EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-048 LHCb-PAPER-2013-005 April 10, 2013

Measurement of the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ branching fraction
at low dilepton mass

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

The branching fraction of the rare decay $B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-}$ in the dilepton mass region from 30 to 1000
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ has been measured by the LHCb
experiment, using $pp$ collision data, corresponding to an integrated
luminosity of 1.0 $\mbox{\,fb}^{-1}$, at a centre-of-mass energy of 7
$\mathrm{\,Te\kern-1.00006ptV}$. The decay mode
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ is utilized as a normalization channel. The
branching fraction ${{\cal B}(B^{0}\rightarrow K^{*0}e^{+}e^{-}})$ is measured
to be

${\cal B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}=(3.1\,^{+0.9\mbox{
}+0.2}_{-0.8\mbox{ }-0.3}\pm 0.2)\times 10^{-7},$

where the first error is statistical, the second is systematic, and the third
comes from the uncertainties on the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow e^{+}e^{-}$
branching fractions.

Submitted to JHEP

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

LHCb collaboration

R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso57, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso58, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, M.
Benayoun8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R.
Bernet39, M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T.
Bird53, A. Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11,
S. Blusk57, V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53,
A. Borgia57, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van
den Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton57, N.H.
Brook45, H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J.
Buytaert37, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A.
Camboni35, P. Campana18,37, D. Campora Perez37, A. Carbone14,c, G.
Carboni23,k, R. Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52,
K. Carvalho Akiba2, G. Casse51, M. Cattaneo37, Ch. Cauet9, M. Charles54, Ph.
Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz 25, K. Ciba37, X.
Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46, J.
Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S. Cunliffe52, R.
Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, I. De Bonis4, K. De
Bruyn40, S. De Capua53, M. De Cian39, J.M. De Miranda1, L. De Paula2, W. De
Silva56, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, D.
Derkach14, O. Deschamps5, F. Dettori41, A. Di Canto11, H. Dijkstra37, M.
Dogaru28, S. Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A.
Dovbnya42, F. Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S.
Easo48,37, U. Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S.
Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch.
Elsasser39, D. Elsby44, A. Falabella14,e, C. Färber11, G. Fardell49, C.
Farinelli40, S. Farry12, V. Fave38, D. Ferguson49, V. Fernandez Albor36, F.
Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16, C.
Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M.
Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas
Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini57, Y. Gao3, J. Garofoli57,
P. Garosi53, J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E.
Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V.
Gligorov37, C. Göbel58, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H.
Gordon54, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37,
E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, O.
Grünberg59, B. Gui57, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou57, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T.
Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann59, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks51, D. Hill54, M. Hoballah5, C.
Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, F. Jing3, M. John54, D.
Johnson54, C.R. Jones46, B. Jost37, M. Kaballo9, S. Kandybei42, M. Karacson37,
T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T. Ketel41, A. Keune38, B.
Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M.
Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G.
Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A.
Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J. van Leerdam40,
J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T.
Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C.
Linn11, B. Liu3, G. Liu37, S. Lohn37, I. Longstaff50, J.H. Lopes2, E. Lopez
Asamar35, N. Lopez-March38, H. Lu3, D. Lucchesi21,q, J. Luisier38, H. Luo49,
F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54,
G. Manca15,d, G. Mancinelli6, U. Marconi14, R. Märki38, J. Marks11, G.
Martellotti24, A. Martens8, L. Martin54, A. Martín Sánchez7, M. Martinelli40,
D. Martinez Santos41, D. Martins Tostes2, A. Massafferri1, R. Matev37, Z.
Mathe37, C. Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, J. McCarthy44, R.
McNulty12, A. Mcnab53, B. Meadows56,54, F. Meier9, M. Meissner11, M. Merk40,
D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez58, S. Monteil5, D. Moran53,
P. Morawski25, M.J. Morello22,s, R. Mountain57, I. Mous40, F. Muheim49, K.
Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R.
Nandakumar48, I. Nasteva1, M. Needham49, N. Neufeld37, A.D. Nguyen38, T.D.
Nguyen38, C. Nguyen-Mau38,p, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin31, T.
Nikodem11, A. Nomerotski54, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero40, S. Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen52, A. Oyanguren 35,o, B.K. Pal57,
A. Palano13,b, M. Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C.
Parkes53, C.J. Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N.
Patrick48, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A.
Pellegrino40, G. Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, K. Petridis52, A. Petrolini19,i, A. Phan57, E.
Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M.
Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, D.
Popov10, B. Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, C.
Prouve7, V. Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, J.H.
Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N.
Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47, A.C. dos Reis1, S.
Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5,
P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37, V. Romanovsky34,
A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz
Valls35,o, G. Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50,
B. Saitta15,d, C. Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R.
Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A.
Sarti18,l, C. Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P.
Schaack52, M. Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I.
Sepp52, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42,
P. Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O.
Shevchenko42, V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T.
Skwarnicki57, N.A. Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P.
Soler50, F. Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49,
P. Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S.
Stone57, B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß59, H. Voss10, R. Waldi59, R. Wallace12,
S. Wandernoth11, J. Wang57, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, F. Xing54, Z. Xing57, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang57, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.

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

##  1 Introduction

The $b\\!\rightarrow s\gamma$ transition proceeds through flavour changing
neutral currents, and thus is sensitive to the effects of physics beyond the
Standard Model (BSM). Although the branching fraction of the
$B^{0}\\!\rightarrow K^{*0}\gamma$ decay has been measured [1, 2, 3] to be
consistent with the Standard Model (SM) prediction [4], BSM effects could
still be present and detectable through more detailed studies of the decay
process. In particular, in the SM the photon helicity is predominantly left-
handed, with a small right-handed current arising from long distance effects
and from the non-zero value of the ratio of the $s$-quark mass to the
$b$-quark mass. Information on the photon polarisation can be obtained with an
angular analysis of the $B^{0}\\!\rightarrow K^{*0}\ell^{+}\ell^{-}$ decay
($\ell=e,\mu$) in the low dilepton invariant mass squared ($q^{2}$) region
where the photon contribution dominates. The inclusion of charge-conjugate
modes is implied throughout the paper. The low $q^{2}$ region also has the
benefit of reduced theoretical uncertainties due to long distance
contributions compared to the full $q^{2}$ region [5]. The more precise SM
prediction allows for increased sensitivity to contributions from BSM. In the
low $q^{2}$ interval there is a contribution from $B^{0}\rightarrow
K^{*0}V(V\rightarrow\ell^{+}\ell^{-})$ where $V$ is one of the vector
resonances $\rho$, $\omega$ or $\phi$; however this contribution has been
calculated to be at most 1% [6]. The diagrams contributing to the
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ decay are shown in Fig. 1.

Figure 1: Dominant Standard Model diagrams contributing to the decay
${B^{0}\rightarrow K^{*0}e^{+}e^{-}}$.

