# On similarity to contractions of class $C_{\cdot 0}$ with finite defects

Maria F. Gamal  St. Petersburg Branch
V. A. Steklov Institute of Mathematics
Russian Academy of Sciences
Fontanka 27, St. Petersburg
191023, Russia<EMAIL_ADDRESS>
###### Abstract.

A criterion on the similarity of a (bounded, linear) operator $T$ on a
(complex, separable) Hilbert space $\mathcal{H}$ to a contraction of class
$C_{\cdot 0}$ with finite unequal defects is given in terms of shift-type
invariant subspaces of $T$. Namely, $T$ is similar to such a contraction if
and only if there exists a finite collection of (closed) invariant subspaces
$\mathcal{M}$ of $T$ such that the restriction $T|_{\mathcal{M}}$ of $T$ on
$\mathcal{M}$ is similar to the simple unilateral shift and the linear span of
these subspaces $\mathcal{M}$ is $\mathcal{H}$. A sufficient condition for the
similarity of an absolutely continuous polynomially bounded operator $T$ to a
contraction of class $C_{\cdot 0}$ with finite equal defects is given. Namely,
$T$ is similar to such a contraction if the (spectral) multiplicity of $T$ is
finite and $B(T)=\mathbb{O}$, where $B$ is a finite product of Blaschke
products with simple zeros satisfying the Carleson interpolating condition (a
Carleson–Newman product).

2020 _Mathematics Subject Classification_. 47A45, 47A15.

###### Key words and phrases:

Similarity, contractions of class $C_{\cdot 0}$, contractions with finite
defects, shift-type invariant subspaces.

##  1\. Introduction

Let $\mathcal{H}$ be a (complex, separable) Hilbert space, and let
$\mathcal{L}(\mathcal{H})$ be the algebra of all (bounded, linear) operators
acting on $\mathcal{H}$. A (closed) subspace $\mathcal{M}$ of $\mathcal{H}$ is
called _invariant_ for an operator $T\in\mathcal{L}(\mathcal{H})$, if
$T\mathcal{M}\subset\mathcal{M}$. Denote by $\operatorname{Lat}T$ the
collection of all invariant subspaces of $T$. It is well known and easy to see
that $\operatorname{Lat}T$ is a complete lattice with the inclusion as a
partial order.

The _multiplicity_ $\mu_{T}$ of an operator $T\in\mathcal{L}(\mathcal{H})$ is
the minimum dimension of its reproducing subspaces:

(1.1) $\mu_{T}=\min\\{\dim E:E\subset\mathcal{H},\ \
\vee_{n=0}^{\infty}T^{n}E=\mathcal{H}\\}.$

For Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, let
$\mathcal{L}(\mathcal{H},\mathcal{K})$ denote the space of (bounded, linear)
transformations acting from $\mathcal{H}$ to $\mathcal{K}$. For
$T\in\mathcal{L}(\mathcal{H})$ and $R\in\mathcal{L}(\mathcal{K})$ set

$\mathcal{I}(T,R)=\\{X\in\mathcal{L}(\mathcal{H},\mathcal{K})\ :\ XT=RX\\}.$

Then $\mathcal{I}(T,R)$ is the set of all transformations $X$ which
_intertwine_ $T$ and $R$. Let $X\in\mathcal{I}(T,R)$. If $X$ is unitary, then
$T$ and $R$ are called _unitarily equivalent_ , in notation: $T\cong R$. If
$X$ is invertible, that is, $X^{-1}\in\mathcal{L}(\mathcal{K},\mathcal{H})$,
then $T$ and $R$ are called _similar_ , in notation: $T\approx R$. If there
exists $X\in\mathcal{I}(T,R)$ such that $\ker X=\\{0\\}$ and
$\operatorname{clos}X\mathcal{H}=\mathcal{K}$, then $T$ is called a
_quasiaffine transform_ of $R$, in notation: $T\prec R$. If $T\prec R$ and
$R\prec T$, then $T$ and $R$ are called _quasisimilar_ , in notation: $T\sim
R$.

An operator $T$ is called _polynomially bounded_ , if there exists a constant
$C>0$ such that $\|p(T)\|\leq C\|p\|_{\infty}$ for every (analytic) polynomial
$p$, where $\|p\|_{\infty}=\max_{|z|\leq 1}|p(z)|$. $T$ is an _absolutely
continuous (a.c.)_ polynomially bounded operator, if and only if the
$H^{\infty}$-functional calculus is well defined for $T$, see [12] or [11].
(The definition of an a.c. polynomially bounded operator is not recalled
here.) An a.c. polynomially bounded operator $T$ is called an _operator of
class $C_{0}$_, if there exists $0\not\equiv\varphi\in H^{\infty}$ such that
$\varphi(T)=\mathbb{O}$. For every operator $T$ of class $C_{0}$ there exists
a _minimal function_ , that is, an inner function $\theta$ such that
$\theta(T)=\mathbb{O}$ and if $\varphi\in H^{\infty}$ is such that
$\varphi(T)=\mathbb{O}$, then $\varphi\in\theta H^{\infty}$, see [3] and [14,
Proposition III.4.4] or [2, Sec. II.1].

An operator $T$ is called a _contraction_ , if $\|T\|\leq 1$. Every
contraction $T$ is polynomially bounded with $C=1$ by the von Neumann
inequality (see, for example, [14, Proposition I.8.3]). If an operator of
class $C_{0}$ is a contraction, then it is called a _contraction of class
$C_{0}$_, see [2] and [14].

A contraction $T\in\mathcal{L}(\mathcal{H})$ is called _of class_ $C_{\cdot
0}$, if $\|(T^{*})^{n}x\|\to 0$ for every $x\in\mathcal{H}$. If $T$ is a
contraction of class $C_{0}$, then $T$ and $T^{*}$ are of class $C_{\cdot 0}$
([14, Proposition III.4.2]). The _defect indices_ (the _defects_ for brevity)
of $T$ are $d_{T}=\dim(I-T^{*}T)\mathcal{H}$ and
$d_{T^{*}}=\dim(I-TT^{*})\mathcal{H}$, see [14, Sec. I.3.1]. If $T$ is a
contraction of class $C_{\cdot 0}$, then $d_{T}\leq d_{T^{*}}$. Furthermore,
$T$ is unitarily equivalent to a compression on the orthogonal complement of
an invariant subspace of the unilateral shift of multiplicity $d_{T^{*}}$.
Consequently, $\mu_{T}\leq d_{T^{*}}$ ([14, Corollary X.3.3]). If $T$ is a
contraction of class $C_{\cdot 0}$ and $N:=d_{T^{*}}<\infty$, then $T$ is of
class $C_{0}$ if and only if $d_{T}=d_{T^{*}}$, see [13]. Contractions $T$ of
class $C_{0}$ with $d_{T^{*}}=N<\infty$ are called _of class_ $C_{0}(N)$. If
$T$ is a contraction of class $C_{\cdot 0}$ and $d_{T}<d_{T^{*}}$, then $T$
has shift-type invariant subspaces, that is, there exist
$\mathcal{M}\in\operatorname{Lat}T$ such that $T|_{\mathcal{M}}\approx S$, and
the closed linear span of such subspaces is the space $\mathcal{H}$ on which
$T$ acts, see [13] and [10] or [14, Sec. IX.3]. In [9], it is proved that the
minimal quantity of shift-type invariant subspaces of $T$ whose _closed
linear_ span is $\mathcal{H}$ is equal to $\mu_{T}$, if $2\leq\mu_{T}$, and is
equal to $2$, if $\mu_{T}=1$. In this paper, it is proved that an operator $T$
is similar to a contraction $R$ of class $C_{\cdot 0}$ with
$d_{R}<d_{R^{*}}<\infty$ if and only if the minimal quantity of shift-type
invariant subspaces of $T$ whose _linear_ span is $\mathcal{H}$ is finite. For
operators of class $C_{0}$ analogous results (with replacing shift-type
invariant subspace by appropriate analogous subspaces) cannot hold true in
general, because operators of class $C_{0}$ can be unicellular. Remember that
an operator $T$ is called _unicellular_ , if $\operatorname{Lat}T$ is totally
ordered by inclusion. On the other side, if $B(T)=\mathbb{O}$ for a
Carleson–Newman product $B$ and $\mu_{T}<\infty$, then $T$ is similar to a
contraction of class $C_{0}(N)$ (with $N\in\mathbb{N}$). Similarity to
contractions of class $C_{0}(1)$, that is, Jordan blocks, is considered in
[5], [6], [7].

The paper is organized as follows. In Sec. 2 the criterion on similarity to
contractions $R$ of class $C_{\cdot 0}$ with $d_{R}<d_{R^{*}}<\infty$ is
proved. In Sec. 3 an example of a unicellular contraction of class $C_{0}(2)$
is given which is not similar to a contraction of class $C_{0}(1)$. In Sec. 4
operators $T$ of class $C_{0}$ such that $B(T)=\mathbb{O}$ for a
Carleson–Newman product $B$ are considered.

The following notation will be used. For a (closed) subspace $\mathcal{M}$ of
a Hilbert space $\mathcal{H}$, by $P_{\mathcal{M}}$ and $I_{\mathcal{M}}$ the
orthogonal projection from $\mathcal{H}$ onto $\mathcal{M}$ and the identity
operator on $\mathcal{M}$ are denoted, respectively. By $\mathbb{O}$ the zero
transformation acting between (maybe nonzero) spaces is denoted.

Symbols $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disc and the unit
circle, respectively. The normalized Lebesgue measure on $\mathbb{T}$ is
denoted by $m$. $H^{2}$ is the Hardy–Hilbert space on $\mathbb{T}$.
$H^{\infty}$ is the Banach algebra of all bounded analytic functions in
$\mathbb{D}$; $\|\varphi\|_{\infty}=\sup_{|z|<1}|\varphi(z)|$ for $\varphi\in
H^{\infty}$. The simple unilateral shift $S$ is the operator of multiplication
by the independent variable on $H^{2}$. For $N\in\mathbb{N}$ denote by
$H^{2}_{N}$ and by $S_{N}$ the orthogonal sum of $N$ copies of $H^{2}$ and
$S_{N}$, respectively. Then $S_{N}$ is the unilateral shift of multiplicity
$N$ acting on $H^{2}_{N}$: $S_{N}\in\mathcal{L}(H^{2}_{N})$ and
$\mu_{S_{N}}=N$. For $N\in\mathbb{N}$, vectors from $H^{2}_{N}$ are columns of
the height $N$ of functions from $H^{2}$.

Let $M,N\in\mathbb{N}$. The space $H^{\infty}(\mathbb{C}^{M},\mathbb{C}^{N})$
is the space of bounded analytic functions on $\mathbb{D}$ whose values are
(bounded, linear) operators acting from $\mathbb{C}^{M}$ to $\mathbb{C}^{N}$.
Let $\Phi\in H^{\infty}(\mathbb{C}^{M},\mathbb{C}^{N})$. Then $\Phi$ has
nontangential boundary values $\Phi(\zeta)$ for a.e. $\zeta\in\mathbb{T}$ with
respect to $m$. Furthermore, $\Phi$ has a representation
$\Phi=[\varphi_{nk}]_{n=1,\ldots,N\atop k=1,\ldots,M}$, where $\varphi_{nk}\in
H^{\infty}$, with respect to some orthonormal bases of $\mathbb{C}^{M}$ and
$\mathbb{C}^{N}$. The operator of multiplication by $\Phi$ acts from
$H^{2}_{M}$ to $H^{2}_{N}$. For $\Phi\in
H^{\infty}(\mathbb{C}^{M},\mathbb{C}^{N})$ set
$\widetilde{\Phi}(z)=\Phi^{*}(\overline{z})$, $z\in\mathbb{D}$. The function
$\Theta\in H^{\infty}(\mathbb{C}^{M},\mathbb{C}^{N})$ is called inner, if the
multiplication by $\Theta$ is an isometry from $H^{2}_{M}$ to $H^{2}_{N}$. The
function $\Theta$ is called $*$-inner, if $\widetilde{\Theta}$ is inner, and
is called inner from both sides, if $\Theta$ is inner and $*$-inner. The
definition of outer and $*$-outer function is not recalled here. For
references, see [14, Sec. V.2.3]

Let $\Theta\in H^{\infty}(\mathbb{C}^{M},\mathbb{C}^{N})$. Then $\Theta$ is
inner if and only if $\Theta^{*}(\zeta)\Theta(\zeta)=I_{\mathbb{C}^{M}}$ for
a.e. $\zeta\in\mathbb{T}$ with respect to $m$, and then $M\leq N$ ([14,
Proposition V.2.2]). For inner function $\Theta\in
H^{\infty}(\mathbb{C}^{M},\mathbb{C}^{N})$ set

$\mathcal{H}(\Theta)=H^{2}_{N}\ominus\Theta H^{2}_{M}\ \ \text{ and }\ \
T_{\Theta}=P_{\mathcal{H}(\Theta)}S_{N}|_{\mathcal{H}(\Theta)}.$

Then $T_{\Theta}$ is a contraction of class $C_{\cdot 0}$ with $d_{T^{*}}\leq
N$. Conversely, for every contraction $T$ of class $C_{\cdot 0}$ with
$d_{T^{*}}<\infty$ there exists an inner function $\Theta\in
H^{\infty}(\mathbb{C}^{d_{T}},\mathbb{C}^{d_{T^{*}}})$ such that $T\cong
T_{\Theta}$. (Such $\Theta$ is called the characteristic function of a
contraction $T$.) For references, see [14, Ch. V, VI], or [2, Sec. V.1], or
[16, Ch. C.1].

