# Brown representability for triangulated categories with a linear action by a
graded ring

Janina C. Letz  Janina C. Letz
Fakultät für Mathematik
Universität Bielefeld
33501 Bielefeld
Germany<EMAIL_ADDRESS>
###### Abstract.

In this paper we give necessary and sufficient conditions for a functor to be
representable in a strongly generated triangulated category which has a linear
action by a graded ring, and we discuss some applications and examples.

###### Key words and phrases:

Brown representability, representable functor, triangulated category, Ext-
finite, graded ring

###### 2020 Mathematics Subject Classification:

18G80

Partly supported by the NSF grant DMS-1700985, and the Alexander von Humboldt
Foundation in the framework of an Alexander von Humboldt Professorship endowed
by the German Federal Ministry of Education and Research.

##  1\. Introduction

Every object $X$ in a category induces a contravariant functor into the
category of sets that sends an object $Y$ to the set
$\operatorname{Hom}(Y,X)$. Any functor that is naturally isomorphic to such a
functor is called _representable_. There are a number of results in various
settings, called Brown representability, when every ‘reasonable’ functor is
representable. The first such result is due to Brown; see [11].

The first Brown representability result for triangulated categories was
established by Neeman [19, Theorem 3.1]. The work on hand was motivated by
[10, Theorem 1.3] and [21, 4.3].

######  Theorem 2.7.

Let $R$ be a $\mathbb{Z}$-graded graded-commutative noetherian ring and
${\mathsf{T}}$ a graded $R$-linear triangulated category, that is strongly
generated, Ext-finite and idempotent complete. Then a graded $R$-linear
cohomological functor
$\mathsf{f}\colon{\mathsf{T}}^{op}\to\operatorname{grMod}(R)$ is graded
representable if and only if $\mathsf{f}$ only takes values in
$\operatorname{grmod}(R)$.

In contrast to the previous works, we characterize the _graded representable_
functors; those are the functors naturally isomorphic to

$\coprod_{d\in\mathbb{Z}}\operatorname{Hom}_{{\mathsf{T}}}(-,\Sigma^{d}X)$

for some object $X$. The result is proved in Section 2. Without the assumption
that $R$ is noetherian and ${\mathsf{T}}$ Ext-finite, we obtain necessary,
through not sufficient, conditions for a functor to be graded representable;
see Corollary 2.19.

The study of representability is motivated by the fact, that the
characterization of representable functors in a triangulated category
${\mathsf{T}}$ yields the existence of a right adjoint functor to a functor
${\mathsf{S}}\to{\mathsf{T}}$. For a nice discussion on this see [20,
Introduction]. In Section 3 we show the same holds in the graded setting.

Finally we discuss some examples where Theorem 2.7 yields new insight: When
$G$ is a finite group and $R$ a commutative noetherian ring, then
$\operatorname{D}_{b}(\operatorname{mod}(RG))$ is Ext-finite as a
$\operatorname{H}^{*}(G,R)$-linear category. In the second example we consider
the action of Hochschild cohomology $\operatorname{HH}^{*}(R/Q)$ on
$\operatorname{D}_{b}(\operatorname{mod}(R))$, when $Q$ is a regular ring and
$R=Q/(\bm{f})$ a quotient by a regular sequence.

##  2\. Representable Functors in the Graded Setting

Let ${\mathsf{T}}$ be a triangulated category with suspension functor
$\Sigma$.

######  2.1.

For objects $X$ and $Y$ in ${\mathsf{T}}$ we write

(2.1.1)
$\operatorname{Ext}^{*}_{{\mathsf{T}}}(X,Y)\colonequals\coprod_{d\in\mathbb{Z}}\operatorname{Hom}_{{\mathsf{T}}}(X,\Sigma^{d}Y)\,.$

When ${\mathsf{T}}=\operatorname{D}(R)$, the derived category of modules over
a ring $R$, and $X$ and $Y$ are $R$-modules viewed as objects in
$\operatorname{D}(R)$ via the natural embedding, then this coincides with the
classical Ext-groups.

######  2.2.

Let $R$ be a $\mathbb{Z}$-graded graded-commutative ring. This means $R$
decomposes as

$R=\coprod_{d\in\mathbb{Z}}R_{d}\,,$

and the Koszul sign rule holds

$rs=(-1)^{de}sr\quad\text{for }r\in R_{d}\text{ and }s\in R_{e}\,.$

We say $r\in R_{d}$ is an homogeneous element of degree $d$.

######  2.3.

A triangulated category ${\mathsf{T}}$ is _graded $R$-linear_ if

  1. (1)

for any objects $X$ and $Y$ in ${\mathsf{T}}$, the abelian group
$\operatorname{Ext}^{*}_{{\mathsf{T}}}(X,Y)$ is a graded $R$-module with the
grading given by the coproduct in Eq. 2.1.1, and

  2. (2)

composition is $R$-bilinear.

This data is equivalent to a ring homomorphism
$R\to\operatorname{Z}({\mathsf{T}})$, where

$\operatorname{Z}({\mathsf{T}})\colonequals\coprod_{d\in\mathbb{Z}}\left\\{\eta\colon\operatorname{id}_{\mathsf{T}}\to\Sigma^{d}\,\middle|\,\eta\Sigma=(-1)^{d}\Sigma\eta\right\\}$

is the _graded center of ${\mathsf{T}}$_. More precisely, a ring homomorphism
$\varphi\colon R\to\operatorname{Z}({\mathsf{T}})$ yields an $R$-action on
$\operatorname{Ext}^{*}_{{\mathsf{T}}}(X,Y)$ via

$\displaystyle
r\cdot-\colon\operatorname{Ext}^{*}_{{\mathsf{T}}}(X,Y)\to\operatorname{Ext}^{*}_{{\mathsf{T}}}(X,\Sigma^{d}Y)=\operatorname{Ext}^{*}_{{\mathsf{T}}}(X,Y)[d]\,,$
$\displaystyle f\mapsto(\Sigma^{d}f)\circ\varphi(r)_{X}=\varphi(r)_{Y}\circ f$

for any homogeneous element $r\in R$. Conversely, any homogeneous element
$r\in R$ yields a natural transformation $\eta$ given by

$\eta_{X}\colonequals r\cdot\operatorname{id}_{X}\colon X\to\Sigma^{|r|}X$

for any $X\in{\mathsf{T}}$. It is straightforward to check, that these
identifications are well-defined and mutually inverse. The graded center has
been studied in a number of works; for example [8, 13].

######  2.4.

We denote by $\operatorname{grMod}(R)$ the category of graded $R$-modules, and
by $\operatorname{grmod}(R)$ its full subcategory of finitely generated
$R$-modules. The $n$th shift $M[n]$ of a graded $R$-module $M$ is given by
$(M[n])_{d}=M_{n+d}$.

The suspension functor of a graded $R$-linear category ${\mathsf{T}}$ in the
first component of $\operatorname{Ext}^{*}_{{\mathsf{T}}}(-,-)$ corresponds to
the negative shift in $\operatorname{grMod}(R)$:

$\operatorname{Ext}^{*}_{{\mathsf{T}}}(\Sigma^{n}X,Y)\cong\operatorname{Ext}^{*}_{{\mathsf{T}}}(X,Y)[-n]\,.$

######  2.5.

A functor $\mathsf{f}\colon{\mathsf{T}}^{op}\to\operatorname{grMod}(R)$ is
_graded $R$-linear_, if

  1. (1)

the induced map
$\operatorname{Ext}^{*}_{{\mathsf{T}}}(X,Y)\to\operatorname{Ext}^{*}_{R}(\mathsf{f}(Y),\mathsf{f}(X))$
is a map of graded $R$-modules, and

  2. (2)

the suspension becomes the negative shift under $\mathsf{f}$, that is

$\mathsf{f}(\Sigma^{n}X)=\mathsf{f}(X)[-n]\,.$

The functor $\mathsf{f}$ is _cohomological_ if $\mathsf{f}$ applied to any
exact triangle yields a long exact sequence of graded $R$-modules.

Without explicitly stating we always assume that a natural transformation
between graded $R$-linear functors respects this structure.

######  Definition 2.6.

A functor ${\mathsf{T}}^{op}\to\operatorname{grMod}(R)$ is _graded
representable_ if it is naturally isomorphic to

$\mathsf{g}_{X}\colonequals\operatorname{Ext}^{*}_{{\mathsf{T}}}(-,X)\colon{\mathsf{T}}^{op}\to\operatorname{grMod}(R)$

for some object $X$ in ${\mathsf{T}}$.

When ${\mathsf{T}}$ is graded $R$-linear, then any graded representable
functor is graded $R$-linear.

A graded $R$-linear functor
$\mathsf{f}\colon{\mathsf{T}}^{op}\to\operatorname{grMod}(R)$ is graded
representable if and only if
$\mathsf{f}_{d}\colon{\mathsf{T}}^{op}\to\operatorname{Mod}(R_{0})$ is
representable for an(y) arbitrary integer $d$. The functors $\mathsf{f}_{d}$
are the degree $d$ part of $\mathsf{f}$, that is
$\mathsf{f}_{d}(X)\colonequals\mathsf{f}(X)_{d}$. Since $\mathsf{f}$ is graded
$R$-linear, the degree $d$ part $\mathsf{f}_{d}$ for an integer $d$ encodes
all the information of $\mathsf{f}$, that is

$\mathsf{f}_{d}(\Sigma^{e}X)=\mathsf{f}(\Sigma^{e}X)_{d}=\mathsf{f}(X)[-e]_{d}=\mathsf{f}(X)_{d-e}=\mathsf{f}_{d-e}(X)\,.$

######  Theorem 2.7.

Let $R$ be a $\mathbb{Z}$-graded graded-commutative noetherian ring and
${\mathsf{T}}$ a graded $R$-linear triangulated category, that is strongly
generated, Ext-finite and idempotent complete. Then a graded $R$-linear
cohomological functor
$\mathsf{f}\colon{\mathsf{T}}^{op}\to\operatorname{grMod}(R)$ is graded
representable if and only if $\mathsf{f}$ is locally finite.

Before we give a proof, we recall some definitions and properties:

######  2.8.

A $\mathbb{Z}$-graded ring $R$ is noetherian, if and only if $R_{0}$ is
noetherian and $R$ is finitely generated as an $R_{0}$-algebra; see for
example [15, Corollaire (2.1.5)] or [12, Theorem 1.5.5]. In particular, such a
ring is bounded below.

