Article
Reference
Search for the Production of Scalar Bottom Quarks in pp Collisions at
√s=1.96 TeV
CDF Collaboration
CLARK, Allan Geoffrey (Collab.), et al.
Abstract
We report on a search for direct scalar bottom quark (sbottom) pair production in pp collisions at √s=1.96 TeV, in events with large missing transverse energy and two jets of hadrons in the final state, where at least one of the jets is required to be identified as originating from a b quark. The study uses a collider detector at Fermilab Run II data sample corresponding to 2.65 fb−1 of integrated luminosity. The data are in agreement with the standard model. In an R-parity conserving minimal supersymmetric scenario, and assuming that the sbottom decays exclusively into a bottom quark and a neutralino, 95% confidence-level upper limits on the sbottom pair production cross section of 0.1 pb are obtained. For neutralino masses below 70 GeV/c2, sbottom masses up to 230 GeV/c2 are excluded at 95% confidence level.
CDF Collaboration, CLARK, Allan Geoffrey (Collab.), et al . Search for the Production of Scalar Bottom Quarks in pp Collisions at √s=1.96 TeV. Physical Review Letters , 2010, vol. 105, p.
7p.
DOI : 10.1103/PhysRevLett.105.081802
Available at:
http://archive-ouverte.unige.ch/unige:38695
Disclaimer: layout of this document may differ from the published version.
1 / 1
Search for the Production of Scalar Bottom Quarks in p p Collisions at ffiffiffi p s
¼ 1:96 TeV
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Y. Zheng,10,eand S. Zucchelli7,6
(CDF Collaboration)
1Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China
2Argonne National Laboratory, Argonne, Illinois 60439, USA
3University of Athens, 157 71 Athens, Greece
4Institut de Fisica d’Altes Energies, Universitat Autonoma de Barcelona, E-08193, Bellaterra (Barcelona), Spain
5Baylor University, Waco, Texas 76798, USA
6Istituto Nazionale di Fisica Nucleare Bologna, I-40127 Bologna, Italy
7University of Bologna, I-40127 Bologna, Italy
8Brandeis University, Waltham, Massachusetts 02254, USA
9University of California, Davis, Davis, California 95616, USA
10University of California, Los Angeles, Los Angeles, California 90024, USA
11University of California, San Diego, La Jolla, California 92093, USA
12University of California, Santa Barbara, Santa Barbara, California 93106, USA
13Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain
14Carnegie Mellon University, Pittsburgh, Pennslyvania 15213, USA
15Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA
16Comenius University, 842 48 Bratislava, Slovakia; Institute of Experimental Physics, 040 01 Kosice, Slovakia
17Joint Institute for Nuclear Research, RU-141980 Dubna, Russia
18Duke University, Durham, North Carolina 27708, USA
19Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
20University of Florida, Gainesville, Florida 32611, USA
21Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, I-00044 Frascati, Italy
22University of Geneva, CH-1211 Geneva 4, Switzerland
23Glasgow University, Glasgow G12 8QQ, United Kingdom
24Harvard University, Cambridge, Massachusetts 02138, USA
25Division of High Energy Physics, Department of Physics, University of Helsinki and Helsinki Institute of Physics, FIN-00014, Helsinki, Finland
26University of Illinois, Urbana, Illinois 61801, USA
27The Johns Hopkins University, Baltimore, Maryland 21218, USA
28Institut fu¨r Experimentelle Kernphysik, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany
081802-2
29Center for High Energy Physics, Kyungpook National University, Daegu 702-701, Korea;
Seoul National University, Seoul 151-742, Korea;
Sungkyunkwan University, Suwon 440-746, Korea;
Korea Institute of Science and Technology Information, Daejeon 305-806, Korea;
Chonnam National University, Gwangju 500-757, Korea;
Chonbuk National University, Jeonju 561-756, Korea
30Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
31University of Liverpool, Liverpool L69 7ZE, United Kingdom
32University College London, London WC1E 6BT, United Kingdom
33Centro de Investigaciones Energeticas Medioambientales y Tecnologicas, E-28040 Madrid, Spain
34Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
35Institute of Particle Physics: McGill University, Montre´al, Que´bec, H3A 2T8, Canada;
Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada;
University of Toronto, Toronto, Ontario, M5S 1A7, Canada;
and TRIUMF, Vancouver, British Columbia, V6T 2A3, Canada
36University of Michigan, Ann Arbor, Michigan 48109, USA
37Michigan State University, East Lansing, Michigan 48824, USA
38Institution for Theoretical and Experimental Physics, ITEP, Moscow 117259, Russia
39University of New Mexico, Albuquerque, New Mexico 87131, USA
40Northwestern University, Evanston, Illinois 60208, USA
41The Ohio State University, Columbus, Ohio 43210, USA
42Okayama University, Okayama 700-8530, Japan
43Osaka City University, Osaka 588, Japan
44University of Oxford, Oxford OX1 3RH, United Kingdom
45Istituto Nazionale di Fisica Nucleare, Sezione di Padova-Trento, I-35131 Padova, Italy
46University of Padova, I-35131 Padova, Italy
47LPNHE, Universite Pierre et Marie Curie/IN2P3-CNRS, UMR7585, Paris, F-75252 France, USA
48University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
49Istituto Nazionale di Fisica Nucleare Pisa, I-56127 Pisa, Italy
50University of Pisa, I-56127 Pisa, Italy
51University of Siena, I-56127 Pisa, Italy
52Scuola Normale Superiore, I-56127 Pisa, Italy
53University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
54Purdue University, West Lafayette, Indiana 47907, USA
55University of Rochester, Rochester, New York 14627, USA
56The Rockefeller University, New York, New York 10021, USA
57Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1, Italy
58Sapienza Universita` di Roma, I-00185 Roma, Italy
59Rutgers University, Piscataway, New Jersey 08855, USA
60Texas A&M University, College Station, Texas 77843, USA
61Istituto Nazionale di Fisica Nucleare Trieste/Udine, I-34100 Trieste, Italy
62University of Trieste/Udine, I-33100 Udine, Italy
63University of Tsukuba, Tsukuba, Ibaraki 305, Japan
64Tufts University, Medford, Massachusetts 02155, USA
65Waseda University, Tokyo 169, Japan
66Wayne State University, Detroit, Michigan 48201, USA
67University of Wisconsin, Madison, Wisconsin 53706, USA
68Yale University, New Haven, Connecticut 06520, USA (Received 16 May 2010; published 19 August 2010)
We report on a search for direct scalar bottom quark (sbottom) pair production in pp collisions at ffiffiffis
p ¼1:96 TeV, in events with large missing transverse energy and two jets of hadrons in the final state, where at least one of the jets is required to be identified as originating from abquark. The study uses a collider detector at Fermilab Run II data sample corresponding to2:65 fb1of integrated luminosity. The data are in agreement with the standard model. In an R-parity conserving minimal supersymmetric scenario, and assuming that the sbottom decays exclusively into a bottom quark and a neutralino, 95%
confidence-level upper limits on the sbottom pair production cross section of 0.1 pb are obtained. For neutralino masses below70 GeV=c2, sbottom masses up to230 GeV=c2are excluded at 95% confidence level.
DOI:10.1103/PhysRevLett.105.081802 PACS numbers: 14.80.Ly, 12.60.Jv
Supersymmetry (SUSY) [1] is an extension of the stan- dard model (SM) that naturally solves the hierarchy prob- lem [2] and provides a possible candidate for dark matter in the Universe. SUSY doubles the SM spectrum of particles by introducing a new supersymmetric partner (sparticle) for each particle in the SM. In a generic minimal super- symmetric extension of the SM (MSSM) and assuming R-parity conservation [1], sparticles are produced in pairs, and the lightest supersymmetric particle is stable and iden- tified as the neutralino~01. Assuming a SUSY particle mass hierarchy such that the scalar bottom quark (sbottom) b~1 decays exclusively as b~1!b~01, the expected signal for direct sbottom pair production is characterized by the presence of two energetic jets from the hadronization of the bottom quarks and large missing transverse energyE6 T [3] from the two lightest supersymmetric particles in the final state. Results on searches in this channel using Tevatron data have been previously reported by both the collider detector at Fermilab (CDF) and D0 experiments in Run II [4–6]. This Letter presents new results based on an almost 10 times larger data sample collected by the CDF experiment, corresponding to2:65 fb1of total luminosity.
