2.6.

[1, Theorem and definition 2.30] Given a natural transformation
${\omega}:{G}\Rightarrow{H}$ between functors from $\mathzc{C}$ to
$\mathzc{C^{\prime}}$ with lifting data
${\alpha}:{T^{\prime}G}\Rightarrow{GT}$ and
${\beta}:{T^{\prime}H}\Rightarrow{HT}$ along the monads $T$ and $T^{\prime}$.
There is an unique ${\bar{\omega}}:{G^{\alpha}}\Rightarrow{H^{\beta}}$ which
lifts $\omega$ (i.e. $U^{T^{\prime}}\bar{\omega}=\omega U^{T}$), if and only
if, $(\omega T)\cdot\alpha=\beta\cdot(T^{\prime}\omega)$.

###  2.2. Monoidal closed categories and Hopf monads

A monoidal category is a category $\mathzc{C}$ endowed with a functor
${\otimes}:{\mathzc{C}\times\mathzc{C}}\rightarrow{\mathzc{C}}$ called
monoidal product, an object $\mathbbm{1}$ in $\mathzc{C}$ called unit object,
and natural isomorphisms called associativity and unit constrains which are
subject to well known axioms of coherence. In virtue of the strictness
theorems, — see [1, Lemma 3.7 and 3.8], [5, Section VII.2] or [3, Section 2.8]
— it is unnecessary to explicitly keep track of the associative or unit
constraints in any monoidal category, as long as you are only interested in
properties up to monoidal equivalence of categories.

One key feature of monoidal categories is that an object $X$, can be regarded
as an endofunctor $X\otimes\mathbf{\qm}$; this kind of functors are called
$\otimes$-representable. The same idea applies to morphisms and natural
transformations. For instance, $\otimes$-representable monads correspond to
monoids. Furthermore, the Eilenberg-Moore category of such a monad is the
category of actions of the monoid.

Bimonads are monads which lift the monoidal product to the Eilenberg-Moore
category. More precisely, a monad $T$ over
$(\operatorname{\mathzc{C}},\otimes.\mathbbm{1})$ is a bimonad if and only if
$C^{T}$ is also monoidal and the forgetful functor
${U^{T}}:{\mathzc{C}^{T}}\rightarrow{\mathzc{C}}$ is strong monoidal. By means
of Theorem 2.4, Bimonads are characterized as opmonoidal monads, see [1,
Theorem 3.19].

A monoidal category is left closed if every $\otimes$-representable functor
$\mathbf{\qm}\otimes X$ has a right adjoint, denoted
${\left[X,\mathbf{\qm}\right]}^{\ell}$. In this case, we call
${\left[X,Z\right]}^{\ell}$ the left internal Hom from $X$ to $Z$. Since the
adjunction $(\eta^{X},\mathbf{\qm}\otimes
X\dashv{\left[X,\mathbf{\qm}\right]}^{\ell},\varepsilon^{X})$, gives a
biyection between $\operatorname{Hom}\left(Y\otimes X,Z\right)$ and
$\operatorname{Hom}\left(Y,{\left[X,Z\right]}^{\ell}\right)$. Borrowing the
terminology from computer science, we define the currying of any ${Y\otimes
X}\xrightarrow{f}{Z}$ as $\ulcorner
f\urcorner={\left[X,f\right]}^{\ell}\eta^{X}_{Y}$, and the uncurrying of
${Y}\xrightarrow{g}{{\left[X,Z\right]}^{\ell}}$ as $\llcorner
g\lrcorner=\varepsilon^{X}_{Z}(g\otimes X)$. It is worth noticing that by the
theorem on adjunctions with a parameter — see [5, Theorem 3 in IV.7.] —
${\left[\mathbf{\qm}_{1},\mathbf{\qm}_{2}\right]}^{\ell}$ is functor from
$\mathzc{C}^{op}\otimes\mathzc{C}$ to $\mathzc{C}$, when $\mathzc{C}$ is left
monoidal closed.

When the functor ${\left[X,\mathbf{\qm}\right]}^{\ell}$ is
$\otimes$-representable, i.e.
${\left[X,\mathbf{\qm}\right]}^{\ell}=\mathbf{\qm}\otimes{X}^{\ell}$ we call
${X}^{\ell}$ the left dual of $X$. A monoidal category is left rigid (also
called compact or autonomous) if every object has a left dual.

Since a bimonad $T$ lifts the monoidal product, for every $T$-action
$M=(X,r_{M})$ the functor ${\mathbf{\qm}\otimes
M}:{\mathzc{C}^{T}}\rightarrow{\mathzc{C}^{T}}$ is a lift of
${\mathbf{\qm}\otimes X}:{\mathzc{C}}\rightarrow{\mathzc{C}}$. The bimonad $T$
is a left Hopf monad if whenever $\mathbf{\qm}\otimes X$ has a right adjoint,
the adjoint lifts to an adjoint of $\mathbf{\qm}\otimes M$. Consequently, if
$\mathzc{C}$ is left monoidal closed (resp. rigid) and $T$ is a left Hopf
monad, then $\mathzc{C}^{T}$ is also left closed (resp. rigid) and the
forgetful functor preserves internal Homs. Left Hopf monads are characterized
by the invertibility of the left fusion operator, see [2, Theorem 3.6] or [1,
Theorem 3.27].

Another key feature of monoidal categories is that, in addition to the usual
duality between a category and its opposite, there is a left-right duality
obtained by exchanging the order in the monoidal product. Implying that there
are right variants for each of the above defined concepts. When the laterality
is omitted it is tacitly understood that both variants holds. For instance, a
rigid category is both left and right rigid.

A gadget for comparing left and right $\otimes$-representable functors is
called a braiding. More precisely, a lax half-braiding on a monoidal category
$\mathzc{C}$ is a pair $(A,\sigma)$ where $A$ is an object and
${\sigma}:{A\otimes\mathbf{\qm}}\Rightarrow{\mathbf{\qm}\otimes A}$ is a
$\otimes$-multiplicative natural transformation. Where
$\otimes$-multiplicative means that $\sigma_{\mathbbm{1}}=\mathrm{id}_{A}$ and

$\sigma_{X\otimes Y}=(X\otimes\sigma_{Y})(\sigma_{X}\otimes Y),$

for arbitrary $X$ and $Y$ in $\mathzc{C}$. The adjective lax is dropped when
$\sigma$ is an isomorphism. A braiding is a $\otimes$-multiplicative choice of
a half braiding for every object in $\mathzc{C}$.

A lax central bimonoid is simultaneously a lax central comonoid and monoid
such that the multiplication and unit are comonoid homomorphisms. Explicitly,
for a lax central bimonoid $A=(A,\sigma,\delta,\mu,\varepsilon,\eta)$:
$(A,\sigma)$ is a lax half-braiding, $(A,\delta,\varepsilon)$ is a comonoid,
$(A,\mu,\eta)$ is a monoid, and they satisfy
$(\mu\otimes\mu)(A\otimes\sigma_{A}\otimes A)(\delta\otimes\delta)=\delta\mu$,
$\varepsilon\otimes\varepsilon=\varepsilon\mu$, $\eta\otimes\eta=\delta\eta$
and $\varepsilon\eta=\mathbbm{1}$. Every lax central bimonoid defines a
bimonad given by the functor $A\otimes\mathbf{\qm}$, see [2, Lemma 5.6].

A lax central bimonoid $H=(H,\sigma,\delta,\mu,\varepsilon,\eta)$ is called
lax Hopf monoid if the fusion morphism $(H\otimes\mu)(\delta\otimes H)$ is an
isomorphism. By the the relation between convolution and fusion morfism, a lax
Hopf monoid has an antipode ${H}\xrightarrow{s}{H}$ such that

$\mu(s\otimes H)\delta=\eta\varepsilon=\mu(H\otimes s)\delta.$

The adjective lax is dropped when both $\sigma$ and $s$ are isomorphisms.

Since this paper is about augmented monads is very significant for us that in
the Hopf case every augmented monad is representable.

######  Theorem 2.7.

[2, Corollary 5.18] There is an equivalence between the categories of
augmented left Hopf monads and lax central Hopf monoids, which restricts to an
equivalence between augmented Hopf monads and central Hopf monoids.

###  2.3. Galois connections induced by functorial relations

This is slight generalization of the Galois connection induced by a relation
relationship on sets. It is completely elementary and should be folklore, but
the author didn’t find any appropriate reference and decided to include brief
but self contained exposition of this.

######  Definition 2.8.

We call functorial relation between the categories $\mathzc{X}$ and
$\mathzc{Y}$ to a functor
${R}:{\mathzc{X}\times\mathzc{Y}}\rightarrow{\mathzc{2}}$, where
$\mathzc{2}=\\{\mathrm{true}\to\mathrm{false}\\}$ is thought as the truth
values of the relation. In other words, for objects $X$ and $Y$ in
$\mathzc{X}$ and $\mathzc{Y}$, respectively, we say that $X$ is related to $Y$
when $R(X,Y)=\mathrm{true}$.

Notice that asserting that $R$ is a functor simply means that for any pair of
morphisms, ${X^{\prime}}\xrightarrow{f}{X}$ and
${Y^{\prime}}\xrightarrow{g}{}Y$, if $X$ is related to $Y$ then $X^{\prime}$
is related $Y^{\prime}$.

######  Definition 2.9.

The $R$-representant for an object $X$ in $\mathzc{X}$ is an object, denoted
$R^{\ast}X$, such that for every $Y$ in $\mathzc{Y}$, $X$ is related to $Y$ if
and only if there is an unique morphism $Y\to R^{*}X$. The $R$-representant
for a $Y$ in $\mathzc{Y}$, denoted $R_{\ast}Y$ is analogously defined.

The next definition generalizes the case of a Galois connection between
partially ordered sets, to adjunctions with a pre-order as its core. Remember
that a pre-order is a category with at most one morphism among any pair of
objects.

######  Definition 2.10.

A Galois connection, between the categories $\mathzc{X}$ and $\mathzc{Y}$, is
a contravariant adjunction
${G}:{\mathzc{Y}}\rightarrow{\mathzc{X}^{op}}\dashv{F}:{\mathzc{X}^{op}}\rightarrow{\mathzc{Y}}$
such that
$\operatorname{Hom}_{\mathzc{Y}}\left(Y,FX\right)\approx\operatorname{Hom}_{\mathzc{X}}\left(X,GY\right)$
has at most one element, for arbitrary $X$ and $Y$.

######  Remark 2.11.

Let
${G}:{\mathzc{Y}}\rightarrow{\mathzc{X}^{op}}\dashv{F}:{\mathzc{X}^{op}}\rightarrow{\mathzc{Y}}$
be a Galois connection.

  1. i.

For any $X$ in $\mathzc{X}$, there is a unique morphism $X\to GFX$. We say
that $X$ is Galois closed if this is an isomorphism. The Galois closed objects
in $\mathzc{X}$ form a pre-order.

  2. ii.

Since evaluating $G$ at ${Y}\xrightarrow{}{FGY}$ gives a morphism
${GFGY}\xrightarrow{}{GY}$, for any $Y$ in $\mathzc{Y}$, $GY$ is Galois closed
in $\mathzc{X}$. Analogously, for any $X$ in $\mathzc{X}$, $FX$ is closed in
$\mathzc{Y}$.

  3. iii.

In conclusion, $F$ and $G$ induce an order-reversing equivalence between
Galois closed objects. This is known as the induced Galois correspondance.

Now we are able to define the Galois connection induced by $R$.

######  Proposition 2.12.

Let $R$ be a functorial relationship between $X$ and $Y$.

  1. i.

If every object in $\mathzc{X}$ has a representant, then mapping $X\mapsto
R^{\ast}X$ defines a functor
${R^{\ast}}:{\mathzc{X}^{op}}\rightarrow{\mathzc{Y}}$.

Analogously, when every object in $\mathzc{Y}$ has a representant there is
${R_{\ast}}:{\mathzc{Y}}\rightarrow{\mathzc{X}^{op}}$.

  2. ii.

If both $R^{\ast}$ and $R_{\ast}$ are defined, then they form a Galois
connection.

###### Proof.

Given a morphism ${X^{\prime}}\xrightarrow{f}{X}$ in $\mathzc{X}$, since $R$
is functorial and $X$ is related to $R^{\ast}X$, then $X^{\prime}$ is related
to $R^{\ast}X$; hence there is an unique $R^{\ast}X\to R^{\ast}X^{\prime}$.

The adjunction plainly recalls that the existence of maps $Y\to R^{\ast}X$ and
$X\to R_{\ast}Y$ are both equivalent to $X$ is related to $Y$. ∎

##  3\. Invariants and stabilizers for augmented monads

This section contains the main result of this paper. It states that in an
augmented monad it is possible to define of invariants and stabilizers, in a
way that naturally establish a Galois correspondence when they exist.

######  Definition 3.1.

Let $T=(T,\mu,\eta)$ be a monad over $\mathzc{C}$. An augmentation of $T$ is a
monad homomorphism ${e}:{T}\Rightarrow{\mathrm{Id}_{\mathzc{C}}}$. This is a
natural transformation which satisfies $e\cdot\eta=\mathrm{Id}_{\mathzc{C}}$
and $e\cdot\mu=e\star e$. A monad endowed with an augmentation is called
augmented monad.

The intuition for an augmentation, is that the morphism
${TX}\xrightarrow{e_{X}}{X}$ is the trivial action for any $X$ in
$\mathzc{C}$; the action which forgets $T$ and leaves $X$ fixed. During this
whole section we will work in an augmented monad $T=(T,\mu,\eta,e)$ over a
category $\mathzc{C}$ and denote by $(\eta,F\dashv U,\varepsilon)$ the
Eilenberg-Moore decomposition of $T$; where
${U}:{\mathzc{C}^{T}}\rightarrow{\mathzc{C}}$ is the forgetful functor from
the category of $T$-actions, and its left adjoint $F$ sends every object in
$C$ to the free action on it. See definition 2.2 for the details.