##  2\. On similarity to contractions of class $C_{\cdot 0}$ with unequal
finite defects

In this section, a criterion on the similarity of an operator $T$ to a
contraction of class $C_{\cdot 0}$ with finite unequal defects (Theorem 2.5)
will be proved.

######  Lemma 2.1.

Let $N\in\mathbb{N}$, $N\geq 2$, and let $\tau_{0n}\subset\mathbb{T}$,
$n=1,\ldots,N$, be measurable sets such that $m(\tau_{0n})>0$ for every $n$,
$1\leq n\leq N$. Then there exist measurable sets $\sigma_{n}$,
$n=1,\ldots,N$, such that $\sigma_{n}\subset\tau_{0n}$, $m(\sigma_{n})>0$,
$\sigma_{n}\cap\sigma_{k}=\emptyset$, if $n\neq k$, $n,k=1,\ldots,N$, and
$m(\cup_{n=1}^{N}\sigma_{n})=m(\cup_{n=1}^{N}\tau_{0n})$.

###### Proof.

The lemma will be proved using induction. Let $N=2$. Without loss of
generality, we may assume that $m(\tau_{01})\leq m(\tau_{02})$. Set
$\tau_{2}=\tau_{02}\setminus\tau_{01}$. If $m(\tau_{2})>0$, then set
$\sigma_{1}=\tau_{01}$ and $\sigma_{2}=\tau_{2}$. If $m(\tau_{2})=0$, then
$m(\tau_{01}\cup\tau_{02})=m(\tau_{01}\cap\tau_{02})$. Every representation
$\tau_{01}\cap\tau_{02}=\sigma_{1}\cup\sigma_{2}$ with $\sigma_{1}$ and
$\sigma_{2}$ such that $m(\sigma_{k})>0$, $k=1,2$, and
$\sigma_{1}\cap\sigma_{2}=\emptyset$ satisfies the conclusion of the lemma in
this case.

If $N>2$, assume that the lemma is proved for all $M$, $2\leq M\leq N-1$.
Without loss of generality, we may assume that $m(\tau_{0N})=\min_{1\leq n\leq
N}m(\tau_{0n})$. Set $\tau_{n}=\tau_{0n}\setminus\tau_{0N}$, $n=1,\ldots,N-1$.
Without loss of generality, we may assume that there exists $1\leq M\leq N$
such that $m(\tau_{n})>0$, $1\leq n\leq M-1$, and $m(\tau_{n})=0$, $M\leq
n\leq N-1$. Thus,
$m(\cup_{n=1}^{N}\tau_{0n})=m(\cup_{n=1}^{M-1}\tau_{n}\cup\tau_{0N})$ and
$(\cup_{n=1}^{M-1}\tau_{n})\cap\tau_{0N}=\emptyset$. By the inductive
hypothesis, there exist measurable sets $\sigma_{n}$, $n=1,\ldots,M-1$, such
that $\sigma_{n}\subset\tau_{n}$, $m(\sigma_{n})>0$,
$\sigma_{n}\cap\sigma_{k}=\emptyset$, if $n\neq k$, $n,k=1,\ldots,M-1$, and
$m(\cup_{n=1}^{M-1}\sigma_{n})=m(\cup_{n=1}^{M-1}\tau_{n})$. Furthermore,
$\tau_{0N}=\cup_{n=M}^{N-1}(\tau_{0N}\setminus\tau_{0n})\cup\cap_{n=M}^{N}\tau_{0n}$.
The relations $m(\tau_{0N})\leq m(\tau_{0n})$ and $m(\tau_{n})=0$, $M\leq
n\leq N-1$, imply $m(\tau_{0N}\setminus\tau_{0n})=0$. Consequently,
$m(\tau_{0N})=m(\cap_{n=M}^{N}\tau_{0n})$. Take a representation
$\cap_{n=M}^{N}\tau_{0n}=\cup_{n=M}^{N}\sigma_{n}$ with $\sigma_{n}$ such that
$m(\sigma_{n})>0$, and $\sigma_{n}\cap\sigma_{k}=\emptyset$, if $n\neq k$,
$n,k=M,\ldots,N$. Then $\sigma_{n}$, $n=1,\ldots,N$, satisfy the conclusion of
the lemma. ∎

######  Lemma 2.2.

Let $N\in\mathbb{N}$, $N\geq 2$, and let $c>0$. Let $\varphi_{n}$,
$n=1,\ldots,N$, be such that $0\not\equiv\varphi_{n}\in H^{\infty}$,
$\|\varphi_{n}\|_{\infty}\leq 1$ for every $n$, $1\leq n\leq N$, and

$\mathop{\mathrm{ess\,inf}}_{\zeta\in\mathbb{T}}\sum_{n=1}^{N}|\varphi_{n}(\zeta)|^{2}=c^{2}>0.$

Then there exist $\Psi\in H^{\infty}(\mathbb{C}^{N},\mathbb{C}^{N})$ and inner
functions $\vartheta_{n}$, $n=1,\ldots,N$, such that $\Psi$ is invertible in
$H^{\infty}(\mathbb{C}^{N},\mathbb{C}^{N})$ and

$\Psi[\varphi_{n}]_{n=1}^{N}=[\vartheta_{n}]_{n=1}^{N}.$

###### Proof.

Take $\delta_{n}$, $n=1,\ldots,N$, such that
$0<\delta_{n}<\|\varphi_{n}\|_{\infty}$ for all $n=1,\ldots,N$, and
$\sum_{n=1}^{N}\delta_{n}^{2}<c^{2}$. Set

$\tau_{0n}=\\{\zeta\in\mathbb{T}:\ |\varphi_{n}(\zeta)|\geq\delta_{n}\\},\ \
n=1,\ldots,N.$

Then $m(\tau_{0n})>0$ for every $n=1,\ldots,N$ and
$m(\cup_{n=1}^{N}\tau_{0n})=1$. By Lemma 2.1, there exist measurable sets
$\sigma_{n}$, $n=1,\ldots,N$, such that $\sigma_{n}\subset\tau_{0n}$,
$m(\sigma_{n})>0$, $\sigma_{n}\cap\sigma_{k}=\emptyset$, if $n\neq k$,
$n,k=1,\ldots,N$, and $m(\cup_{n=1}^{N}\sigma_{n})=1$.

Set $\delta=\min_{1\leq n\leq N}\delta_{n}$. Take $a$, $b>0$ such that
$a^{N-1}>N!b$ and $\delta>a+(N-2)b$. Define outer functions $\kappa_{nk}$,
$n,k=1,\ldots,N$, by the formulas

$|\kappa_{nn}(\zeta)|=\begin{cases}a,\text{ if
}\zeta\in\mathbb{T}\setminus\sigma_{n},\\\ 1,\text{ if
}\zeta\in\sigma_{n},\end{cases}\ \ |\kappa_{nk}(\zeta)|=\begin{cases}b,\text{
if }\zeta\in\mathbb{T}\setminus\sigma_{k},\\\ 1,\text{ if
}\zeta\in\sigma_{k},\end{cases}\ \ n\neq k.$

Set $\kappa=\det[\kappa_{nk}]_{n,k=1}^{N}$. Then

(2.1) $|\kappa(z)|\geq a^{N-1}-N!b,\ \ z\in\mathbb{D},$

and

(2.2)
$\Bigl{|}\sum_{k=1}^{N}\kappa_{nk}(\zeta)\varphi_{k}(\zeta)\Bigr{|}\geq\delta-a-(N-2)b\
\text{ for a.e. }\zeta\in\mathbb{T},\ n=1,\ldots,N.$

Indeed,

$\kappa=\prod_{n=1}^{N}\kappa_{nn}+\sum_{\\{n_{j}\\}_{j=1}^{N}}(-1)^{s}\prod_{j=1}^{N}\kappa_{n_{j}j},$

where $s=0,1$ (depending on $\\{n_{j}\\}_{j=1}^{N}$), and
$\\{n_{j}\\}_{j=1}^{N}$ is a permutation of $\\{1,\ldots,N\\}$ such that
$n_{j}\neq j$ for at least one $j$. (The sum is taken by all such
permutations.) We have $|\prod_{n=1}^{N}\kappa_{nn}|=a^{N-1}$ a.e. on
$\mathbb{T}$. Since $\kappa_{nn}$ are outer, we have
$\prod_{n=1}^{N}\kappa_{nn}$ is outer. Thus,
$\prod_{n=1}^{N}\kappa_{nn}(z)=a^{N-1}$ for every $z\in\mathbb{D}$. Let
$\\{n_{j}\\}_{j=1}^{N}$ be a permutation of $\\{1,\ldots,N\\}$ such that
$n_{j}\neq j$ for at least one $j$, and let $1\leq k\leq N$. Then there exists
$l\neq k$, $1\leq l\leq N$, such that $n_{l}\neq l$. Consequently,
$|\kappa_{n_{l}l}(\zeta)|=b$ for a.e. $\zeta\in\sigma_{k}$, and
$|\prod_{j=1}^{N}\kappa_{n_{j}j}(\zeta)|\leq b$ for a.e. $\zeta\in\sigma_{k}$.
Since $m(\cup_{k=1}^{N}\sigma_{k})=1$, we conclude that
$|\prod_{j=1}^{N}\kappa_{n_{j}j}(\zeta)|\leq b$ for a.e. $\zeta\in\mathbb{T}$.
Consequently, $|\prod_{j=1}^{N}\kappa_{n_{j}j}(z)|\leq b$, $z\in\mathbb{D}$.
Thus,

$|\kappa(z)|\geq\Bigl{|}\prod_{n=1}^{N}\kappa_{nn}(z)\Bigr{|}-\sum_{\\{n_{j}\\}_{j=1}^{N}}\Bigl{|}\prod_{j=1}^{N}\kappa_{n_{j}j}(z)\Bigr{|}\geq
a^{N-1}-N!b,\ \ z\in\mathbb{D}.$

The estimate (2.1) is proved. Let $1\leq n\leq N$. If $\zeta\in\sigma_{n}$,
then

$\Bigl{|}\sum_{k=1}^{N}\kappa_{nk}(\zeta)\varphi_{k}(\zeta)\Bigr{|}\geq|\kappa_{nn}(\zeta)\varphi_{n}(\zeta)|-\sum_{k\neq
n}|\kappa_{nk}(\zeta)\varphi_{k}(\zeta)|\geq\delta_{n}-(N-1)b.$

If $\zeta\in\sigma_{k}$ for $k\neq n$, then

$\Bigl{|}\sum_{l=1}^{N}\kappa_{nl}(\zeta)\varphi_{l}(\zeta)\Bigr{|}\geq|\kappa_{nk}(\zeta)\varphi_{k}(\zeta)|-\sum_{l\neq
k}|\kappa_{nl}(\zeta)\varphi_{l}(\zeta)|\geq\delta_{k}-a-(N-2)b.$

The estimate (2.2) is proved. Set
$\sum_{k=1}^{N}\kappa_{nk}\varphi_{k}=\vartheta_{n}\eta_{n}$, $n=1,\ldots,N$,
where $\vartheta_{n}$ is inner and $\eta_{n}$ is outer. By (2.2),
$1/\eta_{n}\in H^{\infty}$, $n=1,\ldots,N$. Set
$\Psi=[\kappa_{nk}/\eta_{n}]_{n,k=1}^{N}$. Then
$\det\Psi=\kappa/\prod_{n=1}^{N}\eta_{n}$. Consequently, $\Psi$ is invertible
in $H^{\infty}(\mathbb{C}^{N},\mathbb{C}^{N})$. The equality in the conclusion
of the lemma follows from the defiition of $\Psi$. ∎

######  Corollary 2.3.

Lemma 2.2 holds true without assumption $\varphi_{n}\not\equiv 0$ for every
$n=1,\ldots,N$.

