EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2012-092 LHCb-PAPER-2012-004 June 15, 2012

Measurement of the polarization amplitudes and triple product asymmetries in
the $B^{0}_{s}\rightarrow\phi\phi$ decay

The LHCb collaboration 111Authors are listed on the following pages.

Using $1.0~{}\mbox{\,fb}^{-1}$ of $pp$ collision data collected at a centre-
of-mass energy of $\sqrt{\mathrm{s}}=7~{}{\mathrm{TeV}}$ with the LHCb
detector, measurements of the polarization amplitudes, strong phase difference
and triple product asymmetries in the $B^{0}_{s}\rightarrow\phi\phi$ decay
mode are presented. The measured values are

$\displaystyle|A_{0}|^{2}$ $\displaystyle=$ $\displaystyle 0.365\pm
0.022\,({\rm stat})\pm 0.012\,({\rm syst})\,,$ $\displaystyle|A_{\perp}|^{2}$
$\displaystyle=$ $\displaystyle 0.291\pm 0.024\,({\rm stat})\pm 0.010\,({\rm
syst})\,,$ $\displaystyle\cos(\delta_{\parallel})$ $\displaystyle=$
$\displaystyle-0.844\pm 0.068\,({\rm stat})\pm 0.029\,({\rm syst})\,,$
$\displaystyle A_{U}$ $\displaystyle=$ $\displaystyle-0.055\pm 0.036\,({\rm
stat})\pm 0.018\,({\rm syst})\,,$ $\displaystyle A_{V}$ $\displaystyle=$
$\displaystyle 0.010\pm 0.036\,({\rm stat})\pm 0.018\,({\rm syst})\,.$

LHCb collaboration

R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, S.
Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y.
Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F.
Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E.
Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G. Carboni21,k, R.
Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho Akiba2, G. Casse49,
M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph. Charpentier35, N. Chiapolini37,
K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35,
H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, P. Collins35, A.
Comerma-Montells33, A. Contu52, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y.
David38, I. De Bonis4, K. De Bruyn38, S. De Capua21,k, M. De Cian37, J.M. De
Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H.
Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35, O. Deschamps5,
F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1, F. Domingo
Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A.
Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, S. Easo46, U.
Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11, S.
Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, A. Falabella16,e,14, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4,
V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35,
A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A.
Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11,
R. Harji50, N. Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P.
Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41,
P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38,
M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G.
Krocker11, P. Krokovny31, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j,
V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35,
C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33,
N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe12,
C. Matteuzzi20, M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P.
Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35,
A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T.
Nikodem11, A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V.
Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M.
Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53, J. Palacios37, A.
Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C.
Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, S.K.
Paterson50, G.N. Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos
Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrolini19,i, A. Phan53, E.
Picatoste Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R.
Plackett48, S. Playfer47, M. Plo Casasus34, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52,
M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, K. Sobczak5,
F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B. Souza De Paula2, B. Spaan9,
A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37, S.
Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T. Szumlak24, S.
T’Jampens4, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, S. Topp-Joergensen52, N.
Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N.
Tuning38, M. Ubeda Garcia35, A. Ukleja25, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-
Cardona33,n, J. Visniakov34, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A.
Vorobyev27, V. Vorobyev31, H. Voss10, R. Waldi55, S. Wandernoth11, J. Wang53,
D.R. Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, D.
Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,
F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A. Wotton44, K.
Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R. Young47, O.
Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.

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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11

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

##  1 Introduction

In the Standard Model, the flavour-changing neutral current decay
$B^{0}_{s}\rightarrow\phi\phi$ proceeds via a $b\rightarrow s\bar{s}s$ penguin
process. Studies of the polarization amplitudes and triple product asymmetries
in this decay provide powerful tests for the presence of contributions from
processes beyond the Standard Model [1, 2, 3, 4, 5].

The $B^{0}_{s}\rightarrow\phi\phi$ decay is a pseudoscalar to vector-vector
transition. As a result, there are three possible spin configurations of the
vector meson pair allowed by angular momentum conservation. These manifest
themselves as three helicity states, with amplitudes denoted $H_{+1},H_{-1}$
and $H_{0}$. It is convenient to define linear polarization amplitudes, which
are related to the helicity amplitudes through the following transformations

$\displaystyle A_{0}$ $\displaystyle=$ $\displaystyle H_{0}\,,$ $\displaystyle
A_{\perp}$ $\displaystyle=$ $\displaystyle\frac{H_{+1}-H_{-1}}{\sqrt{2}}\,,$
$\displaystyle A_{\parallel}$ $\displaystyle=$
$\displaystyle\frac{H_{+1}+H_{-1}}{\sqrt{2}}\,.$ (1)

The $\phi\phi$ final state can be a mixture of $C\\!P$-even and $C\\!P$-odd
eigenstates. The longitudinal ($A_{0}$) and parallel ($A_{\parallel}$)
components are $C\\!P$-even and the perpendicular component ($A_{\perp}$) is
$C\\!P$-odd. From the V–A structure of the weak interaction, the longitudinal
component,
$f_{L}=|A_{0}|^{2}/(|A_{0}|^{2}+|A_{\perp}|^{2}+|A_{\parallel}|^{2})$, is
expected to be dominant [6, 7, 8]. However, roughly equal longitudinal and
transverse components are found in measurements of $B^{+}\rightarrow\phi
K^{*+}$, $B^{0}\rightarrow\phi K^{*0}$, $B^{+}\rightarrow\rho^{0}K^{*+}$ and
$B^{0}\rightarrow\rho^{0}K^{*0}$ decays at the B-factories [9, 10, 11, 12, 13,
14]. To explain this, large contributions from either penguin annihilation
effects [15] or final state interactions [16] have been proposed. Recent
calculations where phenomenological parameters are adjusted to account for the
data allow $f_{L}$ in the range $0.4-0.7$ [6, 7]. Another pseudoscalar to
vector-vector penguin decay is $B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. A recent measurement by the LHCb
Collaboration in this decay mode has found a value of $f_{L}=0.31\pm 0.12\pm
0.04$ [17].

$\hat{n}_{1}$$\hat{n}_{2}$$B^{0}_{s}$$\Phi$$\theta_{2}$$\theta_{1}$$K^{-}$$K^{+}$$K^{-}$$K^{+}$$\phi_{1}$$\phi_{2}$
Figure 1:  Decay angles for the $B^{0}_{s}\rightarrow\phi\phi$ decay, where
the $K^{+}$ momentum in the $\phi_{1,2}$ rest frame, and the parent
$\phi_{1,2}$ momentum in the rest frame of the $B^{0}_{s}$ meson span the two
$\phi$ meson decay planes, $\theta_{1,2}$ is the angle between the $K^{+}$
track momentum in the $\phi_{1,2}$ meson rest frame and the parent
$\phi_{1,2}$ momentum in the $B^{0}_{s}$ rest frame, $\Phi$ is the angle
between the two $\phi$ meson decay planes and $\hat{n}_{1,2}$ is the unit
vector normal to the decay plane of the $\phi_{1,2}$ meson.