###  3.1. The fix relation

Let $\operatorname{\mathzc{Monad}}_{\mathzc{C}}$ be the category of monads
over $\mathzc{C}$. For brevity, we denote by $S_{h}$ a monad homomorphism
${h}:{S}\Rightarrow{T}$ considered as object of
$\operatorname{\mathzc{Monad}}_{\mathzc{C}}/T$ and by $V_{\alpha}$ a natural
transformation ${\alpha}:{V}\Rightarrow{U}$ from a functor
${V}:{\mathzc{C}^{T}}\rightarrow{\mathzc{C}}$ considered as object of
$\operatorname{\mathzc{Fun}}(\mathzc{C}^{T},\mathzc{C})/U$.

Since the augmentation provides us with a notion of trivial action, we can
compare any action with the trivial one over the same object. Moreover, we can
restrict this comparison to a part of the monad or to a part of the object.
The next definition does both restrictions simultaneously and then performs
the comparison.

######  Definition 3.2.

We say that $S_{h}$ fixes $V_{\alpha}$ when
$(U\varepsilon)\cdot(h\star\alpha)=(eU)\cdot(h\star\alpha).$

Let’s expand the above definition for a given $T$-action
$M=(X,{TX}\xrightarrow{r_{M}}{X})$:

Since $X=UM$ and the action morphism $r_{M}$ equals $U\varepsilon_{M}$, then
the left hand side is obtained by composing $r_{M}$ with
$(h\star\alpha)_{M}=(T\alpha_{M})h_{VM}=h_{X}(S\alpha_{M})$.

On the other hand, $(eU)_{M}=e_{X}$ is the trivial action; therefore, when
$S_{h}$ fixes $V_{\alpha}$ the restriction of the trivial action equals the
restricted action, this is

$r_{M}h_{X}(S\alpha_{M})=e_{X}h_{X}(S\alpha_{M}).$

######  Remark 3.3.

By Corollary 2.5, every monad homomorphism ${h}:{S}\Rightarrow{T}$ corresponds
to a functor ${H}:{\mathzc{C}^{T}}\rightarrow{\mathzc{C}^{S}}$ such that
$U^{S}H=U$, where ${U^{S}}:{\mathzc{C}^{S}}\rightarrow{\mathzc{C}}$ is the
forgetful functor from the category of $S$-actions. In particular, the
augmentation ${e}:{T}\Rightarrow{\mathrm{Id}_{\mathzc{C}}}$ corresponds a
functor ${E}:{\mathzc{C}}\rightarrow{\mathzc{C}^{T}}$ which is a section of
the forgetful functor, i.e. $UE=\mathrm{Id}_{\mathzc{C}}$.

Recalling the definitions, any $X$ in $\mathzc{C}$ gets mapped to
$EX=(X,{TX}\xrightarrow{e_{X}}{X})$, and any
$M=(X,{TX}\xrightarrow{r_{M}}{X})$ in $\mathzc{C}^{T}$ gets mapped to
$HM=(X,{SX}\xrightarrow{r_{M}h_{X}}{X})$.

The next lemma allow us to reinterpret the definition of acting trivially in
terms of the functors defined in the above remark.

######  Lemma 3.4.

The homomorphism ${h}:{S}\Rightarrow{T}$ fixes ${\alpha}:{V}\Rightarrow{U}$ if
and only if there is a natural transformation
${\hat{\alpha}}:{HEV}\Rightarrow{H}$ such that $U^{S}\hat{\alpha}=\alpha$.

###### Proof.

First notice that $H$ and $HEV$ are a lifts of $U$ and $V$, respectively,
along the monads $\mathrm{Id}_{\mathzc{C}^{T}}$ and $S$.

${{\mathzc{C}^{T}}}$${{\mathzc{C}^{S}}}$${{\mathzc{C}^{T}}}$${\mathzc{C}}$$\scriptstyle{U}$$\scriptstyle{U^{S}}$$\scriptstyle{V}$$\scriptstyle{H}$$\scriptstyle{\mathrm{Id}_{\mathzc{C}^{T}}}$$\scriptstyle{HEV}$$\scriptstyle{\alpha}$$\scriptstyle{\bar{\alpha}}$

The lifting data for $H$ over $U$ is $(U\varepsilon)\cdot(hU)$ and for $HEV$
over $V$ is $(e\cdot h)V$.

As an application of Proposition 2.6, there is a lift $\bar{\alpha}$ if and
only if $\alpha\cdot((e\cdot h)V)=(eU)\cdot(h\star\alpha)$ equals
$(U\varepsilon)\cdot(hU)\cdot(S\alpha)=(U\varepsilon)\cdot(h\star\alpha)$;
which is the definition of $S_{h}$ fixes $V_{\alpha}$. ∎

Now we will check that the fix relationship is preserved whenever we restrict
further. In other words that fixing defines a functorial relation between
$\operatorname{\mathzc{Monad}}_{\mathzc{C}}/T$ and
$\operatorname{\mathzc{Fun}}(\mathzc{C}^{T},\mathzc{C})/U$.

Afterwards, we will apply Definition 2.9 and Proposition 2.12, to this
relation, in order to define a Galois connection.

######  Proposition 3.5.

Let ${g}:{S^{\prime}}\Rightarrow{S}$ be any monad homomorphism and
${\beta}:{V^{\prime}}\Rightarrow{V}$ be any natural transformation. If
${h}:{S}\Rightarrow{T}$ fixes ${\alpha}:{V}\Rightarrow{U}$, then ${h\cdot
g}:{S^{\prime}}\Rightarrow{T}$ fixes
${\alpha\cdot\beta}:{V^{\prime}}\Rightarrow{U}$.

###### Proof.

This is obvious from the facts that $(h\cdot
g)\star(\alpha\cdot\beta)=(h\star\alpha)\cdot(g\star\beta)$ and the definition
of fixing $(U\varepsilon)\cdot(h\star\alpha)=(eU)\cdot(h\star\alpha)$. ∎

######  Definition 3.6.

Let ${h}:{S}\Rightarrow{T}$ be a monad homomorphism and
${\alpha}:{V}\Rightarrow{U}$ a natural transformation, where
${U}:{\mathzc{C}^{T}}\rightarrow{\mathzc{C}}$ is the forgetful functor.

  1. (a).

We say that $h$ is the stabilizer of $\alpha$ (or $S$ is the stabilizer of
$V$) if: it fixes $\alpha$ and for every monad homomorphism
${h^{\prime}}:{S^{\prime}}\Rightarrow{T}$ that also fixes $\alpha$ there is an
unique ${g}:{S^{\prime}}\Rightarrow{S}$ such that $h^{\prime}=h\cdot g$.

  2. (b).

We say that $\alpha$ is the $h$-invariant inclusion (or $V$ is the functor of
$S$-invariants) if: it is fixed by $h$ and for every functor
${\alpha^{\prime}}:{V^{\prime}}\rightarrow{U}$ also fixed by $h$ there is an
unique ${\beta}:{V^{\prime}}\rightarrow{V}$ such that
$\alpha^{\prime}=\alpha\cdot\beta$.

The next corollary is an straightforward application of Proposition 2.12, to
the fixing relationship.

######  Corollary 3.7.

Let $\mathzc{M}$ and $\mathzc{N}$ be subcategories of
$\operatorname{\mathzc{Monad}}(\mathzc{C})/T$ and
$\operatorname{\mathzc{Fun}}(\mathzc{C}^{T},\mathzc{C})/U$, respectively.

  1. i.

If every $h$ in $\mathzc{M}$ has an $h$-invariant inclusion in $\mathzc{N}$,
then the mapping $h\mapsto\operatorname{Inv}h$ defines a functor
${\operatorname{Inv}}:{\mathzc{M}^{op}}\rightarrow{\mathzc{N}}$. Analogously,
If every $\alpha$ in $\mathzc{N}$ has an stabilizer in $\mathzc{M}$, then the
mapping $\alpha\mapsto\operatorname{Stab}\alpha$ defines a functor
${\operatorname{Stab}}:{\mathzc{N}}\rightarrow{\mathzc{M}^{op}}$.

  2. ii.

If both $\operatorname{Stab}$ and $\operatorname{Inv}$ are defined, then they
form a Galois connection.

##  4\. Computing invariants via right adjoints

Now that we have an abstract definition of invariants and stabilizers, we face
the concrete — or slightly less abstract — questions of determining when they
exist and how can we compute it. This segment offers an answer for the case of
invariants, by linking them to right adjoints. Remember that $(T,\mu,\eta,e)$
is an augmented monad over $\mathzc{C}$.

First we observe that when the domain monad has a right adjoint we compute
invariants as an equalizer.

######  Lemma 4.1.

Let ${h}:{S}\Rightarrow{T}$ be a monad homomorphism and assume that the monad
${S}:{\mathzc{C}}\rightarrow{\mathzc{C}}$ has a right adjoint,
$(\theta,S\dashv R,\upsilon)$.

For any ${\alpha}:{V}\Rightarrow{U}$, $S_{h}$ fixes $V_{\alpha}$ if and only
if for each $M=(X,r_{M})$ in $\mathzc{C}^{T}$,
$(R(eh)_{X})(\theta_{X})\alpha_{M}=(R(r_{M}h_{X}))(\theta_{X})\alpha_{M}$.

In consequence, when the pair $(R(eh)_{X})(\theta_{X})$ and
$(R(r_{M}h_{X}))(\theta_{X})$ has an equalizer for every $M$, this equalizer
defines the functor of $S_{h}$-invariants.

###### Proof.

Just recall that $S_{h}$ fixes $V_{\alpha}$ when
$r_{M}h_{X}(S\alpha_{M})=e_{X}h_{X}(S\alpha_{M})$ for every $M$, and bend $S$
to $R$ in both sides of the equation. In other words,

$(R((U\varepsilon)\cdot(hU)))\cdot(\theta U)\cdot\alpha=(R(e\cdot
h)U)\cdot(\theta U)\cdot\alpha.$

Since equalizers are funtorial, the last sentence is evident. ∎

The next example illustrate the above lemma in the case of a
$\otimes$-representable monad.

######  Example 4.2.

Let $T=A\otimes\mathbf{\qm}$ for an augmented monoid

$A=(A,{A\otimes
A}\xrightarrow{m}{A},{\mathbbm{1}}\xrightarrow{u}{A},{A}\xrightarrow{\epsilon}{\mathbbm{1}})$

in a monoidal category $(\mathzc{C},\otimes,\mathbbm{1})$. Assume that $A$ is
right closed in $\mathzc{C}$ and that $\mathzc{C}$ has equalizers. The functor
of $A$-invariants $\operatorname{Inv}A:=\operatorname{Inv}T$ sends each action
$M=(X,{A\otimes X}\xrightarrow{r_{M}}{X})$ to the equalizer of
${{X}}$${{{\left[A,X\right]}^{\mathit{r}}.}}$$\scriptstyle{\ulcorner
r_{M}\urcorner}$$\scriptstyle{\ulcorner\epsilon\otimes X\urcorner}$

Moreover, given a monoid homomorphism ${B}\xrightarrow{h}{A}$, with $B$ also
right closed in $C$.

The functor of $B$-invariants
$\operatorname{Inv}B:=\operatorname{Inv}(B\otimes\mathbf{\qm})_{h\otimes\mathbf{\qm}}$
sends each action to the equalizer of
${{X}}$${{{\left[A,X\right]}^{\mathit{r}}}}$${{{\left[B,X\right]}^{\mathit{r}}}.}$$\scriptstyle{\ulcorner
r_{M}\urcorner}$$\scriptstyle{\ulcorner\epsilon\otimes
X\urcorner}$$\scriptstyle{{\left[h,X\right]}^{\mathit{r}}}$

###  4.1. Invariants are right adjoints

Right adjoints not only allow us to compute invariants. In fact, the
$T$-invariants, i.e the invariants for ${\mathrm{id}_{T}}:{T}\Rightarrow{T}$,
are right adjoints of the trivial action functor $E$. Where
${E}:{\mathzc{C}}\rightarrow{\mathzc{C}}^{T}$ is the functor corresponding to
the augmentation ${e}:{T}\Rightarrow{\mathrm{Id}_{\mathzc{C}}}$, remember that
it endows every object in $\mathzc{C}$ with the trivial action given by the
augmentation.

######  Lemma 4.3.

If $E$ has a right adjoint functor
${\Gamma}:{\mathzc{C}^{T}}\rightarrow{\mathzc{C}}$, then $\Gamma$ is the
functor of $T$-invariants.

###### Proof.

Take the adjunction $(\zeta,E\dashv\Gamma,\bar{\gamma})$ and define
${\gamma}:{\Gamma}\Rightarrow{U}$ as $\gamma=U\bar{\gamma}$. Hence by Lemma
3.4, $T$ fixes $\Gamma_{\gamma}$. Moreover by the same lemma, if $T$ fixes
${\alpha}:{V}\Rightarrow{U}$ there is
${\bar{\alpha}}:{EV}\Rightarrow{\mathrm{Id}_{\mathzc{C}^{T}}}$ such that
$\alpha=U\bar{\alpha}$. We need a ${\beta}:{V}\Rightarrow{\Gamma}$ such that
$\alpha=\gamma\cdot\beta=(U\bar{\gamma})\cdot(UE\beta)$ since $U$ is faithful
$\bar{\alpha}=\bar{\gamma}\cdot(E\beta)$, after bending the $E$ we obtain that
$(\Gamma\bar{\alpha})\cdot(\zeta V)=\beta$.