###### Proof.

Denote by $M$ the quantity of $n$ for which $\varphi_{n}\not\equiv 0$. Then
$1\leq M\leq N$. Multiplying the column $[\varphi_{n}]_{n=1}^{N}$ by the
(invertible) $N\times N$-matrix $A_{1}$ whose columns is an appropriate
permutation of the columns of the unit $N\times N$-matrix $I_{N}$, we obtain
the column $[\varphi_{1n}]_{n=1}^{N}$ such that $\varphi_{1n}\not\equiv 0$ for
$n=1,\ldots,M$ and $\varphi_{1n}\equiv 0$ for $n=M+1,\ldots,N$. Note that
$\\{\varphi_{1n}\\}_{n=1}^{N}$ is a permutation of
$\\{\varphi_{n}\\}_{n=1}^{N}$. Let $A_{2}$ be the $N\times N$-matrix whose
elements of first columns with indices $n=M+1,\ldots,N$ are equal to $1$, and
all other elements of $A_{2}$ are equal to $0$. Set
$[\varphi_{2n}]_{n=1}^{N}=(I_{N}+A_{2})[\varphi_{1n}]_{n=1}^{N}$. Then
$\varphi_{2n}=\varphi_{1n}$ for $n=1,\ldots,M$ and $\varphi_{2n}=\varphi_{11}$
for $n=M+1,\ldots,N$. The column $[\varphi_{2n}]_{n=1}^{N}$ satisfy the
assumption of Lemma 2.2. The matrix $\Psi(I_{N}+A_{2})A_{1}$ is the required
matrix. ∎

######  Theorem 2.4.

Let $M,N\in\mathbb{N}$, $N\geq 2$, $1\leq M<N$, and let $\Theta\in
H^{\infty}(\mathbb{C}^{N-M},\mathbb{C}^{N})$ be an inner function. Then there
exist $\mathcal{M}_{n}\in\operatorname{Lat}T_{\Theta}$, $n=1,\ldots,N$, such
that $T_{\Theta}|_{\mathcal{M}_{n}}\approx S$ for every $n$, $1\leq n\leq N$,
and $\mathcal{H}(\Theta)=\mathcal{M}_{1}+\ldots+\mathcal{M}_{N}$ in the
following sense: for every $x\in\mathcal{H}$ there exist
$x_{n}\in\mathcal{M}_{n}$, $n=1,\ldots,N$, such that $x=x_{1}+\ldots+x_{N}$.

###### Proof.

Let $\Theta=\Theta_{1}\Omega$ be the $*$-canonical factorization of $\Theta$,
see [14, (V.4.20)]. Then $\Theta_{1}\in
H^{\infty}(\mathbb{C}^{N-M},\mathbb{C}^{N})$ is inner $*$-outer and $\Omega\in
H^{\infty}(\mathbb{C}^{N-M},\mathbb{C}^{N-M})$ is inner from both sides. By
[13] and [18], there exists an $*$-inner function $\Phi\in
H^{\infty}(\mathbb{C}^{N},\mathbb{C}^{M})$ such that
$\Phi\Theta_{1}=\mathbb{O}$. Let

$\Phi=\left[\begin{matrix}\varphi_{11}&\ldots&\varphi_{1N}\\\
\ldots&\ldots&\ldots\\\ \varphi_{M1}&\ldots&\varphi_{MN}\end{matrix}\right],$

where $\varphi_{kn}\in H^{\infty}$, $k=1,\ldots,M$, $n=1,\ldots,N$. Since
$\Phi$ is $*$-inner, we have $\sum_{n=1}^{N}|\varphi_{kn}(\zeta)|^{2}=1$ for
a.e. $\zeta\in\mathbb{T}$ and every $k=1,\ldots,M$. Let $\Psi\in
H^{\infty}(\mathbb{C}^{N},\mathbb{C}^{N})$ be a function from Corollary 2.3
applied to the functions $\varphi_{1n}$, $n=1,\ldots,N$. Then $\Psi$ is
invertible in $H^{\infty}(\mathbb{C}^{N},\mathbb{C}^{N})$ and

$\Psi[\varphi_{1n}]_{n=1}^{N}=[\vartheta_{n}]_{n=1}^{N}$

for some inner functions $\vartheta_{n}$, $n=1,\ldots,N$. Denote by
$\Psi^{\top}$ the transpose of $\Psi$. Then

$\Phi\Psi^{\top}=\left[\begin{matrix}\vartheta_{1}&\ldots&\vartheta_{N}\\\
\psi_{21}&\ldots&\psi_{2N}\\\ \ldots&\ldots&\ldots\\\
\psi_{M1}&\ldots&\psi_{MN}\end{matrix}\right]$

for some functions $\psi_{kn}\in H^{\infty}$, $k=2,\ldots,M$, $n=1,\ldots,N$.

Let $n$, $1\leq n\leq N$, be fixed. For $h\in H^{2}$ denote by $h_{(n,N)}$ the
column of the height $N$ whose element on $n$th place is equal to $h$ and all
other elements are equal to $0$. Then $h_{(n,N)}\in H^{2}_{N}$, and

(2.3) $\Phi\Psi^{\top}h_{(n,N)}=\left[\begin{matrix}\vartheta_{n}h\\\
\psi_{2n}h\\\ \ldots\\\ \psi_{Mn}h\end{matrix}\right].$

Define $Y_{n}\in\mathcal{L}(H^{2},\mathcal{H}(\Theta))$ by the formula
$Y_{n}h=P_{\mathcal{H}(\Theta)}\Psi^{\top}h_{(n,N)}$, $h\in H^{2}$. Then
$Y_{n}S=T_{\Theta}Y_{n}$ and $\|Y_{n}h\|\geq\|h\|$ for every $h\in H^{2}$.
Indeed,

$\displaystyle Y_{n}Sh$
$\displaystyle=P_{\mathcal{H}(\Theta)}\Psi^{\top}(Sh)_{(n,N)}=P_{\mathcal{H}(\Theta)}S_{N}\Psi^{\top}h_{(n,N)}=P_{\mathcal{H}(\Theta)}S_{N}P_{\mathcal{H}(\Theta)}\Psi^{\top}h_{(n,N)}$
$\displaystyle=T_{\Theta}P_{\mathcal{H}(\Theta)}\Psi^{\top}h_{(n,N)}=T_{\Theta}Y_{n}h,\
\ h\in H^{2}.$

Since $\Phi\Theta_{1}=\mathbb{O}$ and $\Theta=\Theta_{1}\Omega$, we have

$\Phi Y_{n}h=\Phi
P_{\mathcal{H}(\Theta)}\Psi^{\top}h_{(n,N)}=\Phi(\Psi^{\top}h_{(n,N)}-P_{\Theta
H^{2}_{N-M}}\Psi^{\top}h_{(n,N)})=\Phi\Psi^{\top}h_{(n,N)}.$

It follows from the last equality and (2.3) that

$\|Y_{n}h\|=\|\Phi\|\|Y_{n}h\|\geq\|\Phi
Y_{n}h\|=\|\Phi\Psi^{\top}h_{(n,N)}\|\geq\|\vartheta_{n}h\|=\|h\|,\ \ h\in
H^{2}.$

(The equality $\|\Phi\|=1$ holds true, because $\Phi$ is $*$-inner.) Set
$\mathcal{M}_{n}=Y_{n}H^{2}$ and consider $Y_{n}$ as a transformation from
$H^{2}$ to $\mathcal{M}_{n}$. Then
$\mathcal{M}_{n}\in\operatorname{Lat}T_{\Theta}$ and $Y_{n}$ realizes the
relation $T_{\Theta}|_{\mathcal{M}_{n}}\approx S$.

The equality $\mathcal{H}(\Theta)=\mathcal{M}_{1}+\ldots+\mathcal{M}_{N}$
follows from the equality $H^{2}_{N}=\Psi^{\top}H^{2}_{N}$, which is
fulfilled, because $\Psi$ is invertible in
$H^{\infty}(\mathbb{C}^{N},\mathbb{C}^{N})$. ∎

######  Theorem 2.5.

Let $T\in\mathcal{L}(\mathcal{H})$. Then the following are equivalent:

$\mathrm{(i)}$ there exists a contraction $R$ of class $C_{\cdot 0}$ with
$d_{R}<d_{R^{*}}<\infty$ such that $T\approx R$;

$\mathrm{(ii)}$ there exist $N\in\mathbb{N}$ and
$\mathcal{M}_{n}\in\operatorname{Lat}T$, $n=1,\ldots,N$, such that
$T|_{\mathcal{M}_{n}}\approx S$ for every $n=1,\ldots,N$, and
$\mathcal{H}=\mathcal{M}_{1}+\ldots+\mathcal{M}_{N}$ in the following sense:
for every $x\in\mathcal{H}$ there exist $x_{n}\in\mathcal{M}_{n}$,
$n=1,\ldots,N$, such that $x=x_{1}+\ldots+x_{N}$.

Moreover, if $\mathrm{(i)}$ and $\mathrm{(ii)}$ are fulfilled, then

$\displaystyle\min\\{d_{R^{*}}\ :\ R\text{ is a contraction such that
}T\approx R\\}$ $\displaystyle=\min\\{N\in\mathbb{N}\ :\ N\text{ satisfies
assumption }\mathrm{(ii)}\\}.$

###### Proof.

The part (i)$\Rightarrow$(ii) is Theorem 2.4. Indeed, if $R$ is a contraction
of class $C_{\cdot 0}$ with $d_{R}<d_{R^{*}}<\infty$, then $R\cong
T_{\Theta}$, where $\Theta$ satisfies the assumption of Theorem 2.4, see
references in Introduction.

Assume that (ii) is fulfilled. Let
$Y_{n}\in\mathcal{I}(S,T|_{\mathcal{M}_{n}})$ be invertible, $n=1,\ldots,N$.
Let $[h_{n}]_{n=1}^{N}\in H^{2}_{N}$, where $h_{n}\in H^{2}$, $n=1,\ldots,N$.
Set

$Y[h_{n}]_{n=1}^{N}=\sum_{n=1}^{N}Y_{n}h_{n}.$

Then

$\displaystyle\|Y[h_{n}]_{n=1}^{N}\|^{2}$
$\displaystyle\leq\Bigl{(}\sum_{n=1}^{N}\|Y_{n}h_{n}\|\Bigr{)}^{2}\leq
N\Bigl{(}\sum_{n=1}^{N}\|Y_{n}h_{n}\|^{2}\Bigr{)}$ $\displaystyle\leq
N\max_{1\leq n\leq N}\|Y_{n}\|^{2}\sum_{n=1}^{N}\|h_{n}\|^{2}=N\max_{1\leq
n\leq N}\|Y_{n}\|^{2}\|[h_{n}]_{n=1}^{N}\|^{2}.$

Thus, $Y$ is bounded. The equations $YS_{N}=TY$ and $YH^{2}_{N}=\mathcal{H}$
follow from the definition of $Y$ and the assumption
$\mathcal{H}=\mathcal{M}_{1}+\ldots+\mathcal{M}_{N}$. Furthermore, the
equation $YS_{N}=TY$ implies $\ker Y\in\operatorname{Lat}S_{N}$. Set
$\mathcal{K}=H^{2}_{N}\ominus\ker Y$, $Z=Y|_{\mathcal{K}}$ and
$R=P_{\mathcal{K}}S_{N}|_{\mathcal{K}}$. Then $ZR=TZ$,
$Z\mathcal{K}=YH^{2}_{N}=\mathcal{H}$ and $\ker Z=\\{0\\}$. Consequently,
$T\approx R$. Clearly, $R$ is a contraction of class $C_{\cdot 0}$ with
$d_{R^{*}}\leq N<\infty$. The relation $T\approx R$ and the assumptions on $T$
imply the existense of $\mathcal{N}\in\operatorname{Lat}R$ such that
$R|_{\mathcal{N}}\approx S$. By [13], $d_{R}<d_{R^{*}}$.

The equality of minima follows from the proof of the theorem. ∎

######  Remark 2.6.

The representation $x=x_{1}+\ldots+x_{N}$, $x_{n}\in\mathcal{M}_{n}$,
$n=1,\ldots,N$, from the statement (ii) of Theorem 2.5 is not unique in
general. Moreover, if such a representation is unique for every
$x\in\mathcal{H}$, then $T\approx S_{N}$. Indeed, in this case the mapping
$X\colon\mathcal{H}\to\oplus_{n=1}^{N}\mathcal{M}_{n}$,
$Xx=x_{1}\oplus\ldots\oplus x_{N}$, where $x_{n}\in\mathcal{M}_{n}$,
$n=1,\ldots,N$, are from the representation $x=x_{1}+\ldots+x_{N}$, is a well-
defined linear bijection. It is easy to see that $X$ and $X^{-1}$ are bounded
by the Closed Graph Theorem. Consequently,
$T\approx\oplus_{n=1}^{N}T|_{\mathcal{M}_{n}}\approx S_{N}$, where first
similarity is realized by $X$, and second similarity follows from the
assumption $T|_{\mathcal{M}_{n}}\approx S$ for every $n=1,\ldots,N$.

