# Improving Maximum Likelihood Difference Scaling method to measure inter
content scale

###### Abstract

The goal of most subjective studies is to place a set of stimuli on a
perceptual scale. This is mostly done directly by rating, e.g. using single or
double stimulus methodologies, or indirectly by ranking or pairwise
comparison. All these methods estimate the perceptual magnitudes of the
stimuli on a scale. However, procedures such as Maximum Likelihood Difference
Scaling (MLDS) have shown that considering perceptual distances can bring
benefits in terms of discriminatory power, observers’ cognitive load, and the
number of trials required. One of the disadvantages of the MLDS method is that
the perceptual scales obtained for stimuli created from different source
content are generally not comparable. In this paper, we propose an extension
of the MLDS method that ensures inter-content comparability of the results and
shows its usefulness especially in the presence of observer errors.

Index Terms—  Difference scaling, supra-threshold estimation, human
perception, subjective experiment

##  1 Introduction

In general, it is important to perform subjective tests to validate systems
(e.g. coding algorithms), benchmark objectives metrics, or create datasets to
train machine learning or deep learning models. However, collecting these
annotations can be time-consuming and expensive to perform. Due to the
subjectivity of the data, we don’t always have an agreement in the judgment of
people and accurate estimation is important to reduce noise and uncertainty on
collected data. In that case, to boost these subjective tests is essential to
allocate annotation resources on the right stimuli.

Multiple methodologies exist to rate stimuli, with direct estimation like
Absolute Category Rating (ACR) or Double Stimuli Impairment Scale (DSIS), or
indirectly, two-Alternative Forced Choice (2AFC) or pairwise comparison (PC).
PC is a more reliable method because observers only need to provide their
preference on each pair, comparisons are more sensitive, which is important to
improve the discriminability.

PC experiments are generating matrices of values indicating how many times a
stimulus has been preferred over another. These Pair Comparison Matrix (PCM)
can be translated to a continuous scale, using models (e.g. Thurstone[1],
Bradley and Terry [2], Tversky [3]). Due to the pairwise manner of presenting
stimuli, the PCM size, and the number of possible comparisons is growing
quadratically with the number of stimuli, introducing efficiency in a
subjective protocol. A lot of previous works have focused on active sampling
solutions [4, 5, 6, 7, 8, 9] and more recently [10, 11, 12, 13, 14, 15, 16] to
select the most informative pairs, and minimize experimental effort while
maintaining accurate estimations.

In this work, we want to present an extension to the MLDS methodology [17,
18]. MLDS can be used to separately estimate scales of different stimuli sets
with intra-set supra-threshold perceptual differences, see in section 2.1. But
it isn’t possible to estimate a global scale where the stimuli of different
sets can be related, section2.2. These sets of stimuli from different contents
are compared using different sampling strategies, introduced in section 3.
With simulations, objectives are to demonstrate that our propositions can
achieve good discriminability over content with robustness against bad
annotator behavior and with efficiency.

Fig. 1: Example of intra-content trial presented to an observer to rate where
he perceives a greater difference in the quadruplet.

##  2 Maximum Likelihood Difference Scaling

In this section, we explain the principle of MLDS[17, 18] and its extension to
estimate the scale of cross-content perception.

The content $C_{i}$ consists of a reference stimulus $S^{i}_{1}$ and $n-1$
modified versions of this reference stimuli: $S^{i}_{2},...,S^{i}_{n}$. This
set of stimuli $C_{i}=\\{S^{i}_{1},S^{i}_{2},...,S^{i}_{n}\\}$ is pre-ordered
along a physical continuum, with the assumption that larger alteration
introduce higher perceptual difference:

$S^{i}_{1}<S^{i}_{2}<...<S^{i}_{n}$ (1)

The observer is presented during a trial with a quadruplet of stimuli
$(S_{i},S_{j},S_{k},S_{l})$, see figure 1 for an example, and asked to
estimate where the perceptual distance is greater: inside the pair
$(S_{i},S_{j})$ or $(S_{k},S_{l})$. A trial outcome is either 0 or 1,
corresponding to the judgment of

$\left|S_{i}-S_{j}\right|-\left|S_{k}-S_{l}\right|>0.$ (2)

During solving, MLDS estimates scalar values $(\phi^{i}_{1},...,\phi^{i}_{n})$
to predict the observer judgment in each quadruplet.

###  2.1 Intra-content difference scaling

This system of equations can be reduced to a set of linear equations with the
previous assumption and solved with a General Linear Model (GLM), using a link
function $\eta=logit(\pi(x))$ or $probit(\pi(x))$ where $\pi(x)$ is
$\mathbb{P}(Y=1|X_{1}=x_{1},...,X_{n}=x_{n})$ and,

$\pi(x)=F(\phi_{1}X_{1}+\phi_{2}X_{2}+...+\phi_{n}X_{n})$ (3)

where F is the inverse of the link function $\eta$. An example for 5
quadruplets of a content with 7 stimuli (1-7):

$\begin{pmatrix}2&4&5&6\\\ 1&2&3&7\\\ 1&5&6&7\\\ 1&2&4&6\\\
3&5&6&7\end{pmatrix}$ (4)

yield the following matrix,

$X=\begin{pmatrix}0&1&0&-1&-1&1&0\\\ 1&-1&-1&0&0&0&1\\\ 1&0&0&0&-1&-1&1\\\
1&-1&0&-1&0&1&0\\\ 0&0&1&0&-1&-1&1\end{pmatrix}$ (5)

Once the $(\phi^{i}_{1},...,\phi^{i}_{n})$ are estimated, they can be plotted
as a perceptual curve, for example one of the line in fig. 2.