######  2.9.

A graded $R$-linear triangulated category ${\mathsf{T}}$ is _Ext-finite_ if
for all $X,Y\in{\mathsf{T}}$, the graded $R$-module
$\operatorname{Ext}^{*}_{{\mathsf{T}}}(X,Y)$ is finitely generated.

A triangulated category ${\mathsf{T}}$ is _idempotent complete_ if for every
object $X$ in ${\mathsf{T}}$ and every idempotent
$e\in\operatorname{End}_{\mathsf{T}}(X)$, that is $e^{2}=e$, there exists an
object $Y$ and maps

$i\colon Y\to X\quad\text{and}\quad p\colon X\to Y$

such that $p\circ i=\operatorname{id}_{Y}$ and $i\circ p=e$.

######  2.10.

A subcategory ${\mathsf{S}}\subseteq{\mathsf{T}}$ is _thick_ , if it is
triangulated and closed under retracts. Since the intersection of thick
subcategories is thick, there exists a smallest thick subcategory of
${\mathsf{T}}$ containing an object $G$, which we denote by
$\operatorname{thick}(G)$. We say $G$ _finitely builds_ an object $X$ in
${\mathsf{T}}$ when $X\in\operatorname{thick}(G)$.

There is an exhaustive filtration of $\operatorname{thick}_{R}(G)$: Let
$\operatorname{thick}^{1}(G)$ be the smallest full subcategory containing $G$
that is closed under finite coproducts, retracts and suspension. Then

$\operatorname{thick}^{n}(G)\colonequals\left\\{X\in{\mathsf{T}}\,\middle|\,\begin{gathered}\text{there
exists }X^{\prime}\in{\mathsf{T}}\text{ and an exact triangle }\\\ Y\to
X\oplus X^{\prime}\to Z\to\Sigma Y\\\ \text{ such that
}Y\in\operatorname{thick}^{n-1}(G)\text{ and
}Z\in\operatorname{thick}^{1}(G)\end{gathered}\right\\}\,.$

These are full subcategories and form an exhaustive filtration of
$\operatorname{thick}(G)$; cf. [10, 3]. In particular, if $X$ lies in
$\operatorname{thick}(G)$, then there exists an integer $n$, such that
$X\in\operatorname{thick}^{n}(G)$.

A triangulated category ${\mathsf{T}}$ is _strongly generated_ , if there
exists an object $G$ in ${\mathsf{T}}$ and a non-negative integer $n$, such
that ${\mathsf{T}}=\operatorname{thick}^{n}(G)$. The object $X$ is a _strong
generator_ of ${\mathsf{T}}$; cf. [21].

In the reminder of this section, we give a proof of Theorem 2.7. We fix a
$\mathbb{Z}$-graded graded-commutative ring $R$, a graded $R$-linear
triangulated category ${\mathsf{T}}$, and a graded $R$-linear cohomological
functor $\mathsf{f}\colon{\mathsf{T}}^{op}\to\operatorname{grMod}(R)$.

######  Lemma 2.11 (Graded version of Yoneda’s lemma).

For any $X\in{\mathsf{T}}$, the map

$\operatorname{Nat}(\mathsf{g}_{X},\mathsf{f})\to\mathsf{f}_{0}(X)\quad\text{given
by}\quad\eta\mapsto\eta(X)(\operatorname{id}_{X})$

is an isomorphism of abelian groups.

###### Proof.

For $u\in\mathsf{f}_{0}(X)$, we define a natural transformation

$\eta_{u}\colon\mathsf{g}_{X}\to\mathsf{f}\quad\text{as}\quad\eta_{u}(Y)(f)\colonequals\mathsf{f}(f)(u)$

where $Y\in{\mathsf{T}}$ and $f\in\operatorname{Ext}^{*}_{{\mathsf{T}}}(Y,X)$.
Since $u$ is a degree zero element, the map $\eta_{u}(Y)$ is homogeneous. It
is straightforward to verify that this is the inverse of the map in the claim
and both are maps of abelian groups. ∎

In particular, any morphism $f\colon X\to Y$ corresponds to a natural
transformation

$f_{*}\colon\mathsf{g}_{X}\to\mathsf{g}_{Y}$

given by post-composition.

######  2.12.

Adapting the definitions in [21, Section 4], a graded $R$-linear functor
$\mathsf{f}\colon{\mathsf{T}}^{op}\to\operatorname{grMod}(R)$ is

  * •

_locally finitely generated_ , if for every $X$ in ${\mathsf{T}}$ there exists
$Y$ in ${\mathsf{T}}$ and a natural transformation
$\zeta\colon\mathsf{g}_{Y}\to\mathsf{f}$ such that $\zeta(X)$ is surjective,

  * •

_locally finitely presented_ , if it is locally finitely generated and the
kernel of any natural transformation $\mathsf{g}_{Y}\to\mathsf{f}$ is locally
finitely generated, and

  * •

_locally finite_ , if $\mathsf{f}$ only takes values in
$\operatorname{grmod}(R)$.

When $\mathsf{f}\colon{\mathsf{T}}^{op}\to\operatorname{grMod}(R)$ is locally
finitely generated or locally finitely presented, then
$\mathsf{f}_{d}\colon{\mathsf{T}}^{op}\to\operatorname{Mod}(R_{0})$ is locally
finitely generated or locally finitely presented in the sense of [21, Section
4], respectively. The same need not hold for locally finite, for examples see
Section 3.

If ${\mathsf{T}}$ is Ext-finite, then any graded representable functor is
locally finite. Without the assumption that ${\mathsf{T}}$ is Ext-finite, we
can make the following statement:

######  Lemma 2.13.

Any graded representable functor is locally finitely presented.

###### Proof.

It is clear that a graded representable functor is locally finitely generated.
Let $\mathsf{g}_{X}$ be a graded representable functor, and
$\mathsf{g}_{Y}\to\mathsf{g}_{X}$ a natural transformation. By Yoneda’s Lemma
2.11, this corresponds to a morphism $Y\to X$. If we complete this to an exact
triangle $Z\to Y\to X\to\Sigma Z$, the sequence

$\mathsf{g}_{Z}\to\mathsf{g}_{Y}\to\mathsf{g}_{X}$

is exact on ${\mathsf{T}}$. In particular, the kernel of
$\mathsf{g}_{Y}\to\mathsf{g}_{X}$ is locally finitely generated. ∎

######  Lemma 2.14.

If $\mathsf{f}$ is locally finite, then $\mathsf{f}$ is locally finitely
generated.

###### Proof.

Let $X$ be an object in ${\mathsf{T}}$. Then the $R$-module $\mathsf{f}(X)$ is
finitely generated, and we can choose a finite set of homogeneous generators
$x_{1},\dots,x_{n}$ of $\mathsf{f}(X)$ in degrees $d_{1},\dots,d_{n}$. Set

$Y\colonequals\coprod_{j=1}^{n}\Sigma^{d_{j}}X\,.$

For every generator $x_{j}$, we obtain canonical maps

$\Sigma^{d_{j}}X\xrightarrow{i_{j}}Y\xrightarrow{p_{j}}\Sigma^{d_{j}}X$

whose composition is the identity map on $\Sigma^{d_{j}}X$. Let
$y\in\mathsf{f}(Y)$ be the canonical element, for which

$x_{j}=\mathsf{f}(i_{j})(y)\quad\text{for }1\leq j\leq n\,.$

Because of the suspensions introduced in the definition of $Y$, the element
$y$ is homogeneous of degree 0. By Yoneda’s lemma 2.11, the element $y$
corresponds to the natural transformation
$\zeta\colon\mathsf{g}_{Y}\to\mathsf{f}$ with
$\zeta(Y)(\operatorname{id}_{Y})=y$. Then $\zeta(y)(i_{j})=x_{j}$, and
$\zeta(X)$ is surjective. That is $\mathsf{f}$ is locally finitely generated.
∎

In general a locally finite functor need not be locally finitely presented.
This requires further assumptions on $R$ and ${\mathsf{T}}$:

######  Lemma 2.15.

If $R$ is noetherian and ${\mathsf{T}}$ Ext-finite, then a locally finite
functor $\mathsf{f}\colon{\mathsf{T}}^{op}\to\operatorname{grMod}(R)$ is
locally finitely presented.

###### Proof.

By Lemma 2.14 the functor $\mathsf{f}$ is locally finitely generated. Let
$\mathsf{g}_{Y}\to\mathsf{f}$ be a natural transformation. We set

$\mathsf{f}^{\prime}(X)\colonequals\ker(\mathsf{g}_{Y}(X)\to\mathsf{f}(X))\,.$

Since ${\mathsf{T}}$ is Ext-finite, the $R$-module $\mathsf{g}_{Y}(X)$ is
finitely generated. By assumption on $\mathsf{f}$, so is $\mathsf{f}(X)$.
Since $R$ is noetherian, the kernel $\mathsf{f}^{\prime}(X)$ is also finitely
generated. Thus $\mathsf{f}^{\prime}$ is a locally finite functor and by Lemma
2.14 it is locally finitely generated. In particular, $\mathsf{f}$ is locally
finitely presented. ∎

######  2.16.

Let $(\mathsf{f}_{i},\eta_{i})_{i>0}$ be a direct system of cohomological
functors $\mathsf{f}_{i}\colon{\mathsf{T}}\to{\mathsf{A}}$ where
${\mathsf{A}}$ an abelian category and natural transformations
$\eta_{i}\colon\mathsf{f}_{i}\to\mathsf{f}_{i+1}$. Following [21, 4.2.2], a
direct system $(\mathsf{f}_{i},\eta_{i})_{i>0}$ is _almost constant on_ a
subcategory ${\mathsf{S}}$ of ${\mathsf{T}}$, if for every $X\in{\mathsf{S}}$
the sequence

$0\to\ker(\eta_{i}(X))\to\mathsf{f}_{i}(X)\to\operatorname{colim}_{j}\mathsf{f}_{j}(X)\to
0$

is exact for all positive integers $i$.

A direct system $(X_{i},f_{i})_{i>0}$ of objects $X_{i}$ and morphisms
$f_{i}\colon X_{i}\to X_{i+1}$ in ${\mathsf{T}}$ is _almost constant on
${\mathsf{S}}$_, if the induced direct system of functors
$(\mathsf{g}_{X_{i}},(f_{i})_{*})_{i>0}$ is almost constant on ${\mathsf{S}}$.

For almost constant direct systems the following hold; see [21, Proposition
4.13].