The time-dependent differential decay rate for the
$B^{0}_{s}\rightarrow\phi\phi$ mode can be written as

$\frac{d^{4}\Gamma}{d\cos\theta_{1}d\cos\theta_{2}d\Phi
dt}\propto\sum^{6}_{i=1}K_{i}(t)f_{i}(\theta_{1},\theta_{2},\Phi)\,,$ (2)

where the helicity angles $\Omega=(\theta_{1},\theta_{2},\Phi)$ are defined in
Fig. 1. The angular functions $f_{i}(\Omega)$ are [18]

$\displaystyle f_{1}(\theta_{1},\theta_{2},\Phi)$ $\displaystyle=$
$\displaystyle 4\cos^{2}\theta_{1}\cos^{2}\theta_{2}\,,$ $\displaystyle
f_{2}(\theta_{1},\theta_{2},\Phi)$ $\displaystyle=$
$\displaystyle\sin^{2}\theta_{1}\sin^{2}\theta_{2}(1+\cos 2\Phi)\,,$
$\displaystyle f_{3}(\theta_{1},\theta_{2},\Phi)$ $\displaystyle=$
$\displaystyle\sin^{2}\theta_{1}\sin^{2}\theta_{2}(1-\cos 2\Phi)\,,$
$\displaystyle f_{4}(\theta_{1},\theta_{2},\Phi)$ $\displaystyle=$
$\displaystyle-2\sin^{2}\theta_{1}\sin^{2}\theta_{2}\sin 2\Phi\,,$
$\displaystyle f_{5}(\theta_{1},\theta_{2},\Phi)$ $\displaystyle=$
$\displaystyle\sqrt{2}\sin 2\theta_{1}\sin 2\theta_{2}\cos\Phi\,,$
$\displaystyle f_{6}(\theta_{1},\theta_{2},\Phi)$ $\displaystyle=$
$\displaystyle-\sqrt{2}\sin 2\theta_{1}\sin 2\theta_{2}\sin\Phi\,.$ (3)

The time-dependent functions $K_{i}(t)$ are given by [19]

$\displaystyle K_{1}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}A_{0}^{2}[(1+\cos\phi_{s})e^{-\Gamma_{\rm{L}}t}+(1-\cos\phi_{s})e^{-\Gamma_{\rm{H}}t}\pm
2e^{-\Gamma_{s}t}\sin(\Delta m_{s}t)\sin\phi_{s}]\,,$ $\displaystyle K_{2}(t)$
$\displaystyle=$
$\displaystyle\frac{1}{2}A_{\parallel}^{2}[(1+\cos\phi_{s})e^{-\Gamma_{\rm{L}}t}+(1-\cos\phi_{s})e^{-\Gamma_{\rm{H}}t}\pm
2e^{-\Gamma_{s}t}\sin(\Delta m_{s}t)\sin\phi_{s}]\,,$ $\displaystyle K_{3}(t)$
$\displaystyle=$
$\displaystyle\frac{1}{2}A_{\perp}^{2}[(1-\cos\phi_{s})e^{-\Gamma_{\rm{L}}t}+(1+\cos\phi_{s})e^{-\Gamma_{\rm{H}}t}\mp
2e^{-\Gamma_{s}t}\sin(\Delta m_{s}t)\sin\phi_{s}]\,,$ $\displaystyle K_{4}(t)$
$\displaystyle=$ $\displaystyle|A_{\parallel}||A_{\perp}|[\pm
e^{-\Gamma_{s}t}\\{\sin\delta_{1}\cos(\Delta m_{s}t)-\cos\delta_{1}\sin(\Delta
m_{s}t)\cos\phi_{s}\\}$
$\displaystyle-\frac{1}{2}(e^{-\Gamma_{\rm{H}}t}-e^{-\Gamma_{\rm{L}}t})\cos\delta_{1}\sin\phi_{s}]\,,$
$\displaystyle K_{5}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}|A_{0}||A_{\parallel}|\cos(\delta_{2}-\delta_{1})$
$\displaystyle[(1+\cos\phi_{s})e^{-\Gamma_{\rm{L}}t}+(1-\cos\phi_{s})e^{-\Gamma_{\rm{H}}t}\pm
2e^{-\Gamma_{s}t}\sin(\Delta m_{s}t)\sin\phi_{s}]\,,$ $\displaystyle K_{6}(t)$
$\displaystyle=$ $\displaystyle|A_{0}||A_{\perp}|[\pm
e^{-\Gamma_{s}t}\\{\sin\delta_{2}\cos(\Delta m_{s}t)-\cos\delta_{2}\sin(\Delta
m_{s}t)\cos\phi_{s}\\}$ (4)
$\displaystyle-\frac{1}{2}(e^{-\Gamma_{\rm{H}}t}-e^{-\Gamma_{\rm{L}}t})\cos\delta_{2}\sin\phi_{s}]\,,$

where the upper of the $\pm$ or $\mp$ signs refers to the $B^{0}_{s}$ meson
and the lower refers to a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson. Here, $\Gamma_{\rm L}$
and $\Gamma_{\rm H}$ are the decay widths of the light and heavy $B^{0}_{s}$
mass eigenstates,222Units are adopted such that $\hbar=1$. $\Delta m_{s}$ is
the $B^{0}_{s}$ oscillation frequency,
$\delta_{1}=\arg(A_{\perp}/A_{\parallel})$ and
$\delta_{2}=\arg(A_{\perp}/A_{0})$ are $C\\!P$-conserving strong phases and
$\phi_{s}$ is the weak $C\\!P$-violating phase. It is assumed that the weak
phases of the three polarization amplitudes are equal. The quantities
$\Gamma_{\rm H}$ and $\Gamma_{\rm L}$ correspond to the observables
$\Delta\Gamma_{s}=\Gamma_{\rm L}-\Gamma_{\rm H}$ and $\Gamma_{s}=(\Gamma_{\rm
L}+\Gamma_{\rm H})/2$. In the Standard Model, the value of $\phi_{s}$ for this
mode is expected to be very close to zero due to a cancellation between the
phases arising from mixing and decay [20].333The convention used in this
Letter is that the symbol $\phi_{s}$ refers solely to the weak phase
difference measured in the $B^{0}_{s}\rightarrow\phi\phi$ decay. A calculation
based on QCD factorization provides an upper limit of 0.02 rad for $\phi_{s}$
[21, 6]. This is different to the situation in the $B^{0}_{s}\rightarrow
J/\psi\phi$ decay, where the Standard Model predicts
$\phi_{s}(J/\psi\phi)=-2\arg\left(-V_{ts}V_{tb}^{*}/V_{cs}V_{cb}^{*}\right)=-0.036\pm
0.002$ rad [22]. The magnitude of both weak phase differences can be enhanced
in the presence of new physics in $B^{0}_{s}$ mixing, where recent results
from LHCb have placed stringent constraints [23]. For the
$B^{0}_{s}\rightarrow\phi\phi$ decay, new particles could also contribute in
$b\rightarrow s$ penguin loops.

To measure the polarization amplitudes, a time-integrated untagged analysis is
performed, assuming that an equal number of $B^{0}_{s}$ and $\bar{B}_{s}^{0}$
mesons are produced and that the $C\\!P$-violating phase is zero as predicted
in the Standard Model.444In the case of non-zero $\phi_{s}$ deviations from
these formulas are suppressed by a factor of $\Delta\Gamma_{s}/\Gamma_{s}$ and
hence only small variations would be observed on the fitted parameters. In
this case, the functions $K_{i}(t)$ integrate to

$\displaystyle K_{1}$ $\displaystyle=$ $\displaystyle|A_{0}|^{2}/\Gamma_{\rm
L}\,,$ $\displaystyle K_{2}$ $\displaystyle=$
$\displaystyle|A_{\parallel}|^{2}/\Gamma_{\rm L}\,,$ $\displaystyle K_{3}$
$\displaystyle=$ $\displaystyle|A_{\perp}|^{2}/\Gamma_{\rm H}\,,$
$\displaystyle K_{4}$ $\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle
K_{5}$ $\displaystyle=$
$\displaystyle|A_{0}||A_{\parallel}|\cos(\delta_{\parallel})/\Gamma_{\rm
L}\,,$ $\displaystyle K_{6}$ $\displaystyle=$ $\displaystyle 0\,,$ (5)

where the strong phase difference is defined by
$\delta_{\parallel}\equiv\delta_{2}-\delta_{1}=\arg(A_{\parallel}/A_{0})$ and
the time integration assumes uniform time acceptance.