######  Remark 2.7.

In [22], for every $N\in\mathbb{N}$, $N\geq 2$, an example of a contraction
$T$ is constructed such that $d_{T}+1=d_{T^{*}}=N$ and $T\prec S$, but $T$ is
not quasisimilar (and, consequently, not similar) to any contraction $R$ with
$d_{R^{*}}<N$. The relation $T\prec S$ implies $\mu_{T}\leq 2$, see [13]. (For
further results on multiplicity of contractions with finite defects (not
necessarily of class $C_{\cdot 0}$) see [21].) Let $N\in\mathbb{N}$, $N\geq
2$, and let $T$ be a contraction from [22] with $d_{T^{*}}=N$ described just
above. Denote by $\mathcal{H}$ the space on which $T$ acts. The minimal
quantity of shift-type invariant subspaces of $T$ whose _linear_ span is
$\mathcal{H}$ is equal to $N$ by [22] and Theorem 2.5, while the minimal
quantity of shift-type invariant subspaces of $T$ whose _closed linear_ span
is $\mathcal{H}$ is equal to $2$ by [9] and [13]. (For a generalization of
[22] see [17].)

##  3\. On similarity to contractions of class $C_{0}(N)$

Let $N\in\mathbb{N}$. Let $T$ be a contraction of class $C_{\cdot 0}$ with
$d_{T}=d_{T^{*}}=N<\infty$. Equivalently, let $T$ be of class $C_{0}(N)$ (see
[13]). Since $T$ is of class $C_{0}$, there is no shift-type invariant
subspace of $T$, that is, such invariant subspace that the restriction of $T$
on this subspace is similar to $S$. Moreover, it is not possible to obtain a
result similar to Theorem 2.4 even if $S$ will be replaced by contractions
$T_{\vartheta_{n}}$ for inner functions $\vartheta_{n}\in H^{\infty}$,
$n=1,\ldots,N$, different from each other, because contractions of class
$C_{0}$ can be unicellular (the definition is recalled in Introduction). If
$T$ is unicellular and $\mathcal{H}=\mathcal{M}_{1}+\ldots+\mathcal{M}_{M}$
for some $\mathcal{M}_{n}\in\operatorname{Lat}T$, $n=1,\ldots,M$,
$M\in\mathbb{N}$, then $\mathcal{H}=\mathcal{M}_{n}$ for some $n$, $1\leq
n\leq M$. Consequently, a statement analogous to Theorem 2.4 must imply that
every unicellular contraction of class $C_{0}(N)$ must be similar to
$T_{\vartheta}$ for some inner function $\vartheta\in H^{\infty}$. In Example
3.3 below a unicellular contraction of class $C_{0}(2)$ is constructed which
is not similar to $T_{\vartheta}$ for any inner function $\vartheta\in
H^{\infty}$.

On the other side, the following proposition analogous to the statement
(ii)$\Rightarrow$(i) in Theorem 2.5 holds true.

######  Proposition 3.1.

Let $N\in\mathbb{N}$, and let $T\in\mathcal{L}(\mathcal{H})$. Suppose that
there exist inner functions $\vartheta_{n}\in H^{\infty}$ and
$\mathcal{M}_{n}\in\operatorname{Lat}T$, $n=1,\ldots,N$, such that
$T|_{\mathcal{M}_{n}}\approx T_{\vartheta_{n}}$ for every $n=1,\ldots,N$, and
$\mathcal{H}=\mathcal{M}_{1}+\ldots+\mathcal{M}_{N}$ in the following sence:
for every $x\in\mathcal{H}$ there exist $x_{n}\in\mathcal{M}_{n}$,
$n=1,\ldots,N$, such that $x=x_{1}+\ldots+x_{N}$.

Then there exists a contraction $R$ of class $C_{0}(M)$ with $M\leq N$ such
that $T\approx R$.

###### Proof.

Let $Y_{n}\in\mathcal{I}(T_{\vartheta_{n}},T|_{\mathcal{M}_{n}})$ be
invertible, $n=1,\ldots,N$. Let $h_{n}\in\mathcal{H}(\vartheta_{n})$,
$n=1,\ldots,N$. Set

$Y(\oplus_{n=1}^{N}h_{n})=\sum_{n=1}^{N}Y_{n}h_{n}.$

The proof of the boundedness of $Y$ is the same as in the proof of Theorem
2.5. Set

$\Theta=\left[\begin{matrix}\vartheta_{1}&0&\ldots&0\\\
0&\vartheta_{2}&\ldots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\ldots&\vartheta_{N}\end{matrix}\right].$

Then $\mathcal{H}(\Theta)=\oplus_{n=1}^{N}\mathcal{H}(\vartheta_{n})$ and
$T_{\Theta}=\oplus_{n=1}^{N}T_{\vartheta_{n}}$. The equations $YT_{\Theta}=TY$
and $Y\mathcal{H}(\Theta)=\mathcal{H}$ follow from the definition of $Y$ and
the assumption $\mathcal{H}=\mathcal{M}_{1}+\ldots+\mathcal{M}_{N}$.
Furthermore, the equation $YT_{\Theta}=TY$ implies $\ker
Y\in\operatorname{Lat}T_{\Theta}$. Set
$\mathcal{K}=\mathcal{H}(\Theta)\ominus\ker Y$, $Z=Y|_{\mathcal{K}}$ and
$R=P_{\mathcal{K}}T_{\Theta}|_{\mathcal{K}}$. Then $R$ is a contraction of
class $C_{0}(M)$ with $M\leq N$, because $T_{\Theta}$ is a contraction of
class $C_{0}(N)$. Furthermore, $ZR=TZ$,
$Z\mathcal{K}=Y\mathcal{H}(\Theta)=\mathcal{H}$ and $\ker Z=\\{0\\}$.
Consequently, $T\approx R$. ∎

The following lemma is a corollary of results from [14] and [2]. This lemma is
used in Example 3.3.

######  Lemma 3.2.

Let $N\in\mathbb{N}$, and let $\Theta\in
H^{\infty}(\mathbb{C}^{N},\mathbb{C}^{N})$ be inner. Denote by $\Theta^{Ad}$
the algebraic adjoint of $\Theta$. Let $\Theta^{Ad}$ be represented by a
matrix $\Theta^{Ad}=[\theta_{nk}]_{n,k=1}^{N}$ with respect to some
orthonormal basis of $\mathbb{C}^{N}$. Furthermore, let $\vartheta\in
H^{\infty}$ be inner. Then $T_{\Theta}\sim T_{\vartheta}$ if and only if
$\det\Theta=\vartheta$ and the greatest common inner divisor of $\theta_{nk}$,
$n,k=1,\ldots,N$, is $1$. If $T_{\Theta}\approx T_{\vartheta}$, then

$\inf_{z\in\mathbb{D}}\sum_{n,k=1}^{N}|\theta_{nk}(z)|>0.$

###### Proof.

The statement on the quasisimilarity $T_{\Theta}\sim T_{\vartheta}$ is
contained in [14, Theorem X.6.6] (see also [2, Proposition VI.5.11]), so its
proof is omitted. Let $T_{\Theta}\approx T_{\vartheta}$. Let
$X\in\mathcal{I}(T_{\vartheta},T_{\Theta})$ be invertible. By [14, Theorem
VI.3.6], there exist $\Psi_{1}$, $\Psi_{3}\in
H^{\infty}(\mathbb{C},\mathbb{C}^{N})$ and $\Psi_{2}$, $\Psi_{4}\in
H^{\infty}(\mathbb{C}^{N},\mathbb{C})$ such that
$Xf=P_{\mathcal{H}(\Theta)}\Psi_{1}f$ for every $f\in\mathcal{H}(\vartheta)$,
$X^{-1}h=P_{\mathcal{H}(\vartheta)}\Psi_{4}h$ for every
$h\in\mathcal{H}(\Theta)$, $\Psi_{1}\vartheta=\Theta\Psi_{3}$, and
$\Psi_{4}\Theta=\vartheta\Psi_{2}$. Since
$\Theta^{Ad}\Theta=\Theta\Theta^{Ad}=(\det\Theta)\cdot
I_{\mathbb{C}^{N}}=\vartheta I_{\mathbb{C}^{N}}$, we obtain
$\Psi_{4}=\Psi_{2}\Theta^{Ad}$. For every $f\in\mathcal{H}(\vartheta)$ we have

$\displaystyle f$
$\displaystyle=X^{-1}Xf=P_{\mathcal{H}(\vartheta)}\Psi_{4}P_{\mathcal{H}(\Theta)}\Psi_{1}f=P_{\mathcal{H}(\vartheta)}\Psi_{2}\Theta^{Ad}P_{\mathcal{H}(\Theta)}\Psi_{1}f$
$\displaystyle=P_{\mathcal{H}(\vartheta)}\Psi_{2}\Theta^{Ad}(I_{H^{2}_{N}}-P_{\Theta
H^{2}_{N}})\Psi_{1}f=P_{\mathcal{H}(\vartheta)}\Psi_{2}\Theta^{Ad}\Psi_{1}f.$

Therefore, there exists $\psi_{0}\in H^{\infty}$ such that
$\Psi_{2}\Theta^{Ad}\Psi_{1}=1+\psi_{0}\vartheta$. Denote by $\psi_{1n}$ and
$\psi_{2n}$, $n=1,\ldots,N$, the elements of the column $\Psi_{1}$ and the row
$\Psi_{2}$, respectively. Then
$\Psi_{2}\Theta^{Ad}\Psi_{1}=\sum_{n,k=1}^{N}\theta_{nk}\psi_{1k}\psi_{2n}=1+\psi_{0}\vartheta$.
Consequently,

$\inf_{z\in\mathbb{D}}\Bigl{(}\sum_{n,k=1}^{N}|\theta_{nk}(z)|+|\vartheta(z)|\Bigr{)}>0.$

Since $\det\Theta^{Ad}=\vartheta^{N-1}$, we have

$|\vartheta(z)|^{N-1}=|\det\Theta^{Ad}(z)|\leq
N!\Bigl{(}\max_{n,k=1,\ldots,N}|\theta_{nk}(z)|\Bigr{)}^{N},\ \
z\in\mathbb{D}.$

Consequently, if $\inf_{z\in\mathbb{D}}\sum_{n,k=1}^{N}|\theta_{nk}(z)|=0$,
then
$\inf_{z\in\mathbb{D}}(\sum_{n,k=1}^{N}|\theta_{nk}(z)|+|\vartheta(z)|)=0$, a
contradiction. ∎

######  Example 3.3.

Let $\vartheta_{1}$, $\vartheta_{2}\in H^{\infty}$ be two inner functions, and
let $c\in\mathbb{C}$ be such that
$|c|^{2}=\frac{1}{1+|\vartheta_{1}(0)|^{2}}$. Set $\chi(z)=z$,
$z\in\mathbb{D}$, and

(3.1)
$\Theta=\left[\begin{matrix}\frac{c\vartheta_{2}}{2}\frac{(1-\chi)\vartheta_{1}-(1+\chi)\vartheta_{1}(0)}{\chi}&\frac{c}{2}\frac{(1+\chi)\vartheta_{1}-(1-\chi)\vartheta_{1}(0)}{\chi}\\\
&\\\
\frac{\overline{c}\vartheta_{2}}{2}(1+\chi+(1-\chi)\vartheta_{1}\overline{\vartheta_{1}(0)})&\frac{\overline{c}}{2}(1-\chi+(1+\chi)\vartheta_{1}\overline{\vartheta_{1}(0)})\end{matrix}\right].$

Clearly, $\Theta\in H^{\infty}(\mathbb{C}^{2},\mathbb{C}^{2})$.
Straightforward calculation shows that
$\Theta^{*}(\zeta)\Theta(\zeta)=I_{\mathbb{C}^{2}}$ for a.e.
$\zeta\in\mathbb{T}$. Consequently, $\Theta$ is an inner function (see the
references in Introduction). We have $\det\Theta=-\vartheta_{1}\vartheta_{2}$.
Denote the elements of the matrix $\Theta^{Ad}$ by $\theta_{nk}$, $n,k=1,2$.
Then $\Theta=\left[\begin{matrix}\theta_{22}&-\theta_{12}\\\
-\theta_{21}&\theta_{11}\end{matrix}\right]$ and
$\Theta^{Ad}=\left[\begin{matrix}\theta_{11}&\theta_{12}\\\
\theta_{21}&\theta_{22}\end{matrix}\right]$. If there exists a sequence
$\\{z_{l}\\}_{l}\subset\mathbb{D}$ such that $z_{l}\to 1$ and
$\vartheta_{k}(z_{l})\to 0$ for $k=1,2$, then

$\inf_{z\in\mathbb{D}}\sum_{n,k=1}^{2}|\theta_{nk}(z)|=0.$

For $a>0$ set

(3.2) $\alpha_{a}(z)=\exp\Bigl{(}a\frac{z+1}{z-1}\Bigr{)},\ \ z\in\mathbb{D}.$

Now let $a_{k}>0$, and let $\vartheta_{k}(z)=\alpha_{a_{k}}$, $k=1,2$. Then
$\vartheta_{k}(z)\to 0$ for $k=1,2$, when $z\in(0,1)$, $z\to 1$. Furthermore,
the greatest common inner divisor of $\theta_{nk}$, $n,k=1,2$, is $1$. Indeed,
if $\alpha$ is the greatest common inner divisor of $\theta_{nk}$, $n,k=1,2$,
then $\alpha$ is a divisor of $\det\Theta=-\alpha_{a_{1}+a_{2}}$.
Consequently, $\alpha=\alpha_{a_{0}}$ for some $a_{0}>0$. In particular,
$\theta_{11}=\alpha_{a_{0}}g_{0}$ for some $g_{0}\in H^{\infty}$. By
definition,

$\theta_{11}=\frac{\overline{c}}{2}(1-\chi+(1+\chi)\vartheta_{1}\overline{\vartheta_{1}(0)})=\frac{\overline{c}}{2}(1-\chi+(1+\chi)\alpha_{a_{1}}\overline{\alpha_{a_{1}}(0)}).$

It follows $1-\chi=\alpha_{\min(a_{1},a_{0})}g$ for some $g\in H^{\infty}$, a
contradiction.