With the LHCb detector, the $B^{0}\\!\rightarrow K^{*0}\ell^{+}\ell^{-}$
analysis can be carried out using either muons [7] or electrons.
Experimentally, the decay with muons in the final state produces a much higher
yield per unit integrated luminosity than electrons, primarily due to the
clean trigger signature. In addition, the much smaller bremsstrahlung
radiation leads to better momentum resolution, allowing a more efficient
selection. On the other hand, the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ decay
probes lower dilepton invariant masses, thus providing greater sensitivity to
the photon polarisation [5]. Furthermore, the formalism is greatly simplified
due to the negligible lepton mass [8]. It is therefore interesting to carry
out an angular analysis of the decay $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ in
the region where the dilepton mass is less than
1000${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The lower limit is set to
30${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ since below this value the
sensitivity for the angular analysis decreases because of a degradation in the
precision of the orientation of the $e^{+}e^{-}$ decay plane due to multiple
scattering. Furthermore, the contamination from the $B^{0}\\!\rightarrow
K^{*0}\gamma$ decay, with the photon converting into an $e^{+}e^{-}$ pair in
the detector material, increases significantly as $q^{2}\rightarrow 0$.

The first step towards performing the angular analysis is to measure the
branching fraction in this very low dilepton invariant mass region. Indeed,
even if there is no doubt about the existence of this decay, no clear
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ signal has been observed in this region
and therefore the partial branching fraction is unknown. The only experiments
to have observed $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ to date are BaBar [9]
and Belle [10], which have collected about 30 $B^{0}\\!\rightarrow
K^{*0}\ell^{+}\ell^{-}$ events each in the region
$q^{2}<2{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, summing over electron
and muon final states.

##  2 The LHCb detector, dataset and analysis strategy

The study reported here is based on $pp$ collision data, corresponding to an
integrated luminosity of 1.0 $\mbox{\,fb}^{-1}$, collected at the Large Hadron
Collider (LHC) with the LHCb detector [11] at a centre-of-mass energy of 7 TeV
during 2011. The LHCb detector is a single-arm forward spectrometer covering
the pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. It includes a high precision tracking system
consisting of a silicon-strip vertex detector (VELO) surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system has momentum resolution $(\Delta p/p)$ that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter (IP)
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
($p_{\rm T}$). 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 (SPD) and preshower (PS)
detectors, an electromagnetic calorimeter (ECAL) and a hadronic calorimeter.
Muons are identified by a system composed of alternating layers of iron and
multiwire proportional chambers. The trigger [12] consists of a hardware
stage, based on information from the calorimeter and muon systems, followed by
a software stage which applies a full event reconstruction.

For signal candidates to be considered in this analysis, at least one of the
electrons from the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ decay must pass the
hardware electron trigger, or the hardware trigger must be satisfied
independently of any of the daughters of the signal $B^{0}$ candidate (usually
triggering on the other $b$-hadron in the event). The hardware electron
trigger requires the presence of an ECAL cluster with a transverse energy
greater than 2.5 GeV. An energy deposit is also required in one of the PS
cells in front of the ECAL cluster, where the threshold corresponds to the
energy that would be deposited by the passage of five minimum ionising
particles. Finally, at least one SPD hit is required among the SPD cells in
front of the cluster. The software trigger requires a two-, three- or four-
track secondary vertex with a high sum of the $p_{\rm T}$ of the tracks and a
significant displacement from the primary $pp$ interaction vertices (PVs). At
least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and IP $\chi^{2}$ with respect
to the primary interaction greater than 16. The IP $\chi^{2}$ is defined as
the difference between the $\chi^{2}$ of the PV reconstructed with and without
the considered track. A multivariate algorithm is used for the identification
of secondary vertices consistent with the decay of a $b$-hadron.

The strategy of the analysis is to measure a ratio of branching fractions in
which most of the potentially large systematic uncertainties cancel. The decay
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ is used as normalization mode, since it has the same
final state as the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ decay and has a well
measured branching fraction [13, 14], approximately 300 times larger than
${\cal B}(B^{0}\\!\rightarrow K^{*0}e^{+}e^{-})$ in the $e^{+}e^{-}$ invariant
mass range 30 to 1000 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Selection
efficiencies are determined using data whenever possible, otherwise simulation
is used, with the events weighted to match the relevant distributions in data.
The $pp$ collisions are generated using Pythia 6.4 [15] with a specific LHCb
configuration [16]. Hadron decays are described by EvtGen [17] in which final
state radiation is generated using Photos [18]. The interaction of the
generated particles with the detector and its response are implemented using
the Geant4 toolkit [19, *Agostinelli:2002hh] as described in Ref. [21].

##  3 Selection and backgrounds

The candidate selection is divided into three steps: a loose selection, a
multivariate algorithm to suppress the combinatorial background, and
additional selection criteria to remove specific backgrounds.

Candidate $K^{*0}$ mesons are reconstructed in the $K^{*0}\rightarrow
K^{+}\pi^{-}$ mode. The $p_{\rm T}$ of the charged $K$ ($\pi$) mesons must be
larger than 400 (300) ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. Particle
identification (PID) information is used to distinguish charged pions from
kaons [22]. The difference between the logarithms of the likelihoods of the
kaon and pion hypotheses is required to be larger than 0 for kaons and smaller
than 5 for pions; the combined efficiency of these cuts is 88%. Candidates
with a $K^{+}\pi^{-}$ invariant mass within 130
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal $K^{*0}$ mass and a
good quality vertex fit are retained for further analysis. To remove
background from $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})\phi$ and $B^{0}_{s}\\!\rightarrow\phi e^{+}e^{-}$ decays,
where one of the kaons is misidentified as a pion, the mass computed under the
$K^{+}K^{-}$ hypothesis is required to be larger than 1040
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

Bremsstrahlung radiation, if not accounted for, would worsen the $B^{0}$ mass
resolution. If the radiation occurs downstream of the dipole magnet the
momentum of the electron is correctly measured and the photon energy is
deposited in the same calorimeter cell as the electron. In contrast, if
photons are emitted upstream of the magnet, the measured electron momentum
will be that after photon emission, and the measured $B^{0}$ mass will be
degraded. In general, these bremsstrahlung photons will deposit their energy
in different calorimeter cells than the electron. In both cases, the ratio of
the energy detected in the ECAL to the momentum measured by the tracking
system, an important variable in identifying electrons, is unbiased. To
improve the momentum reconstruction, a dedicated bremsstrahlung recovery
procedure is used, correcting the measured electron momentum by the
bremsstrahlung photon energy. As there is little material within the magnet,
the bremsstrahlung photons are searched for among neutral clusters with an
energy larger than 75 $\mathrm{\,Me\kern-1.00006ptV}$ in a well defined
position given by the electron track extrapolation from before the magnet.
Oppositely-charged electron pairs with an electron $p_{\rm T}$ larger than 350
${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and a good quality vertex are used to
form $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ candidates. The $e^{+}e^{-}$ invariant mass is
required to be in the range 30 – 1000
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ or 2400 – 3400
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for the two decay modes,
respectively. Candidate $K^{*0}$ mesons and $e^{+}e^{-}$ pairs are combined to
form $B^{0}$ candidates which are required to have a good-quality vertex. For
each $B^{0}$ candidate, the production vertex is assigned to be that with the
smallest IP $\chi^{2}$. The $B^{0}$ candidate is also required to have a
direction that is consistent with coming from the PV as well as a
reconstructed decay point that is significantly separated from the PV.