In addition, a search for physics beyond the Standard Model is performed by
studying the triple product asymmetries [1, 2, 3] in the
$B^{0}_{s}\rightarrow\phi\phi$ decay. Non-zero values of these quantities can
be either due to $T$-violation or final-state interactions. Assuming $C\\!PT$
conservation, the former case implies that $C\\!P$ is violated.
Experimentally, the extraction of the triple product asymmetries is
straightforward and provides a measure of $C\\!P$ violation that does not
require flavour tagging or a time-dependent analysis.

There are two observable triple products denoted $U=\sin(2\Phi)/2$ and
$V=\pm\sin(\Phi)$, where the positive sign is taken if the $T$-even quantity
$\cos\theta_{1}\cos\theta_{2}\geq 0$ and the negative sign otherwise. These
variables correspond to the $T$-odd triple products

$\displaystyle\sin\Phi$ $\displaystyle=$
$\displaystyle(\hat{n}_{1}\times\hat{n}_{2})\cdot\hat{p}_{1}\,,$
$\displaystyle\sin(2\Phi)/2$ $\displaystyle=$
$\displaystyle(\hat{n}_{1}\cdot\hat{n}_{2})(\hat{n}_{1}\times\hat{n}_{2})\cdot\hat{p}_{1}\,,$
(6)

where $\hat{n}_{i}$ ($i=1,2$) is a unit vector perpendicular to the $\phi_{i}$
decay plane and $\hat{p}_{1}$ is a unit vector in the direction of the
$\phi_{1}$ momentum in the $B^{0}_{s}$ rest frame. The triple products, $U$
and $V$, are proportional to the $f_{4}$ and $f_{6}$ angular functions which,
for $\phi_{s}=0$, vanish in the untagged decay rate for any value of $t$. The
$f_{4}$ and $f_{6}$ angular functions would not vanish in the presence of new
physics processes that cause the polarization amplitudes to have different
weak phases [1]. Therefore, a measurement of significant asymmetries would be
an unambiguous signal for the effects of new physics [1, 3].

The asymmetry, $A_{U}$, is defined as

$A_{U}=\frac{N_{+}-N_{-}}{N_{+}+N_{-}}\,,$ (7)

where $N_{+}$ ($N_{-}$) is the number of events with $U>0$ ($U<0$). Similarly
$A_{V}$ is defined as

$\displaystyle A_{V}=\frac{M_{+}-M_{-}}{M_{+}+M_{-}}\,,$ (8)

where $M+$ ($M_{-}$) is the number of events with $V>0$ ($V<0$). The triple
product asymmetries, $A_{U}$ and $A_{V}$ are proportional to the interference
terms $\mathcal{I}m(A_{\perp}A_{\parallel}^{*})$ and
$\mathcal{I}m(A_{\perp}A_{0}^{*})$ in the decay rate.

The $B^{0}_{s}\rightarrow\phi\phi$ decay mode was first observed by the CDF
Collaboration [24]. More recently, CDF has reported measurements of the
polarization amplitudes and triple product asymmetries in this mode based on a
sample of 295 events [25]. In this Letter, measurements of the polarization
amplitudes, $|A_{0}|^{2}$ and $|A_{\perp}|^{2}$, the strong phase difference,
$\delta_{\parallel}$, and the triple product asymmetries, $A_{U}$ and $A_{V}$,
are presented. The dataset consists of $801\pm 29$ candidates collected in
$1.0~{}\mbox{\,fb}^{-1}$ of $pp$ collisions at the LHC. The Monte Carlo (MC)
simulation samples used are based on the Pythia 6.4 generator [26] configured
with the parameters detailed in Ref. [27]. The EvtGen [28] and Geant4 [29]
packages are used to generate hadron decays and simulate interactions in the
detector, respectively.

##  2 Detector description

The LHCb detector [30] 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 a momentum resolution $\Delta p/p$ that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum. Charged hadrons
are identified using two ring-imaging Cherenkov detectors. 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 muon system composed of
alternating layers of iron and detector stations. The trigger 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.

The software trigger used in this analysis requires a two-, three- or four-
track secondary vertex with a high sum of the transverse momentum, $p_{\rm
T}$, of the tracks, significant displacement from the primary interaction, and
at least one track with $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$; impact parameter $\chi^{2}$
with respect to the primary interaction greater than 16; and a track fit
$\chi^{2}/\rm{ndf}<2$ where ndf is the number of degrees of freedom in the
track fit. A multivariate algorithm is used for the identification of the
secondary vertices [31]. The $B^{0}_{s}\rightarrow\phi\phi$ candidates are
selected with high efficiency either by identifying events containing a $\phi$
meson or using topological information to select hadronic $b$ decays. Events
passing the software trigger are stored for subsequent offline processing.

##  3 Event selection

The $B^{0}_{s}\rightarrow\phi\phi$ channel is reconstructed using events where
both $\phi$ mesons decay into a $K^{+}K^{-}$ pair. The
$B^{0}_{s}\rightarrow\phi\phi$ selection criteria were optimized using a data-
driven approach based on the $\phantom{}{}_{s}\mathcal{P}lot$ technique
employing the four-kaon mass as the unfolding variable [32] to separate signal
($S$) and background ($B$) with the aim of maximizing $S/\sqrt{S+B}$. The
resulting cuts are summarized in Table 1. Good quality track reconstruction is
ensured by a cut on the transverse momentum ($p_{\rm T}$) of the daughter
particles and a cut on the $\chi^{2}/{\rm ndf}$ of the track fit.

Combinatorial background is reduced by cuts on the minimum impact parameter
significance of the tracks with respect to all reconstructed $pp$ interaction
vertices and also by imposing a requirement on the vertex separation
$\chi^{2}$ of the $B^{0}_{s}$ candidate. Well-identified $\phi$ meson
candidates are selected by requiring that the two particles involved are
identified as kaons by the ring-imaging Cherenkov detectors using a cut on the
difference in the global likelihood between the kaon and pion hypotheses
($\Delta\ln{\cal L}_{{\rm\mathit{K}}{\rm\mathit{\pi}}}>0$) and by requiring
that the reconstructed mass of each $K^{+}K^{-}$ pair is within
$12~{}\rm\,MeV\\!/\\!{\it c}^{2}$ of the nominal mass of the $\phi$ meson
[33]. Further signal purity is achieved by cuts on the transverse momentum of
the $\phi$ candidates.