###  2.2 Inter-contents difference scaling

In the previous section, we introduced the MLDS method and how it estimates
perceptual differences inside a set of stimuli from a content. However, this
method can’t directly relate and scale the difference of perception of a group
of contents.

To estimate a scaling factor for each perceptual curve, one perceptual curve
per content, we added inter-content comparisons. Where a quadruplet is
composed of a pair of stimuli from a content $C_{i}$ and a pair from a content
$C_{j}$: $(S^{i}_{a},S^{i}_{b},S^{j}_{c},S^{j}_{d})$. The observer is asked,
similarly as in intra-content comparison, to judge where he perceives a large
perceptual distance:

$\left|S^{i}_{a}-S^{i}_{b}\right|-\left|S^{j}_{c}-S^{j}_{d}\right|>0$ (6)

With the addition of this set of inter-content trials, we can construct a
larger matrix $X$, solved with GLM, and estimate the difference of scales
between a group of perceptual curves, figure 2.

$X=\begin{pmatrix}&\begin{matrix}0&0&0&0&0&0\end{matrix}\\\
X_{1}&\begin{matrix}0&0&0&0&0&0\end{matrix}\\\
&\begin{matrix}0&0&0&0&0&0\end{matrix}\\\
\begin{matrix}0&0&0&0&0&0\end{matrix}&\\\
\begin{matrix}0&0&0&0&0&0\end{matrix}&X_{2}\\\
\begin{matrix}0&0&0&0&0&0\end{matrix}&\\\
\begin{matrix}1&-1&0&0&0&0\end{matrix}&\begin{matrix}-1&1&0&0&0&0\end{matrix}\\\
\begin{matrix}1&0&-1&0&0&0\end{matrix}&\begin{matrix}0&0&-1&0&1&0\end{matrix}\\\
\end{pmatrix}$ (7)

$X$ is composed of submatrices $(X_{1},X_{2})$ of intra-content comparisons
for a content $C_{1}$ and $C_{2}$ and a third matrix of inter-content
comparisons (e.g. last 2 rows in the example above).

Fig. 2: Example of 5 contents with 7 stimuli each and their perceptual curves,
estimated using GLM and different scaling. On the right, the ground truth data
to recover, and on the left estimations produced with simulated annotations of
15 participants, CIs are obtained via bootstrapping [19].

##  3 Experimental Simulations

In this section, we will experiment with the modification that we added to the
MLDS methodology and check how we can select relevant quadruplets to minimize
the experimental effort and be robust to spammer behavior. All comparisons are
performed with simulated annotations.

###  3.1 Quadruplet selection

####  3.1.1 Intra-content quadruplet selection

The quadruplets for intra-content comparisons are generated following the
recommendation of MLDS paper [17, 18], where there are $\binom{L}{4}$
quadruplets for a L levels difference scaling experiment. In our case, we
selected $L=7$ levels to be estimated (a reference stimuli + 6 distorted
versions) yielding 35 quadruplets to evaluate for each content.

####  3.1.2 Inter-content quadruplet selection

In the case of inter-content comparisons, we can create for a first content
$M=\frac{L\times(L-1)}{2}$ pairs to be compared with the $M$ pairs of a second
content, which is not tractable when the number of contents is large: if N is
the number of content, there are $\frac{N\times(N-1)}{2}\times M^{2}$ possible
quadruplets. This will be the baseline fulldesign that we will improve from in
the next section.

Next, we describe some sampling strategies to reduce the fraction of
comparisons to perform and how we validate them in simulations.

###  3.2 Sampling strategies

####  3.2.1 Similar alteration levels comparisons

To reduce the amount of inter-content quadruplets to evaluate, we decided to
compare only the same alteration levels pair in inter-content where a
quadruplet is formed as follows $(S^{i}_{a},S^{i}_{b},S^{j}_{a},S^{j}_{b})$,
with $a$ and $b$ 2 different alteration levels. On top of that to reduce the
number of pairs available in comparison we constrained in a case named
”consecutive design” to have $b=a+1$ and in a second case ”reference design”
$a=1$ where $S^{i}_{1}$ and $S^{j}_{1}$ are the references.

with these 2 cases, and the associated design constraints, we reduce the
number of comparisons to $L-1$ quadruplets: where the $L-1$ pairs, created
from $L$ levels of alterations, of a content A are compared once to the $L-1$
pairs of a content B. Then when comparing N contents together we get
$(L-1)\times\frac{N\times(N-1)}{2}$ quadruplets.

####  3.2.2 k-connection contents comparisons

The second part we can improve on is to reduce comparisons to only in sets of
k contents, instead of the N-1 other available contents: $(L-1)\times(N\times
k)$ comparisons with k the number of compared contents.

The k-connection is performed over all contents $A={A_{1},...,A_{n}}$ by
connecting a content $A_{i}$ to the set of contents
$(A_{i+1},A_{i+2},...,A_{i+k})$ with a modulo operation $(i+k)\%n$, to loop
back on the first contents if needed for the last $k-1$ contents. We
experimented with different values of k and report the most interesting one in
the result section.

####  3.2.3 AFAD: Adaptive Far Apart Discard design

When performing selecting comparisons, it is better to focus on close enough
stimuli to avoid comparing perceptual distances that have a large difference,
see eq.8: if we present these pairs to an observer, at best it will result in
the same judgment as to all other previous observers, or in the worse case
(e.g. spammer or error behavior) be an inverse judgment that will mislead the
solving algorithm.

$(S_{i}-S_{j})>>(S_{k}-S_{l})$ (8)

To avoid these cases, we will discard far apart quadruplets using the
estimation of $\phi^{i}_{j}$ values. After each set of annotations from an
observer, our difference scaling implementation fits and estimates the
difference in the stimuli, then discards the bottom 20% of quadruplets with
the largest difference scale. This strategy is reported as ”AFAD” in the
results section.

