The LHCb collaboration

# First evidence of direct $C\\!P$ violation in charmless two-body decays of
$B^{0}_{s}$ mesons

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, L. Arrabito55, 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, K. de Bruyn38, 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. Clarke47,35, 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, S. De Capua21,k, M. De
Cian37, F. De Lorenzi12, 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. Hartmann56, 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.
Krokovny11, 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. Langenbruch11, 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, 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, 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, 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. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S.
Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35,
A. Ukleja25, P. Urquijo53, 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, H. Voss10, R.
Waldi56, 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 Vrije Universiteit,
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
55CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
56Physikalisches Institut, 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

###### Abstract

Using a data sample corresponding to an integrated luminosity of 0.35
$\mathrm{fb}^{-1}$ collected by LHCb in 2011, we report the first evidence of
$C\\!P$ violation in the decays of $B^{0}_{s}$ mesons to $K^{\pm}\pi^{\mp}$
pairs, $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)=0.27\pm 0.08\,\mathrm{(stat)}\pm
0.02\,\mathrm{(syst)}$, with a significance of 3.3$\sigma$. Furthermore, we
report the most precise measurement of $C\\!P$ violation in the decays of
$B^{0}$ mesons to $K^{\pm}\pi^{\mp}$ pairs, $A_{C\\!P}(B^{0}\rightarrow
K\pi)=-0.088\pm 0.011\,\mathrm{(stat)}\pm 0.008\,\mathrm{(syst)}$, with a
significance exceeding 6$\sigma$.

###### pacs:

Valid PACS appear here

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 |  |
---|---|---
|  | LHCb-PAPER-2011-029
|  | CERN-PH-EP-2012-058

The violation of $C\\!P$ symmetry, _i.e._ the non-invariance of fundamental
forces under the combined action of the charge conjugation ($C$) and parity
($P$) transformations, is well established in the $K^{0}$ and $B^{0}$ meson
systems Christenson:1964fg ; Aubert:2001nu ; Abe:2001xe ; Nakamura:2010zzi .
Recent results from the LHCb collaboration have also provided evidence for
$C\\!P$ violation in the decays of $D^{0}$ mesons Aaij:2011in . Consequently,
there now remains only one neutral heavy meson system, the $B^{0}_{s}$, where
$C\\!P$ violation has not yet been seen. All current experimental measurements
of $C\\!P$ violation in the quark flavor sector are well described by the
Cabibbo-Kobayashi-Maskawa mechanism Cabibbo:1963yz ; Kobayashi:1973fv  which
is embedded in the framework of the Standard Model (SM). However, it is
believed that the size of $C\\!P$ violation in the SM is not sufficient to
account for the asymmetry between matter and antimatter in the Universe
Hou:2008xd , hence additional sources of $C\\!P$ symmetry breaking are being
searched for as manifestations of physics beyond the SM.

In this Letter we report measurements of direct $C\\!P$ violating asymmetries
in $B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{-}\pi^{+}$
decays using data collected with the LHCb detector. The inclusion of charge-
conjugate modes is implied except in the asymmetry definitions. $C\\!P$
violation in charmless two-body $B$ decays could potentially reveal the
presence of physics beyond the SM Fleischer:1999pa ; Gronau:2000md ;
Lipkin:2005pb ; Fleischer:2007hj ; Fleischer:2010ib , and has been extensively
studied at the $B$ factories and at the Tevatron Aubert:2008sb ; Belle:2008zza
; Aaltonen:2011qt . The direct $C\\!P$ asymmetry in the $B^{0}_{(s)}$ decay
rate to the final state $f_{(s)}$, with $f=K^{+}\pi^{-}$ and
$f_{s}=K^{-}\pi^{+}$, is defined as

$A_{C\\!P}=\Phi\\!\left[\Gamma\\!\left(\overline{B}^{0}_{(s)}\rightarrow\bar{f}_{(s)}\right)\\!\\!,\,\Gamma\\!\left(B^{0}_{(s)}\rightarrow
f_{(s)}\right)\right]\\!\\!,$ (1)

where $\Phi[X,\,Y]=(X-Y)/(X+Y)$ and $\bar{f}_{(s)}$ denotes the charge-
conjugate of $f_{(s)}$.

LHCb is a forward spectrometer covering the pseudo-rapidity range $2<\eta<5$,
designed to perform flavor physics measurements at the LHC. A detailed
description of the detector can be found in Ref. Alves:2008zz . The analysis
is based on $pp$ collision data collected in the first half of 2011 at a
center-of-mass energy of $7\\!$ $\mathrm{\,Te\kern-1.00006ptV}$, corresponding
to an integrated luminosity of $0.35~{}\mathrm{fb}^{-1}$. The polarity of the
LHCb magnetic field is reversed from time to time in order to partially cancel
the effects of instrumental charge asymmetries, and about
$0.15~{}\mathrm{fb}^{-1}$ were acquired with one polarity and
$0.20~{}\mathrm{fb}^{-1}$ with the opposite polarity.

The LHCb trigger system comprises a hardware trigger followed by a High Level
Trigger (HLT) implemented in software. The hadronic hardware trigger selects
high transverse energy clusters in the hadronic calorimeter. A transverse
energy threshold of 3.5 $\mathrm{\,Ge\kern-1.00006ptV}$ has been adopted for
the data set under study. The HLT first selects events with at least one large
transverse momentum track characterized by a large impact parameter, and then
uses algorithms to reconstruct $D$ and $B$ meson decays. Most of the events
containing the decays under study have been acquired by means of a dedicated
two-body HLT selection. To discriminate between signal and background events,
this trigger selection imposes requirements on: the quality of the online-
reconstructed tracks ($\chi^{2}$ per degree of freedom), their transverse
momenta ($p_{\mathrm{T}}$) and their impact parameters ($d_{\mathrm{IP}}$,
defined as the distance between the reconstructed trajectory of the track and
the $pp$ collision vertex); the distance of closest approach of the decay
products of the $B$ meson candidate ($d_{\mathrm{CA}}$), its transverse
momentum ($p_{\mathrm{T}}^{B}$), its impact parameter ($d_{\mathrm{IP}}^{B}$)
and the decay time in its rest frame ($t_{\pi\pi}$, calculated assuming the
decay into $\pi^{+}\pi^{-}$). Only $B$ candidates within the $\pi\pi$
invariant mass range 4.7–5.9 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are
accepted. The $\pi\pi$ mass hypothesis is conventionally chosen to select all
charmless two-body $B$ decays using the same criteria.