In order to maximize the signal efficiency while still reducing the high level
of combinatorial background, a multivariate analysis, based on a Boosted
Decision Tree (BDT) [23, *Roe] with the AdaBoost algorithm [25], is used. The
signal training sample is $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ simulated
data. The background training sample is taken from the upper sideband
($m_{B^{0}}>5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) from half of the
data sample. The variables used in the BDT are the $p_{\rm T}$, the IP and
track $\chi^{2}$ of the final state particles; the $K^{*0}$ candidate
invariant mass, the vertex $\chi^{2}$ and flight distance $\chi^{2}$ (from the
PV) of the $K^{*0}$ and $e^{+}e^{-}$ candidates; the $B^{0}$ $p_{\rm T}$, its
vertex $\chi^{2}$, flight distance $\chi^{2}$ and IP $\chi^{2}$, and the angle
between the $B^{0}$ momentum direction and its direction of flight from the
PV. A comparison of the BDT output for the data and the simulation for
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ decays is shown in Fig. 2. The candidates for this
test are reconstructed using a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
mass constraint and the background is statistically subtracted using the sPlot
technique [26] based on a fit to the $B^{0}$ invariant mass spectrum. The
agreement between data and simulation confirms a proper modelling of the
relevant variables. The optimal cut value on the BDT response is chosen by
considering the combinatorial background yield ($b$) on the
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ invariant mass distribution outside the
signal region111The signal region is defined as $\pm
300{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the nominal $B^{0}$ mass.
and evaluating the signal yield ($s$) using the $B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-}$ simulation assuming a visible $B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-}$ branching fraction of $2.7\times 10^{-7}$. The quantity
$s/\sqrt{s+b}$ serves as an optimisation metric, for which the optimal BDT cut
is 0.96. The signal efficiency of this cut is about 93% while the background
is reduced by two orders of magnitude.

Figure 2: Output of the BDT for
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ data (points) and simulation (red line).

After applying the BDT selection, specific backgrounds from decays that have
the same visible final state particles as the $B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-}$ signal remain. Since some of these backgrounds have larger
branching fractions, additional requirements are applied to the
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ candidates.

A large non-peaking background comes from the $B^{0}\\!\rightarrow
D^{-}e^{+}\nu$ decay, with ${D^{-}\rightarrow e^{-}{\overline{\nu}}K^{*0}}$.
The branching fraction for this channel is about five orders of magnitude
larger than that of the signal. When the neutrinos have low energies, the
signal selections are ineffective at rejecting this background. Therefore, the
$K^{*0}e^{-}$ invariant mass is required to be larger than 1900
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, which is 97% efficient on signal
decays. Another important source of background comes from the
$B^{0}\\!\rightarrow K^{*0}\gamma$ decay, where the photon converts into an
$e^{+}e^{-}$ pair. In LHCb, approximately 40% of the photons convert before
the calorimeter, and although only about 10% are reconstructed as an
$e^{+}e^{-}$ pair, the resulting mass of the $B^{0}$ candidate peaks in the
signal region. This background is suppressed by a factor 23 after the
selection cuts (including the 30${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
minimum requirement on the $e^{+}e^{-}$ invariant mass). The fact that signal
$e^{+}e^{-}$ pairs are produced at the $B^{0}$ decay point, whereas conversion
electrons are produced in the VELO detector material, is exploited to further
suppress this background. The difference in the $z$ coordinates, $\Delta z$,
between the first VELO hit and the expected position of the first hit,
assuming the electron was produced at the $K^{*0}$ vertex, should satisfy
$|\Delta z|<30$ mm. In addition, we require that the calculated uncertainty on
the $z$-position of the $e^{+}e^{-}$ vertex be less than 30 mm, since a large
uncertainty makes it difficult to determine if the $e^{+}e^{-}$ pair
originates from the same vertex as the $K^{*0}$ meson, or from a point inside
the detector material. These two additional requirements reject about 2/3 of
the remaining $B^{0}\\!\rightarrow K^{*0}\gamma$ background, while retaining
about 90% of the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ signal. After applying
these cuts, the $B^{0}\\!\rightarrow K^{*0}\gamma$ contamination under the
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ signal peak is estimated to be $(10\pm
3)\%$ of the expected signal yield.

Other specific backgrounds have been studied using either simulated data or
analytical calculations and include the decays $B\rightarrow
K^{\ast}\eta,K^{\ast}\eta^{\prime},K^{\ast}\pi^{0}$ and $\mathchar
28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{\ast}\gamma$, where
$\mathchar 28931\relax^{\ast}$ represents a high mass resonance decaying into
a proton and a charged kaon. The main source of background is found to be the
$B\rightarrow K^{\ast}\eta$ mode, followed by a Dalitz decay
($\eta\rightarrow\gamma e^{+}e^{-}$). These events form an almost flat
background in the mass range
$4300-5250{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. None of these backgrounds
contribute significantly in the $B^{0}$ mass region, and therefore are not
specifically modelled in the mass fits described later.

More generally, partially reconstructed backgrounds arise from $B$ decays with
one or more decay products in addition to a $K^{*0}$ meson and an $e^{+}e^{-}$
pair. In the case of the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ decay, there are two sources for these partially
reconstructed events: those from the hadronic part, such as events with higher
$K^{*}$ resonances (partially reconstructed hadronic background), and those
from the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ part (partially
reconstructed $J/\psi$ background), such as events coming from $\psi{(2S)}$
decays. For the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ decay mode, only the
partially reconstructed hadronic background has to be considered.

##  4 Fitting procedure

Since the signal resolution, type and rate of backgrounds depend on whether
the hardware trigger was caused by a signal electron or by other activity in
the event, the data sample is divided into two mutually exclusive categories:
events triggered by an extra particle $(e,\gamma,h,\mu)$ excluding the four
final state particles (called HWTIS, since they are triggered independently of
the signal) and events for which one of the electrons from the $B^{0}$ decay
satisfies the hardware electron trigger (HWElectron). Events satisfying both
requirements (20%) are assigned to the HWTIS category. The numbers of
reconstructed signal candidates are determined from unbinned maximum
likelihood fits to their mass distributions separately for each trigger
category. The mass distribution of each category is fitted to a sum of
probability density functions (PDFs) modelling the different components.

