Sparse Non-Convex Optimization For Higher Moment Portfolio Management
Farshad Noravesh111Email<EMAIL_ADDRESS>
One of the reasons that higher order moment portfolio optimization methods are
not fully used by practitioners in investment decisions is the complexity that
these higher moments create by making the optimization problem nonconvex. Many
few methods and theoretical results exists in the literature, but the present
paper uses the method of successive convex approximation for the mean-
variance-skewness problem.
Keywords: nonconvex, $\ell_{1}$ regularization, higher moment, portfolio
optimization, portfolio selection, skewness

##  1 Introduction

Although Modern portfolio theory started by Harry Markowitz by the classical
mean-variance optimization, there are many shortcoming on this theory from
different perspectives. The first shortcoming is that it does not care about
higher moments which becomes a critical issure when assets are highly skewed
and having big kurtosis which is far from normal distribution. One of the
obstacles for development of higher order moment portfolio optimization
methods is the nonconvexity that is generated by considering skewness and
kurtosis. There are four major approaches to handles nonconvex optimization
problems. The first groups of methods is based on stochastic gradient descent
and its convergence rates are far from other methods as is described in (Lei
et al., 2020). The second approach is to see noncovex functions as a
difference of some convex functions but this framework does not work for any
class of nonconvex problems. The third approach is to use the idea of
majorization minimization (MM) by iteratively minimize a surrogate function
that upper-bounds the objective function but has the same derivative with it
at the current iteration which is explained in (Sun et al., 2017). The forth
approach is called successive convex approximation and is used in the present
paper to show practitioners how to use it for mean-variance-skewness problem.
This method is very similar to MM method but requires the upper bound function
to be a convex function.

The second shortcoming is the uncertainty around the parameters inside mean
return, covariance and coskewness which also makes the portfolio solutions
very unstable. Also if incorrect parameters are used the solution is only
suboptimal. There are three major approaches to tackle this problem. The first
approach comes from frequentists that try to estimate the covariance matrices
and others using maximum likelihood or any other frequentist approaches. One
may argue that simple averaging could help, but since historical data are
being used, only recent data can be relevant which reduces the overall data
that are taken into considerations and creates big volatilities. The second
approach is the bayesian approach which utilizes some intrinsic structure such
as Fama-French three-factor model as a prior to shrink the mean and covariance
toward the structure. The third approach which is of interest in the present
paper is penalizing the objective function such as (Ho et al., 2015) which
uses the weighted elastic net for penalizing the mean-variance portfolio.
There are huge literature on the fact that why these $\ell_{1}$-norm or
$\ell_{2}$-norm or any combination of them is helpful. For example (Brodie et
al., 2009) mentioned four reasons on the importance of $\ell_{1}$-norm. Apart
from the fact that sparsity limits the number of positions that must be
monitored and liquidated, they noticed that it reduce the sensitivity of the
optimization to the possible collinearities between the assets and therefore
has good stability and this is possible with using only limited amount of
training data.(Fan et al., 2012) showed the following important result which
bounds the risk minimization error by $\ell_{1}$-norm.

$|R(w,\hat{\Sigma})-R(w,\Sigma)|\leq\lVert\hat{\Sigma}-\Sigma\rVert_{\infty}\lVert
w\rVert_{1}^{2}$ (1.1)

where risk is defined by $\Sigma=w^{T}\Sigma w$

##  2 MVS portfolio

Mean-Variance-Skewness portfolio selection and optimization can be defined as
solution of

$\underset{w}{minimize}\
f(w)+g(w)=-\lambda_{1}\phi_{1}(w)+\lambda_{2}\phi_{2}(w)-\lambda_{3}\phi_{3}(w)+\lambda_{4}\lVert
w\rVert_{1}$ (2.1)

where $\phi_{1}$,$\phi_{2}$,$\phi_{3}$ are mean,variance and skewness
respectively and are defined as follows:

$\begin{split}\phi_{1}(w)&=w^{T}\mu\\\ \phi_{2}(w)&=w^{T}\Sigma w\\\
\phi_{3}(w)&=w^{T}\Phi(w\otimes w)\end{split}$ (2.2)

where $\Phi$ is the co-skewness matrix. As is shown in (Zhou & Palomar, 2021),
the gradient and hessian of skewness can be written as:

$\begin{split}\nabla\phi_{3}&=3\Phi(w\otimes w)\\\
\nabla^{2}\phi_{3}&=6\Phi(I\otimes w)\end{split}$ (2.3)

(Byrd et al., 2013) could be used to solve the optimization problem in (2.1)
but that algorithm has big computational complexity since it uses newton
method and in general second order methods are harder than first order methods
although they could be more accurate in some algorithms. Therefore an
algorithm is developed in the next section that is computationally easier and
also much simpler to implement.

##  3 Problem Formulation

If $\lambda_{3}$ in (2.1) is zero, then (Philipp et al., 2020) has used the
$\ell_{1}$-norm regularization to do sparse portfolio selection but skewness
and other higher moments are not handled in their article. If $\lambda_{4}$ in
(2.1) is zero, then successive convex approximation method can be used as is
described in (Zhou & Palomar, 2021), otherwise the optimization functions can
be rearranged like (Yang & Pesavento, 2017) to handle nondifferentiability of
the $\ell_{1}$ penalizing term. (2.1) can be written as

$\begin{split}\underset{w,y}{minimize}\ &f(w)+y\\\ subject\ to\ &g(w)\leq
y\end{split}$ (3.1)

where $g=\lambda_{4}\lVert w\rVert_{1}$. Since f is a nonconvex function,
successfive psudoconvex approximation can be used to sequentially solve
minimization of a set of surogate approximation functions. Psuedoconvex
surogate functions can be used as explained in (Yang & Pesavento, 2017) to
broaden the type of approximate functions that can be chosen. Let
$\tilde{f}(w;w^{t})$ be the approximation of f at iteration t. Thus, the
solution to approximate function of problem in (3.1) around $(w^{t},y^{t})$ is

$(Bw^{t},y^{\star}(w^{t}))=\underset{(w,y):w\in W,g(w)\leq y}{argmin}\
\tilde{f}(w;w^{t})+y$ (3.2)

The step size $\gamma^{t}$ can be easily achieved by the following exact line
search algorithm.