The CDF II detector is described in detail elsewhere [7].
The data are collected using a three-level trigger system that selects events with at least one calorimeter cluster with ET above 100 GeV. The events are then required to have a primary vertex with a z position within 60 cm of the nominal interaction point. Jets are reconstructed from the energy deposits in the calorimeter towers using a cone- basedffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffialgorithm [8] with cone radius R¼
2þ2
p ¼0:4, and the measured EjetT is corrected for detector effects and contributions from multiple pp interactions per crossing at high instantaneous luminosity, as discussed in Ref. [9]. The events are initially selected with E6 T>10 GeV and two jets with transverse energy EjetT >25 GeV and pseudorapidity jjetj<2:0, where at least one of the two jets is required to havejjetj<1:1. Events with additional jets with EjetT >15 GeV and jjetj<2:8are rejected. The effect of the trigger selection is studied using control data samples and parametrized as a function ofE6 T and the transverse energy of the leading jet Ejetð1ÞT [10]. At a givenE6 T, the trigger efficiency increases with increasing Ejetð1ÞT . Finally, at least one of the two leading jets is required to originate from a b-quark jet candidate, using the default CDF b-jet tagging algorithm (SECVTX) [11], based on the presence of a displaced vertex due to the decay of abhadron inside the jet.
The SM QCD multijet background contribution with large E6 T, due to the mismeasurement of the jet energies in the calorimeters, is suppressed by requiring an azimuthal separation ðE6 T jetÞ>0:4 for the two jets in the event. The SM background contributions with energetic electrons [12] fromW andZdecays and reconstructed as jets in the final state are suppressed by requiring
EjetT;em=EjetT <0:9 for each jet, where EjetT;em denotes the electromagnetic component of the jet transverse energy.
In addition, events with isolated tracks with transverse momentum pT;track above 10 GeV=c are vetoed, thus re- jecting backgrounds with W or Z bosons decaying into muons or tau leptons. Beam-related backgrounds and cos- mic rays are eliminated by requiring an average jet elec- tromagnetic fraction fem>0:15 and an average charged particle fractionfch>0:15, as defined in [13], and have no significant effect on SUSY signal and SM background physics samples.
The dominant source of background in the analysis is due to events with a light-flavor jet which is misidentified as a b-jet (mistags). There are also contributions from heavy-flavor QCD multijet events with largeE6 T and pass- ing theðE6 TjetÞrequirement discussed above. In both cases, the background estimation, after final selection cri- teria (see below), is extracted from data using a procedure similar to that described in [11,14], where different data- driven multidimensional parameterizations are employed to estimate the probability for a light-flavor jet in each event to be mistagged, and the probability for a heavy- flavor jet in QCD multijet events to pass the CDF b-tag requirements, respectively.
Simulated event samples are used to determine detector acceptance and reconstruction efficiency, estimate the con- tribution from the rest of the SM backgrounds with heavy- flavor jets in the final state, and compute the number of expected SUSY signal events. Samples of simulatedttand diboson (WW,WZ, andZZ) processes are generated using thePYTHIA6.216 [15] Monte Carlo generator with Tune A [16], and normalized to next-to-leading order (NLO) pre- dictions [17,18]. Samples of simulated Z=ð!lþlÞ þ jets (l¼e, , ), Zð!Þ þ jets, and Wð!lÞ þjets events with light- and heavy-flavor jets are generated using the ALPGEN 2.1 program [19] interfaced with the parton- shower model from PYTHIA. The normalization of the boson plus jets heavy-flavor samples includes an additional multiplicative factor kHF ¼1:40:4 which brings the predicted light- to heavy-flavor relative contributions in agreement with data [11]. The completeZ=þjets and Wþjets samples are then normalized to the measuredZ and W inclusive cross sections [20]. Finally, samples of single top events are produced using the MADEVENTgen- erator [21] and normalized according to NLO predictions [22]. The SUSY samples are generated in the framework of MSSM usingPYTHIA. Massesb~1and~01are fixed, andb~1is only allowed to decay via b~1 !b~01 channel. A total of 106 different samples have been generated with sbottom mass Mb~1 in the range between 80 GeV=c2 and 280 GeV=c2 and neutralino mass M~01 up to 100 GeV=c2. The samples are normalized to NLO predic- tions, as implemented in PROSPINO2[23], using CTEQ6.6 [24] parton distribution functions (PDFs) and renormaliza- tion and factorization scales set to Mb~1. The production 081802-4
cross section depends on Mb~1 and decreases from 50 to 0.01 pb as the sbottom mass increases. The Monte Carlo generated events are passed through a full CDF II detector simulation and reconstructed and analyzed with the same analysis chain as for the data.