  1. 1.

The signal is described by the sum of two Crystal Ball functions [27] (CB)
sharing all their parameters but with different widths.

  2. 2.

The combinatorial background is described by an exponential function.

  3. 3.

The shapes of the partially reconstructed hadronic and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ backgrounds are described by
non-parametric PDFs [28] determined from fully simulated events.

The signal shape parameters are fixed to the values obtained from simulation,
unless otherwise specified.

There are seven free parameters for the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ fit for each trigger category. These include the
peak value of the $B^{0}$ candidate mass, a scaling factor applied to the
widths of the CB functions to take into account small differences between
simulation and data, and the exponent of the combinatorial background. The
remaining four free parameters are the yields for each fit component. The
invariant mass distributions together with the PDFs resulting from the fit are
shown in Fig. 3. The number of signal events in each category is summarized in
Table 1.

A fit to the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ candidates is then
performed, with several parameters fixed to the values found from the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ fit. These fixed parameters are the scaling factor
applied to the widths of the CB functions, the peak value of the $B^{0}$
candidate mass and the ratio of the partially reconstructed hadronic
background to the signal yield. The $B^{0}\\!\rightarrow K^{*0}\gamma$ yield
is fixed in the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ mass fit using the
fitted $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ signal yield, the ratio of efficiencies of the
$B^{0}\\!\rightarrow K^{*0}\gamma$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ modes, and the ratio of branching fractions ${\cal
B}(B^{0}\\!\rightarrow K^{*0}\gamma)/{\cal
B}(B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0})$. Hence there are three free parameters for the
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ fit for each trigger category: the
exponent and yield of the combinatorial background and the signal yield. The
invariant mass distributions together with the PDFs resulting from the fit are
shown in Fig. 4. The signal yield in each trigger category is summarized in
Table 1. The probability of the background fluctuating to obtain the observed
signal corresponds to 4.1 standard deviations for the HWElectron category and
2.4 standard deviations for the HWTIS category, as determined from the change
in the value of twice the natural logarithm of the likelihood of the fit with
and without signal. Combining the two results, the statistical significance of
the signal corresponds to 4.8 standard deviations.

Figure 3: Invariant mass distributions for the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ decay mode for the (left) HWElectron and (right)
HWTIS trigger categories. The dashed line is the signal PDF, the light grey
area corresponds to the combinatorial background, the medium grey area is the
partially reconstructed hadronic background and the dark grey area is the
partially reconstructed ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
background component.

Figure 4: Invariant mass distributions for the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ decay mode for the (left) HWElectron and (right) HWTIS trigger categories. The dashed line is the signal PDF, the light grey area corresponds to the combinatorial background, the medium grey area is the partially reconstructed hadronic background and the black area is the $B^{0}\\!\rightarrow K^{*0}\gamma$ component. Table 1: Signal yields with their statistical uncertainties. Trigger category | $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(e^{+}e^{-})K^{*0}$ | $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$
---|---|---
HWElectron | $5082\pm 104$ | $15.0\,^{+5.1}_{-4.5}$
HWTIS | $4305\pm 101$ | $14.1\,^{+7.0}_{-6.3}$

##  5 Results

The $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ branching fraction is calculated in
each trigger category using the measured signal yields and the ratio of
efficiencies

$\displaystyle{\cal B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}=$
$\displaystyle\frac{N(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})}{N(B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0})}\times r_{\rm sel}\times r_{\rm PID}\times r_{\rm
HW}$ $\displaystyle\times{\cal
B}(B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0})\times{\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow e^{+}e^{-}),$

where the ratio of efficiencies is sub-divided into the contributions arising
from the selection requirements (including acceptance effects, but excluding
PID), $r_{\rm sel}$, the PID requirements $r_{\rm PID}$ and the trigger
requirements $r_{\rm HW}$. The values of $r_{\rm sel}$ are determined using
simulated data, while $r_{\rm PID}$ and $r_{\rm HW}$ are obtained directly
from calibration data samples: ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow e^{+}e^{-}$ and $D^{0}\rightarrow K^{-}\pi^{+}$ from
$D^{*+}$ decays for $r_{\rm PID}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ decays for $r_{\rm HW}$. The values are summarized
in Table 2. The only ratio that is inconsistent with unity is the hardware
trigger efficiency due to the different mean electron $p_{\rm T}$ for the
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ decays.

The branching fraction for the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ decay
mode is taken from Ref. [14] and a correction factor of 1.02 has been applied
to take into account the difference in the $K\pi$ invariant mass range used,
and therefore the different S-wave contributions.

Table 2: Ratios of efficiencies used for the measurement of the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ branching fraction. The ratio $r_{\rm HW}$ for the HWTIS trigger category is assumed to be equal to unity. The uncertainties are the total ones and are discussed in Sec. 6. | HWElectron category | HWTIS category
---|---|---
$r_{\rm sel}$ | $1.03\pm 0.02$ | $1.03\pm 0.02$
$r_{\rm PID}$ | $1.01\pm 0.02$ | $1.03\pm 0.02$
$r_{\rm HW}$ | $1.35\pm 0.03$ | 1

The $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ branching fraction, for each
trigger category, is measured to be

$\displaystyle{\cal B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}_{\text{HWElectron}}$
$\displaystyle=$ $\displaystyle(3.3\,^{+1.1}_{-1.0})\times 10^{-7}$
$\displaystyle{\cal B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}_{\text{HWTIS}}$
$\displaystyle=$ $\displaystyle(2.8\,^{+1.4}_{-1.2})\times 10^{-7},$

where the uncertainties are statistical only.

##  6 Systematic uncertainties

Several sources of systematic uncertainty are considered, affecting either the
determination of the number of signal events or the computation of the
efficiencies. They are summarized in Table 3.

The ratio of trigger efficiencies is determined using a
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ calibration sample from data, which is reweighted
using the $p_{\rm T}$ of the triggering electron in order to model properly
the kinematical properties of the two decays. The uncertainties due to the
limited size of the calibration samples are propagated to get the related
systematic uncertainty shown in Table 2.

The PID calibration introduces a systematic uncertainty on the calculated PID
efficiencies as given in Table 2. For the kaon and pion candidates this
systematic uncertainty is estimated by comparing, in simulated events, the
results obtained using a $D^{*+}$ calibration sample to the true simulated PID
performance. For the $e^{+}e^{-}$ candidates, the systematic uncertainty is
assessed ignoring the $p_{\rm T}$ dependence of the electron identification.
The resulting effect is limited by the fact that the kinematic differences
between the $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ and the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$
decays are small once the full selection chain is applied.