Table 1: Selection criteria for the $B^{0}_{s}\rightarrow\phi\phi$ decay. The abbreviation IP stands for impact parameter and $\mbox{$p_{\rm T}$}^{\phi 1}$ and $\mbox{$p_{\rm T}$}^{\phi 2}$ refer to the transverse momentum of the two $\phi$ candidates. Variable | Value
---|---
Track $\chi^{2}/{\rm ndf}$ | $<~{}5$
Track $p_{\rm T}$ | $>~{}500~{}\rm\,MeV\\!/\\!{\it c}$
Track IP $\chi^{2}$ | $>~{}21$
$\Delta\ln{\cal L}_{{\rm\mathit{K}}{\rm\mathit{\pi}}}$ | $>~{}0$
$|M_{\phi}-M_{\phi}^{\rm{PDG}}|$ | $<~{}12~{}\rm\,MeV\\!/\\!{\it c}^{2}$
$\mbox{$p_{\rm T}$}^{\phi 1},\,\mbox{$p_{\rm T}$}^{\phi 2}$ | $>~{}900~{}\rm\,MeV\\!/\\!{\it c}$
$\mbox{$p_{\rm T}$}^{\phi 1}\cdot\mbox{$p_{\rm T}$}^{\phi 2}$ |  $>~{}2$ GeV${}^{2}/c^{2}$
$\phi$ vertex $\chi^{2}/{\rm ndf}$ | $<~{}24$
$B^{0}_{s}$ vertex $\chi^{2}/{\rm ndf}$ | $<~{}7.5$
$B^{0}_{s}$ vertex separation $\chi^{2}$ | $>~{}270$
$B^{0}_{s}$ IP $\chi^{2}$ | $<~{}15$

Figure 2 shows the four-kaon invariant mass distribution for selected events.
To determine the signal yield an unbinned maximum likelihood fit is performed.
The $B^{0}_{s}\rightarrow\phi\phi$ signal component is modelled by two
Gaussian functions with a common mean. The resolution of the first Gaussian is
measured from data to be $13.9\pm 0.6$ $\rm\,MeV\\!/\\!{\it c}^{2}$. The
relative fraction and resolution of the second Gaussian are fixed to 0.785 and
29.5 $\rm\,MeV\\!/\\!{\it c}^{2}$ respectively, where values have been
obtained from simulation. Combinatorial background is modelled using an
exponential function. Background from $B^{0}\rightarrow\phi K^{*0}$ and
$B^{0}_{s}\rightarrow K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
decays is found to be negligible both in simulation and data driven studies.
Fitting the probability density function (PDF) described above to the data, a
signal yield of $801\pm 29$ events is found.

In addition to the dominant P-wave $\phi\rightarrow K^{+}K^{-}$ component
described in Section 1, other contributions, either from $f_{0}\rightarrow
K^{+}K^{-}$ or non-resonant $K^{+}K^{-}$, are possible. The size of these
contributions, neglecting interference effects, is studied by relaxing the
$\phi$ mass cut to be within $25$ $\rm\,MeV\\!/\\!{\it c}^{2}$ of the nominal
value555This is a larger window than the $\pm 12$ $\rm\,MeV\\!/\\!{\it c}^{2}$
window used in the polarization amplitude and strong phase difference
measurements. and using the $\phantom{}{}_{s}\mathcal{P}lot$ technique in
conjunction with the $\phi$ mass to subtract the combinatorial background.

Figure 2: Invariant $K^{+}K^{-}K^{+}K^{-}$ mass distribution for selected
$B^{0}_{s}\rightarrow\phi\phi$ candidates. A fit of a double Gaussian signal
component together with an exponential background (dotted line) is
superimposed.

The resulting $\phi$ mass distribution is shown in Fig. 3. A fit of a
relativistic P-wave Breit-Wigner function together with a two body phase space
component to model the S-wave contribution is superimposed. In a $\pm
25~{}\rm\,MeV\\!/\\!{\it c}^{2}$ mass window, the size of the S-wave component
is found to be $(1.3\pm 1.2)\%$. Since the S-wave yield is consistent with
zero, it will be neglected in the following section. A systematic uncertainty
arising from this assumption will be assigned.

Figure 3:  Invariant mass distribution of $K^{+}K^{-}$ pairs for the
$B^{0}_{s}\rightarrow\phi\phi$ data without a $\phi$ mass cut. The background
has been removed using the ${}_{s}\mathcal{P}lot$ technique in conjunction
with the $K^{+}K^{-}$ invariant mass. There are two entries per $B^{0}_{s}$
candidate. The solid line shows the result of the fit model described in the
text. The fitted S-wave component is shown by the dotted line.

##  4 Results

The polarization amplitudes ($|A_{0}|^{2}$, $|A_{\perp}|^{2}$,
$|A_{\parallel}|^{2}$), are determined by performing an unbinned maximum
likelihood fit to the reconstructed mass and helicity angle distributions. For
each event, the $\phi$ meson used to define $\theta_{1}$ is chosen at random.
Both the signal and background PDFs are the products of a mass component
described in Section 3 together with an angular component. The angular
component of the signal is given by Eq. 3 multiplied by the angular acceptance
of the detector. The acceptance is determined using the simulation and is
calculated separately according to trigger type, i.e. whether the event was
triggered by the signal candidate or other particles in the event. In total
the fit for the polarization amplitudes has eight free parameters: the signal
angular parameters $|A_{0}|^{2}$, $|A_{\perp}|^{2}$ and
$\cos(\delta_{\parallel})$ defined in Section 1, the fractions of signal for
each trigger type, the resolution of the core Gaussian, the $B^{0}_{s}$ mass
and the slope of the mass background. The sum of squared amplitudes is
constrained such that $|A_{0}|^{2}+|A_{\perp}|^{2}+|A_{\parallel}|^{2}=1$. The
angular distributions for the background have been studied using the mass
sidebands in the data, where mass sidebands are defined to be between $60$ and
$300$ $\rm\,MeV\\!/\\!{\it c}^{2}$ either side of the nominal $B^{0}_{s}$ mass
[33]. With the current sample size these distributions are consistent with
being flat in ($\cos\theta_{1},\cos\theta_{2},\Phi$). Therefore, a uniform
angular PDF is assumed and more complicated shapes are considered as part of
the systematic studies. The values of $\Gamma_{s}=0.657\pm 0.009\pm
0.008{\rm\,ps}^{-1}$ and $\Delta\Gamma_{s}=0.123\pm 0.029\pm
0.011{\rm\,ps}^{-1}$ together with their correlation coefficient of $-0.3$
quoted in [23] are used as a Gaussian constraint. The validity of the fit
model has been extensively tested using simulated data samples. The results
are given in Table 2 and the angular projections are shown in Fig. 4.

Table 2: Measured polarization amplitudes and strong phase difference. The uncertainties are statistical only. The sum of the squared amplitudes is constrained to unity. The correlation coefficient between ${|A_{0}|}^{2}$ and ${|A_{\perp}}|^{2}$ is $-0.47$. Parameter | Measurement
---|---
${|A_{0}|}^{2}$ | 0.365$\pm$ | 0.022
${|A_{\perp}|}^{2}$ | 0.291$\pm$ | 0.024
${|A_{\parallel}|}^{2}=1-(|A_{0}|^{2}+|A_{\perp}|^{2})$ | 0.344$\pm$ | 0.024
$\cos(\delta_{\parallel})$ |  $-$0.844$\pm$ | 0.068

Figure 4:  Angular distributions for (a) $\Phi$, (b) $\cos\theta_{1}$ and (c)
$\cos\theta_{2}$ of $B_{s}^{0}\rightarrow\phi\phi$ events with the fit
projections for signal and background superimposed for the total fitted PDF
(solid line) and background component (dotted line).

Several sources of systematic uncertainty on the determination of the
polarization amplitudes are considered and summarized in Table 3. With the
present size of the dataset, the S-wave component is consistent with zero.
From the studies described in Section 3 and fits to the data including the
S-wave terms in the PDF [34], we consider a maximum S-wave component of $2\%$.
Simulation studies have been performed to investigate the effect of neglecting
an S-wave component of this size. As discussed in Section 1, the integration
that leads to Eq. 5 assumes uniform time acceptance. This is not the case due
to lifetime biasing cuts in the trigger and offline selections. The functional
form of the decay time acceptance is obtained through the use of Monte Carlo
events. The difference between using this functional form in simulation
studies and using uniform time acceptance is taken as a systematic
uncertainty. The uncertainty on the angular acceptance for the signal is
propagated to the observables also using Monte Carlo studies. The analysis was
repeated with an alternative background angular distribution, taken from a
coarsely binned histogram in $(\cos\theta_{1},\cos\theta_{2},\Phi)$ of the
mass sidebands, and the difference taken as a systematic uncertainty. An
additional uncertainty arises from angular acceptance dependencies on trigger
type. This dependency is corrected for using Monte Carlo events, with half of
the effect on fitted parameters assigned as systematic uncertainties. The
total systematic uncertainty is obtained from the sum in quadrature of the
individual uncertainties.