Thus, if $\Theta$ is defined by (3.1) with $\vartheta_{k}(z)=\alpha_{a_{k}}$
for some $a_{k}>0$, $k=1,2$, where $\alpha_{a_{k}}$ are defined by (3.2), then
Lemma 3.2 and the consideration above imply that $T_{\Theta}\sim
T_{\alpha_{a_{1}+a_{2}}}$ and $T_{\Theta}\not\approx
T_{\alpha_{a_{1}+a_{2}}}$. By [2, Exersices IV.1.13 and IV.1.14], $T_{\Theta}$
is unicellular (because $T_{\Theta}\sim T_{\alpha_{a}}$ for some $a>0$; see
also [14, Sec. III.7.2], [15, Sec. IV.3] and [2, Proposition VII.1.21]).

##  4\. On operators of class $C_{0}$ whose minimal function is a
Carleson–Newman product

Remember the definitions. For $\lambda\in\mathbb{D}$ set
$b_{\lambda}(z)=\frac{|\lambda|}{\lambda}\frac{\lambda-z}{1-\overline{\lambda}z}$,
$z\in\mathbb{D}$, if $\lambda\neq 0$, and $b_{\lambda}(z)=z$,
$z\in\mathbb{D}$, if $\lambda=0$. It is well known that if
$\\{\lambda_{n}\\}_{n}\subset\mathbb{D}$, then the product
$B=\prod_{n}b_{\lambda_{n}}$ converges if and only if
$\sum_{n}(1-|\lambda_{n}|)<\infty$ (that is, $\\{\lambda_{n}\\}_{n}$ satisfies
the _Blaschke condition_). The product $B$ is called a _Blaschke product_. Let
a sequence $\\{\lambda_{n}\\}_{n}\subset\mathbb{D}$ be such that
$\lambda_{n}\neq\lambda_{k}$, if $n\neq k$, and
$\sum_{n}(1-|\lambda_{n}|)<\infty$. Set $B_{n}=\prod_{k\neq
n}b_{\lambda_{k}}$. The sequence $\\{\lambda_{n}\\}_{n}$ satisfies the
_Carleson interpolating condition_ (the Carleson condition for brevity), if
$\inf_{n}|B_{n}(\lambda_{n})|>0$. A _Carleson–Newman product_ is a finite
product of Blaschke products with simple zeros satisfying the Carleson
condition.

######  Lemma 4.1.

Let $\Lambda\subset\mathbb{D}$ satisfy the Blaschke condition and the Carleson
condition. Set $B=\prod_{\lambda\in\Lambda}b_{\lambda}$. Let $T$ be a
polynomially bounded operator of class $C_{0}$. Suppose that $B$ is the
minimal function of $T$ and $M=\mu_{T}<\infty$. Set

$\displaystyle\Lambda_{n}$ $\displaystyle=\\{\lambda\in\Lambda\ :\
\dim\ker(T-\lambda I)\geq n\\},\ \
B_{n}=\prod_{\lambda\in\Lambda_{n}}b_{\lambda},\ \ n\in\mathbb{N},$
$\displaystyle\text{ and }\Theta$
$\displaystyle=\left[\begin{matrix}B_{1}&0&\ldots&0\\\ 0&B_{2}&\ldots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\ldots&B_{M}\end{matrix}\right].$

Then $T\approx T_{\Theta}$.

###### Proof.

Set $k(\lambda)=\dim\ker(T-\lambda I)$, $\lambda\in\Lambda$, and

$R=\oplus_{\lambda\in\Lambda}\lambda I_{\mathbb{C}^{k(\lambda)}}.$

It is well known (see, for example, [4, proof of Theorem 2.1], [5, Corollary
3.3], [23, Lemma 2.3]) that $T\approx R$. Since $\ker(R-\lambda I)$ is a
reducing subspace for $R$ and $\mu_{R|_{\ker(R-\lambda I)}}=k(\lambda)$, we
have $k(\lambda)\leq\mu_{R}=\mu_{T}=M$.

Since $T_{B_{n}}\approx\oplus_{\lambda\in\Lambda_{n}}\lambda I_{\mathbb{C}}$
and $T_{\Theta}\cong\oplus_{n=1}^{M}T_{B_{n}}$, we have

$T_{\Theta}\approx\oplus_{n=1}^{M}\oplus_{\lambda\in\Lambda_{n}}\lambda
I_{\mathbb{C}}=R.$

Thus, $T\approx T_{\Theta}$. ∎

In the next two lemmas, one deals with a triangulation of an operator $T$ of
the form (4.1) just below, which is given before the formulations of the
lemmas to avoid repetition:

(4.1) $T=\left[\begin{matrix}T_{1}&*&\ldots&*&*\\\
\mathbb{O}&T_{2}&\ldots&*&*\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
\mathbb{O}&\mathbb{O}&\ldots&T_{N-1}&*\\\
\mathbb{O}&\mathbb{O}&\ldots&\mathbb{O}&T_{N}\end{matrix}\right].$

######  Lemma 4.2.

Let $N\in\mathbb{N}$, and let $\vartheta_{n}$, $n=1,\ldots,N$, be inner
functions. Set $\vartheta=\prod_{n=1}^{N}\vartheta_{n}$. Suppose that $T$ is
an a.c. continuous polynomially bounded operator and
$\vartheta(T)=\mathbb{O}$. Then $T$ has a triangulation of the form (4.1),
where $\vartheta_{n}(T_{n})=\mathbb{O}$, $n=1,\ldots,N$.

###### Proof.

The lemma will be proved using induction. Let $N=2$. Denote by $\mathcal{H}$
the space on which $T$ acts. Set $\mathcal{H}_{1}=\ker\vartheta_{1}(T)$,
$T_{1}=T|_{\mathcal{H}_{1}}$,
$\mathcal{H}_{2}=\mathcal{H}\ominus\mathcal{H}_{1}$, and
$T_{2}=P_{\mathcal{H}_{2}}T|_{\mathcal{H}_{2}}$. Since

$\mathcal{H}_{2}=\operatorname{clos}(\vartheta_{1}(T))^{*}\mathcal{H}=\operatorname{clos}\widetilde{\vartheta}_{1}(T^{*})\mathcal{H}$

and

$\\{0\\}=\operatorname{clos}\widetilde{\vartheta}_{2}(T^{*})\widetilde{\vartheta}_{1}(T^{*})\mathcal{H}=\operatorname{clos}\widetilde{\vartheta}_{2}(T^{*})\mathcal{H}_{2},$

we conclude that
$\widetilde{\vartheta}_{2}(T^{*})|_{\mathcal{H}_{2}}=\mathbb{O}$.
Consequently,
$\mathbb{O}=(\widetilde{\vartheta}_{2}(T^{*})|_{\mathcal{H}_{2}})^{*}=\vartheta_{2}(T_{2})$.

If $N>2$, assume that the lemma is proved for $N-1$. Apply the case $N=2$ to
$T$ and the representation
$\vartheta=(\prod_{n=1}^{N-1}\vartheta_{n})\cdot\vartheta_{N}$. Then

$T=\left[\begin{matrix}R&*\\\ \mathbb{O}&T_{N}\end{matrix}\right],$

where $(\prod_{n=1}^{N-1}\vartheta_{n})(R)=\mathbb{O}$ and
$\vartheta_{N}(T_{N})=\mathbb{O}$. The conclusion of the lemma follows from
the obtained representation of $T$ and the inductive hypothesis applied to
$R$. ∎

The proof of the following lemma is based on [19, Lemma 1].

######  Lemma 4.3.

Suppose that $N\in\mathbb{N}$, $T\in\mathcal{L}(\mathcal{H})$ is a
contraction, and $T$ has a triangulation of the form (4.1). Denote by
$\mathcal{H}_{n}$ the space on which $T_{n}$ acts, $n=1,\ldots,N$. Suppose
that for every $n$, $n=1,\ldots,N$, there exist $M_{n}\in\mathbb{N}$ and
$Y_{n}\in\mathcal{I}(S_{M_{n}},T_{n})$ such that
$Y_{n}H^{2}_{M_{n}}=\mathcal{H}_{n}$. Set $M=\sum_{n=1}^{N}M_{n}$. Then there
exists $Y\in\mathcal{I}(S_{M},T)$ such that $YH^{2}_{M}=\mathcal{H}$.

###### Proof.

The lemma will be proved using induction. Let $N=2$. Then $T$ has the form
$T=\left[\begin{matrix}T_{1}&A\\\ \mathbb{O}&T_{2}\end{matrix}\right]$ for
some transformation $A$. By [19, Lemma 1] applied to $T_{2}$ and $Y_{2}$,
there exists $Z\in\mathcal{L}(H^{2}_{M_{2}},\mathcal{H}_{1})$ such that

$\left[\begin{matrix}I_{\mathcal{H}_{1}}&Z\\\
\mathbb{O}&Y_{2}\end{matrix}\right]\left[\begin{matrix}T_{1}&\mathbb{O}\\\
\mathbb{O}&S_{M_{2}}\end{matrix}\right]=\left[\begin{matrix}T_{1}&A\\\
\mathbb{O}&T_{2}\end{matrix}\right]\left[\begin{matrix}I_{\mathcal{H}_{1}}&Z\\\
\mathbb{O}&Y_{2}\end{matrix}\right].$

Set $Y=\left[\begin{matrix}Y_{1}&Z\\\ \mathbb{O}&Y_{2}\end{matrix}\right]$. It
is easy to see that $Y$ satisfies the conclusion of the lemma for $N=2$.

If $N>2$, assume that the lemma is proved for $N-1$. Write

$T=\left[\begin{matrix}R&*\\\ \mathbb{O}&T_{N}\end{matrix}\right],$

where $R$ has the form (4.1) for $N-1$. Let $W$ be a transformation obtained
by applying the inductive hypothesis to $R$. Apply the case $N=2$ to the
operator $T$ and transformations $W$ and $Y_{N}$. The obtained transformation
satisfies the conclusion of the lemma. ∎

######  Theorem 4.4.

Let $N\in\mathbb{N}$, and let $B_{n}$, $n=1,\ldots,N$, be Blaschke products
with simple zeros satisfying Carleson condition. Set $B=\prod_{n=1}^{N}B_{n}$.
Let $T$ be an a.c. continuous polynomially bounded operator. Suppose that
$B(T)=\mathbb{O}$ and $\mu_{T}<\infty$. Then $T$ is similar to a contraction
with finite defects.