$\gamma^{t}\in\underset{0\leq\gamma\leq
1}{argmin}\\{f(w^{t}+\gamma(Bw^{t}-w^{t})+y^{t}+\gamma(y^{\star}(w^{t})-y^{t}))\\}$
(3.3)

where $y^{t}\geq g(x^{t})$. Now the update can be calculated as follows

$\begin{split}w^{t+1}&=w^{t}+\gamma^{t}(Bw^{t}-w^{t})\\\
y^{t+1}&=y^{t}+\gamma^{t}(y^{\star}(w^{t})-y^{t})\end{split}$ (3.4)

By substituting (2.2) in (2.1) and picking the differentiable part, the
following nonconvex function results

$f(w)=\underbrace{-\lambda_{1}w^{T}\mu+\lambda_{2}\omega^{T}\Sigma\omega}_{\text{convex}}-\underbrace{\lambda_{3}w^{T}\Phi(w\otimes
w)}_{\text{non-convex}}$ (3.5)

The first two terms in (3.5) are convex functions and therefore there is no
need to approximate them by convex functions, the third term which is the
portfolio skewness should be approximated by a second order function as

$\begin{split}\tilde{f}_{ncvx}(w,w^{t})=f_{ncvx}(w^{t})+\nabla
f_{ncvx}(w^{t})^{T}(w-w^{t})&\\\
+\frac{1}{2}(w-w^{t})^{T}\nabla^{2}f_{ncvx}^{t}(w^{t})(w-w^{t})\end{split}$
(3.6)

Approximating the nonconvex part of (3.5) by (3.6) yields

$\begin{split}\tilde{f}_{ncvx}=-\lambda_{3}(w^{t})^{T}\Phi(w^{t}\otimes
w^{t})+3\Phi(w^{t}\otimes w^{t})^{T}(w-w^{t})\\\
+\frac{1}{2}(w-w^{t})^{T}(-\lambda_{3}(6\Phi(I\otimes
w^{t})))(w-w^{t})\end{split}$ (3.7)

Thus, the solution to the following constraint convex optimization problem is
used to update (3.4) for the next update which generates a point for another
successive approximation.

$\begin{split}\underset{w}{minimize}&-\lambda_{1}w^{T}\mu+\lambda_{2}w^{T}\Sigma
w+\tilde{f}_{ncvx}\\\ subject\ to\ &\lVert w\rVert_{1}\leq y\end{split}$ (3.8)

(3.8) can be solved by convex optimization methods such as (Solntsev et al.,
2015) which uses a proximal gradient method since proximal operator for
$\ell_{1}$-norm is easy to evaluate. The full algorithm in the present paper
is summarized in Algorithm 1

Algorithm 1 sparse nonconvex optimization for mean-variance-skewness

Input: given returns of a portfolio having N assets
1: calculate mean,variance and coskewness matrices
2: initialize the weight vector randomly
Loop: until the local minimum of approximate function converges to local
minimum of original function.
3: substitute weights and calculate variables using (2.3)
4: solve (3.8) to find $Bw^{t}$
5: update the weights and y from (3.4)

Figure 1: objective value in terms of successive iterations

Figure 1 shows that the value of function in (2.1) evaluated on weights at
each iteration $t$ is reduced. The $\ell_{2}$-norm of gradient is shown in
Figure 2 and demonstrates how it decays as successive iterations increases.

Figure 2: L2 norm of gradient of objective function in terms of successive
iterations

## References

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-141 LHCb-PAPER-2013-033 October 25, 2013

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

The $C\\!P$-violating asymmetry $a_{\rm sl}^{\rm s}$ is studied using
semileptonic decays of $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons produced in $pp$
collisions at a centre-of-mass energy of 7 TeV at the LHC, exploiting a data
sample corresponding to an integrated luminosity of 1.0 fb-1. The
reconstructed final states are $D^{\pm}_{s}\mu^{\mp}$, with the $D^{\pm}_{s}$
particle decaying in the $\phi\pi^{\pm}$ mode. The $D^{\pm}_{s}\mu^{\mp}$
yields are summed over $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
and $B^{0}_{s}$ initial states, and integrated with respect to decay time.
Data-driven methods are used to measure efficiency ratios. We obtain $a_{\rm
sl}^{\rm s}$ = $(-0.06\pm 0.50\pm 0.36)$%, where the first uncertainty is
statistical and the second systematic.

#  Measurement of the flavour-specific $C\\!P$-violating asymmetry $a_{\rm
sl}^{s}$ in $B^{0}_{s}$ decays

Published in Phys. Lett. B

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

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Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M. Ubeda Garcia37,
A. Ukleja27, D. Urner53, A. Ustyuzhanin52,p, U. Uwer11, V. Vagnoni14, G.
Valenti14, A. Vallier7, M. Van Dijk45, R. Vazquez Gomez18, P. Vazquez
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

The $C\\!P$ asymmetry in $B^{0}_{s}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mixing is a sensitive probe of
new physics. In the neutral $B$ system ($B^{0}$ or $B^{0}_{s}$), the mixing of
the flavour eigenstates (the neutral $B$ and its antiparticle $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$) is governed by a $2\times 2$ complex
effective Hamiltonian matrix [1]

$\begin{pmatrix}\noindent
M_{11}-\frac{i}{2}\Gamma_{11}&M_{12}-\frac{i}{2}\Gamma_{12}\\\
M_{12}^{*}-\frac{i}{2}\Gamma_{12}^{*}&M_{22}-\frac{i}{2}\Gamma_{22}\end{pmatrix},$
(1)

which operates on the neutral $B$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ flavour eigenstates. The mass
eigenstates have eigenvalues $M_{\rm H}$ and $M_{\rm L}$. Other measurable
quantities are the mass difference $\Delta M$, the width difference
$\Delta\Gamma$, and the semileptonic (or flavour-specific) asymmetry $a_{\rm
sl}$. These quantities are related to the off-diagonal matrix elements and the
phase $\phi_{12}\equiv\arg\left(-M_{12}/\Gamma_{12}\right)$ by