##  4 Simulations results

Fig. 3: RMSE results for 25 contents with 7 alteration levels: X axis log
scale RMSE (left: number of annotations, right: number of annotations - log
scale) with a 5% probability of inverting an annotation vote.

Fig. 4: Pearson correlation results for 25 contents with 7 alteration levels:
X axis log scale Pearson correlation (left: number of annotators, right:
number of annotations) with a 5% probability of inverting an annotation vote.

To select and compare the performances of the different quadruple selection
methodologies, we compared them through simulation. To evaluate the
improvement of the proposed method, a Monte Carlo simulation was conducted. 25
contents with 7 levels of alterations were designed whose scores were randomly
selected using the following principle. To construct the scores of each
alteration in a content, we uniformly sample on $[0,5/6]$ 6 values and add a
first value of 0 corresponding to the reference. Then we compute the
cumulative sum over this array of 7 elements to yield the final 7 subjective
scores to retrieve through simulation for a content, starting at 0 and ranging
until a maximum of 5. An example of designed contents can be checked in figure
2, right part.

The following assumptions were made for simulation: 1) each stimulus has a
single score, described above; 2) in each observation, the observed judgment
of a quadruplet follows a Gaussian random variable, X, $N(|ma-mb|-|mc-md|,1)$,
with mi, 4 scalars corresponding to the scores of the stimuli in the
quadruplet and a fix standard deviation of 1 to model stochasticity of the
participant perception; 3) each observer has a 5% probability to produce a
mistake, i.e., inverting his judgment, to test the robustness of the solution
against outlier behavior; and 4) each trial is independent. Raw trial
judgments are obtained by taking the sign of the sampled value from X and
mapping it to the values 0 or 1.

The simulations use our python implementation of the original difference
scaling algorithm, with GLM to convert raw trials judgment to estimated
scores. RMSE and Pearson correlation are reported between the estimated scores
and the designed ones. Each of the simulations was run at least 100 times to
report 95% confidence intervals.

Fig. 5: RMSE results for 25 contents with 7 alteration levels: X axis log
scale RMSE (left: number of annotations, right: number of annotations - log
scale) with a 20% probability of inverting an annotation vote.

Fig. 6: Pearson correlation results for 25 contents with 7 alteration levels:
X axis log scale Pearson correlation (left: number of annotators, right:
number of annotations - log scale) with a 20% probability of inverting an
annotation vote.

In figures 3 and 4, we present the simulation results for the different
sampling strategies that we investigated. It shows that for a fixed number of
trials (e.g $10^{4}$), the original fulldesign performs worse than some
designs and especially AFAD. The proposed design converges faster to smaller
estimation errors and a higher correlation score. To estimate the number of
annotations saved by one design over another, we linearly fitted the data:

$log(numberAnnotations)=A\times log(RMSE)+B$ (9)

We use coefficients A and B to predict the number of annotations needed for
RMSE scores of $R=[0.01,...,0.1]$. Finally, we compute the percentage of
annotators saved for each RMSE score in $R$ and average them. In table 1, we
read that AFAD design needs 39.7% less annotations compared to full design to
achieve the same RMSE value on average.

| full design
---|---
4-connect design | -0.68%
reference design | -26.67%
consecutive design | 228%
AFAD | -39.7%
Table 1:  Distance between curves in percent.

###  4.1 Observer Error Resiliency

When we increase the ratio of annotation error from 5% to 20% in fig. 5-6, the
proposed solution AFAD still estimates better than the other methods. We can
see as well that the estimation error is higher than the one report in fig.
3-4, especially with a small number of annotations.

##  5 Conclusion and Future Work

As we emphasized previously, MLDS methodology has a limitation when we need to
compare perceptual scales of different source stimuli. The extension that we
introduce and the active sampling strategies that we suggested, is providing a
solution to perform a subjective experiment in this paradigm of supra-
threshold estimation. The strategy presents robustness to bad annotator
behavior and can effectively reduce the number of comparisons to perform.

This work could be applied to collect robustly and effectively, in a large-
scale crowdsourcing experiment, human judgments on distorted video patches,
via video coding algorithms.

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2011-167 LHCb-PAPER-2011-014

Measurement of the effective $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime

The LHCb Collaboration111Authors are listed on the following pages.

A measurement of the effective $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime
is presented using approximately 37 $\mbox{\,pb}^{-1}$ of data collected by
LHCb during 2010\. This quantity can be used to put constraints on
contributions from processes beyond the Standard Model in the $B^{0}_{s}$
meson system and is determined by two complementary approaches as