###### Proof.

It is well known (see, for example, [6, Theorem 4.4] or [8, Theorem 3.2],
based on [20], see also [15, Ch. IX], [16, Sec. C.3]) that if $T$ is an a.c.
continuous polynomially bounded operator, $B$ satisfies the assumption of the
theorem, and $B(T)=\mathbb{O}$, then $T$ is similar to a contraction.
Therefore, we may assume that $T$ is a contraction. Denote by $\mathcal{H}$
the space on which $T$ acts. Apply Lemma 4.2 to $T$ and
$B=\prod_{n=1}^{N}B_{n}$. We obtain a triangulation of $T$ of the form (4.1),
where $B_{n}(T_{n})=\mathbb{O}$, $n=1,\ldots,N$. By [2, Corollaries III.5.26
and III.5.27], $M_{n}:=\mu_{T_{n}}\leq\mu_{T}$ for every $n=1,\ldots,N$. Let
$B_{1n}$ be the minimal function of $T_{n}$. Since $B_{1n}$ divides $B_{n}$,
the sequence of zeros of $B_{1n}$ satisfies the Carleson condition. Let
$\Theta_{n}\in H^{\infty}(\mathbb{C}^{M_{n}},\mathbb{C}^{M_{n}})$ be a
function from Lemma 4.1 applied to $T_{n}$, $n=1,\ldots,N$. Then $T_{n}\approx
T_{\Theta_{n}}$. Denote by $\mathcal{H}_{n}$ the space on which $T_{n}$ acts.
There exists an invertible transformation
$X_{n}\in\mathcal{I}(T_{\Theta_{n}},T_{n})$. Set
$Y_{n}=P_{\mathcal{H}(\Theta_{n})}X_{n}$. Then
$Y_{n}\in\mathcal{I}(S_{M_{n}},T_{n})$ and
$Y_{n}H^{2}_{M_{n}}=\mathcal{H}_{n}$.

Set $M=\sum_{n=1}^{N}M_{n}$. Let $Y$ be a transformation from the conclusion
of Lemma 4.3 applied to $T$. Then $YS_{M}=TY$ and $YH^{2}_{M}=\mathcal{H}$.
Set $\mathcal{K}=H^{2}_{M}\ominus\ker Y$ and
$R=P_{\mathcal{K}}S_{M}|_{\mathcal{K}}$. Then $R$ is a contraction with finite
defects and $Y|_{\mathcal{K}}$ realizes the similarity $R\approx T$. ∎

## References

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Three-loop contributions to $b\to s\gamma$ associated with the current-current
operators

Christoph Greuba, Hrachia M. Asatrianb, Hrachya H. Asatryanb,c,
Lukas Borna and Julian Eichera

aAlbert Einstein Center for Fundamental Physics, Institute for Theoretical
Physics,

University of Bern, CH-3012 Bern, Switzerland

bYerevan Physics Institute, 0036 Yerevan, Armenia

cYerevan State University, 0070 Yerevan, Armenia

In a recent work, we calculated all three-loop diagrams contributing to the
decay amplitude for $b\to s\gamma$ where none of the gluons touch the $b$-leg.
In the present paper, we complete the calculation by working out all remaining
three-loop diagrams (of order $\alpha_{s}^{2}$) associated with the current-
current operators $O_{1}$ and $O_{2}$ at the physical value of the charm-quark
mass $m_{c}$. Using the programs AMFlow and DiffExp to solve the differential
equations for the master integrals, we obtained precise numerical results at
23 values for $z=m_{c}^{2}/m_{b}^{2}$, ranging from $z=1/1000$ to $z=1/5$,
along with asymptotic expansions around $z=0$. For certain diagrams, the
asymptotic expansion breaks down in the physical $z$-range, necessitating a
Taylor expansion (which we do around $z=1/10$). In all expansions, we retained
power terms up to $z^{20}$ and included the accompanying $\log(z)$ terms to
all powers for asymptotic expansions. Numerical results for the sum of all
diagrams (including those calculated in the previous paper) are presented in
tabular form, while the mentioned expansions of individual diagram classes are
provided electronically. We note that our results for the asymptotic
expansions around $z=0$ are in good agreement with those recently published by
Fael et al. and Czaja et al..

###### Contents

  1. 1 Introduction
  2. 2 Theoretical framework
    1. 2.1 Form factor decomposition
  3. 3 Differential equations
    1. 3.1 Calculation of boundary conditions for differential equations
    2. 3.2 Solving the differential equations
  4. 4 Results
  5. 5 Summary
  6. A Details on the ancillary file
    1. A.1 Results for the individual contributions of different diagram classes to the form factor $\hat{B}$ in electronic form

##  1 Introduction

Rare $B$-meson decays have been the focus point of theorists and
experimentalists for some time, which is due to the potential they provide for
the tests of the Standard Model (SM) at scales of several hundreds of GeV.
Getting experimental information on rare decays puts strong constraints on the
extensions of the SM, or can lead to disagreements with the SM predictions,
providing evidence for new physics. To make a rigorous comparison between
experiment and theory, one has to get refined theoretical predictions for the
rare decay at hand. For inclusive rare $B$-decays, the perturbative strong
interaction effects result in sizable contributions.

In view of the expected increase in precision for the experimental
measurements of the decay $B\to X_{s}\gamma$, a full next-to-next-to-leading
logarithmic order (NNLL) calculation is necessary to reduce the theoretical
uncertainties and to enable us to perform a rigorous comparison with existing
and future experimental data. The first estimate of the branching ratio at
$O(\alpha_{s}^{2})$, leading to ${\cal B}(B\to X_{s}\gamma)=(3.15\pm
0.23)\times 10^{-4}$, was done in [1], which is consistent with the
experimental averages at the 1.2 $\sigma$ level. An updated version for this
branching ratio, incorporating all results for NNLL contributions and lower-
order perturbative corrections that have been calculated after 2006, was
published in [2] where some of us were involved. The CP- and isospin-averaged
branching ratio reads ${\cal B}(B\to X_{s}\gamma)=(3.36\pm 0.23)\times
10^{-4}$ which is in agreement with the current experimental averages.

It is well known that a part of the $\alpha_{s}^{2}$ contributions in [2] was
obtained via interpolation, i.e., using the results obtained through the large
$m_{c}$ asymptotic expansion on one hand and the results for $m_{c}=0$ on the
other hand. In the process of evaluating $\alpha_{s}^{2}$ corrections directly
at the physical value of $m_{c}$ in [3], the part stemming from diagrams with
closed fermion loops on gluon lines that contribute to the interference of the
current-current and photonic dipole operators was calculated, extending
previous work on such contributions [4, 5, 6].

In [7] we started the computation of virtual $\alpha_{s}^{2}$ corrections to
the decay amplitude $b\to s\gamma$ associated with the current-current
operators111This work can be considered as an extension of our work [10] on
the corresponding $\alpha_{s}^{1}$ corrections., by working out all those
diagrams where no gluons are touching the $b$-quark line. These contributions,
which we denote in the following as “$s$-leg diagrams”, are shown in Figure 1
of [7]. A few months later, two papers were published [8, 9] in which our
results were confirmed and extended to the complete set of virtual correction
diagrams. As these highly non-trivial computations are a crucial ingredient in
the NNLL program, we decided to do in our present work an independent
calculation of the contributions which we did not include in our paper [7]. To
this end, we calculate in the present paper all diagrams of order
$\alpha_{s}^{2}$ in Figure 1 where no gluon touches the $s$-quark line
(referred to as “$b$-leg diagrams”), as well as the “mixed diagrams” in Figure
2, and finally the “bubble diagrams” in Figure 3 where the gluon propagator is
dressed by a fermion-, a gluon- or a ghost-loop. Note that for the “bubble
diagrams” it is sufficient to calculate only the diagrams where the gluon
propagator is dressed by a fermion loop; as has been shown in [11, 12, 13],
the sum of the ghost- and the gluon-loop can be obtained from the massless
fermion loop diagrams by replacing (in the Feynman gauge)

$n_{\ell}\,tr\,\to-
C_{A}\,\left(\frac{5}{4}+\frac{\epsilon}{2}+\frac{\epsilon^{2}}{2}+\frac{\epsilon^{3}}{2}+O(\epsilon^{4})\right)\,,$
(1.1)

where $n_{\ell}$ denotes the number of massless quarks and $C_{A}$ and $tr$
are color factors.

The remainder of this paper is organized as follows: In Section 2 we present
the theoretical framework and a few conventions. In Section 3 we present the
methodology of solving differential equations using the program AMFlow, as
well as checks we have done using the program DiffExp. In Section 4, we
present numerical results for $z\doteq m_{c}^{2}/m_{b}^{2}=0$ and at 23 points
between $z=1/1000$ and $z=1/5$. We also provide the result as an asymptotic
expansion around $z=0$ (which contains non-negative integer and half-integer
powers of $z$ and non-negative integer powers of $\log(z)$). For this formula
we get perfect agreement with the results in [8]. However, as this expansion
breaks down in the physical $z-$range for some diagrams in the “$b$-leg”
class, we also provide a Taylor expansion for this class around the value
$z=1/10$ (which is a typical physical value for $z$). We illustrate with a
plot that the transition between the two mentioned formulas is very smooth. In
Section 5 we summarize our work and give a short outlook. The results for
different classes of diagrams are submitted in electronic form together with
the paper, as described in the appendix.

##  2 Theoretical framework

$B$-meson or $b$-quark decay amplitudes are usually calculated within the Weak
Effective Theory (WET) where the SM particles with EW-scale masses have been
integrated out. The WET Lagrangian then contains QCD and QED interactions, and
a tower of higher dimensional local operators which is typically truncated at
dimension six [14, 15]. The part of the WET Lagrangian which is relevant for
the contributions discussed in this paper is

${\cal L}_{\rm WET}={\cal
L}^{(4)}+\frac{4G_{F}}{\sqrt{2}}V_{ts}^{*}V_{tb}\Big{[}C_{1}O_{1}+C_{2}O_{2}+C_{7}O_{7}\Big{]}\,,$
(2.1)

where

$\displaystyle
O_{1}=(\bar{s}\gamma_{\mu}P_{L}T^{a}c)(\bar{c}\gamma^{\mu}P_{L}T^{a}b)\ ,$
$\displaystyle O_{2}=(\bar{s}\gamma_{\mu}P_{L}c)(\bar{c}\gamma^{\mu}P_{L}b)\
,$ $\displaystyle
O_{7}=\frac{e}{16\pi^{2}}m_{b}(\bar{s}\sigma_{\mu\nu}P_{R}b)F^{\mu\nu}\ .$
(2.2)

${\cal L}^{(4)}$ contains the usual kinetic terms and the mass terms of the
quarks $u,d,s,c,b$ as well as their interactions with the photon and the
gluons. $P_{R,L}=(1\pm\gamma_{5})/2$ stand for the right and left projection
operator, $\sigma_{\mu\nu}\equiv(i/2)[\gamma_{\mu},\gamma_{\nu}]$ and our
convention for the covariant derivative is given by
$D_{\mu}q=(\partial_{\mu}+ieQ_{q}A_{\mu}+ig_{s}T^{A}G^{A}_{\mu})q$;
$m_{b}=m_{b}(\mu)$ denotes the $\overline{\rm MS}$ $b$-quark mass. In our
calculation of order $\alpha_{s}^{2}$ corrections from $O_{1,2}$, the scheme
dependence of $m_{b}$ is a higher order effect. Furthermore, we will neglect
the strange quark mass throughout our paper.

###  2.1 Form factor decomposition

As mentioned in some detail in [7], the decay amplitude ${\cal A}(b\to
s\gamma)=\langle s\gamma|O_{1,2}|b\rangle$ can be written as ${\cal
A}=M_{\mu}\,\varepsilon^{\mu}$, where $\varepsilon^{\mu}$ denotes the
polarization vector of the emitted photon. In the considered limit where the
strange quark mass is put to zero, $M_{\mu}$ is of the form

$M_{\mu}=\bar{u}_{s}(p_{s})P_{R}\left[A\,q_{\mu}+B\,p_{\mu}+C\,\gamma_{\mu}\right]u_{b}(p)\
.$ (2.3)

In this equation $u_{b}(p)$ denotes the Dirac spinor of the $b$-quark with
four-momentum $p$, $u_{s}(p_{s})$ is the analogous quantity for the $s$-quark
with four-momentum $p_{s}$ and $q=p-p_{s}$ is the four-momentum of the emitted
photon. Applying standard algebraic manipulations, the form factors $A$, $B$,
$C$ are given in terms of linear combinations of scalar three-loop
integrals222The form of such integrals can be seen e.g. in Eq. (3.4) in our
previous paper [7].. After performing these integrals and taking into account
that $q^{2}=0$ in our process, these form factors are functions of $m_{b}$ and
$m_{c}$ (and depend also on the renormalization scale $\mu$). Concerning gauge
invariance, it is instructive to look first at the corresponding 2-loop
amplitude: at the bare (i.e. unrenomalized) level, $M_{\mu}$ satisfies the
Ward identity $q^{\mu}\,M_{\mu}=0$, leading to the relation $C=-m_{b}/2\,B$.
Using this relation, the decay amplitude ${\cal A}=M_{\mu}\varepsilon^{\mu}$
can then be written (in bare 2-loop) as

${\cal A}=-\frac{4\pi^{2}}{e\,m_{b}}\,B\,\langle{O}_{7}\rangle_{\rm tree}\,,$
(2.4)

where the tree-level matrix element of the operator ${O}_{7}$ is given e.g. in
[7]. At 3-loop, however, the bare version of the Ward identity is broken as
pointed out in [8]. The Ward identity is restored only after renormalization
(actually, as also pointed out there and checked by one of us, it is the
renormalization of the $b$-quark mass (in the on-shell scheme) which does the
restoration). In our present paper, we only calculate the bare version of the
form factor $B$. When later including the renormalization of the amplitude, we
suggest to calculate the effects of the counterterms on $B$ only and then use
the Ward identity to get the renormalized version of $C$.