Since $E=(\bar{\gamma}E)\cdot(E\zeta)$ then

$\displaystyle\bar{\alpha}$
$\displaystyle=\bar{\alpha}\cdot(\bar{\gamma}EV)\cdot(E\zeta V)$
$\displaystyle=\bar{\gamma}\cdot(E\Gamma\bar{\alpha})\cdot(E\zeta V).$

Therefore $\alpha=U\bar{\alpha}=\gamma\cdot(\Gamma\bar{\alpha})\cdot(\zeta
V)$. ∎

Right adjoints also allow to change the category where one is calculating. Let
${h}:{S}\Rightarrow{T}$ be a monad homomorphism, with correspondent functor
${H}:{\mathzc{C}^{T}}\rightarrow{\mathzc{C}^{S}}$ as in Remark 3.3; in
particular $U=U^{S}H$.

Notice that $(S,\mu^{\prime},\eta^{\prime},eh)$ is also an augmented monad and
instead of computing $h$-invariants in $\mathzc{C}^{T}$, we can try to compute
$S$-invariants in $\mathzc{C}^{S}$ and pull them back. The next two lemmas
show that this strategy is successful when $H$ has a right adjoint.

######  Lemma 4.4.

Let $(\zeta,H\dashv K,\omega)$ be an adjunction with
${H}:{\mathzc{C}^{T}}\rightarrow{\mathzc{C}^{S}}$, coming from a monad
homomorphism ${h}:{S}\Rightarrow{T}$. For any ${\alpha}:{V}\Rightarrow{U}$,
$S_{h}$ fixes $V_{\alpha}$ is equivalent to $S$ fixes
${\hat{\alpha}}:{VK}\Rightarrow{U^{S}}$, where
$\hat{\alpha}=(U^{S}\omega)\cdot(\alpha K)$.

###### Proof.

We will use Lemma 3.4. If $S_{h}$ fixes $V_{\alpha}$, there is
${\bar{\alpha}}:{HEV}\Rightarrow{H}$ such that $\alpha=U^{S}\bar{\alpha}$.
Therefore

$\hat{\alpha}=(U^{S}\omega)\cdot(\alpha K)=U^{S}(\omega\cdot(\bar{\alpha}K)).$

Hence, by the same lemma $S$ fixes $(VK)_{\hat{\alpha}}$.

For the reciprocal, notice that $\alpha=(\hat{\alpha}H)\cdot(V\zeta)$ by
bending $K$ to $H$. Now the proof is completely analogous:

If $S$ fixes $\hat{\alpha}$, there is
${\tilde{\alpha}}:{HEVK}\Rightarrow{\mathrm{Id}_{\mathzc{C}^{S}}}$ such that
$\hat{\alpha}=U^{S}\tilde{\alpha}$. Therefore

$\alpha=(\hat{\alpha}H)\cdot(V\zeta)=U^{S}((\tilde{\alpha}H)\cdot(HEV\zeta)),$

since $U^{S}HE=UE=\mathrm{Id}_{\mathzc{C}}$. The same lemma tells us that
$S_{h}$ fixes $V_{\alpha}$. ∎

######  Corollary 4.5.

Let ${h}:{S}\Rightarrow{T}$ be a monad homomorphism and suppose that
${H}:{\mathzc{C}^{T}}\rightarrow{\mathzc{C}^{S}}$ has a right adjoint. If
${\gamma^{S}}:{\Gamma^{S}}\Rightarrow{U^{S}}$ is the $S$-invariant inclusion,
then ${\gamma^{S}H}:{\Gamma^{S}H}\Rightarrow{U}$ is the $h$-invariant
inclusion.

###### Proof.

With the notations from the previous lemma, $S_{h}$ fixes $V_{\alpha}$ if and
only if $S$ fixes $\hat{\alpha}$. Since $\gamma^{S}$ is the $S$-invariant
inclusion, there is an unique ${\hat{\beta}}:{\Gamma^{S}}\Rightarrow{VK}$ such
that $\hat{\alpha}=\gamma^{S}\cdot\hat{\beta}$. Remembering that
$\alpha=(\hat{\alpha}H)\cdot(V\omega)$ we obtain
$\alpha=((\gamma^{S}\cdot\hat{\beta})H)\cdot(V\omega)=(\gamma^{S}H)\cdot\beta$
where $\beta=(\hat{\beta}H)\cdot(V\omega)$. ∎

We end this segment with a summary of the computation of invariants by means
of right adjoints.

######  Corollary 4.6.

Let $(T,\mu,\eta,e)$ be an augmented monad, and ${h}:{S}\Rightarrow{T}$ a
monad homomorphism. Take ${E}:{\mathzc{C}}\rightarrow{\mathzc{C}}^{T}$ and
${H}:{\mathzc{C}^{S}}\rightarrow{\mathzc{C}^{T}}$ the functors corresponding
to $e$ and $h$, respectively; and assume that they are part of adjunctions
$(\zeta,E\dashv\Gamma,\bar{\gamma})$ and $(\zeta,H\dashv K,\omega)$.

Then $\operatorname{Inv}T=\Gamma_{\gamma}$ with $\gamma=U\bar{\gamma}$,
$\operatorname{Inv}S=\Gamma K_{\hat{\gamma}}$ with
$\hat{\gamma}=(U^{S}\omega)\cdot(U\bar{\gamma}K),$ and
$\operatorname{Inv}S_{h}=\Gamma KH_{\hat{\gamma}H}$ with
${\hat{\gamma}H}=(U^{S}\omega H)\cdot(U\bar{\gamma}KH)$.

###### Proof.

This is Lemma 4.3 applied twice to $E\dashv\Gamma$ and to $HE\dashv\Gamma K$,
and Corollary 4.5. Notice that when $H$ has also a left adjoint $L$ then
$LHE\dashv\Gamma KH$. ∎

##  5\. Computing stabilizers in monoidal closed categories

Currently we don’t know of any procedure for explicitly describing the
stabilizers for augmented monads in categories without additional structure.
We imagine a description in terms of Kan extensions, but we have not succeeded
in formalizing it. However, in the case of a $\otimes$-representable monad
over a monoidal closed category, we can use ends as a tool for the computation
of stabilizers. This is a procedure inpired by Tannakian reconstruction.
encompass augmented Hopf monads in monoidal closed categories.

###  5.1. The end enrichement of functor categories

For this section, we assume familiarity with the basic properties of ends, as
presented for instance in [5, Sections IX. 5-8]. For fixing notation, we
briefly recall the definition.

######  Definition 5.1.

A wedge for a functor
${S}:{\mathzc{D}^{op}\times\mathzc{D}}\rightarrow{\mathzc{C}}$ is a pair
$(W,\xi_{X})_{X\in\mathzc{D}}$ consisting of an object $W$ in $\mathzc{C}$ and
a family of morphisms ${W}\xrightarrow{\xi_{X}}{S(X,X)}$ indexed by the
objects in $\mathzc{C}$ such that for any morphism ${X}\xrightarrow{f}{Y}$ in
$\mathzc{D}$ the diagram

${W}$${{S(X,X)}}$${{S(Y,Y)}}$${{S(X,Y)}}$$\scriptstyle{\xi_{X}}$$\scriptstyle{S(X,f)}$$\scriptstyle{\xi_{Y}}$$\scriptstyle{S(f,Y)}$

commutes.

The end of $S$ denoted

$\int_{\mathzc{D}}S=\int_{X\in\mathzc{D}}S(X,X)=(L,\lambda_{X})_{X\in\mathzc{D}}$

is the universal wedge. This means that for any other wedge,
$(W,\xi_{X})_{X\in\mathzc{D}}$ there is an unique morphism, denoted
${W}\xrightarrow{\int_{X}\xi_{X}}{L}$, such that
$\lambda_{Y}\int_{X}\xi_{X}=\xi_{Y}$ for any $Y$ in $\mathzc{C}$.

The next definition allow us to view the natural transformations as an object
in the co-domain category.

######  Definition 5.2.

Let $V$ and $W$ be two functors from an arbitrary category $\mathzc{D}$ to a
monoidal category $(\mathzc{C},\otimes,\mathbbm{1})$. Assume that $\mathzc{C}$
is left closed over $V$, i.e. for every $D$ in $\mathzc{D}$ the functor
${\mathbf{\qm}\otimes VD}:{\mathzc{C}}\rightarrow{\mathzc{C}}$ has a right
adjoint ${\left[VD,\mathbf{\qm}\right]}^{\ell}$.

The left internal Nat from $V$ to $W$ is defined as the end of the functor
${{\left[V\mathbf{\qm}_{1},W\mathbf{\qm}_{2}\right]}^{\ell}}:{\mathzc{\mathzc{D}^{op}}\times\mathzc{D}}\rightarrow{\mathzc{\mathzc{C}}}$,
and denote it by

${\left[V,W\right]}^{\ell}:=\int_{\mathzc{D}}{\left[V\mathbf{\qm}_{1},W\mathbf{\qm}_{2}\right]}^{\ell}=\int_{D\in\mathzc{D}}{\left[VD,WD\right]}^{\ell}=({\left[V,W\right]}^{\ell},{{\left[V,W\right]}^{\ell}}\xrightarrow{\lambda_{D}}{{\left[VD,WD\right]}^{\ell}})_{D\in\mathzc{D}}.$

######  Remark 5.3.

If the internal Nat ${\left[V,W\right]}^{\ell}$ exists for any pair of
functors, — for instance when $\mathzc{C}$ is complete, left closed and
$\mathzc{D}$ is small — then
$\operatorname{\mathzc{Fun}}(\mathzc{D},\mathzc{C})$ is a left
$\mathzc{C}$-enriched category. The next two propositions are consequences of
this fact.

######  Proposition 5.4.

Let $V$, $V^{\prime}$, $W$ and $W^{\prime}$ be functors from $\mathzc{D}$ to
$\mathzc{C}$ with ${\omega}:{W}\Rightarrow{W^{\prime}}$ and
${\upsilon}:{V^{\prime}}\Rightarrow{V}$ natural transformations.

  * •

If both internal Nats ${\left[V,W\right]}^{\ell}$ and
${\left[V,W^{\prime}\right]}^{\ell}$ exist, then there is a morphism

${{\left[V,W\right]}^{\ell}}\xrightarrow{{\left[V,\omega\right]}^{\ell}}{{\left[V,W^{\prime}\right]}^{\ell}}.$

  * •

Analogously, when ${\left[V,W\right]}^{\ell}$ and
${\left[V^{\prime},W\right]}^{\ell}$ exist, there is

${{\left[V,W\right]}^{\ell}}\xrightarrow{{\left[\upsilon,W\right]}^{\ell}}{{\left[V^{\prime},W\right]}^{\ell}}.$

  * •

Whenever the four internal Nats in the following diagram exist, the arrows
also exist and the diagram is commutative.

${{{\left[V,W\right]}^{\ell}}}$${{{\left[V,W^{\prime}\right]}^{\ell}}}$${{{\left[V^{\prime},W\right]}^{\ell}}}$${{{\left[V^{\prime},W^{\prime}\right]}^{\ell}.}}$$\scriptstyle{{\left[\upsilon,W\right]}^{\ell}}$$\scriptstyle{{\left[V^{\prime},\omega\right]}^{\ell}}$$\scriptstyle{{\left[V,\omega\right]}^{\ell}}$$\scriptstyle{{\left[\upsilon,W^{\prime}\right]}^{\ell}}$

###### Proof.

In the first case, for any $D$ in $\mathzc{D}$ the composition of
${{\left[V,W\right]}^{\ell}}\xrightarrow{\lambda_{D}}{{\left[VD,WD\right]}^{\ell}}$
and
${{\left[VD,WD\right]}^{\ell}}\xrightarrow{{\left[VD,\omega_{D}\right]}^{\ell}}{{\left[VD,W^{\prime}D\right]}^{\ell}}$
defines a wedge for the functor
${\left[V\mathbf{\qm}_{1},W^{\prime}\mathbf{\qm}_{2}\right]}^{\ell}$. Define
${\left[V,\omega\right]}^{\ell}:=\int_{D\in\mathzc{D}}{\left[VD,\omega_{D}\right]}^{\ell}\lambda_{D}$.
∎

######  Proposition 5.5.

Take ${W}:{\mathzc{D}}\rightarrow{\mathzc{C}}$ such that the internal Nat
${\left[W,W\right]}^{\ell}$ exist. There is a canonical monoid structure on
${\left[W,W\right]}^{\ell}$ such that for any $D$ in $\mathzc{D}$ the morphism
${{\left[W,W\right]}^{\ell}}\xrightarrow{\lambda_{D}}{{\left[WD,WD\right]}^{\ell}}$
is a monoid homomorphism.

###### Proof.

Remember that for any $D$ in $\mathzc{D}$, ${\left[WD,WD\right]}^{\ell}$ is a
monoid denoted $\operatorname{End}{[WD]}^{\ell}$. The units
$({\mathbbm{1}}\xrightarrow{\ulcorner\mathrm{id}_{WD}\urcorner}{WD})_{D\in\mathzc{D}}$
make a wedge, therefore take
$\eta=\int_{D\in\mathzc{D}}\ulcorner\mathrm{id}_{WD}\urcorner$ as unit for
${\left[W,W\right]}^{\ell}$.