The fit procedure is validated with pseudo-experiments. Samples are generated
with different fractions or shapes for the partially reconstructed hadronic
background, or different values for the fixed signal parameters and are then
fitted with the standard PDFs. The corresponding systematic uncertainty is
estimated from the bias in the results obtained by performing the fits
described above. The resulting deviations from zero of each variation are
added in quadrature to get the total systematic uncertainty due to the fitting
procedure. The parameters of the signal shape are varied within their
statistical uncertainties as obtained from the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ fit. An alternate signal shape, obtained by studying
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ signal decays in data both with and without a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass constraint is also tried;
the difference in the yields from that obtained using the nominal signal shape
is taken as an additional source of uncertainty. The ratio of the partially
reconstructed hadronic background to the signal yield is assumed to be
identical to that determined from the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ fit. The systematic uncertainty linked to this
hypothesis is evaluated by varying the ratio by $\pm 50\%$. The fraction of
partially reconstructed hadronic background thus determined is in agreement
within errors with the one found in $B^{0}\\!\rightarrow K^{*0}\gamma$ decays
[29]. The shape of the partially reconstructed background used in the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ and the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ fits
are the same. The related systematic uncertainty has been evaluated using an
alternative shape obtained from charmless $b$-hadron decays. The
$B^{0}\\!\rightarrow K^{*0}\gamma$ contamination in the $B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-}$ signal sample is $1.2\pm 0.4$ and $1.5\pm 0.5$ events for
the HWElectron and HWTIS signal samples, respectively. Combining the
systematic uncertainties in quadrature, the branching fractions are found to
be

$\displaystyle{\cal B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}_{\rm
HWElectron}$ $\displaystyle=$ $\displaystyle(3.3\,^{+1.1\mbox{
}+0.2}_{-1.0\mbox{ }-0.3}\pm 0.2)\times 10^{-7}$ $\displaystyle{\cal
B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}_{\rm
HWTIS}$ $\displaystyle=$ $\displaystyle(2.8\,^{+1.4\mbox{ }+0.2}_{-1.2\mbox{
}-0.3}\pm 0.2)\times 10^{-7},$

where the first error is statistical, the second systematic, and the third
comes from the uncertainties on the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow e^{+}e^{-}$
branching fractions [13, 14]. The branching ratios are combined assuming all
the systematic uncertainties to be fully correlated between the two trigger
categories except those related to the size of the simulation samples. The
combined branching ratio is found to be

${\cal B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}=(3.1\,^{+0.9\mbox{
}+0.2}_{-0.8\mbox{ }-0.3}\pm 0.2)\times 10^{-7}.$

Table 3: Absolute systematic uncertainties on the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ branching ratio (in $10^{-7}$) . Source | HWElectron category | HWTIS category
---|---|---
Simulation sample statistics | 0.06 | 0.05
Trigger efficiency | 0.07 | -
PID efficiency | 0.08 | 0.10
Fit procedure | ${}^{+0.09}_{-0.22}$ | ${}^{+0.07}_{-0.23}$
$B^{0}\\!\rightarrow K^{*0}\gamma$ contamination | 0.08 | 0.08
Total | ${}^{+0.17}_{-0.26}$ | ${}^{+0.16}_{-0.27}$

##  7 Summary

Using $pp$ collision data corresponding to an integrated luminosity of 1.0
$\mbox{\,fb}^{-1}$, collected by the LHCb experiment in 2011 at a centre-of-
mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$, a sample of approximately 30
$B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ events, in the dilepton mass range 30
to 1000${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, has been observed. The
probability of the background to fluctuate upward to form the signal
corresponds to 4.6 standard deviations including systematic uncertainties. The
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(e^{+}e^{-})K^{*0}$ decay mode is utilized as a normalization channel,
and the branching fraction $\cal B$($B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$) is
measured to be

${\cal B}(B^{0}\\!\rightarrow
K^{*0}e^{+}e^{-})^{30-1000{\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}}=(3.1\,^{+0.9\mbox{
}+0.2}_{-0.8\mbox{ }-0.3}\pm 0.2)\times 10^{-7}.$

This result can be compared to theoretical predictions. A simplified formula
suggested in Ref. [5] takes into account only the photon diagrams of Fig. 1.
When evaluated in the 30 to 1000${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
$e^{+}e^{-}$ invariant mass interval using $\cal B$($B^{0}\\!\rightarrow
K^{*0}\gamma$) [1, 2, 3], it predicts a $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$
branching fraction of $2.35\times 10^{-7}$. A full calculation has been
recently performed [30] and the numerical result for the $e^{+}e^{-}$
invariant mass interval of interest is $(2.43^{+0.66}_{-0.47})\times 10^{-7}$.
The consistency between the two values reflects the photon pole dominance. The
result presented here is in good agreement with both predictions.

Using the full LHCb data sample obtained in 2011 – 2012 it will be possible to
do an angular analysis. The measurement of the $A_{\mathrm{T}}^{2}$ parameter
[8] thus obtained, is sensitive to the existence of right handed currents in
the virtual loops in diagrams similar to those of Fig. 1. For this purpose,
the analysis of the $B^{0}\\!\rightarrow K^{*0}e^{+}e^{-}$ decay is
complementary to that of the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ mode.
Indeed, it is predominantly sensitive to a modification of $\mathcal{C}_{7}$
(the so-called $\mathcal{C}_{7}^{{}^{\prime}}$ terms) while, because of the
higher $q^{2}$ in the decay, the $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$
$A_{\mathrm{T}}^{2}$ parameter has a larger possible contribution from the
$\mathcal{C}_{9}^{{}^{\prime}}$ terms [31].

## Acknowledgements

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

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[a]Alexander Rothkopf

# Conserving Lattice Gauge Theory for Finite Systems

###### Abstract

In this contribution I discuss a recent proposal of a novel action for lattice
gauge theory for finite systems, which accommodates non-periodic spatial
boundary conditions. Drawing on the summation-by-parts formulation of finite
differences and finite volume strategies of computational electrodynamics, an
action is constructed that implements the proper integral form of Gauss’ law
and exhibits an inherently symmetric energy momentum tensor, all while
realizing automatic ${\cal O}(a)$ improvement. Its central ingredients are
illustrated using Abelian gauge theory as example.

