follows that $\\{T^{n}(f)\\}$ is a uniformly equicontinuous family. Since
$\|T\|\leq 1$, it is also uniformly bounded, so $\\{T^{n}(f)\\}$ is relatively
compact in $E$ by the Arzelá–Ascoli Theorem and thus

$T^{n_{k}}(f)\to g\text{ as }k\to\infty$

for some subsequence $(n_{k})_{k}$ and $g\in K$. By primitivity, dismissing
the trivial case of $f\neq 0$, we must have that $g>0$. We will show that $g$
is constant.

Observe that $g$ is itself also a cluster point of iterates of $T$ applied to
$g$. Indeed,

$\displaystyle\|T^{n_{k+1}-n_{k}}(g)-g\|=\|T^{n_{k+1}-n_{k}}(T^{n_{k}}f-v_{k})-g\|=\|T^{n_{k+1}}f-T^{n_{k+1}-n_{k}}(v_{k})-g\|\leq$
$\displaystyle\|T^{n_{k+1}}f-g\|+\|T^{n_{k+1}-n_{k}}(v_{k})\|\leq\|T^{n_{k+1}}f-g\|+\|v_{k}\|\to
0\text{ as }k\to\infty,$

where $v_{k}=T^{n_{k}}(f)-g\to 0$, so $T^{n_{k+1}-n_{k}}(g)\to g$ as
$k\to\infty$. Suppose that $n_{k+1}-n_{k}$ is bounded. Then $T^{p}(g)=g$ for
some $p$ so that $T^{np}(g)=g$ for arbitrarily large $np$. Otherwise, we have
that $T^{j_{k}}g\to g$ as $k\to\infty$ for $j_{k}\to\infty$. In either case,
we see that $g$ must be constant. Indeed, $\|T^{j}g\|$ is monotonically
decreasing in $j$, since $\|T\|\leq 1$. On the other hand, by primitivity,
$\|T^{j}g\|<\|g\|$ for sufficiently large $j$ and $g$ non-constant. To see
this, suppose that $U\subset\mathcal{A}$ is an open set small enough so that
$g(a)<\|g\|$ for all $a\in U$. By primitivity, for sufficiently large $j$ we
have that $\varrho^{j}(a)$ contains a letter in $U$ for all $a\in\mathcal{A}$,
which implies that $T^{j}(g)(a)<\|g\|$ for all $b\in\mathcal{A}$, that is,
$\|T^{j}g\|<\|g\|$. But $T$ is a contraction so then cannot converge to $g$,
which is a contradiction, so $g$ must be constant.

Recall that $T^{n_{k}}(f)\to g$ as $k\to\infty$. On the other hand,

$\|T^{n+1}(f)-g\|=\|T^{n+1}(f-g)\|\leq\|T^{n}(f-g)\|=\|T^{n}(f)-g\|,$

since $T(g)=g$, so that $\|T^{n}(f)-g\|$ is monotonically decreasing. Hence,
the whole sequence $T^{n}f\to g$ as $n\to\infty$, as required. If $f\notin K$
then $f=f_{+}-f_{-}$, where $f_{+}$ and $f_{-}\in K$, so by the above
$T^{n}(f)=T^{n}(f_{+})-T^{n}(f_{-})\to g_{+}-g_{-}$ as $n\to\infty$ so that
$T$ is strongly power convergent on all of $E$. By Theorem 5.10 the tiling
dynamical system $(\Omega,{\mathbb{R}})$ or, by Corollary 5.11, the subshift
$(X_{\varrho},\sigma)$, is uniquely ergodic. ∎

The following substitution (considered in [27, Exp. 12.1] in the one-sided
shift setting) is an example for which $T$ cannot be quasi-compact, and $r$
cannot be a pole of the resolvent, since $T$ is _not_ uniformly power
convergent, but the subshift is still uniquely ergodic by the above result:

######  Example 6.17.

Let $\mathcal{A}=S^{1}\subseteq{\mathbb{C}}$ and recall the substitution from
Example 3.3: $\varrho(z)=z\ \alpha z$, where $\alpha\in S^{1}$ is irrational
(i.e., $\alpha=e^{2\pi\beta i}$ for $\beta$ irrational). By Kronecker’s
Theorem, by density of the orbit of $[z]\mapsto[\alpha z]$, it is clear that
$\varrho$ is primitive. Also note that
$\Phi=\left\langle\text{id},\cdot\,\alpha\right\rangle$ is an equicontinuous
semigroup because it is generated by group translations.

We do not have that $T^{n}\mapsto P$ uniformly as $n\mapsto\infty$, where $P$
is a projection operator to the eigenline spanned by $\ell$. Indeed, first
note that an eigenmeasure is clearly provided by the Lebesgue measure on the
circle (since each letter map of the substitution is a rotation). Consider a
function $f\in E$, $\|f\|=1$ which has very small integral $c=c_{f}>0$ and yet
has $f(z)=1$ for $z=0$, $\alpha$, $2\alpha$, …, $n\alpha$ (by taking a very
spiky function this is of course possible; in fact, by taking a function with
negative spikes too, we may choose such a function so that $c=0$). Then by
Lemma 4.5, $(M^{n}(f))(0)=2^{n}$ and hence $T^{n}(f)(0)=1$, since all elements
of $\varrho^{n}(0)$ are of the form $[k\alpha]$ for some non-negative $k\leq
n$. Assuming uniform power convergence, we would have $T^{n}f\to
c_{f}\cdot\ell$ uniformly as $n\mapsto\infty$, that is
$\|T^{n}(f)-c_{f}\ell\|<\epsilon$ for sufficiently large $n$ over all
$\|f\|\leq 1$. However, $\|T^{n}(f)-c_{f}\ell\|\geq 1-c_{f}$, so we do not
have such uniform convergence and thus $r$ cannot be a pole of the resolvent,
and $M$ cannot be quasi-compact, by Theorem 6.12. However, whilst $T$ is not
uniformly power convergent, it is strongly power convergent by Theorem 6.16.
$\Diamond$

##  7\. Discrepancy estimates from a spectral gap

Throughout this section we make the following assumptions on $\varrho$:

  * (A1):

$T$ is uniformly power convergent;

  * (A2):

$T$ is irreducible.

The first assumption implies that $1$ is a simple pole of the resolvent [23,
Thm. 2.5]. It then follows from Theorem 4.23 that

  * •

$\sigma_{\mathrm{per}}(T)=\\{1\\}$;

  * •

$\varrho$ admits a (unique) natural length function $0\neq\ell\in K$ with
stretching factor $\lambda=r$.