For the multiplication, compose
${{\left[W,W\right]}^{\ell}\otimes{\left[W,W\right]}^{\ell}}\xrightarrow{\lambda_{D}\otimes\lambda_{D}}{{\left[WD,WD\right]}^{\ell}\otimes{\left[WD,WD\right]}^{\ell}}$
with
${\operatorname{End}{[WD]}^{\ell}\otimes\operatorname{End}{[WD]}^{\ell}}\xrightarrow{\mu_{D}}{\operatorname{End}{[WD]}^{\ell}}$,
verify that this conforms a wedge111This uses that the compositions are
associative and that $\lambda_{D}$ is a wedge. and define
$\mu=\int_{D\in\mathzc{D}}\mu_{D}(\lambda_{D}\otimes\lambda_{D})$. Finally,
the monoid axioms are verified component-wise. ∎

Following the usual terminology we will denote ${\left[W,W\right]}^{\ell}$ by
$\operatorname{End}{[W]}^{\ell}$ and call it the left internal End of the
functor $W$. The next proposition formalizes the idea of restricting the
endomorphisms of $W$ along a functor
${G}:{\mathzc{D}^{\prime}}\rightarrow{\mathzc{D}}$.

######  Proposition 5.6.

Take ${W}:{\mathzc{D}}\rightarrow{\mathzc{C}}$ and
${G}:{\mathzc{D}^{\prime}}\rightarrow{\mathzc{D}}$ such that
$\operatorname{End}{[W]}^{\ell}$ and $\operatorname{End}{[WG]}^{\ell}$ both
exist, then there is a canonical monoid homomorphism
${\operatorname{End}{[W]}^{\ell}}\xrightarrow{|G}{\operatorname{End}{[WG]}^{\ell}}$.

###### Proof.

The map $|G$ is defined by $|G:=\int_{X\in\mathzc{D}^{\prime}}\lambda_{GX}$
when
$\operatorname{End}{[W]}^{\ell}=(\operatorname{End}{[W]}^{\ell},\lambda_{D})_{D\in\mathzc{D}}$.
∎

######  Lemma 5.7.

Take ${W}:{\mathzc{D}}\rightarrow{\mathzc{C}}$ such that
$\operatorname{End}{[W]}^{\ell}$ exists. There is a unique functor
${\overline{W}}:{\mathzc{D}}\rightarrow{\mathzc{C}}^{\operatorname{End}{[W]}^{\ell}}$
which lifts $W$ to the category of actions of
$\operatorname{End}{[W]}^{\ell}$. This means that $W=U\overline{W}$ where
${U}:{\mathzc{C}^{\operatorname{End}{[W]}^{\ell}}}\rightarrow{\mathzc{C}}$ is
the forgetful functor.

###### Proof.

For $D$ in $\mathzc{D}$ define
$\overline{W}D=(WD,\llcorner\lambda_{D}\lrcorner)$ where
${\operatorname{End}{[W]}^{\ell}}\xrightarrow{\lambda_{D}}{{\left[WD,WD\right]}^{\ell}}$
and ${\operatorname{End}{[W]}^{\ell}\otimes
WD}\xrightarrow{\llcorner\lambda_{D}\lrcorner}{WD}$ is obtained by un-
currying. ∎

###  5.2. Tannakian reconstruction of stabilizers

The above construction of internal Nats gives us a path for reconstruction.
Let

Let $(\mathzc{\bar{C}},\otimes,\mathbbm{1})$ be a monoidal category, with all
small limits and left closed over a small subcategory
$\mathzc{C}\hookrightarrow\mathzc{\bar{C}}$, i.e. for every $X$ in
$\mathzc{C}$ the functor ${\mathbf{\qm}\otimes
X}:{\mathzc{\bar{C}}}\rightarrow{\mathzc{\bar{C}}}$ has a right adjoint
${\left[X,\mathbf{\qm}\right]}^{\ell}$. In this case, we denote by
$\operatorname{End}{[\mathzc{C}]}^{\ell}$ the internal End for the inclusion
functor $\mathzc{C}\hookrightarrow\mathzc{\bar{C}}$ which exists in
$\mathzc{\bar{C}}$. Moreover, for a small category $D$ every pair of functors
${V,W}:{\mathzc{D}}\rightarrow{\mathzc{C}}$, after composing with the
inclusion $\mathzc{C}\hookrightarrow\mathzc{\bar{C}}$ has an internal Nat in
$\mathzc{\bar{C}}$, which abusing a little of notation we call
${\left[V,W\right]}^{\ell}$.

From a more general perspective, the next proposition defines a unit for the
Tannakian reconstruction adjunction.

######  Proposition 5.8.

Let $(A,m,u)$ be a monoid in $\mathzc{\bar{C}}$,
$\mathzc{C}^{A}\hookrightarrow\mathzc{\bar{C}}^{A}$ the subcategory of
$A$-actions on $\mathzc{C}$ and ${U}:{\mathzc{C}^{A}}\rightarrow{\mathzc{C}}$
the forgetful functor. There is a monoid homomorphism
${A}\xrightarrow{\rho}{\operatorname{End}{[U]}^{\ell}}$such that the functors
${\overline{U}}:{\mathzc{C}^{A}}\rightarrow{\mathzc{\bar{C}}}^{\operatorname{End}{[U]}^{\ell}}$,
obtained by 5.7, and
${K}:{\mathzc{\bar{C}}^{\operatorname{End}{[U]}^{\ell}}}\rightarrow{\mathzc{\bar{C}}^{A}}$,
corresponding to the monad homomorphism $\rho\otimes\mathbf{\qm}$, compose to
the inclusion $\mathzc{C}^{A}\hookrightarrow\mathzc{\bar{C}}^{A}$.

###### Proof.

For every $A$-action $M=(X,({A\otimes X}\xrightarrow{r_{M}}{X}))$ curry the
action. Notice that for any $A$-action morphism to obtain
${A}\xrightarrow{\ulcorner r_{M}\urcorner}{{\left[X,X\right]}^{\ell}}$.
homomorphism ${M}\xrightarrow{f}{N}$ the diagram

${A}$${{{\left[UM,UM\right]}^{\ell}}}$${{{\left[UN,UN\right]}^{\ell}}}$${{{\left[UN,UM\right]}^{\ell}}}$$\scriptstyle{\ulcorner
r_{N}\urcorner}$$\scriptstyle{{\left[f,UN\right]}^{\ell}}$$\scriptstyle{\ulcorner
r_{M}\urcorner}$$\scriptstyle{{\left[UM,f\right]}^{\ell}}$

commutes because $\ulcorner fr_{M}\urcorner={\left[X,f\right]}^{\ell}\ulcorner
r_{M}\urcorner$ and $\ulcorner r_{N}(Z\otimes
f)\urcorner={\left[f,Y\right]}^{\ell}\ulcorner r_{N}\urcorner$.

Therefore $(A,\ulcorner r_{M}\urcorner)_{M\in\mathzc{C}^{A}}$ is a wedge, and
we obtain a morphism ${A}\xrightarrow{\rho}{\operatorname{End}{[U]}^{\ell}}$
such that $\ulcorner r_{M}\urcorner=\lambda_{M}\rho$. Since
$\overline{U}M=(X,\llcorner\lambda_{M}\lrcorner)$ then
$K\overline{U}M=(X,\llcorner\lambda_{M}\lrcorner(\rho\otimes X))$, and
$\llcorner\lambda_{M}\lrcorner(\rho\otimes
X)=\llcorner\lambda_{M}\rho\lrcorner=r_{M}$. ∎

Now we turn to the question of internal Ends which fix a natural
transformation.

######  Proposition 5.9.

Let $(A,m,u,\epsilon)$ be an augmented monoid in $\mathzc{\bar{C}}$. Then we
have a diagram of monoid homomorphisms

${{A}}$${{\operatorname{End}{[U]}^{\ell}}}$${{\mathbbm{1}}}$${{\operatorname{End}{[\mathzc{C}]}^{\ell}},}$$\scriptstyle{\epsilon}$$\scriptstyle{\rho}$$\scriptstyle{\eta}$$\scriptstyle{|E}$$\scriptstyle{|U}$

where $(|E)(|U)=\mathrm{id}_{\operatorname{End}{[\mathzc{C}]}^{\ell}}$ and
$(|U)\eta\epsilon=\int_{M\in\mathzc{C}^{A}}\ulcorner\epsilon\otimes
UM\urcorner$.

###### Proof.

Notice that the functor ${E}:{\mathzc{C}}\rightarrow{\mathzc{C}^{T}}$, given
by $EX=(X,{A\otimes X}\xrightarrow{\epsilon\otimes X}{X})$ is a section for
$U$. The morphisms $|E$ and $|U$ are given by Proposition 5.6, and
$(|U)(|E)=|(UE)=\mathrm{id}_{\operatorname{End}{[\mathzc{C}]}^{\ell}}$. Lets
check that $(|E)\rho=\eta\epsilon$: For any $X$ in $\mathzc{C}$, consider the
universal wedge components
${\operatorname{End}{[C]}^{\ell}}\xrightarrow{\xi_{X}}{{\left[X,X\right]}^{\ell}}$
and
${\operatorname{End}{[U]}^{\ell}}\xrightarrow{\lambda_{M}}{{\left[UM,UM\right]}^{\ell}}$.
On one hand $\xi_{X}(|E)\rho=\lambda_{EX}\rho=\ulcorner
r_{EX}\urcorner=\ulcorner\epsilon\otimes X\urcorner$ since
$\rho=\int_{M}\ulcorner r_{M}\urcorner$. On the other hand
$\xi_{X}\eta\epsilon=\ulcorner\mathrm{id}_{X}\urcorner\epsilon=\ulcorner\epsilon\otimes
X\urcorner$ since $\eta=\int_{X}\ulcorner\mathrm{id}_{X}\urcorner$. ∎

######  Lemma 5.10.

Take a monoid homomorphism ${B}\xrightarrow{h}{A}$ in $\mathzc{\bar{C}}$ and a
natural transformation ${\alpha}:{V}\Rightarrow{U}$, then
$(B\otimes\mathbf{\qm})_{h\otimes\mathbf{\qm}}$ fixes $V_{\alpha}$ if and only
if $h$ equalizes the diagram
${{A}}$${{\operatorname{End}{[U]}^{\ell}}}$${{{\left[V,U\right]}^{\ell}}.}$$\scriptstyle{\rho}$$\scriptstyle{(|U)\eta\epsilon}$$\scriptstyle{{\left[\alpha,U\right]}^{\ell}}$

###### Proof.

For any $M=(X,r_{M})$ in $\mathzc{C}^{T}$, consider the universal wedge
components $\lambda_{M}$ and $\xi_{M}$ displayed in the next diagram:

${B}$${A}$${{\operatorname{End}{[U]}^{\ell}}}$${{{\left[UM,UM\right]}^{\ell}}}$${{{\left[V,U\right]}^{\ell}}}$${{{\left[VM,UM\right]}^{\ell}}}$$\scriptstyle{(|U)\eta\epsilon}$$\scriptstyle{{\left[\alpha,U\right]}^{\ell}}$$\scriptstyle{\xi_{M}}$$\scriptstyle{\lambda_{M}}$$\scriptstyle{{\left[\alpha_{M},UM\right]}^{\ell}}$$\scriptstyle{\rho}$$\scriptstyle{h}$

On one hand, since $\rho=\int_{M}\ulcorner r_{M}\urcorner$ we obtain

$\displaystyle\xi_{M}{\left[\alpha,U\right]}^{\ell}\rho h$
$\displaystyle={\left[\alpha_{M},UM\right]}^{\ell}\lambda_{M}\rho
h={\left[\alpha_{M},UM\right]}^{\ell}\ulcorner r_{M}\urcorner h$
$\displaystyle=\ulcorner r_{M}(h\otimes\alpha_{M})\urcorner.$

On the other hand, since
$(|U)\eta\epsilon=\int_{M\in\mathzc{C}^{A}}\ulcorner\epsilon\otimes
UM\urcorner$, we obtain

$\displaystyle\xi_{M}{\left[\alpha,U\right]}^{\ell}(|U)\eta\epsilon h$
$\displaystyle={\left[\alpha_{M},UM\right]}^{\ell}\lambda_{M}(|U)\eta\epsilon
h={\left[\alpha_{M},UM\right]}^{\ell}\ulcorner\epsilon\otimes X\urcorner h$
$\displaystyle=\ulcorner(\epsilon\otimes UM)(h\otimes\alpha_{M})\urcorner.$

The result follows because uncurrying the last term in each equation gives the
definition of the fix relation. ∎

## References

  * [1] Gabriella Böhm, _Hopf algebras and their generalizations from a category theoretical point of view_ , Lecture Notes in Mathematics, Springer International Publishing, 2018.
  * [2] Alain Bruguières, Steve Lack, and Alexis Virelizier, _Hopf monads on monoidal categories_ , Advances in Mathematics 227 (2011), no. 2, 745 – 800.
  * [3] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, _Tensor categories_ , Mathematical Surveys and Monographs, vol. 205, American Mathematical Society, 2011.
  * [4] Alexander Grothendieck, _Revêtements étales et groupe fondamental (SGA 1)_ , Lecture Notes in Mathematics, vol. 224, Springer, 1971 (french).
  * [5] Saunders MacLane, _Categories for the working mathematician_ , Graduate Texts in Mathematics, Springer, 1998 (english).