An optimization is performed with the aim to maximize the sensitivity to a SUSY signal across theb~1~01 mass plane. For each of the 106 signal samples considered, the procedure maximizesS= ffiffiffiffi
pB
, whereSdenotes the number of SUSY events andBis the total SM background. As the difference MMb~1M~01 increases, the optimal thresholds on E6 T,Ejetð1ÞT , Ejetð2ÞT , and ½E6 T jetð2Þ for the second leading jet, andHT, defined as HT ¼P
iEjetðiÞT (i¼1–2), increase. The results from the different SUSY samples are combined to define two single sets of thresh- olds (see Table I) that maximize the search sensitivity in the widest range of sbottom and neutralino masses at low M (M <90 GeV=c2) and high M (M >
90 GeV=c2), respectively. As an example, for M~01 ¼ 70 GeV=c2, Mb~1¼193 GeV=c2, and a SUSY cross sec- tion of 0.32 pb, a value S= ffiffiffiffi
pB
¼5:8 is obtained, corre- sponding to a selection efficiency of 4.7%.
A number of control samples in data is considered to test the validity of the SM background predictions. Samples dominated by QCD multijet events or mistags are obtained by reversing the requirements on either E6 T or Ejetð2ÞT , or using a nonoverlapping sample selected with looser b-tagging requirements. Control samples dominated by Z=þjets, Wþjets, and top-quark processes, with highly energetic leptons in the final state, are obtained after requiring eitherEjetT;em=EjetT >0:9for at least one of the jets, or selecting events with isolated tracks with pT;track>
10 GeV=c. Agreement is always observed between the data and the SM predictions.
A detailed study of systematic uncertainties was carried out [10]. The uncertainty on the SM background predic- tions is dominated by the determination of theb-jet mistag rates, which propagates into an uncertainty in the SM prediction between 13% and 11% as M increases. The uncertainty on thekHFvalue applied to the boson plus jets heavy-flavor samples translates into an 11% uncertainty in the SM predictions. The dependence on the amount of initial-state and final-state radiation in the Monte Carlo generated samples for top, boson plus jets, and diboson
contributions introduces a 6% uncertainty on the SM pre- dictions. Other sources of uncertainty on the predicted SM background are: a 3% uncertainty due to the determination of the b-tagging efficiency in the Monte Carlo simulated samples; a 3% uncertainty from the uncertainties on the absolute normalization of the top quark, diboson,W, andZ Monte Carlo generated processes; and a 2.5% uncertainty from the knowledge of the jet energy scale [9]. In addition, uncertainties related to trigger efficiency and the heavy- flavor QCD multijet background contribute less than 1% to the final uncertainty. The total systematic uncertainty on the SM predictions varies between 19% and 18% asM increases. In the case of the MSSM signal, various sources of uncertainty on the predicted cross sections at NLO, as determined using PROSPINO2, are considered: the uncer- tainty due to PDFs is computed using the Hessian method [25] and translates into a 12% uncertainty on the absolute predictions; variations of the renormalization and factori- zation scale by a factor of 2 change the theoretical cross sections by about 26%. Uncertainties on the amount of initial- and final-state gluon radiation in the MSSM Monte Carlo generated samples introduce a 10% uncer- tainty on the signal yields. The 3% uncertainty on the absolute jet energy scale translates into a 9% to 14%
uncertainty on the MSSM predictions. Other sources of uncertainty include: a 4% uncertainty due to the determi- nation of theb-tagging efficiency, and a 2% to 1% uncer- tainty due to the uncertainty on the trigger efficiency. The total systematic uncertainty on the MSSM signal yields varies between 30% and 32% asMincreases. Finally, an additional 6% uncertainty on the quoted total integrated luminosity is also taken into account in both SM and SUSY predictions.