##  1 Motivation

The powerful combination of lattice gauge theory and Monte-Carlo methods in
Euclidean time has over the past decade revealed many vital properties of
strongly interacting matter under extreme conditions. One example is the phase
structure of QCD at finite temperature and more recently also at small to
moderate Baryo-chemical potential, which has been explored in detail (for
recent works see e.g. [1, 2, 3, 4]). Such studies of the phase diagram are
motivated in large part by heavy-ion collision experiments carried out at the
LHC at CERN and formerly RHIC at BNL. Phenomenological models of locally
thermalized nuclear matter in the form of relativistic hydrodynamics, amended
by fist principles input, such as the QCD equation of state from lattice QCD,
have been successful in interpreting the experimental findings in such
collisions of heavy nuclei [5, 6].

One still outstanding puzzle however emerged in the study of intermediate and
small systems, in which tantalizing hints for the existence of a quark-gluon-
plasma (QGP) were observed. These signals are similar to those in a large
systems collision, even though the conditions for a transition of nuclear
matter to the high temperature phase are much less favorable. One line of
inquiry in this context asks, whether the small spatial size of these systems
may mimic effects that in a large system can be ascribed to the existence of a
strongly interacting QGP.

Recent exploratory work in the continuum and on the lattice has revealed that
the presence of a finite volume and non-trivial boundary conditions can indeed
lead to significant deviations from the infinite volume results in a region
that may be of relevance for phenomenology. In Ref.[7] the authors
investigated the extreme example of free thermal scalar field theory, which
due absence of any form of screening, probes the geometry on all length
scales. It was shown that restricting the extent of one or more spatial
directions lead to deviations from the Stefan-Boltzmann behavior in a fashion
that reduces the pressure, a change similar to that induced by finite
interactions. The more directions become constrained, the stronger the effect,
so that in a box geometry of finite extent in all three spatial directions a
reduction of the pressure by around 10% was encountered at a value of
$TL\approx 20$, and an even more significant reduction by 40% at $TL\approx
3-5$. A $TL\approx 20$ is similar to the geometry encountered in a $A+A$
collision at LHC, while $TL\approx 3-5$ corresponds to what one might argue
arises in a $p+A$ collisions.

Obviously free scalar theory may not provide relevant insight for strongly
interacting matter, which is why the results of Ref.[8] are very interesting.
They reveal that reducing the physical length of a single spatial axis (which
led to the weakest deviation in the case of scalar theory) of a purely gluonic
system to values between $1<LT<2$ leads to significant changes in the pressure
associated with that direction, even flipping the sign of the pressure. While
a first important step, this study still relied on periodic boundary
conditions and only restricted one of the spatial directions.

Both studies hint at the relevance of finite volume effects in nuclear matter
under current experimental conditions. In order to systematically explore
these effects of a genuine finite extent of the fireball, created in a
relativistic heavy-ion collision, we thus need to formulate lattice gauge
theory in a way that accommodates non-periodic (e.g. Dirichlet) boundary
conditions. (Note that in a gauge theory such boundary conditions must of
course be formulated in terms of gauge invariant expressions on the boundary
of the spatial volume.)

A second motivation is recent work on the heavy quark potential in classical
statistical gauge theory, where in Ref.[9] for the first time the screened
real-part of that potential was successfully computed. The take home message
of that study was that charges in classical LGT are not introduced by the
evaluation of the Wilson loop observable but have to the treated explicitly in
a modified Gauss’ law.

In turn one may ask whether there are alternatives to evaluating the Wilson
loop and the answer is in the affirmative. Ref.[10] showed how to extract the
potential from the gauge invariant stress tensor, the spatial components of
the energy momentum tensor. The discretization prescription formulated in the
following section sets out to provide a lattice gauge theory which in its
energy momentum tensor accurately reproduces the symmetric field lines of the
simplest possible system, that of a static charge and anti-charge, already on
the classical level and at the same time fulfills Gauss’ law.

Taken together our goal here is to construct a lattice gauge theory, which
accommodates systems with non-trivial boundary conditions and those where
translational invariance is not given, such as in the presence of external
sources.

##  2 Two Challenges

The first challenge I wish to focus on is the treatment of non-periodic
boundary conditions. Let us inspect the standard Wilson action
$S=\frac{1}{g^{2}}\sum_{x}\big{[}\sum_{i}2\frac{a_{s}}{a_{t}}{\rm
ReTr}[P^{1\times 1}_{0i}-1]-\sum_{i,j}\frac{a_{t}}{a_{s}}{\rm ReTr}[P^{1\times
1}_{ij}-1]\big{]}$ expressed in terms of plaquettes

$\displaystyle P^{1\times 1}_{\mu\nu,x}$
$\displaystyle=U_{\mu,x}U_{\nu,x+a_{\mu}{\hat{\mu}}}U^{\dagger}_{\mu,x+a_{\nu}{\hat{\nu}}}U^{\dagger}_{\nu,x}=e^{ia_{\mu}a_{\nu}\tilde{F}_{\mu\nu,x}}+{\cal
O}(a^{2}),$ $\displaystyle\tilde{F}_{\mu\nu}=\Delta^{\rm
F}_{\mu}A_{\nu,x}-\Delta^{\rm F}_{\nu}A_{\mu,x}+i[A_{\mu,x},A_{\nu,x}],$ (1)

which themselves are constructed of link variables $U_{\mu,x}={\rm
exp}[ia_{\mu}A_{\mu,x}]$. As is well known, the above corresponds to a
discretization of the field strength tensor in terms of forward finite
differences $\Delta^{\rm
F}_{\mu}\phi(x)=(\phi(x+a_{\mu}\hat{\mu})-\phi(x))/a_{\mu}$, with $a_{\mu}$
denoting the lattice spacing and $\hat{\mu}$ the unit vector in the $\mu$
direction. In order to distill the essence of the challenge and not be
encumbered by the technical difficulties of non-Abelian gauge theory let us
explore this discretization in the Abelian case first.

Figure 1: (left) sketch of the geometry and the true solution (red) of the
field lines for the example of a circular capacitor at finite potential.
(right) backward finite difference solution (blue) of Gauss’ law with the true
solution as boundary condition.

The quantization of gauge theories requires the formulation of a Hamiltonian
picture, which is amended by a constraint in the form of Gauss’ law and which
singles out physical states within the Hilbert space. Therefore let me focus
on the solutions of the discretized Gauss’ law in the presence of non-periodic
boundary conditions.

On the left of fig. 1 I show as an example the geometry of a circular
capacitor held at a finite potential. The classical potential in the interior
may be computed from solving the Poisson equation, the corresponding field
lines ensue from taking the negative gradient. If this solution, referred to
as "true" in the following, is supplied on the boundary for the solution of
Gauss’ law with backward finite differences (corresponding to the forward FD
of the Wilson action) one obtains the solution shown in blue on the right. It
is easy to see that while the backward FD can produce field strength at the
backward boundary it fails to do so at the forward boundary.