# Dark matter relic density in strongly interacting dark sectors with light
vector mesons

Elias Bernreuther<EMAIL_ADDRESS>Particle Theory Department, Fermilab,
Batavia, IL 60510, USA  Nicoline Hemme<EMAIL_ADDRESS>Institute for
Theoretical Particle Physics (TTP), Karlsruhe Institute of Technology (KIT),
76128 Karlsruhe, Germany  Felix Kahlhoefer<EMAIL_ADDRESS>Institute
for Theoretical Particle Physics (TTP), Karlsruhe Institute of Technology
(KIT), 76128 Karlsruhe, Germany  Suchita Kulkarni  suchita.kulkarni@uni-
graz.at Institute of Physics, NAWI Graz, University of Graz, Universitätsplatz
5, 8010 Graz, Austria

###### Abstract

Stable dark matter particles may arise as pseudo-Goldstone bosons from the
confinement of dark quarks interacting via a non-Abelian gauge force. Their
relic abundance is determined not by annihilations into visible particles but
by dark pion number-changing processes within the dark sector, such as
$3{\pi_{D}}\to 2{\pi_{D}}$. However, if the dark vector mesons ${\rho_{D}}$
are light enough for $3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}$ annihilations to be
kinematically allowed, this process dominates and significantly delays freeze-
out. As a result, the preferred dark matter mass scale increases and bounds
from the Bullet Cluster can be evaded.

††preprint: FERMILAB-PUB-23-744-T, TTP23-057, P3H-23-096

## Introduction.—

An attractive alternative to the paradigm of weakly interacting massive
particles is the idea that dark matter (DM) is part of a strongly interacting
dark sector Kribs:2016cew . At high energies, such a dark sector can be
described in terms of dark quarks interacting via the dark gluons of a non-
Abelian extension of the Standard Model (SM) gauge group. At low energies, on
the other hand, the dark sector confines, and the dark quarks and gluons are
bound in dark mesons and dark baryons. In analogy to SM quantum
chromodynamics, the pseudoscalar mesons, called dark pions, are expected to be
the lightest dark sector state, because they are the pseudo-Goldstone bosons
of chiral symmetry breaking. Indeed, if the dark quark masses are sufficiently
small, the dark pions can be significantly lighter than the confinement scale.
In contrast to the SM, however, the dark pions may be stable, due to either a
$U(1)$ or a parity symmetry Bai:2010qg ; Buckley:2012ky ; Cline:2013zca
(although the latter may be broken through gravitational effects Dondi:2019olm
), making them attractive DM candidates Hochberg:2014kqa ; Hochberg:2015vrg ;
Bernreuther:2019pfb .

Strongly interacting dark sectors are an attractive target for collider
searches due to their striking signature: If a pair of dark quarks is produced
in a hard process, it will generate a shower of dark hadrons, some of which
may decay into SM particles, while others evade detection Strassler:2006im ;
Englert:2016knz ; Pierce:2017taw ; Kribs:2018ilo ; Cheng:2019yai ;
Beauchesne:2019ato ; Butterworth:2021jto ; Cheng:2021kjg ; Bernreuther:2022jlj
; Albouy:2022cin . The results are one or more semi-visible Cohen:2015toa ;
Beauchesne:2017yhh  or emerging Schwaller:2015gea  jets. These signatures have
been explored in a bottom-up way by varying phenomenological parameters such
as the masses of dark mesons and the fraction of invisible particles
Cohen:2017pzm ; Cohen:2020afv ; Bernreuther:2020vhm ; Knapen:2021eip , and
corresponding searches have been carried out by ATLAS ATLAS:2023swa  and CMS
CMS:2021dzg . In order to make a connection to the DM puzzle, however, it
becomes necessary to explore the cosmological history of strongly interacting
dark sectors, and to understand whether the dark pions can be produced in the
right amount to explain observations of the Cosmic Microwave Background
Bernreuther:2019pfb .

A central result in this context is the so-called SIMP (for strongly
interacting massive particle) mechanism Hochberg:2014dra ; Hochberg:2014kqa :
dark pions may participate in number-changing processes, such as
$3{\pi_{D}}\to 2{\pi_{D}}$ via the Wess-Zumino-Witten anomaly. At low
temperatures, these processes lead to the conversion of rest mass into kinetic
energy and hence a depletion of the dark sector. The freeze-out temperature,
when number-changing processes become inefficient, then determines the relic
abundance of dark pions. Comparison to observations leads to a strict upper
bound of $m_{\pi_{D}}\lesssim 100\,\mathrm{MeV}$ in order to avoid an
overabundance of DM Hochberg:2014kqa ; Hansen:2015yaa ; Choi:2017mkk ;
Braat:2023fhn . This bound is in tension with a lower bound on the dark pion
mass obtained from observations of the Bullet Cluster, which constrains the DM
self-interaction cross section Randall:2008ppe ; Robertson:2016xjh ;
Wittman:2017gxn .

In this letter we point out an important modification of the SIMP mechanism:
In addition to number-changing processes involving only dark pions, there may
also be processes involving heavier dark sector states, in particular the
vector mesons, called dark rho mesons Choi:2018iit ; Berlin:2018tvf ;
Beauchesne:2018myj . Indeed, if the dark quarks have masses comparable to the
confinement scale, it is possible to have $m_{\rho_{D}}<2m_{\pi_{D}}$. This
scenario is particularly interesting for collider experiments, since the dark
rho meson in this case cannot decay into dark pions and must instead decay
into SM particles (e.g. via kinetic mixing with the SM photon). The lifetime
of the dark rho mesons can be quite long, such that their decays lead to
displaced vertices in the detector Bernreuther:2020xus ; Bernreuther:2022jlj .

We show that, if the process $3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}$ is
kinematically allowed at low temperatures, it will typically dominate the rate
of number-changing processes. This is due to a favourable velocity dependence
(the process proceeds via $s$-wave, whereas the process involving only pions
proceeds via $d$-wave) and a resonant enhancement if the internal dark pions
can be nearly on-shell. We calculate the relevant cross sections, discuss how
thermal effects determine the width of the resonance, and solve the resulting
Boltzmann equation. Our central result is that the presence of dark rho mesons
relaxes the cosmological bound on the dark pion mass. It thus becomes possible
to realize the SIMP mechanism for heavier dark pions and smaller couplings,
thereby evading the Bullet Cluster constraint.

The remainder of this letter is structured as follows. We first introduce the
strongly interacting dark sector model and discuss the spectrum of dark mesons
expected from non-perturbative calculations. We then present the relevant
Boltzmann equations and the required reaction rates and finally calculate the
dark pion relic density and the Bullet Cluster constraint.

## Model details.—

We start from an $SU(N_{c_{D}})$ gauge group, with $N_{f_{D}}$ mass-degenerate
Dirac fermions $q_{D}$ in its fundamental representation. The high-energy
Lagrangian for such a theory is given by

$\mathcal{L}_{\text{UV}}=-\frac{1}{4}G_{D\mu\nu}^{a}G^{\mu\nu,a}_{D}+\bar{q}_{D}(i\gamma^{\mu}D_{\mu}-M_{q_{D}})q_{D}\,,$
(1)

where $M_{q_{D}}$ is the mass of the dark quarks, $G_{D}^{\mu\nu}$ denotes the
dark gluon field strength tensor and $D_{\mu}$ is the gauge covariant
derivative. While we leave $N_{c_{D}},N_{f_{D}}$ as free parameters, it is
important to note that chiral symmetry breaking only takes place for
$N_{f_{D}}<3N_{c_{D}}$.

At low energies, chiral symmetry breaking leads to confinement. The dynamics
of the resulting pseudo-Nambu-Goldstone Bosons, called dark pions ${\pi_{D}}$,
are described by a chiral Lagrangian. The kinetic term and the mass term are
contained in

$\mathcal{L}_{\text{Ch}}=\tfrac{f_{\pi_{D}}^{2}}{4}\mathrm{Tr}\left(\partial_{\mu}U\partial^{\mu}U^{\dagger}\right)+\left[\tfrac{\mu_{D}^{3}}{2}\mathrm{Tr}\left(M_{q_{D}}U^{\dagger}\right)+\mathrm{h.c.}\right]$
(2)

with the $SU(N_{f_{D}})$ matrix
$U\equiv\exp\left(2i{\pi_{D}}/f_{\pi_{D}}\right)$, the dark pion decay
constant $f_{\pi_{D}}$, and the quark condensate $\mu^{3}_{D}$. In addition,
the terms in Eq. (2) give rise to interactions between even numbers of dark
pions. In the chirally broken phase the dark pions are the lightest mesons in
the spectrum, and they can be stabilised by a suitable discrete or continuous
symmetry.

In addition, there will also be heavier dark mesons in the spectrum, in
particular vector mesons analogous to the SM $\rho$ mesons. Interactions
between dark pions and dark vector mesons are introduced by promoting the
derivative acting on the dark pions in Eq. (2) to the covariant derivative

$D_{\mu}{\pi_{D}}=\partial_{\mu}{\pi_{D}}+ig[{\pi_{D}},{\rho_{D}}_{\mu}]\,,$
(3)

with the ${\pi_{D}}{\pi_{D}}{\rho_{D}}$ coupling $g\approx
m_{\rho_{D}}/(\sqrt{2}f_{\pi_{D}})$ obtained from the KSRF relation
Kawarabayashi:1966kd ; PhysRev.147.1071 . Hence, the chiral Lagrangian of dark
pions and dark vector mesons, expanded up to terms with at most four dark pion
fields, reads

$\displaystyle\mathcal{L}_{\text{Ch}}\supset$
$\displaystyle\mathrm{Tr}\left(D_{\mu}{\pi_{D}}D^{\mu}{\pi_{D}}\right)+m_{\pi_{D}}^{2}\mathrm{Tr}\left(\pi_{D}^{2}\right)+\tfrac{m_{\pi_{D}}^{2}}{3f_{\pi_{D}}^{2}}\mathrm{Tr}\left(\pi_{D}^{4}\right)$
$\displaystyle-\tfrac{2}{3f_{\pi_{D}}^{2}}\mathrm{Tr}\left(\pi_{D}^{2}D_{\mu}{\pi_{D}}D^{\mu}{\pi_{D}}-{\pi_{D}}D_{\mu}{\pi_{D}}{\pi_{D}}D^{\mu}{\pi_{D}}\right)\;.$
(4)

For $N_{f_{D}}\geq 3$, the Wess-Zumino-Witten (WZW) term induces an anomalous
five-point interaction given by

$\mathcal{L}_{\text{WZW}}=\frac{2N_{c_{D}}}{15\pi^{2}f_{\pi_{D}}^{5}}\epsilon^{\mu\nu\rho\sigma}\mathrm{Tr}\left({\pi_{D}}\partial_{\mu}{\pi_{D}}\partial_{\nu}{\pi_{D}}\partial_{\rho}{\pi_{D}}\partial_{\sigma}{\pi_{D}}\right)\;.$
(5)

The free parameters in the above chiral Lagrangian
(${m_{\pi_{D}}},{m_{\rho_{D}}},f_{\pi_{D}}$) need to be calculated using non-
perturbative methods. Along with $N_{c_{D}},N_{f_{D}}$, a QCD-like strongly
interacting theory with mass-degenerate quarks has two additional free
parameters, one mass ratio and one mass scale Albouy:2022cin . Once these
inputs are fixed, the dark meson spectrum can be computed using non-
perturbative methods, e.g. lattice simulations. In the UV the two free
parameters could be considered $M_{q_{D}},M_{q_{D}}/\Lambda_{D}$, which are
traded in for e.g. ${m_{\pi_{D}}},{m_{\pi_{D}}}/f_{\pi_{D}}$ in the chirally
broken phase.

Figure 1: Representative Feynman diagrams for the processes that contribute to
the $3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}$ cross section. Additional diagrams are
obtained from the first diagram through permutations of the initial state.

In the following, we are particularly interested in the process
$3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}$ (see Fig. 1 for the corresponding Feynman
diagrams) and its effect on the relic density of dark pions. For this process
to be allowed in the non-relativistic limit (i.e. for negligible kinetic
energy of the initial state particles), we require
${m_{\rho_{D}}}/{m_{\pi_{D}}}<2$. Incidentally, this condition also implies
that dark rho mesons cannot decay into pairs of dark pions. Nevertheless, dark
rho mesons do in general decay into SM particles (for example through kinetic
mixing with the SM photon Holdom:1985ag ; Babu:1997st , see Refs.
Hochberg:2015vrg ; Kribs:2018ilo ; Katz:2020ywn ; Bernreuther:2022jlj ;
Kulkarni:2022bvh  for explicit constructions). These (inverse) decays can
efficiently transfer energy and entropy between the dark sector and the SM and
keep the two sectors in thermal equilibrium.

It will therefore be convenient to use ${m_{\pi_{D}}}$ and the mass ratio
${m_{\rho_{D}}}/{m_{\pi_{D}}}$ as free parameters for our analysis. Using the
results from Ref. Maris:2005tt  and neglecting the effect of varying
$N_{c_{D}}$ and $N_{f_{D}}$, we obtain an approximate relation between
$f_{\pi_{D}}$, ${m_{\pi_{D}}}$ and ${m_{\rho_{D}}}$:

$\xi\equiv\displaystyle\frac{{m_{\pi_{D}}}}{f_{\pi_{D}}}=7.79\,\displaystyle\frac{{m_{\pi_{D}}}}{{m_{\rho_{D}}}}+0.57\left(\displaystyle\frac{{m_{\pi_{D}}}}{{m_{\rho_{D}}}}\right)^{2}\,,$
(6)

which is valid for $1<{m_{\rho_{D}}}/{m_{\pi_{D}}}\lesssim 20$, where the
lower limit on ${m_{\rho_{D}}}/{m_{\pi_{D}}}$ corresponds to $\xi\sim 8$.