######  Facts 2.17.

Let ${\mathsf{S}}\subseteq{\mathsf{T}}$ be a subcategory closed under
suspension, and $(\mathsf{f}_{i},\eta_{i})_{i>0}$ a direct system that is
almost constant on ${\mathsf{S}}$. Then

  1. (1)

$(\mathsf{f}_{ni+r})_{i\geqslant 0}$ is almost constant on
$\operatorname{thick}^{n}({\mathsf{S}})$ for any $r>0$, and

  2. (2)

$\mathsf{f}_{n+1}\to\operatorname{colim}\mathsf{f}_{i}$ is split surjective on
$\operatorname{thick}^{n}({\mathsf{S}})$.

If the functors $\mathsf{f}_{i}$ are graded $R$-linear, the assumption that
${\mathsf{S}}$ is closed under suspension is redundant.

######  Proposition 2.18.

Let ${\mathsf{T}}$ be a strongly generated, graded $R$-linear triangulated
category and $\mathsf{f}\colon{\mathsf{T}}^{op}\to\operatorname{grMod}(R)$ a
cohomological graded $R$-linear functor. Then $\mathsf{f}$ is locally finitely
presented if and only if $\mathsf{f}$ is a retract of a graded representable
functor.

###### Proof.

We assume $\mathsf{f}$ is locally finitely presented. Let $G\in{\mathsf{T}}$
be a strong generator of ${\mathsf{T}}$ with
$\operatorname{thick}^{d}(G)={\mathsf{T}}$. Then there exist
$A_{1}\in{\mathsf{T}}$ and a natural transformation
$\zeta_{1}\colon\mathsf{g}_{A_{1}}\to\mathsf{f}$ such that $\zeta_{1}(G)$ is
surjective. Inductively we construct a direct system

$\mathsf{g}_{A_{1}}\to\mathsf{g}_{A_{2}}\to\cdots$

with compatible natural transformations
$\zeta_{i}\colon\mathsf{g}_{A_{i}}\to\mathsf{f}$: Assume we have constructed
$A_{i}$ and $\zeta_{i}$ for $i\leq n$. Since $\mathsf{f}$ is locally finitely
presented, there exists

$\mathsf{g}_{B}\to\ker(\mathsf{g}_{A_{n}}\to\mathsf{f})$

that is surjective on $G$. This induces a natural transformation
$\mathsf{g}_{B}\to\mathsf{g}_{A_{n}}$, which by the graded version of Yoneda’s
Lemma 2.11 corresponds to a morphism $f\colon B\to A_{n}$. We complete this
morphism to an exact triangle

$B\to A_{n}\to A_{n+1}\to\Sigma B$

and apply $\mathsf{f}_{0}$, the degree 0 part of $\mathsf{f}$. By the graded
version of Yoneda’s Lemma 2.11 we obtain the exact sequence

$\operatorname{Nat}(\mathsf{g}_{B},\mathsf{f})\leftarrow\operatorname{Nat}(\mathsf{g}_{A_{n}},\mathsf{f})\leftarrow\operatorname{Nat}(\mathsf{g}_{A_{n+1}},\mathsf{f})\,.$

Thus by construction of $B$ there exists a natural transformation
$\zeta_{n+1}$ whose image is $\zeta_{n}$.

By this construction we have

$\ker(\mathsf{g}_{A_{n}}(G)\to\mathsf{f}(G))=\ker(\mathsf{g}_{A_{n}}(G)\to\mathsf{g}_{A_{n+1}}(G))\,.$

Using this and that $\zeta_{1}(G)$ is surjective, it is straightforward to
verify that the direct system is almost constant on $G$. Then the induced
natural transformation
$\operatorname{colim}_{i}\mathsf{g}_{A_{i}}\to\mathsf{f}$ is a natural
isomorphism. By 2.17, the natural transformation

$\mathsf{g}_{A_{d+1}}\to\operatorname{colim}_{i}\mathsf{g}_{A_{i}}\xrightarrow{\sim}\mathsf{f}$

is split surjective, and thus $\mathsf{f}$ is a retract of
$\mathsf{g}_{A_{d+1}}$ on ${\mathsf{T}}$.

For the converse direction, we assume $\mathsf{f}$ is the retract of
$\mathsf{g}_{X}$ for some object $X$. Then we have a canonical projection and
a canonical injection

$\mathsf{g}_{X}\to\mathsf{f}\quad\text{and}\quad\mathsf{f}\to\mathsf{g}_{X}\,,$

respectively. The canonical projection is surjective on ${\mathsf{T}}$, the
canonical injection is injective. In particular, the canonical projection
yields that $\mathsf{f}$ is locally finitely generated. Given a natural
transformation $\mathsf{g}_{Y}\to\mathsf{f}$, its kernel coincides with the
kernel of the composition $\mathsf{g}_{Y}\to\mathsf{f}\to\mathsf{g}_{X}$. By
Lemma 2.13, any representable functor is locally finitely presented, and thus
is $\mathsf{f}$. ∎

######  Corollary 2.19.

If ${\mathsf{T}}$ is additionally idempotent complete, then every locally
finitely presented functor is graded representable.

###### Proof.

Let $\mathsf{f}$ be a locally finitely presented functor. By Proposition 2.18,
it is a retract of a graded representable functor $\mathsf{g}_{X}$. Then the
natural transformation

$\mathsf{g}_{X}\to\mathsf{f}\to\mathsf{g}_{X}$

corresponds to an idempotent $e\colon X\to X$. Since ${\mathsf{T}}$ is
idempotent complete, there exists a retract of $Y$ of $X$, such that $e$
decomposes as the natural inclusion and projection morphism. Then
$\mathsf{f}\to\mathsf{g}_{X}\to\mathsf{g}_{Y}$ is a natural isomorphism, and
$\mathsf{f}$ is graded representable. ∎

###### Proof of Theorem 2.7.

Since ${\mathsf{T}}$ is Ext-finite, any graded representable functor is
locally finite. For the converse, we assume $\mathsf{f}$ is locally finite.
Since $R$ is noetherian and ${\mathsf{T}}$ Ext-finite, we can apply Lemma 2.15
to obtain that $\mathsf{f}$ is locally finitely presented. Then $\mathsf{f}$
is graded representable by Corollary 2.19. ∎

##  3\. Applications

### Adjoint Functors

As explained in [20, Introduction] there is a connection between representable
functors and adjoint functors. In our context we obtain the following:

Let $R$ be a $\mathbb{Z}$-graded graded-commutative ring. A functor
$\mathsf{f}\colon{\mathsf{S}}\to{\mathsf{T}}$ between $R$-linear graded
triangulated categories is _graded $R$-linear_, if it is exact and the induced
map

$\operatorname{Ext}^{*}_{{\mathsf{S}}}(X,Y)\to\operatorname{Ext}^{*}_{{\mathsf{T}}}(\mathsf{f}(X),\mathsf{f}(Y))$

is a map of graded $R$-modules.

######  Lemma 3.1.

Let $R$ be a $\mathbb{Z}$-graded graded-commutative ring, and ${\mathsf{S}}$,
${\mathsf{T}}$ graded $R$-linear triangulated categories. Suppose
${\mathsf{T}}$ is Ext-finite and every cohomological graded $R$-linear functor
${\mathsf{S}}^{op}\to\operatorname{grMod}(R)$, that is locally finite, is
graded representable. Then every graded $R$-linear functor
$\mathsf{f}\colon{\mathsf{S}}\to{\mathsf{T}}$ has a right adjoint.

###### Proof.

We adapt the proof of [20, Theorem 8.4.4]. Given $Y\in{\mathsf{T}}$ we define
a functor $\mathsf{h}\colon{\mathsf{S}}\to\operatorname{grMod}(R)$ by

$\mathsf{h}(-)\colonequals\operatorname{Ext}^{*}_{{\mathsf{T}}}(\mathsf{f}(-),Y)\,.$

This is a graded $R$-linear functor. Since ${\mathsf{T}}$ is Ext-finite, this
functor is locally finite. So by assumption ${\mathsf{h}}$ is graded
representable, that is there exists an object
$\mathsf{f}^{\prime}(Y)\in{\mathsf{S}}$, such that

$\operatorname{Ext}^{*}_{{\mathsf{T}}}(\mathsf{f}(-),Y)\cong\operatorname{Ext}^{*}_{{\mathsf{S}}}(-,\mathsf{f}^{\prime}(Y))\,.$

It remains to verify that $\mathsf{f}^{\prime}$ is a functor and this
isomorphism is natural in both components. Let $f\colon Y\to Z$ be a morphism
in ${\mathsf{T}}$. Then the induced map

$\operatorname{Ext}^{*}_{{\mathsf{S}}}(-,\mathsf{f}^{\prime}(Y))\to\operatorname{Ext}^{*}_{{\mathsf{S}}}(-,\mathsf{f}^{\prime}(Z))$

corresponds to a morphism $\mathsf{f}^{\prime}(Y)\to\mathsf{f}^{\prime}(Z)$ by
Yoneda’s Lemma 2.11. Thus $\mathsf{f}^{\prime}$ is a functor. The above
isomorphism is natural by construction. So $\mathsf{f}^{\prime}$ is a right
adjoint of $\mathsf{f}$. ∎

######  Corollary 3.2.

Let $R$ be a $\mathbb{Z}$-graded graded-commutative noetherian ring and
${\mathsf{S}}$, ${\mathsf{T}}$ Ext-finite graded $R$-linear triangulated
categories. Suppose ${\mathsf{S}}$ is strongly generated and idempotent
complete. Then every graded $R$-linear functor
$\mathsf{f}\colon{\mathsf{S}}\to{\mathsf{T}}$ has a right adjoint. ∎

### Derived Category

Let $R$ be a commutative noetherian ring and $A$ an $R$-algebra that is
finitely generated as an $R$-module. Then $A$ is noetherian; see for example
[18, Theorem 3.7]. The bounded derived category of finitely generated modules
over $A$, denoted by $\operatorname{D}_{b}(\operatorname{mod}(A))$, has a
canonical structure as an $R$-linear category, and the $R$-module
$\operatorname{Hom}_{\operatorname{D}_{b}(\operatorname{mod}(A))}(X,Y)=\operatorname{Ext}^{0}_{R}(X,Y)$
is finitely generated for any $X,Y$. In general, the category
$\operatorname{D}_{b}(\operatorname{mod}(A))$ need not be Ext-finite as an
$R$-linear category. By [6, Corollary 2.10], the category
$\operatorname{D}_{b}(\operatorname{mod}(A))$ is idempotent complete.