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Machikhiliyan4,30, F. Maciuc10, O. Maev29,37, J. Magnin1, S. Malde51, R.M.D.
Mamunur37, G. Manca15,d, G. Mancinelli6, N. Mangiafave43, U. Marconi14, R.
Märki38, J. Marks11, G. Martellotti22, A. Martens7, L. Martin51, A. Martín
Sánchez7, D. Martinez Santos37, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G. McGregor50, R.
McNulty12, C. Mclean14, M. Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S.
Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, S. Monteil5, D. Moran12, P.
Morawski25, R. Mountain52, I. Mous23, F. Muheim46, K. Müller39, R.
Muresan28,38, B. Muryn26, M. Musy35, J. Mylroie-Smith48, P. Naik42, T.
Nakada38, R. Nandakumar45, J. Nardulli45, I. Nasteva1, M. Nedos9, M.
Needham46, N. Neufeld37, C. Nguyen-Mau38,p, M. Nicol7, S. Nies9, V. Niess5, N.
Nikitin31, A. Nomerotski51, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen49, K. Pal52, J. Palacios39, A.
Palano13,b, M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo13,b, C.
Parkes47,37, C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49, S.K.
Paterson49, G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos
Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A.
Petrolini19,i, E. Picatoste Olloqui35, B. Pie Valls35, B. Pietrzyk4, T.
Pilar44, D. Pinci22, R. Plackett47, S. Playfer46, M. Plo Casasus36, G.
Polok25, A. Poluektov44,33, E. Polycarpo2, D. Popov10, B. Popovici28, C.
Potterat35, A. Powell51, T. du Pree23, J. Prisciandaro38, V. Pugatch41, A.
Puig Navarro35, W. Qian52, J.H. Rademacker42, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk40, G. Raven24, S. Redford51, M.M. Reid44, A.C. dos Reis1,
S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5, P. Robbe7, E. Rodrigues47, F.
Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43, S. Roiser37, V. Romanovsky34,
M. Rosello35,n, J. Rouvinet38, T. Ruf37, H. Ruiz35, G. Sabatino21,k, J.J.
Saborido Silva36, N. Sagidova29, P. Sail47, B. Saitta15,d, C. Salzmann39, M.
Sannino19,i, R. Santacesaria22, C. Santamarina Rios36, R. Santinelli37, E.
Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M.
Savrie16,e, D. Savrina30, P. Schaack49, M. Schiller11, S. Schleich9, M.
Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R.
Schwemmer37, B. Sciascia18, A. Sciubba18,l, M. Seco36, A. Semennikov30, K.
Senderowska26, I. Sepp49, N. Serra39, J. Serrano6, P. Seyfert11, B. Shao3, M.
Shapkin34, I. Shapoval40,37, P. Shatalov30, Y. Shcheglov29, T. Shears48, L.
Shekhtman33, O. Shevchenko40, V. Shevchenko30, A. Shires49, R. Silva
Coutinho54, H.P. Skottowe43, T. Skwarnicki52, A.C. Smith37, N.A. Smith48, E.
Smith51,45, K. Sobczak5, F.J.P. Soler47, A. Solomin42, F. Soomro18, B. Souza
De Paula2, B. Spaan9, A. Sparkes46, P. Spradlin47, F. Stagni37, S. Stahl11, O.
Steinkamp39, S. Stoica28, S. Stone52,37, B. Storaci23, M. Straticiuc28, U.
Straumann39, N. Styles46, V.K. Subbiah37, S. Swientek9, M. Szczekowski27, P.
Szczypka38, T. Szumlak26, S. T’Jampens4, E. Teodorescu28, F. Teubert37, C.
Thomas51, E. Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin39, S. Topp-
Joergensen51, N. Torr51, E. Tournefier4,49, M.T. Tran38, A. Tsaregorodtsev6,
N. Tuning23, M. Ubeda Garcia37, A. Ukleja27, P. Urquijo52, U. Uwer11, V.
Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S.
Vecchi16, J.J. Velthuis42, M. Veltri17,g, K. Vervink37, B. Viaud7, I. Videau7,
X. Vilasis-Cardona35,n, J. Visniakov36, A. Vollhardt39, D. Voong42, A.
Vorobyev29, H. Voss10, K. Wacker9, S. Wandernoth11, J. Wang52, D.R. Ward43,
A.D. Webber50, D. Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G.
Wilkinson51, M.P. Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M.
Witek25, W. Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z.
Xing52, Z. Yang3, R. Young46, O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3,
L. Zhang52, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, 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 Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Cracow, Poland
26Faculty of Physics & Applied Computer Science, Cracow, Poland
27Soltan Institute for Nuclear Studies, 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
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2

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
oInstitució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam

##  1 Introduction

The study of charmless $B$ meson decays of the form $B\\!\rightarrow
h^{+}h^{\prime-}$, where $h^{(\prime)}$ is either a kaon, pion or proton,
offers a rich opportunity to explore the phase structure of the CKM matrix and
to search for manifestations of physics beyond the Standard Model. The
effective lifetime, defined as the decay time expectation value, of the
$B^{0}_{s}$ meson measured in the decay channel $B^{0}_{s}\\!\rightarrow
K^{+}K^{-}$ (charge conjugate modes are implied throughout the paper) is of
considerable interest as it can be used to put constraints on contributions
from new physical phenomena to the $B^{0}_{s}$ meson system [1, 2, 3, 4]. The
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ decay was first observed by CDF [5, 6].
The decay has subsequently been confirmed by Belle [7].

The detailed formalism of the effective lifetime in $B^{0}_{s}\\!\rightarrow
K^{+}K^{-}$ decay can be found in Refs. [3, 4]. The untagged decay time
distribution can be written as

$\displaystyle\Gamma(t)$ $\displaystyle\propto$ $\displaystyle\left(1-{\cal
A}_{\Delta\Gamma_{s}}\right)e^{-\Gamma_{L}t}+\left(1+{\cal
A}_{\Delta\Gamma_{s}}\right)e^{-\Gamma_{H}t}\,.$ (1)

The parameter ${\cal A}_{\Delta\Gamma_{s}}$ is defined as ${\cal
A}_{\Delta\Gamma_{s}}=-2{\rm Re}(\lambda)/\left(1+|\lambda|^{2}\right)$ where
$\lambda=(q/p)(\overline{A}/A)$ and the complex coefficients $p$ and $q$
define the mass eigenstates of the $B^{0}_{s}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ system in terms of the flavour
eigenstates (see, e.g., Ref. [8]), while $A$ ($\overline{A}$) gives the
amplitude for $B^{0}_{s}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$) decay to the $C\\!P$ even
$K^{+}K^{-}$ final state. In the absence of $C\\!P$ violation, ${\rm
Re}(\lambda)=1$ and $\rm{Im(\lambda)=0}$, so that the distribution involves
only the term containing $\Gamma_{L}$. Any deviation from a pure single
exponential with decay constant $\Gamma^{-1}_{L}$ is a measure of $C\\!P$
violation.