TABLE I. Minimum values ofE6 T,HTþE6 T,Ejetð1ÞT ,Ejetð2ÞT , and
½E6 Tjetð2Þfor eachMregion.
Minimum values
E6 T HTþE6 T Ejetð1ÞT Ejetð2ÞT ðE6 Tjetð2ÞÞ
GeV radians
lowM 60 165 80 25 0.7
highM 80 300 90 40 0.7
1 10 102
1 10 102
100 200 300
1 10
100 200 300
1 10
200 400 600
200 400 600
[GeV]
ET HT + ET [GeV]
events per bin
SM
Total Syst. Uncertainty
-1) CDF RunII DATA (L=2.65 fb SM + MSSM
M analysis
∆ low
= 123 GeV/c2 b1
M~
= 90 GeV/c2 1 χ∼0
M
M analysis
∆ high
= 193 GeV/c2 b1
M~
= 70 GeV/c2 1 χ∼0
M
/ /
FIG. 1 (color online). MeasuredE6 TandHTþE6 Tdistributions (black dots) for low-M(top) and high-Manalyses (bottom), compared to the SM predictions (solid lines) and the SMþ MSSM predictions (dashed lines). The shaded bands show the total systematic uncertainty on the SM predictions.
Figure1shows the measuredE6 T andHTþE6 Tdistribu- tions compared to the SM predictions after all final selec- tion criteria are applied. For illustrative purposes, the figure indicates the impact of two given MSSM scenarios. The data are in agreement with the SM predictions within uncertainties for each of the two analyses at low and high M. In TableII, the observed number of events and the SM predictions are presented in each case.
The results are translated into 95% confidence level (C.L.) upper limits on the cross section for sbottom pair production at given sbottom and neutralino masses, using a Bayesian approach [26] and including statistical and sys- tematic uncertainties. For the latter, correlations between systematic uncertainties on signal efficiencies and back- ground predictions are taken into account. For each MSSM point considered, observed and expected limits are com- puted separately for both low- and high-Manalyses, and the one with the best expected limit is adopted as the nominal result. Cross sections above 0.1 pb are excluded at 95% C.L. for the range of sbottom masses considered.
Similarly, the observed numbers of events in data are translated into 95% C.L. upper limits for sbottom and neutralino masses, for which the uncertainties on the theo- retical cross sections are also included in the limit calcu- lation, and where both analyses are combined in the same way as for the cross section limits. Figure 2 shows the expected and observed exclusion regions in the sbottom- neutralino mass plane. For the MSSM scenario considered, sbottom masses up to 230 GeV=c2 are excluded at 95%
C.L. for neutralino masses below70 GeV=c2. This analy- sis extends the previous CDF limits [4] on the sbottom mass by more than40 GeV=c2.
In summary, we report results on a search for sbottom pair production inpp collisions at ffiffiffi
ps
¼1:96 TeV, based on2:65 fb1 of CDF Run II data. The events are selected with largeE6 Tand two energetic jets in the final state, and at least one jet is required to originate from abquark. The measurements are in agreement with SM predictions for backgrounds. The results are translated into 95% C.L.
upper limits on production cross sections and sbottom
and neutralino masses in a given MSSM scenario for which the exclusive decay b~1!b~01 is assumed, and signifi- cantly extend previous CDF results.
We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions.