Since $\sigma(M)=r\sigma(T)$ is compact, $r$ is a pole of the resolvent (and
thus isolated) and the only element of the peripheral spectrum of $M$, it
follows that $M$ has a _spectral gap_ , that is, $\sigma(M)$ has a ‘second
largest element’:

$r_{2}\coloneqq\sup\\{|\lambda|\mid\lambda\in\sigma(M),\lambda\neq r\\}<r.$

We fix a (necessarily unique) eigenmeasure $\mu\in K^{\prime}$, which we
normalise here with $\mu(\ell)=1$.

######  Remark 7.1.

By Theorem 6.12 the above conditions are satisfied whenever $\varrho$ is
primitive and $T$ is quasi-compact. This includes any primitive substitution
for which Theorem 6.3 applies, so the constructions to follow apply to those
relevant examples from the last section.  $\Diamond$

The existence of a spectral gap will allow for some control on the discrepancy
on the ‘expected weight’ of a weighted selection of tiles across a large
supertile relative to the ‘actual’ sum of weights. Given $f\in E$ (a ‘weight
function’), $a\in\mathcal{A}$, and $n\in{\mathbb{N}}$ we define

$\mathrm{Exp}(f,a,n)=r^{n}\ell(a)\cdot\mu(f).$ (8)

That is, we simply multiply the length of the $n$-supertile $\varrho^{n}(a)$
by the ‘average weight’ of $f$. Such quantities are prevalent in recent works
on bounded distance equivalence[18, 34].

######  Example 7.2.

For a primitive substitution on a finite alphabet, $\mu$ is represented by the
row vector $f=(f_{\alpha},f_{\beta},\ldots)$ of frequencies $f_{a}>0$ of each
letter $a\in\mathcal{A}$. Then the expected number of tiles of type $b$ in
$\varrho^{n}(a)$ is
$\mathrm{Exp}(\mathds{1}_{b},a,n)=r^{n}\ell(a)\mu(\mathds{1}_{b})=r^{n}\ell(a)(v\cdot
f)=r^{n}\ell(a)f_{b}$, where $v$ is the vector with entry corresponding to $b$
equal to $1$ and all other entries $0$. So we just multiply length of the
$n$-supertile by the frequency of occurrence of $b$. More generally, given a
weighted selection $v=(v_{\alpha},v_{\beta},\ldots)$ of tiles, where we have
one $v_{a}\in{\mathbb{R}}$ for each tile $a\in\mathcal{A}$,

$\mathrm{Exp}(f,a,n)=r^{n}\ell(a)(v\cdot f).$

Here, we are identifying $v$ with an arbitrary function $f\in E$, given by
$f(a)=v_{a}$.  $\Diamond$

We define the _actual_ weighted sum of $f$ across an $n$-supertile
$\varrho^{n}(a)$ by

$\mathrm{Act}(f,a,n)=\sum_{b\triangleleft\varrho^{n}(a)}f(b)=(M^{n}f)(a),$

where the second equality follows from Lemma 4.5. The result below bounds the
discrepancy between the expected and actual weight across all possible
supertiles:

######  Theorem 7.3.

Suppose assumptions (A1) and (A2) hold. There is some function
$\theta\colon{\mathbb{N}}\to{\mathbb{R}}_{>0}$ with $\sqrt[n]{\theta(n)}\to 1$
as $n\to\infty$ so that for all $f\in E$ with $\|f\|\leq 1$, $a\in\mathcal{A}$
and $n\in{\mathbb{N}}$ we have

$|\mathrm{Exp}(f,a,n)-\mathrm{Act}(f,a,n)|\leq\theta(n)(r_{2})^{n}.$

In particular, for any $\epsilon>0$, we have the following for sufficiently
large $n$:

$|\mathrm{Exp}(f,a,n)-\mathrm{Act}(f,a,n)|\leq(r_{2}+\epsilon)^{n}.$

In the above, recall that $r_{2}$ denotes the maximum modulus of element of
$\sigma(M)\setminus\\{r\\}$. Before proving this result, we require a few
technical lemmas regarding the spectrum of the operator restricted to the
subspace

$\Pi\coloneqq\\{f\in E\mid\mu(f)=0\\}.$

By uniform power convergence, we have $T^{n}\to P$ uniformly as $n\to\infty$,
where $P$ is a projection to the eigenline
$\Lambda=\langle\ell\rangle_{\mathbb{R}}$. Hence $\Pi=\ker(P)$ is a closed,
codimension one subspace and we have $E=\Lambda\oplus\Pi$. We let $T_{\Pi}$
denote the restriction to $\Pi$.

######  Lemma 7.4.

We have an inclusion $\sigma(T_{\Pi})\subseteq\sigma(T)$.

###### Proof.

Since $P$ is bounded (and thus continuous), its kernel is closed and so
$T_{\Pi}$ is a bounded operator on a Banach space. For such operators,
$\tau\in\sigma(T)$ if and only if $T-\tau\mathrm{Id}$ is not bijective.
Suppose that $\tau\in\sigma(T_{\Pi})$. First assume that $T-\tau\mathrm{Id}$
is not injective, so that $T(f)=\tau f$ for some $f\in\Pi$. The same $f$
demonstrates that $\tau\in\sigma(T)$.

So suppose instead that $T-\tau\mathrm{Id}$ is not surjective, and thus there
is some $g\in\Pi$ with $Tf-\tau f\neq g$ for any $f\in\Pi$. Assume, on the
other hand, that $\tau\notin\sigma(T)$ so that $Tf-\tau f=g$ for some $f\in
E$. We may write $f=a+b$ for $a\in\Lambda$ and $b\in\Pi$. Then

$T(a+b)-\tau(a+b)=(Ta-\tau a)+(Tb-\tau b)=g\in\Pi.$

But since $\Lambda$ and $\Pi$ are complementary this implies that $Ta-\tau
a=0$. If $a\neq 0$ it follows that $\tau=1$ (since then $a$ is a non-negative
multiple of $\ell$), which we already know is in $\sigma(T)$. Otherwise, we
have that $Tb-\tau b=g$, contradicting that $Tb-\tau b\neq g$ for all
$b\in\Pi$. ∎

We get the following consequence of uniform power convergence:

######  Lemma 7.5.

We have that $1\notin\sigma(T_{\Pi})$.

###### Proof.

Let $f\in\Pi$ be arbitrary; we have $T^{n}f\to P(f)=c\cdot\ell$ as
$n\to\infty$, uniformly in $f$. Clearly $c=0$, since $\mu(T^{n}f)=\mu(f)=0$
for all $n$ and $\mu(c\ell)=c$. Thus $T^{n}_{\Pi}\to 0$ uniformly so
$\|T_{\Pi}^{n}\|<1$ for some $n\in{\mathbb{N}}$. For any bounded operator,
$r(T)\leq\sqrt[n]{\|T^{n}\|}$, so we have $r(T_{\Pi})<1$ and thus
$1\notin\sigma(T_{\Pi})$. ∎

######  Lemma 7.6.

Suppose that $\kappa\in\sigma(T)$ with $\kappa\neq 1$. Then
$\kappa\in\sigma(T_{\Pi})$.