When modelling the decay time distribution shown in Eq. 1 with a single
exponential function in a maximum likelihood fit, it converges to the
effective lifetime given in Eq. 2 [9]. For small values of the relative width
difference
$\Delta\Gamma_{s}/\Gamma_{s}=(\Gamma_{L}-\Gamma_{H})/\left((\Gamma_{L}+\Gamma_{H})/2\right)$,
the distribution can be approximated by Taylor expansion as shown in the
second part of the equation [3]

$\tau_{KK}=\tau_{B^{0}_{s}}\frac{1}{1-y_{s}^{2}}\left[\frac{1+2{\cal
A}_{\Delta\Gamma_{s}}y_{s}+y_{s}^{2}}{1+{\cal
A}_{\Delta\Gamma_{s}}y_{s}}\right]=\tau_{B^{0}_{s}}\left(1+{\cal
A}_{\Delta\Gamma_{s}}y_{s}+\mathcal{O}(y_{s}^{2})\right),$ (2)

where $\tau_{B^{0}_{s}}=2/\left(\Gamma_{H}+\Gamma_{L}\right)=\Gamma_{s}^{-1}$
and $y_{s}=\Delta\Gamma_{s}/2\Gamma_{s}$. The Standard Model predictions for
these parameters are ${\cal A}_{\Delta\Gamma_{s}}=-0.97^{+0.014}_{-0.009}$[3]
and $y_{s}=0.066\pm 0.016$[10].

The decay $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ is dominated by loop diagrams
carrying, in the Standard Model, the same phase as the $B^{0}_{s}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mixing amplitude and hence the
measured effective lifetime is expected to be close to $\Gamma_{L}^{-1}$. The
tree contribution to the $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ decay amplitude,
however, introduces $C\\!P$ violation effects. The Standard Model prediction
is $\tau_{KK}=1.390\pm 0.032~{}{\rm\,ps}$ [3]. In the presence of physics
beyond the Standard Model, deviations of the measured value from this
prediction are possible.

The measurement has been performed using a data sample corresponding to an
integrated luminosity of $37~{}\mbox{\,pb}^{-1}$ collected by LHCb at an
energy of $\sqrt{s}=7$ TeV during 2010. A key aspect of the analysis is the
correction of lifetime biasing effects, referred to as the acceptance, which
are introduced by the selection criteria to enrich the $B$ meson sample. Two
complementary data-driven approaches have been developed to compensate for
this bias. One method relies on extracting the acceptance function from data,
and then applies this acceptance correction to obtain a measurement of the
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime. The other approach cancels the
acceptance bias by taking the ratio of the $B^{0}_{s}\\!\rightarrow
K^{+}K^{-}$ lifetime distribution with that of $B^{0}\\!\rightarrow
K^{+}\pi^{-}$.

##  2 Data Sample

The LHCb detector [11] is a single arm spectrometer with a pseudorapidity
acceptance of $2<\eta<5$ for charged particles. The detector includes a high
precision tracking system which consists of a silicon vertex detector and
several dedicated tracking planes with silicon microstrip detectors (Inner
Tracker) covering the region with high charged particle multiplicity and straw
tube detectors (Outer Tracker) for the region with lower occupancy. The Inner
and Outer trackers are placed after the dipole magnet to allow the measurement
of the charged particles’ momenta as they traverse the detector. Excellent
particle identification capabilities are provided by two ring imaging
Cherenkov detectors which allow charged pions, kaons and protons to be
distinguished from each other in the momentum range 2–100
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The experiment employs a multi-level
trigger to reduce the readout rate and enhance signal purity: a hardware
trigger based on the measurement of the energy deposited in the calorimeter
cells and the momentum transverse to the beamline of muon candidates, as well
as a software trigger which allows the reconstruction of the full event
information.

$B$ mesons are produced with an average momentum of around 100
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and have decay vertices displaced from
the primary interaction vertex. Background particles tend to have low momentum
and tend to originate from the primary $pp$ collision. These features are
exploited in the event selection. In the absolute lifetime measurement the
final event selection is designed to be more stringent than the trigger
requirements, as this simplifies the calculation of the candidate’s acceptance
function. The tracks associated with the final state particles of the $B$
meson decay are required to have a good track fit quality ($\chi^{2}$/ndf $<3$
for one of the two tracks and $\chi^{2}$/ndf $<4$ for the other), have high
momentum ($p>13.5$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$), and at least one
particle must have a transverse momentum of more than 2.5
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The primary proton-proton interaction
vertex (or vertices in case of multiple interactions) of the event is fitted
from the reconstructed charged particles. The reconstructed trajectory of at
least one of the final state particles is required to have a distance of
closest approach to all primary vertices of at least 0.25$\rm\,mm$.

The $B$ meson candidate is obtained by reconstructing the vertex formed by the
two-particle final state. The $B$ meson transverse momentum is required to be
greater than 0.9 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and the distance of
the decay vertex to the closest primary $pp$ interaction vertex has to be
larger than 2.4$\rm\,mm$. In the final stage of the selection the modes
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ and $B^{0}\\!\rightarrow K^{+}\pi^{-}$
are separated by pion/kaon likelihood variables which use information obtained
from the ring imaging Cherenkov detectors.

The event selection used in the relative lifetime analysis is very similar.
However, some selection criteria can be slightly relaxed as the analysis does
not depend on the exact trigger requirements.