Going over to a central finite difference discretization of Gauss’ law appears
to be a straight forward solution to this issue. However, as shown in the
center panel of fig. 2, this choice of discretization, while producing
symmetric field lines, introduces an artificial staggered pattern among them
that does not yield an accurate reproduction of the true solution.

Figure 2: (left) True field solution of the circular capacitor in red.
(center) Central finite difference solution (green) of Gauss’ law with the
true solution as boundary condition, (right) SBP solution of Gauss’ law in
red.

The correct way to treat non-trivial boundary conditions with finite
differences requires instead a consistent treatment of integration and
differentiation embodied in the concept of summation by parts (SBP). One aims
to construct a finite difference operator (see e.g. [11]) to fulfill this
discrete counterpart to integration by parts.

To be concrete: if we discretize the inner product on a real function space
$\int_{0}^{L}dxf(x)g(x)\approx{\cal T}_{0}^{N}[f_{x}g_{x}]$, we require the
finite difference operator to fulfill

$\displaystyle{\cal T}_{0}^{N}[(\Delta^{\rm
SBP}f_{x})g_{x}]\overset{!}{=}-{\cal T}_{0}^{N}[f_{x}($
$\displaystyle\Delta^{\rm SBP}g_{x})]+f_{N}g_{N}-f_{0}g_{0}.$ (2)

The simplest form to implement an SBP finite difference for a first order
derivative is given by the following example: using the matrix $H=a\;{\rm
diag}[\frac{1}{2},1,\ldots,1,\frac{1}{2}]$ to denote the inner product between
the two vectors of discretized functions $f=[f(0),f(a),\ldots,f((N_{x}-1)a)]$,
the corresponding lowest order SBP operator for $N_{x}=4$ reads

$\displaystyle\Delta^{\rm SBP}=a^{-1}\left[\begin{array}[]{cccc}-1&1&0&0\\\
-\frac{1}{2}&0&\frac{1}{2}&0\\\ 0&-\frac{1}{2}&0&\frac{1}{2}\\\
0&0&-1&1\end{array}\right].$ (7)

And indeed, as can be seen in the right panel of fig. 2 the combination of
forward and backward FD on the boundary together with central FD in the
interior leads to a much more accurate reproduction of the field lines for the
Abelian capacitor example.

The second challenge is related to the accurate reproduction of field lines on
the level of the stress tensor, the spatial components of
$T^{\alpha\beta}=\frac{1}{4\pi}\big{(}g^{\alpha\mu}F_{\mu\lambda}F^{\lambda\beta}+\frac{1}{4}g^{\alpha\beta}F_{\mu\lambda}F^{\mu\lambda}\big{)}$.
As concrete example I show a pair of a static charge and anti-charge placed at
a certain distance in the left panel of fig. 3. Diagonalizing the $3\times 3$
spatial submatrix of $T^{\alpha\beta}$ yields a set of eigenvectors, two of
which due to symmetry lie in the plane shown in fig. 3, one is perpendicular
to it. The in-plane eigenvector associated with positive eigenvalues is
plotted and encodes the direction and strength of the field lines between the
charges.

Figure 3: (left) True solution of the eigenvectors of the stress tensor in the
presence of a static charge anti-charge pair (right) solution of the
corresponding backward FD Gauss’ law is given as blue arrows.

The solution of the same scenario based on the backward FD is shown on the
left and one immediately sees the asymmetry it introduces. Again we are in
need of a symmetric discretization, however it is a known shortcoming of
central finite differences to not respect the integral form of Gauss’ law,
i.e. in general

$\displaystyle Q=\int\,dV\,q=\int\,dV\,{\bf\nabla^{\rm FD}E}\neq\int_{\partial
V}\,d{\bf A}\cdot{\bf E}.$ (8)

The artifacts that this introduces are clearly visible among the green arrows
in the left panel of fig. 4. We again encounter an artificially staggered
pattern of the field lines.

Figure 4: (left) Solution of the central FD Gauss’ law for a static charge
anti-charge pair (right) solution of the corresponding finite volume
discretized Gauss’ law. In each panel the true solution is given in the
background as light red arrows for comparison.

A solution to this issue has been developed in the context of computational
electrodynamics and it amounts to going over to a so called finite-volume
discretization. The starting point of which is to consider the integral form
of Gauss’ law

$\displaystyle\int_{x_{i-1/2}}^{x_{i+1/2}}dx\int_{y_{i-1/2}}^{y_{i+1/2}}dy$
$\displaystyle\int_{z_{i-1/2}}^{z_{i+1/2}}dz\big{(}\frac{dE_{x}}{dx}+\frac{dE_{y}}{dy}+\frac{dE_{z}}{dz}\big{)}=\int
d^{3}x\delta^{(3)}({\bf x}-{\bf x}_{0}).$ (9)

and to discretize its LHS using the midpoint rule. This expression is not in
the form of a finite difference. It can be brought into that form if we
combine multiple copies of it, each shifted by one lattice spacing in one of
the spatial directions. The resulting expression

$\displaystyle\sum_{i}\Delta_{i}^{\rm C}E_{i}({\bf x})$
$\displaystyle=\frac{1}{8a^{3}}\Big{[}\sum_{i}\big{(}\delta_{{\bf x}+a\hat{\bf
i},{\bf x}_{0}}+\delta_{{\bf x}-a\hat{\bf i},{\bf x}_{0}}\big{)}-2\delta_{{\bf
x},{\bf x}_{0}}\Big{]}$ (10)

indicates that the central finite difference needs to be amended with a
particular form of distributed sources to fulfill Gauss’ law. The resulting
much improved reproduction of the field lines is shown as red arrows on the
right of fig. 4.

##  3 Solving the two challenges in a non-Abelian setting

In order to achieve a summation-by-parts discretization of lattice gauge
theory, we require a central finite difference scheme for the interior of our
system. Since the discretization there is correct up to ${\cal O}(a^{2})$ we
need to use a discretization for the links of the same order
$\bar{U}_{\mu,x}={\rm
exp}\big{[}ia_{\mu}A_{\mu,x+\frac{1}{2}a\hat{\mu}}\big{]}={\rm
exp}\big{[}ia_{\mu}\frac{1}{2}\big{(}A_{\mu,x}+A_{\mu,x+a\hat{\mu}}\big{)}\big{]}+{\cal
O}(a^{2})$. As put forward in a recent preprint [12] I propose to achieve this
type of symmetric discretization by considering a $2\times 2$ plaquette
centered around the spacetime point $x$, which reads