Two remarks concerning the diagrams leading to the bare version of the form
factor $B$ are in order, which are best illustrated in Figure 1 of [7]: first,
it is quite easy to see that those diagrams which are marked with a cross that
does not carry a number, would only contribute to the form factors $C$.
Therefore only the diagrams with numbered crosses (1-44) had to be worked out.
Second, diagrams which contain a 1-loop quark self-energy on the very left or
very right of the external $b$\- or $s$-leg (i.e. which formally contain a
zero propagator denominator) are omitted from the list of bare diagrams; their
effects can be taken into account later by quark wave function
renormalization. Note that in the present paper we only listed those
(numbered) diagrams which contribute to the bare form factor $B$. For the
calculation of this form factor we employ the method of differential
equations, which is detailed in Section 3.

Figure 1: “$b$-leg diagrams”: List of three-loop contributions to $b\to
s\gamma$ associated with $O_{1}$ and $O_{2}$, where no gluon touches the
$s$-leg. A cross on a quark line represents a possible place where the photon
can be emitted.

Figure 2: “Mixed diagrams”: List of three-loop contributions to $b\to s\gamma$
associated with $O_{1}$ and $O_{2}$, where one gluon touches the $b$-leg and
one touches the $s$-leg. A cross on a quark line represents a possible place
where the photon can be emitted.

Figure 3: List of the “bubble diagrams” contributing to $b\to s\gamma$
associated with $O_{1}$ and $O_{2}$. A cross on a quark line represents a
possible place where the photon can be emitted. The particle running in the
bubble (inserted in the gluon line) is one of the following: a massless
fermion ($u,d,s$-quarks), a fermion with mass $m_{c}$, a fermion with mass
$m_{b}$, a gluon, or a ghost.

##  3 Differential equations

In this section, we explain the details of the calculations of the form factor
$B$ using the method of differential equations. The general outline of this
method is as follows: express each diagram (or sets of diagrams) as a linear
combination of scalar integrals, which in turn are reduced to a smaller number
of master integrals by applying Integration-By-Parts (IBP) identities to the
scalar integrals (see [16, 17, 18, 19, 20, 21, 22, 23, 24]). We use the
program Kira [17] to bring the scalar integrals to a set of master integrals.
In general, there are many different sets (we can call each set a basis) of
master integrals we can reduce to. In [21], an algorithm is suggested which
chooses a “good” basis in the sense that the $\epsilon$ dependence factors out
in the denominators of the relations in which the scalar integrals are
expressed in terms of master integrals; this simplifies further steps in the
calculation, in particular solving the differential equations. We therefore
use this “good” basis throughout our calculations.

Using the IBP identities generated by Kira, one can construct a system of
differential equations for the master integrals with respect to $z$. These
equations are of the form

$\partial_{z}J_{k}(\epsilon,z)=a^{k\ell}(\epsilon,z)J_{\ell}(\epsilon,z),$
(3.1)

where $a^{k\ell}$ are the entries of a $N\times N$ matrix depending on
$\epsilon$ and $z$. The derivatives of the MIs $J_{k}$ are obtained by
differentiating the integrands, which produce new scalar integrals. These
scalar integrals are then subjected to IBP reduction again to express the
derivatives $\partial_{z}J_{k}$ in terms of the MIs $J_{\ell}$ (for more
detail, see [7]).

In [7], two methods were used for solving the differential equations. For most
diagrams, we were able to transform the differential equation into canonical
form [25], where the equations could then be solved in an iterative manner as
an expansion in $\epsilon$. The $z$ dependence in these solutions are
contained in Generalized Polylogarithms (GPLs) [26]. After fixing the
integration constants (using the large $m_{c}$ behavior of the MIs), we were
left with purely analytical precise results. However, for two sets of diagrams
(namely (11, 12) and (13, 14)), we were unable to transform the differential
equations to canonical form. As such, a different method was used to solve
these differential equations: a series expansion was constructed around $z=0$,
by bringing the differential equation matrix first to Fuchsian form [27] and
then to Jordan form. This leads to a simple first-order linear differential
equation, which has $N$ linearly independent fundamental solutions. To fix the
integration constants, the program FIESTA5 [28] was used. This program allows
to numerically calculate the leading terms of the $z$-expansion of the master
integrals directly from their integral representations. This entire process is
discussed in detail in [7]. The problem with this second approach is the
precision of FIESTA5; the authors of [8] state that when comparing with the
results of our “$s$-leg diagrams”, for the sets of diagrams involving FIESTA5
results, the agreement was only 5 digits, while for other sets, it was at
least 10 digits. Given the extra complexity of the classes of diagrams we
solve in the present paper, it is problematic to use the first method (that
is, solving the differential equations analytically) for most sets of
diagrams, forcing us to use the second method which involves FIESTA5. We
found, however, that the results suffer from very low precision; therefore we
use a different method for the solution of our differential equations, which
is discussed in the following subsections.

###  3.1 Calculation of boundary conditions for differential equations

To solve the system of differential equations for a given diagram in the three
classes considered in this paper, we need $N$ boundary conditions, where $N$
is the number of MIs in the diagram. We use the program AMFlow [29] to
calculate these boundary conditions at a specific point. In our application
AMFlow receives as input the value of $m_{c}^{2}$ and $m_{b}^{2}$ (or the
value of $z=m_{c}^{2}/m_{b}^{2}$, when working in units where $m_{b}$ is put
to $1$, as we do), and calculates the given list of master integrals for these
values of the quark masses with very high precision as a Laurent series of
sufficiently high order in $\epsilon$.

In practice, we used the AMFlow package to calculate the boundary values of
the MIs at $z=1/100$.

###  3.2 Solving the differential equations

To solve the differential equations for a given diagram in the three classes
considered in this paper, we use the DESolver package of AMFlow. DESolver
allows us to transport the results for the MIs from $z=1/100$ to any point. We
use this functionality to iteratively obtain the results for the MIs at the
following 23 points:

$\displaystyle\biggl{\\{}\frac{1}{1000},\frac{1}{500},\frac{1}{200},\frac{1}{100},\frac{2}{100},\frac{3}{100},\frac{4}{100},\frac{5}{100},\frac{6}{100},\frac{7}{100},\frac{8}{100},\frac{9}{100},$
$\displaystyle\frac{10}{100},\frac{11}{100},\frac{12}{100},\frac{13}{100},\frac{14}{100},\frac{15}{100},\frac{16}{100},\frac{17}{100},\frac{18}{100},\frac{19}{100},\frac{20}{100}\biggr{\\}}$
(3.2)

To validate these results, we also directly calculated the values of the MIs
at $z=1/10$ using AMFlow. When comparing the directly calculated values at
$z=1/10$ with those obtained via transportation from $z=1/100$, we have
perfect agreement; this consistency makes us confident that the results for
the MIs for each diagram at the 23 points mentioned above are very precise.
This in turn also means that we have very precise numerical results for the
contribution of the three classes of diagrams to the form factor $B$ (see Eq.
(2.3)) at these 23 points.

Besides these numerical results, it is convenient to have also an analytic
expression in terms of an asymptotic expansion around $z=0$ for these
contributions to the form factor $B$. In particular, such an expression also
covers the results for the (CKM unsuppressed) $u$-quark loops in the analogous
process $b\to d\gamma$, where the corresponding $z$ is $m_{u}^{2}/m_{c}^{2}$,
which is basically zero. The asymptotic expansion around $z=0$ can easily be
constructed on a diagram-by-diagram basis with the program DESolver, using as
input the differential equation matrix, as well as the boundary conditions for
the MIs which we have calculated at $z=1/100$. We worked out this asymptotic
expansion for the contribution to the form factor $B$ of all three classes of
diagrams. For the “mixed diagrams” and for the “bubble diagrams”, the obtained
asymptotic formulas coincide very well with the 23 points. Only for “$b$-leg
diagrams”, we have noticed that this expansion breaks down within the physical
$z$-range. This might be related to the presence of poles at $z=1/16$ in
certain entries of the matrices (in diagrams 3-6) which define the
differential equations; however, this “hypothesis”remains uncertain. In Figure
4, we combine the $\epsilon^{0}$ coefficients of the three classes of diagrams
with the “$s$-leg diagrams” computed in [7]; the breakdown of the asymptotic
expansion is clearly seen.

Figure 4: Real part of the $\epsilon^{0}$ coefficient of the $O_{2}$
contribution to $\hat{B}$ as defined in Eq. (4.1). The highlighted points are
the 23 points mentioned in the text. The solid line is the sum of the
expansion around $z=0$ for all four classes of diagrams. As can be seen, the
asymptotic expansion breaks down around the value of $z=0.07$.

Given that the physical value for $z$ is $\sim 1/10$ (i.e. higher than the
value for which the $z=0$ expansion breaks down), we need a different
expansion. To this end, we also worked out a Taylor expansion for the “$b$-leg
diagrams” around $z=1/10$. In Figure 5, we illustrate that the transition
between these two formulas is very smooth. We therefore propose to give the
final results for the “$b$-leg diagrams” in terms of a “switched expansion”,
by switching at $z=0.04$ from the asymptotic expansion to the Taylor
expansion, as suggested by the figure.

Figure 5: Real part of the $\epsilon^{0}$ coefficient of the $O_{2}$
contribution to $\hat{B}$ as defined in Eq. (4.1) for all “$b$-leg diagrams”.
The highlighted points are the 23 points mentioned in the text. The orange
line shows the sum of the asymptotic expansion for all diagrams in Figure 1,
while the blue line shows the corresponing Taylor expansion around $z=1/10$.
The breakdown of the asymptotic expansion near the physical range of $z$, as
well as the breakdown of the Taylor expansion near $z=0$ are clearly visible.
The transition from the asymptotic expansion to the Taylor expansion is very
smooth.

Using these two expansions, we are able to cover all values of $z$ from $0$ up
to and beyond the physical range. The two expansions are given up to order
$z^{20}$ and $(z-1/10)^{20}$, respectively, after having checked that this
accuracy is sufficient. For the asymptotic expansion we have kept the
accompanying $\log(z)$ terms to all powers of $z$. All our results are
presented in the electronic file submitted with the paper.

As mentioned at the beginning of this section, the results for “$s$-leg
diagrams” 11-14 in [7] are not very accurate due to limitations of the FIESTA5
program. Therefore, we have recalculated these four diagrams using our new
method. These new calculations are very precise and were used in the final
results given in the electronic file submitted with the paper.

Note that we have calculated most of the diagrams also using the program
DiffExp [30].

##  4 Results

In this section we present the contributions of all four classes of diagrams
of order $\alpha_{s}^{2}$ to the form factor $B$, both for $O_{1}$ and
$O_{2}$. In fact, we show our results in terms of the the dimensionless
quantity $\hat{B}$, defined according to

$B={\rm pref}\cdot\hat{B}\quad\mbox{with}\quad{\rm
pref}=-\frac{e\,m_{b}}{4\pi^{2}}\,g_{s}^{4}\,\left(\frac{\mu^{2}}{m_{b}^{2}}\right)^{3\epsilon}\,.$
(4.1)

In Tables 1 and 2 the dimensionless form factor $\hat{B}$ is given (for all
powers of $\epsilon$ up to $\epsilon^{0}$) at the 23 points mentioned in Eq.
(3.2) for the operators $O_{1}$ and $O_{2}$, respectively.