###### Proof.

Let $\kappa\in\sigma(T)$ with $\kappa\neq 1$. First assume that $Tf-\kappa
f=0$ for some $f\in E$ and write $f=a+b$ for $a=c\ell\in\Lambda$ and
$b\in\Pi$. Then

$Tf-\kappa f=c(1-\kappa)\ell+(Tb-\kappa b)=0.$

Since $c(1-\kappa)\ell\in\Lambda$, $Tb-\kappa b\in\Pi$, and the subspaces
$\Lambda$ and $\Pi$ are complementary, it is necessary that $\kappa=1$ or
$c=0$. Since we assume that $\kappa\neq 1$, we have that $c=0$ and thus
$f\in\Pi$, so that $\kappa\in\sigma(T_{\Pi})$ too.

So suppose instead that $Tf-\kappa f\neq g$ for all $f\in E$ and some $g\in
E$. Write $g=a+b=c\ell+b$ for $a\in\Lambda$ and $b\in\Pi$. Suppose that
$Tf-\kappa f=b$ for some $f\in\Pi$. Then

$T\left(\frac{c}{1-\kappa}\ell+f\right)-\kappa\left(\frac{c}{1-\kappa}\ell+f\right)=c\ell+(Tf-\kappa
f)=c\ell+b=g,$

contradicting that $Tx-\kappa x\neq g$ for all $x\in E$. Hence $Tf-\kappa
f\neq b$ for all $f\in\Pi$ and so $\kappa\in\sigma(T_{\Pi})$, as required. ∎

######  Corollary 7.7.

We have $\sigma(T)=\sigma(T_{\Pi})\sqcup\\{1\\}$ or, equivalently,
$\sigma(M)=\sigma(M_{\Pi})\sqcup\\{r\\}$.

###### Proof of Theorem 7.3.

We may write

$|\mathrm{Exp}(f,a,n)-\mathrm{Act}(f,a,n)|=|r^{n}\ell(a)\cdot\mu(f)-(M^{n}f)(a)|=|M^{n}(\mu(f)\ell-f)(a)|.$

Since $M=rT$ we thus have

$|\mathrm{Exp}(f,a,n)-\mathrm{Act}(f,a,n)|=r^{n}|T^{n}(\mu(f)\ell-f)(a)|=r^{n}|(T^{n}v)(a)|,$

where $v=\mu(f)\ell-f$. We have that
$\mu(v)=\mu(\mu(f)\ell-f)=\mu(f)(\mu(\ell))-\mu(f)=0$ (since we normalise with
$\mu(\ell)=1$) so $v\in T_{\Pi}$.

By Corollary 7.7 $r(T_{\Pi})=r_{2}/r$. By the Gelfand formula for the spectral
radius,

$\lim_{n\to\infty}\sqrt[n]{\|T_{\Pi}^{n}\|}=r(T_{\Pi}),\text{ hence
}\sqrt[n]{\|T_{\Pi}^{n}\|}\leq\psi(n)(r_{2}/r),$

where $\psi(n)\to 1$ as $n\to\infty$. Taking $n$th powers and substituting
into the above,

$|\mathrm{Exp}(f,a,n)-\mathrm{Act}(f,a,n)|=r^{n}|(T^{n}v)(a)|=r^{n}|(T_{\Pi}^{n}v)(a)|\leq
r^{n}\|T_{\Pi}^{n}\|\|v\|\leq\|v\|(\psi(n))^{n}(r_{2})^{n},$

where $(\psi(n))^{n}$ has $n$th roots converging to $1$ as $n\to\infty$. Note
that $\|v\|=\|\mu(f)\ell-f\|$ is bounded above by a constant $c$, since $\mu$
is a bounded function and $\|f\|\leq 1$, so the discrepancy estimate follows
taking $\theta(n)=c(\psi(n))^{n}$.

Let $\delta>0$ be arbitrary. Then $h(n)=(1+\delta)^{n}$ is such that
$\sqrt[n]{h(n)}=(1+\delta)>\sqrt[n]{\theta(n)}$, and thus $h(n)>\theta(n)$,
for sufficiently large $n$. So for such $n$,

$|\mathrm{Exp}(f,a,n)-\mathrm{Act}(f,a,n)|\leq\theta(n)(r_{2})^{n}\leq
h(n)(r_{2})^{n}\leq(1+\delta)^{n}(r_{2})^{n}=((1+\delta)r_{2})^{n}.$

Setting $0<\delta<\epsilon/r_{2}$ establishes the second result. ∎

##  8\. acknowledgements

The authors would like to thank Michael Baake, Joel Feinstein, Natalie Priebe-
Frank, Dirk Frettlöh, Alexey Garber, Philipp Gohlke, Samuel Petite, Nicolae
Strungaru and Reem Yassawi for valuable discussions. NM is funded by the
German Academic Exchange Service (DAAD) via the Postdoctoral Researchers
International Mobility Experience (PRIME) Fellowship programme. DR
acknowledges the support of the German Research Foundation (DFG) via SFB1283/1
2019-317210226. JW expresses his gratitude to the same SFB of the Faculty of
Mathematics of Bielefeld University where a part of this work was completed.

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-150 LHCb-PAPER-2013-035 21 August 2013

Observation of the decay $B^{0}_{s}\rightarrow\kern
4.14793pt\overline{\kern-4.14793ptD}{}^{0}\phi$

The LHCb collaboration†††Authors are listed on the following pages.

First observation of the decay $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ is reported using $pp$
collision data, corresponding to an integrated luminosity of 1.0
$\mbox{\,fb}^{-1}$, collected by the LHCb experiment at a centre-of-mass
energy of $7$ TeV. The significance of the signal is 6.5 standard deviations.
The branching fraction is measured relative to that of the decay
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ to be

$\frac{{\cal{B}}(B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi)}{{\cal{B}}(B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})}=0.069\pm
0.013~{}(\mathrm{stat})\pm 0.007~{}(\mathrm{syst}).$

The first measurement of the ratio of branching fractions for the decays
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $B^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ is found to be

$\frac{{\cal{B}}(B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})}{{\cal{B}}(B^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0})}=7.8\pm
0.7~{}(\mathrm{stat})\pm 0.3~{}(\mathrm{syst})\pm 0.6~{}(f_{s}/f_{d}),$

where the last uncertainty is due to the ratio of the $B^{0}_{s}$ and $B^{0}$
fragmentation fractions.

Submitted to Phys. Lett. B

© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.