Table 3: Systematic uncertainties on the measured polarization amplitudes and the strong phase difference. Source | $|A_{0}|^{2}$ | $|A_{\perp}|^{2}$ | $|A_{\parallel}|^{2}$ | $\cos\delta_{\parallel}$
---|---|---|---|---
S-wave component | 0.007 | 0.005 | 0.012 | 0.001
Decay time acceptance | 0.006 | 0.006 | 0.002 | 0.007
Angular acceptance | 0.007 | 0.006 | 0.006 | 0.028
Trigger category | 0.003 | 0.002 | 0.001 | 0.004
Background model | 0.001 | – | 0.001 | 0.003
Total | 0.012 | 0.010 | 0.014 | 0.029

Figure 5: Distributions of the $U$ (left) and $V$ (right) observables for the
$B_{s}^{0}\rightarrow\phi\phi$ data in the mass range
$5286.6<M(K^{+}K^{-}K^{+}K^{-})<5446.6~{}\rm\,MeV\\!/\\!{\it c}^{2}$. The
distribution for the background is taken from the mass sidebands, normalized
to the same mass range and is shown by the solid histogram.

The distributions of the $U$ and $V$ triple product observables are shown in
Fig. 5 for the mass range
$5286.6<M(K^{+}K^{-}K^{+}K^{-})<5446.6~{}\rm\,MeV\\!/\\!{\it c}^{2}$. To
determine the triple product asymmetries, the dataset is partitioned according
to whether $U$ ($V$) is less than or greater than zero. Simultaneous fits are
performed to the mass distributions for each of the two partitions
corresponding to each observable individually. In these fits, the mean and
resolution of the Gaussian signal component together with the slope of the
exponential background component are common parameters. The asymmetries are
left as free parameters and are fitted for directly in the simultaneous fit.
The measured values are

$A_{U}$ $~{}=~{}$ |  $-$0.055$\,\,\pm\,$ | 0.036 $\,,$
---|---|---
$A_{V}$ $~{}=~{}$ | 0.010$\,\,\pm\,$ | 0.036 $\,.$

Systematic uncertainties due to the residual effect of the decay time,
geometrical acceptance and the signal and background fit models have been
evaluated and are summarized in Table 4. The effect of the decay time
acceptance has been found using the same method as for the polarization
amplitudes. The impact of angular acceptance on the measured values has been
obtained from simplified simulation studies. The total systematic uncertainty
is conservatively estimated by choosing the larger of the two individual
systematic uncertainties on $A_{U}$ and $A_{V}$. The contributions are
combined in quadrature to determine the total systematic error. Various cross-
checks of the stability of the result have been performed. For example,
dividing the data according to how the event was triggered or by magnet
polarity. No significant bias is observed in these studies.

Table 4:  Systematic uncertainties on the triple product asymmetries $A_{U}$ and $A_{V}$. The total uncertainty is the quadratic sum of the larger of the two components. Source | $A_{U}$ | $A_{V}$ | Final uncertainty
---|---|---|---
Angular acceptance | 0.009 | 0.006 | 0.009
Decay time acceptance | 0.006 | 0.014 | 0.014
Fit model | 0.004 | 0.005 | 0.005
Total |  | $0.018$

##  5 Summary

The polarization amplitudes and strong phase difference in the
$B^{0}_{s}\rightarrow\phi\phi$ decay mode are measured to be

$|A_{0}|^{2}$ $~{}=~{}$ | 0.365$\,\pm\,$ | 0.022 (stat) $\pm\,$ | 0.012 (syst) | $\,,$
---|---|---|---|---
$|A_{\perp}|^{2}$ $~{}=~{}$ | 0.291$\,\pm\,$ | 0.024 (stat) $\pm\,$ | 0.010 (syst) | $\,,$
$|A_{\parallel}|^{2}$ $~{}=~{}$ | 0.344$\,\pm\,$ | 0.024 (stat) $\pm\,$ | 0.014 (syst) | $\,,$
$\cos(\delta_{\parallel})$ $~{}=~{}$ |  $-$0.844$\,\pm\,$ | 0.068 (stat) $\pm\,$ | 0.029 (syst) | $\,,$

where the sum of the squared amplitudes is constrained to be unity. These
values agree well with the CDF measurements [25]. Measurements in other
$B\rightarrow VV$ penguin transitions at the B factories generally give higher
values of $f_{L}$ [9, 10, 11, 12, 13, 14]. It is interesting to note that the
value of $f_{L}$ found in the $B^{0}_{s}\rightarrow\phi\phi$ channel is almost
equal to that in the $B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay [17]. The results are in
agreement with QCD factorization predictions [6, 7], but disfavour the pQCD
estimate given in [8].

The triple product asymmetries in this mode are measured to be

$A_{U}$ $~{}=~{}$ |  $-$0.055$\,\pm\,$ | 0.036 (stat) $\pm\,$ | 0.018 (syst) | $\,,$
---|---|---|---|---
$A_{V}$ $~{}=~{}$ | 0.010$\,\pm\,$ | 0.036 (stat) $\pm\,$ | 0.018 (syst) | $\,.$

Both values are in good agreement with those reported by the CDF Collaboration
[25] and consistent with the hypothesis of $C\\!P$ conservation.

## 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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11institutetext: AIM for Health Lab, Faculty of IT, Monash University,
Melbourne, Australia
%First␣names␣are␣abbreviated␣in␣the␣running␣head.%****␣samplepaper.tex␣Line␣50␣****%If␣there␣are␣more␣than␣two␣authors,␣’et␣al.’␣is␣used.https://www.monash.edu/it/aimh-
lab/home
11email<EMAIL_ADDRESS>22institutetext: Monash Medical AI, Monash
University, Melbourne, Australia 33institutetext: Faculty of Engineering,
Monash University, Melbourne, Australia
44institutetext: Department of Neuroscience, Central Clinical School, Faculty
of Medicine Nursing and Health Sciences, Monash University, Melbourne,
Australia
55institutetext: Department of Neurology, Alfred Health, Melbourne, Australia
66institutetext: Departments of Medicine and Neurology, The University of
Melbourne, Royal Melbourne Hospital, Parkville, Victoria, Australia
77institutetext: Airdoc-Monash Research Lab, Monash University, Melbourne,
Australia

# Privacy-preserving Early Detection of Epileptic Seizures in Videos

Deval  Mehta() 112233 Shobi Sivathamboo 445566 Hugh Simpson 4455 Patrick Kwan
445566 Terence O’Brien 4455 Zongyuan Ge 11223377

###### Abstract

In this work, we contribute towards the development of video-based epileptic
seizure classification by introducing a novel framework (SETR-PKD), which
could achieve privacy-preserved early detection of seizures in videos.
Specifically, our framework has two significant components - (1) It is built
upon optical flow features extracted from the video of a seizure, which
encodes the seizure motion semiotics while preserving the privacy of the
patient; (2) It utilizes a transformer based progressive knowledge
distillation, where the knowledge is gradually distilled from networks trained
on a longer portion of video samples to the ones which will operate on shorter
portions. Thus, our proposed framework addresses the limitations of the
current approaches which compromise the privacy of the patients by directly
operating on the RGB video of a seizure as well as impede real-time detection
of a seizure by utilizing the full video sample to make a prediction. Our
SETR-PKD framework could detect tonic-clonic seizures (TCSs) in a privacy-
preserving manner with an accuracy of 83.9% while they are only half-way into
their progression. Our data and code is available at
https://github.com/DevD1092/seizure-detection.