Offline selection requirements are subsequently applied. Two sets of criteria
have been optimized with the aim of minimizing the expected uncertainty either
on $A_{C\\!P}(B^{0}\rightarrow K\pi)$ or on $A_{C\\!P}(B^{0}_{s}\rightarrow
K\pi)$. In addition to more selective requirements on the kinematic variables
already used in the HLT, two further requirements on the larger of the
transverse momenta and of the impact parameters of the daughter tracks are
applied. A summary of the two distinct sets of selection criteria is reported
in Table 1. In the case of $B^{0}_{s}\rightarrow K\pi$ decays a tighter
selection is needed because the probability for a $b$ quark to decay as
$B^{0}_{s}\rightarrow K\pi$ is about 14 times smaller than that to decay as
$B^{0}\rightarrow K\pi$ Aaltonen:2008hg , and consequently a stronger
rejection of combinatorial background is required. The two samples passing the
event selection are then subdivided into different final states using the
particle identification (PID) provided by the two ring-imaging Cherenkov
(RICH) detectors. Again two sets of PID selection criteria are applied: a
loose set optimized for the measurement of $A_{C\\!P}(B^{0}\rightarrow K\pi)$
and a tight set for that of $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$.

Table 1: Summary of selection criteria adopted for the measurement of $A_{C\\!P}(B^{0}\rightarrow K\pi)$ and $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$. Variable |  $A_{C\\!P}(B^{0}\rightarrow K\pi)$ | $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$
---|---|---
Track quality $\chi^{2}$/ndf | $<3$ | $<3$
Track $p_{\mathrm{T}}\,[\textrm{Ge\kern-1.00006ptV\\!/}c]$ | $>1.1$ | $>1.2$
Track $d_{\mathrm{IP}}\,[\mathrm{mm}]$ | $>0.15$ | $>0.20$
$\mathrm{max}(p_{\mathrm{T}}^{K},\,p_{\mathrm{T}}^{\pi})\,[\textrm{Ge\kern-1.00006ptV\\!/}c]$ | $>2.8$ | $>3.0$
$\mathrm{max}(d_{\mathrm{IP}}^{K},\,d_{\mathrm{IP}}^{\pi})\,[\mathrm{mm}]$ | $>0.3$ | $>0.4$
$d_{\mathrm{CA}}$ $[\mathrm{mm}]$ | $<0.08$ | $<0.08$
$p_{\mathrm{T}}^{B}\,[\textrm{Ge\kern-1.00006ptV\\!/}c]$ | $>2.2$ | $>2.4$
$d_{\mathrm{IP}}^{B}\,[\mathrm{mm}]$ | $<0.06$ | $<0.06$
$t_{\pi\pi}\,[\textrm{ps}]$ | $>0.9$ | $>1.5$
|
---|---
|
Figure 1: Invariant $K\pi$ mass spectra obtained using the event selection
adopted for the best sensitivity on (a, b) $A_{C\\!P}(B^{0}\rightarrow K\pi)$
and (c, d) $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$. Plots (a) and (c) represent
the $K^{+}\pi^{-}$ invariant mass whereas plots (b) and (d) represent the
$K^{-}\pi^{+}$ invariant mass. The results of the unbinned maximum likelihood
fits are overlaid. The main components contributing to the fit model are also
shown.

To estimate the background from other two-body $B$ decays with a misidentified
pion or kaon (cross-feed background), the relative efficiencies of the RICH
PID selection criteria must be determined. The high production rate of charged
$D^{*}$ mesons at the LHC and the kinematic characteristics of the
$D^{*+}\rightarrow D^{0}(K^{-}\pi^{+})\pi^{+}$ decay chain make such events an
appropriate calibration sample for the PID of kaons and pions. In addition,
for calibrating the response of the RICH system for protons, a sample of
$\Lambda\rightarrow p\pi^{-}$ decays is used. PID information is not used to
select either sample, as the selection of pure final states can be realized by
means of kinematic criteria alone. The production and decay kinematics of the
$D^{0}\rightarrow K^{-}\pi^{+}$ and $\Lambda\rightarrow p\pi^{-}$ channels
differ from those of the $B$ decays under study. Since the RICH PID
information is momentum dependent, the distributions obtained from calibration
samples are reweighted according to the momentum distributions of $B$ daughter
tracks observed in data.

Unbinned maximum likelihood fits to the $K\pi$ mass spectra of the selected
events are performed. The $B^{0}\rightarrow K\pi$ and $B^{0}_{s}\rightarrow
K\pi$ signal components are described by single Gaussian functions convolved
with a function which describes the effect of final state radiation on the
mass lineshape Baracchini:2005wp . The background due to partially
reconstructed three-body $B$ decays is parameterized by means of an ARGUS
function Albrecht:1989ga  convolved with a Gaussian resolution function. The
combinatorial background is modeled by an exponential and the shapes of the
cross-feed backgrounds, mainly due to $B^{0}\rightarrow\pi^{+}\pi^{-}$ and
$B^{0}_{s}\rightarrow K^{+}K^{-}$ decays with one misidentified particle in
the final state, are obtained from Monte Carlo simulations. The
$B^{0}\rightarrow\pi^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{+}K^{-}$ cross-
feed background yields are determined from fits to the $\pi^{+}\pi^{-}$ and
$K^{+}K^{-}$ mass spectra respectively, using events selected by the same
offline selection as the signal and taking into account the appropriate PID
efficiency factors. The $K^{+}\pi^{-}$ and $K^{-}\pi^{+}$ mass spectra for the
events passing the two offline selections are shown in Fig. 1.

From the two mass fits we determine respectively the signal yields
$N(B^{0}\rightarrow K\pi)=13\hskip 1.42262pt250\pm 150$ and
$N(B^{0}_{s}\rightarrow K\pi)=314\pm 27$, as well as the raw yield asymmetries
$A_{\mathrm{raw}}(B^{0}\rightarrow K\pi)=-0.095\pm 0.011$ and
$A_{\mathrm{raw}}(B^{0}_{s}\rightarrow K\pi)=0.28\pm 0.08$, where the
uncertainties are statistical only. In order to determine the $C\\!P$
asymmetries from the observed raw asymmetries, effects induced by the detector
acceptance and event reconstruction, as well as due to strong interactions of
final state particles with the detector material, need to be taken into
account. Furthermore, the possible presence of a
$B^{0}_{(s)}-\overline{B}^{0}_{(s)}$ production asymmetry must also be
considered. The $C\\!P$ asymmetry is related to the raw asymmetry by
$A_{C\\!P}=A_{\mathrm{raw}}-A_{\Delta}$, where the correction $A_{\Delta}$ is
defined as

$A_{\Delta}(B^{0}_{(s)}\rightarrow
K\pi)=\zeta_{d(s)}A_{\mathrm{D}}(K\pi)+\kappa_{d(s)}A_{\mathrm{P}}(B^{0}_{(s)}),$
(2)

where $\zeta_{d}=1$ and $\zeta_{s}=-1$, following the sign convention for $f$
and $f_{s}$ in Eq. (1). The instrumental asymmetry $A_{\mathrm{D}}(K\pi)$ is
given in terms of the detection efficiencies $\varepsilon_{\mathrm{D}}$ of the
charge-conjugate final states by
$A_{\mathrm{D}}(K\pi)=\Phi[\varepsilon_{\mathrm{D}}(K^{-}\pi^{+}),\,\varepsilon_{\mathrm{D}}(K^{+}\pi^{-})]$,
and the production asymmetry $A_{\mathrm{P}}(B^{0}_{(s)})$ is defined in terms
of the $\overline{B}^{0}_{(s)}$ and $B^{0}_{(s)}$ production rates,
$R(\overline{B}^{0}_{(s)})$ and $R(B^{0}_{(s)})$, as
$A_{\mathrm{P}}(B^{0}_{(s)})=\Phi[R(\overline{B}^{0}_{(s)}),\,R(B^{0}_{(s)})]$.
The factor $\kappa_{d(s)}$ takes into account dilution due to neutral
$B^{0}_{(s)}$ meson mixing, and is defined as