##  3 Relative Lifetime Measurement

Figure 1: Results of the relative lifetime fit. Left: Fit to the time-
integrated $KK$ mass spectrum. Right: Fit to the $KK$ decay time distribution.
The black points show the total number of candidates per picosecond in each
decay time bin, the stacked histogram shows the $B^{0}_{s}\\!\rightarrow
K^{+}K^{-}$ yield in red (dark) and the background yield in grey (light).

This analysis exploits the fact that the kinematic properties of the
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ decay are very similar to those of
$B^{0}\\!\rightarrow K^{+}\pi^{-}$. The two different decay modes can be
separated using information from the ring imaging Cherenkov detectors. The
left part of Fig. 1 shows the invariant mass distribution of the
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ candidates after the final event
selection. In addition 1,424 $B^{0}\\!\rightarrow K^{+}\pi^{-}$ candidates are
selected. Using a data-driven particle identification calibration method
described in the systematics section, the remaining contamination in the
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ sample from other $B\\!\rightarrow
h^{+}h^{\prime-}$ final states in the analysed mass region is estimated to be
3.8%.

$B$ mesons in either channel can be selected using identical kinematic
constraints and hence their decay time acceptance functions are almost
identical. Therefore the effects of the decay time acceptance cancel in the
ratio and the effective $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime can be
extracted relative to the $B^{0}\\!\rightarrow K^{+}\pi^{-}$ mode from the
variation of the ratio $R(t)$ of the yield of $B$ meson candidates in both
decay modes with decay time :

$R(t)=R(0)e^{-t\left(\tau_{KK}^{-1}~{}-~{}\tau_{K\pi}^{-1}\right)}.$ (3)

The cancellation of acceptance effects has been verified using simulated
events, including the full simulation of detector effects, trigger response
and final event selection. Any non-cancelling acceptance bias on the measured
lifetime is found to be smaller 1$\rm\,fs$.

In order to extract the effective $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$
lifetime, the yield of $B$ meson candidates is determined in bins of decay
time for both decay modes. Thirty bins between -1${\rm\,ps}$ and 35${\rm\,ps}$
are chosen such that each bin contains approximately the same number of $B$
meson candidates. The ratio of the yields is then fitted as a function of
decay time and the relative lifetime can be determined according to Eq. 3.
With this approach it is not necessary to parametrise the decay time
distribution of the background. In order to maximise the statistical
precision, both steps of the analysis are combined in a simultaneous fit to
the $K^{+}K^{-}$ and $K^{+}\pi^{-}$ invariant mass spectra across all decay
time bins. The signal distributions are described by Gaussian functions and
the combinatorial background by first order polynomials. The parameters of the
signal and background probability density functions (PDFs) are fixed to the
results of time-integrated mass fits before the lifetime fit is performed. The
$B^{0}\\!\rightarrow K^{+}\pi^{-}$ yield ($N_{B\rightarrow K\pi}$) is allowed
to float freely in each bin but the $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ yield
($N_{B_{s}\rightarrow KK}$) is constrained to follow

$N_{B_{s}\rightarrow KK}(\bar{t}_{i})=N_{B\rightarrow
K\pi}(\bar{t}_{i})R(0)e^{-\bar{t}_{i}\left(\tau_{KK}^{-1}~{}-~{}\tau_{K\pi}^{-1}\right)},$
(4)

where $\bar{t}_{i}$ is the mean decay time in the $i^{\rm th}$ bin. In total
the simultaneous fit has 94 free parameters and tests using Toy Monte Carlo
simulated data have found the fit to be unbiased to below 1$\rm\,fs$ on the
measured $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime. Each mass fit used in
the simultaneous fit is unbinned and must be split into mass bins in order to
evaluate the fit $\chi^{2}$. Two mass bins are chosen, one signal dominated
and one background dominated, in order to guarantee a minimum of 5-6
candidates in each bin. Using this appraoch the $\chi^{2}$ per degree of
freedom of the simultaneous fit is found to be 0.82. The right part of Fig. 1
shows the decay time distribution obtained from the fit and the fitted
reciprocal lifetime difference is

$\tau_{KK}^{-1}~{}-~{}\tau_{K\pi}^{-1}=0.013~{}\pm~{}0.045~{}\mathrm{(stat)}~{}{\rm\,ps}^{-1}.$

Taking the $B^{0}\\!\rightarrow K^{+}\pi^{-}$ lifetime as equal to the mean
$B^{0}$ lifetime ($\tau_{B^{0}}=1.519\pm 0.007~{}{\rm\,ps}$) [8], this
measurement can be expressed as

$\tau_{KK}=1.490\pm 0.100~{}\mathrm{(stat)}\pm
0.007~{}\mbox{(input)~{}${\rm\,ps}$}$

where the second uncertainty originates from the uncertainty of the $B^{0}$
lifetime.

##  4 Absolute Lifetime Measurement

The absolute lifetime measurement method directly determines the effective
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime using an acceptance correction
calculated from the data. This method was first used at the NA11 spectrometer
at CERN SPS [12], further developed within CDF [13, 14] and was subsequently
studied and implemented in LHCb [15, 16]. The per event acceptance function is
determined by evaluating whether the candidate would be selected for different
values of the $B$ meson candidate decay time. For example, for a $B$ meson
candidate, with given kinematic properties, the measured decay time of the $B$
meson candidate is directly related to the point of closest approach of the
final state particles to the associated primary vertex. Thus a selection
requirement on this quantity directly translates into a discrete decision
about acceptance or rejection of a candidate as a function of its decay time.
This is illustrated in Fig. 2. In the presence of several reconstructed
primary interaction vertices, the meson may enter a decay-time region where
one of the final state particles no longer fulfills the selection criteria
with respect to another primary vertex. Hence the acceptance function is
determined as a series of step changes. These _turning points_ at which the
candidates enter or leave the acceptance of a given primary vertex form the
basis of extracting the per-event acceptance function in the data. The turning
points are determined by moving the reconstructed primary vertex position of
the event along the $B$ meson momentum vector, and then reapplying the event
selection criteria. The analysis presented in this paper only includes events
with a single turning point. The drop of the acceptance to zero when the final
state particles are so far downstream that one is outside the detector
acceptance occurs only after many lifetimes and hence is safely neglected.