$\displaystyle\Delta M$ $\displaystyle\equiv$ $\displaystyle M_{\rm H}-M_{\rm
L}=2|M_{12}|\left(1-\frac{1}{8}\frac{|\Gamma_{12}|^{2}}{|M_{12}|^{2}}\sin^{2}\phi_{12}+....\right),$
$\displaystyle\Delta\Gamma$ $\displaystyle\equiv$ $\displaystyle\Gamma_{\rm
L}-\Gamma_{\rm
H}=2|\Gamma_{12}|\cos\phi_{12}\left(1+\frac{1}{8}\frac{|\Gamma_{12}|^{2}}{|M_{12}|^{2}}\sin^{2}\phi_{12}+....\right),$
$\displaystyle a_{\rm sl}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}(t)\rightarrow
f\right)-\Gamma\left(B(t)\rightarrow\bar{f}\right)}{\Gamma\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}(t)\rightarrow
f\right)+\Gamma\left(B(t)\rightarrow\bar{f}\right)}\simeq\frac{\Delta\Gamma}{\Delta
M}\tan{\phi_{12}}\,,$ (2)

where $B(t)$ is the state into which a produced $B$ meson has evolved after a
proper time $t$ measured in the meson rest frame, and $f$ indicates a flavour-
specific final state. The term flavour-specific means that the final state is
only reachable by the decay of the $B$ meson, and consequently reachable by a
meson originally produced as a $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$
only through mixing. We use the semileptonic flavour specific final state and
thus refer to this quantity as $a_{\rm sl}$. Note that $a_{\rm sl}$ is decay
time independent. Throughout the paper, mention of a specific channel implies
the inclusion of the charge-conjugate mode, except in reference to
asymmetries.

The phase $\phi_{12}$ is very small in the Standard Model (SM), in particular,
for $B^{0}_{s}$ mixing, $\phi_{12}^{s}$ is approximately $0.2^{\circ}$
[2].111This phase should not be confused with the $C\\!P$ violation phase
measured in $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ and $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays, sometimes called $\phi_{s}$ [4]. New physics can
affect this phase [3, 4] and therefore $a_{\rm sl}^{s}$. The D0 collaboration
has reported evidence for a decay asymmetry $A_{\rm sl}^{b}=(-0.787\pm
0.172\pm 0.093)\%$ in a mixture of $B^{0}$ and $B^{0}_{s}$ semileptonic
decays, where the first uncertainty is statistical and the second systematic
[5, *Abazov:2010hv, *Abazov:2010hj]. This asymmetry is much larger in
magnitude than the SM predictions for semileptonic asymmetries in $B^{0}_{s}$
and $B^{0}$ decays, namely $a^{s}_{\rm sl}=(1.9\pm 0.3)\times 10^{-5}$ and
$a^{d}_{\rm sl}=(-4.1\pm 0.6)\times 10^{-4}$ [4]. More recently D0 published
measurements of $a^{d}_{\rm sl}=(0.68\pm 0.45\pm 0.14)\%$ [8], and $a^{s}_{\rm
sl}=(-1.12\pm 0.74\pm 0.17)\%$ [9], consistent both with the anomalous
asymmetry $A_{\rm sl}^{b}$ and the SM predictions for $a^{s}_{\rm sl}$ and
$a^{d}_{\rm sl}$. If the measured value of $A_{\rm sl}^{b}$ is confirmed, this
would demonstrate the presence of physics beyond the SM in the quark sector.
The $e^{+}e^{-}$ $B$-factory average asymmetry in $B^{0}$ decays is
$a^{d}_{\rm sl}=(0.02\pm 0.31)$% [10], in good agreement with the SM. A
measurement of $a_{\rm sl}^{s}$ with comparable accuracy is important to
establish whether physics beyond the SM influences flavour oscillations in the
$B^{0}_{s}$ system.

When measuring a semileptonic asymmetry at a $pp$ collider, such as the LHC,
particle-antiparticle production asymmetries, denoted as $a_{\rm P}$, as well
as detector related asymmetries, may bias the measured value of $a_{\rm
sl}^{s}$. We define $a_{\rm P}$ in terms of the numbers of produced
$b$-hadrons, $N(B)$, and anti $b$-hadrons, $N(\kern
1.79993pt\overline{\kern-1.79993ptB}{})$, as

$a_{\rm P}\equiv\frac{N(B)-N(\overline{B})}{N(B)+N(\overline{B})}~{},$ (3)

where $a_{\rm P}$ may in general be different for different species of
$b$-hadron.

In this paper we report the measurement of the asymmetry between
$D_{s}^{+}X\mu^{-}\overline{\nu}$ and $D_{s}^{-}X\mu^{+}\nu$ decays, with $X$
representing possible associated hadrons. We use the
$D_{s}^{\pm}\rightarrow\phi\pi^{\pm}$ decay. For a time-integrated measurement
we have, to first order in $a_{\rm sl}^{\rm s}$

$A_{\rm
meas}\equiv\frac{\Gamma[{D_{s}^{-}\mu^{+}}]-\Gamma[{D_{s}^{+}\mu^{-}}]}{\Gamma[{D_{s}^{-}\mu^{+}}]+\Gamma[{D_{s}^{+}\mu^{-}}]}=\frac{a_{\rm
sl}^{s}}{2}+\left[a_{\rm P}-\frac{a_{\rm
sl}^{s}}{2}\right]\frac{\int_{t=0}^{\infty}{e^{-\Gamma_{s}t}\cos(\Delta
M_{s}\,t)\epsilon(t)dt}}{\int_{t=0}^{\infty}{e^{-\Gamma_{s}t}\cosh(\frac{\Delta\Gamma_{s}\,t}{2})\epsilon(t)dt}},$
(4)

where $\Delta M_{s}$ and $\Gamma_{s}$ are the mass difference and average
decay width of the $B^{0}_{s}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson system, respectively,
and $\epsilon(t)$ is the decay time acceptance function for $B^{0}_{s}$
mesons. Due to the large value of $\Delta M_{s}$, 17.768 $\pm$0.024 ps-1 [11],
the oscillations are rapid and the integral ratio in Eq. (4) is approximately
0.2%. Since the production asymmetry within the detector acceptance is
expected to be at most a few percent [12, 13, 14], this reduces the effect of
$a_{\rm p}$ to the level of a few $10^{-4}$ for $B^{0}_{s}$ decays. This is
well beneath our target uncertainty of the order of $10^{-3}$, and thus can be
neglected, therefore yielding $A_{\rm meas}$=0.5 $a_{\rm sl}^{\rm s}$.