$\kappa_{d(s)}\\!=\\!\frac{\int_{0}^{\infty}\\!e^{-\Gamma_{d(s)}t}\\!\cos\\!\left(\Delta
m_{d(s)}t\right)\\!\varepsilon(B^{0}_{(s)}\\!\rightarrow\\!K\pi;\,t)\mathrm{d}t}{\int_{0}^{\infty}\\!e^{-\Gamma_{d(s)}t}\\!\cosh\\!\left(\frac{\Delta\Gamma_{d(s)}}{2}t\right)\\!\varepsilon(B^{0}_{(s)}\\!\rightarrow\\!K\pi;\,t)\mathrm{d}t},$
(3)

where $\varepsilon(B^{0}\rightarrow K\pi;\,t)$ and
$\varepsilon(B^{0}_{s}\rightarrow K\pi;\,t)$ are the acceptances as functions
of the decay time for the two reconstructed decays. To calculate $\kappa_{d}$
and $\kappa_{s}$ we assume that $\Delta\Gamma_{d}=0$ and we use the world
averages for $\Gamma_{d}$, $\Delta m_{d}$, $\Gamma_{s}$, $\Delta m_{s}$ and
$\Delta\Gamma_{s}$ Nakamura:2010zzi . The shapes of the acceptance functions
are parameterized using signal decay time distributions extracted from data.
We obtain $\kappa_{d}=0.303\pm 0.005$ and $\kappa_{s}=-0.033\pm 0.003$, where
the uncertainties are statistical only. In contrast to $\kappa_{d}$, the
factor $\kappa_{s}$ is small, owing to the large $B^{0}_{s}$ oscillation
frequency, thus leading to a negligible impact of a possible production
asymmetry of $B^{0}_{s}$ mesons on the corresponding $C\\!P$ asymmetry
measurement.

The instrumental charge asymmetry $A_{\mathrm{D}}(K\pi)$ can be expressed in
terms of two distinct contributions
$A_{\mathrm{D}}(K\pi)=A_{\mathrm{I}}(K\pi)+\alpha(K\pi)A_{\mathrm{R}}(K\pi)$,
where $A_{\mathrm{I}}(K\pi)$ is an asymmetry due to the different strong
interaction cross-sections with the detector material of $K^{+}\pi^{-}$ and
$K^{-}\pi^{+}$ final state particles, and $A_{\mathrm{R}}(K\pi)$ arises from
the possible presence of a reconstruction or detection asymmetry. The quantity
$A_{\mathrm{I}}(K\pi)$ does not change its value by reversing the magnetic
field, as the difference in the interaction lengths seen by the positive and
negative particles for opposite polarities is small. By contrast,
$A_{\mathrm{R}}(K\pi)$ changes its sign when the magnetic field polarity is
reversed. The factor $\alpha(K\pi)$ accounts for different signal yields in
the data sets with opposite polarities, due to the different values of the
corresponding integrated luminosities and to changing trigger conditions in
the course of the run. It is estimated by using the yields of the largest
decay mode, _i.e._ $B^{0}\rightarrow K\pi$, determined from the mass fits
applied to the two data sets separately. We obtain $\alpha(K\pi)=\Phi[N^{\rm
up}(B^{0}\rightarrow K\pi),\,N^{\rm down}(B^{0}\rightarrow K\pi)]=-0.202\pm
0.011$, where “up” and “down” denote the direction of the main component of
the dipole field.

The instrumental asymmetries for the final state $K\pi$ are measured from data
using large samples of tagged $D^{*+}\rightarrow D^{0}(K^{-}\pi^{+})\pi^{+}$
and $D^{*+}\rightarrow D^{0}(K^{-}K^{+})\pi^{+}$ decays, and untagged
$D^{0}\rightarrow K^{-}\pi^{+}$ decays. The combination of the integrated raw
asymmetries of all these decay modes is necessary to disentangle the various
contributions to the raw asymmetries of each mode, notably including the
$K\pi$ instrumental asymmetry as well as that of the pion from the $D^{*+}$
decay, and the production asymmetries of the $D^{*+}$ and $D^{0}$ mesons. In
order to determine the raw asymmetry of the $D^{0}\rightarrow K\pi$ decay, a
maximum likelihood fit to the $K^{-}\pi^{+}$ and $K^{+}\pi^{-}$ mass spectra
is performed. For the decays $D^{*+}\rightarrow D^{0}(K^{-}\pi^{+})\pi^{+}$
and $D^{*+}\rightarrow D^{0}(K^{-}K^{+})\pi^{+}$, we perform maximum
likelihood fits to the discriminating variable $\delta m=M_{D^{*}}-M_{D^{0}}$,
where $M_{D^{*}}$ and $M_{D^{0}}$ are the reconstructed $D^{*}$ and $D^{0}$
invariant masses respectively. Approximately 54 million $D^{0}\rightarrow
K^{-}\pi^{+}$ decays, 7.5 million $D^{*+}\rightarrow
D^{0}(K^{-}\pi^{+})\pi^{+}$ and 1.1 million $D^{*+}\rightarrow
D^{0}(K^{-}K^{+})\pi^{+}$ decays are used. The mass distributions are shown in
Fig. 2 (a), (b) and (c). The $D^{0}\rightarrow K^{-}\pi^{+}$ signal component
is modeled as the sum of two Gaussian functions with common mean convolved
with a function accounting for final state radiation Baracchini:2005wp , on
top of an exponential combinatorial background. The $D^{*+}\rightarrow
D^{0}(K^{-}\pi^{+})\pi^{+}$ and $D^{*+}\rightarrow D^{0}(K^{-}K^{+})\pi^{+}$
signal components are modeled as the sum of two Gaussian functions convolved
with a function taking account of the asymmetric shape of the measured
distribution Aaij:2011in . The background is described by an empirical
function of the form $1-e^{-(\delta m-\delta m_{0})/\xi}$, where $\delta
m_{0}$ and $\xi$ are free parameters.

|
---|---
|
Figure 2: Distributions of the invariant mass or invariant mass difference of
(a) $D^{0}\rightarrow K^{-}\pi^{+}$, (b) $D^{*+}\rightarrow
D^{0}(K^{-}\pi^{+})\pi^{+}$, (c) $D^{*+}\rightarrow D^{0}(K^{-}K^{+})\pi^{+}$
and (d) $B^{0}\rightarrow J/\psi(\mu^{+}\mu^{-})K^{*0}(K^{+}\pi^{-})$. The
results of the maximum likelihood fits are overlaid.