###### Keywords:

epilepsy early detection knowledge distillation

##  1 Introduction

Epilepsy is a chronic neurological condition that affects more than 60 million
people worldwide in which patients experience epileptic seizures due to
abnormal brain activity [17]. Different types of seizures are associated with
the specific part of the brain involved in the abnormal activity [8]. Thus,
accurate detection of the type of epileptic seizure is essential to epilepsy
diagnosis, prognosis, drug selection and treatment. Concurrently, real-time
seizure alerts are also essential for caregivers to prevent potential
complications, such as related injuries and accidents, that may result from
seizures. Particularly, patients suffering from tonic-clonic seizures (TCSs)
are at a high risk of sudden unexpected death in epilepsy (SUDEP) [18].
Studies have shown that SUDEP is caused by severe alteration of cardiac
activity actuated by TCS, leading to immediate death or cardiac arrest within
minutes after the seizure [5]. Therefore, it is critical to accurately and
promptly detect and classify epileptic seizures to provide better patient care
and prevent any potentially catastrophic events.

The current gold standard practice for detection and classification of
epileptic seizures is the hospital-based Video EEG Monitoring (VEM) units
[23]. However, this approach is expensive and time consuming which is only
available at specialized centers [3]. To address this issue, the research
community has developed automated methods to detect and classify seizures
based on several modalities - EEG [7, 30], accelerometer [16], and even
functional neuroimaging modalities such as fMRI [22] and electrocorticography
(ECoG) [24]. Although, there have been developments of approaches for the
above modalities, seizure detection using videos remains highly desirable as
it involves no contact with the patient and is easier to setup and acquire
data compared to other modalities. Thus, researchers have also developed
automated approaches for the video modality.

Initial works primarily employed hand-crafted features based on patient motion
trajectory by attaching infrared reflective markers to specific body key
points [4, 15]. However, these approaches were limited in performance due to
their inability to generalize to changing luminance (night time seizures) or
when the patient is occluded (covered by a bed sheet) [14]. Thus, very
recently deep learning (DL) models have been explored for this task [1, 2, 29,
21, 12]. [29] demonstrated that DL models could detect generalized tonic-
clonic seizures (GTCSs) from the RGB video of seizures. Authors in [21]
radically used transfer learning (from action recognition task) to train DL
networks for distinguishing focal onset seizures (FOSs) from bilateral TCSs
using features extracted from the RGB video of seizures. Whereas, the authors
in [12] developed a DL model to discriminate dystonia and emotion in videos of
Hyperkinetic seizures. However, these developed approaches have two crucial
limitations - (1) As these approaches directly operate on RGB videos, there is
a possibility of privacy leakage of the sensitive patient data from videos.
Moreover, obtaining consent from patients to share their raw RGB video data
for building inter-cohort validation studies and generalizing these approaches
on a large scale becomes challenging; (2) The current approaches consider the
full video of a seizure to make predictions, which makes early detection of
seizures impossible. The duration of a seizure varies significantly among
patients, with some lasting as short as 30 seconds while others can take
minutes to self-terminate. Thus, it is unrealistic to wait until the
completion of a long seizure to make a prediction and alert caregivers.

In this work, we address the above two challenges by building an in-house
dataset of privacy-preserved extracted features from a video and propose a
framework for early detection of seizures. Specifically, we investigate two
aspects - (1) The feasibility of detecting and classifying seizures based only
on _optical flow_ , a modality that captures temporal differences in a scene
while being intrinsically privacy-preserving. (2) The potential of predicting
the type of seizure during its progression by analyzing only a fraction of the
video sample. Our early detection approach is inspired by recent developments
in early action recognition in videos [9, 21, 31, 10, 28, 19]. We develop a
custom feature extractor-transformer framework, named SEizure TRansformer
(SETR) block for processing a single video sample. To achieve early detection
from a fraction of the sample, we propose Progressive Knowledge Distillation
(PKD), where we gradually distill knowledge from SETR blocks trained on longer
portions of a video sample to SETR blocks which will operate on shorter
portions. We evaluate our proposed SETR-PKD framework on two datasets - an in-
house dataset collected from a VEM unit in a hospital and a publicly available
dataset of video-extracted features (GESTURES) [21]. Our experiments
demonstrate that our proposed SETR-PKD framework can detect TCS seizures with
an accuracy of 83.9% in a privacy-preserving manner when they are only half-
way into their progression. Furthermore, we comprehensively compare the
performance of direct knowledge distillation with our PKD approach on both
optical flow features (in-house dataset) and raw video features (public
dataset). We firmly believe that our proposed method makes the first step
towards developing a privacy-preserving real-time system for seizure detection
in clinical practice.

##  2 Proposed Method

In this section, we first outline the process of extracting privacy-preserving
information from RGB video samples to build our in-house dataset. Later, we
explain our proposed approach for early detection of seizures in a sample.

###  2.1 Privacy Preserving Optical Flow Acquisition

Our in-house dataset of RGB videos of patients experiencing seizures resides
on hospital premises and is not exportable due to the hospital’s ethics
agreement111We have a data ethics agreement approved for collection of data at
hospital. To work around this limitation, we develop a pipeline to extract
optical flow information [11] from the videos. This pipeline runs locally
within the hospital and preserves the privacy of the patients while providing
us with motion semiotics of the seizures. An example of the extracted optical
flow video sample can be seen in Fig 1. We use the TV-L1 algorithm [20] to
extract the optical flow features for each video, which we then export out of
the hospital for building our proposed approach. We provide more information
about our dataset, including the number of patients and seizures, annotation
protocol, etc. in section 3.

###  2.2 Early Detection of Seizures in a Sample

Consider an input optical flow video sample $V_{i}$ as shown in Fig 1(a) with
a time period of $T_{i}$, consisting of $N$ frames -
$\\{f_{0},f_{1},...f_{N-1}\\}$, and having a ground truth label of $y_{i}$
$\in$ $\\{0,1,...C\\}$ where is $C$ the total number of categories. Then, the
task of early detection is to build a framework that could classify the
category of the sample correctly by analyzing the least possible partial
segment of the sample. Thus, to define the problem of early detection, we
split the sample $V_{i}$ into $k$ segments -$\\{0,1,...k-1\\}$ starting from
the beginning to the end as shown in Fig 1(b). Here $V^{k-1}_{i}$ corresponds
to the full video sample and the descending segments correspond to the reduced
partial video samples. We build these partial segments by equally adding the
temporal information throughout the sample i.e. the time period for a partial
subset $V^{j}_{i}$ of a sample $V_{i}$ is computed as $(j+1)\times T_{i}/k$.
Thus, the early detection task is to correctly predict the category $y_{i}$ of
the sample $V_{i}$ from the lowest possible ($j$) partial segment $V^{j}_{i}$
of $V_{i}$. In Fig 1, we illustrate our proposed framework where - (a) First,
we build a Seizure Transformer (SETR) block for processing a single optical
flow video sample (b) Later, we employ SETR based Progressive Knowledge
Distillation (SETR-PKD) to achieve early detection in a sample.