The measurement could be affected by a detection charge-asymmetry, which may
be induced by the event selection, tracking, and muon selection criteria. The
measured asymmetry can be written as

$A_{\rm meas}=A_{\mu}^{\rm c}-A_{\rm track}-A_{\rm bkg},$ (5)

where $A_{\mu}^{\rm c}$ is given by

$A_{\mu}^{\rm
c}=\frac{N(D^{-}_{s}\mu^{+})-N(D^{+}_{s}\mu^{-})\times\frac{\epsilon(\mu^{+})}{\epsilon(\mu^{-})}}{N(D^{-}_{s}\mu^{+})+N(D^{+}_{s}\mu^{-})\times\frac{\epsilon(\mu^{+})}{\epsilon(\mu^{-})}}.$
(6)

$N(D^{-}_{s}\mu^{+})$ and $N(D^{+}_{s}\mu^{-})$ are the measured yields of
$D_{s}\mu$ pairs, $\epsilon(\mu^{+})$ and $\epsilon(\mu^{-})$ are efficiency
corrections accounting for trigger and muon identification effects, $A_{\rm
track}$ is the track-reconstruction asymmetry of charged particles, and
$A_{\rm bkg}$ accounts for asymmetries induced by backgrounds.

##  2 The LHCb detector and trigger

We use a data sample corresponding to an integrated luminosity of 1.0
$\mbox{\,fb}^{-1}$ collected in 7 TeV $pp$ collisions with the LHCb detector
[15]. This detector 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 has momentum resolution $\Delta p/p$ that varies from
0.4% at 5$\mathrm{\,Ge\kern-1.00006ptV}$ to 0.6% at
100$\mathrm{\,Ge\kern-1.00006ptV}$.222We work in units with $c$=1. Charged
hadrons are identified using two ring-imaging Cherenkov (RICH) detectors [16].
Photon, electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and pre-shower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers
[17]. The LHCb coordinate system is a right handed Cartesian system with the
positive $z$-axis aligned with the beam line and pointing away from the
interaction point and the positive $x$-axis following the ground of the
experimental area, and pointing towards the outside of the LHC ring.

The trigger system [18] 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. For the $D_{s}\mu$ signal samples, the
hardware trigger (L0) requires the detection of a muon of either charge with
transverse momentum $\mbox{$p_{\rm T}$}>1.64$ GeV. In the subsequent software
trigger, a first selection algorithm confirms the L0 candidate muon as a fully
reconstructed track, while the second level algorithm includes two possible
selections. One is based on the topology of the candidate muon and one or two
additional tracks, requiring them to be detached from the primary interaction
vertex. The second category is specifically designed to detect inclusive
$\phi\rightarrow K^{+}K^{-}$ decays. We consider all candidates that satisfy
either selection algorithm. We also study two mutually exclusive samples, one
composed of candidates that satisfy the second trigger category, and the other
satisfying the topological selection of events including a muon, but not the
inclusive $\phi$ algorithm. Approximately 40% of the data were taken with the
magnetic field up, oriented along the positive $y$-axis in the LHCb coordinate
system, and the rest with the opposite down polarity. We exploit the fact that
certain detection asymmetries cancel if data from different magnet polarities
are combined.

##  3 Selection requirements

Additional selection criteria exploiting the kinematic properties of
semileptonic $b$-hadron decays [19, 20, 21] are used. In order to minimize
backgrounds associated with misidentified muons, additional selection criteria
on muons are that the momentum, $p$, be between 6 and 100 GeV, that the
pseudorapidity, $\eta$, be between 2 and 5, and that they are inconsistent
with being produced at any primary vertex. Tracks are considered as kaon
candidates if they are identified by the RICH system, have $\mbox{$p_{\rm
T}$}>0.3$ GeV and $p>2$ GeV. The impact parameter (IP), defined as the minimum
distance of approach of the track with respect to the primary vertex, is used
to select tracks coming from charm decays. We require that the $\chi^{2}$,
formed by using the hypothesis that each track’s IP is equal to 0, which
measures whether a track is consistent with coming from the PV, is greater
than 9. To be reconstructed as a $\phi$ meson candidate, a $K^{+}K^{-}$ pair
must have invariant mass within $\pm$20 MeV of the $\phi$ meson mass.
Candidate $\phi$ mesons are combined with charged pions to make $D_{s}$ meson
candidates. The sum of the $p_{\rm T}$ of $K^{+}$, $K^{-}$ and $\pi^{\pm}$
candidates must be larger than 2.1 GeV. The vertex fit $\chi^{2}$ divided by
the number of degrees of freedom (ndf) must be less than 6, and the flight
distance $\chi^{2}$, formed by using the hypothesis that the $D^{+}_{s}$
flight distance is equal to 0, must be greater than 100. The $B^{0}_{s}$
candidate, formed from the $D_{s}$ and the muon, must have vertex fit
$\chi^{2}$/ndf $<6$, be downstream of the primary vertex, have $2<\eta<5$ and
have invariant mass between 3.1 and 5.1 GeV. Finally, we include some angular
selection criteria that require that the $B_{s}$ candidate have a momentum
aligned with the measured fight direction. The cosine of the angle between the
$D_{s}\mu$ momentum direction and the vector from the primary vertex to the
$D_{s}\mu$ origin must be larger than 0.999. The cosine of the angle between
the $D_{s}$ momentum and the vector from the primary vertex to the $D_{s}$
decay vertex must be larger than 0.99.

##  4 Analysis method

Signal yields are determined by fitting the $K^{+}K^{-}\pi^{+}$ invariant mass
distributions shown in Fig. 1. We fit both the signal $D^{+}_{s}$ and $D^{+}$
peaks with double Gaussian functions with common means. The $D^{+}$ channel is
used only as a component of the fit to the mass spectrum. The average mass
resolution is about 7.1 $\mathrm{\,Me\kern-1.00006ptV}$. The background is
modelled with a second-order Chebychev polynomial. The signal yields from the
fits are listed in Table 1.