Using the current world average of the integrated $C\\!P$ asymmetry for the
$D^{0}\rightarrow K^{-}K^{+}$ decay bib:hfagbase  and neglecting $C\\!P$
violation in the Cabibbo-favored $D^{0}\rightarrow K^{-}\pi^{+}$ decay
Bianco:2003vb , from the raw yield asymmetries returned by the mass fits we
determine $A_{\mathrm{I}}(K\pi)=(-1.0\pm 0.2)\times 10^{-2}$ and
$A_{\mathrm{R}}(K\pi)=(-1.8\pm 0.2)\times 10^{-3}$, where the uncertainties
are statistical only.

The possible existence of a $B^{0}-\overline{B}^{0}$ production asymmetry is
studied by reconstructing a sample of $B^{0}\rightarrow J/\psi K^{*0}$ decays.
$C\\!P$ violation in $b\rightarrow c\bar{c}s$ transitions, which is predicted
in the SM to be at the $10^{-3}$ level Hou:2006du , is neglected. The raw
asymmetry $A_{\mathrm{raw}}(B^{0}\rightarrow J/\psi K^{*0})$ is determined
from an unbinned maximum likelihood fit to the
$J/\psi(\mu^{+}\mu^{-})K^{*0}(K^{+}\pi^{-})$ and
$J/\psi(\mu^{+}\mu^{-})\overline{K}^{*0}(K^{-}\pi^{+})$ mass spectra. The
signal mass peak is modeled as the sum of two Gaussian functions with common
mean, whereas the combinatorial background is modeled by an exponential. The
data sample contains approximately 25 400 $B^{0}\rightarrow J/\psi K^{*0}$
decays. The mass distribution is shown in Fig. 2 (d). To determine the
production asymmetry we need to correct for the presence of instrumental
asymmetries. Once the necessary corrections are applied, we obtain a value for
the $B^{0}$ production asymmetry $A_{\mathrm{P}}(B^{0})=0.010\pm 0.013$, where
the uncertainty is statistical only.

By using the instrumental and production asymmetries, the correction factor to
the raw asymmetry $A_{\Delta}(B^{0}\rightarrow K\pi)=-0.007\pm 0.006$ is
obtained. Since the $B^{0}_{s}$ meson has no valence quarks in common with
those of the incident protons, its production asymmetry is expected to be
smaller than for the $B^{0}$, an expectation that is supported by
hadronization models as discussed in Ref. Lambert:2009zz . Even conservatively
assuming a value of the production asymmetry equal to that for the $B^{0}$,
owing to the small value of $\kappa_{s}$ the effect of
$A_{\mathrm{P}}(B^{0}_{s})$ is negligible, and we find
$A_{\Delta}(B^{0}_{s}\rightarrow K\pi)=0.010\pm 0.002$.

The systematic uncertainties on the asymmetries fall into the following main
categories, related to: (a) PID calibration; (b) modeling of the signal and
background components in the maximum likelihood fits; and (c) instrumental and
production asymmetries. Knowledge of PID efficiencies is necessary in this
analysis to compute the number of cross-feed background events affecting the
mass fit of the $B^{0}\rightarrow K\pi$ and $B^{0}_{s}\rightarrow K\pi$ decay
channels. In order to estimate the impact of imperfect PID calibration, we
perform unbinned maximum likelihood fits after having altered the number of
cross-feed background events present in the relevant mass spectra according to
the systematic uncertainties affecting the PID efficiencies. An estimate of
the uncertainty due to possible imperfections in the description of the final
state radiation is determined by varying, over a wide range, the amount of
emitted radiation Baracchini:2005wp  in the signal lineshape parameterization.
The possibility of an incorrect description of the core distribution in the
signal mass model is investigated by replacing the single Gaussian with the
sum of two Gaussian functions with a common mean. The impact of additional
three-body $B$ decays in the $K\pi$ spectrum, not accounted for in the
baseline fit — namely $B\rightarrow\pi\pi\pi$ where one pion is missed in the
reconstruction and another is misidentified as a kaon — is investigated. The
mass lineshape of this background component is determined from Monte Carlo
simulations, and then the fit is repeated after having modified the baseline
parameterization accordingly. For the modeling of the combinatorial background
component, the fit is repeated using a first-order polynomial. Finally, for
the case of the cross-feed backgrounds, two distinct systematic uncertainties
are estimated: one due to a relative bias in the mass scale of the simulated
distributions with respect to the signal distributions in data, and another
accounting for the difference in mass resolution between simulation and data.
All the shifts from the relevant baseline values are accounted for as
systematic uncertainties. Differences in the kinematic properties of $B$
decays with respect to the charm control samples, as well as different
triggers and offline selections, are taken into account by introducing a
systematic uncertainty on the values of the $A_{\Delta}$ corrections. This
uncertainty dominates the total systematic uncertainty related to the
instrumental and production asymmetries, and can be reduced in future
measurements with a better understanding of the dependence of such asymmetries
on the kinematics of selected signal and control samples. The systematic
uncertainties for $A_{C\\!P}(B^{0}\rightarrow K\pi)$ and
$A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$ are summarized in Table 2.

Table 2: Summary of systematic uncertainties on $A_{C\\!P}(B^{0}\rightarrow
K\pi)$ and $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$. The categories (a), (b) and
(c) defined in the text are also indicated. The total systematic uncertainties
given in the last row are obtained by summing the individual contributions in
quadrature.

Systematic uncertainty |  $A_{C\\!P}(B^{0}\rightarrow K\pi)$ | $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$
---|---|---
(a)PID calibration | 0.0012 | 0.001
(b)Final state radiation | 0.0026 | 0.010
(b)Signal model | 0.0004 | 0.005
(b)Combinatorial background | 0.0001 | 0.009
(b)3-body background | 0.0009 | 0.007
(b)Cross-feed background | 0.0011 | 0.008
(c)Instr. and prod. asym. ($A_{\Delta}$) | 0.0078 | 0.005
Total | 0.0084 | 0.019

In conclusion we obtain the following measurements of the $C\\!P$ asymmetries:

$A_{C\\!P}(B^{0}\rightarrow K\pi)=-0.088\pm 0.011\,\mathrm{(stat)}\pm
0.008\,\mathrm{(syst)}$

and

$A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)=0.27\pm 0.08\,\mathrm{(stat)}\pm
0.02\,\mathrm{(syst)}.$

The result for $A_{C\\!P}(B^{0}\rightarrow K\pi)$ constitutes the most precise
measurement available to date. It is in good agreement with the current world
average provided by the Heavy Flavor Averaging Group
$A_{C\\!P}(B^{0}\rightarrow K\pi)=-0.098^{+0.012}_{-0.011}$ bib:hfagbase .
Dividing the central value of $A_{C\\!P}(B^{0}\rightarrow K\pi)$ by the sum in
quadrature of the statistical and systematic uncertainties, the significance
of the measured deviation from zero exceeds $6\sigma$, making this the first
observation (greater than 5$\sigma$) of $C\\!P$ violation in the $B$ meson
sector at a hadron collider. The same significance computed for
$A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$ is 3.3$\sigma$, therefore this is the
first evidence for $C\\!P$ violation in the decays of $B^{0}_{s}$ mesons. The
result for $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$ is in agreement with the
only measurement previously available Aaltonen:2011qt .