LHCb collaboration

R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R.
Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M.
Chrzaszcz25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M.
Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E.
Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M.
Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C.
Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y.
David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11,
J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F.
Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F.
Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, P.
Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48, U. Egede52,
V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, A.
Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S. Farry51, D.
Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S.
Filippov32, M. Fiore16, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R.
Forty37, O. Francisco2, M. Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E.
Furfaro23,k, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58,
Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra Tico46, L. Garrido35, C.
Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph.
Ghez4, V. Gibson46, L. Giubega28, V.V. Gligorov37, C. Göbel59, D. Golubkov30,
A. Golutvin52,30,37, A. Gomes2, P. Gorbounov30,37, H. Gordon37, C. Gotti20, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, B.
Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45,
J. Harrison53, T. Hartmann60, J. He37, T. Head37, V. Heijne40, K. Hennessy51,
P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, M. Hess60, A.
Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5, C. Hombach53, P. Hopchev4, W.
Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50,
V. Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E.
Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R.
Jones46, C. Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, W. Kanso6, M.
Karacson37, T.M. Karbach37, I.R. Kenyon44, T. Ketel41, A. Keune38, B.
Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M.
Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G.
Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A.
Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G. Lanfranchi18, C.
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J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T.
Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C.
Linn11, B. Liu3, G. Liu37, S. Lohn37, I. Longstaff50, J.H. Lopes2, N. Lopez-
March38, H. Lu3, D. Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V.
Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G.
Mancinelli6, J. Maratas5, U. Marconi14, P. Marino22,s, R. Märki38, J. Marks11,
G. Martellotti24, A. Martens8, A. Martín Sánchez7, M. Martinelli40, D.
Martinez Santos41, D. Martins Tostes2, A. Martynov31, A. Massafferri1, R.
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Rodriguez59, S. Monteil5, D. Moran53, P. Morawski25, A. Mordà6, M.J.
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B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1,
M. Needham49, S. Neubert37, N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C.
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Nomerotski54, A. Novoselov34, A. Oblakowska-Mucha26, V. Obraztsov34, S.
Oggero40, S. Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M.
Otalora Goicochea2, P. Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, T.
Palczewski27, M. Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C.
Parkes53, C.J. Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N.
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Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L.
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Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M.
Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, A.
Popov34, D. Popov10, B. Popovici28, C. Potterat35, A. Powell54, J.
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G. Punzi22,r, W. Qian4, J.H. Rademacker45, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47,
A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, D.A. Roberts57, E. Rodrigues53, P.
Rodriguez Perez36, S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J.
Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz Valls35, G.
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V. Salustino Guimaraes2, B. Sanmartin Sedes36, M. Sannino19,i, R.
Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A.
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P. Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O.
Shevchenko42, V. Shevchenko30, A. Shires9, R. Silva Coutinho47, M. Sirendi46,
N. Skidmore45, T. Skwarnicki58, N.A. Smith51, E. Smith54,48, J. Smith46, M.
Smith53, M.D. Sokoloff56, F.J.P. Soler50, F. Soomro18, D. Souza45, B. Souza De
Paula2, B. Spaan9, A. Sparkes49, P. Spradlin50, F. Stagni37, S. Stahl11, O.
Steinkamp39, S. Stevenson54, S. Stoica28, S. Stone58, B. Storaci39, M.
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Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26, S. T’Jampens4,
M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E. Thomas37, J. van
Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D. Tonelli37, S. Topp-
Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38, M.T. Tran38, M.
Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M. Ubeda Garcia37,
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Regueiro36, C. Vázquez Sierra36, S. Vecchi16, J.J. Velthuis45, M. Veltri17,g,
G. Veneziano38, M. Vesterinen37, B. Viaud7, D. Vieira2, X. Vilasis-
Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45, A. Vorobyev29, V.
Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C. Wallace47, R. Wallace12, S.
Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wimberley57, J. Wishahi9, W. Wislicki27, M. Witek25, S.A. Wotton46, S.
Wright46, S. Wu3, K. Wyllie37, Y. Xie49,37, Z. Xing58, Z. Yang3, R. Young49,
X. Yuan3, O. Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L.
Zhang58, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A.
Zvyagin37.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37

aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy

##  1 Introduction

Measurements of the decay111The inclusion of charge conjugate processes is
implied, unless otherwise stated. $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ are of particular interest
because they provide information that can be used to determine the CKM angles
$\gamma\,\equiv\mathrm{arg}[-V_{ud}V_{ub}^{*}/(V_{cd}V_{cb}^{*})]$ and
$\beta_{s}\,\equiv\mathrm{arg}[-V_{ts}V_{tb}^{*}/(V_{cs}V_{cb}^{*})]$ without
theoretical uncertainties [1]. Knowledge of these $C\\!P$-violating phases is
crucial to search for new sources of $C\\!P$ violation and unravel subtle
effects of physics beyond the Standard Model, which may appear in flavour-
changing interactions. Their precise measurements are among the most important
goals of flavour physics experiments.

To date, the angle $\gamma$ is the least well-determined angle of the
Unitarity Triangle with an uncertainty of about $10^{\circ}$ [2, 3, 4]. The
current precision is dominated by measurements of time-integrated
$B^{+}\rightarrow DK^{+}$ decay rates, where $D$ indicates a superposition of
$D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays to a
common final state. In these decays, sensitivity to $\gamma$ arises from
direct $C\\!P$ violation in the interference between the $b\rightarrow
c\bar{u}s$ and $b\rightarrow u\bar{c}s$ tree-level amplitudes. As there are no
loop contributions to the decay amplitudes, no theoretical uncertainties
arise. The main limitation is due to the size of the data samples collected by
the experiments. To improve on the precision, it is important to perform
additional measurements from other channels with small theoretical
uncertainties.

The large production cross-section of $B^{0}_{s}$ mesons in $pp$ collisions at
the LHC opens new possibilities for measuring both $\gamma$ and $\beta_{s}$.
For example, the decay $B^{0}_{s}\rightarrow D_{s}^{\pm}K^{\mp}$ is sensitive
to $\gamma+2\beta_{s}$ through measurements of time-dependent decay rates [5,
6]; although the determination of $\gamma$ from this mode requires an
independent measurement of the mixing phase $\beta_{s}$.

The decay $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$, first proposed in 1991 by
Gronau and London for measuring $\gamma$ [7], can also probe $\beta_{s}$ via
measurements of time-dependent decay rates. Nandi and London have shown [1]
that both $\gamma$ and $\beta_{s}$ can be determined without theoretical
uncertainties and ambiguities, using the known sign of $\Delta\Gamma_{s}$, the
decay-width difference between the two $B^{0}_{s}$ mass eigenstates [8].