######  3.3.

In general the question whether $\operatorname{D}_{b}(\operatorname{mod}(A))$
is strongly generated is rather difficult. When $A$ is artinian, then
$\operatorname{D}_{b}(\operatorname{mod}(A))$ is strongly generated by [21,
Proposition 7.37]. When $A=R$ is a commutative notherian ring, then
$\operatorname{D}_{b}(\operatorname{mod}(R))$ is strongly generated when $R$
is either essentially of finite type over a field or over an
equicharacteristic excellent local ring; see [1, Main Theorem] and [16,
Corollary 7.2].

In the following we discuss two examples in which
$\operatorname{D}_{b}(\operatorname{mod}(A))$ is Ext-finite for some
cohomology ring connected to $A$.

### Finite Group over a Commutative Ring

We consider $A=RG$, the group algebra of a finite group $G$.

######  3.4.

The group cohomology of the group algebra $RG$ with coefficients in an
$RG$-complex $M$ is

$\operatorname{H}^{*}(G,M)\colonequals\operatorname{Ext}^{*}_{RG}(R,M)\,.$

When $M=R$ this is a $\mathbb{Z}$-graded graded-commutative ring, and every
$\operatorname{H}^{*}(G,M)$ is a graded $\operatorname{H}^{*}(G,R)$-module. In
particular, for $RG$-complexes $X$, $Y$ the identification

$\operatorname{Ext}^{*}_{\operatorname{D}_{b}(\operatorname{mod}(RG))}(X,Y)=\operatorname{Ext}^{*}_{RG}(X,Y)\cong\operatorname{H}^{*}(G,\operatorname{Hom}_{R}(X,Y))$

holds and the cohomology ring $\operatorname{H}^{*}(G,R)$ acts on any Ext-
module; see for example [9, Proposition 3.1.8]. So the bounded derived
category of finitely generated $RG$-modules
$\operatorname{D}_{b}(\operatorname{mod}(RG))$ is graded
$\operatorname{H}^{*}(G,R)$-linear.

######  3.5.

By [22, 14], the group cohomology ring $\operatorname{H}^{*}(G,R)$ is
noetherian, and $\operatorname{H}^{*}(G,M)$ is finitely generated over
$\operatorname{H}^{*}(G,R)$ for every finitely generated $RG$-module $M$. In
particular, the derived category
$\operatorname{D}_{b}(\operatorname{mod}(RG))$ is Ext-finite as a graded
$\operatorname{H}^{*}(G,R)$-linear triangulated category.

######  Corollary 3.6.

Let $R$ be a commutative notherian ring and $G$ a finite group. If
$\operatorname{D}_{b}(\operatorname{mod}(RG))$ is strongly generated, then a
graded $\operatorname{H}^{*}(G,R)$-linear functor

$\mathsf{f}\colon\operatorname{D}_{b}(\operatorname{mod}(RG))\to\operatorname{grMod}(\operatorname{H}^{*}(G,R))$

is graded representable if and only if $\mathsf{f}$ is locally finite. ∎

### Regular ring modulo a regular sequence

We consider $R=A$ a commutative noetherian ring.

######  3.7.

The category $\operatorname{D}_{b}(\operatorname{mod}(R))$ is Ext-finite over
$R$, if and only if the Ext-modules $\operatorname{Ext}^{*}_{R}(X,Y)$ are
bounded for all $X$ and $Y$ in $\operatorname{D}_{b}(\operatorname{mod}(R))$.
That is precisely when $R$ is regular: When $R$ is regular the Ext-modules are
bounded by definition. For the converse, for every $X$ in
$\operatorname{D}_{b}(\operatorname{mod}(R))$ the Ext-module
$\operatorname{Ext}^{*}_{R}(X,R/\mathfrak{p})$ is bounded and
$X_{\mathfrak{p}}$ has finite projective dimension for any prime ideal
$\mathfrak{p}$ of $R$. Then $X$ has finite projective dimension; see [7, Lemma
4.5] for modules, and [5, Theorem 4.1] and [17, Theorem 3.6] for complexes.

When $R$ is regular, the bounded derived category
$\operatorname{D}_{b}(\operatorname{mod}(R))$ is strongly generated if and
only if $R$ is a strong generator. The later holds precisely when $R$ has
finite global dimension, that is $R$ has finite Krull dimension. Then
Rouquier’s representability theorem [21, Corollary 4.18] applies.

######  3.8.

Suppose $R=Q/(\bm{f})$ is the quotient of a regular ring $Q$ by a regular
sequence $\bm{f}=f_{1},\ldots,f_{c}$. Then there exist cohomological operators
$\bm{\chi}=\chi_{1},\ldots,\chi_{c}$ in degree 2, such that for $X,Y$ in
$\operatorname{D}_{b}(\operatorname{mod}(R))$ the graded modules
$\operatorname{Ext}^{*}_{R}(X,Y)$ are finitely generated over the noetherian
graded ring $R[\bm{\chi}]$; see [4, Theorem (4.2)]. In particular, the
category $\operatorname{D}_{b}(\operatorname{mod}(R))$ is
$R[\bm{\chi}]$-linear and Ext-finite.

The ring of cohomological operators coincides with the Hochschild cohomology

$R[\bm{\chi}]\cong\operatorname{HH}^{*}(R/Q)\colonequals\operatorname{Ext}^{*}_{R\otimes^{\mathbf{L}}_{Q}R}(R,R)\,;$

see [2, Section 3].

######  Corollary 3.9.

Let $R=Q/(\bm{f})$ be the quotient of a regular ring $Q$ by a regular sequence
$\bm{f}=f_{1},\ldots,f_{c}$ with cohomological operators $\bm{\chi}$. If
$\operatorname{D}_{b}(\operatorname{mod}(R))$ is strongly generated, then any
graded $R[\bm{\chi}]$-linear functor
$\mathsf{f}\colon\operatorname{D}_{b}(\operatorname{mod}(R))\to\operatorname{grMod}(R[\bm{\chi}])$
is graded representable if and only if $\mathsf{f}$ is locally finite. ∎

######  3.10.

For Corollaries 3.6 and 3.9 it is crucial that the ring action on the derived
category is graded, since the Ext-modules need not be not bounded. In
particular, Corollaries 3.6 and 3.9 are not consequences of [21, 4.3], but
require Theorem 2.7.

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11institutetext: Department of Engineering Science, University of Oxford,
Oxford, UK
11email<EMAIL_ADDRESS>22institutetext: Department of
Oncology, University of Oxford, UK 33institutetext: Department of Medical
Physics and Clinical Engineering, Oxford University Hospitals NHS FT, Oxford,
UK 44institutetext: Department of Computing, Imperial College London, London,
UK 55institutetext: School of Computer Science, University of Birmingham,
Birmingham, UK

# Quality Control for Radiology Report Generation Models via Auxiliary
Auditing Components

Hermione Warr 11 Yasin Ibrahim 11 Daniel R. McGowan 2233
Konstantinos Kamnitsas 114455

###### Abstract

Automation of medical image interpretation could alleviate bottlenecks in
diagnostic workflows, and has become of particular interest in recent years
due to advancements in natural language processing. Great strides have been
made towards automated radiology report generation via AI, yet ensuring
clinical accuracy in generated reports is a significant challenge, hindering
deployment of such methods in clinical practice. In this work we propose a
quality control framework for assessing the reliability of AI-generated
radiology reports with respect to semantics of diagnostic importance using
modular auxiliary auditing components (ACs). Evaluating our pipeline on the
MIMIC-CXR dataset, our findings show that incorporating ACs in the form of
disease-classifiers can enable auditing that identifies more reliable reports,
resulting in higher F1 scores compared to unfiltered generated reports.
Additionally, leveraging the confidence of the AC labels further improves the
audit’s effectiveness. Code will be made available at:
https://github.com/hermionewarr/GenX_Report_Audit

###### Keywords:

Radiology Report Generation Error Detection Vision Language Modeling
Reliability

##  1 Introduction

Radiologists are facing mounting challenges managing the growing volume of
medical imaging data under resource constraints, necessitating more efficient
and innovative solutions for image analysis and interpretation. Automated (or
partly automated) radiology report generation is one such solution. Since the
introduction of transformers [28], which significantly enhanced language
generation, numerous studies have investigated radiology report generation
from medical images, particularly chest X-rays (CXR) [4, 9, 7, 15, 22, 26].
However, deployment of language models to medical applications faces a variety
of challenges, with a key area being the need for factual accuracy. Various
methods have been developed to address this - particularly useful have been
the introduction of clinical metrics to supplement the standard natural
language processing (NLP) metrics used to evaluate report generation quality
[4, 18]. Previous efforts to improve the quality of the generated reports have
incorporated extra information during training, such as bounding boxes for
improved localisation, or have aimed to refine diagnostic accuracy by
including loss functions that reward clinical accuracy [26, 5, 21, 7].
Additionally, more complex architectures, such as memory-augmented attention,
have been developed to facilitate better image understanding [21, 6].

Besides efforts to improve report generation accuracy, another line of work is
error detection in language modeling. Most work in this area has targeted the
natural language domain. The two main types of uncertainty estimation in large
language models (LLMs) are statistical uncertainty (SU) and in-dialogue
uncertainty (IDU) [27]. The latter relies on large pretrained models to admit
when they don’t know an answer [16, 13]. SU relates certainty to the model’s
output entropy, though in language modelling, it must address the coherence of
entire sentences, as well as on a per token basis [20, 14]. These methods
often depend on extensive labelled datasets in the natural language domain,
which do not adequately capture the specialised terminology and context of
clinical settings [27]. Few works have investigated the possibility of error
detection in reports written by radiologists [30, 29, 12, 8]. To our
knowledge, no previous work has investigated error detection for _automated_
report generation, which is the focus of our study.

We introduce an auditing framework for identifying potential errors in AI-
based radiology report generation, using modular auxiliary auditing components
(AC) for quality control. Our study focuses on chest-Xray (CXR) imaging,
defining ACs as image-based disease classifiers to extract semantics relevant
to diagnosis. We develop a report generation model, GenX, which performs
competitively with previous works and serves as basis for testing our auditing
framework. Reports are audited using ACs that provide disease classification
and confidence levels. Reports with consistent diagnoses across the pipeline
meet our criteria, while inconsistencies flag potential errors. Additionally,
by leveraging the ACs’ classification confidence, we enforce stricter quality
control, requiring consistency with the highest confidence predictions. Our
evaluation on MIMIC-CXR shows a significantly higher F1 score for reports
meeting the audit’s criteria. Results demonstrate that consistency between
report and auxiliary components is a promising auditing mechanism for
automated report generation.