## 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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# Pseudo-Riemannian and Hessian Geometry Related to Monge-Ampère Structures

S. Hronek111Department of Theoretical Physics and Astrophysics, Faculty of
Science, Masaryk University 611 37 Brno, Czech Republic
R. Suchánek222Dept. of Mathematics and Statistics, Faculty of Science, Masaryk
University 611 37 Brno, Czech Republic 333Angevin Laboratory of Mathematical
Research - UMR CNRS 6093, University of Angers, 2 Boulevard Lavoisier, 49045,
Angers CEDEX 0, France.

Abstract.  We study properties of pseudo-Riemannian metrics corresponding to
Monge-Ampère structures on four dimensional $T^{*}M$. We describe a family of
Ricci flat solutions, which are parametrized by six coefficients satisfying
the Plücker embedding equation. We also focus on pullbacks of the pseudo-
metrics on two dimensional $M$, and describe the corresponding Hessian
structures.

†† 2022 Mathematics Subject Classification: primary 53B20; secondary 83C15 ††
Key words and phrases: Einstein equation, General relativity, Hessian
structure, Lychagin-Rubtsov metric, Monge-Ampère structure, Monge-Ampère
equation, Plücker embedding.

##  1 Introduction

Let $T^{*}M\xrightarrow{\pi}M$ be the cotangent bundle over a real, smooth,
two dimensional manifold $M$. Let $\Omega\in\Omega^{2}(T^{*}M)$ be the
canonical symplectic form, which in the Darboux (or canonical) coordinates is
written as

$\displaystyle\Omega=\operatorname{d}x\wedge\operatorname{d}p+\operatorname{d}y\wedge\operatorname{d}q\
.$ (1)

Let us further consider a $2$-form $\alpha\in\Omega^{2}(T^{*}M)$ given by

$\displaystyle\alpha=A\operatorname{d}p\wedge\operatorname{d}y+B(\operatorname{d}x\wedge\operatorname{d}p-\operatorname{d}y\wedge\operatorname{d}q)+C\operatorname{d}x\wedge\operatorname{d}q+D\operatorname{d}p\wedge\operatorname{d}q+E\operatorname{d}x\wedge\operatorname{d}y\
,$ (2)

where $A,B,C,D,E\in C^{\infty}(T^{*}M)$ are smooth functions. If
$\Omega\wedge\alpha=0$, then the pair $(\Omega,\alpha)$ is called a Monge-
Ampère (M-A) structure over $T^{*}M$ [8, 9]. This terminology reflects the
fact that the pairs $(\Omega,\alpha)$ are in correspondence with a subclass of
all nonlinear second-order PDEs, called Monge-Ampère equations, in the
following way. Choose a function $f\in C^{\infty}(M)$. The deRham differential
of $f$ gives rise to a section, $\operatorname{d}f\colon M\to T^{*}M$, and one
can consider the pullback $(\operatorname{d}f)^{*}\alpha\in\Omega^{2}(M)$.
Then the equation

$\displaystyle(\operatorname{d}f)^{*}\alpha=0$ (3)

defines a nonlinear second order PDE with respect to $f$, where the
nonlinearity is given by the determinant of the Hessian matrix of $f$. In the
above chosen coordinates, the equation (3) writes

$\displaystyle
Af_{xx}+2Bf_{xy}+Cf_{yy}+D\left(f_{xx}f_{yy}-{f_{xy}}^{2}\right)+E=0\ ,$ (4)

where $A,\ldots,E$ are the coefficients given in (2), but now depending on
$x,y,f_{x},f_{y}$, instead of $x,y,p,q$. The equation (4) is called a 2D
(symplectic) Monge-Ampère equation (shortly just M-A equation). M-A equations
arise and have rich applications, for example, in differential geometry of
surfaces, integrability of geometric structures, hydrodynamics, acoustics,
variational calculus, Riemannian, CR, or complex geometry [5, 8, 9, 11, 15].
For a detailed exposition of some of these applications and geometric
treatment of M-A equations, especially in 2D and 3D, see [7, 8].

From a different perspective, M-A structure $(\Omega,\alpha)$ yields other
geometric structures on $T^{*}M$, for example, complex, product, Kähler, or
nearly Calabi-Yau structures (depending on various assumptions on the
coefficients $A,\ldots,E$) [4, 5, 7, 8, 9, 14, 15]. We call a M-A structure
non-degenerate, if the Pfaffian of a M-A structure, which is given by the
equation

$\displaystyle\alpha\wedge\alpha=\operatorname{Pf}(\alpha)\Omega\wedge\Omega\
,$ (5)

satisfies $\operatorname{Pf}(\alpha)\neq 0$. The sign of the Pfaffian decides
whether M-A structure gives rise to a complex or product structure, and
whether the M-A equation corressponding to the M-A structure is elliptic or
hyperbolic [4, 8, 9]. We have found that non-degeneracy of M-A structures on
$T^{*}M$ is equivalent to non-degeneracy of certain bilinear forms on $M$.

Inspired by applications in theoretical meteorology [2, 11, 13], we are
interested in a specific family of symmetric bilinear forms, parameterized by
$2$-forms $\alpha$ as given in (2) (or, equivalently, by the corresponding
coefficients $A,\ldots,E$). More concretely, there is a map

$\displaystyle\Omega^{2}(T^{*}M)$ $\displaystyle\to S^{2}(T^{*}M)\ ,$
$\displaystyle\alpha$ $\displaystyle\mapsto g_{\alpha}\ ,$

given locally by

$\displaystyle
g_{\alpha}(X,Y):=\frac{2(\iota_{X}\alpha\wedge\iota_{Y}\Omega+\iota_{Y}\alpha\wedge\iota_{X}\Omega)\wedge\pi^{*}\operatorname{vol}}{\Omega\wedge\Omega}\
,$ (6)

where $X,Y\in\Gamma(TT^{*}M)$, $\iota$ is the interior product, and
$\pi^{*}\operatorname{vol}\in\Omega^{2}(T^{*}M)$ is the pullback of a locally
chosen top form $\operatorname{vol}\in\Omega^{2}(M)$ along the cotangent
bundle projection. Under a mild assumption on $\alpha$, the symmetric form
$g_{\alpha}$ is a pseudo-metric on $T^{*}M$, which is called Lychagin-Rubtsov
metric [3, 5, 6].