An alternative method to measure $\gamma$ using $B^{0}_{s}\rightarrow D\phi$
decays was proposed in Refs. [9, 10], where it was shown that $\gamma$ can be
determined from time-integrated decay rates, in a similar way as from
$B^{+}\rightarrow DK^{+}$ decays, even if $B^{0}_{s}\rightarrow D\phi$ is not
a self-tagged decay mode. The only requirement for the determination is that a
sufficient number of different $D$ final states are included in the
measurement. The time-integrated method does not require flavour-tagging, and
hence makes optimal use of the statistical power of the large $b\overline{}b$
production at LHC. An estimation of the sensitivity with this method shows
that the mode $B^{0}_{s}\rightarrow D\phi$ has the potential to make a
significant impact on the determination of $\gamma$ at LHCb [11].

Figure 1:  Feynman diagrams for the following decays: (a)
$B^{0}_{s}\\!\rightarrow\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}\phi$;
(b) $B^{0}_{s}\rightarrow D^{0}\phi$; (c) $B^{0}\\!\rightarrow\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}K^{*0}$; and (d)
$B^{0}_{s}\\!\rightarrow\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$. The
$B^{0}_{s}\\!\rightarrow\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}\phi$
and $B^{0}_{s}\rightarrow D^{0}\phi$ decay amplitudes interfere when $\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}$ and $D^{0}$ decay to the same
final state.

The observation of the $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ decay and the measurement of
its branching fraction, described in this Letter, are the first steps towards
a programme of $C\\!P$ violation studies with this channel. The branching
fraction is measured relative to the topologically similar decay
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, that was previously observed by
LHCb [12]. In addition, the first measurement of the branching fraction of the
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay relative to the
$B^{0}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$
decay is reported and used to improve on the knowledge of the branching
fraction of the $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay. The Feynman diagrams
corresponding to the $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ and $B^{0}_{s}\rightarrow
D^{0}\phi$ decay amplitudes are shown in Fig. 1. The Feynman diagrams for the
leading $b\rightarrow c$ amplitudes in $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $B^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ decays are also shown in
Fig. 1. Since only $D^{0}\rightarrow K^{-}\pi^{+}$ decays are considered in
this study, all of the measured quantities for the
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$,
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, and $B^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ channels include
contributions from the $B^{0}_{s}\rightarrow D^{0}\phi$, $B^{0}_{s}\rightarrow
D^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, and $B^{0}\rightarrow
D^{0}K^{*0}$ modes, respectively, through the doubly-Cabibbo-suppressed decay
$D^{0}\rightarrow K^{+}\pi^{-}$.

##  2 Event selection

The study reported here is based on $pp$ collision data, corresponding to an
integrated luminosity of 1.0 $\mbox{\,fb}^{-1}$, collected by the LHCb
experiment at a centre-of-mass energy of 7 TeV. The LHCb detector [13] is a
single-arm forward spectrometer covering the pseudorapidity range $2<\eta<5$,
designed for the study of particles containing $b$ or $c$ quarks. The detector
includes a high-precision tracking system consisting of a silicon-strip vertex
detector surrounding the $pp$ interaction region, a large-area silicon-strip
detector located upstream of a dipole magnet with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift
tubes placed downstream. The combined tracking system provides a momentum
($p$) measurement with relative uncertainty that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter (IP)
resolution of 20$\,\upmu\rm m$ for tracks with large transverse momentum
($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov
detectors [14]. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers [15]. The trigger [16] consists of a hardware stage, based on
information from the calorimeter and muon systems, followed by a software
stage, which applies a full event reconstruction.

Simulated signal samples and data control channels are used to optimise the
selection criteria. In the simulation, $pp$ collisions are generated using
Pythia 6.4 [17] with a specific LHCb configuration [18]. Decays of hadrons are
described by EvtGen [19], in which final state radiation is generated using
Photos [20]. The interaction of the generated particles with the detector and
its response are implemented using the Geant4 toolkit [21,
*Agostinelli:2002hh] as described in Ref. [23].

Selected events fulfill one of two hardware trigger requirements: either a
particle from the signal decay deposits enough energy in the calorimeter
system, or one of the particles in the event, not originating from the signal
decay, fulfils any of the trigger requirements (e.g., events triggered by one
or more particles coming from the decay of the other $B$ meson in the
$pp\rightarrow b\bar{b}X$ event). The software trigger requires a two-, three-
or four-track secondary vertex with a large scalar sum of the tracks $p_{\rm
T}$ and significant displacement from the associated primary $pp$ interaction
vertex (PV). At least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and a value of $\chi^{2}_{\rm
IP}>16$, where $\chi^{2}_{\rm IP}$ is defined as the difference between the
$\chi^{2}$ of the PV reconstructed with and without the considered particle. A
multivariate algorithm identifies secondary vertices consistent with the decay
of a $b$ hadron.

Reconstructed tracks are selected with criteria on their $p$, $p_{\rm T}$,
track $\chi^{2}$ per degree of freedom, $\chi^{2}_{\rm IP}$ and particle
identification (PID). Tracks identified as muons are discarded.

The $D^{0}$ mesons are reconstructed in the decay mode $D^{0}\rightarrow
K^{-}\pi^{+}$. Particle identification criteria used to select the daughters
require the difference between the log-likelihoods of the kaon and pion
hypotheses ($\Delta LL_{K\pi}$) to be larger than 0 for the kaon and smaller
than 4 for the pion. The $D^{0}$ meson $\chi^{2}_{\rm IP}$ is required to be
larger than 2 to separate mesons originating from a $B$ decay and those
produced at the PV. In addition, for the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ ($\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$) final states, the charm meson
flight distance with respect to the $B^{0}_{(s)}$ vertex is required to be
larger than 0 with a significance of at least 2 standard deviations in order
to suppress background from $B^{0}_{(s)}$ decays without an intermediate charm
meson, such as the mode $B^{0}\rightarrow K^{-}\pi^{+}K^{*0}$. There is no
corresponding requirement in the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ final state, since the
charmless background is negligible. The $D^{0}$ candidates with invariant mass
within $\pm 20\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known mass
[24] are retained.

The $\phi$ mesons are reconstructed in the mode $\phi\rightarrow K^{+}K^{-}$.
The $p_{\rm T}$ of the kaon daughters is required to be larger than 350
${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and the $\Delta LL_{K\pi}$ of both
daughters to be larger than 3. Candidates are retained if their invariant mass
is within $\pm 10\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known
$\phi$ mass [24].

The $K^{*0}$ mesons are reconstructed in the mode $K^{*0}\rightarrow
K^{+}\pi^{-}$. The $p_{\rm T}$ of the kaon (pion) is required to be larger
than 350 (250) ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. In addition, to reduce
the cross-feed from $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rho^{0}$ and $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}K^{-}$ decays, the $\Delta
LL_{K\pi}$ of the kaon must be larger than 3 and that of the pion smaller than
3. Possible background from protons in the kaon sample, for example from the
decay $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$, is suppressed
by selecting kaon candidates with a difference between the log-likelihoods of
proton and kaon hypotheses, $\Delta LL_{pK}$, smaller than 10. Candidate
$K^{*0}$ mesons with invariant mass within $\pm
50\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known mass [24] are kept.