In principle, one could consider the entire range of above fit. For large
$\xi$, however, the validity of chiral perturbation theory becomes
increasingly dubious and higher-order corrections become relevant
Hansen:2015yaa . In the present work we therefore only consider dark rho meson
masses larger than $1.45\,m_{\pi_{D}}$ (equivalently $\xi\lesssim 5.7$),
corresponding to the range where the GMOR relation Gell-Mann:1968hlm  is
expected to be satisfied Faber:2017alm ; Engel:2014eea ; Gattringer:2008vj ;
DelDebbio:2006cn . This bound implies in particular that the forbidden
annihilation process ${\pi_{D}}{\pi_{D}}\to{\rho_{D}}{\rho_{D}}$ studied in
Ref. Bernreuther:2019pfb  is not relevant for the calculation of the dark pion
relic density. The case $m_{\rho_{D}}>2m_{\pi_{D}}$, on the other hand, has
previously been studied in Ref. Choi:2018iit . In this case, the process
$3{\pi_{D}}\to 2{\pi_{D}}$ may receive a strong resonant enhancement from on-
shell intermediate dark vector mesons.

## Dark pion relic density.—

Let us now turn to the calculation of the dark pion relic density. The
Boltzmann equation describes the evolution of the density of a particle
species as it falls out of equilibrium. Integrating the Boltzmann equation
yields the relic abundance of a stable particle given the contributing freeze-
out processes. If ${\rho_{D}}$ is in equilibrium with the SM bath throughout
and $3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}$ annihilations dominate the depletion of
dark pions, we can use the principle of detailed balance to write the
Boltzmann equation as

$\dot{n}_{\pi_{D}}+3H\,n_{\pi_{D}}=\langle\sigma
v^{2}\rangle_{3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}}n_{\pi_{D}}((n_{\pi_{D}}^{\mathrm{eq}})^{2}-n_{\pi_{D}}^{2})\;,$
(7)

where $n_{\pi_{D}}$ is the dark pion number density and $H$ denotes the Hubble
rate.

In the non-relativistic limit, the thermally averaged cross section is given
by

$\displaystyle\langle\sigma v^{2}\rangle_{3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}}=$
$\displaystyle\frac{|\mathcal{M}|^{2}_{3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}}}{144\pi
S_{\alpha}S_{\beta}m^{3}_{\pi_{D}}}\sqrt{4-5y+y^{2}}\,,$ (8)

with $S_{\alpha}=3!$ and $S_{\beta}=1$ denoting the number of permutations of
identical particles in the initial and final state, respectively, and
$y=m_{\rho_{D}}^{2}/(4m_{\pi_{D}}^{2})$.

Representative diagrams for the process are shown in Fig. 1. The sum of all
diagrams yields the total squared amplitude

$|\mathcal{M}|^{2}_{3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}}=\frac{\kappa\,g^{2}\,m_{\rho_{D}}^{2}m_{\pi_{D}}^{2}(4-y)(1-y)}{f_{\pi_{D}}^{4}\left(64m_{\pi_{D}}^{2}(1-y)^{2}+9\Gamma_{\text{th}}^{2}\right)}\,,$
(9)

with

$\kappa=\frac{2(37N_{f_{D}}^{4}-30N_{f_{D}}^{2}+18)}{3N_{f_{D}}(N_{f_{D}}^{2}-1)^{2}}=\frac{74}{3N_{f_{D}}}\,+\,\mathcal{O}(N_{f_{D}}^{-3})\;,$
(10)

and $\Gamma_{\text{th}}$ being the thermal width of dark pions discussed
below.

As $m_{\rho_{D}}$ approaches $2m_{\pi_{D}}$ from below, the internal dark pion
can go on shell and the cross section grows rapidly. At first sight, this
should lead to a divergence, since the dark pion is stable and therefore does
not have a decay width. In a thermal plasma, however, dark pions do not have
an infinite lifetime, due to interactions with other particles in the plasma.
This leads to a thermal self-energy, the imaginary part of which can be
interpreted as an effective decay width Weldon:1983jn .

We present the derivation of the dark pion thermal width $\Gamma_{\text{th}}$
in appendix A. The main contribution is found to arise from the scattering of
two dark pions. For $m_{\rho_{D}}\approx 2m_{\pi_{D}}$ and low temperatures,
i.e. $x\equiv m_{\pi_{D}}/T\gg 1$, we find

$\displaystyle\Gamma_{\text{th}}=$
$\displaystyle\frac{8\pi(N_{f_{D}}^{2}-1)}{x^{2}}e^{-x}m_{\pi_{D}}^{3}\sigma_{c}\;,$
(11)

with

$\displaystyle\sigma_{\mathrm{c}}$
$\displaystyle\approx\frac{1}{64\pi}\frac{3N_{f_{D}}^{4}-2N_{f_{D}}^{2}+6}{N_{f_{D}}^{2}(N_{f_{D}}^{2}-1)}\frac{m_{\pi}^{2}}{f_{\pi}^{4}}$
$\displaystyle=\frac{3}{64\pi}\frac{m_{\pi}^{2}}{f_{\pi}^{4}}(1+\mathcal{O}(N_{f_{D}}^{-2}))$
(12)

denoting the two-pion scattering cross section in the limit of vanishing
initial velocities.

We emphasize that the thermal width becomes exponentially suppressed at small
temperatures, as a result of the suppressed number density of dark pions that
can participate in scattering. In the temperature range relevant for freeze-
out ($x\approx 20$) the contribution from the dark pion width is therefore
negligible unless $m_{\rho_{D}}$ is extremely close to $2\,m_{\pi_{D}}$.

It is instructive to compare our result to the conventional freeze-out of
SIMPs through the WZW anomaly. The corresponding Boltzmann equation reads

$\dot{n}_{\pi_{D}}+3H\,n_{\pi_{D}}=\langle\sigma v^{2}\rangle_{3{\pi_{D}}\to
2{\pi_{D}}}n_{\pi_{D}}^{2}(n_{\pi_{D}}^{\mathrm{eq}}-n_{\pi_{D}})\;,$ (13)

where the thermally averaged cross section is given by

$\langle\sigma v^{2}\rangle_{3{\pi_{D}}\to
2{\pi_{D}}}=\frac{5\sqrt{5}N_{c_{D}}^{2}\kappa_{3{\pi_{D}}\to
2{\pi_{D}}}\xi^{10}}{1536\pi^{5}m_{\pi_{D}}^{5}x^{2}}\;,$ (14)

with

$\kappa_{3{\pi_{D}}\to
2{\pi_{D}}}=\frac{N_{f_{D}}(N_{f_{D}}^{2}-4)}{(N_{f_{D}}^{2}-1)^{2}}=\frac{1}{N_{f_{D}}}+\mathcal{O}(N_{f_{D}}^{-3})\,.$
(15)

The cross sections for the processes $3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}$ and
$3{\pi_{D}}\to 2{\pi_{D}}$ can both be written as

$\langle\sigma
v^{2}\rangle\equiv\frac{\alpha^{\mathrm{eff}}}{m_{\pi_{D}}^{5}}\;.$ (16)

To understand which process dominates dark pion freeze-out, it is useful to
consider the ratio

$\displaystyle R$ $\displaystyle\equiv\frac{\langle\sigma
v^{2}\rangle_{3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}}}{\langle\sigma
v^{2}\rangle_{3{\pi_{D}}\to
2{\pi_{D}}}}=\frac{\alpha^{\mathrm{eff}}_{3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}}}{\alpha^{\mathrm{eff}}_{3{\pi_{D}}\to
2{\pi_{D}}}}$ $\displaystyle\approx\left(100\text{ --
}400\right)\times\frac{1}{N_{c_{D}}^{2}\,\xi^{4}}\frac{x^{2}}{\sqrt{1-y}}\;,$
(17)

where the range for the numerical factor has been obtained by varying $y$ in
the range $0.6<y<1$ and $N_{f_{D}}$ between 3 and 6, noting that the process
$3{\pi_{D}}\to 2{\pi_{D}}$ does not exist for $N_{f_{D}}=2$
Bernreuther:2019pfb . We find that the ratio $R$ is much larger than unity for
all values that we consider. As expected, $R$ grows rapidly for $y\to 1$ and
also grows with increasing $x$. This is because the process $3{\pi_{D}}\to
2{\pi_{D}}$ proceeds via $d$-wave, whereas the process
$3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}$ proceeds via $s$-wave. We conclude that the
latter process will completely dominate dark pion freeze-out for
$m_{\rho_{D}}<2m_{\pi_{D}}$.

## Results and discussion.—

Figure 2: Solutions of the Boltzmann equation for the (dimensionless) dark
pion number density as a function of inverse temperature. For the red lines
only the process $3{\pi_{D}}\to 2{\pi_{D}}$ is included, whereas the blue
lines include $3{\pi_{D}}\to{\pi_{D}}X_{D}$ with $X_{D}={\pi_{D}},{\rho_{D}}$.
The value of $Y_{\pi_{D}}$ that corresponds to the observed DM relic abundance
is indicated by the dot-dashed black line, while the equilibrium value of
$Y_{\pi_{D}}$ is represented by the dashed grey line.

It is convenient to express the Boltzmann equation in terms of the
dimensionless quantity $Y_{\pi_{D}}=n_{\pi_{D}}/s$, where
$s=2\pi^{2}/45\,g_{\star}T^{3}$ is the total entropy density. The Boltzmann
equation (7) then takes the form

$\frac{dY}{dx}=\frac{s^{2}}{\tilde{H}x}\langle\sigma
v^{2}\rangle\>Y\>(Y_{\mathrm{eq}}^{2}-Y^{2})\;,$ (18)

where following Ref. bringmann2022darksusy  we have introduced the modified
Hubble rate
$\tilde{H}=(\frac{8\pi^{3}}{90}g)^{1/2}\frac{T^{2}}{M_{\mathrm{Pl}}}[1+\frac{1}{3}\frac{\text{d(ln
}g_{\star})}{\text{d(ln T)}}]^{-1}$, with $g$ ($g_{\star}$) denoting the
energy (entropy) degrees of freedom taken from Ref. Drees_2015 . We solve the
Boltzmann equation in log-space.

In Fig. 2 we show the solution of the Boltzmann equation for
$m_{\pi_{D}}=100\,\mathrm{MeV}$, $N_{f_{D}}=N_{c_{D}}=3$ and different values
of $m_{\rho_{D}}/m_{\pi_{D}}$. As expected, the dimensionless dark pion number
density $Y_{\pi_{D}}$ follows the equilibrium value
$Y_{\pi_{D}}^{\text{eq}}=(N_{f_{D}}^{2}-1)m_{\pi_{D}}^{3}(x/(2\pi))^{3/2}\exp(-x)$
until it freezes out at $x_{f}\approx 20$ and becomes constant. For the chosen
parameters, the observed DM relic abundance $\Omega_{D}h^{2}=0.12$ is
approximately reproduced. For comparison, we also show the evolution of
$Y_{\pi_{D}}$ obtained when only the process $3{\pi_{D}}\to 2{\pi_{D}}$ is
considered 111Here we neglect the contribution of the dark vector mesons to
the anomaly terms, which depend on additional unknown parameters and therefore
introduce a further model dependence Choi:2018iit .. For the same masses, the
predicted relic abundance is too large by approximately an order of magnitude.

We can understand this result analytically, by writing Eq. (18) as

$\frac{dY}{dx}=\frac{\lambda_{3\to
2}}{x^{5}}\>Y\>(Y_{\mathrm{eq}}^{2}-Y^{2})\;,$ (19)

with

$\lambda_{3\to
2}=\frac{2\sqrt{5}}{675}\pi^{5/2}\,g^{3/2}\,M_{\mathrm{Pl}}\,m_{\pi_{D}}^{4}\,\langle\sigma
v^{2}\rangle\,,$ (20)

where $g=g_{*}=10.75$ in the temperature range of interest and we can drop the
derivative in the expression for $\tilde{H}$.

Typically, $Y_{\mathrm{eq}}\ll Y$ during freeze-out and the corresponding term
in the Boltzmann equation can be neglected. Treating $\lambda_{3\to 2}$ as a
constant during freeze-out by setting $\lambda_{3\to 2}\equiv\lambda_{3\to
2}(x=x_{f})$, Eq. (19) can be approximately solved analytically, which yields
the asymptotic solution

$Y_{\infty}\approx\sqrt{2}\frac{x_{f}^{2}}{\sqrt{\lambda_{3\to 2}}}\,.$ (21)

Hence, the DM relic abundance scales with the DM mass and effective coupling
(as defined in Eq. (16)) as $\Omega_{D}h^{2}\sim
m_{\pi_{D}}^{3/2}/\sqrt{\alpha_{\mathrm{eff}}}$. Therefore, when
$3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}$ annihilations dominate over $3{\pi_{D}}\to
2{\pi_{D}}$, the preferred dark pion mass scale increases as $m_{\pi_{D}}\sim
R^{1/3}$ relative to the usual expectation for SIMPs annihilating via the WZW
term. For typical values of $\xi$, $m_{\rho_{D}}/m_{\pi_{D}}$ and $N_{c_{D}}$
this corresponds to a factor of 2–3, and even more for $m_{\rho_{D}}\to
2m_{\pi_{D}}$, see Eq. (Dark pion relic density.—).

Figure 3: Combinations of $m_{\pi_{D}}$ and $\xi=m_{\pi_{D}}/f_{\pi_{D}}$ (or
$m_{\rho_{D}}/m_{\pi_{D}}$ via Eq. (6)) that yield $\Omega_{D}h^{2}=0.12$ in
agreement with observations. For the red lines only the process $3{\pi_{D}}\to
2{\pi_{D}}$ is included, whereas for the blue lines
$3{\pi_{D}}\to{\rho_{D}}{\pi_{D}}$ is also taken into consideration. Note that
the process $3{\pi_{D}}\to 2{\pi_{D}}$ does not exist for $N_{f_{D}}=2$. The
grey shaded region is excluded by the Bullet Cluster bound on the DM self-
interaction cross section (evaluated for $N_{f_{D}}\gg 1$).