Figure 1: Invariant mass distributions for: (a) $K^{+}K^{-}\pi^{+}$ and (b) $K^{+}K^{-}\pi^{-}$ candidates for magnet up, (c) $K^{+}K^{-}\pi^{+}$ and (d) $K^{+}K^{-}\pi^{-}$ candidates for magnet down with $K^{+}K^{-}$ invariant mass within $\pm$20 $\mathrm{\,Me\kern-0.90005ptV}$ of the $\phi$ meson mass. The $D^{+}_{s}$ [yellow (grey) shaded area] and $D^{+}$ [red (dark) shaded area] signal shapes are described in the text. The $\chi^{2}$/ndf for these fits are 1.28, 1.25, 1.53, and 1.27 respectively, the corresponding p-values are 7%, 8%, 4%, 7%. Table 1: Yields for $D^{+}_{s}\mu^{-}$ and $D^{-}_{s}\mu^{+}$ events separately for magnet up and down data. These yields contain very small contributions from prompt $D_{s}$ and b-hadron backgrounds.  | magnet up | magnet down
---|---|---
$D^{-}_{s}\mu^{+}$ | $38\,742\pm 218$ | $53\,768\pm 264$
$D^{+}_{s}\mu^{-}$ | $38\,055\pm 223$ | $54\,252\pm 259$

The detection asymmetry is largely induced by the dipole magnet, which bends
particles of different charge in different detector halves. The magnet
polarity is reversed periodically, thus allowing the measurement and
understanding of the size of this effect. We analyze data taken with different
magnet polarities separately, deriving charge asymmetry corrections for the
two data sets independently. Finally, we average the two values in order to
cancel charge any residual effects. We use two calibration samples containing
muons to measure the relative trigger efficiencies of
$D_{s}^{+}\mu^{-}/D_{s}^{-}\mu^{+}$ events, and the relative $\mu^{-}/\mu^{+}$
identification efficiencies. The first sample contains
$b\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})X$ decays triggered independently of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson, and where the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is selected by requiring two
particles of opposite charge have an invariant mass consistent with the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass. This sample is called the
kinematically-selected (KS) sample. The second sample is collected by
triggering on one muon from a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
decay that is detached from the primary vertex. It is called muon selected
(MS) as it relies on the presence of a well identified muon.

In order to measure the relative $\pi^{+}$ and $\pi^{-}$ detection
efficiencies, we use the ratio of partially reconstructed and fully
reconstructed $D^{*+}\rightarrow\pi^{+}D^{0}$, $D^{0}\rightarrow
K^{-}\pi^{+}\pi^{+}(\pi^{-})$ decays. The former sample is gathered without
explicitly reconstructing the $\pi^{-}$ particle, and then the efficiency of
finding this track in the event is measured. The same procedure is applied to
the charge conjugate mode, so the relative $\pi^{+}$ to $\pi^{-}$ efficiency
is measured. A detailed description is given in Ref. [22].

Finally, a sample of $D^{+}(\rightarrow K^{-}\pi^{+}\pi^{+})\mu^{-}$
candidates is obtained using similar triggers to the $D_{s}\mu$ sample. This
sample is used to assess charge asymmetries induced by the software trigger.

The efficiency ratio $\epsilon_{\mu^{+}}/\epsilon_{\mu^{-}}$ in Eq. (6)
accounts for losses due to the muon identification efficiency algorithm and
the trigger requirements. We measure $\epsilon_{\mu^{+}}/\epsilon_{\mu^{-}}$
using the KS and MS calibration samples. There are about 0.6 million KS
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates selected in total,
and about 1.2 million MS ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidates. As the calibration muon spectra are slightly softer than that of
the signal, we subdivide the signal and calibration samples into subsamples
defined by the kinematic properties of the candidate muon. We define five muon
momentum bins: $6-20~{}\mathrm{\,Ge\kern-1.00006ptV}$,
$20-30~{}\mathrm{\,Ge\kern-1.00006ptV}$,
$30-40~{}\mathrm{\,Ge\kern-1.00006ptV}$,
$40-50~{}\mathrm{\,Ge\kern-1.00006ptV}$, and
$50-100~{}\mathrm{\,Ge\kern-1.00006ptV}$. We further subdivide the signal and
calibration samples with two binning schemes. In the first, each $\mu$
momentum bin is split into 10 rectangular regions in $qp_{x}$ and $p_{y}$,
where $q$ represents the muon charge and $p_{x}$ and $p_{y}$ are the Cartesian
components of the muon momentum in the directions perpendicular to the beam
axis. The second grid uses 8 regions of muon $p_{\rm T}$ and azimuthal angle
$\phi$ to reduce the sensitivity to differences in $\phi$ acceptance between
signal and calibration samples. In this case the first and third bins in
$\phi$ are flipped for negative charges, to symmetrize the acceptance in a
consistent manner with the $qp_{x}$ and $p_{y}$ binning. Signal and
calibration yields are determined separately in each of the intervals both for
magnet up and down data. Figure 2 shows the $\mu^{+}\mu^{-}$ invariant mass
distribution for the KS ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ events
in magnet up data.

Figure 2: Invariant $\mu^{+}\mu^{-}$ mass distributions of the kinematically-
selected ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates in magnet
up data, where the red (open) circles represent entries where the muon
candidate, kinematically selected, is rejected and the black (filled) circles
those where it is accepted by the muon identification algorithm. The dashed
lines represent the combinatorial background.

The relative efficiencies for triggering and identifying muons in five
different momentum bins are shown in Fig. 3 for magnet up and magnet down data
using the KS calibration sample. They are consistent with being independent of
momentum. The small difference of approximately 1% between the two samples can
be attributed to the alignment of the muon stations, which affects
predominantly the hardware muon trigger.

Figure 3: Relative muon efficiency as a function of muon momentum determined
using the kinematically-selected ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ sample.

The $D_{s}^{+}\mu^{-}$ final state benefits from several cancellations of
potential instrumental asymmetries that can arise due to the different
interaction cross-sections in the detector material or to differences between
tracking reconstructions of negative and positive particles. The $\mu$ and
$\pi$ charged tracks have very similar reconstruction efficiencies. Using the
partially-reconstructed $D^{*+}$ calibration sample, we found that the
$\pi^{+}$ versus $\pi^{-}$ relative tracking efficiencies are independent of
momentum and transverse momentum [22]. This, along with the fact that
$\pi^{+}$ and $\pi^{-}$ interaction cross-sections on isoscalar targets are
equal, and that the detector is almost isoscalar, implies that the difference
between $\pi^{+}$ and $\pi^{-}$ tracking efficiencies depend only upon the
magnetic field orientation and the detector acceptance. Thus the charge
asymmetry ratios measured for pions are applicable to muons as well. In the
$\phi\pi^{+}\mu^{-}$ final states, the pion and muon have opposite signs, and
thus the charge asymmetry in the track reconstruction efficiency induced by
imperfect $\pi\mu$ cancellation, $A_{\rm track}^{\pi\mu}$, is small. Using the
efficiency ratios $\epsilon_{\pi^{+}}/\epsilon_{\pi^{-}}$ measured with the
$D^{*+}$ calibration sample, we obtain $A_{\rm track}^{\pi\mu}=(+0.01\pm
0.13)$%. A small residual sensitivity to the charge asymmetry in $K$ track
reconstruction is present due to a slight momentum mismatch between the two
kaons from $\phi$ decays arising from the interference with the S-wave
component. It is determined to be $A_{\rm track}^{KK}=(+0.012\pm 0.004)$%. The
efficiency ratios used in determining $A_{\rm track}^{KK}$ are based on
$\epsilon_{\pi^{+}}/\epsilon_{\pi^{-}}$ with a correction derived from the
comparison between the Cabibbo-favoured decays $D^{+}\rightarrow
K^{-}\pi^{+}\pi^{-}$ and $D^{+}_{s}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}$, accounting for additional charge asymmetry induced by $K$
interactions in the detector. Therefore, the total tracking asymmetry is
$A_{\rm track}=(+0.02\pm 0.13)$%.