Neutral $B$ meson candidates are formed from $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ and $\phi$ (or $K^{*0}$)
candidates, which are fitted to a common vertex with the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ constrained to its known mass. In
order to reduce contributions from non-resonant decays,
$B^{0}_{(s)}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}K^{-}$,
$B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$, and
$B^{0}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$
[25, 26], the absolute value of the cosine of the vector-daughter helicity
angle ($\mathrm{cos}\,\theta_{h}$) is required to be larger than 0.4. This
angle is defined between the momentum direction of the $K^{+}$ daughter in the
$\phi\,(K^{*0})$ frame, and the vector meson direction in the $B$ rest frame.
Backgrounds from $B^{0}_{(s)}\rightarrow D^{\mp}_{(s)}h^{\pm}$ ($h=\pi,K$)
decays, are rejected by vetoing candidates with $K^{+}K^{-}\pi^{+}$
($K^{-}\pi^{+}\pi^{+}$ and $K^{+}K^{-}\pi^{+}$) invariant mass within
$\pm\,15\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the $D^{+}_{s}$
($D^{+}$) meson known mass [24].

A boosted decision tree (BDT) [27] suppresses the residual background. Nine
variables are input to the BDT: the decay vertex $\chi^{2}$ of the
reconstructed $B^{0}_{(s)}$ and $D^{0}$ mesons; the $\chi^{2}_{\rm IP}$ of the
$B^{0}_{(s)}$, $D^{0}$, $\phi$ ($K^{*0}$) mesons, and of both the $D^{0}$
daughters; and the $p_{\rm T}$ of the $D^{0}$ and $\phi$ ($K^{*0}$) mesons.
The BDT is optimised and tested using simulated signal events and events
outside of the $D^{0}$ mass signal region for background. Events with BDT
response larger than 0.2 are retained, resulting in a rejection of 74% of the
background, while retaining 84% of the signal. The working point maximises
$N_{\mathrm{s}}/\sqrt{N_{\mathrm{s}}+N_{\mathrm{b}}}$. Here, $N_{\mathrm{s}}$
is the expected $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ signal yield, computed using
simulated events and assuming that the branching fraction is equal to that of
the $B^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ decay (as expected under
SU(3) flavour symmetry), and $N_{\mathrm{b}}$ is the background yield
estimated using data events in the sidebands outside the
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$
signal region ($\pm\,50$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the
$B^{0}_{s}$ known mass [24]). No multiple candidates are found for the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ final state. The fraction of
events with more than one candidate is 0.6% in the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ or $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ invariant mass range of
5150–5600 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and the candidate
retained is chosen randomly.

##  3 Signal yield

Signal yields are determined with an unbinned maximum likelihood fit to the
$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ and the sum of the
$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ invariant mass ($M$)
distributions in the range $5150<M<5600$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The two samples are fitted
simultaneously with a sum of probability density functions (PDFs) modelling
signal and background contributions.

The $B^{0}_{s}$ and $B^{0}$ signals are described by a modified Gaussian
distribution of the form

$f(M;\mu,\sigma,\alpha_{L},\alpha_{R})\propto\mathrm{exp}\left(\frac{-(M-\mu)^{2}}{2\sigma^{2}+\alpha_{L,R}(M-\mu)^{2}}\right),$
(1)

where $\mu$ is the peak position, $\sigma$ the width, and
$\alpha_{L}\,(M<\mu)$ and $\alpha_{R}\,(M>\mu)$ parameterise the tails. The
width and the tail parameters depend on the final state, but are common to the
$B^{0}_{s}$ and $B^{0}$ decays. The $B^{0}$ peak position and width are left
free to vary in the fit with the difference between $B^{0}_{s}$ and $B^{0}$
peak positions fixed to the current world-average value [24]. The tail
parameters are fixed to values determined from simulated events and are
considered among the sources of systematic uncertainty. The recently observed
decay $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}K^{-}$ [25] is expected to
contribute to the $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$
distribution and is modelled with the same modified Gaussian distribution, but
with different peak position, as that used to describe the
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$
decay.

Background from the $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rho^{0}$ decay in the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ (or $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$) final state can arise from
misidentification of one of the pions from the
$\rho^{0}\rightarrow\pi^{+}\pi^{-}$ decay as a kaon. The shape of this cross-
feed contribution is modelled with a Crystal Ball function [28] determined
from simulated events. This background component is absent in the
$B^{0}_{s}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$
mode, since the probability that both pions are misidentified as kaons and
that their invariant mass is inside the narrow $\phi$ mass window is
negligible. For similar reasons, the cross-feed between $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ and
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$
decays is negligible.

The decay $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, where a $\pi^{0}$ or photon from
the $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{*0}$ decay is not
reconstructed, constitutes the main background contribution to the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ final state below the $B^{0}$
mass. Similarly, the decay $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{*0}\phi$ is expected to contribute to
the low-mass background in the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ final state. These decays of a
pseudoscalar to two vector mesons are modelled by a non-parametric PDF [29]
determined from simulation. The mass shape depends on the unknown fraction of
longitudinal polarisation, which is assumed to be identical for the two modes
and is treated as an additional free parameter in the fit.

The remaining combinatorial background is described by a linear function, with
a common slope for the two considered final states, left free to vary in the
fit.

Figure 2:  Invariant mass distributions for (a) $B^{0}_{s}\\!\rightarrow\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}\phi$, and (b)
$B^{0}\rightarrow\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}K^{*0}$ or
$B^{0}_{s}\rightarrow\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$ decays. Data points are shown in
black, the total fitted PDF as solid black line, and the components as
detailed in the legends.

Signal yield ratios are directly determined in the fit to take into account
statistical correlations in the measurement of ratios of branching fractions.
In total, there are 13 free parameters in the fit, including the background
yields of the different components and the overall normalisation. The
invariant mass distributions with the resulting fits are shown in Fig. 2.

The helicity angle distribution of the $\phi$ candidates for the $B^{0}_{s}$
and $B^{0}$ signal is investigated. The sPlot [30] technique is adopted to
assign a weight to the events and determine the signal components, using the
$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ invariant mass as the
discriminating variable. For this purpose, the requirement on
$\mathrm{cos}\,\theta_{h}>0.4$ has been lifted prior to the computation of the
signal weights. The data distributions of $\mathrm{cos}\,\theta_{h}$, shown in
Fig. 3, are compared to the expected distribution of
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$
decays from simulation. The distribution observed for the
$B^{0}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}K^{-}$
decay candidates is consistent with the expectation that this decay is not
dominated by a pseudoscalar-vector quasi-two-body final state.