$O_{1}$ | $\frac{1}{\epsilon^{2}}$ | $\frac{1}{\epsilon^{1}}$ | $\epsilon^{0}$
---|---|---|---
$\frac{1}{1000}$ | $0.02522-0.02553i$ | $0.3025+0.08587i$ | $1.278+2.269i$
$\frac{1}{500}$ | $0.02572-0.02425i$ | $0.3025+0.07716i$ | $1.371+2.079i$
$\frac{1}{200}$ | $0.02716-0.02153i$ | $0.2936+0.05769i$ | $1.447+1.707i$
$\frac{1}{100}$ | $0.02935-0.01845i$ | $0.2732+0.03502i$ | $1.432+1.325i$
$\frac{2}{100}$ | $0.03314-0.01437i$ | $0.2322+0.005243i$ | $1.321+0.8652i$
$\frac{3}{100}$ | $0.03640-0.01156i$ | $0.1946-0.01436i$ | $1.200+0.5751i$
$\frac{4}{100}$ | $0.03931-0.009440i$ | $0.1602-0.02823i$ | $1.088+0.3689i$
$\frac{5}{100}$ | $0.04194-0.007765i$ | $0.1287-0.03832i$ | $0.9862+0.2139i$
$\frac{6}{100}$ | $0.04435-0.006405i$ | $0.09965-0.04568i$ | $0.8934+0.09385i$
$\frac{7}{100}$ | $0.04659-0.005284i$ | $0.07275-0.05097i$ | $0.8089-0.0005617i$
$\frac{8}{100}$ | $0.04867-0.004348i$ | $0.04775-0.05465i$ | $0.7317-0.07520i$
$\frac{9}{100}$ | $0.05062-0.003563i$ | $0.02447-0.05704i$ | $0.6612-0.1340i$
$\frac{10}{100}$ | $0.05245-0.002901i$ | $0.002748-0.05837i$ | $0.5967-0.1797i$
$\frac{11}{100}$ | $0.05418-0.002343i$ | $-0.01753-0.05885i$ | $0.5380-0.2144i$
$\frac{12}{100}$ | $0.05581-0.001873i$ | $-0.03648-0.05863i$ | $0.4846-0.2397i$
$\frac{13}{100}$ | $0.05735-0.001479i$ | $-0.05419-0.05783i$ | $0.4365-0.2568i$
$\frac{14}{100}$ | $0.05881-0.001150i$ | $-0.07074-0.05657i$ | $0.3935-0.2667i$
$\frac{15}{100}$ | $0.06020-0.0008767i$ | $-0.08619-0.05495i$ | $0.3556-0.2703i$
$\frac{16}{100}$ | $0.06151-0.0006529i$ | $-0.1006-0.05305i$ | $0.3228-0.2684i$
$\frac{17}{100}$ | $0.06276-0.0004721i$ | $-0.1140-0.05095i$ | $0.2951-0.2616i$
$\frac{18}{100}$ | $0.06395-0.0003285i$ | $-0.1266-0.04872i$ | $0.2725-0.2507i$
$\frac{19}{100}$ | $0.06508-0.0002173i$ | $-0.1382-0.04643i$ | $0.2554-0.2362i$
$\frac{20}{100}$ | $0.06615-0.0001341i$ | $-0.1490-0.04415i$ | $0.2438-0.2187i$
Table 1: Dimensionless form factor $1000\cdot\hat{B}$ of order $\alpha_{s}^{2}$ for the operator $O_{1}$, given at the 23 points in Eq. (3.2). The three columns are the coefficients of $\epsilon^{-2}$, $\epsilon^{-1}$ and $\epsilon^{0}$, respectively. The value of the $\epsilon^{-3}$ coefficient is $z$-independent and is equal to $-0.01012$ for all points. $O_{2}$ | $\frac{1}{\epsilon^{2}}$ | $\frac{1}{\epsilon^{1}}$ | $\epsilon^{0}$
---|---|---|---
$\frac{1}{1000}$ | $0.08632+0.1532i$ | $0.05402+0.2669i$ | $0.9238-4.765i$
$\frac{1}{500}$ | $0.08334+0.1455i$ | $0.03352+0.2671i$ | $0.3961-4.204i$
$\frac{1}{200}$ | $0.07466+0.1292i$ | $0.02862+0.2737i$ | $-0.1371-3.136i$
$\frac{1}{100}$ | $0.06152+0.1107i$ | $0.06222+0.2850i$ | $-0.2863-2.072i$
$\frac{2}{100}$ | $0.03880+0.08620i$ | $0.1548+0.2984i$ | $-0.1014-0.8485i$
$\frac{3}{100}$ | $0.01922+0.06936i$ | $0.2485+0.3023i$ | $0.1907-0.1181i$
$\frac{4}{100}$ | $0.001802+0.05664i$ | $0.3370+0.2998i$ | $0.4826+0.3712i$
$\frac{5}{100}$ | $-0.01399+0.04659i$ | $0.4195+0.2924i$ | $0.7541+0.7143i$
$\frac{6}{100}$ | $-0.02848+0.03843i$ | $0.4960+0.2815i$ | $1.002+0.9578i$
$\frac{7}{100}$ | $-0.04190+0.03170i$ | $0.5669+0.2679i$ | $1.225+1.129i$
$\frac{8}{100}$ | $-0.05439+0.02609i$ | $0.6325+0.2520i$ | $1.425+1.244i$
$\frac{9}{100}$ | $-0.06609+0.02138i$ | $0.6933+0.2345i$ | $1.604+1.314i$
$\frac{10}{100}$ | $-0.07707+0.01741i$ | $0.7494+0.2158i$ | $1.761+1.348i$
$\frac{11}{100}$ | $-0.08743+0.01406i$ | $0.8012+0.1960i$ | $1.899+1.351i$
$\frac{12}{100}$ | $-0.09721+0.01124i$ | $0.8489+0.1757i$ | $2.017+1.328i$
$\frac{13}{100}$ | $-0.1065+0.008874i$ | $0.8927+0.1549i$ | $2.115+1.284i$
$\frac{14}{100}$ | $-0.1152+0.006897i$ | $0.9328+0.1340i$ | $2.195+1.220i$
$\frac{15}{100}$ | $-0.1235+0.005260i$ | $0.9694+0.1132i$ | $2.256+1.139i$
$\frac{16}{100}$ | $-0.1314+0.003918i$ | $1.003+0.09278i$ | $2.297+1.044i$
$\frac{17}{100}$ | $-0.1389+0.002832i$ | $1.033+0.07284i$ | $2.318+0.9377i$
$\frac{18}{100}$ | $-0.1460+0.001971i$ | $1.060+0.05365i$ | $2.319+0.8212i$
$\frac{19}{100}$ | $-0.1528+0.001304i$ | $1.084+0.03541i$ | $2.299+0.6968i$
$\frac{20}{100}$ | $-0.1592+0.0008049i$ | $1.105+0.01837i$ | $2.256+0.5670i$
Table 2: Dimensionless form factor $1000\cdot\hat{B}$ of order
$\alpha_{s}^{2}$ for the operator $O_{2}$, given at the 23 points in Eq.
(3.2). The three columns are the coefficients of $\epsilon^{-2}$,
$\epsilon^{-1}$ and $\epsilon^{0}$, respectively. The value of the
$\epsilon^{-3}$ coefficient is $z$-independent and is equal to $0.06073$ for
all points.

We also present the values for $z\to 0$ separately, as the value at this point
is not obtained the same way as for the 23 points, but rather from the limit
of the asymptotic expansion. We obtain

$1000\cdot\hat{B}_{O_{1}}(0)=-\frac{0.01012}{\epsilon^{3}}+\frac{0.02486-0.02765i}{\epsilon^{2}}+\frac{0.2802+0.09776i}{\epsilon}+(0.6538+2.6231i)$
(4.2)
$1000\cdot\hat{B}_{O_{2}}(0)=\frac{0.06073}{\epsilon^{3}}+\frac{0.08846+0.1659i}{\epsilon^{2}}+\frac{0.2020+0.2813i}{\epsilon}+(4.3882-5.8342i)$
(4.3)

In Figures 6 and 7 we present the $\epsilon^{0}$ coefficient’s real and
imaginary parts of expansions for $O_{1}$ and $O_{2}$, respectively. Both
figures show the “switched expansions” (with switching at $z=0.04$), as well
as the values at the 23 points.

Figure 6: $\epsilon^{0}$ coefficient of the $O_{1}$ contributions to $\hat{B}$
as defined in Eq. (4.1). The highlighted 23 points are mentioned in the text.
The solid line is constructed by switching at $z=0.04$ from the asymptotic
expansion to the Taylor expansion.

Figure 7: $\epsilon^{0}$ coefficient of the $O_{2}$ contributions to $\hat{B}$
as defined in Eq. (4.1). The highlighted 23 points are mentioned in the text.
The solid line is constructed by switching at $z=0.04$ from the asymptotic
expansion to the Taylor expansion.

In [8], the bare form factors $\hat{B}$ associated with $O_{1}$ and $O_{2}$
(denoted there by $t_{2}^{Q_{1}}$ and $t_{2}^{Q_{2}}$) have recently been
presented as an asymptotic expansion around $z=0$. Modulo an overall (constant
and $\epsilon$-independent) prefactor, these asymptotic expansions perfectly
agree with our asymptotic expansions for $\hat{B}$, and consequently also
break down for $z$ values $\sim 1/10$, corresponding to the physical value.
This aspect, however, has not been mentioned in their paper.

##  5 Summary

In this paper we worked out three-loop diagrams (of order $\alpha_{s}^{2}$)
contributing to the decay amplitude for $b\to s\gamma$ associated with the
current-current operators $O_{1}$ and $O_{2}$ at different values of $m_{c}$.
As a continuation of the work done in [7], we have calculated diagrams where
at least one gluon is touching the $b$-quark line (see Figure 1 and Figure 2)
in the present paper. We have also worked out all three-loop diagrams with
bubbles on the gluon lines (see Figure 3). We have used AMFlow to calculate
the boundary conditions for all diagrams at $z=1/100$, as described in Section
3.1. Using these boundary conditions, we were able to calculate the results at
23 different points between $z=1/1000$ and $z=1/5$ with very high precision.
We have also calculated the asymptotic expansion around $z=0$ for all
diagrams, which are useful in many practical applications. For the “$b$-leg
diagrams”, the asymptotic expansion breaks down at $z\sim 0.07$, therefore we
also worked out a Taylor expansion around $z=1/10$, which corresponds to the
physical value. For most diagrams the same expressions have also been
calculated using DiffExp, and the results using these two programs agree with
very high precision. Our asymptotic expansions have been compared with the
asymptotic expansions in [8], and we noticed very good agreement.

The numerical results at the 23 points in Eq. (3.2) are given for $O_{1}$ and
$O_{2}$ in Tables 1 and 2, respectively. These results and the symbolic
expression for the asymptotic formula up to $z^{20}$ (as well as the symbolic
expression for the Taylor expansion for the “$b$-leg diagrams” up to
$(z-1/10)^{20}$) are given in electronic form in the file ancillary.m which is
submitted together with this paper.

## Acknowledgements

We would like to thank Martijn Hidding for very useful discussions concerning
his excellent program DiffExp. Many thanks also go to Xiao Liu for answering
our questions on his superb program AMFlow always in a very short time. C.G.
would like thank Matteo Fael for very useful discussions related to his work
[8]. Discussions with Valentin Hirschi, Nicolas Schalch and Yannick Ulrich on
various features connected with multiloop diagrams are also greatfully
acknowledged.

The work of C.G. is partially supported by the Swiss National Science
Foundation under grant 200020-204075.

H.M.A. and H.H.A. are supported by the Higher Education and Science Committee
of Armenia Program Grant No. 21AG‐1C084.

##  Appendix A Details on the ancillary file

###  A.1 Results for the individual contributions of different diagram classes
to the form factor $\hat{B}$ in electronic form

In the mathematica file “ancillary.m” (which is included in the submission of
this paper) we give the contributions to the form factor $\hat{B}$ as defined
in Eq. (4.1) for the 4 different classes of diagrams (“$b$-leg diagrams”,
“$s$-leg diagrams”, “mixed diagrams” and “bubble diagrams”).

All the results mentioned in the paper are given in this file. Firstly, the
values at the 23 points in Eq. (3.2) are presented for all 4 classes of
diagrams for both $O_{1}$ and $O_{2}$. For the $O_{1}$ contributions, these
are the expressions BlegPointsO1, SlegPointsO1, MixedPointsO1 and
BubblesPointsO1 for the 4 classes of diagrams, respectively (the same
expressions for $O_{2}$ are named accordingly). The asymptotic expansions are
also presented up to order $z^{20}$ for all classes of diagrams. The
corresponding expressions for $O_{1}$ are BlegAsymO1, SlegAsymO1, MixedAsymO1
and BubblesAsymO1. As mentioned in the paper, the Taylor expansion around
$z=1/10$ is also presented for the “$b$-leg diagrams”, as a function of $u$,
where $u=z-1/10$, up to order $u^{20}$. This expression for $O_{1}$ is
BlegTaylorO1.

Note that these formulas contain the following symbolic constants: $nl=3$
(number of ‘light’ quarks), $nb=1$ (number of quarks with mass equal to
$m_{b}$), $nc=1$ (number of quarks with mass equal to $m_{c}$),
$ca=3,cf=4/3,tr=1/2$ (color factors), $Qd=-1/3$ (charge of down-type quark)
and $Qu=2/3$ (charge of up-type quark). All these constants are given in the
ancillary file.

At the end of the ancillary file, we present the two functions
FormFactorBhatO1 and FormFactorBhatO2, which, for a given value of $z$, return
the form factor $\hat{B}$ corresponding to all 4 classes of diagrams, with all
the symbolic constants inserted, for $O_{1}$ and $O_{2}$, respectively. These
functions automatically choose the correct expansion for the “$b$-leg
diagrams” depending on whether the given value of $z$ is larger than $z=0.04$.

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