##  5 Backgrounds

Backgrounds include prompt charm production, fake muons associated with real
$D_{s}^{+}$ particles produced in $b$-hadron decays, and $B\rightarrow DD_{s}$
decays where the $D$ hadron decays semileptonically. Here $B$ denotes any
meson or baryon containing a $b$ (or $\overline{b}$) quark, and similarly, $D$
denotes any hadron containing a $c$ (or $\overline{c}$) quark. The prompt
background is highly suppressed by the requirement of a well identified muon
forming a vertex with the $D^{+}_{s}$ candidate. The prompt yield is separated
from false $D_{s}$ backgrounds using a binned two-dimensional fit to the mass
and ln(IP/mm) of the $\phi\pi^{+}$ candidates. The method is described in
detail in Ref. [21]. Figure 4 shows the fit results for the magnet-down
$D_{s}^{+}\mu^{-}$ candidate sample. From the asymmetry in the prompt yield
normalized to the overall signal yield in the five momentum bins, we obtain an
asymmetry due to prompt background equal to (+0.14$\pm$0.07)% for magnet up
data, ($-0.05\pm 0.05)$% for magnet down data, with an average value of
(+0.04$\pm$0.04)%.

Figure 4: (a) Spectrum of the logarithm of the IP calculated with respect to
the primary vertex for $D^{+}_{s}$ candidates in combination with muons; the
insert shows a magnified view of the region where the prompt $D^{+}_{s}$
contribution peaks. The blue dashed line is the component coming from $B$
hadron decays, the black dashed line the false $D^{+}_{s}$ background, the red
line the prompt background, (b) the invariant mass distributions for
$D^{+}_{s}\rightarrow\phi\pi$ candidates. These distributions are for the
magnet down sample. (For interpretation of the reference to colour in this
figure legend, the reader is referred to the web version of this Letter.)

Samples of $D^{+}_{s}\pi^{-}X$ and $D^{+}_{s}K^{-}X$ events, where $X$
represents undetected particles from the same decay, are used to infer the
numbers of $D^{+}_{s}$-hadron combinations from $B$ decays that could be
mistaken for $D^{+}_{s}\mu^{-}$ events if the hadron is misidentified as a
muon. Kaons and pions are identified using the RICH. These numbers, combined
with knowledge of the probability that kaons or pions are mistaken for muons,
provide a measurement of the fake hadron background. These misidentification
probabilities are also calculated in the five momentum bins using
$D^{*+}\rightarrow\pi^{+}D^{0}$ decays, with $D^{0}$ decaying into the
$K^{-}\pi^{+}$ final state. The net effect on the asymmetry is below $10^{-4}$
and thus the $D^{+}_{s}$-hadron background can be ignored.

We also consider the background induced by $D_{s}^{+}\mu^{-}$ events deriving
from $b\rightarrow c\bar{c}s$ decays where the $D^{+}_{s}$ hadron originates
from the virtual $W^{+}$ boson and the muon originates from the charmed-hadron
semileptonic decay. These backgrounds are suppressed since the $D$ hadron
travels away from the $B$ vertex prior to its semileptonic decay. As these
decays are of opposite sign to the signal, they cause a background asymmetry
that is proportional to the production asymmetry of the background sources.
The $B^{0}$ production asymmetry has been measured in LHCb to be $(-0.1\pm
1.0)$% [13], and the $B^{+}$ production asymmetry to be $(+0.3\pm 0.9)$% by
comparing $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
and $B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$ decays
[23]. A small subset of this background is from $\mathchar
28931\relax_{b}^{0}$ decays, whose production asymmetry is not well known,
$a_{\rm P}=(-1.0\pm 4.0)$%, but is consistent with zero [24]. The $B^{0}$
final states include $D^{0}$ and $D^{+}$ hadrons, in proportions determined
according to the $D^{*+}/D^{+}$ ratio in the measured exclusive final states.
In addition, we consider backgrounds coming from $B^{0},B^{+}\rightarrow
D^{-}_{s}K\mu^{+}$ decays, that provide a background asymmetry with opposite
sign. We estimate this background asymmetry to be (+0.01$\pm$0.04)%. The
systematic uncertainty includes the limited knowledge of the inclusive
branching fraction of the $b$-hadrons, uncertainties in the $b$-hadron
production ratios, and in the charm semileptonic branching fractions, but is
dominated by the uncertainty in the production asymmetry. By combining these
estimates, we obtain $A_{\rm bkg}=(+0.05\pm 0.05)$%.

##  6 Results

We perform weighted averages of the corrected asymmetries $A_{\mu}^{c}$
observed in each $p_{\rm T}\phi$ and $p_{x}p_{y}$ subsample, using muon
identification corrections both in the KS and MS sample (see Fig. 5). In order
to cancel remaining detection asymmetry effects, the most appropriate way to
combine magnet up and magnet down data is with an arithmetic average [22]. We
then perform an arithmetic average of the four values of $A_{\mu}^{c}$
obtained with the two binning schemes chosen and with the two muon correction
methods, assuming the results to be fully statistically correlated, and obtain
$A_{\mu}^{c}=(+0.04\pm 0.25)$%. The results are shown in Table 2. Finally, we
correct for tracking efficiency asymmetries and background asymmetries, and
obtain

$A_{\rm meas}=(-0.03\pm 0.25\pm 0.18)\%,$

where the first uncertainty reflects statistical fluctuations in the signal
yield and the second reflects the systematic uncertainties. This gives

$a^{s}_{\rm sl}=(-0.06\pm 0.50\pm 0.36)\%.$

Figure 5: Asymmetries corrected for relative muon efficiencies,
$A_{\mu}^{\text{c}}$, examined in the five muon momentum intervals for (a)
magnet up data, (b) magnet down data and (c) average, using the KS muon
calibration method. Then (d) magnet up data, (e) magnet down data and (f)
average, using the MS muon calibration method in the two different binning
scheme.  Table 2: Muon efficiency ratio corrected asymmetry $A_{\mu}^{c}$. The
errors account for the statistical uncertainties in the $B^{0}_{s}$ signal
yields.