Figure 3:  Distribution of the cosine of the helicity angle of the $\phi$
candidates.

The signal yield ratios are corrected for two residual backgrounds that peak
at the mass of the $B^{0}_{s}$ or $B^{0}$ meson and are distributed as the
signal. The first of the two backgrounds is the charmless background due to
the decays $B^{0}_{s}\rightarrow K^{+}\pi^{-}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $B^{0}\rightarrow
K^{+}\pi^{-}K^{*0}$ proceeding without the presence of an intermediate $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ meson. There is no evidence of
such background in the $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ channel. A large fraction of
the charmless background in the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ final state is rejected with
the requirement of a minimal $D^{0}$ flight distance introduced in Sec. 2. The
remaining charmless background is evaluated using candidates from the $D^{0}$
sidebands. The $B$ yields in the $D^{0}$ sidebands above a linear background
are extrapolated to the $D^{0}$ signal region and used to correct the signal.
The uncorrected signal yields and the background contributions are given in
Table 1.

Table 1:  Uncorrected signal yields and the peaking (charmless, S-wave) background yields.  Channel | Signal | Charmless background | S-wave background
---|---|---|---
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ | 43$\,\pm\,$8 | $0\,\pm\,2$ | 2$\,\pm\,3$
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ | 535$\,\pm\,$30 | $4\,\pm\,3$ | 24$\,\pm\,$7
$B^{0}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ | 260$\,\pm\,$24 | $4\,\pm\,3$ | 13$\,\pm\,$6

The other source of peaking background is due to higher mass resonances and
non-resonant $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}K^{-}$,
$B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$, and
$B^{0}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}\pi^{-}$
decays that fall in the $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$, $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, and $B^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ signal regions,
respectively. This contribution is evaluated with fits to the $\phi$ and
$K^{*0}$ background-subtracted mass distributions in a wider range than the
signal window. The background subtraction is performed using the sPlot
technique, with the $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ and
$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ (or $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$) mass as discriminating
variables. A linear PDF describes the S-wave background in the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ final state. A spin-one Breit-
Wigner distribution convolved with a Gaussian resolution function describes
the signal, and an S-wave PDF the non-resonant background. The S-wave
component in the $B^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ and
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ channels takes into account non-
resonant and $K^{*0}(1430)$ resonance contributions and uses experimental
input from the LASS experiment [31, *bib:LASS2]. It is approximately linear in
the region of interest, $\pm\,200{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
around the $K^{*0}$ nominal mass. Potential interference effects between the
S-wave and the P-wave components are covered by the assigned systematic
uncertainty. The $\phi$ and $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ mass distributions are shown in
Fig. 4.

Figure 4:  Background-subtracted distributions of the reconstructed (left)
$\phi$ mass from the $B^{0}_{s}\\!\rightarrow\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}\phi$ decay and (right) $\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$ mass from the
$B^{0}_{s}\\!\rightarrow\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$ decay. The dashed red line
represents the S-wave component, the solid blue line the total fit result.

The background yields, after extrapolation to the $K^{*0}$ and $\phi$ signal
mass windows, are listed in Table 1.

A likelihood ratio test is employed to assess the statistical significance of
the $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ signal, which is given by
$\sqrt{2{\rm ln}(\mathcal{L}_{\rm s+b}/\mathcal{L}_{\rm b})}$ and found to be
7.1 standard deviations. Here $\mathcal{L}_{\rm s+b}$ and $\mathcal{L}_{\rm
b}$ are the maximum values of the likelihoods for the signal-plus-background
and background-only hypotheses, respectively.

The ratios of branching fractions are evaluated from the uncorrected signal
yields, $N$, and the sum of the charmless and non-resonant background yields,
$N^{\rm bkg}$, as

${\cal{R}}_{\phi}\equiv\frac{{\cal B}(B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi)}{{\cal{B}}(B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})}=\frac{N_{B^{0}_{s}\\!\rightarrow\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}\phi}}{N_{B^{0}_{s}\\!\rightarrow\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}\cdot\frac{\left(1-\frac{N^{\rm
bkg}_{B^{0}_{s}\\!\rightarrow\kern
0.99998pt\overline{\kern-0.99998ptD}{}^{0}\phi}}{N_{B^{0}_{s}\\!\rightarrow\kern
0.99998pt\overline{\kern-0.99998ptD}{}^{0}\phi}}\right)}{\left(1-\frac{N^{\rm
bkg}_{B^{0}_{s}\\!\rightarrow\kern
0.99998pt\overline{\kern-0.99998ptD}{}^{0}\kern
0.99998pt\overline{\kern-0.99998ptK}{}^{*0}}}{N_{B^{0}_{s}\\!\rightarrow\kern
0.99998pt\overline{\kern-0.99998ptD}{}^{0}\kern
0.99998pt\overline{\kern-0.99998ptK}{}^{*0}}}\right)}\cdot\frac{\epsilon_{B^{0}_{s}\\!\rightarrow\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}{\epsilon_{B^{0}_{s}\\!\rightarrow\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}\phi}}\cdot\frac{{\cal{B}}(K^{*0}\rightarrow
K^{+}\pi^{-})}{{\cal{B}}(\phi\rightarrow K^{+}K^{-})},$ (2)

and

${\cal{R}}_{K^{*0}}\equiv\frac{{\cal B}(B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})}{{\cal{B}}(B^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0})}=\frac{N_{B^{0}_{s}\\!\rightarrow\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}{N_{B^{0}\\!\rightarrow\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}K^{*0}}}\cdot\frac{\left(1-\frac{N^{\rm
bkg}_{B^{0}_{s}\\!\rightarrow\kern
0.99998pt\overline{\kern-0.99998ptD}{}^{0}\kern
0.99998pt\overline{\kern-0.99998ptK}{}^{*0}}}{N_{B^{0}_{s}\\!\rightarrow\kern
0.99998pt\overline{\kern-0.99998ptD}{}^{0}\kern
0.99998pt\overline{\kern-0.99998ptK}{}^{*0}}}\right)}{\left(1-\frac{N^{\rm
bkg}_{B^{0}\\!\rightarrow\kern
0.99998pt\overline{\kern-0.99998ptD}{}^{0}K^{*0}}}{N_{B^{0}\\!\rightarrow\kern
0.99998pt\overline{\kern-0.99998ptD}{}^{0}K^{*0}}}\right)}\cdot\frac{\epsilon_{B^{0}\\!\rightarrow\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}K^{*0}}}{\epsilon_{B^{0}_{s}\\!\rightarrow\kern
1.39998pt\overline{\kern-1.39998ptD}{}^{0}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}\cdot\left(\frac{f_{s}}{f_{d}}\right)^{-1},$
(3)

where the ratio of the $B^{0}_{s}$ and $B^{0}$ fragmentation fractions is
$f_{s}/f_{d}=0.256\,\pm\,0.020$ [33], the value of the $\phi\rightarrow
K^{+}K^{-}$ branching fraction is 0.489$\,\pm\,$0.005 [24], and
${\cal{B}}(K^{*0}\rightarrow K^{+}\pi^{-})=2/3$. The total efficiencies,
$\epsilon$, account for the geometrical acceptance of the detector, the
reconstruction, the event selection, the PID, and the trigger efficiencies.
All efficiencies are computed from simulated events, except for the PID and
hardware trigger efficiencies, which are obtained from data, using a high-
purity calibration sample of $D^{*+}\rightarrow D^{0}(\rightarrow
K^{-}\pi^{+})\pi^{+}$ decays. The resulting ratios of branching fractions are
${\cal{R}}_{\phi}=0.069\pm 0.013$ and ${\cal{R}}_{K^{*0}}=7.8\pm 0.7$, where
the uncertainties are statistical only.