We explicitly confirm this expectation in Fig. 3, where we plot the
combinations of $m_{\pi_{D}}$ and $\xi$ (or equivalently
$m_{\rho_{D}}/m_{\pi_{D}}$ as related to $\xi$ by Eq. (6)) that yield
$\Omega_{\mathrm{D}}h^{2}=0.12$ via a numerical solution of the Boltzmann
equation for different choices of $N_{f_{D}}$ and $N_{c_{D}}$. As before, we
show for comparison the result when considering $3{\pi_{D}}\to 2{\pi_{D}}$
only, which is robustly excluded by the Bullet Cluster constraint on DM self-
interactions, given by Robertson:2016xjh ; Wittman:2017gxn

$\frac{\sigma_{c}}{m_{\pi_{D}}}\lesssim 2~{}\mathrm{cm^{2}/g}\;,$ (22)

with $\sigma_{c}$ as in Eq. (Dark pion relic density.—). Including the dark
rho mesons in the final state, on the other hand, increases the dark pion mass
scale sufficiently to evade these constraints. Due to the interplay between
$\xi$ and $m_{\rho_{D}}/m_{\pi_{D}}$ in Eq. 9, we find that the preferred
value of $m_{\pi_{D}}$ is largely independent of these parameters and can be
approximately written as $m_{\pi_{D}}\approx
150\,\mathrm{MeV}/N_{f_{D}}^{1/3}$.

In conclusion, we have shown that for strongly interacting dark sectors with
$m_{\rho_{D}}/m_{\pi_{D}}<2$, the dominant process that changes the number
density of dark pions in the early universe is
$3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}$. We emphasize that this process does not
depend on any non-perturbative parameters other than $m_{\pi_{D}}$,
$f_{\pi_{D}}$ and $m_{\rho_{D}}$ and – in contrast to the conventionally
studied process $3{\pi_{D}}\to 2{\pi_{D}}$ – does not rely on the WZW anomaly,
i.e. it also exists for theories with only two light flavours. In contrast to
the recently proposed Co-SIMP mechanism Smirnov:2020zwf ; Parikh:2023qtk , the
process that we consider requires no interactions between the dark sector and
SM particles beyond those needed for thermalisation (which only places a very
weak lower bound on the decay width of the dark rho mesons). As a result, we
obtain a theoretically clean and robust prediction for the dark pion mass that
reproduces the observed DM relic abundance. We have specifically considered
the range $1.45\leq m_{\rho_{D}}/m_{\pi_{D}}<2$ corresponding to $4\lesssim
m_{\pi_{D}}/f_{\pi_{D}}\lesssim 6$, within the validity of chiral perturbation
theory. We find typical values of $m_{\pi_{D}}$ around $100\,\mathrm{MeV}$
reproduce the measured DM relic abundance, with only mild dependence on the
other parameters. Crucially, these dark pion masses satisfy the Bullet Cluster
constraint on DM self-interactions – unlike the masses favoured by freeze-out
via $3{\pi_{D}}\to 2{\pi_{D}}$ – and therefore provide an important benchmark
scenario for further exploration. We emphasize that the required mass spectrum
implies that the dark rho mesons can only decay into SM particles, leading to
exciting signatures at laboratory experiments.

###### Acknowledgements.

We thank Pieter Braat, Maximilian Detering, Gernot Eichmann, Mads Frandsen,
Bror Hjemgaard, Martin Hoferichter, Martin Napetschnig, Marieke Postma and
Fabian Zierler for helpful discussions. NH thanks CP3-Origins for hospitality.
Fermilab is administered by Fermi Research Alliance, LLC under Contract No.
DE-AC02 07CH11359 with the U.S. Department of Energy, Office of Science,
Office of High Energy Physics. NH and FK are supported by the Deutsche
Forschungsgemeinschaft (DFG) through the Emmy Noether Grant No. KA 4662/1-2
and grant 396021762 – TRR 257. SK is supported by FWF research group grant
FG1.

## References

  * (1) G. D. Kribs and E. T. Neil, Int. J. Mod. Phys. A 31 (2016), no. 22, 1643004, [1604.04627].
  * (2) Y. Bai and R. J. Hill, Phys. Rev. D 82 (2010), 111701, [1005.0008].
  * (3) M. R. Buckley and E. T. Neil, Phys. Rev. D 87 (2013), no. 4, 043510, [1209.6054].
  * (4) J. M. Cline, Z. Liu, G. D. Moore, and W. Xue, Phys. Rev. D 90 (2014), no. 1, 015023, [1312.3325].
  * (5) N. A. Dondi, F. Sannino, and J. Smirnov, Phys. Rev. D 101 (2020), no. 10, 103010, [1905.08810].
  * (6) Y. Hochberg, E. Kuflik, H. Murayama, T. Volansky, and J. G. Wacker, Phys. Rev. Lett. 115 (2015), no. 2, 021301, [1411.3727].
  * (7) Y. Hochberg, E. Kuflik, and H. Murayama, JHEP 05 (2016), 090, [1512.07917].
  * (8) E. Bernreuther, F. Kahlhoefer, M. Krämer, and P. Tunney, JHEP 01 (2020), 162, [1907.04346].
  * (9) M. J. Strassler and K. M. Zurek, Phys. Lett. B 651 (2007), 374–379, [hep-ph/0604261].
  * (10) C. Englert, K. Nordström, and M. Spannowsky, Phys. Rev. D 94 (2016), no. 5, 055028, [1606.05359].
  * (11) A. Pierce, B. Shakya, Y. Tsai, and Y. Zhao, Phys. Rev. D 97 (2018), no. 9, 095033, [1708.05389].
  * (12) G. D. Kribs, A. Martin, B. Ostdiek, and T. Tong, JHEP 07 (2019), 133, [1809.10184].
  * (13) H.-C. Cheng, L. Li, E. Salvioni, and C. B. Verhaaren, JHEP 11 (2019), 031, [1906.02198].
  * (14) H. Beauchesne and G. Grilli di Cortona, JHEP 02 (2020), 196, [1910.10724].
  * (15) J. M. Butterworth, L. Corpe, X. Kong, S. Kulkarni, and M. Thomas, Phys. Rev. D 105 (2022), no. 1, 015008, [2105.08494].
  * (16) H.-C. Cheng, L. Li, and E. Salvioni, JHEP 01 (2022), 122, [2110.10691].
  * (17) E. Bernreuther, K. Böse, T. Ferber, C. Hearty, F. Kahlhoefer, A. Morandini, and K. Schmidt-Hoberg, JHEP 12 (2022), 005, [2203.08824].
  * (18) G. Albouy et al., Eur. Phys. J. C 82 (2022), no. 12, 1132, [2203.09503].
  * (19) T. Cohen, M. Lisanti, and H. K. Lou, Phys. Rev. Lett. 115 (2015), no. 17, 171804, [1503.00009].
  * (20) H. Beauchesne, E. Bertuzzo, G. Grilli Di Cortona, and Z. Tabrizi, JHEP 08 (2018), 030, [1712.07160].
  * (21) P. Schwaller, D. Stolarski, and A. Weiler, JHEP 05 (2015), 059, [1502.05409].
  * (22) T. Cohen, M. Lisanti, H. K. Lou, and S. Mishra-Sharma, JHEP 11 (2017), 196, [1707.05326].
  * (23) T. Cohen, J. Doss, and M. Freytsis, JHEP 09 (2020), 118, [2004.00631].
  * (24) E. Bernreuther, T. Finke, F. Kahlhoefer, M. Krämer, and A. Mück, SciPost Phys. 10 (2021), no. 2, 046, [2006.08639].
  * (25) S. Knapen, J. Shelton, and D. Xu, Phys. Rev. D 103 (2021), no. 11, 115013, [2103.01238].
  * (26) ATLAS, G. Aad et al., (2023), 2305.18037.
  * (27) CMS, A. Tumasyan et al., JHEP 06 (2022), 156, [2112.11125].
  * (28) Y. Hochberg, E. Kuflik, T. Volansky, and J. G. Wacker, Phys. Rev. Lett. 113 (2014), 171301, [1402.5143].
  * (29) M. Hansen, K. Langæble, and F. Sannino, Phys. Rev. D 92 (2015), no. 7, 075036, [1507.01590].
  * (30) S.-M. Choi, H. M. Lee, and M.-S. Seo, JHEP 04 (2017), 154, [1702.07860].
  * (31) P. Braat and M. Postma, JHEP 03 (2023), 216, [2301.04513].
  * (32) S. W. Randall, M. Markevitch, D. Clowe, A. H. Gonzalez, and M. Bradac, Astrophys. J. 679 (2008), 1173–1180, [0704.0261].
  * (33) A. Robertson, R. Massey, and V. Eke, Mon. Not. Roy. Astron. Soc. 465 (2017), no. 1, 569–587, [1605.04307].
  * (34) D. Wittman, N. Golovich, and W. A. Dawson, Astrophys. J. 869 (2018), no. 2, 104, [1701.05877].
  * (35) S.-M. Choi, H. M. Lee, P. Ko, and A. Natale, Phys. Rev. D 98 (2018), no. 1, 015034, [1801.07726].
  * (36) A. Berlin, N. Blinov, S. Gori, P. Schuster, and N. Toro, Phys. Rev. D 97 (2018), no. 5, 055033, [1801.05805].
  * (37) H. Beauchesne, E. Bertuzzo, and G. Grilli Di Cortona, JHEP 04 (2019), 118, [1809.10152].
  * (38) E. Bernreuther, J. C. Mejia, F. Kahlhoefer, M. Krämer, and P. Tunney, JHEP 04 (2021), 210, [2011.06604].
  * (39) K. Kawarabayashi and M. Suzuki, Phys. Rev. Lett. 16 (1966), 255.
  * (40) Riazuddin and Fayyazuddin, Phys. Rev. 147 (1966), 1071–1073.
  * (41) B. Holdom, Phys. Lett. B 166 (1986), 196–198.
  * (42) K. S. Babu, C. F. Kolda, and J. March-Russell, Phys. Rev. D 57 (1998), 6788–6792, [hep-ph/9710441].
  * (43) A. Katz, E. Salvioni, and B. Shakya, JHEP 10 (2020), 049, [2006.15148].
  * (44) S. Kulkarni, A. Maas, S. Mee, M. Nikolic, J. Pradler, and F. Zierler, SciPost Phys. 14 (2023), 044, [2202.05191].
  * (45) P. Maris and P. C. Tandy, Nucl. Phys. B Proc. Suppl. 161 (2006), 136–152, [nucl-th/0511017].
  * (46) M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev. 175 (1968), 2195–2199.
  * (47) M. Faber and R. Höllwieser, Prog. Part. Nucl. Phys. 97 (2017), 312–355, [1908.09740].
  * (48) G. P. Engel, L. Giusti, S. Lottini, and R. Sommer, Phys. Rev. D 91 (2015), no. 5, 054505, [1411.6386].
  * (49) C. Gattringer, C. Hagen, C. B. Lang, M. Limmer, D. Mohler, and A. Schafer, Phys. Rev. D 79 (2009), 054501, [0812.1681].
  * (50) L. Del Debbio, L. Giusti, M. Luscher, R. Petronzio, and N. Tantalo, JHEP 02 (2007), 056, [hep-lat/0610059].
  * (51) H. A. Weldon, Phys. Rev. D 28 (1983), 2007.
  * (52) T. Bringmann and J. Edsjö, _Darksusy 6.3: Freeze-in, out-of-equilibrium freeze-out, cosmic-ray upscattering and further new features_ , 2022.
  * (53) M. Drees, F. Hajkarim, and E. R. Schmitz, Journal of Cosmology and Astroparticle Physics 2015 (2015), no. 06, 025–025.
  * (54) Here we neglect the contribution of the dark vector mesons to the anomaly terms, which depend on additional unknown parameters and therefore introduce a further model dependence Choi:2018iit .
  * (55) J. Smirnov and J. F. Beacom, Phys. Rev. Lett. 125 (2020), no. 13, 131301, [2002.04038].
  * (56) A. Parikh, J. Smirnov, W. L. Xu, and B. Zhou, JHEP 08 (2023), 091, [2302.00008].
  * (57) To avoid confusion, we replace $E_{\pi_{D}}$ from above by $E_{1}$.
  * (58) We have confirmed through explicit calculation that this statement is true even for $m_{\rho_{D}}$ close to the two-pion threshold. The reason is that the cross section for pion scattering via ${\rho_{D}}$ meson exchange is finite in the limit $m_{\rho_{D}}\to 2m_{\pi_{D}}$ and $v_{\text{rel}}\to 0$.
  * (59) Note that this expression holds only for $y$ sufficiently close to unity, such that $v_{1}\ll 1$.
  * (60) R. Mertig, M. Bohm, and A. Denner, Comput. Phys. Commun. 64 (1991), 345–359.
  * (61) V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun. 207 (2016), 432–444, [1601.01167].
  * (62) V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun. 256 (2020), 107478, [2001.04407].