$A_{\mu}^{c}~{}~{}[\%]$ | KS muon correction | MS muon correction | Average
---|---|---|---
Magnet | $p_{x}p_{y}$ | $p_{\rm T}\phi$ | $p_{x}p_{y}$ | $p_{\rm T}\phi$ |
Up | $+0.38\pm 0.38$ | $+0.30\pm 0.38$ | $+0.64\pm 0.37$ | $+0.63\pm 0.37$ | $+0.49\pm 0.38$
Down | $-0.17\pm 0.32$ | $-0.25\pm 0.32$ | $-0.60\pm 0.32$ | $-0.62\pm 0.32$ | $-0.41\pm 0.32$
Avg. | $+0.11\pm 0.25$ | $+0.02\pm 0.25$ | $+0.02\pm 0.24$ | $+0.01\pm 0.24$ | $+0.04\pm 0.25$

We consider several sources of systematic uncertainties on $A_{\rm meas}$ that
are summarized in Table 3. By examining the variations on the average
$A_{\mu}^{c}$ obtained with different procedures, we assign a 0.07%
uncertainty, reflecting three almost equal components: the fitting procedure,
the kinematic binning and a residual systematic uncertainty related to the
muon efficiency ratio calculation. We study the effect of the fitting
procedure by comparing results obtained with different models for signal and
background shapes. In addition, we consider the effects of the statistical
uncertainties of the efficiency ratios, assigning 0.08%, which is obtained by
propagating the uncertainties in the average $A_{\mu}^{\rm c}$. The
uncertainties affecting the background estimates are discussed in Sec. 5.
Possible changes in detector acceptance during magnet up and magnet down data
taking periods are estimated to contribute 0.01%. The software trigger
systematic uncertainty is mainly due to the topological trigger algorithm and
is estimated to be 0.05%. These uncertainties are considered uncorrelated and
added in quadrature to obtain the total systematic uncertainty.

Table 3: Sources of systematic uncertainty on $A_{\rm meas}$. Source |  $\sigma(A_{\text{meas}})$[%]
---|---
Signal modelling and muon correction | 0.07
Statistical uncertainty on the efficiency ratios | 0.08
Background asymmetry | 0.05
Asymmetry in track reconstruction | 0.13
Field-up and field-down run conditions | 0.01
Software trigger bias (topological trigger) | 0.05
Total | 0.18

##  7 Conclusions

We measure the asymmetry $a^{s}_{\rm sl}$, which is twice the measured
asymmetry between $D_{s}^{-}\mu^{+}$ and $D_{s}^{+}\mu^{-}$ yields, to be

$a^{s}_{\rm sl}=(-0.06\pm 0.50\pm 0.36)\%.$

Figure 6 shows this measurement, the D0 measured asymmetries in dimuon decays
in 1.96 TeV $p\overline{p}$ collisions of $A_{\rm sl}^{b}=(-0.787\pm 0.172\pm
0.093)$% [5], $a^{d}_{\rm sl}=(0.68\pm 0.45\pm 0.14)\%$ [8], and $a^{s}_{\rm
sl}=(-1.12\pm 0.74\pm 0.17)\%$ [9], and the most recent average from
$B$-factories [10], namely $a_{\rm sl}^{d}=(0.02\pm 0.31)$%. Our result for
$a^{s}_{\rm sl}$ is currently the most precise measurement made and is
consistent with the SM.

Figure 6: Measurements of semileptonic decay asymmetries. The bands correspond
to the central values $\pm$1 standard deviation uncertainties, defined as the
sum in quadrature of the statistical and systematic errors. The solid dot
indicates the SM prediction.

## 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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G}\left(\left\\{fe^{-\frac{\chi}{2}}\left[\frac{2}{z^{3}}\log\left(-4f\,{L}_{\rm
ct}^{2}\right)-\frac{1}{z^{2}}\frac{d}{dz}\left(\log\frac{-fe^{-\chi}}{z^{2}}\right)\right]\right\\}_{z=z_{m1}}\right.$
(245)
$\displaystyle-\left.\left\\{fe^{-\frac{\chi}{2}}\left[\frac{2}{z^{3}}\log\left(-4f\,{L}_{\rm
ct}^{2}\right)-\frac{1}{z^{2}}\frac{d}{dz}\left(\log\frac{-fe^{-\chi}}{z^{2}}\right)\right]\right\\}_{z=z_{m2}}\right)\,,$

where the dependence on $a_{\rm reg}$ simplifies, due to reparametrization
invariance.

###  G.3 After $t_{c}$

After $t_{c}$ the expressions for case $U$ are the same as for case $D$, with
$z_{m}=z_{m1}$, see figures 1 and 2 on the right. To find the time dependence
of action, it is important to evaluate the value of $z_{m}$ as a function of
the boundary time; this can be found by solving the equation

$z^{*}(z_{m})=\frac{t_{b}}{2}\,.$ (246)

Taking time derivative

$\frac{dz_{m}}{dt_{b}}=\frac{1}{2}\frac{1}{\frac{dz^{*}}{dz_{m}}}=\frac{f(z_{m})e^{-\chi(z_{m})/2}}{2}\,.$
(247)

Bulk term: The first two terms $I_{V}^{1,2}$ remain the same as in eq. (237).
The third term is:

$I_{V}^{3}=V_{0}\int^{z_{m}}_{z_{h}}dz\int_{\frac{t_{b}}{2}-z^{*}}^{0}dt\,s(z)=V_{0}\int^{z_{m}}_{z_{h}}dz\left[-\frac{t_{b}}{2}+z^{*}(z)\right]s(z)\,.$
(248)

where $z_{m}$ is the coordinate of the lower joint of the WDW patch (which is
a function of $t_{b}$). The time derivative of the total bulk action is

$\frac{1}{V_{0}}\frac{d\,\Delta
I_{V}}{dt_{b}}=\int^{\infty}_{z_{m}}dz\,s(z)+\frac{d\,z_{m}}{dt_{b}}\left(2z^{*}(z_{m})-t_{b}\right)s(z_{m})=\int^{\infty}_{z_{m}}dz\,s(z)\,,$
(249)

where we used eq. (246).