##  2 Methodology

Figure 1: Proposed error detection pipeline of radiology report generation
using auxiliary auditing components. The standard report generation pipeline
($a$) is followed by the CheXbert labeler, $g_{T}$ ($b$) that extracts
pathology labels, $C_{T}$, from the reports, that are semantically meaningful
for clinical diagnosis. ($c$) Modular image-based audit models, AC, here
disease classifiers, predict disease class labels, $C_{I}$, based on image,
$I$. ($d$) If the labels predicted based on image ($C_{I}$) and report
($C_{T}$) are consistent, the report’s contents are deemed likely reliable. In
case of inconsistency, $C_{I}\neq C_{T}$, the report is flagged as less
reliable, potentially containing an error. If the image-based classifiers have
predictive confidence below threshold $t$, auditing can be deferred to user
due to uncertainty.

###  2.1 Report Generation Model - GenX

The proposed framework for auditing radiology report generation is shown in
Fig. 1. The backbone of the framework is the report generation model. The
generalised vision language modeling problem of generating a textual sequence
T given an input image I can be framed as estimating the conditional
distribution $p(T|I)$, as a product of the conditional probabilities:

$p(T|I)=\prod_{l=1}^{L}p(T_{l}\>|\>T_{<l},\>I;\>\theta),$ (1)

where $T$ is modelled as a sequence of word tokens $\\{T_{1},...,T_{L}\\}$,
with $T_{<l}$ being the set of tokens preceding $T_{l}$, where $L$ is the
number of tokens in the sequence, and $\theta$ the model parameters [22, 1].

An overview of our report generation model, GenX, is shown in Fig. 2. The
model is trained to maximise the probability of generating the target sequence
from the image, $p(T|I)$ from eq 1. Image features
$I_{feat}=f_{I}(I)\in\mathbb{R}^{D_{I}}$, with dimensionality $D_{I}$, are
extracted by image encoder $f_{I}$. A linear layer, $f_{I,emb}$, projects
$I_{feat}$ to the space of token embeddings that will be input to the language
generator, giving $I_{emb}=\\{I_{emb,1}\ ...\
I_{emb,M}\\}=f_{I,emb}(I_{feat})\in\mathbb{R}^{D_{T}\times M}$, a set of $M$
image token embeddings with dimensionality $D_{T}$, similarly to LLaVA [19]. A
tokeniser maps text to a set of $L$ token vectors
$\\{T_{l}\\}_{1:L}\in\mathbb{R}^{L\times V}$, where $V$ is the vocabulary
size. Each $T_{l}$ is embedded by a linear layer to
$T_{l,emb}=f_{T,emb}(T_{l})\in\mathbb{R}^{D_{T}}$. The concatenated embeddings
of image and text tokens $\\{I_{emb,1}\ ...\ I_{emb,M},T_{emb,1}\ ...\
T_{emb,L}\\}$ are passed to a Transformer decoder (Fig. 2b) which predicts a
probability distribution over $V$ vocabulary tokens, expressing which is more
probable in the sequence. The decoder has $N_{layers}$ with Attention blocks
that each has $N_{heads}$ parallel heads [28]. The model is trained using the
common paradigm of diagonal attention masking [28]. A report $T_{G}$ is
generated with auto-regressive inference, starting only with image token
embeddings $I_{emb}$ as input, then predicting the first and each following
word token as the one with highest predicted probability.

Figure 2: Illustration of one inference step for radiology report generation
(a). Embeddings of image and previously generated text tokens are passed to
the autoregressive language model ($b$) which outputs a probability
distribution over the vocabulary, to predict the next word in the sequence.

###  2.2 Auditing Generated Reports via Auxiliary Classifiers

While some types of mistakes in AI-generated radiology reports, such as
grammatical errors, are merely inconvenient, semantic mistakes related to
disease diagnosis can be detrimentalor patient care. Consider a semantic
concept of interest, denoted $c$. We focus on the case where $c$ represents a
specific disease, with the class label $C\\!\in\\!\\{1,0\\}$ indicating the
presence or absence of that disease. To audit whether a generated report
$T_{G}$ is semantically correct about $c$, we need to extract the value
$C_{T}$ described in text $T_{G}$ via a mapping function
$g_{T}\\!:\\!T_{G}\\!\rightarrow\\!c$. This can be any text-based classifier,
or prompt-based LLM that can answer questions in the form "Does this text
suggest evidence or not for disease $c$?", which have become widely available.
Herein we use a pretrained CheXbert text-classifier [25] as $g_{T}$, obtaining
$C_{T}=g_{T}(T_{G})$ (Fig 1b).

We then introduce to the framework an independent model
$g_{I}\\!:\\!I\\!\rightarrow\\!c$, which infers value $C_{I}$ for concept of
interest $c$ from image $I$ (Fig. 1c). We term such models auxiliary auditing
components (AC). In the examined case when $c$ is existence of specific
disease, $g_{I}$ is a classifier predicting $C_{I}\\!=\\!1$ if image $I$ shows
the disease or $C_{I}\\!=\\!0$ otherwise, with model confidence
$p_{AC}(c\\!=\\!C_{I}|I)$. If multiple concepts $c$ are of interest (e.g.
different diseases), an independent AC model can be used per $c$, for ease of
modular development, or a multi-label AC model.

Auditing is performed by comparing values $C_{T}$ and $C_{I}$ about the
concept $c$ of interest (e.g. disease) extracted from the report or the image
respectively (Fig 1d). To consider the contents of the report reliable with
respect to $c$, agreement between the report and AC is required, i.e.
$C_{T}\\!=\\!C_{I}$. We further extend this by taking into consideration the
confidence $p_{AC}$ of the AC. We only consider the auditing successful if it
satisfies the additional requirement that the value $C_{T}=C_{I}$ has been
inferred by the AC with confidence $p_{AC}$ over a pre-determined threshold
$t$. Formally, for a successfully completed auditing we require:

$(C_{I}=C_{T})\land(p_{AC}(c=C_{I}\ |\ I)\geq t),$ (2)

where $\land$ is the logical _and_ operator. The set of generated reports that
pass the audit for a particular concept $c$ is denoted by
$T_{G,pass}^{(c)}=\\{T_{G}|\ \text{Eq.~{}\ref{eq:condition} is }True\\}$. The
remainder, $T_{G,err}^{(c)}\\!=\\!\\{T_{G}|\ \text{Eq.~{}\ref{eq:condition} is
}False\\}$, can be deferred to the human user for review as potentially
inaccurate for that disease.

The framework is designed based on ACs that are modular, independent models.
The design draws inspiration from N-Version programming [3] and the principle
of redundancy for reliability in software engineering. Assuming independence
of components ($GenX$, $g_{T}$, $g_{I}$) with label error rates $e_{C_{T}}$
and $e_{C_{I}}$, the error rate of the auditing framework is $e=e_{C_{T}}\cdot
e_{C_{I}}$, which is lower than the individual error rates of $C_{T}$ and
$C_{I}$, as failure of auditing requires both components to fail
simultaneously to incorrectly satisfy Eq. 2. Modular ACs enable independent
component training and validation prior to integration into the audit
pipeline.

##  3 Experiments and Results

###  3.1 Data

Experiments are performed using MIMIC-CXR [11], containing chest X-rays (CXR),
associated radiology reports, and disease labels for $\sim$228,000 studies of
over 65,000 patients. As common practice, we consider only the frontal-view
images, and only the “findings” section of the reports, which describes
evidence of pathologies and support devices. The labels include 14 classes (12
diseases, 1 support devices, 1 no findings) and can be positive, uncertain,
negative, or not mentioned. We report 5 out of 14 classes separately
(Atelectasis, Cardiomegaly, Consolidation, Edema, Pleural Effusion) to
facilitate comparison with literature [10]. We divide data into splits for
training (226,261), validating hyper-parameters (1,864), and testing
performance (3,595). To train and evaluate the report generation, we further
exclude studies that do not include a “findings” section, resulting in 155,322
training, 1,231 validation and 2,607 testing samples. Finally, to help address
class imbalance and noisy labels, following past works [9, 10, 21, 26], we
consider uncertain disease labels as positive and class no-mention as
negative.

###  3.2 Evaluating the Report Generation Model

Experimental settings: We here show that our report generation model, $GenX$,
is of representative quality in comparison to previous work to serve as
effective backbone for the study of our auditing framework. To build GenX we
use ResNet-50 as the image encoder $f_{I}$, which we pre-train as a multi-
label classifier of the 14 classes described above. Its penultimate layer of
dimension $D_{I}=512$ is then projected to $M\\!=\\!10$ image token embeddings
with $D_{T}\\!=\\!512$. These are concatenated with up to $L\\!=\\!512$ text
embeddings, resulting from a GPT-2 tokeniser ($V\approx 50,257$) [24],
followed by a projection, $D_{T}\\!=\\!512$. The Transformer has
$N_{layers}\\!=\\!8$, $N_{heads}\\!=\\!8$, and dimension $D_{ff}\\!=\\!2048$
of its feed-forward layer. GenX is then trained end-to-end (Fig. 2), learning
parameters of the randomly initialised projections and Transformer, and fine-
tuning the $f_{I}$ encoder.