##  Appendix A Thermal width of dark pions

At zero temperature, the dark pions are stable, and hence their propagator is
given by

$D=\frac{1}{q^{2}-m_{\pi_{D}}^{2}}\;,$ (23)

which diverges for $q^{2}=m_{\pi_{D}}^{2}$. At finite temperature, this
expression is modified to

$D=\frac{1}{q^{2}-m_{\pi_{D}}^{2}-\Pi(q^{2})}\;,$ (24)

where $\Pi(q^{2})$ denotes the self-energy of the dark pion. The real part of
the self-energy amounts to a thermal mass, which shifts the position of the
pole of the propagator. For temperatures small compared to the dark pion mass
(as relevant for freeze-out calculations), this effect is negligible. The
imaginary part of the self-energy, on the other hand, provides a thermal
width, which regulates the divergence of the propagator at finite
temperatures. We can then write the propagator in the usual Breit-Wigner form

$D=\frac{1}{q^{2}-m_{\pi_{D}}^{2}+iE_{\pi_{D}}\Gamma_{\text{th}}}$ (25)

with

$\Gamma_{\text{th}}=-\frac{1}{E_{\pi_{D}}}\text{Im}\,\Pi\;.$ (26)

Note that the thermal width depends on the energy $E_{\pi_{D}}$ of the pion
relative to the plasma (i.e. in the cosmic rest frame).

The thermal width can be interpreted as the rate at which the phase space
distribution approaches equilibrium Weldon:1983jn . In other words,
$\Gamma_{\text{th}}$ is given by the interaction rate of the dark pion with
the plasma. Schematically,

$\displaystyle\text{Im}\,\Pi=-\frac{1}{2(1+f_{\pi_{D}}(E_{\pi_{D}}))}\sum_{X,Y}$
$\displaystyle\int\mathrm{d}\Pi_{X}\mathrm{d}\Pi_{Y}|\mathcal{M}_{\pi+X\to
Y}|^{2}$
$\displaystyle\times(2\pi)^{4}\delta^{4}\left(p_{\pi}+p_{X}-p_{Y}\right)$ (27)

where the sum is over all kinematically allowed processes and
$\mathrm{d}\Pi_{X,Y}$ denote the phase space volume multiplied with the
appropriate distribution function, i.e.

$\mathrm{d}\Pi_{X,i}=\frac{\mathrm{d}^{3}p_{i}}{2E_{i}(2\pi)^{3}}f(E_{i})$
(28)

for a particle in the initial state and

$\mathrm{d}\Pi_{Y,i}=\frac{\mathrm{d}^{3}p_{i}}{2E_{i}(2\pi)^{3}}(1\pm
f(E_{i}))$ (29)

for a boson (positive sign) or fermion (negative sign) in the final state.

Since the pion is the lightest particle of the theory, there are no $1\to n$
processes with real particles in the final state, and hence the leading
diagrams contributing to the thermal width are $2\to 2$ processes, i.e.
${\pi_{D}}{\pi_{D}}\to{\pi_{D}}{\pi_{D}}$ and
${\pi_{D}}{\rho_{D}}\to{\pi_{D}}{\rho_{D}}$. The latter is suppressed relative
to the former by the ratio of phase-space densities
$f_{\rho_{D}}(E_{i})/f_{\pi_{D}}(E_{i})$ and is therefore negligible. The
thermal width is then given by 222To avoid confusion, we replace $E_{\pi_{D}}$
from above by $E_{1}$.

$\Gamma_{\text{th}}(E_{1})=\int\frac{\mathrm{d}^{3}p_{2}}{(2\pi)^{3}}f_{\pi_{D}}(E_{2})\sigma({\pi_{D}}_{1}{\pi_{D}}_{2}\to{\pi_{D}}_{3}{\pi_{D}}_{4})v_{\text{rel}}\;,$
(30)

where we assume that the pions are non-relativistic, such that
$f_{\pi_{D}}(E)\ll 1$ and hence $1+f_{\pi_{D}}(E)\approx 1$. In this case we
recover the standard phase space integral for a $2\to 2$ cross section at zero
temperature.

To obtain the thermal width, we need to calculate the cross section of
${\pi_{D}}{\pi_{D}}\to{\pi_{D}}{\pi_{D}}$ scattering. The matrix element
obtains contributions from the four-pion-contact interaction, and from virtual
${\rho_{D}}$ exchange in $s$-, $t$\- and $u$-channel diagrams. The the latter
contributions are momentum-suppressed and hence subdominant in the non-
relativistic limit 333We have confirmed through explicit calculation that this
statement is true even for $m_{\rho_{D}}$ close to the two-pion threshold. The
reason is that the cross section for pion scattering via ${\rho_{D}}$ meson
exchange is finite in the limit $m_{\rho_{D}}\to 2m_{\pi_{D}}$ and
$v_{\text{rel}}\to 0$.. We can therefore approximate the total cross section
by the cross section for contact interaction

$\displaystyle\sigma_{\mathrm{c}}=\frac{1}{2\pi
s}\frac{1}{f_{\pi_{D}}^{4}}\Bigg{(}$
$\displaystyle\frac{3N_{f_{D}}^{4}-2N_{f_{D}}^{2}+6}{8N_{f_{D}}^{2}(N_{f_{D}}^{2}-1)}m_{\pi_{D}}^{4}\Bigg{.}$
$\displaystyle\Bigg{.}+\frac{N_{f_{D}}^{2}}{N_{f_{D}}^{2}-1}\left(m_{\pi_{D}}^{2}p^{2}+\frac{5}{6}p^{4}\right)\Bigg{)}\;,$
(31)

with $p$ denoting the momentum of each of the incoming pions in the center-of-
mass frame. In the non-relativistic limit we have

$\sigma_{\mathrm{c}}\approx\frac{1}{64\pi}\frac{3N_{f_{D}}^{4}-2N_{f_{D}}^{2}+6}{N_{f_{D}}^{2}(N_{f_{D}}^{2}-1)}\frac{m_{\pi_{D}}^{2}}{f_{\pi_{D}}^{4}}\;.$
(32)

For non-relativistic dark pions, the phase space distribution is given by

$f_{\pi_{D}}(E)=\left(N_{f_{D}}^{2}-1\right)e^{-\left(x+\tfrac{1}{2}xv^{2}\right)}\;,$
(33)

where $x=m_{\pi_{D}}/T$. We therefore need to calculate the integral

$I=\int\mathrm{d}^{3}v_{2}\,\sigma(v_{\text{rel}})v_{\text{rel}}e^{-xv_{2}^{2}/2}\;.$
(34)

To do so, we note that $\mathbf{v}_{2}=\mathbf{v}_{\text{rel}}+\mathbf{v}_{1}$
and obtain

$\displaystyle I$
$\displaystyle=\frac{\pi}{x^{2}}\sigma_{\text{c}}\left[4e^{-w_{1}^{2}/2}+\frac{2\sqrt{2\pi}(1+w_{1}^{2})\text{erf}(w_{1}/\sqrt{2})}{w_{1}}\right]$
$\displaystyle\approx\frac{8\pi}{x^{2}}\sigma_{\text{c}}\;,$ (35)

where we have defined $w_{1}=v_{1}\sqrt{x}$ and the second line holds for
$w_{1}\ll 1$.

When calculating the cross section for $3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}$, we
set the velocities of all initial state particles to zero. The velocity $v$ of
the virtual pion is therefore fixed by energy and momentum conservation:

$v_{1}=\frac{2\sqrt{4-5y+y^{2}}}{3}\;,$ (36)

where $y=m_{\rho_{D}}^{2}/(4m_{\pi_{D}}^{2})$ 444Note that this expression
holds only for $y$ sufficiently close to unity, such that $v_{1}\ll 1$..

Putting everything together, we then obtain Eq. (11). It is worth noting that,
due to the exponential suppression of pions in the plasma, the thermal width
is extremely small and rapidly decreases with temperature. Explicit
calculations show that for any of the points in parameter space that we
consider, the thermal width contributes less than a percent of the total cross
section, and around the time of freeze-out, at $x=x_{f}$, the contribution is
negligible.

##  Appendix B Matrix elements

Here we present the matrix elements corresponding to the
$3{\pi_{D}}\to{\pi_{D}}{\rho_{D}}$ annihilation diagrams shown in Fig. 1.

The matrix element for the contact-interaction contribution is given by

$\displaystyle
i\mathcal{M}=\frac{2ig}{3f_{\pi_{D}}}\epsilon^{\star}_{\mu}(p_{e})($
$\displaystyle p_{d}^{\mu}f_{abcde}+p_{a}^{\mu}f_{bcdae}$
$\displaystyle+p_{b}^{\mu}f_{cdabe}+p_{c}^{\mu}f_{dabce})\;,$ (37)

where $p_{a}$, $p_{b}$, $p_{c}$, $p_{d}$ denote the dark pion momenta, $p_{e}$
is the dark rho momentum, and all momenta are defined as incoming.
$\epsilon^{\star}_{\mu}(p_{e})$ denotes the polarization vector of the dark
rho. The group generators of $SU(N_{f_{D}})$ enter via

$\displaystyle f_{abcde}\equiv\,$ $\displaystyle
T_{abcde}+T_{acbde}+T_{bacde}+T_{bcade}+T_{cabde}+T_{cbade}\;,$ (38)

with

$\displaystyle T_{abcde}\equiv$ $\displaystyle
2\,\mathrm{Tr}[\,T^{a}T^{b}T^{e}T^{c}T^{d}]-2\,\mathrm{Tr}[T^{a}T^{e}T^{b}T^{c}T^{d}]-\mathrm{Tr}[T^{a}T^{b}T^{c}T^{e}T^{d}]+\mathrm{Tr}[T^{a}T^{b}T^{e}T^{c}T^{d}]$
$\displaystyle-\mathrm{Tr}[T^{a}T^{b}T^{d}T^{c}T^{e}]+\mathrm{Tr}[T^{a}T^{b}T^{d}T^{e}T^{c}]]\;.$
(39)

The matrix elements for the contributions with one 4${\pi_{D}}$ vertex and one
${\pi_{D}}{\pi_{D}}{\rho_{D}}$ vertex connected by an internal pion propagator
are given by

$\displaystyle i\mathcal{M}=$
$\displaystyle\frac{-2ig}{3f_{\pi_{D}}^{2}((p_{a}+p_{e})^{2}-m_{\pi_{D}}^{2}+i(E_{a}+E_{e})\Gamma_{\mathrm{th}})}\epsilon^{\star}_{\mu}(p_{e})$
$\displaystyle\times\left[p_{a}^{\mu}(\mathrm{Tr}[T^{a}T^{f}T^{e}]-\mathrm{Tr}[T^{a}T^{e}T^{f}])-(p_{a}^{\mu}+p_{e}^{\mu})(\mathrm{Tr}[T^{f}T^{a}T^{e}]-\mathrm{Tr}[T^{f}T^{e}T^{a}])\right]$
$\displaystyle\times\left[\left(-4p_{b}\cdot p_{d}-4(p_{a}+p_{e})\cdot
p_{c}+2p_{c}\cdot p_{d}+2p_{b}\cdot p_{c}+2(p_{a}+p_{e})\cdot
p_{b}+2p_{d}\cdot(p_{a}+p_{e})-4m_{\pi_{D}}^{2}\right)\mathrm{Tr}[T^{f}T^{b}T^{c}T^{d}]\right.$
$\displaystyle\quad+\left(-4p_{b}\cdot p_{c}-4(p_{a}+p_{e})\cdot
p_{d}+2p_{d}\cdot p_{c}+2p_{b}\cdot p_{d}+2(p_{a}+p_{e})\cdot
p_{b}+2p_{c}\cdot(p_{a}+p_{e})-4m_{\pi_{D}}^{2}\right)\mathrm{Tr}[T^{f}T^{b}T^{d}T^{c}]$
$\displaystyle\quad+\left(-4p_{c}\cdot p_{d}-4(p_{a}+p_{e})\cdot
p_{b}+2p_{b}\cdot p_{d}+2p_{c}\cdot p_{b}+2(p_{a}+p_{e})\cdot
p_{c}+2p_{d}\cdot(p_{a}+p_{e})-4m_{\pi_{D}}^{2}\right)\mathrm{Tr}[T^{f}T^{c}T^{b}T^{d}]$
$\displaystyle\quad+\left(-4p_{c}\cdot p_{b}-4(p_{a}+p_{e})\cdot
p_{d}+2p_{d}\cdot p_{b}+2p_{c}\cdot p_{d}+2(p_{a}+p_{e})\cdot
p_{c}+2p_{b}\cdot(p_{a}+p_{e})-4m_{\pi_{D}}^{2}\right)\mathrm{Tr}[T^{f}T^{c}T^{d}T^{b}]$
$\displaystyle\quad+\left(-4p_{d}\cdot p_{c}-4(p_{a}+p_{e})\cdot
p_{b}+2p_{b}\cdot p_{c}+2p_{d}\cdot p_{b}+2(p_{a}+p_{e})\cdot
p_{d}+2p_{c}\cdot(p_{a}+p_{e})-4m_{\pi_{D}}^{2}\right)\mathrm{Tr}[T^{f}T^{d}T^{b}T^{c}]$
$\displaystyle\quad+\left.\left(-4p_{d}\cdot p_{b}-4(p_{a}+p_{e})\cdot
p_{c}+2p_{c}\cdot p_{b}+2p_{d}\cdot p_{c}+2(p_{a}+p_{e})\cdot
p_{d}+2p_{b}\cdot(p_{a}+p_{e})-4m_{\pi_{D}}^{2}\right)\mathrm{Tr}[T^{f}T^{d}T^{c}T^{b}]\right]\,,$
(40)

and corresponding expressions with $(a\leftrightarrow b)$, $(a\leftrightarrow
c)$, and $(a\leftrightarrow d)$, with the thermal dark pion width
$\Gamma_{\mathrm{th}}$ as given in Eq. (11). To calculate squared matrix
elements we use the FeynCalc Mertig:1990an ; Shtabovenko:2016sxi ;
Shtabovenko:2020gxv  Mathematica package.