GHY term: There is just one contribution, nearby the future singularity , that
we cutoff at $z=z_{\max}$

$I_{\rm
GHY}^{1}=2V_{0}\sqrt{|h|}K(z_{\max})(z^{*}(z_{\max})+\frac{t_{b}}{2})=V_{0}A(z_{\max})\left(z^{*}(z_{\max})+\frac{t_{b}}{2}\right)\,,$
(250)

where $A(z)$ is given in eq. (211) The action rate (including the factor of
$2$) is

$\frac{1}{V_{0}}\frac{dI_{\rm GHY}}{dt_{b}}=A(z_{\rm max})\,,$ (251)

which should be computed in the limit $z_{\rm max}\rightarrow\infty$.

Joint contributions: There is only a contribution which is time dependent,
which is the one located at $z=z_{m}$. The light-like normals are:

$k_{1}=(-\,dt+\frac{1}{f}e^{\frac{\chi}{2}}\,dz)a_{\rm reg}\,,\qquad
k_{2}=(\,dt+\frac{1}{f}e^{\frac{\chi}{2}}\,dz)a_{\rm reg}\,,$ (252)

The join term in eq. (213) then gives

$I_{J}=-\frac{V_{0}}{8\pi
G}\frac{1}{z_{m}^{2}}\log\left|\frac{fe^{-\chi}}{z^{2}a_{\rm
reg}^{2}}\right|_{z=z_{m}}\,,$ (253)

where we took $\eta=1$.

At late time $z_{m}\rightarrow z_{h}$, but the term does not approach to
constant because it is log divergent. The time derivative is:

$\frac{dI_{J}}{dt_{b}}=-\frac{V_{0}}{16\pi
G}\left\\{fe^{-\frac{\chi}{2}}\frac{d}{dz}\left(\frac{1}{z^{2}}\log\frac{-fe^{-\chi}}{z^{2}a_{\rm
reg}^{2}}\right)\right\\}_{z=z_{m}}\,.$ (254)

Null boundaries counterterms: From eq. (222), including a factor of $2$

$\frac{I_{ct}}{dt_{b}}=\frac{V_{0}}{4\pi
G}\left\\{fe^{-\frac{\chi}{2}}\frac{1}{z^{3}}\,\log\left(2a_{\rm
reg}\,{L}_{\rm ct}\,z\,e^{\frac{\chi}{2}}\right)\right\\}_{z=z_{m}}$ (255)

This contribution vanishes at late time, as $z_{m}\to z_{h}$. Combining the
joint term and the counterterm, we find

$\frac{d(I_{J}+I_{ct})}{dt_{b}}=\frac{V_{0}}{16\pi
G}\left\\{fe^{-\frac{\chi}{2}}\left[\frac{2}{z^{3}}\log\left(-4f\,{L}_{\rm
ct}^{2}\right)-\frac{1}{z^{2}}\frac{d}{dz}\left(\log\frac{-fe^{-\chi}}{z^{2}}\right)\right]\right\\}_{z=z_{m}}$
(256)

Note that the dependence on $a_{\rm reg}$ simplifies, due to reparametrization
invariance.

##  Appendix H An estimate of $t_{c}$ for small $\phi_{h}$

This estimate is independent from the mass of the scalar, because is derived
using the behaviour nearby the collapse of the ER bridge, which is universal
in the scalar mass Hartnoll:2020rwq ; Hartnoll:2020fhc . It also applies to
the $q\neq 0$ case.

In the regime of small $\phi_{h}$, we have that the back reaction is small
just up to $z<z_{c}-\epsilon$, where $\epsilon$ should go to zero for
$\phi_{h}\to 0$. In this limit we expect that the collapse of Einstein-Rosen
bridge is very fast in the coordinate $z$, and so we expect that for
$z>z_{c}-\epsilon$ we are already in the Kasner regime. So we may say that:

$\displaystyle z^{*}(z)$ $\displaystyle=$ $\displaystyle z^{*}_{\rm
RN}(z)\qquad{\rm for}\qquad z<z_{i}=z_{c}-\epsilon\,,$ $\displaystyle
z^{*}(z)$ $\displaystyle=$ $\displaystyle z^{*}_{\rm
RN}(z_{c}-\epsilon)+\frac{1}{2}\frac{e^{\frac{\chi_{0}}{2}}}{f_{0}}\left(\frac{1}{z^{2}}-\frac{1}{z_{c}^{2}}\right)\qquad{\rm
for}\qquad z>z_{i}\,,$ (257)

where we have used the Kasner approximation eq. (32) for $z>z_{c}-\epsilon$.
The critical time is given by (147).

We can estimate $\epsilon$ using equation (61), in which
$\tilde{A}=O(\phi_{h}^{2})$ and $\tilde{B},\tilde{C}=O(\phi_{h}^{0})$. The
location of $\epsilon$ is determined by the coordinate in which the terms
proportional to $\log H$ and the one proportional to $H$ are of the same
order. This happens for

$O(\phi_{h})<\epsilon<O(\phi_{h}^{2})\,.$ (258)

Using eq. (224), we can approximate $z^{*}_{\rm RN}$ nearby $z_{c}$ as
follows:

$z^{*}_{\rm
RN}(z_{c}-\epsilon)\approx\frac{z_{h}z_{c}}{z_{c}-z_{h}}\frac{1}{\left(\frac{z_{c}}{z_{h}}+1\right)^{2}+2}\log\frac{\epsilon}{z_{c}}$
(259)

From eq. (257), the critical time is given by

$t_{c}=-2z^{*}(\infty)=2\frac{z_{h}z_{c}}{z_{c}-z_{h}}\frac{1}{\left(\frac{z_{c}}{z_{h}}+1\right)^{2}+2}\log\left(\frac{\epsilon}{z_{c}}\right)+\frac{1}{f_{0}e^{\frac{-\chi_{0}}{2}}}\frac{1}{z_{c}^{2}}\,.$
(260)

From eq. (85), we find that the first term in eq. (260) is subleading compared
to the second, and that the critical time scales as in eq. (148). This
behaviour is consistent with a numerical analysis, see figure 19.

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