$\displaystyle P^{2\times 2}_{\mu\nu,x}=$
$\displaystyle\bar{U}_{\mu,x-a\hat{\mu}-a\hat{\nu}}\bar{U}_{\mu,x-a\hat{\nu}}\bar{U}_{\nu,x+a\hat{\mu}-a\hat{\nu}}\bar{U}_{\nu,x+a\hat{\mu}}\bar{U}^{\dagger}_{\mu,x+a\hat{\nu}}\bar{U}^{\dagger}_{\mu,x-a\hat{\mu}+a\hat{\nu}}\bar{U}^{\dagger}_{\nu,x-a\hat{\mu}}\bar{U}^{\dagger}_{\nu,x-a\hat{\mu}-a\hat{\nu}}$
$\displaystyle=$ $\displaystyle{\rm
exp}\big{[}4iga_{\mu}a_{\nu}\bar{F}_{\mu\nu,x}\big{]}+{\cal
O}(a^{3}),\quad{\rm and\,where}\quad\bar{F}_{\mu\nu,x}={\bf\Delta}^{\rm
C}_{\mu}A_{\nu,x}-{\bf\Delta}^{\rm C}_{\nu}A_{\mu,x}+i[A_{\mu,x},A_{\nu,x}].$

This type of plaquette can now be used to form a gauge invariant classical
action for lattice gauge theory in the interior of the system

$\displaystyle S^{2\times 2}=\sum_{x\notin\partial
V}a_{t}a_{s}^{3}\Big{[}\frac{2}{16a_{t}^{2}a_{s}^{2}}\sum_{i}{\rm
ReTr}\big{[}1-P^{2\times 2}_{0i,x}\big{]}-\frac{1}{16a_{s}^{4}}\sum_{ij}{\rm
ReTr}\big{[}1-P^{2\times 2}_{ij,x}\big{]}\Big{]}.$ (11)

Since the action $S^{2\times 2}$ respects the gauge invariance of the lattice
theory and reduces to the correct action in the continuum limit, it is a
viable candidate to quantize the theory via the path integral framework. In
order to achieve a genuine SBP discretization of the field strength tensor and
in turn of Gauss’ law, we need to amend the action $S^{2\times 2}$ on the
boundaries with expressions that are based on $2\times 1$ and $1\times 2$
plaquettes, where as in the corners the $1\times 1$ plaquettes are deployed.

Actions based on plaquettes with more than unit area have been considered
before in the literature in the context of Symanzik’s improvement program. I
would like to note that in that context $2\times 2$ plaquettes were used
predominantly in combination with the $1\times 1$ plaquette action and they
were considered as oriented in the forward direction. In that fashion they did
not realize the symmetry around the spacetime point $x$ of the present
proposal and also do not lend themselves to constructing a genuine summation
by parts discretization that captures non-trivial boundary conditions.

Having discussed a possible solution to the challenge of non-periodic boundary
conditions, the remaining challenge amounts to implementing the finite volume
discretization of Gauss’ law in the interior of the volume. This, as discussed
in the previous section, is necessitated by the presence of a central finite
difference discretization in the SBP action. While in classical
electrodynamics, formulated on the level of Maxwell’s equations, the
discretization of Gauss’ law may be simply changed by hand, for the quantum
theory we need to formulate an appropriate action. I.e. we need to introduce
the spatial distribution of the sources in the fermion action.

As a first step let me restrict the discussion to heavy fermions where only a
temporal covariant derivative acts on the spinors

$\displaystyle
S_{\psi}=a_{t}a_{s}^{3}\sum_{x}i\psi_{x}^{\dagger}D_{0}\psi_{x}.$ (12)

In the classical equation of motion, the heavy fermions enter as charge
density dynamically via their equal time commutator $J_{0}=\frac{g}{2}{\rm
Tr}\left[\langle[\psi_{x}^{\dagger},\psi_{x}]\rangle\right]$ as shown in
Ref.[13]. Hence in order to realize the staggered pattern of eq. 10, shifts in
spatial directions need to be realized in the covariant derivative. My
proposal is to take inspiration from a finite difference operator $\Delta^{\rm
RN-
SBP}_{t(j)}\phi(x)=(\phi(x+a\hat{0}-a\hat{j})-\phi(x-a\hat{0}-a\hat{j})-\phi(x-a\hat{0}+a\hat{j})+\phi(x+a\hat{0}+a\hat{j}))/4a$,
recently developed in the context of open quantum systems [14]. It displaces
the fermions not only in temporal but also in spatial direction and when
amended by appropriate factors of the link variables can be made into a gauge
invariant and symmetric expression that can serve as operator in the heavy
fermion action

$\displaystyle S_{\psi}$
$\displaystyle=a_{t}a_{s}^{3}\frac{i}{3}\sum_{x,i}\psi_{x}^{\dagger}\bar{D}_{0(i)}\psi_{x}=a_{t}a_{s}^{3}\frac{i}{3}\sum_{x,i}\psi_{x}^{\dagger}\frac{1}{8a}\Big{[}\big{(}U_{0,x}U^{\dagger}_{j,x+a\hat{0}-a\hat{j}}+U^{\dagger}_{j,x-a\hat{j}}U_{0,x-a\hat{j}}\big{)}\psi_{x+a\hat{0}-a\hat{j}}$
(13) $\displaystyle-$
$\displaystyle\big{(}U^{\dagger}_{j,x-a\hat{j}}U^{\dagger}_{0,x-a\hat{0}-a\hat{j}}+U^{\dagger}_{0,x-a\hat{0}}U^{\dagger}_{jx-a\hat{0}-a\hat{j}}\big{)}\psi_{x-a\hat{0}-a\hat{j}}-\big{(}U^{\dagger}_{0,x-a\hat{0}}U_{j,x-a\hat{0}}+U_{j,x}U^{\dagger}_{0,x-a\hat{0}+a\hat{j}}\big{)}\psi_{x-a\hat{0}+a\hat{j}}$
$\displaystyle+$
$\displaystyle(U_{j,x}U_{0,x+a\hat{j}}+U_{0,x}U_{j,x+a\hat{0}}\big{)}\psi_{x+a\hat{0}+a\hat{j}}\Big{]}.$

##  4 Summary

The question of quark-gluon-plasma production in small collision systems, as
well as the precision exploration of the complex potential acting between a
heavy quark-antiquark pair requires a consistent formalism for lattice gauge
theory in the absence of (hypercubic-)translational invariance. On the level
of the discretized classical action, the treatment of Dirichlet boundary
conditions requires finite difference approximations that respect the
summation by parts property, which I here propose to implement in the non-
Abelian case via a novel centered $2\times 2$ plaquette action in the interior
and appropriate combinations of $1\times 2$, $2\times 1$ and $1\times 1$
plaquettes on the boundary. In order to preserve Gauss’ law in the presence of
a central finite difference discretization in the interior I take inspiration
from computational electrodynamics to propose a finite volume discretiztion,
which translates to spatially shifted sources in the finite difference
language. A reparametrization invariant finite difference discretization is
put forward that introduces the appropriate spatial shifts, when deployed in
the action of heavy fermions.

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