Table 1: Report generation performance of our model, GenX, compared to
previous works. To assess reports’ clinical correctness, we report macro and
micro averages of F1 score, recall (R) and precision (P), averaged over the 5
most commonly reported classes (Atelect., Cardiom., Consolid., Edema, Pleural
Eff. cf Sec. 3.1) and over all 14 classes, in format (5 / 14 classes), to
facilitate comparison with literature. Our model performs well in terms of F1
that quantifies clinical correctness, surpassed only by more complex
frameworks using reinforcement [7] or additional labels [21], demonstrating
that GenX is an effective backbone for our auditing study.

| Disease Classif. Metrics (Report) | NLP Metrics
---|---|---
Model | macro average | micro average
|  F1(5/14) |  R(5/14) |  P(5/14) |  F1(5/14) |  R(5/14) |  P(5/14) |  BL-1 |  M |  R-L
R2Gen [4] |  -/27.6 |  -/27.3 |  -/33.3 |  -/- |  -/- |  -/- |  35.3 |  14.2 |  27.7
Ratchet [9] |  38.8/- |  -/- |  -/- |  -/- |  -/- |  -/- |  23.2 |  10.1 |  24.0
$\mathcal{M}^{2}$ Trans [21] |  47.3/30.2 |  55.1/36.0 |  45.8/40.5 |  56.7/- |  65.1/- |  50.3/- |  - |  - |  -
RGRG [26] |  -/- |  -/- |  -/- |  54.7/- |  61.7/- |  49.1/- |  37.3 |  16.8 |  26.4
$A_{t}\texttt{+}$TE+RG[7] |  -/- |  -/- |  -/- |  -/53.7 |  -/49.6 |  -/58.5 |  49.0 |  21.3 |  40.6
GenX |  46.8/30.0 |  43.1/31.9 |  48.0/37.1 |  56.6/46.2 |  58.3/43.1 |  55.0/49.8 |  20.8 |  10.9 |  22.2

Results: Table 1 reports test performance by our report generation model,
along with previous works. NLP metrics BLEU (BL) [23], METEOR (M) [2] and
ROUGE-L (R-L) [17] assess generated language quality. More important for this
study is the assessment of diagnosis-related semantics in generated reports,
using disease-classification metrics such as F1 score, by comparing agreement
of class labels, $C_{T}$, extracted from generated and reference reports via
CheXbert [4]. Our report generator, GenX, achieves NLP metrics similar to
Ratchet [9], which has the most comparable architecture. Importantly, GenX
achieves a high F1 score in comparison to Ratchet, R2Gen [4], even RGRG [26]
that needs additional labelled bounding boxes for training. It approaches F1
score by $\mathcal{M}^{2}$ Trans [21] that trains using a more complex
framework and reinforcement, and is surpassed by $A_{t}\texttt{+}$TE+RG[7]
which uses extra labels for training. We conclude that GenX produces reports
of sufficient quality to serve as effective backbone for the following study
of our auditing method. Two examples of generated reports are shown in Table
2. The generated reports use realistic language, though they can occasionally
miss a diagnosis. Thus a framework that can extract and audit report semantics
is crucial.

Table 2:  Examples of CXRs and their associated radiology report alongside the
AI generated report from GenX.

X-ray |  Reference Report |  Generated Report
---|---|---
|  The lungs are low in volume but without focal consolidation. Diffuse opacities likely reflect mild pulmonary edema. There is no pleural effusion or pneumothorax. The cardiac size and cardiac silhouette are obscured by low lung volumes. The mediastinal and hilar contours appear unremarkable. |  The lung volumes are low. There is mild pulmonary edema. There is no focal consolidation, pleural effusion, or pneumothorax. The cardiomediastinal silhouette is normal.
|  The ET and NG tubes have been removed. A right PICC line terminates in the low SVC. Calcified left lung nodules are unchanged. The lungs are otherwise clear except for left basilar atelectasis. A small left pleural effusion has developed. Moderate cardiomegaly is unchanged. |  The heart is mildly enlarged. There is mild unfolding of the thoracic aorta. There is a moderate-sized hiatal hernia. Streaky left basilar opacities suggest minor atelectasis. Otherwise, the lungs appear clear. There is no definite pleural effusion or pneumothorax. Mild rightward convex curvature is centered along the mid thoracic spine.

###  3.3 Evaluating the Auditing Framework for Generated Reports

Experimental settings: We analyse whether including auxiliary auditing
components, ACs, enable us to quality control the generated reports and filter
out those that are more likely to contain errors. We evaluate on the 5
diseases most commonly reported in the literature (Sec. 3.1), from the
CheXpert competition [10]. For this, we first train 5 independent disease-
specific AC image classifiers ($AC_{1}$), where each is a ResNet-18. We then
apply the framework as described in Sec. 2.2 to the test-split. For each
image, the 5 $AC_{1}$ models predict 5 disease labels $C_{I}^{(c)}$, one per
disease $c$. GenX generates a report for each test image and we extract 5
report-based class labels $C_{T}^{(c)}$ via Chexbert, for each disease. For a
given scan, per disease $c$, we compare image and report based labels,
$C_{I}^{(c)}$ and $C_{T}^{(c)}$, to check if they fulfil Eq. 2. We perform the
check in two settings: First, without considering a confidence threshold for
$p_{AC}$, satisfying only the first condition in Eq. 2; Second, with a
confidence threshold $t=0.8$ where Eq. 2 is fulfilled if
$p_{AC}\\!\geq\\!0.8$, to investigate whether stricter auditing criteria
results in more accurate reports For each disease, the images that fulfil the
above criteria are considered to have successfully passed the audit. For these
images, we calculate average F1 score by comparing their report-based labels
$C_{T}^{(c)}$ with the reference report labels, to determine whether the
average score is higher than that of all unfiltered reports generated by GenX.
Finally, we repeat these experiments using a bigger ResNet-50 AC, trained as a
multi-label image classifier of all 14 classes ($AC_{14}$). This is to
evaluate whether it’s beneficial to integrate a larger AC that learns from all
14 labels available per scan, or independent, disease-specific ACs.

Table 3: Evaluation on test split of our auditing framework. For the 5 most
commonly reported classes in literature, $GenX$ column reports F1 score of all
generated reports ($C_{T}$ vs true label). $AC_{1}$ assesses image-based
classification by 5 disease-specific ACs ($C_{I}$ vs true label).
$GenX\\!+\\!AC_{1}^{t=0}$ and $GenX\\!+\\!AC^{t=0.8}_{1}$ show semantic
factuality of generated reports that passed auditing for the specific disease
against $AC_{1}$, satisfying Eq. 2 without the requirement for confidence
$p_{AC}\\!\geq\\!t$ and when confidence over $t=0.8$ is required,
respectively. Bold shows improvement by auditing over baseline GenX.
Parentheses show percentage of reports that satisfy the auditing per disease.
We also report results when auditing with a single multi-label AC trained for
all 14 classes ($AC_{14}$ and $GenX\\!+\\!AC_{14}^{t=0}$). Developing
independent ACs per disease is easier and more effective in practice.

| F1 score (AC1) | F1 score (AC14)
---|---|---
Disease |  AC1 |  GenX |  GenX+AC${}^{t=0}_{1}$ |  GenX+AC${}^{t=0.8}_{1}$ |  AC14 |  GenX+AC${}^{t=0}_{14}$
Atelectasis |  55.6 |  42.2 |  52.6 (72) |  51.7 (30) |  37.1 |  31.8 (77)
Cardiomegaly |  57.6 |  66.8 |  73.7 (78) |  81.3 (34) |  40.3 |  53.3 (64)
Consolidation |  23.5 |  5.6 |  5.8 (76) |  1.6 (65) |  15.9 |  2.1 (89)
Edema |  62.1 |  55.6 |  64.6 (80) |  74.1 (45) |  53.1 |  60.2 (75)
Pleural Effusion |  71.9 |  63.8 |  74.7 (80) |  83.5 (57) |  61.8 |  64.1 (82)
Average |  54.1 |  46.8 |  54.3 |  58.4 |  41.6 |  42.3

Results: Table 3 shows the results. Class labels $C_{T}$ extracted from
reports generated by GenX achieve an average F1 score of $46.8$ over the 5
diseases. Reports that agree with $AC_{1}$, fulfilling the auditing without
the extra requirement for confidence $p_{AC\\!}\geq\\!t$, achieve average F1
score $54.3$. These benefits are realised over a significant percentage of the
reports (72-80% across the diseases), while the rest would be deferred for
user inspection and error correction in an actual workflow. Reports that pass
the stricter auditing and additionally fulfil the confidence requirement
$p_{AC}\\!\geq\\!0.8$ present even less errors, with average F1 score $58.4$.
This restricts the number of reports that pass the audit (30-65%), in return
for higher reliability. In comparison to the unfiltered reports of GenX, the
reports that fulfil the audit show increased F1-scores for 4 out of 5 studied
diseases. The exception is Consolidation, the class where both GenX and AC
image classifiers have very low performance. Interestingly, for 3 out of 5
diseases, the reports that successfully pass the audit show F1-scores that are
even higher than the auditing $AC_{1}$ classifiers. These results show that
measuring consistency between semantics of reports and auditing classifiers is
a promising framework.

Finally, we find that a single, multi-label classifier, trained for all
14-classes as $AC_{14}$, achieves significantly lower F1 scores for image
classification, which also leads to lower effectiveness of the auditing.
Although $AC_{14}$ is a ResNet-50, larger than ResNet-18s used for the 5
$AC_{1}$ classifiers, it is difficult to develop it at the level of disease-
specific classifiers. This demonstrates the benefits of modularity, with
independently developed AC for each concept of interest.

##  4 Conclusion

We introduced a framework to quality control AI-generated radiology reports
with respect to semantics of interest via auxiliary auditing components. Our
study focused on the task of report generation from chest-Xrays, detecting
potential semantic errors in diagnosis-related report content, using auxiliary
disease classifiers and assessing consistency between labels inferred from
image and report. Experiments show that reports fulfilling the auditing
criteria exhibit fewer errors, with average F1 score that can even exceed the
score of the auditing components. The framework is generic and future work
could explore it for auditing semantic concepts beyond disease classification,
such as for description of pathology location or volume, using regression
models as ACs for such properties.

The limitation of the framework, as with any filtering pipeline for quality-
control, is the inherent trade-off between reducing the number of reports that
successfully pass the audit and can be used for down-stream workflow, in
exchange for their higher reliability. Future work could improve performance
in both aspects by integrating more potent models for inferring image-based
and report-based labels, thereby ensuring greater consistency and reliability
from their comparison.

##  5 Acknowledgements

Hermione Warr and Yasin Ibrahim are supported by the EPSRC Centre for Doctoral
Training in Health Data Science (EP/S02428X/1).

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# EXPLAINING THE CABIBBO ANGLE ANOMALY

A. CRIVELLIN1,2 1Paul Scherrer Institut, CH–5232 Villigen PSI, Switzerland
2Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH–8057 Zürich,
Switzerland

###### Abstract

The determinations of the Cabibbo angle from kaon, pion, $D$ and tau decays
disagree with the one from super-allowed beta decays. The resulting $\approx
3\,\sigma$ deficit in first row (but also first column) CKM unitarity is known
as the Cabibbo angle anomaly (CAA). Determining $V_{ud}$ from beta decays
requires knowledge of the Fermi constant $G_{F}$, usually extracted from muon
decay. However, because new physics might also affect muon decay, and thus the
determination of the Fermi constant, an interesting interplay with the global
electroweak fit arises where $G_{F}$ is a crucial input. In these proceedings
we review the CAA and its different interpretations in terms of physics beyond
the SM, first discussing the corresponding effect in the effective field
theory and then using (simplified) models.