The L-R metrics are the main objects of our interest. They find applications,
for example, in theoretical meteorology [2, 12, 14, 15]. In particular, we
want to mention a paper in progress of I. Roulstone, V. Rubtsov, and M. Wolf,
where an approach motivated by general principles of topological fluid
dynamics is employed. Specifically, the curvature of L-R metric (6) is used to
study geometric aspects of semi-geostrophic incompressible flows in 2D and 3D,
which are associated with the accumulation of vorticity of the flow. These
results were discussed on a series of lectures Monge–Ampère Geometry and the
Navier–Stokes Equations, given by I. Roulstone during Winter School and
Workshop Wisla 22 held in the beginning of February 2022.

###  1.1 Stucture of the paper

After the introduction we focus on properties of L-R metrics over the 4D
cotangent bundle $T^{*}M$ and investigate the Ricci flat case. Using the
Plücker embedding, we show that there is a correspondence between the set
given by Ricci flat $g_{\alpha}$ and points in projective space. We then
consider the pullback of $g_{\alpha}$ along sections of $T^{*}M\to M$ and
describe their basic features, as well as some particular cases, among which
are Hessian structures. Hessian structures are connected to and have
applications in, for example, affine differential geometry, homogeneous
spaces, cohomology theory, statistical manifolds, string theory, etc. [1, 16,
17]. We also find a special subclass of Hessian structures given by the
solutions of certain Monge-Ampère equations. In this case, we compute the
Koszul forms and comment on the related Kähler structure and its Ricci tensor.
We note that more general M-A structures (i.e. $A,B,C$ not necessarily zero)
naturally lead to deformations of the Hessian structures on $M$. We show that
the pullback metrics behave rather independently of the original ones, and
that the non-degeneracy of a special subset of the pullback metrics is
equivalent to non-degeneracy of the corresponding M-A structures.

##  2 Pseudo-Riemannian Lychagin-Rubtsov metric on $T^{*}M$

We start with some basic information about the L-R metric. In the canonical
coordinates, the matrix of $g_{\alpha}$ defined by (6) is

$\displaystyle G_{\alpha}=\begin{pmatrix}2C&-2B&D&0\\\ -2B&2A&0&D\\\
D&0&0&0\\\ 0&D&0&0\end{pmatrix}\ .$ (7)

We note the independency of $G_{\alpha}$ on $E$. From $\det G_{\alpha}=D^{4}$
further follows the non-degeneracy condition is

$\displaystyle\det G_{\alpha}\neq 0\iff D\neq 0\ .$

######  Lemma 2.1.

Let $g_{\alpha}$ be the L-R metric given by (6). Suppose that $\alpha$ given
by (2) satisfies $D\neq 0$. Then $g_{\alpha}$ is a pseudo-Riemannian metric
field on $T^{*}M$ with signature $(2,2)$.

###### Proof.

From (7) we see that the vector fields $\frac{\partial}{\partial
p},\frac{\partial}{\partial q}$ span a $2$-dimensional totally isotropic
subspace. Since $\dim T_{p}T^{*}M=4$, the signature of $g_{\alpha}$ is
necessarily $\left(2,2\right)$. ∎

###  2.1 L-R metric and Ricci flatness condition

We search for the conditions on $\alpha$ so that the L-R metric (6) satisfies
the Ricci flatness condition

$\displaystyle R_{ij}=0\ .$ (8)

The Ricci curvature $R_{ij}$ and the Ricci scalar $R$ of $g_{\alpha}$ are, in
general, complicated expressions with the second order derivatives of the
coefficients $A,B,\ldots$ We focused on the special case $A=B=C=0$. This means
that $\alpha$ reduces to

$\displaystyle\alpha=D\operatorname{d}p\wedge\operatorname{d}q+E\operatorname{d}x\wedge\operatorname{d}y\
,$ (9)

and the corresponding Monge-Ampère equation is

$\displaystyle D\det\operatorname{Hess}(f)=-E\ .$

Although we have restricted our considerations significantly by this choice of
$A,B,C$, the above M-A equation has interesting properties. For example, it
naturally emerges in the context of incompressible fluid dynamics, and is
related to rich geometric structures [8, 7, 2, 15]. Moreover, the pullback of
the L-R metric yields a Hessian structure on $M$, which will be further
discussed in the next section. We proceed with the following lemma, which will
be used in the subsequent result.

######  Lemma 2.2.

Let $g_{\alpha}$ be the L-R metric with $A=B=C=0$. Then the Ricci scalar is

$\displaystyle
R=\frac{3}{D^{3}}\left(-D_{q}D_{y}+2DD_{yq}-D_{p}D_{x}+2DD_{xp}\right)\ .$
(10)

###### Proof.

The proof is given by a direct computation. Note that the factor
$\frac{3}{D^{3}}$ is well-defined, since $G_{\alpha}$ corresponds to a metric
only if $D\neq 0$, due to lemma (2.1). ∎

######  Proposition 2.3.

Let $\alpha\in\Omega^{2}(T^{*}M)$ be given by (9), where $E\in
C^{\infty}(T^{*}M)$ is arbitrary. Then $g_{\alpha}$ is a Ricci flat pseudo-
Riemannian metric if and only if,

$\displaystyle
D(x,y,p,q)=\left(c_{1}+c_{2}x+c_{3}y+c_{4}p+c_{5}q+c_{6}(xp+yq)\right)^{-2}\
,$ (11)

where the constants $c_{i}\in\mathbb{R}$, $i=1,2,\ldots,6$, satisfy

$\displaystyle c_{2}c_{4}+c_{3}c_{5}-c_{1}c_{6}=0\ .$ (12)

###### Proof.

The Ricci curvature is a symmetric tensor and thus, in general, has $10$
independent components. We will split the set of equations corresponding to
$R_{ij}=0,1\leq i,j\leq 4$ into three subsets. The first subset contains four
equations

$\displaystyle 3D_{i}^{2}-2DD_{ii}$ $\displaystyle=0,\quad i\in\\{x,y,p,q\\}\
,$ (13)

as well as the second subset

$\displaystyle\begin{split}3D_{x}D_{y}-2DD_{xy}&=0\ ,\\\
3D_{x}D_{q}-2DD_{xq}&=0\ ,\\\ 3D_{p}D_{y}-2DD_{py}&=0\ ,\\\
3D_{p}D_{q}-2DD_{pq}&=0\ .\end{split}$ (14)

The last two equations are

$\displaystyle\begin{split}-2DD_{yq}+3D_{x}D_{p}-4DD_{xp}&=0\ ,\\\
-2DD_{xp}+3D_{y}D_{q}-4DD_{yq}&=0\ .\end{split}$ (15)

Now we use the following scheme for looking for solutions of the above system
of PDEs. We start with the first equation $3D_{x}^{2}-2DD_{xx}=0$, which can
be easily transformed by multiplying both sides of the equation with the
factor $\frac{1}{DD_{x}}$