####  2.2.1 Processing a Single Sample

Since seizure patterns comprise of body movements, we implement transfer
learning from a feature extractor pre-trained on action recognition task to
extract the spatial features from the optical flow frames. Prior work [21] has
shown that Temporal Segment Networks (TSNs) [27] pretrained on RGB videos of
various actions are effective at extracting features from videos of seizures.
We also utilize TSNs but pretrained on the optical flow modality, since we
have privacy-preserved optical flow frames. The TSNs extract a 1D feature
sequence for each frame $f_{j}$, referred as spatial features in Fig 1(a). The
spatial features are then processed by a linear transformation (1-layer MLP)
that maps them into $motion_{tokens}$ $\in$ $\mathbb{R}^{N\times D}$, where
each token has $D$-dimensions.

Figure 1: Our proposed framework - (a) SEizure TRansformer (SETR) block for a
single optical flow video sample (b) SETR based Progressive Knowledge
Distillation (SETR-PKD) for early detection of seizures in a sample. (Best
viewed in zoom and color).

We leverage transformers to effectively learn temporal relations between the
extracted spatial features of the seizure patterns. Following the strategy of
ViT [6], after extracting the spatial features, we append a trainable class
embedding $class_{embed}$ $\in$ $\mathbb{R}^{D}$ to the motion tokens. This
class embedding serves to represent the temporal relationships between the
motion tokens and is later used for classification ($class_{token}$ in Fig
1(a)). As the order of the $motion_{tokens}$ is not known, we also add a
learnable positional encoding $L_{POS}$ $\in$ $\mathbb{R}^{(N+1)\times D}$ to
the combined $motion_{tokens}$ and $class_{embed}$. This is achieved using an
element-wise addition and we term it as the input $X_{i}$ for the input sample
$V_{i}$.

To enable the interaction between tokens and learn temporal relationships for
input sample classification, we employ the Vanilla Multi-Head Self Attention
(MHSA) mechanism [26]. First, we normalize the input sequence $X_{i}$ $\in$
$\mathbb{R}^{(N+1)\times D}$ by passing it through a layer normalization,
yielding $X^{{}^{\prime}}_{i}$. We then use projection matrices
$(Q_{i},K_{i},V_{i})$ =
$(X^{{}^{\prime}}_{i}W^{Q}_{i},X^{{}^{\prime}}_{i}W^{K}_{i},X^{{}^{\prime}}_{i}W^{V}_{i})$
to project $X^{{}^{\prime}}_{i}$ into queries (Q), keys (K), and values (V),
where $W^{Q/K/V}_{i}$ $\in$ $\mathbb{R}^{D\times D}$ are the projection
matrices for query, key, and value respectively. Next, we compute a dot
product of $Q$ with $K$ and apply a softmax layer to obtain weights on the
values. We repeat this self-attention computation $N_{h}$ times, where $N_{h}$
is the number of heads, and concatenate their outputs. Eq 1, 2 depict the MHSA
process in general.

$A_{i}=Softmax(Q_{i}K_{i})$ (1)

$MHSA(X^{{}^{\prime}}_{i})=A_{i}\times
W^{V}_{i},\>\>\>\>\>\>\>\>\>\>\>\>\>\>X^{{}^{\prime}}_{i}=Norm(X_{i})$ (2)

Subsequently, the output of MHSA is passed to a two-layered MLP with GELU non-
linearity while applying layer normalization and residual connections
concurrently. Eq 3, 4 represent this overall process.

$m^{{}^{\prime}}_{l}=MHSA(X^{{}^{\prime}}_{l-1})+X_{l-1},\>\>\>\>\>\>\>\>\>\>\>\>\>\>l=1...L$
(3)

$m_{l}=MLP(Norm(m^{{}^{\prime}}_{l}))+m^{{}^{\prime}}_{l},\>\>\>\>\>\>\>\>\>\>\>\>\>\>l=1...L$
(4)

where $m_{L}$ $\in$ $\mathbb{R}^{(N+1)\times D}$ are the final output feature
representations and $L$ is the total number of encoding layers in the
Transformer Encoder. Note that the first $\mathbb{R}^{N\times D}$ features
correspond to the $patch_{tokens}$, while the final $\mathbb{R}^{D}$
correspond to the $class_{token}$ of the $m_{L}$ as shown in Fig 1(a). As
mentioned earlier, we then use a one-layer MLP to predict the class label from
the $class_{token}$. We refer to this whole process as a SEizure TRansformer
(SETR) block shown in Fig 1(a).

####  2.2.2 Progressive Knowledge Distillation

To achieve early detection, we use Knowledge Distillation in a Progressive
manner (PKD), starting from a SETR block trained on a full video sample and
gradually moving to a SETR block trained on a partial video sample, as shown
in Fig 1(b). Directly distilling from a SETR block which has seen a
significantly longer portion of the video (say $V^{k-1}_{i}$) to a SETR block
which has only seen a smaller portion of the video sample (say $V^{0}_{i}$)
will lead to considerable mismatches between the features extracted from the
two SETRs as there is a large portion of the input sample that the
$student_{0}$ SETR has not seen. In contrast, our proposed PKD operates in
steps. First we pass the knowledge from teacher ($Teacher_{k-1}$ in Fig 1(b))
SETR trained on $V^{k-1}_{i}$ to a student ($Sub-teacher_{k-2}$) SETR that
operates on $V^{k-2}_{i}$; Later, the $Sub-teacher_{k-2}$ SETR passes its
distilled knowledge to its subsequent student ($Sub-teacher_{k-3}$) SETR, and
this continues until the final $Sub-teacher_{1}$ SETR passes its knowledge to
the bottom most $Student_{0}$ SETR. Since the consecutive segments of the
videos do not differ significantly, PKD is more effective than direct
distillation, which is proven by results in section 3.4.

For distilling knowledge we consider both class token and patch tokens of the
teacher and student networks. A standard Kullback–Leibler divergence
($\mathcal{L}_{KL}$) loss is applied between the probabilities generated from
class token of the teacher and student SETR, whereas a mean squared error
($\mathcal{L}_{MSE}$) loss is computed between the patch tokens of teacher and
student SETR. Overall, a student SETR is trained with three losses -
$\mathcal{L}_{KL}$ and $\mathcal{L}_{MSE}$ loss for knowledge distillation,
and a cross-entropy ($\mathcal{L}_{CE}$) loss for classification, given by the
equations below.

$\mathcal{L}_{KL}=\tau^{2}\sum_{j}q^{T}_{j}(log(q^{T}_{j}/q^{S}_{j}))$ (5)

where $q^{S}_{j}$ and $q^{T}_{j}$ are the soft probabilities (moderated by
temperature $\tau$) of the student and teacher SETRs for the $j^{th}$ class,
respectively.

$\mathcal{L}_{mse}=(\sum^{N}_{i=0}|p^{T}_{i}-p^{S}_{i}\|^{2})/N$ (6)

where $N$ is the number of patches and $p^{T}_{i}$ and $p^{S}_{i}$ are the
patches of teacher and student SETRs respectively.

$\mathcal{L}_{total}=\mathcal{L}_{CE}+\alpha\mathcal{L}_{KL}+\beta\mathcal{L}_{mse}$
(7)

where $\alpha$ and $\beta$ are the weights for $\mathcal{L}_{KL}$ and
$\mathcal{L}_{MSE}$ loss respectively.