##  1 Introduction

The Cabibbo angle parametrizes, up to higher order corrections in the
Wolfenstein parameter, the mixing among the first two generations of quarks
and therefore determines the CKM elements $V_{us}$ and $V_{ud}$. It can be
measured in kaon, pion, tau and beta decays. Concerning the latter, super-
allowed beta decays provide the most precise determination of the Cabibbo
angle, however, its values disagrees with the other determinations. The
significance of this tension crucially depends on the radiative corrections
applied to $\beta$ decays [1], but also on the treatment of tensions between
$K_{\ell 2}$ and $K_{\ell 3}$ decays [2] and the constraints from $\tau$
decays [3], see Ref. [4] for more details. However, in the end, an $\approx
3\,\sigma$ deficit in first row (and less significant in first column) CKM
unitarity [5]

$\displaystyle\big{|}V_{ud}\big{|}^{2}+\big{|}V_{us}\big{|}^{2}+\big{|}V_{ub}\big{|}^{2}=0.9985(5)\,,\qquad\big{|}V_{ud}\big{|}^{2}+\big{|}V_{cd}\big{|}^{2}+\big{|}V_{td}\big{|}^{2}=0.9970(18)\,,$
(1)

seems reasonable. This situation is shown in the left plot of Fig. 1 [6].

As these unitarity relations are dominated by $V_{ud}$, and since tau, pion
and kaon and $D$ decays would require different kinds of new physics
contributions, this suggests that NP should enter the determination of
$V_{ud}$, assuming that the anomaly has, in fact, a beyond the Standard Model
(BSM) origin. Furthermore, as shown in Ref. [4], the sensitivity to a BSM
effect in (super-allowed) $\beta$ decays is enhanced by a factor
$|V_{ud}|^{2}/|V_{us}|^{2}$ compared to kaon, $\tau$, or $D$ decays.

Importantly, for determining the Cabibbo angle from (super-allowed) beta
decays, knowledge of the Fermi constant $G_{F}$, i.e. the effective weak force
at low energies, is necessary. Even though $G_{F}$ is extremely precisely
measured via muon decay [7]

$G_{F}^{\mu}=1.1663787(6)\times 10^{-5}\,\text{GeV}^{-2}\,,$ (2)

at the level of $0.5\,\text{ppm}$, it is not necessarily free of BSM
contributions. In fact, as for any measurement, one can only conclude the
presence or absence of BSM effects by comparing $G_{F}^{\mu}$ to another
independent determination. Therefore, the CAA could not only be explained via
a BSM effect in beta decays, i.e. $d\to ue\nu$ transitions, but also via an
effect in muon decay ($\mu\to e\nu\nu$), changing the value of the Fermi
constant.

Figure 1: Left: Different determinations of $V_{us}$ and $V_{ud}$ leading to
the CAA [6]. Right: Values of $G_{F}$ from the different determinations [8].

In order to determine if muon decay and/or beta decays subsume BSM effects,
one can determine the Fermi constant from the EW precision observables [9]. In
our global fit [8], including the $W$ mass, $\sin^{2}\theta_{W}$, and $Z$-pole
observables [10] etc., we find

$G_{F}^{\text{EW}}\Big{|}_{\text{full}}=1.16716(39)\times
10^{-5}\,\text{GeV}^{-2}\,.$ (3)

As depicted in the right plot of Fig. 1, this value lies above $G_{F}^{\mu}$
by $\approx 2\sigma$, reflecting the known tensions within the EW fit [11,
12]. For comparison with Ref. [9], if one considered only
$\sin^{2}\theta_{W}$, one would obtain

$G_{F}^{\text{EW}}\Big{|}_{\text{minimal}}=1.16728(83)\times
10^{-5}\,\text{GeV}^{-2},$ (4)

consistent with Eq. (3), but with a larger uncertainty. The pull of
$G_{F}^{\text{EW}}$ away from $G_{F}^{\mu}$ is mainly driven by $M_{W}$,
$\sin^{2}\theta_{W}$ from the hadron colliders, $A_{\ell}$, and $A_{\rm
FB}^{0,\ell}$.

We can also translate the CAA into a determination of the Fermi constant by
requiring that it takes the value which restores CKM unitarity:

$G_{F}^{\text{CKM}}=0.99925(25)\times G_{F}^{\mu}=1.16550(29)\times
10^{-5}\,\text{GeV}^{-2}\,.$ (5)

Comparing the three independent determinations of $G_{F}$ in right plot of
Fig. 1, one finds a situation in which $G_{F}^{\text{EW}}$ lies above
$G_{F}^{\mu}$ by $2\sigma$, $G_{F}^{\text{CKM}}$ below $G_{F}^{\mu}$ by
$3\sigma$, and the tension between $G_{F}^{\text{EW}}$ and
$G_{F}^{\text{CKM}}$ amounts to $3.4\sigma$. Therefore, if one wants to
explain the CAA, i.e. the disagreement between $G_{F}^{\rm CKM}$ and
$G_{F}^{\mu}$ by NP in muon decay only, one increases the tension with the EW.
Therefore, models that give a direct effect in beta decays and/or an effect in
the EW fit are welcome.

In these proceedings, we first review the possible solutions to the CAA in
terms of effective operators before we briefly discuss (simplified) new
physics models.

Figure 2: Possible explanations of the CAA in terms of effective operators. a)
and b) change directly the determination of $V_{ud}$ from beta decays while c)
and d) lead to an effect in the Fermi constant that changes indirectly the
values of $V_{ud}$ extracted from (super-allowed) beta decays.

##  2 SMEFT analysis

Among the set of gauge-invariant dimension-$6$ operators [13, 14], there are
four classes that affect muon and/or beta decays [15, 16, 17, 18] and
therefore have the potential to explain the CAA:

  * a)

4-fermion operators affecting beta decays ($u\to de\nu$)

  * b)

Operators generating modified $W$–$u$–$d$ coupling,

  * c)

4-fermion operators affecting muon decays ($\mu\to e\nu\nu$)

  * d)

Operators generating modified $W$–$\mu$–$\nu$ couplings

This is illustrated in Fig. 2.

###  2.1 Four-fermion operators in ${d\to ue\nu}$

The operators $Q_{\ell equ}^{\left(1\right)1111}$ and $Q_{\ell
equ}^{\left(3\right)1111}$ give rise to $d\to ue\nu$ scalar amplitudes which
lead to enhanced effects in $\pi\to\mu\nu/\pi\to e\nu$ (w.r.t $\beta$ decays)
and therefore can only have a negligible impact on $V_{ud}$. Furthermore, the
tensor amplitude generated by $Q_{\ell equ}^{\left(3\right)ijkl}$ has a
suppressed matrix element in $\beta$ decays.

For $Q_{\ell q}^{\left(3\right)1111}$, the CAA prefers $C_{\ell
q}^{\left(3\right)1111}\approx 0.7\times 10^{-3}G_{F}$. Via $SU(2)_{L}$
invariance, this operator generates neutral currents

${{\cal L}_{{\rm{NC}}}}=C_{\ell
q}^{\left(3\right)1111}\left({\bar{d}{\gamma^{\mu}}{P_{L}}d-\bar{u}{\gamma^{\mu}}{P_{L}}u}\right)\bar{e}{\gamma_{\mu}}{P_{L}}e,$
(6)

leading to effects in non-resonant di-electron searches at the LHC.
Interestingly, CMS found an excess of such high-energetic electrons [24] and
also ALTAS observed more events than expected [25], as predicted by an
explanation of the CAA via this operator [26]. This can be seen Fig. 3 where
also $R(\pi)$, testable at the future PIONEER experiment [27], is shown.

Figure 3:  $\Delta\chi^{2}=\chi^{2}-\chi^{2}_{SM}$ as a function of the Wilson
coefficient $[C_{\ell q}^{(3)}]_{1111}$ for $R(\pi)$ (blue), the CAA (green),
the CMS analysis of di-lepton pairs (red) and the combination of them (black)
with the dark orange region showing the $1\sigma$ and $2\sigma$ regions of the
combined fit. The hatched region is excluded at 95% CL from the non-resonant
di-lepton search of ATLAS (not included in the $\chi^{2}$ function). The best
fit point for the combined fit (black) is at $[C_{\ell q}^{(3)}]_{1111}\approx
1.1/(10\,{\rm TeV})^{2}$ where $\chi^{2}-\chi^{2}_{SM}\approx-20$.

###  2.2 Modified ${W}-{u}-{d}$ couplings

There are only two operators modifying $W$ couplings to quarks

$\displaystyle Q_{\phi q}^{\left(3\right)ij}$
$\displaystyle={\phi^{\dagger}}i\overset{\leftrightarrow}{D}^{I}_{\mu}\phi{{\bar{q}}_{i}}{\gamma^{\mu}}{\tau^{I}}{q_{j}},$
$\displaystyle Q_{\phi ud}^{ij}$
$\displaystyle={\phi^{\dagger}}i\overset{\leftrightarrow}{D}_{\mu}\phi{{\bar{u}}_{i}}{\gamma^{\mu}}{d_{j}}.$
(7)

First of all, $Q_{\phi ud}^{ij}$ generates right-handed $W$–quark couplings.
$Q_{\phi q}^{\left(3\right)ij}$ generates modifications of the left-handed
$W$–quark couplings and data prefer

$C_{\phi q}^{\left(3\right)11}\approx 0.7\times 10^{-3}G_{F}.$ (8)

Due to $SU(2)_{L}$ invariance, in general effects in $D^{0}$–$\bar{D}^{0}$ and
$K^{0}$–$\bar{K}^{0}$ mixing are generated. However, in case of alignment with
the down-sector, the effect in $D^{0}$–$\bar{D}^{0}$ is smaller than the
experimental value and thus not constraining as the SM prediction cannot be
reliably calculated.