$\displaystyle\frac{3}{2}\frac{D_{x}}{D}=\frac{D_{xx}}{D_{x}}\ ,$

thus obtaining an equation, which can be integrated to a separable equation

$\displaystyle\frac{3}{2}\ln D=\ln D_{x}+c\ .$

The solution to the last equation is

$\displaystyle D=\frac{1}{(c_{1}(y,p,q)+xc_{2}(y,p,q))^{2}}\ ,$

where $c_{1}(y,p,q),c_{2}(y,p,q)$ are unknown functions. We use the remaining
equations in (13) and (14) to fix the dependencies of $c_{1}(y,p,q)$ and
$c_{2}(y,p,q)$ on the variables $y,p,q$. This leads to

$D(x,y,p,q)=\left(c_{1}+c_{2}x+c_{3}y+c_{4}p+c_{5}q+c_{6}(xp+yq)\right)^{-2}\
.$

Plugging this in the remaining two equations (15), we get the condition (12),
which finishes the proof. ∎

####  2.1.1 Plücker embedding

Following [10], let $V$ be a real vector space, $\dim V=4$. Denote by
$\operatorname{Gr}(2,V)$ the Grassmanian of $2D$ subspaces in $V$, and by
$\mathbb{P}(\Lambda^{2}V)$ the projectivization of the space of exterior
$2$-forms on $V$. Let $p_{12},p_{13},p_{14},p_{23},p_{24},p_{34}$ be
coordinates on $\mathbb{P}(\Lambda^{2}V)$. The Plücker embedding is a map

$\displaystyle\operatorname{Gr}(2,V)\to\mathbb{P}(\Lambda^{2}V)$ (16)

with image given by the equation

$\displaystyle p_{12}p_{34}-p_{13}p_{24}+p_{14}p_{23}=0\ .$ (17)

Now we are ready to formulate a corollary of proposition 2.3, which describes
a link between the solutions $g_{\alpha}$ and points on quadric in the 5D real
projective space.

######  Corollary 2.3.1.

There is a correspondence between the set of metrics $g_{\alpha}\in
S^{2}(T^{*}M)$, which satisfy (11), (12), and the quadric in $\mathbb{R}P^{5}$
given by the image of the Plücker embedding (16). The condition (12) is
equivalent to vanishing of the scalar curvature of $g_{\alpha}$.

###### Proof.

Define

$\displaystyle c_{1}=-p_{14}\ ,\quad c_{2}=p_{12}\ ,\quad c_{3}=-p_{13}\
,\quad c_{4}=p_{34}\ ,\quad c_{5}=p_{24}\ ,\quad c_{6}=p_{23}\ .$

Then (17) is satisfied only if (12) holds. If $g_{\alpha}$ is given by (11),
then its scalar curvature is

$\displaystyle R=-24(c_{2}c_{4}+c_{3}c_{5}-c_{1}c_{6})\ .$

Thus (12) holds only if $R=0$. ∎

##  3 The pullback metric

Given a section of the cotangent bundle $\operatorname{d}f\colon M\rightarrow
T^{*}M$, we can pullback the metric from $T^{*}M$ to $M$ and induce a
(pseudo-)Riemannian structure $(\operatorname{d}f)^{*}g_{\alpha}$ on $M$. In
this section we explore the properties of this metric.

###  3.1 General case

We start with the matrix of the pullback metric
$(\operatorname{d}f)^{*}g_{\alpha}$ in canonical coordinates, which we will
denote $G^{*}_{\alpha}$

$\displaystyle G^{*}_{\alpha}=\begin{pmatrix}2C&-2B\\\
-2B&2A\end{pmatrix}+2D\operatorname{Hess}(f)\ .$ (18)

We remind the reader that we use the same notation for the functions
$A,B,C,D\in C^{\infty}(T^{*}M)$ (i.e. depending on $x,y,p,q$) as we do for the
precompositions with the section of $T^{*}M\to M$ (i.e. depending on
$x,y,f_{x},f_{y}$). The determinant is

$\displaystyle\det
G^{*}_{\alpha}=4(AC-B^{2})+4D\left(Af_{xx}+2Bf_{xy}+Cf_{yy}+Df_{xx}f_{yy}-Df^{2}_{xy}\right)\
.$ (19)

We observe that the non-degeneracy of $g_{\alpha}$, which was governed by the
condition $D\neq 0$, is not directly related to non-degeneracy of
$(\operatorname{d}f)^{*}g_{\alpha}$, which is much more complicated. This
allows for all four situations of degeneracy/non-degeneracy of the pair
$\left(g_{\alpha},(\operatorname{d}f)^{*}g_{\alpha}\right)$.

###  3.2 Special choices of $A,\ldots,E$

We will now discuss certain choices of the coefficients $A,\ldots,E$, leading
to further simplifications and interesting properties of
$(\operatorname{d}f)^{*}g_{\alpha}$.

####  3.2.1 Hessian structures

Suppose that $A=B=C=0$ and $D=\frac{1}{2}$. Then the pullback metric is of the
form

$\displaystyle G^{*}_{\alpha}=\operatorname{Hess}(f)\ ,$ (20)

which is called a Hessian structure on $M$. The theory of Hessian structures
is well studied, for example, see [16], where the Riemannian geometry of the
Hessian structures is investigated. Interesting properties of these structures
are also described, for example, in [1, 17, 16].

In [16], the author defines so-called first and second Kozsul forms,
$a_{i},b_{ij}$, of the Hessian structure, which are derived from the
Christoffel symbols of the corresponding metric as follows

$\displaystyle a_{i}=\Gamma^{k}_{ki},\qquad b_{ij}=\partial_{j}\alpha_{i}\ .$
(21)

We can employ the condition that the M-A equation
$(\operatorname{d}f)^{*}\alpha=0$ is satisfied, which yields

$\displaystyle\det G^{*}_{\alpha}=-E\ .$

In this case, the first and second Kozsul forms simplify to the following nice
form

$\displaystyle a_{i}$ $\displaystyle=\frac{1}{2}\frac{\partial\ln\det
G^{*}_{\alpha}}{\partial x^{i}}=\frac{1}{2}\frac{\partial\ln|E|}{\partial
x^{i}}\ ,$ $\displaystyle b_{ij}$
$\displaystyle=\frac{1}{2}\frac{\partial^{2}\ln|E|}{\partial x^{i}\partial
x^{j}}=\frac{1}{2}\operatorname{Hess}(\ln|E|)\ .$

Especially the second Koszul form is of great importance. Following [16],
given a Hessian structure on $M$, one can define a Kähler structure on $TM$.
Let $(x^{i},\xi^{i})$ be a coordinate system on $TM$ and
$z^{j}=x^{j}+i\xi^{j}$. Then the Hessian structure is

$\displaystyle
g^{T}=(g_{ij}\circ\pi)\operatorname{d}z^{i}\operatorname{d}\overline{z}^{j}\
.$

Then the Ricci tensor of $g^{T}$ is given by the second Koszul form

$\displaystyle R^{T}_{ij}=-\frac{1}{2}b_{ij}\circ\pi\ .$

The authors of this paper are interested in the relationship between the
complex structure given by the above construction, and the complex structure
naturally associated with M-A structures with negative Pfaffian (see (5) for
the definition of the notion).