##  3 Datasets & Experimental Results

###  3.1 In-house and Public Dataset

Our in-house dataset222We plan to release the in-house optical flow dataset
and corresponding code. contains optical flow information extracted from high-
definition (1920x1080 pixels at 30 frames per second) video recordings of TCS
seizures (infrared cameras are used for nighttime seizures) in a VEM unit in
hospital. To annotate the dataset, two neurologists examined both the video
and corresponding EEG to identify the clinical seizure onset ($t_{ON}$) and
clinical seizure offset ($t_{OFF}$) times for each seizure sample. We curated
a dataset comprising of 40 TCSs from 40 epileptic patients, with one sample
per patient. The duration (in seconds) of the 40 TCSs in our dataset ranges
from 52 to 367 s, with a median duration of 114 s. We also prepared normal
samples (no seizure) for each patient by considering the pre-ictal duration
from ($t_{ON}$ \- 300) to ($t_{ON}$ \- 60) seconds, resulting in dataset of 80
samples (40 normal and 40 TCSs). We refrain from using the 60 seconds prior to
clinical onset as it corresponds to the transition period to the seizure
containing preictal activity [25, 13]. We use a 5-fold cross validation (split
based on patients) for training and testing on our dataset.

We also evaluate the effectiveness of our early detection approach on the
GESTURES dataset [21], which contains features extracted from RGB video
samples of seizures. The dataset includes two seizure types - 106 focal onset
seizures (FOS) and 77 Tonic-Clonic Seizures (TCS). In contrast to our in-house
dataset, the features are provided by the authors, and we directly input them
into our SETR block without using a feature extractor. To evaluate our method,
we adopt the stratified 10-fold cross-validation protocol as used in GESTURES.

###  3.2 Training Implementation & Evaluation Metrics

We implement all experiments in PyTorch 1.8.1 on a single A100 GPU. The SETR
block takes in a total of 64 frames ($N$) with 512 1-D spatial feature per
frame, has 8 MHSA heads ($N_{h}$) with a dropout rate of 0.1, 3 encoder layers
($L$), and 256 hidden dimensions ($D$). For early detection, we experiment by
progressively segmenting a sample into -{4,8,16} parts ($k$). We employ a grid
search to select the weight of 0.2 and 0.5 for KL divergence ($\tau$ = 10) and
MSE loss respectively. We train all methods with a batch size of 16, a
learning rate of 1e-3 and use the AdamW optimizer with a weight decay of 1e-4
for a total 50 epochs. For GESTURES dataset, we implement a weighted BCE loss
to deal with the dataset imbalance, whereas for our in-house dataset we
implement the standard BCE loss. We use precision, recall and f1-score for
benchmarking.

###  3.3 Performance for Early Detection

Table 1 shows the benchmarking performance of all techniques with varying
fractions of input video samples on both datasets. We observed three key
findings from the results in Table 1. First, transformer-based methods such as
our proposed SETR-PKD and OaDTR exhibit better performance retention compared
to LSTM-based techniques (RULSTM, Slowfast RULSTM, EgoAKD, GESTURES) with a
reduction in the fraction of input sample. Second, SETR-PKD performance
increases with $k$=8 from $k$=4, but saturates at $k$=16 for in-house dataset,
whereas it achieves the best performance for $k$=4 for GESTURES dataset. The
median seizure length for the in-house dataset and GESTURES dataset is 114
seconds and 71 seconds, respectively. As a result,

Table 1: Benchmarking of different techniques for different fraction {1/4,
1/2, 3/4, Full} of input video sample. The performance is presented as mean of
- {Precision/Recall/F1-score} across the 5-folds & 10-folds for in-house and
GESTURES dataset respectively. (Best viewed in zoom).

Method/Dataset | In-house dataset | GESTURES
---|---|---
1/4 | 1/2 | 3/4 | Full | 1/4 | 1/2 | 3/4 | Full
RULSTM [9] | 0.57/0.56/0.56 | 0.72/0.71/0.71 | 0.79/0.79/0.79 | 0.95/0.93/0.94 | 0.65/0.64/0.64 | 0.71/0.73/0.72 | 0.84/0.85/0.84 | 0.93/0.94/0.93
Slowfast RULSTM [19] | 0.57/0.56/0.56 | 0.73/0.72/0.72 | 0.81/0.80/0.80 | 0.94/0.94/0.94 | 0.67/0.65/0.66 | 0.73/0.72/0.72 | 0.86/0.84/0.85 | 0.97/0.95/0.96
EgoAKD [31] | 0.64/0.65/0.64 | 0.79/0.80/0.79 | 0.89/0.90/0.89 | 0.95/0.94/0.94 | 0.70/0.69/0.69 | 0.80/0.79/0.79 | 0.93/0.90/91 | 0.97/0.94/0.95
OaDTR [28] | 0.66/0.65/0.65 | 0.82/0.83/0.82 | 0.90/0.90/0.90 | 0.95/0.95/0.95 | 0.72/0.69/0.70 | 0.82/0.83/0.82 | 0.91/0.92/0.91 | 0.99/0.99/0.99
GESTURES [21] | 0.59/0.60/0.59 | 0.74/0.73/0.73 | 0.82/0.83/0.82 | 0.94/0.94/0.94 | 0.68/0.66/0.66 | 0.74/0.72/0.73 | 0.86/0.85/0.85 | 0.97/0.99/0.98
SETR | 0.61/0.60/0.60 | 0.75/0.73/0.74 | 0.84/0.83/0.83 | 0.96/0.95/0.95 | 0.67/0.66/0.66 | 0.73/0.74/0.73 | 0.88/0.88/0.88 | 0.98/0.99/0.98
SETR-PKD (k=4) | 0.63/0.62/0.62 | 0.78/0.79/0.78 | 0.89/0.90/0.89 | 0.96/0.95/0.95 | 0.74/0.73/0.73 | 0.86/0.85/0.85 | 0.96/0.95/0.95 | 0.98/0.99/0.98
SETR-PKD (k=8) | 0.70/0.69/0.69 | 0.86/0.84/0.85 | 0.92/0.93/0.92 | 0.96/0.95/0.95 | 0.73/0.74/0.73 | 0.85/0.85/0.85 | 0.95/0.96/0.95 | 0.98/0.99/0.98
SETR-PKD (k=16) | 0.69/0.69/0.69 | 0.85/0.84/0.84 | 0.92/0.92/0.92 | 0.96/0.95/0.95 | 0.72/0.73/0.72 | 0.85/0.84/0.84 | 0.96/0.95/0.95 | 0.98/0.99/0.98

PKD using relatively longer partial segments ($k$=4) is sufficient for
GESTURES, while shorter partial segments ($k$=8) are required for our dataset.
Thus, the optimal value of $k$ for PKD may vary depending on a dataset.
Finally, we observed better performance on the GESTURES dataset, which is
expected given the more detailed and refined features extracted from RGB video
compared to optical flow information.

###  3.4 Progressive v/s Direct Knowledge Distillation

Figure 2: Performance comparison of direct knowledge distillation and
progressive knowledge distillation between SETR blocks for different fractions
of input video sample.

To validate our approach of progressive knowledge distillation in a fair
manner, we conducted an ablation study to compare it with direct knowledge
distillation. Fig 2 shows the comparison of the accuracy of the two approaches
for different fractions of the input video sample on both datasets. The
results indicate that although direct knowledge distillation can increase
performance, it is less effective when the knowledge gap is wide, i.e., from a
SETR block trained on a full input sample to a SETR block trained on a minimal
fraction of the input sample (1/8, 1/4, .. 1/2) compared to when the knowledge
gap is small (5/8, .. 7/8). On the other hand, our SETR-PKD approach
significantly improves performance for minimal fractions of input samples on
both datasets.

##  4 Conclusion

In this work, we show that it is possible to detect epileptic seizures from
optical flow modality in a privacy-preserving manner. Moreover, to achieve
real-time seizure detection, we specifically develop a novel approach using
progressive knowledge distillation which proves to detect seizures more
accurately during their progression itself. We believe that our proposed
privacy-preserving early detection of seizures will inspire the research
community to pursue real-time seizure detection in videos as well as
facilitate inter-cohort studies.

## References

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