###  2.3 Four-fermion operators in ${\mu\to e\nu\nu}$

Disregarding flavor indices, two operators can generate a charged current
involving four leptons:

$\displaystyle
Q_{\ell\ell}^{ijkl}={{\bar{\ell}}_{i}}{\gamma^{\mu}}{\ell_{j}}{{\bar{\ell}}_{k}}{\gamma^{\mu}}{\ell_{l}},\qquad
Q_{\ell e}^{ijkl}$
$\displaystyle={{\bar{\ell}}_{i}}{\gamma^{\mu}}{\ell_{j}}{{\bar{e}}_{k}}{\gamma^{\mu}}{e_{l}}.$
(9)

Note that for $Q_{\ell\ell}^{ijkl}$, e.g. the identities
$Q_{\ell\ell}^{ijkl}=Q_{\ell\ell}^{klij}=Q_{\ell\ell}^{ilkj}=Q_{\ell\ell}^{kjil}$
hold. Taking this into account, we have 9 independent operators that
contribute to ${\mu\to e\nu\nu}$ at tree-level [8]:

  1. 1.

$Q_{\ell\ell}^{2112}$ gives rise to a real and SM-like amplitude. Therefore,
it can give a constructive or destructive effect in the muon lifetime and does
not affect the Michel parameters [19]. In order explain the CAA at the
$1\sigma$ level we need $C_{\ell\ell}^{2112}\approx-1.4\times 10^{-3}G_{F}$.
This Wilson coefficient is constrained by LEP searches for
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ [10]

$-\frac{4\pi}{(9.8\,\text{TeV})^{2}}<C_{\ell\ell}^{1221}<\frac{4\pi}{(12.2\,\text{TeV})^{2}},$
(10)

such that a solution of the CAA is possible.

  2. 2.

Even though $Q_{\ell e}^{2112}$ possess a vectorial Dirac structure, it leads
to a scalar amplitude after using Fierz identities. Its interference with the
SM is usually expressed via the Michel parameter $\eta=\text{Re}\,C_{\ell
e}^{2112}/(2\sqrt{2}G_{F})$, leading to a correction $1-2\eta m_{e}/m_{\mu}$.
The shift in $G_{F}^{\mu}$ is constrained to be at most $0.68\times 10^{-4}$
[20], well below the required effect for solving the CAA.

  3. 3.

The operators $Q_{\ell\ell(e)}^{1212}$ can contribute to muon decay since the
neutrino flavors are not detected. To explain the CAA we need
$|C_{\ell\ell}^{1212}|\approx 0.045\,G_{F}$ or $|C_{\ell e}^{1212}|\approx
0.09\,G_{F}$, such that both solutions are excluded by muonium–anti-muonium
oscillations ($M=\mu^{+}e^{-}$) [21].

  4. 4.

For $Q_{\ell\ell(e)}^{1112}$ again numerical values of
$|C_{\ell\ell(e)}^{1112}|\approx 0.09\,G_{F}$ are preferred (as for all the
remaining Wilson coefficients in this list). Both operators give tree-level
effects in $\mu\to 3e$, e.g.,

$\text{Br}\left[{\mu\to
3e}\right]=\frac{{m_{\mu}^{5}\tau_{\mu}}}{{768{\pi^{3}}}}{\left|{C_{\ell\ell}^{1112}}\right|^{2}}=0.25\bigg{|}\frac{C_{\ell\ell}^{1112}}{G_{F}}\bigg{|}^{2},$
(11)

which exceeds the experimental limit on the branching ratio of $1.0\times
10^{-12}$ [22] by orders of magnitude (the result for $C_{\ell e}^{1112}$ is
smaller by a factor $1/2$).

  5. 5.

The operators $Q_{\ell\ell(e)}^{2212}$ and $Q_{\ell\ell(e)}^{3312}$ contribute
at the one-loop level to $\mu\to e$ conversion and $\mu\to 3e$ and at the two-
loop level to $\mu\to e\gamma$ [23]. Here the current best bounds come from
$\mu\to e$ conversion. Using Table 3 in Ref. [23] we have

$\displaystyle\left|{C_{\ell\ell}^{3312}}\right|<6.4\times
10^{-5}G_{F},\qquad\left|{C_{\ell\ell}^{2212}}\right|<2.8\times 10^{-5}G_{F},$
(12)

again excluding a sizable BSM effect. Similarly, $Q_{\ell e}^{3312}$ and
$Q_{\ell e}^{2212}$ cannot provide a viable solution of the CAA.

  6. 6.

$Q_{\ell\ell(e)}^{2312}$, $Q_{\ell\ell(e)}^{3212}$, $Q_{\ell\ell(e)}^{1312}$,
and $Q_{\ell\ell(e)}^{3112}$ contribute to $\tau\to\mu\mu e$ and $\tau\to\mu
ee$, respectively, which forbid a sizable effect in analogy to $\mu\to 3e$.

###  2.4 Modified ${W}$–${\mu}$–${\nu}$ couplings

Only the operator

$\displaystyle
Q_{\phi\ell}^{(3)ij}={\phi^{\dagger}}i\overset{\leftrightarrow}{D}^{I}_{\mu}\phi{{\bar{\ell}}_{i}}{\gamma^{\mu}}{\tau^{I}}{\ell_{j}}$
(13)

generates modified $W$–$\ell$–$\nu$ couplings at tree level. In order to avoid
the stringent bounds from charged lepton flavor violation, the off-diagonal
Wilson coefficients, in particular $C_{\phi\ell}^{(3)12}$, must be very small.
Since they also do not generate amplitudes interfering with the SM ones, their
effect can be neglected. Because $C_{\phi\ell}^{(3)11}$ affects $G_{F}^{\mu}$
and $G_{F}^{\text{CKM}}$ in the same way and thus has no effect in the CAA.
Therefore, only $C_{\phi\ell}^{(3)22}>0$ can explain the CAA while
$C_{\phi\ell}^{(3)22}>0$, and $|C_{\phi\ell}^{(3)22}|<|C_{\phi\ell}^{(3)11}|$
are preferred in this case by tests of lepton flavor universality tests such
as $\pi(K)\to\mu\nu/\pi(K)\to e\nu$ or $\tau\to\mu\nu\nu/\tau(\mu)\to e\nu\nu$
[28, 4, 29]. However, $C_{\phi\ell}^{(3)ij}$ also affects $Z$ couplings to
leptons and neutrinos, which enter the global EW fit.

##  3 Models addressing the CAA

The following NP models (see Ref. [30] for a complete categorization of tree-
level extensions of the SM) give contributions to the effective operators
discussed in the last subsection, which can explain the CAA.

###  3.1 $W^{\prime}$ boson

A $W^{\prime}$ boson with couplings to left-handed fermions (i.e. the charged
component of an $SU(2)_{L}$ triplet) affects the CAA via:

  * •

Modified $W\mu\nu$ coupling generated by the mixing with the SM $W$, encoded
in $C^{(3)}_{\phi\ell}$. This leads to limited effects in the CAA due to the
stringent constraints from $Z$ decays [31, 32].

  * •

Tree-level effects in $\mu\to e\nu\nu$ via $C_{\ell\ell}^{(3)}$.

  * •

Tree-level effects in $d\to ue\nu$ via $C_{\ell q}^{(3)}$ if the $W^{\prime}$
couples to quarks and leptons [33].

  * •

Modified $W-u-d$ coupling generated by the mixing with the SM $W$ encoded in
$C^{(3)}_{\phi q}$.

###  3.2 Vector-like Leptons

Vector-like leptons (VLL), such as right-handed neutrinos [34], affect
$W-\ell-\nu$ coupling via their mixing with SM leptons and are EW symmetry
breaking. There are 5 representations of vector-like leptons that can couple
to SM leptons and the Higgs and mix with the former after EW symmetry
breaking. However, a single representation alone cannot explain the CAA and
result simultaneously in a good EW fit [31, 32].

###  3.3 Vector-like Quarks

Similar to vector-like leptons, vector-like quarks mix with the SM ones after
EW symmetry breaking an lead to modified $W-u-d$ couplings. Two vector-like
quarks are found of being capable of accounting for the CAA [35].

###  3.4 Singly charged $SU(2)_{L}$ singlet scalar

Being an $SU(2)_{L}\times SU(3)_{C}$ singlet with hypercharge +1, only Yukawa-
type interactions with leptons are allowed. Due to hermicity of the Lagrangian
it has anti-symmetric (i.e. off-diagonal) couplings it results in an effect in
muon decay with the right sign such that in can explain the CAA via an shift
in the Fermi constant [36, 37, 38, 39].

###  3.5 $SU(2)_{L}$ Neutral Vector Boson ($Z^{\prime}$):

A $Z^{\prime}$ boson which is an $SU(2)_{L}$ singlet only interferes with the
SM amplitudes for $\mu\to e\nu\nu$ if it has flavour violating couplings to
leptons. In this case a necessarily constructive effect is generated that can
explain the CAA if other couplings are so small that the bounds from charged
LFV are respected [40].

###  3.6 Leptoquark

Even though LQs can generate tree-level effects in beta decays, in this case,
low energy parity violation or kaon decays, are, in general, more constraining
and prevent a solution of the CAA [41, 42].

##  4 Conclusions and outlook

The Cabibbo angle, parametrizing the mixing among the first two quark
generations can be measured in kaon, tau, pion, $D$ and beta decays. However,
the determination from super-allowed beta decays disagrees with the other
measurements, leading to an $\approx 3\,\sigma$ deficit in first row CKM
unitarity, known as the CAA.

Since for extracting $V_{ud}$ from (super-allowed) beta decays knowledge of
the Fermi constant is necessary. Even though the the Fermi constant is very
precisely measured in muon decay, it is not necessarily free of new physics.
THerefore, there are for ways how one can explain the CAA in terms of NP via
effective operators: a) 4-fermion operators affecting beta decays ($u\to
de\nu$), b) operators generating modified $W$–$u$–$d$ coupling, c) 4-fermion
operators affecting muon decays ($\mu\to e\nu\nu$) and d) operators generating
modified $W$–$\mu$–$\nu$ coupling. In terms of (simplified) BSM models, these
operator can be generated by a) $W^{\prime}$, leptoquarks, b) $W^{\prime}$
(mixing with the SM $W$), vector-like quarks, c) $W^{\prime}$, singly charged
scalar, $Z^{\prime}$ (with lepton flavour violating couplings), d) vector-like
leptons, $W^{\prime}$ (mixing with the SM $W$). Note if only muon decay
receives a BSM contribution, the CAA can be explained, but at the expense of a
enhanced tension within the global EW fit.

## Acknowledgements

I thank the organisers for the invitation to Moriond and for assembling an
extremely interesting program. Support via a Professorship Grant
(PP00P2_176884) of the Swiss National Science Foundation is gratefully
acknowledged.

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