##  4 Systematic uncertainties

Several sources of systematic uncertainties are considered. Those associated
to the trigger and PID selection affect only ${\cal{R}}_{\phi}$ and are mainly
due to systematic uncertainties in the calibration procedure. The ratios of
the efficiencies of the decays $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ and
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ for the trigger and PID are found
to be 0.97$\,\pm\,$0.05 and 1.08$\,\pm\,$0.03, respectively, where the errors
are propagated as systematic uncertainties to ${\cal{R}}_{\phi}$.

Similarly, the uncertainty on the efficiencies of the charm meson flight
distance selection affects only ${\cal{R}}_{\phi}$, where different criteria
are chosen for the $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ and
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ modes. The ratio of the
corresponding efficiencies is found to be 1.27$\,\pm\,$0.03, where the
uncertainty includes a contribution from the difference between data and
simulation. In order to estimate the efficiency in data, the fit to the
invariant mass of the $B$ candidates is performed to data samples selected
with all criteria except that on the flight distance. For this sample, the
charmless background contribution is estimated using events in the upper $D$
mass sideband and subtracted from the signal yields.

The ratio of the efficiencies for the decays $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ and
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ of the remaining selection
criteria is found to be $1.21\,\pm\,0.03$, where the deviation from unity is
mainly due to the different widths and mass windows for the $\phi$ and
$K^{*0}$ resonances. The ratio of the efficiencies for the decays
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $B^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ is found from simulation to
be $1.04\,\pm\,0.01$. The uncertainties on these efficiencies are propagated
as systematic uncertainties due to the selection.

The fit procedure is validated using simulated pseudo-experiments. The fit
bias, relative to the fitted ratio, is evaluated to be 1.4% for
${\cal{R}}_{\phi}$ and 0.2% for ${\cal{R}}_{K^{*0}}$ and is assigned as
systematic uncertainty. The signal model uncertainty is evaluated by varying
the fixed signal parameters by 10%, which is about three times the difference
between data and simulation, as determined by a fit where those parameters are
free to vary. The background shape uncertainty is determined from the bias in
the results obtained by fitting samples generated with an alternative
(exponential) combinatorial background model.

The uncertainties on the charmless background yields given in Table 1 are
assumed to be uncorrelated and are propagated to assign the associated
systematic uncertainty. Similarly, the statistical uncertainties on the S-wave
background yields are propagated to ${\cal{R}}_{\phi}$ and
${\cal{R}}_{K^{*0}}$ to assign respective systematic uncertainties due to the
non-resonant correction.

A summary of the systematic uncertainties is given in Table 2. The uncertainty
on the fragmentation fraction $f_{s}/f_{d}$, which is the dominant systematic
uncertainty for ${\cal{R}}_{K^{*0}}$, is not included, and is listed
separately.

Table 2:  Absolute systematic uncertainties of the measured ratio of branching fractions. The total is obtained as sum in quadrature of the different contributions.  Source | ${\cal{R}}_{\phi}$ | ${\cal{R}}_{K^{*0}}$
---|---|---
Trigger | 0.003 | -
PID | 0.002 | -
Flight distance | 0.002 | -
Selection | 0.002 | -
Simulation statistics | 0.001 | 0.10
Fit bias | 0.001 | 0.03
Signal model | 0.001 | 0.04
Background model | 0.001 | 0.01
Charmless correction | 0.003 | 0.10
Non-resonant correction | 0.004 | 0.22
$\phi$ branching fraction | 0.001 | -
Total | 0.007 | 0.26

##  5 Results and conclusions

The significance of the $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ signal, including systematic
uncertainties, is obtained by scaling the statistical significance with the
ratio of the statistical to the total (statistical and systematic) uncertainty
on the signal yield. It is found to be 6.5 standard deviations. This decay is
therefore observed for the first time.

The ratios of branching fractions are found to be

${\cal{R}}_{\phi}=0.069\pm 0.013~{}(\rm stat)\pm 0.007~{}(\rm syst),$
${\cal{R}}_{K^{*0}}=7.8\pm 0.7~{}({\rm stat})\pm 0.3~{}({\rm syst})\pm
0.6~{}(f_{s}/f_{d}).$

From ${\cal{R}}_{K^{*0}}$ and the value of the $B^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ branching fraction from Ref.
[24], the $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ branching fraction is calculated
to be

${\cal{B}}(B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=[3.3\pm 0.3~{}({\rm stat})\pm
0.1~{}({\rm syst})\pm 0.3~{}(f_{s}/f_{d})\pm
0.5~{}({\cal{B}}(B^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}))]\times 10^{-4}.$

This result is consistent with and improves on the previous determination by
LHCb [12], which is based on an independent data sample. Using the above
results for ${\cal{R}}_{\phi}$, ${\cal{R}}_{K^{*0}}$ and the
$B^{0}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$
branching fraction, the branching fraction for $B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$ is calculated to be

${\cal{B}}(B^{0}_{s}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi)=[2.3\pm 0.4~{}({\rm stat})\pm
0.2~{}({\rm syst})\pm 0.2~{}(f_{s}/f_{d})\pm
0.3~{}({\cal{B}}(B^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}))]\times 10^{-5},$

which takes into account the correlation in the statistical uncertainties
between ${\cal{R}}_{\phi}$ and ${\cal{R}}_{K^{*0}}$ of $-13.6$%. The
correlation between the corresponding systematic uncertainties is negligible.
The central value is about a factor two smaller than the branching fraction
for the $B^{0}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{*0}$ decay and supports the
observation of SU(3) breaking effects in other colour suppressed
$B^{0}_{(s)}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}V$
decays [12], where $V$ is a vector meson. With larger data samples, the
$B^{0}_{s}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\phi$
decay will contribute to the measurements of the $C\\!P$ violating phases
$\gamma$ and $\beta_{s}$.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.

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