(a)

(b)

Figure 2: Decay-time acceptance function for an event of a two-body hadronic
decay. The light blue (shaded) regions show the bands for accepting the impact
parameter of a track. The impact parameter of the negative track (IP2) is too
small in (a) and lies within the accepted range in (b). The actual measured
decay time lies in the accepted region. The acceptance intervals give
conditional likelihoods used in the lifetime fit.

The distributions of the turning points, combined with the decay-time
distributions, are converted into an average acceptance function (see Fig. 3).
The average acceptance is not used in the lifetime fit, except in the
determination of the background decay-time distribution.

The effective $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime is extracted by an
unbinned maximum likelihood fit using an analytical probability density
function (PDF) for the signal decay time and a non-parametric PDF for the
combinatorial background, as described below. The measurement is factorised
into two independent fits.

A first fit is performed to the observed mass spectrum and used to determine
the signal and background probabilities of each event. Events with $B^{0}_{s}$
candidates in the mass range $5272-5800$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ were used, hence reducing the
contribution of partially reconstructed background and contamination of
$B^{0}$ decays below the $B^{0}_{s}$ mass peak. The signal distribution is
modelled with a Gaussian, and the background with a linear distribution. The
fitted mass value is compatible with the current world average [8].

The signal and background probabilities are used in the subsequent lifetime
fit. The decay-time PDF of the signal is calculated analytically taking into
account the per-event acceptance and the decay-time resolution. The decay-time
PDF of the combinatorial background is estimated from data using a non-
parametric method and is modelled by a sum of kernel functions which represent
each candidate by a normalised Gaussian function centred at the measured decay
time with a width proportional to an estimate of the density of candidates at
this decay time [17]. The lifetime fit is performed in the decay-time range of
$0.6-15~{}{\rm\,ps}$, hence only candidates within this range were accepted.
The analysis was tested on the $B^{0}\\!\rightarrow K^{+}\pi^{-}$ channel, for
which a lifetime compatible with the world average value was obtained, and
applied to the $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ channel only once the full
analysis procedure had been fixed. The result of the lifetime fit is

$\tau_{KK}=1.440\pm 0.096~{}\mathrm{(stat)}\;\mathrm{ps}$

and is illustrated in Fig. 3.

Figure 3: Left: Average decay-time acceptance function for signal events,
where the error band is an estimate of the statistical uncertainty. The plot
is scaled to 1 at large decay times, not taking into account the total signal
efficiency. Right: Decay-time distribution of the $B^{0}_{s}\\!\rightarrow
K^{+}K^{-}$ candidates and the fitted functions. The estimation of the
background distribution is sensitive to fluctuations due to the limited
statistics. Both plots are for the absolute lifetime measurement.

##  5 Systematic Uncertainties

Table 1: Summary of systematic uncertainties on the $B^{0}_{s}\\!\rightarrow
K^{+}K^{-}$ lifetime measurements.

Source of uncertainty | Uncertainty on | Uncertainty on
---|---|---
|  $\tau_{KK}$ (fs) |  $\tau_{KK}^{-1}~{}-~{}\tau_{K\pi}^{-1}$(ns-1)
Fit method | 3.2 |
Acceptance correction | 6.3 |  0.5
Mass model | 1.9 |
$B\\!\rightarrow h^{+}h^{\prime-}$ background | 1.9 | 1.4
Partially reconstructed background | 1.9 | 1.1
Combinatorial background | 1.5 | 1.6
Primary vertex association | 1.2 | 0.5
Detector length scale | 1.5 | 0.7
Production asymmetry | 1.4 | 0.6
Minimum accepted lifetime | 1.1 | N/A
Total (added in quadrature) | 8.4 | 2.7
Effective lifetime interpretation | 2.8 | 1.1

$\qquad\qquad\qquad\qquad\Bigg{\\}}$

The systematic uncertainties are listed in Table 1 and discussed below. The
dominant contributions to the systematic uncertainty for the absolute lifetime
measurement come from the treatment of the acceptance correction
($6.3~{}\rm\,fs$) and the fitting procedure ($3.2~{}\rm\,fs$). The systematic
uncertainty from the acceptance correction is determined by applying the same
analysis technique to a kinematically similar high statistics decay in the
charm sector ($D^{0}\\!\rightarrow K^{-}\pi^{+}$ [18]). This analysis yields a
lifetime value in good agreement with the current world average and of better
statistical accuracy. The uncertainty on the comparison between the measured
value and the world average is rescaled by the $B$ meson and charm meson
lifetime ratio. The uncertainty due to the fitting procedure is evaluated
using simplified simulations. A large number of pseudo-experiments are
simulated and the pull of the fitted lifetimes compared to the input value to
the fit is used to estimate the accuracy of the fit. These sources of
uncertainty are not dominant in the relative method, and are estimated from
simplified simulations which also include the systematic uncertainty of the
mass model. Hence a common systematic uncertainty is assigned to these three
sources.