This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Culture, Sports, Science and Technology of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China; the Swiss National Science Founda- tion; the A. P. Sloan Foundation; the Bundesministerium fu¨r Bildung und Forschung, Germany; the World Class University Program, the National Research Foundation of Korea; the Science and Technology Facilities Council and the Royal Society, UK; the Institut National de Physique Nucleaire et Physique des Particules/CNRS; the Russian Foundation for Basic Research; the Ministerio de Ciencia e Innovacio´n, and Programa Consolider-Ingenio 2010, Spain; the Slovak R and D Agency; and the Academy of Finland.
aDeceased.
bWith visitors from University of Massachusetts Amherst, Amherst, MA 01003, USA.
cWith visitors from Universiteit Antwerpen, B-2610 Antwerp, Belgium.
TABLE II. Number of events in data for the two analyses compared to SM predictions, including statistical and systematic uncertainties summed in quadrature.
(2:65 fb1) lowM highM
mistags 51:418:2 18:55:5
QCD jets 7:61:9 1:60:2
top 21:23:4 7:81:3
Z!þjets 27:78:8 10:93:5 Z=!lþlþjets 0:50:2 0:110:04 W!lþjets 22:37:3 7:32:4
WW,WZ,ZZ 3:10:5 1:40:2
SM prediction 133:826:4 47:68:8
Events in data 139 38
0 50 100 150 200 250
0 20 40 60 80 100 120
CDF D0 CDF -1) ) (295 pb ) (310 pb-1 (2650 pb-1 Observed Limit (95% C.L.) Expected Limit (95% C.L.) + Mb
1 χ∼0 < M b1 M~
= 0o θ LEP
0 50 100 150 200 250
0 20 40 60 80 100 120
2] [GeV/c
b1
M~
]2 [GeV/c 10 χ∼M
FIG. 2 (color online). Exclusion plane at 95% C.L. as a func- tion of sbottom and neutralino masses. The observed and ex- pected upper limits from this analysis are compared to previous results from CDF and D0 experiments at the Tevatron in Run II [4,5], and from LEP [27] experiments at CERN with squark mixing angle¼0. The hatched area indicates the kinemati- cally prohibited region in the plane.
081802-6
dWith visitors from University of Bristol, Bristol BS8 1TL, United Kingdom.
eWith visitors from Chinese Academy of Sciences, Beijing 100864, China.
fWith visitors from Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari, 09042 Monserrato (Cagliari), Italy.
gWith visitors from University of California Irvine, Irvine, CA 92697, USA.
hWith visitors from University of California Santa Cruz, Santa Cruz, CA 95064, USA.
iWith visitors from Cornell University, Ithaca, NY 14853, USA.
jWith visitors from University of Cyprus, Nicosia CY- 1678, Cyprus.
kWith visitors from University College Dublin, Dublin 4, Ireland.
lWith visitors from University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom.
mWith visitors from University of Fukui, Fukui City, Fukui Prefecture, Japan 910-0017.
nWith visitors from Kinki University, Higashi-Osaka City, Japan 577-8502.
oWith visitors from Universidad Iberoamericana, Mexico D.F., Mexico.
pWith visitors from University of Iowa, Iowa City, IA 52242, USA.
qWith visitors from Kansas State University, Manhattan, KS 66506., USA
rWith visitors from Queen Mary, University of London, London, E1 4NS, England.
sWith visitors from University of Manchester, Manchester M13 9PL, England.
tWith visitors from Muons, Inc., Batavia, IL 60510, USA.
uWith visitors from Nagasaki Institute of Applied Science, Nagasaki, Japan.
vWith visitors from University of Notre Dame, Notre Dame, IN 46556., USA
wWith visitors from University de Oviedo, E-33007 Oviedo, Spain.
xWith visitors from Texas Tech University, Lubbock, TX 79609., USA
yWith visitors from IFIC(CSIC-Universitat de Valencia), 56071 Valencia, Spain.
zWith visitors from Universidad Tecnica Federico Santa Maria, 110v Valparaiso, Chile.
aaWith visitors from University of Virginia, Charlottesville, VA 22906., USA
bbWith visitors from Bergische Universita¨t Wuppertal, 42097 Wuppertal, Germany.
ccWith visitors from Yarmouk University, Irbid 211-63, Jordan.
ddOn leave from J. Stefan Institute, Ljubljana, Slovenia.
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