####  3.2.2 Deformations of the Hessian structure

Let us now suppose only $D=\frac{1}{2}$, then

$\displaystyle G^{*}_{\alpha}=\begin{pmatrix}2C&-2B\\\
-2B&2A\end{pmatrix}+\operatorname{Hess}(f)\ .$

It is possible to choose suitable functions $A,B,C$ such that the matrix

$\displaystyle\begin{pmatrix}2C&-2B\\\ -2B&2A\end{pmatrix}$

is a Hessian matrix of some function $\epsilon g$, where $\epsilon>0$ is a
scalar. Such a choice gives rise to a deformation of the Hessian structure,
since

$\displaystyle
G^{*}_{\alpha}=\operatorname{Hess}(f)+\epsilon\operatorname{Hess}(g)=\operatorname{Hess}(f+\epsilon
g)\ .$

Notably, all the results of the previous paragraphs holds.

####  3.2.3 Pfaffian and non-degeneracy

So far we did not need to assume that the M-A equation related to the M-A
structure $(\Omega,\alpha)$ is satisfied by $f$. Nevertheless, if we assume
that the M-A equation $(\operatorname{d}f)^{*}\alpha=0$ holds, then it allows
us to significantly simplify the expression (19).

######  Proposition 3.1.

Let $f\in C^{\infty}(M)$ and consider a M-A structure
$(\Omega,\alpha)\in\Omega^{2}(T^{*}M)\times\Omega^{2}(T^{*}M)$. Suppose that
$f$ satisfies the corresponding M-A equation
$(\operatorname{d}f)^{*}\alpha=0$. Then $(\operatorname{d}f)^{*}g_{\alpha}$ is
a metric on $M$ only if $\operatorname{Pf}(\alpha)\neq 0$, which is equivalent
to non-degeneracy of the M-A structure $(\Omega,\alpha)$. Moreover, the
eigenvalues of the matrix of $(\operatorname{d}f)^{*}g_{\alpha}$ in canonical
coordinates are

$\displaystyle\lambda_{1,2}=C+A+D(f_{xx}+f_{yy})\pm
2\sqrt{\operatorname{Pf}(\alpha)}\ .$ (22)

###### Proof.

A coordinate description of the M-A equation $(\operatorname{d}f)^{*}\alpha=0$
is (4). Thus, if $f$ satisfies the equation, then

$\displaystyle
Af_{xx}+2Bf_{xy}+Cf_{yy}+D\left(f_{xx}f_{yy}-{f_{xy}}^{2}\right)=-E\ .$

This implies that (19) becomes

$\displaystyle\det G^{*}_{\alpha}=4(AC-B^{2})-4DE\ .$

Using the equation (5) and the coordinate descriptions (1), (2) of the M-A
structure, one can easily compute $\operatorname{Pf}(\alpha)=-B^{2}+AC-DE$.
Therefore

$\displaystyle\det G^{*}_{\alpha}=4\operatorname{Pf}(\alpha)\ .$

The eigenvalues are solutions of the following equation

$\displaystyle\lambda^{2}-\lambda\left(2C+2A+2Df_{yy}+2Df_{xx}\right)-4(B^{2}-AC)-4DE=0$
(23)

and their determination is a direct computation. ∎

We observe that the signature of $(\operatorname{d}f)^{*}g_{\alpha}$ depends
on two M-A equations. Firstly, to obtain the previous result, we had to assume
that $f$ solves the M-A equation $(\operatorname{d}f)^{*}\alpha=0$. Secondly,
we see that the eigenvalue equation (23) contains the Laplacian expression
$2D(f_{yy}+f_{xx})$. But vanishing of this expression amounts to the Laplace
equation $f_{yy}+f_{xx}=0$, which, in general, is a different M-A equation
then $(\operatorname{d}f)^{*}\alpha=0$.

##  4 Conclusions and Outlook

Motivated by the results of V. Lychagin et. al. [9, 11], and I. Roulstone et.
al. [2, 13, 12], we were interested in (pseudo-)Riemannian and Hessian
structures related to Monge-Ampère structures and the corresponding $2D$
Monge-Ampère equations. More concretely, this paper was focused on the
following questions.

  1. 1.

Are there any non-constant Lychagin-Rubtsov metrics with vanishing Ricci
curvature?

  2. 2.

If the answer to the first question is positive, can we classify all such
Monge-Ampère structures and the corresponding M-A equations?

  3. 3.

What kind of geometry on $M$ yields the pullback of the L-R metric along
sections of $T^{*}M\to M$?

We have presented partial answers to the above questions in the case of 4D
metrics on $T^{*}M$ and their 2D pullbacks on $M$.

The first question is positively answered in proposition 2.3. There is a
family of non-constant L-R metrics, depending on five real parameters
satisfying conditions (11), (12), with $R_{ij}=0$. Considering the second
question, we have found definite answers for the case

$\displaystyle D\det\operatorname{Hess}f=-E\ ,$ (24)

This amounts to suppressing three out of five degrees of freedom of a general
2D (symplectic) M-A equation (3) by choosing $A=B=C=0$. The answer to the last
question is also based on this choice, which naturally leads to 2D Hessian
structures on $M$, their deformations, and a subclass of Hessian structures
determined by the requirement that $f$ is a solution of the M-A equation (24).
We want to emphasize that only some of our results depend on the requirement
that $f$ satisfies certain PDE. The L-R metric and its pullback can be
introduced without any mentioning of the correspondence between M-A structures
and M-A equations, so (24) (or possibly other M-A equation) is an additional
assumption. Aside from the above three questions, we observed an interesting
relation between a family of solutions of the vacuum field equations and the
Plücker embedding, which is described in corollary 2.3.1. Regarding the future
work, we are interested in the following.

  * •

Further properties of the curvature tensor of $g_{\alpha}$ and the relation
between the curvature and the corresponding M-A equations.

  * •

The Hessian structures determined by (24) and aim at establishing further
links between M-A structures and Hessian structures (as well as their
deformations) in dimensions greater then $2$.

  * •

Comparison of the complex structure related to the Koszul form with the
complex structures associated with M-A equations satisfying
$\operatorname{Pf}(\alpha)>0$.

##  5 Acknowledgment

Both authors are grateful for the valuable comments and suggestions of the
reviewer. R. Suchánek is grateful to J. Slovák and V. Rubtsov for their
discussions and advices, as well as to the Masaryk University and to the
University of Angers for their hospitality. The work of R. Suchánek was
financially supported under the project GAČR EXPRO GX19-28628X, and project
MUNI/A/1092/2021, and partly by the Barrande Fellowship program organized by
The French Institute in Prague (IFP) and the Czech Ministry of Education,
Youth and Sports (MYES). Authors are also grateful to the organizers of the
42nd Winter School Geometry and Physics for a wonderful event, which enabled
the authors to work on this subject in a very inspiring atmosphere.

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