The effect of the contamination of other $B\\!\rightarrow h^{+}h^{\prime-}$
modes to the signal modes is determined by a data-driven method. The
misidentification probability of protons, pions and kaons is measured in data
using the decays $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$,
$D^{0}\rightarrow K^{+}\pi^{-}$, $\phi\rightarrow K^{+}K^{-}$ and $\mathchar
28931\relax\rightarrow p\pi^{-}$, where the particle type is inferred from
kinematic constraints alone [19]. As the particle identification likelihood
separating protons, kaons and pions depends on kinematic properties such as
momentum, transverse momentum, and number of reconstructed primary interaction
vertices, the sample is reweighted to reflect the different kinematic range of
the final state particles in $B\\!\rightarrow h^{+}h^{\prime-}$ decays. The
effect on the measured lifetime is evaluated with simplified simulations.

Decays of $B^{0}_{s}$ and $B^{0}$ to three or more final state particles,
which have been partially reconstructed, lie predominantly in the mass range
below the $B^{0}_{s}$ mass peak outside the analysed region. Residual
background from this source is estimated from data and evaluated with a sample
of fully simulated partially reconstructed decays. The effect on the fitted
lifetime is then evaluated.

In the absolute lifetime measurement, the combinatorial background of the
decay time distribution is described by a non-parametric function, based on
the observed events with masses above the $B^{0}_{s}$ meson signal region. The
systematic uncertainty is evaluated by varying the region used for evaluating
the combinatorial background. In the relative lifetime measurement, the
combinatorial background in the $hh^{\prime}$ invariant mass spectrum is
described by a first order polynomial. To estimate the systematic uncertainty,
a sample of simulated events is obtained with a simplified simulation using an
exponential function, and subsequently fitted with a first order polynomial.

Events may contain several primary interactions and a reconstructed $B$ meson
candidate may be associated to the wrong primary vertex. This effect is
studied using the more abundant charm meson decays where the lifetime is
measured separately for events with only one or any number of primary vertices
and the observed variation is scaled to the $B$ meson system.

Particle decay times are measured from the distance between the primary vertex
and secondary decay vertex in the silicon vertex detector. The systematic
uncertainty from this source is determined by considering the potential error
on the length scale of the detector from the mechanical survey, thermal
expansion and the current alignment precision.

The analysis assumes that $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons are produced in equal
quantities. The influence of a production asymmetry for $B^{0}_{s}$ mesons on
the measured lifetime is found to be small.

In the absolute lifetime method both a Gaussian and a Crystal Ball mass model
[20] are implemented and the effect on fully simulated data is evaluated to
estimate the systematic uncertainty due to the modelling of the signal PDF. In
the relative lifetime method this uncertainty is evaluated with simplified
simulations and included in the fitting procedure uncertainty.

In the absolute $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime measurement a
cut is applied on the minimal reconstructed decay time. As the background
decay time estimation will smear this step in the distribution, a systematic
uncertainty is quoted from varying this cut.

There is an additional uncertainty introduced if the result is interpreted
using Eq. 2, as this expression does not take into account detector resolution
and decay time acceptance. This effect was studied using simplified
simulations modelling the acceptance observed in the data and conservative
values of $\Delta\Gamma_{s}$ = 0.1 ${\rm\,ps}$ and ${\cal
A}_{\Delta\Gamma_{s}}$ = -0.6. The observed bias with respect to the
prediction of Eq. 2 is 3 $\rm\,fs$. This effect is labelled “Effective
lifetime interpretation” in Table 1 and is not a source of systematic
uncertainty on the measurement but is relevant to the interpretation of the
measured lifetime.

##  6 Results and Conclusions

The effective $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime has been measured
in $pp$ interactions using a data sample corresponding to an integrated
luminosity of $37\rm\,pb^{-1}$ recorded by the LHCb experiment in 2010. Two
complementary approaches have been followed to compensate for acceptance
effects introduced by the trigger and final event selection used to enrich the
sample of $B^{0}_{s}$ mesons. The absolute measurement extracts the per event
acceptance function directly from the data and finds:

$\tau_{KK}=1.440\pm 0.096~{}\mathrm{(stat)}\pm 0.008~{}\mathrm{(syst)}\pm
0.003~{}(\mathrm{model})~{}{\rm\,ps}$

where the third source of uncertainty labelled “model” is related to the
interpretation of the effective lifetime.

The relative method exploits the fact that the kinematic properties of the
various $B\\!\rightarrow h^{+}h^{\prime-}$ modes are almost identical and
extracts the $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime relative to the
$B^{0}\\!\rightarrow K^{+}\pi^{-}$ lifetime as:

$\tau_{KK}^{-1}~{}-~{}\tau_{K\pi}^{-1}=0.013\pm 0.045~{}\mathrm{(stat)}\pm
0.003~{}\mathrm{(syst)}\pm 0.001~{}(\mathrm{model})~{}{\rm\,ps}^{-1}.$

Taking the $B^{0}\\!\rightarrow K^{+}\pi^{-}$ lifetime as equal to the mean
$B^{0}$ lifetime ($\tau_{B^{0}}=1.519\pm 0.007~{}{\rm\,ps}$) [8], this
measurement can be expressed as:

$\tau_{KK}=1.490\pm 0.100~{}\mathrm{(stat)}\pm 0.006~{}\mathrm{(syst)}\pm
0.002~{}(\mathrm{model})\pm 0.007~{}\mbox{(input)~{}${\rm\,ps}$}.$

where the last uncertainty originates from the uncertainty of the $B^{0}$
lifetime. Both measurements are found to be compatible with each other, taking
the overlap in the data analysed into account.

Due to the large overlap of the data analysed by the two methods and the high
correlation of the systematic uncertainties, there is no significant gain from
a combination of the two numbers. Instead, the result obtained using the
absolute lifetime method is taken as the final result. The measured effective
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ lifetime is in agreement with the
Standard Model prediction of $\tau_{KK}=1.390\pm 0.032~{}{\rm\,ps}$ [3].

## 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 CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, 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 and the Region Auvergne.

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